Datasets:

---

ashishkgpian

/

adspapers_5_10

like 0

[ { "title": "ABSTRACT", "content": "Context. In recent years, several potentially habitable, probably terrestrial exoplanets and exoplanet candidates have been discovered. The amount of CO2 in their atmosphere is of great importance for surface conditions and habitability. In the absence of detailed information on the geochemistry of the planet, this amount could be considered as a free parameter. Aims. Up to now, CO2 partial pressures for terrestrial planets have been obtained assuming an available volatile reservoir and outgassing scenarios. This study aims at calculating the allowed maximum CO2 pressure at the surface of terrestrial exoplanets orbiting near the outer boundary of the habitable zone by coupling the radiative e ff ects of the CO2 and its condensation at the surface. These constraints might limit the permitted amount of atmospheric CO2, independent of the planetary reservoir. Methods. A1Dradiative-convective cloud-free atmospheric model was used to calculate surface conditions for hypothetical terrestrial exoplanets. CO2 partial pressures are fixed according to surface temperature and vapor pressure curve. Considered scenarios cover a wide range of parameters, such as gravity, central star type and orbital distance, atmospheric N2 content and surface albedo. Results. Results show that for planets in the habitable zone around K-, G-, and F-type stars the allowed CO2 pressure is limited by the vapor pressure curve and not by the planetary reservoir. The maximum CO2 pressure lies below the CO2 vapor pressure at the critical point of p crit = 73.8 bar. For M-type stars, due to the stellar spectrum being shifted to the near-IR, CO2 pressures above p crit are possible for almost all scenarios considered across the habitable zone. This implies that determining CO2 partial pressures for terrestrial planets by using only geological models is probably too simplified and might over-estimate atmospheric CO2 towards the outer edge of the habitable zone. Key words. Planets and satellites: atmospheres, Planets and satellites: composition", "pages": [ 1 ] }, { "title": "Atmospheric constraints for the CO 2 partial pressure on terrestrial planets near the outer edge of the habitable zone", "content": "P. von Paris 1 , 2 , 3 , J.L. Grenfell 4 , P. Hedelt 1 , 2 /star , H. Rauer 3 , 4 , F. Selsis 1 , 2 , and B. Stracke 3 Preprint online version: October 15, 2018", "pages": [ 1 ] }, { "title": "1. Introduction", "content": "Given the di ffi culties and challenges of detecting sub-surface life on Earth, any life to be first discovered beyond our own solar system will most likely be restricted to the planetary surface and atmosphere. This is the basis of the concept of the habitable zone (HZ, e.g., Dole 1964, Hart 1978, Kasting et al. 1993). The HZ is defined as the region around a star where a rocky planet with a suitable atmosphere can host liquid water on its surface, a condition motivated by the fact that all life as we know it requires liquid water. Several studies have implied that small, potentially rocky planets are common (e.g., Howard et al. 2010, Wittenmyer et al. 2011, Borucki et al. 2011, Mayor et al. 2011, Cassan et al. 2012, Gaidos et al. 2012). Hence, it is not unreasonable to assume that planets in the HZ of their central stars may also be relatively common. Indeed, some potentially habitable (candidate) super-Earths in or very close to the HZ of their central star have already been discovered (Udry et al. 2007, Mayor et al. 2009, Borucki et al. 2011, Pepe et al. 2011, Bonfils et al. 2011, Anglada-Escud'e et al. 2012, Delfosse et al. 2012, Borucki et al. 2012). Also, Neptune- or Jupiter-like planets have been discovered in the HZ (e.g., Lovis et al. 2006, Fischer et al. 2008, Haghighipour et al. 2010, Tinney et al. 2011) which raises the possibility of habitable satellites around these planets. A simple criterion for the potential habitability of a planet, which is immediately accessible from the discovery data, is its equilibrium temperature, T eq. The equilibrium temperature is calculated by where A is the planetary albedo, F the stellar flux at the orbital distance of the planet and σ the Stefan-Boltzmann constant. As was discussed by, e.g., Selsis et al. (2007) and Kaltenegger & Sasselov (2011), a habitable planet should have T eq /lessorsimilar 270K to avoid a runaway heating of the surface and corresponding loss of the complete surface water reservoir. For low values of T eq near the outer edge of the HZ (e.g., model calculations for GL 581 d suggest T eq ∼ 190K, von Paris et al. 2010), a massive greenhouse e ff ect must be provided by the atmosphere to obtain habitable surface conditions. H2O is the most obvious candidate of radiatively active gases which could provide the necessary greenhouse warming. It provides the bulk of the greenhouse e ff ect on Earth. Furthermore, H2O is by definition present on the surface of a habitable planet. The H2O partial pressure in an atmosphere of a potentially habitable planet is controlled by evaporation (or sublimation) from the surface reservoir, taking into account the water vapor pressure curve. Besides H2O, CO2 is usually considered the most important greenhouse gas for the determination of the outer boundary of the HZ (e.g., Kasting et al. 1993). On Earth, CO2 is controlled by processes such as volcanic outgassing or rock weathering. To estimate CO2 partial pressures for terrestrial exoplanets, up to now only geological models were used (e.g., Elkins-Tanton & Seager 2008, Kite et al. 2009, Kite et al. 2011, Edson et al. 2012, Abbot et al. 2012). Furthermore, the volatile content of habitable zone planets is expected to be highly variable due to orbital migration (e.g., Raymond et al. 2004). For instance, planets originating from the outer planetary system and made of a large fraction of cometary material can migrate to habitable orbital distances, resulting in the so-called ocean-planets (L'eger et al. 2004). Planetary CO2 reservoirs of the order of thousands of bars are certainly plausible, when considering typical solar system values for the composition of the cometary material. It is possible that the silicate carbonate cycle, which regulates the level of atmospheric CO2 on Earth, does not operate on ocean planets in the absence of continents. Such large reservoirs of CO2 are therefore a concern for habitability if totally outgassed into a CO2-rich envelope. Fig. 1 shows the phase diagram of CO2. The critical point lies at T crit = 303K and p crit = 73.8 bar. At a given surface temperature below T crit, the vapor pressure curve actually limits the amount of CO2 which can be outgassed into the atmosphere, independent of the planetary reservoir. It is the aim of this study to quantify this maximum CO2 partial pressure for a range of possible planetary scenarios near the outer edge of the HZ, based on the phase diagram in Fig. 1. In order to put constraints on atmospheric CO2, the interplay between CO2 greenhouse e ff ect, surface temperature and CO2 partial pressure must be taken into account. Therefore, this work will use an atmospheric model which consistently calculates temperature profiles and surface conditions. It will be investigated how di ff erent parameters such as planetary gravity, orbital distance and central star type, N2 pressure and surface albedo influence the behavior of the maximum CO2 partial pressure. The paper is organized as follows: Sect. 2 presents the model and scenarios, Sect. 3 the results and Sect. 4 a discussion. We conclude with Sect. 5.", "pages": [ 1, 2 ] }, { "title": "2.1. Atmosphericmodel", "content": "We used a cloud-free, one-dimensional radiative-convective model to determine the globally averaged atmospheric temperature-pressure profile. The original model was first described by Kasting et al. (1984a) and Kasting et al. (1984b). Further developments were introduced by e.g. Kasting (1988), Kasting (1991), Kasting et al. (1993) Mischna et al. (2000), Pavlov et al. (2000) and Segura et al. (2003). The model version used in this work is taken from von Paris et al. (2008) and von Paris et al. (2010) where more details on the model are given. The model atmospheres are assumed to be composed of N2, H2O, and CO2. Temperature profiles are obtained on 52 model layers, approximately spaced equidistantly in log (pressure). The pressure grid is determined from the surface pressure p surf (variable, see below) up to a pressure of 6.6 × 10 -5 bar (fixed) at the model lid. The model calculates the temperature profile by solving the radiative transfer equation. The radiative fluxes are calculated separately for the stellar (mostly visible) and the planetary (mostly IR) flux. The stellar part of the radiative transfer uses gaseous opacities from Pavlov et al. (2000) and Rayleigh scattering formulations from von Paris et al. (2010). Gaseous opacities in the IR are based on Hitemp data (Rothman et al. 1995) and continuum absorption adapted from Clough et al. (1989) and Kasting et al. (1984a). The purpose of the 1D model used here is to calculate an arbitrary range of temperature-pressure scenarios ranging from the outer to the inner boundary of the HZ. Therefore, we used Hitemp in order to have reliable results for wet, hot atmospheres. The choice of the specific opacity database (e.g., Hitran 2008, Hitran 2004, etc.) for gaseous absorption is not critical for the results presented below, i.e. relatively dry, cold scenarios. If the calculated radiative lapse rate is sub-adiabatic, the model performs convective adjustment, assuming a wet adiabatic lapse rate. This wet adiabatic lapse rate is determined considering either CO2 or H2O as condensing species. The treatment of CO2 condensation for the calculation of the adiabatic lapse rate follows von Paris et al. (2010). We assume that CO2 condensation occurs when the atmosphere is supersaturated with respect to CO2, as described by the super saturation ratio S s: where p CO2 is the partial CO2 pressure and p vap , CO2 the saturation vapor pressure of CO2. The chosen value of S s is motivated by measurements reported in Glandorf et al. (2002). Condensation of an atmospheric constituent can occur when S s is closer to unity than the value chosen here. Note that other studies (e.g., Kasting 1991 or Kasting et al. 1993) assumed S s = 1 which represents the thermodynamic lower limit where condensation could occur. The water profile in the model is calculated based on the relative humidity distribution of Manabe & Wetherald (1967). Above the cold trap, the water profile is set to an isoprofile taken from the cold trap value. Despite the fact that CO2 is allowed to condense, the major atmospheric constituents N2 and CO2 are isoprofiles throughout the entire atmosphere, i.e. are assumed to be well-mixed. The impact of fixing the CO2 mixing ratio at the saturation value on the atmospheric energy budget is expected to be rather small, hence would not change our results by much. A more consistent treatment of CO2 condensation (including an altitude-dependent CO2 profile) would involve vertical mass transport and an atmospheric pressure grid which is not in hydrostatic equilibrium in the region of CO2 condensation. Introducing this into our atmospheric model is beyond the scope of the current work.", "pages": [ 2, 3 ] }, { "title": "2.2. Modelprocedure", "content": "The simulations started with a CO2 partial pressure of 73.8 bar, corresponding to the pressure at the critical point, p crit, and an isothermal temperature profile of 320 K, i.e. higher than the critical temperature of 303 K. The choice of the initial temperature profile is not critical for the final outcome of the simulations. We did not allow for CO2 partial pressures higher than 73.8 bar, even though higher pressures are certainly possible (e.g., Venus). The surface pressure in model iteration step t + 1 is recalculated based on the surface temperature T surf as where p N2 is the fixed background pressure of N2. The water vapor pressure is obtained from with p vap , H2O( T surf) the water vapor saturation pressure at surface temperature and p ocean the ocean reservoir assumed (here, 1 Earth ocean, i.e. 270 bar). The CO2 partial pressure is accordingly calculated as Note that this corresponds to assuming a super-saturation ratio of S s = 1 at the surface, in contrast to S s = 1.34 used for the atmospheric CO2 adiabatic lapse rate (see eq. 2). This is motivated by the fact that atmospheric condensation generally requires S s > 1 (i.e., the presence of condensation nuclei). At the surface, however, atmosphere and reservoir are in equilibrium, hence the partial pressure follows the vapor pressure curve. The mixing ratio of N2 is then adjusted via where CN 2 , t + 1, CN 2 , t are the N2 concentrations and p surf , t + 1, p surf , t the surface pressures at iteration steps ( t + 1) and t . Based on the new value for the surface pressure p surf , the pressure grid on the 52 model levels is then re-calculated. Fig. 2 shows a flow chart of the model to illustrate the model procedure. The CO2 saturation vapor pressure p vap , CO2 is taken from Ambrose (1956). It is divided into two temperature regimes. For T > 216 . 6 K (gas over liquid): For T ≤ 216 . 6 K (gas over solid): If surface temperatures remain above 303 K throughout the entire simulation, the maximum CO2 partial pressure is assumed to lie above the critical pressure. However, if surface temperatures converge to values below 303 K, the corresponding CO2 partial pressure is taken as the maximum possible CO2 pressure for the particular planetary scenario.", "pages": [ 3 ] }, { "title": "2.3. Parametervariations", "content": "We varied five important model parameters: The planetary gravity, related to its mass and radius, the type of the central star and the energy input from the star, related to orbital distance, as well as model surface albedo and N2 partial pressure. Table 1 summarizes the varied parameters. where S 0 is the flux currently received by modern Earth (i.e., S 0 = 1366Wm -2 ) and S is a constant factor related to orbital distance (e.g., for Earth, S = 1). In this study, S was varied from S = 0.2 to S = 0.5. Corresponding orbital distances ranged from 0.21-0.34AU, 0.85-1.35AU, 1.41-2.23AU and 2.67-4.22AU for the M-, K-, G- and F-type stars, respectively (based on Kitzmann et al. 2010). The range of stellar insolation considered here roughly covers the outer limit of the HZ for the stellar types used in this work (e.g., GL 581 d with S = 0.29 and early Mars with S = 0.32, are both potentially habitable) as well as orbits slightly closer to or slightly farther away from the central star.", "pages": [ 3, 4 ] }, { "title": "3. Results", "content": "Fig. 3 shows the maximum partial pressures of CO2 as a function of stellar insolation (hence, orbital distance, see Eq. 9) for the nominal runs of Table 1. Additionally shown as triple dot-dashed line in Fig. 3 is the CO2 partial pressure when using an equilibrium temperature assuming zero albedo (i.e., the maximum equilibrium temperature, T eq , max, see eq. 1). This shows that detailed atmospheric modeling (taking into account the greenhouse effect) is indeed needed to obtain consistent constraints on the CO2 partial pressure. Also indicated in Fig. 3 (by the horizontal plain line) is the boundary between liquid and solid phase of surface CO2, i.e. the triple point pressure of 5.1 bar (see the phase diagram, Fig. 1). For maximum CO2 pressures below 5.1 bar, the atmosphere is in equilibrium with CO2 ice, above 5.1 bar, the formation of (shallow) CO2 oceans is suggested. Fig. 4 shows sample temperatures profile of the simulations, i.e. a 1 g planet at S = 0.35, with a N2 pressure of 1 bar and AS = 0.13. As can be clearly seen, the K- and M-star planets retain their initial CO2 inventory of 73.8 bar (since at the surface, the atmosphere is not saturated with respect to CO2), whereas for the F- and G-star planets, CO2 partial pressures are below the critical pressure, at 10.9 and 23.2 bar, respectively. The upper stratosphere is sensitive to absorption of stellar radiation in the near-IR bands of CO2 and H2O, resulting in about 30 K increase for an M-star planet compared to the F-star planet. Additionally shown in Fig. 4 are the CO2 vapor pressure curve ( S s = 1, dashed line, eq. 5) which intersects the temperature profile (for the F star and the G star) at the surface. Furthermore, Fig. 4 shows the CO2 condensation curve from eq. 2 ( S s = 1.34) indicating the CO2 convective regime. It is clearly seen that the atmospheres of the F-star and the G-star planet are dominated by a CO2 convective regime, followed by a very shallow near-surface H2O convective regime. In contrast, the K- and M-star planets show a relatively extensive lower troposphere dominated by H2O condensation.", "pages": [ 4 ] }, { "title": "3.1. Effectofstellartype", "content": "From Fig. 3, it is clear that with increasing stellar e ff ective temperature (changing stellar type from M to F), the maximum partial pressure of CO2 decreases. Also, the minimum stellar insolation S min for which maximum CO2 pressures above p crit are possible depends sensitively on the stellar type ( S min = 0.25 for the M-star planets and S min ≥ 0.5 for the F-star planets). This is due to the distribution of the stellar energy received by the model planets. With increasing stellar e ff ective temperature, the stellar spectrum is shifted towards lower (bluer) wavelengths, as illustrated by Fig. 5. Broadly, the stellar spectrum can be separated into three regimes, 1) a Rayleigh scattering regime, 2) an absorption regime and 3) a 'window' in between. The Rayleigh scattering regime is here defined as the spectral range where the Rayleigh cross section remains larger than 10 -2 of the maximum value ( λ /lessorsimilar 0.75 µ m). 0 0 The absorption regime starts at about 1.5 µ m where the first strong water and CO2 absorption bands occur. At the high CO2 partial pressures considered in this work, both the Rayleigh scattering regime and the absorption regime are almost entirely optically thick to incoming stellar radiation (i.e., no radiation reaching the surface). In the Rayleigh scattering regime, radiation is reflected back to space (high spectral albedo), whereas in the absorption regime, the radiation is deposited in the upper to middle atmosphere (very low spectral albedo), as illustrated in Fig. 6 for a 2 and 20 bar CO2 atmosphere. Depending on spectral type, the actual percentage of stellar radiation contained in the 'window' changes quite considerably, as illustrated in Fig. 5 (around 50% for the M star, only 30% for the F star). Therefore, the planetary albedo becomes larger for increasing stellar e ff ective temperature (M to F) because of the increasingly important contribution of Rayleigh scattering, and thus surface temperatures and corresponding CO2 partial pressures are lower.", "pages": [ 4, 5 ] }, { "title": "3.2. Effectofplanetarygravity", "content": "The most noticeable e ff ect when changing the planetary gravity g is the e ff ect on atmospheric column density C . At constant pressure p , C and g are related linearly via C ∼ pg -1 . Hence, an increase in gravity leads to a corresponding decrease of atmospheric column density. This leads to three important e ff ects. Firstly, such a decrease in atmospheric column density leads to decreased Rayleigh scattering, hence a lower planetary albedo (see Fig. 6), hence favors surface warming. Furthermore, less atmospheric column density leads to less near-IR absorption of stellar radiation, hence higher albedo (again, see Fig. 6), hence surface warming (more starlight reaches the surface) and stratospheric cooling. On the other hand, a decreased atmospheric column density leads to less greenhouse e ff ect (GHE), hence surface cooling. The net result on surface temperature when combining these three e ff ects (either cooling or warming) depends on the amount of CO2 and the stellar type which determines the planetary albedo and stellar energy distribution (see Figs. 5 and 6). Fig. 7 shows the ratio between calculated CO2 pressures at 1 g and 3 g . At low stellar insolation, hence low CO2 pressures (see Fig. 3), increasing gravity leads to cooler surface temperatures, and consequently lower CO2 partial pressures (i.e., a ratio higher than 1 for all stars except the F star in Fig. 7). This indicates that the impact of the reduced GHE is dominating, in agreement with other studies of optically rather thin planetary atmospheres (e.g., Rauer et al. 2011). In contrast, at higher stellar insolation (and correspondingly higher CO2 pressures), increasing gravity leads to warmer surface temperatures, hence higher CO2 partial pressures (i.e., a ratio lower than 1 in Fig. 7), implying that the decrease of the GHE is compensated by the decrease in planetary albedo. The influence of the stellar type is clearly seen in Fig. 7. For the M-star planet, with very little radiation in the Rayleigh regime (see Fig. 5), the e ff ect of increasing gravity is much higher than for the F-star planet, for which Rayleigh scattering is very important.", "pages": [ 5, 6 ] }, { "title": "3.3. Implicationsforhabitability", "content": "As can be inferred from Fig. 3, our calculations imply that relatively massive CO2 atmospheres of the order of several bars are possible for almost all scenarios, even for planets orbiting far from their central star (stellar insolation S /greaterorsimilar 0.25). At the triple point temperature of water, i.e. 273 K, which permits its liquid phase, the CO2 vapor pressure is about 34 bar (see Fig. 1). Hence, Fig. 3 implies that liquid surface water can be achieved for stellar insolation S 34bar as low as S 34bar = 0.25 for the M-type star and S 34bar = 0.4 for the F-type star, providing a su ffi ciently large source of CO2 is available for outgassing on the planet. This is, however, not the outer edge of the HZ, since surface temperature is not necessarily a monotonic function of CO2 partial pressure (known as the maximum greenhouse effect, e.g., Kasting et al. 1993). The CO2 pressures corresponding to the maximum surface temperatures are therefore expected to be somewhat lower than the maximum CO2 pressures in Fig. 3. Hence, the outer edge of the HZ is most likely located at lower stellar insolation (i.e., farther away from the star), than S 34bar.", "pages": [ 6 ] }, { "title": "3.4. N 2 partialpressure", "content": "The results of the N2 study (Sect. 2.3 and Table 1) are shown in Fig. 8. As expected, for the high CO2 partial pressures found for higher stellar insolation, the e ff ect of varying N2 is negligible, given that CO2 is a much more e ffi cient Rayleigh scatterer than N2. However, for lower stellar insolation, and consequently lower CO2 partial pressures, the e ff ect of N2 becomes discernible. At these lower stellar insolation, the warming e ff ect of adding N2 to the atmosphere is clearly dominating, since the calculated maximum CO2 pressures increase with increasing N2 partial pressure. The e ff ect is rather pronounced (almost a factor of 4 when increasing pN 2 from 0.1 to 10 bar) for the M star since Rayleigh scattering does not contribute greatly to the overall energy budget for these cases (most of the stellar radiation is emitted at wavelengths where Rayleigh scattering is negligible, see Fig. 5). For the F-star simulations, maximum CO2 pressures increase only by about 30%, i.e. warming and cooling e ff ects approximately cancel out.", "pages": [ 6 ] }, { "title": "3.5. Surfacealbedo", "content": "The results of the surface albedo study (Sect. 2.3 and Table 1) are presented in Fig. 9 which shows the decrease in calculated maximum CO2 pressure when increasing the surface albedo. At S = 0.2, the decrease of CO2 pressure is rather large, reaching about a factor of 20 for the M-type star. For a planet orbiting around an F-star, calculations imply maximum CO2 pressures of the order of 0.1 bar, so rather a teneous atmosphere. At S = 0.4, the e ff ect of increasing surface albedo is smaller than at S = 0.2, but still reaches about a factor of 2-3 for the F-type star. Fig. 9 shows that, at S = 0.2, the M-star planet is much more sensitive to a change in surface albedo (a reduction of a factor of about 20 in CO2 pressure) than the F-star planet (a factor of 8), as seen by the steeper slope of the M-star line. The sensitivity is generally increasing for increasing stellar e ff ective temperature (type from M to F). This is due to the larger amount of stellar energy emitted in the window regime (see Sect. 3.1 and Fig. 5). Hence, the response to an increase in surface albedo, which a ff ects principally the window, is more pronounced for the Mstar planet and for lower stellar insolation (and correspondingly lower CO2 partial pressures). For example, at S = 0.4, the reduction for the F-star planet is decreased to about a factor of 2. In order to investigate the combined e ff ect of, e.g., an increase in N2 partial pressure and an increase in surface albedo, we performed some additional test runs with both parameters changed. For the M star case, for example, the e ff ect of N2 was nearly unaltered even at high surface albedo. At S = 0.2, an increase in surface albedo reduced the maximum CO2 pressure from 8.2 to roughly 0.4 bar (see Fig. 9) whereas an increase of N2 partial pressure increased the maximum CO2 from 8.2 to 16.3 bar (see Fig. 8). At high surface albedo and high N2 pressure, the maximum CO2 pressure obtained was 14.3 bar, i.e. nearly as high as for the simulations at low surface albedo.", "pages": [ 7 ] }, { "title": "4.1. H 2 O-CO 2 oceans", "content": "As has been shown above (Fig. 3), for planets orbiting within the HZ of K-G-F stars there is a region of liquid surface CO2 combined with surface temperatures above 273 K, i.e. liquid surface H2O. This means that it is possible to form H2OCO2 oceans. Then, the question of planetary habitability would depend strongly on the pH of the liquid, even though extremophiles on Earth could support quite low pH values (e.g., Rothschild & Mancinelli 2001). A detailed investigation of this interesting issue is however beyond the scope of this work.", "pages": [ 7 ] }, { "title": "4.2. Implicationsofmodelassumptions", "content": "The 1D atmospheric model used in this work is based on relatively few, simple assumptions. Most of these assumptions are physically justified, i.e. the assumption of adiabatic temperature gradients in the troposphere or radiative transfer as the main energy transport mechanism in the upper atmosphere. However, some of them (presence of clouds, greenhouse gases, water profile, etc.) are model-specific, hence need to be discussed further with respect to their possible influence on the results presented above. The model is a cloud-free code, hence the potential impact of CO2 clouds on the climate is neglected. It was shown by several authors that this potential impact could be quite large (e.g., Forget & Pierrehumbert 1997, Mischna et al. 2000, Colaprete & Toon 2003, Wordsworth et al. 2010b, Wordsworth et al. 2011). However, this e ff ect depends sensitively on cloud opacity, cloud coverage and cloud altitude. In addition, the e ff ect of clouds is also probably very dependent on stellar type (see, e.g., Kitzmann et al. 2010 investigating the e ff ect of stellar type for H2O clouds). Investigating this is therefore a subject of further studies. Furthermore, the model atmospheres considered in this work contained only the greenhouse gases CO2 and water. This choice may be restrictive when applied to our own solar system, since other species, such as O3, SO2, CH4, and N2O, have been con- sidered in models of the early Earth or early Mars climate (e.g., Yung et al. 1997, Buick 2007, Haqq-Misra et al. 2008). But given that the concentration of these gases depend on very specific planetary scenarios (e.g., outgassing history, biospheric evolution, etc.), assuming them in the context of exoplanets (without any geological or other constraints) is rather arbitrary. However, the impact on stratospheric temperatures through the absorption of UV (e.g., O3 and SO2) or near-IR (e.g., CH4) stellar radiation is potentially important. Radiative transfer in dense, CO2-dominated atmospheres presents many challenges (e.g., collision-induced absorption, sub-Lorentzian behavior of line wings, etc.). The parametrization of the collision-induced absorption (CIA) used in this study is taken from Kasting et al. (1984b). A recent study (Wordsworth et al. 2010a) presented a revised parametrization, showing that the calculation presented by Kasting et al. (1984b) most likely over-estimates the opacity. In order to estimate the impact of the CIA uncertainties on our results, we performed a sensitivity study with a reduced (by roughly a factor of 2) CIA. The conclusions however did not change qualitatively. At S = 0.2, calculated CO2 maximum pressures around K-, G- and F-stars decreased by less than 50%, for the M-star the maximum CO2 pressure decreased from 8.2 to 3.0 bar. At S = 0.35, results changed less than 20% except for the K-star planet, where a maximum CO2 pressure of 47.3bar was calculated, instead of 73.8 bar (i.e., the critical pressure of CO2, see Fig. 3). Therefore, our calculations (using Kasting et al. 1984b) are likely to be overestimates of the maximum CO2 partial pressures. The model uses a super-saturation of S s = 1.34 to determine the CO2 convective regime (see eq. 2). The choice of S s has been shown to be very important for early Mars climate simulations, e.g. Pollack et al. (1987) (using S s = ∞ ) find significantly higher surface temperatures ( > 30K) than Kasting (1991) (using S s = 1). The assumed S s = 1.34 is based on Glandorf et al. (2002), a value observed for specific conditions (e.g., dust loading available for nucleation) which could be di ff erent on exoplanets (as low as S s = 1, but also possibly significantly higher). In this sense, the calculated maximum CO2 pressures are not necessarily upper limits. To further investigate this, we performed some sensitivity simulations with S s = 1. As expected, calculated maximum CO2 pressures were lower, of the same order of magnitude as for the CIA study mentioned above. However, the main conclusions obtained in this work (i.e., the existence of maximum CO2 pressures far below the critical pressure) were not a ff ected. The relative humidity profile used in this work (Manabe & Wetherald 1967) has been derived from observations of modern Earth. It has been used in many 1D simulations of terrestrial exoplanets, both Earth-like (e.g., Segura et al. 2003, Grenfell et al. 2007a) and not (e.g., von Paris et al. 2010, Wordsworth et al. 2010b). Since the humidity profile is anything but trivial to model in 1D simulations, some authors chose to fix relative humidity at an isoprofile (e.g., Kasting 1991). However, given the large amounts of CO2 in the model atmospheres (73.8 bar at 303 K), the impact of water (42 mbar at 303 K) on atmospheric structure (via near-IR absorption) and surface conditions (via the GHE) is somewhat negligible. Therefore, the choice of the relative humidity profile is probably not important.", "pages": [ 7, 8 ] }, { "title": "4.3. Synchronousrotation", "content": "For planets orbiting very close to their star, tidal locking of the planetary rotation with the orbital period is very likely. The time scale t lock of tidal locking is very sensitive to orbital distance ( t lock ∼ a 6 , a orbital distance, see e.g. Grießmeier et al. 2005). Hence, tidal locking is mainly an issue for the habitability of planets orbiting around M stars due to the closeness of the HZ to the star. It has been argued that for planets with a perpetual nightside, the atmosphere could collapse since the nightside forms a cold trap for the volatiles, which, in the context of this work, could present an alternative way of obtaining maximum CO2 pressures. However, as has been shown by numerous modeling studies (e.g., Joshi et al. 1997, Joshi 2003, Wordsworth et al. 2011, Kite et al. 2011), moderately dense atmospheres containing hundreds of millibars or more of CO2 are su ffi cient to avoid atmospheric collapse by means of atmospheric circulation. Hence, the M-star simulations presented in this work are not thought to be subject to atmospheric collapse induced by synchronous rotation.", "pages": [ 8 ] }, { "title": "5. Conclusions", "content": "We have presented a detailed parameter study to constrain the maximum CO2 partial pressure possible for terrestrial exoplanets, using a 1D cloud-free atmospheric model. Parameters investigated included the central star type, the orbital distance and the planetary gravity. Furthermore, we investigated the influence of N2 partial pressure and the surface albedo on the maximum CO2 partial pressure. Results imply that super-critical atmospheres (i.e., p CO2 ≥ p crit = 73.8 bar) are possible for planets around M stars for stellar insolation corresponding to S crit = 0.25 or higher. For increasingly bluer stars (i.e., higher e ff ective temperatures), this super-critical stellar insolation increases (e.g., S crit > 0.5 for an F-type star). For lower stellar insolation, the calculations presented here imply that there is indeed a maximum CO2 partial pressure, even if the planets are orbiting well within the habitable zone. Nevertheless, massive CO2 atmospheres of the order of bars are still possible for most scenarios. For planets orbiting very far from an F-type central star (e.g., S = 0.2 in this work), CO2 partial pressures could be constrained to be less than 1 bar. The e ff ect of planetary gravity is twofold. For low stellar insolation and corresponding cold surface temperatures, increasing planetary gravity leads to a decrease of maximum CO2 partial pressure due to less atmospheric greenhouse e ff ect. At higher stellar insolation, an increase of planetary gravity increases the calculated maximum CO2 partial pressure because of less Rayleigh scattering. For F-star planets, the e ff ect is up to a factor of 2, depending on stellar insolation, whereas for an M-star planet, a factor of about 4 has been calculated. Increasing the N2 partial pressure leads to warmer surface temperatures for all cases, hence corresponding maximum CO2 pressures are higher. The e ff ect reaches up to a factor of 3 for planets around an M star, upon increasing the N2 partial pressure from 0.1 to 10 bar. The surface albedo has an important e ff ect on the values of the maximum CO2 partial pressure. A higher surface albedo leads to cooler surface temperatures, hence less CO2 in the atmosphere. Decreases of about a factor of 20 have been shown when increasing the surface albedo from 0.13 to 0.4. The presence of CO2 and H2O clouds could alter these results because of their potentially large impact on the planetary energy balance. However, our (clear-sky) results show in a robust way that the composition and the evolution of planetary atmospheres strongly depend on orbital and planetary parameters. Although a consistent model for the determination of CO2 partial pressures must take processes such as sequestration or outgassing into account, our results show that there is a fundamental thermodynamic limit to the amount of CO2 in terrestrial atmospheres, independent of the planetary reservoir. Hence, a more detailed coupling between interior, surface and atmosphere models should be used to accurately predict atmospheric composition of terrestrial planets. Acknowledgements. P.v.P., P.H. and F. Selsis acknowledge support from the European Research Council (Starting Grant 209622: E3ARTHs). This research has been partly supported by the Helmholtz Association through the research alliance 'Planetary Evolution and Life'. We thank M. Godolt and D. Kitzmann for valuable discussions and comments regarding this manuscript.", "pages": [ 8, 9 ] }, { "title": "References", "content": "Abbot, D. S., Cowan, N. B., & Ciesla, F. J. 2012, accepted in Astrophys. J. Ambrose, D. 1956, Trans. Faraday Society, 52, 772 Anglada-Escud'e, G., Arriagada, P., Vogt, S. S., et al. 2012, Astrophys. J. Letters, 751, L16 Bonfils, X., Delfosse, X., Udry, S., et al. 2011, submitted to Astron. Astrophys. Borucki, W. J., Koch, D. G., Basri, G., et al. 2011, Astrophys. J., 736, 19 Borucki, W. J., Koch, D. G., Batalha, N., et al. 2012, Astrophys. J., 745, 120 Buick, R. 2007, Geobiology, 5, 97 Clough, S., Kneizys, F., & Davies, R. 1989, Atm. Research, 23, 229 Elkins-Tanton, L. T. & Seager, S. 2008, Astrophys. J., 685, 1237 Hart, M. H. 1978, Icarus, 33, 23 Howard, A. W., Marcy, G. W., Johnson, J. A., et al. 2010, Science, 330, 653 Joshi, M. 2003, Astrobiology, 3, 415 Kaltenegger, L. & Sasselov, D. 2011, Astrophys. J. Letters, 736, L25 Kasting, J. F. 1988, Icarus, 74, 472 Kasting, J. F. 1991, Icarus, 94, 1 Kasting, J. F., Pollack, J. B., & Crisp, D. 1984b, J. Atmospheric Chem., 1, 403 Kasting, J. F., Whitmire, D. P., & Reynolds, R. T. 1993, Icarus, 101, 108 Kite, E. S., Gaidos, E., & Manga, M. 2011, Astrophys. J., 743, 41 Kite, E. S., Manga, M., & Gaidos, E. 2009, Astrophys. J., 700, 1732 L'eger, A., Selsis, F., Sotin, C., et al. 2004, Icarus, 169, 499 Lovis, C., Mayor, M., Pepe, F., et al. 2006, Nature, 441, 305 Manabe, S. & Wetherald, R. T. 1967, J. Atmosph. Sciences, 24, 241 Mayor, M., Bonfils, X., Forveille, T., et al. 2009, Astron. Astrophys., 507, 487 Mayor, M., Marmier, M., Lovis, C., et al. 2011, submitted to Astron. Astrophys. Mischna, M. A., Kasting, J. F., Pavlov, A., & Freedman, R. 2000, Icarus, 145, 546 Rossow, W. B. & Schi ff er, R. A. 1999, Bull. Americ. Meteor. Soc., 80, 2261 Rothman, L. S., Wattson, R. B., Gamache, R., Schroeder, J. W., & McCann, ed. J. C. Dainty, 105-111 Rothschild, L. J. & Mancinelli, R. L. 2001, Nature, 409, 1092 Segura, A., Kasting, J. F., Meadows, V., et al. 2005, Astrobiology, 5, 706 Segura, A., Krelove, K., Kasting, J. F., et al. 2003, Astrobiology, 3, 689 Selsis, F., Kasting, J. F., Levrard, B., et al. 2007, Astron. Astrophys., 476, 1373 Sotin, C., Grasset, O., & Mocquet, A. 2007, Icarus, 191, 337 Tinney, C. G., Wittenmyer, R. A., Butler, R. P., et al. 2011, Astrophys. J., 732, 31 Udry, S., Bonfils, X., Delfosse, X., et al. 2007, Astron. Astrophys., 469, L43 von Paris, P., Gebauer, S., Godolt, M., et al. 2010, Astron. Astrophys., 522, A23 von Paris, P., Rauer, H., Grenfell, J. L., et al. 2008, Planet. Space Science, 56, 1244 Warren, S. G., Wiscombe, W. J., & Firestone, J. F. 1990, J. Geophys. Res., 95, 14717 Wittenmyer, R. A., Tinney, C. G., Butler, R. P., et al. 2011, Astrophys. J., 738, 81 Wordsworth, R., Forget, F., & Eymet, V. 2010a, Icarus, 210, 992 Wordsworth, R., Forget, F., Selsis, F., et al. 2010b, Astron. Astrophys., 522, A22 Wordsworth, R. D., Forget, F., Selsis, F., et al. 2011, Astrophys. J. Letters, 733, L48", "pages": [ 9 ] } ]

[ { "title": "ABSTRACT", "content": "Context. The features of pulsed γ -ray emission from classical and millisecond pulsars indicate that the high energy radiation processes in their inner magnetospheres occur in a similar way. In the last decade several TeV γ -ray nebulae have been discovered around classical pulsars. The above facts suggest that γ -rays should be produced also in the surroundings of millisecond pulsars. Aims. We discuss a model for the bow shock nebula around the well known Black Widow binary system containing the millisecond pulsar B1957 + 20. This model predicts the existence of a synchrotron X-ray and inverse Compton γ -ray nebula around this system. Wewant to find out whether γ -ray emission from the nebula around B1957 + 20 could be detected by the future and present Cherenkov telescopes. Methods. Using the Monte Carlo method we followed the propagation of relativistic electrons in the vicinity of the pulsar. We calculated the very high energy radiation produced by them in the synchrotron process and the inverse Compton scattering of the Microwave Background Radiation and of the infrared radiation from the galactic disk. We also computed the X-ray emission produced by the electrons in the synchrotron process. Results. Weshow that the hard X-ray tail emission observed from the vicinity of B1957 + 20 can be explained by our model. Moreover, we predict that the TeV γ -ray emission produced by the electrons in the inverse Compton process should be detectable by the future Cherenkov Telescope Array and possibly by the long term observations with the present Cherenkov arrays such as MAGIC and VERITAS. The γ -ray emission from B1957 + 20 is expected to be extended, inhomogeneous, and shifted from the present location of the binary system by a distance comparable to the radius of the nebula. Key words. pulsars: general - stars: binaries: close - radiation mechanisms: non-thermal - gamma-rays: general", "pages": [ 1 ] }, { "title": "High energy emission from the nebula around the Black Widow binary system containing millisecond pulsar B1957+20", "content": "W. Bednarek 1 &J. Sitarek 2 1 2 IFAE, Edifici Cn., Campus UAB, E-08193 Bellaterra, Spain e-mail: [email protected]; [email protected] Department of Astrophysics, University of Ł´od´z, ul. Pomo rska 149 / 153, 90-236 Ł´od´z, Poland Received ; accepted", "pages": [ 1 ] }, { "title": "1. Introduction", "content": "PSR B1957 + 20 was the first millisecond pulsar (MSP) discovered within the binary system belonging to the class of Black Widows (Fruchter et al. 1988). This pulsar has a very small mass companion ( ∼ 0 . 022 M /circledot , van Paradijs et al. 1988) which evaporates under the irradiation from the pulsar magnetosphere. The pulsar has the period of 1.607 ms, the surface magnetic field of ∼ 10 8 G, and the rotational energy loss rate of 7 . 5 × 10 34 erg s -1 . The distance to the binary system is estimated on 2.5 kpc (from the model for Galactic electron density) consistent with the recently established lower limit ∼ 2 kpc (van Kerkwijk et al. 2011). The binary system is compact with the orbital radius of 1 . 5 × 10 11 cm. The companion star has the radius ∼ 10 10 cmand the surface temperature which varies between 2900 K for the unilluminated side to 8300 K for the illuminated side (Fruchter et al. 1995, Reynolds et al. 2007). Therefore, stellar radiation is not expected to create a very strong target for relativistic particles within the binary system. At present, the companion star loses mass at a rather low rate estimated on 10 -10 M /circledot yr -1 (Takata et al. (2012). The importance of the high energy processes in the vicinity of PSR B1957 + 20 has become clear with the discovery of an H α emission nebula (Kulkarni & Hester 1988). This emission is expected to be produced in shocks formed in the interaction of the pulsar wind with the interstellar medium. A clear bow shock has been detected which apex is located at the distance of ∼ 0 . 02 pc from the pulsar. The bow shock appears due to the motion of the binary system with the velocity 220 km s -1 through the interstellar medium (Arzoumanian et al. 1994). The X-ray emission has been also reported from the direction of the binary system in the observations of Chandra (Stappers et al. 2003, Huang & Becker 2007, Huang et al. 2012). This emission comes from the interior of the bow shock creating a tail behind the moving binary system. The length of the tail is ∼ 10 18 cm (Huang et al. 2012). The X-ray emission is well described by a single power law spectrum with the index in the range 2 . 3 -2 . 6 depending on the absorption model. The extended X-ray feature has been interpreted as emission from energetic electrons which radiate on the crossing time scale of this region by the pulsar moving with velocity of 220 km s -1 (Cheng et al. 2006). The Black Widow binary system containing B1957 + 20 was claimed in the past to be a GeV-TeV γ -ray source (Brink et al. 1990). But this early report was not confirmed in the analysis of the EGRET data (Buccheri et al. 1996). In fact, such high energy emission has been suspected already since the discovery of Black Widow pulsars as a result of either the acceleration of particles within the binary system or in the the shock waves of the pulsar wind (e.g. Arons & Tavani 1993, Cheng et al. 2006, Takata et al. 2012). Recently, a pulsed GeV emission from the pulsar B1957 + 20 has been discovered by Fermi (Guillemot et al. 2012). The pulsed spectrum is flat above 0.1 GeV (spectral index close to 2) and extends up to ∼ 4 GeV. The phasogram (light curve folded with the period of the pulsar) shows two well separated peaks. Such structure is also common in the case of classi- pulsars. Therefore, it is expected that the radiation processes in the inner magnetosphere of the millisecond pulsar B1957 + 20 are similar to those occurring in the case of classical pulsars. This strongly indicate that also processes of acceleration of particles in the pulsar wind are expected to occur similarly. Very recently, Wu et al. (2012) reports detection of orbital modulation of the γ -ray emission at energies above ∼ 2.7 GeV from the Black Widow pulsar PSR B1957 + 20. This emission is expected to be produced by electrons from the pulsar wind which comptonize stellar radiation. We investigate the radiation processes in the supposed pulsar wind nebula around the binary system containing PSR B1957 + 20. The synchrotron X-ray and inverse Compton (IC) γ -ray emission is calculated from such nebula for the range of likely parameters. Based on the comparison of the calculated synchrotron spectrum with the observed X-ray emission we conclude on the detectability of the TeV γ -ray emission from the bow shock nebula surrounding PSR B1957 + 20.", "pages": [ 1, 2 ] }, { "title": "2. The nebula around binary system downstream of the bow shock", "content": "Since the proprieties of high energy γ -ray emission from the millisecond pulsars and classical radio pulsars are surprisingly similar (see the first pulsar catalogue, Abdo et al. 2010), it seems clear that the processes occurring in their inner magnetospheres are these same. Therefore, millisecond pulsars should also produce relativistic pulsar winds with the parameters similar to those observed around classical pulsars. However, nebulae around MSPs are expected to have a very complicated structure (and also other proprieties) since many MSPs form compact binary systems which additionally move in the interstellar space with large velocities. In fact, this is the case of the binary system PSR B1957 + 20. The pulsar wind around B1957 + 20 is expected to interact with the induced wind of the low mass companion star within a small solid angle, of the order of ∼ 0.01 sr, corresponding to eclipse time of the pulsar radio emission by the wind of the companion star (Fruchter et al. 1988). Therefore, most of the pulsar wind is expected to escape una ff ected from the binary system. Due to the fast velocity of the binary system, the pulsar wind has to interact with the interstellar medium creating a bow shock. Such bow shock has been detected in H α emission in the case of B1957 + 20. The distance of the apex of the bow shock to the pulsar is estimated on ∼ 0 . 02 pc (Kulkarni & Hester 1988). This bow shock confines the pulsar wind at least in the direction of pulsar's motion. Relativistic electrons in the wind can di ff use mainly in the direction opposite to the pulsar's motion. Cheng et al. (2006) have proposed that the synchrotron radiation from such ultrarelativistic electrons is responsible for the observed X-ray tail extending along the axis of the bow shock in the direction opposite to the pulsar velocity. We intend to perform calculations of the Inverse Compton (IC) γ -ray emission from such relativistic electrons applying general scenario proposed by Cheng et al. (2006). These authors argue that e ffi cient synchrotron emission by electrons can occur on the dynamical time scale of the pulsar crossing the length of the tail estimated on ∼ 10 18 cm (Huang et al. 2012). This dynamical time scale is equal to, where the velocity of the binary system is v bin = 220 km s -1 and the length of the tail is R = 1 R 1 pc. Applying the observed length of the synchrotron emission, we can estimate the optical depth for electrons on the IC scattering of the Microwave Background Radiation (MBR) in the Thomson regime (true for electrons with energies below ∼ 100 TeV) on, τ = c τ dyn n MBR σ T ≈ 0 . 35, where c is the velocity of light, σ T is the Thomson cross section, and n MBR is the photon density of the Microwave Background Radiation. Note however that electrons cool only partially in the region of observed X-ray emission. Many of them escape from this region but continue to interact with the MBR and other soft photon field, producing high energy γ -rays. Therefore, we expect the appearance of the γ -ray nebula in the vicinity of the Black Widow binary pulsar. This nebula should be shifted in respect to the observed location of the binary system in the direction opposite to the pulsar's motion. On the other hand, the energy loss time scale of electrons on the IC scattering in the Thomson regime is, where U rad is the energy density of the soft radiation field equal to 0 . 3 eV cm -3 for the Microwave Background Radiation (MBR) and to ∼ 1 . 5 eV cm -3 for the infrared radiation with characteristic energies ∼ 0 . 01 eV, produced in the galactic disk (e.g. see the values calculated in Hui et al. (2011) based on the GALPROP code developed by Strong & Moskalenko 1998), and m e is the rest mass of an electron. For these energy densities we obtain the energy loss time scales of the order of τ T IC ∼ 6 . 3 × 10 19 /γ e s for the MBR and ∼ 1 . 3 × 10 19 /γ e s for the infrared radiation, where γ e is the Lorentz factor of the electrons. In order to cool the electrons e ffi ciently on the IC process during the dynamical time of the moving pulsar, the emission region should have the diameter of the order of R ≈ 8 . 5 × 10 7 /γ e pc. For example, in the region of 10 pc, electrons with energies larger than ∼ 4 TeV (but below ∼ 100 TeV since the electrons have to interact in the Thomson regime) should be able to produce e ffi ciently γ -rays in the IC process by scattering infrared photons from the galactic disk. Note that, the region of the γ -ray production in the IC process should be clearly shifted from the pulsar position in the direction of the observed tail X-ray emission. This region should be also inhomogeneous with higher energy γ -rays produced closer to the pulsar. In the above estimates we neglected the energy density of stellar photons, in respect to the MBR and infrared radiation at the region of the acceleration of electrons (the shock in the pulsar wind). In fact, the energy density of stellar photons depends on the distance from the star as U /star ≈ 4 . 5 × 10 -5 / D 2 18 eV cm -3 , where the distance from the star is D = 10 18 D 18 cm. It is assumed that the companion star in the binary system PSR 1957 + 20 has the radius 10 10 cm and most of its surface has temperature close to ∼ 3000 K (Fruchter et al. 1995). For these parameters, the electron energy losses are dominated by scattering of the infrared photons for distances above ∼ 5 × 10 15 cm. Note also that the scattering of the optical photons from the star occurs in the Klein-Nishina regime for electrons with energies above ∼ 100 GeV. Therefore, the e ff ects of scattering stellar radiation by the TeV electrons can be safely neglected. The region of the γ -ray production can be also a ff ected by the di ff usion of the electrons in the pulsar wind downstream of the pulsar wind shock. We estimate the di ff usion distance of the electrons, as a function of their energy, and compare it with the time scale corresponding to the dynamical motion of the pulsar. For the Bohm di ff usion approximation, the di ff usion distance is R dif = √ 2 D dif t , where D dif = cR L / 3 is the di ff usion coe ffi cient, R L is the Larmor radius of electrons, B is the magnetic field strength in the considered region, and t is the di ff usion time. If B is fixed on 1 µ G, then D dif ≈ 1 . 5 × 10 19 γ e cm 2 s -1 and R dif ≈ 5 . 5 × 10 9 √ γ e t cm. The spread of the emission region due to the di ff usion process is smaller than that one due to the motion of the pulsar, i.e. R dif < R dyn = v pul t , for the following condition t > 6 . 2 × 10 4 γ e s. We compare this condition with the energy loss time scale on the IC process in the Thomson regime (see Eq. 2 and estimates below). It is found that electrons with energies below ∼ 7 TeV lose energy on production of γ -rays when the ballistic motion of the binary system determines the morphology of the γ -ray source. We conclude that depending on the electron energy, the dimension of the γ -ray source is determined either by the motion of the Black Widow binary system through the interstellar medium or by the di ff usion process of the electrons.", "pages": [ 2, 3 ] }, { "title": "3. Relativistic electrons in nebula", "content": "We estimate the magnetic field strength around the pulsar, above its light cylinder radius, by extrapolating it from the pulsar surface. The magnetic field strength is then given by, where R = 10 18 R 18 cm is the distance from the pulsar, B NS = 10 8 B 8 G is the magnetic field strength on the neutron star surface, P = 10 -3 P ms s is the period of the millisecond pulsar, and σ is the magnetization parameter of the pulsar wind. σ has been estimated in the case of the Crab Nebula on 0.003 (de Jager & Harding 1992) and in the case of the Vela Nebula on ∼ 0.1 (Sefako & de Jager 2003). σ is expected to be in the range 0 . 001 -0 . 01 in the modeling of the Crab Nebula presented by Kennel & Coroniti (1984). The magnetic field given by Eq. 3, is expected to be enhanced at the shock region in the pulsar wind by a factor of ∼ 3. Downstream of the shock, electrons are isotropized and start to radiate e ffi ciently synchrotron radiation. Therefore, the magnetic field in the region downstream of the shock is an important factor which determines the di ff usion of the relativistic electrons and production of the synchrotron radiation. The maximum energies to which the electrons can be accelerated in the pulsar shock region can be estimated from, Note that this simple formula gives the energies of electrons present in the Crab Nebula comparable to those expected from the modelling of its multi-TeV γ -ray spectrum (e.g. de Jager & Harding 1992). As noted above, Chandra has detected the tail behind the pulsar B1957 + 20 in the energy range 0.3-8 keV (Huang et al. 2012). If this emission is due to the synchrotron process from the relativistic electrons, then the Lorentz factors of the electrons can be estimated from, where ε = 8 keV is the energy of synchrotron photons, B and B cr = 4 . 4 × 10 13 G are the magnetic field in the emission region and the critical magnetic field strength. The inspection of the above equations allows us to conclude that the production of the synchrotron photons with ∼ 10 keV energies is possible provided that the Lorentz factors of electrons are at least γ e = 2 . 2 × 10 8 P ms R 1 / 2 18 / ( σ 1 / 4 B 1 / 2 8 ), obtained by substitution of Eq. 3 into Eq. (5). Electrons are accelerated to such energies provided that the magnetic field is strong enough, i.e. the shock in the pulsar wind appears close to the pulsar. For the parameters of PSR B1957 + 20, the distance of the shock has to be below R 18 ≈ 3 . 4 × 10 3 σ 3 / 2 , which equals to R sh ≈ 10 17 -10 20 cm for σ in the range 0 . 001 -0 . 1. This condition is consistent with the observations of the PWNe around classical pulsars. For example, in the case of the Crab Nebula the shock is located at the distance of ∼ 3 × 10 17 cm (Kennel & Coroniti 1984). It is not clear at present in what process electrons reach such large energies. This might be reconnection of the magnetic field or the shock acceleration mechanism. In the second case, the limit on the maximum energies of the electrons have to be consistent with the limit due to the presence of the synchrotron energy losses already during the acceleration process. The maximum energies of the electrons, due to the saturation by the synchrotron energy losses, can be derived from the comparison of the electron acceleration time scale, with the synchrotron energy loss time scale, where ˙ E syn = (4 / 3) cU B σ T E 2 e / m 2 e ≈ 0 . 0027 B 2 E 2 TeV / s, the acceleration e ffi ciency is parametrised by the factor χ = 10 -1 χ -1, and E e is the electron energy in TeV. Energies of the electrons can not be larger than, For the pulsar with the parameters of PSR B1957 + 20, E max is lower than E syn max for the location of the shock at R 18 > 0 . 05 σ 3 / 2 , which corresponds to R sh > 1 . 6 × 10 15 cmfor σ = 0 . 1. Therefore, we conclude that for the expected localizations of the shock in the nebula around the pulsar B1957 + 20 (above ∼ 10 15 cm), the synchrotron energy losses can not limit the acceleration process of the electrons below the maximum possible energies given by Eq. 4.", "pages": [ 3, 4 ] }, { "title": "4. Production of high energy radiation", "content": "We calculate the γ -ray spectra produced by relativistic electrons in the IC scattering of the MBR and the infrared radiation from the galactic disk. These electrons also produce synchrotron emission which can extend up X-ray energy range. It is commonly expected that electrons accelerated at the pulsar wind shock obtain the power law spectrum. We assume that this spectrum has a lower energy cut-o ff at energies corresponding to the Lorentz factor of the pulsar wind, i.e γ w is equal to a few times 10 6 . In our calculations we fix this value on 3 TeV, in agreement with the modelling of the PWNe (Kennel & Coroniti 1984) and recent calculations of the spectra of the electrons leaving the inner magnetospheres of the millisecond pulsars in the frame of the pair starved polar cap model (e.g. Zajczyk et al. 2010). The electrons take a significant part of the energy lost by the millisecond pulsar, which is of the order of ∼ 10%. The spectrum of the electrons extends up to the maximum energy described in Sect. 3. These electrons are accelerated close to the pulsar wind shock and di ff use to the outer region creating a tail trailing behind the pulsar. In this calculations we take the energy density of the infrared galactic disk emission equal to 1 . 5 eV cm -3 . It is assumed that the magnetic field is enhanced by a factor of 3 in the pulsar wind shock and at larger distances continue to drop according to Eq. 3 up to the minimum value B min. This minimum magnetic field strength can be even below the magnetic field strength in the interstellar space (of the order of ∼ 2-6 µ G), since the volume of the pulsar wind nebula is not penetrated by the interstellar medium. We assume that electrons are injected at the distance of the shock from the pulsar, R sh. They slowly di ff use outward according to the Bohm di ff usion model in a decreasing magnetic field. During the di ff usion process, the electrons interact with the background radiation producing GeV-TeV γ -rays in the IC process. We apply the Monte Carlo method in order to determine the energy of the γ -ray photons and the distance from the pulsar at which they are produced. For this purpose we modify the numerical code developed for the interaction and di ff usion of electrons (Bednarek & Sitarek 2007). This code allows us not only to calculate the spectrum of γ -rays produced by electrons but also determine their production sites around the pulsar, i.e. allowing us to study the morphology of the γ -ray source. Since the electrons are immersed in a relatively strong magnetic field, especially close to the pulsar wind shock, we also include in the simulations their synchrotron energy losses during the di ff usion process. We calculate the X-ray spectra produced by these electrons in the synchrotron process. In order to obtain reasonable precision of the IC γ -ray spectra, we simulate the propagation of 1 . 5 × 10 4 electrons per decade of the spectrum. The spectra are obtained within di ff erent regions around the pulsar defined by the radius R Neb. We investigate the dependence of the X-ray and γ -ray spectra on di ff erent parameters which determine the acceleration of the electrons (i.e. the magnetization parameter of the pulsar wind σ , the spectral index of the electrons' spectrum α ', the radius of the pulsar wind shock R sh; and the minimum value of the magnetic field in the nebula B min). As shown in Fig. 2, the TeV γ -ray spectra produced by the electrons in the IC process only weakly depend on the range of the considered parameters. On the other hand, the synchrotron X-ray emission depends on these parameters much stronger (intensity, shape, energy range). The strong dependence of the synchrotron emission is due to the strong dependence of the magnetic field in the vicinity of the pulsar on the assumed parameters of the model. On the other hand, relatively weak dependence of the IC emission is due to the homogeneity of the background radiation field (MBR and infrared galactic background) which is up-scattered by the relativistic electrons. Weconclude that the TeV γ -ray fluxes expected in this model depend rather weakly on the details of the electron spectrum (in the considered range of parameters). However, their intensity is obviously determined by the energy conversion e ffi ciency from the pulsar to the relativistic particles. In contrast, the spectra of the synchrotron radiation in the X-ray range much stronger depend on the spectrum of the electrons and the propagation model. We also investigate the γ -ray production in di ff erent volume around the Black Widow binary system B1957 + 20. The IC γ -ray and the synchrotron X-ray spectra are calculated assuming that this emission is produced within the region with the radius equal to 1.5 pc, 2.5 pc, 5 pc, 10 pc, and 15 pc (see Fig. 3). These dimensions correspond roughly to the angular size of the γ -ray source on the sky equal to 2, 3.4, 7, 14, and 20 arc min for the distance of the source equal to 2.5 kpc. The electrons expand into such a region due to their di ff usion in the nebula. Moreover, the TeV γ -ray source is also expected to be shifted from the present location of the Black Widow binary due to its motion and / or limitted in specific directions by the di ff usion of the electrons confined by the presence of the bow shock. In the case of a source with the radius above ∼ 5 pc, the TeV γ -ray source should appear extended for the telescope array such as MAGIC. Our calculations show that most of the TeV γ -ray emission (i.e. within a factor of two) is already produced within a region with the radius of 5 pc. The shapes of the spectra, produced in specific parts of the γ -ray source, are quite similar since the background radiation field (MBR and infrared), scattered by the relativistic electrons, fills this region homogeneously. Moreover the cooling process of the electrons is not very e ffi cient. The electrons do not usually interact frequently but in a specific interaction lose significant amount of their energy when producing TeV γ -rays. Due to the ine ffi cient cooling, the parts of the spectra at low energies (in the GeV range), produced in the Thomson regime, are very similar. On the other hand, the synchrotron X-ray emission does not depend on the considered radius of the source at energies above a few keV. This can be understood since the hard synchrotron radiation is mainly produced close to the pulsar wind shock within the region with the extend of ∼ 2 pc. There is however an important contribution from the outer nebula to the part of the synchrotron spectrum at lower energies (below a few keV) since these electrons can still produce keV photons in the assumed minimum magnetic field.", "pages": [ 4 ] }, { "title": "5. Comparison with observations of B1957+20", "content": "Finally, we compare the example calculations performed in terms of this modelling with the available observations of the Black Widow binary system B1957 + 20. The X-ray emission, extending along the direction of the motion of the binary, has been detected by Chandra (Stappers et al. 2003, Huang et al. 2012). The X-ray synchrotron emission expected in our model has to be consistent with this observed spectral features. Recently, the pulsed GeV γ -ray emission has been also reported from B1957 + 20 (Guillemot et al. 2012). The IC γ -ray emission, pro- ced in the nebula by relativistic electrons, has to be below this pulsed emission. There are not available any positive detections or the upper limits on the TeV γ -ray emission from this source. We have chosen intermediate parameters of the nebula from the range considered in Sect. 4. The IC and the synchrotron spectra are confronted with the available observations of the binary system containing B1957 + 20 in Fig. 4. We have got good consistency with the level and shape of the X-ray spectrum from the nebula. Note that the X-ray observations put strong constraints on the parameters of the considered model. The emission extending up to ∼ 10 keV requires the presence of electrons with energies at least ∼ 4 . 7 × 10 5 B -1 / 2 µ G GeV(see Eq. 5). On the other hand, the observed X-ray flux constrains the number of the relativistic electrons (which we fix on 10% of the pulsar energy loss rate) and the synchrotron energy loss rate which depends on ∝ B 2 E 2 e . Therefore, it is not so easy to model the observed X-ray spectrum correctly since the change of the parameters have strong e ff ect on the energies and intensity of the emitted synchrotron radiation (see calculations in Fig. 2). We conclude that observed X-ray extended emission put strong constraints on the parameters of the considered model. Having obtained consistency with the observed synchrotron spectrum, we calculate the IC γ -ray spectrum for the same parameters (see the caption of Fig. 4). These spectra are confronted then with the sensitivities of the Cherenkov telescopes. We show the level of the IC emission expected from the region with the radius of ∼ 5 pc, which is shifted from the present location of the binary system by about the same distance in the direction opposite to the movement of the binary due to the motion of the binary system. Therefore, we conclude that the TeV γ -ray source should be extended for the Cherenkov telescopes. The IC γ -ray spectrum is clearly above the 100 hr sensitivity of the future Cherenkov telescope Array (CTA). It is also on the 100 hr sensitivity limit of the MAGIC Cherenkov telescopes. We conclude that even with the present Cherenkov telescopes (MAGIC, VERITAS) the bow shock nebula around the Black Widow millisecond pulsar B1957 + 20 might be detected. Note however that as the source is expected to be extended, the sensitivity of the present Cherenkov telescopes might become worse than the point source sensitivity shown in Figs. 2-4.", "pages": [ 4, 5, 6 ] }, { "title": "6. Conclusion", "content": "Assuming that the millisecond pulsars are able to accelerate electrons to relativistic energies in their vicinity, similarly as observed in the case of nebulae around classical pulsars, we calculate the synchrotron and the IC high energy emission from their nebulae. In fact, the existence of an extended synchrotron nebula has been recently confirmed in the Chandra observations in the case of the Black Widow binary system containing millisecond pulsar B1957 + 20. Therefore, as an example, we consider the bow shock nebula around this object. Note that in contrast to the nebulae around classical pulsars, the soft radiation field in the nebula around B1957 + 20 is not dominated by the synchrotron radiation but by the MBR and infrared radiation from the galactic disk. We have investigated the features of the X-ray and γ -ray spectra for likely range of parameters which determine the nebula, assuming that the propagation of electrons is determined by the di ff usion process and / or the dynamical movement of the binary system. We conclude that the observed extended X-ray emission from the bow shock nebula can be explained by the synchrotron radiation of electrons provided that the energy conversion e ffi ciency from the pulsar to the relativistic electrons is of the order of 10%. The TeV γ -ray emission, produced by the same electrons in the IC scattering process, is expected to be detectable by the future CTA instrument. The predicted emission is also on the level of the 100 hr sensitivity limit of the MAGIC telescopes. However, since the nebula is expected to be extended, due to rather slow cooling process of electrons, the detectability of the TeV γ -ray emission from the nebula around B1957 + 20 may be di ffi cult. Note also that due to the motion of the binary system the TeV γ -ray nebula should be shifted in respect to the direction towards the binary system by the distance comparable to the extend of the source (see also Cheng et al. 2006). Other bow shock nebulae around energetic pulsars should also emit synchrotron and IC high energy emission from their surrounding. However their detectability will strongly depend on the distance to the nebula. It can not be too large since the expected flux will be below detectability of the Cherenkov telescopes. But it should not be too close since the TeV γ -ray nebula will have very large dimensions on the sky which again will make problematic its detectability with the Cherenkov telescopes. For example, the bow shock nebula around nearby Geminga pulsar (at the distance 169 pc) may not be detected by the present Cherenkov telescopes. Due to its small distance, the angular size of the TeV nebula expected in terms of discussed above model, should be of the order of a few degrees, i.e more in accordance with the recent report on the presence of the extended ∼ 20 TeV γ -ray source with diameter (2 . 8 ± 0 . 8) 0 , towards the Geminga pulsar by the MILAGRO observatory (Abdo et al. 2009). However, such nebulae might be detected by the planned CTA which field of view can be as large as 8-9 degrees (Actis et al. 2011). Acknowledgements. We would like to thank the Editor Steven N. Shore and the Referee for useful comments. This work is supported by the grants from the Polish MNiSzW through the NCN No. 2011 / 01 / B / ST9 / 00411 and UMO2011 / 01 / M / ST9 / 01891.", "pages": [ 6, 7 ] }, { "title": "References", "content": "Abdo, A.A. et al. 2009 ApJ 700, L127 Abdo, A.A. et al. 2010 ApJS 187, 460 Actis, M. et al.. 2011 Exp.Astron. 32, 193 Aleksic, J. et al. 2012 APh 35, 435 Arons, J., Tavani, M. 1993 ApJ 403, 249 Arzoumanian, Z. et al. 1994 ApJ 426, L85 Bednarek, W., Sitarek, J. 2007 MNRAS 377, 920 Brink, C. et al. 1990 ApJ 364, L37 Buccheri, R. et al. 1996 A&AS 115, 305 Cheng, K.S., Taam, R.E., Wang, W. 2006 ApJ 641, 427 de Jager, O.C., Harding, A.K. 1992 ApJ 396, 161 Fruchter, A.S. et al. 1988 Nature 333, 237 Fruchter, A.S. et al. 1996 ApJ 443, 21 Guillemot, L. et al. 2012 ApJ 744, 33 Huang, H.H., Becker, W. 2007 A&A 463, L5 Huang, R.H.H. et al. 2012 ApJ, in press (arXiv:1209.5871) Hui, C.Y. et al. 2011 ApJ 726, 100 Kennel, C.F., Coroniti, F.V. 1984 ApJ 283, 694 Kulkarni, S.R., Hester, J.J. 1988 Nature, 335, 801 Reynolds, M.T. et al. 2007 MNRAS 379, 1117 Sefako, R.R., de Jager, O.C. 2003 ApJ 593, 1013 Stappers, B.W. et al. 2003 Science 299, 1372 Strong, A.W., Moskalenko, I.V. 1998 ApJ 509, 212 Takata, J., Cheng, K.S., Taam, R.E. 2012 ApJ 745, 100 van Kerkwijk, M.H. et al. 2011 ApJ 728, 95 van paradijs, J. et al. 1988 Nature 334, 684 Wu, E.M.H. et al. 2012 ApJ, in press (arXiv:1210.7209) Zajczyk, A. et al. 2010, in Proc. High Time Resolution Astrophysics -The Era of Extremely Large Telescopes Agios Nikolaos (Crete Greece), Procceedings of Science published on line: http: // pos.sissa.it / cgi-bin / reader / conf.cgi?confid = 108, id.52", "pages": [ 7 ] } ]

[ { "title": "ABSTRACT", "content": "Context. To address the physics behind the soft X-ray light curve asymmetries in Cygnus X-3, a well-known microquasar. Aims. Observable e GLYPH<11> ects of the jet close to the line-of-sight were investigated and interpreted within the frame of light curve physics. Methods. The path of a hypothetical imprint of the jet, advected by the WR-wind, was computed and its crossing with the lineof-sight during the binary orbit determined. We explore the possibility that physically this 'imprint' is a formation of dense clumps triggered by jet bow shocks in the wind ('clumpy trail'). Models for X-ray continuum and emission line light curves were constructed using two absorbers: mass columns along the line-of-sight of i) the WR wind and ii) the clumpy trail, as seen from the compact star. These model light curves were compared with the observed ones from the RXTE / ASM (continuum) and Chandra / HETG (emission lines). Results. We show that the shapes of the Cygnus X-3 light curves can be explained by the two absorbers using the inclination and true anomaly angles of the jet as derived in Dubus et al. (2010) from gamma-ray Fermi / LAT observations. The clumpy trail absorber is much larger for the lines than for the continuum. We suggest that the clumpy trail is a mixture of equilibrium and hot (shock heated) clumps. Conclusions. A possible way for studying jets in binary stars when the jet axis and the line-of-sight are close to each other is demonstrated. The X-ray continuum and emission line light curves of Cygnus X-3 can be explained by two absorbers: the WR companion wind plus an absorber lying in the jet path (clumpy trail). We propose that the clumpy trail absorber is due to dense clumps triggered by jet bow shocks. Key words. Stars:individual:Cyg X-3 - Stars:binaries:spectroscopic - Stars:winds - Accretion - Stars:neutron - Black hole physics", "pages": [ 1 ] }, { "title": "Possible role of the jet.", "content": "O. Vilhu 1 and D.C. Hannikainen 2 Received 19.6.2012 ; accepted 16.12.2012", "pages": [ 1 ] }, { "title": "1. Introduction", "content": "Cygnus X-3 (4U 2030 + 40, V1521 Cyg) is a high-mass X-ray binary (HMXB) located at a distance of 9 kpc with a close binary orbit (P = 4.8 hour; Hanson et al. 2000; Liu et al. 2007). The compact star is either a neutron star or a black hole and the companion is most probably a WN5-7 type Wolf-Rayet (WR) star (helium star; van Kerkwijk et al. 1992, 1996). However, it is possible that the WR-phenomenon comes from the accretion disc wind, but to our knowledge no such detailed modeling for this source exists. Cygnus X-3 is also a source of relativistic jets (e.g. Mioduszewski 2001; Marti 2001), thus associating Cygnus X-3 with the class of microquasars. Massive winds are generally observed in Wolf-Rayet stars (Langer 1989; Crowther 2007). This wind produces the wellobserved orbital modulation of radiation from the compact star by asymmetric absorption during the orbit along the line-ofsight. The system inclination ( i = the angle between the line-of-sight and the orbital axis) is unknown but probably small. Depending on inclination, the compact star mass is either large (3 - 10 M GLYPH<12> if i = 30 deg) or smaller (1 - 3 M GLYPH<12> if i = 60 deg) (Vilhu et al. 2009, see their Fig. 10). By modeling the X-ray spectra in all states, Hjalmarsdotter et al. (2009) concluded that the compact star is a massive black hole (30 M GLYPH<12> ) indicating a very small inclination. Koljonen et al. (2010) found six distinct X-ray states reminiscent of those seen in black hole transients, also supporting the black hole case. Hot star winds are clumpy rather than homogeneous, consisting of dense condensations inside a rare gas (see papers in Hamann et al. 2008). In Cygnus X-3 the clumps are photoionized by strong X-ray / UVradiation and are consequently highly radiative. Jets destroy clumps when colliding with them (Perucho & Bosch-Ramon 2011). However, the opposite may also be true, such as in star formation regions where clumps instead form when a shock front collides with interstellar clouds (van Breugel et al. 2004). In this paper, we explore the possibility that a bow shock from the jet indeed triggers the formation of clumps and / or makes the existing clumps denser and that these are advected by the WR-wind. The result is a 'clumpy trail', a volume filled by the clumps and advected by the wind. In reality the formation of such a trail is far from being demonstrated but could in principle be investigated by radiative hydrodynamics (see Perucho and Bosch-Ramon 2008). The continuum and line absorptions are sensitive to density, and in di GLYPH<11> erent ways in a photoionised medium. Hence, one can expect observable e GLYPH<11> ects in X-rays if the line-of-sight from the compact star (X-ray source) crosses the clumpy trail region. We use the parameters derived by Dubus et al. (2009) for the jet inclination and true anomaly angles, and compute the intersection of the clumpy trail with the line-of-sight. Two di GLYPH<11> erent bow shock geometries are applied, and the continuum and emission line light curves modeled and compared with the observed ones.", "pages": [ 1, 2 ] }, { "title": "2. Geometry", "content": "Dubus et al. (2010) modeled the Fermi / LAT gamma-ray orbital modulation of Cygnus X-3 by inverse Compton scattering of WR photons on the jet using two models for the compact star given in Szostek & Zdziarski (2008): a neutron star with high orbital inclination (60 deg) and a black hole with small inclination (30 deg). Dubus et al. derived values for the jet inclination and true anomaly angles, and showed that the jet axis is close to the line-of-sight. We used the black hole alternative, supported by the spectral studies of Hjalmarsdotter et al. (2009), Vilhu et al. (2009), and Koljonen et al. (2010). The WR-wind is assumed to be spherically symmetric with density depending on the distance r from the WR-companion center as r GLYPH<0> GLYPH<13> with GLYPH<13> = 2. The rectangular xyz-coordinate system was used where the WR star is in the center and the z-coordinate is parallel to the orbital axis ( see Fig. 1). The vector formalism and definitions given in Dubus et al. (2010) were used. The unit vectors of the jet axis and line-of-sight (observer) in the xyz-coordinate system are: Here GLYPH<2> j is the jet azimuth (true anomaly), GLYPH<30> j the jet polar angle (jet inclination = angle between the z-axis and the jet-axis) and i the orbital inclination (angle between the z-axis and the line-of-sight). The true anomaly is GLYPH<6> 90 deg at conjunctions (orbital phases 0 and 0.5). For the black hole case, Dubus et al. (2010) derived 39 deg and 319 deg for GLYPH<30> j and GLYPH<2> j , respectively. Since the observer has a permanent 270 deg true anomaly ( GLYPH<2> obs ) and 30 deg inclination ( GLYPH<30> obs ), the jet pointing is close to the line-of-sight. Hence, Cygnus X-3 is more likely a 'microblazar' rather than a microquasar. Since the true jet anomaly is somewhat larger than that of the observer, the line-of-sight trails the jet during the orbit. Besides the jet inclination and azimuth, an additional parameter in the modeling is the ratio H / d, where d is the binary separation and H the height along the jet where high energy electrons are released and Compton scattering takes place. Dubus et al. (2010) obtained a best value of 2.66 for this ratio. As an example, for total masses 12 M GLYPH<12> and 28 M GLYPH<12> the binary separation d is 3.2 R GLYPH<12> (2.2 GLYPH<2> 10 11 cm) and 4.2 R GLYPH<12> (4.2 GLYPH<2> 10 11 cm), respectively (the separation depends on total mass as M tot 1 = 3 ). Numerical models for microquasar jets show a bow shock surrounding the jet (Perucho & Bosch-Ramon 2008). We used two simplified geometries to represent the stationary shocked region surrounding the jet: i) cone shells with thickness 0.1d and opening angles between 20-50 deg, and ii) cylinders with thickness 0.1d and radius between 0.2d-0.3d and starting at distances between 0-1.0d from the compact star. Further, we assume that the shocked region immediately transforms the wind material into clumps of equal density with space density N wind / N clump advected radially from the WR-star by velocities 1000-3000 km / s. N wind and N clump are the wind and clump densities, respectively. Hence, the wind mass is conserved in the process. Note that this is just our assumption to explore the possibility of clumps without demonstrating that they really exist. In numerical computations we used the xyz-grid divided into 70 GLYPH<2> 70 GLYPH<2> 70 pixels with pixel size 0.18d, giving 12.6d for the box size. The cone geometry with opening angles less than 20 deg or larger than 50 deg did not produce any orbital modulation (Section 3), nor thinner tubes in the tube geometry. The shell thickness has a minimal impact since we are not interested in the absolute values of mass columns, but only relative values along the orbit. The geometries are shown schematically in Fig. 2 as projections on the orbital plane. The WR-companion is in the center while the compact star is moving in a circular orbit around it (dotted circle wit four phases marked). The observation direction is from below (see also Fig. 4). Note that the projections appear to end before reaching the grid boundaries (0,70) due to small jet inclination (39 deg).", "pages": [ 2 ] }, { "title": "3. The 'clumpy trail' and the two absorbers", "content": "We assume that the bow shock triggers the formation of dense clumps or makes the existing clumps in the WR wind denser. The formation of clumps migh physically similar to star formation when a shock front impinges upon interstellar clouds (van Breugel et al. 2004). Furthermore, we assume that these clumps participate in the wind outflow with velocity between 1000-3000 km / s. Hence, once formed the clumps flow out with the wind. We stress that we just want to explore the possibility of clumps without demonstrating that they really exist. We assume that the clump density (like the wind density) decreases with distance r from the WR-star as r GLYPH<0> GLYPH<13> with GLYPH<13> = 2. The binary separation d = 3.4 R GLYPH<12> used in the computations obviously scales with the wind velocity. An example of the initial jet bow shock from many computations is shown in Fig. 2 for a wind velocity 2000 km / s, a cone with opening angle 20 deg (at orbital phase 0.35), and a tube with radius 0.3d (at orbital phase 0.85). We further assume that the bow shocks transform wind material immediately into dense clumps, to be advected by the wind. The projections of the advected clumps on the orbital plane are shown one hour later in Fig. 3. When coupling similar images over all phases and times we get the outflowing formation we call the 'clumpy trail'. Thus the trail is the whole volume of clumps generated by the wind advection. The crossing of the line-of-sight (from the compact star) and the clumpy trail was then computed numerically. For a fixed orbital phase this intersection is a segment ('strip') of the line-ofsight along which the clump density is weighted by r GLYPH<0> 2 . For the cone in Fig. 2 these strips integrated over all phases are shown in Fig. 4 as projected on the orbital plane (note the small inclination of 30 deg used and twice smaller pixel size used in this plot). The crossing region distance r from the compact star is between 0.3-3d (1-10 R GLYPH<12> ) at the ionization parameter range log( GLYPH<24> ) = 3.75-2.75 erg cm / sec for density 10 12 cm GLYPH<0> 3 ( GLYPH<24> = L / (Nr 2 )). In this plot we use a smaller pixel size than in Fig. 2 (0.09d) to better reveal the details. Mass columns (assumed proportional to optical depths) along the line-of-sight for three cone models (20 deg, 1000 km / s), (20 deg, 2000 km / s), (40 deg, 2000 km / s) and a tube with radius 0.3d are shown in Fig. 5, scaled with their maximum values along the orbit (dotted lines). The average phase at maxima is around 0.35 while the FWHM is approximately 0.25. The solid line is from the individual cone model in Figs. 2 and 3 (opening angle 20 deg, wind velocity 2000 km / s). Amongst the dozen computed models, the latter produced the best fitting results and will be used as the clumpy trail absorber GLYPH<28> (clump) (see Ch 4.) In the same plot (Fig. 5) we include the wind optical depth GLYPH<28> (wind) ( = mass column along the line-of-sight, the solid line with maximum around phase 0). This will be used as the first absorber (wind absorber GLYPH<28> (wind)) in Section 4. Using (mass column) 2 for the optical depth would give a better fit for the continuum light curve around minimum, and it may reflect that GLYPH<13> in reality is larger than 2 (accelerating wind) or else there is an ionization e GLYPH<11> ect in the wind. Using a pure mass column yielded practically the same continuum light curve but with a partial eclipse profile that was slightly too broad. Around GLYPH<30> j = 39 deg the results are not very sensitive on this jet inclination angle, while changing GLYPH<2> j between 290 deg340 deg moves the maximum of GLYPH<28> (clump) between 0.25-0.40 in phase. The key factor in this concept is whether the clumps formed are radiative or not. In our case, the wind (including clumps) is photoionised and very radiative. Hence, the concept should work (from shocks to dense clumps).", "pages": [ 2, 3, 4 ] }, { "title": "3.1. Cooling times of shocked clumps", "content": "Bearing in mind that we have assumed, for the purpose of this paper, that a bow shock triggers clump formation, we nevertheless estimate cooling times for clumps, both in equilibrium with the radiation field as well as heated to 10 7 K. We speculate that hot clumps could exist at short distances from their origin and this is discussed in Section 5. To estimate cooling times we use the photoionisation model of Vilhu et al. (2009), where the compact star luminosity is L x = 2.46 GLYPH<2> 10 38 erg / s. The clump densities probably vary, as do their sizes, but as an example we use a density of 10 13 cm GLYPH<0> 3 . Based on our XSTAR-computations with inclination 30 deg, the wind base density should be around 10 12 cm GLYPH<0> 3 to guarantee a continuum optical depth around unity (at 2 keV). This is what the partial eclipse of the continuum light curve requires (see Section 4). A somewhat higher clump density is then a good guess. The radiation spectrum of a clump in equilibrium with density 10 13 cm GLYPH<0> 3 at a distance of 3 R GLYPH<12> from the compact star (log( GLYPH<24> ) = 3.75) is shown in Fig. 6, as computed with the XSTAR code (Kallman 2006). In this model, the total outward luminosity L out per clump particle equals 4.2 GLYPH<2> 10 GLYPH<0> 9 erg / sec, while its thermal content per particle in equilibrium with the radiation field (temperature approximately 10 5 K) is 1.38 GLYPH<2> 10 GLYPH<0> 11 erg. Hence, the cooling time is short (3.3 msec) but this is counterbalanced by heating from the radiation field (heating = cooling in equilibrium). For a clump with N = 10 12 cm GLYPH<0> 3 the cooling time is 20 msec (with similar equilibrium temperature 10 5 K). Cooling times for hot 10 7 Kclumps (collision dominated) are almost 1000 times longer: 3 sec and 17 sec for densities 10 13 cm GLYPH<0> 3 and 10 12 cm GLYPH<0> 3 , respectively. This is due to a heat content that is 100 times larger and outward luminosities 10 times smaller of the 10 7 Kclumps. Cooling times of 10 8 Kclumps can be several minutes. Hot 10 7 K clumps are not in equilibrium with the surrounding radiation field and try to expand (and cool) after the shock has passed. A crude estimate of the dynamical time scale can be found as follows. The force F across the clump surface area S is P GLYPH<2> S, where P is the gas pressure inside a clump with mass m. From the clump expansion acceleration a = F / m, one can compute that the dynamical expansion time is 3.5 sec and 35 sec for clump sizes 10 8 cm and 10 9 cm, respectively (independent of clump density). Hence, hot 10 7 K clumps can survive at most a few minutes after their formation, depending on their density and size. Their possible role is discussed in Section 5.", "pages": [ 4 ] }, { "title": "4. Modeling of the X-ray continuum and emission line light curves", "content": "Here we compare the observed light curves with models using two absorbers: i) the WR wind, and ii) an absorber lying in the clumpy trail (see Fig. 4). Both optical depths are plotted in Fig. 5 and explained in Section 3. The observed gamma-ray ( Fermi / LAT) light curve is taken from Abdo et al. (2009) using the pdf measuring perimeter tool for graphics (Adobe) and modeled with Dubus et al. (2010) inverse Compton scattering formulas and parameters ( i = 30 deg, GLYPH<30> j = 39 deg, GLYPH<2> j = 319 deg, H / d = 2.66). The same parameter values were used for the clumpy trail computation. Note that Dubus et al. (2010) used phase units where the X-ray minimum occurs at phase 0.25. In this paper we use the X-ray phases where the X-ray minimun occurs at phase 0 (the WR star is between the observer and the compact star). For a jet with GLYPH<30> j = i and GLYPH<2> j = 270 deg, the jet axis and line-of-sight would coincide. The RXTE / ASMlight curve integrated over fifteen years, and scaled inside a specific orbit by the daily mean, was used as a template for the continuum light curve. The ephemeris of Singh et al. (2002) was used to compute the orbits. The data were limited to moderately high states (daily means GLYPH<21> 15 counts / sec), to be more consistent with the Cygnus X-3 spectral state during the Chandra HETG observations used. The emission line light curves (Si XIV Ly GLYPH<11> 6.185 Å, FeXXVI Ly GLYPH<11> 1.780 Å (H-type) and FeXXV 1.859 Å (He-type) ) were taken from Vilhu et al. (2009) and observed during a high state by Chandra / HETG (PI M. McCollough). The number of phase bins used was 30 and 10 for the continuum and lines, respectively. The modeling of the light curves requires implementation of both the wind and clumpy trail absorbers explained in previous sections. We assume that the continuum soft X-ray emission is centered at the compact star (disc). Most line emission (in particular SiXIV) comes from a broad region in the wind, except the high excitation FeXXVI Ly GLYPH<11> emission that probably originates from the compact star disc (see Vilhu et al. 2009). Line absorption in the continuum (at wavelengths below the line) should then be the common factor for all the lines. Let GLYPH<28> (wind) and GLYPH<28> (clump) be the optical depths along the line-of-sight (from the compact star) in the wind and clumpy trail, respectively, and scaled with their maximum values along the orbit (as shown in Fig. 5). As baseline absorbers we adopt those delineated by the heavy solid lines in Fig. 5 and described in Section 3. The model flux is defined with the help of these two absorbers as follows:", "pages": [ 4, 5 ] }, { "title": "F = exp[-A GLYPH<28> (wind) - B GLYPH<28> (clump)] .", "content": "Aand B are constants derived from the fitting to observations and given in Table 1 (using the baseline absorbers). The IDL procedure mpcurvefit : pro 1 was used in the fitting. The F-statistic and the significance (between 0.0-1.0) of the fit was computed with the IDL-procedure fv GLYPH<0> test : pro where 1.0 represents the highest significance. Table 1 shows that the fits are significant for both the continuum and the lines. If the clumpy trail absorber is not used at all (see Table 2) the fits are not acceptable, in particular for the lines. Fig. 7 shows the observations overplotted with model fits. The mean light curve for the lines is shown, but note that Table 1 gives fitting parameters for all the lines. The contributing optical depths, A GLYPH<28> (wind) and B GLYPH<28> (clump), are also shown in the two lower plots by dotted lines (scaled with the continuum wind column A GLYPH<28> (wind)). The scale on the left is the same as for the light curves. The gamma-ray ( Fermi LAT) light curve is shown for comparison in the uppermost panel. It can be seen that most continuum modulation comes from the wind while the small asymmetry around phase 0.3-0.4 is caused by the clumpy trail (B much smaller than A, see Table 1). On the contrary, emission line absorption is enhanced at the clumpy trail (larger B).", "pages": [ 5 ] }, { "title": "5. Discussion", "content": "The continuum light curve is represented by the ASM average light curve over fifteen years limited to daily means GLYPH<21> 15 counts / sec. So far there is no clear evidence that the form of the light curve changes during this time nor depends on spectral state - low (hard) or high (soft) - in a significant way. However, we limited the ASM data to relatively high states to be more consistent with the emission line observations. The emission line light curves were observed during a high state and with a few days' exposure by Chandra HETG (Vilhu et al. 2009, PI M. McCollough). It was assumed that they are representative also for the whole set of ASM observations. This is just an assumption and has no observational verification due to the limited amount of spectral data. The same applies for the Fermi / LAT gamma-ray observations since they are means over one year of observations during flaring periods (Dubus et al. 2010). We assumed that the eventual jet precession has a long time scale (over tens of years). Otherwise the observations could not be compared with the same jet parameters. This appears to be the case, since most imaging radio observations between 1991 Jan 15 - 2001 Sept 15 locate the jet position angle in the NorthSouth direction (0 deg or 180 deg; Mart'ı et al. 2000; Mart'ı et al. 2001; Miller-Jones et al. 2004; Schalinski et al. 1998). However, Tudose et al. (2007) using 2006 May 01 observations locate knots (which might be interpreted as segments of jets or counter jets) at position angles 55 deg - 82 deg. Furthermore, Mioduszewski et al. (2001) locate the position angle during the 1997 Feb 06 observations at 145 deg. We also assumed that the clumpy trail (in the circumbinary region) is more or less the same during all observations (including big flares or micro-flaring). The physical nature of the two absorbers requires more work. At soft X-rays below 10 keV photoelectric absorption is important and depends both on the distance from the X-ray source and wind density. It also depends on the element and line transition in question. Pure electron scattering (depending on the number of particles along the line of sight) contributes in a small way. To illustrate the situation we computed three di GLYPH<11> erent scenarios with the XSTAR code (using the X-ray source model explained in Section 3.1) along the line-of-sight from the X-ray source at phase 0.35 (at the strip-segment 0.3d-3d from the source, see Fig. 4): In all cases the mass columns along the line of sight strip were the same (wind mass conserved). Continuum and line optical depths were computed along the strip using the XSTAR-code. Table 3 gives the integrated depths while Fig. 8 gives the optical depths per 10 10 cm versus distance from th X-ray source (in units of the binary separation). In the plot the clumps are presented by a mixture (50 / 50) of equilibrium and 10 7 K clumps (solid lines). This mixture gives a moderate fit to the B-parameter values of Table 1, provided that the wind base density is twice larger 2 GLYPH<2> 10 12 cm GLYPH<0> 3 . The pure wind case is shown by dashed lines. The dotted lines give the continuum for clumps (upper line) and wind (lower line) cases. Table 3 and Fig. 8 show that the line absorption can indeed be enhaced at the clumpy trail. The iron lines appear to require hot collision dominated clumps (hotter than the equilibrium ones), in particular at short distances from the X-ray source. Replacing the clump density 10 13 cm GLYPH<0> 3 by 10 12 cm GLYPH<0> 3 or 10 14 cm GLYPH<0> 3 does not change much this picture. The short cooling times (see Section 3.1) seem to require some modification to our assumptions: e.g. the jet axis is closer to the line of sight, the jet cone is broader, the bow shock propagates with the wind or clump formation is delayed. It is noteworthy that the presence of the clumpy trail absorber occurs around phase 0.3 (see Fig. 3) which is also where the 9 mHz QPOs were found by Koljonen et al. (2011). Whether these QPOs arise from some sort of flickering when the line of sight passes the moving clumps remains to be solved. In the present paper a formal exponential law with two absorbers was used to model the light curves (Section 4). However, it is significant to note that the location of the second absorber coincides with the clumpy trail region.", "pages": [ 5, 6 ] }, { "title": "6. Conclusions", "content": "We have demonstrated a possible method to study observable e GLYPH<11> ects of jets in microquasars when the jet direction and line-ofsight are close to each other (as in Cygnus X-3). In particular, we showed that the jet in Cygnus X-3 can produce a 'clumpy trail' which crosses the line-of-sight during the binary orbit periodically. Using jet parameters derived by Dubus et al. (2010) from Fermi / LAT observations, we computed the location of the clumpy trail (see Fig. 3). Model light curves were then constructed using the two absorbers in Fig. 5 weighted with constants A and B : i) the WR wind, and ii) a clumpy trail. These light curves were compared with the observed ones. Good agreements were achieved for the soft X-ray continuum ( RXTE / ASM) and emission lines ( Chandra HETG) (see Fig. 7 and Table 1). Wefound that the location of the clumpy trail computed from jet parameters matches well with that of the second absorber required (clumpy trail absorber) which may not be just a coincidence. Although more work is required to clarify the physical nature of this clumpy absorber, we suggest that the clumpy trail consists of a mixture of equilibrium and hot (shock heated) clumps. We note that the results are based on an assumption of constancy of the jet position angle over the past fifteen years (long precession time) which needs to be confirmed. Acknowledgements. We thank the referee for valuable criticism that greatly improved the paper. We are grateful to Dr. Guillaume Dubus for correspondence and permission to use his plot (Fig. 1). We also thank Wiley Publishers for permission to reproduce this figure. We thank Dr. Pasi Hakala for useful comments on the manuscript.", "pages": [ 6 ] }, { "title": "References", "content": "Abdo, A.A. et al. (Fermi-LAT collaboration) 2009, Science, 326, 1512 Crowther, P.A. 2007, 'WR-Stars', Ann.Rev.Astr.Astrophys., 45, 177 Dubus, G., Cerutti, B., & Henri G. 2010, MNRAS, 404, L55 Hamann, W-R., Feldmeier, A., & Oskinova, L.M (editors). 2008, 'Clumping in Hot Star Winds', Proceedings of International Workshop held in Potsdam, Germany, 18-22 June 2007, Potsdam:Univ.-Verl. URN:nbn:de:kolv:517- opus-13981. Hanson, M.M., Still, M.D., & Fender, R.P. 2000, ApJ, 541, 308 Hjalmarsdotter, L., Zdziarski, A.A., Szostek, A., & Hannikainen, D.C. 2009, MNRAS, 392, 251 Kallman T.R. 2006, 'A Spectral Analysis Tool XSTAR', version 2.1kn6, Goddard Space Flight Center, May25, 2006. Koljonen, K.I.I., Hannikainen, D.C., McCollough, M.L., Pooley, G.G., & Trushkin, S.A. 2010, MNRAS, 406, 307 Koljonen, K.I.I., Hannikainen, D.C., McCollough, M.L. 2011, MNRAS, 416, L84 Langer, N. 1989, A&A, 210, 93 Liu, Q.Z., van Paradijs, J., & van den Heuvel, E.P.J. 2007, A&A, 469, 807 Mart'ı, J., Paredes, J.M., & Peracaula, M. 2000, ApJ, 545, 939 Mart'ı, J., Paredes, J.M., & Peracaula, M. 2001, A&A, 375, 476 Miller-Jones, J.C.A., Blundell, K.M., Rupen, M.P., et al. 2004, ApJ, 600, 368 Mioduszewski, A.J., Rupen, M.P., Hjellming, R.M., et al. 2001, ApJ, 553, 766 Perucho, M., & Bosch-Ramon, V. 2008, A&A, 482, 917 Perucho, M., & Bosch-Ramon, V. 2012, A&A, 539, 57 Schalinski, C.J., Johnston, K.J., Witzel, A., et al. 1998, A&A, 329, 504 Szostek, A., & Zdziarski, A.A. 2008, MNRAS, 386, 593 Singh, N.S., Naik, S., Paul, B., et al. 2002, A&A, 392, 161 Tudose, V., Fender, R.P., Garrett, M.A., et al. 2007, MNRAS, 375, L11 van Breugel, W., Fragile, C., Anninos, P., & Murray S. 2004, 'Jet-induced star formation', in IAU Symposium Series, Vol.217, 472. van Kerkwijk, M.H.,Charles, P.A., Geballe, T.R. et al. 1992, Nature, 355, 703 van Kerkwijk, M.H., Geballe, T.R., King, D.L. et al. 1996, A&A, 314, 521 Vilhu, O., Hakala, P., Hannikainen, D.C., McCollough, M., & Koljonen K. 2009, A&A, 501, 679", "pages": [ 6, 7 ] } ]

[ { "title": "ABSTRACT", "content": "Aims. Molecular clouds act as primary targets for cosmic-ray interactions and are expected to shine in γ -rays as a by-product of these interactions. Indeed several detected γ -ray sources both in HE and VHE γ -rays (HE: 100 MeV < E < 100 GeV; VHE: E > 100 GeV) have been directly or indirectly associated with molecular clouds. Information on the local diffusion coefficient and the cosmic-ray population can be inferred from the observed γ -ray signals. In this work we explore the capability of the forthcoming Cherenkov Telescope Array Observatory (CTA) to provide such measurements. Methods. We investigate the expected emission from clouds hosting an accelerator, surveying the parameter space for different modes of acceleration, age of the source, cloud density profile, and cosmic-ray diffusion coefficient. Results. We present some of the most interesting cases for CTA regarding this science topic. The simulated γ -ray fluxes depend strongly on the input parameters. In several cases, we find that it will be possible to constrain both the properties of the accelerator and the propagation mode of cosmic rays in the cloud from CTA data alone. Key words. astroparticle physics - radiation mechanism: non-thermal - ISM: clouds - cosmic-rays - gamma rays: ISM", "pages": [ 1 ] }, { "title": "On the potential of the Cherenkov Telescope Array for the study of cosmic-ray diffusion in molecular clouds", "content": "G. Pedaletti 1 , D. F. Torres 1 , 2 , S. Gabici 3 , E. de O˜na Wilhelmi 4 , D. Mazin 5 , 6 , and V. Stamatescu 5 Received: 17 October 2012 / Accepted: 22 December 2012", "pages": [ 1 ] }, { "title": "1. Introduction", "content": "Emission in HE-VHE γ -rays is expected in spatial coincidence with molecular clouds, resulting from the hadronic interaction between cosmic-ray (CR) particles and the dense material in the cloud acting as a target. Indeed, some MCs have been detected in γ -rays in both the GeV and TeV domain (see, e.g., Aharonian et al 2008; Albert 2007; Aharonian et al 2008; Giuliani et al. 2011; Ackermann et al. 2012; Aharonian 2012). Moreover, it has been suggested that some of the as yet unidentified γ -ray sources might also be MCs illuminated by CRs that escaped from an accelerator located inside the cloud or in its proximity (Montmerle 1979; Aharonian & Atoyan 1996; Gabici et al. 2007; Rodr'ıguez Marrero et al. 2008). In such cases the modeling of the emission involves the parametrization of the diffusion of charged particles. The diffusion coefficient is in general considered to be energy-dependent (Ginzburg & Syrovatskii 1964), with lower energy particles diffusing more slowly than higher energy ones, under the same medium conditions. When those diffused charged particles (protons or heavier nuclei) interact with target material in a density enhancement of the interstellar medium, such as a molecular cloud located in the vicinity of the accelerator, significant γ -ray emission is expected due to the production and subsequent decay of neutral pions. γ -ray emission produced in massive molecular clouds was predicted long ago (see e.g. Black & Fazio 1973; Morfill et al. 1984; Aharonian 1991). The study of this emission is extremely useful in unveiling the physics of CR sources. Due to energy dependent propagation effects the γ -ray spectrum from the molecular clouds may differ significantly from the spectrum observed at (or closer to) the accelerator. This may explain discrepancies in the particle spectral indeces inferred from the same source at different frequencies, even if all particles, leptons and hadrons, are accelerated to the same power-law at the source. In this scenario, a great variety of γ -ray spectra is expected, depending on several parameters, including: the age of the acceleration, the distance between the cloud and the accelerator, the duration of the injection of CRs and their diffusion coefficient. This can produce a variety of different GeV-TeV connections, some of which could explain the observed phenomenology (see, e.g., Funk et al. 2008; Tam et al. 2010). Injection and propagation of CR have been recently put forward to explain a number of the TeV sources currently known, especially those for which there is no, or there is a spatially displaced, GeV counterpart (e.g. see Fujita 2009; Li & Chen 2010; Gabici et al. 2010; Torres et al. 2010; Ohira et al. 2011). Given the expected CTA angular resolution and sensitivity, variations in flux at less than 5 pc bins at 5kpc distances could be resolved. Changes in the CR spectrum could be derived accordingly, leading even to, e.g., the derivation of a diffusion coefficient as a function of energy and position ( D ( E,r )). The measurement of such spatial variability in the diffusion coefficient would be an important result in CR physics. In this paper, the γ -ray emission due to an accelerator inside a molecular cloud is calculated. The expected CTA measurement of such emission is then derived taking into account the simulated CTA response functions. The CTA Observatory is described in Section 2. The calculation of γ -ray emission and the physical parameters of the scenario are described in Section 3. The simplified case of flat density of the target material (i.e. flat density profile of the molecular cloud) is investigated in Section 4 along with the CTA capabilities in distinguishing the parameter space. Small and nearby clouds are investigated in Section 5. Section 6 deals qualitatively with a more realistic case of a peaked density profile. Conclusions are given in Section 7.", "pages": [ 1, 2 ] }, { "title": "2. The Cherenkov Telescope Array (CTA)", "content": "CTA is an international project for the development of the next generation ground-based γ -ray instrument (see Actis et al. 2011). The detection of γ -rays ( E > 10 GeV) with ground-based facilities is possible thanks to the imaging atmospheric Cherenkov technique (Weekes 2003). VHE γ -rays interact with nuclei in the atmosphere producing a cascade of particles, where velocities are larger than the speed of light in the medium, leading to Cherenkov light emission. The resulting Cherenkov light flashes may be imaged by Imaging Atmospheric Cherenkov Telescopes (IACTs) (for a recent review, see Hinton & Hofmann 2009). The shower images are then used to reconstruct the energy and direction of the original particle. Particle cascades can also be initiated by CRs and constitute the main source of background. In this case the showers are generally broader than those initiated by primary photons. Gammahadron separation can be achieved thanks to the differences in size and shape of the shower images. The showers illuminate a pool at the ground level. The radius of the light pool depends on the height of the observatory and on the energy of the primary photon. An array of IACTs allows for better sampling of the Cherenkov light distribution of a given event. From a stereoscopic view of the same event, the reconstruction of the direction of the primary photon and the background rejection are improved with respect to a stand-alone telescope observation. CTAwill significantly advance on the present generation IACTs: it will feature an order of magnitude improvement in sensitivity at the core energy range of 1 TeV, improve in its angular and energy resolution, and provide wider energy coverage, see Actis et al. (2011). Indeed, the array is expected to have an unprecedented sensitivity down to ∼ 50 GeV and above ∼ 50 TeV, establishing a strong link to the satellite-based operations at low energies, namely the Large Area Telescope on board the Fermi satellite, see Atwood et al. (2009) and water Cherenkov experiments at the highest energies (e.g, HAWC, see Goodman 2010). The gain in sensitivity is due to the increase in the number of telescopes. The widening of the explored energy range is due to a combination of different-sized telescopes in different parts of the light pool. Large size telescopes (LST, with a dish of ∼ 23 m) will be placed at the center of the array. Thanks to their large mirror area, dim flashes from the low energy events ( ∼ 50 GeV) are expected to be reconstructed. Tens of medium size telescopes (MST, with a dish of ∼ 11 m) will be placed in a surrounding ring, covering a large fraction of the light pool and thus enhancing the reconstruction of medium energy ( ∼ 1 TeV) events. Finally, the outer regions will be composed of small size telescopes (SST, with a dish of ∼ 7m) enlarging the effective area of the array for the bright but rare high energy events (above ∼ 50 TeV). Both a southern and a northern hemisphere observatory are foreseen.", "pages": [ 2 ] }, { "title": "3. An accelerator inside a cloud", "content": "If a power-law energy spectrum ( J p ( E p ) = KE -γ p ) is assumed for the intensity of primary CRs, the resulting γ -ray spectrum due to hadronic interactions would also follow a power-law spectrum ( F ( E ) ∝ E -Γ ). However, if we consider an energy-dependent diffusion coefficient, the CR spectrum may differ from a simple power-law near the acceleration site. The spectrum of the accelerated CR can be expressed as J p ( E p , r, t ) = ( c/ 4 π ) f , where f ( E p , r, t ) is the distribution function of protons at time t and distance r from the source. The distribution function satisfies the diffusion-loss equation (e.g., Ginzburg & Syrovatskii 1964) where P = -dE p /dt is the continuous energy loss rate of the particles, Q = Q ( E p , r, t ) δ ( R ) is the source function (for injection), and D ( E p ) is the diffusion coefficient. Here, we assumed that the source is point-like and located at the origin of the coordinate system. Solutions to this equation have been extensively studied for different cases, considering either spatially constant diffusion coefficient (Atoyan et al. 1995; Aharonian & Atoyan 1996; Rodr'ıguez Marrero et al. 2008; Gabici et al. 2009) or no CR accelerator near the cloud, i.e. Q=0, (passive clouds where the only γ -ray emission arises from the contribution of the CR background, Gabici et al. 2007). We investigate the case of an accelerator positioned at the center of a molecular cloud. This is an idealized case, but it allows to study the impact of an enhancement of CR content, above and beyond the passive cloud case. The γ -ray emission can be calculated in concentric shells of increasing radius, each shell retaining the footprint of the diffusion coefficient and of the cloud density. The study of such footprint is done for a simple symmetrical and homogeneous system, where expected spectral and morphological behaviors can be shown clearly. The acceleration and diffusion processes are computed following the approach of Aharonian & Atoyan (1996). The diffusion coefficient is assumed to depend on the CR energy only, as: D ( E p ) = D 10 ( E p / 10 GeV) δ cm 2 s -1 . More details of the flux calculation are given in the Appendix A. The resulting flux in γ -rays is mainly dependent on the diffusion coefficient, the age of the accelerator, the type and spectrum of injection of accelerated particles, and on the density and mass of the cloud. An impulsive source of particles corresponds to the case when the bulk of relativistic cosmic-rays are released during times much smaller than the age of the accelerator itself. When the timescales are comparable, the source is referred to as a continuous injector. All the parameters are free and might assume slightly different values to those studied here. The intervals for the values of the parameters adopted in this work are given below: However, we mainly concentrate, as an example, on the case were the injection slope is γ = 2 . 2 and the diffusion coefficient dependence on energy is δ = 0 . 5. This pair of parameters satisfies the observed CR spectrum data. Indeed, with these values, the index of the equilibrium spectrum in the galaxy is expected to be γ + δ = 2 . 7. The typical value of the diffusion coefficient in the galaxy is D 10 = 10 28 cm 2 s -1 (Berezinsky 1990). However, this value is very uncertain and depends on the level of the magnetic turbulence in which particles propagate. For example, higher level of turbulence will in general lead to a suppression of diffusion. The total energy input is taken as W p = 10 50 erg ( η = 1) in the impulsive case, while the energy injection rate in the continuous case is of L p = 10 37 erg s -1 , resulting in the same total input for accelerators with an age of a few hundreds of thousand years. The spectra of γ -rays produced by proton-proton interactions have been computed following Aharonian & Atoyan (1996), where a delta-function approximation has been used to model the interaction cross section.", "pages": [ 2, 3 ] }, { "title": "3.1. CR sea penetration in the cloud", "content": "The CR background (also referred to as 'sea') is ubiquitous in the Galactic plane. For the energy range considered here, it is assumed to be well described by the locally measured differential spectrum as (e.g., see Adriani et al. 2011) Therefore, to compute the total CR content inside the cloud, it is necessary to calculate the degree of penetration of the CR sea in the simulated cloud. The comparison of relevant timescales (pp losses and diffusion timescale) has been done following, e.g., Gabici et al. (2007). The energy loss timescale for proton interactions is where n H is the density of the gas, c is the speed of light, κ is the inelasticity (assumed to be κ ∼ 0.45 throughout the paper) and σ pp = 33 mb is the cross-section of the process. σ pp is mildly dependent on the energy of the particles, however, for the energies considered here, the assumption of energy independence is satisfactory (Aharonian & Atoyan 1996). The diffusion timescale, i.e. the time it takes a CR to diffuse from the edge to the centre of the cloud, can be expressed as where R cloud = 20pc is the radius of the cloud in the example investigated here. For a mass of 10 5 M /circledot , this results in a uniform density of n H = 130 cm -3 , rather typical for a molecular cloud. The timescale of Eq. 4, in the impulsive case, represents the time at which the maximum of particle flux is reached at the distance R cloud (Aharonian & Atoyan 1996). Penetration is ensured for particles of energies higher than the energy resulting from equating Eq. (3) and Eq. (4). For the smaller value of the diffusion coefficient adopted here (i.e. D 10 = 10 26 cm 2 s -1 ), the minimum energy of CR that can penetrate fully the cloud is E p ≈ 2 GeV. This is shown in Fig. 1 as the intersection of the dashed line representing the pp loss timescale and the black solid line representing the diffusion timescale in the case of slow diffusion. For faster diffusion, the minimum energy is even lower. This conclusion does not depend on the preperties of the accelerator, so as to say, it is valid both for continuous and impulsive accelerators, but depends on the diffusion coefficient, the density of the gas and the size of the cloud. Therefore, the CR background density is always added to the CR spectrum from the accelerator itself in the cases shown in Fig. 1. From the latter figure, thus, it is possible to derive the degree of penetration to the core of the cloud for CRs of different energies, assuming a flat density distribution.", "pages": [ 3 ] }, { "title": "4.1. Detectability", "content": "The γ -ray emission has been simulated for a molecular cloud with a mass of M 5 = 10 5 M /circledot and a radius of 20 pc (hence with an average density of n H = 130 cm -3 ) located at a distance of d =1 kpc, with the accelerator parameters given above for a continuous and impulsive source. Giant molecular clouds as close as ∼ 1 kpc distance might be uncommon, however for a list see Dame et al. (1987). The angular extension is ∼ 1 deg in radius. Most of the results can be rescaled to an arbitrary cloud mass and distance by recalling that, for a given CR intensity in the cloud, the expected gamma ray flux is expected to scale as ∝ M 5 /d 2 and the cloud apparent size as ∝ R cloud /d . γ -ray fluxes were calculated for a permutation of age, acceleration mode, and diffusion coefficient as outlined in Section 3. Fig. 2 shows that for such permutations, all the cases can be detected with 50 hrs of CTA observations except for the case of an old impulsive accelerator and fast diffusion. The sensitivity of the instrument is scaled for the extension of the source, as detailed in Sec. 4.2. The calculated integral fluxes are very similar for some of the permutations. For example, in the case of fast diffusion and continuous acceleration, the calculated flux is constant with age (red, green and blue squares at D 10 = 10 28 cm 2 s -1 ). This can be explained by the fact that the bulk of the particles contributing to γ -ray emission at energies above 10 GeV can diffuse out of the cloud in a time smaller than the age of the accelerator. Therefore only the latest generation of accelerated particles contributes to the γ -ray signal and a steady state is reached. Once a spectrum and source morphology are obtained using CTA these would act as diagnostic tools with which to reconstruct the initial parameters of the accelerator.", "pages": [ 3, 4 ] }, { "title": "4.2. Spectral features", "content": "A constraint on the diffusion coefficient can come from the identification of a break in the γ -ray spectrum integrated from the entire cloud region. The break can be related to the minimum energy that can diffuse in the entire cloud over a timescale comparable to the age of the accelerator. By equating Eq. 4, which gives the diffusion timescale, with the age of the accelerator, one obtains the constraints shown in Fig. 3, which correspond to: For scenarios with fast diffusion ( D 10 = 10 28 cm 2 s -1 ) and energy dependence parameter in the range δ = [0 . 3 .. 0 . 6], the corresponding break in γ -ray emission will always be at energies below the CTA energy acceptance. This is accurate for impulsive accelerators. In the continuous acceleration case, new injections of high energy particles will smooth the effect of a break. At energies higher than the break given by Eq. 5, the particle spectrum will follow a powerlaw form composed of the slope of the injection spectrum and the energy dependence of the diffusion coefficient (i.e γ + δ ). Therefore the γ -ray emission will show a powerlaw behavior, thereby reducing the ability to constrain the parameter space from γ -ray data in the cases when E break is below 70 GeV. The reconstruction of a break is indeed a powerful tool. If a break is not present, and the CTA detected spectrum is a simple power-law, a plethora of models will fit the spectral points. An example is given in the following, where the parameters are searched for in a grid for the intervals given in the Appendix A. Let us assume that a molecular cloud with measured mass and distance is detected at TeV energies. Let us further assume that from multiwavelength observations we identified a possible accelerator of CRs responsible for the gamma ray emission and that an estimate of the age of that accelerator is known. In Fig. 4 we show the simulated spectra for the gamma ray emission for such an accelerator, which is assumed to be impulsive and with an age of 10 4 years. The other parameters are varied on a grid (see Appendix A) and the corresponding observed spectrum is simulated from CTA 'responses'. By using a χ 2 minimization procedure, the best fitting model belonging to the parameter space grid is found, along with a representative sample of the models within the χ 2 /dof +1 contour ( dof =degrees of freedom), which is those models with a χ 2 /dof not in excess of unity from the best fit model. The top panel of Fig. 4 shows the case of D 10 = 10 26 cm 2 s -1 , δ = 0 . 4, γ = 2 . 3, η = 1 / 3. Thanks to high flux reached in this case and the presence of a break, the spectrum is reconstructed easily to the intrinsic parameters, with a break in the γ -ray spectrum at E ≈ 2 TeV and slopes Γ 1 = 2 . 3 and Γ 2 = 2 . 7 below and above the break, respectively. However, for lower fluxes, the reconstruction can be more uncertain. This is shown in Fig. 4 (bottom panel), where it can be seen that the intrinsic model ( D 10 = 10 28 cm 2 s -1 , δ = 0 . 6, γ = 2 . 1, η = 1) does not even provide the best fit. Additional and more stringent constraints can come from studying the spatial dependence of the γ -ray spectrum from the inner to the outer region, which depends on the diffusion inside the cloud. We slice the expected emission, projected onto the sky, in concentric shells (see Appendix). We choose the linear size of the shells to match the expected angular resolution of CTA at the lowest energies resolvable by the array (e.g. ∼ 0.25 · at 50 GeV, see Actis et al. 2011), with 5 concentric shells, with shell 1 being the closest to the accelerator. Fig. 5 shows an example of the predicted fluxes and Fig. 6 shows the expected CTA energy spectra. Fig. 5 shows also the expected CTA sensitivity scaled with the size of the source. In order to obtain the CTA sensitivity for an extended source, the point-source sensitivity is taken from Actis et al. (2011) and then scaled with an appropriate energy dependent factor. This factor is related to the optimal cut on the angular size of the source. The ratio between sensitivity for a point-like source (PS) and an extended (EXT) source is Θ EXT / Θ PS . In Fig. 5 we show the PS sensitivity for CTA together with the scaled one for Θ EXT . We show this to illustrate the capability of CTA. However, to calculate the spectral points and profiles from CTA simulated observations, we followed the procedure de- led in Bernlohr et al. (2012). To evaluate the spectral CTA response for each shell, we simulate the background counts coming from a region as extended as the outer border of each shell. Contamination from adjacent shell is not taken into account. The LAT instrument on board of the Fermi satellite will provide at least 5 years worth of data by the time that the full CTA array will be in operation. Figs. 5 and 6 show the expected 5 year point source sensitivity for Fermi/LAT. This is calculated from the 1 year sensitivity in (Atwood et al. 2009), linearly scaled with time at high energies ( > 10 GeV). The linear scaling is expected in the signal limited regime. The simulations show that the total cloud emission, as well as that from some individual shells, are well above the limit of detection with CTA. For the particular case in Fig. 6, the low energy particles still have not diffused out to the outer shells (the last shell is in magenta in the plot). Indeed the concave shape of the spectrum evidences the dominant contribution from the CR background at low energies on the outer shells. The concavity in the spectrum is expected at energies below the energy range of CTA. However, it will be possible to distinguish a hardening of the spectrum for shells with increasing distances from the accelerator. These features can be seen for middle age accelerators (age=10 4 years) only for D 10 = 10 26 cm 2 s -1 . For faster acceleration, the spectra of all shells will be a simple power-law with same photon index, thus not distinguishable.", "pages": [ 4, 5 ] }, { "title": "4.3. Morphology", "content": "The different scenarios can also be disentangled by investigating the morphology and extension of the emission region. We used the fluxes and shell area from the scenarios described above in order to simulate an excess map from the simulated response functions of CTA, as detailed in Bernlohr et al. (2012). We then create a profile from the excess map, integrating azimuthally the counts for bins in increasing angular distance from the center of the map. The profiles are then weighted by the bin area. The shape of such profile depends on the parameters of the simulated scenario, an example of which is shown in Fig. 7. These profiles are easily distinguishable from one another. Extensions of the γ -ray emission depends on the parameters studied here, with some general trends. Older sources are always more extended than younger sources, as the lower energy particles will have diffused further from the center of the cloud and thus from the accelerator. Emission due to continuous accelerators will present steeper profiles due to the freshly accelerated particles in the center of the source. Faster diffusion also leads to a larger extension. The profiles in Fig. 7 are normalized to their respective maximum counts, with flux decreasing with age in the impulsive case and opposite behavior in the continuous case. However we do not expect many of these high flux objects, therefore it is useful to predict the maximum distance at which an object is expected to be detected and resolved, depending on its intrinsic luminosity. The maximum distance ( d max ) at which a source can be detected depends on the sensitivity of the instrument and on the angular size of the source, θ src = ξR cloud /d ( ξ represents the fraction of the cloud radius that contains the 68% of the counts in profiles as shown in Fig. 7). For a source of constant luminosity, d 2 max = L src / (4 πF sens ), where L src = 4 πF src d 2 src is the isotropic luminosity of the source. The sensitivity of the array, for an extended source, is F sens ( d ) = F PS θ src /θ PS = F PS /θ PS × ξR cloud /d , where F PS is the expected point-source sensitivity of the array and θ PS its angular resolution. For CTA, the point-source sensitivity is F PS , CTA ( > 50GeV) ≈ 10 -12 TeV / cm 2 / s and θ PS , CTA ≈ 0 . 25 · its angular resolution, at low energies (Actis et al. 2011). Therefore, for F sens ( d max ), we will have: The maximum distance at which a source can be resolved is when the angular extension of the cloud equals the PSF of the instrument ( θ src ≡ θ PS ), therefore the maximum distance is At this distance we will be able to discriminate between the point-source and extended case, but the exact determination of the extension might need longer observation times. Fig. 8 shows the evolution of the maximum distances of detection and resolvability as a function of the luminosity of the source. As an example, let us consider a passive cloud, i.e. a case where only the CR background is included. With this assumption, the expected signal is (Aharonian 1991; Gabici 2008): where κ is the enhancement factor of CRs, assumed to be unity for passive clouds. Therefore a cloud of 10 5 M /circledot would be detected out to only ∼ 1 kpc, due to its expected isotropic luminosity of L iso ≈ 2 × 10 32 erg s -1 . If such cloud was to be more compact than the ones investigated here, the horizon of its detectability would be larger, according to the scaling given in Eq. 6. It has to be noted that enhancing the CR content of the cloud by a factor κ > 20 would allow the detection of a cloud of such mass in the entire galaxy. For comparison, κ ∼ 20 would correspond to one of the cases exemplified in Fig. 2, specifically a continuous accelerator of 10 4 year age and D 10 = 10 27 cm 2 s -1 .", "pages": [ 5, 6, 7 ] }, { "title": "5. Passive clouds: giants and cloudlets", "content": "Giant molecular clouds (GMC) as close as ∼ 1 kpc distance are uncommon (for a list see Dame et al. 1987). These are passive clouds (i.e. only the CR background is included) that would also be interesting for detection. The feasibility of a detection depends mainly on their mass, distance and most of all extension of the gamma-ray emission, as already discussed in Acero et al. (2012). Here we assume that the extension is ∼ 0.7 of the boundaries listed in Dame et al. (1987) together with the mass and distances given in that paper. The assumption on the extension comes from the fact that we need to consider the 68% containment radius of the γ -ray emission. We estimate the γ -ray flux from Eq. 8, with κ = 1, and we compare it to the integral sensitivity given in ( > 100 GeV, 50 hours; Actis et al. 2011) and its scaling for the extension of the source (i.e. ∝ θ src /θ PS , where θ PS =0.1). The scaling of the sensitivity given here is valid only for an infinite field of view with flat acceptance, while we expect a degradation of sensitivity with increasing angular distance from the center of the field of view (for a discussion see Dubus et al. 2012). Many of the clouds are very extended with respect to the expected angular acceptance of CTA and would need either to be scanned or to be observed in divergent mode, therefore increasing the observation time for a mapping of the entire object (for a general survey in divergent mode see Dubus et al. 2012). Expected fluxes and the corresponding scaled sensitivity are given in Fig. 9. The ranges in extension refer to the cloud boundaries in latitude and longitude. The prospects for detection with CTA are slim due to the very large extension of the clouds. However, there is an abundance of smaller clouds at closer distances, that have been already detected at HE γ -rays. Following the averages derived from the cloudlet population studied in Torres et al. (2005), we simulate a 40 M /circledot cloud at a distance of 150 pc and an extension of 2.8 pc (1.08 · in angular size). The resulting density of the cloudlet is similar to that of the larger cloud studied in the paragraph above, n H = 140 cm -3 . Therefore we expect total penetration of CR sea for energies above E p > 1 GeV for such small radial extensions. However, it is unlikely to have such cloudlets hosting a powerful accelerator. So we consider them also as passive clouds. With this assumption, the expected signal is given by Eq. 8, with κ = 1. This translates for the cloudlet considered into F ( > 1TeV) ∼ 2 × 10 -15 cm -2 s -1 ∼ 0 . 1 /permil CU, where CU stands for Crab Units and represents the integral flux of the Crab nebula above 1 TeV. This flux would not be easily detectable (the expected CTA sensitivity is of ∼ 1 /permil CU ). Nonetheless, a stacking approach on a population of roughly 100 clouds -a list of which is used by Torres et al. (2005)- would lead to a detection also in VHE γ -rays, implying a good test for the isotropy of the CR sea in the local neighborhood.", "pages": [ 7 ] }, { "title": "6. Peaked density profile of the molecular cloud", "content": "In this section we describe qualitatively the expectation for a different density distribution of the target material inside the cloud. Indeed, the density of the molecular cloud might not be constant throughout its extension. Therefore we study the case of a cloud with a density profile of the form where R C is the core radius and n 0 is the density at the center of the cloud. Following the observations by Crutcher (1999) we assume that the cloud magnetic field scales with density as This will in turn affect the value of the CR diffusion coefficient. We parametrize the effect of the magnetic field as: The mass of the interacting target material can be calculated following where p = (1 + r/R C ). The case of α n = 0 reduces to a flat density profile investigated in Section 4. The total mass M is obtained with R 1 = 0 and R 2 = R cloud . If the mass of the cloud is known, the central density can be derived from the formula above. We again assume M = 10 5 M /circledot and R max =20 pc. From this assumption we can calculate the central density depending on the α n chosen (with the core radius R C =0.5 pc fixed). This allows us to calculate for each radius, n H , B, D from Eq. (9). With increasing densities, and correspondingly stronger magnetic fields, diffusion becomes much slower and the timescales for pp losses faster, resulting in a slower penetration of the CR sea. Because of this, the penetration of the CR sea into the cloud is not complete, and we cannot any longer assume that the sea is constant throughout the cloud. To compute the level of penetration of the CR sea into the cloud, we follow here the approach by Protheroe et al. (2008) and treat the penetration as an analog of optical thickness, where the CR intensity I CR = e -τ ∗ ( E,r ) I CR ( E,r ) and τ ∗ ( E,r ) = τ a ( τ a + τ s ) can be decomposed in two terms, analogs of absorption and scattering: Examples of CR penetration as a function of energy are shown in Fig. 10. Comparing the penetration factors and density profiles, we will expect a dominant contribution from the accelerator in the inner shells. The low number density of target material in the outer shell and the correspondent higher diffusion coefficient assure full penetration of the CR background. On the other hand, only a fraction of the accelerated particles will manage to diffuse out to the outermost shells. Indeed, because of the higher densities found in the core of the cloud, the dominant emission in the outer shells will come from the CR background. With these assumptions, we expect to detect a source with a steep surface brightness profile.", "pages": [ 8 ] }, { "title": "7. Concluding remarks", "content": "CTA is the forthcoming array of IACTs, and one of its most important physics goals is the study of the origin and propagation of CR. We have investigated its capabilities for the study of CR diffusion in molecular clouds, by making a phase space exploration of different cases which could be observable with such facility. This complements the study presented by Acero et al. (2012). We have showed theoretical predictions for VHE γ -rays fluxes from an accelerator inside a massive cloud. The simulations using the CTA response for 50h observation time indicate that it will be possible to constrain the diffusion coefficient parameter space. To have the accelerator inside the cloud is an idealized case. However, it allows us to study the impact of CTA observations on expected spectral and morphological features due to an overdensity of CR in the cloud over the CR sea. We expect to be able to detect γ -ray emission from massive molecular clouds with CTA observations. Specifically, for clouds of 10 5 M /circledot and θ src ∼ 0 . 5 · , we expect detection even in the case of passive cloud, i.e. only permeated by the CR background, out to distances of /lessorsimilar 1 kpc. Clouds with the same mass, but with an enhanced population of CR ( κ > 20) could be detected throughout the galaxy. GMC in the Gould Belt are very massive but very extended objects, therefore rendering problematic the prospect of detection for these clouds. Spectral features will aid derivation of constraints, especially the possible identification of breaks. Moreover, emission from extended objects could be divided in subregions. The superior angular resolution and sensitivity of CTA, with respect to the current generation of instruments, will permit the study of spatial spectral evolution with increasing distance from the accelerator, which retains the footprint of the underlying particle distribution and its diffusion. Regarding the source profiles, older sources are expected to have larger extension. A faster diffusion coefficient will have the same effect. It is expected that, if the observed molecular cloud exhibits a peaked density profile, the emission profile will also be peaked. Indeed the outer shell of the molecular cloud might fall below detection with CTA sensitivity, and thus prevent the detection of the full extension of the cloud. Therefore the knowledge of the properties of molecular clouds is important to aid the study of the objects described here. Information on the distribution of the target material will be of great help in order to reconstruct the parameters relative to the acceleration and diffusion of charged particles in the cloud in those cases. At the beginning of operation, CTA might be used to survey the galactic plane. Two types of tests could be performed, even with small integration times for a given position in the sky. The observation of a typical impulsive source like a SNR hosted in a molecular cloud with similar mass and distance as the ones investigated here, even if the age of the accelerator is of the order of 10 5 years, will constrain the diffusion coefficient (provided a precise knowledge of its distance), with detection assured for slow diffusion even with a few hours integration. While a source with the high fluxes investigated here might be rare (and would probably be already detected in TeV even though with much less detail in the spectral reconstruction), we have shown in Section 4.3 that we will be able to probe a significant fraction of the galaxy for even lower fluxes. Therefore, at the completion of the galactic plane survey, we will be able to study a numerous population, thus contributing to the study of the properties of accelerators and the propagation mode of CRs in molecular cloud.", "pages": [ 8, 9 ] }, { "title": "Appendix A: Grid construction", "content": "The sphere of emission has been divided in a regular cartesian grid ( x , y , z ). The approximation is checked with the calculation of the total mass, correctly reproduced. Also the Aharonian & Atoyan (1996) curves are reasonably reproduced by our code. Throught the paper the parameter space investigated is limited in a grid formed by the following parameters: The corresponding flux is calculated for each grid block from the corresponding proton spectrum solving the diffusion equation as in (Aharonian & Atoyan 1996). The emission is then integrated along the line of sight. The integration is on the variable y , where a condition is posed. The maximum accepted is y max = √ R 2 cloud -z 2 -x 2 . Only a quarter of the sphere is calculated, because of isotropy. The emission integrated along the line of sight can then be used to simulate the flux of the chosen projected shells. Throughout the paper we divide the projected emission region in five concentric shells. The projected mass of each of the five shells is shown in Table A.1. Acknowledgements. We acknowledge discussions with Ana Y. Rodriguez-Marrero and many colleagues of the CTA collaboration, especially Sabrina Casanova, Elsa de Cea del Pozo, Daniela Hadasch, Jim Hinton, Konrad Bernloehr, and Abelardo Moralejo. We acknowledge support from the Ministry of Science and the Generalitat de Catalunya, through the grants AYA2009-07391 and SGR2009-811, as well as by ASPERA-EU through grant EUI-2009-04072.", "pages": [ 9 ] }, { "title": "References", "content": "Ackermann, M., Ajello, M., Allafort, A. et al. 2012 ApJ, 755, 22A Acero, F. Bamba, A., Casanova, S. et al. (CTA Consortium) 2012 Astroparticle Physics CTA Special Issue, in press (http://dx.doi.org/10.1016/j.astropartphys.2012.05.024, arXiv:1209.0582) Fujita, Y., Ohira, Y., Tanaka, S. J. & Takahara, F. 2009 ApJ,707L,179F Giuliani, A., Cardillo, M., Tavani, M. et al 2011 ApJ, 742L, 30G Goodman J. A., for the HAWC Collaboration 2010 ASPC, 426, 19G Hinton, J. A., & Hofmann, W. 2009 ARA&A, 47, 523H Li H., & Chen, Y. 2010, MNRAS 409, L35", "pages": [ 9, 10 ] } ]

[ { "title": "ABSTRACT", "content": "We study the hadron-quark phase transition in the interior of hot protoneutron stars, combining the Brueckner-Hartree-Fock approach for hadronic matter with the MIT bag model or the Dyson-Schwinger model for quark matter. We examine the structure of the mixed phase constructed according to di ff erent prescriptions for the phase transition, and the resulting consequences for stellar properties. We find important e ff ects for the internal composition, but only very small influence on the global stellar properties. Key words. dense matter - equation of state - stars:interiors - stars:neutron", "pages": [ 1 ] }, { "title": "Structure of the hadron-quark mixed phase in protoneutron stars", "content": "H. Chen, 1 , 2 G. F. Burgio, 1 H.-J. Schulze, 1 , 3 and N. Yasutake 4 October 11, 2021", "pages": [ 1 ] }, { "title": "1. Introduction", "content": "A protoneutron star (PNS) is formed after the gravitational collapse of the core of a massive star ( M /greaterorsimilar 8 M /circledot ), exploding in a type-II supernova (Shapiro & Teukolsky 1983; Bethe 1990). Although the explosion mechanism is still not fully explained (Burrows 2012), some general features can be considered as robust. In fact, just after the core bounce, the PNS is very hot and leptonrich, and neutrinos are trapped for a few seconds. The following evolution of the PNS is dominated by neutrino di ff usion, causing deleptonization and subsequently cooling. Ultimately, the neutron star (NS) achieves thermal equilibrium and stabilizes at practically zero temperature without trapped neutrinos. The theoretical description of the formation of a PNS requires an accurate treatment of the microphysics of the collapsing matter, in particular of neutrino transport and related processes (Burrows & Lattimer 1986; Prakash et al. 1997; Burrows 2012). Moreover the violent dynamical processes occurring in the contracting-exploding star need to be treated in the framework of general relativity [see Ott (2009) for a recent review]. The physical processes which contribute to the subsequent PNS evolution, such as nuclear and weak interactions and energy and lepton number transport by neutrino di ff usion, are very di ffi cult to include in dynamical simulations. Thus, most simulations of the gravitational core collapse to a PNS end shortly after the core bounce and the launch of the supernova explosion, typically after a few hundreds of milliseconds, and only a few dynamical simulations extend to the first minute of the PNS life (Pons et al. 1999; Fischer & Mueller 2009). During the evolution of a PNS into a NS, a hadron-quark (HQ) phase transition could take place in the central region of the star (Prakash et al. 1995; Lugones & Benvenuto 1998; Pons et al. 1999, 2001; Steiner et al. 2000; Epsztein Grynberg et al. 2000; Nicotra et al. 2006a; Yasutake et al. 2011), and this would alter substantially the composition of the core. In fact the heaviest NS, close to the maximum mass (about two solar masses), are characterized by central baryon densities larger than 1 / fm 3 , as predicted by calculations based on a microscopic nucleonic equation of state (EOS). The study of hybrid stars is also important from another point of view: Purely nucleonic EOS are able to accommodate fairly large (P)NS maximum masses (Baldo et al. 1997; Akmal et al. 1998; Glendenning 2000; Zhou et al. 2004; Li & Schulze 2008), but the appearance of hyperons in beta-stable matter could strongly reduce this value (Glendenning 2000; Schulze et al. 2006; Li & Schulze 2008; Carroll et al. 2009; Ðapo et al. 2010; Schulze & Rijken 2011). In this case the presence of non-baryonic, i.e., 'quark' matter would be a possible manner to sti ff en the EOS and reach larger NS masses (Burgio et al. 2002; Maieron et al. 2004; Kurkela et al. 2010; Weissenborn et al. 2011). Heavy NS thus would be hybrid quark stars. In previous articles (Nicotra et al. 2006b; Burgio & Schulze 2009, 2010; Burgio et al. 2011a) we have studied static properties of PNS using a finite-temperature hadronic EOS including also hyperons (Burgio et al. 2011b) derived within the Brueckner-Bethe-Goldstone theory of nuclear matter (Baldo 1999). An eventual HQ phase transition was modeled within an extended MIT bag model (Nicotra et al. 2006a; Yasutake et al. 2011) or a more sophisticated quark model, the DysonSchwinger model (DSM) (Roberts & Williams 1994; Roberts & Schmidt 2000; Alkofer & von Smekal 2001; Roberts et al. 2007; Chen et al. 2011, 2012). The purpose of the present work is to complement our previous articles by studying details of the HQ phase transition occuring in hybrid stars and their implications for the structure of a PNS, in particular the question whether global (P)NS observables are sensitive to and thus may reveal information on the internal stellar structure. In Sec. II we briefly sketch the theoretical approaches which we use for modeling the hadron and the quark phases, and in Sec. III we describe the corresponding pure phases. The structure of the mixed phase is discussed in Sec. IV, and the results are illustrated in Sec. V. Finally, in Sec. VI we summarize our conclusions.", "pages": [ 1 ] }, { "title": "2. Equations of state", "content": "The EOSs for hadronic matter (HM) and quark matter (QM) that we use in this work, have been amply discussed in previous publications (Burgio et al. 2002; Maieron et al. 2004; Nicotra et al. 2006a,b; Burgio & Schulze 2009, 2010; Burgio et al. 2011b; Chen et al. 2011, 2012), where all necessary details can be found. Our hadronic EOS is obtained from Brueckner-HartreeFock (BHF) calculations of (hyper)nuclear matter (Schulze et al. 1998; Baldo et al. 1998, 2000) based on realistic potentials [the Argonne V 18 nucleon-nucleon (Wiringa et al. 1995) and the Nijmegen NSC89 nucleon-hyperon (Maessen et al. 1989) in this case] supplemented by nucleonic Urbana UIX three-body forces (Carlson et al. 1983; Schiavilla et al. 1986; Pudliner et al. 1997), and extended to finite temperature (Burgio et al. 2011b). We employ two di ff erent representative models for QM, an extended MIT bag model [the model with a density-dependent bag constant of Burgio et al. (2002); Maieron et al. (2004); Nicotra et al. (2006a)] and a Dyson-Schwinger model [the model DS4 of Chen et al. (2011, 2012)], which yield in fact quite di ff erent internal structures of hybrid stars. Those theoretical calculations provide the free energy density of the bulk system (pure HM or QM) as a function of the relevant partial number densities ni and the temperature, f ( { ni } , T ), from which all thermodynamic quantities of interest can be computed, namely, the chemical potentials µ i , pressure p , entropy density s , and internal energy density ε read as where nB is the total baryon number density. These quantities allow to determine the stellar matter composition and the EOS, which is the fundamental input for solving the TolmanOppenheimer-Volko ff equations of (P)NS structure.", "pages": [ 2 ] }, { "title": "3. Pure phases", "content": "In neutrino-trapped beta-stable (hyper)nuclear or quark matter the chemical potential µ i of any particle i = n , p , Λ , Σ -, u , d , s , e , µ, ν e , νµ, . . . is uniquely determined by the conserved quantities baryon number Bi , electric charge Ci , and weak charges (lepton numbers) L ( l ) i , l = e , µ with the corresponding set of independent chemical potentials µ B , µ C , µ L ( e ) , µ L ( µ ) : In this work we neglect muons and muon neutrinos due to their low fractions and negligible impact on global stellar properties, hence use simply ν ≡ ν e , L ≡ L ( e ) . The relations between chemical potentials and partial densities for hadrons and quarks are given by the microscopic models mentioned before, while leptons are treated as free fermions. With such relations, the bulk system in each phase can be solved for a given baryon density, imposing the charge neutrality condition and lepton number con- When the neutrinos ν e are untrapped, the lepton number is not conserved any more, the density and the chemical potential of ν e vanish, and the above equations simplify accordingly.", "pages": [ 2 ] }, { "title": "4. Mixed phase constructions", "content": "Weare interested in the HQ phase transition in PNS and consider therefore the usual oversimplified standard conditions, namely trapped hot matter with a fixed lepton fraction Ye ≡ ( ne + n ν ) / nB = 0 . 4 and either an isentropic, S / A = 2, or an isothermal, T = 40 MeV, temperature profile. One could consider more realistic profiles (Burgio et al. 2011a), but we focus in this work on the di ff erence between phase transition constructions. Afully microscopic treatment of the HQ mixed phase involving finite-size (pasta) structures can only be performed numerically (Tatsumi et al. 2003; Endo et al. 2005, 2006; Maruyama et al. 2007; Yasutake et al. 2009, 2012a,b). One introduces Coulomb energies and surface energies via a HQ surface tension and then minimizes the (free) energy of a Wigner-Seitz (WS) cell, allowing for di ff erent geometrical structures of the quark phase embedded in the hadron phase and vice versa. The output are the optimal size and geometry of the cell, as well as the local distributions of the individual particle species, and also the Coulomb field inside the cell. Some illustrative examples can be found in the given references. This is a very time-consuming and not very transparent numerical procedure. It is therefore convenient to search for reliable approximations to this procedure, and in this article we compare two prescriptions corresponding to two limiting cases of the full numerical procedure, that are termed global charge neutral (GCN) and local charge neutral (LCN) mixed phase. The first procedure is well known as Bulk Gibbs or Glendenning construction (Glendenning 1992, 2001) from the zerotemperature case and corresponds to a 'small' WS cell [compared to the electromagnetic Debye screening length, which is about 5-10 fm (Heiselberg et al. 1993; Heiselberg 1993; Glendenning & Pei 1995; Christiansen & Glendenning 1997; Takatsuka et al. 2006)] caused by a 'small' HQ surface tension. In this case the electromagnetic potential is practically constant throughout the cell, and an electric field does not exist. Consequently the electron density is also constant, while the hadron and quark densities and their electric charges are di ff erent in order to fulfill the conditions of pressure and baryon chemical potential equality at the HQ interface. In the case of neutrino trapping, the neutrino densities have also to be equal in both phases, which together with the equal electron densities implies equal lepton densities nL = ne + n ν (but not lepton fractions Ye = nL / nB ) in both phases. Altogether we have therefore the equality of the intensive thermodynamical quantities in both phases: which together with the general rule Eq. (5) determines the composition of the system for given overall baryon density nB , vanishing electric charge, fixed lepton fraction Ye in the trapped case, and eventually a prescribed entropy profile S / A ( nB ): where χ is the volume fraction occupied by the quark phase and the last equation determines the local temperature. The opposite limiting case (LCN) corresponds to a WS cell that is large relative to the electromagnetic Debye screening length, and a large surface tension. In this situation the electric charges are well screened inside the cell and both QM and HMare locally charge neutral nearly everywhere, except on a small boundary layer near the HQ interface, where a positively charged layer of HM and a negatively charged one of QM are present and create a strong but very localized electric field (Voskresensky 2002; Voskresensky et al. 2003). Consequently there occurs a sharp rise δµ C of the Coulomb potential at the HQ interface and Eq. (11) is modified to such that for example the electron density is now di ff erent in hadron and quark phases. In beta-stable untrapped matter this situation corresponds exactly to the usual Maxwell construction, joining two chargeneutral phases by equality of pressure and baryon chemical potential, that is often employed for simplicity. Including neutrino trapping with microscopic finite-size structures requires always homogeneous neutrino densities, Eq. (9), and therefore, due to the unequal electron densities, in this case the trapping condition becomes a global one, as expressed by Eq. (17). Due to the additional degree of freedom represented by the neutrino density, then the LCN construction is realized in the PNS as an extended mixed phase involving a HQ coexistence region with a continuously varying pressure (Hempel et al. 2009; Pagliara et al. 2009, 2010; Yasutake et al. 2012b). We have explained that the GCN and LCN constructions are in fact idealized scenarios that correspond to two opposite extremes of the microscopic treatment. It has been pointed out in Yasutake et al. (2012a,b) that actually the LCN construction is closer to the full microscopic treatment of finite-size e ff ects than the GCN, and it is therefore of interest to compare the predictions of the two constructions for the internal composition and other properties of PNS, which we will do now. /s32 /s32 /s32 /s32 /s32 /s32", "pages": [ 2, 3 ] }, { "title": "5.1. Internalcomposition", "content": "The relative particle populations are shown as a function of the baryon density in Fig. 1 for the bag model and in Fig. 2 for the DSM, for trapped matter with Ye = 0 . 4 and i) entropy per baryon S / A = 2 (upper panels), ii) temperature T = 40 MeV (middle panels), and iii) untrapped and cold neutron star matter (lower panels). The GCN (left panels) and LCN (right panels) calculations are compared. There are big di ff erences between the MIT and DSM regarding the HQ mixed phase that have been pointed out in Chen et al. (2011, 2012): With the MIT model the HQ phase transition starts at fairly low baryon density and a pure quark phase is reached at not too large density, whereas with the DSM the onset of the mixed phase occurs at higher density and the system remains in the mixed phase even at very large density. Furthermore, hyperons are allowed with the MIT model, where they might appear only in small fractions at low density and are replaced by strange QM at higher density (this can be seen in the central panels of Fig. 1), whereas they prevent any transition to QM with the DSM and have to be excluded by hand in that case. Therefore, these two very di ff erent quark models might be good candidates to reveal important di ff erences between the phase transition constructions that we are examining. In fact, comparing now the results obtained with both prescriptions (left and right panels), we observe behavior in line with the general properties mentioned before. We remind that for cold NS matter (bottom panels) the LCN corresponds to the usual Maxwell construction and the GCN to the bulk Gibbs construction; and it is well known that the density range of the mixed phase with the Gibbs construction is wider than the one with the Maxwell construction, which can be seen in the plots. This behavior remains also in trapped hot matter, where the GCN spans always a wider density range than the LCN (with the MIT) or begins at lower density (with the DSM). In all cases the trapping condition shifts the onset of the mixed phase to slightly higher density. The di ff erences between the LCN and GCN constructions are fairly small for the MIT model, but significant for the DSM: Here the GCN (Bulk Gibbs) mixed phase occurs in a much wider density interval than the LCN (Maxwell) one. Apart from the Maxwell construction for cold NSs, the matter remains in the mixed phase and pure QM is never reached. This variance is due to the qualitatively di ff erent density dependence of the e ff ective bag constant in the MIT and DSM, see Chen et al. (2011, 2012).", "pages": [ 4 ] }, { "title": "5.2. Equationofstate", "content": "In Fig. 3 the EOS p ( nB ) is displayed for the di ff erent stellar configurations, quark models, and mixed phase constructions as before. For comparison also the pure phases (nucleons only, nucleons + hyperons, quarks) are shown. We observe only minor di ff erences between GCN and LCN in the hot and trapped matter, even for the DSM, where the particle fractions are quite di ff erent in both cases. Stronger di ff erences between GCN and LCN appear in the cold case, where a plateau in the pressure shows up for the LCN (Maxwell) calculation. Thus if during the temporal evolution the system would remain in the LCN phase, the extended mixed phase region would gradually disappear. However, one should remember that both LCN and GCN are highly idealized constructions and the true behavior depends on currently uncertain microphysics like the HQ surface tension.", "pages": [ 4 ] }, { "title": "5.3. Stellarstructure", "content": "Once the EOS is known, the stable configurations of a (P)NS can be obtained from the well-known hydrostatic equilibrium equations of Tolman, Oppenheimer, and Volkov (Shapiro & Teukolsky 1983). In the low-density range, where nucleonic clustering sets in, we cannot use the BHF approach, and therefore we join (Burgio & Schulze 2010) the BHF EOS to the finite-temperature or -entropy EOS of Shen et al. (1998a,b), which is more appropriate at densities below nB /lessorsimilar 0 . 07 fm -3 , since it does include the treatment of finite nuclei. Our results for the gravitational mass as a function of the central baryon density, for the di ff erent stellar configurations and using the di ff erent EOS introduced previously, are displayed in Fig. 4. We observe in NS matter the strong softening e ff ect of the hyperons (dash-dotted purple curves) on the purely nucleonic configurations (dotted blue curves), which is however strongly reduced in trapped matter, because the hyperon concentrations remain smaller (Prakash et al. 1997; Burgio et al. 2011b). In the case of the MIT model, the mass-density relations obtained with the GCN or LCN constructions (dash-dot-dotted green vs short-dashed orange curves) are nearly indistinguishable, apart from the unphysical low-mass region, where the Maxwell construction can be recognized in the NS configuration with LCN. The maximum mass of the hybrid stars decreases slightly with respect to both the nucleonic and the hyperonic stars in the trapped cases, whereas it increases (decreases) with respect to cold hyperon (nucleon) NS. In NS this is due to the fact that the hyperon population is suppressed by the onset of quarks, whereas in PNS the trapping reduces the hyperon population. In all cases, with the HQ phase transition, the value of /s32 /s32 /s32 /s32 the maximum mass is about 1.5 M /circledot and thus rather low, as is a general feature of the MIT model (Alford & Reddy 2003). If the DSM is used, the di ff erences between GCN and LCN (solid black vs dashed red curves) are slightly larger, in particular for NS, where the LCN leads to unstable configurations at the onset of the quark phase. Nevertheless the di ff erences between GCN and LCN maximum masses are insignificant in all configurations. In this case the phase transition takes place only if hyperons are excluded from the hadronic phase, and the value of the maximum mass for the hybrid configuration always decreases with respect to the purely nucleonic star. In particular, for PNS the maximum mass is about 1 . 75 M /circledot , while for NS it is slightly smaller. These values for the maximum mass of hybrid (P)NS depend on the choice of the nucleonic three-body force and on the details of the DSM. Larger values could eventually be reached with di ff erent parameter choices (Chen et al. 2011, 2012), in agreement with the current observational data (Demorest et al. 2010), which is in contrast to the case with the MIT model. However, the exact value of the maximum mass of a (P)NS is still an open problem [see also Romani et al. (2012)] and not the purpose of the present article. /s32 /s32 /s32", "pages": [ 4, 5 ] }, { "title": "6. Conclusions", "content": "In this article we discussed the occurrence of a hadron-quark mixed phase in the interior of hybrid (proto)neutron stars. We explained the physical origin and justification of the idealized LCN and GCN phase transition constructions, which represent two opposite limiting cases of the microscopic treatment of electromagnetic finite-size e ff ects, and examined their consequences with two very di ff erent quark models. While indeed the internal composition of hybrid (proto)stars turns out to be very di ff erent with both constructions, the impact on the equation of state and masses is very much reduced, so that these global observables could hardly serve as an indication for the type of phase transition and thus the internal stellar structure. Therefore, for a true understanding of the nature of the mixed phase, detailed microscopic investigations of the finite size effects and their importance for the stellar microphysics (cooling, transport, oscillations, ...) are required.", "pages": [ 5 ] }, { "title": "Acknowledgments", "content": "We acknowledge useful discussions with T. Maruyama and T. Tatsumi. This work was partially supported by CompStar, a Research Networking Programme of the European Science Foundation, and by the MIUR-PRIN Project No. 2008KRBZTR.", "pages": [ 5 ] }, { "title": "References", "content": "Heiselberg, H., Pethick, C. J., & Staubo, E. F. 1993, Physical Review Letters, 70, 1355 Maessen, P. M. M., Rijken, T. A., & de Swart, J. J. 1989, Phys. Rev. C, 40, 2226 Maieron, C., Baldo, M., Burgio, G. F., & Schulze, H.-J. 2004, Phys. Rev. D, 70, 043010", "pages": [ 6 ] } ]

[ { "title": "ABSTRACT", "content": "Context. Energetic feedback from Supernovae and stellar winds can drive galactic winds. Dwarf galaxies, due to their shallower potential wells, are assumed to be more vulnerable to this phenomenon. Metal loss through galactic winds is also commonly invoked to explain the low metal content of dwarf galaxies. Aims. Our main aim in this paper is to show that galactic mass cannot be the only parameter determining the fraction of metals lost by a galaxy. In particular, the distribution of gas must play an equally important role. Methods. We perform 2-D chemo-dynamical simulations of galaxies characterized by di ff erent gas distributions, masses and gas fractions. Results. The gas distribution can change the fraction of lost metals through galactic winds by up to one order of magnitude. In particular, disk-like galaxies tend to loose metals more easily than roundish ones. Consequently, also the final metallicities attained by models with the same mass but with di ff erent gas distributions can vary by up to one dex. Confirming previous studies, we also show that the fate of gas and freshly produced metals strongly depends on the mass of the galaxy. Smaller galaxies (with shallower potential wells) more easily develop large-scale outflows, therefore the fraction of lost metals tends to be higher. Key words. Galaxies: abundances - Galaxies: dwarf - Galaxies: evolution - Galaxies: ISM - Galaxies: jets", "pages": [ 1 ] }, { "title": "The fate of heavy elements in dwarf galaxies - the role of mass and geometry", "content": "S. Recchi 1 /star and G. Hensler 1 /star/star Institute for Astrophysics, University of Vienna, Turkenschanzstrasse 17, A-1180 Vienna", "pages": [ 1 ] }, { "title": "1. Introduction", "content": "Theories of cold dark matter-dominated hierarchical growth of structures predict that dwarf galaxy- (DG-)sized objects are the building blocks for the formation of large galaxies. In spite of their relevance, the most important physical phenomena regulating the birth and evolution of DGs are still obscure to date. For sure, star formation (SF) plays a key role in shaping DGs and determining their fates. Since the binding energy of the interstellar medium (ISM) in early gas-rich DGs is very small (smaller than the explosion energy of just a few Supernovae (SNe)) many authors speculated that a high SF rate in a DG would create a galactic wind and thus produce a transition from a dwarf irregular (dIrr) to a dwarf spheroidal (dSph) or dwarf elliptical (dE) (see e.g. Larson 1974; Vader 1986; Dekel & Silk 1986). From a chemical point of view, the occurrence of a galactic wind right after the formation of the first stars would imply a very limited interval of time during which the metals, restored by dying stars, can pollute the ISM and enrich the following stellar populations. If galactic winds preferentially occur in DGs, these objects (with low masses) will experience a limited chemical enrichment. Therefore, a correlation between stellar mass M (or luminosity L) and metallicity Z of DGs is expected. Indeed, this M-Zcorrelation among DGs exists (see e.g. Skillman et al. 1989; Lee et al. 2006; Kirby et al. 2008; Zhao et al. 2010; Andrews & Martini 2012). A M-Z relation extends also to high redshifts (Erb et al. 2006; Maiolino et al. 2008; Laskar et al. 2011; Wuyts et al. 2012) although, in this case, the galactic masses of the galaxies for which metallicity determinations are available are usually quite high (but see Mannucci et al. 2011). The M-Z relation among DGs corroborates the idea that SN-driven galactic winds play a dominant role in the evolution of these objects. More recently, a general relation between stellar mass, gas-phase metallicity and star formation rate has been found (Mannucci et al. 2010). However, detailed hydrodynamical simulations of DGs showed that the galactic winds, although often able to expel a large fraction of freshly produced metals, are unable to eject an equally large fraction of pristine (i.e. not processed) gas. This is mostly due to the fact that, if the initial DG gas distribution is flattened (as observed in dIrrs), then there is a direction with steeper pressure gradient so that the galactic wind will preferentially expand along that direction, the transport of gas along the other directions being very limited (see e.g. D'Ercole & Brighenti 1999; MacLow & Ferrara 1999 (hereafter MF99); Recchi et al. 2001). This e ff ect can be appreciated by inspecting Fig. 1, where the gas and oxygen distribution of a model galaxy experiencing a galactic wind (taken from the calculations of Recchi et al. 2006) is shown. Most of the disk gas (at least above R = 1 kpc) has not been a ff ected by the galactic wind and it will fall back towards the center of the galaxy once the energy source of the starburst will be exhausted. On the other hand, the oxygen can be easily channelled along the funnel created by the galactic wind. Also very energetic ( E ∼ 10 53 erg) hypernovae exploding in dwarf protogalaxies, although more disruptive than normal SNe, are able to expel only ∼ 10 to 20 % of the baryonic mass originally present in the galaxy (Vasiliev et al. 2008). This result is widely accepted by the astrophysical community, although some authors (e.g. Tenorio-Tagle 1996; Silich & Tenorio-Tagle 1998), believe that low-density gaseous galactic halos can strongly a ff ect the circulation of the metal-rich matter processed by the central starburst. Kiloparsec-scale bipolar outflows are created but, later on, loss of pressure support and interaction with the di ff use halo slow down the expansion of the superbubble and the freshly produced metals can eventually fall back towards the center of the galaxy. Thus, according to this scenario, energetic events associated to a starburst can still create a large-scale outflow but this outflow does not necessarily become a galactic wind (i.e. its velocity does not exceed the escape velocity of the galaxy). It is also worth noticing that, while the chemical yields of exploding massive stars are very large (thus the metallicity of interiors of SN-driven bubbles is up to 40 times the solar value), abundances of the hot gas in galaxies experiencing large-scale outflows (determined from X-ray spectra) cover with high probability a range between only solar and twice solar (see e.g. Martin et al. 2002 for NGC1569 or Ott et al. 2005 for a larger sample of galaxies). From this fact, it can be deduced that the hot SN ejecta becomes mass loaded (by a factor of up to 10) during its expansion, probably due to evaporative and turbulent mixing with colder interstellar clouds. By this, the hot gas loses not only momentum but also energy by enhanced cooling in addition to adiabatic expansion so that the outbreak of the hot gas to a galactic wind could be hampered in many cases. This massloading e ff ect is probably causing the discrepancy between the observed extent and temperature of the extremely cool superbubble in NGC 1705 and its analytic parameter correlation (Hensler et al. 1998). Because of the gas density outside the SF region, the development of galactic outflows or galactic winds considerably depends on the degree of flattening (or on the rotation) of the parent galaxy, an aspect that has not been fully explored in the past (but see Silich & Tenorio-Tagle 2001; Michielsen et al. 2007; Schroyen et al. 2011. See also Strickland & Stevens 2000; Ferrara & Tolstoy 2000; Vasiliev et al. 2008; Recchi et al. 2009). Although flat galaxies are supposed to loose a large fraction of metals as a consequence of bipolar galactic winds, roundish galaxies are not characterized by a direction along which the pressure gradient is significantly steeper, therefore they are more likely to retain much of the metals (Marcolini et al. 2006; Recchi et al. 2007). In the end, this di ff erence in the initial gas distribution could lead to a spread in the observed final metallicity of galaxies with the same mass. Such a spread is indeed observed in the M-Z relation of DGs (Lee et al. 2006; Kirby et al. 2008; Zhao et al. 2010), or in the same correlation at z /similarequal 3 . 5 (Maiolino et al. 2008). In this paper we explore in detail the e ff ect of gas geometry on the development of galactic winds and on the fate of freshly produced metals. This work must be thus seen as a refined study along the lines of the MF99 work, in which variations in the initial distribution of gas in galaxies have not been considered. The second key parameters we consider in this study is the initial mass of the galaxy. As already mentioned, small galaxies (with shallow potential wells), are expected to develop galactic winds more easily than larger galaxies, therefore the mass is certainly a key parameter in determining the fate of pristine gas and freshly produced heavy elements. The dynamical evolution of a galaxy is so complex that the fate of heavy elements cannot depend only on mass and geometry of the parent galaxy. An obvious parameter that strongly a ff ects the development of galactic winds is the luminosity (MF99). In turn, the luminosity depends on the star formation rate (SFR) and on the star formation history (SFH) and it is known, at least in the Local Group, that very large galaxy-togalaxy SFH variations exist (see e.g. Monelli et al. 2010a, 2010b; Hidalgo et al. 2011). Moreover, other details of the ISM structure (for instance the presence of clouds or porosity) and of the feedback prescriptions (e.g. how the SN explosion energy is redistributed, where, on which timescales and at which e ffi ciency etc.) can all play a role in determining the amount of metals carried out of a galaxy by galactic winds and eventually in the final global metallicity of the galaxy. Although the consequences of an inhomogeneous ISM on the development of galactic winds is analyzed in detail in Recchi & Hensler (2007, hereafter RH07), we have undertaken a comprehensive parameter study through detailed chemo-dynamical simulations, the results of which will be presented elsewhere. Here we will focus mainly on 18 basic models, characterized by di ff erent geometries, total baryonic masses and initial gas fractions. The paper is organized as follows: in Sect. 2 a short summary on the literature results concerning the fate of freshly produced metals will be given. In Sect. 3 our numerical scheme will be briefly reviewed and the set-up of the 18 basic models models described. In Sect. 4 the main results of these calculations will be presented, underlying also the e ff ect of some other key parameters not fully explored in this work. In Sect. 5 the results will be discussed and some conclusions will be drawn.", "pages": [ 1, 2 ] }, { "title": "2. A brief overview of literature results", "content": "As already mentioned in the Introduction, several papers in the literature attempted to study the e ff ect of galactic winds on the circulation and redistribution of metals in DGs. The main results of the often-cited work of MF99 are that, even in the presence of a strong galactic wind driven by SNeII, the ejection e ffi ciency of unprocessed gas is always close to zero (with the exception of galaxies with initial baryonic mass ≤ 10 6 M /circledot ). On the other hand, the ejection e ffi ciency of freshly produced heavy elements is always close to one (with the exception of galaxies with initial baryonic mass ≥ 10 9 M /circledot ). D'Ercole & Brighenti (1999) found results similar to the ones of MF99, in the sense that only minor fractions of ISM (but in some cases large fractions of metals) are expelled through galactic winds. They however calculated the evolution of the galaxy for much longer times than MF99, discovering that the center of the galaxy can be replenished with cold gas in a timescale of ∼ 100 Myr (see also Recchi &Hensler 2006). As already mentioned, Silich & Tenorio-Tagle (1998) found instead that in most of the models galactic winds do not develop (mainly due to the presence of a hot gaseous halo surrounding the galaxy). Fragile et al. (2004) studied the e ff ect of SN explosions not localized in the center of the galaxy (as MF99 did) but distributed over radii of up to 80% of the disk radius, discovering that, in this case, radiative losses are more e ff ective and the development of a large-scale outflow is hampered. Also very interesting is the work of Scannapieco & Bruggen (2010), who attempted to model turbulent velocities and turbulent length scales in DGs and to inject SN energy into supersonic turbulence. The wind e ffi ciencies they found are still quite low. Rodr'ıguez-Gonz'alez et al. (2011) addressed the same prob lem (mass and metal ejection e ffi ciencies in DGs) exploring a much more extended set of parameters (in particular they covered a wide range of starburst and galactic gas masses). Their results are overall in agreement with the previously quoted works (with the exception of Silich & Tenorio-Tagle 1998). More recently, Schroyen et al. (2011) explored in detail the e ff ect of rotation on the evolution of DGs. Their focus was to understand the origin of the dichotomy in radial stellar metallicity profiles of DGs and their 'centrifugal barrier mechanism' help explaining these observations. Usually their models do not develop galactic winds (or develop only weak outflows), thus the determination of ejection e ffi ciencies (of pristine gas and freshly produced metals) is not addressed in their paper. The same can be said about the detailed determination of the energy required to expel newly processed matter by Silich & Tenorio-Tagle (2001); their analytic model is in good agreement with the numerical results of MF99 and shows clearly the e ff ect of geometry on the development of galactic winds and on the fate of metals if the intergalactic pressure is low. However, their inferred large intergalactic pressure hampers the development of galactic winds. As this short summary indicates, the past literature mostly focused on the e ff ect of mass and luminosity on the gas and metal ejection e ffi ciencies from galaxies. The chemical evolution is usually not very detailed in these models. Recchi et al. (2001, 2002, 2004, 2006) provided detailed information on the ejection e ffi ciencies of single chemical species produced by SNe (of both Type II and Type Ia) and by winds from intermediatemass stars. RH07 added important information on the e ff ect of clouds. The most significant results of these works can be summarized as follows: A study of the early evolution of tidal DGs (which are assumed to be dark-matter- (DM-)free) and their possible correlation with the satellite DGs surrounding the Milky Way has been also performed (Recchi et al. 2007). These models, in spite of being DM-free, can sustain the energy released by dying stars without experiencing a complete blow-away, for several hundreds of Myr, provided that they keep their initial spherical symmetry. A typical model develops through a network of cavities and filaments due to the patchy distribution of the SF sites. A supershell is formed, but it grows slowly in size and does not quench the SF process. This is because the initial ISM distribution in the galaxy is spherical and there is therefore no preferential direction through which the galactic wind can flow. Under these conditions, either the feedback is able to remove all the gas at once ( blow-away ) or all the gas (or most of it) is retained inside the galaxy. Our study, and other similar ones (e.g. Marcolini et al. 2006), show that it is not easy to get rid of all the gas in an initially spherical galaxy.", "pages": [ 2, 3 ] }, { "title": "3. Description of the model", "content": "This work aims at studying in detail the dynamical and chemical evolution of DGs and the e ff ect of key parameters (in particular baryonic mass, initial gas mass fraction and degree of flattening) on the development of galactic winds and on the fate of freshly produced metals. A follow-up study will extend the parameter space and will concentrate on the details of the SFH and on the e ff ect of boundary conditions. The ultimate goal is to understand the main mechanisms determining the metallicity and the abundance ratios in DGs and, at the same time, infer the degree of metal pollution of the inter-galactic and intra-cluster medium due to these objects.", "pages": [ 3 ] }, { "title": "3.1. Thechemo-dynamicalcode", "content": "As chemo-dynamical code we use basically the one of Recchi et al. (2007). We recall here the basic features of this code. It is a 2-D code in cylindrical coordinates based on a second-order, MUSCL-type upwind scheme (the 1-D version of this scheme is described in Bedogni & D'Ercole 1986). The chemical enrichment of the galaxy is followed in detail; the production (by SNe of Type II, Type Ia and intermediate-mass stars) of 8 chemical elements (H, He, C, N, O, Mg, Si and Fe) is considered and the advection of these elements is followed by means of passive scalar fields. For instantaneous bursts, this scheme is described in detail in Recchi et al. (2001) and its extension to more complex SFHs is explained in Recchi et al. (2004). Since the code keeps correctly track of the evolution of the metallicity in each computational cell, the detailed metallicity-dependent cooling function of Bohringer & Hensler (1989) can be implemented. The heat transport equation is also solved by means of the Crank-Nicolson method (see D'Ercole & Brighenti 1999 for details) using the classical Spitzer-Harm thermal conductivity (Spitzer & Harm 1953; Spitzer 1956). A saturated heat flux (Cowie & McKee 1977) is adopted if the mean free path of electrons is larger than the temperature scalelength. Some improvements of this code are described in Recchi et al. (2007). In particular, metallicity-dependent stellar winds from massive and intermediate-mass stars have been considered and, primary and secondary nucleosynthetic production by stars of any age and any metallicity has been carefully calculated. Recchi et al. (2007) describe also an implementation of SF recipes. The selfconsistent SF module is not adopted in this work where, in analogy with many previous studies (MF99 for instance) the SFH (or the luminosity) is an input of the model and not the result of the galactic evolution. In a work in preparation, models with a self-consistent SFH will be shown and discussed. Moreover, although in Recchi et al. (2007) a self-gravity solver (based on Rieschick & Hensler 2003) has been implemented, we decided not to adopt it in this study neither. The main reason for this choice is that we neglect self-gravity in building the initial equilibrium configuration (see below in the next Subsection). A parallel line of research (Vorobyov et al. 2012; Vorobyov et al., in preparation) analyzes in detail DG models with a consistent implementation of the gas self-gravity. The inclusion of self-gravity in the code would imply that, at the beginning of the simulation, the gas is initially out of equilibrium even without sinks or sources of energy. This would establish an inward gas flow. Since our main aim in this paper is to study the e ff ects of geometry and other parameters on the flow rate of gas and metals, we do not want gas infall to a ff ect our results and we want flows of gas (and metals) to be solely a ff ected by feedback processes. Moreover it is important to notice that most of similar papers already described in Sect. 2 (e.g. MF99, D'Ercole & Brighenti 1999, Strickland & Stevens 2000) also neglect self-gravity.", "pages": [ 3, 4 ] }, { "title": "3.2. Theset-up", "content": "We start with a reliable set-up of a DG, in which the gas is initially in isothermal equilibrium with a spherical DM halo and with the centrifugal force. Analogously to what is done in many similar studies, we neglect self-gravity in building the initial equilibrium configuration. We outline here that this is in general not correct even if most of the mass of the model galaxy is in the form of DM and the correct derivation of an equilibrium configuration is the one outlined by Vorobyov et al. (2012). However, the equilibrium configurations produced by Vorobyov et al. (2012) are quite complex (and also computationally demanding) and it is di ffi cult to obtain two analogous equilibrium configurations, di ff ering only for the degree of flattening of the initial gas distribution. We consider three possible values for the initial baryonic mass of the galaxy (10 7 , 10 8 and 10 9 M /circledot ) with a factor of ∼ 10 more massive DM halos. The exact factor is deduced by the correlation between the dark matter-to-baryon ratio φ and the gas content as adopted by MF99, namely φ = 34 . 7 M -0 . 29 g , 7 , where Mg , 7 is the gaseous mass of the galaxy in units of 10 7 M /circledot ). This correlation is adapted from the work of Persic et al. (1996), and is based on fitting procedures of observed rotation curves in local galaxies. One should warn the reader that it is not clear whether present-day dark matter-to-baryon ratios also correctly reflect the initial conditions of the galaxy, but we nevertheless adopt this correlation in order to facilitate the comparison with the results of MF99. The virial radius of the DM halo is assumed to be: (Madau et al. 2001; Mori et al. 2002). In this formula, M 8 is the halo mass in units of 10 8 M /circledot and zgf is the redshift of galaxy formation. Our reference value for zgf is 8. It is important also to stress that our initial distributions of gas are not artificially truncated at some cut-o ff radii as the models of MF99. Therefore, without sources or sinks of energy, our models preserve their initial configurations (at variance with the models of MF99 which tend to expand). However, since our distribution of gas extends until the edges of the computational box, we need to establish a radius within which we calculate the mass of the galaxy. We take this to be half of the virial radius. Therefore, galaxies with the same nominal baryonic mass are normalized in such a way that the total mass within 0 . 5 · rvir is the same, irrespective of the degree of flattening. This implies that flat galaxies have larger central densities than roundish ones (see below and Table 2). A small stellar disk is also initially present in the galaxy. Its density is such that a Miyamoto-Nagai potential (where Md is the mass of the stellar disk and R , z are the cylindrical radial and vertical coordinates, respectively), is reproduced. The ratio b / a between the scale lengths identifying the Miyamoto-Nagai potential is taken as one of the key parameters that we vary in our models. A small b / a corresponds to a flat model (for b / a → 0 the potential tends to the razor-thin Kuzmin model), whereas if b / a is very large, the galaxy tends to be rounder (for b / a → ∞ the potential tends to the Plummer's spherical potential). It is expected that an initially flat distribution of gas results in an easier development of bipolar outflows, since the pressure gradient along the polar direction is much steeper than that along the galactic disk. As already discussed, along the polar direction a significant fraction of the matter processed in the central SF region can be thus channelled and eventually lost from the galaxy. On the other hand, if the galaxy is initially spherical (or almost spherical), no preferential propagation direction for the superbubble exists and the freshly produced metals are more likely to remain confined inside the galaxy. One of the aims of this work is to confirm this empirical assumption and quantitatively determine the fraction of gas and metal-rich stellar ejecta lost from a DG, as a function of its degree of flattening. We consider three representative values for the ratio b / a : 0 . 2 (flat models, designated with the letter 'F'); 1 (medium models or 'M') and 5 (roundish models or 'R'). However, the mass of this pre-existing stellar disk cannot be established a priori, therefore as another parameter we vary the ratio between the mass of the MiyamotoNagai stellar disk and the total baryonic mass initially present in the galaxy. In particular, we consider two basics sets of models; for the first set (designated with 'H') the initial gas fraction is high (90% of the total baryonic mass of the galaxy) and, consequently, the pre-existing stellar disk represents a small fraction of the mass budget in the galaxy. The second set of models (designated with 'L') is characterized by a much smaller initial gas fraction (60% of the baryonic mass). Therefore, for instance, the model H7M represents a galaxy with high initial gas fraction, 10 7 M /circledot of initial baryonic mass and a medium degree of flattening ( b / a = 1). We consider thus a total of 18 basic models, but we will discuss the dependence of our results on some key parameters in Sect. 4.2. Other key parameters to be varied are the duration and intensity of the SF episode (as already mentioned, the analysis of models with self-consistent SFHs will be deferred to a future work). We assume here for simplicity that the SF is constant (with some pre-defined intensity ψ ) for a period of time ∆ t . The two parameters ψ and ∆ t are allowed to vary, with the constraint that for models of equal mass the same final amount of stars is produced. In particular, we constrain the models 'H' to produce, at the end of the SF period, a mass of newly formed stars which is twice as much as the mass of the initial stellar disk (namely the fraction fN = M ∗ , new / Md is set to be equal to 2). For the second set of models, designated with 'L', we set fN = 0 . 5, namely (since the pre-existing stellar disk is more massive for this set of models) we allow newly formed stars to be only 50% of the pre-existing disk at the end of the SF period. The resulting integrated luminosity of our models is not far from being constant (a constant luminosity has been assumed by MF99). Our reference models have ∆ t = 500 Myr and a SFR ψ such that, after ∆ t , the required fraction fN is obtained. It is obvious that, keeping the starburst duration ∆ t constant, a large starburst luminosity (i.e. a large ψ ) produces a stronger outflow, thus in turn larger ejection e ffi ciencies of gas and metals (see MF99). It is however not obvious if short, intense starbursts are more e ff ective than longlasting, milder SF episodes in getting rid of gas and processed matter. The answer to this question will be sought by considering the dependence of the ejection e ffi ciencies in models with varying ψ and ∆ t but with constant ψ × ∆ t . In particular, a few models have been calculated in which ∆ t is a factor of 10 smaller than the reference value (i.e. ∆ t = 50 Myr) and, consequently, the SFR is ten times more intense. Our reference value for ∆ t of 500 Myr is justified by observational studies indicating that the starbursts in DGs are not as short as previously thought, lasting on average a few hundred Myrs (see e.g. McQuinn et al. 2010a; 2010b). The initial gas metallicity of our reference sets of model galaxies is zero. This is because we do not have a self-consistent evolution of the early phases of our model galaxies and we can only suppose that the chemical enrichment of gas and stars in this early phase will be limited (particularly for the models 'H'). Moreover, we do not expect it to be a fundamental parameter in determining the fate of metals, since the only way a di ff erent metallicity can a ff ect the development of a galactic wind is through the cooling function and the di ff erence between the zero-metal cooling curve and that of a modest metallicity (Z = 10 -2 Z /circledot say) is negligible. We have nevertheless considered also models in which the initial metallicity of the stars (and gas!) is observationally determined (in particular we make use of the M-Z relations observationally determined by Tremonti et al. 2004 for the local Universe). As done by Fragile et al. (2004) and Rodr'ıguez-Gonz'alez et al. (2011), another important parameter in our study is the radius over which the feedback from dying stars is redistributed. It is to expect that, if the 'feedback radius' is very small, the density of the (metal-rich) ejected material will be so high that radiative losses can be very significant and can substantially increase the probability that the ejecta remain locked inside the galaxy (see e.g. Tenorio-Tagle et al. 2007). It is unclear what happens if the feedback radius is larger although, as already mentioned, the study of Fragile et al. (2004) suggests a reduction of ejection e ffi ciencies (of both gas and starburst matter). Our reference feedback radius is RF = 200 pc but models with RF = 50 pc and RF = 1000 pc have been also computed. We have also considered the possibility (as we did in RH07) that the initial distribution of gas is characterized by inhomogeneities. However, for the sake of simplicity, we consider only a random perturbation of the initial galactic density distribution. Namely, once an equilibrium configuration has been obtained, we modify the density ρ ( i , j ) in the grid ( i , j ) in this way: where ε is a small number (corresponding to the largest possible amplitude of the perturbation) and R [ -1 , 1] is a randomly generated number in the range [ -1 , 1]. A more thorough (and selfconsistent) treatment of inhomogeneities and a detailed study of the e ff ect of boundary conditions (gravitational perturbations, external pressure, infall of clouds) is deferred to a future paper. The central resolution of the simulations is 4 pc for the models with 10 8 and 10 9 M /circledot of initial baryonic mass and 2 pc for the models with 10 7 M /circledot of baryons. The resolution decreases outwards with a ratio between adjacent cells of 1.02. The reference values for some key parameters (common to all 18 basic models) are summarized in Table 1 whereas the specific parameter values distinguishing those models are recalled in Table 2. In order to appreciate the distinction between models with di ff erent geometries, Fig. 2 shows the initial gas distribution of the models L8R (upper panel), L8M (central panel) and L8F (lower panel).", "pages": [ 4, 5, 6 ] }, { "title": "4.1. Thereferencemodels", "content": "Asnapshot of the evolution of all 18 reference models (see Table 2) is shown in Fig. 3 (after an evolutionary time of 100 Myr) and in Fig. 4 (after an evolutionary time of 200 Myr). The main features of our model results can be noticed from these two figures: A more quantitative analysis of the results of the models is obtained by inspecting the columns 9-12 of Table 2. Here the ejected fractions /epsilon1 of gas and oxygen are indicated for each model after 200 and 500 Myr. We simply estimate the retained gas fractions as the ratios between the mass of pristine gas contained in 0.5 rvir at the time 200 (500) Myr and the same mass at the beginning of the simulation. To calculate the retained oxygen fraction we divide instead the mass of oxygen within 0.5 rvir by the total amount of oxygen expelled by dying stars until 200 (500) Myr. The ejection e ffi ciencies (of gas and oxygen) can thus be simply estimated as 1 -r f , where r f indicates the corresponding retained fraction. Although our estimate is approximate and can be a ff ected for instance by gas (or oxygen) temporarily leaving the galaxy and being re-accreted later on, it is clear that estimates based on the escape velocities (as in MF99) are approximate as well (ejected gas does not evolve ballistically), thus we keep our simple definition of ejected fractions, bearing in mind the inherent uncertainties, thus focusing on differences in the ejected fractions rather than on absolute values. A close inspection of the ejected fractions reveal the following properties of the models: The last two columns of Table 2 show the oxygen abundance measured as 12 + log(O / H)abundance ratio, where O / H is the abundance ratio in number, of all model galaxies. This abun- Notes. ( a ) : Initial baryonic mass in M /circledot (within 0.5 · rvir ); ( b ) : Ratio between the scale lengths a and b in the initial Miyamoto-Nagai stellar distribution (see Eq. 2); ( c ) : Gas-to-baryon fraction; ( d ) : Ratio between the mass of newly formed stars at the end of the SF period and the mass of the pre-existing disk Md ; ( e ) : Mass of the DM halo (in 10 8 M /circledot ); ( f ) : Virial radius (in Kpc, see eq. 1); ( g ) : Central gas density (in 10 -24 g cm -3 ); ( h ) : Ejected gas fraction after 200 Myr; ( i ) : Ejected oxygen fraction after 200 Myr; ( j ) : Ejected gas fraction after 500 Myr; ( k ) : Ejected oxygen fraction after 500 Myr; ( l ) : Abundance of oxygen measured as 12 + log(O / H), where O / H is the abundance ratio in number, at 200 Myr; ( m ) : Abundance of oxygen measured as 12 + log(O / H), where O / H is the abundance ratio in number, at 500 Myr. dance ratio is a mass-weighted average of the abundance ratios in each computational cell lying within a sphere of radius 0.5 · rvir . Again, this is only an approximate measure of the galactic metallicity and only relative abundance ratios between different model galaxies are relevant. In particular, the present grid of models is not aimed at reproducing the observed metallicities of individual DGs and it can be noticed e.g., that large galaxies do not always show larger O / H ratios compared to smaller galaxies (at variance with the observed M-Z relation; see Lee et al. 2006). It is also worth noticing that, unexpectedly, the models with the highest metallicities are the ones experiencing the strongest galactic winds (models 'L7'), at variance with the general idea that galactic winds can keep the metallicities of galaxies low (see the Introduction). This is due to the fact that these model galaxies expel almost all pristine gas (see Table 2) and therefore the (continuously produced) oxygen is mixed with a very small fraction of unpolluted gas. The average metallicity of the galaxy is thus close to the (very large) metallicity of the stellar ejecta. However, this gas is deemed at leaving the parent galaxy and is anyway too hot to form star; it will thus not increase the average metallicity of the stellar populations in the galaxy. The models attaining the lowest metallicities are the ones (like H7F) developing large-scale outflows but retaining a large fraction of pristine gas bound to the galaxy. See also Sect. 5 for a more extended discussion on the connection between the results of our models and the M-Z relation.", "pages": [ 6, 7, 8 ] }, { "title": "4.2. Wideningtheparameterspace", "content": "As already mentioned in Sect. 3.2, we have considered also models with di ff erent SFHs, di ff erent feedback radii, di ff erent initial metallicities and with perturbed initial distributions of gas. We summarize in this subsections how these parameters a ff ect our findings.", "pages": [ 8 ] }, { "title": "4.2.1. Dependence on the star formation history", "content": "Wehave re-run some models (specifically the ones for the family 'L8') in which the SFH, instead of being constant for 500 Myr as in the standard models, is di ff erent from zero only during the first 50 Myr, but the SFR is ten times more intense than in the standard models. A large-scale outflow quickly develops, even for the roundish model, sustained by the large rate of SNII explosions during the first ∼ 80 Myr. At the end of this phase, energy is still supplied by Type Ia SNe and winds from intermediatemass stars, but at a lower rate. This energy is still able to sustain the outflow for some time, but after a few times 10 8 yr, the gravitational pull prevails over the pressure gradient caused by the hot cavity of gas and some gas begins to recollapse towards the center of the galaxy (see Recchi & Hensler 2006 for a precise assessment of this phenomenon). The result is that the calculated ejection e ffi ciencies (of gas and metals) are larger at 100 or 200 Myr than at 500 Myr. In particular, the ejection e ffi ciencies at 200 Myr are much larger than the ones in the standard models, tabulated in Table 2. Disk-like models (models 'F' and 'M') retain only a few percent of gas and metals, and also the roundish model ('R') is able to expel more than 50% of the initial gas at distances larger than 0.5 · rvir . The large gas ejection e ffi ciencies for these models are due to the fact that, at these high rates of energy release, also the horizontal transport of gas can be e ff ective and some gas is pushed to large distances from the galactic center even along the disk. The disklike 'M' and 'F' models show again metal ejection e ffi ciencies (slightly) larger than gas ejection e ffi ciencies. Consequently, the metallicities of these model galaxies are quite low during the SF episode. However, as soon as the luminosity considerably reduces, this gas is pulled again within 0.5 · rvir (i.e. a fall back is occurring), whereas the material previously channelled along the galactic wind accelerates away of the galaxy due to the steep density gradient. Consequently, after 500 Myr of evolution, the retained gas fractions for these models tend to be larger (by a factor of ∼ 1.5-2) than the corresponding fractions of the standard models (the ones reported in column 11 of Table 2), whereas the retained fraction of metals are only slightly larger.", "pages": [ 9 ] }, { "title": "4.2.2. Dependence on the feedback radius", "content": "Wehave considered (again for the family of models 'L8') a variation of the radius over which energy and metals are injected. As already anticipated in Sect. 3.2, instead of the reference value RF = 200 pc, we have calculated models with RF = 50 pc and RF = 1000 pc. The models with RF = 1000 show very large ejection e ffi ciencies for both gas and metals. This is due to the unfortunate circumstance that 0.5 · rvir for this family of models is only slightly larger than 1000 pc, therefore a moderate energy injection already su ffi ces to push a large fraction of gas at distances larger than 0.5 · rvir . The comparison with the ejection e ffi ciencies of standard models makes thus little sense. We have therefore calculated the ejection e ffi ciencies of these models (and of the corresponding standard models) at r = rvir . These turn out to be ∼ 10% smaller than the ones of corresponding reference models. We confirm therefore that, if the energy is redistributed over a large volume, radiative energy losses are more e ff ective and the final ejection e ffi ciencies are reduced, in agreement with the findings of Fragile et al. (2004). Also the models with RF = 50 pc show slightly lower ejection e ffi ciencies. In this case, the enhanced cooling e ffi ciency is due to the fact that the gas within the feedback radius is now characterized by large densities and large metallicities (see also Tenorio-Tagle et al. 2007). Also in this case, the e ff ect of the feedback radius is quite limited (ejection e ffi ciencies change by ∼ 10-15 %). As also shown by Rodr'ıguez-Gonz'alez et al. (2011; see their fi g. 7) the dependence of the ejection e ffi ciencies on the feedback radius is non-monotonic, although the dependence of our results on this particular parameter is quite limited.", "pages": [ 9 ] }, { "title": "4.2.3. Dependence on the initial metallicity", "content": "Since we consider models in which a stellar disk is already present at the beginning of the simulations, it seems unreasonable to start with a primordial metallicity. We considered therefore models in which the initial metallicity (of stars and gas) is regulated by the mass of the pre-existing disk, namely, according to the M-Z relation, the larger the disk mass, the larger the initial metallicity we must consider. In particular, if we take into account the correlation between mass and metallicity obtained by Tremonti et al. (2004), models of the 'L8' family should have an initial metallicity of about one tenth of solar. Of course, the final metallicities of these models will be much higher than the ones attained by models initially without metals. However, as already mentioned, the only dynamical e ff ect of a di ff erent initial metallicity is to increase the radiative losses and thus to reduce the amount of thermal energy of the galaxy. It turns out that this reduction is quite limited and, therefore, the ejection e ffi ciencies do not change substantially (they are only a few per cent lower than the ejection e ffi ciencies of corresponding models initially without metals). For these models, the metal ejection e ffi ciencies are calculated according only to the amount of freshly produced metals retained or ejected.", "pages": [ 9 ] }, { "title": "4.2.4. Models with random perturbation of the initial gas distribution", "content": "As described in Sect. 3.2, once an equilibrium initial configuration has been obtained, we have perturbed it by means of eq. 3. In particular, we have perturbed the model L8R by amplitudes ε of 1% (mildly perturbed model) and 5% (largely perturbed model). The perturbation increases the turbulence of the model. Mixing between hot and cold regions is enhanced and the net e ff ect is an increase in the radiative energy losses, hence a reduced amount of energy available to drive galactic winds (in other words, the galactic winds become more mass loaded). As a consequence of that, the retained gas fraction of the mildly perturbed model increases (from the final value of 0.442 of the model L8R it increases to 0.547). However, the final retained fraction of freshly produced metals (slightly) decreases for this model (it reduces to 0.345 from the reference value of 0.401). This is due to the fact that it is much more di ffi cult to keep the model close to spherical symmetry if the initial gas distribution is perturbed. Regions of lower pressure can be created randomly within the computational box and the propagation of metals can thus deviate from isotropicity, being faster (and allowing some venting out of metals) along (randomly oriented) directions with steep pressure gradients. This e ff ect is similar to the one analyzed in great detail by RH07 (see also Sect. 2 for a short summary of the main results of this paper). In the largely perturbed model, the final metal ejection e ffi ciency is similar to the one obtained in the mildly perturbed model. However, the venting out (along random directions) involves now also a non-negligible fraction of pristine gas and the final gas retained fraction is 0.443 (very similar to the gas retained fraction of the reference model L8R). The piercing of the supershell along di ff erent directions and not just along the direction perpendicular to the disk of the galaxy can be appreciated by inspecting Fig. 8. It shows the gas and oxygen density distribution after 200 Myr of evolution of the largely perturbed model. Clearly, the perturbation of the initial gaseous distribution a ff ects the ejection e ffi ciencies of gas and metals in a non-linear (and some times non-predictable) way and deserves further studies. It is important to stress that, due to the assumed symmetry and dimensionality of the code, inhomogeneities represent rings of denser gas. It is not clear whether a realistic 3-D distribution of inhomogeneities will produce the same results. Our group is currently running (with the AMR code FLASH) 3-D chemodynamical simulations of DGs with inhomogeneities (Mitchell et al., in preparation). The results of these simulations can shed light on the reliability of our results.", "pages": [ 9, 10 ] }, { "title": "5. Discussion and conclusions", "content": "In this paper we have studied the development of galactic outflows in DG models. Our main focus is to study the dependence of the ejected mass fractions (of gas and freshly produced metals) on the degree of flattening of the galaxy, a problem that has received little attention in the past. A very simple theoretical analysis leads to the conclusion that, in a very flat galaxy, the ejection e ffi ciencies of freshly produced metals must be quite high. In fact, once the over-pressurized gas formed by SN explosions and stellar wind breaks out and forms a bipolar galactic wind, metals can be easily ejected out of the galaxy through this funnel. On the other hand, in a spherical (or almost spherical) galaxy, bipolar outflows cannot be formed because there is no preferential direction, along which the pressure gradient is steeper. Therefore, either the galaxy expels gas isotropically (but blow-away is very di ffi cult in DGs, see e.g. Hensler et al. 2004) or the superbubble of hot (and metal-enriched) gas remains confined inside the galaxy (thus the metal ejection e ffi ciencies are very low). We have considered three di ff erent degrees of flattening for our model galaxies: a roundish model (one for which the ratio between the length scales a and b describing the MiyamotoNagai potential is b / a = 5), a thick-disk model (with b / a = 1) and a thin disk model (with b / a = 0 . 2). Our study confirms the trend described above: ejection e ffi ciencies in roundish galaxies are systematically lower than the ones in disk-like galaxies (see Table 2 or Fig. 7). On the other hand, for most of the analyzed models, transport of gas along the disk is quite limited, therefore (in agreement with many previous studies, e.g. MF99; D'Ercole & Brighenti 1999) we can conclude that ejection e ffi ciencies of gas are (at least for disk-like models) lower than ejection e ffi -ciencies of freshly produced metals. We have also studied in detail the e ff ect of galactic mass on the fate of gas and freshly produced metals, a problem that, on the contrary, has received much more attention in the past (MF99; Rodr'ıguez-Gonz'alez et al. 2011). We confirm th at smaller DGs, with shallower potential wells, favor the development of large-scale outflows, hence the ejection e ffi ciencies (of gas and metals) increase with decreasing galactic masses (see Table 2 or Fig. 6). The increase of metal ejection e ffi ciencies through galactic winds has been often invoked to explain the observed M-Z relation in galaxies (see e.g. Tremonti et al. 2004). The dependence of the obtained metallicity (measured by means of 12 + log (O / H)) of our model galaxies as a function of their initial masses is shown in Fig. 9 for the 'H' set of models. For all three degrees of flattening, a trend of increasing metallicity as a function of mass is clearly visible. However, as outlined also in Sec. 4.1, for the 'L' set of models this trend is much less visible. It is important to notice that the abundances tabulated in Table 2 and plotted in Fig. 9 are mass-averaged means within the whole galactic region. However, two gas phases coexist within this region: a hot gas phase (the galactic wind and the starburst region) and a colder ISM. We expect the hot phase to have much higher metallicities than the cold phase. The metallicity of the galactic wind is expected to play no role in the process of chemical enrichment of the galaxy (this gas is destined to leave the parent galaxy) but is important in the process of chemical enrichment of the intra-cluster medium. To have an idea on the oxygen abundances of these two phases, we distinguish grid points with temperatures above and below 2 · 10 4 K. Gas above this temperature threshold is supposed to be too hot to be detected by optical spectroscopy. The resulting values of 12 + log(O / H) after 500 Myr of evolution for the 'H' family of models is shown in Fig. 10 (red lines: cold phase; blue lines: hot phase). As expected, the abundances in the cold phase are much lower than the abundances in the hot phase. The cold phase abundances show the same trend of the total abundances shown in Fig. 9. This is mainly due to the fact that the calculated mean abundances are mass-weighted and most of the gas mass is colder than 2 · 10 4 K. For some models (in particular the models with log M = 7), the cold phase abundances are significantly lower ( ∼ 0.2 dex) than the total abundances. This is expected because, in some models, the metal-rich hot gas (wind and starburst region) represents a significant fraction of the total gas. On the other hand, models experiencing no (or very limited) galactic winds contain much less hot gas, therefore the average total abundance is less affected by the metallicity of the hot phase and is much closer to the abundance of the cold gas (the di ff erences are a few hundredths of dex). Also the abundances of the hot phase tend to grow with mass (see the blue lines in Fig. 10). High mass galaxies show higher hot phase abundances because, for these models, much of the hot gas is in the starburst region and thus its metallicity is close to the (very high) metallicity of the stellar ejecta. On the other hand, galaxy models experiencing strong galactic winds show smaller hot-phase metallicities because the winds entrain more cold (and metal-poor) gas. This process is usually indicated as mass-loading and is confirmed by many studies (both theoretical and observational) of galactic winds (see e.g. Strickland & Stevens 2000; Tescari et al. 2009; Hopkins et al. 2012; Newman et al. 2012 among many others). Indeed, a clearer correlation can be shown between the ejected oxygen fraction after 500 Myr (12th column in Table 2) and the hot-phase oxygen abundance. This correlation is shown in Fig. 11. Models experiencing galactic winds (thus with high oxygen ejection e ffi ciencies) show also mass-loading, thus dilution of the oxygen abundances in the hot phase. On the other hand, in models with weak (or without) galactic winds most of the hot gas is concentrated in the starburst region, where the abundances closely resemble the (very high) abundances of the stellar ejecta. One should warn the reader that the processes of metal dispersal and mixing are very di ffi cult to model. No hydrodynamical code is able to correctly resolve contact discontinuities that form between hot cavities and cold shells. Numerical di ff usion tends to smear this discontinuity and create regions of intermediate temperature and densities, where metals can cool in a rela- tively short time-scale and then mix with the surrounding shell. In spite of these uncertainties, we believe that 500 Myr is a sufficient time span to cool down most of the produced metals (if they are not channelled in a galactic wind). This is confirmed by some of our previous studies (see in particular Recchi et al. 2001; their sect. 3.2) but also by other studies addressing the specific issue of metal mixing (e.g. de Avillez & MacLow 2002; Pan et al. 2012; Yang & Krumholz 2012), which confirm that a few hundred Myrs is a su ffi cient time to disperse and mix metallicity inhomogeneities. It is also important to stress that, in our results, cooling and mixing of stellar ejecta is mainly due to thermal conduction and to the formation of eddies and vortices. The presence of inhomogeneities (see Sect. 4.2.4) can facilitate both processes, thus enhance the mass loading of the galactic winds. One should also notice that the chosen temperature threshold separating the hot and the cold phase (2 · 10 4 K) is somewhat arbitrary and it has been taken in analogy with what done in some of our previous publications (see in particular Recchi et al. 2006). A di ff erent and maybe better choice could be 4 · 10 4 K (the minimum of the cooling curve for low values of the metallicity - see Bohringer & Hensler 1989, their fig. 1), or even 10 5 K. We have checked how much are our results a ff ected by the choice of the temperature threshold. It turns out that the abundances in the cold phase are almost una ff ected by this choice (for T = 4 · 10 4 Kthe abundances change negligibly whereas for T = · 10 5 K the di ff erences are 0.02 dex at most). This is due to the fact that a small fraction (in mass) of gas has temperatures in the range [2 · 10 4 , 4 · 10 4 ] K or [2 · 10 4 , 10 5 ] K. On the other hand, the variations in the hot phase abundances are more significant. They increase by up to 0.15-0.2 dex if the threshold is 10 5 K. However, the overall correlations shown in Figs. 10 and 11 remain. We must outline once again that the main aim of this paper was to show the e ff ect that the gas distribution can have on the development of galactic winds and, hence, on the fate of gas and freshly produced metals. We have not attempted to reproduce the observed M-Z relation (we will focus on this task in a paper in preparation). For instance, all the models, irrespective of their masses, have the same SFRs per unit available gas mass (hence the same SF e ffi ciencies). Models with initially 10 8 M /circledot of baryonic mass have SFRs ten times larger than models with Mb = 10 7 M /circledot . The metal production rates are also ten times larger. However, these larger fractions of metals are mixed with ten times more pristine gas. Hence, if galactic winds do not play a significant role, the final metallicities of the models should be approximately independent on the galactic mass (see roundish models in Fig. 9). However, it is known from chemical evolution studies (e.g. Matteucci 1994; Pipino & Matteucci 2004) that SF e ffi ciencies must increase with galactic masses in order to reproduce the observed chemical abundances and abundance ratios of galaxies. This is also consistent with the idea of having SF e ffi ciencies increasing with pressure of the ambient di ff use gas, which increases with galactic mass (Elmegreen & Efremov 1997; Harfst et al. 2006; see also Leroy et al. 2008). However, we have noticed in Sect. 4.2.1 that metal ejection e ffi ciencies of galaxies with higher SFRs (but producing at the end of the simulation the same amount of stars) are larger. Consequently, these galaxies tend to have lower metallicities during the SF episode. This qualitative trend is in agreement with the so-called fundamental metallicity relation (FMR: Mannucci et al. 2010), according to which the oxygen abundance in star forming galaxies correlates with the quantity µα = log( M ∗ ) -α log( S FR ), where α is a free parameter chosen to minimize the scatter in the FMR. Although the precise shape of the FMR is debated (for instance the recent paper of Andrews & Martini 2012 find α = 0 . 66, more than twice the value of Mannucci et al.), undoubtedly the SFR plays a very important role in determining the gas-phase metallicity of a galaxy. Galaxies with the same stellar mass but with higher SFRs are characterized by lower metallicities (see e.g. fig. 11 of Andrews & Martini 2012). According to Andrews & Martini (2012), this is probably due to the fact that galaxies with high SFRs are presently experiencing a merging. Major mergers drive in considerable amounts of low metallicity gas from large radii, which dilutes the metallicity of the galaxy and triggers a vigorous SF burst. Since most of the galaxies with large SFR have already large stellar masses, it seems unlikely that their gas components have still a low metallicity. It seems to us more likely that large SFRs drive large outflows, with a consequent significant metal loss, as our models show. What we want to point out here is that, for models with the same mass and SFR, geometry plays a very significant role in determining the fate of freshly produced metals and, consequently, the final metallicity. An inspection of Table 2 shows that, for some models, ejection e ffi ciencies of metals can change by up to an order of magnitude depending on the degree of flattening, being instead the gas ejection e ffi ciencies quite independent on b / a . Consequently, the final metallicities of models with the same mass can vary up to 1 dex depending on the geometry (see Figs. 9 and 10). The large spread observed in the metallicity of DGs with the same masses (see e.g. Lee et al. 2006, their fig. 8) could be due to the e ff ect of gas distribution. Although the parameter α is chosen to minimize the scatter in the FMR, still galaxies with the same stellar mass and SFR show a spread in the metal content (see e.g. figs. 11 and 12 of Andrews & Martini 2012). The gas distribution might be responsible for this spread. A last comment related to the M-Z relation (or to the FMR) concerns the correlation between mass and degree of flattening of DGs. Recent studies suggest that smaller DGs tend to have larger axial ratios (i.e. they tend to be rounder) than DGs with larger masses. This might complicate the interpretation of the M-Z and FMR relations according to our models. However, this e ff ect seems to be quite limited. For instance, S'anchez-Janssen et al. (2010) show (their fig. 1) that the average axial ratios change by at most ∼ 20% between DGs with stellar masses of 10 7 and 10 9 M /circledot , and that the spread in axial ratios is extremely large at all stellar masses. Moreover, Lisker et al. (2009) show that (at least for DGs in clusters) the axial ratio is more related to the DG velocity than to the mass. Acknowledgements. We thank the anonymous referee for very helpful comments and remarks.", "pages": [ 10, 11, 12 ] }, { "title": "References", "content": "Andrews, B. H., & Martini, P. 2012, arXiv:1211.3418 de Avillez, M. A., & Mac Low, M.-M. 2002, ApJ, 581, 1047 Bedogni, R., & Dercole, A. 1986, A&A, 157, 101 Boehringer, H., & Hensler, G. 1989, A&A, 215, 147 Cowie, L. L., & McKee, C. F. 1977, ApJ, 211, 135 Dekel, A., & Silk, J. 1986, ApJ, 303, 39 D'Ercole, A., & Brighenti, F. 1999, MNRAS, 309, 941 Elmegreen, B. G., & Efremov, Y. N. 1997, ApJ, 480, 235 Erb, D. K., Shapley, A. E., Pettini, M., et al. 2006, ApJ, 644, 813 Ferrara, A., & Tolstoy, E. 2000, MNRAS, 313, 291 Fragile, P. C., Murray, S. D., & Lin, D. N. C. 2004, ApJ, 617, 1077 Harfst, S., Theis, C., & Hensler, G. 2006, A&A, 449, 509 Hensler, G., Dickow, R., Junkes, N., & Gallagher, J. S. 1998, ApJ, 502, L17 Hensler, G., Theis, C., & Gallagher, J. S., III. 2004, A&A, 426, 25 Hidalgo, S. L., Aparicio, A., Skillman, E., et al. 2011, ApJ, 730, 14 Hopkins, P. F., Quataert, E., & Murray, N. 2012, MNRAS, 421, 3522 Kirby, E. N., Simon, J. D., Geha, M., Guhathakurta, P., & Frebel, A. 2008, ApJ, 685, L43 Larson, R. B. 1974, MNRAS, 169, 229 Laskar, T., Berger, E., & Chary, R.-R. 2011, ApJ, 739, 1 Lee, H., Skillman, E. D., Cannon, J. M., et al. 2006, ApJ, 647, 970 Leroy, A. K., Walter, F., Brinks, E., et al. 2008, AJ, 136, 2782 Lisker, T., Janz, J., Hensler, G., et al. 2009, ApJ, 706, L124 Mac Low, M.-M., & Ferrara, A. 1999, ApJ, 513, 142 (MF99) McQuinn, K. B. W., Skillman, E. D., Cannon, J. M., et al. 2010, ApJ, 721, 297 McQuinn, K. B. W., Skillman, E. D., Cannon, J. M., et al. 2010, ApJ, 724, 49 Madau, P., Ferrara, A., & Rees, M. J. 2001, ApJ, 555, 92 Maiolino, R., Nagao, T., Grazian, A., et al. 2008, A&A, 488, 463 Mannucci, F., Cresci, G., Maiolino, R., Marconi, A., & Gnerucci, A. 2010, MNRAS, 408, 2115 Mannucci, F., Salvaterra, R., & Campisi, M. A. 2011, MNRAS, 414, 1263 Marcolini, A., D'Ercole, A., Brighenti, F., & Recchi, S. 2006, MNRAS, 371, 643 Martin, C. L., Kobulnicky, H. A., & Heckman, T. M. 2002, ApJ, 574, 663 Matteucci, F. 1994, A&A, 288, 57 Michielsen, D., Valcke, S., & de Rijcke, S. 2007, EAS Publications Series, 24, 287 Monelli, M., Gallart, C., Hidalgo, S. L., et al. 2010, ApJ, 722, 1864 Monelli, M., Hidalgo, S. L., Stetson, P. B., et al. 2010, ApJ, 720, 1225 Mori, M., Ferrara, A., & Madau, P. 2002, ApJ, 571, 40 Newman, S. F., Genzel, R., Forster-Schreiber, N. M., et al.2012, ApJ, 761, 43 Ott, J., Walter, F., & Brinks, E. 2005, MNRAS, 358, 1423 Pan, L., Desch, S. J., Scannapieco, E., & Timmes, F. X. 2012, ApJ, 756, 102 Persic, M., Salucci, P., & Stel, F. 1996, MNRAS, 281, 27 Pipino, A., & Matteucci, F. 2004, MNRAS, 347, 968 Recchi, S., & Hensler, G. 2006, A&A, 445, L39 Recchi, S., & Hensler, G. 2007, A&A, 476, 841 (RH07) Recchi, S., Hensler, G., & Anelli, D. 2009, arXiv:0901.1976 Recchi, S., Hensler, G., Angeretti, L., & Matteucci, F. 2006, A&A, 445, 875 Recchi, S., Matteucci, F., & D'Ercole, A. 2001, MNRAS, 322, 800 Recchi, S., Matteucci, F., D'Ercole, A., & Tosi, M. 2002, A&A, 384, 799 Recchi, S., Matteucci, F., D'Ercole, A., & Tosi, M. 2004, A&A, 426, 37 Recchi, S., Theis, C., Kroupa, P., & Hensler, G. 2007, A&A, 470, L5 Rieschick, A., & Hensler, G. 2003, Ap&SS, 284, 861 Rodr'ıguez-Gonz'alez, A., Esquivel, A., Raga, A. C., & Col'ın, P. 2011, Revista Mexicana de Astronomia y Astrofisica Conference Series, 40, 86 S'anchez-Janssen, R., M'endez-Abreu, J., & Aguerri, J. A. L . 2010, MNRAS, 406, L65 Scannapieco, E., & Bruggen, M. 2010, MNRAS, 405, 1634 Schroyen, J., de Rijcke, S., Valcke, S., Cloet-Osselaer, A., & Dejonghe, H. 2011, MNRAS, 416, 601 Silich, S. A., & Tenorio-Tagle, G. 1998, MNRAS, 299, 249 Silich, S., & Tenorio-Tagle, G. 2001, ApJ, 552, 91 Skillman, E. D., Kennicutt, R. C., & Hodge, P. W. 1989, ApJ, 347, 875 Spitzer, L. 1956, Physics of Fully Ionized Gases, New York: Interscience Publishers Spitzer, L., & Harm, R. 1953, Phys. Rev., 89, 977 Strickland, D. K., & Stevens, I. R. 2000, MNRAS, 314, 511 Tenorio-Tagle, G. 1996, AJ, 111, 1641 Tenorio-Tagle, G., Wunsch, R., Silich, S., & Palouˇs, J. 2007, ApJ, 658, 1196 Tescari, E., Viel, M., Tornatore, L., & Borgani, S. 2009, MNRAS, 397, 411 Tremonti, C. A., Heckman, T. M., Kau ff mann, G., et al. 2004, ApJ, 613, 898 Vader, J. P. 1986, ApJ, 305, 669 Vasiliev, E. O., Vorobyov, E. I., & Shchekinov, Y. A. 2008, A&A, 489, 505 Vorobyov, E. I., Recchi, S., & Hensler, G. 2012, A&A, 543, A129 Wuyts, E., Rigby, J. R., Sharon, K., & Gladders, M. D. 2012, ApJ, 755, 73 Yang, C.-C., & Krumholz, M. 2012, ApJ, 758, 48 Zhao, Y., Gao, Y., & Gu, Q. 2010, ApJ, 710, 663", "pages": [ 12, 13 ] } ]

[ { "title": "ABSTRACT", "content": "Context. Bouncing collisions of dust aggregates within the protoplanetary may have a significant impact on the growth process of planetesimals. Yet, the conditions that result in bouncing are not very well understood. Existing simulations studying the bouncing behavior used aggregates with an artificial, very regular internal structure. Aims. Here, we study the bouncing behavior of sub-mm dust aggregates that are constructed applying di GLYPH<11> erent sample preparation methods. We analyze how the internal structure of the aggregate alters the collisional outcome and determine the influence of aggregate size, porosity, collision velocity, and impact parameter. Methods. We use molecular dynamics simulations where the individual aggregates are treated as spheres that are made up of several hundred thousand individual monomers. The simulations are run on GPUs. Results. Statistical bulk properties and thus bouncing behavior of sub-mm dust aggregates depend heavily on the preparation method. In particular, there is no unique relation between the average volume filling factor and the coordination number of the aggregate. Realistic aggregates bounce only if their volume filling factor exceeds 0 : 5 and collision velocities are below 0 : 1 ms GLYPH<0> 1 . Conclusions. For dust particles in the protoplanetary nebula we suggest that the bouncing barrier may not be such a strong handicap in the growth phase of dust agglomerates, at least in the size range of GLYPH<25> 100 GLYPH<22> m. Key words. Planets and satellites: formation - Protoplanetary disks - Methods: numerical", "pages": [ 1 ] }, { "title": "Bouncing Behavior of Microscopic Dust Aggregates", "content": "A. Seizinger, 1 and W. Kley 1 Institut für Astronomie and Astrophysik, Eberhard Karls Universität Tübingen, Auf der Morgenstelle 10c, D-72076 Tübingen, Germany e-mail: [email protected] Received 18.12.2012; accepted 14.01.2013", "pages": [ 1 ] }, { "title": "1. Introduction", "content": "For the planet formation process, the growth from micron sized dust grains to kilometer sized objects is a key ingredient of the core accretion scenario originally proposed by Pollack et al. (1996). Yet, the question remains unanswered how this process is accomplished in the face of various impediments. First of all, fast inward drift limits the time available to form planetesimals by successive collisions to less than 10 4 years (Weidenschilling 1977). The growth process itself heavily depends on two ingredients: Since the information accessible through direct observations is limited the first aspect is addressed mainly by theoretical work and computer simulations (Flaig et al. 2012). For an overview on the properties of protoplanetary disks we refer to the following reviews by Dullemond et al. (2007) and Armitage (2011). To investigate the collision behavior of dust / icy aggregates various methods are employed. Depending on the size of the aggregates and the desired collision velocity, laboratory experiments are possible. As of today, laboratory experiments provide data of collisions ranging from millimeter- to decimetersized aggregates composed of di GLYPH<11> erent materials (mainly Silicate / Quartz and Ice). A comprehensive summary of laboratory experiments is given by Blum & Wurm (2008). Computer simulations provide a second method to study the collisional behavior of dust or ice aggregates. Collisions of very small, micron sized aggregates have been simulated using a molecular dynamics approach (e.g. Dominik & Tielens 1997; Paszun & Dominik 2009; Wada et al. 2007, 2009). For macroscopic aggregates di GLYPH<11> erent methods such as smoothed particle hydrodynamics (SPH) are employed (e.g. Schäfer et al. 2007; Geretshauser et al. 2010). More recent experiments showed that collisions of mm to cm-sized aggregates often result in bouncing (e.g. Weidling et al. 2009; Heißelmann et al. 2010; Weidling et al. 2012; Jankowski et al. 2012). Extrapolating the results obtained from the various experiments Güttler et al. (2010) devised a model describing the outcome of collision with respect to the collision velocity, and the mass and porosity of the colliding aggregates. Employing this model to simulate the evolution of a swarm of dust aggregates in a protoplanetary disk the so called 'bouncing barrier' emerged (Zsom et al. 2010). As the aggregates grow larger their relative velocities increase. Due to the growing kinetic impact energy aggregates get increasingly compacted during successive collisions. When the aggregates get too compact their collisions do not result in sticking anymore. Instead, they bounce o GLYPH<11> each other and the growth process is stopped. This occurs in the size regime of centimeters. A possible way to overcome the bouncing barrier has been recently suggested by Windmark et al. (2012a). Under the assumption of a few bigger aggregates that act as initial seeds it is possible to grow larger 100 m sized bodies on the timescale of 1 Myr. A possible origin of those seeds has been proposed by Windmark et al. (2012b). Taking into account a Maxwellian velocity distribution they found that low velocity collisions can allow a few aggregates to grow considerably larger than the average of the simulated population. Despite its significant influence on the growth process, bouncing still lacks theoretical understanding of its prerequisites on a microscopical scale. So far, molecular dynamics (MD) simulations result in bouncing only for rather compact aggregates (Wada et al. 2011; Schräpler et al. 2012). According to Wada et al. (2011) an average coordination number of 6 is required for aggregates to bounce o GLYPH<11> each other. However, in laboratory experiments bouncing frequently observed in collisions of aggregates with lower filling factors / coordination numbers for which MD simulations clearly predict sticking. It has been speculated that this discrepancy could result from a size e GLYPH<11> ect or a possible compaction of the outer shell during the handling process of the aggregates used in the laboratory experiments. However, the latter hypothesis has been refuted by recent experiments (Kothe et al. 2012). The aim of this work is to study the influence of the internal structure on the bouncing behavior of sub-mm dust aggregates in greater detail. Using the enormous computing power provided by GPUs we can simulate aggregates consisting of several hundreds of thousands of monomers and thus simulate aggregates in a size range from several microns up to GLYPH<25> 0 : 1 mm in diameter.", "pages": [ 1, 2 ] }, { "title": "2. Interaction model", "content": "To simulate the behavior of of dust aggregates we use a soft sphere discrete element method (SSDEM) approach. The dust aggregates are composed of hundreds of thousands of micron sized spherical grains (monomers). Our interaction model is based on the work of Dominik & Tielens (1997) who combined earlier theoretical work by Johnson et al. (1971); Dominik & Tielens (1995, 1996) into a detailed micro-mechanical model describing the interaction between two monomers. These monomers may establish adhesive contacts when touching each other and kinetic energy is dissipated upon deformation of these contacts. A few years later, Wada et al. (2007) presented a di GLYPH<11> erent approach when deriving nearly the same forces and torques from corresponding potentials. When trying to reproduce the results of laboratory experiments performed by Güttler et al. (2009) on the compression of porous dust cakes Seizinger et al. (2012) observed that the behavior predicted by the model of Dominik & Tielens (1997) was too soft. Since the samples used by Güttler et al. (2009) had been composed of micron-sized, spherical, monodisperse silicate grains their results constituted a perfect possibility to calibrate the model. Introducing two free parameters m r and m s that modify the strength of the rolling and sliding interaction between two monomers Seizinger et al. (2012) were able to obtain excellent agreement between laboratory results and computer simulations. In this work we use the modified interaction model presented by Seizinger et al. (2012) with m r = 8 and m s = 2 : 5. The material parameters are listed in Tab. 1.", "pages": [ 2 ] }, { "title": "3. Sample generation", "content": "In this work we examine the conditions under which bouncing occurs. Apart from the external parameters describing the physics of collisions such as the impact parameter or velocity we study the influence of the internal structure of the aggregate. Examples of such aggregates that have been generated by di GLYPH<11> erent methods are shown in Fig. 1. To study the influence of the aggregate size we use aggregates with diameters in the range of 30 to 100 GLYPH<22> m. Unfortunately, simulations with larger aggregates are infeasible due to the required computational cost, at least for a wider range of parameters. Typically, the volume filling factor GLYPH<30> and the average coordination number n c are used to classify aggregates. In general, the filling factor is given by where N denotes the number of monomers, V p is the volume of a monomer, and V A is the total volume occupied by the aggregate. As we use spherical aggregates V A can be calculated easily from the outer radius of the aggregate. For irregular shaped aggregates there are di GLYPH<11> erent possibilities to define V A. For example Kozasa et al. (1992) determine the size of a porous aggregate based on its radius of gyration whereas Ossenkopf (1993) use the geometric cross section. It is important to keep this ambiguity in mind when comparing the porosities of flu GLYPH<11> y aggregates to other works. In molecular dynamics simulations, the coordination number of a monomer denotes the number of the neighbors it interacts with. Thus, the average coordination number n c is obtained by averaging the number of contacts of each particle. In this work we use three di GLYPH<11> erent types of aggregates: Hexagonal lattice aggregates featuring a regular grid structure, aggregates produced by slowly compacting a porous dust cake, and aggregates generated by successively adding single monomers from randomly chosen directions. These choices have been motivated by the fact that hexagonal lattice aggregates are easy to build and allow for comparison with earlier work by Wada et al. (2011), whereas the static compaction resembles the generation of samples used in laboratory results. The aggregates of the third type are generated algorithmically but their structure remains comparable to the static compaction type (see Sect. 3.4). Because the aggregates within the protoplanetary nebula grow through successive collisions one might expect that their internal structure lies somewhere in between the static compression and the ballistic aggregation cases.", "pages": [ 2 ] }, { "title": "3.1. Hexagonal lattice (CPE)", "content": "Hexagonal-lattice type aggregates (also referred to as hexagonal close packing with extraction (CPE)) may be generated very easily. First, a hexagonal close packing aggregate is generated which features a volume filling factor GLYPH<30> GLYPH<25> 0 : 74 and a coordination number n c GLYPH<25> 12 (due to surface e GLYPH<11> ects nc equals 12 only for aggregates of infinite size). In the second step a suitable number of randomly selected monomers will be removed to achieve the desired volume filling factor. As a result a small number of monomers on the surface may be become disconnected from the main aggregate and will be removed as well. Wada et al. (2011) have already studied the bouncing behavior of this type of aggregates and found that bouncing will occur if the average coordination number is greater than 6. Schräpler et al. (2012) analyzed the relation between the coe GLYPH<14> cient of restitution and the collision velocity in experiments and simulations using CPE-aggregates. Their results agreed well with a theoretical model by Thornton & Ning (1998).", "pages": [ 2, 3 ] }, { "title": "3.2. Ballistic aggregation with migration (BAM)", "content": "The second type of aggregates was originally suggested by Shen et al. (2008) and also studied in the work of Wada et al. (2011). To generate a larger aggregate single monomers are successively shot in from random directions onto the existing aggregate. When the monomer hits the aggregate it will either remain at the position where the first contact has been established or migrate to a position close by where it establishes contacts with two or three monomers. Compared to Shen et al. (2008), we use three di GLYPH<11> erent methods to select the final position of the migrating particle: For a given coordination number the resulting aggregates show a di GLYPH<11> erent filling factor depending on which selection mechanism is employed (see Fig. 2). The first method leads to rather porous aggregates since the monomers typically migrate to positions further outward compared to the case of random migration. Likewise, the resulting aggregates will become even more compact if monomers migrate to the most inward position available. Note that we do not claim that random or center migration are realistic growth processes that accurately describe the growth of dust aggregates in protoplanetary discs. Yet, they constitute a computationally very cheap approach to generate larger aggregates that do not su GLYPH<11> er from the artificial lattice structure like the CPE aggregates described above. Compared to the 'static compaction'-aggregates they additionally o GLYPH<11> er the advantage that they are perfectly relaxated. Since all monomers are in equilibrium distance from each other, there are no attractive or repulsive forces that could lead to a breakup of the aggregate. c", "pages": [ 3 ] }, { "title": "3.3. Static compaction (SC)", "content": "The last type of aggregates used for our the studies is the most computationally expensive. To generate a spherical aggregate of a certain diameter we start with a su GLYPH<14> ciently large, cuboid shaped dust cake generated by random ballistic deposition (RBD). Since RBD-aggregates feature an initial volume filling factor of GLYPH<30> = 0 : 15 we first have to compact the aggregate until we reach the desired filling factor. For this purpose, the aggregate is put into a box of walls that may move towards each other. According to Seizinger et al. (2012) this compaction must be very slow to avoid inhomogeneities. Even if the cake is compacted homogeneously for filling factors above GLYPH<25> 0 : 45 it will get elastically charged and expand when the compacting walls are removed. Therefore the dust cakes needs to be relaxated before removing the walls of the compaction box. For this purpose we disturb the aggregate by modifying the position of the monomers randomly by a very small amount. We keep the disturbed aggregate in a box of fixed size and wait until the kinetic energy induced by the disturbance is damped away by the inelastic monomer interaction. To get rid of kinetic energy below the threshold where the inelastic regime is entered we additionally enforce a viscous damping mechanism. For this purpose, the velocities and angular velocities of the monomers are multiplied by a factor of 1 GLYPH<0> GLYPH<20> in each time step, where GLYPH<20> denotes a damping coe GLYPH<14> cient. In this work we use GLYPH<20> = 0 : 0001. It turns out that a maximum disturbance of a factor of 0 : 001 of the radius of a monomer is su GLYPH<14> cient to stabilize the aggregate without altering its internal structure. Higher values may alter the coordination number significantly which could potentially change the collisional behavior of the aggregates and is therefore unwanted. For fillings factors above GLYPH<30> = 0 : 58 GLYPH<0> 0 : 59 this method does not work anymore. Here the compaction is too close to the random closest packing ( GLYPH<30> GLYPH<25> 0 : 63) and no stable configuration can be reached without rearranging the monomers significantly. After the aggregate has been relaxated the compaction box is removed and a spherical aggregate will be cut out of the compacted cake. As this procedure is computationally very expensive it takes several days to generate larger (50 GLYPH<22> m in diameter and above) aggregates of this type.", "pages": [ 3, 4 ] }, { "title": "3.4. Comparison", "content": "Comparing the relation between GLYPH<30> and n c of the di GLYPH<11> erent types of aggregates described above shows very interesting features: As we can see in Fig. 2 the relation depends considerably on the preparation method. The di GLYPH<11> erent BAM generation methods have been described in Sect. 3.2. From the work presented in this section two important conclusions can be drawn: In laboratory experiments with aggregates composed of micron sized dust grains, it is typically only possible to determine the filling factor but not the coordination number. Thus, one has to be very careful when comparing results from numerical simulations of CPE aggregates with laboratory experiments.", "pages": [ 4 ] }, { "title": "4. Results", "content": "In this section we present our results from various simulations in which we study the influence of the collisions velocity, impact parameter, and aggregate size on the bouncing behavior. All simulations have been performed on NVIDIA GPUs (GTX460, GTX570, Tesla C2070). Depending on the aggregate size and filling factor each simulation took between less than an hour and half a day.", "pages": [ 4 ] }, { "title": "4.1. Growth Factor", "content": "In the following bouncing maps the 'growth factor' GLYPH<13> that is inspired by the four-population model suggested by Geretshauser et al. (2011) is depicted. It is defined by where m largest is the mass of the largest fragment and m tot the total mass of the colliding aggregates. For perfect sticking we obtain GLYPH<13> = 1, for total destruction GLYPH<13> ! 0. In collisions of equal sized aggregates, a value GLYPH<13> = 0 : 5 indicates bouncing. However, during the transition from perfect sticking to fragmentation GLYPH<13> may also become 0 : 5. To distinguish between the two cases we consider the mass ratio GLYPH<13> 2 of the second largest fragment. In the bouncing case it is 0 : 5 as well whereas in the fragmentation case the mass of the second largest fragment is much lower than 0.5 of the total mass as there are a lot of other smaller fragments. Thus, in the bouncing maps presented in this work the green areas indicate sticking, the upper left yellow areas bouncing, and the color gradient from green to yellow to red on the right marks the transition from sticking to fragmentation. Note that GLYPH<13> = GLYPH<13> 2 = 0 : 5 only applies in the case of 'perfect bouncing'. In our simulations we often observe the loss of a few monomers (typically less than 100) which is negligible compared to the total number of monomers of 5 GLYPH<1> 10 4 to 5 GLYPH<1> 10 5 . Thus, we also count collisions as bouncing events if GLYPH<13> and GLYPH<13> 2 are slightly smaller than 0 : 5.", "pages": [ 4 ] }, { "title": "4.2. Hexagonal lattice", "content": "The outcome of head-on collisions of CPE aggregates has already been studied by Wada et al. (2011) who observed bouncing if the coordination number was greater than 6. However, their aggregates were much smaller ( GLYPH<25> 10 4 monomers). As hexagonal lattice aggregates feature a regular lattice structure their orientation is likely to influence the collision behavior. Thus, we first examine the e GLYPH<11> ect the orientation by comparing the case where the aggregates are aligned to their lattice structure (see upper part of Fig. 3) to a random orientation (lower part of Fig. 3)). As we can see in Fig. 4, the orientation of the aggregates is important especially for the transition from sticking to bouncing with increasing filling factor. Looking at the left panel of Fig. 4 and comparing the filling factor with the coordination number in Fig. 2 we can reproduce the n c GLYPH<21> 6-criterion proposed by Wada et al. (2011) for the aligned case. On the other hand, the bounc- ing maps di GLYPH<11> ers significantly for a non aligned orientation (see right panel of Fig. 4). In order to mitigate the e GLYPH<11> ect of the orientation we averaged over three di GLYPH<11> erent orientations to investigate the size dependency of our results. Each map has been generated using 12 di GLYPH<11> erent filling factors and 28 velocities. Thus, 3 GLYPH<1> 336 = 1008 simulations had to be performed in total. Concerning bouncing we could not observe a clear di GLYPH<11> erence between aggregates with a diameter of 30 and 60 GLYPH<22> m(see Fig. 5). However, the velocity v s ! f at which the transition from sticking to fragmentation occurs changes significantly. For the small aggregates we get v s ! f GLYPH<25> 4 ms GLYPH<0> 1 (left panel of Fig. 5). For the bigger aggregates we observe that v s ! f depends on the filling factor. For GLYPH<30> < 0 : 43 we get v s ! f GLYPH<25> 10 ms GLYPH<0> 1 whereas v s ! f GLYPH<25> 5 ms GLYPH<0> 1 for GLYPH<30> > 0 : 43 (right panel of Fig. 5). This can be explained by the reduced capability of compact aggregates to dissipate kinetic energy by restructuring. Taking into account Fig. 2 we see that the transition occurs when the average coordination number n c exceeds a value of 6. A monomer with six or more contacts is fixated rather tightly and and thus the aggregate cannot change its internal structure as easily anymore. In summary it can be said that for hexagonal lattice aggregates we regularly observe bouncing collisions for filling factors above 0 : 5 and collision velocities up to roughly 0 : 2ms GLYPH<0> 1 .", "pages": [ 4, 5 ] }, { "title": "4.3. Ballistic aggregation with migration", "content": "Wada et al. (2011) found that bouncing may occur if n c GLYPH<21> 6 independent of the type of aggregate they used. In Fig. 6, we show the outcome of collisions between two roughly 75 GLYPH<22> m sized BAM aggregates generated by using the shortest migration method described in Sect. 3.2. The corresponding filling factor is between 0 : 36 and 0 : 39. However, we did observe only two bouncing collisions. Since n c = 6 is the maximum value that can be achieved by two times migration we could not investigate what happens at higher coordination numbers. Repeating the setup described above for the center migration case we get similar results as for the shortest migration case shown in Fig. 6. We observe hardly any bouncing events even for aggregates with n c = 6 (which corresponds to a filling factor of 0 : 49 GLYPH<0> 0 : 5). This indicates that the bouncing behavior of BAM aggregates depends more on the filling factor than the coordination number. To achieve higher filling factors we switched to the center migration method (see Sect. 3.2). The corresponding bouncing maps are shown in Fig. 7. It is striking that the bouncing regime is much smaller compared to the CPE aggregates. As before, for larger aggregates the transition from sticking to fragmentation occurs at higher velocities.", "pages": [ 5, 6 ] }, { "title": "4.4. Static compaction", "content": "The bouncing behavior of the aggregates generated by static compaction is quite similar to the BAM aggregates (see Fig. 8). Again, the bouncing regime is considerably smaller compared to the case of hexagonal lattice aggregates and bouncing is observed only in some cases for high filling factors above 0 : 5 and collision velocities below 0 : 1 ms GLYPH<0> 1 . As in the case of the other aggregate types the transition velocity v s ! f from sticking to fragmentation increases with increasing aggregate size. For the the small aggregates with d = 30 GLYPH<22> m we observe v s ! f GLYPH<25> 4 ms GLYPH<0> 1 whereas for d = 60 GLYPH<22> m the transition velocity goes up to v s ! f GLYPH<25> 12 ms GLYPH<0> 1 .", "pages": [ 6 ] }, { "title": "4.5. Size dependency", "content": "To further examine the influence of the aggregate size we performed collisions of 100 GLYPH<22> m-sized CPE and BAM aggregates (using the center migration method as it yields the BAM aggregates with the highest filling factors). For the CPE aggregates we observe slightly more bouncing for filling factors between 0 : 4 to 0 : 5 (see left panel of Fig. 9). However, for BAM aggregates there is no noticeable di GLYPH<11> erence compared to the 60 GLYPH<22> m aggregates (see right panel of Fig. 9). Depending on the filling factor the 100 GLYPH<22> m aggregates consist of up to 350 ; 000 monomers. In order to analyze the size dependency it would be desirable to simulate collisions of even larger aggregates. Unfortunately, this is rendered impossible by the lack of available computing power. Simulating a single collision of two 100 GLYPH<22> msized aggregates took 10 to 20 hours (due to the di GLYPH<11> erent filling factors) on a GPU. Doubling the size would require computing times on the order of weeks for a single collision. For each orientation shown in Fig. 9 48 collisions have been performed.", "pages": [ 6 ] }, { "title": "4.6. Impact parameter", "content": "As a last step we examine the influence of the impact parameter b = 0 : 5. For the collisions, we used the same orientation as for the results shown in the right panel of Fig. 4. Contrary to our expectations we do not observe a significant influence of the impact parameter on the bouncing behavior of CPE aggregates (see Fig. 10). However, fragmentation sets in at considerably lower velocities of v s ! f GLYPH<25> 3 ms GLYPH<0> 1 . In a head on collision the entire aggregate may dissipate the kinetic impact energy by internal restructuring and thus help to avoid fragmentation. This does not apply to o GLYPH<11> set collisions where it is easier to tear away the outer layers without major restructuring of the core of the aggregates. However, comparing the growth map of the collisions between aggregates with a diameter of 30 and 60 GLYPH<22> m we do not observe any significant increase of velocity v s ! f where the transition from sticking to fragmentation occurs (see left and right panel of Fig. 10). As already pointed out by Wada et al. (2009) the increase of v s ! f for larger aggregates is limited to the case of head-on collisions. In contrast, for BAM aggregates we do observe bouncing in a larger regime compared to case of head-on collisions (see Fig. 11). Again, the same samples and orientation as for the head-on collisions (right panel of Fig. 7) have been used.", "pages": [ 6 ] }, { "title": "5. Requirements for bouncing", "content": "Compared to the aggregates generated by BAM or static compaction the bouncing regime of CPE aggregates is significantly larger. A likely explanation for this discrepancy is given by the di GLYPH<11> erent structure of the aggregates. To gain a deeper insight in the physical processes leading to sticking or bouncing it is worthwhile to have a closer look at a single collision. Two aggregates may bounce o GLYPH<11> each other only if there is enough elastic energy left to break the contact area. Thus, a significant amount of the kinetic impact energy must be stored temporarily without being dissipated. If the colliding aggregates penetrate each other too deeply the impact energy is dissipated upon internal restructuring in the area where the contact is established. In this case inelastic sliding and rolling constitute the main dissipation channels (Wada et al. 2011). Thus, the ratio of elastic to dissipated energy of colliding aggregates is the key parameter that determines whether sticking or bouncing will occur. Being able to track the evolution of the di GLYPH<11> erent types of energies over time is the key advantage of the model presented by Wada et al. (2007). To address the di GLYPH<11> erent behavior of BAM and CPE aggregates we compare a bouncing collision of two CPE aggregates with a sticking collision of BAM aggregates. Both aggregates are 60 GLYPH<22> m in diameter and have a filling factor GLYPH<30> GLYPH<25> 0 : 59. The time evolution of di GLYPH<11> erent types of energies and potentials for such collisions is shown in Fig. 12. As expected, in the sticking case most kinetic energy is dissipated by inelastic sliding and rolling (right panel of Fig. 12). Only a small percentage of the impact energy is stored in the elastic regime of the normal U normal and sliding potential U slide (since the elastic energy stored in the rolling and twisting potentials is negligible they are not shown in Fig. 12). We observe an entirely di GLYPH<11> erent situation in the bouncing case: As shown in the left panel of Fig. 12 only about one third of the impact energy is dissipated, whilst roughly half of the kinetic energy is temporarily converted into potential energy. This coincides well with our predictions above. The potential energy that is stored mainly in the normal and sliding interaction is con- verted back into kinetic energy and allows that the colliding aggregates to separate again. Wecan conclude that due to their lattice structure CPE aggregates can convert significantly more impact energy into elastic energy than BAM aggregates. In a compact CPE aggregate the monomers are located in densely packed layers. When the outer monomer of such a layer hits the other aggregate it is pushed inwards and will compress the layer. This way, kinetic energy is converted into potential energy without the occurrence of inelastic restructuring. This mechanism works well in the presence of a regular grid structure as it is the case for CPE aggregates. However, the monomers of BAM aggregates are not arranged in any regular pattern. Thus, they are not likely to bounce unless they are very compact in which case energy dissipation by internal restructuring is hindered because the monomers are locked in their position. It also o GLYPH<11> ers an explanation for the lower impact velocity of BAMaggregates at which we observe the transition from bouncing to sticking. At impact velocities above 0 : 1 ms GLYPH<0> 1 the kinetic impact energy is su GLYPH<14> cient to restructure the monomers in the contact area that had been locked at lower impact velocities. Since the lattice structure of CPE aggregates o GLYPH<11> ers higher resistivity against restructuring their transition velocity from bouncing to sticking is roughly 0 : 3 ms GLYPH<0> 1 .", "pages": [ 6, 7 ] }, { "title": "6. Conclusions", "content": "From the analysis of the statistical properties of the di GLYPH<11> erent samples presented in Sect. 3 we clearly see that the preparation method plays a crucial role when studying the collisional behavior of microscopic dust aggregates. As the relation between the volume filling factor and the average coordination number strongly depends on the preparation method one must be careful when comparing results obtained from computer simulations with laboratory work. In most laboratory experiments, only the volume filling factor can be measured, while the likewise important coordination number remains unknown. Since the more compact aggregates used in laboratory experiments are typically produced by mechanical compression, we expect that their microscopic structure resembles the static compaction aggregates much more closely than the hexagonal lattice type aggregates. For computer simulations, generating large, compact aggregates by static compaction is infeasible due to the additional computational e GLYPH<11> ort. We suggest to use BAM aggregates as an alternative. Their statistical properties are close to the aggregates produced by static compaction, yet they can be generated directly. Additionally, one does not run into problems caused by elastic charging as the generation procedure ensures that BAM aggregates are perfectly relaxed. At least in the various collisions φ Filling factor simulations performed in this work BAM and static compaction aggregates show very similar behavior. Based on the outcome of the simulations presented in this work we can conclude that bouncing collisions of dust aggregates in the size regime below 0 : 1 mm are rare. Unless the aggregates feature a regular lattice structure, which is not likely to be the case for the aggregates in a protoplanetary disk, bouncing requires filling factors greater than 0 : 5 and collision velocities below 0 : 1 ms GLYPH<0> 1 . Even if these prerequisites are met bouncing does not occur very frequently. Additionally, laboratory experiments on dust growth show that the maximum filling factor that is achieved during the growth process is much lower than GLYPH<30> = 0 : 5 (e.g. Teiser et al. 2011). Therefore, the influence of bouncing on the growth process is limited in the sub-mm size regime. Hence, from a microscopic view it remains unclear how cmsized aggregates with filling factors considerably below 0 : 5 are able to bounce o GLYPH<11> each other. The idea of a compacted outer layer (also referred to as a hard shell) has been put forward as a possible explanation. Using SPH-simulations, Geretshauser et al. (in preparation) observed that such a hard shell can indeed lead to bouncing collisions between aggregates with a porous core. Wada et al. (2011) obtained similar results when performing molecular dynamics simulations of collisions of CPE aggre- gates featuring a hard sphere. Langkowski et al. (2008) found that molding an aggregate significantly alters the outcome of a collision experiment. However, Kothe et al. (in prep. ) analyzed aggregates used in their collision experiments with X-ray computer tomography imaging and could not find any compacted outer layers. In Sect. 4.6 we have shown that o GLYPH<11> set collisions result in bouncing somewhat more often than head-on collisions. Depending on the experimental setup head-on collisions will be rare, and in a free collision with many particles setup head-on collisions will be rare as well (e.g. Weidling et al. 2012; Beitz et al. 2012). Thus, the impact parameter helps to resolve some of the discrepancies between numerical simulations and laboratory experiments. Taking into account the di GLYPH<11> erent aggregate types we can only partially confirm the n c GLYPH<21> 6 criterion for bouncing proposed by Wada et al. (2011). It agrees well with our results from collisions of CPE aggregates. However, for BAM aggregates generated by the shortest migration method we observe very little bouncing at n c = 6 (see Sect. 4.3 and Fig. 6). Likewise, for static compaction aggregates or BAM aggregates generated by the random or center migration method there are a few bouncing events where n c is considerably lower than 6. A volume filling factor of GLYPH<30> GLYPH<25> 0 : 5 appears to constitute a lower limit for bouncing. At the present time we do not have an explanation what determines the exact value of the critical filling factor for the onset of bouncing. We have shown that it depends on the amount of energy that can be stored in the normal potential. The energy deposition is a continuous process, and it is to be expected that below a certain density sticking ensues. Numerically, we determined this value to be GLYPH<30> GLYPH<25> 0 : 5 in the sub-mm size regime. Our simulations give insight into the fragmentation threshold as well. For small aggregates (30 GLYPH<22> m) the fragmentation velocity is around 4 m = s. Upon increasing the projectile size the fragmentation threshold rises up to about 10 m = s for the largest particle sizes we considered (60 GLYPH<22> m), and this is independent of the sample generation method. This value is in very good agreement with the findings of SPH simulation for much larger objects (Geretshauser et al. 2011). The shift to a larger fragmentation velocity is caused by the fact that larger particles can dissipate more energy than smaller particles. Upon increasing the filling factor, the fragmentation threshold decreases because the aggregates become much sti GLYPH<11> er and cannot be deformed so easily. However, as shown in Sect. 4.6 this e GLYPH<11> ect only applies to the case of head-on collisions. With respect to the growth of small dust agglomerates in the protoplanetary nebula our results indicate that for more realistic aggregates (BAM-type) bouncing only occurs for very small collision velocities ( < 0 : 1 m / s) and large filling factors > 0 : 5. Thus, the bouncing barrier may not be such a strong handicap in the growth phase of dust agglomerates, at least in the size range of GLYPH<25> 100 GLYPH<22> m. For larger, m -sized particles SPH results indicate bouncing up to 1 m = s. Acknowledgements. A. Seizinger acknowledges the support through the German Research Foundation (DFG) grant KL 650 / 16. The authors acknowledge support through DFG grant KL 650 / 7 within the collaborative research group FOR 759 The formation of planets . We also thank the referee, Koji Wada, for his helpful comments that improved the quality of this paper. A part of the simulations were performed on the bwGRiD cluster, which is funded by the Ministry for Education and Research of Germany and the Ministry for Science, Research and Arts of the state Baden-Württemberg.", "pages": [ 7, 8, 9 ] }, { "title": "References", "content": "Armitage, P. J. 2011, ARA&A, 49, 195 Beitz, E., Güttler, C., Weidling, R., & Blum, J. 2012, Icarus, 218, 701 Blum, J. & Wurm, G. 2008, ARA&A, 46, 21 Dominik, C. & Tielens, A. G. G. M. 1995, Philosophical Magazine, Part A, 72, Flaig, M., Ruo GLYPH<11> , P., Kley, W., & Kissmann, R. 2012, MNRAS, 420, 2419 Geretshauser, R. J., Meru, F., Speith, R., & Kley, W. 2011, A&A, 531, A166 Geretshauser, R. J., Speith, R., Güttler, C., Krause, M., & Blum, J. 2010, A&A, 513, A58 Heißelmann, D., Blum, J., Fraser, H. J., & Wolling, K. 2010, Icarus, 206, 424 Jankowski, T., Wurm, G., Kelling, T., et al. 2012, A&A, 542, A80 Kothe, S., Blum, J., Weidling, R., & Güttler, C. in prep. , submitted to Icarus Kozasa, T., Blum, J., & Mukai, T. 1992, A&A, 263, 423 Langkowski, D., Teiser, J., & Blum, J. 2008, ApJ, 675, 764 Ossenkopf, V. 1993, A&A, 280, 617 Pollack, J. B., Hubickyj, O., Bodenheimer, P., et al. 1996, Icarus, 124, 62 Schäfer, C., Speith, R., & Kley, W. 2007, A&A, 470, 733 Seizinger, A., Speith, R., & Kley, W. 2012, A&A, 541, A59", "pages": [ 9, 10 ] } ]

[ { "title": "ABSTRACT", "content": "Context. Blazars are powerful active galactic nuclei (AGNs) radiating prominently in the whole electromagnetic spectrum, from the radio to the X-ray and gamma-ray bands. Their emission is dominated by synchrotron and inverse-Compton radiation from a relativistic jet originating from an accreting central supermassive black hole. The object IGR J12319 -0749 has recently been identified as a soft gamma-ray source with the IBIS instrument of the INTEGRAL satellite, coincident with a quasar at high redshift ( z = 3.12). Aims. We studied the radio structure of IGR J12319 -0749 to strengthen its blazar identification by detecting a compact radio jet on the milli-arcsecond (mas) angular scale, and to measure its astrometric position accurate to mas level. Methods. We used the technique of electronic very long baseline interferometry (e-VLBI) to image IGR J12319 -0749 with the European VLBI Network (EVN) at 5 GHz on 2012 June 19. Results. IGR J12319 -0749 (J1231 -0747) is a compact radio source, practically unresolved on interferometric baselines up to ∼ 136 million wavelengths. The estimated brightness temperature (at least 2 × 10 11 K) indicates that the radio emission of its jet is Dopplerboosted. The accurate position of the compact radio source is consistent with the positions measured at higher energies. Key words. techniques: interferometric - radio continuum: galaxies - galaxies: active - quasars: individual: IGR J12319 -0749", "pages": [ 1 ] }, { "title": "A compact radio source in the high-redshift soft gamma-ray blazar IGR J12319-0749 (ResearchNote)", "content": "S. Frey 1 , Z. Paragi 2 , K.É. Gabányi 3 , and T. An 4 , 5 Received 22 November 2012; accepted 7 March 2013", "pages": [ 1 ] }, { "title": "1. Introduction", "content": "Bassani et al. (2012) identified the soft gamma-ray source IGR J12319 -0749 found by the INTEGRAL satellite as a radioand X-ray-emitting object. Based on its optical spectrum, the source is a quasar at z = 3.12 (Massetti et al. 2012). From the broad emission lines, the mass of the central black hole is estimated to be 2.8 × 10 9 M /circledot . Bassani et al. (2012) collected several other pieces of evidence that suggest the source is an extreme blazar, a flat-spectrum radio quasar with powerful jets. If this is the case, the source is the second most distant blazar detected by INTEGRAL so far. The X-ray source located within the INTEGRAL error circle has been recently followed up with the Swift satellite's X-ray telescope (XRT), and UV-optical telescope (UVOT). The X-ray image confirms that the object coincides with a National Radio Astronomy Observatory (NRAO) Very Large Array (VLA) Sky Survey (NVSS) radio source (Condon et al. 1998, VLA D configuration, flux density 60.4 mJy at 1.4 GHz). The radio source is also listed as an unresolved object ( < 5 '' ) in the VLA Faint Images of the Radio Sky at Twenty-cm (FIRST) survey catalogue (White et al. 1997, VLA B configuration, 62.9 mJy flux density and 61.0 mJy / beam peak brightness at 1.4 GHz). No other radio measurements (at frequencies other than 1.4 GHz, and with better resolution) are known. The X-ray flux seems variable over a period of a few months (Bassani et al. 2012). From the sparsely available, non-simultaneous data and upper limits, Bassani et al. (2012) tried to reconstruct the spectral energy distribution of IGR J12319 -074, and concluded that it is similar to that of another high-redshift blazar, IGR J22517 + 2218 (at z = 3.668; Falco et al. 1998). Here we report on our high-resolution radio interferometric observation of the radio counterpart of IGR J12319 -0749, using the technique of very long baseline interferometry (VLBI) with the European VLBI Network (EVN). This is an excellent tool for confirming with imaging that a source is indeed a blazar with compact radio emission on milli-arcsecond (mas) angular scale; VLBI is also capable of determining the accurate astrometric position of a compact radio-emitting object.", "pages": [ 1 ] }, { "title": "2. EVN observations and data reduction", "content": "The EVN observation of IGR J12319 -0749(or J1231 -0747, the name derived from the more accurate coordinates of the radio counterpart, and used in this paper hereafter) took place on 2012 June 19 at 5 GHz frequency. We utilized the electronic VLBI (e-VLBI) mode (Szomoru 2008) where, unlike the conventional VLBI, the signals are not recorded at the telescope sites, but are transmitted to the correlator via wide-band optical fibre networks. The real-time correlation of the data took place at the EVN Data Processor at the Joint Institute for VLBI in Europe (JIVE), Dwingeloo, The Netherlands. At the maximum recording rate of 1024 Mbit s -1 , eight antennas of the radio telescope network participated in the experiment: E ff elsberg (Germany), Jodrell Bank Mk2 telescope (UK), Medicina, Noto (Italy), Toru'n (Poland), Onsala (Sweden), Hartebeesthoek (South Africa), and the phased array of the Westerbork Synthesis Radio Telescope (WSRT, The Netherlands). Inter-continental baselines up to the length of ∼ 8100 km were provided by the Hartebeesthoek antenna. Our short exploratory e-EVN experiment (project code RSF06) lasted for 2 h. Eight intermediate frequency channels (IFs) were used in both left and right circular polarisations. The total bandwidth was 128 MHz per polarisation. The target source J1231 -0747 was observed in phasereference mode (e.g. Beasley & Conway 1995). We did not have prior information about the expected correlated flux density of J1231 -0747. For a successful detection, we planned su ffi ciently long coherent integration time to be spent on the source to improve the sensitivity of the observations. Phase-referencing involves regular nodding of the radio telescopes between the target and a nearby bright and compact reference source. We used J1233 -1025 as the phase-reference calibrator, at 2 · . 65 separation from our target. The target-reference cycles of ∼ 6 min allowed us to spend ∼ 4.5 min on the target source in each cycle, leading to ∼ 80 min accumulated integration time on J1231 -0747. The absolute astrometric position of the reference source is listed in the catalogue of the current realisation of the International Celestial Reference Frame (ICRF2, Fey et al. 2009). Phase- ing is suitable for accurately determining the position of the target source with respect to the reference source. The US National Radio Astronomy Observatory (NRAO) Astronomical Image Processing System (AIPS) was used for the data calibration in a standard way (e.g. Diamond 1995). The visibility amplitudes were calibrated using the antenna gains, and the system temperatures measured at the antennas during the experiment. Fringe-fitting (Schwab & Cotton 1983) was performed for the phase-reference calibrator (J1233 -1025), and the fringefinder sources (J1125 + 2610, J1159 + 2914). The data were then exported to the Caltech D ifmap package (Shepherd et al. 1994) for imaging. The conventional mapping procedure involving several iterations of CLEANing (Högbom 1974) and phase (then amplitude) self-calibration resulted in the images and brightness distribution models for the calibrators. Overall antenna gain correction factors (typically ∼ 10%or less) were determined and applied to the visibility amplitudes in AIPS. Then fringefitting was repeated in AIPS, now taking the CLEAN component model of the phase-reference calibrator into account. The residual phase corrections resulting from the non-pointlike structure of the phase-reference calibrator were considered this way. The solutions obtained were interpolated and applied to the target source data. Then the visibility data of J1231 -0747 were also exported to D ifmap for imaging. The phase-referenced image obtained was used for determining the position of the brightness peak with the AIPS task MAXFIT. The right ascension 12 h 31 m 57 s . 68547 and the declination -7 · 47 ' 18 '' . 0901 have accuracies of 0.7 mas and 1 mas, respectively, estimated from the phase-reference calibrator source position accuracy, the targetcalibrator separation, the angular resolution of the interferometer array, and the signal-to-noise ratio. From the phase-referenced data, it turned out that the target source itself is su ffi ciently bright and compact for fringe-fitting. Therefore, we applied the AIPS task FRING for J1231 -0747, as was done earlier for the phase-reference source and the fringefinders. This way the absolute position information is lost for the target. However, the final naturally-weighted image (Fig. 1) obtained in D ifmap with the standard hybrid-mapping cycles of CLEANing, and phase and amplitude self-calibration is somewhat more sensitive than the phase-referenced image.", "pages": [ 1, 2 ] }, { "title": "3. Results and discussion", "content": "Our 5-GHz VLBI image (Fig. 1) shows J1231 -0747 (IGR J12319 -0749) as a compact radio source without any extended feature above the brightness limit of ∼ 0.5 mJy / beam (5 σ ) at mas or 10-mas angular scales. The radio source appears practically unresolved with the e-EVN on baselines up to ∼ 136 million wavelengths (M λ ), i.e. from the European antennas to Hartebeesthoek. To physically characterise the source, we used D ifmap to fit a circular Gaussian brightness distribution model directly to the self-calibrated visibility data. The best-fit model componenthas 84.6 ± 4.2 mJy flux density and 0.24 ± 0.01 mas diameter (full width at half maximum, FWHM). This size can be compared with the minimum resolvable angular size (e.g. Kovalev et al. 2005; Lobanov 2005) that can be obtained in this VLBI experiment, assuming natural weighting Here b ψ is the half-power beam size along a given position angle ψ , and SNR the signal-to-noise ratio, i.e. the ratio between the peak brightness and the image noise. A source is considered unresolved if its measured visibility amplitude at the longest baseline di ff ers from that of a point source of the same flux density by not more than the 1σ uncertainty. This method is often applied to determine the minimum resolvable size of compact VLBI-detected components (e.g. Lee et al. 2008; Savolainen et al. 2008; Reynolds et al. 2009; Abdo et al. 2011; O'Sullivan et al. 2011) In our case, SNR = 829, and the beam size varies between 2.0 mas and 8.8 mas, corresponding to the FWHM of the elongated elliptical Gaussian restoring beam in the direction of its minor and major axes, respectively (Fig. 1). The resulting minimum resolvable angular size is the largest in the direction of the major axis, ϑ lim , maj = 0.29 mas. Our fitted model component size (0.24 mas) is similar but somewhat smaller than this value. With a more conservative approach, we adopt the 3σ image noise level which reduces the signal-to-noise ratio to 276. In this case, the minimum resolvable angular size in the major axis direction becomes ϑ lim , maj = 0.50 mas. We can use our modelfit result as the size estimate in the minor axis direction ( ϑ min = 0.24 mas) since it still exceeds the minimum resolvable size along this position angle. The measured parameters allow us to estimate the apparent brightness temperature ( T b) of the compact radio-emitting region (Condon et al. 1982) in the rest frame of the source where S is the flux density (Jy) and ν the observing frequency (GHz). For J1231 -0747, we obtain T b = (1.42 ± 0.09) × 10 11 K. Because of the unresolved nature of the radio source, this value can be considered a lower limit to the brightness temperature. A reasonable assumption about the intrinsic brightness temperature ( T b , int) of the source would lead to an estimate of the Doppler-boosting factor δ = T b / T b , int (Readhead 1994) in the jet. It is customary to assume the equipartition value, T b , int = T eq /similarequal 5 × 10 10 K (Readhead 1994; Lähteenmäki et al. 1999) as a good approximation of the intrinsic brightness temperature of compact AGN (e.g. Jorstad et al. 2006; Hovatta et al. 2009; Veres et al. 2010; Wu et al. 2012). This is valid if the energy in the radiating particles is equal to the energy stored in the magnetic field. Departure from equipartition is certainly possible for sources in their maximum brightness state, which could increase T b , int (Homan et al. 2006). On the other hand, VLBI studies of large samples found that for the majority of objects the typical T b , int measurements are ∼ 2 -3 × 10 10 K, very close to but somewhat below the canonical equipartition value (e.g. Kellermann et al. 2004; Homan et al. 2006). Bearing in mind that there is always an uncertainty in the determination of Doppler factors from single-epoch brightness temperature measurements, we follow the general practice and adopt T b , int = 5 × 10 10 K for J1231 -0747. In this case δ > ∼ 2 . 8. It can naturally be explained in the framework of the standard orientation-based unified model for radio-loud active galactic nuclei (Urry & Padovani 1995), i.e. with synchrotron emission of the plasma in an approaching relativistic jet pointing close to our line of sight. A rough estimate of the jet inclination angle with respect to the line of sight θ can be given by assuming a typical bulk Lorentz factor found for quasar jets, Γ= 10 (e.g. Kellermann et al. 2004). Using the jet inclination is θ ≈ 14 · , which becomes smaller if the Doppler factor and / or the Lorentz factor are higher than assumed here. The compact high-resolution radio structure of J1231 -0747 and its measured brightness temperature are consistent with the VLBI imaging data for most radio-loud quasars at around z = 3 (e.g. Gurvits et al. 1992, 1994; Frey et al. 1997; Paragi et al. 1999; O'Sullivan et al. 2011). Our VLBI experiment provides the first spectral data point available for the source at 5 GHz. The measured S 5 = 84.6 ± 4.2 mJy is a lower limit to the total flux density of J1231 -0747 if there is any extended emission around the central compact source that is resolved out by the EVN. This value is higher than the 1.4-GHz flux densities from the NVSS and FIRST surveys (consistently ∼ 60 mJy), which may indicate an inverted power-law spectrum of this quasar in the observed GHz frequency range. Note that the emitted (rest-frame) frequencies are higher by a factor of (1 + z ) because of the expansion of the Universe. In our case, ν obs = 5 GHz corresponds to ν em = 20.6 GHz. Since the di ff erent flux density measurements are made at widely separated epochs, it is also possible that J1231 -0747 shows significant flux density variations. Both the flat or slightly inverted radio spectrum and the variability are characteristic to compact blazars.", "pages": [ 2, 3 ] }, { "title": "4. Conclusions", "content": "Our e-EVN observation of the suspected radio counterpart of IGR J12319 -0749 (J1231 -0747), the second-highest redshift soft gamma-ray blazar detected with the INTEGRAL satellite, provided strong additional support for its blazar nature suggested by Bassani et al. (2012). The derived position of the compact radio source is consistent with the less accurate a-priori coordinates taken from the FIRST survey (White et al. 1997) within the errors, thus strengthening the identification of the gammaray source with the radio source, the only compact source within the FIRST beam. We found that J1231 -0747 is a ∼ 85-mJy radio source with sub-mas angular size. The slightly inverted radio spectrum or the flux density variability implied by our measurement at 5 GHz are expected from a blazar. With the usual assumption of energy equipartition between the magnetic field and the relativistic particles in the jet (Readhead 1994), we estimated the lower limit to the brightness temperature of J1231 -0747. It indicates a Doppler-boosted radio jet inclined within ∼ 14 · to the line of sight. The size of the radio source and the parameters of the jet could be better constrained with future higher-resolution VLBI imaging observations, in particular with longer interferometric baselines in the east-west direction. Simultaneous multi-frequency observations with large radio telescopes or interferometers would reveal the radio part of the SED and determine the turnover frequency in the radio spectrum. Flux density monitoring would shed light on the variability properties of this blazar, possibly providing additional constraints on the physical parameters of its jet. From an observational point of view, its su ffi ciently high brightness and compactness make J1231 -0747 an ideal new phase-reference calibrator object for any future VLBI experiment studying nearby weak radio sources. Acknowledgements. We are grateful to the chair of the EVN Program Committee, Tom Muxlow, for granting us short exploratory e-VLBI observing time. We thank the two anonymous referees for their suggestions. The EVN is a joint facility of European, Chinese, South African, and other radio astronomy institutes funded by their national research councils. e-VLBI research infrastructure in Europe is supported by the European Community's Seventh Framework Programme (FP7 / 2007-2013) under grant agreement RI-261525 NEXPReS. The research leading to these results has received funding from FP7 under grant agree- ment No. 283393 (RadioNet3), the Hungarian Scientific Research Fund (OTKA, grant no. K104539), and the China-Hungary Collaboration and Exchange Programme by the International Cooperation Bureau of the Chinese Academy of Sciences. T. An is grateful for the financial support from the National Natural Science Foundation of Science and Technology of China (2013CB837901).", "pages": [ 3, 4 ] } ]

[ { "title": "ABSTRACT", "content": "Context. Sub-millimetre dust emission is routinely used to derive the column density N of dense interstellar clouds. The observations consist of data at several wavelengths but also, with increasing wavelength, of poorer resolution. Procedures have been proposed for deriving higher resolution maps of N . In this paper the main ones are called Methods A and B. Method A uses low-resolution temperature estimates combined with higher resolution intensity data. Method B is a combination of column density estimates obtained using di ff erent wavelength ranges. Aims. Our aim is to determine the accuracy of the proposed methods relative to the true column densities and to the estimates that could be obtained with radiative transfer modelling. Methods. Weused magnetohydrodynamical (MHD) simulations and radiative transfer calculations to simulate sub-millimetre surface brightness observations at the wavelengths of the Herschel Space Observatory . The synthetic observations were analysed with the proposed methods and the results compared to the true column densities and to the results obtained with simple 3D radiative transfer modelling of the observations. Results. Both methods give relatively reliable column density estimates at the resolution of 250 µ mdata while also making use of the longer wavelengths. In case of high signal-to-noise data, the results of Method B are better correlated with the true column density, while Method A is less sensitive to noise. When the cloud has internal heating sources, Method B gives results that are consistent with those that would be obtained if high-resolution data were available at all wavelengths. Because of line-of-sight temperature variations, these underestimate the true column density, and because of a favourable cancellation of errors, Method A can sometimes give more correct values. Radiative transfer modelling, even with very simple 3D cloud models, usually provides more accurate results. However, the complexity of the models that are required for improved results increases rapidly with the complexity and opacity of the clouds. Conclusions. Method B provides reliable estimates of the column density, although in the case of internal heating, Method A can be less biased because of fortuitous cancellation of errors. For clouds with a simple density structure, improved column density estimates can be obtained even with simple radiative transfer modelling. Key words. ISM: clouds - Infrared: ISM - Radiative transfer - Submillimeter: ISM", "pages": [ 1 ] }, { "title": "Comparison of modified black-body fits and radiative transfer modelling", "content": "M. Juvela 1 , J. Malinen 1 , T. Lunttila 1 Department of Physics, P.O.Box 64, FI-00014, University of Helsinki, Finland, [email protected] Received September 15, 1996; accepted March 16, 1997", "pages": [ 1 ] }, { "title": "1. Introduction", "content": "Sub-millimetre and millimetre dust emission data are widely used to map the structure of dense interstellar clouds (Motte et al. 1998; Andr'e et al. 2000). The large surveys of Herschel are currently providing data on, for example, major nearby molecular clouds (Andr'e et al. 2010) and on the whole Galactic plane Molinari et al. (2010). The column densities derived from the emission depend not only on the strength of the emission but also on its spectrum. By covering the far-infrared part of the spectrum, Herschel observations are also sensitive to the dust temperature. Accurate estimates of the column density require accurate values of both temperature and dust opacity. In addition to being a tracer of cloud structure, the dust emission also carries information on the properties of the dust grains themselves. The properties are observed to change between clouds, which is associated with di ff erences in the grain optical properties, the size distributions, the presence of ice mantles, and possibly even temperature-dependent optical characteristics (Ossenkopf & Henning 1994; Stepnik et al. 2003; Meny et al. 2007; Compi'egne et al. 2011). Measurements of dust emission can be complemented with information from other tracers. Both dust extinction and scat- tering can be observed using near-infrared (NIR) wavelengths. These would be valuable because they are independent of that dust temperature that is a major uncertainty in the interpretation of emission data. Additional data would also help constrain the dust properties; see, e.g., Goodman et al. (2009) and Malinen et al. (2012) for a comparison of the use of dust emission and extinction. Dust extinction has been mapped over large areas (Lombardi et al. 2006; Goodman et al. 2009; Schneider et al. 2011), but high resolution observations are expensive and restricted to small fields. The same applies to observations of NIR scattered light that still exist only for a few clouds (Lehtinen & Mattila 1996; Nakajima et al. 2003; Foster & Goodman 2006; Juvela et al. 2008; Nakajima et al. 2008; Malinen et al. 2013). Measurements of scattered light are even rarer in the mid-infrared, and they probably tell more about the dust grains than the column density (Steinacker et al. 2010; Pagani et al. 2010). Therefore, in most cases one must rely on correct interpretation of the dust emission, preferably at farinfrared and longer wavelengths. Below 100 µ m the situation is complicated by the contribution of transiently heated grains and by the sensitivity to the shorter wavelengths, for which the clouds are typically optically thick. In addition to the uncertainty of the intrinsic grain properties, interpretation of dust emission data is a ff ected by two main problems, the e ff ect of noise and the e ff ect temperature variations. The noise is particularly problematic if one tries to determine both the dust temperature and the dust emissivity spectral index β (Shetty et al. 2009a; Juvela & Ysard 2012a). Therefore, most estimates of cloud masses are derived by assuming a constant value of β . However, there is always dust with di ff erent temperatures along the line-of-sight and at different positions within the beam, and this a ff ects the mass estimates (Shetty et al. 2009b; Malinen et al. 2011; Juvela & Ysard 2012b; Ysard et al. 2012). Because the emission of warm dust is stronger than the emission of colder dust of the same mass, the colour temperature derived from the observed intensities overestimates the mass-averaged temperature. The greater the temperature variations are, the more the dust mass is underestimated (Evans et al. 2001; Stamatellos & Whitworth 2003; Malinen et al. 2011; Ysard et al. 2012). The problem could be solved only if the temperature structure of the source were known so that the e ff ect could be determined with modelling. One would like to measure the density and temperature structure of the clouds not only as reliably as possible but also with as high a resolution as possible. The resolution depends on the telescope and the wavelengths used. For Herschel , the resolution varies from less than 8 '' at 100 µ m to ∼ 37 '' at 500 µ m. The standard way to calculate a column density map is to convert all data first to the lowest common resolution. Thus, most of the input data has significantly higher resolution than the result. Therefore, it would be beneficial to find ways to combine the data in a way that, although all wavelengths are used, the final map would retain a resolution better than that of the longest waveband. Juvela et al. (2012c) examined cloud filaments using column densities derived from Herschel 250 µ m surface brightness data at ∼ 20 '' resolution and dust colour temperature at ∼ 40 '' resolution. It was argued that the e ff ective resolution must be better than 40 '' because, on small scales, the temperature changes are small. Palmeirim et al. (2013) presented a better justified method that combined Herschel data at 160, 250, 350, and 500 µ m to produce high resolution column density maps. The methods may raise the question, what is the actual resolution of the maps. Furthermore, if one uses temperature maps of di ff erent resolution, they will be a ff ected di ff erently by the line-ofsight temperature variations and this could be reflected in the results. The aim of this paper is to investigate these questions. In the work we use the results of radiative transfer calculations, where the model clouds are the result of magnetohydrodynamical (MHD) simulations and the clouds may also contain point sources that introduce strong local temperature gradients. We compare the results of the above mentioned and similar methods (see Sect. 3) with the true column densities known from the models. We also examine the accuracy to which the column densities can be determined by carrying out radiative transfer modelling of the data. Such modelling has been applied, also recently, in the examination of the density and temperature structure of dense clouds (e.g. Ridderstad & Juvela 2010; Juvela et al. 2012b; Nielbock et al. 2012; Wilcock et al. 2012). The content of the paper is the following. In Sect. 2 we present the cloud models and the calculation of the surface brightness maps. In Sect. 3 we describe the basic estimation of the colour temperatures and column densities and present the two methods that are used to convert surface brightness data back to high-resolution column density maps. In Sect. 4 we present the main results, comparing the column density esti- mates to the true column densities in the cloud models and to the column densities that would be recovered by higher resolution observations. In Sect. 5 we construct three-dimensional models that are adjusted to reproduce the surface brightness observations and in this way used to estimate the column densities. The column densities of these models are again compared to the column density in the original model cloud. The results are discussed in Sect. 6 where we also draw the final conclusions on the relative merits of the methods used.", "pages": [ 1, 2 ] }, { "title": "2. Simulated observations", "content": "We use surface brightness maps calculated for two MHD models that are described in more detail in Malinen et al. (2011) and Juvela et al. (2012a). Cloud I corresponds to an isothermal magnetohydrodynamical (MHD) simulation carried out on a regular grid of 1000 3 cells (Padoan & Nordlund 2011). The calculations included self-gravity and the snapshot corresponds to situation before any significant core collapse. The cloud is scaled to a linear size of 6 pc and a mean density of n (H) = 222.0cm -3 , giving a mean visual extinction of 2 m . The model is the same that was used in Juvela et al. (2012a) to study filamentary structures. Cloud II was calculated using the adaptive mesh refinement (AMR) code Enzo (Collins et al. 2010). The model has a base grid of 128 3 , four levels of refinement, and an e ff ective resolution of 2048 3 cells. The model has been discussed in Collins et al. (2011) and in Malinen et al. (2011) (called Model II in that paper). As in the case of Cloud I, the MHD calculations assumed an isothermal equation of state. The linear size and the mean density of the model are scaled to 10 pc and 400 cm -3 . This gives an average column density of N (H) = 1.23 × 10 22 cm -2 that corresponds to AV ∼ 6.6 mag . The extinction reaches 20 m in less than 2% of the map pixels. For the radiative transfer modelling, the density fields were resampled onto hierarchical grids. The gridding preserves the full resolution in the dense parts of the model clouds but, to speed up the calculations, the resolution is degraded in the low density regions. Occasionally the greater size of some cells along the line-of-sight produces noticeable artefacts in the surface brightness maps which, however, usually disappear when the data are convolved with the telescope beam. Because the parameters that are being compared (i.e., the true and the estimated column densities) refer to the same discretisation, possible discretisation errors do not directly a ff ect this comparison. The dust temperature distributions and the emerging dust continuum emission were calculated with the radiative transfer code described in Lunttila & Juvela (2012). The clouds are illuminated externally by an isotropic interstellar radiation field (Mathis et al. 1983) and the dust properties correspond to those of the normal di ff use interstellar medium (Draine 2003) with a gas-to-dust ratio of 124 and RV = 3.1. The calculations are described in more detail in Malinen et al. (2011) and Juvela et al. (2012a). We refer to the densest sub-structures of the model clouds as cores. We will also examine a case where the cores of Cloud II, which already are known to be gravitationally bound, have internal heating sources. The properties of the sources and the procedures used in their modelling are described in Malinen et al. (2011). There are 34 sources with luminosities between 2.1 and 82 solar luminosities. For the present study, their main e ff ect is how they modify the three-dimensional distribution of dust temperature and how that is reflected in the surface brightness measurements. With the assumed cloud distance of 500pc, the sources can locally raise the dust colour temperature to 20-30 K when observed at the resolution of 40 '' . We use the radiative transfer modelling to simulate observations by the Herschel Space Observatory (Pilbratt et al. 2010). The calculations result in synthetic surface brightness maps at 160, 250, 350, and 500 µ m. The map size is 1000 × 1000 pixels for Cloud I and 2048 × 2048 pixels for Cloud II. As a default we assume noise levels of 3.7, 1.20, 0.85, and 0.35 MJy sr -1 per beam for 160, 250, 350, and 500 µ m, respectively. However, we also examine cases with noise 0.3 or 3.0 times these values.The pixel size of the maps is set to a value of 2.0 '' . During the analysis the maps are convolved with the assumed beam sizes of 12.0 '' , 18.3 '' , 24.9 '' , and 36.3 '' , for the four bands in the order of increasing wavelength. The values correspond to the approximate beam sizes of Herschel (Poglitsch et al. 2010; Gri ffi n et al. 2010).", "pages": [ 2, 3 ] }, { "title": "3. Analysis methods", "content": "In this section, we present the methods that are used to estimate the dust temperature and column density without resorting to full radiative transfer modelling. In particular, we recount the procedures used in Juvela et al. (2012c) and Palmeirim et al. (2013) to increase the spatial resolution of the column density maps. The methods are explained below and a summary of all the analytical combinations of individual column density estimates are summarised in Table 1.", "pages": [ 3 ] }, { "title": "3.1. Estimationofcolumndensity", "content": "The basic principles of the column density estimation are the same for all methods. The observed intensity I ν is approximated with a modified black body curve The equation assumes that the medium can be described with a single temperature value. The included approximation of the exponential term is valid if the optical depth τ is much smaller than one. This is the case for the models and the wavelengths examined in this paper. The optical depth τ is the product of dust opacity at the frequency in question, κν , and the column density N . Thus, the equation can be used to estimate the column density, provided that the dust temperature T is known. If we assume for the opacity a frequency dependence of κν ∝ ν β with some fixed value of the emissivity spectral index β , the value of T can be estimated with observations of two or more wavelengths, the latter requiring a fit to the observed intensities. We carry out these as least squares fits. Whenever the source contains temperature variations, the colour temperature obtained from these fits is only an approximation of the mass averaged dust temperature (Shetty et al. 2009b; Malinen et al. 2011; Juvela & Ysard 2012b; Ysard et al. 2012). This is one of the main reason why the morphology of the derived column density maps (i.e., column density contrasts) deviate from the reality. The least squares fits are carried out pixel by pixel, weighting the data points according to the observational noise. The fitted temperature and intensity are inserted to Eq. 1 for the calculation of an estimate of N . If data are available at more than two wavelengths, also the dust emissivity spectral index β could be determined. However, in this paper the value of β is kept fixed to the value of 2.0. In the dust model used in the radiative transfer calculations, the spectral index changes only a little as a function of wavelength and is ∼ 2.08 between the wavelengths of 160 µ m and 500 µ m. In the case of real observations, the absolute value of the opacity κν is a major source of uncertainty. In this paper, we are not interested in this factor and simply scale the median of the estimated column density maps to the median of the true column density that is known for the model clouds.", "pages": [ 3 ] }, { "title": "3.2. Higherresolutionestimates:MethodA", "content": "Juvela et al. (2012c) used Herschel observations to estimate the column density in the usual way, convolving all surface brightness data to the resolution of the 500 µ m data. However, the paper also used alternative column density estimates that were obtained combining the colour temperatures at the 40 '' resolution with 250 µ m surface brightness data at a resolution of 20 '' . It was argued that the e ff ective resolution of those maps would be close to 20 '' because the variations in the surface brightness are stronger than the e ff ects of colour temperature variations. It is not clear to what extent this is correct. This also depends on the di ff erence between the colour temperature and the true mass averaged dust temperature. For example, a compact cold core can have a significantly lower physical temperature without a significant e ff ect on the colour temperature that is dominated by emission from warmer regions. This means that although lower resolution of the temperature map does increase errors, these may not always be very significant.", "pages": [ 3 ] }, { "title": "3.3. Higherresolutionestimates:MethodB", "content": "In Palmeirim et al. (2013) a higher resolution column density map was obtained combining column density maps that were derived using di ff erent sets of wavelengths. The data consisted of Herschel at 160, 250, 350, and 500 µ m. One starts by calculating column density maps N (250), N (350), and N (500) that are based on data up to the specified wavelength and convolved to the corresponding resolution. For example, N (350) is based on the 160 µ m, 250 µ m, and 350 µ m maps that are convolved to the resolution of the 350 µ m map, ∼ 25 '' . If one convolves such a column density map to lower resolution, one also obtains estimates for the di ff erence in the structures that are visible in the two versions. We use the notation N ( λ 1 → λ 2) to denote a column density map that is first estimated using data at wavelengths λ ≤ λ 1 and at the resolution of the observations at λ 1 and is then convolved to the resolution of observations at wavelength λ 2. The final estimate of the column densities is obtained as a combination N (500) is the best estimate of column density at low resolution. The other terms add information on structures that are visible at the resolution of 350 µ mdata but not at the resolution of 500 µ m data and finally the structures that are visible at 250 µ mbut not at the resolution of the 350 µ m data. In principle, the method thus provides estimates for the column density at the resolution of the 250 µ m data, ∼ 18 '' . The estimates N (250), N (350), and N (500) will be di ff erent and not only because of the di ff erent resolution. By using di ff erent sets of wavelengths, one will not only have di ff erent noise levels but also the bias of each estimates will be di ff erent (Shetty et al. 2009b,a; Malinen et al. 2011). The biases are related to the temperature distribution of the source. In particular, without data at long wavelengths, one will be relatively insensitive to very cold dust. Of course, if the estimates were identical, one could use directly the N (250) map. With Eq. 2, one can include all the data although, of course, the correction terms [ N (350) -N (350 → 500)] and [ N (250) -N (250 → 350)] (i.e., estimates of small-scale structures) will be progressively more insensitive to cold emission.", "pages": [ 3, 4 ] }, { "title": "3.4. Otherhigherresolutionestimates", "content": "Because the bias of the column density estimates depends on the wavelengths used, it is possible that this particular combination of N (250), N (350), and N (500) is not optimal for the overall accuracy of the results. Therefore, we will later in Sect. 4 also examine some other linear and non-linear combinations. In particular, we will examine which linear combination of the terms N (500), [ N (350) -N (350 → 500)], [ N (250) -N (250 → 350)] gives the best correlation with the true column density. This will be called the Method C although, of course, in the case of real observations the values of these coe ffi cients could not be estimated. Finally, to examine the e ff ects of the wavelengthdependent biases further, we consider a linear combination of N (250 → 500), N (350 → 500), and N (500) where all surface brightness maps are first converted to the resolution of N (500). We call this Method D. Because already all the surface brightness data are convolved to a common resolution, Method D does not try to improve the spatial resolution of the column density map, only the correctness of the low resolution estimates. The final analytical method includes one additional, nonlinear term. The method is denoted by NL and, in addition to the terms already included in Method D, it contains a term [ N (500) -N (250)] 2 . At this point this is examined only as one possible idea of taking the non-linear features introduced by the colour temperature bias into account.", "pages": [ 4 ] }, { "title": "3.5. Radiativetransfermodelling", "content": "As an alternative to the 'analytical' Methods A and B, we will also examine radiative transfer modelling as a tool for the column density determination. We construct a simple threedimensional model for a source and carry out radiative transfer modelling to predict the dust emission at the observed wavelengths. The model is then adjusted until there is a satisfactory correspondence between the observed and the modelled intensities. The column density estimates are then read from the final model cloud. In practice, we construct three-dimensional models where the density distribution is discretised onto a grid of 81 3 cells. The radiative transfer calculations are carried out with a Monte Carlo Program (Juvela & Padoan 2003; Juvela 2005) using the same dust model as in the original calculations that were used to create the synthetic observations of Cloud I and Cloud II. Therefore, we exclude from consideration the errors that would be caused by wrong assumptions of dust opacity (and, consequently, of the dust emissivity spectral index). These errors are not included in the results of Methods A and B either. However, one must note that in the modelling we assume consistency not only at the submillimetre wavelengths but also at the short wavelengths where the dust grains are absorbing most of their energy. As the external radiation field we also used the same Mathis et al. (1983) model as in the original simulations but absolute level of the radiation field is not assumed to be known. The cell size of the constructed 81 3 cell models is set equal to 4 '' . When the observations and the model are compared, the 160 µ mand250 µ mdata are convolved to the resolution of 18.3 '' and the 350 µ m and 500 µ m data to the resolution of the observations, 24.9 '' and 36.3 '' , respectively. The details of the assumed density distribution and the procedures used to update the model clouds are discussed in more detail in Sect. 5. However, because the column densities are adjusted directly based on the 250 µ m data, the modelling should also provide column density estimates at the same resolution.", "pages": [ 4 ] }, { "title": "4.1. ComparisonofMethodsAandB", "content": "Using the surface brightness maps at 160, 250, 350, and 500 µ m resulting from the radiative transfer modelling (Sect. 2) we calculate column density maps using the Methods A and B (see Sect. 3). The results are compared with the true column density of the model clouds and with column density maps that are derived from synthetic observations with a uniform spatial resolution of 18.3 '' and without observational noise.", "pages": [ 4 ] }, { "title": "4.1.1. Estimates on large scales: Cloud I", "content": "We start by examining Cloud I. Because we are mostly interested in the dense structures and because the bias is stronger at higher column densities, we restrict the analysis to regions with true column density N true > 5 × 10 21 cm -2 . Figure 1 compares the results of Methods A and B to the true column density, including the Pearson correlation coe ffi cients. On average, the methods give comparable results. For lower observational noise, Method B gives higher correlation with the true values. However, Method B is also more sensitive to the presence of noise and, in the case of three times the default noise level, the correlation coe ffi cient is higher for Method A. Figure 2 compares the results in a small region centred at a dense clump, i.e., concentrating on the highest column densities. The calculations are carried out with the default observational noise. In the figure, in the neighbourhood of the main peak, Method A now shows up to ∼ 50% greater errors than Method B. This appears to be a direct consequence of the low resolution of the temperature information used by Method A. The figure also suggests that a large fraction of the errors in Method B map result from temperature variations that increase with the lineof-sight column density. Therefore, the availability of surface brightness data at the 18.3 '' resolution would not result in lower errors. Figure 3 shows column density profiles for the same region, again comparing cases with 0.3 times at 3.0 times the default noise. We also examine Method C that is an optimised version of Method B, a linear combination that results in the best correlation with the true column densities at the 18.3 '' resolution. Finally, Method D is a similar linear combination of the N (250 → 500), N (350 → 500), and N (500) maps at the lower 36.3 '' resolution. Figure 4 shows the correlations as scatter plots. In these overall correlations, Method B is performing consistently slightly better than Method A. With its optimised linear coe ffi cients, Method C produces still some improvement that is visible as a smaller scatter in Fig. 4. Even more interesting are the best coe ffi cients that are obtained for the terms N (500), [ N (350) -N (350 → 500)], [ N (250) -N (250 → 350)] (see Table 2). In Method B these are by construction all equal to 1.0. However, in the case of Cloud I, both Method C and Method D have partly negative coe ffi cients. The correlation coe ffi cient with true values is highest for Method D, r = 0.977, partly also because of the lower resolution. The opposite signs and greater magnitude of the N (250) and N (350) coe ffi cients mean that significant corrections are made based on the small di ff erences between the individual column density estimates. The results are partly encouraging, suggesting that it might be possible to improve the accuracy of the column density estimates beyond those obtained assuming a single modified black body. However, the method should perform well for all clouds, not only for the one for which it is originally tuned. We repeated the previous analysis also using β = 1 . 7 instead of β = 2 . 0. This changes the coe ffi cients of Methods C and D by some tens of percent but the correlations with the true column density are only slightly lower (less than 0.01). The main e ff ect of a wrong spectral index value is, of course, bias in the absolute values of the column density estimates.", "pages": [ 4, 5, 6 ] }, { "title": "4.1.2. Estimates for cores: Cloud II", "content": "In this section, we concentrate on the densest structures of Cloud II (the cores) and their environment. We look first at all data between the column densities of 5 × 10 21 cm -2 and 20 × 10 21 cm -2 , the selection being made using the true column densities at a resolution of 18.3 '' . The lower limit is the same as above while the upper limit is ∼ 25% higher than the maximum column density of Cloud I. For these data all Methods A-D give a correlation coe ffi cient r ∼ 0 . 997 for the comparison with the true column density. This is probably due to the fact that in Cloud II these column densities still correspond to extended medium without strong dust temperature variations. In this situation the coe ffi -cients of Methods C and D are not very well defined. In the column density range 20 × 10 21 cm -2 and 100 × 10 21 cm -2 , small di ff erences again appear between the methods. The correlation coe ffi cients are r = 0 . 972 for Method A, r = 0 . 978 for both Method B and Method C, and r = 0 . 987 for Method D. Unfortunately, for Method D, the coe ffi cients found in Cloud I and Cloud II or between the column density intervals of Cloud II are not very similar (see Table 2). For Method C, the variation of the parameter values is smaller. Nevertheless, there is no single set of coe ffi cients that could be used to improve the accuracy of the column density estimates for any cloud. On large scales, the linear combinations of N (250), N (350), and N (500) did produce some improvement in the accuracy but not with universal coe ffi cients. The usual column density estimates should be more biased at the locations of the dense cores where the line-of-sight temperature variations are the greatest. The small scale fidelity of the column density maps is crucial for the interpretation of the core properties. Therefore, we examine separately the neighbourhood of the cores in Cloud II. These are the same gravitationally bound regions as discussed in Malinen et al. (2011). We study pixels that fall within 2 ' radius of the centre of each core. Figure 5 compares the results at the 36.3 '' resolution. All methods tend to underestimate the true column density, especially towards the column density peaks. This produces the strong flattening of column densities above ∼ 10 22 cm -2 and the greatest errors are close to a factor of three. The order of accuracy of Methods A-D is as in the case of the large scale correlations but the di ff erences are more pronounced. At the highest column densities, the estimates given by Method D are twice the values of Methods A and B. Nevertheless, even Method D underestimates the true column density by up to ∼ 50%. For this set of data, the correlation coe ffi cients are 0.914, 0.921, 0.925, 0.964 for Methods A-D, respectively. Malinen et al. (2011) noted that when the cores are extremely dense, internal heating will improve the accuracy of the column density estimates. However, the e ff ect was small for the Cloud II (see their Fig. 11) and depends on how large regions around the cores are examined. We plot in Fig. 6 similar relation but for all pixels within the 2 ' radius. The figure is thus similar to Fig. 5 but shows the situation after the addition of the internal heating sources. The sources only have a little e ff ect on the accuracy of the column density estimates when this is calculated at the resolution of the 500 µ m data, the FWHM corresponding to ∼ 20 pixels. The bias shown by Methods A-C is very similar to Fig. 5 but the correlation coe ffi cients are slightly lower. At the highest column densities, Method D manages to bring the column density estimates up by ∼ 10% but below 100 × 10 21 cm -2 the results show a greater scatter. The coe ffi cients optimised for the cores di ff er from those derived on large scale but, on the other hand, the e ff ect of the internal heating sources remains marginal (see Table 2). Figure 7 shows a 2.9 pc × 2.9 pc piece of Cloud II, looking at the column density estimates obtained from surface brightness data at a resolution of 18.3 '' . The area includes some filamentary structures and six sources that raise the colour temperature locally up to ∼ 25K. With the sources, the column density estimates of the main filament are higher by up to ∼ 10%. This might be related to a higher average dust temperature that decreases the bias associated with line-of-sight temperature variations. At the very location of the sources the column density estimates are 10-30% lower. If the column density maps are convolved to a 2 ' resolution, the internal sources are seen to increase (i.e., improve) the estimates of the average column density of the cores. 0.6 The net e ff ect is negative only for a few of the strongest sources. This is the case for the source in the lower left corner of Fig. 7) where, at the 2 ' resolution, the estimate of the average column density of the region is still by ∼ 7% lower because of the presence of the radiation source. At the 18.3 '' resolution, the e ff ect is 35% towards the source but goes to zero already at the distance of ∼ 20 '' . Figure 9 shows the same area after the addition of an internal heating source. Because of stronger temperature variations, the column density errors are ∼ 30%, the 36.3 '' surface brightness data resulting in only slightly greater errors than the 18.3 '' data. Compared to Method B, the errors of Method A are smaller at lower column densities (because of lower sensitivity to noise), greater in a ring around the main clump, and again smaller at the location of the heating source where the error is below 10%. Figure 10 shows the column density profiles for the main clump. For the quiescent core, the Method B gives a profile that is almost identical to that of high resolution data, this still slightly underestimating the true column density. In Method A, the low resolution of the temperature information results in too low column densities for the quiescent core but in the case of internally heated core this compensates the e ff ect of temperature variations. In the case of internal heating, none of the methods is able to recover the actual column density profile of the clump. The result of Method B also is clearly impacted when the observational noise is increased. /N s) e c r sou ( N", "pages": [ 6, 7, 8 ] }, { "title": "4.2. Non-linearcorrectiontocolumndensityestimates", "content": "In this section we examine non-linear combinations of the N (250), N (350), and N (500) values. The goal is to find a method that would better trace the relative column density variations around the dense cores. Figure 11 shows the radial relative error of the column density estimates for all cores with peak column density exceeding 5 × 10 22 cm -2 . The values are calculated using data at 18.3 '' resolution, i.e., the figure shows only the e ff ect of the line-of-sight temperature variations on the estimates. The column density is underestimated up to 70% or, in the case of internal heating, up to ∼ 60%. For most cores the errors are below 30%. As the simplest extension of Method D, we attempted a nonlinear correction using the formula N NL = p 1 N (250 → 500) + p 2 N (350 → 500) + p 3 N (500) + p 4 [ N (500) -N (250 → 500)] 2 . For the 2 ' neighbourhoods of the gravitationally bound cores of Cloud II, the least-squares solution leads to the parameter values p 1 = -2 . 93, p 2 = -0 . 042, p 3 = 4 . 10, and p 4 = 0 . 62. As suggested by Fig. 12, the greater the di ff erence in N (500) -N (250), the greater the upward correction of the column density estimates. The correction (see Fig. 13) increases the correlation coe ffi cient between the true column density and the estimates from 0.924 for N (500) to 0.976 for N NL. The coe ffi cients are largely determined by the two cores with the highest column densities. However, also in the range N true = (10 -60) × 10 21 cm -2 the bias appears to be somewhat smaller and the correlation coe ffi cient has risen, although only very marginally (from 0.992 to 0.993). Of course, the quadratic term in the formula of N NL does not take the sign of the N (500) -N (250) di ff erence into account. In practice, the results would not change if the di ff erence were replaced with max { 0 . 0 , N (500) -N (250) } . The use of an additional term [ N (500) -N (350)] 2 does not bring any further improvement. When the same model is fitted to the data with internal sources, the parameters are p 1 = -3 . 03, p 2 = 1 . 42, p 3 = 2 . 79, and p 4 = 0 . 48. The parameters p 1, p 3, and p 4 are roughly similar as in the case without sources, but p 2 has increased to a value of 1.42. With these parameters, the correlation with the true column density increases from 0.920 for N (500) to 0.982 for N NL. If we use directly the coe ffi cients derived from the model without internal sources, the correlation coe ffi cient for N NL becomes 0.977. This is still a clear improvement over N (500) but, with these coe ffi cients, N NL overestimates the column densities beyond ∼ 150 × 10 21 cm -2 , the error increasing to ∼ 30% for the highest column densities. However, for these lines-of-sight N (500) underestimates the true values by a factor of three. This shows that the estimated non-linear correction could be useful in general, not only in the cloud where its coe ffi cients were derived.", "pages": [ 8, 9 ] }, { "title": "5. Radiative transfer modelling", "content": "The errors in the predictions of the previous methods are largely related to the temperature variations within the sources and the way the variations are reflected in observations at di ff erent wavelengths. By constructing three-dimensional models of the sources, one should be able to account for these e ff ects. In this section, we examine how well this works in practice, especially in the case of high column densities. Our synthetic observations are themselves based on numerical simulations. In this section, we only use the resulting surface brightness data, not the information about the structure of the sources or the radiation field seen by the individual clumps.", "pages": [ 9 ] }, { "title": "5.1. Themodellingprocedures", "content": "We carry out radiative transfer modelling of the nine cores of Cloud II with the highest column densities. The modelling is done purely on the basis of the 'observed' surface brightness maps, without using any information on the three-dimensional structure of the cloud. Each core separately is described with model of 81 3 4 '' cells, the cell size corresponding to 0.0098 pc. Each model thus covers a projected area of 5 . 3 ' × 5 . 3 ' or 0.79 pc × 0.79pc. The column density maps are shown in Fig. 14 with values obtained from the Cloud II density cube. Note that the term 'observation' refers to the synthetic surface brightness maps obtained from Cloud II and the term 'model' refers to the 81 3 cell models constructed for the regions around the selected cores. Along the line-of-sight, the full extent of Cloud II is 10 pc, muchgreater than the size of the 81 3 cell models. Thus, the structures that are visible within these 5 . 3 ' × 5 . 3 ' maps may be phys- y connected or may be quite distant from each other. The lack of knowledge on the line-of-sight structure is one of the main sources of uncertainty in the modelling. To analyse the observations of Cloud II, we subtract from the surface brightness maps the local background that is defined with the position of minimum N (500) value within the 5 . 3 ' × 5 . 3 ' area. This eliminates the necessity of modelling the extended emission. In some fields surface brightness gradients result in large regions with negative residual signal. However, this has little e ff ect on the actual cores that are much above the background. The line-of-sight density distribution is assumed to be Gaussian, with the FWHM set to 20% of the box size. The value is not tuned separately for each core but it is roughly consistent with the average core sizes in the plane of the sky. In the plane of the sky, the models are optimised by comparing the observed surface brightness maps and the corresponding maps produced by the models. The comparison is restricted to inner 2 ' × 2 ' area because the boundaries are a ff ected by edge e ff ects (because of the flat surfaces subjected to the full external field). The modelled cores are embedded deep within Cloud II that has an average visual extinction of AV ∼ 6.6 mag . This means that the radiation field is di ff erent for each core. In the calculations, this is taken into account by illuminating the model cloud by an attenuated radiation field. The unattenuated radiation field corresponds to three times the Mathis et al. (1983) model of the interstellar radiation field (ISRF), i.e., it is clearly higher than the actual field in the original simulations. The attenuation is parametrised by the visual extinction A ext V and is calculated using the extinction curve of the selected dust model. The attenuation is assumed to take place outside the model volume, by some external di ff use cloud component that would correspond to the di ff use component already subtracted from observations. We carry out radiative transfer calculations to produce synthetic 81 × 81 pixel maps of surface brightness that are compared with the 'observations' of Cloud II. The model column densities are adjusted according to the ratio of the observed and the modelled 250 µ m values. This means that the density in each cell corresponding to a given map pixel is multiplied by the same number that depends on whether the current model is overestimating or underestimating the observed surface brightness. The procedure is iterated until the 250 µ m errors are below 1% for the innermost 2 ' × 2 ' area.", "pages": [ 9, 10 ] }, { "title": "5.1.1. Basic models", "content": "If we do not use any other spectral information, the column density estimates depend on the assumed intensity of the heating radiation. If the same value were used for all cores, the column densities would often be wrong by a factor of two, a result significantly worse than either of Methods A or B (comparing the estimates to the true column density in Cloud II). Therefore, we adjust the attenuation of the external field until the observed and modelled ratios of 160 µ m and 500 µ msurface brightness agree. The ratios are measured as the average within a 30 '' radius of the core. Thus, both the intensity and shape of the emission spectral energy distribution (SED) should be correct at the location of the column density peak. Both attenuation and column densities (one value per map pixel) are adjusted iteratively until the relative errors are below 1%. Figure 15 compares the column densities of the optimised models to the true column densities and to the estimates from Method B. Although the cores are not circularly-symmetric, we plot the azimuthally averaged column densities as the function of distance from the centre of the selected core. The error bars on the true column density correspond to the azimuthal variation. For cores 1-3, the modelling recovers the radial column density profiles quite accurately. For core 6, the fit is worse but still better than for Method B that underestimates the true values by a significant fraction. For core 8, the Method B and modelling are equally close to the truth, while in cores 5 and 9 the modelling overestimates the peak. For the sources with the highest column densities, cores number 4 and 7, the peak is missing for both methods. This is not surprising, as the very compact central object is almost invisible still at the wavelength of 250 µ m. If the external field is raised to five times the Mathis et al. (1983) values, the central column densities decrease by 5-10%. The higher level of the external field is compensated by a greater value of A ext V , which means that energy is absorbed mainly at longer wavelengths. This reduces temperature gradients, also reducing the estimated column density contrasts of the cores by a similar factor of ∼ 10%.", "pages": [ 10 ] }, { "title": "5.1.2. Asymmetric illumination", "content": "In the field number 4 the results of the modelling are completely wrong at greater distances (Fig.15). This is caused by the presence of a strong temperature gradient across the field. The SED is correct at the centre of the field but on one side of the map the same assumption of the external field strength is not enough to produce the observed 250 µ m intensity. The column density increases to the point where the surface brightness saturates, leaving a spot where the 250 µ msurface brightness is underestimated in spite of the model column density being far above the correct value. Fortunately, the problem is plainly visible in the surface brightness maps, i.e., is apparent for the observer. In the problem area, the model underestimates the 160 µ m surface bright- s by more than 50% and overestimates the 500 µ m intensity by ∼ 150%. Temperature gradients are seen in a few fields and these can a ff ect also the central column density because of the mutual shadowing of the regions. To make a first order correction for gradients in the plane of the sky, we added to each model an anisotropic radiation source that covers a circular area of the sky with an opening angle of 45 degrees. The centre of this sky area is in a direction perpendicular to the line of sight and at a position angle where, based on the previous results, the source will help to remove the residual colour gradients. The spectrum of the anisotropic component corresponds to the normal ISRF attenuated by AV = 2 . 0 mag and its intensity is scaled to obtain a solution where, for data within the innermost one arcmin radius, the quantity ∆ S (160 µ m) -∆ S (500 µ m) is no longer correlated with the distance along the original gradient direction. In the formula, ∆ S stands for the di ff erence of the observed and modelled surface brightness values. The e ff ect of asymmetric illumination is in most cores almost unnoticeable on the column density estimates. In core 3, the central column density increases by ∼ 5%while in core 8 it decreases by the same amount. Clear e ff ects are visible only in the case of the cores 4 and 7, the ones with the highest column density (see Fig. 16). In field 4, the model is now much closer to the correct column density values outside the central core. However, the constructed models still miss the high column density peaks of both fields 4 and 7. Figure 17 shows this for core 4. The densest core is invisible in the 160 µ m map, not visible as a separate peak in the 500 µ m surface brightness, and also missed by the model. The model was tuned so that the SED (250 µ m surface brightness and the 160 µ m / 500 µ m colour) on the average match the observations over an one arcmin circle. At the very centre, however, the residual errors are ∼ 10% of the surface brightness and rise above 30% elsewhere in the field. The significance of these residuals suggests that further improvements in the models should be possible.", "pages": [ 10 ] }, { "title": "5.1.3. Varying the line-of-sight mass distribution", "content": "The main problem for the modelling is that the cores are embedded in an optically thick cloud whose line-of-sight extent is two orders of magnitude longer than the typical core size. Any structure seen in a map can be a compact object also in threedimensions, it may be elongated along the line-of-sight or may even consist of several unconnected structures within the 10 pc distance through the model cloud. This impacts the dust temperatures and, consequently, the column density estimates. The previous models consisted of a smooth density distribution with a single FWHM for its line-of-sight extent. We can try to take some of the variations into account by modifying the FWHMvalues for each line-of-sight separately. A small FWHM value would corresponds to a compact and cold region, a greater FWHM to more di ff use region with a higher average temperature. We do not modify the FWHM at the core location where the data are already used to adjust the strength of the external radiation field. For any other line-of-sight, if the model predicts a too cold spectrum (low ratio of 160 µ m and 500 µ m intensi- - ties), the FWHM is increased (and vice versa). The final e ff ect is complicated by shielding between the di ff erent regions. This further modification helps to bring down the errors outside the densest core, e.g., in the field number 4 (see Fig. 18). The surface brightness residuals at 160 µ m and 500 µ m are mostly below 10% but raise up to 20% at the location of the column density peak. The column density estimate of the peak is close to that of the previous models. The situation is similar in field 7 where the peak value of ∼ 120 × 10 21 cm 2 remains far below the correct number of ∼ 800 × 10 21 cm 2 . Of the lower column density cores, the column density predictions were previously most erroneous in field 9 where the peak value was overestimated by ∼ 35%. The adjustment of the line-of-sight extent of the clump has reduced this error only to ∼ 28%. For the other cores, the e ff ects are smaller. This shows that even the variation of the line-of-sight extent, as implemented, is not able to reproduce the complexity of these regions.", "pages": [ 11 ] }, { "title": "5.2. Ellipsoidalmodel", "content": "For the final test, we return to a simple geometry and isotropic illumination. The cloud densities are generated as a threedimensional ellipsoid with Gaussian density profiles and a ratio of 1:2:4 between the FWHM values along the main axes. The density field is sampled on a 81 3 Cartesian grid, first applying three 45 degree rotations to the density distribution. The resulting cloud has peak column densities of 21.5 × 10 21 cm -2 , 31.2 × 10 21 cm -2 , and 37.5 × 10 21 cm -2 towards the three main axes (see Fig. 19). We calculate synthetic surface brightness maps with the default noise and ISRF. Based on these data, the column densities were then estimated with 3D modelling similar to that of Sect. 5.1.1. Thus, in the modelling only the column densities and the external isotropic field were adjusted. The results are compared only with the results of Method B because, as seen in Sect. 4, the di ff erences in the accuracy of Methods A-D are not very significant. When the 3D modelling was done with the correct level of the external fields but keeping its attenuation as a free parameter, the column densities were recovered with an accuracy of a couple of percent. The cloud has a density distribution that is consistent with the assumption of a Gaussian line-of-sight density distribution used in the modelling. However, the width of the distribution is not the same as in the modelling and, furthermore, varies by a factor of four depending on the viewing direction. This suggests that in this case the results are not very sensitive to the uncertainty of the line-of-sight extent. The results of Method B and the 3D modelling are compared further in Fig. 19. In this case, the modelling is done with an external field that is twice as strong as the actual field used to produce the synthetic observations. Therefore, the field needs to be adjusted by introducing significant external attenuation. This correction is not exact because the attenuation changes not only the level but also the SED of the incoming radiation. After removing the shortest wavelengths, the remaining radiation penetrates deeper, making the cloud more isothermal. The resulting errors are visible in Fig. 19g-i where the column density is overestimated in the outer parts of the cloud and correspondingly underestimated at the centre. The errors rise over ∼ 4% only at the centre and only when the cloud is viewed from the direction with the highest column density. The incorrect assumption of the ISRF spectrum is also visible in the residuals at 160 µ m and 500 µ m. Although the average colours are adjusted to be correct, at the cloud centre 160 µ m intensity is overestimated by more than ∼ 5% while the 500 µ mintensity is underestimated by a couple of percent. This information could be used to further improve the accuracy of the model. For Method B the errors are stronger, column density being underestimated by up to ∼ 10% percent. The relative bias is better visible in the radial profiles at the bottom of Fig. 19. The sensitivity to noise is another important point. Unlike in the Cloud I and Cloud II, the observations of the outer parts of the clump are now dominated by noise. This a ff ects Method B results already at ∼ 4 × 10 21 cm -2 , mainly via the N (250) estimates. The modelling results in much lower statistical noise even when it does not use all the available data optimally and the column density distribution is adjusted using the 250 µ mobservations only. The low noise can be understood as a result of the strong intrinsic regularisation of the modelling procedure. In particular, this precludes unphysical temperature variations (i.e., those greater than allowed by the optical depths) and keeps the estimates reasonable even when the signal goes to zero.", "pages": [ 11, 12 ] }, { "title": "6. Discussion", "content": "We have examined di ff erent ways of estimating the column density based on dust emission maps, especially using Herschel data in the bands between 160 µ mand 500 µ m. The methods A-C try to recover the column density at a resolution better than the lowest resolution of the input maps. These aim at a resolution of 18 '' (the resolution of the 250 µ m observations), a factor of two better than the resolution of the 500 µ m data. The radiative transfer models also were constructed in a way that results in column density information at the same resolution. In the tests with the extended emission (Sect. 4.1.1), Method B performed consistently better than Method A, the errors near dense clumps being smaller by up to ∼ 50% (see Figs.2 and 8). The di ff erence remained clear even when estimates were compared at lower resolution, 36.3 '' , corresponding to the resolution of the 500 µ m maps. Therefore, the di ff erence is not limited to direct e ff ects of resolution. Only in the case of internal heating Method A exhibited noticeably smaller bias. In those cases Method B was close to the results that would be obtained if data at all wavelengths were available at the same 18.3 '' resolution. However, these estimates are biased because of the lineof-sight temperature variations that lead to overestimation of the dust temperature and underestimation of the column density. By underestimating the temperature variations that exist on small spatial scales, Method A was actually closer to the true column density at the location of internally heated clumps (see Figs. 9 and 10). In Method B, there is some freedom to select the wavelengths that are used to derive the estimates N (500), N (350), etc. For example, if one assumes that SPIRE data give a more reliable picture of column density on large scales (or that they are less biased by temperature variations), one can base the N (500) estimate on data between 250 µ m and 500 µ m only. The other terms could include also the shorter wavelengths but, because in Method B these are high pass filtered correction terms, they would not be sensitive to large scale artefacts. For example, possible arteficial gradients or high pass filtering of the PACS maps themselves would have only a limited impact on the derived column density maps. We also examined as Method C other linear combinations of the three constituent terms of Method B, N (500), [ N (350) -N (350 → 500)], and [ N (250) -N (250 → 350)]. By selecting optimal multipliers (Methods C and D), it was possible to increase the correlation with the true column density by a small but significant amount. The fact that those multipliers were not very similar for the two examined cloud models suggests that this may not be a viable method for general use. The best correspondence with the true column density was obtained with multipliers that were of di ff erent signs. This shows that, on large scales, the di ff erent biases of the N (250), N (350), and N (500) estimates have a significant e ff ect on the final errors. The di ff erences between the methods were accentuated in the optically thick cores (Sect. 4.1.2). For cores with AV > ∼ 50 m or more, the central column densities can be underestimated by several tens of percent. The strongest errors observed for both Method A and B are a factor of three and the associated column density peaks are hardly visible in the surface brightness maps, even at 250 µ m. The optimised linear combination of N (250), N (350), and N (500) improves the fit at the highest column densities, raising the column density estimates by up to a factor of two. However, because the errors behave in a very non-linear fashion (as a function of column density), this increases the errors at the lower column densities. We also examined the possibility of making a non-linear combination of the N (250), N (350), and N (500) estimates. The bias depends on the wavelengths used and we found, as expected, that N (250) < N (350) < N (500). The di ff erences increase as a function of column density. As a result, a non-linear combination of the estimates resulted in significant improvement in the accuracy of the column density predictions (see Fig. 13). It remains to be established whether the parameters are be stable enough, so that the method could be reliably applied to actual observations. The presence of internal heating sources was already seen to eliminate much of the systematic behaviour of N (250), N (350), and N (500) relative to each other. As the final method, we examined 3D radiative transfer modelling as a way to estimate the column densities. In the case of Cloud II, this turned out to be quite challenging because of the high optical depths. Together with the complex density field this means that the radiation field illuminating the modelled core could be strongly asymmetric. The lack of information about the line-of-sight density structure is always a major problem and in this case, the line-of-sight extent was more that ten times the perpendicular extent of the modelled fields. Thus, also the radiation field could vary significantly along this extent. The dense material was seen to be distributed over long distances. This was in stark contrast with the assumed simple model where, for all lines of sight, the density always peaked in the mid-plane. This maximises the shadowing e ff ect compared to the reality of isolated clumps (Fig. 21) or oblique filaments (Fig.20). In spite of these caveats, the modelling produced fair results. The model column density was adjusted based on the 250 µ m observations and the attenuation of the external field was adjusted according to the 160 µ m / 500 µ m colour in the central region. The accuracy of the results was typically slightly better than for Method B. For the cores with a simple geometry (e.g., cores 1-3, see Fig. 15) the basic modelling produced very accurate density profiles, while Method B underestimated the central column density by ∼ 10%. For the most opaque cores the modelling required the inclusion of an anisotropic radiation field to avoid strong errors outside the central regions for which the radiation field was tuned. Because the di ff erent structures along the line-of-sight may be subjected to quite di ff erent radiation fields (e.g., of di ff erent intensity, SED, and anisotropy), it may be difficult improve the results much further, at least not without exhaustive examination of more complex models. The adjustment of the width of the density distribution along the line-of-sight direction did not produce very significant improvement. It is possible to construct models that (at least in the case of such synthetic observations) reproduce all the observed surface brightness maps to within the observational noise. However, in the case of Cloud II this was already deemed too time consuming. The modelling procedure used in this paper was very simple and, apart from the column densities that were adjusted for each pixel separately, the number of free parameters was small. As a result, the solution was found in just some tens of iterations (run time of the order of one hour per model). Even when the lineof-sight extent of the density distribution was modified, all parameters could be updated relatively independently using heuristic rules based on the observed and modelled surface brightness maps. In more complex models (i.e., more complex parameterisation of the cloud structure) the link between the individual parameters and the surface brightness would be less obvious and the solution would have to be obtained through general optimisation. Depending on the number of parameters, this could be orders of magnitude more time consuming. However, as long as the models still exhibited systematic residuals (e.g., Fig. 18), further improvements remain possible. Cloud II is rather extreme in its opacity. The dense cores in nearby molecular clouds would probably fall between Cloud II and the ellipsoidal cloud of Sect. 5.2 in complexity (Fig. 19). For the ellipsoidal cloud, if the external field was estimated correctly, the modelling recovered the column density to within a couple of percent. If the assumed ISRF was overestimated by a factor of two the errors remained below ∼ 5% and the signature of the wrong radiation field was visible in the surface brightness maps. The same interpretation would be more di ffi cult to make in the case of real, irregularly shaped clouds. However, the results suggest that for most of the clumps detected in nearby clouds one can, with careful modelling, determine the column density profiles to an accuracy of a few percent. Method B, possibly combined with a small bias correction, would result in an almost similar accuracy and with considerably less e ff ort. One must also remember that we did not consider any of the uncertainties that are related to dust properties and are likely to be the dominant errors in the estimates of absolute column density.", "pages": [ 12, 13 ] }, { "title": "7. Conclusions", "content": "We have compared di ff erent, previously presented, methods to calculate column density maps from dust emission, especially using the Herschel wavelengths 160-500 µ m. Method A (Juvela et al. (2012c) uses low resolution temperature estimates combined with higher resolution intensity data. Method B (Palmeirim et al. 2013) uses a combination of column density estimates obtained using di ff erent wavelength ranges. The methods try to recover the column density at a resolution better than that of the lowest resolution input map. We also test other modifications of the methods and compare these with simple radiative transfer modelling that also is used to obtain the column densities. We have found that Acknowledgements. The authors acknowledge the support of the Academy of Finland grant No. 250741. TL acknowledges the support of the Academy of Finland grant No. 132291.", "pages": [ 13, 14 ] }, { "title": "References", "content": "Andr'e, P., Men'shchikov, A., Bontemps, S., et al. 2010, A&A, 518, L102 Andr'e, P., Ward-Thompson, D., & Barsony, M. 2000, Protostars and Planets IV, Collins, D. C., Padoan, P., Norman, M. L., & Xu, H. 2011, ApJ, 731, 59 Collins, D. C., Xu, H., Norman, M. L., Li, H., & Li, S. 2010, ApJS, 186, 308 Compi'egne, M., Verstraete, L., Jones, A., et al. 2011, A&A, 525, A103 Evans, II, N. J., Rawlings, J. M. C., Shirley, Y. L., & Mundy, L. G. 2001, ApJ, 59 Draine, B. T. 2003, ApJ, 598, 1017 557, 193 Foster, J. B. & Goodman, A. A. 2006, ApJ, 636, L105 Goodman, A. A., Pineda, J. E., & Schnee, S. L. 2009, ApJ, 692, 91 Gri ffi n, M. J., Abergel, A., Abreu, A., et al. 2010, A&A, 518, L3 Juvela, M. 2005, A&A, 440, 531 Juvela, M., Malinen, J., & Lunttila, T. 2012a, A&A, 544, A141 Juvela, M. & Padoan, P. 2003, A&A, 397, 201 Juvela, M., Pelkonen, V.-M., Padoan, P., & Mattila, K. 2008, A&A, 480, 445 Juvela, M., Pelkonen, V.-M., White, G. J., et al. 2012b, A&A, 544, A14 Juvela, M., Ristorcelli, I., Pagani, L., et al. 2012c, A&A, 541, A12 Juvela, M. & Ysard, N. 2012a, A&A, 541, A33 Juvela, M. & Ysard, N. 2012b, A&A, 539, A71 Lehtinen, K. & Mattila, K. 1996, A&A, 309, 570 Lombardi, M., Alves, J., & Lada, C. J. 2006, A&A, 454, 781 Lunttila, T. & Juvela, M. 2012, A&A, 544, A52 Malinen, J., Juvela, M., Collins, D. C., Lunttila, T., & Padoan, P. 2011, A&A, 530, A101 + Malinen, J., Juvela, M., M., R., & Pelkonen, V.-M. 2013, A&A, in preparation Malinen, J., Juvela, M., Rawlings, M. G., et al. 2012, A&A, 544, A50 Mathis, J. S., Mezger, P. G., & Panagia, N. 1983, A&A, 128, 212 Meny, C., Gromov, V., Boudet, N., et al. 2007, A&A, 468, 171 Molinari, S., Swinyard, B., Bally, J., et al. 2010, A&A, 518, L100 Motte, F., Andre, P., & Neri, R. 1998, A&A, 336, 150 Nakajima, Y., Kandori, R., Tamura, M., et al. 2008, PASJ, 60, 731 Nakajima, Y., Nagata, T., Sato, S., et al. 2003, AJ, 125, 1407 Nielbock, M., Launhardt, R., Steinacker, J., et al. 2012, A&A, 547, A11 Ossenkopf, V. & Henning, T. 1994, A&A, 291, 943 Padoan, P. & Nordlund, Å. 2011, ApJ, 730, 40 Pagani, L., Steinacker, J., Bacmann, A., Stutz, A., & Henning, T. 2010, Science, 329, 1622 Palmeirim, P., Andr'e, P., Kirk, J., et al. 2013, A&A, 550, A38 Pilbratt, G. L., Riedinger, J. R., Passvogel, T., et al. 2010, A&A, 518, L1 Poglitsch, A., Waelkens, C., Geis, N., et al. 2010, A&A, 518, L2 Ridderstad, M. & Juvela, M. 2010, A&A, 520, A18 + Schneider, N., Bontemps, S., Simon, R., et al. 2011, A&A, 529, A1 + Shetty, R., Kau ff mann, J., Schnee, S., & Goodman, A. A. 2009a, ApJ, 696, 676 Shetty, R., Kau ff mann, J., Schnee, S., Goodman, A. A., & Ercolano, B. 2009b, ApJ, 696, 2234 Stamatellos, D. & Whitworth, A. P. 2003, A&A, 407, 941 Steinacker, J., Pagani, L., Bacmann, A., & Guieu, S. 2010, A&A, 511, A9 + Stepnik, B., Abergel, A., Bernard, J., et al. 2003, A&A, 398, 551 Wilcock, L. A., Ward-Thompson, D., Kirk, J. M., et al. 2012, MNRAS, 424, 716 Ysard, N., Juvela, M., Demyk, K., et al. 2012, A&A, 542, A21", "pages": [ 14 ] } ]

[ { "title": "ABSTRACT", "content": "Aims. We present new results regarding the companion mass-ratio distribution (CMRD) of stars, as a follow-up of our previous work. Methods. We used a maximum-likelihood-estimation method to re-derive the field CMRD power law avoiding dependence on the arbitrary binning. We also considered two new surveys of multiples in the field for solar-type stars and M dwarfs to test the universality of the CMRD. Results. We found no significant di ff erences in the CMRD for M dwarfs and solar-type stars compared with previous results over the common mass ratio and separation range. The new best-fit power law of the CMRD in the field, combining two previous sets of data, is dN / dq ∝ q β , with β = 0 . 25 ± 0 . 29. Key words. binaries: general - stars: formation - stars: low-mass - stars: solar-type", "pages": [ 1 ] }, { "title": "Universality of the companion mass-ratio distribution (ResearchNote)", "content": "M. Reggiani 1 and M. R. Meyer 1 Institute of Astronomy, ETH Zurich, CH-8093 Zurich, Switzerland Preprint online version: July 3, 2018", "pages": [ 1 ] }, { "title": "1. Introduction", "content": "A large portion of stars, both in the field (Raghavan et al. 2010; Janson et al. 2012) and in star-forming regions (Patience et al. 2002), are formed in multiple systems. Therefore understanding multiple star formation is necessary to investigate star formation in general (Goodwin et al. 2007; Duchˆene & Kraus 2013). Because binary properties reflect the main characteristics of binary formation, they may help us determining the most common mechanisms for the formation of multiple stars. In a binary system of stars with masses M 1 and M 2 ( M 1 > M 2), the mass.ratio is conventionally defined as q = M 2 / M 1. Similar to the initial mass function (IMF) for single objects, the companion mass-ratio distribution (CMRD) is the distribution of q values as a function of primary mass. Tidal capture models predict that for each primary star the mass of the secondary is chosen randomly from the single-star mass function, and the CMRD reflects the IMF (McDonald & Clarke 1993; Kroupa et al. 2003). In fragmentation scenarios subsequent continued accretion onto both objects from a common reservoir tends to equalize the masses, resulting in a q distribution peaked toward unity (Bate 2000). Capture is unlikely to be the most relevant binary formation mechanism, but it may still occur during the dissolution phase of star clusters, causing di ff erences in the shape of the CMRD as a function of orbital separation (Moeckel & Bate 2010; Moeckel & Clarke 2011). Motivated by the fact that every theoretical model predicts a di ff erent shape of the mass-ratio distribution and of dependency of the CMRD on the primary mass, we used Monte-Carlo simulations to compare the CMRD for different samples and to study the relationship between the IMF and the CMRD (Reggiani & Meyer (2011), hereafter, RM11). This research note represents a follow-up to RM11, in which we reanalyze the 'universal' CMRD by adopting a di ff erent statistical approach (Section 2) and some new results on the CMRD on the basis of recent datasets (Section 3).", "pages": [ 1 ] }, { "title": "2. Universal companion mass-ratio distribution", "content": "The CMRD appears to be universal over a wide range of q values and primary masses (e.g. Metchev & Hillenbrand 2009). According to RM11, the CMRD follows a single-slope power law dN / dq ∝ q β over the separation range 1-2400 AU and primary mass range 0.25-6.5 M /circledot , and there is no evidence for variation of the CMRD with orbital separation. In previous work we combined samples of M dwarfs (Fischer & Marcy 1992) and G stars (Metchev & Hillenbrand 2009) in the field and intermediate mass stars in ScoOB2 (Kouwenhoven et al. 2005) adopting a χ 2 fit of the combined binned distribution to derive the power law slope, obtaining β = -0 . 50 ± 0 . 29 (Reggiani & Meyer 2011). The choice of the statistical method was motivated by the need of comparing our results with previous studies of the CMRD (e.g. Kouwenhoven et al. 2005; Metchev & Hillenbrand 2009). However, the χ 2 fit of a binned distribution can lead to a biased estimate, in particular for small samples. A more robust analysis is instead achieved through a maximum-likelihood estimation method (Feigelson & Babu 2011). This approach gives a new best-fit power law dN / dq ∝ q β , with β = -0 . 18 ± 0 . 33 to the data described in RM11. Although the two values are consistent with each other within the errors, the new estimate is flatter than previously thought.", "pages": [ 1 ] }, { "title": "3. Updates to the CMRD in the field", "content": "Recently, two new studies of the CMRD for solar-type (Raghavan et al. 2010) and M-dwarf primaries (Janson et al. 2012) in the field have been carried out. Since they represent the most complete samples to date for sun-like stars and M dwarfs, respectively, we applied the same statistical analysis as was presented in RM11 to follow up this preliminary work. In the first study (Raghavan et al. 2010), roughly 200 binaries with primary masses between 0.5-3 M /circledot were considered to determine the CMRD over a wide range of separations (10 -1 - 10 5 AU) and mass ratios (0.02-1). The new CMRD appears to be more peaked toward unity than previously observed and the period distribution is unimodal and roughly log-normal with a peak at around 50 AU. Following the methodology described in RM11, we used a KS test to compare the newly observed CMRDwith the CMRD by Metchev & Hillenbrand (2009), over the common range of mass ratios (0.02-1) and separations (281590 AU). The KS test returns a probability of ∼ 30%, therefore we cannot reject the hypothesis that the data were drawn from the same parent population. However, when we compare the two samples over the common range of mass ratios, irrespective of separation, the probability is only ∼ 1%, pointing toward a change of the CMRD with orbital radius, because that of Raghavan et al. (2010) covers a wider range than that of Metchev & Hillenbrand (2009). We therefore tested the possibility of a variation of the CMRD with angular separation. To do this we considered break points in the angular separation distribution between 10 -1 and 10 5 AU and used a KS test for each of them to determine the probability that the CMRD inside the break point is consistent with the CMRD outside. Because we found probabilities greater than 1% for any possible choice of break point, we conclude that we have no strong evidence for a dependence of the CMRD on angular separation. Moreover, because we do not expect to see random pairing from cluster dissolution models inside 10 4 AU (Kouwenhoven et al. 2010) and these widest binaries are relatively rare, we need larger samples in the future to test these models. The second study (Janson et al. 2012) consists of 85 systems with primary masses between 0.15-0.5 M /circledot , separations in the range 3-227 AU and mass ratios between 0.1 and 1. For M dwarfs, the CMRD appears to be flat and the period distribution is narrower and peaks at lower values than for solar type primaries, indicating a continuous transition from higherto lower-mass stars (Burgasser et al. 2007). In this case as well, we tested the newly observed CMRD with the CMRD from Fischer & Marcy (1992) over the common range of mass ratios and separations. With a probability of ∼ 56%the KS test does not allow us to reject the hypothesis that the newer data were drawn from the same parent sample. Finally, we used the same procedure as we adopted for sun-like stars, but in the range 3-227 AU, to explore the dependence of the observed CMRD on separation. We saw no evidence of this dependence either for this sample. Moreover, we compared the CMRD for solar-type primaries from Raghavan et al. (2010) with the new sample of M-dwarf primary binaries (Janson et al. 2012). The KS test returned a probability of 30% that the two distributions are consistent with each other (Figure 1). Motivated by this result and because the CMRD is independent of angular separation, we combined the two CMRDs over the common range of mass ratios. We again used a maximum-likelihood method to fit the distribution and found a power law dN / dq ∝ q β , with β = 0.25 ± 0.29. While this slope β is formally consistent with the one derived in Section 2 (within the errors), the change in sign is significant. It is also worth mentioning that this fit is consistent with the mass-ratio distribution with power-law exponent β = -0.10 ± 0.58 presented in a recent study of O-type spectroscopic binaries (Sana et al. 2012), whereas the observed CMRD for brown dwarfs ( β ∼ 1.5) points toward a di ff erent formation mechanisms for these objects (Goodwin 2013).", "pages": [ 1, 2 ] }, { "title": "4. Summary", "content": "In this research note we have presented some updates to the study of RM11. First, we adopted a maximum-likelihood es- timation method to re-derive the field CMRD power law, based on the combination of samples (Fischer & Marcy 1992; Metchev & Hillenbrand 2009; Kouwenhoven et al. 2005) described in RM11, to show how the dependence on the bin size can bias the result. Secondly, we analyzed recent binarity studies from the field (Raghavan et al. 2010; Janson et al. 2012) adopting the same methodology as in RM11. The new results from Raghavan et al. (2010) appear to be consistent with Metchev & Hillenbrand (2009) over the common range of mass ratios and angular separations. The recent updates on the M-dwarf CMRD (Janson et al. 2012) are also consistent with past results. The KS test does not allow us to reject the hypothesis that the CMRDs from Raghavan et al. (2010) and Janson et al. (2012) are drawn from the same parent sample. In both studies we uncovered no evidence for a dependence of the CMRD on separation. Therefore we combined the two distributions and obtained a new maximum-likelihood fit to the field CMRD dN / dq ∝ q β , with β = 0.25 ± 0.29. Since the CMRD appears to be independent of separation and dynamical evolution (see also Parker & Reggiani 2013), it represents a measurable parameter of binary stars to focus on when investigating binary formation mechanisms. However, we need larger samples to look for subtle variations of the CMRD with separation. In the future we aim to study the CMRD in other star-forming regions (e.g. the ONC) and test its dependence on separation for wide systems. Acknowledgements. We thank the referee, Simon Goodwin, for his review. We are also grateful to Carolina Bergfors, Stanimir Metchev, Deepak Raghavan, Eric Feigelson, and Richard Parker for the insightful discussions and sharing their data electronically.", "pages": [ 2 ] }, { "title": "References", "content": "Bate, M. R. 2000, MNRAS, 314, 33 Burgasser, A. J., Reid, I. N., Siegler, N., et al. 2007, Protostars and Planets V, 427 Duchˆene, G., & Kraus, A. 2013, arXiv:1303.3028 Feigelson, E. D. & Babu, G. J. 2011, Modern Statistical Methods for Astronomy with R Applications, Cambridge Univ. Press Fischer, D. A., & Marcy, G. W. 1992, ApJ, 396, 178 Goodwin, S. P. 2013, MNRAS, 430, L6 Goodwin, S. P., Kroupa, P., Goodman, A., & Burkert, A. 2007, Protostars and Planets V, 133 Kouwenhoven, M. B. N., Brown, A. G. A., Zinnecker, H., Kaper, L., & Portegies Zwart, S. F. 2005, A&A, 430, 137 Kouwenhoven, M. B. N., Goodwin, S. P., Parker, R. J., et al. 2010, MNRAS, 404, 1835 Kroupa, P., Bouvier, J., Duchˆene, G., & Moraux, E. 2003, MNRAS, 346, 354 Janson, M., Hormuth, F., Bergfors, C., et al. 2012, ApJ, 754, 44 McDonald, J. M., & Clarke, C. J. 1993, MNRAS, 262, 800 Metchev, S. A., & Hillenbrand, L. A. 2009, ApJS, 181, 62 Moeckel, N., & Bate, M. R. 2010, MNRAS, 404, 721 Moeckel, N., & Clarke, C. J. 2011, MNRAS, 415, 1179 Parker, R. J., & Reggiani, M. M. 2013, arXiv:1304.3123 Patience, J., Ghez, A. M., Reid, I. N., & Matthews, K. 2002, AJ, 123, 1570 Raghavan, D., McAlister, H. A., Henry, T. J., et al. 2010, ApJS, 190, 1 Reggiani, M. M., & Meyer, M. R. 2011, ApJ, 738, 60 Sana, H., de Mink, S. E., de Koter, A., et al. 2012, Science, 337, 444", "pages": [ 2, 3 ] } ]

[ { "title": "ABSTRACT", "content": "Recovering the Cosmic Microwave Background (CMB) from WMAP data requires galactic foreground emissions to be accurately separated out. Most component separation techniques rely on second order statistics such as Internal Linear Combination (ILC) techniques. In this paper, we present a new WMAP 9-year CMB map, with 15 arcmin resolution, which is reconstructed using a recently introduced sparse component separation technique, coined Local Generalized Morphological Component Analysis (LGMCA). LGMCA emphasizes on the sparsity of the components to be retrieved in the wavelet domain. We show that although derived from a radically different separation criterion ( i.e. sparsity), the LGMCA-WMAP 9 map and its power spectrum are fully consistent with their more recent estimates from WMAP 9. Key words. Cosmology : Cosmic Microwave Background, Methods : Data Analysis, Methods : Statistical", "pages": [ 1 ] }, { "title": "WMAP 9-year CMB estimation using sparsity", "content": "J. Bobin glyph[star] , F. Sureau, P. Paykari, A. Rassat, S. Basak, and J. -L. Starck Laboratoire AIM, UMR CEA-CNRS-Paris 7, Irfu, SAp/SEDI, Service d'Astrophysique, CEA Saclay, F-91191 GIFSUR-YVETTE CEDEX, France. Received -; accepted -", "pages": [ 1 ] }, { "title": "1. Introduction", "content": "The Cosmic Microwave Background (CMB) is a snapshot of the state of the Universe at the time of recombination. It provides information about the primordial Universe and its evolution to the current state. Our current understanding of our Universe is heavily based on measurements of the CMB radiation. The statistical properties of CMB fluctuations depend on the primordial perturbations from which they arose, as well as on the subsequent evolution of the Universe as a whole. For cosmological models in which initial perturbations are of a Gaussian nature, the information carried by CMB anisotropies can completely be characterized by their angular power spectrum which depends on the cosmological parameters. This makes the precise measurement of the CMB power spectrum a gold mine for understanding and describing the Universe throughout its history. In the estimation of the CMB map, the astrophysical foreground emissions from our galaxy and the extragalactic sources have to be removed. In addition, the instrumental noise hinders the estimation of the CMB map. In the low frequency regime (below 100GHz, i.e. for WMAP channels) the strongest contamination comes from the galactic synchrotron and free-free emission, with the highest contribution at large angular scales. At higher frequencies, dust emissions dominate whereas the synchrotron and free-free emissions are low. The spinning dust is an extra emission which spatially correlates with dust and dominates at low frequencies. Since second-order statistics provide sufficient statistics for a gaussian CMB field, most of the component separation techniques, such as the Internal Linear Combination (ILC), are built upon them to recover the CMB map from the observed sky maps. However, these techniques are not optimal for non-stationary and non-Gaussian components such as the foregrounds (or even non-stationary noise). On the contrary sparsity-based source separation techniques that focus on the higher-order statistics of the components have proven to be highly efficient (Bobin et al. 2007; Bobin et al. 2013). In this paper, we present a new WMAP 9-year CMB estimation based on this sparsity concept, and we compare the results to the official WMAP products. Next section 2 briefly describes the LGMCA method. We then describe in section 3 the processing of WMAP data and the derived LGMCA products are displayed in section 4. Results on WMAP simulations are also presented in Appendix C.", "pages": [ 1 ] }, { "title": "2. Component Separation for CMB maps", "content": "Exploiting the fact that foreground components are sparse in the wavelet domain (i.e. a few wavelet coefficients are enough to represent most of the energy of the component), LGMCA(Bobin et al. 2013) estimates both the components of interest and the mixing matrix by maximizing the level sparsity of each component; it seeks the sparsest sources possible in a wavelet basis. The assumption is that the observed sky is a linear combination of all the components, each resulting from a completely different physical process, and the instrumental noise. The separation principle in this method relies on the different spatial morphologies or structures of the various foregrounds, which translate into different sparsity patterns when transformed to a fixed wavelet dictionary. A linear combination of these components decreases the level of sparsity. Therefore, reconstructing each component from the observed map, by maximizing its sparsity in wavelet space, is an efficient strategy to distinguish between physically different sources. As mentioned in Appendix A, channels resolution variation and spatial variations of component emissions such as dust are taken into account by estimating a mixing matrix per wavelet scale and per area. Full details are given in Bobin et al. (2013) In Appendix C, LGMCA is evaluated on simulated WMAP data and it is found that: Altogether, these results incentivize applying LGMCA to WMAP9 data.", "pages": [ 1, 2 ] }, { "title": "3. Map and Power Spectrum Estimation", "content": "The WMAP satellite has observed the sky in five frequency bands denoted K, Ka, Q, V and W centered on 23 , 33 , 41 , 61 and 94 GHz respectively. The released data includes sky maps obtained with ten differencing assemblies, for nine individual years; per year, there is one map for the K band and one for the Ka band, two for the Q band, two for the V band and four for the W band. These maps are sampled using the HEALPix pixelization scheme at a resolution corresponding to nside of 1024 . At first, we average all the differencing-assembly maps obtained for the same frequency band, which yields five band-averaged maps. However, these maps are not offset-corrected. In order to determine the offset value for a particular frequency band, we use the standard resolution of the 9 -year bandaverage maps of WMAP as reference maps. Offset values are obtained by determining the mean of the difference between the band-averaged map being considered, and the reference map. At the WMAP frequencies, the major sources of contamination in the maps are the synchrotron, free-free, spinning dust and thermal dust. To model the foreground contamination, we use two foreground templates in our analysis: dust at 100 microns, as obtained by (Schlegel et al. 1998) and the composite all-sky H-alpha map of (Finkbeiner 2003). Among these templates, the thermal dust template is the most important one, as it helps in removing the dust emission on small scales, which is otherwise significant in the W channel. As the spinning dust is spatially correlated with thermal dust, the thermal dust template also helps in reducing the spinning dust residuals. It is usually standard to use the 408 MHz synchrotron map of (Haslam et al. 1981) as a template for the synchrotron emission. However, this map has quite a low resolution of about 1 degree. Furthermore, adding this template to LGMCA did not show any improvement for the component separation. LGMCA map: following (Bobin et al. 2013), the LGMCAis applied to the five offset-corrected WMAP maps and the two templates used as extra observations. Details on LGMCA parameters are given in Appendix B. LGMCA power spectrum: using the mixing matrices previously estimated from all data, we can estimate a CMB map from each of the nine individual year data set. From these nine maps, we have derived all 36 possible cross-spectra using the high resolution temperature analysis mask kq85 with ( f sky = 0 . 75) 1 provided by WMAP collaboration. The final CMB power spectrum is then obtained by averaging them and by using a MASTER mask deconvolution (Hivon et al. 2002). In contrast to the CMB and foreground signals, noise is uncorrelated between different years of data and it is therefore considerably reduced in the averaged cross spectrum.", "pages": [ 2 ] }, { "title": "LGMCA Map and Power Spectrum", "content": "Figure 1 shows the official ILC WMAP 9 years CMB map, which has a 1 degree resolution, and Figure 2 shows the LGMCA WMAP CMB map at 15 arcmin resolution and at 1 degree. The 15 arcmin LGMCA map exhibits slight high-frequency structures which are likely related to point sources emission on the galactic center. The two 1 degree maps look very clean, even on the galactic center. Figure 3 features the difference at resolution 1 degree between the CMBmapestimated by LGMCA and the official ILC-based CMB map provided by the WMAP consortium. The difference map shows significant foreground residuals on the galactic center, that are however roughly three times lower than the CMB level. When masking the galactic center with the kq85 WMAP mask (Fsky= 75% ), no significant feature can be seen anymore. The estimated power spectrum is displayed in Figure 4. The errors bars, essentially the cosmic variance and the noise-related variance, are derived from classical power spectrum variance estimators as described in (Greason et al. 2013). Note that except for glyph[lscript] < 32 , the official WMAP 9 years power spectrum (in blue in Fig. 4) is computed from the V and W bands only. In opposite, the spectrum in red in Fig. 4 is derived from the full dataset. Even if the estimation procedure differs slightly, it is remarkable that the two power spectra look very similar. They also tend to depart from each other with a slightly lower third multipole. As well the measurement about the third peak seem to be higher to some extent. However both spectra are compatible at all scales within 2 σ error bars. Sanity check: The LGMCA algorithm computes mixture parameters ( i.e. mixing matrices and their inverse) which are then applied to the data to estimate the CMB map. It is important to note that the final CMB map linearly depends on the input data. This makes it possible to check whether the inversion process may induce some bias at the level of the estimated CMB map and its power spectrum (assuming no calibration or beam errors). For that purpose, we applied exactly the same parameters we computed from the real data to 100 random CMB realizations. These realizations were generated as Gaussian random processes with a power spectrum defined by the WMAP 9-year best-fit theoretical power spectrum. The point spread function of these simulations are chosen as the beams of the 9-year bandaverage maps of WMAP thus providing pure CMB simulations that mimic the CMB part of the WMAP 9 years data. Figure 5 shows in blue the ratio between the estimated and theoretical power spectrum C TT glyph[lscript] /C TT,th glyph[lscript] computed from an average of 100 random CMB simulations. The theoretical power spectrum C TT,th glyph[lscript] of the simulated CMB maps is exactly defined as the WMAP 9-year best- fit theoretical power spectrum. The error bars are defined as the cosmic variance from 100 simulations. If the estimation by LGMCA of the CMB map and more specifically its power spectrum were biased, the C TT glyph[lscript] /C TT,th glyph[lscript] should depart from 1 . Figure 5 shows that there is no statistical evidence of discrepancy from 1 . We therefore conclude that the LGMCA does not introduce any bias at the level of the CMB power spectrum. Any bias of the estimated CMB map will come from remaining noise and foreground contamination. Higher Order Statistics - Non-Gaussianities: Higher order statistics were also computed for the full 9 year CMB maps recovered at 1 degree by ILC and LGMCA to assess potential differences in their distribution. The 75% mask was employed on both maps to avoid computing the higher order statistics in regions contaminated by foreground residuals. Sparse inpainting was then performed to interpolate the signal inside the mask (Starck et al. 2013) in order to avoid artifacts on the wavelet coefficients, and the skewness and kurtosis were calculated for each wavelet scale considering only wavelet coefficients within the mask. These statistics were centered and normalized by processing similarly a set of 100 realizations of noise and CMB according to the Λ -CDM fit provided by the WMAP consortium. Figures 6 shows the skewness and kurtosis versus the wavelet scale, and illustrates that not only both ILC and LGMCA maps are compatible with no non-gaussianities, but also that no significant difference between ILC and LGMCA can be found with these statistics at that resolution.", "pages": [ 2, 3, 4 ] }, { "title": "5. Conclusion", "content": "We have investigated how sparsity could be used for WMAP CMB map reconstruction. Based on WMAP simulations, we have shown that LGMCA provides a low foreground map, and that noise remains the major source of contamination. Then a high resolution ( 15 arcmin) clean CMB map has been computed from the full WMAP 9 years dataset, and its power spectrum was estimated. Remarkably, though LGMCA-based and official WMAP 9 years power spectrum were derived from completely different estimation procedures, they are in very good agreement and compatible within 2 σ error bars. Lastly, non- aussianity tests based on higher order statistics were carried out, and do not show statistically significant departure from gaussianity at resolution 1 degree. The LGMCA and the official WMAP 9 maps essentially differ close to the galactic center where it remains extremely difficult to assess which map is less contaminated by foreground residuals or biases due to chance correlations in between CMB and foregrounds. The LGMCA code is available at http://www.cosmostat.org/lgmca and the LGMCA CMB map as well as the estimated power spectrum are available at http://www.cosmostat.org/product .", "pages": [ 4 ] }, { "title": "Acknowledgement", "content": "This work was supported by the European Research Council grant SparseAstro (ERC-228261).", "pages": [ 4 ] }, { "title": "References", "content": "Bobin, J., Starck, J.-L., Fadili, J., & Moudden, Y. 2007, Image Processing, IEEE Transactions on, 16, 2662 Bobin, J., Starck, J.-L., Sureau, F., & Basak, S. 2013, A&A, 550 Delabrouille, J., Betoule, M., Melin, J.-B., et al. 2012, ArXiv 1207.3675 Finkbeiner, D., Davis, M., & Schlegel, D. 1999, ApJ, 524 Finkbeiner, D. P. 2003, ApJS, 146, 407 Greason, M. R., Limon, M., Wollack, E., et al. 2013, Wilkinson Microwave Anisotropy Probe (WMAP): Nine-Year Explanatory Supplement, Explanatory supplement, The WMAP Science Working Group Haslam, C., Salter, C., Stoffel, H., & W.E., W. 1982, Astronomy and Astrophysics Supplement Series, 47, 1 Haslam, C. G. T., Klein, U., Salter, C. J., et al. 1981, A&A, 100, 209 Hivon, E., Górski, K. M., Netterfield, C. B., et al. 2002, ApJ, 567, 2 Schlegel, D. J., Finkbeiner, D. P., & Davis, M. 1998, ApJ, 500, 525 Starck, J. L., Donoho, D. L., Fadili, M. J., & Rassat, A. 2013, ArXiv 1302.2758, in press Starck, J.-L., Moudden, Y., Abrial P., & Nguyen, M. . 2006, A&A, 446, 1191", "pages": [ 4 ] }, { "title": "The GMCA Framework", "content": "The GMCA (Generalized Morphological Component Analysis) method is based on blind source separation (BSS) (Bobin et al. 2013). In the framework of BSS, each of the five WMAP frequency channels are modeled as a linear combination of n components: where s j stands for the j -th component, a ij is a scalar that models for the contribution of the j -th component to channel i and n i models the instrumental noise. This problem is more conveniently recast into the matrix formulation : In practice, the number of components is set to n = 5 which allows for more degrees of freedom to get a clean CMB map while keeping A invertible. Contrary to standard approaches in astrophysics (see (Bobin et al. 2013) and references therein), the GMCA relies on the sparsity of the components S in the wavelet domain. Taking the data to the wavelet representation only alters the statistical distribution of the data coefficients without affecting its information content. A wavelet transform tends to grab the informative coherence between pixels while averaging the noise contributions, thus enhancing the structure in the data. This allows to better distinguish components that do not share the same sparse distribution in the wavelet domain. In addition, sparsity has the ability to be more sensitive to non-Gaussian processes, which has been shown to improve the foreground separation method. Having A as the mixing matrix and Φ as a wavelet transform, we assume that each source s j can be sparsely represented in Φ ; s j = α j Φ . The multichannel noiseless data Y can be written as where α is a N s × T matrix whose rows are α j . This means the sparsity of the sources in Φ translates into sparsity of the multichannel data Y . The GMCA algorithm seeks an unmixing scheme through the estimation of A , which leads to the sparsest sources S . This is expressed by the following optimization problem (written in the augmented Lagrangian form) where typically p = 0 (or its relaxed convex version with p = 1 ) and ‖ X ‖ F = sqrt ( trace ( X T X ) ) is the Frobenius norm.", "pages": [ 5 ] }, { "title": "Local GMCA", "content": "The Local-GMCA (LGMCA) algorithm (Bobin et al. 2013) has been introduced as an extension of GMCA: resolution that ranges from 13 . 2 arcmin for the W band to 52 . 8 arcmin for the K band. The makes the linear mixture model underlying the GMCA algorithm not valid. It is customary to alleviate this issue by degrading the frequency channels down to a common resolution prior to applying any component separation technique (the official CMB map provided by the WMAP consortium has a resolution of 1 degree). For this purpose, the data are decomposed in the wavelet domain and at each wavelet scales we only use the observations with invertible beams and then degrade the maps to a common resolution. This allows to estimate a CMB map with a resolution of 15 arcmin. The LGMCA algorithm has been implemented and evaluated on simulated Planck data in (Bobin et al. 2013).", "pages": [ 5 ] }, { "title": "Appendix B: LGMCA parameters for WMAP data", "content": "As described in Appendix A, LGMCA mixing matrices are estimated from a set of input channels at a given resolution, in a patch of data at a given wavelet scale. For WMAP data, the parameters used by LGMCA to compute these matrices are described in Table B.1. Figure B.1 displays the filters in spherical harmonics defining the wavelet bands where the derived weights (by inverting these mixing matrices) were applied to.", "pages": [ 5 ] }, { "title": "Appendix C: Simulations", "content": "In this section, the LGMCA algorithm is applied to data simulated by the Planck Sky Model (PSM) developed by J. Delabrouille and collaborators 2 in (Delabrouille et al. 2012). The PSM models the astrophysical foregrounds in the range of frequencies probed by WMAP, the simulated instrumental noise and the beams. In detail, the simulations were obtained as follows. A simulated WMAP dataset is produced for each of the 9 years. This allows to process the simulated data in the exact similar manner as the WMAP data are processed.", "pages": [ 6 ] }, { "title": "Component Separation", "content": "The same templates and parameters as listed in Table B.1 were used for LGMCA. We also implemented an ILC as for the WMAP9 release: first computing the weights in the same regions as for the WMAP9 release, then smoothing them to 1.5 degree and finally applying them to the data at 1 degree in the same regions as defined in the official WMAP9 product. Note that no post-processing was performed to subtract the ILC bias due to foreground propagation as in the official product. This allows to compare on the simulations the relative performance of LGMCA and a localized ILC in pixel space at a resolution of 1 degree.", "pages": [ 6 ] }, { "title": "Recovered Maps and Power Spectra", "content": "The power-spectrum is computed following the procedure described in Section 3. Figure C.1 displays the theoretical power-spectrum in black and the LGMCA estimated power-spectrum in red. The pseudo-spectrum of the input map is shown in blue; these points would correspond to a perfect estimation of the CMB map where only cosmic variance is a source of uncertainty. The larger 1 σ red errors originate from the error coming from the remaining instrumental noise. In this experiment, 75% of the sky coverage is used; the mask we used is a combination of point sources and galactic masks. These two plots show that the power-spectrum of the CMB map estimated after component separation does not show any statistically significant bias. The use of simulations allow for a precise decomposition of the CMB estimation error into its different components : CMB, remaining instrumental noise and foregrounds. For that purpose we apply the inversion parameters estimated with LGMCA independently to the simulated foregrounds and the instrumental noise. The resulting maps give the exact level of contamination of the CMB estimated by LGMCA. Figure C.2 shows the power spectra of the CMB as well as the residual noise and foregrounds that contaminate the estimated map. The resolution of the map estimated with LGMCA is 15 arcmin; therefore the different spectra in Figure C.2 remains at the same resolution and are not deconvolved to infinite resolution. Again, exactly the same sky coverage of 75% is used in this experiment thus quantifying the exact level of foreground contamination of the estimated CMB power spectrum displayed in Figure C.1. The two panels of Figure C.2 first show that the main source of contamination is the remaining instrumental noise which predominates for glyph[lscript] > 600 . This translates to the large error bars of the estimated power spectrum at small scales in Figure C.1. For very lowglyph[lscript] ( glyph[lscript] < 20 ), the contribution of both the remaining noise and foregrounds is less than 1% which is way below the error related to the cosmic vari- ance. In this experiment, the level of foreground contamination seem to be below 1% at all scales. This very low level has to be tempered : the ancillary data, namely the composite all-sky H-alpha map of (Finkbeiner 2003) and the Finkbeiner thermal dust template (Finkbeiner et al. 1999) are also used within the PSM to produce the simulations of the free-free emission and thermal dust emission respectively. We should therefore expect the level of residual foregrounds to be higher when LGMCA is applied to the real WMAP data. Interestingly, we also applied LGMCA with exactly the same parameters except that only the WMAP maps without ancillary data were used. The contamination levels are featured in Figure C.2 in dashed line. If using templates indeed lowers the level of remaining foregrounds, their contribution is still much lower than the level of CMB. Noise is therefore the main source of contamination in the final CMB estimate whether external templates are used or not. Finally, in Figure C.3 all components were propagated using weights computed by ILC and LGMCA. This figure illustrates that LGMCA is more efficient at lowering noise levels (due to the high amplification of noise in the ILC map when the 1 degree deconvolution is performed) and foregrounds contamination (due to localization and the use of templates) compared to the computed ILC map.", "pages": [ 6, 7 ] }, { "title": "Higher Order Statistics - Non-Gaussianities", "content": "The level of non-Gaussianity in the recovered CMB map provides a sanity check to measure and localize any remaining foreground contamination in the recovered CMB map, since in this case the CMB is generated as a Gaussian random field. In this work, we have computed the nonGaussianity levels from the recovered CMB maps for LGMCA and ILC at 1 degree, and for the LGMCA map at 15 arcminutes. The 75 % mask was employed and sparse inpainting was performed to interpolate the signal inside the mask (Starck et al. 2013). The skewness and kurtosis were then computed on the simulation inside these masks on different wavelet scales using an isotropic undecimated wavelet on the sphere (Starck, J.-L. et al. 2006), with the wavelet filters in spherical harmonic space described in Figure C.4. These statistics were then centered on the expected value (computed by propagating only the simulated noise and CMB) and normalized by the standard deviation computed from a set of 100 CMB and noise realizations. These statistics were also computed at different latitude bands for each wavelet scale to assess the level of foreground contamination in the maps at various scales and positions. The comparison between LGMCA and ILC at 1 degree is displayed in Figures C.5 and C.6. For both methods, the skewness and kurtosis are compatible with the error bars due to propagated noise and cosmic variance, with a maximal detection at 2 . 5 σ close to the galactic center. The same tests were also performed for the LGMCA map at the full resolution of 15 arcminutes and are displayed in Figures C.7, C.8 and C.9. The difference observed between the LGMCA non-gaussianity levels and those computed from the simulation without foregrounds is compatible with the errors expected at that resolution.", "pages": [ 7 ] } ]

[ { "title": "ABSTRACT", "content": "The analysis of quickly damped transverse oscillations of solar coronal loops using magneto-hydrodynamic seismology allow us to infer physical parameters that are di GLYPH<14> cult to measure otherwise. Under the assumption that such damped oscillations are due to the resonant conversion of global modes into Alfv'en oscillations of the tube surface, we carry out a global seismological analysis of a large set of coronal loops. A Bayesian hierarchical method is used to obtain distributions for coronal loop physical parameters by means of a global analysis of a large number of observations. The resulting distributions summarise global information and constitute data-favoured information that can be used for the inversion of individual events. The results strongly suggest that internal Alfv'en travel times along the loop are larger than 100 s and smaller than 540 s with 95% probability. Likewise, the density contrast between the loop interior and the surrounding is larger than 2.3 and below 6.9 with 95% probability. Key words. magnetohydrodynamics (MHD) methods: statistical Sun: corona Sun: oscillations", "pages": [ 1 ] }, { "title": "Coronal loop physical parameters from the analysis of multiple observed transverse oscillations", "content": "A. Asensio Ramos and I. Arregui Preprint online version: December 2, 2021", "pages": [ 1 ] }, { "title": "1. Introduction", "content": "The discovery of quickly damped transverse oscillations of solar coronal loops was first reported by Aschwanden et al. (1999) and Nakariakov et al. (1999) using Transition Region and Coronal Explorer (TRACE) observations. The phenomenon was interpreted in terms of the standing linear magnetohydrodynamic (MHD) kink mode of a magnetic flux tube in its fundamental harmonic. The cause of the quick damping of the oscillations has been attributed to the resonant conversion of global motions into localised Alfv'en oscillations at the tube boundary, because of the transverse inhomogeneity of the medium (Goossens et al., 2002). The essence of the kink mode has been found to be of mixed nature Goossens et al. (2009) with a dominant Alfv'enic character (Goossens et al., 2012a). Recent reviews on theoretical aspects of MHD kink waves can be found in Ruderman & Erd'elyi (2009) and Goossens et al. (2011). Observational properties of transverse coronal loop oscillations are presented and discussed by Schrijver et al. (2002) and Aschwanden et al. (2002). MHD seismology (Uchida, 1970; Roberts et al., 1984) uses inversion techniques to infer di GLYPH<14> cult to measure physical parameters combining theory and observations of MHD waves. Coronal seismology applications using transverse loop oscillation have been successful in determining parameters such as the magnetic field strength (Nakariakov & Ofman, 2001), the Alfv'en speed (Arregui et al., 2007; Goossens et al., 2008, 2012b), the transversal density structuring (Verwichte et al., 2006), or the coronal density scale height (Andries et al., 2005; Verth et al., 2008; Arregui et al., 2013). In a previous paper (Arregui & Asensio Ramos, 2011), we pursued the Bayesian analysis of individual coronal loops with the aim of inferring their fundamental parameters. In the Bayesian framework the inference is given by a distribution, the so-called posterior probability distribution, that is a combination of how well the observed data are predicted by the model, the likelihood, and our state of knowledge on the unknowns before considering the data, given by the priors. In that analysis, we demonstrated that the inferred values of the Alfv'en travel time are robust. Additionally, a Bayesian analysis is able to give some information (at least put some constraints) on the transverse inhomogeneity. A result of the work was the fact that the density contrast between the coronal loop and the ambient medium is poorly constrained by the observations. The posterior distribution changes when di GLYPH<11> erent prior distributions are used for this parameter. However, if an independent measurement of the density contrast is available, we demonstrated that the three parameters can be accurately inferred from the period and damping time of the coronal loop oscillations. These previous studies have focused on the inversion of physical parameters using measured wave properties for particular events, on a one-by-one basis, thus obtaining estimates for the local properties of the plasma for each particular event. Also, in most of the studies, the wave properties that are used as input in the inversions consist of quantities that are obtained upon manipulation of parameters obtained at the primary stage of the data analysis. Some examples are the use of the period and damping of the oscillations that are obtained after a fitting of the measured time evolution of the displacement in a sequence of imaging observations, or the phase speed of propagating waves derived from time-distance diagrams for propagating waves. This work represents a substantial step forward along two lines. First, we go here to a more fundamental level than Arregui &Asensio Ramos (2011) and use the displacement curves themselves, as measured by Aschwanden et al. (2002), instead of period and damping times. The reason is that the assumption of a gaussian likelihood function is more appropriate for the displacement curves than for the derived quantities. The noise statistics in the derived quantities is very complicated to obtain, given the non-trivial manipulations that are needed to obtain them. Second, a key issue in Bayesian parameter inference is the use of prior information that accounts for our state of knowledge on the unknowns, before considering the data. This prior knowledge is usually constructed on the basis of informed guesses about, e.g., the values and ranges of variation of physical parameters, physical constraints imposed by the model, etc. In this work, we compute these priors using observational information and obtained from the global analysis of a number of observed events. This is done by performing a fully consistent analysis of a large number of observations using a hierarchical Bayesian framework. In the same way as directly measured properties of transverse loop oscillations can be summarised by performing histograms, from which quantities such as the mean, median or standard deviation can be obtained, the Bayesian hierarchical framework enable us to obtain similar information for the physical parameters that cannot be directly measured and need thus to be inferred. As a result, data-favoured distributions for the unknown parameters are obtained. They can then be used to construct priors based on our current observational information of coronal loops. The layout of the paper is as follows. Section 2 presents our inference approach based on a Bayesian hierarchical model and how this is applied to the observations. The sampling of the posterior and the marginalization is also discussed in this section. Section 3 presents the final results of the paper and we end with the conclusions in Section 4.", "pages": [ 1, 2 ] }, { "title": "2. Hierarchical modeling of coronal loops", "content": "If one has direct observational access to a given physical quantity (for instance, brightness, displacement, etc.), obtaining an estimation of the probability density for this quantity is easily achieved by just counting events in bins. If observational uncertainties can shift events from one bin to another, it is possible to use Bayesian schemes to take this and other e GLYPH<11> ects into account (e.g., the extreme deconvolution technique of Bovy et al., 2011, and references therein). However, when one is interested in a quantity that cannot be directly observed but has to be inferred from observations, the situation is not so straightforward. This is exactly the problem we have in our case, because we do not have direct access to the physical parameters of the oscillating coronal loops. We propose a Bayesian hierarchical scheme to solve this problem. It can be considered as an e GLYPH<14> cient way to estimate a probability density of an unobserved quantity, obtained from many observations of quantities that are non-linearly related to the one of interest. In summary, in this work we impose parametric shapes for the priors for all the parameters of interest and learn the value of these parameters from a large set of observed coronal loop oscillations. The ensuing final priors with their shape inferred from the data summarize all the global information we currently have for the physical properties of coronal loop oscillations. In the following, we first describe the model used to explain the observations and how it depends on the physical parameters of interest. Afterwards, we describe the hierarchical probability model used to explain the complete set of observations and we define the hierarchical priors used in this work. We also explain how to e GLYPH<14> ciently sample the high-dimensional posterior probability distribution function.", "pages": [ 2 ] }, { "title": "2.1. Coronal loops oscillation generative model", "content": "Observing oscillations in coronal loops with the aim of carrying out magneto-hydrodynamical seismology is a very di GLYPH<14> cult task. After an arduous process that requires a detailed analysis of the time evolution of images obtained in coronal lines from space missions, the time variation of the displacement, d ( t ), that describes the motion of the coronal loop apex at di GLYPH<11> erent time steps, is obtained (e.g., Aschwanden et al., 2002). In order to extract information from the time evolution of the displacement, this quantity is modeled as a combination of a systematic motion of the entire loop and a real oscillatory component. Therefore, the generative model 1 for the observations is then: where GLYPH<15> ( t ) represents the uncertainty of the amplitude measurement, while b ( t ) takes into account the presence of any remaining uncertainty produced by non-modeled e GLYPH<11> ects like background loops, wrong estimation of the noise variance, etc. Concerning the standard uncertainty, we assume it has Gaussian statistics, with zero mean and time-independent variance GLYPH<27> 2 n . Such a simplification means that measurements at di GLYPH<11> erent times are completely uncorrelated. Additionally, we use an estimation of GLYPH<27> n obtained directly from the observations. The background component is assumed, for simplicity, to be also Gaussian with zero mean and time-independent variance GLYPH<27> 2 b . The oscillatory component is modeled in terms of a sinusoidal with an exponential decay as follows: where A is the amplitude of the oscillatory part, P is its period, t 0 is a reference initial time that is fixed from the observations, GLYPH<30> 0 is the initial phase of the oscillation and GLYPH<28> d is the damping time scale. The detrending of the oscillatory motion is very di GLYPH<14> cult to carry out and might depend on an undetermined (potentially large) number of factors. For this reason, it is customary to use a simple polynomial function, that absorbs all the unknown e GLYPH<11> ects (Aschwanden et al., 2002): where the coe GLYPH<14> cients ai are obtained for each coronal loop and the order N is adapted to the needed complexity. For simplicity, we fix the values of the ai coe GLYPH<14> cients to those obtained by Aschwanden et al. (2002) because no physical information is extracted from them. The generative model that we have written does not allow us to extract much physical information. The period and damping time are purely observational quantities and we need to relate them with the physical conditions in the loops. To this end, we propose the resonantly damped MHD kink mode interpretation of quickly damped transverse oscillations of coronal loops (Ruderman & Roberts, 2002; Goossens et al., 2002) to explain the observed period P and damping time GLYPH<28> d . This approximation applies to a straight cylindrically symmetric magnetic flux tube with a uniform magnetic field pointing along the axis of the tube. Under the zero plasmaGLYPH<12> approximation, coronal loops can be considered to be density enhancements with a constant internal density, GLYPH<26> i , a constant external density, GLYPH<26> e < GLYPH<26> i , and a non-uniform transitional layer of thickness l that connects both regions. Following Goossens et al. (2008), it is possible to give the following analytical expression for P and GLYPH<28> d under the thin tube and thin boundary approximations: From these considerations, the parameters in which we are interested are the internal Alfv'en travel time, GLYPH<28> A , the density contrast between the tube and the environment, GLYPH<24> = GLYPH<26> i =GLYPH<26> e , and the transverse inhomogeneity length scale in units of the radius of the loop, l = R .", "pages": [ 2, 3 ] }, { "title": "2.2. Hierarchy", "content": "According to the previous model, the oscillatory displacement of the i -th coronal loop is determined by the set of parameters GLYPH<18> i = f GLYPH<28> A ; GLYPH<24> ; l = R ; A ; GLYPH<30> 0 ; GLYPH<27> b g , where we use the vector GLYPH<18> i to compact the notation. The Bayesian analysis performed by Arregui & Asensio Ramos (2011) demonstrated that the constraining power of the observations is very limited. Although GLYPH<28> A can be successfully estimated from the observations (although with relatively large and asymmetric error bars), the situation is much worse for the density contrast and the length scale, with the density contrast being the poorest constrained. Arregui & Asensio Ramos (2011) have shown that the marginal posterior distribution for GLYPH<24> is very close to the assumed prior distribution, meaning that there is almost no information in the observations to constrain GLYPH<24> . The reason why, even in the absence of information for the density contrast, the Alfv'en travel time can be correctly recovered has to be found on the specific shapes of the curves in the threedimensional space ( GLYPH<28> A ; GLYPH<24> ; l = R ) pertaining to constant values of P and GLYPH<28> d , as explained in Arregui et al. (2007). Consider GLYPH<2> = f GLYPH<18> 1 ; GLYPH<18> 2 ; : : : ; GLYPH<18> n g to be a vector of length 6 N that contains all the model parameters for all the observed N loops. In a standard Bayesian approach, the posterior distribution (which encodes the updated information about the model parameters) is given by: where D = f D 1 ; D 2 ; : : : ; Dn g refers to the observed data, the measured time variation of the displacement, d obs ( t ), for all the loops. The function p ( D j GLYPH<2> ) is the likelihood, that measures the probability of getting a set of observed displacements for a given combination of the parameters. Viewed as a function of the parameters GLYPH<2> , the likelihood measures the quality of the parametric model to explain the observations. Finally, the function p ( GLYPH<2> ) is the prior distribution that encodes a-priori information about the model parameters, while p ( D ) is the evidence. Given that p ( D ) does not depend on the model parameters, it is just a multiplicative constant and can be dropped from the calculations. The quantities with physical interest in our problem are GLYPH<28> A , GLYPH<24> and l = R . They are obviously directly unobservable. For this reason, one cannot use the standard histogram to a set of observed coronal loops with the aim of obtaining their general physical properties. It is widely known that Bayesian hierarchical models constitute a very powerful way to overcome this di GLYPH<14> culty (REF). The idea behind hierarchical models is extremely simple. The priors p ( GLYPH<2> ) used in Eq. (5) are made dependent on a set of hyperparameters GLYPH<10> , which are then included in the inference scheme. Formally, the posterior is given by: where we have used the general fact that p ( GLYPH<2> ; GLYPH<10> ) = p ( GLYPH<2> j GLYPH<10> ) p ( GLYPH<10> ). Note that we have dropped the dependence of the likelihood on GLYPH<10> , given that GLYPH<10> are just hyperparameters or, in other words, parameters of the priors. If we make the assumption that there is not any correlation between any two coronal loops from the set of N observations, we can largely simplify Eq. (6). In such a case, the likelihood and the priors can be factorized, so that the posterior distribution simplifies to read: where we have made use of standard probability calculus. Since the global properties of the physical properties are governed by the priors, our aim is to compute the statistical properties of their parameters, GLYPH<10> . Consequently, and although it might seem counterintuitive, all the individual physical parameters GLYPH<2> are nuisance parameters for us and have to be integrated out from the posterior (e.g., Gregory, 2005): where we have made used of the fact that the parameters of one loop do not a GLYPH<11> ect those of another loop. It is this integration operation the one that propagates information from all individual loops simultaneously to the hyperparameters.", "pages": [ 3 ] }, { "title": "2.3. Likelihood", "content": "According to the characteristics of the noise and background components of the generative model displayed in Eq. (1), the likelihood function is a Gaussian. Given that both GLYPH<15> ( t ) and b ( t ) follow the same Gaussian statistics with zero mean although with di GLYPH<11> erent (time-independent) variances, the total likelihood for an individual coronal loop is given by: where mi is the number of time steps measured for the i -th loop and Of importance is to have a good estimation of GLYPH<27> n , the variance of the noise. According to Aschwanden et al. (2002), the process of obtaining the time evolution of the displacement for a given coronal loop is indeed quite complicated. For this reason, we take a conservative approach and use GLYPH<27> n equal to 10% of the maximum absolute displacement in each coronal loop. Our results demonstrate that this number is indeed a lower limit to the actual uncertainty.", "pages": [ 3, 4 ] }, { "title": "2.4. Priors", "content": "In the hierarchical Bayesian scheme, as important as the definition of the likelihood is the definition of suitable priors. As described in the introduction, the idea is that, since the hyperparameters of the priors are learnt from all the data simultaneously , the resulting prior distributions will be then adapted to the data. As a consequence, the prior distributions defined hierarchically are generalizations of the standard calculation of a histogram for quantities that cannot be directly observed, like l = R , GLYPH<28> A and GLYPH<24> . To this end, it is favorable to use general probability distributions that naturally fulfill the boundaries for all the parameters. The first step is to consider the range of variation of the model parameters. After Goossens et al. (2008), we know that the model parameters have to fulfill Additionally, GLYPH<28> A and GLYPH<24> have upper boundaries that do not emerge from the theory but can be estimated based on physical arguments. We use GLYPH<28> max A = 1500 s and GLYPH<24> max = 100, although their precise values are of reduced impact in the final result provided that they are large enough. The graphical model describing the hierarchy that we consider in the analysis of coronal loop oscillations is shown in Fig. 1. The selected priors, which depend on the set of hyperparameters GLYPH<10> = f GLYPH<11>; GLYPH<12>; GLYPH<13>; GLYPH<14>; GLYPH<15> ; GLYPH<17> g , are summarized in Tab. 1. We give more details in the following.", "pages": [ 4 ] }, { "title": "2.4.1. Prior for l = R", "content": "The theory says that the transverse inhomogeneity length scale has to lie in the interval [0 ; 2], so it is advisable to use a prior that automatically fulfills this restriction. We have used a truncated normal distribution, that depends on two parameters, GLYPH<11> and GLYPH<12> , and is given by: Another option that gives very similar results and also depends on two parameters is the scaled Beta prior, defined as where B ( GLYPH<11>; GLYPH<12> ) is the beta function (e.g., Abramowitz & Stegun, 1972), which can be computed in terms of the gamma function as:", "pages": [ 4 ] }, { "title": "2.4.2. Prior for GLYPH<28> A", "content": "The Alfv'en travel time is defined in the interval [0 ; 1 ). A quite general distribution that is naturally defined in this interval is the inverse gamma distribution, which depends on two parameters, GLYPH<13> and GLYPH<14> : This distribution has the advantage of describing variables with skewness with only two parameters. The selection of the inverse gamma distribution is somehow arbitrary and other distributions like the gamma distribution can be chosen. We have verified with a few of them that the results are very robust to the precise selection of the functional form, provided they have su GLYPH<14> cient generality. A few examples of the shape of this prior are shown in Fig. 2.", "pages": [ 4 ] }, { "title": "2.4.3. Prior for GLYPH<24>", "content": "The density contrast is a parameter defined in the interval [1 ; 1 ) and scarce information is available as to what the upper limit can be. For this reason, we choose a shifted inverse gamma distribution, defined as", "pages": [ 5 ] }, { "title": "2.4.4. Prior for GLYPH<27> b", "content": "The standard deviation of the background contribution, GLYPH<27> b , is inferred from the data. Given that it is a scale parameter, it is customary to use a Je GLYPH<11> reys' prior. Given that GLYPH<27> b is defined in the interval [0 ; 1 ) and the Je GLYPH<11> reys' prior is not proper and not well defined at zero, we propose a modified Je GLYPH<11> reys' prior (Gregory, 2005): This prior behaves as a Je GLYPH<11> reys' prior (i.e., as GLYPH<27> GLYPH<0> 1 b ) for GLYPH<27> b GLYPH<29> GLYPH<27> 0 b and as a uniform prior for GLYPH<27> b GLYPH<28> GLYPH<27> 0 b . Consequently, the transition parameter GLYPH<27> 0 b is a lower boundary of the Je GLYPH<11> reys' prior. We choose the small value GLYPH<27> 0 b = 0 : 1. We made sure that this value is su GLYPH<14> ciently small so that the posterior for this parameter peaks at larger values and is therefore not influenced by its actual value. Concerning GLYPH<27> max b , it is made to be very large and its influence on the final results is negligible.", "pages": [ 5 ] }, { "title": "2.4.5. Prior for GLYPH<30> 0 and A", "content": "Without any additional a-priori information, we choose a flat prior for the phase of the oscillation in the interval [ GLYPH<0> GLYPH<25>; GLYPH<25> ]. This uniform prior equals (2 GLYPH<25> ) GLYPH<0> 1 if GLYPH<0> GLYPH<25> GLYPH<20> GLYPH<30> 0 GLYPH<20> GLYPH<25> and zero elsewhere. The amplitude of the oscillation is a scale parameter that is defined in the interval [0 ; 1 ). For this reason, we choose a modified Je GLYPH<11> reys' prior: with A 0 = 10 GLYPH<0> 3 (much smaller than the actual amplitude of the oscillation) and a very large A max .", "pages": [ 5 ] }, { "title": "2.4.6. Priors for hyperparameters", "content": "The hyperpriors for the hyperparameters GLYPH<11> , GLYPH<12> , GLYPH<13> , and GLYPH<15> are all flat in positive real line. For GLYPH<14> and GLYPH<17> , given that they can be considered to be scale parameters, we choose modified Je GLYPH<11> reys' priors with very small transition parameter. However, the final results are very robust and do not depend on the specific hyperpriors.", "pages": [ 5 ] }, { "title": "2.5. Sampling the posterior", "content": "It is clear that the integrals of Eq. (8) cannot be computed analytically. Therefore, it is necessary to rely on numerical techniques. We carry out the integral using a technique based on a Markov Chain Monte Carlo (MCMC; Metropolis et al., 1953; Neal, 1993). Instead of the general Metropolis-Hastings method, we used a Metropolis-within-Gibbs method (Tierney, 1994), which has recently been applied by Sale (2012) for mapping the extinction in the Milky Way using a hierarchical Bayesian model 2 . The reason for using this scheme is that, in principle, the sampling of the posterior distribution function for every coronal loop is independent of the others, except for the presence of the hyperparameters. Therefore, every step of the posterior sampling for each coronal loop can be done independently. After one iteration of each chain is carried out, the hyperparameters can be updated using a standard Metropolis-Hastings rule. This update is then propagated to every coronal loop. The total length of the converged Markov chains is of the order of a few hundred thousands samples. We verified that the Markov chains are converged using standard criteria. Finally, the initial 30% of the chain is discarded to minimize the sample correlation. As well, we use only one sample every three to further reduce the correlation.", "pages": [ 5 ] }, { "title": "2.6. Selection of observations", "content": "Because of the di GLYPH<14> culty of observing oscillations in coronal loops, some of the curves analyzed by Aschwanden et al. (2002) do not really display the behavior that we assume in Sec. 2.1. This poses a problem if the generative model of Eq. (1) does not include the term b ( t ) because no combination of the model parameters yields a fit to the observations whose residual is Gaussian with zero mean and variance GLYPH<27> 2 n . However, the inclusion of GLYPH<27> b into the inference solves this issue. The observed loops for which the observation is far from a damped sinusoidal will display a larger GLYPH<27> b . The total number of coronal loops observed by Aschwanden et al. (2002) is 30. The number of random variables is then 6 N + 6 = 186, the model parameters for each loop, including the standard deviation of the background contribution, plus the hyperparameters.", "pages": [ 5 ] }, { "title": "3.1. Inference about model parameters", "content": "The output of the MCMC code are samples of the model parameters which are distributed according to the joint posterior p ( GLYPH<2> ; GLYPH<10> j D ). To this, we have to add the advantage that the Markov chain for a certain parameter is distributed according to the marginal posterior distribution of this parameter, so the integrals of Eq. (8) are automatically obtained. Figure 3 displays the marginal posteriors for a sample of 5 among the 30 coronal loops that we consider in this work. The upper row shows the marginal posterior for the Alfv'en travel time, which are well constrained in all the cases. The marginal posteriors display a conspicuous peak, although the confidence intervals are clearly asymmetric. This is similar to the findings of Arregui & Asensio Ramos (2011), although in that paper we did not fit the whole time evolution. The second and third rows show the marginal posteriors for the density contrast and for the length scale. It is clear from Eq. (4) that the length scale and the density contrast are intimately related. A fixed value of GLYPH<28> d = P can be obtained with an infinite number of combinations of GLYPH<24> and l = R . Therefore, it is almost impossible to get reliable information for each parameter separately unless a strong a-priori information is available for any of the two (see Arregui et al., 2007; Arregui & Asensio Ramos, 2011). Our results show a very interesting phenomenon that is a direct consequence of the hierarchical scheme. The fact that we assume that the priors for GLYPH<24> and l = R have to be the same for all the observed coronal loops introduces a large amount of information into the inference. This results into very well defined posteriors both for the density contrast and the length scale. The strong constraint imposed by the hierarchical model induces that the density contrast is roughly the same for all loops, and the transverse inhomogeneity length scale is the one changing from loop 5000 to loop. We conclude that, under the assumption that the physical properties of all coronal loops are extracted from common probability distribution functions, the damping time scale is fundamentally determined by the transverse inhomogeneity length scale. The fourth row shows the information inferred for the standard deviation of the background component. Interestingly, GLYPH<27> b is always non-negligible, meaning that none of the observed coronal loops displays a pure damped sinusoidal oscillation. Additionally, the distribution is very well defined in all cases, so that it is possible to reliably characterize this background component. Although GLYPH<28> A , GLYPH<24> and l = R are the physical parameters behind the model, it is possible to compute the marginal posteriors for derived parameters. Using Eq. (4), we have computed the marginal posteriors for the period and the damping time, which are shown in the fifth and sixth rows of Fig. 3. An interesting property of these posteriors is that, although some of the model parameters might not be strongly constrained, P and GLYPH<28> d are very well constrained from the observations. The marginal posteriors are really close to Gaussian, which reinforces the assumption used in Arregui & Asensio Ramos (2011) of a Gaussian likelihood with diagonal covariance matrix. Finally, the lowest row of Fig. 3 displays the measured displacement for each loop and the best fit (roughly equivalent to the least-squares solution, except for the presence of the priors). The black error bars are obtained using the estimated value of GLYPH<27> n , while the red error bars are obtained by adding in quadrature GLYPH<27> n and GLYPH<27> b .", "pages": [ 5, 6, 7 ] }, { "title": "3.2. Global properties of coronal loops", "content": "The hierarchical structure of our model allows us to obtain the general properties of coronal loops. To this end, we show in Fig. 4 the inferred distributions for the hyperparameters that describe the prior distributions described in x 2.4. The first column shows the results for GLYPH<11> and GLYPH<12> , that are the parameters of the truncated normal distribution for l = R . The results indicate that these hyperparameters have very well defined values. The median values are GLYPH<11> med GLYPH<25> 0 : 85 and GLYPH<12> med GLYPH<25> 0 : 36. Likewise, the results for the hyperparameters of the prior for GLYPH<28> A are also well defined, with GLYPH<13> med GLYPH<25> 4 : 4 and GLYPH<14> med GLYPH<25> 870. The situation is less favourable for the hyperparameters of the prior for GLYPH<24> , probably a consequence of the fact that a single inverse gamma distribution is not able to capture the complexity of the global properties of GLYPH<24> over the whole sample of coronal loops. Once the hyperparameters are known, it is possible to use this information to get the global properties of the physical properties of coronal loops. The first approach is to follow what is known as the type-II maximum likelihood approximation. In this case, we simply evaluate the parametric priors defined in x 2.4 at the most probable values of their parameters, obtained from the peaks on Fig. 4. The results are shown as blue lines in Fig. 5. Another way, that fully takes into account the presence of uncertainties in the hyperparameters, is to use the Ns Monte Carlo samples of GLYPH<11> , GLYPH<12> , GLYPH<13> , GLYPH<14> , GLYPH<15> and GLYPH<17> from the posterior to evaluate the following marginalized distributions: The previous expressions are the Monte Carlo approximations to the marginalization of the hyperparameters from the hyperpriors. These distributions are shown as grey lines in Fig. 5. The distributions shown in Fig. 5, which constitute the main result of this paper, represent the underlying distribution from which the values of l = R , GLYPH<28> A and GLYPH<24> have been sampled, under the assumption that this global distribution is shared among all the coronal loops. Consequently, they are generalized histograms of these unobserved quantities, which already take into account any possible degeneracy and uncertainty during the inference process. They represent a data-favored updated prior for the parameters of the model. These priors can be used in the future when making seismological analysis of coronal loops using the resonantly damped magneto-hydrodynamic kink mode interpretation of quickly damped transverse oscillations. Concerning the transverse inhomogenity length scale, the left panel of Fig. 5 demonstrates that roughly all allowed values are possible. However, the slight shift of the distribution shows that there is a small preference for l = R < 1. Concerning the Alfv'en travel time, it is clear from the central panel of Fig. 5 that the most probable value for GLYPH<28> A is GLYPH<24> 160 s, with a median value of GLYPH<24> 212 s. The Alfv'en travel time is below GLYPH<24> 540 s are and above GLYPH<24> 100 s with 95% probability. In a surely oversimplified situation in which the typical length L and density GLYPH<26> of the coronal loop is known with precision, the Alfv'en travel time limits that we obtain might be used to put some general constraints on the magnetic field. Given that: where vA is the Alfv'en velocity and B is the magnetic field. For instance, if L GLYPH<24> 100 Mm and GLYPH<26> GLYPH<24> 10 GLYPH<0> 14 g cm GLYPH<0> 3 , we end up with 6 G . B . 35 G. If the density is an order of magnitude larger, the magnetic field range increases in a factor p 10. The most probable value of the magnetic field, corresponding to the peak of the Alfv'en travel time in Fig. 5 turns out to be B GLYPH<24> 16 G. These figures are just plain estimations based on an unrealistic situation in which the properties of the coronal loop are known. The information gained for the density contrast is also very interesting. We remind that the strong constraint for this param- eter is a direct consequence of the hierarchical scheme, which forced the same distribution for all observed coronal loops. According to our results, the density contrast is above 2.3 and below 6.9 with 95% probability, with a median value of 3.8. Finally, we display in Fig. 6 the comparison between our results and what one would obtain using a simple histogram with the inferred value of the parameters. To this end, we have used the inferred values of GLYPH<28> A and l = R that were obtained by Arregui & Asensio Ramos (2011), complemented with the results of applying the Bayesian formalism presented of Arregui & Asensio Ramos (2011) to the observations collected in Table 1 of Verwichte et al. (2013). A Je GLYPH<11> reys' prior in the range [1 : 2 ; 50] is used for the density contrast and an uncertainty of 10% is used for the period and damping time if no measurement is available. Although the results are somehow comparable, note that the error bars are not taken into account in the histogram. This is of special relevance for l = R and less important for GLYPH<28> A , where the inferred values are less uncertain.", "pages": [ 7, 8 ] }, { "title": "4. Conclusions", "content": "This paper presented the inference of the global physical properties of coronal loops obtained through MHD seismology. We have obtained the inferred distribution of the Alfv'en travel time, size of the transition layer between the surroundings and the coronal loop density enhancement. These distributions are valid under the assumption that the properties of all coronal loops are just realizations of some underlying distributions. The results demonstrate that sharp transitions between the surrounding and the internal media are slightly favored. Additionally, we have found that Alfv'en travel times are in the interval [100 ; 540] s with 95% probability. If the length and density of the coronal loop are known, this poses some constraints on the magnetic field strength in the loop. Likewise, the density contrast between the loop interior and the surrounding is in the interval [2 : 3 ; 6 : 9] with 95% probability. Our contribution improves over our previous approach. First, we make the model closer to the observation, by using a generative model to explain the measured displacements. Second, we use a method that obtains global information for model parameters that cannot be directly measured but need to be inferred. Our results allowed us to construct informative priors that can be used for inversions of individual events. The inference then takes into account prior beliefs extracted from data. Apart from the extraordinary di GLYPH<14> culty of extracting the oscillations in coronal loops, the potential to massively apply MHD seismology techniques is now larger than ever thanks to the continuous observations of the Atmospheric Imaging Assembly (AIA; Lemen et al., 2012) on board the Solar Dynamics Observatory (SDO). Its high-temporal cadence of 12 s and high spatial resolution of GLYPH<24> 0.6 arcsec make them the perfect instrument to follow these oscillatory events and extract reliable physical information from these coronal events. Acknowledgements. We are grateful to Markus J. Aschwanden for kindly providing the measurements of coronal loops oscillations used in this paper. We thank M. J. Mart'ınez Gonz'alez, R. Manso Sainz, M. J. Aschwanden and R. Oliver for useful suggestions to improve the quality of the manuscript. Financial support by the Spanish Ministry of Economy and Competitiveness through projects AYA2010-18029 (Solar Magnetism and Astrophysical Spectropolarimetry) and AYA2011-22846 is gratefully acknowledged. We also acknowledge financial support through the Ram'on y Cajal fellowships and the Consolider-Ingenio 2010 CSD2009-00038 project.", "pages": [ 8, 9 ] }, { "title": "References", "content": "Abramowitz, M. & Stegun, I. A. 1972, Handbook of Mathematical Functions (New York: Dover) A&A, 463, 333 Arregui, I. & Asensio Ramos, A. 2011, ApJ, 740, 44 Arregui, I., Asensio Ramos, A., & D'ıaz, A. J. 2013, ApJ, in press Aschwanden, M. J., Fletcher, L., Schrijver, C. J., & Alexander, D. 1999, ApJ, 520, 880 Gregory, P. C. 2005, Bayesian Logical Data Analysis for the Physical Sciences (Cambridge: Cambridge University Press) Metropolis, N., Rosenbluth, A. W., Rosenbluth, M. N., Teller, A. H., & Teller, E. 1953, J. Chem. Phys., 21, 1087 Nakariakov, V. M. & Ofman, L. 2001, A&A, 372, L53 Nakariakov, V. M., Ofman, L., Deluca, E. E., Roberts, B., & Davila, J. M. 1999, Science, 285, 862 Neal, R. M. 1993, Probabilistic Inference Using Markov Chain Monte Carlo Methods (Dept. of Statistics, University of Toronto: Technical Report No. 0506) Roberts, B., Edwin, P. M., & Benz, A. O. 1984, ApJ, 279, 857 Ruderman, M. S. & Erd'elyi, R. 2009, Space Sci. Rev., 149, 199 Ruderman, M. S. & Roberts, B. 2002, ApJ, 577, 475 Sale, S. E. 2012, MNRAS, 427, 2119 Schrijver, C. J., Aschwanden, M. J., & Title, A. M. 2002, Sol. Phys., 206, 69 Tierney, L. 1994, Annals of Statistics, 22, 2701 Uchida, Y. 1970, PASJ, 22, 341 Verth, G., Erd'elyi, R., & Jess, D. B. 2008, ApJ, 687, L45 Verwichte, E., Foullon, C., & Nakariakov, V. M. 2006, A&A, 452, 615 Verwichte, E., Van Doorsselaere, T., White, R. S., & Antolin, P. 2013, A&A, in press", "pages": [ 9 ] } ]

[ { "title": "ABSTRACT", "content": "Aims. Runaway instability operates in fluid tori around black holes. It a ff ects systems close to the critical (cusp overflowing) configuration. The runaway e ff ect depends on the radial profile l ( R ) of the angular momentum distribution of the fluid, on the dimension-less spin a of the central black hole ( | a | ≤ 1), and other factors, such as self-gravity. Previously it was demonstrated that, for the power-law dependence of the radial angular momentum profile, l ( R ) ∝ R q , non-magnetized tori always become runaway stable for a su ffi ciently high positive value of q . Here we discuss the role of runaway instability within a framework of an axially symmetric model of perfect fluid endowed with a purely toroidal magnetic field. Methods. The gradual accretion of material over the cusp transfers the mass and angular momentum onto the black hole, thereby changing the intrinsic parameters of the Kerr metric. We studied the e ff ect of the plasma parameter β (ratio of gas to magnetic pressure) and other parameters of the model on the evolution of critical configurations that are just on the verge of cusp overflow. Results. By contributing to the total pressure, the magnetic field causes small departures from the corresponding non-magnetic configuration in the early phases of accretion. However, we show that the toroidal magnetic component inside an accretion torus does not change the frequency of its oscillations significantly. We identify these oscillations as the radial epicyclic mode in our example. Nevertheless, these weak e ff ects can trigger the runaway instability even in situations when the purely hydrodynamical regime of the torus is stable. On the other hand, in most cases the stable configuration remains una ff ected, and the initial deviations gradually decay after several orbital periods. We show examples of the torus evolution depending on the initial magnetization β , the slope q , and the spin a . Conclusions. The toroidal magnetic field plays a more important role in the early phases of the accretion process until the perturbed configuration finds a new equilibrium or disappears because of the runaway instability. Key words. Accretion: accretion-discs - black hole physics - instabilities", "pages": [ 1 ] }, { "title": "Effect of the toroidal magnetic field on the runaway instability of relativistic tori (ResearchNote)", "content": "Jaroslav Hamerský 1 , 2 /star and Vladimír Karas 1 Received 18 April 2013; Accepted 16 May 2013", "pages": [ 1 ] }, { "title": "1. Introduction", "content": "Toroidal equilibria of perfect fluid in permanent rotation were introduced a long time ago as an initial step on the way towards an astrophysically realistic description of accretion of gaseous material onto a black hole in active galactic nuclei and black hole binaries (Fishbone & Moncrief 1976; Abramowicz et al. 1978; Pugliese et al. 2013). These axially symmetric and stationary solutions are subject to various types of instability (e.g., Abramowicz & Fragile 2013). Here we concentrate on a global type of instability caused by an overflow of material over the cusp of a critical equipotential surface (Daigne & Mochkovitch 1997; Abramowicz et al. 1998; Korobkin et al. 2013). It was suggested that this may lead to specific features that should be observable in the radiation emitted from an accreting black-hole system (Zanotti et al. 2003). The e ff ect of the mentioned instability can be catastrophic under certain conditions. In particular, a black-hole torus becomes runaway unstable if the angular momentum profile within the torus does not rise su ffi ciently fast with radius (Abramowicz et al. 1998; Lu et al. 2000). The role of general relativity e ff ects on the runaway mechanism was studied in Font & Daigne (2002) in the context of gamma-ray burst sources. These authors found that by allowing the mass of the black hole to grow by accretion, the disc becomes unstable. However, the parameter space of the problem is much richer than what could be taken into account in early works. For example, the self-gravity of the fluid tends to act against the stability of non-accreting tori (Goodman & Narayan 1988; Masuda et al. 1998; Montero et al. 2010; Korobkin et al. 2011). Furthermore, the spin parameter can play a role for accretion onto a rotating black hole. In astrophysically realistic models, an interplay of mutually competing e ff ects have to be taken into account. The role of magnetic fields is known to be essential for accretion. Even the Rayleigh-stable tori (Seguin 1975) with a radially increasing profile, d l / d R > 0, become dynamically unstable because of turbulence in the presence of a weak magnetic field (Balbus & Hawley 1991). Here we aim to clarify the simpler question of the global stability of a rotationally symmetric black-hole accretion tori, taking into account the e ff ect of a large-scale (organized) magnetic field that obeys the same axial symmetry. Komissarov (2006) has developed a suitable analytical (toy) model of such a magnetized torus described by a polytropic equation of state in Kerr metric. In this model the magnetic field only enters the equilibrium solution for the torus as an additional pressure-like term (Pugliese & Montani 2013). We employed this solution as an initial configuration, which we then perturbed and evolved numerically by using a twodimensional numerical scheme (HARM; see Gammie et al. 2003). A complementary approach in the context of gammaray bursts has been developed in Barkov & Baushev (2011), who adopted the same initial configuration of an axially symmetric magnetized torus, which they evolved taking self-gravity and neutrino cooling mechanisms into account. Although the basic idea behind the runaway instability has been well-known since the early papers (see Abramowicz et al. 1983; Wilson 1984) it is connected with the existence of the innermost stable circular orbit around black holes in general relativity - an interplay of di ff erent e ff ects makes the discussion rather complex, and so simplified models have their value for understanding the runaway mechanism in astrophysically realistic systems. Assuming axial symmetry is a useful simplification to explore the origin of runaway instability, although it is a far too strong constraint for any realistic model of an accreting system. Moreover, a purely toroidal structure of the magnetic field and complete negligence of radiative cooling are an oversimplification, which we adopt in this paper. However, these assumptions allow us to concentrate on the particular type of the above-mentioned relativistic instability while non-axisymmetric modes are suppressed. It also helps us to proceed systematically through the parameter space of the model to reveal the dependence on black-hole spin and the magnetic field strength as they act concurrently within the relativistic scheme. In this respect our work is complementary to three-dimensional simulations (e.g., Korobkin et al. 2013; McKinney et al. 2012), which are more complete and, at the same time, more di ffi cult to comprehend. In sec. 2 we summarize our approach to magnetized tori and the numerical scheme used in our simulations. Then we present our results by comparing properties of magnetized and non-magnetized accretion tori that are subject to a weak perturbation from the equilibrium state. In sec. 3 we discuss our results and give a brief conclusion.", "pages": [ 1, 2 ] }, { "title": "2.1. Initialconfiguration", "content": "The magnetized ideal fluid can be described by the energymomentum tensor (e.g., Anile 1989) where w is the specific enthalpy, P g is the gas pressure, and b µ is the projection of the magnetic field vector ( b 2 = b µ b µ ). From the energy-momentum tensor conservation, T µν ; ν = 0, it follows for a purely axially rotating fluid (Abramowicz et al. 1978; 2013) where ut is the covariant component of the four-velocity (subscript 'in' corresponds to the inner edge of the torus), Ω = u ϕ / u t is the angular velocity and l = -u ϕ/ ut is the angular momentum density. The specific enthalpy, w ≡ ρ + P g + U , can be expressed in terms of the internal energy density U , the rest-mass density Article number, page 2 of 9 ρ , and two contributions to the total pressure, P ( ρ ) = P g + P m, where P g is the gas (thermodynamical) part and P m = 1 2 b 2 is the magnetic part. The last term introduces the e ff ect of the magnetic field in the context of our model and the tilde denotes ˜ w ≡ L w , ˜ P m ≡ L P m, where L ( r , ϑ ; a ) ≡ g 2 t ϕ -gtt g ϕϕ is a combination of metric terms, in our case the known function of radius, latitude, and spin of the Kerr metric. By assuming a suitable equation of state and the rotation law of the fluid, eq. (2) can be integrated to obtain the structure of equipotential surfaces of the equilibrium configuration. The last term on the left-hand side of eq. (2) was assumed to be zero in the original solution (Abramowicz et al. 1978), but Komissarov (2006) noticed that its special form allows us to take a purely toroidal magnetic field readily into consideration. The two contributions are assumed to be proportional to each other. Non-vanishing components of the magnetic vector b µ are with A ( r , ϑ ) ≡ g ϕϕ + 2 lgt ϕ + l 2 gtt . To solve eq. (2) and construct the torus, one needs to constrain the angular momentum profile l ( R ). In this way, we obtain the stationary non-accreting con- tion, where all viscous e ff ects are neglected. Therefore, accretion can proceed only due to the relativistic cusp overflow. For constant angular momentum density, eq. (2) can be solved analytically as well as in the magnetic case by imposing proportionality where C is a constant that sets the mutual relation between hydrodynamic and magnetic e ff ects. Since C determines the mutual relation between the hydrodynamic and the magnetic pressure components, its value indirectly influences also the magnetization ratio β (the equipartion state is reached for β near unity, whereas β /greatermuch 1 represents a sub-equipartion magnetic field). Indeed, the thermodynamic and the magnetic pressure terms and the total pressure are directly proportional to each other. This leads to the solution for total pressure in the form where A = 0 . 0039 and ˜ C = C / (1 + C ) are constants, and P = κρ γ is the assumed form of equation of state with the polytropic index γ = 4 / 3 (Abramowicz & Fragile 2013). We focus on critical tori that develop the relativistic cusp at the inner edge, R = R in, where the angular momentum is equal to the Keplerian angular momentum at the corresponding radius in the common equatorial plane of the black hole and the torus ( ϑ = π/ 2). The mass accretion can either bring the system out of the critical configuration and stabilise it with an updated set of model parameters and a modified profile of angular momentum, or the accretion process continues in a runaway mode and leads to a complete destruction of the torus. An exemplary profile of such a torus is shown in Figure 1, where contours of constant mass density are constructed from the analytical form of the solution (5). The contours are overplotted on top of the colour-coded density structure. The black hole is located towards the left side of the plot; the coordinates are defined in such a way that the entire left edge of the plot corresponds to the location of the outer horizon at r = r + . The bottom panel shows the same configuration in Boyer-Lindquist coordinates ( r , ϑ ). The torus surface extends from the inner rim at r = 4 . 5 up to the outer boundary at r = 8 . 4 in the equatorial plane, y = 0. The right edge is at x = 15; Cartesian coordinates are derived from Boyer-Lindquist coordinates, x 2 + y 2 = r 2 ( x = r sin ϑ , y = r cos ϑ ) and scaled with units of GM / c 2 . The colour bar is scaled logarithmically with values of normalised rest-mass density ρ ( x , y ) relative to its maximum value ρ c at the torus centre ( x = R c = 6 . 3, y = 0). In the bottom panel the horizon is located at √ x 2 + y 2 = 1 . 95, so it is hidden behind the left border of the graph. We notice that the constant density and pressure contours from eq. (5) correspond accurately to the colour scale of the image. Since the latter is based on a direct output from the numerical code, we can be confident that the computed structure and the necessary transformations of the coordinates are correct. This is y y y y a reassuring check before we embark on the time evolution of a perturbed configuration, where the analytical calculation is not available. Let us note that the HARM code (Gammie et al. 2003) defines radial and latitudinal coordinates in a way that helps resolving the plunging region near above the black-hole horizon. On the other hand, the sharp inner cusp of the critical configuration is revealed more clearly in standard Boyer-Lindquist coordinates. Figure 1 compares the torus structure in both types of coordinates.", "pages": [ 2, 3, 4 ] }, { "title": "2.2. Timeevolutionofperturbedconfiguration", "content": "We assume that the above-described initial stationary state is pushed out of equilibrium. This leads to the capture of a small amount of material by the black hole, which increases the black-hole mass, and so the accretion occurs. Abramowicz et al. (1998) argued that tori with radially increasing angular momentum density are more stable. Therefore, we started with q > 0 (see Figure 2) and concentrated on the influence of the magnetic field on the accretion rate. The algorithm of the numerical experiment proceeds as follows. At the initial step the mass of the black hole was increased by a small amount, typically by about few percent. After the time step δ t , the elementary mass δ M and angular momen- m δ L = l ( R in) δ M are accreted across the horizon, r = r + ≡ [1 + √ 1 -a 2 ] GM / c 2 . The mass increase δ M is computed as a di ff erence of the mass of torus M d = ∫ V ρ d V at t and t + δ t , where d V = u t √ -gd 3 x is taken over the spatial volume occupied by the torus. The corresponding elementary spin increase is δ a = l δ M / ( M + δ M ). Therefore, at each step of the simulation we updated the model parameters by the corresponding low values of mass and angular momentum changes: M → M + δ M , a → a + δ a . The inner cusp moves accordingly. We employed geometrized units, setting the speed of light and the gravitational constant equal to unity, c = G = 1. This implies the scaling of various quantities with the central blackhole mass M . However, the mass as well as the spin parameter evolve gradually (adiabatically) as the accretion of material proceeds from the torus, M ≡ M ( t ), a ≡ a ( t ). Corresponding quantities in physical units are obtained by the following conversions: Also, To obtain the frequency in physical units [Hz], one uses the relation κ phys = c κ . The geometrized frequencies are scaled by M -1 . Therefore, their numerical values must be multiplied by the factor to find the frequency in [Hz]. Figure 3 shows the dependence of the torus mass on time for di ff erent values β of the ratio between thermodynamical and magnetic pressure (plasma parameter), β ≡ P g / P m, for a torus with the radially increasing distribution of angular momentum, l ( R ) = l K , R = R in [1 + /epsilon1 ( R -R in)] q with q > 0, 0 < /epsilon1 /lessmuch 1. This means that the reference level of the angular momentum density is set to l = const = l K( R in), motivated by the standard theory of thick accretion discs, where the constant value is a limit for stability. A radially growing profile then helps to stabilise the configuration. Unless stated otherwise, we set q = 1, /epsilon1 = 0 . 03 / l ( R in) for definiteness of examples in the simulations. At the inner edge of the torus the angular momentum equals the Keplerian value, and for higher radii it grows to super-Keplerian rotation, taking into account the specific shape of the relativistic Keplerian angular momentum (e.g., Abramowicz & Fragile 2013). Furthermore, following the von Zeipel theorem, in the vertical direction along the constant R = const surface within the torus, the angular momentum is defined by its value in the equatorial plane. The topology of these surfaces is cylindrical except for relativistic deviations that are important only at very small radii (Chakrabarti 1991). From the graph we see that the amount of accreted mass is generally larger for smaller β . The plot also shows that the overall gradually decreasing trend is superposed with fast oscillations. After the initial drop of the torus mass (given by the magnitude of the initial perturbation, δ M /similarequal 0 . 01 M ) phases of enhanced accretion change with phases of diminished or zero accretion. The oscillatory behaviour can be traced by the position of the torus centre, which we discuss below. It resembles the eigenfrequency modes that were proposed as a model for quasiperiodic oscillations in some X-ray binaries (Rezzolla et al. 2003; Montero & Zanotti 2012). A similar accretion history is found also for higher spin values, i.e., closer to extreme rotation, although we always assumed | a | < 1 (we did not consider the possibility of a naked singularity, but see Stuchlík et al.,2010 for a recent discussion of such a possibility). 50 100 150 200 Figure 5 is complementary to Fig. 3. In Figure 5 we compare several cases of di ff erent angular momentum profiles, as characterised by the slope q . One can see that the accretion rate is higher than in the previous example. This plot also confirms that for higher q the amount of accreted mass is diminished, in agreement with prior studies. In fact, for cases with indices q = 0 . 6 and q = 0 . 8 almost the whole torus is accreted. For q = 1 . 0, almost 96% of the initial torus mass is accreted, while for q = 1 . 1 it becomes 88% of the initial mass. A comparison between the two panels of Fig. 5 confirms the general trend, which shows that tori are more stable for higher spin values, i.e., closer to the extremely co-rotating black hole ( a → 1) for otherwise similar parameters. We also followed the oscillations of the point of maximal mass density inside the torus (the torus centre); it shows a behaviour consistent with the above-described evolution of the accreted mass. Figure 6 studies the dependence of accretion rate by plotting the half-mass accretion time as a function of q and β . Furthermore, Figure 7 compares the magnetized vs. non-magnetized tori for the same spin ( a = 0 . 3). In the top panel we show the time dependence of the radial coordinate of the point with the highest mass density R = R c (hence the highest pressure) of these two tori, and in the bottom panel the dependence of the highest mass density is captured as a function of time. In the limit of a non-magnetised slender torus ( R c /greatermuch 1) these oscillations correspond to the situation that has been treated previously by analytical methods (Blaes et al. 2006). Although the amplitude of R c oscillations is quite small in these examples (because the oscillations were initiated by a weak perturbation and the torus centre is relatively far from the black hole), the outer layers of the torus are a ff ected more significantly and can be accreted across the inner edge. In these simulations we neglected self-gravitation of the torus (Goodman & Narayan 1988; Karas et al. 2004). However, when the mass and angular momentum are accreted by the black hole, its parameters M and a are obviously changed. Hence, even for non-selfgravitating tori we need to update the parameters of Kerr space-time metric to achieve a consistent description. In our scheme we changed M and a at each time step, according to accreted mass and its angular momentum content. Then we can watch how this updating influences the accretion process, namely, oscillations of the torus centre, torus mass, and other characteristics. Figure 8 plots the dependence of the central mass density on time for a l ( R ) ∝ R q profile for the non-magnetized case. In an analogous way, we examined the role of di ff erent initial perturbations δ M ( t = 0). From these illustrations one can deduce that for a su ffi ciently steep slope q the torus becomes stabilised with respect to run- away accretion. The steeper q , the longer oscillation period. The (weak) initial perturbation does not significantly influence the oscillation period, only the oscillation amplitude is a ff ected. Qualitatively identical conclusions are obtained for a slightly different value of the polytropic gas index. As mentioned above, the results presented here were computed for γ = 4 / 3; we also computed the same set of plots for γ = 5 / 3 with very similar results, while the model is more sensitive to relatively weak variations of q . Finally, two additional plots reveal the changing parameters of the torus and the black hole in the course of accretion. Figure 9 compares the positions of the torus cusp and the torus centre as functions of the accreted mass for di ff erent initial values of spin a = 0 . 3 , 0 . 5, and 0 . 7. The evolution of metric parameters M and a has a stabilising e ff ect because the critical surface moves inward, so this case corresponds to the situation when the accretion rate and other characteristics oscillate. It does not lead to the runaway behaviour, which is also reflected in the gradually decreasing cusp radius. On the other hand, the solid line is related to the case when the accreted mass contains less angular momentum, and so the impact of increasing the black-hole mass is stronger than the e ff ect of increasing the spin. In this case updating the metric parameters results in tori that tend to be more unstable. The mutual relation between the two radii, i.e. R cusp vs. R c, provides information about the size of the torus as it changes by loosing material onto the black hole and moving in radius, while the black hole grows. Naturally, the mean centre of the torus, R = R c, moves along with the above-mentioned gradual evolution of the inner cusp, R = R cusp, by the mass transfer from the torus onto the black hole. The torus centre obviously satisfies R c( t ) > R cusp( t ) > r + ( t ) at each moment of the evolution. However, the exact mutual relation between these radii depends on details of the particular case, namely, the density and the angular momentum distribution within the torus. Therefore, both the mean R c and the mean R cusp can either approach the centre or recede towards a larger distance, depending on whether the torus shrinks and eventually becomes accreted onto the black hole (which is the case of runaway accretion) or if the partially accreted structure becomes stabilised against more mass transfer and stays away from the black hole. Naturally, the centre radius is influenced also by the fact that the black hole itself evolves its mass and spin. Figure 10 shows the oscillation frequency of the torus, as determined from a sequence of our numerical solutions with different position of the torus centre. The frequency varies gradually in this graph, along with the black-hole dimensionless spin a . Again, this change can be seen as a result of accretion of the material from the torus onto the black hole, which modifies the model parameters including the black-hole spin and the torus centre (as well as the corresponding torus mass, M d, and other characteristics of the system, as explained above). This dependence allows us to unambiguously identify the relevant oscillation mode. We confirm a very precise agreement between the resulting curve in Fig. 10 and the theoretical formula for the radial epicyclic oscillation κ ≡ κ ( M , a ) (e.g. Kato et al. 2008, see eq. (2.105)). The di ff erence between the numerically determined value and the analytical formula for the radial epicyclic frequency is less than 1 per cent, and so the two dependencies are practically indistinguishable in the plot. Naturally, to achieve a consistent solution, the parameters need to be evolved during the accretion process. Nevertheless, we checked that the above-mentioned point about the metric parameters does not influence the oscillation period of a globally stable configuration, i.e., until the perturbation itself remains weak. In other words, while the position of the torus centre and the magnitude of central density di ff er at a level of several percent between di ff erent simulations, the agreement about the oscillation frequency is typically one order of magnitude better. We recall a useful scheme (see Tab. 1 in Font & Daigne 2002) that summarises a competing role of di ff erent agents that influence the stability of geometrically thick accretion tori near black holes. These are partly real physical e ff ects (such as the angular moment profile of the accreted material, rotation of the black hole, and self-gravity), and partly reflect the impact of approximations that are employed to describe the system (such as the Newtonian versus pseudo-Newtonian versus general-relativistic models). As mentioned above, radially growing angular momentum and fast rotation of the black hole tend to stabilise the system against the runaway instability, while selfgravity acts instead against stability. Therefore we can also include the magnetic field as another ingredient into the discussion of stability. However, as demonstrated also above in this paper, with the increasing number of di ff erent factors and the interplay of mechanisms taken into account, the whole pictures becomes more complicated than when one had restricted the discussion to the competition of just two or three degrees of freedom. In the end the outcome of the analysis can depend on detailed values of the parameters, e.g. q vs. a . Moreover, especially the magnetic field can develop di ff erent geometrical structures on vastly di ff erent scales, and so it may be di ffi cult or impossible to characterise the role of magnetic field on the runaway stability in a simple way. In our highly simplified scheme we showed that the e ff ect of the toroidal test magnetic field (embedded in a prescribed manner into the polytropic fluid) just adds to the internal pressure of the fluid. Hence it basically enhances the instability in a similar way as any other contribution that can enhance the pressure above the equilibrium value.", "pages": [ 4, 5, 6, 7 ] }, { "title": "3. Discussion and conclusions", "content": "Within the framework of an axially symmetric magnetized fluid torus model we have extended the previous results on the onset of runaway instability of relativistic configurations near a rotating black hole. We concentrated on systems with radially increasing angular momentum density that are threaded by a purely toroidal magnetic field. We neglected self-gravity of the gaseous material (the mass of the torus was set to be at most several percent of the black-hole mass), nevertheless, we allowed for a gradual change of the Kerr metric mass and spin parameters by accretion over the inner edge. The angular momentum distribution within the torus was also allowed to evolve, starting from the initial power-law profile. The mass transfer influences the location of the cusp of the critical configuration, which can lead to the runaway instability. If the profile of the angular momentum increases su ffi ciently fast with radius (typically, for q /greaterorsimilar 0 . 8), the initial perturbation becomes stabilised by accretion of a small amount of material, whereas for small q the instability causes rapid accretion of the torus. The intensity of the threaded magnetic field influences the process of stabilisation or destruction of the torus because, within the framework of the adopted model, the magnetic pressure adds directly to the gas pressure (plasma parameter β /greaterorsimilar 1). The process of accretion is not perfectly monotonic, instead, there are changing phases of enhanced accretion rate and phases where the mass of torus remains almost constant. The overall gradual decrease of the torus mass is superposed with oscillations that can be seen by following the central density variations on the dynamical time-scale and the position of the centre of the torus. The oscillation amplitude is sensitive to the initial perturbation, but the frequency is not, namely, a small change of the metric coe ffi cients does not a ff ect the oscillation frequency. The toroidal magnetic field plays a more important role in the early phases of the accretion process until the perturbed configuration finds a new equilibrium or disappears because of the runaway instability. If the oscillations become stabilised with time, no significant di ff erences occur from the corresponding non-magnetized case, even when β is near unity (equipartition) and the accreted fraction of the torus material is significant. Acknowledgements. We thank an anonymous referee for helpful suggestions. We acknowledge support from the student project of the Charles University (GAUK 139810; JH) and the collaboration project between the Czech Science Foundation and Deutsche Forschungsgemeinschaft (GACR-DFG 1300070J; VK). The Astronomical Institute has been operated under the program RVO:67985815.", "pages": [ 7, 8, 9 ] } ]

[ { "title": "ABSTRACT", "content": "In this paper we present the tools to optimally extract the Lensing-Integrated Sachs Wolfe (L-ISW) bispectrum signal from future CMBdata. We implement two di ff erent methods to simulate the non-Gaussian CMB maps with the L-ISW signal: a non-perturbative method based on the FLINTS lensing code and the separable mode expansion method. We implement the Komatsu, Spergel and Wandelt (KSW) optimal estimator analysis for the Lensing-ISW bispectrum and we test it on the non-Gaussian simulations in the case of a realistic CMB experimental settings with an inhomogeneous sky coverage. We show that the estimator approaches the Cramer-Rao bound and that Wiener filtering the L-ISW simulations gives a slight improvement on the estimate of f L -ISW NL of ≤ 10%. For a realistic CMB experimental setting accounting for anisotropic noise and masked sky, we show that the linear term of the estimator is highly correlated to the cubic term and it is necessary to recover the signal and the optimal error bars. We also show that the L-ISW bispectrum, if not correctly accounted for, yields an underestimation of the f local NL error bars of /similarequal 4%. A joint analysis of the non-Gaussian shapes and / or L-ISW template subtraction is needed in order to recover unbiased results of the primordial non-Gaussian signal from ongoing and future CMB experiments. Key words. The Cosmic Microwave Background, non-Gaussianity, Lensing, ISW, cosmology", "pages": [ 1 ] }, { "title": "Optimal bispectrum estimator and simulations of the the CMB Lensing-ISW non-Gaussian signal", "content": "A. Mangilli 1 , B. Wandelt 1 , 2 , Franz Elsner 1 , and Michele Liguori 3 Preprint online version: October 3, 2018", "pages": [ 1 ] }, { "title": "1. Introduction", "content": "One of the most relevant mechanisms that can generate nonGaussianity from secondary Cosmic Microwave Background (CMB) anisotropies is the coupling between weak lensing and the Integrated Sachs Wolfe (ISW) (Sachs & Wolfe 1967) and the Rees Sciama (RS) (Rees & Sciama 1968). This correlation gives in fact the leading contribution to the CMB secondary bispectrum with a blackbody frequency dependence (Goldberg & Spergel 1999; Verde & Spergel 2002; Giovi et al. 2005). Weak lensing of the CMB is caused by gradients in the matter gravitational potential that distorts the CMB photon geodesics. The ISW and the RS e ff ects, on the other hand, are related to the time variation of the gravitational potential wells. The relevant mechanism is given by the late ISW, owing to the action of Dark Energy which causes the decay of the gravitational potential wells as the Universe expands. Both the lensing and the ISW e ff ect are then related to the matter gravitational potential and thus are correlated phenomena. This gives rise to a non-vanishing three-point correlation function or, analogously, a non-vanishing bispectrum, its Fourier counterpart. The RS (also referred as the non-linear ISW) arises when the growth of structure in the evolving universe becomes non-linear. Being a second order e ff ect, the RS gives a smaller contribution to the signal with respect to the ISW. The CMB bispectrum arising from the cross correlation between lensing and ISW / RS (from now on referred to as L-ISW) is expected to have an high signal-to-noise ratio from ongoing and future CMB experiments so that it will be detectable in the near future with an high statistical significance (Verde & Spergel 2002; Giovi et al. 2005; Mangilli & Verde 2009; Lewis et al. 2011). A detection would open the possibility to exploit the cosmological information related to the late time evolution encoded in the L-ISW signal. It is useful to stress that a significant detection of the L-ISW signal from ongoing CMB experiments like Planck would be a powerful probe of Dark Energy from CMB alone and it would be a complementary probe of the late time Universe with respect to the large scale structure and the the CMB power spectrum analysis. Moreover, Mangilli & Verde (2009); Hanson et al. (2009) showed that the L-ISW bispectrum can be a serious contaminant problem for the estimation of the primary local non-Gaussianity from future data. Ongoing CMB experiment such as Planck (Ade et al. 2011) and future experiments like COrE (Bouchet et al. 2011) will then require a detailed reconstruction of the L-ISW bispectrum either to be able to correctly remove the L-ISW contribution when estimating the local primary non-Gaussian parameter fNL , or to exploit the cosmological information encoded in the signal; therefore it becomes extremely important to know how to model and simulate it. In this paper we present the formalism and the numerical implementation i ) to generate simulated CMB maps containing the L-ISW signal and ii ) to build and test the optimal estimator for the L-ISW bispectrum, accounting for both the cubic and the linear parts. The linear part for this specific kind of signal, has been here calculated and tested for the first time. As regarding the CMB non-Gaussian simulations, we implemented and tested the L-ISW signal with two methods: the separable mode expansion method (Fergusson et al. 2010; Smith & Zaldarriaga 2011) and the non-perturbative approach described in Sec. 2.1. It is important to have an optimal estimator for the L-ISW bispectrum in order to extract the signal optimally from future data and to disentangle it from other kinds of non-Gaussianities, i.e. the local primary bispectrum, with which it is degenerated. Here, following Komatsu (2010) and Munshi et al. (2011), we implemented the KSW bispectrum estimator (Komatsu et al. 2005) for the L-ISW signal of a full sky, cosmic variance limited CMB experiment and in the case of a more realistic instrumental setting, similar to that of a space-based experiment. Furthermore, for this realistic case, we investigate the statistical detection significance and the impact that the L-ISW bispectrum has on the estimation and on the variance of the primary local non-Gaussian parameter fNL . The outline of this paper is as follows. In section 2 we present the methods to simulate the non-Gaussian CMB maps containing the L-ISW bispectrum signal by the use of both the separable mode expansion method and the non-perturbative covariance method. Section 3 provides the basics to build and implement the optimal estimator for the L-ISW signal, including its linear part. It also includes a discussion regarding the implementation of the Wiener filtered simulations algorithm. In section 4 we present the relevant tests and results. In section 5 we quantify the statistical detection significance of the L-ISW bispectrum and the impact on the error of primary non-Gaussianity fNL due to the presence of the ISW signal. Finally, in section 6, we discuss the results and we summarize the conclusions. Details on the simulations built with the covariance method and on the L-ISW cross correlation coe ffi cients are given in the appendix.", "pages": [ 1, 2 ] }, { "title": "2. Simulated non-Gaussian CMB maps", "content": "In this section, we present the formalism to create simulated CMB maps for the L-ISW bispectrum. We use two di ff erent methods: a no perturbative approach, here named the 'covariance method', and the separable modes expansion method ((Fergusson et al. 2010) and (Smith & Zaldarriaga 2011)). The latter gives an e ffi cient and easy to handle way to generate LISW maps, while the former method provides better insights on the physics related to the L-ISW bispectrum. In this case, in fact, the L-ISW signal is generated starting from the covariance matrix representing the expected correlation between the lensing and the ISW / RS e ff ects.", "pages": [ 2 ] }, { "title": "2.1. Covariancemethod", "content": "The L-ISW correlation is defined by the covariance matrix: and the cross correlation coe ffi cient: Here, C TT /lscript δ/lscript/lscript ' δ mm ' = 〈 a P /lscript m a P ∗ /lscript ' m ' 〉 and C φφ /lscript δ/lscript/lscript ' δ mm ' = 〈 φ L /lscript m φ L ∗ /lscript ' m ' 〉 are respectively the CMB primary temperature power spectrum and the lensing power spectrum, where the lensing potential φ (the gravitational potential projection along the line of sight) is defined by: The term in the numerator, C T φ /lscript , is the power spectrum of the cross correlation between the lensing and the ISW / RS e ff ect, see appendix B for details. After a Cholesky decomposition of the L-ISW correlation matrix C L -IS W , the two new variables t /lscript m and z /lscript m are then defined by where x /lscript m and y /lscript m are two independent random gaussian fields. By definition, the new fields are such that: 〈 t 2 〉 = C φφ /lscript , 〈 z 2 〉 = C TT /lscript and they have the non-zero cross correlation 〈 zt 〉 = C T φ /lscript . As described in appendix A, the map that contains the desired L-ISW bispectrum is then given by the coe ffi cients where a P /lscript m and a L /lscript m are, respectively, the unlensed primary and the lensing angular coe ffi cients and ∆ a L /lscript m = a L /lscript m -a P /lscript m corresponds to the lensing expansion terms only. Note that by construction y /lscript m has the same phases as a P /lscript m ≡ y /lscript m √ C TT /lscript and φ L /lscript m ≡ x /lscript m √ C φφ /lscript the same as x /lscript m , which is necessary for building a map with the wanted bispectrum signal. Figure 1 shows in black the temperature CMB power spectrum of one simulated L-ISW map, C L -IS W /lscript , built from Eq. 6. The non-Gaussian contribution, in blue in the figure, is always subdominant and the C L -IS W /lscript are consistent with the theoretical input ( C TT /lscript ) th (red line) obtained with CAMB (Lewis et al. 2000) 1 . As throughout the paper, the reference cosmological model used is the Λ CDM model with parameter values defined in (Komatsu et al. 2011).", "pages": [ 2, 3 ] }, { "title": "2.2. Separablemodesexpansionmethod", "content": "Following Fergusson et al. (2010) and Smith & Zaldarriaga (2011), the non-Gaussian part of the CMB angular coe ffi cients can be defined starting from a given reduced bispectrum. In the case of the L-ISW signal, the method can be used because this kind of signal is separable, so From the expression of the L-ISW reduced bispectrum in Eq. (13) and by factorizing the /lscript dependence, the explicit form of the non-Gaussian contribution to the a /lscript m from the L-ISW cross correlation is given by: Here, The maps with a δ 2 prefix are given by, e.g., δ 2 P = -∑ /lscript /lscript ( /lscript + 1) a /lscript mY /lscript m ( ˆ n ); they correspond to the maps of Eq. 10 multiplied by the -/lscript ( /lscript + 1) factor. The final solution containing the L-ISW signal is then: where a G /lscript m is the Gaussian part. In Fig. 2, we show the CMB temperature power spectra from the Gaussian and the non-Gaussian map, as defined in Eq. (8). The non-Gaussian contribution is always subdominant as expected.", "pages": [ 3 ] }, { "title": "3. The Optimal KSW estimator for the lensing-ISW/RS bispectrum", "content": "In this section we present the formalism related to the KSW estimator (Komatsu et al. 2005) for the Lensing-Integrated-Sachs Wolfe bispectrum signal.", "pages": [ 3 ] }, { "title": "3.1. Definition", "content": "The a /lscript m probability distribution function (PDF) in the limit of weak non-Gaussianity (i.e. truncated at the bispectrum level) is given by (Babich 2005; Taylor & Watts 2001; Komatsu 2010): where 〈 a /lscript 1 m 1 a /lscript 2 m 2 a /lscript 3 m 3 〉 is the angular bispectrum . Here, we are interested in the L-ISW case, for which the angular bispectrum, parametrized by the amplitude parameter f L -IS W NL , is where is the reduced bispectrum and C T φ /lscript ≡ 〈 φ ∗ /lscript m a L -IS W /lscript m 〉 are the LISW cross-correlation coe ffi cients. According to (Komatsu et al. 2005), for small departure from Gaussianity, the optimal estimator for theL-ISW amplitude parameter is given by: where ( F -1 ) is the inverse of the L-ISW Fisher matrix In the case of a realistic CMB experimental setting, the noise, N /lscript , and the beam window function, w /lscript , are accounted for so that C /lscript = N /lscript + C th /lscript w 2 /lscript . In this case, the bispectrum is also convolved with the beam transfer function w /lscript , B L -IS W /lscript 1 /lscript 2 /lscript 3 ∝ b L -IS W /lscript 1 /lscript 2 /lscript 3 w /lscript 1 w /lscript 2 w /lscript 3 . Given a mask M ( p ), the observed sky fraction fsky is defined as: where Npix = 12 N 2 s is the number of pixels in the map, Ns is the map resolution and the sum p is done over the pixels. ∑ Assuming that the only relevant non-Gaussian contribution is coming from the L-ISW term, which is the case if the local primordial non-Gaussianity is small and foregrounds and point sources have been correctly removed and masked, SL -IS W is given by the data as By factorizing the /lscript i dependence, this becomes where the maps P ( ˆ n ), E ( ˆ n ), Q ( ˆ n ) etc. are the same as defined in Eqs. 10 and, in the case of a realistic experiment, they are convolved with the experimental window function w /lscript so that, for example, P ( ˆ n ) ≡ ∑ /lscript m w /lscript a /lscript mY /lscript m ( ˆ n ). In Eq. 18, the first two lines refer to the cubic part of the estimator, while S L -IS W lin is the linear part which corrects for anisotropies and must be included in the case rotational invariance is not preserved. Details on the analytic expression of the LISW linear term and on its numerical implementation are given, respectively, in the next subsection 3.2 and in Sec. 4.", "pages": [ 3, 4 ] }, { "title": "3.2. Thelinearterm", "content": "The linear term of the estimator is given by By using the explicit form of b L -IS W /lscript 1 /lscript 2 /lscript 3 and by factorizing the /lscript -dependence one obtains where 〈〉 MC indicates the Monte Carlo (MC) averages and the di ff erent maps are defined in Eq. 10 and they are convolved with the experimental window function w /lscript , so that P ( ˆ n ) ≡ ∑ /lscript m w /lscript a /lscript mY /lscript m ( ˆ n ), etc.", "pages": [ 5 ] }, { "title": "3.3. Wienerfilteredmaps", "content": "The optimal bispectrum estimator as described in Eqs. (17, 18) involves products of inverse variance filtered maps, C -1 a = ( S + N ) -1 a , where S and N are the signal and the noise covariance matrix, respectively. A brute force calculation of such an expression is impractical for modern high-resolution experiments as it involves the inversion of two matrices that are too large to be stored and processed as dense systems. In case the noise covariance can be described in terms of a simple power spectrum in spherical harmonic space, the calculation simplifies significantly. However, this approach is no longer exact for experiments with anisotropic noise distribution or reduced sky coverage, leading to an increase in the error bars of the estimates. Here, we use Wiener filtering as a basis for the exact evaluation of terms involving C -1 a . We apply the iterative scheme of (Elsner & Wandelt 2013) to calculate the Wiener filter a WF ≡ S ( S + N ) -1 a , the maximum a posteriori solution in case signal and noise are Gaussian random fields. After a WF has been successfully computed, we finally obtain the inverse variance filtered map by normalizing the spherical harmonic coe ffi cients of the Wiener filter solution by the CMB power spectrum multiplied with the beam window function, C -1 a /lscript m = ( C th /lscript b 2 /lscript ) -1 a WF /lscript m .", "pages": [ 5 ] }, { "title": "4. Results", "content": "In this section we present the results regarding the numerical implementation of the optimal estimator and of the methods presented in sec. 2.1 and 2.2 to build the CMB maps containing the L-ISW bispectrum. In particular, we processed the simulated LISW maps through the estimator pipeline to get the amplitude parameter f L -IS W NL of Eq. 14. We consider two main settings: All runs have been performed at full resolution Nside = 2048 (which corresponds to a map pixel number of 5 . 033 · 10 7 ). The maps in Eqs. (10) are calculated by using the Healpix package (Gorski et al. 2005). The theoretical power spectrum of the temperature-only primary CMB coe ffi cients C /lscript has been generated with the CAMB code for a fiducial Λ CDM cosmological model with parameters corresponding to WMAP7 cosmological parameters (Komatsu et al. 2011). For illustrative purpose, the plots of the maps and of the correspondent /lscript -filters containing the L-ISW cross correlation coe ffi cients C T φ /lscript are shown in Figs. 3 and 4. We built a set of 100 CMB simulations for each of the two methods described in sections 2.2 and 2.1 for a cosmic variance limited CMB experiment with full sky coverage. For the covariance method, we used the FLINTS code (Lavaux & Wandelt 2010) to generate the lensing coe ffi cients a L /lscript m and the lensing potential coe ffi cients φ/lscript m needed to build the non-Gaussian a L -IS W /lscript m as described in Sec. 2.1. In both cases, we analyzed the LISW simulated CMB maps with the L-ISW estimator up to /lscript max = 1000. According to the definition of f L -IS W NL , the expected value is 1 with 1σ error predicted from theory for /lscript max = 1000 of /similarequal 0 . 64. In the case of the separable expansions mode method, the simulations give a mean f L -IS W NL = 1 . 1 with averaged 1σ error /similarequal 0 . 69. With the simulations built with the covariance method, we obtain a mean f L -IS W NL = 1 . 21 with averaged 1σ error of 0.67. The results are summarized in Fig. 6 and Fig. 5, respectively. These estimates are compatible with the theoretical predictions. The error bars are slightly suboptimal because of numerical noise and the fact that we are assuming a diagonal covariance matrix so that ( C -1 a ) /lscript m = a /lscript m / C /lscript . In order to test the estimator on a more realistic case, we built a set of 100 simulations with the separable mode expansion method considering a realistic experimental setting. This consists of a CMB one channel experiment with a Gaussian beam FWHM θ b = 7 ' , a galactic mask with fsky = 0 . 78 and anisotropic noise, as previously described. In this case, we run the estimator up to /lscript max = 1500. The expected theoretical 1σ error on f L -IS W NL for this experimental setting and up to /lscript max = 1500 is /similarequal 0 . 49. This estimate accounts for a /similarequal 10% percent increase in the error bar due to the fact that the lensing is intrinsically non-Gaussian and it gives an extra contribution to the variance, as shown in (Lewis et al. 2011). We get a mean f IS W NL = 1 . 09 with averaged 1σ error /similarequal 0 . 55. In this case we computed both the cubic and linear part of the estimator. In particular, the linear term has been tested with a set of 100 Monte Carlo (MC) averages generated for each map product in equation (Eq. 21). In the presence of anisotropic noise and a sky cut, the linear part of the estimator is necessary to recover the expected estimation of f L -IS W NL and error bars. The linear contribution to f L -IS W NL is strongly anti-correlated with the cubic part. This behavior is summarized in figure Fig. 8. In the plot are shown the linear and the cubic contributions to the total amplitude f L -IS W NL ≡ ( f L -IS W NL ) cubic + ( f L -IS W NL ) linear . We also checked that with 100 MC averages the linear term converges and it is stable: for this specific experimental setting we find that the results do not improve when increasing the MC averages to 200. Finally, in order to test optimality, we Wiener filtered the 100 L-ISW simulations and we processed them through the LISW estimator pipeline. The maps has been produced following (Elsner & Wandelt 2013), as described in section Sec. 3.3. We use as inputs the same experimental settings as described previously. The linear term has been computed with 100 Wiener filtered MC simulations. We found that the improvement with respect to the non Wiener filtered simulations is small ( < 10%) in the case of our particular settings. However, this does not exclude that the Wiener filtering may have a more noticeable impact for a more realistic experimental setting and noise covariance.", "pages": [ 5, 6 ] }, { "title": "5. f NL error estimation", "content": "This section summarizes the results regarding the impact of the L-ISW signal on the error estimation of fNL from the local type non-Gaussianity. If the only contribution to fNL were from the primary local type non-Gaussianity the error on this parameter would be simply given by i.e. the inverse of the Fisher matrix of the local type non-gaussian contribution where fsky refers to the observed sky fraction. The noise, N /lscript , and the beam, b /lscript , can be accounted for so that C /lscript = N /lscript + C th /lscript b 2 /lscript . In this case the bispectrum is also convolved with the beam transfer function b /lscript : B P /lscript 1 /lscript 2 /lscript 3 ∝ b P /lscript 1 /lscript 2 /lscript 3 b /lscript 1 b /lscript 2 b /lscript 3 . However, the L-ISW can be a serious contaminant of the local primary signal (Mangilli & Verde 2009; Hanson et al. 2009), so that it is important to quantify the e ff ect on the expected fNL error as well. If the L-ISW signal is present, the error matrix will be given by a non-diagonal Fisher matrix of the form where is the cross correlation term and F L -IS W is the the Fisher term of the L-ISW signal of Eq. 15. The expected error on the local fNL will be then: i.e. the inverse of the full Fisher matrix containing the cross correlation between the primary local non-Gaussianity and the LISW signal. The di ff erence between the error estimation on fNL -primary with and without the L-ISW contribution is: To quantify the level of correlation between the two signals one can define the correlation coe ffi cient as Wefindthat the e ff ect of the local non-Gaussianity on the L-ISW is negligible. Therefore, the 1 -σ error of the L-ISW amplitude parameter f L -IS W NL is given by σ L -IS W = √ ( F L -IS W ) -1 . For a realistic CMB experiment as described in Sec. 4 and appendix C, we find that the correlation between the two signals is r = 0 . 20 at /lscript max = 1500 and r = 0 . 27 at /lscript max = 2000 and that the expected detection significance of the L-ISW signal (1 /σ L -IS W ) is, respectively, at /similarequal 2 and /similarequal 3 σ . The e ff ect on the fNL local error due to the contamination is in the range between /similarequal 3% to /similarequal 5 % for /lscript max from 1000 to 2000, depending on how much the two signal are correlated. For /lscript max = 2000, if the L-ISW signal is not accounted correctly, the fNL error bars are overestimated by /similarequal 4%.", "pages": [ 6, 7 ] }, { "title": "6. Discussion and conclusions", "content": "We have presented the formalism and the numerical implementation to build the optimal KSW estimator for the Lensing-ISW bispectrum. Moreover, we have tested the estimator on simulated CMB maps containing the Lensing-Integrated Sachs Wolfe (L-ISW) non-Gaussian signal and on the Wiener filtered simulations in order to test optimality. As regarding simulations, we have implemented and tested two methods: a non-perturbative approach to simulate CMB sky maps with the L-ISW signal which is based on the FLINTS lensing code (Lavaux & Wandelt 2010) and the perturbative separable mode expansion method calculated for this specific signal. We provide the analytical expression and the numerical implementation of the linear term of the estimator for this specific kind of bispectrum. For a realistic CMB experimental setting accounting for anisotropic noise and masked sky, the linear term gives a relevant contribution which is highly anti-correlated with the cubic part and it is necessary to recover the signal and optimal error bars. In order to achieve optimality, we also tested the estimator on the Wiener filtered L-ISW simulated CMB maps. In this case we recovered the signal with error bars which saturate the theoretical Cramer-Rao bound, with a small improvement of < 10% with respect to the non Wiener filtered simulations. Finally, we estimate that, if not correctly accounted for, the L-ISW e ff ect has also an impact on f local NL error bars leading to a bias and an overestimation of /similarequal 4%, in agreement with (Lewis et al. 2011). Thus a joint analysis of non-Gaussian shapes and / or L-ISW template subtraction will be needed in order to recover unbiased minimum variance results of the local type primordial non-Gaussian signal. It is important to note that the KSW bispectrum approach to the estimation of the L-ISW is complementary to the lensing reconstruction estimator of (Lewis et al. 2011). In principle, the KSW estimator can o ff er advantages with respect to other methods because the bispectrum has a unique shape and it has been shown to be robust to foreground contamination (Yadav & Wandelt 2010) so it can be measured by using a larger sky fraction. In addition, inclusion of the L-ISW in the framework of bispectrum analysis gives an unified approach to testing for primordial non-Gaussianity. The tools presented in this paper enable the optimal analysis of this important signal from future CMB data. Acknowledgements. This work was supported in part by NSF grants AST 0708849 and AST 09-08902, and by NASA / JPL subcontract 1413479; and through Ben Wandelt's ANR Chaire d'excellence ANR-10-CEXC-004-01. AM acknowledges Guilhem Lavaux for the FLINTS lensing simulations, Licia Verde for useful comments and discussion and the University of Illinois for the use of the curvaton computers.", "pages": [ 7 ] }, { "title": "References", "content": "Ade, P. et al. 2011, Astron.Astrophys., 536, 16464 Babich, D. 2005, Phys.Rev., D72, 043003 Bouchet, F. et al. 2011 Elsner, F. & Wandelt, B. D. 2013, Astron.Astrophys., 549, A111 Fergusson, J., Liguori, M., & Shellard, E. 2010, Phys.Rev., D82, 023502 Giovi, F., Baccigalupi, C., & Perrotta, F. 2005, Phys.Rev., D71, 103009 Goldberg, D. M. & Spergel, D. N. 1999, Phys.Rev., D59, 103002 Gorski, K., Hivon, E., Banday, A., et al. 2005, Astrophys.J., 622, 759 Hanson, D., Smith, K. M., Challinor, A., & Liguori, M. 2009, Phys.Rev., D80, 083004 Junk, V. & Komatsu, E. 2012, Phys.Rev., D85, 123524 Komatsu, E. 2010, Class.Quant.Grav., 27, 124010 Komatsu, E., Spergel, D. N., & Wandelt, B. D. 2005, Astrophys.J., 634, 14 Komatsu, E. et al. 2011, Astrophys.J.Suppl., 192, 18 Lavaux, G. & Wandelt, B. D. 2010, Astrophys.J.Suppl., 191, 32 Lewis, A. 2012, JCAP, 1206, 023 Lewis, A., Challinor, A., & Hanson, D. 2011, JCAP, 1103, 018 Lewis, A., Challinor, A., & Lasenby, A. 2000, Astrophys.J., 538, 473 Mangilli, A. & Verde, L. 2009, Phys.Rev., D80, 123007 Munshi, D., Valageas, P., Cooray, A., & Heavens, A. 2011, Mon.Not.Roy.Astron.Soc., 414, 3173 Rees, M. J. & Sciama, D. W. 1968, Nature, 217, 511 Sachs, R. K. & Wolfe, A. M. 1967, ApJ, 147, 73 Smith, K. M. & Zaldarriaga, M. 2011, Mon.Not.Roy.Astron.Soc., 417, 2 Spergel, D. N. & Goldberg, D. M. 1999, Phys.Rev., D59, 103001 Taylor, A. & Watts, P. 2001, Mon.Not.Roy.Astron.Soc., 328, 1027 Verde, L. & Spergel, D. N. 2002, Phys.Rev., D65, 043007 Yadav, A. P. & Wandelt, B. D. 2010, Adv.Astron., 2010, 565248", "pages": [ 7 ] }, { "title": "Appendix A: The simulated L-ISW CMB bispectrum from the covariance method", "content": "This appendix refers to the covariance method used to build the L-ISW simulated maps and described in Sec. 2.1. It is straightforward o check that the coe ffi cients a L -IS W /lscript m = z /lscript m +∆ a L /lscript m give the wanted bispectrum by calculating 〈 ( a IS W /lscript m ) 3 〉 = 〈 ( z /lscript m + ∆ a L /lscript m ) 3 〉 . The lensing coe ffi cients a L /lscript m can be expressed analytically, at first order in the lensing expansion, as where a P /lscript m the primary and φ L /lscript m the harmonic coe ffi cients of the lensing potential φ L . Since, according to the new variables definition ( z /lscript m , t /lscript m ) of eq. 6, t /lscript m = φ L /lscript m , the a L /lscript m can be written as: Here, for simplifying the notation, f /lscript = ∑ /lscript ' /lscript '' m ' m '' ( -1) m + m ' + m '' G -mm ' m '' /lscript/lscript ' /lscript '' /lscript ' ( /lscript ' + 1) -/lscript ( /lscript + 1) + /lscript '' ( /lscript '' + 1) 2 so that ∆ a L /lscript m = f /lscript a P ∗ /lscript m t /lscript m at first order. The explicit expression for 〈 ( a L -IS W /lscript m ) 3 〉 takes the form: From this the only non-zero term is: From this only survives: By using the definition of r T φ /lscript in Eq. 2 and the approximation r T φ /lscript << 1 for which 1 -( r T φ /lscript ) 2 /similarequal 1, since by construction: a P /lscript m = y /lscript m √ C TT /lscript , t /lscript m = x /lscript m √ C φφ /lscript , 〈 x 2 〉 = 1, 〈 y 2 〉 = 1 and 〈 xy 〉 = 0 we recover the expected signal:", "pages": [ 7, 8 ] }, { "title": "Appendix B: L-ISW cross correlation coefficients", "content": "The definition of the CMB lensing-ISW / RS cross correlation coe ffi cients is (Spergel & Goldberg 1999; Verde & Spergel 2002; Giovi et al. 2005): where, r ( z ) is the co-moving conformal distance and P φ ( k , z ) is the gravitational potential power spectrum which accounts for both the linear and non-linear contributions. The non-linear regime RS contribution to the signal is tiny, in agreement with (Lewis 2012; Junk & Komatsu 2012) 2 . Considering both the linear ISW and the Rees Sciama e ff ect improves the f L -IS W NL variance and signal to noise by few percent ( /similarequal 2%) with respect to the linear only case calculation. In this work we consider both contributions for completeness. As a template, for both the simulations and the estimator, we used the late ISW-lensing cross correlation coe ffi cients of Eq. B.1. This is a good approximation since this e ff ect is the one which gives the main contribution. However, for a detailed description see (Lewis 2012).", "pages": [ 8 ] }, { "title": "Appendix C: The experimental setting", "content": "spectrum is given, respectively, in the bottom and in the upper panels of Fig. C.1. The noise start dominating from /lscript /similarequal 1300.", "pages": [ 9 ] } ]

[ { "title": "ABSTRACT", "content": "An ever-growing observational aggregate of extrasolar planets has revealed that systems of planets that reside in or near mean-motion resonances are relatively common. While the origin of such systems is attributed to protoplanetary disk-driven migration, a qualitative description of the dynamical evolution of resonant planets remains largely elusive. Aided by the pioneering works of the last century, we formulate an approximate, integrable theory for first-order resonant motion. We utilize the developed theory to construct an intuitive, geometrical representation of resonances within the context of the unrestricted three-body problem. Moreover, we derive a simple analytical criterion for the appearance of secondary resonances between resonant and secular motion. Subsequently, we demonstrate the onset of rapid chaotic motion as a result of overlap among neighboring first-order mean-motion resonances, as well as the appearance of slow chaos as a result of secular modulation of the planetary orbits. Finally, we take advantage of the integrable theory to analytically show that, in the adiabatic regime, divergent encounters with first-order mean-motion resonances always lead to persistent apsidal anti-alignment. Key words. Celestial mechanics - Planets and satellites: dynamical evolution and stability - Chaos - HD 82943: Planets and satellites: individual - HD 45364: Planets and satellites: individual", "pages": [ 1 ] }, { "title": "Analytical Treatment of Planetary Resonances", "content": "Konstantin Batygin 1 ; 2 ? and Alessandro Morbidelli 1 ?? August 9, 2018", "pages": [ 1 ] }, { "title": "1. Introduction", "content": "The continued search for extrasolar planets around nearby stars has proven to be a goldmine of discoveries in numerous subfields of planetary astrophysics. Among the disciplines that have benefited the most is the study of orbital dynamics, as the aggregate of known planetary system architectures has grown immensely. Importantly, the observations collectively suggest that the orbital structure of the solar system is a singular example among numerous possible dynamical states. Indeed, orbital configurations that are quite unlike our own exist. Within the currently available observational collection, of particular interest is the class of systems that contains planets that reside in or near mean-motion resonances or loosely speaking, display integer commensurabilities among the orbital periods. The range of parameter space occupied by resonant planets is remarkably vast. Long-term radial velocity monitoring has revealed that giant planets occasionally reside in mean motion resonances at orbital distances exceeding GLYPH<24> 1AU (Wright et al. 2011). At the same time, searches aimed at transiting exoplanets (the Kepler mission in particular) have shown that (near)resonances are quite common among low-mass planets that reside in close proximity to their host stars (Fabrycky et al. 2012). Furthermore, it has been proposed that the giant planets of the solar system once occupied a resonant state (Masset and Snellgrove 2001; Morbidelli et al. 2007), before undergoing a transient dynamical instability that drove the orbits to their current locations (Batygin and Brown 2010; Levison et al. 2011). The prevalence of mean motion commensurabilities among planets is probably not coincidental and is likely to be a result of a physical mechanism. Indeed, it is believed that resonances congregate at an epoch in the dynamical evolution when the protoplanetary nebula is still present. Specifically, interactions between newly formed planets and the gaseous disk, into which they are embedded, leads to a time-irreversible exchange of angular momentum that results in planetary migration (Goldreich and Tremaine 1980; Lin et al. 1996; Crida et al. 2007). Although the particular regime (i.e. rate, direction) of the migration depends upon the planetary mass (Armitage 2010) as well as the thermodynamic properties of the disk (Paardekooper and Papaloizou 2009; Bitsch and Kley 2011), occurrences where migration among planetary pairs is slow and convergent are thought to be common (Terquem and Papaloizou 2007). In such cases, provided that the disk in question is not overwhelmingly turbulent (Adams et al. 2008; Cresswell and Nelson 2008; Ketchum et al. 2011) and the planetary orbits are nearly circular, capture into resonance is essentially guaranteed (Henrard 1982; Peale 1986). An example of resonant capture among giant planets, resulting from disk-dirven migration is shown in Figure (1). Specifically, the figure shows Jupiter and Saturn locked in a 3:2 mean motion resonance, having opened a mutual gap in the protoplanetary disk. The figure depicts a reproduction of the results of Masset and Snellgrove (2001) and Morbidelli and Crida (2007), where all simulation parameters were adopted from the latter study. It is noteworthy that gaseous protoplanetary disks are not the only environments where migrating planets can encounter mean motion resonances. Massive objects embedded in debris disks often undergo planetesimal-driven migration (Fernandez and Ip 1984; Murray et al. 1998; Kirsh et al. 2009). In fact, Malhotra (1995) proposed exactly this process for the origin of the 3:2 mean motion resonance between Neptune and Pluto. Yet another setting where resonant encounters are common is the orbital region occupied by planetary satellites (Peale 1986, 1999). In the context of the planetary satellite problem, migra- tion is usually forced by tidal interactions with the host planet (Goldreich 1963; Goldreich and Soter 1966). An oft-quoted example of a tidally assembled system is the Laplace resonance of the Galilean moons (Goldreich and Soter 1966; Henrard 1983). Systems of resonant planets on orbits that are close to their host stars also interact with the star tidally. However, in such systems, the interplay between the resonant dynamics and the dissipative forces results in a repulsion of the orbits (Batygin and Morbidelli 2012; Lithwick and Wu 2012), rather than a convergence towards nominal commensurability. Quite contrary to the examples described above, encounters with mean motion resonances by divergently migrating planets can never result in capture (Henrard 1991; Murray and Dermott 1999). Instead, passage through resonance leads to an impulsive excitation of the orbital parameters. As an example, such a process is thought to be responsible for the mutual inclinations of the Uranian satellites (Peale 1988; Tittemore and Wisdom 1990). Furthermore, our own Jupiter and Saturn may have once encountered the 2:1 mean motion resonance, jumpstarting the transient dynamical instability of the solar system that helped shape the Kuiper belt (Tsiganis et al. 2005; Levison et al. 2008; Batygin et al. 2011). The long-term evolution of resonant objects can be quite complex. In fact, it is now well known that overlap of resonances gives rise to chaos (Chirikov 1979; Wisdom 1980). In turn, this can result in orbital instabilities. Indeed, the process of chaotic clearing of resonant orbits is illustrated by the lack of objects in the Kirkwood gaps of the Asteroid belt (Wisdom 1983; Henrard and Caranicolas 1990; Murray and Holman 1997). The majority of the work on the chaotic dynamics of mean motion resonances has found its application in the study of the orbital evolution of small bodies with negligible masses (e.g. Asteroids, Kuiper Belt objects, (ir)regular satellites) (Nesvorný et al. 2002; Morbidelli et al. 2008), although chaotic di GLYPH<11> usion of planetary orbits in the outer solar system has also received some attention (Murray and Holman 1999). With a growing aggregate of detected extrasolar planets, (near-)resonant planetary pairs characterized by secondary mass-ratios close to unity have become common. This implies an expanded tally of objects to which the well-studied restricted formalism, where one of the three bodies is taken to be mass-less, is inapplicable. In particular, Figure (2) depicts the mass ratios of the currently known, well-characterized first-order resonant extrasolar planets (Wright et al. 2011) as well as some solar system examples. The sizes of the circles are representative of the planetary orbital radii in units of the primary's physical radius. Green circles denote resonant pairs with a more massive outer planet while blue circles denote systems with a more massive inner planet. Influenced by the emergence of observational detections, a handful of authors have studied the global resonant dynamics of the unrestricted three-body problem (Rivera et al. 2005; Callegari and Yokoyama 2007; Michtchenko et al. 2008). While quantitatively precise, the latter studies are generally tailored to particular systems, characterized by specific resonances and mass ratios. This renders the translation of the results to other systems and the acquisition of an overall understanding of the motion a di GLYPH<14> cult task. Indeed, a more physically intuitive and broadly applicable picture of resonant dynamics is desirable. Here, we shall set out to draw such a picture. As such, the analytical characterization of the global first-order resonant dynamics, is the primary purpose of this work.The paper is organized as follows. In the following section, we formulate a fully analytical, integrable treatment of resonant phenomena. Using the approximate theory, we construct surfaces of section that prove useful as a visual representation of the resonant motion. In section 3, we consider the onset of chaos via overlap of neighboring resonances as well as the incorporation of higher-order secular perturbations into the developed framework. In section 4, we apply the constructed formalism to divergent resonant encounters and examine the characteristic features of post-encounter dynamical states. We summarize and discuss our results in section 5.", "pages": [ 1, 2 ] }, { "title": "2.1. An Integrable Approximation", "content": "Our first aim is to construct an integrable approximation to the first-order resonant motion (i.e. the orbital period ratio P 1 = P 2 GLYPH<25> k = ( k GLYPH<0> 1) ; k 2 Z ) of two massive secondary bodies with masses m 1 and m 2, which orbit a much more massive ( M GLYPH<29> m 1 ; 2) primary. By convention, we shall take the subscripts 1 and 2 to denote the inner and outer orbits respectively. The exact Hamiltonian, H which governs the gravitational three body problem is characterized by six degrees of freedom. Specifically, the canonical heliocentric formulation of H reads (Poincaré 1902; Laskar and Robutel 1995; Morbidelli 2002): where G is the gravitational constant, p is the barycentric linear momentum, r 1, r 2 are the distances between the primary and the secondaries while GLYPH<1> 12 is the distance between the planets. Today, the availability of numerical tools for integration of the Hamiltonian (1) (Wisdom and Holman 1991; Duncan et al. 1998; Chambers 1999) allows for a prompt and precise realization of a given system's orbital evolution. However, any such realization provides a scarce theoretical basis for the characterization of the dynamics. Moreover, as was first pointed out by Poincaré (1902), such solutions may exhibit chaotic motion further obscuring candid interpretation. Consequently, rather than working with the Hamiltonian (1) directly, it is sensible to turn to the classical perturbation methods developed over the last four centuries, in search of a suitable approximation to the Hamiltonian (1). Throughout the following derivation, we shall be aided by numerous preceding contributions to the study of resonance in celestial mechanics. Specifically, we shall follow the pioneering work of Peale (1976) and Sessin and Ferraz-Mello (1984). The calculation will be greatly simplified by a reducing transformation (see Henrard et al. (1986); Wisdom (1986)) and the final Hamiltonian will closely resemble the second fundamental model for resonance (Henrard and Lemaitre 1983). It is useful to begin, (without fear of overstating the obvious) by pointing out that the combination of the first and third as well as second and fourth terms in equation (1) govern the Keplerian motion of the planets. It can be easily shown (Murray and Dermott 1999; Morbidelli 2002) that in terms of orbital elements, this Keplerian part of the Hamiltonian can be written as follows: where a is the semi-major axis. The remaining terms in the Hamiltonian (1) govern the planet-planet interactions and are much smaller in magnitude. Accordingly, it is often called the disturbing function, since it provides small perturbations to the integrable Hamiltonian (2) that are still important in the long term. A qualitative analysis of the dynamics can be performed by expanding the disturbing function as a Fourier series in the orbital angles and a power series of the planetary eccentricities and inclinations (Laskar and Robutel 1995; Laskar and Boué 2010). Accordingly, this procedure allows for the identification of resonant terms, that is, harmonics that vary on a timescale much longer than the orbital timescale in the vicinity of exact commensurability. While such terms are dynamically important and should be retained in the Hamiltonian, short-periodic terms (i.e. those that vary on an orbital timescale) can be readily averaged over and dropped from the Hamiltonian (Murray and Dermott 1999). It is noteworthy that in addition to short-periodic and resonant terms, the disturbing function also contains secular terms which do not depend on the mean longitudes of the planets. The leading secular terms are of order O ( e 2 ; i 2 ), where e and i are the eccentricity and inclination respectively. For the purposes of the construction of a first-order resonant theory, we shall neglect them, along with all resonant terms of order greater than unity in e and i . Additionally, we shall only retain terms that are linear in planetary masses. However, as will be shown in the subsequent sections, these higher-order terms play a crucial role in the onset of chaotic motion. In accord with the above-mentioned linear expansion of the disturbing function, we can approximate the Hamiltonian (1) as where the k : k GLYPH<0> 1 resonant perturbation to the Keplerian motion reads: Following conventional notation, $ denotes the longitude of perihelion and GLYPH<21> = M + $ is the mean longitude, M being the mean anomaly. The quantities f (1) res and f (2) res are of order unity to within a factor of a few and (weakly) depend on the semi-major axis ratio ( a 1 = a 2) only. Their values are tabulated in numerous references and can be easily evaluated numerically with the aid of computer algebra (see for example Callegari and Yokoyama (2007); Laskar and Boué (2010)). At the expense of working in a noninertial reference frame, we had to introduce the indirect term, p 1 GLYPH<1> p 2 = M , into the disturbing function that accounts for fixing the origin on the central body (Laskar and Boué 2010). However, this correction is trivial, given that all indirect terms in H res corresponding to the same harmonics have the same dependence on the actions as the direct ones, meaning that the indirect terms can be accounted for simply by modifying the coe GLYPH<14> cients f res. Although the functional form of the simplified Hamiltonian is given, Keplerian orbital elements do not form a canonically Thus to make further progress, we convert to conjugated set. Poincar'e action-angle variables defined as: where GLYPH<22> = mM = ( M + m ) ' m is the reduced mass. In terms of the Poincar'e variables, the Hamiltonians, H kep and H res take on Article number, page 4 of 21 the following forms respectively (Murray and Dermott 1999): Although we did not explicitly assume coplanar orbits, the linear expansion of the disturbing function contains no terms that depend on the longitudes of the ascending node (the third Poincar'e angle), or the orbital inclinations (related to the third Poincar'e action). This renders these quantities integrals of motion. Therefore, it is evident that the number of degrees of freedom of H has been reduced to four (although the distinct presence of only two harmonics in equation (8) suggests that H can be reduced to a two degrees of freedom system with ease). Because we are interested in near-commensurate planetary motion, it is sensible to expand the Keplerian Hamiltonian around the nominal resonant location. Carrying out the expansion to second order in GLYPH<14> GLYPH<3> = GLYPH<3> GLYPH<0> [ GLYPH<3> ], where [ GLYPH<3> ] is the nominal resonant value of GLYPH<3> , we have: Substituting the definition of GLYPH<14> GLYPH<3> into equation (9) and dropping the dynamically unimportant constant terms, H kep takes on the following remarkably simple form: Here, [ n ] = p G M = [ a ] 3 is the nominal mean motion and [ h ] = [ n ] = [ GLYPH<3> ] = 1 = ( m [ a ] 2 ). In accord with the above approximation, we shall also evaluate H res at [ GLYPH<3> ], as it is already of order O ( e ). Indeed, this step is relevant to the evaluation of the resonant strengths, as the coe GLYPH<14> cients f res can now be considered truly constant. In the limit of very small eccentricities and semi-major axes ratios exceeding the nominal resonant value (that is, assuming that the system remains close to the (pseudo-)resonant equilibrium point, which is in turn taken to be close to the origin of the phase-space portrait), the dynamics governed by Hamiltonians (10) and (8) can be treated linearly. An analysis of this kind has recently been performed by Batygin and Morbidelli (2012) and the resulting equations were used to study the resonant evolution of close-in planets under the e GLYPH<11> ect of tides. In this work, we wish to provide a more general picture of resonant motion that is not limited to the vicinity of any equilibrium point. Consequently, here, we retain the nonlinear coupling of the actions inherent to (8). The functional form of the resonant harmonics can be simplified considerably by employing a canonical contact transformation of variables, arising from the following generating function of the second kind (Sessin and Ferraz-Mello 1984): An application of the transformation equations GLYPH<3> = @ F 2 =@GLYPH<21> yields new action-angle variables: Upon substitution of the new variables into the Hamiltonian and utilizing the resonant relationship ( k GLYPH<0> 1)[ n ]1 = k [ n ]2, the Keplerian Hamiltonian becomes Meanwhile, the resonant contribution to H now takes the form: where Note that GLYPH<20> is no longer present in H . Thus, K is now a constant of motion and the number of degrees of freedom of H has been reduced to three. Physically, the conservation of K arises from the fact that the resonant Hamiltonian depends on the semi-major axis ratio rather than the semi-major axes themselves (recall that K / GLYPH<3> / p a ). Accordingly, Michtchenko et al. (2008) have dubbed K a 'scaling parameter\" (see also their discussion of K 's significance and its relationship to the behavior of the semimajor axes outside the resonant domain). The utility of K can be illustrated intuitively by expressing it in a dimensionless form: For a given mass ratio, we can choose a nominal semi-major axis ratio, [ a ]1 = [ a ]2 and obtain the value of K = [ GLYPH<3> ]2 accordingly. Although K is simply a constant of motion and can in principle take on arbitrary values, without loss of generality, we can choose [ GLYPH<3> ]2 = 1, thereby defining a natural value of K . In this sense, the actual value of K is simply representative of the units in which the semi-major axes are measured. Once the value of K is fixed, both planetary semi-major axes, a 1 and a 2 are unequivocally defined given their ratio, a 1 = a 2. The conservation of K is of additional importance, as it yields the location of nominal mean motion, [ n ], around which we have chosen to expand the Keplerian Hamiltonian (7). In the unrestricted problem, the semi-major axes of both planets deviate away from nominal commensurability during a resonant cycle. The extent of such deviation is dependent on the planetary masses. Thus, it would seem that the nominal locations of the semi-major axes are not defined a-priori. This issue is remedied by the fact that K encapsulates the planetary mass ratio. Consequently, given an (observed) pair of semi-major axes, the conserved value of K can be used to compute their nominal counterparts by setting a 2 = (( k GLYPH<0> 1) = k ) 2 = 3 a 1 in equation (16). For further calculations, we shall drop the first and last terms in the Hamiltonian (13) because they are constant. Doing so does not change the overall picture of the dynamics with the exception of the eliminated ability to derive the time evolution of the individual mean longitudes, GLYPH<21> 1 and GLYPH<21> 2. However, all other information, including the behavior of the resonant critical arguments in equation (8), is retained despite this simplification. The Hamiltonians (13) and (14) are equivalent to (in fact, a trivial transformation away from) those considered by Sessin and Ferraz-Mello (1984). Through a rather involved calculation, utilizing the perturbation method devised by Hori (1966), these authors demonstrated the integrability of the first-order resonant motion. In a subsequent e GLYPH<11> ort, aimed primarily at the elliptic restricted three-body problem, Henrard et al. (1986) and Wisdom (1986) greatly simplified the Sessin and Ferraz-Mello (1984) solution by introducing a canonical transformation that explicitly identifies a novel constant of motion. Here, we shall follow the latter approach. Turning our attention to the resonant contribution to H , let us transform the remaining Poincar'e variables ( GLYPH<0> ; GLYPH<13> ) to mixed secular cartesian coordinates (Murray and Dermott 1999) where y is identified as the coordinate and x is the conjugate momentum. After some manipulation, the resonant Hamiltonian reads: At this point, the Hamiltonian (18) is ready for yet another change of variables. In particular, we introduce the rotation formulated by Henrard et al. (1986); Wisdom (1986) The canonical nature of this transformation can be verified by the Poisson bracket criterion f vi ; uj g ( yj ; xj ) = GLYPH<14> i ; j (Morbidelli 2002). Upon doing so, we can immediately identify v as the coordinate and u as the conjugated momentum. Defining implicit action-angle polar coordinates ( GLYPH<8> ; GLYPH<30> ) as we can re-write the expression for the Hamiltonian (18) in a substantially simplified form: Note that the Hamiltonian (21) does not depend on GLYPH<30> 2. Additionally, recall that e GLYPH<11> ectively, the Keplerian Hamiltonian (13) only depends on GLYPH<2> . Evidently, the newly defined action GLYPH<8> 2 is another constant of motion and the number of degrees of freedom of H has been reduced to two. Reverting to the original variables, the conservation of GLYPH<8> 2 implies that where GLYPH<11> 0 = GLYPH<11> p [ GLYPH<3> ]2 and GLYPH<12> 0 = GLYPH<12> p [ GLYPH<3> ]1. Article number, page 6 of 21 We are only one degree of freedom away from integrability. Fortunately, the final reduction is arguably the simplest. However, before proceeding to the final transformation, let us briefly digress and rescale the total Hamiltonian. An examination of the expressions (13) and (21) reveals that currently, H is parameterized by GLYPH<11> , GLYPH<12> , [ h ]1 and [ h ]2. Following Henrard and Lemaitre (1983), we wish to combine these values into a single parameter, ˆ GLYPH<14> (not to be confused with the Kronecker delta, GLYPH<14> i ; j ). Rescaling the actions and the Hamiltonian (while retaining the angles as before) by a constant factor GLYPH<17> , such that we shall require that the constant in front of the GLYPH<2> 0 2 term in (13) be the same as the constant in (21). In order for this transformation to remain canonical, we must also divide H by GLYPH<17> . The expression for the scaling factor is: Choosing to measure time in units of 3([ h ]1( k GLYPH<0> 1) 2 + [ h ]2 k 2 ) = 2, which in turn allows us to again divide H by the same factor, we obtain an elegant expression for the scaled total Hamiltonian: Accordingly, the parameter ˆ GLYPH<14> reads: We now return to the issue of the final canonical transformation. The change of variables we are after is given by the following generating function of the second kind: Taking the appropriate derivatives as above, we obtain our final action-angle variables: Upon conversion to these variables, we arrive at an integrable one degree of freedom Hamiltonian: Indeed, the newly defined angle ! is no longer present in the Hamiltonian (29), rendering GLYPH<10> our last constant of motion. An examination of the expressions for GLYPH<10> and K reveals that the total angular momentum of the (planar) system, L , is given by Furthermore, it is relevant to note that the planar angular momentum deficit, A , which is conserved far away from mean motion resonances (i.e. in the secular domain - see Laskar (1997)) is conveniently given by At first glance, the transformation (28) may appear odd, because of the implicit choice to introduce the constant of motion GLYPH<9> 2 explicitly into the Hamiltonian. However, this selection is deliberate and will turn out to be useful in the next section, where it will be shown that the conservation of GLYPH<9> 2 is destroyed when higher-order interactions are taken into account (accordingly, the integrability of the Hamiltonian (29) is also compromised at higher order). As already mentioned above, the Hamiltonian (29) is equivalent to the widely-discussed second fundamental model of resonance (Henrard and Lemaitre (1983), Peale (1986), Murray and Dermott (1999) and the references therein) and is therefore closely related to the pendulum model for resonance (Peale 1976; Yoder 1973). In other words, the dynamics of the unrestricted resonant problem exhibits qualitatively similar behavior to the broadly studied restricted problem, although the physical meanings of the involved variables are di GLYPH<11> erent. An important distinction is that in the context of the Hamiltonian (29), the proximity to exact resonance is not given as a single parameter (as it is in the context of the restricted problem), but rather a combination of K ; GLYPH<10> and GLYPH<9> 2. That said, it should be noted that K and GLYPH<10> dictate the angular momentum surface on which the dynamics resides, while GLYPH<9> 2 is related to the initial conditions of the system confined on such a surface. Thus, it makes sense to treat GLYPH<9> 2 as an e GLYPH<11> ective measure of proximity to exact resonance, given fundamental parameters of the system. Level curves of the Hamiltonian for values of K and GLYPH<10> that correspond to the HD 82943 entry in Table (1) (Mayor et al. 2004) and various values of GLYPH<9> 2 are presented in Figure (3). The curves on the panels correspond to di GLYPH<11> erent energy levels and are color-coded in order to highlight the distinct nature of the dynamics they entail. Specifically, the black and blue curves denote resonant and nonresonant trajectories respectively, while the gray curves on panels A, B and C denote critical curves that separate the resonant and nonresonant regions of the phase space. Note that the presence of the gray curves depends on the value of GLYPH<9> 2 that characterizes the plot. For instance, the orange curve on panel D describes libration in absence of a critical curve. Fixed points of the Hamiltonian are marked with black dots, where filled circles correspond to stable equilibria and the converse is true for open circles. A more detailed discussion of the motion described by these dynamical portraits and how they relate to the envisioned behavior of the orbits is presented below.", "pages": [ 2, 3, 4, 5, 6, 7, 8 ] }, { "title": "2.2. A Geometrical Representation of Resonant Dynamics", "content": "Armed with an integrable approximation to resonant motion, we may now take advantage of the various integrals identified above in order to formulate a geometrical representation of the orbital evolution. As a result of the numerous canonical transformations employed in the derivation of the Hamiltonian (29), the final variables (28) are rather serpentine. Consequently, here we shall opt to obtain the solutions as shown above, but subsequently work backwards through the transformations in order to represent the resonant behavior in terms of the Keplerian elements. We begin by defining the representative plane. As already mentioned in the discussion of transformation (12), the actual values of the semi-major axes determine the timescale on which resonant perturbations occur, rather than the form of the interactions themselves (Murray and Dermott 1999). We can therefore use the conservation of K to introduce the semi-major axis ratio, a 1 = a 2 as our first independent variable. Keeping in mind that the problem we consider is e GLYPH<11> ectively planar, it is natural to turn to the definition of the angular momentum, L , for further development. Upon substitution of the definition of K and a 1 = a 2 into equation (30), the conservation of L yields one of the eccentricities as a second independent variable. Because the variables ( GLYPH<9> ; ) implicitly originate from the definition of the vectors ( x ; y ) given by equations (17), they encompass information about the planetary eccentricities as well as the di GLYPH<11> erence in the longitudes of perihelia. Accordingly, a final requirement for the delineation of parameter space in question is a condition on the apsidal angles of the orbits. Clearly, the full time-evolution of the resonant dynamics requires a threedimensional manifold, defined by ( a 1 = a 2 ; e 1 or e2 ; GLYPH<1> $ ). However, a suitable representation of the dynamics can still be obtained by constructing a surface of section, choosing GLYPH<1> GLYPH<13> = (0 ; GLYPH<6> GLYPH<25> ) as an intersection plane 1 . Importantly, in our section, we shall not discriminate based on the direction of the trajectory's encounter with the GLYPH<1> GLYPH<13> = (0 ; GLYPH<6> GLYPH<25> ) plane. As will become clear below, this choice allows us to readily distinguish between apsidally librating and circulating orbits. To summarize the above discussion, we choose to represent the resonant dynamics on a ( a 1 = a 2 ; e cos( GLYPH<1> GLYPH<13> ) = GLYPH<6> e ) surface of section. As expository examples, figures (4) and (5) depict such sections, roughly corresponding to the 2:1 resonant dynamics of the HD 82943 system (Mayor et al. 2004; Lee et al. 2006) and the 3:2 resonant dynamics of the HD 45364 system (Correia et al. 2009; Rein et al. 2010) respectively. The two panels in each figure show both, e 1 and e 2 for completeness, although as mentioned already, this is a redundancy, and upon examination it is clear that the two figures are a simple vertical rotation away from each other. Both of the planetary systems we use as examples here were detected by the radial velocity technique, and are comprised of giant planets around Sun-like stars. Notably, the masses of the planets are comparable ( m 2 = m 1 ' 0 : 9 for HD 82943 and m 2 = m 1 ' 3 : 5 for HD 45364, see also Figure 2) preventing a description of the dynamics within the context of the restricted three-body problem. While it is firmly established that both of these systems are indeed resonant, some uncertainties exist in the orbital fits to the radial velocity data (see Lee et al. (2006), Rein et al. (2010)). We reiterate that for the purposes of this work, we shall only use these planetary pairs as illustrative examples, with little desire to quantify the exact nature of their dynamics. Furthermore, having picked a mass ratio and an angular momentum surface, we shall survey other parameters (e.g. orbital energy, values of GLYPH<9> 2) freely in order to epitomize an approximate yet global, rather than a precise but delimited picture of the dynamics. As a baseline, we shall adopt the (long-term stable) orbital solutions of Lee et al. (2006) for HD 82943 (specifically, Fit 2) and Correia et al. (2009) for HD 45364 (listed in Table 1 for convenience), ignoring the possibility that improved fits to the data may yield somewhat di GLYPH<11> erent orbits. Within the context of each section presented in Figures (4) and (5), an admissible region can be defined by the conservation of L , along with the requirement that the eccentricities remain real. The admissible regions are delineated by bounding orange lines. Meanwhile, straight vertical lines depict the nominal semimajor axes of the shown resonances. In addition to K and L , the Hamiltonian (29) is characterized by conservation of GLYPH<9> 2. Consequently, within the admissible region, the dynamics must reside on contours of GLYPH<9> 2. These contours are shown as black lines with the background color indicating the values (dark blue stands for GLYPH<9> 2 = 0). Note that for the nominal value of a 1 = a 2, each contour of GLYPH<9> 2 is intersected twice. For some contours (i.e. those below level A on Figure 4) both intersections take place at GLYPH<1> GLYPH<13> = GLYPH<6> GLYPH<25> . This means that the dynamics that reside on this GLYPH<9> 2 level are characterized by antialigned libration of the periapsis. For other levels of GLYPH<9> 2, (e.g level B of Figure 4), one intersection occurs for GLYPH<1> GLYPH<13> = 0 and one for GLYPH<1> GLYPH<13> = GLYPH<6> GLYPH<25> . This implies that the dynamics is characterized by circulation of the di GLYPH<11> erence of the periapses. Finally, for higher levels of GLYPH<9> 2, (e.g. level C on the same figure), both intersections occur at GLYPH<1> GLYPH<13> = 0, implying that the dynamics is characterized by aligned apsidal libration. For each combination of K , L and GLYPH<9> 2, there exists an energy level that separates librating and circulating orbits. Accordingly, such energy levels correspond to the maximal attainable libration widths and thereby define the resonant domain in parameter space (Morbidelli 2002). Examples of such energy levels are shown as gray curves in panels A, B, C and as an orange curve in panel D of Figure (3). If an energy level of this sort describes an orbit that passes through a hyperbolic fixed point, (shown as open circles on panels A, B and C of Figure 3), such an energy level is referred to as a separatrix or a critical curve. Strictly speaking, resonant orbits are exclusively those that reside within the croissant-shaped domain encompassed by the separatrix (see for example Delisle et al. (2012)) e.g. the black curves in panels A, B, C in Figure (3). Conversely, if regions of phase-space occupied by librating and circulating orbits are separated by a regular curve (such as the orange curve in panel D of Figure 3), none of the trajectories (even librating ones) are technically resonant, although numerous authors including ourselves (e.g. Poincaré (1902); Henrard and Lemaitre (1983); Batygin and Morbidelli (2012)) have loosely used resonance and libration as synonyms. While in certain applications, this mixing of definitions does not pose significant problems, here, in interest of avoiding confusion, we shall refer to libration in absence of separatrix as a pseudo-resonance and retain the strict definition for true resonant motion. The boundary of the resonant domain is represented in Figures (4) and (5) by white points along contours of GLYPH<9> 2, while the pseudo-resonant domain is bounded by yellow points. In other words, regions confined by white points and shown as thick black curves, correspond to true separatricies such as those shown in panels A, B and C of Figure (3). The regions confined by yellow points and shown as thick orange curves correspond to regular trajectories that separate libration and circulation. These points depict sections of the dynamics at 1 = GLYPH<25> . Accordingly, from Figure (3), it is clear that resonant and pseudo-resonant orbits intersect the 1 = GLYPH<25> line twice, as opposed to circulating orbits that intersect the 1 = GLYPH<25> line only once 2 . Note the deliberate parallel between curves labeled A, B, C, Din Figure 4 and the panels in Figure 3). Indeed, the two figures represent the same dynamics, depicted in di GLYPH<11> erent spaces. The concave part of GLYPH<9> 2 contours, residing to the right of the resonant domain in Figures (4) and (5) (shown as thin black lines) can be identified as the inner circulation region shown in panels A, B, and C of Figure (3), while the parameter space to the left of the resonant domain respectively corresponds to the outer circulation region. Of course, the inner circulation region disappears along with the separatrix. Consequently, the separatrix disappears on a level of GLYPH<9> 2 where the right hand side of the thick black curve reaches the rightmost extreme of the GLYPH<9> 2 contour. Because the two thick black curves on the upper and lower sides of the GLYPH<9> 2 contour are symmetric, the disappearance of the separatrix can be understood as taking place when the right-hand sides of two such curves join. Specifically, the curve labeled C in Figure 4 is close to such a transition. Recall that together, K , L and GLYPH<9> 2 constitute a measure of proximity to exact resonance. Indeed, once defined, we have all the ingredients to construct phase-space diagrams such as those shown in Figure (3). Bearing in mind that each trajectory that resides on such a diagram is characterized by its energy level, we identify the value of H itself as the final geometrical constraint on the dynamics. Contours of H , evaluated at GLYPH<1> GLYPH<13> = (0 ; GLYPH<6> GLYPH<25> ) and sectioned at 1 = GLYPH<25> , are shown as green lines on the surfaces of section. Importantly, the intersections of H contours with GLYPH<9> 2 contours depict the resonant libration width of a given configuration. Notice the inherent shape of H : some contours are concave, while others are convex. The best-fit orbital solutions of Lee et al. (2006) and Correia et al. (2009) are shown on the surfaces of section using white points connected by gray lines. Both solutions reside deep within the resonance signaling qualitative agreement of the integrable model with the true rendition of the dynamics. However, quantitatively, one should not expect the agreement between real resonant dynamics and the integrable approximations to be particularly good, because the convergence of the perturbation series employed here is questionable at best (especially at moderate eccentricities, provided that the orbits may intersect). Additionally, one should keep in mind the fact that thus far, we have neglected any terms in the disturbing function of order e 2 or greater, further spoiling the approximation 3 . Figure (6) illustrates a comparison between the dynamical evolution of the (Correia et al. 2009) fit obtained by numerical integration of Hamiltonian (1) (shown as black lines) using the mercury6 integration package (Chambers 1999) and the analytical solution that arises from the approximate Hamiltonian (29) (shown as red lines). The panels A, B and C depict the time evolution of the resonant angle GLYPH<18> + GLYPH<13> 1 = 3 GLYPH<21> 2 GLYPH<0> 2 GLYPH<21> 1 GLYPH<0> $ 1, the di GLYPH<11> erence of the apsidal angles, GLYPH<1> $ and the eccentricities respectively. Clearly, the characteristic frequencies of the oscillations di GLYPH<11> er by a factor of a few, however the amplitudes are well captured within the context of the approximate model.", "pages": [ 8, 9, 10 ] }, { "title": "3.1. Overlap of Mean Motion Resonances", "content": "The orbital architecture of small bodies in our solar system highlights the fact that resonances may exhibit both, regular and highly chaotic motion. In particular, while Neptune's external 3:2 and 2:1 mean motion resonances are densely populated with Kuiper belt objects (Morbidelli et al. 2008), Jupiter's interior 2:1 and 3:1 resonances, that coincide with Kirkwood gaps of the asteroid belt are cleared out. The removal of resonant asteroids is now understood to be a result of chaotic di GLYPH<11> usion that drives asteroids onto Mars-crossing orbits (Wisdom 1985; Henrard and Lemaitre 1987; Henrard and Caranicolas 1990). The same rationale is applicable to the unrestricted problem we address here. It is well known that overlap among neighboring resonant domains gives rise to chaotic di GLYPH<11> usion (Walker and Ford 1969; Chirikov 1979; Wisdom 1980). Consequently, the approximate (strictly periodic) model derived above is of virtually no use to the description of energy levels that allow the corresponding orbits to penetrate neighboring resonances. In other words, the domain of applicability of the integrable model is in part determined by a given trajectory's proximity to a neighboring separatrix. Although there is no reason why all separatricies should lie on the same energy level (in fact they don't), it can be seen in Figures (4) and (5) that the resonant and pseudo-resonant domains are approximately bounded by the transition between concave and convex energy levels. This similarity can be taken advantage of, to map the approximate locations of the neighboring resonances. This portrayal of the onset of chaos is by no means intended to be precise and is strictly speaking heuristic since the separatricies are obtained by sectioning each resonant dynamics relative to di GLYPH<11> erent critical angles, even if they lie on the same angular momentum surface. However, we do not view this as a significant drawback, since the Chirikov resonance overlap criterion is in itself an approximation that neglects the coupling of the resonances and the resulting deformation of their shape, as well as the generation of higher order secondary resonances (that act to expand the size of the chaotic zone). As an example, consider the level of GLYPH<9> 2 adjacent to the Correia et al. (2009) orbital solution shown in Figure (5). While it was shown in the previous section that the orbital fit itself (roughly corresponding by the second inner most energy level) is moderately well represented by the analytical Hamiltonian (29) (see Figure 6), we can anticipate that the same will not be true of the solution characterized by the second outermost energy level, since the orbit resides in close proximity to the separatricies of the 3:2 and the 4:3 resonances. A straight forward way to account for the e GLYPH<11> ects of both, the 3:2 and the 4:3 mean motion resonances is to construct a Hamiltonian of the form where the Keplerian term is given by equation (7) and the two resonant contributions each take the form of equation (8), with k = 3 ; 4 and appropriately chosen coe GLYPH<14> cients, f res (note that choosing to not expand the Keplerian Hamiltonian around any nominal resonant location further contributes to the nonlinearity of the system and acts to expand the chaotic zone). Such a Hamiltonian possesses four degrees of freedom and four harmonics hindering further simplification. As a result, we integrate the equations of motion that result from the Hamiltonian (32) using conventional numerical methods. The resulting solution exhibits rapid dynamical chaos, as is made evident by the eccentricity and semi-major axis ratio evolution shown in figure (7) with blue lines. Indeed, the timescale for the onset of irregularity is comparable to the orbital timescale. We have repeated the numerical experiment with an N-body simulation as above and confirmed the fully chaotic nature of the configuration in question. Specifically, within the context of the N-body simulation, the stochastic evolution comes to a rapid end GLYPH<24> 40 years into the integration, when the planets collide. The N-body results are shown in the figure with black lines. Meanwhile, the analytical model given by the Hamiltonian (29) predicts regular oscillations for the same configuration, as can be gauged from the red lines shown in the figure. Here we have chosen a somewhat extreme example to demonstrate planetary chaos. However, this exercise highlights the dangers and the associated care that must be taken during application of the simple model described in the previous section.", "pages": [ 10, 11 ] }, { "title": "3.2. Secular modulation of Resonant Motion", "content": "Let us now turn our attention to a region of the resonant domain that is well-separated from the neighboring mean motion resonances. Over a su GLYPH<14> ciently short period of time (which is related to the timescale on which resonant interactions exchange energy and angular momentum between the orbits), the Hamiltonian (29) provides a suitable approximation to the motion. However, if we wish to characterize the behavior of the system over a longer (secular) timescale, we are forced to retain additional terms in the disturbing function (Laskar 1996). This is due to the fact that characteristic resonant frequencies are proportional to / p m = M while secular frequencies are proportional to / m = M , giving rise to an inherent separation of timescales between the integrable Hamiltonian and the secular correction (Henrard and Caranicolas 1990). As such, we extend our perturbation series to account for second-order secular coupling of the orbits 4 : where in terms of Keplerian orbital elements (Murray and Dermott 1999), As before, we shall evaluate H sec at nominal semi-major axes, rendering f sec constant coe GLYPH<14> cients that depend on the semimajor axis ratios only. It should be noted that H sec does not provide the only secular contribution to the dynamics at second order in e . Resonant terms at second order in e , once averaged over a libration or a circulation cycle of 1, also give rise to pure secular terms that can be as large as those given in (33). Here, we opt to discard such terms for the sake of simplicity, especially given that our aim is merely to demonstrate the qualitative impact of secular terms (that is the generation of chaos) on the integrable approximation developed above. Following the same procedure outlined in the derivation of the resonant Hamiltonian, we first revert to Poincare actionangle coordinates. The secular Hamiltonian now takes the form: 4 For k : 1 type resonances, asymmetric resonant librations are possible (Beauge 1994; Ketchum et al. 2012). Consequently, for certain combinations of parameters, the phase-space portrait of the 2:1 mean motion resonance may be topologically di GLYPH<11> erent from that shown in Figure (3). It is noteworthy that Hamiltonian (35) depends only on a single harmonic and can thus be easily transformed into a one degree of freedom Hamiltonian, recognizing the angular momentum deficit, A , as a secular constant of motion. Indeed, in isolation, H sec is integrable and the solution is referred to as the Laplace-Lagrange secular theory (Murray and Dermott 1999). Upon employing the transformation to eccentricity vectors given by equations (17), the linear nature of the equations of motion that arise from H sec becomes apparent: An application of the canonical rotation transformation (19) converts H sec into a more cumbersome form: Finally, combining transformations (20) & (28), and rescaling the time as above, we can express the full Hamiltonian as: The Hamiltonian (39) is characterized by two degrees of freedom, and as will become apparent shortly, exhibits chaotic motion. This implies that no canonical transformation can be found to identify additional constants of motion. However, prior to working with equation (39), it is worthwhile to examine the timescales on which the two degrees of freedom evolve, and identify the relevant regimes of motion, corresponding to commensurability and separation between the characteristic frequencies. Let us fist examine the conditions for commensurability and the generation of secondary resonances.", "pages": [ 11, 12 ] }, { "title": "3.2.1. Secondary Resonances", "content": "In the framework of the unrestricted resonance problem, the numerical simulations of Michtchenko et al. (2008) showed that at very low eccentricities, secular and resonant angles can evolve on comparable timescales, giving rise to secondary resonances. With an integrable approximation to resonant motion in hand, we can examine the criteria for the appearance of secondary resonances analytically. More specifically, we shall aim to find conditions under which the period of resonant libration is close to a low-order integer ratio with the apsidal period. To estimate the former, we expand Hamiltonian (29) as Taylor series in ( GLYPH<9> 1 ; 1) to second order, around the resonant equilibrium point, ([ GLYPH<9> ]1 ; GLYPH<25> ). Dropping constant terms, and defining the variables we have: As long as the barred quantities remain small (that is, the system does not deviate away from equilibrium much), this simplification directly implies nearly-constant eccentricities and apsidal anti-alignment of the orbits i.e. GLYPH<1> GLYPH<13> ' GLYPH<25> . Furthermore, because we are expanding the Hamiltonian around a fixed point, to linear order, d ¯ 1 = dt = 0, meaning This expression automatically defines the nominal value of the action [ GLYPH<9> ]1 for a given combination of ˆ GLYPH<14> , GLYPH<10> and GLYPH<9> 2, while further simplifying the Hamiltonian (41), as now only the quadratic terms remain. Finally, after applying the transformation Hamiltonian (41) reduces to that of a harmonic oscillator: where ' res is immediately identified as the resonant libration frequency. By working back through the canonical transformations outlined in the previous section, it can be easily shown that because d ¯ 1 = dt = 0 to leading order, the average apsidal frequency coincides with that of the cyclic angle 2. An application of Hamilton's equations yields where the latter equality follows from equation (42). Equating the two frequencies, we find that the 1:1 secondary resonance only exists in the unphysical limit of [ GLYPH<9> ]1 ! 0, which corresponds to null eccentricities. However, higher order secondary resonances are indeed permitted at small values of [ GLYPH<9> ]1, in agreement with the work of Michtchenko et al. (2008) (see also Morbidelli and Moons (1993)). Figure (8) shows the two frequencies as a function of [ GLYPH<9> ]1, and the 3:2, 2:1 and the 3:1 secondary resonances are labeled for reference. Having dropped the secular terms from the Hamiltonian (41), we have implicitly limited the scope of the above calculations to systems where 2 circulates. It is however important to note that upon inclusion of secular terms, libration of 2 is possible within a limited range of parameter space, rendering the above calculation inapplicable. Such configurations will be discussed in section 5.", "pages": [ 12, 13 ] }, { "title": "3.2.2. Adiabatic Evolution", "content": "We now turn our attention away from the characteristic domain of secondary resonances, and towards the parameter regime that is more typical of the exoplanetary systems discussed in the previous section. As shown in Figure (8), low-order secondary resonances are not possible if [ GLYPH<9> ]1 is su GLYPH<14> ciently large. In this case, the libration frequency of 1 is much higher than the circulation frequency of the secular angle, 1 GLYPH<0> 2. It is therefore sensible to transform the variables accordingly and employ the separation of timescales between the two degrees of freedom to our advantage (Henrard 1982). The transformation we seek is given by the generating function which yields the variables The new action-angle variables are actually somewhat more intuitive than the previous. Specifically, as can be understood from equation (31), GLYPH<4> = A =GLYPH<17> is simply the re-scaled angular momentum deficit. Meanwhile, a modulation of the action, GLYPH<7> , changes the GLYPH<9> 2 contour on which the resonant dynamics resides in surfaces of section (4) and (5). In terms of the new variables, the Hamiltonian reads: Before considering an example that highlights the onset of chaos through secular modulation, let us reflect on the somewhat satisfactory agreement between the N-body simulation and the analytical treatment of the (Correia et al. 2009) orbital solution shown in Figure (6). A numerical solution of the equations of motion that arises from the Hamiltonian (48) (using the initial conditions listed in Table 1) is shown with purple lines in Figure (6). This solution demonstrates that rather than introducing chaos, the addition of secular terms (unsurprisingly) improves the agreement between the perturbative treatment of the dynamics and the N-body simulation. Specifically, both the amplitude and frequency of oscillations in the apsidal angle GLYPH<1> GLYPH<13> and the eccentricities are decreased compared to the analytical results stemming from the Hamiltonian (29), better matching the N-body calculations. Indeed, the introduction of higher-order terms does not render the entire phase-space chaotic. The dynamical portrait of a two degrees of freedom system, cannot be represented visually in a simple fashion. However, it is still instructive to visualize the behavior of one of the degrees of freedom by freezing the evolution of the second degree of freedom. In particular, here we choose to set GLYPH<7> = const :; GLYPH<29> = 0. This is especially relevant to the dynamics described by the Hamiltonian (48) because the evolution timescales of the two degrees of freedom are well-separated. Maintaining a parallel with the discussion of the previous section, Figure (9) shows surfaces of section of the level curves of the Hamiltonian (48), for the same values of GLYPH<7> = GLYPH<9> 2 as those shown in Figure (3). As before, black, blue and gray curves denote resonant orbits, nonresonant orbits, and separatricies respectively. Pseudo-resonant orbits are marked as orange lines and are shown in panels A and B. As expected, these pseudoresonant trajectories do not circle the center of the figure and therefore imply libration. Filled black dots denote stable equilibria while open circles mark unstable fixed points. Although Figures (9) and (3) show essentially the same dynamical portraits, it can be argued that visualization in terms of the variables (47) is more instructive. Most importantly, the phase-space portrait retains the same location of the separatrix for all values of GLYPH<7> , at the expense of introducing an inadmissible region, marked by a light purple circle centered on the origin. The inadmissible region itself is defined by the condition GLYPH<4> 6 GLYPH<7> . Recalling that GLYPH<4> is related to the angular momentum deficit, the physical interpretation of the inadmissible region is simply the requirement that the eccentricities never acquire a complex component: = ( e ) = 0. An equivalent interpretation of the boundary of the inadmissible region is that it represents a stretched out origin of the panels in Figure (3) and thus corresponds to e 1 = e 2 = 0. An advantage of this representation is that the disappearance of the separatrix can be easily understood to be a result of the changes in GLYPH<7> . Indeed as the value of GLYPH<7> grows from panel C to panel D, the separatrix is engulfed by the inadmissible region, leaving only nonresonant trajectories to fill the phase-space. With this interpretation in mind, it can be intuitively understood why the introduction of secular terms into our model can give rise to chaotic motion. Namely, while GLYPH<7> is a constant of motion in the context of the Hamiltonain (29), it ceases to be constant with the introduction of secular terms. Referring back to Figures (4) and (5), the modulation in GLYPH<7> can be visualized as a vertical translation across contours of GLYPH<9> 2, while confined to a particular energy level, denoted by green lines. Because the evolution timescales of the two degrees of freedom are very distinct, the distortions of the orbit in the ( GLYPH<4> ; GLYPH<24> ) plane that result from the modulation, preserve the adiabatic invariant, defined as (Henrard 1982; Neishtadt 1984): Physically, the action J represents an area occupied by a given orbit. The conservation of J thus implies that any distortion of the orbit in the ( GLYPH<4> ; GLYPH<24> ) plane must be area-preserving. An important exception to this principle, intimately related to the onset of chaos, is that the conservation of J is broken when a trajectory encounters a critical curve. Consider an initially resonant orbit such those depicted by black lines in Figures (9). As long as the modulation of GLYPH<7> is such that the inadmissible region remains far from the resonant orbit (e.g. taking panels A and B as the extremes of the modulation), the resonant region is not a GLYPH<11> ected much. However, if we consider a stronger modulation (e.g. taking panels A and C as the √ √ extremes), it can be immediately seen that the area available for libration may shrink during a modulation cycle. In such a scenario, when the area of the separatrix becomes equal to the area occupied by the trajectory, the trajectory is forced to cross the separatrix, inevitably passing through the unstable (hyperbolic) equilibrium point. This marks the onset of chaotic motion. Following along the same lines of reasoning, one may deduce that if the separatrix disappears and reappears during a modulation cycle (e.g. taking panels C and D as the extremes), a considerable fraction (which depends on the modulation amplitude) of the trajectories may be understood to be chaotic. With a handle on the role that the conservation (or lack thereof) of J plays, a nearly complete picture of the dynamics can be gleamed by sectioning the orbit in the ( GLYPH<4> ; GLYPH<24> ) plane and examining its evolution in the ( GLYPH<7> ; GLYPH<29> ) plane (Wisdom 1985; Henrard and Caranicolas 1990). If the surface of section reveals a closed, regular orbit in the ( GLYPH<7> ; GLYPH<29> ) plane, it automatically implies that J is conserved along the evolutionary path and the separatrix was never encountered. Conversely, if the surface of section in ( GLYPH<7> ; GLYPH<29> ) plane reveals an area-filling manifold, conservation of J is broken as the orbit repeatedly encounters a separatrix in the ( GLYPH<4> ; GLYPH<24> ) plane (Henrard 1982; Morbidelli 2002). Indeed the situation is √ quite analogous to the well-studied problem of a modulated pendulum (Elskens and Escande 1991; Bruhwiler and Cary 1989). A surface of section of the (Correia et al. 2009) orbital solution is shown as a thick purple line in Figure (10). The apparent regularity of the observed motion raises the question if any 'relative\" of the considered orbital fit, sharing the same values of H ; K and GLYPH<10> can exhibit chaos. In order to address this, we surveyed the dynamical evolution of such orbits, characterized by di GLYPH<11> erent values of GLYPH<7> . A few examples of such orbits are plotted as thin purple lines on Figure (10). As can be gathered from the figure, the entire phase-space available to such orbits is occupied by regular trajectories. Evidently, any secular modulation of GLYPH<7> permitted by the orbital energy and angular momentum corresponding to the (Correia et al. 2009) fit is not large enough to drive the orbit through a separatrix. This is not particularly surprising, since an examination of Figure (5) explicitly shows that the energy level on which the (Correia et al. 2009) fit resides never approaches the vicinity of the separatrix. In other words, the (Correia et al. 2009) orbital solution is too deep within the resonance to exhibit chaotic motion. Note further that the circulation of GLYPH<29> seen in Figure (10) is fully consistent with libration of GLYPH<1> $ seen in Figure (6). Indeed, if the asymmetry (that is controlled entirely by secular terms) of the orbit in the ( GLYPH<29>; GLYPH<7> ) plane is not large, and the value of GLYPH<7> (equivalently GLYPH<8> 2) remains su GLYPH<14> - √ ciently small (e.g. GLYPH<8> 2 exceeds the value corresponding to level A in Figure 4), the angle between the apsidal lines of the orbits remains in libration. As already shown within the context of our discussion of resonance overlap, retaining the same starting level of GLYPH<7> as that of the (Correia et al. 2009) fit and pushing the initial condition to an energy level that is close to the separatrix, will indeed result in highly irregular motion. However, the motion will not be irregular as a result of our sought-after e GLYPH<11> ect, the secular modulation. Consequently, in order to demonstrate the onset of chaos due to secular interactions more coherently, let us relocate our discussion to the 2:1 resonance and choose a starting value of GLYPH<7> and an energy level such that the unperturbed solution lies close to the 2:1 separatrix yet far enough away from the 3:2 separatrix for the perturbations from the neighboring resonance to rapidly average out. The initial condition we shall consider lies on a contour of GLYPH<7> directly above the one labeled B in Figure (4) and on the energy level that intersects the contour immediately inside the separatrix. For convenience, the unperturbed version of the starting state in question is labeled by a dashed white line in Figure (4). Naturally, as can be inferred from the figure, the unperturbed solution is characterized by large-amplitude resonant libration in the ( GLYPH<4> ; GLYPH<24> ) plane. Accounting for the secular terms, the evolution of this initial condition exhibits large-scale chaos. The extensive chaotic sea occupied by the solution is shown with opaque red points in the Poincare surface of section (11). However, the phase-space portrait is not entirely occupied by irregular trajectories. A survey of initial conditions permitted by the values of energy and angular momentum reveals the existence of quasi-periodic solutions as well, depicted as purple and black curves. J is conserved at all times along these curves. The dynamics is characterized by resonant libration in the ( GLYPH<4> ; GLYPH<24> ) plane within the regular region on the inside of the chaotic zone (corresponding to purple curves in Figure 11) and by circulation in the ( GLYPH<4> ; GLYPH<24> ) plane on the outside of the chaotic zone (corresponding to black curves). It should be noted that the example considered here was specifically chosen to reside in close proximity to the separatrix. For an arbitrary choice of initial conditions, even if the chaotic zone is permissible by the conservation of angular momentum and energy, it would likely occupy a considerably smaller fraction of phase-space. The boundary between circulation and libration is denoted by the thick brown circle shown in Figure 11). In other words, the thick brown circle is a projection of the separatrix in the ( GLYPH<4> ; GLYPH<24> ) plane onto the ( GLYPH<7> ; GLYPH<29> ) plane. The attribution of the origin of chaos to secular modulation is exemplified by the fact that the projected separatrix hugs the boundary of the chaotic zone on the inside as well as the outside. Thus, any secular trajectory that crosses the projected separatrix is driven to irregularity. It is worth noting that although any orbit that starts out within the chaotic sea will be irregular by definition, unlike the case of mean motion resonance overlap considered above, between encounters with the separatrix, the evolution will be characterized by conservation of J and will therefore be temporarily regular. Thus, a clear di GLYPH<11> erence between the two chaotic regimes can be established. Chaos that arises from secular modulation is described by slow di GLYPH<11> usion that takes place on a secular timescale, while di GLYPH<11> usion that arrises from the overlap of mean motion resonance is fast, characterized by the resonant timescale.", "pages": [ 13, 14, 15, 16 ] }, { "title": "4. Divergent Resonant Encounters", "content": "An interesting and useful application of the theory formulated above is the treatment of divergent encounters of planets with mean motion resonances. As briefly described in the introduction, capture into resonance requires convergent migration (Borderies and Goldreich 1984; Henrard 1991). In contrast, resonant encounters that stem from divergent migration can never lead to capture and instead always yield impulsive excitation of the orbits. Here, we wish to consider the latter scenario and address the translation of post encounter dynamics onto the secular domain. While studying divergent encounters with the 2:1 mean motion resonance by Jupiter and Saturn within the context of the Nice model, Morbidelli et al. (2009) identified that the planets always come out of the resonance locked in an apsidally antialigned state. Furthermore, the apsidal alignment persists indefinitely, unless it is broken by a close encounter with a transiently unstable ice-giant. Although Morbidelli et al. (2009) attributed the origin of the apsidal lock to a fortunious mass-ratio between Jupiter and Saturn, here we assert that this result is largely independent of the planetary masses. Panels A and B in Figure (12) show the time evolution of the eccentricities and the di GLYPH<11> erence in longitudes of perihelia where the 2:1 mean motion resonance is encountered by Jupiter and Saturn as well as a planetary pair with reversed masses i.e. Jupiter residing further from the sun. The solutions are obtained from numerical experiments where divergent migration was implemented via a fictitious force. Specifically, the simulations were performed using the Symba N-body integration software package (Duncan et al. 1998), modified such that in isolation, the outer orbit drifts outwards and the inner orbit drifts inwards with the migration rate decaying as / exp t =GLYPH<28> , choosing GLYPH<28> = 1Myr (Morbidelli et al. 2009). As can be assessed from the figure, the values of the eccentricities acquired by the planets during the resonance passage depend on the planetary mass ratio. Yet the encounter drives the planets to an apsidally anti-aligned state in both cases. Changing the mass ratio to unity does not a GLYPH<11> ect the results.", "pages": [ 16 ] }, { "title": "4.1. Pre-Encounter Evolution", "content": "This behavior can be readily explained in the context of the model developed here. Let us begin by first discussing the preencounter initial conditions. Because orbital migration is usually driven by time-irreversible (e GLYPH<11> ectively dissipative) processes (e.g. interactions with the protoplanetary nebula, tidal interactions, planetesimal scattering), it is natural to assume that planets migrate on circular orbits. As a result, recalling the definitions of the variables (47), we shall set the pre-encounter actions (where planets reside far away from resonance) to GLYPH<4> pre = GLYPH<7> pre = 0. Next, consider the migration rate. Numerical simulations (Tsiganis et al. 2005; Crida et al. 2007; Zhang and Hamilton 2008) suggest that in most cases of interest, the rate of orbital migration is slow compared to the secular interaction timescale, closely related to the evolution timescale of the acton GLYPH<7> . Taking the assumption of slow migration as a guiding principle, we are tempted to define a second adiabatic invariant related to secular motion. However, prior to doing so, we must first examine if the adiabatic approach is viable despite near-null eccentricities, which we showed in the last section can lead to the appearance of secondary resonances. At first glance, adiabatic invariance seems impossible because the criterion given by equations (44) and (45) clearly indicates that the system should be close to the 1:1 secondary resonance (see also Figure 8). However, as already mentioned in the previous section, in the limit of a vanishingly low value of GLYPH<7> , the criterion for secondary resonances must be reevaluated in light of the possibility of a librating rather than circulating GLYPH<29> . The procedure we follow is essentially identical to that outlined in section (3.2.1), but in order to appropriately capture the dynamics, we must work with a two degree of freedom Hamiltonian. Specifically, we shall consider a simplified version of Hamiltonian (48), where the secular terms are dropped with the exception of the harmonic 5 . Assembling the relevant constants to the secular part of the Hamiltonian into a single constant C s (see equation 48), we have: Upon expanding the Hamiltonian in Taylor series to second order around nominal resonance in both degrees of freedom (that is, ( GLYPH<4> ; GLYPH<24> ) is expanded around ([ GLYPH<4> ] ; 0) and ( GLYPH<7> ; GLYPH<29> ) is expanded around ([ GLYPH<7> ] ; 0)), we obtain the following expression: where the barred variables are defined as the deviations away from equilibrium (see equation (40) for an analogous definition). As before, the nominal actions are given by setting the linear terms in the above Hamiltonian to zero: We are now in a position to convert the Hamiltonian (51) into the form of two decoupled harmonic oscillators. However, before doing so let us examine equations (52) in greater detail. Adding the two equations together, we can obtain an expression for [ GLYPH<7> ] = [ GLYPH<4> ]: The latter simplification utilizes the fact that C s GLYPH<28> ˆ GLYPH<14> GLYPH<0> 2([ GLYPH<4> ] +GLYPH<10> ) since the former arises from a higher order perturbation. Note that this expression implies that [ GLYPH<7> ] = [ GLYPH<4> ] is a small parameter. This relationship between [ GLYPH<7> ] and [ GLYPH<4> ] will prove useful in obtaining simplified expressions for the libration frequencies below. Employing a change of variables of the same type as (43) with coe GLYPH<14> cients from equation (51), we transform the Hamiltonian into the desired form: The explicit expressions for the libration frequencies 'GLYPH<24> and 'GLYPH<29> can be made simpler by expanding them to leading order in [ GLYPH<7> ] = [ GLYPH<4> ], which we showed above to be a small parameter: Evidently, the two angles, GLYPH<24> and GLYPH<29> evolve on similar timescales. As in the previous section, we can take advantage of this similitude to easily identify an adiabatic invariant. Let us implicitly define two sets of action angle coordinates In these variables, the Hamiltonian (54) reads: H = 'GLYPH<24> X + 'GLYPH<29> Y : (57) Applying a contact transformation originating from the generating function we obtain the action-angle variables: The Hamiltonian is now explicitly adiabatic: Indeed, the evolution of the angle ˜ z is much slower than that of the angle ˜ w . This allows us to reintroduce the first adiabatic invariant Moreover, assuming that migration occurs more slowly than the evolution of ˜ z , we can define a second adiabatic invariant 6 ˜ ˜ In essence, the system we are concerned with here is subject to the double-adiabatic condition. Namely, I is conserved by construction because the migration rate is taken to be sufficiently slow and J is conserved because the two degrees of freedom are well-separated. Although the secular phase space portrait depicted by the Hamiltonian (48) contains no critical curves, conservation of both, J and I is broken when a separatrix is encountered in ( GLYPH<4> ; GLYPH<24> ) space. Consequently, the doubleadiabatic condition applies before and after, but not during the resonant encounter. However, because the impulsive excitation of the orbits occurs on a resonant timescale, GLYPH<7> (and equivalently, GLYPH<9> 2) itself is conserved across the encounter. At this point, we have enough information to show that after the resonant encounter, the orbits must be anti-aligned. Let us begin by reasoning through the calculation of the impulsive orbital excitation. Throughout the evolution prior to the encounter, J = I = 0. Note that this condition does not imply circular orbits. Instead, it implies that the system resides on a global fixed point, nearest to the origin. As exact resonance is approached, the location of the fixed point on the ( GLYPH<4> ; GLYPH<24> ) plane moves to the right (i.e. acquires a finite value of GLYPH<4> while remaining at GLYPH<24> = 0). In other words, the approach to exact resonance can be viewed as sequential evolution through panels A, B, C and D of Figure (3), where the solution resides on the black dot in the center of the inner circulation zone. As the inner circulation zone contracts, the stable fixed point and the unstable fixed move closer together in phase space.", "pages": [ 16, 17, 18 ] }, { "title": "4.2. Post-Encounter Evolution", "content": "The impulsive excitation occurs when the stable fixed point on which the dynamics resides and the unstable fixed point at the crest of the separatrix join. As long as the resonant encounter takes place at low eccentricities, the phase-space portrait of the system can be visualized, neglecting the second order secular contribution. In the purely resonant framework, this occurs when two of the roots to the cubic equilibrium equation, derived from the Hamiltonian (29) where GLYPH<26> = p 2 GLYPH<4> , are identical (Murray and Dermott 1999). In fact, the bifurcation of the fixed point can be used to calculate the exact semi-major axis ratio at which the encounter occurs. At this point, conservation of J is momentarily broken and the system obtains an orbit defined by the separatrix in the ( GLYPH<4> ; GLYPH<24> ) plane, shown as a gray curve in Figure (13). Because the resonant encounter occurs 'instantaneously\" with respect to the migration timescale, the orbital angular momentum must be conserved across the jump. Consequently, the acquisition of angular momentum deficit (related to GLYPH<4> ) is accompanied with a small jump in the semi-major axis ratio that converts the separatrix into a similarly-shaped regular circulating orbit. The circulating, rather than librating nature of the new orbit is ensured because during divergent migration, the phase-space area occupied by resonant trajectories shrinks, preventing capture (Henrard 1982; Peale 1986). Strictly speaking, this means that the dynamics no longer resides on a fixed point in the ( GLYPH<7> ; GLYPH<29> ) plane because the newly acquired angular momentum deficit changes the dynamical portrait. Indeed, the new trajectory in the ( GLYPH<7> ; GLYPH<29> ) plane envelopes the new coordinates of the fixed point and passes through the pre-encounter equilibrium location. However, it can be argued with some level of rigor that √ change in the fixed point's location will be small and by extension, so will the radius of the post-encounter orbit in the ( GLYPH<7> ; GLYPH<29> ) plane. First, note that neglecting second order terms, the ( GLYPH<7> ; GLYPH<29> ) fixed point always resides at the origin because the Hamiltonian (29) is independent of GLYPH<29> (equivalently, 2). This line of reasoning is a useful starting point but is an oversimplification as it only implies trivial secular dynamics embedded in the transformation (19). In reality (as can be seen in Figure 10), the ( GLYPH<7> ; GLYPH<29> ) fixed point resides somewhat o GLYPH<11> -center. In particular, prior to the encounter, the fixed point in the ( GLYPH<7> ; GLYPH<29> ) plane is obtained from equations (52). After the encounter, the ( GLYPH<7> ; GLYPH<29> ) equilibrium point can be calculated in a similar way, however, taking into account the fact that ( GLYPH<4> ; GLYPH<24> ) no longer resides at an equilibrium point. Consider a modified version of equation (52): where T is the period required to complete a single orbit in the ( GLYPH<4> ; GLYPH<24> ) plane. Note that the above expression simplifies to equation (52b) in the limit where GLYPH<4> = [ GLYPH<4> ] and GLYPH<24> = 0 for all t . As already stated above, immediately after resonance crossing the ( GLYPH<4> ; GLYPH<24> ) trajectory begins circulation (see Figure 13). However, during a single circulation cycle, the ( GLYPH<4> ; GLYPH<24> ) trajectory spends most of its period in close proximity to the ([ GLYPH<4> ] ; 0) fixed point, because that is where the time derivative of GLYPH<24> is minimal. Thus, the solution of equation (64) in GLYPH<7> will be close to that of equation (52). In other words, the equilibrium point in ( GLYPH<7> ; GLYPH<29> ) will not move considerably. Consequently, the dynamics of ( GLYPH<7> ; GLYPH<29> ), which was on the stable equilibrium point before the resonance crossing, will describe a cycle around the new equilibrium point after the crossing. The corresponding radius of the orbit will equal to the displacement su GLYPH<11> ered by the equilibrium point itself, which is small. For all subsequent evolution, as the planets migrate away from resonance, the conservation of both adiabatic invariants is once again in e GLYPH<11> ect. Consequently, because of the conservation of the first adiabatic invariant J , the orbit on the ( GLYPH<4> ; GLYPH<24> ) plane asymptotically approaches a circle centered on the origin, whose area is given by: Moreover, on the ( GLYPH<7> ; GLYPH<29> ) plane, the small radius of the cycle around the equilibrium point will be maintained, thanks to the conservation of the second adiabatic invariant, I . As long as this equilibrium point remains close to the origin, GLYPH<29> may circulate, but the smallness of GLYPH<7> ensures that GLYPH<1> $ librates around GLYPH<25> (see GLYPH<7> = GLYPH<9> 2 ' 0 contours in Figures 4 and 5). In principle, as the planets move away from resonance, the ( GLYPH<7> ; GLYPH<29> ) equilibrium point can move away from the origin. In this case, the small radius of the orbit around the equilibrium point on the ( GLYPH<7> ; GLYPH<29> ) plane implies that GLYPH<29> librates. This, again, ensures the libration of GLYPH<1> $ around GLYPH<25> . As a final point, it is important to comment on the results of an additional numerical experiment reported by Morbidelli et al. (2009). In particular, Morbidelli et al. (2009) showed that if the masses of both, Jupiter and Saturn are reduced by a factor of 100, the post-encounter apsidal alignment among the orbits no longer holds. This phenomenon (although apparently contradictory to the statements made above), can also be understood within the context of our model. Recall that our formulation of resonant encounters specifically assumed the double adiabatic condition. In the low-mass experiment considered by Morbidelli et al. (2009), the conservation of I is broken because the migration timescale is taken to be faster than the longest interaction timescale of the planets. Consequently, we can expect that there exists a tentative cut-o GLYPH<11> in mass below which apsidal alignment cannot endure. The characteristic value of such a cuto GLYPH<11> however is dependent on the migration process in question and will therefore vary among di GLYPH<11> ering astrophysical settings.", "pages": [ 18, 19 ] }, { "title": "5. Conclusion", "content": "In this paper, we have set out to construct a simple geometrical representation of the global dynamics of the unrestricted, first order resonant three-body problem. As the primary purpose of the paper is the delineation of a comprehensive dynamical picture, we have opted to work within the context of analytically tractable, but approximate perturbation theory. Although first-order resonant motion can be apparently complex, here, greatly aided by the pioneering works of Sessin and Ferraz-Mello (1984) as well as Henrard et al. (1986) and Wisdom (1986), we have shown that the essential features of the dynamics is captured within the context of a simple integrable Hamiltonian. The Hamiltonian in question is qualitatively similar to that of a pendulum and more precisely, is related to the second fundamental model for resonance (Henrard and Lemaitre 1983). This highlights a certain kinship between the unrestricted and the restricted three-body problems, as the second fundamental model for resonance has also been applied extensively to the study of the latter. Quantitatively, the formulated theory is only accurate at low eccentricities. Nevertheless, it still provides the much-needed qualitative insight relevant to a broad range of orbital architectures. Indeed, at an age when N-body integration software is freely available (Duncan et al. 1998; Chambers 1999) and computational resources required for problems such as these are abundant, the qualitative understanding that emerges from the theory is of greater importance than the particularities of its direct application. Consequently, the utility of the developed theory is best envisioned as a theoretical supplement to (rather than a replacement of) numerical N-body simulations. Utilizing the various constants of motion that arise within the context of the integrable theory, we have constructed a geometrical characterization of the resonant motion. Indeed, global maps of the dynamics, such as those presented in Figures (4) and (5) provide a visual aid that allows one to instantly assess important features of any particular resonant solution such as the proximity of the system to a separatrix or conversely the depth within the resonance at which a given orbital fit resides. Although the global maps (4) and (5) are restricted by the fact that they portray surfaces of section, combined with corresponding phase-space portraits, such as those presented in Figures (3) and (9), a more comprehensive understanding of the dynamics can be obtained. The applicability of the integrable theory is unavoidably limited. An important, well-known feature of resonant dynamics is its capacity for chaotic motion. Because the nature of the integrable model is inherently regular, in isolation, it is essentially of no use in the chaotic domain. In this work, we emphasized two distinct modes of the onset of chaos. Namely, we considered the rapid irregularity that arises from the overlap of mean motion resonances as well as slow chaos that arises as a result of the secular modulation of the orbit through the separatrix. The first mode dominates in the region of parameter space where neighboring resonant separaticies reside in some proximity to each other. In direct analogy with the restricted problem, for ( k : k GLYPH<0> 1) resonances, the region of parameter space occupied by this e GLYPH<11> ect grows with increasing k . Conversely, chaotic di GLYPH<11> usion near seaparatricies that are isolated from neghboring mean motion resonances is dominated by secular modulation of the resonant dynamics. It is important to recall that beyond the integrable approximation, we only accounted for a limited number of second order secular terms. Obviously, even after averaging out shortperiod terms, the dynamics encapsulated into the residual disturbing function is much richer than the simple model utilized here. This implies that the description of the onset of chaos is far from exhaustive. That said, the method outlined in this paper, namelyintroducing an integrable Hamiltonian by freezing the secular degree of freedom and then studying its evolution in the adiabatic regime, is valid for arbitrary eccentricities and inclinations. Consequently, the largely qualitative account of the onset of chaos presented here should be viewed as a guide to a general methodology rather than a particular model with extended applicability. As an application of the simple theory formulated in this work, we addressed divergent resonant encounters between massive planets. Particularly, we showed that the natural outcome of adiabatic resonant encounters is an apsidally anti-aligned orbital state. Interestingly, this result is largely independent of the planetary masses. Moreover, the preservation of the second adiabatic invariant (related to secular dynamics) ensures that smallamplitude libration around the anti-aligned fixed point persists far away from the resonance. As a consequence of this result, it is tempting to interpret small-amplitude anti-aligned libration of nonresonant planets as a signature of past resonant encounters as well as the associated migration. Indeed, such an interpretation holds great value as an instrument for disentangling the dynamical histories of planetary systems. However, care must be taken when drawing any such conclusion because eccentricity damping (such as that resulting from the dissipative processes that drive divergent orbital migration in the first place) in the secular domain may lead to antialigned orbits independently (Wu and Goldreich 2002; Mardling 2007; Batygin and Laughlin 2011). Furthermore, it is important to note that lack of co-precessing anti-aligned obits in a given system should not be viewed as evidence for lack of past resonant encounters, since resonant encounters in densely populated planetary systems, can lead to orbital instabilities that act to chaotically erase fossilized remnants of past evolution. The lack of apsidal alignment between Jupiter and Saturn suggests that the solar system is in fact, such an example (Morbidelli et al. 2009; Batygin and Brown 2010; Nesvorný and Morbidelli 2012). Although we have solely addressed divergent resonant encounters here, the same model can also be applied to convergent resonant encounters. As discussed above, the outcomes of convergent encounters include both, capture into resonance as well as capture-free orbital excitations (Henrard 1991; Lee and Peale 2002). While the latter scenario is qualitatively similar to the example considered here, in case of successful capture, postencounter evolution can depend strongly on factors such as the orbital migration and eccentricity dissipation rates as well as the strength of external stochastic perturbations (Adams et al. 2008; Rein et al. 2010). Similar factors contribute to the determination of whether capture can occur in the first place (Murray and Dermott 1999). Consequently, astrophysically relevant analysis of convergent resonant encounters within the framework of the model discussed here requires extensive, numerical validation. Owing to the significant associated computational cost of such a project, addressing this issue is far beyond the scope of the current study. However, our investigation aimed at quantifying the various regimes of convergent resonant encounters is already underway and will be published in a subsequent follow up study. We wish to thank Jake Ketchum and Matt Holman for carefully reviewing the manuscript and providing helpful suggestions. We wish to thank Christian Beauge for a very thorough and insightful referee report that greatly enhanced the quality of this manuscript. K.B. acknowledges the generous support from the ITC Prize Postdoctoral Fellowship at the Institute for Theory and Computation, Harvard-Smithsonian Center for Astrophysics.", "pages": [ 19, 20 ] }, { "title": "References", "content": "Adams, F. C., Laughlin, G., Bloch, A. M. 2008. Turbulence Implies that Mean Motion Resonances are Rare. The Astrophysical Journal 683, 1117-1128. Batygin, K., Brown, M. E. 2010. Early Dynamical Evolution of the Solar System: Pinning Down the Initial Conditions of the Nice Model. The Astrophysical Journal 716, 1323-1331. Batygin, K., Laughlin, G. 2011. Resolving the sin(I) Degeneracy in Low-mass Multi-planet Systems. The Astrophysical Journal 730, 95. Laskar, J. 1996. Large Scale Chaos and Marginal Stability in the Solar System. Celestial Mechanics and Dynamical Astronomy 64, 115-162. Lin, D. N. C., Bodenheimer, P., Richardson, D. C. 1996. Orbital migration of the planetary companion of 51 Pegasi to its present location. Nature 380, 606607. Malhotra, R. 1995. The Origin of Pluto's Orbit: Implications for the Solar System Beyond Neptune. The Astronomical Journal 110, 420. Morbidelli, A., Moons, M. 1993. Secular resonances inside mean motion commensurabilities: The 2 / 1 and 3 / 2 cases. Icarus 103, 99-108. Morbidelli, A., Crida, A. 2007. The dynamics of Jupiter and Saturn in the gaseous protoplanetary disk. Icarus 191, 158-171. Morbidelli, A., Tsiganis, K., Crida, A., Levison, H. F., Gomes, R. 2007. Dynamics of the Giant Planets of the Solar System in the Gaseous Protoplanetary Disk and Their Relationship to the Current Orbital Architecture. The Astronomical Journal 134, 1790-1798. Morbidelli, A., Levison, H. F., Gomes, R. 2008. The Dynamical Structure of the Kuiper Belt and Its Primordial Origin. The Solar System Beyond Neptune 275-292. Murray, N., Holman, M. 1997. Di GLYPH<11> usive chaos in the outer asteroid belt.. The Astronomical Journal 114, 1246-1259. Michtchenko, T. A., Beaugé, C., Ferraz-Mello, S. 2008. Dynamic portrait of the planetary 2 / 1 mean-motion resonance - I. Systems with a more massive outer planet. Monthly Notices of the Royal Astronomical Society 387, 747-758. Neishtadt, A. I. 1984. Separation of motions in systems with a rapidly rotating phase. Prikladnaia Matematika i Mekhanika 48, 197-204. Nesvorný, D., Ferraz-Mello, S., Holman, M., Morbidelli, A. 2002. Regular and Chaotic Dynamics in the Mean-Motion Resonances: Implications for the Structure and Evolution of the Asteroid Belt. Asteroids III 379-394. Nesvorný, D., Morbidelli, A. 2012. Statistical Study of the Early Solar System's Instability with Four, Five, and Six Giant Planets. The Astronomical Journal 144, 117.", "pages": [ 20, 21 ] } ]

[ { "title": "ABSTRACT", "content": "Limb darkening is a fundamental ingredient for interpreting observations of planetary transits, eclipsing binaries, optical / infrared interferometry and microlensing events. However, this modeling traditionally represents limb darkening by a simple law having one or two coe ffi cients that have been derived from plane-parallel model stellar atmospheres, which has been done by many researchers. More recently, researchers have gone beyond plane-parallel models and considered other geometries. We previously studied the limbdarkening coe ffi cients from spherically symmetric and plane-parallel model stellar atmospheres for cool giant and supergiant stars, and in this investigation we apply the same techniques to FGK dwarf stars. We present limb-darkening coe ffi cients, gravity-darkening coe ffi cients and interferometric angular diameter corrections from Atlas and SAtlas model stellar atmospheres. We find that sphericity is important even for dwarf model atmospheres, leading to significant di ff erences in the predicted coe ffi cients. Key words. Stars: atmospheres - Stars: late-type - stars: binaries: eclipsing - stars: evolution - planetary systems - techniques: interferometric", "pages": [ 1 ] }, { "title": "Spherically symmetric model stellar atmospheres and limb darkening II: limb-darkening laws, gravity-darkening coefficients and angular diameter corrections for FGK dwarf stars /star", "content": "Hilding R. Neilson 1 and John B. Lester 2 , 3 e-mail: [email protected]", "pages": [ 1 ] }, { "title": "1. Introduction", "content": "One of the great astronomical advances of the past two decades has been the discovery and study of extrasolar planets via the transit method, i.e. from the minute drop of a star's flux as a planet passes in front of it. The transit not only constrains the planet's properties but also the star's properties, such as limb darkening. However, interpreting planetary transits typically assumes that limb darkening can be parametrized by a simple relation (Mandel & Agol 2002) with a few free parameters that can be fit directly from the observations or assumed from model stellar atmospheres. Limb darkening is important not only for understanding planetary transits (e.g. Croll et al. 2011), but also for interpreting optical interferometric observations (e.g. Davis et al. 2000) and microlensing observations (e.g. An et al. 2002) and eclipsing binary light curves (e.g. Bass et al. 2012). Like transit measurements, interferometric and microlensing observations are typically fit by simple limb-darkening laws with coe ffi cients derived from model stellar atmospheres (Al-Naimiy 1978; Wade & Rucinski 1985; van Hamme 1993; Claret 2000; Claret & Bloemen 2011; Claret et al. 2012). However, these simple limb-darkening laws have become less suitable as the observations have improved. For example, Fields et al. (2003) showed that flux-normalized limb-darkening laws fit to Atlas plane-parallel model atmospheres disagreed with microlensing observations. Limb-darkening coe ffi cients derived from plan- etary transit observations with large impact parameters di ff er more from the limb-darkening coe ffi cients from model atmospheres, but the discrepancy still exists when the impact parameter is taken into account (Barros et al. 2012). This discrepancy might be due to a number of physical processes, including granulation, multidimensional convection and / or the presence of magnetic fields in the stellar atmosphere. However, the simplest step is to assume a more realistic geometry for the model stellar atmospheres. Limb-darkening coe ffi cients presented in the literature are based on two forms: plane-parallel model stellar atmospheres computed using the Atlas (Kurucz 1979) and Phoenix code (Hauschildt et al. 1999) and spherically symmetric model stellar atmospheres also computed from the Phoenix code (Sing 2010; Howarth 2011a; Claret & Bloemen 2011; Claret et al. 2012). In particular, Claret & Hauschildt (2003) and Claret et al. (2012, 2013) explored limb darkening using spherically symmetric Phoenix model stellar atmospheres specifically for main sequence stars. They also introduced the concept of 'quasi-spherical' models, defined as the spherically-symmetric intensity profile restricted to inner part of the stellar disk ( µ ≥ 0 . 1), to compare limbdarkening coe ffi cients with those from plane-parallel models. In our previous study (Neilson & Lester 2013, hereafter Paper 1), we presented coe ffi cients for six typical limb-darkening laws fit to the surface intensities for grids of plane-parallel and spherical model atmospheres (Lester & Neilson 2008) representing red giant and supergiant stars. The intensities were for the wavebands of the Johnson-Cousins (Johnson & Morgan 1953; Bessell 2005), CoRot (Auvergne et al. 2009) and Kepler (Koch et al. 2004) filters. We also computed gravity-darkening coe ffi cients and in- terferometric angular diameter corrections. We found that the predicted limb-darkening coe ffi cients computed from spherical model atmospheres di ff er from those computed from planeparallel model atmospheres, which was not unexpected; the height of the atmospheres of red giants and supergiants relative to the stellar radius is many times greater than the relative heights of the atmospheres of main-sequence stars, i.e. the assumed model geometry is important. We found similar di ff erences for the angular diameter corrections as a function of geometry but little di ff erence between gravity-darkening coe ffi cients as a function of geometry. While model atmosphere geometry is clearly important for understanding the extended atmospheres of red giant and supergiant stars, it is not as obvious that geometry also changes predictions for model stellar atmospheres of main sequence dwarf stars (e.g. Claret & Hauschildt 2003). In this work, we explore the role of model atmosphere geometry in understanding limb darkening in dwarf stars and compute limb-darkening coe ffi cients, gravity-darkening coe ffi cients and interferometric angular diameter corrections from grids of model stellar atmospheres of dwarf stars. In Sect. 2, we briefly describe the grids of model atmospheres used. In Sect. 3, we describe various limb-darkening laws and compare predicted limb-darkening coe ffi cients, while in Sect. 4 we compute gravity-darkening coe ffi cients. We present interferometric angular diameter corrections as a function of geometry in Sect. 5 and summarize our results in Sect. 6.", "pages": [ 1, 2 ] }, { "title": "2. Model stellar atmospheres", "content": "The Atlas / SAtlas code was used to compute model stellar atmospheres assuming either plane-parallel or spherically symmetric geometry. Details of the code can be found in Lester & Neilson (2008), Neilson & Lester (2011, 2012) and Paper 1. We computed model stellar atmospheres with parameters 3500 K ≤ T e ff ≤ 8000 K in steps of 100 K, and 4 . 0 ≤ log g ≤ 4 . 75 in steps of 0 . 25. For the spherically symmetric models, which require an additional parameter, such as stellar mass, to characterize the atmosphere, we chose M = 0 . 2 to 1 . 4 M /circledot in steps of 0 . 3 M /circledot . For each model stellar atmosphere we compute intensities at each wavelength for 1000 uniformly spaced values of µ , the cosine of the angle formed by the line-of-sight point on the stellar disk and the disk center, spanning 0 ≤ µ ≤ 1. Typically, Atlas models compute intensities at only seventeen angles (Kurucz 1979), but some have employed 100 µ -points (Claret & Bloemen 2011). We compute intensity profiles for each model atmosphere for the BVRIH and K -bands as well as the CoRot and Kepler -bands. As an example, Fig. 1 shows the Kepler -band intensity profiles for plane-parallel and spherical models with T e ff = 5800 K, log g = 4 . 5 and M = 1 . 1 M /circledot . Using the wavebands outlined above, we compute limb-darkening coe ffi cients, gravitydarkening coe ffi cients and interferometric angular diameter corrections.", "pages": [ 2 ] }, { "title": "3. Limb-darkening laws", "content": "We consider the same six limb-darkening laws as in Paper 1: As in Paper 1, we use a general least-squares fitting algorithm to compute the limb-darkening coe ffi cients for each law in the BVRIH - and K -bands as well as for the CoRot and Kepler -bands. Using the Kepler -band as an example, Figure 2 shows the best-fit limb-darkening coe ffi cient for the linear law (Eq. 1), Fig. 3 shows the coe ffi cients for the quadratic (Eq. 2) and squareroot (Eq. 3) laws, Fig. 4 shows the coe ffi cients for the exponential (Eq. 5) and logarithmic (Eq. 6) laws and Fig. 5 shows the coe ffi cients for the Claret (2000) four-parameter law (Eq. 4). The results shown in Fig. 2 demonstrate how the geometry of the model atmosphere a ff ects the best-fit linear Kepler -band limb-darkening coe ffi cients, with squares representing fits to the spherically symmetric model atmospheres and crosses representing fits to the plane-parallel models. The values of the u -coe ffi cient for the spherical models are larger than those for the planar models, particularly for models with T e ff > 4500 K. At these higher e ff ective temperatures the di ff erence due to geometry, ∆ u Kepler, is ∼ 0 . 3. There is also a greater spread for the spherical model coe ffi cients at a given e ff ective temperature. This is caused by the spherical models being defined by three parameters, with mass and radius being separated, as opposed to the two parameters for plane-parallel model atmospheres, where mass and radius are combined in the surface gravity. At T e ff < 4500 K the u -coe ffi cients computed for both geometries shift to similar values. A likely cause of this change relative to the higher e ff ective temperatures is the shift in dominant opacities from H -to TiO. The more complex limb-darkening laws have similar differences between coe ffi cients from plane-parallel and spherically symmetric models. For the quadratic and square-root laws, the coe ffi cients of the linear term ( a and c , respectively) shows similar behavior as a function of e ff ective temperature as does the u -coe ffi cients, while the coe ffi cients of the non-linear terms ( b and d ) appear correlated to the coe ffi cients of the linear terms, as was seen previously for other laws (Fields et al. 2003; Neilson & Lester 2011, 2012). For the exponential and logarithmic laws, the best-fit coefficients again di ff er as a function of model atmosphere geometry. The limb-darkening coe ffi cients also appear to be correlated for each law. It is notable that the best-fit m -coe ffi cients of the logarithmic law from spherically symmetric models are approximately constant with respect to e ff ective temperature, whereas the non-linear term is not constant. The limb-darkening coe ffi -cients from spherically symmetric models for both exponential and logarithmic laws vary significantly for any given e ff ective temperature, suggesting the coe ffi cients are sensitive to the mass and gravity of a model stellar atmosphere. The best-fit coe ffi cients for the Claret (2000) four-parameter limb-darkening laws do not agree for spherical and planeparallel models. For e ff ective temperatures greater than 4000 K, the limb-darkening coe ffi cient f 1 varies from -2 to + 4 for the spherical models but only from -0 . 5 to 0 . 5 for the plane-parallel models. The dramatic di ff erence is due to the more complex structure of spherically symmetric model intensity profiles, even when considering the smaller atmospheric extensions for models used in this work as opposed to those considered in Paper 1, which indicates that even this more sophisticated limb-darkening law is not ideal for fitting spherically symmetric model intensity profiles. ∑[ ] to measure the di ff erence for every model between the computed intensity distribution and the best fit to those intensities for each limb-darkening law. Unfortunately, as we showed in Paper 1, the computed error depends on how the models are sampled and the number of intensity points. If one fits intensity profiles for µ -points near the center of the stellar disk then the limb-darkening coe ffi cients and predicted errors di ff er from limb-darkening coe ffi cients and errors predicted from a sample of µ -points near the edge of the stellar disk. However, we can predict the relative quality of fits as a function of geometry. We show in Fig. 6 the predicted errors for each limb-darkening law as a function of e ff ective temperature. As expected, Fig. 6 shows that all six limb-darkening laws fit the plane-parallel model atmosphere intensity profiles better than intensity profiles from spherical models. The definition of plane-parallel radiative transfer (Feautrier 1964) assumes that I ( µ ) ∝ e -τ/µ , where τ is the monochromatic optical depth. As µ → 0, then I ( µ ) → 0, i.e. the intensity and the derivative of the intensity, d I / d µ , both change monotonically. These properties allow simple limb-darkening laws to fit plane-parallel model intensity profiles well. eff For spherically symmetric model atmospheres the radiative transfer is calculated for a set of rays along the line-of-sight between the observer and points on the stellar disk. The rays nearer the center of the stellar disk come from depths that are assumed to be infinitely optically thick. The rays farther from the center of the stellar disk penetrate to depths where the optical depth is assumed never to reach infinity (Rybicki 1971; Lester & Neilson 2008), although the rays can reach extremely large optical depths. Rays located toward the limb of the star can penetrate the tenuous outer atmosphere, never reaching large optical depths. As a result, the computed intensity profiles have a point of inflection (see Fig. 1) where the intensity derivative, VI / d µ , is not changing monotonically, which prevents the simple limb-darkening laws from fitting as well. While it is di ffi cult to draw conclusions from the predicted errors, we can reliably state that the linear and exponential limbdarkening laws do not fit the spherical model atmospheres. The predicted errors for those limb-darkening laws range from 0 . 05 to 0 . 2 and are significantly greater than the errors for the fits to plane-parallel models. The best-fitting relations are the squareroot law and the four-parameter limb-darkening law of Claret (2000), which have errors less than 0 . 08. Another thing to note is that based on fits to plane-parallel model atmospheres, Diaz-Cordoves et al. (1995) suggested that the square-root law is more adequate for fitting hotter stars ( T e ff > 8000 K), although they were unclear which law is preferred for cooler stars. For spherical model atmospheres we find that the predicted errors for the square-root limb-darkening law are less than the errors for the quadratic law, making the former the clear preference. Also, the quadratic limb-darkening law is of particular interest because it is the most commonly used limb-darkening law for analyzing planetary transit observations (Mandel & Agol 2002). However, numerous comparisons of quadratic limb-darkening laws fit directly to observations and those fit to model stellar atmospheres suggest disagreement for a number of cases (Howarth 2011b). The results presented here suggest it may be advantageous to consider fitting transit ob- servations with a square-root limb-darkening law or the more accurate four-parameter limb-darkening law.", "pages": [ 2, 3, 4, 5, 6 ] }, { "title": "4. Gravity-darkening coefficients", "content": "Claret & Bloemen (2011) computed wavelength-dependent gravity-darkening coe ffi cients from Atlas plane-parallel model stellar atmospheres based on the analytic relation developed by Bloemen et al. (2011). In Paper 1 we used this same prescription for both plane-parallel and spherically symmetric model stellar atmospheres to compute gravity-darkening coe ffi cients for cool giant stars, and we found that model geometry played a negligible role in determining gravity-darkening coe ffi cients except for T e ff < 4000 K. At the cooler e ff ective temperatures, the spherically symmetric model gravity-darkening coe ffi cients are predicted to be vary significantly, and are up to an orderof-magnitude greater than those predicted from plane-parallel model atmospheres. We repeat that analysis here for our higher gravity model stellar atmospheres. As described by Bloemen et al. (2011), the gravity-darkening coe ffi cient, y ( λ ) for a star is As described in Paper 1, von Zeipel (1924) showed that T e ff ∼ ( g e ff ) β 1 / 4 , where β 1 ≡ d ln T e ff / d ln g . As previously, we assume β 1 = 0 . 2 for models with T e ff < 7500 K and β 1 = 1 otherwise. Using these constant values for β 1 provides only a limited analysis of the gravity-darkening because β 1 is a function of e ff ective temperature, but assuming these two values does enable us to gain some perspective on the role of model atmosphere geometry. The other terms are the partial derivatives of the wavelengthdependent intensity with respect to gravity and e ff ective temperature, respectively. We compute the two intensity derivatives and predicted gravity-darkening coe ffi cients for our grids of plane-parallel and spherically symmetric model atmospheres and plot the predicted values in Fig. 7 for the Kepler waveband. The predicted derivatives and gravity-darkening coe ffi cients are similar to those computed in Paper 1, for which there is little di ff erence between spherically symmetric and plane-parallel model predictions for e ff ective temperatures greater than 4000 K. The spherical and planar predictions then diverge for cooler e ff ective temperatures. However, the range of values for the spherical model predictions is less for the higher gravity models explored in this work relative to the lower gravity models studied in Paper 1.", "pages": [ 6 ] }, { "title": "5. Interferometric angular diameter corrections", "content": "Interferometry provides precise measurements of stellar angular diameters. However, stellar interferometry measures the combination of angular diameter and intensity profile and the two quantities are degenerate. One route to break the degeneracy is to assume a uniform intensity profile and measure the uniform-disk angular diameter. The limb-darkened angular diameter can then be predicted from the uniform-disk angular diameter using corrections computed from stellar atmosphere models (Davis et al. 2000). Another technique for measuring limb-darkened angular diameters is to assume a simple limb-darkening law and coe ffi -cients from model stellar atmospheres to fit the interferometric observations (e.g Boyajian et al. 2012). However, this technique might also predict incorrect angular diameters because planeparallel model atmospheres are typically used for fitting limbdarkening coe ffi cients. We can assess the potential error of assuming plane-parallel limb-darkening coe ffi cients to fit the angular diameter by comparing predicted angular diameter corrections from spherically symmetric model stellar atmospheres with those from plane-parallel models. In Fig. 8 we plot the V - and K -band angular diameter corrections as a function of e ff ective temperature and gravity for both spherical and planar model atmospheres. The V -band corrections vary from 0 . 93 to 0 . 97 for the plane-parallel model atmospheres and from 0 . 92 to 0 . 95 for spherical models. The di ff erence is more apparent if one considers stellar atmospheres with T e ff > 4500 K, where the di ff erence between plane-parallel and spherical model corrections is about 0 . 01 to 0 . 02. This suggests that employing plane-parallel model corrections for measuring stellar angular diameters from interferometric observations will lead to a 1 to 2% underestimate of the angular diameter. Similarly, the K -band corrections also vary as a function of model atmosphere geometry; plane-parallel models suggest values of θ UD /θ LD = 0 . 98 to 0 . 99 while spherical models suggest θ UD /θ LD = 0 . 97 to 0 . 985. Again, using plane-parallel model corrections to fit K -band interferometric observations will underestimate the actual angular diameter by about 1%. Thus, for precision measurements of angular diameters, hence fundamental stellar parameters from optical interferometry, one should employ more physically representative spherical model atmospheres. This appears to be the case even for main sequence stars with large gravities and small atmospheric extensions.", "pages": [ 6 ] }, { "title": "6. Summary", "content": "In this work, we followed up on the study of Paper 1 to measure how model stellar atmosphere geometry a ff ects predicted limbdarkening coe ffi cients, gravity-darkening coe ffi cients and interferometric angular diameter corrections for main sequence FGK dwarf stars. As in Paper 1, we find significant di ff erences between predictions from plane-parallel and spherically symmetric model atmospheres computed with the Atlas / SAtlas codes. The results in this article are surprising because geometry is believed to be not important for stars with smaller atmospheric extension, i.e. main sequence stars with log g ≥ 4. As atmospheric extension gets smaller, defined as the ratio of the atmospheric depth to stellar radius, then it is expected that a spherical model atmosphere should appear more and more like a plane-parallel model atmosphere. However, even for small atmospheric extension models, we find di ff erences in predicted intensity profiles, hence di ff erences in limb-darkening and angular diameter corrections. As in Paper 1, there is negligible di ff erence between gravitydarkening coe ffi cients predicted from planar and spherical model atmospheres. This is because gravity-darkening coe ffi -cients depend heavily on the central intensity of the star, not the entire intensity profile. The central intensity is approximately a function of the e ff ective temperature at τ Ross = 1 according to the Eddington-Barbier relation (Mihalas 1978). Lester & Neilson (2008) showed that the atmospheric temperature structure computed from plane-parallel and spherically symmetric model atmospheres for the same e ff ective temperature and gravity primarily di ff ers closer to the surface, τ Ross < 2 / 3, and converges as τ →∞ . Because the computed temperatures at depth are very similar for the two geometries, the central intensity is also similar for both model geometries, making the gravity-darkening coe ffi cients insensitive to model geometry. However, geometry ∂ ∂ is important for stars with T e ff < 4000 K, which is due to differences in the opacity structure and convection, which lead to changes in the temperature structure. Angular diameter corrections do vary as a function of geometry. The corrections account for the degeneracy between the intensity profile and limb-darkened angular diameter in modeling interferometric observations. Therefore, di ff erences between the intensity profiles of plane-parallel and spherically symmetric model stellar atmospheres lead directly to di ff erences between predicted angular diameter corrections. We find that spherically symmetric model corrections are about 1 to 2% smaller than planar model corrections for the main sequence stars analyzed here. Similarly, we computed limb-darkening coe ffi cients for six di ff erent limb-darkening laws. As in Paper 1, we find that the linear law is least consistent with predicted intensity profiles and that the four-parameter law is best. We also find that the commonly used quadratic limb-darkening law does not fit spherically symmetric model atmosphere intensity profiles as precisely as the similar square-root or four-parameter limb-darkening laws. This suggests that as planetary-transit observations become increasingly precise, the four-parameter law combined with the more physically representative spherically symmetric model stellar atmospheres will be more appropriate for fitting observations or, better still, using intensity profiles directly. The angular-diameter corrections, limb-darkening and gravity-darkening coe ffi cients are publicly available as online tables. Each table has the format T e ff (K), log g and M ( M /circledot ) and then the appropriate variables for each waveband, such as linear limb-darkening coe ffi cients. Tables for plane-parallel model fits do not include mass. Tables of gravity-darkening coe ffi cients also contain values of the intensity derivatives with respect to gravity and e ff ective temperature. For plane-parallel models, values of mass, radius and luminosity are presented as zero in the tables. We list the properties of these tables in Table 1 that are available from CDS. Tabulated grids of the model atmosphere intensity profiles used in this work are also available. Acknowledgements. This work has been supported by a research grant from the Natural Sciences and Engineering Research Council of Canada, the Alexander von Humboldt Foundation and NSF grant (AST-0807664).", "pages": [ 6, 7, 8 ] }, { "title": "References", "content": "Al-Naimiy, H. M. 1978, Ap&SS, 53, 181 An, J. H., Albrow, M. D., Beaulieu, J.-P., et al. 2002, ApJ, 572, 521 Auvergne, M., Bodin, P., Boisnard, L., et al. 2009, A&A, 506, 411 Barros, S. C. C., Pollacco, D. L., Gibson, N. P., et al. 2012, MNRAS, 419, 1248 Bass, G., Orosz, J. A., Welsh, W. F., et al. 2012, ApJ, 761, 157 Bessell, M. S. 2005, ARA&A, 43, 293 Bloemen, S., Marsh, T. R., Østensen, R. H., et al. 2011, MNRAS, 410, 1787 Boyajian, T. S., von Braun, K., van Belle, G., et al. 2012, ApJ, 757, 112 Claret, A. 2000, A&A, 363, 1081 Claret, A. & Bloemen, S. 2011, A&A, 529, A75 Claret, A. & Hauschildt, P. H. 2003, A&A, 412, 241 Claret, A., Hauschildt, P. H., & Witte, S. 2012, A&A, 546, A14 Claret, A., Hauschildt, P. H., & Witte, S. 2013, A&A, 552, A16 Croll, B., Albert, L., Jayawardhana, R., et al. 2011, ApJ, 736, 78 Davis, J., Tango, W. J., & Booth, A. J. 2000, MNRAS, 318, 387 Diaz-Cordoves, J., Claret, A., & Gimenez, A. 1995, A&AS, 110, 329 Feautrier, P. 1964, Comptes Rendus Academie des Sciences (serie non specifiee), 258, 3189 Fields, D. L., Albrow, M. D., An, J., et al. 2003, ApJ, 596, 1305 Hauschildt, P. H., Allard, F., & Baron, E. 1999, ApJ, 512, 377 Howarth, I. D. 2011a, MNRAS, 413, 1515 Howarth, I. D. 2011b, MNRAS, 418, 1165 Johnson, H. L. & Morgan, W. W. 1953, ApJ, 117, 313 Koch, D. G., Borucki, W., Dunham, E., et al. 2004, in Society of Photo-Optical Instrumentation Engineers (SPIE) Conference Series, Vol. 5487, Society of Photo-Optical Instrumentation Engineers (SPIE) Conference Series, ed. J. C. Mather, 1491-1500 Kurucz, R. L. 1979, ApJS, 40, 1 Lester, J. B. & Neilson, H. R. 2008, A&A, 491, 633 Mandel, K. & Agol, E. 2002, ApJ, 580, L171 Mihalas, D. 1978, Stellar atmospheres / 2nd edition / (San Francisco: W. H. Freeman and Co.) Neilson, H. R. & Lester, J. B. 2011, A&A, 530, A65 Neilson, H. R. & Lester, J. B. 2012, A&A, 544, A117 Neilson, H. R. & Lester, J. B. 2013, A&A, 554, A98 Rybicki, G. B. 1971, J. Quant. Spec. Radiat. Transf., 11, 589 Sing, D. K. 2010, A&A, 510, A21 van Hamme, W. 1993, AJ, 106, 2096 von Zeipel, H. 1924, MNRAS, 84, 665 Wade, R. A. & Rucinski, S. M. 1985, A&AS, 60, 471", "pages": [ 8, 9 ] } ]

[ { "title": "ABSTRACT", "content": "Context. Within the sequential accretion scenario of planet formation, planets are build up through a sequence sticking collisions. The outcome of collisions between porous dust aggregates is very important for the growth from very small dust particles to planetesimals. In this work we determine the necessary material properties of dust aggregates as a function the porosity. Aims. Continuum models such as SPH that are capable of simulating collisions of macroscopic dust aggregates require a set of material parameters. Some of them such as the tensile and shear strength are di GLYPH<14> cult to obtain from laboratory experiments. The aim of this work is to determine these parameters from ab-initio molecular dynamics simulations. Methods. We simulate the behavior of porous dust aggregates using a detailed micro-physical model of the interaction of spherical grains that includes adhesion forces, rolling, twisting, and sliding. Using di GLYPH<11> erent methods of preparing the samples we study the strength behavior of our samples with varying porosity and coordination number of the material. Results. For the tensile strength, we can reproduce data from laboratory experiments very well. For the shear strength, there are no experimental data available. The results from our simulations di GLYPH<11> er significantly from previous theoretical models, which indicates that the latter might not be su GLYPH<14> cient to describe porous dust aggregates. Conclusions. We have provided functional behavior of tensile and shear strength of porous dust aggregates as a function of the porosity that can be directly applied in continuum simulations of these objects in planet formation scenarios. Key words. Planets and satellites: formation - Methods: numerical", "pages": [ 1 ] }, { "title": "Tensile & shear strength of porous dust agglomerates", "content": "A. Seizinger, 1 R. Speith, 2 and W. Kley 1 Received 10.06.2013; accepted 23.08.2013", "pages": [ 1 ] }, { "title": "1. Introduction", "content": "The formation of planetesimals, km sized objects that are massive enough for gravity to come into play, constitutes a key step of the core accretion scenario for planet formation proposed by Pollack et al. (1996). However, the earlier growth from mm to km sized bodies is not fully understood yet. Understanding the interplay between porosity, impact velocity, and the size and structure of colliding aggregates in the meter-size regime is crucial to unravel the process of planetesimal formation. Obviously, this size regime renders laboratory experiments impossible. Thus, for years to come astrophysicists will have to rely on computer simulations to obtain the necessary insight into this complex process. Molecular dynamics simulations featuring detailed micromechanical interactions have been employed to study collisions of sub-mm sized dust and ice aggregates (e.g. Dominik & Tielens, 1997; Wada et al., 2007; Ringl et al., 2012). Owing to the high computational demand a di GLYPH<11> erent approach is necessary for the mm to meter size regime. In this regime, SPH simulations have often been utilized to model pre-planetesimal collisions (e.g. Sirono, 2004; Schäfer et al., 2007; Geretshauser et al., 2011). Smoothed particle hydrodynamics (SPH) constitutes a continuum approach that is capable to simulate the collisional behavior of macroscopic aggregates including physical processes such as compaction or fragmentation. Being a continuum approach, SPH requires various material parameters such as the compressive, tensile, and shear strength, and hence a proper calibration is necessary (Geretshauser et al., 2010). Typically, the calibration process is based on comparison with results from laboratory experiments (Güttler et al., 2009). However, not all material parameters have been obtained in this way. E.g. the shear strength is based only on theoretical models and estimations so far. So far, only few laboratory experiments have been performed to investigate the mechanical properties of porous dust aggregates. Blum & Schräpler (2004) measured the tensile strength of highly porous dust aggregates generated by random ballistic deposition. As they used monodisperse, spherical silica grains their experiments are comparable to our simulations. It has been shown both theoretically (Bertini et al., 2009) and experimentally (Blum et al., 2006) that the mechanical properties depend on the shape and size distribution of the grains. Recently, Meisner et al. (2012) presented results from various experiments on the mechanical properties of irregularly shaped quartz aggregates. Determining material parameters directly from molecular dynamics simulations of porous dust aggregates constitutes a tempting alternative. Paszun & Dominik (2008) presented the first attempt to obtain the compressive strength from ab-initio simulations. A few years later, Seizinger et al. (2012) studied the compressive strength in greater detail and especially revealed the di GLYPH<11> erences between static and dynamic compaction processes. The aim of the present work is to extend this approach to determine the tensile and shear strength of porous aggregates. Together with the compressive strength we then can provide continuum simulations (such as SPH) with a complete parameter set describing the transition from elastic to plastic deformation.", "pages": [ 1, 2 ] }, { "title": "2. Interaction model", "content": "We simulate the behavior of porous dust aggregates with a molecular dynamics approach. In this work we use cuboidal shaped aggregates with an edge length of 30 to 60 GLYPH<22> m. Depending on its volume filling factor such an aggregate consists of up to 6 GLYPH<1> 10 4 micron sized spherical grains (monomers). The interaction between individual monomers is based on the work of Dominik & Tielens (1997). Based on earlier theoretical work by Johnson et al. (1971); Dominik & Tielens (1995, 1996), they developed a detailed micro-mechanical model of the interaction of two microscopic spherical grains. When two monomers touch each other, surface forces allow for the creation of an adhesive contact. Upon deformation of these contacts caused by the relative motion of the monomers kinetic energy is dissipated. This approach is favorable for our purpose because the process of internal restructuring of the aggregate is modeled far more realistically compared to simpler hard-sphere models. Later, Wada et al. (2007) derived almost the same interaction laws from corresponding potentials. Seizinger et al. (2012) calibrated the interaction model by comparison with laboratory experiments on the compression of porous dust aggregates (Güttler et al., 2009). They observed that the original model of Dominik & Tielens (1997) was too soft. In order to increase the strength of the aggregates, Seizinger et al. (2012) introduced the rolling and sliding modifiers m r and m s that modify the strength of the corresponding type of interaction. By increasing the rolling interaction by a factor of 8 and the sliding interaction by a factor of 2 : 5 they achieved excellent agreement between numerical simulations and laboratory results. In this work we use the modified interaction model presented by Seizinger et al. (2012) with m r = 8 and m s = 2 : 5 unless stated otherwise. The material parameters are listed in Tab. 1.", "pages": [ 2 ] }, { "title": "3. Sample generation", "content": "In this Section we briefly summarize our sample generation methods. Here we use cuboidal shaped dust aggregates (also referred to as dust cakes) of di GLYPH<11> erent porosities. In principle, we may employ di GLYPH<11> erent methods to generate these samples. As shown in in Fig. 1 the relation between the volume filling factor and the average coordination number n c depends on the selected generation method (Seizinger & Kley, 2013). c The volume filling factor GLYPH<30> is given by where N denotes the number of monomers, V p is the volume of an individual monomer, and V is the total volume occupied by the sample. The coordination number, n c, denotes the number of contacts a monomer has established with its neighbors. The mean n c of the sample is calculated by averaging the number of contacts of each monomer.", "pages": [ 2 ] }, { "title": "3.1. Static compaction", "content": "Static compaction constitutes a method that very closely resembles the generation process of dust cakes in laboratory experiments. Initially, a dust cake is built by random ballistic deposition (RBD) where single monomers are successively dropped onto the existing sample. They come to rest right at the spot where they hit the existing sample. This growth process results in fractal, highly porous aggregates with a volume filling factor of GLYPH<30> = 0 : 15 (Watson et al., 1997). In the second step, the RBD cake is put into a box and compacted until the desired filling factor is reached. However, the compaction must occur slowly enough in order to avoid inhomogeneities (Seizinger et al., 2012). For the material / monomer size used in this work a typical speed of the compacting wall is 1 cm s GLYPH<0> 1 . The simulation time needed to compact an aggregate is proportional to the desired size and filling factor. Since the number of monomers increases with the size and compactness of the final dust cake the computational e GLYPH<11> ort per integration step rises as well. Thus, generating samples by static compaction can become a computationally expensive, time consuming procedure. Elastic loading of dust cakes compressed to filling factors above GLYPH<25> 0 : 45 constitutes another setback of this method (Seizinger & Kley, 2013). Because of the elastic energy stored in the contacts between monomers the aggregate will start to expand once the confining walls of the compaction box are removed. This e GLYPH<11> ect will alter the results of a measurement of the tensile or shear strength significantly. Up to a filling factor of GLYPH<25> 0 : 58 dust cakes may be stabilized in the following way: After slightly disturbing the positions of monomers the aggregate is kept in a box until the energy induced by the disturbance is damped away (Seizinger & Kley, 2013).", "pages": [ 2 ] }, { "title": "3.2. Ballistic aggregation and migration", "content": "The generation procedure of ballistic aggregation and migration (BAM) has been proposed by Shen et al. (2008). A larger aggregate is generated by successively shooting in single monomers from random directions onto the existing aggregate. In Seizinger & Kley (2013) we employed three di GLYPH<11> erent methods to select the final position of a monomer hitting the aggregate: The volume filling factor of the generated aggregate depends on the selection mechanism. For a given coordination number, aggregates generated with 'shortest migration' feature the lowest filling factor whereas the 'center migration' method leads to the most compact aggregates. The relation between the filling factor and the coordination number is displayed in Fig. 1, for the di GLYPH<11> erent preparation methods. For comparison we show the results for the hexagonal close packing as well, see Seizinger & Kley (2013) for more details. After generating a su GLYPH<14> ciently large aggregate a cuboidal shaped dust cake of the desired size is sliced out to be used for the subsequent numerical experiments.", "pages": [ 3 ] }, { "title": "4.1. Setup", "content": "In principle, the procedure to determine the tensile strength of a given sample is simple: After attaching two plates at the top and bottom of the cubic aggregate the plates are pulled apart at constant speed. During this process the pulling force exerted on these plates is measured. At first, the force will increase with the distance between the plates. If a certain distance is exceeded, cracks will form. Thus, the strength of the sample is reduced and the force required to pull the two plates apart decreases. An example showing a typical relation between the pull distance and tension is shown in Fig. 2. It is very similar to experimental data (see Blum & Schräpler, 2004, Fig. 4). The force is determined by summing up the individual interaction forces of all monomers that are in contact with the wall. In case of the tensile strength, only the component in the direction of the motion of the wall (which is equivalent to the normal vector of the wall) is taken into account. To allow easier comparison between di GLYPH<11> erent sized samples we use their base area to normalize the values and plot the corresponding tension instead of the force. In accordance with Blum et al. (2006), we define the maximumtension that is measured during a run as the tensile strength of the sample. The displacement at which force / pressure peaks depends on the porosity of the sample. In a sample with high filling factor monomers are fixed tightly, which hampers internal restructuring. In contrast, in a flu GLYPH<11> y sample individual chains of monomers can be unfolded and thus the material can be stretched out significantly before the formation of cracks sets in. To model this setup in our simulations, the sample is put into a box of flat walls. Before slowly moving the top and bottom wall away from each other we must ensure that a su GLYPH<14> cient number of monomers is in contact with these walls. For this purpose the top and bottom wall are slowly pushed inwards a short distance. For samples with filling factors below GLYPH<30> = 0 : 2 we use a value of one monomer radius whereas for more compact samples we decrease the distance to 0 : 5 GLYPH<0> 0 : 1 monomer radii. When pulling the two plates away from each other another problem arises: The critical force F c required to break a contact between two monomers is given by F c = 3 GLYPH<25>GLYPH<13> R , where GLYPH<13> denotes the surface energy and R the reduced radius of the two particles (Johnson et al., 1971). The wall is modeled as a particle of infinite radius, which means that the reduced radius of a particlewall contact equals twice the reduced radius of a particle-particle contact (Seizinger et al., 2012). Thus, contacts between two particles can be broken more easily than particle-wall contacts. As a result, the monomers that are in contact with one of the plates tend to get ripped o GLYPH<11> the remaining sample (see left panel of Fig. 3). To counter this e GLYPH<11> ect we artificially increase the strength of the adhesion between two monomers dependent on the distance to the plates. To achieve this 'gluing e GLYPH<11> ect' the force / potential of the normal interaction that is responsible for the adhesion is multiplied with a gluing factor GLYPH<20> . To avoid discontinuities in the particle interaction GLYPH<20> is interpolated linearly depending on the distance to the closest plate. Above a threshold of 8 particle radii GLYPH<20> is set to 1 and thus the default JKR interaction is used. As shown in the right panel of Fig 3, this mechanism leads to the rupture somewhere in the middle of the sample rather than just tearing o GLYPH<11> the upper- or lowermost layer of monomers. As the first cracks will form where the aggregate is weakest, the exact location is random owing to the inhomogeneous structure of the aggregate. We tested di GLYPH<11> erent maximum values of GLYPH<20> and found that a value of 2 is su GLYPH<14> cient for our purpose. For GLYPH<20> < 1 : 5, samples do not break in the center anymore. On the other hand, larger values do not alter the measured tensile strength significantly (see Fig. 4).", "pages": [ 3 ] }, { "title": "4.2. Results", "content": "Apart from the wall gluing factor GLYPH<20> there are several other parameters whose influence has to be studied. To determine the influence of the rolling and sliding interaction we performed a series of simulations using m r = m s = 1, which is equivalent to the model of Wada et al. (2007). In this and all of the following simulations we used a wall gluing factor of GLYPH<20> = 2. Apparently, internal restructuring which is governed by rolling and sliding does not play a major role when determining the tensile strength (see the purple squares in Fig. 4 showing results for the model from Wada et al. (2007)). While the sample aggregate is torn apart contacts between the monomers have to be broken. Therefore, we expect that the measured tensile strength depends on the number of contacts that have to be broken as well as the critical force F c which is necessary to break a contact between individual monomers. To check this hypothesis we alter the strength of the normal force by multiplying it with the normal force modifier m n. Indeed, when doubling the strength of the normal interaction (and thus F c) by setting m n = 2, we observe a steeper increase of the tensile strength with the filling factor (blue asterisks in Fig. 4). When determining a linear fit for filling factors below 0 : 5 to the GLYPH<20> = 3 ; m n = 1 and GLYPH<20> = 3 ; m n = 2 simulations we get a slope of 7 : 9 kPa and 15 : 3 kPa, respectively. Their ratio of 15 : 3 = 7 : 9 = 1 : 94 is very close to the value of the normal interaction modifier m n = 2. This strongly suggests that the pull o GLYPH<11> force F c is critical for the measured value of the tensile strength. Independent of GLYPH<20> , m n, or the rolling and sliding interaction there is a striking drop of the tensile strength for filling factors above 0 : 5. To unravel its cause we first used a di GLYPH<11> erent type of samples. Much to our surprise, the critical filling factor at which the tensile strength drops is di GLYPH<11> erent for each type of aggregate and close to the maximum filling factor that may be achieved by a given generation method (see Fig. 5). Apparently, the micromechanical behavior is not governed by the volume filling factor alone. Therefore we take the average coordination number n c into account. From the relation between the filling factor and the coordination number of the di GLYPH<11> erent sample types (see Fig. 1) we find that the drop of the measured tensile strength of BAM aggregates coincides with a value of n c around GLYPH<25> 4 : 5 (see Fig. 6). This points to the influence of the coordination number on the micromechanical properties of the sample aggregates. If the average coordination number is low, the majority of monomers may react to an external stress by rearranging themselves. Thus, a large number of monomers participate in absorbing the external stress and the aggregate exhibits rather ductile behavior. As the coordination number increases, monomers are fixated in their positions more tightly. For compact aggregates the monomers cannot rearrange themselves freely anymore which means that a lower number of monomers has to absorb the applied strain. Therefore, the aggregates become brittle. 10 The e GLYPH<11> ect of brittleness can clearly be seen in case of the hexagonal lattice aggregates. Because of their regular, crystal like structure (see Seizinger & Kley, 2013, Fig. 1a), their capability of internal restructuring is very limited. Thus, contacts break very easily when external strain is applied. As a result, the measured tensile strength is considerably lower compared to BAM or static compaction aggregates. Additionally, the pressure exerted on the sample when slowly pressing on the top and bottom walls su GLYPH<14> ces to disrupt very compact aggregates. Moving the top and bottom walls inwards by a distance of 0 : 5 to 0 : 15 particle radii is necessary to establish a firm contact between the walls and the sample. E.g. in the case of a BAM (center migration) aggregate with GLYPH<30> = 0 : 59 the average coordination number decreased from 5 : 94 to 4 : 95 after moving the top wall inwards by a distance of only 0 : 1 monomer radii. This means that the strength of the sample is lowered during the preparation process. This raises the question whether the transition from ductile to brittle behavior or the disturbance when a GLYPH<14> xing the plates is the dominant e GLYPH<11> ect. Recently, Kataoka et al. (2013) presented a di GLYPH<11> erent approach to determine the compressive strength of highly porous ( GLYPH<30> < 0 : 1) dust aggregates by using periodic boundary conditions. A similar approach might work for the tensile strength as well and would avoid the problem of attaching the plates onto a highly compact sample without lowering its strength. Luckily, measuring the tensile strength of hexagonal lattice aggregates also allows us to circumvent this problem. Because of their regular grid structure the contact between the wall and all particles of the top / bottom layer is established without compacting the sample. Nevertheless, we observe a significant drop of the tensile strength for n c ! 6 (see Fig. 6). This observation allows us to conclude that the disruption caused by a GLYPH<14> xing the plates only plays a secondary role. To explain the discrepancy between BAM and static compaction aggregates as shown in Fig. 5, the process of pulling the plates apart is analyzed more closely. Comparing the dissipated energy of the di GLYPH<11> erent types of aggregates we clearly see that tearing apart the static compaction aggregate requires more energy (solid lines in Fig. 7). Thus, a higher value of the tensile strength is measured. Tracking the number of contacts that have been broken since the start of the simulation provides us with a hint to an explanation of this observation: Compared to the BAM aggregate, less than half as many contacts break while pulling apart the static compaction aggregate (dotted line in Fig. 7) although more kinetic energy is dissipated. This observation might be surprising at first sight, yet it illustrates the importance of the internal structure for the outcome of the measurements. In reaction to the applied strain the internal structure of the aggregates changes, where inelastic rolling accounts for GLYPH<25> 80% and the breaking of contacts for only GLYPH<25> 3% of the total dissipated energy for both aggregate types. In case of the static compaction aggregate the final number of broken contacts is reached much more quickly after pulling the sample roughly 7 : 5 GLYPH<22> m apart. Likewise, the peak of the measured tension is reached earlier. At this point, a lot more energy has been dissipated by internal restructuring (mainly by inelastic rolling) as compared to the BAM aggregate. However, for the BAM aggregates the total number of broken contacts increases much faster. This means that the internal structure of this type of aggregate allows for less restructuring before contacts start to break. As roughly 97% of the dissipated energy are required for restructuring rather than breaking of the contacts we measure a lower tensile strength for BAM aggregates. We think that the reason behind this observation lies in the fact that the tensile strength test is the reversal of the compression process by which the static compaction aggregates have been generated. While being slowly compacted to the desired filling factor su GLYPH<14> cient time had been given to the fractal chains of the initial RBD aggregates to rearrange themselves. As a result, the monomers of the static compaction aggregates adopt a structure that is favorable to withstand an external load. While pulling the plates apart the total number of broken contacts (dotted line in Fig. 7) sometimes decreases. This happens when the connection between two already stretched out parts of the sample breaks and the both parts 'snap back'. We also checked whether there are any preferential directions resulting from the generation process. For this reason we rotated cubic static compaction samples by 90 GLYPH<14> before determining the tensile strength. Reassuringly, we measured the same values (not shown in this work) and may rule out that the direction of the compaction induces any preferred direction in the structure of the sample. As a last step we varied the geometry of the samples to check if the size influences the results. As it can be seen in Fig. 8 the results do not vary significantly if we alter the size of the samples. In Fig. 9 we compare our results with laboratory experiments performed by Blum & Schräpler (2004) and Blum et al. (2006). Taking into account that their samples have been produced by static compaction our results show good agreement with their data for filling factors below GLYPH<30> = 0 : 5. Because of the drop of the tensile strength for n c ! 6 explained in the previous paragraphs we cannot compare our simulations to laboratory experiments for higher filling factors. Earlier compression simulations already indicated that our physical model may not be valid for highly compact aggregates anymore (Seizinger et al., 2012). Luckily, the filling factor regime relevant for the growth processes of planetesimals is below 0 : 5 (e.g. Teiser et al., 2011). To determine the fitting curve depicted in Fig. 9 only data points for filling factors below 0.5 have been taken into account. Because of the significant di GLYPH<11> erence between the static compaction and BAM aggregates we determined two fit curves T SC( GLYPH<30> ) and T BAM( GLYPH<30> ), respectively. We obtain and Based on their generation process the laboratory samples should resemble the static compaction cakes. Indeed, for GLYPH<30> GLYPH<25> 0 : 2 our simulations agree very well with laboratory experiments (Blum & Schräpler, 2004; Blum et al., 2006). However, for higher filling factors the laboratory results lie somewhere between the static compaction and BAM results. In private conversation Jürgen Blum (Braunschweig) pointed out that creating more compact samples in the lab sometimes proved to be a difficult task. Thus, we also determined a fit T ( GLYPH<30> ) to the combined results of the static compaction and BAM aggregates. Using the combined data points from both aggregate types shown in Fig. 9 for a single fit T ( GLYPH<30> ), we find for values of GLYPH<30> < 0 : 5", "pages": [ 3, 4, 5, 6 ] }, { "title": "4.3. Influence of the monomer size", "content": "Most other numerical simulations dealing with Silicates have been performed with monomers with a diameter of 1 : 2 and 1 : 5 GLYPH<22> m because these sizes have been used in laboratory experiments with spherical Silicate grains. Out of curiosity we varied the size of the monomers. In general, it can be said that according to JKR-theory the adhesion forces increase as grains get smaller. Indeed, we find that the tensile strength depends strongly on the size of the monomers (see Fig. 10). Note that our interaction model has not been calibrated for monomer radii other than r p = 0 : 6 GLYPH<22> m. Therefore the rolling and sliding modifiers m r and m s may not have the correct values to properly describe restructuring processes. However, in case of the tensile strength this problem does not arise as we have already seen that it is mainly governed by the normal interaction (see Fig. 4). From Fig. 4 we see that the tensile strength scales linearly with the normal interaction. Altering the monomer size by a factor of 2, for the transition from 0 : 6 GLYPH<22> mto 1 : 2 GLYPH<22> msized monomers the change of the measured tensile strength di GLYPH<11> ers from 2. At first glance this may seem odd as the critical pull o GLYPH<11> force F c depends linearly on the monomer radius. However, the dependence of the normal force acting upon the monomers before they are separated on the monomer radius is non-linear (see Seizinger et al., 2012, Eqs. 2 and 3). Nevertheless, these simulations confirm the importance of the pull o GLYPH<11> force F c that has already been shown in Fig. 4. The results clearly demonstrate the e GLYPH<11> ect of the stickiness of the single monomers on the tensile strength. For future work it would be interesting to perform simulations with aggregates composed of di GLYPH<11> erently sized monomers.", "pages": [ 6, 7 ] }, { "title": "5.1. Setup", "content": "The shear strength of a porous aggregate is determined in a similar way as the tensile strength. As before, two plates are attached to the top and bottom of the sample. During the shearing motion of the plates the force acting on them is tracked. However, in case of the shear strength the direction of motion is perpendicular to the normal of the wall, i.e. tangential to a cuboid surface. During the simulation the vertical positions of the top and bottom wall remain fixed to keep the filling factor constant. This means, similar to the 'fixed walls' used in the work of Seizinger et al. (2012) the normal component of the force acting upon the walls is ignored. As before, the initial base area of the sample is used to normalize the force. The setup closely resembles the tensile strength setup. A firm contact between the sample and the plates is achieved in the same way as described in Sec. 4.1. To prevent the monomers that are in contact with the moving wall from being torn away from the sample, an additional 'gluing e GLYPH<11> ect' is applied to particles that are close to one of the plates. A snapshot taken during a typical simulation is depicted in Fig. 11. As the top plate is slowly moving shearing sets in. With increasing pulling distance cracks will form and reduce the strength of the sample. Thus, we expect a similar shape as for the tensile strength if we plot the tension acting on the moving . The upper plate is slowly moving to the right. Adhesion between particles that are close to one of the plates has been artificially increased. plate with respect to the displacement. Indeed, the example shear strength curve shown in Fig. 12 resembles the curves shown in Fig. 2. Similar to the tensile strength case, we define the shear strength as the maximum tension that is measured during the simulation. Again, the higher the porosity of a sample the larger the necessary displacement at which the force peaks.", "pages": [ 7 ] }, { "title": "5.2. Results", "content": "Owing to the computational demand of the simulations the size of our samples is limited to values below 0 : 1 mm. To study the dependency of our results on the sample size we prepared both BAM and static compaction aggregates with di GLYPH<11> erent edge lengths. For each data point six di GLYPH<11> erent samples with equal statistical properties have been generated. Some of the results are shown in Fig. 13. As we can see, the results of the di GLYPH<11> erent sample sizes do not alter significantly. In order to check whether the length of the sample in direction of the shearing motion is su GLYPH<14> cient we also performed simulations for sample sizes of 80 GLYPH<2> 40 GLYPH<2> 50 GLYPH<22> mand 120 GLYPH<2> 40 GLYPH<2> 50 GLYPH<22> m. Owing to the huge number of particles these simulations took several weeks. Therefore we restricted the values of the filling factor to GLYPH<30> = 0 : 33 and GLYPH<30> = 0 : 49. The deviation to the values obtained from the smaller 40 GLYPH<2> 40 GLYPH<2> 50 GLYPH<22> maggregates was GLYPH<25> 8 GLYPH<0> 10% for GLYPH<30> = 0 : 33 and GLYPH<25> 2 GLYPH<0> 3% for GLYPH<30> = 0 : 49. Thus, we may draw the conclusion that the samples are in fact su GLYPH<14> ciently large. With the exception of the most compact samples ( GLYPH<30> = 0 : 59) the error bars obtained by averaging the results from the six samples are very small. Interestingly, we do not observe a significant di GLYPH<11> erence between the static compaction and BAM aggregates as in the case of the tensile strength. As explained in Sect. 4.2, owing to their generation process the internal structure of the static compaction aggregates is more favorable to counteract external loading / tension. However, this does not apply to shearing motion that results in di GLYPH<11> erent kind of deformation compared to the tensile strength test. Therefore, the two types of samples exhibit similar values for the shear strength. To provide SPH simulations with an easy to implement model for the shear strength we describe the dependency of shear strength S on the filling factor GLYPH<30> with a power law S ( GLYPH<30> ) = a GLYPH<30> b . Using the results from BAM and static compaction aggregates of 40 GLYPH<2> 40 GLYPH<2> 50 GLYPH<22> medge length we obtained (see Fig. 13)", "pages": [ 7, 8 ] }, { "title": "5.3. Comparison with the SPH continuum model", "content": "One objective of the present investigations has been the comparison of the resulting strengths with those adopted in the SPH simulations by Geretshauser et al. (2010). To model shear failure, Sirono (2004) introduced a von Mises yielding criterion in his SPH simulations. The required shear strength is in principle equivalent to the shear strength obtained in our calculations. Güttler et al. (2009) calibrated their SPH model in an extensive process comparing simulation results with laboratory experiments. They found a new representation for the dynamic compressive strength, with GLYPH<30> 1 = 0 : 12 and GLYPH<30> 2 = 0 : 58, and they chose the tensile strength according to Blum & Schräpler (2004), For the shear strength, no experimental data have been available. Therefore the shear strength was adopted following Sirono (2004) according to In Fig. 14 the fit curves of tensile strength T ( GLYPH<30> ), eq. (4), and shear strength S ( GLYPH<30> ), Eq. (5), are compared to the corresponding values of the SPH model, T SPH( GLYPH<30> ), Eq. (7), and S SPH( GLYPH<30> ), eq. (8). As can be seen, the tensile strength curves match rather well. This emphasizes that the present molecular dynamics method is well suited to model highly porous aggregates. The shear strength curves, however, di GLYPH<11> er by nearly one order of magnitude. This indicates that the approach of Sirono (Eq. 8) for the SPH shear strength, which is not based directly on laboratory experiments, might be inappropriate. But during the calibration process it was found already that the SPH simulation results for the chosen reference problems only depend weakly on the exact values of the shear strength (Güttler et al., 2009).", "pages": [ 8 ] }, { "title": "6. Conclusions", "content": "This work supports the observation of Seizinger & Kley (2013) that the sample generation method influences its mechanical behavior significantly. Whereas the bouncing behavior of microscopic dust aggregates di GLYPH<11> ers little for BAM and static compaction aggregates they do behave di GLYPH<11> erently when external strain is applied (see Fig. 5). It is important to keep this in mind when comparing numerical simulations to laboratory results. Observing the transition from ductile to brittle behavior for coordination numbers of GLYPH<25> 6 is very interesting. It certainly influences the outcome of collisions as well. For brittle aggregates fragmentation will play a significantly larger role. In this work we determined simple power laws to describe the relation between the tensile strength (see Eq. 4) or shear strength (see Eq. 5) and the porosity. In combination with earlier work on the the compressive strength (Seizinger et al., 2012) it provides a complete description when the inelastic regime is entered upon deformation of porous dust aggregates. Since the dissipation of the kinetic impact energy is critical, this knowledge is crucial for continuum simulations of collisions of macroscopic porous aggregates. Comparing with a special SPH model, we find that our tensile strength agrees well with the tensile strength adopted in the SPH code. The same holds for the compressive strength as found in earlier work (Seizinger et al., 2012). However, the shear strength di GLYPH<11> ers significantly. Future analysis has to show whether our improved relation for the shear strength will have fundamental impact on the SPH simulation results, or whether the shear strength only alters details in the simulations, as might be indicated by previous work. Acknowledgements. A. Seizinger acknowledges the support through the German Research Foundation (DFG) grant KL 650 / 16. The authors acknowledge support through DFG grant KL 650 / 7. Additional support through the German Research Foundation (DFG) through grant KL 650 / 11 within the Collaborative Research Group FOR 759: The formation of Planets: The Critical First Growth Phase is acknowledged. We thank the anonymous referee for pointing out possible misunderstandings and helping to improve the quality of the paper.", "pages": [ 8, 9 ] }, { "title": "References", "content": "Teiser, J., Engelhardt, I., & Wurm, G. 2011, ApJ, 742, 5 Wada, K., Tanaka, H., Suyama, T., Kimura, H., & Yamamoto, T. 2007, ApJ, 661, 320 Watson, P. K., Mizes, H., Castellanos, A., & Perez, A. T. 1997, Powders & Grains, 109", "pages": [ 9 ] } ]

[ { "title": "ABSTRACT", "content": "Aims. We propose new estimates of the secular aberration drift, mainly due to the rotation of the Solar System about the Galactic center, based on up-to-date VLBI observations and and improved method of outlier elimination. Methods. We fit degree-2 vector spherical harmonics to extragalactic radio source proper motion field derived from geodetic VLBI observations spanning 1979-2013. We pay particular attention to the outlier elimination procedure to remove outliers from (i) radio source coordinate time series and (ii) the proper motion sample. Results. We obtain more accurate values of the Solar system acceleration compared to those in our previous paper. The acceleration vector is oriented towards the Galactic center within ∼ 7 · . The component perpendicular to the Galactic plane is statistically insignificant. We show that an insu ffi cient cleaning of the data set can lead to strong variations in the dipole amplitude and orientation, and statistically biased results. Key words. Astrometry - Reference systems - Techniques: interferometric", "pages": [ 1 ] }, { "title": "On the VLBI measurement of the Solar System acceleration", "content": "O. Titov 1 and S. Lambert 2", "pages": [ 1 ] }, { "title": "1. Introduction", "content": "The accelerated motion of the Solar System in the Universe, mainly due to its rotation about the Galactic center, induces an apparent proper motion of extragalactic objects of a few microseconds of arc per year ( µ as / yr) called secular aberration drift, in the direction of the acceleration vector. This e ff ect shows up in the systematic part of the proper motion field. The detection of the secular aberration drift in the positions of extragalactic radio sources observed by very long baseline interferometry (VLBI) in accordance with the theoretical predictions (see, e.g., Fanselow 1983, Bastian 1995, Eubanks et al. 1995, Sovers et al. 1998, Mignard 2002, Kovalevsky 2003, Kopeikin & Makarov 2006) was announced recently by Titov et al. (2011), hereafter TLG11, in the following, and Xu et al. (2012, 2013). These works provided direct measurements of the Solar System acceleration in the Universe independent from any dynamical model of the Galaxy. TLG11 found an acceleration vector pointing towards the Galactic center ( α G = 266 . 4 · , δ G = -28 . 9 · ) within 10 · and an amplitude in agreement with predictions based on the Galactic parameters derived with other methods. Using a similar observational data set but an alternative estimation method, Xu et al. (2012, 2013) obtained an amplitude close to TLG11 but oriented about 18 · north and 23 · west from the Galactic center, and thus a significant acceleration of the Solar System perpendicularly to the Galactic plane. The authors raised the possibility of a companion star orbiting the Sun to explain the deviation. This current study brings out improved estimates of the secular aberration drift, making use of new VLBI observations since TLG11. Section 2 presents some generalities about the Solar System motion and the expected amplitude of the secular aberration drift. Section 3 recalls the basic equations. In Section 4, we study the sensitivity of the solution to the outliers and show how they can dramatically corrupt the least-squares estimates of the dipole components.", "pages": [ 1 ] }, { "title": "2. The Solar System motion in the Galaxy", "content": "The theoretical e ff ects of the Solar System acceleration on the apparent position of distant bodies are described in many articles (see TLG11 and references therein). Recent estimates of the Galactic parameters based on trigonometric parallaxes of massive star regions (Reid et al. 2009) give distances R to the Galactic center of the order of 8 . 4 ± 0 . 6 kpc and a circular rotation speed V of 254 ± 16 km / s (thus a rotation period of ∼ 200 Myr). Consistent values are obtained by other methods, for instance stellar orbit monitoring (see, e.g., Ghez et al. 2008, Gillesen et al. 2009). The acceleration V 2 / R deduced from these values equals 7 . 9 ± 1 . 6 mm / s / yr. This induces a dipolar proper motion to distant bodies of amplitude 5 . 4 ± 0 . 7 µ as / yr towards the Galactic center. As the Solar System rotates around the Galactic center, it also oscillates around the Galactic plane with an amplitude of 49-93 pc and a period of 52-74 Myr (Bahcall & Bahcall 1985). The Solar System passed through the Galactic plane about 3 Myr ago and is now about 26 pc above it (Majaess et al. 2009). It is therefore moving towards the north Galactic pole and slowing down before going back to the south. The magnitude of the aberration on distant body proper motion resulting from this oscillation can reach ∼ 0.5 µ as / yr when the Solar system reaches the maximum distance from the Galactic plane. When this smaller contribution is added to the main contribution from the Galactic rotation, the apparent direction of the dipole can be displaced by about 5 · from the Galactic center (Fig. 1).", "pages": [ 1 ] }, { "title": "3. The proper motion field parameters", "content": "We decomposed the systematic part of the proper motion field into a dipole, a rotation, and a quadrupole. Equations (8)-(9) of TLG11 appear to be vitiated. We corrected these formulae in the present paper (Eqs. (5)-(6)) and we followed the convention of Mignard & Klioner (2012). Corrected TLG11 values are given later in this paper in Table 1. For a distant body of equatorial coordinates ( α, δ ), the dipole part reads (see, e.g., Mignard & Morando 1990, Mignard & Klioner 2012) where the di are the components of the acceleration vector in unit of the proper motion. In addition to the aberration distortion, there may also be a small global rotation that can be described by the toroidal harmonics of degree 1: The quadrupolar anisotropy of the proper motion field is given by the development of the degree 2 vector spherical harmonics of electric ( E ) and magnetic ( M ) types: In the following, we consider two solutions: the DR solution only estimates dipole and rotation parameters and the DRQ solution estimates the 16 parameters relevant to the dipole, the rotation, and the quadrupole.", "pages": [ 1, 2 ] }, { "title": "4. Data analysis and results", "content": "The VLBI data processing made use of the analysis options extensively described in TLG11. We extend the session list up to the end of February 2013. Due to the major 2011 Tohoku earthquake, several Japanese telescopes in the Tokyo area were displaced by several tens of centimeters. Consequently, these stations were removed from the no-net rotation and translation constraints, as previously done for the Fairbanks and Concepci'on antennas. As in TLG11, we applied a loose constraint of nonet rotation on the radio source coordinates, with the exception of the 39 sources showing significant nonlinear positional variations due to large scale variations in their structure (e.g., 3C84, 3C273B, 3C279, 3C345, 3C454.3, 4C39.25) pointed out in Fey et al. (2010). In TLG11, we emphasized that the constraint should be loose enough to allow the reference frame axes to deform slowly. The analysis returned coordinate time series for 3635 extragalactic radio sources, of which 707 were observed in more than one session. In each time series, data points obtained from less then three delays were removed. Data points whose distance to the mean is higher than a certain threshold T s times the uncertainty were also removed. This elimination is repeated until the χ 2 is reasonably close to unity. The removed outliers are generally associated with unreliable networks or corrupted data. Proper motions were then computed by the least-squares fit to time series longer than 10 sessions weighted by the inverse of the squared errors. (The influence of the minimum number of sessions was checked afterwards and found to be negligible.) Note that the VLBI analysis software package used here allows DR direct computation of proper motions from VLBI delays. Proper motions obtained by this method appear to be very di ff erent from those obtained by fitting to time series. The di ff erence may come from the fact that the software package does not remove bad observations and gets therefore biased estimates. Such discrepancies should be investigated by the community in the future. The next step consisted of fitting the proper motion field parameters. Fitting the dipole and rotation parameters to the data with T s in a range of 10 to 100 gives a dipole of amplitude ∼ 6 µ as / yr oriented towards α ∼ 280 ± 15 · and δ ∼ -35 ± 15 · , which is consistent with the values expected from the theory. However, proper motions adjusted in the previous step can be unreliable due to two major reasons. Firstly, strong, frequently observed radio source have very small formal errors. However, some of them show very large nonlinear positional variations and / or significant apparent motion due to relativistic jets and intrinsic radio source structure (Charlot 1990, Fey et al. 2004). Fortunately, these astrometrically unstable radio sources are well known in advance and could be removed from the sample. Secondly, some radio sources with a small number of observa- it proper motions extremely large compared to the global rms. These radio sources have the same impact on the solution as the frequently observed ones and should be removed from the sample. The problem is that these radio sources are not known in advance. They can only be identified after a preliminary solution by inspecting the proper motion sample. To identify and remove these spurious proper motions, the systematics were first estimated and removed, and then the source with residual velocities larger than a threshold T p times the residual rms were eliminated from the sample. The process was iterated a few times to convergence. To illustrate the usefullness of such an elimination algorithm, we took T s = 90 and estimated the dipole parameters through the obtained velocity field. We obtained 6.3 µ as / yr, α = 275 · , δ = -30 · . Then, we introduced a fake source located at the Vernal point with a spurious proper motion of magnitude 10 mas / yr ± 50 µ as / yr both in right ascension and declination. The fit gave 3.5 µ as / yr, α = 304 · , δ = 2 · . We tried several values of the thresholds T s and T p. Figure 2 displays the results of the di ff erent adjustments with a contour delimiting the region where the χ 2 is close to 1. This occurs for T p preferably close to 7. We took this value to check the sensitivy of the fit to T s (bottom panel of Fig. 2). The region for which χ 2 nears 1 with the lower postfit rms is narrow and located around T s = 90. In this region, the dipole gets oriented towards the Galactic center. Lowering T s leads to a lower formal error on the dipole parameters but also to a χ 2 significantly far from 1. In contrast, larger values of T s (equivalent to imposing no outlier elimination) produce large o ff sets to the Galactic center and the departure of the χ 2 from unity. Table 1 reports the dipole, rotation, and quadrupole parameters obtained using T s = 90 and T p = 7. The rightest column of this table also shows results of Table 3 of TLG11 obtained when using Eqs. (5)-(6) of the present paper. With respect to TLG11, the dipole standard error has improved by about 20% and the postfit rms was reduced by 28%. No statistically significant rotation and quadrupole harmonics were found. Tables 2 and 3 display the correlations between the various parameters which are larger than 0.4 between d 1 and r 2, d 2 and r 1, and d 3 and a E 2 , 0 . The pattern obtained for the dipole is plotted in Fig. 3. The total acceleration of the Solar System barycentre from the DR solution is (9 . 3, 0 . 4, 0 . 3) ± (1.1, 1.1, 1.3) mm / s / yr in the Galactic reference frame. The centripetal acceleration is 9 . 3 ± 1 . 1 mm / s / yr. Assuming R = 8 . 4 kpc, it is equivalent to a circular rotation speed in the Galactic plane of 282 ± 32 km / s. The acceleration exhibits a non statistically significant component perpendicular to the Galactic plane. For the DRQ solution, the acceleration amounts to (10 . 1, 1 . 0, -1 . 3) ± (1.2, 1.3, 1.4) mm / s / yr in the Galactic reference frame, equivalent to a rotation speed of 303 ± 34 km / s. The vertical component is also statistically unsignificant.", "pages": [ 2, 3, 4 ] }, { "title": "5. Conclusion", "content": "This study showed that our previous determination of the secular aberration drift (Titov et al. 2011) is robust after the addition of new VLBI data. The results are consistent with predictions from Galactic models. The quasar proper motion field exhibits a dipole component oriented towards the Galactic center within ∼ 7 · . However, the quadrupole component remains statistically insignificant. A key point of our computation was the elimination of outlier data points and proper motions. We showed that this step must be considered with great care: if bad data are not properly identified and eliminated, they are likely to perturb significantly the estimates and lead to statistically biased results. Acknowledgements. The authors thank Drs. Laura Stanford and Craig Harrison of Geoscience Australia for proof-reading the manuscript and valuable comments. OT has pleasure in acknowledging the financial support from the Paris Observatory which made possible a one-month stay in Paris. This paper has been published with permission of the Chief Executive O ffi cer of Geoscience Australia.", "pages": [ 4 ] }, { "title": "References", "content": "Bahcall, J. N., & Bahcall, S. 1985, Nature, 316, 706 Bastian, U. 1995, In: M. A. C. Perryman & F. Van Leeuwen (Eds.), Proc. RGOESA Workshop on Future Possibilities for Astrometry in Space, ESA SP379, 99 Charlot, P. 1990, AJ, 99, 1309 Eubanks, T. M., Matsakis, D. N., Josties, F. J., et al. 1995, In: E. Hog & P. K. Seidelmann (Eds.), International Astronomical Union (IAU) Symp. 166, Kluwer Academic, Publishers, Dordrecht, 283 Fanselow, J. L. 1983, Observation Model and Parameter Partials for the JPL VLBI Parameter Estimation Software MASTERFIT-V1.0, JPL Publication 83-39 Fey, A. L., Ma, C., Arias, E. F., et al. 2004, AJ, 127, 3587 Fey, A. L., Gordon, D. G., & Jacobs, C. S. (Eds.) 2010, The Second Realization of the International Celestial Reference Frame by Very Long Baseline Interferometry, Presented on behalf of the IERS / IVS Working Group, International Earth Rotation and Reference Systems Service (IERS) Technical Note 35, Frankfurt am Main: Verlag des Bundesamts fur Kartographie und Geodasie Ghez, A. M., Salim, S., Weinberg, N. N., et al. 2008, AJ, 689, 1044 Gillessen, S., Eisenhauer, F., Trippe, S., et al. 2009, ApJ, 692, 1075 Kopeikin, S. M., & Makarov, V. V. 2006, AJ, 131, 1471 Majaess, D. J., Turner, D. G., & Lane, D. J. 2009, MNRAS, 398, 263 Mignard, F., & Morando, B. 1990, In: N. Capitaine & S. D'ebarbat (Eds.), Proc. Journ'ees 1990 Syst'emes de R'ef'erence Spatio-Temporels , Observatoire de Paris, 151 Mignard, F. 2002, In: O. Bienaym'e & C. Turon (Eds.), GAIA: A European Space Projet, EAS Publication Series, 2, 327 Mignard, F., & Klioner, S. 2012, A&A, 574, A59 Perryman, M. A. C., de Boer, K. S., Gilmore, G., et al. 2001, A&A, 369, 339 Reid, M. J., Menten, K. M., Zheng, X. W., et al. 2009, ApJ, 700, 137 Sovers, O. J., Fanselow, J. L., & Jacobs, C. S. 1998, Rev. Mod. Phys., 70, 1393 Titov, O., Lambert, S., & Gontier, A.-M. 2011, A&A, 529, A91 Xu, M. H., Wang, G. L., Zhao, M. 2012, A&A, 544, A135 Xu, M. H., Wang, G. L., Zhao, M. 2013, MNRAS, 430, 2633", "pages": [ 4 ] } ]

[ { "title": "ABSTRACT", "content": "Context. A promising model for normal Type Ia supernova (SN Ia) explosions are delayed detonations of Chandrasekhar-mass white dwarfs, in which the burning starts out as a subsonic deflagration and turns at a later phase of the explosion into a supersonic detonation. The mechanism of the underlying deflagration-to-detonation transition (DDT) is unknown in detail, but necessary conditions have been determined recently. The region of detonation initiation cannot be spatially resolved in multi-dimensional full-star simulations of the explosion. Aims. Wedevelop a subgrid-scale (SGS) model for DDTs in thermonuclear supernova simulations that is consistent with the currently known constraints. Methods. The probability for a DDT to occur is calculated from the distribution of turbulent velocities measured on the grid scale in the vicinity of the flame and the fractal flame surface area that satisfies further physical constraints, such as fuel fraction and fuel density. Results. The implementation of our DDT criterion provides a solid basis for simulations of thermonuclear supernova explosions in the delayed detonation scenario. It accounts for the currently known necessary conditions for the transition and avoids the inclusion of resolution-dependent quantities in the model. The functionality of our DDT criterion is demonstrated on the example of one three-dimensional thermonuclear supernova explosion simulation. Key words. Supernovae: general - hydrodynamics - turbulence - methods: statistical", "pages": [ 1 ] }, { "title": "Numerical implementation", "content": "F. Ciaraldi-Schoolmann 1 , I. R. Seitenzahl 1 , 2 , and F. K. Ropke 2 e-mail: [email protected] Received xxxx xx, xxxx / accepted xxxx xx, xxxx", "pages": [ 1 ] }, { "title": "1. Introduction", "content": "In the Chandrasekhar-mass model for SNe Ia, a thermonuclear burning front (flame) ignites near the center of a white dwarf star when its mass approaches the Chandrasekhar-limit (see Hillebrandt & Niemeyer 2000, for a review on SNe Ia models). In principle, there are two possible modes for this flame to burn through the degenerate material: a supersonic detonation and a subsonic deflagration . The result of the thermonuclear burning process has to be consistent with the main observational features, in particular the observed range in brightness. The origin for the diversity in brightness of SNe Ia are primarily di ff erences in the radioactive 56 Ni produced in the explosion (Truran et al. 1967; Colgate & McKee 1969). According to studies of Contardo et al. (2000), Stritzinger et al. (2006), and Mazzali et al. (2007), any valid model for normal SN Ia explosions should cover a range in the 56 Ni production of ∼ 0 . 4 to 1 . 0 M /circledot . Numerical simulations show that prompt detonations lead to strong explosions that produce almost exclusively iron group elements (Arnett et al. 1971), which is inconsistent with observed spectra. In contrast, pure deflagrations produce not enough iron group elements and release too little energy to explain the bulk of normal SNe Ia (Khokhlov 2000; Gamezo et al. 2003; Ropke et al. 2007). Moreover, Kozma et al. (2005) argue that unburned material left behind by the deflagration near the center of the star leaves imprints in nebular spectra that are not observed in normal SNe Ia. These problems are cured if a detonation triggers sometime during the late deflagration phase. In this delayed detonation scenario (Khokhlov 1991a), the detonation stage leads to a more complete burning of the white dwarf, resulting in an explosion strength and a chemical structure of the ejecta that is more consistent with the observed characteristics of SNe Ia (e.g. Gamezo et al. 2005; Golombek & Niemeyer 2005; Ropke & Niemeyer 2007; Mazzali et al. 2007; Kasen et al. 2009; Ropke et al. 2012; Seitenzahl et al. 2011, 2013). Whether or not a transition of the flame from a subsonic deflagration to a supersonic detonation is possible in SNe Ia has remained an open question since Blinnikov & Khokhlov (1986) first alluded to such a possibility. To understand deflagration-to-detonation transitions (DDTs) in general, the microphysical nature of turbulently mixed flames has to be analyzed. Extensive studies in this field were carried out by Lisewski et al. (2000), Woosley (2007), Aspden et al. (2008), and Woosley et al. (2009). Although these studies do not provide stringent evidence for DDTs in SNe Ia, necessary conditions for such transitions can be derived from them. In particular, their analyses show that strong turbulence must interact with the flame during later stages of the explosion in order to facilitate a DDT. This raises the question of whether su ffi ciently high turbulent velocity fluctuations still occur when the deflagration is close to extinction due to the expansion of the star. The Rayleigh-Taylor instability becomes weaker in the later expansion phase, hence this expansion will ultimately freeze out all turbulent motions (Khokhlov 1995). Ropke (2007) showed that high turbulent velocities, although rare, are indeed still found in late stages of three-dimensional simulations of the deflagration phase. This indicates that the macroscopic conditions for a DDT are met, but it is clear that evidence for DDTs requires to resolve the microscopic mechanism of this transition as well. The length scales on which this process takes place, however, are too small to be resolved in multi-dimensional full-star simulations of the explosion. Therefore, large-scale simulations of the delayed detonation scenario have to invoke some kind of model for DDTs. A simple parameterization is to prescribe a certain fuel density ahead of the flame at which the DDT is triggered (Khokhlov et al. 1997; Hoflich et al. 1998; Gamezo et al. 2005; Townsley et al. 2009; Jackson et al. 2010). This, however, does not account for the important role that turbulence plays in the DDT mechanism. An alternative is to trigger the DDT at patches of the burning front where turbulent eddies first penetrate the internal flame structure (Golombek & Niemeyer 2005; Ropke & Niemeyer 2007). The onset of this so-called distributed burning regime (e.g. Peters 2000) is necessary (Niemeyer & Woosley 1997), but still not su ffi cient for a DDT. Woosley et al. (2009) argue that, in addition to entering the distributed burning regime, particularly high velocity fluctuations are required. In a very simple way this constraint has been implemented in a series of two-dimensional delayed detonation simulations (Kasen et al. 2009). Here, we present a subgrid-scale (SGS) model of DDTs for full-star simulations of the delayed detonation scenario. In particular, we aim at consistency with the microphysical mechanism of this process, as far as known, and independence of the numerical resolution in the simulation. Due to the stochastic nature of turbulence, a SGS model for DDTs cannot provide any proof for a DDT to occur, but it can evaluate a probability for this transition under certain assumptions. This paper is organized as follows. In Section 2 we outline the constraints on DDTs in SNe Ia according to current knowledge. The implementation of the DDT-SGS model in the hydrodynamic code is described in Section 3. The resolution dependence of this model is tested in Section 4. Section 5 gives a summary and an outlook for further applications.", "pages": [ 1, 2 ] }, { "title": "2. Constraints on DDTs in SNe Ia", "content": "Which physical mechanism causes a DDT in unconfined media (as required in the supernova case) remains uncertain, but several possibilities have been suggested. One proposed mechanism for the initiation of a detonation relies on the dissipation and the consequential conversion of turbulent energy into internal energy on the Kolmogorov length scale (Woosley 2007). Here, it is assumed that the rate of dissipating turbulent energy is high enough that the temperature of a region of fuel reaches the ignition point. Provided that a su ffi cient amount of fuel is available (the ignition region is large enough) a detonation may be formed. Another mechanism recently proposed by Charignon & Chi'eze (2013) is based on the amplification of acoustic waves in the steep outer density gradient of the white dwarf. This would trigger the detonation wave far away from the deflagration front. In our work, however, we assume that the deflagration flame itself produces conditions suitable for a DDT and follow the concept of the Zel'dovich Gradient Mechanism (Zel'dovich et al. 1970), even though it has been suggested that the formation of a preconditioned hot spot may not a neccessary prerequisite (Poludnenko et al. 2011; Kushnir et al. 2012). In the Gradient Mechanism, it is assumed that a spontaneous ignition of the fuel in a region with a shallow spatial gradient of induction times leads to a supersonic reaction wave and the build-up of a shock. If the phase velocity of the reaction wave approaches the Chapman-Jouguet velocity it may transition into a detonation. The Gradient Mechanism has been applied to SNe Ia first by Blinnikov & Khokhlov (1986, 1987) and has been further investigated by Khokhlov (1991a,b), Khokhlov et al. (1997) and Niemeyer & Woosley (1997). The most important result of their analyses is that DDTs in SNe Ia can only occur if turbulence approaches an intensity that causes strong mixing of cold fuel and hot burned material. A microphysical study of Lisewski et al. (2000) revealed that the required turbulent velocity fluctuations v ' crit must be higher than 10 8 cms -1 . By analyzing some time steps of a pure deflagration model, Ropke (2007) found a nonvanishing probability of finding such high velocity fluctuations at the flame. Hence, the probability of finding su ffi ciently high velocity fluctuations in the entire late deflagration phase may reach high values. The occurrence of high turbulent velocity fluctuations is attributed to intermittency in the turbulent motions. Weak intermittency in burned regions in the exploding white dwarf was found by Schmidt et al. (2010) by calculating and fitting characteristic scaling exponents of the turbulent velocity field. These exponents were obtained from the computation of high-order velocity structure functions (Ciaraldi-Schoolmann et al. 2009), using the data of a highly resolved numerical simulation, the deflagration model of Ropke et al. (2007). The high velocity fluctuations that Ropke (2007) found in the same model indicate that intermittency at the flame is significantly stronger than in burned regions. However, due to the challenges of performing a detailed analysis of intermittency at a highly wrinkled and folded flame front in full-star simulations, some uncertainties in the origin of these high velocity fluctuations remain. That high velocity fluctuations occur somewhere at the flame is necessary, but not su ffi cient for a DDT. It is important that these fluctuations are located within a certain amount of fuel of the turbulently mixed regions. The minimum amount of fuel X DDT fuel required for ignition and creation of a selfpropagating detonation wave depends on various quantities, such as the fuel density, the chemical composition, as well as the fuel temperature (see Arnett & Livne 1994; Khokhlov et al. 1997; Seitenzahl et al. 2009). Due to these dependencies, one cannot specify a general, constant value for X DDT fuel (but see Seitenzahl et al. 2009, and tables therein). Niemeyer & Woosley (1997) point out that a necessary constraint for a DDT is the burning in the distributed burning regime. In the distributed burning regime, turbulent eddies are able to penetrate the internal flame structure. Under this condition, the nuclear burning time scale τ nuc becomes independent of heat conduction processes and is exclusively given by the dynamics of turbulent eddies. The reason is that these eddies reach the fuel faster than the flame itself and mix it during the turnover into the reaction zone. The eddy turnover time is given by where /lscript is the typical length scale of a turbulent eddy and v ' ( /lscript ) the velocity fluctuation on that scale. Woosley (2007) point out that for a successful DDT, the carbon and the oxygen flame have to be su ffi ciently separated spatially. They argue that this is expected to be the case for fuel densities below ∼ 3 × 10 7 g cm 3 , which covers the density regime we consider here. The distributed burning regime for the canonical composition of equal mass 12 C and 16 O is reached when the fuel density ρ fuel at the flame has declined below ∼ 3 × 10 7 g cm -3 (Niemeyer & Woosley 1997). Recent studies of Woosley (2007) and Woosley et al. (2009) suggest that there are further constraints on triggering detonations. Within the distributed burning regime, it is necessary that the balance between turbulent mixing and nuclear burning becomes disturbed, which is the case for D T = τ eddy( L ) /τ nuc /greaterorsimilar 1, where D T is the turbulent Damkohler number and L the turbulent integral scale. During the burning in this so-called stirred flame regime (Kerstein 2001), the flame becomes significantly broadened until at D T ∼ 1 the flame width δ approaches L which is approximately 10 6 cm (e.g. Woosley 2007). With turbulent intensities typically expected for deflagrations in SNe Ia, the density at which this condition is expected to be met is 0 . 5 /lessorsimilar ρ fuel / (10 7 g cm -3 ) /lessorsimilar 1 . 5 (Woosley 2007). Finally, a DDT region which meets the described constraints concerning v ' crit , X DDT fuel and ρ fuel has to exceed a critical spatial scale /lscript crit, which is of the order of 10 6 cm (e.g. Khokhlov et al. 1997; Dursi & Timmes 2006; Seitenzahl et al. 2009) and hence comparable to the integral scale L . The time scale of mixing the fuel and ash in this region can be estimated with Eq. (1). Assuming that both fuel and ash elements can be carried by a turbulent eddy of size /lscript crit over the distance /lscript crit in a half eddy turnover time, it takes to mix the components. While v ' is well-determined in our model, /lscript crit is uncertain because of the unresolved shape of the temperature gradient (Seitenzahl et al. 2009). Using v ' ( /lscript crit) = 10 8 cms -1 (Lisewski et al. 2000), we find τ eddy1 / 2 ( /lscript crit) = 5 × 10 -3 s and we adopt this typical, fixed value in our model. A region fulfilling all DDT criteria described above must exist for at least this amount of time such that a DDT may occur.", "pages": [ 2, 3 ] }, { "title": "3.1. Three-dimensionalfull-starsimulations", "content": "The hydrodynamicscode that is used to carry out the simulations of this study is based on the PROMETHEUS code (Fryxell et al. 1989) that implements the Piecewise Parabolic Methods (PPM) of Colella & Woodward (1984) to solve the reactive Euler equations in a finite volume approach. The thermonuclear combustion waves are modeled as sharp discontinuities between fuel and ash and are numerically represented with a level set technique following Reinecke et al. (1999). Our implementation follows some basic concepts of large eddy simulations, in which the largest turbulent structures and motions are resolved on the grid scale or above. Turbulence on unresolved scales is calculated with a SGS turbulence model (Schmidt et al. 2006a,b). In our simulations, we use a comoving grid technique (Ropke 2005; Ropke et al. 2006). We discretize our set of model equationson two nested computational grids for which the grid spacing is continuously enlarged to capture the explosion. While an outer inhomogeneous grid follows the overall expansion of the white dwarf, the deflagration flame is tracked with an inner homogeneous Cartesian grid. For the initial composition of the white dwarf, we choose a 12 C and 16 O mixture in equal amounts by mass and set the electron fraction to Ye = 0 . 49886, corresponding to solar metallicity. The white dwarf is assumed to be cold ( T = 5 × 10 5 K). We use an initial central density of 2 . 9 × 10 9 g cm -3 . The initial flame configuration from which the deflagration front evolves equals the setup described in Ropke et al. (2007) with 1600 spherical kernels of radius 2 . 6 km distributed within a sphere of 180 km around the center of the white dwarf. In our full-star simulations, the DDT regions are not resolved, since ∆ ( t ) > /lscript crit for all times, where ∆ ( t ) is the time-dependent resolution of the inner comoving grid. Therefore, we employ a SGS model for DDTs, which models the DDT relevant quantities on unresolved scales.", "pages": [ 3 ] }, { "title": "3.2. Determinationoftheflamesurfacearea", "content": "As described in Section 2, we have to determine the area of the flame where the values of X DDT fuel , ρ fuel and v ' crit are appropriate for a DDT. Here we face the problem that the discontinuity approach of the flame generally prevents us from determining the physical conditions at the flame precisely. Below we show how to obtain an approximation for the physical conditions at the flame front, by considering only grid cells that are approximately split into two equal parts by the flame (resp. the level set). We define X fuel as the mass fraction of unburned material in a grid cell. For the later analysis, we are interested in the quantities at the flame . These are di ffi cult to measure since the flame is numerically represented as a discontinuity and the computational cells intersected by it contain a mixture of fuel and ash. We therefore consider only cells with 1 / 3 ≤ X fuel ≤ 2 / 3. This way we ensure that the flame separates the grid cell into roughly equal size parts of fuel and ash, and the thermodynamic values at the cell center should reasonably approximate the real values at the turbulent flame, instead of being dominated by fuel or ash material. We emphasize that the numerical quantity X fuel is not directly equivalent to the required physical amount of fuel X DDT fuel for triggering a DDT. X DDT fuel cannot accurately be determined on scales /lscript crit < ∆ ( t ) and we cannot evaluate precisely whether the required amount of fuel for a DDT is available. As described in Section 2, we further have to ensure that the flame resides in the distributed burning regime and additionally obeys the constraints described by Woosley (2007). Therefore we additionally limit our analysis to grid cells in the density range of 0 . 5 /lessorsimilar ρ fuel / (10 7 g cm -3 ) /lessorsimilar 1 . 5. We define the number of all grid cells at the flame at a given time t as N flame( t ) and the cells which additionally meet the constraints concerning X fuel and ρ fuel as N ∗ flame ( t ). In the same context we define the flame surface area as A flame( t ) and the part which meets the mentioned constraints as A ∗ flame ( t ), respectively. To determine A flame( t ) we assume that due to the nature of turbulence the flame is similar to a fractal object with fractal dimension D (see Kerstein 1988, 1991; Niemeyer 1995; Blinnikov & Sasorov 1996). We note that compared to an ideal fractal, the wrinkles and curvatures of the flame are not sustained on very small scales. In our model, the DDT occurs shortly after entering the distributed regime. Stricly speaking, the description of the flame as a fractal was established for the flamelet regime only. However, for the specific case we consider here, the flame neither fills the entire star nor a large fraction of its volume. Instead, seen from the large scales resolved in our simulations, the burning is still confined to a narrow sheet, to which we apply our fractal description. The same line of argument was used by Schmidt (2007) to justify a level-set based flame model beyond the flamelet regime. Therefore, for our large-scale simulations, a fractal approach is an acceptable description of the flame for all physical scales directly relevant to our DDT model. If turbulence is driven by the Rayleigh-Taylor instability, D = 2 . 5, whereas for Kolmogorov turbulence without intermittency, a value of D = 2 . 33 is expected (e.g. Kerstein 1988; Sreenivasan 1991; Kerstein 1991; Niemeyer 1995, and references therein). For intermittent turbulence, it is argued that D = 2 . 36 (e.g. Halsey et al. 1986; Sreenivasan 1991). The level set method o ff ers us the opportunity to relate the quantities ∆ ( t ) and N flame( t ) to A flame( t ). Since for every numerical resolution the flame propagates like a thin interface through the grid cells, we assume that the flame surface behaves selfsimilar and is resolution-independent on all considered length scales. We therefore determine the self-similarity dimension defined by where N is the number of self-similar pieces and /epsilon1 the reduction (or zoom) factor. For our purposes we need the number of grid cells N flame1 and N flame2 from two simulations with di ff erent resolutions ∆ 1( t ) and ∆ 2( t ) of the same initial white dwarf model. Then D is given by From here it follows and since A flame( t ) should be equal for both simulations, we identify as the flame surface area. Once D is determined we evaluate A ∗ flame ( t ) with Eq. 6 by using N ∗ flame ( t ) instead of N flame( t ). Since N ∗ flame ( t ) /lessmuch N flame( t ) there are not enough data to derive a reliable value of D for A ∗ flame ( t ) directly. The calculation of D is performed together with a resolution test in Section 4.1.", "pages": [ 3, 4 ] }, { "title": "3.3. Theprobabilitydensityfunctionofturbulentvelocity fluctuations", "content": "The turbulent velocity fluctuations v ' ( /lscript ) are determined by the SGS model of Schmidt et al. (2006a,b). This model has already been applied to a simulation of a pure deflagration in a Chandrasekhar-mass WD (e.g. Ropke et al. 2007), and turbulence properties of this model were analyzed in Ciaraldi-Schoolmann et al. (2009). However, it has not been explicitly tested yet whether the SGS model can properly reproduce the rare high velocity fluctuations at the flame required for a DDT. In this section we perform some test calculations in order to evaluate whether the SGS model can be used for the construction of a DDT model.", "pages": [ 4 ] }, { "title": "3.3.1. Testing the SGS model in reproducing the high velocity fluctuations", "content": "To judge whether the SGS model is capable of modeling the high velocity fluctuations at the flame correctly, we first have to find out how often these fluctuations occur. A commonly used statistical method is the calculation of a probability density function (PDF) of v ' ( /lscript ). By definition, a PDF constitutes a continuous distribution function, but in our case only discrete data are available. However, by sorting and sampling the data into bins, we can construct a histogram of v ' ( /lscript ). Fitting this histograms with an appropriate fit function then gives us an approximated PDF of v ' ( /lscript ). This procedure has already been performed by Ropke (2007). The result shows clearly a slow decline of the histogram toward higher velocity fluctuations, indicating a nonvanishing probability of finding su ffi ciently high velocity fluctuations for a DDT. However, an open question is whether the slow decline seen in the histogram is of physical origin, or whether it is an artifact of turbulence- or flame-modeling. To investigate this, we developed an algorithm that derives the velocity fluctuations from the resolved velocity field of the hydrodynamic flow. This allows us to compare the histogram that contains the data of these resolved fluctuations with the histogram that contains the values v ' ( /lscript ) of the SGS model. The resolved velocity field v ( r ) of the hydrodynamic flow is a superposition of the turbulent velocity fluctuations and the bulk expansion of the white dwarf, where the latter contribution points in radial direction. We have to subtract the bulk expansion from v ( r ) to obtain the pure fluctuating part v turb( r ). For details on how the turbulent velocity fluctiations are calculated see Ciaraldi-Schoolmann et al. (2009). To compare v turb( r ) = | v turb( r ) | with v ' ( /lscript ), we have to take into account that the SGS model returns a value on the scale ∆ ( t ) and that the quantity v turb( r ) has to be considered on the same scale. We thus determine the average absolute velocity di ff erences | v turb[ ∆ ( t )] | of neighboring grid cells, which is given by where v turb( r ) is the velocity fluctuation in the chosen grid cell and v turbi ( r + d ) is the velocity fluctuation in the i -th of the N adjacent grid cells (note that | d | = ∆ ( t )). The described procedure has been performed with a Monte-Carlo based program for a total number of randomly chosen 10 6 di ff erent grid cells, where for a larger number of cells, no change in the results was found. We then construct a histogram of | v turb[ ∆ ( t )] | . In Fig. 1(a) the histograms of | v turb[ ∆ ( t )] | and v ' [ ∆ ( t )] that contain the data in the vicinity of the flame are shown. The simulation is based on a grid with 512 3 cells and the histograms shown are for t ≈ 0 . 9 s as an illustrative example. This instant corresponds to the late deflagration phase, when turbulence is strong and a ff ects the structure and propagation of the flame significantly. We see in both histograms a slowing decline toward higher velocity fluctuations, which shows that the decline in the histogram of v ' [ ∆ ( t )] is no artifact of SGS turbulence model. Another possibility, however, is that it is caused by our levelset based flame model and the flame-flow coupling on the resolved scales. We therefore repeat the analysis described above using a fixed length scale of | d | = 4 ∆ ( t ). Even though the turbulence model calculates quantities on the grid scale, in this case a rescaling of the velocity fluctuation from ∆ ( t ) to 4 ∆ ( t ) is not required for evaluating the presence of the highest velocity fluctuations in the tail of the histogram. For | d | = 4 ∆ ( t ) we impose the additional constraint X fuel ≤ 0 . 5 to avoid counting cells containing mainly fuel far ahead of flame. This result is also shown in Fig. 1(a). We can identify again a slow decline toward high velocity fluctuations similar to the histogram of | v turb[ ∆ ( t )] | , and hence also to that of v ' [ ∆ ( t )]. Thus, the slow decline seems to originate not only from computational cells that are intersected by the flame but it persists in a certain region away from it. This indicates that it is not an artifact of the modeling but is rather due to intermittency in the turbulent flow field near the flame.", "pages": [ 4 ] }, { "title": "3.3.2. Rescaling of the velocity fluctuations", "content": "Since our simulation code uses a comoving grid technique, we rescale the value of v ' [ ∆ ( t )] to v ' ( /lscript crit) with /lscript crit = 10 6 cm (see Section 2). The rescaled velocity fluctuations are given by where the scaling exponent α depends on the mechanism which drives the turbulence. We assume incompressible and isotropic Kolmogorov turbulence (Kolmogorov 1941), where α = 1 / 3. We note, however, that Ciaraldi-Schoolmann et al. (2009) found in burned regions a transition of the turbulence driving mechanism at a certain length scale (see also Niemeyer & Woosley 1997). This length scale is of the same order of magnitude as /lscript crit and it separates the regime of small-scale isotropic Kolmogorov turbulence from Rayleigh-Taylor instability driven anisotropic turbulence on large scales. For the latter, α = 1 / 2. These considerations take the entire turbulent velocity field into account that has well-defined statistical properties, but for a DDT only the strong turbulent velocity fluctuations are important. Turbulence is most intense in trailing patches of the Rayleigh-Taylor 'mushroom caps', where strong shear instabilities occur (see Ropke 2007). The scaling properties of an intermittent velocity field for scales /lscript /greaterorsimilar /lscript crit in such regions at the flame front are not known. We can estimate the di ff erence f di ff between the scaling relations of a Kolmogorov- and Rayleigh-Taylor instability driven turbulence. Using Eq. 8 we find For highly resolved simulations, where ∆ ( t ) ≈ /lscript crit, the di ff erence is negligible. We perform simulations with 256 3 and 512 3 grid cells and find for the late deflagration phase where DDTs are expected ∆ ( t ) ≈ 4 × 10 6 cm for the lower resolved and ∆ ( t ) ≈ 2 × 10 6 cm for the higher resolved simulation, leading to uncertainties of about 26% and 12%, respectively. To check to what extent these deviations a ff ect the rescaled values of the high velocity fluctuations, we compare the histograms of v ' ( /lscript crit) with both scaling exponents α = 1 / 3 and α = 1 / 2. Since we implement a DDT model we take now only grid cells into account that meet certain DDT constraints, hence the data N ∗ flame ( t ) is used for the histogram construction. The result is shown in Fig. 1(b,c) for the late deflagration phase at t ≈ 0 . 9 s. The agreement of both histograms is excellent, particularly in the high resolution case. We note that intermittency may slow down the decrease of the velocity fluctuations towards smaller scales compared to the scaling given in eq. 8, or, if it dominates the scaling behavior, it may change the trend completely. Our model would still be a good approximation in the first case. Comparing the histograms in Fig. 1(a) suggests that indeed the velocity fluctuations still decrease with scale, but a more rigorous verification is not possible with our simulations. While studying intermittency e ff ects in ash regions is possible based on the computation of structure functions of the velocity field (Schmidt et al. 2010; Ciaraldi-Schoolmann et al. 2009), for geometrical reasons such functions cannot easily be determined at the flame front itself.", "pages": [ 5, 6 ] }, { "title": "3.3.3. Fitting the data of the histogram", "content": "To calculate the probability of finding su ffi ciently high velocity fluctuations for a DDT, we apply a fit to the histogram of v ' ( /lscript crit) to obtain an approximated PDF (see also Ropke 2007). Since for a DDT only the high velocity fluctuations are of interest, we are justified in restricting our fit to the right of the maximum of the histogram. The fit should further be motivated by an appropriate distribution function that can explain the intermittent behavior in turbulence at the flame. Schmidt et al. (2010) used a lognormal distribution of an intermittency model of Kolmogorov (1962) and Oboukhov (1962) to fit characteristic scaling exponents that where obtained from the computation of high order velocity correlation functions. This detailed analysis revealed that the intermittency in ash regions is weaker than predicted in the log normal model. In contrast, Ropke (2007) found that a lognormal fit fails to reproduce the distribution of the high velocity fluctuations at the flame, since it declines faster toward larger v ' ( /lscript crit) than the velocity data of the histogram. This result suggests that intermittency at the flame is fundamentally di ff erent than in ash (see also the discussion in Schmidt et al. 2010). In Fig. 1(d) we show histograms of v ' ( /lscript crit) that contain the data N ∗ flame ( t ) and the data in ash regions in the late deflagration phase at t ≈ 0 . 9 s (again, this instant is chosen as an illustrative example here). The simulation was run with 512 3 grid cells. There is a significant di ff erence between the shapes of the PDFs. The slow decline of the histogram that contains the data in ash regions appears almost linear in the log-normal illustration, while the histogram that contains the data in the vicinity of the flame has a significant positive curvature after its maximum. This is further evidence that turbulence near the flame has stronger intermittency than in ash regions. As of yet there is no physically motivated model for explaining intermittency at a deflagration front in white dwarfs. Consequently, an empirical distribution function has to be used to fit the slow decline of the histogram of v ' ( /lscript crit) at the flame front. Here we follow Ropke (2007) and use an ansatz of the form This geometric function is able to fit the right part of the histogram over a large range and a 1, a 2 and a 3 are the three fitting paramters. The probability P [ v ' ( /lscript crit) ≥ v ' crit ]( t ) of finding velocity fluctuations of at least v ' crit is given by where Γ is the upper incomplete gamma function. We note that the DDT instant determined below is not really a special point in the time evolution of the PDF. When a detonation is triggered in the model, the parts of the deflagration flame that are directly attached to the quickly spreading detonation front are excluded from the determination of the PDF.", "pages": [ 6 ] }, { "title": "3.4. ThedetonationareaandtheDDTcriterion", "content": "In Section 3.2 we defined A ∗ flame ( t ) as the part of the flame that meets the required conditions for a DDT concerning the quantities ρ fuel and X fuel. The probability of finding su ffi ciently high velocity fluctuations at this restricted flame surface area was derived separately in the previous section. We define now as the part of the flame surface area that can potentially undergo a DDT (see also Ropke 2007). This quantity has to exceed a critical value A crit that is required for a DDT. We assume that a DDT region has a smooth two-dimensional geometry and use therefore A crit = /lscript 2 crit = 10 12 cm 2 . For A det( t ) > A crit, we finally check whether this condition holds for at least τ eddy1 / 2 ( /lscript crit) to ensure a su ffi cient mixing (see Section 2). If this is true, our DDT criterion is met and detonations are initialized. The number of DDTs N DDT in our model is given by where N DDT is always rounded down to the next integer. We note that both quantities A ∗ flame ( t ) and particularly P [ v ' ( /lscript crit) ≥ 10 8 cm s -1 ]( t ) may rise steeply within τ eddy1 / 2 ( /lscript crit), hence we often get N DDT > 1. The minimum time between two DDTs is given by τ eddy1 / 2 ( /lscript crit), since, after a successful DDT, the time for A det( t ) > A crit is restarted. The same holds for the case A det( t ) < A crit happens before τ eddy1 / 2 ( /lscript crit) is reached. We still have to decide on the location where detonations are initialized. Since the high turbulent velocity fluctuations are crucial for a DDT, we chose those N DDT grid cells from N ∗ flame ( t ) that contain the highest values of v ' ( /lscript crit). In analogy to the deflagration ignition, detonations are set by initializing an additional level set that propagates supersonically at the appropriate detonation speed (see Fink et al. 2010) through the white dwarf matter. A shortcoming of this DDT model is that it does not assess whether there is indeed a 'connected' region of size 10 12 cm 2 that fulfills the requirements for a DDT. The probability P [ v ' ( /lscript crit) ≥ v ' crit ]( t ) and the flame surface area A ∗ flame ( t ) are determined from all (possibly disconnected) grid cells suitable for a DDT. Therefore they do not provide any information on localized areas. They rather are global quantities. The same holds for τ eddy1 / 2 ( /lscript crit), since here we also use a uniform value. From a computational point of view, we emphasize that the inclusion of τ eddy1 / 2 ( /lscript crit) is also important to keep the DDT criterion independent of resolution. Since the maximum time step ∆ CFL of our code is given by the Courant-Friedrichs-Lewy (CFL) condition (Courant & Friedrichs 1948), the time steps of higher resolved simulations are shorter than for lower resolved ones. Applying our criterion without a time-dependent variable would mean that higher resolved simulations get an enhanced chance for a successful detonation, simply because it tests for DDTs more frequently. We note that in our simulations ∆ CFL is usually much shorter than τ eddy1 / 2 ( /lscript crit).", "pages": [ 6, 7 ] }, { "title": "4. The fractal dimension of the flame and resolution test in one full-star model", "content": "To test the resolution dependence of the implemented DDT criterion we apply it to the deflagration model described in Section 3.1 and run it with a resolution of 256 3 and 512 3 grid cells. Unfortunately, we cannot perform a detailed resolution study, since simulations with more than 512 3 grid cells are computationally too expensive, while the DDT model cannot be applied for very low resolved simulations due to insu ffi cient data for fitting the histogram of v ' ( /lscript crit). The quantities and the corresponding threshold values of the DDT criterion shall be summarized here: 1 / 3 ≤ X fuel ≤ 2 / 3, 0 . 5 /lessorsimilar ρ fuel / (10 7 g cm -3 ) /lessorsimilar 1 . 5, v ' crit = 10 8 cm s -1 , A crit = 10 12 cm 2 and τ eddy1 / 2 ( /lscript crit) = 5 × 10 -3 s. One parameter still undetermined is the fractal dimension of the flame, which we now derive from the resolution test.", "pages": [ 7 ] }, { "title": "4.1. Thefractaldimensionoftheflame", "content": "In Fig. 2(a) we show N flame( t ). The thick black curve is the result for the lower resolved simulation and the thick red (dashed) curve for the higher resolved one, respectively. The other curves are theoretically expected results for the higher resolved simulation, if a certain fractal dimension of the flame is assumed. These curves can be calculated from N flame1( t ) and the known resolutions ∆ 1( t ) and ∆ 2( t ) of the simulations, by specifying a value for D in Eq. 5. We see that the curves for D = 2 and D = 3 are not consistent with the data, which shows that the flame is indeed a fractal. In Fig. 2(b) the fractal dimension D (calculated from Eq. 4) is shown as function of time. A necessary constraint in our criterion is that ρ fuel must be in a certain range (see Section 2). At approximately t = 0 . 8 s, the first grid cells at the flame front approach ρ fuel = 1 . 5 × 10 7 g cm -3 , while most part of the flame resides at higher densities. We see that at this time D ≈ 2 . 5, but DDTs will occur later when a su ffi ciently large part of the flame surface area meets the DDT constraints and τ eddy1 / 2 ( /lscript crit) has elapsed. At these times D < 2 . 5. In agreement with theory (e.g. Halsey et al. 1986; Sreenivasan 1991), we use a constant value of D = 2 . 36 for our DDT model (see Section 3.2). In Fig. 2(c), we show the quantity A ∗ flame ( t ) (calculated from Eq. 6 with D = 2 . 36) for both simulations. Since the curves are in a good agreement, the choice of D = 2 . 36 is justified. We next discuss some caveats concerning the determination of the fractal dimension of the flame. It is obviously a rough approximation to take D as a constant, since we see in Fig. 2(b) that this quantity declines continuously until t ≈ 1 . 2 s. Moreover, one could expect that models with a di ff erent deflagration phase may have another curve progression of D . Therefore, a determination of D for every delayed detonation simulation that has a di ff erent evolution of the deflagration would be necessary. However, at the time when DDTs occur, we assume that turbulence in the deflagration phase is fully developed and obeys well defined statistical properties. Hence a significant deviation of D for di ff erent deflagrations seems to be unlikely, but it cannot be ruled out completely. Finally the results may be di ff erent, if higher resolutions with more data are used for the determination of D , so that a convergence of D may be obtained at very high resolutions only. Apart from the limited computational resources that prevent simulations at such high resolutions, we mention in Section 3.2 that the flame is no ideal fractal, so that at some length scale Eq. 4 becomes inappropriate to derive D . We note that the values for D in Fig. 2(b) may also indicate that the flame is a ff ected by di ff erent mechanisms and instabilities that drive the turbulence, since 2 . 3 /lessorsimilar D /lessorsimilar 2 . 5 are expected for di ff erent instabilities at the flame front (see Section 3.2). In any case, the chosen value of D = 2 . 36 is appropriate for our purposes, since both curves of A ∗ flame ( t ) in Fig. 2(c) are in a good agreement.", "pages": [ 7 ] }, { "title": "4.2. Theprobabilityoffindinghighvelocityfluctuations", "content": "In Fig. 2(d), a histogram of v ' ( /lscript crit) with the fit according to Eq. 10 through the data for both simulations is shown at t = 0 . 9 s. Since we are mainly interested in the high velocity fluctuations, the starting point of the fit is at twice the velocity at the maximumofthe corresponding histogram. We see that the histograms and the approximated PDFs are in very good agreement. For the lower resolved simulation, however, the data contains more scatter and there is an earlier cuto ff toward higher velocity fluctuations. This is the result of a coarser binning due to less data in lower resolved simulations. The probability P [ v ' ( /lscript crit) ≥ 10 8 cm s -1 ]( t ) is shown in Fig. 3(a) for both simulations. The good agreement of both curves reflects that the approximated PDFs are largely independent of resolution. The highest values are found for 0 . 95 s < t < 1 . 00 s, when the intermittency in the turbulence is most pronounced.", "pages": [ 7 ] }, { "title": "4.3. Thedetonationarea", "content": "The quantity A det( t ) is shown in Fig. 3(b) for both resolutions. Since A det( t ) is calculated from A ∗ flame ( t ) and P [ v ' ( /lscript crit) ≥ 10 8 cm s -1 ]( t ), it is also independent of resolution. This manifests in Fig. 3, in which the overall shape of the curve of A det( t ) is very similar to that of P [ v ' ( /lscript crit) ≥ 10 8 cm s -1 ]( t ). In particular the strong variations of P [ v ' ( /lscript crit) ≥ 10 8 cm s -1 ]( t ) can be identified again. This indicates that the change of the flame surface area A ∗ flame ( t ) for a given interesting time interval τ eddy1 / 2 ( /lscript crit) is much smaller than the fast temporal variations of the probability P [ v ' ( /lscript crit) ≥ 10 8 cm s -1 ]( t ). To see this, compare A ∗ flame ( t ) in Fig. 2(c) with Fig. 3(a). For our resolution test we have chosen A crit = 10 12 cm 2 , and we see that this value is exceeded by A det( t ). Therefore, DDTs will occur if A det( t ) > A crit for at least τ eddy1 / 2 ( /lscript crit). This condition is indeed reached in both simulation, where the first DDTs are initialized at approximately 0 . 92 s. This time is marked with a dot at the curve of A det in Fig. 3(b). For the lower resolved simulation we find A det ≈ 1 . 72 × 10 12 , hence N DDT = 1. In the higher resolved simulation we find A det ≈ 2 . 12 × 10 12 , hence here detonations are initialized already in two grid cells. Note that these cells are located at di ff erent deflagration plumes and that they are spatially disconnected. As long as A det( t ) > A crit for τ eddy1 / 2 ( /lscript crit), new DDTs commence at later time steps. This happens in our simulations, since A det( t ) > A crit for most of the time in the interval 0 . 92s /lessorsimilar t /lessorsimilar 1 . 07s. However, there are a few interruptions: In some time steps the condition A det( t ) < A crit occurs before τ eddy1 / 2 ( /lscript crit) was reached, preventing some DDTs. The maximum of N DDT is 10 for the lower resolved and 12 for the higher resolved simulation, which is reached at t ≈ 0 . 96 s for both cases. The last DDT occurs at t ≈ 1 . 06 s in a single grid cell in both simulations. In Fig. 4, the time of the first DDT is visualized for both simulations. The deflagration flame, represented by the level set function, is shown as a transparent iso-surface. While the number of DDTs is largely resolution-independent, their localization is generally quite di ff erent. In Fig. 4, the first DDT occurs at di ff erent places at the deflagration flame (both figures show the same viewing angle). The reason is that the exact number and locations of the grid cells in which the highest velocity fluctuations occur di ff ers. In subsequent papers of this series we will show that di ff erent distribution of DDT spots have an impact on the 56 Ni production rate in the detonation phase.", "pages": [ 7, 8 ] }, { "title": "5. Conclusion", "content": "We introduced the first subgrid-scale model for implementing deflagration-to-detonation transitions (DDT) in a hydrodynamic code for large-scale simulations of Type Ia supernova (SN Ia) explosions. The model includes the current knowledge on DDTs in SNe Ia and can be summarized as follows: We first ensure that a su ffi cient number of grid cells at the flame have a certain fuel 0.85 0.90 0.95 1.00 1.05 1.10 1.15 time [s] (b) fraction and are in a certain fuel density range. From the number and size of these cells we determine a suitable flame surface area for DDTs, where we assume that the flame can be considered as a fractal. Simultaneously, we construct a histogram of the turbulent velocity fluctuations in the above-mentioned cells, where we rescale these fluctuations from the grid scale to the critical length scale of a DDT region by assuming Kolmogorov turbulence. Then we estimate the probability of finding su ffi ciently high velocity fluctuations for a DDT, by applying a fit function to the histogram. This probability multiplied with the flame surface area that is suitable for a DDT constitutes a potential deto- nation area, which we compare with the required critical size of a DDT region. When the potential detonation area exceeds this critical size for at least half of an eddy turnover time, the DDT constraints are fulfilled. In this case detonations are initialized in the grid cells at the flame surface area suitable for a DDT that contain the highest velocity fluctuations. The number of initialized detonations equals the ratio of the potential detonation area to the critical size of a DDT region. Although our model refers to the initiation of the detonation via the Zel'dovich gradient mechanism, we note that other proposed mechanisms for forming a detonation out of a 13 10 12 10 11 10 ] 2 [cm det A A crit 3 [256 ] 3 ] [512 turbulent deflagration burning regime (Poludnenko et al. 2011; Kushnir et al. 2012) would require a similar parameterization of the DDT-SGS model. In all cases, the critical quantity is the strengh of turbulence. However, the models of Poludnenko et al. (2011) and Kushnir et al. (2012) require turbulence speeds close to sonic, which we do not observe in our simulations of deflagrations in white dwarfs. The velocity fluctuations of 10 8 cms -1 assumed in our DDT-SGS model correspond to Mach numbers in the density range in which DDTs are expected of ∼ 0 . 3 with respect to the fuel material ( ∼ 0 . 1 with respect to the ashes). We showed that the DDT-SGS model is largely resolution independent. Assuming that the DDT region has a smooth twodimensional geometry we found in a specific deflagration model that the criterion is met, indicating that the necessary constraints for DDTs in SNe Ia were appropriate. Our model includes a global criterion, since the histogram of v ' ( /lscript crit) and A ∗ flame ( t ) do not provide any information of local areas. Therefore, a shortcoming of our model is that we cannot fully ensure that there is indeed a compact region that obeys the necessary constraints for a DDT. For testing our DDT model, we used one specific simulations of the deflagration phase in a thermonuclear explosion of a Chandrasekhar-mass WD. The evolution of the turbulent deflagration depends strongly on the ignition scenario of the flame, which is currently unknown. Certain turbulent deflagrations will meet a given DDT criterion more frequently, which will consequently a ff ect the occurrence of DDTs. Therefore, the ignition scenario of the deflagration is another crucial model parameter for simulations of delayed detonations. An analysis of the importance of the ignition scenario on the DDT SGS-model will be the subject of a future publication. The values of the DDT parameters are not well known and have been kept constant or fixed in a certain range in our DDT model. For this reason, we intend to perform a parameter study by varying all DDT quantities. These future studies will reveal further insights into the relevance and constraints of delayed detonations in Chandrasekhar-mass white dwarfs. Acknowledgements. This work was partially supported by the Deutsche Forschungsgemeinschaft via the Transregional Collaborative Research Center TRR 33 'The Dark Universe', the Excellence Cluster EXC153 'Origin and Structure of the Universe', the Emmy Noether Program (RO 3676 / 1-1) and the graduate school 'Theoretical Astrophysics and Particle Physics' GRK 1147. FKR also acknowledges financial support by the ARCHES prize of the German Ministry of Education and Research (BMBF) and by the Group of Eight / Deutscher Akademischer Austausch Dienst (Go8 / DAAD) GermanAustralian exchange program.", "pages": [ 8, 9, 10 ] }, { "title": "References", "content": "Arnett, D. & Livne, E. 1994, ApJ, 427, 330 Arnett, W. D., Truran, J. W., & Woosley, S. E. 1971, ApJ, 165, 87 Hillebrandt, W. & Niemeyer, J. C. 2000, ARA&A, 38, 191 Khokhlov, A. M. 1991b, A&A, 246, 383 Kushnir, D., Livne, E., & Waxman, E. 2012, ApJ, 752, 89", "pages": [ 10 ] } ]

[ { "title": "ABSTRACT", "content": "Context. The weak gravitational lensing distortion of distant galaxy images (defined as sources) probes the projected large-scale matter distribution in the Universe. The availability of redshift information in galaxy surveys also allows us to recover the radial matter distribution to a certain degree. Aims. To improve quality in the mass mapping, we combine the lensing information with the spatial clustering of a population of galaxies that trace the matter density with a known galaxy bias (defined as tracers). Methods. We construct a minimum-variance estimator for the 3D matter density that incorporates the angular distribution of galaxy tracers, which are coarsely binned in redshift. Merely all the second-order bias of the tracers has to be known, which can in principle be self-consistently constrained in the data by lensing techniques. This synergy introduces a new noise component because of the stochasticity in the matter-tracer density relation. We give a description of the stochasticity noise in the Gaussian regime, and we investigate the estimator characteristics analytically. We apply the estimator to a mock survey based on the Millennium Simulation. Results. The estimator linearly mixes the individual lensing mass and tracer number density maps into a combined smoothed mass map. The weighting in the mix depends on the S / N of the individual maps and the correlation, R , between the matter and galaxy density. The weight of the tracers can be reduced by hand. For moderate mixing, the S / N in the mass map improves by a factor ∼ 2 -3 for R /greaterorsimilar 0 . 4. Importantly, the systematic o ff set between a true and apparent mass peak distance (defined as z -shift bias) in a lensing-only map is eliminated, even for weak correlations of R ∼ 0 . 4. Conclusions. If the second-order bias of tracer galaxies can be determined, the synergy technique potentially provides an option to improve redshift accuracy and completeness of the lensing 3D mass map. Herein, the aim is to visualise the spatial distribution of cluster-sized mass peaks. Our noise description of the estimator is accurate in the linear, Gaussian regime. However, its performance on sub-degree scales depends on the details in the galaxy bias mechanism and, hence, on the choice of the tracer population. Nonetheless, we expect that the mapping technique yields qualitatively reasonable results even for arcmin smoothing scales, as observed when this technique is applied to the mock survey with two di ff erent tracer populations. Key words. Gravitational lensing:weak - (Cosmology:) large-scale structure - (Cosmology:) dark matter - Methods: data analysis", "pages": [ 1 ] }, { "title": "Improving three-dimensional mass mapping with weak gravitational lensing using galaxy clustering", "content": "Patrick Simon Argelander-Institut f¨ur Astronomie, Universit¨at Bonn,Auf dem H¨ugel 71, 53121 Bonn, Germany e-mail: [email protected] Received June 15, 2021", "pages": [ 1 ] }, { "title": "1. Introduction", "content": "The weak gravitational lensing e ff ect is a well-established tool to infer properties of the projected large-scale matter distribution (e.g. Munshi et al. 2008; Schneider 2006a,b). These therein exploited coherent shear distortions of distant galaxy images (defined as sources) result from the continuous deflection of light bundles by the intervening fluctuations in the large-scale gravitational field, which are most prominent and detectable around galaxy clusters. The lensing distortions probe the total matter content in the Universe, which makes them an excellent tool for studying the dark matter component, an essential ingredient of the standard cosmological model of cold dark matter with a cosmological constant ( Λ CDM, e.g., Dodelson 2003). The shear distortion pattern can be translated into a map of projected matter fluctuations. Early non-parametric mapping algorithms, which were refined later to obtain optimised methods for finite fields, achieved this only on the basis of a catalogue of source angular positions and ellipticities (e.g. Kaiser & Squires 1993; Seitz & Schneider 2001). With the advent of distance indicators of galaxies in wide field galaxy surveys, the purely geometric relation between shear magnitude and source (and lens) distance was incorporated into a new three-dimensional (3D) lensing algorithm to also recover information on the radial dis- tribution of matter (Hu & Keeton 2002; Bacon & Taylor 2003; Simon et al. 2009; VanderPlas et al. 2011; Leonard et al. 2012). The best studied methodologies so far utilise linear inversion techniques, such as Wiener filtering or a radial matter-density eigenmode decomposition with a suppression of low signal-tonoise (S / N) modes. Owing to the relatively sparse and noisy sampling of the survey area with background sources, however, the resulting maps are usually very noisy, and significant detections are basically restricted to mass peaks of a galaxy cluster scale that has only moderate redshift accuracy. Moreover, the linear inversion utilises a radial smoothing with a broad smoothing kernel that (a) smears out localised peaks in a radial direction and (b) biases the peak distances (known as z -shift bias; Simon et al. 2009), which potentially renders the resulting maps hard to interpret. To attain more realistic 3D maps, the radial elongation of peaks inside the map can be mended by regularising the inversion (Leonard et al. 2012), or by finding the maximum likelihood positions of one or a few individual mass peaks along the line-of-sight (l.o.s.) given the radial smoothing kernel and radial density profile in the map (Simon et al. 2012). However, this does not alleviate the principle problem of noisy maps and inaccurate peak distances. It merely provides more realistic estimators for the 3D mass map. Moreover, the noise properties of the maps are likely to be complex in regularised, non-linear methods. On the other hand, galaxy positions themselves are also tracers of the 3D matter density field and could therefore be employed to add extra information to the matter density maps that are obtained from 3D lensing. However, there are two complications here: (i) galaxies trace the matter density field only up to a systematic mismatch, which is generally dubbed galaxy bias , and (ii) a sampling by galaxy positions is a ff ected by shot-noise (e.g. Dekel & Lahav 1999; Mart'ınez & Saar 2002). The strategy of this paper is to refine the minimum-variance estimator in Simon et al. (2009) (STH09 hereafter) for the 3D matter density by adding the galaxy clustering information to the map making process. Since the minimum-variance estimators (Zaroubi et al. 1995) require second-order statistics of the input data to be specified, only the second-order bias parameters of the galaxy tracers have to be known (Gaussian bias or linear stochastic bias; Dekel & Lahav 1999). The galaxy bias as a function of scale and redshift could in principle be acquired in a self-consistent approach from the data by using lensing techniques (Schneider 1998; van Waerbeke 1998; Pen et al. 2003; Fan 2003; Jullo et al. 2012; Simon 2012), or with lesser certainty from simulations (Yoshikawa et al. 2001; Somerville et al. 2001; Weinberg et al. 2004). We therefore assume that it is basically known. The galaxy noise covariance within the minimum-variance estimator takes care of the galaxy sampling shot-noise. The outline of this paper is as follows. The Sections 2 and 3 present the details of the algorithm and a formalism to quantify its noise properties. We discuss the algorithm in the context of an idealised survey and then apply it to simulated data. In Section 4, we give details of the fiducial survey and the mock data. The results on the expected performance of the algorithm are presented in Section 5 and discussed in the final Section 6.", "pages": [ 1, 2 ] }, { "title": "2. Independent reconstructions", "content": "We first consider the reconstruction of the matter density field and galaxy-number density field separately. The next section combines both into one 3D mass map.", "pages": [ 2 ] }, { "title": "2.1. Matterdensityonlensplanes", "content": "We briefly summarise here the formalism already presented in STH09. We adopt the exact notation that is employed therein. For more details, we refer the reader to this paper. We split the source catalogue into i = 1 . . . N z sub-samples where a redshift probability distribution (p.d.f.) is known. The complex ellipticities (Bartelmann & Schneider 2001) of the sources belonging to the i th sub-sample are binned on a 2D grid that covers the field-of-view of the survey area. This ellipticity grid is denoted by the vector /epsilon1 ( i ) , whose elements are the sorted pixel values of the grid. Every sub-sample uses the same grid geometry. The paper assumes that the weak lensing approximation is accurate enough for the lensing catalogue on the whole. That is, for the given source redshift and in the l.o.s. direction θ i , the complex ellipticity, /epsilon1 s , is an unbiased estimator of the shear distortion, γ = γ 1 + i γ 2, where /epsilon1 i denotes the intrinsic (unlensed) complex ellipticity of a source image. Moreover, we assume a flat sky with a Cartesian coordinate frame. We slice the light-cone volume, where the matter distribution is reconstructed, into N lp slices. Within the slices we approximate the matter density contrast as constant along the lineof-sight. Every grid pixel defines a solid angle associated with a l.o.s. direction θ . Thus, the fluctuations of the matter density field inside a slice are fully described by the angular distribution of mean density contrasts on a plane (lens plane) and the width of the slice. The matter density contrast on a lens plane, δ ( i ) m , is binned with the same angular grid as the source ellipticities. We represent the grids, /epsilon1 ( i ) and δ ( i ) m , as vectors of equally ordered pixel values. We refer to a particular pixel by δ ( i ) m ( θ j ), where θ j is the position of the pixel on the sky. Therefore, our algorithm represents the 3D-matter density contrast as an approximation by a discrete set of lens planes, which numerically limits the radial resolution, and a discrete set of pixels on the sky, limiting the angular resolution. The complete sets of ellipticity planes and lens planes are combined inside vectors of grids: respectively. The brackets, which group together the vector arguments, should be understood as big vectors that are obtained by piling up all embraced vectors on top of each other. In the weak lensing regime, the (pixelised) lensing convergence κ ( i ) ( θ j ) in the lowest-order Born approximation is the weighed projection of the density contrast on the lens planes: where the coe ffi cients Qij express the response of the i th convergence plane κ ( i ) to the density contrast in the j th lens plane. Namely, where The function p ( i ) χ ( χ ) denotes the p.d.f. of sources in comoving distance χ of the i th source sub-sample, and [ χ j , χ j + 1[ sets the comoving radial boundaries of the j th matter slice. We use D H : = c / H 0 for the Hubble radius and f K( χ ) for the (comoving) angular diameter distance. The projection from a grid vector in δ m-space to a grid vector in κ -space is hence denoted by the operator Q that is acting on δ m. The next step connects the convergence planes κ to the shear planes by a convolution of the lensing convergence on the grid which introduces the operator P γκ to map κ ( i ) to the corresponding shear plane γ ( i ) (Hu & Keeton 2002). In this sense, P γκ performs a linear transformation from κ - to γ -space. Using this compact notation, we express the linear relation between the matter density (contrast) on the lens planes and the observed, binned ellipticity planes as: Here, an additional vector n γ denotes the binned intrinsic ellipticties of the sources of all source sub-samples. In the language of lensing, we consider this the noise term that dilutes the shear signal P γκ Q δ m. For the scope of this paper, possible correlations between shear and intrinsic shapes are ignored (Hirata & Seljak 2004). According to STH09, minimum-variance estimator of δ m in Eq. (8) is then As the only input, the minimum-variance filter requires the signal covariance S δ = 〈 δ m δ t m 〉 , which specifies the presumed two-point correlation between pixel values of δ ( i ) m ( θ ) on the lens plane(s) and the noise covariance N γ = 〈 n γ n t γ 〉 , which quantifies the shear pixel noise variance and the correlation of noise between di ff erent pixels. Pixels that contain no sources have infinite noise. For the signal covariance, correlations between pixels that belong to di ff erent lens planes are set to zero. We note here that the signal covariance does not need to be the true signal covariance in the data, although the reconstruction may be sub-optimal as to map noise when it is not. The signal covariance determines the degree of smoothing in the 3D map. The smoothing is uniquely defined by the linear transformation and can be utilised for a comparison of the map δ m , est to a theoretical matter distribution δ m , th by B δ δ m , th (Simon et al. 2012). The radial smoothing is characterised by a radial point-spread function (p.s.f.) of the filter (STH09). After smoothing with the radial p.s.f., a peak in the true matter distribution δ m , th does not necessarily peak at the same distance on average as in the smoothed map, which gives rise to the so-called redshift bias or z -bias. Inside the filter, the constant α ∈ [0 , 1] tunes the level of smoothing by rescaling the noise covariance. From a practical point of view, the Wiener filter consists of a series of linear operators that is applied step-by-step from the right to the left on the grids (Appendix B of STH09). Within this process, the signal covariance, S δ , is a convolution or, equivalently, a multiplication in Fourier space of Fourier modes, ˜ f ( /lscript ), of the i th lens plane with the angular signal power spectrum, P ( i ) δ ( /lscript ), which is implicitly defined by We approximate the power spectrum by using Limber's equation in Fourier space: where ∆ χ i : = χ i + 1 -χ i , ˜ F ( /lscript ) is the Fourier transform of the pixel window function, P 3d( k , χ ) is the 3D matter-density power spectrum at radial distance χ for wave-number k , and δ D( x ) is Dirac's delta function (Kaiser 1992). We denote the Fourier transforms of flat fields, f ( θ ), on the sky by ˜ f ( /lscript ), which is defined by", "pages": [ 2, 3 ] }, { "title": "2.2. Galaxynumbersdensitiesonlensplanes", "content": "To improve the information in the 3D matter map and to possibly alleviate the z -shift bias, we add the information gained from galaxy positions, which also probe the matter distribution (defined as tracers). In this section, however, we first visit the problem of mapping the spatial galaxy number densities. For this purpose, we estimate the number density of galaxies projected onto the previously defined lens planes. Hence, we slice the full true 3Dgalaxy distribution into N lp distance slices with distance limits [ χ i , χ i + 1[. The galaxies are counted within each slice and angular grid pixel of the solid angle A ω . Thereby, we receive the galaxy number density n ( i ) g ( θ j ) = N ( i ) ( θ j ) / A ω in the l.o.s. direction θ j of the i th slice, where N ( i ) ( θ j ) is the number of counted galaxies. We compile the galaxy-number density values inside a grid vector n ( i ) g , and we then arrange all grids inside a vector of grids: This number density distribution of galaxies is what the following scheme seeks to recover from a galaxy sample with inaccurate distance information. Towards this goal, we split the observed galaxy sample utilising their redshift estimators, z est ∈ [ z ( χ i ) , z ( χ i + 1)[, into N lp sub-samples with known radial p.d.f. p ( i ) f ( χ ); z ( χ ) denotes the redshift corresponding to χ . By projecting the i th sample onto a 2D grid on the sky, one obtains the observed number density distribution where f mask ∈ { 0 , 1 } flags mask pixels ( = 0 for mask), and is the probability that a galaxy inside η ( i ) g belongs to the slice j . Owing to the redshift errors and masking, the observed distribution on the lens planes, η ( i ) g ( θ k ), does not exactly match the true distribution n ( i ) g ( θ k ). Therefore, 0 ≤ Gij ( θ k ) ≤ 1 denotes the expected fraction of galaxies on the j th lens plane that is mapped onto the grid η ( i ) g . Because of masking, the total number of galaxies is not necessarily conserved; that is ∑ N lp i = 1 Gij ( θ k ) /nequal 1. By a proper arrangement of the elements Gij ( θ k ) inside a matrix G , the e ff ect of Gij ( θ k ) on the entire 3D grid n g can be written as where We presume that galaxies sample an underlying smooth galaxy number density by a discrete Poisson process (e.g., Mart'ınez & Saar 2002). Therefore, the observable galaxy counts sample the underlying galaxy number density n g up to shot-noise, which is here formally expressed by the noise component φ g. By analogy with the matter density δ m, we can find an minimum-variance filter to estimate the true distribution of galaxies on the lens planes; namely As before, S g = 〈 n g n t g 〉 is the signal covariance, which is the angular clustering two-point correlation function of the galaxies on the lens planes, and N g = 〈 φ g φ t g 〉 denotes the shot-noise covariance. The degree of smoothing by the Wiener filter is tunable by using β ∈ [0 , 1], which does not need to equal the parameter α in Eq. (9). For the Poisson shot-noise covariance, we adopt a diagonal noise covariance, [ N g] i j = 0 for i /nequal j , with [ N g] ii = ¯ η ( k ) g ( θ l ) for unmasked grid pixels θ l , and infinite noise otherwise. The Wiener filter in the given form requires the inverse noise covariance, such that elements with infinite noise on the diagonal are zero. By ¯ η ( k ) g ( θ l ), we denote the estimated mean number density of galaxies in pixel θ l of the k th sub-sample (see next section). As for the matter density Wiener filter, a practical implementation of the Wiener filter in Eq. (19) consists of a series of linear operations applied to η g. The e ff ect of S g is to multiply every angular mode ˜ f ( /lscript ) of the i th lens plane with the prior galaxy power spectrum P ( i ) g ( /lscript ), which we define relative to the matter power spectrum using the galaxy bias factor b ( i ) ( /lscript ) ≥ 0 (e.g., Tegmark &Peebles 1998): where ¯ n ( i ) g denotes the true mean number density of galaxies on the i th lens plane. For this definition of the bias factor, the shotnoise contribution to the galaxy power spectrum is excluded as it is already accounted for in φ g. The angular bias factor b ( i ) ( /lscript ) is related to the 3D bias factor b ( k , χ ), where k is the comoving 3D wave-number, by a projection that is approximated by Limber's equation: P ( i ) δ ( /lscript ) is given by Eq. (12). For this approximation, we assume that the number density of galaxies stays constant as function of χ inside the slice.", "pages": [ 3, 4 ] }, { "title": "2.3. Truemeangalaxynumbers", "content": "The true galaxy number densities ¯ n ( i ) g in Eq. (20) have to be derived from the data itself. For an estimator of ¯ n ( i ) g , we go back to Eq. (17), which relates the observed number of galaxies, η g, to the true number on the lens planes, n g. For an ensemble average of this relation, we expect wherein all elements ¯ n ( i ) g ( θ j ) equal the same number ¯ n ( i ) g owing to the statistical homogeneity of the galaxy-number density fields, hence Summing over all pixels with N p in total of the i th tracer sample yields where averages G i j over the area of the grid. Inverting the former equation, gives For an unbiased estimator of X ( i ) g on the right hand side, we insert the observed galaxy number densities, which is ¯ η ( i ) g ( θ k ) = η ( i ) g ( θ k ). The value of ¯ η ( i ) g ( θ k ), which is utilised for the noise covariance N g in the foregoing section, is computed from Eq. (23) and the estimated ¯ n ( i ) g . In the simple case of negligible redshift errors, we find G i j ( θ k ) = δ K i j f mask( θ k ), where δ K i j denotes the Kronecker symbol. In this case, we consequently find for the number N of unmasked pixels. Moreover, we find ¯ n ( i ) g = N p N ( i ) g / ( N Ω ) for a number N ( i ) g of galaxies within the i th sub-sample and a survey area Ω . Thus, the galaxy number density N ( i ) g / Ω is scaled up by N p / N to account for the mask. However, the estimator in Eq. (26) has one caveat, since ¯ η ( i ) g is basically a convolution of ¯ n ( i ) g with the redshift error of galaxies. A deconvolution through ¯ G -1 possibly results in oscillating and negative values for ¯ n ( i ) g . We therefore regularise Eq. (26) by a constrained solution of ¯ n ( i ) g that maximises the likelihood: under the condition that ¯ n ( i ) g ≥ 0 for all i . We determine this solution numerically. The additional covariance N η can be used to give di ff erent weights to the observed ¯ η ( i ) g values, such as by weighing the number of galaxies in each galaxy sample in order to account for the galaxy shot-noise. For equal weights, we simply set N η = 1 .", "pages": [ 4 ] }, { "title": "3. Combined reconstruction", "content": "In this section, we combine the information on the 3D matter density in the lensing data and the galaxy distribution.", "pages": [ 4 ] }, { "title": "3.1. Minimum-varianceestimator", "content": "Up to now, we have considered the galaxy number density and matter density fields separately. However, η g contains information about δ m and vice versa, as galaxies trace the matter distribution to a certain degree. On a statistical level, this relation is reflected by a non-vanishing cross-correlation, for pairs of pixels on the same lens plane, which has not entered our formalism thus far. Slices are thought to be wide enough, such that correlations between pixels belonging to di ff erent lens plane are negligible. We combine the δ m- and n g-grids inside one new vector, Eqs. (8) and (17) relate s to the observed shear and the tracer number density grids, according to where the combined noise vector is In this compact notation, the action of a matrix on a product vector v = [ v 1 , v 2] is defined as In this sense, the projection matrix R is Following the usual assumptions of a minimum-variance filter, the optimal filter for estimating s from d in this combined problem is which uses the short-hand notations, The galaxy shot-noise φ g and the intrinsic ellipticities of the sources, which are comprised in n γ , are assumed to be uncorrelated. By choosing di ff erent tuning parameters α /nequal β , the impact of the Wiener smoothing can be adjusted independently for the matter and galaxy map. The novelty of the combined reconstruction is that tracer number and matter density maps exchange information, if the cross-correlation matrix S δ g is non-vanishing. In a practical implementation of the filter (37), we apply step by step linear operations to the grids stored inside d as before. As with the previous operators S δ and S g, the application of S δ g amounts to a multiplication of angular grid modes with the cross-correlation power spectrum, P ( i ) δ g ( /lscript ), determined by (See the next section for details on the implementation.) We define P ( i ) δ g ( /lscript ) with respect to the matter power spectrum P ( i ) δ ( /lscript ) by employing the galaxy-matter cross-correlation factor r ( i ) ( /lscript ) (Tegmark & Peebles 1998). The angular function r ( i ) ( /lscript ) is approximately related to the 3D correlation factor r ( k , χ ) according to where k /lscript : = /lscript/ f K( χ ). To understand the mode of operation of the minimumvariance filter in Eq. (37), it is instructive to recast it into the mathematically equivalent form: where N -1 δ n g : = R † N -1 αβ R . Step-1 involves no Wiener smoothing to construct the maps; no matrix S is involved in this step. As this is usually too noisy, we apply an additional smoothing to these maps by virtue of the Wiener filter in Step-2. This filter linearly combines and averages pixel values in the maps based upon the expected S / N in the unbiased maps. It is Step-2, the analogue of the matrix B δ in Sect. 2.1, that introduces biases into the maps, especially through a radial smoothing. Moreover, only Step-2 formally mixes pixels from the mass map and the tracer number density map by means of the o ff -diagonal matrix S δ g. Therefore, Step-1 makes independent mass and tracer maps that are only later combined in Step-2, according to our prior knowledge of their correlation. Setting α = β = 0 results in a unity matrix for Step-2 or no smoothing. Analogous to a lensing-only reconstruction, the Wiener filter thus applies a radial and transverse smoothing to the map to increase the signal-to-noise ratio. The smoothing makes the maps biased estimators of the matter and galaxy-number density fields. The smoothing is, however, uniquely defined by which and can be applied to theoretical maps of the matter and galaxy number density for a quantitative comparison to the data.", "pages": [ 4, 5 ] }, { "title": "3.2. Fourierspacerepresentation", "content": "For shear and galaxy number noise homogeneous over infinite grids with no gaps, the estimator in Eq. (37) takes a simple form in Fourier space. Under these idealistic conditions, the angular modes of all lens planes combine to which are only linear functions of the η g- and γ -modes of the same /lscript ; there is no mixing between modes of di ff erent /lscript . Therefore, a reconstruction is then done most easily in Fourier space by where ˜ d ( /lscript ) = [ ˜ γ ( /lscript ) , ˜ η g ( /lscript ) ] are the observable input grids. The tuned covariance matrix of the (homogeneous) noise is where ¯ n ( i ) s is the mean source number density of the i th source sample (out of in total Nz ); σ 2 /epsilon1 = 〈 /epsilon1 i [ /epsilon1 i ] ∗ 〉 is their intrinsic shape noise variance, and ¯ η ( i ) g is the Poisson shot-noise power (white noise). Possible noise contributions owing to intrinsic alignments of sources are ignored here, hence ˜ N αβ has no o ff -diagonal elements. Furthermore, one has where D ( /lscript ) = /lscript / /lscript ∗ (Kaiser & Squires 1993). For /lscript = 0, we set D ( /lscript ) = 0. Here, G does not depend on θ k . The signal covariance is Because of the diagonal structure of the last three matrices, the matrix ˜ S /lscript acting on a vector ˜ v ( /lscript ) actually only mixes the matter and tracer density modes from the same lens plane and of the same wave vector /lscript . Thus, rearranging the modes inside ˜ v ( /lscript ) and pairing together matter and tracer density modes of the same lens plane render ˜ S /lscript a diagonal block matrix, such that with 2 × 2-blocks on the diagonal. This structure is useful when implementing the action of S in (37) in practise. Clearly, modes will not a ff ect each other when r ( i ) ( /lscript ) = 0 with no improvement by the synergy of lensing and galaxy tracers.", "pages": [ 5, 6 ] }, { "title": "3.3. Radialpointspreadfunction", "content": "The radial p.s.f. is the average sight-line profile of a single mass peak in the smoothed matter map. Ideally, the p.s.f. spikes at the true mass peak redshift (no z -shift bias). In reality, however, a z -shift bias is one of the main nuisances in Wiener reconstructions with 3D lensing data. We assume a homogeneous survey, where the choice of the l.o.s. direction θ is irrelevant. We hence arbitrarily pick θ = 0 as a reference direction and omit the pixel index θ in the following. We consider a singular test peak with a profile of δ m( χ ) = A p δ D( χ -¯ χ i ) in the un-smoothed map; A p is the peak amplitude. It is located at the distance ¯ χ i : = ( χ i + χ i + 1) / 2 of the i th lens plane. For circular pixels with angular radius Θ s, the pixel value of this peak is in the un-smoothed map or in the Fourier space, By Jn ( x ), we denote the spherical Bessel functions of the first kind. Because of the linearity of the reconstruction algorithm, the peak amplitude is unimportant for the shape of the radial p.s.f. We therefore simply set A p = 1. Unlike the discussion in STH09 for calculating the radial p.s.f., we also have to factor in the tracer number density on the i th lens plane here (and same direction θ ). As this is a random variable for r ( /lscript ) /nequal 1, we define the p.s.f. as the radial density profile in the smoothed map given a matter peak ˜ δ ( i ) m ( /lscript ) = ˜ F ( Θ s /lscript ) on the i th lens plane that is marginalised over the tracer density ˜ n ( i ) g ( /lscript ). This is associated with the mass peak. This conditional mean tracer number density is given by is taken over all realisations of the tracer density field and P ( x , y ) denotes the bivariate p.d.f. of the tracer number density x and the matter density y . The expression on the r.h.s. in the Eq. (55) is exact only for Gaussian statistics, which is assumed here as lowest-order approximation (Appendix A). For di ff ering statistics, such as a log-normal tracer density field (Coles & Jones 1991), we have to expect deviations from this expression. Evidently, the conditional average will vanish if the correlation factor is r ( i ) ( /lscript ) = 0. The average tracer number density about a mass peak vanishes in this case. According to this definition, the radial p.s.f. equals the average sight-line density profile (analogous to Eq. 77 of STH09): where the Wiener filter ˜ W /lscript is given in Eq. (60) and the vectors ˜ δ m( /lscript ) and ˜ n g( /lscript ) ( N lp elements) vanish everywhere except in their i th element that equals 1 and ¯ n ( i ) g r ( i ) ( /lscript ) b ( i ) ( /lscript ), respectively. The elements of the vector ¯ δ m( Θ s) encapsulate the radial p.s.f. of the matter map, and the radial p.s.f. of the tracer number density map in the case of ¯ n g( Θ s). The former is the focus in the following.", "pages": [ 6 ] }, { "title": "3.4. Mapsignal-to-noise", "content": "With the estimator (44) at hand, we forecast the S / Nofthe matter and tracer number density modes as a function of angular wavenumber /lscript . To this end, we compare the cosmic average power spectrum, P s( /lscript ) : = ˜ W /lscript ˜ S /lscript ˜ W † /lscript , (58) of the reconstructed matter and galaxy-number density modes on the lens planes to noise in the reconstruction from shape noise and tracer sampling noise, which is In this reconstruction, the Wiener filter ˜ W /lscript uses the true signal power ˜ S /lscript present in the data. As pointed out earlier, this is not a necessity but is required for an optimal minimum-variance filter. For α = β = 0 (neither smoothing nor mixing), the noise covariance is P n( /lscript ) = ˜ X /lscript .", "pages": [ 6 ] }, { "title": "3.5. Galaxy-stochasticitynoise", "content": "The noise covariance P n( /lscript ) contains only a part of the statistical uncertainty in a combined reconstruction; namely, this is the noise originating from the unknown intrinsic source galaxy shapes and galaxy sampling noise. In the presence of stochasticity between matter and tracer density, however, there is a random scatter in the sample-noise-free tracer density for a given matter density field that gives rise to the additional noise covariance P gsn, (called random biasing field in Dekel & Lahav 1999). Contrary to Poisson shot-noise, this galaxy-stochasticity noise (GSN) is also present, if the number of tracer galaxies were infinite. This is a new feature compared to reconstruction techniques relying only on the 3D lensing signal. Possible realisations of a galaxynumber density field for a given matter density field on the lens planes depend on the details of the physics behind the galaxy bias. Consequently, a precise estimate of the GSN level can only be provided if the galaxy bias scheme is known. For a first-order estimate of GSN, we assume Gaussian fluctuations in the galaxy number and matter density on every lens plane. In this Gaussian approximation, the bivariate p.d.f. of modes of the matter density contrast, ˜ δ ( i ) m ( /lscript ) , and a galaxy number density, ˜ n ( i ) g ( /lscript ), are fully determined by the variance P ( i ) δ ( /lscript ) of ˜ δ ( i ) m , the variance [¯ n ( i ) g b ( i ) ( /lscript )] 2 P ( i ) δ ( /lscript ) of ˜ n ( i ) g , and the cross-correlation coe ffi cient r ( i ) ( /lscript ) of both. From this the variance of a galaxy tracer mode about a fixed matter density mode follows: See Appendix A for details. The random biasing field is an independent Gaussian realisation with power spectrum P ( i ) gsn . The essential parameter for this random scatter is r ( /lscript ), which vanishes for | r ( /lscript ) | = 1, but reaches a maximum in amplitude for r ( /lscript ) = 0. On the other hand, a smaller r ( /lscript ) also results in a reduction of the mixing of matter and tracer density modes by the minimumvariance filter. On the extreme end for r ( /lscript ) = 0, the filter does not make use of any tracer information at all for the matter density map. The total noise power per angular mode in the mass map can hence then be approximated by For each lens plane, we translate these /lscript -dependent GSN levels to the noise variance on the map pixel scale by virtue of the integral (STH09) and likewise to compute 〈 [ δ ( i ) m ( Θ s)] 2 〉 for the signal power [ P s( /lscript )] ii in Eq. (58). The resulting ratios of GSN and signal power are for pixels in our fiducial mass map. For this estimate of f ( i ) gsn , we take the cosmic average P ( i ) δ ( /lscript ). This certainly underestimates the matter fluctuations in galaxy cluster regions. On the other hand, the GSN scales, at least for Gaussian random fields, linearly with the amplitude of the actual matter density fluctuations, as in Eq. (62), such that the ratio f ( i ) gsn should be a robust approximation for Gaussian fields with more matter clustering.", "pages": [ 7 ] }, { "title": "3.6. Correctionforgalaxy-stochasticitynoise", "content": "In practise, we estimate the S / N of the synergy reconstructions by dividing pixel values δ m , est( θ ) in the map by the pixel variance in noise realisations of the map. We obtain the noise realisations by randomising the source ellipticities and the tracer positions in accordance with their completeness and redshift errors G . However, the noise realisations do not include the GSN but only contributions of σ shot from galaxy shape- and tracer sampling noise. In this section, we propose a GSN correction factor that is applied to this S / N map. The correction factor is based on the foregoing f ( i ) gsn and the variance σ shot in the noise realisations. For each lens plane of the map the pixel variance σ 2 all has three independent components, where σ s is the variance in the matter density signal, σ gsn is the GSN variance, and σ shot is the source shape- and tracer shotnoise variance. On the right hand side, we have substituted the GSN variance by the signal variance and f gsn. A S / N map that accounts for both σ shot and σ gsn is where δ m , est( θ ) /σ shot on the right hand side is the S / N invoking shot-noise only, as produced by randomising the catalogues. For the correction factor inside the brackets, the signal variance σ s can be estimated by employing Eq. (65) with an appropriate Wiener-filtered signal power spectrum. In addition, the shotnoise variance, σ shot, is determined by Eq. (65) with the Wiener filter noise power spectrum P n( /lscript ) inside the integral. For a signal variance σ s /lessmuch σ shot, the correction factor is roughly unity, which is always the case for a cosmic average matter density power spectrum and our fiducial survey. As we are mainly targeting galaxy cluster regions with lensing cartography, however, a fiducial value of σ s with higher variance than a cosmic average is likely. To obtain a more realistic fiducial value, we construct an alternative signal power spectrum for σ s, assuming (i) Gaussian fluctuations, (ii) randomly scattered haloes with an average number density ¯ n sis and (iii) haloes with an average singular isothermal sphere (SIS) matter density profile (STH09): in Fourier space and SIS velocity σ v. Therefore, the matter power spectrum for the i th lens plane is described by which we insert into Eq. (58) and Eq. (65) to calculate the pixel signal-variance σ s (Appendix B).", "pages": [ 7 ] }, { "title": "3.7. Clustersignal-to-noise", "content": "We now consider the significance with which a single mass peak at a given radial distance can be detected in a synergy reconstruction. For a fiducial mass peak, we adopt a SIS-like matter over-density ˜ δ ( i ) sis that is fully contained inside the i th lens plane, as in Eq. (69). The associated average number density of tracers is on the level of a Gaussian approximation ˜ n ( i ) sis ( /lscript ) = n ( i ) g r ( i ) ( /lscript ) b ( i ) ( /lscript ) ˜ δ ( i ) sis ( /lscript ), which is analogous to the rationale in Sect. 3.3, and vanishes for all other lens planes j /nequal i . When we combine this peaked mass model ˜ δ sis( /lscript ) and the tracer density model, ˜ n sis( /lscript ), in [ ˜ δ sis( /lscript ) , ˜ n sis( /lscript )], we acquire the average map response [ δ sis( Θ s) , n sis( Θ s)] in a smoothed map by Eq. (57), where Θ s is the transverse smoothing kernel size. The vector δ sis( Θ s) exhibits the expected mass map response to the central pixel of a SIS peak in the map. This signal is compared to the expected noise level inside a pixel. Relevant contributions to noise are (i) sample and shot noise, σ ( i ) shot , (ii) the GSN variance σ ( i ) gsn , and (iii) interference σ ( i ) cn by intervening matter density fluctuations on lens planes that do not host the fiducial SIS peak. The sources of noise (i) and (ii) are detailed in the Sect. 3.4 and 3.5. For (ii), we additionally assume that the interfering matter density power on all lens planes j /nequal i , which do not host the SIS, is given by the cosmic average P ( j ) δ ( /lscript ) in Eq. (62), whereas we have Eq. (70) as a GSN model for the i th plane . We determine the pixel variance σ ( i ) cn in (iii) by the signal covariance P s( /lscript ), as noted in Eq. (58), whose diagonals [ P s( /lscript )] ii are inserted into Eq. (65). Finally, the radial S / N profile of the SIS peak in the map is δ ( i ) sis ( Θ s) / √ [ σ ( i ) s ] 2 + [ σ ( i ) gsn ] 2 + [ σ ( i ) cn ] 2 . As a theoretical S / N of the detection, we pick the lens plane index i at maximum S / N, which may not correspond to the true distance of the SIS peak due to the z -shift bias.", "pages": [ 8 ] }, { "title": "4. Survey parameters", "content": "We consider an idealised survey with homogeneous noise and a G that is independent of the pixel position to discuss the impact of a joint reconstruction in the following sections. This section defines the fiducial cosmology and binning details of the idealised survey. Moreover, we generate mock data to which the reconstruction algorithm is applied. The mocks utilise a N-body simulation of the large-scale dark matter density field populated with semi-analytical galaxies.", "pages": [ 8 ] }, { "title": "4.1. Fiducialparametersoftheidealisedsurvey", "content": "As fiducial cosmology, we use a standard flat Λ CDM model with the matter-density parameter Ω m = 0 . 27, where baryons are Ω b = 0 . 046 and a shape parameter of Γ = 0 . 19. The normalisation of the matter fluctuations within a sphere of radius 8 h -1 Mpc at a redshift of zero is σ 8 = 0 . 8. For the spectral index of the primordial matter power spectrum, we use n s = 0 . 96. With these parameters, we construct a fiducial 3D matter power spectrum according to Smith et al. (2003) which is then used to model the signal covariance S /lscript . ˜ For the fiducial survey, we split the source galaxy catalogue into Nz = 20 equally sized redshift slices of width ∆ z = 0 . 1, which span the redshift range of z = 0 . . . 2. For the sources, we neglect the e ff ect of redshift errors greater than the width of the redshift slices, such that the true p.d.f. p ( i ) z ( z ) of sources of the i th slice is well-described within z ∈ [ zi , zi + 1] by the p.d.f. of redshift estimates of the full sample, where z 0 = 0 . 57, λ = 1 . 5, and zi = ( i -1) ∆ z . We represent the reconstruction volume by N lp = 10 lens planes between z = 0 and z = 2 that are centred within slices of moderate width ∆ z lp = 0 . 2. The total number density of sources on the sky is ¯ n = 30 arcmin -2 with an intrinsic shape noise of σ/epsilon1 = 0 . 3. To support the matter density reconstruction, we include fiducial galaxy tracers with known galaxy bias. For simplicity, their p.d.f. of redshift estimates is identical to pz ( z ). Contrary to the sources, however, we now also emulate the e ff ect of redshift errors by adopting a root-mean-square (r.m.s.) accuracy of σ z ( z ) = 0 . 04(1 + z ) (Gaussian errors), which is built into G in Eq. (15). The slicing scheme for the tracers is equivalent to the scheme of the sources. From this, we compute the average number density of tracers ¯ n ( i ) g and ¯ η ( i ) g from Eq. (23) for each redshift slice, and the observed redshift distributions p ( i ) f ( z ) by piecewise convolving the p.d.f. pz ( z ) with a Gaussian kernel of the r.m.s. σ z ( z ). For low redshifts, we have ¯ η ( i ) g ≈ ¯ n ( i ) g , but we find di ff erences at higher redshifts where ∆ z lp ≈ σ z . For the fiducial survey, we reduce the total number of tracers to 10 percent of the sources, ¯ n g = 3 arcmin -2 , since a reconstruction realistically requires a specifically selected tracer population for an accurately known bias. Here, the tracers are clustered as matter with b ( /lscript ) = 1 for all redshifts, but, more relevantly, we assume a slight stochasticity on all scales, namely, r ( /lscript ) = 0 . 8. A high correlation, r /greaterorsimilar 0 . 5, for various galaxy populations is expected from theoretical models (e.g., Guzik & Seljak 2001) and observed for some cases (Hoekstra et al. 2002; Simon et al. 2007; Jullo et al. 2012).", "pages": [ 8 ] }, { "title": "4.2. N-bodymockdata", "content": "For a realistic application of the methodology we employ the Millennium Simulation data set, a state-of-the-art N-body cold dark matter simulation with fiducial parameters of Ω m = 0 . 25, ΩΛ = 0 . 75, Γ = 0 . 21, n s = 1, and σ 8 = 0 . 9 (Springel 2005; Springel et al. 2005). In the simulation, haloes of dark matter were populated with galaxies according to a semi-analytic recipe, as described in Guo et al. (2011) 1 . We select galaxies with SDSS magnitudes of m r < 25 as a set of observable galaxies with known redshifts; Fig. 1 displays the redshift distributions of all magnitude limited galaxy samples. For the simulated survey, we use galaxies from a 1 × 1 deg 2 field and galaxies down to a redshift of z = 2, yielding an average density of ∼ 25 sources per square arcmin. The mean redshift of the sources is ¯ z = 1 . 0. Each source galaxy is equipped with a shear signal corresponding to its angular position and redshift. The shear signal is estimated by ray-tracing through a series of simulation snapshots in the direction of a source (Hilbert et al. 2009). For the intrinsic shape noise we adopt a variance of the ellipticity of σ/epsilon1 = 0 . 3. We further subdivide the total galaxy sample into red ( m u -m r > 2 . 2) and blue galaxies ( m u -m r ≤ 2 . 2) to be used as galaxy tracers for the reconstruction technique of the mass map. We use only tracers below or equal z = 1 to aid the reconstruction, which provides a density of ∼ 10 blue and ∼ 5 red tracers per square arcmin. For the mapping, all galaxy samples are split into redshift slices of width ∆ z = 0 . 1 within the regime 0 ≤ z < 1, and a width of ∆ z = 0 . 2 within 1 ≤ z < 2 for the sources. Similar to the idealised fiducial survey we add Gaussian errors to the tracer redshifts with σ z ( z ) = 0 . 04(1 + z ). The mapping methodology requires the specification of second-order galaxy bias parameters { b ( /lscript ) , r ( /lscript ) } of the tracer samples as a function of the angular scale /lscript and redshift. We acquire estimators of the angular galaxy bias parameters by applying the methodology of Schneider (1998) and van Waerbeke (1998) to our simulated galaxy catalogues separately for each tracer redshift slice. We average the results thereof over all simulated 128 one-square-degree fields. Herein, we set the intrinsic shape noise to zero, as we do not attempt to account for uncertainties in bias parameters here. This lensing technique has already successfully been applied to real lensing data, as seen in Simon et al. (2007). We refer the reader to the latter article for the method details, which are irrelevant here. Figure 2 summarises the galaxy bias results of our tracer samples, including errorbars due to cosmic variance and sampling variance. In the following, we take the mean of all fields. To determine these measurements, we employed, as in Simon et al. (2007), a polynomial filter for the aperture statistics. These statistics probe the angular second-order galaxy bias averaged over a /lscript -band centred on /lscript cen ≈ 4 . 25 /θ ap, where θ ap is the aperture radius in radians. The top x -axes values in the figure denote the values of θ ap that correspond to /lscript cen (bottom x -axes). The red tracers are more strongly clustered than matter, where b ( /lscript ) > 1, and highly correlated with the matter density field of r ( /lscript ) ≈ 1 on scales larger than a few arcmin. Blue tracers, on the other hand, are less clustered and less well correlated in both cases. The correlation factor r ( /lscript ) can exceed values of | r ( /lscript ) | = 1, because it is defined here and in the aforementioned references in terms of the tracer power spectrum P ( i ) g ( /lscript ) from which the Poisson shot-noise 1 / ¯ n ( i ) g has been subtracted. In the framework of a halo model and on scales dominated by haloes that are populated on average by 〈 N 〉 < 1 galaxies, the shot-noise subtraction may lead naturally to r ( /lscript ) > 1, because galaxies can trace the matter distribution inside haloes by a sub-Poisson sampling process with a variance 〈 N ( N -1) 〉 1 / 2 < 〈 N 〉 (Guzik & Seljak 2001; Seljak 2000). The presence of central galaxies has a similar impact. We clearly observe this e ff ect here for small angular scales in the simulation. The Wiener filters in Eq. (37) or Eq. (44) diverge for | r ( /lscript ) | > 1, because the signal matrix S becomes singular. This indicates that our minimum-variance Ansatz, presuming sampling by a Poisson process, breaks down where the sub-Poisson e ff ects become significant. To avoid this problem specific to small angular scales, we use more smoothing of Θ s = 2 arcmin and clip correlation factors at r ( /lscript ) = 0 . 9. The latter a ff ects the filter artificially by reducing the mixing for clipped modes and adding less information from the tracer clustering to the mass map. Note that we can always reduce the mixing inside the Wiener filter by adopting a lower correlation factor than in the data.", "pages": [ 8, 9 ] }, { "title": "5. Results", "content": "In this section, we present our results for the S / N and radial p.s.f. in the idealised survey, and demonstrate the methodology for mock data based on a N-body simulation as blueprint for a realistic survey. For the idealised survey, we set α = 0 . 01 to be consistent with Fig. 11 of STH09 for a lensing-only reconstruction with a transverse filter. Generally, the parameter α must not be too close to unity, as this results in too much radial smoothing, which moves basically all mass peaks to the middle of the reconstruction volume (no radial information). Adjusting the tuning parameters below unity means that we scale the noise covariance towards less noise in the Wiener filter. Note that this does not mean that we obtain less noise in the reconstruction. In contrast, the Wiener filter applies less smoothing, which yields more noise in the map, but less bias. For the synergy reconstruction, we adopt β = 0 . 1. A parameter β greater than α is a means to down-weigh the impact of the tracers in the joint reconstruction, which is desirable if the details of the galaxy bias are not accurately known.", "pages": [ 10 ] }, { "title": "5.1. Radialpointspreadfunction", "content": "The resulting p.s.f. of the idealised fiducial survey and a pixel size of Θ s = 1 arcmin is depicted in Fig. 3 for the cases r ( /lscript ) = 0 , 0 . 8. Owing to the Wiener smoothing, the mass peaks are generally radially smeared, and their amplitude is suppressed, especially for very small and high redshifts. Compared to the lensingonly technique ( r ( /lscript ) = 0), however, adding tracers with r /nequal 0 to the map-making process clearly improves the p.s.f.: The radial profiles are narrowed and more pronounced; the amplitudes are less suppressed. The peak maximum of the p.s.f. (apparent redshift) for given mass peak redshift (true redshift) determines the z -bias. The bias as a function of tracer correlation coe ffi cient r ( /lscript ) is explored by Fig. 4. We essentially find no z -shift bias for r ( /lscript ) ≥ 0 . 4.", "pages": [ 10 ] }, { "title": "5.2. Signal-to-noiseofmap", "content": "For the idealised survey, Fig. 5 depicts the S / N of the lens plane density modes as a function of angular scale and lens plane redshift. In the left panel, we have the matter density modes; the right panel shows the tracer number density modes. Di ff erent line styles correspond to di ff erent lens planes with the thin lines to reconstructions with mode mixing switched o ff , or r ( /lscript ) = 0, and the thick lines to the joint reconstruction. In addition, the black thin lines in the left panel depict the S / N in a map with no radial smoothing ( α = 0) and no mixing. Clearly, a lensingonly map absolutely requires some radial smoothing, which is seen here by comparing the low S / N of the thin black lines to the boosted S / N in the red lines. The impact of a moderate mixing on the S / N of the tracer number density maps (right panel) is small, which is most prominently on the small angular scales. This changes slightly if we choose an even larger tuning parameter β (that is not shown): A larger β scales up the shot-noise of the tracers inside the Wiener filter, which attributes even more weight to the lensing data in the joint reconstruction. As the S / N of the tracers in the data is actually higher than that of the shear, this will result in a decreased S / Nfor the galaxy-number density maps in comparison to a reconstruction with no mixing; the joint reconstruction is not optimal as to the map noise.", "pages": [ 10, 11 ] }, { "title": "5.3. Galaxy-stochasticitynoise", "content": "Figure 6 shows the estimated ratios f gsn of the pixel GSNvariance and pixel signal-variance for lens planes of increasing redshift. The map smoothing scale is Θ s = 1 arcmin. We find the GSN on a pixel scale to be most prominent for r ( /lscript ) ∼ 0 . 8, which declines for correlations greater or weaker than that; between r ( /lscript ) = 0 . 6 -0 . 8, there is only little change, and in the absence of stochasticity, where r ( /lscript ) = 1, f gsn vanishes. The dependence on lens plane redshift is marginal; most of the change occurs below z /lessorsimilar 0 . 4. The GSN increases with the bias factor b ( /lscript ) of the tracers. Overall, typical figures for f ( i ) gsn are below 30%, but can be above this level for strongly clustered tracers.", "pages": [ 11 ] }, { "title": "5.4. Clustersignal-to-noise", "content": "In Fig. 7 we plot the S / Ndetection of a SIS mass peak as a function of peak redshift. The peak has the mass of a large galaxy cluster with M 200 = 6 . 6 × 10 14 M /circledot h -1 , or σ v = 10 3 kms -1 , at z = 0. Until redshift value z ∼ 0 . 6 this peak is visible above a 3 σ limit when only 3D lensing information and a tuning of α = 0 . 01 are used; see the black solid line with r = 0. The S / Nscales with M 2 / 3 200 as discussed in STH09. The S / N detection improves when we combine the lensing information with the galaxy tracer information, adopting β = 0 . 1; see lines with r > 0. The S / N im- t provement is greater for higher correlation factors r ( /lscript ) or more clustering b ( /lscript ) of the tracers. Unless we have extreme cases of high correlations, where r ∼ 0 . 9, and strong clustering, where b ∼ 3, the S / N enhancement is only moderate between the factors of 2 -3. The GSN model adopts a Gaussian approximation with ¯ n sis = 1 deg -2 . For this approximation, shape noise and sampling noise are still the dominating source of pixel noise, such that a scaling of the S / N detection ∝ M 2 / 3 200 is also found for the synergy technique within ∼ 10% accuracy. We verified this within the mass range 5 × 10 13 M /circledot h -1 ≤ M 200 ≤ 10 15 M /circledot h -1 . When we consider both r = 0 . 8 and b = 1 for the GSN correction factor, we find that the S / Nlevels in a randomised map have to be reduced to ∼ 75% at z = 0 . 15, 90% at z = 0 . 25, and /greaterorsimilar 93% at all other values (Sect. 3.6). These figures are typical values for r ( /lscript ) ∈ [0 , 1] and b ( /lscript ) ∈ [0 , 3]. Therefore, the GSN is a small e ff ect in the Gaussian regime and mostly relevant at redshifts z /lessorsimilar 0 . 3.", "pages": [ 11, 12 ] }, { "title": "5.5. N-bodymockdata", "content": "In Fig. 8, we show the simulated mass mapping of one N-body simulated field in di ff erent versions. The field has been randomly selected from the set of 128 one square degree fields. The top left panel displays transparent matter density contrast iso-surfaces of the data without source galaxy shape noise and without synergy. This ideal map has subsequently been transversely smoothed with a Gaussian kernel of 2 arcmin r.m.s. size. All maps in this figures are subject to the same angular smoothing. The top right panel corresponds to a map based on the lensing catalogue only but now with shape-noise of variance σ/epsilon1 = 0 . 3 and α = 0 . 05. This map and the following other two depict iso-surfaces of S / N based on 500 noise realisations that do not include GSN. The two bottom panels are the mass maps that include both the noisy lensing data and the information from galaxy tracers, which are either red galaxies with m u -m r > 2 . 2 or blue galaxies with m u -m r ≤ 2 . 2. The galaxy catalogues are flux-limited with m r ≤ 25 . 0. In these maps, we set the tuning parameters to α = 0 . 05 and β = 0 . 5. Mass peaks of more than ∼ 3 σ in the lensing-only map are designated by numbers between 1 and 6 . Mass peaks along the same line-of-sight and that are closest in redshift obtain the same number in the noise-free map. The distribution of mass peaks in the noise-free map was confusing and needed to be viewed on a computer display from di ff erent view angles to identify possible matches, especially at higher redshifts where a redshift slice of the light cone contains more volume. The complex 5 comprises a series of peaks that are connected by lines to guide the eye. Significant mass peaks in the combined reconstruction that are not visible in the lensing-only map are given capital letters between A and F . Their possible matches are also indicated in the noise-free map. The question mark in 1? indicates that the match to 1 in the noise-free map is uncertain. By 5 / B , we mean that the peak is located between the peaks 5 and B , which are both along the same l.o.s. but at di ff erent redshifts. All maps recover the prominent structures 2 and 3 at low redshifts but fail to significantly recover C and X , which both appear prominent in the noise-free map. The benefit from adding tracer information is mostly visible at higher redshifts, at z /greaterorsimilar 0 . 5, where more individual structures are lifted above the 3 σ threshold; peaks are less elongated in this regime as well. In particular, B and 5 are resolved when using red tracers, whereas the lensing-only map merges both together at an intermediate redshift. The blue tracers do not recover 5 but at least B at lower redshift. When using red tracers overall the S / N in the map is higher because red galaxies are more strongly clustered than blue galaxies and stronger correlated with the matter density field. The blue tracers render the original lensing-only map modestly in comparison by shifting the lensing signal in 5 / B to B , correcting 6 , weighing down 7 , and adding a couple of new features of A , D , and E that are insignificant in the lensing-only map. The redshift o ff set of peaks can be as high as ∆ z ≈ 0 . 2, as for seen for 1? (blue), or F (red).", "pages": [ 12 ] }, { "title": "6. Discussion", "content": "A synergy of 3D lensing data and galaxy clustering information can potentially alleviate the notorious z -shift bias in 3D lensing mass maps, provided the distribution of the tracers is statistically correlated with the underlying mass-density field. This can be seen in Fig. 3, which compares the radial p.s.f. for uncorrelated tracers to the p.s.f. in a synergy reconstruction with highly correlated tracers. The synergy produces a mass map in which the p.s.f. now peaks on average at the redshift of the original mass peaks and in which mass peaks are less smeared out in radial direction (width of p.s.f). Moreover, the z -shift bias is already fixed for relatively loosely correlated tracers with r ( /lscript ) > 0 . 4, as the additional Fig. 4 shows. Mixing the tracer and lensing information therefore promises to be an e ff ective technique to address the z -shift bias. The 3D mass mapping with gravitational lensing is essentially a tool for the visualisation of the spatial distribution of mass peaks on a galaxy-cluster mass scale; a moderate synergy with tracers improves the accuracy of the distance estimates and the detection rate at greater distances. Figure 7 displays the change in S / Nof cluster-sized mass peaks in a synergy map with moderate mixing ( α/β = 0 . 1). In the case of r /greaterorsimilar 0 . 4, we expect a S / N enhancement by a factor 2 -3. Strongly clustered tracers with a bias of b ∼ 3 are an exception here as they yield even more enhancement. They, however, should not be utilised in a reconstruction, because large density fluctuations clearly cannot obey a Gaussian statistics, which is the underlying assumption of the GSN treatment in the figure; Gaussian density fluctuations, δ , require a symmetric distribution about δ = 0, whereas large fluctuations 〈 δ 〉 2 /greatermuch 1 are bound to have a skewed distribution due to the constraint δ ≥ -1. The S / N improvement at larger distances is underlined by Fig. 8. An increase in the S / N and less radial smearing, which is visible in Fig. 3, at the same time results in a higher redshift accuracy of the mass peaks, because radial profiles of mass peaks are distinguishable more easily (Simon et al. 2012). Therefore, the benefit from our new algorithm is also a higher redshift accuracy instead of a more complete visualisation of the spatial distribution of cluster-sized masses. Based on this, the search for lensing mass peaks can be supported by galaxy tracers, and lensing mass models of clusters can be refined by accounting for possible alignments of peaks close to a single l.o.s.", "pages": [ 12 ] }, { "title": "combined: blue tracers", "content": "Our synergy technique is linear and for this reason has limited applicability on sub-degree scales due to a potentially nonlinear galaxy bias. The red thin lines in the left panel of Fig. 5 display the S / N of matter density modes in the 3D mass map before synergy. Compare this to the thin red lines in the right panel, which exhibit the S / Nof the tracer number density modes that is roughly ten times higher. This basically quantifies the information on the matter density field as encoded in the tracer distribution, if there is no stochastic galaxy bias and if the ex- act mapping between tracer and matter density is known (deterministic galaxy bias; e.g., Mann et al. 1998). This seems to favour a large weight for the galaxy tracers in a synergy reconstruction, and this would result in a S / N boost compared to a lensing mass map. In reality, however, galaxy bias is stochastic, non-linear and possibly even non-local (e.g., Tegmark & Bromley 1999; Yoshikawa et al. 2001; Hoekstra et al. 2002; Dekel & Lahav 1999; Matsubara 1999), which is not properly accounted for in a linear filter: Our filter assumes by construc- tion a linear relation between tracer and matter density, which is a Gaussian bias, and a Poisson process by which tracers sample the matter density field. The former can be seen by the fact that only a linear mixing of both fields is possible within the filter. Gaussianity is a valid assumption on (smoothing) scales, where density fluctuations are small, 〈 δ 2 〉 /lessmuch 1, or on large scales beyond ∼ 10 Mpc. Hence it is a fair assumption on angular scales larger than ∼ 45 arcmin (15 arcmin) at z ∼ 0 . 2(0 . 8) but, on the other hand, is prone to bias the mass maps on smaller angular scales. Moreover, sub-Poisson sampling processes become relevant on small angular scales, as indicated by the shot-noise corrected correlation factor of r ( /lscript ) > 1 in Fig. 2. To reduce bias on these scales we weigh down the tracer information by adopting small values of α/β /lessorsimilar 0 . 1 at the expense of map S / N, and we smooth the map with a kernel of several arcmin size. To further relax this problem, it is also conceivable to exclude tracer information by setting r ( /lscript ) = 0 at low redshifts of z /lessorsimilar 0 . 3 where the lensing information is highest, as seen in Fig. 7. Despite these issues, we conclude that a moderate mixing and smoothing yields qualitatively sensible results from the reconstructions in Fig. 8 with two di ff erent tracer samples. Nevertheless, giving less weight to the tracers adds less information to the mass map, so that realistically only a modest S / N improvement is feasible with the synergy method on non-linear scales. In addition, there remains an uncertainty in the GSN due to non-linear stochastic galaxy bias that can only be quantified by more accurate modelling. In contrast to red galaxy tracers, blue galaxy tracers lead to modest but presumably more reliable improvements in 3D mapping. We draw this conclusion from Fig. 8 that shows the combined reconstructions with red and blue galaxies in comparison. Clearly, including red galaxy clustering information adds more S / Nto the map than blue galaxies. This can be explained by Fig. 7 by considering that red galaxies are both more clustered and more strongly correlated with the matter density field in terms of r ( /lscript ) as seen Fig. 2. On the other hand, the assumption of a Gaussian bias model is less appropriate for red galaxies than for blue galaxies because of their greater density fluctuations 〈 δ 2 〉 . Typical blue galaxies exhibit density fluctuations smaller by a factor of ∼ 5 on arcmin scales, which in theory is even less than matter ( b < 1), and their number density is higher, which reduces the shot-noise error. Furthermore, blue galaxies are frequently field galaxies so that they also map out the large-scale matter distribution outside of clusters unlike red galaxies, which are preferentially found in galaxy clusters (Postman & Geller 1984; Zehavi et al. 2011). Considering the unknowns of the galaxy bias scheme, a blue tracer population is hence presumably the more favourable choice.", "pages": [ 13, 14 ] }, { "title": "Acknowledgements", "content": "I thank Stefan Hilbert and Jan Hartlap for providing simulated galaxy and shear catalogues, based on the Millennium Simulation, which were utilised to demonstrate the reconstruction algorithm. The Millennium Simulation databases used in this paper and the web application providing online access to them were constructed as part of the activities of the German Astrophysical Virtual Observatory. I also acknowlegde the useful comments by Stefan Hilbert and the anonymous referee on the paper. The work in this was supported by the European DUEL Research-Training Network (MRTN-CT-2006-036133) and the Deutsche Forschungsgemeinschaft in the framework of the Collaborative Research Center TR33 'The Dark Universe'.", "pages": [ 14 ] }, { "title": "References", "content": "Bacon, D. J. & Taylor, A. N. 2003, MNRAS, 344, 1307 Bartelmann, M. & Schneider, P. 2001, Physics Reports, 340, 291 Coles, P. & Jones, B. 1991, MNRAS, 248, 1 Dekel, A. & Lahav, O. 1999, ApJ, 520, 24 Dodelson, S. 2003, Modern cosmology, ed. Dodelson, S. Fan, Z. 2003, ApJ, 594, 33 Guo, Q., White, S., Boylan-Kolchin, M., et al. 2011, MNRAS, 413, 101 Guzik, J. & Seljak, U. 2001, MNRAS, 321, 439 Hilbert, S., Hartlap, J., White, S. D. M., & Schneider, P. 2009, A&A, 499, 31 Hirata, C. M. & Seljak, U. 2004, Phys. Rev. D, 70, 063526 Hoekstra, H., van Waerbeke, L., & Gladders, M. D. e. 2002, ApJ, 577, 604 Hu, W. & Keeton, C. R. 2002, Phys. Rev. D, 66, 063506 Jullo, E., Rhodes, J., Kiessling, A., et al. 2012, arXiv:1202.6491 Kaiser, N. 1992, ApJ, 388, 272 Kaiser, N. & Squires, G. 1993, ApJ, 404, 441 Lemson, G. & Virgo Consortium, t. 2006, ArXiv Astrophysics e-prints Leonard, A., Dup'e, F.-X., & Starck, J.-L. 2012, A&A, 539, A85 Mann, R. G., Peacock, J. A., & Heavens, A. F. 1998, MNRAS, 293, 209 Mart'ınez, V. J. & Saar, E. 2002, Statistics of the Galaxy Distribution, ed. Mart'ınez, V. J. & Saar, E. (Chapman and Hall / CRC) Matsubara, T. 1999, ApJ, 525, 543 Munshi, D., Valageas, P., van Waerbeke, L., & Heavens, A. 2008, Phys. Rep., 462, 67 Pen, U.-L., Lu, T., van Waerbeke, L., & Mellier, Y. 2003, MNRAS, 346, 994 Postman, M. & Geller, M. J. 1984, ApJ, 281, 95 Schneider, P. 1998, ApJ, 498, 43 Schneider, P. 2006a, in Saas-Fee Advanced Course 33: Gravitational Lensing: Strong, Weak and Micro, ed. G. Meylan, P. Jetzer, P. North, P. Schneider, C. S. Kochanek, & J. Wambsganss, 1-89 Schneider, P. 2006b, in Saas-Fee Advanced Course 33: Gravitational Lensing: Strong, Weak and Micro, ed. G. Meylan, P. Jetzer, P. North, P. Schneider, C. S. Kochanek, & J. Wambsganss, 269-451 Seitz, S. & Schneider, P. 2001, A&A, 374, 740 Seljak, U. 2000, MNRAS, 318, 203 Simon, P. 2012, A&A, 543, A2 Simon, P., Hetterscheidt, M., Schirmer, M., et al. 2007, A&A, 461, 861 Simon, P., Heymans, C., & Schrabback, e. 2012, MNRAS, 419, 998 Simon, P., Taylor, A. N., & Hartlap, J. 2009, MNRAS, 399, 48 Smith, R. E., Peacock, J. A., Jenkins, A., & et al. 2003, MNRAS, 341, 1311 Somerville, R. S., Lemson, G., Sigad, Y., et al. 2001, MNRAS, 320, 289 Springel, V. 2005, MNRAS, 364, 1105 Springel, V., White, S. D. M., Jenkins, A., et al. 2005, Nat, 435, 629 Tegmark, M. & Bromley, B. C. 1999, ApJ, 518, L69 Tegmark, M. & Peebles, P. J. E. 1998, ApJ, 500, 79 van Waerbeke, L. 1998, A&A, 334, 1 VanderPlas, J. T., Connolly, A. J., Jain, B., & Jarvis, M. 2011, ApJ, 727, 118 Weinberg, D. H., Dav'e, R., & Katz, N. e. 2004, ApJ, 601, 1 Yoshikawa, K., Taruya, A., & Jing, Y. P. e. 2001, ApJ, 558, 520 Zaroubi, S., Ho ff man, Y., & Fisher, K. B. e. 1995, ApJ, 449, 446 Zehavi, I., Zheng, Z., Weinberg, D. H., et al. 2011, ApJ, 736, 59", "pages": [ 14 ] }, { "title": "Appendix A: Gaussian galaxy bias", "content": "Let ˜ κ g = ˜ n g / ¯ n g and ˜ δ m be the real part of the Fourier coe ffi cients of the galaxy tracer number density contrast and matter density fluctuations, respectively, on a given lens plane and for a given angular mode /lscript . In the Gaussian regime the bivariate p.d.f. of both is given by where the matter variance is σ 2 = 〈 ˜ δ 2 m 〉 , { b , r } are the Gaussian bias parameters, and all means 〈 ˜ κ g 〉 = 〈 ˜ δ m 〉 = 0 vanish. The same relation holds for the imaginary parts of the Fourier coe ffi cients; the real and imaginary parts are independent. The conditional p.d.f. is therefore also a Gaussian, namely with a mean of and variance of The variance in ˜ δ m is given by the matter power spectrum P δ ( /lscript ) for the lens plane, the solid angle A fov of the plane, and Therefore, we expect an average tracer number density of br ˜ δ m with r.m.s. variance b √ 1 -r 2 σ for a fixed matter density mode ˜ δ m. The latter gives rise to the GSN in the Gaussian case (Dekel &Lahav 1999).", "pages": [ 14, 15 ] }, { "title": "Appendix B: SIS power spectrum", "content": "A set of N h haloes with positions θ j on the i th lens plane and an average, axial-symmetric matter density contrast, δ sis( | θ | ), produces the combined density contrast where the second equation on the right hand side is the Fourier transform of δ ( i ) m ( θ ). Averaging the two-point correlator of the density in Fourier space over all halo positions results in Weignore the clustering of the haloes over the field-of-view A fov, so that the two-halo term in the second sum vanishes, and where ¯ n sis : = N h / A fov expresses the mean number density of haloes. Therefore, we obtain in this scenario", "pages": [ 15 ] } ]

[ { "title": "ABSTRACT", "content": "Aims. We explore the evolution of supermassive black holes (SMBH) centered in a circumnuclear disk (CND) as a function of the mass supply from the host galaxy and considering di GLYPH<11> erent star formation laws, which may give rise to a self-regulation via the injection of supernova-driven turbulence. Methods. A system of equations describing star formation, black hole accretion and angular momentum transport in the disk was solved self-consistently for an axisymmetric disk in which the gravitational potential includes contributions from the black hole, the disk and the hosting galaxy. Our model extends the framework provided by Kawakatu & Wada (2008) by separately considering the inner and outer part of the disk, and by introducing a potentially non-linear dependence of the star formation rate on the gas surface density and the turbulent velocity. The star formation recipes are calibrated using observational data for NGC 1097, while the accretion model is based on turbulent viscosity as a source of angular momentum transport in a thin viscous accretion disk. Results. We find that current data provide no strong constraint on the star formation recipe, and can in particular not distinguish between models entirely regulated by the surface density, and models including a dependence on the turbulent velocity. The evolution of the black hole mass, on the other hand, strongly depends on the applied star formation law, as well as the mass supply from the host galaxy. We suggest to explore the star formation process in local AGN with high-resolution ALMA observations to break the degeneracy between di GLYPH<11> erent star formation models. Key words. Accretion, accretion disks - Black hole physics - Galaxies: nuclei - Quasars: general - Stars: formation", "pages": [ 1 ] }, { "title": "Star formation and accretion in the circumnuclear disks of active galaxies", "content": "Stephanie Wutschik 1 , Dominik R. G. Schleicher 1 , and Thomas S. Palmer II 2 Received date / Accepted date", "pages": [ 1 ] }, { "title": "1. Introduction", "content": "Supermassive black holes are observed in the centers of virtually all galaxies, and their properties are tightly correlated to the mass of the stellar bulge and its velocity dispersion (Magorrian et al. 1998; Ferrarese & Merritt 2000; Gebhardt et al. 2000; Graham et al. 2001; Merritt & Ferrarese 2001; Tremaine et al. 2002; Häring & Rix 2004). The correlation between the stellar mass and its velocity dispersion indeed suggests a link between the evolution of galaxies and their central black hole, or between the star formation rates and the black hole accretion rate. While the presence of supermassive black holes in the local Universe can be understood in terms of Eddington accretion, the latter appears much more di GLYPH<14> cult for supermassive black holes at z & 6 (Shapiro 2005). Black holes with more than 10 9 M GLYPH<12> have however been confirmed even beyond z = 6, both with the Sloan Digital Sky Survey (SDSS) (Fan et al. 2001; Fan et al. 2003, 2004, 2006a,b), the Canada-France high-redshift Quasar survey (Willott et al. 2007) or the UKIDSS Large Area Survey (Venemans et al. 2007). The currently highest-redshift detection corresponds to a 2 GLYPH<2> 10 9 M GLYPH<12> black hole at z = 7 : 085 (Mortlock et al. 2011), about 760 million years after the Big Bang. The formation of supermassive black holes thus likely requires a mechanism of e GLYPH<14> cient accretion and angular momentum transport. While high-redshift black holes are particularly challenging from a theoretical perspective, active galactic nuclei (AGN) in our local environment provide a relevant test case in order to probe the mechanisms for accretion and angular momentum transport in realistic systems. Observations reported the occurrence of circumnuclear disks and starburst rings in many of these systems. NGC 1068, the prototype AGN in our local neighborhood, has been explored in the stellar light (Telesco et al. 1984; Davies et al. 2007), but also in molecular gas and dust continuum by Schinnerer et al. (2000); Galliano et al. (2003); Galliano et al. (2005), finding a 3 kpc-scale ring of molecular gas with intense star formation activity. In a similar manner, NGC 1097 shows a molecular ring combined with intense star formation activity on scales of GLYPH<24> 700 pc (Hummel et al. 1987; Telesco & Gatley 1981; Kotilainen et al. 2000), and the presence of dense gas has been inferred via CO and HCN observations by Kohno et al. (2003) and Hsieh et al. (2008). The latter indicate typical densities of 10 3 cm GLYPH<0> 3 and temperatures of GLYPH<24> 100 K, comparable to the conditions in starburst galaxies (Wild et al. 1992; Aalto et al. 1995; Loenen et al. 2010). Indeed, more recent studies by Hsieh et al. (2011) have for the first time resolved individual giant molecular cloud complexes, allowing to test the correlation between molecular gas and star formation on more local scales. With ALMA 1 , accretion in these galaxies has now been probed down to scales of 40 pc (Fathi et al. 2013). Radio observations suggest the presence of strong magnetic fields with up to 60 GLYPH<22> G, which may thus contribute to the accretion process (Beck et al. 1999, 2005). Indeed, circumncuclear disks and rings appear to be ubiquitos in local AGN, and can thus be expected to play a fundamental role for black hole accretion (Davies et al. 2007). A systematic study concerning the correlation between the black hole accretion rate and the star formation rate has been pursued by Diamond-Stanic & Rieke (2012). While such a correlation appears to exist on all scales, it appears particularly strong below 1 kpc, i.e. on scales corresponding to the circumnuclear disk, where they report a scaling relation of SFR / ˙ M 0 : 8 BH . A strong connection between star formation and black hole accretion has further been indicated by Herschel 2 observations of Mrk 231 (van der Werf et al. 2010), as an analysis of the highJ CO lines provides indications both for strong photon-dominated regions (PDRs) and X-ray dominated regions (XDRs). The latter does correspond to the feedback from stars and a supermassive black hole. While Herschel was probing this phenomenon for nearby AGN, we note here that ALMA may provide the prespective of detecting X-ray dominated regions even at high redshift (Schleicher et al. 2010). While observational probes become more di GLYPH<14> cult at high redshift, the presence of dust and molecular gas in z > 4 quasars has nevertheless been reported already by Omont et al. (1996b,a); Carilli et al. (2002). The first detections in a z = 6 : 42 quasar have been pursued by Walter et al. (2004); Riechers et al. (2007); Walter et al. (2009), revealing the presence of a molecular gas mass of 4 : 5 GLYPH<2> 10 10 M GLYPH<12> . Even for the highest-redshift black hole at z = 7 : 085, dust and [CII] emission have been reported by Venemans et al. (2012). Particularly well-suited for the study of the circumnuclear accretion disks in high-redshift quasars are lensed systems, which reveal the dynamics in the central 100 GLYPH<0> 500 pc. An especially well-studied system is APM 08279 + 5255 at z = 3 : 9, where CO and HCN observations indicate warm molecular gas with densities of GLYPH<24> 10 5 cm GLYPH<0> 3 (Weiß et al. 2007; Riechers et al. 2010), while detections of water lines indicate an Eddingtonlimited starburst on similar scales (van der Werf et al. 2011). Similar conditions have been found for a lensed quasar at z = 4 : 1 (Riechers et al. 2008) and the Cloverleaf quasar at z = 2 : 56, where warm molecular gas was detected both in CO (Bradford et al. 2009) and HCN (Riechers et al. 2011). Based on these even though limited observations, we may thus conclude that circumnuclear starburst rings may play a significant role in connecting black hole growth and star formation both at low and high redshift. Such a connection has indeed been suggested also in theoretical models (Thompson et al. 2005; Levin 2007; Kawakatu & Wada 2008; Vollmer et al. 2008; Kawakatu & Wada 2009; Kumar & Johnson 2010), as stellar feedback may drive strong supersonic turbulence, which may drive the accretion via turbulent viscosity (e.g. Shakura 1973; Duschl & Strittmatter 2011). Indeed, enhanced accretion has been observed in numerical simulations in the presence of highly supersonic turbulence (Hobbs et al. 2011), and more realistic approaches aim to self-consistently inject turbulent energy via supernova-explosions (Wada et al. 2009). Other attempts have focused on the impact of black hole feedback on nearby starforming clouds (Hocuk & Spaans 2011), including the potential e GLYPH<11> ects of magnetic fields (Hocuk et al. 2012). While it is known that turbulence injected via stellar feedback may enhance the accretion, numerical simulations allow to explore only a limited range of conditions, while previous semianalytic models required simplifying assumptions concerning http: // sci.esa.int / science- the structure of the disk. As high-resolution observations with ALMA will allow to probe the structure of these disks, we aim here to extend the model of Kawakatu & Wada (2008); Kawakatu & Wada (2009) by including a more detailed description of the interior structure, and by considering a potentially nonlinear dependence of the star formation rate on the gas surface density and the turbulent velocity. Of course, such an attempt still requires assumptions to be made, which can however be validated both with upcoming observations as well as numerical simulations. In section 2, we provide the outline of our model, which is then applied in section 3 to di GLYPH<11> erent star formation models. We report here the impact of di GLYPH<11> erent star formation descriptions, as the latter are still uncertain both from a theoretical and an observational point of view. A discussion of these results is then provided in section 4, and the main conclusions are summarized in section 5.", "pages": [ 1, 2 ] }, { "title": "2. Outline of the model", "content": "The concept of this model is based on a self-gravitating gaseous disk around a SMBH. We supply the system with dusty gas at a time-dependent rate ˙ M sup from a host galaxy with a surface density GLYPH<6> host, including gas and stars, where the dust to gas ratio is similar to the solar neighbourhood. The accumulating gas and dust grains form a disk around the central seed black hole with mass M BH as described in Fig. 1. All components are rotating around the central black hole with an angular velocity GLYPH<10> ( r ) = GM BH = r 3 + G GLYPH<25>= r ( GLYPH<6> disk + GLYPH<6> host). In this calculation, our model assumes that the disks are dominated by the gas component, which we expect to be a good assumption in the active accretion phase and during intense starbursts, requiring ubiquitous molecular gas for intense star formation activity. This assumption becomes less accurate at late times when the stellar mass becomes comparable and the molecular gas reservoir is exhausted. We are however not predominantly interested in these phases, and we have checked that the resulting mass ratios remain essentially unchanged if we account for the additional stellar mass in the angular velocity. We assume an isothermal disk with a constant sound speed. We take into account that only part of the disk might be gravitationally unstable, separately considering the evolution of the stable part of the disk (inner disk) and the self-gravitating part (outer diks). We apply di GLYPH<11> erent star formation models and consider the amount of turbulence introduced into the gas by supernovae, which strongly influences the accretion onto the black hole. For the accretion process we adopt the formula for a viscous accretion disk as formulated by Pringle (1981, see Lodato (2008) for a review).", "pages": [ 2, 3 ] }, { "title": "2.1. Radial disk structure", "content": "The outer radius of the disk is defined by r out given in Eq. (1) and marks the farthest reach of the dominion of the gravitational potential of the system with respect to the host galaxy. If the disk mass M disk = R r out r in 2 GLYPH<25> r 0 GLYPH<6> disk( r 0 ) dr 0 with the inner radius of the disk r in dominates the gravitational potential inside the disk, we obtain r out from GM disk = r out = GLYPH<25> G GLYPH<6> host r out. On the other hand, if the gravitational potential inside the disk is dominated by the black hole, we obtain r out from GM BH = r out = GLYPH<25> G GLYPH<6> host r out. There M disk = M gas + M GLYPH<3> is the sum of the gaseous mass M gas and the stellar mass M GLYPH<3> . The inner radius of the disk is defined by r in which is assumed to be determined by the sublimation radius of silicon dust. This assumption is supported by observational results (Suganuma et al. 2006). If the AGN luminosity L AGN heating up the dust grains to the sublimation temperature of 1500 K, equals the Eddington luminosity L Edd = 4 GLYPH<25> c GM BH m p =GLYPH<27> T with the proton mass m p and the Thomson cross section GLYPH<27> T, the inner radius can be computed to r in = 3 pc p M BH GLYPH<2> 10 GLYPH<0> 8 . We note that the calculated inner radius is a maximum radius. Especially during slow accretion phases the actual inner radius might be closer to the central black hole. As gas is supplied to the system the disk will become gravitationally unstable eventually, giving rise to star formation which will trigger supernova feedback. To identify the region of instability we use the Toomre-function Q ( r ) to calculate the radius, at which the disk is just stable, i.e. at which Q ( r ) = 1. This we call the critical radius r c so that the disk is self-gravitating at radii r > r c, which essentially determines the region of star formation and supernova feedback. The Toomre function is determined as the fraction of the critical surface density GLYPH<6> crit( r ) over the gas surface density GLYPH<6> gas( r ) where GLYPH<6> crit is determined via There, GLYPH<20> 2 ( r ) = 4 GLYPH<10> 2 ( r ) + 2 GLYPH<10> ( r ) GLYPH<1> r d GLYPH<10> ( r ) = d r is the epicyclic frequency and c s is the sound speed. We find the critical radius r c at Q ( r ) = GLYPH<6> crit( r ) = GLYPH<6> gas( r ) = 1, where the surface density of the gas is equal to the critical surface density. The gas surface density is calculated for the inner and the outer disk: where GLYPH<6> 0 is computed from M g ; o = 2 GLYPH<25> R r out r c r GLYPH<6> out( r ) d r and GLYPH<13> is a free parameter. For the inner disk GLYPH<13> GLYPH<3> is obtained from M g ; i = 2 GLYPH<25> R r c r in r GLYPH<6> in( r ) d r . In case of a totally stable disk, i.e. no outer disk exists, GLYPH<6> 0 is calculated from M g ; i = 2 GLYPH<25> R r out r in r GLYPH<6> 0 ( r = r c) GLYPH<0> GLYPH<13> d r .", "pages": [ 3 ] }, { "title": "2.2. Vertical structure", "content": "We have two regimes which determine the vertical structure, i.e. the scaleheight of the disk. The first regime is the subsonic turbulent regime, where the thermal pressure is greater than the turbulent pressure and therefore determines the scaleheight. We assume hydrodynamical equilibrium so that GLYPH<6> gas( r ) GLYPH<1> c 2 s = GLYPH<6> gas( r ) GLYPH<1> g GLYPH<1> h therm( r ) where g = GM BH GLYPH<1> h ( r ) = r 3 + GLYPH<25> GLYPH<16> GLYPH<6> gas( r ) + GLYPH<6> host GLYPH<17> is the local gravity. There the second term can be neglected for smaller radii, where the gravitational potential is dominated by the central black hole. In this case, we obtain expression (5) for the thermal scaleheight and the turbulent velocity is equal to the sound velocity throughout the subsonic turbulent region. The second regime is the supersonic turbulent regime, where the turbulent pressure determines the scaleheight. Observations indicate, that active galactic nuclei often show violent star formation in the galactic nucleus as well, e.g. LaMassa et al. (2013); Santini et al. (2012). An intense star formation can trigger feedback mechanisms, which can deposit significant amounts of energy in the surrounding ISM. In this model we consider supernovae, which will deposit part of their thermal energy into the ISM as kinetic energy. Due to the vertical hydrostatic balance (Shetty & Ostriker 2012), E turb( r ) = GLYPH<6> gas( r ) GLYPH<1> g GLYPH<1> h ( r ) with E turb( r ) the turbulent energy per unit area at radius r , we obtain the turbulent scaleheight from equation (6). On the other hand, the turbulent velocity v turb( r ) is obtained directly from E turb( r ) = GLYPH<6> gas( r ) GLYPH<1> v 2 turb ( r ). In our model, the evolution of the turbulent energy is described by the di GLYPH<11> erential equation where ˙ E inj( r ) is the rate at which energy is injected into the medium and ˙ E dis( r ) = E turb( r ) = t dis( r ) is the energy dissipation rate with the dissipation time scale t dis = h turb( r ) =v turb( r ). This implies, that the dissipation timescale is equal to a crossing time. This equation is solved numerically via the common fourthorder Runge-Kutta method for each time t . For the energy-injection we calculate the energy input via supernova explosions and obtain where ˆ ˙ GLYPH<24> GLYPH<3> ( r ; t GLYPH<0> T SN) is the local star formation rate per unit area at time t GLYPH<0> T SN with T SN GLYPH<25> 10 6 yr being the average life time of massive stars, that explode into core-collaps supernovae, i.e. stars with initial masses greater than eight solar masses 3 . f SN = 7 : 9 GLYPH<2> 10 GLYPH<0> 3 is the supernova rate per solar mass of formed stars 3 , GLYPH<17> SN is a heating e GLYPH<14> ciency and E SN = 10 51 erg is the thermal energy typically injected by core-collapse supernovae. For easier calculation of the local supernova rate, we use a modified star formation rate surface density ˜ ˙ GLYPH<24> GLYPH<3> ( r ; t GLYPH<0> T SN) = A ( r = r c) GLYPH<0> GLYPH<18>GLYPH<13> GLYPH<0> GLYPH<21>GLYPH<15> where the integration of ˜ ˙ GLYPH<24> GLYPH<3> ( r ; t GLYPH<0> T SN) over the whole disk yields the absolute star formation rate ˆ ˙ M GLYPH<3> ( t GLYPH<0> T SN) and where ˆ ˙ M GLYPH<3> ( t GLYPH<0> T SN) is obtained from the integration of the original star formation rate per unit area over the currently star forming region, as the typical disk properties are expected to evolve over longer timescales than a typical supernova explosion. The local energy injection rate per unit area is then given as", "pages": [ 3, 4 ] }, { "title": "2.3. Star formation", "content": "The star formation rate in the disk ˙ M GLYPH<3> can be calculated by integrating the local star formation rate per unit area over the whole star forming region: R r out r min 2 GLYPH<25> r GLYPH<1> ˙ GLYPH<24> GLYPH<3> ( r ) d r where r min is either the inner radius r in ( r c GLYPH<20> r in) or the critical radius r c ( r c > r in). As we want to study the influence of the star formation model on the evolution of the system, we parametrize the star formation rate per unit area as follows: There GLYPH<9> is the normalization constant and 0 GLYPH<20> GLYPH<18> GLYPH<20> 2 and 0 GLYPH<20> GLYPH<15> GLYPH<20> 1 are free parameters. This allows us to apply a great diversity of both velocityindependent ( GLYPH<15> = 0) and self-regulating ( GLYPH<15> = 1) star formation models e.g. the models suggested by Kawakatu & Wada (2008) and Elmegreen & Burkert (2010). Table 1 lists the star formation models we applied in this work, where U1 rouhly corresponds to the model studied by Kawakatu & Wada (2008) and S4 corresponds to the model proposed by Elmegreen & Burkert (2010). To obtain the total star formation rate we need to calculate the integral of ˙ GLYPH<24> GLYPH<3> ( r ) from the critical radius r c to the outer radius of the disk r out. In order to carry out this integration, we approximate the turbulent velocity as a power law in Eq. (12). We obtain Article number, page 4 of 13 where we choose v 0 = c s and GLYPH<21> = 0 in case of (sub-)sonic turbulence. This leads to eq. (15), which gives the star formation rate at time t integrated over the disk. The free parameters GLYPH<18> , GLYPH<13> and GLYPH<15> have to be chosen carefully to ensure that GLYPH<18>GLYPH<13> + GLYPH<15> GLYPH<21> , 2.", "pages": [ 4 ] }, { "title": "2.4. Black hole accretion", "content": "In order to accrete matter from the circumnuclear disk onto the central black hole angular momentum has to be transported by some mechanism. In this model we will follow the GLYPH<11> -viscosityprescription by Shakura (1973) in the approximation for a thin disk, as formulated by Pringle (1981). Equation (16) gives the expression for accreted matter at radius R , as given by Kawakatu &Wada (2008). There, the viscosity GLYPH<23> ( r ) = GLYPH<11> v ( r ) GLYPH<1> h ( r ) is i) determined by supersonic turbulence in case of a self-gravitating disk or ii) caused by magneto-rotational instabilities, resulting into subsonic turbulence. In the first case, GLYPH<11> is of the order of unity, v ( r ) is the turbulent velocity v turb( r ) and h ( r ) is the turbulent scaleheight h turb( r ), determined by the turbulent pressure caused by stellar feedback. In the second case, the scaleheight is dominated by thermal pressure. The turbulent velocities are comparable to sound velocity or smaller, resulting into GLYPH<11> GLYPH<25> 0 : 01 GLYPH<0> 0 : 5 (see Kawakatu & Wada 2008, and references therein). To calculate the accretion rate of the black hole, we evaluate Eq. (16) at the inner radius of the disk r in, assuming that all matter crossing that radius will fall into the black hole eventually.", "pages": [ 4 ] }, { "title": "2.5. Matter transport inside the disk", "content": "In general, we assume that the matter infall from the host galaxy is significantly less e GLYPH<14> cient than the redistribution inside the disk. Therefore we assume, that the exponent of the gas surface density power law is not influenced by the infalling matter. As we consider the inner and outer disk seperately with individual power law surface densities, we need to take into account the transport of the gas from the outer to the inner disk and vice versa. The matter exchange consists of two contributions: i) a physical transport of matter from the outer to the inner disk due to accretion processes and ii) a purely geometrical transport process due to the evolution of the critical radius. i) If the disk is only partially self-gravitating, we assume, that the surface density in the inner and outer disk obey power laws with di GLYPH<11> erent exponents (see sect. 2.1). As the disk is supplied with additional gas from the host galaxy, this matter flows into the outer disk. However, the inner part of the disk is able to accrete matter from the outer disk with the same transport mechanism as the black hole accretes matter from the inner disk. We take this matter transport into account by calculating the disk accretion rate at times t , evaluating Eq. (16) at the critical radius r c. ii) If matter in the outer disk, which is gravitationally unstable, becomes stable, the critical radius moves outward. In this case, the now-stable matter has to be moved from the outer gas reservoir to the inner gas reservoir and the gas surface densities of both inner and outer disk have to be recalculated. To determine the amount of gas that has to become part of the inner disk, we calculate the integral of the gas surface density from the former critical radius to the updated critical radius, using the power law of the outer disk. In case of an inward directed critical radius (gravitationally stable matter becomes self-gravitating) the integration is calculated with the surface density power law of the inner disk and the matter is moved from the inner gas mass to the outer gas mass. The general formulation where GLYPH<12> = GLYPH<13> if r c( t 0) < r c( t ) and GLYPH<12> = GLYPH<13> GLYPH<3> if r c( t 0) > r c( t ), results into a positive value if gas has become stable ( r c( t 0) < r c( t )) and into a negative value, if gas has become self-gravitating ( r c( t 0) > r c( t )). So we obtain the updated gas masses M g ; o( t ) = M g ; o( t 0) GLYPH<0> GLYPH<1> M gas and M g ; i( t ) = M g ; i( t 0) + GLYPH<1> M gas.", "pages": [ 4, 5 ] }, { "title": "2.6. Model parameters", "content": "To model the system of the central black hole embedded in a circumnuclear disk we need to specify several free parameters. Table 2 gives the parameters which were chosen to correspond to those utilized by Kawakatu & Wada (2008) for better comparability. While the value of GLYPH<11> is uncertain in both regimes, we chose GLYPH<11> = 1 in order to have a smooth transition between the two different accretion processes. As the contribution of subsonic accretion to the final black hole mass is negligible with accretion rates two orders of magnitude smaller than supersonic accretion, the specific choice of GLYPH<11> in the stable phase of the disk has no impact on our final results.", "pages": [ 5 ] }, { "title": "3.1. Dynamical evolution of the system with unregulated star formation", "content": "As described in section 2.3 we apply di GLYPH<11> erent star formation models which are parametrized by the three parameters GLYPH<18> , GLYPH<15> and GLYPH<9> . For unregulated star formation we have GLYPH<15> GLYPH<17> 0 so that the star formation model is insensitive to an increased turbulent velocity (compare Eq. 11). GLYPH<18> is a free parameter, whereas GLYPH<9> has been chosen in such manner that the final stellar masses are roughly equal for all pairs of ( GLYPH<18>; GLYPH<9> ), to ensure comparability of the different models. Table 1 lists the studied models of unregulated star formation where U1 is equivalent to the model proposed by Kawakatu & Wada (2008) who chose GLYPH<18> = 1 and GLYPH<9> = 3 GLYPH<2> 10 GLYPH<0> 8 . All models show a characteristic behaviour in the evolution of the star formation rate, in concordance with the results of Kawakatu & Wada (2008). As shown in Fig. 2 the overall star formation rate rises continuously as long as gas is supplied to the disk. When the supply stops, the star formation rate quickly decreases and finally drops to zero, when the circumnuclear gas disk becomes completely stable, i.e. when r c GLYPH<21> r out. For the models with higher GLYPH<18> the star formation rate evolves less steeply and breaks o GLYPH<11> later than for lower GLYPH<18> . This behaviour results into similar final stellar masses for all models of approximately 5 GLYPH<2> 10 7 M GLYPH<12> , as shown in Fig. 3, which is a consequence of the normalization. Although star formation rates are similar for all models the accretion rates strongly depend on the chosen star formation model. Figure 4 shows the evolution of the accretion rates for the five studied models until time t = 10 9 yr. For U1 the evolution of the accretion rate resembles the results of Kawakatu & Wada (2008) although the drop down to what they called low accretion phase happens later in our model. We explain this behaviour by the fact that in our model the forming stars inject supernova-energy only 10 6 yr after they have formed and the injected energy dissipates more slowly (see section 2.2), thereby delaying the ceasing of supersonic turbulence which in our model is the main driver of e GLYPH<14> cient accretion. The obtained accretion rates of model U1 are one order of magnitude lower than in the paper of Kawakatu & Wada (2008). The origin of this discrepancy is not fully clear, as it remains even when we solve the equations as provided by Kawakatu & Wada (2008). The overall behavior is however similar, consisting of an e GLYPH<14> cient accretion phase during the gas supply and a significant drop when the supply is shut down. We show here that the accretion rate rises more steeply with higher GLYPH<18> , while for very small GLYPH<18> , the rise happens very late (for GLYPH<18> = 1) or not at all (for GLYPH<18> = 0 : 5), before the gas contents of the circumnuclear disk starts to decrease. The transition to the low accretion phase happens significantly later than in the model by Kawakatu & Wada (2008), roughly at the same time, when star formation completely ceases. By then, almost all injected turbulent energy has been dissipated so that the drop to the low-accretion phase is not as deep. The variable accretion rates result in strongly parameter-dependent final masses for the SMBH in the center of the gas disk. Di GLYPH<11> erent from the results of Kawakatu & Wada (2008), for small GLYPH<18> the black hole barely grows to 2 GLYPH<2> 10 6 M GLYPH<12> , due to the considerably lower accretion rates. Only for larger GLYPH<18> the SMBH grows up to a mass", "pages": [ 5, 6 ] }, { "title": "of GLYPH<25> 2 GLYPH<2> 10 7 M GLYPH<12> .", "content": "The gas mass evolves similar to what Kawakatu & Wada (2008) obtain, as shown in Fig. 6. The gas contents of the disk are continuously increasing as long as gas is supplied from the hosting galaxy. When the supply ceases the gas mass quickly decreases, mainly due to still forming stars. As soon as star formation ceases the gas content decreases to a few 10 6 M GLYPH<12> for all models.", "pages": [ 6 ] }, { "title": "3.2. Dynamical evolution with self-regulated star formation", "content": "For self-regulated star formation we have GLYPH<15> GLYPH<17> 1 so that star formation is sensitive to an increased turbulent velocity (compare Eq. 11). GLYPH<18> is a free parameter, whereas GLYPH<9> has been chosen in such manner that the final stellar masses are roughly equal for all pairs of ( GLYPH<18>; GLYPH<9> ) to ensure comparability of the di GLYPH<11> erent models. Tab. 1 lists the studied models of unregulated star formation where S4 is equivalent to the model proposed by Elmegreen & Burkert (2010) who chose GLYPH<18> = 2 and GLYPH<9> = 1 : 4 GLYPH<2> 10 GLYPH<0> 16 . The initial phase of the evolution shows some oscillatory behavior as a result of supernova feedback, as here the injected turbulence strongly suppresses star formation activity, which reduced the feedback at later stages. However, when the evolutionary timescale of the system becomes longer than the injection timescale of the supernova feedback, these oscillations adjust and a smooth transition is obtained (see Fig. 7). The star formation starts decreasing when the supply from the host galaxy stops after 10 8 years. Even for higher GLYPH<18> star formation carries on only little longer ( < 10 7 yr). The final stellar masses are roughly 7 GLYPH<2> 10 7 M GLYPH<12> for all models as shown in Fig. 8. The accretion rates clearly depend on model parameters, as can be seen in Fig. 9. Similar to the star formation rate the accretion rates show initially an oscillatory behaviour, which is anticyclic to the star formation rate. This latter is expected, as the accretion onto the black hole is proportional to the square of the turbulent velocity (compare section 2.4). Considering the accretion rates for model S4 we notice that the first oscillation is overlaid by another oscillation which is unique for this choice of parameters. The cause for the additional oscillation is the over-e GLYPH<14> cient accretion, which depletes the gas reservoir faster than it is refilled by the gas supply, which can also be seen in Fig. 11. Comparing Figs. 7 and 9 it can be noticed, that the models with lower star formation during the period between 10 7 and 10 8 yr have the higher accretion rates during that interval. As gas masses are generally low for all models (see Fig. 11), i.e. < GLYPH<24> 10 % of the supplied mass over the whole period, this seems to indicate a competition for gas between star formation and black hole accretion. For the di GLYPH<11> erent models we obtain final black hole masses of a few 10 7 M GLYPH<12> with higher masses for greater GLYPH<18> . As shown in Fig. 10 the dependence of the black hole mass on model parameters is significant but not as strong as for unregulated star formation models (compare Fig. 5). The evolution of the gas masses is slightly dependend on model parameters as shown in Fig. 11. However, the final gas masses are insensitive to the choice of parameters and amount to approximately 5 GLYPH<2> 10 6 M GLYPH<12> for all models. For model S4 we observe several oscillations between t = 10 6 yr and t = 2 GLYPH<2> 10 6 yr which are caused by the overe GLYPH<14> cient accretion onto the black hole as mentioned above. In the same period the gas mass of model S3 shows a single dip, supposedly for the same reasons. For low GLYPH<18> we do not observe such an overreaction to the onset of supernova-feedback, because the star formation per unit area follows a less steep power law which results into a less steep power law for the turbulent velocity and therefore also for the scale height. As the accretion rate is linear dependend on the turbulent velocity as well as the scale height at the inner disk radius r in, where both reach their highest values, the accretion rate reacts strongly to a steeper power law for velocity and scale height. Table 3 gives an overview of the mass balance for all models. As expected black hole masses rise with higher GLYPH<18> as steeper power laws for the gas surface density and thereby for the star formation per unit area result into higher accretion rates onto the central black hole. The disk mass is reduced accordingly where we like to mention that most of the final disk mass is contributed by stars, as final gas masses are smaller than stellar masses by approximately one order of magnitude for all models (compare Figs. 3 with 6 and 8 with 11). Notes. The first column identifies the model (see Tab. 1), second and third column denote the mass of the black hole and the disk. The third column displays the mass ratio M BH = M disk.", "pages": [ 6, 7, 8 ] }, { "title": "3.3. Radial disk structure", "content": "The evolution of the characteristic radii is quite similar for the di GLYPH<11> erent star formation models. While the outer radius for all models is dominated by the disk mass, therefore by the matter supply from the hosting galaxy, the inner radius more or less traces the growth of the SMBH (see sec. 2.1). As can be seen from Figs. 12 and 13 most of the time the circumnuclear disk is either completely self gravitating or completely stable against self-gravitation. The self-regulating star formation seems to result into longer transitional phases where the critical radius r c lies inside the disk, especially after the gas supply from the host galaxy has ceased.", "pages": [ 8 ] }, { "title": "4.1. Influence of the mass supply on the black hole growth", "content": "We studied the influence of the mass supply from the host galaxy on the final black hole mass and stellar mass of the system. We applied supply rates ˙ M sup = 1 : : : 100 M GLYPH<12> yr GLYPH<0> 1 exemplary on the four models U1, U4, S1 and S4 (see Table 1) with a run-time of 10 9 yr. As displayed in Fig. 14 final stellar masses as well as final black hole masses follow a power law M final = M 0 GLYPH<1> ( ˙ M sup = [ M GLYPH<12> yr GLYPH<0> 1 ]) x with model-dependent exponent x . The stellar mass x is roughly the same for all models whereas for the black hole masses the exponents di GLYPH<11> er significantly for the various models, as shown in Table 4. Notes. first column: studied model, for model parameters see Table 1; second and third column: power law-exponents x for the final black hole mass and the final stellar mass, see also Fig. 14; fourth column: power law-exponents for the mass ratio M BH = M GLYPH<3> In general, the final stellar masses are approximately linearly dependend on the mass supply rate for all models. The dependence of the final black hole masses on the mass supply rate is weaker and di GLYPH<11> ers significantly between self-regulated star formation and unregulated star formation (compare especially models U1 and S1 with GLYPH<18> = 1 : 0 for both models) as well as between di GLYPH<11> erent GLYPH<18> . We note that the exponent is rising with rising GLYPH<18> , but more steeply for the unregulated star formation (see Table 4). Consequently the mass ratio M BH = M GLYPH<3> depends on the mass supply rate as well.", "pages": [ 8, 9 ] }, { "title": "4.2. Influence of the seed mass", "content": "As there is a great discussion in the literature about the possible seeds of SMBHs and processes to form very massive seeds, e.g. Davies et al. (2011); Johnson et al. (2013); Latif et al. (2013a,b); Schleicher et al. (2013); Van Borm & Spaans (2013), we studied the influence of the seed mass on the final black hole mass. For seed masses from 100 M GLYPH<12> to 10 6 M GLYPH<12> we evolved the models U1 and S1 (see Table 1) for 1 Gyr and compared final masses of the SMBH and the stars. The results as shown in Table 5 make clear, that in our model even very massive seeds do not enhance the growth of the SMBH. This is in agreement with the findings of Montesinos Armijo & de Freitas Pacheco (2011), who found that the seed mass influenced the final SMBH mass only weakly, albeit they did not include star formation into their simulation. Notes. first column: mass of black hole seed; second and third column: final masses of the black hole and the stars for model U1; fourth and fifth column: final masses of the black hole and the stars for model S1; for model parameters see Table 1 In contrast to expectations very massive seeds even result into a slightly reduced amount of accreted matter and at the same time slightly enhance star formation. The reason for this non-intuitive behaviour lies in the considerably larger disk, as a very massive seed black hole dominates the gravitational potential on larger scales. This means, that the gaseous disk from the very beginning reaches out to several parsecs rather than to a few 10 GLYPH<0> 1 pc, which results into considerably lower gas surface densities at the inner radius, as the same amount of gas is distributed over a larger disk area. As the accretion rate is linearly dependend on the gas surface density the larger disk leads to accretion rates which are smaller by two orders of magnitude. These differences hold until the gravitational potential is dominated by the disk mass, i.e. until M disk > M BH. However, we note here that the environmental conditions of the first seed black holes are still unclear, and the approximations employed in this study are not necessarily appropriate for the growth of low-mass black holes in highly metal-poor environments. The growth of seed black holes provided by di GLYPH<11> erent mechanisms should ideally be pursuit in numerical simulations which account for the specific conditions in the local environments.", "pages": [ 9 ] }, { "title": "4.3. Nuclear star formation law: implications from observations", "content": "As our calculations have shown that the black hole growth depends significantly on the adopted star formation law, we explore whether the latter can be constrained from the detailed observations available for NGC 1097. We further discuss the observational status of other systems to assess whether significant restrictions for our model can be obtained.", "pages": [ 9 ] }, { "title": "NGC 1097", "content": "In order to evaluate the ability of our parametrized star formation law to calculate realistic star formation rates, we applied it to observational data of NGC 1097 (Hsieh et al. 2011), a nearby Seyfert 1 galaxy with a prominent star burst ring, and compared the obtained results with the observed star formation rates. The authors gave estimates of the gas masses of individual giant molecular cloud associations (GMAs), their central coordinates, diameters and velocity distributions as well as star formation rates per unit area, deduced from the Pa GLYPH<11> -luminosities (Hsieh et al. 2011). To apply the star formation model we considered annuli covering the complete region up to a radius of 1500 pc, and calculated the contribution of each GMA to the di GLYPH<11> erent annuli as well as the average velocity distribution v o . We then calculated a theoretical star formation rate per unit area for each annulus with inner radius Rin and outer radius R out: Here, GLYPH<6> o denotes the gas surface density in the annulus, calculated from the contributions of the GMAs to the di GLYPH<11> erent annuli divided by the area of the annulus A o = GLYPH<25> ( R 2 out GLYPH<0> R 2 in ). In Fig. 15 we compare the observational data with the two di GLYPH<11> erent star formation models U1 and S4 with GLYPH<9> = 7 GLYPH<2> 10 GLYPH<0> 16 (see Table 1), corresponding to the models of Kawakatu & Wada (2008) with a third of their star formation e GLYPH<14> ciency (diamond) and Elmegreen & Burkert (2010) with a star formation e GLYPH<14> ciency of 5 % (circle) instead of 1 %. Both star formation models roughly trace the observed values for the anuli with high star formation rates. Annuli with very low rates cannot be reproduced well by both models. This may indicate that the star formation e GLYPH<14> ciencies vary for the different annuli. It is noticable that, in order to fit the models to the observed values, we have to apply a smaller star formation e GLYPH<14> ciency for the model of Kawakatu & Wada (2008), where the model of Elmegreen & Burkert (2010) needs to be supplied with a larger star formation e GLYPH<14> ciency than originally proposed. The latter is not surprising, as we consider a starburst region where the star formation e GLYPH<14> ciency should be higher than average, while the star formation model of Elmegreen & Burkert (2010) was suggested for general star formation in galactic disks. For NGC 1097 Davies et al. (2007) estimated the dynamical mass in the central 20 - 40 pc from the kinematics of the stars and gas to 1 : 4 GLYPH<2> 10 8 M GLYPH<12> , where 1 : 2 GLYPH<2> 10 8 M GLYPH<12> are contributed by the central black hole (Lewis & Eracleous 2006). This implies a mass of gas and stars of 2 GLYPH<2> 10 7 M GLYPH<12> . This results into a mass ratio of M BH = M GLYPH<3> GLYPH<25> 25. On scales of 250 pc, Hsieh et al. (2011) give a similar value of 3 : 2 GLYPH<2> 10 8 GLYPH<2> M GLYPH<12> for the molecular gas, while the stellar mass corresponds to GLYPH<24> 3 : 0 GLYPH<2> 10 9 M GLYPH<12> . The ratio of stellar mass to disk mass is then of the order GLYPH<24> 30, which is consistent with the expectations in most of our models except U1 (see Fig. 14).", "pages": [ 9, 10 ] }, { "title": "Other objects", "content": "For di GLYPH<11> erent AGN, a certain range of black hole - to stellar mass ratios has been reported in the literature, as detailed in Table 6. NGC1068 is a case of a relatively small SMBH in the center of a very massive disk with a mass ratio similar to those we obtain for our models with unregulated star formation. Davies et al. (2007) found that this AGN shows no current star formation but had experienced a star burst GLYPH<25> 2 GLYPH<0> 3 GLYPH<2> 10 8 yr ago on a possible time scale of 10 8 yr. This starburst must have bound the majority of the available gas thereby producing the very small black hole-tostellar mass ratio, which is similar to those we obtained for our model with unregulated star formation (i.e. GLYPH<15> = 0). For NGC 3227 the comparison with observational data is more di GLYPH<14> cult, as the gas mass estimate is connected with great uncertainties of 2 GLYPH<0> 20 GLYPH<2> 10 7 M GLYPH<12> where the stellar mass itself is estimated to be 2 GLYPH<2> 10 7 M GLYPH<12> (Davies et al. 2006). In a later paper the authors suggest that the starburst was confined to the inner 12 pc around the SMBH (Davies et al. 2007). This means, that the gas component might not belong to the original circumnuclear disk. Therefore we compare the black hole mass only to the stellar component of the dynamical mass estimate. The resulting mass ratio is considerably higher than those obtained by our models. Still, the black hole mass of NGC 3227 as well as the stellar mass are both of the same order as the final masses of model S4 (see Table 3). For NGC 3783 the estimate of the disk mass is better constrained but still allowing for several interpretations (Davies et al. 2007). Assuming the higher value of 7 GLYPH<2> 10 7 M GLYPH<12> for the disk mass gives a mass ratio consistent with our results for model S4. For the smallest disk mass estimate the mass ratio is comparable to that for NGC 3227, which would be inconsistent with our model. An observational verification to distinguish between these scenarios would thus be valuable for our understanding of black hole accretion. For NGC 7469 the stellar component was estimated from the stellar K -band luminosity (Davies et al. 2007). The black holeto-stellar mass ratio is comparable to those obtained by our models with self-regulated star formation as are the absolute masses. Davies et al. (2007) suggest an age of 10 8 yr for the most recent star burst with a duration of approximately 10 7 yr where the authors argue against continuous star formation. We do not see such behaviour in our models, which assume a constant mass supply rates during the first 10 8 years. Their observations might thus indicate a strong variation of the mass supply in this particular galaxies, which could produce such an intense and short-lived starburst. How such a large stellar mass could build up during the available time may indeed be a subject of further investigation. For the observed local AGN the accretion rates are of similar order of 10 GLYPH<0> 2 M GLYPH<12> yr GLYPH<0> 1 , where in our models accretion rates can be of order unity when accretion is e GLYPH<14> cient. As none of the discussed objects are currently in a starburst phase, such a discrepancy may be expected. In fact, our models for the post-starburst phase show that the accretion rate may drop rapidly in the absence of a mass supply. For the observed active AGN, the mass supply rate to the center of the galaxy appears to be already reduced but di GLYPH<11> erent from zero. Our comparison shows that a one-to-one match of models and observations is non-trivial. The reasons are both the limited information in observational data, but also the large parameter space in the models, as well as their time evolution. Considering the case of NGC 1097 (see Fig. 15), the overall behavior of the star formation rate is reproduced both with the models of Elmegreen & Burkert (2010) and Kawakatu & Wada (2008). However, we also note that significant fluctuations exist and for both models, some annuli deviate from the theoretical expectation. The latter may indicate that the star formation law is more complex and the e GLYPH<14> ciency may change depending on the environmental conditions. To further improve on understanding the relation between star formation and black hole accretion, a first important step would be a more solid understanding of the star formation law. In this respect, Shetty et al. (2013) recently discussed evidence for a non-universal Kennicutt-Schmidt relation, while a potential explanation for such non-universality was given by Federrath (2013) based on the properties of supersonic turbulence. In addition, the structure of the disk can be probed in further detail with ALMA, yielding high-resolution data on surface densities and gas kinematics. Such data, together with an indication of the current star formation rate, would strongly constrain the current Notes. For each object (column 1), in column 2 the black hole mass is given as estimated from reverberation mapping except for NGC 1068, for which the dynamical mass estimate is given. Column 3 gives the disk mass inside the radius denoted in column 4, either estimated from the dynamical mass or from the K-band luminosity. Column 5 gives the e GLYPH<14> ciency of accretion luminosity GLYPH<15> L = L acc = L Edd from observation, from which the accretion rate ˙ M BH = GLYPH<15> L =GLYPH<15> M GLYPH<1> (1 GLYPH<0> GLYPH<15> M) GLYPH<1> M BH =GLYPH<28> can be calculated, where GLYPH<15> M is the mass accretion e GLYPH<14> ciency with a typical value of 0 : 3 and GLYPH<28> is the characteristical accretion timescale with a value of GLYPH<25> 0 : 23 Gyr (see Shapiro 2005). ( a ) stellar mass only References. (1) Khorunzhev et al. (2012); (2) Davies et al. (2007); (3) Davies et al. (2006); (4) Davies et al. (2004) state of the disk and thus provide a strong constraint on potential accretion models.", "pages": [ 10, 11 ] }, { "title": "5. Conclusions", "content": "In this paper, we present a semi-analytic model based on the previous work of Kawakatu & Wada (2008); Kawakatu & Wada (2009) describing the evolution of a black hole centered in a selfgravitating gaseous disk, that is time-dependently supplied with matter from the hosting galaxy. We have extended their model by separately considering the inner and outer disk and by considering the impact of a non-linear relation between the star formation rate, the gas surface density and the turbulent velocity. The aim of the model is to investigate the importance of star formation in the disk for the accretion process, i.e. how the mass of the SMBH depends on the way star formation is modelled. The accretion process is described via viscous accretion in the thin disk approximation, where angular momentum is transported by kinetic viscosity. The main source of the viscosity is supersonic turbulence which is generated by stellar feedback. There we limited our model to the contribution of supernovae, as Cen (2012) found, that AGN-feedback is less important than stellar feedback. Although Davies et al. (2012) state, that slow stellar winds as produced by AGB play an important role for the accretion process, we find that compared to supernovae stellar winds from OB stars as well as AGB stars inject negligable amounts of turbulent energy. Therefore we do not consider the contribution of stellar winds in our model. We note that, even when solving only the equations given in Kawakatu & Wada (2008), our accretion rates during the initial 10 8 yrs is di GLYPH<11> erent by about one order of magnitude. Here this discrepancy could not be resolved in full. Ideally, this point should thus be addressed through an independent calculation for instance based on a more detailed 3D modeling. Our model is able to grow SMBHs with masses of up to 10 7 M GLYPH<12> with mass supply rates of 1 M GLYPH<12> yr GLYPH<0> 1 . These results support the claims of de Xivry et al. (2011) and Schawinski et al. (2011), that (low-mass) SMBHs can form from accretion only. Given enough gas supplied to the nuclear region, our model is able to produce SMBHs of 10 9 M GLYPH<12> on similar time scales. Such large SMBHs have been observed at redshifts z ' 6 (e.g. Fan et al. 2001) and z ' 7 (Mortlock et al. 2011), approximately one billion years and 800 million years after the big bang respectively. This means that even these objects could have formed from accretion only, although our results do not exclude the occurence of major merger events during the evolution of the hosting galaxy, which might even be necessary to fuel the nuclear region with enough gas. In our model, we consider the accretion process on small scales (several tens of parsecs), as the influence of large scale structures and dynamics is highly debated in the literature. For instance Nayakshin et al. (2012) conclude that SMBHs are not fed by large scale (100 pc to several kpc) gaseous structures such as disks or bars. Also Jahnke & Macciò (2011) explain the observed M BHM bulge scaling relation by a hierarchical assembly of black hole and stellar mass rather than some kind of largescale co-evolution of SMBH and the hosting galaxy. DiamondStanic & Rieke (2012) found observational evidence for a strong correlation between black hole growth and star formation on smaller scales ( r < 1kpc) whereas the correlation on larger scales seems to be weak. Nevertheless, in order to form a SMBH large amounts of gas have to be supplied to the nuclear region even larger amounts of gas have to be avaiable in the hosting galaxy. In agreement with Montesinos Armijo & de Freitas Pacheco (2011) we found that final SMBH masses do not depend significantly on the seed mass, i.e. our model produces the same masses for black hole seeds of 10 2 to 10 5 M GLYPH<12> where for seed masses of 10 6 M GLYPH<12> the amount of accreted matter is slightly reduced. As higher seed masses result into larger disks the gas surface density is considerably lower as the same amount of matter is spread over a larger area. This leads to significantly smaller accretion rates until the total mass of gas and stars has reached the mass of the central black hole. This e GLYPH<11> ect is only visible for very large seed masses as for smaller seeds the relevant time scales are negligible compared to the total time scale. The latter implies that the growth of black holes of any mass is strongly regulated by the ambient surface densities, consistent with the findings of Shin et al. (2012). To obtain such high surface densities at early times and close to the black hole, it is however important that the gas can collapse without e GLYPH<14> cient fragmentation (Latif et al. 2013a,b). It is thus conceivable that the formation mechanism and the ambient densities are closely linked, thus providing both massive seeds and intense accretion. Apart from the gas surface density the turbulent velocity of the gas plays a very important role for the accretion process. As in our model turbulence is mainly driven by stellar feedback we do obtain high accretion rates only if stars are formed in the nuclear disk. The model therefore indicates a close connection between AGN activity and star formation as observed for several objects and discussed in the literature (e.g. Hopkins 2012; Santini et al. 2012; Kumar & Johnson 2010; Davies et al. 2007). We also stress that the choice of the star formation model, and in particular the self-regulation via turbulent velocities, has a strong impact on our results. Understanding the impact of turbulence on the star formation process in turbulent, self-gravitating disks is thus of central importance to understand what regulates the accretion of supermassive black holes, and may in fact determine the link between the star formation rates and black hole accretion rates that was derived in observational studies. While the study of Hsieh et al. (2011) already provided high-resolution data for the giant molecular cloud complexes in NGC 1097, similar studies are required for a larger number of active galaxies under di GLYPH<11> erent conditions, to probe star formation on di GLYPH<11> erent mass scales and in environments with di GLYPH<11> erent degrees of rotation and turbulence. The latter may allow to break the current degeneracy in the data and to derive a star formation law that can be applied in models of the circumnuclear disks. We note in particular that a Bayesian approach, as applied by Shetty et al. (2012, 2013) to di GLYPH<11> erent environments, may help to reliably constrain such star formation models.", "pages": [ 11, 12 ] }, { "title": "Appendix A: Average lifetime of massive stars", "content": "To calculate the energy output by supernovae, we need to estimate the average lifetime of the relevant stellar population, i.e. of stars with masses between 8 and 150 solar masses. As is known from stellar evolution theory, a star spends most of its lifetime on the main sequence, for which the relations between mass M , luminosity L and lifetime GLYPH<28> are well known: For the sun, mass and luminosity are well known, from which the lifetime on the main sequence has been deduced as GLYPH<28> GLYPH<12> = 10 10 yr. With this one obtains for a star of known mass the relation To estimate the average lifetime of a given population we integrate the product of lifetime and relative abundance over the relevant mass range: where N ( M ) dM is the initial mass function. For the relevant stars we obtain ¯ GLYPH<28> = 8 : 7 GLYPH<2> 10 5 yr GLYPH<25> 10 6 yr. The supernova rate per solar mass of formed stars is the fraction of the stellar population bound to explode into supernovae (i.e. the population of stars with M GLYPH<3> GLYPH<21> 8 M GLYPH<12> ) over the total stellar population. Acknowledgements. We thank W. Schmidt for fruitful discussions on the topic. D.S. and S.W. thank for financial support by the SFB 963 'Astrophysical Flow Instabilities and Turbulence' (project A12) via the German Science Foundation (DFG). T. S. P. is grateful for financial support by the RISE program of the German Academic Exchange Service (DAAD). This work was partly inspired by the COST conference Black Holes: From Quantum To Gravity (April 2012). DRGS further acknowledges funding from the Deutsche Forschungsgemeinschaft (DFG) in the Schwerpunktprogramm SPP 1573 'Physics of the Interstellar Medium' under grant SCHL 1964 / 1-1. We thank the anonymous referee for valuable comments on our manuscript.", "pages": [ 12 ] }, { "title": "References", "content": "Aalto, S., Booth, R. S., Black, J. H., & Johansson, L. E. B. 1995, A&A, 300, 369 Carilli, C. L., Cox, P., Bertoldi, F., et al. 2002, ApJ, 575, 145 Cen, R. 2012, ApJ, 755, 28 Davies, R. I., Burtscher, L., Dodds-Eden, K., & de Xivry, G. O. 2012, J. Phys.: Conf. Ser., 372, 7 Davies, R. I., Müller Sánchez, F., Genzel, R., et al. 2007, ApJ, 671, 1388 Davies, R. I., Tacconi, L. J., & Genzel, R. 2004, ApJ, 602, 148 Davies, R. I., Thomas, J., Genzel, R., et al. 2006, ApJ, 646, 754 de Xivry, G. O., Davies, R. I., Schartmann, M., et al. 2011, MNRAS, 417, 2721 Diamond-Stanic, A. M. & Rieke, G. H. 2012, ApJ, 746, 168 Diamond-Stanic, A. M. & Rieke, G. H. 2012, ApJ, 746, 168 Duschl, W. J. & Strittmatter, P. A. 2011, MNRAS, 413, 1495 Elmegreen, B. G. & Burkert, A. 2010, ApJ, 712, 294 Fan, X., Hennawi, J. F., Richards, G. T., et al. 2004, AJ, 128, 515 Fan, X., Narayanan, V. K., Lupton, R. H., et al. 2001, AJ, 122, 2833 Fan, X., Strauss, M. A., Becker, R. H., et al. 2006a, AJ, 132, 117 Fan, X., Strauss, M. A., Richards, G. T., et al. 2006b, AJ, 131, 1203 Fan, X., Strauss, M. A., Schneider, D. P., et al. 2003, AJ, 125, 1649 Federrath, C. 2013, ArXiv e-prints 1307.1467 Ferrarese, L. & Merritt, D. 2000, ApJ, 539, L9 Galliano, E., Alloin, D., & Granato, G. L. 2003, A&A, 412, 615 Galliano, E., Pantin, E., Alloin, D., & Lagage, P. O. 2005, MN- RAS, 363, L1 Gebhardt, K., Bender, R., Bower, G., et al. 2000, ApJ, 539, L13 Graham, A. W., Erwin, P., Caon, N., & Trujillo, I. 2001, ApJ, 563, L11 Häring, N. & Rix, H.-W. 2004, ApJ, 604, L89 Hobbs, A., Nayakshin, S., Power, C., & King, A. 2011, MNRAS, 413, 2633 Hocuk, S., Schleicher, D. R. G., Spaans, M., & Cazaux, S. 2012, A&A, 545, A46 Hocuk, S. & Spaans, M. 2011, A&A, 536, A41 Hopkins, P. F. 2012, MNRAS, 420, L8 Hsieh, P.-Y., Matsushita, S., Lim, J., Kohno, K., & SawadaSatoh, S. 2008, ApJ, 683, 70 Hsieh, P.-Y., Matsushita, S., Liu, G., et al. 2011, ApJ, 736, 129 Hummel, E., van der Hulst, J. M., & Keel, W. C. 1987, A&A, 172, 32 Jahnke, K. & Macciò, A. V. 2011, ApJ, 734, 92 Latif, M. A., Schleicher, D. R. G., Schmidt, W., & Niemeyer, J. C. 2013b, MNRAS, submitted (ArXiv e-prints 1309.1097) Levin, Y. 2007, MNRAS, 374, 515 Lewis, K. T. & Eracleous, M. 2006, ApJ, 642, 711 Lodato, G. 2008, arXiv: 0801.3848v Merritt, D. & Ferrarese, L. 2001, MNRAS, 320, L30 Mortlock, D. J., Warren, S. J., Venemans, B. P., et al. 2011, Nat, 474, 616 Nayakshin, S., Power, C., & King, A. R. 2012, ApJ, 753, 15 Omont, A., McMahon, R. G., Cox, P., et al. 1996a, A&A, 315, 1 Omont, A., Petitjean, P., Guilloteau, S., et al. 1996b, Nature, 382, 428 Pringle, J. E. 1981, ARA&A, 19, 137 Riechers, D. A., Walter, F., Brewer, B. J., et al. 2008, ApJ, 686, 851 Santini, P., Rosario, D. J., Shao, L., et al. 2012, A&A, 540, 1 Schawinski, K., Treister, E., Urry, C. M., et al. 2011, ApJ, L31 Schinnerer, E., Eckart, A., Tacconi, L. J., Genzel, R., & Downes, D. 2000, ApJ, 533, 850 Shakura, N. I. 1973, SvA, 16, 756 Shapiro, S. L. 2005, ApJ, 620, 59 Shapiro, S. L. 2005, ApJ, 620, 59 Shetty, R., Beaumont, C. N., Burton, M. G., Kelly, B. C., & Klessen, R. S. 2012, MNRAS, 425, 720 Shetty, R., Kelly, B. C., & Bigiel, F. 2013, MNRAS, 430, 288 Shetty, R. & Ostriker, E. C. 2012, ApJ, 754, 2 Shin, M.-S., Ostriker, J. P., & Ciotti, L. 2012, ApJ, 745, 13 Suganuma, M., Yoshii, Y., Kobayashi, Y., et al. 2006, ApJ, 639, 46 Telesco, C. M., Becklin, E. E., Wynn-Williams, C. G., & Harper, D. A. 1984, ApJ, 282, 427 Vollmer, B., Beckert, T., & Davies, R. I. 2008, A&A, 491, 441 Wada, K., Papadopoulos, P. P., & Spaans, M. 2009, ApJ, 702, 63 Walter, F., Carilli, C., Bertoldi, F., et al. 2004, ApJ, 615, L17 Walter, F., Riechers, D., Cox, P., et al. 2009, Nature, 457, 699 Weiß, A., Downes, D., Neri, R., et al. 2007, A&A, 467, 955 Wild, W., Harris, A. I., Eckart, A., et al. 1992, A&A, 265, 447", "pages": [ 12, 13 ] } ]

[ { "title": "ASERA: A Spectrum Eye Recognition Assistant for Quasar Spectra", "content": "Hailong YUAN a , Haotong ZHANG a , Yanxia ZHANG a , Yajuan LEI a , Yiqiao DONG a , Yongheng ZHAO a a Key Laboratory of Optical Astronomy, National Astronomical Observatories, Chinese Academy of Sciences, 20A Datun Road, Chaoyang District, Beijing, China, 100012. Email: [email protected]", "pages": [ 1 ] }, { "title": "Abstract", "content": "Spectral type recognition is an important and fundamental step of large sky survey projects in the data reduction for further scientific research, like parameter measurement and statistic work. It tends out to be a huge job to manually inspect the low quality spectra produced from massive spectroscopic survey, where the automatic pipeline may not provide confident type classification results. In order to improve the efficiency and effectiveness of spectral classification, we develop a semi-automated toolkit named ASERA, A S pectrum E ye R ecognition A ssistant. The main purpose of ASERA is to help the user in quasar spectral recognition and redshift measurement. Furthermore it can also be used to recognize various types of spectra of stars, galaxies and AGNs (Active Galactic Nucleus). It is an interactive software allowing the user to visualize observed spectra, superimpose template spectra from the Sloan Digital Sky Survey (SDSS), and interactively access related spectral line information. It is an efficient and user-friendly toolkit for accurate classification of spectra observed by LAMOST (the Large Sky Area Multi-object Fiber Spectroscopic Telescope). The toolkit is available in two modes: a Java standalone application and a Java applet. ASERA has a few functions, such as wavelength and flux scale setting, zoom in and out, redshift estimation, spectral line identification, which helps user to improve the spectral classification accuracy especially for low quality spectra and reduce the labor of eyeball check. The function and performance of this tool is displayed through the recognition of several quasar spectra and a late type stellar spectrum from the LAMOST Pilot survey. Its future expansion capabilities are discussed.", "pages": [ 1 ] }, { "title": "1. Introduction", "content": "The Large Sky Area Multi-object Fiber Spectroscopic Telescope (LAMOST) is a special reflecting Schmidt telescope specialized for conducting spectroscopic surveys with a wide field of view and a large aperture (Wang et al., 1996). One of the key scientific goals of LAMOST is the extragalactic spectroscopic survey of the large scale structure of the Universe and the physics of galaxies and quasars (Wu et al., 2011). The pilot survey (Cui et al., 2012) performed from October 2011 to June 2012 and the regular survey started in September 2012. There have been already millions of targets observed, including thousands of quasar candidates. Then to recognize quasars via spectra becomes essential for critical candidate confirmation and follow up scientific work. Spectra with high signal-to-noise ratio (SNR) are easily classified and the physical parameters can be determined with high accuracy using the LAMOST data processing pipeline (Luo et al., 2012). However there are still a large number of spectra with low SNR and probably some defects (e.g., skylight residual, splice connecting red part and blue part). Before the automatic pipeline is upgraded to be intelligent enough, eyeball check is in great need and enough astronomical knowledge is necessary. In SDSS quasar survey, visual inspection has been widely used to ensure the reliability of spectral identifications (Pˆ a ris et al. 2012). However in a large sky spectra survey, the quantity of spectra is very large. In order to reduce human efforts, we developed a spectrum eye recognition toolkit which provides a flexible platform to help identifying quasar spectra and estimating their redshifts at the same time. We call this toolkit ASERA. Both fits-formatted and imageformatted spectral files are supported. The input source can be placed in a local storage device, or distributed on the internet, described by a URL name. Since the software is developed using the Java programming language, it can be either started as a desktop application or accessed via a web browser, after deploying it as a Java applet. ASERA is initially dedicated to experienced spectrum analysts. It can also be used by teachers, undergraduate students and amateur astronomers. In the following sections, we first describe the detailed design and realization of the toolkit. Then several examples and figures are presented to explain how to use this toolkit on the spectra from LAMOST. In the end we discuss the system error on the estimated redshift z and the following upgrade plans to extend the functionality of ASERA.", "pages": [ 2, 3 ] }, { "title": "2. ASERA Development Status", "content": "The basic design idea of this toolkit is to overlay the quasar spectral template on the observed spectrum. With the help of the spectral identification experience, researchers try to superimpose these two spectra by choosing appropriate redshift and flux scale ratio. The first important step is to choose a quasar template.", "pages": [ 3 ] }, { "title": "2.1. The quasar spectrum template", "content": "Due to the similar wavelength coverage and spectral resolution of LAMOST and SDSS, a median composite quasar spectrum generated by a sample of over 2200 quasars from SDSS is applied as the standard quasar template (Vanden Berk et al., 2001). The spectrum, as shown in Figure 1, has over 80 identified emission lines within the band of wavelength from about 900 to 9000 angstroms. Eight most distinct emission lines are listed in Table 1. /s32 In this toolkit, the composite quasar spectrum can be transformed to various shapes by adjusting the redshift and the flux scale via the mouse. By comparing the observed spectrum with the transformed composite spectrum, /s32 researchers can easily figure out the best fit and provide the apt redshift value if the observed target is a quasar.", "pages": [ 3, 4 ] }, { "title": "2.2. Redshift calculation", "content": "Since the spectrum point is described by wavelength and flux density, a simple linear transformation is imported to get the pixel coordinate, and inversely to get spectrum point from pixel coordinate. We construct a linear pixel transformation function from the observed target spectrum firstly and then apply it to the template. In our toolkit, a default pixel transformation function is applied for the LAMOST spectrum products. To use this toolkit for image spectra from other spectral surveys, the user can choose the starting and end points in the pixel frame by simple mouse click, and tag them with the correct wavelength value. Then the new transformation can be established on these two points. The first step for plotting the template spectrum is to define the redshift z . We will choose one of the emission lines in Table 1 as the reference line and place it on the observed target spectrum. The redshift z of a wavelength which is shifted from λ 0 (from Table 1) to λ (calculated using pixel transformation function inversely) is defined as On the contrary, given the redshift value z , the wavelength λ 0 of a composite quasar spectrum can be shifted to λ defined by Then the whole template spectrum can be plotted at the specified redshift and flux scale. The user can find the most possible spectral type of the target and its corresponding redshift by changing and adjusting the template spectrum interactively.", "pages": [ 4, 5 ] }, { "title": "2.3. Supported spectral formats and locations", "content": "The current version of this toolkit supports both image-formatted and fits-formatted spectral files produced by LAMOST. File types are identified by their suffixes. Files ending with 'PNG', 'JPG', 'JPEG', 'WBMP', 'GIF' and 'BMP' are treated as images. The image file is plotted directly on a plane as the background and then the template spectrum as the foreground. The pixel transformation mentioned in the subsection 2.2 is used to calculate wavelength of any pixel point. Files ending with 'FITS' or 'FIT' are treated as LAMOST fits products. The java fits library 'nom.tam.fits' is imported to provide I/O for FITS image and binary tables. Then the spectrum is obtained by referring to the fits header definition of the LAMOST spectra. In the latest version, 'fits' files from SDSS are also supported. The data file can be placed in a local storage device or on the internet. The Uniform Resource Locator (URL) is used to describe both the exact location and accessing protocol. The standard edition of Java Develop Kit supports several protocol types, such as 'file', 'http', 'ftp' and 'gopher'. The 'file', 'ftp' and 'http' protocol type have been tested for the current version. Here are some examples of the URL external format string used in Windows platform: ftp://user:[email protected]/pdr/fits/20111024/F5902/spec-55859-F5902 sp16-249.fits http://data.lamost.org/pdr/fits/20111024/F5902/spec-55859-F5902 sp16-249.fits file://F: \\ B55878 \\ spec-55878-B87808 sp02-002.fits", "pages": [ 5 ] }, { "title": "2.4. Installation", "content": "The local installation of the toolkit takes merely no time since the latest version is provided as a single JAR file. Java Virtual Machine (JVM) is required to run the toolkit. Once the JVM is installed, the toolkit can be easily launched on all the dominating operation systems such as Windows, Linux and Mac-OS. Users can also start the toolkit as an applet from a simple web navigator such as Internet Explorer and Firefox. Any other software component can extend the capability of this toolkit by an HTTP link to the server. To enable this ability, the program should be firstly deployed as a Java applet in a dynamic web server, such as Apache Tomcat Server and Apache HTTP Server. The benefit for this deployment type is to save time for the client side user since the web server manager need handle the upgrade of the software release. However the weakness is the dependence of net access to the web server. Here is the example code for embedding the applet in a Java Server Page (JSP) file, namely 'index.jsp': < applet code='FittingApplet.class' archive='ASERA.jar' width='900' height='720' / > In this condition, the JAR library file should be placed in the same directory as the JSP file.", "pages": [ 5, 6 ] }, { "title": "2.5. The graphical interface", "content": "ASERA offers a single main window for displaying and manipulating the data, as shown in Figure 2. The combined functional regions are discussed in the following paragraphs. Region 1: the input data resource path and the path specification buttons. Region 2: the target and template spectrum in the centric viewport. The final redshift z of the template spectrum is printed at the lower-left corner. A set of spectral lines are plotted together with the template spectrum at the same time. Region 3: the most important information of the target including RA, DEC, target name, target type, SNR, magnitude, et al. Region 4: a button group for scaling and shifting flux range of the viewport for the target spectrum. It helps the user to acquire a proper flux density range for inspection. Region 5: a button group for scaling and shifting wavelength range of the viewport for both the target spectrum and the template spectrum. It helps the user to acquire a proper wavelength range for inspection. Region 6: the mouse right click popup menu. It provides a set of functions including template spectrum selection, visible absorption/emission lines selection, redshift reference line selection, image saving, pixel-wavelength conversion starting and end point specification. The current pixel-wavelength conversion status is displayed in the bottom line of the main window. The adjustment of the flux density of the template spectrum is handled by mouse. A left mouse button click event will replace the flux zero point and the reference line of the template to the clicked position. Then the redshift z will be recalculated and the spectrum will be repainted. A left mouse button drag event will shift the flux zero point and the reference line according to the drag distance. The mouse scrolling event will change the flux scale. In addition, the toolkit provides a spectrum selector and a FITS header viewer component, as shown in Figure 3. The spectrum selector enables the user to open a batch of spectra at one time and presents them in a tree like component. Currently there are three approaches to generate a spectral file list. The first is to scan the local directory containing fits and image files. The second is to parse the textual or XML VOTable file containing spectral URLs. The third is to construct URLs by querying a MySQL Database. Especially, the URL locating the VOTable can be either a static XML file or a dynamic web service, for example the SSAP (Simple Spectral Access Protocol) (Dolensky & Tody, 2004) server. The common compressed file formats, such as ZIP and GZIP, are recognized. The FITS header viewer provides a detailed FITS header information for users.", "pages": [ 6, 7 ] }, { "title": "3.1. Quasar spectral recognition", "content": "In the pilot survey about 400 plates were observed including several thousands of quasar candidates. SNR of some spectra are very low thus most of the features can't be recognized except the broad emission line. Some spectra have a little bit higher SNR but suffer from sky emission line residuals. Automatic program always fail to classify those spectral types and determine the redshifts with high confidence. With this toolkit, users can simplify the discrimination process under the guide of their rich spectral recognition experience. In Figure 4, we pick a spectrum processed by an early version pipeline to test our toolkit. The observed data is shown in black; the green line is the fitting result of the pipeline; the blue line is plotted by this toolkit; the blue vertical line represents the position of the MgII emission line. Apparently, the spectrum was misclassified as 'star' by the pipeline, but can be identified as 'quasar' with the help of this toolkit. By means of this toolkit, several emission lines are easily found, meanwhile the redshift can be obtained handily and is printed at the lower-left corner of Figure 4. Besides, the spectrum of the same source observed by SDSS is presented for comparison. The spectrum from SDSS is apparently identified as a quasar while the spectrum from LAMOST is difficult to recognize. To further demonstrate the feasibility of this toolkit, two objects misclassified as 'star' by the pipeline are identified as 'quasar' with the help of this toolkit, as shown in Figure 5.", "pages": [ 8 ] }, { "title": "3.2. Spectral recognition of other types of objects", "content": "By importing other templates, ASERA can be used to recognize spectra from various types of celestial bodies. The SDSS has provided 33 typical spectral templates in 'http://www.sdss.org/dr5/algorithms/spectemplates/', in- cluding various types of stars, galaxies and quasars. The template wavelength has already been transformed to rest frame, the same as the composite quasar spectrum mentioned in section 2.1. For templates whose redshifts are not absolutely zero, their wavelength is recalculated when loaded. In Figure 6, we show an example of ASERA to recognize an M-type star spectrum from the LAMOST pilot survey. The difference between the recognition of quasars and stars is that the later needs little redshift adjustment.", "pages": [ 9, 10 ] }, { "title": "4. Discussion", "content": "In this toolkit, the redshift z is calculated from the pixel coordinate, thus the redshift systematic error is here λ 0 is the wavelength of the emission line that we choose from Table 1 as the reference line , k is the wavelength difference between two adjacent pixels and ∆N is the difference of pixel between the point we choose and the ideal correct point. For example when the wavelength varies from 3700 to 9100 and the pixel width is 765, k is about 7.05882. Taken the lines in Table 1 as examples, with an assumed ∆N of 1, the system errors vary from 0.001075 to 0.005804. In future, we are ready to update the toolkit in several approaches. Firstly we will import more spectral templates of other types of celestial objects together with an interface to load a user specified template. Secondly we will extend the supported spectral data formats from most important survey services, besides LAMOST and SDSS. The astronomical data is complicated and the format is hard to be unified. The IVOA has already released many data representation and accessing protocols to facilitate the communication but time is needed for popularization and application. The VOTable from LAMOST data release server can be recognized currently but we need to extend the access to the online spectral service using the Simple Spectral Access Protocol (SSAP) proposed by the IVOA. Spectra in VOTable format will also be recognized and processed.", "pages": [ 11, 12 ] }, { "title": "5. Conclusions", "content": "To improve the efficiency and effectiveness of spectral classification, ASERA, a spectrum eye recognition assistant, is developed using Java programming language, especially designed for quasar spectral recognition. The toolkit includes a graphical interactive interface with both the target spectrum and the template spectrum plotted, a group of convenient viewport adjustment functions to provide entire or partial inspection of the spectrum arbitrarily, and various spectral templates helping users to identify the target spectrum by eye. Via choosing a suitable redshift z interactively, an artificial spectrum can be generated from a composite spectrum from Sloan Digital Sky Survey (SDSS). By comparing the generated spectrum with the target spectrum, taking the human experience as reference, users can finally recognize whether the target spectrum is a quasar or not, without being hampered by the partial abnormal or low SNR spectra. At the same time, ASERA may estimate the redshift z of the recognized quasar spectrum. Several quasar spectra from the LAMOST Pilot survey are tested to show the advantage of this toolkit in handling low SNR spectra with skylight residual or stray light. ASEAR can be used to recognized various types of stars, galaxies and AGNs by importing their related template spectra. The systematic error of the redshift calculation is discussed. The toolkit will be publicly available as soon as possible and user may contact the author for a trial edition at present. In the future, FITS spectral files besides LAMOST and SDSS, will be supported further. Also, we will realize the access to the online spectral service using the Simple Spectral Access Protocol (SSAP) proposed by the IVOA. In addition, spectra in VOTable format will also be recognized and processed.", "pages": [ 12 ] }, { "title": "Acknowledgments", "content": "This paper is funded by the National Natural Science Foundation of China under grant Nos.10778724, 11178021 and No.11033001. We acknowledge LAMOST and SDSS databases. Guoshoujing Telescope (the Large Sky Area Multi-Object Fiber Spectroscopic Telescope LAMOST) is a National Major Scientific Project built by the Chinese Academy of Sciences. Funding for the project has been provided by the National Development and Reform Commission. LAMOST is operated and managed by the National Astronomical Observatories, Chinese Academy of Sciences.", "pages": [ 12, 13 ] }, { "title": "References", "content": "Cui, X.-Q., Zhao, Y.-H., Chu, Y.-Q., et al. 2012, RAA, 12, 1197 Dolensky, M., & Tody, D. 2004, Proceedings of the SPIE, 5493, 262 Luo, A.-L., Zhang, H.-T., Zhao, Y.-H., et al. 2012, RAA, 12, 1243 Pˆ a ris, I., Petitjean, P., Aubourg, ´ E , et al. 2012, A&A, 548, 66 Vanden Berk, D. E., Richards, G. T., Bauer, A., et al. 2001, AJ, 122, 549 Wang, S.-G., Su, D.-Q., Chu, Y.-Q., Cui, X.-Q., & Wang, Y.-N. 1996, Appl. Opt., 35, 5155 Wu, X.-B., & LAMOST Extragalactic Survey LEGAS Team. 2011, American Astronomical Society Meeting Abstracts #218, #123.07", "pages": [ 13 ] } ]

[ { "title": "Anna Nakonieczna 1 and Marek Rogatko 2", "content": "Institute of Physics Faculty of Mathematics, Physics and Computer Science Maria Curie-Skłodowska University 20-031 Lublin, pl. Marii Curie-Sklodowskiej 1, Poland Abstract. Our research focus on gravitational collapse of electrically charged scalar field in dilaton gravity and in the presence of phantom coupling. We examine dynamical behaviour of the scalar field coupled to Maxwell field when gravitational interactions have form consistent with the low-energy limit of the string theory. Moreover, we allow the evolving fields to have negative sign in front of the respective kinetic term of the Lagrangian. The main aim of our studies is to investigate in what manner does the phantom nature of either Maxwell or dilaton fields (or both of them) affect the outcomes of the collapse. It turns out that the influence is crucial to the obtained spacetime structures. Negative kinetic energy of one (or both) of the fields delays, changes the course or even prevents the collapse. Keywords: dynamical collapse, dilaton gravity, phantom coupling PACS: 04.25.dg, 04.40.-b", "pages": [ 1 ] }, { "title": "INTRODUCTION", "content": "Dark energy is a highly predominating constituent of the Universe. Hence, an interesting issue arises: in what manner does its presence affect the processes, which are driven by gravitational force? In order to answer this question the properties of dark energy ought to be unraveled. Although it seems to be so widespread, the problem of its exact nature is still unresolved. There are numerous diverse theoretical models which attempt to describe and explain the properties of the considered unknown constituent of the Universe. Dark energy is described by the equation of state w = P r -1 , where P and r correspond to its pressure and density, respectively [1]. Although the exact value of the barotropic index w is still unknown, observations do not exclude the possibility of w < -1 [2]. In this particular case, the so-called phantom fields provide a theoretical model for dark energy [3, 4, 5]. Our research concern the influence of phantom nature of evolving fields on dynamical gravitational collapse. We are particularly interested in simulating the collapse of electrically charged scalar field, which allows us to draw conclusions about outcomes of the realistic process taking place in nature. Astrophysical black holes are electrically neutral and rotating [6]. The non-zero angular momentum results in axial symmetry of these objects. Within the framework of the general theory of relativity the structure of spacetime containing such a realistic black hole is described by the Kerr metric [7, 8]. Unfortunately, investigating gravitational collapse which leads to the formation of the Kerr black hole is extremely difficult due to analytical and numerical obstacles. For this reason some simplifications ought to be introduced. Basing on striking similarities in causal structures of spacetimes containing Kerr and Reissner-Nordström black holes the latter spacetime may be regarded as a toy model for the former one [7, 9]. Since it is spherically symmetric the calculations simplify considerably. The dynamical Reissner-Nordström spacetime stems from the collapse of electrically charged scalar field [10]. Hence spacetime structure emerging from the evolution of such a field actually imitates a structure resulting from the realistic gravitational collapse. We simulate the dynamical behaviour of complex scalar field coupled to Maxwell field when gravitational interactions take form of dilaton gravity and phantom coupling of Maxwell and dilaton fields is possible. We are primarily interested in singular spacetimes since examining the dynamical evolution of fields under the influence of gravity allows us to describe internal structures of the objects contained within them properly.", "pages": [ 1 ] }, { "title": "EINSTEIN-MAXWELL-DILATON THEORY WITH PHANTOM COUPLING", "content": "The action, which allows us to investigate gravitational collapse in the considered theory, represents complex scalar field coupled to Maxwell field in dilaton gravity. Moreover, it takes phantom coupling of Maxwell and dilaton fields into account. In the string frame it has the following form: where L ( SF ) stands for Lagrangian density while f is the dilaton field, y - complex scalar field, F mn ≡ A n , m -A m , n stands for the electromagnetic field tensor, where A m is the Maxwell field. The covariant derivative of the scalar field is defined as D m ≡ GLYPH<209> ( SF ) m + ieA m . Coupling constants e and a characterize couplings between the scalar field and Maxwell and dilaton fields, respectively. Phantom constants x 1 and x 2 refer to dilaton and Maxwell fields, respectively, and are equal to + 1 or -1. In the latter case the respective field is phantom. The equations of motion for the considered fields are obtained via variational principle. During their derivation the string frame is exchanged for the Einstein frame according to conformal transformation g ( EF ) mn = e -2 f g ( SF ) mn . Due to the fact that the evolving fields are massless, the collapse is simulated in double null coordinates and the assumed line element has the form ds 2 = -a ( u , v ) 2 dudv + r ( u , v ) 2 d W 2 , where a ( u , v ) is an arbitrary function, r ( u , v ) is the radial function and d W 2 stands for the line element on the unit sphere. Einstein equations obtained in accordance with the above assumptions and conditions are given by The equations of motion of the complex scalar field are as follows: while the equation of motion of the dilaton field has the following form: On account of gauge freedom of electromagnetic potential A m → A m + GLYPH<209> m L , where L is an arbitrary scalar function, Maxwell equations may be written as The set of equations (3)-(11) describes the gravitational collapse in Einstein-Maxwell-dilaton theory with phantom coupling. Because of its complicated structure it has to be solved numerically. The suitable algorithm was described in [11].", "pages": [ 2 ] }, { "title": "SPACETIME STRUCTURES", "content": "The initial conditions for analysed evolutions consist of profiles for complex scalar field and dilaton field. They were chosen as trigonometric type with an amplitude ˜ p y and gaussian type with an amplitude ˜ p f , respectively [12]. Since the dynamical gravitational collapse is universal, its outcomes do not depend on the specific form of initial profiles. They are also independent of the values of an electric coupling constant e and a parameter d provided that these are not equal to zero. For this reason, these two quantities were fixed in all the conducted evolutions. On the other hand, it turns out that structures of spacetimes emerging from the collapse depend on initial amplitudes of the collapsing fields ˜ p y and ˜ p f as well as on the value of dilatonic coupling constant a . We considered two values of it, namely a = -1 and a = 0. The former refers to the low-energy string theory, while the latter causes the evolution to run in the presence of uncoupled dilaton field. In order to examine the roles of evolving fields and couplings among them in the considered gravitational collapse it is sufficient to describe its outcomes when initial amplitude of complex scalar field is constant and the amplitude of dilaton field varies. An amplitude ˜ p y was set as equal to 0 . 6 in all the evolutions. The types of all phantom evolutions, which will be analysed, are characterized in Table 1.", "pages": [ 2, 3 ] }, { "title": "Phantom Maxwell field", "content": "Taking the above into account, it may be stated that phantom nature of Maxwell field prevents the formation of a Cauchy horizon and supports the emergence of the simplest spacetime structure containing a black hole. The results of an EMD -collapse are identical for both considered values of a . They are shown in Fig.2. Regardless of the initial value of dilaton field amplitude all emerging spacetimes are singular and Schwarzschild-type. For small values of the amplitude the spacetime structure is typical. For its bigger values the collapse runs in two stages, what results in an appearance of a tempo- glyph[negationslash] glyph[negationslash] glyph[negationslash] rary horizon in spacetime. The joint collapse of phantom Maxwell field coupled to the complex scalar one and non-phantom dilaton field leads to the formation of a simplest spacetime structure describing a black hole. Since phantom Maxwell field favours the emergence of such a structure, this result indicates that non-phantom dilaton field either also supports the process or is unable to counteract effectively.", "pages": [ 3 ] }, { "title": "Phantom dilaton field", "content": "The ED -collapse represents the evolution of phantom scalar field under the influence of Einstein gravity. It turns out that regardless of an initial value of the field's amplitude the resulting spacetime is non-singular. On account of the fact that dynamical Schwarzschild spacetime emerges from the collapse of non-phantom scalar field [13], the obtained result means that phantom nature of the scalar field prevents singularity formation. A collection of singular spacetime structures, which are obtained during the dynamical gravitational collapse in EMD -theory for a = -1, is presented in Fig.3. For small values of dilaton field amplitude the initially formed central singularity, which is surrounded by an apparent horizon coinciding at v → ¥ with the event horizon situated along u = const . , bifurcates and forms two wormhole throats during the course of the collapse. Finally, there exists a dynamical wormhole within the glyph[negationslash] glyph[negationslash] glyph[negationslash] glyph[negationslash] glyph[negationslash] emerging spacetime [12]. For bigger initial values of the amplitude of dilaton field the spacetime is singular, but the singularity is not surrounded by the event horizon, i.e. a naked singularity forms. For even larger values of the dilaton field amplitude the resulting spacetime is non-singular. In contrast, the collapse of an electrically charged scalar field in the presence of phantom dilaton field when a = 0 does not lead to singular spacetimes. Summarizing the above findings it may be stated that the tendency of phantom dilaton field to prevent the singularity formation is stronger in the uncoupled case, i.e. for a = 0, than for a = -1.", "pages": [ 3, 4 ] }, { "title": "Phantom Maxwell and dilaton fields", "content": "The structures of singular spacetimes, which stem from the EMD -collapse for a equal to -1 and 0, are depicted in Fig.4. In both cases the considered process leads to the formation of singular spacetimes for all initial values of dilaton field amplitude. For the former value of dilatonic coupling constant Schwarzschild-type spacetimes emerge, while for the latter naked singularities are observed. These results confirm the above conclusions concerning the role of particular fields and couplings in the analysed collapse.", "pages": [ 4 ] }, { "title": "CONCLUSIONS", "content": "We examined the results of a gravitational collapse of electrically charged scalar field in dilaton gravity with a possibility of phantom coupling of Maxwell and dilaton fields to gravity. The conclusions regarding the role of particular fields and couplings in its course are the following:", "pages": [ 4 ] } ]

[ { "title": "Łukasz Nakonieczny 1 and Marek Rogatko 2", "content": "Institute of Physics Maria Curie-Skłodowska University 20-031 Lublin, pl. Marii Curie-Skłodowskiej 1, Poland Abstract. According to the Big Bang Theory as we go back in time the Universe becomes progressively hotter and denser. This leads us to believe that the early Universe was filled with hot plasma of elementary particles. Among many questions concerning this phase of history of the Universe there are questions of existence and fate of magnetic monopoles and primordial black holes. Static solution of Einstein-Yang-Mills system may be used as a toy model for such a black hole. Using methods of field theory we will show that its existence and regularity depend crucially on the presence of fermions around it. Keywords: primordial black holes, fermions PACS: 04.50.+h", "pages": [ 1 ] }, { "title": "INTRODUCTION", "content": "Recent observational data strongly suggest the existence of dark matter (DM) [1]. One of a few remaining DM candidates compatible with the Standard Model (SM) are primordial black holes (PBH) [2]. These black holes were formed in the early Universe during gravitational collapse of density fluctuation. The existence of the PBH was first proposed in [3, 4, 5] and their formation during inflation or phase transition in the early Universe was discussed in [6, 7, 8]. Another type of objects that could be formed during phase transition in early Universe are topological defects like cosmic strings and magnetic monopoles [9]. If these objects really formed during evolution of the Universe we should consider the possibility of interactions among them. Within the framework of field theory a simple model of the system that allows the existence of magnetic monopoles may be given by Yang-Mills (YM) theory with SU ( 2 ) gauge group. Particle like solutions of Einstein-Yang-Mills (EYM) system describing the magnetic monopole were first constructed in [10]. On the other hand, solutions describing a black hole with magnetic monopole in this context were presented in [11]. There was also shown in [12] that finiteness of the black hole mass demands absence of an electric part of YM field. Another important component of the early Universe whose interaction with PBH we should consider is fermionic matter. To describe these interactions we use methods of field theory. As a toy model of PBH we take the aforementioned spherically symmetric YM black hole. As a representation of fermionic matter we use Dirac field. Using dimensional reduction and bosonization we will analyze the influence of fermions on the considered PBH.", "pages": [ 1 ] }, { "title": "DIRAC EQUATION IN MAGNETIC YANG-MILLS BLACK HOLE BACKGROUND", "content": "A general form of a spherically symmetric and time independent metric is where r is an usual radial coordinate and d Ω 2 = d θ 2 + sin 2 ( θ ) d φ 2 is a standard metric on two dimensional sphere. The location of an event horizon of a black hole is given by the largest positive root of A 2 ( r ) , which is denoted as rH . Actually for analyzing fermions equations of motion it is more convenient to introduce the so-called tortoise coordinate r ∗ . This coordinate is given by the following relation: Our metric expressed in coordinates ( t , r ∗ , θ , φ ) is where we introduce a function C to keep in mind that now we have r = r ( r ∗ ) . From now on, to simplify our notation we will omit explicit writing of arguments of functions. During our computations we will use tetrad formalism. The local ortogonal tetrad is defined by the relation g µν = e i µ e j µ η i j , where e i µ is an element of the tetrad, g µν and η i j are metric tensors in curved and flat spacetimes respectively. We use the following signature convention: η 00 = -1, η 11 = η 22 = η 33 =+ 1. The general form of static and spherically symmetric Yang-Mills potential may be written as where ai =( a 0 , a 1 ) and w represent electric and magnetic parts of YM field, n i is a unit vector normal to the sphere, λ is a coupling constant, ε i jk stands for a totally antisymmetric Levi-Civita symbol, and τ k are generators of SU ( 2 ) gauge group represented by Pauli matrices. As the representation for four dimensional gamma matrices we choose Weyl basis where I is a unit two dimensional matrix and σ i are three Pauli matrices. Gamma matrices in curved spacetime are given by the standard relation: γ µ = e µ k γ k . Representing a spinor ψ as ψ = ( ψ L ψ R ) we may write the Dirac equa- tion as where operators / D ± are given by [13, 14] In the above formula a bar over a quantity represents three dimensional vector, a dot - scalar multiplication, × - vector multiplication, and D S 2 is Dirac operator on S 2 . We are mainly interested in fermions in s-wave sector, which is described by the lowest eigenvalue of D S 2 . The action of operators ¯ n · ¯ τ , ¯ n · ¯ σ × ¯ τ , ¯ σ · ¯ n and D S 2 on these states may be found in [13]. We may use this knowledge to integrate over the angles in four dimensional Dirac action to obtain effective two dimensional theory in ( t , r ∗ ) plane. On the other hand, dimensional reduction of the YM action gives us the following Lagrangian: where da = ∇ a -iBa , f ab = ∂ aBb -∂ b Ba , and Ba = e i a ai . From now on latin letters from the beginning of alphabet will label indexes from curved two dimensional ( t , r ∗ ) spacetime and those from the middle from flat spacetime.", "pages": [ 1, 2 ] }, { "title": "MASSLESS FERMIONS", "content": "In the massles case we see from (6) that chiralities decouple and effective two dimensional theory contains two fermionic fields connected to ψ L and ψ R . The effective Lagrangian for the field ψ R is where V = w C . In this formula GR is two dimensional spinor field connected to ψ R by rescaling and multiplication by appropriate σ matrices [15]. Two dimensional flat spacetime gamma matrices are given by the following relations: The Lagrangian for the field connected to ψ L differs from (9) only by a sign of a term proportional to Ba . Analyzing equations of motion for fermions in curved spacetime is a highly nontrivial and difficult task. However, in case at hand since we can express our problem in form of effective two dimensional theory we may use bosonization technique [16]. This technique is well defined for flat spacetime two dimensional problems and also for asymptotically flat spacetimes [17, 18]. The merit of bosonization is that we express a fermionic sector of our theory in terms of complex scalar field. Basic bosonization formulas in our case are given below After bosonization from (9) we obtain the following scalar Lagrangian: from which we derive an equation of motion for φ R field Then, we derive an equation of motion for scalar field connected to the ψ L sector of our original fermionic theory analogically: Equations (15) and (16) are highly nonlinear and we were not able to find a solution in terms of known special functions in the whole spacetime. Nevertheless, we may find some useful information about their solutions by analyzing them in two asymptotic regions, namely in near horizon region and in large r ∗ region. In the large r ∗ region we have that Taking this into account the equation (15) turns into After dropping a term proportional to O ( r -1 ) we get a free wave equation A regular solution to this equation may be expressed as a plane wave where c 0 is an integration constant. On the other hand, in the near horizon region we have where κ is surface gravity of a Yang-Mills black hole. The equation (15) in this region takes the form Because as we approach a black hole horizon r ∗ →-∞ , we may drop a term proportional to e 2 κ r ∗ in the above equation. From this we see that a regular solution is again a time dependent plane wave given by where c 1 is some other integration constant. On the basis of this analysis we may conclude that a regular solution to the considered scalar field equation will be time dependent. But both scalar fields represent fermionic currents and their time dependence ultimately means that fermionic fields will also be time dependent. To see how this may influence YM field we use equations of motion for electric and magnetic parts of this field in the presence of fermions. After using bosonization formulas (11) these equations read [15] Conclusions that stem from the above equations are the following. First, let us remind that equations (15) and (16) differ only by sign in front of the term proportional to Ba and we assume that initially a black hole has only magnetic charge ( Ba = 0). Second, from equation (24) we see that in this case contributions of scalar fields cancel each other and Ba = 0 is still a valid solution to (24). Third, from equation (25) we see that there is a nonzero contribution of scalar fields to magnetic part of YM field. But since these fields are time dependent so should be the resulting w . On the other hand, through Einstein equations, this results in time dependence of metric tensor elements and ultimately means that the assumption of staticity of Yang-Mills black hole is destroyed in the presence of massless fermions.", "pages": [ 2, 3 ] }, { "title": "MASSIVE FERMIONS", "content": "For massive fermions we see from (6) that chiralities are mixed up. To use bosonization in this case we need to make an additional assumption about the form of ψ R and ψ L . We use the following ansatz: where GL and GR are two dimensional spinor fields connected with ψ L and ψ R respectively [15]. Having this in mind an effective two dimensional fermionic Lagrangian is as follows: where like in massless case V = w C . Using the same bosonization formulas as in the massless case we arrive at the following scalar Lagrangian: The equation of motion for scalar field in this case is In the large r ∗ limit, after dropping terms proportional to O ( r -1 ) , we obtain a sine-Gordon equation for which a regular time dependent and decaying at infinity (kink type) solution is given by On the other hand, the same type of analysis like in the massless case revealed that in the near horizon limit we again have a free wave equation with a regular time dependent solution in form of a plane wave: Nowwewill discuss the influence of massive fermions on a YM black hole. Yang-Mills equations of motion in the presence of fermions (after bosonization) are the following: From equation (33) we see that, even if we initially set Ba = 0, fermions give a nonzero contribution to the electric part of YM field. This means that contrary to the massless case the presence of fermions leads to dyonic structure of a black hole. But as was shown in [12] dyonic Yang-Mills black hole necessarily has infinite mass. On the other hand, form equation (34) we see that time dependent fermions give a nonzero contribution to the magnetic part of YM field. Through Einstein equations this leads to a time dependent line element and the destruction of staticity of a black hole.", "pages": [ 3, 4 ] }, { "title": "CONCLUSIONS", "content": "Now we shall present a short summary of our results concerning the influence of fermions on a primordial black hole modeled by a magnetic Yang-Mills black hole. First we will discuss the massless case. By asymptotic analysis we find an evidence that regular solutions to equations (15) and (16) should be time dependent. These equations describe bosonized massless fermions and give a nonzero contribution to a magnetic part of Yang-Mills field. Because their solutions are time dependent the resulting magnetic part of YM field will be time dependent and, through Einstein equations, elements of a metric tensor will also be time dependent. In massive case bosonized fermions are described by solutions to equation (29). The same type of analysis like in massless case also reveals the destruction of staticity of our black hole. Moreover, massive fermions will lead to the formation of a dyonic black hole. But, as was shown in [12], the presence of an electric part of YangMills field leads to infinite mass of the resulting black hole. In conclusion we may say that the presence of fermions and their interaction with the considered PBH will lead to its destruction through mechanisms described above.", "pages": [ 4 ] } ]

[ { "title": "C. J. A. P. Martins", "content": "Centro de Astrofísica, Universidade do Porto, Rua das Estrelas, 4150-762 Porto, Portugal Abstract. The observational evidence for the acceleration of the universe demonstrates that canonical theories of gravitation and particle physics are incomplete, if not incorrect. A new generation of astronomical facilities will shortly be able to carry out precision consistency tests of the standard cosmological model and search for evidence of new physics beyond it. I describe some of these tests, focusing on the universality of nature's fundamental couplings and the characterization of the properties of dark energy. I will also comment on prospects for forthcoming ESA and ESO facilities in which the CAUP Dark Side team is involved. Keywords: Fundamental cosmology; dark energy; fundamental couplings; consistency tests PACS: 98.80.Cq, 98.80.-k, 98.80.Jk, 98.70.Vc", "pages": [ 1 ] }, { "title": "THE DARK SIDE OF THE UNIVERSE", "content": "In the middle of the XIX century Urbain Le Verrier and others mathematically discovered two new planets by insisting that the observed orbits of Uranus and Mercury agreed with the predictions of Newtonian physics. The first of these- which we now call Neptune-was soon observed by Johann Galle and Heinrich d'Arrest. However, the second (dubbed Vulcan) was never found. We now know that the discrepancies in Mercury's orbit were a consequence of the fact that Newtonian physics can't adequately describe Mercury's orbit, and accounting for them was the first success of Einstein's General Relativity. Over the past several decades, cosmologists have mathematically discovered two new components of the universe-which we have called dark matter and dark energy-which have so far not been directly detected. Whether they will prove to be Neptunes or Vulcans remains to be seen but even their mathematical discovery highlights the fact that the standard Λ CDM paradigm, despite its phenomenological success, is at least incomplete. Something similar applies to particle physics, where to some extent it is our confidence in the standard model that leads us to the expectation that there must be new physics beyond it. Neutrino masses, dark matter and the size of the baryon asymmetry of the universe all require new physics, and-significantly-all have obvious astrophysical and cosmological implications. Recent years have made it clear that further progress in fundamental particle physics will increasingly depend on progress in cosmology. One must therefore carry out explicit consistency tests of the standard cosmological model and search for evidence of new physics beyond it.", "pages": [ 1 ] }, { "title": "FUNDAMENTAL SCALAR FIELDS", "content": "After a quest of several decades, the recent LHC evidence for a Higgs-like particle [1, 2] finally provides strong evidence in favour of the notion that fundamental scalar fields are part of Nature's building blocks. A pressing follow-up question is whether the associated field has a cosmological role (or indeed if there is some cosmological counterpart). At the same time, fundamental scalar fields already play a key role in most paradigms of modern cosmology. Inter alia they are routinely invoked to describe Even more important than each of these paradigms is the fact that they don't occur alone: whenever a scalar field plays one of the above roles, it will also leave imprints in other contexts that one can look for. Three obvious examples can be given do not in fact occur for realistic defect networks. Although this complementary point is often overlooked, it will be crucial for future consistency tests.", "pages": [ 1, 2 ] }, { "title": "VARYING FUNDAMENTAL COUPLINGS", "content": "Nature is characterized by a set of physical laws and fundamental dimensionless couplings, which historically we have assumed to be spacetime-invariant. For the former this is a cornerstone of the scientific method (indeed, it's hard to imagine how one could do science at all if it were not the case), but for the latter it is only a simplifying assumption without further justification. These couplings determine the properties of atoms, cells, planets and the universe as a whole, so it's remarkable how little we know about them-in fact we have no 'theory of constants', that describes their role in physical theories or even which of them are really fundamental. If they vary, all the physics we know is incomplete. Fundamental couplings are expected to vary in many extensions of the current standard model. In particular, this will be the case in theories with additional spacetime dimensions, such as string theory. Interestingly, the first generation of string theorists had the hope that the theory would ultimately predict a unique set of laws and couplings for low-energy physics. However, following the discovery of the evidence for the acceleration of the universe this hope has been pragmatically replaced by an 'anything goes' approach, sometimes combined with anthropic arguments. Regardless of the merit of such approaches, experimental and observational tests of the stability of these couplings may be their best route towards a testable prediction. It goes without saying that a detection of varying fundamental couplings will be revolutionary: it will immediately prove that the Einstein Equivalence Principle is violated (and therefore that gravity can't be purely geometry), that there is a fifth force of nature, and so on. Moreover, even improved null results are important and useful. The simple way to understand this is to realize that the natural scale for cosmological evolution of one of these couplings (driven by a fundamental scalar field) would be Hubble time. We would therefore expect a drift rate of the order of 10 -10 yr -1 . However, current local bounds, coming from atomic clock comparison experiments, are 6 orders of magnitude stronger [3, 4]. Recent astrophysical evidence from quasar absorption systems [5] observed with HIRES/Keck and UVES/VLT suggests a parts-per-million spatial variation of the finestructure constant α at low redshifts; although no known model can explain such a result without considerable fine-tuning, it should also be said that there is also no identified systematic effect that can explain it [6]. One possible cause for concern (with these and other results) is that almost all of the existing data has been taken with other purposes in mind (and subsequently reanalized for this purpose), whereas this kind of measurements needs customized analysis pipelines and wavelength calibration procedures beyond those supplied by standard pipelines [7]. An ongoing ESO UVES Large Programme dedicated to fundamental physics will soon provide further measurements [8]. This is the only large program (so far) dedicated to varying constants, carrying out observations of a selected sample with optimized methodology. Although the programme was selected before the α dipole was known (and it is therefore not optimized for it), it will be able to test it. In the short term the PEPSI spectrograph at the LBT can also play a role here, and in the longer term a new generation of high-resolution, ultra-stable spectrographs like ESPRESSO (for the VLT) and CODEX (or its future incarnation, for the E-ELT) will significantly improve the precision of these measurements and should be able to resolve the current controversy. A key technical improvement will be that ultimately one must do the wavelength calibration with laser frequency combs. In theories where a dynamical scalar field yields varying α , the other gauge and Yukawa couplings are also expected to vary. In particular, in Grand Unified Theories the variation of α is related to that of energy scale of Quantum Chromodynamics, whence the nucleon masses necessarily vary when measured in an energy scale that is independent of QCD (such as the electron mass). It follows that we should expect a varying proton-toelectron mass ratio, µ = mp / me , which can be probed with molecular Hydrogen [9] and other molecules. These use the fact that molecular vibrational and rotational transitions depend on the reduced mass of the molecule, and the dependence is different for different transitions. Obviously, the specific relation between α ( z ) and µ ( z ) will be completely model-dependent, but this is a blessing rather than a curse: astrophysical measurements of both provide us with a unique discriminating tool between different unification models. It's worth emphasizing that while molecular Hydrogen measurements do probe µ , those involving more complicated molecules are probing an effective nucleon-toelectron mass ratio, and this will be proportional to µ (the proportionality factor being a pure number) only in the absence of composition-dependentforces. But again, this is provides us with a unique opportunity: by simultaneously doing these measurements with several molecules ( H 2, HD , ammonia, etc), which occasionally may be found in the same cloud, one will ultimately be able to constrain possible composition-dependent couplings. This is a direct astrophysical test of the Equivalence Principle, which is not feasible with current facilities but should be within the reach of ESPRESSO and CODEX. At much higher redshifts, the Cosmic Microwave Background is an ideal, clean probe of a varying finestructure constant. A changed α will impact the ionization history of the universe: the energy levels and binding energies are shifted, and the Thomson cross-section is proportional to α 2 . Having said this, bounds are relatively weak due to degeneracies with other cosmological parameters, and the percent barrier has only recently been broken [10]. For the reasons explained above it is too naive to consider variations of a single quantity, and fortunately the CMB data is becoming good enough for coupled variations to be constrained, despite the degeneracies; for example [11] constrained simultaneous variations in α and the gravitational sector. The latter can be interpreted as constraints on Newton's gravitational constant G if one chooses units in which the particle masses are fixed. A cosmological constant is negligible at recombination, but a dynamical, tracking scalar field can induce significant α variations. An example is the class of early dark energy models [12], linearly coupled to electromagnetism [13]. One can in fact constrain the coupling between the putative scalar field and electromagnetism, independently (and on a completely different scale) from what is done in local tests [14]. The local bound is (conservatively) at the 10 -3 level, and our constraint is only about 20 times weaker, which is a testimony to the CMB sensitivity. (As a comparison, lensing constraints on the Eddington parameter γ [15] are currently 2500 times weaker than those from the Cassini bound [16].) The recent CMB measurements from WMAP and arcminute angular scales (from ACT and SPT) suggest that the effective number of relativistic degrees of freedom is larger than the standard value of and inconsistent with it at more than two standard deviations. We have recently shown [17] that, if one assumes this standard value, these same CMB datasets significantly improve previous constraints on α , with hinting also to a more than two-sigma deviation from the current, local, value. A significant degeneracy is present between α and N eff, and when variations in the latter are allowed the constraints on α are consistent with the standard value. Again it's worth stressing that deviations of either parameter from their standard values would imply the presence of new, currently unknown physics. Forthcoming Planck data should improve these constraints. Moreover, once this data is available, and additional set of Equivalence Principle tests will also become possible. Although QSO spectroscopy and the CMB are by now the two standard methods to probe the stability of fundamental couplings, they are merely the tip of the iceberg. Many compact astrophysical objects can also be used for this purpose, and in particular to test for their possible environmental dependence. Recent progress includes work on Naturally all these systems are sensitive to several dimensionless couplings (and not just α ) so they will soon provide us with further opportunities to constrain unification scenarios.", "pages": [ 2, 3 ] }, { "title": "DYNAMICAL DARK ENERGY", "content": "Observations suggest that the universe is dominated by an energy component whose gravitational behavior is quite similar to that of a cosmological constant. Its value is so small that a dynamical scalar field is arguably a more likely explanation. Such a field must be slowrolling (which is mandatory for p < 0) and be dominating the dynamics around the present day. It follows [21] that couplings of this field to the rest of the model lead to potentially observable long-range forces and time dependencies of the constants of nature. The above point is worth emphasizing because it is often misunderstood. Any scalar field couples to gravity. It couples to nothing else only if there is an ad-hoc global symmetry which suppresses couplings to the rest of the Lagrangian. (In that case, only derivatives and derivative couplings will survive.) However, such symmetries are hard to come by. Specifically, quantum gravity effects don't respect global symmetries, and there are no unbroken global symmetries in string theory. Therefore, if one goes to great lenghts to justify a coupling one is in fact reversing the burden of proof: the expectation is that scalars in the theory will couple to the rest of the world in any manner not prevented by symmetry principles. (It is the absence of couplings that requires proper justification.) Standard observables such as supernovae are of limited use as dark energy probes [22]. A clear detection of varying w ( z ) is crucial, given that we know that w ∼-1 today. Since the field is slow-rolling when dynamically important (close to the present day), a convincing detection of a varying w ( z ) will be tough at low redshift, and we must probe the deep matter era regime, where the dynamics of the hypothetical scalar field is fastest. Varying fundamental couplings are ideal for probing scalar field dynamics beyond the domination regime [13]: such measurements can presently be made up to redshift z ∼ 4, and future facilities such as the E-ELT may be able to significantly extend this redshift range. Importantly, even null measurements of varying couplings can lead to interesting constraints on dark energy scenarios [23, 24]. Wehaverecently studied [25], using Principal Component Analysis techniques, the impact of ESPRESSO and CODEX in constraining dark energy through measurements of varying fundamental couplings. In the case of CODEX, a reconstruction using quasar absorption lines is expected to be more accurate than using supernovae data (its key advantage being huge redshift lever arm), and even ESPRESSO can provide a significant contribution. Since the two types of measurements probe different (but overlapping) redshift ranges, combining them leads to a more complete picture of the evolution of the equation of state parameter, and these can realize the prospect of a detailed characterization of dark energy properties almost all the way up to redshift 4. Although the most obvious way to proceed is to combine the two datasets, we should also point out that they can be used separately to provide independent reconstructions. Comparing the two reconstructions will in itself provide a consistency test, specifically for the assumption on the coupling between the scalar field and electromagnetism. In this case one can also obtain a measurement for the coupling parameter, which can be compared to that obtained from the CMB (as discussed above) and those obtained from local tests. Again, null results can also be constraining-see [23, 24] for tight constraints on a broad range of slow-roll models. Dark energy reconstruction using varying fundamental constants does in principle require a mild assumption on the field coupling, but there are in-built consistency checks, so that inconsistent assumptions can be identified and corrected. An explicit example of an incorrect assumption that leads to an observation inconsistency is discussed in [26]. In this regard CODEX at the EELT, with its ability to carry out the Sandage-Loeb test [27, 28], will play a crucial role [29]. Interesting synegies also exist between these ground-based spectroscopic methods and Euclid, which need to be further explored.", "pages": [ 3, 4 ] }, { "title": "THE QUEST FOR REDUNDANCY", "content": "Whichever way one finds direct evidence for new physics, it will only be believed once it is seen through multiple independent probes. This was manifest in the case of the discovery of the recent acceleration of the universe, where the supernova results were only accepted by the wider community once they were confimed through CMB, large-scale structure and other data. It is clear that history will repeat itself in the case of varying fundamental couplings and/or dynamical dark energy. It is therefore crucial to develop consistency tests-in other words, astrophysical observables whose behaviour will also be non-standard as a consequence of either or both of the above. The temperature-redshift relation, is a robust prediction of standard cosmology; it assumes adiabatic expansion and photon number conservation, but it is violated in many scenarios, including string theory inspired ones. At a phenomenological level one can parametrize deviations to this law by adding an extra parameter, say Measurements of the SZ effect at resdshifts z < 1, combined with spectroscopic measurements at redshifts z ∼ 2 -3 yield the direct constraint [30] Our recent work [31, 32] has shown that forthciming data from Planck, ESPRESSO and CODEX will lead to much stronger constraints: We emphasize that estimates of all these gains rely on quite conservative on the number of sources (SZ clusters and absorption systems, respectively) where these measurements can be made. If the number of such sources increases, future constraints can be correspondingly stronger. The distance duality relation, is an equally robust prediction of standard cosmology; it assumes a metric theory of gravity and photon number conservation, but is violated if there's photon dimming, absorption or conversion. At a similarly phenomenological level one can parametrize deviations to this law by adding an extra parameter, say In this case, current constraints are [33] and improvements are similarly expected from Euclid, the E-ELT and JWST. In fact, in many models where photon number is not conserved the temperature-redshift relation and the distance duality relation are not independent. With the above parametrizations it's easy to show [32] that but one can in fact further show that a direct relation exists for any such model, provided the dependence is in redshift only (models where there are frequencydependent effects are more complex). This link allows us [32] to use distance duality measurements to further constrain β , and we recently found up to a redshift z ∼ 3, which is a 40% improvement on the previous constraint. With the next generation of space and ground-based experiments, these constraints can be further improved (as discussed above) by more than one order of magnitude.", "pages": [ 4, 5 ] }, { "title": "OUTLOOK", "content": "The observational evidence for the acceleration of the universe demonstrates that canonical theories of cosmology and particle physics are incomplete, if not incorrect. Several few-sigma hints of new physics are emergine, but so far these are smoke without a smoking gun; it's time to actively search for the gun. The forthcoming generation of high-resolution ultrastable spectrographs will play a key role in this endeavour, by enabling a new generation of precision consistency tests of the standard cosmological paradigm and its extensions. Some further exciting possibiblites, only mentioned briefly in this contribution, include direct astrophysical Equivalence Principle tests and E-ELT measurements of the redshift drift. Last but not least, there are important synergies between ground and space facilities, and in particular between the E-ELT andEuclid, which require further study: together, they will be a unique tool to study fundamental physics and gravity.", "pages": [ 5 ] }, { "title": "ACKNOWLEDGMENTS", "content": "I am grateful to Mariusz Dabrowski and the rest of the Multicosmofun'12 organizers for their hospitality and for organizing such an enjoyable and productive meeting. This work has been done in the context of the grant PTDC/FIS/111725/2009(The Dark Side of the Universe, funded by FCT), with additional support from grants PESSOA 2012/2013 Proc. 441.00 (Testing Fundamental Physics with Planck, finded in Portugal by FCT) and PP-IJUP2011-212 (Astrophysical Tests of Fundamental Physics, funded by U. Porto and Santander Totta). Many interesting discussions with the rest of CAUP's Dark Side team and our collaborators elsewhere have shaped my views on this subject, and are gratefully acknowledged. The work of CJM is supported by a Ciência2007 Research Contract, funded by FCT/MCTES (Portugal) and POPH/FSE (EC).", "pages": [ 5 ] }, { "title": "REFERENCES", "content": "063515 (2012).", "pages": [ 6 ] } ]

[ { "title": "Classical and quantum Big Brake cosmology for scalar field and tachyonic models", "content": "A.Yu. Kamenshchik 1 and S. Manti 2", "pages": [ 1 ] }, { "title": "Abstract", "content": "We study a relation between the cosmological singularities in classical and quantum theory, comparing the classical and quantum dynamics in some models possessing the Big Brake singularity - the model based on a scalar field and two models based on a tachyon-pseudo-tachyon field . It is shown that the effect of quantum avoidance is absent for the soft singularities of the Big Brake type while it is present for the Big Bang and Big Crunch singularities. Thus, there is some kind of a classical - quantum correspondence, because soft singularities are traversable in classical cosmology, while the strong Big Bang and Big Crunch singularities are not traversable.", "pages": [ 1 ] }, { "title": "1 Introduction", "content": "The cosmological singularities constitute one of the main problems of modern cosmology. 'Traditional' or 'hard' singularities such as the Big Bang and the Big Crunch singularities are associated with the zero volume of the universe (or of its scale factor), and with infinite values of the Hubble parameter, of the energy density and of the pressure. The discovery of the cosmic acceleration stimulated the development of 'exotic' cosmological models of dark energy; some of these models possess the so called soft or sudden singularities characterized by the finite value of the radius of the universe and of its Hubble parameter. Such singularities sometimes arise quite unexpectedly in some dark energy models. One of examples of such singularities is the Big Brake singularity arising in a specific tachyon model [1]. Tachyons (Born-Infeld fields) is a natural candidate for a dark energy because their pressure is negative. The toy tachyon model [1], proposed in 2004, has two particular features: tachyon field transforms itself into a pseudo-tachyon field; the evolution of the universe can encounter a new type of singularity - the Big Brake singularity. The Big Brake singularity is a particular type of the so called 'soft' cosmological singularities - the radius of the universe is finite, the velocity of expansion is equal to zero, the deceleration is infinite. Amusingl, the predictions of the model do not contradict observational data on supenovae of the type Ia [2, 3] and moreovoer, the Big Brake singularity is indeed a very special one - it is possible to cross it [3]. Naturally some open questions arise: what can we say about other soft singularities - is it possible to cross them ? An attempt to study this question was undertaken in [4]. Antoher question is as follows: What can tell us the Quantum cosmology on the Big Brake singularity and other soft singularities ? This question was studied in [5] and here we present briefly the main results of this work.", "pages": [ 1, 2 ] }, { "title": "2 Description of the tachyon model", "content": "We consider a flat Friedmann universe The tachyon Lagrange density is The energy density The pressure The Friedmann equation is as usual The equation of motion for the tachyon field is In our model [1] where k and Λ > 0 are the parameters of the model. The case k > 0 is more interesting. In this case some trajectories (cosmological evolutions) finish in the infinite de Sitter expansion. In other trajectories the tachyon field transforms into the pseudotachyon field with the Lagrange density, energy density and positive pressure given by What happens with the Universe after the transformation of the tachyon into the pseudotachyon ? It encounters the Big Brake cosmological singularity.", "pages": [ 2, 3 ] }, { "title": "3 The Big Brake cosmological singularity and other soft singularities", "content": "The Big Brake singularity is characterized by the following formulae: If ˙ a ( t BB ) = 0 it is a more general soft singularity. /negationslash", "pages": [ 3 ] }, { "title": "4 Crossing the Big Brake singularity and the future of the universe", "content": "At the Big Brake singularity the equations for geodesics are regular, because the Christoffel symbols are regular (moreover, they are equal to zero). Is it possible to cross the Big Brake ? Let us study the regime of approaching the Big Brake. Analyzing the equations of motion we find that approaching the Big Brake singularity the tachyon field behaves as Its time derivative s ≡ ˙ T behaves as the cosmological radius is its time derivative is and the Hubble variable is All these expressions can be continued in the region where t > t BB ,which amounts to crossing the Big Brake singularity. Only the expression for s is singular but this singularity is integrable and not dangerous. Once reaching the Big Brake, it is impossible for the system to stay there because of the infinite deceleration, which eventually leads to the decrease of the scale factor. This is because after the Big Brake crossing the time derivative of the cosmological radius and Hubble variable change their signs. The expansion is then followed by a contraction, culminating in the Big Crunch singularity.", "pages": [ 4 ] }, { "title": "5 The classical and quantum dynamics in the scalar field model with a soft singularity", "content": "One of the simplest cosmological models revealing the Big Brake singularity is the model based on the anti-Chaplygin gas with an equation of state Then At a = a ∗ = ( B A ) 1 / 6 the universe encounters the Big Brake singularity. The scalar field model reproducing the cosmological evolution of the model based on the anti-Chaplygin gas has the potential [6] We shall study the model with a more simple potential, which has basically the same qualitative behaviour We shall study first the classical dynamics of this model. Here are the main results of our analysis. For completeness we can add that the evolutions with the negative values of the scalar field belong to two classes - first, an infinite expansion beginning from the Big Bang and second, the evolutions obtained by the time reversion of those of the first class, which are contracting and end in the Big Crunch singularity.", "pages": [ 4, 5 ] }, { "title": "6 Quantum dynamics - the Wheeler - DeWitt equation", "content": "Applying the Hamiltonian formalism to the system Gravity + Scalar field, we obtain the super-Hamiltonian constraint and the implementation of the Dirac quantization procedure gives the WheelerDeWitt equation Requiring the normalizability of the wave function of the universe we come to the conclusion that this wavefunction should vanish at ϕ → 0. This value classically corresponds to a soft singularity. Does it indicate the presence of a quantum avoidance of the singularity ? No. The physical sense has the wave function depending on the physical degrees of freedom, obtained after the gauge fixing choice, which simultaneously introduces the time parameter [7]. If we choose the Hubble parameter as a new time parameter τ ≡ -H , then its conjugated is a 3 . The reduction of the initial set of variables to the smaller set of physical degrees of freedom implies the appearance of the Faddeev-Popov determinant which is equal to the Poisson bracket of the gauge-fixing condition and the constraint. This Faddeev-Popov determinant will be proportional to the potential, which is singular at ϕ = 0. Thus, the quantum probability to find the universe, crossing the soft singularity is different from zero. The 'hard' Big Bang and Big Crunch singularities a = 0 correspond to ϕ = ∞ . To provide the normalizability of the wave function one should have the integral on the values of the scalar field ϕ convergent, when | ϕ | → ∞ . That means that, independently of details connected with the gauge choice, not only the wave function of the universe but also the probability density of scalar field values should decrease rather rapidly when the absolute value of the scalar field is increasing. Thus, in this case, the effect of the quantum avoidance of the classical singularity is present.", "pages": [ 5, 6 ] }, { "title": "7 The quantum cosmology of the tachyon model", "content": "The Wheeler-DeWitt equation for the system Gravity + Pseudotachyon is Let us first consider the case of the constant potential W ( T ) = W 0 . The Big Brake singularity corresponds to | p T | = a 3 √ W 0 . It is convenient to work in the momenta representation ψ ( a, p T ). Then requirement of the well-definiteness of becomes algebraic and implies It does not mean that the probability of finding of the universe crossing the Big Brake is equal to zero because the corresponding Faddeev-Popov determinant contains a singular factor One can see that in the case of the trigonometrical potential there is no need to require even the disappearance of ψ ( a, T ) at the Big Brake. Thus, there is no a quantum avoidance effect of the Big Brake singularity in the tachyon models. There is the effect of quantum avoidance for the Big Bang and Big Crunch singularities, because these singularities correspond to and the corresponding probability density should tend to zero.", "pages": [ 6, 7 ] }, { "title": "8 Conclusions and discussion", "content": "It wass shown that the effect of quantum avoidance is absent for the soft singularities of the Big Brake type while it is present for the Big Bang and Big Crunch singularities. Thus, there is some kind of a classical - quantum correspondence, because soft singularities are traversable in classical cosmology, while the strong Big Bang and Big Crunch singularities are not traversable. We are grateful to A.O. Barvinsky and C. Kiefer for fruitful discussions and to P.V. Moniz and M. Bouhmadi-Lopez for useful correspondence. This work was partially supported by the RFBR grant 11-02-00643.", "pages": [ 7 ] } ]

[ { "title": "Michał Kamionka", "content": "Astronomical Institute, University of Wrocław ul. Kopernika 11, 51-622 Wrocław, Poland. e-mail: [email protected] Abstract. Newconstraints on previously investigated Palatini cosmological models [1] have been obtained by adding Gamma Ray Burst (GRB) data [2]. Keywords: PACS: 98.80.-k, 04.50.Kd modified gravity, cosmological simulations, dark energy theory, cosmic singularity", "pages": [ 1 ] }, { "title": "COSMOLOGYFROMTHE GENERALIZED EINSTEIN EQUATIONS", "content": "Recently, we have investigated cosmological applications and confronted them against astrophysical data the following class of gravitational Lagrangians: within the first-order Palatini formalism [1]. Here L d = -1 2 g µν ∂µφ∂νφ is a scalar (dilaton-like) field Lagrangian non-minimally coupled to the curvature and Lmat represents perfect fluid Lagrangian for a dust (non-relativistic) matter. The numerical parameters α , β , δ , σ are to be determined by astrophysical data. Applying (Palatini) variational principle compiled with flat FLRW metric one arrives to general Friedmann equation: where H = ˙ a a denotes the Hubble parameter related to the FLWR cosmic scale factor. This reconstructs the Λ CDM model under the choice f = R -2 Λ , F = 0, which is the limit α = 0, δ = -1, β = 2 Λ . Setting further Λ = 0 leads to Einstein-de Sitter (decelerating) universe. We want to recall that the generalized Friedmann equation under the form: (which is always the case for the Palatini formalism) leads to one-dimensional particle like Newton-type dynamics which is fully described by the effective potential V ( a ) = -1 2 a 2 G ( a ) . This relevant property allows us to compare various cosmological models on the level at the effective potential functions and the corresponding phase-space diagrams. Particularly, the dynamics of Λ CDM model is described by V Λ CDM = -1 2 ( Λ a 2 + η a -1 ) where η is a density parameter for the dust matter. As it was shown in [1] the equation (2) leads to two classes of cosmological models implemented by different solutions of generalized Einstein equations.", "pages": [ 1 ] }, { "title": "Model I", "content": "Solving equations of motion by one obtains generalized Friedmann equation under the form where are dimensionless (density like) parameters.", "pages": [ 1 ] }, { "title": "Model II", "content": "Another cosmological model can be determined by which leads to where now Both models have Ω 0 , m , Ω 0 , α , Ω 0 , β , δ as free parameters. By the normalization condition H ( 0 ) = H 0, only three of them are independent ( H 0 denotes the Hubble constant).", "pages": [ 1, 2 ] }, { "title": "FITTING PARAMETERS OF THE MODELS", "content": "In order to estimate the parameters of our models we use a sample of N = 557 supernovae (SNIa) data [3], the observational H ( z ) data [4], the measurements of the baryon acoustic oscillations (BAO) from the SDSS luminous red galaxies [5], information from CMB [6] and, as an adition to [1], information coming from observations of GRB [2]. The entire likelihood function LTOT is characterized by: We have assumed flat prior probabilities for all model's parameters. We also assumed that H 0 = 74 . 2 [ kms -1 Mpc -1 ] [8]. The likelihood function is defined in the following way: where: σ i is the total measurement error, µ obs i = mi -M is the measured value ( mi -apparent magnitude, M -absolute magnitude of SNIa), µ theor i = 5log 10 DLi + M = 5log 10 dLi + 25, M = -5log 10 H 0 + 25 and DLi = H 0 dLi , where dLi is the luminosity distance given by dLi =( 1 + zi ) c ∫ z i 0 dz ' H ( z ' ) (with the assumption k = 0). In this paper the likelihood as a function independent of H 0 has been used (which is obtained after analytical marginalization of formula (11) over H 0). For the H ( z ) data the likelihood function is given by: where H ( zi ) is the Hubble function, Hi denotes observational data. For BAO A parameter data the likelihood function is characterized by: where A theor = √ Ω m , 0 ( H ( z A ) H 0 ) -1 3 [ 1 z A ∫ z A 0 H 0 H ( z ) dz ] 2 3 and A obs = 0 . 469 ± 0 . 017 for zA = 0 . 35. We also use constraints coming from CMB temperature power spectrum, ie. CMB R shift parameter [7], which is related to the angular diameter distance ( DA ( z ∗ ) ) to the last scattering surface: The likelihood function has the following form: where R obs = 1 . 725 and σ -2 A = 6825 . 27 for z ∗ = 1091 . 3 [6]. The likelihood function for GRB data is defined as: The mode of joined posterior pdf as well as mean (together with 68% credible interval) of marginalized posterior pdf were calculated, by means of Markov Chains Monte Carlo analysis, using free accessible CosmoNest code [9] which has been modified for our purpose. The results are presented on fig. 2,3. The numerical values of best fitted parameters for two our models as well as for Λ CDM are collected in table 1: the previous estimations without the GRB data (i.e. SNIa, H(z) and BAO and CMB) are shown in top part of the table. The new estimations including the GRB data occupy bottom part of the table. Quality of the estimation can be visualized on the Hubble's diagram (fig. 1). Both of our models are in good agreement in the observational data.", "pages": [ 2 ] }, { "title": "CONCLUSIONS", "content": "In this paper we continued and completed analysis of new cosmological models which were previously described and investigated in our paper [1]. Adding GRB data [2] allowed us to obtain better constraints of parameter Ωα which wasn't present previously. As it can be seen on the potential plots (fig. 4,5, both models dynamically mimics Λ CDM model from the Big Bang singularity until the present time. Discrepancies will appear in the near future. Both of our models predict the final finite size and finite time singularities (at a = 1 . 673 for the model I, and at a = 1 . 559 for the model II). However, comparing with our previous simulations, adding new GRB data has changed properties of the model II (Big Bounce is now replaced by Big Bang). /Minus 4", "pages": [ 2, 3 ] }, { "title": "ACKNOWLEDGMENTS", "content": "M.K. is supported by the Polish NCN grant PRELUDIUM 2012/05/N/ST9/03857.", "pages": [ 3 ] }, { "title": "models I - the parameters estimated without GRB data", "content": "indicators similar to Type Ia Supernovae?',(2012), arXiv:1205.2954 . anisotropy experiments', Mon. Not. Roy. Astron. Soc. 291 , L33-L41 (1997), astro-ph/9702100 . P. Mukherjee, D. Parkinson, P. S. Corasaniti, A. R. Liddle, M. Kunz, 'Model selection as a science driver for dark energy surveys', Mon. Not. Roy. Astron. Soc. 369 , 17251734 (2006), astro-ph/0512484 ; -", "pages": [ 4 ] } ]

[ { "title": "Jason Kumar", "content": "Department of Physics and Astronomy, University of Hawaii, Honolulu, HI, 96822 USA Abstract. We consider the impact of relaxing some typical assumptions about dark matter interactions, including isospininvariance, elastic scattering and contact interactions. We show that detection strategies with neutrino detectors, gamma-ray searches, new direct detection experiments and collider searches can all provide complementary information. We argue that data from many such strategies may be necessary to gain a more complete understanding of dark matter interactions. Keywords: dark matter PACS: 95.35.+d", "pages": [ 1 ] }, { "title": "INTRODUCTION", "content": "The idea that experiments could search for dark matter beyond its gravitational effects was proposed almost 30 years ago [1]. Since that time, the search for dark matter has become an experimental reality, and dark matter research is a field which is well beyond its infancy. Data is arriving from a variety of detectors using different strategies, and some very interesting hints which may suggest the presence of dark matter have been seen. WIMPs of the MSSM are one of the most appealing dark matter candidates, and dark matter search results are often interpreted with this model in mind. Aside from the theoretical appeal of LSP WIMPs, this model is relatively constrained, bringing with it a number of assumptions which simplify the tasks of interpreting the data and comparing data from different detectors. Typical assumptions which one makes include: But as new data comes in, and potential hints of dark matter are seen by some detectors, it has becoming increasingly clear that it may be difficult to reconcile the data from all of these experiments under the assumptions given above. It is thus necessary to consider how the role of different detection strategies changes when the above assumptions are relaxed. 1 We will argue that, in the absence of the above assumptions, it is necessary to combine the results of several experiments using novel detection strategies in order to study the nature of dark matter interactions.", "pages": [ 1 ] }, { "title": "DARK MATTER INTERACTIONS", "content": "It is useful to briefly review the general structure of non-relativistic dark matter-nucleus scattering. We allow for the possibility that the outgoing dark sector particle may have a mass m X ' which is different from the dark matter mass mX (we will assume m X ' ≥ mX ). But since we assume that both incoming and outgoing particles are non-relativistic, we have d mX ≡ m X ' -mX glyph[lessmuch] mX . We may then write the reduced mass as m A = mXm A / ( mX + m A ) for both the incoming and outgoing system, where m A is the nucleus mass. For cold dark matter, one expects dark matter to coherently scatter off the nucleus as a whole. The dark matter-nucleus differential scattering cross-section is where v is the relative velocity of the incoming particles, pout = m A v √ 1 -2 d mX / m A v 2 is the outgoing momentum in center of mass frame, M is the matrix element for dark matter-nucleus scattering, N is the number of initial states, and the sum is over all initial and final states. The nuclear form factor is given by F A ( ER ) , and the maximum (minimum) nucleus recoil energy which is kinematically possible in two-body scattering is given by E + ( E -): The differential rate of scattering events is then given by [3] where h T , X are the number densities of the target material and of dark matter, respectively. The dark matter velocity distribution is given by f ( u ) , where u is the velocity of a dark matter particle relative to the detector when it is far from the solar system. Note that u and v can be different, since an infalling dark matter particle will gain speed before scattering due to the gravitational potential; we then have ( 1 / 2 ) mX ( v 2 -u 2 ) = | Vgrav ( r ) | , where Vgrav ( r ) is the gravitational potential energy of the dark matter at r . Usually u ≈ v , unless scattering occurs within the sun. The lower integration limit is given by Emin = max [ E th , E -] , where E th is the threshold recoil energy. Thus far we have relied only on kinematics, making no assumptions about dark matter particle physics. Henceforth, we will focus only on the case where dark matter interactions are spin-independent. In that case, the dark matter coupling scales as the number of nucleons and M GLYPH<181> [ fpZ + fn ( A -Z )] , where fp , n parameterize the relative strength of dark matter interactions with protons and neutrons, respectively. If scattering arises through exchange of a single mediating particle with mass M ∗ , then we may write the matrix element as where G is a dimension-2 function of the masses and Mandelstam variables. The momentum transfer in the hard scattering process, q , is give by q 2 ∼ m 2 X , -2 m A ER , m 2 X for s -, t - and u -channel exchange, respectively. For t -channel exchange, the shape of the recoil spectrum will depend on whether the propagator is dominated by the messenger mass or the momentum transfer. Since | t | < 4 m 2 A v 2 , scattering interactions will be short-ranged if M ∗ > ∼ 1 GeV. We will also assume that scattering is velocity-independent, which is the case if there are no cancelations in the leading order term of the matrix element. G is determined, up to coupling constants and other numerical factors, by the normalization of the incoming and outgoing states and by the derivatives (if any) in the matrix element. We find that G GLYPH<181> mXm A for fermionic dark matter, or for scalar dark matter which couples to the mediating particle through a derivative. For scalar dark matter (examples include [4]) without a derivative coupling, we can also have terms of the form G GLYPH<181> M ∗ m A . If dark matter scatters elastically, then we may set d mX = 0. If | t | glyph[lessmuch] M 2 ∗ , then we may write where m p is the dark matter-proton reduced mass and s p is the dark matter-proton spin-independent scattering crosssection. One can make motivated assumptions about h X , f ( u ) (from astrophysics) and F A ( ER ) (from nuclear physics). The only parameter left is fn / fp ; given a choice of this parameter, a bound on the event rate can be directly translated into a bound on s p . Conversely, an excess event rate can be directly translated into a preferred region for s p .", "pages": [ 1, 2 ] }, { "title": "ISOSPIN-VIOLATING DARK MATTER", "content": "Typically, bounds are presented in terms of a 'normalized-to-nucleon\" scattering cross-section, s Z N . This is the dark matter-nucleon scattering cross-section which would be inferred under the assumption fn = fp , and provides an easy way to compare the results of experiments using different targets. But the quantity which actually should agree among different experiments is the dark matter-proton scattering cross-section, s p , which is related to s Z N by the relation The sum is over isotopes labeled by i , and h i is the natural abundance of the i th isotope (we have assumed that F A i ( ER ) does not vary much between isotopes). FZ thus depends only on fn / fp , and as expected, FZ = 1 for fn / fp = 1. For isospin-violating dark matter (IVDM) [5, 6], fn / fp is a parameter which is set by the details of the particle physics model. Figure 1 [6] shows the favored regions of DAMA [7, 8] (3 s ), CoGeNT [9, 10] (90% CL) and CRESST [11] (2 s ), and 90% CL exclusion contours from CDMS [12, 13], XENON10 [14], XENON100 [15, 16, 17], SIMPLE [18] and COUPP [19]. The left panel plots these regions for fn / fp = 1, while the right panel assumes fn / fp = -0 . 7. One can see the dramatic effect of isospin-violating interactions on the relative sensitivity of various direct detection experiments. For fn / fp = -0 . 7, the CoGeNT and DAMA preferred regions are brought closer into alignment, while the bounds from XENON10/100 are weakened, and no longer exclude the entire DAMA and CoGeNT regions. However, it is clear that this is not a complete solution. The marginal tension between bounds from CDMS (Soudan) and the preferred region of CoGeNT is not altered by isospin-violating interactions, since both detectors use a germanium target. Moreover, although the choice fn / fp = -0 . 7 would alleviate the tension with the xenon-based experiments, it creates tension with bounds from the SIMPLE experiment. And there is no choice of fn / fp for which the signal regions from DAMA, CoGeNT and CRESST are all consistent. There are many experimental uncertainties with the data at low-mass, some of which will be resolved soon. CoGeNT is gaining a better understanding of their surface-area contamination [20], which will likely move their preferred signal region to higher mass and lower cross-section. The understanding of the response of sodium and xenonbased detectors to low-energy recoils is improving. New germanium-based (Majorana, SuperCDMS) and xenon-based (LUX) detectors will soon provide even higher sensitivity at low-mass. It seems clear that the possibility of isospinviolating interactions will have a significant impact on the interpretation of this new data. If one treats fn / fp as a free parameter of the particle physics model, then one finds that one must use multiple direct detection experiments to get a handle on dark matter interactions. We may define the ratio which is the ratio of normalized-to-nucleon cross-sections which would be inferred by two detectors, using materials with Z 1 and Z 2 protons, under the assumption of isospin-invariant interactions. Signals at two dark matter experiments provide an experimental measurement of R and can be used to solve for fn / fp . If a detector using a material with Z 1 protons finds a dark matter signal, R [ Z 1 , Z 2 ] determines the range of sensitivity a second detector (using a material with Z 2 protons) would need to either potentially confirm or definitively exclude this signal for any choice of fn / fp . If an element has multiple isotopes, no choice of fn / fp can result in total destructive interference between proton and neutron couplings for all isotopes. For example, marginalizing over fn / fp , we find max { R [ Z 1 = Ge , Z 2 = Xe ]( fn / fp ) } ∼ 22; a xenon-based detector would need at most a factor of 22 greater sensitivity to confirm a signal at a germanium-based detector, for any choice of fn / fp . Data from LUX could thus provide a much more definitive test of the signals from DAMA, CoGeNT and CRESST, assuming the low-energy response can be well understood. Although IVDM is often discussed in relation to low-mass dark matter, it is a possibility which one must consider for any dark matter signal. Figure 2 [21] shows the sensitivities of XENON1T [22], Super-CDMS [23], MiniCLEAN, DEAP-3600 and CLEAN [24], for fn / fp = 1 , -0 . 7 , or -0 . 82. Also plotted in figure 2 is the estimated sensitivity of IceCube/DeepCore [25] with 180 days of data (assuming annihilation to the hard channel).", "pages": [ 2, 3, 4 ] }, { "title": "Dark Matter Searches with Neutrino Detectors", "content": "IceCube/DeepCore and other neutrino detectors search for the neutrino flux arising from dark matter annihilation in the core of the sun. If the sun is in equilibrium, bounds on the dark matter annihilation rate can be directly translated into bounds on the rate at which dark matter is captured by the sun through scattering off solar nuclei. 2 In turn, this yields a bound on the dark matter-proton scattering cross-section. The dark matter capture rate can also be derived from eqn. 3, where the threshold energy E th is the minimum recoil energy needed for dark matter to be captured. Dark matter capture occurs when a dark matter particle scatters and loses enough energy to become confined to an orbit around the sun. After many subsequent scatterings, it will eventually settle to the core of the sun, where it annihilates. But since many-body effects can be important for dark matter in large-radius orbits, one often requires captured dark matter to be confined to an orbit which will not exceed a maximum distance r 0 from the sun (typically taken to be the Jupiter-sun distance). The minimum recoil energy necessary for capture is thus given by where vesc . ( r 0 ) is the escape velocity for a particle at distance r 0 from the sun. One can then determine the capture rate from scattering against each element by integrating the differential capture rate (eq. 3) throughout the volume of the sun, accounting for the densities of different elements, as well as the sun's gravitational potential. If one computes the capture rate for each isotope of each element under the assumption of isospin-invariant interactions, then one obtains the capture rate for any choice of fn / fp by simply rescaling by the factor [ Z +( fn / fp )( A -Z )] 2 / A 2 . As shown in figure 2, when there is destructive interference between proton and neutron scattering, neutrino detectors can provide a nice complementary probe with a sensitivity which is comparable to that of direct detection experiments. This is because many direct detection experiments use heavy nuclei which have many more neutrons than protons; scattering in the sun is largely off lighter elements (including hydrogen), with fewer neutrons. Neutrino detectors can also provide sensitivity to low-mass dark matter which is comparable to that of direct detection experiments [26, 27, 28]. This can easily be understood from eq. 3. For mX ∼ 5 -10 GeV many direct detection experiments satisfy mX glyph[lessmuch] m A , E + ∼ 2 m 2 X v 2 / m A , E th = fixed, implying that the event rate vanishes for small mX . For neutrino detectors, however, there are many relatively light elements with mass comparable to the dark matter, implying E + ∼ 2 mXv 2 , while E th = ( 1 / 2 ) mXu 2 < E + . In fact, neutrino detectors can be sensitive to dark matter as light as 4 GeV; lighter dark matter will evaporate out of the sun [3]. We will consider a search for dark matter annihilation utilizing a 1 kT liquid scintillation (LS) detector with 2135 live-days of data (roughly the specifications of KamLAND). Neutrino detectors search for the charged lepton produced from ¯ n , n by a charged-current interaction. It has recently been realized that LS detectors can also be used for dark matter searches, because the direction of the charged lepton track in the scintillator can be determined from the timing of the first photons reaching the PMTs [29]. Moreover, the charged lepton flavor can be determined with very high efficiency. For low-energy nm (arising from low-mass dark matter annihilation), the muons produced by a chargedcurrent interaction will be short-ranged; there is thus no great advantage in searching for muons which may have been produced outside the detector volume. Instead, we focus on a search for e ± produced from electron (anti-)neutrinos, which have the advantage of a much smaller atmospheric neutrino background. In figure 3 [27] we plot the sensitivity which can be achieved with KamLAND's current data set (assuming annihilation to ¯ tt ), as well as the DAMA, CoGeNT and CRESST signal regions and the CDMS, XENON10 and XENON100 exclusion curves, assuming fn / fp = 1 , -0 . 7. We can see that KamLAND, with its current data set, can potentially probe the signals of DAMA and CoGeNT if fn / fp ∼-0 . 7.", "pages": [ 4, 5 ] }, { "title": "Complementary Bounds from Indirect Detection and the LHC", "content": "Indirect detection and collider bounds provide interesting complementary tests of IVDM models. The Xq → Xq scattering matrix element is related to the XX → ¯ qq annihilation matrix element and the ¯ qq → XX production matrix element by crossing symmetry. As a result, if an assumption is made about the form and flavor dependence of the scattering matrix element, then bounds on the total annihilation cross-section 〈 s A v 〉 or on the production cross-section can be directly translated into a bound on s p . The sensitivity of direct detection experiments using heavy targets, such as xenon, is minimized for the case of partial destructive interference between fn and fp . In that case, a signal seen at CoGeNT corresponds to a large value of s p ( O ( 10 -2 ) pb). This implies a large coupling to up- and down-quarks, which destructively interfere. The dark matter annihilation cross section to up- and down-quarks, or production crosssection from ¯ qq initial states, can then be greatly enhanced. A key point to note, though, is that the kinematics of these complementary processes can be different. For the annihilation process, the momentum transfer is roughly 2 mX , while for the production process, the momentum transfer is typically > 2 mX . We will consider the case where dark matter-nucleus scattering is spin- and velocity-independent, and arises from tree-level t -channel exchange with quarks. Furthermore, we will only consider the case where the resulting annihilation matrix element is not p -wave suppressed. For scalar dark matter (real or complex), these conditions are only satisfied by exchange of a scalar, while for Dirac fermion dark matter one must exchange a vector (these conditions can thus not be satisfied for Majorana fermion dark matter). In Table 1, we list the energy dependence of the scattering, annihilation and creation matrix elements, as well as the effective scattering operator in the limit where scattering is a contact interaction ( | t | glyph[lessmuch] M 2 ∗ ). If we assume that dark matter couples only to up- and down-quarks in the limit of elastic -1 contact scattering, then for any given choice of the contact operator and fn / fp , we can directly relate the annihilation and production cross-sections to the scattering cross-section.", "pages": [ 5, 6 ] }, { "title": "Fermi-LAT Searches of Dwarf Spheroidals", "content": "Tight bounds can be placed on dark matter annihilation in dwarf spheroidal galaxies with data from the FermiLAT [30, 31]. These bounds can be phrased in terms of the quantity where s A is the total annihilation cross-section and B f and dN f / dE are the branching fraction and photon spectrum, respectively, for annihilation to final state f . For a threshold energy E thr = 1 GeV, the 95% CL bound is F PP ≤ 5 . 0 + 4 . 3 -4 . 5 × 10 -30 cm 3 s -1 GeV -2 [31], where the uncertainties in the bound arise from systematic uncertainties in the density profile of the dwarf spheroidal galaxies. We computed dN f / dE for low-mass dark matter annihilation to the up- and down-quark channels (the two channels are essentially identical) using the Hawaii Open Supercomputing Center (HOSC). The corresponding bounds on 〈 s A v 〉 are plotted in the left panel of figure 4 [32], along with bounds from BESS-Polar II measurements of the anti-proton flux [33]. Systematic uncertainties can weaken these Fermi bounds by up to a factor of 2, or strengthen them by up to a factor of 10. In contrast, BESS-Polar II bounds may be weakened by up to a factor of 50 [34] due to systematic uncertainties in cosmic ray propagation, solar modulation, etc. These bounds can be directly translated into bounds on s p , under the assumption | t | , m 2 X glyph[lessmuch] M 2 ∗ . For the case when dark matter is a complex scalar ( fn / fp ∼-0 . 7), these bounds are plotted in the right panel of figure 4. If dark matter is a real scalar or Dirac fermion, then these bounds are tightened by a factor of 2 or ∼ 4, respectively. We can see that these constraints would rule out an IVDM model which could match the DAMA and CoGeNT data. But this is subject to the specific assumptions which we have made. For example, if dark matter couples to up- (a) p T > 120 GeV / E T > 120 GeV (b) p T > 350 GeV / E T > 300 GeV and down-quarks through an elastic contact operator which permits spin-independent, velocity-independent scattering, but only p -wave suppressed annihilation, then models which could potentially match the DAMA and CoGeNT data would be unconstrained by Fermi bounds. Another possibility is that dark matter may couple to Standard Model matter through a relatively light mediator with mass M ∗ ∼ 1 GeV. In this case, scattering interactions would still be short-ranged, but dark matter annihilation would have an additional M 4 ∗ / ( 2 mX ) 4 glyph[lessmuch] 1 suppression.", "pages": [ 6, 7 ] }, { "title": "Monojet searches at the LHC", "content": "An interesting way to bound direct dark matter production is with monojet or monophoton searches at colliders [35, 36]. Much work has been done on this search strategy [37]. The key uncertainties in setting this bound arise from the flavor structure of dark matter-quark couplings, and the energy dependence of the ¯ qq → XX matrix element. For example, if one assumes that the dark matter coupling to quarks is proportional to the quark mass, then a large contribution to the LHC event rate will typically come from couplings to ¯ cc , ¯ bb , even though these couplings yield a relatively small contribution to the scattering cross-section. Similarly, if the matrix element scales with negative powers of the energy, the dark matter production cross-section at the LHC will also be suppressed, since the energy scale of the dark matter production process is > 2 mX due to the phase space suppression at the production threshold. Fermionic dark matter is tightly constrained by LHC monojet/monophoton bounds, since the relativistic normalization of the dark matter state will introduce additional positive powers of the energy of the process, as seen in Table 1. On the other hand, models with M ∗ ∼ 1 GeV will have negative powers of the process energy from the propagator, and are very weakly constrained. In Table 2 [32], we indicate bounds on s p derived from ATLAS monojet searches ( pp → XX j ) with 1 fb -1 of data [35], if we assume dark matter couples only to first generation quarks through operators which permit elastic contact spin-independent velocity-independent scattering and s -wave annihilation. We assume fn / fp = -0 . 7. These bounds arise from analyses using two possible cuts on / E T and on the pT of the leading jet, as listed. The bounds on scalar dark matter are not competitive, but bounds on Dirac fermion dark matter can be competitive with (and in some cases exceed) bounds from direct detection experiments and Fermi. However, if dark matter interacts though a low-mass mediator ( M ∗ < ∼ 1 GeV), then bounds from the LHC will be weakened by several orders of magnitude.", "pages": [ 7 ] }, { "title": "LONG-RANGE INTERACTIONS", "content": "If dark matter scatters through a very low-mass mediator ( M 2 ∗ glyph[lessmuch] 2 m A E th ), then the differential scattering cross-section scales like ( 2 m A ER ) -2 instead of M -4 ∗ . As a result, light elements will tend to have enhanced scattering cross-sections. For example, the event rate at neutrino detectors is expected to receive a significant enhancement relative to leading direct detection experiments, due to the presence of many light elements in the sun. In the case of elastic long-range interactions, unlike elastic contact scattering, we cannot parameterize bounds on dark matter-nucleon scattering in terms of the total cross-section, which is infinite (an example is Rutherford scattering). Instead, we can parameterize the dark matter-nucleus differential scattering cross-section as: where C is a constant which normalizes the strength of the mediator coupling to dark matter ( gX ) and to the proton ( gp ) to the proton charge; C = g 2 X g 2 p / e 4 . Figure 5 [28] shows the sensitivity of a 1 kT LS detector with 2135 days of data, assuming annihilation exclusively to either the t , b , c , n (flavor-independent) or g channels. The neutrino spectra arising from dark matter annihilation to these channels was simulated using the HOSC cluster. We also plot the estimated sensitivity of CDMS, if their current bounds are reinterpreted as bounds on long-range dark matter interactions. Note that the CDMS sensitivity is derived under the assumption that the detector's efficiency is constant above the threshold; as a result of these and other assumptions [28], the CDMS sensitivity curve should only be regarded as an estimate. Nevertheless, it is clear that neutrino detectors have a sensitivity to dark matter models with long-range interactions which can rival leading direct detection experiments. A detector with a 51 kT liquid argon target [38] could achieve the same sensitivity as that given in figure 5 with ∼ 17 days of data.", "pages": [ 7, 8 ] }, { "title": "INELASTIC DARK MATTER (IDM)", "content": "If dark matter scattering is inelastic [39], then neutrino detectors can provide interesting detection possibilities which are complementary to direct detection experiments. For iDM, we again cannot characterize the bounds from experiments in terms of the dark matter-nucleon scattering cross-section, because there are kinematic regimes where dark matter-nucleon scattering is forbidden, while dark matter can still scatter off heavier elements. The reason is easy to see; for inelastic scattering to occur, the total initial kinetic energy in center-of-mass frame must exceed d mX . This yields the constraint ( 1 / 2 ) m A v 2 > d mX , and as m A decreases, v must increase in order for this constraint to be satisfied. We can instead parameterize the dark matter-nucleus differential scattering cross-section as where the squared dark matter-nucleus scattering matrix element can be written as m 2 X m 2 A [ Z + ( A -Z )( fn / fp )] 2 | F A ( ER ) | 2 × I . I is a quantity which, at leading order, is thus independent of the target nucleus mass and the relative velocity. In figure 6 [28] we plot the dark matter capture rate as a function of mX (in the low-mass regime) for I / 32 p = 10 -4 pb GeV -2 , for each element in the sun. As expected, we see that, as d mX increases, the capture rate from scattering off light elements decreases. Interestingly, we find that the capture is significant even for relatively large d mX . Indeed, dark matter capture in the sun is possible even for values of d mX which would render inelastic scattering on earth kinematically impossible. The reason is that dark matter gains significant kinetic energy as it falls towards the sun, allowing inelastic scattering even for large d mX . The escape velocity at the surface of the sun is ∼ 600 km / s; one would expect the range of d mX accessible to neutrino detectors via dark matter capture in the sun to be roughly a factor of 10 larger than that accessible to earth-based direct detection experiments.", "pages": [ 8, 9 ] }, { "title": "CONCLUSION", "content": "We have seen that, by relaxing some of the assumptions typically made regarding dark matter interactions, one can dramatically change the interpretation of data from direct detection, indirect detection and collider experiments. This suggests that, to gain a definitive understanding of the properties of dark matter, one may need to combine data from a variety of different detectors. Moreover, one finds that some reasonable dark matter models are best probed with novel detection strategies. Neutrino detectors, gamma-ray searches of dwarf spheroidal galaxies, and collider searches can all play an important complementary role. All of these new possibilities for dark matter interactions have interesting implications for upcoming detectors. An illustrative example is the D 3 experiment [40], which is under construction at the University of Hawaii. This experiment is a low-threshold gas time projection chamber which uses ionization to image the track made by a recoiling gas nucleus. An initial plan is for this detector to use a fluorocarbon target. One can thus see that, for IVDM models with destructive interference, detectors such as D 3 can have enhanced sensitivity as compared to other direct detection experiments. For example, one can compare the normalized-to-nucleon cross-section which would be seen by a fluorine-based detector to that of a germanium-based detector if fn / fp = -0 . 7, in which case one finds s Z = F N = 4 . 2 × s Z = Ge N . Another interesting possibility is the use of a hydrocarbon target. This could be especially interesting in the case where dark matter interactions are long-ranged. As we have seen, the squared scattering matrix element scales as m -2 A in the case of long-range interactions, implying that a detector with a hydrogen target would receive an O ( 10 3 ) enhancement in sensitivity relative to a detector with a germanium target. As more data comes in from a variety of current and upcoming experiments, it will be important to keep in mind the effect which relaxing typical theoretical assumptions can have on a coherent interpretation of the data.", "pages": [ 9 ] }, { "title": "ACKNOWLEDGMENTS", "content": "We are grateful to J. L. Feng, Y. Gao, D. Marfatia, K. Richardson, M. Sakai, D. Sanford, S. Smith and L. Strigari, for collaboration. We are grateful to HOSC and we thank the Center for Theoretical Underground Physics and Related Areas (CETUP* 2012) in South Dakota for its hospitality and for partial support during the completion of this work. This work is supported in part by the Department of Energy under Grant DE-FG02-04ER41291.", "pages": [ 9 ] } ]

[ { "title": "Structure and prospects of the simplest SO ( 10 ) GUTs", "content": "Michal Malinský 1 ∗ , Stefano Bertolini † and Luca Di Luzio ∗∗ ∗ AHEP Group, Instituto de Física Corpuscular - C.S.I.C./Universitat de València, Edificio de Institutos de Paterna, Apartado 22085, E 46071 València, Spain † INFN, Sezione di Trieste, SISSA, via Bonomea 265, 34136 Trieste, Italy ∗∗ Institut für Theoretische Teilchenphysik, Karlsruhe Institute of Technology (KIT), D-76128 Karlsruhe, Germany Abstract. We recapitulate the latest results on the class of the simplest SO ( 10 ) grand unified models in which the GUTscale symmetry breaking is triggered by an adjoint Higgs representation. We argue that the minimal survival approximation traditionally used in the GUT- and seesaw-scale estimates tends to be blind to very interesting parts of the parameter space in which some of the intermediate-scale states necessary for non-supersymmetric unification of the SM gauge couplings can be as light as to leave their imprints in the TeV domain. The stringent minimal-survival-based estimates of the B -L scale are shown to be relaxed by as much as four orders of magnitude, thus admitting for a consistent implementation of the standard seesaw mechanism even without excessive fine-tuning implied by the previous studies. The prospects of the minimal renormalizable SO ( 10 ) GUT as a potential candidate for a well-calculable theory of proton decay are discussed in brief. Keywords: Grand unification, SO(10), neutrino masses PACS: 12.10.-g, 12.60.Jv, 12.15.Ff", "pages": [ 1 ] }, { "title": "INTRODUCTION", "content": "With the next generation of large-volume proton-decay searches and neutrino experiments currently in the R&D phase (in particular, LBNE [1], LENA [2] and Hyper-K [3]) there are good prospects to push the current lower bounds on the proton lifetime to the unprecedented level of 10 35 years. On the theory side, the new information may be, at least in principle, used for further testing of the grand unification paradigm; however, this would require a very good grip on the proton lifetime predictions supplied by specific GUTs. Unfortunately, the quality of the existing estimates is rather limited even in very simple scenarios, see FIGURE 1, and it is namely due to the low accuracy of the leading-order methods used in most of the relevant calculations. On the other hand, consistent next-to-leading-order (NLO) proton lifetime estimates are parametrically more difficult: First, at the NLO level, the GUT scale MG must be determined at two-loops; this, however, requires a detailed understanding of the one-loop theory spectrum. Second, the flavour structure of the relevant baryon-number-violating (BNV) currents must be constrained by the existing data to a maximum attainable degree. Third, one has to account for several classes of almost irreducible uncertainties related to the Planck-scale physics (such as, e.g., gravity smearing of the gauge unification pattern [4, 5]) which are often comparable to the NLO effects. Therefore, the only foreseeable way to overcome this conundrum is to focus on the simplest possible GUTs. In contrast to the minimal SU ( 5 ) Georgi-Glashow model [16] which was shown to be incompatible with the electroweak data already back in mid 1980's, the history of the minimal SO ( 10 ) GUTs is rather non-linear and even after almost 40 years it is still lively and evolving. Interestingly, this can be partly attributed also to the fact that, in the SO ( 10 ) context, the very meaning of minimality is not entirely agreed upon. This owes namely to the relatively large number of potentially viable symmetry breaking chains in SO ( 10 ) characterized by different effective scenarios emerging at intermediate scales. Let us recall that this is not the case in SU(5) simply because there the need to preserve rank reduces the set of Higgs representations available for the GUT symmetry breaking to just few. In a certain sense, this is not the case in supersymmetric theories either because the rigidity of the MSSM gauge unification pattern calls for a single-step breaking where most of the details of the GUT-scale dynamics remain obscured. Thus, besides very special features like natural R-parity conservation etc., the main distinctive characteristics of many models is namely their flavour structure. Hence, with the spectacular failure [17, 18] of the simplest potentially realistic renormalizable SUSY SO ( 10 ) [19, 20] (advocated by many to be even the very minimal SUSY GUT [21]), and, in particular, with no signs of SUSY at the LHC so far, the community's attention naturally drifts back to nonsupersymmetric GUTs. In this review, we shall comment in brief on the status of the simplest non-SUSY SO ( 10 ) scenarios and on the latest developments including, in particular, the new upper limits on the seesaw scale obtained recently in the work [22] and possible future prospects of accurate proton lifetime calculations in this scenario.", "pages": [ 1, 2 ] }, { "title": "THE MINIMAL SO ( 10 ) GRANDUNFICATION", "content": "The simplest multiplet that can consistently support spontaneous breaking of the SO ( 10 ) gauge symmetry in the SM direction is the 45-dimensional adjoint representation. Together with either the 16-dimensional spinor or the 126dimensional self-dual part of the maximally antisymmetric tensor the models based on the combinations 45 ⊕ 16 or 45 ⊕ 126 are often regarded to as the minimal renormalizable realizations of the Higgs mechanism in the SO ( 10 ) GUTs. In this respect, it is important to recall that this is not the case in SUSY where the F -flatness conditions align the VEV of 45 H along that of 16 H which, although providing the desired rank reduction, leaves a full SU(5) as an unbroken subgroup. Remarkably enough, in the non-supersymmetric Higgs model based on 45 ⊕ 16 or 45 ⊕ 126 the SU(5) trap can not be entirely avoided either.", "pages": [ 2 ] }, { "title": "The tree-level curse of the minimal SO(10) GUTs", "content": "The point is that there are two states in the scalar spectrum of either of the two variants of the minimal model that can be simultaneously non-tachyonic only in a narrow region of the parameter space which, unfortunately, happens to support only SU(5)-like symmetry-breaking chains. Indeed, the masses of the color-octet and the SU ( 2 ) L -triplet components of 45 H are at the tree level given by [23, 24, 25] where a 2 is a coupling in the relevant scalar potential (see, e.g., [22]) and ω BL and ω R are the two independent SM-compatible VEVs in 45 H (with τ 2 denoting the second Pauli matrix) which, if hierarchical enough, break the SO ( 10 ) gauge symmetry along two different symmetry breaking chains Given that, it is clear that the right hand sides (RHS) of both equations in (1) are positive if and only if -2 ≤ ω BL / ω R ≤ -1 / 2, i.e., when there is essentially no hierarchy between ω R and ω BL , otherwise the corresponding vacuum is unstable. Obviously, in such a case, 〈 45 H 〉 is almost homogeneous and the descent is SU ( 5 ) -like; this, however, conflicts with the gauge unification constraints as in the Georgi-Glashow model.", "pages": [ 2, 3 ] }, { "title": "The quantum salvation", "content": "Until recently, the argument above was taken as a no-go for any potential viability of the minimal SO ( 10 ) model including the adjoint scalar as a Higgs field responsible for the initial SO ( 10 ) symmetry breakdown. However, as shown in [26] this was premature because the hierarchy between ω BL and ω R may be stabilized by quantum effects because loop corrections such as those diplayed in FIGURE 2 provide non-negligible positive contributions to the RHS of eq. (1). A thorough effective potential analysis [26] in the simplest 45 ⊕ 16 variant yields (in the notation of [26]) where the 'logs' denote the typically sub-leading logarithmic corrections. Hence, for small-enough a 2 in (1) the two problematic states may have non-tachyonic masses even for a large hierarchy between ω R and ω BL , thus avoiding the tree-level SU(5) trap. Let us also note that, up to the obvious differences in the O ( 1 ) factors, the same dynamical mechanism works in the 45 ⊕ 126 Higgs model.", "pages": [ 3 ] }, { "title": "SEESAW SCALE IN THE MINIMAL RENORMALIZABLE SO ( 10 ) GUT", "content": "However, the vacuum stability was not the only issue that plagued the SO ( 10 ) GUTs for years. The enormous progress in neutrino physics in the last two decades pinned the light neutrino masses into the sub-eV domain with the upper bound (namely, from cosmology and double-beta-decay searches) in the 1 eV ballpark. In the seesaw picture, this typically translates into a lower bound on the scale of the underlying dynamics somewhere in the 10 12 -13 GeVdomain. This, however, was long ago claimed to be incompatible with the basic features of the symmetry-breaking pattern in the minimal SO(10) models.", "pages": [ 3 ] }, { "title": "Seesaw scale in the minimal survival approximation", "content": "Without any detailed information about the scalar spectrum of a theory under consideration, the best one can do in order to study the relevant gauge coupling unification patterns is to employ the minimum survival hypothesis (MSH) [27], i.e., to assume that the components of the unified-theory multiplets cluster around the specific symmetrybreaking scales. As rough as this approximation sounds, it often gives a qualitatively good first look at the salient features of the unification pattern. In the non-SUSY SO ( 10 ) framework, the 'natural' positions of the seesaw and grand unification scales have been, under this assumption, studied in [28, 29, 30] and later reviewed in [31]. In order to retain a grip on neutrinos and keep the theory well under control, in what follows we shall focus entirely on the 45 ⊕ 126 realization of the Higgs mechanism in the minimal SO ( 10 ) GUTscheme in which the type-I+II seesaw mechanism is supported at the renormalizable level. For more information about this framework an interested reader is referred to the relevant literature [22]. The resulting constraints on the unification and intermediate scales obtained in this scenario (in the minimal-survival approximation) are displayed in FIGURE 3. Remarkably enough, for both descends of interest there turn out to be stringent upper limits on the seesaw scale in the minimal survival picture well below 10 11 GeV and, moreover, for the chains passing through the intermediate SU ( 4 ) C ⊗ SU ( 2 ) L ⊗ U ( 1 ) R stage, the upper limit for MG is in the region which tends to be problematic from the proton lifetime perspective. This, however, implies that seesaw scale is far outside the 10 12 -14 GeV domain favoured by the light neutrino masses unless the Dirac neutrino mass terms are artificially suppressed. Although there is nothing a-priori wrong about this option we shall not entertain it here. Rather than that, we shall attempt to do better than the naive estimates above by exploiting the main drawbacks of the minimumsurvival approach: First, the MSH does not reflect many important features of realistic spectral patterns (such as, e.g., splitting among different components of multiplets below the relevant symmetry-breaking scales). Second, it is totally ignorant of special regions of the parameter space where the scalar spectrum exhibits unexpected features such as, e.g., accidentally light states deep in the desert. However, these are exactly the cases when the unification picture can be altered considerably.", "pages": [ 3, 4 ] }, { "title": "Consistency beyond minimal survival", "content": "Beyond the minimum-survival approximation, the only guiding principle left for an adventurous parameter-space explorer is the overall consistency of the theory. This has several basic aspects: Non-tachyonic scalar spectrum. First, all potentially interesting regions of the parameter space should support stable (or at least metastable) vacua. Since the full-featured vacuum stability analysis is very difficult, we shall stick only to the necessary condition, i.e., that there should be no tachyonic states in the scalar spectrum. Let us point out that, for each such vacuum configuration at hand, one can obtain other viable settings by, e.g., rescaling all dimensionful parameters in the scalar potential by a common factor. Similarly, it is clear that fiddling around with the mass of an accidentally light state within a range well below the typical mass-scale of all other heavy states does not destabilize specific vacua either because such variations correspond to only very small shifts in the fundamental parameters of the theory. These two 'degrees of freedom' can subsequently be used as an efficient tool for reducing the complexity of the numerical analysis of consistent unification patterns. Current proton decay limits. Another obvious constraint on the parameter space of the minimal SO ( 10 ) GUT comes from the proton decay; in particular, the current best limit for the p → e + π 0 mode from Super-Kamiokande [33] should be accommodated. In what follows we shall use this together with two assumed future limits that HyperKamiokande (HK) [3] may reach by 2025 and 2040 (if built): Furthermore, we shall for simplicity neglect all the details related to the flavour structure of the baryon-numberviolating currents so that the numbers above translate directly to the bounds on the position of the GUT scale. In the relevant plots (namely, FIGUREs 5 and 6), the points falling between these limits will be distinguished by a simple colour-code where the light grey is used for proton lifetimes between 8 . 2 × 10 33 and 9 × 10 34 years, dark grey corresponds to lifetimes between 9 × 10 34 and 2 × 10 35 years and black points yield more than 2 × 10 35 years. Big-bang nucleosynthesis. Third, accidentally light coloured states should not be too-long-lived otherwise their late decays may interfere with the highly successful classical big-bang-nucleosynthesis (BBN) account of the light elements' abundances. Actually, as we shall see, this is not a problem here because the accidentally light multiplets in all fully consistent cases originate in 126 H and, thus, couple directly to the SM matter fields through the same Yukawa couplings that give rise to, e.g, right-handed neutrino masses. Thus, all the light remnants should decay well before the BBN epoch. Consistent unification patterns. The simple constraints above are enough to filter out all but two qualitatively different settings with a single accidentally light scalar multiplet well below the B -L symmetry-breaking scale: a scenario with a very light colour octet ( 8 , 2 , + 1 2 ) and another scheme with an intermediate-mass-scale colour sextet ( 6 , 3 , + 1 3 ) . The typical shapes of the gauge unification patterns in these two cases are shown in FIGURE 4. The results of a detailed numerical scan over extended regions supporting these solutions are given in FIGURE 5. Interestingly, the mass range of the octet solution (on the left panel in FIGURE 5) can stretch as low as to the TeV domain so, in principle, it can even leave its imprints in the LHC searches; however, the sextet is not allowed below roughly 10 9 GeV.", "pages": [ 4, 5 ] }, { "title": "Seesaw scale upper limits in consistent scenarios", "content": "Finally, the allowed ranges for the B -L -breaking VEV (denoted σ ) in these two scenarios are depicted in FIGURE 6. Remarkably enough, the naive MSH-based upper bounds on the seesaw scale are in both cases relaxed by as much as four orders of magnitude as they stretch up to about 10 14 GeV in the light-octet case and up to almost 5 × 10 14 GeV in the case of the light sextet. This, however, makes the implementation of the standard seesaw mechanism possible even without resorting to the excessive fine-tuning in the Yukawa sector implied in previous studies.", "pages": [ 5 ] }, { "title": "CONCLUSIONS AND OUTLOOK", "content": "Even though the minimal SO(10) models have been recently revived as consistent quantum field theories free of inherent tachyonic instabilities, this beautiful and simple framework has never been rehabilitated as a potentially realistic and predictive GUT scheme. This was namely due to the old studies of the relevant gauge unification patterns which revealed a generic tendency for the B -L symmetry breaking scale to be confined below 10 11 GeV, apparently too low for a reasonable implementation of the seesaw mechanism for neutrino masses. However, all these early studies based on the minimal survival hypothesis suffer from a generic incapability to account for the effects of accidentally light multiplets with masses detached from any specific symmetry breaking scale. As we argued, a closer inspection of the unification constraints reveals a much wider room for the B -L breaking VEV stretching up to the 10 14 GeV ballpark which, in turn, allows the seesaw picture to be implemented without excessive fine-tuning. Besides that, the minimal renormalizable SO(10) scenario has several other interesting features which make it an interesting candidate for a further theoretical scrutiny: Possible LHC imprints of the light octet scenario. Remarkably enough, the same pair of scalars that we identified as powerful 'running helpers' in the minimal SO(10) framework was recently singled out in the work [34] from a totally different perspective. There, the apparent enhancement in the H → γγ rate indicated by the current LHC data was shown to be attributable to just this couple of states if any of them falls into the vicinity of the TeV scale. However, at the current level of accuracy only the octet ( 8 , 2 , + 1 2 ) can be light enough in the minimal SO ( 10 ) to play any role in the Higgs physics because the sextet is not allowed below about 10 9 GeV, see FIGURE 5. Nevertheless, the proton-decay limits used in cutting the low-mass-sextet region therein are rather naive and it can happen that a more-sophisticated analysis including flavour effects opens a bigger room for the light sextet too. Suppression of the Planck-scale induced unification-smearing effects in the minimal SO ( 10 ) GUTs. Besides simplicity, the SO ( 10 ) models in which the GUT-scale symmetry breaking is driven by a VEV of the 45-dimensional adjoint representation have another very interesting feature. This has to do with the general fragility of grand unification with respect to the Planck-scale ( M Pl ) effects which, given the proximity of MG and M Pl , may not be entirely negligible. Concerning their possible impact on, e.g., proton lifetime estimates, the most important of these is namely the Planck-scale induced violation of the canonical normalization of the heavy gauge fields [4, 5] due to the higherorder corrections to the gauge kinetic form emerging already at the d = 5 level: L ( 5 ) /owner Tr [ F µν HF µν ] / M Pl ; here H is any scalar in the theory which can couple to a pair of adjoint representations of a specific GUT symmetry group, i.e., any field appearing in the symmetric part of the decomposition of their tensor product. For a GUT-scale VEV of H , this induces a percent-level effect which, after a suitable redefinition of the gauge fields, leads to similar-size shifts in the GUT-scale matching conditions. Such a 'unification smearing effect' can, subsequently, play a significant role in an accurate NLO GUT-scale determination which, in turn, further adds to the existing theoretical uncertainties in the absolute proton lifetime estimates. However, if in the SO(10) the GUT-scale symmetry breaking is triggered by the VEV of 45 H , this problem is absent because Tr [ F µν 45 HF µν ] = 0 due to the fact that 45 is not in the symmetric part of the 45 ⊗ 45 decomposition [recall that ( 45 ⊗ 45 ) sym = 54 ⊕ 210 ⊕ 770]. Thus, the minimal SO ( 10 ) scheme with the adjoint-driven Higgs mechanism is uniquely robust with respect to this class of quantum gravity effects. This makes the symmetry-breaking analysis more reliable and, hence, admits in principle for a strong reduction of this type of theoretical uncertainties in the proton lifetime estimates. Proton lifetime at the next-to-leading-order level. The simplicity of the minimal SO ( 10 ) scenario advocated in this study, together with its rather unique robustness with respect to the Planck-scale-induced unification smearing effects make this class of models particularly suitable for a possible next-to-leading-order (NLO) proton lifetime analysis. That, however, is far from trivial. To this end, let us just note that the main source of the large theoretical uncertainties in the existing proton lifetime estimates, cf. FIGURE 1, is the inaccuracy of the GUT-scale determination, partly due to the uncertainties in the low-energy inputs (especially in α s ( MZ ) ) and, in particular, the limited precision of the one-loop approach - given the logarithmic nature of the renormalization-group evolution, both these errors are exponentially amplified in the resulting proton decay amplitude. The only way to keep such uncertainties under control is thus a careful two-loop renormalization-group calculation including, as a prerequisite, the one-loop spectrum of the theory resulting from a dedicated analysis (like, e.g., that in [26]) together with the proper one-loop matching [35, 36] conditions. In this respect, the minimal renormalizable SO ( 10 ) GUT of our main concern here can be the scenario in which a decisive NLO proton lifetime analysis can be just at the verge of tractability. GUT-scale baryogenesis. The option of a GUT-scale baryogenesis in SO(10) models, recently revived in [37], is another interesting feature of the model under consideration. On the technical side, however, the amount of thus generated baryon asymmetry depends on the size of the quartic coupling η 2 in the scalar potential, cf. formula (3) in reference [22], which, unfortunately, turns out to be one of the most elusive of all the theory parameters - as it was argued in [22], η 2 does not enter the tree-level scalar spectrum and, so, the one-loop unification pattern does not impose any constraints on it. However, this coupling does take part in the decomposition of the light Higgs boson in terms of its defining components in 10 H ⊕ 126 H and, thus, it may be constrained indirectly by the flavour structure of the effective theory. Hence, a dedicated proton lifetime analysis advocated above (which, inevitably, must include a detailed account of the flavour structure of the model) may, as one of its by-products, provide also a better grip on the GUT-scale baryogenesis in the minimal SO ( 10 ) GUT.", "pages": [ 5, 6, 7, 8 ] }, { "title": "Acknowledgments", "content": "M.M. is grateful for the invitation to CETUP'12 and support during his stay in Lead, SD. S.B. is partially supported by MIUR and the EU UNILHC-grant agreement PITN-GA-2009-237920. The work of L.DL. was supported by the DFGthroughthe SFB/TR 9 'Computational Particle Physics'. The work of M.M. is supported by the Marie Curie Intra European Fellowship within the 7th European Community Framework Programme FP7-PEOPLE-2009-IEF, contract number PIEF-GA-2009-253119,by the EU Network grant UNILHC PITN-GA-2009-237920,by the Spanish MICINN grants FPA2008-00319/FPA and MULTIDARK CAD2009-00064 (Consolider-Ingenio 2010 Programme) and by the Generalitat Valenciana grant Prometeo/2009/091.", "pages": [ 8 ] } ]

[ { "title": "Benoît Revenu", "content": "École des Mines de Nantes - Université de Nantes - CNRS/IN2P3 4 rue Alfred Kastler, BP20722, 44307 Nantes CÉDEX 03, FRANCE Abstract. In this paper, I present a review of the main results obtained in the last 10 years in the field of radio-detection of cosmic-ray air showers in the MHz range. All results from all experiments cannot be reported here so that I will focus on the results more than on the experiments themselves. Modern experiments started in 2003 with CODALEMA and LOPES. In 2006, small-size autonomous prototypes setup were installed at the Pierre Auger Observatory site, to help the design of the Auger Engineering Radio Array (AERA). We will discuss the principal aspects of the radio data analysis and the determination of the primary cosmic ray characteristics: the arrival direction, the lateral distribution of the electric field, the correlation with the primary energy, the emission mechanisms and the sensitivity to the composition of the cosmic rays. Keywords: Cosmic rays, Radio-detection PACS: 95.55.Jz,95.85.Ry,96.50.sd", "pages": [ 1 ] }, { "title": "1. INTRODUCTION", "content": "In 1971, Allan [1] gave a review of the latest results, at that time, concerning the detection of extensive air showers initiated by high-energy cosmic rays. The initial motivations for studying the radio signal was the possibility to build an array of widely spaced receivers to detect extensive air showers at the highest energies (above 10 EeV), in a cheap way. It was also proposed to use the radio signal in conjonction with a particle array to get additional information on the longitudinal profile, providing valuable constraints on the nature of the primary cosmic ray. Using the data available at that time, Allan concluded that the main signal should be polarized in the direction of the vector v × B , where v is the direction of the shower axis and B the direction of the geomagnetic field. The mechanism responsible for this specific polarization is the geomagnetic contribution due to the Lorentz force acting on each secondary charged particles, in particular the electrons and positrons. It was predicted that the charge excess contribution should be less important but still detectable, mainly for incoming directions close to the direction of the geomagnetic field. It is also stated that the electric field amplitude extrapolated on the shower axis, over the range 32-55 MHz, is proportional to the primary energy. Finally, for a given event observed by several detectors, the electric field amplitude decreases exponentially with the axis distance. The equation proposed to fully describe the electric field en observed by a single detector at a distance R of the shower axis is: in m V m -1 MHz -1 , where EP is the primary energy, a the angle between the shower axis and the geomagnetic field, q the zenith angle, R the distance between the detector and the shower axis and R 0 the attenuation length of the electric field. The lateral distribution function (LDF) proposed by Allan depends on the axis distance leading to an azimuthal invariance of the electric field with respect to the shower axis. This formula was used as a starting point at the beginning of the years 2000 for the design of the CODALEMA [2] and LOPES [3] experiments. These first modern experiments were triggered by an array of particle detectors in order to search a posteriori for a radio counterpart to a shower candidate. The next challenge consisted in setting up fully autonomous and independent radio detectors to test the possibility to build a full stand-alone radio array. A first type of such radio array was installed in 2006 at the center of the surface detector (SD) of the Pierre Auger Observatory, in the scope of the Auger Engineering Radio Array (AERA), experiment which finishes its phase 1 in 2012. AERA benefits from the SD and FD (fluorescence detector) reconstructions, providing high-quality super-hybrid events. The LOFAR [4] experiment also detects the radio emission of air showers. The complementary particle detector, LORA [5], helps in triggering and identifying the cosmic rays detected by LOFAR. We present in this overview the updated results of the field. We will discuss the lateral distribution function, the correlation with the primary energy and we will insist on the current situation concerning the emission mechanisms that can be determined through the polarization of the total electric field.", "pages": [ 1 ] }, { "title": "2. DETECTION OF THE ELECTRIC FIELD EMITTED BY EAS", "content": "The secondary electrons and positrons created during the development of the air shower form a pancake-shape particle front moving at the speed of light. The thickness of the shower front is of the order of 1 m on the shower axis, up to ∼ 10 m far from the axis. These particles suffer a systematic opposite drift caused by the Lorentz force due to the Earth's magnetic field generating a coherent emission of electromagnetic waves in the 1-500 MHz range. The amplitude of the resulting macroscopic current and its variation depends on the number of charges particles of the shower and therefore, can provide information on the longitudinal profile. The electric field produced this way is expected to have a polarization following the direction of the Lorentz force, v × B . This mechanism is known as the geomagnetic contribution as discussed in the introduction. Another mechanism is due to the variation of the excess of electrons due to the annihilation of positrons and knock-on electrons, known as the charge-excess contribution. This corresponds to the Askaryan [6] effect in the air, leading to a radiallypolarized electric field, also in the MHz domain. It is therefore possible to disentangle these two contributions through the different polarizations of the associated electric fields. Various simulation codes are available, a review is given in this conference, see [7]. This electric field is usually detected by systems composed of antennas, precise time tagging system (GPS for instance) and the signal is digitized by fast ADCs in order to be able to compute the Fourier spectrum up to some hundreds of MHz. The electric field is filtered in a frequency band above the AM ( ∼ 20 MHz) and below the FM (80 MHz). Following the antenna used, the filtering can be done in more restricted bands accordingly to the frequency response of the device used. In general, the electric field is measured in, at least, two horizontal orthogonal polarizations; this permits to reconstruct the polarization angle in the horizontal plane and to check for the underlying emission mechanism. The two first modern radio experiments, CODALEMA and LOPES, led to many progresses in the understanding of the emission of the electric field by EAS. These experiments detect the EAS initiated by cosmic rays with an energy between 10 16 and 10 18 eV and follow the same principle: they use a particle array which triggers the radio array. The particle array for CODALEMA is an array of 17 scintillators and in the case of LOPES, the particle array is the KASCADEGrande experiment. LOPES is installed close to the city of Karlsruhe which is an electromagnetically noisy environment. The solution adopted by LOPES is to use interferometric technics, using as an input the high- quality information provided by KASCADE-Grande. A digital beam-forming is applied to inter-phase the time series of all LOPES antennas using the arrival direction given by KASCADE-Grande. Then, a cross-calibration procedure is used to estimate the global amplitude of the electric field. For high signal-to-noise ratio events, the electric field can be estimated for each antenna, this permits to compute the LDF of these events. On the contrary, the CODALEMA experiment is installed in the radio observatory of Nançay which is a protected area in the sense that the electromagnetic emissions of the neighbourhood is controlled. This permits to use the data of the radio array of CODALEMA independently of the data of the particle array, once the EAS is clearly identified and validated by the particle array. Each radio detector provides the electric field as a function of time with a high sampling rate (typically 500 Ms/s or 1 Gs/s) so that it's possible to measure the time of the maximum of the signal.", "pages": [ 2 ] }, { "title": "3. ARRIVAL DIRECTION", "content": "From the measurements of the electric field by several detectors, it is possible, in the very same way than with a particle array, to reconstruct the incoming direction of the EAS. Provided the ground coordinates of at least three non-aligned detectors and the time of transit of the electromagnetic wave of the shower in these detectors has been measured, we can triangulate to estimate q and f , the zenith and azimuth angles of the shower, respectively. For higher multiplicity events, we can compute the radius of curvature of the electromagnetic wave assuming a spherical front. The time resolution of GPS receivers allow to reach an angular resolution of the order of a fraction of degree, even for low-multiplicity events. Artificial sources are commonly used to time-calibrate the radio detectors. For instance, the detection of airplane transients represents a gold mine because it's possible to inter-calibrate the detectors, to study the antenna lobe sensitivity and also to compute the angular resolution. It has been proven that this angular resolution can reach 0 . 5 · , even for a small number of detectors spaced by 140 m only [8]. Octocopter flights have been used for instance in LOFAR, LOPES and AERA. The arrival direction estimation is important at two levels: it permits to identify - and suppress - anthropic events (mainly coming from the horizon, apart for airplanes) and to identify cosmic ray events by a comparison with the arrival direction given by the particle array. If the directions agree within a certain angle (typically less than 20 · ) and the time of the shower is the same within a certain time window (less than 100 ns for instance), then the event is seen by both arrays.", "pages": [ 2 ] }, { "title": "4. LATERAL DISTRIBUTION FUNCTION", "content": "The LDF describes the electric field amplitude as a function of the distance to the shower axis. The starting point for the function describing the LDF is an exponential, as first proposed by Allan. Most of the radio-detected events are well described by an exponential profile (see Eq. 1). Nevertheless, a non-negligible fraction of events (of the order of 20%) present a flattening for detectors close to the shower axis, in particular for inclined showers, as reported by the LOPES collaboration in [9]. The same observation holds for the CODALEMA data and the LOFAR collaborations. A very nice example of such event is given in [4]. This event was detected by 5 LOFAR stations corresponding to more than 200 independent measurements of the electric field at different axis distances; the radio pulse power as a function of axis distance is presented in Figure 1. The flattening of the profile is very clear for antennas located at less than 100 m from the shower axis. The flattening of the electric field profiles can be understood as the effect of the air refractive index. At the sea level, n ∼ 1 . 0003 and this value decreases with increasing altitude; the consideration of realistic values of the air refractive index in the simulations permits to reproduce events with a flat profile close to the shower axis which is the consequence of a Cerenkov ring (see [7], [10], [11], [12]). One should be cautious when using the usual 1D exponential profile, in particular when considering the electric field measurements close to the shower axis. Moreover, the electric field should not exhibit azimuthal invariance with respect to the shower axis. This property is predicted by all the simulation codes and has also been observed in the CODALEMA data [13]. More complicated 2D LDFs should be used in the future analyses but at the time of this conference, no analysis using 2D LDF has been presented. The minimization of the c 2 based on the model s ( d ) = s 0 exp ( -d / d 0 ) gives the core position through the axis distance d = | n × CA | where n is the shower axis and CA is the vector between the core position and the position of antenna A . The minimization also provides the attenuation length d 0 and the on-axis signal s 0. The on-axis signal can be defined as the actual electric field if the amplitudes s ( d ) have properly been deconvoluted for the antenna response (arrival direction and frequency gain). This deconvolution is mandatory in order to be able to study the correlation between electric field and energy of the primary cosmic ray. Using simulations, it has been demonstrated that the shower-to-shower fluctuations are minimized when considering the electric field at a distance of ∼ 110 m of the shower axis. It could be therefore interesting to consider this value instead of the on-axis electric field [14].", "pages": [ 3 ] }, { "title": "5. CORRELATION WITH THE PRIMARY ENERGY", "content": "Hybrid arrays usually use the particle detector array for the energy estimation. As stated before, the LOPES experiment relies on the KASCADE-Grande reconstruction, the CODALEMA experiment uses the scintillator array and the AERA experiment depends on the Auger SD and/or FD reconstruction. Allan proposed a linear relation between the on-axis electric field and the primary energy. This relation is confirmed by the LOPES 30 data, using the pulse height in the east-west (EW) polarization: e EW GLYPH<181> E 0 . 95 ± 0 . 04 p (see [15]). More recently, the AERA experiment reported the correlation between the electric field and the primary energy determined by the Auger SD and FD. The signal is recorded in the three polarizations EW, north-south (NS) and vertical (V). The AERA radio detectors have been fully calibrated and the measurements are deconvoluted to estimate the 3D electric field vector [16]. Then the Hilbert envelope is computed for the three directions and the total signal strength is defined as the maximum of the 3D Hilbert envelope [17]. The electric field is finally interpolated at 110 m of the shower axis, where the energy resolution is maximum, and the correlation with the primary energy is compatible with a linear relation. The AERA energy correlation plot will be published in a forthcoming paper. The Pierre Auger collaboration also reported a positive correlation at 99.99% CL between the on-axis electric field and the primary energy, obtained with the (pre-AERA) RAuger setup, as can be seen in Figure 2 extracted from [18]. It has been reported during this conference the quasilinear correlation of the CODALEMA data with the primary energy using raw data (not deconvoluted for the antenna response) [19]. Due to the different estimators used in the community (either the on-axis electric field or its value at ∼ 110 m), one should be cautious when comparing the slopes of the correlations reported by several experiments.", "pages": [ 3, 4 ] }, { "title": "6. EMISSION MECHANISMS AND POLARIZATION", "content": "The geomagnetic contribution to the total electric field is dominant, as stated in the 1960s and confirmed with much more statistics and better data some years ago by CODALEMA [20], [21] and LOPES [15]. In the southern hemisphere, the RAuger prototype reported that the arrival directions of the detected air showers were in good agreement with a v × B effect. Using the AERA data, we studied the correlation in the horizontal plane of the expected polarization angle f G of the geomagnetically-induced electric field with the measured polarization angle f P . The correlation is excellent, as can be seen in Figure 3 (extracted from [22]). As discussed in the introduction, the electrons in excess relatively to the positrons implies a net electric field whose polarization pattern is radial in the shower transverse plane. At first order, the total electric field is oriented following v × B but it is possible to search for second order effect, that can be revealed either by a change in the polarization angle or by an enhanced electric field amplitude due to constructive interferences. Figure 4 presents the polarization patterns for a vertical shower in a geomagnetic field with a NS component. In the last year, there have been two different approaches to detect such an additional contribution, compatible with the charge-excess contribution.", "pages": [ 4 ] }, { "title": "6.1. Evidence for a radial contribution through polarization angle studies", "content": "The polarization angle of the total electric field is estimated using the measurements along the EW direction ( x -axis) and the NS direction ( y -axis). To detect an electric field contribution incompatible with the geomagnetic mechanism (i.e. having an orientation not aligned along v × B ), we first construct an observable R characterizing the deviation from a pure geomagnetic electric field, for each radio detector. For that, we first rotate the coordinate system so that the new x -axis, x ' , is aligned along the direction of v × B projected in the horizontal plane and the new y -axis, y ' , is perpendicular to x ' . R quantifies the relative signal strength in the direction perpendicular to the geomagnetic expectation. By construction, a pure geomagnetic electric field leads to R = 0 and an electric field with no geomagnetic component has R = ± 1. For a given set of detected showers, we can compute R data for the data (we used the data of a preAERA prototype) and compute the same factor R simu for the same simulated showers (same geometry and detector array). The simulations used here are MGMR [23] and REAS3 [24] and were run using realistic showers (having an excess of electrons) and for showers forced to have the same number of electrons and positrons. The results showed that the correlation is much stronger when including the charge-excess mechanism in the shower. Figure 5 presents the correlation between R data and R simu with the charge-excess taken into account for the simulation code MGMR for example. Quantitatively, the reduced c 2 decreases from 6.4 (4.7) to 2.7 (3.0) when taking into account the charge-excess in the simulated showers with REAS3 (MGMR). The conclusion of these results is that the electric field emitted by the showers detected by this pre-AERA prototype is in better agreement with the simulations when the charge-excess mechanism is included in the simulation codes. The relative influence of both prototypes depends strongly on the shower arrival direction and on the relative position of the observer with respect to the shower core. More details on this analysis can be found in [25]. The same analysis is currently performed using the AERA data and more refined simulation codes.", "pages": [ 4, 5 ] }, { "title": "6.2. Evidence for a radial contribution through core shift", "content": "The CODALEMA collaboration reported in [13] the measurement of a shift toward the east of the shower core position estimated using the radio data (the radio core) with respect to the shower core estimated using the particle array data (the particle core). The particle core position is obtained using the Nishimura-Kamata-Greisen (NKG) lateral distribution (see [20] for the details). The radio core position is obtained by fitting an exponential function of the type s ( d ) = s 0 exp ( -d / d 0 ) where the core coordinates are hidden in the axis distance d . The shift to the east is characterized by the quantity D c = x c , radio -x c , particle . This shift is interpreted as the result of the superposition of the geomagnetic and charge-excess electric field components, as presented in Figure 4. For this example of a vertical shower, an observer located at the east of the shower core will observe a constructive superposition of the two components of the electric field, contrary to an observer located at the west of the shower core. The radio core will then be reconstructed, in this example, to the east of the actual shower core, defined as the intersection of the shower axis with the ground. The shift strongly depends on the incoming direction through the geomagnetic component. It is therefore interesting to study the shift as a function of the EW component of v × B (because CODALEMA measures only the EW polarization). To check the influence of the charge-excess, simulations were run (using the code SELFAS2 [26]) with and without the charge-excess mechanism. With no charge-excess, no core shift is observed. Therefore, the core shift is an evidence for the charge-excess contribution. For the selected CODALEMA data set, the dependence of D c with ( v × B ) EW is presented in Figure 6 and compared with expectations from simulations with SELFAS2, ran on showers having the same characteristics than the selected showers detected by CODALEMA. In the last two years, there have been two evidences of a non-geomagnetic contribution to the total electric field. The additional contribution appears to be radially polarized in the transverse shower plane, as it is the case for the charge-excess contribution. More data on a wider energy range will be needed to confirm if this contribution can be associated unambiguously to the charge-excess contribution (Askaryn affect in the air). The AERA experiment should be able to study in great details this contribution.", "pages": [ 5 ] }, { "title": "7. SENSITIVITY OF THE RADIO SIGNAL TO THE NATURE OF THE PRIMARY", "content": "The accurate determination shower by shower of the composition of the cosmic rays at ultra-high energies is the next challenge to be taken up. Since some years, using simulations, the radio signal is expected to be correlated to the nature of the primary cosmic ray (see [27], [14]). From simulations, it is possible to estimate the value of the atmospheric depth of maximum development of showers X max, using the LDF of radio profiles. The LOPES collaboration report an uncertainty on X max of the order of 150 g cm -2 . This large value is dominated by the very noisy environment in Karlsruhe and one could expect (using simulations) a much better resolution - around 30 g cm -2 -inquiet sites. For comparison, the Auger FD has a resolution of 20 g cm -2 [28]. We can expect, by combining data from radio and the SD, to reach a resolution on X max close to that of the FD. Another possible method to constrain the nature of the primary has been presented during this conference in [29]. The principle of this method is to study the slope (spectral index) of the frequency spectra between 40 and 60 MHz. Simulations show that this spectral index depends not only on the shower geometry but also on the nature of the primary cosmic ray, at a level of 10% according to MGMR. The dependence on the shower geometry has been confirmed on the AERA data and the possibily to constrain the composition with this method is still under study. Experimentally, the LOPES collaboration reported [30] a first evidence, at a level of 3 . 7 s , of the sensitivity of the radio signal to the nature of the primary cosmic ray through the longitudinal profile development. The slope of the radio lateral distribution (using an exponential profile with an estimation of the amplitude at an axis distance of 100 m) is correlated to the mean muon pseudorapidity which is in turn correlated to the shower development: the pseudorapidity is small (large) when the muons are produced at low (high) atmospheric heights. The slope is defined as the factor a in the lateral distribution function: s ( d ) = s 100 exp ( -a ( d -100 m )) , where s is the electric field amplitude expressed in m V m -1 MHz -1 in the band [ 40 -80 ] MHz and d is the axis distance. Figure 7 shows this correlation using a selection of 59 events.", "pages": [ 6 ] }, { "title": "8. CONCLUSION", "content": "The most recent and important result of the field is the evidence for a secondary electric field, radially polarized with respect to the shower transverse plane. This electric field could well be the contribution of electrons in excess in the shower, known as the Askaryan effect in the air. The confirmation of such observation and its quantification would be a major advance in the understanding of the emission processes of electric field in air showers. The radio signal, supposed to be correlated to the longitudinal profile of the shower, is very promising in the estimation of the composition of ultra-high energy cosmic rays. The AERA experiment, located at the north-west of the Auger SD, it installed at an ideal site because showers can be detected by many detectors: the Infill Auger SD, the regular Auger SD, the regular Auger FD, AMIGA (dedicated to the muonic contents of the shower, [25]) and HEAT [25] which permits to detect the fluorescence light at higher elevation angles corresponding to lower energy showers. All these detectors will study in great details the showers in the energy range 10 17 to 10 18 . 5 , providing high-quality data to study the transition from a galactic to an extra-galactic origin of cosmic rays. The phase 2 of AERA, providing a total of 161 stations spread over 20 km 2 is starting in March 2013.", "pages": [ 6, 7 ] } ]

[ { "title": "Timo Karg", "content": "DESY, Platanenallee 6, 15738 Zeuthen, Germany Abstract. Acoustic neutrino detection is a promising technique to instrument the large volumes required to measure the small expected flux of ultra-high energy cosmogenic neutrinos. Using ice as detection medium allows for coincident detection of neutrino interactions with acoustic sensors, radio antennas and optical light sensors with the benefit of cross calibration possibilities or independent measurements of the the same event. We review the past development of the field and discuss its current status and challenges. Results from site exploration studies, mainly by the South Pole Acoustic Test Setup (SPATS) which has been codeployed with the IceCube neutrino telescope at South Pole, and current physics results are presented. Current ideas for the design, calibration, and deployment of acoustic sensors for new projects are shown. The possible role of the acoustic technique in future in-ice neutrino detectors is discussed. Keywords: ultra-high energy neutrinos, acoustic detection, Antarctica, SPATS PACS: 07.64.+z, 92.40.Vq, 95.55.Vj, 98.70.Sa", "pages": [ 1 ] }, { "title": "INTRODUCTION", "content": "In the year 2012 we celebrate the 100 th anniversary of the discovery of cosmic rays. But even after a hundred years of research many questions about the origin, acceleration, and composition of ultra-high energy cosmic rays remain unanswered. The multi-messenger approach, combining the information gained from electromagnetic radiation from radio to TeV photons, charged cosmic rays, and neutrinos promises to resolve these problems. Neutrinos are ideal messengers in the sense that they are undeflected by magnetic fields during their propagation and that they rarely interact, preserving their initial direction and energy until detected at Earth. Ultra-high energy (UHE; E n glyph[greaterorsimilar] 100 PeV) neutrinos, offer a very rich physics program, including astrophysics, cosmology, particle physics, and physics beyond the Standard Model: overburden, e.g. at different zenith angles with an underground detector, will allow us to determine the neutrino absorption in the Earth and thus probe the neutrino nucleon cross section at high center-ofmass energies [6]. To achieve all these goals we need to measure UHE neutrinos with reasonable statistics and good energy and direction resolution. This requires a detector with a volume ≥ 100 km 3 . There are different experimental techniques to build such large scale detectors which are currently pursued either in running experiments or as feasibility studies. Radio detection experiments are looking for short radio pulses in the hundreds of MHz to GHz frequency range emitted by the electromagnetic cascade generated in a neutrino interaction. These experiments can be embedded in radio-transparent, homogeneous media like ice [9] or salt, or use balloons or satellites to observe large natural ice volumes. Also, observations of the Moon with radio telescopes are employed to look for neutrino interactions in the lunar regolith [10]. Extensive air shower experiments can detect UHE neutrinos either as highly inclined, 'young' air showers, where the primary neutrino has penetrated deep into the atmosphere before interacting, or as up-going air showers from Earth-skimming neutrinos. The HiRes detector [11] and the Pierre Auger Observatory [12] have used these methods to set upper limits on the flux of UHE neutrinos. Finally, acoustic neutrino detectors are searching for ultrasonic pressure pulses generated in the instantaneous heating and expansion of the medium induced by electromagnetic and hadronic cascades. Water [13], ice, salt, and permafrost soil [14] have been discussed as detection media. In this work we review the development, status and perspectives of acoustic neutrino detection in ice.", "pages": [ 1, 2 ] }, { "title": "A BRIEF HISTORY OF ACOUSTIC NEUTRINO DETECTION IN ICE", "content": "Acoustic neutrino detection in liquids is based on the thermo-acoustic model [15, 16]: When a neutrino of any flavor interacts via a charged- or neutral current interaction, a hadronic and/or electromagnetic cascade develops at the interaction vertex, which carries a significant amount of the neutrino energy. In a dense medium this energy is dissipated in a volume of typically 10 m in length and a few centimeters in diameter. This leads to an instantaneous heating of the cascade volume. The corresponding rapid expansion of the volume propagates as an ultrasonic shock wave perpendicular to the cascade axis and can be measured as a short (i.e. with a broad frequency spectrum), bipolar pressure pulse with a duration of several ten microseconds. The details of the pulse depend on the material properties of the medium and on the modeling of the cascade energy deposition density. The first ideas about acoustic detection of particles in liquids date back to the 1950s [17]. It was then revived in the 1970s and studied in great detail in the context of the DUMANDproject, leading to detailed calculations of the expected acoustic signals from the thermo-acoustic model [18, 19] and first measurements with a proton beam from an accelerator dumped in water [20]. With the design and construction of the AMANDA optical Cherenkov neutrino telescope at South Pole interest in acoustic neutrino detection in ice began. Ice, in contrast to water, allows for the propagation of longitudinal (pressure, p) sound waves and transverse (shear, s) waves. The formalism of the thermo-acoustic model can be expanded to the case of solid media and predicts the excitation of mainly pressure waves by neutrino interactions (cf. e.g. [21]); shear waves can be generated at impurities in the crystal structure of the medium. Acoustic signals predicted by the thermo-acoustic model scale, for equal energy deposition densities, with the thermo-elastic properties of the detection medium. Due to the nearly equal matter densities of water and ice the energy deposition density from a neutrino interaction is very similar. From detailed calculations, taking into account the elastic properties of ice, the amplitudes of neutrino-induced thermo-acoustic signals in ice are expected to have amplitudes which are larger by a factor of about four compared to water [21]. This is supported by measurements in the Aachen Acoustic Laboratory [23] where laser-induced acoustic pulses in water and ice have been studied [22]. Figure 1 shows that the scaling of the signal amplitude from water to ice is compatible with expectations. It can also be seen that in ice, due to the larger speed of sound, higher frequency signals are generated. Phenomenological studies of the ice acoustic properties predicted an acoustic attenuation length of several kilometers [24, 25] and low background noise [25], which would allow for very large, sparsely instrumented detection volumes. Subsequent studies favored the radio technique as being more sensitive than acoustics [26]. In the same article (Ref. [26]) the possibility of hybrid detection, using several complementary techniques (radio and optical) is discussed. In the following years experimental limits on the flux of high energy neutrinos became more stringent and theoretical flux predictions decreased accordingly. It became clear that detector volumes ≥ 100 km 3 are required which are difficult to achieve with optical Cherenkov detectors. Since radio and acoustic signals were expected to have similar attenuation lengths, hybrid radio-acoustic detectors were discussed and simulation studies showed very promising results [27]. However, it was clear that the predicted acoustic properties of the ice need to be tested by in-situ measurements. Different piezoelectric sensors for use in ice were developed and characterized [28] and sound generation in ice by an accelerator proton beam was studied [29]. These efforts led to the construction and deployment of the South Pole Acoustic Test Setup (SPATS) that will be discussed later in this work.", "pages": [ 2 ] }, { "title": "MEASURING ACOUSTIC WAVES IN ICE", "content": "Building a large acoustic detector with reasonable energy and direction resolution requires to fully understand the sensor response in-situ. Sensor sensitivity is not just a single number to convert incident pressure to output voltage measured at the sensor, but is a function of incident wave direction, wave mode, temperature, and possibly other environmental parameters. Since no pre-calibrated sensors for ice are commercially available that can be used for relative calibration, extensive studies have been performed to use the reciprocity calibration method, that does not require a reference receiver, in ice [30]. The insitu calibration of sensors deep in natural glacial ice is even more challenging due to the limited possibilities of access to the detectors. In the SPATS project (cf. next section) it has been tried to factorize the problem in the laboratory: SPATS sensors have been absolutely calibrated in water at 0 · C before deployment [31] and the angular response of the sensor has been determined at different frequencies [32]. It is not obvious whether the calibration results can be transferred to operation conditions, where the sensors are frozen in the deep ice at South Pole. There, they are subject to low temperatures of approx. -50 · C, increased static pressure, and a different sensor-medium interface (ice to steel). The different effects have been studied separately in the laboratory: Assuming that the influences of the environmental effects are independent, it has been concluded that for the SPATS sensors the sensitivity in ice will be increased by a factor of 1 . 5 ± 0 . 4 compared to the pre-deployment calibration in water [34]. This factor takes into account the uncertainties from the temperature and pressure measurements. It is under investigation how naturally occurring transient noise events and artificial calibration transmitters can be utilized for sensor relative calibration and angular response measurements in-situ [35].", "pages": [ 3 ] }, { "title": "SITE EXPLORATION", "content": "Another important step towards a large scale detector embedded in a natural detection medium is the full understanding of the signal propagation properties and backgrounds therein. For an acoustic experiment this means the determination of the sound speed depth profile, the attenuation length, the noise level, and possible transient backgrounds. The sound speed profile determines possible refraction during the signal propagation that impedes accurate vertex reconstruction up the existence of multiple solutions. The attenuation length and noise level will determine the detector geometry required to achieve a given neutrino energy threshold. Transient noise sources need to be identified and characterized to separate them from neutrino induced events. The measurement of the sound speed profile and transient backgrounds are easier to accomplish in the sense that they only rely on time information. The attenuation length and noise level measurement depend on amplitude informa- tion and are thus subject to the calibration challenge discussed in the previous section.", "pages": [ 3, 4 ] }, { "title": "SPATS - Hardware", "content": "To carry out these measurements in the Antarctic ice at the South Pole, the site of the IceCube neutrino observatory, the South Pole Acoustic Test Setup (SPATS) has been designed and is successfully operated since January 2007 [31]. SPATS consists of four vertical strings that are deployed in the upper 500 m part of IceCube bore holes after the installation of the optical IceCube string. Horizontal baselines between 125 m and 543 m are covered. Each SPATS string is instrumented with seven stages, each containing an acoustic receiver and a transmitter. The SPATS sensor is made from a steel housing with three piezoceramic disks pressed to the inner wall at 120 · separation for full azimuthal coverage. The signals are amplified in the sensor module and the differential analogue signal is transmitted via twisted pair cable to the surface where it is digitized and time stamped in a StringPC. The data from all four strings are collected by a Master-PC housed in the IceCube counting house and are prepared for satellite transmission to the IceCube central data storage. At two positions an alternative sensor type, HADES, is installed, where the piezoceramic element is cast in resin and mounted below the steel hous- g. HADES is used for systematic studies of the sensor medium coupling. The SPATS transmitter consists of a piezoceramic ring cast in resin and frozen directly into the ice. It is connected to a high voltage pulser which is protected in a steel housing and steered by the String-PC. A schematic overview of the SPATS hardware is shown in Fig. 3. SPATS is complemented by a mobile acoustic transmitter, called 'pinger', which can be lowered into freshly drilled water filled IceCube holes while continuously emitting acoustic pulses with high stability. The pinger is retrieved from the hole after operation.", "pages": [ 4 ] }, { "title": "SPATS - Results", "content": "SPATS has measured the speed of sound depth profile in the Antarctic ice at the South Pole in the depth range from 80 m to 500 m using the retrievable pinger over horizontal baselines of 125 m [37]. The pinger emits longitudinal waves that propagate through the water column in the drill hole and are then transmitted into the ice. When the incidence on the water-ice interface is non-normal, part of the waves energy is transferred into a shear wave. Thus, the sound speed profile for pressure and shear waves could be determined. The speed of sound is found to be increasing in the top 200 m of the ice where a gradual transition from a snow/air mixture occurs (firn layer) and is found to be constant below that depth. The best fit values [37] for the sound speed v at 375 m depth and its gradient g at this depth are for pressure waves and for shear waves Since the gradient is compatible with zero only very little refraction is expected below the firn layer. Pinger data have also proven very valuable to determine the signal amplitude attenuation length [38] in the frequency range from 10 to 30 kHz. This analysis requires the comparison of the signal observed at different distances. To reduce systematic uncertainties from the sensor absolute calibration and angular response, attenuation lengths are derived for each sensor channel. For this the pinger, which produces highly reproducible pulses, is deployed in different IceCube drill holes at increasing distances, but aligned in direction, so that only a small range of the angular response of the sensor is probed. Averaging over all available sensor channels leads to an signal amplitude attenuation length [38] of This value is much smaller than the several kilometers initially expected from theoretical calculations [25]. To study this discrepancy a modified pinger has been constructed that emits gated sine bursts at different frequencies (30, 45, and 60 kHz). It has been used to study the frequency dependence of the attenuation length. Absorption, if it would be the main attenuation mechanism, is expected to be frequency independent, whereas the scattering coefficient would increase GLYPH<181> f 4 , where f is the frequency of the signal [25]. The data from the modified pinger are analyzed with the same methods as described above [36]. Figure 4 shows the attenuation coefficient measured in the different sensor channels for 30 kHz and 45 kHz as well as the mean value and spread of the data points. The signal-to-noise ratio for 60 kHz has been too poor to extract an attenuation length since the transmit response of the pinger piezoelectric element is small at this frequency. It can be seen that the two values are compatible with a frequency independent attenuation length and that an f 4 frequency dependence is hard to reconcile with the data. SPATS unbiased noise data, that are recorded every hour for 0 . 1 s have been used to study the background noise level. It has been shown that the noise is Gaussian and that the RMS is very stable over time [34]. Using the corrections on the sensor calibration discussed in the previous section, the absolute noise level below the firn layer has been estimated to be 14 mPa, corresponding to the signal expected from a 10 11 GeV neutrino interacting at a distance of 1000 m to the sensor [34]. However, the systematic uncertainty on the noise level is still large, making additional measurements with sensors precalibrated in ice desirable. An analysis of transient events triggering all four strings of the SPATS detector revealed only man-made sources: all events were reconstructed to either refreezing IceCube drill holes or Rodriguez Wells, large caverns in the ice used to circulate the water for the IceCube hot water drill system [34]. The absence of unidentified transient noise sources in the deep ice allowed setting an upper limit on the flux of ultra-high energy neutrinos. Figure 5 compares the limit set by SPATS to other experiments and the expected cosmogenic neutrino flux. i It has to be kept in mind that none of the acoustic experiments shown there were designed as neutrino detectors, but were built for site-exploration (SPATS) or parasitically use existing military hydrophone arrays to search for neutrino induced signals (SAUND II, ACoRNE).", "pages": [ 4, 5, 6 ] }, { "title": "A PATH FORWARDS", "content": "The SPATS results show that acoustic neutrino detection in the glacial ice at the South Pole is feasible. The acoustic attenuation length is shorter than for radio signals but of comparable magnitude which opens the possibility to design a hybrid radio/acoustic detector. Due to the intrinsically higher energy threshold of the acoustic technique a possible scenario is to use the radio sub-array to trigger the acoustic sub-array up to energies where the acoustic detector becomes fully efficient by itself. In this case a single in-time hit in an acoustic sensor can already be highly significant since all known backgrounds produce either only radio emission or only sound emission. The construction of such a detector at the South Pole will require the deployment of a few hundred strings per 100 km 2 instrumented area (estimating a spacing of 500 m between strings based on the measured attenuation lengths for radio and acoustic) reaching below the firn layer. A design which is scalable in size is desirable so that, once the magnitude of the flux of ultra-high energy neutrinos is measured, the detector can be expanded in size to accumulate sufficient event statistics to answer the physics questions discussed in the introductory section. The installation of a new detector consisting of several hundred to a few thousand strings will also require new techniques for drilling and deployment. One possibility that is being discussed is the use of autonomous drillingand/or melting-probes similar to the IceMole [40] prototype. To minimize human intervention in the drilling and deployment procedure, the sensors and cable would be stored in and deployed from the disposable icecraft which would remain in the ice at the bottom of the string after deployment. This allows for the hole to refreeze immediately after the passage of the probe. A schematic of the procedure is shown in Fig. 6. A large area detector at a remote site will also require new concepts for calibration, communication and power supply. The large overall extent and the large distances between the components prohibit a fully cabled design. Wireless data transmission and power generation at the site of the component by e.g. wind or solar power are required. Ideally one would have a combination of wind and solar power since beyond the polar circles solar power is unavailable half of the year. Valuable lessons can be learned from large area experiments which are already in operation, like the Pierre Auger Observatory [41].", "pages": [ 6, 7 ] }, { "title": "CONCLUSIONS", "content": "Ultra-high energy neutrinos offer a vast physics program covering astrophysics, cosmology, particle physics, and physics beyond the Standard Model. The acoustic neutrino detection technique has made large advances over the last few years: sensors have been designed and their behavior in ice is largely understood. The acoustic properties of the South Pole glacier have been measured and found to be suitable for neutrino detection. It is expected that acoustic can play an important part in a future hybrid neutrino telescope in ice. To realize this a strong R&D program has to be established and first promising studies have already been presented.", "pages": [ 7 ] }, { "title": "ACKNOWLEDGMENTS", "content": "T.K. is supported by the 'Helmholtz Alliance for Astroparticle Physics HAP' funded by the Initiative and Networking Fund of the Helmholtz Association.", "pages": [ 7 ] } ]

[ { "title": "ABSTRACT", "content": "We use the pulsational properties of the RR Lyrae variables in the globular cluster NGC 1851 to obtain detailed constraints of the various sub-stellar populations present along its horizontal branch. On the basis of detailed synthetic horizontal branch modeling, we find that minor helium variations ( Y ∼ 0.248-0.280) are able to reproduce the observed periods and amplitudes of the RR Lyrae variables, as well as the frequency of fundamental and first-overtone RR Lyrae stars. Comparison of number ratios amongst the blue and red horizontal branch components and the two observed subgiant branches also suggest that the RR Lyrae variables originated from the progeny of the bright subgiant branch. The RR Lyrae variables with a slightly enhanced helium ( Y ∼ 0.270-0.280) have longer periods at a given amplitude, as is seen with Oosterhoff II (OoII) RR Lyrae variables, whereas the RR Lyrae variables with Y ∼ 0.248-0.270 have shorter periods, exhibiting properties of Oosterhoff I (OoI) variables. This correlation does suggest that the pulsational properties of RR Lyrae stars can be very useful for tracing the various sub-populations and can provide suitable constraints on the multiple population phenomenon. It appears of great interest to explore whether this conclusion can be generalized to other globular clusters hosting multiple populations. Subject headings: globular clusters: general - globular clusters: individual(NGC 1851) stars: abundances, distances, Population II", "pages": [ 1 ] }, { "title": "THE HORIZONTAL BRANCH OF NGC1851: CONSTRAINTS FROM ITS RR LYRAE VARIABLES", "content": "Andrea Kunder 1 , Maurizio Salaris 2 , Santi Cassisi 3 , Roberto de Propris 1 , Alistair Walker 1 , Peter B. Stetson 4 , M'arcio Catelan 5,6 , P'ıa Amigo 5,6 E-mail: [email protected] accepted to AJ", "pages": [ 1 ] }, { "title": "1. INTRODUCTION", "content": "Especially following the detection of two distinct subgiant branches (SGBs) in the color-magnitude diagram (CMD) of NGC1851 (Milone et al. 2008), attempts to piece together the formation history of this cluster have become alluring. One promising explanation of the split between the bright SGB (SGBb) and faint SGB (SGBf) is that the two subpopulations differ in age by about 1 Gyr, and this scenario has been discussed in a number of studies (Milone et al. 2008; Carretta et al. 2011a,b; Gratton et al. 2012a). Another valid explanation is that the SGB splitting is due to differing C+N+O contents and that the two SGBs are nearly coeval (Cassisi et al. 2008; Ventura et al. 2009). The horizontal branch (HB) of NGC1851 is also bimodal, with both a prominent red HB clump and a blue tail. From the morphology of the HB and the main sequence (MS), strong helium variations within the cluster do not seem likely (Salaris et al. 2008; D'Antona et al. 2009), and recent spectroscopy of the blue HB stars suggests minor helium enhancements (Gratton et al. 2012a). Lastly, the red giant branch (RGB) is known to harbor different 1 NOAO-Cerro Tololo Inter-American Observatory, Casilla 603, La Serena, Chile populations (Grundahl et al. 1999; Calamida et al. 2007; Lee et al. 2009; Han et al. 2009). The stellar distribution along the HB of globular clusters is commonly used to understand their formation and evolution (Gratton et al. 2010; Dotter et al. 2010), and previous papers dealing with the modeling of the HB of NGC1851 (Salaris et al. 2008; Gratton et al. 2012a) used synthetic HB models to obtain scenarios of the formation this clusters bimodal horizontal branch. In this paper we will approach a more detailed investigation of the portion of the HB dealing with the instability strip (IS), by discussing the case of the RR Lyrae period distribution in NGC 1851. Period distributions of RR Lyrae stars have been shown to place strong constraints in the framework of canonical HB evolution. For example, the problem of the peaked distribution of the RR Lyrae periods in M3 (Castellani & Tornambe 1981; Rood & Crocker 1989) has challenged model predictions as nicely described by Catelan (2004). Recent studies have come to explain both the M3 period distribution and HB morphology as the consequence of a range of initial He together with a uniform total RGB mass loss (with a very small spread) (Caloi & D'Antona 2008), or a suitable bimodal mass-loss efficiency along the RGB but a single initial He abundance (Castellani et al. 2005). The RR Lyrae properties of NGC 1851 have been used to describe this cluster as 'truly an unusual Oosterhoff type I object' (Downes et al. 2004). This is largely because NGC1851 is at the extreme end of the OoI-type clusters, with its RR Lyrae variables having not only a longer than average period for its Oosterhoff class, but also a ratio of first overtone RR Lyrae (RR1) to fundamental mode RR Lyrae (RR0) stars more in line with OoII-type GCs (Walker 1998). The suggestion has also been made that the RR Lyrae variables can be divided into two subgroups based on their Ca uvby photometry (Lee et al. 2009), although they acknowledge that their sample is small, and the apparent bi-modality may merely reflect a calcium metallicity spread in the variables. Further suggestions that this cluster may be different than other Galactic GCs comes from its phase-space distribution, which indicates it may be associated with the Canis Major dwarf (Frinchaboy et al. 2004; Martin 2004, however see also L'opez-Corredoira et al. 2007 who show that the signatures of the Canis Major dwarf can be fully accounted for by Galactic models without new substructures). Reproducing theoretically both the morphology of its unusual CMD as well as the period distribution of its RR Lyrae variables is an important step in piecing together the formation scenario of NGC 1851.", "pages": [ 1, 2 ] }, { "title": "2.1. Sample and Completeness", "content": "The most complete study of the RR Lyrae variables in NGC1851 was carried out by Walker (1998). He presents 33 variables in a 13.6 arcmin 2 area centered on this cluster in the B , V and I passbands, 30 of which are RR Lyrae stars. Recently Sumerel et al. (2004) discovered 19 additional variables and Downes et al. (2004) reported eleven variables, all within 40' of the cluster center. There is overlap in these two samples, as described by the 2011 update of NGC1851 in the Clement et al. (2001) catalog, and most of the new discoveries are RR Lyrae stars, although the classification for a handful of these stars is still uncertain. Neither Sumerel et al. (2004) nor Downes et al. (2004) provide calibrated mean magnitudes or amplitudes for their variables, making it difficult to use these stars in our analysis. As there is no indication that the Walker (1998) sample is incomplete at distances greater than 40' from the cluster center, we limit our sample of RR Lyrae stars with which to compare our HB models to this outer region. Of the 29 RR Lyrae stars studied by Walker (1998) that are greater than 40' from the center, 25 have both unblended magnitudes and well determined periods. Thus our sample of variables is 25/29 or 86% complete (outside the inner core). The position on the sky of our sample of variables is shown in Figure 1, and the central 40' is designated by a circle. Light curves for these stars are presented by Walker (1998) in the BVI passbands, so for these 25 RR Lyrae variables, robust mean magnitudes, periods and amplitudes are available. The edges of the instability strip, judged by the measured colors of variables near the strip boundaries, were also determined by Walker (1998), as well as the RR1 - RR0 boundary.", "pages": [ 2 ] }, { "title": "2.2. Period-Amplitude Diagram", "content": "Clement & Shelton (1999a) show that the position of an RR Lyrae variable in the period-amplitude (PA) diagram is not a function of metal abundance, but rather of Oosterhoff type, and derive PA relations for OoI and OoII-type RR Lyrae stars empirically. More recently Cacciari et al. (2005) study the PA plane of 3 typical OoI-type GCs, 3 typical OoII-type GCs and 3 intermediate types and find that there is a unique periodamplitude relation independent of metallicity for RR0 variables in OoI-, OoII- and intermediate-type clusters. The periods and V -amplitude of our sample of RR0 Lyrae variables is shown in Figure 2, and the periodamplitude relation of typical OoI and OoII-type systems is over-plotted. We note that although many of the RR Lyrae stars in NGC 1851 have periods and amplitudes that cause them to fall near the OoI PA relation, there are a number of stars following the OoII PA relation. It is well known that the Blazhko effect, or other effects such as a rapidly changing period, can cause scatter in the PA plane (Clement & Shelton 1999a). The Blazhko effect causes the amplitude of light variation to vary over timescales longer than the basic pulsation period. The Walker (1998) RR Lyrae variables were observed over an ample time frame (126 total frames observed over 15 nights during a 1.5 year time span), so determining the amplitudes using the average light curves of the RR Lyrae stars is straightforward. Nevertheless a visual determination of the change in amplitudes in each RR0 is obtained and shown as an error-bar in Figure 2. The change from the average light curve amplitude to the maximum Blazhko amplitude ranges from ∼ 0.1 - 0.3 mag, and photometric uncertainties lead to amplitude uncertainties of ∼ 0.01-0.05 mag. We conclude that even when taking amplitude variations into account, the variables are both OoI and OoII-type. In comparison, Figure 2 shows the periods and V -amplitudes of 1097 RR0 Lyrae variables in 39 Galactic globular clusters. The data for this diagram come from Table 1, where each cluster is listed along with the number of RR0 Lyrae stars within the cluster that have well determined periods and V -amplitudes. For completeness, the [Fe / H] from Carretta et al. (2009), HB-type and Oosterhoff type for each GC are also given. This sample of RR0 Lyrae variables are divided by their position in the period-amplitude plane following the lines that Clement & Shelton (1999a) derived for Oosterhoff I and Oosterhoff II RR0 stars (see Figure 2). Here an OoI RR Lyrae variable is defined by and an OoII-type RR Lyrae variable by where A ( V ) is the V -amplitude and P is the period. We define an Oosterhoff ratio for each GC, which is simply the number of OoI-type RR0 Lyrae stars compared to the total number of RR0 Lyrae stars in the GC, OoI RR0 / Tot RR0 . We note that Cacciari et al. (2005) showed for the M3 RR0 Lyrae stars, a quadratic PA relation is a closer fit than a linear one. However, their relation does not approximate high amplitude RR0 Lyrae variables well (A V > 1.5 mag), largely because such variables are absent in M3. As our comprehensive sample includes a handful of such stars (one OoI- and three OoII-type RR Lyrae), the Cacciari et al. (2005) relation is not used here. Figure 3 shows the histogram of the Oosterhoff ratio of the GCs in our sample. Most of these clusters lack a complete sample of RR Lyrae variables and at least some of the RR Lyrae amplitudes are likely affected by the Blazhko effect or other light curve 'noise'. But even with these caveats, it is clear from the figure that the Oosterhoff ratio splits the GCs into two groups; the OoI-type clusters have RR Lyrae variables with shorter periods for a given amplitude and hence have larger Oosterhoff ratios (with respect to the OoII-type clusters). Further, there is an absence of clusters falling in the 'gap'. We therefore believe that our Oosterhoff ratio is useful to distinguish between OoI- and OoII-type GCs. Moreover, this ratio may be used to evaluate the degree for which a GC is a typical OoI- or OoII-type cluster. The majority of OoI-type GCs have a RR Lyrae star population in which OoI RR0 / Tot RR0 > 0.8 (i.e., 80% of the variables can be defined by Equation 1). In contrast, the Oosterhoff ratio of NGC 1851 is 0.74. Equation 1 and 2 are defined somewhat arbitrarily, and therefore we experimented with a variety cuts (where we modified the zero-points and slopes) to distinguish between OoI- and OoII-type stars. The Oosterhoff ratio of NGC 1851 varied between ∼ 0.63-0.75, a percentage that is smaller than 83% of the rest of the OoI-type clusters. This indicates that NGC1851 contains variables with more OoII-like periods and V -amplitudes than the majority of the other Milky Way OoI-type GCs. The other three OoI GCs that have comparatively small Oosterhoff ratios are NGC 362 (0.71), NGC 4590 (0.58) and NGC 6362 (0.67). It has been noted that the RR Lyrae variables of NGC 362 and NGC 1851 are remarkably similar in the period-amplitude diagram, suggesting similar masses and luminosities. This is also seen here, as they have very similar Oosterhoff ratios. Recently it was found that like NGC 1851, NGC 362 also has a split sub-giant branch, although the SGBf component includes only a few percent of the total number of SGB stars of the cluster (just ∼ 2-3%, A. Milone private communication). New photometry of the NGC 1851 RR Lyrae variables, in which periods and amplitudes of the inner RR Lyrae stars have been obtained, suggest that NGC 1851 has a slightly higher Oosterhoff ratio of ∼ 0.80 (Amigo et al 2012, in preparation). We therefore suspect that the Oosterhoff ratio of NGC 1851, whereas still lower than the majority of the OoI-type clusters, is not as abnormal as the globular clusters NGC 362, NGC 4590 and NGC6362, discussed above.", "pages": [ 2, 3 ] }, { "title": "2.3. First Overtone vs. The Fundamental Mode", "content": "The number ratio of first-overtone RR Lyrae stars to total RR Lyrae stars, N 1 /N tot , is usually given to quantify the frequency of the different RR Lyrae pulsators. In general OoII-type GCs have about 2-4 times the frequency of RR1 stars as OoI-type GCs. This is thought to be related to the transition temperature between the instability strip for first overtone pulsation and fundamental one. For example, moving the transition from RR0 to RR1 variables toward lower temperatures (i.e., transforming fundamental in first overtone pulsators) has the twofold effect of increasing the periods of the RR0 Lyrae population, as well as increasing the relative number of first overtones. This is discussed in detail by Castellani, Caputo & Castellani (2003). NGC1851 has a large N 1 /N tot with respect to most OoI-type GCs. This ratio strongly depends on the completeness of the sample, and because NGC 1851 is too crowded for ground-based observations to resolve, the Walker (1998) sample is incomplete at distances close to the core. Assuming the Walker (1998) RR Lyrae sample is complete at distances larger than 40' (see Figure 1), N 1 /N tot = 0.27, one of the largest for OoI-type GCs. For example, from the Castellani et al. (2003) compilation of 32 clusters with 12 or more pulsators and well recognized period and pulsation modes, only 3 of the 17 OoI GCs have an N 1 /N tot greater than 0.27. Recent studies have discovered RR Lyrae stars closer than 40' from the center. Downes et al. (2004) find N 1 /N tot = 0.54, and using Clement's catalog, the value of this ratio is ∼ 0.40. These ratios are somewhat uncertain, however, because of the lack of mode identification for some of these newly discovered pulsators. The periods listed in Sumerel et al. (2004) are 'tentative' and the periods derived by Downes et al. (2004) do not cover a full period for any of their objects. Further, because these stars lack amplitudes, their position in a PA diagram also can not be used as a diagnostic to identify fundamentals and first overtones. Although their is no consensus on the value of N 1 /N tot yet, it is clear that this ratio is at least 0.27, and likely even larger. A N 1 /N tot larger than 0.27 is also consistent with the results from Amigo et al. (2012, in preparation).", "pages": [ 4 ] }, { "title": "3. SYNTHETIC HB MODELING", "content": "Previous synthetic HB models for NGC 1851 have been presented by Catelan et al. (1998), and more recently by Salaris et al. (2008), Han et al. (2009) and Gratton et al. (2012a). Salaris et al. (2008) compared their simulations to the HST observations by Milone et al. (2008) and found two satisfactory scenarios to reproduce the CMD of HB stars. In both of these models, the blue HB, red HB and variable stars are predicted to come from the SGBf, and the stars from the SGBb are confined to the red portion of the observed sequence. This inference was based on the number ratio SGBf/SGBb=55:45 determined in Milone et al. (2008). Salaris et al. (2008) also found that the initial He abundance of HB stars had to be relatively uniform to reproduce the CMD derived by Milone et al. (2008). In these data the measured magnitudes and colors of the RR Lyrae population are at random phases; therefore the portion of the observed HB crossing the IS could not be used for detailed con- straints on the models. However, they did verify that recent theoretical pulsation models of RR Lyrae stars (Di Criscienzo et al. 2004a) predict an instability strip for NGC1851 too red by ∼ 0.03 -0.04 mag in ( F 606 W -F 814 W ) compared to the Milone et al. (2008) data. Han et al. (2009), on the other hand, found from their UVI photometry that the RR Lyrae variables could come from both the SGBb and the SGBf. They construct two population models for NGC 1851. In the first, the second generation population is more enhanced in metallicity but not in helium (∆Z=0.0004 and ∆age = 0.1Gyr), and in the second, both metal and helium abundances are enhanced (∆Z=0.0004, ∆Y=0.05, and ∆age = 0.1Gyr). They find that their ∆Z-only model is in conflict with the observed CMDs of NGC 1851, but that their ∆Z+∆Y model is in good agreement with the observations from the MS to the HB. In this model, the RR Lyrae variables exhibit different He abundances. Because their RR Lyrae variable sample was found at a random phase of pulsation, the RR Lyrae colors could not be used as a stringent constraint in their models and the RR Lyrae star periods were not discussed. More recently Gratton et al. (2012a) considered the revised SGBb/SGBf ratio determined by Milone et al. (2009) and introduced new spectroscopic constraints; they find a small difference in the iron content between the SGBb and SGBf, and argue that an age spread of ∼ 1.5 Gyr is the most viable explanation for the splitting SGB. They also find that the RHB stars separate into two groups depending on their O and Na abundances, and that the BHB stars are slightly helium enriched as compared to the RHB stars. Hence to satisfy these constraints, the HB is modeled with four different components, with the IS originating from the SGBb. That each SGB hosts multiple generations of stars is shown from spectroscopy of stars on the double SGB (Lardo et al. 2012), making it likely that multiple components may be needed to model the HB. Our own synthetic HB calculations described below are aimed at answering the following question: what is the most straight-forward way to reproduce the RR Lyrae instability strip of NGC 1851 - and in particular the pulsational properties of its RR Lyrae variables? We seek to provide a simple and attractive explanation for the cluster HB and IS morphology, keeping the number of free parameters to a minimum, yet still reproducing the RR Lyrae star properties that make this cluster stand out as having an unusual Oosterhoff type.", "pages": [ 4 ] }, { "title": "3.1. Synthetic HB models", "content": "The HB evolutionary tracks used here are from the BaSTI stellar library (Pietrinferni et al. 2004, 2006, 2009) and have already been described by Salaris et al. (2008) and Cassisi et al. (2008). They are also the same that Gratton et al. (2012a) employed. Briefly, evolutionary tracks are for a normal α -enhanced ([ α /Fe]=0.4) metal mixture, with [Fe / H]= -1.31 dex and Y =0.248. The HB tracks were interpolated among the models with Y =0.248 and additional BaSTI models with Y =0.300, to determine HB tracks for intermediate values of Y , at the same iron content. Similarly, to determine HB tracks with a milder CNO-enhancement, an interpolation between the reference set and the models with the CNO sum enhanced by 0.3 dex (Pietrinferni et al. 2009) is used for a portion of the synthesis, as in Gratton et al. (2012a). Hence, the [Fe / H], α -enhancement and CNOenhancement is consistent with spectroscopic results from Carretta et al. (2011a) and Gratton et al. (2012a). We wish to remind the reader that, as long as the CNO sum is unchanged, the effect of the observed CNONa anticorrelations (overimposed to a standard α -enhanced metal mixture) on the evolutionary tracks and isochrones is negligible, and standard α -enhanced models are adequate to represent the whole cluster population. Only an enhancement of the CNO sum requires the calculation of appropriate models. On the other hand CNONa anticorrelations even at constant CNO affect the bolometric corrections of filters like B and U (Sbordone et al. 2011) at low effective temperatures, but not longer wavelength filters. The four HB components described by Gratton et al. (2012a) are used as a starting points for our calculations. Objects from our synthetic HB that fall within the observed IS from Walker (1998) are considered RR Lyrae variables (this region is labeled in Figure 4) and their period is calculated from the pulsation equation given by Di Criscienzo et al. (2004a). The intensity mean magnitudes and colors given by Walker (1998) are used as a comparison to the synthetic HB, because the static magnitudes and colors from stellar evolution models are represented better by intensity-averaged quantities (Di Criscienzo et al. 2004a). Although the Walker (1998) observations include the BVI passbands, we employ only the V and I magnitudes, because - as discussed before - they are not affected by the observed CNONa abundance anticorrelations. In addition to the observed V and ( V -I ) distribution of the HB stars in the CMD, and the observed (B :V :R) (blue :variable :red HB) ratio of (B :V :R) = (33 ± 8:10 ± 5:56 ± 11) (in line with the results by Catelan et al. 1998; Saviane et al. 1998), we impose as a further constraint on our simulation the observed distribution of the RR Lyrae periods. As in Gratton et al. (2012a) we adopted E(B -V)=0.02 mag (Walker 1998) and fixed the apparent distance modulus to (m -M) V =15.56 mag by matching the observed mean magnitude of the RHB with our synthetic counterpart. The (B :V :R) ratio of our 'best fit' simulation is (27 :9 :64), consistent with the observed value. For reasons that will become clear in the discussion that follows, we consider a preliminary reference age of 11 Gyr for the progenitors of the RHB stars. This implies, for the assumed metallicity and a 'normal' Y =0.248, an initial mass of 0.86M /circledot for the stars at the tip of the RGB. The HB components are described below. (1) As in Gratton et al. (2012a), the majority of the RHB population is modeled with normal CNO abundance, a normal Y =0.248, and a Gaussian mass distribution with < M > = 0 . 67 ± 0 . 005 M /circledot . This corresponds to a total mean mass loss ∆M=0.19 M /circledot along the RGB, for the assumed 11 Gyr age. (2) A smaller RHB subpopulation, that is Ba-rich, makes up ∼ 10% of the HB population. It is modelled, as in Gratton et al. (2012a), employing a Gaussian mass distribution with < M > = 0 . 65 ± 0 . 004 M /circledot (corresponding to ∆M ∼ 0.21 M /circledot ) an 0.15 dex enhanced CNO abundance, and normal Y =0.248. If we assume that the mean total mass loss has to be constant among all cluster RGB stars - and equal to ∆M=0.19 M /circledot as determined for the rest of the RHB component - the mean value of the mass for this HB sub-population implies an age ∼ 1 Gyr older for the progenitors of this HB component. (3) The horizontal part of the BHB, including the RR Lyrae instability strip, makes up ∼ 10% of the cluster stellar content. This component is the focus here, and is the only one that is modified from Gratton et al. (2012a). In particular, instead of adopting a constant He abundance Y =0.265, the helium content for stars between the blue end of the red clump and the beginning of the BHB blue tail has a continuous distribution between Y =0.248 and 0.280. A simple, flat probability distribution for Y and constant ∆M=0.19 M /circledot (for an age of 11 Gyr) with a 1 σ Gaussian spread of 0.005 M /circledot - as for the RHB stars - for all RGB progenitors provide a good match to the observed RR Lyrae periods, as discussed below. The mean He abundance in the IS is < Y > =0.271, close to the constant abundance Y =0.265 employed by Gratton et al. (2012a) for this component, and the mean mass is < M > = 0 . 634 M /circledot . It is worth noting that the observed HB distribution of these stars is well matched by both this simulation and the simulation by Gratton et al. (2012a). This spread in He content is necessary to reproduce the observed period distribution. (4) The blue tail of the BHB population makes up ∼ 20% of the HB stars. As in Gratton et al. (2012a) this component is modeled with normal CNO abundance and Y =0.28. The Gaussian mass distribution has < M > =0 . 59 ± 0 . 005 M /circledot , that for an age of 11 Gyr would correspond to a mean ∆M=0 . 22 M /circledot . If the mean total mass loss is instead fixed at ∆M=0.19 M /circledot , this value of < M > implies that the progenitors of the BHB blue tail stars are ∼ 1.5 Gyr older that the RHB component with normal CNO and Y . Notice that the constraint on the progenitor Y is weaker for BHB stars (see discussion in Gratton et al. 2012a) and a small spread of order 0.01 may be present. These results from the HB synthetic modeling can be interpreted in terms of the progeny of the SGBb and SGBf subpopulations (the ratios ∼ 2/3 and ∼ 1/3 of the total SGB population, respectively, are adopted as determined by Milone et al. 2009) as follows: (1) The sum of the fraction of stars in the blue tail of the BHB and in the mildly CNO-enhanced Ba-rich RHB component is ∼ 35% of the total HB population. If we consider as a reference the CMD of 11 Gyr old SGB stars with 'normal' Y and CNO abundance, the progenitors of these two HB components will be distributed along a fainter SGB than the reference one. In the case of the progenitors of the BHB component this is an age effect, for a change of Y does not have a major effect on the SGB luminosity. For the Ba-rich RHB progenitors the reason is the slightly higher age and the mildly enhanced CNO abundance, that act both in the direction of producing a fainter SGB. As a result, both the Ba-rich RHB stars and the blue tail HB progenitors display an approximately coincident SGB, that we tentatively identify as the SGBf in the cluster CMD. (2) The sum of the fraction of stars in the horizontal BHB (including the IS) and the RHB stars with normal composition amounts to ∼ 65% of the total HB population. We identify their progenitors as the stars harbored by the SGBb in the cluster CMD. To conclude this section, it is worth noting that the exact value of the assumed reference age (11 Gyr) is not critical. Had a different age been assumed, i.e. 10 or 12 Gyr, the previous conclusions will still be valid. The only difference is that all values of ∆M would need to be shifted downwards (or upwards) by ∼ 0.02 M /circledot - to keep the mass distribution along the HB unchanged - but the interpretation of the results would be identical. Finally, the different chemical composition - and small age differences - assigned to the SGBb and SGBf populations do not affect substantially SGB and RGB timescales; as a consequence, the number ratio SGBb/SGBf will be approximately equal to the number ratio of their HB progeny.", "pages": [ 4, 5, 6 ] }, { "title": "3.2. Comparison With Observations", "content": "This synthetic HB model is shown compared to the observed one in Figure 4 where an observational scatter of σ V,I = 0.01 mag is assumed. The four components are high-lighted for clarity and the RR Lyrae region is labeled. Our full synthetic HB model reproduces two peculiarities in the CMD of NGC 1851 pointed out by Brocato et al. (1999), namely the clump of stars near the red edge of the HB and the slightly tilted HB (∆ tilt V ∼ 0.1 mag). Features such as these are present also in NGC6362 (which has an RR Lyrae population with properties very similar to NGC 1851), and in the extreme cases of NGC 6388 and NGC 6441 (∆ tilt V ∼ 0.5 mag). The focus here concerns the component that includes the instability strip, which comes from the SGBb. As Walker (1998) mentioned, the ZAHB is very cleanly defined and is not horizontal, being slightly brighter at bluer colors. This is reproduced in our synthetic HB model by stars that range in helium abundance from Y =0.25-0.28, and range in mass from 0.61 to 0.65M /circledot . The theoretical periods and pulsation amplitudes from the RR Lyrae variables in our synthetic HB model are compared to the observed periods and amplitudes in Figure 5. Here the observed periods come from 28 RR0 and 18 RR1 variables as determined by Walker (1998) and Sumerel et al. (2004) to encompass all the data available in the literature (see the 2011 update of NGC 1851 in Clement et al. 2001). We also compare the theoretical periods with 27 RR0 and 18 RR1 variables as determined by Amigo et al. (2012, in preparation). These authors derive periods for the recently identified inner RR Lyrae variables based on light curves with ∼ 200-300 points in each of the B -V - and I -passbands. The observed amplitudes come exclusively from the Walker (1998) RR0 Lyrae sample, as Sumerel et al. (2004) present instrumental magnitudes only. The amplitudes from the Amigo et al. (2012, in preparation) sample are not used, because unlike when determining periods, amplitudes can be affected by crowding and blending issues (e.g., Majaess et al. 2012) and we do not have a feel for how/if blending affects their (preliminary) amplitude determinations. Marconi et al. (2011) provided a detailed comparison between the impact of the He abundance on the pulsation properties of RR Lyrae stars and concluded that He content marginally affects the pulsation behavior of RR Lyrae stars. They noted that the increase in the average pulsational period associated with the He increase is only due to the brighter luminosities which character- ize He-enhanced evolutionary models. As a consequence, from a theoretical point of view the impact of an Heenhancement on the pulsation properties of RR Lyrae stars can be directly taken into account when adopting evolutionary tracks for the appropriate He abundance and pulsational model predictions obtained for a canonical He abundance. The periods of the RR1 variables are fundamentalized via log P 0 ∼ logP + 0 . 127, where P 0 is the fundamental mode period, and the theoretical periods are calculated for all HB stars falling within the observed IS using the Di Criscienzo et al. (2004a) RR Lyrae pulsation models. The mean observed RR0 Lyrae period in NGC1851 is < P 0 > = 0.571 (Catelan 2009a), and from our model < P 0 > = 0.569 ± 0.006 d is calculated, where the uncertainty is the error in the mean. The number of RR Lyrae stars in the simulations is ∼ 5 times larger than the number of observed RR Lyrae stars. In this way, in the synthetic HB model, the effect of statistical fluctuations in the number of objects at a given magnitude and color is minimized. The observed minimum fundamentalized period of the first overtones and the minimum fundamentalized period from the synthetic HB is very similar, P F ∼ 0.34 days. It has been shown that the minimum fundamentalized period is a robust observable to constrain the evolutionary properties of RR Lyrae stars (Bono et al. 1995), so it is especially encouraging that the observed and theoretical values agree. We have performed a Kolmogorov-Smirnov (KS) test to establish whether one can reject the null hypothesis that the two samples of periods (observed and synthetic) come from the same distribution. From a comparison between the combined Walker (1998) and Sumerel et al. (2004) sample and the synthetic RR Lyrae stars, the KStest returns a probability P=0.86, well above the default threshold P th =0.05 below which one rejects the null hypothesis. When the Amigo et al. period distribution is compared with our theoretical one, if we assume that both samples are drawn from the same parent population, the samples will differ by this amount or more 94% of the time. On this basis, we find that the synthetic periods from our simulated HB and the observed periods agree well with each other. Theoretical pulsation amplitudes are also determined from the Di Criscienzo et al. (2004a) relations, although three points should be taken into account: 1. Theoretical amplitudes depend on the the mixinglength, l/H p , (where l/H p is the pressure scale-height), which is uncertain and may change from the blue to the red edge of the IS (Marconi et al. 2003). Pulsation amplitudes are affected by l/H p in the sense that a larger value of l/H p translates to smaller pulsation amplitudes as a consequence of the larger efficiency of convective motions and, hence larger quenching to the pulsation mechanism provided by convection. data provided by Di Criscienzo et al. (2004a), we do not expect a great accuracy in our A ( V ) estimates. In addition, Di Criscienzo et al. (2004a) have shown that for P < 0 . 68 d, pulsation model predictions for the V -amplitude are still more affected by a change in the adopted mixing length value (see lower panel of the quoted figure): for increasing mixing length values they predict a significant decrease in the dependence of the V -amplitude on the pulsation period. When using a l/H p =2.0, the Walker (1998) V -amplitude distribution is similar to that observed, although the theoretical amplitudes appear to be ∼ 0.1 mag larger. This is not completely surprising considering the scatter in the Di Criscienzo et al. (2004a) V -amplitude relation as well as the magnitude corrections discussed above. Extrapolating linearly between l/H p =1.5 and 2.0, an increase in the mixing length of 0.1 would cause a decrease of the theoretical amplitudes by ∼ 0.08 mag. Such a decrease would provide a satisfactory agreement between theory and observations. A 0.1 change in l/H p is well within the uncertainties in the mixing length calibration, and not nearly large enough to affect the predicted pulsation periods (Bono & Stellingwerf 1994; Di Criscienzo et al. 2004b; Marconi & Degl'Innocenti 2007). An l/H p =2.0 was also used by Bono et al. (2007) to derive a visual distance modulus from nonlinear convective models of RR Lyrae stars to NGC 1851 of ( m -M ) V =15.52 ± 0.11 mag, which is similar to the distance modulus adopted here. In contrast, when using a l/H p =1.5, Bono et al. (2007) find a distance modulus of ( m -M ) V =15.40 ± 0.12. Therefore we conclude that using a larger value of l/H p , ( l/H p ∼ 2.0), provides a consistent comparison between our synthetic HB and pulsational predictions. In general, the RR Lyrae variables with Y < 0.27 fall in the OoI area of the PA diagram, whereas the RR Lyrae variables with Y > 0.27 fall close to the OoII line. Assuming that the period-amplitude diagram can be effectively used to classify RR Lyrae stars into an Oosterhoff type, this means that He and Oosterhoff type are correlated in this cluster. This is not completely unexpected, as an increase in He makes RR Lyrae variables brighter and, as a consequence, higher helium abundance makes their pulsational period longer (Bono et al. 1997; Marconi 2009). On a general ground, for a given total mass the HB stars with Y < 0.27 are redder than those with Y > 0.27. The red part of the IS, where the fundamental mode RR Lyrae stars reside, is consequently more populated, and a smaller N 1 /N tot is obtained. Our synthetic HB yields N 1 /N tot ∼ 0.1 for the stars with Y < 0.27. This is a ratio that is seen for the majority of the OoI-type GCs. In contrast, the blue part of the horizontal branch contains more stars with higher helium abundances, and as the first overtone RR Lyrae variables reside in the blue part of the IS, a larger N 1 /N tot is obtained. Our synthetic HB yields N 1 /N tot ∼ 0.45 for stars with Y > 0.27, a ratio more in line with OoII-type GCs. The observed ratio of first overtone to total RR Lyrae variables for NGC 1851 in our sample is N 1 /N tot ∼ 0.30 and is easily explained (and reproduced with our synthetic HB) by the spread in Y along the RR Lyrae instability strip. We find that simulations using a constant helium for the portion of the HB containing the IS (as in Gratton et al. 2012a) do not fit the constraints given by the NGC 1851 RR Lyrae variables as well. For example, adopting Y =0.265 results in an N 1 / N tot = 0.11 (versus the observed N 1 / N tot = 0.28). We note that Milone et al. (2008) provide an upper limit to a possible dispersion in helium abundance of ∆Y=0.026 between the two SGBs in NGC1851, a value close to the spread assumed in the synthetic HB presented here. Other estimates of the He spread in NGC1851 are slightly larger, i.e., ∆Y=0.04 (Ventura et al. 2009), ∆Y=0.05 (Han et al. 2009) or ∆Y=0.048 (Gratton et al. 2010).", "pages": [ 6, 7, 8 ] }, { "title": "4. DISCUSSION AND CONCLUSIONS", "content": "The population distribution of the stars along the HB has been modeled assuming the presence of (at least) four populations with differing helium contents. In our simulations, the only parameter we vary is the initial He abundance of the HB progenitors, keeping the same total RGB mass loss for all components. Both the RR Lyrae period distribution as well as the number ratio of first overtone RR Lyrae to total RR Lyrae stars, N 1 / N tot , provides constraints pertaining to the component of the HB containing the IS. It is straight forward to reproduce the observed distribution of RR Lyrae stars inside the instability strip with minor He variations ( Y ∼ 0.2480.280) and from a HB subpopulation corresponding to the progeny of a fraction of the SGBb stellar population. Therefore, the IS of NGC 1851 belongs to a second generation (SG) of stars. That a SG exists within the SGBb component is in agreement with recent spectroscopy of the SGB stars, showing that each SGB hosts multiple subgenerations of stars (Lardo et al. 2012). D'Antona & Caloi (2008) have also postulated that the longer periods of the NGC 1851 RR Lyrae may indicate that these variables may belong to the SG, and our synthetic HB strengthens this notion. Chemical anomalies in GCs suggest that self-enrichment is a common feature among GCs. The quasi-constancy of heavy metals in most GCs leads to the assumption that abundance variations are not or scarcely affected by SN ejecta, but involves formation of a second generation of stars from matter processed into the FG stars. The SG will most probably show a spread in He (D'Antona & Caloi 2008) because the self-enriched material may come from different progenitors that have different chemical peculiarities, or may be diluted in different fractions with matter from the first generation (FG). We remark that such a helium spread is an essential ingredient in order to reproduce the pulsation properties of the RR Lyrae population as a whole. Simulations using constant He across the IS give synthetic period distributions that do not match the observed one as well and result in the observed N 1 / N tot being lower than what is observed. Actually, one can note that variations of He in 19 GCs have also been deduced by Bragaglia et al. (2010) from 1400 RGB stars with abundance determinations. As discussed in Gratton et al. (2010), a starto-star spread in the He abundance may explain many aspects of the horizontal branches of GCs. It is worth pointing out that there have been suggestions of problems in the late stages of HB evolution in current HB tracks (Catelan 2009b; Valcarce & Catelan 2008). Moreover, Catelan (2009b) show that in the case of NGC1851, over-luminous stars on the blue HB could be interpreted by an underestimate of the luminosity evolution along the HB rather than in terms of a moderate level of helium enrichment. Here we do not attempt to resolve this ambiguity for NGC 1851; rather, we assume that the evolutionary tracks adopted represent the HB evolution accurately, and remind the reader that our comparisons are ultimately subject to both theoretical and observational uncertainties. The pulsation periods and amplitudes from the RR Lyrae variables resulting from variations in He along the IS have different characteristics. The RR Lyrae variables with a 'normal' helium have periods and amplitudes, as well as a N 1 /N tot ratio, that is inline with OoI-type GCs. In contrast, the RR Lyrae variables with slightly enhanced He (0.27 < Y < 0.28) have longer periods and a higher ratio of N 1 /N tot , indicative of RR Lyrae variables in OoII-type GCs. In the absence of spectroscopy of the RR Lyrae variables in NGC 1851, the synthetic horizontal part of the HB and RR Lyrae instability strip presented here is the simplest one that reproduces the available observations with the smallest amount of free parameters. New observations of the RR Lyrae variables may require more complex modeling, however, and would be particularly interesting. Oosterhoff-I clusters tend to be more metal-rich and host fainter RR Lyrae variables than OoII clusters (Caputo et al. 2000). As the metallicity has an effect on the absolute magnitude of an RR Lyrae, it has been difficult to disentangle whether the metallicity difference alone is affecting the brightness differences, or whether there are differences in the intrinsic magnitudes of RR Lyrae variables in OoI and OoII globular clusters caused by something other than just metallicity (like evolution or helium). In this cluster, where an internal spread in [Fe / H] is small at most, our results indicate that a difference in helium abundance in the RR Lyrae variables is affecting where in the PA diagram the RR Lyrae star falls. Our model consists of a BHB that is He-enriched ( Y ∼ 0.28) yet older than the RHB ( Y =0.248). This can be explained if the cluster was formed by a process such as a merger with populations that differ in He and age. Such a scenario has already been discussed by e.g. Carretta et al. (2011b) and Bekki & Yong (2012). Hence the BHB would not be a second generation (SG) of stars originating from the same population as the RHB (the SGBb). Since this paper was submitted, results from an intermediate resolution spectroscopic analysis of the two SGBs by Gratton et al. (2012b) indicate that the [Fe / H] difference between the SGBb and SGBf is ∼ 0.07 dex, the SGBf being more metal rich. The RR Lyrae stars in our scenario are the progeny of SGBb; hence this metallicity difference does not affect our results. We find that for BHB stars hotter than the IS a 0.1 dex increase in [Fe / H] at fixed (V -I) changes the HB masses (at fixed Y ) by ∼ 0.01 M /circledot , and M V changes by ∼ 0.01 mag. Therefore the effect of such a [Fe / H] difference between the two SGBs has a negligible effect on our HB modeling. We have shown that a spread in He reproduces the pulsational properties of the RR Lyrae sample as a whole, indicating the presence of a SG of stars in NGC 1851. Our analysis therefore demonstrates that RR Lyrae properties in a given GC can provide suitable constraints on the multiple population phenomenon in that GC. It is worth carrying out more studies of this kind to investigate further this connection with the occurrence of the multiple population phenomenon, especially in GCs with a sizable population of RR Lyrae stars and in which the stellar chemical patterns are well known. The authors thank Aaron Dotter for helpful discussions. S.C warmly thanks PRIN INAF 2009 'Formation and early evolution of massive star clusters' (P.I.: R. Gratton) and PRIN INAF 2011 'Multiple populations in Globular Clusters: their role in the Galaxy assembly' (P.I.: E. Carretta) for financial support. Support for M.C. and P.A. is provided by the Chilean Ministry for the Economy, Development, and Tourism's Programa Iniciativa Cient'ıfica Milenio through grant P07-021-F, awarded to The Milky Way Millennium Nucleus; by Proyecto Fondecyt Regular #1110326; by the BASAL Center for Astrophysics and Associated Technologies (PFB-06); and by Proyecto Anillo ACT-86. We would like to thank the anonymous referee whose thorough report has led to substantial improvements to this paper.", "pages": [ 8, 9 ] }, { "title": "REFERENCES", "content": "David R., Bond, H.E. & Onken, C. 2001, AJ, 121, 318 Arellano Ferro, A., Figuera Jaimes, R., Giridhar, S., Bramich, D.M., Hernandez Santisteban, J. V., & Kuppuswamy, K. 2011, arXiv1106.1880 Bekki, K. & Yong, D. 2012, MNRAS, 419, 2063 Benko, J.M., Bakos, G.A. & Nuspl, J. 2006, MNRAS, 372, 1657 Bono, G. & Stellingwerf, R.F. 1994, ApJS, 93, 233 Bono, G., Caputo, F., & Stellingwerf, R.F. 1995, ApJS, 99, 263 Bono, G., Caputo, V., Castellani, V., & Marconi, M. 1997, A&AS, 121, 327 Bono, G., Caputo, F., & Di Criscienzo, M. 2007, A&A, 476, 779 Borissova, J., Catelan, M. & Valchev, T. 2001, MNRAS, 324, 77 Bragaglia, A., Carretta, E., Gratton, R., DOrazi, V., Cassisi, S. & Lucatello, S. 2010, A&A, 505, 139 Brocato, E., Castellani, V., Raimondo, G., & Walker, A.R. 1999, 527, 230 Cacciari, C. 1979, AJ, 84, 1542 Cacciari, C., Corwin, T. M., & Carney, B. W. 2005, AJ, 129, 267 Calamida, A., Bono, G., Stetson, P. B., et al. 2007, ApJ, 670, 400 Caloi, V. & D'Antona, F. 2008, ApJ, 673, 847 Kaluzny, J., Olech, A., Thompson, I. B., Pych, W., Krzemi'nski, Olech, A., Kaluzny, J., Thompson, I. B., Pych, W., Krzemi'nski, W. & Schwarzenberg-Czerny, A. 2001, MNRAS, 321, 421 Papadakis, I.; Hatzidimitriou, D.; Croke, B. F. W.; Papamastorakis, I. 2000, AJ, 119, 851", "pages": [ 9, 10 ] } ]

[ { "title": "ABSTRACT", "content": "We present time-resolved spectroscopy and photometry of CSS 120422:111127+571239 (= SBS 1108+574), a recently discovered SU UMatype dwarf nova whose 55-minute orbital period is well below the CV period minimum of ∼ 78 minutes. In contrast with most other known CVs, its spectrum features He I emission of comparable strength to the Balmer lines, implying a hydrogen abundance less than 0.1 of long period CVs-but still at least 10 times higher than than in AM CVn stars. Together, the short orbital period and remarkable helium-to-hydrogen ratio suggest that mass transfer in CSS 120422 began near the end of the donor star's main-sequence lifetime, meaning that the system is probably an AM CVn progenitor as described by Podsiadlowski, Han, & Rappaport (2003). Moreover, a Doppler tomogram of the H α line reveals two distinct regions of enhanced emission. While one is the result of the stream-disk impact, the other is probably attributable to spiral disk structure generated when material in the outer disk achieves a 2:1 orbital resonance with respect to the donor. Subject headings: stars: cataclysmic variables - stars: dwarf nova - stars: evolution - stars: individual(CSS 120422:111127+571239) - stars: individual(SBS 1108+574)", "pages": [ 1 ] }, { "title": "A New Sub-Period-Minimum CV with Partial Hydrogen Depletion and Evidence of Spiral Disk Structure", "content": "C. Littlefield, 1 P. Garnavich, 1 A. Applegate, 5 K. Magno, 1 R. Pogge, 6 J. Irwin, 2 G. H. Marion, 2 J. Vink'o, 3 , 4 R. Kirshner 2", "pages": [ 1 ] }, { "title": "1.1. Cataclysmic Variables and the Period Minimum", "content": "Cataclysmic variables (CVs) are close binary systems in which a white dwarf accretes matter from a companion star which overflows its Roche lobe. Very few CVs have orbital periods below the so-called period minimum, which has been observed to be ∼ 78 minutes (Hellier 2001). As Kolb & Baraffe (1999) explain, the period minimum is a consequence of emerging electron degeneracy in the donor star and the strong, inverse relationship between donor-star density and a CV's orbital period. When a non-degenerate donor sheds some of its mass, its density increases, prompting the CV's orbital period to shrink. A degenerate donor, however, has exactly the opposite response to mass loss, so once the secondary star becomes degenerate, it will have reached its maximum density-and thus the shortest orbital period possible for the system. Thereafter, the CV will evolve toward a longer orbital period, its period having 'bounced' off the minimum. Nevertheless, a handful of CVs, some with orbital periods as short as several minutes, defy the 78-minute period minimum. Almost every CV in this category is an AM CVn-type system, featuring a rich helium spectrum lacking even the slightest hint of hydrogen. Heliumenhanced stars are denser than their hydrogen-dominated counterparts, so they adhere to a shorter period minimum (Augusteijn et al. 1996). There are three theoretical avenues of formation for these unusual stars (Nelemans et al. 2010). First, the distance between two detached white dwarfs might decrease as a result of gravitational radiation, causing the lower-mass companion to eventually overflow its Roche lobe. Second, a low-mass, helium-fusing star in an interacting binary might lose so much mass that its core would be unable to sustain the necessary temperature and pressure for continued helium fusion, producing a semi-degenerate core. Third, if mass transfer were to commence near the end of the main-sequence lifetime of the donor star, the secondary star would shed its hydrogen envelope, exposing the relatively dense, hydrogen-deficient interior of the star. Spectroscopically, the helium-to-hydrogen ratio would gradually increase as the system continued to evolve. Unlike the first two models, the evolved-main-sequence-donor model predicts the presence of some hydrogen in the donor star (Podsiadlowski, Han, & Rappaport 2003; Breedt et al. 2012). Oddly, a few CVs are below the period minimum, but with strong hydrogen lines in their spectra, they cannot be AM CVn stars. Instead, these stars belong to the SU UMa sequence of CVs; consequently, while in outburst, they display superhumps-periodic photometric oscillations attributed to a precessing, eccentric accretion disk-whose period is within several percent of the orbital period. However, the secondary stars in the sub-period-minimum SU UMa systems are not the garden-variety donor stars found in most CVs above the minimum. In two of these CVs, EI Psc (Thorstensen et al. 2002) and V485 Cen (Augusteijn et al. 1996), the He I 667.8-nm-to-H α ratio is elevated with respect to CVs above the period minimum, and the secondary star is plainly visible in optical and near-infrared spectra. A third system below the minimum, SDSS J150722.30+523039.8, has very weak helium lines and shows no trace of the donor, leading Littlefair et al. (2007) to conclude that the secondary is a brown dwarf. Another sub-period-minimum star, CSS 100603:112253-111037 (hereinafter, CSS 100603), is a hybrid of these three systems. Like EI Psc and V485 Cen, its spectrum has strong emission from both hydrogen and neutral helium. Like the brown-dwarf system, though, the donor star is invisible in spectra, and the estimated mass ratio is quite low. Breedt et al. (2012) argue that CSS 100603 is probably an AM CVn progenitor in which the donor star is an evolved main-sequence star. Here, we present observations of CSS 120422, a very short period system similar to CSS 100603, but with several interesting differences.", "pages": [ 2, 3 ] }, { "title": "1.2. CSS 120422:111127+571239", "content": "On 2012 April 22, the Catalina Sky Survey (Drake et al. 2009) detected an outburst of CSS 120422:111127+571239 (hereinafter, CSS 120422), a previously undiscovered CV. With a peak observed brightness of V ≈ 15.4, the system brightened by at least 4 magnitudes from its typical brightness in the Catalina data. 1 Even prior to the discovery of its very short orbital period, the system garnered attention because it is remarkably blue during quiescence, even for a CV (Kato 2012). The VSNET collaboration (Kato et al. 2004) immediately launched an extensive observing campaign, finding an initial superhump period of 56 minutes, well below the 78-minute period minimum for hydrogen-rich CVs (Kato et al. 2012). Although most known CVs below the minimum are AM CVn systems, spectroscopy revealed a hydrogen-dominated spectrum, eliminating the AM CVn hypothesis (Garnavich et al. 2012). After the system returned to quiescence, Pavlenko et al. (in preparation; preliminary findings discussed in Kato et al. 2012) reported unusually strong helium lines in CSS 120422's quiescent spectrum.", "pages": [ 3 ] }, { "title": "2.1. VATT Photometry", "content": "We observed CSS 120422 with the Vatican Advanced Technology Telescope (VATT) and VATT4K CCD imager on 2012 June 15, 16 and 18 (UT), performing time-resolved photometry through a Bessell B filter on the first two nights a Bessell R filter on the final night. Clouds prevented observations on June 17. The CCD pixels were binned 2 × 2, and only the central 500 rows were read out in order to reduce the overhead on each exposure. Generally exposures were 60 sec long with a readout time of 11 sec, but seeing degraded during the second half of the final night, so the exposure time was increased to 100 sec. The CCD images were bias-subtracted and flat-field-corrected using dithered exposures taken during twilight. We used a star 2 arcmin south of CSS 120422 as a comparison star (USNO-B1.0 position RA=11:11:26.83 DEC=+57:12:38.9 J2000). The SDSS photometric catalog magnitudes for the comparison star convert to B = 16 . 23 ± 0 . 05 and R = 14 . 62 ± 0 . 04 mag. Located 1 arcmin west of the comparison star, the check star has SDSS converted magnitudes of B = 19 . 1 and R = 18 . 0. Aperture photometry using DAOPHOT in IRAF produced the light curves shown in Fig. 1.", "pages": [ 4 ] }, { "title": "2.2. FLWO Spectroscopy", "content": "We obtained spectra of CSS 120422 with the 1.5m Tillinghast telescope at the Fred L. Whipple Observatory on 2012 May 12 and 13 (UT), roughly three weeks after the initial detection of the outburst. The FAST spectrograph utilized a 300 line mm -1 grating and covered the spectral range between 360 nm and 750 nm. On May 12, a single 30 minute exposure with a 3-arcsec-wide slit was obtained. During the following night, a series of nine images, each with an exposure time of 420 sec, were taken through a 2-arcsec-wide slit. Seeing was approximately 1.5 arcsec, and the slit was set to a position angle of 90 · on both nights. Table 1 gives a log of the spectral observations. The CCD images were bias-subtracted and flat-field-corrected using normalized internal lamps. Extracted using the 'twod' package in IRAF, 2 the spectra were wavelength- calibrated using images of an internal HeNeAr lamp interspersed with the stellar spectra. The CSS 120422 spectra, having all been obtained at an airmass less than 1.15, were fluxcalibrated using a spectrum of Feige 34 taken at an airmass of 1.03. The combined spectrum, displayed in Figure 2, reveals a blue continuum with a conspicuous Balmer jump. Weak emission lines of H α and He I at both 587.5 nm and 667.8 nm are visible in addition to broad Balmer absorption features with emission cores, a spectrum typical of a hydrogen-accreting dwarf nova declining from outburst (Garnavich et al. 2012). Because the system was returning to quiescence, its absorption-dominated outburst spectrum was in the process of transitioning back into an emission-line spectrum during our observations. Consequently, the simultaneous presence of emission and absorption undoubtedly diluted the strengths of many lines.", "pages": [ 4, 5, 6 ] }, { "title": "2.3. LBT Spectroscopy", "content": "We obtained spectra of CSS 120422 with the Large Binocular Telescope (LBT) and Multi-Object Dual Spectrograph (MODS; Pogge et al. 2010) on 2012 June 16 (UT). The sixteen 400-second exposures employed a 1.0 arcsec slit oriented to the parallactic angle in order to minimize slit losses. The spectra cover a time period of about 1 hour and 15 minutes, or about 1.3 orbits of the binary. Table 1 presents a log of the LBT spectral observations. The red and blue MODS CCD spectra were bias-subtracted and flat-field-corrected using normalized internal lamp illumination images. As with the FLWO data, the spectra were extracted using the 'twod' package in IRAF. We used an internal Neon lamp on the red side and an Argon lamp for the blue channel for wavelength calibration, and the star HZ 44 served as the spectrophotometric standard for the flux calibration. The brightness of CSS 120422 in the LBT data is approximately 2.8 mag fainter than the FLWO spectra obtained in May. The combined LBT spectrum appears in Figure 3. Similar to the May spectra, a strong blue continuum is obvious, but by June, a wide variety of emission lines had appeared. Double-peaked Balmer and He I emission lines which originate in the accretion disk are evident, as are weaker lines of intermediate-mass elements, especially Si II and Ca II. The H α profile is well fit by two Gaussians separated by 1200 km s -1 , each with a full width at half maximum (FWHM) of 940 km s -1 . The center of mass of the line is shifted to the red by 50 km s -1 . The near-equal strength of the He I and Balmer lines is particularly striking. In normal, hydrogen-rich CVs, the He I emission is much weaker than the hydrogen lines, and in AM CVn systems, He I dwarfs the negligible hydrogen emission. However, in CSS 120422, the average total flux of the He I 587.5 nm line is 20% higher than that of H β . To put this in perspective, the H β line is typically three times as strong as the He I 587.5-nm feature in CVs. As alluded to earlier, other systems, such as CSS 100603 and EI Psc, exhibit He I-to-H ratios comparable to that of CSS 120422. Williams & Ferguson (1982) modeled the line strengths in several long-period CVs and found that they were somewhat depleted in hydrogen compared to the Sun. Specifically, the He I 5876-to-H β flux ratio observed in long-period CVs was about 0.2 ± 0 . 1, implying a hydrogen abundance [H]=log 10 (H/He)/Solar(H/He)= -3 . 5 according to Williams & Ferguson (1982). Nagel, Rauch, & Werner (2009) concluded that significant H α emission lines are present at [H]= -5 and that AM CVn stars must have [H] < -6 for hydrogen emission to be undetectable (see Fig. 6). With some interpolation, these studies indicate that in CSS 120422, which features a He I 587.6-nm line of similar strength to the H α line, the hydrogen abundance is [H] ≈ -4 . 5. We also measured this line ratio for CSS 100603 from the public SDSS spectrum and find a hydrogen abundance that is similar to CSS 120422. The He I 587.6-nm line was not measured by Thorstensen et al., but the He I 667.8-nm line strength suggests that EI Psc has a hydrogen abundance which is a factor of three higher in comparison with these recently discovered objects. In CSS 120422, Ca II emission from the infrared triplet and HK lines is also quite prominent. Additionally, we identify Si II 634.7/637.1 nm lines, both of which have been observed in the AM CVn star CP Eri (Groot et al. 2001). The feature at 923 nm may be due to Mg II, but the line expected at 788.5 is not detected (Marsh et al. 1991). Alternatively, it might be a Paschen line. There is also a weak He II 468.6 nm feature blended with the He I 471.3 nm line, and on the red side of the He I 587.5 nm line, there is weak emission which may be due to low-velocity Na I. CSS 120422's spectrum contains more than just this forest of emission lines. In particular, the H β and H γ lines both sit within broad absorption troughs in the continuum. (To a lesser extent, this is also true of the H α line, too.) In other short-period CVs, the WD is responsible for producing comparable features at these wavelengths (Rodr'ıguez-Gil et al. 2005), so we attribute the absorption in CSS 120422 to pressure-broadened absorption from the WD's photosphere. These absorption lines suggest that the WD must contribute much of the system's overall optical flux (e.g. Patterson, Thorstensen, & Knigge 2008), likely indicating a low mass-transfer rate during quiescence. If this inference is correct, it would account for the unusually blue quiescent color noted by Kato (2012). A quiescent accretion disk will usually be cooler than the WD, causing the overall flux of the system to be redder, but in a system with a low rate of mass transfer, the disk will be relatively tenuous and dim, enabling the WD's blue continuum to overwhelm the disk's meager contribution.", "pages": [ 6, 7, 8, 9, 10 ] }, { "title": "3.1.1. Orbital Period, Ephemeris, & Light Curve Morphology", "content": "We removed a linear trend from each night of VATT photometry and created a power spectrum from all three nights using the phase-dispersion-minimization (PDM) algorithm (Stellingwerf 1978). PDM scans a user-selected period range, calculates a phase plot for each trial period, divides the phase plot into a user-specified number of bins, and quantifies the total amount of scatter in each bin. The best candidate periods will have the least amount of scatter in their phase plots, a technique which is well-suited to analyzing non-sinusoidal signals, such as the waveform of CSS 120422's orbital period. In the PDM power spectrum (see Fig. 5), the strongest signal is at 55.36 ± 0 . 13 minutes, but there is also a competing alias at 53.28 ± 0 . 12 minutes. Using a completely independent dataset, Kato et al. (2012) report the longer of these two periods as the orbital period and find no evidence of the shorter period (T. Kato, private communication). We show the phase plot of all three nights of data (both B and R bands) in Figure 5 using the ephemeris of where phase zero corresponds with the peak in the light curve. The phased light curve is very saw-toothed, with a gradual rise, a sharp peak, and then a rapid drop in brightness. When the data are binned to improve the signal-to-noise ratio, the waveform displays a weakly double-humped structure, with two distinct maxima per orbital cycle. In this regard, the light curve resembles that of WZ Sge-type CVs. The maximum at phase zero and the broad minimum centered on phase ∼ 0.25 are both readily apparent, but the maximum seen near phase ∼ 0.55 and the ensuing minimum around phase ∼ 0.7 are much more subtle. The phase plot generated by Kato et al. (in preparation) is generally similar, but the double-peaked structure is somewhat difficult to discern. The morphologies of the individual light curves are remarkably variable from night-tonight and even from cycle-to-cycle. For example, the light curve from 2012 June 15 should have contained three minima similar to the one centered on photometric phase ∼ 0.2 in the phase plot in Figure 5, but the second one is nearly non-existent. Likewise, on the following night, an expected peak in the light curve did not occur and was followed instead by a comparatively gentle decrease in flux. In the very next cycle, however, the peak returned, and the post-peak dip had a different shape. Clearly, the source of the orbital modulation, which we believe to be tidally-induced spiral structure in the outer disk (see Section 4.2.1), is capable of significant variability on timescales of less than one hour.", "pages": [ 10, 11 ] }, { "title": "3.1.2. Inferred Mass Ratio", "content": "Patterson (1998) reported an empirical relation between a CV's mass ratio ( q = m 2 / m 1 ) and its fractional superhump period excess ( /epsilon1 = [ P sh /P orb ] -1). Although the exact physical processes underlying this /epsilon1 -q correlation are unclear, it provides approximate estimates of q using just the orbital and superhump periods, making it a particularly useful technique for analyzing a system whose mass ratio cannot be ascertained by other means. Relying upon a detailed survey of superhumps, Kato et al. (2009) updated Patterson's original formula slightly, obtaining /epsilon1 = 0 . 16 q + 0 . 25 q 2 , where /epsilon1 is determined using the shortest P sh rather than the mean P sh . Applying the quadratic formula and rejecting the negative root (to ensure a positive value of q ) yields a more convenient, explicit function: q = -0 . 32+2 √ 0 . 0256 + /epsilon1. Using this formula, a P sh of 55.971 minutes (Kato et al. 2012), and our P orb of 55.36 minutes, we obtain q = 0 . 06 ± 0 . 01. However, since the /epsilon1 -q relation in other systems is only approximate, the uncertainty for q is probably greatly underestimated. The correlation between /epsilon1 and q underscores that the 53-minute signal in our photometry is an alias of the true period at 55 minutes. If the signal at 53 minutes were the actual orbital period, then q = 0 . 24, a value which would be difficult to reconcile with the complete nondetection of the donor and the low mass ratios of other short-period CVs. Additionally, it would be inconsistent with the evidence of spiral shocks produced by disk material at the 2:1 resonance which, as Section 4.2.1 explains, could form only if q /lessorsimilar 0 . 1. Kato et al. (2012) point out that a mass ratio of 0.06 is not nearly as extreme as the mass ratios of other short-period dwarf novae, suggesting an unusually dense and massive secondary star which has shed much of its hydrogen envelope via mass transfer. In Section 4.1, we reach a similar conclusion based on our interpretation of the system's spectrum.", "pages": [ 11, 12 ] }, { "title": "3.2. Spectroscopy", "content": "The double-peaked Balmer and He I emission lines in the LBT spectra exhibit a radialvelocity variation over the photometric period (Figure 7). In contrast to many AM CVn stars and CSS 100603, these lines do not show stationary, central peaks in our trailed spectra, but there is an 'S-wave' present between the peaks of each of these lines. The single-peaked He II emission shows a particularly prominent S-wave for approximately half of the orbit, a feature which we attribute to the bright spot created by the stream-disk collision. The non-appearance of the He II line for over half of the orbit implies that the inclination of the system is sufficiently high that the bright spot rotates behind the disk as seen from Earth.", "pages": [ 12 ] }, { "title": "3.2.1. Equivalent Width Variation", "content": "We estimated the equivalent width (EW) of several of the brightest lines using the 'splot' routine in IRAF. Each line was measured several times with slightly different continuum values and the resulting differences used to estimate an error. Only H α showed significant variation above the noise, and the result is shown in Figure 6 along with the phase plot. The H α EW varies by 20%, with the lowest EW occurring at photometric phase ∼ 0.8, when the orbital modulation nears its peak flux.", "pages": [ 13 ] }, { "title": "3.2.2. Radial Velocities", "content": "To avoid influencing the measurement of radial velocity variations with changes in the line shapes, we estimated the line velocities using the method described by Shafter (1983). The Shafter method multiples the red and blue wings of an emission line by a pair of Gaussian functions separated by a certain number of ˚ Angstroms. The wavelength of the Gaussians are shifted until the difference in flux measured by the two functions is zero. This wavelength is assumed to be the central wavelength of the emission line. This is repeated for all the spectra, at which point it is possible to construct a radial velocity curve. The best separation between the Gaussian functions is determined by fitting each set of radial velocity curves with a sinusoidal func- tion and choosing the one that gives the lowest χ 2 parameter. This method was applied to the H α line and the fit parameters as a function of Gaussian separation are shown in Figure 8. The minimum χ 2 occurred at a full Gaussian separation of 61 ˚ A, and the χ 2 parameter rises sharply when the separation is increased. The minimum χ 2 was 12 for sixteen spectra and four fit parameters (amplitude, K 1 ; systemic velocity, γ ; orbital period, P ; phase, φ ). We find that K 1 = 75 ± 4 km s -1 , γ = 41 ± 3 km s -1 , and P = 52 . 2 ± 2 . 3 min, a value within 1.4 σ of the 55.36-minute photometric period. At first, our estimate of K 1 = 75 ± 4 km s -1 might seem to contradict the small mass ratio that we have inferred, as it is impossible to obtain so large a value for the WD's radial-velocity am- plitude using q = 0 . 06. Nevertheless, K 1 in a CV can often be significantly different than the true orbital motion of the WD (e.g. U Geminorum: Long & Gilliland 1999). Since an accretion disk's emission is often asymmetric, techniques which rely upon a measurement of the wings of a spectral line will frequently produce inaccurate radial velocities for the WD (e.g. Appendix A.2 in Hellier 2001). For example, although CSS 100603 has a mass ratio of just q = 0 . 017, K 1 in that system is 69 . 4 ± 2 . 9 km s -1 for the H α line, a number which would be unreasonably high if it actually represented the WD's motion. The resulting radial velocity curve is shown in Figure 7 where the phase has been set to match the photometric phase. The maximum brightness in the orbital modulation occurs at the maximum disk redshift.", "pages": [ 13, 14 ] }, { "title": "4.1. Evolutionary Track of CSS 120422", "content": "Several attributes of CSS 120422 distinguish it from most other sub-period-minimum systems with detectable hydrogen. For example, EI Psc and V485 Cen both contain prominent secondary stars which are relatively easy to detect spectroscopically. The prevailing explanation for the secondary stars in these two systems is that they are evolved stars which have shed their outer envelopes. Both are abnormally luminous for such short orbital periods (Uemura et al. 2002). While EI Psc and V485 Cen both exhibit helium enrichment like CSS 120422, the non-detection of the donor in CSS 120422 out to 950 nm suggests that it is cool and dim, in agreement with the system's very low mass ratio. While the combination of an invisible donor and a sub-minimum period could be the signature of a brown-dwarf secondary, the greatly elevated levels of helium in CSS 120422 do not support this scenario. Brown dwarfs, by definition, lack the mass necessary to enter the main sequence, so they should not be particularly rich in helium. Thus, the helium enhancement in EI Psc and V485 Cen is inconsistent with the presence of brown-dwarf donors in those systems (Politano 2004); indeed, in the spectra of brown-dwarf CVs, the He I emission is quite subdued (e.g., Littlefair et al. 2007; Unda-Sanzana et al. 2008). Although the He I/H α ratio is sensitive to both pressure and temperature, the intense He I lines in CSS 120422 disfavor the possibility that the secondary star is a brown dwarf. Indeed, at first blush, CSS 120422 is almost identical to CSS 100603, the system reported by Breedt et al. (2012). In both of these sub-period-minimum systems, the accretion disk shows high levels of both hydrogen and helium, and the late-type donor is spectroscopically undetectable. In their theoretical examination of AM CVn progenitors, Nelemans et al. (2010) identify the presence of hydrogen as conspicuous evidence favoring the evolved-CV channel over the double-white-dwarf and helium-star channels of AM CVn formation; if the secondary were either a white dwarf or a helium star, hydrogen would likely be undetectable. Thus, both of these systems are excellent candidates for AM CVn progenitors evolving pursuant to the evolved-main-sequence-donor model. In neither system is the donor fully degenerate yet. To reach this conclusion about CSS 100603, Breedt et al. (2012) relied upon models of fully degenerate helium stars, and we took a similar approach with CSS 120422. Specifically, we used Equation 15 in Verbunt & Rappaport (1988) (a mass-radius relation for a fully degenerate helium star) and set it equal to Equation 6 in Knigge (2006) (a formula for the radius of a Roche-lobe-filling star given its mass and orbital period). Together, these formulae reveal that if the secondary were a degenerate helium star with an orbital period of 0.9227 hours, its mass would be 0.0083M /circledot -which, given the mass ratio of 0.06, implies an unreasonably low WD mass of 0.14M /circledot . Alternatively, if we adopt 0.83M /circledot as the mass of the primary, 3 the expected mass of the secondary would be 0.05M /circledot according to the mass ratio. A helium star of this mass would have a radius of just 0.03R /circledot , making it considerably smaller than the radius of its Roche lobe (0.08R /circledot ). Based on these considerations, we conclude that the donor in CSS 120422 is semi-degenerate, and the system will continue to evolve toward the shorter orbital periods which characterize the majority of AM CVn stars. While it is likely that these two systems are the products of very similar evolutionary processes, CSS 120422 does have several features which differentiate it from its cousin. The two most obvious dissimilarities are that CSS 120422 is an additional ∼ 10 minutes below the period minimum and has a mass ratio ∼ 4 times greater than Breedt's CV. Moreover, CSS 120422 shows more extensive heavy-element enrichment, especially the Si II 634.7/637.1nm doublet and the near-infrared Ca II triplet, both of which are weak or non-existent in CSS 100603. As Section 4.2 explains, we also find that CSS 120422's disk is decidedly non-uniform-especially in the H α wavelength-and likely contains spiral structure. No comparable features have been reported in CSS 100603.", "pages": [ 15, 16, 17 ] }, { "title": "4.2. Non-Uniform Disk Emission", "content": "The H α spectroscopy shows evidence of multiple emission regions on the disk. In the trailed H α spectra in Fig. 9, the most prominent feature is the bright spot's classic S-wave, which oscillates between the H α line's two peaks over the course of the orbit. This Swave vanishes as the bright spot moves from zero-velocity to maximum blueshift, only to reappear abruptly as the bright spot attains its maximum blueshift. During the bright spot's invisibility, a blueshifted absorption-like feature appears, but we suspect that it is the mere absence of emission rather than true absorption. In addition, near the maximum redshift of the bright spot, an even stronger emission feature appears in the blue wing of the H α line. As the bright spot transitions from maximum redshift to zero radial velocity, the blueshifted emission feature also moves toward zero velocity, but its intensity plummets dramatically. The appearance of the H β and He I lines (also in Fig. 9) is somewhat less remarkable. The S-wave from the bright spot remains apparent, but the second emission feature is very subtle compared to the H α trailed spectrum. As with the H α line, the bright spot vanishes in these wavelengths as it moves from zero radial velocity to maximum blueshift. The velocity information contained within a trailed spectrum can be used to reconstruct an indirect image of the disk in velocity coordinates (as opposed to spatial coordinates). Known as Doppler tomography, this technique essentially generates an inside-out image of the disk (Marsh & Horne 1988). Using the Doppler tomography algorithm of Spruit (1998), we find that the competing S-waves in the H α trailed spectrum correspond with two dis- inct emission regions located on opposite sides of the disk (Fig. 10). The bright spot in the -V x ,+V y quadrant, which has a noticeably higher velocity than the other emission, is almost certainly attributable to the shock created by the stream-disk interaction, but the emission in the +V x , -V y quadrant-which consists of both intermediate- and low-velocity components-presents a more of a mystery, one which we address in the following two subsections.", "pages": [ 17, 18, 19 ] }, { "title": "4.2.1. Tidally-Induced Spiral Shocks", "content": "In other CVs, there have been at least two major proposed explanations for non-uniform emission of this sort. One possibility is that the outer disk contains spiral arms produced by the tidal torque of the secondary. Spiral shocks facilitate angular momentum transfer in the disk, and in CVs with very low mass ratios, spirals can be permanent features in the disk because material in the outer disk attains a 2:1 orbital resonance with the donor (Lin & Papaloizou 1979). In CVs with relatively high mass ratios, tidal forces dissipate the outer disk before it can expand to the 2:1 resonance, but calculations by both Lin & Papaloizou (1979) and Osaki & Meyer (2002) found that when q /lessorsimilar 0 . 08, the 2:1 resonance is inside the tidal truncation radius. Observationally, several CVs have shown strong evidence of spiral structure stemming from the 2:1 resonance (e.g. Aviles et al. 2010). We favor the presence of spiral shocks in as the source of the features observed in our Doppler tomogram. Given that q = 0 . 06, the 2:1 resonance in CSS 120422 rests within the tidal truncation radius, making it possible for permanent spiral structure to develop as described in Lin & Papaloizou (1979). We further note that the overall appearance of our tomogram is very similar to a simulated tomogram of a disk with spiral arms caused by disk material at this very resonance (Aviles et al. 2010). To test more quantitatively for the presence of disk material at the 2:1 resonance, we again adopted a WD mass of 0.83M /circledot (see our footnote 3) and computed the Keplerian velocity of disk material with an orbital period of exactly half of the system's orbital period. According to the formula the corresponding resonance velocity for these parameters is 750 km s -1 , which we plot as a circle around the origin in the tomogram. Although there are obvious pitfalls with simply assuming the WD's mass, the overlaid circle intersects much of the intense, non-uniform emission, furnishing circumstantial evidence that the disk in CSS 120422 did extend to the 2:1 resonance when we observed the system. Just as importantly, Kunze & Speith (2005) calculated that spiral structure in such a system would produce a double-peaked light curve and presented a simulated phase plot which matches ours with uncanny accuracy. Kunze and Speith predict that both the minima and maxima will be unequal, with the deepest minimum coming at phase ∼ 0.25 and a weaker one at phase ∼ 0.75; the global maximum occurs at phase zero, and the weaker peak at phase ∼ 0.55. 4 Compared with the corresponding features in Kunze's and Speith's simulated light curve, the secondary maximum and minimum in our phase plot (Fig. 5) are somewhat feeble in appearance. This relatively minor disparity might be because the simulations assume a higher inclination than the one actually observed in CSS 120422.", "pages": [ 19, 20 ] }, { "title": "4.2.2. Accretion-Stream Overflow", "content": "Alternatively, non-uniform disk emission might be the result of an accretion stream which overflows the disk after its initial collision, a mechanism which has received a great deal of attention in theoretical studies (e.g., Armitage & Livio 1996, 1998). The overflowing stream would cloak portions of the disk, producing absorption at certain photometric phases and spectroscopic velocities, as it accelerated toward the inner disk. The reimpact of the stream with the disk, in turn, would produce an inner hotspot which, according to Armitage & Livio (1998), would be much more apparent in a system with a low-luminosity accretion disk. The emission in the +V x , -V y quadrant of our tomogram, therefore, might be such a feature. A shortcoming of the stream-overflow hypothesis is that the inner bright spot should have a noticeably higher velocity than the outer bright spot because the WD's strong gravity would significantly accelerate the overflowing stream. We observe the reverse; the second bright spot in the +V x , -V y quadrant has a lower velocity than the stream-disk interaction. Thus, the lower-than-expected velocity weighs against the possibility of an overflowing accretion stream. Furthermore, an overflowing stream would probably produce absorption at most phases, something which we do not observe. Though we cannot rule out the possibility of stream overflow, spiral shocks elegantly weave the spectroscopy, photometry, and mass ratio of CSS 120422 into a reasonably coherent theory of the system, one which is less speculative than the stream-overflow model.", "pages": [ 20 ] }, { "title": "5. Conclusion", "content": "We have reported photometry and spectroscopy of CSS 120422:111127+571239, a CV with an orbital period over 20 minutes below the period minimum. While the system is hydrogen-rich, its helium-to-hydrogen ratio is much higher than in typical SU UMa-type CVs, and the donor is completely invisible in our spectra. We identify spectroscopic and photometric periods of 52.2 minutes and 55.36 minutes, respectively. Using the 55-minute period and the previously reported superhump period (Kato et al. 2012), we estimate a mass ratio of q = 0 . 06. Furthermore, Doppler tomography reveals two distinct regions of intense H α emission on the disk, consistent with spiral shocks produced when material in the outer disk reaches a 2:1 resonance with the secondary. Drawing upon theoretical light curves of low-mass-ratio CVs, we suspect that these spiral arms are responsible for the intermittently double-peaked orbital modulation in the photometry, which is reminiscent of the variation observed in WZ Sge stars. The best explanation for the short orbital period and the elevated helium abundance is that CSS 120422 is a progenitor of an AM CVn system following the evolved-CV track, similar to the system reported by Breedt et al. (2012). The discovery of two systems of this type in such rapid succession substantiates theoretical predictions (e.g. Nelemans et al. 2010) that the evolved CV channel of evolution can contribute significantly to the galactic AM CVn population. 5", "pages": [ 21 ] }, { "title": "Acknowledgments", "content": "This paper has benefitted from the comments and suggestions of the two referees, to whom we are grateful. We thank Taichi Kato for sharing with us a power spectrum of his photometry of CSS 120422 and helping us to rule out the 53-minute orbital period. R.K. received support from N.S.F. grant AST-1211196. K.M. and A.A. received funding for this research through the Research Experience for Undergraduates program offered by the Department of Physics at the University of Notre Dame. The results presented here are partially based on observations made with the VATT: the Alice P. Lennon Telescope and the Thomas J. Bannan Astrophysics Facility. We thank Richard Boyle and the Vatican Observatory Research Group for providing time on the VATT for this project. The MODS spectrographs were built with funding from the NSF grant AST-9987045 and the NSF Telescope System Instrumentation Program (TSIP), with additional funds from the Ohio Board of Regents and the Ohio State University Office of Research. The LBT is an international collaboration among institutions in the United States, Italy and Germany. LBT Corporation partners are: The University of Arizona on behalf of the Arizona university system; Istituto Nazionale di Astrofisica, Italy; LBT Beteiligungsgesellschaft, Germany, representing the Max-Planck Society, the Astrophysical Institute Potsdam, and Heidelberg University; The Ohio State University, and The Research Corporation, on behalf of The University of Notre Dame, University of Minnesota and University of Virginia. Facilities: LBT, VATT, FLWO", "pages": [ 21, 22 ] }, { "title": "REFERENCES", "content": "Armitage, P. J. & Livio, M. 1996, ApJ, 470, 1024 Armitage, P. J. & Livio, M. 1998, ApJ, 493, 898 Augusteijn, T., van der Hooft, F., de Jong, J. A., & van Paradijs, J. 1996, A&A, 311, 889 Aviles, A., Zharikov, S., Tovmassian, G., et al. 2010, ApJ, 711, 389 Breedt, E., G¨ansicke, B. T., Marsh, T. R., et al. 2012, MNRAS, 425, 2548 Drake, A. J., Djorgovski, S. G., Mahabal, A., et al. 2009 ApJ, 696, 870 Garnavich, P., Littlefield, C., et al. 2012, The Astronomer's Telegram, 4112, 1 Groot, P. J., Nelemans, G., Steeghs, D., & Marsh, T. R. 2001, ApJ, 558, L123 Hellier, C., 2001, Cataclysmic Variable Stars: How and Why They Vary. London: Springer- Praxis Kato, T. 2012, VSNET-alert 14475 Kato, T., et al. 2012, arXiv:1210.0678v1 Kato, T., Uemura, M., Ishioka, R., et al. 2004, PASJ, 56, 1 Kato, T., et al. 2009, PASJ, 61, 395 Knigge, C. 1988, MNRAS, 373, 484 Kolb, U. & Baraffe, I. 1999, MNRAS, 309, 1034 Kunze, S. & Speith, R. 2005, in ASP Conf. Ser. 330, The Astrophysics of Cataclysmic Variables and Related Objects, eds. J. M. Hameury & J. P. Lasota (San Francisco, CA: ASP), 389 Lin, D. N. C. & Papaloizou, J. 1979, MNRAS, 186, 799 Littlefair, S. P., Dhillon, V. S., Marsh, T. R., et al. 2007, MNRAS, 381, 827 Long, K. S. & Gilliland, R. L. 1999, ApJ, 511, 916 Marsh, T. R. & Horne, K. 1988, MNRAS, 235, 269 Marsh, T. R., Horne, K., & Rosen, S. 1991, ApJ, 366, 535 Nagel, T., Rauch, T., & Werner, K. 2009,A&A, 499, 773 Nelemans, G., Yungelson, L. R., van der Sluys, M. V., and Tout, C. A. 2010, MNRAS, 401, 1347 Osaki, Y. & Meyer, F. 2002, A&A, 383, 574 Patterson, J. 1998, PASP, 110, 1132 Patterson, J., Thorstensen, J., & Knigge, C. 2008, PASP, 120, 510 Podsiadlowki, P., Han, Z., & Rappaport, S. 2003 MNRAS, 340, 1214 Pogge, R. W., Atwood, B., Brewer, D. F., et al. 2010, Proc. SPIE, 7735 Politano, M. 2004, ApJ, 604, 817 Rodr'ıguez-Gil, P., Gansicke, B. T., Hagen, H.-J., et al. 2005, A&A, 431, 269 Shafter, A. W. 1983, ApJ, 267, 222 Spruit, H. C. 1998, arXiv:astro-ph/9806141v1 Stellingwerf, R. F. 1978, ApJ, 224, 953 Thorstensen, J. R., Fenton, W. H., Patterson, J. O., et al. 2002, ApJ, 567, 49 Uemura, M., Kato, T., Ishioka, R., et al. 2002, PASJ, 54, 599 Unda-Sanzana, E., Marsh, T. R., Gansicke, B. T., et al. 2008, MNRAS, 388, 889 Verbunt, F. & Rappaport, S. 1988, ApJ, 332, 193 Williams, R. E. & Ferguson, D. H. 1982, ApJ, 257, 672 Zorotovic, M., Schreiber, M. R., & Gansicke, B. T. 2011, A&A, 536, 42", "pages": [ 22, 23, 24 ] } ]

[ { "title": "ABSTRACT", "content": "We reported the monitoring results on spectrophotometry, photometry and imaging of comet 103P/Hartley 2 obtained at Lulin (1m), Calar Alto (2.2m) and Beijing Astronomical (2.16m) Observatory from April to December 2010. We found that a dust feature at sunward direction was detected starting from the end of September until the beginning of December (our last observation from the Lulin and Calar Alto observatory). Two distinct sunward jet features in the processed images were observed on October 11 and after October 29 until November 2. In parallel, the CN images reveal two asymmetrical jet features which are nearly perpendicular to the Sun-nucleus direction and this asymmetrical features implies that the comet was in a nearly side-on view in late-October and early-November. Additional to the jet features, the average result of the C 2 -to-CN production rate ratio ranges from 0.7 to 1.5 which places 103P/Hartley 2 as being of typical cometary chemistry. We found that the r h dependence for the dust production rate, Af ρ (5,000 km), is -3.75 ± 0.45 before perihelion and is -3.44 ± 1.20 during post-perihelion period. We detected the higher dust reddening is around the optocenter and getting bluer outward along the sunward jet feature and concluded that the former one, higher dust reddening, could be associated with strong jet activity and the latter one, the lowering of the reddening, might imply the optical properties changed or could be associated with outburst. The average dust color did not appear to vary significantly as the comet passed through perihelion. Subject headings: Comets: individual: 103P/Hartley 2, gas, dust, coma structures", "pages": [ 2 ] }, { "title": "Long-Term Monitoring of Comet 103P/Hartley 2", "content": "Z.-Y. Lin Institute of Astronomy, National Central University No. 300, Jhongda Rd, Jhongli City, Taoyuan County, Taiwan, 32001 [email protected]", "pages": [ 1 ] }, { "title": "L.M. Lara", "content": "Instituto de Astrof´ısica de Andaluc´ıa (CSIC), Glorieta de la Astronom´ıa s/n, ES-18008 Granada, Spain [email protected] and W.-H. Ip Institute of Astronomy, National Central University No. 300, Jhongda Rd, Jhongli City, Taoyuan County, Taiwan, 32001 Received ; accepted Based on observations collected at the Centro Astron'omico Hispano Alem'an (CAHA) at Calar Alto, operated jointly by the Max-Planck Institut fur Astronomie and the Instituto de Astrof'ısica de Andaluc'ıa (CSIC), at Lulin observatory operated by the Institute of Astronomy, National Central University in Taiwan, and at Xinglong station inaugurated by National Astronomical Observatory (BAO), Beijing", "pages": [ 1 ] }, { "title": "1. Introduction", "content": "Comet 103P/Hartley 2, hereafter referred to as Hartley 2, was first spotted by M. Hartley on March 16, 1986. It has a semi-major axis of a = 3 . 47AU, eccentricity e = 0 . 695, and inclination i = 13 . 617 and an orbital period of 6.46 years. Its low eccentricity made it a suitable target for the extended mission of NASA's Deep Impact spacecraft after the impact experiment at comet 9P/Tempel 1 on July 4, 2005. The mission to Hartley 2 was renamed EPOXI and given two missions, Extrasolar Planet Observation and Characterization(EPOCh), and Deep Impact Extended Investigation (DIXI). The EPOXI flyby observations at a closest distance of 694 km on November 4, 2010, brought a wealth of information on the outgassing activity, shape and surface structure of this small Jupiter family comet (A'Hearn et al., 2011). For example, the strong outflows of the CO 2 -rich jet from the sun-lit end of the bowling-pin shaped and the H 2 O-rich jet in the waist region came as a surprise. How would they be connected to the large-scale jet structures observed in the coma? How would the outgassing process be modulated by the rotation of the comet nucleus? In fact, based on the time variability of the CN coma morphology and millimeter/sub-millimeter spectra, the rotation period of Hartley 2 has been found to be increasing from 16.7 hr in August, 2010, to 18.4 hr in the first half of November and then to nearly 19 hr in late November (Samarasinha et al. 2011; Knight and Schleicher 2011; Meech et al. 2011; Waniak et al. 2012). Such time variations of the nucleus rotation period together with the close-up measurements by the EPOXI mission demonstrate the complex nature of the surface outgassing process. In anticipation of the scientific opportunity to compare the large-scale coma structures and gas production rates of Hartley 2 with the EPOXI results, we have made a long-term monitoring program from April to December, 2010, using imaging with both broadband and narrowband filters, and long-slit spectrophotometry. This cooperative effort involved observations at the Lulin Observatory in Taiwan, the Calar Alto Observatory in Spain, and the Beijing Astronomical Observatory in China. The paper is organized as follows. In Section 2, we will explain the observational procedures, instruments and analysis methods. In Section 3, the derived morphology and gas production of the CN coma and jets will be described. In Section 4, we will describe the dust jets and the structure of the dust coma during this period. A summary of the major characteristics of the large-scaled structures of the gas and dust comas of Harley 2 is given in Section 5.", "pages": [ 3, 4 ] }, { "title": "2. Observations, instruments and data analysis", "content": "Imaging: The bulk of the photometric imaging observations was done by using the Lulin One-meter Telescope (LOT) at Lulin observatory. In our first image of Hartley 2 on April 24, 2010, when the comet was 2.42 AU away from the Sun and 2.36 AU from the Earth, only a diffuse coma of 5' diameter was visible with 10-min exposure time. There was no tail feature. In the monitoring program, an Asahi R broadband filter and the narrowband filters of the Rosetta filter set were used. The specifications of these narrowband filters are given as λ c / ∆ λ both in nm, λ c being the central wavelength and ∆ λ the band width: CN (387/5nm), C 2 (512.5/12.5nm), blue continuum BC (443/4nm), and red continuum RC (684/9nm). Because of the consideration of the signal-to-noise ratio, the narrowband filters were used only in October and November 2 just before the EPOXI close encounter. The camera used on LOT from April to November was PI 1300B which has a pixel scale of 0.516 arcsec and a field of view of 11 . 2 × 11 . 6 arcmin. In late-November 2010, there was a cooling problem with PI1300B. We, therefore, switched to U42 CCD which has 2k × 2k pixels and a field of view of 12 . 17 × 11 . 88 arcmin. The telescope was always operated with non-sidereal tracking so as not to produce trail in the comet images. Typical integration were 600s ∼ 900s for the narrowband filters and 30s ∼ 300s for the broadband", "pages": [ 4 ] }, { "title": "R filter.", "content": "Table 1 is the observational log of our program. Standard procedure of data reduction was applied. It began with dark current subtraction and flat-field correction of all image frames. This was then followed by the subtraction of the night sky contribution. For the observations obtained before late-September, the night sky levels were determined directly areas of the CCD frames that do not contain contributions from the cometary emission. However, the sky-background of those images taken from late-September to early-November was all influenced by the cometary coma. Therefore, we took sky background images positioned at about 0.5 degrees away from the comet center. The extinction coefficients of the narrowband and broadband R filters were determined for all nights with photometric sky conditions, using the photometric stars like Feige 110 and GD71, observed at different airmasses during the night. For example, the first order extinction coefficient (in units of magnitudes per air mass) measured by Feige 110 with observing airmass range from 1.1 to 1.7 for R-filter on October 29 is 0.10 and for CN and C 2 are 0.39 and 0.14, respectively. These data were used to convert the measured counting rates into physical units and the detail has been described in Lin et al. (2007b). Because the CN images contain 29% contribution from the continuum in the blue range while the C 2 images have as much as 93%, the net CN and C 2 gas coma images need to go through the subtraction procedure according to the following formulas: CN= CN - 0.29 BC , C = C obs - 0.93 BC . obs obs 2 2 obs In addition to Lulin observations, the coma activity of 103P/Hartley 2 was also monitored continuously in R-band from the Calar Alto Observatory (near Almeria, Spain) from July 14 to December 26, 2010 (see Table 1). We used the CAFOS imaging camera (2k × 2k pixels, pixel size: 0'53, FOV 18 ' × 18 ' ) which was mounted on the 2.2 m telescope. In our observations, only the central 1k x 1k pixels were used, providing thus a FOV of 9 ' × 9 ' . Appropriate bias and flat field frames were taken each night. If photometric conditions prevailed, photometric standard stars were observed at airmass similar to the comet observations. Table 1 contains the observations log for the complete dataset. Notice that a larger number of night mentioned in Table 1 is for the Af ρ estimation and some of them are not used to enhance the structures in the coma because the SNR was too low. Spectroscopy: Spectroscopic measurements were planned once every month using CAFOS with grism B400 (see http://www.caha.es/alises/cafos/cafos22.pdf) which renders an spectral range between 3,200 and 8,800 ˚ A with a wavelength scale of 9.4 ˚ A/pixel. The slit of the spectrograph was oriented in the north-south direction, giving dust and gas profiles at different cross-cuts through the coma, depending on the PA of the sun-comet vector on the sky. For absolute calibration, observations of appropriate spectrophotometric standard stars were acquired. All comet observations were done with telescope tracking on Hartley 2. With the exception of Nov. 5, 2010, all observations were done in service mode of the Calar Alto Observatory. Details on the images and spectra reduction and calibration can be found in Lara et al. (2001, 2011a) and they will not be repeated here. If the gas coma covered the whole slit, the sky level was estimated from the edges of the frame. Otherwise, the background could be measured directly by using regions near the edges of the frame. Besides spectra obtained from the Calar Alto Observatory, spectroscopic observations were also performed on October 9 and October 11 at the Beijing Astronomical Observatory using the 2.16m telescope in the spectral range between 3,600 ˚ A to 8,400 ˚ A at a dispersion 4.8 ˚ A/pixel. The spectroscopic data were reduced following the standard procedures including bias and flat-field corrections and cosmic ray removal. Wavelength calibration was performed based on helium-argon lamps exposed at both the beginning and the end of the observations every night. Flux calibration of all spectra were conducted based on observations of at least one of the spectral standard stars, i.e., HD19445 and the atmospheric extinction effect was corrected by the mean extinction coefficients measured by the Beijing-Arizona-Taiwan-Connecticut (BATC) multicolor survey. See Lin et al. (2007a) for more detailed information.", "pages": [ 5, 6, 7 ] }, { "title": "3.1. CN jets", "content": "In order to study the visibility of faint structures of the gas coma of comet Hartley 2, an image enhancement technique was applied to the present set of images. The method used here is the azimuthally averaged profile division, a detailed description which can be found in Lin et al. (2012). This method was applied to all images taken in the CN filter, in continuum filters and in the R-band filter. To estimate the rotational phase from CN morphology, a lot of observing data have to be acquired in consecutive night. However, the images obtained in our observing nights with less temporal coverage were not enough to estimate and display rotational period due to snapshot observations, poor weather and telescope tracking problem. We, therefore, use the known periodicity to estimate the rotational phase in our images. However, we have to face several problems: a non-principal axis rotation of comet Hartley 2 and a rapid change of the viewing geometry might cause different periodicities between rotational cycles. A specific phase is really only applicable to a short stretch of data if we adopt known periodicity such as 18.15 hr in mid-October and 18.7hr in early-November from Knight and Schleicher (2011) or 18.22 hr around perihelion from Harmon et al. (2011). Notice that those ground-based observations have error bars between 0.01 and 0.3. The most robust rotation period at present is from the EPOXI spacecraft lightcurve given in Belton et al. (2013). This gives a spin period of 18.40 hr at encounter and states that it was increasing by 1.3 minutes/day. As the data acquired with the Rosetta filter set spread around one month, it is appropriate to use the midpoint of the observational time interval for this period of time. We extrapolate the rotation period back to midpoint assuming the rotation period was steadily changing during this time frame. Therefore, the rotation period quoted in this work is 18.11 hr on October 21.5 UT which refers to the midpoint of the Oct 10-Nov 2 data. The zero phase is set at 11:40 UT on October 10, 2010. In Figure 1, we can see that the morphology of the CN coma extended almost perfectly along the east-west direction in early October and the north-south direction around Hartley 2's perihelion. The CN images all showed clear asymmetries before performing the image enhancement. One of these unprocessed CN images is given by contour plot in Figure 1 (top-left panel). The variations in between early-October and around its perihelion in the CN jet features are related to the spin period of the comet nucleus, the changing viewing geometry and non-principal axis rotation as has already been reported by Samarasinha et al. (2011), Knight and Schleicher (2011), Lara et al. (2011b) and Waniak et al. (2012). The processed CN images from the observations between October 11 and November 2 revealed two jets in the coma of comet Hartley 2. The CN jet features being nearly perpendicular to the Sun-tail direction not only varied smoothly during a night but showed similar morphology near its perihelion even though the rotational state was different. We compared the morphology of the CN jet features with those presented by Knight and Schleicher (2011), and Samarasinha et al. (2011) and found that the CN jet features of Hartley 2 did not show the spiral-like structure in early-October but was compatible with the observations obtained by Knight and Schleicher (2011) and Lara et al. (2011b) in late-October. The reason could be the observing geometry, i.e., whether it is observed from the face-on or side-on. Knight and Schleicher (2011) confirmed this effect from the images that revealed the face-on spiral structures in August and September. Furthermore, we found these two CN jet features to be asymmetrical. One of them is always brighter than the other, possibly because it is facing toward the Earth. For example, the southern jet of the images obtained from October 25 to October 27 is slightly stronger than the northern jet of those images. Such asymmetrical features have also been reported in earliler works by Samarasinha et al. (2011), Lara et al. (2011b) and Waniak et al. (2012).", "pages": [ 7, 8, 9 ] }, { "title": "3.2. Gas production rates", "content": "In order to determine the gas production rates, the mean radial emission profiles of CN and C 2 were derived from the images with the continuum subtracted. Regarding the spectra of the comet acquired at Calar Alto Observatory and Beijing Astronomical Observatory, they are also used to investigate the CN, C 3 , C 2 and NH 2 profiles in the North-South direction and to derive the production rates of these gaseous species. The spectral regions and the subtraction of the underlying continuum in the gas emission bands were done as described by Lara et al. (2001). The conversion of the emission band fluxes into column densities made use of fluorescence efficiency factors ( g -factors) for C 3 , C 2 and NH 2 (A'Hearn et al. 1995), whereas the g -factors of the CN molecule was calculated for the heliocentric distance and velocity of 103P/Hartley 2 on every date from the set of values given by Schleicher (2010). The gas production rates are obtained by means of the Haser (1957) model for isotropic emission of cometary neutral molecules and their daughter molecules and radicals. The parameter used for the parent velocity is v p = 0.85 r -0 . 5 h kms -1 (Fray et al. 2005) and for the daughter velocity it is 1 kms -1 . For the corresponding set of parameters in the Haser model, we produced theoretical column density profiles for each species by varying the production rate until the best match between observations and theoretical predictions is achieved. The results of nightly averages for Q(CN), Q(C 3 ), Q(C 2 ) and Q(NH 2 ) are summarized in Table 2. Table 2 also contains the average gas production rates obtained from the images acquired in one night together with the aperture size we have considered to derive Q. The variation of production rates seen in multiple measurements during a night were less than 5% that is reflected in the uncertainties in Table 2. Our results on Q(C 2 ) are less numerous as there were tracking problems at LOT from October 10 ∼ 11 and October 25 ∼ 27, whereas the long-slit spectroscopic measurements could provide Q(C 2 ) at other dates thus spanning larger heliocentric distances. Our Lulin, BAO and CA results in Table 2 show that there is no significant variation of Q(CN) from mid-October to early-November. This result is consistent with the Lara et al. (2011b) and Mumma et al. (2011) results that assumed that HCN is the main parent species of CN and that expected variation of Q(HCN) around the perihelion is not very large. Notice that we used the mean radial profile to estimate the gas production rate from the images obtained from Lulin observatory. However, if we averaged the radial profile in the north-south direction where the CN jet feature exists, the derived Q(CN) would be larger in a factor of two to three when compared with the azimuthally averaged radial profile. Figure 2 shows the logarithm of the production rate for CN and C 2 as a function of the heliocentric distance (r h ). The data points include those obtained by Lara et al. (2011b), Knight and Schleicher (2013) and the Lulin and CA results (this work) for pre-perihelion and post-perihelion observations during the 2010 apparition are presented here. We used the linear fitting in the log-log scale to estimate the slope of the r h dependence of the gas production rate, Q ∼ r -α h , and the slopes ( α ) of CN and C 2 are 4.57 and 4.84 before perihelion and 3.21 and 3.42 after perihelion, respectively. The corresponding slopes are significantly steeper than the average value estimated for Jupiter-family comets, i.e. Q(gas) ∼ r -2 . 7 h (A'Hearn et al. 1995). Additionally to this, the average C 2 -to-CN production rate ratio is 0 . 7 ∼ 1 . 5 which places 103P/Hartley 2 as being 'typical' in terms of cometary chemistry defined by A'Hearn et al (1995) . Our measurement is consistent with the results from the spectroscopic observations (Lara et al. 2011b) and the narrow-band photometry observations (Knight and Schleicher 2013).", "pages": [ 9, 10 ] }, { "title": "4.1. Jet feature in dust coma", "content": "We describe the morphology and evolution of the coma structures that can be treated with routine procedures, i.e. Larson-Sekanina algorithm (Larson and Sekanina 1984). In case of doubt, we used additional techniques, such as azimuthal median profile division and Adaptive Laplace filter (Bohnhardt and Birkle 1994) to clearly separate morphological features from artifacts. Figure 3 compares the jet structure and dust tail feature on October 11 obtained by using three different image enhancement methods: (a) the Larson-Sekanina filtering, (b) the azimuthal median profile, and (c) the adaptive Laplace filtering. In spite of some differences in their appearances, the presence of two jets in the sunward quarter is common to all numerical treatments. It is therefore clear that the jet features are real and not artifacts associated with the image processing procedures. Figure 4 is a summary of the R-band images enhanced by the Larson-Sekanina filtering method to bring out the inhomogeneous structures in the dust coma of 103P/Hartley 2. It can be seen that from April until July, 2010, no clear sign of dust features could be found. However, beginning in August 1, a dust tail of diffuse structure (labeled T in Figure 4) began to appear in the anti-sunward direction. On September 29, a short jet (indicated by arrows in Figure 4) in the sunward direction can be seen. Hereafter, this sunward jet feature can be detected in all our images obtained at Lulin and Calar Alto Observatory. It is interesting to note that two distinct sunward dust structures are visible after October 29 lasting until November 2. Around the same time, from November 2 to November 4, Mueller et al. (2013) also reported seeing two separate continuum features in sunward direction. Afterwards, only a single jet could be seen in the sunward direction that became fainter and fainter as Hartley 2's heliocentric and geocentric distances increased. The sunward jet features showed relatively little variation during a night but its shape and position angle slightly changed from night to night until October 11 when two distinct jet features apparently emerged from the sunward direction (Figure 5). In order to examine the existence of this extremely faint jet feature and to distinguish it from the trail of a background star, we transformed the enhance image into polar coordinates ρ -θ where ρ is the projected cometocentric distance from the nucleus and θ is the azimuth (position angle). At several distances ρ from the nucleus, we analyzed the resulting azimuthal profile. In Fig. 5 (right panel) we show the azimuthal profile at ρ = 5 , 000 km. It can be seen that this faint jet (referred as main feature in the figure) appears on Oct. 11.76 and persists until Oct. 11.87, that is ∼ 0.7hr later (bottom panel in Fig. 5). It points towards the Sun and it does not display significant changes. On Oct. 11.84, a new faint feature appears nearly perpendicularly to the Sun-comet line. It is interesting to note that the position angle of the secondary jet is roughly the same as that of the CN jet features shown in Figure 1 (pointing to the east-south direction in the top-middle panel). At first, one could think that icy grains mixed with the dust grains of this weaker jet could provide the partial fuel to the CN gas jet. However, the gas jets persist for most of a rotation period (Knight and Schleicher 2011, Samarasinha et al. 2011) and are clearly being released over an extended period of time as the nucleus rotates. Thus, the CN jets cannot mainly come from this faint jet feature if it is only active for a few hours as found here. That switching phenomenon may also be explained as a projection effect due to the comet nucleus rotation. For Oct. 28 and 29, we obtained a series of images from Lulin and Calar Alto observatories that provide insight into how the sunward feature evolved throughout ∼ 1.4 rotation cycles. Representative images from these nights are shown in Fig. 6, with each panel enhanced by the Larson-Sekanina filtering method. Notice that the position angle (PA) of the Sun during these two days is near 97 · . Setting the zero phase at 11:40 UT on October 10 and using a period of 18.11 hr (see the Section 3), the rotational phase can be easily estimated in these three images (see the bottom right corner of Figure 6). A dust jet (labeled J and marked with an arrow in Figure 6) can be seen in the sunward direction whose shape slightly changes as the rotational phase change from 0.22 (on Oct. 28.68 UT) to 0.82 (Oct. 29.13 UT). Thirteen hours later, (rotational phase of 0.57, Oct. 29.70 UT) two dust jet features emanating in the sunward direction can be seen. One of them, labeled J1, is close to the position of Sun (PAs ∼ 85 · ) and the other, labeled J2, lays at the PAs ∼ 130 · . Owing to the similar PAs, we consider the possibility that J1 feature might have the same source region as seen from the previous two images (Oct. 28.68 and Oct. 29.13). Under this assumption, J2 feature is new. Another possibility is J1 feature might be associated with the cometary rotational effect, i.e. local sunrise accompanied by temperature increase turns that jet on. This localized temperature difference in the regions of waist and the sun-lit end of the nucleus have been addressed by Belton et al (2013). The J2 feature which has a collimated-like shape is the persistent feature we detected on Oct. 28.68 and Oct. 29.13 although PA and shape changed between those two dates. We note that the brightness of J2 feature is higher than that of J1 feature and this higher intensity could be related to the dusty ice, or to an outburst from the surface of the comet nucleus. To understand their interrelationship better, our images need to be interpreted in the context of a larger image series that displays the time evolution of the jet structure over two or more rotational period. On the tailward side, only the dust tail was readily visible starting in August, 2010. Dust tail was found to point approximately in the antisolar direction. As expected, it appears to be curved slightly counterclock-wise.", "pages": [ 11, 12, 13 ] }, { "title": "4.2. The properties of dust coma", "content": "We used Af ρ (A'Hearn et al. 1984) to characterize the dust activity of the comet, the derived values acquired with broadband R-filter from April to November 2010 are presented in Figure 7. Except for the night on October 29, the average values estimated every photometric night were all measured within a projected distance of 5,000km. Notice that Af ρ shows a weak dependence on the ρ , projected distance from nucleus, from 5,000km to 20,000km and the variation was found to be less than approx. 5-8%. The reason why we used 5,000km for uniform radius is to reduce the influence of star trails in field of view. The Afρ values steadily increased with decreasing heliocentric distance, although there was not a noticeable increase when the second jet appeared on Oct. 11.64 UT or at the perihelion. The Afρ value on October 29.77 ∼ 29.85 UT increased from 155 cm to 174 cm in two hours, and at the same time the dust jet seen in Figure 6 (right panel) was more prominent on this night than on any of the other nights and a relatively weak secondary jet feature was also detected. Possible causes for this deviation might include the changes in the physical properties of the grains as they travel outward (i.e. loss of volatiles or fragmentation), the action of solar radiation pressure modifying the straight trajectories of small particles inside the field of view, or a long-lasting population of large particles (Schleicher et al. 1998). Furthermore, the power law index of the r h dependence for the dust, Af ρ (5,000 km), is -3 . 75 ± 0 . 45 before perihelion and is -3 . 44 ± 1 . 20 post-perihelion. This is result is completely consistent with Knight and Schleicher's (2013) when using A( θ )f ρ . The derived Af ρ values for the narrowband filter can be taken to estimate the color of the cometary dust (Jewitt and Meech 1987) as the normalized gradient of the Af ρ product between the blue (BC, λ 0 = 4,430 ˚ A) and red (RC, λ 0 = 6,840 ˚ A) continuum filters. The dust color can be converted to a percentage of reddening per 1,000 ˚ A and is defined by the following relation: The summarized results in Table 3 indicate that the averaging dust color within the innermost 5,000 km of the coma did not appear to vary significantly with heliocentric distance. This behavior of the averaging dust color seems to indicate that the innermost coma do not introduce significant changes on the size distribution and/or overall properties of dust grains. As we found a jet feature that switches on and off from our images in Figure 4 to Figure 6, we analyzed the entire flux-calibrated images acquired with BC and RC narrowband filters instead of integrating whole flux in the innermost 5,000 km. The resulting two-dimensional dust color map can be seen in Figure 8 (the third column). Figure 8 displays the dust coma of comet 103P/Hartley 2 from October 10 to November 2 imaged in BC and RC narrowband filters (first two column), the dust color map (the third column) and azimuthal median profile subtracted RC filter images (the fourth column) that displays the jet activity in the dust coma. The data here presented pertaining to October and November give an extremely reddened dust, with a normalized color ∼ 30-45 %, within a radius of ∼ 50 -100 km measured from the optocenter of the images. This red dust could be associated with strong jet activity. The sunward jet feature might give rise to higher dust abundances at closer cometocentric distances (i.e. near the optocenter). These dust grains are initially large with a reddening of ∼ 30 -40%/1,000 ˚ A, while travelling out they split up and show bluer at ∼ 500 km with a dust reddening of ∼ 10-15 %. In comparison with tailward direction, the color variation is 5% to 10%. The decrease in the dust reddening means that the optical properties of the dust grains change as the dust grains move outward or this blueing of the dust could be also associated with an outburst (Bonev et al., 2002). A possible explanation for color variation is that the larger dust grains mixed with the icy grains dominate the scattering behavior at close distance around the nucleus. When these larger dust grains move outwards, they break up or sublimate into the small sub-micron particles resulting in a bluer continuum due to their smaller sizes (Lara et al. 2011b).", "pages": [ 14, 15, 16 ] }, { "title": "5. Summary", "content": "We observed the comet Hartley 2 at the Lulin Observatory in Taiwan, the Calar Alto Observatory in Spain, and the Beijing Astronomical Observatory in China, from April to December, 2010 using both broadband and narrowband filters, and long-slit spectrophotometry. The results are summarize below. dust reddening is found around the optocenter and could be associated with a stronger jet activity. The dust color is getting bluer outwards along the sunward jet which implies that the optical properties of the dust grains change with ρ . The average dust color did not appear to vary significantly when the heliocentric distance decreased to perihelion. This work was based on observations obtained at Taiwan's Lulin Observatory. We thank the staff members and Yu-Chi Cheng for their assistances with the observations. We greatfully acknowledge valuable discussions with the referee. The research was supported by project AyA2009-08011 of the Ministerio de Ciencia e Innovacion. Zhong Yi Lin acknowledges a post-doctoral grant awarded by the Junta de Andalucia through project number P07-TIC-274. This work was also supported by grant number NSC 99-2923-M-008-002-MY3 for the Formosa Program (NSC-CSIC).", "pages": [ 16, 17 ] }, { "title": "REFERENCES", "content": "A'Hearn, M.F., Schleicher, D. G., Millis, R. L., Feldman, P. D., Thompson, D. T. 1984, AJ, 89, 579 A'Hearn, M.F., Millis, R. L., Schleicher, D. G., Osip, D. J., Birch, P. V. 1995, Icarus, 118, 223 A'Hearn, M.F. et al. 2011, Science, 332, 1396 Belton, M.J.S. et al. 2013, Icarus, 222, 595 Bohnhardt, H., and Birkle, K. 1994, A&A, 107, 101 Bonev, T., Jockers, K., Petrova, E., Delva, M., Borisov, G., Ivanova, A. 2002, Icarus, 160, 419 Fray, N., B'enilan, Y., Cottin, H., Gazeau, M.-C., Crovisier, J. 2005, P&SS, 53, 1243 Jewitt, D.C., and Meech, K.J. 1987, ApJ, 317, 992 Harmon, J.K., Nolan, M.C., Howell, E. S., Giorgini, J. D., Taylor, P.A. 2011, ApJ, 734, L2 Haser, L. 1957, Bulletin de la Societe Royale des Sciences de Leige, 43, 740 Knight, M.M. and Schleicher, D.G. 2011, AJ, 141, 183 Knight, M.M. and Schleicher, D.G. 2013, Icarus, 222, 691 Lara, L.M., Schulz, R., Stwe, J. A., Tozzi, G. P. 2001, Icarus, 150, 124 Lara, L.M.,Lin, Z.-Y., Rodrigo, R., Ip, W.-H. 2011a, A&A, 525, 36L Lara, L.M., Lin, Z.-Y., Meech, K. 2011b, A&A, 532, 87L Larson, S. M. and Sekanina, Z. 1984, AJ, 89, 571 Lin, Z.-Y. Lin, Chang, C.-P., Ip, W.-H. 2007a, AJ, 133, 1861 Lin, Z.-Y., Weiler, M., Rauer, H., Ip, W.-H.. 2007b, A&A, 469, 771 Lin, Z.-Y., Lara, L. M., Vincent, J. B., Ip, W.-H. 2012, AJ, 537, 101L Meech, K.J. et al. 2011, ApJ, 734, L1 Mumma, M.J. et al., 2011, ApJ, 734, L7 Mueller, B. E. A., Samarasinha, .H., Farnham, T.L., A'Hearn, M.F. 2013, Icarus, 222, 799 Samarasinha, N.H., Mueller, B. E. A., A'Hearn, M. F., Farnham, T. L., Gersch, A. 2011, ApJ, 734, L3 Schleicher, D.G. 1998, Icarus, 132, 397 Schleicher, D.G. 2010, AJ, 140, 973 Soderblom, L.A. 2002, Science, 296, 1087 Waniak, W., Borisov, G., Drahus, M., Bonev, T. 2012, A&A, 543, A32 T able 1. Log of o bserv ations Sky Obs. Data scale Pix α .A. P ∆ h r UT Date April 24 19:56-20:17 2.424 2.363 256.2 24.2 884.4 R Lulin Phot. Ma y 11 18:39-18:48 2.283 2.026 252.1 26.3 758.2 R Lulin Phot. Ma y 15 18:58-19:10 2.249 1.948 251.0 26.7 729.1 R Lulin P art. cloudy Ma y 16 19:07-19:36 2.240 1.928 250.8 26.8 721.6 R Lulin P art. cloudy Ma y 20 18:34-18:56 2.206 1.851 249.7 27.1 692.7 R Lulin P art. cloudy July 14 17:01-18: 40 1.721 0.924 226.9 29.0 345.8 R Lulin P art. cloudy July 14 23:00-23: 49 1.719 0.921 226.7 29.0 354.0 R CA Phot. July 22 01:25-02: 36 1.656 0.825 221.7 29.0 316.1 R CA P art. phot. July 30 02:22-02: 55 1.585 0.725 215.2 29.2 278.7 R CA Phot. August 1 17:30-18:42 1.562 0.694 212.9 29.3 259.7 R Lulin Phot. August 19 14:00-20:15 1.409 0.503 195.5 31.1 188.3 R Lulin P art. cloudy August 20 01:33-01:44 1.406 0.500 195.1 31.1 192.2 R CA Phot. August 20 13:17-20:01 1.402 0.494 194.5 31.2 184.9 R Lulin Phot. August 21 19:27-20:43 1.392 0.483 193.3 31.4 180.8 R Lulin Phot. August 25 22:46-00:23 1.359 0.445 189.3 32.3 174.5 R,S CA Phot. Sky Obs. Data scale Pix α .A. P ∆ h r UT Date Sky Obs. Data scale Pix α .A. P ∆ h r UT Date y cloud art. P CA R 47.6 55.6 268.6 0.125 1.060 03:53-04:38 25 er Octob Phot. Lulin R+N 47.2 55.9 269.8 0.126 1.059 18:09-18:39 25 er Octob Phot. Lulin N 47.9 56.4 271.8 0.128 1.059 16:09-21:24 26 er Octob Phot. Lulin N 48.7 56.5 273.7 0.130 1.059 16:24-26:56 27 er Octob Phot. Lulin N 49.4 57.4 275.3 0.132 1.059 17:50-21:28 28 er Octob y cloud art. P CA R 51.1 57.5 275.9 0.133 1.059 03:06-03:20 29 er Octob Phot. Lulin R+N 50.5 57.7 276.9 0.135 1.059 16:24-21:00 29 er Octob cloudy art. P Lulin R 3 54. 58.6 280.9 0.145 1.060 -21:12 19:14 1 er b em v No Phot. Lulin R+N 1 56. 58.7 282.0 0.150 1.061 -20:31 17:58 2 er b em v No cloudy art. P Lulin R 7 55. 58.7 281.9 0.149 1.061 -16:28 16:04 3 er b em v No cloudy art. P Lulin R 8 58. 58.8 284.2 0.157 1.064 -21:09 18:07 5 er b em v No phot. art. P CA R,S 3 62. 58.8 285.4 0.162 1.065 -05:45 01:25 5 er b em v No phot. art. P CA R,S 80.3 56.1 294.0 0.210 1.090 01:18-01:47 16 er b em v No Phot. Lulin R 60.3 53.2 298.8 0.239 1.112 19:07-19:37 21 er b em v No cloudy art. P Lulin R 74.5 46.4 308.9 0.295 1.116 16:24-18:48 2 er b Decem", "pages": [ 18, 19, 20, 21, 22 ] }, { "title": "T able 1-Con tin ued", "content": "extended the is .A. P U; A in distances tric cen helio and tric cen geo the are h r and ∆ - Note. ards w to North from measured , plane-of-sky er's observ the in seen as ector v radius Sun-target the is N filter, band oad br the is R angle). er (Sun-comet-observ angle phase the is α East. cometary filter set and S ref ers to long-slit sp ectra. 2. 103P/Hartley comet of es rat duction pro Gas 2. able T )/Q(CN) 2 Q(C 2 NH 3 C 2 C CN erture Ap atory Observ UT Date", "pages": [ 23, 24 ] } ]

[ { "title": "ABSTRACT", "content": "We present the results of HCN/HCO + /HNC J = 4-3 transition line observations of the nearby starburst galaxy NGC 1614, obtained with ALMA Cycle 0. We find that high density molecular gas, traced with these lines, shows a velocity structure such that the northern (southern) side of the nucleus is redshifted (blueshifted) with respect to the nuclear velocity of this galaxy. The redshifted and blueshifted emission peaks are offset by ∼ 0.6 '' at the northern and southern sides of the nucleus, respectively. At these offset positions, observations at infrared > 3 µ m indicate the presence of active dusty starbursts, supporting the picture that high-density molecular gas is the site of active starbursts. The enclosed dynamical mass within the central ∼ 2 '' in radius, derived from the dynamics of the highdensity molecular gas, is ∼ 10 9 M /circledot , which is similar to previous estimates. Finally, the HCN emission is weaker than HCO + but stronger than HNC for J = 4-3 for all starburst regions of NGC 1614, as seen for J = 1-0 transition lines in starburst-dominated galaxies. Subject headings: galaxies: active - galaxies: nuclei - galaxies: starburst - submillimeter: galaxies", "pages": [ 1 ] }, { "title": "HIGH-DENSITY MOLECULAR GAS PROPERTIES OF THE STARBURST GALAXY NGC 1614 REVEALED WITH ALMA", "content": "Masatoshi Imanishi 1,2 Subaru Telescope, 650 North A'ohoku Place, Hilo, Hawaii, 96720, U.S.A. and Kouichiro Nakanishi 1,2 Joint ALMA Observatory, Alonso de Cordova 3107, Vitacura 763-0355, Santiago de Chile AJ", "pages": [ 1 ] }, { "title": "1. INTRODUCTION", "content": "Luminous infrared galaxies (LIRGs) show strong infrared emission, with infrared (8-1000 µ m) luminosities of L IR > 10 11 L /circledot , created by energy sources hidden behind dust. They are mostly found in gas-rich galaxy mergers (Sanders & Mirabel 1996). Molecular gas in galaxy mergers is largely influenced by merger-induced physical processes, and obtaining observational constraints on the spatial distribution and dynamics of molecular gas in merging LIRGs is important to our understanding of gas-rich galaxy merger processes. (Sub)millimeter interferometric observations of rotational J-transition lines of molecular gas are a powerful tool for this purpose. High-sensitivity, high-spatialresolution ( < a few arcsec) interferometric observations of merging LIRGs using bright CO molecular lines have been widely performed (Downes & Solomon 1998; Bryant & Scoville 1999; Trung et al. 2001; Evans et al. 2002; Downes & Eckart 2007). These CO observations at lowJ transitions (J = 1-0 and 2-1) have effectively traced the low-density (10 1 -3 cm -3 ) molecular gas properties in great detail due to the low dipole moment of CO ( µ ∼ 0.1 debye). However, in merging LIRGs, the fraction of high density ( > 10 4 cm -3 ) molecular gas is much higher than in normal quiescent star-forming galaxies (Solomon et al. 1992; Gao & Solomon 2004), and it is within such high-density gas that stars are actually born. Thus, it is vital to obtain observational constraints of the properties of high-density molecular gas in merging LIRGs if we are to understand the essential physical processes in gas-rich Electronic address: [email protected] galaxy mergers. Observations of molecular gas with high dipole moments, such as HCN, HCO + , and HNC ( µ > 3 debye), can effectively probe high-density molecular gas. However, these molecular lines are generally much fainter than the bright CO lines, and the spatial information of dense gas is still limited to nearby, very bright merging LIRGs only (Aalto et al. 1997; Casoli et al. 1999; Imanishi et al. 2004; Nakanishi et al. 2005; Imanishi et al. 2006b; Imanishi & Nakanishi 2006; Imanishi et al. 2007; Iono et al. 2007; Wilson et al. 2008; Imanishi et al. 2009; Aalto et al. 2009; Sakamoto et al. 2009, 2013). From these previously performed observations, it has been shown that the spatial distribution of high-density molecular gas is significantly different from that of lowdensity molecular gas, in that high-density gas is more concentrated in the nuclear regions of galaxies (Iono et al. 2004; Nakanishi et al. 2005; Imanishi et al. 2007; Wilson et al. 2008; Sakamoto et al. 2013). In merging LIRGs, it is in the nuclear regions where very violent processes, including starbursts (= active star-formation) and active mass accretion onto a supermassive black hole (SMBH) occur. Additionally, feedback to the surrounding interstellar medium and entire galaxy, if present, originates from these regions (Hopkins et al. 2005; Springel et al. 2005; Di Matteo et al. 2005; Hopkins et al. 2006). Spatially resolved interferometric observations using dense gas tracers are of particular importance in investigating the interesting nuclear regions of merging LIRGs. Starburst and active galactic nucleus (AGN) activity powered by a mass-accreting SMBH can have different effects/feedback on the surrounding dense molecular gas at the merging LIRG's nuclei. It is proposed that starburst and AGN activity could be distinguishable based on the line flux ratios of dense molecular gas tracers (Kohno 2005; Imanishi et al. 2004, 2006b, 2007, 2009; Krips et al. 2008) because (1) an AGN has an energy source with a much higher emission surface brightness than a starburst, and thus it can heat the surrounding dust and gas to higher temperature; and (2) an AGN emits stronger X-rays than a starburst does. Both of these factors could alter the chemical compositions of molecular gas in AGNs compared with starbursts (Meijerink & Spaans 2005; Lintott & Viti 2006; Harada et al. 2010); therefore, AGNs and starbursts could exhibit different molecular line flux ratios. High-spatial-resolution interferometric observations of multiple dense gas tracers can thus be used to scrutinize the processes deep inside the obscuring dust and gas in merging LIRG nuclei. With the advent of ALMA, such observations are now feasible. NGC 1614 (z = 0.016; L IR = 10 11 . 6 L /circledot ) is a wellstudied, nearby LIRG. Merging signatures are clearly seen in the optical and near-infrared (1-2.5 µ m) images as long, prominent tails around a single nucleus (AlonsoHerrero et al. 2001; Rothberg & Joseph 2004; Haan et al. 2011). It is classified as a starburst galaxy through optical spectroscopy (Veilleux et al. 1995; Kewley et al. 2001; Yuan et al. 2010). The infrared 2.5-30 µ m spectrum of NGC 1614 is typical of a starburst-dominated galaxy with no AGN signature (Brandl et al. 2006; BernardSalas et al. 2009; Imanishi et al. 2010b; Vaisanen et al. 2012). The luminosities of the starburst-generated 3.3 µ m polycyclic aromatic hydrocarbon (PAH) emission feature and the Br α (4.05 µ m) hydrogen emission, measured through slitless spectroscopy and calculated relative to the total infrared (8-1000 µ m) luminosity, are both as high as in starburst-dominated galaxies (Imanishi et al. 2010b), suggesting that the observed luminosity of NGC 1614 is totally accounted for by the detected starbursts, with no need for a significant AGN contribution. The infrared K -band (2.2 µ m) spectrum shows a strong stellar-origin 2.3 µ m CO absorption feature (Ridgway et al. 1994; Alonso-Herrero et al. 2001) and so supports the starburst-dominated scenario. Highspatial-resolution infrared 8-20 µ m imaging observations reveal spatially extended, but compact ( ∼ 2 '' ), starburstheated dust continuum emission (Miles et al. 1996; Soifer et al. 2001; Diaz-Santos et al. 2008; Imanishi et al. 2011), and the measured emission surface brightness is within the range explained by star formation, again requiring no significant contribution from an energetically important AGN (Soifer et al. 2001; Imanishi et al. 2011). The Pa α (1.88 µ m) emission from star-forming HII-regions is dominated by nuclear ∼ 3 '' area (Alonso-Herrero et al. 2001), which suggests that most of starburst activity in NGC 1614 is concentrated in the nuclear regions within < a few arcsec. Although ALMA is a very powerful tool for unveiling the molecular gas distribution in detail, it is not sensitive to spatially extended structure beyond the maximum scale, which is ∼ 6 '' at ∼ 350 GHz in ALMA Cycle 0. The spatially resolved, but intrinsically compact ( ∼ 2 '' ) nuclear emission morphology and the known starburstdominated nature make NGC 1614 an ideal target to investigate the spatial distribution and dynamics of dense molecular gas in merging LIRG nuclei, as well as to obtain a template of line flux ratios of dense gas tracers in starburst-dominated galaxies, during ALMA Cycle 0. We thus performed ALMA band 7 (275-373 GHz) observations of NGC 1614 to study the emission properties of HCN J = 4-3, HCO + J = 4-3, and HNC J = 4-3 lines. The basic information of NGC 1614 is summarized in Table 1. Throughout this paper, we adopt H 0 = 71 km s -1 Mpc -1 , Ω M = 0.27, and Ω Λ = 0.73 (Komatsu et al. 2009), where 1 '' corresponds to ∼ 320 pc at the distance of NGC 1614 (z = 0.016).", "pages": [ 1, 2 ] }, { "title": "2. OBSERVATIONS AND DATA ANALYSIS", "content": "All observations were made during ALMA Cycle 0 within the program 2011.0.00020.S (PI = M. Imanishi). Observation details are described in Table 2. We adopted the widest 1.875 GHz band mode, and the total channel number was 3840. For NGC 1614 (z = 0.016), HCN J = 4-3 ( ν rest = 354.505 GHz), and HCO + J = 4-3 ( ν rest = 356.734 GHz) lines are simultaneously observable in ALMA band 7. Four frequency setups can cover four different frequencies at the same time. Two were used to observe HCN J = 4-3 (central frequency was set as ν center = 348.922 GHz) and HCO + J = 4-3 lines ( ν center = 350.920 GHz), and the remaining two were used to measure the continuum flux level ( ν center = 337.106 GHz and 338.681 GHz). The net on-source exposure time for the HCN/HCO + J = 4-3 line observation of NGC 1614 was ∼ 26 min. The frequency of the HNC J = 4-3 line ( ν rest =362.630 GHz) is separated from the HCN J = 4-3 and HCO + J = 4-3 lines, and we required independent observations. The HNC line was covered with one spectral window ( ν center = 356.920 GHz), and an additional second spectral window was used to probe the continuum emission ( ν center = 345.079 GHz). The net on-source exposure time for the HNC J = 4-3 line observation of NGC 1614 was ∼ 25 min. We started data analysis from calibrated data provided by the Joint ALMA Observatory. We first checked the visibility plots to see if the signatures of the emission lines were recognizable. The presence of HCN, HCO + , and HNC J = 4-3 lines were evident in the visibility plot, whereas signatures of other molecular lines were not clearly seen. We then selected channels that were free from strong line emission to estimate the continuum flux level. We subtracted this continuum level and performed the task 'clean' for molecular emission lines. The 'clean' procedure was applied also to the continuum data. We employed 40-channel spectral binning ( ∼ 17 km s -1 ) and 0.3 '' pixel -1 spatial binning in this clean procedure.", "pages": [ 2 ] }, { "title": "3. RESULTS", "content": "Continuum emission properties are shown in Figure 1 and Table 3. Figure 2 displays the integrated intensity (moment 0) maps of molecular lines and spectra within the beam size at the continuum peak positions. Continuum emission is found to be well subtracted in the spectra (Figure 2 right panels), and so the moment 0 maps in Figure 2 (left) should reflect the properties of individual molecular gas emission lines. The peak flux, rms noise level, and synthesized beams in individual maps are summarized in Table 4 (row denoted 'all'). The peak positions of HCN J = 4-3 and HCO + J = 4-3 agree with that of continuum 'a', whereas that of HNC J = 4-3 is two pixels (0.6 '' ) displaced to the south from the continuum 'b' peak. CS J = 7-6 line at ν rest = 342.883 GHz is covered in a spectral window during the observations of HCN/HCO + J = 4-3 line, but it is not clearly detected in the spectrum (within the beam size) at the nucleus, which is defined from the continuum 'a' emission peak (Figure 2, right). Because the signals of the CS J = 7-6 emission are not clearly seen, to create the moment 0 map of CS J = 7-6, we refer to the velocity profile of the brightest HCO + J = 4-3 emission line at the nucleus and sum signals with v opt ≡ c ( λ -λ 0 )/ λ 0 = 4600-4925 [km s -1 ]. No clear CS J = 7-6 emission line is seen ( < 3 σ ) in the moment 0 map (Figure 2, left and Table 4). This undetected CS J = 7-6 line will not be used in the following discussion. The moment 0 maps in Figure 2 display spatially extended structures compared with the beam size, particularly for HCN and HCO + J = 4-3 lines. We created spectra of the HCN, HCO + , and HNC J = 4-3 lines, integrated over all regions with significant signal detection ( ∼ 3 '' × 3 '' ), in Figure 3. All show double-peaked emission profiles, as seen in the 12 CO J = 2-1, 12 CO J = 3-2, and 13 CO J = 2-1 lines (Wilson et al. 2008). We applied double Gaussian fits. The derived parameters are summarized in Table 5 (rows denoted with 'all'). For HCO + J = 4-3 emission, a triple Gaussian fit was also applied because three emission peaks were seen. The molecular line luminosities, integrated over regions of significant signal detection (corresponding to 'all' in Table 5) are summarized in Table 6, where we adopted equations (2) and (3) of Solomon & Vanden Bout (2005). Previously obtained high-spatial-resolution images of starburst indicators, such as the Pa α emission line (1.88 µ m), the PAH emission feature (3.3 µ m), the infrared 8.7 µ m continuum, and the radio 5-8.4 GHz (3.6-6 cm) continuum, have revealed the presence of ring-shaped circumnuclear starbursts with radii of 0.5-1.0' (Neff et al. 1990; Alonso-Herrero et al. 2001; Diaz-Santos et al. 2008; Olsson et al. 2010; Vaisanen et al. 2012). A similar ring pattern is discernible in the high-spatial-resolution (0.5' × 0.4') CO J = 2-1 molecular line map (Konig et al. 2013). Our ALMA map has a resolution of 1.5' × 1.3'. In this map, particularly in the brightest HCO + J = 4-3 line, where the highest S/N ratios are achieved, the emission is more elongated toward the north-south direction than the east-west direction, suggesting that a larger amount of high-density molecular gas is distributed at the northern and southern part of the nucleus. Figure 4 displays the intensity weighted mean velocity (moment 1) and intensity weighted velocity dispersion (moment 2) maps of HCN and HCO + J = 4-3 lines. The dense molecular gas in the northern (southern) part of the nucleus is clearly redshifted (blueshifted) relative to the nuclear velocity of NGC 1614 (v opt = 4800 [km s -1 ] for z = 0.016). The contours of emission of the red component with v opt > 4800 [km s -1 ] and the blue component with v opt < 4800 [km s -1 ] are displayed in Figure 5. The emission properties of the red and blue components are summarized in Table 4 (rows denoted with 'red' and 'blue'). The emission peak positions of the red and blue components are shifted by two pixels (0.6 '' ) to the north and south direction, respectively, from the nucleus. For HNC J = 4-3 emission in Figure 5, the blue component is much brighter than the red component, which could explain the slight peak offset in the HNC J = 4-3 moment 0 map (integrated over all ve- ty components with significant signal detection) with respect to the nuclear position (Table 3). Spectra within the beam size, at the nuclear position and the peak position of the red and blue components, are shown in Figure 6. As expected, spectra at the red (blue) component peaks are redshifted (blueshifted), relative to those at the nuclear position for all the HCN, HCO + , and HNC J = 4-3 lines. Gaussian fits were applied to the spectra. The resulting parameters are shown in Table 5 (rows denoted with 'peak', 'red', and 'blue'). The line widths of these dense molecular gas tracers are generally larger in the blue component than the red component, which has also been seen in ionized gas (De Robertis & Shaw 1988). This indicates that the southern part of the nucleus is more turbulent than the northern part. Figure 7 presents the position-velocity diagram along the north-south direction, passing through the nucleus. The channel map of the brightest HCO + J = 4-3 emission line is shown in Figure 8. The dense molecular gas in the northern region has greater velocity (more redshifted) than the southern molecular gas.", "pages": [ 2, 3 ] }, { "title": "4.1. Spatial distribution of dense molecular gas", "content": "The red and blue components of the dense molecular gas show emission peaks at ∼ 0.6 '' north and south of the nucleus, respectively, roughly corresponding to the northern and southern parts of the previously identified circumnuclear starburst ring with a radius of 0.5-1 '' (Neff et al. 1990; Alonso-Herrero et al. 2001; Diaz-Santos et al. 2008; Olsson et al. 2010; Vaisanen et al. 2012). Vaisanen et al. (2012) investigated the spatial distribution of the 3.3 µ mPAH emission feature, and found particularly strong PAH emission at the northern and southern parts of the ring. The 3.3 µ m PAH emission is a good indicator of starburst activity (Moorwood 1986; Imanishi & Dudley 2000; Imanishi et al. 2006a, 2008, 2010b). In contrast, the Pa α (1.88 µ m) emission, originating in HII regions, is strongest at the eastern and western sides of the circumnuclear starburst ring (Alonso-Herrero et al. 2001). How are these two kinds of starbursts related? One scenario involves an age difference. The eastern and western starburst regions are probed with the tracers of HII regions where plenty of ionizing ( λ < 912 ˚ A) UV photons, usually dominated by short-lived massive Ostars, are needed. In contrast, the 3.3 µ m PAH emission mostly comes from photo-dissociation regions between HII regions and molecular gas, where PAHs are excited by non-ionizing ( λ > 912 ˚ A) stellar UV photons (Sellgren 1981). For the PAH-exciting non-ionizing UV continuum, the contribution from less massive stars than O stars is higher than that for ionizing UV photons inside HII regions. Thus, the PAH emission feature is sensitive not only to very young O-stars dominant starbursts but also to slightly aged starbursts where massive O-stars have mostly died, but less massive stars (e.g., B-stars) still survive and emit a sufficient quantity of non-ionizing PAH-exciting UV photons. Diaz-Santos et al. (2008) showed that the 8.7 µ minfrared dust continuum, relative to the HII region tracer Pa α line (1.88 µ m), is enhanced at the northern and southern starburst ring. Because non-ionizing UV continuum emission can contribute significantly to the infrared continuum emission but cannot do so for the Pa α emission, the observed spatial variation in the 3.3 µ m PAH to Pa α and 8.7 µ m to Pa α flux ratio is explainable under the scenario that the typical starburst age is older at the northern and southern ring than at the eastern and western ring (Diaz-Santos et al. 2008). If the circumnuclear starburst ring is formed by nuclear starbursts that are progressing outward (AlonsoHerrero et al. 2001), the age difference in starbursts at different positions of the ring needs to be explained. Dust extinction is another possibility because Pa α emission at 1.88 µ m can be more highly flux-attenuated than the 3.3 µ m PAH emission and the 8.7 µ m dust continuum. Radio free-free continuum emission from HII regions is less susceptible to foreground dust extinction and could help to determine whether Pa α emission is significantly affected by dust extinction. High-spatialresolution radio continuum maps at 5 GHz (6 cm) and 8.4 GHz (3.6 cm) are available (Neff et al. 1990; Olsson et al. 2010) and Olsson et al. (2010) ascribe the radio 5 GHz and 8.4 GHz emission in NGC 1614 to free-free emission from HII-regions in young starbursts. Because the effects of free-free absorption inside HII regions are smaller at 8.4 GHz than at 5 GHz, the 8.4 GHz radio emission map is taken as the better probe of the true spatial distribution of HII regions. In the radio 8.4 GHz map, strong emission is detected in the eastern and western starburst ring (Olsson et al. 2010; Konig et al. 2013), as seen in Pa α line map (AlonsoHerrero et al. 2001), confirming that luminous HII regions are present at those locations. However, at 8.4 GHz, bright emission is seen also at the northern starburst ring, where Pa α emission is not strong (Konig et al. 2013). Similarly, when compared with the eastern starburst ring, the southern starburst ring is more conspicuous at 8.4 GHz than in Pa α (Konig et al. 2013). The comparison of the radio 8.4 GHz (3.6 cm) and 1.88 µ m Pa α emission indicates that HII-regions at the northern and southern parts of the starburst ring, unveiled by the radio 8.4 GHz emission, are not sufficiently distinguished by Pa α emission. These northern and southern regions of the starburst ring are the locations where dense molecular gas is distributed, according to our ALMA data. Given that dust coexists with dense molecular gas, dust extinction is a natural explanation for the small Pa α to 8.4 GHz flux ratio at the northern and southern starburst rings. Weak dust extinction is reported for starburst regions in NGC 1614 compared with other general starburst galaxies (Alonso-Herrero et al. 2001), based on near-infrared observations at λ < 2 µ m. This could be due partly to the fact that observations at λ < 2 µ m, including Pa α emission, selectively trace emission from less dusty starburst regions at the eastern and western ring and do not properly probe the dusty starbursts at the northern and southern parts of the ring due to flux attenuation by dust extinction.", "pages": [ 3, 4 ] }, { "title": "4.2. Dynamics of dense molecular gas", "content": "Our ALMA data show that the high-density molecular gas in the northern part of the nucleus is redshifted and that gas in the southern part is blueshifted with respect to the nuclear velocity of this galaxy. This suggests the rotation of dense molecular gas along the east-west axis (Neff et al. 1990). A similar velocity pattern was found previously in the ionized gas maps (De Robertis & Shaw 1988) and the lower density molecular gas probed with CO J = 3-2 and J = 2-1 (Wilson et al. 2008; Konig et al. 2013). The observed velocity dispersion is highest in the nuclear region with ∼ 80 km s -1 (Figure 4), although it may be affected by the beam smearing of a rotating motion at the center. This value is similar to those measured through near-infrared spectroscopy (Shier et al. 1994) and with (sub)millimeter CO J = 3-2 and CO J = 2-1 emission lines (Wilson et al. 2008). The rotational motion found in our moment 1 maps (Figure 4) can be used to derive the dynamical mass inside the rotating dense molecular gas disk. We used the HCO + J = 4-3 line because it is brighter than HCN J = 4-3, and so the achieved S/N ratios are higher. In Figure 4, the rotational velocity is v ∼ 100 km s -1 at 1.5-2 '' (or r = 480-640 pc at z = 0.016) from the nucleus. The derived dynamical mass within 1.5-2 '' radius is M dyn = rv 2 /G/sin(i) 2 = 1.5-2.5 × 10 9 M /circledot , where the inclination angle i = 51 · is adopted (De Robertis & Shaw 1988; Alonso-Herrero et al. 2001). This mass is comparable to the previously estimated values (Shier et al. 1994; Alonso-Herrero et al. 2001; Olsson et al. 2010).", "pages": [ 4 ] }, { "title": "4.3. Flux ratios of dense gas tracers", "content": "Figure 9 is a plot of HCN-to-HCO + and HCN-to-HNC flux ratios at the J = 4-3 transition, derived from the spectra at the nucleus, red component peak position, blue component peak, and all regions with significant signal detection. In all data, HCN J = 4-3 flux is smaller than that of HCO + J = 4-3, but higher than HNC J = 4-3 flux. Using the J = 1-0 transitions of HCN, HCO + , and HNC, the possibility of distinguishing the hidden energy sources of merging LIRG's dusty nuclei is suggested (Kohno 2005; Imanishi et al. 2004, 2006b, 2007; PerezBeaupuits et al. 2007; Krips et al. 2008; Imanishi et al. 2009; Costagliola et al. 2011). In general, HCN-to-HCO + flux ratios are small ( < 1), and HCN-to-HNC flux ratios are large ( > 1) in starburst-dominated galaxies, whereas HCN-to-HCO + flux ratios can be high ( > 1) in AGNs. AGNs could enhance HCN flux relative to HCO + , due possibly to HCN abundance enhancement by AGN radiation (Harada et al. 2010) and/or infrared radiative pumping of HCN (Sakamoto et al. 2010). The low HCNto-HCO + flux ratios obtained in the starburst-dominated galaxy NGC 1614 at J = 4-3 are similar to other starburst galaxies at J = 1-0. To obtain a physical interpretation of the observed J = 4-3 flux ratio, we need to know the excitation conditions. The HCN J = 1-0 flux of NGC 1614 was found to be 7.2 [Jy km s -1 ] by Gao & Solomon (2004) based on single dish telescope observations. Under thermal excitation, the HCN J = 4-3 flux is expected to be 16 times higher than J = 1-0, and so ∼ 115 [Jy km s -1 ]. Our ALMA observations provide an observed HCN J = 4-3 flux from all signal-detected regions of 2.8 [Jy km s -1 ]. The smaller flux of our ALMA data could partly be caused by missing flux, as our ALMA observations are insensitive to spatially extended emission with > 6 '' . Scoville et al. (1989) and Wilson et al. (2008) estimated that in NGC 1614, the nuclear ( < several arcsec) CO J = 1-0 and J = 3-2 emission can account for > 30% and > 45% of the total flux measured with single-dish telescopes. Because the HCN J = 4-3 line traces higher density molecular gas ( > 10 6 cm -3 ) than do CO J = 1-0 and J = 3-2 lines and because higher-density gas is more concentrated in the nuclear region, our ALMA HCN J = 4-3 data should recover > 45% of the total flux. Even assuming a missing flux of a factor of ∼ 2, the HCN J = 4-3 flux of NGC 1614 is 5.6 [Jy km s -1 ], only < 5% of the expected flux (115 Jy km s -1 ) for thermal excitation. Thus, HCN J = 4-3 line is significantly sub-thermally excited in NGC 1614, as observed in nearby galaxies at > 100 pc scale (Knudsen et al. 2007). Since the critical density of HCN J = 4-3 (n crit ∼ 2 × 10 7 cm -3 ) is higher than that of HCO + J = 4-3 (n crit ∼ 4 × 10 6 cm -3 ) (Meijerink et al. 2007), HCO + J = 4-3 is more easily excited than HCN J = 4-3 in starbursts. In an AGN, the emission surface brightness is higher than starburst activity, so the surrounding dust and gas are heated to a higher temperature, which may help to excite HCN J = 4-3 more than starburst activity. A high HCN-to-HCO + J = 4-3 flux ratio could be a good diagnostic of an AGN, simply because of the high excitation of HCN J = 4-3 in an AGN, even without an HCNabundance enhancement (Harada et al. 2010). HCN-toHCO + flux ratios of known AGN-important galaxies are being measured in our ALMA program (Imanishi et al., in preparation), and they tend to show higher HCN-toHCO + J = 4-3 flux ratios than NGC 1614, the template starburst galaxy (see also Imanishi et al. 2010a; Sakamoto et al. 2010; Iono et al. 2013). Since the J = 4-3 lines of HCN and HCO + are at higher frequencies (shorter wavelengths) than the lower J transition lines, the empirical energy diagnostic method, if established at J = 4-3, is applicable to more distant merging LIRGs using ALMA. This advantage is strengthened if HCN excitation is generally thermal up to the J = 43 transition in AGN-important galaxies because HCN flux increases proportional to the square of frequency, partly compensating for the increase in Earth's atmospheric background emission at higher-frequency ALMA bands. However, if the excitation at HCN J = 4-3 is sub-thermal, then HCN J = 3-2 or J = 2-1 lines may be better tracers of AGN in terms of actually obtainable S/N ratios with ALMA. Additional molecular line transition data at J = 3-2 and 2-1 for starburst-dominated galaxies and AGN-important galaxies are needed (1) to distinguish whether high HCN-to-HCO + flux ratios at J = 4-3 in AGN-important galaxies are due to HCN abundance enhancement and/or more HCN J = 4-3 excitation than starbursts and (2) to identify the J transition lines that are practically the best diagnostic for separating AGNs from starburst-dominated galaxies.", "pages": [ 4, 5 ] }, { "title": "5. SUMMARY", "content": "We performed HCN, HCO + , and HNC J = 4-3 line observations of the well-studied starburst galaxy NGC 1614 to trace the properties of the high-density molecular gas. Our results are summarized as follows: shifted and blueshifted, respectively, with respect to the nuclear velocity of this galaxy. When the emission is separated into the red and blue components, the red and blue components are strongest at 0.6 '' north and south of the nucleus for all of the HCN, HCO + , and HNC J = 4-3 lines. We thank E. Mullar and H. Nagai for their useful advice on ALMA data analysis. M.I. is supported by Grants-in-Aid for Scientific Research (no. 22012006). This paper makes use of the following ALMA data: ADS/JAO.ALMA#2011.0.00020.S . ALMA is a partnership of ESO (representing its member states), NSF (USA), and NINS (Japan), together with NRC (Canada) and NSC and ASIAA (Taiwan), in cooperation with the Republic of Chile. The Joint ALMA Observatory is operated by ESO, AUI/NRAO, and NAOJ. P., Robertson, B., & Springel, V. 2005, ApJ, 630, 705 Kohno, K. 2005, in AIP Conf. Ser. 783, The Evolution of Starbursts, ed. S. Huttemeister, E. Manthey, D. Bomans, & K. Weis (New York: AIP), 203 (astro-ph/0508420) Komatsu, E., et al. 2009, ApJS, 180, 330 Lintott, C., & Viti, S. 2006, ApJ, 646, L37 Sellgren, K. 1981, ApJ, 245, 138 Solomon, P. M., & Vanden Bout, P. A. 2005, ARA&A, 43, 677 Note . - Col.(1): Object name. Col.(2): Redshift. Col.(3)-(6): f 12 , f 25 , f 60 , and f 100 are IRAS fluxes at 12 µ m, 25 µ m, 60 µ m, and 100 µ m, respectively, taken from Sanders et al. (2003). Col.(7): Decimal logarithm of infrared (8 -1000 µ m) luminosity in units of solar luminosity (L /circledot ), calculated with L IR = 2 . 1 × 10 39 × D(Mpc) 2 × (13.48 × f 12 + 5.16 × f 25 + 2 . 58 × f 60 + f 100 ) [ergs s -1 ] (Sanders & Mirabel 1996). Note . - Col.(1): Observed molecular line. Col.(2): Observing date in UT. Col.(3): Number of antennas used for observations. Cols.(4), (5), and (6): Bandpass, flux, and phase calibrators used for our NGC 1614 observations, respectively.", "pages": [ 5, 6, 7 ] }, { "title": "Continuum emission of NGC 1614", "content": "Note . - Col.(1): Continuum 'a' and 'b' data were taken during observations of HCN/HCO + J = 4-3 and HNC J = 4-3, respectively. Col.(2): Central frequency of the continuum in [GHz]. Col.(3): Peak signal value in the continuum map in [mJy beam -1 ], and detection significance, relative to the rms noise, in parentheses. Col.(4): The coordinate of the continuum emission peak in J2000. Col.(5): The rms noise (1 σ ) of the continuum map in [mJy beam -1 ]. Col.(6): The synthesized beam of the continuum map. Position angle is 0 · along the north-south direction, and increases in the counter-clockwise direction. Note . - Col.(1): Molecular line. Col.(2): Velocity component. The notations 'all', 'red', and 'blue' mean all velocity components with significant signal detection, red component with v opt ≡ c ( λ -λ 0 )/ λ 0 > 4800 km s -1 , and blue component with v opt < 4800 km s -1 , respectively. Col.(3): Peak flux in the integrated intensity (moment 0) map in [Jy beam -1 km s -1 ] and detection significance relative to the rms noise, in parentheses. Col.(4): The rms noise level (1 σ ) in the moment 0 map in [Jy beam -1 km s -1 ]. Col.(5): Synthesized beam of the moment 0 map. Position angle is 0 · along the north-south direction and increases in the counter-clockwise direction. a The peak position of the HNC J = 4-3 emission, integrating over all velocity components with significant signal detection, is two pixels (0.6 '' ) south of the continuum 'b' peak, which could be explained by the stronger blue HNC emission component compared with the red component (Figure 5). See text in § 3. b The HCN J=4-3 to CS J=7-6 flux ratio is > 3.7. This lower limit is lower than the ratios found in AGNs, and is comparable to those observed in starburst galaxies (Izumi et al. 2013). Note . - Col.(1): Molecular line. Col.(2): Position and area for spectral extraction. 'all' means spectra integrated over all regions of significant signal detection ( ∼ 3 '' × 3 '' ). The terms 'peak', 'red', and 'blue' denote spectra within the beam size at the peak position of the continuum, red, and blue molecular line components, respectively. The coordinates of the red and blue peaks are (04 34 00.01, -08 34 44.3) and (04 34 00.01, -08 34 45.5) in J2000, respectively, for all of the HCN, HCO + , and HNC. For the 'peak', the continuum 'a' peak coordinate (Table 3) is used for HCN and HCO + , and the continuum 'b' peak (Table 3) is used for HNC. Cols. (3)-(6): Gaussian fits of the detected molecular emission lines. For 'all', double Gaussian fits are applied because emission lines are double peaked. For 'all' of HCO + , a triple Gaussian fit is also applied for comparison. Col.(3): Central velocity of the Gaussian fits in [km s -1 ]. Col.(4): Peak flux of the Gaussian fits in [mJy]. Col.(5): Full width at half maximum (FWHM) of the Gaussian fits in [km s -1 ]. Col.(6): Total line flux, based on the Gaussian fits, in [Jy km s -1 ]. a Wilson et al. (2008) derived the flux to be > 14 ± 3 [Jy km s -1 ] based on the Submillimeter Array (SMA) data, which probe emission over the spatial extent of < 7 '' . Note . - Col.(1): Molecular line. Col.(2): Luminosity in units of [L /circledot ]. Col.(3): Luminosity in units of [K km s -1 pc 2 ].", "pages": [ 7, 8, 9 ] } ]

[ { "title": "ABSTRACT", "content": "In an attempt to widen access to the study of magnetic fields in stellar astronomy, I present MOOGStokes, a version of the MOOG one-dimensional LTE radiative transfer code, overhauled to incorporate a Stokes vector treatment of polarized radiation through a magnetic medium. MOOGStokes is a suite of three complementary programs, which together can synthesize the disk-averaged emergent spectrum of a star with a magnetic field. The first element (a pre-processing script called CounterPoint) calculates for a given magnetic field strength, wavelength shifts and polarizations for the components of Zeeman sensitive lines. The second element (a MOOG driver called SynStokes derived from the existing MOOG driver Synth) uses the list of Zeeman shifted absorption lines together with the existing machinery of MOOG to synthesize the emergent spectrum at numerous locations across the stellar disk, accounting for stellar and magnetic field geometry. The third and final element (a post-processing script called DiskoBall) calculates the disk-averaged spectrum by weighting the individual emergent spectra by limb darkening and projected area, and applying the effects of Doppler broadening. All together, the MOOGStokes package allows users to synthesize emergent spectra of stars with magnetic fields in a familiar computational framework. MOOGStokes produces disk-averaged spectra for all Stokes vectors ( I , Q , U , V ), normalized by the continuum. MOOGStokes agrees well with the predictions of INVERS10 a polarized radiative transfer code with a long history of use in the study of stellar magnetic fields. In the non-magnetic limit, MOOGStokes also agrees with the predictions of the scalar version of MOOG. Subject headings: Techniques: polarimetric, spectroscopic - Stars: magnetic fields", "pages": [ 1 ] }, { "title": "MODIFICATION OF THE MOOG SPECTRAL SYNTHESIS CODES TO ACCOUNT FOR ZEEMAN BROADENING OF SPECTRAL LINES", "content": "Casey P. Deen 1,2 Draft version August 3, 2018", "pages": [ 1 ] }, { "title": "1. INTRODUCTION", "content": "In many areas of stellar astrophysics, the effects of magnetic fields are small and may be safely ignored. However, certain classes of stars (Ap stars, flare stars, active M dwarfs, young stellar objects, etc. . . ) have nonnegligible magnetic fields. Observationally, these strong magnetic fields can significantly affect the equivalent widths of many spectral features normally used to determine stellar physical properties. A model spectrum with physical parameters ( T eff , log g, [Fe / H]) but which omits the effect of the magnetic field will not match the observed spectrum of a magnetic star with the same physical parameters. More perniciously, a non-magnetic synthetic spectrum with different physical parameters will likely provide a better fit to the observed data, injecting a bias into any study of these parameters or quantities derived from these parameters (spectral type, age, mass etc. . . ). The increasing sensitivity of current and future infrared spectrographs (CRIRES (Kaeufl et al. 2004), TripleSpec (Wilson et al. 2004), SpeX (Rayner et al. 2003) XSHOOTER (Vernet et al. 2011), IGRINS (Yuk et al. 2010), GMTNIRS (Lee et al. 2010), etc. . . ) permit observations of cooler, more embedded objects. Unfortunately, magnetic fields affect infrared spectra more than visible spectra (the magnitude of the magnetic broadening grows as λ 2 , whereas Doppler broadening scales with λ ). Magnetic fields affect line shapes 1 Max Planck Institut fur Astronomie, Konigstuhl 17, D-69117 Heidelberg, Deutschland 2 Department of Astronomy University of Texas at Austin, 1 University Station, 78712, Austin, TX, USA in high resolution spectra, and affect equivalent widths of strong lines visisble in low resolution spectra, potentially biasing studies of objects only accessible at infrared wavelengths. Therefore, accurate studies of magnetic stars require a spectral synthesis code which can handle magnetic effects. Zeeman (1897) qualitatively described the splitting (in wavelength and polarization) of a spectral line under the influence of an external magnetic field. Honl (1925) developed the quantum mechanical formulas to describe the Zeeman effect on the spectrum of a parcel of emitting/absorbing material under the influence of a uniform magnetic field as a function of magnetic field strength, geometry, quantum mechanical properties of the transition, and Stokes ( I , Q , U , and V ) polarizations. In a stellar photosphere, the picture is not so simple. The observed spectrum of a star with a magnetic field is a complicated combination of light produced at different depths of the photosphere, in various polarizations, from different locations across the stellar disk, and at different orientations to the magnetic field. To complicate matters further, the propagation of polarized light in a magnetic medium requires careful attention to the Stokes parameters. The evolution through the stellar atmosphere of each Stokes parameter depends on the other Stokes parameters as well as on polarized opacities, requiring any code hoping to solve this system of coupled differential equations to treat each of the Stokes parameters as vector quantities. Radiative transfer codes developed for the synthesis of large spectral regions (MOOG (Sneden 1973), SPECTRUM (Gray & Corbally 1994), Synspec (Hubeny et al. 1985), Synth (Piskunov 1992), etc. . . ), use various algorithms (Feautrier 1964; Edmonds 1969) to solve the equation of radiative transfer and calculate only the emergent intensity (Stokes I ), and do not account for the interplay between the Stokes vectors. Motivated by the study of magnetic fields in sunspots, Unno (1956) offered a solution to the Stokes vector equation of radiative transfer through a stellar atmosphere for a normal Zeeman triplet. Rachkovsky (1962) amended these basic equations to include magneto-optical effects with the addition of the Faraday-Voigt anomalous dispersion profile. Beckers (1969) then generalized the theory to anomalous Zeeman patterns. Further investigations into the radiative transfer of polarized radiation through a magnetic medium largely focused on improving computation speed via mathematically complex matrix calculations (Auer et al. 1977; Rees et al. 1989), calculating effects due to deviations from local thermodynamic equilibrium (LTE) (Landi Degl'Innocenti 1976; Auer et al. 1977; Socas-Navarro et al. 2000), and determining a quantum mechanical basis (Landi Degl'Innocenti & Landi Degl'Innocenti 1972) for the basic equations (Unno 1956; Rachkovsky 1962). Many of these early formulations of the radiative transfer were useful primarily for detailed studies of single (or small numbers of) absorption lines. Building on this early work, there are several more recent codes that account for the effects of magnetic fields and polarized radiative transfer (COSSAM (Stift 2000), Zeeman2 (Landstreet 1988), and Synthmag (Piskunov 1999)) which have been used to study stars with magnetic fields (Weiss et al. 2000; Valenti & Johns-Krull 2001; Stift & Alecian 2001; Landstreet et al. 2008; Silvester et al. 2012). MOOG (Sneden 1973) is a widely used onedimensional LTE radiative transfer code with a suite of drivers often used to analyze stellar spectra. The MOOG driver synth uses stellar atmosphere models together with atomic and molecular line parameters to produce high resolution synthetic emergent spectra. In this paper, I present a customization of MOOG called MOOGStokes, which permits MOOG to account for the major effects of Zeeman splitting of spectral lines. MOOGStokes traces the polarized Stokes components though a magnetic stellar photosphere with a uniform magnetic field, producing a disk-averaged spectrum suitable for comparison with observed spectra. Building the vector radiative transfer package into the familiar framework of MOOG lowers the potential barrier into studies of magnetic fields to an existing broad community of stellar spectroscopists. Additionally, while Zeeman broadening of absorption lines in infrared spectra can make accurate determinations of T eff and log g impossible for scalar codes (codes which do not solve the Stokes vector equation of radiative transfer), armed with a polarized radiative transfer Zeeman code, the same data can not only constrain temperature and surface gravity more accurately, but also provide a measure of magnetic field strength. Finally, all spectral synthesis codes make certain assumptions (e.g. their chosen analytic approximation for solving the equation of radiative transfer) and a code which makes different assumptions from other polarized radiative transfer codes, or calculates relevant quantities in different manners can serve as a foil to help elucidate the consequences of those assumptions and the robustness of results (Wade et al. 2001). In the subsequent paper I give a brief introduction to the theoretical concepts involve in polarized radiative transfer in § 2. In § 3, I describe the algorithmic structure of the MOOGStokes suite of software. § 4 describes the results of various verification tests and benchmarks, and in § 5, I discuss possible applications of MOOGStokes, including an illustration of T eff bias caused by neglecting the effects of magnetic fields.", "pages": [ 1, 2 ] }, { "title": "2. THEORETICAL BACKGROUND", "content": "Before describing the MOOGStokes algorithm in detail, I summarize the relevant theoretical concepts and formulas involved in accounting for the effects of magnetic fields in a synthetic spectrum. This summary is not intended to be an exhaustive theoretical introduction, but should be sufficient to allow discussion of the algorithms in MOOGStokes. There are excellent and thorough discussions of this material in Landi Degl'Innocenti (1976), Rees et al. (1989), and Piskunov & Kochukhov (2002) (hereafterward referred to as PK02).", "pages": [ 2 ] }, { "title": "2.1. The Zeeman Effect", "content": "/negationslash The Zeeman effect describes the behavior of spectral absorption or emission lines under the influence of an external magnetic field. In the absence of a magnetic field, the states associated with different magnetic quantum numbers ( M J ) of an atomic state (corresponding to the eigenvalues of the angular momentum vector J projected along the z axis) are degenerate in energy. An external magnetic field breaks the degeneracy between the eigenstates into 2 J +1 sublevels (denoted by M J = J, J -1 , . . . , -( J -1) , -J ). For a transition between two atomic states, the total number of Zeeman components into which a spectral line will split depends on the J of the upper state, J of the lower state, and the electric dipole selection rules (∆ J = ± 1 , 0, ∆ J = 0 if J = 0, ∆ M j = ± 1 , 0, P f = -P i ). The shift in energy for a given eigenvalue M J is given by ∆ E = gµ B BM J , where g is the Land'e factor, µ B = e /planckover2pi1 2 m e is the Bohr magneton, and B is the strength of the magnetic field. The shift in energy for a photon emitted between state E up ( M J up ) and E low ( M J low ) is therefore ∆ E γ = ∆ E up -∆ E low = µ B B ( g up M J up -g low M J low ). Zeeman components with ∆ M J = 0 are known as π components, and correspond classically to a charge oscillating along the axis of the magnetic field. π components emit radiation linearly polarized the direction of the magnetic field. Zeeman components with ∆ M J = 1( -1) are known as σ components, and correspond classically to a charge in a right (left) hand circular orbit around the magnetic field vector (according to the IAU definition of Stokes V). σ components emit right (left) hand circularly polarized radiation along the field lines as well as linearly polarized radiation in transverse directions. In the following discussion, quantities related to π components are denoted by a p subscript, while quantities related to σ + , σ -components are denoted by ( b )lue and ( r )ed, signifying the direction of wavelength shift. Equation 1 (adapted from equations 1-3 in section 4 16 of Condon & Shortley (1935)) describes the relative intensities ( A b,p,r ) of Zeeman components as a function of ∆ J and ∆ M J (Ornstein & Burger 1924; Honl 1925) of the initial (lower) state.", "pages": [ 2 ] }, { "title": "2.2. Polarized Radiative Transfer", "content": "While the Zeeman effect describes the effect of a magnetic field on electric dipole transitions and the resultant radiation, still more theory is required to describe the transfer of that radiation through a magnetized stellar photosphere. In the following discussion, I adopt the geometrical convention described in Figure 1 (derived from Figure 2 of PK02), where γ is the angle the local magnetic field makes with the observer's line of sight, where χ is the clocking angle of the magnetic field projected onto the plane of the sky as measured from the x direction, and where θ is the viewing angle measured between the line-of-sight and the local normal to the stellar surface. The different polarizations of the Zeeman component require the use of the Stokes vector I = ( I, Q, U, V ) T , where I , Q , U , and V describe respectively the total intensity, two orthogonal linear polarization intensities, and one circular polarization intensity. The rmT emphasizes that I is a column vector. The equation of radiative transfer now takes vector form: where µ = cos θ , τ is the optical depth, K is the opacity matrix, and J is the emission vector, each computed at the wavelength of interest. The opacity matrix K (Equation 3, first introduced by Unno (1956) and later modified by Rachkovsky (1962)) describes the interaction between the intensities of different Stokes components. where κ C is the continuum opacity, κ 0 is the line opacity due to non-hydrogen absorbers/emitters, and κ I = κ C + κ 0 Φ I is the total opacity. The Φ matrix is comprised of Φ (Equation 4) and Ψ (Equation 5) matrix elements: The Φ matrix elements (Equations 4) are comprised of absorption profiles ( φ b,p,r , Equation 6), while the Ψ matrix elements (Equations 5) are composed of anomalous dispersion profiles ( ψ b,p,r , Equation 7) resulting from magneto-optical effects. SynStokes calculates φ and ψ profiles for all Zeeman components that contribute significantly at the current wavelength. All κ , Φ I,Q,U,V , Ψ Q,U,V , φ b,p,r , and ψ b,p,r implicitly depend upon the current wavelength of interest. The A coefficients reflect the relative strengths of different Zeeman components (Equation 1), and are normalized so that: In the nonmagnetic case, there are no wavelength shifts, meaning φ b,p,r = φ 0 (the absorption profile of the unsplit line), and all Φ Q,U,V and Ψ Q,U,V are zero due to the opposite signs of the π and σ components. In Φ I , however, the π and σ components add, giving Φ I = 1 2 ( φ 0 sin 2 γ + φ 0 ( 1 + cos 2 γ )) = φ 0 , and the opacity matrix K collapses to the non-magnetic scalar case ( K = κ C + κ 0 = κ I ) H and F are respectively the Voigt and Faraday-Voigt functions which describe the absorption and anomalous dispersion profiles as a function of line damping coefficients ( a ) and distance from the absorption line center ( v ) in terms of the Doppler width for the current absorber species. The Φ matrix is also used to calculate the emission vector J , given by Equation 9, assumes both the continuum and line emission are in local thermodynamic equilibrium, and calculates S continuum = S lines = B ( λ, T j ), where B is the Planck function, and T j is the temperature of the j th layer of the atmosphere. where e 0 = (1 , 0 , 0 , 0) T is a column vector. As described in the subesquent section, MOOGStokes now uses the quantities and variables discussed here to solve the Stokes vector equation of radiative transfer (Equation 2) and calculate the emergent spectrum, tracing the Stokes vectors from the base of the photosphere to the top using the Diagonal Lambda Element Operator (DELO) method.", "pages": [ 3, 4 ] }, { "title": "3. DESCRIPTION OF ALGORITHM", "content": "MOOGStokes contains three packages. A absorption line pre-processor, a MOOG driver, and a diskintegration post-processor. The pre-processor (named CounterPoint, see Section 3.1) calculates the number, polarization, and wavelength shift of Zeeman components into which each line splits, given a magnetic field strength. The MOOG driver (named SynStokes, see Section 3.2) takes the Zeeman-split line list together with a model atmosphere and magnetic field geometry, and calculates the emergent spectrum at many different locations across the stellar disk. The post-processing algorithm (named DiskoBall, see Section 3.3) then weights the emergent spectrum of each location on the disk by its projected area and limb darkening, applies a Doppler shift due to stellar rotation, and calculates the diskaveraged spectrum as seen by an observer. Algorithm 1 shows a pseudocode representation of the suite of three programs.", "pages": [ 4 ] }, { "title": "3.1. CounterPoint: Absorption Line Pre-Processor", "content": "I have developed a Python code called CounterPoint to pre-process absorption lines and account for the Zeeman effect prior to input into MOOG. Given a magnetic field strength and quantum mechanical constants for the transition ( J up , J low , g up , g low , log gf ), the program calculates the number and polarization into which the line will split, the relative intensities of the components, and the component energy (wavelength) shifts. From an initial linelist retrieved from VALD (Kupka et al. 2000) CounterPoint produces an entry in a MOOGreadable line list for each Zeeman component, containing the following information: central wavelength, relative oscillator strength, atomic species and ionization state, excitation potential, damping factors if known (i.e. Γ vdW , Γ Stark , Γ Rad ), and change in magnetic quantum number (∆ M J ). Because the intensity of a spectral line is directly proportional to the oscillator strength, I change the log gf value of each Zeeman component to match the relative intensities predicted by quantum mechanics (Ornstein & Burger 1924; Honl 1925), similar to common practices in studies of hyperfine structure. I perform the normalization (see Equation 8) here so that the sum of the oscillator strengths of all like-polarized components for a line equals the oscillator strength of the unsplit line. The first part of algorithm 1 describes the logic of the CounterPoint program. As an example, CounterPoint calculates the Zeeman splitting of a singly ionized iron line under the influence of a 5 . 0 kG magnetic field in the following manner: The 4923 . 927 ˚ A Fe line is a dipole transition between a 6 S 5 2 lower state ( J = 5 2 ) and a 6 P 3 2 upper state ( J = 3 2 ), with an oscillator strength of log gf = -1 . 320 and a lower state energy of 2 . 891 eV. The Land'e g factors of the lower and upper states are 2 . 0 and 2 . 4, respectively. The magnetic field will split the lower state into 6 levels, and the upper state into 4 levels. Application of the electric dipole selection rules determines that there will be four π components, and eight σ components. The relative intensities and relative oscillator strengths of the π and σ components are given in Table 1.", "pages": [ 4 ] }, { "title": "3.2. SynStokes: Polarized Radiative Transfer Calculator", "content": "I describe here the general framework of SynStokes, the FORTRAN driver I added to the MOOG spectral synthesis programs. The second part of algorithm 1 describes in pseudocode the algorithm for the MOOG driver 'SynStokes': SynStokes (SynStokes.f 1 ) begins by reading in a parameter file (Params.f 2 ). The parameter file describes (among other common MOOG inputs) the number of regions into which the stellar surface will be divided, the desired starting and ending synthesis wavelength, the desired model atmosphere file, and the location of the linelists. MOOGStokes then reads in the model atmosphere (Inmodel.f 2 ) and list of absorption lines (Inlines.f 2 ). After calculating the absorber number density (Eqlib.f 3 ), it calculates the populations of absorbers at each layer of the atmosphere, line center opacities (Nearly.f 3 ), and polarized opacities (CalcOpacities.f 1 ). SynStokes then creates a list of lines which contribute significantly to the opacity (Wavegrid.f 1 ), neglecting lines which are too weak due to insufficient opacity in the supplied model atmosphere (e.g. an O II line will not contribute significant opacity in a cool atmosphere, as negligible amounts of oxygen will be ionized). SynStokes will finely sample the region around strong lines to resolve their shapes, and coarsely sample regions with no strong lines. Then the program divides the stellar surface into different regions. For each re- gion, the program calculates the local orientation of the magnetic field relative to the observer (CalcGeom.f 1 ), and calculates the emergent spectrum (ComplexVoigt.f 1 Spline.f 1 , SplineDriver.f 1 , Curfit.f 1 , DELOQuad.f 1 ), one wavelength point at a time, writing the each spectrum to the output file. The program then moves on to the next region. To keep track of the additional variables required by polarzied radiative transfer, MOOGStokes adds or modifies the following COMMON blocks to the existing MOOG: Angles.com 1 , Atmos.com 2 , Linex.com 2 , and Stokes.com 1 . In the current implementation, SynStokes adopts a uniform radial geometry for the magnetic field, primarily for its simplicity. In future versions of the code, the user will be able to specify other magnetic field geometries, as well as multi-temperature atmospheres (as in the case of a stellar spot).", "pages": [ 4, 5 ] }, { "title": "3.2.1. Model Atmosphere", "content": "The model atmosphere gives SynStokes the temperature, pressure, and density profiles of the photospheric region as functions of the optical depth. I opt not to include the effects of magnetic pressure on the atmospheric structure. Future versions of the code may explicitly address the issue of magnetic pressure.", "pages": [ 5 ] }, { "title": "3.2.2. Absorption Line List", "content": "The original scalar version of MOOG, using a routine in Inlines.f, reads information regarding absorption lines from a file, where each line of the file contains the following information: Wavelength of the transition λ (in ˚ A or µ m), atomic (or molecular) species, energy of the lower state (in electron volts), oscillator strength of the transition, van der Waals damping coefficient, and molecular dissociation energy (only for molecular lines). I modified the Inlines routine to accept three additional paramters: ∆ M J , Γ Rad , and Γ Stark . The change in angular momentum ∆ M J as calculated by CounterPoint, allows SynStokes to identify π and σ -, σ + Zeeman components. Γ Rad and Γ Stark are damping constants related to radiative and Stark broadening, respectively.", "pages": [ 5, 6 ] }, { "title": "3.2.3. Equilibrium Calculations and Line Center Opacities", "content": "SynStokes then uses the existing machinery of MOOG to solve the classical Boltzmann and Saha equations to calculate the number of absorbers of each species and lower energy state at each layer of the atmosphere, assuming LTE. The number of absorbers in turn allows the calculation of the line-center opacity κ 0 for each line in the line list. A non-LTE calculation would include radiative and density effects (Socas-Navarro et al. 2000), and would slightly affect the line opacity and source function, but the LTE approximation is frequently made by other stellar radiative transfer codes (Landstreet 1988; Stift 2000; Piskunov & Kochukhov 2002).", "pages": [ 6 ] }, { "title": "3.2.4. Dividing the Stellar Surface into Different Regions", "content": "The emergent spectrum of a magnetic star is a function of the geometry of the magnetic field, the stellar photosphere, and the observing angle. SynStokes provides the user two strategies for synthesizing disk-averaged spectra. The first strategy (described in PK02) divides the stellar surface into a number of approximately equal-area tiles. The user can specify the inclination and clocking angle of the stellar rotation axis, and can control the tile size by specifying in the MOOG parameter file the number of total tiles and number of latitude belts. For each tile, SynStokes calculates the angles γ and χ for the center of the tile, as well as the viewing angle θ (see Figure 1). γ and χ are calculated using the orientation of the local magnetic field (assumed to be uniform and radial, and hence θ = γ ) relative to the observer. If the viewing angle θ implies that the center of the tile is visible to the observer (cos θ > 0), SynStokes calculates the emergent spectrum. This first strategy produces emergent spectra for all four Stokes parameters, but due to the large number of tiles necessary for calculation of an accurate average flux, can be quite slow. The second strategy available to the user is to divide the stellar disk into a number of annuli, and calculate the emergent Stokes I and V spectra for each annulus at the stellar equator. While significantly faster, this strategy only produces disk-averaged spectra for Stokes I and V (due to the azimuthal dependence of Stokes Q and U).", "pages": [ 6 ] }, { "title": "3.2.5. Calculation of Line Opacities, Opacity Matrix K , and Source Function J", "content": "In order to calculate K for a wavelength λ and atmospheric layer j , SynStokes first calculates the total φ b,p,r (Equations 4) and ψ b,p,r opacities (Equations 5) summed over all lines in the line list which contribute significant opacity at λ . As the original scalar version of MOOG contained only a formula for the Voigt profile (Kurucz 1970), I include in SynStokes the algorithm from Huml'ıˇcek (1982) to calculate both the Voigt and Faraday-Voigt profiles. Equipped with the individual polarized opacities and the geometry of the magnetic field, SynStokes calculates the individual elements of the opacity matrix K (see Equation 3) and of the emission vector J (see Equation 9). SynStokes then constructs a sequence of opacity matrices and emission vectors calculated at each layer of the atmosphere.", "pages": [ 6 ] }, { "title": "3.2.6. Calculation of Optical Depths", "content": "SynStokes must also keep track of the line and continuum optical depths to calculate an emergent spectrum. The optical depth given in the model atmosphere file is a reference optical depth (often measured at either a reference wavelength, or a Rosseland mean opacity). MOOG converts this reference optical depth to an optical depth at the current wavelength λ and atmospheric layer j using κ C and κ ref . However, MOOG does not calculate the physical depth through the atmosphere (a quantity required by the DELO algorithm in § 3.2.7), so SynStokes must caclculate this quantity. Adopting dτ l = -κ I dz for the line optical depth and dτ C = -κ C dz for the continuum optical depth at the current wavelength λ , SynStokes converts the reference optical depth given in the model atmosphere to a physical depth into the photosphere by integrating the equivalent equation for the reference optical depth z = -∫ τ ref 0 1 κ rmref d τ . Calculation of κ I and κ C then allows SynStokes to calculate line and continuum optical depths at the current atmospheric layer and wavelength λ .", "pages": [ 6 ] }, { "title": "3.2.7. DELO Integration", "content": "The scalar version of MOOG employes a formal integrative methodology to obtain contribution functions which are subsequently used to calculate the emergent intensity at a given wavelength (Edmonds 1969; Sneden 1973). To account for the exchange of light between different Stokes vectors, it becomes necessary to solve the Stokes vector equation of radiative transfer (Equation 2) through the atmosphere of the star. The most straight-forward method of solving this system of differential equations is a brute-force Runge-Kutta algorithm (Landi Degl'Innocenti 1976). However, while accurate, Runge-Kutta algorithms are quite computationally intensive. Even given the numerous folding-length times of Moore's Law between the computers available to Landi Degl'Innocenti (1976) and the computers of today, the time required to synthesize all but the smallest of wavelength intervals becomes prohibitively long. For this reason, I adopt the Diagonal Element Lambda Operator (DELO) method of Rees & Murphy (1987). This method slightly modifies equation 2, allowing the propagation of the Stokes I vector to be treated as a linear relation between adjacent points in the stellar atmosphere (see Equations 10). To improve the accuracy of this algorithm, I use the quadratic formula ( Z ) for the emission vector S ' employed by Olson & Kunasz (1987), Kunasz & Auer (1988), Socas-Navarro et al. (2000), and PK02. For a more detailed discussion of the DELO algorithm and definitions of the constants contained in equations 10, I refer the reader to the excellent treatments in Rees & Murphy (1987), Socas-Navarro et al. (2000), and in PK02. SynStokes sub-samples each decade of τ ref by steps of 0 . 05 dex, to trace each Stokes vector from the base (log τ ref ∼ 2) to the top (log τ ref ∼ -5). I assume the radiation originating at the base of the photosphere to be initially unpolarized and in LTE ( I = B ( T j ) , Q = U = V = 0). I then calculate the emergent continuum by setting all non-continuum sources of opacity to zero and performing the same DELO integration. The MOOGStokes implementation of the DELO algorithm uses the ATLAS and LAPACK linear algebra packages (Anderson et al. 1999; Blackford et al. 2002; Whaley & Petitet 2005), as well as the Numerical Recipes implementation of cubic splines (Press et al. 1992). SynStokes calculates an emergent spectrum for each tile of stellar surface visible to the observer. SynStokes saves each individual spectrum (in Stokes I, Q, U, V, and continuum), along with a description of its geometry ( θ, γ, χ ) for post-processing (as described in the subsequent section).", "pages": [ 6, 7 ] }, { "title": "3.3. DiskoBall: Disk Integration Post-Processor", "content": "Once SynStokes has calculated emergent spectra from the stellar tiles visible to the observer, the individual spectra must then be averaged together to produce the final disk-averaged spectrum. For this purpose, I have created a Python post-processing script called DiskoBall. DiskoBall reads emergent spectra from the output files created by SynStokes, and combines them into a single disk-averaged spectrum. If the spectra correspond to tiles covering the entire stellar surface, DiskoBall calculates the disk-averaged spectrum by weighting the flux of each tile by the limb darkening ( § 3.3.1) and projected area of the tile ( § 3.3.2), and shifting the wavelength by the appropriate Doppler velocity ( § 3.3.3), given the source geometry (inclination and clocking angle). DiskoBall then saves a composite spectrum for each Stokes parameter. If instead, the spectra correspond to annuli, DiskoBall produces a composite spectrum by convolving the spectrum of each annulus with a rotation kernel and weighting by annular area and limb darkening (3.3.1)(Valenti & Piskunov 1996).", "pages": [ 7 ] }, { "title": "3.3.1. Limb Darkening", "content": "I use the simple power-law limb darkening prescription from Hestroffer & Magnan (1998), primarily for its simplicity. As the limb darkening effect is wavelength dependent, Diskoball calculates the limb darkening coefficient for the mean wavelength, and applies it to the entire spectrum. While newer limb darkening laws can produce more accurate results, the Hestroffer & Magnan (1998) law provides sufficient accuracy for the initial release of the program.", "pages": [ 7 ] }, { "title": "3.3.2. Projected Area", "content": "DiskoBall weights the emergent spectrum coming from each one of the surface tiles by the projected area, as seen by the observer. The projected area is the surface area (d α · d φ ) multiplied by the cosine of the viewing angle ( µ = cos θ ).", "pages": [ 7 ] }, { "title": "3.3.3. Doppler Broadening", "content": "DiskoBall allows the user to provide a rotational velocity v for the requested composite spectrum. Together with the source geometry, DiskoBall calculates the appropriate v sin i for the spectrum from each element on the stellar disk and applies the corresponding red or blue shift, before coadding with the other spectra. This feature allows the user to re-process a single output of SynStokes with any number of arbitrary stellar rotational velocities.", "pages": [ 7 ] }, { "title": "4. VERIFICATION", "content": "Before trusting the predictions of any spectral synthesis code, the code must reproduce to high accuracy the calculations of other well-tested spectral synthesis codes under identical input conditions. To verify that the code can accurately synthesize spectra when no magnetic fields are present, I check the emergent Stokes I calculated by MOOGStokes against that of its predecessor, MOOG. After successfully proving MOOGStokes introduces no major deviations, I then check the code against the magnetic profiles provided in Wade et al. (2001).", "pages": [ 7 ] }, { "title": "4.1. Non-Magnetic Verification", "content": "MOOGStokes uses the existing software framework of MOOG to support a completely different spectral synthesis engine. To verify that the chassis of MOOG is properly connected to its new engine, I have tested that MOOGStokes produces the same result as the original MOOG in the non-magnetic limit. For this test, I synthesize emergent central intensity (Figure 2, µ = 1 . 0) produced by the Fe II line described in Table 1 with an ATLAS9 model atmosphere (Castelli & Kurucz 2003). There are small differences on the order of 0 . 1% of the continuum. These differences are likely due to small numerical differences between the DELO and contribution function algorithms, both approximate analytical solutions to the radiation transport equation. The profiles produced by the two codes agree to a level where errors in the spectrum will be dominated by uncertainties in the parameters of the transitions being modeled.", "pages": [ 7 ] }, { "title": "4.2. Comparison to Wade et al. (2001)", "content": "The most important test of a polarized radiative transfer code is a synthesis of a transition under the influence of a strong magnetic field. Figure 4 of Wade et al. (2001) shows Stokes IQUV profiles produced by the Fe II line described in Table 1 under the influence a magnetic fields of strength 0 . 1, 5 . 0, and 20 . 0 kG , as calculated by the INVERS10 polarized radiative transfer code. INVERS10 makes use of the more accurate Feautrier algorithm (Feautrier 1964; Auer et al. 1977), while MOOGStokes uses the quadratic DELO algorithm (Socas-Navarro et al. 2000). Figure 3 shows a comparison of the predictions of MOOGStokes with the profiles provided in Wade et al. (2001). While the differences between MOOGStokes and INVERS10 ( ∼ 0 . 25%) are larger than those between MOOG and MOOGStokes ized radiative transfer. The result of these modifications, MOOGStokes, is sufficient for the study of the behavior of absorption line shapes and equivalent widths under the influence of changes in the physical parameters of the photosphere ( T eff , log g , and B ). I have attempted to make the interface of MOOGStokes similar and complementary to that of the original MOOG program, allowing observers and stellar spectroscopists already familiar with MOOG to make the transition to studying magnetic fields. ( ∼ 0 . 1%), they are still small enough that other uncertainties in the creation of synthetic spectra (model atmospheres, oscillator strengths, etc. . . ) will dominate the errors in any comparisons to real observations.", "pages": [ 7, 8 ] }, { "title": "5. DISCUSSION", "content": "I have described the necessary steps to modify an existing scalar spectral synthesis code to account for the major effects produced by Zeeman splitting and polar- As intimated in the introduction, neglecting the effects on spectra of strong magnetic fields can affect conclusions drawn from the spectra. The damping wings and widths of certain absorption lines are frequently used to determine certain physical parameters (surface gravity, microturbulence, v sin i , etc. . . ). The equivalent widths of other lines are often used to constrain other parameters (effective temperature, metallicity, etc. . . ). Weak optically thin lines, which are in the linear portion of the curve of growth change their shapes under the influence of a magnetic field, but do not change appreciably in equivalent width. Strong optically thick lines change shape as well, but also increase in equivalent width, due to the saturation of the individual Zeeman components in the logarithmic portion of the curve of growth. While changes in line shape only become noticeable at high spectral resolution, changes in equivalent widths of strong lines affect spectra of all resolutions (and hence the aforementioned properties derived from them). As an illustration of the magnitude of the effect that strong magnetic fields can have on the appearance of the emergent spectrum of a star as observed by a normal spectrograph (Stokes I), Figure 4 shows a comparison between three synthetic spectra of the sodium doublet at 2 . 2 µ m convolved to R = ∆ λ λ = 2000. For late-type stars, the equivalent width of the sodium doublet is often used in determining spectral type and veiling (excess continuum emission due to hot circumstellar dust). The first spectrum (black solid line) is a spectrum generated with parameters appropriate for a low mass young stellar object ( T eff = 4000 K , log g = 4 . 0, and average magnetic field strength of 2 . 0 kG ). Converting effective temperature to spectral type, this corresponds to a spectral type of roughly K7 (Mamajek 2013; Luhman et al. 2003). The second spectrum (red dashed line) shows a spectrum generated with parameters appropriate for a young non-magnetic K7 star ( T eff = 4000 K , log g = 4 . 0, and no magnetic field). The third spectrum (blue dotted line) is a spectrum of a young non-magnetic M1.5 star ( T eff = 3600 K , log g = 4 . 0, no magnetic field). The equivalent width of the sodium doublet in the M1.5 star matches the equivalent width of the magnetic young star better than the K7 star, even though its effective temperature is 400 K cooler. Astronomers often determine ages and masses of young stellar objects by comparing their locations on the HR diagram to evolutionary models (Baraffe et al. 1998; Palla & Stahler 1999). An error in effective temperature of the magnitude displayed in this example can result in errors in the derived age of several million years and errors several tenths of M /circledot in the derived stellar mass, introducing biases into studies of young stellar object properties (i.e. lifetimes of circumstellar disks, initial mass functions, etc. . . ). While magnetic fields can make determining stellar parameters from individual or small numbers of absorption lines difficult, not all lines are affected equally by the Zeeman effect. In subsequent investigations, I will use MOOGStokes, along with this fact, as tools to determine physical parameters ( T eff , log g, magnetic field strength) of stars with magnetic fields.", "pages": [ 8, 9 ] }, { "title": "5.1. Further Work and Acknowledgments", "content": "Further versions of the MOOGStokes code will address non-radial, non-uniform magnetic fields, and temperature variations caused by spotting across the disk of the star. During the development of this code, I became indebted to many experts in radiative transfer, polarized or otherwise. Chris Sneden, Rob Robinson, Dan Jaffe, John Lacy, Christopher Johns-Krull, Cornelis Dullemond, Oleg Kochukhov, and Juan Manuel Borrero all provided invaluable advice and suggestions. I wish to also thank the anonymous referee, whose comments and suggestions improved the manuscript. This work was begun under a NASA USRA SOFIA Grant. All portions of the MOOGStokes package (CounterPoint, SynStokes, and DiskoBall) are available upon request from the author or from the author's website.", "pages": [ 9 ] }, { "title": "REFERENCES", "content": "Anderson, E., Bai, Z., Bischof, C., et al. 1999, LAPACK Users' Guide, 3rd edn. (Philadelphia, PA: Society for Industrial and Applied Mathematics) Auer, L. H., Heasley, J. N., & House, L. L. 1977, ApJ, 216, 531 Baraffe, I., Chabrier, G., Allard, F., & Hauschildt, P. H. 1998, A&A, 337, 403 Beckers, J. M. 1969, Sol. Phys., 10, 262 Castelli, F., & Kurucz, R. L. 2003, in IAU Symposium, Vol. 210, Modelling of Stellar Atmospheres, ed. N. Piskunov, W. W. Weiss, & D. F. Gray, 20P Edmonds, J. F. N. 1969, J. Quant. Spec. Radiat. Transf., 9, 1427 Feautrier, P. 1964, SAO Special Report, 167, 80 Hubeny, I., Stefl, S., & Harmanec, P. 1985, Bulletin of the Astronomical Institutes of Czechoslovakia, 36, 214 Kaeufl, H.-U., Ballester, P., Biereichel, P., et al. 2004, in Society of Photo-Optical Instrumentation Engineers (SPIE) Conference Series, Vol. 5492, Society of Photo-Optical Instrumentation Engineers (SPIE) Conference Series, ed. A. F. M. Moorwood & M. Iye, 1218-1227 Kunasz, P., & Auer, L. H. 1988, J. Quant. Spec. Radiat. Transf., 39, 67 Kupka, F. G., Ryabchikova, T. A., Piskunov, N. E., Stempels, H. C., & Weiss, W. W. 2000, Baltic Astronomy, 9, 590 Lee, S., Yuk, I.-S., Lee, H., et al. 2010, in Society of Photo-Optical Instrumentation Engineers (SPIE) Conference Series, Vol. 7735, Society of Photo-Optical Instrumentation Engineers (SPIE) Conference Series Luhman, K. L., Stauffer, J. R., Muench, A. A., et al. 2003, ApJ, 593, 1093 Valenti, J. A., & Piskunov, N. 1996, A&AS, 118, 595 Vernet, J., Dekker, H., D'Odorico, S., et al. 2011, A&A, 536, A105 Wade, G. A., Bagnulo, S., Kochukhov, O., et al. 2001, A&A, 374, 265 Weiss, W. W., Ryabchikova, T. A., Kupka, F., et al. 2000, in Astronomical Society of the Pacific Conference Series, Vol. 203, IAU Colloq. 176: The Impact of Large-Scale Surveys on Pulsating Star Research, ed. L. Szabados & D. Kurtz, 487-488 Whaley, R. C., & Petitet, A. 2005, Software: Practice and Experience, 35, 101, http://www.cs.utsa.edu/~whaley/papers/spercw04.ps Wilson, J. C., Henderson, C. P., Herter, T. L., et al. 2004, in Society of Photo-Optical Instrumentation Engineers (SPIE) Conference Series, Vol. 5492, Society of Photo-Optical Instrumentation Engineers (SPIE) Conference Series, ed. A. F. M. Moorwood & M. Iye, 1295-1305 Yuk, I.-S., Jaffe, D. T., Barnes, S., et al. 2010, in Society of Photo-Optical Instrumentation Engineers (SPIE) Conference Series, Vol. 7735 Zeeman, P. 1897, Nature, 55, 347", "pages": [ 9, 10 ] } ]

[ { "title": "The peculiar light curve of the Symbiotic Star AX Per of the last 125 years", "content": "Elia M. Leibowitz and Liliana Formiggini Wise Observatory and School of Physics and Astronomy, Raymond and Beverly Sackler Faculty of Exact Sciences, Tel Aviv University, Tel Aviv 69978, Israel [email protected]  ", "pages": [ 1 ] }, { "title": "abstract", "content": "We analyze the last 125 years optical light curve of the symbiotic star AX Per through some remarkable correlations that we discovered in its power spectrum. The data were assembled from the literature and from the AAVSO database. A series of 6 major outbursts dominate the light curve. They are presented in the power spectrum as 13 harmonics of the fundamental frequency f a =1/P a =1/23172 day -1 . We refer to them as the \"red\" frequencies. Oscillations with the binary periodicity of the system P b =1/f b =681.48 d are also seen in the light curve, with particularly large amplitudes during outbursts. The f b peak in the power spectrum is accompanied by 13 other peaks on each side, to which we refer as the \"blue\" frequencies. A distinct structure in the frequency distribution of the blue peaks, as well as in their peak power are best interpreted as reflecting beating of the 13 \"red\" frequencies with the binary one. We suggest, following others, that the major outbursts of the system result from events of intense mass loss from the giant star. Mass accretion onto the hot component, partially through the L1 point of the system, took place in the last 125 years at a rate that oscillated with the 13 first harmonics of the f a frequency. The binary orbit is slightly eccentric and periastron passages induced modulation of the L1 accretion at the binary frequency. Hence the f b oscillations in the brightness of the star of amplitude that is modulated by the \"red\" frequencies of the system. Keywords: binaries: symbiotic -- stars: individual (AX Per)  ", "pages": [ 2 ] }, { "title": "1. Introduction", "content": "AX Per is one of the well studied prototypes of the class of symbiotic stars. The stellar system consists of a giant of type M4.5 III (Murset & Schmid 1999) and a hot component, most probably a white dwarf, that are in a binary orbit around each other with a period of some 682 d, This has been established photometrically as well as in radial velocity measurements (Mikolajewska & Kenyon 1992; Fekel et al. 2000). A thick wind is emanating from the red giant. The gaseous medium is heated by the UV radiation of the hot component, producing the bright emission lines that characterize the spectrum of the object( Iijima 1988; Mikolajewska &Kenyon 1992; Ivison et al. 1993, Skopal et al. 2001). The light-curve (LC) of the star is characterized by brightness variations of amplitude 0.1-0.4 mag alternating with epochs of activity characterized by variations reaching amplitudes of up to 3 mag. The binary period of the system has long been recognized as a major modulating agent of the optical output. The light oscillations are mostly attributed to the well known reflection effect in binary systems, by the giant star )Formiggini & Leibowitz 1990). Skopal (1994) has shown that in the geometry of the system, heating and ionization of the thick wind emanating from the giant by collision with the wind of the hot component and by its UV radiation are even more efficient in introducing variations with the binary cycle. In a few cycles of the binary period, eclipses of the hot component have also been identified (Skopal 1994, Skopal et al., 2001, 2011). These interpretations may be adequate for explaining the oscillations during quiescence states but not the wild fluctuations during the outbursts of the system. Also the   energy source, as well as the physical process or processes that give rise to these outbursts are far from being understood. More details about the present status of our knowledge about the system, observational as well as in theory, can be found in a number of excellent investigations in this system performed in the last 25 years (e.g. Mikolajewska & Kenyon 1992; Ivison et al. 1993; Skopal 1994; Skopal et al. 2001, 2011). AX Per is one of the not too many stars the brightness of which started to be monitored systematically already in the 19th century. We are therefore fortunate in being able to construct the LC of the system that extends over the last 125 years. This work utilizes this rich treasure of data, so far not well exploited, in order to extract new knowledge about this system. This paper is the 7th one in our long lasting project of studying symbiotic stars through their historical light-curves (Formiggini & Leibowitz 1994, 2006, 2012 ; Leibowitz & Formiggini 2006, 2008, 2011).", "pages": [ 3, 4 ] }, { "title": "2. Data", "content": "AX Per photometric behavior was first discussed by Lindsay (1932) who, analyzing Harvard plates, noticed variations with period P~ 650 d. The Harvard plates photographic magnitudes, digitized from the figure of Lindsay are our first set of data. The total time covered is from November 1887 up to June 1932, corresponding to JD 241 0590 - 242 6900. The data up to JD 242 3600 represent single observations, after this date they are 25 d means. From now on we omit the 2 first digits, 24, from the JD dates. We designate them as jd.   Another set of photographic data from the Harvard plates, based on a redetermination of the comparison stars magnitudes, covers the time interval from jd 23000 up to 31460 (PayneGaposchkin 1946). These are also presented as a plot, from which we extracted a second data set for our use. Further photographic observations (Wenzel 1956) were retrieved from fig 4 of Skopal et al. (2000). This set is overlapped by a set of photographic estimates by the Association Francaise des Observateurs d'Etoiles Variables (AFOEV) from jd 32085 to 40290. Eye-visual regular monitoring of AX Per brightness by the AFOEV and by the American Association of Variable Stars Observers (AAVSO( started around the year 1968. The data gathered by the AFOEV are consistent with those of the AAVSO, and in our analysis we used the AAVSO ones updated to May 5 2013 =jd 56417, binned into 30 d bins. We note that a few AFOEV data between jd 41272.5 and jd 41317.2 show a brightening of the star by more than one magnitude. Following an inquiry with the AFOEV, these points in the AFOEV LC should be discarded as spurious (E. Schweitzer, private communication). Jurdana-Sepic and Munari (2010) have published 6 B magnitude values and one V magnitude value extracted from historical plates of Asiago observatory between the years 1976 and 1985. Finally, we should note that the Mjalkovskij (1977) data were not used owing to the scale discrepancy shown both in the visual and in the photographic estimates.  ", "pages": [ 4, 5 ] }, { "title": "2.1 Scaling the data", "content": "The photographic data, called from different papers and data bases, were checked for scale consistency. In fact, the new estimation of the comparison stars (Payne-Gaposchkin 1946) results in a difference of magnitudes between the two data sets that were retrieved from the Harvard plates. Analyzing the data after jd ~ 26000 when the system was in quiescence, we empirically estimated that the old Lyndsay's (1932) magnitudes are .24 mag fainter than those of Payne-Gaposchkin (1946). The Lyndsay's data were shifted and merged with the PayneGaposchkin's ones. As previously noted, the Wenzel (1956) data agree with the AFOEV photographic ones and the merging does not required any scale correction. The final data set of photographic estimates covers the time from jd=10590 to 40290. The photographic set of data ends during a quiescence state of AX Per when also a few AAVSO data points are available. Using these short overlapping portions of the LC, the AAVSO data were shifted empirically, in order to connect the photographic and the eye-visual data. Fig. 1a presents the full LC of AX Per so obtained, covering the last 125 years of the star history. The y axis displays the magnitudes in normal units: nmag(t)=mean(m)-m(t), where m(t) are photographic magnitudes. The displayed LC is essentially the one presented in Skopal et al (2001, 2011), except for the omission of the spurious AFOEV points mentioned above, and the extension of the LC up to May 2013. Figure 1b presents the outcome of an application of the running mean operator on the LC with a running window width of 682 days. We shall refer to this curve in Section 3.6. Frame c is the difference between the observed LC and the running mean curve. It shows in particular the strong correlation between the overall brightness of the system and the amplitude of its variation with the binary orbital frequency (see next section).  ", "pages": [ 6 ] }, { "title": "3.1 Power Spectrum", "content": "Figure 2a presents the power spectrum (PS) of the LC seen in Fig. 1a. Two clusters of peaks characterize the periodogram. One group is at the low frequency (\"red\") end, corresponding to periods of the order of 1700 d and longer. The second cluster (\"blue\") is centered around a   distinct peak near the frequency f b = 1.47x10 -3 d -1 , corresponding to the well known binary period of the system. Here we adopt for this periodicity the value P b = 681.48 d, a choice that will be explained in Section 3.5. The left heavy vertical line marks the high frequency edge of a frequency band that contains the 13 high \"red\" peaks in the PS. Figure 2b is zoom on this passband in which the peak frequencies are marked by the short vertical lines. The two other heavy vertical lines in rame a mark respectively the low and the high frequency edges of two pass bands on the left and on the right of the binary frequency f b , of width equal to that of the \"red\" band. Frame c is zoom on the frequency between these two vertical lines. The short low vertical lines mark the frequency f b and the peak frequencies of the highest 13 peaks in each of the two bands on the left and on the right of it. We refer to them as the b1 and the b2 peak sequences.   ", "pages": [ 7, 8 ] }, { "title": "3.2 The \"red\" cluster of peaks", "content": "Even an eye examination of Figure 2 reveals that to a very good approximation, the thirteen 'red' peaks are distributed along the frequency axis with a constant interval between any two neighbors. By the least squares procedure, with the peak power values as weights, we found the set of frequencies such that any 2 of them are separated by an integral number of a constant value, that best fit these 13 \"red\" peak frequencies. The constant value, termed \"quantum\" by Broadbent (1955) is found to be f a = 4.329x10 -5 d -1 , corresponding to the period P a = 23100 d. The long vertical lines in the \"red\" band in Fig. 2b mark the frequencies k/23100 d -1 for all integers l <= k <= 13. We adopt the value P a =23172 d for reasons explained in Section 3.5 In view of the above we shall refer to the 13 \"red\" peaks in the PS as harmonics of the fundamental frequency f a . We note, however, that this does not necessarily imply that 1/ f a is a genuine periodicity of the AX Per system, since the observations cover no more than twice the length of this time interval.", "pages": [ 9 ] }, { "title": "3.3 The \"blue\" cluster of peaks", "content": "We applied the same least square procedure to find the \"quantum\", an integral number of which separates any two of the set of 27 peak frequencies in the \"blue\" PS of the star, marked by the vertical short lines in Fig. 2c. The \"quantum\" so found is again f a , exactly the same as the one among the \"red\" sequence. The long vertical lines in frame c mark the frequencies of the best fitted set of frequencies with this \"quantum\" number.   We perform a bootstrap type test of the null hypothesis that the 13 peaks on each side of f b are distributed randomly along the \"blue\" section of the frequency axis of the PS. We consider the standard deviation S of the differences between the observed peak frequencies (short vertical lines in Fig. 2) and the predicted frequencies (long lines) as a measure of the quality of the least square fitting. Within each one of the two \"blue\" passbands around f b we created 16000 random distributions of 13 frequency values, with the constraint that no two numbers among them will be closer to each other than the theoretical resolution of the PS set by the total length of the LC. Applying the least squares procedure on each one of these distributions of 27 numbers we found only 249 cases with S value as small as, or smaller than the one associated with the 27 peaks in the PS of the observed LC. The probability that the observed 26 \"blue\" peaks around f b are due to random noise in the LC can therefore be rejected at a 98.4% statistically significant level. The equality of the \"quantum\" values in the \"red\" and in the \"blue\" sections of the PS reduces the probability of randomness in the distribution of the \"blue\" peaks to practically zero.", "pages": [ 9, 10 ] }, { "title": "3.4 Correlation in power", "content": "We now consider the values of maximum power of the 26 peaks of the b1 and the b2 sequences defined above. We combine the powers of the nearest peaks to f b in the two sequences, then the powers of the second nearest peaks to f b in b1 and b2, etc. We thus obtain 13 values of power of the 13 pairs of peaks around f b . Figure 3a is a plot of these 13 power values vs. the powers of the 13 \"red\" peaks seen in figure 2. The line in the figure is the regression line between these two sets of 13 power values. A bootstrap test on a sample 10000 sets indicates that the null hypothesis that the two sets are uncorrelated can be rejected at 99.96% probability. The slope of the line is 0.347 +- 0.168, where the quoted uncertainty indicates the 99%   confidence interval around the slope value. Somewhat lower but still statistically significant correlations are found also between the 13 powers of the \"red\" sequence and the powers of the b1 and the b2 sequences considered separately. We emphasize at this point that these highly improbable features of the PS are uncovered in the periodogram of the raw LC of the star. In the process of their discovery we have made no assumption concerning the presence of any periodical content in the LC. Even the fact that P b is the well known binary period of the system has not been used in this analysis . ", "pages": [ 10, 11 ] }, { "title": "3.5 Beats", "content": "The most natural and straightforward interpretation of the peculiar characteristics of the PS presented in the previous 2 sections is that they reflect beating of the 13 frequencies of the \"red\" sequence with f b , the binary frequency of the system. A light curve of oscillations at a frequency f b and at the 13 first harmonics of frequency f a that are also beating with the frequency f b , can be expressed mathematically as follows (1) The first sum on the right hand side of the above formula represents free oscillations at the 13 first harmonics of the frequency f a . The second sum represents the beats of these harmonics with the frequency f b . We may also view this sum as representing an oscillator of frequency f b , modulated by the 13 harmonics of f a . The last term in the formula is required in order to produce in the PS the prominent peak of the f b frequency itself. This free f b oscillator is not necessarily in phase with the modulated one.   By least squares fitting of the Y(t) function to the observed LC we find the values of the two frequencies f a and f b , the amplitude of the free oscillator and the phases of the two f b oscillators. The least squares procedure also yields the values of the amplitudes and phases of the 13 \"red\" harmonics, as well as those of the 13 harmonics of the same frequencies that modulate the modulated oscillator. Table 1 lists the values of f a , f b , the amplitude A free of the free oscillator, and the phases  mod and  free of the modulated and the free oscillators. Here phase zero is at jd 51135, the time of maximum radial velocity of the giant as reported by Fekel et al (2000). The uncertainty values are estimated on the basis of bootstrap calculations. We consider the residuals of the observed minus theoretical Y value on each time point of the observations as a bank of \"errors\". We then create a pseudo-observed LC by considering the Y(t) values on each time point, to which we add one number chosen randomly from the error bank. We apply our least squares procedure on this pseudo-observed LC and obtain pseudo-observed values of the 5 parameters of Table 1. We perform these calculations on a sample of 200 such pseudo-observed LCs. The dispersion of the 200 values so obtained for each parameter defines the corresponding uncertainty number presented in Table 1.   The quality of the fit of the Y(t) function to the observed LC can be appreciated qualitatively by comparing the observed LC, displayed again as the upper curve in Figure 4, to the lower curve which is the best fitted Y function. The PS of the lower curve is almost indistinguishable from the PS shown in Figure 2. In the least squares procedure, all the A and B parameters are treated as entirely independent, free parameters. We find that the resulting B k amplitudes are linearly correlated with the resulting A j amplitudes, as expected if the LC of the star does indeed represent physically beating frequencies. We point out that our presentation of the theoretical LC in fig. 4 and the demonstration of its good fit to the observation are not intended to be regarded as evidence for the validity of our beating model. The figure serves only for showing that the LC of the star may well be simulated by a beating model with an appropriate choice of parameters. The evidence for the beats lies entirely in the peculiar features and correlations that we discover in the PS of the raw data that are clearly non random at a highly statistical significant level.  ", "pages": [ 12, 13, 14 ] }, { "title": "3.6 Comments on the fitted curves", "content": "  its 13 harmonics, the f b frequency and the corresponding beat frequencies. The features in the lower PS must therefore represent interferences among these frequencies, due to the final length of the LC and to the non equal sampling of it. This suggests that the nearly identical spectral features in the PS of the real data seen in the upper curve reflect similar interferences among the same basic frequencies, implying that these frequencies are indeed underlying the observed LC. 3. Skopal at al (2012) performed precise photoelectric photometry of AX Per during some parts of the time interval jd 53000 < t < 56000. They report on large oscillations of the system that are significantly out of phase of the P b periodicity that is prevalent almost always in the LC, at quiescence phases as well as during outbursts. These out of phase oscillations are also apparent in the AAVSO data. The getting out of phase of the f b variability during this time interval is well described by the theoretical Y(t), although not with the details of the structure that the LC assumed at that time. It is even predicted by the function Y(t) when fitted to the observed LC from which the data points of the above time interval are omitted. It seems that the out of phase variations at that time result from interference of frequencies involved with the Y(t) function.  ", "pages": [ 15, 16 ] }, { "title": "3.7 Macro vs. micro presentation", "content": "Our present work is an attempt to understand the gross features of the optical photometric LC of AX Per. It addresses the time variations in the brightness of the star of the order of 0.5 mag and larger in brightness, and of time-scale of 450 d and longer. The system is varying also on shorter time-scale and with smaller amplitudes in brightness. The out of phase oscillations mentioned in the previous section is one example. Also Skopal et al. (2012) as well as the LC of the star presented in this work show that the P b periodicity is hardly being preserved on the \"micro\" scale. This is apparent in the varying shape of the P b cycle even at quiescence states of the star. It is also manifested by the fact that the time interval between apparent neighboring minima is varying by up to 100 days, and even more, around the P b periodicity. For example, the time intervals between the apparent minima of 7 consecutive P b cycles between jd 46500 and 51500 are 602, 678, 678, 693, 723 and 662 d. These different values are independent of the binning operation that is applied on the LC. We obtain nearly the same numbers when we consider the AAVSO LC binned into 10 or 30 d binning. Minima measured photoelectrically show similar excursions away from their expected times of occurrence. It should be noted, however, that some variation from a strict binary cycle may be attributed to variations in the ionization structure of the giant wind (Skopal 1998). In figure 1c one can also identify variability on time scale shorter and amplitude smaller than the variations on the binary cycle periodicity. The 681.5 d periodicity is, however, a robust feature of the system. The centers of the \"micro\" variability excursions preserve their periodicity and phasing throughout the last 125 years of the star history.   In this work we are concerned only with the 'macro' variability of the system. An underlying assumption in our approach is that the gross features of the LC carry information about the clocks that control the timing of the major events in the dynamics of the system. We assume that the effects of these clocks may indeed be decoupled from the micro-variations without losing this information. All variations to which we refer as 'micro' have to do with the physical processes that transform the gravitational energy released in the accretion process and possibly also in nuclear reaction on the surface of the white dwarf to the optical output of the system. These are taking place in various components of the system such as winds of the two stars, an inter-binary and circumbinary nebulosity, gaseous streams and possibly also an accretion disk. They depend naturally on the dynamics and kinematics of these components, as well as on details of the thermodynamical processes within them. In view of their relatively smaller effects on the light curve these processes may be regarded as perturbations on the more orderly temporal behavior of the basic energy generation processes in the system. The detailed study of these perturbations is beyond the scope of this paper.", "pages": [ 17, 18 ] }, { "title": "3.8 The relation with the spectroscopic ephemeris", "content": "If one assumes that the orbit of the AX Per binary system is circular, maximum radial velocity of the giant occurs a quarter of a cycle after the times of inferior conjunction of that star. In Section 3.5 Table 1 we reported that the phases of minimum light of the modulated and the free oscillators precede the phase of maximum radial velocity as given by Fekel et al. (2000) by less than 0.25. Note, however, that for P b =681.48 d, the 2 very deep minimum points measured during two major outbursts, at jd 24389 and jd 47555 (see fig. 1a), are indeed at phase 0.75.  ", "pages": [ 18 ] }, { "title": "4. Discussion", "content": "In this section we propose a qualitative physical model for the AX Per symbiotic system that is likely to be a realization of the mathematical beating model suggested by the observations.", "pages": [ 19 ] }, { "title": "4.1 The Source of the continuum radiation", "content": "The main sources contributing to the optical continuum of AX Per are represented by the hot component, the giant and the nebula (Skopal, 2005). The hot component of the system and its close environment are undergoing partial and some time total eclipses by the giant of the system, giving rise to the modulations at the binary period of the system. These variations are also augmented by the reflection effect from the surface of the giant and from its thick wind (see Section 1). However, the periodic variations in the aspect ratio of the revolving AX binary system can hardly explain some basic observed features of the LC of the star. As noted in Section 3.7, the observed times of minimum light of individual cycles make frequent excursions that may reach a distance of more than 100 days away from the nominal times of a strictly P b periodicity. In phase space these differences are sometimes as large as 0.15. It is hard to see how minima that are due to eclipses and to the reflection effect, being mostly geometrical in origin, can differ that much from the strictly periodic times of inferior conjunction. Furthermore, the amplitude of light variations due to an eclipse of the light source or to varying aspects of a reflecting medium of its light are to first approximation some given fraction of the source light intensity. The measured amplitude in magnitude units of variations   due to these mechanisms alone should be roughly independent of light intensity of the source. This is clearly not the case in AX Per where the amplitude in magnitude units of the P b variations during outbursts may be 3 or more times the mean amplitude during quiescence (see figure 1c). Also the amplitude at quiescence intervals is also varying considerably with time. The system must therefore include also an additional light source that is fluctuating at the binary frequency, nearly in phase with the eclipse and the reflection effects. The amplitude of this fluctuating source is strongly modulated by the same 13 \"red\" frequencies that gave the LC of the star the outbursts structure that it had in the last 125 years (Fig 1b).", "pages": [ 19, 20 ] }, { "title": "4.2 Outbursts", "content": "The major energy source of the radiation of AX Per is in an accretion of material, from the wind and possibly also from the atmosphere of the giant star onto the hot component of the system. This process releases large quantities of gravitational energy. It may also supply fuel to an ongoing nuclear burning process on the surface of the white dwarf. . This has been proposed already more than 20 years ago by Mikolajewska & Kenyon (1992) and Ivison et al. (1993). The outburst episodes would then be events of extra material that is being ejected from the atmosphere of the giant star (Skopal et al 2011). The rate of mass loss from the giant is most likely controlled by an intrinsic parameter or parameters of this star. Skopal et al suggested, for example, that during one active phase of the system that they have studied, the radius of the giant increased by more than 10% (from 102 R O to 115 R O ). Variations in the mass loss rate may also be associated with variations in a global magnetic field in the outer layers of the star.   To date, there is no direct observational indication of a global magnetic field on the surface of the giant of AX Per.. However, non detection is by no means a detection of the absence (see for example the detection of magnetic field in Arcturus, the presence of which was unrecognized until very recently; Sennhauser & Berdyugina 2011). Cyclic or quasi-cyclic variability in the intensity, perhaps also in polarity, of stellar magnetic fields are ubiquitous, not only in our sun , but also in a wide range of different stars, including red giants (see for example Dal, Sipahi, Ozdarcan 2012, and many references therein). Also, in a recent paper, Olah et al. (2013) report on photometric observations in 3 K giant stars that vary on time-scale of years, possibly with periods of 5 or 10 years. These authors suggest that the photometric variability is a testimony on surface magnetic field variations of the same time-scale. We note in this connection that in the series of papers of the present authors listed in Section 1, we have pointed out the possibility and produced at least circumstantial evidence for the presence of magnetic fields in the atmosphere of the cool components of a few well known symbiotic systems. The outbursts structure of the LC is accordingly most probably an optical manifestation of variations in the value of the relevant intrinsic parameter of the giant star, which we assume for the rest of our discussion to be the radius of the star. The PS then shows that during the last 125 years the giant was pulsating as an harmonic oscillator of the 13 frequencies of the \"red\" peaks of the PS. According to Skopal et al (2001) if the inclination angle of the orbital plane is the radius of a tidal lobe filling giant is 102 while if the radius is 170 . In fact, Mikolajewska & Kenyon (1992) have also suggested that the orbital inclination might   be approx. 70 degrees with the red giant filling its tidal lobe. It would therefore seem that at each mode of the giant pulsations, accretion of matter through the L1 point of the system may have taken place. Alternatively or in addition, each mode of this pulsation created a corresponding mode of oscillations in the intensity of the wind emanating from the giant and consequently also in the rate of wind accretion of matter onto the hot component and also through the L1 point. The combined effect of the 13 frequencies oscillations in the accretion rate gave rise to the observed series of outbursts of the system.", "pages": [ 20, 21, 22 ] }, { "title": "4.3 The modulated oscillator", "content": "Accretion through the L1 point of the system is thus modulated with the frequency of any one of the star pulsation modes. If the binary orbit is non circular, the excess mass flow through the L1 point will be also modulated at the binary frequency. This will be translated to variations in the luminosity of the system at this frequency, with maximum light that is reached around a certain specific phase after periastron (Lajoie & Sills 2011). Details of the process of the release of energy, gravitational and/or nuclear, associated with varying accretion rate onto the hot component have been discussed by Sokoloski et al. (2006), in the context of another well studies symbiotic star, Z And. This mechanism may be suggested as the source of the very large amplitudes of the binary oscillations during outbursts. We may describe the process in a slightly different language. Due to a slight eccentricity of the binary orbit, the mass accretion through the L1 point of the system is modulated at the binary frequency. The mass flowing towards the L1 point is supplied by the giant wind. Therefore the amplitude of f b oscillations in the accretion rate is modulated by the 13 frequencies of the giant pulsations. This is the physical origin of the second sum on the right hand side of equation (1).   Fekel et al. (2000) have analyzed all the radial velocity data for the giant star of AX Per that were available at the time of their work. They concluded that the binary orbit of this system is circular, following the precepts outlined by Lucy & Sweeney (1971). These authors have found however an orbital solution with a non zero eccentricity. We have reanalyzed the radial velocity curve of AX Per as given in Table 8 in Fekel et al. (2000). Using formula (1) in Lucy & Sweeney (1971), we find a best fit orbital solution with eccentricity e=0.052. However, as noted by Fekel et al. we also find that due to the noise in the data, a circular solution is quite consistent with the data. In other words, the null hypothesis that the orbit is circular cannot be rejected by the presently available data. But as stated by Feigelson & Babu (2013), \"... the null hypothesis can be rejected at a given level of significance, but the null hypothesis can not formally be accepted\". Also in a recent paper Lucy (2013) has presented a new Bayesian inference for orbital eccentricities of single line spectroscopic binaries. One of his conclusions is that \"Systems assigned e = 0 [according to these earlier tests] should preferably have upper limits e U computed\". In Table A.3 in Lucy's paper it is also shown that upper limits for eccentricities could be significantly larger than the values derived by the least squares procedure. Thus a slightly non circular orbit of the AX Per binary system is certainly not ruled out by the presently available observations. Lajoie and Sills (2011) have shown theoretically that at least for some binary systems, an eccentricity of the order of 0.1 is enough for modulating an accretion process at the binary frequency.  ", "pages": [ 22, 23 ] }, { "title": "4.4 The two oscillators", "content": "According to our proposition, two distinct oscillations are responsible for the variations of the brightness of AX Per at the binary period, the varying aspect ratio due to the system binary revolution and the varying mass flow through L1 due to the system eccentricity. This fact was so far elusive because the resulting two f b oscillations in the light of the system have nearly the same phase. The slight difference between these two phases, as presented in Table 1, could be real but analysis shows that it is statistically not significant. The coincidence of near equality of the phases of the two oscillators may be regarded as a weakness of our model, but one has to bear in mind that such close phases occur with a probability that is no smaller than any other specific phase difference between the two. What enabled us to disentangle the effects of the two distinct oscillators in spite of their nearly equal phases is our analysis of the 125 years LC of the star in its entirety, and not only of a limited section of it, as was done so far by most studies of the system variability.", "pages": [ 24 ] }, { "title": "4.5 The free oscillator", "content": "The free oscillator term, the last one on the right hand side of expression (1), represents the variability that is due to eclipses and the reflection effect. The amount of the hot component light that is blocked by the giant, and the amount reflected from the giant surface or from any other fixed material element in the rotating binary system, at any given phase are determined primarily by the geometrical size of the giant star and perhaps also of an optically thick wind   around it. To first approximation, these geometrical parameters are independent of the luminosity of the system. This is why this P b variability is represented by an un-modulated, free oscillator. The phase of minimum light of the un-modulated oscillator that is due to eclipses should be the mark of inferior conjunction of the giant. Unfortunately, it cannot be determined by photometric measurements due to the effect of reflection that does not necessarily attain its maximum brightness level at superior conjunction of the giant. Also in general, the eclipse effect is being masked by interference with the oscillations of the modulated oscillator. The later is dominating the LC because of its larger amplitudes. The cycle of this oscillator is much less well defined on the \"micro\" level. This is due to the nature of the chain of physical processes that transform the variations in the gravitational/nuclear releases of energy to optical light fluctuations. Note, however, that when circumstances in the system allow identification of an eclipse, as is was probably the case at the two dates mentioned in Section 3.8, the eclipse had indeed preceded the phase of maximum radial velocity of the giant by quarter of a cycle.", "pages": [ 24, 25 ] }, { "title": "5. Summary and conclusion", "content": "We discover a certain structure in the PS of the LC of the symbiotic star AX Per as well as a tight linear relation between the powers of its apparent peaks. These features are non random at a very high statistically significant confidence level. The outbursts of the system that characterize the long term structure of the LC can be decomposed into 13 interfering harmonic oscillations of periods of thousands of days. We suggest that these 13 \"red\" frequencies are those of oscillations in the value of some fundamental parameter of the giant atmosphere, a   most likely one is the radius of the star. These oscillations drive oscillations at the same frequencies in the rate of mass loss from the giant and therefore also of the accretion rate onto the hot component. This is the origin of the series of the outbursts of the star. Some of the accretion onto the hot component is by overflow of material through the L1 point of the system. Due to a slight eccentricity of the binary orbit, this flow of matter is also modulated by the binary frequency. The beat of the 13 \"red\" frequencies with the binary one gives the \"blue\" section of the PS of the LC its observed peculiar structure. It also explains the linear relation that is found between the power of the \"blue\" peaks in the PS and that of the \"red\" peaks. If the giant star continues to oscillate with the f a frequency also into the future, one could predict that the outburst that the system is presently undergoing will reach its peak luminosity sometime around jd 56840 (June/July 2014).", "pages": [ 25, 26 ] }, { "title": "Acknowledgments", "content": "We acknowledge with thanks the American Association of Variable Stars Observers for the data we extracted from their International Database that made this research possible. We also thank E. Schweitzer for providing us with useful information regarding the AFOEV data set. We also thank an anonymous referee for some useful suggestions.  ", "pages": [ 26 ] }, { "title": "References", "content": "    ", "pages": [ 27, 28 ] } ]

[ { "title": "ABSTRACT", "content": "To explain the multi-wavelength light curves (from radio to X-ray) of HST-1 in the M87 jet, we propose an hourglass model that is a modified two-zone system of Tavecchio & Ghisellini (hereafter TG08): a slow hourglass-shaped or Laval nozzleshaped layer connected by two revolving exponential surfaces surrounding a fast spine, through which plasma blobs flow. Based on the conservation of magnetic flux, the magnetic field changes along the axis of the hourglass. We adopt the result of TG08the high-energy emission from GeV to TeV can be produced through inverse Compton by the two-zone system, and the photons from radio to X-ray are mainly radiated by the fast inner zone system. Here, we only discuss the light curves of the fast inner blob from radio to X-ray. When a compressible blob travels down the axis of the first bulb in the hourglass, because of magnetic flux conservation, its cross section experiences an adiabatic compression process, which results in particle acceleration and the brightening of HST-1. When the blob moves into the second bulb of the hourglass, because of magnetic flux conservation, the dimming of the knot occurs along with an adiabatic expansion of its cross section. A similar broken exponential function could fit the TeV peaks in M87, which may imply a correlation between the TeV flares of M87 and the light curves from radio to X-ray in HST-1. The Very Large Array (VLA) 22 GHz radio light curve of HST-1 verifies our prediction based on the model fit to the main peak of the VLA 15 GHz radio light curve. Subject headings: galaxies: active - galaxies: jets - radiation mechanisms: nonthermal", "pages": [ 1 ] }, { "title": "AN HOURGLASS MODEL FOR THE FLARE OF HST-1 IN M87", "content": "Wen-Po Liu 1 , 2 , Guang-Yao Zhao 2 , 3 , Yong Jun Chen 2 , 4 , Chun-Cheng Wang 5 , and Zhi-Qiang Shen 2 , 4", "pages": [ 1 ] }, { "title": "1. INTRODUCTION AND OBSERVATION CONSTRAINTS", "content": "As is well known, HST-1 is the innermost knot of the M87 jet, located ∼ 80 pc from the core (Biretta et al. 1999). The multi-wavelength light curves of HST-1 have been previously studied using radio data (Chang et al. 2010), optical and UV data (Perlman et al. 2003; Madrid 2009) and X-ray data (Harris et al. 2003, 2006, 2009). Chen et al. (2011) investigated the radio polarization and spectral variability of HST-1, and Perlman et al. (2011) researched the optical polarization and spectral variability of the M87 jet. Cheung et al. (2007) argued that HST-1 may be the site of the flaring TeV gamma-ray emission reported by the H.E.S.S. (Aharonian et al. 2006). A study of particular note was by Abramowski et al. (2012), who released 10 yr of multi-wavelength observations of M87 and the very high energy γ -ray flare of 2010. The quasi-simultaneous spectrum (from the radio to X-ray band; e.g., Marshall et al. 2002; Waters & Zepf 2005; Perlman & Wilson 2005; Harris et al. 2006; Cheung et al. 2007) and polarization observations (e.g., Perlman et al. 1999, 2011; Chen et al. 2011) of the M87 knots demonstrate the nature of synchrotron radiation. From the multi-wavelength light curves of HST-1 (Fig. 1), we could obtain some physical constrains on HST-1: the light curves show two big flares, with the main peaks of the light curves around the year 2005.30 and the second ones around the year 2007. Here, we discuss a pure (or single) process (Doppler e ff ect). For synchrotron emission in the case of a moving sphere, with observed fluxes I ν, obs = δ 3 + α I ν (Dermer 1995; δ is the Doppler factor of HST-1 and α = ( p -1) / 2, where p is the spectral index of the particles), the change of the Doppler factor may explain the change in the light curve. However, Harris et al. (2006) suggested a modest beaming synchrotron model with a Doppler factor of three or four, while Wang & Zhou (2009) obtained the Doppler factor of HST-1 to be 3 . 57 ± 0 . 51 through a synchrotron model fitting. The HST-1 complex could be model fitted with multiple components (e.g., Cheung et al. 2007; Giroletti et al. 2012); in this case, the speed of one component in HST-1 is obtained through a longer monitoring of the same one, but the velocity estimate usually has fairly large uncertainties. Giroletti et al. (2012) reported the apparent velocities of very long baseline interferometry (1.7 GHz Very Long Baseline Array, VLBA and 5 GHz European VLBI Network, EVN) components in HST-1 with high precision ( < 2%, this is an unprecedented accuracy for determining the apparent velocities of components in HST-1) during the decay period of the HST-1 flare, which implies that components in HST-1 have uniform motion with high precision. Hence, the variation range of the Doppler factor may be very small and the change in the Doppler factor in HST-1 may not explain the order of the flux change. Excluding the aforementioned single mechanism, we believe that the flare of HST-1 may be correlated with some complex processes which include a changing magnetic field strength. In Section 2, we describe in detail our model for HST-1. In Section 3, we present and discuss the fitting results of this model to the main peak in the multi-wavelength light curves of HST-1. A summary is provided in Section 4.", "pages": [ 2, 3 ] }, { "title": "2. The Model", "content": "Tavecchio & Ghisellini (2008, TG08) suggested a two-zone scenario in subparsec-scale jets to explain the TeV emissions in M87: a slow hollow cylindrical layer (the velocity relative to the M87 core is vl ) surrounds a fast cylindrical zone (the velocity relative to the M87 core is vs , vs /greatermuch vl , and the velocity of the inner zone relative to the cylindrical layer is v ). The high energy from GeV to TeV could be produced through inverse Compton by the two-zone, and the photons from radio to X-ray are mainly radiated by the fast inner zone. If this subparsec-scale structure is located within HST-1, HST-1 may be a TeV emission source. However, the model of TG08 could not explain the multi-wavelength light curves (or flare) from radio to TeV. We found that a modified scheme of the TG08 model could achieve this; here we only discuss the light curves of the inner blob from radio to X-ray. We believe that the slow layer may be an hourglass-shaped or Laval nozzle-shaped layer connected by two revolving exponential surfaces (Fig. 2). Considering magnetic field conservation, we believe that the magnetic field will change along the axis of the hourglass. We assumed that the length of the inner blob along the jet axis may be smaller than the radius Rs of its cross section, which may be subparsec-scale, and may remain unchanged (the blob may be constrained in a series of shocks along the jet axis). If the axis coordinate and radius of the hourglass nozzle are xn and Rn , respectively, then the layer radius R (we assumed that R ∼ Rs ) as a broken exponential function of the axis coordinate is where k and k ' are constant. Considering magnetic field conservation, B ∝ R -2 , the magnetic field B along the axis of the hourglass will be   where Bn is the magnetic field of the hourglass nozzle. When a blob travels down the axis of the first bulb in the hourglass, because of magnetic field conservation, its cross section experiences an adiabatic compression (compression velocity u = -( dR / dt ) = Rnkve k ( xn -x ) ), which results in particle acceleration and the brightening of HST-1 (this may explain why Perlman et al. 2011 found no evidence for the motion of the flaring blob of HST-1 in their optical data from Hubble Space Telescope ( HST ) observations but Giroletti et al. 2012 found no prominent stationary components in their radio data from VLBA and EVN observations). When the blob moves into the second bulb of the hourglass, considering magnetic field conservation, the dimming of the knot will occur along with an adiabatic expansion of its cross section ( u = -( dR / dt ) = -Rnk ' ve k ' ( x -xn ) ). When a blob approaches the hourglass nozzle, the energy gains of the particles in the blob are ( ∂ E /∂ t ) = α 2 E , where α 2 = (2 / 3)( u / R ) = (2 / 3) kv > 0 (Pacholczyk 1970). The compression timescale τ = ( R / u ) = (2 / 3 α 2). Amodified continuous injection model, which considers that source particles (a broken power law which could be interpreted as partial loss or escape of high-energy particles in the acceleration region) are continuously injected into an adjacent radiation region from an acceleration region, can be fit to the spectral energy distribution of the outer jet in M87 (Liu & Shen 2007; Sahayanathan 2008 presented a similar model with two spectral indices). We suppose that the radiation mechanism of the inner jet is the same as that of the outer jet. When the source particles (a broken power law) have been injected for the time interval of t 1, the radiation region of a blob in HST-1 moves into the hourglass layer. Then, the initial condition of the particle spectra (this is a relic of a past process and may not be ignored for the next process) in HST-1 at the beginning of the compression process is (Liu & Shen 2007; Sahayanathan 2008) where N ∗ , q ∗ 1 , and q ∗ 2 refer to the entire radiation region in the blob taken as an entity (but N , q 1, and q 2 are applied to a unit volume); θ is the pitch angle between the magnetic field and the particle; Eb is the break energy of a broken power law (here, we assumed that Eb < (1 /β 0 t 1); Sahayanathan 2008); p 1 and p 2 denote particle spectrum indices that may be di ff erent (e.g., Sahayanathan 2008); β 0 = bB 2 ⊥ ; b is a constant; and B ⊥ represents the component of the magnetic field perpendicular to the velocity of the particle. Because the acting timescale of the compression process is shorter than that of the particle injection process, which may be comparable with the kinetic timescale of HST-1 relative to the M87 core, we can ignore the influence of particle injection in the compression process. Next, we consider synchrotron radiation and an adiabatic compression of the radiation region in the blob, and the kinetic equation is (Kardashev 1962) Under the initial conditions given in Equation (3), we have where a 2 = ∫ t 0 α 2 dt , q ∗ represents q ∗ 1 or q ∗ 2 , λ is a substitute for p 1 or p 2 + 1, and we have assumed that E /lessmuch ( e a 2 / ∫ t 0 β e a 2 dt ) and α 2 = const in the formula (5). Hence, Considering the conservation of magnetic flux, B ∝ R -2 , R = R 0 e -kvt = R 0 e -(3 / 2) α 2 t , where R 0 is the initial radius of the blob. Hence, β = β 0 e 6 α 2 t . The factors [1 -( β/ 7 α 2) E (1 -e -7 α 2 t )] p 1 -2 and [1 -( β/ 7 α 2) E (1 -e -7 α 2 t )] p 2 -1 in Formula (6) are close to 1, because of the following reasons. (1) E /lessmuch (7 α 2 /β (1 -e -7 α 2 t )), and so 1 -( β/ 7 α 2) E (1 -e -7 α 2 t ) → 1. (2) p 1 , p 2 ∼ 2 in the M87 jet (Perlman & Wilson 2005; Liu & Shen 2007; Perlman et al. 2011), hence p 1 -2 ∼ 0 and p 2 -1 ∼ 1. Formula (6) is thus close to and when this is converted to unit volume (entire volume V ∝ R 2 ∝ e -3 α 2 t ), Next, we consider the increase in magnetic field strength and the time dependence of blob length along the line of sight ( ∝ e -(3 / 2) α 2 t ). If the distributions of particles are isotropic, we can derive the flux formula of the synchrotron model: where ν B 1 and ν B 2 are break frequencies. When the radiation region of the blob in HST-1 moves away from the hourglass and into the adjacent bulb, deceleration of particles in the blob may occur. Then, α ' 2 = -(2 / 3) k ' v < 0, the expansion timescale τ ' = -(2 / 3 α ' 2 ), and the formula for the particle spectrum will become where t 2 is the acting timescale of the compression process, β ' ∝ e 6 α ' 2 ( t -t 2) . Now, we consider the decrease of magnetic field strength and the time dependence of blob length along the line of sight ( ∝ e -(3 / 2) α ' 2 ( t -t 2) ). With this consideration, the flux expression would be changed to where ν B 1 and ν ' B 2 are break frequencies. According to Formulae (9) and (11), the flux f changes exponentially with time in our model, i.e., for low frequency, where t peak denotes the peak time of the light curve. Further, for high frequency,", "pages": [ 3, 4, 5, 6 ] }, { "title": "3. FITTING RESULTS AND DISCUSSION", "content": "Now, we apply the above hourglass model to the multi-wavelength light curves of HST-1 in the M87 jet. Here, the beaming factor of a blob is assumed to be constant. The data we used are plotted in Fig. 1. The units of the radio data are 1 Jy. VLBA 15 GHz radio data from Chang et al. (2010), in which relative uncertainty is assumed to be 5%, are plotted as up triangles and down triangles that show the upper limits of fluxes. Very Large Array (VLA) 15 GHz radio data from Harris et al. (2009) and Abramowski et al. (2012) are plotted as squares. The units of the UV data are 1 mJy (Madrid 2009); these data are plotted as circles. For the X-ray data, we use the flux density integrated from 0.2 to 6 keV (Harris et al. 2006, 2009), and the units are 10 -11 erg cm -2 s -1 ; these data are plotted as diamonds. The X-ray data after 2005 August 6 were estimated by assuming that the correction factor for the (unknown) e ff ective area (Harris et al. 2006) is the same as that for 2005 August 6. Harris et al. (2006) estimate that the resulting uncertainties are of the order of 15%. Note that because of the unknown e ff ective area, the uncertainties of the X-ray fluxes are larger than those in the counts, which may reduce (or smooth) the individual oscillations of the X-ray fluxes. In our model, as the time dependence of the e ff ective area is considered, the resulting uncertainties of the X-ray fluxes may be larger than 15%. Based on the shape trends of the radio, UV, and X-ray light curves in HST-1, as shown in Fig. 1, we select a common shape section (from about the year 2003 to 2006.60), which may be the main peak in our fitting area. We used the weighted least-squares method to fit our model to this main peak, with Equation (12) corresponding to the radio and UV light curves and Equation (13) corresponding to the X-ray light curve. Although there is a peak time in our model, its exact position was unknown in the fitting of our model to the light curve. Hence, we first arbitrarily divided the light curve data during the chosen period into two groups to perform the least-squares method, in which the sum of the reduced chi square χ 2 ν for the two parts is the least. Then we can calculate the corresponding least χ 2 ν by changing the division of the two groups. All reasonable combinations of the two groups (e.g., the fitting peak time should lie between the two groups) are considered before we obtain the best fit with a minimal χ 2 ν . These best-fit parameters for each light curve are shown in Fig. 1 by blue solid lines. Note that the reduced chi square value depends on uncertainties of the data. Our model can well fit the main peak in the multi-wavelength light curves of HST-1 as shown in Fig. 1. It satisfies the aforementioned first constraint for the multi-wavelength light curves of HST-1: the main peak time is around the year 2005.30, as shown in Fig. 1 and Table 1. The stratified e ff ect of radiation regions in the outer knots of M87 has been verified by Perlman et al. (1999), Marshall et al. (2002), and Perlman & Wilson (2005). The flattened peak section in the radio light curve can be explained by the greater length of the radio radiation region along the jet axis than the UV and X-ray region. Abramowski et al. (2012) showed that the VLA 22 GHz radio light curve of HST-1 is consistent with the 15 GHz light curve within the error range, which implies that our model can also explain the VLA 22 GHz light curve. In other words, the observation verifies our prediction based on the model fit to the main peak of the VLA 15 GHz light curve. The aforementioned case is a single component; however, a complex structure that contains multiple components may be more reasonable, as in this case, each peak corresponds to a component passing through the hourglass nozzle. We also use the aforementioned method to fit our model to the possible secondary peak of the multi-wavelength light curves. The best fits for our model are plotted in Fig. 1 in red dotted and dot-dashed lines. A similar broken exponential function could fit the TeV peaks in M87 (Abramowski et al. 2012), which may imply a correlation between the TeV flares of M87 and the light curves from radio to X-ray in HST-1. Further, the maximum of the TeV flares of M87 was coincident with the peak of light curves from radio to X-ray in HST-1 observed in 2005 (Cheung et al. 2007); this may imply that the observable TeV flux density was produced through inverse Compton as a blob passed through the hourglass nozzle. The detailed generation process of the TeV flare may be very complicated. The estimates of p 1 and p 2 in HST-1 require simultaneous broad observational data, which are scarce. Liu & Shen (2007) found that the averaged spectral index of outer knots in M87 is about 2.36 through model fitting. If we assumed that the spectral index of HST-1 is similar to that of outer knots (i.e., p 1 , p 2 ≈ 2 . 36), we could derive some physical parameters of HST-1 from our model fits, as shown in Table 2. Based on the timescale of ∼ 5.6 yr, the local size of the component is smaller than a parsec. The derived α 2 agrees well with the derived α ' 2 , and the common value is around 0.12, which implies that the hourglass-shaped layer may be symmetrical with respect to the nozzle.", "pages": [ 6, 7, 8 ] }, { "title": "4. CONCLUSION", "content": "Wepropose a modified two-zone system of TG08: a fast blob passes through a slow hourglassshaped layer that is connected by two revolving exponential surfaces. Mainly, the emissions from radio to X-ray are radiated by the fast blob. Because of magnetic flux conservation, the brightening and dimming of HST-1 could be explained as adiabatic compression and expansion, respectively, of a blob passing through the outer layer. The observable TeV flux density may be produced through inverse Compton as a blob passes through the nozzle. The VLA 22 GHz radio light curve of HST-1 was used to verify our model.", "pages": [ 8 ] }, { "title": "5. ACKNOWLEDGMENT", "content": "We are grateful to Prof. D. E. Harris for his help in the VLA 15 GHz radio light curve of HST1. We acknowledge the support from the National Natural Science Foundation of China (NSFC) through grants U1231106 and 11273042, the Science and Technology Commission of Shanghai Municipality (12ZR1436100), and the Scientific Research Foundation of the Civil Aviation University of China (09QD15X). This work is partly supported by the China Ministry of Science and Technology under the State Key Development Program for Basic Research (2012CB821800), NSFC(grants 10625314, 11121062, 11173046, 11033007, 10973012, and 11073019), the CAS / SAFEA International Partnership Program for Creative Research Teams, the Strategic Priority Research Program on Space Science, the Chinese Academy of Sciences (Grant No. XDA04060700), and 973 program 2007CB815405.", "pages": [ 8, 9 ] }, { "title": "REFERENCES", "content": "s e v r u C t h g i L y a R i h c d e c u d e r e h t t a h t e t o N . s e s e r . 0 n a 2 α ) 2 / 2 α ) 9 M e h t o t ) 6 2 ( 5 3 . 1 4 0 . 0 + 4 - 11 - ) 7 1 ( 9 9 . 8 0 1 1 . . 0 - 6 2 . 5 0 0 2 0 ± a p 0 2 9 0 . 0 ± 3 d e l F i t s ( 5 p 2 + 1 . 3 f o r t h e n a n d t h e o M r o f s r e t e m a r a P . 1 e l b a T · · · y a r - X 4 0 . 0 ± 8 8 . 0 V U 6 0 . 0 ± 0 8 . 0 ) z H G 5 1 A L V ( o i d a R 3 1 . 0 ± 2 0 . 1 ) z H G 5 1 A B L V ( o i d a R f o r e b m u e r a u q s i h c d e c u d e r : 7 n m u l o C - . e t o N a e p n i a m s e i t n i a t r e c n u a t a d n o s d n e p e d e u l a v e r a u q s · · · · · · · · · 0 + 2 ) 5 ( 6 6 . 0 6 0 4 . . 0 - 7 3 . 5 0 0 2 0 + 4 ) 2 ( 0 8 . 0 4 6 3 2 . . 0 - 7 6 . 5 0 0 2 0 + 2 1 . 0 - 4 0 . 5 0 0 2 ) f o d ( 2 ν χ ) r y ( e m i T k a e P 2 / 2 α ) 4 + 1 p 5 ( d n a B e v a W s t ) r y ( ' 7 2 . 0 ± - 12 - 7 8 . 0 ± 9 6 . 0 ± 7 1 . 3 ± n o i s n a p x t d t e m a r a P 1 1 0 . 0 ± 3 1 y a r - X 2 1 0 . 0 ± 1 1 V U 2 1 0 . 0 ± 0 1 6 2 0 . 0 ± 3 1 G 5 1 A B L V ( o i d a R h t , ' 2 α , 2 α a h t i w 1 e l b d 2 . T h e 0 . 0 . H z ) 0 . H z ) 0 . m e t e r s ( f r o m T a e l b a T G 5 1 A L V ( o i d a R r a p e h T - . e t o N e v i r e d ) ' τ e l a c s e m i t 2 α s r e t e m a r a P", "pages": [ 11, 12 ] } ]

[ { "title": "J. Nevalainen", "content": "1 Tartu Observatory, 61602 T˜oravere, Estonia 2 Department of Physics, University of Helsinki, Finland Received , accepted Published online later Key words galaxies: clusters: general - instrumentation: spectrographs - intergalactic medium - techniques: spectroscopic - X-rays: galaxies: clusters The hierarchical formation of clusters of galaxies by accretion of material releases gravitational energy which dissipates into the intracluster gas. The process heats the material and generates gas turbulence and bulk motions and thus kinetic pressure. Mapping the velocity fields of the moving subunits would enable a new diagnostics tool for cluster formation studies and unbiased X-ray mass estimates. The required spatially resolved high resolution spectroscopy is not currently available. I demonstrate here the feasibility of detecting and mapping the velocities of the bulk motions using the Doppler shift of the Fe XXV K α line with the proposed ATHENA satellite. c © 2012 WILEY-VCH Verlag GmbH&Co.KGaA, Weinheim", "pages": [ 1 ] }, { "title": "1 Introduction", "content": "During its life cycle, a cluster of galaxies experiences collisions and mergers with other clusters and subunits. Eventually the relaxation processes take place and the intracluster material approaches hydrostatic and virial equilibrium. Consequently, there will be bulk motions in the cluster material at different distance scales and velocity levels, depending on the magnitude of the event and at which phase we observe the cluster. Mapping the velocity fields of the bulk motions would open a new tool for studying the dynamics of the clusters. Comparing the velocity maps with those predicted by cosmological simulations would be useful for the study of the formation of the large scale structure and provide constraints on cosmological parameters. Combining with the measured galaxy density field, the bulk velocity maps could be used to test whether the fundamental picture of gravitational collapse is correct (Branchini et al. 2001; da Costa et al. 1998; Dor'e et al., 2003). The bulk motions produce kinetic pressure which can reach a level of 10% level of that of the thermal gas even in most relaxed clusters (Lau et al. 2009). If not accounted for, the kinetic pressure may bias the hydrostatic X-ray mass estimates low by ∼ 10% at r 500 . Thus, it would be important to measure this component in clusters in order to obtain unbiased mass estimates for the cosmological applications. In case of recent, strong collisions which happen close to the plane of the sky, the merger shocks can be observed by the X-ray morphology, as in the rare cases of A520 (Markevitch et al. 2005), A754 (Macario et al. 2011) and A2146 (Russell et al. 2012). However, in most lines of sight the merger features are hidden by the projection. This allows the possibility of measuring the radial bulk motions via the Doppler shift of the emission lines. Currently the constrains on radial bulk motions are rather poor due to the lack of spatially resolved high spectral resolution instruments. The proposed ATHENA mission would have carried such an instrument, an X-ray Microcalorimeter Spectrometer XMS. In this paper I will examine the expected quality of bulk motion measurements with a mission approximating the capabilities of the proposed ATHENA instruments, i.e. an ATHENA-like mission.", "pages": [ 1 ] }, { "title": "2.1 Range of velocities", "content": "It is widely accepted that clusters of galaxies form by merging of smaller structures of matter accreted along large scale structure filaments. The velocities of the accretion flows are assumed to reach a level of 1000 km s -1 (e.g. Frenk et al. 1999). A collision of two clusters or protoclusters of similar mass produces a strong merger. The related bulk motion velocities can reach several 1000 km s -1 as in the case of the Bullet cluster (e.g. Markevitch et al. 2002). Simulations of Nagai et al. (2002;2003) indicate that ongoing minor mergers may produce bulk velocities up to ∼ 1000 km s -1 . The released gravitational energy is dissipated into the intracluster material which eventually approaches the hydrostatic equilibrium. However, simulations indicate that even in the most relaxed clusters there are residual bulk motions present throughout the cluster volume at ∼ 100 km s -1 level (e.g. Lau et al., 2009; Nagai et al. 2003).", "pages": [ 1 ] }, { "title": "2.2 Doppler shift", "content": "The radial components of the bulk motions of 100 - 1000 km s -1 correspond to 2-20 eV shift of the Fe XXV and XXVI K α emission line centroid energies ( ∼ 6 keV). It is challenging to use the currently most powerful X-ray instruments at 6 keV (XMM-Newton/EPIC, Chandra/ACIS and SUZAKU/XIS CCDs) for these measurements due to limitations in the energy resolution ( ∼ 100 eV). The Gaussian centroid can still be determined with precision better than 100 eV, assuming that the instrument gain is accurately calibrated. This requirement can be relaxed if one considers relatime motions between the main cluster and the moving part. In the following I discuss only the statistical precision of the line centroid determination, and not the calibration issues nor the uncertainties of the cosmic redshift measurements.", "pages": [ 2 ] }, { "title": "2.3 Observational constraints", "content": "Suzaku/XIS instruments have been used to place upper limits for the bulk motion velocities at ∼ 1000 km s -1 level in several clusters: A2319 (Sugawara et al. 2009), Centaurus (Ota et al., 2007), AWM7 (Sato et al. 2008). A first significant detection of a bulk motion has been achieved with Suzaku for the merging subclump in A2256 (Tamura et al. 2011). The radial velocity difference between the main cluster and the subclump is 1500 ± 300 ± 300 km s -1 (where the two sets of uncertainties refer to statistical and systematical ones, respectively).", "pages": [ 2 ] }, { "title": "3 ATHENA-like missions", "content": "ATHENA (Advanced Telescope for High ENergy Astrophysics) was one of three L-class (large) missions being considered by the European Space Agency in the Cosmic Vision 2015-2025 plan. In May 2012 the Jupiter mission Jupiter Icy Moons Explorer (JUICE) was chosen for launch. However, ESA has committed to continue supporting technology developments for a future large X-ray facility. At the time of writing this paper (Oct 2012) there was an understanding that ESA would shortly appoint a small team from the community to provide input based on the the ATHENA study team activities. In this paper I used the responses and background estimates for the X-ray Microcalorimeter spectrometer (XMS) and a wide field imager (WFI) as reported in the ATHENA assessment report used by the ESA Space Science Advisory Committee (the ATHENA Yellow Book, Barcons et al., 2012).", "pages": [ 2 ] }, { "title": "3.1 X-ray Microcalorimeter Spectrometer XMS", "content": "The requirement for the energy resolution of XMS is 3 eV at 6 keV. With its 100-1000 times higher effective area at 0.5 keV, compared to those of the current high resolution instruments onboard XMM-Newton and Chandra, and spatial resolution of 10 arcsec, XMS would enable spatially resolved high resolution spectroscopy. This would yield a breakthrough in mapping the spectral properties for extended sources like clusters. Also, the bandpass extends to 12 keV which allows the measurement of the Fe XXV K α emission line, prominent in clusters of galaxies. The downside of XMS is the relatively small FOV (2.3x2.3 arcmin 2 ) which renders the velocity mapping of a whole cluster challenging.", "pages": [ 2 ] }, { "title": "3.2 Wide Field Imager WFI", "content": "Even though the energy resolution of WFI (150 eV at 6 keV) is not sufficient to resolve the components of Fe K α complex, it can still determine very precisely the centroid of the Gaussian distribution of the total line emission (see Fig. 1) due to high photon statistics. This is due to the very large effective area ( ∼ 0.5 m 2 ) of the X-ray telescopes onboard ATHENA. Thus also WFI is a powerful tool for mapping the cluster velocities.", "pages": [ 2 ] }, { "title": "4 Improvement with ATHENA-like missions", "content": "I examine here the capability of the proposed ATHENA instruments for measuring the expected bulk motions in clusters. I used the MEKAL model (Kaastra 1992) in the XSPEC package to model the cluster emission (bremsstrahlung continuum + collisional excitation lines) adopting the metal abundances of Grevesse & Sauval (1998). I used the current estimates of the instrument responses (Barcons et al., 2012) to simulate the cluster spectra, including the instrument background and 90% resolved cosmic X- Notes. ( a ) Distance from the cluster centre. ( b ) Spatial resolution. ray background. Then I fitted the simulated data in the 5.57.5 keV band (excluding the channels where the Fe XXVI K α emission is significant) with a model consisting of a power-law component for the continuum and a Gaussian line for the Fe XXV K α emission. The 1 σ statistical uncertainty of the Gaussian centroid then yielded the estimate for the statistical precision of the velocity measurement, as summarised in Table 1.", "pages": [ 2, 3 ] }, { "title": "4.1 A2256 and the subclump", "content": "In order to obtain a real-life example of the performance of XMS and WFI for measuring the velocities of a nearby minor merger, I used the Suzaku results for the A2256 and the subclump (Tamura et al. 2011) to simulate spectra using an exposure time of 100 ks. A fit to the simulated XMS data (see Fig. 2) yielded a statistical uncertainty of the redshift ( σ z ∼ 10 -6 ) corresponding to a velocity precision at a level of ∼ 1 km s -1 , i.e. a 1500 σ detection for the clump motion. Measurement is very precise because many line features are resolved and each centroid gives weight to χ 2 . Dividing the XMS emission of the subclump into boxes with width of 30 arcsec (i.e. 5 × 5 pixel map for the full XMS FOV) yielded a statistical precision level of ∼ 10 km s -1 . This level of detail would provide a breakthrough in modelling the dynamics of the merging subclumps. Since the flux of A2256 subclump is comparable to that of the central region of similar size in A2256 (Tamura et al., 2011), the above calculations yield an approximate estimate of the expected velocity mapping precision level in the bright nearby cluster centres with XMS. A spectral fit to the data from a 100 ks WFI simulation of A2256 main cluster and subclump (see Fig. 1) yielded a statistical precision for Fe XXV K α shift corresponding to velocity precision of 60 km s -1 , i.e. a detection at ∼ 25 σ level.", "pages": [ 3 ] }, { "title": "4.2 Mapping the whole cluster", "content": "In order to derive general conclusions about the performance of ATHENA-like missions for the velocity mapping in clusters, one should simulate spectra with a grid of representative values for cluster temperatures, metal abundances and redshifts and realistic exposure times. Also, one should consider a range of luminosities for the moving regions and their velocities and directions of motion. In this work I limited the complexity of the above approach by using an exposure time of 100ks for a cluster with kT = 5 keV and z=0.1, assuming that all of the emission in the line of sight of a studied region originates from the moving subunit. I further assumed that the luminosity of the moving subunit at a given distance from the cluster centre can be estimated with a a β - model for the surface brightness with β = 2/3 and r core = 0.1 r 500 . This may be an underestimate of the actual signal since the likely enhancement of the emission due to the subunit is not accounted for. I adopted a bolometric luminosity L bol (r 500 ) = 7 × 10 44 erg s -1 within r 500 from L-T relation of Pratt et al. (2009). Using r 500 - T relation of Vikhlinin et al. (2006) I adopted r 500 = 10 arcmin.", "pages": [ 3 ] }, { "title": "4.2.1 XMS", "content": "A single XMS pointing covers only a fraction of the full cluster volume: at z = 0.1 the r 500 radius for a cluster with kT = 5 keV is ∼ 1 Mpc (Vikhlinin et al., 2006) and covers an area of ∼ 300 arcmin 2 , i.e. ∼ 50 higher than the FOV of XMS. Thus it is not feasible to cover the full r = r 500 region in nearby clusters with XMS. Limiting the mapping into the central r=0.25 r 500 region would require ∼ 10 pointings which might be feasible for a few clusters. Single XMS pointings at larger radii might still be useful, and I thus calculate estimates for these observations in the following. I simulated XMS spectra with the above generic cluster (see Section 4.2) within the full FOV i.e. ∼ (0.2 r 500 ) 2 . Fitting the simulated spectra yielded a statistical precision of ∼ 200 km s -1 at r=0.25 r 500 . At r=0.5 r 500 ∼ 90% of the total signal in the 5.5-7.5 keV band is due to background emission which degrades the statistical precision of velocity to ∼ 400 km s -1 and at r 500 the data are so noisy due to the background that reliable velocity measurements cannot be obtained. If the above cluster was located at z=1.0, a single central XMSpointing would cover the cluster out to ∼ 0.5 r 500 . The received flux would be similar as that of the z=0.1 cluster at 0.5 r 500 , when using the full XMS FOV. Thus, with a central 100ks XMS pointing it is feasible to obtain a single bulk motion measurement at z = 1.0 assuming the full cluster within 0.5 r 500 is moving with a radial velocity component higher than 400 km s -1 .", "pages": [ 3, 4 ] }, { "title": "4.2.2 WFI", "content": "Fitting the simulated data using the generic cluster described in Section 4.2 showed that in the centre the bulk velocity can be mapped with angular resolution of ∼ 0.1 r 500 with statistical precision of ∼ 100 km s -1 . At r=0.25 r 500 the lower cluster flux requires a larger extraction region to achieve similar statistical quality as in the centre. Using an extraction box with size of ( ∼ 0.2 r 500 ) 2 yields a velocity precision of ∼ 200 km s -1 . At a distance of 0.5 r 500 from the centre, the background dominates (as in the case of XMS) and the velocity precision degrades to ∼ 800 km s -1 level when using an extraction box size of ∼ 0.3 r 500 ). Repeating the exercise with a very hot cluster (kT = 10 keV) improves the continuum signal but the Fe XXV emission decreases and thus the velocity precision does not improve significantly.", "pages": [ 4 ] }, { "title": "5 Conclusions and discussion", "content": "The simulations showed that for a nearby (z ≤ 0.1) cluster with kT = 5 keV, using a 100 ks exposure using the instruments proposed for ATHENA, one could The above estimates for the level of statistical precision of velocity measurements with XMS and WFI indicate that an ATHENA-like mission would enable for the first time the mapping of radial components of the bulk motions due to recent minor and major mergers in nearby (z ≤ 0.1) clusters of galaxies within the central 0.25 r 500 . Also the residual motions due to past mergers can be mapped in these central regions. Single velocity measurements at the distance of 0.5 r 500 for a cluster at z=0.1 at the level of ∼ 400 km s -1 can be achieved with 100ks off-axis pointings using XMS. At higher distances the mapping is more limited due to the background but the average motion within the central 0.5 r 500 up to z=1.0 can be measured with a central 100ks pointing with XMS, if the radial velocity component is bigger than ∼ 400 km s -1 . These measurements would open a new tool to study cluster dynamics. This would be complementary to the current analyses of intracluster shocks and sloshing and thus give a better handle on the 3-dimensional dynamics of clusters. This would yield a breakthrough in modelling the cluster physics and its implications to cosmology. The kinetic Sunyaev Zeldovic (kSZ) effect as measured by WMAP (Kashlinsky et al., 2008; Osborne et al., 2012) has been used to constrain bulk motions of clusters. However these results have been controversial due to relatively weak kSZ signal compared to the thermal one. Thus, one must combine a large sample of clusters and assume a similar motion for all the clusters in the sample. Mak et al. (2011) calculate that a sample of 400 clusters would allow Planck to detect a coherent 500 km s -1 bulk motion. Clearly ATHENA would make improvement here, being able to measure the velocities in individual clusters with better precision and spatial resolution.", "pages": [ 4 ] }, { "title": "References", "content": "Barcons, X., Barret, D., Decourchelle, A. et al., arXiv:1207.2745 Branchini, E., Freudling, W., Da Costa L.N et al.: 2001, MNRAS 326, 1191 Sato, K., Matsushita, K., Ishisaki, Y. et al.: 2008, PASJ, 60, 333 Sugawara, C., Takizawa, M. & Nakazawa, K.: 2009, PASJ, 61, 1293 Sun, M., Murray, S. S., Markevitch, M. et al.: 2002, ApJ 565, 867 Tamura, T., Hayashida, K., Ueda, S. et al.: 2011, PASJ, 63S1009 Vikhlinin, A., Kravtsov, A., Forman, W. et al., 2006, ApJ, 640, 691", "pages": [ 4, 5 ] } ]

[ { "title": "D'aniel Barta", "content": "email: [email protected] Institute for Particle and Nuclear Physics, Wigner Research Centre for Physics, Hungarian Academy of Sciences, Konkoly-Thege Mikl'os 'ut 29-33., H-1121 Budapest, Hungary September 8, 2021", "pages": [ 1 ] }, { "title": "Abstract", "content": "We investigate insterstellar gas spheres by determining the metric functions, the material distribution, and the features of particle orbits in terms of stability and geodesics. An exact solution of the Einstein's equations for interstellar gas clouds is derived that is compatible with the results of recent astronomical measurements. The solution determines the distribution of pressure and density, and it is suitable to describe the energy, speed, trajectory, and further relevant physical features of the cloud's particles. We describe the spacetime inside the nebula and give the density profile and the geodesics of particles. We find that circular orbits are stable and the cloud rotates rigidly by an angular velocity that is inversely proportional to the radius.", "pages": [ 1 ] }, { "title": "1 Introduction", "content": "The general relativistic gravitational fields are described by Einstein's equations. Although due to their non-linearity one unawares encounter difficulties in solving them for spherically symmetric static gas cloud. [Burlankov (1993)] and [Goldman (1978)] has studied perfect fluid spacetimes and introduced generating functions by choices of new variables in order to replace the second order ODEs with algebraic equations. For retain generality, [Fodor (2000)] used no equation of state either. By specializing the problem for ideal gas, one may make use of the linear relation between pressure and density. One may also characterize the matter by the stress-energy tensor of perfect fluid, while keeping in mind the fact that the gasous medium is compressible contrary to the perfect fluids. In addition, one should impose a few physical criteria for the maximal mass of the nebula, boundary condition on the pressure and density, as well as for the sound of speed in the medium. Evidently, the corresponding quantities interpreted in the context of general relativity must be identical with the results of [Bohigas (1988)] and [Kritsuk et al. (2011)] in the classical limit. [Hobson et al. (2006)] lend assistance to calculate the geodetics and further derived quantities.", "pages": [ 1, 2 ] }, { "title": "2 Basic properties of a self-gravitating spherically symmetric static cloud", "content": "Consider an isolated interstellar nebula remote from any other matter, and assume that the hydrostatic pressure is balanced by the cloud's self-gravitation. The mass within a distance r from the centre of the cloud is given by The most dense and heavy of all nebulae are the giant molecular clouds composed by mostly gas and some dust. For the sake of simplicity, assume that the considered medium consists only of cold neutral gas. In this case collisions between these low-energy particles are rare and weak, and have no significant effect on the system. Hence the material of the cloud can be realistically regarded as ideal gas. Since temperature is nearly constant ( 10 -20 K for molecular clouds), equation of state becomes a linear relation of pressure p and density ρ , the coefficient c s is the isothermal sound speed within the gaseous medium. On the basis of the isothermal equation of state (2) and (1), the total mass of the cloud M R is expressed by the average pressure ¯ p (see the following Table 1.) as M R = 4 πR 3 ¯ p/ 3 c 2 s . By the comparsion of these with a given radial pressure distribution p ( r ) , the value of average pressure can be written as It is important to state that the following criteria must be met for real physical systems: These conditions can be easily justified: the first one expresses the simple fact that the density and pressure disappears, the second one says that they do not change on the border of the cloud. So the matter does not suddenly vanish on the border, but steadily aligns into the environment.", "pages": [ 2, 3 ] }, { "title": "3 Field equations for the compact gas cloud", "content": "The metrics of a general stationary spherically symmetric configuration can be written in area coordinates as where ν and λ are functions of the radial coordinate r . Consider the above described spacetime region is filled with ideal gas, u ν is the contravariant velocity vector of gas particles. The stress-energy tensor is equivalent with the stress-energy tensor T µν = ( ρ + p ) u µ u ν + pg µν for perfect fluid. Calculating the Einstein's equation G µν = 8 πT µν , one obtains the mass density, the radial and the angular directional pressure as where the prime denotates derivates with respect to the radial coordinate r . [Fodor (2000)] has shown that the set of differential equations (7) can be reduced to algebraic ones with integration required only for one metric function but not the physical variables ρ and p . From this point on we slightly modify Fodor's method and apply it to isothermal ideal gas. Due to the isotropic configuration, p ≡ p r = p ϑ implies one can require one more field equation by extracting the last equation from the second one. Regarding the coefficient of d e -λ / d r , it turns out to be practical to introduce a pair of new variables Then the field equation (8) reduces to a second order algebric equation in β , namely For any function α the quadratic equation (10) is solved by the real roots where the discriminant must be non-negative. The only physically relevant solution as [Fodor (2000)] has already shown is β + , since its non-positive counterpart always belongs to a non-positive, hence non-physical mass density. The metric functions belonging to β are formally given by the definitions (9) as where the constant ν 0 determines the scaling of the time coordinate t . One can also calculate the pressure and density by substituting functions α and β into the first two field equations of Eq. (7). The simple, but still realistic choice for the generating function α is the ratio of two polynomials of the radial coordinate r . The lowest degree form which is physically valid for a compact fluid or gaseous sphere is where A and B are positive constants associated by inverse first and second power of distance dimensions. It is advisable to introduce a further new nonnegative real constant C 2 = 2 B/A 2 -2 and use it in place of constant B . In order to eliminate the square root appeared in Eq. (11) while expressing β , a new radial variable defined by will be introduced. Then the centre gets into ξ c = arcsinh (2 C/ 3) , and the spatial infinity ξ ∞ = arcsinh ( C/ 2) , and the new variable is restricted by 0 < ξ ∞ ≤ ξ ≤ ξ c . Through (14) the generating functions α and β become The equations (12) provide the metric functions 1 and the inner Schwarzschild metrics appears to be by using the charasteric ξ as radial coordinate. The constant A corresponds to a constant conformal transformation of the metrics. From Eq. (7) both the density and the pressure are expressable by a ratio of two polynomials of hyperbolic function of the radial coordinate ξ . Moving away from the centre of the cloud due to the conditions (4), the pressure monotonously tends to zero at r = R , on the border of the cloud. Any choice of constants A and C satisfies the restriction (5) on the speed of sound in the medium. One takes C 2 = 2 B/A 2 -2 into consideration and assumes B glyph[lessmuch] 1 glyph[lessmuch] C , then p and ρ vanish simultaneously at r = R if and only if B = 4 /R 2 . This restriction implies A = 8 c 2 s /R . By eliminating the variable ξ via the transformation (15), one can formulate the functions of state in terms of polynomials of the natural radial coordinate r , see Fig. 1. As it was required, if B glyph[lessmuch] 1 glyph[lessmuch] C then the equation of state is nearly linear for every r ≤ R , therefore fixes the last constant as C = 4 /c 2 s . Consequently, they differ from one another only by a constant factor, thus verifying the legitimacy of the isothermal equation of state (2). The obtained density and pressure distribution correspond with the classical results of [Bohigas (1988)] (see his Fig. 6.) and [Kritsuk et al. (2011)] , hence the metric functions consistent with the distributions must be valid. Similarly, the metric functions ν and λ can be expressed as functions of the radial coordinate r . The line element is immediately obtaind by the insertion of the metric functions into the general form of Schwarzschild metrics (6).", "pages": [ 3, 4, 5 ] }, { "title": "4 Lagrangian function and geodetics of the particles", "content": "In the following sections the geodetics in the Schwarzschild geometry in several textbooks, like [Hobson et al. (2006)] originally studied the behaviour of massive particles and photons will be briefly reviewed according to our special geometry. For the Schwarzschild metrics (18) the relativistic Lagrangian function L = g µν ˙ x µ ˙ x ν of the particles in the investigated nebula is where the dot denotates derivates with respect to the proper time coordinate τ . By substituting this form for L into the Euler-Lagrange equations d d τ ( ∂L ∂ ˙ x µ ) -∂L ∂x µ = 0 , result the geodetic equations. Since the eqation for µ = 3 is satisfied by ϑ = π/ 2 , it is sufficient to keep only the set of three equations independent of ϑ : The two simplest equations are derived immediately since the Lagrangian is not an explicit function of t or ϕ . The appearing constants L t and L ϕ proportional to the total energy and the angular momentum of the particles. It is expedient to replace the complicated second equation of Eq. (20) by the first integral g µν ˙ x µ ˙ x ν = -1 of the geodetic equation, since the worldline of a massive partice is timelike. In our case, it takes the form By substituting the two original expressions of (20) into (21), one obtain the combined energy equation for the radial coordinate valid inside the cloud. Outside of the cloud the customary equation ˙ r 2 +(1 -2 M/r ) L 2 ϕ /r 2 -2 M/r = ( L 2 t -1) governs the motion of particles. Note that the right-hand side is a constant of motion, L t ∝ E as previously stated. The constant of proportionality is fixed by requiring E = m 0 for a particle at rest at r = ∞ , m 0 is the mass of the particle at rest. Letting r → ∞ and ˙ r = 0 in the equation, L 2 t = 1 thus is required. Hence, one must has L t = E/m 0 where E is the total energy of the particle in its orbit. The shape of a particle orbit is given by using the last equation of Eq. (20) to express ˙ r in the (22) as Furthermore, if one parametrizes Eq. (21) by ˜ r ≡ 1 /r , one obtains ( d˜ r d ϕ ) 2 + ˜ r 2 e -λ = 1 L 2 ϕ ( L 2 t e -ν -1 ) e -λ . Finally, the differentiation with respect to ϕ provides the orbits for a particle in the equatorial plane ϑ = π/ 2 where the prime denotes derivates with respect to the radial coordinate r . The particle orbits have two special cases, namely the radial motion where ˙ ϕ = 0 and the circular motion where ˙ r = 0 . Since the density of the nebula is constant in time, one might ignore the radial motion and focus on investigating the latter motion.", "pages": [ 5, 6, 7, 8 ] }, { "title": "5 Circular motion on bounded and stabil orbits, velocity of the gas particles", "content": "In the equatorial plain for circular motion, one has r = constant, and thus ˙ r = r = 0 . This restriction in accordance of (23) imposes ˜ r ' = ˙ ˜ r/L ϕ ˜ r 2 = 0 ; consequently ˜ r '' is zero too. Setting ˜ r = 1 /r = constant in the equation of orbits (24), one has L 2 ϕ = 1 2 r 3 L 2 t ( ν ' + λ ' ) ν ' e -ν . Beside replacing the differentials of the metrics functions of Eq. (18) in the energy equation (22) in addition to the condition ˙ r = 0 , one can identify the constants of motion as It has been shown that E = L t m 0 is the total energy of a particle of rest mass m 0 in a circular of radius r . Subsequently one can circumscribe the bounded orbits by requiring E < m 0 , so as long as L t = 1 . The limits on r for the orbit to be bound is given by 1 = c s 2 ( 1 + c 2 s r 2 4 R 2 ) which is satisfied when The first and third geodetic equations in Eq. (20) immediately shows that the components of 4-velocity of a particle are simply in the coordinate system ( t, r, ϑ, ϕ ) . The geodetic equations specify the circular trajectory ϕ ( τ ) and the orbital period T = 2 π/ ˙ ϕ , which according to (26) is T = 4 πR/c s by substituting L ϕ from (26). Although r is not the radius of the orbit, it is readily conceivable that the spatial distance travelled in one complete revolution is 2 πr , just as in the Newtonian case. Instead of parametrizing ϕ in the proper time, one can alternatively describe it by d ϕ/ d t = c 2 s / 4 R in terms of coordinate time t , thus the 4-velocity is", "pages": [ 8 ] }, { "title": "6 Conclusions", "content": "For cold isotermal molecular clouds in gravitational and thermal equilibrium, the pressure correlates linearily to the density. Their distribution can expressed by a decreasing function of radius in terms of only the speed of sound in the medium and the size of the cloud. The profils correspond with astrophysical measurments. The metric functions provides us Lagrangians that determines the geodetics of the particles; all the circular orbits are stable, thus the cloud rotate rigidly and theoretically it remains stable permanently. The value of the four-velocity of a particle slightly differs from the one observed in an ordinary Schwarzschild spacetime, but the angular velocity is inversely proportional to the radius.", "pages": [ 9 ] }, { "title": "References", "content": "[Girichidis et al. (2010)] Girichidis, P., Federrath, C., Banerjee R., Klessen R. S.: 2010, MNRAS, Vol. 413, No. 4, 2741-2759 [(Ferri'ere (2001))] Ferri'ere, K.: 2001, Rev. of Mod. Phys. 73, 1031-1066 [Burlankov (1993)] Burlankov, D. E.: 1993, Theor. and Math. Phys. 95, 455 [Goldman (1978)] Goldman, S. P.: 1978, Astrophys J. 226, 1079 [Fodor (2000)] Fodor, G.: 2000, arXiv:gr-qc/0011040v1 [Bohigas (1988)] Bohigas, J.: 1988, A&A, Vol. 205, No. 1-2, 257-266. [Kritsuk et al. (2011)] Kritsuk, A. G., Norman, M. L., Wagner, R.: 2011, APJL, Vol. 727, No. 1 [Hobson et al. (2006)] Hobson, M. P., Efstathiou, G. P., Lasenby, A. N.: 2006, Cambridge: Cambridge University Press, General Relativity: An Introduction for Physicists, 205-217", "pages": [ 9 ] } ]

[ { "title": "Studies of active galactic nuclei with CTA", "content": "A. Reimer 1 and M. Bottcher 2 1 Institut fur Theoretische Physik, and Institut fur Astro- und Teilchenphysik, Leopold-Franzens-Universitat Innsbruck Technikerstraße 25, A-6020 Innstruck, Austria [email protected] 2 Astrophysical Institute Department of Physics and Astronomy Ohio University, Clippinger # 251B Athens, OH 45701, USA [email protected]", "pages": [ 1 ] }, { "title": "Abstract", "content": "In this paper, we review the prospects for studies of active galactic nuclei (AGN) using the envisioned future Cherenkov Telescope Array (CTA). This review focuses on jetted AGN, which constitute the vast majority of AGN detected at gamma-ray energies. Future progress will be driven by the planned lower energy threshold for very high energy (VHE) gamma-ray detections to ∼ 10 GeV and improved flux sensitivity compared to current-generation Cherenkov Telescope facilities. We argue that CTA will enable substantial progress on gamma-ray population studies by deepening existing surveys both through increased flux sensitivity and by improving the chances of detecting a larger number of low-frequency peaked blazars because of the lower energy threshold. More detailed studies of the VHE gamma-ray spectral shape and variability might furthermore yield insight into unsolved questions concerning jet formation and composition, the acceleration of particles within relativistic jets, and the microphysics of the radiation mechanisms leading to the observable high-energy emission. The broad energy range covered by CTA includes energies where gamma-rays are una ff ected from absorption while propagating in the extragalactic background light (EBL), and extends to an energy regime where VHE spectra are strongly distorted. This will help to reduce systematic e ff ects in the spectra from di ff erent instruments, leading to a more reliable EBL determination, and hence will make it possible to constrain blazar models up to the highest energies with less ambiguity. Keywords: Active galactic nuclei - Gamma-rays - Jets", "pages": [ 1 ] }, { "title": "1. Introduction", "content": "Active galactic nuclei (AGN) are extragalactic sources of enhanced activity that are powered by the release of gravitational energy from a supermassive central black hole. Energy linked to the black hole spin (e.g., Blandford & Znajek, 1977) or rotating accretion disks (e.g., Blandford & Payne, 1982) may be instrumental for forming prominent jets which transport material from the innermost region of the AGN to kpc-, sometimes even Mpc-scale distances with relativistic speed. Such jets are usually identified through the detection of bright non-thermal radio emission as observed in radio-loud AGN. Only a small percentage ( ∼ 10 %) of all AGN are known to be radio-loud 1 . In the vicinity of the central region of an AGN matter is accreted from a disk onto the black hole, line-emitting clouds of material (the so-called broad-line region, BLR, and narrow line region, NLR) swirl at pc to kpc distances from the central engine, dusty material surrounding the accretion disk may imprint thermal signatures in the infrared part of the AGN spectrum, and the prominent jets of material in case of radio-loud AGN dominate the non-thermal radiative power in such systems (see Fig. 1). The radiation from material which moves relativistically with speed β Γ c (with Γ = 1 / √ 1 -β 2 Γ being the bulk Lorentz factor) along the jet axis is beamed into an angle ∼ 1 / Γ around the direction of propagation. Because of this beaming e ff ect mostly those AGN whose jet axes are close to alignment with our line of sight (i.e., blazars) are favourably detected as sources of high-energy (gamma-ray) emission. However, also some mis-aligned AGN (i.e., radio galaxies) can be detected, if they are su ffi ciently nearby. Blazars therefore o ff er an excellent opportunity to study jet physics of massive black hole systems, and through population studies also their evolution over cosmic time. Because the bolometric radiative energy output of AGN jets is often dominated by the gamma-ray regime 2 , the observed peak flux ( ν F ν ) pk in this band, together with a knowledge of the bulk Lorentz factor Γ , provides a robust lower limit for the overall jet energetics constrained by the total radiative power, L jet > L rad, with where d L is the luminosity distance. Such limit is not only crucial for constraining jet formation scenarios and the overall particle and field content of a jet including its impact for searches for the sources of the ultrahigh energy cosmic rays, but also for, e.g., investigating the jet's feedback on its environment. Comparing disk and jet energetics may give important clues on the physical connection between disk accretion and jetted outflows. Because these jets form in the vicinity of the strong gravitational fields of massive, probably rotating, black holes, studying events occuring close to the central engine may contribute to understanding jet formation. Size scales of the emission region of the order of the Schwarzschild radius are implied by extreme variability observed e.g., down to a few minutes time scales at TeV energies (Aharonian et al., 2007; Albert et al., 2007a), in a few radio-loud AGN, and this might imply a location of the emission region very close to the central black hole. On the other hand, the observation of systematic variations of the optical polarization over several days associated with a gamma-ray flare (e.g., Abdo et al., 2010f), and distinct gamma-ray flares coinciding with the peak polarization of the mm-core (Jorstad et al., 2009) seem to favour rather pc-scale distances of the emission region relative to the central engine. This highlights the current debate regarding the location of the emission region. Studying the gamma rays from jets within the multifrequency context o ff ers a view towards the global structure and composition of magnetized relativistic outflows, which provide constraints on the dominant radiation mechanisms. Monitoring the transition from flaring events to the quiescent phases together with the estimates on the overall flaring duty cycles may provide hints on the origin of variability. Gamma rays probe the highest energy particles present in these jets, and therefore are relevant for our understanding of how charged particles are accelerated in jet plasmas, e.g., via shocks, and / or turbulence and / or magnetic reconnection. This may also have implications for our understanding of the origin of ultrahigh energy cosmic rays. In this article, we review the prospects of CTA to facilitate progress in our understanding of the AGN phenomenon and its related physics including the large-scale impact of the associated jets.", "pages": [ 1, 2, 3 ] }, { "title": "2. CTA and the population of AGN", "content": "According to the unification scheme of radio-loud AGN (e.g., Urry & Padovani, 1995), flat spectrum radio-quasars (FSRQs) and BL Lac objects (commonly referred to as 'blazars') are sources which are observed under a small viewing angle with respect to the jet axis. Those observed at large viewing angles are classified as Fanaro ff Riley I and II radio galaxies (Fanaro ff &Riley, 1974). Hence, they are commonly considered as the parent populations of blazars 3 . Jetted AGN are oberved as sources of radiation across the electromagnetic spectrum, from the radio band up to very-high energy (VHE: E > 100 GeV) gamma-ray energies. The blazar class is subdivided into observationally weak-lined AGNs, identified as BL Lac objects, and strong-lined ones, called flat-spectrum radio quasars (FSRQs) (Landt et al., 2004). The latter show signatures of a bright accretion disk (e.g., a 'blue bump') and strong emission lines, whereas the former are lacking such features. However, the lack of thermal (accretion disk) and emission line features in the spectra of objects typically classified as BL Lac objects may be - at least in some cases - due to the bright non-thermal continuum of jet emission outshining those features rather than their actual absence (Giommi et al., 2012). The spectral energy distributions (SEDs) of jetted AGN consist generally of two broad components (see, e.g., Fig. 7). The low-energy component is commonly attributed to synchrotron radiation from relativistic electrons, and possibly positrons, in a relativistically moving emission region ('blob') in the jet. The origin of the high-energy emission is still a matter of debate, depending strongly on the overall jet composition (see below). Spectrally, blazars can be classified according to the frequency of the synchrotron peak in their broadband SED, independent of the optical emission-line based characterization of being a BL Lac object or a quasar (Abdo et al., 2010c). Low-synchrotronpeaked (LSP) blazars have their low-energy peak at ν s , peak < 10 14 Hz, intermediate-synchrotron-peaked (ISP) blazars at 10 14 ≤ ν s , peak ≤ 10 15 Hz, and high-synchrotron-peaked (HSP) blazars at ν s , peak > 10 15 Hz. Considering the BL Lac population only, we shall distinguish low-frequency-peakedBL Lacs (LBLs), intermediate-frequency-peakedBL Lacs (IBLs) and high-frequency-peaked BL Lacs (HBLs), correspondingly. The census of gamma-ray detected blazars has recently experienced a dramatic increase from less than 100 blazars detected by the EGRET instrument onboard the Compton Gamma-Ray Observatory and Cherenkov telescopes to nearly 10 3 high-latitude objects detected by the Large Area Telescope (LAT) on board the Fermi Gamma-Ray Space Telescope (Ackermann et al., 2011), that have been associated with AGN. Only a few non-blazar AGN are detected at gamma-ray energies to date: nearly a dozen radio galaxies, a few Narrow-Line Seyfert 1 (NLS1) galaxies and a few unusual sources that escaped a convincing identification so far (Ackermann et al., 2011). Until now, there is no convincing case of a gamma-ray detection of a 'classical' radio-quiet Seyfert galaxy. An upper limit for the GeV luminosity of hard X-ray selected radio-quiet Seyferts as a class is currently probing the level of about 1 % of their bolometric luminosity, corresponding to a few 10 -9 ph cm -2 s -1 in the 0.1 - 10 GeV band (Ackermann et al., 2012). With CTA at its predicted sensitivity at low energies it will be possible to extend this energy range to several tens to hundreds of GeV at a comparable energy flux level. This would probe whether there exists a smooth extension of radio-loud low-luminosity AGN towards the Seyfert population 4 both spectrally as well as regarding their luminosity function, or whether those are distinct source classes. The low number ratio of FR II to FR I radio galaxies detected at gamma-ray energies to date is surprising in the framework of the unification scheme. Though the doubling of the Fermi survey time from one to two years has increased the overall number of detected gamma-ray emitting AGN by ∼ 50 %, the relative number of LAT FSRQs to LAT BL Lacs has decreased from ∼ 0 . 9 to ∼ 0 . 8 5 . The number of gamma-ray detected misaligned AGN has not changed significantly, and neither has the FR II to FR I number ratio (Ackermann et al., 2011). Three of the four radio galaxies detected at VHE gamma-ray energies (M87, Cen A, NGC 1275) belong to the FR I class (the fourth one, IC 310, is of unknown class, possibly a head-tail radio galaxy: Aleksic et al., 2010). A deeper survey of FR I and FR II radio galaxies with CTA at energies > 10 GeV is expected to increase the sample of VHE gamma-ray emitting radio galaxies and may lead to the detection of a few FR II radio galaxies. This might facilitate the determination of the ratio of VHE gamma-ray emitting FR II to FR I sources over a broader energy range. Such studies might reveal whether the less prominent gamma-ray emission from FR IIs is indicative of less e ffi cient particle acceleration in their jets, a di ff erence in jet structure (e.g., Chiaberge, et al. (2000)) and / or beaming pattern between FR Is / BL Lacs and FR IIs / FSRQs (e.g., Dermer (1995)), whether possibly γγ absorption in the dense nuclear radiation fields of these generally more powerful sources (Reimer, 2007; Liu et al., 2008; Sitarek & Bednarek, 2008; Poutanen & Stern, 2010; Roustazadeh & Bottcher, 2010, 2011) plays a role in suppressing observable gamma-ray emission, or whether FRII / HERGsare intrinsically less numerous as a consequence of being located at the high-luminosity end of an overall radio galaxy luminosity function (Giommi et al., 2012). Among the LAT-detected BL Lacs the high-synchrotron peaked sources (HSPs) are the largest subclass, which is also the AGN subclass that is mostly detected in the VHE-regime by current Atmospheric Cherenkov Telescope (ACT) instruments. The (nearly permanent) survey observation mode of Fermi -LAT has triggered many follow-up observations of selected flaring AGN also with H.E.S.S., MAGIC and VERITAS. Until a few years ago, almost all AGNs detected by ground-based Cherenkov telescopes were HSPs, primarily because of their harder GeV gamma-ray spectra (see Fig. 3), indicating higher gamma-ray peak frequencies than other blazar subclasses. However, due to their permanently improving flux sensitivity and decreasing threshold energies, more than 40 blazars of all subtypes (FSRQs, all types of BL Lac objects: LBLs, IBLs, HBLs) have meanwhile been detected in VHE gamma-rays, covering the redshift range 0.03 to at least 0.536, thereby nearly doubling the census of VHE blazars during the past couple of years 6 . This may indicate that with CTA it will be possible to significantly expand the population of low-frequency peaked VHE AGN, both FSRQs and BL Lac objects, in addition to the HSP population. A systematic unbiased study may reveal then the required environment and jet properties that allows particles and photons to reach high energies. In particular, future observation of a larger number of BL Lac - FSRQ transition objects in the VHE gamma-ray band, accompanied by multifrequency coverage, may provide more insight in this regard. So far only few (e.g., 3C 279 whose thermal components may be overwhelmed by a strong non-thermal flux in a bright state (Pian et al., 1999); or the BL Lac prototype, BL Lacertae, which shows occasionally broad lines (e.g., Capetti, Raiteri & Buttiglione, 2010)) of such objects have been detected at VHEs when they reached an elevated flaring state (Albert et al., 2008; Bottcher et al., 2009; Albert et al., 2007b). E.g., while monitoring 3C 279 in 2006 in a dedicated multifrequency campaign (Bottcher et al., 2007, see Fig. 4) this source transitioned to an overall high state observed in the optical as well as at X-rays. During this state, in February 2006, a VHE signal from 3C 279 was detected by the MAGIC telescope (Albert et al., 2008). The observations of a larger sample of such objects with CTA might uncover a particular spectral and / or variability pattern with possible relations to other frequency bands that may help to finally reveal the conditions in the jet that allows charged particles to reach extreme energies. A similar consideration can be applied to LBL HBL transition AGN. Broadband studies of such objects, including the important VHE regime, may shed light on the physical origin of such behaviour, and will help to determine to what extent LBLs and HBLs are fundamentally di ff erent AGNs. Both the expected increased sensitivity of CTA and extension of the available energy range towards tens of GeV will also facilitate studies of the AGN population at VHE gamma-rays to very large redshifts. The so far highestredshift source detected at VHEs is 3C 279 at z = 0 . 537. The redshift range of AGNs covered by Fermi -LAT extends to z < 3 . 1 (unchanged from the first to the second year of Fermi exposure; see Fig. 2), above ∼ 20 GeV it is z < 3 (Ackermann et al., 2011), while FSRQs are known to exist up to z ∼ 5 . 5 (e.g., Q 0906 + 6930: Romani (2006)). With CTA, a new quality of the study of AGN evolution over cosmic time will be possible. The VHE range is important as it provides an undiluted view on the pure jet. Proposed cosmological evolution scenarios (Bottcher & Dermer, 2002; Cavaliere & D'Elia, 2002) consider a gradual depletion of the circum-nuclear matter and radiation fields over cosmic time thereby turning highly-accreting into pure non-thermal jet systems. This would suggest a transition from external-Compton to synchrotron-self-Compton dominated high-energy emission in the framework of leptonic emission scenarios or from photo-pion dominated to proton-synchrotron dominated high-energy emission in the framework of hadronic emission scenarios (see § 4). If this scenario is correct, a systematic study of the sub-GeV - TeV spectra of the various subclasses of blazars should therefore reveal a gradual transition from multi-component gamma-ray emission in accretion-dominated blazars to featureless single-component gamma-ray emission in pure jet sources. For the first time, it will be possible to build large, well-defined, statistically complete 7 and unbiased 8 samples at VHEs which allow us to derive population properties like the VHE Log(N)-Log(S) distribution for the various types of AGN, the luminosity function at VHE gamma rays and compared to other wavelengths, and to study their cosmological evolution. This will extend our knowledge on the origin of the extragalactic gamma-ray background (e.g., Abdo et al. (2010e)) up to the highest photon energies, and its impact on the evolution of the intergalactic medium and structure formation (Puchwein et al., 2011). As the AGN population detected at VHE gamma rays will penetrate to larger redshifts, predominantly the high luminosity tail of this population will be detected. In particular, verifying the existence or non-existence of a highluminosity HSP population and its broadband spectral properties will be interesting as this would contradict the traditional understanding of the blazar sequence (Fossati et al., 1998) 9 . According to this picture, a sequence of blazar subclasses has been proposed to be linked to their bolometric luminosity, black hole mass, accretion disk luminosity and accretion mode (e.g., Ghisellini et al., 2011). It has been suggested that FSRQ activity is powered by accretion at a high Eddington ratio ( ˙ M / ˙ M Edd /greaterorsimilar 10 -2 ), which might be related to dense circum-nuclear environments. The corresponding dense circum-nuclear radiation fields are expected to leave their imprints in two-component gamma-ray spectra as well as potentially γγ absorption features, if the gamma-ray production zone is located within the broad-line region of the AGN (Poutanen & Stern, 2010). At the same time, e ffi cient radiative cooling of relativistic particles in these dense radiation fields might then impede their acceleration to very high energies, resulting in SED peaks at low frequencies. On the other hand, BL Lac objects are suspected to be powered by radiatively ine ffi cient accretion at low rates ( ˙ M / ˙ M Edd /lessmuch 10 -2 ), possibly - at least in part - due to larger black-hole masses (and hence larger ˙ M Edd). If the jet power correlates positively with the accretion rate (e.g., Rawling & Saunders, 1991), this implies a lower power in the jets produced in these objects, compared to FSRQs. At the same time, the circum-nuclear radiation fields are expected to be very dilute, with only minor impact on the formation of the high-energy (gamma-ray) emission. The search for high-luminosity (high-redshift) BL Lac objects with high synchrotron and gamma-ray peak frequencies with CTA, in combination with on-going monitoring by Fermi -LAT promises progress in verifying the existence of and understanding the origin of the blazar sequence, or whether the peak energy is intrinsically unrelated to the blazar luminosity (Giommi et al., 2012). The redshift of these objects, if lacking as argued by Giommi et al. (2012), could be constrained using UV-to-NIR photometry (Rau et al., 2012), or limits inferred from the shape of the deabsorbed spectrum if the extragalactic background light (EBL) and its evolution were known (e.g. Abdo et al., 2010a; Prandini, Mariotti & Tavecchio, 2011).", "pages": [ 3, 4, 5, 7 ] }, { "title": "3. The extragalactic background light and blazar spectra", "content": "VHE gamma-rays from sources at cosmological distances will be attenuated through γγ absorption on the extragalactic background light (EBL; e.g., Dwek & Krennrich, 2005; Stecker et al., 2006; Franceschini et al., 2008; Gilmore et al., 2009; Finke et al., 2010). The SED of the EBL has two maxima: one at ∼ 1 µ m due to star light from cool stars, and one at ∼ 100 µ m due to cool dust (see Fig. 5). A direct measurement of this background is extremely di ffi cult because of bright foreground emissions (both within our solar system and our Galaxy). The recent measurements of unexpectedly hard VHE gamma-ray spectra from blazars at relatively high redshifts (see, e.g., Fig. 6) has led to the conclusion that the intensity of the EBL must be near the lower limit set by direct galaxy counts (e.g., Aharonian et al., 2006; Abdo et al., 2010d), or that the gamma-ray signal might be contaminated by ultra-high energy cosmic ray-induced photons (e.g., Essey & Kusenko, 2010). A more exotic alternative explanation that has been proposed is that VHE γ -ray photons may be converted to axion-like particles when interacting with magnetic fields either in the vicinity of the blazar or in intergalactic space. Those particles would be able to travel to Earth unaffected by the EBL, and may be re-converted to γ -rays in interactions with Galactic magnetic fields (De Angelis et al., 2007; Simet et al., 2008). Even assuming that EBL absorption is not circumvented, details of the spectral shape and, in particular, the cosmological evolution of the EBL are still uncertain. Indirectly, the EBL and its cosmological evolution can be studied by analyzing simultaneous broadband SEDs of VHE gamma-ray blazars at various known redshifts. In particular, simultaneous Fermi -LAT and ground-based VHE gamma-ray spectra are crucial for such an analysis. However, this requires an a priori knowledge of the sourceintrinsic SED throughout the GeV - TeV energy range. The uncertainties and ambiguities in blazar jet models (see § 4) currently preclude definite conclusions about the EBL based on blazar SED modeling alone. An observational challenge in such studies lies in the often vastly di ff erent integration times over which gamma-ray spectra in the Fermi -LAT energy range are measured (typically several weeks), compared to VHE gamma-ray spectra, often extracted from a few hours of good data from ground-based ACTs. This often leads to mis-matches in the spectral shapes and flux normalizations, which complicates or impedes any meaningful theoretical interpretation. With the reduced energy threshold of CTA, down to ∼ 10 GeV, it will be possible to determine the shape of the gamma-ray spectrum from energies at which EBL absorption is negligible (typically below a few tens of GeV) out to > 100 GeV energies where the spectrum might be significantly a ff ected by EBL absorption, depending on redshift. The significant overlap with Fermi -LAT could then potentially also allow for a more reliable cross-calibration between LAT and ground-based ACTs. Given the often very moderate variability of the gamma-ray spectral indices of many LAT-detected blazars (Abdo et al., 2010b), the cross-calibration with CTA might then allow for the construction of reliable, truly simultaneous gamma-ray SEDs through the LAT and VHE gamma-ray energy ranges. Note that limited correlated flux variability between the GeV and TeV energy range of prominent TeV-blazars has been observed so far (e.g., during intensive campaigns performed on Mkn 501 (Abdo et al., 2011d), Mkn 421 (Abdo et al., 2011c), or PKS 2155-304 (Aharonian et al., 2009)). Simultaneous multiwavelength data sets at lower wavelengths may then be used to constrain SED models for a meaningful study of EBL absorption e ff ects at the highest energies. We caution, however, that the overlap between the operations of the LAT and CTA could be extremely limited which leads to a correspondingly lower scientific return in this regard.", "pages": [ 7, 8, 9 ] }, { "title": "4. The physics of extragalactic jets", "content": "Active galactic nuclei are thought to be systems that are powered by the release of gravitational energy. How, where and in which form this energy is released, and especially the physics governing to the formation, acceleration and collimation of relativistic jets and the conversion of jet power into radiative power is poorly understood (for a review of the current status of the field, see, e.g., Bottcher, Harris & Krawczynski, 2012). The observed links (see § 1) between enhanced emission at high photon energies and changes in the polarization properties in the emission region may indicate an important impact of the magnetic field topology and strength on the broadband spectral variability behaviour of jetted AGN and possibly on the intrinsic acceleration of jet knots (e.g., by magnetic driving: Vlahakis & Konigl, 2004). As we will outline below, studies of the SEDs and variability of blazars with CTA, Fermi -LAT, and co-ordinated observations at lower frequencies will be crucial to gain insight into these issues.", "pages": [ 9 ] }, { "title": "4.1. Radiative processes in extragalactic jets", "content": "Depending on the jet's relativistic matter composition two types of emission models have emerged during the last decade. Leptonic models consider relativistic electrons and positrons as the dominating emitting relativistic particle population, while in hadronic 10 emission models the relativistic jet material is composed of relativistic protons and electrons (for a recent review of blazar emission models, see Bottcher, 2010; Reimer, 2012). In both scenarios, cold (i.e., non-relativistic) pairs and / or protons may exist as well, allowing charge neutrality to be fulfilled. There is meanwhile mounting evidence that jets of powerful AGN have to be energetically and dynamically dominated by protons and / or ions (see e.g., Celotti & Ghisellini, 2008), albeit little is known about their spectral distribution (cold, relativistic) and number density with respect to the electrons. In both leptonic and hadronic models, the low-frequency emission is produced as synchrotron radiation of relativistic electrons in magnetic fields in the emission region, which is moving with relativistic speed corresponding to a bulk Lorentz factor Γ along the jet. For ease of computation, the magnetic field is typically assumed to be tangled (i.e., randomly oriented), and the electron distribution is assumed to be isotropic in the co-moving frame of the emission region. In leptonic models, the high-energy emission is produced via Compton upscattering of soft photons o ff the same ultra-relativistic electrons which are producing the synchrotron emission. Both the synchrotron photons produced within the jet (the SSC process: Marscher & Gear, 1985; Maraschi et al., 1992; Bloom & Marscher, 1996), and external photons (the EC process) can serve as target photons for Compton scattering. Possible sources of external seed photons include the accretion disk radiation (e.g., Dermer et al., 1992; Dermer & Schlickeiser, 1993), reprocessed optical - UV emission from circumnuclear material (e.g., the BLR: Sikora et al., 1994; Dermer et al., 1997), infrared emission from warm dust (Bla˙zejowski et al., 2000), or synchrotron emission from other (faster / slower) regions of the jet itself (Georganopoulos & Kazanas, 2003; Ghisellini & Tavecchio, 2008). Relativistic Doppler boosting allows one to choose model parameters in a way that the γγ absorption opacity of the emission region is low throughout most of the high-energy spectrum (i.e., low compactness). However, at the highest photon energies, this e ff ect may make a non-negligible contribution to the formation of the emerging spectrum (Aharonian et al., 2008) and re-process some of the radiated power to lower frequencies. The resulting VHE gammaray cut-o ff or spectral break, and associated MeV - GeV emission features may be revealed by high-resolution, simultaneous Fermi and CTA observations. Hadronic models consider a significant ultra-relativistic proton component in addition to primary ultra-relativistic electrons, to be present in the AGN jet. The charged particles interact with magnetic and photon fields. In heavy jet models the interaction of protons / ions with matter (via e.g., relativistic blast waves (Pohl & Schlickeiser, 2000), star / cloud-jet interaction (Bednarek, 1999; Beall & Bednarek, 1999; Araudo et al., 2010), jet-red giant interaction: (Barkov et al., 2010)) may dominate. However, such models (e.g., Reynoso et al., 2011) do often not predict rapid flux variability. Particle-photon interaction processes in hadronic models include photomeson production, Bethe-Heitler pair production for protons, and inverse Compton scattering of pairs. An inevitable by-product of hadronic interactions is the production of neutrinos. The target photon fields for such processes include internal jet synchrotron photon fields (Mannheim & Biermann, 1992; Mucke & Protheroe, 2001; Mucke et al., 2003), and fields external to the jet such as direct accretion disk radiation (Bednarek & Protheroe, 1999), jet or accretion disk radiation reprocessed in the BLR (Atoyan & Dermer, 2003), the radiation field of a massive star in the vicinity of the jet (Bednarek & Protheroe, 1997) or infrared radiation by warm dust (e.g., Dermer et al., 2012). The secondary particles and photons from interactions of ultra-relativistic hadrons in general initiate synchrotron and / or Compton-supported pair cascades which redistribute the power from very high to lower energies (e.g., Mucke et al., 2003). For high magnetic field strengths, any IC component is in general strongly suppressed, leaving the proton-initiated radiation as the dominating high energy emission component. Figure 7 compares a steady-state leptonic (SSC & EC) fit to a corresponding hadronic fit of the SED of the IBL 3C66A detected in VHE gamma-rays by VERITAS in 2008 (Acciari et al., 2009; Abdo et al., 2011a). Both leptonic and hadronic models provide excellent fits to the simultaneous SEDs obtained during the prominent 2008 October gamma-ray flare, with plausible physical parameters. Because hadronic interactions convert some protons into neutrons 11 via charge exchange, collimated neutron beams may form (Eichler & Wiita, 1978; Atoyan & Dermer, 2003) which can transport a significant portion of the initial energy to large distances from the black hole. When such powerful jets interact with the intergalactic medium, large amounts of their power and momentum are expected to be deposited into the surroundings as huge lobes. The good angular resolution of CTA may permit the imaging of such extended emission, and will provide valuable information about the total power stored in jets, which in turn may constrain jet formation scenarios and jet composition. Because of the suppression of the Compton cross section in the Klein-Nishina regime 12 and e ffi cient radiative (synchrotron + Compton) cooling, leptonic models are typically hard-pressed to explain hard (energy spectral index α /lessorsimilar 1, where F ν ∝ ν -α ) gamma-ray spectra extending to E /greaterorsimilar 1 TeV after correction for γγ absorption by the EBL (e.g., Aharonian et al., 2006). Detailed spectral measurements in the GeV - TeV regime through simultaneous observations by Fermi -LAT and CTA are expected to reveal the signatures of radiative cooling of leptons and / or KleinNishina e ff ects in leptonic models, or of proton-synchrotron emission and ultra-high-energy induced pair cascades in hadronic models. These might therefore distinguish between leptonic and hadronic models. Simultaneous multi-wavelength coverage will be crucial to put meaningful constraints on models. In this context, e.g., Bottcher et al. (2009) have demonstrated that the extension of the gamma-ray emission of the FSRQ 3C 279 into the VHE regime (Albert et al., 2008) poses severe problems for homogeneous, leptonic one-zone models, and may favor hadronic models, or multi-zone models. The lowered energy threshold of CTA compared to current ACTs promises the detection of VHE gamma-ray emission from a larger number of low-frequency peaked blazars (including FSRQs), which will allow for similar studies on a larger sample of LSP blazars. The radiative cooling time scales are generally expected to be much shorter for leptons than for hadrons. Therefore, measurements of rapid variability (e.g., Aharonian et al., 2007; Albert et al., 2007a, see also Fig. 8) might be an indication for a leptonic origin of (at least parts of) the gamma-ray emission from blazars exhibiting variability on subhour time scales. Variability on a few minutes time scale has been observed at VHEs from few blazars both of HSP and LSP type (e.g., PKS 2155-304 (Aharonian et al., 2007), Mkn 501 (Albert et al., 2007a), PKS 1222 + 216(Aleksic et al., 2011)) so far. This implies extremely large bulk Doppler factors if interpreted within a homogeneous emission model, or TeV emitting sub-structures within the jet such as filaments, reconnection zones (Giannios et al., 2009), etc. For example, the spine-sheath picture (Ghisellini et al., 2005) of a jet envisions an ultra-fast spine surrounded by a slower sheath. If the jet points almost towards the observer, radiation from the strongly beamed fast spine dominates the observed spectrum, while the radiation from the sheath contributes only weakly. In AGN where the jet is more inclined to the sight line the spine appears as a dim source while the radiation from the slower sheath becomes dominant. In order to test this behaviour a larger sample of rapidly varying sources, both blazars and radio galaxies, at VHEs is required. With current technology, only the brightest of such sources can be detected, and only in extreme flaring states. The increased sensitivity of CTA compared to present-generation ACT facilities will allow for the extension of the study of rapid gamma-ray variability to a large sample of sources and to more quiescent states. Variability information in addition to high resolution spectra is particularly important for unambiguously constraining the parameter space im emission models since in many cases (see, e.g., Fig. 7), pure snap-shot SED modeling is unable to distinguish between a leptonic and a hadronic origin of the gamma-ray emission.", "pages": [ 9, 10, 11, 12 ] }, { "title": "4.2. Probing particle acceleration using CTA", "content": "Both the SED shape and multi-wavelength variability patterns in blazar emission can provide constraints on the mode of particle acceleration in the jets of AGN. The shape of the high-energy end of the particle spectrum - which will be directly reflected in the shape of the high-energy end of the gamma-ray emission - will provide valuable information about the competition between radiative (and possibly adiabatic) losses, escape, and energy gain at those energies (e.g., Protheroe & Stanev, 1999). The decreased energy threshold and improved sensitivity of CTA over current ACTs will enable detailed studies of the shape of the high-energy cut-o ff s of blazar spectra (including LSP blazars) and, in particular, trace the cuto ff in sources not yet detected at VHEs. Di ff erent particle acceleration scenarios (e.g., di ff usive shock acceleration at relativistic shocks, first-order Fermi acceleration, perpendicular vs. oblique shocks, di ff usive acceleration in shear layers) and di ff erent magnetic field topologies predict characteristically di ff erent spectral indices in the resulting particle spectra (e.g., Ostrowski & Bednarz, 2002; Stawarz & Ostrowski, 2002; Ellison & Double, 2004; Stecker et al., 2007, see also Fig. 9). These will be directly reflected in the spectral indices of the non-thermal synchrotron and gamma-ray emission of blazars. E.g., some HBLs at low fluxes possess very hard photon spectral indices (see Fig. 3) in the LAT energy range, implying hard particle spectra of the accelerated particle population. CTA might probe the required acceleration conditions in a systematic way. Simultaneous multiwavelength observations, including at the highest energies, will be helpful to probe potential mis-matches between the low-energy (synchrotron) and high-energy (gamma-ray) SEDs. In leptonic models, such spectral-index mis-matches typically require multi-component gamma-ray emission scenarios, if they can be re-conciled with these models at all. In hadronic models, they might be explained through di ff erent acceleration modes (and hence, di ff erent particle spectral indices) for electrons and protons. In addition to simultaneous snap-shot SEDs, spectral variability can provide crucial insight into the particle acceleration and cooling mechanisms in AGN jets (e.g., Kirk et al., 1998; Chiaberge & Ghisellini, 1999; Li & Kusunose, 2000; Bottcher & Chiang, 2002). Detailed measurements of spectral variability have so far been restricted to lowerenergy observations (e.g., X-rays: Takahashi et al., 1996), or to the brightest gamma-ray AGN only (e.g., 3C 454.3 at LAT-energies: Abdo et al., 2011b). The improved sensitivity of CTA in the > 100 GeV regime might enable the study of precision spectral variability and persistent long-term variability patterns in this energy range for a large sample of sources. In particular, this will provide a probe of the dynamics of the highest-energy particles in LSP blazars in which the high-energy end of the synchrotron component is often not observationally accessible because it is (a) located in the UV / soft X-ray regime, which is notoriously di ffi cult to observe, and (b) overlapping with (and often overwhelmed by) the low-energy end of the high-energy emission.", "pages": [ 12, 13 ] }, { "title": "5. Concluding remarks", "content": "This surely incomplete list of topics discussed above reveals the potential of CTA for significant progress in the field of AGN research. Improvements in sensitivity and energy coverage will allow for the study of a much larger population of AGN, although we caution that the here important GeV energy range as is currently provided by the Fermi -LAT instrument may be available at the time of CTA operations only to an extremely limited extent. This will enable to tackle a large range of topics from population studies and questions of cosmological evolution of AGN via studies of the formation and composition of extragalactic jets and the microphysics of the production of high energy emission in relativistic jets, to studies of the Extragalactic Background Light, which will shed light on the broader issues of cosmological galaxy evolution and structure formation. Most exciting, as CTA will enlarge the dynamical flux range and explore the high-redshift universe at VHEs, unexpected, possibly surprising, phenomena may challenge current theoretical concepts, and trigger to deepen our understanding of the extragalactic sky. This review might provide some insight into possible ways that observations by CTA - coordinated with simultaneous observations at other wavelengths - might lead to progress in the study of some of the most pressing questions of the VHE sky.", "pages": [ 13, 14 ] }, { "title": "Acknowledgements", "content": "We like to thank Chuck Dermer, Benoit Lott, Marco Ajello and Paolo Giommi for providing excellent comments on this work which improved this manuscript. MB acknowledges support from NASA through Astrophysics Theory Program grant NNX10AC79G and Fermi Guest Investigator Grants NNX10AO49G and NNX11AO20G. AR acknowledges support by Marie Curie IRG grant 248037 within the FP7 Program.", "pages": [ 14 ] }, { "title": "References", "content": "Abdo, A. A., et al., 2010a, ApJ, 708, 1310 Abdo, A. A., et al., 2010b, ApJ, 715, 429 Abdo, A. A., et al., 2010c, ApJ, 716, 30 Abdo, A. A., et al., 2010d, ApJ, 723, 1082 Abdo, A. A. et al. 2010e, ApJ, 720, 435 Abdo, A. A. et al. 2010f, Nature, 463, 919 Abdo, A. A., et al., 2011a, ApJ, 726, 43 Abdo, A. A., et al., 2011b, ApJ, 733, 26 Abdo, A. A. et al. 2011c, ApJ, 736, 131 Abdo, A. A. et al. 2011d, ApJ, 727, 129 Acciari, V. A., et al., 2009, ApJ, 693, L104 Ackermann, M., et al., 2011, ApJ, 743, 171 Ackermann, M., et al., 2011, ApJ, 747, 104 Aharonian, F. A., et al., 2006, Nature, vol. 440, Issue 7087, p. 1018 Aharonian, F. A., et al., 2007, ApJ, 664, L71 Aharonian, F. A., Khangulyan, D., & Costamante, L., 2008, MNRAS, 387, 1206 Aharonian, F. et al. 2009, ApJL, 696, L150 Albert, J., et al., 2008, Science, vol. 320, Issue 5584, p. 1752 Albert, J., et al., 2007, ApJ, 669, 862 Albert, J., et al., 2007, ApJ, 666, L17 Aleksic, J., et al., 2010, ApJ, 723, L207 Aleksic, J., Antonelli, L.A., Antoranz, P. et al., 2011, ApJ, 723, L207 Araudo, A. T., Bosch-Ramon, V., Romero, G. E., 2010, A&A, 522, 97 Atoyan, A. M., & Dermer, C. D., 2003, ApJ, 586, 79 Baring, M. G., 2009, AIP Conf. Proc., 1133, 294 Barkov, M. V., Aharonian, F. A., Bosch-Ramon, V., 2010, ApJ, 724, 1517 Beall, J.H. & Bednarek, W., 1999, ApJ, 510, 188 Bednarek, W., & Protheroe, R. J., 1997, MNRAS, 287, 9 Bednarek, W., & Protheroe, R. J., 1999, MNRAS, 302, 373 Bednarek, W., 1999, Plasma Turbulence and Energetic Particles in Astrophysics, Proceedings of the International Conference, Cracow (Poland), 5-10 September 1999, Eds.: Michal Ostrowski, Reinhard Schlickeiser, Obserwatorium Astronomiczne, Uniwersytet Jagiellonski, Cracow 1999, p. 360 Blandford, R. D., & Znajek, R. L., 1977, MNRAS, 179, 433 Blandford, R. D., & Payne, D. G., 1982, MNRAS, 199, 883 Bla˙zejowski, M., et al., 2000, ApJ, 545, 107 Bloom, S. D., & Marscher, A. P., 1996, ApJ, 461, 657 Bottcher, M., & Chiang, J., 2002, ApJ, 581, 127 Bottcher, M., & Dermer, C. D., 2002, ApJ, 564, 86 Bottcher, M., et al., 2007, ApJ, 670, 968 Bottcher, M., Reimer, A., & Marscher, A. P., 2009, ApJ, 703,1168 Bottcher, M., 2010, in proc. 'Fermi Meets Jansky', Bonn, Germany, 2010, Eds.: T. Savolainen, E. Ros, R. W. Porcas, & J. A. Zensus; p. 41; arXiv:1006.5048 Bottcher, M., Harris, D. E., & Krawczynski, H. (Eds.), 2012: 'Relativistic Jets from Active Galactic Nuclei', Wiley-VCH Chiaberge M., Celotti A., Capetti A., Ghisellini G., 2000, A&A, 358, 104 Capetti, A., Raiteri, C.M. & Buttiglione, S. 2010, A&A, 516, 59 Cavaliere, A., & D'Elia, V., 2002, ApJ, 571, 226 Celotti, A., & Ghisellini, G., 2008, MNRAS, 385, 283 Chiaberge, M., & Ghisellini, G., 1999, MNRAS, 306, 551 De Angelis, A., Roncadelli, M., & Mansutti, O., 2007, Phys. Rev. D., 76, 121301 Dermer, C. D., Schlickeiser, R., & Mastichiadis, A., 1992, A&A, 256, L27 Dermer, C. D., & Schlickeiser, R., 1993, ApJ, 416, 458 Jorstad, S., Marscher, A., D'Arcangelo, F., Harrison, B., 2009, Fermi Symposium 2009, eConf Proceedings C091122 Liu, H. T., et al., ApJ, 688, 148 Mannheim, K., & Biermann, P. L., 1992, A&A, 253, L21 Maraschi, L., Celotti, A., & Ghisellini, G., 1992, ApJ, 397, L5 Marscher, A. P., & Gear, W. K., 1985, ApJ, 298, 114 Mucke, A., et al., 2003, Astropart. Phys., 18, 593 Pian, E., Urry, C. M., Maraschi, L., et al. 1999, ApJ, 521, 112 Pohl, M., & Schlickeiser, R., 2000, A&A, 354, 395 Ostrowski, M., & Bednarz, J., 2002, A&A, 394, 1141 Padovani, P., 2007, ApSS, 309, 63 Poutanen, J., & Stern, B., 2010, ApJ, 717, L118 Prandini E., Mariotti M. & Tavecchio F., 2011, in press; arXiv1111.0913 Protheroe, R. J. & Stanev, T., 1999, Astropart. Phys., 10, 185 Puchwein E., Pfrommer C., Springel V., et al. 2011, MNRAS, submitted; arXiv1107.3837 Rawling, S., & Saunders, R., 1991, Nature, 349, 138 Rau, A., Schady, P., Greiner, J., et al., 2012, A&A, 538, 26 Reimer, A., 2007, ApJ, 665, 1023 Reimer, A., 2012, Journal of Physics: Conference Series, 355, 012011 Reynoso, M. M., Medina, M. C., Romero, G. E., 2011, A&A, 531, 30 Romani, R., 2006, ApJ, 132, 1959 Roustazadeh, P., & Bottcher, M., 2010, ApJ, 717, 468 Roustazadeh, P., & Bottcher, M., 2011, ApJ, 728, 134 Sikora, M., Begelman, M., & Rees, M., 1994, ApJ, 421, 153 Simet, M., Hooper, D., & Serpico, P. D., 2008, Phys. Rev. D., 77, 063001 Sitarek, J., & Bednarek, W., 2008, MNRAS, 391, 624 Stawarz, L., & Ostrowski,M., 2002, ApJ, 578, 763 Stecker, F. W., Malkan, M. A., & Scully, S. T., 2006, ApJ, 648, 774 Stecker, F. W., Baring, M. G., & Summerlin, E. J., 2007, ApJ, 667, L29 Takahashi, T., et al., 1996, ApJ, 470, L89 Urry, C. M., & Padovani, P., 1995, PASP, 107, 803 Vlahakis, N. & Konigl, A., 2004, ApJ, 605, 656", "pages": [ 14, 15 ] } ]

[ { "title": "T. Bretz and M. Ribordy", "content": "High Energy Physics Laboratory, EPFL, CH-1015 Lausanne, Switzerland", "pages": [ 1 ] }, { "title": "Abstract", "content": "This paper discusses the construction of high-performance ground-based gamma-ray Cherenkov telescopes with a Davies-Cotton reflector. For the design of such telescopes, usually physics constrains the field-of-view, while the photo-sensor size is defined by limited options. Including the effect of light-concentrators in front of the photo sensor, it is demonstrated that these constraints are enough to mutually constrain all other design parameters. The dependability of the various design parameters naturally arises once a relationship between the value of the point-spread functions at the edge of the field-of-view and the pixel field-of-view is introduced. To be able to include this constraint into a system of equations, an analytical description for the point-spread function of a tessellated Davies-Cotton reflector is derived from Taylor developments and ray-tracing simulations. Including higher order terms renders the result precise on the percent level. Design curves are provided within the typical phase space of Cherenkov telescopes. The impact of all design parameters on the overall design is discussed. Allowing an immediate comparison of several options with identical physics performance allows the determination of the most cost efficient solution. Emphasize is given on the possible application of solid light concentrators with their typically about two times better concentration allowing the use of small photo sensors such as Geiger-mode avalanche photo diodes. This is discussed in more details in the context of possible design options for the Cherenkov Telescope Array. In particular, a solution for a 60 mm 2 photo sensor with hollow cone is compared to a 36 mm 2 with solid cone. Keywords: TeV Cherenkov astronomy, Davies-Cotton design parameters, photo-sensors, Winston cone", "pages": [ 1 ] }, { "title": "1. Introduction", "content": "The next generation Cherenkov Telescope Array (CTA) will be constituted of three types of imaging atmospheric Cherenkov telescopes: small-, medium- and large-sized telescopes. The array will be dedicated to the observation of the high energy gamma-ray sky with unprecedented sensitivity [5] over a broad range of energies (0.01 TeV glyph[lessorsimilar] E GLYPH<13> glyph[lessorsimilar] 100 TeV). This instrument will enable astronomers and astro-particle physicists to refine models of gamma-ray sources and underlying non-thermal mechanism at work, to study the origin and the composition of the cosmic rays up to the knee region, question the nature of the dark matter, etc. These kind of telescopes detect Cherenkov light emitted along developing showers in the atmosphere nearly uniformly illuminating the ground over an area of about 50'000 m 2 . Shower maxima of a gamma-ray induced electromagnetic shower occur at altitudes comprised between GLYPH<24> 7 km and GLYPH<24> 12 km, for a primary energy ranging from GLYPH<24> 100 GeV to GLYPH<24> 100 TeV. The photon density on the ground depends on the energy of the gamma ray, its incidence direction and the distance from the resulting shower axis. The displacement of the image centroid in the telescope camera is a function of the angular distance of the shower core, defined by shower impact distance and altitude of the shower and is typically about 1.5° for an energy of 1 TeV and an impact parameter of 150 m, and 3° to 3.5° for energies around 100 TeV with an impact parameter of 300 m. As very high energy showers penetrate deeply in the atmosphere and generate a large amount of light, they can be observed at a relatively large distance from the main light pool already with relatively small reflectors translating into the necessity of a deployment of telescopes with large field-of-view for the exploration of very high energy showers. In general, the required telescope field-of view in order to record contained shower images ranges between 3° and 10° depending on the energy range of interest. Apromising design for wide field observations is the DaviesCotton telescope [17]. Its advantage is a point-spread function enabling a larger field-of-view than a parabolic design. The Schwarzschild-Coud'e design (e.g. [16]) is not considered further, as it presents technical and cost challenges compared to a conservative CTA proposal using a prime optics design (see also the comments in section 2.1). It is demonstrated that once the field-of-view of the camera pixels and of the whole instrument together with the photosensor technology is fixed by means of physics arguments, only one parameter is left free. This parameter can be fixed as well, relating the worst optical resolution in the field-of-view, i.e. at the edge of the field-of-view, with the size of a pixel. Having a reasonable cost model at hand, even the most cost-efficient single telescope or telescope design operating in an array can be derived. Knowing the anticipated angular size of a pixel from physics constraints, a requirement on the point-spread function of the telescope can be determined. If the point-spread function is identified by the root-mean-square of the light distribution, requiring a pixel diameter four times the root-mean-square ensures that most of the light from a point source at infinity is concentrated in a single pixel at the edge of the field-of-view. 1 If simulations show that such a containment is not necessary in terms of angular resolution of the shower origin or background suppression, also a smaller value like twice the rootmean-square can be considered, c.f. [1]. If this requirement is not optimized or not met, either the physics outcome is worsened or the camera has more pixels than necessary and will not be cost-efficient. The CTA array layout will consist of large size telescopes (LST, primary mirror diameter GLYPH<24> 24 m) in the center, and successively surrounded with an increasing number of mediumsized (MST, GLYPH<24> 12 m) and small-sized (SST, GLYPH<24> 6 m) telescopes in order to instrument a ground surface area comprised between 4 km 2 and 10 km 2 . Telescope spacings, sizes and field-of-views reflect the energy range to be explored by a certain type of telescope. The LST sub-array will be primarily focusing on the observation of the high energy gamma-ray sky with great precision below about 100 GeV, the inter-telescope spacing will be small, about 60 m and the single telescope field-of-view limited to about 3° - 4°. The SST sub-array will conversely be optimized for multi-TeV observations, up to around 100 TeV allowing for large spacing of up to 300 m - 500 m with relatively modest reflector size. The camera pixel field-of-view of these different telescope types will span between about 0.08° - 0.1° (LST) up to 0.2° - 0.3° (SST), linked to the different shower intensity with its intrinsic fluctuations and the concurrent necessity of keeping the number of camera pixels reasonably low. Consequently, this study will focus on reflector diameters in the range of a meter to about 30 m and on pixel field-of-views in the range between GLYPH<24> 0.08° and GLYPH<24> 0.3°. and is extended to off-axis angles of the incoming light up to more than 5°. In the following, first a description is derived for the relations between the existing design parameters of a Cherenkov telescopes, e.g. focal length and reflector diameter. Then, by including a semi-analytical treatment for the optical quality of a generalized Davies-Cotton reflector, this description becomes applicable for the design of real telescopes. At the end, the influence of the variation of the parameters on the optimized design is discussed.", "pages": [ 1, 2 ] }, { "title": "2. Optimization", "content": "The light collection is one of the most important parameter of a Cherenkov telescope and can be improved by, e.g., an increase of the photo-detection efficiency of the photo-sensors or a rescaling of the system, i.e., a corresponding rescaling of the reflective area and photo-sensor size. As current technology (photo-multiplier tubes, hybrid photo-detectors or silicon photo-multipliers) only offer a limited choice of photo-sensor sizes, the Cherenkov telescope design parameter phase space is reduced. Given a particular photo-sensor type, the addition of a light concentrator in front of the photo-sensor is the only way to increase the light collection efficiency. In the following, it will be shown that the relationship between the characteristic parameters of an optimal telescope design (pixel field-of-view and linear size, reflector diameter and focal distance) is fully constrained, once the technological requirement (photo-sensor size and the light-guide material) and the physics requirements (the pixel angular size and the fieldof-view) are frozen. This result is obtained by imposing restrictions on the value of the point-spread function at the edge of the field-of-view in order to keep adequate image quality and thus analysis potential.", "pages": [ 2 ] }, { "title": "2.1. Light concentrators", "content": "The theorem of Liouville states that the maximum concentration theoretically achievable is defined by maintaining the phase space, i.e. the product of solid angle, defined by the incoming light rays directions, light ray momentum squared and the surface area crossed by the light rays. The theorem of Liouville is applicable to the case of a Winston cone with entrance area A placed in front of a photo-detector of area A 0 , corresponding to the exit area of the cone. A is defined provided A 0 , the solid angle GLYPH<10> defined by the light rays entering the cone and the solid angle GLYPH<10> 0 defined by the light rays leaving the cone. Winston has shown in [6] that the maximum concentration factor for a rotationally symmetric light concentrator is where n and n 0 denote the refractive index of the media in front of and inside the optical system with n GLYPH<25> 1 in air. GLYPH<30> corresponds to the maximum angle at which a light ray enters the system related to the solid angle GLYPH<10> . If the system is not axisymmetric or the angular acceptance of the photo detector is smaller than GLYPH<10> 0 = 2 GLYPH<25> (as assumed in Eq. 1), C max has to be adapted accordingly. Besides the increase of the light collection area, the use of cones enables a partial screening of the night-sky light pollution corresponding to GLYPH<30> being larger than the angle of light rays coming from the edge of the reflector. Simulations [7] and recent concentration efficiency measurements [8] of solid cones ( n 0 GLYPH<25> 1.4) designed for the FACT camera [9] demonstrated that their shape is nearly ideal, that the concentration factor reaches a value close to C max and the geometric loss is only of the order of a few percent excluding absorption loss 2 . In the case of hollow cones, Fresnel reflection losses have to be considered at the surface of the photo-sensor. If the camera is sealed with a protective window, which is usually the case, also losses at the window surface need to be taken into account. By choosing a material for the cones and the protective window with a similar refractive index than the material of the photo-sensor light entrance, these losses can be omitted. Combined with the almost perfect reflectivity of solid cones due to total reflection (limited only by the surface roughness), solid cone usually outperform hollow cone. The concentration factor achieved with the Winston cones is fundamentally similar to the size reduction of the focal plane in a Schwarzschild-Coud'e design and linked to the conservation of the space-momentum phase space according to the Liouville theorem: the conversion of the spatial into momentum phase space, by means of Winston cones or secondary optics. While cones reduce the acceptance of the incoming light rays from a large area at the entrance to a large angular acceptance and small area at the cone exit, the secondary mirror optics leads to a similar spatial compression and angular widening of the light rays at the photo-sensor and thus a reduced plate-scale. While cones are non-imaging devices, the secondary optics is imaging. Hence, in the Schwarzschild-Coud'e design, it is possible to attain excellent optical resolution, in terms of Cherenkov telescope requirements, with a field-of-view as large as 15°. Both designs enable compression of the photo sensitive area by factors larger than ten w.r.t. their primary optics design.", "pages": [ 2, 3 ] }, { "title": "2.2. Connection to the optical system", "content": "In the case of a Cherenkov telescope, the light entering the cone comes from a reflector visible under a maximum angle GLYPH<30> 3 . The opening angle of the light at the entry of the cone is therefore well defined by the properties of the optical system, i.e. by the diameter of the reflector D and the focal length F , f = F = D . Combining this with Eq. 1 yields For instance, taking the FACT values, n 0 = 1 : 4 and f = 1 : 4, we obtain the theoretically maximal achievable concentration factor C max = 17 : 3, i.e. the linear size of the entrance area can ideally be larger than four times the linear photo-sensor size.", "pages": [ 3 ] }, { "title": "2.3. The optical system", "content": "In addition, the optical system defines the zoom factor or plate-scale, i.e. the field-of-view corresponding to a physical area in the focal plane. The correspondence between the angular size # and the linear size GLYPH<14> on the focal plane is or in the limit of small # , Cameras in Cherenkov telescopes are pixelized due to the use of photo-detectors. To increase the light collection efficiency further, and to maintain symmetry, these pixels are usually aligned on a hexagonal grid, i.e. in closed package geometry. In recent years, MAGIC has exploited the photon arrival time extracted from the measured pulse and demonstrated significant improvements in the sensitivity [10]. The technique, taking into account the change of the arrival time between neighboring pixels, performs best, if all neighbors are at an identical distance from the central pixel. Consequently, the ideal shape of a pixel is hexagonal. The distance GLYPH<14> on the camera surface is the distance between two parallel sides of a hexagon, its area is Combining with Eq. 3, the plate-scale formula Eq. 4 and Eq. 6, is obtained, which translates the close relationship between the pixel field-of-view # , the focal length F and the reflector diameter D , once the technological parameters fixed: photo detector size A 0 and light concentrator material n 0 . Defining a constant related to these properties and rewriting Eq. 7 (tan # GLYPH<25> # for # GLYPH<28> 1°) as it is immediately apparent that the focal length F of the system is a direct consequence of the pixel field-of-view and the reflector diameter, if the properties of the photon detector and the material of the cones are known. For typical Cherenkov telescopes, F = D is between unity and two. Below unity the resolution becomes too coarse and above two, not only the number of pixels and hence the price of a camera becomes too high, but also the camera holding structure becomes mechanically complex and hence disproportionally expensive. This constraint on F = D applied to Eq. 9 yields Precisely the choice of F and D defines the optical quality of a mirror system. At the same time, the size of a single pixel defines a natural constraint on the optical quality of a system, i.e., F = D should be chosen such that the point-spread function at the edge of the camera is within a limit well defined by the pixel's field-of-view.", "pages": [ 3 ] }, { "title": "2.4. Optical quality", "content": "The light collection area is important for a Cherenkov telescope and typical reflector sizes range from a few to ten or twenty meters. However, with the current technology, it is not possible to produce large mirrors with the requested quality at a reasonable cost. Furthermore, optical systems compiled from a single mirror suffer large aberration effects at large off-axis angles, while a wide field-of-view is necessary for the observation of multi-TeV showers up to large impact parameters, as well as for extended sources. Therefore, segmented mirrors are in use. The layout providing the best optical quality for segments of identical focal length is the so-called Davies-Cotton layout [15], where the single spherical mirrors are located on a sphere with radius F and focused to a point at 2 F . The relevant quantity which influences the on-axis and offaxis optical quality is the focal ratio F = D . The optical quality improves with larger values. This scale invariance statement is true only as long as the optical quality of a single mirror can be neglected against the optical quality of the whole system, which is generally the case at the edge of the camera. To be able to constrain the optical point-spread function, a relation between the tessellation, the focal ratio and the resulting point-spread function is needed for a given maximum inclination angle of the light, i.e. at the edge of the field-of-view of the camera. In Appendix A.1, a formalism describing the point-spread function of an ideal Davies-Cotton reflector, i.e. a reflector with infinite tessellation is presented. The point-spread function is described by the root-mean-square of the light distribution. With the help of ray-tracing simulation, a reasonably good description of a real tessellated Davies-Cotton reflector is derived from this analytical approach in Appendix A.2. Including the correction factor which describes the deviation of the analytical approach from the simulations, a good description of the point-spread function is obtained. It is shown that the point-spread function P of an ideal Davies-Cotton can be expanded into a polynomial in GLYPH<11> i GLYPH<1> f GLYPH<0> j . Given the incident angle GLYPH<11> GLYPH<28> 1 of the incoming ray and f in the range between one and two, the polynomial is hypothesized into the more simpler form with coefficients c 0 and c 1 and N the tessellation number as described in Appendix A.2. This parametrization is found to match the ray-tracing simulation without loss of precision. The coefficient c 0 can directly be deduced as the result of Eqs. A.11 at f = 1 and c 1 is derived by a fit. An example for the coefficients c 0 and c 1 for selected GLYPH<11> is shown in Fig. 1.", "pages": [ 3, 4 ] }, { "title": "2.5. Result", "content": "As discussed in the introduction, it is required that the pointspread function is small compared to the pixel field-of-view at the edge of the field-of-view, so that the light of a point source is well contained in one pixel. Defining the ratio r between both, this requirement can be expressed as Combined with Eq. 11, the focal ratio f can now be expressed as Including this in Eq. 9 yields with a correction factor g defined as The absolute focal length F can now be calculated using Eq. 9. To deduce the effective reflective area from Eq. 14, the shadow of the camera on the reflector has to be taken into account. To calculate the fraction of the reflector shadowed, the ratio of their areas is calculated. Conversion of GLYPH<11> from an angle to a length yields approximately R = F GLYPH<1> GLYPH<11> for small values of GLYPH<11> ( GLYPH<28> 10°). Expressing the focal length by Eq. 13 the fraction of the camera shadow on the reflector is derived. If a real camera housing is significantly larger than the photo sensitive area itself, a correction factor should be included. Now the effective light collection area of the optical system can be deduced as If real setups should be compared, like e.g. Davies-Cotton and Schwarzschild-Coud'e, also other sources of light-losses must be included, such as geometrical efficiency of the cones (light-loss at the edge of the mirror), total mirror reflectivity, cone transmission or reflection losses, or photo detection efficiency.", "pages": [ 4 ] }, { "title": "2.6. Discussion", "content": "The relation given in Eq. 14 includes several parameters which are subject to change. For simplicity, a standard setup has been defined to which altered setups are compared. Silicon photo-detectors are a recent and very promising technology. Therefore, a silicon photo-detector with a sensitive area of 36 mm 2 is chosen as a benchmark device. Such devices are already commercially available with acceptable properties. Although their sensitive area is still rather small compared to photo-multipliers, by increasing their light-collection with solid light concentrators, their light-collection area becomes reasonably large. Such light-concentrators still maintain a reasonable weight and length in term of absorption. Typical Plexiglas materials have a refractive indices of the order of n 0 =1.4 and are used hereafter as a reference. As the light-collection area of a telescopes scales directly with the photo-sensitive area, the most obvious use of small photo sensors is a small telescope sensitive mostly to high energetic showers. At high energies, the collection area of a telescope array is of prime importance due to rapidly decreasing ] ] fluxes. Due to the bright light-pool of high energy showers, telescopes with relatively small reflectors can be operated with large spacing of, e.g., 400 m or more, c.f. [2]. Since such spacings demand a large camera field-of-view, a field-of-view of 9° diameter is chosen as a reference. A typical reflector for a Cherenkov telescope with DaviesCotton layout enables the manufacturing of a primary reflector tessellated into spherical mirror of identical focal lengths. From the scaling with the tessellation number as derived in A.11, it can be concluded that a layout with only three mirrors on the diagonal ( N =3) has still a significantly worse optical quality than a reflector with five mirrors on the diagonal. Although the point-spread function at the center of the camera is clearly dominated by the mirror size, the relative influence almost vanishes at higher off-axis angles. Since the solution with N =3 still shows a degradation of more than 10% compared to the solution with N =5 even at the highest simulated off-axis angles, it is discarded. On the other hand, a further increase of the tessellation number (individual mirror size over primary reflector diameter) does not significantly improve the optical quality. Consequently, choosing N =5 is a good compromise and already close to the optimum achievable. Comparable results were obtained in [4] although using a third order approximation overestimating the optical quality. It must be noted that the simulation does not take the pointspread function of the individual mirrors nor any possible misalignment into account which must be added quadratically to the result. However, for the solutions discussed here this can be neglected, c.f. [3]. In general, alignment errors can be kept minimal and individual mirrors can be machined with a pointspread function small compared to the point-spread function at the edge of the camera. On average, all Davies-Cotton designs with a reasonable F = D have a root-mean-square of the light distribution in the tangential direction about two times larger than in the sagittal direction. Ideally, the sagittal root-mean-square at the edge of the camera should fit a fourth of the pixel's field-of-view. This ensures that in the sagittal direction 95% of the light is contained within one pixel diameter and roughly 68% in the tangential direction. However, since the point-spread function is not Gaussian and has long tails in tangential direction exact numbers for the light content might slightly differ. For convenience, all following plots show dots for F = D = { 1, 1.25, 1.5, 1.75, 2 } . Fig. 2 shows the reflective area versus the pixel's field-ofview for comparison in the standard case with and without shadowing for different camera field-of-views. Since the effect is comparably small and the mirror diameter is more expressive, in the following all plots show the mirror diameter rather than the reflective surface in the non-obstructed case. The effects of changing different input parameters w.r.t. to the previously described benchmark configuration are shown in Fig. 3 and discussed below. Changing the camera field-of-view (Fig. 3, top plot). Changing the camera's field-of-view basically shifts the valid range along the line, i.e. the range corresponding to F = D =[1.0, 2.0]. That means that it is possible to build telescopes identical in optical quality, pixel's field-of-view and mirror diameter, but different field-of-view resulting simply in a different focal length of the system. In short: Changing the field-of-view only changes the focal length. Changing the optical quality (Fig. 3, middle plots). A change in the requirement on the optical quality r directly influences F = D , and therefore also shifts the range of reasonable F = D almost linearly in # (left plot). Changing the tessellation (right plot) is like changing the requirement on the optical quality. While the difference in optical quality between a Davies-Cotton layout with three mirrors on the diagonal and five mirrors is still significant, all other layouts give identical results within a few percent. In short: Any tessellation number GLYPH<21> 5 gives similar results. Changing the requirement on the optical quality only changes the focal length. Changing the photo sensitive area (Fig. 3, bottom left plot). Since the constant k is directly proportional to the size A 0 of the photon detector, the mirror area is directly proportional to the size of the photo sensor. If the size of the photon sensor is limited, a simple way to increase the field-of-view of a single pixel is to sum the signal of several photon counters to a single signal. To maintain a hexagonal, i.e. most symmetric layout, summing the signal of three, four or seven photon sensors seems appropriate. In short: Assuming an optimized light-concentrator, the photo sensor's physical size defines the scale of the system. Changing the light concentrator (Fig. 3, bottom right plot). Another way to increase the reflective area is an increase of the refractive index of the light concentrator entering quadratically. Using solid cones made from a Plexiglas material with a typical refractive index in the order of 1.4 allows to increase the achievable reflective area by a factor of two compared to hollow cones. Since the length of a typical light concentrator for an exit of 1 mm diameter is in the order of 3 mm - 4 mm, weight and light-attenuation, which is dependent on the length of the material crossed, will define a natural limit on the sensor size for which a solid cone is still efficient. For comparison reasons not only solid ( n 0 =1.4) cones but also intentionally less efficient hollow cones ( n =1.0) are shown. Non-optimum hollow cones are typically used in current Cherenkov telescopes, in which the sensitive area of standard photo-detectors (PMTs) is not a limiting factor. In short: Increasing the refractive index, quadratically increases the reflective area of the system. Another interesting aspect for the final performance of a telescope is the collection of background photons from the diffuse night-sky background. Here, Eq. 17 leads to an interesting conclusion. Since the rate of the night-sky background photons per channel scales with the effective reflective area and the solid angle corresponding to the field-of-view of the pixels, the nightsky background rate R is proportional to yielding For the range of F = D = [1 ; 2], Eq.13 and Eq. 15 yield a correction factor g between 1.0625 and 1.25. With them, Eq. 19 can be transformed into R / c = k , with Eq. 8 into R / c 0 A 0 n 0 2 . It is immediately apparent that the night-sky background rate scales with the physical entry area of the pixel. Assuming only reasonable camera field-of-views between 3° and 13° diameter, the coefficient c 0 = 2 = p 3 c is between 9.6 and 14.2. This can be interpreted such that the night-sky background rate per pixel can be considered constant within GLYPH<6> 10% in the first order along the lines of an optimized telescope. The dependence of c 0 on F = D and the camera field-of-view is shown in Fig. 4. Quantitatively the night-sky background rate r NSB is given by where GLYPH<31> ( GLYPH<23> ) and f ( GLYPH<23> ) are the photo-sensor's photo-detection efficiency and the night-sky background intensity, respectively. For silicon photo detectors (MPPC [12]), as used in the FACT camera, and the night-sky background at La Palma [13], r NSB is about 150 MHz (much larger than the device dark count rate) given a reflective area of 10 m 2 and a pixel field-of-view of 0.2°. If a cutoff in the photo-detection efficiency is introduced at 650 nm (PMT-like behavior), this can be further reduced. In general, the night-sky background rate is not a main problem in Cherenkov astronomy, as the combined trigger requirements of signal among nearest pixel neighbors and within a short time lead to its very efficient suppression. General considerations. Existing telescopes are usually underdesigned, i.e. the photo-detectors are larger than necessary or the light-concentrators do not reach the maximum possible concentration. Onthe contrary, currently so-called Silicon Photo-multipliers have proven their potential in Cherenkov telescopes [11]. These silicon based photo-detectors usually have a very limited area, but used in an optimized setup, their effective physical light collection area, i.e. entry of the light concentrator, can be much larger. If prices of photo-sensors are compared, this has to be taken into account. Not the price per mm 2 physical sensitive area, but the price per cone entrance area or field-of-view, has to be considered. If cheap enough, the signal of several photodetectors, equipped with individual light concentrators, could even be summed. For a comparison, in terms of effective reflective area, the transmission losses of solid cones and their gain from avoiding Fresnel reflection has to be taken into account, as well as the reflection losses of hollow cones. A note on timing. For an ideal Davies-Cotton reflector, the arrival time distribution of an instantaneous parallel beam flash is practically flat. More precisely, it is linearly decreasing, but looks flat on the small interval. The number of photons in the arrival time interval [ T ; T + GLYPH<14> T ] is N ( T ; GLYPH<14> T ) / T . Its width GLYPH<14> T is given by D = c GLYPH<1> fDC (0 : 5 ; 0), where c is the speed of light. The interval is of the order 1.1 ns for a 4 m class reflector ( F = D GLYPH<24> 1 : 5, up to slightly less than 4.5 ns for a 12 m reflector considering F = D GLYPH<24> 1 : 2. This short time spread is not a problem for the observation of showers with a small size telescope as it is still small compared to the Cherenkov light flash duration. For medium and large size telescopes, a slightly different mirror arrangement should be chosen if time spread matters. By a mirror arrangement, intermediate between a spherical (Davies-Cotton) and a parabolic design, the time spread can considerably be improved, maintaining the point-spread function almost completely. While the point-spread function is dominated by the majority of the mirrors, i.e. outermost mirrors, the time spread is dominated by the ones with the largest DeltaT mirrors, i.e. innermost mirrors. Consequently, moving the innermost mirrors closer to a parabola immediately improves the time-spread while the effect on the point-spread function is rather limited. Ideally, mirrors on a parabola with adapted focal lengths are used, but might be a cost issue. With adapted focal lengths, all mirrors are placed at correct focal distance, so that, a similar point-spread function than for the Davies-Cotton arrangement can be expected. Remarks about CTA. Recent results of FACT [11] show that a reflector in the order of 3.5 m diameter can give already reasonable physics results with current analysis and detector technology. Therefore, a 4 m diameter reflector for SST is assumed. For physics reason, the field-of-view is supposed to be between 9° and 12° (leading to a reasonable F = D between 1.5 and 1.8 assuming an optical quality of 4 and a tessellation number of 5). Requiring a pixel field-of-view in the order of 0.26°, possible solutions could be solid cones with a 36 mm 2 G-APD or hollow cones with a 60 mm 2 G-APD, see also Fig. 5. The manufacturing of 60 mm 2 G-APDs is under discussion with Hamamatsu. A rough estimate shows that a solid cone for such a device would be about three times longer than for a 9 mm 2 as used in FACT. Considering the transmission loss of 10% in the FACT cones [8], which is a very conservative estimate, such cones would have a loss in the order of 35%. Since solid cones avoid the loss from Fresnel reflection at the sealing surface and the G-APD surface, the real light loss would only be around 27% assuming that the hollow cone has a reflectivity of 100% which in reality is not true. Keeping the pixel field-of-view constant, the gain in reflective area corresponds to the refractive index of the cone material squared. In the case of a refractive index of typical Poly(methyl methacrylate) PMMA of 1.4 this is a gain of GLYPH<25> 100% reflective area, which outperforms the transmission loss significantly. Assuming that the manufacturing of a 36 mm 2 G-APD would be as easy as of a 60 mm 2 G-APD, one can compare a solution with a 36 mm 2 G-APD and a solid cone and a 60 mm 2 hollow cone (assuming perfect reflectivity). In this case, the transmission loss of the solid cone is around 13% compared to 8% Fresnel loss for the hollow cone. On the other hand, the solution with the hollow cone yields a 15% smaller reflective area (same pixel field-of-view) or 27% more pixels (same reflective area). Assuming further that the price of the camera scales with the price of each channels, a reduction of the number of channels by almost 30% reduces the costs for the camera significantly. Since the costs are also dominated by the price for the photodetectors, and the price of G-APDs, in the first order, scales with the sensitive area, it can be estimated that the price for the 36 mm 2 G-APDs would be almost a factor of two lower than for the larger ones. In Figure 5 possibly solutions for MST and LST designs are shown using G-APDs and solid cones. On both cases it is convenient to sum at least three, or even seven, pixels into one readout channel to keep the ratio F = D low for construction reasons.", "pages": [ 4, 5, 6, 8, 9 ] }, { "title": "3. Conclusion", "content": "The Davies-Cotton design with its simplicity as compared to non validated dual optic systems is assuredly a good option for a wide field-of-view, up to 10° - 12°, high energy Cherenkov telescope. With this study, it is possible to scan a wide phase space of the design of Cherenkov telescopes or telescope arrays. This was achieved by a description of the optical performance of Davies-Cotton reflectors and introduction of the effect of lightconcentrators. In particular, this study provides an analytical description of the optical performance of a tessellated DaviesCotton reflector precise enough to enable performance studies without the need for dedicated simulations. By including the effect of the light-collector into the system of equations, the available phase space of design parameters is reduced to a single parameter, once the photon detector has been chosen and either the pixel field-of-view or the camera field-of-view has been fixed by physics constraints. While the choice of photo sensor is usually defined by the availability on the market, constraints on the camera field-of-view are a result of the physics targets. If these two parameters are fixed, the whole available phase space of possible solution can now be scanned by changing a single input parameter. It can, for example, be convenient to scan a reasonable range of the focal ratio F = D and derive all other parameters accordingly. From the result, the most cost efficient solution, or the one performing best in sense of physics targets can be chosen. For the Cherenkov telescope array, several design options were presented. It could be shown that for the small size telescope, considering a camera field-of-view of 9° to 12°, a four meter reflector is enough if 36 mm 2 senors are topped with solid cones to achieve a pixel field-of-view in the order or 0.25° to 0.3° at reasonable F = D . An alternative solution are hollow cones with correspondingly larger sensor area, which is disfavored because of the costs dominated by the sensor. An example plot which easily allows to determine reasonable options from the available phase space is shown in Fig. 6. The reflector diameter can easily be re-scaled linearly with the photo sensor size and the refractive index of the cone material. For the medium size and large size telescopes, the most reasonable solution using small sensors would be the summation of three and seven, respectively. Equipped with different sum-stages, these modules could be applied in any telescopes. Larger silicon based sensors, expected soon on the marked, would allow a single-channel/single-sensor solution. Using several small sensors in one channel has the advantage that the application of solid cones is possible in terms of weight and transmission and costs for photo sensors can be kept low due to their at least two times higher compression ratio.", "pages": [ 9, 10 ] }, { "title": "Appendix A.1. Ideal Davies-Cotton reflector", "content": "The Davies-Cotton design [15] is known to be promising for wide field prime-focus telescopes and was studied earlier analytically [16] and through simulations [17]. However, the parametrizations are moderately accurate and non existing for tessellated reflectors. Here, parametrizations are provided, accurate at the percent level up to 12° field-of-view, for the ideal (non-constructable) Davies-Cotton telescope and accurate to a few percent for a realistic Davies-Cotton telescope with arbitrary tessellation of the reflector. Prime-focus telescope design. The major issue of the design of a telescope is the reflector and its optical performance. Since design parameters like the field-of-view of a single pixel or the field-of-view of the whole camera are closely related to the reflectors optical performance, it is important to understand the relation between the reflector design and its performance. Unfortunately, neither spherical nor parabolic mirrors can provide both, good optical point spread function for on-axis and inclined rays, at the same time, because the distance between any point on the mirror surface to the focal point does not match the local focal length defined by the local radius of curvature. Furthermore, in the case of a spherical mirror, also the shape of the mirror surface is not ideal compared to a parabolic mirror. The parabolic shape ensures that parallel rays from infinity are well focused into a single point (due to the definition of a parabolic surface) while in the spherical case this is not the case. That means that in both cases rays hitting the mirror far off its center have their focal point not at the focal plane. In the case of a spherical mirror they also miss the focal point ( aberration ). Consequently, the ideal mirror would be a combination of two properties: A mirror surface which is shaped such that it has the right focal distance at any point, but at the same time any point is correctly oriented, so that focal distance and direction are correct. Since local normal vector and local curvature cannot be disentangled such a mirror can only be a theoretical construction. Tessellating the reflector into individual mirrors, this behavior can be approximated, as shown by Davies and Cotton, if the reflector is built from several spherical mirrors which are placed on a sphere around the focal point. In this case, the reflector can have the correct focal distance locally and, at the same time, the mirror elements can be oriented such that they correctly focus to the focal point. Apart from an improved optical performance for inclined rays, the production of several small and identical mirrors is also much more cost efficient than the production of a single large mirror. Since any optical system can always be linearly scaled, in the following a scale factor is chosen such that the reflector diameter corresponds to unity, which is identical to defining f = F = D with F being the focal length and D the diameter of the mirror. Spherical reflector. The spherical mirror has its focal point at half its radius of curvature f . Its surface is given by z = f sph( x ; y ) and its normal vector by ~ n sph( x ; y ): Ideal Davies-Cotton reflector. The ideal Davies-Cotton reflector has a non constructable surface. Its shape z = f DC( x ; y ) is spherical with radius of curvature f , but its normal vectors are defined to intercept at location ~ F = (0 ; 0 ; 2 f ). Formally, the surface equation f DC( x ; y ) and the surface normal vectors ~ n DC( x ; y ) are Practically, this equations describes infinitely small mirror elements placed on a sphere, oriented accordingly. Taylor development. To have the root-mean-square of the projection of reflected rays on the focal plane along x and y coinciding with tangential and sagittal resolutions, a rotation around z is performed without loss of generality. An incoming ray with vector ~ v = (0 ; sin GLYPH<30>; cos GLYPH<30> ) will therefore be reflected on the surface in the direction ~ vr = ~ v GLYPH<0> 2( ~ v GLYPH<1> ~ n ) ~ n = n 2 and intercept the (non curved) focal plane at r = p x 2 + y 2 , generally yielding For the ideal Davies-Cotton this takes the explicit form It is straightforward to numerically calculate the image centroid ( ¯ GLYPH<24> ; ¯ GLYPH<17> ) and the resolution ( GLYPH<1> GLYPH<24> ; GLYPH<1> GLYPH<17> ) of such a telescope and to estimate the contribution of various terms to the resolution with a Taylor development of X and Y in terms of x , y and GLYPH<30> . The development of terms of the form x i y j GLYPH<30> k , with i + j GLYPH<20> 5 and k GLYPH<20> 3 was found to be sufficient for a percent precision in the resolution parameters. The tangential and sagittal barycenter in the focal plane (the image centroid) for a uniform beam on the primary surface are given by and the corresponding resolution in term of root-mean-square: The upper integration bound in r originates from the fact that the optical system was scaled to meet a reflector mirror of d =1, hence r =1 = 2. Taylor development of the above formulas brings the desired result Coefficients given in Table A.1. For the ideal Davies-Cotton, only leading terms c GLYPH<24> i ; j f j GLYPH<30> i and c GLYPH<17> i ; j f j GLYPH<30> i are retained, at f = 1 and a maximum off-axis angle of the incoming rays of GLYPH<30> max=5°, i j terms such that c GLYPH<24> i ; j GLYPH<30> i = (2 f j GLYPH<1> GLYPH<24> ) > 10 GLYPH<0> 3 and c GLYPH<17> i ; j GLYPH<30> i = (2 f j GLYPH<1> GLYPH<17> ) > 10 GLYPH<0> 3 . As we apply the same conditions to spherical prime-focus design, less terms are present at higher order, i.e. less spherical aberration. To mirror the result for the ideal Davies-Cotton, several non leading terms are added giving a consistent picture for both developments as shown in Table A.1. Comparisons. A comparison between exact (numerically calculated) results and the presented limited Taylor development for both designs is presented in Fig. A.7. In [16] and [17], 3 rd order developments for the DaviesCotton and sherical mirror, respectively, have been discussed. The obtained coefficients are repeated here for completeness in table A.1. Both solutions show up to 20% fractional error GLYPH<14> GLYPH<1> GLYPH<17> , e.g. at f = 1. While in [17], terms x i y j GLYPH<30> k in the development were kept only to the 3 rd order, i.e. i + j GLYPH<20> 3, here terms were kept up to i + j GLYPH<20> 5 and k GLYPH<20> 3. At the expense of the introduction of more terms, consequently, the precision of the presented development is about ten times better as illustrated in Fig. A.8. Obscuration. In the above considerations, the shadow of a possible detector in the focal plane has been neglected. By changing the lower bound from r = 0 to r = 5° GLYPH<1> f in expressions A.8, obscuration can easily be quantified. Fig. A.9 shows that obscuration degrades the resolution parameters by about 1.5% at f = 1 up to about 6% at f = 2, 5° off-axis.", "pages": [ 10, 11, 12 ] }, { "title": "Appendix A.2. Parameterization for a tessellated reflector", "content": "A realistic implementation of the non-constructable DaviesCotton telescope consists in introducing a reflector made of multiple individual spherical mirrors. The tessellation number is defined as the the number N of mirrors in the diagonal. In the limit N = 1 , it is identical to the ideal Davies-Cotton design. In practice, N is a number glyph[lessorsimilar] 30. Simulation. The effective parametrization is presented as a correction to the limited Taylor development derived earlier. The correction is implemented through ray-tracing simulation performed with the MARS software (described in [18, 19, 20]), which do fully reproduce the results obtained earlier in the case of a spherical and ideal Davies-Cotton reflectors. Although an ideal Davies-Cotton reflector cannot be build in reality, it can be simulated easily. Simulations enable the use of arbitrary tessellation, since analytical solution being not very well suited for this task. For the simulation the following properties have been used: An example for such a reflector is given in Fig. A.10 Simulations have been carried out for N between 1 and 79 in steps of two, in the range 1 GLYPH<20> F = D GLYPH<20> 2 in steps of 0.1 and for rays with off-axis angles comprised in 0° GLYPH<20> GLYPH<30> GLYPH<20> 6.5° in steps of 0.5°. Empirically, it could be found that introducing a dependence on the tessellation number, the formulas given in Appendix A.1 for the spherical mirror and the ideal Davies-Cotton mirror could be unified. For this, a linear dependence at 0 th order in GLYPH<30> , and a quadratically at order GLYPH<30> i for i , 0, has been introduced. Additionally, an effective rescaling f e GLYPH<11> = f = w is needed to reach an accuracy about 5% in the whole simulated range. The root-mean-square of a tessellated Davies-Cotton can then be written as The coefficients si ; j and di ; j are the ones given in Table A.1 for the spherical and the ideal Davies-Cotton, respectively. The tessellation number N can here be interpreted as a parameter describing the transition from a single spherical mirror to an ideal Davies-Cotton reflector. The rescaling factor w can be interpreted as the deviation of the shape from the ideal case. Its value was determined by minimizing the residual, i.e. GLYPH<31> 2 , between simulated point-spread function and approximated root-mean-square. The differences of the sagittal and tangential residual are minimized independently for each N . Its value is depicted in Fig. A.11. The introduction of this scale factor effectively reduced the residual from a maximum of 12% to less than 5% for tessellation numbers smaller than 40. Fig. A.12 shows the distribution of the residuals for different tessellation numbers. Note that for the case N = 1 the simulated single mirror is of hexagonal shape while the analytical approximation de- scribes a disc-like mirror. For cases N GLYPH<29> 1 the properties of the simulated reflector converge to the ideal Davies-Cotton. While the presented development was calculated for a disc shaped reflector, here, the simulated Davies-Cotten converge to an ideal hexagon. Consequently, in both cases the rescaling factor is expected to be different from unity. In general, it is not expected to obtain a perfect match between the analytical approximation and the simulation, because simulations will always take into account effect which cannot be easily described analytically, like rays lost between individual mirrors. From Eqs. A.11 it is evident that for rays with small incident angles the point-spread function is dominated by the 0 th -order term which decreases fast with high tessellation number. At higher incident angles the point-spread function is dominated by higher order terms which only turn from the spherical to the ideal Davies-Cotton solution for increasing tessellation numbers. In general the dominating term for reasonable incident angles is the 0 th -order term. Consequently, the point-spread function dramatically improves for N > 1 but for N GLYPH<21> 5 changes become unimportant. Hence, for practical purposes a single mirror and the case of N = 3 can be excluded while for most practical purposes N = 5 will already be enough.", "pages": [ 12, 13, 14 ] }, { "title": "Appendix A.3. Summary", "content": "It is possible to describe the optical quality of a set of well defined Davies-Cotton reflectors quite well in a single analytical formula. Even the real Davies-Cotton might be slightly different, e.g. different mirror or reflector shapes or obscuration by the focal plane instrumentation, this gives a very good estimate of the optical performance.", "pages": [ 14 ] } ]

[ { "title": "Improving photon-hadron discrimination based on cosmic ray surface detector data", "content": "G. Ros a , A. D. Supanitsky b , G. A. Medina-Tanco c , L. del Peral a , M. D. Rodr'ıguez-Fr'ıas a Space and Astroparticle Group, Dpto. F'ısica y Matem'atica s, Universidad de Alcal'a Ctra. Madrid-Barcelona km. 33. Alcal'a de Henares, E-28871 (Spain). b Instituto de Ciencias Nucleares, UNAM, Circuito Exteriror S / N, Ciudad Universitaria, M'exico D. F. 04510, M'exico. a Instituto de Astronom'ıa y F'ısica del Espacio, IAFE, CONIC ET-UBA, Argentina c", "pages": [ 1 ] }, { "title": "Abstract", "content": "The search for photons at EeV energies and beyond has considerable astrophysical interest and will remain one of the key challenges for ultra-high energy cosmic ray (UHECR) observatories in the near future. Several upper limits to the photon flux have been established since no photon has been unambiguously observed up to now. An improvement in the reconstruction e ffi ciency of the photon showers and / or better discrimination tools are needed to improve these limits apart from an increase in statistics. Following this direction, we analyze in this work the ability of the surface parameter Sb , originally proposed for hadron discrimination, for photon search. Semi-analytical and numerical studies are performed in order to optimize Sb for the discrimination of photons from a proton background in the energy range from 10 18 . 5 to 10 19 . 6 eV. Although not shown explicitly, the same analysis has been performed for Fe nuclei and the corresponding results are discussed when appropriate. The e ff ects of di ff erent array geometries and the underestimation of the muon component in the shower simulations are analyzed, as well as the Sb dependence on primary energy and zenith angle. Keywords: Cosmic Rays, Photon Discrimination, Cherenkov Detectors, Sb parameter", "pages": [ 1 ] }, { "title": "1. Introduction", "content": "Photons at EeV energies and higher are thought to be typically produced as decay secondaries in our cosmological neighborhood. They come from higher-energy cosmic rays (nucleon or nucleus) that interact with matter or background photons producing neutral pions and neutrons. A typical case is the Greisen, Zatsepin and Kuzmin (GZK) process (see e.g. Ref. [1]) where a proton above EGZK /revasymptequal 60 EeV interacts with the cosmic microwave background (CMB) photons losing energy and, in the most probable case, producing a neutral pion that almost immediately decay into 2 photons of about 10% each of the initial proton energy. Neutrons could also be produced in the GZK interaction with ∼ 80% of the initial energy and later decay producing an electron and a new proton with around 10 and 90% of the neutron energy respectively. If the initial proton energy is /greaterorsimilar 10 20 eV , the secondary electron could finally produce a photon of EeV energies through inverse Compton. Also, if UHE photons are generated in cosmologically distant sources, the flux is expected to steepen above the energy threshold of the GZK process since their attenuation length is only of the order of a few Mpc at such high energies. The AGASA Collaboration on the other hand, reported a flux of UHECRs with no apparent steepening above EGZK [2]. Motivated by these measurements, many theoretical models were proposed that are able to create particles of the observed energy at relatively close distances from the Earth. These models involve super heavy dark matter (SHDM), topological defects, neutrino interactions with the relic neutrino background (Zbursts), etc. These are called top-down models since the UHE particle is a consequence of the decay or annihilation of a more energetic entity (see Ref. [3] for a review). A key signature of these models is a substantial photon flux at the highest energies. Thus, the search for UHE photons was highly stimulated. Recently, the suppression in the spectrum has been confirmed by Auger [4] and HiRes [5], but its origin is still unknown and compatible with a subdominant contribution of these top-down models. The present status is that no observation of photons has been claimed above 10 18 eV by any experiment. The main candidates reported by both older experiments, like AGASA [6] and Yakutsk [7], or the newer Pierre Auger Observatory (Auger hereafter) [8] and Telescope Array (T.A.) [9], are all compatible with the expected fluctuations of a pure sample of very deep proton shower events. The most stringent upper limits to the photon flux have been established by Auger (0 . 4%, 0 . 5%, 1 . 0%, 2 . 6%, 8 . 9% for energy above 1, 2, 3, 5, 10 EeV using hybrid data [10] and, 2 . 0%, 5 . 1%, 31% for energy above 10, 20, 40 EeV using surface data [8]) . Despite the fact that no photons have been unambiguously identified up to now, a relatively small fraction of photons in the primary flux cannot be ruled out, and their detection would have profound implications in our understanding of the nature and origin of UHECRs. In fact, recent upper limits in the photon fraction constrain SHDM models in such a way that cos- mic rays originated in these scenarios could only contribute in a subdominant way to the total flux. In addition, these limits are close to the predicted photon flux caused by the GZK interaction in certain models, whose detection would support the extragalactic origin of UHECRs and bring independent clues on their composition (see Ref. [11] for a review). Also, more stringent limits on EeV photons reduce corresponding systematic uncertainties in the reconstruction of the energy spectrum [12] and the derivation of the proton-air cross-section [13], and a ff ect the interpretation of the observed elongation rate [14]. Auger and the Telescope Array are the experiments that can currently detect EeV photons. Both are hybrid observatories with a ground array of detectors and fluorescence telescopes. At these energies, cosmic rays interact with Earth's atmosphere producing extensive air showers (EAS). EAS initiated by photon primaries are expected to develop deeper in the atmosphere compared to hadrons, producing larger values of Xmax , the maximum of shower development measurable by the fluorescence telescopes. On the other hand, the surface detector exploits the fact that photon showers are characterized by a smaller number of secondary muons and a more compact footprint at ground. Several observables have been applied to surface data, mainly related with the spatial and temporal structure of the shower front at ground [8, 9]. A new surface parameter, called Sb , was proposed for proton-iron discrimination in Ref. [15]. It is sensitive to the combined e ff ects of the di ff erent muon and electromagnetic components on the lateral distribution function. In this work, we optimize Sb for photon searches and analyze its specific properties for photon primaries. The energy calibration with the surface detector is di ff erent for hadron and photon primaries, so the calculation of an upper photon limit from pure surface information is a complex issue. The interpolated signal at a certain distance to the shower axis is used as energy estimator ( S 1000 in Auger [4] and S 800 in Telescope Array [16]) for both primaries but, comparing hadron and photon showers of the same primary energy and zenith angle, the di ff erence in the energy estimator is about a factor of 2 above 10 18 . 5 eV, on average. Therefore, while the energy calibration for hadron primaries is done by using hybrid events, i.e. events seen by the fluorescence telescopes and the surface detectors simultaneously, pure Monte Carlo (MC) methods are used in case of photon-induced showers (see Ref. [8, 17] for Auger and Ref. [9] for T.A.). This energy scale di ff erence is unavoidable for surface detector alone since it is a consequence of the di ff erent physics involved in hadron and pure electromagnetic showers. An unbiased measurement of the energy is possible if only hybrid events are used, since the primary energy is directly obtained from the longitudinal profile measured by the fluorescence telescopes. We assume here that the primary energy is the one used to simulate the showers (MC energy) since the problem of the di ff erent energy scales for pure surface events is beyond the scope of this work.", "pages": [ 1, 2 ] }, { "title": "2. Semi-analytical calculation", "content": "In this section an improved version of the semi-analytical calculation developed in Ref. [15] is introduced, in order to more deeply understand the behavior of the Sb parameter. The parameter Sb [15], is defined as, where the sum extends over all triggered stations N , r 0 is a reference distance (1000 m in the case of Auger), si is the signal measured in the i th station, and ri is the distance of this station to the shower axis. The discrimination power between protons ( p ) and photons ( γ ) of the parameter Sb can be estimated by using a merit factor defined as, where E [ S A b ] and Var [ S A b ] are the mean value and the variance of S A b , respectively, with A = p , γ . The calculation of the merit factor of Sb corresponding to protons and photons, by using a semi-analytical approach, requires the knowledge of the lateral distribution function (LDF), the signal as a function of the distance to the shower axis, for both protons and photons. Figure 1 shows the LDFs, obtained from simulations of the showers impinging on Auger water Cherenkov surface detectors (see section 3.2 for details), corresponding to proton and photon primaries of energy in the interval [10 19 , 10 19 . 1 ] eV and zenith angle θ , such that 1 ≤ sec θ ≤ 1 . 25, i.e. θ ∈ [0 · , 36 . 87 · ]. Also shown are the LDFs corresponding to muons and to the electromagnetic particles (mainly electrons, positrons and photons). Solid lines correspond to the fits of the simulated data with a NKG-like function [18], where rs = 700 m and r 0 = 1000 m, and S 0, β and α are free fit parameters. For the fits of the LDFs corresponding to the total and electromagnetic signal, the condition α = β is used, i.e. α is considered as a free parameter just for the fit corresponding to the muon signal. As expected, from figure 1 it can be seen that the muon component of the photon showers is much smaller than the corresponding one to protons. Following Ref. [15] the distribution function for a given configuration of distances to the shower axis and signals (in a given event) can be written as, where ri is the distance to the shower axis of the i th station (the first station, r 1, is the closest one) and S ( ri ) is the average LDF evaluated at ri . Note that, in this case, the Gaussian distribution corresponding to the deposited signal in each station used in Ref. [15] is replaced by a Poissonian distribution which is more suitable for small values of the total signal. Here f ( r 1 , . . . , rN ) is the distribution function of the random variables ri with i = 1 . . . N , which depends on the incident flux and the geometry of the array. From the definition of Sb and Eq. (4) the following expressions for the expectation value and the variance of Sb are obtained, where see Ref. [15] for details. Here fE ( S ( ri )) and fV ( S ( ri )) correspond to the mean value of si and s 2 i respectively, where it is assumed that the stations included in the Sb calculation are such that smin ≤ si ≤ smax , where smin corresponds to a trigger condition and smax to a saturation level. Taking smin = 3 VEM and assuming that for si ≥ smax the Poissonian distribution can be approximated by a Gaussian, the following expressions are obtained, where Following Ref. [20] it is assumed that smax = 1221 VEM. The calculation of the expectation value and the variance of Sb for proton and photon primaries requires the knowledge of the distribution function f ( r 1 , . . . , rN ) which is very di ffi cult to obtain analytically. Therefore, a very simple Monte Carlo simulation is used instead. A triangular grid of 1500 m of distance between detectors, like the one corresponding to Auger, is first considered. The impact points are distributed uniformly in the central triangle of the array and the arrival directions of the primaries are simulated following an isotropic flux such that 1 ≤ sec θ ≤ 1 . 25. The merit factor η is calculated from Eqs. (2,5,6), the fitted proton and photon LDFs and the position of the stations obtained from the Monte Carlo simulations. Figure 2 shows the comparison between the merit factor η as a function of b , obtained by using the semi-analytical approach and a simplified Monte Carlo simulation, proposed in Ref. [20] and also tested in Ref. [15], which includes the simulation of the impact points of the showers, the arrival direction and also the Poissonian fluctuations of the signal in each station. Note that the proton and photon LDFs used in both calculations are the same. From the figure, it can be seen that, as expected, η as a function of b obtained from the two di ff erent methods are in very good agreement. Also note that the maximum value of η is obtained for b /simequal 2 . 8, very close to b = 3. η", "pages": [ 2, 3, 4 ] }, { "title": "2.1. Influence of fluctuations on the discrimination power of S b", "content": "The discrimination power of Sb is dominated by two type of fluctuations, the ones corresponding to the distance of the stations to the shower axis, which come from the uniform distribution of the impact points of the showers over the array area, and the ones originated by the detection of the particles that reach a given station, i.e. signal fluctuations. The semi-analytical approach allow us to isolate the contributions of the di ff erent sources of fluctuations that generate the maximum of the curve of η as a function of b . Let us consider the case in which we freeze a realization of the spatial distributions of the stations with respect to the shower core position, then Eqs. (5,6) become, where E [ ri ] is the expectation value of the distance to the shower axis of the i th station. Line labeled as (a) of figure 3 corresponds to η as a function of b calculated under this approximation. It can be seen that η decreases for increasing values of b . The signal corresponding to the stations that are far from the shower axis presents larger fluctuations, therefore, when b increases, the weight of these stations also increases making η to decrease. Let us consider the other important case in which the fluctuations of the signal are switched o ff . In this case Eqs. (5,6) become, η where Line labeled as (b) of figure 3 corresponds to η as a function of b calculated by using Eqs. (16,17). It can be seen that for small and for large values of b , η is small. For values of b close to zero the most important contribution to Sb comes from the signal of the station closest to shower core. Therefore, due to the fast variation of the LDF with the distance to the shower axis, the fluctuations on the position of the first station are translated into very large fluctuations of the signal, decreasing drastically the discrimination power of Sb . The same happens for larger values of b but in this case the farthest station is the important one. Note that the dominant e ff ect for the increase of η in the regions of b where the curves (a) and (b) di ff er significantly from the exact value comes from the decrease of the variance. For the case in which the fluctuations on the positions of the stations are frozen the di ff erence between the mean values is larger than the exact one for small values of b . However in the case where the signal fluctuations are frozen the di ff erence between the mean values is smaller than the exact one for large values of b . Also note that comparing the expression of the variance for the two cases considered, Eqs. (15) and (17), with the exact expression, Eq. (6), it can be seen that the first term of the variance for the exact case has to do with the signal fluctuations and the second one with the fluctuations on the distance of the stations to the shower axis. Line labeled as (c) in the figure 3 corresponds to the calculation of η in which the variance of Eq. (6) is calculated by just considering the first term. It can be seen that, for values of b larger than the corresponding to the maximum, this term is dominated by the fluctuations of the signal. Line labeled as (d) in the figure corresponds to the calculation of η in which the variance of Eq. (6) is calculated by just considering the second term. In this case it can be seen that from b = 0 up to values close to the maximum, the behavior of η is dominated by the fluctuations on the position of the stations combined with the fast variation of the LDFs with r . Therefore, the formation of the maximum in η as a function of b appears due to these two e ff ects. Note that, the fluctuations on the position of the stations also contribute to the calculation of η corresponding to line (c) and the fluctuations on the signal also contribute to the calculation of η corresponding to the line (d), i.e. the exact value of the maximum cannot be obtained by just combining the cases in which these two kind of fluctuations are isolated.", "pages": [ 4, 5 ] }, { "title": "2.2. Modifying the muon content of showers", "content": "There is experimental evidence about a deficit in the muon content of the simulated showers [21, 22, 23]. The hadronic interaction models at the highest energies cannot completely describe the observations. Therefore, the muon content of the showers is modified artificially, in order to study its influence on the discrimination power of Sb . For that purpose, the LDFs corresponding to the total signal, for both protons and photons, are obtained combining the fits of the LDFs corresponding to the electromagnetic and muon components (see figure 1) in such a way that, S ( r ) = Sem ( r ) + f µ S µ ( r ), where f µ = 1 corresponds to the prediction of QGSJET-II. Figure 4 shows η as a function of b for di ff erent values of f µ , from f µ = 0 . 2 to f µ = 1 . 8 in steps of ∆ f µ = 0 . 1. It can be seen that the maximum value reached by η increases with f µ . This is due to the fact that the di ff erence between the mean value of Sb for protons and the corresponding one to photons increases with f µ , as in the case of proton and iron primaries (see Ref. [15] for details). Also, when f µ increases the total signal increases, reducing the fluctuations of the Sb parameter. Note that, bopt , the value that maximize η decreases with f µ going from ∼ 3 for f µ = 0 . 2 to ∼ 2 . 6 for f µ = 1 . 8. η", "pages": [ 5 ] }, { "title": "3. Shower and detector simulations", "content": "In this Section detector simulations are performed in order to analyze the most relevant properties of the Sb parameter. For the calculation of Sb , at least 3 triggered stations in the event are needed to assure the geometrical reconstruction of the shower axis. Therefore, the e ffi ciency, i.e. the fraction of events that fulfills this requirement, is almost 100% above the energy threshold of the corresponding array, highlighting a major advantage of the Sb parameter. In a real experiment no quality cut on Sb is needed except that it could be convenient to require a minimum number of active (not necessarily triggered) detectors during the event (for example ≥ 4 were imposed in Ref. [10]) or to examine individually the few events selected as photon candidates to avoid a possible underestimation of Sb due to a missing or non-operating station which would mimic the behavior of a primary photon.", "pages": [ 5 ] }, { "title": "3.1. S b optimization for di ff erent array sizes and geometries", "content": "The detection of the extensive air showers by a surface array of water Cherenkov tanks is here simulated by using our own simulation program described previously in Section 2 and Ref. [20]. The geometry of the array and the distance between detectors are easily modified in order to study their e ff ect on η ( Sb ). Thus, triangular and square grids are considered varying the array spacing from 500 to 1750 meters. The error in the merit factor, ∆ η , is calculated assuming Poissonian errors and is given by, where Np and N γ are the number of events in each population (here Np = N γ = 10 4 are used). Figure 5 shows the merit factor η as a function of b for di ff erent array sizes corresponding to a triangular and square grids. η increases as the array spacing decreases as expected, since the LDF is sampled in more points as the array becomes denser. η is slightly larger for the triangular grid since the number of triggered stations is also larger for this geometry. b /similarequal 3 . 0 is the optimum value for most of the arrays considered, independent of the geometry.", "pages": [ 5 ] }, { "title": "3.2. More realistic simulations", "content": "In what follows, we perform a more realistic simulation in order to treat more accurately the tank response and to take into account the shower to shower fluctuations and experimental uncertainties such as the shower reconstruction. The simulation of the atmospheric showers is performed with the AIRES Monte Carlo program (version 2.8.4a) [24] with either QGSJET-II-03 or [19] Sibyll 2.1 [25] as the hadronic interaction model (HIM). The simulation of the tank response and the shower reconstruction are performed with the O ffl ine Software provided by the Pierre Auger Collaboration [26]. The simulation is done for a triangular grid of water Cherenkov detectors of 1 . 5 km of spacing, as in Auger. The primary energy goes from log( E / eV) = 18 . 50 to 19 . 60 in steps of ∆ log( E / eV) = 0 . 05. 1000 events are simulated per each HIM and energy bin. The zenith angle follows an isotropic distribution from 0 · to 60 · while the azimuth is selected randomly from a uniform distribution in the interval from 0 · to 360 · . The library called MaGICS [27] can be linked to AIRES in order to simulate the conversion of photons in the geomagnetic field. However, we do not have to deal with photon splitting, because only a negligible fraction of inclined showers convert at most latitudes of interest below 50 EeV [28]. The results are very similar for both HIM, so most are only shown for QGSJET-II-03 unless otherwise stated.", "pages": [ 5, 6 ] }, { "title": "3.3. S b optimization for log( E / eV ) in [18 . 5 , 19 . 6] and θ in [0 · , 60 · ]", "content": "The value of b that maximizes the merit factor η as a function of the logarithm of the primary energy, bopt , is shown in figure 6 for three zenith angle bins. In case of vertical showers with log( E / eV) = 19 -19 . 1, bopt /revasymptequal 3 in agreement with the semianalytical calculation (figure 2). In the bottom panel, the bands that represent a 5% variation in η are added showing the reliability of Sb as a discriminator, even for a non-optimal selection of the index b . From figure 7 it can be seen that η ( S 3) /revasymptequal η ( Sbopt ) for all energies and zenith angles analyzed, except for low energy pri- ries in the small range with sec ( θ ) > 1 . 67 ( θ > 53 · ). Therefore, we conclude that b = 3 is an optimum choice for the whole energy and zenith angle ranges analyzed, maintaining the simplicity of the parameter. Although the merit factor is a good parameter to measure the statistical discrimination power of a variable, it carries by itself few information on the existence, shape and strength of tails of the distribution functions of the parameters. Since those tails can be also important from the point of view of the definition and understanding of the quality cuts, we include in figure 8 an example of the S 3 distribution functions for protons and photons in the energy range from log( E / eV) = 19 . 05 -19 . 10 and 1.00 < sec( θ ) < 1.33, where it can be seen that photon tails with proton-like behavior are statistically negligible but do exist. Despite the fact that only protons have been considered so far in the analysis, a sizable fraction of heavier nuclei cannot be discarded at the highest energies [14]. However, although not shown in this paper for brevity, equivalent calculations considering a pure iron composition show that η ( S 3) for photon-iron discrimination is larger than for photon-proton discrimination. Therefore, Sb , particularized for b = 3, can be used in general for photon-hadron discrimination with similar, or even better results, regardless of the exact UHECR mass composition.", "pages": [ 6 ] }, { "title": "3.4. S 3 dependence with primary energy and zenith angle", "content": "Figure 9 shows the relation between S 3 and the primary energy. An almost linear relation is found, in agreement with Ref. [15] where only hadrons were considered. Note that the result is almost independent of the hadronic interaction model and that the slope is smaller for photons compared to hadrons. The dependence of S 3 with the zenith angle of the incoming shower for primary photons is quite complex, as shown in the top panel of figure 10. While the dependence with sec( θ ) is stronger as the energy increases, the shape is similar, showing a maximum that slowly increases from 35 · to 50 · over a decade of energy. The θ dependence of Sb can be qualitatively understood by considering a simplified physical situation. Let us assume that the LDF follows a power-law, S ( r ) = S 1000 ( r r 0 ) -β , where r 0 = 1000 m and β is the slope. If b = β , then Sb = N × S 1000, where N is the number of candidate stations. The dependence of N × S 1000 with zenith angle is shown in the bottom panel of figure 10. N is expected to increase with θ since the shower footprint at ground becomes larger and more elongated. On the other hand, S 1000 decreases with θ due to the larger attenuation in the atmosphere. The combination of these two e ff ects roughly explain the existence of this maximum. In the case of hadrons, Sb has in general a small dependence on zenith angle, which is more manifest for quasi vertical showers at the lowest energies (c.f., [15]). In any case, as it is shown in figure 11, such a dependence does not hinder the discrimination power of the parameter, unless the error in energy estimate is unrealistically large ( ∆ log( E / eV ) > 0 . 35 or ∆ E > 50%).", "pages": [ 7, 8 ] }, { "title": "4. Conclusions", "content": "We have applied the proposed Sb parameter, obtained from the information given by an array of water Cherenkov detectors, to photon-hadron discrimination. By means of an improved semi-analytical calculation we have shown that, as in the case of proton-iron discrimination, there is a well defined value of the Sb exponent that maximizes its discrimination capability. We have found that at E /simequal 10 19 eV the optimum value of the exponent b is /simequal 3. We have demonstrated that the fluctuations on the position of the stations, combined with the very fast variation of the LDFs with distance, are responsible for the decrease of the merit factor at small values of b . On the other hand, we have shown that the fluctuations of the signal measured in each station are dominant at large values of b , decreasing the merit factor in this range. Therefore, the maximum of η is attained in the transition between these two regimes. Experimental data suggest an excess of muons in the showers with respect to the prediction of current hadronic interaction models. By means of the semi-analytical calculation we have studied the e ff ects on the Sb discrimination power when the muon content of the showers is modified. We have found that, the optimal value of the exponent b is still close to 3 when the muon content of the showers is modified and that the discrimination power of S 3 is actually enhanced when the muon content of the showers increases. This result is generalized by using two complementary and independent approaches. First, using our own simple MC program [20] of the shower detection and reconstruction, we have demonstrated that b /simequal 3 is the value that maximizes the merit factor for many di ff erent arrays, varying the geometry (triangular and square unitary cells) and the distance between detectors for a large range of separations (from 500 to 1750 m). Second, using a set of full numerical simulations, with a realistic tank response and taking into account the shower to shower fluctuations and experimental uncertainties, we have demonstrated that b = 3 is close to the optimum value in the whole energy range from 10 18 . 5 to 10 19 . 0 eV and zenith angles from 0 · to 60 · . Furthermore, we have also shown that the discrimination power of Sb is not significantly a ff ected even if a suboptimal value of b is used. Additionally, since the UHECR flux likely includes a sizable fraction of heavier primaries besides protons, the same analysis has been performed assuming the opposite scenario, i.e. a pure iron background. The discrimination power of S 3 is even larger in this case, confirming the fact that S 3 can be used as a composition discriminator regardless of the exact hadron composition. We have demonstrated that S 3 is almost linearly dependent on the primary energy. The zenith angle dependence for photon primaries has been qualitatively understood in terms of the evolution of the number of triggered stations and S 1000 with the primary zenith angle. In the case of hadrons, S 3 has in general a small dependence on zenith angle which does not hinder the discrimination power of the parameter, unless the error in energy estimate is unrealistically large ( ∆ log( E / eV ) > 0 . 35 or ∆ E > 50%). The calculation of an upper photon limit from pure surface information is a great challenge since, as commented previously, the energy reconstruction method introduces a composition-dependent bias. This problem could be overcome if only hybrid events are considered. Then, our results suggest that Sb combined with fluorescence observables (mainly Xmax as in Ref. [10]) could improve the upper limits to the photon flux in the whole energy range of the experiments with a unified treatment since Sb is almost full-e ffi cient above the energy threshold of the corresponding array with a large discrimination power.", "pages": [ 8, 9 ] }, { "title": "5. Acknowledgments", "content": "All the authors have greatly benefited from their participation in the Pierre Auger Collaboration and its profitable scientific atmosphere. Extensive numerical simulations were made possible by the use of the UNAM super-cluster Kanbalam and the UAH-Spas cluster at the Universidad de Alcal'a. We want to thank the Pierre Auger Collaboration for allowing us to use the Auger O ffl ine packages in this work, C. Bleve and B. Zamorano for fruitful discussions and J. A. Morales de los R'ıos for the maintenance of the UAH-Spas cluster. We also thank the support of the MICINN Consolider-Ingenio 2010 Programme under grant MultiDark CSD2009-00064, Astomadrid S2009 / ESP1496, and EPLANET FP7-PEOPLE-2009-IRSES. This work is partially supported by Spanish Ministerio de Educaci'on y Ciencia under the projects FPA2009-11672, Mexican PAPIIT-UNAM through grants IN115707-3, IN115607, IN115210 and CONACyT through grants 46999-F, 57772, CB2007 / 83539. ADS is member of the Carrera del Investigador Cient'ıfico of CONICET, Argentina.", "pages": [ 9 ] } ]

[ { "title": "Ionization Efficiency Study for Low Energy Nuclear Recoils in Germanium", "content": "D. Barker a , W.-Z. Wei a , D.-M. Mei a , ∗ , and C. Zhang a , b b College of Sciences, China Three Gorges University, Yichang 443002, China", "pages": [ 1 ] }, { "title": "Abstract", "content": "We used the internal conversion ( E 0 transition) of germanium-72 to indirectly measure the low energy nuclear recoils of germanium. Together with a reliable Monte Carlo package, in which we implement the internal conversion process, the data was compared to the Lindhard ( k =0.159) and Barker-Mei models. A shape analysis indicates that both models agree well with data in the region of interest within 4%. The most probable value (MPV) of the nuclear recoils obtained from the shape analysis is 17.5 ± 0.12 (sys) ± 0.035 (stat) keV with an average path-length of 0.014 µ m. Key words: Nuclear Recoil, Ionization Efficiency, Dark Matter Detection PACS: 95.35.+d, 07.05.Tp, 25.40.Fq, 29.40.Wk", "pages": [ 1 ] }, { "title": "1 Introduction", "content": "Understanding detector response to low energy nuclear recoils is imperative to the interpretation of experimental results from a detector designed to search for WIMPs (weakly interacting massive particles), a dark matter candidate. Direct detection of low mass WIMPs occurs in the low energy region of detectors with a threshold down to sub keV. Since the threshold energy represents the visible energy in the detector, understanding the ionization efficiency of the detector response to low energy nuclear recoils is crucial to calculating the recoil energy. An example of the need for this requirement is the claim of experimental evidence for dark matter by CoGeNT [1] that has been unverified by (D.-M. Mei). CDMS II [2]. Both experiments use germanium as the target material. Thus, similar results are expected if the detection thresholds for both experiments were determined using a standardized ionization efficiency, which accurately accounts for all processes that occur at a low energy range. This standardized ionization efficiency must also be validated with measurements in combination with reliable Monte Carlo simulations. Two different approaches can be used for modeling ionization efficiency in germanium detectors [3,4]. One model, traditionally used for a number of different detector materials, is proposed by Lindhard et. al. [3]: where k = 0.133 Z 2 / 3 A -1 / 2 , g ( glyph[epsilon1] ) = 3 glyph[epsilon1] 0 . 15 +0 . 7 glyph[epsilon1] 0 . 6 + glyph[epsilon1] , and glyph[epsilon1] = 11.5 E r Z -7 / 3 for a given atomic number, Z , mass number, A , and recoil energy, E r . However, this model has not been proved accurate at low energies as the theoretical derivation has uncertainties in this region [3]. Another model designed for low energy interactions in germanium was proposed by Barker and Mei [4]. This model takes into account the fraction of nuclear stopping power that contributes to the ionization efficiency at low energies [4]. The Barker-Mei model can be expressed as: This model is valid for recoil energies, E r , from 1 keV to 100 keV. However, the Barker-Mei model has not been experimentally verified. The purpose of this paper is to address that issue. A comparison between the two models and existing data was performed [4], and Fig. 1 shows both models together with available experimental data. As shown in Fig. 1, the two models agree in the low energy region but disagree with the available data points in the same region. Further validation of the Lindhard and Barker-Mei models, using more measurements and more accurate Monte Carlo simulation, was necessary. Taking exact nuclear recoil measurements can be challenging at lower energies when systematic errors are introduced by a variety of sources. For example, thermal neutrons and elastic/inelastic scattering have their own uncertainties. Utilizing thermal neutrons requires the full absorption of out-going gamma rays measured by another detector in coincidence. Without this additional measurement, Compton scattering from out-going gamma rays within the germanium detector can contaminate the visible energy. In a measurement of neutron elastic scattering, a Monte Carlo simulation must be incorporated to exclude multiple scatters. It is also necessary to precisely measure the scattering angle and time-of-flight of the out-going neutrons. With inelastic scattering, the Compton scattering due to de-excitation of gamma rays in the detector can contaminate the signal. All of these techniques need to be implemented with an accurate Monte Carlo that reduces systematic errors in order to obtain a reliable ionization efficiency. However, current popular simulation tools often need to be tuned to simulate inelastic scattering processes. Therefore, we desired a simple method to accurately measure nuclear recoils in germanium. The E 0 transition of germanium-72 ( 72 Ge(n,n ' e)), which is the internal conversion process for this nucleus, was chosen. The E 0 transition of 72 Ge(n,n ' e) is induced when neutrons inelastically scatter off a 72 Ge nucleus. After this collision, the 72 Ge nucleus is left in an excited state. When the nucleus returns to ground state, it does not directly produce a gamma-ray (as is common for other nuclei), but interacts electromagnetically with the inner shell electrons and causes one to be emitted from the atom [14]. The physics process for the internal conversion of 72 Ge can be generalized as: where 72 Ge ∗ denotes the excited 0 + state. The total detectable energy from the 72 Ge ∗ (0 + ) to 72 Ge(0 + ) transition is 691.6 keV, which includes energy from the X-ray. This energy is well within the visible range of traditional detectors, and both electron and X-ray can be detected with a consistency of approximately 100%. The characteristic energy, 691.6 keV, is superimposed with the nuclear recoil of a 72 Ge nucleus to form a quasi-triangular shape, which is distinguishable from other processes in the spectrum. Thus, we can observe low energy nuclear recoils without taking low energy nuclear recoil measurements by extrapolating with the known quasi-triangular fit. This quasi-triangular shape, starting at 691.6 keV, has been studied in great detail [15,20,21,22,23]. The tail of the quasi-triangular shape comes from a combination of the 691.6 keV energy deposition and nuclear recoil energy due to neutron scattering. Depending on the incident neutron energy and scattering angle [15], nuclear recoil energy can be expressed as: where E r is the nuclear recoil energy, E n is the neutron energy, M is the nucleus mass, m is the neutron mass, and θ is the scattering angle between the incident neutron and the recoil nucleus. The quasi-triangular shape is created by adding the energy from the nuclear recoils to the 691.6 keV energy deposition: where E is the observed energy and glyph[epsilon1] is the ionization efficiency. The nuclear recoil energies can be determined using a Monte Carlo simulation with applied ionization efficiency, if an agreement with the measurements is obtained. To validate the ionization efficiency models proposed by Lindhard and BarkerMei, a Geant4.9.2-based Monte Carlo simulation package [16], corrected for the internal conversion processes, was used. This simulation package was verified using a well-calibrated 60 Co radioactive source with the same experimental setup. Utilizing the spectrum measurements with a substantiated Monte Carlo simulation, we compared the unique quasi-triangular shape induced by the 691.6 keV electrons and nuclear recoils in the data to the Monte Carlo simulations for the two models. Shape analysis was used to verify the quasi-triangular shape of spectra in the data and Monte Carlo simulations by analyzing data points in the region of interest. We found a good agreement between the measurements and the Lindhard (with k = 0.159) and Barker-Mei models. In this paper, we corroborate the ionization efficiency models proposed by Lindhard et al and Barker-Mei with a neutron induced E 0 transition for 72 Ge. The experimental design is discussed in Section 2, followed by data analysis in Section 3. The Monte Carlo simulation is demonstrated and validated in Section 4, and the comparison with data described in Section 5. Section 6 summarizes our results.", "pages": [ 1, 2, 3, 4, 5 ] }, { "title": "2 Experimental Design", "content": "The germanium detector used in our experiment was an old coaxial detector from Princeton Gamma Tech, model RG11B/C [17]. Its linearity of energy response between consecutive calibrations was within 0.35%. However, the full width at half maximum (FWHM) was 7.1 keV at the 1173 keV 60 Co peak, a factor of two worse in energy resolution than a new germanium detector. Nevertheless, the detector was sufficient for measuring neutron induced internal conversion. We used an 241 Am9 Be (AmBe) source with neutron energies ranging from ∼ 1.0 to 11.2 MeV, at a frequency of 100 Hz [18]. An AmBe neutron source produces neutrons in four discrete groups: n 0 , n 1 , n 2 , and n 3 , which populate the ground state, the 4.443 MeV level, the 7.65 MeV level, and the 9.64 MeV level of the 12 C product nucleus, respectively [24,25]. The n 1 group neutrons, accompanied by 4.443 MeV gamma rays, dominate the production of neutrons in an AmBe source [24,25]. This feature allowed us to set up a coincidence measurement using sodium-iodide (NaI) detectors (Bicron model number 3M3/3 [19]) with a threshold of above 1 MeV. Utilizing this coincidence method, the NaI detectors measured gamma rays while neutrons were detected with the germanium detector. This coincidence pattern required that the NaI and Ge detectors both trigger in order for an event to be recorded, suppressing random background events generated by gamma rays from surrounding materials. We took approximately 22 days of data with three NaI detectors and 13 days of data with two NaI detectors. Due to low neutron emission, the source was placed on the center of the Ge detector cap and held in place by electrical tape. When using three NaI detectors, they were placed on the right, the left, and directly in front of the germanium detector. When using two NaI detectors, they were placed on the right and on the left of the germanium detector. Fig. 2 shows the experimental set-up. The data acquisition was performed using a National Instruments PXI-1031 system [26] and Igor Pro 4.07 software [27]. Each run lasted approximately 4.5 days. The data from each individual run was added consecutively off-line using analysis code from the Root software package [28]. Evidence of the 691.6 keV E 0 transition peak can be seen in Fig. 3.", "pages": [ 5 ] }, { "title": "3.1 Data Analysis", "content": "After 35 days, the number of events in the region of interest (675 ∼ 765 keV) was approximately 500. This is consistent with the known neutron emission from the AmBe source and the coincidence method, which suppresses random background events from gamma rays, as verified by the Monte Carlo simulation. Because of the small sample size of valid data in the region of interest, the bin size in the histogram was set to 6 keV in order to mitigate statistical fluctuation. As shown in Fig. 3, there are several peaks near the region of interest. We identified these peaks in order to help understand the processes that took place within our detector as well as possible sources of contamination. The first peak can be fitted using a standard Gaussian distribution and linear background distribution. This is given by: where p 0 = 37 ± 1.4 is the normalization constant, p 1 = 508 ± 0.2 keV is the center value, p 2 = 4.7 ± 0.2 keV is the Gaussian width, p 3 = -65 ± 12 is a constant, p 4 = 0.2 ± 0.02 is the slope, and E is the energy in keV. The peak at 511 keV is mainly from the annihilation of e -e + pairs induced by cosmic rays passing through the surrounding materials with other minor contributions. Most notable is the 4.443 MeV induced e -e + in the surrounding materials. The positrons can annihilate with electrons producing 511 keV gamma rays, which enter the germanium detector. The fitted central value of 508 keV is slightly lower than the expected 511 keV, but still within the margin of error for the given bin size. The remaining peaks can be fitted using a Moyal distribution and linear background. The Moyal distribution used is [29]: where R is a constant and E mpv is the most probable value of energy deposition in the detector. R and E mpv are physically significant parameters. The value of R is related to the reciprocal of the stopping power and can be used to calculate average path length for particles in the medium. The most probable value of energy, E mpv , is the most common energy deposition in that region. These parameters are further discussed in Section 3.3 and used to interpret the experimental data. The peak around 560 keV, is likely caused by events from 76 Ge( n, n ' γ ) inelastic scattering. Because of the small number of events, the fitting function was only partially accurate. The second peak at around 600 keV has the fitted parameters, R = 0.06 ± 0.02 and E mpv = 609 ± 3.6 keV. This peak is likely the combination of inelastic scattering from 74 Ge( n, n ' γ ) and the neutron capture on 73 Ge, 73 Ge( n, γ ). Finally, the peak of interest, 691.6 keV, exhibits the internal conversion of 72 Ge( n, n ' e ). The fitted parameters are, R = 0.06 ± 0.02 and E mpv = 705 ± 3.0 keV. All fits in the region of interest are shown in Fig. 4. We also took a background spectrum using the coincidence method to identify any random coincidence. This spectrum is also shown in Fig. 4. The background data was taken using two sodium iodide detectors (see right of Fig. 2) for 4.6 days. After the background spectrum was collected, we subtracted it from the AmBe data sets as shown by Fig. 5. The 511 keV peak was significantly reduced, veri- fying our conjecture on the main origin of this peak as induced pair production from cosmic rays in the surrounding materials. The subtracted data set was also fitted using a Moyal distribution and linear background. It was found that the most probable value of energy deposition in the detector for the E 0 transition was 696 ± 1.4 keV. Thus the percent difference of the AmBe data with and without background subtraction is 1.3%.", "pages": [ 6, 7, 8, 9 ] }, { "title": "3.2 Error Analysis", "content": "We have identified the following sources of systematic error for our data analysis: the energy scale and the energy resolution. The associated energy scale error is 3 keV, which is from the fit of the known 511 keV peak. The center value of the fitted function was 508 keV which is a difference of 3 keV from the known value of the peak. This shift in energy is from the energy calibration used to convert to the energy scale. As stated previously, there was a 0.35% error in successive energy calibrations. Using a more accurate energy scale to produce a peak at 511 keV caused other energy regions to become less accurate. For energy resolution, the value from the 662 keV peak of 137 Cs was used, it is nearest of the calibration sources to our region of interest. The energy resolution for this peak was 3.45 keV. Adding two errors in quadrature, since they are independent of each other, causing the resulting systematic error to be 4.6 keV. Thus, the larger bin size of 6 keV was used to accommodate for this error. Statistically, there are approximately 500 events in the region of interest (675 keV ∼ 765 keV) which gives a statistical error of 4.5%. For data with the background subtracted out, the error associated with the most probable value is 696 ± 4.6 (sys) ± 1.4 (stat).", "pages": [ 9 ] }, { "title": "3.3 Interpretation of Results", "content": "Electrons and nuclear recoils that travel through germanium with a high momentum lose energy by exciting and ionizing the germanium atoms. The average amount of energy lost can be calculated with the Bethe-Bloch equation [30]. However, the energy transfer is not a continuous process. It occurs through random collisions during which various amounts of energy can be transferred. The energy loss in the detector can be described by the Moyal distribution, Eq. 7, as shown in Fig. 4. As shown in Eq.7, R is the reciprocal of the density, ρ , times the average path-length of charged particles in the detector, d , times the parameter, K , as given by the Bethe-Block formula: where K = 4 πN a m e c 2 r 2 e z 2 Z A 1 β 2 is related to the stopping power, with constants N a , Avogadro's number; m e , the mass of electrons; c , the speed of light; r e , the Bohr radius; z , the electron charge; Z , atomic number of target; A , the atomic mass number of target; and β , the speed of charged particles divided by c . The density of germanium is ρ = 5.323 g/cm 3 . Since electronic stopping power is different from nuclear stopping power, the value of R is very different for pure electronic recoils than nuclear recoils. Thus, the value of R can be used to derive the average path-length for a given electronic recoil or nuclear recoil. However, the value of R from our measurements is a combination of electronic and nuclear recoils, and it cannot be used to directly determine the average path length for either. Nevertheless, we can use the most probable value of energy deposition, which is related to the stopping power multiplied by the average path length, to determine the average path length. Given E mpv = 696 ± 4.6 (sys) ± 1.4 (stat) keV, obtained from the fitted function in Fig. 5, is a convolution of the 691.6 keV energy deposition and the nuclear recoils, then we contend that 696 keV = dE dX · d · ρ , where dE dX is the mass stopping power in keV cm 2 /g. Since there are both electronic and nuclear recoils, we can rewrite this as: where dE dX e = 1301 keV cm 2 /g [31] and dE dX e · d e · ρ = 691.6 keV; thus d e = 691.6 /( dE dX e · ρ ) = 0.1 cm, which is the average path length in germanium for electrons with an energy of 691.6 keV. Therefore, we have and From Eq.11, we can conclude that the most probable nuclear recoils that we measured in the detector have an average of 4.4 ± 0.007 (sys) ± 0.002 (stat) keV electronic equivalent energy, which corresponds to approximately 17.5 ± 0.12 (sys) ± 0.035 (stat) keV nuclear recoil energy from ionization efficiency in the Barker-Mei model. To determine d n , we can use dE dX n = 2341720 keV cm 2 /g for a nuclear recoil of 17.5 keV [4]. Thus, d n = 17.5 keV /( dE dX n · ρ ) = 0.014 µ m is the average path length in germanium for 17.5 keV nuclear recoils. This determines the range of low energy nuclear recoils in a germanium detector.", "pages": [ 9, 10, 11 ] }, { "title": "4 Monte Carlo Simulation of E 0", "content": "An accurate Monte Carlo simulation is needed to determine the validity of the Lindhard and Barker-Mei models when compared to collected data. Two crucial steps were taken before creating a Monte Carlo simulation that would determine nuclear recoil energy: 1) the E 0 transition in Geant4.9.2 [16] was fixed and 2) the Monte Carlo simulation was verified with a well-known gamma-ray source.", "pages": [ 11 ] }, { "title": "4.1 Fixing the E 0 transition in Geant4.9.2", "content": "By studying the inelastic scattering processes that contribute to internal conversion, we found that the E 0 transition code is included in Geant4, but is missing neutron data for 72 Ge. Specifically, Geant4 does not provide γ / e transition data in each energy level (Data Type = 12) and has no cross section data corresponding to the E 0 transition. In order to make the internal conversion process available in our Geant4 simulation, we created a γ / e ratio in the database for 72 Ge with Geant4.9.2 several years ago (see our correction in Ref. [32]). In addition, we calculated all the cross-sections of the 72 Ge( n, n ' ) reaction using TALYS [33], a reliable software for the simulation of nuclear reactions. We then converted the transition cross-section data into the transition ratio in Geant4.9.2 as shown in Fig. 6 (we utilize Geant4.9.2 for this work because the E 0 transition problem was amended several years ago). The independence of this ratio can be cross-checked by looking at the reactions in different energy scales. Eventually, the TALYS data was converted to Geant4 format where it could be used in the internal conversion process (see the supplemental data used in Geant4.9.2 from Ref. [32]). During the E 0 transition, a characteristic X-ray, or Auger electron, is produced simultaneously with the conversion electron. This process was not included in the Geant4 code. However, since this is a complicated process that involves binding energy, it would require editing for all the elements in the Geant4 package. For the sake of simplicity, we combined the X-ray and conversion electron together, since they both contribute to the total energy deposition (in a solid germanium detector only). A more general solution will be required when considering other materials. A simulation was created from the Geant4.9.2 and G4NDL3.13 packages that includes the additional transition ratio data. For further information, readers can refer to Ref. [37] for detailed geometry and experimental framework. After the missing data was added, the E 0 transition of 72 Ge( n, n ' e ) could be simulated using Geant4.9.2. This makes the Geant4.9.2 simulation more reliable for dark matter searches utilizing germanium detectors.", "pages": [ 11, 12 ] }, { "title": "4.2 Verification of the Monte Carlo Simulation", "content": "To obtain reliable results from our Monte Carlo, it was essential to verify the simulation. 60 Co, with an original radioactivity of 1.0 µ Ci, was used to validate the simulation. The 60 Co source was mounted on the center of the Ge detector cap as shown in Fig. 2 (right). The corresponding geometry used in the Monte Carlo simulation is shown in Fig. 7. Since the germanium detector is aged, the Monte Carlo geometry was modified to include a dead layer of thickness 2.5 mm (see Fig. 7, left). We also implemented a smearing process by applying energy resolution to the spectrum of deposited energy in the active germanium. When fitting the energy resolution, these four peaks were identified: 1173 keV, 1332 keV, 2506 keV and 662 keV. The first three peaks are gamma rays from the 60 Co source and the last peak is a gamma ray from 137 Cs source. The best fit function is presented in Eq. 12: where E is the energy in keV. Under the square root, the first term, 0.3, is the percentage of energy resolution, for a new Ge detector; the second and third terms are due to noise and the age of the Ge detector. The percentage of relative energy resolution as a function of energy is plotted in Fig. 8 Using this fitted function, we applied the energy resolution to the range 600 keV ∼ 2510 keV and obtained agreement between the experimental data and Monte Carlo simulation as shown in Fig. 9. This validates that our Monte Carlo can be reliably used for gamma-ray simulation.", "pages": [ 12, 13 ] }, { "title": "5 Monte Carlo Simulation of Nuclear Recoils", "content": "Using the modified Geant4.9.2 package, we performed simulations to verify the accuracy of the Lindhard and Barker-Mei models with an AmBe neutron source. The experimental setup for our Ge and NaI detectors, as shown in Fig. 2, was simulated according to the dimension and material information provided by the manufacturer [34]. Since the 72 Ge( n, n ' e ) reaction causes a quasi-triangular shape in the data, we expect to see this same feature in our Monte Carlo simulation if the Lindhard and Barker-Mei models are accurate. In order to provide an accurate simulation, the geometry, as well as the AmBe source generator, need to be implemented correctly.", "pages": [ 13, 14 ] }, { "title": "5.1 AmBe Neutron Generator", "content": "Because the AmBe source was placed very close to the germanium detector, gamma rays emitted from the source have a greater chance of entering the germanium detector and causing contamination in the region of interest (675 keV ∼ 765 keV). Thus, in the simulation it is necessary to account for all potential gamma rays emitted from the AmBe source. Two reactions, shown in Eqs. 13 and 14, occur in the AmBe source: Eq. 13 shows the decay of 241 Am to 237 Np, which causes the emission of alpha particles and gamma rays. Eq. 14 shows the reaction between an alpha particle and 9 Be. In Eq. 14, the energy of gamma rays emitted depends on the resulting state of 12 C, which is 4.443 MeV (for the 1st excited state), 7.65 MeV (for the 2nd excited state) and 9.64 MeV (for the third excited state). Using the gamma ray energy and recoil energy of 12 C, the resulting neutron energy can be calculated by applying energy conservation to Eq. 14. Fully absorbed gamma rays (from Eq. 14) will not be in the region of interest (675 keV ∼ 765 keV) since their energies are at a few MeV. However, the Compton continuum of their interaction can contribute to the region of interest. In addition, gamma rays (from Eq. 13) with energies 26.34 keV (a branching ratio of 2.4%), 59.54 keV (a branching ratio of 35.9%) and 722.01 keV (a branching ratio of 0.000196%) [35] can contribute to the region of interest by occurring in coincidence with the E 0 transition (26.34 keV and 59.54 keV) or by becoming fully absorbed (722.01 keV). We generated these three gamma rays in our Monte Carlo simulation along with gamma rays at energies of 4.443 MeV, 7.65 MeV, and 9.64 MeV caused by the transitions in the excited 12 C nucleus. The simulated neutron energy spectrum from the AmBe neutron source is shown in Fig. 10. The spectrum agrees with the prediction from Marsh et. al. [36]. After this validation, the AmBe source generator was implemented in the Monte Carlo simulation.", "pages": [ 15 ] }, { "title": "5.2 Verification of the E 0 Transition in Simulation", "content": "The simulated results are presented as a spectrum of energy deposited in the germanium detector after the application of smearing, which is the process of 12 Gaussian and Moyal distributions with fitted parameters from our data (Section 3) have been incorporated into the smearing of different peaks in the region of interest. We used the model proposed by Lindhard et al. [3] ( k = 0.159) to determine the ionization efficiency for a germanium detector. Fig. 11 shows the simulated energy deposition spectrum after smearing. The E 0 , 691.6 keV, transition is visible, as shown in Fig. 11. Appearance of this peak indicates that the internal conversion process has been successfully implemented in our Geant4.9.2.", "pages": [ 15, 16 ] }, { "title": "6 Monte Carlo Shape Analysis", "content": "After successfully simulating the AmBe neutrons in our Geant4.9.2, we collected the energy deposited in the germanium detector. This collected energy spectrum was compared with our data after the application of smearing (Eq. 12). Two models utilizing ionization efficiency functions were applied to the Monte Carlo spectrum: Lindhard k = 0.159 and the Barker-Mei model. Normalization was applied to the energy range 675 to 765 keV. By overlaying the Monte Carlo energy spectrum and the collected data, we were able to perform a shape analysis on the characteristic E 0 transition. This is demonstrated in Fig. 12. The shape analysis was performed on a bin-to-bin basis by comparing the data and two Monte Carlo simulations, which correspond to the two models. The difference in shape between the data and the two Monte Carlo simulations was calculated using ( data -MC ) ( data + MC ) / 2 . Table 1 shows the results of this comparison in which the largest difference is shown to be less than 4%. Unfortunately, we were not able to include the neutron capture lines of some specific even nuclei, such as 70 Ge, in our Geant4.9.2 package due to the lack of adequate cross-sections. Thus, at 708.2 keV and 747.7 keV, there is a discrepancy between the collected data and Monte Carlo simulations due to the neutron capture of 70 Ge( n, γ ) [37]. There is also inconsistency in the energy range 650 ∼ 680 keV that is likely due to 115 In( n, γ ) and 206 Pb( n, n ' γ ) [37] processes, which are not included in the Geant4.9.2 package. Since a good agreement between the data and two models was achieved in the shape analysis, we extracted an average visible nuclear recoil energy, E vr , from each bin using the difference between the measured visible energy and the 72 Ge ∗ (0 + ) to 72 Ge(0 + ) transition energy, 691.6 keV. The corresponding nuclear recoil energy, E r , was determined utilizing the Monte Carlo simulation. Table 2 displays the extracted results. Using E vr E r , Fig. 13 shows the extracted ionization efficiency from the data and Monte Carlo simulation. Note that this is not a direct measurement of ionization efficiency, but an extraction using the shape analysis.", "pages": [ 16, 17, 18 ] }, { "title": "7 Conclusion", "content": "We took measurements using an AmBe neutron source, incident on a germanium detector, for a total of 35 days. The characteristic quasi-triangular shape located at 691.6 keV represents the E 0 transition of germanium-72. All peaks in the region of interest have been identified, thereby confirming source related and environmental backgrounds. The unique quasi-triangular shape induced by neutrons can be described using the Moyal distribution with the fitted parameters R = 0.06 ± 0.01 (stat) and E mpv = 696 ± 4.6 (sys) ± 1.4 (stat) keV, after background subtraction. Utilizing these parameters, we derived the most probable value for nuclear recoils as 4.4 ± 0.007 (sys) ± 0.002 (stat) keV electronic equivalent energy, which corresponds to a nuclear recoil energy of 17.5 ± 0.12 (sys) ± 0.035 (stat) keV. The average path length in the germanium detector for 17.5 keV nuclear recoils was approximately 0.014 µ m. AMonte Carlo simulation employing a corrected Geant4.9.2 package was modified to duplicate the same experimental setup and AmBe neutron source. The Lindhard ( k = 0.159) and Barker-Mei models were used to apply ionization efficiency to the energy spectrum and were compared to the experimental measurements. A bin-to-bin shape analysis was performed, and the difference between the measurements and two models were calculated. We obtained a percent difference that was less than 4% for the collected data and two Monte Carlo simulations (see Table 1). Using the shape analysis, we calculated the most probable values for visible nuclear recoils (as shown in Table 1), extracted the average visible nuclear recoil energy (see Table 2), and obtained the corresponding nuclear recoil energy from the Monte Carlo simulation (see Table 2). The extracted average ionization efficiency is shown in Fig. 13. These values are in agreement with the Lindhard and Barker-Mei models in the energy range of 1 to 100 keV. Therefore, the Lindhard model (with k = 0.159) and Barker-Mei model can be used to determine the ionization efficiency in germanium detectors for 1 to 100 keV nuclear recoil energy.", "pages": [ 19, 20 ] }, { "title": "Acknowledgments", "content": "The authors wish to thank Iseley Marshall and Angela A. Chiller for their careful reading of this manuscript. Additionally, the authors would like to thank Rupak Mahapatra for his comments and suggestions. This work was supported in part by NSF PHY-0758120, DOE grant DE-FG02-10ER46709, the Office of Research at the University of South Dakota and a 2010 research center support by the State of South Dakota.", "pages": [ 20 ] } ]

[ { "title": "Isaac Saba", "content": "Ruhr-Universitt-Bochum, 44780 Bochum, Germany", "pages": [ 1 ] }, { "title": "Julia Becker Tjus", "content": "Ruhr-Universitt-Bochum, 44780 Bochum, Germany", "pages": [ 1 ] }, { "title": "Francis Halzen", "content": "Department of Physics, University of Wisconsin, Madison, WI-53706, USA", "pages": [ 1 ] }, { "title": "Abstract", "content": "Active galactic nuclei (AGN) are believed to be the source of ultra high energy cosmic rays (UHECRs, E > 10 18 eV). Particles are assumed to be accelerated in the accretion disk and the plasma jets, produced due to conservation of angular momentum, to the highest energies, where they interact with each other and produce pions, which decay among others in neutrinos. For a known cosmic ray spectral behavior, the main parameters in the calculation of the neutrino flux from proton-proton interactions are the target density n H and the ratio of electrons to protons f e . Using most recent neutrino flux limits from IceCube point source searches, we set limits on the target densities for 33 FR-I galaxies. The densities are shown to be smaller than 30 cm -3 to 2 · 10 3 cm -3 , depending on the source and when using a fixed electron to proton ratio of f e = 0 . 1. This implies that some cosmic ray acceleration sites, especially those close to the core of the AGN, can already be excluded, or else that the ratio of electrons to protons deviates significantly from the commonly used value of 0.1. For Centaurus A (Cen A) and Messier 87 (M 87) we use Fermi observations to model the γ -flux, the neutrino flux and the resulting target density. The detection of these neutrinos will help to find information about acceleration processes in the source. Keywords: Active galactic nuclei, FR-I galaxies, Inelastic proton-proton interaction, Target density, Centaurus A, Messier 87", "pages": [ 1 ] }, { "title": "1. Introduction", "content": "Active Galactic Nuclei (AGN) are the most powerful permanent objects known. The observed luminosities range from 10 41 erg s -1 , measured for nearby galaxies, up to 10 47 erg s -1 for distant galaxies. The prevalent picture is a supermassive black hole (SMBH), located in the center of the host galaxy, with gravitational energy as the source of the luminosity. The emission is spread widely across the electromagnetic spectrum, often peaking in the ultra-violet, but with significant luminosity in the X-ray and infrared bands. The emitted power varies on time scales of years, days or minutes. Due to angular momentum conservation, plasma is ejected and forms bipolar collimated jets, strong radio sources if the host galaxy is elliptical, or weak radio sources if the host galaxy is a gas rich spiral. AGN are already observed and identified as sources of high energy γ -rays and additional observations indicate that the arrival direction of the highest energetic CR might correlate with the position of Cen A and M 87 [1, 2]. High energy neutrinos in AGN are produced in coincidence with high energy γ -rays when high energetic protons interact with a local target and produce secondaries, which decay among others into neutrinos. The exact mechanism of the energy and momentum transfer in AGN is still under debate. Matter which is attracted by the SMBH, the central engine of the AGN cannot directly fall into the black hole, since it possesses angular momentum. A disk of matter is formed within which magnetic viscosity transfers angular momentum outward and mass inward. In the core region protons can be accelerated to high energies via reconnection of the magnetic field to high energies [3]. The highly energetic protons can interact with other protons and photons, producing secondary particles as high energetic neutrinos, which can leave their point of origin almost unimpeded to be detected on Earth. A detailed approach to proton-photon and proton-proton interaction in the jet can be found in [4]. Here we focus on proton-proton (p-p) interaction in the core region and concentrate on the muon neutrino flux calculation for 33 FR-I galaxies. Assuming that the electrons lose all their energy to synchrotron radiation, the radio luminosity is equal to the electron luminosity Furthermore protons and electrons are expected to be accelerated at the same site, meaning that the proton luminosity can be determined by assuming a constant ratio f e = L e /L p between radio and proton luminosity [5]. Using this assumption, the proton luminosity can be estimated from radio observations of individual sources. A further astrophysical parameter is the target density n H , which determines the optical depth τ pp of the p-p interaction. It is given by where I 0 is the initial intensity and I the observed one. For the calculation the optical depth is given by where n H is the target density, σ inel ( E p ) the inelastic cross section for proton proton interaction and R the is the size of the interaction region. For the considered parameter space (emission from the core region of the AGN, R < 100 kpc, with densities n H < 2 · 10 3 cm -3 ), Equation (2) can be approximated by a linear behavior of the optical depth. For Cen A and M 87 there are detailed γ -ray observations available, which will be used to normalize the γ -flux and to determine the neutrino flux and the resulting target densities n H at a given fraction f e [6]. This paper is constructed in the following manner. In section 2 the main calculation will be introduced. In section 3, the results for the neutrino fluxes and the derived limits on the target density for 33 FR-I galaxies will be presented. The observations of γ -rays from Cen A and M 87 provides the possibility of deriving an exact value for n H , which is done in section 4. Conclusions are presented in section 5.", "pages": [ 1, 2, 3 ] }, { "title": "2. Modeling neutrino and γ -ray spectra from FR-I using the sources radio luminosity", "content": "High energy neutrinos can be produced through inelastic proton-proton interactions, where high energy protons interact with ambient protons and produce pions which decay into neutrinos. The muon neutrino flux at Earth is given by [7] Here, /epsilon1 osc considers the oscillation, n H is the target density in cm -3 , σ inel is the inelastic proton-proton cross section in mb, given by with E th the threshold energy for the π + production. Neutrinos at the sources are created with the ratio Due to oscillation of neutrinos from the distant source to Earth the ratio to be expected here is meaning /epsilon1 osc = 1 / 3 for our calculation [8]. The function F ν ( x, E ν /x ) gives the muon neutrino spectrum for a fixed proton energy E p and is divided into three summands, F 1 , F 2 and F 3 . The first one denotes the muon neutrino spectrum of neutrinos from direct pion decay, F 2 gives the spectrum of neutrinos produced by muon decay and F 3 considers the electron neutrinos produced [7]. Further, d N p / (d A d E p ), the incident proton spectrum in units of cm -2 TeV -1 , is given by Here, A p is the normalization of the spectrum in units of cm -2 TeV -1 , E 0 is the cut-off energy and p is the spectral index. In the following calculations, we use p = 2. While the cosmic ray spectrum might indeed deviate from an E -2 behavior (see e.g. [9], IceCube point source limits are usually given for an E -2 -spectrum only [10]. In future work, it will be interesting to investigate the effect of different spectral indices as well, when IceCube limits are provided for other cases as well. For the two sources Cen A and M 87, where γ -ray measurements indicate a deviation from an E -2 -behavior, we actually do use the observed values, as we do not rely on IceCube limits in that case. The cosmic ray normalization is connected to the total cosmic ray energy W p via The cosmic ray emission is isotropic, meaning that only the fraction (4 πd L ( z ) 2 ) -1 , with d L representing the redshift dependent luminosity distance, reaches the Earth. Due to the scaling law of the cross section with the nuclei number, the results apply for different cosmic ray compositions. We perform the calculations for protons only for simplicity, but expect the same results for a heavier composition. The minimum energy of the cosmic rays is the threshold energy for the π + production, E min p ≈ 1 . 2 · 10 -3 TeV, and E max p is the maximum cosmic ray energy. E min p ≈ 10 9 TeV. The total proton energy of a single source is given by In Equation (13) we assume a constant proton luminosity L p . The parameter t H is the Hubble time and E ( z ) is given by [11] In Equation (14) we assume a ratio f e between radio and proton luminosity, given by Considering the mentioned assumptions the normalization is", "pages": [ 3, 4, 5 ] }, { "title": "3. Limits on the target densities for 33 FR-I galaxies", "content": "Fanaroff and Riley divided radio galaxies according to the correlation between morphology and luminosity into two groups, FR-I and FR-II [12]. FR-I are brightest towards the center, while FR-II are brightest at outermost part of the jet. Later it was discovered that this behavior is correlated with a critical radio luminosity at 178 MHz, L 178 = 2 · 10 26 W / Hz, dividing the radio sources in FR-I and FR-II, where FR-I have lower and FR-II higher luminosities [13]. Detailed observation showed that the morphology is correlated to the energy transport in the sources. FR-I sources have bright knots along the jets, while FR-II have faint jets but bright hot spots at the end of the lobes, indicating that they appear to be able to transport energy efficiently to the ends of the lobes. FR-I sources on the other hand are inefficient in the way that a large amount of energy is radiated. Considering the AGN unification scheme, FR-I galaxies are assumed to be the misaligned counterparts of BL Lacs [14], meaning that the non thermal beamed emission from the relativistic jets should be present in radio galaxies. Observation show that the ratio of nuclear luminosities of FR-I and BL-Lacs show a correlation with the orientation of FR-I galaxies, supporting the assumption that they are correlated with each other. In this paper we focus on FR-I galaxies presented [13]. The selection includes 33 FR-I galaxies with a given radio luminosity L radio at 178 MHz. The redshift ranges between z = 0 . 0037 -0 . 29. Using the radio luminosity, the redshift z , considering the cosmology ( H 0 = 75 km s -1 Mpc -1 Ω M = 0 . 27 , Ω Λ = 0 . 73) and f e the normalization A p and the muon neutrino flux are calculated. The latter can then be used to calculate the neutrino flux in dependence of the target density n H .", "pages": [ 5 ] }, { "title": "3.1. Results", "content": "Table 1 provides the upper limits for the target densities of the 33 FR-I galaxies presented in [13]. Here, the radio luminosity of each single source, the cosmology dependent parameters, luminosity distance d L ( z ) the lookback time, and the constant ratio f e = 0 . 1, were used to compute the normalization A p . The resulting muon neutrino fluxes are calculated and matched in normalization to the IceCube limit, resulting in an upper limit to the target density. Including the knowledge on neutrino flux limits provided by IceCube [10] allows thus to set an upper limit on the target density. As IceCube limits are only provided directly for a fixed number of sources, we use the declination dependent sensitivity from [10]. Since we have detailed information about the high energy part of CRs, we can use observations of data to calculate the proton target density. The two assumptions we use is that a) UHECRs originate from AGN and b) protons are accelerated in the jet in the same way as electrons. Taking this assumption we calculate n H , which ranges between 20 cm -3 for 3C 028 and 1500 cm -3 for 3C 386 for f e = 0 . 1. Comparing our results with the model of Kazanaz and Elliason [15], where a spherically symmetric accretion shock, accelerating a fraction of the inflowing plasma to the highest energies is considered, the proton density is given by Here, ˙ m := ˙ M/ (1 M /circledot yr -1 ) is the accretion rate, M 9 := M/ (10 9 M /circledot ) the black hole mass in units of 10 9 solar masses. The parameter x := r/r S , with the Schwarzschild radius r S , gives the radial distance. Comparing Equation (18) with our results indicates that accelerated protons might originate from a maximum orbit x max ≈ 3 · 10 4 for ˙ m = 0 . 1, from x max ≈ 5 · 10 4 for ˙ m = 0 . 2 and from x max ≈ 9 · 10 4 for ˙ m = 0 . 5 and from a minimum orbit x min ≈ 2000 for ˙ m = 0 . 1, from x min ≈ 3000 for ˙ m = 0 . 2 and from x min ≈ 6000 for ˙ m = 0 . 5. Considering the jet disk model [16, 17], the target density in the jet in the observers frame, is given by Here, L 46 is the disk luminosity in units of 10 46 erg/s, q j/ 1 is the ratio between jet power and disk luminosity, Z pc is the distance from the origin in parsec and Γ is the boost factor of the plasma. The parameters for this model are Γ ≈ 10, L 46 ≤ 10 -3 and q j/ 1 ≈ 0 . 15 for Z pc < 1. The resulting densities of n H ≤ 1700 ( Z pc / 0 . 1) -2 cm -3 are well consistent with the limits derived here and would only start to become inconsistent for Z pc /lessmuch 0 . 1, a distance smaller than 10 2 r s . So far, it is not clear, how well the jet disk scenario works for FR-I galaxies, though, as their accretion disks are extremely faint and difficult to observe [18].", "pages": [ 5, 6 ] }, { "title": "4. Cen A and M 87", "content": "For Cen A and M 87 we use the Fermi LAT observations to normalize the γ -ray flux, which fixes the neutrino flux directly. Thus IceCube limits are not needed in this case. The Large Area telescope is a pair conversion γ -ray telescope, covering the energy range from 20 MeV to more than 300 GeV [6]. Due to its vicinity to Earth, Cen A has been well studied over the entire electromagnetic spectrum, from radio to γ -rays. Observations performed by experiments like the Auger observatory indicate that the origin of the highest energy CRs ( E ≥ 10 19 eV) could correlate with the angular position of Cen A [19]. M 87 one of the nearest ( d = 16 Mpc) and best studied radio galaxies, just like Cen A is known for its bright arcsec-scale jet. It contains an SMBH with a mass of ∼ (3 -6) · 10 9 M /circledot . At TeV energies M 87 is detected by H.E.S.S. [20], MAGIC[21] and VERITAS [22]. Due to the short observation period of MAGIC, VERITAS and H.E.S.S. in comparison to Fermi LAT we focus on Fermi observations for M87. A further reason is the time variability of the γ -flux for E > 730 GeV [20], while Fermi gives due to the longer observation period time averaged fluxes. A satisfactory answer to the question of the source of high energy γ -rays is not yet found. For Cen A we also focus on Fermi observations, since Fermi LAT can resolve the inner region of Cen A [23]. A further reason is the γγ absorption, making it unlikely that the γ -ray flux measured by H.E.S.S. and Fermi, originates from the same region [1]. First we give an estimate on the target density for the two sources, by assuming that γ -rays are produced in the inner region, leading to the condition Here, λ is the mean free path and D source is the diameter of the source. Using this assumption excludes the other AGN from [6], since the origin of the observed γ -rays cannot be absolutely restored to the inner region of the sources. For the observed energy range, the cross section remains almost constant see (Equation (5)) Assuming that the cores are spherical, using the distance to the source, the diameter of the cores can be estimated, as For M 87 the 95% confidence error radius r 95% = 0 · . 086 is used [24]. Considering our assumptions, the mean free path has to fulfill the following relation, It has to be noticed that the assumptions used here are simplifications. For example, the geometry can be very complex due to physical processes, like clumping, so that the assumption of spherical geometry might not be sufficient to describe the real shape of the source. This has an influence on the target density, Equation (26). Furthermore we assume γ -production only in the inner region. Nevertheless the above calculation provides a first order approximation of the required target densities. We compare the results from the estimate with the results, using our model explained in section 2. Therefore, we use the radio luminosity of the inner region to calculate the normalization. The fraction 1 / 10 = L radio core /L radio total , observed for Cen A given in [25] is assumed to be the same for M 87. Using a power law function, with Γ for the spectral index and E 0 for the cut-off energy, a power-law spectrum of primary cosmic rays is used to model the γ -spectrum: Here, we used E 0 = 10 8 TeV as the maximum energy, which is needed to explain the observed flux of ultra-high energy cosmic rays. The Fermi data for Cen A and M 87 require spectral indices of Γ CenA = 2 . 5 and Γ M87 = 2 . 21. Considering the cosmology, the radio core luminosity and the distances, we calculate the normalization A p , the γ -ray flux (Figure 2) and the densities for f e =0.1: As can be seen for Cen A the density estimated from the core size, Equation (27) and the one calculated within the model presented, Equation (30) are in agreement. For M 87 the situation is different. Uncertainties like the unknown core size or the unknown core luminosity have a large influence on the estimate. To eliminate these uncertainties, the core size has to be resolved and the fraction of the core to the whole radio luminosity has to be known. For this purpose, improved measurements for the core size of M 87 will help. Additionally, the detailed observations of a larger sample of AGN in the future can contribute to receive a statistical sample of the fraction of core to total luminosity of FR-I sources. If the variance is of physical nature, on the other hand, the ratio f e might deviate significantly from f e = 0 . 1.", "pages": [ 6, 9, 10 ] }, { "title": "4.0.1. The density and muon neutrino spectrum for Cen A and M 87", "content": "The muon neutrino flux can be derived directly from the fit to the γ -measurement and compared with the limits given in [10]. As one can see, the flux is below the current sensitivity of the IC40 for both sources. It has to be mentioned that the uncertainties in the assumptions are significant, so the results have to be interpreted carefully. For M 87 the sensitivity is about one order of magnitude and for Cen A two orders of magnitude higher than the flux in the corresponding energy range. Long-term observations with IceCube and possible future extensions might be able to yield a first significant signal from M 87. In the case of Cen A, the measured flux is simply too steep to be obesrvable by high-energy neutrino telescopes. Other emission regions, like outer parts of Cen A, might be more interesting and existing H.E.S.S. measurements reveal the possibility of such sources with a rather flat spectrum [26]. Future and more sophisticated experiments like KM3NET will help to obtain better observations, since it is optimized for the southern hemisphere and therefor sensitiv to lower TeV energies [27]. Since the neutrino flux is derived from the γ -measurements, uncertainties in γ -observations have an influence on the estimates of the neutrino flux and hence an influence on the difference between calculated neutrino flux and the sensitivity of the detector. The more precise the γ -observations are, the more precise are the calculated neutrino fluxes, resulting in an improved comparison between sensitivity and the flux. For this reason experiments with improved equipment like more sophisticated satellites and the future ground based Cherenkov detector array (CTA) will help to obtain more detailed γ -ray observations. If the spectrum of Cen A turns out to be significantly flatter than it is currently indicated by Fermi measurements, the central source might still be interesting for neutrino telescopes. /gl48/gl88/gl82/gl81o/gl49/gl72/gl88/gl87/gl85/gl76/gl81/gl82o/gl41/gl79/gl88/gl91o/gl50/gl73o/gl48s-", "pages": [ 12, 13 ] }, { "title": "5. Conclusions", "content": "In this paper, we derive limits on the product n H /f e / (0 . 1) for 33 FR-I galaxies, using the radio luminosity at 178 MHz and neutrino flux limits. The target density can be limited to be between 20 cm -3 and 1500 cm -3 , for a fixed f e = 0 . 1 for the sources. An explanation is the normalization of the sources. Two objects at the same location at the sky but different normalization, differ in the target density, since it is A ν ∝ n H , resulting in A ν 1 /A ν 2 = n H 2 /n H 1 for the two objects. Two objects with a different location but approximately same normalization, the ratio is n H 1 /n H 2 ∝ α 1 /α 2 , with α for the sensitivity of the detector for the object. Considering different models of the distribution of matter in AGN, we conclude that the innermost core models for proton acceleration can be excluded, as densities there are expected to be of the order of 10 9 cm -3 . An alternative explanation would be that the ratio of protons to electrons must be significantly lower than expected from standard theory [5, 28]. Gamma-ray observations exist for Cen A and M 87, the two closest and best studied AGN and they are used in this paper to estimate the average density of the emission region in case of hadronic interaction processes. For our calculation we used Fermi observations, since the observation period by Fermi reflects the average flux over a long-time measurement, rather than individual short-term measurements by Imaging Air Cherenkov Telescopes like MAGIC, VERITAS and H.E.S.S. Those provide interesting insights on the flaring behavior of the sources, but no reliable measurement of an average flux at this point. We find that the density required for gamma-ray emission in the core of Cen A and M 87 has to be of the order of 10 -100 cm -3 . The exact value again depends on the ratio of protons to electrons accelerated at the source. In the first part we assumed that γ -rays are only produced in the inner region and that the inner region is approximated by a spherical symmetry. We estimated the target density and compared these results with the densities we obtained by using the model explained in section 2. For Cen A the differences between the estimate and the model are in agreement within the uncertainty of the measurements, while for M 87 the situation is different. The differences can explained by our assumptions. Due to physical process, like clumping, the geometry can be very complex. Additionally γ -ray production might not only takes place in the inner region of the objects. In the case of M 87 neither radio luminositiy observations of the core region nor observations of the core region exist. This leads to a bigger difference between estimate and model. Future measurements will give more detailed information about the inner region of the sources and thus help with the identification of the sources of ultra-high energy cosmic rays.", "pages": [ 14 ] }, { "title": "Acknowledgements", "content": "We acknowledge generous constantly support from many scientists. We would like to thank Florian Schuppan, Matthias Mandelartz for their contribution to finish the paper. We also thank Peter Biermann and Anthony Brown for the very helpful discussions we had. Furthermore we would like to thank the IceCube collaboration for grantig information about the detector. IS and JBT furthermore acknowledge support from the DFG grant BE-3714-1, 'cosmic ray tracers from gas-rich active galaxies', as well as from the Research Department of Plasmas with complex Interactions Bochum.", "pages": [ 14, 15 ] }, { "title": "[1] A. A. Abdo et al.", "content": "Fermi Large Area Telescope View of the Core of the Radio Galaxy 1 Centaurus A The Astrophysical Journal, Volume 719, Issue 2, pp. 1433, 2010", "pages": [ 16 ] }, { "title": "[3] W. Bednarek & R. J. Protheroe", "content": "Gamma-ray and neutrino flares produced by protons accelerated on accretion disk surface in active galactic nuclei Monthly Notices of the Royal Astronomical Society, Volume 302, pp. 373, 1999", "pages": [ 16 ] }, { "title": "[4] J. K. Becker & P. Biermann", "content": "Neutrinos from active black holes sources of ultra high energy cosmic rays Astroparticle Physics, Volume 31, pp. 138, 2009", "pages": [ 16 ] }, { "title": "[5] R. Beck & M. Krause", "content": "Revised equipartition and minimum energy formula for magnetic field strength estimates from radio synchrotron observations Astronomische Nachrichten, Volume 326, pp. 414, 2005", "pages": [ 16 ] }, { "title": "[6] A. A. Abdo et al.", "content": "Fermi Large Area Telescope Observations of Misaligned AGN The Astrophysical Journal, Volume 720, pp. 912 , 2012", "pages": [ 16 ] }, { "title": "[7] S. R. Kelner & F. A. Aharonian", "content": "Energy spectra of gamma rays, electrons and neutrinos produced at proton proton interactions in the very high energy regime Physical Review D, Volume 74, 2006", "pages": [ 16 ] }, { "title": "[8] J. K. Becker", "content": "High-energy neutrinos in the context of multimessenger astrophysics Physics Reports, Volume 458, pp. 173, 2008", "pages": [ 16 ] }, { "title": "[9] A. Meli, J. K. Becker, J. J. Quenby", "content": "On the origin of ultra high energy cosmic rays: subluminal and superluminal relativistic shocks Astronomy and Astrophysics, Volume 492, pp. 323, 2008", "pages": [ 16 ] }, { "title": "[10] R. Abbasi et al.", "content": "Time-Integrated Searches for Point-like Sources of Neutrinos with the 40-String IceCube Detector The Astrophysical Journal, Volume 732, pp. 16, 2011", "pages": [ 16, 17 ] }, { "title": "[11] D. W. Hogg", "content": "Distance measures in cosmology arXiv:astro-ph/9905116v4, 1999", "pages": [ 17 ] }, { "title": "[12] B. L. Fanaroff & J. M. Riley", "content": "The morphology of extragalactic radio sources of high and low luminosity Monthly Notices of the Royal Astronomy Society, Volume 167, pp. 31, 1974", "pages": [ 17 ] }, { "title": "[13] M. Chiaberge, A. Capetti, A. Celotti", "content": "The HST view of FR I radio galaxies: evidence for non thermal nuclear sources Astronomy and Astrophysics, Volume 349, pp. 77, 1999", "pages": [ 17 ] }, { "title": "[14] C. M. Urry & P. Padovani", "content": "Unified Schemes For Radio-Loud Active Galactic Nuclei Publications of the Astronomical Society of the Pacific, Volume 107, pp. 803, 1995", "pages": [ 17 ] }, { "title": "[15] D. Kazanas & D. C. Ellison", "content": "The Central Engine Of Quasars And Active Galactic Nuclei: Hadronic Interactions Of Shock-Accelerated Relativistic Protons The Astrophysical Journal, Volume 304, pp. 178, 1986", "pages": [ 17 ] }, { "title": "[16] H. Falcke & P. Biermann", "content": "The jet-disk symbiosis I. Radio to X-ray emission models for quasars Astronomy and Astrophysics, Volume 293, pp. 665, 1995", "pages": [ 17 ] }, { "title": "[17] H. Falcke", "content": "Black Hole, Jet, and Disk: The Universal Engine Jets from Stars and Galactic Nuclei. Proceedings of a Workshop Held at Bad Honnef, Germany, 3 - 7 July 1995. SpringerVerlag Berlin Heidelberg New York. Also Lecture Notes in Physics, Volume 471", "pages": [ 17 ] }, { "title": "[18] H. Falcke, M. A. Malkan, P. L. Biermann", "content": "The jet-disk symbiosis II. Interpreting the radio/UV correla- tions in quasars Astronomy and Astrophysics, Volume 298, pp. 375, 1995", "pages": [ 17 ] }, { "title": "[19] J. Abraham et al.", "content": "Correlation of the Highest-Energy Cosmic Rays with Nearby Extragalactic Objects Science, Volume 318, pp. 938, 2007", "pages": [ 17 ] }, { "title": "[20] F. Aharonian et al.", "content": "Fast Variability of Tera-Electron Volt γ Rays from the Radio Galaxy M87 Science, Volume 314, pp. 1424, 2006", "pages": [ 18 ] }, { "title": "[21] J. Albert et al.", "content": "Very High Energy Gamma-Ray Observations of Strong Flar- ing Activity in M87 in 2008 February The Astrophysical Journal, Volume 685, pp. 23, 2008", "pages": [ 18 ] }, { "title": "[22] V. A. Acciari", "content": "Observation of Gamma-Ray Emission from the Galaxy M87 above 250 GeV with VERITAS The Astrophysical Journal, Volume 679, pp. 397, 2008", "pages": [ 18 ] }, { "title": "[23] A. A. Abdo et al.", "content": "Fermi Gamma-Ray Imaging of a radio Galaxy Science, Volume 328, pp. 725, 2010", "pages": [ 18 ] }, { "title": "[24] A. A. Abdo et al.", "content": "The Fermi-LAT collaboration Fermi Large Telescope Gamma-Ray Detection Of The Radio Galaxy M 87 Bulletin of American Astronomical Society, Volume 36, pp. 1112, 2010", "pages": [ 18 ] }, { "title": "[25] H. Alvarez, J. Aparici, P. Reich", "content": "The radio continuum spectrum of Centaurus A's large-scale components Astronomy and Astrophysics, Volume 355, pp. 863, 2000", "pages": [ 18 ] }, { "title": "[26] F. Aharonian", "content": "Discovery of Very High Energy γ -Ray Emission from Centaurus a with H.E.S.S. The Astrophysical Journal Letters, Volume 695, pp. 40, 2009", "pages": [ 18 ] }, { "title": "[27] A. Kappes", "content": "KM3NeT: A Next Generation Neutrino Telescope in the Mediterranean Sea arXiv:0711.0563, 2007", "pages": [ 18 ] }, { "title": "[28] R. Schlickeiser", "content": "Cosmic Ray Astrophysics ISBN 3-540-66465-3 Springer-Verlag, 2003", "pages": [ 18 ] } ]

[ { "title": "Bayesian Approach for Counting Experiment Statistics applied to a Neutrino Point Source Analysis", "content": "D. Bose, L. Brayeur, M. Casier, K. D. de Vries, G. Golup and N. van Eijndhoven Vrije Universiteit Brussel, Dienst ELEM Pleinlaan 2, B-1050 Brussels, Belgium", "pages": [ 1 ] }, { "title": "Abstract", "content": "In this paper we present a model independent analysis method following Bayesian statistics to analyse data from a generic counting experiment and apply it to the search for neutrinos from point sources. We discuss a test statistic defined following a Bayesian framework that will be used in the search for a signal. In case no signal is found, we derive an upper limit without the introduction of approximations. The Bayesian approach allows us to obtain the full probability density function for both the background and the signal rate. As such, we have direct access to any signal upper limit. The upper limit derivation directly compares with a frequentist approach and is robust in the case of low-counting observations. Furthermore, it allows also to account for previous upper limits obtained by other analyses via the concept of prior information without the need of the ad hoc application of trial factors. To investigate the validity of the presented Bayesian approach, we have applied this method to the public IceCube 40-string configuration data for 10 nearby blazars and we have obtained a flux upper limit, which is in agreement with the upper limits determined via a frequentist approach. Furthermore, the upper limit obtained compares well with the previously published result of IceCube, using the same data set. Key words: Neutrino Astronomy, Neutrino Telescopes, Active Galactic Nuclei, Bayesian Statistics.", "pages": [ 1 ] }, { "title": "1 Introduction", "content": "Bayesian approaches are gaining more and more popularity in scientific analyses [1,2,3,4,5]. In this paper we discuss a formalism following Bayesian inference [6,7] to analyse signals in a generic counting experiment and for illustration we apply it to a point source analysis using data from a neutrino telescope. A frequentist approach is based on the long run relative frequency of occurrences in identical repeats of an experiment. Consequently, this can only provide the probability for a certain outcome under the assumption of a specific hypothesis. On the other hand, the Bayesian approach allows us to directly compute the probability of any particular hypothesis or parameter value based on observations. One of the great strengths of Bayesian inference is the ability to incorporate relevant prior information in the analysis. This provides a mechanism for a statistical learning process that automatically takes previous results into account. Concerning a signal detection we will analyse the performance of the presented Bayesian approach and we will compare it to methods frequently found in literature. One of the standard test statistics is the frequentist method developed by Li and Ma [8]. While this method is based on the detection of an excess of number of events in a given angular window, the presented Bayesian method is sensitive to the distribution of the angular distance between the arrival direction of the observed events and the source position. In case no signal is observed, upper limits for the signal flux from the investigated point sources are determined. In the light of a flux upper limit determination we will discuss a method following the Bayesian framework and a comparison will be made to the standard frequentist methods developed by Feldman and Cousins [9] and Rolke et al. [10]. The Feldman-Cousins method introduces the likelihood ratio as an ordering principle, when determining the acceptance interval from which one derives the confidence interval. This construction resolves the issue of empty confidence intervals when the interval is entirely in the non-physical region. The Feldman-Cousins construction also resolves the 'flip-flop' problem, which can lead to under coverage. This arises when one wants to report an upper limit if the evidence for a signal is below a certain threshold, but a central confidence interval otherwise. The procedure outlined by Rolke et al. can deal with several nuisance parameters which include uncertainties (for example background expectations or signal efficiencies) by means of the Profile Likelihood method. However, this method has the disadvantage that large samples are assumed and thus applying it to cases with low counting statistics might lead to under coverage. The Bayesian method presented here will be applied to the study of Active Galactic Nuclei (AGN), which are, together with Gamma Ray Bursts (GRBs), among the leading candidates for the sources of ultra-high energy cosmic rays (UHECRs), as outlined in recent reviews such as [11,12]. On time scales of order of minutes or even seconds GRBs are transient phenomena, whereas AGN in general can be regarded as steady sources over time periods even exceeding several years. In case hadronic acceleration takes place in these objects, also an accompanying high-energy neutrino flux is expected due to the decay of secondary particles produced in the interactions of accelerated hadrons with the ambient photon field or matter [13]. Kilometer-scale neutrino detectors have the sensitivity to measure the predicted neutrino flux. The IceCube Neutrino Telescope [14], located at the geographic South Pole, was completed at the end of 2010 and has been taking data since 2006. In the Northern Hemisphere a kilometer-scale detector called KM3Net is being proposed for construction [15]. There are smaller telescopes, one located in the Mediterranean Sea, named ANTARES which has been collecting data since 2008 [16] and another one in Lake Baikal, Russia, which has been operating its NT200+ configuration since 2005 [17]. These telescopes detect high energy neutrinos ( E > 10 GeV) by observing Cherenkov radiation in ice or water from secondary particles produced in neutrino interactions. In the following we introduce the basis of the Bayesian formalism and present the test statistic that we will use to assess consistency of the data with the null hypothesis, being an isotropic distribution of the events. In Section 4, we describe the procedure to determine upper limits in case no significant signal is observed. Subsequently, in Section 5, we apply the method to IceCube public data [18] investigating the neutrino flux from the ten closest blazars (a subclass of AGN). Finally, a summary of our results and conclusions are provided in Section 6.", "pages": [ 1, 2 ] }, { "title": "2 Bayesian Formalism", "content": "We denote by p ( A,B | C ) the probability for hypothesis A and B to be true under the condition that C is true. The product rule [1], p ( A,B | C ) = p ( A | C ) p ( B | A,C ) = p ( B | C ) p ( A | B,C ), directly yields Bayes' theorem [19]: Bayes' theorem is extremely powerful in hypothesis testing. Consider a hypothesis H , some observed data D and prior information I . Bayes' theorem can then be rewritten as: where p ( H | D,I ) ≡ Posterior probability of hypothesis H. p ( H | I ) ≡ Prior probability of hypothesis H. p ( D | H,I ) ≡ Likelihood function, L ( H ) . p ( D | I ) ≡ Normalization factor. From Eq. (2) it is seen that the Bayesian formalism automatically provides a learning process. The first step is to encode our state of knowledge before analyzing the data into a prior probability p ( H | I ). This is then converted into a posterior probability p ( H | D,I ) when a new data set is analysed. We will apply this approach to the analysis of astrophysical point sources. This method will enable us to obtain the full Probability Density Function (PDF) for the source rate (i.e. the number of signal events per unit of time arriving from the source) from which we can derive the corresponding flux or any upper limit.", "pages": [ 2, 3 ] }, { "title": "3 Assessment of Significance", "content": "As outlined above, Bayesian inference allows us to make statements about the probability of various hypotheses in the light of obtained data. Following the developments described in [20], we quantify the degree to which data support a certain hypothesis and as such make an assessment of the significance. To quantify our degree of belief in a certain hypothesis H , one can use the so called evidence [1] where H indicates hypothesis H to be false. Due to the log 10 , this evidence is in a decibel scale. One can calculate [20] that the evidence e ( H | D,I ) for any alternative to hypothesis H based on the data D and prior information I is constrained by the observable Thus, ψ provides a reference to quantify our degree of belief in H . Let us now consider an experiment where the probabilities p k corresponding to the various outcomes A k on successive trials are independent and stationary. Such experiments belong to the so called Bernoulli class B m [1]. The probability p ( n 1 ....n m | B m , I ) of observing n k occurrences of each outcome A k after n trials is therefore given by the multinomial distribution: In terms of the observable ψ we then obtain [20] for each H ∈ B m : which is an exact expression. In case the data are represented in histogram form, the above can be applied if n is the total number of entries, m represents the number of bins, n k is the number of entries in bin k and p k is the probability for an entry to fall in bin k . Once the various p k are known, the ψ -value corresponding to a certain observed distribution can easily be obtained using Eq. (6). In our investigation of neutrino point sources, the total number of events n will populate a histogram according to the angular difference α of each neutrino arrival direction with respect to the source location in the sky. The p k values are determined assuming an isotropic background and taking into account the solid angle effect. By generating randomly the same total number of events ∗ following an isotropic distribution, we obtain the distribution of ψ for an isotropic background. Comparing the value of ψ for the data with the obtained background ψ -distribution, we determine the P-value or significance of our measurement. A source detection is claimed if the consistency of the data with the null hypothesis has a P-value smaller than 5 . 7 × 10 -7 , corresponding to a 5 σ effect in case of a positive single sided Gaussian distribution. It is important to note that as ψ is sensitive to the distribution of the data in the histogram, it is binning-dependent. To avoid this binning effect in the final P-value, the bin size is chosen such that there is only one or zero event per bin. In this special, quasi-unbinned approach, Eq. (6) would simplify in the following expression where the sum is running over the total number of events n . The p k probabilities are then the crucial factor that will differentiate between background events following an isotropic distribution and signal events from a source. The signal events will be located at angular distances around the source position following a Gaussian distribution with its standard deviation given by the experimental angular resolution convoluted with the solid angle effect. On the other hand, the test statistic developed by Li and Ma [8] is based on the absolute value of the number of events in the on-source ( N on ) and off-source ( N off ) angular windows. Consequently, both test statistics behave differently as a function of the considered angular window. As we will show in Section 5, the Li-Ma test statistic performs better than ψ for small angular windows while ψ is more sensitive at larger windows. Note that in Section 5 the complete expression of Eq. (6) is used instead of the simplified Eq. (7) since for our background distributions it occurs that n k > 1 in a small fraction of cases, which we control to be at maximum 5%. In case the observation does not lead to a significant detection, we determine an upper limit on the signal strength. To obtain such an upper limit we will use an exact analytical expression following a Bayesian approach using a uniform prior. The details of this procedure are provided in the following section.", "pages": [ 3, 4 ] }, { "title": "4 Bayesian upper limit determination", "content": "To obtain an upper limit for a possible source flux, we first have to determine the upper limit for the source rate. Using Bayesian inference we obtain the full posterior PDF for the source rate and from that we can derive the corresponding flux PDF, via the concept of effective area, as explained hereafter. In any experiment where events are detected at a known rate and independently of the time since the last event, the PDF for the number of observed events is described by a Poisson distribution. Our approach to obtain an upper limit follows the one outlined in [6,7], except that we apply the exact analytical expression without any approximation. The data consist of on-source and off-source measurements, where the off-source data consist only of background events and the on-source data are a mix of background and source ( i.e. signal) events. We start with the off-source analysis and subsequently use the obtained information as background prior information in the on-source analysis. This implies that the current study is completely data driven and as such is a model independent search for a possible source signal.", "pages": [ 4 ] }, { "title": "4.1 Off-source measurements", "content": "Consider the case that in an off-source measurement N off background events have been recorded over a time interval T off with a constant background rate b . Using Eq.(2) we obtain the posterior background PDF by: In the above equation the likelihood function p ( N off | b, I ) is given by the Poisson distribution corresponding to the measurement of N off background events over a time span T off at a constant rate b : Since the integrated PDF amounts to 1 ( i.e. ∫ b max b min p ( b | N off , I ) d b = 1), the normalisation factor p ( N off | I ) appearing in Eq. (8) is given by: As mentioned before, p ( b | I ) is the prior PDF for the background rate. For our analysis we use a uniform prior [7], which is given by The uniform prior attributes the same probability to every value of the background rate within the indicated range, reflecting that we do not favour a particular value of the actual background rate. This prior also has the advantage that the derived upper limits are directly comparable to classical frequentist upper limits [5]. To cover the full range of possible background rates, the minimum value of the rate b is taken to be zero and Eq. (11) can be written as Using this expression for p ( b | I ) together with Eqs. (9) and (10) we obtain an analytical expression for the normalisation factor Solving the above equation (for details see Appendix A1) we get: where γ ( a, x ) is the Incomplete Gamma Function. Substitution of Eqs. (9), (12) and (14) in Eq. (8) yields which represents the posterior background rate PDF. In [6,7] the Incomplete Gamma Function is then approximated to γ ( N off +1 , b max T off ) ≈ Γ( N off +1) = N off !. This approximation is valid for T off b max /greatermuch N off and cannot be applied in general when the number of events and time window are small (see [20] for an example). We therefore work with the complete analytical expression.", "pages": [ 4, 5 ] }, { "title": "4.2 On-source measurements", "content": "Consider the case that in an on-source measurement N on events, consisting of signal and background, have been recorded over a time interval T on with a constant signal rate s and background rate b . Following Eq. (2) the joint probability of source and background is given by: Using the product rule [1] we can write the above equation as, where p ( b | I ) is the prior probability for the background rate, which is in our case the posterior background PDF obtained from the off-source measurement reflected in Eq. (15). The likelihood function p ( N on | s, b, I ) is the Poisson distribution for the combined signal and background rate ( s + b ). The normalisation constant p ( N on | I ) is obtained, as outlined in the previous subsection, by integrating the numerator of Eq. (17). It is important to note that since the source rate s and the background rate b are independent, we can write p ( s | b, I ) = p ( s | I ). Like for the background case discussed before, we use a uniform prior for p ( s | I ), i.e. As mentioned before, the uniform prior attributes the same probability to every value of the signal rate within the indicated range, reflecting that we do not favour a particular value of the actual signal rate and also allows us to directly compare the derived upper limits with the classical frequentist results [5]. By substituting in Eq. (17) the expressions of Eq. (18) for p ( s | b, I ), (15) for p ( b | I ) and (9) for a total rate ( s + b ) we obtain the joint PDF for the source and background rates: However, we are interested in the posterior PDF for the source rate alone, independent of the background. The Bayesian formalism allows us to obtain this posterior PDF by marginalisation [1] of the joint PDF, Eq. (19), with respect to the background i.e. Solving the above integral we obtain an exact expression for the source rate posterior PDF (for details see Appendix A2): where u max ≡ b max ( T on + T off ) and N ≡ N on + N off . As explained before we use this exact analytical expression without the approximation used in [6,7]. In case no significant source signal is observed, Eq. (21) allows us to derive any upper limit for the source rate. As an example, the 90% source rate upper limit s u.l. is given by:", "pages": [ 5, 6 ] }, { "title": "5 Application of the method to a neutrino point source analysis", "content": "The approach presented in the current paper is not exploiting a potential source variability when it comes to providing upper limits to a possible source rate, as outlined in Section 6. Consequently the analysis presented here is tailored for sources with a steady rate within the detection time. Our primary goal is the analysis of AGN which, at the time scale considered here, may be regarded to be steady sources of high-energy neutrinos if hadronic acceleration takes place in these sites. By selecting a small region around every well known source location, several sky patches are defined from which data were collected to search for a possible deviation from the background 'noise'. To validate the analysis procedure described in this report, we use the public data [18] of the muon neutrino candidate events recorded by the IceCube Neutrino Observatory [14,21] in its 40 string configuration (IC40), that collected data during the season 2008-2009. Our analysis is performed on ten nearby blazars (a special class of AGN with one of the jets pointing in the direction of the Earth) following the approach described in the previous sections. The blazars were selected from the online 'Roma BZCAT Multi-frequency Catalogue of Blazars' [22] and are listed in Table 1. These blazars are chosen to be nearby, i.e. with a small redshift, and in such a way that their respective angular windows are not overlapping. We limit ourselves to sources in the Northern hemisphere to reduce the atmospheric muon background for the IceCube measurements.", "pages": [ 6 ] }, { "title": "5.1 Assessment of significance", "content": "As outlined in [20], we stack the recorded events within a given angular window centered on each of these ten blazars according to their angular distance α from the actual blazar position. As mentioned in the previous sections, we need an expression for the various probabilities p k of Eq. (6) to derive the ψ -value of the signal. In our case, the background is isotropic and consequently the probabilities have to be consistent with the solid angle effect within the selected cone, i.e. : where w is the width of each bin in our stacked histogram and α max is the size of the angular window. To determine the angular window size for which our test-statistic is most sensitive, we have generated 266 (being the number of events of the actual observation as outlined hereafter) isotropically distributed events plus a signal of 20 events. These 20 events were generated such that their angular distance α from the source position follows a Gaussian distribution (with standard deviation of 1 degree) convoluted with the solid angle effect. The chosen standard deviation of the Gaussian distribution is the angular uncertainty of the IceCube track reconstruction [21,23]. As we mentioned before, our test statistic is sensitive to the distribution of events as a function of the distance to the source. To avoid the dependence on the position generation in our determination of the most optimal angular window size, we have repeated the procedure 10 times for each value of α for the signal events and present the mean P-values in Fig. 1. From Fig. 1 it is seen that our ψ test statistic does not perform well on small angular windows. This is due to the fact that the difference of the individual probabilities per bin ( p k ) for small angular windows is not large enough to distinguish a source-like distribution from an isotropic background ( i.e. a larger angular window is needed to see the 'shape' of the excess). For an angular window of 4 degrees, we see that the sensitivity obtained with ψ becomes optimal. We also include in Fig. 1 a comparison to the P-value obtained with the standard Li-Ma method. As mentioned in Section 3, Li-Ma is a test statistic based on the total number of events in the on-source ( N on ) and off-source ( N off ) angular windows. Li-Ma performs better than ψ for small angular windows (a large N on /N off ratio). When comparing the smallest P-values of each test statistic, we see that both are similar. Fixing α max to 4 degrees, we use the IceCube public data and we obtain the stacked distribution of events presented in Fig. 2. These stacked data comprise 266 events recorded over a time period of 375.5 days [24]. Using Eqs. (6) and (23) for a number of entries N on = 266, the data represented in Fig. 2 yield ψ observed = 10621 dB. As explained in Section 3, the ψ distribution in the case of an isotropic background is obtained by randomly generating 10 6 times the same number of events as in the on-source region. The distribution is presented in Fig. 3. Comparison of the actual observation ψ observed with the background distribution ψ bkg gives a P -value of 0 . 15. Consequently, we will proceed to give an upper limit on the signal strength.", "pages": [ 6, 7, 8 ] }, { "title": "5.2.1 Determination of the background rate", "content": "To determine the number of events in the off-source region ( N off ) we perform measurements in 4 · regions of the sky, shifted from the various blazars positions only in right ascension, keeping the declination constant due to the declination dependence of the IceCube detection efficiency. The specific IC40 configuration of IceCube is also right ascension dependent, so we make shifts of 180 · in right ascension to eliminate the right ascension dependence. The IC40 sample has been taken over a detector live time period of 375.5 days, so that both the exposures for on-source, T on , and off-source, T off , amount to 375.5 days. The stacked off-source measurements yield a total of 265 events. The posterior background rate PDF is obtained by substitution of the previously mentioned values of T off and N off in Eq. (15) and by using a sufficiently large value b max = N off T off · 100 =0.8 mHz. The resulting background rate PDF is shown in Fig. 4.", "pages": [ 9 ] }, { "title": "5.2.2 Determination of the source rate", "content": "The posterior source rate PDF is obtained by inserting the previously mentioned values of T on , T off , N on and N off in Eq. (21) and by using a sufficiently large value s max = 1 Hz. The resulting source rate PDF is shown in Fig. 5. Using the PDF shown in Fig. 5 and applying Eq. (22), we obtain the 90% upper limit for the source rate: To compare this Bayesian method with the frequentist approach, we have also determined the 90% upper limit for the source rate using the Feldman-Cousins [9] and Rolke et al. [10] methods. The values obtained are the following: We see that the Bayesian approach is equal to the Rolke et al. method and is more conservative than the Feldman-Cousins method. To further test the upper limit calculation, we have generated source signals (or under-fluctuations) by increasing (or decreasing) the number of events in the on-source region, while keeping the same number of background events as in the data. In Fig. 6 we plot the rate upper limits obtained with the Bayesian and frequentist methods as a function of the difference of the number of events between the on-source and off-source regions ( N on -N off ). We also show the actual rate for the case of a positive difference of N on -N off . The Bayesian upper limits are similar to the Rolke et al. limits for a small difference of N on -N off but the former is more restrictive when this difference increases. When comparing to the Feldman-Cousins results, the Bayesian limits are more conservative for low N on -N off and tend to the Feldman-Cousins limits as this difference grows, as expected from the fact that we use an uniform prior. The decrease of the slope when the difference of the number of events is negative shows that the Bayesian method is better protected against under-fluctuations. This effect is shown in Fig. 7, where we consider the background fluctuations by generating isotropic distributions of events in the sky and compute each time N ' on = N on + N s and N off (with N s , the generated source events). Fig. 7 shows the computed event rate upper limit for the Bayesian and Feldman-Cousins methods as a function of the generated N s events. We see that the decrease of the slopes of the upper limit determinations (Fig. 6) result in an upper limit that can fall below the actual generated rate. This problem occurs less often for the Bayesian method because the decrease of the slope is less steep compared to Feldman-Cousins. Moreover, as we assume an uniform prior, the Bayesian limit is equal to the Feldman-Cousins for large over-fluctuations and this translates to Fig. 7 by having the same values for the largest upper limits of the rate. Note that the obtained rate does not take into account the reconstruction efficiency. The latter is taken into account by converting the source rate upper limit into a flux upper limit by means of the so called Effective Area, A eff , which is defined as For the current analysis we use the angle averaged Effective Area determined from a simulated E -2 spectrum [24], taking into account the observed energy estimate for each individual observed event [18]. The median value corresponds to A eff = 2 . 2 × 10 6 cm 2 over the considered energy range. Our analysis is performed on a circular area of 4 · centered on each of the 10 sources, representing in total 10 × 0 . 0153 sr = 0 . 153 sr. From the result of s u . l . and taking the effective area and the size of the onsource region into account we arrive at a 90% upper limit for an E -2 signal flux of Φ u . l . = s u . l . A eff · 0 . 153 = ----", "pages": [ 9, 10, 11 ] }, { "title": "3 . 6 × 10 12 TeVs 1 cm 2 sr 1 .", "content": "However, this flux upper limit does not take into account the effect of neutrino oscillations. At the source, astrophysical models predict a flavor ratio of ν µ : ν e : ν τ = 2 : 1 : 0. Assuming maximum oscillation we expect to observe at Earth ν µ : ν e : ν τ = 1 : 1 : 1. A tiny fraction of the ν τ will produce a muon which might also be detected in IceCube and as such have entered our event sample. However, we will neglect this effect since it is marginal and would require a special simulation which is beyond the scope of this paper. So our final value for the 90% upper limit for a E -2 signal flux is: For consistency checking, we can compare our limit to the result published by the IceCube Collaboration [24], which was obtained with a different analysis concerning a search for a diffuse high-energy neutrino flux in the full Northern hemisphere: Φ u . l . = 8 . 9 × 10 -12 TeV s -1 cm -2 sr -1 , which is comparable to our result. In that analysis the same data set was used and since the ten blazars we studied are randomly located in the sky and have not been selected based on any (astro)physical characteristics, the 4 · windows around them represent a fair sample of the sky which can be used to compare to a diffuse search.", "pages": [ 11 ] }, { "title": "6 Conclusion and Outlook", "content": "In this paper we have discussed a statistical method to analyse point sources using data from a neutrino telescope following Bayesian inference. Using the observable ψ , we have indicated how to assess the significance of a possible signal in the data by comparing it to the ψ distribution expected for an isotropic background. We have shown how to obtain upper limits for the corresponding flux in case the observation does not lead to a significant signal detection. Our calculations are similar to [6,7] but we have made no approximations in the final results and thus this method can be applied to low counting observations. Applying this method we have analysed the public IceCube 40-string configuration data for 10 nearby blazars located in the Northern sky. From our analysis it was also seen that the on-source data is consistent with an isotropic background only hypothesis. Therefore we have determined a 90% upper limit, which has been compared to the upper limits obtained using the same data set but applying the Feldman-Cousins and the Rolke et al. methods. Simulating a signal from a source, by artificially changing the number of events in the on-source region, we have shown that the Bayesian limits are similar to the Rolke et al. calculations for small difference in the number of events between the on-source and off-source region and tend to the Feldman-Cousins limits as this difference in the number of events increases. We have shown that in the case of under-fluctuations in the background the Bayesian method is better protected. It is our intention to extend the current method also for non-steady sources like for instance GRBs. Apart from providing a signal significance for discovery [20], this should also provide a mechanism to accurately determine flux upper limits. For flaring sources we do not know the time window in which the neutrinos are emitted. If we take a time window large enough to cover all possible scenarios for neutrino emission, we would obtain a rate which is not the actual one (because of the existence of time intervals with and without neutrino emission within our time window). The proper extension of the method is currently under study.", "pages": [ 11, 12 ] }, { "title": "Acknowledgements", "content": "The authors would like to thank the IceCube Collaboration for providing the public data used in this report to evaluate our analysis method. This research was performed with financial support from the Odysseus programme of the Flemish Foundation for Scientific Research (FWO) under contract number G.0917.09.", "pages": [ 12 ] }, { "title": "Appendix A1", "content": "The normalisation factor of the posterior background rate PDF given in Eq. (13) may be written as: The integral part can be expressed as the so-called Incomplete Gamma function given by: Using this expression, we can rewrite Eq. (24) as follows: which is the expression reflected in Eq. (14).", "pages": [ 13 ] }, { "title": "Appendix A2", "content": "According to Eq. (20) the posterior PDF for the source rate alone is given by: where Substitution of the various expressions given in Section 4 yields: Combination of Eqs. (28) and (29) yields: Considering the numerator of Eq. (30), we obtain: Using the Newtonian Binomial, ( a + b ) n = ∑ n i =0 n ! i !( n -i )! a n -i b i , we find: Using the Incomplete Gamma function, Eq. (25), and simplifying the above equation, we finally find where u max = b max ( T on + T off ) and N = N on + N off . Applying the same procedure, we obtain for the denominator: Which finally gives, after simplification, for the posterior source rate probability density function:", "pages": [ 13, 14 ] } ]

[ { "title": "ABSTRACT", "content": "Recent Trends in the Study of Compact Objects: Theory and Observation ASI Conference Series, 2013, Vol. **, pp **-** Edited by Santabrata Das, Anuj Nandi and Indranil Chattopadhyay", "pages": [ 1 ] }, { "title": "Nucleosynthesis inside accretion disks and outflows formed during core collapse of massive stars", "content": "Indrani Banerjee 1 ∗ 1 Department of Physics, Indian Institute of Science, Bangalore 560012, India Received - ; accepted - Abstract. We investigate nucleosynthesis inside the gamma-ray burst (GRB) accretion disks and in the outflows launched from these disks mainly in the context of Type II collapsars. We report the synthesis of several unusual nuclei like 31 P, 39 K, 43 Sc, 35 Cl and various isotopes of titanium, vanadium, chromium, manganese and copper in the disk. We also confirm the presence of iron-group and α -elements in the disk, as shown by previous authors. Much of these heavy elements thus synthesized are ejected from the disk and survive in the outflows. While emission lines of several of these elements have been observed in the X-ray afterglows of GRBs by BeppoSAX, Chandra, XMM-Newton etc., Swift seems to have not found these lines yet. Keywords : accretion, accretion disks - gamma rays: bursts - collapsars -nucleosynthesis - abundance", "pages": [ 1 ] }, { "title": "1. Introduction", "content": "We plan to investigate nucleosynthesis in the accretion disks formed by the Type II collapsars where the accretion rate ( ˙ M ) is: 0 . 0001 M /circledot s -1 /lessorsimilar ˙ M /lessorsimilar 0 . 01 M /circledot s -1 , when M /circledot indicates solar mass, as this regime is the ideal site for the synthesis of heavy elements. These disks are predominantly advection dominated. However, neutrino cooling becomes important in the inner disk where the temperature and density are higher. We also consider nucleosynthesis in the outflows from these disks and report that many of the heavy elements thus synthesized in the disk do survive in the outflow. Moreover, depending on the abundance of 56 Ni synthesized in the outflow, we can predict whether the outflow will lead to an observable supernova explosion or not.", "pages": [ 1 ] }, { "title": "2. Disk and Outflow Models", "content": "The accretion disk formed in a Type II collapsar is modelled within the framework suggested by Kohri et al. (2005) where the electron degeneracy pressure and the evolving neutron to proton ratio are appropriately calculated. Height-averaged equations based on a pseudo-Newtonian framework as suggested by Mukhopadhyay (2002) is used . Following Fujimoto et al. (2004), we adopt a spherically expanding, onedimensional and adiabatic outflow model to investigate nucleosynthesis in the outflow. Since ˙ M is very high, it is always possible that the matter may get deposited onto the accretion disk which favors outflow. Outflows may also be due to magnetic centrifugal force and viscosity. We use well tested nuclear network code as has been used by Mukhopadhyay & Chakrabarti (2000). We have modified this code further by increasing the nuclear network and including reaction rates from the JINA Reaclib Database, https: // groups.nscl.msu.edu / jina / reaclib / db / (Cybert et al. 2010). We use He-rich and Si-rich abundances as the initial conditions of nucleosynthesis at the outer disk. We also consider outflow from various radii of ejection, Rej , with Rej < 200 R g , R g being Schwarzschild radius and evaluate the abundance evolution in the outflow assuming the initial composition the same as in the accretion disk at Rej .", "pages": [ 3 ] }, { "title": "3. Nucleosynthesis inside accretion disks and outflows", "content": "Figures 1(a) and 1(c) illustrate the abundance evolution in the accretion disk around a 3 M /circledot Schwarzschild black hole accreting at ˙ M = 0 . 001 M /circledot s -1 , with the viscosity parameter α = 0 . 01 and the composition of the accreting gas at the outer disk similar to the pre-supernova He-rich and Si-rich layer respectively. They depict that the disks comprise of several zones characterized by dominant elements. In Fig. 1(a) the region ∼ 1000 -300 R g , is mainly the 40 Ca, 44 Ti and 48 Cr rich zone. This is because unburnt 36 Ar undergoes α -capture reaction to give rise to 40 Ca through 36 Ar( α, γ ) 40 Ca, which undergoes partial α -capture to give rise to 44 Ti and 48 Cr. Inside this region, the temperature and density in the disk favor complete photodisintegration of 44 Ti and 48 Cr resulting in the formation of 40 Ca, 36 Ar, 32 S and 28 Si, as is evident from Fig. 1(a). Subsequently, 28 Si and 32 S start burning, which favors formation of iron-group elements via photodisintegration rearrangement reactions (Clayton 1968). Therefore, in the range ∼ 300 -80 R g , there is a zone overabundant in 56 Ni, 54 Fe, 32 S and 28 Si. Inside this zone, all the heavy elements photodisintegrate to 4 He, neutron and proton. In Figure 1(c) the disk has a huge zone rich in 28 Si and 32 S extending from 1000 R g to 250 R g . Inside this radius, silicon burning commences and soon the disk becomes rich in 54 Fe, 56 Ni and 58 Ni. Inside ∼ 70 R g , all the heavy elements again get photodisintegrated to α -s and free nucleons. Another remarkable feature in the He-rich and Si-rich disks is that inside ∼ 100 R g , the abundances of various elements start becoming almost identical as if once threshold density and temperature are achieved, the nuclear reactions follow only the underlying disk hydrodynamics. Banerjee & Mukhopadhyay (2013) gives the details of the nuclear reactions. On increasing ˙ M ten times we find that the individual zones in both disks shift outward retaining similar composition as is in the low ˙ M cases described above. Next we consider outflow from 40 R g , which lies in the He-rich zone of the aforementioned He-rich disk. The abundance evolution in the outflow is shown in Fig. 1(b). We find that 56 Ni is copiously synthesized along with isotopes of copper and zinc. Presence of 56 Ni in the outflow signifies that it will result in an observable supernova explosion. Figure 1(b) also depicts that on changing the initial velocity of ejection the final abundances of the nucleosynthesis products change significantly. More 56 Ni is synthesized when the velocity of ejection is low (see Fig. 1(b)) because then the temperature drops slowly in the ejecta which facilitates greater recombination of alphas to nickel. Figure 1(d) depicts the abundance evolution in the outflow from 180 R g of the above mentioned Si-rich disk. We choose this radius of ejection because outflow from the He-rich zone yields similar results as in Fig. 1(b). Outflow from the Si-rich zone remains rich in 28 Si and 32 S. 56 Ni is hardly synthesized and there will be no observable supernova explosion.", "pages": [ 3, 4 ] }, { "title": "4. Summary and conclusions", "content": "Apart from the synthesis of iron-group and α -elements we report for the first time, to the best of our knowledge, that several unusual nuclei like 31 P, 39 K, 43 Sc, 35 Cl and various uncommon isotopes of titanium, vanadium, chromium, manganese and copper are synthesized in the disk. Several of these heavy elements survive in the outflow from these disks, and when 56 Ni is abundantly synthesized in the outflow, there is always a supernova explosion.", "pages": [ 4 ] }, { "title": "Acknowledgements", "content": "I would like to thank Banibrata Mukhopadhyay for suggesting the problem and discussing throughout the course of this work. This work was partly supported by the ISRO grant ISRO / RES / 2 / 367 / 10-11.", "pages": [ 4 ] }, { "title": "References", "content": "Banerjee, I., & Mukhopadhyay, B. 2013, RAA, 13,1063; arXiv:1305.1755 Clayton D.D., 1968, Principles of Stellar Evolution and Nucleosynthesis (New York: McGraw-Hill) Cybert R.H., Amthor A.M., Ferguson R., 2010, ApJS, 189, 240 Fujimoto S.-I., Hashimoto M., Arai K., Matsuba R., 2004, ApJ, 614, 847 Kohri K., Narayan R., Piran T., 2005, ApJ, 629, 341 Matsuba R., Arai K., Fujimoto S.-I., Hashimoto M., 2004, PASJ, 56, 407 Mukhopadhyay B., 2002, ApJ, 581, 427 Mukhopadhyay B., Chakrabarti S.K., 2000, A &A, 353, 1029", "pages": [ 4 ] } ]

[ { "title": "A multi-wavelength investigation of newly discovered planetary nebulae in the Large Magellanic Cloud: Central stars", "content": "Warren Reid 1 , 2 1 Department of Physics and Astronomy, Macquarie University, North Ryde, Sydney, NSW 2109 Australia 2 Centre for Astronomy, Astrophysics and Astrophotonics, Macquarie University, North Ryde, Sydney, NSW 2109 Australia Abstract. Having completed my search for faint PNe in the LMC, including the outer 64 deg 2 area not covered in the original UKST survey, I now have the most complete number of PNe within any galaxy with which to assess stellar parameters. I present preliminary estimates for planetary nebula central star temperatures for 688 LMC PNe using the excitation class parameter derived from emission lines in the nebula. These are then compared to a photoionisation model in order to evaluate the contribution of metallicity when determining stellar temperatures using only emission lines. I include measurements from my latest confirmatory spectroscopic observations which have yielded a further 110 new LMC PNe while confirming the 102 previously known PNe in the outer LMC. These observations, providing low and medium resolution spectra from 3650Å to 6900Å, have been added to my comparable data for PNe in the central 25deg 2 of the LMC. The combined data were used to measure fluxes in preparation for a number of projects related to luminosity functions, chemical abundances, central star properties and LMC kinematics. Here I provide a preliminary look at the range of derived central star e ff ective temperature estimates. I also show a correlation between the central star temperatures and the expansion velocity of the nebula.", "pages": [ 1 ] }, { "title": "1. Introduction", "content": "In recent years considerable progress has been made in understanding the evolutionary sequence of planetary nebulae (PNe). The evolution of the photoionised nebula needs to be understood with regard to the processes leading to its ejection, mass / density relation, chemical composition and the post-AGB evolution of the central star. The central star in particular is the driving force, both ejecting the nebula and then releasing fast winds, driven by radiation pressure, which compress and accelerate the pre-ejected material, creating thin, ionised shells. Since a strong link has been observationally established between the parameters of the central star and those of the surrounding nebula (eg. Dopita et al. (1987, 1988), Dopita & Meatheringham (1990), Schmidt-Voigt & Koppen (1987), Stasi'nska (1989)), it follows that certain parameters of the central star can be determined indirectly by measuring key emission lines in the nebula. This is especially useful in the LMC where the central star cannot be directly observed. Over the past couple of years we have used both the UKST H α and short red maps of the central 25deg 2 region of the LMC to uncover over 460 candidate PNe. These were labeled as 'true', 'likely' and 'possible' depending on the quality of images and confirmatory spectra obtained. To these were added the 169 PNe that were previously catalogued in that area. Spectroscopically confirmed results including calibrated fluxes, luminosity functions and radial velocities were published in Reid & Parker (2006a,b, 2010a). I have now extended our survey to the outer regions of the LMC mainly using the [O iii ], [S ii ] and H α images provided by the Magellanic Cloud Emission Line Survey (MCELS). From the 1,000 or so candidates selected for spectroscopic followup, I identified 110 newly discovered and 101 previously known PNe. The complete sample, comprising 749 LMC PNe spanning the entire galaxy, has the advantage of being at a near common, known distance (49.2 kpc, Reid & Parker (2010a)) with low reddening, yet close enough to be studied in detail. It is currently the most complete PN sample in existence for any galaxy (Reid 2012). The objective of this preliminary work is to compare the temperature of the central stars to the excitation and expansion velocity of the nebulae. This allows me to investigate the evolution of both the nebula and central star as it evolves into a white dwarf.", "pages": [ 1, 2 ] }, { "title": "2. Observational data for the LMC PNe", "content": "Follow-up spectroscopy was mainly performed on the AAT using AAOmega which comprises 400 fibres placed by robotics across a 2 degree field of view. Three nights of observations in February 2010 plus three field observations in February 2012 provided coverage of the most concentrated outer areas. For more extended outer areas of the LMC where the density of candidates was too low for AAOmega I used 6dF on the UK Schmidt telescope. This instrument operates essentially the same way but covers a larger, 6 degree area of the sky while using only 150 fibres. Flux calibration was conducted using the method described in Reid & Parker (2010a) where data counts are calibrated to fluxes from HST observations for the same objects. This method has proved very reliable and allows the whole dataset to be homogeneously calibrated. Additional spectroscopy for PNe in the inner main bar regions was obtained using FLAMES on the VLT, the 1.9m telescope at the South African Astronomical Observatory and the 2.3m telescope at Siding Spring Observatory. While the long-slit spectra were reduced using standard IRAF tasks, the FLAMES multi-fibre data were flux calibrated using the method described for AAOmega and 6dF data (Reid & Parker 2010a).", "pages": [ 2 ] }, { "title": "3. PN central star temperatures", "content": "Without the ability to individually pinpoint and observe the central stars of LMC PNe, I use photoinisation models (Dopita et al. 1992; Reid & Parker 2010b) that demonstrate that for optically thick PNe in the Magellanic Clouds, the excitation class parameter is related to stellar temperature. The equation to estimate low excitation is given by: while the high excitation PNe are estimated by Using this definition, a transformation from excitation class to stellar e ff ective temperature ( T e ff ) was made using: which is based on the transformation given in Dopita et al. (1992) but adjusted to match the Zanstra temperatures published by Villaver et al. (2003, 2007) (see Reid & Parker (2010b)). For average abundance levels within the LMC, this equation provides a useful transformation to stellar temperatures. Dopita et al. (1992) also expected this relation to work well, having tested it using 66 of the brightest PNe in the LMC, but predicted the relationship would break down for low excitation PNe. The reason given for this was the strong dependency of the [O iii ] / H β ratio on metallicity as well as upon stellar temperature. In order to correct for any over-dependency on the metallicity introduced by using the [O iii ] / H β ratio, Dopita et al. (1992) constructed a grid, based on covering a range of stellar temperatures and metallicities using the generalised modeling code MAPPINGS (Binette et al. 1985). They use an ionisation parameter defined as Q = N Ly -c / 4 π 〈 r 2 〉 N H where N Ly -c is the number of Lyman continuum photons emitted by the central star, 〈 r 2 〉 is the mean radius of the ionised nebula and N H is the nebula's hydrogen particle density. By adopting a high value for Q (2 × 10 8 cm s -1 ), they simulate stellar luminosity and nebula gas pressure typical of the brighter PNe in the LMC as well as those in the Galactic Bulge. The resulting grids, encompassing abundances from 0.1 to 2.0 times solar, each with a set of temperatures between 35,000 and 140,000 K, encompass the maximum luminosity range for PNe in both the H β and [O iii ] λ 5007 lines. Importantly, although these grids have been available for 20 years, they have not been tested against medium to faint and evolved PNe in the LMC, typical of those that would be found in the 0.0 < E < 5.0 excitation bracket. With our improvements to the original formulas given for excitation class and temperature, I need to investigate whether our new temperature estimates agree with the temperatures found from the modeled grid of Dopita et al. (1992).", "pages": [ 2, 3 ] }, { "title": "3.1. Results", "content": "I compared central star temperatures for high, medium and low excitation PNe, derived using our formulae (equations 1 & 2) with central star temperatures acquired using the modeled grid of Dopita et al. (1992). The grid relies on the [O iii ] λ 5007 / H β ratio and the electron temperature ( T e ) in order to produce an estimate of log (Z) and ( T e ff ). For low excitation PNe, the similar reliance on the [O iii ] λ 5007 / H β ratio means that the only di ff erence will be introduced by T e . For low excitation PNe I find an exponential fit between temperatures derived directly from the excitation class (equation 3) and those derived from the grid of Dopita et al. (1992). In order to show this relation, a curve has been fitted to the data (black circles) in Figure 1. For comparison I also show the results for medium to high excitation PNe (red-filled boxes) and low excitation which do not fit the grid (green triangles). High excitation PNe do not correlate to central star temperatures derived using excitation class (equation 2). Clearly, the reason is that high excitation PNe require the use of the HeII λ 4686 line in order to obtain T e ff estimates. The [O iii ] λ 5007 / H β ratio and T e alone do not measure su ffi cient levels of excitation to permit the estimation of high central star temperatures. This result agrees with the warning given by Dopita et al. (1992) in which they find that the grid is not very useful for determining stellar temperatures where T e ff > 90,000 K and log [Z] < -0.5. Although an exponential correlation is found for most low excitation PNe, there is a subgroup that return higher T e ff . Using the grid, low excitation PNe with T e higher than 12,000 K and log (Z) less than -1.0 have increasingly higher T e ff estimates than those found using equation 1. For this reason I suggest that the grid is not useful for estimating T e ff where ( T e ) are greater than 12,000 K, even though the grid allows the estimation of T e ff using T e up to 15000 K. The exponential curve for those low excitation PNe with T e below 12,000 K follows the form: where T e ff [ grid ] is the stellar e ff ective temperature found from the grid and T e ff [ E ] is the stellar e ff ective temperature found from equations 1 and 3 for low excitation PNe. At low T e ff , the grid and excitation class produce near equivalent results but as T e ff increases, T e has the e ff ect of exponentially decreasing T e ff estimates produced by the model. Our previous comparisons of equations 1, 2 & 3 with T e ff estimates using the Zanstra method (Reid & Parker 2010b) show a good correlation where the nebulae are optically thick. In this case there is an increasing decline in grid temperature estimates where they are compared to Zanstra and excitation (equation 3) temperature estimates. Furthermore, with T e greater than 12,000 K the grid produces a number of the low excitation PNe with inflated T e ff . This is presumably the result of an over correction for the e ff ect of metallicity within the central star. Since there is little correlation between T e and any method used to produce a T e ff estimate, I have decided to use equations 1, 2 & 3 alone to estimate my central star e ff ective temperatures for this presentation. My central star e ff ective temperatures are shown in Figure 2 where the temperatures range from 28,000 K to 291,000 K with a mean of 90,300 K. Since there is a correlation between excitation class and T e ff , it follows that there is also a moderate correlation between T e ff and the expansion velocity of the surrounding nebula. In Figure 3 I show the derived expansion velocity of the nebula versus the T e ff from equation 3. This correlation was first discovered by Dopita et al. (1985) and later improved using a two parameter fit which included the excitation class and the H β flux (Dopita & Meatheringham 1990). The equation for estimating the expansion velocity is given as equation 3.2 in Dopita & Meatheringham (1990). With a strong relationship between excitation class, the H β flux and the Zanstra temperature of the central star (Morgan 1984), the position of a PN on plots such as Figure 3, representing the relationship between the nebula expansion velocity and T e ff will depend principally on the optical density, mass of the nebula and intrinsic properties of the central star. Since the most massive stars achieve the highest temperatures, the excitation class should also follow the mass of the star. Massive central stars fade rapidly (as seen in the brightest 4 magnitudes of the PNLF (Reid & Parker 2010a)) so when low H β fluxes are associated with high-excitation nebulae we can confidently assume the presence of a massive central star. Such stars drive high expansion velocities in the nebula, delivering high energy and making them more e ffi cient at ionising the surrounding AGB wind.", "pages": [ 3, 4, 5, 6 ] }, { "title": "References", "content": "Binette, L., Dopita, M. A., & Tuohy, I. R. 1985, ApJ, 297, 476 Dopita, M., Ford, H., Lawrence, C., & Webster, B. 1985, ApJ, 296, 390 Dopita, M., Jacoby, G., & Vassiliadis, E. 1992, ApJ, 389, 27 Dopita, M., & Meatheringham, S. 1990, ApJ, 357, 140 Dopita, M., Meatheringham, S., Webster, B., & Ford, H. 1988, ApJ, 327, 639 Dopita, M., Meatheringham, S., Wood, P., Webster, B., Morgan, D., & Ford, H. 1987, ApJ, 315, 107 Morgan, D. 1984, MNRAS, 209, 241 Reid, W. 2012, IAUS, 283, 227 Reid, W., & Parker, Q. 2006a, MNRAS, 365, 401 -2006b, MNRAS, 373, 521 -2010a, MNRAS, 405, 1349 -2010b, PASA, 27, 187 Schmidt-Voigt, M., & Koppen, J. 1987, å, 174, 211 Stasi'nska, G. 1989, A&A, 213, 274 Villaver, E., Stanghellini, L., & Shaw, R. 2003, ApJ, 597, 298 -2007, ApJ, 656, 840", "pages": [ 6 ] } ]

[ { "title": "ABSTRACT", "content": "Origins of the Expanding Universe: 1912-1932 ASP Conference Series, Vol. 471 Michael J. Way and Deidre Hunter, eds. c © 2013 Astronomical Society of the Pacific", "pages": [ 1 ] }, { "title": "Joseph S. Tenn", "content": "Department of Physics & Astronomy, Sonoma State University, Rohnert Park, CA, 94928, USA Abstract. When V. M. Slipher gave the 1933 George Darwin lecture to the Royal Astronomical Society, it was natural that he spoke on spectrographic studies of planets. Less than one-sixth of his published work deals with globular clusters and the objects we now call galaxies. In his most productive years, when he had Percival Lowell to give him direction, Slipher made major discoveries regarding stars, galactic nebulae, and solar system objects. These included the first spectroscopic measurement of the rotation period of Uranus, evidence that Venus's rotation is very slow, the existence of reflection nebulae and hence interstellar dust, and the stationary lines that prove the existence of interstellar calcium and sodium. After Lowell's death in 1916 Slipher continued making spectroscopic observations of planets, comets, and the aurora and night sky. He directed the Lowell Observatory from 1916 to 1954, where his greatest achievements were keeping the observatory running despite very limited sta ff and budget, and initiating and supervising the 'successful' search for Lowell's Planet X. However, he did little science in his last decades, spending most of his time and energy on business endeavors.", "pages": [ 1 ] }, { "title": "1. Introduction", "content": "Vesto Melvin Slipher, always referred to and addressed as 'V. M.' (Giclas 2007; Hoyt 1980b) came to Flagsta ff in August 1901, two months after completing his B.A. in mechanics and astronomy at Indiana University, because his professor, Wilbur Cogshall, had persuaded Percival Lowell to hire him temporarily. He arrived at age 25 and stayed there 53 years. After retirement he lived 15 more years in Flagsta ff . I will discuss his life in Flagsta ff , which I divide into five parts, and his research on the night sky, the aurora, planets, comets, stellar radial velocities, variable stars, and interstellar gas and dust. We have heard from others about his early life and his work on globular clusters and galaxies. The closest thing to a published biography is William G. Hoyt's Biographical Memoir (Hoyt 1980b).", "pages": [ 1 ] }, { "title": "2. Great Achievements under Lowell, 1901-1916", "content": "In his early years at Lowell Observatory, Slipher was not his own man. This was fortunate, as together he and his employer made a formidable team. As John S. Hall wrote in an obituary in Sky and Telescope (Hall 1970), Slipher and Lowell had complementary temperaments. The latter was brilliant, enthusiastic, and a driving personality. ... Slipher, on the other hand, was deliberate, fastidious, patient, and showed a high order of technical knowledge. Lowell knew what he wanted, and Slipher provided it. Lowell, of course, was primarily interested in the solar system, with special emphasis on Mars. He wanted Slipher to find chlorophyll, as well as oxygen and water there. As we have heard, he asked Slipher to obtain a spectrogram of a spiral nebula because he thought of it as a newly-forming solar system. However, he allowed Slipher to spend some of his time on his own pursuits, and Slipher was interested in the thenfashionable fields of measuring stellar radial velocities and discovering spectroscopic binary stars. His first publication, in the Astronomical Journal in 1902 (Lowell 1902), was a report of measurements of the variable velocity of zeta Herculis with the new spectrograph. In the paper he compares his average radial velocity of -74.4 with the -74.6 reported earlier by Lick Observatory's W. W. Campbell, who was swiftly becoming recognized as the world's leading astronomical spectroscopist. (As a teacher I would take points o ff for not specifying units. This is especially bad because Slipher sometimes discussed velocities in miles per second. I had to go to Campbell's article (Campbell 1902) to find that the velocities were in 'km', at the time the standard abbreviation for km / s. It is possible that young Slipher was unaware of this convention.) Volume 1 of the Lowell Observatory Bulletin , dating from 1903 to 1911, shows that Slipher was already an important member of the small Lowell team. The volume contains 62 articles, 13 of them by Slipher alone and one co-authored by him. There are 38 by Lowell, some of them including spectroscopic observations by Slipher, and 10 by other members of the sta ff . Slipher's publications in this early volume present spectroscopic observations of stars, including spectroscopic binary stars, standard velocity stars, and stars of variable radial velocity, of the Moon and planets, and of Halley's comet. He also began an extensive study of the Crab Nebula, which he never published. Among his most important discoveries in this period were two involving the interstellar medium. In 1909 he published an account (Slipher 1909a) of the selective absorption of light in space, proof that there were calcium ions in the interstellar medium between the Sun and a number of stars in Scorpius, Orion, Ophiuchus, and Perseus. In each case the sharp, weak calcium lines remained stationary while lines from the binary stars shifted back and forth. This confirmed a hypothesis made earlier by Johannes Hartmann of Potsdam, who found stationary calcium and sodium lines in Nova Persei in 1901 (Vogel 1901) and calcium again in the single-line spectroscopic binary delta Orionis in 1904 (Hartmann 1904). According to historian Daniel Seeley (Seeley 1973, p. 83), 'Hartmann set the stage for investigations into interstellar gas but Slipher provided the first real progress - his observations indicated that the interstellar lines were not a singular phenomenon and his interpretation proved to be accurate.' However, Seeley also notes, 'Slipher's interpretations of the stationary line data, published in a Lowell Observatory Bulletin , either were not widely known or were ignored.' 1 Slipher found the first reflection nebula, evidence of what we now call interstellar dust, in 1912 (Slipher 1912b). He noted that he found the spectrum of the cloud surrounding Merope, a star in the Pleiades, to be identical to that of the star, and that this could be explained by assuming that 'the nebula is disintegrated matter similar to what we know in the solar system, in the rings of Saturn, comets, etc., and ... it shines by reflected star light.' However, he ended this paper, published in December 1912 while he was in the midst of obtaining his measurement of the huge velocity of approach of the Andromeda Nebula, with The observation of the nebula in the Pleiades has suggested to me that the Andromeda Nebula and similar spiral nebulae might consist of a central star enveloped and beclouded by fragmentary and disintegrated matter which shines by light supplied by the central sun. This conception is in keeping with spectrograms of the Andromeda Nebula made here and with Bohlin's value for its parallax. Also of considerable importance - it was cited on the awarding of two of his gold medals in the 1930s (Stratton 1933; Einarsson 1935) - was his work on the planets. As early as 1903 he showed that his spectrograph could measure the rotation period of Mars (Slipher 1903). He obtained a period of 25 h 35 min, 'or just one hour longer than the true period.' At a time when many thought the rotation period of Venus was about 24 hours he showed that it had to be far longer than that. In fact the rotation was too slow to measure. The following year he published spectrograms of Uranus and Neptune, and compared them with the purely solar radiation from the Moon. By 1906 he had added Jupiter and Saturn. After experimenting with new sensitizing dyes on his plates, he found a combination which allowed him to be the first to extend his spectrograms past 7000 Ångstroms into the red, so in 1909 he p rovided completely new analyses of the spectra of all four major planets (Figure 2). He found new absorption bands in these planets, stronger in the more distant ones, and he failed to find evidence of oxygen in any of them. He was able to show that his spectrograph could detect the rotation of Uranus (Slipher 1912a). He had tried in 1903 without success. Six years later Lowell pointed out to him that the line of sight component of the rotational motion had increased, and he tried again. By 1911 he obtained seven good spectrograms, and his results were published the following year. Both he and Lowell measured the plates, without knowing their orientation. His final result was a rotation period of 10h 50 min, not particularly good by today's standards (the current accepted value is 17h 14min), but the first to be measured. It was during this period that Wilbur Cogshall, who had taught V. M. astronomy at Indiana University, wrote him (Cogshall 1908a) and suggested that the University might award him a Ph.D. for research he had done at Lowell. V. M. was enthusiastic, replying, 'Your letter was received a few days its content surprised me for the P.H.D. degree has been furthest from my thoughts. ... I hardly feel deserving of the honor, ....' (Slipher 1908). He sent what he considered 'by far, my best work' - a published paper on the spectra of the planets - to serve as his thesis, but almost lost the degree when Lowell declined to allow him to go to Bloomington in June 1908 to defend his thesis. Cogshall suggested (Cogshall 1908b) that Lowell was o ff ended that Slipher was even asked to defend. The patrician Bostonian thought that the degree should simply be conferred. Slipher received the degree a year later, with all residence and course requirements waived. By the time of Lowell's death the day after Slipher's 41st birthday, V. M. had begun examining the night sky with heroic exposures (Slipher 1916), discovering what he called the permanent aurora, with a greenish line known from aurorae present in all his spectrograms. He had also added observations of nebulae and interpretations involving the interstellar medium to his published work. And of course he had observed what we now call galaxies, but this gathering has heard plenty about that.", "pages": [ 1, 2, 3, 4 ] }, { "title": "3. Lowell's Death Brings New Responsibilities and Cares, 1916-1926", "content": "Lowell's unexpected death on 12 November 1916, just eight years after his marriage (Figure 3) and one year after appointing V. M. assistant director and designated successor, was a disaster for his observatory and its sta ff . For the next decade successive trustees fought legal battles with Lowell's widow over the estate. She received half the income, and it was a struggle to keep the observatory open. As acting director, V. M. had to be extremely parsimonious. Slipher had married in 1904 and his children were nine and five when Lowell died. He was justifiably concerned with supporting his family. The daily ration of milk from the Observatory cow, Venus, was helpful but not su ffi cient. Slipher started buying rental properties and eventually bought a number of ranches. He owned and operated a furniture store at one time. He also took part in civic activities, serving as president of the school board when his children were in school and joining with others to found the Museum of Northern Arizona. He served as chairman of the board of Flagsta ff 's premier hotel, the Monte Vista. Founded by public subscription, including a major donation from author Zane Grey, it was built by the city in 1927. Meanwhile the Sliphers raised their family on Mars Hill (Figure 4). Despite these distractions, Slipher continued to be astronomically productive. It was during this period that he published observations of the spectra of both galactic and extragalactic nebulae and measured rotation speeds of planets. He made his only forays into solar astronomy, leading eclipse expeditions to Kansas in 1918 (Slipher 1922) and Baja California in 1923 to photograph the spectrum of the solar corona, and he continued the Lowell-inspired search for water, oxygen and chlorophyll on Mars (Slipher 1924). He also published his first investigations of the spectra of lightning, the aurora, and the night sky (Slipher 1916, 1917, 1919).", "pages": [ 5 ] }, { "title": "4. The Last Productive Years, 1926-33", "content": "Slipher was appointed Director of the Lowell Observatory in 1926, after the final settling of the Lowell estate (Smith 1994). Mrs. Lowell continued to receive some of the income until her death in 1954. Correspondence between V. M. and trustee Roger Lowell Putnam shows that the Observatory was frequently in the red, and paychecks were not always issued on time. The trustee often helped with an extra check for $500 or $1000, and he even got his mother to pay the salary so that the observatory could have a secretary. The Observatory sta ff lost heavily when Flagsta ff 's only bank failed in 1932 (Giclas 1987). During this period Slipher published his most extensive work on the night sky, zodiacal light, and the aurora, and he published additional spectroscopic observations of Venus, Mars, and comets. Every time C. E. Kenneth Mees of Eastman Kodak came up with an emulsion that was sensitive a little farther out into the infrared Slipher used it to extend his planetary observations. One of his night sky spectrograms involved exposures totaling 147 hours! He also continued his observations of nebulae and studies of the interstellar medium. His most famous work during this period was again an e ff ort to carry on the work of his master. Lowell had spent years computing orbits (with much of the tedious calculation done by assistants, especially Elizabeth Williams) and attempting to make a prediction that would lead to the discovery of a ninth planet that would account for the perturbations of Neptune. In his highly mathematical 1915 book, Memoir on a TransNeptunian Planet (Lowell 1915), he called it Planet X. He had employed as many as five computers in Flagsta ff and Boston and had hired three successive 'Lawrence Fellows' to observe in Flagsta ff searching for the planet, but he died without knowing whether it existed. In his book he had suggested two regions of the ecliptic, in opposite directions, where the planet might be found. In 1927 glass disks for a 13-inch refractor became available, and Slipher suggested to the trustee that they be purchased and made into a telescope to resume the search for Planet X. Trustee Guy Lowell personally purchased the disks and planned to have the telescope made, but he died later that year. At this point another member of the Lowell family, Percival's brother A. Lawrence Lowell, then president of Harvard, stepped in and funded the building of the telescope. It arrived in Flagsta ff in 1929 and was erected on a mounting built by the observatory's longtime instrument maker, Stanley Sykes. Designed from the start for the planet search, the Lawrence Lowell telescope (Figure 5) produced highly defined star images over 14 x 17-inch plates. It could record 50,000 to 500,000 stars in a one-hour exposure. The plan was to photograph every field along the ecliptic, starting with the areas suggested by Percival Lowell, to repeat a few days later, and then to 'blink' the plates in order to find objects that moved. Blinking the plates was incredibly tedious work. Slipher hired 23-year-old high school graduate Clyde Tombaugh (Figure 6) to do it, and the rest is history. 2 It is to Slipher's credit that Pluto is universally recognized as having been discovered by young Tombaugh. Had it been found at one of several other major observatories at the time, the director would have claimed credit. This is characteristic of Slipher's modesty. It is to Slipher's discredit that, knowing that the Lowell Observatory lacked the expertise to compute the orbit of the newly-discovered planet, he delayed announcing the positions so that his former teacher, John A. Miller of Sproul Observatory, could come to Flagsta ff and lead the computation of the first orbit. Three days after the public announcement of the discovery (itself held until Lowell's birthday), a telegram to the Trustee signed 'Lowell Sta ff ' reported (Lowell Sta ff 1930): Impressed with vital importance to Observatory that our discovery announcement be followed soonest possible by best determined orbit our observations can give because orbit will demonstrate much about nature and status of new wanderer that we telegraphed Professor Miller Director Sproul observatory asking him come Flagsta ff and help us work best possible orbit. Miller experienced with orbits loyal friend. Plans kept confidential. Trustee Roger Putnam replied (Putnam 1930), 'Frankly, I am very uncertain as to the ethics of when and what should be released, and will leave that to your judgment. I can't help feeling that having gotten the whole world stirred up, we have got to give them the information they want, but you know that sort of thing much better than I do.' Putnam was right, but Slipher delayed announcing more positions for four weeks while he and his colleague, C. O. Lampland, frantically worked their slide rules under Miller's direction until they had an orbit. This infuriated many in the astronomical community, such as the ace orbit-computers at Berkeley, who could have produced an orbit much more quickly (Giclas 1987). Slipher defended himself in a letter to the Trustee (Slipher 1930): We have been severely criticised for not giving out positions that others might comput [sic] the orbit, and this will no doubt not stop for a while yet. However, unpleasant as that has been it seemed our clear duty to make use of our materila [sic] for the orbit as it was more useful to us than it could be made to others without still more delay. Of course others could have done the orbit quicker than we did it, but we did it as carefully as possible. To have followed the other policy would have meant a considerable sacrifice to the Observatory. This is characteristic of Slipher's intense loyalty to Lowell Observatory and the memory of Percival Lowell.", "pages": [ 6, 7, 8, 9 ] }, { "title": "5. The Doldrums, 1934-54", "content": "The year 1933, when he turned 58, was essentially the last year that Slipher published his own original research. His five-page article on 'Spectra of the Night Sky, the Zodiacal Light, the Aurora, and the Cosmic Radiations of the Sky' appeared in the Transactions of the American Geophysical Union and was reprinted in the Journal of the Royal Astronomical Society of Canada (Slipher 1933a). It reports on many years of work, including the use of a newly-designed spectrograph to photograph the spectra of five regions of the sky at once. He gave the George Darwin Lecture to the Royal Astronomical Society after accepting the RAS Gold Medal that same year. The lecture was on spectroscopic studies of the planets and summed up his work, mostly completed long before (Slipher 1933b). After that the Lowell Observatory slowly declined. For many years, three men V. M., C. O. Lampland, and V. M.'s younger brother, E. C. Slipher (Figure 7) - dominated the observatory. Occasionally, a younger man, such as Henry Giclas in 1931, would be hired to a subordinate position, but the three senior astronomers jealously guarded the telescopes. All had been seriously wounded by the criticism from other astronomers, especially Lick Observatory directors W. W. Campbell and W. H. Wright, of work done at the Lowell Observatory. These eminent astronomers had developed an intense distaste for Percival Lowell and anything associated with him. Work coming from Lowell's observatory was automatically suspect. 3 V. M. had gotten into an exchange of criticisms with Campbell over his claim to have detected water on Mars in 1909, and after losing this battle he became even more reticent than he had been. He was very careful to check his work many times and to get repeated observations before going public. He devoted more and more time to his business a ff airs, and less to research. Meanwhile his brother, E. C., spent much of his time on politics, and Lampland puttered around without completing anything. V. M. published his last Observatory Report in 1933. It covered the years 1930-1932. The next to appear from the Lowell Observatory was for the years 1952-1954. Although signed by V. M., it was probably written by Albert G. Wilson, who was assistant director at the time. The most significant papers with Slipher's name on them after 1933 were of a totally di ff erent character from his other work. There were six of them, published in Nature and the Physical Review , and they contained astrophysical observations and theory far beyond Slipher's abilities. 4 They were written by Arthur Adel, who had been hired in 1933 by the trustee over the opposition of the senior astronomers. Adel was to work at his alma mater, the University of Michigan, and do infrared studies that would relate Slipher's spectra to conditions on the planets. Adel built a 22.5-m long high pressure cell and put up to 40 atmospheres of carbon dioxide in it. Later he filled his tube with ammonia and methane. He was able to duplicate some of the spectra that Mt. Wilson astronomers had observed in Venus in 1932 and that V. M. had observed in Jupiter many years earlier. He did this entirely by himself in Ann Arbor for $1000 per year, which even in 1933 was not much. Adel used Slipher's published data but got nothing new from Slipher. Nevertheless, he put Slipher's name on the papers as co-author. In 1987 Adel told Robert Smith in an oral history interview (Adel 1987), I had to do that, and neither he nor Lampland nor E. C. Slipher, none of them really knew what I was doing, had a real understanding of it. ... They didn't know anything about infrared spectroscopy. They didn't know anything about spectroscopy. They really didn't know anything about this work that I was doing, or the work I did in Ann Arbor. When Adel was appointed to a lowly position in Flagsta ff by the trustee, V. M. treated him very badly. And when Adel showed that the carbon dioxide bands in the spectrum of Venus could be photographed with the 24-inch refractor and thus could have been discovered by Slipher before they were found by Adams and Dunham at Mt. Wilson (Adams & Dunham 1932), he was barred from all the telescopes (Adel 1987). According to Henry Giclas (Giclas 1987, 1990), Slipher resisted applying for grants and couldn't be bothered with the complications of payroll, social security, etc. Giclas was appointed executive secretary in 1953 and took over all the business a ff airs. Trustee Roger Putnam pushed for grants, and the first, from the Weather Bureau, was obtained in 1948 and included funds to measure the variation in the solar constant as well as meteorology of planetary atmospheres. Later this project was taken over by the Air Force. The appointment of Harold Johnson in July 1948, initially to work on the Weather Bureau project, was a turning point. Although very di ffi cult to get along with and constantly complaining, he was a competent, energetic young scientist, and he accounted for nearly all of the Observatory's publications in the early 1950s. He quit and went to Yerkes after one year, but was hired back in August 1952 by the trustee over V. M.'s objections. Slipher's last scientific publication was a brief abstract in 1939 announcing that he had re-observed Hubble's variable nebula, NGC 2261, and found that its spectrum had not changed since his observations of 1916-17 (Slipher 1939). He also wrote an occasional letter asserting his priority on something done long before.", "pages": [ 9, 10, 11 ] }, { "title": "6. Retirement, 1954-69", "content": "After Lampland died in December 1951, trustee Roger Putnam finally stepped in to make some changes. On the advice of John Duncan he selected 33-year-old Albert G. Wilson, who had been directing the National Geographic Palomar Sky Survey since completing his Ph.D. in mathematics at Caltech. It seems that the primary criterion for the appointment was that Wilson was acceptable to the Slipher brothers. Wilson came as assistant director in 1953 and took over as director on V. M.'s 79th birthday, 11 November 1954, when the old man finally retired (Figure 8). Wilson's directorship was short and unhappy. After a rebellion from the younger astronomers, especially Harold Johnson and Henry Giclas, and the breakup of his marriage, he left in January 1957 and returned to California and a career in industry (Tenn 2007). Slipher remained in Flagsta ff although he moved o ff Mars Hill into one of his houses. His wife, Emma, died in 1961. Frances Wilson, the ex-wife of Slipher's successor, returned to Flagsta ff and became Slipher's 'private secretary and companion' according to Henry Giclas (Giclas 1990). V. M. Slipher died 8 November 1969, three days before he would have turned 94. His will (Slipher 1967) stated that 'During the latter years of my lifetime, FRANCES M. WILSON has devoted herself to my business a ff airs, and it is my desire from my Estate to make provision for her.' He left her $3000 per year for life, and he made her executrix of his estate. Aside from the endowment to support her he left his wealth to the V. M. Slipher Trust with a bank as trustee and Arthur Adel 5 as Advisory Trustee. A portion of the income was to provide scholarships for worthy students pursuing scientific studies at Arizona's three public universities. After that 50% of income went to the National Academy of Sciences for Astronomy. There was a great deal of property, including ranches and cattle.", "pages": [ 11, 12 ] }, { "title": "7. Conclusion", "content": "Although he received prestigious awards in his lifetime, including the 1935 Bruce Medal of the Astronomical Society of the Pacific and the three mentioned in the obituary below, Slipher is probably underrated today. I gave a talk (Tenn 2005) at a meeting of the Historical Astronomy Division of the American Astronomical Society in 2006 titled 'Why Does V. M. Slipher Get So Little Respect?' My current conclusion is that the most important reasons are Slipher's brief obituary in Physics Today (Anonymous 1970), which mentions only the discovery of Pluto among his accomplishments, makes this clear. It reads, in its entirety: Vesto M. Slipher, director of the Lowell Observatory until 1952 [sic], died 8 Nov. at 93. Slipher had been at the observatory since 1901 and became director in 1926. He supervised work that led to the discovery in 1930 of Pluto. Among the honors received by Slipher were the Lalande Prize and gold medal of the Paris Academy of Sciences (1919), the Draper Medal of the National Academy of Sciences (1932) and the Royal Astronomical Society gold medal (1932). Acknowledgments. I thank Lauren Amundson, Antoinette Beiser, and Martin Hecht of the Lowell Observatory Archives for documents and images and Traci Lehman for one image. I benefited from helpful discussions with Arthur Adel (1994), Frank Edmondson (2005), Henry L. Giclas (2007), and Albert G. Wilson (2005, 2012). I also appreciate David DeVorkin's helpful comments on this article. This research has made extensive use of NASA's Astrophysics Data System.", "pages": [ 12, 13 ] }, { "title": "References", "content": "Adams, W. S., & Dunham, T., Jr. 1932, Absorption Bands in the Infra-Red Spectrum of Venus, PASP, 44, 243 Adel, A. 1987, Interview of Arthur Adel by Robert W. Smith on 12 August 1987, http://www.aip.org/history/ohilist/5000.html . Niels Bohr Library and Archives, American Institute of Physics, College Park, MD, USA Adel, A., & Slipher, V. M. 1934a, Concerning the Carbon Dioxide Content of the Atmosphere of the Planet Venus, Physical Review, 46, 240 -1934b, On the Identification of the Methane Bands in the Solar Spectra of the Major Planets, Physical Review, 46, 240 -1934c, The Atmospheres of the Giant Planets, Nat, 134, 148 -1934d, The Constitution of the Atmospheres of the Giant Planets, Physical Review, 46, 902 -1935, Di ff erence Bands in the Spectra of the Major Planets, Physical Review, 47, 651 Adel, A., Slipher, V. M., & Barker, E. F. 1935, The Absorption of Sunlight by the Earth's Atmosphere in the Remote Infrared Region of the Spectrum, Physical Review, 47, 580 Anonymous 1970, V. M. Slipher, died 1969 Nov. 8, Physics Today, 23, 101 Campbell, W. W. 1902, Six Stars Whose Velocities in the Line of Sight are Variable, ApJ, 16, 114 Cogshall, W. 1908a, Letter to V. M. Slipher, dated 18 February 1908. Lowell Observatory Archives -1908b, Letter to V. M. Slipher, dated 20 October 1908. Lowell Observatory Archives Einarsson, S. 1935, The Award of the Bruce Gold Medal to Dr. Vesto Melvin Slipher, PASP, 47, 5 Giclas, H. L. 1987, Interview of Henry Giclas by Robert W. Smith on 12 August 1987, http://www.aip.org/history/ohilist/5022.html . Niels Bohr Library and Archives, American Institute of Physics, College Park, MD, USA -1990, Reminiscences by Henry Giclas, Lowell Observatory Archives. Unpublished -2007, personal communication Hall, J. S. 1970, V. M. Slipher's Trailblazing Career, S&T, 39, 84 Hartmann, J. 1904, Investigations on the Spectrum and Orbit of Delta Orionis, ApJ, 19, 268 Hoyt, W. G. 1980a, Planets X and Pluto (University of Arizona Press) -1980b, Vesto Melvin Slipher 1875-1969, Biographical Memoirs of the National Academy of Sciences, 52, 410 Hubble, E. 1929, A Relation between Distance and Radial Velocity among Extra-Galactic Neb- ulae, Proceedings of the National Academy of Sciences, 15, 168 Lowell, P. 1902, On the Variable Velocity of Zeta Herculis in the Line of Sight, AJ, 22, 190 -1915, Memoir on a Trans-Neptunian Planet, Lowell Observatory Memoirs, v. 1, no. 1 Lowell Sta ff 1930, telegram to R. L. Putnam, dated 16 March 1930. Lowell Observatory Archives Putnam, R. L. 1930, Letter to V. M. Slipher, dated 17 March 1930. Lowell Observatory Archives Seeley, D. H. 1973, Ph.D. thesis, Boston University Graduate School Stratton, F. J. M. 1933, Society Business: Presidential Address on Presenting the Gold Medal to Dr. V. M. Slipher, MNRAS, 93, 476", "pages": [ 13, 14 ] } ]

[ { "title": "ABSTRACT", "content": "**Author**", "pages": [ 1 ] }, { "title": "Evolution of Giant Molecular Clouds in Nearby Galaxies", "content": "Jin Koda 1 1 Stony Brook University Abstract. Our knowledge of GMC evolution in galactic disks has advanced tremendously in past several years. Studies were limited to local, predominantly atom-rich small galaxies, but have now been expanded to typical spiral galaxies with a rich molecular content. The evolution appears quite di GLYPH<11> erent between the two environments. GMCs exist almost exclusively along HI spiral arms and filaments in the disks of local small galaxies (LMC, M33), suggesting that GMCs form and end their short lives there. However, in a more molecular-rich environment (MW, M51), GMCs are present everywhere independent of HI structures. Indeed, the molecular gas fraction remains high and almost constant during arm passage into the next inter-arm region. The gas remains molecular, presumably in GMCs, for a long time. A transitional case has been found recently in the central region of the atom-rich galaxy M33 - GMCs do not coincide with HI there. Evolution of the physical conditions of molecular gas from spiral arms to inter-arm regions is also being revealed in molecule-rich galaxies. An increase of the CO J = 2-1 and 1-0 line ratio in spiral arms in M51 suggests density and / or temperature increases by a factor of 2-3 in GMCs in the arms, compared to their counterparts in the inter-arm regions. An analysis of high-resolution Milky Way survey data revealed that the fraction of dense (or warm) clumps increases dramatically in the spiral arms. The formation, evolution, and lifetime of GMCs in galaxies are of critical importance to our understanding of interstellar matter (ISM) and star formation. The ISM evolves during galactic rotation. Therefore knowing the distribution of GMCs in galactic disks and their relations to galactic structures is essential. This presentation was given at a celebration of the 30th anniversary of the Nobeyama Radio Observatory (NRO). I summarize recent progress in the field of GMC evolution with a particular emphasis on recent results from NRO.", "pages": [ 1 ] }, { "title": "1. Brief History", "content": "The textbook picture of ISM phases posits that GMCs are assembled in spiral arm shocks from di GLYPH<11> use interarm HI gas and then photo-dissociated back into the atomic phase by OB star formation within the spiral arms (Figure 1 left ). This picture predicts a rapid gas-phase change across spiral arms - from atomic to molecular and back into atomic after spiral arm passage. In retrospect, observational support for this standard picture left some unanswered questions. There were intense debates on the distribution and lifetime of GMCs in the MW in 1980's, which were never fully resolved. Indeed, two major Galactic plane CO ( J = 1-0) surveys arrived at contradictory results. The Columbia survey (and the Harvard survey later) found very little CO emission in the inter-arm regions in the longitude-velocity ( l -v ) diagram, suggesting that GMCs survive only for the duration of spiral arm passage (an order of GLYPH<24> 20-30 Myr; Cohen et al. 1980; Dame et al. 2001). On the other hand, the Massachusett-Stony Brook survey found an abundant population of GMCs even in the inter-arm regions, concluding that GMCs live for the order of the Galactic rotation timescale ( glyph[greaterorsimilar] 100 Myr; Sanders et al. 1985; Scoville & Wilson 2004). Early interferometric observations found only few GMCs in the inter-arm regions in the grand-design spiral galaxy M51 (Rand & Kulkarni 1990), apparently supporting the textbook picture. However, interferometric observations have a well-known disadvantage in studies of extended nearby galaxies. Arrays with glyph[lessorsimilar] 10 antennas (e.g., Nobeyama, OVRO, BIMA, and PdBI arrays) miss most of spatial information, especially in extended inter-arm regions, being biased toward confined emission along the spiral arms. Mapping GMCs in inter-arm regions, especially in presence of bright spiral arms, was di GLYPH<14> cult even when sensitivity was not an issue.", "pages": [ 1, 2 ] }, { "title": "2.1. Molecule-rich Galaxies", "content": "Acombination of modern arrays with GLYPH<21> 15 antennas and single-dish telescopes permits high-quality molecular gas imaging. An abundant GMC population in the inter-arm regions was found in the molecule-rich galaxy M51, using the Combined Array for Research in Millimeter Astronomy (CARMA) and the Nobeyama Radio Observatory 45m (NRO45) telescope (Koda et al. 2009). The combination of CARMA and NRO45 provided an unprecedented high-image quality, reconstructing the full emission including the most extended component (Koda et al. 2011). In M51, the majority of the gas remains molecular from arm entry through the interarm region and into the next spiral arm passage. The molecular gas fraction varies only little azimuthally. The most massive GMCs (giant molecular associations GMAs) appear only along spiral arms (Figure 2), suggesting that they are first assembled from pre-existing smaller GMCs entering the spiral arms, and then broken up as the gas flows through the arms. The GMAs and their H2 molecules are not fully dissociated into atomic gas as predicted in stellar feedback scenarios, but are fragmented into smaller GMCs upon leaving the spiral arms. The remnants of GMAs are detected as the chains of GMCs that emerge from the spiral arms into interarm regions. The observed kinematic shear within the spiral arms is su GLYPH<14> cient to unbind the GMAs against self-gravity. Koda et al. (2009) discussed that the evolution of GMCs is driven by large- scale galactic dynamics - their coagulation into GMAs is due to spiral arm streaming motions, and their fragmentation as they leave the arms is due to shear. Therefore, the lifetime of molecular gas appears to be an order of the Galactic rotation timescale ( glyph[greaterorsimilar] 100 Myr), while GMCs can be coagulated and / or fragmented to the next-generation GMCs without passing through the atomic phase. A similar GMC mass segregation between spiral arms and inter-arm regions is found in the Milky Way, another molecule-rich galaxy. GMCs exist also in the inter-arm regions, but their masses are about an order of magnitude smaller than their counterparts in the spiral arms (Koda et al. 2006). The same tend is being found in other moleculerich spiral galaxies in the CARMA-Nobeyama Nearby-galaxies (CANON) CO( J = 1-0) survey, in which 29 local spiral galaxies are observed with both CARMA and NRO45.", "pages": [ 2, 3 ] }, { "title": "2.2. Atom-rich Galaxies", "content": "Comparisons of molecule-rich and atom-rich galaxies provide another clue in understanding GMC evolution. Smaller local (dwarf) galaxies rich in atomic gas (e.g., LMC and M33) show much fewer GMCs in their disks (Fukui et al. 2009; Engargiola et al. 2003). The GMCs are almost exclusively associated with HI spiral arms and filaments and are mostly absent in the inter-arm regions. These observations immediately indicate their short lifetimes (an arm-crossing timescale of GLYPH<24> 30 Myr). Kawamura et al. (2009) found a similar lifetime of 20-30 Myr in LMC, by translating the fractions of GMCs with and without young star clusters to their lifetimes using cluster ages as a gauge. Miura et al. (2012) also found a similar lifetime of 20-40 Myr in M33 by comparing CO( J = 3-2) data and star clusters. The short lifetime of GMCs appears to be common in these atom-rich galaxies. In retrospect, the observations of extragalactic GMCs beyond the MW were limited to these closest, pre- dominantly atom-rich galaxies; this bias might have contributed to the textbook picture of GMC evolution. As a side note, these relatively-short lifetimes may still be at odds with the even shorter dissipation timescale of internal velocity dispersions (a few Myr; Mac Low 1999). It is di GLYPH<14> cult to maintain the velocity dispersions even for the relatively short lifetime of 20-30 Myr .", "pages": [ 3, 4 ] }, { "title": "2.3. A Transitional Case", "content": "An interesting transitional case has been found recently in the central 4 kpc region of M33 (Tosaki et al. 2011, see Figure 3). In this atom-rich galaxy, GMCs are located predominantly in HI spiral structures in most of the disk (Engargiola et al. 2003), indicating that GMCs form and die there. However, they do not coincide with HI structures in the central r < 2 kpc, just as in the disks of molecule-rich galaxies. GMCs are decoupled from the HI distribution as if they are entities that survive through almost a galactic rotation period. The molecular gas fraction increases in this central region, although the dominant phase of gas there is still atomic (Tosaki et al. 2011). There are many parameters potentially responsible for this transition, including the amount of gas, stellar gravitational potential, metallicity, radiation field, dynamical environment (such as shear), etc. Obviously, a much larger sample is necessary to isolate the cause.", "pages": [ 4 ] }, { "title": "3. GMCEvolution", "content": "Molecular gas dominates even in the inter-arm regions in the MW and M51. Evidence is being accumulated that the physical conditions of molecular gas vary systematically as the gas enters spiral arms.", "pages": [ 4 ] }, { "title": "3.1. M51", "content": "The CO J = 2 GLYPH<0> 1 and 1 GLYPH<0> 0 line ratio ( R 2 GLYPH<0> 1 = 1 GLYPH<0> 0) varies systematically in M51 (Koda et al. 2012, Figure 4 left ). R 2 GLYPH<0> 1 = 1 GLYPH<0> 0 rises clearly from a typical low value of < 0 : 7 (and often 0.4-0.6) in the inter-arm regions to a higher value of > 0 : 7 (0.8-1.0) in the spiral arms, particularly at the leading (downstream) edge of the molecular arms. These high and low R 2 GLYPH<0> 1 = 1 GLYPH<0> 0 are similar to those in Galactic GMCs with and without OB star formation, respectively (Sakamoto et al. 1997). Thus, the physical conditions of molecular gas evolve during arm passage. The current spatial resolution of this image ( GLYPH<24> 1 kpc) almost certainly blends multiple GMCs. Assuming all the unresolved GMCs share similar physical conditions, we can apply a simple one-zone analysis. Note that virtually all CO emission should come from unresolved GMCs, since the CO molecule is easily photo-dissociated without self-shielding. It is also di GLYPH<14> cult to collisionally excite CO emission outside GMCs, as the critical density is about the average density within GMCs ( GLYPH<24> 300 cm GLYPH<0> 3 ). A Large Velocity Gradient radiative transfer calculation provides insight into the changes in the physical conditions. The R 2 GLYPH<0> 1 = 1 GLYPH<0> 0 variations indicate that cold and low density gas ( glyph[lessorsimilar] 10 K, glyph[lessorsimilar] 300 pc) is required for the interarm GMCs but this gas must become warmer and / or denser, by a factor of 2-3, in the more active star forming spiral arms (Figure 4 right ). We cannot separate the increases in temperature and density with the two line analysis. However, the enhanced R 2 GLYPH<0> 1 = 1 GLYPH<0> 0 at the arm downstream may suggest that the main cause is stellar heating. Indeed, most star formation is found at the downstream side (Louie et al. 2013), and R 2 GLYPH<0> 1 = 1 GLYPH<0> 0 is higher in areas of high 24 GLYPH<22> m dust surface brightness (which is an approximate tracer of star formation rate surface density). Some upstream regions also show high R 2 GLYPH<0> 1 = 1 GLYPH<0> 0, possibly indicating density increases before star formation. The sizes of GMCs need to be shrunk only by 20-30% as they enter spiral arms in order to increase GMC densities by a factor of 2-3.", "pages": [ 5 ] }, { "title": "3.2. The Milky Way", "content": "We can resolve molecular gas structure at small scales in the Milky Way disk and study its evolution in relation to the spiral structure. Sawada et al. (2012b) demonstrated remarkable variations of molecular gas structure between inter-arm regions and spiral arms (Figure 5). Their line of sight, l GLYPH<24> 38 GLYPH<14> , samples the molecular gas in both the Sagittarius arm and the inter-arm regions, and hence the regions can be distinguished in velocity channel maps. The inter-arm emission appears mostly featureless and extended, while the spiral arm shows a lot of clumps ( GLYPH<24> 1 pc in size; either dense or / and warm). The velocity channel of the highest clump fraction coincides with those of H II regions and high CO 3-2 / 1-0 intensity ratio integrated over the field (i.e., warm on average). It also appears o GLYPH<11> set from the molecular spiral arm velocity, indicating that the clumps are at its downstream side. Therefore, bright and spatially confined structures develop in a spiral arm, leading to star formation at downstream side, while extended molecular emission dominates in the inter-arm region. To quantify the development of the small-scale molecular structure, Sawada et al. (2012b) introduced the brightness distribution index (BDI), with typical boundary brightness temperatures to enclose the extended and lumpy components ( T 0, T 1, T 2, T 3). The BDI measures the fractional contribution of spatially confined bright molecular emission over faint emission extended over large areas. This relative quantity is largely independent of the amount of molecular gas and of any conventional, pre-conceived structures, such as cores, clumps, or giant molecular clouds. Sawada et al. (2012a) confirmed the same evolution over the MW disk. They applied the BDI to the entire inner MW disk in the northern hemisphere, using archival data from the Boston University-Five College Radio Astronomy Observatory 13 CO J = 1-0 Galactic Ring Survey (Figure 6; Jackson et al. 2006). The structured molecular gas, traced by higher BDI, appear continuously along the spiral arms in the l GLYPH<0> v diagram. The high-BDI gas generally coincides with areas with a high population of H II regions, while there is also some high-BDI gas with no / little signature of ongoing star formation. These results support the evolutionary sequence in which unstructured, extended gas transforms itself into a structured state on encountering the spiral arms, followed by star formation and an eventual return to the unstructured state after spiral arm passage.", "pages": [ 5, 6, 7 ] }, { "title": "4. Summary and Future Prospects", "content": "We saw that the evolution of GMCs depends strongly on the parent galactic environment. Atom-rich, small (dwarf) galaxies show GMCs almost exclusively on HI spiral arms and filaments, suggesting their lifetimes as short as an arm-crossing time of 20-30 Myr. On the other hand, in molecule-rich spiral galaxies GMCs are found almost everywhere, including the inter-arm regions. The lifetimes there could be as long as a galactic rotation timescale of GLYPH<24> 100 Myr (they stay molecular throughout galactic rotation, though GMCs can be coagulated and / or fragmented into next-generation GMCs). The textbook picture of the gas evolution predicts a rapid atomic / molecular phase change across spiral arms, but this needs to be revised based on new observations. We saw clear evidence for the evolution of physical conditions and structure of molecular gas in galactic disks. The molecular gas and GMCs become denser and / or warmer in spiral arms, and small pc-scale structures (dense or warm clumps) develop during spiral arm passage. With ALMA, we are at an exciting moment for furthering understanding of the evolution of the ISM in galaxies. Extragalactic GMCs will be easily identified and resolved. Multi-line analyses become possible for individual GMCs with an ALMA sensitivity. We have not yet seen the full imaging capability of ALMA, as most early observations are designed solely on sensitivity. Complete synthesis observations with the full ALMA array, in conjunction with short-spacing data from the total power telescopes and compact array, will revolutionize image fidelity, and enable imaging of highly-complex molecular structures in nearby galaxies. Acknowledgments. JK acknowledges support from the NSF through grant AST1211680 and NASA through grant NNX09AF40G, a Herschel Space Observatory grant, and an Hubble Space Telescope grant.", "pages": [ 7, 8 ] }, { "title": "References", "content": "Cohen, R. S., Cong, H., Dame, T. M., & Thaddeus, P. 1980, ApJ, 239, L53 Dame, T. M., Hartmann, D., & Thaddeus, P. 2001, ApJ, 547, 792. Engargiola, G., Plambeck, R. L., Rosolowsky, E., & Blitz, L. 2003, ApJS, 149, 343. Fukui, Y., Kawamura, A., Wong, T., Murai, M., Iritani, H., Mizuno, N., Mizuno, Y., Onishi, T., Hughes, A., Ott, J., Muller, E., Staveley-Smith, L., & Kim, S. 2009, ApJ, 705, 144. Jackson, J. M., Rathborne, J. M., Shah, R. Y., Simon, R., Bania, T. M., Clemens, D. P., Chambers, E. T., Johnson, A. M., Dormody, M., Lavoie, R., & Heyer, M. H. 2006, ApJS, 163, 145. Kawamura, A., Mizuno, Y., Minamidani, T., Filipovi'c, M. D., Staveley-Smith, L., Kim, S., Mizuno, N., Onishi, T., Mizuno, A., & Fukui, Y. 2009, ApJS, 184, 1. Koda, J., Sawada, T., Hasegawa, T., & Scoville, N. Z. 2006, ApJ, 638, 191. Koda, J., Sawada, T., Wright, M. C. H., Teuben, P., Corder, S. A., Patience, J., Scoville, N., Donovan Meyer, J., & Egusa, F. 2011, ApJS, 193, 19 Koda, J., Scoville, N., Hasegawa, T., Calzetti, D., Donovan Meyer, J., Egusa, F., Kennicutt, R., Kuno, N., Louie, M., Momose, R., Sawada, T., Sorai, K., & Umei, M. 2012, ApJ, 761, 41. Koda, J., Scoville, N., Sawada, T., La Vigne, M. A., Vogel, S. N., Potts, A. E., Carpenter, J. M., Corder, S. A., Wright, M. C. H., White, S. M., Zauderer, B. A., Patience, J., Sargent, A. I., Bock, D. C. J., Hawkins, D., Hodges, M., Kemball, A., Lamb, J. W., Plambeck, R. L., Pound, M. W., Scott, S. L., Teuben, P., & Woody, D. P. 2009, ApJ, 700, L132. Louie, M., Koda, J., & Egusa, F. 2013, ApJ, 763, 94. Mac Low, M.-M. 1999, ApJ, 524, 169. Miura, R. E., Kohno, K., Tosaki, T., Espada, D., Hwang, N., Kuno, N., Okumura, S. K., Hirota, A., Muraoka, K., Onodera, S., Minamidani, T., Komugi, S., Nakanishi, K., Sawada, T., Kaneko, H., & Kawabe, R. 2012, ApJ, 761, 37. Rand, R. J., & Kulkarni, S. R. 1990, ApJ, 349, L43 Sakamoto, S., Hasegawa, T., Handa, T., Hayashi, M., & Oka, T. 1997, ApJ, 486, 276 Sanders, D. B., Scoville, N. Z., & Solomon, P. M. 1985, ApJ, 289, 373 Sawada, T., Hasegawa, T., & Koda, J. 2012a, ApJ, 759, L26. Sawada, T., Hasegawa, T., Sugimoto, M., Koda, J., & Handa, T. 2012b, ApJ, 752, 118. Scoville, N. Z., & Wilson, C. D. 2004, in The Formation and Evolution of Massive Young Star Clusters, edited by H. J. G. L. M. Lamers, L. J. Smith, & A. Nota, vol. 322 of Astronomical Society of the Pacific Conference Series, 245 Tosaki, T., Kuno, N., Onodera, S. M., Rie, Sawada, T., Muraoka, K., Nakanishi, K., Komugi, S., Nakanishi, H., Kaneko, H., Hirota, A., Kohno, K., & Kawabe, R. 2011, PASJ, 63, 1171.", "pages": [ 8 ] } ]

[ { "title": "Solar-like stars observed by Kepler: an incredible adventure", "content": "R.A. Garc'ıa Abstract The NASA Kepler mission -in flight since March 2009- is producing an enormous number of high-quality continuous light curves. Now, and for the first time ever, we are able to do ensemble asteroseismology, i.e., to do an asteroseismic analysis with a statistically significant sub-sample of solar-like stars covering a wide range of stellar characteristics. In the present work, I highlight some of the most recent studies carried out using these data.", "pages": [ 1 ] }, { "title": "1 Introduction", "content": "Kepler [7] is a planet-hunter mission to look for Earth-size planets around solar-like stars in the habitable zone. The photometric stability and the quiet environment provided by the Earth trailing heliocentric orbit offers a great opportunity to perform asteroseismic observations of thousands of stars (after some specific processing of the light curves [17]) among the 150,000 observed by Kepler at a single star field in the Cygnus-Lyra region of the galaxy. Kepler is planned to work for 3.5 years with a possible extension to 6 years. With its telescope of 0.95-m aperture, Kepler monitored 100,000 stars at any time with a cadence of 29.42 min (long-cadence measurements). A subsample of 512 stars can be observed at a much faster cadence of 58.85s (short-cadence measurements; Nyquist frequency of ∼ 8.5 mHz) allowing for more precise transit timings [19]. This running mode is of particular interest for asteroseismology because it allows us to study stochastically-excited oscillations in main-sequence solar-like stars and subgiants [10]. During the first year of operations, the working group 1 of the Kepler Asteroseismic Science Consortium (KASC http://astro.phys.au.dk/KASC/ ) observed more than 2000 solar-like stars looking for stellar pulsations with 1-month long time series reporting R.A. Garc'ıa on behalf of the WG#1 Team Laboratoire AIM, CEA/DSM-CNRS-Universit'e Paris Diderot; IRFU/SAp, Centre de Saclay, 91191, Gif-sur-Yvette, France, e-mail: [email protected] their existence in more than 550 stars [13]. From the second year on, around 60 stars have been continuously observed for a year and additional ∼ 150 stars have been observed for at least 3 continuous months. All this observational set constitutes an ensemble in which we can perform very accurate asteroseismic analyses. Moreover, stellar evolution studies of solar-like stars are completed by the analysis of several thousands of red giants measured in the Kepler long-cadence mode (e.g. [5], [23]).", "pages": [ 1, 2 ] }, { "title": "2 Some initial results", "content": "Asteroseismology is a powerful tool to look inside the stars and put strong constraints in structure and evolution models (see e.g. [30], [24], [29]). In pulsating stars that host planets, asteroseismology allows us to also place very tight constraints on the exoplanetary systems (e.g. [15], [1], [2]). Depending on the signal-to-noise ratio of the modes -which also seems to depend on the magnetic activity of the star [11]- on one hand we can have access to only some global seismic parameters (e.g. ∆ν , ν max, A max [35]) of the stars using a new generation of automatic procedures that have been set up for solar-like pulsating stars (e.g. [20], [22],[26], [31]) including red giants [21]. From these measurements we were able, for example, to verify [34] the log g and the radius of the Kepler Input Catalog [8] inferred from the scaling relations based on solar values ([25] and [12]). On the other hand -when the signal-to-noise ratio is high enough- we can characterize very precisely the individual p-mode parameters (e.g. [28], [9]). In Fig. 1 the 'echelle diagram of KIC 11395018 -observed for more than 8 continuous months- is shown. We clearly identify the ridges of the modes l=2, 0, and 1 from left to right respectively. A nice feature of the power spectrum is the presence of a mixed l=1 mode at 740.29 µ Hz that bumps the modes in the surrounding orders. Indeed mixed modes are very interesting. They behave like acoustic modes near the surface of the stars and as g modes in the interior. They have a very high sensitivity to the core of the stars. In the case of the red giants, many mixed modes can be measured (see e.g. [3]). They can be used to determine the status of the core (if they are already burning He or not [6]) and even to track the internal differential rotation ([4], Deheuvels et al. in preparation). In the case of the subgiants, where only a few mixed modes are available, their position in frequency depends very much on the age of the star and can be used to further constrain the models [14]. The high number of stars available to the asteroseismic research allows us to build evolutionary sequences of stars -of similar masses and compositions- and study the evolution of the physical processes governing their interiors as a function of time, like a collection of snapshots of the life of a star (see [33] for more details). With the possible extension of the Kepler mission to 6 years, the future of the solar-like studies with Kepler is amazing. In particular, having several years of continuous observations of a several dozens of stars will allow us to study their dynamics: rotation [18] and magnetic fields. Indeed we will be able to study the surface magnetic activity [27] at the same time as the internal one [16], and the internal structure including the characteristics of the convective zones. The physical processes governing the dynamos will be better constrained and we will improve our knowledge of the solar dynamo, which still hides its secrets (e.g. [32]). Acknowledgements RAG wishes to thank the Kepler team. Funding for this Discovery mission is provided by NASAs Science Mission Directorate. This work has received funding from the European Communitys Seventh Framework Program (FP7/2007-2013) under grant agreement No. 269194 and the NSF under Grant No. NSF PHY05-51164.", "pages": [ 2, 3 ] } ]

[ { "title": "Two-Dimensional Quantum Geometry 1", "content": "J. Ambjørn a,b and T.Budd a a The Niels Bohr Institute, Copenhagen University Blegdamsvej 17, DK-2100 Copenhagen Ø, Denmark. email: [email protected], [email protected] b Institute for Mathematics, Astrophysics and Particle Physics (IMAPP) Radbaud University Nijmegen, Heyendaalseweg 135, 6525 AJ, Nijmegen, The Netherlands.", "pages": [ 1 ] }, { "title": "Abstract", "content": "In these lectures we review our present understanding of the fractal structure of two-dimensional Euclidean quantum gravity coupled to matter. PACS: 04.60.Ds, 04.60.Kz, 04.06.Nc, 04.62.+v. Keywords: quantum gravity, lower dimensional models, lattice models.", "pages": [ 1 ] }, { "title": "1 Introduction", "content": "A noble task in ancient, pre-AdS/CFT time was to find a non-perturbative definition of Polyakov's bosonic string theory. The formal partition function was defined by the path integral: Here [ g αβ ] represents a continuous 2d geometry of some fixed topology. Assume that the set of piece-wise linear geometries one can obtain by gluing together equilateral triangles with link length a is uniformly dense in the set of continuous 2d geometries when a → 0. Each such geometry can be identified with an abstract triangulation. By placing the matter field X µ ( ξ ) in the center of each triangle and using the natural discretized version of the matter Lagrangian in (1) we obtain a lattice regularization of the action, for which the lattice spacing a acts as a UV cut-off. Summing over the abstract triangulations provides a lattice regularization of the integral over geometries in (1), coined Dynamical Triangulations (DT) [1, 2, 3]. If the assumption about the denseness of these triangulations in the set of continuous geometries holds, we expect to obtain the continuum path integral in the limit a → 0. Of course, it is to be expected that one has to renormalize the bare coupling constants entering in the lattice partition function to recover the continuum results. If we work in units where the lattice spacing a is put to one, we obtain the dimensionless DT partition function for the bosonic string, where the overall sum is over triangulations T with N T triangles and the sum in the exponent is over pairs /triangle , /triangle ' of neighboring triangles.", "pages": [ 2 ] }, { "title": "1.1 The free particle", "content": "To understand how to obtain the continuum limit of (2), it is useful to study the simpler system of a free particle. In this case the propagator G ( X ν , X ' ν ) has the path integral representation where X ν (0) = X ν and X ν (1) = X ' ν , and [ g ] is the geometry of a world line, i.e. d/lscript 2 = g ( ξ ) dξ 2 and ∫ dξ √ g = /lscript . The structure of eq. (3) is quite similar to that of eq. (1). The path integral is discretized by dividing the worldline in n equal steps (the equivalent of the equilateral triangles for DT) and using dimensionless variables: with x (0) = x and x ( n ) = x ' . One can perform the Gaussian integrations: Introducing µ c = 1 2 d log(2 π ), we get leading to Performing a mass renormalization and a scaling, we obtain the standard proper time representation of the free relativistic propagator The explicit, well-defined path integral representation (4) of the free particle is useful for analyzing simple basic properties of the propagator. Let us just mention one such property, the exponential decay of the propagator for large distances. Why can the propagator not fall of faster than exponentially at large distances? The answer is found by looking at Fig. 1. The set of paths from x to y has as a subset the set of paths intersecting the straight line connecting x and y at a point z . A path in this subset is a union of a path from x to z and from z to y . Since the action for such a path is the sum of the actions of the path from x to z and the path from z to y , it is not difficult to show i.e. The subadditivity of -log G ( x, y ) implies that there exits a positive constant m such that i.e. The constant m is the mass of the particle (which can be zero in special cases).", "pages": [ 2, 3, 4 ] }, { "title": "1.2 The bosonic string", "content": "One can also perform the Gaussian integration in the string case: where ∆ T N is the combinatorial Laplacian on the dual φ 3 -graph. The prime indicates that the constant zero mode is projected out in the determinant. We find and In the scaling limit µ → µ c one may identify Equation (16) is valid for geometries with fixed topology of the sphere, but Z ( µ ) generalizes naturally to surfaces with n boundaries { γ i } of fixed length L i on which the coordinates x µ are fixed. In particular, in the limit L i → 0 we obtain the n -point function G ( x 1 , . . . , x n ; µ ) for spherical string world sheets with n marked points at prescribed positions x 1 , . . . , x n . A basic property of the two-point function G ( x 1 , x 2 ; µ ) is subadditivity. The argument is essentially the same as for the particle, except that random surfaces are involved instead of random walks, as illustrated in Fig. 2. Therefore we find Similarly we may consider the planar 'Wilson loop' G ( γ L 1 × L 2 , µ ), corresponding to the partition function with one boundary γ L 1 × L 2 of length 2 L 1 +2 L 2 corresponding to a rectangular loop in R d with sides of length L 1 and L 2 . As illustrated in Fig. 3, G ( γ L 1 × L 2 , µ ) is subadditive both in L 1 and L 2 , and therefore we obtain 2 where A ( γ L 1 × L 2 ) = L 1 L 2 is the area of the loop, and σ ( µ ) is known as the string tension. However, the dominant worldsheet surfaces look completely different from the nice surfaces shown in Fig. 2 and Fig. 3. The reason for this is shown in Fig. 4. It is seen from the figure that, while the mass of the two point function scales to zero at a critical point, which is needed if one wants a continuum limit, this is not the case for the string tension σ ( µ ) ([5] or [4], theorem 3.6). The consequence is that the physical string tension scales to infinity as µ → µ c : An infinite string tension implies that any surface with finite area is forbidden unless it is dictated by some imposed boundary conditions. A typical surface with no area contributing to the two-point function G ( x 1 , x 2 , µ ) is shown in Fig. 5. Such surfaces are called branched polymer (BP) surfaces. They have only one mass excitation corresponding to a free particle, since one basically obtains a random walk representation corresponding to the free particle by scaling away the branches decorating the shortest path from x 1 to x 2 for a given surface connecting x 1 and x 2 . In the case of the Wilson loop we are summing over surfaces where the boundary is fixed. Therefore we have a minimal-area surface stretching to the boundary. The fluctuations around this surface, however, are again branched polymers, as shown in Fig. 6, and are nothing like the surface in Fig. 3. The conclusion is that the bosonic string theory defined through a regulated path integral where all surfaces have positive weight does not exist. The reason that we do not obtain the standard bosonic string, despite such a well-defined procedure, is that the two-point function of the standard bosonic string has tachyonic mass excitations, which are excluded by our construction and which make standard bosonic string theory sick.", "pages": [ 4, 5, 6, 7 ] }, { "title": "2 Non-critical string theory", "content": "However, interpreting the string world sheet as 2d space-time, we can view Polyakov's bosonic string theory in d dimensions as 2d gravity coupled to d massless scalar fields, i.e. to a conformal field theory with central charge c = d . Therefore, as another route towards the bosonic string, we can study 2d quantum gravity coupled to (conformal) field theories. Surprisingly this theory, called non-critical string theory, has a rich structure as long as the central charge c ≤ 1. The regularized version of such a theory is typically obtained as follows: assume we have a conformal field theory originating from a field theory on a regular lattice. Usually the lattice theory has a critical point with a secondorder phase transition and the continuum conformal field theory is then defined at the critical point. This lattice field theory can usually be transfered from a regular lattice to a random one, hence, also to the random lattice appearing in the DT formalism. Including a summation over different lattices in ensemble averages is what is called an annealed average in the context of condensed matter physics. Here it will play the role of integrating over 2d geometries, as for the bosonic string. The partition function of 2d gravity coupled to matter can be written as where Z T N (matter) is the matter partition function on a fixed triangulation T N . A typical example is the Ising model coupled to DT [6], The partition function scales as Here µ c ( β ) appears as the critical 'cosmological' constant for the geometries, such that one obtains universes with infinitely many triangles when µ → µ c from above. This is similar to the situation for the free particle and the bosonic string and we clearly want to take that limit in order to recover continuum physics from the lattice theory. However, it also follows from (23) that it has the interpretation as the the free energy density of spins in the annealed ensemble. /negationslash The model has a phase transition at a critical β c , the transition being third order rather than the standard second order phase transition [6]. At the transition point γ ( β ) jumps from -1/2 to -1/3. The interpretation is as follows: on a regular lattice the Ising spin system also has a phase transition at a certain critical temperature β c . The transition is a second order transition and at the transition point the spin system describes the continuum conformal field theory of central charge c = 1 / 2. The lattice theory, defined on the annealed average of lattices, describes at its critical point the c = 1 / 2 conformal field theory coupled to 2d quantum gravity, the average over the DT lattices being the path integral over geometries. It is not surprising that the transition can change from a second order to a third order transition, the randomness of the lattices and the averaging over different lattices making it more difficult to build up large critical spin clusters at the phase transition point. Maybe it is more surprising that there is a transition at all. But it is known to be the case, since one can solve the model analytically. One finds that the critical spin exponents have changed compared to Onsager exponents on a regular lattice. Thus the continuum conformal field theory has changed due to the interaction with 2d quantum gravity. Further, as we mentioned, the exponent γ ( β ) jumps at β c . The exponent γ ( β ) as it appears in (24) reflects average fractal geometric properties of the ensemble of random geometries appearing in the path integral. Thus a change in the exponent reflects that the conformal field theory back-reacts on the geometry and changes its fractal properties, something we will discuss in detail below. Away from β c the Ising model is not critical, and the lattice spins couple only weakly to the lattice. For all β = β c one has γ ( β ) = -1 / 2 and this can then be viewed as the exponent for 'pure 2d Euclidean gravity' without matter fields.", "pages": [ 8, 9 ] }, { "title": "2.1 Continuum formulation", "content": "One can study 2d quantum gravity coupled to matter fields entirely in the continuum. Just like for the partition function (1) for the bosonic string, we can write formally where ψ represents some matter field. A partial gauge fixing, to the so-called conformal gauge g αβ = e φ ˆ g ( τ i ) leads to where S L ( φ, ˆ g ) is fixed by the requirement that Z (ˆ g ) is independent of ˆ g , namely [8] Even for c = 0 we have a non-trivial theory. The c = 0 partition function can be obtained explicitly at the regularized level simply by counting the triangulations, since there are no matter fields. A slightly non-trivial structure can be imposed by n boundaries of lengths /lscript n , as illustrated in Fig. 7 for the case n = 3. Also in that case the counting can be done and the continuum limit taken. The continuum definitions of the n -loop functions are Formally (29) counts each continuous geometry (defined by an equivalence class of metrics [ g αβ ( ξ )]) with weight one. Eq. (30) defines the partition function for universes with fixed boundary lengths /lscript i and with a cosmological constant Λ. Eq. (31) defines the partition function for universes with boundary cosmological constants Λ B i and bulk cosmological constant Λ, i.e. the partition function where both the lengths of the boundaries and the size of the universe are allowed to fluctuate, controlled by the various cosmological constants. From a 'counting perspective' one can view W (Λ B 1 , . . . , Λ B n , Λ) as the generating function for W ( /lscript 1 , . . . , /lscript n , V ), the number of continuous geometries with n boundaries of lengths /lscript i . Of course, to perform any real counting one has to introduce a regularization such that one starts out with a finite number of geometries, and for this purpose the DT-formalism is perfect. As an example we can write the regularized DT version of W (Λ B , Λ), i.e. the 1-loop function, as such that W ( z 1 , g ) is the generation function for W ( l 1 , k ), the number of triangulations with k triangles and a boundary with l 1 links. As with most counting problems, it is easier first to find the generating function W ( z 1 , g ) and then by inverse (discrete) Laplace transformations to find the numbers W ( l 1 , k ). The result of this counting ([7], see [4], Chapter 4, for a review) is that after the continuum limit is taken, using the techniques of renormalization of the bare lattice cosmological constant µ and boundary cosmological constants λ i (appearing in (32)), one obtains the expression Starting out from the continuum Liouville theory the same result has been reproduced. In this sense the agreement shows that the DT lattice regularization works perfectly (and even allows one to perform certain analytic calculation with less effort than using the continuum formulation, something very rare for a lattice regularization). It also gives additional confidence in the continuum Liouville calculations, which rely on certain bootstrap assumptions about conformal invariance.", "pages": [ 9, 10, 11 ] }, { "title": "3 The fractal structure of 2d QG", "content": "While eq. (33) is an amazing formula, basically counting the number of continuous 2d geometries with the topology of a sphere with n boundaries, it tells us little about the 'typical' 2d continuous geometry one encounters in the path integral. In order to probe such a geometry we need some specific reference to distance. One could be worried that it makes no sense to talk about distance in a theory of quantum gravity, i.e. a theory of fluctuating geometry, since it is precisely the geometry that defines distance. However, the key message of the following is that it does make sense to talk about geodesic distance even in a such a theory. Let us define the two-point function G ( R ; V ) of geodesic distance R for surfaces of fixed volume V by where A ( g ) ≡ ∫ d 2 x √ g ( x ) and D g ( x, y ) denotes the geodesic distance between x and y in the geometry defined by the metric g αβ ( x ). The defining formula (34) is valid for any matter field ψ coupled to 2d quantum gravity. In principle it is also valid in a higher dimensional theory of quantum gravity provided one includes in S [ g, ψ ] the Einstein action (or whatever one uses as the action). In two dimensions the Einstein action is topological and we may drop it. It might be convenient not to keep V fixed, but rather to consider the two-point function for the ensemble of universes with a fixed cosmological constant Λ, i.e. These two-point functions probe the geometries in the following way. Denote the 'area' of a spherical shell at geodesic distance R from point x by which, of course, depends both on the chosen geometry g αβ and the point x . Let us denote the diffeomorphism invariant average of S V ( x ; R ) by The quantum average of S V ( R ) over all geometries is then related to G ( R ; V ) by where Z V is the corresponding partition function of 2d quantum gravity coupled to matter, i.e. the rhs of (34) but with the integral (and integrand) over x, y removed. For a smooth 2d geometry we have while in general we define the fractal dimension, or Hausdorff dimension, d h for the quantum average by The partition function scales as Z ( V ) ∼ V γ ( c ) -3 , where the string susceptibility γ ( c ) is a function of the central charge c of the matter field coupled to the geometry and is known to be given by [8, 9] In the absence of matter fields, i.e. c = 0, we have γ = -1 / 2, and the scaling is seen to agree with (33) for n = 0. Therefore we can determine d h from the functional form of G ( R ; V ) or G ( R ; Λ). Remarkably, there is a simple and closed formula for G ( R ; Λ) for c = 0, obtained again by counting triangulations, namely [11] This can be turned into an expression for G ( R ; V ) by an inverse Laplace transformation, which may plugged into (40), leading to where F ( x ) is a hypergeometric function falling off for large x as e -x 4 / 3 . Note that, while G ( R ; V ) falls of faster than exponentially as a function of R , this is not possible for G ( R ; Λ) because of arguments of subadditivity of the kind already used for the two-point function of the bosonic string. Comparing (43) to (40), we conclude that 2d continuous geometry is fractal with Hausdorff dimension d h = 4 [10, 12]. This is in some sense similar to the situation for the free particle, where one is summing over continuous path from x to y in R d . There a typical path is not a one-dimensional object, but is fractal with d h = 2. The difference is that for the geometries we have no embedding space R d with respect to which we can define a distance. This makes it the more remarkable that one still has a concept of geodesic distance that survives the averaging over all geometries. How is it possible that d h = 4? The reason d h can be larger than 2 is that S V ( x ; R ) is almost surely not connected, as is illustrated in Fig. 8. In fact, one can show [11] that the number of connected components of S V ( x ; R ) with length /lscript between /lscript and /lscript + d/lscript is given by in the limit V →∞ . Thus the number of components with small /lscript diverges for /lscript → 0. Of course, in the DT formalism there is a cut-off in the sense that the smallest loop length consists of a single link (of length a , the UV cut-off). In the presence of such a cut-off (44) leads to again leading to the conclusion that d h = 4.", "pages": [ 12, 13, 14, 15 ] }, { "title": "3.1 The central charge different from zero", "content": "/negationslash For c = 0 (and c ≤ 1) no detailed calculations exist like the ones reported above. However, there exists a remarkable formula derived by Watabiki [13] for d h for any c ≤ 1: The formula was derived by applying scaling arguments, which we will briefly summarize, to diffusion on two-dimensional geometries in quantum Liouville theory. Let Φ n [ g ] be a functional of the metric which is invariant under diffeomorphisms and assume that classically Φ n [ λg ] = λ -n Φ[ g ] for constant λ . According to the KPZ relations the quantum average then satisfies [14, 8, 13] One now applies this to the operator which appears when we study diffusion on a smooth manifold with metric g µν . The diffusion kernel is It has short distance behavior The return probability is defined in terms of the diffusion kernel as These equations are trivially correct for a smooth geometry g αβ ( x ), and they link the dimension of Φ 1 [ g ] to the dimension of D ( x, x 0 ): Of course, this link is trivial in the sense that Dim[ D ( x, x 0 )] = 1 and Dim[Φ 1 [ g ]] = -2 by construction. Watabiki now conjectured that (52) survives the quantum averaging, where we know from (47) how the dimension of Φ 1 [ g ] changes. Thus one obtains leading to (46) if we declare that Dim[ V ] = 2, such that", "pages": [ 15, 16 ] }, { "title": "3.2 Is the Watabiki formula correct?", "content": "/negationslash One may be worried about the previous derivation of d h ( c ), since the result implies that a typical spacetime is fractal, while the basic relation used, namely (51), is valid only on smooth spacetimes. But not only that: numerical simulations [15] seem to show that the diffusion distance R ( t ) scales like 〈 R 2 ( t ) 〉 ∼ t 2 /d h , rather than like in (50). Anomalous diffusion is normal on fractal spacetimes, but it makes the Watabiki derivation problematic. Nevertheless, the predicted d h (0) is clearly correct and it might be that d h ( c ) is also correct for c = 0. This is what we have tried to test using numerical methods to measure d h ( c ). We have found it convenient to use 2d spacetimes with toroidal topology. These have the virtue that their shortest non-contractible loop is automatically a geodesic curve [18]. Thus in the discretized case we only have to look for such loops. Further, the harmonic forms which are important tools for analytic manifolds have very nice discretized analogies, and we can use the these to construct a conformal mapping from the abstract triangulation to the complex plane [16, 17]. We have shown an example of such a map in Fig. 9. Since the shortest non-contractible loop is a geodesic we expect An amazing qualitative test of this is shown in Fig. 10, where we use the harmonic map mentioned to map two abstract triangulations corresponding to c = 0 and c = -2 and 150000 triangles into the complex plane. Already just by looking at the figures one can basically verify qualitatively (55). A quantitative check of 〈 L 〉 N ∼ N 1 /d h for c = -2 is shown in Fig. 11, where we have averaged over many configurations for a fixed size N of the triangulation, and performed the measurements of the shortest non-contractible loops for different sizes N . Formula (46) seems very well satisfied numerically for c = -2. Recall that the partition function for the (regularized) bosonic string embedded in d dimensions is given by eq. (14): it can be viewed as a conformal field theory with central charge c = d coupled to 2d quantum gravity. As we have seen, the theory degenerates into BP for c > 1. However, from (14) it is clear that we can formally perform an analytic continuation to c < 1. A special case is c = -2 because then the triangulations are weighted precisely by the determinant of the graph Laplacian, which can be represented as a sum over spanning trees of the given triangulations. This fact was used in the numerical simulations reported above and allowed us to sample very large triangulations and to obtain great numerical accuracy [16]. More generally one can sample from the partition function for any fixed real value c by explicitly evaluating the determinant in a Monte Carlo simulation [19]. This can, of course, only be done efficiently for relatively small triangulations. However, it turns out that to study DT for large negative c /lessmuch -2 and to obtain a qualitative verification of formula (46), one only requires such small triangulations. In particular, the formula tells us that d h → 2 for large negative c , indicating that nice smooth geometries should dominate in that limit. This is illustrated in Fig. 12. The situation for c > 0 is more difficult and until recently numerical simulations could not really determine d h ( c ) properly for c > 0. Matter correlation functions gave agreement with Watabiki's formula, but geometric measurements agreed better with d h = 4 for 0 < c < 1. Recently simulations have been performed of DT on the torus coupled to the Ising model ( c = 1 / 2) and the 3-states Potts model ( c = 4 / 5) [20]. In addition to the shortest non-contractible loop length /lscript 0 , also the length /lscript 1 of the second shortest independent loop was analyzed (see Fig. 13), yielding data with little discretization 'noise'. The probability distributions for the lengths /lscript i are expected, for large N , to be of the form By measuring the distributions for various N 's and attempting to 'collapse' the distributions to the common, universal functions F i ( x i ) we can determine d h . Typically, one chooses reference distributions, here chosen to be interpo- lations of the loop length distributions for N = 8000, to which the data for the other system sizes is fitted. In Fig. 13 the reference distributions P N ( /lscript 0 ) and P N ( /lscript 1 ) are plotted for both the Ising model and the 3-states Potts model. It is seen that the second shortest loop distributions contain less very short loops, which is probably why their lengths have less discretization effects and show better scaling. The best fits of d h for the data are shown in Fig. 15 and summarized in the following table. ±", "pages": [ 16, 17, 19, 20 ] }, { "title": "4 Matter correlation functions", "content": "We have seen that the two-point functions G ( R ; Λ) and G ( R ; V ) are good probes of the quantum geometry of 2d spacetime and allowed us to define the concept of an average geodesic distance. Also for matter correlators 〈 φ ( x ) φ ( y ) 〉 the first obvious question one can ask is whether it makes any sense to talk about such correlators as functions of distance, and which distance should one use if we are integrating over all geometries? It is natural to define the diffeomorphism invariant matter correlators as the following generalization of eq. (34) for G ( R ; V ): It is a non-local definition of a matter correlator, but there exists no diffeomorphism invariant local definition. Assume we consider a conformal field theory in flat spacetime and let φ ( x ) be a primary operator with scaling dimension ∆ 0 . We thus have the following behavior of the φ -φ correlator If we take the quantum average as in (57) the geodesic distance R scales anomalously and we expect for dimensional reasons that | x -y | -2 is replaced by R -d h . However, we also know from KPZ scaling that the scaling dimension ∆ 0 of φ will be changed after coupling to 2d quantum gravity such that where one observes that c → -∞ implies ∆( c ) → ∆ 0 in agreement with the earlier observation that Watabiki's formula shows that d h ( c ) → 2 for c →-∞ . For a finite spacetime volume V we finally expect a behavior which alternatively can be written as For a given conformal field theory g φ ( x ) is a universal finite size function with g φ (0) = const. > 0 and g φ ( x ) falling of at least exponentially fast for x > 1. The formula (61) is convenient to use in the DT regularization where V ∼ N T and the geodesic distance R ∼ /lscript , the link distance between two vertices: We note that eq. (62) has for form of a standard finite size scaling relation. One can thus apply the formula to the Ising model or 3-states Potts model and measure the spin-spin correlation functions for various values of N . Collapsing these correlation functions to universal functions g φ ( x ) for either the Ising model ( c = 1 / 2) or the 3-states Potts model ( c = 4 / 5) allow us to determine both d h ( c ) and ∆( c ) for these values of c . One finds a d h ( c ) in agreement with watabiki's formula as mentioned earlier (but not with the same precision as with the method described in the last section), and one finds a ∆( c ) in good agreement with the KPZ formula (59) [21]. The result of collapsing the data to a (best possible) universal function g φ ( x ) is shown in Fig. 10 for the Ising model. It works very well for an impressive range of lattice sizes. Remarkably, finite size scaling works even better on the DTensemble of lattices than on a fixed lattice. Somehow the random lattices average out finite lattice artifacts.", "pages": [ 21, 22 ] }, { "title": "5 Conclusions", "content": "Two-dimensional quantum gravity is a nice playground for testing to what extent it makes sense to talk about non-trivial diffeomorphism invariant theories of fluctuating geometry. We have here focused on the very simplest question: if one integrates over the fluctuating geometries as one should do in a path integral representation of a quantum theory, how can one at all talk about concepts like distances and correlation functions falling off with this distance. In this context 2d quantum gravity is the perfect theory for such tests. It has no propagating gravitational degrees of freedom, but it is maximally quantum , the reason precisely being that the Einstein action in two dimensions is trivial. Every geometry carries therefore the same weight in the path integral, as exemplified by eq. (29), i.e. formally it corresponds to a /planckover2pi1 →∞ limit. If we want to study ordinary field theories (like conformal field theories) and not just esoteric topological field theories, we cannot avoid the clash between the integration over all geometries and the need to have some concept of distance. However, as we have seen some aspects of geodesic distance remarkably survive this quantum average over geometries, despite the fact that geodesic distance is a awfully non-local notion. Although the geodesic distance picks up an anomalous dimension due to quantum fluctuations, it maintains its role as the distance which can be used in the correlators between fields. In higher dimensions there might not exist a well-defined, stand-alone theory of quantum gravity. The UV problems for such a theory might be too severe. This question is still up in the air, and it might well be that the metric degrees of freedom we have in classical GR are not the fundamental degrees of freedom one should use in the UV regime. However, the studies reported here show that conceptually there seems to be no problem with a theory of 'fluctuating' geometries per se and even in the most radical such one, namely 2d quantum gravity, one can maintain many of the concepts we know from flat spacetime.", "pages": [ 23, 24 ] }, { "title": "Acknowledgments", "content": "The authors acknowledge support from the ERC-Advance grant 291092, 'Exploring the Quantum Universe' (EQU). JA acknowledges support of FNU, the Free Danish Research Council, from the grant 'quantum gravity and the role of black holes'. JA thanks his collaborators K. Anagnostopoulos, B. Durhuus, T.Jonsson, J. Jurkiewicz and Y. Watabiki for many discussions on the topics covered here. They cannot be blamed for any mistakes (conceptional or otherwise) in this article.", "pages": [ 24 ] } ]

[ { "title": "IceCube Observatory: Neutrinos and the Origin of Cosmic Rays", "content": "Paolo Desiati 1 , 2 , for the IceCube Collaboration 3 3 http://icecube.wisc.edu Corresponding author: [email protected]", "pages": [ 1 ] }, { "title": "Abstract", "content": "The completed IceCube Observatory, the first km 3 neutrino telescope, is already providing the most stringent limits on the flux of high energy cosmic neutrinos from point-like and diffuse galactic and extra-galactic sources. The nondetection of extra-terrestrial neutrinos has important consequences on the origin of the cosmic rays. Here the current status of astrophysical neutrino searches, and of the observation of a persistent cosmic ray anisotropy above 100 TeV, are reviewed. Keywords: Neutrinos - Cosmic Rays - Anisotropy.", "pages": [ 1 ] }, { "title": "1 Introduction", "content": "One hundred years after their discovery, the origin of the cosmic rays is still a mystery. The current leading model is that cosmic rays are accelerated in diffusive shocks. In this case Supernova Remnants (SNRs) in our Galaxy could be the major source of cosmic rays up to about 10 15 -10 17 eV. The SNR energy output in the Galaxy can provide the energy budget necessary to maintain the presently observed population of galactic cosmic-rays. In particular, in order to achieve such high energies it is expected that acceleration occurs during the relatively short period in the SNR evolution between the end of free expansion and the beginning of the so-called Sedov phase. This period is about 10 3 years from the explosion when the shock velocity is high enough to allow for efficient acceleration. At energies in excess of about 10 17 eV, Active Galactic Nuclei (AGN) and Gamma Ray Bursts (GRB) could play an important role in the origin of the extra-galactic cosmic rays. Since cosmic rays are deflected by magnetic fields, it is not possible to associate them to their sources. However, if hadronic particles are accelerated, a fraction of them would interact within their sources or in surrounding molecular clouds to produce mesons. The mesons eventually decay into high energy γ rays and neutrinos with an energy spectrum ∼ E -2 of the accelerated cosmic rays. The remaining hadronic particles propagate until their detection on Earth. Detection of γ rays and neutrinos from individual galactic or extra-galactic source candidates of cosmic rays, or from extended molecular clouds, is there- a method to indirectly probe the origin of cosmic rays. During the last decade, detection of γ rays from galactic sources has been successfully achieved by satellite experiments such as AGILE and Fermi up to 10 and 100 GeV, respectively. Imaging Cherenkov Telescope Arrays such as MAGIC, VERITAS and H.E.S.S., and water Cherenkov detectors such as Milagro have made measurements up to O(10 TeV). High energy direct emission from old SNRs appears to be inconsistent with hadronic acceleration 1 . It is interesting, however, that delayed secondary γ ray emissions can be produced by the most energetic particles that escaped the acceleration region when they propagate through molecular clouds that surround the star forming regions [1]. With this mechanism, indirect evidence of hadronic acceleration is present even when SNR are several 10 4 years old. In fact, the detection of an extended emission of TeV γ rays from the Galactic Center by H.E.S.S., which is attributed to cosmic rays accelerated by SNR G0.9+0.1 interacting with the surrounding clouds, might provide the first evidence of hadronic acceleration [2]. The most compelling evidence currently comes from low energy γ ray emission from the regions surrounding the intermediate-age SNR W44. AGILE observations in the energy range of 50 MeV - 10 GeV [3] and Fermi observations up to 100 GeV [4] show that while leptonic models fail to describe simultaneously γ and radio emissions without requiring too large circumstellar densities, the hadronic models are consistent with experimental constraints from radio, optical, X and γ rays observations. Although the γ ray energy spectrum is consistent with a proton spectral index of 3 and a low energy cut-off of approximately 10 GeV 2 , the hadronic origin of the observed emission is considered likely. The observed steep spectrum and low energy cut-off may be caused by suppression of efficient particle acceleration in the dense environment of this source [5]. Ion-neutral collisions in the weakly ionized dense gas surrounding the remnant lead to a softer spectrum as well as to damping of the plasma Alfv'en waves that form the shock. The resulting poor particle confinement leads to a low energy cutoff [6]. Other than the specific properties of single objects, evidence of an instance of hadronic acceleration is a very important step towards the discovery of the origin of cosmic rays. However, this would not mean that all galactic cosmic rays are necessarily accelerated in SNR. If cosmic ray acceleration occurs predominantly on a larger scale, such as in superbubbles [7] or in the Galaxy cluster medium where particles could be accelerated to ultra-high energies [8], the search for the origin of cosmic rays should concentrate on extended sources or diffuse fluxes. While the TeV γ ray horizon is limited within our Galaxy, because of absorption in the infrared and microwave cosmic background, the GeV γ emissions can be observed within about 100 Mpc making it possible to search for extragalactic sources of cosmic rays. On the other hand, detection of neutrinos from individual sources are an efficient and unambiguous probe for the high energy hadronic acceleration mechanism, and therefore for the sources of cosmic rays. However, the very same property that makes neutrinos an excellent cosmic messenger also makes them difficult to detect. Thus large instrumented volume of target matter is required to capture sufficient event statistics. The IceCube Neutrino Observatory (see Fig. 1), completed in December 2010, is currently the only km 3 scale neutrino telescope collecting data. The observatory consists of an array of 5,160 optical sensors arranged along 86 cables (or strings) between 1,450 and 2,450 meters below the geographic South Pole, where the antarctic ice is particularly transparent. IceCube includes a surface shower array, IceTop, and a dense instrumented core with a lower energy threshold, DeepCore. The surface array, IceTop, is 81 stations each consisting of two tanks of frozen clean water with each tank containing two optical sensors. IceTop, using events in coincidence with the deep IceCube array, provides the measurement of the spectrum and mass composition of cosmic rays at the knee and up to about 10 18 eV. The DeepCore subarray, consisting of 6 densely instrumented strings located at the bottom-center of IceCube, lowers the observatory neutrino energy threshold to about 10 GeV. DeepCore uses the surrounding IceCube instru- mented volume as a veto for the background of cosmic ray induced through-going muon bundles, thus enhancing the detection of down-going neutrinos within the Deep Core volume. Veto rejection power in excess of 10 8 has been achieved [9]. The basic detection component of IceCube is the Digital Optical Module (DOM) which consists of a 10-inch Hamamatsu photomultiplier tube (PMT) and its own data acquisition (DAQ) circuitry enclosed in a pressure-resistant glass sphere. The DOMs detect, digitize and timestamp the signals from Cherenkov radiation photons. Their main DAQ board is connected to the central DAQ in the IceCube Laboratory at the surface, where the global trigger is determined [11]. The construction of IceCube started in 2004 and physics quality data taking commenced in 2006. With this early data the observatory is providing the most stringent limits on the flux of high energy neutrinos from extra-terrestrial origin, and therefore strong constraints on the models of individual sources of cosmic rays and unidentified diffuse sources. At the same time, IceCube has accumulated a large number of cosmic ray induced neutrinos produced in the atmosphere, making it possible to probe the combined effect of hadronic interaction models, cosmic ray spectrum and composition on the neutrino spectrum up to a few hundred TeV [10]. In the search for high energy neutrinos, the large exposure of IceCube makes it possible to collect an unprecedented number of events in the form of bundles of high energy muons generated in the cosmic ray induced extensive air showers. Although these events represent an overwhelming background in the neutrino searches, they make it possible, for the first time, to determine the degree of anisotropy of cosmic rays from a few TeV to several PeV of particle energy. The persistence of a cosmic ray anisotropy at high energy raises the question of the responsible mechanism. The notion that cosmic ray anisotropy might be connected to the distribution of nearby and recent supernovae is intriguing, and might thus provide a new probe into the origin of the cosmic rays. On the other hand the complex energy-dependent topology suggests that non-diffusive processes in the local interstellar medium most probably play an important role.", "pages": [ 1, 2, 3 ] }, { "title": "2 Physics Results", "content": "If the signals from detected Cherenkov photons satisfy specific trigger conditions, an event is defined and recorded by the surface data acquisition system. Online data filtering at the South Pole reduces the event volume to about 10% of the trigger rate, based on a series of reconstruction and filter algorithms aimed to select events based on directionality, topology and energy [15]. The filter makes it possible to transfer data via satellite from the experimental site for prompt physics analyses.", "pages": [ 3 ] }, { "title": "2.1 Atmospheric neutrinos", "content": "Of the events that trigger IceCube, the vast majority are muon bundles produced by the impact of primary cosmic rays in the atmosphere. Only a small fraction of the detected events ( ∼ 10 -5 ) are muons produced by the charged current interaction of atmospheric muon neutrinos. The easiest way to reject the down-going muon bundle background is to exclusively select well reconstructed up-going events, since these can only be produced by neutrinos crossing the Earth and interacting in the matter surrounding the detector. Depending on the detector configuration and on the specific reconstruction algorithms and event selection utilized, the atmospheric neutrino sample is characterized by a directional resolution of better than 1 · above 1 TeV. The corresponding resolution in the estimation of the muon energy is about 0.2-0.3 (in log10 of the energy) for crossing tracklike events, and about 0.1 or better for contained cascade-like events. Typically, 30%-40% of the upgoing events survive the selection with a background contamination of less than about 1% (see Tab. 1). The atmospheric neutrino sample collected by IceCube over the years is the largest ever recorded and currently reaches energies near 400 TeV (see Fig. 2). For the first time the precision of this measurement is providing a powerful tool to constrain the effects of high energy hadronic interaction models that represent our present knowledge of the cosmic ray induced extensive air showers and the spectrum and composition of primary cosmic rays [10].", "pages": [ 3 ] }, { "title": "2.2 Search for astrophysical ν 's", "content": "Atmospheric neutrinos represent an irreducible background for the search of high energy astrophysical neutrinos. If hadronic acceleration is the underlying process of high energy cosmic ray production and γ ray observations in galactic and extra-galactic sources, the charged mesons could produce enough neutrinos to be observed in a detector the size of IceCube. Fig. 3 shows the sensitivity (90% CL) of IceCube for the full-sky search of steady point sources of E -2 muon neutrinos as a function of declination, along with that of other experiments. The extension of the point source search to the southern hemisphere is made possible by a high energy event selection that rejects the background down-going events by five orders of magnitude, and restricts neutrino energies to above 100 TeV. Still dominated by high energy large muon bundles, this makes the southern hemisphere poor in atmospheric neutrinos yielding a low neutrino detection sensitivity. Nevertheless, this provides IceCube with a full-sky view that complements coverage of the neutrino telescopes in the Mediterranean. The figure shows the sensitivities from IceCube and other observatories (interpreted as the median upper limit we expect to observe from individual sources across the sky) along with upper limits from selected sources. The sensitivity is reaching the level of current predictions for flux from astrophysical sources ( i.e. below 10 12 × E -2 TeV cm -2 s -1 ) although the discovery potential, defined to be 5 σ for 50% of the trials, is typically a factor of three higher than the sensitivity. Therefore constraints on the parameters of hadronic acceleration models are starting to develop. Searches for neutrinos from transient [30] and periodic [31] sources have also been performed. In particular, a time window scan for transient sources (with no external triggers) shows that the discovery potential drops by a factor of 2 if searching for 1 day duration flares. A particular search for transient sources is that for neutrinos from GRB. For the first time, the IceCube Observatory has provided a definitive test of the GRB models with the most stringent constraints. Fig. 4 shows the upper limits obtained with the data collected by the 40-string configuration of IceCube and by the combined data of the 40- and 59-string configurations [32]. For each detector configuration, a list of GRBs detected during the corresponding physics runs was compiled and the predicted neutrino flux was calculated based on the γ ray spectrum shown in [34]. The corresponding stacked neutrino flux was used to search for events collected within the time window in which 5% to 95% of the fluence is recorded ( i.e. T 90 ). The upper limit is about 3 times below the predicted flux of the Waxman & Bahcall model, challenging the hypothesis that GRB are the sources of Ultra High Energy Cosmic Rays (UHECR). This result has profound consequences for the predicted flux of neutrinos produced by the interaction of UHECR with the cosmic microwave background, the so-called cosmogenic neutrinos, as well as for the GeV-TeV γ ray background flux (see for instance [35, 36]). It is important to note that it was recently shown that the fireball model with refined assumptions yields a 10 times smaller predicted flux (see [37, 38]). There is the possibility that the bulk of cosmic rays does not originate from individual sources, but from large-scale acceleration processes in superbubbles or even Galaxy clusters. In addition, unresolved sources of cosmic rays over cosmological times are expected to have produced detectable fluxes of diffuse neutrinos. Since shock acceleration is expected to provide an ∼ E -2 energy spectrum, harder than the ∼ E -3 . 7 of the atmospheric neutrinos, the diffuse flux is expected to dominate at high energy where the sensitivity is strongly dependent on the experimental quality of the selected events. Fig. 5 shows a collection of sensitivities and upper limits (90% CL) for an E -2 flux of ν µ + ¯ ν µ , from AMANDA, Antares and various IceCube configurations compared to the experimental and theoretical flux of the atmospheric neutrinos and various models of astrophysical neutrinos. The most recent results lie below the Waxmann & Bahcall neutrino bound [42], again indicating IceCube's potential for discovering the origin of cosmic rays. In the Ultra High Energy range (UHE), above ∼ 10 6 GeV, IceCube is reaching a competitive sensitivity as well. At this level one begins to reach current models of cosmogenic neutrino production (see Fig. 6) that are simultaneously constrained by the current observations of UHECRs and the GeV γ rays by Fermi-LAT [49]. Taking into account that UHECR mass composition is a key ingredient for the absolute flux and spectral shape of cosmogenic neutrinos [35], its large uncertainty still weighs pro- ndly on current models. This means that although the IceCube sensitivity to UHE neutrinos is currently the best ever achieved below 10 10 GeV it might be still far from the actual flux. From this point of view, the current developments toward a radio array in Antarctica, such as Askaryan Radio Array (ARA) [51] is a natural extension toward the highest energies. It is worth noting that the preliminary sensitivity for an arbitrary spectrum, shown in Fig. 6, has a minimum just above 1 PeV, where no significant cosmogenic neutrino flux is expected. In the experimental analysis performed on data collected during 2010-12, where events with a large number of detected photons were selected, two events were found on a background of conventional atmospheric neutrinos of 0.3. The events deposited an energy in the detector of about 1 PeV, and further study is underway to determine their nature. One possible hypothesis is that these events represent an upper fluctuation of the prompt neutrino production in the atmosphere from the decay of heavy charm mesons.", "pages": [ 3, 4, 5 ] }, { "title": "2.3 Cosmic ray anisotropy", "content": "The large number of muon bundle events collected by IceCube (about 10 10 -10 11 each year, depending on the detector configuration) makes it possible to study the arrival direction distribution of the cosmic rays at a level of about 10 -5 . The bundles of highly collimated atmospheric muons share the same direction as the parent cosmic ray particle. Since this study does not require highly well reconstructed muon directions, all collected and reconstructed events with a median angular resolution of about 3 · are used. Using full simulation of cosmic ray induced extensive air shower we find that the median particle energy of the IceCube data sample is about 20 TeV. With these data IceCube provides the first high statistics determination of the anisotropy of galactic cosmic rays in the southern hemisphere in the multi-TeV energy range. The large scale anisotropy observed by IceCube [52] appears to complement the observations in the northern hemisphere, providing for the first time an all-sky view of TeV cosmic ray arrival directions. The sky map obtained by subtracting an averaged map (over a scale of 30 · -60 · ) from the data [53], shows significant small angular scale structures in the cosmic ray anisotropy, similarly to observations in the northern hemisphere [54, 55]. Another interesting result obtained by IceCube is the persistence of the anisotropy at an energy in excess of 100 TeV. At such energies a different structure is observed that can be interpreted in terms of a different phase [56] as already reported by the EAS-TOP shower array in the northern hemisphere for the first time [58]. The observation at high energy was recently confirmed by the preliminary result from the IceTop shower array [57]. The change of the anisotropy pattern at about 100 TeV may suggest that the heliosphere could have an effect in flipping the apparent direction of the anisotropy. In fact, at about 100 TeV the cosmic rays' gyro-radius in the 3 µ G local interstellar magnetic field is of the order of magnitude of the elongated heliosphere. Below this energy scale the scattering processes on the heliospheric perturbations at the boundary with the interstellar magnetic field might be the dominant processes affecting the global cosmic ray arrival distribution and the small angular structure as well (see [59] where a review of other proposed models is also given). The Milagro observation of a likely harder than average cosmic ray spectrum from the localized excess region toward the direction of the heliotail, the so-called region B in [54] and also observed by ARGO-YBJ shower array [55], have triggered astrophysics interpretations (see [60, 61, 62]). However, this may suggest that some type of re-acceleration mechanism associated with cosmic ray propagation in the turbulent heliospheric tail might occur [63, 64]. On the other hand, the TeV cosmic ray anisotropy is a tracer of the local interstellar magnetic field, and it might indicate cosmic ray streaming along the magnetic field lines due to the Loop I shell expanding from the ScorpionCentaurus Association [65]. If the local propagation effects on the cosmic ray anisotropy below 100 TeV are dominant, at higher energy it is reasonable to believe that the persistent anisotropy might be a natural consequence of the stochastic nature of cosmic ray galactic sources, in particular nearby and recent SNRs [66, 67, 68].", "pages": [ 5, 6 ] } ]

[ { "title": "Observations of Anomalous Microwave Emission from HII regions", "content": "Clive Dickinson Jodrell Bank Centre for Astrophysics School of Physics & Astronomy University of Manchester Oxford Road, Manchester, M13 9PL (U.K.) October 10, 2018", "pages": [ 1 ] }, { "title": "Abstract", "content": "In this brief review, I give a summary of the observations of Anomalous Microwave Emission (AME) from HII regions. AME has been detected in, or in the vicinity of, HII regions. Given the difficulties in measuring accurate SEDs over a wide range of frequencies and in complex environments, many of these detections require more data to confirm them as emitting significant AME. The contribution from optically thick free-free emission from UCHII regions may be also be significant in some cases. The AMEemissivity, defined as the ratio of the AME brightness to the 100 µ mbrightness, is comparable to the value observed in high-latitude diffuse cirrus in some regions, but is significantly lower in others. However, this value is dependent on the dust temperature. More data, both at high frequencies ( > ∼ 5 GHz) and high resolution ( ∼ 1 ' or better) is required to disentangle the emission processes in such complex regions.", "pages": [ 1 ] }, { "title": "1 Introduction", "content": "HII regions refer to the environment around the most (O and B type) massive stars, which are hot enough to produce intense UV radiation that can ionize the gas around them. HII regions typically form within large molecular clouds, often in clusters (due to triggered star formation), and are therefore are also associated with significant amounts of dust grains. Anomalous Microwave Emission (AME), if due to electric dipole radiation from spinning dust [8], requires a large column of dust grains (with a population of the smallest dust grains or PAHs) and a mechanism for rotationally exciting these grains e.g. plasma drag, photons etc. For these reasons, HII regions may be a good place to look for AME. In fact, there is evidence that photodissociation regions (PDRs) typically found around the edges of HII regions/molecular clouds might be good AME emitters [2, 14]. Counter-arguments include the depletion of PAHs close in the centre of HII regions and the fact that they strongly emit in other forms of continuum emission, notably free-free (thermal bremmsstrahlung) and thermal dust radiation. In this brief review, I give an overview of the the continuum radiation and current observations of AME from HII regions. I will discuss some issues with measuring AME from HII regions, including calibration, the contribution from ultracompact (UCHII) regions and the definition of emissivity.", "pages": [ 1, 2 ] }, { "title": "2.1 The SEDs of classical HII regions", "content": "The general form of HII region SEDs (radio to the far infrared continuum) is thought to be well understood. Fig. 1 shows the SED of the well-known Orion nebula (M42) HII region, measured by a number of different experiments including Planck [13]. The spectral shape is typical of HII regions. At radio wavelengths (frequencies ∼ 1 GHz to ∼ 100 GHz) it is dominated by free-free emission from warm ( T e ∼ 10 4 K) ionized gas. This is usually expressed in terms of the free-free opacity, τ ff , which at radio wavelengths can be approximated by [4]: The intensity is proportional to the Emission Measure, defined as EM = ∫ n 2 e dl , the integral of the square of the electron density along the line of sight. Above a certain 'turnover' frequency (i.e. τ ff < 1; optically thin), free-free emission has an almost flat flux density spectrum of α ≈ -0 . 1 ( S ∝ ν α ). At lower frequencies, and particularly for more dense (young and compact HII regions) with EM >> 10 7 pccm -6 , the emission becomes optically thick ( τ ff > 1) and has a spectrum of α = +2. At frequencies ≥∼ 100 GHz, black-body emission from dust grains at T d ∼ 10-100 K dominates. The thermal dust spectrum is often parameterised by a modified black-body function, S ∝ ν β +2 B ( ν, T d ), with typical values of the emissivity index of β ∼ +1 . 8 in the Rayleigh-Jeans tail (corresponding to α = +3 . 8) and a peak at ∼ 3000 GHz (100 µ m). Also, the dust temperatures around HII regions are typically warmer ( ∼ 30-80 K) compared to the diffuse cirrus ( T d ∼ 18 K) [9].", "pages": [ 2 ] }, { "title": "2.2 Observations of AME in HII regions", "content": "To search for AME, one is essentially looking for excess emission at frequencies ∼ 30 GHz. AME is usually detected at frequencies in the range ∼ 10-60 GHz where the non-AME components are weaker. Also, theoretical models of spinning dust tend to peak at frequencies near 30 GHz [8]. Table 1 summarises the observations of AME from HII regions to-date 1 . We list the frequency range, approximate angular scales (of the experiment or of the source, whichever is largest), the AME significance level ( σ AME ) and the emissivity ( E ), defined as the AME brightness relative the 100 µ m brightness, in units of µ K/(MJy/sr). The list is ordered in terms of their approximate AME detection significance level.", "pages": [ 3 ] }, { "title": "3.1 Reliability of detections", "content": "The measurement of AME in HII regions is clearly a difficult task. It must be remembered that measuring accurate flux densities over a wide range of frequencies, particularly for extended regions in the presence of complicated backgrounds (as is often the case for Galactic HII regions), is exceptionally difficult. For low angular resolution observations, the free-free and dust emission regions will be coincident and therefore may well be a small fraction of the total flux. To subtract free-free to say 1% precision relies on having absolute flux scales that are good to this accuracy (most astronomical data are accurate to a few % and many older data are good to ∼ 10 % or worse!). Perhaps even more problematic is the comparison of data with a wide range of angular resolutions, and especially interferometric data compared to single-dish data, where the response to different angular scales can vary and is difficult to quantify (unless a detailed model of the source is available). Given the above issues, one must be cautious given that the majority of the detections listed in Table 1 are not hugely significant (i.e. they are ∼ 5 σ or below). One of the most clear detections comes from the source G159.6-18.5 within the Perseus molecular cloud. However, this is actually a very weak HII region relative to the dust-correlated emission from the larger surrounding area (the Planck AME source is actually located at G160.26-18.62). The free-free emission is therefore a very small fraction ( ≈ 10 %) of the total large-scale flux. The environment is therefore somewhat different to the other HII regions and perhaps should not be compared with the other HII regions. From the other HII regions listed in Table 1, several of them are likely to be spurious detections. LPH96 201.663+1.643 was one of the first claimed detections of AME [10] based on a rising spectrum from 5 to 10 GHz. However, it was later shown that this result is likely to be spurious when no AME was observed at 31 GHz with an upper limit of 24 % (2 σ ) [5]. Indeed, private communication with Doug Finkbeiner revealed that follow-up observations of this source with the GBT did not confirm the spectral rise seen in early observations. Similarly, an analysis of the SED of W40 using WMAP 1-year data [11] suggested that W40 may have a significant AME excess at 33 GHz. However a re-analysis using WMAP 7-year data, combined with higher resolution CBI data, could not confirm any significant deviations from an optically thin free-free spectrum [18]. The detection of RCW49 [6] could be contested based on the reliability and scarcity of low frequency (1-15 GHz) data. If one were to remove the data point at 14.7 GHz, which happens to be lower than the other data, then the significance of the detection at 31 GHz is reduced to ∼ 2 σ . Finally, analyses involving averaging the results from a sample of HII regions (e.g. [6, 20]) can be misleading since systematics errors (e.g. calibration, background subtraction etc) can become dominant. Clearly, more data at a range of frequencies and angular resolution are required to confirm and improve the accuracy for the quantification of AME.", "pages": [ 3, 4 ] }, { "title": "3.2 Compact vs extended regions and the contribution from UCHII", "content": "The SED of an evolved diffuse HII region ( EM << 10 6 pccm -6 ) will typically be optically thin above ∼ 1 GHz. However, very compact HII regions, with EM > 10 7 cmpc -6 (ultracompact (UCHII) and hypercompact (HCHII)) can have turnover frequencies of ∼ 15 GHz and higher. These would be difficult to detect at lower frequencies. A nearby ionized region at T e ∼ 10 000 K with angular size ∼ 1 '' could have a maximum flux density of up to ∼ 10 Jy at 30 GHz although most are at < 1 Jy [22]. It is therefore possible that AME (or a portion of it) from HII regions could be produced by UCHII regions that turnover at ∼ 15-40 GHz. Estimating the contribution from UCHII is somewhat difficult. Methods include using high resolution radio data to extrapolate flux densities of point sources assuming a given EM and angular size (e.g. Perrott et al., this issue) or the use of H α [4] and/or Radio Recombination Line(s) data [1]. Another way is to use an empirical relation between the ratio of 100 µ mflux density, S 100 µ m , and 2 cm (15 GHz) radio flux density, S 2 cm , from [12] who measured S 100 µ m /S 2 cm values between 1000 and 360000, with no UCHII regions below 1000; the median value was ∼ 3000-5000. We apply this method for the Perseus, S140 and S235 AME regions to estimate the maximum UCHII flux density, S max , assuming S 100 µ m /S 2 cm = 1000. To identify UCHII candidates within the vicinity of these HII regions, we use the colourcolour relation of [23] who found that UCHII regions tend to have IRAS colour ratios of log 10 ( S 60 /S 12 ) ≥ 1 . 30 and log 10 ( S 25 /S 12 ) ≥ 0 . 57. Although this method was found to be very useful for finding the majority of UCHII regions, it also selects a large fraction of nonUCHII regions, such as cloud cores with lower mass stars [16]. This therefore serves to be a very conservative upper limit to the contribution of UCHII, and is more likely to be a significant over-estimate by factors of several.", "pages": [ 4, 5 ] }, { "title": "3.3 AME emissivity of HII regions", "content": "The first detections of dust-correlated AME originate from CMB experiments measuring the sky at high Galactic latitudes, and thus authors have often calculated the dust 'emissivity' in terms of the radio brightness relative to a dust template map. This has led to the use of the IRAS 100 µ m map being used as a predictor of the AME amplitude with the emissivity defined in these terms; specifically, in units of µ K/(MJy/sr). Typical values for diffuse cirrus at high Galactic latitudes are ∼ 10 µ K/(MJy/sr) with variations of a factor of ∼ 2 [3]; this corresponds to approximately 1 Jy at 33 GHz for every 3000 Jy at 100 µ m. Table 1 lists the AME dust emissivities, E , in terms of the AME brightness temperature relative to the 100 µ m brightness, converted to units µ K/(MJy/sr). Although the uncertainties are large, it appears that the AME emissivity are comparable to the high latitude value, but on average are lower compared to the high latitude value (and lower still than the Perseus AME region). More strikingly, upper limits from the compact sample observed by AMI [17] and also upper limits from W40 suggest the AME emissivities are at least an order of magnitude lower still. This surprising result can be understood in terms of the different environments in the vicinity of HII regions. The most important is the lack of smallest dust grains (PAHs) which are known to be depleted inside HII regions (e.g. [15]). If AME is due to spinning dust grains this would severely reduce the AME brightness since the very smallest grains produce most of the spinning dust flux. Other factors may also be contributing such as the interstellar radiation field and distribution of ions. Finally, we point out that although the 100 µ m emissivity is a convenient quantity, it can be significantly biased in regions with a higher than average dust temperature. The 100 µ m intensity is very sensitive to the dust temperature. For example, compared to an average dust temperature of 18.1 K, for a value of 22 K the 100 µ m intensity is a factor of 4 times higher, while for 30 K it is a factor of 23 times higher (see Tibbs et al., this issue). HII regions are known to have warmer dust, typically 30-80 K, and thus the AME emissivity will naturally be lower. This may explain the apparently lower values observed in more compact regions where this will be pronounced. A better definition of emissivity would be to use the column density or, equivalently, the thermal dust optical depth [11].", "pages": [ 5, 6 ] }, { "title": "4 Conclusions", "content": "HII regions are an interesting place to look for AME. So far, there have been a number of detections from HII regions (or in the vicinity of HII regions). However, measuring accurate SEDs over a wide frequency range, in addition to the complex environment and the presence of bright continuum (e.g. free-free, thermal dust) emission makes this is a very difficult task. Furthermore, the presence of optically thick free-free emission from UCHII regions may be contributing to a portion of the AME for some regions. Nevertheless, the AME emissivity is comparable to the more robust detections on molecular clouds and diffuse cirrus, although on average it is lower; in some HII regions there are only upper limits on AME. More data, particularly at higher resolutions and high frequencies ( > 5 GHz) are needed. Acknowledgements. CDacknowledges an STFC Advanced Fellowship, an EC Marie-Curie IRG grant under the FP7, and an ERC Starting Grant (No. 307209).", "pages": [ 6 ] } ]

[ { "title": "ABSTRACT", "content": "Hindawi Publishing Corporation Advances in Astronomy Volume /two.fitted/zero.fitted/one.fitted/three.fitted, Article ID /six.fitted/two.fitted/seven.fitted/eight.fitted/six.fitted/seven.fitted, /one.fitted/four.fitted pages http://dx.doi.org//one.fitted/zero.fitted./one.fitted/one.fitted/five.fitted/five.fitted//two.fitted/zero.fitted/one.fitted/three.fitted//six.fitted/two.fitted/seven.fitted/eight.fitted/six.fitted/seven.fitted", "pages": [ 1 ] }, { "title": "Fernando Fabián Rosales-Ortega", "content": "Instituto Nacional de Astrof'ısica, ' Optica y Electr'onica, Luis E. Erro /one.fitted, /seven.fitted/two.fitted/eight.fitted/four.fitted/zero.fitted Tonantzintla, PUE, Mexico Correspondence should be addressed to Fernando Fabi'an Rosales-Ortega; [email protected] Received /seven.fitted August /two.fitted/zero.fitted/one.fitted/three.fitted; Accepted /two.fitted/seven.fitted September /two.fitted/zero.fitted/one.fitted/three.fitted Academic Editor: Jos'eManuelV'ılchez Medina Copyright © /two.fitted/zero.fitted/one.fitted/three.fitted Fernando Fabi'an Rosales-Ortega. /T_his is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. A new generation of wide-/field emission-line surveys based on integral /field units (IFU) is allowing us to obtain spatially resolved information of the gas-phase emission in nearby late-type galaxies, based on large samples of HII regions and full two-dimensional coverage. /T_hese observations are allowing us to discover and characterise abundance di/fferentials between galactic substructures and new scaling relations with global physical properties. Here I review some highlights of our current studies employing this technique: (/one.fitted) the case study of NGC /six.fitted/two.fitted/eight.fitted, the largest galaxy ever sampled with an IFU; (/two.fitted) a statistical approach to the abundance gradients of spiral galaxies, which indicates a universal radial gradient for oxygen abundance; and (/three.fitted) the discovery of a new scaling relation of HII regions in spiral galaxies, the local mass-metallicity relation of star-forming galaxies. /T_he observational properties and constrains found in local galaxies using this new technique will allow us to interpret the gas-phase abundance of analogue high-z systems.", "pages": [ 1 ] }, { "title": "1. Introduction", "content": "/T_he study of the interstellar medium (ISM), like many other areas of astrophysics, has undergone a remarkable acceleration in the /flow of data over the last few years. Large surveys such as the /two.fitteddFGRS [/one.fitted], SDSS [/two.fitted], GEMS [/three.fitted], or COSMOS [/four.fitted], to name a few, have revolutionised our understanding of the Universe and its constituents as they have enabled us to study the global properties of a large number of objects, allowing for meaningful statistical analysis to be performed, together with a broad coverage of galaxy subtypes and environmental conditions. /T_he nebular emission arising from extragalactic objects has played an important role in this new understanding. Nebular emission lines have been, historically, the main tool at our disposal for the direct measurement of the gas-phase abundance at discrete spatial positions in low redshi/f_t galaxies. /T_hey trace the young, massive star component in galaxies, illuminating and ionizing cubic kiloparsec-sized volumes of ISM. Metals are a fundamental parameter for cooling mechanisms in the intergalactic and interstellar medium, starformation, stellar physics, and planet formation. Measuring the chemical abundance in individual galaxies and galactic substructures, over a wide range of redshi/f_ts, is a crucial step to understanding the chemical evolution and nucleosynthesis at di/fferent epochs, since the heavy atomic nuclei trace the evolution of past and current stellar generations. /T_his evolution is dictated by a complex array of parameters, including the local initial gas composition, star-formation history (SFH), gas infall and out/flows, radial transport and mixing of gas within discs, stellar yields, and the initial mass function. Although it is di/fficult to disentangle the e/ffects of the various contributors, determinations of current elemental abundance constrain the possible evolutionary histories of the existing stars and galaxies, and the interaction of galaxies with the intergalactic medium. /T_he details of such a complex mechanism are still observationally not well established and theoretically not well developed and threaten our understanding of galaxy evolution from the early Universe to the present day. /T_he relevance of the study of the ISM in the local Universe cannot be underestimated, since it actually constitutes the bases of the methods and calibrations employed to derive abundance and their relations with global galaxy parameters in high redshi/f_t galaxies (e.g., [/five.fitted, /six.fitted]), objects that are typically solely identi/fiable by their emission line spectra. Nearby galaxies o/ffer a unique opportunity to study the SFH-ISM coupling on a spatially resolved basis, over large dynamic ranges in gas density and pressure, metallicity, dust content, and other physically relevant parameters of gas and dust.However,mostoftheobservationstargetingnebular emission in nearby galaxies have been made with multibroadband and narrow-band imaging in the optical and nearinfrared, or single-aperture or long-slit spectrographs, resulting in samples of typically a dozen or fewer HII regions per galaxy. /T_hese observations have been used to derive the properties of their dominant stellar populations, gas content, and kinematics (e.g., [/seven.fitted-/nine.fitted]). Nevertheless, despite many e/fforts, it has been di/fficult to obtain a complete picture of the main properties of these galaxies, especially those ones thatcanonlyberevealedbyspectroscopicstudies(likethe nature of the ionization and/or the metal content of the gas). /T_his is because previous spectroscopic studies only sampled a very few discrete regions in these complex targets (e.g., [/one.fitted/zero.fitted, /one.fitted/one.fitted]), or used narrow-band imaging of speci/fic /fields to obtain information of star-forming regions and the ionized gas (e.g., [/nine.fitted]), and in many cases they were sampling very particular types of regions [/one.fitted/two.fitted-/one.fitted/five.fitted]. Integrated spectra over large apertures were required to derive these properties in a more complete way (e.g., dri/f_t-scanning ,[/one.fitted/six.fitted]), but even in these cases, only a single integrated spectrum is derived, and the spatial information is lost. On the other hand, although large spectroscopic surveys like the /two.fitteddFGRS or the SDSS do provide a large number of objects sampled and vast statistical information, they are generally limited to one spectrum per galaxy, thus missing all the radial information and spatially resolved properties of the galaxy. /T_hese surveys have been successful to describe the integrated properties and relations of a large number of galaxies along a wide redshi/f_t range. But galaxies are complex systems not fully represented by a single spectrum or just broad band colours. Disc and spheroidal components are structurally and dynamically di/fferent entities with di/fferent SFH and chemical evolution. A main drawback of this techniqueisthatitleadstoaperturebiasthatisdi/fficulttocontrol, as the area covered to integrate the spectra corresponds to di/fferent physical scales at di/fferent redshi/f_ts (e.g., SDSS), and also the physical mechanisms involved in ionizing the gas may be very di/fferent within the sampled area, as this wouldincluderegionswithemissionduetodi/ffuseionized gas (DIG), shocks, or AGN/LINER activity. /T_he advent of Multi-Object Spectrometers (MOS) and Integral Field Spectroscopy (IFS) instruments with large /fields of view (FoV) now o/ffers us the opportunity to undertake a new generation of surveys, based on a full twodimensional (/two.fittedD) coverage of the optical extent of nearby galaxies. /T_he /first application of IFS to obtain spatially resolved, continuously sampled spectroscopy of certain portions of nearby galaxies was due to the SAURON project [/one.fitted/seven.fitted, /one.fitted/eight.fitted]. SAURON was speci/fically designed to study the kinematics and stellar populations of a sample of nearby elliptical and lenticular galaxies. /T_he application of SAURON to spiral galaxies was restricted to the study of spiral bulges [/one.fitted/nine.fitted]. However, IFS was rarely used in a 'survey mode' to investigate sizeable samples. /T_here were several reasons for the lack of a systematic study targeting galaxies in the local Universe using IFS that could cover a substantial fraction of their optical sizes. /T_he reasons included small wavelength coverage, /fibre-optic calibration problems, but mainly the limited FoV of the instruments available worldwide. Most IFUs have a FoV of the order of arcsec, preventing a good coverage of the target galaxies on the sky in a reasonable time, even with a mosaicking technique. Furthermore, in some cases the emission lines used in chemical abundance studies were not covered by the restricted wavelenght range of the instruments. Moreover, the complex data reduction and visualisation imposed a further obstacle. In order to /fill this gap, in the last few years we started a major observational programme aimed at studying the /two.fittedD properties of the ionized gas and HII regions in a representative sample of nearby face-on spiral galaxies using IFS. /T_he spatially resolved information provided by these observations is allowing us to test and extend the previous body of results from small-sample studies, while at the same time it opens up a new frontier of studying the /two.fittedD gas abundance on discs and the intrinsic dispersion in metallicity, progressing from a one-dimensional study (radial abundance gradients) to a /two.fittedD understanding (distributions), allowing us atthesametimetostrengthenthediagnosticmethodsthat are used to measure HII region abundance in galaxies. Here we present the highlights of our current studies employing this large spectroscopic database: (/one.fitted) the case of NGC/six.fitted/two.fitted/eight.fitted, the largest galaxy ever sampled with IFS; (/two.fitted) an IFSbased statistical approach to the abundance gradients of spiral galaxies; and (/three.fitted) the discovery of a new scaling relation of HII regions in spiral galaxies and how we use it to to reproducewith remarkable agreement-the mass-metallicity relation of star-forming galaxies.", "pages": [ 1, 2 ] }, { "title": "2. A IFS Sample of Nearby Disc Galaxies", "content": "/T_he studies here described were performed using IFS data of a sample of nearby disc galaxies. /T_he observations were designed to obtain continuous coverage spectra of the whole surface of the galaxies. /T_hey include observations from the PPAKIFSNearby Galaxies Survey: PINGS [/two.fitted/zero.fitted], and a sample of face-on spiral galaxies from M'armol-Queralt'oetal.[/two.fitted/one.fitted], as part of the feasibility studies for the CALIFA survey [/two.fitted/two.fitted, /two.fitted/three.fitted], a legacy project which aims to observe a statistically complete sample of ∼ /six.fitted/zero.fitted/zero.fitted galaxies in the local Universe; all projects are carried out at the Centro Astron'omico Hispano-Alem'an of Calar Alto, Spain. PINGS represented the /first attempt to obtain continuous coverage spectra of the whole surface of a representative sample of late-type galaxies in the nearby Universe. /T_his /first sample includes normal, lopsided, interacting and barred spirals with a good range of galactic properties and starforming environments with available multiwavelength public data (e.g., see Figure /one.fitted). /T_he second sample consists of visually classi/fied face-on spirals from M'armol-Queralt'oetal.[/two.fitted/one.fitted] extracted from the SDSS DR/four.fitted imaging sample selecting galaxies brighter than r < /one.fitted/five.fitted./seven.fitted/five.fitted mag with redshi/f_ts in the range /zero.fitted./zero.fitted/zero.fitted/five.fitted Both samples were observed with the PMAS spectrograph [/two.fitted/five.fitted]inthePPAKmode[/two.fitted/six.fitted, /two.fitted/seven.fitted] on the /three.fitted./five.fitted m telescope in Calar Alto with similar setup, resolutions, and integration times, covering their optical extension up to ∼ /two.fitted./four.fitted e/ffective radii within a wavelength range ∼ /three.fitted/seven.fitted/zero.fitted/zero.fitted-/seven.fitted/zero.fitted/zero.fitted/zero.fitted ˚ A. /T_he PPAK /fiber bundle consists of /three.fitted/eight.fitted/two.fitted /fibers of /two.fitted./seven.fitted arcsec diameter each. Of these /three.fitted/eight.fitted/two.fitted /fibers, /three.fitted/three.fitted/one.fitted (the science /fibers) are concentrated in a single hexagonal bundle covering a /field-of-view of 74 × 64 arcsec 2 , with a /filling factor of ∼ /six.fitted/zero.fitted%. /T_he sky background is sampled by /three.fitted/six.fitted additional /fibers, distributed in /six.fitted bundles of /six.fitted /fibers each, along a circle ∼ /seven.fitted/two.fitted arcsec from the center of the instrument FoV. In the case of PINGS, the observations consisted of IFU spectroscopic mosaics for /one.fitted/seven.fitted spiral galaxies within a maximumdistance of /one.fitted/zero.fitted/zero.fitted Mpc; the average distance of the sample is /two.fitted/eight.fitted Mpc (for /u1D43B 0 =73 kms -1 Mpc -1 ). Most of the objects in PINGS could not be covered in a single pointing with IFS instruments, so a new observing-reduction technique hadtobedevelopedtoperformaccuratemosaickingofthe targets. /T_he spectroscopic mosaicking was acquired during aperiodofthreeyearsandthe/finaldatasetcomprises more than /five.fitted/zero.fitted /zero.fitted /zero.fitted /zero.fitted individual spectra, covering in total an observed area of nearly /eight.fitted/zero.fitted arcmin 2 , and an observed surface withoutprecedentsbyaIFSstudyuptothatpoint(thecase study of NGC /six.fitted/two.fitted/eight.fitted presented in Section /three.fitted is based in the data of this survey). For the second sample, the galaxies were observed over /fi/f_teen nights in several observing runs. /T_he main di/fference is that, for the latter sample, a single pointing strategy using a dithering scheme was applied, while,forthelargestgalaxiesofthePINGSsurvey,amosaic comprising di/fferent pointings was required. /T_his is due to the di/fferences in projected size, considering the di/fferent redshi/f_t range of both samples: the PINGS galaxies correspond to /u1D467 ∼ 0.001 -0.003 , while, for the face-on spirals, it is /u1D467∼ 0.01 -0.025 . /T_herefore, in both survey samples, the data extent corresponds to about ∼ /two.fitted e/ffective radii for all galaxies (/T_he e/ffective radius is classically de/fined as the radius at which one half of the total light of the system is emitted). So the /final sample comprises /three.fitted/eight.fitted objects, with a redshi/f_t range between ∼ /zero.fitted./zero.fitted/zero.fitted/one.fitted and /zero.fitted./zero.fitted/two.fitted/five.fitted. Although this sample is by no means a statistical subset of the galaxies in the local Universe, it is a representative sample of face-on, mostly quiescent, and spiral galaxies at the considered redshi/f_t range (see Figure /one.fitted). Data reduction was performed using R/three.fittedD [/three.fitted/one.fitted], obtaining asanoutputadatacubeforeachgalaxy,witha/finalspatial sampling between /one.fitted-/two.fitted arcsec/pixel, which translates to a linear physical size between a few hundreds of parsecs to ∼ /one.fittedkpc. Usingthisdatabasewecataloguedmorethan ≈ /two.fitted/five.fitted/zero.fitted/zero.fitted HII regions with good spectroscopic quality in all /three.fitted/eight.fitted galaxies, representing one of the largest and more homogeneous /two.fittedD spectroscopic HII region surveys ever accomplished. NGC 628 N 0 /uni0394 RA (arcsec) (a) F/i.sc/g.sc/u.sc/r.sc/e.sc /two.fitted: (a) Spatial map of the /fibres within the IFS mosaic of NGC /six.fitted/two.fitted/eight.fitted where nebular emission was detected. Blue /fibres indicate regions above a S/N threshold for a proper abundance analysis, and grey /fibres correspond to a di/ffuse emission. /T_he size and position of the /fibres (at real scale) are displayed in the standard NE-positive orientation. /T_he crosshairs mark the central reference point of the IFS mosaic. /T_he colour intensity of each /fibre in the blue sample has been scaled to the /flux intensity of H /u1D6FC for that particular spectrum. (b) Oxygen abundance map of NGC /six.fitted/two.fitted/eight.fitted derived by applying the O/three.fittedN/two.fitted calibrator [/two.fitted/eight.fitted] to the emission line maps of the galaxy. /T_he /figure shows a clear gradient in metallicity, with more abundant regions in the inner part or the galaxy. Figure adapted from S'anchez et al. [/two.fitted/nine.fitted]andRosales-Ortegaetal.[/three.fitted/zero.fitted]. /T_he discussion presented in Sections /four.fitted and /five.fitted isbasedonthese databases. /T_he primary scienti/fic objectives of these surveys were to use the /two.fittedD IFS observations to study the small and intermediate scale variation in the line emission and stellar continuum by means of pixel-resolved maps across the discs of nearby galaxies, as described in the following sections. 3. NGC 628: A Case Study of IFS-Based Nebular Emission Studies NGC/six.fitted/two.fitted/eight.fitted (or M /seven.fitted/four.fitted) is the largest galaxy in projected angular size ( ∼ /one.fitted/zero.fitted./five.fitted × /nine.fitted. /five.fitted arcmin 2 , /u1D467∼ /zero.fitted./zero.fitted/zero.fitted/two.fitted/one.fitted/nine.fitted ∼ /nine.fittedMpc) of the PINGSsample.DuetothelargesizeofNGC/six.fitted/two.fitted/eight.fittedcompared to the FoV of the PPAK instrument (/seven.fitted/two.fitted × /six.fitted/four.fitted arcsec 2 )a mosaicking scheme was adopted, employing /three.fitted/four.fitted di/fferent pointings. /T_he initial pointing was centered on the bulge of the galaxy. Consecutive pointings followed a concentric ringshaped pattern, adjusted to the shape of the PPAK bundle (see Figure /one.fitted). /T_he observations of this galaxy spanned a period of threeyears./T_heareacoveredbyalltheobservedpositions accounts approximately for /three.fitted/four.fitted arcmin 2 , making this galaxy the widest spectroscopic survey ever made on a single nearby galaxy. /T_he spectroscopic mosaic contains /one.fitted/one.fitted/zero.fitted/nine.fitted/four.fitted individual spectra. With such dimensions, this galaxy allows us to study the /two.fittedD metallicity structure of the disc, the second order properties of its abundance distribution, and-as a very important byproduct-a complete /two.fittedD picture of the underlying stellar populations of the galaxy. Note that the linear physical scale thatasinglePPAK/fibresamplesattheassumeddistance of the galaxy is ∼ /one.fitted/two.fitted/zero.fitted pc. /T_his scale can be compared to the physical diameter of a well-known HII region in our Galaxy, that is, the Orion nebula ( /u1D437∼ /eight.fitted pc), or to the extent of what is considered prototypes of extragalactic giant HII regions, such as /three.fitted/zero.fitted Doradus ( /u1D437∼ /two.fitted/zero.fitted/zero.fitted pc) or NGC /six.fitted/zero.fitted/four.fitted ( /u1D437∼ /four.fitted/six.fitted/zero.fitted pc). /T_he area sampled by an individual /fibre in the mosaic would subtend a fraction of a typical giant HII region in NGC /six.fitted/two.fitted/eight.fitted, but the same area would fully encompass a number of small andmediumsizeHIIregionsofthegalaxy(seeFigure /two.fitted). /T_he IFS analysis of NGC /six.fitted/two.fitted/eight.fitted was taken as a case study in order to explore di/fferent spectra extraction and analysis methodologies, taking into account the signal-to-noise of the data, the /two.fittedD spatial coverage, the physical meaning of the derived results, and the /final number of analysed spectra. /T_he analysis performed on this object represents an example of the potential and extent of studies based on IFS on nearby galaxies. In the /first paper of the series ([/two.fitted/nine.fitted], herea/f_ter Paper I), we present a study of the line emission and stellar continuum of NGC /six.fitted/two.fitted/eight.fitted by means of pixel-resolved maps across the disc of the galaxy. /T_his study includes a qualitative description of the /two.fittedD distribution of the physical properties inferred from the line intensity maps and a comparison of these properties with both the integrated spectrum of the galaxyandthespatiallyresolvedspectra.Inthesecondarticle ([/three.fitted/zero.fitted], herea/f_ter Paper II), we present a detailed, spatially c) e s c r c (a e D /uni0394 200 100 0 -100 -200 E 100 -100 -200 c) e s c r (a /uni0394/u1D6FF resolved spectroscopic abundance analysis, based on di/fferent spectral samples extracted from the area covered by the IFS observations of NGC /six.fitted/two.fitted/eight.fitted, and we de/fine a spectra selection methodology specially conceived for the study of the nebular emission in IFU-based spectroscopic observations. /T_his allows us to derive the gas chemistry distribution across the surface of the galaxy with unprecedented detail. In the third paper of the series (S'anchez-Bl'azquez et al., submitted; herea/f_ter Paper III), we present a stellar population analysis of the galaxy, a/f_ter applying spectral inversion methods to derive /two.fitted -dimensional maps of star-formation histories and chemical enrichment. In Paper I, spatially resolved maps of the emission line intensities and physical properties were derived for NGC /six.fitted/two.fitted/eight.fitted. Contrary to previous attempts to perform a /two.fittedD wide-/field analysis based on narrow-band (or Fabry-Perot) imaging, which only allowed a basic analysis of the physical parameters and/or required assumptions on the line ratios included within individual /filters (e.g., H /u1D6FC ), the emission line maps presented in this paper were constructed from individual (deblended) emission lines at any discrete spatial location of the galaxy, where enough signal-to-noise was found. /T_his fact allowed investigating the point-to-point variation of the physical properties over a considerable area on the galaxy. Extinction, ionization, and metallicity-sensitive indicator maps were derived from reddening corrected emission line maps. In general, they show that the ionized gas in these spiral galaxies exhibits a complex structure, morphologically associated with the star-forming regions located along the spiral arms. /T_he (thermal) ionization is stronger along the spiral arms, associated with the HII regions, and more intense in the outer than in the inner ones. Indeed, the surface SFR is an order of magnitude stronger in the outer HII regions, at distance larger than ∼ /one.fitted/zero.fitted/zero.fitted arcsec (/four.fitted./five.fitted kpc), than in the inner ones. Considering that in these outer regions there is a lower mass density, the growing rate of stellar mass is considerably larger there than in the inner ones. /T_herefore, the growth of the galaxy is dominated by the inside-out process. /T_he spatially resolved distribution of the abundance shows a clear gradient of higher oxygen metallicity values from the inner part to the outer part of the galaxy, and along the spiral arms (see right-panel of Figure /two.fitted). However, in some instances, the value of the oxygen abundance (and other physical properties like extinction and the ionization parameter) varies within what would be considered a classical well-de/fined HII region (or HII complex), showing some level of structure. Indeed, the /two.fittedD character of the data allows us to study the small-scale variation of the spectra within a given emitting area. /T_he values of the emission line ratios measured using di/fferent extraction apertures vary considerably as a functionoftheaperturesize,andthescatterofthecentral valueislargerthanthestatisticalerrorinthemeasurements, re/flecting that this might in fact be a physical e/ffect. By constructing /two.fittedD maps of the oxygen abundance distributions, wefound that the /two.fittedD metallicity structure of the galaxy varies depending on the metallicity calibrator employed in order to derive the oxygen abundance. Di/fferent calibrators /find regions of enhanced log(O/H) at spatial positions which are not coincident among them. /T_his implies that the use of di/fferent empirical calibrations does not only re/flect in a linear scale o/ffset but may introduce spurious inhomogeneities. /T_his information is usually lost in a simple radial abundance gradient,andthatmightberelevantwhenconstructinga chemical evolution model based on a particular abundance determination (see Figure /three.fitted). /T_he emission line maps presented in Paper I proved to be useful in describing the general /two.fittedD properties of the galaxy. More robust conclusions were presented in Paper II, where we analysed speci/fic individual regions across the disc of the galaxy, either by taking individual spectra above as a certain S/N threshold, or by coadding spectra with the same physical properties and comparing the results in the /two.fittedD context. With the/firstmethodwewereabletoidentifyregionsofinterstellar di/ffuse emission (see le/f_t panel of Figure /three.fitted), while with the second we created a classic catalogue of HII regions from a purely geometrical principle, that is, by coadding /fibres considered to belong to the same morphological region. Some highlights of this study (which also apply to the rest of the PINGS galaxies analysed so far) are the following. (/one.fitted) Despite the large number of spectra contained in the original observed mosaic, the /final number of /fibres containing analysable spectra of enough signal-tonoise for a spectroscopic study of the ionized gas represents only a reduced percentage of the total number of /fibres contained in the full IFS mosaic. For the particular case of NGC /six.fitted/two.fitted/eight.fitted, less than /one.fitted/zero.fitted% of the total area sampled by the IFU observations is considered of su/fficient quality. (/two.fitted) Independently of the abundance calibrator used, the metallicity distribution of NGC /six.fitted/two.fitted/eight.fitted is consistent with a nearly /flat distribution in the innermost regions of the galaxy ( /u1D70C//u1D70C 25 < 0.2 ), a steep negative gradient for 0.2 /uni2272 /u1D70C//u1D70C 25 <1 , and a shallow or nearly constant distribution beyond the optical edge of the galaxy, that is, implying a multimodality of the abundance gradientofNGC/six.fitted/two.fitted/eight.fitted./T_hesamefeatureisobserved for the N/O versus /u1D70C distribution. /T_he existence of this feature may be related to the di/fferences in the /two.fittedD gas surface density and star-formation rate between the inner and outer disc which inhibits the formation of massive stars in the outer regions, causing a lack of chemical evolution in the outer disc compared with the inner regions. (/three.fitted)/T_heobserveddispersioninthemetallicityatagiven radius is neither a function of spatial position, nor due tolowS/Nofthespectra,andshowsnosystematic dependence on the ionization conditions of the gas, implying that the dispersion is real and is re/flecting a true spatial physical variation of the oxygen content (see Figure /three.fitted). (/four.fitted) /T_he values of the oxygen abundance derived from the integrated spectrum for each calibrator equal the abundance derived from the radial gradient at a radius /u1D70C ∼ 0.4/u1D70C 25 , con/firming for this galaxy the previous results obtained for other objects, that is, that the integrated abundance of a normal disc F/i.sc/g.sc/u.sc/r.sc/e.sc /three.fitted: (a) Radial abundance gradient derived for NGC /six.fitted/two.fitted/eight.fitted based on the PINGS HII region catalogue (green symbols), and HII regions from the literature (black symbols) using the O/three.fittedN/two.fitted calibrator. /T_he horizontal grey lines correspond to the abundance derived using the integrated spectrum as reported in Paper I. /T_he top /u1D44B -axis values correspond to the projected radii in arcsec for the radial average data. Note the /flattening of the gradient for innermost regions of the galaxy and for radii >/u1D70C 25 ,thatis,amultimodalityoftheabundancegradient.(b)/two.fittedD distribution of the oxygen abundance derived from the IFS H II regions catalogue of NGC /six.fitted/two.fitted/eight.fitted (plus selected HII regions from the literature), for the KK/zero.fitted/four.fitted (top-le/f_t) metallicity calibrators. /T_he shape and colours of the symbols correspond to the di/fference /uni0394 [/one.fitted/two.fitted + log(O/H)] ≡/uni0394 log (O/H) between the abundance obtained on each HII region with respect to the characteristic abundance /one.fitted/two.fitted + log(O/H) /u1D70C=0.4/u1D70C 25 of the same calibrator, grouped into bins of /zero.fitted./zero.fitted, ± /zero.fitted./one.fitted, /zero.fitted./two.fitted dex (e.g., +/zero.fitted./one.fitted dex = /zero.fitted./zero.fitted/five.fitted ≤/uni0394 log (O/H) < /zero.fitted./one.fitted/five.fitted). /T_he large symbol in red colour stands for the location of the HII region with the maximum amount of /one.fitted/two.fitted + log(O/H) measured for that calibrator. /T_he grey thick lines de/fine the operational spiral arms of the galaxy. /T_he dotted circle corresponds to the size of the optical radius /u1D70C 25 . Figure adapted from Rosales-Ortega et al. [/three.fitted/zero.fitted]. F/i.sc/g.sc/u.sc/r.sc/e.sc /three.fitted: (a) Radial abundance gradient derived for NGC /six.fitted/two.fitted/eight.fitted based on the PINGS HII region catalogue (green symbols), and HII regions from the literature (black symbols) using the O/three.fittedN/two.fitted calibrator. /T_he horizontal grey lines correspond to the abundance derived using the integrated spectrum as reported in Paper I. /T_he top /u1D44B -axis values correspond to the projected radii in arcsec for the radial average data. Note the /flattening of the gradient for innermost regions of the galaxy and for radii >/u1D70C 25 ,thatis,amultimodalityoftheabundancegradient.(b)/two.fittedD distribution of the oxygen abundance derived from the IFS H II regions catalogue of NGC /six.fitted/two.fitted/eight.fitted (plus selected HII regions from the literature), for the KK/zero.fitted/four.fitted (top-le/f_t) metallicity calibrators. /T_he shape and colours of the symbols correspond to the di/fference /uni0394 [/one.fitted/two.fitted + log(O/H)] ≡/uni0394 log (O/H) between the abundance obtained on each HII region with respect to the characteristic abundance /one.fitted/two.fitted + log(O/H) /u1D70C=0.4/u1D70C 25 of the same calibrator, grouped into bins of /zero.fitted./zero.fitted, ± /zero.fitted./one.fitted, /zero.fitted./two.fitted dex (e.g., +/zero.fitted./one.fitted dex = /zero.fitted./zero.fitted/five.fitted ≤/uni0394 log (O/H) < /zero.fitted./one.fitted/five.fitted). /T_he large symbol in red colour stands for the location of the HII region with the maximum amount of /one.fitted/two.fitted + log(O/H) measured for that calibrator. /T_he grey thick lines de/fine the operational spiral arms of the galaxy. /T_he dotted circle corresponds to the size of the optical radius /u1D70C 25 . Figure adapted from Rosales-Ortega et al. [/three.fitted/zero.fitted]. galaxy correlates with the characteristic gas-phase abundance measured at /u1D70C ∼ 0.4/u1D70C 25 . (/five.fitted) While trying to /find axisymmetric variations of the metallicity content in the galaxy, we found slight variations between the central oxygen abundance and slopes for both the geometrical (quadrants) and morphological (arms) regions of the galaxy. However, these small variations fall within the expected errors involved in strong-line empirical calibrations (see Figure /four.fitted). If the radial trends in the ionization parameter and metallicity abundance were somewhat distinct, this would indicate that, to a certain extent, the physical conditions and the star-formation history of di/fferent-symmetric regions of the galaxy would have evolved in a di/fferent manner. Likewise, [/three.fitted/two.fitted ] found no evidence for signi/ficant large-scale azimuthal variations of the oxygen abundance across the whole disk of M /one.fitted/zero.fitted/one.fitted and marginal evidence for the existence of moderate deviations from chemical abundance homogeneity in the interstellar medium of this galaxy. In the case of the stellar populations, in Paper III we derive maps of the mean (luminosity and mass weighted) age and metallicity that reveal a negative age gradient and thepresenceofstructuressuchasanuclearring,previously seen in molecular gas (see Figure /five.fitted). /T_he disc is dominated in mass by an old stellar component at all radii sampled by the IFS data, while the percentage of young stars increases with radius, as predicted in an inside-out formation scenario, where outer parts of the disc formed later due to the increasing timescales for gas infall with radius. We also detect an inversion of the metallicity gradient at the very centre of the galaxy ( ∼ /one.fitted kpc), where apparently there exists a ring of old stars at this distance, with a trend to younger ones at the very center. Similar results are found in the Milky Way (MW) using Open Clusters and Cepheids, that is, a clear bimodal gradient for the older population, with a /flat outer plateau, and a more continuous gradient for the younger population (e.g., [/three.fitted/three.fitted-/three.fitted/six.fitted]). /T_his behaviour has also been reported in other galaxies, mostly Sa/S/zero.fitted, where the inner regions of their bulges present bluer colors, consistent with younger stellar populations (e.g., [/three.fitted/seven.fitted]). /T_he relevance of this study regarding the nebular emission is that the young component shows a metallicity gradient thatisverysimilartothatofthegas,andthatis/flatter than that of the old stars. Although the metallicity gradients for the young stars and the gas also show a break, this is much less prominent than for the old stars. /T_he position of the break is more coincident with the corotation radius of the oval distortion than that of the spiral pattern, which is F/i.sc/g.sc/u.sc/r.sc/e.sc /four.fitted: (a) BPT diagnostic diagram for HII regions coded according to the geometric position (quadrants) with respect to the an arbitrary axis drawn across the galaxy surface. /T_he locus of di/fferent sectors does not show a clear trend or do not populate a clearly visible region on any diagram, compared to the rest of the quadrants. Points from all the regions are equally distributed within the cloud of points on each diagnostic diagram, indicating that the emission line ratios of the HII regions are not a function of azimuthal angle across the disc. (b) Radial gradients of the ionization parameter log /u1D462 for morphologically selected HII regions of NGC /six.fitted/two.fitted/eight.fitted. /T_he panel shows the log /u1D462 versus /u1D70C relation for the regions belonging to the north and south spiral arms of the galaxy. /T_he di/fference between the two spiral arms resides in the slope of the gradient of log /u1D462 , for the north arm, and the values of log /u1D462 increase moderately with galactocentric distance, while, for the south arm, the ionization parameter increases with a steeper slope, although within the errors of the linear /fittings. Figure adapted from Rosales-Ortega et al. [/three.fitted/zero.fitted]. F/i.sc/g.sc/u.sc/r.sc/e.sc /four.fitted: (a) BPT diagnostic diagram for HII regions coded according to the geometric position (quadrants) with respect to the an arbitrary axis drawn across the galaxy surface. /T_he locus of di/fferent sectors does not show a clear trend or do not populate a clearly visible region on any diagram, compared to the rest of the quadrants. Points from all the regions are equally distributed within the cloud of points on each diagnostic diagram, indicating that the emission line ratios of the HII regions are not a function of azimuthal angle across the disc. (b) Radial gradients of the ionization parameter log /u1D462 for morphologically selected HII regions of NGC /six.fitted/two.fitted/eight.fitted. /T_he panel shows the log /u1D462 versus /u1D70C relation for the regions belonging to the north and south spiral arms of the galaxy. /T_he di/fference between the two spiral arms resides in the slope of the gradient of log /u1D462 , for the north arm, and the values of log /u1D462 increase moderately with galactocentric distance, while, for the south arm, the ionization parameter increases with a steeper slope, although within the errors of the linear /fittings. Figure adapted from Rosales-Ortega et al. [/three.fitted/zero.fitted]. beyond the radius sampled by our data. We speculate about thepossibleorigenofthisbreak,thepossibilitiesbeingdue to star-formation variation with the spiral pattern speed or that is due to radial mixing produced by either the spiral arms, the oval distortion, or a coupling of both. We argue that NGC /six.fitted/two.fitted/eight.fitted could represent a good example of secular evolutionduetothepresenceofadissolvingbar.Inthis scenario, the strong bar has funneled large amounts of gas into the central regions while radial /flows induced in the disc have /flattened the O/H gradient. Nuclear starbursts resulting from the gas sinking into the center contributed to the bulge's growth until enough mass was accreted to dissolve the bar by dynamical instabilities. /T_he oval distortion observed in the central region could be the remains of the bar. Forthcoming studies analysing a sample of galaxies with di/fferent masses and showing di/fferent morphological features (e.g., bars of di/fferent strength, spiral arms with di/fferent morphologies, etc.) using, for example, the CALIFA survey that will help to elucidate the importance of the di/fferent mechanisms producing radial mixing in the galaxy discs. 4. Hints of a Universal Abundance Gradient IFS o/ffers the possibility to analyse and study a single object in great detail, such as the case of NGC /six.fitted/two.fitted/eight.fitted described above. However, it also o/ffers the unique chance of studying the spectroscopic properties of thousands of HII regions in a homogeneous way. We used our catalogue of HII regions introduced in Section /two.fitted to characterize the radial trends and the physical properties of the HII regions of the galaxy sample. However, contrary to the case of NGC /six.fitted/two.fitted/eight.fitted where the HII regions on the disc of the galaxy were basically selected and extracted by-hand, the HII regions in these galaxies were detected, spatially segregated, and spectrally extracted using HIIexplorer [/three.fitted/nine.fitted], a new automatic procedure to detect HII regions, based on the contrast of the H /u1D6FC intensity maps extracted from the data cubes. Once detected, the algorithm provides with the integrated spectra of each individual segmented region. /T_his change of paradigm is totally necessary when working with thousands of HII regions, contrary to the case of a handful of targets in classic long-slit spectroscopy. We detected a total of /two.fitted/five.fitted/seven.fitted/three.fitted HII regions with good spectroscopic quality. /T_his is by far the largest spatially resolved, nearby spectroscopic HII region survey ever accomplished. /T_he emission lines were decoupled from the underlying stellar population using FIT/three.fittedD [/four.fitted/zero.fitted], following a robust and well-tested methodology [/two.fitted/zero.fitted, /two.fitted/nine.fitted]. Extinction-corrected, /flux intensities of the stronger emission lines were obtained and used to select only star-forming regions based on typical BPT diagnostic diagrams. /T_he /final F/i.sc/g.sc/u.sc/r.sc/e.sc /five.fitted: Mean age /two.fitted-D maps weighted by the mass (a) and by the light (b) of the stars. /T_he di/fferent regions correspond to a Voronoitessellation binning scheme performed to the IFS mosaic of NGC /six.fitted/two.fitted/eight.fitted. Figure adapted from S'anchez-Bl'azquez et al. (submitted). F/i.sc/g.sc/u.sc/r.sc/e.sc /five.fitted: Mean age /two.fitted-D maps weighted by the mass (a) and by the light (b) of the stars. /T_he di/fferent regions correspond to a Voronoitessellation binning scheme performed to the IFS mosaic of NGC /six.fitted/two.fitted/eight.fitted. Figure adapted from S'anchez-Bl'azquez et al. (submitted). sample comprises /one.fitted/eight.fitted/nine.fitted/six.fitted high-quality, spatially resolved HII regions/aggregations of disc galaxies in the local Universe [/three.fitted/nine.fitted]. It is well known that di/fferent spectroscopic properties of HII regions show strong variations across the area of disc galaxies. In particular, some of these parameters (e.g., oxygen abundance, EW[H /u1D6FC ], etc.), show a strong radial gradient, that in average indicates that more evolved, metal rich, stellar populations are located in the center of galaxies, and less evolved, metal poor ones are in the outer ones. Despite the several di/fferent studies describing these observational events, there is a large degree of discrepancy between the actual derived parameters describing the gradients: (i) slope of the gradient, (ii) average value and dispersion of the zeropoint, and (iii) scale length of the truncation. In general, this is mostly due to di/fferent observational biases and the lack of a proper statistical number of analysed HII regions per galaxy. For each galaxy of our sample we derived the correlation coe/fficient, the slope, and the zero point of a linear regression for a number of parameters showing radial distributions across the discs of the galaxies. For those properties showing a strong correlation, we investigated if the gradient was universal within our range of explored parameters. We found that, for the equivalent width of H /u1D6FC and the oxygen abundance,theslopesofthegradientsareconsistentwitha Gaussian distribution; that is, the dispersion of values found for each individual galaxy is compatible with the average one, not showing strong statistical deviations. /T_his implies that we can de/fine a characteristic value for the slope and that we do not /find a population of galaxies with slopes inconsistent with this normal distribution. /T_he right panel of Figure /six.fitted shows the radial density distribution for the oxygen abundance derived using the O/three.fittedN/two.fitted indicator [/two.fitted/eight.fitted], once scaled to the average value at the e/ffective radius for each galaxy. /T_he radial distance was normalised to the e/ffective radius of each galaxy. /T_hesolidlineshowstheaveragelinearregressionfoundfor eachindividualgalaxy./T_hered-dashedlineshowstheactual regression found for all the HII regions detected for all the galaxies. Our results seem to indicate that there is a universal radial gradient for oxygen abundance and the equivalent width of H /u1D6FC when normalized with the e/ffective radii of the galaxies; that is, they present a radial gradient that, statistically, has the same slope for all the galaxies in our sample. /T_he derived slopes for each galaxy are compatible with a Gaussian random distribution and are independent of the morphology of the analysed galaxies (barred/nonbarred, grand-design//flocculent). /T_his is one of the most important results in the abundance gradients of spiral galaxies, obtained thanks to the use of IFS. 5. The Local Mass-Metallicity Relation /T_he existence of a strong correlation between stellar mass and gas-phase metallicity in galaxies is a well-known fact. /T_he mass-metallicity ( M -Z) relation is consistent with more massive galaxies being more metal-enriched; a/f_ter the seminal work on this relationship by Lequeux et al. [/four.fitted/one.fitted], it was /firmly established observationally by Tremonti et al. ([/four.fitted/two.fitted], herea/f_ter T/zero.fitted/four.fitted) using the SDSS. However, there has been no Average linear /fit (all galaxies) -0.12 dex/ R Linear /fit to all HII regions -0.11 dex/ R Mean value at radial bins ∼0.15 R e Average abundance of solar neighborhood at R sun F/i.sc/g.sc/u.sc/r.sc/e.sc /six.fitted: Radial oxygen abundance density distribution for the wholeHIIregionspectroscopicsamplediscussedinthetext./T_he /first contour indicates the mean density, with a regular spacing of four times this value for each consecutive contour. /T_he light-blue solid circles indicate the mean value (plus 1-/u1D70E errors) for each consecutive radial bin of ∼ 0.15 /u1D445 /u1D452 . /T_he average error of the derived oxygen abundance is shown by a single error bar located at the topright side of the panel. /T_he solid-orange square indicates the average abundance of the solar neighbourhood, at the distance of the Sun to the Milky-Way galactic center. /T_he lines represent linear /fits to all galaxies (black) and all HII regions (dotted red) independently, showing a universal slope ∼-/zero.fitted./one.fitted dex/ /u1D445 /u1D452 . Figure is adapted from Rosales-Ortega et al. [/three.fitted/eight.fitted]. major e/ffort to test the M -Z relation using spatially resolved information. We used our IFS observations in order to test the distribution of mass and metals within the discs of the galaxies. We derived the (luminosity) surface mass density ( Σ Lum , /u1D440 /uni2299 pc -2 ) within the area encompassed by our IFSsegmented HII regions, using the prescriptions given by Bell and de Jong [/four.fitted/three.fitted]toconvert /u1D435 -/u1D449 colors into a /u1D435 -band massto-light ratio ( /u1D440//u1D43F ). /T_he le/f_t panel of Figure /seven.fitted shows the striking correlation between the local surface mass density and gas metallicity for our sample of nearby HII regions, that is, the local M -Z relation, extending over ∼ /three.fitted orders of magnitude in Σ Lum and a factor ∼ /eight.fittedinmetallicity[/three.fitted/eight.fitted]. /T_he notable similarity with the global M -Z relation can be visually recognised with the aid of the blue lines which stand for the [/four.fitted/two.fitted]/fit( ± /zero.fitted./two.fitted dex) to the global M -Z relation, shi/f_ted arbitrarily both in mass and metallicity to coincide with the peak of the HII region M -Z distribution. Other abundance calibrations were tested obtaining the same shape (and similar /fit) of the relation. In addition, we /find the existence of a more general relation between mass surface density, metallicity, and the equivalent width of H /u1D6FC , de/fined as the emission-line luminosity normalized to the adjacent continuum /flux, that is, a measure of the SFR per unit luminosity [/four.fitted/four.fitted]. /T_his functional relation is evident in a /three.fittedD space with orthogonal coordinate axes de/fined by these parameters, consistent with | EW ( H /u1D6FC)| being e e inversely proportional to both Σ Lum andmetallicity,asshown in Figure /eight.fitted.AsdiscussedinRosales-Ortegaetal.[/three.fitted/eight.fitted], we interpret the local M -Z-EW(H /u1D6FC ) relation as the combination of (i) the well-known relationships between both the mass and metallicity with respect to the di/fferential distributions of these parameters found in typical disc galaxies, that is, the inside-out growth, and (ii) the fact that more massive regions form stars faster (i.e., at higher SFRs), thus earlier in cosmological times. In order to test whether the global M -Z relation observed by [/four.fitted/two.fitted]usingSDSSdataisare/flection(aperturee/ffect)ofthe local HII region mass-density versus metallicity relation, we perform the following exercise. We simulate a galaxy with typical /u1D440 /u1D435 and /u1D435 -/u1D449 values drawn from /flat distributions in magnitude ( -/one.fitted/five.fitted to -/two.fitted/three.fitted) and colour ( ∼ /zero.fitted./four.fitted -/one.fitted). A redshi/f_t isassumedforthemockgalaxy,drawnfromaGaussian distribution with mean ∼ /zero.fitted./one.fitted and /u1D70E = 0.05 ,witharedshi/f_t cut 0.02 < /u1D467 < 0.3 in order to resemble the SDSS [/four.fitted/two.fitted] distribution. /T_he mass of the galaxy is derived using the integrated /u1D435 -band magnitudes, /u1D435 -/u1D449 colours, and the average /u1D440//u1D43F ratio following Bell and de Jong [/four.fitted/three.fitted]. /T_he metallicity ofthemockgalaxyisderivedusingthelocal M -Z relation within an aperture equal to the SDSS /fiber (/three.fitted arcsec), that is, the metallicity that corresponds to the mass density surface at this radius. /T_he process is repeated over /one.fitted/zero.fitted,/zero.fitted/zero.fitted/zero.fitted times in order to obtain a reliable distribution in the mass and metallicity of the mock galaxies. /T_he right panel of Figure /eight.fitted shows the result of the simulation, that is, the distribution of the mock galaxies in the M -Z parameter space. We reproduce-with a remarkable agreement-the overall shape of the global M -Z relation assuming a local M -Z relation and considering the aperture e/ffect of the SDSS /fiber. /T_he overlaid lines correspond to the [/four.fitted/two.fitted] /fit (black) and the Kewley and Ellison [/four.fitted/five.fitted] ± /zero.fitted./two.fitted dex relation (blue), for which the agreement is extremely good over a wide range of masses. /T_he result is remarkable considering that we are able to reproduce the global M -Z relation over a huge dynamical range, using a local M -Z relation derived from a galaxy sample with a restricted range in mass (/nine.fitted./two.fitted < log /u1D440 Lum < /one.fitted/one.fitted./two.fitted) and metallicity (/eight.fitted./three.fitted < /one.fitted/two.fitted + log(O/H) < /eight.fitted./nine.fitted),indicatedbytherectangleshownintheright panel of Figure /eight.fitted. /T_herefore, by using the power of IFS applied to a sample of nearby galaxies we demonstrate the existence of a local relation between the surface mass density, gas-phase oxygen abundance, and | EW ( H /u1D6FC)| in ∼ /two.fitted/zero.fitted/zero.fitted/zero.fitted spatially resolved HII regions of the Local Universe. /T_he projection of this distribution in the metallicity versus Σ Lum plane-the local M -Z relation-shows a tight correlation expanding over a wide rangeinthisparameterspace.Weusethelocal M -Z relation to reproduce the global M -Z relation by means of a simple simulation which considers the aperture e/ffects of the SDSS /fiber at di/fferent redshi/f_ts. Note that the 'local' M -Z-| EW ( H /u1D6FC)| relation is conceptually di/fferent from the 'global' M -Z-SFR relation proposed by Lara-L'opez et al. ([/four.fitted/six.fitted], dubbed FP), Mannucci et al. ([/four.fitted/seven.fitted], dubbed FMR), or Hunt et al. [/four.fitted/eight.fitted], based on the integrated spectra of galaxies (the basic di/fference between these relations is the proposed shape in the /three.fittedD distribution, that is, a F/i.sc/g.sc/u.sc/r.sc/e.sc /seven.fitted: (a) /T_he relation between surface mass density and gas-phase oxygen metallicity for ∼ /two.fitted/zero.fitted/zero.fitted/zero.fitted HII regions in nearby galaxies, the local M -Z relation. /T_he /first contour stands for the mean density value, with a regular spacing of four time this value for each consecutive contour. /T_he blue circles represent the mean (plus /one.fitted /u1D70E error bars) in bins of /zero.fitted./one.fitted/five.fitted dex. /T_he red dashed-dotted line is a polynomial /fit to the data. /T_he blue lines correspond to the [/four.fitted/two.fitted]relation( ± /zero.fitted./two.fitted dex) scaled to the relevant units. Typical errors for Σ Lum and metallicity are represented. (b) Distribution of HII regions along the local M -Z relation for three galaxies of the sample at di/fferent redshi/f_ts. /T_he size of the symbols is linked to the value of | EW ( H /u1D6FC)| , being inversely proportional to Σ Lum and metallicity as shown. Figure is adapted from Rosales-Ortega et al. [/three.fitted/eight.fitted]. F/i.sc/g.sc/u.sc/r.sc/e.sc /seven.fitted: (a) /T_he relation between surface mass density and gas-phase oxygen metallicity for ∼ /two.fitted/zero.fitted/zero.fitted/zero.fitted HII regions in nearby galaxies, the local M -Z relation. /T_he /first contour stands for the mean density value, with a regular spacing of four time this value for each consecutive contour. /T_he blue circles represent the mean (plus /one.fitted /u1D70E error bars) in bins of /zero.fitted./one.fitted/five.fitted dex. /T_he red dashed-dotted line is a polynomial /fit to the data. /T_he blue lines correspond to the [/four.fitted/two.fitted]relation( ± /zero.fitted./two.fitted dex) scaled to the relevant units. Typical errors for Σ Lum and metallicity are represented. (b) Distribution of HII regions along the local M -Z relation for three galaxies of the sample at di/fferent redshi/f_ts. /T_he size of the symbols is linked to the value of | EW ( H /u1D6FC)| , being inversely proportional to Σ Lum and metallicity as shown. Figure is adapted from Rosales-Ortega et al. [/three.fitted/eight.fitted]. F/i.sc/g.sc/u.sc/r.sc/e.sc /eight.fitted: (a) /three.fittedD representation of the local M -Z-EW(H /u1D6FC ) relation. /T_he size and color scaling of the data points are linked to the value of log Σ Lum (i.e. low-blue to high-red values). /T_he projection of the data over any pair of axes reduces to the local M -Z, M -EW(H /u1D6FC ), and metallicity-EW(H /u1D6FC ) relations. An online /three.fittedD animated version is available at http://tinyurl.com/local-mass-metallicity. (b) Distribution of simulated galaxies in the M -Z plane assuming a local M -Zrelationandconsideringtheaperturee/ffectoftheSDSS/fiber,asexplainedin the text. /T_he contours correspond to the density of points, while the circles represent the mean value (plus /one.fitted /u1D70E error bars) in bins of /zero.fitted./one.fitted/five.fitted dex. /T_he black line stands for the [/four.fitted/two.fitted] /fitting, while the blue lines correspond to the Kewley and Ellison [/four.fitted/five.fitted] ± /zero.fitted./two.fitted dex relation. /T_he rectangle encompasses the range in mass and metallicity of the galaxy sample. Figure is adapted from Rosales-Ortega et al. [/three.fitted/eight.fitted]. F/i.sc/g.sc/u.sc/r.sc/e.sc /eight.fitted: (a) /three.fittedD representation of the local M -Z-EW(H /u1D6FC ) relation. /T_he size and color scaling of the data points are linked to the value of log Σ Lum (i.e. low-blue to high-red values). /T_he projection of the data over any pair of axes reduces to the local M -Z, M -EW(H /u1D6FC ), and metallicity-EW(H /u1D6FC ) relations. An online /three.fittedD animated version is available at http://tinyurl.com/local-mass-metallicity. (b) Distribution of simulated galaxies in the M -Z plane assuming a local M -Zrelationandconsideringtheaperturee/ffectoftheSDSS/fiber,asexplainedin the text. /T_he contours correspond to the density of points, while the circles represent the mean value (plus /one.fitted /u1D70E error bars) in bins of /zero.fitted./one.fitted/five.fitted dex. /T_he black line stands for the [/four.fitted/two.fitted] /fitting, while the blue lines correspond to the Kewley and Ellison [/four.fitted/five.fitted] ± /zero.fitted./two.fitted dex relation. /T_he rectangle encompasses the range in mass and metallicity of the galaxy sample. Figure is adapted from Rosales-Ortega et al. [/three.fitted/eight.fitted]. surface or a plane). However, the obvious parallelism between these two scaling relations deserves a discussion. While the 'local' M -Z-| EW ( H /u1D6FC)| relation is related to the intrinsic physics involved in the growth of the galaxy disc in an insideout scenario, the existence of the 'global' M -Z-SFRrelation is explained, according to Mannucci et al. [/four.fitted/seven.fitted], by the interplay of infall of pristine gas and out/flow of enriched material at di/fferent redshi/f_ts epochs, supporting the smooth accretion scenario, where galaxy growth is dominated by continuous accretion of cold gas in the local Universe. However, S'anchez et al. [/four.fitted/nine.fitted], using CALIFA data, found no secondary relation of the mass and metallicity with the SFR other than the one induced by the primary relation of this quantity with the stellar mass. /T_he same was found with respect to the speci/fic SFR rate. /T_he results by S'anchez et al. [/four.fitted/nine.fitted]agreewitha scenario in which gas recycling in galaxies, both locally and globally, is much faster than other typical timescales, such like that of gas accretion by in/flow and/or metal loss due to out/flows. In essence, late-type/disc-dominated galaxies seem to be in a quasi-steady situation, with a behavior similar to the one expected from an instantaneous recycling/closed-box model. In this scenario, the inner regions of the galaxy form /first and faster, increasing the gas metallicity of the surrounding interstellar medium. As the galaxy evolves and grows with time, the star-formation progresses radially creating a radial metallicity gradients in the disk of spirals. Mass is progressively accumulated at the inner regions of the galaxy, raising the surface mass density and creating a bulge, with corresponding high metallicity values but low SSFR (low | EW ( H /u1D6FC)| ), that is, an 'inside-out' galaxy disk growth. In such a case, the local M -Z relation would re/flect a more fundamental relation between mass, metallicity, and starformation e/fficiency as a function of radius, equivalent to alocal downsizing e/ffect, similar to the one observed in individual galaxies. Following this reasoning, the origin of the global M -Z relation can be explained as the combined e/ffect of the existence of the local M -Zrelation,anaperturebiasdue to the di/fferent /fibers covering factors of the spectroscopic surveys from which the FMR and FP were derived (as secondorder e/ffect), and a possible selection of a bias of the galaxy populations which are most common at a particular redshi/f_t, and may not re/flect the physics of how galaxies evolve. Supporting evidence in favour of the inside-out scenario of galaxy growth comes from the recent analysis of the spatially resolved history of the stellar mass assembly in galaxies of the local Universe [/five.fitted/zero.fitted]. In summary, the existence of the M-ZSFR relation could also be interpreted as a scaled-up version of the local M-Z-sSFR relation in the distribution of starforming regions across the discs of galaxies as described in Rosales-Ortega et al. [/three.fitted/eight.fitted]andcon/firmedbyS'anchez et al. [/four.fitted/nine.fitted]; that is, the relationship is not primary, but obtained from the sum of a number of local linear relations (and their deviations) with respect to the galaxy radius. 6. Conclusions /T_he emergence of a new generation of instrumentation, that is, multiobject and integral /field spectrometers with large /fields of view, capable of performing emission-line surveys based on samples of hundreds of spectra in a /two.fittedD context, arerevolutionisingthemethodsandtechniquesusedtostudy the gas-phase component of star-forming galaxies in the nearby Universe (objects which were typically studied with smallsamplesbasedonlong-slitspectroscopy).Anewbody of results is coming out from these studies, opening up a new frontier of studying the /two.fittedD structure and intrinsic dispersion of the physical and chemical properties of the discs of nearby spiral galaxies. In this paper we review some of the projects that in the last years tackled for the /first time the problem of obtaining spatially resolved spectroscopic information of the gas in nearby galaxies. PINGS represented the /first endeavour to obtain full /two.fittedD coverage of the discs of a sample of spiral galaxies in the nearby Universe. /T_he PINGS sample covered di/fferent galaxy types, including normal, lopsided, interacting, and barred spirals with a good range of galactic properties and star-forming environments, with multiwavelength public data. /T_he spectroscopic data set comprises more than /five.fitted/zero.fitted /zero.fitted/zero.fitted/zero.fitted individual spectra, covering an observed area of nearly /eight.fitted/zero.fitted arcmin 2 ,anobservedsurface without precedents by an IFS study by the time. /T_heIFSanalysisofNGC/six.fitted/two.fitted/eight.fitted,thelargestspectroscopic mosaic on a single galaxy, was taken as an example of the new methodology and analysis that could be performed with alargespectroscopicdatabaseforasingleobject./T_hecon-tribution of PINGS also resides in de/fining a self-consistent methodology in terms of observation, data reduction and analysis for present and future IFS surveys of the kind, as well as a whole new set of visualization and analysis so/f_tware thathasbeenmadepublictothecommunity(e.g.,[/five.fitted/one.fitted, /five.fitted/two.fitted]). Despite all the complexities involved in the observations, data reduction, and analysis, PINGS proved to be feasible. In less than a three-year period, it was possible to build a comprehensive sample of galaxies with a good range of galactic properties and available multiwavelength ancillary data, maximising both the original science goals of the project and the possible archival value of the survey. In fact, the science case of the PINGS project was the inspiration for the ongoing CALIFA survey. /T_he face-on spirals from M'armolQueralt'oetal.[/two.fitted/one.fitted] were part of the feasibility studies for the CALIFA survey. On completion, CALIFA will be the largest and the most comprehensive wide-/field IFU survey of galaxies carried out to date. It will thus provide an invaluable bridge between large single aperture surveys and more detailed studies of individual galaxies. /six.fitted./one.fitted. Results from Other IFU Projects on Star-Forming Galaxies. Other projects have followed this initiative; for example, the Mitchell spectrograph instrument at McDonald Observatory (a.k.a VIRUS-P) is currently used to carry out two small IFS surveys, namely, VENGA [/five.fitted/three.fitted] and VIXENS [/five.fitted/four.fitted]. VENGA (VIRUS-P Exploration of Nearby Galaxies) is an integral /field spectroscopic survey, which maps the disks of /three.fitted/zero.fitted nearby spiral galaxies, in a very similar manner to PINGS in terms of spectral coverage, resolution, and area sampled (/three.fitted/six.fitted/zero.fitted/zero.fitted ˚ A-/six.fitted/eight.fitted/zero.fitted/zero.fitted ˚ A, ∼ /five.fitted ˚ A FWHM, ∼ /zero.fitted./seven.fitted R 25 )althoughwitha di/fferent spatial resolution (/five.fitted./six.fitted arcsec FWHM). /T_heir targets span a wide range in Hubble type, star-formation activity, morphology, and inclination. Likewise PINGS, the VENGA group used the data cubes of their observations to produce /two.fittedD maps of the star-formation rate, dust extinction, electron density, stellar population parameters, the kinematics and chemical abundance of both stars and ionized gas, and other physical quantities derived from the /fitting of the stellar spectrum and the measurement of nebular emission lines. /T_heir /first results focus on (/one.fitted) the spatially resolved starformationlawofNGC/five.fitted/one.fitted/nine.fitted/four.fittedwheretheygivesupporttothe evidence for a low, and close to constant, star-formation e/fficiency (SFE = /u1D70F -1 ) in the molecular component of the interstellar medium [/five.fitted/five.fitted] and (/two.fitted) using IFS observations of NGC/six.fitted/two.fitted/eight.fitted,theymeasuretheradialpro/fileofthe 12 CO(/one.fitted/zero.fitted) to H 2 conversion factor ( /u1D44B C /u1D442 )inthisgalaxyandstudy howchangesin /u1D44B C /u1D442 follow changes in metallicity, gas density, and ionization parameter [/five.fitted/six.fitted],andalsotheyusetheIFS data to propose a new method to measure the inclination of nearlyface-onsystemsbasedonthematchingofthestellar and gas rotation curves using asymmetric dri/f_t corrections [/five.fitted/three.fitted]. In the case of VIXENS (VIRUS-P Investigation of the eXtreme ENvironments of Starburst), their goal of our survey is to investigate the relation between star-formation and gas content in the extreme environments of interacting galaxy pairs and mergers on spatially resolved scales of 0.2 -0.8 kpc, by using IFS of /one.fitted/five.fitted interacting/starburst galaxies. VIXENS will make extensive use of multiwavelength data in order to investigate the star-formation in this object, including data from Spitzer, GALEX, IRAM, CARMA archival CO, and Hi maps. /T_hese projects and datasets are clearly focused on speci/fic science questions, adopting correspondingly optimized sample selection criteria and also observing strategies. Other surveys in the local Universe using the power of IFS for a detailed study of nearby galaxies include the next generation surveys like Sydney university AAO MOS IFU [/five.fitted/seven.fitted] (SAMI) and Mapping Nearby Galaxies at APO, PI: Kevin Bundy, IPMU (MaNGA), or the new generation instrumentation for Very Large Telescope (VLT, ESO) like Multi Unit Spectroscopic Explorer [/five.fitted/eight.fitted] (MUSE), which aim at studying the the chemical and dynamical evolution history and dark matter contents of galaxies, the physical role of environment in galaxy evolution, when, where, and why does star-formation occur, and so forth, based on spatially resolved spectroscopic surveys of 10 4 -10 5 galaxies. /T_he continuous coverage spectra provided by the imaging spectroscopy technique employed in these projects are already allowing us to study the small and intermediate linear scale variation in line emission and the gas chemistry for a statistically representative number of galaxies of the nearby Universe. /T_he primary motivation common to all of these observational e/fforts is to use this information to link the properties of high redshi/f_t galaxies with those we see around us today and thereby understand the physical processes at play in the formation and evolution of galaxies. /T_he power and importance of all these projects resides in the fact that they will provide an observational anchor of the spatially resolved properties of the galaxies in the local Universe, which will have a potential impact in the interpretation of observed properties at high redshi/f_t from new generation facilities, suchastheJamesWebbSpaceTelescope(JWST),theGiant MagellanTelescope(GMT),ortheEuropeanExtremelyLarge Telescope (E-ELT), projects that will hopefully revolutionise the understanding of our Universe in future years. Acknowledgments BasedonobservationscollectedattheCentroAstron'omico Hispano-Alem'an (CAHA) at Calar Alto, operated jointly by the Max-Planck Institut fur Astronomie and the Instituto de Astrof'ısica de Andaluc'ıa (CSIC) Fernando Fabi'an RosalesOrtega acknowledges the Mexican National Council for Science and Technology (CONACYT) for /financial support under the Programme Estancias Posdoctorales y Sab'aticas al Extranjero para la Consolidaci'on de Grupos de Investigaci'on, /two.fitted/zero.fitted/one.fitted/zero.fitted-/two.fitted/zero.fitted/one.fitted/two.fitted. /T_he author also acknowledges /financial support for the ESTALLIDOS collaboration by the Spanish Ministerio de Ciencia e Innovaci'on under Grant AYA/two.fitted/zero.fitted/one.fitted/zero.fitted-/two.fitted/one.fitted/eight.fitted/eight.fitted/seven.fitted-C/zero.fitted/four.fitted/zero.fitted/three.fitted. References [/one.fitted] S. Folkes, S. Ronen, I. Price et al., '/T_he /two.fitteddF galaxy redshi/f_t survey: spectral types and luminosity functions,' Monthly Notices of the Royal Astronomical Society ,vol./three.fitted/zero.fitted/eight.fitted,no./two.fitted,pp./four.fitted/five.fitted/nine.fitted-/four.fitted/seven.fitted/two.fitted, /one.fitted/nine.fitted/nine.fitted/nine.fitted. [/two.fitted ] D. G. York, J. Adelman, J. E. Anderson et al. et al., '/T_he sloan digital sky survey: technical summary,' /T_he Astronomical Journal ,vol./one.fitted/two.fitted/zero.fitted,no./three.fitted,article/one.fitted/five.fitted/seven.fitted/nine.fitted,/two.fitted/zero.fitted/zero.fitted/zero.fitted. [/three.fitted] H.-W.Rix,M.Barden,S.V.W.Beckwithetal.,'GEMS:galaxy evolution from morphologies and SEDs,' Astrophysical Journal , vol. /one.fitted/five.fitted/two.fitted, no. /two.fitted, pp. /one.fitted/six.fitted/three.fitted-/one.fitted/seven.fitted/three.fitted, /two.fitted/zero.fitted/zero.fitted/four.fitted. [/four.fitted] N. Scoville, H. Aussel, M. Brusa et al., '/T_he cosmic evolution survey (COSMOS): overview,' Astrophysical Journal, Supplement Series ,vol./one.fitted/seven.fitted/two.fitted,no./one.fitted,pp./one.fitted-/eight.fitted,/two.fitted/zero.fitted/zero.fitted/seven.fitted. [/five.fitted] T.Nagao,R.Maiolino,andA.Marconi,'Gasmetallicitydiagnostics in star-forming galaxies,' Astronomy and Astrophysics , vol./four.fitted/five.fitted/nine.fitted,no./one.fitted,pp./eight.fitted/five.fitted-/one.fitted/zero.fitted/one.fitted,/two.fitted/zero.fitted/zero.fitted/six.fitted. [/six.fitted] R. Maiolino, T. Nagao, A. Grazian et al., 'AMAZE: I. /T_he evolution of the mass metallicity relation at z > /three.fitted,' Astronomy and Astrophysics ,vol./four.fitted/eight.fitted/eight.fitted,no./two.fitted,pp./four.fitted/six.fitted/three.fitted-/four.fitted/seven.fitted/nine.fitted,/two.fitted/zero.fitted/zero.fitted/eight.fitted. [/seven.fitted] R. C. Kennicutt and P. W. Hodge, 'H II regions in NGC /six.fitted/two.fitted/eight.fitted. III. H-alpha luminosities and the luminosity function,' /T_he Astrophysical Journal ,vol./two.fitted/four.fitted/one.fitted,article/five.fitted/seven.fitted/three.fitted,/one.fitted/nine.fitted/eight.fitted/zero.fitted. [/eight.fitted] J. Belley and J.-R. Roy, '/T_he abundance gradients across the spiral galaxies NGC /six.fitted/two.fitted/eight.fitted and NGC /six.fitted/nine.fitted/four.fitted/six.fitted,' Astrophysical Journal , vol./seven.fitted/eight.fitted,no./one.fitted,pp./six.fitted/one.fitted-/eight.fitted/five.fitted,/one.fitted/nine.fitted/nine.fitted/two.fitted. [/nine.fitted] P.A.Scowen,J.J.Hester,J.S.Gallagher,E.Wilcots,andT.W.Idt, 'HST WFPC-/two.fitted observations of typical star formation in M/one.fitted/zero.fitted/one.fitted,' in Proceedings of the /one.fitted/eight.fitted/nine.fittedth AAS Meeting ,vol./two.fitted/eight.fitted,p./one.fitted/three.fitted/six.fitted/zero.fitted,Bulletin of the American Astronomical Society, /one.fitted/nine.fitted/nine.fitted/six.fitted. [/one.fitted/zero.fitted] J. R. Roy and J.-R. Walsh, 'Imaging spectroscopy of H II regions inthebarredSpiralgalaxyNGC,' Monthly Notices of the Royal Astronomical Society , vol. /two.fitted/three.fitted/four.fitted, article /nine.fitted/seven.fitted/seven.fitted, /one.fitted/nine.fitted/eight.fitted/eight.fitted. [/one.fitted/one.fitted] R. C. Kennicutt Jr. and D. R. Garnett, '/T_he composition gradient in M/one.fitted/zero.fitted/one.fitted revisited. I. H II region spectra and excitation properties,' Astrophysical Journal Letters ,vol./four.fitted/five.fitted/six.fitted,no./two.fitted,pp. /five.fitted/zero.fitted/four.fitted-/five.fitted/one.fitted/eight.fitted, /one.fitted/nine.fitted/nine.fitted/six.fitted. [/one.fitted/two.fitted] M. L. McCall, P. M. Rybski, and G. A. Shields, '/T_he chemistry of galaxies. I: the nature of giant extragalactic HII regions,' /T_he Astrophysical Journal ,vol./five.fitted/seven.fitted,no./one.fitted,/one.fitted/nine.fitted/eight.fitted/five.fitted. [/one.fitted/three.fitted] L. Van Zee, J. J. Salzer, M. P. Haynes, A. A. O'Donoghue, andT.J.Balonek,'SpectroscopyofoutlyingHIIregionsin spiral galaxies: abundances and radial gradients,' Astronomical Journal , vol. /one.fitted/one.fitted/six.fitted, no. /six.fitted, pp. /two.fitted/eight.fitted/zero.fitted/five.fitted-/two.fitted/eight.fitted/three.fitted/three.fitted, /one.fitted/nine.fitted/nine.fitted/eight.fitted. [/one.fitted/four.fitted] M. Castellanos, A. I. D'ıaz, and E. Terlevich, 'A comprehensive study of reported high-metallicity giant HII regions. I: detailed abundance analysis,' Monthly Notices of the Royal Astronomical Society , vol. /three.fitted/two.fitted/nine.fitted, no. /two.fitted, pp. /three.fitted/one.fitted/five.fitted-/three.fitted/three.fitted/five.fitted, /two.fitted/zero.fitted/zero.fitted/two.fitted. [/one.fitted/five.fitted] M. Castellanos, ' A. I. D'ıaz, and G. Tenorio-Tagle, 'On the large escape of ionizing radiation from giant extragalactic HII regions,' Astrophysical Journal Letters ,vol./five.fitted/six.fitted/five.fitted,no./two.fitted,pp.L/seven.fitted/nine.fittedL/eight.fitted/two.fitted, /two.fitted/zero.fitted/zero.fitted/two.fitted. [/one.fitted/six.fitted] J. Moustakas and R. C. Kennicutt Jr., 'An integrated spectrophotometric survey of nearby star-forming galaxies,' Astrophysical Journal ,vol./one.fitted/six.fitted/four.fitted,no./one.fitted,pp./eight.fitted/one.fitted-/nine.fitted/eight.fitted,/two.fitted/zero.fitted/zero.fitted/six.fitted. [/one.fitted/seven.fitted] R. Bacon, Y. Copin, G. Monnet et al., '/T_he SAURON project. I: the panoramic integral-/field spectrograph,' Monthly Notices of the Royal Astronomical Society ,vol./three.fitted/two.fitted/six.fitted,no./one.fitted,pp./two.fitted/three.fitted-/three.fitted/five.fitted,/two.fitted/zero.fitted/zero.fitted/one.fitted. [/one.fitted/eight.fitted] P. T. de Zeeuw, M. Bureau, E. Emsellem et al., '/T_he SAURON project. II: sample and early results,' Monthly Notices of the Royal Astronomical Society , vol. /three.fitted/two.fitted/nine.fitted, no. /three.fitted, pp. /five.fitted/one.fitted/three.fitted-/five.fitted/three.fitted/zero.fitted, /two.fitted/zero.fitted/zero.fitted/two.fitted. [/one.fitted/nine.fitted] K. Ganda, J. Falc'on-Barroso,R.F.Peletieretal.,'Late-type galaxies observed with SAURON: two-dimensional stellar and emission-line kinematics of /one.fitted/eight.fitted spirals,' Monthly Notices of the Royal Astronomical Society ,vol./three.fitted/six.fitted/seven.fitted,no./one.fitted,pp./four.fitted/six.fitted-/seven.fitted/eight.fitted,/two.fitted/zero.fitted/zero.fitted/six.fitted. [/two.fitted/zero.fitted] F. F. Rosales-Ortega, R. C. Kennicutt, S. F. S'anchez et al., 'PINGS: the PPAK IFS nearby galaxies survey,' Monthly Notices of the Royal Astronomical Society ,vol./four.fitted/zero.fitted/five.fitted,pp./seven.fitted/three.fitted/five.fitted-/seven.fitted/five.fitted/eight.fitted,/two.fitted/zero.fitted/one.fitted/zero.fitted. [/two.fitted/one.fitted] E. M'armol-Queralt'o, S. F. M'armol, R. A. Marino et al., 'Integral /field spectroscopy of a sample of nearby galaxies: I. sample, observations, and data reduction,' Astronomy and Astrophysics , vol. /five.fitted/three.fitted/four.fitted, article A/eight.fitted, /two.fitted/zero.fitted/one.fitted/one.fitted. [/two.fitted/two.fitted] S. F. Sanchez, R. C. Kennicutt, A. Gil de Paz et al. et al., 'CALIFA, the calar alto legacy integral /field area survey,' Astronomy and Astrophysics ,vol./five.fitted/three.fitted/eight.fitted,article/eight.fitted,/two.fitted/zero.fitted/one.fitted/two.fitted. [/two.fitted/three.fitted] B.Husemann,K.Jahnke,S.F.Sanchezetal.etal.,'CALIFA, the calar alto legacy integral /field area survey,' Astronomy & Astrophysics ,vol./five.fitted/four.fitted/nine.fitted,article/eight.fitted/seven.fitted,/two.fitted/five.fittedpages,/two.fitted/zero.fitted/one.fitted/three.fitted. [/two.fitted/four.fitted]M.A.Bershady,M.A.W.Verheijen,R.A.Swaters,D.R. Andersen, K. B. Westfall, and T. Martinsson, '/T_he diskmass survey.I:overview,' Astrophysical Journal Letters ,vol./seven.fitted/one.fitted/six.fitted,no. /one.fitted, pp. /one.fitted/nine.fitted/eight.fitted-/two.fitted/three.fitted/three.fitted, /two.fitted/zero.fitted/one.fitted/zero.fitted. [/two.fitted/five.fitted] M. M. Roth, A. Kelz, T. Fechner et al., 'PMAS: the Potsdam multi-aperture spectrophotometer. I: design, manufacture, and performance,' Publications of the Astronomical Society of the Paci/fic ,vol./one.fitted/one.fitted/seven.fitted,no./eight.fitted/three.fitted/two.fitted,pp./six.fitted/two.fitted/zero.fitted-/six.fitted/four.fitted/two.fitted,/two.fitted/zero.fitted/zero.fitted/five.fitted. [/two.fitted/six.fitted] M. A. W. Verheijen, M. A. Bershady, D. R. Andersen et al., '/T_he DiskMassproject;sciencecaseforanewPMASIFUmodule,' Astronomische Nachrichten ,vol./three.fitted/two.fitted/five.fitted,no./two.fitted,pp./one.fitted/five.fitted/one.fitted-/one.fitted/five.fitted/four.fitted,/two.fitted/zero.fitted/zero.fitted/four.fitted. [/two.fitted/seven.fitted] A. Kelz, M. A. W. Verheijen, M. M. Roth et al., 'PMAS: the potsdam multi-aperture spectrophotometer. II: the wide integral /field unit PPak,' Publications of the Astronomical Society of the Paci/fic ,vol./one.fitted/one.fitted/eight.fitted,no./eight.fitted/three.fitted/nine.fitted,pp./one.fitted/two.fitted/nine.fitted-/one.fitted/four.fitted/five.fitted,/two.fitted/zero.fitted/zero.fitted/six.fitted. [/two.fitted/eight.fitted] M. Pettini and B. E. J. Pagel, '[O III]/[N II] as an abundance indicator at high redshi/f_t,' Monthly Notices of the Royal Astronomical Society ,vol./three.fitted/four.fitted/eight.fitted,no./three.fitted,pp.L/five.fitted/nine.fitted-L/six.fitted/three.fitted,/two.fitted/zero.fitted/zero.fitted/four.fitted. [/two.fitted/nine.fitted] S. F. S'anchez, F. F. Rosales-Ortega, R. C. Kennicutt et al., 'PPAK Wide-/field integral /field spectroscopy of NGC /six.fitted/two.fitted/eight.fitted. I: the largest spectroscopic mosaic on a single galaxy,' Monthly Notices of the Royal Astronomical Society , vol. /four.fitted/one.fitted/zero.fitted, no. /one.fitted, pp. /three.fitted/one.fitted/three.fitted-/three.fitted/four.fitted/zero.fitted, /two.fitted/zero.fitted/one.fitted/one.fitted. [ /three.fitted /zero.fitted ] F. F. Rosales-Ortega, A. I . D'ıaz, R. C. Kennicutt, and S. F. S 'anchez, 'PPAK wide-/field integral /field spectroscopy of NGC/six.fitted/two.fitted/eight.fitted. II: emission line abundance analysis,' Monthly Notices of the Royal Astronomical Society ,vol./four.fitted/one.fitted/five.fitted,no./three.fitted,pp./two.fitted/four.fitted/three.fitted/nine.fitted-/two.fitted/four.fitted/seven.fitted/four.fitted, /two.fitted/zero.fitted/one.fitted/one.fitted. [/three.fitted/one.fitted] S. F. S'anchez, 'Techniques for reducing /fiber-fed and integral-/field spectroscopy data: an implementation on R/three.fittedD,' Astronomische Nachrichten ,vol./three.fitted/two.fitted/seven.fitted,p./eight.fitted/five.fitted/zero.fitted,/two.fitted/zero.fitted/zero.fitted/six.fitted. [/three.fitted/two.fitted] Y. Li, F. Bresolin, and R. C. J. Kennicutt, 'Testing for azimuthal abundance gradients in M/one.fitted/zero.fitted/one.fitted,' /T_he Astrophysical Journal ,vol. /seven.fitted/six.fitted /six.fitted , article /one.fitted/seven.fitted, /two.fitted /zero.fitted /one.fitted /three.fitted . [/three.fitted/three.fitted] Andrievsky, S. M. Bersier, D. Kovtyukh et al., 'Using Cepheids to determine the galactic abundance gradient. II: towards the galactic center,' Astronomy and Astrophysics ,vol./three.fitted/eight.fitted/four.fitted,pp./one.fitted/four.fitted/zero.fitted/one.fitted/four.fitted /four.fitted , /two.fitted /zero.fitted /zero.fitted/two.fitted . [/three.fitted/four.fitted] S. Pedicelli, G. Bono, B. Lemasle et al., 'On the metallicity gradient of the Galactic disk,' Astronomy and Astrophysics ,vol. /five.fitted/zero.fitted/four.fitted, no. /one.fitted, pp. /eight.fitted/one.fitted-/eight.fitted/six.fitted, /two.fitted/zero.fitted/zero.fitted/nine.fitted. [/three.fitted/five.fitted] J. R. D. L'epine, P. Cruz, J. Scarano et al., 'Overlapping abundance gradients and azimuthal gradients related to the spiral structure of the Galaxy,' Monthly Notices of the Royal Astronomical Society ,vol./four.fitted/one.fitted/seven.fitted,no./one.fitted,pp./six.fitted/nine.fitted/eight.fitted-/seven.fitted/zero.fitted/eight.fitted,/two.fitted/zero.fitted/one.fitted/one.fitted. [/three.fitted/six.fitted] B. Lemasle, P. Francois, K. Genovali et al. et al., 'Galactic abundance gradients from Cepheids: alpha and heavy elements in the outer disk,' In Press. http://xxx.tau.ac.il/abs//one.fitted/three.fitted/zero.fitted/eight.fitted./three.fitted/two.fitted/four.fitted/nine.fitted. [/three.fitted/seven.fitted] J.-M. Deharveng, R. Jedrzejewski, P. Crane, M. J. Disney, and B. Rocca-Volmerange, 'Blue stars in the center of the S/zero.fitted galaxy NGC/five.fitted/one.fitted/zero.fitted/two.fitted,' Astronomy and Astrophysics , vol. /three.fitted/two.fitted/six.fitted, no. /two.fitted, pp. /five.fitted/two.fitted/eight.fitted/five.fitted/three.fitted/six.fitted, /one.fitted/nine.fitted/nine.fitted/seven.fitted. [/three.fitted/eight.fitted]F.F.Rosales-Ortega,S.F.Sanchez,J.Iglesias-Paramoetal., 'A new scaling relation for HII regions in spiral galaxies: unveiling the true nature of the mass-metallicity relation,' /T_he Astrophysical Journal , vol. /seven.fitted/five.fitted/six.fitted, article L/three.fitted/one.fitted, /two.fitted/zero.fitted/one.fitted/two.fitted. [/three.fitted/nine.fitted] S. F. S'anchez, F. F. Rosales-Ortega, R. A. Marino et al. et al., 'Integral/fieldspectroscopyofasampleofnearbygalaxies,' Astronomy and Astrophysics ,vol./five.fitted/four.fitted/six.fitted,articleA/two.fitted,/two.fitted/zero.fitted/one.fitted/two.fitted. [/four.fitted/zero.fitted] S. F. S'anchez, N. Cardiel, M. A. W. Verheijen, S. Pedraz, and G. Covone, 'Morphologies and stellar populations of galaxiesinthecoreofAbell/two.fitted/two.fitted/one.fitted/eight.fitted,' Monthly Notices of the Royal Astronomical Society ,vol./three.fitted/seven.fitted/six.fitted,no./one.fitted,pp./one.fitted/two.fitted/five.fitted-/one.fitted/five.fitted/zero.fitted,/two.fitted/zero.fitted/zero.fitted/seven.fitted. [/four.fitted/one.fitted] J. Lequeux, M. Peimbert, J. F. Rayo, A. Serrano, and S. TorresPeimbert, 'Chemical composition and evolution of irregular and blue compact galaxies,' Astronomy and Astrophysics ,vol./eight.fitted/zero.fitted, pp./one.fitted/five.fitted/five.fitted-/one.fitted/six.fitted/six.fitted,/one.fitted/nine.fitted/seven.fitted/nine.fitted. [/four.fitted/two.fitted] C.A.Tremonti,T.M.Heckman,G.Kau/ffmannetal.,'/T_heori-gin of the mass-metallicity relation: insights from /five.fitted/three.fitted,/zero.fitted/zero.fitted/zero.fitted starforming galaxies in the sloan digital sky survey,' Astrophysical Journal Letters ,vol./six.fitted/one.fitted/three.fitted,no./two.fitted,pp./eight.fitted/nine.fitted/eight.fitted-/nine.fitted/one.fitted/three.fitted,/two.fitted/zero.fitted/zero.fitted/four.fitted. [/four.fitted/three.fitted] E. F. Bell and R. S. de Jong, 'Stellar mass-to-light ratios and the Tully-Fisher relation,' Astrophysical Journal Letters ,vol./five.fitted/five.fitted/zero.fitted,no. /one.fitted,pp./two.fitted/one.fitted/two.fitted-/two.fitted/two.fitted/nine.fitted,/two.fitted/zero.fitted/zero.fitted/one.fitted. [/four.fitted/four.fitted] R. C. Kennicutt, 'Star formation in galaxies along the Hubble sequence,' ARA Stronomy and Astrophysics ,vol./three.fitted/six.fitted,article/one.fitted/eight.fitted/nine.fitted, /one.fitted/nine.fitted/nine.fitted/eight.fitted. [/four.fitted/five.fitted] L. J. Kewley and S. L. Ellison, 'Metallicity calibrations and the mass-metallicity relation for star-forming galaxies,' Astrophysical Journal Letters ,vol./six.fitted/eight.fitted/one.fitted,no./two.fitted,pp./one.fitted/one.fitted/eight.fitted/three.fitted-/one.fitted/two.fitted/zero.fitted/four.fitted,/two.fitted/zero.fitted/zero.fitted/eight.fitted. [/four.fitted/six.fitted] M. A. Lara-L'opez, J. Cepa, A. Bongiovanni et al., 'A fundamental plane for /field star-forming galaxies,' Astronomy and Astrophysics ,vol./five.fitted/two.fitted/one.fitted,no./two.fitted,articleL/five.fitted/three.fitted,/two.fitted/zero.fitted/one.fitted/zero.fitted. [/four.fitted/seven.fitted] F. Mannucci, G. Cresci, R. Maiolino, A. Marconi, and A. Gnerucci, 'A fundamental relation between mass, star formation rate and metallicity in local and high-redshi/f_t galaxies,' Monthly Notices of the Royal Astronomical Society ,vol./four.fitted/zero.fitted/eight.fitted,no. /four.fitted, pp. /two.fitted/one.fitted/one.fitted/five.fitted-/two.fitted/one.fitted/two.fitted/seven.fitted, /two.fitted/zero.fitted/one.fitted/zero.fitted. [/four.fitted/eight.fitted] L. Hunt, L. Magrini, D. Galli et al. et al., 'Scaling relations of metallicity, stellar mass and star formation rate in metal-poor starbursts.I:afundamentalplane,' Monthly Notices of the Royal Astronomical Society ,vol./four.fitted/two.fitted/seven.fitted,no./two.fitted,pp./nine.fitted/zero.fitted/six.fitted-/nine.fitted/one.fitted/eight.fitted,/two.fitted/zero.fitted/one.fitted/two.fitted. [/four.fitted/nine.fitted] S. F. S'anchez, F. F. Rosales-Ortega, B. Jungwiert et al. et al., 'Mass-metallicity relation explored with CALIFA,' Astronomy and Astrophysics ,vol./five.fitted/five.fitted/four.fitted,article/five.fitted/eight.fitted,/two.fitted/zero.fitted/one.fitted/three.fitted. [/five.fitted/zero.fitted] E. Perez, R. Cid Fernandes, R. M. Gonzalez Delgado et al. et al., '/T_he evolution of galaxies resolved in space and time: an insideout growth view from the CALIFA survey,' /T_he Astrophysical Journal Letters ,vol./seven.fitted/six.fitted/four.fitted,no./one.fitted,/two.fitted/zero.fitted/one.fitted/three.fitted. [/five.fitted/one.fitted] F. F. Rosales-Ortega, 'PINGSo/f_t: an IDL visualisation and manipulation tool for integral /field spectroscopic data,' New Astronomy ,vol./one.fitted/six.fitted,pp./two.fitted/two.fitted/zero.fitted-/two.fitted/two.fitted/eight.fitted,/two.fitted/zero.fitted/one.fitted/one.fitted. [/five.fitted/two.fitted] F. F. Rosales-Ortega, S. Arribas, and L. Colina, 'Integrated spectra extraction based on signal-to-noise optimization using integral /field spectroscopy,' Astronomy and Astrophysics ,vol. /five.fitted/three.fitted/nine.fitted, article A/seven.fitted/three.fitted, /two.fitted /zero.fitted/one.fitted /two.fitted. [/five.fitted/three.fitted] G. A. Blanc, T. Weinzirl, M. Song et al. et al., '/T_he virusP exploration of nearby galaxies (venga): survey design, data processing, and spectral analysis methods,' /T_he Astronomical Journal ,vol./one.fitted/four.fitted/five.fitted,article/one.fitted/three.fitted/eight.fitted,/two.fitted/zero.fitted/one.fitted/three.fitted. [/five.fitted/four.fitted] A.L.Heiderman,N.J.I.Evans,K.Gebhardtetal.,'/T_heVIRUS-P Investigation of the extreme environments of starbursts (VIXENS): survey and /first results,' in Proceedings of the Frank N.BashSymposiumonNewHorizonsinAstronomy , /two.fitted/zero.fitted/one.fitted/one.fitted. [/five.fitted/five.fitted] G. A. Blanc, A. Heiderman, K. Gebhardt, N. J. Evans, and J. Adams, '/T_he spatially resolved star formation law from integral /field spectroscopy: virus-p observations of NGC /five.fitted/one.fitted/nine.fitted/four.fitted,' Astrophysical Journal Letters ,vol./seven.fitted/zero.fitted/four.fitted,no./one.fitted,pp./eight.fitted/four.fitted/two.fitted-/eight.fitted/six.fitted/two.fitted,/two.fitted/zero.fitted/zero.fitted/nine.fitted. [/five.fitted/six.fitted] G. A. Blanc, A. Schruba, N. J. I. Evans et al. et al., '/T_he virus-P exploration of nearby glaxies (venga): the X co gradient in NGC /six.fitted/two.fitted /eight.fitted ,' /T_he Astrophysical Journal ,vol./seven.fitted/six.fitted/four.fitted,article/one.fitted/one.fitted/seven.fitted,/two.fitted/zero.fitted/one.fitted/three.fitted. [/five.fitted/seven.fitted] S. M. Croom, J. S. Lawrence, J. Bland-Hawthorn et al., '/T_he Sydney-AAOMulti-object Integral /field spectrograph,' Monthly Notices of the Royal Astronomical Society ,vol./four.fitted/two.fitted/one.fitted,no./one.fitted,pp./eight.fitted/seven.fitted/two.fitted/eight.fitted/nine.fitted/three.fitted , /two.fitted /zero.fitted/one.fitted /two.fitted. [/five.fitted/eight.fitted] R. Bacon, S. M. Bauer, R. Bower et al. et al., '/T_he second generation VLT instrument MUSE: science drivers and instrument design,' in Groundbased Instrumentation for Astronomy ,A.F. M. Moorwood and I. Masanori, Eds., Proceedings of the SPIE, pp. /one.fitted/one.fitted/four.fitted/five.fitted-/one.fitted/one.fitted/four.fitted/nine.fitted, Observatoire de Lyon, Lyon, France, /two.fitted/zero.fitted/zero.fitted/four.fitted. [/five.fitted/nine.fitted] A. M. N. Ferguson, J. S. Gallagher, and R. F. G. Wyse, '/T_he Extreme Outer Regions of Disk Galaxies. I. Chemical Abundances of HII Regions,' Astronomical Journal ,vol./one.fitted/one.fitted/six.fitted,article /six.fitted/seven.fitted/three.fitted, /one.fitted/nine.fitted/nine.fitted/eight.fitted. ISRN Thermodynamics ISRN Thermodynamics Hindawi Publishing Corporation http://www.hindawi.com ISRN High Energy Physics Hindawi Publishing Corporation http://www.hindawi.com Volume 2013 Volume 2013 Hindawi Submit your manuscripts at http://www.hindawi.com ISRN Condensed Matter Physics ISRN Condensed Matter Physics Hindawi Publishing Corporation http://www.hindawi.com Volume 2013 ISRN Astronomy and Astrophysics ISRN Astronomy and Astrophysics Hindawi Publishing Corporation http://www.hindawi.com Volume 2013 Hindawi Publishing Corporation http://www.hindawi.com Volume 2013 Advances in Condensed Matter Physics Hindawi Publishing Corporation http://www.hindawi.com Volume 2013 Advances in Condensed Matter Physics Both samples were observed with the PMAS spectrograph [/two.fitted/five.fitted]inthePPAKmode[/two.fitted/six.fitted, /two.fitted/seven.fitted] on the /three.fitted./five.fitted m telescope in Calar Alto with similar setup, resolutions, and integration times, covering their optical extension up to ∼ /two.fitted./four.fitted e/ffective radii within a wavelength range ∼ /three.fitted/seven.fitted/zero.fitted/zero.fitted-/seven.fitted/zero.fitted/zero.fitted/zero.fitted ˚ A. /T_he PPAK /fiber bundle consists of /three.fitted/eight.fitted/two.fitted /fibers of /two.fitted./seven.fitted arcsec diameter each. Of these /three.fitted/eight.fitted/two.fitted /fibers, /three.fitted/three.fitted/one.fitted (the science /fibers) are concentrated in a single hexagonal bundle covering a /field-of-view of 74 × 64 arcsec 2 , with a /filling factor of ∼ /six.fitted/zero.fitted%. /T_he sky background is sampled by /three.fitted/six.fitted additional /fibers, distributed in /six.fitted bundles of /six.fitted /fibers each, along a circle ∼ /seven.fitted/two.fitted arcsec from the center of the instrument FoV. In the case of PINGS, the observations consisted of IFU spectroscopic mosaics for /one.fitted/seven.fitted spiral galaxies within a maximumdistance of /one.fitted/zero.fitted/zero.fitted Mpc; the average distance of the sample is /two.fitted/eight.fitted Mpc (for /u1D43B 0 =73 kms -1 Mpc -1 ). Most of the objects in PINGS could not be covered in a single pointing with IFS instruments, so a new observing-reduction technique hadtobedevelopedtoperformaccuratemosaickingofthe targets. /T_he spectroscopic mosaicking was acquired during aperiodofthreeyearsandthe/finaldatasetcomprises more than /five.fitted/zero.fitted /zero.fitted /zero.fitted /zero.fitted individual spectra, covering in total an observed area of nearly /eight.fitted/zero.fitted arcmin 2 , and an observed surface withoutprecedentsbyaIFSstudyuptothatpoint(thecase study of NGC /six.fitted/two.fitted/eight.fitted presented in Section /three.fitted is based in the data of this survey). For the second sample, the galaxies were observed over /fi/f_teen nights in several observing runs. /T_he main di/fference is that, for the latter sample, a single pointing strategy using a dithering scheme was applied, while,forthelargestgalaxiesofthePINGSsurvey,amosaic comprising di/fferent pointings was required. /T_his is due to the di/fferences in projected size, considering the di/fferent redshi/f_t range of both samples: the PINGS galaxies correspond to /u1D467 ∼ 0.001 -0.003 , while, for the face-on spirals, it is /u1D467∼ 0.01 -0.025 . /T_herefore, in both survey samples, the data extent corresponds to about ∼ /two.fitted e/ffective radii for all galaxies (/T_he e/ffective radius is classically de/fined as the radius at which one half of the total light of the system is emitted). So the /final sample comprises /three.fitted/eight.fitted objects, with a redshi/f_t range between ∼ /zero.fitted./zero.fitted/zero.fitted/one.fitted and /zero.fitted./zero.fitted/two.fitted/five.fitted. Although this sample is by no means a statistical subset of the galaxies in the local Universe, it is a representative sample of face-on, mostly quiescent, and spiral galaxies at the considered redshi/f_t range (see Figure /one.fitted). Data reduction was performed using R/three.fittedD [/three.fitted/one.fitted], obtaining asanoutputadatacubeforeachgalaxy,witha/finalspatial sampling between /one.fitted-/two.fitted arcsec/pixel, which translates to a linear physical size between a few hundreds of parsecs to ∼ /one.fittedkpc. Usingthisdatabasewecataloguedmorethan ≈ /two.fitted/five.fitted/zero.fitted/zero.fitted HII regions with good spectroscopic quality in all /three.fitted/eight.fitted galaxies, representing one of the largest and more homogeneous /two.fittedD spectroscopic HII region surveys ever accomplished. NGC 628 N 0 /uni0394 RA (arcsec) (a) /T_he discussion presented in Sections /four.fitted and /five.fitted isbasedonthese databases. /T_he primary scienti/fic objectives of these surveys were to use the /two.fittedD IFS observations to study the small and intermediate scale variation in the line emission and stellar continuum by means of pixel-resolved maps across the discs of nearby galaxies, as described in the following sections.", "pages": [ 2, 3, 4 ] }, { "title": "3. NGC 628: A Case Study of IFS-Based Nebular Emission Studies", "content": "NGC/six.fitted/two.fitted/eight.fitted (or M /seven.fitted/four.fitted) is the largest galaxy in projected angular size ( ∼ /one.fitted/zero.fitted./five.fitted × /nine.fitted. /five.fitted arcmin 2 , /u1D467∼ /zero.fitted./zero.fitted/zero.fitted/two.fitted/one.fitted/nine.fitted ∼ /nine.fittedMpc) of the PINGSsample.DuetothelargesizeofNGC/six.fitted/two.fitted/eight.fittedcompared to the FoV of the PPAK instrument (/seven.fitted/two.fitted × /six.fitted/four.fitted arcsec 2 )a mosaicking scheme was adopted, employing /three.fitted/four.fitted di/fferent pointings. /T_he initial pointing was centered on the bulge of the galaxy. Consecutive pointings followed a concentric ringshaped pattern, adjusted to the shape of the PPAK bundle (see Figure /one.fitted). /T_he observations of this galaxy spanned a period of threeyears./T_heareacoveredbyalltheobservedpositions accounts approximately for /three.fitted/four.fitted arcmin 2 , making this galaxy the widest spectroscopic survey ever made on a single nearby galaxy. /T_he spectroscopic mosaic contains /one.fitted/one.fitted/zero.fitted/nine.fitted/four.fitted individual spectra. With such dimensions, this galaxy allows us to study the /two.fittedD metallicity structure of the disc, the second order properties of its abundance distribution, and-as a very important byproduct-a complete /two.fittedD picture of the underlying stellar populations of the galaxy. Note that the linear physical scale thatasinglePPAK/fibresamplesattheassumeddistance of the galaxy is ∼ /one.fitted/two.fitted/zero.fitted pc. /T_his scale can be compared to the physical diameter of a well-known HII region in our Galaxy, that is, the Orion nebula ( /u1D437∼ /eight.fitted pc), or to the extent of what is considered prototypes of extragalactic giant HII regions, such as /three.fitted/zero.fitted Doradus ( /u1D437∼ /two.fitted/zero.fitted/zero.fitted pc) or NGC /six.fitted/zero.fitted/four.fitted ( /u1D437∼ /four.fitted/six.fitted/zero.fitted pc). /T_he area sampled by an individual /fibre in the mosaic would subtend a fraction of a typical giant HII region in NGC /six.fitted/two.fitted/eight.fitted, but the same area would fully encompass a number of small andmediumsizeHIIregionsofthegalaxy(seeFigure /two.fitted). /T_he IFS analysis of NGC /six.fitted/two.fitted/eight.fitted was taken as a case study in order to explore di/fferent spectra extraction and analysis methodologies, taking into account the signal-to-noise of the data, the /two.fittedD spatial coverage, the physical meaning of the derived results, and the /final number of analysed spectra. /T_he analysis performed on this object represents an example of the potential and extent of studies based on IFS on nearby galaxies. In the /first paper of the series ([/two.fitted/nine.fitted], herea/f_ter Paper I), we present a study of the line emission and stellar continuum of NGC /six.fitted/two.fitted/eight.fitted by means of pixel-resolved maps across the disc of the galaxy. /T_his study includes a qualitative description of the /two.fittedD distribution of the physical properties inferred from the line intensity maps and a comparison of these properties with both the integrated spectrum of the galaxyandthespatiallyresolvedspectra.Inthesecondarticle ([/three.fitted/zero.fitted], herea/f_ter Paper II), we present a detailed, spatially c) e s c r c (a e D /uni0394 200 100 0 -100 -200 E 100 -100 -200 c) e s c r (a /uni0394/u1D6FF resolved spectroscopic abundance analysis, based on di/fferent spectral samples extracted from the area covered by the IFS observations of NGC /six.fitted/two.fitted/eight.fitted, and we de/fine a spectra selection methodology specially conceived for the study of the nebular emission in IFU-based spectroscopic observations. /T_his allows us to derive the gas chemistry distribution across the surface of the galaxy with unprecedented detail. In the third paper of the series (S'anchez-Bl'azquez et al., submitted; herea/f_ter Paper III), we present a stellar population analysis of the galaxy, a/f_ter applying spectral inversion methods to derive /two.fitted -dimensional maps of star-formation histories and chemical enrichment. In Paper I, spatially resolved maps of the emission line intensities and physical properties were derived for NGC /six.fitted/two.fitted/eight.fitted. Contrary to previous attempts to perform a /two.fittedD wide-/field analysis based on narrow-band (or Fabry-Perot) imaging, which only allowed a basic analysis of the physical parameters and/or required assumptions on the line ratios included within individual /filters (e.g., H /u1D6FC ), the emission line maps presented in this paper were constructed from individual (deblended) emission lines at any discrete spatial location of the galaxy, where enough signal-to-noise was found. /T_his fact allowed investigating the point-to-point variation of the physical properties over a considerable area on the galaxy. Extinction, ionization, and metallicity-sensitive indicator maps were derived from reddening corrected emission line maps. In general, they show that the ionized gas in these spiral galaxies exhibits a complex structure, morphologically associated with the star-forming regions located along the spiral arms. /T_he (thermal) ionization is stronger along the spiral arms, associated with the HII regions, and more intense in the outer than in the inner ones. Indeed, the surface SFR is an order of magnitude stronger in the outer HII regions, at distance larger than ∼ /one.fitted/zero.fitted/zero.fitted arcsec (/four.fitted./five.fitted kpc), than in the inner ones. Considering that in these outer regions there is a lower mass density, the growing rate of stellar mass is considerably larger there than in the inner ones. /T_herefore, the growth of the galaxy is dominated by the inside-out process. /T_he spatially resolved distribution of the abundance shows a clear gradient of higher oxygen metallicity values from the inner part to the outer part of the galaxy, and along the spiral arms (see right-panel of Figure /two.fitted). However, in some instances, the value of the oxygen abundance (and other physical properties like extinction and the ionization parameter) varies within what would be considered a classical well-de/fined HII region (or HII complex), showing some level of structure. Indeed, the /two.fittedD character of the data allows us to study the small-scale variation of the spectra within a given emitting area. /T_he values of the emission line ratios measured using di/fferent extraction apertures vary considerably as a functionoftheaperturesize,andthescatterofthecentral valueislargerthanthestatisticalerrorinthemeasurements, re/flecting that this might in fact be a physical e/ffect. By constructing /two.fittedD maps of the oxygen abundance distributions, wefound that the /two.fittedD metallicity structure of the galaxy varies depending on the metallicity calibrator employed in order to derive the oxygen abundance. Di/fferent calibrators /find regions of enhanced log(O/H) at spatial positions which are not coincident among them. /T_his implies that the use of di/fferent empirical calibrations does not only re/flect in a linear scale o/ffset but may introduce spurious inhomogeneities. /T_his information is usually lost in a simple radial abundance gradient,andthatmightberelevantwhenconstructinga chemical evolution model based on a particular abundance determination (see Figure /three.fitted). /T_he emission line maps presented in Paper I proved to be useful in describing the general /two.fittedD properties of the galaxy. More robust conclusions were presented in Paper II, where we analysed speci/fic individual regions across the disc of the galaxy, either by taking individual spectra above as a certain S/N threshold, or by coadding spectra with the same physical properties and comparing the results in the /two.fittedD context. With the/firstmethodwewereabletoidentifyregionsofinterstellar di/ffuse emission (see le/f_t panel of Figure /three.fitted), while with the second we created a classic catalogue of HII regions from a purely geometrical principle, that is, by coadding /fibres considered to belong to the same morphological region. Some highlights of this study (which also apply to the rest of the PINGS galaxies analysed so far) are the following. galaxy correlates with the characteristic gas-phase abundance measured at /u1D70C ∼ 0.4/u1D70C 25 . In the case of the stellar populations, in Paper III we derive maps of the mean (luminosity and mass weighted) age and metallicity that reveal a negative age gradient and thepresenceofstructuressuchasanuclearring,previously seen in molecular gas (see Figure /five.fitted). /T_he disc is dominated in mass by an old stellar component at all radii sampled by the IFS data, while the percentage of young stars increases with radius, as predicted in an inside-out formation scenario, where outer parts of the disc formed later due to the increasing timescales for gas infall with radius. We also detect an inversion of the metallicity gradient at the very centre of the galaxy ( ∼ /one.fitted kpc), where apparently there exists a ring of old stars at this distance, with a trend to younger ones at the very center. Similar results are found in the Milky Way (MW) using Open Clusters and Cepheids, that is, a clear bimodal gradient for the older population, with a /flat outer plateau, and a more continuous gradient for the younger population (e.g., [/three.fitted/three.fitted-/three.fitted/six.fitted]). /T_his behaviour has also been reported in other galaxies, mostly Sa/S/zero.fitted, where the inner regions of their bulges present bluer colors, consistent with younger stellar populations (e.g., [/three.fitted/seven.fitted]). /T_he relevance of this study regarding the nebular emission is that the young component shows a metallicity gradient thatisverysimilartothatofthegas,andthatis/flatter than that of the old stars. Although the metallicity gradients for the young stars and the gas also show a break, this is much less prominent than for the old stars. /T_he position of the break is more coincident with the corotation radius of the oval distortion than that of the spiral pattern, which is beyond the radius sampled by our data. We speculate about thepossibleorigenofthisbreak,thepossibilitiesbeingdue to star-formation variation with the spiral pattern speed or that is due to radial mixing produced by either the spiral arms, the oval distortion, or a coupling of both. We argue that NGC /six.fitted/two.fitted/eight.fitted could represent a good example of secular evolutionduetothepresenceofadissolvingbar.Inthis scenario, the strong bar has funneled large amounts of gas into the central regions while radial /flows induced in the disc have /flattened the O/H gradient. Nuclear starbursts resulting from the gas sinking into the center contributed to the bulge's growth until enough mass was accreted to dissolve the bar by dynamical instabilities. /T_he oval distortion observed in the central region could be the remains of the bar. Forthcoming studies analysing a sample of galaxies with di/fferent masses and showing di/fferent morphological features (e.g., bars of di/fferent strength, spiral arms with di/fferent morphologies, etc.) using, for example, the CALIFA survey that will help to elucidate the importance of the di/fferent mechanisms producing radial mixing in the galaxy discs.", "pages": [ 4, 5, 6, 7 ] }, { "title": "4. Hints of a Universal Abundance Gradient", "content": "IFS o/ffers the possibility to analyse and study a single object in great detail, such as the case of NGC /six.fitted/two.fitted/eight.fitted described above. However, it also o/ffers the unique chance of studying the spectroscopic properties of thousands of HII regions in a homogeneous way. We used our catalogue of HII regions introduced in Section /two.fitted to characterize the radial trends and the physical properties of the HII regions of the galaxy sample. However, contrary to the case of NGC /six.fitted/two.fitted/eight.fitted where the HII regions on the disc of the galaxy were basically selected and extracted by-hand, the HII regions in these galaxies were detected, spatially segregated, and spectrally extracted using HIIexplorer [/three.fitted/nine.fitted], a new automatic procedure to detect HII regions, based on the contrast of the H /u1D6FC intensity maps extracted from the data cubes. Once detected, the algorithm provides with the integrated spectra of each individual segmented region. /T_his change of paradigm is totally necessary when working with thousands of HII regions, contrary to the case of a handful of targets in classic long-slit spectroscopy. We detected a total of /two.fitted/five.fitted/seven.fitted/three.fitted HII regions with good spectroscopic quality. /T_his is by far the largest spatially resolved, nearby spectroscopic HII region survey ever accomplished. /T_he emission lines were decoupled from the underlying stellar population using FIT/three.fittedD [/four.fitted/zero.fitted], following a robust and well-tested methodology [/two.fitted/zero.fitted, /two.fitted/nine.fitted]. Extinction-corrected, /flux intensities of the stronger emission lines were obtained and used to select only star-forming regions based on typical BPT diagnostic diagrams. /T_he /final sample comprises /one.fitted/eight.fitted/nine.fitted/six.fitted high-quality, spatially resolved HII regions/aggregations of disc galaxies in the local Universe [/three.fitted/nine.fitted]. It is well known that di/fferent spectroscopic properties of HII regions show strong variations across the area of disc galaxies. In particular, some of these parameters (e.g., oxygen abundance, EW[H /u1D6FC ], etc.), show a strong radial gradient, that in average indicates that more evolved, metal rich, stellar populations are located in the center of galaxies, and less evolved, metal poor ones are in the outer ones. Despite the several di/fferent studies describing these observational events, there is a large degree of discrepancy between the actual derived parameters describing the gradients: (i) slope of the gradient, (ii) average value and dispersion of the zeropoint, and (iii) scale length of the truncation. In general, this is mostly due to di/fferent observational biases and the lack of a proper statistical number of analysed HII regions per galaxy. For each galaxy of our sample we derived the correlation coe/fficient, the slope, and the zero point of a linear regression for a number of parameters showing radial distributions across the discs of the galaxies. For those properties showing a strong correlation, we investigated if the gradient was universal within our range of explored parameters. We found that, for the equivalent width of H /u1D6FC and the oxygen abundance,theslopesofthegradientsareconsistentwitha Gaussian distribution; that is, the dispersion of values found for each individual galaxy is compatible with the average one, not showing strong statistical deviations. /T_his implies that we can de/fine a characteristic value for the slope and that we do not /find a population of galaxies with slopes inconsistent with this normal distribution. /T_he right panel of Figure /six.fitted shows the radial density distribution for the oxygen abundance derived using the O/three.fittedN/two.fitted indicator [/two.fitted/eight.fitted], once scaled to the average value at the e/ffective radius for each galaxy. /T_he radial distance was normalised to the e/ffective radius of each galaxy. /T_hesolidlineshowstheaveragelinearregressionfoundfor eachindividualgalaxy./T_hered-dashedlineshowstheactual regression found for all the HII regions detected for all the galaxies. Our results seem to indicate that there is a universal radial gradient for oxygen abundance and the equivalent width of H /u1D6FC when normalized with the e/ffective radii of the galaxies; that is, they present a radial gradient that, statistically, has the same slope for all the galaxies in our sample. /T_he derived slopes for each galaxy are compatible with a Gaussian random distribution and are independent of the morphology of the analysed galaxies (barred/nonbarred, grand-design//flocculent). /T_his is one of the most important results in the abundance gradients of spiral galaxies, obtained thanks to the use of IFS.", "pages": [ 7, 8 ] }, { "title": "5. The Local Mass-Metallicity Relation", "content": "/T_he existence of a strong correlation between stellar mass and gas-phase metallicity in galaxies is a well-known fact. /T_he mass-metallicity ( M -Z) relation is consistent with more massive galaxies being more metal-enriched; a/f_ter the seminal work on this relationship by Lequeux et al. [/four.fitted/one.fitted], it was /firmly established observationally by Tremonti et al. ([/four.fitted/two.fitted], herea/f_ter T/zero.fitted/four.fitted) using the SDSS. However, there has been no Average linear /fit (all galaxies) -0.12 dex/ R Linear /fit to all HII regions -0.11 dex/ R Mean value at radial bins ∼0.15 R e major e/ffort to test the M -Z relation using spatially resolved information. We used our IFS observations in order to test the distribution of mass and metals within the discs of the galaxies. We derived the (luminosity) surface mass density ( Σ Lum , /u1D440 /uni2299 pc -2 ) within the area encompassed by our IFSsegmented HII regions, using the prescriptions given by Bell and de Jong [/four.fitted/three.fitted]toconvert /u1D435 -/u1D449 colors into a /u1D435 -band massto-light ratio ( /u1D440//u1D43F ). /T_he le/f_t panel of Figure /seven.fitted shows the striking correlation between the local surface mass density and gas metallicity for our sample of nearby HII regions, that is, the local M -Z relation, extending over ∼ /three.fitted orders of magnitude in Σ Lum and a factor ∼ /eight.fittedinmetallicity[/three.fitted/eight.fitted]. /T_he notable similarity with the global M -Z relation can be visually recognised with the aid of the blue lines which stand for the [/four.fitted/two.fitted]/fit( ± /zero.fitted./two.fitted dex) to the global M -Z relation, shi/f_ted arbitrarily both in mass and metallicity to coincide with the peak of the HII region M -Z distribution. Other abundance calibrations were tested obtaining the same shape (and similar /fit) of the relation. In addition, we /find the existence of a more general relation between mass surface density, metallicity, and the equivalent width of H /u1D6FC , de/fined as the emission-line luminosity normalized to the adjacent continuum /flux, that is, a measure of the SFR per unit luminosity [/four.fitted/four.fitted]. /T_his functional relation is evident in a /three.fittedD space with orthogonal coordinate axes de/fined by these parameters, consistent with | EW ( H /u1D6FC)| being e e inversely proportional to both Σ Lum andmetallicity,asshown in Figure /eight.fitted.AsdiscussedinRosales-Ortegaetal.[/three.fitted/eight.fitted], we interpret the local M -Z-EW(H /u1D6FC ) relation as the combination of (i) the well-known relationships between both the mass and metallicity with respect to the di/fferential distributions of these parameters found in typical disc galaxies, that is, the inside-out growth, and (ii) the fact that more massive regions form stars faster (i.e., at higher SFRs), thus earlier in cosmological times. In order to test whether the global M -Z relation observed by [/four.fitted/two.fitted]usingSDSSdataisare/flection(aperturee/ffect)ofthe local HII region mass-density versus metallicity relation, we perform the following exercise. We simulate a galaxy with typical /u1D440 /u1D435 and /u1D435 -/u1D449 values drawn from /flat distributions in magnitude ( -/one.fitted/five.fitted to -/two.fitted/three.fitted) and colour ( ∼ /zero.fitted./four.fitted -/one.fitted). A redshi/f_t isassumedforthemockgalaxy,drawnfromaGaussian distribution with mean ∼ /zero.fitted./one.fitted and /u1D70E = 0.05 ,witharedshi/f_t cut 0.02 < /u1D467 < 0.3 in order to resemble the SDSS [/four.fitted/two.fitted] distribution. /T_he mass of the galaxy is derived using the integrated /u1D435 -band magnitudes, /u1D435 -/u1D449 colours, and the average /u1D440//u1D43F ratio following Bell and de Jong [/four.fitted/three.fitted]. /T_he metallicity ofthemockgalaxyisderivedusingthelocal M -Z relation within an aperture equal to the SDSS /fiber (/three.fitted arcsec), that is, the metallicity that corresponds to the mass density surface at this radius. /T_he process is repeated over /one.fitted/zero.fitted,/zero.fitted/zero.fitted/zero.fitted times in order to obtain a reliable distribution in the mass and metallicity of the mock galaxies. /T_he right panel of Figure /eight.fitted shows the result of the simulation, that is, the distribution of the mock galaxies in the M -Z parameter space. We reproduce-with a remarkable agreement-the overall shape of the global M -Z relation assuming a local M -Z relation and considering the aperture e/ffect of the SDSS /fiber. /T_he overlaid lines correspond to the [/four.fitted/two.fitted] /fit (black) and the Kewley and Ellison [/four.fitted/five.fitted] ± /zero.fitted./two.fitted dex relation (blue), for which the agreement is extremely good over a wide range of masses. /T_he result is remarkable considering that we are able to reproduce the global M -Z relation over a huge dynamical range, using a local M -Z relation derived from a galaxy sample with a restricted range in mass (/nine.fitted./two.fitted < log /u1D440 Lum < /one.fitted/one.fitted./two.fitted) and metallicity (/eight.fitted./three.fitted < /one.fitted/two.fitted + log(O/H) < /eight.fitted./nine.fitted),indicatedbytherectangleshownintheright panel of Figure /eight.fitted. /T_herefore, by using the power of IFS applied to a sample of nearby galaxies we demonstrate the existence of a local relation between the surface mass density, gas-phase oxygen abundance, and | EW ( H /u1D6FC)| in ∼ /two.fitted/zero.fitted/zero.fitted/zero.fitted spatially resolved HII regions of the Local Universe. /T_he projection of this distribution in the metallicity versus Σ Lum plane-the local M -Z relation-shows a tight correlation expanding over a wide rangeinthisparameterspace.Weusethelocal M -Z relation to reproduce the global M -Z relation by means of a simple simulation which considers the aperture e/ffects of the SDSS /fiber at di/fferent redshi/f_ts. Note that the 'local' M -Z-| EW ( H /u1D6FC)| relation is conceptually di/fferent from the 'global' M -Z-SFR relation proposed by Lara-L'opez et al. ([/four.fitted/six.fitted], dubbed FP), Mannucci et al. ([/four.fitted/seven.fitted], dubbed FMR), or Hunt et al. [/four.fitted/eight.fitted], based on the integrated spectra of galaxies (the basic di/fference between these relations is the proposed shape in the /three.fittedD distribution, that is, a surface or a plane). However, the obvious parallelism between these two scaling relations deserves a discussion. While the 'local' M -Z-| EW ( H /u1D6FC)| relation is related to the intrinsic physics involved in the growth of the galaxy disc in an insideout scenario, the existence of the 'global' M -Z-SFRrelation is explained, according to Mannucci et al. [/four.fitted/seven.fitted], by the interplay of infall of pristine gas and out/flow of enriched material at di/fferent redshi/f_ts epochs, supporting the smooth accretion scenario, where galaxy growth is dominated by continuous accretion of cold gas in the local Universe. However, S'anchez et al. [/four.fitted/nine.fitted], using CALIFA data, found no secondary relation of the mass and metallicity with the SFR other than the one induced by the primary relation of this quantity with the stellar mass. /T_he same was found with respect to the speci/fic SFR rate. /T_he results by S'anchez et al. [/four.fitted/nine.fitted]agreewitha scenario in which gas recycling in galaxies, both locally and globally, is much faster than other typical timescales, such like that of gas accretion by in/flow and/or metal loss due to out/flows. In essence, late-type/disc-dominated galaxies seem to be in a quasi-steady situation, with a behavior similar to the one expected from an instantaneous recycling/closed-box model. In this scenario, the inner regions of the galaxy form /first and faster, increasing the gas metallicity of the surrounding interstellar medium. As the galaxy evolves and grows with time, the star-formation progresses radially creating a radial metallicity gradients in the disk of spirals. Mass is progressively accumulated at the inner regions of the galaxy, raising the surface mass density and creating a bulge, with corresponding high metallicity values but low SSFR (low | EW ( H /u1D6FC)| ), that is, an 'inside-out' galaxy disk growth. In such a case, the local M -Z relation would re/flect a more fundamental relation between mass, metallicity, and starformation e/fficiency as a function of radius, equivalent to alocal downsizing e/ffect, similar to the one observed in individual galaxies. Following this reasoning, the origin of the global M -Z relation can be explained as the combined e/ffect of the existence of the local M -Zrelation,anaperturebiasdue to the di/fferent /fibers covering factors of the spectroscopic surveys from which the FMR and FP were derived (as secondorder e/ffect), and a possible selection of a bias of the galaxy populations which are most common at a particular redshi/f_t, and may not re/flect the physics of how galaxies evolve. Supporting evidence in favour of the inside-out scenario of galaxy growth comes from the recent analysis of the spatially resolved history of the stellar mass assembly in galaxies of the local Universe [/five.fitted/zero.fitted]. In summary, the existence of the M-ZSFR relation could also be interpreted as a scaled-up version of the local M-Z-sSFR relation in the distribution of starforming regions across the discs of galaxies as described in Rosales-Ortega et al. [/three.fitted/eight.fitted]andcon/firmedbyS'anchez et al. [/four.fitted/nine.fitted]; that is, the relationship is not primary, but obtained from the sum of a number of local linear relations (and their deviations) with respect to the galaxy radius.", "pages": [ 8, 9, 10, 11 ] }, { "title": "6. Conclusions", "content": "/T_he emergence of a new generation of instrumentation, that is, multiobject and integral /field spectrometers with large /fields of view, capable of performing emission-line surveys based on samples of hundreds of spectra in a /two.fittedD context, arerevolutionisingthemethodsandtechniquesusedtostudy the gas-phase component of star-forming galaxies in the nearby Universe (objects which were typically studied with smallsamplesbasedonlong-slitspectroscopy).Anewbody of results is coming out from these studies, opening up a new frontier of studying the /two.fittedD structure and intrinsic dispersion of the physical and chemical properties of the discs of nearby spiral galaxies. In this paper we review some of the projects that in the last years tackled for the /first time the problem of obtaining spatially resolved spectroscopic information of the gas in nearby galaxies. PINGS represented the /first endeavour to obtain full /two.fittedD coverage of the discs of a sample of spiral galaxies in the nearby Universe. /T_he PINGS sample covered di/fferent galaxy types, including normal, lopsided, interacting, and barred spirals with a good range of galactic properties and star-forming environments, with multiwavelength public data. /T_he spectroscopic data set comprises more than /five.fitted/zero.fitted /zero.fitted/zero.fitted/zero.fitted individual spectra, covering an observed area of nearly /eight.fitted/zero.fitted arcmin 2 ,anobservedsurface without precedents by an IFS study by the time. /T_heIFSanalysisofNGC/six.fitted/two.fitted/eight.fitted,thelargestspectroscopic mosaic on a single galaxy, was taken as an example of the new methodology and analysis that could be performed with alargespectroscopicdatabaseforasingleobject./T_hecon-tribution of PINGS also resides in de/fining a self-consistent methodology in terms of observation, data reduction and analysis for present and future IFS surveys of the kind, as well as a whole new set of visualization and analysis so/f_tware thathasbeenmadepublictothecommunity(e.g.,[/five.fitted/one.fitted, /five.fitted/two.fitted]). Despite all the complexities involved in the observations, data reduction, and analysis, PINGS proved to be feasible. In less than a three-year period, it was possible to build a comprehensive sample of galaxies with a good range of galactic properties and available multiwavelength ancillary data, maximising both the original science goals of the project and the possible archival value of the survey. In fact, the science case of the PINGS project was the inspiration for the ongoing CALIFA survey. /T_he face-on spirals from M'armolQueralt'oetal.[/two.fitted/one.fitted] were part of the feasibility studies for the CALIFA survey. On completion, CALIFA will be the largest and the most comprehensive wide-/field IFU survey of galaxies carried out to date. It will thus provide an invaluable bridge between large single aperture surveys and more detailed studies of individual galaxies. /six.fitted./one.fitted. Results from Other IFU Projects on Star-Forming Galaxies. Other projects have followed this initiative; for example, the Mitchell spectrograph instrument at McDonald Observatory (a.k.a VIRUS-P) is currently used to carry out two small IFS surveys, namely, VENGA [/five.fitted/three.fitted] and VIXENS [/five.fitted/four.fitted]. VENGA (VIRUS-P Exploration of Nearby Galaxies) is an integral /field spectroscopic survey, which maps the disks of /three.fitted/zero.fitted nearby spiral galaxies, in a very similar manner to PINGS in terms of spectral coverage, resolution, and area sampled (/three.fitted/six.fitted/zero.fitted/zero.fitted ˚ A-/six.fitted/eight.fitted/zero.fitted/zero.fitted ˚ A, ∼ /five.fitted ˚ A FWHM, ∼ /zero.fitted./seven.fitted R 25 )althoughwitha di/fferent spatial resolution (/five.fitted./six.fitted arcsec FWHM). /T_heir targets span a wide range in Hubble type, star-formation activity, morphology, and inclination. Likewise PINGS, the VENGA group used the data cubes of their observations to produce /two.fittedD maps of the star-formation rate, dust extinction, electron density, stellar population parameters, the kinematics and chemical abundance of both stars and ionized gas, and other physical quantities derived from the /fitting of the stellar spectrum and the measurement of nebular emission lines. /T_heir /first results focus on (/one.fitted) the spatially resolved starformationlawofNGC/five.fitted/one.fitted/nine.fitted/four.fittedwheretheygivesupporttothe evidence for a low, and close to constant, star-formation e/fficiency (SFE = /u1D70F -1 ) in the molecular component of the interstellar medium [/five.fitted/five.fitted] and (/two.fitted) using IFS observations of NGC/six.fitted/two.fitted/eight.fitted,theymeasuretheradialpro/fileofthe 12 CO(/one.fitted/zero.fitted) to H 2 conversion factor ( /u1D44B C /u1D442 )inthisgalaxyandstudy howchangesin /u1D44B C /u1D442 follow changes in metallicity, gas density, and ionization parameter [/five.fitted/six.fitted],andalsotheyusetheIFS data to propose a new method to measure the inclination of nearlyface-onsystemsbasedonthematchingofthestellar and gas rotation curves using asymmetric dri/f_t corrections [/five.fitted/three.fitted]. In the case of VIXENS (VIRUS-P Investigation of the eXtreme ENvironments of Starburst), their goal of our survey is to investigate the relation between star-formation and gas content in the extreme environments of interacting galaxy pairs and mergers on spatially resolved scales of 0.2 -0.8 kpc, by using IFS of /one.fitted/five.fitted interacting/starburst galaxies. VIXENS will make extensive use of multiwavelength data in order to investigate the star-formation in this object, including data from Spitzer, GALEX, IRAM, CARMA archival CO, and Hi maps. /T_hese projects and datasets are clearly focused on speci/fic science questions, adopting correspondingly optimized sample selection criteria and also observing strategies. Other surveys in the local Universe using the power of IFS for a detailed study of nearby galaxies include the next generation surveys like Sydney university AAO MOS IFU [/five.fitted/seven.fitted] (SAMI) and Mapping Nearby Galaxies at APO, PI: Kevin Bundy, IPMU (MaNGA), or the new generation instrumentation for Very Large Telescope (VLT, ESO) like Multi Unit Spectroscopic Explorer [/five.fitted/eight.fitted] (MUSE), which aim at studying the the chemical and dynamical evolution history and dark matter contents of galaxies, the physical role of environment in galaxy evolution, when, where, and why does star-formation occur, and so forth, based on spatially resolved spectroscopic surveys of 10 4 -10 5 galaxies. /T_he continuous coverage spectra provided by the imaging spectroscopy technique employed in these projects are already allowing us to study the small and intermediate linear scale variation in line emission and the gas chemistry for a statistically representative number of galaxies of the nearby Universe. /T_he primary motivation common to all of these observational e/fforts is to use this information to link the properties of high redshi/f_t galaxies with those we see around us today and thereby understand the physical processes at play in the formation and evolution of galaxies. /T_he power and importance of all these projects resides in the fact that they will provide an observational anchor of the spatially resolved properties of the galaxies in the local Universe, which will have a potential impact in the interpretation of observed properties at high redshi/f_t from new generation facilities, suchastheJamesWebbSpaceTelescope(JWST),theGiant MagellanTelescope(GMT),ortheEuropeanExtremelyLarge Telescope (E-ELT), projects that will hopefully revolutionise the understanding of our Universe in future years.", "pages": [ 11, 12 ] }, { "title": "Acknowledgments", "content": "BasedonobservationscollectedattheCentroAstron'omico Hispano-Alem'an (CAHA) at Calar Alto, operated jointly by the Max-Planck Institut fur Astronomie and the Instituto de Astrof'ısica de Andaluc'ıa (CSIC) Fernando Fabi'an RosalesOrtega acknowledges the Mexican National Council for Science and Technology (CONACYT) for /financial support under the Programme Estancias Posdoctorales y Sab'aticas al Extranjero para la Consolidaci'on de Grupos de Investigaci'on, /two.fitted/zero.fitted/one.fitted/zero.fitted-/two.fitted/zero.fitted/one.fitted/two.fitted. /T_he author also acknowledges /financial support for the ESTALLIDOS collaboration by the Spanish Ministerio de Ciencia e Innovaci'on under Grant AYA/two.fitted/zero.fitted/one.fitted/zero.fitted-/two.fitted/one.fitted/eight.fitted/eight.fitted/seven.fitted-C/zero.fitted/four.fitted/zero.fitted/three.fitted.", "pages": [ 12 ] }, { "title": "References", "content": "Hindawi Publishing Corporation http://www.hindawi.com Hindawi Publishing Corporation http://www.hindawi.com Volume 2013 Volume 2013", "pages": [ 15 ] }, { "title": "Hindawi", "content": "Submit your manuscripts at http://www.hindawi.com Hindawi Publishing Corporation http://www.hindawi.com Volume 2013 Hindawi Publishing Corporation http://www.hindawi.com Volume 2013", "pages": [ 15 ] } ]

[ { "title": "Unbiased acceleration measurements with an electrostatic accelerometer on a rotating platform", "content": "Benjamin Lenoir a , Bruno Christophe a , Serge Reynaud b a Onera - The French Aerospace Lab, 29 avenue de la Division Leclerc, F-92322 Châtillon, France b Laboratoire Kastler Brossel (LKB), ENS, UPMC, CNRS, Campus Jussieu, F-75252 Paris Cedex 05, France Published in Advances in Space Research 51 (2012) 188-197 doi: 10.1016/j.asr.2012.08.012 6 January 2013", "pages": [ 1 ] }, { "title": "Abstract", "content": "The Gravity Advanced Package is an instrument composed of an electrostatic accelerometer called MicroSTAR and a rotating platform called Bias Rejection System. It aims at measuring with no bias the non-gravitational acceleration of a spacecraft. It is envisioned to be embarked on an interplanetary spacecraft as a tool to test the laws of gravitation. MicroSTAR is based on Onera's experience and inherits in orbit technology. The addition of the rotating platform is a technological upgrade which allows using an electrostatic accelerometer to make measurements at low frequencies with no bias. To do so, the Bias Rejection System rotates MicroSTAR such that the signal of interest is separated from the bias of the instrument in the frequency domain. Making these unbiased low-frequency measurements requires post-processing the data. The signal processing technique developed for this purpose is the focus of this article. It allows giving the conditions under which the bias is completely removed from the signal of interest. And the precision of the unbiased measurements can be fully characterized: given the characteristics of the subsystems, it is possible to reach a precision of 1 pm s -2 on the non-gravitational acceleration for an integration time of 3 h. Keywords Electrostatic accelerometer; Rotating platform; Bias rejection; Absolute measurement; Modulation PACS 02.50.Ey; 04.80.Cc; 06.30.Gv; 07.87.+v", "pages": [ 1 ] }, { "title": "1 Introduction", "content": "The experimental tests of gravitation are in good agreement with its current theoretical formulation referred to as General Relativity (Will, 2006). But contrary to the quantum description of the three other fundamental interactions, it is a classical theory, which suggests that another description of gravitation lies beyond General Relativity. From the experimental point of view, there are still open windows for deviations from General Relativity at short range (Adelberger et al., 2003) and at long range (Jaekel and Reynaud, 2005). Galactic and cosmic observations also challenge General Relativity. The rotation curves of galaxies and the relation between redshifts and luminosities of supernovae, which are interpreted as manifestations of 'dark matter' and 'dark energy' respectively (Frieman et al., 2008; Bertone et al., 2005), may also be seen as a hint that General Relativity could be an imperfect description of description at these large scales (Aguirre et al., 2001; Nojiri and Odintsov, 2007). In this context, testing General Relativity at the largest possible scales is essential. For man-made instruments, the Solar System can be used as a laboratory for gravitational experiments. NASA performed such a test of gravitation with the Pioneer 10 and 11 missions. The outcome was a signal now known as the Pioneer anomaly (Anderson et al., 1998, 2002b; Turyshev et al., 2011). The long term variations of this signal may be explained as an anisotropic thermal effect (Bertolami et al., 2010; Rievers et al., 2010; Rievers and Lämmerzahl, 2011; Turyshev et al., 2012) but periodic anomalies have also been identified (Lévy et al., 2009; Courty et al., 2010). Nevertheless, the Roadmap for Fundamental Physics in Space issued by ESA in 2010 (Fundamental Physics Roadmap Advisory Team, 2010) stresses the importance of testing gravitation at large scales with missions to the outer planets. To do so, it recommends the development of accelerometers compatible with spacecraft tracking at the 10 pm s -2 level. Several missions have already been proposed (Anderson et al., 2002a; Dittus et al., 2005; Johann et al., 2008; Bertolami and Paramos, 2007; Christophe et al., 2009; Wolf et al., 2009) with the aim of improving the knowledge of the gravitational field in the Solar System. Many of them propose to embark an accelerometer which will measure the non-gravitational forces acting on the spacecraft in order to distinguish unambiguously the non-gravitational accelerations from gravitational effects. In practice, accelerometers measure a combination of the spacecraft non-gravitational acceleration and additional terms (Carbone et al., 2005, 2007). It will be assumed in this article that these additional terms either are negligible or can be corrected, such that the external signal will be referred to as the spacecraft non-gravitational acceleration (Lenoir et al., 2011b). This article deals with the Gravity Advanced Package, an instrument proposed on Laplace mission (Biesbroek, 2008), which is designed to make measurements of the non-gravitational acceleration of a spacecraft with no bias. It provides an additional observable which measures the departure of the spacecraft from geodesic motion. To do so, post-processing is required, which is the subject of this article. It allows retrieving separately the acceleration without bias and the bias of the instrument, these quantities being referred to as 'post-processed' quantities. In a first part, the Gravity Advanced Package will be presented as well as its performances and the measurement principle. Then the post-processing method will be described and conditions will be given so that the bias can be effectively removed from the measurement. These conditions will allow deriving measurement procedures to make unbiased measurements. The emphasis will then be put on the characterization of the post-processed quantities, i.e. the quantities without bias, and on their uncertainty. This paper is focused on the performances of the accelerometer. The constraints in the integration of the instrument in the spacecraft with the aim of preserving these performances requires additional work. The OSS mission (Christophe et al., 2012) proposes a spacecraft design which takes into account these concerns.", "pages": [ 1, 2 ] }, { "title": "2 Instrument principle, design and performance", "content": "The Gravity Advanced Package is made of an electrostatic accelerometer, called MicroSTAR, which can be rotated with the Bias Rejection System. This technological upgrade allows removing the bias introduced by MicroSTAR.", "pages": [ 2 ] }, { "title": "2.1 Overview of the instrument", "content": "MicroSTAR is a 3-axes electrostatic accelerometer (Josselin et al., 1999) based on ONERA's expertise in this field (Touboul et al., 1999; Hudson et al., 2007; Touboul and Rodrigues, 2001). In orbit technology (CHAMP, GRACE and GOCE missions) is used with improvements to reduce power consumption, size and mass. The core of the accelerometer is composed of a proof mass inside a cage made of six identical plates. The motion of the proof mass with respect to the cage is detected by capacitive measurement. A control loop adjusts the potentials of the electrodes in order to keep the proof mass at the center of the cage. The numerical values of these potentials, which are the outputs of the instrument, are proportional to the components of the acceleration of the proof mass with respect to the cage on each axis of the accelerometer. The Bias Rejection System is a rotating platform composed of a rotating actuator and a high resolution angular encoder working in closed loop operation. Piezo-electric technology is envisioned for the actuator. It allows designing a device with no need for any kind of gear, so as to reduce mass and volume. The piezo-electric motor is operated in a slip-stick mode. Finally, even if piezo-electric motors have a non-zero torque in the power-off mode, a blocking system will be implemented to prevent unwanted motion during launch and maneuvers.", "pages": [ 2 ] }, { "title": "2.2 Performance of the electrostatic accelerometer", "content": "The performance of MicroSTAR is measured via the power spectrum density of the noise on the measured acceleration (Lenoir et al., 2011b, Fig. 4). The analytic formula S n (in m 2 s -4 Hz -1 ) as a function of frequency, for a measurement range equal to 1 . 8 × 10 -4 m s -2 , is with K = 5 . 7 × 10 -11 m . s -2 . Hz -1 / 2 . In addition, the instrument has a bias. It corresponds to the deterministic low-frequency variations of the accuracy of the accelerometer. It is due to the gold wires which are used to keep the polarization of the proof mass constant and to the geometrical imperfections of the instrument. In previous missions relying on electrostatic accelerometers (CHAMP, GRACE and GOCE), this bias was not a problem since the measurement bandwidth was 0.1-100 mHz. On the contrary, for the application foreseen in this article, very low-frequencies measurements are to be made and it is therefore required to remove the bias from the measurements.", "pages": [ 3 ] }, { "title": "2.3 Measurement principle", "content": "The measurements made by MicroSTAR along its three axes x , y and z , which are supposed to be orthogonal 1 , are where δk 1 κ , k 2 κ , b κ and n κ ( κ ∈ { x ; y ; z } ) are respectively the scale factors, the quadratic factors, the bias and the noise on each axis. The bias has a deterministic temporal variation whereas the noise is a null-mean stationary stochastic process whose PSD is given by Eq. (1). The quantities a κ are the components of the non-gravitational acceleration in the reference frame of MicroSTAR. As far as orbit reconstruction is concerned, these quantities are not the ones of interest since the Bias Rejection System rotates MicroSTAR with respect to the spacecraft. The spacecraft is supposed to be stabilized along the three axes. The transformation matrix P moves a vector from the spacecraft reference frame (whose axes are X , Y and Z ) to the accelerometer reference frame. In its simplest form (but without any loss of generality), the expression of P is where θ is a monitored angle, which measures the rotation of the accelerometer with respect to the spacecraft. Considering only the plane perpendicular to the axis of rotation of the accelerometer, Eq. (2) becomes The measurements on the axes y and z are combinations of the quantities a Y and a Z . These are the quantities needed so as to measure the impact of non-gravitational forces on the trajectory of the spacecraft. This fact associated with the possibility to give the angle θ any possible time variation allows measuring a Y and a Z without bias. On the contrary, on the axis x , there is no possibility with this instrument to remove the bias from the measurement m x so as to retrieve a X . To do so, another rotating platform would be required. It is not the topic of this article but the method developed here can be applied to this more complex setup. In practice, the axes Y and Z will be in the orbit plane, in which non-gravitational forces are expected to impact the trajectory of the spacecraft. In the following, it will be assumed that N measurements are made with a sampling frequency called f s . It corresponds to a time step called δt = 1 /f s . The scale and quadratic factors will be supposed to be constant. In Eq. (4), there are therefore 4 N unknowns ( a Y , a Z , b y , b z at each sampling time) and 2 N measurements ( m y , m z ) spoiled by noise ( n y , n z ). In the rest of this article, for each of these eight quantities as well as for θ , the notation x will be a vector of M N, 1 ( R ) whose components are the values of x at each sampling time and x k is the value of x at the sampling time k × δt .", "pages": [ 3, 4 ] }, { "title": "3 Signal processing method", "content": "The relations between the measurements made by the instrument, m y and m z , and the non-gravitational acceleration in the spacecraft reference frame, a Y and a Z , has been expressed. In this section, the data processing method is presented. In particular, conditions are derived in order to remove the bias from the measurements.", "pages": [ 4 ] }, { "title": "3.1 Linearization of the problem", "content": "A first step is to linearize Eq. (4) such that they can be written in a matrix form. To do so, it is assumed that k 2 y = k 2 z = 0 . This hypothesis will be shown in paragraph 4.1 not to be restrictive in the framework presented here. The two following diagonal matrices, belonging to M N ( R ) , allow writing Eq. (4) in the matrix form with and The set of solutions for this system is infinite. It is the affine space X p + ker( J ) , where X p is a given solution of the linear equation. This formal resolution gives no useful information on the non-gravitational acceleration since it provides an infinite number of solutions.", "pages": [ 4 ] }, { "title": "3.2 Generalized noise", "content": "Before going further, it is necessary to consider the matrix J more carefully. When solving Eq. (6), J is supposed to be perfectly known. It is however not the case since the knowledge of the angle θ involved in the definition of J may suffer a bias and noise. There is a discrepancy between the true value of the rotation angle θ ∗ and the measured one θ : where b θ is a bias and δθ a random process (whose mean value is equal to zero). Using the same notations, this leads to a noise described by δ Λ c and δ Λ s on the matrices Λ c and Λ s 2 . The impact of the bias on the precision of the measurement has been assessed in (Lenoir et al., 2011b) and it has been shown that b θ must be smaller than 10 -5 rad in order to meet the expected performances. In order to take into account the impact of the noise δθ , it is possible to introduce a generalized noise: the quantities n y and n z in equation (6) are replaced by ˜ n y = n y + ˆ n y and ˜ n z = n z + ˆ n z with This additional noise depends on the non-gravitational accelerations a Y and a Z and on δ Λ c and δ Λ s . As a result, the smaller the magnitude of the external acceleration is, the smaller the noise due to the uncertainty on θ is. It can be used to derive the requirements on δθ such that the predominant source of uncertainty is MicroSTAR and not the Bias Rejections System. To have such a result, one needs S ˆ n ( f ) /lessmuch S n ( f ) around the modulation frequency 1 /τ (cf. section 4.1), where S ˆ n ( f ) is the power spectrum density of the noise ˆ n due to the rotating platform (cf. Eq. (10)). Assuming that a y ≈ a z ≈ a NG , we have ˆ n ≈ δθa NG . This leads to the following requirement : To compute a NG , it is assumed that the main contributor is solar radiation pressure and that the spacecraft is at one astronomical unit (called d 0 ) from the Sun. The power carried by solar photons by surface unit at this distance is approximately equal to P = 1 . 366 × 10 3 Wm -2 (Willson and Mordvinov, 2003). Considering a ballistic coefficient equals to C B = 0 . 1 m 2 kg -1 , which is the order of magnitude for Laplace mission (Biesbroek, 2008), the non-gravitational acceleration is equal to a NG = C B P/c = 4 . 6 × 10 -7 ms -2 at one astronomical unit, where c is the speed of light. Taking the minimum value of S n , the requirement on S δθ reads: In the rest of the article, it will be assumed that this condition is verified and only the noise of MicroSTAR will be considered.", "pages": [ 4, 5 ] }, { "title": "3.3 Conditions for bias rejection", "content": "The general approach presented above to solve the linear system does not give useful information on the non-gravitational acceleration or on the bias of the instrument. Since it is impossible to obtain the value of the unknown quantities at each sampling time, it is necessary to narrow the information retrieved from the data. A possibility is to look for the projection of the vectors a Y and a Z on a vector subspace (of dimension p a ≤ N ) whose basis is made of the column of a matrix V a ∈ M N,p a ( R ) , which are supposed to be orthogonal for the usual scalar product on R N . As a result, the goal is to find the numerical values of V ' a a Y and V ' a a Z knowing m y and m z ( M ' is the matrix transpose of M ). In this article, the choices of V a will allow retrieving the mean value of the acceleration without bias and the slope of the acceleration over one modulation period. But other choices of V a can be made to retrieve for example sinusoidal variations of the signal. Under the following four conditions on the bias, the angle θ and the projection matrix V a and assuming that δk 1 y = δk 1 z = δk 1 , the unbiased values of the external signal can be recovered: Calling v k the k -th column of V a , the conditions (13) can be expressed in the frequency domain where F δt is the discrete time Fourier transform and 〈·〉 is the usual scalar product. This equation means that the bias and the modulated signal must be orthogonal in the frequency domain. It is a priori not possible to know whether conditions (13) are fulfilled since the temporal evolution of the bias of the instrument is not controlled. However, as already mentioned, the bias corresponds deterministic low frequency variations. It is therefore possible to assume that b y and b z belongs to a vector subspace defined by the columns of ˆ V b ∈ M N, ˆ p b ( R ) ( ˆ p b ≤ N ). Given this hypothesis, conditions (13) come down to The results presented in this section can be found by solving equation (6) with a modified least square method. The matrix J , which is unknown (because of the scale factors), is replaced by the matrix ˜ J ∈ M N, 2( p a + p b ) with V b ∈ M N,p b ( R ) . And it is assumed that a Y and a Z belong to the subspace generated by ˆ V a ∈ M N, ˆ p a and that b y and b z belong to the subspace generated by ˆ V b . Section 5.3 will build on this approach.", "pages": [ 5, 6 ] }, { "title": "4 Unbiased measurements of non-gravitational acceleration", "content": "Based on the conditions (16) required for a correct demodulation and given some assumptions on the matrices V a and ˆ V b , it is possible to design a calibration signal, i.e. a pattern for the angle θ , which allows for completely removing the bias from the measurements.", "pages": [ 6 ] }, { "title": "4.1 Choice of a calibration signal", "content": "The calibration signal looked for will be periodic, with a period called τ . First, some practical concerns restrict the possible pattern. Because it can be assumed that rotating the accelerometer will induce vibrations and therefore spoil the measurements, the angle θ will have to be constant when the measurements are done. As result, calibration signals such that θ ( t ) = 2 πft , where f is an angular frequency, are forbidden. Moreover, because the accelerometer may not be perfectly centered on the rotating plate, the rotation induces Coriolis and Centrifugal forces which spoil the signal. Finally, rotating constantly may lead to a quicker breakdown of the instrument. Another practical concern, which appears if no slip ring is used, is about the wires between the accelerometer and the spacecraft. Because of them, it is not possible to rotate the accelerometer indefinitely. Therefore, the angle θ will have to stay in the interval [0; 2 π ] . To go further, it is necessary to be more specific on the matrices V a and ˆ V b . First, constant values of the non-gravitational acceleration during each modulation period will be looked for and the bias of the instrument will be supposed to be, for each period, an affine function of temperature. Therefore, where 1 q is a matrix of M q, 1 ( R ) whose coefficients are 1 , and T is a matrix of M N, 1 ( R ) made of the values of the temperature at each sampling time. The integer q is the number of sampling points in one period. It is assumed that τ and f s are such that τf s is an integer and q = τf s . In this approach, the variation of temperature will be assumed to be driven by the heat generated by the rotating platform: at each rotation, heat is generated and induces a temporary increase of temperature. Figure 1 shows two examples of calibration signals which fulfill the conditions (16) under the previous assumptions for the bias, the non-gravitational acceleration and the temperature. Let us consider the signal of Fig. 1(a) and go back to the assumptions made previously on the linear and quadratic factors 3 . In Eq. (4), the quadratic terms are constant because these two equalities are always true: sin( θ ) = 0 and [cos( θ )] 2 = 1 . Therefore, the quadratic terms behave as a bias which will be separated from the non-gravitational acceleration. Concerning the assumption on the equality of the scale factors, it has to be noticed that Λ c 2 = Id N and Λ s = 0 . Therefore, the derivation leading to Eq. (14) still hold without the assumption on the scale factors. On the contrary, for the signal of Fig. 1(b) as well as for any signal for which θ has values different from 0˚ and 180˚ , these remarks on the scale and quadratic factors do not apply. As a conclusion, only signals for which measurements are made when θ = 0˚ and θ = 180˚ should be considered. The previous hypothesis made on V a and ˆ V b allows to derive simple calibration signals. They are however restrictive because it is assumed that during a modulation period the signal and the bias are constant (with a temperature dependence for the bias). To go further, it is possible to design a calibration signal assuming that the bias is for each period an affine function of time but does not depend on temperature (Lenoir et al., 2011a). With this signal the mean and the slope of the non-gravitational acceleration on each modulation period will be recovered. In this case, where t q is a matrix of M q, 1 ( R ) such that t q k = ( k -q/ 2) δt . Figure 2 shows a calibration signal which fulfill conditions (16). Note that the remarks made on the scale and quadratics factors hold for this calibration signal. Contrary to the calibration signals of Fig. 1, the pattern in this case depends on the masking time which is introduced in the following section. In the rest of this article this calibration signal will be used. In case the bias of MicroSTAR does not belong to the subspace generated by ˆ V b , then the signal of interest is not perfectly recovered: it is spoiled by the quantities V ' a Λ ν b κ ( ν ∈ { c ; s } , κ ∈ { y ; z } ).", "pages": [ 6, 7 ] }, { "title": "4.2 Masking", "content": "As mentioned in the previous paragraph, measurements made when the accelerometer is rotating are not considered for data reduction because they may be spoiled by unwanted signals. Therefore, during postprocessing, the data acquired when the accelerometer is rotating must not be taken into account. This will be refered to as 'masking'. To introduce this masking feature in the signal processing, let consider the diagonal matrix M ∈ M N ( R ) defined by: M kk = 1 if ˙ θ k = 0 and ¨ θ k = 0 , and M kk = 0 otherwise 4 . Then in Eq. (14), the matrix V a is replaced by ˜ V a = MV a . The duration of masking is a key parameter in the precision of the post-processed quantities: the longer it is, the more data points are lost and the uncertainty increases (cf. section 5.4). The total duration of masking during one period is called T M .", "pages": [ 8 ] }, { "title": "5 Demodulated quantities", "content": "The demodulation signals introduced in the previous section allows to retrieve unbiased measurements of the non-gravitational acceleration of the spacecraft. The focus will be now to characterize these postprocessed quantities in term of uncertainty.", "pages": [ 8 ] }, { "title": "5.1 Autocorrelation of the non-gravitational acceleration mean", "content": "The calibration signal of Fig. 2 allows to recover affine variations of the external signal on each modulation period. In term of spacecraft navigation, the goal of the instrument is to measure the impact of nongravitational forces on the dynamics of the spacecraft. And the variation of momentum during one modulation period ----→ ∆ p NG of the spacecraft due to the non-gravitational forces --→ F NG is equal to the mean of the non gravitational forces times the modulation period: where t 0 is an arbitrary time and 〈·〉 τ is the mean during a duration τ . Let us call c i = Λ c ˜ v i ∈ M N, 1 ( R ) the i th column of the matrix Λ c ˜ V a , ̂ a Y i = (1 + δk 1 y ) ˜ v ' i a Y /q the i th component of the column vector (1+ δk 1 y )( ˜ V ' a ˜ V a ) -1 ˜ V ' a a Y , and ̂ a Zi = (1+ δk 1 z ) ˜ v ' i a Z /q the i th component of the column vector (1 + δk 1 z )( ˜ V ' a ˜ V a ) -1 ˜ V ' a a Z . As a result, only the mean values of the external signal are of interest, and the subsequent analysis will be restricted to the matrix V a defined by equation (18). Under the assumption introduced previously, the demodulated acceleration are defined by Eq. (14). In order to have normalized quantities, it is necessary, as in the least square method, to multiply this equation on the left by ( ˜ V ' a ˜ V a ) -1 . Under the assumptions considered here, this matrix is diagonal with all the coefficients equal to q = | ˜ v i | 2 , where ˜ v i ∈ M N, 1 ( R ) is i th column of the matrix ˜ V a . q is independent of the index i . The quantities ̂ a Y i and ̂ a Zi are the means of the non-gravitational acceleration of the spacecraft for the modulation period i along the axes Y and Z respectively. Under the assumption made earlier, the accuracy of the measurements is perfect, i.e. their expected values is equal to the true values. Concerning the precision, assuming that n y and n z are independent and have the same power spectrum density, S n defined by equation (1), the covariances between the post-processed quantities are and Cov( ̂ a Y i , ̂ a Zj ) = 0 (22) The result given by equation (21) is true only if the signal has been filtered before digitization by a perfect low-pass filter with a cut-off frequency of f s / 2 so as to avoid aliasing. According to Fig. 3, the integral of equation (21) select the noise power spectrum density at the frequency 1 /τ and approximately integrate it on an bandwidth 1 /τ for i = j . In order to minimize the absolute value of the covariance, it is therefore necessary to select the noise at the frequencies where it is minimum, i.e. for f ∈ [10 -2 ; 2 × 10 -1 ] Hz, which correspond approximately to modulation period between 5 s and 100 s. Too short modulation periods are impossible to implement in practice. Therefore, in the following, a modulation period equal to 10 min will be considered. The main interest of the demodulation process is to know the mean acceleration over a modulation period τ . This process gives birth to two new discrete-time quantities ̂ a Y i and ̂ a Zi indexed formally by i ∈ Z . It is possible to introduce the autocorrelation function R d [ k ] which is the same for both quantities and which is defined by R d [ k ] = Cov( ̂ a Y i + k , ̂ a Y i ) = Cov( ̂ a Zi + k , ̂ a Zi ) . (23) Fig. 4 shows that the autocorrelation function is close to the one of a white noise. This means that the post-processed quantities are approximately independent. In term of power spectrum density, this corresponds to a level of 10 -10 m s -2 Hz -1 / 2 with a cut-off frequency equal to 8 . 3 × 10 -4 Hz. Since the uncertainty on the demodulated accelerations is known and characterized, it is now possible to use them to gain more information on the non-gravitational accelerations.", "pages": [ 8, 9 ] }, { "title": "5.2 Further characterization of the non-gravitational acceleration", "content": "In order to increase the precision, it may be interesting to know the mean acceleration over periods of time longer than the modulation period. To do so, one needs to average the demodulated accelerations over the period of time of interest. Figure 5 shows the uncertainty on the mean acceleration for different integration time. As the noise is nearly white, the uncertainty on the mean decreases as 1 / √ T where T is the integration time. It is also possible to look for sinusoidal variations of the non gravitational acceleration with a known frequency f ∗ . The goal is to find the coefficients α and β of the time varying signal α cos(2 πf ∗ t ) + β sin(2 πf ∗ t ) using the values of the non-gravitational acceleration for each modulation period. According to the Nyquist-Shannon theorem, it is not possible to recover sinusoidal variations at frequencies higher than half the sampling frequency, i.e. f ∗ ≥ 1 / (2 τ ) . Conversely, when 1 / ( τf ∗ ) becomes too large, the uncertainty diverges. The value for which this happens depends on the number of post-processed points used to fit the sinusoidal variation: the more points are used, the easier it is to fit low frequency signals. For τ = 10 min and f ∗ the frequency related to the revolution period of the Earth 5 , the frequency ratio is 1 / ( τf ∗ ) = 144 . In this particular configuration, one obtains with 60 points and for a modulation period of 10 minutes (which corresponds to 10 hours of measurement): These values show that it is possible to obtain, in this configuration, the amplitude of the sinusoid with a precision better than 1 pm s -2 .", "pages": [ 10 ] }, { "title": "5.3 Generalized least square/optimal filtering", "content": "In section 3.3, it was mentioned that the process described until now corresponds to a least square (LS) method. This method provides estimates with a minimum variance only when the noise is white. However, the noise of MicroSTAR does not fall in this category. The generalized least square (GLS) method (Cornillon and Matzner-Løber, 2007) provides an estimate with minimum variance whatever the measurement noise is. This method is similar to the optimal filtering technique (Papoulis, 1977, p. 325): the first one is expressed in the time domain whereas the second one is express in the frequency domain. It is possible to express the components of the inverse of the covariance matrix V GLS = ( ˜ J ' Ω -1 ˜ J ) -1 using the power spectrum density of the noise instead of its covariance matrix: if v and w are two column vectors, then One may process the data form the accelerometer using the generalized least square method. However, in the specific case of the problem studied here, the gain is rather small. Indeed, Fig. 3 shows that, for the calibration signal considered, the Discrete Time Fourier Transform is peaked around the frequency 2 /τ and the noise PSD does not vary much on the interval [10 -3 ; 10 -1 ] Hz. As a result, using Parseval theorem, Therefore, the autocorrelation function plotted in Fig. 4 is nearly the same as the one obtained with the GLS approach: the autocorrelation function obtained with the GLS approach is similar to the one of a Gaussian noise and its value for k = 0 is 1 . 71 × 10 -23 m 2 s -4 instead of 1 . 84 × 10 -23 m 2 s -4 for Fig. 4. The difference in the level of precision on the post-processed quantities is also visible on Fig. 6 in section 5.4.", "pages": [ 10, 11 ] }, { "title": "5.4 Optimization of the masking time and calibration period", "content": "Until now, only one modulation period ( τ = 10 min) and one masking time ( T M = 200 s) have been considered. But since their value impact the uncertainty on the demodulated accelerations, it is legitimate to choose these values such that the uncertainty is minimized. demodulated accelerations are independent, which has been shown to be true. As what was already said, the smaller the masking time is, the smaller the uncertainty is. However, instrumental constraints do not allow to rotate MicroSTAR too fast. For a given masking time, Fig. 6 gives the optimal modulation period. For example, it shows that the set of parameters used until now ( τ = 10 min and T M = 200 s) is 'optimal' for the GLS approach, i.e. τ = 10 min gives the minimum uncertainty for a masking time of 200 s.", "pages": [ 11, 12 ] }, { "title": "6 Conclusion", "content": "The Gravity Advanced Package, developed to improve orbit reconstruction of interplanetary probes in order to test General Relativity, relies on a technological progress with allows using an electrostatic accelerometer to make measurements with no bias. Indeed, the addition of a rotating platform allows modulating the non-gravitational acceleration while keeping the bias at low frequencies. The data acquired need to be processed in order to obtain the measurement with no bias. This data processing was the topic of this article. The first result obtained was conditions under which the bias is completely removed from the signal of interest. These conditions allowed designing calibration signals, i.e. a time-pattern for the rejection angle. Then the uncertainties on the unbiased non-gravitational acceleration were computed. It was shown that it is possible to recover the mean acceleration over each period of modulation and to have access to sinusoidal variations of this accelerations with some restriction on the pulsation of the signal. Finally, a method was presented to optimize the modulation period and the masking time so as to reach the minimum uncertainty. It has been shown that several parameters influence the precision on the post-processed quantities: the modulation time, the masking time and the integration time. It is possible to choose a set of parameters, which are technologically speaking reasonable, leading to precision below 1 pm s -2 on mean quantities as well as on the amplitude of sinusoidal variations. This precision is expected to improve orbit reconstruction significantly.", "pages": [ 12 ] }, { "title": "Acknowledgements", "content": "The authors are grateful to CNES (Centre National d'Études Spatiales) for its financial support.", "pages": [ 12 ] } ]

[ { "title": "Short Term Topological Changes of Coronal Holes Associated with Prominence Eruptions and Subsequent CMEs", "content": "H. Guti´errez 1, ∗ Space Research Center, University of Costa Rica L. Taliashvili 2, ∗ Space Research Center, University of Costa Rica Z. Mouradian 3, ∗∗ Observatoire de Paris-Meudon, LESIA", "pages": [ 1 ] }, { "title": "Abstract", "content": "We study the short-term topological changes of equatorial and polar coronal hole (CH) boundaries, such as a variation of their area and disintegration, associated to reconnection with nearby (within 15 · distance) quiescent prominence magnetic fields leading to eruptions and subsequent Coronal Mass Ejections (CMEs). The examples presented here correspond to the recent solar minimum years 2008 and 2009. We consider a temporal window of one day between the CH topological changes and the start and end times of prominence eruptions and onset of CMEs. To establish this association we took into account observational conditions related to the instability of prominence/filaments, the occurrence of a CME, as well as the subsequent evolution after the CME. We found an association between short-term local topological changes in CH boundaries and the formation/disappearance of bright points near them, as well as, between short-term topological changes within the whole CH and eruptions of nearby quiescent prominences followed by the appearance of one or more CMEs. Keywords: coronal hole; coronal mass ejection; filament; magnetic field", "pages": [ 1, 2 ] }, { "title": "1. Introduction", "content": "Coronal holes (CHs) are areas which are seen dark in X-rays and extreme ultraviolet (Waldmeier, 1975; Bohlin, 1977; Wang et al., 1996). Their plasma temperature and density is much lower than that of the ambient corona; so, they are low emission zones (Harvey and Recely, 2002; Raju et al., 2005). One of the important characteristics of CHs is their magnetic field, which is characterized by being mainly unipolar (Wang et al., 1996; Harvey and Recely, 2002; Raju et al., 2005) with open magnetic field lines (Bohlin, 1977; Wang et al., 1996; Harvey and Recely, 2002); this means that its direction is primarily radial and the tangential component can be considered zero for distances greater than 2.5Rs (Wang, 2009), with Rs being the solar radius. Closed magnetic field lines can keep the plasma within coronal loops; however, since the field is open in CHs, the plasma can escape into the interplanetary medium making up the solar wind. It has been shown that the emergence of a small bipolar region from below the solar surface and its interaction with the preexisting open field in the coronal hole is a prime candidate to trigger reconnection and the consequent launching of jets along field lines in the corona (Moreno et al., 2008). Magnetic reconnection, which may occur between open and closed magnetic field lines or between open lines (Kahler and Hudson, 2002) is of great importance in the study of CHs, especially near their boundaries. The process of continuous reconnection at CH boundaries is known as 'interchange reconnection' (Wang and Sheeley, 1993; Wang et al., 1996; Fisk, 2005; Raju et al., 2005; Edmondson et al., 2010). Wang and Sheeley (2004) have proposed two kinds of interchange reconnection, the first occurs when two open field regions of the same polarity are separated by photospheric flux of the opposite polarity and the reconnection takes place between the open flux domains and the underlying pair of loop systems. The second involves stepwise displacements within a region of single magnetic polarity and occurs when an open field line exchanges footpoints with a closed field line rooted next to it and the reconnection happens at the apex of the closed loop. Hence, coronal magnetic reconnection near CH boundaries can be responsible for their shape, magnetic topology and evolution. Moreover, the appearance/disappearance of bright points (BPs) associated with short term evolution of CH boundaries (Nolte at el., 1978; Davis, 1985; Kahler and Moses, 1990; Madjarska et al., 2004), as well as CH topological changes associated with CMEs (Gonzalez et al., 1996) or with filament eruptions and subsequent CMEs (Bravo, 1995; Gopalswamy et al., 2006; Jiang et al., 2007; Taliashvili et al., 2009) reported in previous studies can be explained by the associated magnetic field configuration. Regarding the origin of CMEs, statistical studies show that 70% of prominence eruptions are associated with CMEs (Munro et al., 1979; Pojoga and Huang, 2003), moreover CHs may also be considered as signatures of CMEs (Bravo, 1996; Thompson et al., 2000). In addition, filament eruptions are associated with the formation of small adjacent CHs and/or topological fluctuations of CHs (Harvey and Sheeley, 1979; Taliashvili et al., 2008, 2009); and/or increased CH area (Bravo, 1996; Taliashvili et al., 2008, 2009); and/or decrease and/or its complete disappearance (Taliashvili et al., 2008, 2009). Using H α movies and spectroheliograms taken at Observatoire de ParisMeudon, Taliashvili et al. (2009) reviewed 42 quiescent solar filaments/ prominences eruptions during two minimum periods of solar activity (1985-1986 and 1994). These authors found that the majority of prominence/filament eruptions ( ∼ 91%) are associated with the presence of adjacent CHs and subsequent CMEs. Magnetic reconnection is probably the mechanism responsible for the interaction between CHs and prominences located close to their boundaries, which may result in the onset of CMEs. However, the characteristics of this process is not yet clear. Recent studies show that nearly 15 · distance between the prominences and CH boundaries could be considered as a possible critical distance for their interaction (Taliashvili et al., 2009). The Sun is the primary driver of space weather; the most severe solar activity occurs around the cycle maximum, but its influence on the magnetosphere and ionosphere continues through the solar minimum (Cole, 2003). Major events occur due to CMEs and high-speed solar wind streams associated with CHs (Feldman et al., 1978; Echer et al., 2005; Cole, 2003; Schwenn, 2006), as the solar wind speed originating in CHs is of about 750-800 km/s (Wang et al., 1996; Raju et al., 2006). It is important to note that both the CHs as well as their offspring, the high-speed solar wind streams, are representative of an inactive or quiet Sun (Feldman et al., 1976; Bame et al., 1977; Schwenn, 2006). Recent studies of the slow decline of solar Cycle 23 and slow rise of Cycle 24 show that the low solar activity lasted from about 2006 to the end of 2009, with 2008 and 2009 being particularly quiet years (de Toma, 2011), which is reflected in several solar irradiance observations, such as solar UV and EUV irradiance, and radio flux at 10.7 cm wavelengths (Tsurutani et al., 2011; Solomon et al., 2010; Hathaway, 2010). Moreover, in 2007 and 2008, the effect of multiple, large, and long-lived CHs resulted in regular and recurrent solar wind streams and fast wind above 550 km/s, whereas in 2009, the disappearance of the low-latitude CHs shifted the sources of the solar wind to higher latitudes, mostly to the edges of the polar CHs with a drop of about 20% in the mean speed and the almost total disappearance of fast solar wind. Generally, the continuous presence of multiple, low-latitude CHs during 2007 and 2008 and the very low magnetic flux emergence, made the minimum between Cycles 23 and 24 different from the two previous minima (de Toma, 2011). In this work we study the possible association between CH topological changes and nearby prominence eruptions followed by CMEs as well as by sequences of CMEs. We analyze three events involving the evolution of the boundaries of CHs and of their surrounding regions (within 15 · distance), taking a temporal window of 12 days. We identify their topological changes from one day before until one day after a CME occurrence. The events took place during the recent solar minimum years, 2008 and 2009. The short-term topological changes of studied CHs are accompanied by the disappearance of nearby quiescent filaments, as well as subsequent CMEs.", "pages": [ 2, 3, 4 ] }, { "title": "2. Data sources and working method", "content": "The method used in this work is similar to the one reported by Taliashvili et al. (2008, 2009). We consider a period of 12 days of continuous observations of a CH evolution (far from big flares) including the pre- and post-CME stages; thus, we study CMEs not associated to flares. The selected quiescent filaments/prominences are located close to CH boundaries or its surrounding regions (within 15 · distance). Our study comprises from August 24 to September 4 of 2008, December 3-14, 2008 and May 20-31, 2009. During these time periods we study one equatorial and 4 polar CHs, 3 filaments and 7 associated CMEs. This method allows us to identify every CH topological change before and after a non-flare CME. The identification and daily evolution of each coronal hole is based on Xray images supplied by Hinode (XRT, http://darts.isas.jaxa.jp/solar/hinode/), EUV images from EIT/SOHO (http://sohowww.nascom.nasa.gov/), EUVI/STEREO (http://stereo.gsfc.nasa.gov/), SECCHI/STEREO movies (http:// secchi.nrl.navy.mil/index.php?p=movies) and magnetograms from the Wilcox Solar Observatory (WSO, http://wso.stanford.edu/) and Michelson Doppler Imager (MDI) instrument on the SOHO (http://sohowww.nascom. nasa.gov/). The analysis of quiescent filaments located close to a CH boundary is done using observations from the Global High Resolution H-alpha Network (http://swrl.njit.edu/ghn web/), EUV images ( λ =304 ˚ A) obtained by EIT/ SOHOand EUVI/STEREO, and SECCHI/EUVI-304 ˚ Asynoptic maps (http:// secchi.nrl.navy.mil/synomaps/). We identify the associated CMEs considering the position angles of CHs, prominences and CMEs; the starting times of prominence eruptions and CMEs and, CH topological changes within 1 day before and after a CME. CMEs are identified using CME catalogs, images and movies from STEREO/ COR1 and SOHO/LASCO.", "pages": [ 4, 5 ] }, { "title": "3.1. Event 1. August, 24-September, 04 2008. CH1+F1", "content": "During this period we observe a polar coronal hole (CH1), located between N90 · -N60 · , with negative polarity (based on WSO magnetograms) and a maximum latitudinal width of about 30 · . The filament (F1) at ∼ N55 · (at least 30 · long) extends at a distance of ∼ 10 · , almost parallel to the southward boundary of CH1. F1 becomes unstable that starts disappearing on August 29 at 04:04 UT. This disappearance lasts ∼ 35 hours based on data from STEREO-A/EUVI304 ˚ A and STEREO-B/EUVI-304 ˚ A. Twelve hours before and ∼ 5h after the starting time of F1 disappearance, SOHO/LASCO reports two narrow CMEs, at Position Angle (PA) = 355 · , Width (W) = 9 · and PA=357 · , W=9 · , respectively. Due to their small widths (W < 10 · ) we are not including this type of narrow CMEs in our study. Additionally, ∼ 10h and ∼ 36h after the starting time of F1 disappearance, STEREO-B/COR1 reports two CMEs, CME1.1 (at 14:06 UT) and CME1.3 (at 16:06 UT), and STEREO-A/COR1 reports CME1.2 (21:30 UT). Based on STEREO-A/B/COR1 observations we estimate the PA and W for CME1.1, CME1.3 and CME1.2, this information is shown in Table 1. The spatial location and time of these three CMEs are well correlated with the long duration disappearance of F1. The disappearance of F1 is accompanied by changes in the nearby polar coronal hole. A small section of CH1 that is located close to central meridian in EIT (Figure 1; visible at the east by STEREO-A, Figure 2), starts to increase in latitudinal width one day before F1 disappearance (on August 28, 00:00 UT. Figure 1 and August 28, 00:05 UT. Figure 2) and extends up to ∼ N50 · during the next day; moreover, its longitudinal width is ∼ 40 · (August 29, 06:05 UT. Figure 2). These changes last for ∼ 8h (August 29, 12:05 UT. Figure 2). Subsequently, this section begins to decrease in latitudinal width and reaches again ∼ N60 · (August 30, 00:05 UT. Figure 2), whereas its longitudinal width continues to grow (up to ∼ 110 · ) (August 30, 18:06 UT. Figure 2), which leads to the growth of the area of this section (as shown on August 30, 00:00 UT in Figure 1 and August 30, 12:05 UT in Figure 2). The starting and ending times of this process approximately coincide with the onset of CME1.1 and CME1.3 respectively. The consecutive growth and decrease in latitudinal width and the growth in longitudinal width of this eastern section of CH1 coincides with the appearance and disappearance of bright point (BP1) near its western boundary (it appears on August 28, ∼ 07:00 UT and disappers on August 30, ∼ 01:30 UT, see for example Figure 2 on August 29, 06:05 UT) and with the eruption of filament (F1) in progress. BP1 is associated with a small magnetic dipole observed by SOHO/MDI that disappears almost simultaneously with BP1. Similar observations, such as X-ray bright points associated with pairs of opposite-polarity photospheric magnetic fragments, have been reported by several authors (for example, Priest et al. (1994)). BP1 lifetime agrees with the short-term magnetic reorganization, specially from August 28 (22:15 UT) to 29 (00:29 UT), during which the negative polarity CH1 grows and subsequently, ∼ 3h before F1, a magnetic corridor with negative polarity forms between the polar and equatorial regions that extends from BP1 surrounding, near the F1 (Figure 3). Whereas, the western part of CH1 (that is only seen by STEREO-A, Figure 2) decreases from August 26, followed by the appearance of BP2 close to its eastern boundary. This BP2 disappears almost simultaneously with the start of the eruption of F1 and with the process that initiates the decrease of the western section of CH1. After the onset of CME1.2, this section disappears completely almost simultaneously. Due to the position of BP2 at the far side of the Sun, we can not study its magnetic evolution. Therefore, the total CH1 area decreases continuously from August 27 reaching its minimum value on August 28 ( ∼ 24h before the onset of F1 disappearance), later it increases again until August 29 (approximately 10h after the onset of F1 eruption and close to the onset of CME1.1). Subsequently, the total area of CH1 decreases slightly just before the onset of CME1.1 and CME1.2. The period after the onset of the CMEs on August 29 coincides with the growth in longitudinal width of CH1, which reaches its maximum width few hours after CME1.3 starting time. These variations of the area of CH1 are accompanied by morphological variations, from an irregular shape to an elliptical one. The magnetic corridor disappears and the magnetic configuration recovers after the disappearance of BP1 and CMEs (Figure 3). F1 disappearance starts in the eastern hemisphere of the Sun as observed by STEREO-A and appears as a west limb eruption when viewed by STEREO-B. The eruption of F1 is disperse and its direction crosses the solar disk westward. This non-radial eruption propagates in two directions from STEREO-A perspective (Figure 4). The primary direction is southward then it moves away from the eastern part of CH1 and curves westward or to the western CH1 boundary. The motion of F1 coincides with the decrease of the western part of CH1 and the growth of the eastern part of CH1. The maximum area of eastern section and the minimum of the weastern section, are observed simultaneously at the ending time of F1 eruption. The primary direction of F1 is well-correlated with CME1.1 (STEREO-B) that is followed after 7.5h by CME1.2 (STEREO-A). The position angles of both CMEs are displaced ∼ 20 · southward from F1 position angle. The propagation of the CME is not radial in either case, and in the second is also deflected south- . Regarding the last CME1.3 that starts ∼ 36h after the starting time of F1 eruption (STEREO-B), we consider that it is associated to the eruption of the remnant portion of F1 due to the well correlated position angles of CME1.3 and a small section of F1. However, we also consider that CME1.3 could be associated to another activity on the far side of the Sun, which is impossible to observe since the separation between STEREO-A/B (70 · ) is insufficient. In general, three West limb CMEs observed by both COR1B (CME1.1, Figure 5.a and CME1.3, Figure 5.c) and COR1-A (CME1.2, Figure 5.b) are in good correlation with the moving F1 prominence material.", "pages": [ 5, 6, 7, 8, 9 ] }, { "title": "3.2. Event 2. December 3-14, 2008. CH2+CH3+F2", "content": "During this period we analyze a system formed by one polar (CH2) and one equatorial coronal hole (CH3) separated by ∼ 10 · . CH2 is located between N90 · - N50 · , with a maximum of ∼ 40 · latitudinal width; CH3 is located at ∼ N30 · , with a maximum of ∼ 18 · longitudinal and ∼ 25 · latitudinal widths, both on negative polarity field. Moreover, the filament (F2), at ∼ N55 · (at least ∼ 30 · length) extends from ∼ N50 · to ∼ N35 · , between both coronal holes, with a maximum separation of ∼ 10 · eastward of the CHs. F2 starts erupting on December 12 at 00:06 UT, this eruption lasts ∼ 8 hours based on observations of STEREO-A/EUVI-304 ˚ A and STEREOB/EUVI-304 ˚ A. About 4.5h after the starting time of F2 disappearance, STEREO-B/COR1 reports CME2.B (at 04:35 UT) and after ∼ 20 min, STEREO -A/COR1 reports the same CME, CME2.A (at 04:55 UT). Based on COR1/ STEREO-B/A image analyses we estimate the PA and W for both CMEs (see Table 1). Additionally, about 1h after the ending time of F2 disappearance, a CME starts, reported by SOHO/LASCO (CME2.L, at 08:54 UT, PA=297 · , Table 1); based on LASCO images, it expands and reaches W ∼ 115 · at 10:54h. We consider that these three CMEs correspond to the same CME (CME2) observed by LASCO and COR1 aboard STEREO-A/B. The spatial location and time appearance of these LASCO/STEREO CMEs are well correlated with the long duration F2 disappearance. On December 05 we observe two coronal holes, CH2 and CH3 separated by ∼ 25 · (Figure 6) that start merging, during the next four days their separation reduces up to ∼ 5 · (December 9, 12:00 UT), later they separate again. About 23h before F2 disappearance, the separation between both CHs is maximum (December 11, 00:05 UT. Figure 6). Simultaneously, close to the eastern boundary of CH3 a BP, BP1, appears (December 11, 12:05 UT. Figure 6) and stays there for less than a day. Moreover, just a few hours before the starting time of F2 disappearance, another BP, BP2, (December 12, 00:05 UT. Figure 6) is formed near by, which starts weakening after the end of the filament eruption (and after CME2) and is not observed within the next ∼ 23h; at the same time, small sections of CH3 around BP1 appear fragmented. In addition, at the time F2 disappearance begins, both CHs start to merge and form a single coronal hole (December 12, 00:05 UT. Figure 6). This reconnection occurs close to the location of BP1 that disappears during this process. Later, ∼ 12 hours after the onset of F2 disappearance (and after CME2), the connection between the CHs starts to disappear (December 12, 12:05 UT. Figure 6), but within the next 12h they form a single coronal hole for a second time. This CH starts to expand (December 13, 00:05 UT. Figure 6), reaches its maximum area on December 13 (12:05 UT Figure 6) and remains stable during the two following days. BP2 formation and disappearance, agrees in time with the CH disconnection and reconnection during December 12. Each of bright points (BP1 and BP2) is associated with a small magnetic dipole observed by SOHO/MDI that disappears simultaneously with BP1 and BP2 respectively. Figure 7 shows the evolution of F2 eruption in STEREO-A/EUVI-304 ˚ A images. The trajectory of this eruption is not radial and moves away from CH2+CH3. Similarly, the direction of CME2.A observed by COR1-A is not radial and is deflected southward (Figure 8.c). The position angle of CME2.A is displaced ∼ 20 · southward from F2 position angle. However, STEREOB/EUVI-304 ˚ A does not detect the deflection of F2. The CME deflection is lower in COR1 B and LASCO (Figure 8.a and 8.b). Panasenco et al. (2011) have analyzed the deflection of this CME.", "pages": [ 9, 10, 11 ] }, { "title": "3.3. Event 3. May 20-31, 2009. CH4+F3", "content": "During this period we study one polar, positive polarity CH (CH4), reaching S45 · and a polar filament F3 at ∼ S50 · (at least 35 · long) that extends almost parallel to the equator, with a maximum of ∼ 15 · separation from the southern boundary of CH4. F3 starts disapeparing on May 29 at 02:06 UT, this process lasts ∼ 14h based on 304 ˚ A/STEREO-A and 304 ˚ A/STEREO-B observations. About 30min before and ∼ 7.5h after F3 disappearance starting time, two LASCO/CMEs are observed, narrow CME at 01:31 UT (PA=211 · and W=8 · ) and CME3.2 at 09:30 UT (see Table 1). As for the first event, we do not consider the narrow CME1 in this study. Additionally, ∼ 1.75h and ∼ 13h after the starting time of F3 disappearance, COR1/STEREO-B and COR1/STEREO-A report CME3.1 (at 03:45 UT) and CME3.3 (at 15:10 UT). Based on the corresponding COR1/STEREO-A/B images, we estimate their PA and W in Table 1. Additionally, we include CME3.2 (observed by LASCO) and CME3.3 (observed by STEREO-A) a continuation of CME3.1 (observed by STEREO-B). The spatial location and time of these three CMEs is well correlated with the long duration F3 disappearance that is accompanied by variations in the nearby polar coronal hole. On May 27 a section of CH4, located at ∼ S47 · starts to grow toward the equator, reaching ∼ S37 · (May 28, 12:00 UT. Figure 9) and from the next day on it decreases up to ∼ S45 · (May 29, 00:00 UT. Figure 9) just 2h before the starting time of F3 disappearance. The CH stays at this latitude during the next few hours. Two associated bright points emerge, first BP1 on May 28 on the western border of CH4 that is best seen after CME3.2 (May 29, 12:00 UT. Figure 9); the second BP2 appears at approximately the same time as the onset of CME3.1 (May 30, 00:00 UT. Figure 9). About 12h after the onset of F3 disappearance ( ∼ 1h before the onset of CME3.3), BP1 intensifies and simultaneously CH4 fragments in two main sections around this BP1; the maximum separation is observed in the early hours on May 31 (00:05 UT. Figure 9). BP2 is close to the southern boundary of the southern section that observed separately during the next ∼ 6h (May 31, 00:05 UT. Figure 9) and then reconnects again (May 31, 06:05 UT. Figure 9). BP1 persists till June 02, while BP2 lasts until May 31. On the other hand, the growth of BP2 throughout May 29 is associated with a decrease of the entire CH4 area; this process continues accompanied by the fragmentation of the CH and lasts until May 31 (00:05 UT. Figure 9). After a few hours these fragments join again. Due to the polar positions of the bright points, unfortunately we could not observe the magnetic configuration associated with them in magnetograms. F3 disappearance starts on the west hemisphere of the Sun observed by STEREO-A, and as a west limb prominence eruption viewed by STEREOB. Figure 10 shows the evolution of F3 disappearance/eruption observed by STEREO-A. F3 eruption crosses the solar disk to the NW, and after ∼ 7h is observed as a west limb prominence eruption, always moving toward the equator. The direction of the moving F3 eruption coincides with the STEREO/LASCO consecutive three west limb CMEs; first CME3.1 is observed by COR1-B, after ∼ 6h, the second LASCO/CME3.2 starts and finally, after ∼ 6h, the COR1-A/CME3.3. The eruption of F3 (as seen from STEREO-A) is deflected to the North (Figure 10). CME3.1 and CME3.3 do not follow a radial direction, they deflect northward (Figure 11.a and 11.c). The difference between the position angles of CME3.1 and the prominence is ∼ 15 · , whereas the one between CME3.3 and the prominence is ∼ 12 · . CME3.2 is not deflected and the difference between the position angles of CME3.2 and F3 is ∼ 23 · .", "pages": [ 12, 13, 14 ] }, { "title": "4. Discussion and Conclusions", "content": "We study the detailed evolution of three different systems composed by coronal holes and nearby (within 15 · distance from their boundaries) quiescent filaments: CH1+F1, CH2+CH3+F2 and CH4+F3, that include polar CHs as well as equatorial ones. For each system we analyze the short-term CH topological changes during a period of 12 days associated with the disappearance of nearby filaments, especially before and/or after the disappearance. Moreover, we study the associated CMEs observed at similar position angles (in total 7 CMEs, excluding those with less than 10 · width) as the studied prominences and nearby coronal holes. For all the studied events we observed the ejection of CMEs after the onset of the filament eruptions, within 36 hours of the starting time of prominence eruptions. We observe that during a prominence eruption at the northern hemisphere, a section of the CH (event 1) or the entire CH (event 2), located in opposite direction to this eruption, grows in extension; whereas the other section during the event 1, which is located towards the moving erupting prominence, decreases. Additionally, for event 2, the increase of the CH areas is more pronounced, probably due to the merging of both CHs. For both events, we observe associated bright points (two BPs for each events) before the starting time of prominence eruptions and their disappearance after the eruption. In addition, at least one of the BPs is associated to the visible reduction and fragmentation of the CH. In the case of event 1, the BP (located at the eastern section of CH1) lifetime coincides with the growth of this eastern section and with the period of time between the onset of CME1.1 and CME1.3, whereas the disappearance of the BP (located at the western section of CH1) coincides with the decrease of the western section and with CME1.2 starting time; after the onset of CME1.2, this western section disappears completely. For event 2, the appearance of one of the BPs coincides with the maximum separation between both CHs (CH2 and CH3); then, it disappears after the CME2 onset, whereas the lifetime of the other BP coincides with the time period between the disconnection and reconnection of the CHs (CH2+CH3) and it disappears before the CME2 onset. Regarding event 3, located at the South Pole, we observe the appearance of a BP (BP1) before the starting time of prominence eruption followed by the appearance of other BP (BP2) and two CMEs (CME3.1, CME3.2), then BP1 intensifies, simultaneously CH4 fragments in two main sections around this BP; this is followed by CME3.3. After these 3 CMEs the BPs disappear. Each of the bright points observed near the CH (event 1 and event 2) boundaries is associated with a small magnetic dipole that appears and disappears simultaneously with BP. The most evident topological changes of the CHs are seen after the onset of the CMEs. For event 1 (North Pole CH), between the starting times of CME1.1 and CME1.3, the eastern section of CH1 (which lays in the same direction as CME1.1 and CME1.3) decreases its latitudinal width and increases its longitudinal width, whereas the small part of its western section disappears before the onset of the west limb CME1.2. For event 2, CH2 (North Pole CH) and CH3 (equatorial CH) appear separated after CME2 onset. Regarding event 3 (South Pole CH), CH4 disintegrates after CME3.2 onset, and after CME3.3 onset the separation between these segments reaches its maximum. We have found that short-term topological changes in the entire CHs are associated with nearby quiescent prominence eruptions, particularly when a separation between the quiescent filaments and the CH boundary is less than or equal to 15 · , followed by the ejection of one or more CMEs (Taliashvili et al., 2009). Similar results regarding the separation distance (of ∼ 10 · ) between the filament and CH boundaries, associated with prominence eruption and subsequent CME were reported recently by Panasenco et al. (2011). In addition, our observations indicate that BP appearance or disappearance is related with the starting processes of nearby filament eruptions, which can be caused by disturbances of the CH environment. These disturbances are most likely the result of magnetic reconnection between magnetic field lines associated with BPs and the surrounding of CH boundaries followed by the reorganization the magnetic field, which results in different BP lifetimes and CH boundary or area changes and reaches the foot points of nearby filaments, contributing in this way to their destabilization, eruption, and subsequent CME ejection. There are several proposed models and observational evidences related to magnetic reconnection at (or close to) the CH boundaries and related with visible appearance/disappearance of BPs, filament eruptions, short-term topological changes in the entire CHs and subsequent formation of CMEs (Kahler and Moses, 1990; Kahler and Hudson, 2002; Bravo, 1995, 1996; Madjarska et al., 2004; Wang and Sheeley, 2004; Fisk, 2005). Specially, magnetic bipoles emerging within a CH or small loops formed in the CH during interchange reconnection could increase along a CH lifetime and ultimately lead to the fragmentation and diffusion of the CH (Krista et al., 2011). Interchange reconnection at CH boundaries is supported by several observational results (Krista et al., 2011), e.g., the prevalence of very small loops inside CHs and larger loops outside CHs (Wiegelmann and Solanki, 2004) and the boundary displacements observed due to the emergence and disappearance of bright small-scale loops in the form of BPs (Madjarska and Wiegelmann, 2009; Subramanian, 2010). In addition, we observe that the direction of the erupted filaments near CHs and associated CMEs is almost non-radial. Both structures (regarding the three studied events) are moving toward the equator. In addition, the difference between the central position angles of both ranges between 15 · and 20 · . These observations are consistent with the results reported by Gopalswamy et al. (2003); Cremades and Bothmer (2004); Gopalswamy et al. (2009); Panasenco et al. (2011); they found that the CMEs generally move away from the open magnetic field regions and their the deflection is probably due to the fact that at lower coronal heights they are guided by the open field along which the fast solar wind flows. In this study we are not considering long-term topological variation of coronal holes in regard to different stages of the solar cycle and the associated involvement of prominence eruptions and subsequent CMEs, which we plan to pursue in an upcoming project. Acknowledgments. We are grateful to the Hinode, STEREO, SOHO and Global High Resolution H-alpha Network for open access to their data sets. Hinode is a Japanese mission developed and launched by ISAS/JAXA, with NAOJ as domestic partner and NASA and STFC (UK) as international partner. It is operated by these agencies in co-operation with ESA and the NSC (Norway). LASCO and EIT are part of SOHO, SOHO is a project of international cooperation between ESA and NASA. The LASCO CME catalog is generated and maintained at the CDAW Data Center by NASA and The Catholic University of America in cooperation with the Naval Research Laboratory. The STEREOmission is supported by NASA, PPARC (UK), DRL (Germany), CNES (France), and USAF. The SECCHI data used here were produced by an international consortium of the Naval Research Laboratory (USA), Lockheed Martin Solar and Astrophysics Lab (USA), NASA Goddard Space Flight Center (USA), Rutherford Appleton Laboratory (UK), University of Birmingham (UK), Max-Planck-Institut for Solar System Research (Germany), Centre Spatiale de Li'ege (Belgium), Institut d'Optique Theorique et Appliqu'e (France), Institut d'Astrophysique Spatiale (France). The 'COR1 Preliminary Events List' was generated by O. C. St. Cyr prior to September 2007, and is being maintained now by Hong Xie. Wilcox Solar Observatory is currently supported by NASA and data used in this study was obtained via the web site, courtesy of J.T. Hoeksema. We are grateful to M. S'anchez and T. Roinishvili to improve English. This study was performed as a partial requirement for the PhD Degree of Sciences at the University of Costa Rica. Special thanks are owed to anonymous referees for constructive comments that helped to improve the quality of the paper.", "pages": [ 14, 15, 16, 17 ] }, { "title": "Bibliography", "content": "Wang, Y.-M., Sheeley, N.R. Understanding the Rotation of Coronal Holes. Astrophys. J. 414, 916-927, 1993. Wiegelmann, T., Solanki, S.K. Why Are Coronal Holes Indistinguishable From the Quiet Sun in Transition Region Radiation? SOHO 15 Coronal Heating 575, 35-40, 2004.", "pages": [ 22 ] } ]

[ { "title": "The Case for Massive, Evolving Winds in Black Hole X-ray Binaries", "content": "Joey Neilsen 1, ∗ Boston University Department of Astronomy, 725 Commonwealth Avenue, Room 416D, Boston, MA 02215", "pages": [ 1 ] }, { "title": "Abstract", "content": "In the last decade, high-resolution X-ray spectroscopy has revolutionized our understanding of the role of accretion disk winds in black hole X-ray binaries. Here I present a brief review of the state of wind studies in black hole X-ray binaries, focusing on recent arguments that disk winds are not only extremely massive, but also highly variable. I show how new and archival observations at high timing and spectral resolution continue to highlight the intricate links between the inner accretion flow, relativistic jets, and accretion disk winds. Finally, I discuss methods to infer the driving mechanisms of observed disk winds and their implications for connections between mass accretion and ejection processes. Keywords: accretion, accretion disks; black hole physics; stars: winds, outflows; X-rays: binaries; X-rays: individual (GRO J1655-40)", "pages": [ 1 ] }, { "title": "1. Introduction", "content": "In the last 20 years, we have seen the discovery of a multitude of highly-ionized absorbers in moderate and high-resolution X-ray spectra of black hole and neutron star X-ray binaries (e.g. Ebisawa 1997; Kotani et al. 1997; Brandt & Schulz 2000; Kotani et al. 2000a,b; Lee et al. 2002; Sidoli et al. 2001, 2002; Schulz & Brandt 2002; Parmar et al. 2002; Boirin & Parmar 2003; Boirin et al. 2004, 2005; Miller et al. 2004, 2006a,b, 2008, 2011; Neilsen & Lee 2009; Neilsen et al. 2011, 2012a; Neilsen & Homan 2012; Ueda et al. 2004, 2009; Martocchia et al. 2006; Kubota et al. 2007; Blum et al. 2010; Reynolds & Miller 2010; King et al. 2012b; D'ıaz Trigo et al. 2006, 2007, 2009, 2012; Diaz Trigo & Boirin 2012). Often these absorbers are blueshifted, indicative of hot outflowing gas, i.e. accretion disk winds. The prevalence of disk winds in X-ray binaries suggests that these outflows may play a crucial role in the physics of accretion and ejection around compact objects. In this brief review, I discuss some recent developments in the influence of ionized disk winds around black holes.", "pages": [ 1 ] }, { "title": "2. Black Hole Accretion Disk Winds and the DiskJet Connection", "content": "Much of the recent work on accretion and ejection processes in black hole outbursts has focused on radio/X-ray correlations (e.g. Gallo et al. 2003; Corbel et al. 2003; Fender & Belloni 2004; Fender et al. 2004, 2009, although see e.g. Gallo et al. 2012 and references therein for lingering questions about the precise nature of these correlations). Briefly, we now know that typical black hole 1 Einstein Fellow, Boston University transients emerge from quiescence in X-ray hard states that produce steady, compact jets. They rise in luminosity in this (probably radiatively inefficient; e.g. Esin et al. 1997) hard state, until at some point they undergo a transition towards a much softer state, possibly dominated by a radiatively-efficient disk. This transition has also been associated with major relativistic plasma ejections and the disappearance of steady jets. Eventually, the luminosity falls and they return to quiescence via the hard state. Over the last decade, this canonical picture of the 'diskjet connection' has proved to be a fruitful way to characterize accretion and ejection processes around stellar-mass black holes, and has become the backbone of our understanding of the spectral and timing behavior of black hole transients. But this story cannot be complete, for it fails to describe or account for the presence or the influence of another mode of mass ejection: highly-ionized accretion disk winds, whose behavior in outburst is only now becoming clear. Just three years after the launch of Chandra , Lee et al. (2002) argued that winds could be associated with the accretion disk, although they were not confined to diskdominated states. Miller et al. (2008) confirmed that in both GRO J1655-40 and GRS 1915+105, absorption lines were stronger in spectrally soft states (see also Neilsen & Lee 2009). They suggested that higher ionizing flux might be responsible for the changes in the winds, but left open the possibility that other (e.g. geometric) changes might be required as well. Thus it remained unclear how or why winds might change on outburst time scales: were they steady, passive bystanders that simply responded to variations in the ionizing flux, or did they play a role in outbursts, appearing and disappearing just like jets?", "pages": [ 1 ] }, { "title": "2.1. A Case Study in Evolving Winds", "content": "With two high-resolution Chandra HETGS observations of accretion disk winds separated by less than three weeks (Miller et al. 2008; Neilsen & Homan 2012), the 2005 outburst of the microquasar GRO 1655-40 presents an ideal backdrop against which to test the hypothesis that winds do not evolve during outburst. The Chandra and RXTE spectra are shown in the left and right panels of Figure 1, respectively. The first observation (shown in black) took place during a hard state, while the second observation (shown in blue) occurred during a much softer state. And while the first observation contained an Fe xxvi absorption line near 7 keV, the second provided an extremely rich absorption line spectrum that has been studied in great detail (Miller et al. 2006a; Netzer 2006; Miller et al. 2008; Kallman et al. 2009; Neilsen & Homan 2012; see § 3 for a discussion of the origin of this wind). Here, let us consider the question: why are the two Chandra absorption line spectra so different? Are the differences driven by changes in the photoionizing flux from the hard state to the soft state, or did the wind physically evolve over those 20 days? Our detailed analysis (Neilsen & Homan 2012) indicates that the wind must have evolved significantly between the two Chandra observations. This argument can be understood both qualitiatively and quantitatively: small cross-sections above 10 keV). Since the soft Xray spectra of the two observations are quite similar, we conclude that the change in the relevant ionizing flux is negligible and cannot, in and of itself, explain the observed differences in the lines. To summarize briefly, after Lee et al. (2002): 'ionizing flux is only part of the solution.' Based on our careful treatment of photoionization, we find compelling qualitative and quantitative evidence for significant physical changes in the accretion disk wind during the 2005 outburst of GRO J1655-40. In the following section, we argue that broad parallels between this source and other black holes support the conclusion that evolving winds may be an extremely common, if not universal, phenomenon.", "pages": [ 2 ] }, { "title": "2.2. Ubiquitous, Massive Evolving Winds", "content": "As noted above, Lee et al. (2002) and Miller et al. (2008) pointed out that accretion disk winds seem to be associated with the accretion disk and/or spectrally soft states. Recent work by Ponti et al. (2012) provides convincing evidence for these earlier claims: their archival study of Chandra HETGS, XMM-Newton , and Suzaku observations of stellar mass black holes in outburst demonstrates that accretion disk winds are preferentially detected in softer states 2 . In particular, winds are ubiquitous along the high-luminosity branch of the hardness-intensity diagram after the spectrally hard state (see Lee et al. 2002; Miller et al. 2008; Neilsen & Homan 2012 for rare cases of weak winds at the high luminosity end of spectrally hard states). In some cases, stringent upper limits have been placed on the existence of hard state winds (e.g. Blum et al. 2010; Miller et al. 2012). Ponti et al. (2012) suggest that a static absorber with a variable ionization parameter may be unlikely to explain completely the observed behavior of winds in black hole outbursts, although it is also noted that ionization effects may be important and that only detailed photoionization studies can confirm this suggestion. While this is certainly true, black hole wind variability studies on time scales from seconds (Neilsen et al. 2011) to hundreds or thousands of seconds (Lee et al. 2002; Miller et al. 2006b) to weeks and years (Neilsen & Homan 2012; Blum et al. 2010; Miller et al. 2012) have all required changes in the wind density. It therefore seems likely that the observed outburst behavior of winds will also require such changes, in which case we can conclude that disk winds are preferentially but not exclusively launched at high luminosity, around or after the time the black hole begins to exit the spectrally-hard state. To test this interpretation, we undertook a Chandra HETGS campaign to catch this phase of a new outburst; the resulting spectra of 4U 1630-47 are shown in Figure 3. The results will be published in detail in future work (Neilsen et al. 2013), but suffice it to say here that with the robust detection of a strong outflow, this campaign was remarkably successful. It should be noted that Kubota et al. (2007) detected a wind in a similar phase of a prior outburst of 4U 1630-47, so our new detection confirms that wind behavior is predictable. We conclude that winds are reliably launched during this outburst phase in black hole X-ray binaries; confirming their absence or weakness in harder and less luminous states will be the subject of future work. If winds were simply ionized gas along the line of sight, such a conclusion might be interesting but relatively insignificant. In reality, there is now a large and growing body of evidence indicating that disk winds in stellar mass black holes may be extremely massive. In fact, as early as a decade ago, it was discovered that wind mass loss rates ˙ M w could be comparable to black hole accretion rates ˙ M acc (Lee et al., 2002). More recently Neilsen & Lee (2009) suggested that radiatively/thermally-driven winds could deplete the mass of the disk enough to suppress relativistic jets. In a few exceptional cases (e.g. the 'heartbeat' state of GRS 1915+105, Neilsen et al. 2011; IGR J170913624, King et al. 2012b), detailed studies have found mass loss rates in excess of (10 -20) M acc ! These remarkable results, too, are supported by the results of Ponti et al. (2012), who find that ˙ M w is typically at least twice ˙ M acc , and approaches 10 ˙ M acc at high Eddington ratio. If winds are truly as massive as these results suggest, it begins to seem significant that they are preferentially launched at the same phase of black hole outbursts when we observe the disappearance of steady jets and major changes in the structure of the accretion flow. Could it be that disk winds are indeed the mechanism by which jets are suppressed and state transitions take place, as suggested by Neilsen & Lee (2009) and Neilsen et al. (2011)? At present, the data cannot rule out this interpretation, but with careful tracking of ˙ M w going into this state transition, it may be possible to shed new light on this important question in the near future.", "pages": [ 2, 3 ] }, { "title": "3. On Inferring Wind Driving Mechanisms", "content": "As noted in § 2, in the last few years there have been a number of developments that suggest deep connections between the X-ray luminosity or accretion rate, the state of the accretion flow, and the behavior of accretion disk winds. The significance of such connections is not entirely clear, however: Miller et al. (2012) (see also Miller et al. 2008) have argued that these some of these connections can be explained in terms of a magnetic field configuration that changes during outburst, while other authors have presented interpretations based on radiatively- and thermally-driven winds (Neilsen & Lee 2009; Ueda et al. 2009, 2010; Neilsen et al. 2011, 2012a; Ponti et al. 2012). Because the detailed implications of these massive evolving winds depend heavily on their formation physics, it is critical to draw robust conclusions about the mechanisms that produce them. To this end, we often take advantage of the fact that the well-known driving mechanisms (radiation pressure, Compton heating, and MHD) typically operate in different regimes of density, ionization, and distance from the black hole (e.g. Proga & Kallman 2002; Miller et al. 2008). For example, since radiation pressure is most commonly transmitted via UV resonance absorption lines, it may be ineffective when the gas has little or no opacity in the UV (e.g. at very high ionization, ξ /greaterorsimilar 10 3 ergs cm s -1 ; Proga & Kallman 2002). In contrast, Compton heating may produce highly-ionized outflows, but because they require a large surface area of gas to be heated to the point that the sound speed exceeds the escape speed, thermally-driven winds are only expected at large distances from the black hole (10 4 -10 5 r g ; Begelman et al. 1983; Woods et al. 1996). In addition, simulations at varying luminosities and spectral shapes (e.g. Woods et al. 1996) have shown that thermal driving tends to produce outflows with small mass fluxes and/or gas densities ( n /lessorsimilar 10 12 cm -3 for the specific case of GRO J1655-40; Luketic et al. 2010). MHD processes like the Blandford-Payne mechanism, however, may produce dense outflows at small radii (Blandford & Payne, 1982).", "pages": [ 3, 4 ] }, { "title": "3.1. Absorption Lines, Density, and GRO J1655-40", "content": "In principle, then, it is possible to use the observed properties of winds to infer their launching mechanisms. The difficulty is that many physical factors influence the observability of lines. For example, the X-ray luminosity sets the ionization parameter of the gas: while the shape of the ionizing spectrum and the gas density determine which ions are visible at any ξ (Kallman & Bautista 2001). The density and geometry of the wind determine the equivalent hydrogen column density of the absorber: and the column density of each individual ion follows from N H , the chemical abundances A i , and the ionization balance x i : Finally, the ion column densities are folded into the equivalent width W λ of each line via the curve of growth 3 : Here λ and f ji are the line wavelength and oscillator strength, respectively. Given the complexity of the connections between the gas properties, the radiation field, and illumination patterns, it is not immediately obvious how any observed wind is produced. However, for a wind that is both very highly ionized ( ξ /greaterorsimilar 10 3 ergs cm s -1 ) and sufficiently dense that its implied radius (c.f. Equation 1) is well inside the radius where thermal driving is effective (i.e. the Compton radius, R << R C ∼ 10 11 -12 cm), the natural conclusion is that MHD processes likely play a role in its launching. The classic example of this argument in black hole Xray binaries comes from Miller et al. (2006a), who first published the extraordinary Chandra HETGS absorption line spectrum of GRO J1655-40 (Figure 1). Photozioniation analysis indicated a characteristic ionization parameter of ξ /greaterorsimilar 10 4 ergs cm s -1 , an order of magnitude too high for line driving to be effective. In addition to many other lines, they detected two Fe xxii absorption lines at 11.77 ˚ A and 11.92 ˚ A whose ratio can be used as a density diagnostic. Their analysis and subsequent studies (Miller et al. 2008; Kallman et al. 2009) led to the conclusion that the density must have been at least n /greaterorsimilar 10 14 cm -3 , placing the absorber some three orders of magnitude inside the Compton radius, where thermal driving cannot operate. Based on the high optical depth in the wind and possible saturated lines, Netzer (2006) argued for a lower density ( n e ∼ 10 13 cm -3 ) and ionization parameter ( ξ ∼ 10 3 ), which would have implied a more distant wind consistent with thermal driving. Miller et al. (2008) subsequently argued that Netzer's model significantly overpredicted soft X-ray absorption lines, and according to simulations of thermally-driven winds tailored to GRO J1655-40 (Luketic et al. 2010), even the lower density proposed by Netzer (2006) is an order of magnitude too high for thermal driving to be the dominant launching mechanism. By process of elimination, Miller et al. (2006a, 2008); Kallman et al. (2009) concluded that the dense wind in GRO J1655-40 must be powered by magnetic processes 4 .", "pages": [ 4, 5 ] }, { "title": "3.2. MHD Winds in Black Hole X-ray Binaries?", "content": "But how are we to understand the disk-wind-jet coupling in light of the apparent variations in wind formation physics between different systems? Radiative/thermal driving has been invoked to explain most winds in black hole systems, but there is convincing evidence for an MHD wind in GRO J1655-40. The continuing debate over the accretion disk wind launching mechanism begs the question: is there a single, universal process that governs the interaction between disks, winds, and jets in black hole X-ray binaries? Can the behavior of inflows and outflows be unified in such disparate systems? As our understanding of accretion and ejection physics evolves, it can be instructive to revisit prior observations and the conclusions we draw from them. Perhaps the most salient development comes from Reynolds (2012), hereafter R12, who demonstrates that the wellknown launching mechanisms (i.e. Compton heating, magnetocentrifugal acceleration, and radiation pressure) can be distinguished not only by the ionization and the density, but also the optical depth of the winds they produce. With a focus on Compton-thick winds, the forbidden regime of parameter space for each mechanism is clearly set out in terms of the optical depth τ, the Eddington ratio λ, the ratio f v of the wind's terminal velocity to the escape velocity at the launch radius, and the distance to the black hole r (in units of gravitational radii r g ). How does this analysis inform the interpretation of the dense, highly-ionized wind in GRO J1655-40, which had a column density N H ∼ 10 24 cm -2 (near the Comptonthick limit; Miller et al. 2006a, 2008; Kallman et al. 2009)? Following R12, let us consider each launching mechanism in turn: The soft-state wind of GRO J1655-40 provides an interesting illustration of this constraint. During this particular observation, we estimate the Eddington ratio to be about λ ≈ 0 . 06 (Neilsen & Homan 2012). The ratio f v is much harder to determine, since only the line-of-sight velocity can be measured. Kallman et al. (2009) found a blueshift of ∼ 375 km s -1 , which is much less than the escape velocity at plausible launch radii ( v esc ∼ 5000 -14 , 000 km s -1 ; f v ∼ 0 . 03 -0 . 07). Normally, one might suppose that f v ∼ 1 and attribute the small blueshift to a velocity primarily perpendicular to the line of sight, but it is difficult to see how this scenario is consistent with the small solid angle of the wind (see below). If we take the velocities at face value and accept that the wind remains bound to the black hole, we find λ ∼ (6 -40) × 2 f 2 v , i.e. there is no shortage of momentum in the radiation field. However, this is not a sufficient condition to launch the wind, since the radiation force on the gas is still required to exceed the force of gravity (C. Reynolds 2012, private communication). Thus, despite an abundance of momentum flux, radiation pressure cannot explain the dense, highly-ionized wind in GRO J1655-40 (see also Miller et al. 2006a). and lends itself nicely to an analytic constraint on the production of Compton-thick outflows. As shown by R12, Compton-thick magnetocentrifugal winds can only be produced at radii i.e. where Here Ω is the solid angle of the wind, η is the radiative efficiency of the accretion flow, and /pi1 ∼ 2 -3 is the ratio of the size of the acceleration zone of the wind to the launch radius (for more details, see R12 and references therein). For the dense wind in GRO J1655-40, the observed column density N H = 10 24 ± 0 . 02 cm -2 implies an optical depth 5 τ ∼ 0 . 67, from which we estimate τ/λ ∼ 11 ± 2 . Miller et al. (2006a) use upper limits on emission line strengths to place the tight constraint Ω < 4 π/ 9 . If we allow /pi1 /greaterorsimilar 1 and use the smallest plausible radius r ∼ 970 r g , Equation 6 for the allowed parameter space for magnetocentrifugal winds becomes τ/λ < 2 . 0 . That is, given the small solid angle of the wind and its relatively large distance from the black hole, the optical depth of the wind is at least ∼ 5 × too large for it to be launched by magnetocentrifugal processes. However, we stress that other MHD mechanisms (see Miller et al. 2008 and references therein) have not been ruled out. In short, based on the constraints presented in Reynolds (2012), we find that the dense, highly-ionized wind in GRO J1655-40 cannot be driven by magnetocentrifugal effects. We therefore confirm the original suggestion of Miller et al. (2008) that the wind is driven by some other MHD process, like magnetic pressure (e.g. Proga 2003). It should be noted that R12 uses a simplified model of the wind and driving mechanisms, and that the geometry of observed winds may be somewhat more complicated. For example, as detailed by Giustini & Proga (2012) and references therein, the bulk properties of winds (e.g. density, ionization, velocity, etc.) may be strong, non-monotonic functions of position, even for outflows with simple streamlines. This is clearly an area where future theoretical work can continue to improve the accuracy and robustness of inferences from observations. In any case, more detailed study of the remarkable wind in GRO J1655-40 is forthcoming. Finally, we can return to the question at hand: is there a single, universal process that governs the interaction between disks, winds, and jets in black hole X-ray binaries? To the best of our knowledge, we can trace the origin of most winds from stellar-mass black holes back to the radiation field of the inner accretion flow (e.g. Lee et al. 2002; Kubota et al. 2007; Neilsen & Lee 2009; Ueda et al. 2009, 2010; Neilsen et al. 2011, 2012a). This seems to be a compelling argument for the scenario presented in § 2, in which accretion disk winds play an integral role in the evolution of black hole outbursts by virtue of their connections to the radiation field. However, given that there is one clear case of a magneticallydriven wind, and no hard evidence that other winds are not driven by magnetic fields, this question should still give us pause. Are the results from detailed studies of one wind observed in GRO J1655-40 applicable to the class of black hole binaries as a whole? King et al. (2013) find evidence of a three-way correlation between jet power, wind power, and bolometric luminosity over eight orders of magnitude in black hole mass. This seems to indicate that outflows and inflows could be regulated by a common process related to the mass accretion rate, but the underlying physics of this regulation is still open for discovery.", "pages": [ 5, 6 ] }, { "title": "4. Conclusions", "content": "In the last few years, a significant effort has been devoted to understanding the physics and behavior of accretion disk winds in black hole X-ray binaries, with important developments coming from both archival studies and new observations. By extrapolating from the specific (exceptional) case of GRO J1655-40 to the ensemble of winds studied by Ponti et al. (2012), we have come to understand that highly-ionized winds are ubiquitous around stellar mass black holes, that they may evolve significantly in outburst, and that they may carry away a significant fraction of the inflowing gas. It is worth reiterating that winds are not necessarily confined to specific states per se , but appear to evolve continuously during outbursts. While their formation physics is complex and challenging to discern, and while MHD processes may be important, it seems that most (but not all) known accretion disk winds are consistent with radiative or thermal driving (Lee et al. 2002; Kubota et al. 2007; Neilsen & Lee 2009; Ueda et al. 2009, 2010; Neilsen et al. 2011, 2012a; Diaz Trigo & Boirin 2012 come to a similar conclusion for neutron star binaries). At once driven by and ionized by the luminosity of the central engine, these massive outflows may require radiation for their existence, but their substantial influence may ultimately be their undoing. By draining vast quantities of mass from the accretion disk, they may not only suppress relativistic jets and cause or facilitate state transitions as seen in GRS 1915+105 (with tentative evidence in other systems; Ponti et al. 2012), but they may also cripple the ability of the disk to launch a wind! In the future, by taking advantage of our ever-growing understanding of the spectral/timing behavior of X-ray binaries in outburst, we will track these evolving, massive, ionized outflows in order to continue shedding new light on the physics of accretion and ejection around black holes.", "pages": [ 6, 7 ] }, { "title": "Acknowledgements", "content": "I thank the anonymous referees, whose comments enhanced the context and clarity of the paper, as well as Chris Reynolds and Jon Miller for comments that substantially improved the discussion of wind driving mechanisms. This work was supported by the National Aeronautics and Space Administration through the Smithsonian Astrophysical Observatory contract SV3-73016 to MIT for support of the Chandra X-ray center, which is operated by the Smithsonian Astrophysical Observatory for and on behalf of the National Aeronautics and Space Administration under contract NAS8-03060, and by NASA through the Einstein Fellowship Program, grant PF2-130097.", "pages": [ 7 ] }, { "title": "References", "content": "Balbus, S. A., & Hawley, J. F. (1991). A powerful local shear instability in weakly magnetized disks. I - Linear analysis. II - Nonlinear evolution. ApJ , 376 , 214-233. Bautista, M. A., & Kallman, T. R. (2001). The XSTAR Atomic Database. ApJS , 134 , 139-149. Begelman, M. C., McKee, C. F., & Shields, G. A. (1983). Compton heated winds and coronae above accretion disks. I Dynamics. ApJ , 271 , 70-88. Blandford, R. D., & Payne, D. G. (1982). Hydromagnetic flows from accretion discs and the production of radio jets. MNRAS , 199 , 883-903. Blandford, R. D., & Znajek, R. L. (1977). Electromagnetic extraction of energy from Kerr black holes. MNRAS , 179 , 433-456. Blum, J. L., Miller, J. M., & Cackett, E., et al. (2010). Suzaku Observations of the Black Hole H1743-322 in Outburst. ApJ , 713 , 1244-1248. Boirin, L., M'endez, M., D'ıaz Trigo, M., et al. (2005). A highlyionized absorber in the X-ray binary 4U 1323-62: A new explanation for the dipping phenomenon. A&A , 436 , 195-208. Boirin, L., & Parmar, A. N. (2003). Discovery of narrow X-ray absorption features from the low-mass X-ray binary X 1254-690 with XMM-Newton. A&A , 407 , 1079-1084. Boirin, L., Parmar, A. N., Barret, D., et al. (2004). Discovery of Xray absorption features from the dipping low-mass X-ray binary XB 1916-053 with XMM-Newton. A&A , 418 , 1061-1072. Brandt, W. N., & Schulz, N. S. (2000). The Discovery of Broad P Cygni X-Ray Lines from Circinus X-1 with the Chandra HighEnergy Transmission Grating Spectrometer. ApJL , 544 , L123L127. Corbel, S., Nowak, M. A., Fender, R. P., et al. (2003). Radio/Xray correlation in the low/hard state of GX 339-4. A&A , 400 , 1007-1012. Diaz Trigo, M., & Boirin, L. (2012). Disc atmospheres and winds in X-ray binaries. ArXiv e-prints , 1210.0318. D'ıaz Trigo, M., Parmar, A. N., Boirin, L., et al. (2006). Spectral changes during dipping in low-mass X-ray binaries due to highlyionized absorbers. A&A , 445 , 179-195. D'ıaz Trigo, M., Parmar, A. N., Boirin, L., et al. (2009). Variations in the dip properties of the low-mass X-ray binary XB 1254-690 observed with XMM-Newton and INTEGRAL. A&A , 493 , 145157. D'ıaz Trigo, M., Parmar, A. N., Miller, J., et al. (2007). XMMNewton and INTEGRAL spectroscopy of the microquasar GRO J1655-40 during its 2005 outburst. A&A , 462 , 657-666. Ebisawa, K. (1997). X-Ray Spectroscopy of Binary Sources with ASCA - Black Hole Candidates and Super-Soft Sources. In F. Makino & K. Mitsuda (Ed.), X-Ray Imaging and Spectroscopy of Cosmic Hot Plasmas (pp. 427-431). Fender, R., & Belloni, T. (2004). GRS 1915+105 and the Disc-Jet Coupling in Accreting Black Hole Systems. ARA&A , 42 , 317-364. Fender, R. P., Belloni, T. M., & Gallo, E. (2004). Towards a unified model for black hole X-ray binary jets. MNRAS , 355 , 1105-1118. Fender, R. P., Homan, J., & Belloni, T. M. (2009). Jets from black hole X-ray binaries: testing, refining and extending empirical models for the coupling to X-rays. MNRAS , 396 , 1370-1382. Gallo, E., Fender, R. P., & Pooley, G. G. (2003). A universal radio-, Kallman, T., & Bautista, M. (2001). Photoionization and HighDensity Gas. ApJS , 133 , 221-253. Kotani, T., Kawai, N., & Matsuoka, M., et al. (1997). ASCA observations of galactic jet systems. In C. D. Dermer, M. S. Strickman, & J. D. Kurfess (Ed.), Proceedings of the Fourth Compton Symposium (pp. 922-926). volume 410 of American Institute of Physics Conference Series . Kubota, A., Dotani, T., Cottam, J., et al. (2007). Suzaku Discovery of Iron Absorption Lines in Outburst Spectra of the X-Ray Transient 4U 1630-472. PASJ , 59 , 185-198. Luketic, S., Proga, D., & Kallman, T. R., et al. (2010). On the Properties of Thermal Disk Winds in X-ray Transient Sources: A Case Study of GRO J1655-40. ApJ , 719 , 515-522. Martocchia, A., Matt, G., Belloni, T., et al. (2006). The XMMNewton view of GRS 1915+105. A&A , 448 , 677-687. Miller, J. M., Maitra, D., & Cackett, E. M., et al. (2011). A Fast X-ray Disk Wind in the Transient Pulsar IGR J17480-2446 in Terzan 5. ApJL , 731 , L7-L11. ApJ , 601 , 450-465. Miller, J. M., Raymond, J., & Fabian, A. C., et al. (2012). The Disk-wind-Jet Connection in the Black Hole H 1743-322. ApJL , 759 , L6-L9. Miller, J. M., Raymond, J., & Homan, J., et al. (2006b). Simultaneous Chandra and RXTE Spectroscopy of the Microquasar H1743322: Clues to Disk Wind and Jet Formation from a Variable Ionized Outflow. ApJ , 646 , 394-406. Miller, J. M., Raymond, J., & Reynolds, C. S., et al. (2008). The Accretion Disk Wind in the Black Hole GRO J1655-40. ApJ , 680 , 1359-1377. Mirabel, I. F., & Rodr'ıguez, L. F. (1999). Sources of Relativistic Jets in the Galaxy. ARA&A , 37 , 409-443. Neilsen, J., & Homan, J. (2012). A Hybrid Magnetically/Thermally Driven Wind in the Black Hole GRO J1655-40? ApJ , 750 , 27-34. Neilsen, J., & Lee, J. C. (2009). Accretion disk winds as the jet suppression mechanism in the microquasar GRS 1915+105. Nature , 458 , 481-484. Neilsen, J., Petschek, A. J., & Lee, J. C. (2012a). Accretion disc wind variability in the states of the microquasar GRS 1915+105. MNRAS , (p. 502-511). Neilsen, J., Ponti, G., Coriat, M., et al. (2013). Winds, Jets, and State Transitions in the 2011-2012 Outburst of 4U 1630-47. in preparation . Neilsen, J., Remillard, R. A., & Lee, J. C. (2011). The Physics of the 'Heartbeat' State of GRS 1915+105. ApJ , 737 , 69-88. Netzer, H. (2006). A Thermal Wind Model for the X-Ray Outflow in GRO J1655-40. ApJL , 652 , L117-L120. Parmar, A. N., Oosterbroek, T., Boirin, L., et al. (2002). Discovery of narrow X-ray absorption features from the dipping low-mass Xray binary X 1624-490 with XMM-Newton. A&A , 386 , 910-915. Ponti, G., Fender, R. P., & Begelman, M. C., et al. (2012). Ubiquitous equatorial accretion disc winds in black hole soft states. MNRAS , 422 , L11-L15. Proga, D. (2003). Numerical Simulations of Mass Outflows Driven from Accretion Disks by Radiation and Magnetic Forces. ApJ , 585 , 406-417. Proga, D., & Kallman, T. R. (2002). On the Role of the Ultraviolet and X-Ray Radiation in Driving a Disk Wind in X-Ray Binaries. ApJ , 565 , 455-470. Reynolds, C. S. (2012). Constraints on Compton-thick Winds from Black Hole Accretion Disks: Can We See the Inner Disk? ApJL , 759 , L15-L19. Reynolds, M. T., & Miller, J. M. (2010). Chandra Grating Spectroscopy of the Be/X-ray Binary 1A 0535+262. ApJ , 723 , 17991805. Schulz, N. S., & Brandt, W. N. (2002). Variability of the X-Ray P Cygni Line Profiles from Circinus X-1 near Zero Phase. ApJ , 572 , 971-983. Sidoli, L., Oosterbroek, T., Parmar, A. N., et al. (2001). An XMMNewton study of the X-ray binary MXB 1659-298 and the discovery of narrow X-ray absorption lines. A&A , 379 , 540-550. Ueda, Y., Honda, K., Takahashi, H., et al. (2010). Suzaku Observation of GRS 1915+105: Evolution of Accretion Disk Structure during Limit-cycle Oscillation. ApJ , 713 , 257-268. Ueda, Y., Murakami, H., & Yamaoka, K., et al. (2004). Chandra High-Resolution Spectroscopy of the Absorption-Line Features in the Low-Mass X-Ray Binary GX 13+1. ApJ , 609 , 325-334. Ueda, Y., Yamaoka, K., & Remillard, R. (2009). GRS 1915+105 in 'Soft State': Nature of Accretion Disk Wind and Origin of X-ray Emission. ApJ , 695 , 888-899. Woods, D. T., Klein, R. I., & Castor, J. I. (1996). X-Ray-heated Coronae and Winds from Accretion Disks: Time-dependent Twodimensional Hydrodynamics with Adaptive Mesh Refinement. ApJ , 461 , 767-804.", "pages": [ 7, 8 ] } ]

[ { "title": "Modern cosmology: Interactive computer simulations that use recent observational surveys", "content": "Jacob Moldenhauer 1 ∗ , Larry Engelhardt 1 , Keenan M. Stone 1 , Ezekiel Shuler 1 1 Department of Physics and Astronomy, Francis Marion University, Florence, South Carolina 29506 (Dated: October 29, 2018) We present a collection of new, open-source computational tools for numerically modeling recent large-scale observational data sets using modern cosmology theory. Specifically, these tools will allow both students and researchers to constrain the parameter values in competitive cosmological models, thereby discovering both the accelerated expansion of the universe and its composition (e.g., dark matter and dark energy). These programs have several features to help the non-cosmologist build an understanding of cosmological models and their relation to observational data: a built-in collection of several real observational data sets; sliders to vary the values of the parameters that define different cosmological models; real-time plotting of simulated data; and χ 2 calculations of the goodness of fit for each choice of parameters (theory) and observational data (experiment). The current list of built-in observations includes several recent supernovae Type Ia surveys, baryon acoustic oscillations, the cosmic microwave background radiation, gamma-ray bursts, and measurements of the Hubble parameter. In this article, we discuss specific results for testing cosmological models using these observational data. These programs can be found at http://www.compadre.org/osp/ items/detail.cfm?ID=12406 .", "pages": [ 1 ] }, { "title": "I. INTRODUCTION", "content": "In recent decades, the field of cosmology-both observational data and theoretical models-has provided two very significant insights regarding the nature of our universe. One discovery, which earned the 2011 Nobel Prize in Physics 1-3 , is that our universe is not just expanding; it is expanding at a rate that is increasing with time. (i.e., the matter in the universe is experiencing a repulsion that overcomes the attractive force of gravity.) The other major discovery is that-in order to successfully model the (numerous) recent astronomical observations-our universe must be composed of mostly dark matter and dark energy, with only 4% ordinary matter (e.g., atoms) 4 . Clearly, these results are of intrinsic interest and should be understood by people outside the field of cosmology. In fact, several recent articles in nonspecialist journals have discussed these latest findings 5-7 . Unfortunately, resources have not existed that allow the broader physics community (non-cosmologists) to appreciate how cosmological observations inform cosmological models, ultimately leading to these new insights. In the present work, we seek to address this need. In order to help non-cosmologists to understand how recent observations lead to the discoveries described above, we provide a medley of open-source, user-friendly cosmological modeling programs, which we will refer to as ' COSMOEJS ' 8 . These programs allow the user to immediately become an amatuer cosmologist by fitting theoretical models to the actual experimental data and visually observing how well each model agrees with the various observational data. Key features of COSMOEJS include a built-in collection of several real observational data sets; sliders to vary the values of the parameters that define different cosmological models; real-time plotting of simulated data; and χ 2 calculations of the goodness of fit for each choice of parameters (theory) and observational data (experiment). Taking advantage of these features, COSMOEJS has already been used with a variety of noncosmologist audiences to bridge the gap between modern cosmology and mainstream physics. In Sec. II we describe some specific examples of modeling cosmology with experimental observations using COSMOEJS , and Sec. III contains a summary of our results. We also provide a brief introduction to cosmology in the Appendix A, (a more detailed introduction is provided in supplementary material 9 ). The relevant mathematical quantities and equations are introduced in subsection A 1 of the Appendix, and the relevant experimental observations are introduced in subsection A 2 of the Appendix.", "pages": [ 1 ] }, { "title": "II. MODELING COSMOLOGY", "content": "COSMOEJS allows non-cosmologists to simulate the expansion of the universe using various models and to compare the simulated results to experimental observations. Recent articles in non-specialized physics journals have discussed some of the complexities associated with modeling cosmology 5-7 , but these articles only considered specific scenarios from the small subset of cosmological models that permit exact analytical solutions. Moreover, it would be very difficult for a non-cosmologist to recreate or extend those results. Some web-based 11 and mobile device applications 12 do provide 'cosmology calculators' for calculating times and distances of simple models, but do not compare to data, or allow for the diversity of models contained COSMOEJS . Very powerful numerical simulations for testing cosmological models and constraining the values of model parameters do already exist 13,14 , but these tools have a steep learning curve, making them impractical for use by non-specialists. COSMOEJS addresses all of these needs. It provides direct comparisons between theory and experiment in the form of both plots and numbers, and it accurately carries out the complex mathematics without requiring technical expertise from the user. This allows the user to focus on developing a high-level understanding of cosmology, without technical (mathematical and computational) distractions. The process of using COSMOEJS is very straightforward, and we encourage the user to download the program from Ref. 8 in order to recreate and modify the plots that are discussed later in this section. Using COSMOEJS consists of five steps: (1) loading observational data, (2) selecting values for the model parameters, (3) calculating and plotting theoretical observables, (4) assessing the goodness of fit both visually and numerically, and (5) plotting the expansion of the universe for the user-defined model. For step (1), the user can select up to 18 different experimental datasets, and a drop-down menu is provided to simplify the process of loading data (see subsection A 2 of the Appendix for a description of these experimental data) For step (2), users can use sliders to adjust parameter values, subsequently changing from one model to another. For step (3), the calculations are carried out using Romberg's method of approximating integrals, and the user can easily vary the number of partitions to test for convergence of the numerical integration. For step (4), each time that the user changes the model's parameters, several plots are generated; and for each plot, χ 2 is automatically calculated to provide a quantitative measure of the goodness of fit. Finally, for step (5), the size of the universe can be plotted versus time to see what type of universe results from each set of parameters; i.e., is the expansion of the universe constant, or accelerating, or decelerating?", "pages": [ 1, 2 ] }, { "title": "A. Fitting the model", "content": "In this subsection, we demonstrate the modeling capabilities of COSMOEJS by comparing experimental data (astronomical observations) with theoretical curves for a few specific examples that do not permit analytical solutions. The experimental data consist of measurements of Type 1a supernovae (SNe), the Hubble parameter, H ( z ), gamma ray bursts (GRB), baryon acoustic oscillations (BAO), and the cosmic microwave background (CMB). (Each of these observations is described in subsection A 2 of the Appendix.) The simulated data in this section consist of three physically different models, each described by a different set of parameter values. Specifically, in each model, the universe is chosen to have the same expansion rate today, but different fractions of matter, Ω m , and dark energy, Ω Λ : { Ω m , Ω Λ } = { 0 . 01 , 0 . 99 } , { 0 . 27 , 0 . 73 } , and { 1 . 0 , 0 . 0 } . Throughout this section data are plotted versus red- shift. It is important to note that redshift, z , can be used as a measure of time. 29 Light from nearby objects experiences very little redshift ( z ≈ 0), and this light was also emitted very recently (in terms of cosmological timescales). Light from far-away objects experiences a larger redshift, and this light was emitted longer ago. 30 We take advantage of this redshift/time relationship in multiple ways. Given a theoretical model [Eq. (A1) in subsection A 1 of the Appendix], COSMOEJS uses redshift values to calculate and display both the age of the universe today and the 'look-back' time, which refers to how long ago the light was emitted that is observed to have a certain redshift. Also, the experimental data are measured using redshift (x-axis), which provides a means of 'dating' these observational data once a particular model has been selected. In Fig. 1, distance modulus, µ , is plotted versus z for two different types of observations, SNe and GRB; and these observations are compared to the three different models. ( µ is a normalized measure of the distance to an observed object.) We note that all three models provide a reasonably good fit to some of the data in Fig. 1, but there are also differences between the three curves that are clearly visible, and if the plot were scaled in (zoomed in) for low redshift, we would see more differences. For a model with more matter (more gravity), the matter density would use gravity to try to pull the universe together, subsequently slowing the expansion rate, so the objects in the universe would be closer together (bottom curve). Conversely, a universe with too much dark energy would expand the universe too quickly, and objects would be at greater distances than what is observed (top curve). In Fig. 2, the expansion rates, H ( z ), of different galaxies are plotted as a function of their redshift, z . These data are compared with the same three models that are used for Fig. 1, and again, all three the models have been defined to have the same expansion rate today, i.e., H ( z ) ≡ H 0 for z = 0. From these data it is clear that the middle curve (Ω m = 0 . 27, Ω Λ = 0 . 73) gives a better fit than the other two models. This plot correctly displays that a model universe comprised of mostly dark energy (bottom curve) would have a slightly increasing expansion rate as a function of redshift for the range of redshift seen in Fig. 2, whereas a universe with mostly matter would have an expansion rate that drastically decreases with increasing time (decreasing z ). In Figure 3, the three models show a clear difference when compared to the BAO ratio. Physically, the BAO ratio reflects the size of the sound horizon, r s , for the early universe baryon decoupling (see subsection A 2 of the Appendix) to its effective distance, D v in the galaxies today. In Figs. 1 and 2, the two extreme models (upper and lower curves) at least fit some of the data, but they do not come close to the extremely precise (small error bars) BAO ratio data. With the addition of this third observation, it is obvious that only the middle curve fits well to all of the complementary data sets. (By 'complementary,' we mean that the cosmological parameters must be consistent with different observations that constrain different theoretical observables.) The user can easily vary additional parameters, subsequently adding to the complexity of the models, by simply adjusting the sliders for the different parameter values. For example, by varying the value of the current expansion rate, H 0 , while looking at the same compositions, the changing of the expansion rate will uniformly scale all of the theoretical data points. In addition, the user can choose different spatial curvatures for the uni- verse, Ω k , and different types of dark energy { w 0 , w a } models.", "pages": [ 2, 3 ] }, { "title": "B. Cosmological interpretation", "content": "For the cosmological interpretation of these fits, we proceed to plotting the evolution of the expansion of the universe versus to time, t (in Gyrs = 10 9 years), rather than redshift, z . Redshift is a model-independent measurement, which makes it an ideal quantity for the fitting that was done in Sec. II A. However, redshift has a non-linear, model-dependent relationship with time, which makes it very difficult to physically interpret data that are plotted versus z . 31 For this reason, we now plot the expansion of the universe versus time for each of the three models that were studied in Sec. II A. Specifically, we plot the dimensionless ratio a ( t ) /a ( t today ), where a ( t ) represents the expansion function or radius (size) of the universe. It is also referred to as the 'scale factor,' and mathematically, it is defined as a = 1 / (1 + z ). 32 In Figure 4, we provide plots of all three models studied in Sec. II A. These plots can be interpreted to give physical insight to the expansion at a particular time. We know from the fitting in Sec. II A that the model with all matter density did not match the data, and if we look at its expansion, we can see the slope decreases monotonically with increasing time, so the rate of the universe's expansion is decreasing, which would be caused by the gravitational pull of the matter dominating the universe. This slowed expansion yields an age of only ∼ 9 Gyrs. For the other extreme case studied in Sec. II A, we see a very early inflection point where the expansion rate went from decreasing to accelerating, yielding a much older universe of ∼ 28 Gyrs. This acceleration is caused by the domination of the dark energy as the universe expands. Finally, for the model which fit all of the observations in Sec. II A, we see an inflection point at ∼ 9 Gyrs, which is caused by the domination of the dark energy as the universe expands and the gravitational pull from matter is weakened. This model calculates a current age of the universe to be ∼ 14 Gyrs. According to the combinations of observations in the current literature 4,15,17,18 , the best-fit values for the parameters in COSMOEJS are { H 0 ∼ 70 . 0 km/(s Mpc), Ω b ∼ 0 . 04, Ω c ∼ 0 . 23, Ω m ∼ 0 . 27, Ω Λ ∼ 0 . 73, Ω k ∼ 0 . 0, w 0 ∼ -1 . 0, w a ∼ 0 . 0 } , representing the so-called Λ Cold Dark Matter (ΛCDM) model. With COSMOEJS , it is possible to find these values by systematically trying different sets of parameters with combinations of the data sets. Then, a physical interpretation of the model's fit throughout its evolution can be made to compare with the cosmological observations 33 .", "pages": [ 3, 4 ] }, { "title": "III. CONCLUSIONS AND FUTURE WORK", "content": "COSMOEJS is a powerful new tool for cosmology education, and it is also precise enough to perform research grade calculations for testing most cosmological dark energy models. They also allow the user to select inputs for parameters that are perhaps not scientifically accepted. This allows the user to discover how parameters influence the shape of the curve for a particular theoretical model, thereby understanding the physical interpretation of a model's fit to the data. Variations of the programs have been used for science outreach and for classroom illustration. Future versions of the programs will involve a minimization method for the fitting of the cosmological models to the data sets to provide best-fit cosmological parameters. However, this will involve a different fitting method for each survey. We decided not to provide minimization in this version because this would distract from the pedagogical value of the program. Namely, when trying to find a best-fit model with a minimization routine, the user is not required to understand the physical interpretation of one fit over another. The fit is obtained by statistically comparing one model fit to another. Also, if the statistical fit is biased in some way, as explained in the examples above for χ 2 fits, then the best fit could have unphysical parameter constraints. In the current version of the simulation, we are more concerned with understanding the physical interpretation of fitting particular cosmological models to data sets. COSMOEJS allows non-specialists to manipulate cosmological models via their parameters and learn how to fit the model to experimental data sets. This manual process of changing parameter values also allows the user to see what parameter values do not fit the data. The programs are useful for not only learning about cosmology but also data fitting itself, both visually and numerically. The programs will continue to receive updates and modifications for new, more precise data sets as these become publicly available. Using the ΛCDM model with this version of COSMOEJS , we find excellent fits to all the data sets with { H 0 = 70 . 0 km/(s Mpc), Ω b = 0 . 045, Ω c = 0 . 225, Ω m = 0 . 27, Ω Λ = 0 . 73, Ω k = 0 . 0, w 0 = -1 . 0, w a = 0 . 0 } .", "pages": [ 4 ] }, { "title": "Appendix A: Cosmological Background", "content": "In this appendix, we include a brief tutorial into cosmology. In subsection A 1 of the Appendix, we discuss the mathematics of the theoretical cosmology in COSMOEJS , followed by a description of the observations in subsection A 2 of the Appendix. For a more detailed description of cosmology, see the supplementary information in Ref. 9. The cosmic acceleration of the universe can be explained by a cosmological constant, or some other form of repulsive dark energy, i.e. a negative pressure and a negative equation of state, or by an extension or modification to gravity at cosmological scales of distances 20 . In the context of general relativity (GR), to account for this dark energy effect, the addition of a Λ term (cosmological constant) to Einstein's Field Equations (EFE) can be used to derive equations of motion with a cosmological constant of the desired value consistent with the dynamics of Friedmann-Lemaitre-Robertson-Walker (FLRW) universe 34 . We provide a means of testing this commonly accepted model of the Universe and others with observations of SNe Type Ia, gamma-ray bursts (GRB), baryon acoustic oscillations (BAO), the distance to the last scattering surface of the cosmic microwave background (CMB) radiation, and measurements of the Hubble expansion rate parameter, H ( z ), thereby deriving the parameters for the standard model in cosmology.", "pages": [ 4, 5 ] }, { "title": "1. Mathematics", "content": "In this section, we define the mathematics behind the theoretical models involved in COSMOEJS . Specifically, the programs assume a big bang physical universe, a mathe- matical model according to general relativity (GR), and a uniformly distributed spacetime in all directions 35 . From these assumptions, the programs numerically integrate an equation of motion for the dynamical evolution of the expansion rate of the universe 36 . This equation of motion can be expressed in terms of the Hubble expansion rate, H ( z ), as a function of redshift, z . [The theoretical details of the integration of H ( z ) for a particular observation are described in Ref. 9, and the numerical implementation is shown in Fig. 1 of the appendix in Ref. 10.] Explicitly, we use This equation contains all of the parameters that can be varied in COSMOEJS . (All parameters are dimensionless except H 0 ). Briefly, these parameters and their currently accepted values are: As the values of these parameters change, Eq. (A1) describes different types of evolutions for the universe. The details of these parameters are further explained in the next paragraph. The Hubble Constant parameter, H 0 , represents the current value of the expansion rate for the universe. The k in Ω k appearing in Eq. (A1) represents the three types of curvature for the spacetime of the universe as open, flat, or closed ( k = -1 , 0 , 1, respectively). See Fig. 5. This is an inherent curvature of the empty spacetime itself, devoid of any matter or energy. However, as can be seen, the model does not use k directly, but rather the fractional curvature density parameter, Ω k . The total energy density of the universe is split up into fractional pieces to represent its different compositional quantities for matter, dark energy and curvature. The total energy density, Ω 0 , as measured today ( z = 0), accounts for the sum total of all of the matter and energy in the universe. A critically dense universal model (typically accepted in cosmology 37 ), indicating that all matter and energy are accounted for, can be described by the relation Note the curvature parameter, k , has a negative sign ( -) originating from the GR spacetime equations of motion 21 . However, the fractional curvature parameter, Ω k , has the relationship 19 such that The fractional matter density, Ω m , represents the total matter density in the universe. It can be separated into its constituents, Ω m = Ω b + Ω c , into fractional baryon density, Ω b , and fractional cold dark matter density, Ω c , when the observation can constrain the distinction (only BAO and CMB observations 27 ). Note, for all values of Ω m that were used in Sec. II, we kept the constituents the same proportionate percentages of Ω m as the currently accepted values listed above. The Λ represents the cosmological constant, the simplest model of dark energy. For changing the model of dark energy, the model uses an equation of state parameterized as w ( z ) = w 0 + w a [ z/ (1 + z )] 23 . The equation of state for dark energy includes, w 0 , which is the value measured today ( z = 0), and its derivative, w a , that allows dark energy to evolve in time, z , as the universe evolves. For the special condition when w a = 0 . 0, the equation of state is constant, i.e. the density of dark energy does not change with time. Additionally, when w 0 = -1, this corresponds to the cosmological constant model.", "pages": [ 5, 6 ] }, { "title": "2. Experimental Observations", "content": "Within the COSMOEJS package, we include 18 different experimental datasets for five different types of measurements: Type Ia supernovae (SNe), gamma-ray bursts (GRB), measurements of the Hubble expansion rate parameter, H ( z ), baryon acoustic oscillations (BAO), and the cosmic microwave background (CMB) radiation. In cosmology, the evolution of the universe is modeled on scales too large to measure the evolution of a single galaxy or galaxy cluster from its formation to the present. Instead, cosmology combines observations of different galaxies-and the phenomena contained therein-at different times and distances in their evolution to piece together the dynamics of the universe. These different observations and surveys provide independent and complementary measurements of the expansion history of the universe and its composition. According to the big bang theory, the universe began from an initial state of extremely high temperature, density and energy. When the universe had expanded and cooled enough for the photons to decouple from the primordial soup of energy, they no longer scattered and were free to propagate throughout the universe, allowing for their detection. This surface is as far back as scientists can currently make measurements because all the measurements involve some frequency of light. These high energy photons have been stretched with the expansion of the universe into microwaves. So, now the temperature of the universe has cooled to ≈ 3 K. Working backwards, this gives a temperature of ≈ 380 , 000 K for the last scattering surface. We use the three fitting parameters for amplitude, and locations of the acoustic peaks of the CMB temperature power spectrum 4 : 1) the acoustic scale, l a ( z ∗ ), 2) the shift parameter, R ( z ∗ ), 3) the redshift of the surface of last scattering (SLS) of the CMB, z ∗ . (For an accessible introduction to the CMB power spectrum and how temperature oscillations become acoustic peaks, see Ref. 16.) Physically the acoustic scale and shift parameter are the size, shape and position of the acoustic peaks of the CMB power spectrum for different values of the SLS redshift. The size, shape and positions of these acoustic peaks are very powerful in determining the values of the cosmological parameters, due to the complexity of the many peaks in the CMB power spectrum and their ratios to each other. The SNe are standard 'candles' (similar luminosity) used to form a redshift-distance relation, µ ( z ) (distance modulus), to measure the rate of the expansion of the universe. According to these measurements, galaxies at large distances, in which the SNe reside, are receding less rapidly than Hubble's law would predict. This translates to a slower expansion rate in the past, and that the nearby, later time SNe are expanding faster than the more distant, older SNe. Therefore, we are observing an accelerating universe. GRBs are added to fill the large void of redshift between the highz SNe and the redshift of the CMB's last scattering surface, z ≈ 1089. SNe are subject to extinction from the dust of the interstellar medium, however, GRBs are much brighter and, due to the high energy of gamma-ray photons, are rarely affected by the dust. The measurement of the GRB extends the redshift-distance relationship to higher redshift, although, there is a redshift range of overlapping measurements for comparison and consistency. The measurements of the Hubble Parameter, H ( z ) are an independent measurement of the expansion history of the universe. All of the other cosmological observations given in this article require an integration, but H ( z ) is an exact evaluation and comparison of Eq. (A1) to the experimental data. In fact, H ( z ) is actually a direct measurement of the differential age of the universe, ∆ z/ ∆ t , in other words measuring how the age of the universe changes as the redshift changes using the age differences of old elliptical galaxies that are passively evolving 17 . They are used as standard 'clocks' to directly probe the Hubble parameter. We compare the ratio of the sound horizon at the drag epoch, r s ( z d ), or when the baryons decoupled from the primordial universe to its effective distance, D v ( z ) in the galaxy redshift surveys. This decoupling occurs at a slightly later time and lower redshift than the photon decoupling because the baryons get 'stuck' in gravitational potential wells. The correlations in the galaxy redshift surveys consistently have a 'bump' corresponding to the standard 'ruler' measurement of the BAO. This measures the expansion of the primordial sound horizon in the galaxy redshift surveys. As an example of this physical ratio, the sound horizon at decoupling in the range of r s = 153 . 19 Mpc, and effective distance D v ( z = 0 . 57) = 2026 . 49 Mpc, for a ratio of r s /D v = 0 . 076, 18 . Physically, this reflects the size and shape of the acoustic peak, and how it has evolved with the expansion of the universe. The BAO is specially suited for constraining { Ω b , Ω c } with galaxy clusters because of the sensitivity of the sound horizon redshift to these parameters. Equation (A1) can be used to compare several dark energy cosmological models to these observations by allowing different values for the parameters, { Ω m (= Ω b + Ω c ) , Ω Λ , Ω k , H 0 , w 0 , and w a } . While a model may fit one observation, cosmology involves the entire evolution of the universe, so it is important to use all the cosmo- logical observations, covering several redshift and cosmological epochs, to find a best-fit model.", "pages": [ 6, 7 ] }, { "title": "Acknowledgments", "content": "The authors wish to thank W. Christian and F. Esquembre for adding new features to EJS for these programs upon request. We also thank D. Jokisch for useful comments. JM would especially like to thank M. Ishak, W. Rindler and The University of Texas at Dallas Cosmology, Relativity and Astrophysics group for his training in computational and theoretical cosmology. Part of this work was completed under the SC Space Grants Palmetto Academy Program, Award NNG05G168G. release 7 galaxy sample,' Mon. Not. R. Astron. Soc.Volume 401 , pp. 2148-2168 (2010), arXiv:0907.1660; H. Wei, 'Observational constraints on cosmological models with the updated long gamma-ray bursts,' J. Cosmol. Astropart. Phys.JCAP1008:020, pp. 020-1 - 020-24 (2010); F. Beutler, et al., 'The 6df galaxy survey: Baryon acoustic oscillations and the local Hubble Constant,' Mon. Not. R. Astron. Soc. 416 pp. 3017-3032, (2010) arXiv:1106.3366; C. Blake et al., 'The WiggleZ dark energy survey: the growth rate of cosmic structure since redshift z = 0 . 9,' Mon. Not. R. Astron. Soc., 415 , pp. 28762891, (2011); C. Blake et al., 'The WiggleZ dark energy survey: testing the cosmological model with baryon acoustic oscillations at z = 0 . 6,' Mon. Not. R. Astron. Soc., 415 , pp. 2892-2909, (2011).", "pages": [ 7, 8 ] } ]

[ { "title": "PENROSE TYPE INEQUALITIES FOR ASYMPTOTICALLY HYPERBOLIC GRAPHS", "content": "MATTIAS DAHL, ROMAIN GICQUAUD, AND ANNA SAKOVICH Abstract. In this paper we study asymptotically hyperbolic manifolds given as graphs of asymptotically constant functions over hyperbolic space H n . The graphs are considered as unbounded hypersurfaces of H n +1 which carry the induced metric and have an interior boundary. For such manifolds the scalar curvature appears in the divergence of a 1-form involving the integrand for the asymptotically hyperbolic mass. Integrating this divergence we estimate the mass by an integral over the inner boundary. In case the inner boundary satisfies a convexity condition this can in turn be estimated in terms of the area of the inner boundary. The resulting estimates are similar to the conjectured Penrose inequality for asymptotically hyperbolic manifolds. The work presented here is inspired by Lam's article [22] concerning the asymptotically Euclidean case. Using ideas developed by Huang and Wu in [19] we can in certain cases prove that equality is only attained for the anti-de Sitter Schwarzschild metric.", "pages": [ 1 ] }, { "title": "1. Introduction", "content": "In 1973, R. Penrose conjectured that the total mass of a space-time containing black holes cannot be less than a certain function of the sum of the areas of the event horizons. Black holes are objects whose definition requires knowledge of the global space-time. Hence, given Cauchy data (which are the only data needed to define the total mass of a space-time), finding event horizons would require solving the Einstein equations. As a consequence, in the current formulation of the Penrose conjecture, event horizons are usually replaced by the weaker notion of apparent horizons. We refer the reader to [9, Chapter XIII] for further details. The classical Penrose conjecture takes the following form: Let ( M,g,k ) be Cauchy data for the Einstein equations, that is a triple where ( M,g ) is a Riemannian 3-manifold and k is a symmetric 2-tensor on M . Assume that ( M,g,k ) satisfies the dominant energy condition where µ and J are defined through Assume further that ( M,g,k ) is asymptotically Euclidean. A compact oriented surface Σ ⊂ M is called an apparent horizon if Σ satisfies where H g is the trace of the second fundamental form of Σ computed with respect to the outgoing normal ν of Σ, that is S ( X,Y ) = 〈∇ X ν, Y 〉 for any vectors X and Y tangent to Σ, and tr Σ k is the trace of k restricted to the tangent space of Σ for the metric induced by g . Hence viewing ( M,g,k ) as immersed in a space-time, the expansion of Σ in the future outgoing light-like direction vanishes. We assume that Σ is outermost, that is Σ contains all other apparent horizons in its interior. Note that Σ may be disconnected. See [2] for further details. Then the Penrose conjecture takes the form where | Σ | denotes the area of Σ and m is the mass of the manifold ( M,g ). Further, equality should hold only if the exterior of Σ is isometric to a hypersurface in the exterior region of a Schwarzschild black hole with k equal to the second fundamental form of this hypersurface. This conjecture can be generalized to higher dimensional manifolds. All the statements are the same except for the inequality which in n dimensions reads where ω n -1 is the volume of the unit ( n -1)-sphere. Two major breakthroughs in the proof of this inequality were obtained almost simultaneously by Huisken, Ilmanen [21] and Bray [4] for 3-manifolds. They both deal with the time-symmetric case, i.e. when k = 0. The result of Bray was extended to higher dimensions in [6]. We refer the reader to the excellent reviews [23] and [5] for further details. Recently, Lam [22] gave a simple proof of the time-symmetric Penrose inequality for a manifold which is a graph of a smooth function over R n . His proof was extended by Huang and Wu in [20] to give a proof of the positive mass theorem (including the rigidity statement) for asymptotically Euclidean manifolds which are submanifolds of R n +1 . More general ambient spaces were considered by de Lima and Gir˜ao in [11]. The Penrose conjecture can be generalized to space-times with negative cosmological constant. Up to rescaling, we can assume that the cosmological constant Λ equals -n ( n -1) 2 . Restricting ourselves to the time-symmetric case, the dominant energy condition then reads The lower bound for the mass (defined in Section 2.1) is then conjectured to be given by the mass of the anti-de Sitter Schwarzschild space-time (see Section 2.3), In this paper, we prove weaker forms of this inequality for manifolds which are graphs over the hyperbolic space H n when we endow the manifold H n × R with a certain hyperbolic metric. See Theorem 2.1. After the first version of this article appeared on arXiv, de Lima and Gir˜ao posted an article dealing with another case of the asymptotically hyperbolic Penrose inequality [13]. Rigidity was addressed by de Lima and Gir˜ao in [14] and by Huang and Wu in [19]. The approach used in [19] does not require any further assumption and we shall extend it to our context in Section 5. The outline of this paper is as follows. In Section 2.1, we define the mass of a general asymptotically hyperbolic manifold. We explicit the anti-de Sitter Schwarzschild metric in Section 2.3. In Section 3 we prove that the scalar curvature of a graph has divergence form (Equation (7)) and that its integral is related to the mass (Lemma 3.2). In Section 4, we prove the first part of Theorem 2.1. Rigidity is addressed in Section 5. Acknowledgements. We thank Julien Cortier and Hubert Bray for helpful conversations. We are also grateful to Gerhard Huisken for enlightening discussions on the Aleksandrov-Fenchel inequalities and to Lan Hsuan-Huang for pointing us to the article [19]. Further, we want to give a special thanks to Christophe Chalons and Jean-Louis Tu who helped us with the proof of the results stated in Appendix A. A note. After this paper was finished the articles [12] by de Lima and Gir˜ao, and [7] by Brendle, Hung, and Wang appeared on arXiv. In the first of these papers an Alexandrov-Fenchel type inequality for hypersurfaces in hyperbolic space is stated, which together with Proposition 4.1 implies the Penrose inequality (1) for graphs. Certain steps of the proof seem to need further clarification, for example the convergence of hypersurfaces to round spheres under the inverse mean curvature flow. However, combining with arguments of the second paper [7] the result should follow. Note also that a special case of [7, Theorem 2] follows from our formula (13) in Section 4.2.", "pages": [ 2, 3 ] }, { "title": "2. Preliminaries", "content": "Set N := { V ∈ C ∞ ( H n ) | Hess b V = V b } . This is a vector space with a basis of the functions where x 1 , . . . , x n are the coordinate functions on R n restricted to S n -1 . If we consider H n as the upper unit hyperboloid in Minkowski space R n, 1 then the functions V ( i ) are the restrictions to H n of the coordinate functions of R n, 1 . The vector space N is equipped with a Lorentzian inner product η characterized by the condition that the basis above is orthonormal, η ( V (0) , V (0) ) = 1, and η ( V ( i ) , V ( i ) ) = -1 for i = 1 , . . . , n . We also give N a time orientation by specifying that V (0) is future directed. The subset N + of positive functions then coincides with the interior of the future lightcone. Further we denote by N 1 the subset of N + of functions V with η ( V, V ) = 1, this is the unit hyperboloid in the future lightcone of N . All V ∈ N 1 can be constructed as follows. Choose an arbitrary point p 0 ∈ H n . Then the function", "pages": [ 4 ] }, { "title": "is in N 1 .", "content": "A Riemannian manifold ( M,g ) is said to be asymptotically hyperbolic if there exist a compact subset and a diffeomorphism at infinity Φ : M \\ K → H n \\ B , where B is a closed ball in H n , for which Φ ∗ g and b are uniformly equivalent on H n \\ B and where e := Φ ∗ g -b and r is the (hyperbolic) distance from an arbitrary given point in H n . The mass functional of ( M,g ) with respect to Φ is the functional on N defined by Proposition 2.2 of [10] tells us that these limits exist and are finite under the asymptotic decay conditions (2a)-(2b). If Φ is a chart at infinity as above and A is an isometry of the hyperbolic metric b then A · Φ is again such a chart and it is not complicated to verify that If Φ 1 , Φ 2 are charts at infinity as above, then from [17, Theorem 2.3] we know that there is an isometry A of b so that Φ 2 = A · Φ 1 modulo lower order terms which do not affect the mass functional. The mass functional H Φ is timelike future directed if H Φ ( V ) > 0 for all V ∈ N + . In this case the mass of the asymptotically hyperbolic manifold ( M,g ) is defined by Further if H Φ is timelike future directed we may replace Φ by A · Φ for a suitably chosen isometry A so that m = H Φ ( V (0) ). Coordinates with this property are called balanced . The positive mass theorem for asymptotically hyperbolic manifolds, [10, Theorem 4.1] and [31, Theorem 1.1], states that the mass functional is timelike future directed or zero for complete asymptotically hyperbolic spin manifolds with scalar curvature Scal ≥ -n ( n -1). In [1, Theorem 1.3] the same result is proved with the spin assumption replaced by assumptions on the dimension and on the geometry at infinity. To shorten notation we now fix for the rest of the paper. As a model of H n +1 we take H n × R equipped with the metric Let Ω be a relatively compact open subset and let f : H n \\ Ω → R be a continuous function which is smooth on H n \\ Ω. We consider the graph Define the diffeomorphism Φ : Σ → H n \\ Ω by Φ -1 ( p ) = ( p, f ( p )). The push-forward of the metric induced on Σ is We will study the situation when the graph Σ is asymptotically hyperbolic with respect to the chart Φ, that is when satisfies (2a)-(2b) and Note that Condition (2a) is a consequence of the following condition: that is to say that df belongs to a certain weighted Sobolev space. If this holds we say that f is an asymptotically hyperbolic function and Σ is an asymptotically hyperbolic graph . We define f to be balanced at infinity if Φ are balanced coordinates at infinity. In this case the mass of Σ is given by m = H Φ ( V ) with V = V (0) . In this paper we will prove the following theorem which gives estimates similar to the Penrose inequality for asymptotically hyperbolic graphs. In certain cases this theorem also describes the situation when equality is attained. For exact formulations see Theorem 4.2, Theorem 4.4, and Theorem 5.13. Theorem 2.1. Let Ω ⊂ H n be a relatively compact open subset of H n with smooth boundary. Assume that Ω contains an inner ball centered at the origin of radius r 0 . Let f : H n \\ Ω → R be an asymptotically hyperbolic function which is balanced at infinity. Assume that f is locally constant on ∂ Ω and that | df | → ∞ at ∂ Ω so that ∂ Ω is a horizon ( H g = 0 ). Further assume that the scalar curvature of the graph of f satisfies Scal ≥ -n ( n -1) . Then the mass m of the graph is bounded from below as follows. and If equality holds and df ( η )( x ) → + ∞ as x → ∂ Ω where η is the outward normal of the level sets of f then the graph of f is isometric to the t = 0 slice of the anti-de Sitter Schwarzschild space-time. Note that since f is locally constant on ∂ Ω, the areas of ∂ Ω computed using the metric b and using the metric induced on the graph are equal. where σ is the standard round metric on the sphere S n -1 . See for example [23, Section 6]. The horizon is the surface ρ = ρ 0 ( m ), where ρ 0 = ρ 0 ( m ) is the unique solution of Its area is given by A m = ω n -1 ρ n -1 0 , hence multiplying the previous formula by ρ n -2 0 , we get This justifies the form of the right-hand side of (1). Restricting to the slice t = 0, we get the following Riemannian metric. We want to explicit the spatial metric (4) as the induced metric of a graph Σ AdS-Schw . By rotational symmetry, we choose the point ρ = 0 as the origin and f = f ( ρ ). In this coordinate system, the reference hyperbolic metric b is given by The function V is given by V = √ 1 + ρ 2 . Hence we seek a function f satisfying Note that when ρ is close to ρ 0 , this forces ∂f ∂ρ = O (( ρ -ρ 0 ) -1 2 ). Hence we can set Similarly, when ρ → ∞ , f converges to a constant. This contrasts with the Euclidean case where the Schwarzschild metric written as a graph is a half parabola in any radial direction, see [22].", "pages": [ 4, 5, 6, 7 ] }, { "title": "3. Scalar curvature of graphs in H n +1", "content": "3.1. Computation of scalar curvature. Let f : H n \\ Ω → R be a smooth function. Recall that we defined its graph as where F ( x, s ) := f ( x ) -s . For vector fields X and Y on H n the vector fields X = X + ∇ X f∂ 0 and Y = Y + ∇ Y f∂ 0 are tangent to Σ. We use coordinates on H n to parametrize Σ. Recall that we identify H n +1 with H n × R with the metric b = b + V 2 ds ⊗ ds . When using coordinate notation, latin indices i, j, . . . ∈ { 1 , . . . , n } denote (any) coordinates on H n while a zero index denotes the s -coordinate on R . Greek indices go from 0 to n , hence denote coordinates on H n +1 . The Christoffel symbols of b are collected in the following Lemma.", "pages": [ 7 ] }, { "title": "Lemma 3.1.", "content": "The induced metric on Σ is given by The second fundamental form S of Σ is given by Using component notation we get The metric g and its inverse are given by We compute the mean curvature of Σ, or The norm of the second fundamental form of Σ is given by We compute each term separately. First Next, and finally Hence and By taking the trace of the Gauss equation for Σ, we finally arrive at the following formula for the scalar curvature Scal of Σ In view of [22, Proof of Theorem 5] and [17, Definition 3.3], we compute with e = V 2 df ⊗ df . We have .", "pages": [ 8, 9, 10, 11 ] }, { "title": "Further,", "content": "div b V div V | 1 2 df 1 2 ( 2 b e - (∆ f ) V d tr 2 -| b e - Hess f · ) + (tr 4 V b e ) dV Hess f, df dV +4 V df, dV ∆ f | | [ ( ∇ e ( 2 2 ) Comparing this formula with Equation (6) we get | 2 〈 3 V, - [ 1 + V = 1 + V 2 | df ) ] ⊗ 〉 2 〈 〉 where e = V 2 df ⊗ df . 3.2. A mass formula. We now integrate Formula (7) from the previous section over an outer domain under the additional condition that f is locally constant on the boundary. Lemma 3.2. Let Ω ⊂ H n be a relatively compact open subset of H n with smooth boundary. Let f : H n \\ Ω → R be an asymptotically hyperbolic function which is locally constant on ∂ Ω and such that df = 0 at every point of ∂ Ω . Then /negationslash Proof. Let ν denote the outgoing unit normal to ∂ Ω and let ν r = ∂ r be the normal to the spheres of constant r . From Formula (7) we get Here we used that e = V 2 df ⊗ df satisfies (3) to replace the factor 1 1+ V 2 | df | 2 by 1 in the limit of the outer boundary integral. We next compute the integral over ∂ Ω. We will do the calculations assuming that ν = ∇ f |∇ f | , the case ν = -∇ f |∇ f | is similar. The last two terms are and the first two give We next use the following formula for the Laplacian of f on ∂ Ω, Since f is locally constant on ∂ Ω we obtain Hence, It then suffices to note that dµ g = √ 1 + V 2 | df | 2 dµ b to prove Formula (8). /square", "pages": [ 12, 13 ] }, { "title": "4. Penrose type inequalities", "content": "4.1. Horizon boundary. From now on we assume that | df | → ∞ at ∂ Ω, it then follows that the boundary is a minimal hypersurface, or a horizon. This can be seen by taking the double over the boundary of the graph of f . The double is then a C 1 Riemannian manifold for which the original boundary is the fixed point set of a reflection, and thus the boundary is minimal. It is not hard to prove that there can be no other minimal surface in the graph which encloses ∂ Ω. From Lemma 3.2 we conclude the following proposition. Proposition 4.1. Let Ω ⊂ H n be a relatively compact open subset of H n with smooth boundary. Let f : H n \\ Ω → R be an asymptotically hyperbolic function which is locally constant on ∂ Ω and such that | df | → ∞ at ∂ Ω . Further assume that Scal ≥ -n ( n -1) . Then Applying the Hoffman-Spruck inequality or the Minkowski formula we get estimates of the boundary term in (9) and conclude the following Theorem. Theorem 4.2. Let Ω ⊂ H n be a relatively compact open subset of H n with smooth boundary. Assume that Ω contains an inner ball centered at the origin of radius r 0 . Let f : H n \\ Ω → R be an asymptotically hyperbolic function which is locally constant on ∂ Ω and such that | df | → ∞ at ∂ Ω . Further assume that Scal ≥ -n ( n -1) and that ∂ Ω has non-negative mean curvature H ≥ 0 . Then and Proof. The Hoffman-Spruck inequality, [18, 26, 30], applied to a compact hypersurface M of hyperbolic space H n tells us that for any smooth non-negative function h on M . Here Setting h ≡ 1 and M = ∂ Ω in (12) yields (10). The estimate (11) follows from the Minkowski formula in hyperbolic space, see [24, Equation (4')] with the point a = (1 , 0 , . . . , 0) (note that in the cited article the mean curvature is defined as an average and not a sum). /square Neither of the inequalities (10) and (11) is optimal, so we do not get a characterization of the case of equality in the corresponding Penrose type inequalities. Lemma 4.3. Let ν be the outgoing unit normal to ∂ Ω . The second fundamental form of ∂ Ω with respect to the metric b is given by where ψ is a defining function for ∂ Ω such that ∇ ψ = ν . Further, we have where ∇ T V is the gradient of V for the metric induced by b on ∂ Ω . Proof. The second fundamental form of ∂ Ω with respect to the metric ˜ b is given by We compute ˜ ˜ At the center point p 0 of normal coordinates for the metric b the difference between the two Christoffel symbols is given by where we used that Hess b V = V b and 1 + | dV | 2 = V 2 in the last line. Further, we have Hence, We take the trace of this formula with respect to the metric ˜ b . For this we select an orthogonal basis ( e 1 , . . . , e n -1 ) of T p 0 ∂ Ω for the metric b such that e k ∈ ker dV for k ≥ 2. An orthogonal basis for the metric b is then given by Thus, we find Next we note that ( ∇ e 1 V ) 2 = | dV | 2 -〈 dV, ν 〉 2 is the norm of dV restricted to the tangent space of ∂ Ω. Hence the measure d µ induced on ∂ Ω by b is given by ˜ where dµ is the measure induced on ∂ Ω by b . Finally we conclude The assumption ˜ S > 0 is equivalent to where this inequality is to be understood as an inequality between quadratic forms on T∂ Ω. This notion of convexity is not invariant under the action of isometries of the hyperbolic space. Since | dV | < V , it is natural to replace this assumption by This new assumption is equivalent to the definition of h-convexity (see for example [3]). Assuming that Ω is h-convex, we get the following inequality from (13). We estimate the first term of the right-hand side by the Aleksandrov-Fenchel inequality, see [16, Theorem 2], [22, Lemma 12], [28] or [8]. /square Equality in the first inequality here implies that ∂ Ω is a round sphere in the Euclidean metric ˜ b , equality in the second inequality tells us that it must be centered at the origin. To estimate the second term of (14), we rely on [3, Theorem 2]. Assuming that the origin is the center of an inner ball of Ω and denoting by l the distance from the origin, we have, for any point p ∈ ∂ Ω, where τ = tanh r 0 2 and r 0 is the radius of an inner ball of Ω. Hence, setting t = tanh l 2 ( p ), we have It is also easy to check that the equality holds if and only if Ω is the ball of radius r 0 centered at the origin. Combining the last two estimates, we get the following inequality: From Proposition 4.1 and Inequality (15), we immediately get the following theorem. Theorem 4.4. Let Ω be a non-empty h-convex subset of H n admitting an inner ball centered at the origin of radius r 0 . Let f : H n \\ Ω → R be an asymptotically hyperbolic function such that f is locally constant on ∂ Ω , | df | → ∞ at ∂ Ω . Assume that the scalar curvature Scal of its graph is larger than -n ( n -1) . Then Moreover, equality holds in (16) if and only if Scal = -n ( n -1) and ∂ Ω is round sphere centered at the origin. We make a couple of remarks concerning this theorem. Remark 4.5 . so (16) coincides with the standard Penrose inequality (1) in this case. Thus this term may be thought of as a volume integral. Compare with [29]. Let V p := cosh d b ( p, · ). Changing the origin p of hyperbolic space leads to considering the function It is fairly straightforward to see that this function is proper and strictly convex. So there exists a unique point p 0 such that, choosing p 0 as the origin, this integral is minimal. Obviously, p 0 ∈ Ω. From symmetry considerations this point can be seen to coincide with the center of an inner ball for many Ω's. where Ω i are the connected components of Ω and r i is the inner radius of Ω i .", "pages": [ 14, 15, 16, 17, 18 ] }, { "title": "5. Rigidity", "content": "In this section we will prove the rigidity statement concluding Theorem 2.1. The scheme of the proof we give differs very little from [19]. As a first step, we prove the following proposition which is similar to [19, Theorem 3]. Proposition 5.1. Let f : H n \\ Ω → R be a function satisfying the assumptions of Theorem 2.1 and let Σ be its graph. Assume further that Ω is convex. Then the mean curvature H of Σ does not change sign. The proof of this proposition requires several preliminary results. The main observation is the fact that the assumption Scal ≥ -n ( n -1) is equivalent to ∣ ∣ S ∣ ∣ 2 ≤ H 2 . This follows at once from the Gauss equation. In particular, any point p ∈ Σ such that H ( p ) = 0 has S ( p ) = 0. We denote by Σ 0 the set of such points, where int(Σ) = Σ \\ ( ∂ Ω × R ). Lemma 5.2. Let Σ ' 0 be a connected component of Σ 0 . Then Σ ' 0 lies in a codimension 1 hyperbolic subspace tangent to Σ at every point of Σ ' 0 . Proof. Let V (0) , . . . , V ( n ) be as in Section 2.1 and let ν be the unit normal vector field of Σ in H n +1 . For any vector X ∈ T Σ at a point of Σ ' 0 we have where S ( X ) denotes the Weingarten operator which is zero by assumption. From [25, Theorem 4.4] we conclude that dV ( i ) ( ν ) is constant on Σ ' 0 . If we consider H n +1 as the unit hyperboloid in Minkowski space R n +1 , 1 , then the V ( i ) are the coordinate functions of R n +1 , 1 restricted to H n +1 so ν is a constant vector in R n +1 , 1 . Further, ν is tangent to H n +1 so it is orthogonal to the position vector in R n +1 , 1 . This means that ν is everywhere orthogonal to a linear subspace W ⊂ R n +1 , 1 . We conclude that Σ ' 0 ⊂ W ∩ H n +1 /similarequal H n . /square The next result is taken from [20, Proposition 2.1]. Lemma 5.3 (A matrix inequality) . Let A = ( a ij ) be a symmetric n × n matrix. Set Then we have for each 1 ≤ k ≤ n . In particular, /negationslash /negationslash where equality holds if and only if A is diagonal and all a ii are equal for i = 1 , . . . , n , i = k . /negationslash Proposition 5.4. Let Σ and s 0 be given. Assume that s 0 is a regular value for f on Σ . Set Σ( s 0 ) = Σ ∩ f -1 ( s 0 ) . Let ν be the unit normal vector field of Σ in H n +1 , let η be the unit normal vector field to Σ( s 0 ) in H n ×{ s 0 } and let H ( s 0 ) be the mean curvature of Σ( s 0 ) in H n ×{ s 0 } computed with respect to η . Then Equality holds at a point in Σ( s 0 ) if and only if Proof. Let p be a point in Σ( s 0 ). We compute the second fundamental form of Σ( s 0 ) in H n +1 at p in two different ways. Let e 1 ∈ T p Σ be a unit vector field orthogonal to T p Σ( s 0 ). We denote by S 0 the second fundamental form of Σ( s 0 ) in H n +1 . This is a symmetric bilinear form on T p Σ( s 0 ) taking values in the normal bundle N p Σ( s 0 ) ⊂ T p H n +1 . Further, we denote by S 1 the second fundamental form of Σ( s 0 ) in Σ computed with respect to the vector e 1 . Since H n × { s 0 } is totally geodesic in H n +1 , we have Similarly, Hence, taking the scalar product of the last two equalities with ν , we get Let { e 2 , . . . , e n } be an orthonormal basis of T Σ( s 0 ), then { e 1 , . . . , e n } is an orthonormal basis of T p Σ. Set Then, using the notation of Lemma 5.3, we have Proposition 5.4 now follows from Lemma 5.3. /square The proof of Proposition 5.1 will also require following two lemmas, analogous to [19, Lemma 3.3 and Lemma 3.4]. Lemma 5.5. Let W be an open subset of H n , possibly unbounded. Let p ∈ ∂W , and let B ( p ) be a geodesic open ball in H n centered at p . Consider f ∈ C 2 ( W ∩ B ( p )) ∩ C 1 ( W ∩ B ( p )) and let H denote the mean curvature of its graph. If f = C and | df | = 0 on ∂W ∩ B ( p ) , where C is a constant, and H ≥ 0 on W ∩ B ( p ) then either f ≡ C in W ∩ B ( p ) , or Proof. If f ≡ C then there is nothing to prove. Suppose therefore that f /negationslash≡ C and assume to get a contradiction that f ( x ) ≤ C everywhere in W ∩ B ( p ). We first note that in fact f < C everywhere in W ∩ B ( p ). Indeed, let q ∈ W ∩ B ( p ) be such that f ( q ) = C . Then q is an interior maximum point of f in W ∩ B ( p ), whereas in W ∩ B ( p ), see Section 3.1. By the Hopf strong maximum principle it follows that f ≡ C in W ∩ B ( p ), which is a contradiction. /negationslash Now suppose that B ( p ) = B r ( p ) is the ball of radius r around p . Fix a point q ∈ B r/ 2 ( p ) and define r ' := sup { r | B r ( q ) ⊂ W } . It is clear that B r ' ( q ) ⊂ W ∩ B ( p ) and B r ' ( q ) ∩ ∂W = ∅ . Consequently, there is a point s ∈ ∂W such that the interior sphere condition holds at s . Then by the Hopf boundary lemma [15, Lemma 3.4], we have | df | > 0 at s , which is a contradiction. We conclude that f > C holds somewhere in W ∩ B ( p ). /square Definition 5.6. Let W be a bounded subset of H n and let W be its closure. A point p ∈ ∂W is called convex if there is a geodesic ( n -1)-sphere S in H n passing through p such that W \\ { p } is contained in the open geodesic ball enclosed by S . Note that every bounded set in H n \\ Ω has at least one convex point. This follows from the assumption that Ω is convex. We only sketch the proof of this fact leaving the details to the reader. Choose a point p ∈ W and let q be the projection of p onto ∂ Ω. Then the hyperbolic subspace passing through q and orthogonal to the geodesic joining p to q cuts H n in two half-spaces, a 'left' one containing Ω and a 'right' one containing p . Then if O ' is located very far on the left side of the geodesic ( qp ), it is clear that the smallest sphere S centered at O ' containing Ω ∪ W has a non-trivial intersection with ∂W . Any point in S ∩ ∂W is then a convex point. Lemma 5.7. Let W be an open bounded subset of H n and let p ∈ ∂W be a convex point. Suppose that f ∈ C n ( W ∩ B ( p )) ∩ C 1 ( W ∩ B ( p )) is such that f = C and | df | = 0 on ∂W ∩ B ( p ) for some constant C . If the graph of f has scalar curvature Scal ≥ -n ( n -1) , then its mean curvature H must change sign in W ∩ B ( p ) , unless f ≡ C in W ∩ B ( p ) . Proof. Suppose on the contrary that H does not change sign and f /negationslash≡ 0. By possibly reversing sign and adding a constant to f we may assume that H ≥ 0 and that C = 0. Let S r be a geodesic ( n -1)-sphere of radius r as in Definition 5.6, centered at a point O ' ∈ H n , and such that S r ∩ W = { p } . Let µ be a positive number strictly less than the distance from W \\ B ( p ) to S r . Then for every sphere S r ' of radius r ' ∈ ( r -µ, r ) and centered at O ' we obviously have S r ' ∩ W ⊂ B ( p ). Let f 0 be a continuous function on B ( p ) such that f 0 = f on W ∩ B ( p ) and f 0 = 0 on B ( p ) \\ W . Define the function for r ' ∈ [ r -µ, r ]. It is easy to check that g is continuous and satisfies g ( r ) = 0. Next, we observe that by Lemma 5.5 the ball B µ ( p ) contains a point q such that f 0 ( q ) = ε > 0. By the Morse-Sard theorem [27, Theorem 7.2] we may assume that each connected component of the corresponding level set of f 0 inside W ∩ B ( p ) is a C n hypersurface. It is clear that g ([ r -µ, r ]) = [0 , ε ' ], where ε ≤ ε ' , and hence /negationslash is well-defined. Then S r 0 ∩ Σ ε = ∅ , whereas S r ' ∩ Σ ε = ∅ for r 0 < r ' ≤ r , thus S r 0 is tangent to Σ( ε ) at some interior point q . Let U be the open subset of W ∩ B ( p ) bounded by S r 0 and ∂W , then q ∈ ∂U . We have f ( q ) = ε > f ( x ) for any x ∈ U , H ≥ 0 holds in U , and the interior sphere condition is obviously satisfied at q ∈ S r 0 . Since η = -∇ f | df | is orthogonal to ∂U at q , it is easy to conclude by the Hopf boundary lemma that η is the inward pointing normal to ∂U . Hence η is the outward pointing normal for both S r 0 and Σ( ε ) at q . By the comparison principle, the mean curvature H ( ε ) of Σ( ε ) satisfies H ( ε ) > 0 at q . On the other hand, since the scalar curvature of the graph of f is nonnegative, by Proposition 5.4 at q we have Here 〈 ν, η 〉 < 0 since ν = ( ∇ f, -V -2 ) √ V -2 + | df | 2 , H ≥ 0, and if H = 0 then H ( ε ) = 0. This means that H ( ε ) ≤ 0 at q , which is a contradiction. Hence H must change sign in W ∩ B ( p ). /square Proof of Proposition 5.1. We assume by contradiction that H changes sign, both sets { H > 0 } and { H < 0 } are nonempty in Σ. Our first observation is that each connected component of these two sets is unbounded. Indeed, let Σ + be a bounded connected component of { H > 0 } and let ∂ 0 Σ + be its outer boundary component. By Lemma 5.2 we know that ∂ 0 Σ + lies in an n -dimensional hyperbolic subspace Π. We view H n +1 as Π × R with the metric b + V 2 d ˜ s ⊗ d ˜ s , and we let W be a subset of { ˜ s = 0 } bounded by ∂ 0 Σ + . Then in some neighborhood of ∂W we can write Σ + as the graph of a function u such that u = 0 and | du | = 0 on ∂W . Now, considering a sufficiently small ball B ( p ) around p ∈ ∂W , we immediately arrive at the contradiction, since H must change sign in W ∩ B ( p ) by Lemma 5.7. The component Σ + is the graph of f over some open subset W of H n . Moreover, there is an unbounded component ∂ 0 W of the boundary ∂W such that f = C and | df | = 0 on ∂ 0 W . By Lemma 5.5 there exists q ∈ W such that f ( q ) = C + ε for some ε > 0. By the Morse-Sard theorem we know that there is an ε such that We have just seen that if Σ + is a connected component of { H > 0 } then it must be unbounded, and the same is clearly true for a connected component Σ -of { H < 0 } . Moreover, it follows by Proposition A.1 in Appendix A that one of the connected components of its boundary ∂ Σ + is unbounded, and the same holds for ∂ Σ -. Let us denote such an unbounded component by ∂ 0 Σ + . By Lemma 5.2 we know that ∂ 0 Σ + lies in an n -dimensional hyperbolic subspace Π tangent to Σ at every point of ∂ 0 Σ + . Since Σ is asymptotically hyperbolic, f tends to a constant value C at infinity, so the fact that ∂ 0 Σ + is unbounded forces Π to coincide with the plane { s = C } . C + ε is a regular value of f , so that the corresponding level set f -1 ( C + ε ) = { p | f ( p ) = C + ε } is a C n hypersurface with | df | > 0 at each point. Suppose that U is a connected component of { H ≥ 0 } in H n which contains W . Then, using Proposition A.1 and the fact that f tends to C at infinity, it is easy to check that if some connected component of f -1 ( C + ε ) intersects U , then it is contained in U . It is also obvious that f -1 ( C + ε ) ∩ U is nonempty and bounded, so we can find a point p in this set which is at the largest distance d from the origin O of H n . Let Σ( C + ε ) be the connected component of f -1 ( C + ε ) which contains p . Then the geodesic sphere of radius d centered at O touches Σ( C + ε ) at p , and there are no points x such that f ( x ) ≥ C + ε in { r > d } ∩ U . Arguing as in the proof of Lemma 5.7, we can show that η := -∇ f | df | = ∂ r at p , that is, ν is an outgoing normal to Σ( C + ε ). The mean curvature H ( C + ε ) is then positive at p , whereas Proposition 5.4 tells us that H ( C + ε ) ≤ 0 at p , which is a contradiction. /square Let f be as in Theorem 2.1. We recall the expressions for g , S , H , and Scal obtained in Section 2.2, and rewrite them as functions of the arguments Df and D 2 f , where Df and D 2 f denote the Euclidean gradient and the Euclidean Hessian respectively: Following [19, Section 4], we will now prove maximum principles for the scalar curvature equation Scal( Df,D 2 f ) + n ( n -1) = 0. The lemma below concerns ellipticity of this equation.", "pages": [ 18, 19, 20, 21, 22, 23 ] }, { "title": "Lemma 5.8.", "content": "Proof. A straightforward computation gives /square Proposition 5.9. Let f be as in Theorem 2.1. Suppose that the scalar curvature Scal and the mean curvature H of its graph satisfy Scal ≥ -n ( n -1) and H ≥ 0 Then the matrix ( Hg ij -S ij ) is positive semi-definite everywhere in H n \\ Ω . Proof. We work at a point p ∈ H n \\ Ω. Since Hg ij -S ij = ∑ k ( Hδ j k -S j k ) g ik , where g ik is positive definite, we only need to show that ( Hδ j k -S j k ) is positive semi-definite. After possibly rotating the coordinates, we may assume that S = ( S j k ) = diag( λ 1 , . . . , λ n ). Then, in the notation of Lemma 5.3, we have By Lemma 5.3 it follows that for k = 1 , . . . , n . If σ 1 ( S ) = H > 0, since σ 2 ( S ) = 1 2 (Scal + n ( n -1)) ≥ 0, it is obvious that σ 1 ( S | k ) ≥ 0 for every k = 1 , . . . , n . Otherwise if H = 0 then S = 0 and hence σ 1 ( S | k ) = 0. This proves that σ 1 ( S | k ) ≥ 0. /square In the next two propositions we prove versions of the maximum principle for the scalar curvature equation, the first one for points in the interior and the second one for points on the boundary. Proposition 5.10. Let f i : H n \\ Ω → R , i = 1 , 2 , be two functions satisfying the assumptions of Theorem 2.1. Suppose that f 1 ≥ f 2 in H n \\ Ω , and that f i , i = 1 , 2 , satisfy the inequalities in H n \\ Ω . If the matrix ( Hg ij -S ij ) is positive definite in H n \\ Ω for either f 1 or f 2 , and if f 1 = f 2 at some point of H n \\ Ω , then f 1 ≡ f 2 in H n \\ Ω . Proof. We consider the scalar curvature operator as Scal( p, ξ ) ∈ C 1 ( R n , R n × R n ). Then where and Note that by Lemma 5.8 we have If f 1 = f 2 at p ∈ H n \\ Ω, then p is a local minimum point of f 1 -f 2 , hence Df 1 = Df 2 at p . Consequently, a ij is positive definite at p . By continuity, a ij is positive definite in some open neighborhood U of p in H n \\ Ω. Then f 1 ≡ f 2 in U by the Hopf strong maximum principle. It follows that the set { p ∈ H n \\ Ω | f 1 ( p ) = f 2 ( p ) } is both open and closed in H n \\ Ω. Since H n \\ Ω is connected, we conclude that f 1 ≡ f 2 everywhere H n \\ Ω. /square Proposition 5.11. Let f i : H n \\ Ω → R , i = 1 , 2 , be functions satisfying the assumptions of Theorem 2.1. Suppose that f 1 ≥ f 2 ≥ C in H n \\ Ω , and that f i , i = 1 , 2 , satisfy the inequalities in H n \\ Ω . If the matrix ( Hg ij -S ij ) is positive definite in H n \\ Ω for either f 1 or f 2 , and if f 1 = f 2 = C on ∂ Ω , then f 1 ≡ f 2 in H n \\ Ω . Proof. Let Σ i denote the graph of f i , i = 1 , 2. Take p ∈ ∂ Σ 1 = ∂ Σ 2 ⊂ { s = C } , and let ν ( p ) be the common normal to Σ i , i = 1 , 2, at this boundary point. Suppose that Π is the hyperbolic subspace orthogonal to ν ( p ), then Π is isometric to H n . Let B r ( p ) be a geodesic ball of radius r in Π centered at p , and let U = B r ( p ) ∩{ s > C } . If r is sufficiently small, we can write Σ i near p as the graph of ˜ f i : U → R , i = 1 , 2, in U × R with the metric b + V 2 d ˜ s ⊗ d ˜ s , where b is the hyperbolic metric on U , and ˜ s is the coordinate along the R -factor. It is obvious that ∇ ˜ f i = 0 at p for i = 1 , 2. We also have f 1 ≥ f 2 in U , and Moreover, either ˜ f 1 or ˜ f 2 has positive definite matrix ( Hg ij -S ij ) at p . Arguing as in the proof of Proposition 5.10, we see that ( f 1 -f 2 ) satisfies /negationslash where we may assume (after decreasing r ) that a ij is positive definite on U . If we assume that ˜ f 1 > ˜ f 2 in U then by the Hopf boundary lemma we have ∇ ( ˜ f 1 -˜ f 2 )( p ) = 0, a contradiction. Consequently, ˜ f 1 ( q ) = ˜ f 2 ( q ) at some interior point q ∈ H n \\ Ω. Application of Proposition 5.10 completes the proof. /square We recall that ρ := sinh( r ). The hyperbolic metric b takes the form Proposition 5.12. The second fundamental form of the graph given by (5) is given by and the function V = cosh( r ) = √ 1 + ρ 2 . In particular, the principal curvatures of the graph Σ are -n -2 2 √ 2 mρ -n 2 with multiplicity 1 and √ 2 mρ -n 2 with multiplicity n -1 . The mean curvature H is given by In particular, the quadratic form is positive definite. Proof. Straightforward calculations. /square We are now ready to prove the result on rigidity for the case of equality in the last inequality of Theorem 2.1. From Theorem 4.4 we know that in this case Scal = -n ( n -1) and ∂ Ω ⊂ H n is a round sphere centered at the origin. The result thus follows from the next theorem. Theorem 5.13. Let f : H n \\ Ω → R be an asymptotically hyperbolic function which satisfies the assumptions of Theorem 2.1 and such that the graph of f has constant scalar curvature Scal = -n ( n -1) . Also assume that ∂ Ω is a round sphere centered at the origin and that df ( η )( x ) →∞ as x → ∂ Ω where η is the outward normal of the level sets of f . Then the graph of f is isometric to the t = 0 slice of the anti-de Sitter Schwarzschild space-time, as described in Section 2.3. Proof. By adding a constant to f we assume that f = 0 on ∂ Ω. From Proposition 5.1 we know that H does not change sign. Proposition 5.4 together with the fact that H is positive on ∂ Ω tells us that H ≥ 0 on the boundary, and thus H ≥ 0 everywhere. The maximum principle applied to H together with df ( η ) → + ∞ at ∂ Ω tells us that lim sup x →∞ f ( x ) > 0. Since f is an asymptotically hyperbolic function we conclude that lim x →∞ f ( x ) = C where 0 < C < ∞ . Let f AdS-Schw be the asymptotically hyperbolic function whose graph is isometric to the t = 0 slice of anti-de Sitter Schwarzschild space-time, with mass parameter m such that its horizon is exactly the sphere ∂ Ω. This function vanishes on ∂ Ω and has lim x →∞ f AdS-Schw = C 0 where 0 < C 0 < ∞ . If C ≤ C 0 we set u λ = f AdS-Schw + λ for λ ≥ 0. If λ is large enough then u λ > f . We decrease λ until finally u λ ( p ) = f ( p ) at a point p , possibly p = ∞ . If p is an interior point then Proposition 5.10 tells us that u λ ≡ f , if p is a boundary point then Proposition 5.11 tells us that u λ ≡ f . There is however one more situation to consider, namely when u λ > f and lim x →∞ ( u λ -f ) = 0. Since both the graph of u λ and the graph of f have Scal = -n ( n -1), arguing as in the proof of Proposition 5.10 we conclude that u λ -f satisfies the equation In this case, the Hopf strong maximum principle tells us that u λ -f attains its positive maximum either at an interior point or at a point of ∂ Ω. Let us denote this point by q and suppose that ( u λ -f )( q ) = β > 0. Clearly, f ≥ u λ -β , and f ( q ) = ( u λ -β )( q ). By either Proposition 5.10 or Proposition 5.11 we conclude that u λ -β ≡ f . In any case we have found that f and f AdS-Schw differ by a constant, which is the conclusion of the theorem. /square If C > C 0 we set v λ = f AdS-Schw -λ for λ ≥ 0. For λ large enough we have v λ < f and we decrease λ until v λ hits f . Arguing as above it is easy to show that v λ ≡ f . Appendix A. A property of unbounded open subsets of R n In this appendix we will prove the following result on the boundary components of an unbounded open subset of R n . Proposition A.1. Let H : R n → R , n ≥ 2 , be a continuous function which takes both positive and negative values. Assume that each connected component of H -1 ((0 , ∞ )) and H -1 (( -∞ , 0)) is unbounded. Then there is a connected component of H -1 (0) which is unbounded. To prove the proposition we use the following lemma. Lemma A.2. Let K ⊂ R n , n ≥ 2 , be compact and connected. Let U be the unbounded connected component of R n \\ K . Then U ε := { x ∈ U | d ( x, K ) < ε } is connected. /negationslash Proof. Let F := R n \\ U . This set is closed and bounded and therefore compact. We show that F is connected. Let f : F →{ 0 , 1 } be continuous. Then f is constant on K . Take x ∈ F \\ K . For 0 = a ∈ R n consider the half-line { x + ta | 0 ≤ t } . Let t 0 be the smallest number so that x + t 0 a ∈ K . Then the line segment { x + ta | 0 ≤ t ≤ t 0 } is a subset of F , and we conclude that f must be constant on F so F is connected. Next define F ε := { x ∈ R n | d ( x, F ) < ε } . Since F ε = ∪ p ∈ F B ε ( p ) this is a connected set. Note that F ε = U ε ∪ F . The Mayer-Vietoris sequence for homology tells us that from which we conclude that U ε is connected. /square Proof of Proposition A.1. Let V be a connected component of H -1 ((0 , ∞ )). Let V ' ⊂ R n be the image of V when compactifying R n with a point at infinity and then removing a point p lying in an unbounded component of R n \\ V . The set V ' is open, bounded and connected, so the closure K := V ' is compact and connected. Let ∂ ∞ K be the part of the boundary of K facing the unbounded component of R n \\ K . Since the intersection of a nested sequence of compact connected sets is connected we conclude from the Lemma that ∂ ∞ K is connected. Going back to V this means that the union ∂ ∞ V ∪{∞} is connected, where ∂ ∞ V is the part of the boundary facing the component of R n \\ V containing p . From this we see that all components of ∂ ∞ V must be unbounded, since if there was a bounded component this would remain disconnected from the others when adding the point at infinity. Finally, every component of ∂ ∞ V is contained in some connected component of H -1 (0), and those components of H -1 (0) are therefore unbounded. /square", "pages": [ 24, 25, 26, 27, 28 ] }, { "title": "References", "content": "[10] P. T. Chru'sciel and M. Herzlich, The mass of asymptotically hyperbolic Riemannian mani- folds , Pacific J. Math. 212 (2003), no. 2, 231-264. Institutionen for Matematik, Kungliga Tekniska Hogskolan, 100 44 Stockholm, Sweden E-mail address : [email protected] Laboratoire de Math'ematiques et de Physique Th'eorique, UFR Sciences et Technologie, Facult'e Franc¸ois Rabelais, Parc de Grandmont, 37200 Tours, France E-mail address : [email protected] Institutionen for Matematik, Kungliga Tekniska Hogskolan, 100 44 Stockholm, Sweden E-mail address : [email protected]", "pages": [ 28, 29 ] } ]

[ { "title": "Ellery Ames ∗ , Florian Beyer † , James Isenberg ‡ , and Philippe G. LeFloch §", "content": "February 2013 (final version)", "pages": [ 1 ] }, { "title": "Abstract", "content": "We set up the singular initial value problem for quasilinear hyperbolic Fuchsian systems of first order and establish an existence and uniqueness theory for this problem with smooth data and smooth coefficients (and with even lower regularity). We apply this theory in order to show the existence of smooth (generally not analytic) T 2 -symmetric solutions to the vacuum Einstein equations, which exhibit AVTD (asymptotically velocity term dominated) behavior in the neighborhood of their singularities and are polarized or half-polarized.", "pages": [ 1 ] }, { "title": "1 Introduction", "content": "Fuchsian formulations have proven to be very useful for studying the behavior of cosmological spacetimes in the neighborhood of their singularities. Introduced into general relativity almost fifteen years ago by Kichenassamy and Rendall [30], these formulations have been used primarily as a tool for showing that within certain families of solutions of the Einstein equations (defined primarily by the invariance of each member of the family under a fixed isometry group), there is a large collection of solutions which exhibit AVTD (asymptotically velocity term dominated) behavior. Roughly speaking, a spacetime shows AVTD behavior if, in a neighborhood of its cosmological singularity, the evolution of the spacetime metric field of the solution approaches the evolution of a model metric field which (relative to some choice of spacetime coordinates) satisfies a system of ordinary differential equations (ODEs) deduced from the Einstein equations by suppressing spatial derivatives relative to time derivatives. The detection of AVTD behavior has proven to be a very useful step towards verifying that the strong cosmic censorship conjecture holds for certain families of solutions of the Einstein equations [18, 35]. Fuchsian formulations are effective in studying the possible presence of AVTD behavior since they are designed specifically to handle singular systems of partial differential equations (PDEs) or, equivalently, PDE systems in the neighborhood of their singularities. While it is not easy to identify the location of singularities in generic spacetime solutions of Einstein's equations, for certain isometry-defined families of solutions -e.g. spatially homogeneous solutions, Gowdy solutions, T 2 -symmetric solutions, and many families of U (1)-symmetric solutions- one can use areal coordinates or special forms of harmonic coordinates to locate the singularities. If the Einstein equations are then reduced relative to these symmetries and expressed in terms of these coordinates, then the resulting PDE system takes a singular form in the neighborhood of the singularity which is amenable to a Fuchsian formulation and analysis in the form of a singular initial value problem -presuming that various further conditions are met. Most of the earlier applications of Fuchsian formulations to families of solutions of Einstein's equations have presumed that the spacetimes are analytic [30, 24, 26, 4]. This is not surprising, since Fuchsian formulations for generic systems of PDEs were initially developed with analytic PDE systems in mind [29, 31]. It is important, however, to extend studies of AVTD behavior and strong cosmic censorship beyond analytic spacetimes and to consider if they also hold for spacetime solutions which are smooth, but not necessarily analytic. There are two sets of results (prior to our work) known to the authors concerning the existence of solutions to quasilinear Fuchsian PDEs in smooth or finite differentiability regularity classes. 1 In the first of these, proposed by Claudel and Newman [19], the main result is that if a number of quite restrictive technical conditions are satisfied by the PDE system, then the Cauchy problem is well-posed for data specified at the singular time. As noted in [34], these restrictive conditions are not generally satisfied by the PDE systems corresponding to the Einstein equations for the Gowdy spacetimes, for the T 2 -symmetric spacetimes, or for most other families of spacetimes under consideration; hence the Claudel and Newman results are not useful for our present purposes. The second set of results concerning smooth solutions of Fuchsian systems are those proven by Rendall. In [34], he develops a Fuchsian-based approach that is applicable to both semilinear and quasilinear equations, and he uses it to establish the existence of a class of smooth T 3 Gowdy spacetimes which exhibit AVTD behavior. In Rendall's approach, one performs a series of reduction steps in order to obtain a symmetric hyperbolic system, and one then proves the existence of smooth solutions using a sequence of analytic solutions to a sequence of analytic 'approximate equations.' Although this method has successfully been applied by Clausen [20] to the family of polarized T 2 -symmetric spacetimes, and has also been used by St˚ahl [37] in studying S 3 and S 2 × S 1 Gowdy spacetimes, it has proved difficult to apply in more general cases, such as for spacetimes with only one Killing vector field [26], or with no symmetries [4]. Our goal in the present work is to develop a general Fuchsian formulation for analyzing smooth (but not analytic) solutions to quasilinear hyperbolic PDEs which can be fairly directly applied to polarized and half-polarized T 2 -symmetric solutions to the Einstein equations and can be applied to polarized and half-polarized U (1)-symmetric solutions as well. Two of the authors of this work, Beyer and LeFloch in [13], have carried out this program for semilinear hyperbolic systems and have applied their formulation to T 3 Gowdy solutions. Therein, Beyer and LeFloch set up a second-order Fuchsian formulation for smooth semilinear systems. In the present paper, in addition to generalizing to smooth quasilinear hyperbolic systems, we also work with Fuchsian systems in first-order form. One of the motivations for the semilinear work and its application to the T 3 Gowdy spacetimes was that the approximation scheme which plays a key role in the existence proof can also be used as the basis for a robust method for numerical simulations. This numerical approach has been developed and implemented in [12, 15] (see also [3]) as a tool for the numerical exploration of Gowdy solutions. Since our analysis in the present paper involves a similar approximation scheme, we expect to be able to carry out numerical investigations of singular initial value problems in more general classes of equations in future work. An outline of this paper is as follows. We begin the discussion of our results in Subsection 2.2, where we consider a general class of first-order quasilinear Fuchsian systems and, then, formulate the singular initial value problem for such systems. Next, in the same subsection, we introduce the class of first-order quasilinear Fuchsian systems in symmetric hyperbolic form and, for such systems, state an existence and uniqueness result (in Theorem 2.4, below) which holds in, both, infinite differentiability and finite differentiability classes. In Subsection 2.3, we carry out the details of the proof of this result. This theorem holds for a broad class of asymptotic data specified on the singularity. Next, in Subsection 2.4, we discuss special choices for this data, which we call 'ODE leading-order term', and then state and establish an alternative existence and uniqueness result (Theorem 2.21) for the singular initial value problem. This alternative theorem is useful in applications, as we show later in Subsections 2.4.4 and 3.3.3. Interestingly enough, ODE-leading-order asymptotic data also play a useful role as approximate solutions. In the second part of this paper, in Section 3, we apply our theoretical results and study polarized and half-polarized T 2 -symmetric solutions of Einstein's vacuum field equations. To this end, in Subsection 3.2, we define this family of spacetimes, and then, in the same subsection, we write out the Einstein equations in terms of areal coordinates. Next, in Subsection 3.3.1, we discuss the concept of AVTD behavior and discuss what one needs to do in order to check whether a set of polarized or half-polarized T 2 -symmetric solutions do exhibit AVTD behavior. In Subsections 3.3.2 and 3.3.3, we are in a position to rely on our results in Section 2 and we establish that our conditions therein indeed hold true for the spacetimes under consideration. On one hand, we rely on Theorem 2.4 and establish the existence of a parametrized family of polarized and halfpolarized T 2 -symmetric solutions with, both, finite or infinite order of differentiability and with the expected AVTD behavior. On the other hand, we rely on Theorem 2.21 and show that, provided attention is restricted to smooth ( C ∞ ) solutions, then the family of T 2 -symmetric with AVTD behavior can be extended to include those with a wider ('optimal') range of the 'asymptotic velocities' (labeled by k , below, as defined and discussed in Section 3.3). In Section 4, we conclude and discuss the relevance of Fuchsian formulations for numerical simulations, as well as the application of the proposed formulation in order to tackle more general families of spacetimes.", "pages": [ 2, 3, 4 ] }, { "title": "2.1 Objective of this section", "content": "For a given (say, first-order) PDE system P [ ψ ] = 0, the (regular) Cauchy problem involves finding a solution ψ = ψ ( t, x ) to this system such that, at some chosen value t 0 of the time, the solution satisfies the initial condition ψ ( t, x 0 ) = φ ( x ), for some specified initial data function φ = φ ( x ). If the Cauchy problem is well-posed, then for any appropriate choice of φ , the solution ψ exists for some open interval I containing t 0 . Here, we are interested in the singular initial value problem rather than the regular one. That is, rather than seeking solutions to P [ ψ ] = 0 which agree with specified initial data at a chosen time, we seek for solutions which become (in general) singular as t approaches some fixed value t ∞ , and which agree with some specified fall-off data as one approaches the singularity at t ∞ . As for the Cauchy problem, one can prove existence and uniqueness theorems for the singular initial value problem; these theorems guarantee that for any appropriate choice of the 'asymptotic data', there is a solution to P [ ψ ] = 0 which exists for t approaching t ∞ , and which asymptotically matches the prescribed asymptotic data", "pages": [ 5 ] }, { "title": "2.2 Quasilinear first-order symmetric hyperbolic Fuchsian systems", "content": "Before making this notion of singular initial value problem precise, and proving existence and uniqueness results, we carefully define the class of PDEs we shall consider here. While the theory we develop here can be generalized to a much wider class of background spacetime manifolds 1 (see, for example [2], in which we generalize our results to spacetime manifolds (0 , δ ] × T n ), let us presume for now that we work on the cylinder spacetime (0 , δ ] × T 1 , for some small parameter δ , with coordinates t ∈ (0 , δ ] and x ∈ T 1 . The singularity is presumed to occur at t = 0 (hence we set t ∞ = 0 in the earlier notation); correspondingly, it is useful to work with the 'singular time differential operator' The general form of the first-order PDE systems under consideration reads in which the unknown is a vector-valued spacetime function u : (0 , δ ] × T 1 → R d for some integer d ≥ 1 and some real δ > 0. Here S 1 = S 1 ( t, x, u ), S 2 = S 2 ( t, x, u ), and N = N ( t, x, u ) are specified d × d -matrix-valued maps of the spacetime coordinates ( t, x ) and the unknown u (but is independent of its derivatives), while f = f ( t, x, u ) is a specified R d -valued map of ( t, x ) and the unknown u (but again is independent of its derivatives). The specific requirements for the functions S 1 , S 2 , N and f are fixed precisely below; see in particular Definition 2.2. For notational convenience, we often leave out the arguments ( t, x ), instead we use the short-hand notation S 1 ( u ), S 2 ( u ), N ( u ), and f ( u ). Notationally, an object such as S 1 ( u ) may be interpreted as a map u ↦→ S 1 ( u ) between two function spaces (as further discussed below). In this context we often write S 1 ( u )( t, x ), and we do the same for S 2 ( u ) and N ( u ). Observe that, in principle, one could absorb the term N ( u ) u into the source f ( u ); however in view of the conditions on these terms that we will introduce below, it is important to keep these two terms separate. A system of this form (2.1) (noting especially the use of the singular operator D ) will be referred to as a quasilinear first-order Fuchsian system . Before formulating the singular initial value problem for such systems, we wish to define functional norms and the corresponding function spaces which include built-in specifications of the asymptotic behavior of the functions in time. We state these definitions first (here) for vector-valued functions, and then (below) for matrix-valued functions. The definitions are parametric: For the vector-valued function case, we specify as parameters i) a non-negative integer q , and ii) a fixed smooth 1 vector-valued function µ : T 1 → R d . Then using µ (which we label as an exponent vector ) to construct the corresponding diagonal matrix we define the norm for vector-valued functions w = w ( t, x ). Here, || · || H q ( T 1 ) denotes the standard q -order Sobolev norm on T 1 . Based on (2.3), we define the Banach space X δ,µ,q ( T 1 ) -also simply written as X δ,µ,q - as the completion of the set of all functions w ∈ C ∞ ((0 , δ ] × T 1 ) for which this norm is finite, and we denote by B δ,µ,q,r ⊂ X δ,µ,q the closed ball of radius r > 0 (measured with the given norm) and center 0. To handle the class of functions which are infinitely differentiable, we define the space In order to compare two function spaces X δ,µ,q and X δ,ν,q , we write ν > µ if, for each index i = 1 , . . . , d and for all x ∈ T 1 , the components of ν and µ satisfy the inequality ν i ( x ) > µ i ( x ). Clearly, X δ,ν,q ⊂ X δ,µ,q if ν > µ . We use analogously defined norms and functions spaces in order to control d × d matrix-valued functions such as S 1 , S 2 , and N . More specifically, in this case we choose as a parameter a fixed d × d matrix-valued valued function ζ (labeled as an exponent matrix ) which depends smoothly x ∈ T 1 , and we define the corresponding norm as for matrix-valued functions S = S ( t, x ). We denote the corresponding Banach space by X δ,ζ,q . Based on these function spaces of matrix-valued functions, we define (for r > 0) B δ,ζ,q,r ⊂ X δ,ζ,q and X δ,ζ, ∞ (as above), and we note that X δ,ξ,q ⊂ X δ,ζ,q if ξ > ζ. As noted above, the singular initial value problem associated with a system such as Eq. (2.1) consists of choosing a set of 'asymptotic data', and seeking for solutions which asymptotically approach that data. Using the function spaces just defined, we make this idea precise as follows. Definition 2.1. Given the parameters δ, µ and q as above, and a chosen function u 0 : (0 , δ ] × T 1 → R d , the singular initial value problem consists of seeking a solution u = u 0 + w to Eq. (2.1) whose remainder w belongs to X δ,µ,q ( T 1 ) . The function u 0 , which we refer to as the leading-order term , constitutes the asymptotic data, and is (a priori) of unspecified regularity. Regarding the solution function w , if it is to be considered a 'remainder', then in comparison with u 0 it should be of higher order in t as one approaches t = 0. Observe that the exponent vector µ , which parametrizes the function space X δ,µ,q ( T 1 ), controls the order of the singularity of the remainder w at t = 0; roughly speaking, each component of w is of corresponding component order O ( t µ ) if w ∈ X δ,µ,q . Hence, the components of µ are sometimes referred to as the remainder exponents, with µ collectively labeled the (remainder) exponent vector . Generally, we assume here and below that exponent vectors are smooth. Also, for a given exponent vector µ and a given scalar /epsilon1 , we use the notation µ + /epsilon1 to indicate a new exponent vector obtained by adding /epsilon1 to each component of µ . We now discuss the conditions on S 1 ( u ), S 2 ( u ), N ( u ), and f ( u ) in (2.1) which, together with further conditions on the space of leading-order terms and the space of remainder functions, are sufficient for establishing the well-posedness of the singular initial value problem. The main set of conditions needed is included in the following definition. Definition 2.2. Fix positive constants δ and s , a pair of non-negative integers q 0 and q (possibly + ∞ ), and an exponent vector µ : T 1 → R d , together with a leading-order term u 0 : (0 , δ ] × T 1 → R d (with so far unspecified regularity). The system Eq. (2.1) is called a quasilinear symmetric hyperbolic Fuchsian system around u 0 if, for each x ∈ T 1 , there exist a matrix S 1 , 0 ( u 0 )( x ) that is positive definite and symmetric and independent of t , a matrix S 2 , 0 ( u 0 )( x ) that is symmetric and independent of t , and a matrix N 0 ( u 0 )( x ) that is independent of t , all defining matrix-valued functions in the Sobolev space H q 0 ( T 1 ) ; and if there exists a smooth vector function β : T 1 → R d with strictly positive components, such that for every δ ' ∈ (0 , δ ] , each of the'remainder matrices' considered as an operator of the form (for example) w ↦→ S 1 , 1 ( u 0 + w ) , maps all functions w ∈ B δ ' ,µ,q,s to elements in B δ ' ,ζ,q,r , in which ζ is some exponent matrix with strictly positive entries, and r > 0 is some constant. It is furthermore required that S 1 , 1 ( u 0 + w ) and S 2 , 1 ( u 0 + w ) are symmetric matrices for all w ∈ B δ,µ,q,s . Before discussing further conditions which are needed in order to obtain existence and uniqueness for the singular initial value problem, we note the following remarks: Quasilinear symmetric hyperbolic Fuchsian systems are therefore 'essentially' semilinear (described by the coefficients S 1 , 0 , S 2 , 0 and N 0 ), up to 'quasilinear perturbations' (given by S 1 , 1 , S 2 , 1 and N 1 ), which decay as t → 0 with a rate controlled by ζ . The purely semilinear case has been treated earlier within a second-order framework by Beyer and LeFloch [12, 13, 14]. (See also [11, 15]). We discuss a collection of useful technical tools in the appendix (primarily in Section B) which allow us to check if the conditions of Definition 2.2 are satisfied for a given problem. The remaining conditions we consider concern the coupling between the components of an R d -valued function u = u ( t, x ), presumed to satisfy Eq. (2.1), and the effects of these couplings on the asymptotic behavior of the components as t approaches the singularity. Definition 2.3. Given the singular initial value problem (Definition 2.1) for a specified quasilinear symmetric hyperbolic system (Definition 2.2) with specified leading-order term u 0 and specified function space X δ,µ,q , the system Eq. (2.1) is called block diagonal with respect to µ , provided the following commutation conditions hold for all u = u 0 + w with w ∈ X δ,µ,q , and provided the same condition holds for all relevant spatial derivatives of S 1 ( u ) , S 2 ( u ) , and N ( u ) . (Recall that R [ µ ] is defined in Eq. (2.2) .) This block diagonality condition is used in the derivation of energy estimates (in Section 2.3, below) and thus plays a major role in the proof of existence and uniqueness. Roughly speaking, this condition guarantees that the 'principal part operator' 1 takes block diagonal form for v , and that each block is associated with only one component of the exponent vector µ . Recall that the components of µ determine the order of the singularity in t at t = 0 for the components of the remainder of the singular initial value problem. Hence, this condition requires that the principal part may only be coupled within those components of the solution whose remainders behave the same at t = 0. Note that the condition does allow all of the matrices S 1 , S 2 and N to depend on all components of u . It also allows for arbitrary coupling in the source-term, which we write as (or, in short form, as F [ w ], whenever it does not lead to confusion). We are now ready to state our main existence and uniqueness results for the singular initial value problem associated with first-order quasilinear symmetric hyperbolic Fuchsian systems. Our hypotheses below include conditions on the matrix which we refer to as the energy dissipation matrix and depends on the space coordinate x , only. Theorem 2.4 (Existence theory for symmetric hyperbolic Fuchsian systems) . Suppose that Eq. (2.1) is a quasilinear symmetric hyperbolic Fuchsian system around a leadingorder term u 0 (with the choice of the parameters 1 δ , s , µ , q , and q 0 as specified in Definition 2.2) and is block diagonal with respect to µ . Suppose that q ≥ 3 and q 0 = q +2 . Then there exists a unique solution u to Eq. (2.1) whose remainder w := u -u 0 belongs to X ˜ δ,µ,q with Dw ∈ X ˜ δ,µ,q -1 for some ˜ δ ∈ (0 , δ ] , provided the following structural conditions are satisfied: (ii) The map F ( u 0 ) : w ↦→ F ( u 0 )[ w ] -̂ L ( u 0 + w )[ u 0 ] (2.7) is well-defined, and for every δ ' ∈ (0 , δ ] , it maps w ∈ B δ ' ,µ,q,s to X δ ' ,ν,q for some exponent vector ν > µ . and for all w, ˜ w ∈ B δ ' ,µ,q,s . If all of these conditions are satisfied for all q ≥ 3 , then there exists a unique solution u of Eq. (2.1) such that u -u 0 and D ( u -u 0 ) , both, belong to 2 X ˜ δ,µ, ∞ . Section 2.3, below, is devoted to the proof of this theorem. Observe that, in the hypothesis of this theorem, the regularity required for S 1 , 0 , S 2 , 0 , and N 0 (specified by q 0 ) slightly differs from that required for S 1 , 1 , S 2 , 1 , and N 1 (specified by q ). The same observation can be made regarding the asymptotic data u 0 (implicitly specified by Condition (ii)) and the solution u (specified by q ). These gaps arise in the course of our proof, in particular in obtaining the energy estimates for the Cauchy problem Lemma 2.8. It is not clear whether this discrepancy in regularity could be eliminated by another method of proof, and in any case it disappears in the 'smooth' case, if q and q 0 are both infinite. In formulating this theorem, we require that the source term operator w ↦→ F [ w ], and with it the source term function f ( t, x, u ) in Eq. (2.1), be defined on the domain (0 , δ ] × T 1 × U , where U is an open neighborhood of the origin in R d . In the same way as for the coefficient matrices S 1 , S 2 and N , we find that the parameters δ , µ and s must be compatible with U . We note that Condition (ii) also restricts the regularity of the leading-order term u 0 . We also note that the time of existence of the solutions, specified by ˜ δ , could a priori be very small. Indeed, a smaller choice of the parameter s (which may be necessary in order to fit into the domains of the coefficient functions of Eq. (2.1)) generally leads to a shorter guaranteed time interval of existence. In its applications, Theorem 2.4 often allows one to find an open set of values for the exponent vector µ for which the singular initial value problem admits unique solutions. A lower bound for this set 1 can originate in Condition (i), while an upper bound is usually determined by Condition (ii). Both bounds on the set of allowed values for µ provide useful information on the problem. The upper bound for µ specifies the smallest regularity space and, hence, the most precise description of the behavior of w (in the limit t ↘ 0), while the lower bound for µ determines the largest space in which the solution u is guaranteed to be unique. Observe that this uniqueness property must be interpreted with care: under the conditions of our theorem, there is exactly one solution w in the space X ˜ δ,µ,q , although we do not exclude the possibility that another solution may exist in a larger space, for example, in X ˜ δ, ˜ µ,q with ˜ µ < µ . Note that if a given system does not satisfy our hypothesis above, there is sometimes a systematic method which allows one to 'improve' a leading-order term u 0 ; cf. the discussion of (order-n)-leading-order terms in Section 2.4 and, in particular, Theorem 2.21. We also remark that results analogous to those stated in Theorem 2.4 for D ( u -u 0 ) can be derived for higher-order time derivatives of the solution.", "pages": [ 5, 6, 7, 8, 9, 10, 11 ] }, { "title": "2.3.1 Outline of the argument", "content": "Before carrying out the details of the proof of the existence and uniqueness Theorem 2.4, we outline the basic strategy and the basic steps of the proof. We start by working with a linear version of the PDE system. We consider the Cauchy problem for this linear system, verifying that the conditions we have assumed as part of the hypothesis of Theorem 2.4 guarantee local existence and uniqueness of solutions for this Cauchy problem, with appropriate levels of regularity. We then use these results pertaining to the Cauchy problem for the linear system and establish that unique solutions to the singular initial value problem for the linear system exist in a neighborhood of the singularity. This is done using the solutions of sequences of Cauchy problems with the initial time for the j 'th element of this sequence set at t j , and with t j approaching zero, the time of the singularity. To show that the limit of such a sequence of solutions exists, and satisfies the singular initial value problem, we work with the linear PDE system in a weak form, and we also employ a family of energy functionals. To proceed from solutions of the singular initial value problem for the linear system to solutions for the full quasilinear system of Theorem 2.4, we use a standard fixed point iteration argument for a sequence of linearized equations and their singular initial value problems. Observe that arguments similar to those used here have been applied in [12] in order to establish existence and uniqueness results for the singular initial value problem for semilinear (second order) Fuchsian PDEs.", "pages": [ 11, 12 ] }, { "title": "2.3.2 The singular initial value problem for linear PDEs", "content": "The linear systems we consider here are essentially those of Theorem 2.4 (see Eq. (2.1)) with S 1 , S 2 , and N set to be independent of u , and with f set to be linear in u . More specifically, we introduce the following definition. Definition 2.5. Suppose that δ and r are positive reals, q and q 0 are non-negative integers, µ : T 1 → R d is an exponent vector, and ζ : T 1 → R d × d is an exponent matrix such that ζ > 0 . The system Eq. (2.1) is called a linear symmetric hyperbolic Fuchsian system if the following conditions are satisfied: where S 1 , 0 is symmetric and positive definite at every spatial point, where S 1 , 1 , S 2 , 0 , and S 2 , 1 are symmetric, and in addition, the maps S 1 , 0 , S 2 , 0 and N 0 belong to H q 0 ( T 1 ) , while S 1 , 1 , S 2 , 1 and N 1 are d × d -matrix-valued functions in B δ,ζ,q,r . Here, β : T 1 → R d is a smooth vector function with strictly positive components. (ii) The constant δ is sufficiently small so that S 1 is uniformly positive definite 1 . (iii) The source term is linear in the sense that with f 0 ∈ X δ,ν,q and the matrix F 1 satisfying R [ µ ] F 1 R [ µ ] -1 ∈ B δ,ζ,q,r . Here ν is an exponent vector with ν > µ . In this definition, we note the condition ν > µ . It is used primarily in the proof of Proposition 2.10, to enforce the needed rapid decay of the source term f 0 ( t, x ) as t → 0. It is clear from this definition that a linear symmetric hyperbolic Fuchsian system is a special case of a quasilinear symmetric hyperbolic system, with the leading-order term u 0 = 0 (this is no loss of generality for linear systems). Both in the linear and in the quasilinear case, we consider the functions S 1 , 1 , S 2 , 1 , N 1 , and F 1 to be perturbations of S 1 , 0 , S 2 , 0 , N 0 , and f 0 . An important step in our analysis is to seek uniform estimates for these perturbations. It turns out that such estimates can only be obtained if the perturbations are bounded. This is the reason for introducing the balls with radius r above, B δ,ζ,q,r , which can be considered as those spaces in which we seek the perturbations. In carrying out the proof, it is important that we keep careful track of which quantities the constants arising in various estimates are allowed to depend upon. To make this precise, it is useful to have the following definition. Definition 2.6. Suppose that Eq. (2.1) is a linear symmetric hyperbolic Fuchsian system for a chosen set of the parameters δ, µ, ζ, q, q 0 and r . Suppose that a particular estimate (e.g., the energy estimate Eq. (2.16) ), involving a collection C of constants, holds for solutions of Eq. (2.1) under a certain collection of hypotheses H . The constants C are defined to be uniform with respect to the system and the estimate so long as the following conditions hold: Recalling our definition above (see Eq. (2.5)) of the principal part operator ̂ L , we define the linear principal part operator by In terms of this operator, the linearized PDE system Eq. (2.1) may be written in the form L [ w ] = f 0 + F 1 w . In summary, the parameters δ , µ and q determine the space X δ,µ,q for the remainder of the solution of the singular initial value problem with leading-order term u 0 , while δ , ζ , q and r fix the space B δ,ζ,q,r of the perturbations of the coefficients. The parameter q 0 determines the order of differentiability of the 'leading-order' coefficient matrices S 1 , 0 , S 2 , 0 and N 0 . Suppose that Eq. (2.1) is a linear symmetric hyperbolic Fuchsian system (for parameters δ , r , q , q 0 , ζ , µ ; cf. Definition 2.5). We first consider the Cauchy problem ; that is, we prescribe initial data v [ t 0 ] specified at some t 0 ∈ (0 , δ ) and we seek solutions on [ t 0 , δ ] × T 1 which agree with v [ t 0 ] at t = t 0 . It is useful at this stage for us to make the temporary assumption that S 1 , 1 , S 2 , 1 , N 1 and F 1 are C ∞ ((0 , δ ] × T 1 ) functions contained in their respective function spaces, as discussed in Definition 2.5. (If S 1 , 1 , S 2 , 1 , N 1 and F 1 satisfy this smoothness assumption, then Eq. (2.1) is said to have smooth coefficients 1 .) Given such a linear symmetric hyperbolic system Eq. (2.1) with smooth coefficients and if in addition q 0 ≥ 2 and also f 0 ∈ X δ,ν,q is smooth, then it is a standard result (see, e.g., Chapter 16 in [40]) that the Cauchy problem is well-posed in the sense that for initial data v [ t 0 ] ∈ H q 0 ( T 1 ), there is a unique solution v : [ t 0 , δ ] × T 1 → R d to this Cauchy problem with v ( t 0 ) = v [ t 0 ] and with v ( t, · ) ∈ H q 0 ( T 1 ) for all t ∈ [ t 0 , δ ]. It is crucial for the following discussion that indeed this solution exists for the full interval [ t 0 , δ ], regardless of the choice of t 0 ∈ (0 , δ ). This is true as a consequence of the positivity of S 1 on (0 , δ ]; cf., Condition (ii) in Definition 2.5. In fact, this statement about the Cauchy problem remains true if the matrices S 1 , 1 , S 2 , 1 , N 1 , f 0 and F 1 are not required to be smooth, but are only required to have q 0 spatial derivatives. Such a relaxation is, however, not useful for our arguments; we use a more general continuation argument below by which we recover the non-smooth case. We also note that although this assumption that Eq. (2.1) has smooth coefficients implies that S 1 , 1 , S 2 , 1 , N 1 , f 0 , F 1 are differentiable to all orders, it does not not guarantee that all derivatives have controlled asymptotic behavior for t ↘ 0. This control holds only for a set of derivatives given by q , as labeled by the relevant function space. To establish control over the solutions to the Cauchy problem for the linear version of the PDE (2.1) and the regularity of these solutions, we now introduce a two-parameter family of explicitly time-dependent energies: Presuming that the remainder exponent vector µ is fixed, for any pair of positive constants κ and γ we define the energy E µ,κ,γ for a function w : [ t 0 , δ ] × T 1 → R d (with w ( t, · ) ∈ L 2 ( T 1 ) for each t ∈ [ t 0 , δ ]) as follows: where S 1 is the matrix appearing in Eq. (2.1). We emphasize again that, unlike standard definitions of energy, the energy functionals E µ,κ,γ [ w ]( t ) defined here depend on time explicitly, and not just through the time dependence of w ( t, x ). Note that it readily follows from this definition, and from the conditions assumed to hold for S 1 in Definition 2.5, that there exist uniform (in the sense of Definition 2.6) constants C 1 and C 2 such that for any L 2 ( T 1 ) function w ( t, x ), one has (for all t ) These energies, as is usually the case, have been defined in such a way (including the presence of the factor e -κt γ ) that for solutions of the Cauchy problem for Eq. (2.1), which we label v ( t, x ), the growth of the energies is controlled. Explicitly, we obtain the following estimate. Lemma 2.7 (Basic energy estimates for the Cauchy initial value problem) . Suppose that for some choice of the parameters δ, µ, ζ, q, q 0 , and r , with q = 0 and q 0 = 2 , and for u 0 = 0 , Eq. (2.1) is a linear symmetric hyperbolic Fuchsian system with smooth coefficients and with f 0 both smooth and contained in X δ,ν,q for some ν > µ . Suppose also that the system is block diagonal with respect to µ , that the energy dissipation matrix Eq. (2.6) is positive definite for all x ∈ T 1 and, in addition, that DS 1 , 1 and ∂ x S 2 , 1 are contained in B δ,ξ, 0 ,s for some constant s > 0 and some exponent matrix ξ with strictly positive entries. Then there exist positive constants κ , γ , and C such that for any initial data v [ t 0 ] ∈ H 2 ( T 1 ) specified at some t 0 ∈ (0 , δ ] , the solution of the Cauchy problem v for this system and this initial data satisfies the energy estimate for all t ∈ [ t 0 , δ ] . The constants C , κ , and γ may be chosen to be uniform 1 and do not depend on f 0 . In particular, if one replaces v [ t 0 ] specified at t 0 by any v [ t 1 ] specified at any time t 1 ∈ (0 , t 0 ] , then the energy estimate holds for the same constants C , κ , γ . Before proving this lemma, we make a few remarks: I) Lemma 2.7 does not imply that the energy estimate Eq. (2.16) holds for t < t 0 ; in particular, it need not hold for t ↘ 0. II) The well-posedness of the Cauchy problem which is used implicitly in the proof of Lemma 2.7 requires sufficiently high regularity on the coefficients (see for example [40]); this gives rise to the condition q 0 = 2 stated in the hypotheses. III) We remind the reader that the condition that the coefficients be smooth does not imply either the q = 0 condition or the conditions that DS 1 , 1 , ∂ x S 2 , 1 ∈ B δ,ξ, 0 ,s . While the smoothness condition implies the existence of all derivatives, the latter are statements about the behavior of the lowest derivatives in the limit t ↘ 0. It may appear that since Lemma 2.7 focuses on the Cauchy problem at times t 0 > 0 only, control of behavior near t = 0 is not necessary. However, such control is in fact needed to obtain an energy estimate with constants which are uniform and independent of t 0 . IV) In view of the norm equivalence (2.15) stated above, the estimate Eq. (2.16) can be rewritten as We observe that for this version of the energy estimates, all of the constants C , κ and γ are absorbed into the constant C ; every change of the former constants is therefore reflected in a corresponding change of the latter one. V) For some of the following discussion it is important to note that the particular values of the parameters of the perturbations space ζ and r (and also ξ and s ) do not play an essential role in this lemma: if we change from one perturbation space B δ,ζ,q,r to another one B δ, ˜ ζ,q, ˜ r (and ˜ ξ and ˜ s ), the same result is obtained with possibly different, but still uniform, constants C , γ , κ . This is true for all of the following results. ˜ ˜ ˜ Proof. The basic idea of the proof is to compute DE [ v ]( t ), then bound the terms on the right hand side and finally integrate the equation in time. For simplicity we write E [ v ] in place of E µ,κ,γ [ v ]. Computing 1 DE [ v ], and using the symmetry of the matrix S 1 , we obtain We first analyze the fourth term on the right hand side of this expression, which we label I . Using the fact that v is a solution of equation Eq. (2.1), using the block diagonal condition (Definition 2.3), and integrating by parts, we calculate Using the Holder inequality, we may then estimate the first term in this expression as follows: We now argue that for appropriate choices of κ and γ , all the other terms besides this one can be neglected in a certain sense. First, we use the properties of the linear symmetric hyperbolic Fuchsian system to expand the coefficient matrices S 1 , S 2 , N into terms which are O (1) at t → 0, and terms which decay as a power of t . We thereby obtain where we use the expansion for S 2 to write The first integral on the right hand side of this inequality for DE [ v ] is negative definite if the energy dissipation matrix M 0 = N 0 -S 1 , 0 ( D R [ µ ]) R [ µ ] -1 (see Eq. (2.6)) is positive definite, and so can be neglected. All of the terms in the second integral on the right hand side of this inequality decay as some positive power of t . We also know that as a consequence of Definition 2.5, the matrix S 1 is positive definite uniformly. It is at this point that we use the factor of e -κt γ which appears in the definition of the energy functionals. The scheme is to choose κ and γ in such a way that the second integral in the estimate above is negative definite. This can be achieved if we choose γ small enough and κ large enough so that the negative definite S 1 -term dominates all of the other terms in the second integral on (0 , δ ]. To see that the constants κ and γ may be chosen so that they are independent of the functions S 1 , 1 , S 2 , 1 , N 1 and F 1 and are therefore uniform in the sense of Definition 2.6, we recall that by assumption (see Definition 2.5), S 1 , 1 , S 2 , 1 and N 1 , and R [ µ ] F 1 R [ µ ] -1 are contained in the ball B δ,ζ,q,r . Hence these functions all must have finite norms bounded by r . Since the role played by S 1 , 1 , S 2 , 1 , N 1 and F 1 in determining the constants κ and γ depends strictly on the norms of these functions, we may choose a fixed set of the constants such that the inequality holds for any S 1 , 1 , S 2 , 1 , N 1 and F 1 contained in these balls. In total, we obtain which implies that Then using the norm equivalence Eq. (2.15), we may rewrite this as To integrate this inequality, it would be useful to divide both sides by √ E [ v ]( t ). However, since the L 2 norm of v may vanish in special cases, we use the following strategy. We set E /epsilon1 := E + /epsilon1 for some constant /epsilon1 > 0 (see, for instance, [36, Page 59]), and we check that (2.18) holds if we replace E by E /epsilon1 . Then dividing, and using 1 √ E /epsilon1 ∂ t E /epsilon1 = 2 ∂ t √ E /epsilon1 , we obtain after rescaling the constant C . We now integrate both sides over ∫ t t 0 ds , thereby obtaining where we note that the constant C changes from the second to the third line of this calculation. Taking the limit /epsilon1 → 0 finishes the proof that the inequality (2.16) holds. It also follows directly that the constant C is uniform. In order to derive the solution of the singular initial value problem from a sequence of solutions of the Cauchy problem, we need estimates involving higher order spatial derivatives. We obtain these as follows. Lemma 2.8 (Higher-order energy estimates for the Cauchy initial value problem) . Suppose that a linear symmetric hyperbolic Fuchsian system has been chosen which satisfies all of the conditions of the energy estimate Lemma 2.7, except that 1 (rather than q = 0 and q 0 = 2 ) q is an arbitrary integer greater than one, and q 0 = q +2 . Then there exists a pair of positive constants C and ρ such that for every sufficiently small /epsilon1 > 0 , the solution v of the Cauchy initial value problem with initial data v [ t 0 ] ∈ H q 0 ( T 1 ) specified at t 0 satisfies (for all t ∈ [ t 0 , δ ] ) The constants C (which in general differs from ˜ C in Eq. (2.17) ) and ρ are uniform 2 in the sense of Definition 2.6 and do not depend on f 0 . If we replace v [ t 0 ] specified at t 0 by any v [ t 1 ] specified at any t 1 ∈ (0 , t 0 ] , then the same estimate holds, for the same constants C and ρ . Observe that it is necessary (as stated in the hypothesis of this lemma) that the solution (and hence the data and coefficients) be contained in H q +2 if we wish to obtain an energy estimate for q spatial derivatives. The reason for this requirement is made clear in the course of the proof. The main difference between the hypotheses of Lemma 2.7 and 2.8 is that we require stronger control of the behavior of spatial derivatives of S 1 , S 2 , N , f 0 , and F 1 in the limit t ↘ 0 in Lemma 2.8 (i.e., q ≥ 1 as opposed to q = 0 in Lemma 2.7), while we presume smoothness for S 1 , S 2 , N , f 0 , and F 1 in both lemmas. Proof. This lemma is proven by taking q spatial derivatives of Eq. (2.1), reorganizing the resulting equations into a linear symmetric hyperbolic Fuchsian system for the q 'th order derivative of v , applying Lemma 2.7 to that system, and then carrying out a number of estimates needed to derive Eq. (2.19) from the inequality resulting from this application. We discuss some of the details for the q = 1 case here; the q > 1 cases are similar. Presuming that v ( t, x ) is the solution to the Cauchy problem for the linear system (before differentiating) with initial data v [ t 0 ] (contained in H q 0 ( T 1 )) specified at t 0 , we carry out the differentiation and obtain the following PDE system for ∂ x v : where and ̂ F 1 := F 1 -t∂ x S 2 + t∂ x S 1 S -1 1 S 2 . Here, we interpret v as a given function so that ̂ f 0 can be considered as a source term function, and ∂ x v as the unknown. This PDE system is clearly of the desired form Eq. (2.1) (with ∂ x v ∈ H q 0 -1 ( T 1 ) for each value of t ). However, it is not a linear symmetric hyperbolic Fuchsian system with respect to the same exponent vector µ : the term ∂ x Nv in ̂ f 0 violates Condition (iii) of Definition 2.5 since it is in X δ,µ,q -1 rather than in X δ,ν,q -1 for some ν > µ . However, Eq. (2.20) is a linear symmetric hyperbolic Fuchsian system if we choose ̂ µ := µ -/epsilon1/ 2 as the remainder exponent vector for any scalar constant /epsilon1 > 0. One verifies that Eq. (2.20) has block diagonal form with respect to ̂ µ and also that the energy dissipation matrix is positive definite if /epsilon1 is sufficiently small. Consequently, this system Eq. (2.20) satisfies the hypothesis of Lemma 2.7. It follows that there exist uniform (in the sense above) constants ̂ C , ̂ κ and ̂ γ (which generally differ from the ones for the original equation) such that ∂ x v satisfies the energy estimate (for all t ∈ [ t 0 , δ ]) To derive the q = 1 version of the estimate (2.19) from the energy estimate (2.22), we first note two useful inequalities. Letting f : (0 , δ ] × T 1 → R d denote any function for which the following norms are finite, we find 1 that, for all t ∈ (0 , δ ], Here, the constant C > 0 may depend on µ and /epsilon1 , but, in particular, is independent of t . Observe that the use of µ -/epsilon1 on the left hand side of Eq. (2.23) and of µ -/epsilon1 2 on the right hand side, is needed to dominate the terms of the form log t which are picked up on the left hand side when R [ µ ] is differentiated in space if µ is not constant (as a result of the H 1 ( T 1 ) norm). We also readily check that We now work on inequality (2.22): Observe first that inequality (2.16) from Lemma 2.7 holds for ̂ µ = µ -/epsilon1 2 so long as /epsilon1 is sufficiently small; hence we may add the left hand side of Eq. (2.16) (with µ -/epsilon1 2 ) to that of Eq. (2.22), and the right hand side of Eq. (2.16) (again with µ -/epsilon1 2 ) to that of Eq. (2.22). If we now use (i) the norm equivalence Eq. (2.15) on both sides to replace energy terms by terms involving norms of R [ · ], (ii) the definition of the Sobolev norm ‖ · ‖ H 1 ( T 1 ) to combine terms on each side, and (iii) the inequalities (2.23) and (2.24), then we obtain the following inequality: It remains to substitute in the definition of ̂ f 0 from Eq. (2.21). Noting the properties of the functions on the right hand side of Eq. (2.21), we determine that there exists a uniform constant C (in the sense above) such that, for all t ∈ (0 , δ ], Here in the second step, the constant C has been inconsequentially changed. Combining these last two inequalities, we obtain the desired result Eq. (2.19) with q = 1 by setting ρ = /epsilon1/ 2. This concludes the proof that the inequality (2.19) holds for the case q = 1. The proof for q > 1 proceeds very similarly. The argument that the constants C and ̂ δ may be chosen so that the inequality holds for all S 1 , 1 , S 2 , 1 , N 1 , and F 1 contained in B δ,ζ,q,r is essentially the same as that used in proving Lemma 2.7. We remark that while the introduction of /epsilon1 into the estimate Eq. (2.19) is certainly needed, one can choose this /epsilon1 to be arbitrarily small. One might worry that as one proceeds from q = 1 to higher values, the necessary value of /epsilon1 grows and causes trouble. However, since the incremental value needed for each step is arbitrarily small, one sees that the total value of /epsilon1 needed for arbitrary differentiability values can be kept small (below any chosen positive value). With these results for the Cauchy problem for linear symmetric hyperbolic Fuchsian systems established, we now set out to use solutions of the Cauchy problem to establish the existence of solutions to the singular initial value problem. We do this via an approximation scheme which works as follows: We first choose a monotonically decreasing sequence of times t n ∈ (0 , δ ] which converges to zero. Then for each n , we construct a function v n : (0 , δ ] × T 1 → R n which vanishes on (0 , t n ], and which is equal on ( t n , δ ] to the solution of the Cauchy problem with zero initial data at t n . One readily checks that for every choice of µ , one has v n ∈ C 0 ((0 , δ ] × T 1 ) ∩ X δ,µ, 0 . The central result of this section is that if certain hypotheses hold, then the sequence ( v n ) - whose elements we label approximate solutions - converges to a solution of the singular initial value problem for the linear system with vanishing leading term. The first step in showing this convergence is to set up the formalism to work with weak solutions to the linear system. To do this, we define a test function for this system to be any smooth function φ : (0 , δ ] × T 1 → R d for which there is a T ∈ (0 , δ ], such that φ ( t, x ) = 0 for all t > T . We then define the operators L and F acting on functions w ∈ X δ,µ, 0 via 1 and where φ is an arbitrary test function 2 . These operators are well-defined for w ∈ X δ,µ, 0 so long as the system Eq. (2.1) is a linear symmetric hyperbolic Fuchsian system for parameters δ , µ , ζ , r , q and q 0 as in Definition 2.5. We now define w to be a weak solution of Eq. (2.1) with vanishing leading term provided it satisfies, for all test functions φ , Here, we note the discussion of distributional time derivatives in Section A of the appendix. Before proceeding to show that weak solutions exist, we establish the following useful technical result. Lemma 2.9. Suppose that Eq. (2.1) satisfies the conditions to be a linear symmetric hyperbolic Fuchsian system for a fixed set of parameters δ , µ , ζ , δ , r , q , q 0 as per Definition 2.5, and is block diagonal with respect to µ . Then for every test function φ , the maps 〈L [ · ] , φ 〉 and 〈F [ · ] , φ 〉 are bounded linear functionals on X δ,µ, 0 . Proof. To prove this lemma it is sufficient to show that each term in 〈L [ w ] , φ 〉 is bounded by C || w || δ,µ, 0 , for some positive constant C and for every w ∈ X δ,µ, 0 . We demonstrate this for the first term, ∫ δ 0 〈R [ µ ] S 1 w,Dφ 〉 L 2 dt . Using Holder's inequality, the spatial continuity 3 of S 1 and the block-diagonal property, we find that The constant C , which is used to estimate both the contributions from S 1 and from φ , is uniform in the sense defined above. Other terms in 〈L [ w ] , φ 〉 follow similarly, and the same arguments hold for the 〈F [ · ] , φ 〉 operator. We now determine that, for a given linear symmetric hyperbolic Fuchsian system with certain conditions holding, the singular initial value problem with zero leading term has a weak solution. In the proof of this result, we show that these solutions can be obtained as a limit of approximate solutions of the Cauchy problem, as described above. Proposition 2.10 (Existence of weak solutions of the linear singular initial value problem with smooth coefficients) . Suppose that Eq. (2.1) satisfies the same conditions as stated in Lemma 2.7 and hence is a linear symmetric hyperbolic Fuchsian system (with smooth coefficients) for δ , µ , ζ , r , q , and q 0 as per Definition 2.5 with q = 0 and q 0 = 2 . Then there exist weak solutions w : (0 , δ ] × T 1 → R d to the singular initial value problem (with vanishing leading term) which are elements of X δ,µ, 0 . Observe that Proposition 2.10 is the most general existence result which we obtain for linear equations, in the sense that only minimal control of the behavior of the coefficients of the equation is required (i.e., q = 0, q 0 = 2 as in Lemma 2.7). We discuss higher regularity of the solutions under stronger regularity assumptions in Proposition 2.12 below. We also note that while Proposition 2.10 provides sufficient conditions for the existence of solutions, it tells us nothing regarding uniqueness. To obtain uniqueness, we need to impose stronger assumptions on the coefficients of the PDE system; see Proposition 2.14 below. Proof. As described above, we choose a sequence ( t n ) converging to zero, and the corresponding sequence of approximate solutions ( v n ) ∈ C 0 ((0 , δ ] , H q 0 ( T 1 )) ∩ X δ,µ, 0 . We seek to show that the sequence ( v n ) forms a Cauchy sequence in X δ,µ, 0 . Defining ξ mn := v m -v n , we readily see that From the energy estimate for the Cauchy problem Lemma 2.7 on each subinterval, we then derive where in the last inequality we have used the energy/norm equivalence Eq. (2.15) above, and we have also used the fact that the (linear) PDE system for v m -v n has a vanishing source term f 0 . Recalling the definition of the norm || · || δ,µ,q , noting the monotonicity of ∫ t t m s -1 ||R [ µ ] f 0 || 2 L ds , and noting the equality ξ mn ( t n , · ) = v m ( t n , · ) for t = t n , we now have To complete the argument that we have a Cauchy sequence, it is useful to introduce which is well-defined so long as f 0 ∈ X δ,ν, 0 for ν > µ . Choosing /epsilon1 > 0 as a lower bound for the gap between ν and µ among all components, we see that there must exist a constant C such that G ( t ) ≤ Ct /epsilon1 ; thence, we have from which it easily follows that ( v ) is a Cauchy sequence in the Banach space X . n δ,µ, 0 Since it has been established (in Lemma 2.9) that P = L-F is a continuous operator on X δ,µ, 0 , to show that the limit of the Cauchy sequence ( v n ) is a weak solution of the system of interest, it is sufficient to show that the limit of the sequence of reals ( 〈P [ v n ] , φ 〉 ) is zero for all test functions φ . Choosing any v n in our sequence, we know from its definition that v n vanishes on (0 , t n ] and is a solution to the equation 〈P [ · ] , φ 〉 = 0 on [ t n , δ ]. Recalling the definition of P , we calculate on this latter interval, for any test function φ , Straightforward calculation then shows that from which it follows (from the properties of G ( t )), that we have a weak solution. Based on this existence result for weak solutions, we would like to define a map which, for a fixed choice of S 1 , S 2 , N and F 1 , maps any smooth function f 0 ∈ X δ,ν, 0 to a weak solution w ∈ X δ,ν, 0 of 〈P [ v n ] , φ 〉 = 0 . Then as a next step, we would like to extend this map to all f 0 of X δ,ν, 0 , and thereby show that weak solutions exist for all f 0 ∈ X δ,ν, 0 , and not just for those f 0 which are smooth. While the lack of a uniqueness result for weak solutions is an impediment to defining the desired map, we can get around this by provisionally defining an operator of this sort which maps a smooth f 0 to the solution of the weak solution which is obtained as the limit of the sequence ( v n ) (as discussed in the proof of Proposition 2.10). We do this now, noting that the definition makes sense only after we have established that we get the same limit for any choice of the sequence of times t n ). We then establish an estimate for this operator, and use this estimate to extend the operator to all of X δ,ν, 0 . Proposition 2.11. Presuming the hypotheses listed in Proposition 2.10, there exists an operator H : X δ,ν, 0 → X δ,µ, 0 which maps a smooth source function f 0 to the weak solution w of 〈P [ w ] , φ 〉 = 0 which is obtained as the limit of the sequence of approximate solutions ( v n ) corresponding to a choice of a monotonic sequence of times ( t n ) converging to zero. This operator is well-defined (independent of the choice of the sequence ( t n ) ) and satisfies the estimate for all smooth f 0 ∈ X δ,ν, 0 . The positive constants C and ρ are uniform. The operator extends to all (not necessarily smooth) f 0 ∈ X δ,ν, 0 , with the estimate (2.30) holding for all such f 0 with the same constants. Indeed, this extended operator H maps all f 0 ∈ X δ,ν, 0 to weak solutions of Eq. (2.1) . The last paragraph in this proposition generalizes the existence result in Proposition 2.10 to all, not necessarily smooth, source terms f 0 ∈ X δ,ν, 0 . We note, however, that otherwise the system is still assumed to have smooth coefficients in the sense defined above. Proof. We presume initially (as part of the hypothesis of smooth coefficients) that f 0 is smooth; i.e., f 0 ∈ C ∞ ((0 , δ ] × T 1 ) ∩ X δ,ν, 0 . To show that H is a well-defined map from C ∞ ((0 , δ ] × T 1 ) ∩ X δ,ν, 0 to X δ,µ, 0 , independent of the choice of time sequence, we choose a pair of such sequences ( t 1 n ) and ( t 2 m ) with their corresponding sequences ( v 1 n ) and ( v 2 m ) of approximate solutions, and from the union of the two time sequences we construct a third time sequence ( t l ). As is the case for ( v 1 n ) and ( v 2 m ), the combined sequence of approximate solutions ( v l ) must be a Cauchy sequence, so 1 || v 1 n -v 2 m || δ,µ, 0 must vanish in the limit n, m → ∞ . Then labeling w 1 as the limit of the first sequence and w 2 as the limit of the second, we calculate It easily follows that w 1 and w 2 are equal in X δ,µ, 0 . To prove the estimate for H (restricted to smooth f 0 ), we let ( v n ) be a sequence of approximate solutions with limit w = H ( f 0 ), and then based on Eq. (2.29) we determine that || w -v 1 || δ,µ, 0 ≤ CG ( t 1 ) ≤ CG ( δ ). It then follows that If we now apply the energy estimates to show that || v 1 || δ,µ, 0 ≤ ˜ CG ( δ ), we deduce that for some adapted constant C . To relate G ( δ ) to the source term, we check that for some ρ > 0 so long as µ < ν . It then follows from multiplying both sides by s -1 and integrating over ∫ δ 0 that The estimate Eq. (2.30) is then a consequence. To extend the domain of H from C ∞ ((0 , δ ] × T 1 ) ∩ X δ,ν, 0 to X δ,ν, 0 , we note that this first space is dense in the second by definition. Hence, for any f 0 ∈ X δ,ν, 0 , we can find a sequence of functions f 0 ,j ∈ C ∞ ((0 , δ ] × T 1 ) ∩ X δ,ν, 0 which converges to f 0 . It follows as a consequence of the estimate Eq. (2.30) that there is a unique continuous extension of H to the full space X δ,ν, 0 . The extended operator, which we refer to with the same symbol H , is continuous and satisfies the same estimate. The continuity of the extended operator H and the continuity of 〈P [ w ] , φ 〉 = 0 with respect to w easily implies that H maps any f 0 ∈ X δ,ν, 0 , even those which are not smooth, to weak solutions. To proceed from weak solutions to strong solutions of the singular initial value problem for these linear systems (while still keeping the smoothness assumption for the coefficients S 1 , 1 , S 2 , 1 , N 1 and F 1 ), we need to determine the regularity of these weak solutions. We do this in the following proposition, and thereby prove the existence of strong solutions. Proposition 2.12 (Regularity of solutions for smooth coefficients) . Suppose that all of the conditions of Proposition 2.10 hold, with the exception that q ≥ 1 and q 0 = q + 2 . Then, weak solutions w of the singular initial value problem (whose existence has been checked in Proposition 2.10) are differentiable in time 1 and hence are strong solutions of Eq. (2.1) , with w ∈ X δ,µ,q and Dw ∈ X δ,µ,q -1 . As well, the solution operator H defined in Proposition 2.10 maps X δ,ν,q to X δ,µ,q , and satisfies for all (not necessarily smooth) f 0 ∈ X δ,ν,q . The constants C > 0 and ρ > 0 are uniform in the sense of Definition 2.6 (but may depend in particular on q ). Observe (without pursuing the details here) that an estimate similar to Eq. (2.31) can also be proven for the time derivative of the solution. Additional regularity assumptions on the time derivatives of the coefficients of the equation also allow one to prove corresponding statements regarding higher order time derivatives D k ' w for k ' ≥ 2. Proof. Using w to denote the solution to the singular initial value problem whose existence is established in Proposition 2.10 (as an element of X δ,µ, 0 ), and using ( v n ) to denote the sequence of approximate solutions which converges to w , we first note that it follows from their definitions 1 that the v n are contained in C 0 ((0 , δ ] , H q ( T 1 )) - even in X δ,µ,q - under the hypothesis of Proposition 2.12. Hence, in the same way as we have used the energy estimates in Lemma 2.7 to show that ( v n ) is a Cauchy sequence in X δ,µ, 0 for the proof of Proposition 2.10, we can now use the energy estimates of Lemma 2.8 to show that ( v n ) is a Cauchy sequence in X δ,µ -/epsilon1,q for an arbitrarily small constant /epsilon1 > 0. We then show that the limit of this Cauchy sequence equals w above. The solution w is hence in X δ,µ -/epsilon1,q and we get the estimate Now, if the equation is of linear symmetric hyperbolic Fuchsian form for a choice of µ , as we have assumed so far, it is also of linear symmetric hyperbolic Fuchsian form for a choice of ̂ µ := µ + /epsilon1 if /epsilon1 > 0 is sufficiently small in comparison to ν -µ . Moreover, the assumption that the system is block diagonal with respect to µ also implies that this is the case with respect to ̂ µ ; the same is true for the condition involving the energy dissipation matrix. Hence, we can apply the argument in the previous paragraph based on this choice of ̂ µ . This leads to the conclusion that, in fact, the solution w is in X δ, ̂ µ -/epsilon1,q = X δ,µ,q (as opposed to X δ,µ -/epsilon1,q above) and possibly after a slight change of the constants C and ρ . Next we show that the solution w is differentiable in time. We define ̂ v n := S -1 1 ( -S 2 t∂ x v n -Nv n + f 0 + F 1 v n ) . We know that ̂ v n ∈ X δ,µ,q -1 and ̂ v n ( t ) = Dv n ( t ) for all t ∈ [ δ I , δ ] for any δ I ∈ (0 , δ ) and for n sufficiently large; we cannot choose δ I = 0 here since the time derivative of v n is in general not defined at t = 0. Moreover, we find from the definition and the convergence of the sequence v n that || ̂ v n -̂ v m ‖ δ,µ,q -1 ≤ C ‖ v n -v m ‖ δ,µ,q → 0 for a uniform constant C > 0. Hence there exists ̂ v ∈ X δ,µ,q -1 such that ̂ v n → ̂ v . The estimate also holds if we restrict the time interval to [ δ I , δ ] as above and hence we find that Dv n ( t ) = ̂ v n ( t ) → ̂ v ( t ) uniformly at every t ∈ [ δ I , δ ]. It is then a standard result that w is differentiable in t at every t ∈ [ δ I , δ ] and further that ̂ v ( t ) = Dw ( t ). Since δ I can be chosen arbitrarily small, it follows that for all t ∈ (0 , δ ], w is differentiable in t and v = Dw . Consequently, we find that Dw = v ∈ X δ,µ,q -1 . ̂ ̂ To argue that w is a strong solution, we start from the fact that w is a solution of the weak equation whose integral representation can be integrated by parts in both time (using Eq. (A.4) in the appendix) and space. We may then choose a suitable sequence of test functions, for example those which are used as mollifiers in Lemma A.1 in the appendix, so that the resulting system converges pointwise almost everywhere to one of the components of Eq. (2.1) evaluated at one point ( t, x ). Doing this for every component and for every point ( t, x ) ∈ (0 , δ ] × T 1 , we determine that w is actually a solution of the strong equation almost everywhere. To this point, we have assumed throughout our analysis that the matrices S 1 , 1 , S 2 , 1 , N 1 and F 1 are smooth; i.e., we have not thus far allowed these matrices to be general elements of the spaces B δ,ζ,q,r from Definition 2.5. If we wish to use our current (linear) results as a tool for proving that there are (unique) solutions to the singular initial value problem for the (nonlinear) quasilinear system, we need to generalize these linear results to include the possibility that the matrices listed above are not smooth (since, in the quasilinear case, these matrices are functions of the solutions, which may not a priori be smooth). Before carrying out this generalization of the existence (and uniqueness) results for the linear singular initial value problem, we note that we can at this stage assume that the term F 1 vanishes. This simplification does not constitute an essential loss of generality since in our work below, we replace the linear source term function f 0 by a general quasilinear expression shortly; the resulting expression then incorporates the effects of the term F 1 . We recall that the F 1 -term plays a convenient role in our verification that the higher order energy estimates of Lemma 2.8 hold. Such a term is generated in the linear equation for ∂ x v , which we obtain by taking a spatial derivative of the equation for v . Proposition 2.13 (Existence for the linear singular initial value problem for non-smooth coefficients) . Suppose that Eq. (2.1) is a linear symmetric hyperbolic Fuchsian system for δ , µ , ζ , q , q 0 and r as in Definition 2.5 (with F 1 = 0 ) for q 0 = q + 2 and q ≥ 2 (not necessarily with smooth coefficients), and suppose that it is block diagonal with respect to µ . Suppose that the energy dissipation matrix Eq. (2.6) is positive definite. Then there exists a solution w : (0 , δ ] × T 1 → R d to the singular initial value problem with zero leading order term such that w ∈ X δ,µ,q and Dw ∈ X δ,µ,q -1 . The solution operator H : f 0 ↦→ w maps X δ,µ,q into X δ,ν,q , and satisfies for some positive uniform constants C and ρ . Observe that this result also holds in the case that q 0 and q are both infinite: If the conditions of this proposition are satisfied for all integers q 0 = q + 2 and q ≥ 2, then w ∈ X δ,µ, ∞ and Dw ∈ X δ,µ, ∞ . However, the q -parametrized sequence of constants C and ρ occurring in the estimate of the solution operator may in general be unbounded as q →∞ . Proof. The basic idea is to approximate the non-smooth coefficients S 1 , 1 , S 2 , 1 and N 1 by a sequence of smooth ones and then to apply Proposition 2.12 to obtain a sequence of approximate solutions. The main work is then to prove that this sequence converges to the solution of the system with non-smooth coefficients in an appropriate sense. Step 1: A sequence of approximate solutions. We presume that a linear symmetric hyperbolic Fuchsian system with parameters δ , r , q , q 0 , µ and ζ , and with coefficient matrices S 1 , 1 , S 2 , 1 and N 1 in the perturbation space B δ,ζ,q,r has been specified. We assume that this system is block diagonal with respect to µ and that the energy dissipation matrix is positive definite. According to the definition of the space B δ,ζ,q,r , there exist sequences ( S 1 , 1 , [ j ] ), ( S 2 , 1 , [ j ] ) and ( N 1 , [ j ] ) of smooth elements in B δ,ζ,q,r which converge to S 1 , 1 , S 2 , 1 and N 1 (in a way described below). We thus obtain a sequence of linear principal part operators (with smooth coefficients) and hence a sequence of systems of the form L [ j ] [ w ] = f 0 . For each j , this is a linear symmetric hyperbolic Fuchsian system for δ , µ , q , q 0 , ζ and r with smooth coefficients. It is clear that the sequences can be chosen so that, for each j , the block diagonal condition with respect to µ is satisfied, and the energy dissipation matrix is positive definite for each equation. Clearly as well, each S 1 , 1 , [ j ] is differentiable in time and DS 1 , 1 , [ j ] is bounded (in a sense which we make more precise below). Hence, for each j , Proposition 2.12 implies the existence of a solution operator H [ j ] , and therefore a sequence w [ j ] ∈ X δ,µ,q defined by w [ j ] := H [ j ] [ f 0 ]. Step 2: Uniformity of the sequence of coefficients. To study the convergence properties of these approximate solutions w [ j ] in more detail we make a special choice of the sequences ( S 1 , 1 , [ j ] ), ( S 2 , 1 , [ j ] ) and ( N 1 , [ j ] ) as in Lemma A.1 in the appendix (where we replace the two indices i and j by just one index j ). The advantage of this choice is that we will be able to argue that ‖R [ ˜ ζ ] DS 1 , 1 , [ j ] ( t, · ) ‖ L 2 is uniformly bounded in j and t under the hypotheses of Proposition 2.13, which will be important for the following argument. The slight disadvantage, however, as stated in Condition (iv) of Lemma A.1, is that the convergence is guaranteed only with respect to a norm ‖ · ‖ δ, ˜ ζ,q for any exponent matrix ˜ ζ smaller than ζ ; fortunately we will see below that this is not significant. Let us choose such an exponent matrix ˜ ζ with strictly positive entries. Let us moreover suppose for the moment that a uniform bound for ‖R [ ˜ ζ ] DS 1 , 1 , [ j ] ( t, · ) ‖ L 2 has been found (which we show shortly). By setting ξ = ˜ ζ and choosing some uniform value of the (function space ball) radius s in Condition (ii) of Proposition 2.12, we are allowed to apply Proposition 2.12 in such a way that each of the approximate equations L [ j ] [ w ] = f 0 is a perturbation of one common equation in the perturbation space B δ, ˜ ζ,q,r . A particular consequence is then that we obtain an estimate for the operators H [ j ] of the form Eq. (2.31) with C independent of j . To establish this uniform bound of ‖R [ ˜ ζ ] DS 1 , 1 , [ j ] ( t, · ) ‖ L 2 , we use Eqs. (A.1) and (A.2) from Lemma A.1 in the appendix to obtain where ̂ S 1 , 1 is the extension introduced in Lemma A.1. We wish to estimate this expression in the L 2 -norm. The first term can be estimated in the L 2 -norm by C ( ˜ ζ ) ‖ S 1 , 1 , [ j ] ‖ δ, ˜ ζ,q with a constant determined by ζ . The second term is treated as follows 1 ∣ ∣ where we have used the definition of the extension ̂ S 1 , 1 in the second line. The properties of the kernel φ imply that the term with the L 1 y -norm is unity (independently of x ). Since q ≥ 1, we can use Sobolev embedding to estimate the term with the L ∞ y -norm by the H q y -norm and hence the sup s -term by the norm ‖·‖ δ, ˜ ζ,q . As a consequence, all quantities are independent of x , and therefore we find that the second term of R [ ˜ ζ ] DS 1 , 1 , [ j ] above can in total be estimated in the L 2 -norm as less than or equal to ∣ ∣ for some constant C > 0 which is, in particular, independent of t and j . Hence, we only need to estimate the last integral: This takes the form x · φ ( x ) for all x ∈ R . This has the property that it vanishes for all | x | ≥ 1 and is bounded for all | x | ≤ 1. Summarizing, we have thus confirmed that for every j , and for a constant C , which, in particular, does not depend on j . Step 3: Convergence of the sequence ( w [ j ] ) in X δ,µ,q -1 . We seek to show that the sequence ( w [ j ] ) converges 1 in X δ,µ,q -1 . We do this by showing that ( w [ j ] ) is a Cauchy sequence: Setting ξ [ ij ] := w [ i ] -w [ j ] , we derive the equation where we interpret the right hand side as a source term for this linear equation for ξ [ ij ] . One readily checks that Eq. (2.33) is a linear symmetric hyperbolic system for the same parameters as above, but with differentiability index q -1 (since the source term incorporates one spatial derivative); hence, so long as q ≥ 2, we may apply Proposition 2.12 also to this equation. We thus obtain, as a consequence of Eq. (2.31), where the (scalar) constant ˜ ζ min is the minimal value of all components of ˜ ζ at all x ∈ T 1 (note that ˜ ζ min is positive), and where the generic constant C represents the constant in Eq. (2.31) (for ν replaced by µ + ˜ ζ min ). It is crucial here that the constant C does not depend on the index i ; this is a consequence of the uniformity of the constant in Eq. (2.31). If we now expand out the definition of L [ j ] , using the block diagonality conditions, the Sobolev embedding (for spatial dimension one and q ≥ 2) and the Moser inequality (stated for instance in Proposition 3.7 in Chapter 13 of [40]), we find that Step 4: The limit w is a solution of the original equation. Standard arguments of the sort carried out in the proof of Proposition 2.10 show that w is a weak solution of the system Lw = f 0 with non-smooth coefficients. Since each w [ j ] is differentiable in time and a strong solution of the equation, we can solve each equation for Dw [ j ] . Relying on an argument similar to that used in the proof of Proposition 2.12, we see that it follows that w is differentiable in time, with Dw ∈ X δ,µ,q -2 , and therefore the limit w is actually a strong solution of the equation. It immediately follows that ( w [ j ] ) is a Cauchy sequence in X δ,µ,q -1 , and hence the sequence has a limit w ∈ X δ,µ,q -1 . Step 5: The limit w is in X δ,µ,q . We now show that w is in fact in X δ,µ,q (and not only in X δ,µ,q -1 ) and consequently Dw ∈ X δ,µ,q -1 (and not only in X δ,µ,q -2 ). It follows from its definition that the sequence ( w [ j ] ) is contained in X δ,µ,q , and furthermore, as a consequence of the operator estimate Eq. (2.31), we have that Here, the constant C is independent of the index j as a result of our discussion of uniformity above. We thus find that the sequence is uniformly bounded in X δ,µ,q . If we now fix a time t 0 ∈ (0 , δ ), then the sequence ( w [ j ] ( t 0 )) is bounded in the Hilbert space H q ( T 1 ). Since the sequence ( w [ j ] ( t 0 )) is also convergent in the Hilbert space H q -1 ( T 1 ), we can apply Corollary C.3 from the Appendix. We hence find that the limit w ( t 0 ) is in H q ( T 1 ). We consider the function w , which is a strong solution in X δ,µ,q -1 with w ( t 0 ) ∈ H q ( T 1 ), to be a strong solution of the Cauchy problem of the linear symmetric hyperbolic equation with the 'initial data' w ( t 0 ) contained in H q ( T 1 ). Given that the coefficients of the system are contained in C 0 ((0 , δ ] , H q ( T 1 )), the standard theory of linear symmetric hyperbolic equations (see [40]) implies that w , and therefore R [ µ ] w (recall that R [ µ ] is in C ∞ ((0 , δ ] × T 1 )) is a continuous map (0 , δ ] → H q ( T 1 ). In fact, this latter map is bounded as a consequence of taking the limit j →∞ of Eq. (2.34); i.e., w ∈ ̂ X δ,µ,q (see the appendix). Replacing µ by µ + ˜ /epsilon1 in all of the previous steps, we see that the same arguments go through as long as ˜ /epsilon1 > 0 is sufficiently small. We therefore find that R [ µ + ˜ /epsilon1 ] w is a bounded continuous map (0 , δ ] → H q ( T 1 ). It then follows from Lemma A.2 in the appendix that w ∈ X δ,µ,q . Step 6: Properties of the solution operator H . We have thus extended the solution operator H to the case of non-smooth coefficients by the above limit procedure. It can thus be checked that the estimate Eq. (2.31) still holds with uniform constants. Having obtained a comprehensive existence result for the singular initial value problem of linear symmetric hyperbolic Fuchsian equations, we now show uniqueness of these solutions. Proposition 2.14 (Uniqueness for the linear singular initial value problem) . Suppose that all of the conditions of Proposition 2.13 hold for a chosen singular initial value problem (with zero leading-order term). The solution for this problem is unique in X δ,µ,q . Proof. We consider w and ˜ w to be a pair of (generally different) solutions to the same singular initial value problem, and we define ω := w -˜ w to be the difference between the two. It follows that ω is a solution of the same equation with vanishing source-term f 0 , with ω ( t ) being an element of H 2 ( T 1 ) for every time t ∈ (0 , δ ]. Choosing any t 0 ∈ (0 , δ ], we can also consider ω | ( t 0 ,δ ] to be the unique solution of the Cauchy initial value problem (for the same linear PDE system) with initial data ω ( t 0 ). Since the solution ω together with the coefficients have H 2 -regularity and since S 1 is guaranteed to be positive definite on the whole time interval, we may apply the energy estimate Eq. (2.17). We obtain (replacing µ by µ -/epsilon1 , which is allowed for any /epsilon1 > 0) for all t ∈ ( t 0 , δ ], with the constant C independent of t . Observe that it follows from the definition of R [ µ ] that we can rewrite the right hand side of Eq. (2.35) as so that Eq. (2.35) takes the form We now take the limits of Eq. (2.37) as t 0 → 0, noting that the left hand side of the equation and the constants are unchanged by taking this limit. Since R [ µ ] ω is a bounded map from (0 , δ ] to L 2 ( T 1 ), the limit as t 0 → 0 of the right hand side of Eq. (2.35) vanishes. It thus follows that for all t ∈ (0 , δ ], Then since R [ µ -/epsilon1 ]( t, x ) is bounded positive at any fixed t on (0 , δ ], we deduce that ω ( t, x ) = 0 at all t ; uniqueness follows.", "pages": [ 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32 ] }, { "title": "2.3.3 The nonlinear theory", "content": "The results obtained in Section 2.3.2 pertain exclusively to linear systems. In this section, we use those results together with a fixed point iteration procedure to prove Theorem 2.4, which establishes existence and uniqueness of solutions to the singular initial value problem for the nonlinear system Eq. (2.1) with a (no longer necessarily vanishing) leading-order term u 0 . To start, it is useful to rewrite Eq. (2.1) in a convenient form. Recalling the definition (see Eq. (2.5)) of the operator ̂ L ( u )[ v ] := S 1 ( u ) Dv + S 2 ( u ) t∂ x v + N ( u ) v , we may write Eq. (2.1) in the form ̂ L ( u )[ u ] = f ( u ). Despite the nonlinear nature of ̂ L , this operator is linear in the sense that ̂ L ( u )[ v 1 + v 2 ] = ̂ L ( u )[ v 1 ] + ̂ L ( u )[ v 2 ]. Hence if we let u 0 denote a chosen leading-order term (satisfying the hypotheses of Theorem 2.4), if we set u = u 0 + w , and if we recall the definition (see Eq. (2.7)) then Eq. (2.1) takes the form The linear analysis discussed in Section 2.3.2 does not apply to the nonlinear equation Eq. (2.38) directly. Nevertheless, if we linearize this equation by fixing ˜ w ∈ B δ,µ,q,s (for some s > 0) and write ̂ L ( u 0 + ˜ w )[ w ] = F ( u 0 )[ ˜ w ] , (2.39) then (presuming the hypothesis of Theorem 2.4) the techniques of Section 2.3.2 are applicable. In applying these techniques, we assume that δ has been chosen sufficiently small so that Condition (ii) of Definition 2.5 is satisfied; this implies no loss of generality since, as we see below, the argument leading to the proof of Theorem 2.4 requires further shrinkage of the time interval. It is important to note that, for every ˜ w ∈ B δ,µ,q,s , the hypothesis of Theorem 2.4 implies the existence of common quantities ζ and r (as in Definition 2.2) so that S 1 , 1 ( u 0 + ˜ w ) , S 2 , 1 ( u 0 + ˜ w ) and N 1 ( u 0 + ˜ w ) are all contained B δ,ζ,q,r . 32 Moreover, if ˜ w ∈ B δ ' ,µ,q,s for any δ ' < δ , then the same statement regarding S 1 , 1 , S 2 , 1 and N 1 holds for the same common ζ and r , but with δ replaced by δ ' . Replacing the right-hand side of this equation by a fixed function φ ∈ X δ,ν,q , we readily check that the linear system ̂ L ( u 0 + ˜ w )[ w ]( t, x ) = φ ( t, x ) , (2.40) is of linear symmetric hyperbolic Fuchsian form (Definition 2.5) for q and q 0 = q +2 and for a sufficiently large constant r . Hence, it follows from Propositions 2.13 and 2.14 that the system has a unique solution w ∈ X δ,µ,q (we only require q ≥ 2 at this stage of the proof), and we can define the corresponding solution operator H ( u 0 + ˜ w ) which maps the source term φ to the solution w = H ( u 0 + ˜ w )[ φ ]. The case φ = F ( u 0 )[ ˜ w ] corresponds to Eq. (2.39); thus we compose H ( u 0 + ˜ w ) with F ( u 0 ) to define the operator G ( u 0 ) as follows: G ( u 0 )[ ˜ w ] := H ( u 0 + ˜ w )[ F ( u 0 )[ ˜ w ]] . Hence, w = G ( u 0 )[ ˜ w ] ∈ X δ,µ,q is the unique solution of the singular initial value problem of Eq. (2.39). In terms of G ( u 0 ), we see that w is a solution of the singular initial value problem for the nonlinear equation (2.1) with leading-order term u 0 if and only if it satisfies w = G ( u 0 )[ w ]; i.e., if and only if w is a fixed point of G ( u 0 ). The operator G ( u 0 ) is the key to the following fixed point iteration argument. We define the sequence of functions ( w N ) by setting w 0 = 0, and defining w N +1 = G ( u 0 )[ w N ] for N ∈ N . To control this sequence, we need uniform bounds; i.e., we wish to show that each element of the sequence is contained in B δ,µ,q,s . Suppose that this is true for w 0 , . . . , w N . It follows from the hypothesis of Theorem 2.4 (given that w 0 = 0) that The constant C > 0 does not depend on N . Using the definition of w N +1 , together with Eq. (2.31), we have that where ˜ C and ρ > 0 are constants which also do not depend on N . Combining, we obtain We recall that the uniformity of the constants implies that the same estimate holds with the same constants if we choose to formulate the same singular initial value problem in terms of a constant ¯ δ ∈ (0 , δ ) instead of δ itself. Since we have supposed that w N is contained in B δ,µ,q,s - that is, ‖ w N ‖ δ,µ,q ≤ s - we can find such a sufficiently small ¯ δ so that ¯ δ ρ ˜ CC ≤ 1 / 2 and ¯ δ ρ ˜ C ‖ F ( u 0 )[0] ‖ δ,ν,q ≤ s/ 2 , while preserving the bound ‖ w N ‖ ¯ δ,µ,q ≤ s . This can be done, since ‖ F ( u 0 )[0] ‖ ¯ δ,ν,q ≤ ‖ F ( u 0 )[0] ‖ δ,ν,q . For this diminished choice ¯ δ , we thus determine that ‖ w N +1 ‖ ¯ δ,µ,q ≤ s . Since the above estimates do not depend on the index N , it follows that the whole sequence is bounded, and we have ( w N ) ⊂ B ¯ δ,µ,q,s . We now consider an arbitrary pair of functions w,v ∈ B ¯ δ,µ,q,s , and we calculate the following estimate for the norm of the difference of the operator G ( u 0 ) acting on each of these: Note that, for reasons discussed below, we work with ‖ · ‖ ¯ δ,µ,q -1 rather than ‖ · ‖ ¯ δ,µ,q . It follows from the hypothesis of Theorem 2.4 and from Eq. (2.31) that the first term on the right hand side of this estimate satisfies the inequality with the Lipschitz constant C smaller than unity so long as we allow a further decrease in ¯ δ ; the argument for this is the same as for the semilinear case [13, 12]. This controls this first term. To estimate the second term on the right hand side, we set It follows from the definition of H ( u 0 + w ) that Therefore, setting ̂ L ( u 0 + w )[ w A ] and ̂ L ( u 0 + v )[ w B ] equal, and using the linear property of the operator ̂ L ( u 0 + w ) noted above, we derive The right hand side of this equation is similar to a term which appears in Eq. (2.33); thus we can treat it using similar techniques to those used in the proof of Proposition 2.13. In doing this, we rely on the hypothesis for Theorem 2.4, and we use the condition q ≥ 3 in order to guarantee that the source term of Eq. (2.41) has at least two spatial derivatives. We thus obtain for a constant ̂ C which may depend on s , but not on the particular choice of ¯ δ . Again using the hypothesis of Theorem 2.4, we see that for a choice of a possibly even smaller ¯ δ , which we now label ˜ δ , we can control this second term from the right hand side of the estimate for ‖ G ( u 0 )[ w ] -G ( u 0 )[ v ] ‖ ˜ δ,µ,q -1 via a term of the form C‖ w -v ‖ ˜ δ,µ,q -1 for C ∈ (0 , 1). We thus determine that indeed the operator G ( u 0 ) is a contraction mapping on B ˜ δ,µ,q -1 ,s (for sufficiently small ˜ δ ). It follows from standard arguments that the sequence ( w N ) has a unique limit w , contained in B ˜ δ,µ,q -1 ,s , which is a fixed point for G ( u 0 ) and hence is a weak solution. The sequence ( w N ) ⊂ B ˜ δ,µ,q,s is bounded in X ˜ δ,µ,q , but to this stage is known only to converge in X ˜ δ,µ,q -1 to w . This situation is similar to that encountered in the proof of Proposition 2.13. A similar argument involving Corollary C.3 and the standard Cauchy problem of hyperbolic equations implies that w is indeed an element of X ˜ δ,µ,q . To show that w is the remainder of a strong solution of the singular initial value problem, it remains for us to check that w is differentiable in time. The definition of the sequence ( w N ) shows that for each integer N , Dw N exists and is contained in X ˜ δ,µ,q -1 . Furthermore, this sequence converges in X ˜ δ,µ,q -1 by a similar argument as in the proof of Proposition 2.12 using the Condition (iii), and the positivity of ζ . Since this convergence is uniform in time, it follows that w is differentiable at each t and that Dw ( t ) is the limit of ( Dw N ( t )) at each t . It follows from this limiting procedure that u is indeed a strong solution to the singular initial value problem; similar arguments have been used before also in the proof of Proposition 2.13. The uniqueness of this solution w follows from the uniqueness of the fixed point for the contraction mapping G ( u 0 ). In order to complete the proof of Theorem 2.4, we must consider the case q = ∞ . We do this inductively in q . It is important to notice here that all of the constants in the previous estimates may depend on q and hence we may have to adapt the choice of ˜ δ in each induction step. It is thus possible that the sequence of these constants ( ˜ δ q ) tends to zero as q →∞ . To show that this possibility is avoided we use the result that any solution to the Cauchy problem for symmetric hyperbolic systems with a bounded first spatial derivative can be extended to a common time interval. Let us fix any q ≥ 3. Theorem 2.4 (with finite q ) shows that there exists a solution w ∈ X ˜ δ,µ,q for some ˜ δ ∈ (0 , δ ]. Let t ∗ ∈ (0 , ˜ δ ], and consider the regular Cauchy problem with data w ( t ∗ , x ). Since q ≥ 2, the Sobolev inequalities guarantee that the first spatial derivative is bounded on [ t ∗ , ˜ δ ], and we may apply Proposition 1.5 in Chapter 16 of [40] to show that there exists a ˜ δ 2 > ˜ δ such that the solution may be extended as a H q -solution to (0 , ˜ δ 2 ]. The same argument applied to any other value of q ≥ 3 implies that the solution can be extended as H q -solutions to the same time interval (0 , ˜ δ 2 ]. For q = ∞ , we therefore find a unique solution w on the same time interval in X ˜ δ 2 ,µ, ∞ (and hence Dw ∈ X ˜ δ 2 ,µ, ∞ ).", "pages": [ 32, 33, 34, 35 ] }, { "title": "2.4 Existence and uniqueness results based on ODE-leading-order terms", "content": "Definition 2.2 of a quasilinear symmetric hyperbolic Fuchsian system, as well as the conditions which the singular initial value problem for such a system must satisfy if we wish to apply Theorem 2.4 and thereby guarantee the existence of a solution, involve the specified leading-order term u 0 just as crucially as they involve the exponent vector µ and the functions S 1 , S 2 and N appearing in Eq. (2.1). In some applications, it is not easy to determine which choices of u 0 (if any) lead to these conditions being satisfied. Here we discuss an approach which starts with the choice of a leading-order term of a very restricted type (which we label 'ODE'-leading-order terms), and provides an alternate set of criteria for the existence of solutions to the singular initial value problem (with an ODE-leading-order term). This approach, presuming the criteria are satisfied, also systematically produces a sequence (possibly finite) of improved leading-order terms, which effectively serve as progressively higher order approximations to the solution of the singular initial value problem. We detail this approach here. As we see in Section 3.3.3 below, the ODE-leading-order term approach is very useful in our analysis of the T 2 -symmetric spacetimes. In particular, this approach plays a crucial role in our use of Fuchsian methods to obtain an optimal collection of T 2 -symmetric solutions of Einstein's equations with AVTD behavior.", "pages": [ 35, 36 ] }, { "title": "2.4.1 The ODE-Fuchsian operator and ODE-leading-order terms", "content": "We start by defining the differential operator L ODE ( u 0 )[ · ], which plays a central role in carrying out this approach. Presuming that we are working with a specified quasilinear symmetric hyperbolic Fuchsian system Eq. (2.1) with specified (as yet arbitrary) leadingorder term u 0 and with specified parameters δ , µ , q 0 and q , we define the ODE-Fuchsian operator as follows: Here we note that since (by Definition 2.2) S 1 , 0 is invertible, it follows that L ODE ( u 0 )[ · ] is well-defined. We also note that, since L ODE ( u 0 )[ · ] does not involve any spatial derivatives, it is essentially a parametrized set of ordinary differential operators (one for each point x ∈ T 1 ) rather than a partial differential operator (hence the 'ODE' label). Although not necessary yet at this stage, we assume q ≥ 3 and q 0 = q +2 (consistent with Theorem 2.4) in all of what follows. In particular this guarantees that all maps (including their first spatial derivatives) are continuous with respect to x (as follows from the Sobolev embedding theorem). For example, the ODE operator above is well-defined at every spatial point x under this condition. We wish to write the Fuchsian PDE system Eq. (2.1) in terms of the operator L ODE ( u 0 )[ · ]. Recalling the operational form ̂ L ( u 0 + w )[ u 0 + w ] = F ( u 0 )[ w ] for Eq. (2.1), and noting that we can relate the operators L ( u 0 + w )[ · ] and L ODE ( u 0 )[ · ] as follows we find that we can write out Eq. (2.1) in the form if we define the term on the right hand side of Eq. (2.44) as The expression F ODE ( u 0 )[ · ] is well-defined so long as δ is sufficiently small so that S 1 ( u 0 + w ) is invertible for any given w in B δ,µ,q,s for some s > 0; we presume this is the case in all of what follows. As noted above, a key aspect of this approach is the selection of a special class of leading-order terms. Definition 2.15. A leading-order term u 0 is an ODE-leading-order term if it satis- fies the condition Expanding out the expression for L ODE ( u 0 )[ · ], we see that an ODE-leading-order term u 0 must satisfy Du 0 ( t, x ) + ˜ N ( x ) u 0 ( t, x ) = 0, where ˜ N := S -1 1 , 0 ( u 0 ) N 0 ( u 0 ) is (by definition) independent of t . For those very special cases in which ˜ N is independent of u 0 , Eq. (2.46) is a parametrized set of linear ODEs, which can be readily solved for u 0 . More generally, Eq. (2.46) is nonlinear and therefore not so easy to analyze. We are interested here only in those cases in which we can establish that solutions to Eq. (2.46) exist. For those cases, we proceed to seek solutions of the singular initial value problem for the system Eq. (2.1) with ODE-leading-order term u 0 ; we call this the ODE-singular initial value problem. Observe that the solution to an ODE-singular initial value problem, if obtained, behaves in a way which suggests that as t → 0, the spatial derivative terms in Eq. (2.1) become negligible. Indeed, this is true for the AVTD solutions of the Einstein field equations which we treat in Section 3 below. It is useful for our work below to notice that under the assumption Eq. (2.46), a combination of Eqs. (2.45) and (2.43) yields where F ( u 0 )[ w ] is defined in Eq. (2.7).", "pages": [ 36, 37 ] }, { "title": "2.4.2 (Order n)-leading-order terms", "content": "We now use the ODE-leading-order term u 0 (presuming that it exists) and the ODEFuchsian operator L ODE ( u 0 )[ · ] to generate a (possibly finite) sequence ( u n ) of 'qualitative' solutions (in a sense described shortly) to the corresponding ODE-singular initial value problem; these ( u n ) play an important role in establishing a set of conditions which are sufficient to show that this ODE-singular initial value problem does admit a solution (see Theorem 2.21 below). We first consider the x -parametrized set of linear inhomogeneous ODEs where u 0 is a fixed ODE-leading-order term and f 0 is a specified inhomogeneity (whose regularity we discuss below). If we use W ( t, x ) to denote a fundamental matrix for the linear homogeneous equation L ODE ( u 0 )[ v ]( t, x ) = 0, and if we let ( u ∗ , 1 ( x ) , . . . , u ∗ ,d ( x )) represent free data for the initial value problem at t 0 ∈ (0 , δ ), then the general solution to Eq. (2.48) may be formally written as follows: We may then formally define the operator which, if it exists, maps a given source function f 0 to the particular solution w = H ODE ( u 0 )[ f 0 ] of Eq. (2.48) determined by ( u ∗ , 1 ( x ) , . . . , u ∗ ,d ( x )) = 0. We notice that the definition of this operator is invariant if the fundamental matrix W is replaced by an equivalent fundamental matrix W ↦→ W · M for any invertible d × d -matrix M ∈ H q 0 . To proceed, we need to identify conditions which are sufficient for the existence of H ODE ( u 0 )[ · ]. Noting that we may always choose the free data ( u ∗ , 1 ( x ) , . . . , u ∗ ,d ( x )) in such a way that the first term in Eq. (2.49) equals (the already specified) u 0 , we wish to also show that these same conditions are sufficient to guarantee that the second term in Eq. (2.49), i.e., H ODE ( u 0 )[ f 0 ], is higher order in time as t approaches 0 and therefore serves as a remainder term (in the sense of Definition 2.1) for the singular initial value problem. We state the needed conditions in the following lemma, which is readily checked. Lemma 2.16 (Existence and properties of H ODE ( u 0 )[ · ]) . Suppose that a quasilinear symmetric hyperbolic Fuchsian system Eq. (2.1) has been chosen (Definition 2.2) for, in particular, a fixed sufficiently small parameter δ , for differentiability indices q ≥ 3 and q 0 = q +2 , and for an ODE-leading-order term u 0 (Eq. (2.46) ). Suppose that S -1 1 , 0 N 0 is of Jordan normal form. Then H ODE ( u 0 )[ · ] is well-defined on the domain X δ, ˜ ν,q for every smooth exponent vector ˜ ν , so long as each component of ˜ ν is strictly larger than the real part of the negative of the corresponding diagonal element (eigenvalue) of S -1 1 , 0 N 0 . The target space of H ODE ( u 0 )[ · ] is X δ, ˜ µ,q for any smooth exponent vector ˜ µ < ˜ ν , and one has the estimate where C > 0 depends only on the eigenvalues of S -1 1 , 0 N 0 , on the dimension d of the firstorder system, on the choices of q , ˜ ν , and on the difference between ˜ ν and ˜ µ ; the constant ρ > 0 only depends on the difference between ν and µ , and on q . ˜ ˜ In particular, we may choose ˜ µ arbitrarily close to ˜ ν , and we then check that the remainder w = H ODE ( u 0 )[ f 0 ], (the second term in Eq. (2.49)) is of higher order in t near t = 0 (as measured by ˜ µ ) as the order of f 0 (measured by ˜ ν ) becomes large. As well, as the difference between ˜ µ and ˜ ν diminishes, one may have to choose the constant C to be larger, and the constant ρ to be smaller. Some comments about this lemma are in order. For each quasilinear symmetric hyperbolic Fuchsian system and each choice of leading-order term u 0 , there exists an invertible matrix T ∈ H q 0 such that TS -1 1 , 0 N 0 T -1 is in Jordan normal form. We assume in the following that such a transformation has been applied to the system, and hence that S -1 1 , 0 N 0 is in Jordan normal form. The fact that the matrices S 1 and S 2 in the principal part are in general not symmetric after such a transformation has been carried out is not important for the arguments that follow. Moreover, for simplicity we assume for each exponent vector here that those of its components which correspond rto the same Jordan block of S -1 1 , 0 N 0 have the same value. As a consequence of this result, we may formally define the following sequence: Definition 2.17 ((Order n)-leading-order sequence) . Suppose q ≥ 3 . With w 0 = 0 , we formally set for all positive integers 1 ≤ n ≤ q -2 . The (order n)-leading-order terms are then defined by for 0 ≤ n ≤ q -2 . To turn this formal specification of the (order-n)-leading-order-sequence into a definition, we need to state sufficient conditions for the composition on the right hand side of Eq. (2.51) to be well-defined for each n . We do this in the proposition below. This proposition also proves that the sequence is characterized by certain properties which are relevant to the two roles which it plays: i) an increasingly accurate sequence of approximations to the solution of the singular initial value problem with ODE-leadingorder-term u 0 (presuming that such a solution exists); and ii) a sequence of 'new', and 'better' leading-order terms which can be used to define new singular initial value problems (closely tied to the original) for which we can prove the existence of solutions. The following proposition states the manner in which the first use makes sense, and provides the first step towards proving that the second use works. Proposition 2.18 (Existence and properties of the (order n)-leading-order terms u n ) . Let q ≥ 3 and q 0 = q + 2 . Suppose that a quasilinear symmetric hyperbolic Fuchsian system Eq. (2.1) has been chosen satisfying Definition 2.2 for an ODE-leading-order term u 0 satisfying Eq. (2.46) , for fixed parameters δ (sufficiently small) and µ , and for all differentiability indices q ' in the interval [3 , q ] . Here we require that the exponent matrix ζ , whose existence (in specifying the function spaces containing S 1 , 1 , S 2 , 1 , and N 1 ) is a necessary part of the definition of a quasilinear symmetric hyperbolic Fuchsian system (see Definition 2.2), can be written as ζ ij = ξ i for some vector-valued exponent ξ with strictly positive entries. Suppose that the matrix S -1 1 , 0 ( u 0 ) N 0 ( u 0 ) is given in Jordan normal form, and suppose in addition that the following conditions are satisfied for all δ ' ∈ (0 , δ ] and all integers q ' ∈ (3 , q ] : and, there exists r > 0 , so that for ζ defined in terms of ξ , as above. (iii) For all w ∈ B δ ' ,µ,q ' ,s/ 2 and ω ∈ B δ ' , ̂ µ,q ' ,s/ 2 for any exponent vector ̂ µ which satisfies ̂ µ ≥ µ and with respect to which the system is block diagonal, there exists a constant r > 0 for which Moreover, there exists a constant C > 0 such that the norm of each of these quantities can be bounded as follows, with analogous inequalities holding for S 1 , S 2 , and N . Then, the sequence u n specified in Definition 2.17 is well-defined and, for some ˜ δ ∈ (0 , δ ] and constants γ > 0 , one has for all 1 ≤ n ≤ q -2 . Moreover, the residual 1 of u n , defined by is contained in X δ,µ + nκγ,q -n . We make a few remarks here concerning some of the details of this proposition. First, we observe that as a consequence of the definition of this sequence, presuming that we start with an ODE-leading-order term u 0 of a certain order of differentiability, we find that the first term of the sequence u 1 retains that regularity (up to order q ), while the rest of the elements of the sequence ( u 2 , u 3 ... ) generally do not. This is true because, since w = 0 is smooth, it follows from the formula Eq. (2.47) that F ODE ( u 0 )[ w 0 ] has q derivatives; the same is then true for w 1 . But then since F ODE ( u 0 ) maps X δ,µ,q to X δ, ̂ ν,q -1 , we see that w 2 has only q -1 derivatives. The same loss of a derivative occurs for each successive element of the sequence. Secondly, we remark that the modified block diagonal conditions for µ are a slight generalization of Definition 2.3. In particular, it is not necessary that R [ µ ] commute with S 2 here. We have chosen a formulation of Condition (i) which applies directly our applications - see Sections 2.4.4 and 3. Condition (i) can, however, be generalized to match more general situations. Thirdly, we note that Condition (iii) can be checked in our applications using the tools in the appendix; see Section B. Finally, we note that for the special choice of ζ ij = ξ i used here, the space of matrixvalued functions X δ,ζ,q can be equivalently written as X δ,ξ,q . This latter space-a Banach space of matrix-valued functions X δ,ξ,q with a vector-valued exponent ξ -is defined in essentially the same way as above in Eq. (2.3); the key difference is that the norm used to define this space is the H q -norm of R [ ξ ] S - the matrix product of the matrix R [ ξ ] formed from ξ (see Eq. (2.2)) times S . The motivation for specializing the matrix-valued exponent ζ in this way and thence introducing the new notation X δ,ξ,q for matrix-valued functions is to be able to express Condition (iii) of Proposition 2.18 in a natural way. The proof of Proposition 2.18 depends upon tight control of F ODE ( u 0 )[ · ] as given by Eq. (2.47), and tight control of the inverse of S 0 . We obtain this needed control using the following two lemmas. Lemma 2.19. If the hypothesis for Proposition 2.18 holds (presuming as usual that δ is sufficiently small so that S 1 ( u 0 + w ) is invertible for all w ∈ B δ,µ,q,s and presuming that δ ' and q ' satisfy the conditions stated in that hypothesis), it follows that for some r > 0 , S -1 1 ( u 0 + w ) ∈ B δ ' , 0 ,q ' ,r for all w ∈ B δ ' ,µ,q ' ,s . Moreover, the operator given by maps B δ ' ,µ,q ' ,s into B δ ' ,ζ,q ' ,r for some constant r > 0 , and this operator satisfies the difference condition for some exponent vector ζ > 0 , and for some constant C > 0 it satisfies the inequality for all w ∈ B δ ' ,µ,q ' ,s/ 2 and all ω ∈ B δ ' , ̂ µ,q ' ,s/ 2 ; here ̂ µ is any exponent vector which satisfies the inequality µ ≥ µ , and for which the system is block diagonal. ̂ The proof of this lemma relies on i) the fact that the inversion of invertible matrices is a smooth map, ii) the fact that both R [ µ ] and R [ˆ µ ] commute with S 1 , and iii) Proposition 3.9 in Chapter 13 of [40]; we omit the details here. Lemma 2.20. If the hypothesis for Proposition 2.18 holds, then there exist positive constants γ and r so that for all constants δ ' ∈ (0 , δ ] and all integers q ' ∈ [3 , q ] , and for all exponent vectors ̂ µ ≥ µ with respect to which the system is block diagonal, we have that F ODE ( u 0 )[ w ] ∈ B δ ' ,µ + γ,q ' -1 ,r and Further, there exists a constant C > 0 such that for all w ∈ B δ ' ,µ,q ' ,s/ 2 and all ω ∈ B δ ' , ̂ µ,q ' ,s/ 2 , we have Proof of Lemma 2.20. The first statement is easily obtained by multiplying the expression Eq. (2.47) for F ODE ( u 0 )[ · ] by the quantity S 1 ( u 0 + w ) and then using the facts that R [ µ ] commutes with S 1 ( u 0 + w ), and that S 1 ( u 0 + w ) is in B δ, 0 ,q,r for some r > 0, and also applying Lemma B.2 and Lemma 2.19. To prove the rest, we multiply this same expression for F ODE ( u 0 )[ · ] by S 1 ( u 0 + w ), and then calculate Applying arguments of the sort used to verify Proposition 2.18 and Lemma 2.19 together with the estimates included in the hypothesis, we obtain the conclusion. We now proceed to prove Proposition 2.18: Proof of Proposition 2.18. We first show that the sequence ( w n ) (and the corresponding sequence ( u n )) is well-defined at least for finitely many sequence elements. It follows from Condition (ii) of the hypothesis that F ( u 0 )[0] ∈ B δ,ν,q,r for some r > 0. Noting (see Eq. (2.47)) that F ODE ( u 0 ) evaluated at 0 reduces to S -1 1 , 0 F ( u 0 )[0], we infer from Lemma 2.16 and Lemma B.2 that the term w 1 is hence well-defined and is contained in X δ,µ,q . It then follows from Lemma 2.20 and Lemma 2.16 (whose hypotheses are satisfied) that the operator H ODE ( u 0 )[ · ] is well-defined, and consequently that w n is well-defined for all 2 ≤ n ≤ q -2. These functions are all elements of X δ,µ,q -( n -1) . Using the estimate for the operator H ODE ( u 0 )[ · ] stated in Lemma 2.16, we verify that if we shrink the time interval (0 , δ ] to (0 , ˜ δ ] as stated in the hypothesis of the Proposition under consideration, then we can show that finitely many of the sequence elements stay in a ball of fixed radius. We have thus verified the first statement appearing in Eq. (2.54). Note that for convenience, in the remainder of this proof we continue to write δ instead of ˜ δ ; however, we reserve the right to repeatedly shrink the time interval as necessary (a finite number of times). We next argue by induction that the second statement in Eq. (2.54) holds. We presume that the differentiability index q is sufficiently large so that there exist nontrivial n ≤ q -2. To initialize the induction, we note that for n = 1, this statement says that u 1 -u 0 ∈ B δ,µ,q,s/ 2 . Noting that, by definition, u 1 -u 0 = w 1 = H ODE ( u 0 )[ F ODE ( u 0 )[0]], and recalling from above that this term is contained in B δ,µ,q,s/ 2 , we verify the initialization. To continue the induction argument, we suppose now that for some positive integer m ‖ w m +1 -w m ‖ δ,µ ( m +1) ,q -m = ‖ H ODE ( u 0 )[ F ODE ( u 0 )[ w m ] -F ODE ( u 0 )[ w m -1 ]] ‖ δ,µ ( m +1) ,q -m ≤ Cδ ρ ‖ F ODE ( u 0 )[ w m ] -F ODE ( u 0 )[ w m -1 ] ‖ δ,ν ( m +1) ,q -m ≤ Cδ ρ C ‖ w m -w m -1 ‖ δ,ν ( m +1) -γ,q -( m -1) . In carrying out this calculation (with γ being the quantity hypothesized in Lemma 2.20), we note that the operator H ODE ( u 0 )[ · ] is well-defined here according to Lemma 2.16 and Lemma 2.20 since w m -w m -1 ∈ B δ,µ ( m ) ,q -( m -1) ,s/ 2 with µ ( m ) ≥ µ . Finally we note that the constants C and ρ may depend in particular on q and m , but this dependence is not a problem for carrying out our argument since we are only interested in finitely many sequence elements. To complete the induction argument, we verify that since we have assumed (as part of the induction) that w m -w m -1 ∈ B δ,µ ( m ) ,q -( m -1) ,s/ 2 , it follows that so long as ν ( m +1) -γ < µ ( m ) holds, we have the final right hand side of the above inequality finite. Therefore the initial left hand side must be finite, and this holds for any µ ( m +1) , so long as µ ( m +1) < µ ( m ) + γ . We satisfy these conditions by choosing µ ( m +1) = µ + mκγ for any κ < 1. Noting that this is the case for all m , with κ chosen independently of m , we conclude that Eq. (2.54) holds, after having identified κγ with γ to simplify the notation. It remains to verify that Eq. (2.55) holds for the residuals of the sequence ( u n ). Using Eqs. (2.44), (2.46) and (2.51) we calculate Res[ u n ] = ̂ L ( u 0 + w n )[ u 0 + w n ] -F ( u 0 )[ w n ] = -S 1 ( u 0 + w n ) ( F ODE ( u 0 )[ w n ] -F ODE ( u 0 )[ w n -1 ]) . Since w n -w n -1 ∈ B δ,µ +( n -1) γ,q -n +1 ,s/ 2 , it follows from Lemma 2.20 that Res[ u n ] ∈ X δ,µ + nγ,q -n . 2.4.3 (Order n)-singular initial value problem Proposition 2.18 shows that, so long as we can find an ODE-leading-order term u 0 and so long as certain conditions hold, the difference u n +1 -u n behaves like a power of t near t = 0, with this power increasing monotonically with n . It is hence meaningful to consider, in addition to the ODE-singular initial value problem with leading-order term u 0 , a sequence of (order n)-singular initial value problems which use (order n)-leadingorder terms u n ( n ≤ q -2). In view of the relationship between u 0 and the sequence ( u n ), we may write the same solution u of a given singular initial value problem either in the form u = u 0 + w for a remainder w in X δ,µ,q , or, as u = u n + ω for a remainder ω in X δ, ̂ µ,q with ̂ µ increasing suitably with n . The same can be done for any of the u m ( m ≤ n ) in the (order n)-leading-order term sequence. We now use the (order n)-leading-order terms to argue that, at least for the smooth case ( q = ∞ ), if the conditions of Proposition 2.18 are met, the ODE-singular initial value problem, and correspondingly the (order n)-singular initial value problem have solutions. Theorem 2.21 (Existence and uniqueness for the ODE-singular initial value problem) . Suppose that a quasilinear symmetric hyperbolic Fuchsian system with ODE-leadingorder term u 0 has been chosen which satisfies the hypotheses of Proposition 2.18 for all finite values of (differentiability index) q . Then, for some sufficiently small δ 1 ∈ (0 , δ ] and for a sufficiently large n , there exists a unique solution u of Eq. (2.1) with u -u n , D ( u -u n ) ∈ X δ 1 ,µ + nγ, ∞ , where u n is the (order n)-leading-order term defined in Definition 2.17 for this system. This solution u is also the only solution of the ODEsingular initial value problem with u -u 0 ∈ X δ 1 ,µ, ∞ . This result states conditions which are sufficient for the ODE-singular initial value problem (with leading-order term u 0 ) to admit (unique) solutions. In doing so, Theorem 2.21 provides a potentially very useful alternative to Theorem 2.4 of Section 2.2. Observe in particular that the hypothesis for Theorem 2.21 does not require that the energy dissipation matrix be positive definite with respect to µ . Here we state and prove Theorem 2.21 only for the infinite differentiability case ( q = ∞ ). This smoothness restriction plays a role in the proof, since it allows one to always choose n large enough so that Condition (i) of Theorem 2.4 for the singular initial value problem with leading-order term u n is satisfied. If one tries to prove a result like Theorem 2.21 for finite differentiability order, then there is an upper bound for the possible choice of n , and consequently one may not be able to choose it large enough to satisfy Condition (i). However, in certain circumstances, a large but finite order of differentiability is in fact sufficient to carry through the proof. Proof. The basic idea of the proof is to reformulate the system using u n for the leadingorder term in place of u 0 , and then verify that the hypothesis of Theorem 2.4 (in the case q = ∞ ) is satisfied if n is chosen sufficiently large. To carry this through, we first argue that the system Eq. (2.1), which for the ODE-singular initial value problem can be written as can also be written as 0 = ̂ L ( u n + ω )[ ω ] -F ( u n )[ ω ] , (2.58) where we recall the definition Eq. (2.5) of the principal part operator ̂ L and the definition Eq. (2.7) for the operator F ( u n )[ · ]. Here we use w for the remainder term corresponding to u 0 and we use ω for the remainder term corresponding to u n (hence u 0 + w = u n + ω ). To show this equivalence, we note the relations ̂ L ( u n + ω )[ v ] = ̂ L ( u 0 + w )[ v ] and F ( u n )[ ω ] = F ( u 0 )[ w ], and then using these we calculate 0 = ̂ L ( u 0 + w )[ u 0 + w ] -F ( u 0 )[ w ] , (2.57) 0 = ̂ L ( u 0 + w )[ u 0 + w ] -F ( u 0 )[ w ] = ̂ L ( u n + ω )[ u n + ω ] -F ( u n )[ ω ] = ̂ L ( u n + ω )[ u n ] + ̂ L ( u n + ω )[ ω ] -F ( u n )[ ω ] = ̂ L ( u n + ω )[ ω ] -F ( u n )[ ω ] . The equivalence of Eq. (2.57) and Eq. (2.58) immediately follows. ˜ µ < µ ( n ) < µ +( n -1) γ, (2.59) We now choose a sequence of exponent vectors µ ( n ) which satisfy ˜ and which are consistent with the block diagonal condition for Eq. (2.58); we note that this is possible for all sufficiently large integers n . Examining the singular initial value problem corresponding to Eq. (2.58), we verify that for any given sufficiently large integer n , this PDE system, together with u n as leading order term and exponent vector ˜ µ ( n ) , satisfies the conditions to be a quasilinear symmetric hyperbolic Fuchsian system. We also verify, based on Eq. (2.6), that for sufficiently large n (and therefore sufficiently large µ +( n -1) γ ) the exponent vectors ˜ µ ( n ) can be chosen large enough to guarantee that the energy dissipation matrix M 0 is positive definite. Consequently, this system satisfies Condition (i) of the hypothesis of Theorem 2.4. To check that Conditions (ii) and (iii) of Theorem 2.4 are also satisfied, we examine the operator F ( u n ). Using Eqs. (2.43) and (2.47) together with Definition 2.17, we calculate F ( u n )[ ω ] = S 1 ( u 0 + w ) ( F ODE ( u 0 )[ w ] -F ODE ( u 0 )[ w n -1 ]) + S 2 ( u 0 + w ) t∂ x ω + S 1 ( u 0 + w ) ( S -1 1 ( u 0 + w )( N ( u 0 + w ) -N 0 ( u 0 )) + ( S -1 1 ( u 0 + w ) -S -1 1 , 0 ( u 0 ) ) N 0 ( u 0 ) ) ω. Combining the assumptions for S 1 , S 2 and N which are stated in the hypothesis of Proposition 2.18 (and therefore included in the hypothesis of Theorem 2.21) with By comparing the first line of this expression with Eq. (2.56), we notice that all spatial derivative terms cancel; hence there is no loss of regularity in this expression as is the case for the operator F ODE ( u 0 ) itself. We therefore get estimates analogous to those in Lemma 2.20, with q -1 replaced by q . the upper bound stated in Eq. (2.59), we readily show that all of the conditions of Theorem 2.4 hold for q = ∞ . Then the consequent application of Theorem 2.4 shows that so long as n is sufficiently large, there exists exactly one solution u = u n + ω with ω ∈ X δ 1 , ˜ µ ( n ) , . ˜ ω = w n + -w n + ω = ( w n +1 -w n ) + . . . +( w n + -w n + -1 ) + ˜ ω. ∞ We wish to show next that for such a fixed chosen value of n , in fact ω ∈ X δ 1 ,µ + nγ, ∞ . To show this, we consider an integer n + which is large enough so that ˜ µ ( n + ) > µ +( n -1) γ . Applying the same argument as above, but now with n + instead of n (and hence using u n + as the leading-order term), we obtain a solution ˜ u = u n + + ˜ ω which has the property that ˜ ω ∈ X δ 1 , ˜ µ ( n + ) , ∞ . Uniqueness of the singular initial value problem with respect to u n implies that u equals u . Moreover, we have ˜ Given that w n +1 -w n ∈ X δ 1 ,µ + nγ, ∞ , we obtain the desired result ω ∈ X δ 1 ,µ + nγ, ∞ . To conclude the proof of this theorem, we must show that any solution ̂ u of the form ̂ u = u 0 + ̂ w with ̂ w ∈ X δ 1 ,µ, ∞ must equal the solution u . To show this, it is useful to write ̂ u = u n + ̂ w -w n , where u n is defined by Eq. (2.52) and w n is defined by Eq. (2.51). Then if we can verify that ̂ w -w n ∈ X δ 1 , ˜ µ ( n ) , ∞ , it follows from uniqueness that ω = ̂ w -w n , and hence that ̂ u = u . We make this verification by using induction to show that, in fact, ̂ w -w m ∈ X δ 1 ,µ + mγ, ∞ holds for every non-negative integer m . In the case m = 0, we have ̂ w -w 0 = ̂ w ∈ X δ 1 ,µ, ∞ which implies the claim for m = 0. Suppose the claim has been shown for m = m 0 ≥ 1. We know that w is a solution of the equation while w m 0 +1 is a solution of ̂ L ODE ( u 0 )[ ̂ w ] = F ODE ( u 0 )[ ̂ w ] , L ODE ( u 0 )[ w m 0 +1 ] = F ODE ( u 0 )[ w m 0 ] . Taking the difference, we obtain We can write this formally as L ODE ( u 0 )[ ̂ w -w m 0 +1 ] = F ODE ( u 0 )[ ̂ w ] -F ODE ( u 0 )[ w m 0 ] . ̂ w -w m 0 +1 = H ODE ( u 0 )[ F ODE ( u 0 )[ ̂ w ] -F ODE ( u 0 )[ w m 0 ]] . Now, the fact that w -w m 0 ∈ X δ 1 ,µ + m 0 γ, ∞ implies that F ODE ( u 0 )[ w ] -F ODE ( u 0 )[ w m 0 ] ∈ X δ 1 ,µ +( m 0 +1) γ, ∞ (Lemma 2.20). Consequently (see Lemma 2.16), the operator H ODE ( u 0 )[ · ] is well-defined. This completes the proof. 2.4.4 An example: the Euler-Poisson-Darboux equation We consider now the example of the Euler-Poisson-Darboux equation (see also [2] for another example) D 2 u ( t, x ) -t 2 u xx ( t, x ) = f 0 ( t, x ) . (2.60) Here, u ( t, x ) is the unknown (assumed to be a scalar function), and f 0 ( t, x ) is a specified scalar function. The Euler-Poisson-Darboux equation is second order, and in previous work by two of the authors [13] on semilinear second-order Fuchsian systems, it has been shown that this equation admits unique solutions to the singular initial value problem with leading-order term u 0 ( t, x ) = u ∗ ( x ) log t + u ∗∗ ( x ) , (2.61) (for arbitrary functions u ∗ and u ∗∗ ) so long as f 0 = O ( t ̂ ν ) with ̂ ν > 0. We seek to show that we obtain these same results using the first-order methods which we have developed here. In particular, this example demonstrates the usefulness of the techniques discussed in Section 2.4, thereby serving as a linear warmup example with which we can explore some of the issues which arise below in our discussion of the application of these methods to the fully nonlinear T 2 -symmetric Einstein's vacuum equations in Section 3. To apply the first-order theory developed in this paper, we first convert this equation into a first-order system by setting u 1 := u, u 2 := Du, u 3 := t∂ x u, U := ( u 1 , u 2 , u 3 ) T . (2.62) Eq. (2.60) then takes the form of a first-order evolution system S 1 DU + S 2 t∂ x U + NU = f, (2.63) with S 1 = Diag(1 , 1 , 1) , S 2 =   0 0 0 0 0 -1 0 -1 0   , N =   0 -1 0 0 0 0 0 0 -1   , f =   0 f 0 0   , plus a constraint equation ∆ u := u 3 /t -∂ x u 1 = 0 . (2.64) Observe that in working with the Euler-Poisson-Darboux system in this first-order form, one first treats the components u 1 , u 2 , and u 3 as independent functions whose evolution is determined by Eq. (2.63). This means that we solve the singular initial value problem of this system with respect to a leading-order term motivated by Eq. (2.61). Then, in a second step, we identify u 1 with the original variable u and consider the two remaining relations Eq. (2.62) as constraints: the one involving the time derivative is automatically implied by the first of Eqs. (2.63) (the evolution equation for u 1 ), while the one involving the spatial derivative gives rise to the condition ∆ u ≡ 0 in Eq. (2.64). Let us start with the first step. One readily verifies that this evolution system is of (quasilinear) symmetric hyperbolic Fuchsian form for any choice of leading-order term, and hence our theory can, in principle, be applied. Our approach is to find a leading-order term for the first-order variables which is consistent with Eq. (2.61) and which, in addition, is an ODE-leadingorder term. We easily determine that the general solution to Eq. (2.46) for Eq. (2.63) takes the form U 0 = ( C 1 + C 2 log t, C 2 , C 3 t ) T , (2.65) for the spatially-dependent parameters C 1 ( x ), C 2 ( x ) and C 3 ( x ). However, we see that this leading-order term can only be consistent with Eqs. (2.61) and (2.62) in the special case u ∗ = 0 and C 2 = 0. Hence, this approach for finding a leading-order term fails. We circumvent this problem as follows. For a specified function u ∗ (which is at least second order differentiable; we specify its necessary regularity more precisely below), we define ̂ u := u -u ∗ ( x ) log t, (2.66) and work with the evolution equation for ̂ u rather than that for u . Substituting Eq. (2.66) into Eq. (2.60), we obtain D 2 ̂ u -t 2 ̂ u xx = t 2 log tu '' ∗ + f 0 ( t, x ) , where u '' ∗ indicates the second derivative of u ∗ . Now, setting we obtain the evolution equation ̂ u 1 := ̂ u, ̂ u 2 := D ̂ u, ̂ u 3 := t∂ x ̂ u, ̂ U := ( ̂ u 1 , ̂ u 2 , ̂ u 3 ) T , for the same matrices S 1 , S 2 , N as above, but with S 1 D ̂ U + S 2 t∂ x ̂ U + N ̂ U = ̂ f, (2.67) ̂ f = ( 0 , f 0 + t 2 log t u '' ∗ , 0 ) T . ∆ ̂ u := ̂ u 3 /t -∂ x ̂ u 1 = 0 . (2.68) Choosing the ODE-leading-order term for the ̂ U formulation to be of the same form as Eq. (2.65), we have ̂ U 0 = ( C 1 + C 2 log t, C 2 , C 3 t ) T , but now (in view of Eq. (2.61)) we are led to choose the parameter functions in the form C 1 = u ∗∗ , C 2 = 0 and C 3 = u ' ∗∗ ; hence In terms of ̂ u , the constraint Eq. (2.64) takes the form ̂ U 0 ( t, x ) = ( u ∗∗ ( x ) , 0 , tu ' ∗∗ ( x ) ) T . (2.69) The function u ∗ appearing in Eq. (2.66) together with the function u ∗∗ introduced here together comprise the full range of free data suggested by Eq. (2.61). Both play the role of asymptotic data functions. Having found a suitable representation of the equations and the leading-order term, we write the unknown ̂ U of the evolution system as ̂ U = ̂ U 0 + W , and look for sufficient conditions for the existence of solutions to the singular initial value problem in this form, with W as a remainder term. To enforce the remainder falloff properties, we choose an exponent vector µ = ( µ 1 , µ 2 , µ 3 ) and, in view of Eq. (2.69), we require that µ 1 , µ 2 > 0 and µ 3 > 1. We first seek to prove existence of solutions using Theorem 2.4. To satisfy the block diagonality condition of Theorem 2.4 we must set µ 1 = µ 2 = µ 3 . We therefore simplify the notation by writing the exponent vector as ( µ, µ, µ ) for some smooth scalar function µ which, from above considerations, must be greater than one. Observe here that, while this equality of all components of the exponent vector is necessary to satisfy the hypothesis of Theorem 2.4, it does appear to be an artificial restriction. Under reasonable regularity assumptions, we might rather expect that if the first and second components are O ( t µ ), then the third component of W should be O ( t µ +1 log t ); the log t factor may arise from derivatives of t µ since µ is generally not constant. In any case, we readily verify that the energy dissipation matrix M 0 =   µ -1 0 0 µ 0 0 0 µ -1   , is positive definite so long as µ > 1. Calculating F ( ̂ U 0 )[ W ] = F ( ̂ U 0 )[ W ] -̂ L ( ̂ U 0 + W )[ U 0 ] = ( 0 , f 0 + t 2 (log t u '' ∗ + u '' ∗∗ ) , 0 ) T , Given any such solution of the first-order evolution system, our next step is to identify ̂ u 1 with u -log( t ) u ∗ and then, if the remaining constraint ∆ ̂ u ≡ 0 is satisfied, to conclude that u is actually a solution of the original second-order equation Eq. (2.60) with leadingorder term u 0 = u ∗ log t + u ∗∗ and with remainder w = w 1 (the first component of the vector W ) in X δ 1 ,µ,q . To determine if the constraint is satisfied, we use the evolution equation Eq. (2.67) to calculate the time derivative of the constraint violation term ∆ ̂ u , obtaining we now suppose that W ∈ X δ, ( µ,µ,µ ) ,q and f 0 ∈ X δ, ̂ ν,q for ̂ ν > 1. Then F ( ̂ U 0 )[ W ] ∈ X δ, ( ν,ν,ν ) ,q if u ∗ , u ∗∗ ∈ H q +2 ( T 1 ), where ν = ̂ ν , if ̂ ν < 2, or, we have ν < 2, if ̂ ν ≥ 2. Choosing q ≥ 3, we verify that Theorem 2.4 implies the existence of solutions of the evolution system ̂ U = ̂ U 0 + W with W ∈ X δ 1 , ( µ,µ,µ ) ,q for δ 1 sufficiently small 1 and for an exponent µ ∈ (1 , min { 2 , ̂ ν } ). For any specified set of the asymptotic data u ∗ and u ∗∗ , we find that the solution is unique for remainders in the space X δ 1 , ( µ,µ,µ ) ,q . D ∆ ̂ u = 0 . (2.70) We then note that (i) if we construct ∆ ̂ u using ̂ U 0 from Eq. (2.69), we get ∆ ̂ U 0 = 0; and (ii) if we combine the evolution equation Eq. (2.70) with the leading order term ∆ ̂ U 0 as well as q ≥ 3 and other appropriate choices of µ , etc., then we find that ∆ ̂ u satisfies a singular initial value problem which satisfies the hypothesis of Theorem 2.4. Noting that ∆ ̂ u = 0 is a solution to this singular initial value problem, and recalling that Theorem 2.4 implies that solutions are unique, we see that indeed, the constraint ∆ ̂ u = 0 must be satisfied. While this approach to analyzing the singular initial value problem for the EulerPoisson-Darboux system does produce a solution, it is unsatisfactory for two reasons. First, it does not allow us to treat the case in which f 0 ∈ X δ, ̂ ν,q for ̂ ν < 1. Second, if ̂ ν > 1, this approach does not exclude the possible existence of other solutions u with remainders w in X δ,µ,q for µ < 1. Both of these issues are resolved if we use an alternative approach based on Theorem 2.21 and the use of (order n)-leading order terms. In doing this, we pay a price in that we must require a that the spatial derivative parameter q is infinite. If we are to work with Theorem 2.21, a key requirement is that we start with an ODEleading-order term; we have already fulfilled this requirement by our choice of ̂ U 0 . We now have the advantage that we do not need to impose the block diagonal condition, but only the modified block diagonal conditions, see Condition (i) in Proposition 2.18, and also not the positivity of the energy dissipation matrix in choosing the remainder exponent vector µ ; we may work with µ = ( µ 1 , µ 2 , µ 3 ) for any µ 1 , µ 2 > 0 and 1 < µ 3 < µ 2 +1, thereby permitting the full range of values of µ for which the singular initial value problem is meaningful. Notice that the upper bound for µ 3 is implied by Eq. (2.53) and is related to the observation above that a spatial derivative of a spatially dependent power of t may introduce additional log t -terms. Proceeding, we suppose that we have chosen some f 0 ∈ X δ, ̂ ν, ∞ with ̂ ν > 0. Any choice of µ satisfying the above conditions is consistent with Condition (ii) of Proposition 2.18 (as part of Theorem 2.21) if µ 2 < min { 2 , ̂ ν } . Choosing u ∗ , u ∗∗ ∈ C ∞ ( T 1 ), we then verify straightforwardly that Condition (iii) of Proposition 2.18 is satisfied. We conclude that there exists a solution ̂ U of the evolution system with ̂ U -̂ U n ∈ X δ 1 , ( µ 1 ,µ 2 ,µ 3 )+ nγ, ∞ for some constants δ 1 > 0 and a sufficiently large integer n . This solution is unique, with the remainder U U contained in X . ̂ -̂ 0 δ 1 , ( µ 1 ,µ 2 ,µ 3 ) , ∞ Having verified the existence of solutions to the first-order evolution system, we wish to show again that the corresponding solution is actually a solution of the original secondorder equation by considering the constraint Eq. (2.68). This can be done essentially as discussed above. To illustrate the use of the leading-order term approach, we choose the source term in the form f 0 ( t, x ) = f ∗ ( x ) t 1 / 2 for a smooth function f ∗ , and calculate and ̂ U 1 = ̂ U 0 +   4 f ∗ ( x ) t 1 / 2 + 1 4 t 2 (log tu '' ∗ + u '' ∗∗ -u '' ∗ ) 2 f ∗ ( x ) t 1 / 2 + 1 2 t 2 (log tu '' ∗ +2 u '' ∗∗ -u '' ∗ ) 0   ̂ U 2 = ̂ U 1 +    0 0 4 t 3 / 2 log tf ' ∗ + 1 4 t 3 ( log tu (3) ∗ + u (3) ∗∗ -u (3) ∗ )    . One may continue to calculate the sequence, and one verifies (in accord with the last statement in Proposition 2.18) that the residuals corresponding to this sequence are contained in X spaces of monotonically increasing exponent. 3 T 2 -symmetric vacuum Einstein spacetimes 3.1 Objective of this section As noted in the Introduction, one of the main motivations for this work is to explore the singular regions of certain classes of solutions of the Einstein gravitational field equations. In particular, as a step towards studying the strong cosmic censorship conjecture in families of solutions characterized by relatively large isometry groups, we use the Fuchsian formulations developed here to show that there are large sets of solutions in these families which exhibit AVTD behavior in a neighborhood of their singularity. We work here with spacetimes which are characterized by a spatially-acting T 2 isometry group, but do not have the further restriction of a non-vanishing 'twist', which defines the familiar Gowdy spacetimes. Following convention, we refer to them as the ' T 2 -symmetric spacetimes'; if they also satisfy the Einstein equations, we call them ' T 2 -symmetric solutions'. While much is known regarding the Gowdy spacetimes, including a proof that strong cosmic censorship holds for the Gowdy spacetimes with T 3 spatial topology [35] and for polarized Gowdy spacetimes with any allowed spatial topology [18], much less is known about the T 2 -symmetric solutions. For both the Gowdy and T 2 -symmetric families, the presence of the T 2 isometry effectively reduces the analysis to that of a PDE system on a 1 + 1 dimensional manifold. One notable difference, however, is that while the Gowdy PDE system is semilinear, that of the T 2 -symmetric solutions is quasilinear. The first work showing that there are (non-polarized) Gowdy spacetimes with AVTD behavior is that of Kichenassamy and Rendall [30] which uses Fuchsian methods to show that this is true for analytic Gowdy solutions on T 3 . The later work of Rendall [34] shows this for Gowdy spacetimes which are smooth, again using Fuchsian methods (adapted to smooth solutions rather than analytic solutions). Fuchsian methods have been used [24] to verify that there are analytic polarized T 2 -symmetric solutions with AVTD behavior. Here, we use the results presented above to show the same for T 2 -symmetric solutions (polarized and half-polarized) which are not analytic. 3.2 T 2 -symmetric spacetimes The family of vacuum T 2 -symmetric spacetimes is characterized by a T 2 isometry group which acts effectively on each spacetime in the family, with the generating Killing vector fields being everywhere spacelike. We assume that each such spacetime is the maximal globally hyperbolic development of an initial data set on a compact Cauchy surface, with the data invariant under an effective T 2 action. One more condition distinguishes the spacetimes we consider here from the Gowdy subfamily. Let Y and Z be the generators of the T 2 isometry. The Gowdy subfamily is characterized by the assumption that the distribution defined by the tangent planes orthogonal to the generators Y and Z is integrable. This condition is usually expressed as the vanishing of the two twists K Y and K Z . 1 We work here with T 2 -spacetimes with at least one non-vanishing twist. Chru'sciel has shown [17] that the vacuum Einstein equations force the twists to be constants, and that the condition of non-vanishing twist implies the Cauchy surfaces must have T 3 topology. Such spacetimes can be foliated by areal coordinates, in which the time coordinate labeling each symmetry group orbit is equal to the area of that orbit. This coordinate system conveniently locates the singularity at t = 0 except in the special case of flat Kasner, as is shown by Isenberg and Weaver in [27]. Local existence for these coordinates is shown by Chru'sciel, [17], and global existence is proved by Berger et. al. in [8], and further clarified in [27]. Let y, z be coordinates on T 2 , and let x be the remaining spatial coordinate, which takes values in S 1 . The metric can be written [17] in the form 2 g = e 2( η -U ) ( -αdt 2 + dx 2 ) + e 2 U ( dy + Adz + ( G 1 + AG 2 ) dx ) 2 + e -2 U t 2 ( dz + G 2 dx ) 2 , where all the metric functions { η, U, α, A, G 1 , G 2 } depend only on t and x . If both twist constants vanish, then the function α can be chosen to be a constant, in which case the above metric reduces to the Gowdy metric [22]. The polarized class of T 2 -symmetric spacetimes results from setting A equal to a constant in the initial data (or, equivalently, assuming that the dot product of the generators Y, Z is initially the same at all spatial points 3 ), and verifying that this condition is preserved under evolution. While the polarized spacetimes are characterized by a geometric condition, another subclass we consider, called the half-polarized T 2 -symmetric spacetimes, is defined by a restriction on the asymptotic behavior of the fields (see Section 3.3.1). /negationslash Before writing down the Einstein vacuum equations, we make a few further coordinate choices to simplify the presentation. Without loss of generality we choose the generators such that K Y = 0 , K Z ≡ K = 0. This can be achieved by choosing an appropriate linear combination of any generators for the T 2 action. It is sufficient to consider K > 0 since the transformation K → -K preserves all conditions imposed thus far. Next we choose coordinates y, z on T 2 so that Y = ∂ y and Z = ∂ z . This can be done without changing the form of the metric above. Implementing these simplifications, and using the short-hand notation U t := ∂ t U for derivatives, we write the Einstein equations as the following system of PDEs, which includes a set of second order equations U tt + U t t -αU xx = α x U x 2 + α t U t 2 α + e 4 U 2 t 2 ( A 2 t -αA 2 x ) , (3.1) A tt -A t t -αA xx = α x A x 2 + α t A t 2 α -4 A t U t +4 αA x U x , (3.2) α η α η α 2 α 4 t ( -) -4 t η tt -αη xx = x x 2 + t t 2 α -x 4 α + xx 2 -U 2 t + αU 2 x , (3.3) + e 4 U 2 A 2 t αA 2 x 3 e 2 η α 4 K 2 , a set of first-order equations η t = tU 2 t + tαU 2 x + e 4 U 4 t ( A 2 t + αA 2 x ) + e 2 η 4 t 3 αK 2 , (3.4) η x = 2 tU t U x + e 4 U 2 t A t A x -α x 2 α , (3.5) α t = -e 2 η t 3 α 2 K 2 , (3.6) plus a set of auxiliary equations G 1 t = e 2 η √ αAKt -3 , G 2 t = -e 2 η √ αKt -3 . (3.7) Here, the auxiliary equations originate from the definition of the twist constants K Y and K Z and from the 'gauge' simplification K Y = 0 noted above. Observe that the T 2 -symmetric Einstein system reduces to the Gowdy system in the standard areal coordinates if we set K = 0, α ≡ 1, G 1 ≡ 0, and G 2 ≡ 0. The Einstein equations in the Gowdy class are semilinear and a Fuchsian analysis with analytic asymptotic data has been carried out by Kichenassamy and Rendall [30], and with smooth asymptotic data by Rendall [34] and by Beyer and LeFloch [12]. 3.3 Existence of AVTD solutions to the Einstein vacuum equations 3.3.1 AVTD behavior and heuristics What is the behavior of a singular solution to Einstein's equations near the singularity? In principle the behavior could be very complicated for a solution to a system of nonlinear PDE such as the Einstein equations. In [32, 6, 7] Belinskii, Khalatnikov, and Lifshitz (BKL) propose that generically the spacetime dynamics near the singularity is vacuum dominated, local, and oscillatory. According to this picture, an observer traveling toward the singularity (either backward or forward in time, depending upon the location of the singularity) would experience an infinite sequence of Kasner epochs, and each observer at different spatial points would experience a different, generally unrelated, sequence. Numerical simulations of T 2 -symmetric spacetimes [5, 9, 33] support this picture, except perhaps at points where spikes occur. Whether the complicated behavior found near spikes, and the apparent prevalence of spikes, invalidates the BKL picture for general T 2 -symmetric solutions is far from clear. However, for the restricted family of polarized T 2 -symmetric solutions, numerical simulations indicate that a special form of BKL behavior occurs near singularities-asymptotically velocity term dominated, or AVTD, behavior-which is not dominated by what happens near spikes. In a spacetime with AVTD behavior, each observer experiences only a finite sequence of Kasner epochs in the approach to the singularity [25, 21, 23], and the limiting spacetime is different for each observer. While there are no analytical studies of inhomogeneous cosmological solutions which either confirm or deny the presence of general BKL behavior, as noted above there has been a significant amount of such work supporting the generic presence of AVTD behavior in restricted families of solutions. Studies based on singular initial value problem formulations of Fuchsian PDEs are particularly well-adapted to doing this, since they involve specifying a choice of asymptotic behavior (a Kasner evolution independently at each point), and showing that there are solutions of the equations which approach this asymptotic behavior. If we can show that the Einstein equations for the polarized T 2 -symmetric spacetimes, together with certain choices of the leading order term, satisfy the conditions of the hypothesis of either Theorem 2.4 or Theorem 2.21, then we have confirmation that there are such spacetimes which have AVTD behavior. Observe that finding solutions in a given family of spacetimes with AVTD behavior does not imply that there are not solutions in that same family with a very different form of asymptotic behavior. However, since numerical simulations support AVTD behavior being generic among polarized T 2 -symmetric solutions, there have been no searches for alternative forms of asymptotic behavior among them. The name 'asymptotically velocity term dominated' refers to the fact that the leading order terms are chosen as asymptotic solutions of the 'velocity term dominated' (VTD) system, which is formed from the Einstein equations by dropping terms with spatial derivatives. This step encodes the local aspect of the BKL proposal. It can be shown [24, 20] that the following expansions for the metric functions below asymptotically solve this VTD system in the limit t → 0. We write these expansions in terms of asymptotic data { k, U ∗∗ , A ∗ , A ∗∗ , η ∗ , α ∗ , G 1 ∗ , G 2 ∗ } with the regularity of the data specified below. U ( t, x ) = 1 2 (1 -k ( x )) log t + U ∗∗ ( x ) + . . . , (3.8) A ( t, x ) = A ∗ ( x ) + A ∗∗ ( x ) t 2 k ( x ) + . . . , (3.9) η ( t, x ) = 1 4 (1 -k ( x )) 2 log t + η ∗ ( x ) + . . . , (3.10) α ( t, x ) = α ∗ ( x ) + . . . , (3.11) G 1 ( t, x ) = G 1 ∗ ( x ) + . . . , (3.12) G 2 ( t, x ) = G 2 ∗ ( x ) + . . . . (3.13) Of particular importance here is the function k . It determines the Kasner exponents p 1 , p 2 , p 3 of the local Kasner solutions which are approached at any spatial point p 1 = ( k 2 -1) / ( k 2 +3) , p 2 = 2(1 -k ) / ( k 2 +3) , p 3 = 2(1 + k ) / ( k 2 +3) . We recall here that a T 2 -symmetric solution is defined to be polarized if the two Killing vectors corresponding to the T 2 isometry can be chosen to be orthogonal everywhere. This is the case if and only if the metric coefficient A ≡ const . A solution with AVTD behavior has this property if and only if the asymptotic data corresponding to A satisfy the conditions A ∗∗ ≡ 0 and A ∗ ≡ const . Since A ∗ ≡ const can be gauged to A ∗ ≡ 0, we see that in the polarized case, there is effectively no free asymptotic data to choose which relates to A . There is an interesting relationship between the polarization condition and the sign of k : Examining equations (3.8)-(3.13), we find that if a solution is not polarized and has AVTD behavior, then there is power law blow-up at the singularity if and only if k is negative. Yet if that spacetime is polarized, then regardless of the sign of k , there is no power law blow-up at the singularity. The polarization condition is relevant to our application of our Fuchsian results to T 2 -symmetric solutions since, as we see below, our results cannot be applied unless the condition ∂ x A ∗ = 0 holds for the asymptotic data. For polarized T 2 -symmetric solutions, this restriction on A ∗ is automatic. It is important to note, however, that requiring ∂ x A ∗ = 0 does not restrict us to polarized solutions. We may consider asymptotic data which has this restriction on A ∗ , but has no restriction on A ∗∗ . T 2 -symmetric solutions which are AVTD and which have asymptotic data of this sort are known to exist, and have been called 'half-polarized' 1 [20]. Extending the results of both [24] (analytic and polarized) and [20] (higher regularity), we show here that there are large families of both half-polarized and polarized T 2 -symmetric solutions which are smooth or of even lower regularity, and which have AVTD behavior near their cosmological singularities. A general (neither polarized nor half-polarized) T 2 -symmetric solution, were it to be AVTD, would have asymptotic data with both A ∗ and A ∗∗ non-vanishing and nonconstant. Based on numerical and heuristic considerations, however, it is expected that spacetimes with non-constant A ∗ do not generally show AVTD behavior. Rather, these are expected to show Mixmaster-like BKL behavior at the t = 0 singularity, or behavior which is even more complicated (with strong spike influence). We do not address this issue here. We now discuss two applications of our Fuchsian results which verify AVTD behavior in T 2 -symmetric solutions. For the first one, Theorem 3.1, we make only minimal assumptions regarding the regularity of the asymptotic data. The price to pay for this is that the result does not cover the full expected range for the function k = k ( x ) in Eqs. (3.8) - (3.13). For the second result Section 3.3.3, Theorem 3.6, we add regularity restrictions, but we do get the expected full range of allowed values for k . 3.3.2 Existence of low regularity solutions with AVTD behavior The low regularity result, which we formulate, discuss, and prove in this subsection, is an application of Theorem 2.4 to the polarized and half-polarized solutions of the T 2 -symmetric equations. Theorem 3.1 (First result: AVTD (half)-polarized T 2 -symmetric vacuum solutions finite differentiability) . Suppose one chooses a twist constant K ∈ R , a pair of asymptotic data constants A ∗ and η 0 , and a set of asymptotic data functions k, U ∗∗ , α ∗ ∈ H q +2 ( T 1 ) (with α ∗ ( x ) > 0 ), A ∗∗ ∈ H q +1 ( T 1 ) and G 1 ∗ , G 2 ∗ ∈ H q ( T 1 ) for any q ≥ 3 , which satisfy the integrability condition 1 ∫ 2 π 0 ( (1 -k ( x )) U ' ∗∗ ( x ) -1 2 (log α ∗ ) ' ( x ) ) dx = 0 , together with, at each point x ∈ T 1 , either (ii) k ( x ) > 1 + √ 6 or k ( x ) < 1 -√ 6 for A ∗∗ ≡ 0 (the polarized case ). (i) k ( x ) > 1 + √ 6 for arbitrary A ∗∗ (the half-polarized case ), Then there exists a δ > 0 , and a T 2 -symmetric solution U , A , η , α , G 1 , G 2 of Einstein's vacuum field equations with twist K of the form ( U, A, η, α, G 1 , G 2 ) = ( U 0 , A 0 , η 0 , α 0 , G 1 , 0 , G 2 , 0 ) + W. Here, the leading-order term ( U 0 , A 0 , η 0 , α 0 , G 1 , 0 , G 2 , 0 ) is given by Eqs. (3.8) -(3.13) , with η ∗ ( x ) := η 0 + ∫ x 0 ( (1 -k ( X )) U ' ∗∗ ( X ) -1 2 (log α ∗ ) ' ( X ) ) dX. (3.14) The remainder W is contained in X δ,µ,q (and DW ∈ X δ,µ,q -1 ) for any exponent vector µ = ( µ 1 , µ 2 , µ 3 , µ 4 , µ 5 , µ 6 ) with 1 < µ 1 ( x ) < min { 2 , ( k ( x ) -3)( k ( x ) + 1) / 2 } , (2 k ( x ) + √ 1 + 4 k ( x ) 2 ) / 2 < µ 2 ( x ) < 1 + 2 k ( x ) , 0 < µ 3 ( x ) < µ 1 ( x ) , 0 < µ 4 ( x ) , µ 5 ( x ) , µ 6 ( x ) < ( k ( x ) -3)( k ( x ) + 1) / 2 . (3.15) This solution is unique among all solutions with the same leading-order term U 0 and with remainder W ∈ X δ,µ,q . Observe that by taking time derivatives of the Einstein field equations, we can also obtain corresponding statements about the behavior of a certain number of time derivatives D m W of the remainder function W . We do not elaborate on this any further here. This result, based on Theorem 2.4, does not imply uniqueness of the solutions within the whole class of solutions of interest: For a given choice of asymptotic data, Theorem 3.1 determines that there is exactly one solution with remainder W in spaces X δ,µ,q with µ given by Eq. (3.15). The full class of remainders compatible with the leadingorder behavior Eqs. (3.8)-(3.13) however corresponds to exponents µ 1 , µ 2 -2 k, µ 3 , µ 4 , µ 5 , µ 6 > 0 . Hence, for given asymptotic data there may exist further solutions in such a larger space. Strict uniqueness can be explored further using techniques involving (order n)-leading order terms. We return to this issue in Section 3.3.3 below; the price which we have to pay for strict uniqueness is that we need to require higher differentiability for the asymptotic data. In proving Theorem 3.1, it is useful to arrange the T 2 -symmetric Einstein vacuum equations, Eqs. (3.1)-(3.7), as well as the field variables, in a certain hierarchical form: Eqs. (3.1), (3.2), (3.4) and (3.6) together form a coupled evolution system (which we label the 'main evolution equations') for the variables U, A, η , and α . Eq. (3.5) serves as a constraint equation for this system, while Eq. (3.3) is effectively redundant, and can be ignored. The remaining equations Eqs. (3.7) are evolution equations for G 1 and G 2 , and can be handled after the analysis of the main evolution equations. We proceed now to focus on the main evolution equations, with the primary existence result for them - the main step toward a proof of Theorem 3.1 - being Proposition 3.2. Main evolution equations. To rewrite the main evolution equations as a first order symmetric hyperbolic Fuchsian system, it is useful to define certain new variables. Some of the choices of these variables are motivated by considerations in [24], others by the discussion above in Section 2.4.4. First, we set ξ := ∂ x α, (3.16) whose evolution equation is obtained by taking the spatial derivative of Eq. (3.6) and by substituting any occurrence of η x by the constraint Eq. (3.5). One obtains ξ t = -e 2 η t 4 αK 2 ( tξ + α ( e 4 U A x A t +4 t 2 U x U t )) . In all other evolution equations we use Eq. (3.6) to eliminate α t and replace α x by ξ . Next, we find that for both U and η , it is useful to replace the given variable by that which involves the subtraction of the indicated log term in the asymptotic VTD expansions Eq. (3.8)-(3.13): We set ̂ η := η -1 4 (1 -k ) 2 log t and set ̂ U := U -1 2 (1 - )) log t ; compare this to our approach in Section 2.4.4. Adding a few other minor modifications, we are led to define the following set of first-order variables: u 1 = ̂ U, u 2 = D ̂ U, u 3 = t∂ x ̂ U, (3.17) u 4 = A, u 5 = DA, u 6 = t∂ x A, (3.18) u 7 = ̂ η, u 8 = α, u 9 = ξ. (3.19) Observe that, at this stage, k ( x ) is an arbitrary function (introduced in Eqs. (3.8)(3.13)), with no restrictions. In terms of the new set of the variables, the main evolution system Eqs. (3.1), (3.2), (3.4) and (3.6) can be written in symmetric hyperbolic form as follows: Du 1 -u 2 =0 , (3.20) Du 2 -u 8 t∂ x u 3 = 1 2 tu 9 ( u 3 -1 2 t log tk ' ) + 1 2 e 4 u 1 t -2 k ( u 2 5 -u 8 u 2 6 ) (3.21) -1 4 e 2 u 7 t 1 / 2(1 -k ) 2 -2 u 8 K 2 (1 -k +2 u 2 ) -1 2 t 2 log tk '' u 8 , u 8 Du 3 -u 8 t∂ x u 2 -u 8 u 3 =0 , (3.22) Du 4 -u 5 =0 , (3.23) -2 Du 5 -2 ku 5 -u 8 t∂ x u 6 = -4 u 5 u 2 + 1 2 tu 9 u 6 +2 u 8 u 6 (2 u 3 -t log tk ' ) (3.24) 1 e 2 u 7 t 1 / 2(1 -k ) 2 -2 u 8 u 5 K 2 , u 8 Du 6 -u 8 t∂ x u 5 -u 8 u 6 =0 , + 1 4 t -2 k e 4 u 1 ( u 2 5 + u 8 u 2 6 ) + 1 4 e 2 u 7 t 1 / 2(1 -k ) 2 -2 u 8 K 2 , (3.25) Du 7 =(1 -k ) u 2 + u 2 2 + 1 4 u 8 (2 u 3 -t log tk ' ) 2 (3.26) Du 8 = -e 2 u 7 t 1 / 2(1 -k ) 2 -2 u 2 8 K 2 , (3.27) · ( (1 -k +2 u 2 )(2 u 3 -t log tk ' ) u 8 t + t -1 -2 k u 5 u 6 u 8 e 4 u 1 + u 9 ) , Du 9 = -e 2 u 7 t 1 / 2(1 -k ) 2 -2 u 8 K 2 (3.28) S 1 Du + S 2 t∂ x u + Nu = f [ u ] , (3.29) S 1 ( u ) = Diag(1 , 1 , u 8 , 1 , 1 , u 8 , 1 , 1 , 1) , (3.30) or equivalently as where S 2 ( u ) =               0 0 0 0 0 0 0 0 0 0 0 -u 8 0 0 0 0 0 0 0 -u 8 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 -u 8 0 0 0 0 0 0 0 -u 8 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0               , (3.31) N ( u ) =               0 -1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 -u 8 0 0 0 0 0 0 0 0 0 0 -1 0 0 0 0 0 0 0 0 -2 k 0 0 0 0 0 0 0 0 0 -u 8 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0               . (3.32) Note that we have multiplied the third and sixth equations by u 8 . The source-term vector f is given by the right-hand sides of the evolution system Eqs. (3.20)-(3.28). The reason for keeping this particular form of the matrix N ( u ) (and not absorbing some of its entries into the source-term) becomes clear shortly. AVTD solutions of the main evolution system. We now show as an application of Theorem 2.4, and as a step towards proving Theorem 3.1, that there exist unique solutions to the singular initial value problem for the main evolution system (3.29)(3.32), with AVTD leading-order term u 0 =( u 1 , 0 , u 2 , 0 , u 3 , 0 , u 4 , 0 , u 5 , 0 , u 6 , 0 , u 7 , 0 , u 8 , 0 , u 9 , 0 ) = ( U ∗∗ , 0 , tU ' ∗∗ , A ∗ + A ∗∗ t 2 k , 2 kA ∗∗ t 2 k , 0 , η ∗ , α ∗ , ξ ∗ ) . (3.33) Although not needed for our present argument, we note (by inspecting Eq. (2.46)) that this choice of u 0 is an ODE-leading-order term; cf. Section 2.4.4. To check that we have a quasilinear symmetric hyperbolic system, we need to specify an exponent vector along with the PDE system and a leading order term. Looking ahead to the conditions of block diagonality, we choose µ = ( µ 1 , µ 1 , µ 1 , µ 2 , µ 2 , µ 2 , µ 3 , µ 4 , µ 4 ) , (3.34) and expect to construct remainders in spaces X δ,µ,q with µ given by µ 1 , µ 3 , µ 4 > 0 , µ 2 > 2 k. We then find, after replacing u 8 by α ∗ + w 8 , that so long as we choose α ∗ > 0, and so long as we require that all of the asymptotic data functions be contained in some H q ( T 1 ) (which we fix below), we indeed have a quasilinear symmetric hyperbolic system, which in addition does satisfy the block diagonality condition. Before continuing the argument that the hypothesis of Theorem 2.4 is satisfied, we state our result. Proposition 3.2. For any twist constant K ∈ R , for any Sobolev differentiability index q ≥ 3 , and for any choice of the asymptotic data functions such that A ∗ is an arbitrary constant, α ∗ ( x ) > 0 , k, U ∗∗ , α ∗ ∈ H q +2 ( T 1 ) , A ∗∗ ∈ H q +1 ( T 1 ) and η ∗ ∈ H q ( T 1 ) , and k satisfies (at each x ∈ T 1 ) either (ii) k ( x ) > 1 + √ 6 or k ( x ) < 1 -√ 6 (for A ∗∗ ≡ 0 the polarized case), (i) k ( x ) > 1 + √ 6 (for arbitrary A ∗∗ the half-polarized case), there exists a δ 1 ∈ (0 , δ ] , and a unique solution of the first order main evolution system Eqs. (3.29) -(3.32) with leading-order term u 0 and remainder w ∈ X δ 1 ,µ,q (and Dw ∈ X δ 1 ,µ,q -1 ) so long as the exponent vector µ given by Eq. (3.34) satisfies the following inequalities for all x ∈ T 1 : 1 < µ 1 ( x ) < min { 2 , ( k ( x ) -3)( k ( x ) + 1) / 2 } , 1 2 ( 2 k ( x ) + √ 1 + 4 k ( x ) 2 ) < µ 2 ( x ) < 1 + 2 k ( x ) , 0 < µ 3 ( x ) < µ 1 ( x ) , 0 < µ 4 ( x ) < 1 2 ( k ( x ) -3)( k ( x ) + 1) . Observe here that the inequality just stated for µ 2 is not required to hold in the case of a polarized solution, since in that case A is not a dynamical variable, and this condition is vacuous. Although here and below we list results for the polarized and half-polarized cases together for compactness, the reader focusing on the polarized case may ignore all references to µ 2 and to w 4 , w 5 and w 6 . As noted above, this proposition is an application of Theorem 2.4 to Eqs. (3.29)(3.32). In the next lemma we verify that under the assumptions of Proposition 3.2 the Condition (i) of Theorem 2.4 is satisfied. The first condition follows directly from the definition of the energy dissipation matrix M 0 . Lemma 3.3. The energy dissipation matrix M 0 defined in Eq. (2.6) corresponding to Eqs. (3.29) -(3.32) , to the leading-order term u 0 given by Eq. (3.33) and to the exponent vector µ of the form Eq. (3.34) is positive definite at every x , provided that α ∗ ( x ) > 0 , µ 1 ( x ) > 1 , µ 2 ( x ) > max { 1 , k ( x ) + 1 2 √ 1 + 4 k ( x ) 2 } , µ 3 ( x ) , µ 4 ( x ) > 0 , hold for all x ∈ T 1 . The next lemma establishes Conditions (ii) and (iii) of Theorem 2.4. Lemma 3.4. The operator F ( u 0 ) corresponding to Eqs. (3.29) -(3.32) , to the leadingorder term u 0 given by Eq. (3.33) , and to the exponent vector µ of the form Eq. (3.34) satisfies Condition (ii) and (iii) of Theorem 2.4 for some exponent vector ν > µ , for some sufficiently small δ > 0 , and for a choice of the differentiability index q ≥ 3 , so long as α ∗ and η ∗ are functions in H q ( T 1 ) , A ∗∗ is contained in H q +1 ( T 1 ) , k and U ∗∗ are elements of H q +2 ( T 1 ) , and if at each point x ∈ T 1 , the following inequalities hold for µ and k : max { 0 , 1 -( k ( x ) -3)( k ( x ) + 1) / 2 } < µ 1 ( x ) < min { 2 , ( k ( x ) -3)( k ( x ) + 1) / 2 } , 2 k ( x ) < µ 2 ( x ) < min { 1 + 2 k ( x ) , µ 1 ( x ) + 2 k ( x ) } , 0 < µ 3 ( x ) < µ 1 ( x ) , 0 < µ 4 ( x ) < min { ( k ( x ) -3)( k ( x ) + 1) / 2 ,µ 1 ( x ) -1 + ( k ( x ) -3)( k ( x ) + 1) / 2 } , and 3 < k ( x ) in the half-polarized case, . 3 < k ( x ) or k ( x ) < -1 in the polarized case In both the polarized and the half-polarized cases, it follows from the two inequalities stated above for µ 1 that k ( x ) must either satisfy k ( x ) > 1 + √ 5 or k ( x ) < 1 -√ 5 . Proof. If the operator F ( u 0 ), defined in Eq. (2.7), is written out explicitly, it consists of products of asymptotic data functions, and components of the unknown function w (or products involving exponential functions of these). All of the multiplicands in these products are, by hypothesis, contained in designated function spaces (of the form X δ,µ,q ). Thus, to check Condition (ii) of Theorem 2.4, we primarily need to know the multiplication algebra of spaces such as X δ,µ,q . The result we need is provided by Lemma B.1 in the appendix. To check Condition (iii), we need results concerning Lipschitz properties of products and exponential functions of elements of the spaces X δ,µ,q . Lemma B.3 and Lemma B.5 provide these needed results. Proof of Proposition 3.2. If we wish to use Theorem 2.4 to show that the system discussed in Proposition 3.2 admits solutions with the stated properties, it is sufficient that i) the asymptotic data functions, which appear in the leading-order matrices S 1 , 0 , S 2 , 0 and N 0 , (i.e. the functions α ∗ and k ), be contained in H q +2 ( T 1 ) (with q ≥ 3); and ii) we choose the function k ( x ) so that the hypotheses of both of the above lemmas are satisfied. We readily check that exponent functions µ 1 , µ 2 , µ 3 and µ 4 , which satisfy the combined inequalities, can be found if and only if k ( x ) > 1 + √ 6 in the half-polarized case, and either k ( x ) > 1 + √ 6 or k ( x ) < 1 -√ 6 in the polarized case. The full set of Einstein's vacuum field equations. Thus far, we have constructed solutions u of the main evolution equations for the T 2 -symmetric system with the leading-order behavior Eq. (3.33), according to Proposition 3.2. Given such a solution u , we may ask under what conditions is this a solution of the full set of Einstein's vacuum field equations, Eqs. (3.1)-(3.7), with U = u 1 + 1 2 (1 -k ) log t , A = u 4 , η = u 7 + 1 4 (1 -k ) 2 log t , and α = u 8 . Proposition 3.5. For any solution of Proposition 3.2 with asymptotic data satisfying all the conditions in Theorem 3.1, the full set of Einstein's vacuum field equations Eqs. (3.1) - Eq. (3.6) are satisfied, and Eqs. (3.7) can be solved for G 1 and G 2 as stated in Theorem 3.1. Proof. Since the equations for G 1 and G 2 , Eq. (3.7), are semi-decoupled from the rest, we ignore them (as well as G 1 and G 2 ) to start, and focus on the subsystem Eqs. (3.1)(3.6). To monitor the extent to which this subsystem is satisfied by fields which satisfy the main evolution equations, it is useful to define the following set. Based on Eq. (3.3), we define H := -η tt + αη xx + ξη x 2 + α t η t 2 α -ξ 2 4 α + α xx 2 -U 2 t + αU 2 x + e 4 U 4 t 2 ( A 2 t -αA 2 x ) -3 e 2 η α 4 t 4 K 2 (3.35) and, based on the constraint Eq. (3.5), C 1 := -η x +2 tU t U x + e 4 U 2 t A t A x -α x 2 α . (3.36) Based on the constraints which stem from the definition of the new variables which allow us to rewrite the original system in first order, we define C 2 := α x -ξ, (3.37) and C 3 := u 2 /t -∂ t u 1 , C 4 := u 3 /t -∂ x u 1 , (3.38) C 5 := u 5 /t -∂ t u 4 , C 6 := u 6 /t -∂ x u 4 . (3.39) The constraint-violation terms C 3 , C 4 , C 5 and C 6 are the easiest to handle. Arguing as we do in the discussion of ∆ u in Section 2.4.4, we find that the evolution equation for each of these, induced by the main evolution system, takes the form DC 3 = 0 , DC 4 = 0, etc. Then, since the form of the leading order term for the main system implies that each of these terms must asymptotically vanish, it follows that each must vanish for all time. Observe that this determination that C 3 , C 4 , C 5 , and C 6 all vanish allows us to freely substitute in the consequences of their vanishing in the analysis of H , C 1 , and C 2 . Such substitution is very useful. We now focus on H , C 1 and C 2 , noting that a solution u of Eqs. (3.20) - (3.28) is a solution of Eqs. (3.1)-(3.6) (with the above replacements) if and only if H,C 1 , and C 2 vanish identically. Presuming that u is a solution of Eqs. (3.20) - (3.28), we calculate H = -u 8 C 1 ,x -1 2 u 8 ,x C 1 + 1 4 ( e 4 u 1 t -1 -2 k u 5 u 6 +( k -1 -2 u 2 )(log tk ' -2 t -1 u 3 ) ) C 2 , (3.40) which tells us that so long as we can show that C 1 and C 2 vanish, it follows that H vanishes as well. We may therefore focus on C 1 and C 2 , for which the evolution equations take the following form: DC 1 = -1 2 K 2 e 2 u 7 t ( k -3)( k +1) / 2 C 1 -( u 2 3 -1 4 e 4 u 1 t -2 k u 2 6 + t log t u 3 k ' -1 4 t 2 log 2 t ( k ' ) 2 ) C 2 , DC 2 =2 e 2 u 7 K 2 t ( k -3)( k +1) / 2 u 2 8 C 1 . Under our hypothesis (which implies in particular that the coefficients here are continuous functions of x ), this set of evolution equations for C 1 and C 2 can be treated as an essentially independent system of linear homogeneous ODEs at each spatial point. Noting that (for q ≥ 3) the coefficients on the right side of both equations are well-behaved as t → 0 and converge to zero at every spatial point. Hence this system for the unknowns C 1 and C 2 is of the form Eq. (2.1) with S 1 the identity matrix, and S 2 and N 2 the zero matrices. The singular initial value problem of the form C 1 = C 1 ∗ + w 1 , C 2 = C 2 ∗ + w 2 with w 1 , w 2 ∈ X δ,µ,q has a unique solution for every prescribed C 1 ∗ and C 2 ∗ and for every sufficiently small µ > 0. The definition of the quantities C 1 and C 2 in terms of the variables U , A , η , ξ and α according to Eqs. (3.36) and (3.37) implies uniqueness for all constraint violations which are compatible with solutions of Proposition 3.2. In particular, the unique solution corresponding to C 1 ∗ = C 2 ∗ = 0 is C 1 ≡ C 2 ≡ 0. It remains to determine how the condition C 1 ∗ ≡ C 2 ∗ ≡ 0 relates to the choice of the asymptotic data functions k , U ∗∗ , A ∗ , A ∗∗ , η ∗ , α ∗ and ξ ∗ . For a solution u as above, the functions C 1 and C 2 defined by Eqs. (3.36) and (3.37) converge uniformly in space at t = 0, and we obtain C 1 ∗ = -η ' ∗ +(1 -k ) U ' ∗∗ -α ' ∗ 2 α ∗ , C 2 ∗ = α ' ∗ -ξ ∗ . It follows that C 2 ∗ = 0 if and only if ξ ∗ = α ' ∗ , (3.41) and C 1 ∗ = 0 if and only if, for an arbitrary constant η 0 , η ∗ ( x ) = η 0 + ∫ x 0 ( (1 -k ( X )) U ' ∗∗ ( X ) -1 2 (log α ∗ ) ' ( X ) ) dX. (3.42) In particular, the spatial topology implies that we must choose the asymptotic data k , U ∗∗ and α ∗ such that ∫ 2 π 0 ( (1 -k ( x ' )) U ' ∗∗ ( x ' ) -1 2 (log α ∗ ) ' ( x ' ) ) dx ' = 0 . We thus conclude that a solution u of Proposition 3.2 is a solution of Eqs. (3.1)(3.6) (with the above replacements) if and only if the asymptotic data functions satisfy Eqs. (3.41) and (3.42). It only remains to solve Eqs. (3.7) for G 1 and G 2 . The right-hand sides of these two equations have the asymptotic behavior (for t → 0) of a power of t larger than -1 uniformly in space. Hence the right-hand sides are integrable in t at t = 0 at every spatial point, and the general solution is G 1 ( t, x ) = G 1 ∗ ( x ) + ∫ t 0 e 2 η ( t ' ,x ) √ α ( t ' , x ) A ( t ' , x ) Kt '-3 dt ' , G 2 ( t, x ) = G 2 ∗ ( x ) -∫ t 0 e 2 η ( t ' ,x ) √ α ( t ' , x ) Kt '-3 dt ' . The functions G 1 -G 1 ∗ and G 2 -G 2 ∗ are contained in X δ 1 ,µ 5 ,q and X δ 1 ,µ 6 ,q , respectively, for any choice of exponents 0 < µ 5 ( x ) , µ 6 ( x ) < 1 / 2( k ( x ) -3)( k ( x ) + 1), if u is a solution of Proposition 3.2. We therefore can take G 1 ∗ , G 2 ∗ ∈ H q ( T 1 ). 3.3.3 Optimal existence and uniqueness result The result we prove in Section 3.1 allows for relatively rough asymptotic data, but consequently sacrifices some of the expected range (based on numerical and heuristic studies [9, 5, 33]) of allowed values for the asymptotic velocity k = k ( x ). In this section, we consider only smooth asymptotic data, and can then prove a result which increases the range for k . Theorem 3.6 (Optimal result: AVTD (half)-polarized T 2 -symmetric solutions - infinite differentiability) . Suppose one chooses a twist constant K ∈ R , a pair of asymptotic data constants A ∗ and η 0 , and a set of asymptotic data functions k, U ∗∗ , α ∗ (with α ∗ ( x ) > 0 ), A ∗∗ , G 1 ∗ , G 2 ∗ ∈ C ∞ ( T 1 ) which satisfy the integrability condition together with, at each point x ∈ T 1 , either ∫ 2 π 0 ( (1 -k ( x )) U ' ∗∗ ( x ) -1 2 (log α ∗ ) ' ( x ) ) dx = 0 , (i) k ( x ) > 3 for arbitrary A ∗∗ (the half-polarized case ), or (ii) k ( x ) > 3 or k ( x ) < -1 for A ∗∗ ≡ 0 (the polarized case ). Then there exists a δ > 0 , and a T 2 -symmetric solution U , A , η , α , G 1 , G 2 of Einstein's vacuum field equations with twist K of the form ( U, A, η, α, G 1 , G 2 ) = ( U 0 , A 0 , η 0 , α 0 , G 1 , 0 , G 2 , 0 ) + W. Here, the leading-order term ( U 0 , A 0 , η 0 , α 0 , G 1 , 0 , G 2 , 0 ) is given by Eqs. (3.8) -(3.13) and Eq. (3.14) . The remainder W is contained in X δ,µ, ∞ (and DW ∈ X δ,µ, ∞ ) for some exponent vector µ = ( µ 1 , µ 2 , µ 3 , µ 4 , µ 5 , µ 6 ) with µ 1 , µ 2 -2 k, µ 3 , µ 4 , µ 5 , µ 6 > 0 . (3.43) This solution is unique among all solutions with the same leading-order term u 0 and with remainder W ∈ X δ,µ, ∞ . Comparing this result with Theorem 3.1, we see the differences in the hypothesized regularity of the asymptotic data, and in the allowed range of the asymptotic velocity k ( x ) in the two results. As we find in proving this result, in fact one does not need C ∞ data; data of 'sufficiently high differentiability' is enough. One may ask if the reduced range for k ( x ) for rough data is a real effect, or an artifact of our method of proof (which remains an open question). Observe one other important difference between the two theorems: Theorem 3.6 provides a stronger result regarding the uniqueness of solutions to the singular initial value problem for these systems: While in Theorem 3.1, there could in principle exist more than one solution for a given set of asymptotic data (since µ 1 has to be larger than one), we find that according to Theorem 3.6, there is exactly one solution for the remainder functions in the full space of interest given by Eq. (3.43). We note, however, that there may be two solutions for the same asymptotic data which differ by a factor which goes to zero faster than every power of t at t = 0. The proof of Theorem 3.6 is in many ways similar to that of Theorem 3.1, but with the big difference that the latter involves the application of Theorem 2.4, while Theorem 3.6 is obtained by applying Theorem 2.21. In both cases, these results for Fuchsian singular initial value problems (Theorem 2.4 or Theorem 2.21) are applied to the main evolution system for T 2 -symmetric solutions. The portion of the proof of both Theorem 3.6 and Theorem 3.1 which shows that the existence of a solution for the main evolution system implies the existence of a proof to the full system (since one can choose the asymptotic data in such a way that the constraints are necessarily satisfied) is essentially the same for the two cases. Hence, to prove Theorem 3.6, all we need is the following proposition (with the rest of the proof taken care of by the arguments appearing in the proof of Proposition 3.5). Proposition 3.7. For any twist constant K ∈ R , and for any choice of the asymptotic data functions such that A ∗ is an arbitrary constant, α ∗ ( x ) > 0 , k, U ∗∗ , α ∗ , η 0 ∈ C ∞ ( T 1 ) , and either (i) k ( x ) > 3 (for arbitrary A ∗∗ the half-polarized case), or (ii) k ( x ) > 3 or k ( x ) < -1 (for A ∗∗ ≡ 0 the polarized case), for all x ∈ T 1 , there exists a δ 1 ∈ (0 , δ ] , and a unique solution of the main evolution system Eqs. (3.29) -(3.32) with leading-order term u 0 and remainder w ∈ X δ 1 ,µ, ∞ (and Dw ∈ X δ 1 ,µ, ∞ ), so long as the exponent vector µ given by µ = ( µ 1 , µ 1 , ˜ µ 1 , µ 2 , µ 2 , µ 2 , µ 3 , µ 4 , µ 4 ) satisfies the following inequalities for all x ∈ T 1 , 0 < µ 1 ( x ) < min { 2 , ( k ( x ) -3)( k ( x ) + 1) / 2 } 2 k ( x ) < µ 2 ( x ) < min { 1 + 2 k ( x ) , µ 1 ( x ) + 2 k ( x ) } , 0 < µ 3 ( x ) < µ 1 ( x ) , 0 < µ 4 ( x ) < ( k ( x ) -3)( k ( x ) + 1) / 2 , and ˜ µ 1 strictly smaller than, but sufficiently close to 1 + µ 1 . One of the key differences between the hypothesis here and that of Proposition 3.2 is the chosen form for the exponent vector µ . The form we use here - in particular, the choice of ˜ µ 1 for the third component - allows for the wider range of k ( x ). We can choose this form here, but not in Proposition 3.2, since here we do only need to satisfy the modified block diagonal conditions in Theorem 2.21 (as part of Proposition 2.18). Note that we could have carried out a similar modification for the sixth component of µ , but it turns out to be unnecessary; the regularity of the spatial derivative of the A variable follows automatically. Proof. The proof of this result is based on the use of (order n)-leading-order terms, which are developed and discussed in Section 2.4. and, in particular, in Theorem 2.21. Recalling that the leading order term u 0 (which takes the form u 0 with components given by Eq. (3.33)) is in fact an ODE-leading-order term, we check the conditions in the hypothesis of Theorem 2.21, noting that a portion of these conditions appear in the hypothesis of Proposition 2.18. We check that the matrix S -1 1 , 0 ( u 0 ) N 0 ( u 0 ) is in Jordan normal form, and that our choice of µ is strictly larger than the negatives of the corresponding diagonal elements of S -1 1 , 0 N 0 . Then, Condition (ii) of Proposition 2.18 (as part of Theorem 2.21), can be checked in essentially the same way as is done in the proof of Lemma 3.4, but now with the inequalities listed in the hypothesis of Proposition 3.7. Condition (iii) of Proposition 2.18 (as part of Theorem 2.21) follows by repeated applications of Lemma B.4. 4 Concluding remarks Our results here show that there is a large collection of smooth, polarized and halfpolarized T 2 -symmetric solutions of the Einstein vacuum equations which exhibit AVTD behavior in a neighborhood of their singularities. What can we show further? Numerical and heuristic studies of T 2 -symmetric solutions [9, 5, 33] strongly indicate that AVTD behavior is not found in such spacetimes unless they satisfy a polarization condition. These studies do support the conjecture that AVTD behavior occurs generically in polarized T 2 -symmetric solutions. While Fuchsian methods of the sort developed here are not expected to be effective in determining such genericity, further numerical explorations of the polarized T 2 -symmetric solutions could be very useful. Among the issues which might be explored numerically is whether the distinction in the results we have obtained for solutions of finite differentiability and those which are C ∞ is significant in any sense. Observe that one distinguishing feature of our approach here is that an approximation scheme is at the core of the method. This scheme can be implemented for numerical computations straightforwardly and contains useful built-in convergence and error estimates. In earlier work [12, 14, 15], building on [3], we implemented this scheme in the context of semilinear symmetric hyperbolic Fuchsian equations of second-order and have obtained very accurate simulations for Gowdy solutions. We expect to get similarly good results for the polarized T 2 -symmetric solutions, which we have studied here. Of particular interest would be to explore the issue of whether the 'optimal domain' for the asymptotic velocity k ( x ) > 3 (or k ( x ) < -1) can only be obtained for smooth solutions (as suggested by our discussion in Section 3) and to see what might happen if we try to construct such a solution with lower differentiability. It is expected, based on numerical simulations [10], that polarized (and half-polarized) U (1)-symmetric solutions exhibit AVTD behavior. Moreover, Fuchsian methods [26] confirm this, at least for analytic solutions. The methods we have developed here, generalized to PDEs on higher dimensional manifolds (this is done for T n in [2]), should be applicable to the polarized and half-polarized U (1)-symmetric solutions, showing that smooth solutions of this type also exhibit AVTD behavior. Acknowledgements F.B. was partially supported by a special assistance grant of the University of Otago during 2011. The authors E.A. and J.I. are partially supported by NSF grant PHY0968612. E.A., J.I., and P.LF. thank the University of Otago for sponsoring their visits to Dunedin while some of this research was carried out. P.LF was also supported by the Agence Nationale de la Recherche through the grant ANR SIMI-1-003-01 (Mathematical General Relativity. Analysis and geometry of spacetimes with low regularity). References [1] R. A. Adams. Sobolev spaces . Academic Press, 1975. [2] E. Ames, F. Beyer, J. Isenberg, and P. G. LeFloch. Quasi-linear symmetric hyperbolic Fuchsian systems in several space dimensions. Proc. Conference 'Complex Analysis and Dynamical Systems', V, Akko, Israel, May 2011. [3] P. Amorim, C. Bernardi, and P. G. LeFloch. Computing Gowdy spacetimes via spectral evolution in future and past directions. Class. Quantum Grav. , 26(2):025007, 2009. [4] L. Andersson and A. D. Rendall. Quiescent cosmological singularities. Comm. Math. Phys. , 218(3):479-511, 2001. [5] L. Andersson, H. van Elst, W. C. Lim, and C. Uggla. Asymptotic silence of generic cosmological singularities. Phys. Rev. Lett. , 94(5):051101, 2005. [6] V. A. Belinskii, I. M. Khalatnikov, and E. M. Lifshitz. Oscillatory approach to a singular point in the relativistic cosmology. Adv. Phys. , 19(80):525-573, 1970. [7] V. A. Belinskii, I. M. Khalatnikov, and E. M. Lifshitz. A general solution of the Einstein equations with a time singularity. Adv. Phys. , 31(6):639-667, 1982. [8] B. K. Berger, P. T. Chru'sciel, J. Isenberg, and V. Moncrief. Global foliations of vacuum spacetimes with T 2 isometry. Ann. Phys. , 260(1):117-148, 1997. [40] M. E. Taylor. Partial differential equations III - Nonlinear equations , Applied Mathematical Sciences, Vol. 117. Springer, New York, NY, 2nd edition, 2011. Appendices A Properties of the spaces X δ,µ,q In this section we list further basic properties of the spaces X δ,µ,q which are defined in Section 2.2 as the completion of the normed vector spaces ( C ∞ ((0 , δ ] × T 1 ) , ‖·‖ δ,µ,q ), cf. Eq. (2.3). Recall that δ > 0 is a constant, µ is an exponent vector and q is a non-negative integer. We now also define the spaces ̂ X δ,µ,q as the set of maps f : (0 , δ ] → H q ( T 1 ) with the property that R [ µ ] f is bounded and continuous; cf., Eq. (2.2). If we endow ̂ X δ,µ,q with the norm ‖ · ‖ δ,µ,q , then ̂ X δ,µ,q are Banach spaces. Note that if f ∈ ̂ X δ,µ + /epsilon1,q for some /epsilon1 > 0, then R [ µ ] f : (0 , δ ] → H q ( T 1 ) is uniformly continuous. Lemma A.1. Let f ∈ ̂ X δ,µ,q ; i.e., R [ µ ] f : (0 , δ ] → H q ( T 1 ) is bounded and continuous. Let ̂ f be defined as follows By definition, all functions in X δ,µ,q can be approximated by smooth functions. Functions in ̂ X δ,µ,q , however, can be approximated by a particularly useful sequence of smooth functions as follows. ̂ f ( t ) = { f ( t ) , t ∈ (0 , δ ] , R [ µ ] -1 ( t ) R [ µ ]( δ ) f ( δ ) , t ∈ [ δ, ∞ ) . ( R [ µ ] f ) i,j ( t, x ) := ∫ ∞ 0 ∫ T 1 ( R [ µ ] ̂ f )( s, y ) 1 α i φ ( x -y α i ) 1 α j φ ( s -t α j ) dy ds. (A.1) Let φ : R → R be smooth with φ ( x ) > 0 for all | x | < 1 and φ ( x ) = 0 for all | x | ≥ 1 , with ∫ R φ ( x ) dx = 1 . Let ( α i ) be a sequence of positive numbers with limit 0 . For any integers i, j , we set Then ( R [ µ ] f ) i,j has the following properties: (i) ( R [ µ ] f ) i,j ∈ C ∞ ((0 , δ ] × T 1 ) for all integers i, j . (ii) The function f i,j := R [ µ ] -1 ( R [ µ ] f ) i,j (A.2) has the property that In particular, for any given integers i, j , one has f i,j ∈ ̂ X δ,µ,q ∩ X δ,µ,q for all integers i, j. ‖ ( R [ µ ] f ) i,j ( t, · ) ‖ H q ( T 1 ) ≤ C ‖ f ‖ δ,µ,q , for all t ∈ (0 , δ ] , for a constant C > 0 independent of t (but possibly dependent on i , j ). (iii) ( R [ µ ] f ) i,j ( t, x ) -→ R [ µ ] f ( t, x ) for i, j →∞ at a.e. ( t, x ) ∈ (0 , δ ] × T 1 . (iv) If f is such that R [ µ ] f : (0 , δ ] → H q ( T 1 ) is a uniformly continuous map (e.g., if f ∈ ̂ X δ,µ + /epsilon1,q for some /epsilon1 > 0 ), then ‖ f i,j -f ‖ δ,µ,q → 0 for i, j →∞ . Proof. Observe that R [ µ ] ̂ f is a bounded continuous map (0 , ∞ ) → H q ( T 1 ) since R [ µ ]( t ) ̂ f ( t ) = R [ µ ]( δ ) f ( δ ) for all t ≥ δ . We obtain ̂ f ∈ ̂ X ∞ ,µ,q and ‖ ̂ f ‖ ∞ ,µ,q = ‖ f ‖ δ,µ,q . The first two properties of the lemma can be proven by standard arguments. The third one follows from Lebesgue's Differentiation Theorem. We only discuss the fourth property. If we fix any t ∈ (0 , δ ], then 1 we calculate ∥ ∥ ∥ ∞ 0 T 1 ( R [ µ ] ̂ f )( s, y ) -( R [ µ ] f )( t, x ) 1 α i φ x -y α i 1 α j φ s -t α j dy ds ∥ ∥ ∥ H q x ( T 1 ) , ‖R [ µ ] ( |R [ µ ] -1 ( R [ µ ] f ) i,j ( t, x ) -f ( t, x ) ) ‖ H q x ( T 1 ) = ‖ ( R [ µ ] f ) i,j ( t, x ) -( R [ µ ] f )( t, x ) ‖ H q x ( T 1 ) = ∥ ∫ ∫ ( ) ( ) ( ) ∥ ( R [ µ ] ̂ f )( s, y ) -( R [ µ ] f )( t, x ) = ( R [ µ ] ̂ f )( s, y ) -( R [ µ ] ̂ f )( s, x )+( R [ µ ] ̂ f )( s, x ) -( R [ µ ] f )( t, x ) , and therefore obtain the estimate as a consequence of the condition that ∫ R φ ( x ) dx = 1. Now we write ‖R [ µ ] ( |R [ µ ] -1 ( R [ µ ] f ) i,j ( t, x ) -f ( t, x ) ) ‖ H q x ( T 1 ) (A.3) ≤ ∥ ∥ ∥ ∥ ∫ ∞ 0 ∫ T 1 ( ( R [ µ ] ̂ f )( s, y ) -( R [ µ ] ̂ f )( s, x ) ) 1 α i φ ( x -y α i ) 1 α j φ ( s -t α j ) dy ds ∥ ∥ ∥ ∥ H q x ( T 1 ) + ∥ ∫ ∞ 0 ∫ T 1 ( ( R [ µ ] f )( s, x ) -( R [ µ ] f )( t, x ) ) 1 α i φ ( x -y α i ) 1 α j φ ( s -t α j ) dy ds ∥ q 1 . ∥ ∥ ∥ ̂ ∥ ∥ ∥ H x ( T ) Writing the first term on the right hand side of Eq. (A.3) as I , we estimate I ≤ ∫ ∞ 0 ∥ ∥ ∥ ∥ ∫ T 1 ( ( R [ µ ] ̂ f )( s, y ) -( R [ µ ] ̂ f )( s, x ) ) 1 α i φ ( x -y α i ) dy ∥ ∥ ∥ ∥ H q x ( T 1 ) 1 α j φ ( s -t α j ) ds. ∥ ∥ ∥ ∥ ∫ T 1 ( ( R [ µ ] ̂ f )( s, y ) -( R [ µ ] ̂ f )( s, x ) ) 1 α i φ ( x -y α i ) dy ∥ ∥ ∥ ∥ H q x ( T 1 ) ≤ g i ( s ) , Now, it is a standard result for mollifiers that for every s ∈ (0 , ∞ ) where lim i →∞ g i ( s ) = 0 at every s , and for every integer i , the function g is continuous. Since R [ µ ] f is uniformly continuous, this function g i extends to the interval [0 , ∞ ) with the same properties. Since R [ µ ]( t ) ̂ f ( t ) = R [ µ ]( δ ) ̂ f ( δ ) for all t ≥ δ , it follows that there is a sequence ( ̂ g i ) with limit 0, such that g i ( s ) ≤ ̂ g i for all s ∈ [0 , ∞ ). Consequently, I can be estimated by a sequence ( a i ), which (i) is independent of j , (ii) is independent of t , and (iii) goes to zero in the limit i →∞ . We now discuss the second term of the right hand side of Eq. (A.3), which we label as II . The integral over y is trivial, so consequently II ≤ ∫ ∞ 0 ∥ ∥ ∥ ( R [ µ ] ̂ f )( s, x ) -( R [ µ ] f )( t, x ) ∥ ∥ ∥ H q x ( T 1 ) 1 α j φ ( s -t α j ) ds. The term involving the H q -norm is a uniformly continuous function in s and t . Hence, from Lebesgue's Differentiation Theorem and the definition of ̂ f , it follows that the s -integral converges to 0 for j →∞ , independently of t ∈ (0 , δ ] and i . This completes the proof of the fourth property. We can now use Lemma A.1 to relate the spaces X δ,µ,q and ̂ X δ,µ,q . Lemma A.2. Fix a constant δ > 0 , an exponent vector µ , and a non-negative integer q ; then for all /epsilon1 > 0 , ̂ X δ,µ + /epsilon1,q ⊂ X δ,µ,q ⊂ ̂ X δ,µ,q . Proof. The inclusion X δ,µ,q ⊂ ̂ X δ,µ,q follows easily from the fact that each element in X δ,µ,q is the limit of a Cauchy sequence in ( C ∞ ((0 , δ ] × T 1 ) , ‖·‖ δ,µ,q ), whose elements are in particular bounded continuous maps (0 , δ ] → H q ( T 1 ), and the convergence is uniform in time. To check the inclusion ̂ X δ,µ + /epsilon1,q ⊂ X δ,µ,q , let a function f be given in ̂ X δ,µ + /epsilon1,q . Hence f satisfies the condition of the previous lemma, in particular that of Condition (iv). It follows that f ∈ X δ,µ,q . We also wish to comment on time derivatives of functions in X δ,µ,q and ̂ X δ,µ,q . Let f ∈ ̂ X δ,µ,q . We say that f is differentiable in time t if the (bounded continuous) map R [ µ ] f : (0 , δ ] → H q ( T 1 ) is differentiable in the sense of a map between Banach spaces (Frechet derivatives). Its time derivative (multiplied by t ) D ( R [ µ ] f ) can then be considered to be a map (0 , δ ] → H q ( T 1 ), and we set Df := R [ µ ] -1 ( D ( R [ µ ] f ) -D R [ µ ] f ). If this map is continuous, then we call f continuously differentiable in t . If this is the case for f and if in addition R [ µ ] Df is bounded, then we have Df ∈ X δ,µ,q . ̂ Now, let f ∈ ̂ X δ,µ,q be continuously differentiable. Then Df is the distributional time derivative of f in the following sense. Let φ be any test function with the properties as in Section 2.3.2. Choose /epsilon1 > 0. Then we clearly have that ∫ δ /epsilon1 ∂ t ( t 〈R [ µ ] f, φ 〉 L 2 ( T 1 ) ) dt = -/epsilon1 〈R [ µ ] f, φ 〉 L 2 ( T 1 ) ∣ ∣ ∣ t = /epsilon1 . Hence, the boundary term vanishes in the limit /epsilon1 → 0. The following integrals are meaningful for /epsilon1 = 0, and hence we obtain ∫ δ 0 〈R [ µ ] Df,φ 〉 L 2 ( T 1 ) dt = -∫ δ 0 ( 〈R [ µ ] f, Dφ 〉 L 2 ( T 1 ) + 〈R [ µ ] f + D R [ µ ] f, φ 〉 L 2 ( T 1 ) ) dt. (A.4) The reader should compare this with the expressions for weak solutions in Section 2.3.2. B On products of functions We readily check the following results which are useful in dealing with products of functions and their relationship to the function spaces X δ,µ,q . Lemma B.1. Let f ∈ X δ,µ 1 ,q and g ∈ X δ,µ 2 ,q be two functions (0 , δ ] × T 1 → R , for some constant δ > 0 , some smooth exponents µ 1 and µ 2 , and an integer q ≥ 1 . Then f · g is in X δ,µ 1 + µ 2 ,q and, for some constant C > 0 , ‖ f · g ‖ δ,µ 1 + µ 2 ,q ≤ C ‖ f ‖ δ,µ 1 ,q · ‖ g ‖ δ,µ 2 ,q . Observe that the condition q ≥ 1 (for one spatial dimension) is essential here. Proof. An essential part of the proof of this lemma is the general estimate ‖ f · g ‖ H q ≤ C ( ‖ f ‖ H q ‖ g ‖ L ∞ + ‖ g ‖ H q ‖ f ‖ L ∞ ) , for arbitrary functions f and g in H q ∩ L ∞ ; see Proposition 3.7 in Chapter 13 of [40]. The Sobolev inequalities for q ≥ 1 in one spatial dimension then imply ‖ f · g ‖ H q ≤ C ( ‖ f ‖ H q ‖ g ‖ H q ) . Working with this inequality, we see that the lemma follows easily if we choose a sequence ( f i ) which converges to f in the function space X δ,µ 1 ,q , and a sequence ( g i ) which converges to g in X δ,µ 2 ,q , and then write f i · g i -f · g = f i · ( g i -g ) + g · ( f i -f ) . Another important result is the following. Lemma B.2. Let w be a d -vector-valued function in X δ,µ,q for some exponent d -vector µ , a constant δ > 0 , and an integer q ≥ 1 . Let S be a d × d -matrix-valued function so that R [ µ ] · S · R [ -µ ] is an element of X δ,ξ,q for an exponent d × d -matrix ξ of the form ξ ij = ζ i where ζ is an exponent d -vector. Then, the d -vector-valued function S w is in X δ,ζ + µ,q and ‖ S w ‖ δ,ζ + µ,q ≤ C ‖R [ µ ] · S · R [ -µ ] ‖ δ,ξ,q ‖ w ‖ δ,µ,q , for some constant C > 0 . This lemma is proved essentially in the same way as Lemma B.1. Lemma B.3. Suppose that δ > 0 , s > 0 and r > 0 are constants, n , d and q integers with d ≥ 1 and q ≥ 1 , µ an exponent d -vector, and ν 1 and ν 2 exponent scalars. Let functions g 1 , g 2 : U → R be given where U is an open subset of R d . Suppose that g 1 maps all functions w : (0 , δ ] × T 1 → R d in B δ,µ,q,s to elements g 1 ( w ) in B δ,ν 1 ,q,r . Moreover suppose that there is a constant C 1 > 0 with ‖ g 1 [ w 1 ] -g 1 [ w 2 ] ‖ δ,ν 1 ,q ≤ C 1 ‖ w 1 -w 2 ‖ δ,µ,q , for all w 1 , w 2 : (0 , δ ] × T 1 → R d in B δ,µ,q,s . Let us also assume that g 2 maps all functions w : (0 , δ ] × T 1 → R d in B δ,µ,q,s to elements g 2 ( w ) in B δ,ν 2 ,q,r and that there is a constant C 2 > 0 with ‖ g 2 [ w 1 ] -g 2 [ w 2 ] ‖ δ,ν 2 ,q ≤ C 2 ‖ w 1 -w 2 ‖ δ,µ,q , for all w 1 , w 2 : (0 , δ ] × T 1 → R d in B δ,µ,q,s . Now, consider h := g 1 · g 2 , w ↦→ h ( w ) . Then, there exists a ρ > 0 (which is smaller the smaller r is) so that h maps all functions w : (0 , δ ] × T 1 → R d in B δ,µ,q,s to elements h ( w ) in B δ,ν 1 + ν 2 ,q,ρ . Moreover, there exists a constant C > 0 with ‖ h [ w 1 ] -h [ w 2 ] ‖ δ,ν 1 + ν 2 ,q ≤ C ‖ w 1 -w 2 ‖ δ,µ,q , for all w 1 , w 2 : (0 , δ ] × T 1 → R d in B δ,µ,q,s . Proof. If w ∈ B δ,µ,q,s , then g 1 ( w ) ∈ B δ,ν 1 ,q,r and g 2 ( w ) ∈ B δ,ν 2 ,q,r . Lemma B.1 implies that h ( w ) = g 1 ( w ) g 2 ( w ) ∈ X δ,ν 1 + ν 2 ,q and ‖ h ( w ) ‖ δ,ν 1 + ν 2 ,q ≤ C ‖ g 1 ( w ) ‖ δ,ν 1 ,q ‖ g 2 ( w ) ‖ δ,ν 2 ,q ≤ Cr 2 , where C > 0 is the constant in Lemma B.1. This allows us to set ρ = Cr 2 and hence establishes that h ( w ) ∈ B δ,ν 1 + ν 2 ,q,ρ . Regarding the Lipschitz estimate, we find ‖ t -( ν 1 + ν 2 ) ( h [ w 1 ]( t ) -h [ w 2 ]( t )) ‖ H q ≤ C ‖ t -ν 1 ( g 1 [ w 1 ]( t ) -g 1 [ w 2 ]( t )) ‖ H q ‖ t -ν 2 g 2 [ w 1 ]( t ) ‖ H q + C ‖ t -ν 1 g 1 [ w 2 ]( t ) ‖ H q ‖ t -ν 2 ( g 2 [ w 1 ]( t ) -g 2 [ w 2 ]( t )) ‖ H q . (B.1) Then we can use the individual Lipschitz estimates for g 1 and g 2 in order to establish this result. While Lemma B.3 is adequate for the proof of Theorem 2.4, to prove Theorem 2.21 we require a stronger result, which we present here. Lemma B.4. Suppose that q ≥ 1 . Let g 1 and g 2 be functions satisfying all the conditions of Lemma B.3 with exponents ν 1 , ν 2 for all x ∈ T 1 . Suppose that, in addition, one has the following: For all w ∈ B δ,µ,q,s/ 2 with ω ∈ B δ, ̂ µ,q,s/ 2 for some exponent vector ̂ µ which satisfies ̂ µ ≥ µ , there exist scalar exponents γ 1 , γ 2 , independent of ̂ µ , such that g 1 ( w + ω ) -g 1 ( w ) ∈ X δ, ̂ µ + γ 1 ,q , g 2 ( w + ω ) -g 2 ( w ) ∈ X δ, ̂ µ + γ 2 ,q , and ‖ g 1 [ w + ω ] -g 1 [ w ] ‖ δ, ̂ µ + γ 1 ,q ≤ ̂ C 1 ‖ ω ‖ δ, ̂ µ,q , ‖ g 2 [ w + ω ] -g 2 [ w ] ‖ δ, ̂ µ + γ 2 ,q ≤ ̂ C 2 ‖ ω ‖ δ, ̂ µ,q , h ( w + ω ) -h ( w ) ∈ X δ, ̂ µ + γ,q , ‖ h [ w + ω ] -h [ w ] ‖ δ, ̂ µ + γ,q ≤ ̂ C ‖ ω ‖ δ, ̂ µ,q This follows from a more detailed analysis of Eq. (B.1). To handle the exponential function, we rely on the following result. for constants ̂ C 1 , ̂ C 2 > 0 . Then the function h = g 1 · g 2 has the following property. We can choose a scalar exponent γ smaller or equal than min { ν 1 + γ 2 , ν 2 + γ 1 } (independently of ̂ µ ), such that for all w ∈ B δ,µ,q,s/ 2 and ω ∈ B δ, ̂ µ,q,s/ 2 , one has and for a constant ̂ C > 0 . Lemma B.5. Pick constants δ > 0 , s > 0 , an integer q ≥ 1 , and an exponent µ > 0 . Let g ( i ) := exp · Π i , where Π i : R d → R is the projection to the i th component of d -vectors. Then, for every function w : (0 , δ ] × T 1 → R in B δ,µ,q,s , there exists an r > 0 , so that the composed function g ( i ) · w : (0 , δ ] × T 1 → R is in B δ, 0 ,q,r . Moreover, for all w 1 , w 2 ∈ B δ,µ,q,s , there exists a constant C > 0 , so that ‖ g ( i ) ( w 1 ) -g ( i ) ( w 2 ) ‖ δ, 0 ,q ≤ C ‖ w 1 -w 2 ‖ δ,µ,q . In addition, for every scalar exponent ˆ µ ≥ µ and every w ∈ B δ,µ,q,s/ 2 and ω ∈ B δ, ˆ µ,q,s/ 2 , it follows that g ( i ) ( w + ω ) -g ( i ) ( w ) is in X δ, ˆ µ,q and the estimate ‖ g ( i ) ( w + ω ) -g ( i ) ( w ) ‖ δ, ˆ µ,q ≤ C ‖ ω ‖ δ, ˆ µ,q , holds. Proof. This follows from Proposition 3.9 in Chapter 13 of [40] applied to g ( i ) ( w ) -1, together with the Taylor theorem for the exponential function. C Duality and convergence results Sobolev spaces and duality Following [16, Chapter VI] or [36], one defines the Sobolev space H s ( R n ) for any s ∈ R as the set of temperate distributions u such that ̂ u (1 + | ξ | 2 ) s/ 2 ∈ L 2 ( R n ), where ̂ u := F u is the Fourier transform (in the sense of temperate distributions) of u . The norm defined by ‖ u ‖ s := ‖ ̂ u ( ξ )(1 + | ξ | 2 ) s/ 2 ‖ L 2 ξ ( R n ) 75 turns this space into a Banach space. If s = q for any non-negative integer q , then H s ( R n ) is equivalent to the standard ( p = 2) Sobolev space H q ( R n ). For general s ∈ R , the space H s ( R n ) is in fact a Hilbert space for the scalar product 〈 u, v 〉 s := ∫ R n ̂ u ( ξ )(1 + | ξ | 2 ) s/ 2 ̂ v ( ξ )(1 + | ξ | 2 ) s/ 2 dξ. Let u ∈ H -s ( R n ) and v ∈ H s ( R n ) for any s ∈ R . Then the dual pairing between H s ( R n ) and H -s ( R n ), is well-defined, as a consequence of the inequality | ( u, v ) | ≤ ∣ ∣ ∣ ∣ ∫ R n ̂ u ( ξ )(1 + | ξ | 2 ) -s/ 2 ̂ v ( ξ )(1 + | ξ | 2 ) s/ 2 dξ ∣ ∣ ∣ ∣ ≤ ‖ u ‖ -s ‖ v ‖ s . (C.2) ( u, v ) := ∫ R n ̂ u ( ξ ) ̂ v ( ξ ) dξ, (C.1) By means of this pairing, we can identify H -s ( R n ) with H s ( R n ) ∗ (the dual space) as follows. For every u ∈ H -s ( R n ), the map ( u, · ) : H s ( R n ) → R is a bounded linear functional, i.e., an element of H s ( R n ) ∗ . Conversely, according to the Riesz representation theorem, there exists a unique element w φ ∈ H s ( R n ) for each element φ ∈ H s ( R n ) ∗ such that φ ( v ) = 〈 w φ , v 〉 s 〈 w φ , v 〉 s = ∫ R n ̂ w φ ( ξ )(1 + | ξ | 2 ) s/ 2 ̂ v ( ξ )(1 + | ξ | 2 ) s/ 2 dξ = ∫ R n ̂ v φ ( ξ ) ̂ v ( ξ ) dξ, for all v ∈ H s ( R n ) . The last expression can be written as where ̂ v φ := ̂ w φ ( ξ )(1+ | ξ | 2 ) s is the Fourier transform of v φ := F -1 ( ̂ w φ ( ξ )(1+ | ξ | 2 ) s ). We have v φ ∈ H -s ( R n ), since ̂ v φ (1+ | ξ | 2 ) -s/ 2 = ̂ w φ ( ξ )(1+ | ξ | 2 ) s/ 2 ∈ L 2 ( U ). By means of the pairing above, we have thus constructed a unique element v φ ∈ H -s ( R n ) corresponding to each φ ∈ H s ( R n ) ∗ . In this sense, we can therefore identify H -s ( R n ) with H s ( R n ) ∗ for every s ∈ R . The following result concerns the relationship between Sobolev spaces of different indices. Proposition C.1. For every s ∈ R and σ ≥ 0 , the space H s + σ ( R n ) is a dense subset of H s ( R n ) . Proof. We first show that H s + σ ( R n ) is indeed a subset of H s ( R n ) for σ ≥ 0. Suppose that u ∈ H s + σ ( R n ). Calculating the ‖ · ‖ s norm of u , we obtain from which it follows that u ∈ H s ( R n ). To check that H s + σ ( R n ) is a dense subset, it is sufficient to note (see, e.g., [16]) that C ∞ 0 ( R n ) (the space of smooth functions with compact support) is dense in both H s ( R n ) and H s + σ ( R n ). ‖ u ‖ 2 s = ∫ R n | ̂ u ( ξ ) | 2 (1 + | ξ | 2 ) s dξ ≤ ∫ R n | ̂ u ( ξ ) | 2 (1 + | ξ | 2 ) s + σ dξ = ‖ u ‖ 2 s + σ < ∞ , Convergence results in Sobolev spaces One can use this dense inclusion property (Proposition C.2) together with the duality properties discussed above to derive certain convergence and closedness-type results for sequences in Sobolev spaces. We first discuss a result of this sort for Sobolev spaces on R n , and then do the same for Sobolev spaces on T 1 . Proposition C.2. Choose s, s 0 ∈ R so that 0 ≤ s 0 < s . Let ( w m ) be a bounded sequence in H s ( R n ) in the sense that there exists a constant C > 0 so that ‖ w m ‖ s ≤ C , for all integer m . Moreover, suppose that ( w m ) converges to some w ∈ H s 0 ( R n ) ; i.e., ‖ w m -w ‖ s 0 → 0 . Then, w is contained in H s ( R n ) . Proof. The boundedness of the sequence implies the existence of a subsequence of ( w m ) (which for simplicity we identify with ( w m )) which converges weakly. Hence, as a consequence of the Riesz Representation Theorem and the above dual pairing in Eq. (C.1), there exists an element ˜ w ∈ H s ( R n ), so that, for every Y ∈ H -s ( R n ), ( Y, ˜ w -w m ) → 0 (C.3) We wish to show that w = ˜ w and hence that w ∈ H s ( R n ). To do this, we consider an arbitrary X ∈ H -s 0 ( R n ) and the dual pairing | ( X, ˜ w -w ) | ≤ | ( X, ˜ w -w m ) | + | ( X,w -w m ) | , where ˜ w -w is considered as an element of H -s 0 ( R n ), and where we have used the triangle inequality. Since X ∈ H -s 0 ( R n ) ⊂ H -s ( R n ) according to Proposition C.1, we can consider the first term on the right hand side as a pairing between H s ( R n ) and H -s ( R n ), and hence Eq. (C.3) implies that this term can be made arbitrarily small by choosing m sufficiently large. The second term is considered as a pairing between H s 0 ( R n ) and H -s 0 ( R n ) so that Eq. (C.2) yields | ( X,w -w m ) | ≤ ‖ X ‖ -s 0 ‖ w -w m ‖ s 0 . Also this term can be made arbitrarily small by choosing m sufficiently large. Hence, we have found that ( X, ˜ w -w ) = 0 for all X ∈ H -s 0 ( R n ). Now, the Riesz representation theorem implies that for every X ∈ H -s 0 ( R n ) there exists precisely one ˜ X ∈ H s 0 ( R n ) for which In particular, we may choose ˜ X = ˜ w -w , which implies that ˜ w -w = 0. Corollary C.3. Choose non-negative integers q and q 0 so that q 0 < q . Let ( w m ) be a bounded sequence in H q ( T 1 ) , in the sense that there exists a constant C > 0 so that ‖ w m ‖ H q ( T 1 ) ≤ C , for all integers m . Moreover, suppose that ( w m ) converges to some w ∈ H q 0 ( T 1 ) ; i.e., ‖ w m -w ‖ H q 0 ( T 1 ) → 0 . Then, w is contained in H q ( T 1 ) . 0 = ( X, ˜ w -w ) = 〈 ˜ X, ˜ w -w 〉 H s 0 ( R n ) . Proof. We formulate the proof so that it can be easily generalized to general smooth orientable, connected compact Riemannian manifolds M in any dimension n . For this paper, the relevant special case is M = T 1 . Let (( U i , φ i )) be a collection of coordinate charts, i.e., open subsets U i ⊂ M and homeomorphisms φ i : V i → U i where V i := φ -1 i ( U i ) are open subset of R n , which cover M , i.e., M = ⋃ i U i . It follows from compactness that we can assume that there are N such coordinate charts. Let ( τ i ) be a subordinate partition of unity. Then we find that ( w m ) is a bounded sequence in H q ( M ) if and only if for all i = 1 , . . . , N , we have that ( w m · φ i ) is a bounded sequence in H q ( V i ). Moreover, ‖ w m -w ‖ H q 0 ( T 1 ) → 0 for some w ∈ H q 0 ( M ) if and only if for all i = 1 , . . . , N , we have that ‖ w m · φ i -w · φ i ‖ H q 0 ( V i ) → 0 (since w · φ i ∈ H q 0 ( V i )). Now, the Stein Extension Theorem (Theorem 5.24 in [1]) implies the existence of total extension operators E i (Definition 5.17 in [1]), which are linear maps E i from functions defined on V i to functions defined on R n with the following property: If f ∈ H r ( V i ) for any non-negative integer r , then 1. ( E i f ) | V i = f almost everywhere, 2. E i f is in H r ( R n ), and there exists a constant C > 0, so that ‖ E i f ‖ H r ( R n ) ≤ C ‖ f ‖ H r ( V i ) . Hence, we find that ( w m ) is a bounded sequence in H q ( M ) if and only if for all i = 1 , . . . , N , we have that ( E i ( w m · φ i )) is a bounded sequence in H q ( R n ). Moreover, ‖ w m -w ‖ H q 0 ( T 1 ) → 0 for some w ∈ H q 0 ( M ) if and only if for all i = 1 , . . . , N , we have that ‖ E i ( w m · φ i ) -E i ( w · φ i ) ‖ H q 0 ( R n ) → 0 (since E i ( w · φ i ) ∈ H q 0 ( R n )). It follows from Proposition C.2, that E i ( w · φ i ) ∈ H q ( R n ). Hence, w · φ i ∈ H q ( V i ). Since this is true for all i = 1 , . . . , N , it follows that w ∈ H q ( M ). In carrying out this calculation (with γ being the quantity hypothesized in Lemma 2.20), we note that the operator H ODE ( u 0 )[ · ] is well-defined here according to Lemma 2.16 and Lemma 2.20 since w m -w m -1 ∈ B δ,µ ( m ) ,q -( m -1) ,s/ 2 with µ ( m ) ≥ µ . Finally we note that the constants C and ρ may depend in particular on q and m , but this dependence is not a problem for carrying out our argument since we are only interested in finitely many sequence elements. To complete the induction argument, we verify that since we have assumed (as part of the induction) that w m -w m -1 ∈ B δ,µ ( m ) ,q -( m -1) ,s/ 2 , it follows that so long as ν ( m +1) -γ < µ ( m ) holds, we have the final right hand side of the above inequality finite. Therefore the initial left hand side must be finite, and this holds for any µ ( m +1) , so long as µ ( m +1) < µ ( m ) + γ . We satisfy these conditions by choosing µ ( m +1) = µ + mκγ for any κ < 1. Noting that this is the case for all m , with κ chosen independently of m , we conclude that Eq. (2.54) holds, after having identified κγ with γ to simplify the notation. It remains to verify that Eq. (2.55) holds for the residuals of the sequence ( u n ). Using Eqs. (2.44), (2.46) and (2.51) we calculate Since w n -w n -1 ∈ B δ,µ +( n -1) γ,q -n +1 ,s/ 2 , it follows from Lemma 2.20 that", "pages": [ 37, 38, 39, 40, 41, 42, 43 ] }, { "title": "2.4.3 (Order n)-singular initial value problem", "content": "Proposition 2.18 shows that, so long as we can find an ODE-leading-order term u 0 and so long as certain conditions hold, the difference u n +1 -u n behaves like a power of t near t = 0, with this power increasing monotonically with n . It is hence meaningful to consider, in addition to the ODE-singular initial value problem with leading-order term u 0 , a sequence of (order n)-singular initial value problems which use (order n)-leadingorder terms u n ( n ≤ q -2). In view of the relationship between u 0 and the sequence ( u n ), we may write the same solution u of a given singular initial value problem either in the form u = u 0 + w for a remainder w in X δ,µ,q , or, as u = u n + ω for a remainder ω in X δ, ̂ µ,q with ̂ µ increasing suitably with n . The same can be done for any of the u m ( m ≤ n ) in the (order n)-leading-order term sequence. We now use the (order n)-leading-order terms to argue that, at least for the smooth case ( q = ∞ ), if the conditions of Proposition 2.18 are met, the ODE-singular initial value problem, and correspondingly the (order n)-singular initial value problem have solutions. Theorem 2.21 (Existence and uniqueness for the ODE-singular initial value problem) . Suppose that a quasilinear symmetric hyperbolic Fuchsian system with ODE-leadingorder term u 0 has been chosen which satisfies the hypotheses of Proposition 2.18 for all finite values of (differentiability index) q . Then, for some sufficiently small δ 1 ∈ (0 , δ ] and for a sufficiently large n , there exists a unique solution u of Eq. (2.1) with u -u n , D ( u -u n ) ∈ X δ 1 ,µ + nγ, ∞ , where u n is the (order n)-leading-order term defined in Definition 2.17 for this system. This solution u is also the only solution of the ODEsingular initial value problem with u -u 0 ∈ X δ 1 ,µ, ∞ . This result states conditions which are sufficient for the ODE-singular initial value problem (with leading-order term u 0 ) to admit (unique) solutions. In doing so, Theorem 2.21 provides a potentially very useful alternative to Theorem 2.4 of Section 2.2. Observe in particular that the hypothesis for Theorem 2.21 does not require that the energy dissipation matrix be positive definite with respect to µ . Here we state and prove Theorem 2.21 only for the infinite differentiability case ( q = ∞ ). This smoothness restriction plays a role in the proof, since it allows one to always choose n large enough so that Condition (i) of Theorem 2.4 for the singular initial value problem with leading-order term u n is satisfied. If one tries to prove a result like Theorem 2.21 for finite differentiability order, then there is an upper bound for the possible choice of n , and consequently one may not be able to choose it large enough to satisfy Condition (i). However, in certain circumstances, a large but finite order of differentiability is in fact sufficient to carry through the proof. Proof. The basic idea of the proof is to reformulate the system using u n for the leadingorder term in place of u 0 , and then verify that the hypothesis of Theorem 2.4 (in the case q = ∞ ) is satisfied if n is chosen sufficiently large. To carry this through, we first argue that the system Eq. (2.1), which for the ODE-singular initial value problem can be written as can also be written as 0 = ̂ L ( u n + ω )[ ω ] -F ( u n )[ ω ] , (2.58) where we recall the definition Eq. (2.5) of the principal part operator ̂ L and the definition Eq. (2.7) for the operator F ( u n )[ · ]. Here we use w for the remainder term corresponding to u 0 and we use ω for the remainder term corresponding to u n (hence u 0 + w = u n + ω ). To show this equivalence, we note the relations ̂ L ( u n + ω )[ v ] = ̂ L ( u 0 + w )[ v ] and F ( u n )[ ω ] = F ( u 0 )[ w ], and then using these we calculate We now choose a sequence of exponent vectors µ ( n ) which satisfy ˜ and which are consistent with the block diagonal condition for Eq. (2.58); we note that this is possible for all sufficiently large integers n . Examining the singular initial value problem corresponding to Eq. (2.58), we verify that for any given sufficiently large integer n , this PDE system, together with u n as leading order term and exponent vector ˜ µ ( n ) , satisfies the conditions to be a quasilinear symmetric hyperbolic Fuchsian system. We also verify, based on Eq. (2.6), that for sufficiently large n (and therefore sufficiently large µ +( n -1) γ ) the exponent vectors ˜ µ ( n ) can be chosen large enough to guarantee that the energy dissipation matrix M 0 is positive definite. Consequently, this system satisfies Condition (i) of the hypothesis of Theorem 2.4. To check that Conditions (ii) and (iii) of Theorem 2.4 are also satisfied, we examine the operator F ( u n ). Using Eqs. (2.43) and (2.47) together with Definition 2.17, we calculate Combining the assumptions for S 1 , S 2 and N which are stated in the hypothesis of Proposition 2.18 (and therefore included in the hypothesis of Theorem 2.21) with By comparing the first line of this expression with Eq. (2.56), we notice that all spatial derivative terms cancel; hence there is no loss of regularity in this expression as is the case for the operator F ODE ( u 0 ) itself. We therefore get estimates analogous to those in Lemma 2.20, with q -1 replaced by q . the upper bound stated in Eq. (2.59), we readily show that all of the conditions of Theorem 2.4 hold for q = ∞ . Then the consequent application of Theorem 2.4 shows that so long as n is sufficiently large, there exists exactly one solution u = u n + ω with ω ∈ X δ 1 , ˜ µ ( n ) , . ∞ We wish to show next that for such a fixed chosen value of n , in fact ω ∈ X δ 1 ,µ + nγ, ∞ . To show this, we consider an integer n + which is large enough so that ˜ µ ( n + ) > µ +( n -1) γ . Applying the same argument as above, but now with n + instead of n (and hence using u n + as the leading-order term), we obtain a solution ˜ u = u n + + ˜ ω which has the property that ˜ ω ∈ X δ 1 , ˜ µ ( n + ) , ∞ . Uniqueness of the singular initial value problem with respect to u n implies that u equals u . Moreover, we have ˜ Given that w n +1 -w n ∈ X δ 1 ,µ + nγ, ∞ , we obtain the desired result To conclude the proof of this theorem, we must show that any solution ̂ u of the form ̂ u = u 0 + ̂ w with ̂ w ∈ X δ 1 ,µ, ∞ must equal the solution u . To show this, it is useful to write ̂ u = u n + ̂ w -w n , where u n is defined by Eq. (2.52) and w n is defined by Eq. (2.51). Then if we can verify that ̂ w -w n ∈ X δ 1 , ˜ µ ( n ) , ∞ , it follows from uniqueness that ω = ̂ w -w n , and hence that ̂ u = u . We make this verification by using induction to show that, in fact, ̂ w -w m ∈ X δ 1 ,µ + mγ, ∞ holds for every non-negative integer m . In the case m = 0, we have ̂ w -w 0 = ̂ w ∈ X δ 1 ,µ, ∞ which implies the claim for m = 0. Suppose the claim has been shown for m = m 0 ≥ 1. We know that w is a solution of the equation while w m 0 +1 is a solution of Taking the difference, we obtain We can write this formally as ̂ w -w m 0 +1 = H ODE ( u 0 )[ F ODE ( u 0 )[ ̂ w ] -F ODE ( u 0 )[ w m 0 ]] . Now, the fact that w -w m 0 ∈ X δ 1 ,µ + m 0 γ, ∞ implies that F ODE ( u 0 )[ w ] -F ODE ( u 0 )[ w m 0 ] ∈ X δ 1 ,µ +( m 0 +1) γ, ∞ (Lemma 2.20). Consequently (see Lemma 2.16), the operator H ODE ( u 0 )[ · ] is well-defined. This completes the proof.", "pages": [ 44, 45, 46 ] }, { "title": "2.4.4 An example: the Euler-Poisson-Darboux equation", "content": "We consider now the example of the Euler-Poisson-Darboux equation (see also [2] for another example) Here, u ( t, x ) is the unknown (assumed to be a scalar function), and f 0 ( t, x ) is a specified scalar function. The Euler-Poisson-Darboux equation is second order, and in previous work by two of the authors [13] on semilinear second-order Fuchsian systems, it has been shown that this equation admits unique solutions to the singular initial value problem with leading-order term (for arbitrary functions u ∗ and u ∗∗ ) so long as f 0 = O ( t ̂ ν ) with ̂ ν > 0. We seek to show that we obtain these same results using the first-order methods which we have developed here. In particular, this example demonstrates the usefulness of the techniques discussed in Section 2.4, thereby serving as a linear warmup example with which we can explore some of the issues which arise below in our discussion of the application of these methods to the fully nonlinear T 2 -symmetric Einstein's vacuum equations in Section 3. To apply the first-order theory developed in this paper, we first convert this equation into a first-order system by setting Eq. (2.60) then takes the form of a first-order evolution system with plus a constraint equation Observe that in working with the Euler-Poisson-Darboux system in this first-order form, one first treats the components u 1 , u 2 , and u 3 as independent functions whose evolution is determined by Eq. (2.63). This means that we solve the singular initial value problem of this system with respect to a leading-order term motivated by Eq. (2.61). Then, in a second step, we identify u 1 with the original variable u and consider the two remaining relations Eq. (2.62) as constraints: the one involving the time derivative is automatically implied by the first of Eqs. (2.63) (the evolution equation for u 1 ), while the one involving the spatial derivative gives rise to the condition ∆ u ≡ 0 in Eq. (2.64). Let us start with the first step. One readily verifies that this evolution system is of (quasilinear) symmetric hyperbolic Fuchsian form for any choice of leading-order term, and hence our theory can, in principle, be applied. Our approach is to find a leading-order term for the first-order variables which is consistent with Eq. (2.61) and which, in addition, is an ODE-leadingorder term. We easily determine that the general solution to Eq. (2.46) for Eq. (2.63) takes the form for the spatially-dependent parameters C 1 ( x ), C 2 ( x ) and C 3 ( x ). However, we see that this leading-order term can only be consistent with Eqs. (2.61) and (2.62) in the special case u ∗ = 0 and C 2 = 0. Hence, this approach for finding a leading-order term fails. We circumvent this problem as follows. For a specified function u ∗ (which is at least second order differentiable; we specify its necessary regularity more precisely below), we define ̂ u := u -u ∗ ( x ) log t, (2.66) and work with the evolution equation for ̂ u rather than that for u . Substituting Eq. (2.66) into Eq. (2.60), we obtain D 2 ̂ u -t 2 ̂ u xx = t 2 log tu '' ∗ + f 0 ( t, x ) , where u '' ∗ indicates the second derivative of u ∗ . Now, setting we obtain the evolution equation for the same matrices S 1 , S 2 , N as above, but with ∆ ̂ u := ̂ u 3 /t -∂ x ̂ u 1 = 0 . (2.68) Choosing the ODE-leading-order term for the ̂ U formulation to be of the same form as Eq. (2.65), we have ̂ U 0 = ( C 1 + C 2 log t, C 2 , C 3 t ) T , but now (in view of Eq. (2.61)) we are led to choose the parameter functions in the form C 1 = u ∗∗ , C 2 = 0 and C 3 = u ' ∗∗ ; hence In terms of ̂ u , the constraint Eq. (2.64) takes the form The function u ∗ appearing in Eq. (2.66) together with the function u ∗∗ introduced here together comprise the full range of free data suggested by Eq. (2.61). Both play the role of asymptotic data functions. Having found a suitable representation of the equations and the leading-order term, we write the unknown ̂ U of the evolution system as ̂ U = ̂ U 0 + W , and look for sufficient conditions for the existence of solutions to the singular initial value problem in this form, with W as a remainder term. To enforce the remainder falloff properties, we choose an exponent vector µ = ( µ 1 , µ 2 , µ 3 ) and, in view of Eq. (2.69), we require that µ 1 , µ 2 > 0 and µ 3 > 1. We first seek to prove existence of solutions using Theorem 2.4. To satisfy the block diagonality condition of Theorem 2.4 we must set µ 1 = µ 2 = µ 3 . We therefore simplify the notation by writing the exponent vector as ( µ, µ, µ ) for some smooth scalar function µ which, from above considerations, must be greater than one. Observe here that, while this equality of all components of the exponent vector is necessary to satisfy the hypothesis of Theorem 2.4, it does appear to be an artificial restriction. Under reasonable regularity assumptions, we might rather expect that if the first and second components are O ( t µ ), then the third component of W should be O ( t µ +1 log t ); the log t factor may arise from derivatives of t µ since µ is generally not constant. In any case, we readily verify that the energy dissipation matrix is positive definite so long as µ > 1. Calculating Given any such solution of the first-order evolution system, our next step is to identify ̂ u 1 with u -log( t ) u ∗ and then, if the remaining constraint ∆ ̂ u ≡ 0 is satisfied, to conclude that u is actually a solution of the original second-order equation Eq. (2.60) with leadingorder term u 0 = u ∗ log t + u ∗∗ and with remainder w = w 1 (the first component of the vector W ) in X δ 1 ,µ,q . To determine if the constraint is satisfied, we use the evolution equation Eq. (2.67) to calculate the time derivative of the constraint violation term ∆ ̂ u , obtaining we now suppose that W ∈ X δ, ( µ,µ,µ ) ,q and f 0 ∈ X δ, ̂ ν,q for ̂ ν > 1. Then F ( ̂ U 0 )[ W ] ∈ X δ, ( ν,ν,ν ) ,q if u ∗ , u ∗∗ ∈ H q +2 ( T 1 ), where ν = ̂ ν , if ̂ ν < 2, or, we have ν < 2, if ̂ ν ≥ 2. Choosing q ≥ 3, we verify that Theorem 2.4 implies the existence of solutions of the evolution system ̂ U = ̂ U 0 + W with W ∈ X δ 1 , ( µ,µ,µ ) ,q for δ 1 sufficiently small 1 and for an exponent µ ∈ (1 , min { 2 , ̂ ν } ). For any specified set of the asymptotic data u ∗ and u ∗∗ , we find that the solution is unique for remainders in the space X δ 1 , ( µ,µ,µ ) ,q . We then note that (i) if we construct ∆ ̂ u using ̂ U 0 from Eq. (2.69), we get ∆ ̂ U 0 = 0; and (ii) if we combine the evolution equation Eq. (2.70) with the leading order term ∆ ̂ U 0 as well as q ≥ 3 and other appropriate choices of µ , etc., then we find that ∆ ̂ u satisfies a singular initial value problem which satisfies the hypothesis of Theorem 2.4. Noting that ∆ ̂ u = 0 is a solution to this singular initial value problem, and recalling that Theorem 2.4 implies that solutions are unique, we see that indeed, the constraint ∆ ̂ u = 0 must be satisfied. While this approach to analyzing the singular initial value problem for the EulerPoisson-Darboux system does produce a solution, it is unsatisfactory for two reasons. First, it does not allow us to treat the case in which f 0 ∈ X δ, ̂ ν,q for ̂ ν < 1. Second, if ̂ ν > 1, this approach does not exclude the possible existence of other solutions u with remainders w in X δ,µ,q for µ < 1. Both of these issues are resolved if we use an alternative approach based on Theorem 2.21 and the use of (order n)-leading order terms. In doing this, we pay a price in that we must require a that the spatial derivative parameter q is infinite. If we are to work with Theorem 2.21, a key requirement is that we start with an ODEleading-order term; we have already fulfilled this requirement by our choice of ̂ U 0 . We now have the advantage that we do not need to impose the block diagonal condition, but only the modified block diagonal conditions, see Condition (i) in Proposition 2.18, and also not the positivity of the energy dissipation matrix in choosing the remainder exponent vector µ ; we may work with µ = ( µ 1 , µ 2 , µ 3 ) for any µ 1 , µ 2 > 0 and 1 < µ 3 < µ 2 +1, thereby permitting the full range of values of µ for which the singular initial value problem is meaningful. Notice that the upper bound for µ 3 is implied by Eq. (2.53) and is related to the observation above that a spatial derivative of a spatially dependent power of t may introduce additional log t -terms. Proceeding, we suppose that we have chosen some f 0 ∈ X δ, ̂ ν, ∞ with ̂ ν > 0. Any choice of µ satisfying the above conditions is consistent with Condition (ii) of Proposition 2.18 (as part of Theorem 2.21) if µ 2 < min { 2 , ̂ ν } . Choosing u ∗ , u ∗∗ ∈ C ∞ ( T 1 ), we then verify straightforwardly that Condition (iii) of Proposition 2.18 is satisfied. We conclude that there exists a solution ̂ U of the evolution system with ̂ U -̂ U n ∈ X δ 1 , ( µ 1 ,µ 2 ,µ 3 )+ nγ, ∞ for some constants δ 1 > 0 and a sufficiently large integer n . This solution is unique, with the remainder U U contained in X . ̂ -̂ 0 δ 1 , ( µ 1 ,µ 2 ,µ 3 ) , ∞ Having verified the existence of solutions to the first-order evolution system, we wish to show again that the corresponding solution is actually a solution of the original secondorder equation by considering the constraint Eq. (2.68). This can be done essentially as discussed above. To illustrate the use of the leading-order term approach, we choose the source term in the form f 0 ( t, x ) = f ∗ ( x ) t 1 / 2 for a smooth function f ∗ , and calculate and One may continue to calculate the sequence, and one verifies (in accord with the last statement in Proposition 2.18) that the residuals corresponding to this sequence are contained in X spaces of monotonically increasing exponent.", "pages": [ 47, 48, 49, 50, 51 ] }, { "title": "3.1 Objective of this section", "content": "As noted in the Introduction, one of the main motivations for this work is to explore the singular regions of certain classes of solutions of the Einstein gravitational field equations. In particular, as a step towards studying the strong cosmic censorship conjecture in families of solutions characterized by relatively large isometry groups, we use the Fuchsian formulations developed here to show that there are large sets of solutions in these families which exhibit AVTD behavior in a neighborhood of their singularity. We work here with spacetimes which are characterized by a spatially-acting T 2 isometry group, but do not have the further restriction of a non-vanishing 'twist', which defines the familiar Gowdy spacetimes. Following convention, we refer to them as the ' T 2 -symmetric spacetimes'; if they also satisfy the Einstein equations, we call them ' T 2 -symmetric solutions'. While much is known regarding the Gowdy spacetimes, including a proof that strong cosmic censorship holds for the Gowdy spacetimes with T 3 spatial topology [35] and for polarized Gowdy spacetimes with any allowed spatial topology [18], much less is known about the T 2 -symmetric solutions. For both the Gowdy and T 2 -symmetric families, the presence of the T 2 isometry effectively reduces the analysis to that of a PDE system on a 1 + 1 dimensional manifold. One notable difference, however, is that while the Gowdy PDE system is semilinear, that of the T 2 -symmetric solutions is quasilinear. The first work showing that there are (non-polarized) Gowdy spacetimes with AVTD behavior is that of Kichenassamy and Rendall [30] which uses Fuchsian methods to show that this is true for analytic Gowdy solutions on T 3 . The later work of Rendall [34] shows this for Gowdy spacetimes which are smooth, again using Fuchsian methods (adapted to smooth solutions rather than analytic solutions). Fuchsian methods have been used [24] to verify that there are analytic polarized T 2 -symmetric solutions with AVTD behavior. Here, we use the results presented above to show the same for T 2 -symmetric solutions (polarized and half-polarized) which are not analytic.", "pages": [ 51 ] }, { "title": "3.2 T 2 -symmetric spacetimes", "content": "The family of vacuum T 2 -symmetric spacetimes is characterized by a T 2 isometry group which acts effectively on each spacetime in the family, with the generating Killing vector fields being everywhere spacelike. We assume that each such spacetime is the maximal globally hyperbolic development of an initial data set on a compact Cauchy surface, with the data invariant under an effective T 2 action. One more condition distinguishes the spacetimes we consider here from the Gowdy subfamily. Let Y and Z be the generators of the T 2 isometry. The Gowdy subfamily is characterized by the assumption that the distribution defined by the tangent planes orthogonal to the generators Y and Z is integrable. This condition is usually expressed as the vanishing of the two twists K Y and K Z . 1 We work here with T 2 -spacetimes with at least one non-vanishing twist. Chru'sciel has shown [17] that the vacuum Einstein equations force the twists to be constants, and that the condition of non-vanishing twist implies the Cauchy surfaces must have T 3 topology. Such spacetimes can be foliated by areal coordinates, in which the time coordinate labeling each symmetry group orbit is equal to the area of that orbit. This coordinate system conveniently locates the singularity at t = 0 except in the special case of flat Kasner, as is shown by Isenberg and Weaver in [27]. Local existence for these coordinates is shown by Chru'sciel, [17], and global existence is proved by Berger et. al. in [8], and further clarified in [27]. Let y, z be coordinates on T 2 , and let x be the remaining spatial coordinate, which takes values in S 1 . The metric can be written [17] in the form 2 where all the metric functions { η, U, α, A, G 1 , G 2 } depend only on t and x . If both twist constants vanish, then the function α can be chosen to be a constant, in which case the above metric reduces to the Gowdy metric [22]. The polarized class of T 2 -symmetric spacetimes results from setting A equal to a constant in the initial data (or, equivalently, assuming that the dot product of the generators Y, Z is initially the same at all spatial points 3 ), and verifying that this condition is preserved under evolution. While the polarized spacetimes are characterized by a geometric condition, another subclass we consider, called the half-polarized T 2 -symmetric spacetimes, is defined by a restriction on the asymptotic behavior of the fields (see Section 3.3.1). /negationslash Before writing down the Einstein vacuum equations, we make a few further coordinate choices to simplify the presentation. Without loss of generality we choose the generators such that K Y = 0 , K Z ≡ K = 0. This can be achieved by choosing an appropriate linear combination of any generators for the T 2 action. It is sufficient to consider K > 0 since the transformation K → -K preserves all conditions imposed thus far. Next we choose coordinates y, z on T 2 so that Y = ∂ y and Z = ∂ z . This can be done without changing the form of the metric above. Implementing these simplifications, and using the short-hand notation U t := ∂ t U for derivatives, we write the Einstein equations as the following system of PDEs, which includes a set of second order equations a set of first-order equations plus a set of auxiliary equations Here, the auxiliary equations originate from the definition of the twist constants K Y and K Z and from the 'gauge' simplification K Y = 0 noted above. Observe that the T 2 -symmetric Einstein system reduces to the Gowdy system in the standard areal coordinates if we set K = 0, α ≡ 1, G 1 ≡ 0, and G 2 ≡ 0. The Einstein equations in the Gowdy class are semilinear and a Fuchsian analysis with analytic asymptotic data has been carried out by Kichenassamy and Rendall [30], and with smooth asymptotic data by Rendall [34] and by Beyer and LeFloch [12].", "pages": [ 51, 52, 53 ] }, { "title": "3.3.1 AVTD behavior and heuristics", "content": "What is the behavior of a singular solution to Einstein's equations near the singularity? In principle the behavior could be very complicated for a solution to a system of nonlinear PDE such as the Einstein equations. In [32, 6, 7] Belinskii, Khalatnikov, and Lifshitz (BKL) propose that generically the spacetime dynamics near the singularity is vacuum dominated, local, and oscillatory. According to this picture, an observer traveling toward the singularity (either backward or forward in time, depending upon the location of the singularity) would experience an infinite sequence of Kasner epochs, and each observer at different spatial points would experience a different, generally unrelated, sequence. Numerical simulations of T 2 -symmetric spacetimes [5, 9, 33] support this picture, except perhaps at points where spikes occur. Whether the complicated behavior found near spikes, and the apparent prevalence of spikes, invalidates the BKL picture for general T 2 -symmetric solutions is far from clear. However, for the restricted family of polarized T 2 -symmetric solutions, numerical simulations indicate that a special form of BKL behavior occurs near singularities-asymptotically velocity term dominated, or AVTD, behavior-which is not dominated by what happens near spikes. In a spacetime with AVTD behavior, each observer experiences only a finite sequence of Kasner epochs in the approach to the singularity [25, 21, 23], and the limiting spacetime is different for each observer. While there are no analytical studies of inhomogeneous cosmological solutions which either confirm or deny the presence of general BKL behavior, as noted above there has been a significant amount of such work supporting the generic presence of AVTD behavior in restricted families of solutions. Studies based on singular initial value problem formulations of Fuchsian PDEs are particularly well-adapted to doing this, since they involve specifying a choice of asymptotic behavior (a Kasner evolution independently at each point), and showing that there are solutions of the equations which approach this asymptotic behavior. If we can show that the Einstein equations for the polarized T 2 -symmetric spacetimes, together with certain choices of the leading order term, satisfy the conditions of the hypothesis of either Theorem 2.4 or Theorem 2.21, then we have confirmation that there are such spacetimes which have AVTD behavior. Observe that finding solutions in a given family of spacetimes with AVTD behavior does not imply that there are not solutions in that same family with a very different form of asymptotic behavior. However, since numerical simulations support AVTD behavior being generic among polarized T 2 -symmetric solutions, there have been no searches for alternative forms of asymptotic behavior among them. The name 'asymptotically velocity term dominated' refers to the fact that the leading order terms are chosen as asymptotic solutions of the 'velocity term dominated' (VTD) system, which is formed from the Einstein equations by dropping terms with spatial derivatives. This step encodes the local aspect of the BKL proposal. It can be shown [24, 20] that the following expansions for the metric functions below asymptotically solve this VTD system in the limit t → 0. We write these expansions in terms of asymptotic data { k, U ∗∗ , A ∗ , A ∗∗ , η ∗ , α ∗ , G 1 ∗ , G 2 ∗ } with the regularity of the data specified below. Of particular importance here is the function k . It determines the Kasner exponents p 1 , p 2 , p 3 of the local Kasner solutions which are approached at any spatial point We recall here that a T 2 -symmetric solution is defined to be polarized if the two Killing vectors corresponding to the T 2 isometry can be chosen to be orthogonal everywhere. This is the case if and only if the metric coefficient A ≡ const . A solution with AVTD behavior has this property if and only if the asymptotic data corresponding to A satisfy the conditions A ∗∗ ≡ 0 and A ∗ ≡ const . Since A ∗ ≡ const can be gauged to A ∗ ≡ 0, we see that in the polarized case, there is effectively no free asymptotic data to choose which relates to A . There is an interesting relationship between the polarization condition and the sign of k : Examining equations (3.8)-(3.13), we find that if a solution is not polarized and has AVTD behavior, then there is power law blow-up at the singularity if and only if k is negative. Yet if that spacetime is polarized, then regardless of the sign of k , there is no power law blow-up at the singularity. The polarization condition is relevant to our application of our Fuchsian results to T 2 -symmetric solutions since, as we see below, our results cannot be applied unless the condition ∂ x A ∗ = 0 holds for the asymptotic data. For polarized T 2 -symmetric solutions, this restriction on A ∗ is automatic. It is important to note, however, that requiring ∂ x A ∗ = 0 does not restrict us to polarized solutions. We may consider asymptotic data which has this restriction on A ∗ , but has no restriction on A ∗∗ . T 2 -symmetric solutions which are AVTD and which have asymptotic data of this sort are known to exist, and have been called 'half-polarized' 1 [20]. Extending the results of both [24] (analytic and polarized) and [20] (higher regularity), we show here that there are large families of both half-polarized and polarized T 2 -symmetric solutions which are smooth or of even lower regularity, and which have AVTD behavior near their cosmological singularities. A general (neither polarized nor half-polarized) T 2 -symmetric solution, were it to be AVTD, would have asymptotic data with both A ∗ and A ∗∗ non-vanishing and nonconstant. Based on numerical and heuristic considerations, however, it is expected that spacetimes with non-constant A ∗ do not generally show AVTD behavior. Rather, these are expected to show Mixmaster-like BKL behavior at the t = 0 singularity, or behavior which is even more complicated (with strong spike influence). We do not address this issue here. We now discuss two applications of our Fuchsian results which verify AVTD behavior in T 2 -symmetric solutions. For the first one, Theorem 3.1, we make only minimal assumptions regarding the regularity of the asymptotic data. The price to pay for this is that the result does not cover the full expected range for the function k = k ( x ) in Eqs. (3.8) - (3.13). For the second result Section 3.3.3, Theorem 3.6, we add regularity restrictions, but we do get the expected full range of allowed values for k .", "pages": [ 53, 54, 55 ] }, { "title": "3.3.2 Existence of low regularity solutions with AVTD behavior", "content": "The low regularity result, which we formulate, discuss, and prove in this subsection, is an application of Theorem 2.4 to the polarized and half-polarized solutions of the T 2 -symmetric equations. Theorem 3.1 (First result: AVTD (half)-polarized T 2 -symmetric vacuum solutions finite differentiability) . Suppose one chooses a twist constant K ∈ R , a pair of asymptotic data constants A ∗ and η 0 , and a set of asymptotic data functions k, U ∗∗ , α ∗ ∈ H q +2 ( T 1 ) (with α ∗ ( x ) > 0 ), A ∗∗ ∈ H q +1 ( T 1 ) and G 1 ∗ , G 2 ∗ ∈ H q ( T 1 ) for any q ≥ 3 , which satisfy the integrability condition 1 together with, at each point x ∈ T 1 , either Then there exists a δ > 0 , and a T 2 -symmetric solution U , A , η , α , G 1 , G 2 of Einstein's vacuum field equations with twist K of the form Here, the leading-order term ( U 0 , A 0 , η 0 , α 0 , G 1 , 0 , G 2 , 0 ) is given by Eqs. (3.8) -(3.13) , with The remainder W is contained in X δ,µ,q (and DW ∈ X δ,µ,q -1 ) for any exponent vector µ = ( µ 1 , µ 2 , µ 3 , µ 4 , µ 5 , µ 6 ) with This solution is unique among all solutions with the same leading-order term U 0 and with remainder W ∈ X δ,µ,q . Observe that by taking time derivatives of the Einstein field equations, we can also obtain corresponding statements about the behavior of a certain number of time derivatives D m W of the remainder function W . We do not elaborate on this any further here. This result, based on Theorem 2.4, does not imply uniqueness of the solutions within the whole class of solutions of interest: For a given choice of asymptotic data, Theorem 3.1 determines that there is exactly one solution with remainder W in spaces X δ,µ,q with µ given by Eq. (3.15). The full class of remainders compatible with the leadingorder behavior Eqs. (3.8)-(3.13) however corresponds to exponents Hence, for given asymptotic data there may exist further solutions in such a larger space. Strict uniqueness can be explored further using techniques involving (order n)-leading order terms. We return to this issue in Section 3.3.3 below; the price which we have to pay for strict uniqueness is that we need to require higher differentiability for the asymptotic data. In proving Theorem 3.1, it is useful to arrange the T 2 -symmetric Einstein vacuum equations, Eqs. (3.1)-(3.7), as well as the field variables, in a certain hierarchical form: Eqs. (3.1), (3.2), (3.4) and (3.6) together form a coupled evolution system (which we label the 'main evolution equations') for the variables U, A, η , and α . Eq. (3.5) serves as a constraint equation for this system, while Eq. (3.3) is effectively redundant, and can be ignored. The remaining equations Eqs. (3.7) are evolution equations for G 1 and G 2 , and can be handled after the analysis of the main evolution equations. We proceed now to focus on the main evolution equations, with the primary existence result for them - the main step toward a proof of Theorem 3.1 - being Proposition 3.2. Main evolution equations. To rewrite the main evolution equations as a first order symmetric hyperbolic Fuchsian system, it is useful to define certain new variables. Some of the choices of these variables are motivated by considerations in [24], others by the discussion above in Section 2.4.4. First, we set whose evolution equation is obtained by taking the spatial derivative of Eq. (3.6) and by substituting any occurrence of η x by the constraint Eq. (3.5). One obtains In all other evolution equations we use Eq. (3.6) to eliminate α t and replace α x by ξ . Next, we find that for both U and η , it is useful to replace the given variable by that which involves the subtraction of the indicated log term in the asymptotic VTD expansions Eq. (3.8)-(3.13): We set ̂ η := η -1 4 (1 -k ) 2 log t and set ̂ U := U -1 2 (1 - )) log t ; compare this to our approach in Section 2.4.4. Adding a few other minor modifications, we are led to define the following set of first-order variables: u 7 = ̂ η, u 8 = α, u 9 = ξ. (3.19) Observe that, at this stage, k ( x ) is an arbitrary function (introduced in Eqs. (3.8)(3.13)), with no restrictions. In terms of the new set of the variables, the main evolution system Eqs. (3.1), (3.2), (3.4) and (3.6) can be written in symmetric hyperbolic form as follows: u 8 Du 3 -u 8 t∂ x u 2 -u 8 u 3 =0 , (3.22) u 8 Du 6 -u 8 t∂ x u 5 -u 8 u 6 =0 , (3.25) or equivalently as where Note that we have multiplied the third and sixth equations by u 8 . The source-term vector f is given by the right-hand sides of the evolution system Eqs. (3.20)-(3.28). The reason for keeping this particular form of the matrix N ( u ) (and not absorbing some of its entries into the source-term) becomes clear shortly. AVTD solutions of the main evolution system. We now show as an application of Theorem 2.4, and as a step towards proving Theorem 3.1, that there exist unique solutions to the singular initial value problem for the main evolution system (3.29)(3.32), with AVTD leading-order term Although not needed for our present argument, we note (by inspecting Eq. (2.46)) that this choice of u 0 is an ODE-leading-order term; cf. Section 2.4.4. To check that we have a quasilinear symmetric hyperbolic system, we need to specify an exponent vector along with the PDE system and a leading order term. Looking ahead to the conditions of block diagonality, we choose and expect to construct remainders in spaces X δ,µ,q with µ given by We then find, after replacing u 8 by α ∗ + w 8 , that so long as we choose α ∗ > 0, and so long as we require that all of the asymptotic data functions be contained in some H q ( T 1 ) (which we fix below), we indeed have a quasilinear symmetric hyperbolic system, which in addition does satisfy the block diagonality condition. Before continuing the argument that the hypothesis of Theorem 2.4 is satisfied, we state our result. Proposition 3.2. For any twist constant K ∈ R , for any Sobolev differentiability index q ≥ 3 , and for any choice of the asymptotic data functions such that A ∗ is an arbitrary constant, α ∗ ( x ) > 0 , k, U ∗∗ , α ∗ ∈ H q +2 ( T 1 ) , A ∗∗ ∈ H q +1 ( T 1 ) and η ∗ ∈ H q ( T 1 ) , and k satisfies (at each x ∈ T 1 ) either there exists a δ 1 ∈ (0 , δ ] , and a unique solution of the first order main evolution system Eqs. (3.29) -(3.32) with leading-order term u 0 and remainder w ∈ X δ 1 ,µ,q (and Dw ∈ X δ 1 ,µ,q -1 ) so long as the exponent vector µ given by Eq. (3.34) satisfies the following inequalities for all x ∈ T 1 : Observe here that the inequality just stated for µ 2 is not required to hold in the case of a polarized solution, since in that case A is not a dynamical variable, and this condition is vacuous. Although here and below we list results for the polarized and half-polarized cases together for compactness, the reader focusing on the polarized case may ignore all references to µ 2 and to w 4 , w 5 and w 6 . As noted above, this proposition is an application of Theorem 2.4 to Eqs. (3.29)(3.32). In the next lemma we verify that under the assumptions of Proposition 3.2 the Condition (i) of Theorem 2.4 is satisfied. The first condition follows directly from the definition of the energy dissipation matrix M 0 . Lemma 3.3. The energy dissipation matrix M 0 defined in Eq. (2.6) corresponding to Eqs. (3.29) -(3.32) , to the leading-order term u 0 given by Eq. (3.33) and to the exponent vector µ of the form Eq. (3.34) is positive definite at every x , provided that hold for all x ∈ T 1 . The next lemma establishes Conditions (ii) and (iii) of Theorem 2.4. Lemma 3.4. The operator F ( u 0 ) corresponding to Eqs. (3.29) -(3.32) , to the leadingorder term u 0 given by Eq. (3.33) , and to the exponent vector µ of the form Eq. (3.34) satisfies Condition (ii) and (iii) of Theorem 2.4 for some exponent vector ν > µ , for some sufficiently small δ > 0 , and for a choice of the differentiability index q ≥ 3 , so long as α ∗ and η ∗ are functions in H q ( T 1 ) , A ∗∗ is contained in H q +1 ( T 1 ) , k and U ∗∗ are elements of H q +2 ( T 1 ) , and if at each point x ∈ T 1 , the following inequalities hold for µ and k : and In both the polarized and the half-polarized cases, it follows from the two inequalities stated above for µ 1 that k ( x ) must either satisfy k ( x ) > 1 + √ 5 or k ( x ) < 1 -√ 5 . Proof. If the operator F ( u 0 ), defined in Eq. (2.7), is written out explicitly, it consists of products of asymptotic data functions, and components of the unknown function w (or products involving exponential functions of these). All of the multiplicands in these products are, by hypothesis, contained in designated function spaces (of the form X δ,µ,q ). Thus, to check Condition (ii) of Theorem 2.4, we primarily need to know the multiplication algebra of spaces such as X δ,µ,q . The result we need is provided by Lemma B.1 in the appendix. To check Condition (iii), we need results concerning Lipschitz properties of products and exponential functions of elements of the spaces X δ,µ,q . Lemma B.3 and Lemma B.5 provide these needed results. Proof of Proposition 3.2. If we wish to use Theorem 2.4 to show that the system discussed in Proposition 3.2 admits solutions with the stated properties, it is sufficient that i) the asymptotic data functions, which appear in the leading-order matrices S 1 , 0 , S 2 , 0 and N 0 , (i.e. the functions α ∗ and k ), be contained in H q +2 ( T 1 ) (with q ≥ 3); and ii) we choose the function k ( x ) so that the hypotheses of both of the above lemmas are satisfied. We readily check that exponent functions µ 1 , µ 2 , µ 3 and µ 4 , which satisfy the combined inequalities, can be found if and only if k ( x ) > 1 + √ 6 in the half-polarized case, and either k ( x ) > 1 + √ 6 or k ( x ) < 1 -√ 6 in the polarized case. The full set of Einstein's vacuum field equations. Thus far, we have constructed solutions u of the main evolution equations for the T 2 -symmetric system with the leading-order behavior Eq. (3.33), according to Proposition 3.2. Given such a solution u , we may ask under what conditions is this a solution of the full set of Einstein's vacuum field equations, Eqs. (3.1)-(3.7), with U = u 1 + 1 2 (1 -k ) log t , A = u 4 , η = u 7 + 1 4 (1 -k ) 2 log t , and α = u 8 . Proposition 3.5. For any solution of Proposition 3.2 with asymptotic data satisfying all the conditions in Theorem 3.1, the full set of Einstein's vacuum field equations Eqs. (3.1) - Eq. (3.6) are satisfied, and Eqs. (3.7) can be solved for G 1 and G 2 as stated in Theorem 3.1. Proof. Since the equations for G 1 and G 2 , Eq. (3.7), are semi-decoupled from the rest, we ignore them (as well as G 1 and G 2 ) to start, and focus on the subsystem Eqs. (3.1)(3.6). To monitor the extent to which this subsystem is satisfied by fields which satisfy the main evolution equations, it is useful to define the following set. Based on Eq. (3.3), we define and, based on the constraint Eq. (3.5), Based on the constraints which stem from the definition of the new variables which allow us to rewrite the original system in first order, we define and The constraint-violation terms C 3 , C 4 , C 5 and C 6 are the easiest to handle. Arguing as we do in the discussion of ∆ u in Section 2.4.4, we find that the evolution equation for each of these, induced by the main evolution system, takes the form DC 3 = 0 , DC 4 = 0, etc. Then, since the form of the leading order term for the main system implies that each of these terms must asymptotically vanish, it follows that each must vanish for all time. Observe that this determination that C 3 , C 4 , C 5 , and C 6 all vanish allows us to freely substitute in the consequences of their vanishing in the analysis of H , C 1 , and C 2 . Such substitution is very useful. We now focus on H , C 1 and C 2 , noting that a solution u of Eqs. (3.20) - (3.28) is a solution of Eqs. (3.1)-(3.6) (with the above replacements) if and only if H,C 1 , and C 2 vanish identically. Presuming that u is a solution of Eqs. (3.20) - (3.28), we calculate which tells us that so long as we can show that C 1 and C 2 vanish, it follows that H vanishes as well. We may therefore focus on C 1 and C 2 , for which the evolution equations take the following form: Under our hypothesis (which implies in particular that the coefficients here are continuous functions of x ), this set of evolution equations for C 1 and C 2 can be treated as an essentially independent system of linear homogeneous ODEs at each spatial point. Noting that (for q ≥ 3) the coefficients on the right side of both equations are well-behaved as t → 0 and converge to zero at every spatial point. Hence this system for the unknowns C 1 and C 2 is of the form Eq. (2.1) with S 1 the identity matrix, and S 2 and N 2 the zero matrices. The singular initial value problem of the form C 1 = C 1 ∗ + w 1 , C 2 = C 2 ∗ + w 2 with w 1 , w 2 ∈ X δ,µ,q has a unique solution for every prescribed C 1 ∗ and C 2 ∗ and for every sufficiently small µ > 0. The definition of the quantities C 1 and C 2 in terms of the variables U , A , η , ξ and α according to Eqs. (3.36) and (3.37) implies uniqueness for all constraint violations which are compatible with solutions of Proposition 3.2. In particular, the unique solution corresponding to C 1 ∗ = C 2 ∗ = 0 is C 1 ≡ C 2 ≡ 0. It remains to determine how the condition C 1 ∗ ≡ C 2 ∗ ≡ 0 relates to the choice of the asymptotic data functions k , U ∗∗ , A ∗ , A ∗∗ , η ∗ , α ∗ and ξ ∗ . For a solution u as above, the functions C 1 and C 2 defined by Eqs. (3.36) and (3.37) converge uniformly in space at t = 0, and we obtain It follows that C 2 ∗ = 0 if and only if and C 1 ∗ = 0 if and only if, for an arbitrary constant η 0 , In particular, the spatial topology implies that we must choose the asymptotic data k , U ∗∗ and α ∗ such that We thus conclude that a solution u of Proposition 3.2 is a solution of Eqs. (3.1)(3.6) (with the above replacements) if and only if the asymptotic data functions satisfy Eqs. (3.41) and (3.42). It only remains to solve Eqs. (3.7) for G 1 and G 2 . The right-hand sides of these two equations have the asymptotic behavior (for t → 0) of a power of t larger than -1 uniformly in space. Hence the right-hand sides are integrable in t at t = 0 at every spatial point, and the general solution is The functions G 1 -G 1 ∗ and G 2 -G 2 ∗ are contained in X δ 1 ,µ 5 ,q and X δ 1 ,µ 6 ,q , respectively, for any choice of exponents 0 < µ 5 ( x ) , µ 6 ( x ) < 1 / 2( k ( x ) -3)( k ( x ) + 1), if u is a solution of Proposition 3.2. We therefore can take G 1 ∗ , G 2 ∗ ∈ H q ( T 1 ).", "pages": [ 56, 57, 58, 59, 60, 61, 62, 63, 64 ] }, { "title": "3.3.3 Optimal existence and uniqueness result", "content": "The result we prove in Section 3.1 allows for relatively rough asymptotic data, but consequently sacrifices some of the expected range (based on numerical and heuristic studies [9, 5, 33]) of allowed values for the asymptotic velocity k = k ( x ). In this section, we consider only smooth asymptotic data, and can then prove a result which increases the range for k . Theorem 3.6 (Optimal result: AVTD (half)-polarized T 2 -symmetric solutions - infinite differentiability) . Suppose one chooses a twist constant K ∈ R , a pair of asymptotic data constants A ∗ and η 0 , and a set of asymptotic data functions k, U ∗∗ , α ∗ (with α ∗ ( x ) > 0 ), A ∗∗ , G 1 ∗ , G 2 ∗ ∈ C ∞ ( T 1 ) which satisfy the integrability condition together with, at each point x ∈ T 1 , either Then there exists a δ > 0 , and a T 2 -symmetric solution U , A , η , α , G 1 , G 2 of Einstein's vacuum field equations with twist K of the form Here, the leading-order term ( U 0 , A 0 , η 0 , α 0 , G 1 , 0 , G 2 , 0 ) is given by Eqs. (3.8) -(3.13) and Eq. (3.14) . The remainder W is contained in X δ,µ, ∞ (and DW ∈ X δ,µ, ∞ ) for some exponent vector µ = ( µ 1 , µ 2 , µ 3 , µ 4 , µ 5 , µ 6 ) with This solution is unique among all solutions with the same leading-order term u 0 and with remainder W ∈ X δ,µ, ∞ . Comparing this result with Theorem 3.1, we see the differences in the hypothesized regularity of the asymptotic data, and in the allowed range of the asymptotic velocity k ( x ) in the two results. As we find in proving this result, in fact one does not need C ∞ data; data of 'sufficiently high differentiability' is enough. One may ask if the reduced range for k ( x ) for rough data is a real effect, or an artifact of our method of proof (which remains an open question). Observe one other important difference between the two theorems: Theorem 3.6 provides a stronger result regarding the uniqueness of solutions to the singular initial value problem for these systems: While in Theorem 3.1, there could in principle exist more than one solution for a given set of asymptotic data (since µ 1 has to be larger than one), we find that according to Theorem 3.6, there is exactly one solution for the remainder functions in the full space of interest given by Eq. (3.43). We note, however, that there may be two solutions for the same asymptotic data which differ by a factor which goes to zero faster than every power of t at t = 0. The proof of Theorem 3.6 is in many ways similar to that of Theorem 3.1, but with the big difference that the latter involves the application of Theorem 2.4, while Theorem 3.6 is obtained by applying Theorem 2.21. In both cases, these results for Fuchsian singular initial value problems (Theorem 2.4 or Theorem 2.21) are applied to the main evolution system for T 2 -symmetric solutions. The portion of the proof of both Theorem 3.6 and Theorem 3.1 which shows that the existence of a solution for the main evolution system implies the existence of a proof to the full system (since one can choose the asymptotic data in such a way that the constraints are necessarily satisfied) is essentially the same for the two cases. Hence, to prove Theorem 3.6, all we need is the following proposition (with the rest of the proof taken care of by the arguments appearing in the proof of Proposition 3.5). Proposition 3.7. For any twist constant K ∈ R , and for any choice of the asymptotic data functions such that A ∗ is an arbitrary constant, α ∗ ( x ) > 0 , k, U ∗∗ , α ∗ , η 0 ∈ C ∞ ( T 1 ) , and either for all x ∈ T 1 , there exists a δ 1 ∈ (0 , δ ] , and a unique solution of the main evolution system Eqs. (3.29) -(3.32) with leading-order term u 0 and remainder w ∈ X δ 1 ,µ, ∞ (and Dw ∈ X δ 1 ,µ, ∞ ), so long as the exponent vector µ given by satisfies the following inequalities for all x ∈ T 1 and ˜ µ 1 strictly smaller than, but sufficiently close to 1 + µ 1 . One of the key differences between the hypothesis here and that of Proposition 3.2 is the chosen form for the exponent vector µ . The form we use here - in particular, the choice of ˜ µ 1 for the third component - allows for the wider range of k ( x ). We can choose this form here, but not in Proposition 3.2, since here we do only need to satisfy the modified block diagonal conditions in Theorem 2.21 (as part of Proposition 2.18). Note that we could have carried out a similar modification for the sixth component of µ , but it turns out to be unnecessary; the regularity of the spatial derivative of the A variable follows automatically. Proof. The proof of this result is based on the use of (order n)-leading-order terms, which are developed and discussed in Section 2.4. and, in particular, in Theorem 2.21. Recalling that the leading order term u 0 (which takes the form u 0 with components given by Eq. (3.33)) is in fact an ODE-leading-order term, we check the conditions in the hypothesis of Theorem 2.21, noting that a portion of these conditions appear in the hypothesis of Proposition 2.18. We check that the matrix S -1 1 , 0 ( u 0 ) N 0 ( u 0 ) is in Jordan normal form, and that our choice of µ is strictly larger than the negatives of the corresponding diagonal elements of S -1 1 , 0 N 0 . Then, Condition (ii) of Proposition 2.18 (as part of Theorem 2.21), can be checked in essentially the same way as is done in the proof of Lemma 3.4, but now with the inequalities listed in the hypothesis of Proposition 3.7. Condition (iii) of Proposition 2.18 (as part of Theorem 2.21) follows by repeated applications of Lemma B.4.", "pages": [ 64, 65, 66 ] }, { "title": "4 Concluding remarks", "content": "Our results here show that there is a large collection of smooth, polarized and halfpolarized T 2 -symmetric solutions of the Einstein vacuum equations which exhibit AVTD behavior in a neighborhood of their singularities. What can we show further? Numerical and heuristic studies of T 2 -symmetric solutions [9, 5, 33] strongly indicate that AVTD behavior is not found in such spacetimes unless they satisfy a polarization condition. These studies do support the conjecture that AVTD behavior occurs generically in polarized T 2 -symmetric solutions. While Fuchsian methods of the sort developed here are not expected to be effective in determining such genericity, further numerical explorations of the polarized T 2 -symmetric solutions could be very useful. Among the issues which might be explored numerically is whether the distinction in the results we have obtained for solutions of finite differentiability and those which are C ∞ is significant in any sense. Observe that one distinguishing feature of our approach here is that an approximation scheme is at the core of the method. This scheme can be implemented for numerical computations straightforwardly and contains useful built-in convergence and error estimates. In earlier work [12, 14, 15], building on [3], we implemented this scheme in the context of semilinear symmetric hyperbolic Fuchsian equations of second-order and have obtained very accurate simulations for Gowdy solutions. We expect to get similarly good results for the polarized T 2 -symmetric solutions, which we have studied here. Of particular interest would be to explore the issue of whether the 'optimal domain' for the asymptotic velocity k ( x ) > 3 (or k ( x ) < -1) can only be obtained for smooth solutions (as suggested by our discussion in Section 3) and to see what might happen if we try to construct such a solution with lower differentiability. It is expected, based on numerical simulations [10], that polarized (and half-polarized) U (1)-symmetric solutions exhibit AVTD behavior. Moreover, Fuchsian methods [26] confirm this, at least for analytic solutions. The methods we have developed here, generalized to PDEs on higher dimensional manifolds (this is done for T n in [2]), should be applicable to the polarized and half-polarized U (1)-symmetric solutions, showing that smooth solutions of this type also exhibit AVTD behavior.", "pages": [ 66, 67 ] }, { "title": "Acknowledgements", "content": "F.B. was partially supported by a special assistance grant of the University of Otago during 2011. The authors E.A. and J.I. are partially supported by NSF grant PHY0968612. E.A., J.I., and P.LF. thank the University of Otago for sponsoring their visits to Dunedin while some of this research was carried out. P.LF was also supported by the Agence Nationale de la Recherche through the grant ANR SIMI-1-003-01 (Mathematical General Relativity. Analysis and geometry of spacetimes with low regularity).", "pages": [ 67 ] }, { "title": "A Properties of the spaces X δ,µ,q", "content": "In this section we list further basic properties of the spaces X δ,µ,q which are defined in Section 2.2 as the completion of the normed vector spaces ( C ∞ ((0 , δ ] × T 1 ) , ‖·‖ δ,µ,q ), cf. Eq. (2.3). Recall that δ > 0 is a constant, µ is an exponent vector and q is a non-negative integer. We now also define the spaces ̂ X δ,µ,q as the set of maps f : (0 , δ ] → H q ( T 1 ) with the property that R [ µ ] f is bounded and continuous; cf., Eq. (2.2). If we endow ̂ X δ,µ,q with the norm ‖ · ‖ δ,µ,q , then ̂ X δ,µ,q are Banach spaces. Note that if f ∈ ̂ X δ,µ + /epsilon1,q for some /epsilon1 > 0, then R [ µ ] f : (0 , δ ] → H q ( T 1 ) is uniformly continuous. Lemma A.1. Let f ∈ ̂ X δ,µ,q ; i.e., R [ µ ] f : (0 , δ ] → H q ( T 1 ) is bounded and continuous. Let ̂ f be defined as follows By definition, all functions in X δ,µ,q can be approximated by smooth functions. Functions in ̂ X δ,µ,q , however, can be approximated by a particularly useful sequence of smooth functions as follows. Let φ : R → R be smooth with φ ( x ) > 0 for all | x | < 1 and φ ( x ) = 0 for all | x | ≥ 1 , with ∫ R φ ( x ) dx = 1 . Let ( α i ) be a sequence of positive numbers with limit 0 . For any integers i, j , we set Then ( R [ µ ] f ) i,j has the following properties:", "pages": [ 70 ] }, { "title": "(ii) The function", "content": "has the property that In particular, for any given integers i, j , one has for a constant C > 0 independent of t (but possibly dependent on i , j ). (iii) ( R [ µ ] f ) i,j ( t, x ) -→ R [ µ ] f ( t, x ) for i, j →∞ at a.e. ( t, x ) ∈ (0 , δ ] × T 1 . (iv) If f is such that R [ µ ] f : (0 , δ ] → H q ( T 1 ) is a uniformly continuous map (e.g., if f ∈ ̂ X δ,µ + /epsilon1,q for some /epsilon1 > 0 ), then Proof. Observe that R [ µ ] ̂ f is a bounded continuous map (0 , ∞ ) → H q ( T 1 ) since R [ µ ]( t ) ̂ f ( t ) = R [ µ ]( δ ) f ( δ ) for all t ≥ δ . We obtain ̂ f ∈ ̂ X ∞ ,µ,q and ‖ ̂ f ‖ ∞ ,µ,q = ‖ f ‖ δ,µ,q . The first two properties of the lemma can be proven by standard arguments. The third one follows from Lebesgue's Differentiation Theorem. We only discuss the fourth property. If we fix any t ∈ (0 , δ ], then 1 we calculate as a consequence of the condition that ∫ R φ ( x ) dx = 1. Now we write ∥ ∥ ∥ ̂ ∥ ∥ ∥ H x ( T ) Writing the first term on the right hand side of Eq. (A.3) as I , we estimate Now, it is a standard result for mollifiers that for every s ∈ (0 , ∞ ) where lim i →∞ g i ( s ) = 0 at every s , and for every integer i , the function g is continuous. Since R [ µ ] f is uniformly continuous, this function g i extends to the interval [0 , ∞ ) with the same properties. Since R [ µ ]( t ) ̂ f ( t ) = R [ µ ]( δ ) ̂ f ( δ ) for all t ≥ δ , it follows that there is a sequence ( ̂ g i ) with limit 0, such that g i ( s ) ≤ ̂ g i for all s ∈ [0 , ∞ ). Consequently, I can be estimated by a sequence ( a i ), which (i) is independent of j , (ii) is independent of t , and (iii) goes to zero in the limit i →∞ . We now discuss the second term of the right hand side of Eq. (A.3), which we label as II . The integral over y is trivial, so consequently The term involving the H q -norm is a uniformly continuous function in s and t . Hence, from Lebesgue's Differentiation Theorem and the definition of ̂ f , it follows that the s -integral converges to 0 for j →∞ , independently of t ∈ (0 , δ ] and i . This completes the proof of the fourth property. We can now use Lemma A.1 to relate the spaces X δ,µ,q and ̂ X δ,µ,q . Lemma A.2. Fix a constant δ > 0 , an exponent vector µ , and a non-negative integer q ; then for all /epsilon1 > 0 , Proof. The inclusion X δ,µ,q ⊂ ̂ X δ,µ,q follows easily from the fact that each element in X δ,µ,q is the limit of a Cauchy sequence in ( C ∞ ((0 , δ ] × T 1 ) , ‖·‖ δ,µ,q ), whose elements are in particular bounded continuous maps (0 , δ ] → H q ( T 1 ), and the convergence is uniform in time. To check the inclusion ̂ X δ,µ + /epsilon1,q ⊂ X δ,µ,q , let a function f be given in ̂ X δ,µ + /epsilon1,q . Hence f satisfies the condition of the previous lemma, in particular that of Condition (iv). It follows that f ∈ X δ,µ,q . We also wish to comment on time derivatives of functions in X δ,µ,q and ̂ X δ,µ,q . Let f ∈ ̂ X δ,µ,q . We say that f is differentiable in time t if the (bounded continuous) map R [ µ ] f : (0 , δ ] → H q ( T 1 ) is differentiable in the sense of a map between Banach spaces (Frechet derivatives). Its time derivative (multiplied by t ) D ( R [ µ ] f ) can then be considered to be a map (0 , δ ] → H q ( T 1 ), and we set Df := R [ µ ] -1 ( D ( R [ µ ] f ) -D R [ µ ] f ). If this map is continuous, then we call f continuously differentiable in t . If this is the case for f and if in addition R [ µ ] Df is bounded, then we have Df ∈ X δ,µ,q . ̂ Now, let f ∈ ̂ X δ,µ,q be continuously differentiable. Then Df is the distributional time derivative of f in the following sense. Let φ be any test function with the properties as in Section 2.3.2. Choose /epsilon1 > 0. Then we clearly have that Hence, the boundary term vanishes in the limit /epsilon1 → 0. The following integrals are meaningful for /epsilon1 = 0, and hence we obtain The reader should compare this with the expressions for weak solutions in Section 2.3.2.", "pages": [ 70, 71, 72, 73 ] }, { "title": "B On products of functions", "content": "We readily check the following results which are useful in dealing with products of functions and their relationship to the function spaces X δ,µ,q . Lemma B.1. Let f ∈ X δ,µ 1 ,q and g ∈ X δ,µ 2 ,q be two functions (0 , δ ] × T 1 → R , for some constant δ > 0 , some smooth exponents µ 1 and µ 2 , and an integer q ≥ 1 . Then f · g is in X δ,µ 1 + µ 2 ,q and, for some constant C > 0 , Observe that the condition q ≥ 1 (for one spatial dimension) is essential here. Proof. An essential part of the proof of this lemma is the general estimate for arbitrary functions f and g in H q ∩ L ∞ ; see Proposition 3.7 in Chapter 13 of [40]. The Sobolev inequalities for q ≥ 1 in one spatial dimension then imply Working with this inequality, we see that the lemma follows easily if we choose a sequence ( f i ) which converges to f in the function space X δ,µ 1 ,q , and a sequence ( g i ) which converges to g in X δ,µ 2 ,q , and then write Another important result is the following. Lemma B.2. Let w be a d -vector-valued function in X δ,µ,q for some exponent d -vector µ , a constant δ > 0 , and an integer q ≥ 1 . Let S be a d × d -matrix-valued function so that R [ µ ] · S · R [ -µ ] is an element of X δ,ξ,q for an exponent d × d -matrix ξ of the form ξ ij = ζ i where ζ is an exponent d -vector. Then, the d -vector-valued function S w is in X δ,ζ + µ,q and for some constant C > 0 . This lemma is proved essentially in the same way as Lemma B.1. Lemma B.3. Suppose that δ > 0 , s > 0 and r > 0 are constants, n , d and q integers with d ≥ 1 and q ≥ 1 , µ an exponent d -vector, and ν 1 and ν 2 exponent scalars. Let functions g 1 , g 2 : U → R be given where U is an open subset of R d . Suppose that g 1 maps all functions w : (0 , δ ] × T 1 → R d in B δ,µ,q,s to elements g 1 ( w ) in B δ,ν 1 ,q,r . Moreover suppose that there is a constant C 1 > 0 with for all w 1 , w 2 : (0 , δ ] × T 1 → R d in B δ,µ,q,s . Let us also assume that g 2 maps all functions w : (0 , δ ] × T 1 → R d in B δ,µ,q,s to elements g 2 ( w ) in B δ,ν 2 ,q,r and that there is a constant C 2 > 0 with for all w 1 , w 2 : (0 , δ ] × T 1 → R d in B δ,µ,q,s . Now, consider h := g 1 · g 2 , w ↦→ h ( w ) . Then, there exists a ρ > 0 (which is smaller the smaller r is) so that h maps all functions w : (0 , δ ] × T 1 → R d in B δ,µ,q,s to elements h ( w ) in B δ,ν 1 + ν 2 ,q,ρ . Moreover, there exists a constant C > 0 with for all w 1 , w 2 : (0 , δ ] × T 1 → R d in B δ,µ,q,s . Proof. If w ∈ B δ,µ,q,s , then g 1 ( w ) ∈ B δ,ν 1 ,q,r and g 2 ( w ) ∈ B δ,ν 2 ,q,r . Lemma B.1 implies that h ( w ) = g 1 ( w ) g 2 ( w ) ∈ X δ,ν 1 + ν 2 ,q and where C > 0 is the constant in Lemma B.1. This allows us to set ρ = Cr 2 and hence establishes that h ( w ) ∈ B δ,ν 1 + ν 2 ,q,ρ . Regarding the Lipschitz estimate, we find Then we can use the individual Lipschitz estimates for g 1 and g 2 in order to establish this result. While Lemma B.3 is adequate for the proof of Theorem 2.4, to prove Theorem 2.21 we require a stronger result, which we present here. Lemma B.4. Suppose that q ≥ 1 . Let g 1 and g 2 be functions satisfying all the conditions of Lemma B.3 with exponents ν 1 , ν 2 for all x ∈ T 1 . Suppose that, in addition, one has the following: For all w ∈ B δ,µ,q,s/ 2 with ω ∈ B δ, ̂ µ,q,s/ 2 for some exponent vector ̂ µ which satisfies ̂ µ ≥ µ , there exist scalar exponents γ 1 , γ 2 , independent of ̂ µ , such that g 1 ( w + ω ) -g 1 ( w ) ∈ X δ, ̂ µ + γ 1 ,q , g 2 ( w + ω ) -g 2 ( w ) ∈ X δ, ̂ µ + γ 2 ,q , and This follows from a more detailed analysis of Eq. (B.1). To handle the exponential function, we rely on the following result. for constants ̂ C 1 , ̂ C 2 > 0 . Then the function h = g 1 · g 2 has the following property. We can choose a scalar exponent γ smaller or equal than min { ν 1 + γ 2 , ν 2 + γ 1 } (independently of ̂ µ ), such that for all w ∈ B δ,µ,q,s/ 2 and ω ∈ B δ, ̂ µ,q,s/ 2 , one has and for a constant ̂ C > 0 . Lemma B.5. Pick constants δ > 0 , s > 0 , an integer q ≥ 1 , and an exponent µ > 0 . Let g ( i ) := exp · Π i , where Π i : R d → R is the projection to the i th component of d -vectors. Then, for every function w : (0 , δ ] × T 1 → R in B δ,µ,q,s , there exists an r > 0 , so that the composed function g ( i ) · w : (0 , δ ] × T 1 → R is in B δ, 0 ,q,r . Moreover, for all w 1 , w 2 ∈ B δ,µ,q,s , there exists a constant C > 0 , so that In addition, for every scalar exponent ˆ µ ≥ µ and every w ∈ B δ,µ,q,s/ 2 and ω ∈ B δ, ˆ µ,q,s/ 2 , it follows that g ( i ) ( w + ω ) -g ( i ) ( w ) is in X δ, ˆ µ,q and the estimate holds. Proof. This follows from Proposition 3.9 in Chapter 13 of [40] applied to g ( i ) ( w ) -1, together with the Taylor theorem for the exponential function.", "pages": [ 73, 74, 75 ] }, { "title": "Sobolev spaces and duality", "content": "Following [16, Chapter VI] or [36], one defines the Sobolev space H s ( R n ) for any s ∈ R as the set of temperate distributions u such that ̂ u (1 + | ξ | 2 ) s/ 2 ∈ L 2 ( R n ), where ̂ u := F u is the Fourier transform (in the sense of temperate distributions) of u . The norm defined by turns this space into a Banach space. If s = q for any non-negative integer q , then H s ( R n ) is equivalent to the standard ( p = 2) Sobolev space H q ( R n ). For general s ∈ R , the space H s ( R n ) is in fact a Hilbert space for the scalar product Let u ∈ H -s ( R n ) and v ∈ H s ( R n ) for any s ∈ R . Then the dual pairing between H s ( R n ) and H -s ( R n ), is well-defined, as a consequence of the inequality By means of this pairing, we can identify H -s ( R n ) with H s ( R n ) ∗ (the dual space) as follows. For every u ∈ H -s ( R n ), the map ( u, · ) : H s ( R n ) → R is a bounded linear functional, i.e., an element of H s ( R n ) ∗ . Conversely, according to the Riesz representation theorem, there exists a unique element w φ ∈ H s ( R n ) for each element φ ∈ H s ( R n ) ∗ such that for all v ∈ H s ( R n ) . The last expression can be written as where ̂ v φ := ̂ w φ ( ξ )(1+ | ξ | 2 ) s is the Fourier transform of v φ := F -1 ( ̂ w φ ( ξ )(1+ | ξ | 2 ) s ). We have v φ ∈ H -s ( R n ), since ̂ v φ (1+ | ξ | 2 ) -s/ 2 = ̂ w φ ( ξ )(1+ | ξ | 2 ) s/ 2 ∈ L 2 ( U ). By means of the pairing above, we have thus constructed a unique element v φ ∈ H -s ( R n ) corresponding to each φ ∈ H s ( R n ) ∗ . In this sense, we can therefore identify H -s ( R n ) with H s ( R n ) ∗ for every s ∈ R . The following result concerns the relationship between Sobolev spaces of different indices. Proposition C.1. For every s ∈ R and σ ≥ 0 , the space H s + σ ( R n ) is a dense subset of H s ( R n ) . Proof. We first show that H s + σ ( R n ) is indeed a subset of H s ( R n ) for σ ≥ 0. Suppose that u ∈ H s + σ ( R n ). Calculating the ‖ · ‖ s norm of u , we obtain from which it follows that u ∈ H s ( R n ). To check that H s + σ ( R n ) is a dense subset, it is sufficient to note (see, e.g., [16]) that C ∞ 0 ( R n ) (the space of smooth functions with compact support) is dense in both H s ( R n ) and H s + σ ( R n ).", "pages": [ 75, 76 ] }, { "title": "Convergence results in Sobolev spaces", "content": "One can use this dense inclusion property (Proposition C.2) together with the duality properties discussed above to derive certain convergence and closedness-type results for sequences in Sobolev spaces. We first discuss a result of this sort for Sobolev spaces on R n , and then do the same for Sobolev spaces on T 1 . Proposition C.2. Choose s, s 0 ∈ R so that 0 ≤ s 0 < s . Let ( w m ) be a bounded sequence in H s ( R n ) in the sense that there exists a constant C > 0 so that ‖ w m ‖ s ≤ C , for all integer m . Moreover, suppose that ( w m ) converges to some w ∈ H s 0 ( R n ) ; i.e., ‖ w m -w ‖ s 0 → 0 . Then, w is contained in H s ( R n ) . Proof. The boundedness of the sequence implies the existence of a subsequence of ( w m ) (which for simplicity we identify with ( w m )) which converges weakly. Hence, as a consequence of the Riesz Representation Theorem and the above dual pairing in Eq. (C.1), there exists an element ˜ w ∈ H s ( R n ), so that, for every Y ∈ H -s ( R n ), ( Y, ˜ w -w m ) → 0 (C.3) We wish to show that w = ˜ w and hence that w ∈ H s ( R n ). To do this, we consider an arbitrary X ∈ H -s 0 ( R n ) and the dual pairing | ( X, ˜ w -w ) | ≤ | ( X, ˜ w -w m ) | + | ( X,w -w m ) | , where ˜ w -w is considered as an element of H -s 0 ( R n ), and where we have used the triangle inequality. Since X ∈ H -s 0 ( R n ) ⊂ H -s ( R n ) according to Proposition C.1, we can consider the first term on the right hand side as a pairing between H s ( R n ) and H -s ( R n ), and hence Eq. (C.3) implies that this term can be made arbitrarily small by choosing m sufficiently large. The second term is considered as a pairing between H s 0 ( R n ) and H -s 0 ( R n ) so that Eq. (C.2) yields Also this term can be made arbitrarily small by choosing m sufficiently large. Hence, we have found that ( X, ˜ w -w ) = 0 for all X ∈ H -s 0 ( R n ). Now, the Riesz representation theorem implies that for every X ∈ H -s 0 ( R n ) there exists precisely one ˜ X ∈ H s 0 ( R n ) for which In particular, we may choose ˜ X = ˜ w -w , which implies that ˜ w -w = 0. Corollary C.3. Choose non-negative integers q and q 0 so that q 0 < q . Let ( w m ) be a bounded sequence in H q ( T 1 ) , in the sense that there exists a constant C > 0 so that ‖ w m ‖ H q ( T 1 ) ≤ C , for all integers m . Moreover, suppose that ( w m ) converges to some w ∈ H q 0 ( T 1 ) ; i.e., ‖ w m -w ‖ H q 0 ( T 1 ) → 0 . Then, w is contained in H q ( T 1 ) . Proof. We formulate the proof so that it can be easily generalized to general smooth orientable, connected compact Riemannian manifolds M in any dimension n . For this paper, the relevant special case is M = T 1 . Let (( U i , φ i )) be a collection of coordinate charts, i.e., open subsets U i ⊂ M and homeomorphisms φ i : V i → U i where V i := φ -1 i ( U i ) are open subset of R n , which cover M , i.e., M = ⋃ i U i . It follows from compactness that we can assume that there are N such coordinate charts. Let ( τ i ) be a subordinate partition of unity. Then we find that ( w m ) is a bounded sequence in H q ( M ) if and only if for all i = 1 , . . . , N , we have that ( w m · φ i ) is a bounded sequence in H q ( V i ). Moreover, ‖ w m -w ‖ H q 0 ( T 1 ) → 0 for some w ∈ H q 0 ( M ) if and only if for all i = 1 , . . . , N , we have that ‖ w m · φ i -w · φ i ‖ H q 0 ( V i ) → 0 (since w · φ i ∈ H q 0 ( V i )). Now, the Stein Extension Theorem (Theorem 5.24 in [1]) implies the existence of total extension operators E i (Definition 5.17 in [1]), which are linear maps E i from functions defined on V i to functions defined on R n with the following property: If f ∈ H r ( V i ) for any non-negative integer r , then Hence, we find that ( w m ) is a bounded sequence in H q ( M ) if and only if for all i = 1 , . . . , N , we have that ( E i ( w m · φ i )) is a bounded sequence in H q ( R n ). Moreover, ‖ w m -w ‖ H q 0 ( T 1 ) → 0 for some w ∈ H q 0 ( M ) if and only if for all i = 1 , . . . , N , we have that ‖ E i ( w m · φ i ) -E i ( w · φ i ) ‖ H q 0 ( R n ) → 0 (since E i ( w · φ i ) ∈ H q 0 ( R n )). It follows from Proposition C.2, that E i ( w · φ i ) ∈ H q ( R n ). Hence, w · φ i ∈ H q ( V i ). Since this is true for all i = 1 , . . . , N , it follows that w ∈ H q ( M ).", "pages": [ 77, 78 ] } ]

[ { "title": "A conformal approach for the analysis of the non-linear stability of pure radiation cosmologies", "content": "Christian Lübbe ∗ ,1,2 and Juan Antonio Valiente Kroon † ,2 1 Department of Mathematics, University of Leicester, University Road, LE1 8RH, United Kingdom. 2 School of Mathematical Sciences, Queen Mary, University of London, Mile End Road, London E1 4NS, United Kingdom. August 29, 2018", "pages": [ 1 ] }, { "title": "Abstract", "content": "The conformal Einstein equations for a tracefree (radiation) perfect fluid are derived in terms of the Levi-Civita connection of a conformally rescaled metric. These equations are used to provide a non-linear stability result for de Sitter-like tracefree (radiation) perfect fluid Friedman-Lemaître-Robertson-Walker cosmological models. The solutions thus obtained exist globally towards the future and are future geodesically complete. PACS: 04.20.Ex, 04.20.Ha, 98.80.Jk", "pages": [ 1 ] }, { "title": "1 Introduction", "content": "The conformal Einstein field equations have proven a powerful tool to analyse the stability and the global properties of vacuum, electro-vacuum and Yang-Mills-electro-vacuum spacetimes -see e.g. [9, 10, 11, 18, 19, 20]. By contrast, to the best of our knowledge, there has been no attempt to make use of conformal methods to analyse similar issues in spacetimes whose matter content is given by a perfect fluid. In this article we make a first step in this direction. We discuss the stability and the global properties of a class of cosmological spacetimes having as a source a perfect fluid with tracefree energy-momentum tensor. The solutions we construct are non-linear perturbations of a Friedman-Lemaître-Robertson-Walker (FLRW) reference spacetime. The present analysis is to be regarded as a first step in the development of conformal methods for the discussion of cosmological models whose matter content is described by a perfect fluid. Hence, we restrict our attention to the simplest case from the point of view of conformal methods: perturbations of a traceless prefect fluid cosmological model with compact spatial sections of positive constant curvature. Generalisation of our analysis to more general background solutions and equations of state will be discussed elsewhere. The problem of the non-linear stability of FLRW cosmologies and the exponential decay of perturbations is considered in [23]. In that reference, a frame formulation of the Einstein-perfect fluid system [12] is used to obtain a suitable symmetric hyperbolic evolution system for which the Kreiss-Lorenz theory can be readily applied -see [17]. The results obtained hold for a large class of equations of state, but not very stiff ones -like the pure radiation case discussed in the present article. More recently, the problem of the non-linear stability of the irrotational Euler-Einstein system for de Sitter-like spacetimes has been analysed in [24]. This analysis shows that FLRW background solutions with pressure ˜ p and density ˜ ρ related by a barotropic equations of state of the form ˜ p = γ ˜ ρ with 1 < γ < 4 3 are future asymptotically stable under small irrotational perturbations. An extension of this analysis to the case of fluids with non-zero vorticity has been given in [26]. It is notable that the case of a pure radiation perfect fluid cannot be covered by the analysis of [23, 24, 26]. By contrast, from the point of view of conformal methods, the pure radiation perfect fluid case turns out to be one of the simplest scenarios to be considered. Finally, it should be mentioned that conformal methods have been used to pose an initial value problem for the Einstein-Euler system at the Big Bang for a class of cosmological models with isotropic singularities -see [1]. The methods used in that work do not allow, however, to obtain global existence assertions towards the future. Our main result can be stated as follows: Theorem. Suppose one is given Cauchy initial data for the Einstein-Euler system with a de Sitter-like cosmological constant and equation of state for pure radiation. If the initial data is sufficiently close to data for a FLRW cosmological model with the same equation of state, value of the cosmological constant and spatial curvature k = 1 , then the development exists globally towards the future, is future geodesically complete and remains close to the FLRW solution. A detailed and technically precise version of this result is given in Theorem 2. Remark 1. Similar future global existence and stability results can be obtained using the methods of this article for a FLRW background solution with pure radiation equation of state, de Sitter-like or vanishing cosmological constant, λ , and k = 0 , -1 . These models expand indefinitely towards the future, and remarkably, their scale factor can be computed explicitly -see [14]. In the cases with λ = 0 , minor technical modifications need to be introduced to account for a null conformal boundary. The stability of these models will be discussed elsewhere by means of different (conformal) methods. Remark 2. The restriction of our analysis to the case of perfect fluids with traceless energymomentum tensor is technical: in this case the equation of conservation of energy momentum transforms homogeneously under conformal transformations. In the case of perfect fluids with an energy-momentum tensor with non-vanishing trace a regularisation of the rescaled equations of motions must be carried out. The analysis for the wave equation in [2, 15] may be a guide for this type of generalisation of our analysis.", "pages": [ 1, 2 ] }, { "title": "Structure of the article", "content": "The article is organised as follows: Section 2 provides a summary of the tensorial conventions to be used in the present article. Furthermore, in Subsection 2.2 a discussion of the procedure of how to coordinatise and introduce frame fields of the 3-sphere, S 3 is provided. Section 3 provides general remarks concerning perfect fluid cosmological models and a summary of the properties of the background solutions required in our subsequent analysis. These are summarised in Proposition 1. Section 4 gives a brief summary of the conformal Einstein field equations with matter. Section 5 provides a discussion of the Euler equations in the context of the conformal field equations. In Section 6 we discuss gauge considerations and the procedure leading to a hyperbolic reduction of the conformal field equations. The keys steps in this procedure have been discussed extensively elsewhere, so that this discussion is kept to a minimum. In particular, Subsection 6.2 provides a summary of the structural properties of the conformal evolution equations while Subsection 6.3 analyses the issue of the propagation of the constraints. Section 7 casts the FLRW background as a solution of the conformal field equations of Section 4, and analyses some of its properties. Finally, Section 8 is concerned with our main result -the existence and stability result for perfect fluid cosmologies with a de Sitter-like cosmological constant as given in Theorem 2.", "pages": [ 2 ] }, { "title": "2.1 Index and curvature conventions", "content": "Throughout this article we work with a spacetime ( ˜ M , ˜ g µν ) , where ˜ g µν , ( µ, ν = 0 , 1 , 2 , 3 ) is a Lorentzian metric with signature (+ , -, -, -) . We will denote by ˜ ∇ the Levi-Civita connection of ˜ g µν -that is, the unique torsion-free connection that preserves the metric ˜ g µν . In the sequel, ˜ R µνλρ , ˜ R µν and ˜ R will denote, respectively, the Riemann curvature tensor, the Ricci tensor and the Ricci scalar of the Levi-Civita connection ˜ ∇ . The conventions for the curvature used in this article are such that As a consequence of our signature conventions, then λ < 0 corresponds to de Sitter-like values of the cosmological constant, while λ > 0 corresponds to anti-de Sitter-like values. While µ, ν, . . . denote spacetime tensorial indices, α, β, . . . denote spatial tensorial ones. Most of our discussion will be based on a frame formalism in which i, j, . . . denote spacetime indices ranging 0 , . . . , 3 . Similarly, a, b, . . . will denote spatial indices ranging 1 , 2 , 3 . Spinorial expressions and arguments will be used routinely, and we will follow the conventions of [21]. Consequently, the indices A, B, . . . will be spinorial ones.", "pages": [ 3 ] }, { "title": "2.2 Coordinates and vector fields on the 3-sphere", "content": "The present analysis will be concerned with spacetimes which are conformal to manifolds with topology I × S 3 where I is an open interval on R . In what follows, the manifold S 3 will always be thought of as the following submanifold of R 4 : The restrictions of the functions x A , A = 1 , 2 , 3 , 4 on R 4 to S 3 will again be denoted by x A . The vector fields on R 4 are tangent to S 3 . In the sequel, they will always be considered as vectors on S 3 . The vector fields { c r } ≡ { c 1 , c 2 , c 3 } constitute a globally defined frame on S 3 which is orthonormal with respect to the standard metric of S 3 . Moreover, the frame { c 1 , c 2 , c 3 } can be completed with a vector c 0 which is orthonormal to the standard metric on I × S 3 , { c s } ≡ { c 0 , c 1 , c 2 , c 3 } . Let ( M , g µν ) be a spacetime such that the manifold M is diffeomorphic to R × S 3 . A map Φ defined on an open subset U ⊂ M will be said to be a cylinder map if it maps U diffeomorphically onto a set I × S 3 , such that the sets Φ -1 ( { τ }× S 3 ) are spacelike Cauchy hypersurfaces of M and the curves I /owner τ → Φ -1 ( τ, p ) ⊂ M , p ∈ S 3 are timelike with respect to the metric g µν . The cylinder map will be used to pull-back to U the coordinates ( τ, x A ) ≡ ( τ, x ) in I × S 3 . Furthermore, one can use Φ to pull-back to U the frame fields c s defined in the previous paragraph. For simplicity of notation, such pull-back will be denoted again by c s .", "pages": [ 3 ] }, { "title": "3 General remarks about FLRW cosmological models", "content": "A cosmological model ( ˜ M , ˜ g µν , ˜ u µ ) is a representation of the universe at a particular averaging scale. It is defined by a Lorentzian metric ˜ g µν on the manifold ˜ M and by a family of fundamental observers whose congruence of worldlines is represented by the timelike 4-velocity ˜ u µ -usually taken to be the matter 4-velocity. It is usually assumed that this congruence is expanding at some time. These assumptions together with a specification of the matter content are used to determine the dynamics of the universe. In what follows, it will be assumed that the interaction between geometry and matter is described by the Einstein field equations and the energy-momentum conservation equation As already mentioned, the conventions for the cosmological constant λ used in the present article are such that in vacuum, the case λ < 0 describes a de Sitter-like spacetime, while the case λ > 0 corresponds to an anti-de Sitter-like one. Our discussion will be concerned with energy-momentum tensors of perfect fluids for which where ˜ ρ , ˜ p and ˜ u µ denote, respectively, the density, pressure and 4-velocity of the cosmological fluid. The fluid 4-velocity ˜ u µ is timelike and satisfies the normalisation condition ˜ u µ ˜ u µ = 1 . The background solution whose non-linear stability will be considered in the present article belongs to the family of so-called Friedman-Lemaître-Robertson-Walker (FLRW) cosmological models. The FLRW models are homogeneous and isotropic. Their line element is usually given in the form where a ( t ) is the so-called scale factor . This metric automatically defines a perfect fluid energymomentum tensor. When k = 0 the spatial sections are flat, if k < 0 the spatial sections have negative curvature, while if k > 0 , the spatial sections have positive curvature. The present analysis is concerned with FLRW cosmologies with spatial sections of positive curvature ( k = 1 ) for which coordinates can be introduced such that: with the standard line element of S 3 in polar coordinates. If the cosmological fluid satisfies the barotropic equation of state ˜ p = ( γ -1)˜ ρ , where 1 ≤ γ ≤ 2 is a constant, then the evolution of the scale factor a ( t ) is governed by the Friedmann equation : where c is a constant. In what follows we will only be concerned with the case γ = 4 3 corresponding to the so-called traceless perfect fluid (pure radiation). Furthermore, we assume λ < 0 . Equation (7) admits a static (i.e. time independent solution) in which the values of the scale factor and the cosmological constant are related by: /negationslash /negationslash In the dynamical case, under the assumptions γ = 4 3 , λ < 0 , the Friedmann equation (7) can be integrated explicitly -see e.g. [14]. Different types of solutions are obtained, depending on the relative value of λ with respect to λ 0 as given in equation (8), where a 0 = 0 is now the value of the scale factor at some fiduciary time t = t 0 = 0 . The relevant properties for the analysis of these solutions are summarised in the following proposition: /negationslash Proposition 1. For a FLRW cosmology with k = 1 , γ = 4 3 and λ < 0 , λ = λ 0 , the scale factor, a ( t ) , is a smooth, non-vanishing and monotonically increasing function for t ∈ [ t 0 , ∞ ) , with t = t 0 > 0 and a 0 = a ( t 0 ) > 0 . Furthermore, and one has the limits as t →∞ . The pressure for these models is given by where ˜ ρ 0 = ˜ ρ ( t 0 ) . In particular, one has that ˜ ρ → 0 as t →∞ . The proof of this proposition follows from direct inspection of the explicit solutions -see e.g. [14], page 78. Remark 3. A similar type of result can be obtained for FLRW models with γ = 4 3 , λ ≤ 0 and k = -1 , 0 . Again, see [14].", "pages": [ 3, 4, 5 ] }, { "title": "4 The conformal field equations with matter", "content": "The stability of the solutions to the Einstein equations described by the metric ˜ g µν corresponding to the line element (5) will be analysed in terms of a conformally related ( unphysical) metric g µν . This strategy leads to consider the conformal Einstein field equations. The idea of vacuum conformal Einstein field equations expressed in terms of the Levi-Civita connection ∇ of the metric g µν and associated objects was originally introduced in [4, 5, 6]. The generalisation of these conformal equations to physical spacetimes containing matter was discussed in [10]. More recently, a more general type of vacuum conformal equations -the extended conformal Einstein field equations - expressed in terms of a Weyl connection ˆ ∇ has been introduced -see [11].", "pages": [ 5 ] }, { "title": "4.1 Conformal rescalings", "content": "All throughout we assume that the two metrics ˜ g µν and g µν are conformally related to each other via where Θ is a non-negative scalar field -the conformal factor. The Christoffel symbols ˜ Γ µ ρ ν and Γ µ ρ ν of the associated Levi-Civita connections ˜ ∇ and ∇ are related by where Υ λ = Θ -1 ∇ λ Θ and S µν ρλ is the conformally invariant tensor", "pages": [ 5 ] }, { "title": "4.2 Curvature tensors", "content": "In a 4-dimensional spacetime the Schouten tensor , P µν , of the connection ∇ is defined by The Schouten tensor of the connection ˜ ∇ is defined by a similar expression involving the physical Ricci tensor and scalar. The tensors ˜ P µν and P µν are related by We can thus decompose the Riemann curvature tensor, R µ νλρ , of the connection ∇ into its irreducible parts as where C µ νλρ denotes the conformally invariant Weyl tensor . As ∇ is a Levi-Civita connection it satisfies the first and second Bianchi identities : In our discussion of the conformal field equations with matter we will make use of the physical and unphysical Cotton-York tensors ˜ Y λρν and Y λρν given, respectively, by The tensor Y λρν appears in the once contracted Bianchi identity Finally, it is noticed that the twice contracted Bianchi identity takes the form where P = g λν P λν .", "pages": [ 5, 6 ] }, { "title": "4.3 Frame and spinor formulations", "content": "In what follows, consider a frame field { e i } , i = 0 , . . . , 3 which is orthogonal with respect to the metric g µν . By construction one has that In order to discuss the extended conformal Einstein field equations, it will be convenient to regard, for the moment, the connection ∇ only as a metric connection with respect to g µν -i.e. ∇ λ g µν = 0 . Under this assumption, the connection ∇ could have torsion, and thus, it would not be a Levi-Civita connection. The connection coefficients, Γ i k j , of ∇ with respect to the frame e k are defined by the relation As a consequence of having a metric connection, the connection coefficients satisfy The torsion, Σ i k j , of the connection ∇ is defined by If Σ i k j = 0 so that the connection ∇ is the unique Levi-Civita connection of g µν , the connection coefficients acquire the additional symmetry Related to the g -orthonormal frame e k we will consider a normalised spinor dyad { δ A } , A = 0 , 1 , such that where σ k AA ' are the constant van der Waerden symbols. In the sequel, a space spinor formalism will be introduced -see e.g. [25]. To this end, we consider a timelike spinor τ AA ' which in terms of the dyad { δ A } can be expressed as In particular, one has the normalisation condition τ AA ' τ AA ' = 2 . The space spinor formalism allows to turn primed indices in spinorial expressions into unprimed ones by suitable contractions with τ A A ' -see [10, 18, 19, 20] for more details. We simply recall that the space spinor decomposition of a spinor u AA ' is given by where where", "pages": [ 6, 7 ] }, { "title": "4.4 The conformal field equations with tracefree matter", "content": "In our subsequent discussion it will be convenient to distinguish between the geometric curvature r k lij -i.e. the expression of the curvature related to the connection coefficients Γ i j k - and the algebraic curvature R k lij -i.e. the decomposition of the curvature in terms of irreducible components given by equation (12). One has that Following [13], in the sequel it will be convenient to introduce the variables Furthermore, we also consider the following zero quantities -cfr. [10]: The interpretation of the zero quantities (19a)-(19d) is as follows: the zero quantity given by (19a) measures the torsion of the connection ∇ ; that of (19b) relates the expression of the curvature of ∇ with its decomposition in terms of irreducible components. Equations (19c) and (19d) measure the deviation from the fulfillment of the once contracted Bianchi identity. Finally, equations (19e), (19f) and (19g) bring into play the definitions (18b) and (18c) and give rise to differential conditions for the fields Θ , d i and s . The conformal Einstein field equations with matter are then given by These equations yield differential conditions for the frame coefficients e i , the spin coefficients Γ i j k , the components of the Schouten tensor P ij , the rescaled Weyl tensor d k lij , the conformal factor Θ , the 1-form d i , and the scalar s , respectively. As discussed in e.g. [13], equation (19h) has the role of a constraint which holds by virtue of the other conformal field equations if it is satisfied on some initial hypersurface. It is noticed that as the torsion, Σ i k j , is being introduced as a zero quantity, it can be consistently set to zero in the geometric curvature appearing in the definition for the zero quantity Ξ k lij -equation (19b). Equations (20a)-(20b) need to be complemented with the energy-momentum conservation equation (4). Its particular details will depend on the matter model under consideration. Remark 4. Using a direct generalisation of the arguments presented in [5, 4] one can show that a solution to the conformal Einstein field equations with matter (20a)-(20b) and (4) give rise to a solution to the physical Einstein-matter system (3)-(4) -see also Theorem 3.1 in [6]. Remark 5. As a result of the conformal rescaling (9), the conformal equations (20a)-(20b) have a built-in conformal freedom which needs to be specified in order to deduce suitable evolution equations for the conformal fields. Further gauge freedom in equations (20a)-(20b) is concerned with the partial specification of the frame e k and the choice of coordinates. These will be specified by the choice of suitable gauge source functions.", "pages": [ 7, 8 ] }, { "title": "5 Perfect fluids in the context of the conformal approach", "content": "In this section we present a discussion of the relativistic equations describing a perfect fluid which is geared towards our particular applications.", "pages": [ 8 ] }, { "title": "5.1 The energy-momentum tensor and its transformation rules", "content": "Given the spacetime ( ˜ M , ˜ g µν ) , the energy-momentum tensor for a perfect fluid with 4-velocity ˜ u i , pressure ˜ p , and density ˜ ρ has the form In order to perform a discussion of the perfect fluid in the conformally rescaled (unphysical) spacetime one introduces unphysical versions of the physical fields. More precisely, one defines Following the approach used in the discussion of geometric fields, we will work directly with the frame components T ij ≡ e i µ e j ν T µν and u i ≡ e i µ u µ with respect to a g -orthonormal frame e i . Thus T ij = ( ρ + p ) u i u j - pη ij . We observe that ˜ g (˜ u, ˜ u ) = 1 implies that g ( u, u ) = 1 . Now, using u i = η ij u j , u i = η ij u j , our signature convention implies A computation using the standard transformation rules for the covariant derivatives of conformally rescaled metrics yields Consequently, the (physical) equation for the conservation of energy-momentum implies an analogous equation for the (unphysical) conformally rescaled spacetime only if the energy-momentum tensor ˜ T ij is tracefree -see [10]. Notice that ˜ T ≡ ˜ η ij ˜ T ij = 0 if and only if T ≡ η ij T ij = 0 . A quick computation shows that for a perfect fluid the tracefreeness of the energy-momentum tensor implies ρ -3 p = 0 -in other words γ = 4 3 . Hence This class of perfect fluids is usually referred to as pure radiation . In the present article, our analysis will be restricted to the case of tracefree perfect fluids. The unphysical energy-momentum tensor for this class of perfect fluids reduces to As a consequence of the definition of the 4-velocity u i it follows that These identities will be used to rewrite the component u 0 and its derivatives in terms of the spatial components u a and their derivatives. This procedure will be central for the construction of a symmetric hyperbolic system for the matter variables. It is also noticed that equation (24a) implies This expression shows that if u i u i = 1 at some point in a fluid flow line, then u i u i = 1 in the whole flow line.", "pages": [ 8, 9 ] }, { "title": "5.2 The energy conservation equation and the equations of motion", "content": "A direct computation shows that the conservation equation (22) implies This equation can be split into components parallel and orthogonal to u i : where These equations are the conformal versions of the equation of energy conservation and the equations of motion -see e.g. [3]. It is noticed that equations (26a) and (26b) can be combined to give This equation will be used in the sequel to eliminate the gradient of the unphysical density from certain expressions.", "pages": [ 9 ] }, { "title": "5.3 A symmetric hyperbolic system for the fluid fields", "content": "The equations of conservation of energy and motion will be used to construct a symmetric hyperbolic system of evolution equations for the unphysical density ρ and the spatial components of the unphysical velocity u a . The procedure used here follows the presentation given in [3]. In the sequel, it should be understood that, consequently with equation (24a), Substituting identity (24b) into (26a) gives where the extra factor 1 / 4 ρ has been included to ensure symmetric hyperbolicity. Similarly, from equation (26b) one deduces In order to obtain suitable evolution equations for the spatial components of the 4-velocity, we consider the combination or equivalently A direct inspection shows that: Lemma 1. Equations (28) and (29) constitute a symmetric hyperbolic system for the fields ρ and u a . One also has that: Lemma 2. A solution ( ρ, u a ) to the evolution equations (28) and (29) implies a solution ( ρ, u 0 , u a ) to equation (25) with u 0 = √ 1 -u a u a . Proof. We need to show Z j = 0 . The definition of u 0 implies that (24b) and (24c) hold. Now, given a solution to (28) and (29), the right hand side of (28) can be rewritten so as to yield u j Z j = 0 . Substitution into the left hand side of (29) gives Contracting with u a and using (24a), as well as u 0 ≥ 1 gives first Z 0 = 0 and then Z a = 0 . Hence a solution to (28) and (29) satisfies (25). Remark 6. Let u AA ' denote the spinorial counterpart of the 4-velocity vector u µ . The spinor u AA ' can be split using the spinor τ AA ' as done in (17). This implies where σ a AB denote the spatial Infeld-van der Waerden symbols. It follows that (25) implies a symmetric hyperbolic system for the spinorial components u and u AB . The explicit form of these equations will not be required in our subsequent analysis.", "pages": [ 9, 10 ] }, { "title": "5.4 The Cotton-York tensor of a traceless perfect fluid spacetime", "content": "The matter field quantities feedback into the geometric part of the conformal field equation through the physical Cotton-York tensor ˜ Y µνλ . In what follows, the latter is expressed in terms of tensors, however the frame and spinor component versions are easily derived from these equations. For a tracefree energy momentum tensor the physical Schouten tensor is given by ˜ P ij = 1 2 ˜ T ij so that Rewriting this expression in terms of unphysical quantities one obtains for T ijk = Θ -2 ˜ Y ijk that The last two terms in this expression are polynomial in ρ and the components u i . The first term, however, contains derivatives of u i and ρ that would enter the principal part of the Cotton-York and Bianchi equations. The fluid equations cannot be used to eliminate these derivatives. In order to get around this difficulty, we introduce new variables ρ k and u ij and corresponding zero quantities q k and y ij via Observe that if q k = 0 and y ij = 0 , one then has that u ij u j = 0 and y ij u j = 0 , so that one can write Furthermore, from u ij u j = 0 , it also follows that Finally, if q k = 0 , y ij = 0 , then the first term of T ijk can be written as", "pages": [ 10, 11 ] }, { "title": "5.4.1 A symmetric hyperbolic system for ρ k and u k a", "content": "The evolution equations for ρ and u a are derived from equation (25). Taking derivatives of (25) and commuting them gives: where all terms with at most one derivative of ρ or u k have been gathered in V kj . In view of this discussion, in the sequel we will consider the field equation for ρ k and u ij given by the following zero quantity: with From the equation Z ij = 0 one derives, in analogy to (26b) and (26a), that where (32b) has been used and Finally, we rewrite (35a) in the form where Similarly, the combination leads to the evolution equation where ˆ X k a is a combination of and terms from expression (32a). In analogy to Lemma 1 one can readily verify that: Lemma 3. If y ij = 0 , then equations (36) and (37) constitute a symmetric hyperbolic system for the fields ρ k and u k a . A similar argument to the one leading to Lemma 2 yields: Lemma 4. Let y ij = 0 . A solution ( ρ k , u k a ) to the evolution equations (36) and (37) implies a solution to equation (34) .", "pages": [ 11, 12 ] }, { "title": "5.4.2 The subsidiary equations for the fluid variables", "content": "In this section we derive evolution equations for the zero quantities q i and y ij . These subsidiary equations will be of relevance in the discussion of the propagation of the constraints -see Section 6.3. Subtracting equation (33) from equation (34) gives Now, using substitutions like one can deduce that all individual terms in sums of W kj -V kj in equation (38) contain at least one zero quantity. Repeating the discussion for the evolution equations for ( ρ k , u ij ) with ( q k , y ij ) one finds that where all terms in Y k and Y k a contain zero quantities. The evolution equations (39) and (40) constitute a symmetric hyperbolic system for the independent components of q k and y ij .", "pages": [ 12 ] }, { "title": "5.4.3 Final remarks", "content": "/negationslash As a consequence of the analysis in the previous subsections one has that the components T ijk of the tensor T µνλ with respect to the frame e i are polynomial expressions of the unknowns ρ , ρ j , u i and u ij . If desired, the dependence with respect to ρ k can be eliminated using equation (27). As long as ρ = 0 , the fields ρ , ρ j , u i and u ij satisfy symmetric hyperbolic equations 1 . Finally, it is noticed that if the fields ρ , ρ j , u i and u ij are regular at the points where Θ = 0 , then T ijk is also regular -and consequently, also equations (19c) and (19d) are formally regular.", "pages": [ 13 ] }, { "title": "6 A symmetric hyperbolic reduction of the conformal field equations", "content": "In the previous section it has been shown how the equations of motion for the fluid variables and their derivatives lead to a system of symmetric hyperbolic equations independently of geometric gauge considerations. The purpose of this section is to briefly discuss a reduction procedure for the geometric unknowns. Our treatment is inspired on the one given in [10], but it also combines ideas from [9, 19].", "pages": [ 13 ] }, { "title": "6.1 Gauge freedom", "content": "As mentioned previously, the conformal Einstein field equations (20a) and (20b) are endowed with three classes of gauge freedom: conformal, coordinate and frame. In what follows, we briefly discuss a procedure for fixing this freedom.", "pages": [ 13 ] }, { "title": "6.1.1 Conformal gauge freedom", "content": "As already mentioned, the conformal Einstein field equations (20a) and (20b) admit certain freedom in the specification of the representative, g µν , of the conformal class [˜ g µν ] which will be used as the unphysical metric -see e.g. [13] and references therein for more details. Assuming for a moment that one has a solution to the conformal Einstein field equations with matter, it follows then by contraction of indices in equation (11) together with the tracefreeness of the energy momentum tensor that As discussed in, e.g. [13], this equation can always be solved locally so that the condition holds. This condition fixes the conformal freedom in the equations (20a) and (20b).", "pages": [ 13 ] }, { "title": "6.1.2 Coordinate and gauge freedom", "content": "In order to fix the coordinate and frame gauge freedom, we make use of the notion of gauge source functions -see [7, 10, 13]. The gauge source freedoms will be chosen so as to render symmetric hyperbolic evolution equations for the geometric unknowns. In what follows we encounter equations of the form where the dots denote an arbitrary set of indices -cfr. equations (19a)-(19c). The spinorial equivalents of the above equation are given by and its complex conjugate. Now, an equation of the form is well known to imply a symmetric hyperbolic evolution system for the independent components of M AA ' -see e.g. [7]. Note, however, that equation (42) contains no information about the skew term which can be specified arbitrarily. Thus, by adding (44) with a convenient choice of a gauge source function, F ··· , to (42) one obtains an equation of the form of (43), from where a symmetric hyperbolic system can be extracted -see e.g. [7, 13]. The previous discussion will be implemented in the field equations (19a), (19b) and (19c). These equations provide differential conditions for the fields e µ i , P ij and Γ i j k . Let e µ AA ' , P AA ' BB ' and Γ AA ' BB ' CC ' denote the spinorial counterparts of these fields. As a consequence of the metricity of the connection, instead of working with Γ AA ' BB ' CC ' , we will consider a spinorial field ' For convenience, define the gauge source functions where the second equality in the definition of F BB ' follows from the twice contracted Bianchi identity for the unphysical connection ∇ . Motivated by their value in the reference solution (the conformal FLRW solution) the gauge source functions will be fixed by the conditions Notice, in particular, that the last condition is consistent with the conformal gauge condition (41). As discussed in [7] -see also [10]- a particular choice of the coordinate and frame gauge functions F µ and F ( BC ) fixes the coordinates and frame 2 . The spacetimes to be considered in the present analysis have the topology of R × S 3 . Given an initial manifold S for the spacetime, then there is a diffeomorphism Φ : S → S 3 which allows to pull-back coordinates from S 3 to S . These coordinates on the initial manifold S will be used as the initial value of the spatial part of the spacetime coordinates. The time coordinate will be set initially to zero. The initial value of the frame e i is set by choosing on S some arbitrary orthonormal spatial frame e a (with respect to the 3-metric of S ). The e 0 vector is set to coincide initially with the (spacetime) normal to S .", "pages": [ 13, 14 ] }, { "title": "6.2 The evolution equations", "content": "The hyperbolic reduction of the matter variables has already been discussed in Sections 5.3 and 5.4.1. In what concerns the evolution equations for the geometric variables, we follow the procedure indicated in [10]. This consists of a rewriting the spinorial version of the conformal field equations (20a)-(20b) in terms of space spinors so that the resulting equations contain only unprimed indices. In order to encompass the full information of the field equations, one has to include into the set of equations their Hermitian conjugates. If the fields and equations are then decomposed into their irreducible parts, then the equations split in a natural way into symmetric hyperbolic evolution and constraint equations. This procedure is straightforward, but involves lengthy computations, most of which can now be implemented in a computer algebra system like the suite xAct for Mathematica 3 . Γ AA BC such that The required evolution equations have already been deduced in [10]. Their detailed form will not be required here. Instead we present a summary of their key structural properties. In what follows let where only the independent irreducible components of the spinors are taken into account. In terms of these objects, the evolution equations have the form In equations (47a)-(47d), E denotes the 5 × 5 unit matrix, while A s [ φ ] , A s [ /rho1 ] , A s [ ψ ] denote smooth symmetric matrix-valued functions of their respective arguments. In particular, A 0 [ φ ] ( 0 ) = 0 and A 0 [ /rho1 ] , A 0 [ ψ ] are positive definite if ρ > 0 . In addition, B [ υ ] , B [ φ ] , B [ /rho1 ] and B [ ψ ] are smooth matrixvalued functions of υ . Finally, M [ υ ] , M [ φ ] and M [ ψ ] are non-linear vector-valued functions of their respective arguments. These functions are smooth if u 0 = 0 . For convenience of the discussion, in the sequel, we define Remark 7. The deduction of the evolution equations (47a)-(47d) assumes the choice of gauge source functions given in (46).", "pages": [ 14, 15 ] }, { "title": "6.3 Propagation of the constraints", "content": "An analysis of the so-called subsidiary equations describing the propagation of the zero quantities in terms of which the geometric part of the conformal field equations -see equations (20a)(20b)- is expressed has been given in [10]. This lengthy analysis is succinctly summarised in the following lemma: Lemma 5. If the unphysical energy-momentum tensor, T ij , satisfies and the expressions can be rewritten in terms of matter zero quantities, then the geometric zero quantities in (48) satisfy a subsidiary system which is symmetric hyperbolic and homogeneous in the zero quantities. A lengthy computation assuming the form for the energy-momentum tensor given by equation (23) and taking into account the expression (30) for the rescaled Cotton-York tensor, shows that the expressions (49) in Lemma 5 can be rewritten as a homogeneous expressions of the matter zero quantities q k , y ij , Z kj and Q kj defined by equations (31), (34) and (38), respectively. The analogue of Lemma 5 for the matter zero quantities is given by Lemmas 2 and 4. The purpose of the analysis of the propagation of the constraints is to establish the following reduction theorem which follows directly from the symmetric hyperbolicity of the subsidiary systems and their homogeneity with respect to the zero quantities. /negationslash Theorem 1. A smooth solution w of the propagation equations (47a) -(47b) which satisfies the constraint equations on a spacelike hypersurface S defines in the domain of dependence of S a solution to the conformal Einstein field equations with matter model given by a traceless perfect fluid.", "pages": [ 15, 16 ] }, { "title": "7 The traceless perfect fluid FLRW cosmology as a solution to the conformal Einstein field equations", "content": "The purpose of the present section is to cast the traceless perfect fluid FLRW cosmology with λ < 0 in a form in which its character as a solution to the conformal Einstein field equations with matter becomes manifest.", "pages": [ 16 ] }, { "title": "7.1 The FLRW cosmology on the Einstein cylinder", "content": "One of the most important properties characterising FLRW cosmologies is their conformal flatness. This shows that as in the case of the Minkowski, de Sitter and anti-de Sitter spacetimes, these solutions admit a conformal representation in which the unphysical spacetime ( M , g µν ) is given by the so-called Einstein cylinder (or Einstein cosmos) . The Einstein cosmos is given by the manifold M E = R × S 3 with a metric given by the line element where, again, dσ 2 is the standard line element of S 3 . The manifold S 3 will be coordinatised in the way indicated in Subsection 2.2. A g -orthonormal frame ˚ e k can be defined on M E by completing the frame { c 1 , c 2 , c 3 } discussed also in Subsection 2.2 with the vector ˚ e 0 = ∂ τ . Setting, for convenience, c 0 ≡ ∂ τ , one can write ˚ e k = ˚ e k s c s with ˚ e k s = δ k s -the components of ˚ e k with respect to the basis c s . In order to relate the FLRW line element (6) with that of the Einstein cosmos, equation (50), one introduces the change of coordinate This naturally leads to the following choice of conformal factor: so that g E = ˚ Θ 2 ˜ g F , where ˜ g F is given by equation (6). For the class of FLRW cosmologies covered by Proposition 1 one has that ˚ Θ → 0 as t →∞ . Moreover, there exists a (finite) positive constant τ ∞ such that ˚ Θ( τ ∞ ) = 0 . Notice also that τ = 0 for t = t 0 . A direct computation using the line element (50), the frame ˚ e k and the conformal factor (51) gives the following expressions for the unknowns of the conformal field equations: where ' denotes differentiation with respect to τ , and a 0 and ˜ ρ 0 are the values of the scale factor and the physical pressure at the initial time τ = 0 . Notice that the unphysical density for this model is constant. Spinorial versions of the above expressions can be readily obtained by contraction with the constant spacetime Infeld-van der Waerden symbols σ i AA ' , or their space spinor version σ a AB . The explicit expressions will not be required in our subsequent analysis. Following the notation of Lemma 7, we collect the independent spinorial components of the fields in (52a)-(52c) in a vectorial unknown which we denote by ˚ w . A direct computation using the expressions (52a)-(52c) shows that, for this solution, the gauge source functions F µ , F ( AB ) and F AA ' as defined by (45a)-(45c) are given by This computation justifies the choice of gauge source functions made in (46). Recalling that d / d τ = a d / d t , and using the limits given in Proposition 1, it follows that Accordingly, the expressions given in (52a), (52b) and (52c) define a smooth solution to the conformal Einstein field equations (20a)-(20b) for τ ∈ [0 , τ ∞ ] . In fact, this solution extends, at least locally, beyond τ = τ ∞ . This can be easily seen to be the case by using the values of the solution (52a)-(52c) as the initial value for a Cauchy problem on the slice τ = τ ∞ . The symmetric hyperbolicity of the evolution equations implies that the solution to this initial value problem exists for τ ∈ [ τ ∞ , τ ∞ + δ ) for some δ > 0 . From the expression for ˚ d k in (52b) and Proposition 1 it follows that ˚ d 0 < 0 at τ ∞ . Thus, by continuity, δ can be chosen such that ˚ Θ < 0 on ( τ ∞ , τ ∞ + δ ) . In summary we have: Lemma 6. There exists δ > 0 such that the expressions (52a) -(52c) give rise to a solution to the evolution equations implied by the conformal Einstein field equations (20a) -(20b) on [0 , τ ∞ + δ ) . Furthermore ˚ Θ < 0 in ( τ ∞ , τ ∞ + δ ) .", "pages": [ 16, 17 ] }, { "title": "Initial data for a FLRW cosmology", "content": "The expressions in (52a)-(52c) naturally induce an initial data set for the conformal Einstein field equations which we denote by ˚ w 0 . Notice that there is no need for performing a pull-back in this construction as the fields in (52a)-(52c) are all scalars.", "pages": [ 17 ] }, { "title": "7.2 Structure of the conformal boundary", "content": "The structure of the conformal boundary for the solution to the conformal Einstein field equations described by (52a)-(52c) follows directly by inspection. By construction at τ = τ ∞ one has that ˚ Θ = 0 . From the limits (53) -see also equation (19h)- one has that: so that the future conformal boundary I + ≡ { p ∈ M E | ˚ Θ = 0 } is spacelike.", "pages": [ 17 ] }, { "title": "8 Existence and stability results", "content": "The purpose of the present section is to provide our main results. These concern the global existence of solutions to conformal Einstein field equations with matter source given by a perfect fluid in the case λ < 0 , γ = 4 3 which can be regarded as non-linear perturbations of the reference FLRW solution described by Proposition 1. We also provide results concerning the structure of the conformal boundary for these solutions. Altogether these results show the non-linear stability towards the future of the reference FLRW cosmological model.", "pages": [ 17 ] }, { "title": "8.1 An Ansatz for the solution", "content": "We will consider solutions to the evolution equations (47a)-(47d) of the form w = ˚ w + ˘ w , where ˚ w as defined in Section 7.1, and ˘ w describes a non-linear perturbation from the reference solution ˚ w . The fields in ˚ w are interpreted as the pull-back of the original fields on M E under a cylinder map. In what follows let w 0 = ˚ w 0 + ˘ w 0 be an initial data set for the system of evolution equations (47a)-(47d) prescribed on the initial manifold S -as discussed in Subsection 8.2. It will be assumed that w 0 satisfies the conformal constraint equations. The vector ˚ w 0 is to be interpreted as the pull-back of the smooth map relating S 3 and the initial manifold S . A direct inspection gives rise to the following result: /negationslash Lemma 7. For ˘ w sufficiently close to 0 and as long as ρ > 0 and u 0 = 0 , the equations (47a) -(47d) imply a symmetric hyperbolic evolution system for the independent components of w . The matrix valued functions A s , B are smooth functions of their arguments. Furthermore, the entries of the matrix A 0 ( ˚ w ) are bounded from below by 1 / √ 2 . Finally, ˘ w = 0 is a solution of equation (54) .", "pages": [ 17, 18 ] }, { "title": "8.2 Constructing initial data for the conformal evolution equations", "content": "In the sequel, it will be assumed that one has a solution ( S , ˜ h αβ , ˜ K αβ , ˜ ρ, ˜ u α ) to the (physical) λ < 0 Einstein-perfect fluid constraint equations with S having the topology of S 3 and the perfect fluid satisfying a barotropic equation of state with γ = 4 3 . In equations (55a)-(55b) ˜ D β and ˜ r denote the Levi-Civita covariant derivative and the Ricci scalar of the intrinsic 3-metric ˜ h αβ of S . ˜ K αβ is a symmetric 3-dimensional tensor corresponding to the extrinsic curvature of S with respect to the ˜ g -unit normal ˜ n µ . Furthermore, ˜ µ ≡ ˜ n µ ˜ n ν ˜ T µν , while ˜ j β corresponds to the pull-back to S of ˜ j λ ≡ ˜ n µ ˜ h λ ν ˜ T µν . For a perfect fluid with a tracefree energy-momentum tensor, a direct computation gives that where ˜ u ‖ ≡ ˜ u µ ˜ n µ and ˜ u β corresponds to the pull-back of ˜ h µ ν ˜ u ν to S . In particular, if on S one has that ˜ n µ and ˜ u ν are aligned -as in the case of the FLRW cosmologies- then ˜ µ = ˜ ρ and ˜ j β = 0 . In general, however, we will consider perfect fluid configurations for which ˜ n µ and ˜ u ν are not aligned. Using a generalisation of the procedure for vacuum spacetimes described in, say, [6] one can construct a solution to the conformal constraint equations implied on S by equations (20a)-(20b). Following the notation introduced in Section 6.2 we denote the independent components of such a solution by w 0 .", "pages": [ 18 ] }, { "title": "8.3 The main result", "content": "In what follows, given m ∈ N , let || · || m denote the Sobolev-like norm on the space C ∞ ( S 3 , R N ) of smooth R N valued functions on S 3 for some non-negative integer N -see e.g. [8, 18] for precise definitions. Furthermore, let H m ( S 3 , R N ) be the Hilbert space obtained as the completion of the space C ∞ ( S 3 , R N ) in the norm || · || m . Using the cylinder map between M E and M , one can apply the norm || · || m to evaluate the norm of functions on the unphysical initial hypersurface S . Furthermore, the vector ˘ w = ˘ w ( τ, x ) can be regarded as a function of τ which takes values in H m ( S 3 , R N ) . Our main result is the following theorem: Theorem 2. Suppose m ≥ 4 . Let S denote a 3-dimensional manifold diffeomorphic to S 3 , and let w 0 = ˚ w 0 + ˘ w 0 be initial data for the conformal evolution equations (47a) -(47d) constructed from some physical initial data set, ( S , ˜ h αβ , ˜ K αβ , ˜ ρ, ˜ u α ) , for the Einstein field equations with λ < 0 and matter source given by a traceless perfect fluid ( γ = 4 / 3) . There exists ε > 0 such that if || ˘ w 0 || m < ε then the initial data set w 0 determines a unique solution, w , to the evolution equations (47a) -(47d) which exists on [0 , τ ∗ ] with τ ∗ > τ ∞ . The solution w is of class C m -2 ([0 , τ ∗ ] × S 3 ) is such that: Remark 8. The above theorem, and in particular part (iv), amounts to a non-linear stability result for the λ < 0 , γ = 4 3 FLRW cosmological models, in the sense that sufficiently small perturbations of data for the FLRW solution give rise to (future) global solutions to the Einstein field equations with the same asymptotic structure as the reference solution. Remark 9. Note that no consideration of the vorticity of the radiation fluid was required for the derivation. The vorticity of the fluid can be calculated from the components of u ij . Proof. Existence, uniqueness and the smoothness of the solutions to equations (47a)-(47d) follow from the properties of the equation (54) provided in Lemma 7 and an extension of the general existence and stability Theorem by Kato [16] provided in [8] -see also [18]. In particular, if ε is sufficiently small, one obtains a common existence time τ ∗ > τ ∞ for all initial data with || ˘ w 0 || m < ε . Part (iv) follows from the same result. Now, the Reduction Theorem , Theorem 1 ensures that if the conformal constraint equations are satisfied on S , one obtains a C m -2 solution on [0 , τ ∗ ] × S 3 to the λ < 0 conformal Einstein perfect fluid equations with γ = 4 3 -this shows part (i). In order to show part (ii), one observes that because ˚ Θ < 0 in ( τ ∞ , τ ∞ + δ ) , then if ε is sufficiently small one has that Θ < 0 at, say, τ = τ ∞ + δ/ 2 . As Θ > 0 at τ = 0 , then there is a τ + for which Θ = 0 . By reducing, if necessary ε one has that such τ is unique, and hence, the function τ + ( x ) is well defined. If follows from (19e) and (19h) that Θ = 0 implies d k d k = ∇ k Θ ∇ k Θ = -1 3 λ > 0 . Hence Θ = 0 , respectively τ = τ + ( x ) , defines a regular spacelike hypersurface I + . /negationslash /negationslash For part (iii) one notices that a solution of the vacuum conformal Einstein field equations implies a solution to the vacuum Einstein field equations -see e.g. [6, 10]. If ω k denotes the dual cobasis of the frame e i , 〈 ω k , e i 〉 = δ k i , then the unphysical metric is given by g = η ij ω i ⊗ ω j . If ε is sufficiently small, one has that det ( g ) = 0 on [0 , τ ∗ ] as det (˚ g ) = 0 . Thus ˜ g = Θ -2 g is well defined on ˜ M . The matter fields ˜ ρ and ˜ u µ are defined via the formulae in (21). An adaptation of this argument to our setting gives the desired result. Geodesic completeness follows from an analysis of the geodesic equations and standard perturbative arguments for ordinary equations given that the background solution is future geodesically complete.", "pages": [ 18, 19 ] }, { "title": "Acknowledgements", "content": "Part of this research was carried out at the Erwin Schrödinger International Institute for Mathematical Physics of the University of Vienna, Austria, during the course of the programme 'Dynamics of General Relativity: Numerical and Analytical Approaches' (July-September, 2011) and the workshop 'Cartan connections, geometry of homogeneous spaces, and dynamics' (July 2011). The authors thank the organisers for the invitation to attend these programmes and the institute for its hospitality. We have profited from interesting discussions with Prof. H. Friedrich. C.L. would like to thank Queen Mary University of London for a Visiting Fellowship.", "pages": [ 20 ] } ]

[ { "title": "MATIAS F. DAHL", "content": "Abstract. We study Maxwell's equations on a 4-manifold where the electromagnetic medium is modelled by an antisymmetric ( 2 2 ) -tensor with real coefficients. In this setting the Fresnel surface is a fourth order polynomial surface in each cotangent space that acts as a generalisation of the light cone determined by a Lorentz metric; the Fresnel surface parameterises electromagnetic wave-speeds as a function of direction. The contribution of this paper is a pointwise description of all electromagnetic medium tensors that satisfy the following conditions: (i) κ is invertible, We show that there are only three classes of mediums with these properties and give explicit expressions in local coordinates for each class. We will study the pre-metric Maxwell's equations. In this setting Maxwell's equations are written on a 4-manifold N and the electromagnetic medium is described by an antisymmetric ( 2 2 ) -tensor κ on N . Then the electromagnetic medium κ determines a fourth order polynomial surface in each cotangent space called the Fresnel surface F and it acts as a generalisation of the light cone determined by a Lorentz metric; the Fresnel surface parameterises wave-speeds as a function of direction [Rub02, HO03, PSW09]. At each point in spacetime N , the electromagnetic medium depends, in general, on 36 free components. In this work we assume that the medium is skewon-free . Then there are only 21 free components and such medium describe non-dissipative medium. For example, in skewon-free medium Poynting's theorem holds under suitable assumptions. The above means that in the pre-metric setting we have two descriptions of electromagnetic medium: First, we have the ( 2 2 ) -tensor κ that contains the coefficients in Maxwell's equations. On the other hand, we also have the Fresnel surface F , which describes the behaviour of a wavespeed for a propagating electromagnetic wave. If κ is known we can always compute F by an explicit equation (see equation (10)). A more challenging question is to understand the converse dependence, or inverse problem: If the Fresnel surface F | p is known for some p ∈ N , what can we say about κ | p ? Essentially this asks that if the behaviour of wave speed for an electromagnetic medium is known, what can we say about the medium? These questions are of theoretical interest, but also of practical interest as they relate to understanding measured data in engineering applications like traveltime tomography in anisotropic medium. We will here only study the problem at a point p ∈ N since the dependence will never be unique. For example, the Fresnel surface F is always invariant under scalings and inversions of κ [HO03, Dah11a]. In general, these are not the only invariances, and for a general κ the relation between κ and F does not seem to be very well understood. A natural first task is to characterise those mediums κ for which the Fresnel surface F is the light cone of a Lorentz metric g . This question was raised in [OH99, OFR00]. A partial solution was given in [OFR00], and (in skewon-free mediums with real coefficients) the complete solution was given in [FB11]. The result is that if the Fresnel surface is a light cone, then κ is necessarily proportional to a Hodge star operator (plus, possibly, an axion component proportional to the identity). For an alternative proof, see [Dah11a], and for related results, see [OR02, LH04, Iti05]. The main contribution of this paper is Theorem 2.1. It gives a pointwise characterisation of all electromagnetic medium tensors κ with real coefficients such that The first two assumptions imply that κ is essentially in one-to-one correspondence with an area metric . Area metrics also appear when studying the propagation of a photon in a vacuum with a first order correction from quantum electrodynamics [DH80, SWW10]. The Einstein field equations have also been generalised into equations where the unknown field is an area metric [PSW07]. For further examples, see [PSW09, SWW10] and for the differential geometry of area metrics, see [SW06, PSW07]. For Maxwell's equations, the interpretation of condition (iii) is that differently polarised waves can propagate with different wave speeds. In such medium one should expect that propagation of electromagnetic waves is determined by null-geodesics of two Lorentz metrics. A typical example of such medium is a uniaxial crystal For partial results describing when the Fresnel surface factorises, see [RS11] and in 3 dimensions, see [Kac04, Dah06]. In Theorem 2.1 we show that there are only three medium classes with the above properties and we give explicit expressions in local coordinates for each class. Of these classes, the first is a generalisation of uniaxial medium and the last seems to be unphysical; heuristic arguments suggest that Maxwell's equations are not hyperbolic in the last class. The main idea of the proof is as follows. We will use the normal form theorem for area metrics derived in [SWW10], which pointwise divides area metrics into 23 metaclasses and gives explicit expressions in local coordinates for each metaclass. This result was also used in [FB11], and by [SWW10] we only need to consider the first 7 metaclasses. For each of these metaclasses, the Fresnel surface can be written as F | p = { ξ ∈ R 4 : f ( ξ ) = 0 } for a homogeneous 4th order polynomial f : R 4 → R with coefficients determined by κ | p . Since κ | p is birefringent, f factorises as into homogeneous 2nd order polynomials f ± : R 4 → R . By identifying coefficients we obtain a system of polynomial equations in coefficients of f and f ± . In the last step we eliminate the coefficients in f ± from these equations whence we obtain constraints on f (and hence on κ ) that much be satisfied when κ is birefringent. To eliminate variables we use the technique of Grobner bases , which was also used in [Dah11a]. A limitation of Theorem 2.1 is that the explicit expression is only valid at a point. The reason for this is that the decomposition in [SWW10] essentially relies on the Jordan normal form theorem for matrices, which is unstable under perturbations. Another limitation is that we do not allow for complex coefficients in κ . Therefore mediums like chiral medium are not included in the mediums in Theorem 2.1. This paper relies on computations by computer algebra. For information about the Mathematica notebooks for these computations, please see the author's homepage.", "pages": [ 1, 2, 3 ] }, { "title": "1. Maxwell's equations", "content": "By a manifold M we mean a second countable topological Hausdorff space that is locally homeomorphic to R n with C ∞ -smooth transition maps. All objects are assumed to be smooth and real where defined. Let TM and T ∗ M be the tangent and cotangent bundles, respectively. For k ≥ 1, let Λ k ( M ) be the set of antisymmetric k -covectors, so that Λ 1 ( N ) = T ∗ N . Also, let Ω k l ( M ) be ( k l ) -tensors that are antisymmetric in their k upper indices and l lower indices. In particular, let Ω k ( M ) be the set of k -forms. Let C ∞ ( M ) be the set of functions. The Einstein summing convention is used throughout. When writing tensors in local coordinates we assume that the components satisfy the same symmetries as the tensor.", "pages": [ 3 ] }, { "title": "1.1. Maxwell's equations on a 4 -manifold. On a 4-manifold N , Maxwell's equations read", "content": "where d is the exterior derivative on N , F, G ∈ Ω 2 ( N ) are the electromagnetic field variables, and j ∈ Ω 3 ( N ) is the source term. By an electromagnetic medium on N we mean a map We then say that 2-forms F, G ∈ Ω 2 ( N ) solve Maxwell's equations in medium κ if F and G satisfy equations (1)-(2) and Equation (3) is known as the constitutive equation . If κ is invertible, it follows that one can eliminate half of the free variables in Maxwell's equations (1)-(2). We assume that κ is linear and determined pointwise so that we can represent κ by an antisymmetric ( 2 2 ) -tensor κ ∈ Ω 2 2 ( N ). If in coordinates { x i } 3 i =0 for N we have and F = F ij dx i ⊗ dx j and G = G ij dx i ⊗ dx j , then constitutive equation (3) reads Then at each point on N , a general antisymmetric ( 2 2 ) -tensor κ depends on 36 free real components. In the main result of this paper (Theorem 2.1) we will assume that κ is skewon-free , that is, κ ( u ) ∧ v = u ∧ κ ( v ) for all u, v ∈ Ω 2 2 ( N ) whence κ has only 21 free components. Physically, such medium describe non-dissipative medium; if κ is time-independent, then Poynting's theorem holds under suitable assumptions [Dah10, Proposition 3.3]. Let us also note that if N is orientable, then invertible skewon-free mediums are essentially in a one-to-one correspondence with area metrics. See [SWW10] and [Dah11b, Proposition 2.4]. The medium is called axion-free if trace κ = 0 [HO03]. By a pseudo-Riemann metric on a manifold M we mean a symmetric ( 0 2 ) -tensor g that is non-degenerate. If M is not connected we also assume that g has constant signature. By a Lorentz metric we mean a pseudo-Riemann metric on a 4-manifold with signature ( -+++) or (+ ---). The light cone of a Lorentz metric is defined as For p ∈ N we define N p ( g ) = N ( g ) ∩ T ∗ p ( N ). If g is a pseudo-Riemann metric on a orientable 4-manifold N , then the Hodge operator of g induces a skewon-free ( 2 2 ) -tensor that we denote by κ = ∗ g . Moreover, if locally g = g ij dx i ⊗ dx j , and κ is written as in equation (4), then where det g = det g ij , g ij is the ij th entry of ( g ij ) -1 , and ε l 1 ··· l n is the Levi-Civita permutation symbol . We treat ε l 1 ··· l n as a purely combinatorial object (and not as a tensor density). Let also ε l 1 ··· l 4 = ε l 1 ··· l 4 . If { x i } 3 i =0 are local coordinates for a 4-manifold N , and J ∈ O we define dx J = dx J 1 ∧ dx J 2 . A basis for Ω 2 ( N ) is given by { dx J : J ∈ O } , that is, This choice of basis follows [HO03, Section A.1.10] and [FB11]. If κ ∈ Ω 2 2 ( N ) is written as in equation (4) and J ∈ O , then where κ J I = κ J 1 J 2 I 1 I 2 . We will always use capital letters I, J, K, . . . to denote elements in O . Let b be the natural bijection b : O →{ 1 , . . . , 6 } . Then we identify coefficients { κ J I : I, J ∈ O } for κ with the 6 × 6 matrix A = ( κ J I ) IJ defined as κ J I = A b ( I ) b ( J ) for I, J ∈ O [Dah11b]. In overlapping coordinates { ˜ x i } 3 i =0 , these coefficients transform as and components G ijkl 0 define a tensor density G 0 on N of weight 1. The TammRubilar tensor density is the symmetric part of G 0 and we denote this tensor density by G [Rub02, HO03, Dah11a]. In coordinates, G ijkl = G ( ijkl ) 0 , where parenthesis indicate that indices ijkl are symmetrised with scaling 1 / 4!. Using tensor density G , the Fresnel surface at a point p ∈ N is defined as The Fresnel surface is a fundamental object when studying wave propagation in Maxwell's equations. It is clear that in each cotangent space, the Fresnel surface F | p is a fourth order polynomial surface, so it can have multiple sheets and singular points. There are various ways to derive the Fresnel surface; by studying a propagating weak singularity [OFR00, Rub02, HO03], using a geometric optics [Iti09, Dah11a], or as the characteristic polynomial of the full Maxwell's equations [SWW10]. Classically, the Fresnel surface can be seen as the dispersion equation for a medium, so that it constrains possible wave speed(s) as a function of direction. If κ = f ∗ g for a Lorentz metric g and a non-zero function f ∈ C ∞ ( N ), then the Fresnel surface is the light cone of g . The converse is also true (assuming that κ is skewon-free and axion-free) [HO03, FB11, Dah11a]. The medium given by κ = f ∗ g is known as non-birefringent medium . For such medium propagation speed does not depend on polarisation. Definition 1.1. Suppose N is a 4-manifold, κ ∈ Ω 2 2 ( N ) and F is the Fresnel surface for κ . If p ∈ N we say that F | p decomposes into a double light cone if there exists Lorentz metrics g + and g -defined in a neighbourhood of p such that glyph[negationslash] and N p ( g + ) = N p ( g -). We then also say that κ | P is birefringent . glyph[negationslash] We know that two Lorentz metrics are conformally related if and only if they have the same light cones [Ehr91]. The condition N p ( g + ) = N p ( g -) thus only exclude non-birefringent mediums, which for skewon-free mediums are well understood (see above). When F | p decompose into a double light cone as in Definition 1.1 a physical interpretation is that the medium is birefringent . That is, differently polarised electromagnetic waves can propagate with different wave speeds. Common examples of such mediums are uniaxial crystals like calcite [BW80, Section 15.3]. To prove of the next four propositions we will need some terminology from algebraic geometry. If k = R or k = C , we denote by k [ x 1 , . . . , x n ] the ring of polynomials k n → k in variables x 1 , . . . , x n . Moreover, a non-constant polynomial f ∈ k [ x 1 , . . . , x n ] is irreducible if f = uv for u, v ∈ k [ x 1 , . . . , x n ] implies that u or v is a constant. For a polynomial r ∈ k [ x 1 , . . . , x n ], let V ( r ) = { x ∈ k n : r ( x ) = 0 } be the variety induced by r , and let 〈 r 〉 = { fr : f ∈ k [ x 1 , . . . , x n ] } be the the ideal generated by r . For what follows the necessary theory for manipulating these objects can, for example, be found in [CLO92]. The next proposition can be seen as a reformulation of the Brill equations that characterise when a homogeneous polynomial factorises into linear forms [Bri10]. glyph[negationslash] Proposition 1.2. Suppose Q ∈ C 4 × 4 is a symmetric non-zero matrix and f ∈ C [ ξ 0 , . . . , ξ 3 ] is the polynomial f ( ξ ) = ξ t · Q · ξ for ξ = ( ξ 0 , . . . , ξ 3 ) ∈ C 4 . Then f is irreducible in C [ ξ 0 , . . . , ξ 3 ] if and only if adj Q = 0 . In Proposition 1.2, adj Q is the adjugate matrix of all cofactor expansions of Q , and ξ t is the matrix transpose. glyph[negationslash] Proof. The result follows from the following three facts: First, if f = uv for polynomials u, v ∈ C [ ξ 0 , . . . , ξ 3 ] then u and v are linear. (To see this, we know that Q (0) = 0, so we may assume that u (0) = 0. For a contradiction, suppose that v (0) = 0. Then df | 0 = 0 implies that du | 0 = 0, but then u = 0, so Q = 0.) Second, by [Bri10, Example 2] polynomial f is a product of linear forms or f has a multiple factor if and only if G f ( ξ, η, ζ ) = 0 for all ξ, η, ζ ∈ C 4 . Here G f is the Gaeta covariant defined as and ( Df ) a ( b ) = d dt ( f ( a + tb )) | t =0 is the directional derivative. Third, by computer algebra we have adj Q = 0 if and only if G f = 0. glyph[square] In [Mon07] it is proven that the light cone of a Lorentz metric can not contain a vector subspace of dimension ≥ 2. The next proposition generalise this result to double light cones. In the proof of Theorem 2.1 we will use this result to show that medium tensors in the last 16 metaclasses in the classification of [SWW10] can not decompose into a double light cone. In [SWW10] this property was used to show that these last metaclasses are neither hyperbolic, so in these metaclasses, Maxwell's equations are not well-posed. Proposition 1.3. Suppose g ± are Lorentz metrics on a 4 -manifold N . If Γ ⊂ T p N is a non-empty vector subspace such that Γ ⊂ N ( g + ) ∪ N ( g -) , then dimΓ ≤ 1 . Proof. We may assume that dimΓ ≥ 1. Let us first prove the result in the special case that g + and g -are conformally related (after [Mon07, Proposition 2]). Let { x i } 3 i =0 be coordinates around p such that g + | p = ± diag( -1 , 1 , 1 , 1), whence we can identify T p M and R ⊕ R 3 . Let π be the Cartesian projection onto the first component. Then the restriction π | Γ : Γ → R satisfies ker π | Γ = { 0 } and dimrange π | Γ = 1, and the claim follows. For general g + and g -let us show that dim Γ ≥ 2 leads to a contradiction. If dimΓ ≥ 2, we can find linearly independent u, v ∈ Γ such that span { u, v } ⊂ N ( g + ) ∪ N ( g -). We may further assume that u ∈ N ( g + ). Let For w ∈ T ∗ N let us write ‖ w ‖ 2 = g + ( w,w ). If U is empty, then span { u, v } ⊂ N ( g -) and the result follows from the special case. Otherwise there exists a θ 0 ∈ U so that ‖ cos θu +sin θv ‖ 2 = 0 for all θ in some neighbourhood I 0 glyph[owner] θ 0 . Differentiating gives By computing the Wronskian of sin 2 θ, cos 2 θ , it follows that 0 = ‖ u ‖ 2 = ‖ v ‖ 2 and g + ( u, v ) = 0. Thus span { u, v } ⊂ N ( g + ), but this contradicts the special case. glyph[square] The next proposition gives the pointwise description of the Tamm-Rubilar tensor density at points p ∈ N where the the Fresnel surface decomposes into a double light cone. Let us emphasize that the result is pointwise. For example, in equation (12) the two sides have different transformation rules. Proposition 1.4. Suppose N is a 4 -manifold, κ ∈ Ω 2 2 ( N ) , and the Fresnel surface of κ decomposes into a double light cone at p ∈ N . If { x i } 3 i =0 are coordinates around p and G ijkl and g ± = g ± ij dx i ⊗ dx j are as in Definition 1.1, then Proof. Let f ± , γ : C 4 → C be polynomials f ± ( ξ ) = g ij ± ξ i ξ j and γ ( ξ ) = G ijkl ξ i ξ j ξ k ξ l for ξ = ( ξ i ) 3 i =0 ∈ C 4 . For ideals I ± = 〈 f ± 〉 we then have 〈 f + f -〉 = I + ∩ I -whence equation (11) implies that V ( 〈 γ 〉 ) = V ( I + ∩ I -) and passing to ideals gives The Strong Nullstellensatz implies that [CLO92, p. 175] where √ I be the radical of an ideal I and we used identities √ I ∩ J = √ I ∩ √ J and I ⊂ √ I valid for any ideals I and J [CLO92, Proposition 16 in Section 4.3]. By Proposition 1.2, f ± are irreducible polynomials, so I ± are prime ideals and I ± = √ I ± . Thus and γ = pf + f -for some polynomial p : C 4 → C . Computing the polynomial degree of both sides implies that p is a non-zero constant p = C ∈ C . By Proposition 1.3, we can find a ξ ∈ R 4 so that ξ glyph[negationslash]∈ V ( f + ) ∪ V ( f -) = V ( γ ) whence C ∈ R \\{ 0 } . glyph[square] For two Lorentz metrics g and h we know that their light cones N ( g ) and N ( h ) coincide if and only if g and h are conformally related [Ehr91]. The next proposition gives an analogous uniqueness result for Fresnel surfaces that decompose into a double light cone. Proposition 1.5. Suppose N is a 4 -manifold, κ ∈ Ω 2 2 ( N ) and the Fresnel surface of κ decomposes into a double light cone at p ∈ N . Suppose furthermore that equation (11) holds for Lorentz metrics g ± and for Lorentz metrics h ± . Then there exists constants C ± ∈ R \\{ 0 } such that exactly one of the following conditions hold: g ± = C ± h ± at p or g ∓ = C ± h ± at p . Proof. The result follows by Propositions 1.2 and 1.4 and since any polynomial has a unique decomposition into irreducible factors [CLO92, Theorem 5 in Section 3.5]. glyph[square] The next example shows that unique decomposition is not true when F only decompose into second order surfaces. Example 1.6. In coordinates { x i } 3 i =0 for R 4 , let κ be the skewon-free ( 2 2 ) -tensor determined by the 6 × 6 matrix ( κ J I ) IJ = diag( -1 , 1 , 0 , -1 , 1 , 0). Then κ has Fresnel surface It is clear that F has multiple factorisations into quadratic forms, and by Proposition 1.3, F does not factorise into a double light cone. ✷", "pages": [ 3, 4, 5, 6, 7 ] }, { "title": "2. Non-dissipative media with a double light cone", "content": "Theorem 2.1 below is the main result of this paper. To formulate the theorem we first need some terminology. Suppose L : V → V is a linear map where V is a n -dimensional real vector space. If the matrix representation of L in some basis is A ∈ R n × n and A is written using the Jordan normal form we say that L has Segre type [ m 1 · · · m r k 1 k 1 · · · k s k s ] when the blocks corresponding to real eigenvalues have dimensions m 1 ≤ · · · ≤ m r and the blocks corresponding to complex eigenvalues have dimensions 2 k 1 ≤ · · · ≤ 2 k s . Moreover, by uniqueness of the Jordan normal form, the Segre type depends only on L and not on the basis. For a ( 2 2 ) -tensor κ on a 4-manifold, we define the Segre type of κ | p as the Segre type of the linear map Ω 2 ( N ) | p → Ω 2 ( N ) | p in basis (7). By counting how many ways a 6 × 6 matrix can be decomposed into Jordan normal forms, it follows that there are only 23 Segre types for a ( 2 2 ) -tensor. A main result of [SWW10] are simple normal forms in local coordinates for each of these Segre types under the assumption that κ | p is skewon-free and invertible. In the below proof we will use the restatement of this result in [Dah11b]. In the proof of Theorem 2.1 we will eliminate variables in systems of polynomial equations. Suppose V ⊂ C n is the solution set to polynomial equations f 1 = 0 , . . . , f N = 0 where f i ∈ C [ x 1 , . . . , x n ]. If I is the ideal generated by f 1 , . . . , f N , the elimination ideals are the polynomial ideals defined as Thus, if ( x 1 , . . . , x n ) ∈ V then p ( x k +1 , . . . , x n ) = 0 for any p ∈ I k , and I k contain polynomial consequences of the original equations that only depend on variables x k +1 , . . . , x n . Using Grobner basis, one can explicitly compute I k [CLO92, Theorem 2 in Section 3.1]. In the proof of Theorem 2.1, we will use the built-in Mathematica routine GroebnerBasis for such computations. Theorem 2.1. Suppose N is a 4 -manifold and κ ∈ Ω 2 2 ( N ) . Furthermore, suppose that at some p ∈ N Then κ | p must have Segre type [11 11 11] , [22 11] or [11 11 11] . for some for some α 1 , α 2 , α 3 ∈ R and β 1 , β 2 , β 3 > 0 . For i ∈ { 1 , 2 , 3 } let where i ' and i '' are defined such that { i, i ' , i '' } = { 1 , 2 , 3 } and i ' < i '' . Then, for exactly one i ∈ { 1 , 2 , 3 } we have and equation (11) holds for Lorentz metrics where α 1 , α 2 ∈ R and β 1 > 0 . Then α 1 = α 2 and β 1 = β 2 , and equation (11) holds for Lorentz metrics . (iii) Metaclass IV: If κ | p has Segre type [11 11 11] , there are coordinates { x i } 3 i =0 around p such that glyph[negationslash] glyph[negationslash] for some α 1 , α 2 , α 3 , α 4 ∈ R and β 1 , β 2 > 0 . Then α 1 = α 2 , β 1 = β 2 , α 4 = 0 , α 2 3 = α 2 4 and equation (11) holds for Lorentz metrics where Proof. Metaclass I. The local expression for κ | p is given by [Dah11a, Theorem 3.2]. Then the Tamm-Rubilar tensor density for κ | p satisfies where C = β 1 β 2 β 3 and D 0 is given explicitly in terms of α 1 , . . . , β 3 , and implicitly D 0 satisfies By Proposition 1.4, there are real symmetric matrices A = ( A ij ) 3 i,j =0 and B = ( B ij ) 3 i,j =0 such that glyph[negationslash] Writing out these equations shows that A 00 B 00 = 1. Hence A 00 is non-zero, and by rescaling A and B , we may assume that A 00 = 1. This substitution simplifies the equations so that by polynomial substitutions we may eliminate all variables in B and variable D 0 . This results in a system of polynomial equations that only involve D 1 , D 2 , D 3 and the variables in A . Further eliminating the variables in A using a Grobner basis, gives constraints on D 1 , D 2 , D 3 . By equation (13) we know that D 1 , D 2 , D 3 ≥ 2, whence these constraints imply that there exists a unique i ∈ { 1 , 2 , 3 } such that conditions (14) hold. (To see that i is unique it suffices to note that if D i = D j = 2 for i = j then D 1 = D 2 = D 3 = 2 whence D 0 = 0 and equation (18) holds for the single light cone A = B = g -1 0 with g 0 = diag {-1 , 1 , 1 , 1 } . By Proposition 1.4 and unique decomposition into irreducible factors this gives a contradiction.) Equations (14) and (17) imply that D 0 = 0. The result follows since equation (18) holds with A = g -1 + and B = g -1 -, where g ± are the matrices in the theorem formulation. Metaclass II. As in Metaclass I, there are matrices A and B such that the Fresnel polynomial satisfies equation (18) (with C = 1). As in Metaclass I we can eliminate variables in B . Further eliminating all variables in A by a Grobner basis implies that α 1 = α 2 and β 1 = β 2 . Then a a direct computation shows that equation (18) holds with A = g -1 + , B = g -1 -and C = β 1 . Computer algebra shows that g ± have Lorentz signatures. Metaclass III. If κ | p is in Metaclass III, there are coordinates { x i } 3 i =0 around p such that where α 1 ∈ R and β 1 > 0. Decomposing the Fresnel polynomial as in equation (18) (with C = 1) gives a system of polynomial equations for the variables in A , B and κ | p . Computing the Grobner basis for these equations implies that β 1 = 0. Thus κ | p can not be in Metaclass III. glyph[negationslash] Metaclass IV. Let us first note that α 4 = 0 since otherwise span { dx 1 | p , dx 2 | p } ⊂ F | p which is not possible by Proposition 1.3. Then the Tamm-Rubilar tensor density satisfies where C = β 1 β 2 α 4 , D 0 is determined explicitly in terms of α 1 , . . . , β 2 , D 1 is defined in equation (15), D 3 ≥ 2 is defined in equation (13) and glyph[negationslash] glyph[negationslash] By decomposing and eliminating variables as in Metaclass I, it follows that that D 0 = 0 and D 3 = 2. Thus we have proven that α 1 = α 2 and β 1 = β 2 whence D 1 = D 2 and equation (18) holds with A = g -1 + , B = g -1 -and C as above. Moreover, g ± both have Lorentz signatures. Condition α 2 3 = α 2 4 follows since det κ | p = 0. Metaclass V. If κ | p is in Metaclass V, there are coordinates { x i } 3 i =0 around p such that glyph[negationslash] where α 1 , α 2 , α 3 ∈ R and β 1 > 0. We may assume that α 3 = 0, since otherwise span { dx i | p } 3 i =1 ⊂ F | p . Decomposing and eliminating variables as in Metaclass I gives that β 1 is purely complex. Thus κ | p can not be in Metaclass V. Metaclass VI. If κ | p is in Metaclass VI, there are coordinates { x i } 3 i =0 around p such that for some α 1 , . . . , α 5 ∈ R and β 1 > 0. We may assume that α 4 and α 5 are non-zero since otherwise span { dx i , dx 2 } ⊂ F | p for some i ∈ { 0 , 1 } as in Metaclass IV. Then the Tamm-Rubilar tensor density satisfies where C = β 1 α 4 α 5 and D 0 , D 1 , D 2 , D 3 ∈ R are defined in terms of α i and β 1 . By decomposing the Fresnel tensor as in equation (18) and eliminating variables using a Grobner basis, it follows that there exists a σ ∈ {± 1 } such that and moreover, equation (18) holds for A = g -1 + , B = g -1 -and C as above, where Since g + does not have a Lorentz signature for any σ ∈ {± 1 } and D 3 ∈ R , Proposition 1.4 and unique factorisation imply that κ | p can not be in Metaclass VI. Metaclass VII. If κ | p is in Metaclass VII, there are coordinates { x i } 3 i =0 around p such that glyph[negationslash] for some α 1 , . . . , α 6 ∈ R . We may assume that α 4 , α 5 , α 6 = 0 since otherwise span { dx i | p , dx j | p } ⊂ F | p for some i, j ∈ { 0 , 1 , 2 } as in Metaclass IV. Then the Tamm-Rubilar tensor density satisfies where C = α 4 α 5 α 6 , constants D 1 , D 2 , D 3 ∈ R are given by and D 0 ∈ R is given explicitly in terms of α 1 , . . . , β 3 , and implicitly D 0 satisfies glyph[negationslash] Decomposing the Tamm-Rubilar tensor density as in equation (18) and eliminating variables using a Grobner basis, gives polynomial equations for D 0 , D 1 , D 2 , D 3 . Let us consider the cases D 0 = 0 and D 0 = 0 separately. If D 0 = 0, there exists an i ∈ { 1 , 2 , 3 } and a σ ∈ {± 1 } such that where the last condition is a consequence of equation (22). Suppose i = 1. Then Proposition 1.4 implies that for some invertible symmetric matrices A,B ∈ R 4 × 4 with Lorentz signatures we have where matrices L ± ∈ C 4 × 4 are defined as Since L ± are invertible, equation (24), Proposition 1.2 and unique factorisation imply that L ± are real and have Lorentz signatures. Thus | D 2 | ≥ 2 and det L ± < 0, but this contradicts equation (25), which implies that glyph[negationslash] A similar analysis for i = 2 , 3 shows that the case D 0 = 0 is not possible. If D 0 = 0 it follows that there exists σ 1 , σ 2 , σ 3 ∈ {± 1 } and distinct i, j, k ∈ { 1 , 2 , 3 } such that glyph[negationslash] where the last equation follows from equation (22). If ( i, j ) = (1 , 2) then k = 3 and Proposition 1.4 implies that for some invertible symmetric matrices A,B ∈ R 4 × 4 with Lorentz signatures, equation (24) holds for matrices L ± ∈ C 4 × 4 defined as Since both sides in equation (24) should decompose into the same number of irreducible factors, it follows that ξ t · L ± · ξ are irreducible in C [ ξ 0 , . . . , ξ 3 ]. Thus equation (24) and unique factorisation imply that L ± are real and have Lorentz signatures, so det L ± < 0. However, this contradicts equation (27) which implies that The cases ( i, j ) = (1 , 3), (2 , 3) are excluded by the same argument by using metrics respectively. Thus κ | p can not be in metaclasses VII. Metaclasses VIII-XXIII. (Following [SWW10, Lemma 5.1].) Let A = ( κ J I ) IJ be the 6 × 6 matrix that represents κ | p in some coordinates { x i } 3 i =0 around p . Then the Jordan normal form of A has a block of dimension d ∈ { 2 , . . . , 6 } that corresponds to a real eigenvalue λ ∈ R \\{ 0 } . By considering unit vectors in the normal basis, we can find non-zero e 1 , e 2 ∈ Λ 2 ( N ) | p so that κ ( e 1 ) = λe 1 and κ ( e 2 ) = λe 2 + e 1 . Writing out κ ( e 1 ) ∧ e 2 = e 1 ∧ κ ( e 2 ) implies that e 1 ∧ e 1 = 0, so e 1 = η 1 ∧ η 2 for some linearly independent η 1 , η 2 ∈ Λ 1 ( N ) | p [Coh05, p. 184]. Let W = span { η 1 , η 2 } . For all ξ ∈ W we then have whence Theorem 3.3 in [Dah11a] implies that W ⊂ F | p and Proposition 1.3 implies that κ | p can not be in metaclasses VIII-XXIII. glyph[square] glyph[negationslash] In the proof of Theorem 2.1 the assumption that κ is invertible is only used to show that α 2 3 = α 2 4 in Metaclass IV and to exclude Metaclasses VIII-XXIII. It would be interesting to see if these last metaclasses can be excluded also for noninvertible κ by other arguments. Regarding this question it should be emphasized that here κ is real. For complex coefficients κ , the setting becomes more involved. For example, in [Dah11a, Example 5.3] it is shows that for complex κ the Fresnel surface can be a single light cone even if κ is not invertible. Also, chiral medium would be an example of a medium with complex coefficients in κ and with a double light cone. In chiral medium right and left hand circularly polarised plane waves propagate with different wavespeeds. Let us note that if we set ξ 0 = 0 in equation (16) we obtain the ternary quartic studied in [Tho16] and for this polynomial, D 2 0 in equation (17) is one of the factors in the discriminant. Let us make some comments regarding the three mediums derived in Theorem 2.1. A first observation is that for each Metaclass in Theorem 2.1, the light cones are parameterised by only one parameter: D i ' = D i '' ≥ 2 in Metaclass I, β 1 > 0 in Metaclass II, and D 1 ∈ R in Metaclass IV. Let consider each metaclass under the assumption that Theorem 2.1 holds. Metaclass I. In Metaclass I we assume that conditions (14) holds for only one i ∈ { 1 , 2 , 3 } . In terms of α 1 , . . . , β 3 conditions (14) are equivalent to When α 1 = α 2 = α 3 = 0 this medium reduces to a uniaxial medium, where wave propagation is well understood. Let us also note that if D i = D j = 2 for two distinct i, j ∈ { 1 , 2 , 3 } , then α 1 = α 2 = α 3 and β 1 = β 2 = β 3 . Then κ | p = -β 1 ∗ g + α 1 Id for the locally defined Lorentz metric g = diag( -1 , 1 , 1 , 1) and the Fresnel surface is the single light cone Metaclass II. For Metaclass II, the Fresnel polynomial G ijkl ξ i ξ j ξ k ξ l is a function of ξ 0 , ξ 2 1 + ξ 2 2 , ξ 3 . It is therefore motivated to project F | p onto ξ 1 = 0, and we can plot F | p as a surface in R 3 . Figure 1 show this projection for three different values of β 1 . From the figures (or from metrics g ± ) we see that the light cones N ( g ± ) coincide in the limit β 1 →∞ . By a coordinate transformation we can put the local representation of κ | p into a more symmetric form. Let { ˜ x i } 3 i =0 be coordinates defined as ˜ x i = ∑ 3 j =0 L ij x j where L = ( L ij ) is the Jacobian matrix where w = √ 1 + 4 β 2 1 . The motivation for these coordinates is that they diagonalize g + . Then the 6 × 6 matrix ( ˜ κ J I ) IJ that represents κ | p in { ˜ x i } 3 i =0 coordinates is 2.1. Metaclass IV. For Metaclass IV, the Fresnel polynomial is also a function of ξ 0 , ξ 2 1 + ξ 2 2 , ξ 3 . We may therefore visualize the Fresnel surface in the same way as in Metaclass II. See Figure 2. Suppose α 1 = α 2 = α 3 = 0. If we treat x 0 as time and { x i } 3 i =1 as space coordinates, and write Maxwell's equations using vector fields E , D , B , H , then κ | p represents medium For an electromagnetic medium to be physically relevant, the Fresnel polynomial G ijkl ξ i ξ j ξ k ξ l should be hyperbolic polynomial [SWW10]. This is a necessary condition for Maxwell's equations to form a predictive theory, that is, a necessary condition for Maxwell's equations to be solvable forward in time. The above 3D projections and the argument in [SWW10] suggests that Metaclass II is hyperbolic for all β 1 > 0 while Metaclass IV is never hyperbolic for any D 1 ∈ R . This is also supported by some numerical tests. Acknowledgements. The author gratefully appreciates financial support by the Academy of Finland (project 13132527 and Centre of Excellence in Inverse Problems Research), and by the Institute of Mathematics at Aalto University.", "pages": [ 7, 8, 9, 10, 11, 12, 13, 14, 15 ] }, { "title": "References", "content": "[Bri10] [BW80] [CLO92] D. Cox, J. Little, and D. O'Shea, Ideals, varieties, and algorithms , Springer, 1992. [Coh05] P. M. Cohn, Basic algebra: Groups, Rings, and Fields , Springer, 2005. [Dah06] M. Dahl, Electromagnetic Gaussian beams using Riemannian geometry , Progress In Electromagnetics Research 60 (2006), 265-291. [Dah10] , Electromagnetic fields from contact- and symplectic geometry , preprint (2010). [Dah11a] , Determining electromagnetic medium from the Fresnel surface , arXiv: 1103.3118 (2011). [Dah11b] [DH80] [Ehr91] [FB11] , A restatement of the normal form theorem for area metrics , preprint (2011). I.T. Drummond and S.J. Hathrell, QED vacuum polarization in a background gravitational field and its effect on the velocity of photons , Physical Review D 22 (1980), no. 2, 343-355. P. Ehrlich, Null cones and pseudo-Riemannian metrics , Semigroup forum 43 (1991), no. 1, 337-343. A. Favaro and L. Bergamin, The non-birefringent limit of all linear, skewonless media and its unique light-cone structure , Annalen der Physik 523 (2011), no. 5, 383-401. Foundations of classical electrodynamics: Charge, flux, [HO03] F.W. Hehl and Y.N. Obukhov, and metric , Progress in Mathematical Physics, Birkhauser, 2003. [Iti05] Y. Itin, Nonbirefringence conditions for spacetime , Physical Review D 72 (2005), no. 8, 087502. E. Briand, Covariants vanishing on totally decomposable forms , Liaison, Schottky Problem and Invariant Theory (M.E. Alonso, E. Arrondo, R. Mallavibarrena, and I. Sols, eds.), Progress in Mathematics, vol. 280, Birkhauser Basel, 2010, pp. 237-256. M. Born and E. Wolf, Principles of optics , Cambridge University Press, 1980. Matias Dahl, Aalto University, Mathematics, P.O. Box 11100, FI-00076 Aalto, Finland URL : http://www.math.tkk.fi/~fdahl/", "pages": [ 15, 16 ] } ]

[ { "title": "H. Gohar and K. Saifullah", "content": "Department of Mathematics, Quaid-i-Azam University, Islamabad, Pakistan Electronic address: [email protected] Abstract: We study quantum tunneling of scalar particles from black strings. For this purpose we apply WKB approximation and Hamilton-Jacobi method to solve the Klein-Gordon equation for outgoing trajectories. We find the tunneling probability of outgoing charged and uncharged scalars from the event horizon of black strings, and hence the Hawking temperature for these black configurations.", "pages": [ 1 ] }, { "title": "1. Introduction", "content": "Black holes are objects in this universe with such a strong gravitational field that even light cannot escape from them. The important breakthrough in the field of black hole physics occurred when Stephen Hawking showed that quantum mechanically black holes emit radiations [1, 2]. Due to the strong gravitational field and vacuum fluctuations at the event horizon of the black hole, virtual particles-anti particles are created. Here we can have three types of scenarios: (a) both the particles fall into the hole, (b) both of them escape from the event horizon, and (c) one particle falls into the hole while the other escapes. The particle that escapes appears as the Hawking radiation. The negative energy particle that falls into the black hole reduces the mass, charge and the angular momentum of the black hole. As a result, the black hole shrinks. This particle must go into the black hole to conserve energy. After Hawking's discovery these thermal radiations have been studied for different black bodies. There are different methods to derive Hawking radiation and Hawking temperature. These can be studied, for example, by calculating the Bogoliubov transformation [1, 3] between the initial and final states of ingoing and outgoing radiation. The Wick rotation method [4, 5] is also used for investigating Hawking radiation. Recently, the black hole tunneling method ([6]- [20]), anomaly method [21] and the technique of dimensional reduction [22] have been used to investigate Hawking radiation and Hawking temperature. The radiation spectrum from black holes contains all types of particles including scalar particles [16, 17]. Here, we have used the black hole tunneling method to derive the tunneling probability of scalar particles from black strings ([23]- [26]). In order to do this we solve the Klein-Gordon equation by using WKB approximation and complex path integration. As a result we obtain Hawking temperature also.", "pages": [ 2 ] }, { "title": "2. Black strings", "content": "The Einstein field equations have a large number of solutions. Here, we discuss some special solutions of these equations, which are exact with negative cosmological constant, called black strings or cylindrical black holes. A four dimensional metric with g µν ( µ, ν = 0 , 1 , 2 , 3) is given by [25] where g mn and φ are metric functions, m,n = 0 , 1 , 2 , x µ = ( t, r, θ, z ) and z is the Killing coordinate. We will write a cylindrically symmetric metric by taking the θ coordinate also in Killing direction from Eq. (2.1). We consider the Einstein-Hilbert action in four dimensions with a negative cosmological constant in the presence of an electromagnetic field. The total action is given by Here, S is the Einstein-Hilbert action in four dimensions, S em is the action for electromagnetic field, R is the Ricci scalar, g the determinant of the metric tensor, Λ the cosmological constant, G the gravitational constant, and the Maxwell tensor F µν is given by where A ν is vector potential and is given by A ν = -h ( r ) δ 0 ν , h ( r ) being an arbitrary function of the radial coordinate r . Here, we take the solution of the EinsteinMaxwell equations with cylindrical symmetry. The line element for static charged black string with negative cosmological constant in the presence of electromagnetic field becomes [25, 26] where Here, Q is the linear charge density per unit length of the z line and M is mass per unit length of the z line of black string. The event (outer) horizon can be found by putting α 2 r 2 -b αr + c 2 α 2 r 2 = 0 and is given by [25] where If we put Q = 0 in Eq. (2.4) we obtain the simplest case of black strings, which contains only one parameter, which is the mass of black string. The line element for this case is given by The outer horizon for this black string can be found by putting α 2 r 2 -b αr = 0 , which gives", "pages": [ 2, 3, 4 ] }, { "title": "3. Quantum tunneling of scalar particles from black strings", "content": "In order to work out the tunneling probability of scalar particles from the event horizon of black strings we use the Klein-Gordon equation for a scalar field Ψ given by Using WKB approximation, we assume an ansatz of the form for Eq. (3.1) which can be written as Now by using Eq. (3.2) in Eq. (3.3) and evaluating term by term in the highest order of /planckover2pi and dividing by the exponential term and multiplying by /planckover2pi 2 , we get Keeping in view the Killing fields, ∂ t , ∂ θ and ∂ z , of the background spacetime, we separate the variables and consider a solution for Eq. (3.4) of the form where E, J 1 , J 2 and K are constants. Here, we are only considering the radial trajectories. Using Eq. (3.5) in Eq. (3.4) yields after simplification Integrating this and substituting the values of g µν gives, where We have to integrate Eq. (3.7) around the pole at the event horizon, r + = b 1 3 /α, We use the residue theory for semi circle yielding As f ( r + ) = 0 , the above equation reduces to which implies that where The probabilities of crossing the horizon from inside to outside and outside to inside are given by [8, 10] We know that the probability of any incoming particles crossing the horizons and entering the black hole is one, so it is necessary to set in the above equations. From Eq. (3.10), we have This means that the probability of a particle tunneling from inside to outside the horizon is Thus, by using Eq. (3.11), (by choosing /planckover2pi1 = 1) , we get This is the probability of the outgoing scalar particle from the event horizon, r = r + . We can find the Hawking temperature, by comparing Eq. (3.18) with the Boltzmann factor [8, 10], Γ = exp( -βE ) , where E is the energy of the particle and β is the inverse of Hawking temperature. Thus we get or", "pages": [ 4, 5, 6 ] }, { "title": "4. Quantum tunneling of scalar particles from charged black strings", "content": "To study the quantum tunneling from charged black strings (Eq. (2.4)) for scalar field Ψ , we use the charged Klein-Gordon equation Proceeding as before, we apply WKB approximation and assume the field of the form given in Eq. (3.2). Substituting this in Eq. (4.1) and keeping terms only in the leading order of /planckover2pi1 and dividing by exponential term and multiplying by /planckover2pi 2 gives Again assuming a solution of the type of Eq. (3.5) for the above equation and solving for W ( r ) we get where Using the complex integration techniques, the integral around the simple pole at the event horizon given by Eq. (2.10) yields This implies that where Thus the tunneling probability of scalar particles from the charged black string comes out to be From this, we can find the Hawking temperature by comparing this with the Boltzmann factor of particle energy where f ' ( r + ) is given in Eq. (4.7). So which is consistent with the literature [20, 27].", "pages": [ 6, 7 ] }, { "title": "5. Conclusion", "content": "In this paper we have studied Hawking radiation of scalar particles from uncharged and charged black strings. By using Hamilton-Jacobi method we have solved the charged and uncharged Klein-Gordon equations. In order to do this we have employed WKB approximation to Klein-Gordon equation to derive the tunneling probability of outgoing particles. At the end, by comparing with the Boltzmann factor of energy for the particles, we have derived the Hawking temperature for these black configurations. These results are found to be consistent with the literature. If we put Q = 0 in Eq. (4.11), the temperature reduces to that for the uncharged case in Eq. (3.20).", "pages": [ 8 ] } ]

[ { "title": "Bianchi type VI cosmological models: A ScaleCovariant study", "content": "Mohd. Zeyauddin 1 . Bijan Saha 2 1 Bogoliubov Laboratory of Theoretical Physics, Joint Institute For Nuclear Research, Dubna - 141980, Moscow Region Russia E-mail: 2 [email protected], 2 Laboratory of Information Technologies, Joint Institute For Nuclear Research, Dubna - 141980, Moscow Region Russia E-mail: 1 [email protected], Abstract A model for an anisotropic Bianchi type VI universe in a Scale Covariant theory of gravitation (Canuto et al. 1977) is analyzed. Exact solutions to the corresponding field equations are found under some specific assumptions. A finite singularity is found in the model at the initial time t = 0. All the physical parameters are studied and thoroughly discussed. The model behaves like a big bang singular model o f the universe. Key words : Cosmology. Bianchi type VI model. Scale Covariant theory.", "pages": [ 1 ] }, { "title": "1 Introduction", "content": "Canuto et al. (1977) have formulated a Scale-Covariant theory of gravitation by associating the mathematical operation of scale transformation with the physics of using different dynamical systems to measure space-time distances. A Scale-Covariant theory provides the necessary theoretical framework to sensibly discuss the possible variation of the gravitational constant G without compromising the validity of general relativity. In this theory, we measure physical quantities in atomic units whereas Einstein's field equations in gravitational units. If we consider d ¯ s 2 = ¯ g ij dx i dx j , the line element in Einstein units, the corresponding line element in any other units (in atomic units) will be written as The metric tensor in the two systems of units are related by a conformal transformation where the metric ¯ g ij giving macroscopic metric properties and g ij giving microscopic metric properties. Here we consider the gauge function φ as a function of time. Friedmann-Robertson Walker(FRW) space-time models are widely acceptable as a good approximation of the present stage of the evolution of the universe although it is spatially homogeneous and isotropic in nature. However, the large scale matter distribution in the observable universe, largely manifested in the form of discrete structures, does not exhibit a high degree of homogeneity. Also the recent space investigations detect anisotropy in the cosmic microwave background. So the recent experimental data support the existence of an anisotropic phase that approaches an isotropic phase. These theoretical arguments (Saha 2004) lead one to consider models with an anisotropic background. Bianchi type space-times play a vital role in understanding and description of the early stages of evolution of the universe. Bianchi type VI (Saha 2004) space-time is inhomogeneous and anisotropic. Scale-Covariant theory in different Bianchi space-times has been studied so far by several authors. Shri Ram et al. (2009) have studied a spatially homogeneous Bianchi type V cosmological model in Scale-Covariant theory of gravitation. Reddy et al. (2007) have developed a cosmological model with negative constant deceleration parameter in Scale-Covariant theory of gravitation. Beesham (1986) has obtained a solution for Bianchi type I cosmological model in the Scale-Covariant theory. Higher dimensional string cosmologies in Scale-Covariant theory of gravitation have been investigated by Venkateswarlu and Kumar (2004). Reddy et al. (1993) have presented the exact Bianchi type II, VIII and IX cosmological models in Scale-Covariant theory of gravitation. In this paper, we obtain exact solution to the field equations of Scale-Covariant theory for Bianchi type VI space-time metric.", "pages": [ 2, 3 ] }, { "title": "2 Field Equations, Metric and General Expressions", "content": "Canuto et al. (1977) transformed the general Einstein's field equations by using the conformal transformations equations (1) and (2) as follows: where for any scalar φ, φ µ = φ ,µ . Here comma denotes ordinary partial differentiation whereas a semi-colon denotes a covariant differentiation. The Bianchi VI space-time metric is given as with the scale factors A , B , C being functions of time only. Here m , n are some arbitrary constants. Here the source of gravitational field is considered as a perfect fluid. So for a perfect fluid, the energy momentum tensor is given by where ρ is the energy-density, p the pressure and u µ is the four velocity vector of the fluid following u µ u µ = 1. The general formulas of certain physical parameters for the metric equation (5) are given as follows: The expansion scalar is given by where a dot( . ) denotes differentiation with respect to time t . The shear scalar has the form We also introduce generalized Hubble parameter H : with H 1 = ˙ A A , H 2 = ˙ B B and H 3 = ˙ C C are the directional Hubble parameters in the directions of x , y and z respectively. Let us introduce the function V and average scale factor a : It should be noted that the parameters H , V and a are connected by the following relation The field equations (3) and (4) to the metric equation (5) for perfect fluid equation (6), are given as following set of equations Here we have used definition (10). The Bianchi identity reads From equation (17), we find the following relation between the metric functions A , B , C as with the integration constant k . Taking into account the definition (10), from equation (19), we can write the scale factors B and C in terms of A and V , such that Summing equations (13), (14), (15) and 3 times equation (16), in view of the equation (10) for volume scalar, we obtain a non-linear differential equation as Taking into account that the perfect fluid obeys the equation of state p = γρ, (0 < γ < 1), the equation (18) becomes where a 0 is an integration constant. We consider the gauge function φ (Canuto et al. (1977) and Shri Ram et al. (2009)) as where α and φ 0 are arbitrary constants. Now in view of equations (23), (24) equation (22) reduces to where ρ 0 is an arbitrary constant. As we can see, there are two unknown functions A and V in the above equation (25). Let us demand an additional assumption relating to these two variables. So we consider here that the scale factor A is related to the volume scalar V with the relation A = √ V (Saha 2004). This assumption provide us the exact solutions to the field equations at the same time leaving the spacetime anisotropic. Note that such an assumption imposes restrictions on the metric functions. Now, in what follows, we try to find an exact solution of the field equations in Scale-Covariant theory with the help of the equation (25).", "pages": [ 3, 4, 5, 6 ] }, { "title": "3 Exact Solutions", "content": "Under the assumption A = √ V , we obtain the following equation for V , by solving the differential equation (25) as where v 0 = 12 πG (1 -γ ) ρ 0 +2( m 2 + n 2 -mn ) > 0, v 1 and v 2 are integration constants. It should be noted that in case of a non-zero v 2 , V is non-trivial even at t = 0, which imposes that v 2 is essentially positive. For v 2 = 0 we have the model, when V becomes zero at the initial time, i.e., V ∣ ∣ v 2 =0 ,t =0 = 0. We also have a relationship between γ and α as α = -3 γ 3 γ +1 , α ∈ ( -3 / 4 , 0). The equation (26), in view of (10), gives the following expressions of the scale factors A , B and C as follows: and where B 0 = k 1 / ( m + n ) and C 0 = (1 /k ) 1 / ( m + n ) . The expressions for the gauge function φ and the average scale factor a are given by and Using the above expressions in equations (7)-(9), the expansion scalar θ , shear scalar σ 2 and the Hubble parameter H are written as, and Now the value of the energy-momentum tensor ρ and the pressure p can be found as follows: and We now investigate the behavior of the above cosmological model by analyzing the different physical parameters. The above set of exact solutions shows that the expansion scalar θ , shear scalar σ 2 and the Hubble parameter H are infinite at the time t = 0. At the same time t = 0, all the directional Hubble's parameters are also infinite. The pressure and density both will be infinite at this epoch at t = 0 iff v 2 = 0. These characteristics of different and The directional Hubble parameters can be obtained as /negationslash physical parameters identify the existence of singularity in the model at the initial time t = 0. Now one can also observe that all these parameters θ , σ 2 , H , H 1 , H 2 , H 3 , ρ and p are become zero at the large time t →∞ , even for v 2 = 0. That is all these physical parameters are decreasing functions of time. Therefore this model describes a continuously expanding and shearing universe with the singularity at t = 0. This model gives an empty space for large time. Let us now study the behavior of the volume scalar and the scale factors A , B , C in this model. From Figure 1 , it can be seen that the volume scalar V ( α = -1 / 2 , -1 / 4) is the increasing function of time. That is V is zero at t = 0 and it takes infinite value at t →∞ . As A is a function of V , namely A = √ V the behavior of A is almost the same as that of V . As far as B and C are concerned, depending on the values of m and n they either expands rapidly or slowly. These variations can be observed through the Figures 2, 3, 4, 5 and 6 respectively.", "pages": [ 6, 7, 8, 9 ] }, { "title": "4 Conclusion", "content": "In this paper, we have obtained an exact solution for the field equations of Scale-Covariant theory of gravitation in Bianchi type VI line element of the universe. Under some specific assumptions, exact solutions to the corresponding field equations are found. It is found that one of the metric functions ( A ) is an expanding one with acceleration whereas depending on the choice of the parameters two other metric functions B and C expand either with acceleration or deceleration. The model in question does not allow isotropization of the initial anisotropic space-time. All the physical and kinematical parameters have been thoroughly discussed. The solution so obtained, represents a continuously expanding and shearing model of the universe with the singularity at the initial time t = 0. This model gives an empty space for large time.", "pages": [ 9, 10 ] }, { "title": "5 Acknowledgments", "content": "This work is partially supported by a joint Romanian-LIT, JINR, Dubna Research Project 4163-6-12/13, theme no. 05-6-1060-2005/2013.", "pages": [ 10 ] }, { "title": "References", "content": "Canuto, V. et al.: Phys.Rev.D. 16 , 6(1977) Canuto, V., Hsieh, S.H., Adams, P.J.: Phys.Rev.Lett. 39 , 8(1977) Saha, B.: Phys.Rev.D. 69 , 124006(2004) Ram, S., Verma, M.K., Zeyauddin, M.: Chin.Phys.Lett. 26 , 089802(2009) Reddy, D. R. K., Naidu, R. L., Adhav, K. S.: Astrophys. Space.Sci. 307 , 365(2007) Beesham, A.: Class.Quantum.Grav. 3 , 481(1986) Venkateswarlu, R., Kumar, P.K.: Astrophys. Space Sci. 298 , 403(2005) Reddy, D.R.K., Patrudu, B.M., Venkateswarlu, R.: Astrophys. Space Sci. 204 , 155(1993)", "pages": [ 11 ] } ]

[ { "title": "energy", "content": "M. Malekjani 1 ∗ ,A. Khodam-Mohammadi † 1 1 Department of Physics, Faculty of Science, Bu-Ali Sina University, Hamedan 65178, Iran", "pages": [ 1 ] }, { "title": "Abstract", "content": "A new model of dark energy namely 'ghost dark energy model' has recently been suggested to interpret the positive acceleration of cosmic expansion. The energy density of ghost dark energy is proportional to the hubble parameter. In this paper we perform the statefinder diagnostic tool for this model both in flat and non-flat universe. We discuss the dependency of the evolutionary trajectories in s -r and q -r planes on the interaction parameter between dark matter and dark energy as well as the spatial curvature parameter of the universe. Eventually, in the light of SNe+BAO+OHD+CMB observational data, we plot the evolutionary trajectories in s -r and q -r planes for the best fit values of the cosmological parameters and compare the interacting ghost model with other dynamical dark energy models. We show that the evolutionary trajectory of ghost dark energy in statefinder diagram is similar to holographic dark energy model. It has been shown that the statefinder location of ΛCDM is in good agreement with observation and therefore the dark energy models whose current statefinder values are far from the ΛCDM point can be ruled out.", "pages": [ 1 ] }, { "title": "I. INTRODUCTION", "content": "Nowadays it is strongly believed that our universe expands under an accelerated expan- sion. The various cosmological data gathered from SNe Ia [1], WMAP [2], SDSS [3] and X-ray [4] experiments have provided the main evidences for this cosmic acceleration. Within the framework of standard cosmology, a dark energy component with negative pressure is responsible for this acceleration. Up to now many theoretical models have been proposed to interpret the behavior of dark energy. The first and simple candidate is the Einstein's cosmological constant with the time - independent equation of state w Λ = -1. The cosmological constant suffers from tow deep theoretical problems namely the 'fine-tuning' and 'cosmic coincidence'. In addition to cosmological constant, dynamical dark energy model with timevarying equation of state have been investigated to interpret the cosmic acceleration. The scalar field models such as quintessence [5], phantom [6], quintom [7], K-essence [8], tachyon [9] and dilaton [10] together with interacting dark energy models such as holographic [11] and agegraphic [12] models are the examples of dynamical dark energy models. The interacting dark energy models have been constructed within the framework of quantum gravity, by introducing the new degree of freedom or by modifying the theory of gravity [13-15]. Recently, the Veneziano ghost dark energy has been attracted a deal of attention in the dynamical DE category. The Veneziano ghost is proposed to solve the U (1) problem in lowenergy effective theory of QCD [16] and has no contribution in the flat Minkowski spacetime. In curved spacetime, however, it makes a small energy density proportional to Λ 3 QCD H , where Λ QCD is QCD mass scale and H is Hubble parameter. This small vacuum energy density can be considered as a driver engine for evolution of the universe. It is worthwhile to mention that this model is totally arisen from standard model and general relativity. Therefore one needs not to introduce any new parameter or new degree of freedom and this fact is the most advantages of ghost DE. With Λ QCD ∼ 100 Mev and H ∼ 10 -33 ev , the right order of observed DE density can be given by ghost DE. This numerical coincidence also shows that this model gets ride of fine tuning problem [17, 18] Many authors have already suggested DE model with energy density as ρ = αH [19]. Recent observational data gathered from the Abell Cluster A586 support the interaction between dark matter and dark energy [20]. However the strength of this interaction is not clearly identified [21]. Since many theoretical dark energy models have been proposed to explain the accelerated expansion of the universe, therefore the sensitive test which can differentiate between these models is required. The Hubble parameter, H = ˙ a/a , (first time derivative) and the deceleration parameter q = -aH 2 /a (second time derivative) are the geometrical parameters to describe the expansion history of the universe. ˙ a > 0 or H > 0 means the expansion of the universe. Also a > 0, i.e. q < 0, indicates the accelerated expansion of the universe. Since the various dark energy models give H > 0, q < 0 at the percent time, the Hubble and deceleration parameters can not discriminate dark energy models. For this aim we need a higher order of time derivative of scale factor. Sahni et al. [22] and Alam et al. [23], by using the third time derivative of scale factor, introduced the statefinder pair { s,r } in order to remove the degeneracy of H and q at the present time. The statefinder pair is given by Depending the statefinder diagnostic tool on the scale factor indicates that the statefinder parameters are geometrical. The scale factor a ( t ) can be expanded near the present time t 0 as follows where we consider a ( t ) = 1 and H , q , r are the present values of the Hubble parame- 0 0 0 0 ter, deceleration parameter and former statefinder parameter, respectively. Up to now, the various dark energy models have been studied from the viewpoint of statefinder diagnostic. These models have different evolutionary trajectories in { s, r } plane, therefore the statefinder tool can discriminate these models. The well known ΛCDM model is related to the fixed point { s=0,r=1 } in the s -r plane [22]. The distance of the current value of statefinder pair { s 0 , r 0 } for a given dark energy model from the fixed point { s=0,r=1 } is a valuable criterion to a model. In addition, the distance of current statefinder values of a given dark energy model from the constrained observational value { s 0 , r 0 } is a good tool to test a model. The dynamical dark energy models that have been investigated by statefinder diagnostic tool are: the quintessence DE model [22, 23] , the interacting quintessence models [24, 25], the holographic dark energy models [26, 27] , the holographic dark energy model in non-flat universe [28], the phantom model [29], the tachyon [30], the generalized chaplygin gas model [31], the interacting new agegraphic DE model in flat and non-flat universe [32, 33], the agegraphic dark energy model with and without interaction in flat and non-flat universe [34, 35], the new holographic dark energy model [36] and the interacting polytropic gas model [39]. In this work we investigate the interacting ghost dark energy model by statefinder diagnostic tool. The statefinder can be applied to diagnose different cases of the model, including different model parameters and different contributions of spatial curvature. First, we perform the statefinder diagnostic in flat universe in sect. II, then we generalize our work to the non flat universe in sect. III. In sect.IV, the statefinder diagnostic has been discussed based on recent observational data. This work is concluded in sect. V.", "pages": [ 2, 3, 4 ] }, { "title": "II. INTERACTING GHOST DARK ENERGY MODEL IN FLAT UNIVERSE", "content": "Let us first consider the interacting ghost dark energy in the flat Friedmann-RobertsonWalker (FRW) universe. The corresponding Friedmann equation in this case is given by where H and M p are the Hubble parameter and the reduced Planck mass, respectively. The energy density of ghost dark energy is given by [50] where α is a constant of the model. The dimensionless energy densities are defined as Using (5), the Friedmann equation (3) can be written as In a universe dominated by interacting dark energy and dark matter, the total energy density, ρ = ρ m + ρ Λ , satisfies the following conservation equation However, by considering the interaction between dark energy and dark matter, the energy density of dark energy and dark matter does not conserve separately and the conservation equation for each component is given by where Q represents the interaction between dark matter and dark energy. It is worth noting that in equation (8) the right hand side of (8), same as left hand side, should be as a function of inverse of time. The simple choice is that the interaction quantity Q can be considered as a function of Hubble parameter H such as one of the following forms: (i) Q ∝ Hρ Λ , (ii) Q ∝ Hρ m and (iii) Q ∝ H ( ρ m + ρ Λ ). One can assume the above three forms as Q = Γ ρ Λ , where for case (i) Γ = 3 b 2 H , for case (ii) Γ = 3 b 2 H Ω m Ω Λ and for case (iii) Γ = 3 b 2 H 1 Ω Λ . The parameter b is a coupling constant indicating the strength of interaction between dark matter and dark energy [52]. The interaction between dark energy and dark matter is also studied in [53]. Here we assume the third form of interaction for Q . Taking the time derivative from both side of Friedmann equation (3) and using (6, 8, 9) as well as the relation p Λ = w Λ ρ Λ , one can obtain Inserting the third form of interaction term Q = Γ ρ Λ = 3 b 2 H 1 Ω Λ ρ Λ in the right hand side of (9) and using the relations (4), (10), the equation of state for interacting ghost dark energy in the flat universe can be obtained as In the limiting case of non-interacting flat universe (i.e., b = 0 and Ω k = 0), Eq.(11) reduces to which is in agreement with [51]. At the early time when Ω Λ << 1, we can see w Λ = -1 / 2 and at the late time when Ω Λ ∼ 1, one can see w Λ = -1. Therefore the ghost dark energy mimics the cosmological constant at the late time. The evolution of EoS parameter of ghost model has been studied in [51]. It has been shown that the interacting ghost dark energy model can cross the phantom divide for b 2 > 0 . 1. Using (10), the deceleration parameter q in this model can be obtained as Using (13) and ˙ Ω Λ = H Ω ' Λ yields where prime denotes the derivative with respect to ln a . Tacking the time derivative of (10) and using (5), (9) and (4) we obtain We now find the statefinder parameters { s, r } for the interacting ghost dark energy model in the flat universe. From the definition of q and H , the parameter r in (1) can be written as Substituting the relations (13) and (16) in (17), the parameter r is obtained as Inserting Eqs. (13) and (18) in the parameter s of (1) obtains At the late time ( when Ω Λ → 1 ), by inserting w Λ = -1 and therefore w ' Λ = 0, the relations (18) and (19) reduce to the constant values ( r = 1, s = 0) which refers the statefinder parameters of standard ΛCDM model in the flat universe. Therefore, from the viewpoint of statefinder diagnostic, the ghost dark energy mimics the cosmological constant at the late time. It is clear that at the early time ( when Ω Λ → 0) we have q = 1 / 2 which is equal to the value of deceleration parameter obtained in CDM model. Therefore in ghost model, the decelerated expansion phase ( q > 0) at the early time can be achieved. At the late time ( when Ω Λ ∼ 1 and w Λ = -1), we see that q = -1, which represents the accelerated expansion ( q < 0) in dark energy dominated universe, as expected. Tacking the time derivative of dark energy density parameter in (5) and using the ghost dark energy density (4), we have By numerical solving of Eqs. (18) and (19), we obtain the evolutionary trajectory of interacting ghost dark energy in the statefinder plane. It should be noted that in Eqs. (18) and (19) the evolution of w Λ and Ω Λ are governed by Eqs. (15) and(11), respectively. In statefinder plane, the horizontal axis is defined by the parameter s and vertical axis by the parameter r . In this diagram, the standard ΛCDM model corresponds to the fixed point { r = 1 , s = 0 } . In Fig.(1), we plot the evolutionary trajectories of ghost dark energy model in the flat universe in s -r plane for different illustrative values of interaction parameter b . Here we adopt the current values of cosmological parameters Ω Λ and Ω m as 0 . 7 and 0 . 3, respectively. The standard Λ CDM fixed point { r = 1 , s = 0 } is indicated by star symbol in this diagram. The colored circles on the curves show the present values of statefindr pair { s 0 , r 0 } . By expanding the universe, the trajectories in s -r plane start from right to left. The parameter r decreases, then increases to the constant value r = 1 at the late time. While the parameter s deceases from the positive value at the early time to the constant value s = 0 at the late time. Different values of interaction parameter b result the different evolutionary trajectories in s -r plane. Hence the statefinder analysis can discriminate the interacting ghost dark energy model with different interaction parameter. For larger value of b , the present values of s 0 and r 0 decreases. The distance of the point { s 0 , r 0 } form the Λ CDM fixed point { s = 0 , r = 1 } becomes larger for larger values of interaction parameter b . Fig.(1) also shows that the interacting ghost dark energy model mimics the ΛCDM model at the late time. This behavior of ghost dark energy is similar to the holographic [26-28], new agegraphic [32, 33], chaplygin gas [37], generalized chaplygin gas [31] and yang- mils [38] models of dark energy in which they also mimic the ΛCDM model at the late time. Unlike the above models, the agegraphic dark energy model [34, 35] and polytropic gas model [39] mimic the ΛCDM model at the early stage of the evolution of the universe. The evolutionary trajectories of holographic dark energy under granda-Oliveros IR cut-off (new holographic model) [36] and also tachyon dark energy model [30] in s -r plane pass through the ΛCDM fixed point at the middle of the evolution of the universe. The other interesting note is that the evolution of ghost dark energy model in s -r plane is similar to the evolution of holographic model of dark energy with the model parameter c = 1 in this plane (i.e., see Fig.(3) of [27] and upper panel of Fig.(1) in [28]). Also, it is of interest to discuss the dynamical behavior of ghost dark energy in q -r plane. In q -r plane, we use the geometrical quantity q instead of the parameter s at the horizontal axis. In Fig.(2), by solving Eqs.(13) and (18), the evolutionary trajectories of ghost dark energy in flat universe is plotted for different values of interaction parameter b in q -r plane. Same as statefinder analysis, the q -r analysis can discriminate different dark energy models. By expanding the universe, the trajectories start from right to left. The parameter r decrease, then increases to the constant value r = 1 at the late time. While the parameter q decreases from the positive value ( indicating the decelerated expansion) at the early time to the negative value (representing the accelerated expansion) at the late time. Here we see the different evolutionary trajectories for different interaction parameters b . The current value { q 0 , r 0 } can also be affected by interaction parameter. Increasing the interaction parameter b causes both the parameters r and q becomes smaller.", "pages": [ 4, 5, 6, 7, 8 ] }, { "title": "III. INTERACTING GHOST DARK ENERGY MODEL IN A NON FLAT UNIVERSE", "content": "In this section we generalize our work in previous section to the non flat universe. The Friedmann equation in this case is given by where k = 1 , 0 , -1 is a spatial curvature parameter corresponding to the closed, flat and open universe, respectively. The dimensionless energy densities of dark energy and dark matter have been defined in (5) and dimensionless energy density corresponding to the spatial curvature is given as Ω k = k a 2 H 2 . Therefore the Friedmann equation (20) in terms of dimensionless parameters is written as Same as previous section, here in the non flat universe, we consider the third form of interaction between dark matter and dark energy Q ∝ H ( ρ m + ρ Λ ). Using Eqs. (20) and (21), this form of interaction in non flat universe can be written as Q = Γ ρ Λ , where Γ = 3 b 2 H 1+Ω k Ω Λ . Taking the time derivative of both side of Friedmann equation (20) and using (21, 8, 9) as well as the relation p Λ = w Λ ρ Λ , one can obtain where Ω k is given by Inserting the interaction term Q in the right hand side of continuity equation (9) and using the relations (4), (22), the equation of state for interacting ghost dark energy in the non flat universe can be obtained as In the limiting case of flat universe (i.e., Ω k = 0), Eq.(24) reduces to (11), as expected. Using (22), the deceleration parameter q in non flat case can be obtained as The evolution of dark energy density in non flat universe is obtained by tacking the time derivative of (5) and using the ghost dark energy density (4) Using (25) and ˙ Ω Λ = H Ω ' Λ results where q is defined in (25). Tacking the time derivative of Eq. (22) and using (5), (9), (23) and (24) results Inserting Eqs. (25) and (28) in Eq. (17), the former statefinder parameter r for interacting ghost dark energy in the non flat universe is obtained as Following [55], we consider the parameter s in the non flat universe as follows where Ω t = 1 + Ω k is a total energy density as defined in Friedmann equation. Obviously, in the limiting case of flat universe, i.e., Ω k = 0, the above definition is reduced to (1). Substituting Eqs. (25) and (29) in (30) gets In the limiting case of flat universe, the above equations for the statefinder parameter { s, r } are reduced to those obtained in previous section. Here in this section, we focus on the contribution of spatial curvature on the evolution of ghost dark energy in the s -r and q -r planes. For this aim we need to solve numerically the relations (25, 29 and 31). Note that in these equations the dynamics of EoS parameter w Λ , density parameter Ω Λ and spatial curvature parameter Ω k are given by (24), (5) and (23), respectively. In Fig.(3), we plot the statefinder diagram for different contribution of spatial curvatures. The selected curves are plotted by fixing Ω m 0 = 0 . 30, Ω Λ0 = 0 . 70 and varying Ω k 0 = 0 . 02, Ω k 0 = 0 . 00 and Ω k 0 = -0 . 02 corresponding to the closed, flat and open universe, respectively. A closed universe with a small positive curvature ( Ω k = 0 . 02) is compatible with some observations [56]. Here we ignore the interaction between dark matter and dark energy and focus only on the effect of contribution of spatial curvature on the evolution of trajectories in statefinder plane. By expanding the universe, the trajectories evolve from right to left. The parameter r decreases, then increases and reaches to the constant value r = 1 at the late time. The parameter s decreases forever. The different contributions of spatial curvature exhibit the different features in the s -r plane. The colored circles on the curves are the today's value of { s 0 , r 0 } for different spatial curvatures. One can see that the today's value { s 0 , r 0 } of interacting ghost dark energy with different spatial curvatures is discriminated. We can clearly identify the distance from a given dark energy model to the standard flat-ΛCDM model by using the r(s) evolution diagram. Fig.(3) shows that in the closed universe the distance of the present value { s 0 , r 0 } from the location of ΛCDM fixed point { s = 0 , r = 1 } is shorter compare with other spatial curvatures. The holographic dark energy model from the viewpoint of statefinder diagnostic analysis in the non flat universe has already been investigated in [28]. By comparing Fig.(3) with upper panel of Fig.(1) of [28], we see the similarity of evolutionary trajectories between ghost dark energy model and the holographic model of dark energy (with the model parameter c = 1) in non flat universe. Fig.(4) shows the evolutionary trajectories of interacting ghost dark energy in q -r plane for different contributions of spatial curvature of the universe. By expanding the universe the trajectories evolve from right to left, the parameter r first decreases then increases and the parameter q decreases from the positive value at the early time (indicating the decelerated phase of expansion) to the negative value at the at late time ( denoting the accelerated phase). In q -r plane, the interacting ghost model is discriminated for different contribution of spatial curvatures. The current value of statefinder pair { q 0 , r 0 } is also distinguished in different spatial curvatures of the universe. The value of { q 0 , r 0 } is larger in closed universe (Ω k = 0 . 02) compare with flat (Ω k = 0 . 00) and open (Ω k = -0 . 02) universe.", "pages": [ 9, 10, 11, 12 ] }, { "title": "IV. INTERACTING GHOST MODEL AND OBSERVATIONAL CONSTRAINS", "content": "It is clear that constraining the parameterized model against the observational data is model dependent. Hence some doubts usually remain on the validity of the constraints on the derived quantities such as the present day values of the deceleration parameter and the age of the universe. In order to solve this problem, we use the cosmography, i.e. the expansion of scale factor in Taylor series with respect to the cosmic time. For this aim, the s following functions 1 a q = - d 2 a 2 dt H (33) - 2 which are namely the Hubble, deceleration, jerk, snap and lerk parameters, respectively are introduced. The present values of the above parameters can be used to describe the evolution of the universe. For example, q 0 < 0 indicates the current accelerated expansion of the universe and also r 0 allows to discriminate between different dark energy models. Using the Union2 SNeIa data set [59] and the BAO data from the analysis of the SDSS seventh release [60] adding a prior on h from the recent determination of the Hubble constant by the SHOES team [61] and the age of passively evolving galaxies [58], the present values of the above cosmographic parameters are constrain observationally by using the Markov Chain Monte Carlo method [57]. The best fit values of the cosmographic parameters are: { h = 0 . 718, q 0 = -0 . 64, r 0 = 1 . 02, k 0 = -0 . 39, l 0 = 4 . 05 } (see table I of [57] for more details). Inserting the present values of q 0 = -0 . 64 and r 0 = 1 . 02 in Eq. (1), the present value of statefinder parameter s is obtained as s 0 = -0 . 006. Therefore, observationally, the best fit value of the current statefinder pair is { s 0 = -0 . 006 , r 0 = 1 . 02 } . In this section we compare the present value of statefinder parameters { s,r } of interacting ghost dark energy that has been constrained observationally in [51] with the above best fit value of current statefinder pair. For this aim we use the best fit constrained values of the cosmological parameters Ω m 0 = 0 . 35, Ω Λ0 = 0 . 75 and b 2 = 0 . 08 in the ghost dark energy model that have recently been obtained in [51] by using the data of Supernova type Ia (SNIa) Gold sample, shift parameter of Cosmic Microwave Background radiation (CMB) and the Baryonic Acoustic Oscillation (BAO) peak from Sloan Digital Sky Survey (SDSS). In Fig.(5) the evolutionary trajectories of interacting ghost dark energy in s -r plane (upper panel) and in q -r plane (lower panel) are plotted for the above best fit values of cosmological parameters Ω m 0 , Ω Λ0 and b 2 . In s -r diagram, the evolutionary trajectory starts from { s = 0 . 86 , r = 0 . 67 } at the past time, reaches to the { s = 0 . 08 , r = 0 . 74 } at the present time (circle point) and ended at { s = 0 , r = 1 } at the future. The best fit observational value { s 0 = -0 . 006 , r 0 = 1 . 02 } in flat universe is indicated by red-star symbol in this diagram. In q -r diagram, the evolutionary trajectory starts from { q = 0.4, r = 1 } at the past ( corresponds to the decelerated expansion of the universe), reaches to { q = -0 . 6 , r = 0 . 74 } at the present time and ended at { q = -1 , r = 1 } at the late time ( corresponds to the accelerated expansion). The best fit observational value { q 0 = -0 . 64 , r 0 = 1 . 02 } is also indicated by red-star symbol in this diagram. Now we compare the present value { s 0 , r 0 } of constrained interacting ghost dark energy model with other models of dark energy which have been constrained and discussed from the viewpoint of statefinder diagnostic. This comparison includes the interacting ghost model, holographic, new holographic and generalized chaplygin gas models of dark energy. These models have been constrained by astronomical data of SNe+CMB+BAO+OHD experiments and also have been discussed in s -r diagram based on the constrained values of cosmological and model parameters. This comparison also includes the standard ΛCDM model as well as the best fit observational value { s 0 = -0 . 006 , r 0 = 1 . 02 } in flat universe. The holographic dark energy model with the constrained values ( c = 0 . 84, Ω m 0 = 0 . 29, Ω k 0 = 0 . 02, where c is the model parameter of holographic dark energy) obtains the today's statefinder pair as { s 0 = -0 . 102 , r 0 = 1 . 357 } [28]. The new holographic dark energy model with the constrained values (Ω b h 2 = 0 . 0228, Ω m 0 = 0 . 2762, Ω k 0 = 0 . 0305, Ω Λ0 = 0 . 6934, α = 0 . 8824, β = 0 . 5016, where α and β are the parameters of model) results the today's statefinder pair as { s 0 = -0 . 13 , r 0 = 1 . 46 } [36]. The generalized chaplygin gas dark energy ( GCG model) in the flat universe with the constrained values ( A s = 0 . 76, α = 0 . 033, Ω b h 2 = 0 . 0233, H 0 = 69 . 97, where A s and α are the parameters of the model) gives the today's statefinder pair as { s 0 = -0 . 007 , r 0 = 1 . 026 } [31]. Note that the GCG model is constrained in the flat universe, but other models are constrained in general non-flat universe. Fig.(6) shows the location of the present statefinder pair { s 0 , r 0 } for the above constrained models as indicated in legend. The standard ΛCDM model and also the best fit observational value { s 0 = -0 . 006 , r 0 = 1 . 02 } in flat universe are indicated by black and red star symbols, respectively. One can conclude that the ΛCDM model { s = 0 , r = 1 } has a shortest distance to the best fit observational value { s 0 = -0 . 006 , r 0 = 1 . 02 } compare to other dynamical dark energy models. Also, the interacting ghost dark energy model has a shorter distance from { s 0 = -0 . 006 , r 0 = 1 . 02 } compare with the holographic and new holographic dark energy models. Among the dynamical dark energy model, the GCG model has a shortest distance from the location of observational value in s -r plane.", "pages": [ 12, 13, 14, 15, 16 ] }, { "title": "V. CONCLUSION", "content": "Summarizing this work, we investigated the interacting ghost dark energy model in statefinder s -r and q -r diagrams. The statefinder analysis can discriminate the interacting ghost dark energy model for different values of interaction parameter as well as the different spatial curvatures of the universe. Like holographic [26-28], new agegraphic [32, 33], chaplygin gas [37], generalized chaplygin gas [31] and yang-mils [38] models of dark energy, the ghost dark energy model mimics the standard ΛCDM model at the late time. The evolution of ghost dark energy model in s -r plane is similar to holographic model of dark energy with the model parameter c = 1. Different values of interaction parameter obtains the different evolutionary trajectories in s -r and q -r planes. The evolutionary trajectories r ( s ) and r ( q ) for interacting ghost dark energy model in different closed, flat and open universe has also been investigated. We have shown that different contribution of spatial curvatures give the different evolutionary trajectories in s -r and q -r . The spatial curvature can also influence the present value of statefinder parameters { s 0 , r 0 } and { q 0 , r 0 } in these planes. Eventually, we performed the statefinder diagnostic for the interacting ghost model constrained by observational data. We conclude that the ΛCDM model { s = 0 , r = 1 } has a shortest distance to the best fit observational value { s 0 = -0 . 006 , r 0 = 1 . 02 } compare with other dynamical dark energy models. Therefore the models of dark energy whose curerent statefinder values locate far from the ΛCDM point can be ruled out. The interacting ghost dark energy model has a shorter distance from { s 0 = -0 . 006 , r 0 = 1 . 02 } compare with the holographic and new holographic dark energy models. Among the above dynamical dark energy models, the GCG model has a shortest distance from the location of observational statefinder pair (i.e., { s 0 = -0 . 006 , r 0 = 1 . 02 } ). Prog. Theor. Phys. 66 , 1789 (1981); 515524.", "pages": [ 18, 19, 21, 23 ] } ]

[ { "title": "PERIODIC ORBITS FOR A CLASS OF GALACTIC POTENTIALS", "content": "FELIPE ALFARO, JAUME LLIBRE, AND ERNESTO P ' EREZ-CHAVELA Abstract. In this work, applying general results from averaging theory, we find periodic orbits for a class of Hamiltonian systems H whose potential models the motion of elliptic galaxies.", "pages": [ 1 ] }, { "title": "1. introduction and statement of the main results", "content": "Galactic dynamics is a branch of Astrophysics whose development started only around sixty years ago, when it was possible to have a view of the physical world beyond the integrable and near integrable systems [5]. Even the importance of the analysis of galactic potentials, the global dynamics of galaxies is not a simple question and represents a big challenge for the researches in the field [2]. Most of the work in the analysis of galaxies is numerical, in this paper we present an analytical technique, the averaging theory, which allows to find periodic orbits of a differential system. In the last years, great quantity of the research on galactic dynamics has been focused on models of elliptical galaxies. In most of these models the terms in the potential are of even order, so we have adopted this fact in the Hamiltonian system that we are analyzing. Another important point that appears in these kind of potentials is that the existence of periodic orbits is a useful tool for constructing new and more complicated self consistent models. One way to identify periodic orbits is to localize the central fixed points on the surfaces of constant energy. In [11], the authors study the localization of periodic orbits and their linear stability for a particular two-component galactic potential. In fact, in our days the study of individual orbits in some galactic potentials is a new branch of galactic dynamics (see for instance the articles [4, 6, 8]) . The calculation of particular orbits in some analytical potentials modeling elliptical galaxies, indicates that relatively small symmetry breaking corrections can increase dramatically the number of stochastic orbits, showing the importance of the study of perturbations of simple models [7]. The class of potentials studied in this paper have not chosen with the aim of modeling some particular galaxies, our objective is to study systems which are generic in their basic properties. In [12], the authors study the galactic potential These kind of potentials are important in the modeling of elliptic galaxies, as for instance we can mention the potentials V L = log (1 + X 2 + Y 2 /q ) and V C = √ 1 + X 2 + Y 2 /q -1, where the parameter q gives the eccentricity of the elliptic galaxy. In this paper we deal with the Hamiltonian and its respective Hamilton's equation the matrix of the linear part of this system at the origin of coordinates is In order to obtain periodic orbits for these kind of potentials we will apply averaging theory, in this way we re-parametrize the coordinates by the factor √ ε for ε positive small enough (a similar change of coordinates has been used in [9]), that is we do the change with eigenvalues ± i, ± i/q where i = √ -1. After straightforward computations we get the Hamilton's equations which have the same linear part at the origin than the previous one, the structure of the new Hamiltonian is identity with (1) in the new variables. Our goal is to study which periodic orbits for ε = 0 (the unperturbed system) persists for ε positive and small enough (the perturbed system). By the form of the matrix M we observe the necessity to split the analysis for the periodic orbits in two cases In the second one, the orbits have period 2 πq , each periodic orbit on this plane is of the form When q is an irrational number our main result is: Theorem 1.1. For q an irrational number, we have that in every energy level H = h > 0 the Hamiltonian system (2) has So, for q irrational, we obtain that in every energy level H = h > 0 the perturbed Hamiltonian system has at least 2 periodic orbits. Remark 1.1. We note that the periodic orbits found in the statements of Theorem 1.1 are in fact degenerate Hopf bifurcations periodic orbits, since they born from the equilibrium point localized at the origin of coordinates. Unfortunately we cannot obtain periodic solutions when q is a rational number, see Remark 3.1. The paper is organized as follows. In section 2 we present the theorem from averaging theory necessary to prove our main result. In section 3 we give the proof of Theorem 1.1.", "pages": [ 1, 2, 3 ] }, { "title": "2. Some results from averaging theory", "content": "In order to have a self contained paper, in this section we present the basic results from the averaging theory that are necessary for proving the main results of this paper. 2.1. Results from averaging theory. We consider the problem of the bifurcation of T -periodic solutions from the differential system where the functions F 0 , F 1 : R × Ω → R n are of class C 2 functions, T -periodic in the first variable, and Ω is an open subset of R n . When ε = 0 we get the unperturbed system One of the main assumptions on the above system is that it has a submanifold of periodic solutions. A solution of system (5), for ε sufficiently small is given using the averaging theory. For a general introduction to the averaging theory see the books of Sanders and Verhulst [14], and of Verhulst [15]. Let x ( t, z ) be the solution of the unperturbed system (6) such that x (0 , z ) = z . We write the linearization of the unperturbed system along the periodic solution x ( t, z ) as In what follows we denote by M z ( t ) some fundamental matrix of the linear differential system (7), and by ξ : R k × R n -k → R k the projection of R n onto its first k coordinates; i.e. ξ ( x 1 , . . . , x n ) = ( x 1 , . . . , x k ). Theorem 2.1. Let V ⊂ R k be open and bounded, and let β 0 : Cl( V ) → R n -k be a C 2 function. We assume that /negationslash /negationslash We consider the function F : Cl( V ) → R k If there exists a ∈ V with F ( a ) = 0 and det (( d F /dα ) ( a )) = 0 , then there is a T -periodic solution ϕ ( t, ε ) of system (5) such that ϕ (0 , ε ) → z a as ε → 0 . Theorem 2.1 goes back to Malkin [10] and Roseau [13], for a shorter proof see [3].", "pages": [ 3, 4 ] }, { "title": "3. Proof of the main Theorem", "content": "In this section we give the proof of our main result Theorem 1.1. The fix value h of the total energy is determined by the initial periodic orbit, which in our case for the periodic orbit PO 1 it corresponds to h = 1 2 ( p 2 x 0 + x 2 0 ), choosing the sign + for p x , and expanding around ε = 0 we obtain The equations of motion on the energy level H = ( p 2 x 0 + x 2 0 ) / 2 are In order to apply Theorem 2.1 to system (11), let /negationslash The set Ω = { ( x, y, p y ) | q ( p 2 x 0 -x 2 + x 2 0 ) -y 2 -p 2 y = 0 } is an open subset of R 3 . Clearly the above functions are of class C 2 (Ω) . The set V of Theorem 2.1 is given by Let x ( t, z ) be the solution of the unperturbed system (6) such that x (0 , z ) = z . The variational equations of the unperturbed system along the periodic solution PO 1 are where y is a 3 × 3 matrix. The fundamental matrix M ( t ) of the differential system (13) such that M (0) is the identity matrix of R 3 takes the simple form whose inverse is given by An easy computation shows that /negationslash We observe that this matrix has a couple of zeros in the upper right corner of size 1 × 2; the determinant of the 2 × 2 matrix which appears in the lower right corner is 4 sin 2 ( π/q ) = 0 because q is an irrational number. Consequently all the assumptions of Theorem 2.1 are satisfied. Therefore we must compute the simple zeroes of the function F defined in Theorem 2.1. A straightforward computations shows that therefore we get Let In the last equality we have gotten p x 0 from the energy relation h = ( p 2 x 0 + x 2 0 ) / 2. So the solutions of f 1 = 0 are x 0 = ± √ 2 h , which are simple zeroes. On the other hand we can verify that both zeroes generate the same periodic orbit. Then doing the rescaling (3) we obtain statement (a) of Theorem 1.1. For the proof of statement (b) , as in the previous case we fix the value of the total energy as H = ( p 2 x 0 + x 2 0 ) / 2 q = h determined by the initial periodic orbit PO 2 . Computing p y from the equation H = h we obtain we choose the sign + for p y and expand arounf ε = 0 getting Now, we write the equations of motion on the energy level H = ( p 2 x 0 + x 2 0 ) / 2 q in the order ( y, x, p y ), they are given by the system In order to apply Theorem 2.1 to system (17) we are using the same notations and definitions (with the obvious changes) than in the previous case. Let x ( t, z ) be the solution of the unperturbed system (6) such that x (0 , z ) = z . The variational equations of the unperturbed system along the periodic solution PO 2 are where y is a 3 × 3 matrix. The fundamental matrix M ( t ) of the differential system (18) such that M (0) is the identity matrix of R 3 takes the simple form whose inverse is given by An easy computation shows that /negationslash We observe that this matrix has two zeros in the upper right corner of size 1 × 2; the determinant of the 2 × 2 matrix which appears in the lower right corner is 4 sin 2 ( πq ) = 0 because q is an irrational number. Consequently all the assumptions of Theorem 2.1 are satisfied. Therefore we must compute the simple zeroes of the function F defined in Theorem 2.1. A straightforward computations shows that therefore we have Let In the last equality we have gotten p x 0 from the energy relation h = ( p 2 x 0 + x 2 0 ) 2 q . So the solutions of f 1 = 0 are y 0 = ± √ 2 hq , which are simple zeroes. On the other hand we can verify that both zeroes generate the same periodic orbit. Doing the rescaling (3) we get the statement (b) of Theorem 1.1. Therefore we have proved that for q an irrational number, in every energy level h > 0, the Hamiltonian system (2) has at least 2 periodic orbits, so Theorem 1.1 holds. Remark 3.1. Using the methods of averaging theory studied in this paper, we could not obtain any periodic orbit for the Hamiltonian system (2) when q is a rational number. We have tried to get some information in two different ways, using cartesian coordinates as in statement (a) and using a modified kind of polar coordinates in two different planes. In the first way we have obtained the variational equations, but unfortunately we could not solve them. In the second way we have obtained that one of the equations that we must solve for obtain the periodic solutions is identically zero.", "pages": [ 4, 5, 6, 7, 8 ] }, { "title": "References", "content": "Academia de Matem'aticas, UACM, Plantel Casa Libertad, M'exico, D.F., 09620 M'exico E-mail address : [email protected] Departament de Matem'aticues, Universitat Aut'onoma de Barcelona, Bellaterra, Barcelona 08193, Catalonia, Spain E-mail address : [email protected] Departamento de Matem'aticas, UAM-Iztapalapa, Av. Rafael Atlixco 186, M'exico, DF 09340, Mexico E-mail address : [email protected]", "pages": [ 8 ] } ]

[ { "title": "ABSTRACT", "content": "Historically the velocity scatter seen on local Hubble plots has been attributed to the peculiar velocities of individual galaxies. Although most galaxies also have uncertainties in their distances, when galaxies with accurate distances are used recent studies have found that these supposed peculiar velocities may have preferred, or discrete, values. Here we report the interesting result that when these discrete components are identified and removed from the radial velocities of the SNeIa galaxies studied in the Hubble Key Project, there is evidence for a residual oscillation, or ripple, superimposed on the Hubble flow. This oscillation has a wavelength near 40 Mpc and, because its amplitude is small compared to that of the scatter in velocities, it becomes visible only after the discrete components are removed. This result is interesting because even if this ripple has been produced by a selection effect, the fact that it is only revealed after the discrete velocities are removed implies that the discrete velocities are real. Alternatively, if no selection effect can be identified to explain the ripple, then both the discrete velocities and the ripple together become very difficult to explain by chance and these results could then have interesting cosmological consequences. Subject headings: galaxies: Cosmology: distance scale - galaxies: Distances and redshifts - galaxies: quasars: general", "pages": [ 1 ] }, { "title": "Interesting Evidence for a Low-Level Oscillation Superimposed on the Local Hubble Flow", "content": "M.B. Bell 1", "pages": [ 1 ] }, { "title": "1. Introduction", "content": "It has been demonstrated (Tifft 1996, 1997), (and related papers) that there appear to be discrete 'velocity periods' present in the redshifts of galaxies. The most obvious of these was found in common spirals and showed discrete velocity components near 36, 72, 145, 290, etc., km s -1 . In each of the other period groups detected, the velocities showed this same octave related, or doubling nature. More recently we have found evidence that the extrapolation of Tifft's periods to higher components, using this doubling relation, leads to discrete velocity components that appear to be visible in the radial velocities of all galaxies whose distances are accurately known (Bell and Comeau 2003a,b; Bell et al. 2003, 2004). These small non-cosmological redshift compo- nents in galaxies introduce a scatter in the Hubble plot that is much larger than can be explained by the errors of measurement. This scatter has been explained historically by peculiar velocities, although in some cases this has been questioned simply because of their large size (Russell 2005a,b,c). If the quantization found in these velocities is real it would appear to rule out the peculiar velocity interpretation. In this paper we report a remarkable result in which a low-level ripple is clearly seen to be present in the residual Hubble plot after the discrete velocities are identified and removed from the velocities of the SNeIa galaxies studied in the Hubble Key Project. Because its amplitude is small compared to the scatter in velocities it is not visible before the discrete velocity components are removed. One model that might explain this type of oscillation in the Hubble flow has been discussed previously by Morikawa (1990, 1991).", "pages": [ 1 ] }, { "title": "2. Discrete Velocities Defined in Previous Work", "content": "The discrete 'velocity' components in galaxies, found by Tifft (1996, 1997) and recently confirmed by us, can be expressed by the relation: Here the integral values of the quantum number N correspond to the different 'velocity' periods identified by Tifft, and the quantum number m represents the number of halvings (via 2 -m ) below the relevant maximum intrinsic redshift component z iQ [ N,n max ] in each N -group (Bell and Comeau 2003a,b). The maximum for galaxies is equal to the minimum found previously for quasars in each relavent group (Bell 2002a,b,c,d, 2007). Until a physical explanation of the discrete components becomes known and it is determined if these are discrete velocities as suggested by Tifft, or preferred redshifts, in which case they might be related to the atom itself, or even some systematic effect, it is not possible to explain what the N and m quantum numbers might be due to. The discrete components in galaxies get quite small as m increases and because of this cannot be resolved at high m -values. Values for the larger (lowm ) discrete redshift components in galaxies, and their velocity equivalents, obtained using several independent galaxy groups containing 138 galaxies with accurate Tulley-Fisher distances, are listed in Bell and Comeau (2003b, Table 4) for N = 1 to 6. For each value of N the relevant z iQ [ N,n max ] value is given by the highlighted value in Bell (2002d, Table 2). The period group found by Tifft to be associated with common spiral galaxies, corresponds to the lowest, N = 1, group of discrete velocities. Because we looked at galaxies that were more distant than those studied by Tifft we were also able to conclude that the discrete components are superimposed on top of the Hubble flow.", "pages": [ 2 ] }, { "title": "3. Analysis of Type IA Supernovae Data", "content": "A preliminary analysis of the SNeIa data was reported previously (Bell and Comeau 2003b) and it followed the analysis used for spiral galaxies (Bell and Comeau 2003a,b) where a minimum was sought in the RMS deviation in source velocities, calculated relative to the nearest discrete velocity line superimposed on the Hubble flow. This analysis technique is explained in more detail by Bell et al. (2003, 2004). In Fig 1 the RMS deviations in V CMB velocities, relative to the nearest discrete velocity line in Fig 2, are plotted vs H o , using the 36 SNeIa galaxies listed in Table 6 of Freedman et al. (2001). The shape of this curve and a demonstration on how the dip at H o is produced when discrete components are present are discussed in detail by Bell et al. (2004). A clear best-fit feature is visible here at H o = 58 km s -1 Mpc -1 , as was found for the 138 spiral galaxies studied previously (Bell and Comeau 2003b; Bell et al. 2003). We have shown previously (Bell and Comeau 2003a; Bell et al. 2003) that there is nothing in our analysis procedure that can always produce an RMS dip at the same value of H o = 58 in random data. The fact that a similar dip at H o =58 was found in the several independent galaxy groups we looked at therefore makes this result very difficult to explain by chance. The minimum at H o = 67 in Fig 1 is due to the overall shape of the V CMB vs Distance source distribution. It is not produced by the presence of intrinsic components and appears in all source distributions, even randomly generated ones where there are no discrete velocity components present. It can be predicted to be slightly lower than the value that would be obtained by fitting a single straight line to the data (as is done in the Hubble Key Project which found a value of H o = 72). In Fig 2, the V CMB velocities of the 36 SNeIa galaxies are plotted vs distance. The slope of the discrete velocity lines is determined by H o and their discrete velocity values are defined by equation 1 above. Here it can be seen that the RMS dip at H o = 58 is obtained when several sources in each [ N,m ]-group fall along the discrete redshift lines over an extended range of distances. This is particularly obvious in Fig 2 for the [ N,m ] = [2,5], [5,8] and [5,9] lines, where 23 of the 32 sources below 300 Mpc fall along these three lines. Because of this it is possible to identify which N -group the sources are from. Unlike the other groups of galaxies studied, at least 60 percent of these sources appear to be N = 2 sources, while the remaining 40 percent are N = 5 sources. At the same time these sources lie at much greater distances, starting near 60 Mpc, where the previously studied groups leave off, and extending to 450 Mpc. The N values may relate to the type of galaxy that generates Type Ia SNe in the redshift range studied here. Tifft studied closer galaxies and found that N = 1 and 2 were most common. However, until an explanation for the discrete components is found this can only be speculation. It is also interesting to note that in some respects the plot in Fig 2 is similar to that in Fig 2 of Bell (2007), where intrinsic velocity or redshift components appear to increase with more distant objects.", "pages": [ 2, 3 ] }, { "title": "4. Hubble Plot After Removal of Discrete Velocity Components", "content": "Table 1 lists the SNeIa sources in col 1, with their distances and V CMB velocities in cols 2 and 3 respectively. Col 4 gives the intrinsic redshift [ N,m ] (and its associated 'discrete velocity') obtained for each source in the best-fit situation for H o = 58. Col 5 lists the Hubble velocity V H after removal of the discrete velocities. In Fig 3 the Hubble velocities (V H ) are plotted vs distance. A linear regression on the data (all 36 points) gave a slope of 58.26 (std. err. 0.33) km s -1 Mpc -1 . Because there are only four sources between 300 and 450 Mpc, and because the distance uncertainties may increase in proportion to distance, only those 32 sources closer than 300 Mpc have been considered for further analysis. It is important to understand here that the problem related to the increasing distance uncertainty is not one of fitting to the ripple, whose amplitude may also increase with distance, but one of fitting to the discrete velocity lines whose separations do not increase. When the distance uncertainties ap- proach the spacing between these lines the result becomes ambiguous. Although there are many more SNeIa galaxies with redshifts between z = 0.4 and 0.8 measured in the Supernova Cosmology Project (Perlmutter et al. 1999), these have not been included because they lie at redshifts where cosmological corrections need to be taken into account, and these can be confused with the intrinsic redshifts considered here. In Fig 3, where the velocities no longer contain discrete intrinsic components, there is now evidence for a low-level modulation superimposed on top of the otherwise linear Hubble slope. It has been approximated by the sinusoid (solid curve) to which all sources can be reasonably well fitted. Since there is nothing in our analysis that could have produced this modulation, it appears to indicate that there is a low-level, sinusoidal oscillation in the expansion rate of the local Universe. Changes in the expansion rate are not new, since it is now well accepted, at least over a much longer time scale, that the expansion rate of the Universe is currently accelerating. Investigators have argued that ripples in the Hubble flow might someday be detected (Davidson 2004, see for example) and others have already reported periodic density clumping with redshift (Broadhurst et al. 1990; Mo et al. 1992; Hartnett and Hirano 2008) (see below for further discussion). In Fig 4, a slope of 57.9 has been removed from the V H vs distance plot in Fig 3. Two velocities have been plotted for SN 1993ae, which is located at a distance of 71.8 Mpc. The open circle is the velocity obtained if this source is fitted to the nearest N =2group ( N,m =2,6). For this assignment it clearly does not fit the curve. However, if this source is fitted to the nearest grid line ( N,m = 1,4), it fits the curve well. It is therefore assumed that this one source is an N = 1 group object. Fig 4 clearly shows the presence of a sinusoidal fluctuation in the residual velocity (V R ) with increasing cosmological distance. The best-fit wavelength of this modulation is λ g =39.6 Mpc (period = ∼ 1 . 3 × 10 8 yrs, frequency ν g ∼ 2 . 4 × 10 -16 Hz). Its best-fit phase for a sine wave with 39.6 Mpc wavelength is 3.18 radians, with redshift peaks at 29.7, 69.7, 109.7, etc., ± 0 . 3 Mpc. We also note that the sources used here are located in all directions on the sky, confirming the isotropic nature of these results. The amplitude of the oscillation may also increase with distance, at least initially. The observed ripple, with a wavelength near 40 Mpc, is similar to one of the periods reported previously by Hartnett and Hirano (2008). Their distance spacing of (31.7 ± 1 . 8)h -1 corresponds to 44 ± 8 Mpc for H o = 72 ± 8 (h = 0.72 ± 0 . 08), which is the relevant value for H o to be used if the discrete components have not been removed. The spacing they find between peaks for this component is thus in good agreement with our 40 Mpc ripple. The ripple cannot be detected before the intrinsic components have been removed, but it can easily be seen by eye after the discrete components are removed, both before and after the Hubble slope is removed (Figs 3 and 4). There is no obvious sign of this ripple before the discrete components are removed because the scatter in velocities before removal of the discrete components is significantly larger than it is after removal of the discrete components. This is demonstrated more clearly in Fig 5 where V CMB is plotted versus distance, both before and after removal of the intrinsic components. There is no obvious sign of the ripple before the intrinsic components are removed. It is important to point out that to our knowledge no one has previously reported a velocity oscillation. Previous results using much larger samples have found density ripples from a periodic clumping with redshift (Broadhurst et al. 1990; Mo et al. 1992; Hartnett and Hirano 2008). When a velocity oscillation is found there is every reason to suspect that there might also be density clumping with the same period. This was pointed out by Morikawa (1990), who states that an oscillation of the Hubble parameter will cause an apparent density fluctuation. If there is a density ripple present with a similar period in other source samples its detection would not have been prevented by the presence of discrete velocity components if the distances are accurately known. The presence of intrinsic redshift components in the radial velocities of galaxies (velocity components that lie only on the high velocity side of the Hubble flow as here) will result in a Hubble constant that is too high if they are not taken into account. This has led to a suggested, revised value for the Hubble constant of H o = 58 km s -1 Mpc -1 (Bell and Comeau 2003b; Bell et al. 2003). This is ∼ 20% lower than the value reported by the Hubble Key Project, but in good agreement with the value found for intermediate epoch galaxies using the Sunyaev-Zel'dovich effect (Jones et al. 2001; Mason et al. 2001; Reese et al. 2002, 2003).", "pages": [ 3, 4, 6 ] }, { "title": "5. Do Selection Effects Play a Role?", "content": "If some identifiable selection effect can be found that would produce an oscillation in velocity that increases in amplitude as a function of distance it would be interesting because it would mean that since the removal of the discrete components is necessary before the ripple becomes visible, the discrete components are then unlikely to be random. If no selection effect can be found it also makes the ripple very difficult to explain if it is not real since it would then require that two random components have worked together in just the right way to produce these results by chance, which would be highly unlikely.", "pages": [ 7 ] }, { "title": "6. Discussion", "content": "There are several questions that need to be addressed before concluding our investigation. Does the presence of the ripple in the raw V CMB data produce the dip seen in Fig 1? Does its presence affect the dip? Is the ripple somehow produced by the reduction procedure? The data analysis used here contains three main steps as follows: 1) The best RMS fit of the data to the nearest intrinsic line is calculated as the Hubble slope in Fig 2 is varied in unit steps from 45 to 75. These values are plotted versus H o where an RMS dip will be detected when data points align with intrinsic components (Fig 1). 2) The discrete components obtained for the best-fit H o = 58 results are subtracted from the V CMB values to obtain the result in Fig 3. This involves a simple linear subtraction that should not produce a ripple. 3) The Hubble slope is subtracted from the data in Fig 3 to obtain the residual velocity V R plotted in Fig 4. Again this involves a simple linear subtraction that should not produce a ripple. To investigate whether or not the data reduction process produced the ripple we repeated the reduction process after first removing the ripple in Fig 4 from the raw V CMB values. This time no ripple was detected. The ripple was then subtracted from the data in Fig 4 and the result was compared to the result obtained from the second processing. The two were found to be identical, as expected. This showed that the ripple is only found when one is present in the data and we can conclude that the ripple is a real feature that was not created by our analysis procedure. However, it is also important to show that it was not the presence of the ripple in the raw data that produced the dip seen at H o = 58 in Fig 1. If the ripple is removed from the raw V CMB data before processing is the dip affected? Does it disappear completely? Answering these questions will also give us an idea of how robust our analysis is. In order to investigate this we have included in Fig 6 the RMS fit obtained after first removing the ripple in Fig 4 from the raw V CMB data. The dip is still clearly present and it is therefore obvious that the presence or absence of the ripple in the raw data does not affect the fitting of the discrete components. This is most likely because the ripple components are small relative to the separation of the discrete components. It also indicates that the dip at H o =58 is produced entirely by the presence of the discrete components in the data. However, as can be seen in Fig 6 there are slight differences between this curve and the curve in Fig 1. It can be assumed that there are also small pe- culiar velocities and distance uncertainties present in the data. If these affect the analysis differently with and without the ripple being present they may introduce differences that are truly random. If so, their effect can be minimized by averaging the curves in Fig 1 and Fig 6. This will minimize the effect of random fluctuations but should not affect any real features. This has been done in Fig 7 where it is seen that while the dip at H o = 58 is unaffected, some of the weaker features are reduced. We conclude that the dip at H o = 58 is a real feature that is likely to have been produced by the presence of discrete components in the V CMB data as we have argued previously. This conclusion is strengthened by the fact that a dip has been seen in other source samples at the same H o = 58 Hubble slope.", "pages": [ 7, 8 ] }, { "title": "7. Conclusions", "content": "We have identified what we believe are discrete components in the radial velocities of the 36 Type Ia Supernovae galaxies studied in the Hubble Key Project. These are defined by the same relation that also defines those identified previously by Tifft and by us in several other independent galaxy groups. As in our previous work this result is obtained for a Hubble constant of H o = 58 kms -1 Mpc -1 . We show that when these components are removed from the redshifts of the SNeIa galaxies there is evidence for a low-level sinusoidal oscillation superimposed on the Hubble flow. It is isotropic in nature and has a period of 40 Mpc. We have been unable to identify any systematic effect that could have produced this observed oscillation in the Hubble flow. If one can be identified it will mean that the oscillation is not a real oscillation in the Hubble flow. However, it will also mean that the discrete components must then be real since they have to be removed before the selection effect is visible and this is unlikely to happen if the discrete components are random ones. If no selection effect can be identified that can explain the ripple, then both the discrete velocities and the ripple together become very difficult to explain by chance and these results could then have significant cosmological consequences.", "pages": [ 8 ] }, { "title": "8. Acknowledgements", "content": "I thank D.R. McDiarmid for continued support and helpful comments, and S.P. Comeau for assistance with the data analysis, and especially, for writing the computer program that made it possible to do RMS fitting of several parallel lines to the data simultaneously.", "pages": [ 8 ] }, { "title": "REFERENCES", "content": "Bell, M.B.: ApJ, 566, 705 (2002a) Bell, M.B.: ApJ, 567, 801 (2002b) Bell, M.B.: (2002c) astro-ph/0208320 Bell, M.B.: (2002d) astro-ph/0211091 Bell, M.B. and Comeau, S.P.: (2003a), astro-ph/0305060 Bell, M.B. and Comeau, S.P.: (2003b), astro-ph/0305112 Bell, M.B., Comeau, S.P., and Russell, D.G.: (2003), astro-ph/0312132 Bell, M.B., Comeau, S.P. and Russell, D.G.: (2004), astro-ph/0407591 Bell M.B.: ApJ, 667, L129 (2007), arXiv:0704.1631 Broadhurst, T.J., Ellis, R.S., Koo, D.C., & Szalay, A.S. 1990, Nature, 343, 726 Davidson, A.: (2004), arxiv:gr-qc/0409059v1 Freedman, W.L. et al.: ApJ, 553, 47 (2001) Hartnett, J.G. and Hirano, K.: Ap&SS, 318, 13, (2008) Jones et al.: (2001), astro-ph/0103046 Mason,B.S., Myers, S.T., and Readhead, A.S.C.: ApJ, 555, L11, (2001) Mo, H.J., Deng, Z.G., Xia, X.Y., Schiller, P., & Borner, G. 1992, A&A, 257, 1 Morikawa, M. 1990 ApJ, 362, L37 Morikawa, M. 1991 ApJ, 369, 20 Perlmutter, S. et al.: ApJ, 517, 565, (1999) Reese et al.: ApJ, 581, 53, (2002) Reese, E.D.: (2003), astro-ph/0306073 Russell, D.G.: Ap&SS, 299, 387, (2005a) Russell, D.G.: Ap&SS, 299, 405, (2005b) Russell, D.G.: Ap&SS, 298, 577, (2005c) Tifft, W.G.: apj, 468, 491, (1996) Tifft, W.G.: apj, 485, 465, (1997)", "pages": [ 8, 9 ] } ]

[ { "title": "Effects in the Anomalistic Period of Celestial Bodies due to a Logarithmic Correction to the Newtonian Gravitational Potential", "content": "Omiros Ragos 1 , Ioannis Haranas 2 and Ioannis Gkigkitzis 3 1 Dept. of Mathematics, University of Patras, GR-26504 Patras, Greece e-mail: [email protected] M3J 1P3, Canada e-mail: [email protected] 3 Departments of Mathematics and Biomedical Physics, East Carolina University 124 Austin Building, East Fifth Street Greenville NC 27858-4353, USA e-mail: [email protected]", "pages": [ 1 ] }, { "title": "Abstract", "content": "We study the motion of a secondary celestial body under the influence of the logarithmic corrected gravitational force of a primary one. This kind of correction was introduced by Fabris et al. (2009). We derive two equations to compute the rate of change of the periastron w.r.t. the eccentric anomaly and its total variation over one revolution, In a kinematical sense, this influence produces an apsidal motion. We perform numerical estimations for Mercury and for the companion star of the pulsar PSR 1913+16. We also consider the case of the artificial Earth satellite GRACE-A, but the results present a low degree of reliability from a practical standpoint. Key words: Logarithmic potential, Gauss' planetary equations, periastron time, anomalistic period, Keplerian period.", "pages": [ 1 ] }, { "title": "1 Introduction", "content": "In order to explain the difference between the theoretically predicted and the observed position of Mercury's perihelion and its rate of precession, several theories have been proposed. These theories are related to modified versions of the Newtonian potential. Following this direction, Mücket and Treder (1977) introduced a logarithmic correction to the gravitational potential per unit mass. Various authors considered the same potential. Mioc et al. (1991) adopted it in order to estimate the difference between the nodal and Keplerian periods, as well as the changes of the orbital elements over a nodal period. Next, Diacu (1992) examined the validity of the Mücket-Treder gravitational law in the case of a three-body problem. Mioc (2004) has worked out the symmetries of the Mücket-Treder's two-body problem. In more recent works logarithmic potentials have been used by various researches in investigating the motion of galaxies, the existence and influence of dark matter and the applicability of long-range modified gravity models on the motion of the planets of our solar system. Van Moorsel (1987) found that the data obtained from the observation of the motion of some binary galaxies indicate the presence of dark matter. The influence of this matter can be approximated by a logarithmic potential. Kinney et al. (2001) studied the consequences of adopting the existence of a non-gravitational logarithmic potential instead of that of dark matter in order to explain the discrepancies between the dynamical mass measures of objects such as galaxies and clusters and the observed distribution of luminous matter. Kirillov (2006), while studying the bias relation between visible and dark matter in the case that the structure of the universe does not match that of the Friedman space, he justified that, when a galaxy is near a dark matter point source, a logarithmic-like term should be added to the Newton's potential. Iorio et al. (2008a) worked on the secular precessions of the longitudes of the perihelia of some planets of our solar system and examined if they are compatible with those predicted by long-range modified gravity models. Among others, they studied the results of adopting a logarithmic-type correction to the gravitational potential instead of considering the effect of dark matter. Fabris et al. (2009) analyzed the rotation curves of some spiral galaxies moving within a logarithmically corrected Newtonian potential. On the other hand, many contributions have been published either on studying the perturbations that affect the orbital elements of celestial bodies or on explaining and modeling the discrepancies between the predictions of the Newton's and/or Einstein's gravitation theory and the available observations on these elements. See, for example, Iorio (2005 ; 2007a), Adkins et al. (2007), Schmidt (2008), Ruggiero (2010), Xu (2011) and Haranas et al. (2011a). Post-Newtonian effects on the anomalistic period have been investigated, too. Iorio (2007b) considered a two-body system in eccentric orbits and examined the post-Newtonian relativistic gravitoelectric part of the precession of the mean anomaly which is not produced by the variation of the orbital period. Li (2010) studied the results of applying three relativity gravitation theories in expressing the post-Newtonian effects in the variation of the periastron passage time for binary stars. Later (2011), he examined the influence of the gravitational radiation damping on this time. Haranas et al (2011b) worked on the effects of a Yukawatype potential in the anomalistic period of celestial bodies. Last, we must mention that general relativity also predicts the well known gravitomagnetic clock effect in the anomalistic period of a particle orbiting a (slow) spinning main body. This relative literature is certainly too vast to be cited. So, we quote just a review paper written by Iorio et al. (2011a). In the present work we opt to use a logarithmic correction to the gravitational Newtonian potential in order to calculate the anomalistic time of a secondary body orbiting a primary one. This correction can be expressed as a modification of the Newtonian potential energy per unit mass by the term: (Fabris et al. 2009; Iorio et al. 2008a) where M is the mass of the primary, G is the Newtonian gravitational constant, r is the radial distance of the secondary body from the primary one,  is a parameter with dimension of inverse length ( 1 L  ) and 0 r is an arbitrary parameter with dimension of length ( L ). It has been found that a 'concordance' value for this parameter is 1 . 0    Kpc -1 . The total acceleration acting on the secondary is: where   1 r G G r   . Therefore, the presence of the non-Newtonian term can be considered as converting G into a space-varying Newtonian gravitational constant (Iorio 2011b; Haranas et al. 2011a) . In this paper we evaluate our findings using the planet Mercury, the companion star of the pulsar PSR 1913+16, and the artificial satellite GRACE-A. Finally, we compare our results to those obtained by applying a Yukawa-type correction in Haranas et al. (2011b).", "pages": [ 1, 2, 3 ] }, { "title": "2 Rate of change and variation per revolution of the periastron time", "content": "We consider the unperturbed relative orbit of the secondary body, a Keplerian ellipse. Let a be the semimajor axis, e the eccentricity, n the mean motion, and M the mean anomaly of this orbit. First, we will express the rate of change of the periastron time 0  in terms of the true anomaly f . The mean anomaly is defined by where t is the time variable. We differentiate Eq. (3) with respect to t and obtain: Using also that, on the unperturbed Keplerian orbit of the secondary, Kepler's third law is given by 2 3 G n a  M , the time rate of change of the mean motion is found to be: In the presence of a perturbation, the rates of change of the orbital elements can be expressed by means of Gauss' planetary equations: For the semimajor axis and the mean anomaly they read: where R and T are the radial and transverse components of the perturbing acceleration. In our case, ln 0 T  while Substituting Eqs. (5)-(7) into (4) we obtain that, for 0 1 e   : where Then, we use the well known relations (see, e.g., Murray and Dermott, 1999) to express Eqs. (8)-(9) in terms of the eccentric anomaly E We obtain that: and The logarithmic correction effect on 0 T over a whole revolution of the secondary is obtained by integrating 0 / dT dE over the interval [0,2 ]  . Then, the change of the anomalistic period per revolution: If we can measure the change in the anomalistic period per revolution for a given body in an elliptical orbit, we can then write that the coupling constant  is given by the following expression:", "pages": [ 3, 4 ] }, { "title": "3 Numerical results", "content": "First, we proceed with the calculation of the variation of the anomalistic period of SIRIUS companion  CMaB: Mass of primary = 2.02 M sun, semimajor axis of companion aB = 19.80 AU, (Skemer and Close, 2011) e = 0.5923, n = 3.0  10 -9 rad/s, we obtain that:  TYuk for Sirius companion We use lamda and alpha 15 10 94 . 4    m, and  = 10 10 12 10 9 . 4 , 10 6 . 4 , 10 2 . 4       . First, we proceed with the calculation of the variation of the anomalistic period of Mercury. For this planet, we have used the following orbital parameters: 57909083 a  km, 0.205 e  , 7 8.07 10 n    rad/s. For the primary, 30 1.99 10    M M kg. Applying 1 . 0    Kpc -1 to Eq. (18), we obtain that: Next, we estimate the change of the anomalistic period of the companion star of the pulsar PSR 1913+16. The orbital parameters for this star are 6 1.9501 10 a   km, 0.617, e  and 4 1.575 10 n    rad/s. The primary's mass is 1.387 .  M M Then: Finally, we calculate the variation of the perigee passage time of the artificial Earth satellite GRACE-A. We have used that, for this satellite, 6876.4816 a  km, 0.00040989 e  and 0.001100118 n  rad/s (http://www.csr.utexas.edu/grace/) and, for the primary, 24 5.9722 10     M M kg. Using Eq. (18) we find that: We should note here that the degree of reliability of the results concerning GRACE-A is low from a practical standpoint, because of the very small eccentricity. It is known that, for quasicircular orbits, the position of the periastron (hence the periastron time) cannot be accurately determined. However, our results are still of some interest as regards the order of the perigee time variation. A sensitivity analysis for post-Newtonian effects on the GRACE-A and B spacecrafts was worked out by Iorio (2012). In Figures 1, 2 and 3, we present the variation of rate of change of the anomalistic time w.r.t to the eccentric anomaly E of Mercury, the companion star of the pulsar PSR 1913+16 and GRACE-A. Figure 1 for Mercury indicates that there exist two values of the eccentric anomaly for which this rate is zero. Solving numerically the equation: that results from the R.H.S. of Eq. (17), we obtain that these values are 63.293899 E  and 330.688702 E  . The maximum of 0 / dT dE can be found by solving: that comes from the derivative of the L.H.S. of Eq. (23). This maximum is at 215.422386 . E  Similarly, for the pulsar PSR 1913+16, 0 / dT dE is zero at 32.773697 E  and 359.551488 E  while its maximum is achieved at 306.101830 . E  Finally, for GRACE-A, 0 / dT dE is zero at 89.929590 E  and 270.070591 E  , while its maximum occurs at 180.000091. E  Figures 4 and 5 present the variation of the anomalistic periods of Mercury and the companion star of the pulsar PSR 1913+16 as functions of the eccentric anomaly E and the radial orbital distance . r Several works concerning the effects of Yukawa-type potentials on orbital elements have been published (see, for example, Kokubun 2004; Iorio 2008b). In Haranas et al. (2011), the authors dealt with the anomalistic time change due to such a correction to the Newtonian potential. Comparing the results of the present work with those of the aforementioned publication, we see the following: For Mercury, the change of the anomalistic period because of the logarithmic correction and that by the Yukawa correction with a coupling constant 10 Yuk 3.57 10     are connected through the relation 0 ln 0 Yuk 2.220 . T T    In the case of the companion star of the pulsar PSR 1913+16 and for 11 Yuk 6.409 10 ,     the corresponding relation is 0 ln 0 Yuk 0.544 T T    For GRACE-A and a Yukawa coupling constant in the range 12 10 Yuk 4.2 10 3.184 10        , we obtain that 0 Yuk 0 ln 0 Yuk 0.000155 0.0180 . T T T     ", "pages": [ 5, 6 ] }, { "title": "4 Summary and concluding remarks", "content": "Using a logarithmic correction to the Newtonian gravitational potential as in Fabris et al. (2009) and Iorio et al. (2008a), we derive an eccentric anomaly-dependent equation that estimates rate of change of the anomalistic period of a secondary body orbiting a primary one. By using the integral of this equation over a whole revolution, the contribution of the logarithmic correction to the change of the periastron time can be calculated. This variation was estimated for some concrete astronomical cases. Its observational detection can constitute a possible test for the action of post-Newtonian type forces on the solar system bodies or on other celestial objects. A logarithmic correction is by no means the only kind of correction to be considered in the modification of the Newtonian gravitational potential. For example, general relativistic corrections as well as quantum corrections can be also examined but that is another topic that we are going to deal with in the nearest future. Acknowledgements The authors would like to thank the anonymous reviewer for his valuable comments and suggestions that helped to improve this manuscript considerably.", "pages": [ 10 ] }, { "title": "References", "content": "Adkins, G.S., McDonnell, J.: Orbital precession due to central-force perturbations, Phys. Rev. D 75 (8), id. 082001 (2007). Iorio, L.: Astronomical constraints on some long-range models of modified gravity, Advances in High Energy Physics 2007 , Article ID 90731 (2007a). Iorio, L.: The post-Newtonian mean anomaly advance as further post-Keplerian parameter in pulsar binary systems, Astrophys. Space Sci. 312 (3-4), 331 (2007b). Iorio, L., Lichtenegger, H.I.M., Ruggiero, M.L. Luca; Corda, C.: Phenomenology of the Lense-Thirring effect in the solar system, Astrophys. and Space Sci. 331 (2), 351 (2011a). Murray, C.D. and Dermott, S.F., Solar System Dynamics, Cambridge University Press (1999).", "pages": [ 11, 12 ] } ]

[ { "title": "ABSTRACT", "content": "I use the fact that the radiation emitted by the accretion disk of supermassive black hole can heat up the surrounding gas in the protogalaxy to achieve hydrostatic equilibrium during the galaxy formation. The correlation between the black hole mass M BH and velocity dispersion σ thus naturally arises. The result generally agrees with empirical fittings from observational data, even with M BH ≤ 10 6 M /circledot . This model provides a clear picture on how the properties of the galactic supermassive black holes are connected with the kinetic properties of the galactic bulges. Subject headings: Galaxies, galactic center, supermassive blackholes, velocity dispersion", "pages": [ 1 ] }, { "title": "Shaping the relation between the mass of supermassive black holes and the velocity dispersion of galactic bulges", "content": "M. H. Chan Department of Physics and Institute of Theoretical Physics, The Chinese University of Hong Kong, Shatin, New Territories, Hong Kong, China [email protected]", "pages": [ 1 ] }, { "title": "1. Introduction", "content": "In the past decade, some observations have led to some tight relations between the central supermassive blackhole (SMBH) masses M BH and velocity dispersions σ in the bulges of galaxies. These relations can be summarized as log( M BH /M /circledot ) = β log( σ/ 200 km s -1 )+ α . For 10 6 M /circledot ≤ M BH ≤ 10 9 M /circledot , the values of α and β have been estimated several times in the past 12 years: originally ( α, β )=(8 . 08 ± 0 . 08, 3 . 75 ± 0 . 3) (Gebhardt et al. 2000) and (8 . 14 ± 1 . 3, 4 . 80 ± 0 . 54) (Ferrarese and Merritt 2000), then (8 . 13 ± 0 . 06, 4 . 02 ± 0 . 32) (Tremaine et al. 2002), and more recently (8 . 28 ± 0 . 05, 4 . 06 ± 0 . 28) (Hu 2008), (8 . 12 ± 0 . 08, 4 . 24 ± 0 . 41) (Gultekin et al. 2009), (8 . 29 ± 0 . 06, 5 . 12 ± 0 . 36) (McConnell et al. 2011) and (8 . 13 ± 0 . 05, 5 . 13 ± 0 . 34) (Graham et al. 2011). Generally speaking, empirical fittings show that α ≈ 8 and β ≈ 4 -5. These relations correspond to all morphological type galaxies. One can separate the fittings into different groups such as the early-type and late-type or elliptical and spiral. For example, McConnell et al. (2011) obtain ( α, β ) = (8 . 38 , 4 . 53) and (7 . 97 , 4 . 58) for the early-type and late-type galaxies respectively if they are fitted separately. The slopes are shallower than the combined one ( β = 5 . 12). Moreover, the slope β and the scatter of the M BH -σ relation are still subject to debate, particularly at the low mass ends. Recently, Xiao et al. (2011) obtain a new M BH -σ relation with low BH masses (below 2 × 10 6 M /circledot ). They find a zero point α = 7 . 68 ± 0 . 08 and slope β = 3 . 32 ± 0 . 22, which indicate β may be smaller for lower BH masses. Also, Wyithe (2006) obtained a better fit by using a logquadratic form log( M BH /M /circledot ) = α + β log( σ/ 200 km s -1 ) + γ ' [log( σ/ 200 km s -1 )] 2 with α = 8 . 05 ± 0 . 06, β = 4 . 2 ± 0 . 37 and γ ' = 1 . 6 ± 1 . 3. Therefore, it is reasonable to doubt that the relation is not simply given by M BH ∝ σ β with β just a constant for all galaxies. The M BH -σ relation has been derived by recent theoretical models (Silk and Rees 1998; Adams et al. 2001; MacMillan and Henriksen 2002; Robertson et al. 2005; Murray et al. 2005; King 2005; McLaughlin et al. 2006; Power et al. 2011; Nayakshin et al. 2012). However, these models contain various assumptions and fail to explain the relations in the small SMBH mass regime ( β ≈ 3 . 3). In this article, I present a model to get an exact M BH -σ relation, which can explain the parameters α and β in the empirical fitting in both small SMBH regime and apply to different types of galaxies. I use the fact that the strong radiation of the accretion disk of a SMBH can heat up the surrounding gas so that hydrostatic equilibrium of the latter is maintained. The cooling of the surrounding gas is mainly given by recombination, bremsstrahlung radiation and the adiabatic expansion of the gas. Without any other assumptions, the exact M BH -σ relation is naturally obtained. In the following, I will first present the details of the model. Then I will fit our model with the data of σ and M BH from 198 galaxies and show that it generally agrees with the empirical fitting.", "pages": [ 1, 2 ] }, { "title": "2. The Accretion model of supermassive black hole and the M BH -σ relation", "content": "It is commonly believed that all SMBHs accompany with accretion disks to emit high energy radiation during their formation. The luminosity of the disk is mainly come from the rest mass energy of the mass accretion. The accretion luminosity can be expressed as where 0 . 05 ≤ η ≤ 0 . 4 is the efficiency of the process. Let f Ed = L disk /L Ed , where L Ed = 1 . 5 × 10 38 ( M BH /M /circledot ) erg s -1 is the Eddington limit of accretion, we have The accretion disk of SMBH provides a large number of photons to heat up the surrounding gas in the protogalaxy during the galaxy formation. Assume that the power is mainly transmitted to the protogalaxy within a scale radius R through radiation by compton scattering and photoionization. The optical depth of the gas is τ = nσ ph R /lessmuch 1, where n is the number density of the hot gas and σ ph is the effective cross section of the interaction of photons and hot gas particles, which is closed to the Thomson cross section σ Thom for zero metallicity. Therefore, the total power that can be transmitted to the protogalaxy is just L disk τ . In equilibrium, the heating rate is equal to the cooling rate by bremsstrahlung radiation Λ B , recombination Λ R , and adiabatic expansion Λ a (Katz et al. 1996): where Λ B 0 = 1 . 4 × 10 -27 erg cm 3 s -1 , Λ R 0 = 3 . 5 × 10 -26 erg cm 3 s -1 , T , p , V , are the temperature, pressure and volume of the gas within R , respectively. The term pdV/dt can be written as pdV/dt ≈ pV 2 / 3 ( γkT/m g ) 1 / 2 and p = nkT (Muno et al. 2004; Chan and Chu 2008), where γ ≈ 5 / 3 is the adiabatic index and m g is the mean mass of a gas particle. The Virial relation of the effective total mass of hot gas M g and T within R is given by (Sarazin 1988) where f 1 is the virial constant. After the galactic bulge is formed, assuming spherical symmetry and by Virial theorem again, one can get where f 2 is another virial constant. From Eqs. (3), (4) and (5), and assuming nV = M g /m g , we get where Take f 1 ≈ 1 (isothermal distribution) and f 2 ≈ 1 / 5 (Cappellari et al. 2006), and combine with Eq. (2), we have where σ 200 = σ/ 200 km s -1 . Therefore, assuming τ ∼ 0 . 005, the last term dominates for σ ≥ 60 km s -1 , which agrees with the observed range of β . The fitting parameters f Ed and τ are mainly controlled by the empirical fitting parameters α and β of the observational data respectively. In Fig. 1, we get an empirical fit by using the data obtained from Greene and Ho (2006); Xiao et al. (2011); McConnell et al. (2011). The effective cross section due to metallicity may contribute to a factor of 2-3 in Eq. (11). By using the cross sections of some major metals (carbon, nitrogen, oxygen, silicon) calculated from Daltabuit and Cox (1972) and assuming metallicity of a protogalaxy is about 10 -3 solar metallicity (Jappsen et al. 2009), the effective ccross section is 2 × 10 -24 cm -2 , which is about 3 σ Thom . Also, the estimation of the M BH is not too reliable for M BH ≤ 10 6 M /circledot . Therefore, the fitting parameters are just an order of magnitude estimation. In Fig. 1, the functional form of Eq. (11) generally matches the observational data. The best fitted parameters are f Ed = 50 and τ = 0 . 005, with 4.9% rms error. By fitting with the form log( M BH /M /circledot ) = β log σ 200 + α , we can get ( α, β ) = (8 . 2 , 4 . 5) with 5.1% rms error. Therefore, two functional forms can fit the data equally well. However, if we neglect the first two terms in the right hand side of Eq. (11), the best fitted line is M BH / 10 8 M /circledot = 4 . 5 σ 5 200 with 12% rms error. As mentioned above, the first two terms are significant when M BH or σ is small. Therefore, the effective slope of the log M BH -log σ relation is shallower ( β < 5). That means the effect of cooling by recombination and bremsstrahlung radiation should be considered especially in galaxies with low velocity dipersion. Our result is consistent with the recent observations which indicate that many supermassive black holes may involve a long period of moderate super-Eddington accretion ( f Ed ∼ 10) during their formation (Kawaguchi et al. 2004; Brian and Zhao 2004; Wang et al. 2008). On the other hand, the central number density in the Milky Way is about 0 . 1 -0 . 5 cm -1 (Muno et al. 2004), which corresponds to τ ∼ 0 . 001 -0 . 005 in the bulge. The best fitted τ is also consistent with the observational data.", "pages": [ 2, 3, 4 ] }, { "title": "3. Discussion", "content": "In this article, I present a new model to explain the M BH -σ relation in galaxies. The M BH -σ relation is not simply a power-law form M BH ∝ σ β , but partially depends on σ 3 and σ 5 . This exact form of the M BH -σ relation agrees with the recent observational data, especially in the small M BH regime. In this model, we can obtain 3 ≤ β ≤ 5 and α ≈ 8, which is consistent with the observational data of the grouped galaxies (small SMBH: α ≈ 7 . 7, β ≈ 3 . 3; early-type: α ≈ 8 . 4, β ≈ 4 . 5; late-type: α ≈ 8 . 0, β ≈ 4 . 6) (Xiao et al. 2011; McConnell et al. 2011). In general, this model suggests that larger τ and f Ed result in smaller slope β and normalization constant α in the relation. Thus, if similar galaxies have similar bulge structure and accretion disks, then the M BH -σ relation of this particular type may be tighter. It generally agrees with the observation that the M BH -σ relation in elliptical galaxies only is less scattered (Graham et al. 2011). In this model, the evolution pattern of the supermassive black hole does not affect the function form of the relation. The only physics here is the energy balance of the gas between the heating by the radiation from accretion disk and the cooling by the free-free emission, recombination and the adiabatic expansion of the gas particles. If the black hole is still significantly accreting, the energy given out would be balanced by the cooling of gas, which gives the Eq. (11). When the black hole's activity is switched off, the relation between the kinematic properties of the bulge and the M BH has already been established, which remains unchanged in Eq. (11). Therefore, the exact relation between M BH and σ can definitely apply in both active and non-active galaxies. All the parameters obtained ( f Ed ∼ 10 and τ ∼ 0 . 001) are consistent with the theoretical estimation and observation. Generally, our result supports the moderate super-Eddington accretion during the SMBH formation. The variations of f Ed and τ within the groups of galaxies may result in the observed scatter in the M BH -σ fittings. All the above results arose from existing natural physical laws without any extra assumptions. This model provides a clear picture on how the properties of the galactic supermassive black holes are connected with the kinetic properties of the galactic bulges.", "pages": [ 4, 5 ] }, { "title": "4. Acknowledgement", "content": "I am grateful to the referee for helpful comments on the manuscript.", "pages": [ 5 ] }, { "title": "REFERENCES", "content": "Adams, F. C. , Graff, D. S. and Richstone, D. O., 2001, ApJ, 551 , L31. Brian, W. and Zhao, Y. 2004, MNRAS, 352 , 823. Cappellari, M. et al. 2006, MNRAS, 366 , 1126. Chan, M. H. and Chu, M.-C. 2008, MNRAS, 389 , 297. Daltabuit, E. and Cox, D. P. 1972, ApJ, 177 , 855. McConnell, N. J., et al. 2011, Nature, 480 , 215. Ferrarese, L. and Merritt, D. 2000, ApJ, 539 , L9. Gebhardt, K., et al. 2000, ApJ, 539 , L13. Graham A. W., Onken, C. A., Athanassoula, E. and Combes, F. 2011, MNRAS, 412 , 2211. Greene, J. E. and Ho, L. C. 2006, ApJ, 641 , L21. Gultekin, K., et al. 2009, ApJ, 698 , 198. Hu, J. 2008, MNRAS, 386 , 2242. Jappsen, A.-K., Mac Low, M.-M., Glover, S. C. O., Klessen, R. S. and Kitsionas, S. 2009, ApJ, 694 , 1161. Katz, N., Weinberg, D. H. and Hernquist, L. 1996, ApJS, 105 , 19. Kawaguchi, T., Aoki, K., Ohta, K. and Collin, S. 2004, Astron. Astrophys., 420 , L23. King, A. R. 2005, ApJ, 635 , L121. MacMillan, J. D. and Henriksen, R. N. 2002, ApJ, 569 , 83. McLaughlin, D. E. et al. 2006, ApJ, 650 , L37. Muno, M. P. 2004, ApJ, 613 , 326. Murray, N., Quataert, E. and Thompson, T. A. 2005, ApJ, 618 , 569. Nayakshin, S., Power, C. and King, A. R. 2012, ApJ, 753 , 15. Power, C., Zubovas, K., Nayakshin, S. and King, A. R. 2011, MNRAS, 413 , L110. Robertson, B. H. et al. 2005, ApJ, 641 , 90. Sarazin, C. L. 1988, X-ray Emission from Clusters of Galaxies (UK: Cambridge). Silk, J. and Rees, M. J. 1998, Astron. Astrophys., 331 , L1. Tremaine, S. et al. 2002, ApJ, 574 , 740. Wang, J.-M., Chen, Y.-M., Yan, C.-S. and Hu, C. 2008, ApJ, 673 , L9. Wyithe, J. S. B. 2006, MNRAS, 365 , 1082. Xiao, T., et al. 2011, ApJ, 739 , 28.", "pages": [ 5, 6, 7 ] } ]

[ { "title": "A graphical analysis of the systematic error of classical binned methods in constructing luminosity functions", "content": "Zunli Yuan 1,2,3 • Jiancheng Wang 1,2 Abstract The classical 1 /V a and PC methods of constructing binned luminosity functions (LFs) are revisited and compared by graphical analysis. Using both theoretical analysis and illustration with an example, we show why the two methods give different results for the bins which are crossed by the flux limit curves L = L lim ( z ). Based on a combined sample simulated by a Monte Carlo method, the estimate φ of two methods are compared with the input model LFs. The two methods give identical and ideal estimate for the high luminosity points of each redshift interval. However, for the low luminosity bins of all the redshift intervals both methods give smaller estimate than the input model. We conclude that once the LF is evolving with redshift, the classical binned methods will unlikely give an ideal estimate over the total luminosity range. Page & Carrera (2000) noticed that for objects close to the flux limit φ 1 /V a nearly always to be too small. We believe this is due to the arbitrary choosing of redshift and luminosity intervals. Because φ 1 /V a is more sensitive to how the binning are chosen than φ PC . We suggest a new binning method, which can improve the LFs produced by the 1 /V a method significantly, and also improve the LFs produced by the PC methods. Our simulations show that after adopting this new binning, both the 1 /V a and PC methods have comparable results. Keywords galaxies: luminosity function, mass function - galaxies: quasars: general Zunli Yuan Email: [email protected]", "pages": [ 1 ] }, { "title": "1 Introduction", "content": "From shortly after the first quasar was found until the present, considerable effort has been spent in obtaining samples to investigate their luminosity distribution as a function of redshift, known as the luminosity function (LF). The LF is very important because its shape and evolution provide constraints on the nature of activity and the cosmic evolution of quasars/active galactic nuclei (AGNs). Up to now many statistical approaches have been proposed to investigate the LFs. These include parametric techniques which assume analytical form for the LFs, and non-parametric methods which usually need binning the data (see Johnston 2011, for an overall review). Among the non-parametric methods, the 1 /V a estimator (see Avni & Bahcall 1980; Eales 1993; Ellis et al. 1996) is the most classical binned method and is particularly prevalent for its simplicity. Although more than four decades have passed since its original version (i.e., the famous 1 /V max estimator, Schmidt 1968) was presented, the 1 /V a method is not outdated and continues to be widely used in the literature (see Civano et al. 2011; Mao et al. 2012; Marchesini et al. 2012; McAlpine & Jarvis 2011; Padovani et al. 2011; Patel et al. 2012; Hiroi et al. 2012; Yuan & Wang 2012; March˜a & Caccianiga 2013, for latest use). On the other hand, authors have pointed out that the 1 /V a method introduces a significant error for objects close to the flux limit (e.g., Page & Carrera 2000; Cara & Lister 2008). Page & Carrera (2000) presented an improved method (hereafter the PC method) of constructing the binned LF. According to the result based on a Monte Carlo simulation, the authors believed their method is superior to the 1 /V a method in many aspects. However, their simulation was performed using a luminosity function which is unchanging with redshift (i.e. no evolution) and a single flux limit. The assumption of no evolution is too particular to be able to represent more general situations. Furthermore, they focused on the situation of single sample (a single flux limit), and the discussion on multiple samples was not sufficient. In this paper, we revisit the 1 /V a and PC methods to find the reason of systematic error using graphical analysis. The situation when multiple samples are combined to obtain a LF is particularly discussed. By the way, in recent years some more sophisticated and rigorous methods have emerged (e.g., Schafer 2007; Kelly et al. 2008; Christlein et al. 2009; Takeuchi 2010; Johnston et al. 2012). Nevertheless, it needs time for the new methods to be recognized and in widespread use. During this time, specifying deficiencies of the old methods could be helpful. Throughout the paper, we adopt a Lambda Cold Dark Matter cosmology with the parameters Ω m = 0.27, Ω Λ = 0.73, and H 0 = 71 km s -1 Mpc -1 .", "pages": [ 1, 2 ] }, { "title": "2 Methods", "content": "The differential LF φ ( L, z ) is defined as the number density of target sources per unit comoving volume per unit luminosity interval, i.e. More often it is defined in terms of log L . In this paper we do not differentiate L and log L strictly.", "pages": [ 2 ] }, { "title": "2.1 The 1 /V a method", "content": "The 1 /V a method originates from the celebrated paper by Avni & Bahcall (1980), which generalized V/V max (Schmidt 1968) for multiple samples. Here we consider two flux-limited samples observed at the same frequencies. For simplicity, we assume the two samples to be not overlapping in survey regions, as shown in Fig. 1. Sample D (D denotes deep) is assumed to be deeper in all the sample frequencies than Sample B (B denotes bright). Let S D lim and S B lim denote their flux limits respectively, then S D lim < S B lim . If N objects appear in the interval ∆ L ∆ z ( L 1 < L < L 2 , z 1 < z < z 2 ) around the bin center ( L, z ), the LF is estimated as According to the scenario of Avni & Bahcall (1980), the sample B and D can be combined into a 'coherent sample'. It can be regarded as a single sample, in which each object is allowed to be distributed anywhere within the total volume. Thus the available volume V a for any object i in sample B and D can be calculated as where Ω S ( z 1 , z iS max ) is the effective survey area in steradians of the S th survey. We have Ω S ( z 1 < z iS max ) = Ω S and Ω S ( z 1 ≥ z iS max ) = 0. Ω B and Ω D are the solid angles subtended by B and D samples on the sky respectively. z iS top is defined as However, one must keep in mind that the sources with fluxes between S D lim and S B lim (represented by red solid circles in Fig. 1, hereafter red sources) can never appear in sample B. In the L -z plane (shown in Fig. 2), these sources are located between the curves L = L B lim ( z ) and L = L D lim ( z ), which represent the flux limits of survey B and D respectively. On the other hand the sources with fluxes above S B lim (represented by black solid squares in Fig. 1, hereafter black sources) may appear both in sample B and D. The above discussion helps us to distinguish the actual surveyed regions corresponding to red and black sources.", "pages": [ 2 ] }, { "title": "2.2 The PC method", "content": "The key point of the 1 /V a method is that it takes into account the contribution of object i to the number den- sity of the bin ∆ L ∆ z as 1 / (∆ LV i a ). Page & Carrera (2000) presented an improved method of constructing the binned LF (also see Tzanavaris & Georgantopoulos 2008; Yencho et al. 2009). That is, the LF at the center of a bin with a luminosity interval L 1 and L 2 and a redshift interval z 1 and z 2 can be estimated as where N is the number of sources detected within the bin, and the double integral corresponds to the shaded area of bin 1 in Fig. 2. The key point of the PC method is to consider the actual surveyed region of a bin as a four-dimensional polyhedron in the volume-luminosity space, and the φ PC for this bin is the four-numberdensity. /s32", "pages": [ 2, 3 ] }, { "title": "2.3 Comparison of the two methods", "content": "Here we take into account three situations represented by bin 1, bin 2, bin 3 in Fig. 2. For them, the shaded regions are the regions of the volume-luminosity plane in the interval ∆ L ∆ z that has been surveyed (also see Page & Carrera 2000), and are marked by 1, 2 ' ,2 '' and 3. For bin 2, because either red or black sources appear in it, 2 ' and 2 '' are used to represent different surveyed regions. In the volume-luminosity space, these shaded regions are four-dimensional polyhedrons illustrated in Fig. 3(a). Their four-volumes can be calculated as Therefore, according to section 2.2, the estimated LF by the PC method φ PC for bin 1, bin 2 and bin 3 are calculated as N 1 /A 1 , N 2 / ( A 2 ' + A 2 '' ) and N 3 /A 3 , where N 1 , N 2 and N 3 are the numbers of sources detected within bin 1, bin 2 and bin 3, and A 1 , A 2 ' , A 2 '' and A 3 are defined by Eq.(5)-(8). It is emphasized that N 2 is the sum of red and black sources detected within bin 2. In the use of the 1 /V a method, the available volume V a of every source should be calculated. The calculation of V a for a source depends on its location in a bin. In Fig.2, a few example sources with different locations and status are labeled as i, j, k, m , and n respectively. /s32 The area equal to [∆ LV i a ] for an object i (represented by a red spot) is shown in Fig. 4(a). This area is clearly not the same as that of the surveyed region (the shaded region of bin 1, marked as 1 in Fig. 2). The deviation depends on the position of an object i in bin 1. If the object locates at the left margin of bin 1 (see Fig. 4(b)), the area equal to [∆ LV i a ] (the shaded region in Fig. 4(b)) will be much smaller than that of the surveyed region (the shaded region of bin 1, in Fig. 2), and vice versa (see Fig. 4(c)). Hence φ 1 /V a and φ PC will give different estimates of φ for this bin. For a source k (represented by the red solid circle labeled 'k' in Fig. 2) in bin 2, when z 1 < z kB max < z 2 and z 2 < z kD max , we have Although source m (represented by the black square labeled 'm' in Fig. 2) is a black one, its situation is The total surveyed region of bin 2 is the sum of 2 ' and 2 '' . The area equal to [∆ LV j a ] (see Fig. 3(b), j specifically) is clearly smaller than that of the surveyed region (Fig. 3(a), 2 ' +2 '' ). Consequently, the density contribution of a source like j to bin 2 is positive, and possibly leads φ 1 /V a to give a exaggerated estimate for bin 2. The area equal to [∆ LV k a ] as well as [∆ LV m a ] are approximations of the surveyed region of bin2. Overall, φ 1 /V a possibly gives a higher estimate of φ than φ PC for this bin. The area equal to [∆ LV n a ] for an object n is the same as that of the surveyed region and hence φ 1 /V a and φ PC will give the same estimate of φ for this bin. To sum up, the two methods give different results for the bins which are crossed by the flux limit curves L = L lim ( z ). /s32 /s32", "pages": [ 3, 4 ] }, { "title": "2.4 Multiple Samples", "content": "Practically many samples with different depth, extent and degrees of completeness are combined to obtain a LF that spans a wide range of luminosities and redshifts. When the sources from different samples are plotted on the redshift-luminosity plane and binned to use the 1 /V a or PC methods, much more bins are crossed by one or more flux limit curves. Some bins are divided into multiple regions by the flux limit curves L = L lim ( z ) (see Fig. 5 for a example). In this case, the area equal to [∆ LV i a ] for an object i is a gross approximation to that of the surveyed region (see Fig. 6). φ PC and φ 1 /V a will give different estimate of φ for these bins.", "pages": [ 5 ] }, { "title": "2.5 Applying the methods to real data", "content": "In this section we apply the 1 /V a and PC methods to the combined sample established by Yuan & Wang (2012). Fig. 5 shows the L -z plane of the combined sample. The four sub-samples, MRC1 (McCarthy et al. 1996), MS4 (Burgess & Hunstead 2006), BRL (Best et al. 1999) and 3CRR (Laing et al. 1983) are described in the paper of Yuan & Wang (2012). For comparison, the radio luminosity function (RLF) at 408 MHz estimated by the two methods are plotted together for all the redshift bins in Fig. 7. It is not surprising that the two methods give the same results at the bright end of the RLFs, corresponding to the situation of bin 3 discussed in section 2.3. It is clear that φ 1 /V a and φ PC give different estimates at the faint end and middle of the RLFs. φ 1 /V a trends to give a smaller estimate than φ PC at the faint end, while it gives a larger estimate in the middle of the RLFs. Especially for the high redshift bins, φ 1 /V a gives a significantly larger estimate than φ PC in the middle of the RLFs. Because the situation like that for source j in bin 2 (discussed in section 2.3) is more prevalent for high redshift bins. In this section we use a combined sample simulated by a Monte Carlo method to further compare the 1 /V a and PC methods. The simulation is performed using a double-power-law model radio luminosity function which is changing with redshift in the form of (1 + z ) k . The Monte Carlo simulation produces four flux-limited samples containing more than 1 000 000 sources. The flux limits of the four simulated samples are 0.6 Jy, 1.2 Jy, 2.2 Jy, and 3.0 Jy respectively, and the solid angles subtended by them are 0.6 Sr , 1.2 Sr , 2.2 Sr and 3.0 Sr respectively. Without loss reality, a random spectral index following a normal distribution with an average of 0.7 is arranged for all the simulated sources. The four simulated samples are then combined into a 'coherent sample' and binned LFs are produced for it in a range of redshift intervals using both methods. These are shown in Fig. 8, φ 1 /V a on the left and φ PC on the right. The input model LFs take values of z = 0 . 12 , 0 . 35 , 0 . 75 , 1 . 25 , 1 . 75 , 2 . 25 and are shown as dashed lines. As expected from section 2.3, the two methods give identical and ideal estimate for the high luminosity points of each redshift interval. However, for the low luminosity bins of all the redshift intervals both methods give smaller estimate than the input model. This situation is especially serious for the 1 /V a method. For the lowest luminosity bins of the 1 . 0 < z < 1 . 5 , 1 . 5 < z < 2 . 0 , 2 . 0 < z < 2 . 5 redshift intervals φ 1 /V a is significantly smaller than the input model. The above results are not in completely agreement with Page & Carrera (2000). Their simulation showed that φ PC is always a good representation of the input model over the total luminosity range. This is because their simulation was performed using a LF which is unchanging with redshift (i.e. no evolution). Once the LF is evolving with redshift, the classical binned methods will unlikely give an ideal estimate over the total luminosity range. In Fig. 9 we show how this happens. In Fig.9 (a), bin1,bin2,bin3 are three bins located at different luminosity. Each bin is further divided into nine sub-bins. We assume each sub-bin is small enough and the four-volume involved is unit. The number noted in each sub-bin represents the source number. Thus the variation of number across sub-bins indicates how the density changes along luminosity and redshift. According to the PC method, the density (i.e., LF) at the center of bin2 and bin3 is estimated as 10 and 4 respectively. This is in agreement with the input model density at z mid (see Fig.9 (b)). Bin1 is divided into two parts by the flux limit curve L = L lim ( z ). Owing to the positive evolution of density along redshift and decline /s49/s48 /s52/s48/s56 /s32 /s32 of density along luminosity, the left part of bin1 probably contains more sources than the right part. However, the sources in left part can not be observed because the flux is limited. Therefor, according to the PC method, the density at the center of bin1 is estimated as 10 . 55 (the total source number in right part of bin1 divided by its four-volume) which is significantly smaller than the input model density. It can be expected that the more intensely the LF evolves with redshift, the more significant the error of φ PC for bin1 is. The above discussion also applies to the 1 /V a method.", "pages": [ 5, 6, 7 ] }, { "title": "2.7 A simple rule of thumb to determine the redshift and luminosity intervals", "content": "The accuracy of binned LF also depends on how the redshift and luminosity bins are divided. The bins are supposed to be small enough, but can not be too small to include only a few objects. In general, the equal intervals of luminosity are used, e.g. ∆ L = 0 . 3 or 0 . 5. While in the literature, the redshift intervals are commonly chosen something arbitrarily, e.g. 0 . 0 < z < 0 . 2, 0 . 2 < z < 0 . 5, 0 . 5 < z < 1 . 0; 0 . 0 < z < 0 . 15, 0 . 15 < z < 0 . 4, 0 . 4 < z < 0 . 7 etc. We believe this may lead the bins located at the faint end to enclose /s32 very few objects inside(see Fig. 10 (a)) and cause bias with small number statistics. Moreover, in these bins, objects necessarily gather at the right margins (see Fig. 4(c)). Thus the area equal to [∆ LV a ] of these bins will be much larger than that of the surveyed regions, causing a significantly small estimate of φ 1 /V a . To tackle this issue, we take the redshift intervals to be determined by the intersecting points of luminosity grid-line and flux limit curve (see Fig. 10 (b)). Alternatively, if one persists in dividing redshift intervals randomly, the luminosity intervals should be determined by the intersecting points of redshift grid-line and flux limit curve (see Fig. 10 (c)). Fig. 11 shows the binned LFs produced by the 1 /V a and PC methods for our simulated data with the red- /s32 /s32 /s32 /s32 /s32 /s32 /s49/s48 /s82 shift and luminosity intervals determined according to the rule suggested above (Fig. 10 (c) specifically). It is noticed that adopting the rule of thumb indeed improved the binned LFs to some extent, especially for φ 1 /V a . Obviously, φ 1 /V a is more sensitive to how the intervals are chosen than φ PC . Page & Carrera (2000) noticed that for objects close to the flux limit φ 1 /V a nearly always to be too small. We believe this is due to the arbitrary choosing of redshift and luminosity intervals. If the intervals are chosen appropriately, φ 1 /V a is only slightly larger than φ PC at the faint end of LFs. And it is hard to say the estimate of PC method is markedly better than the 1 /V a method. In this sense, the improvement of the PC method over the 1 /V a method is probably slight.", "pages": [ 7, 8 ] }, { "title": "3 Non-parametric Methods", "content": "A non-parametric method is necessary when investigating the LFs, as it constructs the LFs directly from the data and makes no assumptions about the form of the LFs. Examples of non-parametric methods include the classical Φ = N/V method (Christensen 1975), the φ/ Φ method (Turner 1979), the Step-Wise Max Likelihood method (Efstathiou et al. 1988) and the C -method (Lynden-Bell 1971), as well as the 1 /V a and PC methods discussed here. The C -method (also see Willmer 1997; Fan et al. 2001; Choloniewski 1987) is believed to have the advantage over the 1 /V a method as it does not require any binning of data, and rests on a strong mathematical foundation (see Woodroofe 1985). The key assumption in the C -method is that the luminosities and redshifts are independent, but researches on the LFs of AGNs have indicated that there are both luminosity and density evolutions and the independence assumption is incorrect (e.g., Pei 1995; Willott et al. 2001). Some authors (Efron & Petrosian 1992; Maloney & Petrosian 1999) developed the C -method to remove the correlation of luminosities and redshifts by defining new independent variables (say L ' ≡ L/g k ( z ) and z , where the function g k ( z ) describes the luminosity evolution). Then the C -method can be used for the ( L ' , z ) data. In this sense, the improved C -method is no longer a non-parametric method as g k ( z ) is parameter dependent. Usually, a simple form g k ( z ) = (1 + z ) k is used, where k is a free parameter. Then a so-called test statistic τ is performed to determine k by making τ ( k ) = 0 (see Maloney & Petrosian 1999; Singal et al. 2011). In most cases, the (1+ z ) k form is too simplistic. E.g., Pei (1995) used a Gaussian form and two free parameters were introduced. Consequently, the unknown luminosity evolution g k ( z ) restricts the C -method. Notably, in recent years more rigorous approaches, although not all of which are non-parametric, have been /s32 /s32 proposed. E.g., Schafer (2007) developed a powerful semi-parametric approach which is built on a nonparametric extension of maximum likelihood called local likelihood modeling. It is expected that a new method needs time to be recognized and in widespread use. Besides, when combining multiple samples to estimate the LFs, the truncation boundary (as discussed in Schafer 2007) of data in the L -z plane is further complicated. The new non-parametric methods are also supposed to deal with this challenge.", "pages": [ 8, 9 ] }, { "title": "4 Summary", "content": "The classical 1 /V a and PC methods of constructing binned luminosity functions (LFs) are revisited and compared by graphical analysis. The two methods give different estimate of φ for the bins which are crossed by the flux limit curves L = L lim ( z ). Using the combined sample established by Yuan & Wang (2012), we show that φ 1 /V a trends to give a smaller estimate than φ PC at the faint end of LFs, while it gives a larger estimate in the middle of the LFs. Using a combined sample simulated by a Monte Carlo method, the estimate of two methods are compared with the input model LFs. The two methods give identical and ideal estimate for the high luminosity points of each redshift interval. However, for the low luminosity bins of all the redshift intervals both methods give smaller estimate than the input model. We conclude that once the LF is evolving with redshift, the classical binned methods will unlikely give an ideal estimate over the total luminosity range. Page & Carrera (2000) noticed that for objects close to the flux limit φ 1 /V a nearly always to be too small. We believe this is due to the arbitrary choosing of redshift and luminosity intervals. We noticed that φ 1 /V a is more sensitive to how the binning are chosen than φ PC . We suggest a new binning method, which can improve the LFs produced by the 1 /V a method significantly, and also improve the LFs produced by the PC methods. Our simulations show that after adopting this new binning, both the 1 /V a and PC methods have comparable results.", "pages": [ 9 ] }, { "title": "Acknowledgments", "content": "We are grateful to the referee for very useful comments that improved this paper. We acknowledge the financial supports from the National Natural Science Foundation of China 11133006, 11163006, 11173054, the National Basic Research Program of China (973 Program 2009CB824800), and the Policy Research Program of Chinese Academy of Sciences (KJCX2-YW-T24).", "pages": [ 9 ] }, { "title": "References", "content": "March˜a, M. J. M., & Caccianiga, A. 2013, Mon. Not. R. Astron. Soc., 709", "pages": [ 9 ] } ]

[ { "title": "Statefinder diagnostic of logarithmic entropy corrected holographic dark energy with Granda-Oliveros IR cut-off", "content": "A. Khodam-Mohammadi 1 , Antonio Pasqua 2 , M. Malekjani 1 , Iuliia Khomenko 3 , M. Monshizadeh 4 Abstract In this work, we have studied the logarithmic entropy corrected holographic dark energy (LECHDE) model with Granda-Oliveros (G-O) IR cutoff. The evolution of dark energy (DE) density Ω ' D , the deceleration parameter, q , and equation of state parameter (EoS), ω Λ , are calculated. We show that the phantom divide may be crossed by choosing proper model parameters, even in absence of any interaction between dark energy and dark matter. By studying the statefinder diagnostic and ω Λ -ω ' Λ analysis, the pair parameters { r, s } and ( ω Λ -ω ' Λ ) is calculated for flat GO-LECHDE universe. At present time, the pair { r, s } can mimic the ΛCDM scenario for a value of α/β /similarequal 0 . 87, which is lower than the corresponding one for observational data ( α/β = 1 . 76) and for Ricci scale ( α/β = 2). We find that at present, by taking the various values of ( α/β ), the different points in r -s and ( ω Λ -ω ' Λ ) plans are given. Moreover, in the limiting case for a flat dark dominated universe at infinity ( t → ∞ ), we calculate { r, s } at G-O scale. For Ricci scale ( α = 2, β = 1) we obtain { r = 0 , s = 2 / 3 } . A. Khodam-Mohammadi 1 , Antonio Pasqua 2 , M. Malekjani 1 , Iuliia Khomenko 3 , M. Monshizadeh 4 4 Physics Department, Faculty of Science, Islamic Azad University, Hamedan branch, Iran [email protected]. [email protected] [email protected]. [email protected] [email protected]", "pages": [ 1 ] }, { "title": "1 Introduction", "content": "It is widely accepted among cosmologists and astrophysicists that our universe is experiencing an accelerated expansion. The evidences of this accelerated expansion are given by numerous and complementary cosmological observations, like the SNIa (Perlmutter et al., 1999; Astier et al., 2006), the CMB anisotropy, observed mainly by WMAP (Wilkinson Microwave Anisotropy Probe) (Bennett et al., 2003; Spergel et al., 2003), the Large Scale Structure (LSS) (Tegmark et al., 2004; Abazajian et al., 2004, 2005) and X-ray (Allen et al., 2004) experiments. In the framework of standard Friedmann-LemaitreRobertson-Walker (FLRW) cosmology, a missing energy component with negative pressure (known as Dark Energy (DE)) is the source of this expansion. Careful analysis of cosmological observations, in particular of WMAP data (Bennett et al., 2003; Spergel et al., 2003; Peiris et al., 2003) indicates that almost 70 percent of the total energy of the universe is occupied by DE, whereas DM occupies almost the rest (the barionic matter represents only a few percent of the total energy density). The contribution of the radiation is practically negligible. The nature of DE is still unknown and many candidates have been proposed in order to describe it (see (Copeland et al., 2006; Padmanabhan, 2003; Peebles, & Ratra, 2003) and references therein for good reviews). The time-independent cosmological constant Λ with equation of state (EoS) parameter ω = -1 is the earliest and simplest DE candidate. However, cosmologists know that Λ suffers from two main difficulties: the fine-tuning and the cosmic coincidence problems (Copeland et al., 2006). The former asks why the vacuum energy density is so small (about 10 -123 times smaller than what we observe) (Weinberg, 1989) and the latter says why vacuum energy and DM are nearly equal today (which represents an incredible coincidence if no internal connections between them are present). Alternative candidates for DE problem are the dynamical DE scenarios with no longer constant but timevarying ω . It has been shown by observational data analysis of SNe-Ia that the time-varying DE models give a better fit compared with a cosmological constant. A good review about the problem of DE, including a survey of some theoretical models, can be found in (Li et , 2011). as (Banerjee, & Majhi, 2008a,b; Banerjee, & Modak, 2009): where ˜ α and ˜ β are dimensionless constants. These corrections can appear in the black hole entropy in Loop Quantum Gravity (LQG). They can also be due to quantum fluctuation, thermal equilibrium fluctuation or mass and charge fluctuations. The quantum corrections provided to the entropy-area relationship leads to curvature correction in the Einstein-Hilbert action and viceversa (Cai et al., 2009; Nojiri, & Odintsov, 2001; Zhu, & Ren, 2009). Using the corrected entropy-area relation given in Eq. (2), the energy density ρ Λ of the logarithmic entropy-corrected HDE (LECHDE) can be written as (Wei, 2009): where γ 1 and γ 2 are two dimensionless constants. In the limiting case of γ 1 = γ 2 = 0, Eq. (3) yields the well-known HDE density. The second and the third terms in Eq. (3) are due to entropy corrections: since they can be comparable to the first term only when L is very small, the corrections they produce make sense only at the early evolutionary stage of the universe. When the universe becomes large, Eq. (3) reduce to the ordinary HDE. It is worthwhile to mention that the IR cut-off L plays an important role in HDE model. By assuming particle horizon as IR cut-off, the accelerated expansion can not be achieved (Hsu, 2008), while for Hubble scale, event horizon, apparent horizon and Ricci scale, this fact may be achieved (Sheykhi, 2010; An important advance in the study of black hole theory and string theory is the suggestion of the so called holographic principle: according to it, the number of degrees of freedom of a physical system should be finite, it should scale with its bounding area rather than with its volume ('t Hooft, 1993) and it should be constrained by an infrared cut-off (Cohen et al., 1999). The Holographic DE (HDE), based on the holographic principle proposed by (Fischler, & Susskind, 1998), is one of the most interesting DE candidates and it has been widely studied in literature (Enqvist et al., 2005; Shen et al., 2005; Zhang, & Wu, 2005; Zhang, 2006; Sheykhi, 2010; Huang, & Li, 2004; Hsu, 2004; Guberina et al., 2005, 2006; Gong, 2004; Elizalde et al., 2005; Jamil, & Farooq, 2010a; Karami et al., 2011; Setare, & Jamil, 2010a; Sheykhi et al., 2012; Jamil, & Farooq, 2010b; Pasqua et al., 2012; Setare, 2006, 2007a,b,c,d,e,f; Setare, & Vagenas, 2008; Setare, & Jamil, 2010b, 2011; Khodam-Mohammadi, & Malekjani, 2011a; Sheykhi, 2009). The HDE model have also been constrained and tested by various astronomical observations (Enqvist et al., 2005; Shen et al., 2005; Zhang, & Wu, 2005, 2007; Feng et al., 2005; Kao et al., 2005; Micheletti, 2010; Wang, & Xu, 2010; Zhang, 2009) as well as by the anthropic principle (Huang, & Li, 2005). Applying the holographic principle to cosmology, the upper bound of the entropy contained in the universe can be obtained (Fischler, & Susskind, 1998). Following this line, (Li, 2004) suggested the following constraint on the energy density: where c is a numerical constant, L indicates the IR cut-off radius, M p = (8 πG ) -1 / 2 /similarequal 10 18 GeV is the reduced Planck mass ( G is the gravitational constant) and the equality sign holds only when the holographic bound is saturated. Obviously, in the derivation of HDE, the black hole entropy (denoted with S BH ) plays an important role. As it is well known, S BH = A/ (4 G ), where A ≈ L 2 is the area of the horizon. However, this entropy-area relation can be modified Duran, & Pavon, 2011; Nojiri, & Odintsov, 2006; Pavon, & Zimdahl, 2005; Zimdahl, & Pavon, 2007). Recently, Granda and Oliveros (G-O), proposed a new IR cut-off for HDE model, namely 'new holographic DE', which includes a term proportional to . H and one proportional to H 2 (Granda, & Oliveros, 2009, 2008). Despite of the HDE based on the event horizon, this model depends on local quantities, avoiding in this way the causality problem. The investigation of cosmological quantities such as the EoS parameter ω Λ , deceleration parameter q and statefinder diagnosis have attracted a great deal of attention in new cosmology. Since the various DE models give H > 0 and q < 0 at the present time, the Hubble and deceleration parameters can not discriminate various DE models. A higher order of time derivative of scale factor is then required. Sahni et al. (Sahni, & Shtanov, 2003) and Alam et al. (Alam et al., 2003), using the third time derivative of scale factor a ( t ), introduced the statefinder pair { r,s } in order to remove the degeneracy of H and q at the present time. The statefinder pair is given by: The Friedmann equation for non-flat universe dominated by DE and DM has the form: where ρ Λ and ρ m are, respectively, the energy densities of DE and DM. We also define the fractional energy densities for DM, curvature and DE, respectively, as: Many authors have been studied the properties of various DE models from the viewpoint of statefinder diagnostic (Khodam-Mohammadi, & Malekjani, 2011b; Malekjani, & Khodam-Mohammadi, 2010; Malekjani et al., 2011a,b; Malekjani, & Khodam-Mohammadi, 2012, 2013). This paper is organized as follows. In Section 2, we describe the physical contest we are working in and we derive the EoS parameter ω Λ , the deceleration parameter q and Ω ' Λ for GO-LECHDE model. In Section 3, the statefinder diagnosis and ω -ω ' analysis of this model are investigated. We finished our work with some concluding remarks.", "pages": [ 1, 2, 3 ] }, { "title": "2 cosmological properties", "content": "The energy density of GO-LECHDE in Planck mass unit (i.e. M P = 1) is given by where we defined Γ = 1 + 1 3 L -2 GO (2 γ 1 log L GO + γ 2 ) for simplicity. The Granda-Oliveros IR cutoff given by (Granda, & Oliveros, 2009; Khodam-Mohammadi, 2011): where α and β are two constant. The line element of FLRW universe is given by: where t is the cosmic time, a ( t ) is a dimensionless scale factor (which is function of the cosmic time t ), r is referred to the radial component, k is the curvature parameter which can assume the values -1 , 0 and +1 which yield, respectively, a closed, a flat or an open FLRW universe and ( θ, ϕ ) are the angular coordinates. where ρ cr = 3 H 2 represents the critical energy density. Recent observations reveal that Ω k ∼ = 0 . 02 (Spergel et al., 2007), which support a closed universe with a small positive curvature. Using the Friedmann equation given in Eq. (9), Eqs. (10), (11) and (12) yield: In order to preserve the Bianchi identity or the local energy-momentum conservation law, i.e. ∇ µ T µν = 0, the total energy density ρ tot = ρ Λ + ρ m must satisfy the following relation: where ω tot ≡ p tot /ρ tot represents the total EoS parameter. In an non-interacting scenario of DE-DM, the energy densities of DE and DM ρ Λ and ρ m are preserved separately and the equations of conservation assume the following form: The derivative with respect to the cosmic time t of L GO is given by: Using Eq. (17), the derivative with respect to the cosmic time t of the energy density ρ Λ given in Eq. (3) can be written as: Differentiating the Friedmann equation given in Eq. (9) with respect to the cosmic time t and using Eqs. (12), (13), (16) and (18), we can write the term α ˙ H H 2 + β H 2 H 3 as: where u = ρ m /ρ Λ = Ω m / Ω Λ = (1 + Ω k ) / Ω Λ -1 is the ratio of energy densities of DM and DE. Using the expression of L GO given in Eq. (7) and the energy density of DE given in Eq. (8), we obtain that the term ˙ H H 2 can be written as: Therefore, Eq. (18) yields: Differentiating the expression of Ω Λ given in Eq. (12) with respect to the cosmic time t and using the relation ˙ Ω Λ = H Ω ' Λ , we obtain the evolution of the energy density parameter as follow: The dot and the prime denote, respectively, the derivative with respect to the cosmic time t and the derivative with respect to x = ln a . Finally, using Eqs. (12), (15) and (21), the EoS parameter ω Λ and the deceleration parameter (defined as q = -1 -˙ H H 2 ) as functions of Ω Λ and Γ are given, respectively, by: We can easily observe that the EoS parameter ω Λ and the deceleration parameter q given, respectively, in Eqs. (23) and (24) are related each other by the following relation: Moreover, using Eqs. (12) and (24), we can derive that: From Eqs. (15) and (16), the evolution of u is governed by: At Ricci scale, i.e. when α = 2 and β = 1, Eqs. (23) and (24) reduce, respectively, to: and the evolution of the energy density parameter given in Eq. (22) reduces to: By choosing the proper model parameters, it can be easily shown that the equation of state parameter ω Λ given in Eqs. (23) and (28), may cross the phantom divide. Moreover, from Eqs. (24) and (29), we can see that the transition between deceleration to acceleration phase can be happened for various model parameters. In a flat dark dominated universe, i.e. when γ 1 = γ 2 = 0 or at infinity ( t → ∞ ), Ω Λ = 1, Ω k = 0 and u = 0, we find that the Hubble parameter H reduces to: Moreover, the EoS parameter ω Λ and the deceleration parameter q given in Eqs. (23) and (24) reduce, respectively, to: Also in this case the phantom wall can be achieved for α ≤ 1 , β > 0. In Ricci scale in this limit, Eqs. (32), (33) reduce to which corresponds to an expanding universe without any acceleration.", "pages": [ 3, 4 ] }, { "title": "3 Statefinder diagnostic", "content": "We now want to derive the statefinder parameters { r, s } for GO-LECHDE model in the flat universe. The Friedmann equation given in Eq. (9) yields, after some calculations: Taking the time derivation of Eq. (35) and using Eq. (22), we obtain: Using the definition of H (i.e. H = ˙ a/a ), the statefinder parameter r given in Eq. (4) can be written as: Substituting Eqs (20), (24) and (36) in Eqs. (37) and (5), pair parameters { r, s } can be written: At early time, when ω Λ → 0, the pair relations (38) show that that statefinder parameters tends to { r = 1 , s = 0 } , which coincides with the location of the ΛCDM fixed point in r -s plane. Using Eq. (23), the evolution of EoS parameter ω Λ can be written as: where from Eqs. (12) and (17), the term ( L ' GO L GO ) can be calculated as: At present epoch of the Universe (Ω Λ ≈ 0 . 72, u ≈ 0 . 4), the EoS parameter ω Λ given in Eq. (25) reduces to: Then, the universe exists in accelerating phase (i.e q < 0) if ω Λ < -0 . 47 and the phantom divide ω Λ = -1, may be crossed provided that q /lessorsimilar -0 . 5. This condition implies ˙ H H 2 /greaterorsimilar -0 . 58 and, from Eq. (26), we derive: By inserting the above quantities in Eqs. (22) and (40), we have ω ' Λ /greaterorsimilar -1 . 86 ( α/β -3 / 2), which gives: Recently, Wang and Xu (Wang, & Xu, 2010) have constrained the new HDE model in non-flat universe using observational data. The best fit values of ( α, β ) with their confidence level they found are α = 0 . 8824 +0 . 2180 -0 . 1163 (1 σ ) +0 . 2213 -0 . 1378 (2 σ ) and β = 0 . 5016 +0 . 0973 -0 . 0871 (1 σ ) +0 . 1247 -0 . 1102 (2 σ ) . Using these values, the pair parameters { r, s } , at present epoch, become { r = 2 . 77 , s = -0 . 55 } , which are far from ΛCDM model values (i.e., { r = 1 , s = 0 } ). Moreover, it shows that s < 0 , which corresponds to a phantom-like DE. However, in order to mimic these parameters to ΛCDM scenario at present epoch, the ratio of α/β must be approximately 0 . 87, which is lower than the value obtained with observational data. At Ricci scale (i.e., when α/β = 2), at present time, pair parameters assume the values { r = 3 . 25 , s = -0 . 70 } . It is worthwhile to mention that by increasing the value of α/β from 0.87, the distance from ΛCDM fixed point in r -s diagram become longer. In the limiting case of t →∞ or for ordinary new HDE ( γ 1 = γ 2 = 0 , Γ = 1), in flat dark dominated universe ( u = 0 , Ω Λ = 1), we find that: At Ricci scale ( α = 2 , β = 1), Eqs. (47) and (48) reduce, respectively, to : Moreover the ω -ω ' analysis is another tool to distinguish between the different models of DE (Wei, & Cai, 2007). In this analysis the standard ΛCDM model corresponds to the fixed point ( ω Λ = -1 , ω ' Λ = 0). At present time, for α/β = 0 . 87 which corresponds toΛCDM fixed point in r -s diagram, ( ω Λ = -1 , ω ' Λ = 1 . 17). for the observational quantities, ( α/β = 1 . 76), we find: ( ω Λ = -1 , ω ' Λ = -0 . 48), and for Ricci scale these are ( ω Λ = -1 , ω ' Λ = -0 . 93). Therefore we see that ω ' Λ become smaller for higher value of α/β at present.", "pages": [ 4, 5, 6 ] }, { "title": "4 Conclusion", "content": "In this paper, we have extended the work made by Granda and Oliveros (Granda, & Oliveros, 2009) to the logarithmic entropy corrected HDE (LECHDE) model. This model has been arisen from the black hole entropy which may lie in the entanglement of quantum field between inside and outside of the horizon. We obtained the evolution of energy density Ω ' Λ , the deceleration parameter q and EoS parameter ω Λ of the new LECHDE model for non-flat universe. We saw that, by choosing the proper model parameters, the equation of state parameter ω Λ may cross the phantom divide and also the transition between deceleration to acceleration phase could happen. At last, we studied the GO-LECHDE model from the viewpoint of statefinder diagnostic and ω Λ -ω ' Λ analysis, which is a crucial tool for discriminating different DE models. Also, the present value of { r, s } can be viewed as a discriminator for testing different DE models if it can be extracted from precise observational data in a model-independent way. The studying at present time, when ω Λ remains around the phantom wall, ω Λ ≈ -1 and our universe evolves in acceleration phase, pair values of { r, s } was calculated with respect to model parameters α, β . By using the observational data which was obtained by Wang and Xu (Wang, & Xu, 2010), where α/β = 1 . 76, we obtained { r = 2 . 77 , s = -0 . 55 } . For Ricci scale, which has α/β = 2, the pair value assume the values { r = 3 . 25 , s = -0 . 7 } . Also, choosing α/β = 0 . 87, we found { r = 1 , s = 0 } which is corresponds to ΛCDM scenario. We shaw that increasing value of α/β , conclude the ascending distance from ΛCDM fixed point. In the limiting case, at infinity, for flat dark dominated universe at Ricci scale, we found { r = 0 , s = 2 / 3 } , which corresponds to an expanding universe without any acceleration ( q = 0). In ω Λ -ω ' Λ analysis at present time, we found that the higher value of α/β obtains the smaller value of ω ' Λ . In this model the statefinder pairs is determined by parameters α, β, γ 1 , γ 2 . These parameters would be obtained by confronting with cosmic observational data. Giving the wide range of cosmological data available, in the future we expect to further constrain our model parameter and test the viability of our model.", "pages": [ 6 ] } ]

[ { "title": "Phantom Instability of Viscous Dark Energy in Anisotropic Space-Time", "content": "Hassan Amirhashchi 1 , 2 1 Young Researchers Club, Mahshahr Branch, Islamic Azad University, Mahshahr, Iran 2 Laboratory of Computational Sciences and Mathematical Physics, Institute for Mathematical Research, University Putra Malaysia, 43400 Serdang, Selangor D.E., Malaysia 1 E-mail:[email protected]", "pages": [ 1 ] }, { "title": "Abstract", "content": "Phantom dark energy is a proposal that explains the current observations that mildly favor the equation of state of dark energy ω de crossing -1 at 68% confidence level. However, phantom fields are generally ruled out by ultraviolet quantum instabilities. To overcome this discrepancy, in this paper we propose a mechanism to show that how the presence of bulk viscosity in the cosmic fluid can temporarily drive the fluid into the phantom region ( ω < -1). As time is going on, phantom decays and ultimately ω de approaches to -1. Then we show these quintessence and phantom descriptions of non-viscous and viscous dark energy and reconstruct the potential of these two scalar fields. Also a diagnostic for these models are performed by using the statefinder pairs { s, r } . All results are obtained in an anisotropic space-time which is a generalization of FLRW universe. Keywords : Bianchi Type I Model, Dark Energy, Phantom, Statefinder PACS number: 98.80.Es, 98.80-k, 95.36.+x", "pages": [ 1 ] }, { "title": "1 Introduction", "content": "It is a very well known fact that our universe is experiencing an accelerating expansion at the present time (Perlmutter et al. 1999; Riess et al. 1998, 2001; Tonry et al. 2003; Tegmark et al. 2004). It is believed that an exotic form of energy with negative pressure called dark energy is responsible for the current observed accelerating expansion of the universe (Tegmark et al. 2004; Bennet et al. 2003; Spergel et al. 2003; Abazajian et al. 2004). According to the recent observations we live in a nearly spatially flat Universe composed of approximately 4% baryonic matter, 22% dark matter and 74% dark energy. However, the observational data are far from being complete. It is not even known what is the current value of the dark energy effective equation of state (EoS) parameter ω ( de ) = p ( de ) /ρ ( de ) which lies close to -1: it could be equal to -1 (standard ΛCDM cosmology), a little bit upper than -1 (the quintessence dark energy) or less than -1 (phantom dark energy). One of the main candidate for dark energy is cosmological constant Λ, which has pressure p ( de ) = -ρ ( de ) . Although, cosmological constant can explain the current acceleration phase of universe, it would suffer from many serious theoretical problems, such as the fine-tuning and the coincidence problems. Another candidate for dark energy is provided by introducing scalar fields. An important class of scalar fields are known as 'quintessence' with -1 3 > ω > -1 (Ratra and Peebles. 1988; Wetterich 1988; Turner and white 1997; Caldwell et al. 1998; Liddle and Scherrer 1999; Steinhardt et al. 1999) in which the scalar field mimics the perfect fluid and hence could lead to a solution for coincidence problem. However, quintessence scenario of dark energy is not in accurate consistent with recent observations as ω < -1 has been favored by recent observations (Knop et al. 2003; Riess et al. 2004; Alam et al. 2004; Hannestad and E. Mortsell 2004). To get ω < -1, a new class of scalar field models with negative kinetic energy, known as 'phantom field' models have been suggested (Caldwell 2002). Nevertheless, in this case the universe shows some very strange properties (Carroll et al. 2003; Cline et al. 2004; Buniy and Hsu 2006; Buniy et al. 2006). For example, since the energy density of phantom field is unbounded from below, the vacuum becomes unstable against the production of positive energy fields hence these fields are generally ruled out by ultraviolet quantum instabilities (Carroll et al. 2003). Another problem is the future finite singularity called Big Rip (Caldwell et al. 2003) which leads to the occurrence of negative entropy (Brevik et al. 2004). Therefore, on the one hand observations mildly favors models with ω crossing -1 near the past and on another, models with ω < -1 are unstable from theoretical point of view. In this paper we suggest a simple mechanism to overcome this discrepancy by introducing bulk viscosity in the cosmic fluid. First, viscosity causes dark energy which is varying in quintessence to pass the phantom divided line (PDL) and drop it to phantom region. Next, since viscosity is a decreasing function of time, it will die out and ω will leave phantom region and tend to -1 at late time. Hence the problem of future singularity (big rip) will never occur in this scenario. It has been shown in refs (McInnes 2002; Barrow 2004) that, an ideal cosmic fluid, i.e. non-viscous, give raise to the occurrence of a singularity of the universe in the far future called big rip. The singularity problem can be modified or soften via following two methods. The first is the effect of quantum corrections due to the conformal anomaly (Brevik and Odintsov 1999; Nojiri and Odintsov 2003, 2004) and second, is to consider the bulk viscosity of the cosmic fluid (for example see (Misner 1968; Padmanabhan and Chitre 1987; Brevik and Hallanger 2004). The viscosity theory of relativistic fluids was first suggested by Eckart, Landau and Lifshitz (Eckart 1940; Landau and Lifshitz 1987). The introduction of viscosity into cosmology has been investigated from different view points (Grøn 1990; Barrow 1986; Zimdahl 1996; Maartens 1996). The astrophysical observations also indicate some evidences that cosmic media is not a perfect fluid (Jaffe et al. 2005), and the viscosity effect could be concerned in the evolution of the universe (Brevik and Gorbunova 2005; Brevik et al. 2005; Cataldo et al. 2005). It was also argued that a viscous pressure can play the role of an agent that drives the present acceleration of the Universe (Zimdahl et al. 2001; Balakin et al. 2003). The possibility of a viscosity dominated late epoch of the Universe with accelerated expansion was already mentioned by Padmanabhan and Chitre (Padmanabhan and Chitre 1987). Brevik and Gorbunova (2005), Oliver et al (2011), Chen et al (2011), Jamil and Farooq (2010), Cai et al (2010), Setare (2007a, 2007b, 2007c), Setare et al (2007), Setare and Saridakis (2009), Setare et al (2009), Amirhashchi et al (2011 a, 2011b, 2011 c, 2013), Pradhan et al (2011a, 2011b, 2011c), Saha et al (2012), and Sheykhi and Setare (2010) have studied viscous and non-viscous dark energy models in different contexts. Recently, viscous dark energy and generalized second law of thermodynamics has been studied by Setare and Sheykhi (2010). To be general, we use generalized FLRW equations by considering an anisotropic metric as the line-element of the universe. The reason for this choice of metric is behind the fact that because of high symmetry, FLRW models are infinitely improbable in the space of all possible cosmologies. The high symmetry involved in FLRW models requires a very high degree of fine tuning of initial conditions which is extraordinary improbable. Moreover, we can always ask that does the universe necessarily have the same symmetries on very large scales outside the particle horizon or at early times? The plan of our paper is as follows: In section 2 we give the metric and field equations. In section 3 we drive the generalized FLRW equations by solving the field equations of section 2. The general form of non-viscous and viscous dark energy equation of state parameter EoS are given in section 4. We suggest a correspondence between the non-viscous and viscous dark energy scenario and the quintessence and phantom dark energy model in section 5. In section 6, a statefinder diagnostic has been presented. In section 7 we apply our general results to a toy model in order to test the proposed mechanism. Our results are summarized in section 8.", "pages": [ 1, 2 ] }, { "title": "2 The Metric and Field Equations", "content": "We consider the Bianchi type I space-time in the orthogonal form as where A ( t ) , B ( t ) and C ( t ) are functions of time only. The Einstein's field equations ( in gravitational units 8 πG = c = 1) read as where T ( m ) i j and T ( de ) i j are the energy momentum tensors of barotropic matter and dark energy, respectively. These are given by and and where ρ ( m ) and p ( m ) are, respectively the energy density and pressure of the perfect fluid component or matter while ω ( m ) = p ( m ) /ρ ( m ) is its EoS parameter. Similarly, ρ ( de ) and p ( de ) are, respectively the energy density and pressure of the DE component while ω ( de ) = p ( de ) /ρ ( de ) is the corresponding EoS parameter. We assume the four velocity vector u i = (1 , 0 , 0 , 0) satisfying u i u j = -1. In a co-moving coordinate system ( u i = δ i 0 ), Einstein's field equations (2) with (3) and (4) for B-I metric (1) subsequently lead to the following system of equations: If we consider a = ( ABC ) 1 3 as the average scale factor of Bianchi type I model, then the generalized mean Hubble's parameter H defines as The Bianchi identity G ; j ij = 0 leads to T ; j ij = 0. Therefore, the continuity equation for dark energy and baryonic matter can be written as", "pages": [ 2, 3 ] }, { "title": "3 Friedmann-Like Equations", "content": "In this section, we derive the general solution for the Einstein's field equations (5)-(8). Subtracting Eq. (5) from Eq. (6), Eq. (6) from Eq. (7), and Eq. (5) from Eq. (7) we obtain First integral of Eqs. (11), (12) and (13) leads to and where k 1 , k 2 and k 3 are constants of integration. By taking integral from Eqs. (14), (15) and (16) we get and where, d 1 , d 2 and d 3 are constants of integration. Now, we can find all metric potentials from Eqs. (17), (19) as follow and Here where Therefore, one can write the general form of Bianchi type I metric as Using eqs. (20)-(22) in eqs. (5)-(8) we can write the analogue of the Friedmann equation as and Here ρ = ρ m + ρ de , p = p m + p de and K = b 1 b 2 + b 1 b 3 + b 2 b 3 . Note that K denotes the deviation from isotropy e.g. K = 0 represents flat FLRW universe. Thus, when the universe is sufficiently large, almost at the present time, the space-time (1) behaves like a flat FLRW universe.", "pages": [ 3, 4 ] }, { "title": "4 Dark Energy Equation of State", "content": "In this section we obtain the general form of the equation of state (EoS) for the viscous and non viscous dark energy (DE) ω de in Bianchi type I space-time when there is no interaction between dark energy and Cold Dark Matter(CDM) with ω m = 0. In this case the conservation equation (10) for dark and barotropic fluids can be written separately as and Eq.(27)leads to Using eqs. (24), (28) in eqs. (7), (8) we obtain the energy density and pressure of dark fluid as and respectively. Therefore, the equation of state parameter (EoS) of DE in it's general form is given by where q = -a aH 2 is the deceleration parameter, Ω m 0 is the current value of matter density and L = b 2 2 + b 2 3 + b 2 b 3 is a positive constant (Note that K + L = 0). From eq. (31) we see that at present time (i.e L = 0 , q = -0 . 55 , Ω m 0 = 0 . 24 , a = 1), approximately, ω de pf = -0 . 92. At late time, EoS parameter is given by here subscript ' pf ' refers to 'perfect fluid'. According to the observations deceleration parameter is restricted as -1 ≤ q < 0. Therefore, from eq. (32) we observe that at the best approximation the minimum value of ω de pf is -1 i.e EoS of non-viscous DE can not cross phantom divided line (PDL). In another word, non-viscous dark energy can be described by quintessence ( ω de > -1) rather than phantom ( ω de < -1) field. In other hand models with ω de crossing -1 near the past have been mildly favored by the analysis on the nature of dark energy from recent observations (for example see (Astier et al. 2006)). SNe Ia alone favors a ω larger than -1 in the recent past and less than -1 today, regardless of wether using the prior of a flat universe (Alam et al. 2004; Astier et al. 2006) or not (Dicus and Repko 2004). In what follows we show that the possibility of crossing PDL will be available in a universe dominated by viscous DE instead of perfect DE. In Eckart's theory (Eckart 1940) a viscous dark energy EoS is specified by Here Π = -ξ ( ρ de ) u i ; i is the viscous pressure and H = u i ; i 3 is the Hubble's parameter and subscript ' vf ' referees to 'viscous fluid'. On thermodynamical grounds, in conventional physics ξ has to be positive. This is a consequence of the positive sign of the entropy change in an irreversible process (Nojiri and Odintsov 2003). In general, ξ ( ρ de ) = ξ 0 ( ρ de ) τ , where ξ 0 > 0 and τ are constant parameters. Note that, here we have to assume τ > 0 since for negative τ this form of bulk viscosity does not allow our models to cross PDL. A power-law expansion for the scale factor can be achieved for τ = 1 2 (Barrow 1987, 1988). It has been shown by Goliath and Ellis (1999) that some Bianchi models isotropise due to inflation. Substituting eq. (33) in eq. (31) by considering the above description we obtain the EoS parameter of viscous DE as where Ω de = ρ de 2 and α = 1 τ 3 H -. From eq. (34) we observe that the EoS of viscous DE at present time (i.e L = 0 , q = -0 . 55 , H 0 = 70 , Ω m 0 = 0 . 24 , Ω de 0 = 0 . 76 , a = 1), approximately is which clearly cross the PDL for appropriate values of α and ξ 0 . As mentioned before, phantom fields are generally plagued by ultraviolet quantum instabilities. Naively, any phantom model with ω de < -1 should decay to ω de = -1 at late time. As mentioned in (Carroll et al. 2003), this ensures that there is no future singularity (Big Rip); rather, the universe eventually settles into a de Sitter phase. Here we highlight since ξ ( ρ de ) = ξ 0 ( ρ de ) τ , and ρ de is a decreasing function of time in an expanding universe we conclude that the bulk viscosity dies out as time goes on and viscous phantom DE is an unstable state (as expected) and EoS of DE tends to -1 at late time (de-Sitter Universe).", "pages": [ 4, 5, 6 ] }, { "title": "5 Correspondence Between Dark Energy And Scalar Fields", "content": "It is believed that the current accelerated expansion is driven by a dynamical scalar field φ with potential V ( φ ). These models introduce a scalar field φ that is minimally coupled to gravity. As it is shown in previous section, one can generate quintessence and phantom fields from non-viscous and viscous fluids in an anisotropic universe respectively. Quintessence and phantom fields are generally given by the action The energy density and pressure of scalar field (DE) are given by and where /epsilon1 = ± 1. /epsilon1 = 1 is referred to as quintessence whereas /epsilon1 = -1 is referred to as phantom. From eqs. (29), (30) and eqs. (37), (38) we find the general form of the scalar field φ and potential V ( φ ) as and Note that putting ξ 0 = 0 and /epsilon1 = 1 in eqs. (39), (40) we get the scalar field and potential of quintessence. Also for sufficiently large time, the asymptotic behavior of φ and V ( φ ) is given by and respectively. Eq. (41) clearly shows that the only possible scenario at far future is the phantom scenario as /epsilon1 = 1 (quintessence) gives an imaginary φ . It is worth to mention that at late time i.e a →∞ which implies ξ 0 → 0, the potential asymptotically tends to vanish and φ = constant.", "pages": [ 6 ] }, { "title": "6 Statefinder Diagnostic", "content": "V. Sahni and coworkers (2003) have recently introduced a pair of parameters { r, s } called ' statefinders ', which are useful to distinguish different types of dark energy. The statefinders were introduced to characterize primarily flat universe models with cold dark matter (dust) and dark energy. They were defined as Here the formalism of Sahni and coworkers is extended to permit curved universe models. If we suppose that dark energy does not interact with dark matter (as we assumed), then the statefinder pair can be further expressed as where Ω = Ω m + Ω de . The statefinder is a 'geometrical' diagnostic in the sense that it depends upon the expansion factor and hence upon the metric describing space-time. If the dark energy is due to a scalar field the equation of state factor w de is given by by taking differentiation we get Using the equation of motion for the scalar field in eq. (47) and inserting the result into (44) we obtain (note that ˙ V = V ' ˙ φ, V ' = dV ( φ ) dφ ) Furthermore, from Raychaudhuri's equation we find where σ ij is the shear tensor which is given by Therefore, the statefinder s is also obtained as To study the behavior of viscous DE more precisely we consider a toy model in the next section.", "pages": [ 6, 7 ] }, { "title": "7 Test Model", "content": "To examine our above general results we present a worked example in this section. For this propose we assume the following scale factor By assuming a time varying deceleration parameter one can generate such a scale factor (Amirhashchi et al. 2011). It has also been shown that this scale factor is stable under metric perturbation (Chen and Kao 2001). In terms of redshift the above scale factor is In this case one can find the DE energy density ρ de , the bulk viscosity ξ ( ρ de ), deceleration parameter q , and average anisotropy parameter A m as ω de vf = L sinh - ( t ) ρ m 0 sinh - ( t ) - where /triangle H i = H i -H ( i = 1 , 2 , 3) and H 1 = ˙ A A , H 1 = ˙ B B , and H 3 = ˙ C C are the directional Hubble's parameters in the directions of x, y and z respectively. By using eq. (54) in eqs. (34), (39), and (40) and after simplification the EoS of viscous dark energy ω de vf , scalar field φ and the potential V ( φ ) are obtained as 1 3 - ( +2tanh 1 + L sinh - ( t ) + 1 + L sinh - 4 ( t ) cosh - ( t ) 4 - ( t ) cosh - Ω m 0 sinh - ( t ) 2 3 ( t ) ) 2 2 6 3 1 α We can re-write eqs. (60)-(62) in term of redshift as The behavior of EoS parameter, ω de , in terms of redshift z is shown in Fig. 3. It is observed that the EoS parameter is a decreasing function of z and the rapidity of its decrease depends on the value of ξ 0 . We see that in absence of bulk viscosity the EoS always varying in quintessence region (red line/solid line) whereas in presence of viscosity EoS cross PDL and varying in phantom region. But at the later stage of evolution it tends to the same constant value i.e ω de = -1 independent of the value of ξ 0 . This behavior clearly shows that the phantom phase i.e ω de < -1 is an unstable phase and there is a transition from phantom to the cosmological constant phase at late time. As we mention above, the phantom phase instability of the universe is because of the fact that the viscosity dies out as time is passing. The matter density Ω m and dark energy density Ω de can be easily calculated as Also from above two equations we obtain the total energy density as The variation of density parameters Ω m and Ω de with redshift z have been shown in Fig. 4. Here, we observe that Ω de increases as redshift decreases and approaches to 1 at late time whereas Ω m decreases as z decreases and approaches to zero at late time. For our model, the parameters { r, s } can be explicitly written in terms of cosmic time t or redsfift z as /negationslash and Trajectories in s -r plane corresponding to different cosmological models are shown in figure 6. The dots in the diagram locate the current values of the statefinder pairs { s, r } . From this figure we see explicitly that the ingredient parameter L (or K ) makes the model evolve along different trajectories on the s -r plane. It is worth to mention that the cold dark matter with a cosmological constant (Λ CDM ) diagrams (spatially flat) corresponds to the fixed point { s, r } Λ CDM = { 0 , 1 } . From eqs. (69) and (70) we obviously see that { s, r } = { 0 , 1 } at late time i.e z = -1.", "pages": [ 7, 8, 9, 10 ] }, { "title": "8 Concluding Remarks", "content": "Phantom field models have been suggested in order to provide a theoretical support for the recent observation that mildly favor the EoS of DE crossing -1 near the past. A lot of studies have been done in this regard and many phantom field models have been proposed. Some of these models are evolving from quintessence to phantom called quintom. However, theses models suffer from two major problems i.e. (1) Instability of phantom field and (2) finite future singularity (big rip). In this paper we proposed a simple mechanism to alleviate these problems by introducing a special form of bulk viscosity i. e. Π = -3 ξ 0 H ( ρ de ) τ in the cosmic fluid. In this mechanism first, viscosity causes dark energy which is varying in quintessence to pass phantom divided line (PDL) and drop it to the phantom region but since viscosity is a decreasing function of time, as time is passing it dies out and ω de leaves phantom region and tends to -1 at late time. Hence the problem of future singularity (big rip) does not occur in this scenario. To test the impact of the anisotropy parameter ( L ), we perform a statefinder diagnostic on this scenario. This diagnostic shows that the statefinder parameters can probe the anisotropy of the model. May be future SNAP would be capable of probing this effect. In summary, The general form of the EoS parameter of viscous and non-viscous dark energy has been investigated in this paper. It is found that the presence of bulk viscosity causes our universe to get to the darker region i.e phantom temporarily. It is worth to mention that since our anisotropic model behaves as isotropic FLRW universe at late time, as a result, the phantom does not survive in isotropic universe as well. Our results fulfil the theoretical requirement argued by Carroll et all (2003) which state that, to avoid the big rib problem, all phantom models should decay to cosmological constant at late time. Moreover, since we have not restricted our study to the maximally symmetric FLRW space-times, our results seems to be more general than those obtained on the bases of this isotropic universes.", "pages": [ 10, 11 ] }, { "title": "Acknowledgments", "content": "Author would like to thank Laboratory of Computational Sciences and Mathematical Physics, Institute for Mathematical Research, Universiti Putra Malaysia for providing facility where this work was done. Author also would like to acknowledge the anonymous referee for fruitful comments.", "pages": [ 11 ] }, { "title": "References", "content": "[35] Jamil, M., Umar Farooq, M.: Int. J. Theor. Phys. 49 , 42 (2010) [58] Setare, M.R.: Phys. Lett. B 654 , 1 (2007c)", "pages": [ 12, 13 ] } ]

[ { "title": "Rotation Curve of Galaxies by the Force Induced by Mass of Moving Particles", "content": "Kyuwook Ihm 1,* and Kyoung-Jea Lee 2 1 Beamline research division, Pohang Accelerator Laboratory, Pohang, Kyungbuk 790-784, Korea 2 Department of physics, POSTECH, Pohang, Kyungbuk 790-784, Korea We suggest that there is a novel force which is generated by the mass of relatively moving particles. The new force which we named Mirinae Force is a counterpart of the magnetic force operating between electrically charged moving particles. Instead of using the conventional dark matter, we applied the mirinae force to a particular model system of the spiral galaxy in which most of the galaxy's mass is located within the central region where some portion of the inner mass is in revolving motion at a relativistic speed. The calculation yielded three important results that illustrate the existence of mirinae force and validate the proposed model: First, the mirinae force in this model explains why most of the matters in the galactic disk are in the circular motion which is similar to cycloid. Second, the mirinae force well explains not only the flat rotation curve but also the varied slope of the rotation curve observed in the spiral galaxies. Third, at the flat velocity of 220 Km/s, the inner mass of the Milky Way calculated by using the proposed model is 6.0 × 10 11 M ʘ , which is very close to 5.5 × 10 11 M ʘ ( r <50 Kpc, including Leo I) estimated by using the latest kinematic information. This means that the mirinae force well takes the place of the dark matter of the Milky Way. General relativity is a theory which has been considered to describe experimental results related to gravity and cosmology most successfully to date. However, there are indications that the theory is incomplete because, for example, the problem of quantum gravity and the question of the reality of spacetime singularities remain open [1]. The rotation curve of galaxies is one of the questions under debate. There have been many attempts to explain the flat rotation curve of the galaxies regardless of distance which is in a discrepancy with the Newtonian prediction [2, 3]. Many studies have tried to find a correct form of the gravitational law or the distribution of hidden dark matter that fits well with observed data. Although these efforts have been rather phenomenologically successful, two important questions still remain: first, do we correctly take all the factors affecting the rotation curve of the galaxy into account? Second, do we have a correct theory of gravity? In this study we suggest a novel force generated between relatively moving particles by their mass as a counterpart of the magnetic force between the electrically charged particles that are in motion, and we named it 'mirinae force'. If we pay attention to the fact that magnetic field is induced by a variation of electric field in a spacetime without the magnetic monopole, the expectation of alternative force field generated by a variation of gravitational field in the identical spacetime looks reasonable regardless of what is mass. Physical phenomena related to mass and charge have always been described in a separated path way probably because interactions between mass and charge have never been observed. However, mass and charge are not completely independent from each other. Unless the mass of the test particle is known, one cannot determine the amount of electric charge by measuring the acceleration of a test particle driven by Coulomb force generated by another particle with known value of q. Mirinae force is derived by defining a single complex value called the grand-mass whose unit is kg. The grand-mass is assumed to characterize all the physical phenomena of particles related to both mass and electric charge. Grand-mass is defined as where m and q are conventional mass and charge of a stationary particle in a inertial frame, and α is matching constant with an unit of kg/C. We first demonstrate that the grand-mass of the particles well describes the Newtonian mechanics including gravitational and Coulomb interactions and the momentum-energy conservation of special relativity. However, the description of the force between moving charged particles using grand-mass causes a novel force generated by mass between the moving particles as a counterpart of the magnetic force. This force is what we call mirinae force. We will show that the mirinae force is so small that its effects can be important only in a system of astronomical scale. If a huge mass flow exists near the galactic center, the mirinae force field would be produced inducing a circular motion of matters in the galactic disk, which is similar to a cycloid. This model based on the concept of mirinae force explains well the flat rotation curve found in a spiral galaxy, such as the Milky Way and M31, and it gives an orbit velocity and inner mass well accorded with recent estimation. With the grand-mass the momentum which is conserved in all inertial reference frames can be written as where, γ = (1v 2 / c 2 ) -1/2 is Lorentz factor of a particle with velocity v . The energy-momentum conservation law still holds true. If all mass terms are replaced by the grand-mass, M , the result is reduced to E 2 -( pc ) 2 =( mc 2 ) 2 . The grand-force is defined as In the expression of the grand-force, the nontrivial term is only mass related real part of F m. This is identical to the fact that in a system with an isolated single particle with mass, m , and charge, q , the observer can measure only the inertial mass m (Fig. 2a). The observer cannot notice the existence of electric charge included in a single particle system. This situation is well described by the imaginary representation of charge related term in (3). The measureable physical value of an isolated particle cannot be related with charge, q , because the charge related force term is in the imaginary part. The grand-force acting on a stationary ( γ = 1) particle M 1 located at ݎԦ ଵ due to another particle M 2 at ݎԦ ଶ can be expressed as, (Fig. 2b), where G is the gravitational constant of 6.67×10 -11 [m 3 /s 2 Kg]. The constant G α 2 shown in the second term of ] when compared with that of Coulomb force. real force part should be equal to 1/4 πε 0 = 8.99×10 9 [m 3 Kg/s 2 C 2 This gives α = 1.161×10 10 [Kg/C], which show that the unit of the grand-mass in (1) is Kg. The directions of the two force terms in the real value part of (4) well accord with that of gravitational and Coulomb forces, i.e., attractive and repulsive. Interestingly, the charge related term is involved with the mass related force term, ܨ Ԧ ௠ భ in (4). When there is a flow of particles with mass and charge, one can define the grand-mass current density as where ρ m , ρ q and v are the density of mass, charge and the average drift velocity of particles at point r at time t . If identical particles with density n are in the flow, then the grand-mass current density is The magnetic force can be described by the grand-mass current density of moving particles. To do so, the electric current in the magnetic force law need to be replaced by grand-mass current, and constants of the resulting equation should be determined. The simplest example of Ampère's law is the force per unit length between two straight parallel current carrying conductors, which is expressed as (7) with magnetic constant K . Considering the identical system with two parallel straight lines carrying grand-mass current IM 1 and IM 2, and replacing magnetic constant K with undefined one, KGM , then we have Comparing second term in (8) with Ampère's force in (7), one can have KGM = -μ 0/4 πα 2 . Following the way that the magnetic constant K in (7) determines the magnitude of the magnetic field generated by electric current in the Biot-Savart law, the force field generated by a grand-mass current density can be found using KGM as The equation (9) has two terms. The real value term, ܤ ሬԦ ௠ , is induced by the mass flow and named mirinae field. The imaginary second term is induced by the electric current, and ܤ ሬԦ ௤ (= ܤ ሬԦ ஻ିௌ ) represents Biot-Savart law. Note that the direction of the mirinae field is opposite to that of the magnetic field generated by positively charged moving particles. Now, the Lorentz force can be replaced by grand-Lorentz force expressed by the grand-mass as In the eq. (11) BG is replaced by (9). The grand-Lorentz force has two real terms in which the first real term is mirinae force and the second real term is the Lorentz force. Imaginary term corresponds to the imaginary force appeared in (3). To test the magnitude of the mirinae force compared to the Lorentz force, consider a grand-Lorentz force acting on a moving electron due to the grand-force field produced by flowing electrons. In this case the ratio of charge to mass is constant, i.e., Σ = e / m e. Then, the mass current density of electrons is (see the first term in (6)) From (10) and (12) the mirinae field is Then, the real term of grand-Lorentz force on a moving electron with velocity, v , is from (11) and (13) This shows that the grand-Lorenz force on a moving electron is almost the same as the Lorentz force. The magnitude of mirinae force is ( m e/e α ) 2 =2.39×10 -43 of Lorentz force. This shows why the mirinae force had not been found in the laboratories system on earth until now. However, for further discussion the mirinae force has to show some persuasive and meaningful physical examples. Due to such a small magnitude of the mirinae force when compared with Lorentz force, meaningful effects of the mirinae force should be sought among the systems in astronomical scale. It is known that 70% of galaxies exhibit a variety of spiral structures. Among them galaxies with a central bulge in their disk have a rotation curve which is flat from centre to edge [3, 4]. However, it was expected that these galaxies would have a rotation curve that slopes down as the distance increases in the inverse square root relationship, in the same way as other systems with most of their mass in the centre, such as the solar system of planets. The solution of this discrepancy has been sought by introducing dark matter or modified Newtonian dynamics (MOND) because kinetics in most the galactic region except for the vicinity of the black hole can be described by the non-relativistic gravitational theory [5-7]. Unfortunately, these methods have not been always successful in describing the shape of the rotation curves and have not shown a deeper physical understanding [8]. We propose a novel model in which dark matter is excluded whereas most of the mass ( mC ) is assumed to be near the galaxy center and some portion of them revolves at relativistic velocity to the same revolving direction as that of matter on the galactic disk. To simplify the calculation, the circular mass flow around the center of galaxy is considered as a circular loop of radius a carrying the particle flow, lying in the x-y plane, and centered at the origin (Fig. 2). If the net electric charge is zero, the grand-mass current is IM = Im (+ i α Iq =0). The current density has only one component in the ϕ direction, which can be expressed using delta function as From equation (9) the force field generated by grand-mass flow has only mass term (mirinae field) The solution can be easily found following the technique shown in the current loop problem solved in classical electrodynamics [9]. For rough estimation, assuming the case of r >> a , the solution is found as (17). The mirinae field on the galactic disk, corresponding to x -y plane in Fig. 2, can be obtained from (17) at θ = π /2 as (18). where the constant C ≡ μ 0Im π a 2 /4 πα 2 [m 3 /s] in (18). Now, the grand-Lorentz force acting on the moving particle M' = m' (+ i α q' =0) on the galactic disk at distance r with velocity ݒԦ is If a particle M' is moving with the velocity, ݒԦݎ̂ , as shown in figure 2a, the mirinae force acts on it toward ߶ ෠ direction as indicated in (19), inducing a circular motion of particle M' with the same revolving direction of as that of the mass current in a loop. It is notable that, in case of the Lorentz force a charged particle, M'' (= m'' + i α q'' ), with velocity ݒԦݎ̂ is forced to move in the opposite revolving direction to that of electric current of the ring. Consider another case where a particle M' (= m' ) is in circular motion around the galaxy center with velocity ݒԦ߶ ෠ at distance r (Fig. 2b). The particle M' is attracted toward -ݎ̂ direction by the gravitational force induced by a huge centered grand-mass MC (= mC ), and repulsed toward ݎ̂ direction by the centrifugal force and the mirinae force due to െ൫ݒԦൈߠ ෠ ൯ൌݒݎ̂ in (19). Assuming that the orbital motion of particle M' is stable, i.e., dr / dt ≈ 0, we have the relation written as The orbit velocity of M' satisfying the relation (20) can be obtained with (18) as If there is no mass flow ( C= 0), the velocity of (21) describes the rotation curve of the planets in the solar system according to Newtonian mechanics. Figure 3 shows the rotation curves of (21) for the cases that C = 1×10 44 ~ 1×10 49 when the inner mass is mC = 5×10 11 M ʘ . The rotation curve at C ~ 1.0×10 47 (m 3 /s) shows a nearly flat velocity in the wide distance ranges and has a flat velocity value of 220~250 Km/s similar to that of the Milky Way and M31 [10-12]. However, as the orbit distance decreases below 7 Kpc, the velocities start to be out of the permissible range of the observed values. This is originated from over simplified calculation model in Figure 2. In the real galactic system the gravitational field as well as mirinae field varies significantly depending on the distance, especially in the region of the galactic bulge, because the inner mass and mass flow are spatially distributed [4, 12]. Therefore, detailed calculation of the rotation curve near the galactic center must take into account the spatial distribution of mass and mass flow. Variation of C value by a factor of 10 from C =1×10 47 (m 3 /s) makes the curve profile completely different, indicating that the profile of rotation curve is sensitively dependent on the C value. The curve profile can be estimated by the slope of rotation curve at a specific distance, r , as a function of C value, written as To show the detailed dependence of the curve profile on the C , dv / dr at r =15 Kpc and mC = 5×10 11 M ʘ is plotted as a function of C using filled squares in Figure 4. The corresponding velocities from (21) are shown by circles in Figure 4. The orbit velocity of stars when the rotation curve is flat, i.e., flat velocity, can be obtained using the specific C value at which dv / dr is zero. As shown in Figure 4, the C value causing the flat rotation curve is C ≈ 6.4×10 46 m 3 /s at which the orbital velocity corresponds to 250 Km/s. The C values above 1.0×10 49 m 3 /s also induce the slope to be zero, but the region of those values can be excluded from our discussion because in this region the orbit velocity is also zero, which is not the case of the spiral galaxy. It is interesting note that the curve slope has both the negative and positive values in the vicinity of C ( dv / dr =0)= 6.4×10 46 m 3 /s, which accords well with the observation that showed various slopes of rotation curve in the spiral galaxies [7]. The mirinae field in the form of C / r 3 of (18) is derived from the model as described in Figure 2. The magnitude of C (= μ 0Im π a 2 /4 πα 2 ) represents the strength of the mirinae field at specific distance of orbit. This model assumed that some portion of inner mass is in a circular motion with relativistic velocity, which is simplified as a circular loop carrying grad-mass with radius a . Mass current Im in C is given by λγ v , where λγ (= γ ×Mass of loop/2 π a ) is relativistic line density of mass which is in revolving motion at mean velocity v along the loop. Assuming that 1% of mC , ~ 5.0×10 9 M ʘ (= mass of the loop), is in the circular motion at relativistic speed at γ =1.88×10 7 , and the radius of loop is a =100 pc, then we have C = 6.41×10 46 [m 3 /s] at which dv / dr =0 as shown in Figure 4. In case of the electron in relativistic motion the kinetic energy at γ =1.88×10 7 corresponds to about 9 TeV. Note that the loop is imaginary system which represents all possible mass flow generating mirinae field near the galactic center including the black hole which is generally believed to be a supermassive one. However, due to the lack of reliable information on the distribution of the inner mass and mass flow near the galactic center the detailed discussion on the relevance of the parameters of the loop could not be made here. In Figures 3 and 4 we assumed mC = 5×10 11 M ʘ and showed that the magnitude of C and orbit velocity at specific distance can be obtained by the additional condition dv / dr = 0. The C value that satisfy the condition of dv / dr = 0 in (22) and the corresponding orbit velocity are expressed as Figure 5 shows the dependence of v ( mc , r =15 Kpc) and C ( mc , r =15 Kpc) on the inner-mass, mC , when dv / dr =0. In most of the spiral galaxies, the flat orbit velocities of stars fall in the range of 150 ~ 300 Km/s [7]. Within this range the inner mass can be estimated by (23), and, as shown in Figure 5, interestingly it is in the range of 1.5×10 11 ~7×10 11 M ʘ , which is very similar to the estimated results using the kinematic information for the spiral galaxies without taking account of dark matter [10, 12, 13]. For example, at orbit velocity of 220 Km/s, the inner mass is estimated to be 6.0×10 11 M ʘ from Figure 5 or (23); and the orbit velocity of the Milky Way is about 220 Km/s and the mass ( r <50 Kpc, including Leo I) is estimated to be 5.5×10 11 M ʘ by using the latest kinematic information [12]. This result means that the mirinae force well takes the place of the dark matter of the Milky Way. We suggested the existence of a mirinae force field generated by the mass current density similar to the magnetic force. When the concept of grand-mass is developed further in theory, one can obtain an equation on the gravitational-mirinae field which is a counterpart of the electromagnetic field derived from the Maxwell equation. We expect that the enormous mass flow in circular motion near the galaxy center would emit gravitational-mirinae wave at a speed of light analogous to the electromagnetic wave emitted from the accelerated charged particles. In summary, we proposed that a novel force, named mirinae force, is operating between relatively moving particles by their mass. This force is derived as a counterpart of the magnetic force by introducing a grand-mass defined as M=m+i α q [kg]. To show the relevance of the grand-mass in the physical system, we proposed a model of a spiral galaxy in which some portion of the matter in the massive galactic center is revolving at a relativistic speed. The mirinae force in this model explains why most of the matters in the galactic disk are in the circular motion in a manner similar to the cycloid. Calculations in this model explain the various curve profiles of the orbit velocity reported by other researchers, including the flat rotation curve in the spiral galaxies. The inner mass of the the Milky Way estimated by this model gives 6.0×10 11 M ʘ which is very close to 5.5×10 11 M ʘ ( r <50 Kpc, including Leo I) estimated by using the latest kinematic information.", "pages": [ 1, 2, 3, 4, 5, 6, 7, 8, 9 ] }, { "title": "Acknowledgements.", "content": "This work was supported by Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education, Science and Technology (Grant No. 2011-0026550).", "pages": [ 10 ] }, { "title": "REFERENCES", "content": "Figure 1. Particles with grand-mass (a) The force exerted on a particle with grand-mass M in the complex space. In the system composed of an isolated single particle it is possible to measure only the inertial mass m from the real term of grand-force. (b) Interactions between two particles with grand-mass M 1 and M 2. Figure 2. The proposed model for the spiral galaxy. Most of the mass, MC = mC , of galaxy is near the galactic center, and some portion of the inner mass is in revolving motion, which is replaced by the loop carrying the grand-mass at relativistic speed. In this model the direction of the mirinae field is െߠ ෠ on the x -y plane to which the galactic disk belongs. (a) If the velocity of the star is െݒݎ̂ , then the star is forced toward ߶ ෠ direction, resulting in a circular motion. (b) If the velocity of the star is ݒ߶ ෠ , then the star feels the repulsive mirinae force toward ݎ̂ direction, as well as centrifugal force. Figure 3. Rotation curves acquired from (21) when mC = 5×10 11 M ʘ and C = 1.0×10 44 ~1.0×10 49 [m 3 /s]. C value represents the strength of the mirinae force at distance r produced by the mass carrying loop with radius a . Figure 4. Slopes( dv / dr ) of rotation curves at r =15 Kpc when mC = 5×10 11 M ʘ as a function of C values. In the C range of 2.0×10 45 ~2.0×10 47 [m 3 /s] the slopes of the rotation curve are changed from negative to positive values, showing that the varied curve profiles are made in this range. At Slopes( dv / dr )=0, i.e., the flat rotation curve, the flat velocity is 250 Km/s. Figure 5. Flat velocities and C values as a function of the inner mass, mC ., when the condition of dv / dr = 0 is satisfied at r =15 Kpc. In this condition, if the flat velocity is given, the inner mass, mC , and C value are exclusively determined. The inner mass of the spiral galaxies with flat velocity in the range of 150~300 Km/s are estimated to be in the ranges of 1.5×10 11 ~ 6.5×10 11 M ʘ . 1 e 1 e 2 e 2 e 3 e 4 e 5 e", "pages": [ 11, 12, 13, 14, 15, 16, 17, 18 ] } ]

[ { "title": "G-corrected holographic dark energy model", "content": "M. Malekjani 1,2 · M. Honari-Jafarpour 1", "pages": [ 1 ] }, { "title": "Abstract", "content": "Here we investigate the holographic dark energy model in the framework of FRW cosmology where the newtonian gravitational constant, G , is varying with cosmic time. Using the complementary astronomical data which support the time dependency of G , the evolutionary treatment of EoS parameter and energy density of dark energy model are calculated in the presence of time variation of G . It has been shown that in this case, the phantom regime can be achieved at the present time. We also calculate the evolution of G -corrected deceleration parameter for holographic dark energy model and show that the dependency of G on the comic time can influence on the transition epoch from decelerated expansion to the accelerated phase. Finally we perform the statefinder analysis for G - corrected holographic model and show that this model has a shorter distance from the observational point in s -r plane compare with original holographic dark energy model. keyword: Cosmology, dark energy, holographic model, gravitational constant.", "pages": [ 1 ] }, { "title": "1 Introduction", "content": "Nowadays, it is widely believed that the cosmos is experiencing an accelerated expansion. This idea and M. Malekjani M. Honari-Jafarpour E-mail: [email protected]. E-mail: [email protected]. 2 Research Institute for Astronomy & Astrophysics of Maragha (RIAAM)- Maragha, Iran, P. O. Box: 55134-441. belief came into existence after collection of data from 'Type Ia supernova' in 1998 (Perlmutter et al 1998). Also the other data from WMAP (Bennett et al 2009),SDSS (Tegmark et al 2004) and X-ray (Allen et al 2004) experiments support this accelerated expansion. In the framework of standard cosmology, the existence of dark energy with negative pressures is essential to interpret the cosmic acceleration. Hence, dark energy scenario has got a lot of attention in modern cosmology both from theoretical and observational point of view. Observationally, The result of SNeIa experiment shows that dark energy occupies about 72% of the total energy of our universe, dark matter and baryons about 28% of the total energy of the universe(Perlmutter et al 1998). Although the nature of dark energy is still un-known, but the ultimate fate of the current universe is determined by this mysterious component. Till now, some theoretical models have been proposed to interpret the behavior of the dark energy.The first and the simplest one is Einstein's 'cosmological constant' (Sahni & Starobinsky 2003) which, of course, has two problems called fine -tuning and cosmic coincidence. The cosmological constant has the fixed equation of state w Λ = -1, while the dynamics of current expansion can be explained by dynamical dark energy models with time varying equation of state. The scalar fields such as quintessence (Wetterich 1998), phantom (Caldwell 2002) or the combination of both which is called quintom (Elizalde et al 2004) are examples of dynamical models. The other dynamical dark energy models which interprets the current acceleration of expansion are constructed based on quantum gravity theory(Witten 2002). The holographic dark energy (HDE) model is one of the the proposed models based on the holographic principle in quantum gravity (Horava & Minic 2000). According to the holographic principle, a short distance ultra-violet (UV) cut-off is related to the long distance infra-red (IR) cut-off, due to the limit set by the formation of a black hole (Horava & Minic 2000). The holographic principle indicates that the zero-point energy of a system with size L should not be exceed from the mass of black hole with the same size. From the above principle, the energy density of HDE model in cosmology can be described as: Where L is the cosmic horizon and c is a numerical constant of order unity and G is a Newton's gravitational constant. The length scale L has an essential role in the definition of energy density of HDE model. Therefore the various model of HDE have been constructed for different of infrared (IR) cutoff length. For example the simple choice of IR cutoff is the Hubble length which leads to wrong equation of state for DE (Horava & Minic 2000). However in the presence of interaction between dark matter and DE the HDE model with Hubble radios IR cutoff can derive the accelerated expansion and also solve the coincidence problem (Pavon & Zimdahl 2005). The other choice for IR cutoff is the particle horizon. In this case, like Hubble length, the accelerated expansion cannot be achieved Pavon & Zimdahl (2005). Another choice is the event horizon where the cosmic acceleration can be interpreted in this case (Zhou et al 2007). Nojiri and Odintsov (2006) investigated the holographic DE model by assuming IR cutoff depends on the Hubble rate, particle and future horizons, span of life of the universe and cosmological. In this generalized form of HDE the phantom regime can be achieved and also the coincidence problem is demonstrated. Unification of early phantom inflation and late time acceleration of the universe is the other feature of this model. Recently, the HDE model has been constrained by various astronomical observations(Huang & Gong 2004; Zhang & Wu 2005; Wu et al. 2008; Enqvist et al. 2005). In addition, there are some theoretical and observational supports indicating that Newton's gravity constant varies and changes with cosmic time. The first theoretical idea in this respect is the pioneering work of Dirac(Dirac 1938), and then the idea of Dyson(Dyson 1972). Also, the Branse-Dicke framework in the Physics predicts the variability and fluctuation of G(Brans & Dicke 1961). Moreover the varying behavior of G in Kaluza-Klein theory was associated with a scalar field appearing in the metric component corresponding to the 5-th dimension and its size variation(Kaluza, et al 1921).In this theory, a scalar field paired with gravity by a new parameter replaces the quantum gravity. The variabil- ity of G with time is also supported from the observational viewpoint. The observational data collected by Type Ia Supernova (Gaztanaga et al 2002),HulseTaylor Binary(Damour et al 1988), astro-seismological data from pulsating white dwarf stars (Benvenuto 2004; Biesiada & Malec 2004), helio-sesmiological (Guenther 1998) and the Big Bang Nuclei-synthesis data(Copi et al 2004) support a variable value for G with time. We refer to these observations in the section 3 of the paper. Here in this work we consider the HDE model with time varying G , the so-called G -corrected HDE model, in spatially flat FRW universe. We consider the event horizon as an IR cut-off in relation (1). In this concern some other works have been done in which the HDE model has been considered with time dependency of G , i.e., (Jiano et al 2009). Here we obtain the equation of state w d as well as deceleration parameter q and statefinder pair { s, r } for G -corrected HDE model in FRW universe and also solve the related equations numerically by using the observational values for G ( t ). It is clear that constraining a given model against the observational data is model dependent. Therefore some doubts usually remain on the validity of the constraints on cosmological parameters. In order to solve this problem, we use the cosmography, i.e. the expansion of scale factor in Taylor series with respect to the cosmic time. The first term of Taylor series is the Hubble parameter ( H = da adt ), the second term is the deceleration parameter ( q = -d 2 a aH 2 dt 2 ), the third term is the jerk parameter ( r = d 3 a aH 3 dt 3 ), the forth term is snap ( k = d 4 a aH 4 dt 4 ) and the fifth term is lerk parameter ( l = d 5 a aH 5 dt 5 ). The present values of the above parameters can be used to describe the evolution of the universe. For example q 0 < 0 indicates the current accelerated expansion of the universe and also r 0 allows to discriminate between different dark energy models. Since Hubble's parameter which corresponds to the first derivative of the scale factor (˙ a ) and the deceleration parameter which corresponds to the second derivative of the scale factor (a ) can not distinguish between the different models, we need a higher derivative of scale factor. Sahni et al.(Sahni et al 2003)and Alam et al.(Alam et al 2003b), by using the third time derivative of scale factor, introduced the statefinder pair { s, r } in order to diagnosis the treatment of dark energy models. The statefinder pair in spatially flat universe is given by: The statefinder parameters s and r are the geometrical parameters, because they only depend on the scale factor. Up to now, different dark energy models have been investigated from the viewpoint of statefinder diagnostic. These models have different evolutionary trajectories in { s, r } plane, therefore the statefinder tool can discriminate these models. The well known Λ -CDM model corresponds to the fixed point { s = 0 , r = 1 } in the s -r plane (Sahni et al 2003). The distance of the current value of statefinder pair { s,r } of a given dark energy model from the fixed point { s = 0 , r = 1 } is a valuable criterion to examine of model. Also the recant investigation by observational data resulted the best fit value for statefinder in flat universe as { s obs = -0 . 006 , r obs = 1 . 02 } (M. Malekjani & Khodam-Mohammadi 2012c). The other dark energy models which have been studied from the viewpoint of statefinder diagnostic are : corrected Friedmann equation for zero-zero component of field equation in flat geometry can be obtained as follows Since the value of ˙ G/G is small particularly in the late time accelerated universe, therefore we have ignored the higher time derivative of G (i.e., G/G ) and also larger powers than one (i.e., ( ˙ G/G ) 2 , ...). The equation (4) for standard model with time varying gravitational constant can also be obtained from Branse- Dicke gravity by assuming ( w = 0 and ψ = 1 /φ ( t )) in equation (2) of (Banerjee & Pavon 2007). Here w is the Branse-Dicke parameter and ψ is BranseDicke scalar field. If we consider the derivative of G according to ln a the above G -corrected Friedman equation can be re-written as: where prime is derivative with respect to x = ln a . Assuming the event horizon as an IR cut-off as The energy density of HDE model in Eq.(1) is written as the quintessence DE model (Sahni et al 2003; Alam et al 2003b) , the interacting quintessence models (Zimdahl & Pavon 2004; Zhang 2005a), the holographic dark energy models (Zhang 2005b; Zhang et al 2007) , the holographic dark energy model in non-flat universe (Setare et al 2007), the phantom model (Chang et al 2007), the tachyon (Shao & Gui 2007), the generalized chaplygin gas model (Malekjani et al 2011a), the interacting new agegraphic DE model in flat and non-flat universe (Zhang 2010; Khodam-Mohammadi & Malekjani 2010), the agegraphic dark energy model with and without interaction in flat and non-flat universe (Wei & Cai 2007; Malekjani & Khodam-Mohammadi 2010), the new holographic dark energy model (Malekjani et al 2011b), the interacting polytropic gas model (Malekjani & Khodam-Mohammadi 2012a) and the interacting ghost dark energy model (M. Malekjani & Khodam-Mohammadi 2012b). The paper is organized as follows: In section (2) the G -corrected HDE model has been presented in falt FRW universe and the equation of sate w d , deceleration parameter q and statefinder pair { s, r } have been calculated in the presence of time variation of G . In section (3) we present the numerical results and in section (4) the paper is concluded.", "pages": [ 1, 2, 3 ] }, { "title": "2 The G-corrected HDE model in a FRW cosmology", "content": "The Hilbert-Einstein action with time varying gravitational constant, G ( t ) = G 0 φ ( t ), is Here we assume the scalar function φ ( t ) for time dependency of G ( t ). Also G 0 is usual gravitational constant and L m is the lagrangian of matter field. By variation of above action with respect to metric g µν the first In terms of dimensionless energy densities where the ρ c = 3 H 2 8 πG ( t ) is the critical energy density, the corrected Friedman equation(5) can be written as this equation is look like to the Friedman equation in the non-flat universe : Ω m +Ω d = 1 -Ω k . Based on observational data described in introduction we consider the negative values for ˙ G G . Therefore the added term ' G/G in right hand side of (9) can be interpreted as non-flatness parameter Ω k in non-flat universe. In addition the evolution of Hubble parameter in terms of scale factor in G -corrected flat universe including dark matter and dark energy can be calculated from Eq.(4) as follows where H 0 is the present value of Hubble parameter. The conservation equations for dark matter and dark energy are given by: Taking the time derivative of (7) by using ˙ R h = 1+ HR h and (7) in relation (12) we obtain the equation of state for G-corrected HDE model as follows Also, taking the derivative of (13) with respect to x = ln a , we obtain Here we have ignored the terms including ( ' G/G ) 2 and ( ' G/G ) 3 and also ' ' G/G . In what follows which we derive and calculate, we keep only the first- order correction of G (i.e., ' G/G ). Now, derivative of Ω d = ρ d ρ c = c 2 H 2 R 2 h yields the evolutionary equation for dark energy density for Gcorrected HDE model as follows In addition taking the time derivative of corrected Friedman equation (4) obtains Therefore the equation of motion for energy density of G -corrected HDE, i.e., Eq.(15) is written as The deceleration parameter q = -1 -˙ H/H 2 which represents the decelerated or accelerated phase of the expansion of the universe, by using (13)and (16), is written for G-corrected HDE model as For completeness, we now derive the statefinder pair { s, r } in G-corrected HDE model. For this aim, by time derivative of (16), we first obtain Inserting (16) and (19) in r = H H 3 +3 ˙ H H 2 +1 we obtain the following equation for the parameter r of statefinder pair From (2), by using (18) and (20) we also obtained the parameter s in G-corrected HDE model as follows In the limiting case of time-independent gravitational constant G (i.e., ' G = 0) all the above relations reduce to those obtained for original holographic dark energy (OHDE) model in (Zhang 2005).", "pages": [ 3, 4 ] }, { "title": "3 Numerical result", "content": "There are many astronomical observations which show the time dependency of Newtonian gravitational constant. All these data are in agreement with Dyson idea who pointed out that G varies in the length of cosmic age H -1 . Based on the observational data from WMAPfive-year observations the present value of Hubble parameter is H 0 = 6 . 64 × 10 -11 yr -1 (Bennett et al 2009; Zhang & Wu 2009). Moreover the astronomical observations are in the line of Dirac's theory in which G ( t ) ∝ t -1 as follows (Cetto et al 1986) where k 1 and k 2 are constant. The data gathered from SNeIA data yields the best rang for variation of G as: -10 -11 yr -1 ≤ ˙ G G ≤ 0 (Gaztanaga et al 2002) and the data obtained from Binary Pulsar PSR1913 determines the range of variation of ˙ G G as: -(1 . 10 ± 1 . 07) × 10 -11 yr -1 < ˙ G G < 0 (Damour et al 1988). The data obtained from Helio-sesmiological determines the best range -1 . 6 × 10 -12 yr -1 < ˙ G G < 0 (Guenther 1998). Another estimation for ˙ G G has been done through astro-seismological data obtained from pulsating white dwarf star which yields the best range of variation as: -2 . 5 × 10 -10 yr -1 ≤ ˙ G G ≤ +4 . 5 × 10 -10 yr -1 (Benvenuto 2004). In (Biesiada & Malec 2004), the range of ˙ G G was determined as ˙ G G ≤ +4 . 1 × 10 -11 yr -1 . It should be noted that all the above range of ˙ G G are calculated for z ≤ 3 . 5. Finally from the observational data of Big Bang nuclei-synthesis, we have -4 . 0 × 10 -13 yr -1 < ˙ G G < +3 × 10 -13 yr -1 (Copi et al 2004). More details for the variation of G with cosmic time can be seen in (Ray & Mukhopadhyay 2007). In previous section we calculated the effect of variation of G on the HDE model in terms of ' G G . Therefore, we change the time derivative to derivative with respect to x = ln a as ˙ G G = H ' G G where ' G G is a dimensionless numerical value, because the dimensions of Hubble Parameter is same as ˙ G G . Here we call this numerical value as α = ' G G . In this work we use the SNeIa observational data -10 -11 yr -1 ≤ ˙ G G ≤ 0 which covers the other observational range of ˙ G G . We also use the present value H 0 = 6 . 64 × 10 -11 yr -1 based on observational data from WMAP five-year observations (Bennett et al 2009; Zhang & Wu 2009). The parameter α , using by these observational data can be obtained as | α | ∼ 0 . 10. Therefore we choose the illustrative values α = -0 . 1 , 0 , 0 . 1 which are in the order of the observational value. At follows we calculate the evolution of cosmological quantities: EoS parameter, energy density, deceleration parameter and statfinder pair of G-corrected HDE model and obtain the effect of parameter α on the evolution of these cosmological quantities.", "pages": [ 4, 5 ] }, { "title": "3.1 EoS parameter", "content": "By solving (13), we show the evolution of EoS parameter of G-corrected HDE as a function of redshift in Fig.(1). Here we fix the holographic parameter c = 0 . 87. Note that for this value the original HDE model without G correction can not enter the phantom regime. The black solid curve relates to original HDE model without G correction. The red- dashed curve is indicated for α = 0 . 1 and blue- dotted- dashed line represents α = 0 . 1. Here we see that the G-corrected HDE model can enter to phantom regime when α < 0, i.e. blue-dashed line. Hence one can conclude that the G-corrected HDE model can cross the phantom divide without a need of interaction between dark matter and dark energy. Also, the G-corrected HDE model crosses that phantom line ( w d = -1) from up ( w d > -1) to below ( w d < -1). This behavior of G-corrected HDE model is in agreement with recent observations in which the universe transits from quintessence regime ( w d > -1) to the phantom regime ( w d < -1) at the near past (Alam et al 2004). Here we calculate the evolution of energy density of G-corrected HDE model as a function of redshift parameter from the early time up to late time by solving equation (15). In Fig.(2), we plot the evolution of energy density Ω d versus of redshift for different illustrative values of α . We see that at the early times Ω d → 0 and at the late times Ω d → 1, meaning the dark energy dominated universe at the late time. In this figure by fixing c = 0 . 87 the parameter α is varied as illustrative values -0 . 1 , 0 . 0 , +0 . 1 corresponding to dotted-dashed -blue, solid -black and dashed -red curves , respectively. We see that in the past times the dark energy becomes more dominant for positive values of α and at the late times the dark energy dominated universe can be achieved sooner for negative values. Here we study the expansion phase of the universe by calculating the evolution of deceleration parameter q in G-corrected HDE model. By solving equation (18) and using (15), we plot the evolution of q versus redshift parameter z in Fig.(3). We see that the parameter q start from q = 0 . 50, representing the CDM model at the early time. Then the parameter q becomes negative, representing the accelerated expansion phase of the universe at recent epochs. Therefore the G-corrected HDE model can interpret the decelerated phase of the expansion of the universe at the early times and accelerated phase later. we fix the parameter c = 0 . 8 and for the different illustrative value of the α = -0 . 1 , 0 . 0 , +0 . 1 corresponding to dotted-dashed -blue, solid -black and dashed -red curves , respectively. We see for negative value of α , the accelerated expansion can be achieved sooner than the original HDE model( α = 0 . 0) and also positive value of α .", "pages": [ 5, 6 ] }, { "title": "3.4 statefinder diagnosis", "content": "The statefinder pair { s, r } for G-corrected HDE model is given by relations (20) and (21). In statefinder plane, the horizontal axis is defined by the parameter s and vertical axis by the parameter r . In Fig.(4), by putting (13),(14)and (15)in (20) and (21) and solving them ,we obtain the evolutionary trajectories of G-corrected HDE model in s -r plane for different values of parameter α . By expanding the universe, the evolutionary trajectories evolve from right to left. The parameter r decreases then increases, while the parameter s decreases forever. The trajectories cross the Λ -CDM fixed point { s = 0 , r = 1 } at the near past. In the other words, the G-corrected HDE model has mimicking the ΛCDM model at the near past. The present values of the cosmographic parameters, introduced in introduction, have been observationally constrained using the Markov Chain Monte Carlo method in (Capozziello et al 2011) as follows: H 0 = 0 . 718, q 0 = -0 . 64, r 0 = 1 . 02 , k 0 =?0 . 39, l = 4 . 05 . Using q 0 = -0 . 64 and r 0 = 1 . 02, we calculate the present value of statefinder parameter s as s 0 = -0 . 006. Hence the observational point s 0 = -0 . 006 , r 0 = 1 . 02 in s-r diagram is very close to ΛCDM fixed point s 0 = 0 , r 0 = 1. The observational point is indicated by green star in Fig. (4). Here we fix the holographic parameter c = 0 . 87 and vary α as α = -0 . 1 , 0 . 0 , +0 . 1 corresponding to dotted-dashed blue, solid -black and dashed -red curves, respectively. We see that different values of α result different trajectories in s -r plane. Therefore the G-corrected HDE model in s -r plane is discriminated for different values of α . The colored circles on the curves represent the today's value of statefinder parameters { s 0 , r 0 } of the model. We also see that for positive values of α , the distance of { s 0 , r 0 } from the observational point { s obs = -0 . 006 , r obs = 1 . 02 } is shorter and for negative values of α and longer for positive values of α compare with original HDE model.", "pages": [ 7 ] }, { "title": "4 conclusion", "content": "In summary, we extended the holographic dark energy (HDE) model by assuming the time dependency of Newtonian gravitational constant, G , in standard model of cosmology. Here we obtained the G -corrected Friedman equation in flat universe. Regarding, the astronomical data from type Ia Supernova (Gaztanaga et al 2002),Hulse-Taylor Binary (Damour et al 1988), astro-seismological data from pulsating white dwarf stars (Benvenuto 2004; Biesiada & Malec 2004), helio-seismological data (Guenther 1998) and the Big Bang Nuclei-synthesis data(Copi et al 2004), we obtained the parameter | α | = G ' G = 0 . 10. The evolution of EoS parameter, deceleration parameter and energy density parameter of HDE model in the presence of G correction have been calculated. We showed that the G correction can affect the evolution of above parameters at the present time up to near past and is negligible at the early times. It was shown that for an illustrative value of holographic parameter c in which the original HDE model can not cross the phantom line, the G - corrected HDE model can achieve the phantom regime and cross the phantom line from up ( w d > -1) to below ( w < -1) in agreement with recent observations (Alam et al 2004). The parameter α can also influence on the transition from decelerated expansion to the accelerated expansion. We showed that for α > 0 the transition from q > 0 to q < 0 earlier and for α < 0 later compare with original HDE model. Finally we performed the statefinder diagnosis analysis for G -corrected HDE model and showed that the G correction can affect on the evolutionary trajectories of the model in s -r plane. We concluded that for α > 0, the distance of present value { s 0 , r 0 } from the observational point is shorter and for α < 0 is longer compare with original HDE model. We are grateful to A. Khodam-Mohammadi for helpful Acknowledgment discussions.", "pages": [ 7 ] }, { "title": "References", "content": "Zhang X., & Wu, F. Q., Phys. Rev. D 72 , 043524 (2005).", "pages": [ 8 ] } ]

[ { "title": "Generalized holographic dark energy model described at the Hubble length", "content": "M. Malekjani ∗ 1, 2 1 Department of Physics, Faculty of Science, Bu-Ali Sina University, Hamedan 65178, Iran 2 Research Institute for Astronomy and Astrophysics of Maragha (RIAAM), Maragha, Iran We generalize the holographic dark energy model described in Hubble length IR cutoff by assuming a slowly time varying function for holographic parameter c 2 . We calculate the evolution of EoS parameter and the deceleration parameter as well as the evolution of dark energy density in this generalized model. We show that the phantom line is crossed from quintessence regime to phantom regime which is in agreement with observation. The evolution of deceleration parameter indicates the transition from decelerated to accelerated expansion. Eventually, we show that the GHDE with HIR cutoff can interpret the pressureless dark matter era at the early time and dark energy dominated phase later. Nowadays we are strongly believed that our universe experiences an accelerated expansion. The complementary astronomical data gathered from SNe Ia [1], WMAP [2], SDSS [3] and X-ray [4] experiments confirm this cosmic acceleration. Within the framework of general relativity (GR), a dark energy component with negative pressure is introduced to explain this acceleration. Dark energy scenario have got a lot of attention in modern cosmology. In recent years a plenty theoretical models have been investigated to interpret the dynamical properties of dark energy. One can see [5, 6] for a review of dark energy models. The holographic dark energy (HDE) model is one of these models to explain a dark energy scenario. This model is constructed based on the holographic principle in quantum gravity [7]. In quantum gravity, a short distance ultra- violet (UV) cut-off is related to the long distance infra-red (IR) cut-off, due to the limit set by the formation of a black hole [7]. Based on the holographic principle, the energy of a system with size L does not exceed the energy of black hole with the same size, i.e., where m p is the reduced plank mass. By saturating the inequality (1), the energy density of HDE model in cosmology is identified by [8] where c 2 is a numerical constant of order unity and the factor 3 was introduced for convenience. An interesting feature of HDE is that it has a close connection with the space-time foam [9]. Another features of HDE model can be found in Section 3 of [10]. From the observational point of view, the HDE model has been constrained by various astronomical observation [13, 15, 16, 29]. The HDE model has been also investigated widely in the literature [14, 17]. For a recent review on different HDE models and their consistency check with observational data see [18]. From the theoretical point of view there are some motivations leading to the form of HDE model [19]. It should be noted that various HDE models have been investigated via assuming different IR cutoffs. The simple choice for IR cutoff is the Hubble radius, i.e., L = H -1 . In this case, the accelerated expansion of the universe can not be achieved and we get a wrong equation of state for this model [7]. However, in the presence of interaction between dark matter and dark energy, the HDE model can derive the cosmic acceleration and also, in this case, the cosmic coincidence problem can be solved [20]. Event horizon is the another choice for IR cutoff. Although, in this case the accelerated expansion can be achieved, but the generalized second law (GSL) does not satisfy in a universe enveloped by event horizon IR cutoff [21]. The other choice for IR cutoff is the particle horizon. In this case, the HDE model can not also obtain the late time accelerated expansion [7]. Here same as [22] we assume Hubble horizon as an IR cutoff for HDE model. In this case the GSL is also satisfied in the interacting accelerating universe [22]. Therefore the Hubble horizon is preferred from thermodynamical point of view. It is worthwhile to mention that the parameter c 2 in HDE model has an essential role in characterizing the properties of HDE model. For example, the HDE model can behave as a phantom or quintessence dark energy models at the future for the values of c 2 bigger or smaller than 1, respectively [23]. In all above references the HDE model was assumed to have a constant value for holographic parameter c 2 . However there are no strong evidences telling us that c 2 should be a constant parameter. In general the term c 2 can be assumed as a function of time.By slowly vary function with time, ˙ ( c 2 ) /c 2 is upper bounded by the Hubble expansion rate, i.e., In this case the time scale of the evolution of c 2 is shorter than H -1 and one can be satisfied to consider the time dependency of c 2 [24]. It has been also shown that the parameter c 2 can not be constant for all times during the evolution of the universe [24]. As was mentioned above, in the presence of interaction between dark matter and dark energy the HDE model with the Hubble horizon IR cutoff can solve the the coincidence problem and late time accelerated expansion. However, another alternative approach instead of interaction between dark components is that the holographic parameter c 2 varies slowly with time to solve the coincidence problem and explain late time acceleration [26].It has been shown that the interacting model of dark energy in which the coincidence problem is alleviated can be recast as a noninteracting model in which the holographic parameter c 2 evolves slowly with time [26]. In the line of above studies, we consider the HDE model with time-varying holographic parameter c 2 ( z ), namely: generalized holographic dark energy (GHDE, hereafter). We also consider the Hubble horizon as an IR cutoff (HIR, hereafter). We investigate the EoS parameter of the model as well as the deceleration parameter and discuss the density evolution of dark energy in this model. Let us start with flat Friedmann-Robertson-Walker (FRW) universe. In this case the first Friedmann equation is given by where ρ m and ρ d are, respectively, the energy densities of pressureless dark matter and dark energy and m p is the reduced planck mass. For Hubble radius IR cutoff, L = H -1 , the energy density of GHDE model from (2) can be given by where the holographic parameter is considered as a function of redshift. We now define the dimensionless energy density parameters as According to these definitions, the first Friedmann equation in spatially flat universe can be written as follows The conservation equations for pressureless dark matter and dark energy, respectively, are given by Taking the time derivative of Friedmann equation (4) and using (7, 8, 9), one can obtain Also it is obvious to see that differentiating Eq.(5) with respect to time yields Inserting (11) and (5) in conservation equation for dark energy (9) and using (10), we find the equation of state, w d , for GHDE model with Hubble length as where prime represents the derivative with respect to ln a . It is clear that the above relation reduce to w d = 0 for constant holographic parameter c . Hence, as expected, the HDE model in HIR gets to wrong equation of state for dark energy which can no describe the expanding universe. As was mentioned before, this problem for HDE model can be solved, if we consider the interaction between dark matter and dark energy (see [22] for more detail). In is worthwhile to mention that from (12) one can see that the GHDE model in which the holographic parameter c is considered as a function of redshift can get w d < 0 in HIR without assuming the interaction parameter. For this aim we use the Wetterich parametrization in which the holographic parameter c is considered in terms of redshift as follows [28] Putting c 1 = 0, the above holographic parameter reduces to c = c 0 indicating the constant value for HDE model. At the present time: z → 0, c ( z ) → c 0 , and at the early time: z →∞ , c ( z ) → 0. Hence the holographic parameter varies slowly from zero to c 0 during the history of the universe. Also to have positive energy density for dark energy, ρ d ≥ 0, we should take the condition: c 0 > 0 and c 1 ≥ 0. In numerical procedure, we chose matter density parameter Ω m = 1 -c 2 0 , and dark energy density parameter Ω d = c 2 0 , indicating the spatially flat universe. In Fig.(1), by solving (12) and using (13), we plot the evolution of EoS parameter, w d , in terms of redshift z for different illustrative values of c 0 and c 1 . Here we see that the EoS parameter, w d of GHDE with HIR can transit from quintessence regime ( w d > -1) to phantom regime ( w d < -1). The observations favor dark energy models which cross the phantom line w = -1 from up ( w d < -1) to down ( w d < -1) in near past [29]. Therefore this model is compatible with observations. Contrary the GHDE model HIR, the EoS parameter for interacting HDE model with HIR is constant during the history of the universe ( see Eq.(8) of [22]). However, Sheykhi showed that by applying some restrictions on the interaction parameter b and model parameter c , the EoS parameter of the HDE model with HIR can behave as a quintessence or a phantom type dark energy. But, neither the quintessence nor the phantom alone can fulfill the transition from w d > -1 to w d < -1. We now calculate the evolution of energy density of GHDE model described at HIR. From Eq. (6), the dark energy density of GHDE equals to square of varying holographic parameter, Ω d = c ( z ). Using (13), in Fig.(2) the evolution of dark energy density is plotted in terms of redshift for some illustrative values of model parameters c 0 and c 1 . At the early time ( z → ∞ ) the parameter Ω d → 0 which represents the dark matter dominated universe at the early time. Then the parameter Ω d increases to its present value c 0 which indicates the dark energy dominated epoch. The important note is that in standard HDE model under HIR cutoff, since the model parameter c is constant therefore the energy density Ω d = c 2 has no evolution during the history of the universe, i.e., Ω d is constant from early time to present time. Unlike the standard HDE model, in GHDE model with HIR the parameter Ω d increases from zero at the early time and tends to its present value at the present epoch. This behavior of GHDE model can interpret the decelerated expansion at the early time dark matter dominated universe and also the accelerated expansion at the dark energy dominated epoch. For completeness, we calculate the deceleration parameter q for GHDE model with HIR. The positive value of deceleration parameter ( q > 0) indicates the decelerated phase of expansion and the negative value ( q < 0) represents the accelerated phase of expansion of the universe. The parameter q is defined as Inserting (10) in (14) and using (12) as well as Ω d = c 2 , the parameter q for GHDE with HIR can be obtained as In the limiting case of standard HDE model with constant value of c , the parameter q reduces to q = 1 / 2 which describes the decelerated phase and can not represents the accelerated phase. However, including the interaction parameter and some restrictions on the interaction parameter b and model parameter c in standard HDE model with HIR can result the negative value for deceleration parameter q [22], but since the parameter c is constant therefore the transition from decelerated to accelerated expansion can not be achieved. In Fig.(3), by solving (15) and using (13), we plot the parameter q in GHDE with HIR model as a function of redshift parameter z . We see that at the early times the parameter q is 1 / 2 indicating the decelerated phase at dark matter-dominated universe. Then the parameter q reaches to negative values representing the accelerated phase at dark energy-dominated background. This property of GHDE with HIR is consistent with this observational fact that the universe has entered to the accelerated phase at past times [30]. In summery, we considered the generalized holographic dark energy model in spatially flat universe described in Hubble length an an IR cutoff (GHDE with HIR cutoff). The holographic parameter c generally is not constant and can be assumed as a function of cosmic redshift. The standard HDE model described by HIR cutoff gets to wrong equation of state for dark nergy [7]. The observations favor dark energy models which cross the phantom line w = -1 from quintessence regime ( w d < -1) to phantom regime ( w d < -1) in near past [29] and also the models in which the deceleration parameter transit from positive value to negative value [30]. Although, including the interaction between dark matter and dark energy in standard HDE model described by HIR cutoff can solve the coincidence problem and late time accelerated expansion [21, 22], but the EoS parameter of this model behaves as a quintessence or phantom model and can not transit from quintessence regime ( w d > -1) to phantom regime ( w d < -1)[22]. Also in the context of interacting HDE with HIR cutoff the deceleration parameter q is negative for all times in the history of the universe and therefore can not explain the transition from decelerated to accelerated expansion. However, in the case of GHDE with HIR cutoff, we obtained the EoS parameter as well as the deceleration parameter and evolution of dark energy density. Here we assumed the holographic parameter c 2 varies slowly with time instead of adding the interaction term. We showed that in this model the phantom line is crossed from up ( w d > -1) to down ( w d < -1) which is in agreement with observation [29]. Also it has been shown that in this model the evolution of deceleration parameter q indicates the decelerated phase at the early time ( q > 0) and accelerated phase at later ( q < 0). The evolution of energy density of dark energy represents the pressureless dark matter-dominated universe at the early time and dark energy-dominated phase at the present time.", "pages": [ 1, 2, 3, 4, 5, 6, 7 ] } ]

[ { "title": "ABSTRACT", "content": "We build a theoretical model to study the origin of the globular cluster metallicity bimodality in the hierarchical galaxy assembly scenario. The model is based on empirical relations such as the galaxy mass-metallicity relation [O / H] -Mstar as a function of redshift, and on the observed galaxy stellar mass function up to redshift z ∼ 4. We make use of the theoretical merger rates as a function of mass and redshift from the Millennium simulation to build galaxy merger trees. We derive a new galaxy [Fe / H] -Mstar relation as a function of redshift, and by assuming that globular clusters share the metallicity of their original parent galaxy at the time of their formation, we populate the merger tree with globular clusters. We perform a series of Monte-Carlo simulations of the galaxy hierarchical assembly, and study the properties of the final globular cluster population as a function of galaxy mass, assembly and star formation history, and under different assumptions for the evolution of the galaxy mass-metallicity relation. The main results and predictions of the model are the following. 1) The hierarchical clustering scenario naturally predicts a metallicity bimodality in the galaxy globular cluster population, where the metal-rich subpopulation is composed of globular clusters formed in the galaxy main progenitor around redshift z ∼ 2, and the metal-poor subpopulation is composed of clusters accreted from satellites, and formed at redshifts z ∼ 3 -4. 2) The model reproduces the observed relations by Peng et al. (2006) for the metallicities of the metal-rich and metal-poor globular cluster subpopulations as a function of galaxy mass; the positions of the metal-poor and metal-rich peaks depend exclusively on the evolution of the galaxy mass-metallicity relation and the [O / Fe], both of which can be constrained by this method. In particular, we find that the galaxy [O / Fe] evolves linearly with redshift from a value of ∼ 0 . 5 at redshift z ∼ 4 to a value of ∼ 0 . 1 at z = 0. 3) For a given galaxy mass, the relative strenght of the metal-rich and metal-poor peaks depends exclusively on the galaxy assembly and star formation history, where galaxies living in denser environments and/or early types galaxies show a larger fraction of metal-poor clusters, while galaxies with a sparse merger history and/or late type galaxies are dominated by metal-rich clusters. 4) The globular cluster metallicity bimodality disappears for galaxy masses around and below Mstar ∼ 10 9 M /circledot , and for redshifts z > 2. Subject headings: Galaxies: star clusters: general - Galaxies: formation - Galaxies: evolution - Galaxies: stellar content - Galaxies: structure - Galaxy: globular clusters: general", "pages": [ 1 ] }, { "title": "THE METALLICITY BIMODALITY OF GLOBULAR CLUSTER SYSTEMS: A TEST OF GALAXY ASSEMBLY AND OF THE EVOLUTION OF THE GALAXY MASS-METALLICITY RELATION", "content": "CHIARA TONINI Centre for Astrophysics and Supercomputing, Swinburne University of Technology, VIC 3122, Melbourne, Australia Draft version June 26, 2018", "pages": [ 1 ] }, { "title": "1. INTRODUCTION", "content": "Globular cluster (GC) systems in galaxies have become a useful tool to study the mechanisms of galaxy formation. Thanks to a rise in the level of details in observations, now we can gain insight into the colour, metallicity and abundance gradients of such systems for a large number of galaxies, and build statistically solid scaling relations between GC and galaxy properties. GCs are for the most part old objects, with ages estimated to be > 10 Gyr (Brodie et al. 2005, Strader et al. 2005, Peng et al. 2006). Therefore, not only they have survived any violent event in the assembly of their host galaxy, but they also provide a chemical record of the galaxies where they were formed (Pota et al. 2012). Coupled with the fact that they are very luminous, they make for excellent probes of the fossil records of galaxies and shed light on the mechanisms of galaxy assembly and star formation history. Of particular interest is the metallicity distribution of GCs in galaxies. Galaxies of all morphologies have a GC population with an average metallicity that correlates with the galaxy stellar mass or luminosity (as first shown by Brodie & Huchra 1991; Lotz et al. 2004, Peng et al. 2006). In addition, most galaxy GC systems exhibit a colour bimodality (Zepf & Ashman 1993, Ostrov et al. 1993, Whitmore et al. 1995, Elson & Santiago 1996, Peng et al. 2006, Spitler et al. 2006, Strader et al. 2006, Larsen et al. 2001). This is driven by a metallicity bimodality, with bluer GCs being more metal-poor and redder GCs being more metal-rich, while both populations are old ( > 10 Gyr) (Forbes et al. 2001, 1997a,b, 2011, Peng et al. 2006, Strader et al. 2005, 2006, Côté et al. 1998, Puzia et al. 2005, Pierce et al.2006, Brodie et al. 2005, Brodie & Strader 2006). Although there is some debate in the literature (see Yoon et al. 2006), this result has been confirmed spectroscopically (Brodie et al. 2005, 2012, Cohen et al. 2003, Strader et al. 2005, Alves-Brito et al. 2011, Usher et al. 2012). The well-defined metal-rich and metal-poor GC sequences separately follow two galaxy stellar mass - GC metallicity relations [Fe / H]GC -Mstar, of which the metal-rich one is stronger and tighter, while the metal-poor one is weaker and exhibits a larger scatter (Larsen et al. 2001, Strader et al. 2006, Peng et al. 2006, Côté et al. 1998). There is increasing evidence that these features of GC systems are universal, from giant ellipticals to dwarfs (Strader et al. 2006), although some galaxies show an even more complex situation, with multiple metallicity peaks (see for instance Peng et al. 2006, Blom et al. 2012). Intriguingly, the differences between blue/metal-poor and red/metal-rich GCs also extend to their dynamical properties, as shown in recent observations (Pota et al. 2012). The two subpopulations have different spatial distributions inside the host galaxies: the metal-rich GCs are more centrally concen- trared, with a radial distribution profile that follows closely the spheroidal stellar component of the galaxy, while the metal-poor GCs show a more extended distribution, and is likely physically associated with the stellar halo (Bassino et al. 2006, Goudfrooij et al. 2007, Peng et al. 2008, Forbes et al. 2012, Pota et al. 2012; the M87 data of Strader et al. 2011 represent the best example of the close spatial coupling of metal-rich GCs with galaxy starlight and the more extended distribution of metal-poor GCs). Correspondingly, the kinematics of the metal-rich subpopulation follows that of the main stellar component, including rotation (Strader et al. 2011), while the metal-poor subpopulation shows larger velocity dispersion and small or null net rotation. A scenario has been proposed where GCs are formed in gas-rich (major) merger events; at high redshift (z > 4 -5), early mergers of smaller hosts produce metal-poor GCs, while later mergers of more evolved galaxies in high density environments produce metal-rich GCs (Muratov & Gnedin 2010, Kravtsov & Gnedin 2005, Bekki et al. 2007, 2008). These models however encounter a number of problems; there is no clear prediction about any metallicity bimodality or galaxy mass-GC metallicity relations, and the resulting ages of the metal-rich GCs are too young (Muratov & Gnedin 2010), an ad hoc mechanism is needed to shut off blue/metal-poor GC formation (Bekki et al. 2008, Beasley et al. 2002), and an analysis of the observed GC abundance and metallicity gradients is not compatible this kind of formation mechanism (Arnold et al. 2011). Alternatively it has been proposed that, rather than originating from two main epochs or modes of GC formation, the GC chemo-dynamical bimodality can stem from the galaxy assembly history, without invoking mergers as the GC formation mechanism. In this scenario the metal-rich GC subpopulation is formed together with the bulk of the galaxy stellar component in an early violent dissipative phase, and during a later slower phase the metal-poor GC subpopulation is accreted, via minor mergers (Forbes et al. 2011, 1997a,b, Arnold et al. 2011, Masters & Ashman 2010), or via stripping of GCs from satellites (Côté et al. 1998, 2000). The main difference with the merger scenario is that GCs of different metallicities are formed in different galaxies, and then brought together by galaxy assembly, rather than being formed in the same galaxy at different stages of the galaxy evolution. In this work we call this the \"assembly scenario'.", "pages": [ 1, 2 ] }, { "title": "1.1. This work", "content": "In this work we want to put the \"assemby scenario' in the context of the hierarchical structure formation theory, and investigate whether the GC metallicity bimodality indeed originates from the hierarchical nature of galaxy assembly. In other words, is the GC metallicity bimodality a natural prediction of hierarchical clustering? To answer this question, we build a model to produce the assembly history of galaxies and their GC population, in a series of Monte-Carlo simulations. We base our model galaxy properties on observed scaling relations as a function of redshift, such as the galaxy mass-metallicity relation [O / H] -Mstar relation, and the galaxy stellar mass function. We assume that galaxies at z = 0 were formed through a combination of local (in-situ) star formation and accretion of satellite galaxies in a series of merger episodes spanning the lifetime of the galaxy; the merger rates are obtained from the Millennium simulation. We populate each galaxy in the merger tree with GCs, assuming that they share the metallicity of the main stellar component of their parent galaxy at the epoch of their formation. When a satellite is accreted, so is its GC population. We investigate under what conditions the final GC population shows the metallicity bimodality, and follows the observed metal-rich and metal-poor galaxy stellar mass - GC metallicity relations, as well as the observed galaxy mass - GC number abundance relation (Peng et al. 2006, 2008, Strader et al. 2006). The novelty of this analysis is that it provides constraints and predictions 1) on the galaxy [Fe / H] -Mstar relation as a function of redshift, 2) on the galaxy assembly and star formation history, and 3) on the evolution of the GC bimodality, and ultimately it presents a method to test the hierarchical galaxy formation. This paper is organised as follows. In Sections 2 and 3 we present the model: in Section 2 we describe the galaxy assembly and the globular cluster formation, and the Monte Carlo simulation; in Section 3 we present the derivation of the fiducial galaxy mass-metallicity relation. In Section 4 we present our results for the globular cluster metallicity distribution and its implications to constrain galaxy formation, and in Section 5 we discuss our findings. Section 6 is a summary of our conclusions.", "pages": [ 2 ] }, { "title": "2. THE MODEL: GALAXY ASSEMBLY AND GLOBULAR CLUSTER FORMATION", "content": "Consider a galaxy of stellar mass M0 at redshift z = 0. This object represents the final stage of a merger tree , i.e. a system of independent progenitor galaxies which were accreted and contributed to all the mass components (dark matter, gas, stars, GCs) that now characterise the galaxy. At any given time, we identify the main progenitor in the merger tree as the most massive galaxy that is present in the tree, while we (improperly) call satellites the rest of the objects. For any given galaxy at z = 0, we build a Monte Carlo simulation with N realisations of the merger tree, i.e. N different assembly histories. We performed numerical tests on N in a range N=[10 , 10 6 ], finding convergence of our results for N ≥ 100. The plots in this work are made with N = 10 3 . Galaxies evolve depending on the mass of the host dark matter halo and on the density of the surrounding environment. In the hierarchical clustering framework smaller objects virialise earlier (see for instance Frenk & White 2012), so they contain older, metal-poor stars. Their cycle of star formation and feedback is less efficient, and supernovae winds are more effective in expelling metals from the galaxy, factors that contribute to slow down the rise of metallicity in their stellar populations. At the same time, in more massive galaxies the deeper potential wells render supernovae winds less effective in expelling metals, and the enhanced ability to retain gas allows for sustained star formation and more stellar generations. As a consequence, at all redshifts a monothonic positive mass-metallicity correlation Mstar -[Fe / H] is in place for all galaxies in the merger tree. The derivation of this relation from the observed [O / H] -Mstar relation will be described in detail in the next Section. We assume the appearance of a globular cluster population in a galaxy is an event of a relatively short duration, and in general not associated with the quiescent star formation phase, but indicative of a particularly intense evolutionary phase. This assumption is sustained by a number of observations. First, the observed masses of GCs can reach up to M ∼ 10 6 M /circledot , requiring very intense bursts of star forma- on. Secondly GCs, which in general are well described by single stellar populations (SSP, i.e. coheval ensembles of stars that share the same metallicity), are for the most part old objects, with ages > 10 Gyr (as referenced in the Introduction). Fittingly, the observed GC ages put the epoch of their appearance squarely at the peak of the cosmic star formation history, determined to be at redshifts z ≥ 2 -4 (Hopkins & Beacom 2006, Bouwens et al. 2009). Third, the mean metallicity of the GC population is observed to be higher in more massive galaxies, with a Mstar -[Fe / H]GC parallel to that of the galaxy mass-metallicity relation (see for instance Larsen et al. 2001, Peng et al. 2006, Côté et al. 1998), in support of the idea that the GC population is closely related to the main stellar component and is similarly affected by halo mass and environment, i.e. more massive galaxies form their bulk of their stars later, from more enriched gas. We assume that the old globular cluster population (ages > 10 Gyr) were formed in galaxies at the peak of their star formation activity, likely L ∗ galaxies, at all redshifts z > 2. Note that, following this assumption, the older globular clusters formed at higher redshifts in smaller systems, and with a lower metallicity. This is in accord with observations, that estimate metal-poor GCs to be about 1 -2 Gyr older than metalrich GCs (Dotter et al. 2011, Puzia et al. 2005, Woodley et al. 2010), although the precision of the age measurement for extragalactic GCs is too low to confidently discriminate ages differences at this level (Strader et al. 2005). We assume as likely candidates for the formation sites of GCs either the massive star-forming clumps observed in high redshift (z > 2) galaxies (Shapiro et al. 2010), or the central regions of galaxies subject to episodes of violent dissipative collapse. In both cases, the clumpiness and turbulence of the gas plays a fundamental role in boosting the star formation and producing GCs, along with the galaxy main stellar component (Shapiro et al. 2010). The frequency of globular clusters TN is defined as the number of GCs per unit galaxy mass of 10 9 M /circledot , and at z ∼ 0 it is constrained by observations (Peng et al. 2008; see also Spitler et al. 2008, Rhode et al. 2007, Rhode 2012). In the galaxies in the merger tree, TN depends on the interplay of different factors, like the mean gas density (which depends on the depth of the galactic potential well), the metallicity, the feedback regime, and the competing 'regular' star formation that feeds the main stellar component. In lack of other observational constraints, we assume that the redshift z ∼ 0 observed relation TN -Mstar holds at all redshifts, so that the total number of local GCs that each galaxy produces is NGC = TN(Mstar) × Mstar. In addition, galaxies below Mstar = 10 9 M /circledot = Mmin do not form globular clusters, consistently with the observed TN -Mstar relation (Peng et al. 2008), which yields NGC < 10 for Mstar ∼ 10 9 M /circledot (see also Muratov & Gnedin, 2010). We also assume that, once formed, the local GCs stabilise themselves in dynamical equilibrium with the galaxy, and therefore remain kinematically coupled with the main stellar component. We follow the evolution of the merger tree from redshift zin, when the galaxy main progenitor forms its local globular clusters. We assume the epoch is zin ∼ 2. The main progenitor is likely to have a rich gas component and is near the peak of its star formation history; it has a stellar mass M1 and a mean total metallicity Z1, which follows the galaxy mass metallicity relation Mstar -[Fe / H] at z ∼ zin. In each MonteCarlo realisation, we assume that the metallicity of the locally formed GCs is peaked aroung Z1, with a gaussian distribution with σ = 0 . 2 (consistent with Côté et al. 1998, Bekki et al. 2008), which takes into account a non-instantaneous mixing of the metals, and the fact that the GC formation covers a short but finite time-span, in which the mean galaxy metallicity can vary. The number of local GCs in the main progenitor is NGC = TN(M1) × M1. The main progenitor is the most massive galaxy in the merger tree, and is the last one to have its globular cluster population in place. At this point in time, the satellites in the merger tree, which by definition have masses Mi < M1, have already formed their own GCs, and we put such epoch of formation around z ∼ 3 -4 (see also Shapiro et al. 2010). Each satellite metallicity follows the galaxy Mstar -[Fe / H] at that epoch, and in each satellite of mass Mi the GC metallicity is peaked around the current galaxy mean metallicity Zi, while the number of GCs is NGCi = TN(Mi) × Mi. We assume a gaussian distribution of the GC metallicities in each satellite, peaked around Zi with σ = 0 . 3, which takes into account the combined effect of the scatter in the GC metallicity in each satellite ( ∼ 0 . 2), plus an additional uncertainty ( ∼ 0 . 2) due to the scatter in the star formation histories of satellites (sensitive to environment for instance), which affect the satellite's metallicity and the exact epoch of GC formation. The main progenitor M1 evolves into the z = 0 galaxy M0 through two main channels: by accreting stellar mass in the form of satellites, and by forming stars locally. If we define as MSF the mass in stars that are formed inside the main progenitor at any time after the GC formation (including mergertriggered star formation), then the stellar mass accreted from satellites is Msat = M0 -M1 -MSF. Msat is the sum of the stellar mass present in all satellites at redshift zin, under the assumption that the satellite TN remains constant (i.e. satellites do not have a prolongued star formation history aftet the GC formation). The ratios M1 / M0 and MSF / M0 are free parameters in the model, and they constrain the assembly and star-formation history. For each galaxy characterised by (M0 , M1 , MSF / M1), we run a Monte-Carlo simulation of N realisations of the galaxy merger history, from zin to z = 0. In each run, we randomise the metallicity distribution of the main progenitor's GCs around Z1. We build the merger tree based on the observed stellar mass function (SMF) of Marchesini et al. (2009), and the theoretical merger rates obtained from the Millennium simulation (Springel et al. 2005, Fakhouri et al. 2010). In each realisation, we randomise both the mass of the accreted satellites and the redshift of accretion; after sampling a random redshift in the interval [z ∼ 4 -0], we interpolate the observed stellar mass function to that redshift, and we sample a random satellite mass from it with an acceptancerejection algorithm. This provides us with a series of merger candidates; each of them is weighted with the mean merger rate, which represents the probability for a merger to happen, given the mass of the main progenitor M, the ratio between the masses of the satellite and the main progenitor /epsilon1 , and the redshift: where the best-fit parameters are characterised as ( α,β,γ,η ) = (0 . 133 , -1 . 995 , 0 . 263 , 0 . 0993) and (A , /epsilon1 0) = (0 . 0104 , 9 . 72 × 10 -3 ) (Fakhouri et al. 2010). From Fig. (1) in Fakhouri et al. (2010) it is evident that the merger rate increases with increasing redshift and decreasing halo mass, and hugely favours small ratios /epsilon1 << 1 between the satellite's and the main progenitor's masses. At each timestep in our merger history, we add the weighted mass of each satellite to the stellar mass of the main progenitor M, which grows in time, and we continue until the total accreted mass is equal to Msat. Each merged satellite carries a population of NGCi = TN(Mi) × Mi globular clusters, with metallicity centered around Zi and randomised in each run. This completes one realisation in the Monte Carlo simulation and represent one of the N merger histories that we build for each galaxy. For each merger history, we obtain a total GC metallicity distribution that is given by the superposition of the contributions from the main progenitor and all the satellites. After N realisations, we produce a mean of the total GC metallicity distribution. In addition, we also explore a scenario where new globular clusters can be created in gas-rich merger events. In this case, we consider that at a redshift znew a merger event is characterised by a gas mass Mgas that is turned into stars and globular clusters, producing Nnew new globular clusters of random metallicity peaked around Znew (the metallicity of the gas), with a gaussian distribution with σ = 0 . 2. For a given final galaxy mass, the model uses 2 free parameters. The ratio M1 / M0 between the mass of the main progenitor at the epoch when it forms its local GCs and the final stellar mass of the galaxy is the assembly parameter ; the ratio MSF / M0 between the mass of the stars formed locally in the evolving main progenitor after the GC formation and the final stellar mass of the galaxy is the star formation history parameter .", "pages": [ 2, 3, 4 ] }, { "title": "3. THE MODEL: EVOLUTION OF THE GALAXY MASS-METALLICITY RELATION AND GLOBULAR CLUSTER METALLICITY", "content": "The main source of systematic uncertainty in the model comes from the redshift evolution of the galaxy massmetallicity relation Mstar -[Fe / H]. Although this is in principle constrained by observations, we feel that there currently is a lack of consensus on the evolution of Mstar -[Fe / H] at the level of precision required for this investigation. For this reason, to assign a metallicity to the galaxies in the merger tree and their globular cluster systems, we build a fiducial Mstar -[Fe / H] relation as a function of redshift, and we explore the consequences of varying this relation on the model. As a sanity check, to obtain the metallicity of the satellite GCs wealso make use of the total Mstar -[Fe / H]GC relation of Peng et al. (2006). The GC metallicity as a function of galaxy mass is provided in terms of [Fe / H], which is a proxy for the total metallicity (Fig. 14 of Peng et al. 2006, Shapiro et al. 2010). The galaxy metallicity on the other hand is often measured in terms of the quantity 12 + log(O / H); in particular we consider the relations provided by Maiolino et al. (2008) up to z ∼ 3 . 5 for the AMAZE(Assessing the Mass-Abundance redshift[-Z] Evolution) program. In the lower redshift bin, this relation is consistent with the one provided by Tremonti et al. (2004) for a sample of 53000 galaxies in the Sloan Digital Sky Survey. The Maiolino relations can be parameterised as follows: and for redshifts z = (0 . 07 , 0 . 7 , 2 . 2 , 3 . 5) the parame- ters are M0 = (11 . 18 , 11 . 57 , 12 . 38 , 12 . 76 / 12 . 87) and k0 = (9 . 04 , 9 . 04 , 8 . 99 , 8 . 79 / 8 . 9) (Maiolino et al. 2008). To obtain a [Fe / H] estimate from the quantity log(O / H) we need to establish the 12 + log(O / H) solar value, and the [O / Fe] or alternatively [ α/ Fe] values as a function of galaxy mass and redshift. These quantities are degenerate in producing the final [Fe / H]. The solar oxygen abundancy is determined to be 12 + log(O / H) = 8 . 66 (Erb et al. 2006), but other works put it at 12 + log(O / H) = 8 . 9 (as discussed for instance in Liu et al. 2008); unfortunately, the spread in the adopted value of the solar oxygen abundancy significantly increases the uncertainty in the calculation of the galaxy Mstar -[Fe / H] relation. In lack of direct spectroscopic observations, the determination of [O / Fe] as a function of [O / H], of galaxy mass and of redshift depends on models of both stellar and galaxy evolution, and there is currently no consensus on the conversion [O / H] into [Fe / H] (A. Pipino et al. in preparation, and private communication). A determination of [ α/ Fe] as a function of galaxy mass at z ∼ 0 is provided by Thomas et al. (2005), for a sample of 124 early-type galaxies. The scatter is substantial, and the mass range does not include galaxies below 10 10 M /circledot . The relation is parameterised as follows: If we use this prescription to convert the z ∼ 0 Maiolino and Tremonti relations, as [Fe / H] = [O / H] -[ α/ Fe], we obtain Mstar -[Fe / H] relations that are consistent (inside the scatter) with the one provided by Thomas et al. (2005) in the range Mstar = [10 10 -10 12 ]M /circledot , provided that [Fe / H] and [O / Fe] are reasonable proxies for [Z / H] and [ α/ Fe] respectively. The comparison yields values [ α/ Fe] ∼ 0 . 1 for Mstar ∼ 10 10 M /circledot and [ α/ Fe] ∼ 0 . 18 for Mstar ∼ 10 11 M /circledot . We use the Maiolino et al. (2008) relations to obtain the [Fe / H] -Mstar relations at higher redshifts, but we need to make an assumption about the redshift dependence of [ α/ Fe]. Such dependence is very uncertain and not all factors responsible for the variation of [ α/ Fe] are currently understood; for instance, a progressively top-heavy IMF at higher redshift would cause an excess of oxygen that would speed up the [ α/ Fe] evolution. For this reason, we choose to calibrate our [ α/ Fe] vs redshift relation empirically. Shapiro et al. (2010) use an estimated [ α/ Fe] ∼ 0 . 3 to obtain a relation at redshift z ∼ 2 from data of 12 + log(O / H) from Erb et al. (2006). A comparison with the z ∼ 2 relation we obtain from Maiolino et al. (2008) via the Thomas et al. (2005) prescription, shows us that we need to assume that [ α/ Fe] evolves by 0.2 dex in order for the two relations to match. We then extrapolate this evolution linearly with redshift, and obtain values [ α/ Fe] ∼ (0 . 1 , 0 . 2 , 0 . 3 , 0 . 5) for redshifts z ∼ (0 , 1 , 2 , 3 . 5). In Fig. (1) we plot our fiducial galaxy [Fe / H] -Mstar relations up to redshift z ∼ 3 . 5. On the same Figure, we plot the observed relations between the galaxy stellar mass and the GC metallicity [Fe / H]GC -Mstar obtained by Peng et al. (2006, their Fig. 14). These are shown as the straigh lines : black for the average GC metallicity in each galaxy, red for the metal-rich GCs, and blue for the metal-poor GCs. We provide a rough estimate of the error in the [Fe / H] -Mstar relations from the scatter in the Mstar -12 + log(O / H)relation ( ∼ 0 . 2 dex, Tremonti et al. 2004), the scatter in [ α/ Fe] at redshift 0 ( ∼ 0 . 1 dex, Thomas et al. 2005) and the uncertainty in the solar value of 12 + log(O / H) ( ∼ 0 . 2 dex); we obtain an uncertainty σ ∼ 0 . 3 dex on [Fe / H] for any given stellar mass. Note that this estimate does not take into account the errors in the galaxy mass estimates, nor the error increase in the metallicity measurements at higher redshifts, and nonetheless σ ∼ 0 . 3 is of the same order of the systematic errors induced by our choice of the [ α/ Fe] evolution. We will explore the consequences of varying these relations in the next Section. Once we have the galaxy [Fe / H] -Mstar relation in place as a function of redshift, we use it to assign a metallicity to all the globular clusters in the merger tree: the GCs formed in a galaxy of mass Mstar at a redshift z have a mean metallicity corresponding to the galaxy [Fe / H] at that redshift, according to the derived relations. In addition, in each galaxy the GC metallicity is assumed to have a gaussian distribution around the mean value, with σ ∼ 0 . 3 dex.", "pages": [ 4, 5 ] }, { "title": "4. RESULTS", "content": "Fig. (2) shows the globular cluster metallicity distribution, in a galaxy of mass M0 = 10 11 M /circledot at redshift z = 0, with parameters M1 / M0 = 0 . 3 and MSF / M0 = 0 (i.e. 70% of the final stellar mass come from accreted satellites, and there is no additional star formation in the main progenitor after the GC formation), averaged over N = 1000 Monte Carlo realisations of the galaxy formation history. The thick black line shows the total distribution, while the red line shows the distribution for the clusters that were formed locally in the main progenitor at z = zin ∼ 2, and the blue line shows the distrubution for the clusters formed in satellites at an epoch z ∼ 3 -4, and that merged with the main progenitor. The dashed lines + shaded areas show the values of the metal-rich and metal-poor GC metallicity [Fe / H] and their 1 σ uncertainties for a galaxy of mass M0 from the Peng et al. (2006) relations. The model galaxy shows a sharp bimodality in the globular cluster metallicity distribution. The metal-rich peak of the metallicity distibution is entirely dominated by local GCs, formed in the main progenitor at z ∼ 2. The metal-poor peak is entirely dominated by satellite GCs, accreted via the hierar- ical assembly. The positions of both peaks are consistent with the observed galaxy stellar mass - GC metallicity relations by Peng et al. (2006) for metal-rich and metal-poor globular clusters. The number of GCs in various realisations of this galaxy scatters around the value TN ∼ 6 interpolated from Peng et al. (2008) for a galaxy of mass Mstar ∼ 10 11 M /circledot , staying in the (rather large, ± 5) observed scatter limits. For a given set of history parameters M1 / M0 and MSF / M0, the final value of TN depends mainly on the assumption about the minimum stellar mass of a galaxy that can form globular clusters (here Mlimit = 10 9 M /circledot ). Note that a variation of a factor 10 in the mass limit, such that Mlimit = 10 8 M /circledot , yields a value TN ∼ 50, one order of magnitude off the Peng et al. (2008) relation. The positions of the peaks in the GC metallicity distribution are determined by the galaxy [Fe / H] -Mstar relation, given that we can constrain the ages of the GCs from observations, and under the assumption that the GC metallicity is connected to the instantaneous metallicity of the galaxy where they were formed. But is the bimodality just a result of these choices, or is it an intrinsic feature of our mass assembly scenario? Fig. (3) shows the GC metallicity distribution of a galaxy characterised by (M0 = 10 11 M /circledot , M1 / M0 = 0 . 5, MSF = 0) , this time under a very conservative assumption: the GC metallicity in all the objects in the merger tree (main progenitor and satellites) is just taken from the average [Fe / H]GC -Mstar relation of Peng et al. (2006) ( thick black line in Fig.1). Although both the local and the accreted GCs obey the same average relation, they are still separated in metallicity, the distribution of which shows two distinct peaks, albeit at the wrong values. The reason why the metallicity bimodality is still present is that the hierarchical mass assembly is governed by the halo merger rate, which greatly suppresses merger events of high mass ratios (larger than 1 : 10) (as evident in Fakhouri et al. 2010), so that it is highly improbable that a galaxy merges with objects of similar mass, and therefore similar metallicity. This feature alone is what drives the bimodality in the GC metallicity distribution. Therefore, a metallicity bimodality in the GC population is a direct prediction of the hierarchical clustering scenario . Notice also that the metal-poor peak in Fig. (3) is almost at the right value of [Fe / H], while the metal-rich peak is off by ∼ 0 . 5 dex towards the metal-poor side. The slope of the average [Fe / H]GC -Mstar relation in Fig.1 suggests that the number of metal-poor GCs is highly dominant in low-mass galaxies. This happens because these are intrinsically metal-poor galaxies; in addition note that, as their stellar mass is small, in their assembly history they are only able to accrete smaller objects that are devoid of globular clusters (given the existence of Mlimit), therefore their GC population is not bimodal, and their average metallicity peaks exactly where the metalpoor peak is located. On the other hand, the more massive a galaxy is, the richest its assembly history is, with a merger tree with enough mass range to sustain a varied secondary GC population, so its GC population is more likely to be bimodal. Therefore, a massive galaxy always has a secondary, metalpoor GC population, and the average GC metallicity deviates from both peaks. This point is addressed in the next Figure. Fig. (4) illustrates the difference in the GC total metallicity distribution of galaxies with final stellar masses M0 = 10 11 , 5 × 10 10 , 10 10 , 5 × 10 9 M /circledot , all characterised by the history parameters M1 / M0 = 0 . 3, MSF / M0 = 0. The bimodality in the GC metallicity distribution is evident at all masses in this mass range. As expected, the more massive a galaxy is, the richest is its GC population, in both the metal-rich and the metal-poor component. However, notice that, although the history parameters are the same in all cases, the relative con- tribution of the two peaks varies, with the metal-poor peak becoming less and less significant relative to the metal-rich peak for lower-mass galaxies, in accord with Peng et al. (2008) and Shapiro et al. (2010). In the lowest mass bin, the relative height of the metal-rich and metal-poor peaks is reversed; given that the mass limit for GC formation is 10 9 M /circledot , this galaxy is for the most part accreting satellites that don't contribute to the GC population, with the rare exception of major mergers (in this case, Msat > 10 9 M /circledot ). If we assume that the globular cluster formation is hampered in low-mass galaxies, i.e. that galaxies below a mass threshold cannot produce globular clusters, then the model predicts that the GC metallicity bimodality ceases to exist slightly above that mass threshold. In such galaxies, the GC population is unimodal and entirely composed of locally-formed GCs. On the other hand, the overall GC metallicity decreases following the galaxy mass, and as a result, the GC population in low mass galaxies is metal-poor, again in accord with Peng et al. (2008). The galaxy assembly history determines the fraction of the final mass that is accreted from the merger tree, and therefore the fraction of globular clusters that are formed outside the main progenitor and which we have shown to compose the metal-poor peak. Fig. (5) shows the relative height of the metal-rich and metal-poor peaks generated in different assembly histories, parameterised as M1 / M0. For a galaxy of final stellar mass M0 = 10 11 M /circledot , the panels from left to right show the GC metallicity distribution for M1 / M0=(0 . 2 , 0 . 3 , 0 . 5 , 0 . 6) respectively. In all cases, MSF = 0. As expected, a galaxy with a poor merger history (such as the case M1 / M0 = 0 . 6 for instance) shows a GC metallicity distribution dominated by the local metal-rich population. The model therefore predicts that the presence of a very strong metal-rich GC component is a sign of a sparse merger history. For a given galaxy mass, the richness of the merger tree depends on enviromnent; hence the model predicts that galaxies in low-density environments have, for a given mass, a GC population that is more metalrich dominated than galaxies of the same mass living in the centre of clusters. So far we have analysed the simplified case of galaxies with MSF = 0. However, for the majority of galaxies the star formation does not stop at z ∼ 2, and a significant part of the final stellar mass is formed at later times. In this case, a significant fraction of the galaxy stellar mass is not associated with formation or accretion of globular clusters. To account for this stellar component, we vary the value of the star formation history parameter MSF / M0. Fig. (6) shows the GC metallicity distribution for a galaxy of final mass M0 = 10 11 M /circledot , where the stellar mass is contributed by 1) the main progenitor at the epoch of GC formation in proportion of M1 / M0 = 0 . 3 (the local GC population is associated with this component), 2) stars formed locally in the galaxy after the epoch of GC formation, in quantity MSF / M0 = (0 . 2 , 0 . 3 , 0 . 5 , 0 . 6) ( panels from left to right ), and 3) stars accreted from satellites, in quantity M0 -M1 -MSF = Msat. Note that a higher value of MSF / M0 implies a smaller value of Msat, i.e. a poorer merger history. Fig. (6) shows that, as the galaxy growth becomes more dominated by local star formation and the contribution of the mass accreted by satellites is smaller, the GC population becomes more and more dominated by local globular clusters, even if most of the stellar component is not directly associated with the globular clusters themselves. The model predicts that, in galaxies with an active star formation history after the GC formation (i.e. at z < 2), the relative strenght of the metal-rich and metal-poor peaks of the GC metallicity distribution is biased towards the metal-rich GCs, for a given galaxy mass. If we consider the star formation history as associated with morphology, then the model predicts that, for a given galaxy mass M0 and total number of GCs, late-type galaxies have a GC metallicity distribution with a stronger metal-rich peak than early-type galaxies. Note that the results in Figs. (5, 6) show that, given the final mass of the galaxy M0, the final number of globular clusters in the galaxy depends on the value of the assembly history parameter M1 / M0 and the star formation history parameter MSF / M0. The large scatter in the value of the GC frequency per unit mass TN for a given galaxy mass seen in Peng et al. (2008) is likely to be due to the variety of histories for galaxies in each mass bin. The final value of TN decreases for a decreasing value of Msat. Note that the assembly and star formation history parameters have instead no effect on the position of the peaks, which are entirely determined by the evolution of the galaxy [Fe / H] -Mstar relation. The number of GCs, together with the relative abundance of the metal-rich and metal-poor components, can therefore be used to constrain the assembly and star formation history of the galaxy. As discussed in the Introduction, the scenario in which globular cluster are generally formed in gas-rich mergers cannot reproduce the properties and the scaling relations of the GC population. However, if globular clusters indeed form during violent bursts of star formation, it is physically possible that some of them indeed are formed in mergers at all redshifts, a fact that would explain the presence of intermediateage or young GCs in some galaxies (Kissler-Patig et al. 1998, Puzia et al. 2005, Strader et al. 2003, 2004b, Woodley et al. 2010, Brodie & Strader 2006 and references therein). Fig. (7) shows the effect of a gas-rich merger event, where new GCs are formed, on the GC metallicity distribution. In this example, in the same galaxy portrayed in Fig. (2), we introduce a gas-rich merger event that triggers the formation new GCs, in quantity ∼ 30% of the local GC population of the main progenitor, with intermediate metallicities peaked around [Fe / H] ∼ -0 . 8 with a gaussian of width σ = 0 . 2 dex ( green line ). This plot shows that the creation of new GCs in merger events introduces a stochastic variation of the GC metallicity distribution, that leads to the formation of tertiary peaks, in positions determined by the metallicity of the gas perturbed/carried by the merger. The number of newlyformed GCs depends on the available gas mass and the star formation rate in the merger-triggered bursts, as well as the efficiency of GC formation vs star formation. It is clear from this plot that, if we consider the formation of GC in gas-rich merger events, the GC metallicity distribution becomes more complex. The stochasticity of such events allows for any shape of the final metallicity distribution: a prolongued history of gas-rich mergers contributes to the dilution of the bimodality. Such mechanism can explain the number of 'exotic' GC metallicities distributions found by a number of authors, including Blom et al. (2012) and Peng et al. (2006), with a number of galaxies that either show one or more tertiary peaks, or a non-bimodal GC metallicity distribution (it should be noted that such a scenario needs to be confirmed with dynamical analysis; for instance, Blom et al. 2012 show data of a galaxy with an intermediate-metallicity GC subpopulation that rotates with the main body of the galaxy). Note that major gas-rich mergers are good candidates to provide very intense bursts of star formation, during which new globu- lar clusters can be formed. If a galaxy undergoes an assembly history devoid of any gas-rich mergers, it is hard to envisage another mechanism that is able to provide a strong enough perturbation of the gas in the galaxy to trigger very intense bursts of star formation (as per Shapiro et al. 2010), which can create a tertiary GC population. Therefore, we can consider tertiary peaks in the GC metallicity distribution as clear signatures of major gas-rich merger events in the past history of the galaxy. The total number of GCs and the relative height of the metal-rich and metal-poor peaks depend on the galaxy mass and the galaxy assembly and star formation history, while the positions of the peaks depend on the determination of the galaxy mass-metallicity relation as a function of redshift. To explore this point further, we study the GC metallicity distribution resulting from the [Fe / H] -Mstar relation obtained in Section 3, under different prescriptions. In particular, we describe two examples of variations of the recipes described in Section 3, that affect 1) the normalisation and 2) the evolution of the galaxy [Fe / H] -Mstar relation: 1) is a variation of the value for the solar 12 + log(O / H), and 2) is a variation of the prescription for the evolution of the galaxy [ α/ Fe] value. Fig. (8) shows the GC metallicity distribution of a galaxy characterised by (M0 = 10 11 M /circledot , M1 / M0 = 0 . 3, MSF = 0) ( right panels ) resulting from two different sets of galaxy [Fe / H] -Mstar relations as a function of redshift ( left panels ), which are derived following the model of Section 3. In the upper left panel , the galaxy [Fe / H] -Mstar relation is obtained from the evolution of 12 + log (O / H), but assuming that the galaxy [O / Fe] does not evolve with redshift, but has a con- stant value of ∼ 0 . 1 like in the local universe (with a solar oxygen value 12 + log(O / H)=8 . 66 as in our fiducial relation). In the upper right panel , the resulting galaxy GC metallicity distribution shows a somewhat diminished bimodality, and the positions of the peaks are definitely off the observed values obtained by Peng et al. (2006; dotted lines and shaded areas ). Both peaks are centered around too high metallicities, and the problem is worse for the metal-poor peak; if we assume that the galaxy [O / Fe] does not evolve with redshift but mantains the local value, then we are overestimating the galaxy metal content by a factor that is proportional with redshift, and the globular clusters formed in small galaxies at high redshifts are the ones that are affected the most. metallicity distribution does not change in shape (the relative height and position of the peaks is the same) but both peaks are shifted towards lower [Fe / H] values, because the oxygen content [O / H] yields a lower total [Fe / H] content. The model predicts that the positions of the metal-rich and metal-poor peaks of the GC metallicity distribution are exclusively dependent on the galaxy mass-metallicity relation as a function of redshift. In the case where globular clusters are indeed fossil records of the metallicity of their parent galaxy at the time when they formed, then through this model the GC metallicity distribution, and in particular the position of the metal-rich and metal-poor peaks, can be used to constraint the evolution of the galaxy [Fe / H] -Mstar relation and the evolution of the galaxy [O / Fe].", "pages": [ 5, 6, 7, 8, 9 ] }, { "title": "5. DISCUSSION", "content": "The results described in the previous Section show that the hierarchical galaxy assembly directly predicts a metallicity bimodality of the globular cluster populations in galaxies. Pre- The lower left panel of Fig. (8) portrays the galaxy [Fe / H] -Mstar relations as obtained in Section 3, with the same evolution of [O / Fe] used so far in this work, but with a different solar oxygen value, 12 + log(O / H) = 8 . 9 (Liu et al. 2008). In this case, the evolution of the [Fe / H] -Mstar relation is not affected, but its normalisation has changed. The resulting GC vious studies like Shapiro et al. (2010) on the contrary, argued that the hierarchical mass assembly would blur the correlations between galaxies and their GCs. The model presented here predicts globular clusters to have a bimodal metallicity distribution, with the metal-rich and the metal-poor peaks following the [Fe / H]GC -Mstar relations described in Peng et al. (2006), and determined by the evolution of the galaxy massmetallicity relation. The model also predicts that the relative strenght of the metal-rich and metal-poor peaks depends on the assembly and star formation history of the galaxy. Moreover, the model predicts that the metallicity bimodality disappears for masses below Mstar ∼ 10 9 M /circledot and for redshifts z > 2. The main mechanism at work in producing the GC bimodality is the existence of the galaxy mass-metallicity relation, coupled with the natural behaviour of hierarchical mass assembly to strongly favour minor mergers over major mergers. In this scenario, all globular clusters form with the same mechanism and share the metallicity of their parent galaxy; the hierarchical build-up of galaxies then assembles the GC populations so that the GCs of a satellite become part of the accreted, metal-poor GC component of a bigger galaxy of higher metallicity (in a generalisation of the scenario of Shapiro et al. 2010). Note that this mechanism still works in the case of satellites with bimodal GC distributions: both the metal-rich and the metal-poor subpopulations of a satellite are metal-poor compared to the central galaxy, and will contribute to its final metal-poor GC population. Under the assumption that GC formation is a rapid process, closely linked to the properties of the galaxy at the time of the event, the fact that GCs of different metallicities form in different galaxies, at slightly different times, is enough for the hierarchical galaxy formation scenario to naturally produce the correct GC metallicity distribution. The observational facts imply this very clearly. For instance, a comparison of the metal-poor globular cluster [Fe / H]GC -Mstar relation with the galaxy mass-metallicity relation ( blue straight line in Fig. (1)) shows that, for any given galaxy mass above Mstar > 10 9 M /circledot , the galaxy is always more metal-rich that its own metal-poor globular clusters (unless they were formed at z > 4). This implies that these globular clusters must have formed in smaller galaxies and they were then accreted. Note that the alternative scenario of a multi-phase GC formation in each single galaxy (like for instance Beasley et al. 2002), where metal-poor GCs are formed first and metal-rich GCs are formed later in mergers, necessarily implies some form of systematic segregations of metals and an ad hoc shut-down of the formation of the metal-poor component, in order to produce both the correct metallicity and the bimodality. In fact, if a galaxy experiences a prolongued phase of local GC formation, the bimodality is destroyed by the galaxy chemical self-enrichment. The results presented in this work are based on theoretical merger trees extracted from the Millennium simulation. Among the cosmological parameters, the value of σ 8 can affect these results, in the sense that a lower clustering power would produce a delayed mass accretion and sparser merger histories. While the bimodality would remain unchanged, the height of the metal-poor peak would be affected. However, the magnitude of the effect would be smaller than the scatter in the observed values of TN and the scatter between different Monte Carlo runs, and the results presented in this work would remain unaffected. In any case, this would be an interesting avenue of investigation. The bimodality in the GC metallicity distribution is solid against variations of our initial conditions, such as any assumption about the metallicity we assume for the galaxy or the GC themselves (Figs. 2, 3, 8). Thorugh hierarchical galaxy assembly, it is rather hard to produce a GC metallicity distribution that is not bimodal. Nonetheless, the position of the peaks is not recovered correctly if we assume that the local and accreted GC populations form at the same time, given the galaxy mass-metallicity relations described in this work. This is in accord with observations that determine the metal-poor GCs to be 1 -2 Gyr older than the metal-rich GCs. A secondorder sophistication of this model would be to introduce an analytic relation between the galaxy mass and the epoch GC formation, but in reality this epoch is likely to vary with environment and the fluctuations of the local star formation rate. A scatter in the epoch of formation would mirror a scatter in the galaxy metallicity, via the evolution of the galaxy massmetallicity relation; this effect has been mimicked in this work by the introduction of a scatter in the GC metallicity of the satellites. In this model, old globular clusters are considered fossil records of the galaxy where they were formed, and their finally metallicity distribution is exclusively a result of the hierarchical galaxy assembly. The factors that affect the final metallicity distributions are 1) the evolution of the galaxy mass-metallicity relation, which completely determines the position of the metal-rich and metal-poor peaks, and 2) the merger history (therefore the environmental density) and the star formation history, which completely determine the relative strenghts of the metal-poor and metal-rich peaks. These constraints are independent and orthogonal , with no degeneracy between them. Provided we know the GC ages, we can use this model to test and constrain the evolution of the galaxy [Fe / H] -Mstar relation and the evolution of [ α/ Fe], through the positions of the metal-rich and metal-poor peaks. In order to reproduce the Peng et al. (2006) [Fe / H]GC -Mstar relations, the model favours a value of [ α/ Fe] = 0 . 5 at redshift z = 4 and a linear evolution down to [ α/ Fe] = 0 . 1 at z = 0, a solar value 12 + log(O / H) ∼ 8 . 66, and the evolution of the galaxy [Fe / H] -Mstar relation plotted in Fig. (1). Note however that, even if we can rely on spectroscopy for the determination of the globular cluster metallicity with good precision, there are still significant uncertainties on the GC age determination. In this work we have used the observed estimates for the average GC ages, but if we consider their uncertainties, combined with the current size of the uncertaintiy on the galaxy massmetallicity relation, then there is a substantial degeneracy between redshift and galaxy metallicity in the determination of the positions of the metal-rich and metal-poor peaks, an issue that will be solved with higher precision observations from the next-generation instruments and surveys. Unrelated to the particular values of the metal-rich and metal-poor peaks, the model predicts the relative height of the peaks to give an insight into the assembly and star formation history of the galaxy: for a given galaxy mass, a dominant metal-rich peak indicates a quiet merger history and/or a significant growth of the galaxy through local star formation, as opposed to a mass growth driven by accretion of satellites. Therefore, the relative abundance of metal-poor and metalrich GCs is correlated with environmental density and morphology, with isolated late-type galaxies of given mass M0 being more metal-rich GC dominated than early types of the same mass living in dense environments. The current state of observations of GC populations in late-type galaxies does not yet provide definitive constraints, but this is certainly an area where more numerous and improved observations are called for. This is of particular interest when attempting to constrain the galaxy star formation history, in that this method is complementary to the SED-fitting technique, which is affected by a number of systematic uncertainties (Tonini et al. 2009, 2010, 2011, 2012, Henriques et al. 2011, Maraston et al. 2010, Pforr et al. 2012). The hierarchical galaxy mass assembly naturally leads to bimodality in the globular cluster metallicity distribution. A bimodal distribution is sign of a two-phase galaxy formation, with an intense dissipative phase that leads to the formation of the core of the galaxy and the local GCs, and a second phase of accretion of satellites and the acquisition of a secondary GC population. This regularity breaks down at very small masses, when galaxies are not massive enough to accrete GCs from their merger tree: their satellites are in fact so small, that they cannot form their own GCs. In this case the GC distribution is unimodal. On the other hand, the smaller a galaxy is, the least probable it is that it actually has an extended merger tree, so even when the few satellites contribute with globular clusters, the metal-poor peak is subdominant or negligible. Note that this behaviour is not in disagreement with the TN -Mstar relation presented in Peng et al. (2008). For masses below 10 9 M /circledot , even 1 globular cluster will yield a value TN ∼ 10. This shows that our limit mass for the production of globular clusters, Mlimit ∼ 10 9 , is a realistic prediction. In relation to the number of globular clusters per galaxy mass TN predicted by the model, the values we obtain are in the range of the scatter observed by Peng et al. (2008). We argue that in observations such scatter arises from the variety of assembly and star formation histories that generated the galaxies in the sample. In the model, the scatter between different Monte Carlo runs is of the same order of magnitude, and we argue that such a scatter generates from the scatter in the mass function of the merger tree, incorporating all the possible assembly paths to build up the final galaxy mass. The final value of TN depends on the assembly and history parameters M1 / M0 and MSF / M0, where richer merger histories strenghten the metal-poor peak and raise the total number of globular clusters. The model also predicts that the GC bimodality is a function of redshift. The more time a galaxy has to accrete satellites after the GC formation, the more rich its secondary population will be. Therefore, we expect the GC metallicity bimodality to disappear by redshift z ∼ 2 and above.", "pages": [ 9, 10, 11 ] }, { "title": "6. SUMMARY AND CONCLUSIONS", "content": "We have performed a series of Monte-Carlo simulations of the assembly history of galaxies and studied the metallicity distribution of their globular cluster systems. To do so, we built a model for the assembly of the globular cluster population following the hierarchical galaxy assembly, based on empirical scaling relations such as the galaxy mass-metallicity relation [O / H] -Mstar, and on the observed galaxy stellar mass function up to redshift z ∼ 4. We also made use of the theoretical merger rates as a function of mass and redshift from the Millennium simulation, to build merger trees for a set of final galaxies. By determining the galaxy [Fe / H] -Mstar relation for all galaxies in each merger tree, and by assuming that globular clusters share the metallicity of their original parent galaxy at the time of their formation, we populated the merger tree with globular clusters. The hierarchical assembly of the final galaxy creates a globular cluster population composed by the local GCs formed in the main progenitor and those accreted from the merger tree. We conclude that:", "pages": [ 11 ] }, { "title": "ACKNOWLEDGEMENTS", "content": "CT would like to thank the anonymous Referee for her/his comments and suggestions, which improved the clarity of this work. CT would also like to thank Jeremy Mould, Duncan Forbes, Eric Peng, Lee Spitler, Marie Martig, Vincenzo Pota, Chris Usher and Darren Croton for the interesting discussions and for their useful comments.", "pages": [ 11 ] }, { "title": "REFERENCES", "content": "Alves-Brito, A., Hau, G. K. T., Forbes, D. A., Spitler, L. R., Strader, J., Brodie, J. P. & Rhode, K. L., 2011, MNRAS, 417, 1823 Arnold, J. A., Romanowsky, A. J., Brodie, J. P., Chomiuk, L., Spitler, L. R., Strader, J., Benson, A. J. & Forbes, D. A., 2011, Apj, 736, 26 Bassino L. P., Richtler T., Dirsch B., 2006, MNRAS, 367, 156 Beasley, M. A., Baugh, C. M., Forbes, D. A., Sharples, R. M. & Frenk, C. S., 2002, MNRAS, 333, 383 Bekki, K., Yahagi, H., & Forbes, D. A., 2007, MNRAS, 377, 215 Bekki, K., Yahagi, H., Nagashima, M. & Forbes, D. A., 2008, MNRAS, 387, 1131 Blom, C., Spitler, L. R. & Forbes, D. A., 2012, MNRAS, 420, 37 Bouwens, R. J., Illingworth, G. D., Franx, M., Chary, R.-R., Meurer, G. R., Conselice, C. J., Ford, H., Giavalisco, M. & van Dokkum, P., 2009, ApJ, 705, 936 Brodie, J. P. & Huchra, J. P., 1991, ApJ, 379, 157 Peng, E. W., Jordan, A., Cote, P., Blakeslee, J. P., Ferrarese, L., Mei, S., West, Brodie, J. P., Strader, J., Denicolo, G., Beasley, M. A., Cenarro, A. J., Larsen, S. S., Kuntschner, H. & Forbes, D. A., 2005, AJ, 129, 2643 Brodie, J. P. & Strader, J., 2006, ARA&A, 44, 193B Brodie, J. P., Usher, C., Conroy, C., Strader, J., Arnold, J. A., Forbes, D. A. &Romanowsky, A. J., 2012, arXiv:1209.5390 Bruzual, G. & Charlot, S., 2003, MNRAS, 344, 1000B Cohen, J. G., Blakeslee, J. P., & Côté, P. 2003, ApJ, 592, 866 Côté, P., Marzke, R. O. & West, M. J., 1998, ApJ, 501, 554 Côté, P., Marzke, R. O., West, M. J. & Minniti, D., 2000, ApJ, 533, 869 Dotter A., Sarajedini, A. & Anderson, J., 2011, ApJ, 738, 74 Elson, R. A. W. & Santiago, B. X., 1996, MNRAS, 280, 971 Erb, D. K., Shapley, A. E., Pettini, M., Steidel, C. C., Reddy, N. A. & Adelberger, K. L., 2006, ApJ, 644, 813 Fakhouri, O., Ma, C.-P. & Boylan-Kolchin, M., 2010, MNRAS, 406, 2267 Forbes D. A., Brodie, J. P. & Huchra, J., 1997, AJ, 113, 887F (a) Forbes D. A., Ponman, T. & O'Sullivan, E., 2012, MNRAS, 425, 66 Frenk, C. S. & White, S. D. M., 2012, AnP, 524, 507F G. & Tonini, C., 2011, MNRAS, 415, 3571H Hopkins, A. M. & Beacom, J. F., 2006, ApJ, 651, 142 Kissler-Patig, M., Forbes, D. A. & Minniti, D., 1998, MNRAS, 298, 1123 Kravtsov, A. V. & Gnedin, O. Y., 2005, ApJ, 623, 650 Lotz, J. M., Miller, B. W. & Ferguson, H. C., 2004, ApJ, 613, 262 LabbÃl', Ivo & Wuyts, S., 2009, ApJ, 701, 1765 Masters, C. E. & Ashman, K. M., 20 M. J., Merritt, D., Milosavljevic, M. & Torny, J. L., 2006, ApJ, 639, 95", "pages": [ 11, 12 ] } ]

[ { "title": "ABSTRACT", "content": "We discover that the mass of dark matter particles m DM is imprinted in phase-correlations of the cosmic density field more significantly than in the 2-point correlation. In particular, phase-correlations trace m DM out to scales about five times larger than the 2-point correlation. This result relies on a new estimator /lscript ( r ) of pure phase-information in Fourier space, which can be interpreted as a parameterfree and scale-invariant tracer of filament-like structure. Based on simulated density fields we show how m DM can, in principle, be measured using /lscript ( r ), given a suitably reconstructed density field.", "pages": [ 1 ] }, { "title": "A ROBUST MEASURE OF COSMIC STRUCTURE BEYOND THE POWER-SPECTRUM: COSMIC FILAMENTS AND THE TEMPERATURE OF DARK MATTER", "content": "D. Obreschkow 1 , C. Power 1 , M. Bruderer 2 , and C. Bonvin 3 , 4 1 International Centre for Radio Astronomy Research (ICRAR), M468, University of Western Australia, 35 Stirling Hwy, Crawley, WA 6009, Australia 2 Institut fur Theoretische Physik, Albert-Einstein Allee 11, Universitat Ulm, 89069 Ulm, Germany 3 Kavli Institute for Cosmology Cambridge and Institute of Astronomy, Madingley Road, Cambridge CB3 OHA, UK and 4 DAMTP, Centre for Mathematical Sciences, Wilberforce Road, Cambridge CB3 OWA, UK (Dated: February 27, 2018) ApJ, accepted 21/11/2012", "pages": [ 1 ] }, { "title": "1. INTRODUCTION", "content": "The model of a flat and nearly scale-free universe dominated by dark energy and cold dark matter (ΛCDM), passed stringent empirical tests of the new millennium. The six free model parameters were found simultaneously consistent (Komatsu et al. 2011) with the temperature fluctuations in the cosmic microwave background (CMB, Larson et al. 2011) measured by the Wilkinson Microwave Anisotropy Probe (WMAP, Bennett et al. 2003a,b), the baryon acoustic oscillations (BAOs) in the late-time large-scale structure (LSS) derived from galaxy redshift surveys (SDSS: Percival et al. 2010; 2dFGRS: Percival et al. 2007; WiggleZ: Blake et al. 2011), and distance measurements based on type Ia supernovae (SNe, Hicken et al. 2009; Kessler et al. 2009). This phenomenal success of the ΛCDM cosmology contrasts with our ignorance regarding the nature of its dark constituents. Crucial properties of these constituents, such as the particle mass of dark matter, are covertly imprinted in the sub-cluster structure of the LSS (Smith & Markovic 2011; Schneider et al. 2012). Yet, this information is not readily accessible to measurements. For one thing, the actual LSS is not directly observable due to the invisibility of dark matter, redshiftspace distortions (Kaiser 1987), and general relativistic effects (Bonvin & Durrer 2011; Challinor & Lewis 2011; Yoo et al. 2009). For another, the information in the LSS is masked by a random component originating from quantum state reduction in the primordial universe. To filter out the random component, the observed LSS is usually subjected to statistical measures that are independent of cosmic randomness up to a volumedependent shot noise, known as cosmic variance. In the case of a statistically homogeneous and isotropic universe, the infinite family of isotropic n -point correlation functions ( n -PCFs) removes all randomness, but preserves all information (Fry 1985). However, so far no finite set of statistical measures is known, which exclusively and exhaustively describes the information imprinted in the cosmic density field. Most cur- rent studies bypass this issue by considering only the isotropic 2-PCF ξ 2 ( r ) or, equivalently, the power spectrum p ( k ), where r and k denote the separation scale and wave-number. In doing so, important information is lost; e.g., subtle structural features, such as cosmic filaments, become indistinguishable from spherical features. Some studies improve on those drawbacks by invoking higher-order correlations (Fry & Peebles 1978; Suto & Matsubara 1994; Takada & Jain 2003) and alternative statistical measures, such as the fractal correlation dimension (Scrimgeour et al. 2012), void distribution functions (White 1979), and various shapefinders (Babul & Starkman 1992; Luo & Vishniac 1995; Sahni et al. 1998; Arag'on-Calvo et al. 2007; Bond et al. 2010; Sousbie 2011). However, the benefit of these measures in addition to ξ 2 ( r ) is often limited, since they are heavily correlated to ξ 2 ( r ) in terms of ensembles. To truly avoid this issue one must refer to statistical estimators that only measure information not yet contained in ξ 2 ( r ) (e.g. Watts et al. 2003). The aim of this work is to introduce a new statistical estimator of the cosmic density field, which is based solely on the phases of the Fourier spectrum of the density field, but not on its amplitudes, since the latter are already fully captured via ξ 2 ( r ). This requirement combined with the requirement of statistical homogeneity and isotropy naturally leads to a measure, which we will call the line-correlation function /lscript ( r ). We show that, in a limited sense, this function can be interpreted as a proxy for cosmic 'filamentary' on length scales 2 r . Unlike the 2-PCF, phase-correlations are independent of linear growth of LSS. Measures of phase-correlations such as /lscript ( r ) are therefore particularly sensitive to the gravitational non-linear growth of the dark matter dominated cosmic web (Watts et al. 2003). In this work, we therefore chose to explore the dependence of /lscript ( r ) on different 'temperatures' of dark matter. Based on a series of large numerical N -body simulations, both with CDM and warm dark matter (WDM), we find that /lscript ( r ) depends sensitively on the mass of dark matter particles m DM . Our results suggest that /lscript ( r ) constrains m DM an order of magnitude better than ξ 2 ( r ). Moreover, /lscript ( r ) depends on m DM on scales about five times larger than ξ 2 ( r ) - a pivotal result, since the complex baryon physics masking the footprint of dark matter properties becomes less important with increasing scales. The article proceeds as follows. Section 2 summarizes established concepts regarding cosmic structure and clarifies the meaning of n -PCFs and poly-spectra. Section 3 motivates and formally defines the line-correlation function and presents geometrical interpretations. A range of cosmological applications, namely the measurement of m DM , is then considered in Section 4, based on a series of N -body dark matter simulations. Section 5 summarizes the key results and discusses their potential application to observed data.", "pages": [ 1, 2 ] }, { "title": "2. COSMIC STRUCTURE AND CORRELATION FUNCTIONS", "content": "This section reviews the statistical nature of cosmic LSS and summarizes the concepts of correlation functions and spectral analysis (details in section 3 of Bernardeau et al. 2002).", "pages": [ 2 ] }, { "title": "2.1. Cosmic density field and its statistical symmetry", "content": "We consider a flat Euclidean universe, consistent with BAO measurements assuming a cosmological constant (Percival et al. 2010; Blake et al. 2011), with a mass density field where m denotes the mass, V the comoving volume, and r ∈ R D the position in D spatial dimensions; for illustrative purposes we consider both D = 2 and D = 3. To simplify the notation, we omit the implicit timedependance of ρ in eq. (1). According to the Big Bang theory, ρ ( r ) evolved from a dense, maximally symmetric state under the action of physical laws that are spatially homogeneous and isotropic. Complex substructure then grew from seeds of reduced symmetry, known as quantum-fluctuations, caused by quantum state reduction in the inflating primordial universe (Leon et al. 2011). In the current view, quantum state reduction decreases the spatial symmetry, but maintains homogeneity and isotropy in the sense that the outcome probabilities of the process conserve the symmetry of the evolution operator (McWeeny 2002; Obreschkow et al. 2007). This weaker, probabilistic symmetry is referred to as statistical homogeneity and isotropy. The resulting conjecture that ρ ( r ) is statistically homogeneous and isotropic is known as the 'cosmological principle' and is supported by modern redshift surveys (e.g. SDSS: Gong 2010; Sylos Labini & Baryshev 2010; WiggleZ: Scrimgeour et al. 2012). We express the statistical homogeneity and isotropy explicitly by writing ρ ( r ) as where t i ∈ R D are random translation vectors and R i ∈ O ( D ) are rotation matrices of the orthogonal group (det( R i ) = ± 1). The generating functions g i ( r ) ≥ 0 are defined such that all cosmological information is encoded in g i ( r ), while all quantum randomness is absorbed in the variables t i and R i . By definition, eq. (2) thus separates non-random variables { g i ( r ) } from random ones { t i , R i } . This separation is useful when constructing statistical measures that isolate the information. Given the compelling observational evidence for the large-scale homogeneity, thus non-fractal structure, of the universe (e.g. Scrimgeour et al. 2012) we can define a universal average density ¯ ρ and the density perturbation field This field then satisfies δ V → 0 as V → ∞ , with δ V ≡ V -1 d V δ ( r ) being the average density perturbation. ∫ Let us comment on a few points. First, a corollary of the spatial homogeneity is that the total mass ∫ V d V ρ ( r ) = V ρ V is proportional to V as V → ∞ . In other words, the fractal dimension converges to 3 in this limit (Scrimgeour et al. 2012). Second, it is a common misconception that statistical homogeneity and isotropy only concern large-scale ( /greaterorsimilar 100Mpc) averages. For example, a universe with all mass concentrated around the nodes of a Cartesian grid with 1 Mpc spacings would satisfy ρ V → ¯ ρ as V → ∞ , but violate statistical homogeneity as stated in eq. (2). Third, although our universe seems statistically homogeneous and isotropic, observational proxies of ρ ( r ), such as redshift-surveys, can violate this statistical symmetry. A famous example is the fingers-of-God effect (Kaiser 1987) originating from a Doppler-shift contamination in the observed redshifts - a potentially useful feature for observational cosmology as emphasized by Raccanelli et al. (2012).", "pages": [ 2 ] }, { "title": "2.2. n-point correlation functions", "content": "Given a density field ρ ( r ) [eq. (2)] that mixes information with random translations (homogeneity) and rotations (isotropy), how can we extract the information? In a first step, statistical homogeneity is exploited by averaging over all translations. This is the key idea behind the correlation functions, which are spatial averages of product functions. The n -point density correlation function ( n -PCF) is defined as where r n ≡ 0. In particular, the 2-PCF reads In a second step, statistical isotropy is exploited by averaging over all rotations R ∈ O ( D ). This leads to the 'isotropic' n -PCFs, where S{ r 1 , ..., r n -1 } denotes a unique representation of the shape defined by the n -point set { 0 , r 1 , ..., r n -1 } regardless of its orientation. In the case of n = 2, this shape reduces to the distance r ≡ | r 1 | . The resulting isotropic 2-PCF is by far the most common statistical measure of LSS, as justified in Section 2.4. /negationslash /negationslash We emphasize that Ξ n and ξ n here refer to single realizations of the density field and not ensembles of fields, i.e. Ξ n = 〈 Ξ n 〉 and ξ n = 〈 ξ n 〉 , where 〈 〉 denotes the ensemble average. Furthermore, these n -PCFs refer to the perturbation field δ ( r ) rather than ρ ( r ). Our 2-PCF and 3-PCF are therefore identical to those called the 'reduced' 2-PCF and 3-PCF by Peacock (1999) and the 'connected parts' of 2-PCF and 3-PCF by Bernardeau et al. (2002). The family of the isotropic n -PCFs ξ n is statistically complete (Fry 1985) in that it contains all the information contained in the density field ρ ( r ). This information is contaminated by the cosmic variance 〈 ( ξ n -〈 ξ n 〉 ) 2 〉 , which can be calculated for any function ξ n (Szapudi 2001) and vanishes as V →∞ .", "pages": [ 2, 3 ] }, { "title": "2.3. Fourier space representations", "content": "Since the correlation functions Ξ n are convolution integrals over t ∈ R D , they can be computed more efficiently in Fourier space. Using the standard Fourier transform FT : δ ( r ) ↦→ ˆ δ ( k ) in cosmology and its inverse (IFT), and expressing all δ ( r ) in eq. (4) as IFT[ ˆ δ ( k )], we find (details in Appendix A) where k ∈ R D is the wavevector and the complex-valued functions are called 'poly-spectra'. Thus, for any n ≥ 2, the correlation function Ξ n is equal to the IFT (generalized to n -1 variables) of the poly-spectrum P n . The most common poly-spectra are the (real) power spectrum P ( k ) ≡ P 2 ( k ) and the bi-spectrum B ( k , q ) ≡ P 3 ( k , q ), /negationslash Like in the case of the n -PCFs, the spectra P ( k ) = 〈 P ( k ) 〉 and B ( k , q ) = 〈 B ( k , q ) 〉 here refer to a single density field, not to ensembles thereof. These spectra are the FTs of the 2-PCF and 3-PCF, respectively, /negationslash The relations between Ξ n and P n imply similar relations between the isotropic correlation functions ξ n and rotationally symmetrized poly-spectra, called 'isotropic' poly-spectra, Fig. 1 depicts the hierarchy from the density field down to isotropic correlation functions and their equivalents in Fourier space. Because of their importance explicit expressions for ξ 2 and ξ 3 are given in Appendix B.", "pages": [ 3 ] }, { "title": "2.4. Cosmological importance of ξ 2 ( r )", "content": "In the current view (Bennett et al. 2011; Komatsu et al. 2011), the primordial density fluctuations imprinted in the CMB are consistent with a Gaussian random field (GRF). A GRF results from a random (Poissonian) superposition of infinitely many plane or spherical waves with vanishing phasecorrelation. The evidence for the Gaussianity of the CMB thus supports the physical interpretation that the primordial density fluctuations derive from de-correlated quantum fluctuations producing a bath of spherical sound waves. A key property of a GRF is that its information, i.e., its non-randomness, is entirely contained in the isotropic 2-PCF ξ 2 ( r ). Thus, as far as current measurements can tell, all the cosmological information of the CMB is contained in ξ 2 ( r ), or, equivalently, in its isotropic power spectrum p ( k ).", "pages": [ 3 ] }, { "title": "3. PHASE-INFORMATION AND LINE-CORRELATIONS", "content": "This section introduces the 'line-correlation' function, a new estimator of phase-information of cosmic structure.", "pages": [ 3 ] }, { "title": "3.1. What is phase-information?", "content": "For a statistically isotropic density field δ ( r ), ξ 2 ( r ) in eq. (7) contains the same information as the full 2-PCF Ξ 2 ( r ). The reversible mapping between Ξ 2 ( r ) in eq. (12) and the amplitudes | ˆ δ ( k ) | then implies that ξ 2 ( r ) measures all the cosmological information contained in the amplitude field | ˆ δ ( k ) | . All additional information, not captured by ξ 2 ( r ), must therefore reside in the phasesfactors The information contained in these phase-factors is called phase-information and it can take the form of phase-phase correlations 〈 ˆ /epsilon1 ( k )ˆ /epsilon1 ( q ) 〉 and/or amplitudephase correlations 〈| ˆ δ ( k ) | ˆ /epsilon1 ( q ) 〉 . Unlike the primordial universe, the local universe does indeed contain a significant amount of phase-information, as evidenced by its clearly non-vanishing isotropic 3-PCF (e.g. 2dFGRS: Croton et al. 2004; Gazta˜naga et al. 2005; SDSS: Nichol et al. 2006; Mar'ın 2011; numerical simulations: Barriga & Gazta˜naga 2002). Therefore, phaseinformation measurements of the late-time LSS promise to be a pivotal cosmological probe (Watts et al. 2003).", "pages": [ 3, 4 ] }, { "title": "3.2. Concept of line-correlations", "content": "/negationslash Fig. 2 shows an example of a density field δ ( r ) with the corresponding fields ˆ δ ( k ) = FT[ δ ( r )], ˆ /epsilon1 ( k ) = ˆ δ ( k ) / | ˆ δ ( k ) | , and /epsilon1 ( r ) = IFT[ˆ /epsilon1 ( k )]. In this example, δ ( r ) is a statistically homogeneous and isotropic 2D density field constructed on the basis of eq. (2). The generating functions g ( r ) are identical elongated 2D Gaussian distributions. The mapping ˆ δ ( k ) ↦→ ˆ /epsilon1 ( k ) removes all information stored in the amplitudes | ˆ δ ( k ) | and therefore all 2-point correlations. Thus, the filamentary structure of /epsilon1 ( r ) shown in Fig. 2 exclusively represents phase-information of δ ( r ). A natural way to measure phase-information is to use higher-order correlations ξ n ( n ≥ 3). However, this choice may be problematic since the estimators ξ n ( n ≥ 3) and ξ 2 are strongly correlated in the sense that they correlate, i.e. cov( ξ 2 , ξ n ) = 0, even if there are no amplitude-phase correlations, i.e. 〈| ˆ δ ( k ) | ˆ /epsilon1 ( q ) 〉 = 0, across the considered ensemble of density fields. This strong correlation between ξ n ( n ≥ 3) and ξ 2 is due to the fact that the poly-spectra p n depend directly on the amplitudes | ˆ δ ( k ) | in addition to the phases ˆ /epsilon1 ( k ). Thus, constraints on cosmological parameters obtained from ξ n ( n ≥ 3) are generally strongly correlated to constraints obtained from ξ 2 , which can lead to serious statistical difficulties. Alternative estimators, which are fully defined by phase-phase correlations 〈 ˆ /epsilon1 ( k )ˆ /epsilon1 ( q ) 〉 without explicit dependence on the amplitudes | ˆ δ ( k ) | , are here called measures of pure phase-information. Such estimators, must be defined exclusively upon the phase-factors ˆ /epsilon1 ( k ), or, equivalently, on the field /epsilon1 ( r ) ≡ IFT[ˆ /epsilon1 ( k )]. /negationslash By construction, /epsilon1 ( r ) exhibits vanishing 2-point correlations for all r = 0 an thus the simplest meaningful measure of pure phase-information of δ ( r ) must be based on 3-point correlations of /epsilon1 ( r ). As illustrated in Fig. 2, the removal of 2-point correlation tends to collapse elongated structures to line segments. Therefore, the most natural 3-PCF to consider is that of three points on a straight line. For simplicity we chose these points to be equidistant. Using the explicit expressions for ξ 3 ( r ) ≡ Ξ 3 ( r , -r ) | r | = r given in eq. (B6) and substitut- δ ( k ) for ˆ /epsilon1 ( k ), we then obtain the modified 3-PCF with the kernel According to eq. (8), ξ ∗ 3 ( r ) is identical to Unfortunately, ξ ∗ 3 ( r ) is an ill-defined function. (Mathematically, it is a distribution.) This can be seen when expressing /epsilon1 in eq. (18) as a discrete IFT. To do so, we adopt the standard discretization scheme explained in Appendix D and illustrated in Fig. 11. In this scheme the density field is represented on a squared ( D = 2) or cubic ( D = 3) box with side-length L , N D grid cells spaced by ∆ r = L/N , and periodic boundary conditions. Its Fourier space becomes a regular box of side-length 2 π/ ∆ r and N D cells spaced by ∆ k = 2 π/L . The IFT then reads /epsilon1 ( r ) = ∑ k e i k · r ˆ /epsilon1 ( k ). A particular discretization is fully defined by L and ∆ r , and any physical function f ( r ) should converge both as ∆ r → 0 and L →∞ . Yet, both requirements are violated by ξ ∗ 3 ( r ). Second, as L → ∞ , we face a similar challenge: the density of modes per volume of Fourier space increases as L D due to the mode spacing ∆ k = 2 π/L . If L is longer than the longest physical correlation lengths, then increasing L corresponds to adding sub-modes with uncorrelated phases. Hence, we are again in the random phase case, where | /epsilon1 ( r ) | = | ∑ k e i k · r ˆ /epsilon1 ( k ) | ∝ L D/ 2 ; ξ ∗ 3 ( r ) then diverges as L 3 D/ 2 . To avoid this, while keeping ξ ∗ 3 ( r ) dimensionless, we must divide ξ ∗ 3 ( r ) by ( L/r ) 3 D/ 2 . First, as ∆ r → 0, the number of independent modes with wavelengths shorter than a fixed r diverges as ∆ r -D , since max {| k |} = π/ ∆ r . The ∼ ∆ r -D modes with wavelengths shorter than the characteristic scale of δ ( r ) exhibit vanishing amplitudes and thus random phase-factors ˆ /epsilon1 ( k ). Therefore, | /epsilon1 ( r ) | = | ∑ k e i k · r ˆ /epsilon1 ( k ) | ∝ ∆ r -D/ 2 , if ∆ r → 0 . According to eq. (18), ξ ∗ 3 ( r ) hence diverges as ∆ r -3 D/ 2 and becomes infinitely dominated by random noise. To avoid this divergence, we must limit the number of high-frequency modes in eq. (16). By virtue of the Nyquist-Shannon theorem, it is natural to impose | k | ≤ π/r and | q | ≤ π/r . In signal processing terminology, this mode-truncation is called a low-pass filter with a spherical top-hat kernel. With this modification ξ ∗ 3 ( r ) becomes independent of ∆ r , if ∆ r < r/ 2.", "pages": [ 4 ] }, { "title": "3.3. The isotropic line-correlation function /lscript ( r )", "content": "Based on the conceptual discussion of Section 3.2, we now define the 'isotropic line-correlation function' of the density perturbation field δ ( r ) as where w D ( x ) is given in eq. (17). Like the bispectrum B ( k , q ), given in eq. (11), the line-correlation function /negationslash /lscript ( r ) = 〈 /lscript ( r ) 〉 here refers to a particular realization of δ ( r ) rather than an ensemble of fields. If some values of ˆ δ ( k ) vanish, then | B ( k , q ) | = 0 and /lscript ( r ) diverges. In practice, this only happens for k = 0, where ˆ δ ( k ) strictly vanishes by virtue of δ ( r ) = 0; for this case we adopt ˆ /epsilon1 (0) = ˆ δ (0) / | ˆ δ (0) | ≡ 0. The discretized version of the function /lscript ( r ) for the case of a periodic Cartesian grid with side-length L reads (see Appendix D) Given the definition of /lscript ( r ) in eq. (19), a list of basic properties of can be derived: /negationslash", "pages": [ 4, 5 ] }, { "title": "3.4. Physical interpretation of /lscript ( r )", "content": "The correlation function /lscript ( r ) is a measure of pure phase-information, i.e. it depends only on phase-phase correlations 〈 ˆ /epsilon1 ( k )ˆ /epsilon1 ( q ) 〉 . In turn, the 2-PCF ξ 2 ( r ) depends only on amplitude-amplitude correlations 〈| ˆ δ ( k ) || ˆ δ ( q ) |〉 . The latter is often used as a measure of clustering. Similarly, /lscript ( r ) can be interpreted as a measure of elongated structures, such as cosmic filaments, in a sense specified in the following. Fig. 3 shows four examples of a 2D random field δ ( r ), constructed by using eq. (2): (a) a GRF made of plane waves, (b) a random field with circular kernels, (c) a random field using filament-like kernels, and (d) a superposition of the GRF (a) and the filamentary field (c). The fields (a)-(c) are designed to exhibit the same 2-PCF ξ 2 ( r ), thus illustrating that ξ 2 ( r ) cannot distinguish between wave-like, spherical, and filamentary substructure (see also Fig. 1 by Coles 2005). In contrast, ξ 3 ( r ) vanishes for the GRF, but still exhibits a similar shape for the spherical and filamentary fields. In principle, the full isotropic 3-PCF ξ 3 ( | r | , | s | , ∠ r , s ) can distinguish spherical from filamentary structure (e.g. Nichol et al. 2006), but its geometrical interpretation is difficult. Only the linecorrelation function /lscript ( r ) clearly separates the filamentary density field (c) from the fields (a) and (b). Moreover, the shape of /lscript ( r ) describes the straight filaments quantatively: /lscript ( r ) exhibits a bump, /lscript ( r ) > 0 . 2, roughly on the Correlation scale r/L interval r ∈ [ σ x , σ y ] between the short and the long characteristic filament radius (see caption of Fig. 3) . Hence, for a density field composed exclusively of straight filaments, /lscript ( r ) measures the filamentarity on length scales 2 r . This feature of /lscript ( r ) remains true, even if the filamentary field (c) is superposed with the GRF (a), such as illustrated in Fig. 3d. A more systematic account of how aspherical substructure is imprinted in /lscript ( r ) is provided in Fig. 4. This analysis relies on a 3D density field constructed via eq. (2), with the generating functions g ( r ) being 50 identical spheroids. These spheroids are fully characterized by their prolateness q , defined as the ratio between the poleradius r p and the equator-radius r e , and by their average radius r 0 = ( r p r 2 e ) 1 / 3 . The left panel in Fig. 4 shows a projection of the 3D density field for the case of q = 8. The middle panel shows /lscript ( r ) for three selected values of q , confirming that /lscript ( r ) is positive for r > min { r e , r p } and peaks in the interval r ∈ [ r e , r p ]. For r < min { r e , r p } , /lscript ( r ) undergoes a series of oscillations. In the case of perfectly spherical ( q = 1) substructure /lscript ( r ) does not vanish; however, the integral ∫ /lscript ( r )d r nearly vanishes for q = 1 and increases for both q < 1 and q > 1. We thus see that /lscript ( r ) is particularly sensitive to aspherical substructure on scales ∼ 2 r , be it oblate, such as cosmic sheets, or prolate, such as the more common cosmic filaments. The line-correlation function /lscript ( r ) can also be used to characterize a mixture of aspherical structures. An illustration for the case of a 2D density field is shown in Fig. 5. This density field is again constructed via eq. (2), by randomly superposing two sizes of filaments. In this case /lscript ( r ) is double-peaked with each peak approximately measuring the width and length of one type of filament. Next, we investigate the dependence of /lscript ( r ) and ∫ /lscript ( r )d r on the size and density of substructure, e.g. the length and density of filaments. Let us consider a density field constructed via eq. (2) by superposing m identical generating functions g ( r ) of a characteristic linear scale r 0 and a characteristic volume V 0 , e.g., spheroids with r 0 = ( r p r 2 e ) 1 / 3 and V 0 = (4 π/ 3) r 3 0 . We define the fillingfactor of this density field as f ≡ mV 0 /L 3 . If the filling factor is increased by increasing the number of objects m , then the increasing number of random translations and rotations in eq. (2) amplifies the phase-noise as √ m . Hence the amplitude of the line-correlation /lscript ( r ) falls (on average) as 1 / √ m . If, on the other hand, the characteristic scale r 0 is changed by a factor u while maintaining f constant, i.e., by varying the number of objects per unit volume, the situation looks as follows. Let ˜ r 0 ≡ ur 0 be the new length scale, and ˜ δ ( r ) the corresponding density field. Since /lscript ( r ) is independent of the box size L as L →∞ , we may choose the new density field ˜ δ ( r ) to be defined on a cubic volume with side-length ˜ L = uL , such that the number of objects ˜ m in the total volume remains the same, i.e., ˜ m = m . In this way we find ˜ δ ( u r ) = δ ( r ), given the same choice of random translations and rotations for δ ( r ) and ˜ δ ( r ). The line-correlation associated with ˜ δ then becomes ˜ /lscript ( r ) = /lscript ( r/u ). As a consequence, if in a density field δ ( r ) with line-correlation /lscript ( r ) the substructure-scale r 0 is stretched by a factor u and the filling factor f is varied by a factor v , the line-correlation scales as It follows that the rescaled function √ f /lscript ( r/r 0 ) is independent of f and r 0 (up to shot noise) and thus characteristic of a particular type of substructure, e.g., spheroids of a certain prolateness q . This also implies that with κ being a constant depending on the shape of the substructure, but not on its scale and filling factor. This relation is strongly supported by the numerical example in Fig. 6. In summary, /lscript ( r ) is sensitive to aspherical substructure on scales 2 r . In the special case of a filamentary field, /lscript ( r ) measures the characteristic scales of the filaments. The dependance of /lscript ( r ) on the prolateness of aspherical substructure is shown in Fig. 4. The dependance of /lscript ( r ) on the size and filling factor of substructure is given in eq. (21), and illustrated in Fig. 6.", "pages": [ 5, 6, 7 ] }, { "title": "4. APPLICATION TO COSMIC STRUCTURE", "content": "This section studies the line-correlation /lscript ( r ) of simulated cosmic density fields, using both CDM and WDM, i.e., dark matter with a finite particle mass.", "pages": [ 7 ] }, { "title": "4.1. Cosmological simulation", "content": "We have run a suite of cosmological N -body simulations, following the formation and evolution of LSS in a cubic box of comoving side-length L = 100 h -1 Mpc containing 512 3 ≈ 1 . 34 · 10 8 particles, from an initial redshift of z init = 199 to z = 0. Following Komatsu et al. (2011), we adopt matter and dark energy density parameters of Ω 0 = 0 . 273 and Ω Λ = 0 . 727, a Hubble parameter of h = 0 . 705 (defined by H 0 = 100 h kms -1 Mpc -1 ), a primordial spectral index of n spec = 0 . 95 and a normalization σ 8 = 0 . 812. Initial conditions were created using standard techniques (e.g. Power et al. 2003) - a statistical realization of a GRF is generated in Fourier space, with variance given by the linear matter power spectrum, and the Zel'dovich approximation is used to compute initial particle positions and velocities. The power spectrum for the CDM model is obtained by convolving the primordial isotropic power spectrum p ( k ) ∝ k n spec with the transfer function appropriate for our chosen set of cosmological parameters, computed using the Boltzmann code CAMB (cf. Lewis et al. 2000). Following Bode et al. (2001), the initial power spectra for our WDM models are obtained by filtering the CDM power spectrum with the transfer function where µ is a function of the WDM particle mass (labeled ' α ' in eq. (A9) of Bode et al. 2001) and ν = 1 . 2 is a numerical constant. Eq. (23) mimics the free-streaming of WDM particles by preferentially suppressing highfrequency modes. All simulations were run using the parallel TreePM code GADGET2 (Springel et al. 2005) with constant comoving gravitational softening /epsilon1 = 4 kpc and individual and adaptive time steps for each particle, ∆ t = η √ /epsilon1/a , where a is the magnitude of a particle's gravitational acceleration and η = 0 . 05 determines the accuracy of the time integration. In order to study spatial correlations, the particles are discretized onto a regular grid of N 3 cells (here N = 400), as described in Appendix D. The correlation functions are then computed via eqs. (D4).", "pages": [ 7, 8 ] }, { "title": "4.2. Line-correlation /lscript ( r ) in a Λ CDM universe", "content": "Correlation scale r/L structure, namely tidal cosmic filaments. We find that the late-time ( z = 0) line-correlation /lscript ( r ) is well approximated by the power-law shown as dashed-line in Fig. 7c. A power-law behavior had to be expected for it is equivalent to /lscript ( r ) being scale-free in the sense that /lscript ( r 1 ) //lscript ( r 2 ) is constant if r 1 /r 2 is constant - a general feature of LSS on scales 10 kpc < r < 10 Mpc (e.g. the weak-lensing analysis of the Sloan Digital Sky Survey (SDSS), see Fig. 10 of Sheldon et al. 2004). Interestingly, the power-law of eq. (24) is less steep than the observed cosmic 2-PCF ξ 2 ( r ) ∝ r -1 . 79 (Sheldon et al. 2004), consistent with the interpretation that /lscript ( r ) senses structural features extend- to comparatively large scales, such as filaments. Fig. 7c also illustrates the numerical convergence of /lscript ( r ) for an increasing number of grid cells N 3 (different solid lines). The convergence is ensured by the normalization factor in front of the integral in eq. (19) as explained in Section 3.2. Finally, Fig. 7d explores the effect of cosmic variance on /lscript ( r ). The four solid lines correspond to four different random realizations of the initial density field. We notice that cosmic variance dominantly affects the amplitude of the power-law, but only marginally affects its slope. From these examples we estimate that, for a box size L = 100 h -1 Mpc, the normalization of 0 . 73 in eq. (24) has a statistical uncertainty of about 10%, while the exponent of -1 . 15 is accurate to about 3%.", "pages": [ 8, 9 ] }, { "title": "4.3. Measuring the 'temperature' of dark matter", "content": "What cosmological information can be extracted from a local measurement of /lscript ( r )? Since /lscript ( r ) only depends on correlations between the phases ˆ /epsilon1 ( k ), it is strictly insensitive to linear growth, defined as a uniform growth of the amplitudes | ˆ δ ( k ) | . Therefore the physics associated with linear growth, such as that dictating the baryon acoustic scale, remains invisible to /lscript ( r ). However, /lscript ( r ) depends on the physics associated with non-linear growth, namely on local gravitational interactions and hence on the properties of dark matter. We therefore chose to examine the variations of /lscript ( r ) with the dark matter particle mass m DM , using the WDM simulations described in Section 4.1. An illustration of two analogous density fields with CDM and WDM is provided in Fig. 8. Fig. 9 shows the functions ξ 2 ( r ) and /lscript ( r ) for four values of m DM , here expressed in units of energy, e.g. 1 keV ∼ = 1 keVc -2 ≈ 1 . 78 · 10 -33 kg. Fig. 9 confirms that ξ 2 ( r ) only weakly depends on our array of m DM , and that this dependance is restricted to scales r < 1 h -1 Mpc, in good agreement with several recent studies (e.g. Smith & Markovic 2011). By contrast, /lscript ( r ) exhibits a much stronger dependance on m DM , and this dependence extends to scales of about 5 h -1 Mpc. The m DM -dependencies displayed in Fig. 9 can be reversed to infer m DM from ξ 2 ( r ) and /lscript ( r ). To simplify the notations, let f ( r ) be a generic placeholder for ξ 2 ( r ) and /lscript ( r ). When analyzing the variations of log 10 f ( r ) as a function of m DM (for r > 0 . 5 h -1 Mpc and m DM < 1 keV), we find them to be linear in m -1 DM within the uncertainties of cosmic variance. It follows that, at any r , the true particle mass m true DM can be detected against a hypothetical particle mass m hyp DM , e.g. against CDM ( m hyp DM = ∞ ), with a signal-to-noise ratio proportional to ∆ m -1 DM = ( m hyp DM ) -1 -( m true DM ) -1 . In other words, this signal-to-noise ratio can be expressed as s ( r ) | ∆ m -1 DM | , where s ( r ) denotes the signal-to-noise per unit of m -1 DM . We here approximate s ( r ) as where N ( r ) represents the noise, defined as the standard deviation of log 10 f ( r ) due to cosmic variance. We estimate N ( r ) as the root-mean-square of log 10 f CDM ( r ) over four random realizations of a CDM run. This numerical estimate is more reliable than analytical estimates based on the number of Fourier modes, if the latter are phasecorrelated. The normalization factor in the denominator of eq. (25) comes from the fact that the values of m -1 DM in CDM and WDM at 0 . 1 keV differ by 10 keV -1 . When measuring m DM based on f ( r ), we consider ∆ m -1 DM ( r ) to be the difference between a measurement of m -1 DM , at a specific scale r , and its true value. In the Gaussian approximation, the probability distribution φ ( r ) of ∆ m -1 DM ( r ) is then proportional to exp[ -∆ m -2 DM s 2 ( r ) / 2]. Note that the exponent is dimensionless, as it should be. Combining the measurements on the scales r ∈ [ r min , r max ] associated with independent Fourier modes k = π/r , the total probability distribution of ∆ m -1 DM becomes φ ( r min ) · ... · φ ( r max ). Hence the standard-deviation of a measurement of m -1 DM on scales r ∈ [ r min , r max ] becomes Numerical estimates of σ ( r min , r max ) for various intervals [ r min , r max ] and for both correlation functions ( ξ 2 and /lscript ) are given in Tab. 1. Fig. 10 shows the probability distribution of ∆ m -1 DM for the case of a measurement based on the interval [0 . 5 h -1 Mpc , 5 h -1 Mpc]. The following points are worth stressing. First, the values in Tab. 1 and probability distributions in Fig. 10 correspond to a perfect measurement of the density field δ ( r ), since they only account for the fundamental limitations associated with cosmic variance. They ignore potentially large measurement uncertainties and observational biases. Second, the values in Tab. 1 are specific to our box-size L = 100 h -1 Mpc. The noise N ( r ) in eq. (25) scales as V -1 / 2 , where V = L 3 is the volume of the considered density field. Hence standard-deviations of m -1 DM for any other volume V can be obtained as [(100 h -1 Mpc) 3 /V ] 1 / 2 σ . Third, the standard-deviation of m DM rather than m -1 DM is given by m 2 DM σ . The key result from this analysis is that, in the latetime universe, the particle mass of dark matter m DM is much better constrained by phase-phase correlations, measured via /lscript ( r ), than by amplitude-amplitude correlations, measured via ξ 2 ( r ) or p ( k ). Furthermore, m DM affects scales five times larger in /lscript ( r ) than in ξ 2 ( r ). This finding is crucial, since the smaller the scale required to measure m DM , the more the result will be entangled with the uncertainties of complex baryon physics, such as feedback from supernovae, active black holes, and photoionization (Kim et al. 2012). The reason for this advantage of /lscript ( r ) over ξ 2 ( r ) is twofold. First, phase-phase correlations vanish in the primordial CDM/WDM power spectrum and are strictly independent of linear growth. They therefore represent exquisite tracers of non-linearly grown structure, which in turn depends substantially on m DM . This supports the finding that /lscript ( r ) is more sensitive on m DM than ξ 2 ( r ) for a given scale r . Second, /lscript ( r ) is particularly sensitive to cosmic filaments, while ξ 2 ( r ) traces spherical structures, such as clusters. However, cosmic filaments extend to Numerical values of the standard deviations σ ( r min , r max ) of a measurement of m -1 DM in a cosmic volume V = (100 h -1 Mpc) 3 . larger scales than galaxy clusters and they seem to better preserve the primordial free-streaming scale (see Fig. 8), perhaps because filaments are second order effects of the gravitational field (Pen et al. 2012). This might explain why /lscript ( r ) traces m DM to larger scales than ξ 2 ( r ). Caution is nonetheless indicated. Despite the advantage of /lscript ( r ) over ξ 2 ( r ), the case presented in Fig. 9 suggests that particle masses m DM > 1 keV will remain difficult to distinguish from CDM on scales larger than 0 . 5 h -1 Mpc. However, particles lighter than 1 -2 keV seem inconsistent with the Lymanα forest (Boyarsky et al. 2009). To probe particle masses /greatermuch 1 keV, /lscript ( r ) will have to be measured on smaller scales (eq. (3) in Schneider et al. 2012), not yet studied in this work. Those scales will still be larger than those required by ξ 2 ( r ), but baryon physics will become important.", "pages": [ 9, 10, 11 ] }, { "title": "5.1.1. /lscript ( r ) - a measure of pure phase-information", "content": "This work introduced the isotropic line-correlation function /lscript ( r ), defined for a density field δ ( r ) via eq. (19). Unlike conventional n -PCFs, /lscript ( r ) is defined exclusively upon the spectral phases ˆ /epsilon1 ( k ). Thus /lscript ( r ) only measures phase-phase correlations 〈 ˆ /epsilon1 ( k )ˆ /epsilon1 ( q ) 〉 . By contrast, the 2-PCF only depends on amplitude-amplitude correlations 〈| ˆ δ ( k ) || ˆ δ ( q ) |〉 , while the n -PCFs ( n ≥ 3) depend on amplitude-amplitude correlations 〈| ˆ δ ( k ) || ˆ δ ( q ) |〉 , amplitude-phase correlations 〈| ˆ δ ( k ) | ˆ /epsilon1 ( q ) 〉 , and phasephase correlations 〈 ˆ /epsilon1 ( k )ˆ /epsilon1 ( q ) 〉 . It follows that /lscript ( r ) is independent of ξ 2 ( r ) and p ( k ), up to statistical dependencies between the amplitudes | ˆ δ ( k ) | and phases ˆ /epsilon1 ( q ).", "pages": [ 11 ] }, { "title": "5.1.2. /lscript ( r ) - a parameter-free measure of cosmic filaments", "content": "In Section 3.4 we have established that /lscript ( r ) is, in a limited sense, a measure of aspherical structure, such as straight filaments. This measure is statistical in that it cannot identify the individual filaments. Hence typical applications of /lscript ( r ) are studies of cosmic structure rather than investigations of galaxies in particular environments. For the latter, sophisticated 'filamentfinders' have been developed, i.e., algorithms able to convert 3D galaxy distributions into skeletons of filaments (e.g. Bond et al. 2010; Sousbie 2011). On the downside, the relationship between those algorithms and robust measures of cosmic structure, such as correlation functions, is unknown and/or complex (Pogosyan et al. 2009). Moreover, filament-finders always require free parameters, such as user-defined thresholds and scales. By contrast, the definition of /lscript ( r ) is parameter-free and scale-invariant; it is even independent of the numerical grid, if the cell-size is much smaller than r . The relationship between filament-finders and /lscript ( r ) is therefore analogous to that between group-finders and ξ 2 ( r ): the former allow an explicit identification of individual structural components, while the latter represent mathematically robust measures of spatial statistics.", "pages": [ 11 ] }, { "title": "5.1.3. /lscript ( r ) - a thermometer for dark matter", "content": "By calculating /lscript ( r ) for simulated cosmic density fields (Section 4) we demonstrated that /lscript ( r ) is more sensitive to variations of m DM than the 2-PCF ξ 2 ( r ). Therefore, measurements of m DM are significantly more accurate when based on /lscript ( r ) than based on ξ 2 ( r ) (e.g. Fig. 10). Moreover, /lscript ( r ) depends on m DM out to scales at least five times larger than ξ 2 ( r ). This result is pivotal since smaller scales are those more affected by uncertain baryon physics (Kim et al. 2012) masking the footprint of dark matter properties.", "pages": [ 11 ] }, { "title": "5.2. Prospects of using real data", "content": "This work paves the way towards an enhanced analysis of existing an future redshift surveys, for example to better constrain the particle mass of dark matter m DM . Any comparison between simulated and observed LSS is challenged by differences between visible matter and underlying dark matter. These effects are rather small on scales > 1 h -1 Mpc. However, measurements on smaller scales require a precise reconstruction of the actual dark matter density field, for example using weaklensing data, and/or a modeling of visible LSS, for example using mock-skies based on semi-analytic modeling (e.g. Blaizot et al. 2005; Obreschkow et al. 2009). Moreover, real surveys do not come in the shape of a cubic box, but in a truncated survey volume with varying selection criteria across the volume. To deal with such masked data, our idealized formulation of the line-correlation function will need to be transcribed into a form applicable to a generic survey-mask. To do so, one might apply an approach similar to that of Landy & Szalay (1993), which essentially consists in comparing the correlations in the observed density field against those in a random field with an identical surveymask. Finally, observational data is subject to redshift-space distortions (Kaiser 1987), leading to elongated structures along the line-of-sight in the reconstructed 3D density field ('fingers-of-God'). These spurious prolate features add to the line-correlation /lscript ( r ). This effect will require additional modeling and/or an evaluation of /lscript ( r ) separately along radial and transverse directions, as typically done for the 2-PCF (Chuang & Wang 2012). On scales larger than the redshift distortion scale, the radial and transverse parts of /lscript ( r ) might also be used to constrain dark energy in a way analogous to the classical AlcockPaczynski test (Alcock & Paczynski 1979).", "pages": [ 11 ] }, { "title": "5.3. Closing words", "content": "Above all, this work demonstrates the enormous potential of phase-information. Further investigations of this information may unveil a wealth of applications, extending far beyond the case of CDM versus WDM. In light of future redshift surveys, the time seems ripe for phaseinformation to become a standard tool in observational cosmology. Ultimately, this field would tremendously benefit from a complete estimator of cosmic structure, i.e., a function F ( δ ( r )) that exclusively and exhaustively describes the information contained in a statistically homogeneous and isotropic density field δ ( r ).", "pages": [ 11 ] }, { "title": "ACKNOWLEDGEMENTS", "content": "Part of the research presented in this paper was undertaken as part of the Survey Simulation Pipeline (SSimPL; http://www.astronomy.swin.edu.au/SSimPL/ ). D.O. was supported by the Research Collaboration Award 12105012 of the University of Western Australia. C.B. is supported by the Herchel Smith fund and by Kings Col- lege Cambridge. We thank the anonymous referee for a very constructive report.", "pages": [ 11, 12 ] }, { "title": "A. GENERALIZED CONVOLUTION THEOREM", "content": "We here define the Fourier transform (FT) and the corresponding inverse Fourier transform (IFT) as in Peacock (1999) apart from sign of i , where k ∈ R D is the wavevector. Note that the reality of δ ( r ) implies ˆ δ ( -k ) = ˆ δ ∗ ( k ), where the asterisk is the complex conjugate. Substituting δ for eq. (A1b) in eq. (4), we find Rearranging the terms, Solving the integral over t gives where δ 3 is Dirac's delta distribution in 3D. Finally, remembering that r n ≡ 0, which readily reduces to eq. (8).", "pages": [ 12 ] }, { "title": "B. EXPLICIT EXPRESSIONS FOR ξ 2 AND ξ 3", "content": "This paragraph explicits the Fourier space expressions of two isotropic correlation functions, which will be used in the rest of this work. First, to find the Fourier equivalent of ξ 2 ( r ), we substitute Ξ 2 ( r ) in eq. (7) for eq. (12), which implies where w D ( kr ) ≡ e i k · r | r | = r is a weighting function. A quick calculation expanded in Appendix C shows that where J 0 ( x ) is the 0-th order Bessel function. In eq. (B1), w ( kr ) only depends on the integration variable k = | k | . The remaining D -1 integration variables only act on P ( k ). Performing this integration of P ( k ) leads to V ∫ (2 π ) D 0 where p ( k ) ≡ p 2 ( k ) = | ˆ δ ( k ) | 2 | k | = k is the isotropic power-spectrum and S D ( k ) denotes the surface area of the D -sphere, i.e., ∞ ξ 2 ( r ) = d k S D ( k ) w D ( kr ) p ( k ) , (B3) Next, we consider the particular isotropic 3-PCF ξ 3 ( r ) for three equidistant points on a straight line, i.e., After substituting Ξ 3 for eq. (13), a derivation analogous to that in Appendix C then leads to Note that | k -q | = √ k 2 + q 2 -kq cos θ , where θ ≡ ∠ ( k , q ), only depends on the three coordinates ( k, q, θ ). Thus, we can first integrate B ( k , q ) over the remaining 2 D -3 coordinates, which leads to where b ( k, q, θ ) ≡ B ( k , q ) | k | = k, | q | = q, ∠ ( k , q )= θ is the isotropic bi-spectrum and j D ( θ ) is the Jacobian", "pages": [ 12, 13 ] }, { "title": "C. ROTATIONAL AVERAGE OF exp( i k · r )", "content": "In two dimensions, r is expressed in polar coordinates r = | r | and θ , where θ is the angle between k and r , such that k · r = kr cos( θ ). Then, By symmetry, the real part of the integral is equal to twice the integral from 0 to π , and, by anti-symmetry, the imaginary part of the integral vanishes, In three dimensions, r is expressed in spherical coordinates r , ϕ , and θ , where θ is the angle between k and r , such that k · r = kr cos( θ ). Then, By anti-symmetry, the imaginary part of the integral vanishes and", "pages": [ 13 ] }, { "title": "D. NUMERICAL DISCRETIZATION", "content": "For computational purposes, we adopt the standard numerical model: (i) the universe is described in a finite cubic box Ω ⊂ R D of side-length L and volume V = | Ω | = L D ; (ii) this box satisfies periodic boundary conditions; (iii) the density perturbation field δ ( r ) is represented on a regular Cartesian grid of N 3 cubic cells, such that the cells have side-lengths ∆ r = L/N and volumes ∆ V = ( L/N ) D . This model is valid as long as we consider correlations on scales larger than ∆ r and significantly smaller than L . The corresponding Fourier space discretization follows directly from the periodicity condition, which states that each mode k = ( k 1 , ..., k D ) must satisfy k j L ∈ 2 π N ∀ j . Therefore, the Fourier cell spacing equals ∆ k = 2 π/L . The side-length of the Fourier box hence becomes N ∆ k = 2 πN/L . This numerical discretization is illustrated in Fig. 11 in two dimensions ( D = 2). We are free to choose the origins, both in direct and in Fourier space. In our convention, shown in Fig. 11, the sets of discrete vectors r and k become where floor( x ) is defined as the largest integer less than or equal to x . The rules for the mapping between continuous integrals and discrete sums, both in direct and Fourier space, follow directly from the expressions for ∆ r and ∆ k . They read where the values of the functions f ( r ) and ˆ f ( k ) on the right-hand side are cell averages. Using the mapping rules of eqs. (D2a) and (D2b), the FT and IFT of eqs. (A1a) and (A1b) become the discrete FT (DFT) and the inverse DFT (IDFT), respectively, By virtue of the same rules, ξ 2 ( r ) in eq. (B1), ξ 3 ( r ) in eq. (B6), and /lscript ( r ) in eq. (19) become These are the three functions, which we calculated in the examples, e.g. Fig. 3.", "pages": [ 13, 14 ] }, { "title": "REFERENCES", "content": "Alcock C., Paczynski B., 1979, Nature, 281, 358 Arag'on-Calvo M. A., Jones B. J. T., van de Weygaert R., van der Hulst J. M., 2007, A&A, 474, 315 Babul A., Starkman G. D., 1992, ApJ, 401, 28 Barriga J., Gazta˜naga E., 2002, MNRAS, 333, 443 Bennett C. L., et al., 2003a, ApJ, 583, 1 -, 2003b, ApJS, 148, 1 -, 2011, ApJS, 192, 17 Bernardeau F., Colombi S., Gazta˜naga E., Scoccimarro R., 2002, Phys. Rep., 367, 1 Blaizot J., Wadadekar Y., Guiderdoni B., Colombi S. T., Bertin E., Bouchet F. R., Devriendt J. E. G., Hatton S., 2005, MNRAS, 360, 159 Blake C., et al., 2011, MNRAS, 418, 1707 Bode P., Ostriker J. P., Turok N., 2001, ApJ, 556, 93 Bond N. A., Strauss M. A., Cen R., 2010, MNRAS, 409, 156 Bonvin C., Durrer R., 2011, Phys. Rev. D, 84, 063505 Boyarsky A., Lesgourgues J., Ruchayskiy O., Viel M., 2009, Journal of Cosmology and Astroparticle Physics, 5, 12 Challinor A., Lewis A., 2011, Phys. Rev. D, 84, 043516 Chuang C.-H., Wang Y., 2012, MNRAS, 426, 226 Coles P., 2005, ArXiv Astrophysics e-prints Croton D. J., et al., 2004, MNRAS, 352, 1232 Fry J. N., 1985, ApJ, 289, 10 Fry J. N., Peebles P. J. E., 1978, ApJ, 221, 19 Gazta˜naga E., Norberg P., Baugh C. M., Croton D. J., 2005, MNRAS, 364, 620 Gong Z.-Y., 2010, Chinese Astronomy and Astrophysic, 34, 23 Hicken M., Wood-Vasey W. M., Blondin S., Challis P., Jha S., Kelly P. L., Rest A., Kirshner R. P., 2009, ApJ, 700, 1097 Kaiser N., 1987, MNRAS, 227, 1 Kessler R., et al., 2009, ApJS, 185, 32 Kim H.-S., Power C., Baugh C. M., Wyithe J. S. B., Lacey C. G., Lagos C. D. P., Frenk C. S., 2012, MNRAS, 214 Komatsu E., et al., 2011, ApJS, 192, 18 Landy S. D., Szalay A. S., 1993, ApJ, 412, 64 Larson D., et al., 2011, ApJS, 192, 16 Leon G., Landau S. J., Sudarsky D., 2011, ArXiv e-prints Lewis A., Challinor A., Lasenby A., 2000, ApJ, 538, 473 Luo S., Vishniac E., 1995, ApJS, 96, 429 Mar'ın F., 2011, ApJ, 737, 97 McWeeny R., 2002, Symmetry: An Introduction to Group Theory and Its Applications. Dover Publications Nichol R. C., et al., 2006, MNRAS, 368, 1507", "pages": [ 14 ] } ]

[ { "title": "ABSTRACT", "content": "Despite the importance of their size evolution in understanding the dynamical evolution of globular clusters (GCs) of the Milky Way, studies are rare that focus specifically on this issue. Based on the advanced, realistic Fokker-Planck (FP) approach, we predict theoretically the initial size distribution (SD) of the Galactic GCs along with their initial mass function and radial distribution. Over one thousand FP calculations in a wide parameter space have pinpointed the best-fit initial conditions for the SD, mass function, and radial distribution. Our best-fit model shows that the initial SD of the Galactic GCs is of larger dispersion than today's SD, and that typical projected half-light radius of the initial GCs is ∼ 4.6 pc, which is 1.8 times larger than that of the present-day GCs ( ∼ 2.5 pc). Their large size signifies greater susceptibility to the Galactic tides: the total mass of destroyed GCs reaches 3-5 × 10 8 M /circledot , several times larger than the previous estimates. Our result challenges a recent view that the Milky Way GCs were born compact on the sub-pc scale, and rather implies that (1) the initial GCs are generally larger than the typical size of the present-day GCs, (2) the initially large GCs mostly shrink and/or disrupt as a result of the galactic tides, and (3) the initially small GCs expand by two-body relaxation, and later shrink by the galactic tides. Subject headings: Galaxy: evolution - Galaxy: formation - Galaxy: kinematics and dynamics - globular clusters: general - methods: numerical", "pages": [ 1 ] }, { "title": "INITIAL SIZE DISTRIBUTION OF THE GALACTIC GLOBULAR CLUSTER SYSTEM", "content": "1 1,2 3 4 Jihye Shin , Sungsoo S. Kim , Suk-Jin Yoon , and Juhan Kim Draft version March 5, 2022", "pages": [ 1 ] }, { "title": "1. INTRODUCTION", "content": "Whereas the present-day mass functions (MFs) of globular cluster (GC) systems, which are nearly universal among galaxies (Brodie & Strader 2006; Jorn'an et al. 2007), are approximately log-normal with a peak mass M p ≈ 2 × 10 5 M /circledot , the MFs of the young massive star cluster (YMC) systems follow a simple power-law distribution (Whitmore & Schweizer 1995; Zhang & Fall 1999; de Grijs et al. 2003, among others). Motivated by such a difference between GCs and YMCs, numerous studies have examined the dynamical evolution of the GC MFs to determine whether the initial MFs of GC systems resemble those of YMC systems (Gnedin & Ostriker 1997; Baumgardt 1998; Vesperini 1998; Fall & Zhang 2001; Parmentier & Gilmore 2007; Shin, Kim, & Takahashi 2008, among others). In particular, Shin, Kim, & Takahashi (2008, Paper I hereafter) surveyed a wide range of parameter space for the initial conditions of the Milky Way GCs, and considered virtually all internal/external processes: two-body relaxation, stellar evolution, binary heating, galactic tidal field, eccentric orbits and disc/bulge shocks. They found that the initial GC MF that best fits the observed GC MF of the Milky Way is a log-normal function with a peak at 4 × 10 5 M /circledot and a dispersion of 0.33, which is quite different from the typical MFs of YMCs. Using the outcome of N -body calculations, [email protected] 1 Department of Astronomy & Space Science, Kyung Hee University, Yongin, Kyungki 446-701, Republic of Korea 2 School of Space Research, Kyung Hee University, Yongin, Kyungki 446-701, Republic of Korea 3 Department of Astronomy and Center for Galaxy Evolution Research, Yonsei University, Seoul 120-749, Republic of Korea 4 Center for Advanced Computation, Korea Institute for Advanced Study, 87 Hoegiro Dondaemun-gu, Seoul 130-722, Republic of Korea Gieles & Baumgardt (2008) found that the aspect of mass loss in GCs varies with the tidal filling ratio /Rfractur ≡ r h /r J , where r h is the half-mass radius and r J is the Jacobi radius. More specifically, the mass loss of GCs in the 'isolated regime' ( /Rfractur < 0 . 05) is driven mostly by the two-body relaxation, which induces the formation of binaries in the core and causes GCs to expand. On the other hand, the mass loss of GCs in the 'tidal regime' ( /Rfractur > 0 . 05) is influenced by the galactic tides as well, which enables stars in the outer envelope to easily escape (evaporation). Thus, the cluster size ( r h ) is as important as the cluster mass ( M ) and the galactocentric radius ( R G ) in determining the dynamical evolution of GCs. Can YMCs tell us something about the typical initial size of the Milky Way GC system? Observations show that the projected half-light radius R h of YMCs (ages up to 100 Myr) in the local group ranges between ∼ 2 and ∼ 30 pc with a mean value of ∼ 8 pc (Portegies Zwart, McMillan, & Gieles 2010), which is a few times larger than that of the present-day Milky Way GCs, R h ∼ 2 . 5 pc. However, GCs could have formed in different environments and/or by different mechanisms from the YMCs. Perhaps the best way to estimate the typical size of the GCs is to trace them back to their initial state by calculating their dynamical evolution. In this paper, we study the dynamical evolution of the Galactic GCs and identify the most probable initial conditions not only for the MF and radial distribution (RD), but also the size distribution (SD). Using the same numerical method and procedure as in Paper I, we perform Fokker-Planck (FP) calculations for 1152 different initial conditions (mass, half-mass radius, galactocentric radius and orbit eccentricity), and then search a wide-parameter space for the most probable initial distribution models that evolve into the present-day Galactic GC distributions. The paper is organized as follows. Section 2 describes the properties of the observed GCs, again which we compare our model results. Section 3 presents models and initial conditions for FP calculations, and Section 4 analyzes the aspects of the size evolution of GCs. We synthesize our FP results in Section 5 to construct the GC system, and examine common features of the best-fit MF, RD, and SD models in Section 6. We discuss characteristics of the final best-fit SD models of the Galactic GCs in Section 7. Finally, conclusions are presented in Section 8.", "pages": [ 1, 2 ] }, { "title": "2. PRESENT-DAY GC PROPERTIES", "content": "When comparing FP calculations to the present-day Galactic GCs, we consider the 'native' GCs only, i.e., 'old' halo and bulge/disc clusters, which are believed to be created when a protogalaxy collapses while 'young' halo clusters are thought to be formed in external satellite galaxies (Zinn 1993; Parmentier et al. 2000; Mackey & van den Bergh 2005). Our native GC candidates do not include six objects that belong to the Sagittarius dwarf, seven objects whose origins remain unknown, two objects that have no size information, and fifteen objects that are thought to be the remnants of dwarf galaxies (Lee, Gim, & Casetti-Dinescu 2007). The total number of our present-day Galactic native GCs is 93, and their observed properties, such as luminosity L , R h , and R G , were obtained from the database compiled by Harris (1996). Figure 1 shows scatter plots between observed L , R h , and R G values for the 93 Galactic native GCs. The L , R h , and R G values range between 3.9 × 10 3 -5.0 × 10 5 L /circledot , 0.3-16 pc, and 0.6-38 kpc, where the mean values are located at 7.2 × 10 4 L /circledot , 2.5 pc, and 4.1 kpc, respectively. The correlation between R h and R G is tighter than the other two correlations (see Figure 1 b ). This tight R h -R G correlation could be just a result of the initially tight correlation between R h and R G , or it could be due to the preferred disruption of large GCs near the Galactic center (Vesperini & Heggie 1997; Baumgardt & Makino 2003). Another possible cause is the expansion of initially small GCs up to r J , which is roughly proportional to R 2 / 3 G for a given GC mass. One of the goals of this paper is to determine which of these possibilities is more feasible. Previous studies on the evolution of the GC system assumed a certain constant mass-to-light ( M/L ) ratio, and converted the observed L to M when comparing their numerical values with observations. But the conversion of GC luminosity function (LF) to GC MF using a constant M/L ratio may lead to MFs in error because low-mass stars, which have higher M/L ratios than the high-mass stars, preferentially evaporate from the cluster and this causes the M/L ratio of the cluster to evolve with time (Kruijssen & Portegies Zwart 2009). For the same reason, there is not a linear relationship between R h and r h among different GCs. Thus, we transform M to L , instead of L to M , using the stellar mass-luminosity relation of the Padova model (Marigo et al. 2008) with a metallicity of [Fe/H] = -1 . 16, which is the mean value for the Galactic native GCs. Our FP calculations, which will be described later, show that the present-day GCs can have M/L ratios ranging between 1.2 and 2.5 and r h /R h ratios ranging between 1.0 and 2.5. We use dynamical properties such as M and r h when constructing the initial distributions of the Galactic GC system and when calculating the dynamical evolution, while observed quantities, L and R h , are used when comparing our FP results with the observations. We adopt the anisotropic FP model used in Paper I, which was originally developed by Takahashi & Lee (2000, and references therein). The model integrates the orbit-averaged FP equation of two (energy-angular momentum) dimensions and considers multiple stellar mass components, three-body and tidal-capture binary heating, stellar evolution, tidal fields, disk/bulge shocks, dynamical friction, and realistic (eccentric) cluster orbit (see Kim & Lee (1999) for the tidal binary heating and Paper I for the detailed implementation of dynamical friction and realistic orbits). The model implements the Alternating Direction Implicit (ADI) method developed by Shin & Kim (2007) for integrating the twodimensional FP equation with better numerical stability. Parameters for our FP survey are the following four initial cluster conditions: M , r h , apocenter distance of the cluster orbit R a , and cluster orbit eccentricity e . We choose eight M values from 10 3 . 5 to 10 7 M /circledot , six r h values from 10 -1 to 10 1 . 5 pc, and six R a values from 10 0 to 10 1 . 67 kpc, all equally spaced on the logarithmic scale. For the eccentricity, we choose e = 0, 0.25, 0.5, and 0.75. We perform FP calculations for all possible combinations of these four parameters, thus the total number of cluster models considered in the present study amounts to 1152. For the initial stellar mass function (IMF) within each cluster, we adopt the model developed by Kroupa (2001) with a mass range of 0.08-15 M /circledot , which is realized by 15 discrete mass components in our FP model. Each mass component follows the stellar evolution recipe described by Schaller et al. (1992). The stellar density and velocity dispersion distributions within each cluster follow the King model (King 1966) with a concentration parameter W 0 = 7 and with neither initial velocity anisotropy nor initial mass segregation. We use only one value for W 0 , thus the tidal cut-off radius r t of the King profile is proportional to r h , while r J varies depending on M and R G . Therefore, the Roche lobe filling ratio ( r t /r J ) and /Rfractur of our FP models are functions of r h , M , and R G . The aspects of mass and size evolution from our FP model are in a good agreement with those from N -body methods. A comparison of mass evolution between our FP calculations and the N -body simulations performed by Baumgardt & Makino (2003) for clusters on eccentric orbits with initial masses larger than 10 4 M /circledot shows good agreement of cluster lifetimes within ∼ 25%. For a comparison of size evolution, we run a set of N -body simulations using Nbody4 code (Aarseth 2003) with M = 10 4 M /circledot , R G = 8 . 5 kpc, and r h = 1, 3, and 5 pc (these correspond to /Rfractur = 0.04, 0.11, and 0.18), and find that the r h evolutions of the two models agree well within ∼ 20% during the entire cluster lifetimes (see Figure 2). Due to the expulsion of the remnant gas from star formation in the pre-gas-expulsion cluster, some of the low-mass pre-gas expulsion clusters can quickly disrupt, and even the surviving low-mass pre-gasexpulsion clusters will lose a significant fraction of their mass within the first several Myr and rapidly expand (Baumgardt & Kroupa 2007; Parmentier & Gilmore 2007). Since our FP model does not consider the effect of gas expulsion, our initial GC models are to be regarded as models at several Myr after cluster formation.", "pages": [ 2, 3 ] }, { "title": "4. SIZE EVOLUTION OF INDIVIDUAL GLOBULAR CLUSTERS", "content": "The three main drivers of GC size evolution are the two-body relaxation, the mass loss by stellar evolution, and the galactic tides. In this section, we discuss the size evolution of individual GCs with a subset of our FP calculations. Figure 3 shows the ratios between r h values at the present time (13 Gyr) and at the beginning from our FP calculations as a function of r h, 0 and M 0 for two different R G, 0 values (subscripts 0 denote the initial value, hereafter). Two-body relaxation causes GC core to collapse and the subsequent formation of dynamical binaries in the core makes the whole cluster expand. For GCs that have undergone core collapse in the early phase of evolution, the size of the post-corecollapse expansion follows a scaling relation r h ∝ M -1 / 3 0 t 2 / 3 (Goodman 1984; Kim, Lee, & Goodman 1998; Baumgardt, Hut, & Heggie 2002), and thus for a given initial mass and epoch, r h /r h, 0 is simply proportional to r -1 h, 0 . Figure 3 indeed shows that the size of the GCs with the same M 0 tend to converge to a single value ( r h /r h, 0 ∝ r -1 h, 0 ), if the GCs have small t rh, 0 (log t rh, 0 / yr /lessorsimilar 9). Mass loss by stellar evolution causes GCs to adiabatically expand to maintain virialization, and the GC sizes evolve following r h /r h, 0 ∝ M 0 /M when the stellar evolution is the main driver of the GC size evolution (Hills 1980). The combination of Kroupa IMF and the stellar evolution recipe described by Schaller et al. (1992) yields a mass loss of ∼ 40% within 13 Gyr. Thus, GCs would expand by a factor of ∼ 1 . 67 as a result of the stellar evolution, if two-body relaxation or the galactic tides are relatively less important in driving the size evolution. Indeed, clusters with log t rh, 0 / yr /greaterorsimilar 9 and /Rfractur < 0 . 05 have r h, 13 /r h, 0 values between 1 and 2. While stellar evolution and two-body relaxation cause clusters to expand, galactic tides make clusters shrink in general. A cluster extending farther than r J (overfilling; r t > r J ) loses stars outside r J within a few dynamical timescales, and this naturally causes the mean size of the cluster to decrease. Since r J ∝ R G ( M/M G ) 1 / 3 where M G is an enclosed mass of the Milky Way in a given R G , the size decrease caused by the galactic tides takes place mostly while the cluster approaches R p . The cluster re-expands somewhat by two-body relaxation while approaching R a (Baumgardt & Makino 2003), but its size gradually decreases while repeating orbital motions. We find that clusters with 0 . 4 < r t /r J < 1 can also shrink moderately as a result of the galactic tides even if it underfills, and clusters initially with r t /r J < 0 . 4 (or /Rfractur < 0 . 05; i.e., 'isolated' GCs) can gradually move into the 'tidal' regime as they lose mass or expand by stellar evolution or two-body relaxation. Figure 3 shows that GCs with larger /Rfractur 0 are smaller at 13 Gyr for a given M 0 and R G, 0 , as expected. Among various initial GC parameters, r h, 0 is the most important parameter in the size evolution caused by twobody relaxation ( r h /r h, 0 ∝ M -1 / 3 0 r -1 h, 0 ) and that resulting from galactic tides ( /Rfractur 0 ∝ M -1 / 3 0 R -2 / 3 G r h, 0 for a flat rotation curve). For this reason, initially small GCs gen- ly expand (by two-body relaxation), while initially large GCs generally shrink (by the galactic tides) as they evolve. The size evolution of intermediate GCs is determined by more than one dynamical effect, and some GCs can even maintain their initial size over their whole lifetime.", "pages": [ 3, 4 ] }, { "title": "5. SYNTHESIS OF FOKKER-PLANCK CALCULATIONS", "content": "As discussed in Section 3, we performed a total of 1152 FP calculations with different initial cluster conditions in four-dimensional parameter space, M , r h , R a , and e . The goal of the present study is to find the initial distribution of these variables that best describe the observed GCs. For the initial MF model, we adopt a Schechter function, and for the initial RD model, we use a softened power-law function, We assume that the initial MF is independent of initial R G . For the sake of simplicity, we do not parameterize the distribution for e , and adopt the fixed isotropic distributions, i.e., dN ( e ) ∝ e de . Unlike M , the R G of each FP model evolves by oscillating between R p and R a , and thus the model RD at 13 Gyr constructed from our population synthesis may suffer from significant random noise. To reduce this noise, we build a model RD by summing the probability distributions between R p and R a that are given by the orbital information at 13 Gyr, and we call this a phase-mixed RD. Hereafter, RDs in this paper refer to the phase-mixed RD. For initial SDs, we use six distribution models (see Table 1). Models 1, 2, and 3 represent a Gaussian distribution of r h , ρ h (mean density within r h ), and /Rfractur , respectively, implying that the initial GCs have the preferred initial r h , ρ h , and /Rfractur , with dispersions. The initial r h of Model 1 does not correlate with the initial M or R G , while Models 2 and 3 have initial correlations of r h ∝ M 1 / 3 ρ -1 / 3 h and r h ∝ M 1 / 3 R G 2 / 3 /Rfractur . In Models 4, 5, and 6, the initial r h is determined by powers of initial M and/or R G . Note that the power of Model 6 ( r h ∝ M 0 . 615 ) corresponds to that of the masssize relation derived from the Faber-Jackson relation for early-type galaxies (Faber et al. 1989; Ha¸segan et. 2005; Gieles et al. 2010). Once the calculations of the 1152 FP models are done, the aforementioned sets of initial MF, RD, and SD models are used to search for the best-fit parameters in five to seven dimensional space, depending on the SD models (Models 1-6). For this, we synthesize our 1152 FP calculations with appropriate weights to produce a given initial MF, RD, and SD, and find a set of parameters that best fit the present-day MF, RD, and SD for each of the six SD models. When finding the best set of parameters for each SD model, we minimize the sum of χ 2 values from all of the L , R G , and R h histograms, which are constructed by using eight bins between 10 4 and 10 5 . 8 L /circledot for L , nine bins between 10 0 and 10 1 . 6 kpc for R G , and nine bins between 10 -0 . 6 and 10 1 . 2 pc for R h , all equally spaced on a logarithmic scale. Recall that we use dynamical (theoretical) properties M and r h for set- ing the initial distributions, while observable quantities such as L and R h are used for comparing the models and observations.", "pages": [ 4, 5 ] }, { "title": "6. BEST-FIT INITIAL DISTRIBUTION OF THE GALACTIC GLOBULAR CLUSTER SYSTEM", "content": "The best-fit parameter sets that minimize the χ 2 values between observations and our calculations are presented in Table 2 for the six SD models. We examine the characteristics of our best initial MFs, RDs and SDs in turn.", "pages": [ 5 ] }, { "title": "6.1. Initial Mass Function", "content": "The best-fit α values for all six SD models are quite low, ranging between 0.01 and 0.07. The best-fit log M s / M /circledot values for all six SD models are similar to each other, having values between 5.8 and 5.9. Note that Schechter functions with such small α values are similar to log-normal functions, while those of α /greaterorsimilar 2 are closer to power-law functions. Thus, our small α values suggest that log-normal functions better describe the initial MF of the Galactic GC system than power-law functions (see Figure 4 a ), and this result is consistent with the result of Paper I. One way to explain the log-normallike initial MF is expulsion of the remnant gas due to star formation in the pre-gas-expulsion cluster, which can quickly alter a power-law MF into a log-normal-like MF (Parmentier & Gilmore 2007). Another possible mechanism resulting in a rapid change in the initial MF is the collisions of clusters with dense clouds or other clusters during the early phase of the galaxy (Elmegreen 2010).", "pages": [ 5 ] }, { "title": "6.2. Initial Radial Distribution", "content": "Initial RDs from the best-fit parameter sets for all six SD models have similar β values (4.0-4.5) but a rather wide range of R s values (0.3-3.6 kpc), and this is consistent with the result of Paper I ( β = 4 . 2 and R s = 2 . 9 kpc). Figure 4 b shows that most of the GCs that disrupt before 13 Gyr are located in the bulge regime ( R G, 0 < 3 kpc), and most of the GCs formed in the bulge do not survive until now. We find that only 0.1-8.4 % of the total GC mass initially inside 3 kpc remains in GCs at 13 Gyr, and the total stellar mass that escaped from the GCs inside 3 kpc during the last 13 Gyr amounts to 5 × 10 7 -3 × 10 8 M /circledot , depending on the SD model.", "pages": [ 5 ] }, { "title": "6.3. Initial Size Distribution", "content": "The initial SDs from our best-fit parameter sets are of larger dispersion than the present-day SDs for all six SD models (see Figure 4 c ). The initial SDs evolve into the narrower present-day SDs by two main effects: (1) expansion of GCs with small r h, 0 , which normally have small t rh, 0 and/or small /Rfractur 0 , due to two-body relaxation, and (2) shrinkage (evaporation) of large r h, 0 GCs, which normally have large t rh, 0 and/or large /Rfractur 0 , due to the Galactic tides. Figure 5 shows that the SDs of initially small GCs (upper panels) indeed shift to the larger r h region and those of initially large GCs (lower panels) shift to the smaller r h region after 13 Gyr. Three p -values (significance levels) for χ 2 tests of LFs, RDs, and SDs are acceptably high, except for Model 3, which has relatively small χ 2 p -values for RDs and SDs (see Table 2). However, the high p -values from the χ 2 tests do not necessarily guarantee that the models with the best-fit parameters restore the observed correlation between L , R G , and R h as well. Thus, we implement Student's t -tests to see if our models with the best-fit parameters agree with the observed R G dependence of SDs (the R h -R G correlation), the L dependence of SDs (the R h -L correlation), and R G dependence of LFs (the L -R G correlation). For the R h -R G correlation, we calculate χ 2 for the difference of 〈 log R h 〉 and σ log R h between the model and the observation as follows: where subscripts o and m stand for the observation and the model, respectively, subscript j represents the equal number R G bins, and 〈 . . . 〉 denotes the averaged values. The same calculation is applied to R h -L and L -R G correlation as well. We find that Models 3-6 have t -test p -values that are too small ( /lessorsimilar 1%) for at least one of the R h -R G , R h -L , and L -R G correlation. For this reason, we reject Models 3-6 as being a plausible initial SD candidate. Hereafter, we call SD models 1 and 2 'the final best-fit SD models'. Figures 6 and 7 show our two remaining best-fit SD models, a Gaussian distribution of r h (model 1; r h,c = 6 . 4 pc, σ r h = 2 . 7 pc) and a Gaussian distribution of ρ h (model 2; ρ h,c = 690 M /circledot pc -3 , σ ρ h = 4 . 6 M /circledot pc -3 ). Note that r h, 0 values are not correlated with the R G, 0 in either model. This implies that the r h, 0 of GCs probably does not depend on the strength of the galactic tides. Therefore, we interpret the observed, present-day R h -R G correlation (see Figure 1 b ) as an outcome of a preferential disruption of the larger GCs at smaller R G due to the Galactic tides.", "pages": [ 5, 7 ] }, { "title": "7. DISCUSSION", "content": "The typical R h, 0 value from our final best-fit SD models (Models 1 and 2) is ∼ 4 . 6 pc ( r h, 0 ∼ 7 pc), and this is 1.8 times larger than that of the present-day GCs ( ∼ 2 . 5 pc). This result is rather different from a recent argument by Baumgardt et al. (2010) that most GCs were born compact with r h, 0 < 1 pc. Our result implies that GCs initially have a rather wide SD, the typical value of which is similar to that of YMCs in parsec scale, and have evolved to have a narrower SD with a smaller mean value. We also find that GCs formation favors a 'tidal' environment over an 'isolated' environment. The number of tidal GCs ( /Rfractur 0 > 0 . 05) at 0 Gyr from our final best-fit SD models is approximately five times larger than that of isolated GCs ( /Rfractur 0 < 0 . 05). The ratio of tidal to isolated GCs, however, drastically decreases as GCs evolve because tidal GCs are more easily disrupted, and this ratio becomes ∼ 0 . 2 at 13 Gyr. Figure 8 shows the initial SDs of the GCs that survive until 13 Gyr in Models 1 and 2. We find that these initial SDs are broader ( σ ( R h ) = 2 . 1 and 2.5 pc, respectively) and centered at higher values ( R h = 4.1 and 4.0 pc) than the currently observed SD ( σ ( R h ) = 1 . 2 pc, R h = 2 . 5 pc). Thus, the overall size of the GCs were larger at birth than now by a factor of ∼ 2 even when only the surviving GCs are considered. The initial total masses in GCs ( M t, 0 ) of the final best-fit SD models are 2.8 × 10 8 M /circledot (Model 1) and 5.3 × 10 8 M /circledot (Model 2), and the masses that have left the GCs during the lifetime of the Galaxy (∆ M t ) are 2.5 × 10 8 M /circledot (Model 1) and 5.0 × 10 8 M /circledot (Model 2). These give ∆ M t /M t, 0 values of 0.89 and 0.94 for Models 1 and 2, respectively. Our ∆ M t values are several times larger than previous estimates made by Baumgardt (1998, 4.0-9.5 × 10 7 M /circledot ), Vesperini (1998, 5.5 × 10 7 M /circledot ), and Paper I (1.5-1.8 × 10 8 M /circledot ). Our larger ∆ M t values are due to the facts that (1) we consider virtually all disruption mechanisms in the calculations for the dynamical evolution of individual GCs, and (2) we use more a flexible initial r h distribution, which can have a relatively larger fraction of GCs with a large r h (larger GCs are more vulnerable to the galactic tide). Note that ∆ M t will be larger if one considers the clusters that have been disrupted in the process of remnant gas expulsion. We note that contrary to the finding in the present paper, detailed dynamical modeling of individual clusters shows that at least some of the clusters must have started with a very small size. For example, Monte Carlo calculations by Heggie & Giersz (2008) and Giersz & Heggie (2009, 2011) find 0.58 pc, 0.40 pc, and 1.9 pc as best-fit initial r h values for the observed current states of M4, NGC 6397, and 47 Tuc, respectively. These values are several times smaller than the typical initial r h found for the Galactic GC system from our calculations, ∼ 7 pc. However, we also note that the Monte Carlo models used for these three clusters all assume circular cluster orbits while M4 and NGC 6397 have moderate to high orbit eccentricities (0.82 and 0.34, respectively). We have performed several FP calculations for these two clusters and find that consideration of appropriate eccentric orbits can increase the best-fit initial r h by a factor of 3-5.", "pages": [ 7, 8 ] }, { "title": "8. SUMMARY", "content": "We have calculated the dynamical evolution of Galactic GCs using the most advanced and realistic FP model, and searched a wide parameter space for the best-fitting initial SD, MF, and RD models that evolve into the present-day distribution. We found the initial MF of the Galactic GC system is similar to the log-normal function rather than the power-law function, and the RD of the GC system undergoes significant evolution inside R G = 3 kpc through the strong Galactic tides. We also found that the initial SD of the GC system evolves to narrower present-day SDs through two effects: shrinkage of large GCs by the galactic tides and expansion of small GCs by two-body relaxation. The typical initial projected half-mass radius from the final best-fit model, ∼ 4 . 6 pc, is 1.8 times larger than that of the present-day value, ∼ 2 . 5 pc. The ratio of 'tidal' GCs to 'isolated' GCs is ∼ 5 at 0 Gyr and decreases down to ∼ 0 . 2 at 13 Gyr. Since tidal GCs are found to be dominant in the beginning, one might expect the initial size of the GCs to be correlated with the Jacobi radius, i.e., to be a function of the galactocentric radius. However, our final best-fit SD models (Models 1 and 2) do not seem connected to the galactocentric radius. This implies that the GC formation process favors a certain size and density, regardless of the tidal environment. Such a R G -independent initial SD evolves into a present-day SD, which shows a tight r h -R G correlation through evaporation and two-body relaxation. We thank Holger Baumgardt and Mark Gieles for helpful discussion. This work was supported by Basic Science Research Program (No. 2011-0027247) through the National Research Foundation (NRF) grant funded by the Ministry of Education, Science and Technology (MEST) of Korea. This work was partially supported by WCU program through NRF funded by MEST of Korea (No. R31-10016). J.S. deeply appreciates Koji Takahashi for the help with his FP models. S.J.Y. acknowledges support by the NRF of Korea to the Center for Galaxy Evolution Research and by the Korea Astronomy and Space Science Institute Research Fund 2011 and 2012. S.J.Y. thanks Daniel Fabricant, Charles Alcock, Jay Strader, Nelson Caldwell, Dong-Woo Kim, and Jae-Sub Hong for their hospitality during his stay at Harvard-Smithsonian Center for Astrophysics as a Visiting Professor in 20112012.", "pages": [ 8 ] }, { "title": "REFERENCES", "content": "A., Lynden-Bell, D., & Terlevich, R. J. 1989, ApJS, 69, 763 D. 2006, ApJ, 627, 203", "pages": [ 8 ] }, { "title": "Best-fit parameters for initial GC distributions", "content": "Note . -The p value for the χ 2 test ( t -test) is the probability of having a χ 2 ( t ) value that is larger than the value obtained from our χ 2 ( t ) test between the model and the observation, whose degree of freedom is 8 or 9 (4). r h and r h,c are in units of pc, and ρ h and ρ h,c are in units of M /circledot / pc 3 .", "pages": [ 9 ] } ]

[ { "title": "ABSTRACT", "content": "In order to determine what processes govern the assembly history of galaxies with rotating disks, we examine the stellar mass Tully-Fisher relation over a wide range in redshift partitioned according to whether or not galaxies contain a prominent bulge. Using our earlier Keck spectroscopic sample, for which bulge/total parameters are available from analyses of HST images, we find that bulgeless disk galaxies with z > 0 . 8 present a significant offset from the local Tully-Fisher relation whereas, at all redshifts probed, those with significant bulges fall along the local relation. Our results support the suggestion that bulge growth may somehow expedite the maturing of disk galaxies onto the TullyFisher relation. We discuss a variety of physical hypotheses that may explain this result in the context of kinematic observations of star-forming galaxies at redshifts z = 0 and z > 2. Subject headings: galaxies: evolution - galaxies: fundamental parameters - galaxies: kinematics and dynamics - galaxies: spiral", "pages": [ 1 ] }, { "title": "UNCOVERING DRIVERS OF DISK ASSEMBLY: BULGELESS GALAXIES AND THE STELLAR MASS TULLY-FISHER RELATION", "content": "Sarah H. Miller 1,2,3 , Mark Sullivan 4,1 , & Richard S. Ellis 2 Draft version June 9, 2021", "pages": [ 1 ] }, { "title": "1. INTRODUCTION", "content": "A major goal in galaxy evolution studies is to fundamentally understand the evolving dynamical and morphological forms of galaxies (Roberts 1969). The favored method of tracking the assembly of stellar mass as a fraction of the total mass in rotationally-supported galaxies is the redshift-dependent Tully-Fisher (TF) relation, which was first explored at z ∼ 1 by Vogt et al. (1996, 1997). Subsequent studies of the stellar mass ( M ∗ )-TF relation at intermediate-to-high redshifts revealed scatters ∼ 3 × larger than that of the local relation (e.g., Conselice et al. 2005; Kassin et al. 2007; Vergani et al. 2012). This increased scatter was initially thought to represent a weaker coupling between stellar and dynamical mass, precluding detailed studies of the evolution in either slope or normalization. However, we showed in Miller et al. (2011) and Miller et al. (2012) (hereafter, M11 and M12, respectively) with data of improved signal/noise and refined modeling techniques, that the M ∗ -TF relation is actually well-established at z glyph[similarequal] 1 with a scatter comparable to that seen in the local relation. Moreover, in M12, we demonstrated that the relation holds for most disk galaxies since z glyph[similarequal] 1 . 7, thereby posing a challenge of how to explain the rapid evolution in kinematic behavior since z ∼ 2 where star-forming galaxies are morphologically-irregular and dispersion dominated (Forster Schreiber et al. 2006; Law et al. 2007; Forster Schreiber et al. 2009). To the extent that a TF relation can be examined at z glyph[similarequal] 2 (Cresci et al. 2009; Gnerucci et al. 2011), a normalization increase of 0.4 dex is seen over glyph[similarequal] 1 Gyr to z glyph[similarequal] 1.5, in contrast to only 0.02 ± 0.02 dex over the subsequent 9 Gyr. Since in M12 the TF scatter is observed to decline by 60% from z glyph[similarequal] 1.7 to z glyph[similarequal] 1, in this present paper we [email protected] seek to examine whether this might arise from physical properties governing the evolution onto the TF relation. We target our attention on the morphological appearance of each galaxy, specifically the bulge-to-total ratio. Bulgeless disks representing at least 15% of local galaxy populations (Kormendy et al. 2010) provide an interesting challenge for hierarchical ΛCDM galaxy formation (which leads to inevitable bulge-growth without substantial feedback: Robertson et al. 2006; Governato et al. 2010). We test whether the high redshift M ∗ -TF relation can be better understood when tracking the mature, bulge-dominated population of galaxies separately from the evolving population of bulgeless systems experiencing a more secular formation process. Throughout the paper we adopt a Chabrier (2003) initial mass function and a Ω Λ = 0.7, Ω m = 0.3, H 0 = 70 km s -1 Mpc -1 cosmology. All magnitudes refer to those in the AB system (Oke 1974).", "pages": [ 1 ] }, { "title": "2. DYNAMICAL DATA AND STELLAR MASSES", "content": "The key measurements required to follow the evolving M ∗ -TF relations are disk kinematics as parameterized through rotation curve model fits, and stellar mass estimates derived from multi-band photometric data. Our earlier papers (M11, M12) describe the relevant data and their reduction in considerable detail so we provide only a brief summary here. Our spectroscopic sample was selected from Hubble Space Telescope (HST) Advanced Camera for Surveys (ACS) imaging data in various survey fields complete to an apparent magnitude of i =22.5 and is morphology inclusive, containing irregular and merging systems as well as regular spirals with and without bulges. Keck spectroscopy was undertaken for 236 galaxies with 0.2 < z < 1.3 at a median spectral resolution of 30 km s -1 using the DEep Imaging Multi-Object Spectrograph(DEIMOS Faber et al. 2003) and, subsequently, 70 1 . 0 glyph[lessorsimilar] z < 1 . 7 galaxies were targeted at a median resolution of 57 km s -1 with the Low Resolution Imaging Spectrograph (LRIS Oke et al. 1995) equipped with a redsensitive CCD. An unique aspect of both spectroscopic campaigns was the use of long exposure times (4-8 hrs) essential for tracking the rotation curves to the flattening radius (see M1 for details). Rotation curves were derived using various emission lines (H α , [O ii ], and [O iii ] depending on the galaxy redshift. As discussed in M11, we account for position-dependent dispersion and emission brightness profile, convolved with the seeing, and adopt an arctangent function. We use inclination-corrected fiducial velocity measurements at 3.2 times the disk scale radius. The final sample for consideration here comprises 171 galaxies for which rotation curves could be determined (this is all galaxies except for spectrally compact or passive galaxies: see M11 & M12 for details). Stellar mass estimates are determined using a combination of ground-based K -band infrared imaging, multiband optical photometry, and spectroscopic redshift information using the spectral energy distribution (SED) fitting technique first utilized by Brinchmann & Ellis (2000). Measured magnitudes in multiple bands were applied using a Bayesian code based on the precepts discussed in Kauffmann et al. (2003), and later Bundy et al. (2005). Using probability distribution functions that incorporate uncertainties in the photometry, the stellar mass uncertainty is better than 0.2 dex for 83% of our sample.", "pages": [ 1, 2 ] }, { "title": "3. MORPHOLOGICAL DATA", "content": "Our primary goal is to investigate the possible role bulge formation may play in the apparent rapid evolution of the M ∗ -TF relation from z glyph[similarequal] 2 to z glyph[similarequal] 1. We facilitate this investigation with galfit (Peng 2010). As we required disk scale lengths for earlier applications, the bulge-to-disk decomposition procedure described is similar to that in M11, M12, and Miller (2012) and so only briefly discuss the procedure here. We run galfit on each galaxy 1000 times, varying the initial parameters in gaussian distributions based on their SExtractor (Bertin 1996) values. For each object we attempt to fit a deVaucouleurs bulge profile and an exponential disk component, where the fit parameters are the center position, total magnitude m tot , effective radius R e (scale radius, r s , for an exponential disk), S'ersic index n (fixed to n = 4 for deVaucouleurs and n = 1 for disk), axis ratio q , and position angle φ . Where physical bulge solutions are not found, we re-fit the galaxy with an index-varying single S'ersic component (indices typically lie between 1 < n < 4). Such cases generally represent disk galaxies which are bulgeless and/or irregular. Disk sizes, inclinations and position angles were taken from best-fit disk components if more than one component was fit. Final parameter uncertainties from the Monte Carlo distributions are better than 5% on average, and we add these uncertainties in quadrature to the photometric errors from galfit . The scale radii, position angles and inclinations are typically measured better than 10%. Uncertainties are propagated through to TF parameters, resulting in larger errors for those galaxies which are difficult to constrain. In the DEIMOS sample, ∼ 40% were adequately fit using a 2-component decomposition, and ∼ 60% benefitted from a single n -varying S'ersic profile fit. In the LRIS sample, the relevant percentages were ∼ 63% and ∼ 37%, respectively. This serves as a good indication of the morphological distribution of our sample; less than half are well-formed spirals with a clear bulge ( § 2). Where HST data is available in multiple bands we compared galfit runs between bands to test for differences in the scale radius determination as a function of redshift. The scale radii are consistent among the bands indicating no significant redshift-dependent bias (less than 5% in the DEIMOS sample and < 10% for that of LRIS). In order to maximize signal/noise we use the galfit results from the reddest available filter (F814W or F850LP). Acrucial issue affecting classification at high redshift is the 'morphological k-correction' - the change in apparent morphology with increasing redshift following the drift blue ward in rest-frame wavelength. This is potentially troublesome for z > 1 where the the F814W and F850LP images sample the younger star-forming regions rather than the older, redder populations that dominate the stellar mass at lower redshift. As such, there is a danger of underestimating the bulge contribution. The HST near infrared Wide Field Camera 3 (WFC3/IR) provides a F160W filter, which at 1 < z < 2 provides rest-frame optical light and is therefore ideal for the bulge-to-disk decompositions we seek. While deep WFC3/IR F160W imaging from the CANDELS survey (Koekemoer et al. 2011) is available for one-fifth of our sample, the majority of the combined LRIS and DEIMOS samples are unfortunately in GOODS North (the WFC3/IR coverage of which will not complete for at least another year). However, for the purposes of this paper we seek only to demonstrate that use of the ACS data to classify the sample broadly into bulge-less and bulge-dominated subsets does not induce significant biases. As we discuss below, we will split our overall sample according to a dividing bulge-to-total ratio ( B/T ) = 0.1. With this division, we find, for the data with present WFC3/IR coverage, that morphological classifications into these two categories are consistently made between the WFC3/IR and ACS data for 85% of the total sample.", "pages": [ 2 ] }, { "title": "4. RESULTS", "content": "We now examine the stellar mass Tully-Fisher ( M ∗ -TF) relation partitioned by morphology, in terms of the bulge/total ratio, B/T . To facilitate this investigation, we separate our sample according to the HST-derived galfit results into galaxies with prominent bulges and those without (bulgeless disks and irregulars) as described above. We plot the M ∗ -TF relation in four redshift bins (0 . 2 < z ≤ 0 . 5, 0 . 5 < z ≤ 0 . 8, 0 . 8 < z ≤ 1 . 2, 1 . 2 < z ≤ 1 . 7) ensuring nearly equal sub-samples and look back time intervals (see Fig. 1). Using the method described in M11, we fit inverse linear regressions to each subsample and z -bin using a fixed slope (of 3.70), the value of which was derived by fitting a free slope to the entire sample. In the two highest redshift bins (0 . 8 < z ≤ 1 . 2, 1 . 2 < z ≤ 1 . 7), we see bulgeless disks are significantly offset in the stellar mass (y-axis) normalization from that of the local relation by -0 . 23 ± 0 . 06 dex and -0 . 34 ± 0 . 07 dex, respectively. In contrast, disks with significant bulges do not deviate significantly from the local relation, nor in fact do bulgeless disks in the two lower redshift bins (Fig. 2). The presence of a bulge appears to secure a disk galaxy on the M ∗ -TF relation to within a scatter of 0.2 dex. The question then arises as to whether increased scat- ter around the total M ∗ -TF relation from z glyph[similarequal] 1 to z glyph[similarequal] 1 . 7 can be accounted for largely via the inclusion of bulgeless galaxies. In M12, the scatter from z ∼ 1 to z glyph[similarequal] 1 . 7 increased up to 60%, whereas scatter across the 3 lower bins does not significantly evolve. Since the bulge-separated relations in the highest redshift bins have tighter relations separately than when both samples are combined, it seems likely that increased scatter can be attributed to the zero-point shift of the bulgeless relation. The paucity of lower mass galaxies in the 0 . 8 < z ≤ 1 . 2 z -bin arising from the K -band magnitude limit applied for the DEIMOS sample likely complicates this inference (TF scatter increases to lower masses, e.g., Begum et al. 2008). We note that the LRIS sample forming the basis of the highest z -bin was not K -band limited. To allow for various differences between sample sizes and distributions, we quantify the offset significance for the bulgeless subsample using a Monte Carlo 'boot-strap' analysis. We fit two TF relations to randomly-selected subsamples ( × 10,000, with replacement) of our full data sample (across all redshifts), ensuring subsamples of the same size (N) for each z -bin (N=21/22, 34/15, 26/20, 21/12, for bulgeless/bulge-dominated sets respectively). With each pair of randomly selected subsamples, we measure their relative normalization offset, and fit Gaussians to the resulting histograms of the offset distributions (where the FWHM of each distribution is 0.082, 0.099, 0.086, 0.112 dex for each bin in increasing redshift). We also conduct an additional boot-strap analysis of N=10,000 where we select subsamples at random within redshift bins (and also with replacement). This latter method, which accounts for redshift-dependence in the errors, results in broader normalization offset distributions (where FWHMs are 0.098, 0.110, 0.131, 0.132, respectively), and these distributions are plotted in the top left corner of each redshift-bin panel of Fig. 1. The true normalization offset observed in each bin relative to the bootstrapped distribution of the latter method yields offset significances of 1.19 σ , 0.88 σ , 3.14 σ , 3.07 σ for each bin in increasing redshift. Together this translates to a confidence interval of greater than 99 . 8% that the relation of the bulgeless galaxies are offset at highz due to a genuine effect rather than random error or scatter. The slight decline in significance in the highest bin is due to the reduced number of galaxies with bulges in that bin, even though in real terms the offset of the bulgeless relation is greatest in the highest z -bin ( -0 . 34 ± 0.07).", "pages": [ 2, 3, 4 ] }, { "title": "5. DISCUSSION", "content": "In light of our results, we explore explanations of the offset co-evolution of stellar mass and total mass assembly in bulgeless galaxies from those which experienced earlier bulge growth. We first consider in § 5.1 a favored picture in the literature of z ∼ 2 studies, and then turn in § 5.2 and § 5.3 to what may be a more self-consistent and simple picture for our results.", "pages": [ 4 ] }, { "title": "5.1. Clump formation and migration", "content": "At z ∼ 2, low ratios of rotation-to-dispersion velocity support in disks ( V / σ < 2 -5, Forster Schreiber et al. 2006, 2009) have been interpreted in a simple picture whereby disks fragment and the resulting clumps migrate to the centers of galaxies to form bulges (Noguchi 1999; Immeli et al. 2004; Romeo et al. 2010). Importantly, the velocity dispersion measured from such studies comes from the ionized gas. The dispersion in the emission lines could be dominated by energy or momentum driven stellar feedback, rather than dispersion from dynamical pressure (as would be traced by the stars and cold gas). In our sample, while there is a spread in V/σ values from 2-15 across all redshift bins (90% never rise above V/σ =10), no clear trend in σ itself exists with respect to morphology or redshift. These values are more similar to the locally observed spread in V/σ of ionized gas than those found in massive, star- ing galaxies at z > 2 (0 < V/σ < 5, Genzel et al. 2008). Furthermore, the clump migration picture does not explain why bulgeless galaxies arrive on the M ∗ -TF relation by intermediate redshift without forming bulges. In the clump migration picture, gas-rich bulgeless and barless disks would continue to suffer gravitational instabilities and clump formation until a bulge had formed and supposedly stabilized the disk. Some other mechanism is needed to explain the stabilization of bulgeless disks that have yet to form a substantial bulge/bar. Additionally, central stellar velocity dispersions can support an increased stellar mass in the form of an accumulating bulge without significantly changing the rotational velocity of the surrounding disk. This would shift galaxies above the TF relation as their bulges grow without increasing rotational support, which is not supported by our results.", "pages": [ 4 ] }, { "title": "5.2. An underestimation of gas masses in highz bulgeless disks", "content": "A simpler explanation for our results is that bulgeless galaxies have higher gas fractions in their disks as compared to the rest of the sample. By adding total gas masses to our stellar masses, we may find the baryonic Tully-Fisher is universal at all redshifts in rotationallysupported galaxies. To explore this possibility without direct gas mass measurements, we conduct the following exercise. We compare the gas estimates from the empiricallybased analytical method from M11 (Method 1), to the estimates based on the Kennicutt-Schmidt (K-S) relation (Method 2). Described in detail in M11, Method 1 uses local gas fractions based on a galaxy's stellar mass and self-consistently adds the integrated specific starformation rate from the galaxy's redshift (accounting for a gas recycling fraction based on the assumed Chabrier IMF). In Method 2, the K-S gas masses are determined from the rest-frame B -magnitude surface brightnesses, which are used to estimate a SFR density and thus a gas mass according to the inverted K-S relation (Kennicutt 1998). This comparison reveals that the high redshift bulgeless disks have a significantly diminished B-band magnitude per area compared to the rest of the sample. That is to say Method 1 results in 1.11 times more gas than Method 2 with a 10% scatter in bulgeless disks, whereas this factor is 1.43 in the rest of the sample. If we increase the bulgeless disk gas mass estimates 30% so that they align with the rest of the sample, then a universal baryonic TF relation is restored. Physically, a correction of this nature to our gas mass estimates suggests less metal-enriched gas in bulgeless disks, which is less efficient at forming stars. The relative youth of the bulgeless disks may be due to the lack of enriching outflows re-condensing at z ∼ 1 (but do by z ∼ 0). A lengthened 'fountain' duty-cycle could reflect a shallower gravitational potential and lower starformation surface densities (i.e., Oppenheimer & Dav'e 2008; Finlator & Dav'e 2008, etc.), where outflows are slower to re-condense and metals more likely to escape in supernova-driven winds. Also the simultaneous growth of supermassive black holes (SMBHs) at the centers of galaxies with their bulge mass is well known(Ferrarese & Merritt 2000). SMBHs are likely fed by gas that has sunk to the centers of galaxies via disk instabilities and/or mergers, a process which would similarly grow a bulge. It is then unsurprising that galaxies with no central bulge may have a much larger fraction of their gas still in their disks. While the predictions of this exercise await testing via direct gas mass observations, it does suggest interesting implications for a universal baryonic-TF relation (so far confirmed only locally, e.g. McGaugh 2012). Looking towards the future, these predictions can be tested by determining the molecular and neutral hydrogen components of these galaxies, via facilities such as the Atacama Large Millimeter/submillimeter Array (ALMA), or future radio facilities (e.g., MeerKAT, or ultimately the Square Kilometer Array, SKA).", "pages": [ 4, 5 ] }, { "title": "5.3. A z -dependent transition mass", "content": "Locally, low-mass bulgeless galaxies ( M ∗ < 10 9 M glyph[circledot] ) tend to fall below the extrapolated M ∗ -TF relation from M ∗ > 10 9 M glyph[circledot] (Matthews et al. 1998; Stark et al. 2009). Since this offset is similar to that seen in our high-redshift bulgeless galaxies, it suggests that probing further down the stellar-mass function at 0 . 2 < z < 0 . 8 may uncover a similar transition to an offset of bulgeless disks at an intermediate mass between that observed at z ∼ 0 and at z > 1. The physical significance of an evolving transition mass for bulgeless galaxies could be understood via the 'downsizing' concept (e.g. Cowie et al. 1996) or an evolving 'mass floor' in galaxy formation theory (e.g. Bouch'e et al. 2010). These models seek to explain why more massive galaxies formed earlier and faster than lower mass galaxies, regardless of environment (appropriate for our study since the role of environment would be subtle in our field sample). These models are ultimately driven by the cosmic decline in accretion rate, shutting down assembly of massive galaxies first by quickly consuming their reservoirs. A combination of the evolving ultraviolet background with photo-ionizing radiation from the first stars could create a transition mass, above which the cooling efficiency is relatively higher, and below which a lack of self-shielding keeps smaller, thinner disks from remaining neutral (keeping molecular gas collapsing to form GMCs). Galaxies with bulges maintaining thicker disks in thicker, steeper potential-wells could self-shield, and thus form stars more efficiently than thin, bulgeless disks being adversely affected by photo-ionizing radiation in shallower potential-wells. In attempting to better understand drivers of disk assembly from our results, we note a number of tensions regarding the picture where disks settle from the migration of large clumps into central bulges. Rather, a more self-consistent picture could be provided by an universal baryonic-TF relation, where better accounting of gas in highz disks could explain the offsets seen in tracking stellar mass with the total rotational support in galaxies. This may also predict a redshift-dependent transition mass which lowers with the age of the universe, below which bulgeless disks assemble their mass offset from the locally-defined M ∗ -TF relation, but not the baryonic-TF relation. SHM thanks the Rhodes Trust and BFWG for supporting this work. MS acknowledges support from the Royal Society. We thank K. Bundy for stellar mass estimates and spectral reduction, as well as A. Newman for spectral reduction, and helpful discussions with T. Treu. Facilities: Keck I (LRIS), Keck II (DEIMOS), HST.", "pages": [ 5 ] }, { "title": "REFERENCES", "content": "Begum, A., Chengalur, J. N., Karachentsev, I. D., et al. 2008, MNRAS, 386, 138 Bertin, E., & Arnouts, S. 1996, A&AS, 117, 393 Bouch'e, N., Dekel, A., Genzel, R., et al. 2010, ApJ, 718, 1001 Brinchmann, J. & Ellis, R. S. 2000, ApJ, 536, L77 Bruzual, G. & Charlot, S. 2003, MNRAS, 344, 1000 Bundy, K., Ellis, R. S., & Conselice, C. J. 2005, ApJ, 625, 621 Chabrier, G. 2003, PASP, 115, 763 Conselice, C. J., Bundy, K., Ellis, R. S., et al. 2005, ApJ, 628, 160 Cowie, L. L., Songaila, A., Hu, E. M., et al. 1996, AJ, 112, 839 Cresci, G., Hicks, E. K. S., Genzel, R., et al. 2009, ApJ, 697, 115 Faber, S. M., Phillips, A. C., Kibrick, R. I., et al. 2003, SPIE Conference Series, Vol. 4841, ed. M. Iye & A. F. M. Moorwood, 1657-1669 Ferrarese, L. & Merritt, D. 2000, ApJ, 539, L9 Finlator, K. & Dav'e, R. 2008, MNRAS, 385, 2181 Forster Schreiber, N. M., Genzel, R., Bouch'e, N., et al. 2009, ApJ, 706, 1364 Forster Schreiber, N. M., Genzel, R., Lehnert, M. D., et al. 2006, ApJ, 645, 1062 Genzel, R., Burkert, A., Bouch'e, N., et al. 2008, ApJ, 687, 59 Gnerucci, A., Marconi, A., Cresci, G., et al. 2011, A&A, 528, A88+ Governato, F., Brook, C., Mayer, L., et al. 2010, Nature, 463, 203 Immeli, A., Samland, M., Westera, P. et al. O. 2004, ApJ, 611, 20 Kassin, S. A., Weiner, B. J., Faber, S. M., et al. 2007, ApJ, 660, L35 Kauffmann, G., Heckman, T. M., Tremonti, C., et al. 2003, MNRAS, 346, 1055 Kennicutt, Jr., R. C. 1998, ApJ, 498, 541 Koekemoer, A. M., Faber, S. M., Ferguson, H. C., et al. 2011, ApJS, 197, 36 Kormendy, J., Drory, N., Bender, R., et al. 2010, ApJ, 723, 54 Kriek, M., Labb'e, I., Conroy, C., et al. 2010, ApJ, 722, L64 Law, D. R., Steidel, C. C., Erb, D. K., et al. 2007, ApJ, 656, 1 Mannucci, F., Cresci, G., Maiolino, R., et al. 2010, MNRAS, 408, 2115 McGaugh, S. S. 2012, AJ, 143, 40 Miller, S. H., Bundy, K., Sullivan, et al. 2011, ApJ, 741, 115 Miller, S. H., Ellis, R. S., Sullivan, M. et al. 2012, ApJ, 753, 74 Noguchi, M. 1999, ApJ, 514, 77 Peng, C. 2010, Bulletin of the AAS, Vol. 42, 578-+ Roberts, M. S. 1969, AJ, 74, 859 Romeo, A. B., Burkert, A., & Agertz, O. 2010, MNRAS, 407, 1223 Stark, D. V., McGaugh, S. S., & Swaters, R. A. 2009, AJ, 138, 392 Vergani, D., Epinat, B., Contini, T., et al. 2012, ArXiv e-prints, astro-ph/1202.3107 Vogt, N. P., Forbes, D. A., Phillips, et al. 1996, ApJ, 465, L15+ Vogt, N. P., Phillips, A. C., Faber, S. M., et al. 1997, ApJ, 479, L121+ Zibetti, S., Gallazzi, A., Charlot, S. et al. 2012, ArXiv e-prints", "pages": [ 5 ] } ]

[ { "title": "ABSTRACT", "content": "We have compiled a large sample of 151 high redshift ( z = 0 . 5 -4) galaxies selected at 24 µ m ( S 24 > 100 µ Jy) in the GOODS-N and ECDFS fields for which we have deep Spitzer IRS spectroscopy, allowing us to decompose the mid-infrared spectrum into contributions from star formation and activity in the galactic nuclei. In addition, we have a wealth of photometric data from Spitzer IRAC/MIPS and Herschel PACS/SPIRE. We explore how effective different infrared color combinations are at separating our mid-IR spectroscopically determined active galactic nuclei from our star forming galaxies. We look in depth at existing IRAC color diagnostics, and we explore new colorcolor diagnostics combining mid-IR, far-IR, and near-IR photometry, since these combinations provide the most detail about the shape of a source's IR spectrum. An added benefit of using a color that combines far-IR and mid-IR photometry is that it is indicative of the power source driving the IR luminosity. For our data set, the optimal color selections are S 250 /S 24 vs. S 8 /S 3 . 6 and S 100 /S 24 vs. S 8 /S 3 . 6 ; both diagnostics have ∼ 10% contamination rate in the regions occupied primarily by star forming galaxies and active galactic nuclei, respectively. Based on the low contamination rate, these two new IR color-color diagnostics are ideal for estimating both the mid-IR power source of a galaxy when spectroscopy is unavailable and the dominant power source contributing to the IR luminosity. In the absence of far-IR data, we present color diagnostics using the WISE mid-IR bands which can efficiently select out high z ( z ∼ 2) star forming galaxies.", "pages": [ 1 ] }, { "title": "GOODSHERSCHEL : SEPARATING HIGH REDSHIFT ACTIVE GALACTIC NUCLEI AND STAR FORMING GALAXIES USING INFRARED COLOR DIAGNOSTICS", "content": "Allison Kirkpatrick 1 , Alexandra Pope 1 , Vassilis Charmandaris 2,3,4 , Emmanuele Daddi 5 , David Elbaz 6 , Ho Seong Hwang 7 , Maurilio Pannella 6 , Douglas Scott 8 , Bruno Altieri 9 , Herve Aussel 6 , Daniela Coia 9 , Helmut Dannerbauer 10 , Kalliopi Dasyra 6 , Mark Dickinson 11 , Jeyhan Kartaltepe 11 , Roger Leiton 6,12 , Georgios Magdis 13 , Benjamin Magnelli 14 , Paola Popesso 14 , Ivan Valtchanov 9 to be submitted to ApJ", "pages": [ 1 ] }, { "title": "1. INTRODUCTION", "content": "In the current narrative of galaxy evolution, star formation and the growth of supermassive black holes are intertwined. The star formation rate density of the Universe peaks from z ∼ 1 -3 (e.g., Bouwens et al. 2009; Magnelli et al. 2011; Murphy et al. 2011), an epoch in which the black holes within the center of massive galaxies are simultaneously building up their mass (Wall et al. 2005; Kelly et al. 2010). There is compelling evidence that the growth of black holes and the buildup of stellar mass is linked, though the processes that regulate, and ultimately quench, the simultaneous growth of the stellar mass and black hole are not yet fully disentagled (e.g., Mullaney et al. 2012). To study the properties of active galactic nuclei (AGN) and star-forming galaxies at z ∼ 1 -3, when the star formation rate in the most massive galaxies begins to decline, it is necessary to first identify systems likely harboring an AGN. Because AGN are much more luminous in the X-ray than star-forming galaxies, X-ray detection provides one of the best means of identifying an AGN (e.g., Alexander et al. 2003). Deep field surveys with the Chandra X-ray Observatory have exposed an AGN population out to z ∼ 5 (e.g., Brandt et al. 2001; Giacconi et al. 2002). However, detailed X-ray spectral analysis shows that the majority of sources are obscured by gas and dust (see Brandt & Hasinger 2005 for a review), and there is likely a sizeable fraction of AGN that remain undetected by Chandra surveys, as evidenced by the fact that nearly half of the X-ray background is unresolved at > 6 keV (Worsley et al. 2005), and the ratio of observed obscured to unobscured AGN at high redshifts is lower than what is found for comparably luminous AGN in the local Universe (e.g., Treister & Urry 2005). Due to the incompleteness of surveys conducted at X-ray wavelengths, we must employ alternate methods insensitive to dust obscuration to identify the presence of an AGN. The infrared portion of the spectral energy distribution (SED) shows clear signs of both AGN and star formation (SF) activity. Based on the presence of polycyclic aromatic hydrocarbons and continuum thermal dust emission, mid-infrared spectra can be decomposed into the relative contributions of SF and AGN activity (e.g., Laurent et al. 2000; Armus et al. 2007; Sajina et al. 2007; Pope et al. 2008; Kirkpatrick et al. 2012a). At 1 . 6 µ m, SF dominated galaxies will exhibit a stellar bump due to emission from older stellar populations, whereas as the dust surrounding luminous AGN heats up, it will radiate into the near-IR and midIR, creating a pure power-law spectrum at these wavelengths. Finally, the peak and shape of the far-IR emission depends on the temperature of the dust, which becomes warmer as the AGN grows more luminous (e.g., Haas et al. 2003; Sanders et al. 1988). Due to AGN signatures from the near-IR to the far-IR, infrared colorselection techniques are a promising way to select out AGN missed by X-ray surveys, or when X-ray data are unavailable. In this work, we explore the IR color space for a sample of 151 high redshift luminous infrared galaxies (LIRGs, L IR = 10 11 -10 12 L /circledot ) and ultra luminous infrared galaxies (ULIRGs, L IR > 10 12 L /circledot ) at high redshift ( z ∼ 0 . 5 -3 . 5) with deep Spitzer mid-IR spectroscopy and a suite of multiwavelength photometry spanning 3 . 6 -500 µ m. Several studies have explored using Spitzer IRAC colors as a means to separate AGN (Lacy et al. 2004; Stern et al. 2005; Donley et al. 2012). For the first time, we explore these various diagnostics using a sample of mid-IR spectroscopically determined AGN and SF galaxies. In light of the emerging Wide-field Infrared Survey Explorer (WISE) photometry, we also discuss how well mid-IR diagnostics separate our spectroscopic AGN and SF sources. The focus of the paper, however, lies in combining near-, mid-, and far-IR photometry to distinguish our AGN from the SF galaxies. The far-IR is tracing the bulk of the IR luminosity, and if a galaxy's mid-IR power source is also affecting the far-IR, we expect that combining both portions of the spectrum will be a useful diagnostic. We build on a previous paper (Kirkpatrick et al. 2012a, hereafter Paper I) in which we separate our sample into AGN and SF dominated, based on mid-IR spectral decomposition, and analyze the average IR SEDs of each, including mid-IR spectral features, IR luminosity, and dust temperatures. We have spectroscopically determined the nature of the mid-IR power source for our individual sources, and we have created SEDs that represent the average features of our SF and AGN sources. Now we look for advantageous color-color cuts that can be used to select out AGN candidates when spectroscopy is unavailable. We explore different combinations of colors using both the photometry of our individual sources, as well as redshift tracks from the composite SEDs, to determine color combinations that separate AGN from SF galaxies. The paper is laid out as follows: in Section 2, we describe details of our sample, the mid-IR spectral decomposition of individual sources, and the composite spectral energy distributions (SEDs) we create from this sample. In Section 3, we explore the efficacy of existing IRAC color selection techniques for high redshift sources. In Section 4, we present new color diagnostics based on combining photometry from the Spitzer Space Telescope and the Herschel Space Observatory . We apply our new diagnostics to the broader GOODS-N and ECDFS fields in Section 5, and we end with our conclusions in Section 6. Throughout this paper, we assume a standard cosmology with H 0 = 70 kms -1 Mpc -1 , Ω M = 0 . 3 and Ω Λ = 0 . 7.", "pages": [ 1, 2 ] }, { "title": "2. DATA", "content": "A full description of our sample and the composite SEDs we create from it is given in Paper I. Here, we summarize the main details and results.", "pages": [ 2 ] }, { "title": "2.1. Multiwavelength data", "content": "Our sample consists of 151 high redshift galaxies from the Great Observatories Origins Deep Survey North (GOODS-N) and Extended Chandra Deep Field Survey (ECDFS) fields. We include all sources in these fields that were observed with the Spitzer IRS. While this sample contains a diverse range of sources depending on the goals of each individual observing program, the overlying selection criteria is that each source must be detected at 24 µ mwith a flux of S 24 /greaterorsimilar 100 µ Jy, since anything fainter will not be observable with the IRS in less than 10 hours. More details on this database of IRS sources in GOODSN and ECDFS can be found in our data paper (Pope et al. in preparation). The GOODS fields have been extensively surveyed and are rich in deep multiwavelength data including: ground-based imaging in the near-IR ( J and K bands) from VLT/ISAAC (Retzlaff et al. 2010) and CFHT/WIRCAM (Wang et al. 2010; Lin et al. 2012); Chandra 2 Ms X-ray observations (Alexander et al. 2003; Luo et al. 2008); 3.6, 4.5, 5.8, 8.0 µ m from the Infrared Array Camera (IRAC) on Sptizer ; IRS peak-up observations at 16 µ m(Teplitz et al. 2011) and MIPS imaging at 24 and 70 µ m (Magnelli et al. 2011). Recently, GOODSN and GOODS-S have been surveyed with the GOODSHerschel Open Time Key Program (P.I. David Elbaz, Elbaz et al. 2011) using both the PACS and SPIRE instruments providing deep photometry at five far-IR wavelengths: 100, 160, 250, 350, and 500 µ m. For the present study, we combine space-based imaging from Spitzer and Herschel to obtain 12 photometric bandwidths spanning the near-IR to the far-IR. For sources lacking a detection at the Herschel wavelengths, we extract a measurement of the flux density and associated uncertainty for each source directly from our images. The images are in units of mJy/beam, so we use the 24 µ m prior positions to find the appropriate pixels for each galaxy. We do not take a measurement when a source looks too blended on the image itself. In Paper I, we present complete Herschel photometry for our IRS sources, as well as indicating which sources have measurements and which have detections at the Herschel wavelengths. We performed spectral decomposition of the Spitzer IRS mid-IR spectrum for each source in order to disentangle the AGN and SF components. We follow the technique outlined in detail in Pope et al. (2008) which we summarize here. We fit the individual spectra with a model comprised of three components: (1) The SF component is represented by either the local starburst composite of Brandl et al. (2006) or simply the mid-IR spectrum of the prototypical starburst M82 (Forster Schreiber et al. 2003) - with the SNR, wavelength coverage and spectral resolution of our high redshift spectra both give equally good fits to the SF component of our galaxies. (2) The AGN component is determined by fitting a pure power-law with the slope and normalization as free parameters. (3) An extinction curve from the Draine et al. (2003) dust models is applied to the AGN component. The extinction curve is not monotonic in wavelength and contains silicate absorption features, the most notable for our wavelength range being at 9 . 7 µ m. We fit all three components simultaneously and integrate under the PAH and continuum components to determine the fraction of the mid-IR luminosity ( ∼ 5 -12 µ m) from SF and AGN activity, respectively. For each source, we quantify the strength of the AGN in terms of the percentage of the total mid-IR luminosity coming from the AGN continuum component. Based on this mid-IR spectral decomposition, we find that 38 (25%) out of our sample of 151 galaxies are dominated ( ≥ 50% of luminosity) in the mid-IR by an AGN.", "pages": [ 2, 3 ] }, { "title": "2.2. Galaxy Classifications", "content": "To more thoroughly compare the mid-IR spectral properties and full IR SEDs within our sample, we divide our sample into four primary subsamples based on the results of the mid-IR spectral decomposition. First, each galaxy is classified as either SF- or AGN- dominated based on having < 50% and > 50% AGN contribution in the midIR, respectively. We further divide the SF galaxies into two bins: z ∼ 1 ( z < 1 . 5) and z ∼ 2 ( z > 1 . 5). The AGN sources are likewise separated into two bins: those with measurable 9 . 7 µ m silicate absorption (hereafter referred to as silicate AGN) and those without (hereafter referred to as featureless AGN). We are unable to further classify four AGN sources as they lack spectral coverage in the relevant range (9 -10 µ m) to determine whether they exhibit silicate absorption. We refer to these as unclassifiable AGN in the relevant figures. Our four primary subsamples are listed in Table 1 along with their median redshifts and observed frame median 24, 100, and 8 µ m flux densities. While the majority of our sources that are classified as SF-dominated based on the mid-IR spectra have a negligible ( < 20%) contribution from an AGN, the AGN dominated sources exhibit varying degrees of concurrent SF activity. The featureless AGN (lacking silicate absorption) primarily have a very strong AGN continuum accounting for 80 -100% of the mid-IR emission, whereas the silicate AGN have a more uniform distribution of AGN fraction (50 -100%) with some silicate AGN also having weak PAH features.", "pages": [ 3 ] }, { "title": "2.3. Composite SEDs", "content": "To assess the average properties of our primary four subsamples of galaxies (due to the small number, we do not examine the unclassifiable AGN in detail), we created composite SEDs from 0 . 3 -600 µ mrest-frame by combining data from ground-based near-IR; Spitzer IRAC, IRS, and MIPS (24 µ m, 70 µ m); Herschel PACS and SPIRE. We reject any source with less than 3 measurements or detections in the far-IR bandwidths, or any source without mid-IR data in the range 6 . 4 -7 . 5 µ m, as this is the range used to normalize the individual SEDs. After normalization, we stacked flux densities in bin sizes of ∼ 0 . 1 µ m below 20 µ m. Above 20 µ m, we fit all data points with a two-temperature modified blackbody. Full details of these composites are found in Paper I. The composites are shown in Figure 1 with each composite offset on the y-axis to allow for easier comparisons. Composite SEDs are publicly available 15 . The two SF composites are remarkably similar in shape and have most emission from cold dust. The featureless AGN composite is nearly a pure power-law until ∼ 20 µ m, and then is relatively flat from ∼ 20 -100 µ m. The silicate AGN SED is a power-law in the near-IR, has weak PAH features and silicate absorption at 9 . 7 µ m, has warm dust emission around 20 µ m, and has a cold dust component peaking at the same wavelengths as the SF SEDs. The difference in shapes between the two AGN SEDs and the SF SEDs suggests that IR color diagnostics could be useful for separating AGN from SF galaxies.", "pages": [ 3 ] }, { "title": "3. IRAC COLOR-COLOR DIAGNOSTICS", "content": "Some of the most well used IR color diagnostics for separating AGN and SF galaxies are presented in Lacy et al. (2004) and Stern et al. (2005) and utilize IRAC colors. The motivation behind an IRAC selection technique is that, at these wavelengths, luminous AGN should have a monotonically increasing SED, and these power-law colors will separate AGN from SF galaxies in colorspace. With our high redshift photometry and spectroscopy, we are able to apply these diagnostics to a large sample of mid-IR spectroscopically determined AGN and SF galaxies. Lacy et al. (2004) and Stern et al. (2005) define IRAC color-color regions to separate AGN based on large surveys of low redshift ( z /lessorsimilar 0 . 7) galaxies. As we move to higher redshift ( z ∼ 2), the IRAC bandwidths begin to probe the stellar bump; our composite AGN and SF SEDs exhibit different shapes in the λ < 4 µ m region of the spectrum, indicating that IRAC color diagnostics might also be useful at higher redshift. We apply the diagnostics of Stern et al. (2005) and Lacy et al. (2004) We list the upper and lower quartile values in brackets next to each calculated median. a We list the number of sources in each sub-sample that are used to create the composite SEDs in parentheses. We do not include sources with an incomplete IR SED when creating the composites to avoid biasing our results (See § 2.3). to our sample (Figure 2). Sources in our sample that we determined through mid-IR spectral decomposition to be dominated by an AGN ( > 50%) in the mid-IR are plotted in blue (see the online version for color figures), and sources dominated by star formation are plotted in green. In the left panel of Fig. 2, we plot our sources in the IRAC color-color space defined in Stern et al. (2005) with the gray shaded region being the area defined as AGN-dominated by the authors. Our SF sources separate cleanly according to redshift and largely avoid the gray shaded region. We have also overplotted the redshift tracks of our z ∼ 1 SF composite SED and our z ∼ 2 SF composite SED. We calculate the redshift tracks of each of our composite SEDs by convolving the SEDs with the IRAC bandpass filters (and MIPS, PACS, SPIRE, and WISE filters in § 4) at the appropriate wavelengths for a given redshift. The convolution acts to smooth out much of the noise in our spectra as the individual filters span a fairly large wavelength range. The z ∼ 2 SF track contaminates the AGN region around z ∼ 1, and in fact, all of the z ∼ 1 SF galaxies (green filled circles) occupying the shaded region have a redshift between 1 and 1.5. The divergence of the two SF tracks around z ∼ 1 is due to the fact that the 3 . 6 µ m filter is tracing the bluest portion of the stellar bump, which has differing strengths relative to the other IRAC filters for the two SF composite spectra (see Fig. 1 and the right panel of Fig. 2 ). Our individual z ∼ 1 sources have colors that are located around both the z ∼ 1 and z ∼ 2 tracks indicating a spread in the obscuration of the IRAC colors for these sources. The spread of the individual photometry points in relation to the composite SED is illustrated in Paper I. The AGN sources (blue symbols) are not completely constrained to the gray shaded region, though our redshift track from the featureless AGN composite SED indicates they should be on average, illustrating a lack of homogeneity in the near- and mid-IR photometry among mid-IR classified AGN. Most AGN in our sample do not have a clear stellar bump in the near-IR which restricts them to the shaded region (only 8% (2) of the AGN sources in the grey region have a visible stellar bump). Of the outlying high redshift AGN in this plot, the majority (70%) possess a weak visible stellar bump which pushes them into the regions of the color space occupied by SF galaxies. Eleven (50%) of the silicate AGN, 2 (17%) of the featureless AGN, and 1 (25%) of the unclassifiable AGN lie outside the gray region. One of the advantages of this study is that we are able to investigate how well the mid-IR power source determines the near- and far-IR power source. Clearly, our mid-IR dominated AGN do not uniformly have IRAC colors indicative of an AGN. Furthermore, 11 of the AGN contaminating the SF region do not have far-IR colors expressive of an AGN (see § 4). The middle panel of Fig. 2 is the color space defined by Lacy et al. (2004) where the shaded region is used to identify AGN. The vast majority of our sources occupy this region, regardless of their power source diagnosed from the mid-IR spectrum (see also Donley et al. 2008, 2012). The SF sources again show a clean redshift separation as the IRAC channels sample the stellar bump. As neither subsample of AGN similarly exhibits a redshift separation, we do not plot them with different symbols according to redshift. The redshift tracks of the featureless AGN and silicate AGN composite SEDs occupy the same portion of the graph, so we only plot the silicate AGN SED track in orange; the track lies in the upper portion of the graph. We overplot both SF SED redshift tracks ( z ∼ 1 in red and z ∼ 2 in purple) as they show an interesting separation. At low redshifts ( z < 2), the z ∼ 1 SF SED track lies just outside the shaded region, while the z ∼ 2 SF SED lies inside it, but after z = 2, the tracks lie on top of each other. This difference in the color tracks between our z ∼ 1 and z ∼ 2 composite SEDs is likely due to the different intrinsic L IR of each subsample. The z ∼ 1 sources are on average LIRGs ( L IR ∼ 4 × 10 11 L /circledot ) and therefore have less dust to obscure the IRAC colors than our z ∼ 2 ULIRG SED composite; adding more dust to a SF galaxy will cause it to shift towards the top-left of this plot, which is exactly the shift we see between our z ∼ 1 and z ∼ 2 SED tracks. At z ≥ 3, the tracks approach the area occupied by the AGN, but as none of our SF sources possess a redshift this high, we are unable to determine if our sources follow our tracks into the upper portion of the graph. Both of the SF SEDs accurately trace the redshift separation exhibited by our sources. In the right panel of Fig. 2, we show on our composite SEDs the effective wavelengths of each of the IRAC bandwidths at redshifts 1, 2, and 3 (blue, green, and red, respectively). These bands straddle the stellar bump over this redshift range. Both AGN SEDs have a power-law shape over these bands causing little evolution in IRAC colorspace. The high degree of contamination in these IRAC colorcolor diagnostics by high redshift SF galaxies motivated Donley et al. (2012) to create more restrictive criteria than originally presented in Lacy et al. (2004) and Stern et al. (2005). Donley et al. (2012) uses a sample of AGN identified at optical and X-ray wavelengths to construct the new criteria, and now we are able to test these criteria using mid-IR spectroscopic AGN. We apply the new color cut presented in Donley et al. (2012) to our sample in Figure 3. The authors determined the gray shaded region was the most effective at selecting out AGN with power-law spectra in the IRAC bandwidths, and indeed, all of our sources that meet the more restrictive criteria have flux densities such that S 3 . 6 < S 4 . 5 < S 5 . 8 < S 8 . 0 . Based on our high redshift sources, we propose the simple IRAC color cuts of (solid lines in Fig. 3) for selecting potential power-law AGN candidates, similar to the Donley et al. (2012) criteria. The IRAC color selection techniques, even using the more restrictive cuts presented in Donley et al. (2012), still miss a large fraction (39% of the present sample) of mid-IR spectroscopically confirmed AGN, specifically most of the more obscured silicate AGN. Furthermore, such diagnostics cannot conclusively determine if an AGN is significantly contributing to the bolometric luminosity of a galaxy. Since an IRAC color diagnostic applied at high redshift ( z /greaterorsimilar 1.3) is necessarily based on separating sources into AGN- or SF-dominated based on the shape of the spectrum in the near-IR, such diagnostics might not be the most desirable for determining the dominate power source of dust obscured galaxies in the mid-IR and far-IR regime.", "pages": [ 3, 4, 5 ] }, { "title": "4. NEW COLOR-COLOR DIAGNOSTICS", "content": "Our large sample of high redshift SF and AGN sources, identified with deep mid-IR spectroscopy, and wealth of multiwavelength photometry allows us to define new color-color diagnostics that are well suited to uncovering galaxies harboring an AGN as revealed in the mid-IR spectrum, and galaxies with a bolometrically important 250 24 AGN. We seek to combine multiple portions of the IR spectrum that are probing the physical nature of each galaxy with different pieces of information. At 3 . 6 µ m, SF galaxies at z ∼ 1 -2 will have a stellar bump due to the underlying stellar population, while this effect might be washed out in luminous AGN, producing a power-law shape. Similarly, at these redshifts, the 24 µ m filter will straddle the PAH complexes at 7.7 and 12 . 7 µ m, which will be weakened or absent in a bright AGN. However, there is a caveat that at z ∼ 1 . 5, the silicate absorption feature present in some AGN spectra, falls into the 24 µ m bandwidth which can produce colors that mimic SF galaxies. On the other hand, the bandwidth is large enough that we do not expect this to be a significant source of contamination. The far-IR should have a different shape based on the relative amounts of cold and warm dust emission present, and SF galaxies will have relatively more cold dust than AGN, while AGN have an increased amount of warm dust (Paper I). Finally, the 8 µ m filter, at z ∼ 1 -2, covers a relatively featureless portion of the spectrum, so when combined with filters that are tracing features, should act as a base to distinguish between AGN and SF systems. We used available photometry from Herschel and Spitzer ranging from 3 . 6 -350 µ m (observed) and explored every possible color combination. In addition, we also look at where the composite SEDs lie to get a sense of where we should expect to find AGN and SF galaxies, on average. For color-color plots in which the AGN were well separated from the SF sources, we calculated the contamination rates of each region. We find that a color diagnostic spanning the full range of the IR spectrum does the best job of separating both AGN with pure power-law spectrum from 1 -10 µ m and AGN with silicate absorption from SF galaxies at high redshift. We define two new color-color diagnostics which, based on having a low contamination and clarity of separation, are the optimal diagnostics to employ when a full suite of IR photometry exists.", "pages": [ 5, 6 ] }, { "title": "4.1. S 250 /S 24 vs. S 8 /S 3 . 6", "content": "We find that combining longer wavelength photometry from Herschel with mid-IR photometry from Spitzer MIPS/IRAC provides the most reliable separation between our mid-IR classified AGN and SF galaxies since they probe the widest range of dust properties affected by AGN and SF activity. The AGN are loosely separated by the presence or absence of silicate absorption at 9 . 7 µ m in this color-color space. We plot both subsamples according to redshift (filled symbols are z ∼ 1 and open symbols are z ∼ 2). The featureless AGN do not contaminate the SF region and lie further to the left (lower S 250 / S 24 ) than the silicate AGN, due to an excess of warm dust emission in these sources. We overplot the redshift track computed from the featureless AGN SED. The track has some slight evolution along the x-axis with redshift, consistent with the individual data points. The silicate AGN (filled and open squares, according to redshift), on the other hand, show no evolution along the x-axis with redshift but show some spread along the y-axis. The fact that some silicate AGN sources contaminate the SF region can be attributed to two effects, namely that some individual sources possess a weak stellar bump and some sources have more relative cold dust emission in the far-IR. The silicate AGN SED redshift track exhibits no evolution in either direction and so is plotted as a star. Both color axes produce a separation of AGN and SF sources. Based on the location of our sources, we calculate the separation line to be drawn as the bold solid line in Fig. 4. In the absence of all four wavelengths, we find that the majority of our AGN sources satisfy either of the following color criteria shown as the dashed lines: The separation along individual axes is particularly useful if only an upper limit for S 250 is available when searching for AGN. We quantify the contamination of the SF and AGN regions (separated by the diagonal solid line): 10% of the sources (11 of 111 total) in the SF region are AGN and 2 of the 22 sources (9%) in the AGN region are SF dominated. None of the contaminating AGN are power-law dominated. Some of the individual silicate AGN have a stellar bump (41%), causing contamination along the S 8 /S 3 . 6 axis. Futhermore, several of our silicate AGN (8, or 42%) have significant amount of cold dust emission in the far-IR, producing S 250 /S 24 flux density ratios greater than 1.3. The high S 250 /S 24 flux density ratio correlates with the presence of a stellar bump in the silicate AGN, with 32% possessing both. Finally, it is worthwhile to note that the contaminating AGN are significantly fainter at 24 µ m (median S 24 = 220 µ Jy) than the AGN lying in the upper region of the graph (median S 24 = 1240 µ Jy). Indeed, all but one of the properly identified AGN have S 24 > 300 µ Jy. Based on our sample of 24 µ m bright sources, our diagnostic is optimized to select AGN with S 24 > 300 µ Jy, though if the physics is similar in lower luminosity AGN, they will have similar SEDs and colors, in which case our diagnostics can also be used to select out AGN dominated sources. S 250 , in comparison with S 24 , is an indicator of the relative amount of cold dust in a galaxy. As an AGN becomes more powerful, the relative amount of warm dust increases, so that the ratio of S 250 /S 24 decreases. For galaxies lacking a significant amount of warm dust, the cold dust will be the dominant contributor to the L IR . Low S 250 /S 24 ratios in our mid-IR AGN indicate that the warm dust, indicative of an AGN, is significantly contributing to the far-IR emission, and accordingly, the AGN is an important contributor to the bolometric luminosity. Our diagnostic is therefore more powerful than the IRAC diagnostics presented in § 3 since it selects AGN significantly contributing to the total IR emission. Our new diagnostic is a definite improvement over the IRAC selection criteria presented in Lacy et al. (2004) and Stern et al. (2005), though photometry is scarcer. In our sample of 151 galaxies, 99% have IRAC data so both Lacy et al. (2004) and Stern et al. (2005) can be applied, while only 88% have the relevant wavelengths to satisfy our new diagnostic. We verify the more restrictive IRAC criteria of Donley et al. (2012), shown in Fig. 3, with our IRS sample, whose mid-IR power source has been determined via spectral decomposition. Our new diagnostic is a slight improvement over the revised IRAC criteria presented in Donley et al. (2012). In the present case, 35% of our AGN sources are misclassified as SF galaxies, while according to the restricted IRAC criteria (equations (1) and (2)), 39% are misclassified. In addition, with the new criteria, we correctly identify two sources as AGN that were misclassified using only the IRAC colors. Though the improvements gained in recovering mid-IR AGN by combining colors covering the entire IR spectrum is only slight over using IRAC colors alone, an added strength of our new criteria lies in the fact that mid-IR data from the Wide-field Infrared Survey Explorer (WISE) can be easily substituted for Spitzer data. The WISE 22 µ m and 3.4 µ m channels correspond to Spitzer 3.6 µ m and 24 µ m bandwidths. We have used the redshift tracks of our templates to determine that the WISE 12 µ mchannel is the optimal substitute for Spitzer 8 µ m (see § 4.4). The middle panel of Fig. 4 illustrates the different SED tracks shown on the left panel. 250 µ m, 24 µ m, 8 µ m, and 3 . 6 µ m at z = 1, 2, 3 are indicated on the composite SEDs. S 250 traces the far-IR peak of each template while S 24 traces the mid-IR emission - the different ratios of mid-IR to far-IR emission (or warm to cold dust) in AGN and SF sources produces the observed color separation in S 250 /S 24 . At the lower wavelengths, S 8 /S 3 . 6 remains fairly constant for both high redshift AGN SED templates. As some of our individual AGN sources do show signs of weak SF activity as well as AGN activity, several of our AGN possess a noticeable stellar bump which causes a spread in S 8 /S 3 . 6 . Of the 11 AGN sources contaminating the SF region, 60% possess a visible stellar bump compared with only 14% of AGN sources in the AGN region. The z ∼ 1 SF composite SED illustrates how a stellar bump would cause a change in S 8 /S 3 . 6 with redshift. In the right panel of Fig. 4, we overplot other SED templates in our new color-color space, namely the SEDs of local galaxies and high redshift submillimeter galaxies (SMGs, Pope et al. 2008). These local SEDs come from combining all known data on these sources including available IRAS (12, 25, 60, 100 µ m), Spitzer (24, 70, 160 µ m), and SCUBA (850 µ m) photometry as well as mid-IR spectroscopy (e.g., Forster Schreiber et al. 2003; Armus et al. 2007). Mrk 231 (red) lies in the AGN region we defined, although contrary to our high redshift AGN SED templates, it does exhibit some evolution along the y-axis. M 82 (blue) does not enter the SF region until z = 1 which is most likely due to the fact that it's SED peaks at a lower IR wavelength than the majority of our high redshift sources (see Paper I). Both NGC 6240 (orange) and the high redshift SMG composite (purple) lie in the SF region except at very low redshifts ( z /lessorsimilar 0 . 5) and follow the same general redshift evolution as our sources, that is, increasing S 250 /S 24 color with increasing redshift. The SMG composite reaches even higher S 250 /S 24 colors than our sample which is consistent with their selection at submm wavelengths. The consistency of these other local and high redshift templates with our new color diagnostics reinforces our confidence in applying this color selection to a wider range of IR luminous, 24 µ m bright galaxies at high redshift. 100 24", "pages": [ 6, 7, 8 ] }, { "title": "4.2. S 100 /S 24 vs. S 8 /S 3 . 6", "content": "In the absence of longer wavelength SPIRE data, we find that we can substitute S 100 for S 250 and we still see a nice separation between the SF and AGN galaxies. In the left panel of Figure 5, we use the colors S 100 /S 24 and S 8 /S 3 . 6 to define a region that separates our high redshift SF and AGN dominated galaxies (diagonal solid line). The line of separation is where mid-IR AGN lie above the line. We plot the uncertainties on our redshift tracks as the hashed lined regions. We opt to not to plot these uncertainties in the other plots presented in this work (Figs. 2 and 4) for the sake of clarity, and the ranges covered by the uncertainties in this plot are indicative of the spread in the previous figures. As discussed in detail in Paper I, the uncertainties on the composites were calculated by a bootstrapping technique, which indicates how the scatter in the data points affects the calculated median luminosity by resampling with replacement. The uncertainties on our composites are not calculated directly from the intrinsic scatter in the data, but are the standard deviation of the calculated luminosity after resampling the data 10,000 times. Therefore, it is not surprising that uncertainties on the template tracks do not encompass the full spread of all data points, particularly the silicate AGN. The SF region has only a 12% contamination (11 of 92 total galaxies) by AGN sources. The SF dominated systems show a clear separation with redshift. We overplot the redshift track of the z ∼ 1 SF composite SED in red, and it traces out the evolution exhibited by our individual sources. There is only one SF source that lies inside the AGN region (causing a contamination rate of 7%), and it not only has a high redshift ( z = 2 . 57) but also has a 47% AGN contribution to the mid-IR. The silicate AGN exhibit no redshift evolution in this color space, but the featureless AGN display a weak separation. The AGN composite SED tracks (orange and pink) do not move much vertically with redshift since these sources have a simple power-law shape and lack a stellar bump in the rest-frame near-IR. The middle panel of Fig. 5 shows three of our composite SEDs with lines illustrating where the relevant photometry bandwidths lie at a given redshift. The featureless AGN have a relatively flat spectrum from 20 -100 µ m, so the 100 µ m flux does not change with redshift whereas the 24 µ m flux decreases, causing the evolution along S 100 /S 24 . This is not the case for the silicate AGN. The slope from the mid-IR to the far-IR is relatively constant, producing little change in S 100 /S 24 at increasing redshifts. We have presented a simple IR color-color plot that separates our high redshift AGN and SF sources. In the right panel of Fig. 5, we test our diagnostic by overplotting the redshift tracks of local templates and a high redshift SMG SED. The local AGN Mrk 231 (red dashed line) lies well outside the SF region while the starburst M82 (blue) and local ULIRG NGC 6240 (orange) lie inside it for the most part. The SMG redshift track (purple) also lies in the SF region as expected since most SMGs are star formation dominated (e.g., Pope et al. 2008). Substituting S 160 instead of S 100 also works well for separating out the AGN (it has the same contamination rates mentioned above). However, the SF sources do not have a strong redshift separation, although, since S 160 is still probing the cold dust at lower redshifts, there is a stronger separation along the x-axis. The S 100 /S 24 vs. S 8 /S 3 . 6 colorspace can be used to select SF galaxies based on redshift as well as looking for AGN. The z < 1 . 5 SF galaxies have higher S 100 /S 24 ratio since the 100 µ m filter is tracing the cold dust. At redshift of 2, the 100 µ m filter is now tracing the warm dust, and the 24 µ m flux density is boosted by the 8 µ m PAH complex, producing a lower S 100 /S 24 ratio, similar to what is seen for the AGN. The similar S 100 /S 24 colors for both z 2 SF galaxies and AGN keeps this color alone from being an accurate indicator of the bolometrically important power source, unlike S 250 /S 24 .", "pages": [ 8, 9 ] }, { "title": "4.3. Far-IR Color Selection", "content": "As an AGN becomes luminous enough to dominate the mid-IR spectrum, it can heat the dust in the host galaxy causing a shift in the SED to warmer average dust temperatures and an increased importance of warm dust to the bolometric luminosity. Based on this, we might expect that just the Herschel PACS and SPIRE colors can can be used to preferentially select AGN sources. Hatziminaoglou et al. (2010) searched for a separation using S 350 /S 250 and S 500 /S 350 and found that in the SPIRE bands, their sample of AGN were indistinguishable from the non-active star forming galaxies. As many of our sources are not detected or blended at 500 µ m, we instead combine S 350 /S 250 with S 160 /S 100 and also find that it does not separate the mid-IR classified AGN and SF sources. Fig. 1 illustrates that at rest frame wavelengths greater than 40 µ m, the far-IR portion of the silicate AGN SED and both SF SEDs are all broadly consistent in shape explaining the lack of spread in colors. The less pronounced differences in far-IR portions of the composite SEDs is also reflected by the failure of S 250 thresholds alone to preferentially select AGN from the larger GOODS survey of galaxies (discussed below; see Table 3 and § 5). Our AGN sources, particularly the featureless AGN, are significantly brighter than our SF sources at 24 µ mand 8 µ m. Therefore, it is not surprising that we found the largest separation between the AGN and SF sources when combining mid-IR with far-IR photometry. We caution that with only far-IR information it is difficult to determine the impact of the AGN on the full IR SED. The most reliable selection of AGN candidates come from combining data from 3 . 6 µ m to 250 µ m ( § 4.1, § 4.2), which provides the most detail about the shape of IR spectrum.", "pages": [ 9 ] }, { "title": "4.4. Mid-IR Color Selection", "content": "There is now an abundance of mid-IR photometry as a result of two prominent mid-IR space telescopes: Spitzer and the Wide-field Infrared Survey Explorer (WISE). Past studies have explored using Spitzer color combinations to separate out mid-IR selected AGN and SF systems (e.g., Ivison et al. 2004; Pope et al. 2008), and emerging studies show that WISE colors can effectively separate IR luminous AGN (Eisenhardt et al. 2012; Stern et al. 2012; Yan et al. 2012). We are motivated by these studies to investigate how well the WISE photometry can separate our high redshift sources, particularly the SF galaxies. WISE has four transmission filters centered at 3.4, 4.6, 12, and 22 µ m, and though WISE photometry is less sensitive than Spitzer data, it has the advantage that it is an all-sky survey and can be used to search for high-redshift objects in regions of the sky not previously well-studied. We do not have WISE photometry for our sources, but we can use the WISE transmission filters to create synthetic photometry using the IRS spectra and the appropriate transmission filters at 12 and 22 µ m. For the 3.4 and 4.6 µ m filters, we substitute the appropriate IRAC photometry. We have applied a small correction to the IRAC photometry (0.94 for the 3 . 6 µ m filter and 1.04 for the 4 . 5 µ mfilter), which we have calculated by comparing the responses of our composite SEDs to each of the WISE and IRAC transmission filters. In addition, we calculate the redshift tracks of our composite SEDs by convolving with the appropriate WISE filters, and we plot our synthetic photometry and redshift tracks in Figure 6. Colors combining the first three channels, 3.4, 4.6, and 12 µ m, are capable of selecting hyperluminous ( L IR > 10 13 L /circledot ) galaxies, particularly AGN/QSOs (Eisenhardt et al. 2012; Stern et al. 2012; Yan et al. 2012). Stern et al. (2012) found that the WISE color cut [3 . 4] (AB) -[4 . 6] (AB) > 0 . 8 separates luminous AGN, which have been classified as such based on meeting the criteria in Stern et al. (2005, see Fig. 2). As a complement to these techniques, we would like to separate luminous SF galaxies at high redshift. We find that only combining the first three WISE channels does not effectively separate our SF sources. The strongest separation of our tracks is produced by combining all four WISE channels as shown in Fig. 6. We overplot the AGN criterion of [3 . 4] (AB) -[4 . 6] (AB) > 0 . 8 as the grey dashed line. Our AGN redshift track confirms the Stern et al. (2012) criterion for AGN with z ∼ 0 . 5 -1 . 0 and z ∼ 3. However, none of our sources have a power-law slope steep enough in the appropriate wavelength range to meet this criterion. Fig. 6 illustrates that the strongest separation of our sources and redshift tracks is for SF galaxies at z ∼ 2 due to prominent PAH features lying in the 22 µ mtransmission filter. The sensitivity depths at 12 and 22 µ m are ∼ 1 and ∼ 6 mJy, respectively (Wright et al. 2010). Given the sensitivity limits, at a redshift of ∼ 2, a SF galaxy would need to be at least a ULIRG and probably even a hyper-LIRG ( L IR > 10 13 L /circledot ) to be detected by WISE. A benefit of this color selection technique is that few, if any, extragalactic sources with luminosities less than the ULIRG threshold should occupy the same region as the z ∼ 2 ULIRGs. Therefore, our WISE color selection method is useful for selecting the brightest SF galaxies at z ∼ 2. In the absence of WISE photometry, Spitzer photometry can be used in similar combinations, with either 8 µ m or 16 µ m substituting for S 12 (e.g., Ivison et al. 2004; Pope et al. 2008).", "pages": [ 9, 10 ] }, { "title": "4.5. X-ray emission in AGN", "content": "One of the motivating reasons for IR diagnostics of a galaxy's power source is to search for moderately to heavily obscured AGN missed by X-ray surveys. We use Chandra surveys of the GOODS fields (Alexander et al. 2003; Luo et al. 2008) to determine if our diagnostics select galaxies that are obscured and lack an X-ray detection. For S 250 /S 24 vs. S 8 /S 3 . 6 , the X-ray detection fraction is 75% for AGN lying above the solid line and drops to 36% for AGN lying in the region dominated by SF galaxies. For S 100 /S 24 vs. S 8 /S 3 . 6 , the X-ray detection fraction for AGN above the solid line is 79%, comparable to the detection fraction for the S 250 /S 24 vs. S 8 /S 3 . 6 diagnostic. However, in the SF region, the X-ray detection fraction of AGN sources is 60%. Approximately 40% of the X-ray undetected AGN lie above the solid lines in both diagnostics, making our color selection criteria useful for selecting galaxies dominated by an AGN in the IR regardless of the level of obscuration. It is interesting to note that ∼ 60% of our mid-IR dominated AGN would not be classified as such based on either X-ray data or the S 250 /S 24 and S 8 /S 3 . 6 colors. The AGN embedded in these galaxies are not having a strong enough effect on the far-IR and near-IR portions of the spectrum to distinguish them from SF galaxies on the basis of colors, although the AGN are having an effect on the midIR emission. This could be a product of viewing angle, or possibly even an evolutionary sequence where as the AGN grows more luminous, it will be reflected in its IR colors (see Paper I for a full discussion).", "pages": [ 10 ] }, { "title": "5. APPLICATION OF NEW COLOR-COLOR DIAGNOSTIC TO ALL GOODSHERSCHEL GALAXIES", "content": "We now apply our new diagnostics defined above to broadly separate SF and AGN dominated galaxies in the whole GOODSHerschel survey (Elbaz et al. 2011). We plot all galaxies in GOODS-N and GOODS-S detected in all four bands on our two new color-color plots in Figure 7. We use the color cuts derived in Sections 4.1 and 4.2 (equations (3) and (6)) to separate sources dominated by AGN and SF activity in the mid-IR. Using the S 250 /S 24 vs. S 8 /S 3 . 6 plot (top panels), we have a total of 665 galaxies with detections in all four bandwidths in GOODS-N and GOODS-S, of which 58 (9%) lie in the AGN region. For S 100 /S 24 vs. S 8 /S 3 . 6 (bottom panels), we have 988 sources with detections in all four badwidths, of which 94 (10%) lie in the AGN region. Both diagnostic plots lead to a similar fraction of GOODSHerschel sources being AGN dominated in the mid-IR. It has been found that bright 24 µ m flux density correlates with mid-IR AGN indicators at high redshift and in local ULIRGs (Desai et al. 2007; Dey et al. 2008; Donley et al. 2008, respectively). Given this trend, we look to quantify the fraction of galaxies that are AGN dominated in the mid-IR as a function of flux density using our new color-color plots. In Fig. 7, we plot all GOODSHerschel galaxies with different colored symbols depending on their flux density (see legend for each panel). Our diagnostic was determined using 24 µ Jy bright sources, and applied to our own sample, the diagnostic primarily recovered AGN with S 24 > 300 µ Jy. Furthermore, the contribution of the AGN to the midIR has been shown to increase with increasing 24 µ m flux density, particularly at higher redshift ( z > 0 . 6 Brand et al. 2006). Therefore, it is perhaps not surprising that only one fainter source ( S 24 < 200 µ Jy) is found in the AGN region in Fig. 7. It is clear that a larger fraction of sources are in the AGN region of the colorcolor plot for the brightest 24 µ m and 8 µ m flux densities, whereas the 250 µ m and 100 µ m flux densities do not make as much of a difference to the AGN fraction. We quantify the fraction of sources in each flux bin that are classified as AGN and SF dominated based on these color-color plots in Tables 3 and 4. Our results show that the presence of an AGN is only visible as an excess in the mid-IR, while the far-IR photometry alone is largely insensitive to the physics of the nuclear power source. We therefore conclude that the flux limit of a mid-IR survey has a big effect on the fraction of sources which have a significant AGN whereas far-IR/submm fluxes correlate less strongly with AGN fraction. Using the data in Table 3, we plot the percentage of galaxies which are classified as AGN and SF greater than a given 24 µ m flux density in Figure 8. The fraction of AGN sources shows a steady increase with 24 µ m flux density. Above S 24 ∼ 750 µ Jy, the majority of galaxies selected at these wavelengths will be AGN dominated. In Table 1, we compare the intrinsic luminosities of our composite SEDs. Despite our AGN SEDs having an L IR below that of the z ∼ 2 SF SED, the AGN subsamples have a higher median S 24 flux density, particularly the featureless AGN. This effect is due to the differing ratios of far-IR to mid-IR luminosity. With the possible exception of z ∼ 2, where the brightest PAH complex falls in the 24 µ mfilter, at all redshifts AGN should comprise the majority of galaxies above S 24 > 750 µ mdespite possibly being less intrinsically luminous in the IR than SF galaxies (see Paper I for a direct comparison of our SEDs).", "pages": [ 10 ] }, { "title": "6. CONCLUSIONS", "content": "We have combined deep photometry from 3 . 6 -500 µ m with Spitzer mid-IR spectroscopy for 151 high redshift ( z > 0 . 5) (U)LIRGs in order to explore the relative positions of high redshift mid-IR AGN and SF galaxies in IR colorspace. We start by applying color-color diagnostics based on IRAC colors alone (Lacy et al. 2004; Stern et al. 2005) to our high redshift sample and find that there is significant contamination of SF sources in the AGN regions; however, our mid-IR spectroscopic Note . - Data corresponds to the top panels of Fig. 7 which shows GOODS-N and GOODS-S field galaxies plotted on our color-color diagnostic defined in Fig. 4. AGN are those lying above the solid line indicated on the plot. In this table, we calculate the percentage (number) of AGN and SF sources in a given flux density range for S 250 , S 24 , and S 8 . sample confirms the new IRAC color diagnostics from Donley et al. (2012). Adding 24 µ mphotometry does not effectively separate all of the AGN, but it does produce a separation of the SF galaxies according to redshift. The same separation of SF galaxies is produced by plotting our redshifted SED tracks in color-color graph using the WISE transmission filters. Our high redshift AGN and SF sources exhibit different Percentage of AGN and SF field galaxies according to S strength. Note . - Data corresponds to the bottom panels of Fig. 7 which shows GOODS-N and GOODS-S field galaxies plotted on our color diagnostic presented in Fig. 5. AGN sources (94 total) are those lying above the solid line indicated on the plot. In this table, we calculate the percentage (number) of AGN and SF sources in a given flux density range for S 100 , S 24 , and S 8 . spectral features and SED shape in the near-, mid-, and far-IR/submm wavelengths. We explore combining photometry from all three IR ranges for an optimal selection technique. In addition to photometry, we explore where AGNorSF sources should lie on average in different color combinations by using redshift tracks of our composite SEDs. We present two new color-color diagnostics combining Spitzer mid-IR and Herschel far-IR photometry that can be used to estimate whether a galaxy harbors an AGN in the absence of IRS spectroscopy. The optimal color diagnostics for bright S 24 sources spanning the full IR spectrum are S 250 /S 24 vs. S 8 /S 3 . 6 and S 100 /S 24 vs. S 8 /S 3 . 6 . These diagnostics have a low contamination rate of ∼ 10% in each of the SF and AGN occupied regions, and the contaminating AGN chiefly have S 24 < 300 µ Jy. Moreover, the S 250 /S 24 color estimates if an AGN is significantly contributing to the bolometric output of a galaxy. The mid-IR photometry from WISE can also be used in conjunction with Herschel photometry to separate out AGN. When only limited data is available, either of the colors S 250 /S 24 or S 8 /S 3 . 6 can be used alone to select AGN. We apply our new color-color diagnostics to the entire GOODSHerschel survey to determine the fraction of sources which are dominated by AGN and SF activity. We find roughly 10% of GOODSHerschel galaxies have a significant AGN. We also confirm that a higher fraction ( /greaterorsimilar 50%) of the brightest sources, S 8 > 200 µ Jy or S 24 > 600 µ Jy, have colors indicative of a bolometrically significant AGN. The 100 µ m and 250 µ m fluxes show a much weaker dependence on AGN fraction, though when combined with S 24 , S 250 effectively separates out the AGN, indicating that the amount of cold dust emission relative to mid-IR emission is a good indicator of the IR power source. We conclude then that far-IR fluxes or colors alone cannot be used to determine the nature of the IR power source, and the most effective method is to combine mid-IR and far-IR data. We thank the anonymous referee for the careful reading of this paper and the insightful comments provided. This work is based in part on observations made with Herschel Space Observatory , a European Space Agency Cornerstone Mission with significant participation by NASA, and the Spitzer Space Telescope , which is operated by the Jet Propulsion Laboratory, California Institute of Technology under a contract with NASA. Support for this work was provided by NASA through an award issued by JPL/Caltech. Laurent, O., Mirabel, I. F., Charmandaris, V., Gallais, P., Madden, S. C., Sauvage, M., Vigroux, L., and Cesarsky, C. 2000, A&A, 359, 887 Lin, L., et al. 2012, ApJ, 756, 71 Luo, B., et al. 2008, ApJS, 179, 19 Magnelli, B., Elbax, D., Chary, R.-R., Dickinson, M., Frayer, D. T., & Willmer, C. N. A. 2011, A&A, 528, A35 Mullaney, J. R., et al. 2012, MNRAS, 419, 95 Murphy, E. J., Chary, R.-R., Dickinson, M., et al. 2011, ApJ, 732, 126 Pope, A., Chary, R.-R., Alexander, D. M., et al. 2008, ApJ, 675, 1171 Retzlaff, J., Rosati, P., Dickinson, M., et al. 2010, A&A, 511, A50 Sajina, A., Yan, L., Armus, L., Choi, P., Fadda, D., Helou, G., Spoon, H. 2007, ApJ, 664, 713 Sanders, D. B., Soifer, B. T., Elias, J. H., Neugebauer, G., and Matthews, K. 1988, ApJ, 328, 35 Stern, D., et al. 2005, ApJ, 631, 163 Stern, D., et al. 2012, ApJ, 753, 30 Teplitz, H. I., Chary, R., Elbaz, D., et al. 2011, AJ, 141, 1 Treister, E. and Urry, C. M. 2005, ApJ, 630, 115 Wall, J. V., Jackson, C. A., Shaver, P. A., Hook, I. M., and Kellermann, K. I. 2005, A&A, 434, 133 Wang, W.-H., Cowie, L. L., Barger, A. J., Keenan, R., C., and Ting, H. C. 2010, ApJS, 187, 251 Worsley, M. A., Fabian, A. C., Bauer, F. E., Alexander, D. M., Hasinger, G., Mateos, S., Brunner, H., Brandt, W. N., and Schneider, D. P. 2005, MNRAS, 357, 1281 Wright, E., et al. 2010, AJ, 140, 1868 Yan, L., et al. 2012, AJ, submitted (arXiv: 1209:2065)", "pages": [ 10, 11, 12, 13 ] } ]

[ { "title": "ABSTRACT", "content": "We calculate the absolute intensity and anisotropies of the Lymanα radiation field present during the epoch of reionization. We consider emission from both galaxies and the intergalactic medium (IGM) and take into account the main contributions to the production of Lymanα photons: recombinations, collisions, continuum emission from galaxies and scattering of Lyman-n photons in the IGM. We find that the emission from individual galaxies dominates over the IGM with a total Lymanα intensity (times frequency) of about (1 . 43 -3 . 57) × 10 -8 erg s -1 cm -2 sr -1 at a redshift of 7. This intensity level is low so it is unlikely that the Lymanα background during reionization can be established by an experiment aiming at an absolute background light measurement. Instead we consider Lymanα intensity mapping with the aim of measuring the anisotropy power spectrum which has rms fluctuations at the level of 1 × 10 -16 [erg s -1 cm -2 sr -1 ] 2 at a few Mpc scales. These anisotropies could be measured with a spectrometer at near-IR wavelengths from 0.9 to 1.4 µ m with fields in the order of 0.5 to 1 sq. degrees. We recommend that existing ground-based programs using narrow band filters also pursue intensity fluctuations to study statistics on the spatial distribution of faint Lymanα emitters. We also discuss the cross-correlation signal with 21 cm experiments that probe HI in the IGM during reionization. A dedicated sub-orbital or space-based Lymanα intensity mapping experiment could provide a viable complimentary approach to probe reionization, when compared to 21 cm experiments, and is likely within experimental reach. Subject headings: cosmology: theory - large scale structure of Universe - diffuse radiation", "pages": [ 1 ] }, { "title": "INTENSITY MAPPING OF LYMAN-ALPHA EMISSION DURING THE EPOCH OF REIONIZATION", "content": "Marta B. Silva 1 , 2 , Mario G. Santos 1 , Yan Gong 2 , Asantha Cooray 2 and James Bock 3 , 4 1 CENTRA, Instituto Superior T'ecnico, Technical University of Lisbon, Lisboa 1049-001, Portugal 2 Department of Physics & Astronomy, University of California, Irvine, CA 92697 3 4 Jet Propulsion Laboratory (JPL), National Aeronautics and Space Administration (NASA), Pasadena, CA 91109, USA Draft version June 5, 2018", "pages": [ 1 ] }, { "title": "1. INTRODUCTION", "content": "The epoch of reionization (EoR) is a crucial stage in the history of galaxy formation, signaling the birth of the first luminous objects, during which the universe went from completely neutral to almost completely ionized (Barkana & Loeb 2001). This phase has been largely unexplored so far, although current observations suggest it was reasonably extended (Komatsu et al. 2011; Fan et al. 2006) and a wide variety of observational avenues are being explored to probe it. In particular the 21-cm line of neutral hydrogen is now understood to be a promising tool to study reionization and to understand the formation and evolution of galaxies during that epoch (see e.g. Furlanetto et al. 2006). It is also now becoming clear that we need complimentary data in order to obtain extra insight into the sources of reionization. Such complimentary data could also aid in the interpretation of the HI signal by allowing ways to pursue cross-correlations and providing ways to reduce systematics and foregrounds encountered in 21-cm observations. Recently, intensity mapping of other atomic and molecular lines at high redshifts, in particular CO and CII (Gong et al. 2012, 2011; Lidz et al. 2011; Visbal & Loeb 2010), has been proposed as a probe of reionization. In this work we study the viability of also using intensity mapping of the Lymanα (Lyα ) line as an additional probe. For this study we include several Lyα emission mechanisms involving both individual sources of emission such as galaxies and the emission and scattering associated with the intergalactic medium (IGM). We consider both the integrated intensity and anisotropies of the Lyα line and suggest the latter as a new probe of reionization. In particular we suggest that it will be possible to measure the amplitude of the Lyα intensity fluctuations with a narrow-band spectrometer either from the ground with a suppression of atmospheric lines or from the orbital/sub-orbital platform. The Lyα line, corresponding to transitions between the second and first energy level of the hydrogen atom, has a rest wavelength of approximately λ Ly α = 1216 ˚ A. The signal present during reionization is observable in near-IR wavelengths today. Existing imaging observations made with narrow-band filters on 10m class telescopes focus on individual galaxy detections and are limited to a handful of narrow atmospheric windows at nearIR wavelengths. Given the strength of the line, it has now been seen in galaxies at z ≈ 6 . 98 (Iye et al. 2006), z ≈ 8 . 2 (Salvaterra et al. 2009) and z ≈ 8 . 6 (Lehnert et al. 2010), reaching well into the epoch of reionization. Deep narrow-band surveys of high redshift Lyα emitters have led to detections of a sufficient number of galaxies at redshifts 5 . 7, 6 . 6, 7 . 0 and 7 . 7 to allow constraints on the bright-end of the Lyα luminosity function (LF) and its redshift evolution (e.g. Ouchi et al. 2008; Ota et al. 2010; Taniguchi et al. 2005; Iye et al. 2006; Shibuya et al. 2011). Observations of the Lyα LF indicate a decrease in the Lyα intensity from redshift 5 . 7 to 7 . 0. This would require a strong evolution of the Lyα emitters population, which is not predicted by most recent galaxy evolution models (Ota et al. 2010; Shibuya et al. 2011), or could be explained as the result of an increase in the fraction of IGM neutral hydrogen that would absorb or scatter Lyα photons from the observed galaxies (Haiman et al. 2000; Ota et al. 2008). The scattering of Lyα photons by neutral hydrogen in the ISM (interstellar medium) and the IGM is expected to disperse the photons in both frequency and direction (Santos 2004). Such scattering could considerably decrease the Lyα intensity per frequency bin from an individual galaxy, making the detection of most of the high redshift galaxies impossible with current instruments. Exact calculations related to scattering are a difficult problem to solve analytically and in simulations the scattering problem requires ray tracing of photons through the neutral medium in a simulation box (Zheng et al. 2010). While scattering makes individual galaxies dimmer, intensity mapping of the Lyα line at high redshifts can be an improvement over the usual experiments that make detections of Lyα emission from point sources and are only sensible to the strongest Lyα emitters. These are likely to be some of the brightest star-forming galaxies, however, any dust that is present in such galaxies, especially during the late stages of reionization, is likely to suppress the Lyα line. An experiment targeting the integrated emission will be able to measure all the sources of Lyα photons in a large region and will be sensitive to the extended, low surface brightness Lyα emission that is now known to generally form around star-forming regions (e.g., Steidel et al. 2011; Zheng et al. 2011). The anisotropy power spectrum of Lyα intensity then would be a probe of the Lyα halos around star-forming galaxies present during reionization. The cross-correlation with the 21-cm data could provide a direct test on the presence of neutral hydrogen in the extended Lyα halo. The paper is organized as follows: in the next section we estimate the contribution to the Lyα emission from galaxies. In section 3 we analyze the contributions to the Lyα emission from the IGM. In section 4 we calculate the intensity of the Lyα signal as well as its power spectrum using a modified version of the code SimFast21 (Santos et al. 2010, 2011). In section 5 we discuss the correlation of Lyα intensity maps with the 21 cm signal and finally in section 6 we comment on the experimental feasibility of measuring the Lyα intensity power spectrum.", "pages": [ 1, 2 ] }, { "title": "2. LYMANα EMISSION FROM GALAXIES", "content": "The observed Lyα flux is mainly the result of line emission from hydrogen recombinations and collisional excitations in the interstellar clouds or in the IGM powered respectively by UV emission or UV and X-ray emission from galaxies. High energy photons emitted by stars ionize hydrogen that then recombines to emit a rich spectrum of lines including a Lyα photon (Gould & Weinberg 1996; Fernandez & Komatsu 2006). Moreover, the electron ejected during this ionization heats the ISM or the IGM, increasing the probability of Lyα photon emission caused by collisional excitation (Gould & Weinberg 1996; Cantalupo et al. 2008). There is also a small contribution to the lyman alpha flux originated in the continuum emission from stars between the Lyα line and the Lyman-limit (Chuzhoy & Zheng 2007; Barkana & Loeb 2005) plus Lyα from continuum free-free or free-bound emission as well as 2-photon emission during recombinations. This continuum will also make contributions to a given observation from lower redshifts besides the 'Lyα ' redshift (Cooray et al. 2012) which will confuse the Lyα signal. However, due to the smoothness of that continuum across frequency, we expect it should be possible to remove this contribution, for instance, by fitting a smooth polynomial in frequency for each pixel. Another source of Lyα emission in the universe is cooling of gas that has suffered in-fall into a dark matter halo potential well. Several studies show that much of this cooling is made in the form of Lyα emission (Haiman et al. 2000; Fardal et al. 2001; Dijkstra et al. 2006a,b; Dayal et al. 2010; Latif et al. 2011). Cold gas is used by galaxies as fuel to form stars so there is a relation between the star formation rate (SFR) of a galaxy and the Lyα flux emitted as gas cools in that galaxy. Since emission of Lyα radiation is closely connected with the star formation, the contribution from the several mechanisms by which Lyα radiation is emitted in galaxies and in the IGM can be related to the SFR of individual galaxies or galaxy samples. In order to calculate the emission of Lyα radiation from the IGM during the EoR we also need to know the ionized fraction of hydrogen as well as the temperature of the gas in the IGM. Unfortunately both these quantities are poorly constrained at z ≥ 6 (Larson et al. 2011; Ouchi et al. 2010; Zahn et al. 2011). Since hydrogen ionization should be a consequence of stellar ionization/X-ray emission, we can in principle estimate it by following the SFR history and making sure that the resulting evolution of hydrogen ionized fraction is consistent with current constraints on the CMB optical depth. In order to obtain the SFR of galaxies at the high redshifts during the epoch of reionization we make use of parametrizations that reproduce a correct reionization history. Our parametrizations are non linear in a similar way to the relations found in the Guo et al. (2011) and in the De Lucia & Blaizot (2007) galaxies catalogs derived respectively from the high resolution Millennium II (Boylan-Kolchin et al. 2009) and Millennium I (Springel et al. 2005) simulations. Such relations, when available from observations, make an improvement on the models instead of relying purely on theoretical calculations and semi-numerical simulations to predict all of the observations (Mesinger & Furlanetto 2007; Santos et al. 2010). There are additional sources of radiation contributing to the Lyα emission, such as a strong non-local sources of ionizing photons as expected from quasars, which would emit a large amount of energy in X-ray photons that would be able to ionize several neutral atoms giving origin to a locally strong Lyα emission from recombinations. However, since the number of quasars is very small compared to the number of normal galaxies at the redshifts we are considering, we will neglect their contribution in the following calculations. We encourage future works on Lyα intensity to see if the shape of the power spectrum and other statistics can be used to choose between reionization histories that involve both galaxies and quasars. In the following sub-sections we discuss in more detail the four processes for Lyα emission from galaxies: recombinations, excitations/relaxations, gas cooling, and photon emission from continuum processes. Assuming ionizing equilibrium, the number of recombinations in galaxies are expected to match the number of ionizing photons that are absorbed in the galaxy and does not escape into the IGM. Depending on the temperature and density of the gas, a fraction of the radiation due to these recombinations is emitted in the Lyα line. In the interstellar gas, most of the neutral hydrogen is in dense clouds with column densities greater than 3 × 10 18 cm -2 . These clouds are optically thick to Lyα radiation and Lyman photons are scattered in the galaxy several times before escaping into the IGM. Such multiple scatterings increase the probability of absorption. Assuming that these clouds are spherical and that the gas temperature is of the order of 10 4 K, Gould & Weinberg (1996) used atomic physics to study the probability of the Lyα emission per hydrogen recombination. They estimated that a fraction f rec ≈ 66% of the hydrogen recombinations would result in the emission of a Lyα photon and that most of the other recombinations would result in two-photon emission. These fractions should change with the temperature and the shape of the cloud, but such variations are expected to be small. Other calculations yield fractions between 62% and 68% according to the conditions in the cloud. In this paper we have chosen to use a value of f rec = 66% since the overall uncertainty on this number is lower than the uncertainty on the number of hydrogen recombinations. The absorption of Lyα photons by dust is difficult to estimate and changes from galaxy to galaxy, Gould & Weinberg (1996) estimated that for a cloud with a column density N ∼ 10 19 cm -2 , the dust in the galaxy absorbs a fraction f dust ≈ 4% of the emitted Lyα photons before they reach the galaxy virial radius however recent observations of high redshift galaxies indicate a much higher f dust . In this study we will use a redshift parameterization for the fraction of Lyα photons that are not absorbed by dust f Ly α = 1 -f dust that is double the value predicted by the study made by Hayes et al. (2011): where C dust = 3 . 34 and ξ = 2 . 57. The Hayes et al. (2011) parameterization was made so that f Ly α gives the difference between observed Lyα luminosities and Lyα luminosities scaled from star formation rates assuming that the Lyα alpha photons emitted in galaxies are only originated in recombinations. The high redshift observations used to estimate f Ly α are only of massive stars while the bulk of Lyα emission is originated in the low mass stars that cannot be detected by current surveys. According to several studies (Forero-Romero et al. 2011), f Ly α decreases with halo mass, so it is possible that it is being underestimated in Hayes et al. (2011) which is why we decided to use a higher f Ly α . Our results can however be easily scaled to other f dust evolutions. The number of Lyα photons emitted in a galaxy per second, ˙ N Ly α , that reach its virial radius is therefore given by where A He = 4 -4 Y p 4 -3 Y p accounts for the fraction of photons that go into the ionization of helium ( Y p is the mass fraction of helium), ˙ N ion is the rate of ionizing photons emitted by the stars in the galaxy and f esc is the fraction of ionizing photons that escape the galaxy into the IGM. The ionizing photon escape fraction depends on conditions inside each galaxy and is difficult to estimate, especially at high redshifts. The precise determination of its value is one of the major goals of future observations of high redshift galaxies at z glyph[greaterorsimilar] 7. This parameter can be measured from deep imaging observations or can be estimated from the equivalent widths of the hydrogen and helium balmer lines. The ionizing photon escape fraction dependence with the galaxy mass and the star formation rate, as a function of redshift, has been estimated using simulations that make several assumptions about the intensity of this radiation and its absorption in the interstellar medium. However, for the halo virial mass range, 10 8 M glyph[circledot] to 10 13 M glyph[circledot] , and during the broad redshift range related to the epoch of reionization, there are no simulations that cover the full parameter space. Moreover the limited simulations that exist do not always agree with each other (Gnedin et al. 2008; Wise & Cen 2009; Fern'andez-Soto et al. 2003; Siana et al. 2007; Haardt & Madau 2012). Razoumov & Sommer-Larsen (2010) computed the escape fraction of UV radiation for the redshift interval z = 4 to z = 10 and for halos of masses from 10 7 . 8 to 10 11 . 5 M glyph[circledot] using a high-resolution set of galaxies. Their simulations cover most of the parameter space needed for reionization related calculations and their escape fraction parameterization is compatible with most of the current observational results. Thus, we use it for our calculations here. According to Razoumov & Sommer-Larsen (2010) simulations, the escape fraction of ionizing radiation can be parameterized as: where M is the halo mass, α and β are functions of redshift (Table 1). The number of ionizing photons emitted by the stars in a galaxy depends on its star formation rate, metallicity and the stellar initial mass function (IMF). Making reasonable assumptions for these quantities we will now estimate ˙ N ion . Since this UV emission is dominated by massive, short lived stars, we can assume that the intensity of ionizing photons emitted by a galaxy is proportional to its star formation rate. In terms of the star formation rate in one galaxy, where Q ion is the average number of ionizing photons emitted per solar mass of star formation. This can be calculated through: where Ψ( M ) = KM -α is the stellar IMF, K is a constant normalization factor and α is the slope of the IMF. In our calculation we used a Salpeter IMF, with α = 2 . 35. t glyph[star] ( M ) is the star lifetime and Q glyph[star] ( M ) its number of ionizing photons emitted per unit time. The values of Q glyph[star] and t glyph[star] were calculated with the ionizing fluxes obtained by Schaerer (2002) using realistic models of stellar populations and non-LTE atmospheric models, appropriated for POP II stars with a Z glyph[star] = 0 . 02 Z glyph[circledot] metallicity. Assuming that ionizing photons are only emitted by massive OB stars sets a low mass effective limit for the mass of stars contributing to the UV radiation field of a galaxy. This limit is a necessary condition for the star to be able to produce a significant number of ionizing photons. For the stellar population used for this work we take M min ≈ 7 M glyph[circledot] (Schaerer 2002; Shull et al. 2011). The integration upper limit is taken to be M max = 150 M glyph[circledot] . In this paper we calculated Q ion using the parameterization values published in Schaerer (2002). The number of ionizing photons per second emitted by a star as a function of its mass is given by: where x = log 10 ( M glyph[star] /M glyph[circledot] ) and the star's lifetime in years is given by: The use of these parameters results in Q ion ≈ 5 . 38 × 10 60 M -1 glyph[circledot] . In Shull et al. (2011) it has been suggested the use of a different model for stellar atmosphere and evolution (R. S. Sutherland & J. M. Shull, unpublished) which yields Q ion ≈ 3 . 97 × 10 60 M -1 glyph[circledot] . This may imply that the stellar emissivity we calculated is an overestimation and that consequently our Lyα flux powered by stellar emission may be overestimated by about 35%. This is comparable to other large uncertainties, such as the ones in the parameters f esc and f dust . The Lyα luminosity is calculated assuming that the Lyα photons are emitted at the Lyα rest frequency, ν 0 = 2 . 47 × 10 15 Hz with an energy of E Ly α = 1 . 637 × 10 -11 erg. To proceed, we will assume that the SFR for a given galaxy is only a function of redshift and the mass of the dark halo associated with that galaxy. The Lyα luminosity due to recombinations in the interstellar medium, L GAL rec , can then be parameterized as a function of halo mass and redshift as", "pages": [ 2, 3, 4 ] }, { "title": "2.2. Lymanα emission from excitations/relaxations", "content": "The kinetic energy of the electron ejected during the hydrogen ionization heats the gas and assuming thermal equilibrium this heat is emitted as radiation. Using atomic physics, Gould & Weinberg (1996) estimated that for a cloud with an hydrogen column density of ≈ 10 19 cm -2 , the energy emitted in the form of Lyα photons is about 60% for ionizing photons with energy E ν lim < E ν < 4 E ν lim and ≈ 50% for photons with energy E ν > 4 E ν lim , where E ν lim = 13 . 6 eV is the Rydberg energy. The remaining of the energy is emitted in other lines. Using the spectral energy distribution (SED) of galaxies with a metallicity Z = 0 . 02 Z glyph[circledot] from the code of Maraston (2005) we estimated that the average ionizing photon energy is E ν = 21 . 4 eV and that more than 99% of the photons have an energy lower than 4 E ν lim . According to the Gould & Weinberg (1996) calculation, the fraction of energy of the UV photon that is emitted as Lyman alpha radiation due to the collisional excitations/relaxations is given by: For a cloud with the properties considered here this yields an energy in Lyα per ionizing photon of E exc ≈ 2 . 14 eV or 3 . 43 × 10 -12 erg. This results in an average of 0 . 16 Lyα photons per ionizing photon. Finally, the Lyα luminosity due to excitations in the ISM, L GAL exc , is then: where again it is assumed to be a function of the star formation rate.", "pages": [ 4 ] }, { "title": "2.3. Lymanα emission from gas cooling", "content": "During the formation of galaxies, gas from the IGM falls into potential wells composed mainly by dark matter which collapsed under its own gravity. The increase in the gas density leads to a high rate of atomic collisions that heats the gas to a high temperature. According to the study of Fardal et al. (2001) most of the gas in potential wells that collapses under its own gravity never reaches its virial temperature and so a large fraction of the potential energy is released by line emission induced by collisions and excitations from gas with temperatures T K < 2 × 10 4 K. At this temperature approximately 50% of the energy is emitted in Lyα alone. From Fardal et al. (2001) we can relate the luminosity at the Lyα frequency due to the cooling in galaxies to their baryonic cold mass, M bar cool , using: where both the luminosity and the mass are in solar units. To relate this baryonic cold mass to a quantity we can use in our models, we used the relation between cold baryonic mass and the halo mass from the galaxies in the Guo et al. (2011) catalog. From the equation above, we can then obtain an expression for the luminosity, which can be fitted by: with M in units of M glyph[circledot] . The relation between the cold gas mass and the mass of the halo shows very little evolution with redshift during reionization. Thus we expect the relation in equation 13 to only depend on redshift due to the redshift evolution of f Ly α .", "pages": [ 4 ] }, { "title": "2.4. Contributions from continuum emission", "content": "Continuum emission can also contribute to the Lyα observations. These include stellar emission, freefree emission, free-bound emission and two photon emission. Photons emitted with frequencies close to the Lyn lines should scatter within the ISM and eventually get re-emitted out of the galaxy as Lyα photons. Otherwise they will escape the ISM before redshifting into one of the Lyn lines and being reabsorbed by a hydrogen atom. The fraction of photons that scatter in the galaxy can be estimated from the intrinsic width of the Lyman alpha line which has ≈ 4 ˚ A (Jensen et al. 2012). We calculated the stellar contribution assuming an emission spectrum for stars with a metallicity of Z glyph[star] = 0 . 02 Z glyph[circledot] estimated with the code from Maraston (2005) that can be approximated by the emission of a black body with a temperature of 6 . 0 × 10 4 K for hν < 13 . 6eV. The number of stellar origin Lyα photons per solar mass in star formation obtained with this method is: We note that we are not accounting for the higher opacity at the center of the Lyα line which should push the photons out of the line center before exiting the star and so we may be overestimating the stellar Lyα photon emission. Free-bound emission and free-free emission are respectively originated when free electrons scatter off ions with or without being captured. Following the approach of (Fernandez & Komatsu 2006), the free-free and freebound continuum luminosity can be obtained using: where V sphere is the volume of the Stro mgren sphere which can be roughly estimated using the ratio between the number of ionizing photons emitted and the number density of recombinations in the ionized volume, ε ν is the total volume emissivity of free-free and freebound emission, n p is the number density of protons (ionized atoms) and α i is the case A or case B recombination coefficient (see Furlanetto et al. (2006)). The volume emissivity estimated by (Dopita & Sutherland 2003) is given by: where γ c is the continuum emission coefficient including free-free and free-bound emission given in SI units by: In here, x n = Ry/ ( k B T K n 2 ) ( k B is the Boltzmann constant, n is the level to which the electron recombines to and Ry = 13 . 6 eV is the Rydberg unit of energy), ¯ g ff ≈ 1 . 1 -1 . 2 and g fb ( n ) ≈ 1 . 05 -1 . 09 are the thermally average Gaunt factors for free-free and free-bound emission (Karzas & Latter 1961, values from). The initial level n ' is determined by the emitted photon frequency and satisfies the condition cR ∞ /n ' 2 < ν < cR ∞ / ( n ' -1) 2 where R ∞ = 1 . 1 × 10 7 m -1 is the Rydberg constant. The continuum luminosity per frequency interval ( L ν ) is related to the Lyα luminosity emitted from the galaxies by: L cont = L ν × dν (4 ˚ A ) = f Lyα Q Lyα E Lyα SFR ( M,z ), where Q Lyα is the number of emitted lyman alpha photons per solar mass in star formation. We then obtain Q free -free Ly α = 2 . 13 × 10 53 M -1 glyph[circledot] for free-free emission and Q free -bound Ly α = 2 . 22 × 10 55 M -1 glyph[circledot] for free-bound emission. During recombination there is also the probability of two photon emission and although this photons have frequencies below the lyman alpha frequency there is a small fraction of them of Q 2 -photon Ly α that are emitted so close to the lyman alpha line that are included in the lyman alpha intrinsic width. The number of lyman alpha photons that can be originated due to two photon emission during recombination is given by: where P(y)dy is the normalized probability that in a two photon decay one of them is the range dy = dν/ν lyα and 1 -f lyα ≈ 1 / 3 is the probability of 2 photon emission during an hydrogen n=2 → 1 transition. The probability of two photon decay was fitted by Fernandez & Komatsu (2006) using Table 4 of Brown & Mathews (1970) as: Finally, the different contributions to the total Lyα luminosity from galaxies due to continuum emission, L GAL cont = L stellar cont + L free -free cont + L free -bound cont + L 2 -photon cont , are given by: for stellar emission, for free-free emission, for free-bound emission and for 2-photon emission. Note that here we are only considering the part of the continuum emission from galaxies that could contribute to the same 'Lyα redshift'. There will be a continuum emission spectrum with frequencies below the Lyα line from the mechanisms above that will contribute to the same observation from lower redshifts and will generate a 'foreground' to the Lyα signal that needs to be removed. This should be possible due to the smoothness of this background across frequency, in the same manner as foregrounds of the 21-cm signal are removed (e.g. Wang et al. 2006).", "pages": [ 5, 6 ] }, { "title": "2.5. Modeling the relation between star formation rate and halo mass", "content": "Simulations of galaxy formation and observations indicate that the star formation of a halo increases strongly for small halo masses but at high halo masses ( M glyph[greaterorsimilar] 10 11 M glyph[circledot] ) it becomes almost constant (Conroy & Wechsler 2009; Popesso et al. 2012). In order to better estimate and constrain the SFR of a halo we used three non linear SFR versus Halo Mass parameterizations that are in good agreement with different observational constraints. In Sim1 we adjusted the SFR to reproduce a reasonable reionization history and a Lyα Luminosity Function evolution compatible with different observational constraints, in Sim2 we adjusted the SFR vs halo mass relation to the parameterizations from the Guo et al. (2011) galaxies catalogue (low halo masses) and the De Lucia & Blaizot (2007) galaxies catalogue (high halo masses). Sim2 results in an early reionization history with an optical depth to reionization compatible with the low bound of the current observational constraints. Finally Sim3 has the same halo mass dependence as Sim2 but evolves with redshift in a similar way to the De Lucia & Blaizot (2007) and to the Guo et al. (2011) galaxy catalogues. We parametrized the relations between the SFR and halo mass as: where a = 2 . 8, b = -0 . 94, d = -1 . 7 , c 1 = 1 × 10 9 M glyph[circledot] and c 2 = 7 × 10 10 M glyph[circledot] for Sim1 , where a = 2 . 59, b = -0 . 62, d = 0 . 4, e = -2 . 25, c 1 = 8 × 10 8 M glyph[circledot] , c 2 = 7 × 10 9 M glyph[circledot] and c 3 = 1 × 10 11 M glyph[circledot] for Sim2 and where a = 2 . 59, b = -0 . 62, d = 0 . 4, e = -2 . 25, c 1 = 8 × 10 8 M glyph[circledot] , c 2 = 7 × 10 9 M glyph[circledot] and c 3 = 1 × 10 11 M glyph[circledot] for Sim3 . halo In figure 2, the strong decline in the observational SFRD from z ≈ 8 to z ≈ 10, imposed by the observational point at z = 10 . 3, was obtained with the observation of a single galaxy using the Hubble Deep Field 2009 two years data (Bouwens et al. 2011a; Oesch et al. 2012). It was argued in Bouwens et al. (2011b), based on an analytical calculation, that even with such low SFRD at high redshifts it was possible to obtain an optical depth to reionization compatible with the value obtained by WMAP ( τ = 0 . 088 ± 0 . 015) (Komatsu et al. 2011). However, this derivation would imply a high escape fraction of ionizing radiation and that reionization would end at z ≈ 8 which is hard to reconcile with the constraints from observations of quasars spectra (Mesinger & Haiman 2007; Zaldarriaga et al. 2008). Our SFRDs are considerably higher than the current observational constraints, although the difference can be explained by a systematic underestimation of the SFR in observed galaxies. Moreover, current observations only probe the high mass end of the high redshift galaxies mass function which will underestimate the SFR density (also the obtained SFRs have very high error bars due to uncertainties in the correction due to dust extinction, the redshift and the galaxy type). In the following sections the results shown were obtained using Sim1 unless stated otherwise.", "pages": [ 6, 7 ] }, { "title": "2.6. Total Lymanα luminosity: comparison with observations", "content": "In the previous sections we calculated the Lyα luminosity as a function of the SFR for several effects. The commonly used 'empirical' relation between these two quantities is (Jiang et al. 2011) and it is based on the relation between SFR and the H α luminosity from Kennicutt (1998a) and in the line emission ratio of Lyα to H α in case B recombinations calculated assuming a gas temperature of 10 4 K. This empirical relation gives the Lyα luminosity without dust absorption (we have labeled it K98 for the remainder of the paper). Our relation between luminosity and star formation is mass dependent (both from the escape fraction as well as due to the expression from the cooling mechanism), so in order to compare it with the result above, we calculate: where the average 〈 x 〉 of quantity x is done over the halo mass function for the mass range considered. The results are presented in table 2 for a few redshifts. Although our Lyα luminosities per SFR are slightly higher, at least for low redshifts, we point out that the 'empirical' relation is based on a theoretical calculation that only accounts for Lyα emission due to recombinations. Moreover the observational measurements of H α and Lyα are primarily made at low redshifts, where the absorption of Lyα photons by dust in galaxies is expected to be high. Our relation has the advantage of evolving with redshift since it accounts for the evolution of the escape fraction of ionizing photons and for the evolution of the escape fraction of Lyα photons. This z -dependence is not present in the standard empirical relation. This redshift evolution of the UV photons escape fraction is a consequence of the increase in the number of massive galaxies with more clumpy structure as the redshift decreases. The star forming regions of massive galaxies are embedded in clumps and therefore it becomes more difficult for the ionizing photons to escape from such dense regions (Razoumov & Sommer-Larsen 2010; Yajima et al. 2011). The redshift evolution of the relation presented in equation 28 justifies why a theoretical calibration between Lyα luminosity and the SFR of a galaxy is useful for our work. To check the consistency between our theoretical estimation of the Lyα luminosity and the existing observations during reionization we show in figure 3 the luminosity function (LF) using two of the star formation rate vs. halo mass parameterizations presented in section 2.5. This prediction is then compared to Lyα luminosity functions of photometric identified objects in Shimasaku et al. (2006) and in Kashikawa et al. (2006) near the end of the reionization epoch. Our luminosity functions were calculated assuming a minimum halo mass of 8 × 10 8 M glyph[circledot] which corresponds to a minimum luminosity of 3 . 72 × 10 36 erg s -1 for Sim1 , 4 . 49 × 10 36 erg s -1 for Sim2 and 6 . 22 × 10 36 erg s -1 for Sim3 . The agreement between our LFs and observations is reasonable for Sim1 however our Sim2 overpredicts the abundance of high luminosity Lyα emitters. This difference can be due to sample variance or a result of the high sensitivity of theoretical predictions to several parameters in our model. We point out that the luminosity range relevant for this comparison falls in a halo mass range outside the one for which the escape fraction of UV radiation we are using was estimated, so we could easily get a better fit between observations and Sim2 by reducing this escape fraction for high halo masses. This difference could also be related with the choice of halo mass function. Here we choose the Sheth-Tormen halo mass function (Sheth & Tormen 1999) which has been shown to fit low-redshift simulations more accurately, but it is yet to be established the extent to which such a halo mass function can reproduce the halo distribution during reionization. Other possible explanation for this difference is the existence of a small amount of neutral gas in the IGM which would severely decrease the observed Lyα luminosity from galaxies. Also, we could have decreased the high luminosity end of our luminosity functions if we had use an Lyα escape fraction that decreased with halo mass such as the one used in (Forero-Romero et al. 2011). We do not consider a model fit to the data to optimize various parameters in our model given that the current constraints on the observed Lyα LFs have large overall uncertainties, especially considering variations from one survey to another.", "pages": [ 7, 8 ] }, { "title": "2.7. Lymanα Average Intensity", "content": "In this section and the next one we will attempt to estimate the intensity and power spectrum of the Lyα signal using an analytical model. In section 4 we will improve the estimation by doing the same calculation using a semi-numerical simulation. The total intensity of Lyα emission can be obtained from the combined luminosity of Lyα photons associated with different mechanisms described in the previous sub-sections, such that: where dn/dM is the halo mass function (Sheth & Tormen 1999), M is the halo mass, M max = 10 13 M glyph[circledot] , M min = M OB , D L is the proper luminosity distance and D A the comoving angular diameter distance. Finally, y ( z ) = dχ/dν = λ Ly α (1 + z ) 2 /H ( z ), where χ is the comoving distance, ν is the observed frequency and λ Ly α = 2 . 46 × 10 -15 m is the rest-frame wavelength of the Lyα line. The evolution of the Lyα intensity predicted by this calculation is shown in figure 4 together with the scaling expected under the 'empirical' relation from Kennicutt (1998a) combined with an assumption related to the gas temperature. The intensities of Lyα emission from different sources are presented in table 3 for several redshifts. These intensities can be extrapolated to other SFRDs, assuming that the only change is in the amplitude of the SFR halo mass relations presented in figure 1 by using the coefficients in table 4. The intensities from emission at z ≈ 7, 8 and 10 are 9 . 5 × 10 -9 , 3 . 5 × 10 -9 and 3 . 2 × 10 -10 erg s -1 cm -2 sr -1 ,", "pages": [ 8 ] }, { "title": "TABLE 3", "content": "respectively. Such an intensity is substantially smaller than the background intensity of integrated emission from all galaxies (around 1 × 10 -5 erg s -1 cm -2 sr -1 (Madau & Pozzetti 2000), or from the total emission of galaxies during reionization, estimated to be at most 1 × 10 -6 erg s -1 cm -2 sr -1 (Yan et al. 2012).", "pages": [ 8 ] }, { "title": "2.8. Lymanα intensity power spectrum", "content": "The Lyα emission from galaxies will naturally trace the underlying cosmic matter density field so we can write the Lyα line intensity fluctuations due to galaxy clustering as where ¯ I GAL is the mean intensity of the Lyα emission line, δ ( x ) is the matter over-density at the location x , and b Ly α is the average galaxy bias weighted by the Lyα luminosity (see e.g. Gong et al. 2011). Using one of the relations between the SFR and halo mass from section 2.5 we can calculate the luminosity and obtain the Lyman alpha bias following Visbal & Loeb (2010): where b ( z, M ) is the halo bias and dn/dM the halo mass function (Sheth & Tormen 1999). We take M min = 10 8 M glyph[circledot] /h and M max = 10 13 M glyph[circledot] /h . The bias between dark matter fluctuation and the Lyα luminosity, as can be seen in figure 5, is dominated by the galaxies with low Lyα luminosity independently of the redshift. We can then obtain the clustering power spectrum of Lyα emission as where P δδ ( z, k ) is the matter power spectrum. The shotnoise power spectrum, due to discretization of the galaxies, is also considered here. It can be written as (Gong et al. 2011) The resulting power spectrum of Lyα emission from galaxies is presented in figure 6. At all scales presented the Lyα intensity and fluctuations are dominated by the recombination emission from galaxies.", "pages": [ 8, 9 ] }, { "title": "3. LYMANα EMISSION FROM THE IGM", "content": "The Lyα emission from the IGM is mostly originated in recombinations and collisions powered by the ionizing background. These processes are similar to the ones described inside the galaxies, although, since the physical conditions of the gas in the IGM are different from those in the ISM, the intensity of Lyα emission can no longer be connected to the ionizing photon intensity using the previous relations. The biggest challenge in doing these calculations is to connect the IGM ionizations and heating of the gas to the emission of ionizing radiation and the star formation rate assumed in the previous sections. Moreover, in the IGM, we also have to take into account the contribution of continuum radiation from stars between the Lyα and the Lyman limit which redshifts into the Lyα line. In a schematic view, we have to take into account the following processes, The proper calculation of all these processes will require simulations which we will address in section 4. In the following sub-sections we review the contributions through analytical calculations in order to get a better understanding of the dominating effects.", "pages": [ 9, 10 ] }, { "title": "3.1. Lymanα emission from hydrogen recombinations", "content": "The UV radiation that escapes the interstellar medium into the intergalactic medium ionizes low density clouds of neutral gas. Part of the gas in these clouds then recombines giving rise to Lyα emission. The radiation emitted in the IGM is often referred to as fluorescence (Santos 2004). The comoving number density of recombinations per second in a given region, ˙ n rec , is given by: where α changes between the case A and the case B recombinations coefficient, n HII = x i n b (1 -Y p ) 1 -3 / 4 Y p is the ionized hydrogen comoving number density ( x i is the ionization fraction, n b the baryonic comoving number density). The free electron density can be approximated by n e = x i n b . The recombination coefficients are a function of the IGM temperature, T K . The case A comoving recombination coefficient is appropriate for the highly ionized low redshift Universe Furlanetto et al. (2006), while the case B comoving recombination coefficient is appropriate for the high redshift Universe. The use of a larger recombination coefficient when the process of hydrogen recombination is close to its end accounts for the fact that at this time, ionizations (and hence recombinations) take place in dense, partially neutral gas (Lyman-limit systems) and the photons produced after recombinations are consumed inside this systems so they do not help ionizing the IGM (see: eg. Furlanetto et al. 2006). The fraction of Lyα photons emitted per hydrogen recombination, f rec , is temperature dependent so we used the parameterization for f rec made by Cantalupo et al. (2008) using a combination of fits tabulated by Pengelly (1964) and Martin (1988) for T K > 10 3 and T K < 10 3 respectively: The luminosity density (per comoving volume) in Lyα from hydrogen recombinations in the IGM, glyph[lscript] IGM rec , is then given by", "pages": [ 10 ] }, { "title": "3.2. Lymanα emission from excitations in the IGM", "content": "The UV radiation that escapes the galaxies without producing ionization ends up ionizing and exciting the neutral hydrogen in the IGM and heating the gas around the galaxies. The high energetic electron released after the first ionization spends its energy in collisions/excitations, ionizations and heating the IGM gas until it thermalizes (Shull & van Steenberg 1985). We estimated the contribution of the direct collisions/excitations to the Lyα photon budget and concluded that it is negligible. The Lyα luminosity density due to the collisional emission (radiative cooling in the IGM), glyph[lscript] IGM exc , is given by: where n HI = n b (1 -x i ) (1 -Y p ) 1 -3 / 4 Y p is the neutral hydrogen density, x i is the IGM ionized fraction and q Ly α is the effective collisional excitation coefficient for Lyα emission which we calculated in the same way as Cantalupo et al. (2008), but using different values for the gas temperature and IGM ionization fraction. Considering excitation processes up to the level n = 3 that could eventually produce Lyα emission, the effective collisional excitation coefficient is given by: The collisional excitation coefficient for the transition from the ground level (1) to the level ( nl ) is given by where Ω(1 , nl ) is the temperature dependent effective collision strength, ω 1 is the statistical weight of the ground state, E 1 ,n is the energy difference between the ground and the nl level and k B is the Boltzmann constant.", "pages": [ 10 ] }, { "title": "3.3. Scattering of Lyman-n photons emitted from galaxies", "content": "Continuum emission of photons, by stars, from Lyα to the Lyman-limit travels until it reaches one of the Lyn lines where it gets scattered by neutral hydrogen. Most of this scattering will have as end result the production of Lyα photons which eventually redshift out of the line. Since a considerable fraction of this photons only reach a given Lyn frequency in the IGM this Lyα emission is formed as a flux that decays with r 2 around the star that emitted the continuum photons so it appears diluted in frequency in line observations of point sources (Chen & Miralda-Escud'e 2008). This continuum photons are much less likely to be absorbed by the dust in the ISM than photons originated in recombinations. In intensity mapping the frequency band observed is much larger than in line observations so in principle all the continuum Lyα photons can be detected. Using the Spectral Energy Distribution (SED) made with the code from Maraston (2005) we estimated that the number of photons emitted by stars between the Lyα plus the lyman alpha equivalent width and the Lyman-limit is equivalent to Q IGM Lyn = 9 . 31 × 10 60 M -1 glyph[circledot] s -1 . The higher frequency photons are absorbed by hydrogen atoms as they reach the Lyman beta frequency, reemitted and suffer multiple scattering until they reach the Lyα line. The fraction of the continuum photons emitted close to the Lyα line have already redshifted to lower frequencies before reaching the IGM so they will not be scattered by the neutral hydrogen in the IGM and will not contribute to the radiative coupling of the 21 cm signal (they are already included in the calculation of the Lyα emission from galaxies). The intensity of this emission was calculated with a stellar emissivity that evolves with frequency as ν -α with α = 0 . 86 and normalized to Q IGM Lyn . The Lyα luminosity density originated from continuum stellar radiation and emitted in the IGM, glyph[lscript] IGM cont , is then approximately given by: where the SFRD is in units of M glyph[circledot] per second. Note that in section 4, this calculation is done through a full simulation.", "pages": [ 10, 11 ] }, { "title": "3.4. Lymanα Intensity", "content": "We calculated the intensities for the several Lyα sources in the IGM from their luminosity densities using: The luminosity and so the intensity of Lyα emission in the IGM depends on local values of the hydrogen ionized fraction, the gas temperature and the gas density. These parameters are correlated with each other and so theoretical calculations of the average intensity made with the average of this parameters may be misleading. Since this emission is dominated by overdense regions a clumping factor of a few units is usually assumed in theoretical calculations. However we decided to estimate this intensity without using a clumping factor since its effect can be extrapolated from the intensity without clumping. The intensity of Lyα emission due to recombinations or collisions in the IGM is shown in figure 7 as a function of the hydrogen ionized fraction for different values of the gas temperature. Even for a fixed average IGM ionized fraction, the intensity of Lyα emission is the result of emission from several regions and so all the values shown in figure 7 are relevant. As can be seen in figure 7, the intensity of Lyman alpha due to recombinations and collisions in the IGM is very sensitive to the gas temperature and to the fluctuations in the IGM ionized fraction. Numerical simulations predict that the temperatures in the hydrogen gas in the IGM can vary in the range 5000 K to 20000 K (Dav'e et al. 2001; Smith et al. 2011). The theoretical intensities of Lyα emission in galaxies and in the IGM shown in figure 8 indicate that unless the IGM clumping factor is very high, or the Lyα photon escape fraction is very low, the Lyman alpha intensity from the IGM at z = 7 is lower than the emission from galaxies. i At higher redshifts the SFRD will decrease causing the Lyα intensities from galaxies and from the IGM to decrease. The escape fraction of UV photons from galaxies increases as the redshift increases which will contribute negatively to the intensity of emission in galaxies and positively to the intensity of emission in the IGM. At high redshifts the IGM ionized fraction is small which contributes to a strong decrease in the intensity of emission from the IGM compared to the intensity at z = 7.", "pages": [ 11, 12 ] }, { "title": "4. LYMANα INTENSITY AND POWER SPECTRUM USING NUMERICAL SIMULATIONS", "content": "The intensity of Lyman alpha emission in the IGM at a given time and a given region is proportional to the ionized fraction, the gas temperature and the matter density in that region. Since these three quantities are correlated, the use of average values in the Lyα intensity calculation highly underestimates the emission in the more overdense regions. Also the evolution of the average of the IGM ionized fraction is poorly known during the Epoch of Reionization (EOR). Some of these problems can be resolved using a computational code able to produce simulations of the IGM ionized fraction, the gas temperature and the matter density in a volume high enough to properly represent our Universe. The use of simulations has an additional advantage of allowing the calculation of the 3 dimensional power spectra of Lyman alpha emission in the IGM without the need for assuming a bias relation with the underlying dark matter distribution. In this section we will estimate the inhomogeneous Lyα intensity from Galaxies and the IGM using a modified version of the SimFast21 code (Santos et al. 2010). Given a set of astrophysical and cosmological parameters, this code is able to consistently produce 3 dimensional simulations of the dark matter density field, the ionization field, the SFRD, the scattering of Lyn photons in the IGM, the X-ray heating of the IGM and even 21 cm spin and brightness temperature fluctuations for the several redshifts of the EoR. A proper calculation of all the heating and cooling mechanisms would add a high level of complexity to this calculation and would require a small redshift step in the IGM fraction calculation so we assumed a constant temperature in ionized regions of 10000K. Moroever, the results from our calculations can be easily extrapolated to account for a higher temperature. For example for a temperature of 20000K the number of recombinations in the IGM would decrease by a factor of 1.7 and the number of collisions would increase more than two orders in magnitude. Assuming that the clumping of the IGM is not very high, and so Lyα recombination emission dominates over collisional emission during most of the EOR, than this higher temperature would cause a small decrease in the intensity of emission in the IGM and the Reionization period would be less extended than what we predict in section 4.1. We made a few modifications to the SimFast21 code in order to provide a consistent description of the ionization history and its relations to the Lyα emission, which we now describe.", "pages": [ 12 ] }, { "title": "4.1. IGM Ionized fraction calculation", "content": "In the previous version of the SimFast21 code, the IGM ionized fraction was computed assuming that at each redshift the ionization state of a region could be estimated from the collapsed mass in that region assuming a linear relation between collapsed mass and ionizing power. So a given spherical region of radius R is considered ionized if (Furlanetto et al. 2006): where M coll is the collapsed mass which corresponds to the total mass in halos in that region, M tot is the total mass in the region and ζ is an ionizing efficiency parameter. This efficiency parameter tries to include all the ionizations and recombinations produced by a halo as a function of its mass but has no actual physical meaning although its use is somewhat justified by the large uncertainty in the astrophysical quantities involved in the determination of the relation between halo mass and ionizing efficiency and in the adjustment of this parameter in order to reproduce a reionization history compatible with observations. In order to calculate the Lyα field however, we need to include the recombinations in the IGM explicitly, as well as directly relate the ionization process to the emitted stellar radiation. We therefore modified the SimFast21 code to include these improvements. This new method allows a non linear relation between collapsed mass and ionizing power and all the parameters involved in the calculation have values based in current astrophysical constraints. Also, the size of ionized regions is now set by the volume at which the total ionizing emissivity of the sources it contains equals the number of recombinations so that the system is in equilibrium. For each redshift the implementation of this method was done with the following steps: The UV ionizing rates of the halos, ˙ N IGM ion were then put in three dimensional boxes. ionizing rate was equal or higher than its recombination rate. With this method HII bubbles are always fully ionized:", "pages": [ 12, 13 ] }, { "title": "4.2. Intensity from recombinations and collisions in the IGM", "content": "The SimFast21 code was built to calculate the IGM ionized state assuming two types of regions: one fully ionized (inside the HII bubbles) and other fully neutral. The intensity of Lyα emission in the IGM due to recombinations is a smooth function of the IGM ionized fraction and is dominated by emission from fully ionized regions (see figure 7) so the output os the SimFast21 code is good enough to estimate this intensity. Collisions between electrons and neutral hydrogen atoms can also lead to Lyα emission, however as was explained in section 3.2, collisional Lyα emission only occurs in partly ionized regions, mainly in the the edge of HII bubbles, so the estimation of this emission requires a more detailed description of the IGM ionized state than the one given by the limited resolution of semi numerical simulations. Collisions are most important in regions where the IGM ionized fraction is locally close to 0.5 and the temperatures are high. Since high temperature regions are likely to be highly ionized we can deduce with the help of figure 7, that Lyα emission from recombinations is dominant over Lyα emission from collisions in the IGM.", "pages": [ 13 ] }, { "title": "4.3. Intensity from the scattering of Lyman-n photons in the IGM", "content": "The IGM Lyα intensity from scattering of Lyn photons emitted from galaxies can also be calculated using data from the code SimFast21. This code uses Equation 10 in Santos et al. (2010) to calculate the spherical average of the number of Lyα photons, J α , hitting a gas element per unit proper area per unit time per unit frequency per steradian. The Lyman alpha intensity originated from these continuum photons is given by:", "pages": [ 13 ] }, { "title": "4.4. Results", "content": "Using the prescriptions described in the previous sections we ran simulations Sim1 , Sim2 and Sim3 with a volume of 54 3 h -3 Mpc 3 and 1800 cells from redshift 14 to redshift 6. The obtained IGM ionization fractions, at redshift seven, where x i =0.86 for simulation Sim1 and x i =1.0 for simulations Sim2 and Sim3 . These values are consistent with the current most likely values for this parameter, 0 . 8 ≤ x i ( z = 7) ≤ 1 . 0 (Mitra et al. 2012). The IGM ionized fraction evolution for Sim2 and for Sim3 (see figure 9), resulted in optical depths to reionization of 0 . 073 and 0 . 082. This optical depths are consistent with the value obtained by WMAP ( τ = 0 . 088 ± 0 . 015) (Komatsu et al. 2011). The optical depth correspondent to Sim1 is 0.66 which is lower than the current observational constraints. Based in the optical depth constraint Sim2 and Sim3 have the most likely reionization histories and the IGM ionized fraction evolution obtained with Sim1 can be seen as a lower bound. The intensities of Lyman alpha emission from galaxies at redshift seven obtained with the SimFast21 code are similar to the more theoretical estimates summarized in table 3.", "pages": [ 13 ] }, { "title": "TABLE 5", "content": "Surface brightness (in observed frequency times intensity) of Lyα emission from the different sources in the IGM at z ≈ 7 , z ≈ 8 and z ≈ 10 . The intensity values found in tables 3 and 5 and the theoretical estimations plotted in figure 8 indicate that for the Lyα intensity from the IGM to reach a value close to the emission from galaxies at z = 7 would require a very large absorption of Lyα photons by dust in galaxies. The resulting power spectra of Lyα emission in galaxies and in the IGM obtained with the SimFast21 code are presented in figure 10 for z = 7 and for z = 10. We repeated the Lyα power spectra calculation for several redshifts during the EOR and plotted the Lyα power spectra as a function of redshift for several k in figure 11. We calculated the intensity of Lyα emission from galaxies and from the IGM (intensities are shown in figure 12), and found that according with our assumptions and as already previously seen, the Lyα emission from galaxies is dominant over the Lyα emission from the IGM at least during the redshift interval from z = 6 to z = 9. Since the star formation halo mass relation is not very constrained, we can use the results obtained with Sim1", "pages": [ 13 ] }, { "title": "TABLE 6", "content": "Surface brightness (in observed frequency times intensity) in units of [ergs -1 cm -2 sr -1 ] of Lyα emission from Galaxies at z ≈ 7 , z ≈ 8 and z ≈ 10 for Sim1 , Sim2 and Sim3 .", "pages": [ 14 ] }, { "title": "TABLE 7", "content": "Surface brightness (in observed frequency times intensity) in units of [ergs -1 cm -2 sr -1 ] of Lyα emission from the IGM at z ≈ 7 , z ≈ 8 and z ≈ 10 for Sim1 , Sim2 and Sim3 . Surface brightness (in observed frequency times intensity) in units of [ergs -1 cm -2 sr -1 ] of total Lyα emission at z ≈ 7 , z ≈ 8 and z ≈ 10 for Sim1 , Sim2 and Sim3 . and Sim3 as the lower and upper bounds to the expected Lyα intensity. The evolution of the Lyα intensity from galaxies, from the IGM and total intensity can be seen respectively in tables 6 7 and 8 for simulations Sim1 , Sim2 and Sim3 . A map of the total Lyα intensity in galaxies and in the IGM is presented in figure 13 for z = 7.", "pages": [ 14 ] }, { "title": "5. CROSS-CORRELATION BETWEEN Lyα AND 21-CM OBSERVATIONS", "content": "Observations of the 21 cm signal from the EOR will suffer from contamination by foregrounds and systematic effects. Since both 21 cm line emission and Lyα line emission trace neutral hydrogen, these two lines are expected to be strongly correlated. The cross correlation of these two lines can be used as an extra method to probe the evolution of the IGM ionized hydrogen fraction. In particular the power spectra of this cross correlation will have a discontinuity at a scale that is related to the average bubble size and hence the average ionization fraction in the Universe. During the EoR, the 21 cm signal from galaxies is much smaller than the emission from the IGM so it is safe to neglect both this galaxy emission and the shot noise emission in the cross-correlation. Since the Lyα emission from galaxies is dominating over the IGM for most redshifts, we can just concentrate on the Lyα -galaxy/21cm-IGM cross-correlation when analyzing the cross-power spectrum. The cross correlation between the 21 cm signal and the Lyα line in galaxies is therefore given by: where I 21 is the average intensity of 21 cm emission, P x i δ ( z, k ) is the cross correlation power spectra between the ionized field and the matter density fluctuations, P δδ ( z, k ) is the power spectra of matter density fluctuations and we are assuming that the Lyα emission is a biased tracer of the underlying dark matter field. In figure 14 we show the cross-correlation power spectrum between the total Lyα emission and the 21 cm signal at redshifts 7, 8, 9 and 10. For simulation Sim1 this redshifts correspond to ionizing fractions of x i = 0 . 86, x i = 0 . 56, x i = 0 . 35 and x i = 0 . 23 for redshifts 7, 8, 9 and 10 respectively. In figure 14 the scale at which P Ly α, 21 ( k ) goes from negative to positive is determined by the average size of the ionized regions. For small scales the correlation is positive since fluctuations from both lines should be proportional to the underlying density fluctuations but for large scales (small k ) the correlation is negative since the 21 cm line and the lyman alpha line are characteristic of neutral gas and ionized gas respectively (and there will be an extra negative contribution from the ionised bubbles).", "pages": [ 15 ] }, { "title": "6. OBSERVATIONS", "content": "Current observations related to Lyα emission are restricted to narrow-band imaging of Lyα emitters during reionization and the direct detection of individual emitters. This has led to order ∼ 200 secure detections at z > 5, but systematic uncertainties remain on the fraction that are arising at low redshifts and associated with [OIII]/[OII] lines, among others. Due to narrow atmospheric windows, observations in near-IR bands necessary to cover the reionization epoch are also limited to multiple discrete bands. In any case, existing data could be used for a statistical study such as the power spectrum to extract properties of Lyα emitters that remain below the 5 σ level of individual source/line identifications. Given that detections do exist at the bright-end and our predictions are consistent with the Lyα LFs derived from observational measurements, it is likely that a modest improvement in existing technology and programs will lead to an experiment with sufficient sensitivity to measure the Lyα anisotropy power spectrum during reionization over a broad range of redshifts. The main limitation, unfortunately, is that existing groundbased observations are very limited to small fields of view with narrow-bands in the redshift. Note that from the ground we expect a noise ( νI ) of ∼ 2 . 5 × 10 -3 erg cm -2 sr -1 (assuming we can avoid the OHlines, otherwise, the intensity will be ∼ 1 . 0 × 10 -1 erg cm -2 sr -1 ). From space, the main contamination will be the zodiacal light, which will have a value ∼ 5 × 10 -4 erg cm -2 sr -1 . It is possible that a dedicated experiment from the ground can be conceived to improve our understanding of reionization through detailed Lyα mapping over a broad range of redshifts using specific instruments and filters that suppress the atmospheric contamination. Because of this strong atmospheric contamination, suborbital and/or orbital experiments may however offer a better option. The predictions we have made here can be used as a guide in designing such instruments and experiments. In figure 15 we show the expected errors at z = 7 (central wavelength of 0.975 µ m) for a dedicated compact space-borne template instrument to study Lyα EOR fluctuations. We consider a 20 cm aperture and a spectrometer with resolution R = λ/ ∆ λ = 200. The imaging will be done using a 2048x2048 HgCdTe detector array in order to cover in one pointing a field of view of 45x45 arcmin with a resolution of 10 arc-second pixels on the sky and a spectral range from 0.85 to 1.10 µ m. We took a survey area of 20 deg 2 and a total observation time of 2900 hours. This example shows that Lyα EOR science is well within the reach of our modest template instrument. The calculated sensitivities achieved on the deep fields are sufficient to detect Lyα in broad ∆ k/k bins ranging from k = 0 . 01 to 10 h/Mpc in both clustering and Poisson fluctuations. Ideally the spectral resolution would match the maximum k available in the angular direction; however higher spectral resolution requires longer integration times needed to realize photon noise limited sensitivity, which tends to degrade the instrument sensitivity. The angular resolution does not affect surface brightness sensitivity directly, but does determine the depth to which lower-redshift galaxies may be masked using a deep ancillary continuum galaxy survey. Although the continuum emission from galaxies can in principle be removed by looking at the signal across the frequency direction, as explained before, contamination from other lines at lower redshifts does poses a problem to the detection of the Lyα signal, in particular from the H α line. The most straightforward way to remove this contamination would be by masking the pixels where these lowz galaxies are found, either from the observation itself or using another, high sensitivity, continuum observation. For this approach, the angular resolution of the Lyα experiment has to be good enough in order to have enough pixels left after the masking. Therefore, this instrument is required to have higher angular resolution than spectral resolution. Figure 15 also shows the expected contamination from the H α line from galaxies at z = 0 . 5 (black dots). This was calculated following the same approach as for the Lyα line and using the H α to SFR relation taken from Kennicutt et al. (1994) and Kennicutt (1998b). Removing low-z galaxies down to a mass of ∼ 6 . 6 10 10 M glyph[circledot] , corresponding to a cut in Luminosity L > 1 . 0 × 10 40 erg/s, would bring this contamination below the Lyα signal (black crosses). Using the H α Luminosity function from Geach et al. (2010) normalized to the SFR density at z = 0 . 5 we get an expected angular density of about 25 H α emitters per square degree per band, which would mean that only ∼ 0.98% of the pixels would be masked. Note that the rejection of interloping low-redshift galaxies requires a full treatment that is beyond the scope of this paper. Foreground rejection may also be significantly enhanced by simultaneously detecting additional EOR spectral features beyond Lyα , which are produced by interlopers with very low probability. Combining these Lyα measurements with other EOR observations (CO, C+ and particularly HI 21 cm) offers additional information on EOR star-formation, metallicity, and ionization history. The possibility of constructing an experiment in a near-IR band to measure the Lyα flux in order to correlate it with the 21 cm signal was also explored by Wyithe et al. (2007). Although, they used simple models to estimate the fluctuations in each of these two lines, they also considered several foregrounds that will contaminate the observations and concluded that it is possible to remove enough foregrounds that the intensity of radiation emitted from galaxies can be constrained from the cross correlation.", "pages": [ 15, 16 ] }, { "title": "7. SUMMARY", "content": "In this paper we took into account the main contributions to Lyα emission from recombinations, collisions, continuum emission in galaxies and scattering of Lymann photons to calculate the intensity of Lyα emission from galaxies and from the IGM during the EOR. We started by theoretically calculating the intensities using astrophysical data from several observational results and then implemented the calculation in a simulation using a modified version of the code SimFast21 to obtain the spatial fluctuations of Lyα emission. The simulation allowed to calculate the Lyα emission taking into account the spatial fluctuations of the different astrophysical parameters, which represents an improvement over theoretical calculations that only use the average values. Our simulations showed that to achieve optical depths compatible with the WMAP constraints the high SFRD required imply that for reasonable values of UV and Lyα escape fraction the intensity of Lyα emission from galaxies is dominant over the emission from the IGM. By testing different SFR halo mass parameterizations we constrained the intensity of Lyα emission from galaxies to be about (1 . 43 -3 . 57) × 10 -8 and (4 . 55 -9 . 73) × 10 -11 erg -1 cm -2 sr -1 at redshift 7 and 10, respectively which is dominant over the intensity of Lyα emission from the IGM at z = 7 (about 1 . 6 × 10 -5 ) but less at z = 10 (1 . 1 × 10 -10 erg s -1 cm -2 sr -1 ). Since the intensity levels we found are lower than the extragalactic background intensity from galaxies and so are too low to be detected with an experiment aiming the absolute background intensity, we propose an intensity mapping experiment which will allow to measure the Lyα power spectrum. For reasonable astrophysical conditions the process of hydrogen reionization was done by UV radiation originated in galaxies with luminosities below the high redshift observational threshold. In this work we showed the different ways by which UV emission is connected to Lyα emission and so we stress how it would be useful to use intensity mapping of Lyα emission to probe the overall intensity of UV radiation. Lyα emission can also be connected to the 21 cm signal from the Epoch of Reionization, since the continuum photons above the Lyα line that redshift to this line in the IGM contribute to the radiative coupling of the 21 cm signal to the gas temperature. The cross correlation of the Lyα and the 21-cm lines can be used to reduce systematics and foregrounds encountered with 21-cm observations. In particular the discontinuity of the cross correlation power spectra will provide constrains in the evolution of the IGM ionized fraction. In previous studies we have discussed the use of CO molecular and CII fine-structure atomic lines to complement 21-cm data in the attempt to probe the IGM during reionization. Our study shows that Lyα intensity mapping is also a viable approach to probe reionization and is within the experimental reach over the coming decade. This work was supported by FCT-Portugal with the grant (SFRH/BD/51373/2011) for MBS and under grant PTDC/FIS/100170/2008 for MBS and MGS. AC and YG acknowledge support from NSF CAREER AST-0645427 and NASA NNX10AD42G at UCI. MBS was a long-term Visiting Student at UCI, supported by NSF CAREER AST-0645427, when this work was initiated and she thanks the Department of Physics and Astronomy at UCI for hospitality during her stay.", "pages": [ 16, 17 ] }, { "title": "REFERENCES", "content": "Haiman, Z., Spaans, M., & Quataert, E. 2000, ApJ, 537, L5 Hayes, M., Schaerer, D., Ostlin, G., et al. 2011, ApJ, 730, 8 Iye, M., Ota, K., Kashikawa, N., et al. 2006, Nature, 443, 186 Jensen, H., Laursen, P., Mellema, G., et al. 2012, ArXiv e-prints Jiang, L., Egami, E., Kashikawa, N., et al. 2011, ApJ, 743, 65 Karzas, W. J., & Latter, R. 1961, ApJS, 6, 167 -. 1998b, ApJ, 498, 541 Razoumov, A. O., & Sommer-Larsen, J. 2010, ApJ, 710, 1239 Salvaterra, R., Della Valle, M., Campana, S., et al. 2009, Nature, 461, 1258", "pages": [ 17, 18 ] } ]

[ { "title": "ROTATION RATES OF THE CORONAL HOLES AND THEIR PROBABLE ANCHORING DEPTHS", "content": "K. M. Hiremath and Hegde, M. Indian Institute of Astrophyscis, Bengaluru-560034 May 16, 2018", "pages": [ 1 ] }, { "title": "Abstract", "content": "For the years 2001-2008, we use full-disk, SOHO/EIT 195 ˚ A calibrated images to determine latitudinal and day to day variations of the rotation rates of coronal holes. We estimate the weighted average of heliographic coordinates such as latitude and longitude from the central meridian on the observed solar disk. For different latitude zones between 40 o north - 40 o south, we compute rotation rates, and find that, irrespective of their area, number of days observed on the solar disk and latitudes, coronal holes rotate rigidly. Combined for all the latitude zones, we also find that coronal holes rotate rigidly during their evolution history. In addition, for all latitude zones, coronal holes follow a rigid body rotation law during their first appearance. Interestingly, average first rotation rate ( ∼ 438 nHz ) of the coronal holes, computed from their first appearance on the solar disk, match with rotation rate of the solar interior only below the tachocline.", "pages": [ 1 ] }, { "title": "1 INTRODUCTION", "content": "Solar coronal holes (CH) are large regions in the solar corona with low density plasma (Krieger et al. 1973; Neupert & Pizzo, 1974; Nolte et al. 1976; Zirker 1977; Cranmer 2009 and references therein; Wang 2009) and unipolar magnetic field structures (Harvey & Sheeley 1979; Harvey et al. 1982), distinguished as dark features in EUV and X-ray wavelength regimes. During the solar maximum, CH are distributed at all latitudes, while at solar minimum, CH mainly occur near the polar regions (Madjarska & Wiegelmann 2009). In addition to sunspot activity and magnetic activity phenomena that strongly influence the Earth's climate (Hiremath 2009 and references there in), there is increasing evidence that, on short time scales, occurrences of solar coronal holes trigger responses in the Earth's upper atmosphere and magnetosphere (Soon et al. 2000; Lei et al. 2008; Shugai et al. 2009; Sojka et al. 2009; Choi et al. 2009; Ram et al. 2010; Krista 2011; Verbanac et al. 2011). Physics of solar cycle and activity phenomena is not well understood (Hiremath 2010 and references therein). In order to understand the solar cycle and activity phenomena, an understanding of rotational structure of the solar interior and the surface are necessary. On the other hand, rotation rate of the interior and the surface are coupled with the rotation rate of the solar atmosphere, especially the corona. Although there is a general consensus regarding the interior rotation as inferred from the helioseismology (Dalsgaard & Schou 1988; Thompson et al. 1996; Antia et al. 1998; Thompson et al. 2003 and references therein; Howe 2009; Antia & Basu 2010), surface rotation rates as derived from sunspots (Newton & Nunn 1951; Howard et al. 1984; Balthasar et al. 1986; Shivaraman et al. 1993; Javaraiah 2003), Doppler velocity (Howard & Harvey 1970; Ulrich et al. 1988; Snodgrass & Ulrich 1990) and magnetic activity features (Wilcox & Howard 1970; Snodgrass 1983; Komm et al. 1993), there is no such consensus (see also Li et al. 2012) on the magnitude and form of rotation law for features in the corona. For example, by using coronal holes as tracers (Wagner 1975; Wagner 1976; Timothy & Krieger 1975; Bohlin 1977) and large scale coronal structures (Hansen et al. 1969; Parker et al. 1982; Fisher & Sime 1984; Hoeksema 1984; Wang et al. 1988; Weber et al. 1999; Weber & Sturrock 2002), previous studies show that corona rotates rigidly while other studies (Shelke & Pande 1985; Obridko & Shelting 1989; Navarro-Peralta & Sanchez-Ibarra 1994; Insley et al. 1995) indicate differential rotation. In addition to using coronal holes as tracers, X-ray bright points (Chandra et al. 2010; Kariyappa 2008; Hara 2009), coronal bright points (Karachik et al. 2006; Brajˇsa et al. 2004; Wohl et al. 2010), and SOHO/LASCO images have been used for the computation of rotation rates and yield a differentially rotating corona. Recent studies using radio images at 17 GHz (Chandra et al. 2009) and synoptic observations of the O VI 1032 ˚ A spectral line from the SOHO/UVCS telescope (Mancuso & Giordano 2011), however, suggest that the corona rotates rigidly. As part of an ISRO (Indian Space Research Organization) funded project, the present study utilizes SOHO/EIT 195 ˚ A calibrated images for understanding the following four objectives : (i) to check for latitudinal dependency of rotation rates of the coronal holes, (ii) to study rotation rates of CH during their first appearance on the observed disk, (iii) irrespective of their latitude, to study day to day variation of rotation rates of coronal holes and, (iv) to estimate probable anchoring depths of coronal holes. In section 2, we present the data used and method of analysis, and the results of that analysis in section 3. In section 4, we present the discussion on cause for rigid body rotation rate of the coronal holes and estimate their probable anchoring depths with our conclusions.", "pages": [ 1, 2 ] }, { "title": "2 DATA AND ANALYSIS", "content": "For the period 2001 to 2008, we use full-disk SOHO (Solar and Heliospheric Observatory)/EIT images (Delaboudini'ere et al . 1995) that have a resolution of 2.6 arc sec. per pixel in a bandpass around 195 ˚ A to detect coronal holes. The period studied includes both intense activity near solar maximum and the descent of solar activity parameters such as 10.7 cm flux to values of ∼ half of their values around that maximum. The obtained images are in FITS format and individual pixels are in units of data number (DN). DN is defined to be output of the instrument electronics which corresponds to the incident photon signal converted into charge within each CCD pixel (Madjarska & Wiegelmann 2009). We consider coronal holes that appear and disappear between 40 o north - 40 o south latitude of the visible solar hemisphere. Using the SolarSoft eit prep routine (Freeland & Handy 1998), we background subtracted, flat-fielded, degridded and normalized the images. As this calibration involves exposure normalization of the images, now onwards unit of DN is DN/sec. We used the occurrence dates and position of CH from the ' spaceweather.com' website. As the ' spaceweather.com' is not designed for scientific use, we use readily available occurrence dates of CH only. By using approximate position (heliographic coordinates) of CH from this website, we separate a region from the SOHO/EIT images for further analysis and extraction of relevant physical parameters as described below. CH is also confirmed if it has a bimodal distribution in the intensity histogram. In order to extract physical parameters of CH from the EIT images, we use FV interactive FITS file editor (http://heasarc.gsfc.nasa.gov/docs/software/ftools/fv/). Depending upon shape of the CH, from the FV editor, a circle or an ellipse is drawn covering the whole region of CH and, average DN (intensity) (that is set as a threshold for detecting the boundary) of CH is computed for detecting the boundary (private communications with Prof. Aschwanden). Similar to Karachik & Pevtsov (2011), for some of the coronal holes, threshold is modified to match the visually estimated boundary. This method yields results consistent with the previous intensity histogram methods (Krista & Gallagher 2009; Krista 2011; de Toma 2011 and references there in). After determining the boundary of CH, we employed SolarSoft coordinate routines to compute the central meridian distance ( l i ) (heliographic longitude from the central meridian) and latitude ( θ i ) of individual pixels within the CH. Fig 1(a) shows a full disk, solar image with a typical CH close to the center and in the north-east quadrant, while Fig 1 (b) represents the same CH with its threshold DN contour map. In Fig 2(a), DN histogram of the CH is presented. The bimodal distribution in the histogram confirms the DN values in the CH region (Krista & Gallagher 2009; Krista 2011). We summed the total number of pixels and total DN within the CH boundary, which in turn allowed us to compute average heliographic coordinates such as latitude ( θ ) and central meridian distance (L) of CH as follows: where θ i , l i , and DN i (for i = 1 , n , n is number of pixels) are the latitude, the central meridian distance, and DN values of individual pixels. This method of finding the average heliographic coordinates of CH is equivalent to a method in physics of finding the center of mass of an arbitrary geometrical shape. As the average heliographic coordinates of CH are weighted by the intensity (DN counts) of the relevant pixel, thus one can argue that more weight is given to brighter pixels. However, this argument can not be valid as the intensity is weighted in the denominator also (see above equation 1) and, hence, whatever higher weights given to the brighter pixels in the numerator are also equally compensated by the higher weights in the denominator. We also checked with another weighting that emphasizes areas darker than the image mean, ( i.e., ( n ∑ i =1 DN i /N ) -DN i )) and obtained the same results of average heliographic coordinates suggesting that weighted average used in equation (1) is correct and is not biased towards the brighter pixels. For computation of heliographic coordinates of CH, we also used weights with inverse of DN (1/DN) and without weights ( i.e., simple averages) in equation 1 and the results for three typical CH are presented in Table 1. Negative sign for the longitude indicates the CH that are on the eastern side of the central meridian and negative sign for the latitudes indicates the CH that are in the southern hemisphere. One can notice from this table that irrespective of weighted and non-weighted averaging, computed heliographic coordinates of CH are nearly same. Following the previous method (Hiremath 2002) of computation of rotation rates of sunspots, daily siderial rotation rates Ω j of the CH are computed as follows where L j , L j +1 are average longitudes of the CH for the two consecutive days t j and t j +1 respectively, j = 1 , 2 , ..n -1, n is number of days of appearance of CH on the visible solar disk and, δ Ω is a correction factor for the orbital motion of the Earth around the Sun. Strictly speaking, this correction factor is due to orbital motion of the SOHO spacecraft around the sun. Compared to the distance between the sun and earth, the distance between the SOHO satellite and the earth is very small and hence orbital distances of earth and the satellite are almost same and hence the correction factor δ Ω is ∼ 1 deg/day. For the present work, this approximation is sufficient. However, if one wants to find the long term ( ∼ 11 yrs) variation of rotation rates, correction factor δ Ω should be computed accurately (Roˇsa et.al. 1995; Wittmann 1996; Brajˇsa et al. 2002). From the first and second day appearances of CH, one can compute the rotation rate Ω 1 that we call as first rotation rate . Similarly for other successive days, rotation rates Ω 2 , Ω 3 , etc., are computed. For each computed rotation rate of CH, the respective latitude is assigned as the average of two latitudes corresponding to the two longitudes. We also compute standard deviation and error bars of the average heliographic coordinates and rotation rates. Here onwards computation of rotation rates of CH from equation (2) is called as First Method .", "pages": [ 2, 3, 5 ] }, { "title": "3 RESULTS", "content": "We followed the following criteria in selecting CH data: (i) In order to avoid projection effects (especially coronal holes near both the eastern and the western limbs), we considered only the coronal holes that emerge within 65 · central meridian distance, (ii) the coronal hole must be compact, independent, not elongated in latitude, and, (iii) during its passage across the solar disk it should not merge with other coronal holes. For the period of observations from 2001 to 2008, a total of 113 CH satisfy these criteria. We define the term τ of a CH as total number of days observed on same part of the solar disk satisfying the afore mentioned criteria. Suppose we assume that CH decay due to magnetic diffusion only, as the dimension L of CH is very large (from the following section 3.1, one can note that area A is ∼ 10 20 cm 2 ), magnetic diffusion time scale τ ( L 2 η ∼ A πη , where η is magnetic diffusivity and area A of CH is assumed to be a circle; magnetic diffusivity in the corona is considered to be ∼ 10 13 cm 2 sec -1 (Krista 2011; Krista et.al 2011)) is estimated to be ∼ 2 months. Hence, there is a possibility that CH might have reappeared again on the visible disk and might have diffused in the solar atmosphere. Hence, actual life span of CH must be of longer duration. In Fig 2(b), for different τ , we present occurrence number of CH considered for this study. During their evolutionary passage over the solar disk, we compute rotation rates and assign respective latitudes. If the CH exists for n days, then its τ is n days and, total number of rotation rates is ( n -1). Rotation rates of non-recurrent CH that appear and disappear on the visible disk are computed. According to above definition, and in the present data set (see Fig 2(b)), we find 4 CH that appear for 10 days, 13 CH for 9 days and so on. Integrated over all latitudes and in both the hemispheres, we determined a total of 683 rotation rates.", "pages": [ 5, 7 ] }, { "title": "3.1 Average Rotation Rates : Variations With Respect to Latitude and Area", "content": "During their passage over the solar visible disk, daily rotation rates of CH are computed. In both the hemispheres, for each latitude bin of 5 · , we collect rotation rates and compute average rotation rates with their respective standard deviations σ and the errors ( σ √ N , where N is number of rotation rates). We present the results in Fig 3 (a) that illustrate the variation of average rotation rates of the coronal holes for different latitudes. To be on the safer side from the projectional effects, we also compute average rotation rates of coronal holes that emerge within 45 · central meridian distances and are illustrated in Fig 3(b). For the sake of comparison with helioseismic inferred rotation rates, in both the plots, we include a frequency scale on the right hand side of the vertical axis. For different latitude bins, observed rotation rates are subjected to a least square fit of the form Ω( θ ) = Ω 0 +Ω d sin 2 θ (where θ is latitude, Ω 0 &Ω d are constant coefficients to be determined). There is every possibility that as the errors in determination of centers of CH propagate to the rotation rates and hence rotation rates determined from the first method effectively enhance the error in the second coefficient (Ω d ) yielding rigid body rotation rates of CH. Moreover, drawback of the first method is also reflected in Fig 3(b) where unlikely asymmetrical rotation profile in both the hemisphere is obtained. In order to minimize such propagating errors in the rotation rates of CH determined by the first method, we compute rotation rates of CH in the following way and define as a Second Method . In this method, as suggested by the referee, we fit all the daily centroid positions of the individual coronal holes by a first degree polynomial and, computed second coefficient (slope) represents the rotation rate. For each computed rotation rate of CH, the respective latitude is assigned by averaging all the latitudes of CH during its passage. As described in the previous paragraph we binned the rotation rates, computed the average rotation rates, standard deviations and error bars respectively. For different latitude bins, average rotation rates are subjected to a linear least square fit and the results are presented in Fig 4. From both the rotation laws, compared to the first coefficient, magnitude of small second coefficient (Ω d = -0 . 81( ± 1 . 58) in Fig 3(a) or Ω d = -0 . 51( ± 1 . 64) in Fig 4(a)) suggests that CH rotate rigidly . As sunspots show different rotation rates for the small and big areas (Hiremath 2002), it is interesting to know whether similar variations in rotation rates exist in case of coronal holes. As CH evolve, their area also changes and question arises: for which area during the evolutionary passage, rotation rate has to be considered. For this purpose, we adopt the following method. Daily areas and rotation rates of CH are computed. For all the days of CH ' existence, average area and rotation rates are computed. Further, irrespective of their latitude and τ , rotation rates are collected for the area bins (0-1) × 10 20 cm 2 , (1-2) × 10 20 cm 2 , etc, and mean rotation rates are computed. In Fig 5(a), we present occurrence number of CH for different area bins. Whereas, irrespective of their latitudes and τ , for different area bins, Fig 5(b) illustrates the mean rotation rates of CH. It is important to note from Fig 5(b) that, unlike sunspots, for different areas, all the coronal holes (as the second coefficient is almost zero, i.e., (0 . 36 ± 0 . 20) × 10 -21 ) rotate rigidly . This important result implies that all the CH must originate from same region of the solar interior that rotate rigidly.", "pages": [ 7 ] }, { "title": "3.2 Average Rotation Rates : Variations With Respect to τ and Daily Evolution", "content": "In order to check dependency of rotation rates of CH with respect to number of observed days τ , daily rotation rates are computed during their evolution. As described in section 3, if CH has τ of n days, we have ( n -1) rotation rates. Irrespective of their areas and the latitude, for each τ , rotation rates are collected and average rotation rate is computed and the results are illustrated in Fig 6(a). We find that, rotation rates of CH are independent of τ . Further, irrespective of their area and τ , we combined daily rotation rates for all the latitudinal bins; we present the resulting daily average rotation rates in Fig 6(b). If coronal holes rotate rigidly and are independent of latitude, then the integrated rotation rates for all the latitudes should remain constant. For example, let us consider the rotation law (red continuous line) over plotted on Fig 6(b). From this law, when one computes the difference between rotation rates of the first day and the 10th day, the difference is found to be ∼ 0.1 degree/day, approximately same magnitude as the formal uncertainty in the value for each bin, once again strongly suggesting that, for all the days during their evolutionary passage, coronal holes rotate rigidly .", "pages": [ 9 ] }, { "title": "3.3 Comparison of Rotation Rates of CH With Other Activity Indices", "content": "Compared to rotation rates obtained by other surface activity indices (Figures 7 and 8), (i) coronal holes rotate almost like a rigid body and, (ii) on average, coronal holes rotate slower ( ∼ 440 nHz) than the rotation rates of other activity indices over the latitude range -40 to +40. The ratio R = | Ω d Ω 0 | of the two coefficients of each rotational law gives a sense of whether the rotation is rigid or differential. For example, if one computes this ratio for sunspots ( R sunspot ) and for coronal holes ( R coronal hole ), it is clear that R sunspot /greatermuch R coronal hole , as can also be seen from the fifth 3 First rotation rates for the CMD (+45 · to -45 · ) 4 Snodgrass (1992); 5 Snodgrass (1983); 6 Newton & Nunn (1951); 7 Brajˇsa et.al. 2002; 8 Antia and Basu (2010) *CMD-Central Meridian Distance column of Table 2. In this table, goodness of fit χ 2 is also given in the last column. Small value of χ 2 (typically χ 2 should be ≤ (N-n), where N is total number of data points and n is degrees of freedom, in this case n = 2) implies fit is very good. At least compared with any features lower in the solar atmosphere, it is clear that CH rotate rigidly .", "pages": [ 9, 10, 12 ] }, { "title": "3.4 First Rotation Rates : Variations With Respect to Latitude and Number of Observed Days τ", "content": "In the previous subsections, on the basis of small magnitude of second coefficient (as illustrated in Figures 3 and 4) in the rotation law and the ratio R of CH, we concluded that CH rotate rigidly. Although second coefficient is small, it is not completely negligible to conclude unambiguously that CH rotate rigidly. That means a small contribution to the second coefficient due to differential rotation can not be ruled out. This result can be interpreted as follows. The rotation rates of CH presented in the previous sections are combination of rotation rates of CH that are anchored at different parts of the interior during their evolutionary passage on the visible disk. That means if CH are originated only in the convective envelope and raised their anchoring feet towards the surface, owing to differentially rotating convection zone and similar to magnitudes of rotation rates of sunspots, one should get a reliable and large magnitude of second coefficient in the rotation law. On the other hand, if the CH are originated in the radiative core and raised their anchoring feet towards surface, during their first appearance on the surface, one should get combined contribution (from the differential and rigidly rotating regions) to the second coefficient. That means if one computes the first rotation rates Ω 1 of CH during their first appearance on the surface for different latitudes and number of days ( τ ) observed on the disk, one should get unambiguously negligible contribution from the second coefficient of the rotation law. In order to test this conjecture, first rotation rates Ω 1 of CH are computed as follows. Again, we consider CH that are born between +65 · to -65 · from the central meridian. From the first and second day computed longitudes (from the central meridian) of CH and by using first method, first rotation rates are computed. Each first rotation rate is collected in 5 · latitude bins and average of the first rotation rates is computed and, for different latitudes, are illustrated in Fig 9(a). Similarly, for different τ , first rotation rates are collected and average of first rotation rates is computed and the results are presented in Fig 9(b). It is important to note that, according to our conjecture, we find that magnitude of the second coefficient has a negligible contribution to the rotation law that leads to inevitable conclusion that CH must rotate rigidly . From all these results, finally we conclude unambiguously that, independent of their area, number of observed days ( τ ) and latitude, CH rotate rigidly during the evolutionary passage on the solar disk . However, it is interesting to note from the present and previous studies (Wagner 1975; Wagner 1976; Timothy & Krieger 1975; Bohlin 1977) that although whole coronal hole structure rotates rigidly, individual coronal bright points (CBP) that are embedded in the coronal holes rotate differentially (Karachik et.al. 2006). As pointed by these authors, coronal bright points in the corona might be influenced by the surrounding differentially rotating plasma. However, it is not clear how CBP are influenced by the differential rotation of the surrounding plasma.", "pages": [ 12, 13 ] }, { "title": "4 DISCUSSION AND CONCLUSIONS", "content": "In contrast to other persistent solar features of the corona, then, why do coronal holes rotate rigidly? Many observations (Madjarska et.al 2004; Subramanian et.al 2010; Tian et.al. 2011; Yang et.al 2011; Krista 2011; Krista et.al 2011; Crooker & Owens 2011; Madjarska et.al 2012) suggest magnetic reconnection at the coronal hole boundaries (CHBs) as the cause of rigid body rotation. Pevtsov & Abramenko (2010) conclude that coronal holes ' rotation rate is almost like rotation rate of sunspots and the CH are 'analogous to a grass fire, which supports itself by continuously propagating from one patch of dry grass to the other'. That means coronal hole constantly changes its footprint moving from one available polarity to the other. This implies that area of coronal hole will depends on size of available polarity footprint, and it can either decrease or increase depending on size of photospheric magnetic field patch. This also suggests that, on average, difference in the coordinates at the eastern and western boundaries should remain constant yielding a rigid body rotation rate (as suggested by the previous studies). Thus one can argue that coronal holes are surface phenomena. If the coronal hole is a surface phenomenon and if it constantly changes its footprint moving from one available polarity to other, area of coronal hole depends on size of available polarity footprint. Hence, area should either decreases or increases with a result that, on average, area with respect to time must be nearly constant. In order to test this conjecture, in Fig 10, we illustrate measured areas (that are corrected for projection) of CH that have τ of 4 days and 5 days (upper panel) and, 6 days and 10 days (lower panel) respectively. Dates of occurrence of these individual coronal holes that are presented in the upper panel of are : 6th Nov to 9th Nov 2001 (2 · to 41 · East of central meridian); 8th May to 12th May 2004 ( 30 · East to 12 · west of central meridian) and, dates of occurrence of coronal holes that are presented in the lower panel are: 21st Aug to 26th Aug 2003 (45 · East to 11 · west of central meridian); 22nd Dec to 31st Dec 2005 (50 · East to 62 · west of central meridian) respectively. One can notice from Fig 10 that, contrary to expectation (that area of coronal hole nearly remains constant during its evolution), on average, coronal holes ' area smoothly decrease (upper panel) continuously or increase like sunspots ' area evolutionary curve, reaches maximum area and then smoothly decreases (lower panel). From these figures, we can not find other expected signatures for the reconnection, viz., substantial daily variations of areas of CH during their evolution. This does not mean that there is no magnetic reconnection at the boundaries. However, in the following, we show that magnetic reconnection alone can not be sufficient for explanation of dynamics (rigid body rotation) and area evolution of the coronal holes. Hence, coronal holes must be deep rooted rather than mere surface phenomena. Interestingly, similar to Bohlin's (1977) study, we also find the same order ( ∼ 10 14 cm 2 /sec ) of average growth (or decay) of CH. One would also expect, magnetic reconnection at the boundary of CH might have a substantial contribution for the enhancement of the average intensity (DN counts). In order to check this expectation, for the same CH presented in Fig 10, we compute the daily average DN counts n DN i (= i =1 N , where N is total number of pixels) and are illustrated in Fig 12. Obvious fact from Fig 10 and 12 is that as the area of CH increases, average DN counts (intensity) decrease and vice versa . However, according to our expectation, coronal holes do not show any transient and substantial increase in the intensity during their daily evolutionary passage on the solar disk. Off course, as CH is embedded in the atmosphere where closed field lines due to active regions coexist and hence, it is natural to expect reconnection at the boundary of a CH due to oppositely directed field lines. Possible reason for the null detection of magnetic reconnection from our data set is due to low temporal resolution of daily data used in this analysis. In fact, with a high temporal resolution of CH data set, majority of previous studies (Wang et al. 1998; Madjarska et al. 2004; Raju et.al 2005; Aiouaz 2008; Madjarska & Wiegelmann 2009; Subramanian et al. 2010; Edmondson et.al. 2010; Krista 2011; Yang et.al. 2011; Madjarska et.al 2012) show the evidences of reconnection, although other studies have lack of such a evidence (Kahler & Hudson 2002; Kahler et al. 2010). If we go by the majority of results that rigid rotation rates of the coronal holes is due to magnetic reconnection at the coronal hole boundaries, then one would expect that the shape (area) of the coronal hole during their disk passage must remains constant. As most of these majority of studies used short ( ∼ hours) duration data set, question arises whether CH maintain their shape (and hence their areas) through out disk passage (as one can see from our analysis, most of CH exist more than 5 days on the solar disk). One can notice from the area-time plots (Fig 10), during ( ∼ days) its disk passage, CH do not maintain their shape and hence rigid body rotation rate of CH is not due to interchange reconnection. As the previous studies use high temporal, short duration ( ∼ hours) data set and during such time scales (as the CH has a large dimension) obviously one gets constant shape and hence conclusion (that rigid rotation rates of CH is due to magnetic reconnection) is right. However, again we stress from the results presented in Fig 10 that, on long duration ( > 5 days), CH do not maintain their shape and rigid body rotation rate of CH is not due to magnetic reconnection alone at their boundaries. Rigid body rotation rate of CH is likely due to their deep rooted anchoring of their feet and subsequently raising towards the surface and then to the atmosphere. ∑ As for area evolution of the coronal hole, question arises as to which is the dominant physical process that dictates temporal variation of area and hence removal of magnetic flux of the coronal hole? Is it due to magnetic diffusion (whose diffusion time scale is ∼ L η 2 ) or magnetic reconnection at the coronal hole boundaries? Similar to sunspots ' area evolution curve (Hiremath 2010), formation and growth part of area evolution of CH are not understood. However, in order to answer afore mentioned queries, we consider decay part of the area evolution curve with following two physical reasonings: (i) if area evolution of CH is dominated by magnetic diffusion, then its area must varies as ∼ t -1 / 2 (where t is time variable) and, (ii) if area evolution of CH is mainly due to magnetic reconnection, annihilation of magnetic flux due to reconnection of opposite magnetic field lines at the boundaries of the coronal hole leads to an exponential decrease of area with time. If coronal hole is considered to be cylindrical magnetic flux tube with uniform magnetic field structure, from magnetic induction equation (with diffusive dominated term), it is instructive to show that equation for rate of change of magnetic flux φ is dφ dt = η d 2 φ dz 2 (where φ = ∫ r 0 B z Adr is magnetic flux of coronal hole flux tube, A (= 2 πr 2 ) is area, t is time variable, η is magnetic diffusivity, B z is a uniform magnetic field structure along the z direction and r is radius of flux tube). From the results (Krista and Gallagher 2009; CHARM algorithm from solarmonitor.org) illustrated in Fig 11a, absolute magnitude of B z of 10 days CH (during decay part of its area evolution as presented in Fig 10) is found to be nearly independent of time (number of observed days). Using this observational information and assumption that magnetic field structure of coronal hole is also uniform spatially along r direction, it can be easily shown from the rate of change of magnetic flux equation that dA dt = η d 2 A dz 2 and whose solution is obtained as A ∼ t -1 / 2 on diffusion time scales. In order to test these afore mentioned two reasonings, for example, decay part of 10 days area evolution curve is subjected to diffusion and exponential fits. After linearizing the two laws, least- square fits are performed and the result is illustrated in Fig 11b. Compared to exponential fit, for the decay part of area evolution curve, least-square fit for law of diffusion yields very low value of χ 2 with the expected decay index of ∼ -0.5. Hence, during decay part of its evolution of area, coronal hole is consistent with the first reasoning and area evolution of CH is mainly dictated by magnetic diffusion. However, persistent magnetic reconnection at the boundaries of CH during their evolution can not be neglected. Thus, it is reasonable to conclude that both the magnetic diffusion and the reconnection processes control the evolution of area of CH during their passage on the solar disk . Another important result from this study is that why coronal holes rotate with a magnitude of ∼ 438 nHz during their first appearance, where as other active regions, approximately at the same height in the corona, have a magnitude of rotation rate similar to rotation rate of sunspots. Moreover, similar to sunspots, coronal holes are likely to be three dimensional structures whose dynamical evolution is not only controlled by the surface activity, but also related to the solar interior dynamics where roots of CH might be anchored, probably below base of convection zone. This idea that CH probably might be originated below base of the convection zone is not a new one. In fact, nearly three decades back, Gilman (1977) came to the conclusion that CH ' origin and formation may not be due to so called 'dynamo mechanism' that apparently explains the genesis of sunspot cycle. While discussing the origin of XBP (X-ray bright points), Golub et. al. (1981) came to the conclusion that XBP and coronal holes probably might be originated below base of the convection zone. Recently, Jones (2005) also expressed similar doubt that origin of CH is in the convection zone and concludes that their roots must be further deeper below base of convection zone. Very recently, by investigating the formation of isolated, non-polar coronal holes on the remnants of four decaying active regions at the minimum/early ascending phase of sunspot activity, Karachik et. al. (2010) came to a similar conclusion that, during their first appearance, CH might be deeply rooted. Hence, on the basis of these two important results ((i) first rotation rates of CH during their initial appearance and during evolutionary passage and, (ii) magnitude of rotation rates ( ∼ 438 nHz)), we suggest a possibly naive but plausible reasonable proposition in the following way. Compared to other activity indices such as x-ray bright points (XBP), coronal holes are very large ( ∼ 10 times the typical big sunspot) and it is not unreasonable to suggest that their roots may be anchored very deep below the surface. In case of coronal XBP, from the nature of their differential rotation rates, Hara (2009) has conjectured that their roots might be anchored in the convective envelope, as helioseismic inferences (Antia et al. 1998; Antia & Basu 2010) show that whole convective envelope is rotating differentially. On the other hand, the present and previous studies (Wagner 1975; Wagner 1976; Timothy & Krieger 1975; Bohlin 1977) strongly suggest that the rotation rate of coronal holes is independent of latitude, number of days ( τ ) observed on the disk and area. As for the anchoring depths, during their first appearance in the corona and owing to its magnetic nature (Gurman et al. 1974; Bohlin 1977; Levine 1977; Bohlin & Sheeley 1978; Stenflo 1978; Harvey & Sheeley 1979; Harvey et al. 1982; Shelke & Pande 1984; Obridko & Shelting 1989; Zhang et al. 2006; Fainshtein 2010), we expect that a coronal hole might isorotates with the solar plasma, so its rotation rate during its first appearance and the rotation rate at the anchoring depth must be identical. It is interesting to note that the average rotation rate ( ∼ 438 nHz), we have measured in coronal holes (Fig 13) is similar to that of the average rotation rate of the solar plasma inferred by helioseismology (Antia & Basu 2010; rotation rate of the solar interior averaged over one solar cycle is kindly provided by Prof. Antia) at a depth of ∼ 0 . 62( ± 0 . 10) R /circledot . Hence, during first appearance of the coronal hole, it is reasonable to suggest that the depth of anchoring of CH might be around 0 . 62( ± 0 . 10) R /circledot . If we simply identify the rotation rates found here with the internal rotation rate at a given depth, we find a match only inside the radiative interior, at a depth of 0 . 62( ± 0 . 10) R /circledot solar radii. In future, helioseismology may give further inferences on the anchoring depths of coronal holes. We know, however, of no currently accepted model of magnetic field generation that could anchor coronal structures to such a depth in the interior. With a caveat that unless a consistent and acceptable theoretical model of CH that supports of our proposition (that during their first appearance, roots of CH might be anchored in the radiative core), our proposed idea remains mere a conjecture only. To conclude this study, we used SOHO/EIT 195 ˚ A calibrated images to determine the latitudinal and day to day variations of rotation rates of the coronal holes. We found that: (1) irrespective of their areas and number of days ( τ ) observed on the disk, for different latitude zones, rotation rates of CH follow a rigid body rotation law, (2) CH also rotate rigidly during their evolution history and, (3) during their first appearance, CH rotate rigidly with a constant angular velocity ∼ 438 nHz which only matches depth around 0 . 62( ± 0 . 10) R /circledot , in the radiative interior. This result is so counterintuitive that we can only conclude that we do not understand why CH rotate rigidly at that rate.", "pages": [ 13, 14, 15, 17 ] }, { "title": "Acknowledgements", "content": "Authors are grateful to an anonymous referee for the invaluable comments and suggestions that substantially improved the results and presentation of the manuscript. Authors are also grateful to Dr. J. B. Gurman for giving useful information on the SOHO data, for going through the earlier version of this manuscript and, for giving useful ideas. Hiremath is thankful to former Director, Prof. Siraj Hasan, Indian Institute of Astrophysics, for encouraging this ISRO funded project. This work has been carried out under 'CAWSES India Phase-II program of Theme 1' sponsored by Indian Space Research Organization(ISRO), Government of India. SOHO is a mission of international cooperation between ESA and NASA.", "pages": [ 17 ] }, { "title": "References", "content": "Av. DN counts Av. DN counts 50 48 46 44 42 40 38 36 0 1 2 3 4 5 34 32 30 28 26 24 22 20 0 1 2 3 4 5 6 7 DN(t)=[41.04 ( 0.68)t DN(t)=[31.30 ( 8.71) + 4.22 ( ± ± 3.07) t ] ± Day(t) 8.26) - 7.97 ( 5.05) t + 0.83( ± Day(t) 40 35 30 25 20 15 0 2 4 6 8 10 Day(t) ± ± ± ± 2 ] Av. DN counts DN(t)=[31.77 ( 4.87) - 5.97 ( 2.00) t + 0.61( 2 0.18) t ]", "pages": [ 18 ] } ]

[ { "title": "ABSTRACT", "content": "The Planck collaboration has recently published precise and resolved measurements of the SunyaevZel'dovich effect in Abell 1656 (the Coma cluster of galaxies), so directly gauging the electron pressure profile in the intracluster plasma. On the other hand, such a quantity may be also derived from combining the density and temperature provided by X-ray observations of the thermal bremsstrahlung radiation emitted by the plasma. We find a model-independent tension between the SZ and the X-ray pressure, with the SZ one being definitely lower by 15 -20%. We propose that such a challenging tension can be resolved in terms of an additional, non-thermal support to the gravitational equilibrium of the intracluster plasma. This can be straightforwardly included in our Supermodel, so as to fit in detail the Planck SZ profile while being consistent with the X-ray observables. Possible origins of the nonthermal component include cosmic-ray protons, ongoing turbulence, and relativistic electrons; given the existing observational constraints on the first two options, here we focus on the third. For this to be effective, we find that the electron population must include not only an energetic tail accelerated to γ /greaterorsimilar 10 3 responsible for the Coma radiohalo, but also many more, lower energy electrons. The electron acceleration is to be started by merging events similar to those which provided the very high central entropy of the thermal intracluster plasma in Coma. Subject headings: cosmic background radiation - galaxies: clusters: individual (Abell 1656) - X-rays: galaxies: clusters", "pages": [ 1 ] }, { "title": "THE PLANCK SUNYAEV-ZEL'DOVICH VS. THE X-RAY VIEW OF THE COMA CLUSTER", "content": "R. Fusco-Femiano 1 , A. Lapi 2 , 3 , A. Cavaliere 2 , 4 4 1 IAPS-INAF, Via Fosso del Cavaliere, 00133 Roma, Italy. 2 Dip. Fisica, Univ. 'Tor Vergata', Via Ricerca Scientifica 1, 00133 Roma, Italy. 3 SISSA, Via Bonomea 265, 34136 Trieste, Italy. and INAF, Osservatorio Astronomico di Roma, via Frascati 33, 00040 Monteporzio, Italy. Accepted by ApJL", "pages": [ 1 ] }, { "title": "1. INTRODUCTION", "content": "We are motivated by the recent, spatially resolved measurements with the Planck satellite by Ade et al. (2012) of the Sunyaev-Zel'dovich (1980; SZ) effect in Abell 1656, the very rich, nearby cluster in Coma Berenices at z = 0 . 023. The thermal SZ effect describes how the temperature of crossing CMB photons is modulated by the Compton upscattering off the hot electrons in the intracluster plasma (ICP). Its strength is given by the Comptonization parameter y ≡ ( σ T /m e c 2 ) ∫ d /lscript p e ( r ) integrated along lineof-sights across the cluster. It directly probes the electron thermal pressure p e ≈ p (2 + 2 X ) / (3 + 5 X ) ≥ 0 . 5 p , here written in terms of the ICP pressure p ; with the cosmic hydrogen abundance X ≈ 0 . 76, their ratio reads p e /p = 0 . 52. Compared with previous observations including WMAP 's (see Komatsu et al. 2011, and references therein), the resolved Planck data improve the SZ probing of the cluster core and extend it into the outskirts, providing a handle to the complex astrophysical processes in the ICP to be discussed here. On the other hand, the ICP pressure p ≈ nk B T/µ (with the mean molecular weight µ ≈ 0 . 60) may be also derived from combining the density n and temperature T provided by X-ray observations of the thermal bremsstrahlung radiation emitted by the plasma. The overall trend emerging from the Planck data is toward a deficit in the SZ relative to the X-ray pressure, see Ade et al. (2012). In detail, to fit the SZ data these authors based on empirical formulae suggested by numerical simulations (see Nagai et al. 2007) and by X-ray analyses (see Arnaud et al. 2010). These formulae pro- 'universal' pressure profile for the whole cluster population, or a specific version for unrelaxed clusters. However, when applied to fit the precise Planck SZ data these formulae perform inadequately, as discussed by Ade et al. (2012); specifically, the first version turns out to overshoot the data in the core, and both to appreciably undershoot them in the outskirts, well beyond the quoted uncertainties. Aimed modifications of the parameter values in the fitting formulae, that include suppression of unphysical central divergencies, can improve the SZ fits at the cost of inconsistencies with the X-ray pressure. As discussed by the above authors, this is also the case with multiparametric fitting formulae of the type proposed by Vikhlinin et al. (2006) for the X-ray observables. Thus we are stimulated to use an orthogonal approach, provided by the Supermodel (SM; Cavaliere et al. 2009); this yields a direct link between the X-ray and the SZ observables (see Lapi et al. 2012). We specify below how the SM is based on a physically motivated run of the entropy k ( r ) ≡ k B T ( r ) /n 2 / 3 ( r ) that underlies the ICP support in the gravitational potential well provided by the cluster dark matter, to approach hydrostatic equilibrium. The SM improves in a number of respects upon the classic isothermal β -model (Cavaliere & Fusco-Femiano 1976), adopted by Mohr et al. (1999) and Churazov et al. (2012) to closely describe the central X-ray brightness profile in Coma. In fact, the SM incorporates updated, weakly cusped distributions of the dark matter (see Lapi & Cavaliere 2009). It also accurately describes the central conditions both in cool, and in non-cool core clusters as Coma (see Molendi & Pizzolato 2001). Fi- ally, it also describes the region beyond a few 10 2 kpc where T declines outwards while n drops, so that the entropy gradually rises as shown by many X-ray data (e.g., Snowden et al. 2008; Cavagnolo et al. 2009; Pratt et al. 2010; Walker et al. 2012), and as expected on basic astrophysical grounds.", "pages": [ 1, 2 ] }, { "title": "2. THE SUPERMODEL VIEW", "content": "In fact, the spherically-averaged entropy profile is to rise from a central level k c into an outer ramp with slope a toward the virial boundary R , following the pattern (Voit 2005; Lapi et al. 2005) Specifically, a central baseline k c ∼ 10 2 keV cm 2 is produced during the early collapse and virialization of the cluster core; then the intergalactic gas is condensed to levels n ∼ 10 -3 cm -3 in step with the general overdensities around 200 over the average background, while it is heated up to temperatures k B T ≈ GM/ 10 R ∼ a few keVs. These conditions imply thermal pressures p ≈ 2 nk B T , of order a few 10 -11 erg cm -3 . The baseline k c may be subsequently lowered by radiative cooling, or enhanced by AGN feedback (see Lapi et al. 2003; Fabian 2012) and deep, energetic mergers (see Markevitch & Vikhlinin 2007; McCarthy et al. 2007). These processes, respectively, produce current conditions of the cool core kind with k c ∼ 10 keV cm 2 , or of the non-cool core kind with k c > 10 2 keV cm 2 like in Coma (Cavaliere et al. 2011). At the outer end, the slope a ≈ 1 and the boundary value k R ∼ several 10 3 keV cm 2 are originated as entropy is continuously produced by strong virial accretion shocks (for observational evidence in Coma, see Brown & Rudnick 2011; Markevitch 2012), and then is conserved and stratified as the infalling gas is compressed into the gravitational potential well (Tozzi & Norman 2001; Cavaliere et al. 2011). The SM formalism simply consists in inserting the en- tropy run of Eq. (1) in the differential equation for hydrostatic equilibrium of the ICP (see Cavaliere et al. 2009 for details); this is easily integrated to obtain linked runs of the thermal pressure of the density n ( r ) ∝ [ p ( r ) /k ( r )] 3 / 5 , and of the temperature T ( r ) ∝ p 2 / 5 ( r ) k 3 / 5 ( r ). Here p R is the value at the virial boundary, while M ( < r ) is the gravitational mass distribution mainly contributed by the dark matter (see Lapi & Cavaliere 2009). In Fig. 1 we show how the SM fits the projected profiles of X-ray brightness S X ∝ n 2 T 1 / 2 found by Churazov et al. (2012), and of the emission-weighted temperature T measured by Snowden et al. (2008) and Wik et al. (2009) in Coma; the reader is referred to Fusco-Femiano et al. (2009) for details. From these fits we extract values (with their 1 -σ uncertainty, and consistent with the last reference) of the parameters k c ≈ 535 ± 180 keV cm 2 , a ≈ 1 . 3 ± 0 . 2, and k R ≈ 5050 ± 225 keV cm 2 specifying the entropy pattern in Eq. (1). Based on such values, Coma turns out to be an extreme HE (high entropy) cluster both in the core and in the outskirts , according to the classification used in Cavaliere et al. (2011). As discussed there, such conditions call for impacts of several energetic mergers down to the center that deposit energies of order 10 64 ergs, and for strong accretion shocks standing at the virial boundary with Mach numbers M∼ 10 that produce outer temperatures k B T R ∼ 5 keV. Both processes are in tune with the rich, supercluster environment that surrounds Coma. From Eq. (2) we find the radial pressure profile p ( r ) with no recourse to delicate deprojections. Thence we obtain the thermal electron pressure p e , and compute the profile of the corresponding SZ Comptonization parameter y (cf. § 1). In Fig. 2 we express our re- ult in terms of the equivalent Rayleigh-Jeans decrement ∆ T ≡ -2 y T CMB of the CMB temperature T CMB ≈ 2 . 73 K, to compare with the Planck measurements as presented by Ade et al. (2012), see their Fig. 4.", "pages": [ 2, 3 ] }, { "title": "3. COMPARING THE SZ AND X-RAY VIEWS OF COMA", "content": "Fig. 2 highlights a deficit in the values of | ∆ T | as measured by Planck , relative to those expected from the Xray pressure. The discrepancy appears to be remarkably sharp at the center, well beyond the uncertainties budget presented by Ade et al. (2012). Such a SZ vs. X-ray mismatch goes also beyond the uncertainties affecting the entropy parameters from the X-ray fits (cf. shaded areas in Figs. 1 and 2), with effects damped out by the weak dependence on k ( r ) of the integral term in Eq. (2). The mismatch may be marginally alleviated if one relied on the X-ray data from ROSAT instead of XMM-Newton that with its higher-resolution instruments may enhance the clumpiness effects, so biasing high the brightness (see Churazov et al. 2012). On the other hand, we recall from § 1 that an analogous SZ vs. X-ray mismatch is obtained with quite different fitting tools by Ade et al. (2012). Thus the tension turns out to be model-independent , and calls for a physical explanation in terms of a pressure contribution adding to the thermal value p . In this context, the SM formalism is endowed with an extra gear (Cavaliere et al. 2011), i.e., its ability to straightforwardly include in the equilibrium a nonthermal component δp to yield the total pressure p + δp . The result can be simply described in terms of Eq. (2), with p and k rescaled to In the above paper we have discussed in one particular instance (focused on turbulence in cluster outskirts) how the pressure δp ( r ) can be physically characterized in terms of a normalization provided by the infall kinetic energy seeping through the virial shocks to drive turbulence, and of a dissipative decay scale. For Coma a decay scale is not needed, and a nearly uniform δp/p applies to a good approximation. Thus the net outcome is to lower the normalization applying to the thermal pressure at the virial radius, to read p R ∝ (1 + δp/p ) -1 . Resolving the tension between the SZ vs. the X-ray data requires δp/p ≈ 15% (up to 20% for XMMNewton , which may however include a 5% bias due to clumpiness, see above). The outcome is illustrated in Fig. 2 by the solid line; we remark that while the SZ profile from the SM has not been derived from a formal fit, yet it turns out to represent well the Planck data over their whole radial range. In particular, the thermal pressure derived with the SM is now lower by ≈ 15%, as in fact sensed by the SZ effect. Note that a uniform δp/p implies the density n ∝ ( p/k ) 3 / 5 to be closely unaltered, while the temperature normalization is affected as T R ∝ p R /n R ∝ (1+ δp/p ) -1 ; this amounts to a minor recalibration in the strength of the virial shocks (see § 2; also Cavaliere et al. 2009; 2011). The resulting temperature profile stays close to that in Fig. 1 (bottom panel) with the thermal component of the central entropy recalibrated to k c ≈ 470 from the previous value around 540 keV cm 2 . To sum up, the thermal electron pressure is related to the equilibrium pressure ˆ p by With δp/p ≈ 15 -20%, this boils down to p e ≈ 0 . 45 -0 . 42 ˆ p , definitely lower than the bound 0 . 5 p pointed out in § 1. Note that sensible variations in the ICP metallicity Z ≈ 0 . 4 ± 0 . 03 measured in Coma by Sato et al. (2011) would bias only by a few percents the electron pressure inferred from the X-ray bremsstrahlung radiation, as discussed by Churazov et al. (2012).", "pages": [ 3, 4 ] }, { "title": "4. DISCUSSION AND CONCLUSIONS", "content": "Next we discuss the physical nature of such a nonthermal pressure contribution δp to the overall equilibrium. Iapichino et al. 2011); it is widely held to accelerate with moderate efficiency supra-thermal electrons in the plasma to mildly relativistic energies giving rise to steep distributions (see Schlickeiser et al. 1987; Sarazin & Kempner 2000; Blasi et al. 2007; Brunetti & Lazarian 2011). However, in Coma the density fluctuations caused by ongoing subsonic turbulence have been constrained by Churazov et al. (2012; see their § 5.2 and 5.3) to be less than 5% on scales 30 -300 kpc. The corresponding indirect estimates of current turbulent velocities /lessorsimilar 450 km s -1 would fall short of providing the additional pressure required to relieve the SZ vs. Xray tension. The actual turbulence velocities will be directly probed with the upcoming ASTRO-H mission ( http://www.astro-h.isas.jaxa.jp/ ). lue is substantially smaller than the required δp ≈ 0 . 15 p ≈ several 10 -12 erg cm -3 , relativistic electrons can provide interesting candidates if their energy distribution extends steeply toward a lower end γ 1 /lessorsimilar 10 2 . Such an extension is consistent with the radio spectrum retaining a slope α ≈ 1 . 2 or somewhat steeper, as observed down to frequencies ν ≈ 31 MHz (see Henning 1989); the corresponding electron distribution is to rise toward low energies as γ -s with slope s ≡ 2 α +1 ≈ 3 . 4. Existing data (cf. Henning 1989) also show that at lower frequencies the radio flux in Coma is still sustained, and may even feature a steeper component, as found in other clusters (see van Weeren et al. 2012); LOFAR will soon clear the issue (see http:/www.lofar.org/ ). The amount of non-thermal pressure implied by the above electron population may be estimated as δp ≈ γ 1 m e c 2 n rel ( γ 1 ) / 3 ∝ γ 2 -s 1 , and refined with the full expressions including mildly relativistic electrons as given by Enßlin & Kaiser (2000, their Appendix A). Using the value 2 × 10 40 erg s -1 of the radiohalo luminosity at 100 MHz and the profile given by Brunetti et al. (2012), we compute that a non-thermal contribution δp/p ≈ 15% would indeed obtain on extending a straight electron distribution down to γ 1 ∼ a few. On the other hand, a slope sustained against the fast Coulomb losses (e.g., Sarazin 1999; Petrosian & East 2008) requires such electrons cannot be drawn from the thermal pool, but rather to have been injected over a few 10 7 yr by the action of mergers, or by AGNs (like the current sources associated with NGC 4869 and NGC 4874), or by cosmic-ray interactions (see Brunetti et al. 2012). These electrons are widely held to be accelerated via turbulence and lowM shocks, recently driven by mergers deep and energetic but already on the way of dissipating, so as to meet the constraints set by Churazov et al. (2012) and recalled above. We have stressed at the end of § 2 that similar merging events over timescales of Gyrs are independently required for providing the top level k c ≈ 500 keV cm 2 of the central entropy in Coma. Energy distributions steep down to γ 1 ∼ a few imply a density n ∼ 10 -5 cm -3 in trans-relativistic electrons, and so provide an upper bound for gauging the actual lowγ electron population via the tail of the SZ effect at very high frequencies /greaterorsimilar 1 THz, and the displacement of the thermal null at 217 GHz; these spectral distorsions are illustrated in Fig. 3 (see also Rephaeli 1995). Such features in the SZ spectrum are within the reach of sensitive instrumentations like ALMA (see http://www.almaobservatory.org/ ). In the range γ < 10 2 the electron distribution will be progressively flattened down by Coulomb interactions over timescales < 10 -1 Gyr. But a 'silent pool' of cooling electrons with γ ∼ 10 2 can be replenished and piled up since their lifetimes top at about 1 Gyr. With a cumulative density n ∼ 10 -7 cm -3 resulting from many mergers, these electrons can yield a non-thermal contribution δp/p ≈ 15%. Their synchrotron and relativistic bremsstrahlung radiations would easily escape detection (Sarazin 1999; Sarazin & Kempner 2000), while their collective contribution to pressure is probed just by the thermal SZ effect. Note that to sustain such a silent pool would require a rather high energy dissipation rate by mergers. In summary, the intriguing physical conditions featured by the inner ICP in the Coma cluster include both a thermal and a non-thermal component, to be probed via three observational channels across the electromagnetic spectrum: the bremsstrahlung emission in X rays, the thermal and relativistic SZ effects in microwaves, and the diffuse synchrotron radiation in the radio band. On comparing the first two views, we have found a modelindependent tension between the SZ Planck data and the X-ray observations. In fact, a similar mismatch has been also found by Ade et al. (2012) with their empirical fitting formulae, concerning not only spherically-averaged profile, but also 3 out of 4 angular sectors probed in detail. On large scales, the tension is hard to explain out in terms of overall asphericities given their limited impact in Coma, as discussed by De Filippis et al. (2005). On small scales, the narrow shock jumps reported in selected sectors by Planck (Ade et al. 2012) are diluted by lineof-sight projection and azimuthal averaging. On intermediate scales of order several 10 2 kpc, the presence in the ICP of substructures (see Ade et al. 2012) and fluctuations (see Khedekar et al. 2012) may contribute to locally bias the X-ray pressure; however, given the constraints on the density fluctuations in Coma (Churazov et al. 2012), we expect these effects to be limited to about 5%. We have instead proposed that the SZ vs. X-ray tension can be resolved in terms of a physical condition, i.e., non-thermal support δp/p ∼ 15 -20% yielding a lower thermal electron pressure p e ≈ 0 . 52 ˆ p/ (1 + δp/p ) ≈ 0 . 45 -0 . 42 ˆ p , see Eq. (4). The spherical SM proceeds to provide a pressure profile that straightforwardly incorporates the non-thermal contribution to pressure, and fits in detail the SZ shape while being consistent with the resolved X-ray observables (see Figs. 1 and 2). We stress that, at variance with the fitting formulae used by Ade et al. (2012), the SM pressure profile also features a slow decline for r /greaterorsimilar 0 . 3 R in agreement with the Planck SZ data. Related features are also pleasingly apparent in Fig. 1 of Lapi et al. (2012), where the SM is found to perform considerably better than the fitting formulae by Arnaud et al. (2010) in reproducing the stacked SZ profile from several clusters observed with the South Pole Telescope . These successes support our view that the discrepancy between X rays and SZ effect in Coma is not dominated by specific asymmetries. Given the current constraints on cosmic-rays and on turbulence (pending direct measurements of the turbulent velocities), we have discussed the additional nonthermal pressure in terms of a mildly relativistic electron population with γ in the range from a few to 10 2 . We stress such a component to constitute a natural byproduct of the intense merger activity independently required for yielding the very high central entropy in the thermal intracluster plasma of the Coma cluster. We thank our referee for constructive comments. Work supported in part by ASI, INAF, and MIUR. We thank E. Churazov, J. Gonzalez-Nuevo, P. Mazzotta, and P. Natoli for useful discussions. A.L. thanks SISSA for warm hospitality.", "pages": [ 4, 5 ] }, { "title": "REFERENCES", "content": "Ackermann, M., Ajello, M., Allafort, A., et al. 2010, ApJ, 717, L71 Ade, P.A.R., et al. (The Planck Collaboration) 2012, A&A, submitted [preprint arXiv:1208.3611] Arnaud, M., Pratt, G. W., Piffaretti, R., et al. 2010, A&A, 517, 92 Blasi P., Gabici S., & Brunetti G., 2007, Int. Journ. Modern Phys. A, 22, 681 Cavaliere, A., Lapi, A., & Fusco-Femiano, R. 2011, ApJ, 742, 19 Cavaliere, A., Lapi, A., & Fusco-Femiano, R. 2009, ApJ, 698, 580 Cavaliere, A., & Fusco-Femiano, R. 1976, A&A, 49, 137 De Filippis, E., et al. 2005, ApJ, 625, 108 Enßlin, T. A., & Kaiser, C. R. 2000, A&A, 360, 417 Fabian, A.C. 2012, ARA&A, 50, 455 Fusco-Femiano, R., Cavaliere, A., & Lapi, A. 2009, ApJ, 705, 1019 Giovannini, G., Feretti, L., Venturi, T., Kim, K.-T., & Kronberg, P.P. 1993, ApJ, 406, 399 Iapichino, L., et al. 2011, MNRAS, 414, 2297 Khedekar, S., Churazov, E., Kravtsov, A., et al. 2012, MNRAS, submitted [arXiv::1211.3358] Komatsu, E., Smith, K. M., Dunkley, J., et al. 2011, ApJS, 192, 18 Lapi, A., Cavaliere, A., & Fusco-Femiano, R. 2012, ApJ, 745, L15 Lapi, A., & Cavaliere, A. 2009, ApJ, 695, L125 Lapi, A., Cavaliere, A., & Menci, N. 2005, ApJ, 619, 60 Lapi, A., Cavaliere, A., & De Zotti, G. 2003, ApJ, 597, L93 Markevitch, M. 2012, Proc. of the 12 th Marcel Grossmann Meeting on General Relativity, ed. T. Damour, R. T. Jantzen and R. Ruffini (Singapore: World Scientific), p. 397 Markevitch, M., & Vikhlinin, A. 2007, Phys. Rep., 443, 1 McCarthy, I.G., et al. 2007, MNRAS, 376, 497 Nagai, D., Kravtsov, A. V., & Vikhlinin, A. 2007, ApJ, 668, 1 Petrosian, V., & East, W.E. 2008, ApJ, 682, 175", "pages": [ 6 ] } ]

[ { "title": "Waiting Times of Quasi-homologous Coronal Mass Ejections from Super Active Regions", "content": "Yuming Wang ∗ , Lijuan Liu, Chenglong Shen, Rui Liu, Pinzhong Ye, and S. Wang CAS Key Laboratory of Geospace Environment, Department of Geophysics and Planetary Sciences, University of Science and Technology of China, Hefei, Anhui 230026, China", "pages": [ 1 ] }, { "title": "2 Data Preparation", "content": ". . . . . . . . . . . . . . . . .", "pages": [ 1 ] }, { "title": "4 Summary and discussion", "content": "In summary, by investigating 281 quasi-homologous CMEs originating from 28 CME-rich super ARs over the last solar cycle, we find a two-component distribution of their waiting times with the separation of the two components at about 18 hours and the peak waiting time of the first component at about 7 hours. These results suggest a close physical connection between quasi-homologous CMEs which fall in the first component. Furthermore, the likelihood of occurrences of two or more fast CMEs within 18 hours decreases as CME speed increases. A limit likelihood of about 20% is reached when CME speed is larger than 1200 km s -1 . The correlation analysis among CME waiting times, CME speeds and previous CME occurrence rates shows us the statistical evidence that the free energy input is not a direct cause of quasi-homologous CMEs. It is well known that that the free energy stored in ARs may be much higher than that could be consumed by one single CME [e.g., Sun et al. , 2012]. Thus the direct cause of quasi-homologous CMEs is not the quick re-fill of free energy after preceding CMEs, but the perturbation by preceding CMEs, which may lower the threshold of eruption or trigger instabilities to cause the next CME. Pre-eruption flux rope is precisely balanced by outward force from inner core field and inward force from overlying arcades [e.g., Torok and Kliem , 2005; Wang and Zhang , 2007; Liu , 2007]. A CME may reduce the constraint of its nearby flux rope system by removing overlying arcades, and cause the balance broken. As shown in the numerical simulation by Torok et al. [2011], which was designed to study the physical mechanism of a global sympathetic eruptions on 2010 August 1 [ Schrijver and Title , 2011], the second and third eruptions were actually caused by preceding eruptions. In their eruption processes, the preceding eruption caused the overlying arcades reduced through reconnection, and then instability developed. A similar result was obtained in the simulation by Bemporad et al. [2012], in which the second CME was caused by the rearrangement of coronal magnetic field after the first CME. Connecting the above picture to the peak waiting time of 7 hours, we may speculate that the 7-hour waiting time probably characterizes the average time scale of the growth of instabilities. In our previous work [ Chen et al. , 2011b], we proposed three scenarios to interpret the peak waiting time. Here we may tentatively narrow down them to the last two, in which quasi-homologous CMEs probably hatched from a long magnetic flux system or different magnetic flux systems in one AR. A simple/small AR should be difficult to frequently produce CMEs. A detailed investigation on this point is worthy to be carried out in future work. Acknowledgments. We acknowledge the use of the data from SOHO LASCO, EIT and MDI and the CDAW CME catalog. SOHO is a project of international cooperation between ESA and NASA. This work is supported by grants from CAS (Key Research Program KZZD-EW-01 and 100-Talent Program), NSFC (41131065, 40904046, 40874075, 41121003, 41274173 and 41222031), 973 key project (2011CB811403), MOEC (20113402110001) and the fundamental research funds for the central universities. RL is also supported by NSF (AGS1153226).", "pages": [ 4, 5 ] }, { "title": "1 Introduction", "content": "1 2 2 2 3 3 3 Magnetic free energy is thought to be the energy source of coronal mass ejections (CMEs). Active regions (ARs) carry a huge amount of free energy and therefore are the most probable place where CMEs come out. Lots of efforts have been devoted to the triggering mechanisms of CMEs. Flux emergence, shear motion and mass loss all could be the initial cause of an isolated CME [e.g., Forbes and Priest , 1995; Amari et al. , 1966; Chen and Shibata , 2000; Manchester , 2003]. No matter which one takes effect, the determinative factor of the CME's launch is the force balance between the inner core field and the outer overlying arcades [e.g., Wang and Zhang , 2007; Liu , 2007; Schrijver , 2009]. Free energy stored in the source region will be consumed when a CME launches [e.g., Sun et al. , 2012]. The picture of isolated CMEs is somewhat clear. However, it is still a question how CMEs could lift successively in a limited region within a relatively short interval. Usually the energy accumulation is a gradual process in time scale of hours to days [e.g., LaBonte et al. , 2007; Li et al. , 2010], while a CME is a sudden process releasing accumulated energy in minutes. Why and how could some ARs frequently produce CMEs? Does the occurrences of successive CMEs from the same AR mean that the source AR accumulate free energy quickly? The waiting time distribution of quasihomologous CMEs contains clues. Homologous CMEs were defined by Zhang and Wang [2002] after the definition of homologous flares [ Woodgate et al. , 1984]. Strictly speaking, homologous CMEs must originate from the same region, have similar morphology, and be associated with homologous flares and EUV dimmings. Here, we use the term 'quasi-homologous' to refer to successive CMEs originating from the same ARs within a short interval, but may have different morphology and associates. A previous study on 15 CME-rich ARs during the ascending phase of the last solar cycle from 1998 to 1999 have suggested that quasi-homologous CMEs occurred at a pace of about 8 hours, and there was at most one fast CME within 15 hours [ Chen et al. , 2011b]. These results are important for space weather prediction, and did imply that the accumulation rate of free energy in an AR may not support such frequently occurrences of quasi-homologous CMEs, and the triggering mechanisms of the first and the following CMEs are probably different. Three scenarios were proposed to interpret the averagely 8-hour waiting time of quasi-homologous CMEs. Before deepening our understanding of such a phenomenon, we need to check if a similar waiting time distribution of quasi-homologous CMEs could be obtained for the whole solar cycle. In this paper, we extend the period of interest to the whole solar cycle 23 from 1996 to 2006. Instead of searching all ARs and the associated CMEs, which are too many to be identified manually, we investigate super ARs that were reported in literatures. Super ARs are those with larger area, stronger magnetic field and more complex pattern, and thought to be the representative of CME producers. In the following section, we present the selected data and the method. In Sec.3, an analysis of waiting times of quasi-homologous CMEs from these super ARs during the last solar cycle is performed. Finally, conclusions and discussion is given in the last section.", "pages": [ 1, 2 ] }, { "title": "2.1 Super ARs and Associated CMEs", "content": "Super ARs were studied by many researchers [ Bai , 1987, 1988; Tian et al. , 2002; Romano and Zuccarello , 2007; Chen et al. , 2011a]. It was first defined by Bai [1987, 1988] as a region producing four and more major flares. In most studies, super ARs were selected based on several parameters, such as the largest area of sunspot group, the soft X-ray flare index, the 10.7 cm radio peak flux, the short-term total solar irradiance decrease, the peak energetic proton flux, the geomagnetic Ap index, etc. No matter which one or more criteria are used, most selected super ARs are CMEproductive (that could be seen at the last paragraph of this sub-section). In our study, we focus on super ARs during solar cycle 23. Instead of identifying super ARs by ourselves, we simple use existent lists of super ARs in literatures. To our knowledge, there are three lists regarding to super ARs in solar cycle 23. The first one is given by Tian et al. [2002], who found 16 super ARs from 1997 to 2001 base on their selection criteria. The second one is given by Romano and Zuccarello [2007], which contains 26 super ARs from 2000 to 2006. The last one is in paper by Chen et al. [2011a], in which 12 super ARs were identified during the last solar cycle. Since Chen et al. [2011a] used stricter criteria, the last list is actually a subset of the other two. Totally, we have 37 super ARs from 1996 to 2006. To identify the CMEs originating from these super ARs, we examine imaging data from Large Angle and Spectrometric Coronagraph (LASCO, Brueckner et al. 1995) and Extreme Ultraviolet Imaging Telescope (EIT, Delaboudini'ere et al. 1995) onbard Solar and Heliospheric Observatory (SOHO). The identification process is the same as that applied by Wang et al. [2011] and Chen et al. [2011b]. The CMEs listed in the CDAW LASCO CME catalog (refer to http://cdaw.gsfc.nasa.gov/CME list/, Yashiro et al. 2004) are our candidates. Through a careful identification, it is found that a total of 285 CMEs are associated with these super ARs. Figure 1 shows the distribution of the CME productivity of super ARs, in which the numebr of super ARs almost linearly decreases with increasing CME number though there is a sharp decrease below the CME productivity of 3. It should be mentioned that there are 7 super ARs with too many large data gaps in LASCO and/or EIT observations, and therefore their CME productivity cannot be obtained. Except them, there were 28 super ARs producing 3 or more CMEs (called CME-rich ARs), among which 14 super ARs generated at least 10 CMEs. The other 2 super ARs produced only one or two CMEs though sporadic data gaps existed. This fact suggests that not all of super ARs are CME productive. But it is definite that super ARs are more likely to be CME productive. Chen et al. [2011b] identified 108 ARs during 1997-1998 and found that only 14% of these ARs produced 3 or more CMEs. This percentage is much lower than that for super ARs, which is about 93% (28/30). In this study we focus on the 28 CME-rich ARs, which produced 281 CMEs in total. A list of all the CMEs associated with these CME-rich super ARs can be retrieved from http://space.ustc.edu.cn/dreams/quasi-homologous cmes/.", "pages": [ 2 ] }, { "title": "2.2 Waiting Times", "content": "As long as there is no large data gap, we tentatively believe that all CMEs originating from a super AR of interest are recognized based on combined observations from SOHO LASCO and EIT. The waiting time of each CME is obtained according to the times of first appearance of the CME and its preceding one from the same super AR in the field of view of LASCO/C2. However, data gaps exist, and some CMEs may missed. If there was a large data gap between two CMEs from the same super ARs, the waiting time of the second CME cannot be obtained. Here, all data gaps less than 3 hours are ignored, because it is almost impossible for a CME to stealthily escape the field of view of LASCO in 3 hours. Before analyze the waiting times of these CMEs from the super ARs, it has to be noted that there are probably about 32% of frontside CMEs missed by SOHO [ Ma et al. , 2010; Wang et al. , 2011]. Of course, these missed CMEs might be generally weak and faint. The statistical study by Chen et al. [2011b] have suggested that the properties of ARs have effects on the CME productivity, but do little with the kinetic properties of CMEs. Thus, it is possible that some CMEs originating from the super ARs are missed in our study, though such CMEs might be very weak and erupt in a gradual manner. So far. it is hard to evaluate how significant an influence this error will cause, and one may bear it in mind that the following analysis is performed with a bias of normal to strong CMEs.", "pages": [ 2, 3 ] }, { "title": "3.1 Waiting Time Distribution", "content": "The average value of the waiting times is about 17.8 hours. The waiting time distribution is shown in Figure 2. Similar to that shown in Figure 10 of the paper by Chen et al. [2011b], the distribution consists of two components. One component locates less than 18 hours and looks like a Gaussian distribution, and the other beyond 18 hours. For the first component distribution, the peak waiting time is about 7 hours. In Chen et al. [2011b], the separation of the two components of the distribution is near 15 hours, and the first component distribution peaked near 8 hours, which are both slightly different than those obtained here. These slight differences might be caused by the solar cycle variation. An interesting result in Chen et al. [2011b] is that any AR cannot produce two or more CMEs faster than 800 km s -1 within 15 hours. In other words, the time intervals between fast CMEs are longer than 15 hours. If this result obtained during the last solar minimum also holds for the whole solar cycle, we could expect that any AR cannot produce two or more CMEs faster than a certain speed threshold within 18 hours. However, such a speed threshold cannot be found. The blue line in Figure 2 shows the waiting time distribution for CMEs faster than 1200 km s -1 . Note that all the slower CMEs are ignored when we calculate waiting times for CMEs faster a certain speed threshold. Some fast CMEs did occur in the same ARs within 18 hours. For example, there were four CMEs from the super AR 10720 on 2005 January 15 at 06:30 UT, 23:06 UT, on January 17 at 09:30 UT and 09:54 UT, respectively, which were all faster than 2000 km s -1 . The first two CMEs were separated by about 16.6 hours, and the other two by about only 24 minutes. These fast CMEs caused ground-level enhancement (GLE) event [e.g., Grechnev et al. , 2008]. Although a similar result cannot be obtained, we find that the likelihood for an AR producing two or more fast CMEs within 18 hours is much smaller than normal. For all CMEs, 68% of the waiting times are shorter than 18 hours, while for CMEs faster than 1200 km s -1 , the fraction decreases to only about 18%. The dependence of the likelihood on the CME speed threshold is given in Figure 3. Generally, the likelihood monotonically decreases as the speed threshold increases. When the threshold reaches to about 1200 km s -1 , the likelihood stops decreasing and stays between 15%25%, suggesting a limit likelihood of approximate 1/5. The waiting time distribution for all CMEs from 1999 February to 2001 December was investigated by Moon et al. [2003], which is significantly different from the distribution for quasi-homologous CMEs obtained here (see Figure 1 in their paper). This difference reveals that the occurrence of CMEs follows a Poisson process [ Scargle , 1998; Wheatland , 2000], but that of quasi-homologous CMEs does not. In a statistical view, we may conclude that there are tight physical connections between quasi-homologous CMEs, but for CMEs from different source regions, the connection is quite loose.", "pages": [ 3 ] }, { "title": "3.2 Role of Free Energy Input in Causing Quasi-Homologous CMEs", "content": "Sufficient free energy is a necessary condition for an AR to produce CMEs. Generally, the accumulation rate of free energy could be approximately represented by the magnetic helicity injection rate, which is another important parameter in evaluating the productivity of ARs. Magnetic helicity measures the twists, kinks and inter-linkages of magnetic field lines, which indicate the complexity and non-potentiality of a magnetic system. The close relationship between the free energy and magnetic helicity could be seen from their formulae [ Kusano et al. , 2002]. Thus it is not surprising that a higher injection rate of magnetic helicity often implies a higher probability of an eruptive activity, as suggested by many studies [e.g., Zhang et al. , 2006; LaBonte et al. , 2007]. However, it is still questionable if free energy input is a direct cause of quasi-homologous CMEs. Some studies did show that CMEs do not always occur during a quick injection of magnetic helicity or free energy, even if the stored free energy in an AR was much higher than that required for a CME [e.g., D'emoulin and Pariat , 2009; Vemareddy et al. , 2012]. This issue is addressed here in a statistical perspective from two aspects. First, we investigate the correlation between the CME speeds and waiting times. If free energy input is a direct cause, it is expected that there is some regulation between CMEs' speeds and their waiting times, as a long waiting time may lead to more free energy in an AR. This expectation is established under the assumption that the injection rate of free energy or magnetic helicity varies in a relatively small range for different ARs, This assumption is statistically true based on previous studies. For example, the statistical study by Park et al. [2010] suggested that the magnetic helicity fluxes in 378 ARs observed by SOHO/MDI were on the order of about 10 40 Mx h -1 , especially for those ARs with large magnetic flux (see, e.g., Fig.1, 3 and 4 in their paper). The value does not change much even if deriving from higher-resolution data from SDO/HMI, e.g., the helicity injection rate in AR 11158 and 11166 [ Vemareddy et al. , 2012]. Second, we check if the waiting time of a CME depends on the CME occurrence rate in the past 18 hours before its preceding CME. Figure 5a shows the scattering plot between them. Apparently, a low or high CME occurrence rate may lead to a short waiting time of the next CME, and a long waiting time tends to appear when the CME occurrence rate is around 0.1 per hour. However, similar to the previous one, this distribution is also just a manifestation of probability, and contains less physical meaning. If we randomly associate the CME waiting times with the occurrence rates, a possible distribution of the data points is like that shown in Figure 5b, which is statistically same as that in Figure 5a. Thus the CME waiting time is independent on the previous CME occurrence rate. Both results suggest that free energy input is not a direct cause of quasi-homologous CMEs though sufficient free energy is a necessary condition for an AR to produce CMEs.", "pages": [ 3, 4 ] }, { "title": "References", "content": "Amari, T., J. F. Luciani, J. J. Aly, and M. Tagger, Plasmoid formation in a single sheared arcade and application to coronal mass ejections, Astron. & Astrophys. , 306 , 913, 1966. Bai, T., Distribution of flares on the sun - superactive regions and active zones of 1980-1985, Astrophys. J. , 314 , 795-807, 1987. Bai, T., Distribution of flares on the sun during 1955-1985 - 'hot spots' (active zones) lasting for 30 years, Astrophys. J. , 328 , 860-874, 1988. Bemporad, A., F. Zuccarello, C. Jacobs, M. Mierla, and S. Poedts, Study of multiple coronal mass ejections at solar minimum conditions, Sol. Phys. , 281 , 223-236, 2012. Brueckner, G. E., R. A. Howard, M. J. Koomen, C. M. Korendyke, D. J. Michels, J. D. Moses, D. G. Socker, K. P. Dere, P. L. Lamy, A. Llebaria, M. V. Bout, R. Schwenn, G. M. Simnett, D. K. Bedford, and C. J. Eyles, The large angle spectroscopic coronagraph (LASCO), Sol. Phys. , 162 , 357-402, 1995. Chen, A. Q., J. X. Wang, J. W. Li, J. Feynman, and J. Zhang, Statistical properties of superactive regions during solar cycles 19-23, Astron. & Astrophys. , 534 , A47, 2011a. Chen, C., Y. Wang, C. Shen, P. Ye, J. Zhang, and S. Wang, Statistical study of coronal mass ejection source locations: 2. role of active regions in cme production, J. Geophys. Res. , 116 , A12,108, 2011b. Chen, P. F., and K. Shibata, An emerging flux trigger mechanism for coronal mass ejections, Astrophys. J. , 545 , 524-531, 2000. Delaboudini'ere, J.-P., G. E. Artzner, J. Brunaud, and et al., EIT: Extreme-ultraviolet imaging telescope for the SOHO mission, Sol. Phys. , 162 , 291-312, 1995. D'emoulin, P., and E. Pariat, Modelling and observations of photospheric magnetic helicity, Adv. Space Res. , 43 , 1013-1031, 2009. Forbes, T. G., and E. R. Priest, Photospheric magnetic field evolution and eruptive flares, Astrophys. J. , 446 , 377, 1995. Grechnev, V. V., V. G. Kurt, I. M. Chertok, A. M. Uralov, H. Nakajima, A. T. Altyntsev, A. V. Belov, B. Y. Yushkov, S. N. Kuznetsov, L. K. Kashapova, N. S. Meshalkina, and N. P. Prestage, An extreme solar event of 20 January 2005: Properties of the flare and the origin of energetic particles, Sol. Phys. , 252 , 149-177, 2008. Kusano, K., T. Maeshiro, T. Yokoyama, and T. Sakurai, Measurement of magnetic helicity injection and free energy loading into the solar corona, Astrophys. J. , 577 , 501-512, 2002. Woodgate, B. E., M.-J. Martres, J. Smith, J. B., K. T. Strong, M. K. McCabe, M. E. Machado, V. Gaizauskas, R. T. Stewart, and P. A. Sturrock, Progress in the study of homologous flares on the sun. II, Adv. Space Res. , 4 , 11-17, 1984.", "pages": [ 5, 6 ] } ]

[ { "title": "ABSTRACT", "content": "Over the last decade, substantial efforts have been devoted to understanding the stability properties, transport phenomena, and long-term evolution of weakly collisional, magnetized plasmas which are stratified in temperature. The insights gained via these studies have led to a significant improvement of our understanding of the processes that determine the physical evolution and observational properties of the intracluster medium (ICM) permeating galaxy clusters. These studies have been carried out under the assumption that the ICM is a homogeneous medium. This, however, might not be a good approximation if heavy elements are able to sediment in the inner region of the galaxy cluster. Motivated by the need to obtain a more complete picture of the dynamical properties of the ICM, we analyze the stability of a weakly collisional, magnetized plane-parallel atmosphere which is stratified in both temperature and composition. This allows us to discuss for the first time the dynamics of weakly collisional environments where heat conduction, momentum transport, and ion-diffusion are anisotropic with respect to the direction of the magnetic field. We show that, depending on the relative signs and magnitudes of the gradients in the temperature and the mean molecular weight, the plasma can be subject to a wide variety of unstable modes which include modifications to the magnetothermal instability (MTI), the heat-flux-driven buoyancy instability (HBI), and overstable gravity modes previously studied in homogeneous media. We also find that there are new modes which are driven by heat conduction and particle diffusion. We discuss the astrophysical implications of our findings for a representative galaxy cluster where helium has sedimented. Our findings suggest that the core insulation that results from the magnetic field configurations that arise as a natural consequence of the HBI, which would be MTI stable in a homogeneous medium, could be alleviated if the mean molecular weight gradient is steep enough, i.e., ( ∇ µ ) /µ > ( ∇ T ) /T . This study constitutes a first step toward understanding the interaction between magnetic turbulence and the diffusion of heavy elements, and its consequences for the long-term evolution and observational signatures of the ICM in galaxy clusters. Subject headings: galaxies: clusters: intracluster medium - instabilities - magnetohydrodynamics", "pages": [ 1 ] }, { "title": "THE STABILITY OF WEAKLY COLLISIONAL PLASMAS WITH THERMAL AND COMPOSITION GRADIENTS", "content": "MARTIN E. PESSAH 1 AND SAGAR CHAKRABORTY 1 , 2 1 Niels Bohr International Academy, Niels Bohr Institute, Blegdamsvej 17, DK-2100 Copenhagen Ø, Denmark; [email protected] 2 Department of Physics, Indian Institute of Technology, Kanpur, U.P.-208016, India; [email protected] Draft version March 1, 2019", "pages": [ 1 ] }, { "title": "1. INTRODUCTION", "content": "Despite the fact that magnetic fields in galaxy clusters are too weak to be mechanically important, they can play a fundamental role in the dynamical stability of the dilute gas by channeling the transport of heat, momentum, and particles. The weakly collisional character of the hot intracluster medium (ICM), which is generically characterized by stable entropy gradients according to Schwarzschild's criterion (Piffaretti et al. 2005; Cavagnolo et al. 2009), enables the action of magnetic instabilities that are sensitive to temperature gradients (Balbus 2000, 2004). In particular, the magnetothermal instability (MTI) exhibits the fastest growing modes when magnetic field lines are orthogonal to a temperature gradient parallel to the gravitational field (Balbus 2001), whereas the heat-flux-driven buoyancy instability (HBI) does so when magnetic field lines are parallel to a temperature gradient which is anti-parallel to the gravitational field (Quataert 2008). While the landscape of thermal instabilities that render homogeneous, dilute plasmas unstable has been well explored (Kunz 2011), and even extended to account for the effects of cosmic rays (Chandran & Dennis 2006; Sharma et al. 2010), very little is known about the effects that composition gradients can have on the stability of the dilute ICM. If magnetic fields do not prevent the efficient diffusion of ions (Narayan & Medvedev 2001; Chuzhoy & Nusser 2003; Chuzhoy & Loeb 2004) then the gradients in mean molecular weight can be as important as the gradients in tempera- ture (see Section 7 and Qin & Wu 2000; Peng & Nagai 2009; Shtykovskiy & Gilfanov 2010; Bulbul et al. 2011) and provide another source of free energy to feed instabilities. In order to obtain a more complete picture of the stability properties of the ICM, it is thus important to relax the assumption of a homogeneous medium. As a first step toward understanding the role of composition gradients in the stability of dilute plasmas, such as the ICM, we analyze the stability of a weakly magnetized plane-parallel atmosphere where magnetic fields play a key role by channeling the conduction of heat, transport of momentum, and the diffusion of ions. Our analysis generalizes previous studies on the MTI (Balbus 2001), the HBI (Quataert 2008), and overstable gravity modes (Balbus & Reynolds 2010), and reveals the subtle roles played by the temperature and the composition gradients in determining the stability of the plasma. The outline of the paper is as follows. In Section 2 we describe the plasma model for a dilute binary mixture of ions. In Section 3 we perform the linear mode analysis and we obtain the general dispersion relation that governs the linear dynamics of a weakly magnetized medium which is stratified in temperature and composition. We analyze in detail the stability of the plasma in the regimes where conduction across a given scale is, respectively, fast and slow compared to the dynamical timescale in Sections 4 and 5. In Section 6 we describe the physics driving the most relevant instabilities. We discuss the astrophysical implications of this study in Section 7.", "pages": [ 1 ] }, { "title": "2.1. General Considerations for the Plasma Model", "content": "In order to highlight the physical phenomena that emerge when composition gradients are accounted for, we focus our attention on a dilute binary mixture (e.g., hydrogen and helium) 1 in a fixed gravitational field described by 2 Here, the Lagrangian and Eulerian derivatives are related via d/dt ≡ ∂/∂t + v · ∇ , ρ is the mass density, v is the fluid velocity, g is the gravitational acceleration, γ is the adiabatic index, and I stands for the 3 × 3 identity matrix. The symbols ⊥ and ‖ refer respectively to the directions perpendicular and parallel to the magnetic field B , see Figure 1, whose direction is given by the versor ˆ b ≡ B /B = ( b x , 0 , b z ) . The first term on the right-hand side of Equation (4) accounts for entropy production due to viscous heating in a weakly collisional magnetized plasma (see, e.g., Hollweg 1985). These equations have been considered in previous works investigating the dynamics of the weakly collisional ICM with a single ion species, i.e., in the case where the concentration of the other ion species is c = 0 , see, e.g., Kunz (2011); Parrish et al. (2012); Kunz et al. (2012), and references therein. Equations (1)-(5) describe the dynamics of a dilute binary mixture in the low-collisionality regime and they differ from standard MHD in three important respects. ( i ) In a weakly collisional magnetized plasma the pressure tensor P ≡ p ⊥ I + ( p ‖ -p ⊥ ) ˆ b ˆ b is anisotropic. If the frequency of ion collisions ν ii in single ion species magnetofluid is large compared to the rate of change d/dt of all the fields involved, then the anisotropic part of the pressure tensor is small compared to its isotropic part P ≡ 2 p ⊥ / 3+ p ‖ / 3 and (see, e.g., Hollweg 1985; Schekochihin et al. 2005) ∣ ∣ The anisotropic component of the pressure tensor in the momentumequation gives rise to the phenomenon known as Braginskii viscosity. For small pressure anisotropy, 3 this contribution is usually written as where η 0 is the largest of the coefficients in the viscous stress tensor derived by Braginskii (1965). In order to account for the effects of collisions between ions of different species in the binary mixture, we replace the ν ii by an effective ion-ion collision frequency ν eff ii , which we define in Appendix A. ( ii ) Heat flows mainly along magnetic field lines, because the electron mean free path is large compared to its Larmor radius. This process is modeled by the second term on the right-hand side of Equation (4) via where T is the plasma temperature, assumed to be the same for ions and electrons, and χ is the thermal conductivity predominately due to electrons (Spitzer 1962; Braginskii 1965), ( iii ) The composition of fluid elements can change due to particle fluxes. Considering the flux of particles on the right-hand side of Equation (5) ensures that the diffusion of ions is mainly along magnetic field lines. This is a good approximation when the plasma is dilute enough for the ion mean free path to be large compared to the ion Larmor radius. The concentration c is related to the mean molecular weight µ via where µ i and Z i , with i = 1 , 2 , are the molecular weights and the atomic numbers for the two ion species. The isotropic part of the pressure tensor is thus where k B is the Boltzmann constant and m H is the atomic mass unit.", "pages": [ 2 ] }, { "title": "2.2. Initial Background State", "content": "We consider a weakly magnetized, plane-parallel atmosphere in a constant gravitational field g ≡ -g ˆ z . The background magnetic field is weak enough that the mechanical equilibrium of the atmosphere, with scaleheight H , is maintained via dP/dz = -gρ . We assume that the medium is stratified in density, temperature, and composition along the vertical z -direction. In the equilibrium state, all the particles in the plasma are assumed to be described by a Maxwellian distribution with the same temperature, so that p ‖ ≡ p ⊥ initially. /negationslash /negationslash In general, the background heat and particle fluxes do not vanish, i.e., ˆ b ·∇ T = 0 and ˆ b ·∇ c = 0 , unless the magnetic field and the background gradients are orthogonal. The existence of a well-defined steady state, i.e., ∇· Q s = ∇· Q c = 0 , demands that the background fluxes should be linear functions of the distance along the direction of the magnetic field. However, even if this condition is not strictly satisfied, the dynamics of the modes that we consider is unlikely to be significantly affected if the local dynamical timescale is short compared to the timescale in which the entire system evolves (see also Quataert 2008).", "pages": [ 3 ] }, { "title": "2.3. Validity of the Braginskii-MHD Approximation", "content": "If the pressure anisotropy grows beyond | p ‖ -p ⊥ | /P /similarequal β -1 , where the plasma β ≡ v 2 th /v 2 A , v th ≡ (2 P/ρ ) 1 / 2 is the thermal speed, and v A ≡ B/ (4 πρ ) 1 / 2 is the Alfv'en speed, the Braginskii-MHD approximation embodied in Equations (1)(4) becomes ill-posed. This is because the viscous term introduced in Equation ( 7 ) not only fails to damp all the kinetic energy available at the viscous scale but also triggers various fast-growing, micro-scale plasma instabilities, such as mirror and firehose (see Schekochihin et al. 2005, 2008 and references therein). The growth rates of these instabilities are of the order of γ /similarequal k ‖ v th | p ‖ -p ⊥ | /P and thus they can dominate the plasma dynamics at very small scales if | p ‖ -p ⊥ | /P /greaterorsimilar β -1 . This poses a challenge in numerical simulations addressing the non-linear dynamics of the BraginskiiMHD equations since these instabilities grow formally at the grid scale and some procedure must be devised in order to capture their effects (see Kunz et al. 2012 for a detailed discussion). One possibility, which would prevent the micro-instabilities from operating at once, is to ignore the effects of pressure anisotropies and the associated Braginskii viscosity. This was the approach followed in the seminal papers on the MTI (Balbus 2001) and the HBI (Quataert 2008), which showed that both instabilities grow on the dynamical timescale set by ω -1 dyn /similarequal ( H/g ) 1 / 2 . However, because the timescales involved in viscous processes are only a factor of a few longer than the dynamical timescales on which both the HBI and the MTI operate, accounting for small pressure anisotropies can affect the range of wavenumber over which these instabilities operate, as well as their growth rates (Kunz 2011). Furthermore, the timescales involved in processes related to ion-diffusion are only a factor of a few larger than those involved in viscous processes (see below). Since our aim is to understand the interplay of the various processes involved in determining the stability of a medium stratified in both temperature and composition, we retain the term accounting for Braginskii viscosity, Equation ( 7 ), in the momentum Equation ( 2 ). We argue next that Equations (1)-(4) provide an adequate framework to analyze the dynamics of small amplitude perturbations of the stratified atmosphere described. The equilibrium background state over which we perform the stability analysis is such that p ‖ ≡ p ⊥ , and thus there is an initial period of time for which the pressure anisotropy will remain small enough that these plasma-micro instabilities can be ignored. Kunz et al. (2012) estimate that the amplitude to which the fluctuations in the magnetic field can grow before these instabilities set in, and thus Equations (1)-(4) remain self-consistent, is roughly given by δB ‖ /B /similarequal H/ ( βλ mfp ) , where λ mfp stands for the mean free path between particle collisions. We can estimate this value for the ICM as follows. The plasma β increases from /similarequal 10 2 in the inner cluster regions to /similarequal 10 4 in the outer parts, while the ratio H/λ mfp decreases from 10 3 -10 2 in the cluster core to 10 2 -10 in the outer region. Therefore, the ratio H/ ( βλ mfp ) is larger than unity in the central regions of a typical galaxy cluster and decreases outward to roughly 10 -2 . We thus conclude that, for the sake of performing a linear mode analysis, which is only formally valid when the fluctuations of all the physical variables are small, e.g., δB/B /lessmuch 1 , the Braginskii-MHD Equations (1)-(4) describes the problem under consideration self-consistently. These micro-instabilities are likely to play an important role in the subsequent non-linear dynamics, but addressing this regime is beyond the scope of this study.", "pages": [ 3 ] }, { "title": "3.1. Linearized Equations", "content": "The modes of interest have associated timescales that are long compared to the sound crossing time and it thus suffices to work in the Boussinesq approximation (Balbus 2001; Quataert 2008). In this limit, the equations for the linear perturbations δ /similarequal e σt + i k · x become 4 Here, we have defined the anisotropic viscosity coefficient where ν eff ii is an effective collision rate for the binary mixture (see Appendix A), and the thermal diffusion coefficient, 4 The effects of Braginskii viscosity in the thermal evolution of the plasma, which appear in Equation (4) as proportional to ( p ‖ -p ⊥ ) 2 , are of higher order and thus they do not contribute to the linear analysis. We have also introduced the Brunt -Vaisala frequency, N 2 , which, in a medium stratified in density, temperature, and composition, is given by Note that, in agreement with the Boussinesq approximation, the velocity perturbations satisfy k · δ v = 0 and the fluctuations in density, temperature, and mean molecular weight are related via", "pages": [ 3, 4 ] }, { "title": "3.2. Relevant Timescales Across a Mode", "content": "Because of the several physical processes that play a role in the stability of the dilute atmosphere, the dispersion relation corresponding to Equations (13)-(16) is rather involved. It is thus useful to understand the hierarchy of the timescales involved in the dynamics of a single Fourier mode in order to make sensible approximations. The analysis below applies to the range of local modes with wavevectors parallel to the magnetic field for which the fluid approach is valid, i.e., H -1 < k ‖ < λ -1 mfp , or where we have defined the Knudsen number The inverse timescales characterizing the diffusion of heat, momentum, and particles along magnetic field lines are For a given mode, the ratio between these timescales is independent of the direction of the wavevector characterizing the perturbation and the background magnetic field and is set by plasma processes. Because heat conduction is mostly due to electrons, while viscous processes are dominated by the dynamics of ions, it could be expected that the associated timescales would satisfy τ -1 c /greatermuch τ -1 d . However, this is not the case and a simple estimate leads to τ -1 c /similarequal 6 τ -1 v (Kunz 2011). It could be argued that the timescales involved in viscous and diffusion processes should be of the same order because it is mainly the ion dynamics what determines both of them. A detailed analysis of the diffusion coefficient for a binary mixture of ions (see Appendix A) shows that τ -1 d /similarequal 9 τ -1 v for primordial composition ( c /similarequal 0 . 25 or µ /similarequal 0 . 6 ) and decreases toward τ -1 d /similarequal 3 τ -1 v for the compositions expected at the inner core of galaxy clusters according to recent models for helium sedimentation (Bulbul et al. 2011). Since we will be mostly concerned with the two regimes τ -1 c /greatermuch ω dyn or ω dyn /greatermuch τ -1 c , as long as the ratio τ -1 d /τ -1 v is not too small, its particular value will not affect our main conclusions, and we will thus consider that τ -1 d /similarequal τ -1 v . On the other hand, whether the timescales set by plasma processes are fast or slow compared to the dynamical timescale ω -1 dyn ≡ ( H/g ) 1 / 2 depends not only on the wavelength of the mode but also on the direction of the wavevector characterizing the perturbation with respect to the background magnetic field. In particular, as shown in (Kunz 2011), the timescale characterizing conduction across a mode with parallel wavenumber k ‖ is related to the dynamical timescale via where we have assumed γ = 5 / 3 in Equation (23). Thus conduction is faster than the dynamical time, i.e., τ -1 c /ω dyn > 1 , if k ‖ ( λ mfp H ) 1 / 2 > 1 / 3 . If the wavelength of the mode is shorter than this by a factor of τ -1 c /τ -1 v /similarequal 6 , e.g., k ‖ ( λ mfp H ) 1 / 2 /greatermuch 1 , then viscous and diffusive processes are also faster than the dynamical timescale. Therefore, as a useful approximate criterion, whether k ‖ ( λ mfp H ) 1 / 2 is much larger or smaller than unity defines whether the timescales associated with plasma processes, for that given mode, are shorter or longer than the dynamical time. We will thus consider two different regimes which we refer to as the 'fast' and 'slow' conduction limit, where the timescales associated with the modes considered satisfy, respectively,", "pages": [ 4 ] }, { "title": "3.3. The Weak Magnetic Field Limit", "content": "All the timescales related to plasma processes discussed above depend only on the direction of a given wavevector with respect to the magnetic field. The only time scale that depends explicitly on the strength of the field is the one associated with the Alfv'en frequency ω A ≡ k · v A . In order to keep the problem tractable, and given that we are already dealing with four different timescales, we will focus on the case where the magnetic field is so weak that its only physical role is to channel the flux of heat and ions. The advantage of this limit is that it allows us to address the anisotropic dynamics of the weakly collisional magnetized medium without introducing explicitly the timescale associated with ω A . In what follows we focus our attention on modes for which magnetic tension is unimportant and thus ω A /lessmuch min { τ -1 c , ω dyn } . This approximation will be valid for two different ranges of parallel wavenumbers depending on whether k ‖ ( λ mfp H ) 1 / 2 is much larger or smaller than unity. For the modes for which conduction is faster than the dynamical timescale, i.e., k ‖ ( λ mfp H ) 1 / 2 /greatermuch 1 , we must require ω A /lessmuch ω dyn /lessmuch τ -1 c . Using the definitions v th = ( gH ) 1 / 2 , β = v 2 th /v 2 A , and K n = λ mfp /H , we obtain that magnetic tension is negligible provided that and thus For the modes for which conduction is slow compared to the dynamical timescale, i.e., k ‖ ( λ mfp H ) 1 / 2 /lessmuch 1 , we must require ω A /lessmuch τ -1 c /lessmuch ω dyn . This is satisfied if and therefore The plasma β ranges from 10 4 in the outskirts of the ICM down to 10 2 in the centers of cool core clusters (Carilli & Taylor 2002), and the product βK n ranges from 10 3 in the outskirts of the ICM decreasing to 10 -1 in the inner regions. Thus the effects of magnetic tension can be important in the inner cluster regions. We address the implications of neglecting magnetic tension when analyzing the stability of the ICM in further detail in Section 7.", "pages": [ 4, 5 ] }, { "title": "3.4. General Dispersion Relation for the Dilute, Weakly Magnetized Medium", "content": "The dispersion relation corresponding to the set of equations for the linear perturbations (13)-(16) is where the coefficients A i are given by while the B i read Here we have defined and the two quantities which appear naturally when thermal and composition gradients are considered. The dispersion relation (33) is identical to the one derived in (Kunz 2011) in the limit in which ω A , dµ/dz , and D vanish. Note that in the limit of a constant composition gradient, i.e., dµ/dz → 0 , both N 2 Tµ and N 2 T/µ → gd ln T/dz , which is the logarithmic gradient that plays an important role in the stability of a homogeneous, dilute, weakly magnetized medium.", "pages": [ 5 ] }, { "title": "4. THE FAST CONDUCTION LIMIT", "content": "We first consider the stability of the modes for which conduction is faster than the dynamical time, i.e., τ -1 c /greatermuch ω dyn . This is the regime that corresponds to the well-studied HBI and MTI.", "pages": [ 5 ] }, { "title": "4.1. Limit of No Ion-Diffusion", "content": "As a first step toward understanding the effects of composition gradients in the behavior of the HBI and the MTI, we neglect the diffusion of ions along magnetic field lines by setting D = 0 . Because we are considering the timescales for ion-diffusion and viscous processes to be of the same order, i.e., τ -1 v /similarequal τ -1 d , we also ignore here the effects of viscosity and set ν ‖ = 0 for consistency. For the modes for which τ -1 c /greatermuch ω dyn , the dispersion relation (33) yields a (fast) decaying solution, σ ≈ -τ -1 c , together with the two slow modes In a homogeneous plasma, these slow modes contain the wellknown HBI and MTI, depending on the direction of the background magnetic field, i.e., The conditions for the excitation of the HBI and the MTI are thus Note that in both cases, the fastest growing modes are those with wavevectors perpendicular to the gravitational field. Consider a magnetic field parallel to the gravitational field, i.e., b z = 1 , and thus k 2 x + k 2 y = k 2 ⊥ . The modified version of the modes that become HBI-unstable in a homogeneous medium is given by Equation (49). In the medium stratified in composition, these modes become Therefore, neglecting viscous and diffusion processes in a medium which is stratified in the mean molecular weight /negationslash leads to modes that are unstable to a Heat- and Particle-fluxdriven Buoyant Instability (HPBI) if The threshold temperature gradient for instability can be negative if the mean molecular weight increases with height. The combination of temperature and composition gradients that is HPBI-unstable is shown in panel (a) of Figure 2 where, for the sake of convenience, we have defined dimensionless variables ( d ln µ/d ln P, d ln T/d ln P ) in terms of the logarithmic pressure gradient d ln P/dz ≡ -1 /H .", "pages": [ 5, 6 ] }, { "title": "4.1.2. Overstable Modes ( D = 0 )", "content": "Balbus & Reynolds (2010) showed that temperature gradients that are stable to the HBI can nevertheless be subject to overstable gravity modes. This result can be extended to include non-vanishing composition gradients, i.e., there is a range of modes with τ -1 c > ω dyn and σ /similarequal ω dyn that can become overstable when the heat- and particle-flux-driven buoyancy instability (HPBI) does not operate. Calculating these modes requires retaining higher order terms in the dispersion relation, which becomes In this regime, we can treat the first and third terms on the left-hand side as perturbations and extend the solutions (54) to contain corrections of order ω 2 dyn /τ -1 c : Therefore, modes that are stable according to the HPBIstability criterion, i.e., N 2 Tµ < 0 (Equation [55]), can become overstable if N 2 Tµ < -N 2 . In terms of the dimensionless variables introduced earlier, these requirements become The combination of temperature and composition gradients that is subject to overstability is shown in dark gray in panel (a) of Figure 2.", "pages": [ 6 ] }, { "title": "4.1.3. Magneto-Thermo-Compositional Instability ( D = 0 )", "content": "In order to understand how the MTI is modified in the presence of composition gradients we consider a horizontal magnetic field along the x -axis, i.e., b x = 1 and focus on the modes for which τ -1 c /greatermuch ω dyn . In the presence of a gradient in the mean molecular weight, Equation (49) gives the generalization of the modes that become MTI-unstable Thus a non-vanishing gradient in the mean molecular weight sets an upper bound for the temperature gradients that are magneto-thermo-compositional instability (MTCI)-unstable Panel (d) in Figure 2 shows a graphical representation of the region of parameter space that is subject to unstable MTCI modes in a medium that is stratified in composition and temperature.", "pages": [ 6 ] }, { "title": "4.2. Ion-Diffusion Along Magnetic Field Lines", "content": "We now analyze the effects of including ion-diffusion induced by the background composition gradients. Since, for a given mode, viscous and diffusion timescales are of the same order, i.e., τ -1 v /similarequal τ -1 d , we also consider the effects of anisotropic viscosity for consistency. /negationslash", "pages": [ 6 ] }, { "title": "4.2.1. Heat-and Particle-Flux-Driven Buoyant Instability ( D = 0 )", "content": "Starting from the general dispersion relation in the case where b z = 1 , it can be seen that, if we consider modes for which conduction is faster than any other timescale, there is a fast decaying solution σ = -τ -1 c which can be used to self- nsistently obtain three more modes satisfying However, the leading order solutions to this dispersion relation depend on whether the dynamical timescale is fast or slow with respect to the diffusion and viscous timescales across the mode. We thus need to consider these two cases separately. Slow diffusion . The modes for which ω dyn > τ -1 v /similarequal τ -1 d contain the generalization of the HBI for non-vanishing composition gradients 5 which grows dynamically if N 2 Tµ > 0 , or dT/dz > -dµ/dz . This is the same condition for the onset of the HPBI in Equation (55) in the absence of diffusion. It is worth mentioning that even if the combination of temperature and composition gradients is such that N 2 Tµ < 0 (and thus the plasma is HPBI-stable), overstable modes can be excited just as in the non-diffusive case. In fact, the condition for overstability can be shown to be exactly similar to (58), provided that ω dyn > max { τ -1 c τ -1 v , τ -1 c τ -1 d } . There also exists a slower mode driven by ion-diffusion which grows in the region of parameter space where N 2 T/µ /N 2 Tµ < 0 or, in terms of the background gradients, wherever d ln T/d ln P < | d ln µ/d ln P | , as it is shown in panel (b) of Figure 2. Note that these modes can grow on a diffusion timescale even if the system is HPBI stable, i.e., N 2 Tµ < 0 , provided that N 2 T/µ > 0 . As we show below, this latter requirement becomes the deciding one for those modes for which diffusion is not slow compared to the dynamical time. Fast diffusion . For the modes for which τ -1 v /similarequal τ -1 d > ω dyn , there are two solutions that decay on the diffusion and viscous timescales, i.e., σ ≈ -τ -1 d and σ ≈ -τ -1 v k 2 ⊥ /k 2 , and a third one which can become unstable if N 2 T/µ > 0 . This region of parameter space in temperature and composition gradients is unstable to a diffusive version of the Heat- and Particle-fluxdriven Buoyant Instability (D-HPBI) and it is shown in panel (c) of Figure 2. In summary, we conclude that when finite viscous and diffusion timescales are considered, there can be unstable modes /negationslash driven by diffusion (with ω dyn > τ -1 d ) even if the temperature and the composition gradients do not satisfy inequality (55), i.e., they are stable to the HPBI in the absence of diffusion, provided that d ln T > d ln µ . Note that this is the very requirement for the existence of unstable modified HBI modes when finite diffusion timescales are relevant (with τ -1 d > ω dyn ) and differs from the condition (55). We shall discuss the physical reason behind this change in condition of instability in Section 6. /negationslash", "pages": [ 6, 7 ] }, { "title": "4.2.2. Magneto-Thermo-Compositional Instability ( D = 0 )", "content": "/negationslash In order to understand how ion-diffusion driven by a composition gradient affects the MTI, we consider D = 0 and modes for which τ -1 c /greatermuch ω dyn when b x = 1 . In this case, the dispersion relation factorizes and leads to a fast decaying solution σ = -τ -1 c together with For the modes satisfying ω dyn > τ -1 v , the three solutions to this cubic equation correspond to a decaying viscous mode /negationslash and a pair of roots that contain the generalization of the MTI in the presence of a mean molecular weight gradient and D = 0 which grows dynamically if N 2 T/µ < 0 , or dT/dz > dµ/dz . This is the same condition for the onset of the MTCI in Equation (60) in the absence of diffusion. As we found before, a non-vanishing gradient in the mean molecular weight sets a upper bound for the temperature gradients that are MTIunstable. This temperature gradient can be positive when the composition gradient is negative (i.e., mean molecular weight increasing with height). Note that this statement is independent of the value of the diffusion coefficient D , i.e., and thus of whether or not ions diffuse effectively along magnetic field lines on a dynamical timescale. For the modes such that τ -1 v > ω dyn , the three solutions to Equation (66) correspond to another viscously damped mode together with a generalization of the modes identified as Alfv'enic-MTI in Kunz (2011), i.e., These modes grow dynamically if N 2 T/µ < 0 , which corresponds again to the combination of temperature and composition gradients satisfying inequality (60).", "pages": [ 7 ] }, { "title": "5. THE SLOW CONDUCTION LIMIT", "content": "We now analyze the stability of the modes whose associated timescales satisfy ω dyn /greatermuch τ -1 c > τ -1 v /similarequal τ -1 d . For a homogeneous plasma, these modes encompass the overstable g -modes studied in Balbus & Reynolds (2010). The set of Equations (13)-(20) allows us to address the behavior of these, as well as other new modes, in the presence of a nonvanishing gradient in the mean molecular weight and account self-consistently for the diffusion of ions along magnetic field lines.", "pages": [ 7, 8 ] }, { "title": "5.1. Heat and Ion Diffusion Along Vertical Magnetic Fields", "content": "In the case where the background magnetic field is parallel to the gravitational field, b z = 1 , the dispersion relation (33) reduces to with coefficients The Routh-Hurwitz stability criteria that predict exclusively negative real parts for the roots of the quartic polynomial with real coefficients require a 1 > 0 , a 1 a 2 -a 3 > 0 , a 1 a 2 a 3 -a 2 1 a 4 -a 2 3 > 0 , and a 4 > 0 . The first of these conditions is trivially satisfied, while the other three imply, respectively, In the absence of diffusion, only the conditions (75) and (76) need to be met in order to ensure stability, while the condition (77) should also be required for finite diffusion timescales. To leading order, two of the solutions of Equation (71) are given by σ ≈ ± ia 1 / 2 2 +( a 3 -a 1 a 2 ) / 2 a 2 , i.e., which correspond to gravity modes. In the absence of a gradient in the mean molecular weight, these reduce to the g -modes discussed in Balbus & Reynolds (2010). The third root is given by σ ≈ -a 3 /a 2 , i.e., and corresponds to a mode driven by conduction. Assuming that N 2 > 0 , g -modes are overstable if the condition (76) is not satisfied, while conduction modes are unstable if Equation (75) is not fulfilled. The fourth solution consists of a mode driven by ion-diffusion which is unstable if either criterion (75) or (77) is unfulfilled. It is useful to understand what types of modes can be excited in the different regions of the parameter space spanned by the gradients in temperature and composition. The RouthHurwitz stability criteria take simple forms when expressed in terms of the dimensionless gradients defined in terms of the pressure. The classical requirement for stability against buoyancy, i.e., N 2 > 0 becomes while the conditions (75) and (76) become, respectively, If ions can diffuse along magnetic field lines, in addition to requiring that the gradients in temperature, pressure, and mean molecular weight satisfy the inequalities (82) and (83), the inequality (77) must also be satisfied, i.e., Panel (a) in Figure 3 shows the regions of parameter space where g -modes in Equation (78) are stable, overstable, or unstable (gray area). Panel (b) shows that the modes in Equation (79), which are driven by conduction, can be either stable or unstable. Note that in the region of parameter space where both gravity and conduction modes are overstable/unstable they both grow with comparable rates. Panel (c) in Figure 3 shows that the modes in Equation (80), which are driven by diffusion, can be either stable or unstable (gray). Their growth rates are estimated to be an order of magnitude smaller than either g -modes or conduction modes. The importance of these diffusion modes resides in that they can become unstable in regions of parameter space which are stable against g -modes and conduction modes.", "pages": [ 8, 9 ] }, { "title": "5.2. Heat and Ion Diffusion Along Horizontal Magnetic Fields", "content": "If the background magnetic field is perpendicular to the thermal and the composition gradients, i.e., b x = 1 , the dispersion relation (33) becomes 6 where the coefficients are subject to the same considerations employed in deriving the approximate expressions for the coefficients a i . The Routh-Hurwitz stability criteria require b 1 > 0 , b 1 b 2 -b 3 > 0 , b 1 b 2 b 3 -b 2 1 b 4 -b 2 3 > , and b 4 > 0 . The first condition is trivially satisfied, while, in the limit under consideration, i.e., ω dyn /greatermuch τ -1 c > τ -1 v , the other three conditions become, respectively, The inequality (90) is always satisfied, since it can be written as Therefore, the only independent condition required for stability is N 2 T/µ > 0 , or d ln T/dz > d ln µ/dz . Two of the approximate solutions to the dispersion relation (85) are given by σ ≈ ± ib 1 / 2 2 +( b 3 -b 1 b 2 ) / 2 b 2 , i.e., /negationslash 6 In the absence of diffusion, the only result that is modified in this section is that the root σ = -τ -1 d for D = 0 becomes σ = 0 for D = 0 . These correspond to gravity modes, which cannot become overstable on account of Equation (93). The other two solutions correspond to a conduction and a viscous (decaying) mode, which are, respectively, We conclude that when the magnetic field is perpendicular to the temperature and the composition gradients, the stability of g -modes requires only that N 2 > 0 , whereas the stability of conduction modes requires also This is illustrated in panel (d) in Figure 3, which shows the regions of parameter space where g -modes in Equation (94) are stable or unstable (gray). Panel (e) shows that the modes in Equation (95), which are driven by conduction, can be either stable or unstable. These regions are significantly different from the corresponding regions in panel (b), for which the direction of the background magnetic field is parallel to the direction of the temperature and the composition gradients. Note that, unlike the case where b z = 1 , g -modes and conduction modes cannot be simultaneously unstable when b x = 1 .", "pages": [ 9 ] }, { "title": "6. THE PHYSICS DRIVING UNSTABLE MODES", "content": "In previous sections we have seen that the criteria for instability result from an intricate interplay between the thermal and the composition gradients. The final expressions for the inequalities that must be satisfied for the onset of unstable modes depend implicitly on whether conduction across the associated scale is faster or slower than the dynamical timescale, the direction of the background field, and the ability of the ions to diffuse along magnetic field lines. We now analyze in detail the eigenmodes corresponding to some of the most relevant instabilities and shed light on the physical phenomena that play a role in determining the stability of the plasma. This exercise is similar in spirit to the ones presented in Quataert (2008) and Balbus (2001) to highlight the physics of the HBI and the MTI but emphasizes the effects of composition gradients and ion-diffusion along magnetic field lines. For the sake of simplicity we assume that the perturbations under consideration correspond to modes with k y = 0 . In this case, the components of the Lagrangian displacement, ∂ ξ /∂t = δ v , are related via ξ x = -( k z /k x ) ξ z .", "pages": [ 9 ] }, { "title": "6.1. Heat- and Particle-Flux Driven Buoyancy Instability", "content": "Let us first consider a background magnetic field with b z = 1 and focus on perturbations with wavelengths such that the associated timescales satisfy τ -1 c /greatermuch ω dyn /greatermuch τ -1 v /similarequal τ -1 d . For these modes, ion-diffusion along the magnetic field lines is inefficient. In order to understand the effect of a composition gradient, we retain the dominant terms in Equations (15) and (16), which leads to Because the fluctuations in density, temperature, and mean molecular weight are related via Equation (20), this implies that the relative change in density of a fluid element which is vertically displaced by ξ z is given by", "pages": [ 9 ] }, { "title": "D-HPBI", "content": "It is easier to understand the physics behind this equation by analyzing first the effects of each of the two terms on the righthand side separately. In a homogeneous medium, dµ/dz = 0 , the term proportional to the temperature gradient is responsible for the onset of the HBI when dT/dz > 0 . This is illustrated in panel (a) in Figure 4, which shows that a fluid element that is displaced with ξ z > 0 ( ξ z < 0 ) is effectively heated (cooled) by the increased (decrease) flux of heat due to the convergence (divergence) of field lines that results from the displacement of the fluid element. This causes the fluid element to expand (contract), and thus attain a density which is lower (higher) than the surrounding medium. This leads to the runaway process known as the HBI. An isothermal environment, dT/dz = 0 , stratified in composition is unstable if dµ/dz > 0 , as shown in panel (b) in Figure 4. Because the diffusion of ions along magnetic field lines is inefficient, the mean molecular weight of a fluid element that is displaced with ξ z > 0 ( ξ z < 0 ) is lower (higher) than the surrounding medium. This results in a displaced fluid element with a density which is lower (higher) than the density of the surrounding medium, which will thus rise (sink). If the temperature and the composition gradients are both positive (negative) then these arguments act in consonance and lead to the conclusion that the plasma is unstable (stable). On the other hand, if the temperature and the composition gradients have different signs it follows that if dT/dz > 0 ( dµ/dz > 0 ) is steep enough then the expansion induced in an upwardly displaced fluid element can offset the stabilizing effects of d ln µ/dz < 0 ( d ln T/dz < 0 ) and the plasma will be unstable, giving rise to an HPBI. The arguments outlined here can also be derived from the equation of motions for the Lagrangian displacement ξ z . The buoyancy force per unit volume on a vertically displaced fluid element produces an acceleration given by and thus, according to Equation (98), This leads to an instability if N 2 Tµ > 0 , in agreement with the results of Section 4.", "pages": [ 10 ] }, { "title": "6.2. Diffusive Heat- and Particle-Flux Driven Buoyancy Instability", "content": "The effects of ion-diffusion are not negligible for the modes for which the associated timescales satisfy τ -1 c > τ -1 v /similarequal τ -1 d /greatermuch ω dyn . To leading order, the changes in the temperature and the mean molecular weight in a fluid element which is vertically displaced by ξ z in a background magnetic field with b z = 1 are given by Note that because the dominant terms in Equation (16) are both proportional to D , the fractional change in the mean molecular weight is independent of the value of the diffusion coefficient. The fractional change in density is thus The role played by the background temperature gradient is identical to the one discussed in the absence of ion-diffusion, and this situation is shown for the sake of clarity in panel (c) in Figure 4. However, the contribution from the mean molecular weight gradient is now the opposite. This difference in sign is due to the manifest role played by ion-diffusion as an effective agent to tap into the free energy available in the background particle flux needed to maintain the composition gradient. In order to understand the role played by ion-diffusion let us focus on a background with constant temperature, dT/dz = 0 . The term proportional to the composition gradient is responsible for the onset of the D-HPBI when dµ/dz < 0 . This is illustrated in panel (d) in Figure 4, which shows that the mean molecular weight of a fluid element that is displaced with ξ z > 0 ( ξ z < 0 ) will decrease (increase) because of the decrease (increase) in the flux of particles out of (into) it due to the diverge (convergence) of field lines that results from the displacement of the fluid parcel. This causes the density of the fluid element to decrease (increase) with respect to the surrounding medium, leading to a runaway process. A background temperature gradient with dT/dz > 0 will reinforce this process leading to enhanced buoyancy. In general, when both the temperature and the composition gradients are non-zero, the condition for D-HPBI is d ln T/dz < d ln µ/dz . This is reflected in the equation of motion for the Lagrangian displacement which has exponentially growing solutions when N 2 T/µ > 0 , as we have seen before.", "pages": [ 10, 11 ] }, { "title": "6.3. Magneto-Thermo-Compositional Instability", "content": "Let us now consider a background horizontal magnetic field with b x = 1 and modes for which conduction is faster than any other timescale. When a fluid element is displaced by ξ z from its equilibrium position, the fluctuations in temperature and composition are given by It is important to note that this relationship between the fractional change in the mean molecular weight and the Lagrangian displacement holds regardless of whether the dynamical timescale is fast or slow with respect to the viscous and diffusion timescales. The only difference is that in the former case the leading order terms in Equation (16) that lead to δµ = -( dµ/dz ) ξ z are independent of the coefficient D , while in the opposite limit, the dominant terms are those proportional to D . In either case, the relative change in density becomes In this case, the term proportional to the temperature gradient is responsible for the onset of the MTI when dT/dz < 0 in a homogeneous medium, i.e., dµ/dz = 0 . In panel (e) in Figure 4 we consider the situation where τ -1 d /greatermuch ω dyn . Under this condition, a fluid element that is displaced with ξ z > 0 ( ξ z < 0 ) is effectively heated (cooled) by conduction along the magnetic field lines which have been distorted by the displacement of the fluid element. This causes the fluid element to expand (contract), and thus attain a density which is lower (higher) than the surrounding medium. This leads to the runaway process known as the MTI. The term proportional to the composition gradient is responsible for the onset of the MTCI when dµ/dz > 0 in an isothermal environment. This is illustrated in panel (f) in Figure 4, which shows that because of the effective diffusion of ions along distorted magnetic field lines, fluid elements that have been displaced upward (downward) ξ z > 0 ( ξ z < 0 ) maintain the mean molecular weight corresponding to the equilibrium value at their original position. This implies that the fluid element is immersed in a medium that is relatively denser (lighter) and it will thus rise (sink). When both gradients are non-zero their relative magnitudes determine whether the plasma is buoyantly unstable according to Equation (105). This can also be seen in the equation of motion which has exponentially growing solutions when N 2 T/µ < 0 .", "pages": [ 11 ] }, { "title": "6.4. Overstable Modes", "content": "The physics driving overstable modes, which are present when b z = 1 , is more subtle, but some insight can be gained by analyzing directly the equations of motion for the corresponding Lagrangian displacements. Using Equations (57) and (78) together with the equations for the perturbations (13)-(16) we find, to first order, The physics behind the first terms on the right-hand side of each of these equations is readily recognized. The term proportional to N 2 Tµ is responsible for the HPBI if N 2 Tµ > 0 , while the term proportional to N 2 is responsible for BruntViasala oscillations, i.e., stable g -modes, if N 2 > 0 . In a medium where N 2 Tµ < 0 and N 2 > 0 but N 2 + N 2 Tµ < 0 , the anisotropic heat flow along magnetic field lines tends to monotonically increase the restoring force acting on the oscillating fluid parcel and leads to overstability.", "pages": [ 11 ] }, { "title": "7. ASTROPHYSICAL IMPLICATIONS", "content": "Throughout this paper, we have studied the various instabilities that can be present in the parameter space spanned by ( d ln µ/d ln P, d ln T/d ln P ) without imposing restrictions on the relative values of the gradients involved. We can now frame our results in the context provided by observations and theoretical models addressing the temperature and composition structure of galaxy clusters.", "pages": [ 11 ] }, { "title": "7.1. Temperature and Composition Profiles in the ICM from Models and Observations", "content": "Modern X-ray observatories, such as Chandra and XMMNewton , have made it possible to obtain the temperature profiles of a large number of galaxy clusters, see, e.g., Vikhlinin et al. (2006); Leccardi & Molendi (2008). This has enabled to show that, quite generically, the gas temperature increases with radius, reaching values of 5 - 10 keV at 100 kpc, and decreases toward the outer cluster regions. In several cases the temperature changes by a factor of two or three on distances comparable to the cluster radius. In spite of the small mass ratio between hydrogen and helium, because helium is relatively abundant in the primordial gas from which clusters form, its sedimentation over the life-time of the cluster has the potential to produce important gradients in the mean molecular weight profile. Obtaining observational evidence to quantify the helium abundances that would result form this sedimentation is very hard because helium is completely ionized at the characteristic temperatures of typical galaxy clusters. However, this information is key in order to derive physical cluster properties, such as gas mass, total mass and gas mass fraction (see, e.g., Qin & Wu 2000), and even cluster distances (Markevitch 2007) from X-ray observations. This has highlighted the need to understand the efficiency of this process on theoretical and observational grounds (Fabian & Pringle 1977; Abramopoulos et al. 1981; Gilfanov & Syunyaev 1984; Qin & Wu 2000; Chuzhoy & Nusser 2003; Chuzhoy & Loeb 2004; Tamura et al. 2004; Ettori & Fabian 2006; Peng & Nagai 2009). Some of these estimates predict an overabundance by up to a factor of a few with respect to the primordial value µ /similarequal 0 . 6 . Since the realization that the stability properties of a weakly magnetized medium depends critically on its thermal structure, a large number of works have been devoted to study the HBI and the MTI. In spite of the fact that both instabilities grow on a dynamical timescale, the onset of the most relevant modes depends explicitly on the local values of the thermal gradients. Throughout this study, we have seen that if the composition of the plasma is not homogeneous, it is the combination of both the thermal and the mean molecular weight gradients that decides whether the plasma is stable or not. In order to understand how important the contributions from each of these gradients can be, we consider the schematic representation of the temperature and the mean molecular weight profiles of the representative galaxy cluster shown in Figure 5. The temperature profile shown resembles the results obtained by observations (Vikhlinin et al. 2006), whereas the mean molecular weight profile is akin to the helium sedimentation models discusses in Bulbul et al. (2011), which are based on analytical models for the physical properties of the ICM introduced in Bulbul et al. (2010). Although very crude, these representative profiles allow us to provide an estimate for the values of the logarithmic gradients for the temperature, ( ∇ T ) /T , and the mean molecular weight, ( ∇ µ ) /µ , which play a role in determining the stability of the dilute ICM. If the peak in the temperature profile occurs at a larger radius than the peak in the mean molecular weight profile, as shown in Figure 5, then there are three distinct regions defined by the signs of the temperature and the composition gradients. Each of these regions of the ICM will have asso- different characteristic values, which can be estimated according to Here, ∆ T ≡ T out -T in stands for the difference between two values across the characteristic scale L ≡ r out -r in , and ¯ T ≡ ( T out + T in ) / 2 is the associated mean value, with similar definitions for the mean molecular weight. Using the information available in Figure 5, we estimate these characteristic values for the different ICM regions in Section 7.3.", "pages": [ 11, 12 ] }, { "title": "7.2. Applicability of Approximations in the ICM", "content": "Before addressing the stability of the different regions of the ICM, we comment on two of the approximations that we have made on the geometry and strength of the magnetic field, which have allowed us to gain physical insight while keeping the problem tractable. We have studied in detail two special cases for the orientation of the background magnetic field, viz., either parallel, B = B ˆ z , or perpendicular, B = B ˆ x , to the gravitational field g . These two configurations, which have received a lot of attention in the related literature, have the advantage of not only simplifying the mathematics involved but also exposing in a clean way the physics driving the HBI, the MTI, as well as the generalizations that result from including composition gradients. While beyond the scope of this paper, accounting for more general geometries is clearly necessary in order to describe more realistic situations. An important simplification in our study is the assumption that the magnetic field is so weak that the Alfv'en frequency is much smaller than any other inverse timescale involved. It is worth mentioning that this is the regime explored by a number of numerical studies addressing both fundamental aspects of the MTI and the HBI (Parrish & Stone 2005, 2007; Parrish & Quataert 2008; Latter & Kunz 2012), as well as the implications that these instabilities have for the long-term evolution of the ICM (Bogdanovi'c et al. 2009; Ruszkowski & Oh 2010; Parrish et al. 2008, 2009, 2010, 2012; McCourt et al. 2011, 2012; Kunz et al. 2012). For the modes for which magnetic tension cannot be neglected, the Alfv'en timescale can become comparable or even faster than the dynamical and the conduction timescales and neglecting ω A in the dispersion relation (33) is not a good approximation. For the modes for which magnetic tension is important, the explicit dependence on ω A might introduce new stability criteria which have not been captured by our analysis. Furthermore, magnetic tension could affect the growth rates of the instabilities. Kunz (2011) provides a summary of the stabilizing effects provided by magnetic tension on HBI- and MTIunstable modes in a homogeneous medium. The main physical effect introduced by a non-zero Alfv'en frequency is to provide a cut-off for the growth of unstable modes at parallel wavenumbers such that k ‖ v A is comparable to the growth rate of the most unstable modes. For the magnetic field geometries that we analyzed, this must also be the case even in the presence of a composition stratification. The reason for this is that, when either b x = 1 or b z = 1 , all the contributions introduced by the mean molecular weight gradient appear in the form d ln T/dz ± d ln µ/dz . Thus, while the growth rates and range of unstable modes are affected because of the changes in the background composition, the effects of non-negligible magnetic tension on these modes, i.e., the existence of a cut-off parallel wavenumber, must be similar to what has been found in the case of a homogeneous medium. Since the plasma β Note . -The various ICM regions are defined in Figure 5. For convenience, in this table we have defined the dimensionless parallel wavenumber, ˜ k ‖ = k ‖ ( λ mfp H ) 1 / 2 . in galaxy clusters varies with radius, whether neglecting magnetic tension is a sensitive approximation for a local stability analysis or not, depends on the conditions present in the region of the ICM under consideration. We address this issue in detail below.", "pages": [ 12, 13 ] }, { "title": "7.3. Stability of ICM regions", "content": "The analysis of Figures 2 and 3 allows us to understand the implications that a mean molecular weight gradient can have for the various regions of a representative galaxy cluster as depicted in Figure 5. These regions correspond to different quadrants in the ( d ln µ/d ln P, d ln T/d ln P ) plane as shown in panel (a) of Figure 6, which we have denoted as inner, intermediate, and outer ICM. In what follows we will assume, as suggested by observations of galaxy clusters, that the ICM is buoyantly stable according to the classical stability criterion N 2 > 0 (Piffaretti et al. 2005; Cavagnolo et al. 2009).", "pages": [ 13 ] }, { "title": "7.3.1. Outer ICM", "content": "In this region the temperature and the mean molecular weight gradients are both negative. Because the criteria for stability are different whether conduction is fast or slow compared to the dynamical timescale, we consider these two cases separately. Fast conduction . If the magnetic field lines are parallel to the gravitational field, i.e., b z = 1 , gravity modes can become overstable if d ln T/d ln P > ( γ -1) / (2 γ ) , panel (a) in Figure 2. A gradient in the mean molecular weight alone is unable to stabilize these modes and can drive unstable modes driven by diffusion if |∇ µ | /µ > |∇ T | /T , panels (b) and (c) in Figure 2. For magnetic field configurations that are perpendicular to the gravitational field, i.e., b x = 1 , this region is unstable to the MTCI provided that |∇ µ | /µ < |∇ T | /T , panel (d) in Figure 2. This means that the outskirts of galaxy clusters that would be considered prone to the MTI (if they were homogeneous) would remain stable if the gradient in mean molecular weight is steep enough. This is not the case for the particular profiles shown in Figure 5 but this does not imply that this is not the case in general. Slow conduction . In the absence of ion-diffusion, gravity modes can become overstable if bz = 1. These modes cannot be stabilized by means of a gradient in the mean molecular weight alone, panel (a) in Figure 3. Furthermore, when iondiffusion is efficient, it can drive unstable modes if |∇ µ | /µ > |∇ T | /T , panel (c) in Figure 3. For b x = 1 there can be unstable modes driven by conduction if |∇ µ | /µ < |∇ T | /T , panel (d) in Figure 3. We can provide a crude estimate of the impact that a composition gradient would have on the growth rates of the various instabilities discussed by estimating the temperature and the composition gradients shown in Figure 5. For the inner ICM region, the characteristic scale is L /similarequal 0 . 8 Mpc, while ∆ T /similarequal 4 keV, ¯ T /similarequal 7 keV, ∆ µ /similarequal 0 . 1 , and ¯ µ /similarequal 0 . 65 . Thus according to Equations (109), the characteristic logarithmic gradients in this inner region are given by ∣ ∣ These order-of-magnitude estimates, based on the representative values drawn from Figure 5, show that the instabilities with growth rates σ 2 ∝ -ln( T/µ ) , such as the generalization of the MTI, Equations (59) and (68), will be 15% slower with respect to the homogeneous case. Regarding the validity of our assumptions of weak magnetic fields, panel (b) of Figure 6 shows that magnetic tension is unimportant for the range of modes of interest, i.e., K 1 / 2 n < k ‖ ( λ mfp H ) 1 / 2 < K -1 / 2 n . Thus, our approximation of setting ω A /similarequal 0 is fully justified in this region.", "pages": [ 13 ] }, { "title": "7.3.2. Intermediate ICM", "content": "In this region the temperature and the mean molecular weight gradients have different signs, 7 with ∇ µ < 0 and ∇ T > 0 according to the profiles shown in Figure 5. /negationslash When b z = 1 , this region is unstable due to the HPBI, which grows on the dynamical timescale if |∇ µ | /µ < |∇ T | /T . This instability can be prevented if the mean molecular weight is steep enough, panel (a) in Figure 2. When ion-diffusion is efficient this region is unstable due to the D-HPBI, panel (c) in Figure 2. This magnetic field configuration is also prone to unstable modes for which conduction is slow. In this case there are unstable modes driven by conduction if D = 0 and |∇ µ | /µ < |∇ T | /T , while there are unstable modes driven by ion-diffusion if D = 0 and |∇ µ | /µ > |∇ T | /T , panels (b) and (c) in Figure 2, respectively. If b x = 1 , this region is stable whether conduction is fast or slow compared to the dynamical timescale, panels (d) in Figure 2 and (e) in Figure 3. For this intermediate ICM region, the inspection of Figure 5 provides L /similarequal 0 . 15 Mpc, ∆ T /similarequal 2 keV, ¯ T /similarequal 8 keV, ∆ µ /similarequal 0 . 1 , and ¯ µ /similarequal 0 . 75 . Thus the characteristic logarithmic gradients are given by ∣ ∣ Therefore, the instabilities with growth rates for which σ 2 ∝ ln( Tµ ) , such as the generalization of the HBI in the absence of ion-diffusion, Equation (54), will grow of the order of 40% slower with respect to the homogeneous case. On the other hand, the instabilities with growth rates σ ∝ ln( T/µ ) , such as the generalization of the HBI with ion-diffusion, Equation (64), will be 30% faster. As shown in panel (c) in Figure 6, magnetic tension is important for some of the modes for which conduction is fast but not for the modes for which conduction is slow.", "pages": [ 13, 14 ] }, { "title": "7.3.3. Inner ICM", "content": "In this region the temperature and the mean molecular weight gradients are both positive. We consider again the limits in which conduction is fast or slow separately. Fast conduction . For b z = 1 , this region is unstable to the HPBI regardless of whether the mean molecular weight gradient is smaller or larger than the temperature gradient, panel (a) in Figure 2. Furthermore, if ion-diffusion is efficient it can also drive unstable modes, panels (b) and (c) in Figure 2. In a homogeneous medium with b x = 1 , this inner region is sta- ble against the MTI. However, there can be unstable MTCImodes if |∇ µ | /µ > |∇ T | /T , panel (d) in Figure 2. In the homogeneous case when ∇ T > 0 , the HBI tends to re-orient the magnetic field in the radial direction, which results in a field configuration which is stable against the MTI, i.e., b x /similarequal 1 . When ion-diffusion is not efficient, this core insulation could be alleviated by the MTCI if the mean molecular weight gradient is steep enough, i.e., ( ∇ µ ) /µ > ( ∇ T ) /T . Slow conduction . This region can be subject to instabilities driven by both heat conduction and ion-diffusion. When b z = 1 and ion-diffusion is inefficient, there are unstable modes driven by heat conduction regardless of the relative magnitude of the temperature and the mean molecular weight profiles, panel (b) in Figure 3, whereas there are unstable modes driven by diffusion if |∇ µ | /µ > |∇ T | /T , panel (c) in Figure 3. In the case with b x = 1 , there can be unstable modes driven by heat conduction if |∇ µ | /µ > |∇ T | /T , panel (e) in Figure 3. According to Figure 5, the inner ICM region is charac- terized by L /similarequal 0 . 05 Mpc, ∆ T /similarequal 3 keV, ¯ T /similarequal 5 . 5 keV, ∆ µ /similarequal 0 . 1 , and ¯ µ /similarequal 0 . 8 , and thus The analysis of Figure 6 shows that the range of modes for which it is sensible to carry out a local mode analysis within the fluid model embodied in Equations (1)-(5) increases as the inverse Knudsen number increases toward smaller radii. However, due to the increase in the strength of the background magnetic field, the range of modes for which it is sensible to neglect magnetic tension, decreases. Because the relatively low values of β in the inner core region, magnetic tension is important for all the modes of interest. Therefore, our conclusions for this region should be considered with caution. ∣ ∣ Therefore, the instabilities with growth rates for which σ 2 ∝ ln( Tµ ) will grow of the order of 10% faster with respect to the homogeneous case, while the instabilities with growth rates σ 2 ∝ ln( T/µ ) will be 15% slower.", "pages": [ 14, 15 ] }, { "title": "7.3.4. Summary and Outlook", "content": "During the last few years, there has been substantial numerical work for understanding the long-term evolution of the MTI and the HBI and their implications for the gas dynamics in the ICM permeating galaxy clusters (Bogdanovi'c et al. 2009; Parrish et al. 2008, 2009, 2010, 2012; McCourt et al. 2011, 2012; Kunz et al. 2012). All of this work has been done under the assumption that the ICM is homogeneous and thus the temperature gradient provides the only source of energy to feed instabilities. Even though it is hard to quantify concentration gradients from observations, some heavy element sedimentation is expected (Narayan & Medvedev 2001; Chuzhoy & Nusser 2003; Chuzhoy & Loeb 2004; Ettori & Fabian 2006). Indeed, current theoretical models suggest that helium sedimentation can significantly alter the composition profile throughout the cluster and give rise to mean molecular weight gradients which are comparable in magnitude to the temperature gradients, with |∇ T | /T /similarequal |∇ µ | /µ (see, e.g., Bulbul et al. 2011, and Figure 5). This work discusses for the first time the effects that composition gradients can have for the stability of a weakly collisional magnetized medium which is stratified in both temperature and composition. We have found that, depending on the wavelength of the modes under consideration, the plasma can be subject to a wide variety of unstable modes. These include: and the generalization of the HBI (Quataert 2008) in We have also found the generalization of the overstable gravity modes discussed in Balbus & Reynolds (2010), see Equations (57) and (78), as well as other new modes which are driven by conduction and diffusion. This study constitutes a first step toward the long-sought goal of understanding in a self-consistent way the effects of magnetic turbulence on the diffusion of heavy elements and its consequences for the observational signatures and longterm evolution of galaxy clusters. This will only be possible through numerical studies involving realistic models for the microphysics of weakly collisional, multi-component plasmas. Addressing this problem will require to sort out several details, including how to properly handle plasma microinstabilities, many of which are still the subject of active research in homogeneous settings. We thank Matthew Kunz, Henrik Latter, Aldo Serenelli, and Shantanu Mukherjee for useful discussions. We are grateful to the anonymous referee for a detailed and thoughtful report that helped us improve the final version of this manuscript significantly. M.E.P is grateful to the Knud Højgaard Foundation and the Villum Foundation for their generous support. S.C. acknowledges support from the Danish Research Council through FNU Grant No. 505100-50 - 30,168.", "pages": [ 15 ] }, { "title": "A. ION DIFFUSION IN A BINARY MIXTURE", "content": "The ratio between the timescales associated with viscous and diffusion processes is τ -1 v /τ -1 d = 3 ν ‖ /D . Here, the coefficient ν ‖ = v 2 th / (2 ν eff ii ) denotes the kinematic Braginskii viscosity associated with a binary mixture of ions, where ν eff ii is an effective ion-ion collision rate which can be estimated as follows. The Braginskii viscosity for a single species of ions is η 0 = ρv 2 th / (2 ν ii ) , where ν ii is the ion-ion collision frequency Here n i , m i and q i are the number density, the mass, and the charge of the ion respectively; and ln Λ refers to the corresponding Coulomb logarithm. In a binary mixture, the effective viscosity coefficient (ignoring the contribution from electrons) is given by where the collision frequency between ions of species i and j is and m ij ≡ m i m j / ( m i + m j ) is the reduced ion mass. We can thus define the effective ion-ion collision frequency as The coefficient governing the diffusion of species 2 into species 1 (e.g., helium into hydrogen) is given by (Bahcall & Loeb 1990) Therefore the ratio τ -1 v /τ -1 d = 3 ν ‖ /D is given by which is shown in Figure 7 as a function of the concentration c (left panel) and as a function of the mean molecular weight µ (right panel). In the inner regions of the ICM, where c /similarequal 0 . 6 or µ /similarequal 0 . 8 (see Figure 5), viscous and diffusion processes take place on comparable timescales, i.e., τ -1 v /τ -1 d /similarequal 3 , while τ -1 v /τ -1 d /similarequal 9 for a primordial mixture of helium and hydrogen, i.e., c /similarequal 0 . 25 or µ /similarequal 0 . 6 . It should be kept in mind that in carrying out this calculation we have assumed that all the ratios between the Coulomb logarithms are of order unity and the expressions for the transport coefficients along the magnetic field lines are identical to the ones that are valid in the absence of the magnetic field.", "pages": [ 15, 16 ] }, { "title": "REFERENCES", "content": "Abramopoulos, F., Chanan, G. A., & Ku, W. H.-M. 1981, ApJ, 248, 429 Bahcall, J. N., & Loeb, A. 1990, ApJ, 360, 267 Balbus, S. A. 2000, ApJ, 534, 420 -. 2001, ApJ, 562, 909 -. 2004, ApJ, 616, 857 Balbus, S. A., & Reynolds, C. S. 2010, ApJ, 720, L97 Bogdanovi'c, T., Reynolds, C. S., Balbus, S. A., & Parrish, I. J. 2009, ApJ, 704, 211 Braginskii, S. I. 1965, Reviews of Plasma Physics, 1, 205 Bulbul, G. E., Hasler, N., Bonamente, M., & Joy, M. 2010, ApJ, 720, 1038 Bulbul, G. E., Hasler, N., Bonamente, M., Joy, M., Marrone, D., Miller, A., & Mroczkowski, T. 2011, A&A, 533, A6 Carilli, C. L., & Taylor, G. B. 2002, ARA&A, 40, 319 Cavagnolo, K. W., Donahue, M., Voit, G. M., & Sun, M. 2009, ApJS, 182,", "pages": [ 16 ] } ]

[ { "title": "ABSTRACT", "content": "We investigate the alignment processes of spinning black holes and their surrounding warped accretion disks in a frame of two different types of feeding at the outer boundaries. We consider (1) fixed flows in which gas is continually fed with a preferred angular momentum, and (2) free flows in which there is no gas supply and the disks diffuse freely at their outer edges. As expected, we find that for the cases of fixed flows the black hole disk systems always end up aligning on timescales of several 10 6 yr, irrespective of the initial inclinations. If the initial inclination angles are larger than π/ 2, the black hole accretion transits from retrograde to prograde fashion, and the accreted mass onto the black holes during these two phases is comparable. On the other hand, for the cases of free flows, both alignments and anti-alignments can occur, depending on the initial inclinations and the ratios of the angular momentum of the disks to that of the black holes. In such cases, the disks will be consumed within timescales of 10 6 yr by black holes accreting at the Eddington limit. We propose that there is a close connection between the black hole spin and the lifetime for which the feeding persists, which determines the observable episodic lifetimes of active galactic nuclei. We conclude that careful inclusion of the disk feeding at the outer boundaries is crucial for modeling the evolution of the black hole spin. Subject headings: accretion, accretion disks - black hole physics - galaxies: active", "pages": [ 1 ] }, { "title": "EVOLUTION OF WARPED ACCRETION DISKS IN ACTIVE GALACTIC NUCLEI. I. ROLES OF FEEDING AT THE OUTER BOUNDARIES", "content": "YAN-RONG LI 1 , JIAN-MIN WANG 1,2 , CHENG CHENG 1 AND JIE QIU 1 The Astrophysical Journal, 2013, 764, 16", "pages": [ 1 ] }, { "title": "1. INTRODUCTION", "content": "The current hierarchical framework of galaxy formation and evolution predicts that, in addition to secular evolution, repeat galaxy mergers trigger activities of supermassive black holes (SMBHs) located at the galactic center and hence their growth (e.g., Benson & Bower 2010). As such, fuel feedings channeled onto SMBHs most likely proceed at episodic and random phases (e.g., Martini 2004; King & Pringle 2006; Wang et al. 2006, 2008, 2009; Li et al. 2010). This picture is further reinforced by various lines of observations, such as misaligned active galactic nuclei (AGNs) with respect to their host galaxies (Kinney et al. 2000; Gallimore et al. 2006; Muñoz Marín et al. 2007; Shen et al. 2010), and as well as by numerical simulations (e.g., Hopkins et al. 2012 and references therein). Previous studies on the cosmological evolution of the radiative efficiency ( η ) of accretion according to the Sołtan's (1982) argument and formulated by the η -equation, found that η decreases with cosmic time since redshift z ∼ 2, strongly implying that random accretion takes place upon the SMBHs (Wang et al. 2009; Li et al. 2011, 2012; see also the discussion of Zhang et al. 2012). A similar inference has recently been highlighted from the semi-analytical modeling on SMBH demography by Volonteri et al. (2012). In this context, warped accretion disks are ubiquitous in AGNs. A misaligned accretion disk around the central spinning black hole undergoes Lense-Thirring precession arising from the gravitomagnetic effects, of which the rate falls off rapidly with the radius to the hole (e.g., Hartle 2003). The presence of viscosity combined with such differential precession will induce the inner portion of the disk to align or anti-align its orbital angular momentum with the spin axis of the hole out to a transition radius, beyond which the disk retains its initial incli- nation (Bardeen-Petterson effect; Bardeen & Petterson 1975). In the meantime, associated with disk precession, the black hole suffers an equal and opposite gravitomagnetic torque that causes it to precess as well. Under mutual interaction and warp propagation in the disk, this composite system tends to restore full axisymmetry with the entire disk ultimately aligned or anti-aligned with the hole (King et al. 2005). The characteristic timescale and the final state of the system (alignment or anti-alignment) depend on the black hole mass and spin and the properties of the accretion disk. Based on the angular momentum conservation , King et al. (2005) demonstrated that the disk will end up anti-aligned with the hole if the angular momentum of the hole ( J h) dominates over that of the disk ( J d). This refreshes previous analytical studies on the grounds that the disk has an infinite extension and is continuously fed (Scheuer & Feiler 1996; Martin et al. 2007), and has been incorporated into the black hole spin modeling in the context of hierarchical cosmology (King et al. 2008; Lagos et al. 2009; Fanidakis et al. 2011; Barausse 2012). However, while J h is accurately defined, the meaning of J d is not as well defined in King et al. (2005), leading to its interpretation varying among subsequent studies. It is commonplace that, in the outer region, the accretion disk likely becomes self-gravitating, fragments into clumps, and probably manifests itself as a star-forming region (e.g., Paczynski 1978; Shlosman & Begelman 1987; Collin & Zahn 1999; Goodman 2003; Wang et al. 2010). The feedback from the stellar radiation or the resultant supernova explosions embedded in the disk acts to excite turbulence that continues to deprive gas of angular momentum (Wada & Norman 2002; Kawakatu & Wada 2008; Kumar & Johnson 2010; Wang et al. 2010). In this case, it is expected that gas will be continually fueled into the accretion disk from the outer region (e.g., circumnuclear disk or torus) for the duration of the AGN's lifetime. The practical implications are twofold: an outer edge of the disk limited by its own self-gravity exists so that the angular momentum of the disk can be properly defined, and there is a gas supply with specified angular momentum at this outer edge. With the above inference, a comprehensive study of the alignments/anti-alignments between black holes and their warped accretion disks is essential, in particular considering the increased focus on modeling the black hole spin across cosmic time. In this work, we revisit this issue by numerically solving the evolution equation set of black holes and accretion disks in a practical sense. The paper is organized as follows: Section 2 describes the evolution equation set for a black hole and its warped accretion disk. Section 3 surveys the properties of warped accretion disks from the theoretical point of view. Section 4 shows the numerical scheme and sets up the boundary conditions and initial conditions for solving the evolution equation set. The results are then presented in Section 5. The implications of our results on SMBH spin evolution and the uncertainties in our calculations are discussed in Section 6. Conclusions are summarized in Section 7.", "pages": [ 1, 2 ] }, { "title": "2. BASIC EVOLUTION EQUATIONS", "content": "The dynamics of warped disks have been studied extensively through theoretical analysis (e.g., Papaloizou & Pringle 1983; Pringle 1992; Papaloizou & Lin 1995; Ogilvie 1999, 2000; Lubow et al. 2002) and numerical simulations (e.g. Larwood et al. 1996; Nelson & Papaloizou 1999, 2000; Dotti et al. 2010; Lodato & Price 2010; Nixon & King 2012; Nixon et al. 2012; McKinney et al. 2013). Adopting the standard α -prescription of disk viscosity, warp propagation in the disk is characterized into two regimes depending on the relative importance of pressure forces and viscous forces. For a nearly inviscid or sufficiently thick disk such that the dimensionless viscosity coefficient α is smaller than the disk aspect ratio h = H / R , the warp propagates as bending waves at a velocity related to the speed of sound (Papaloizou & Lin 1995); whereas in the regime α > h , the warp evolves in a diffusive fashion. Here H is the semithickness of the disk. We focus our attention on a thin and viscous accretion disk with a Keplerian rotation ( α > h ), the most likely case for accretion disks in AGNs (Pringle 1999). The evolution of such warped disks can be completely described by the angular momentum density of the disk L ( R , t ) as (Ogilvie 1999) where L = Σ R 2 Ω , /lscript = L / L denotes the direction vector, Σ is the mass surface density, Ω is the angular velocity, T LT is the gravitomagnetic torque responsible for the Lense-Thirring precession, and VR is the radial velocity given by ∣ ∣ Here the internal viscous torque of the disk fluid has been distinguished into three kinds associated with three effective viscosities, respectively (Bardeen & Petterson 1975; Ogilvie 1999). The first viscosity, ν 1, governs the usual azimuthal shear due to the differential rotation; the second viscosity, ν 2, governs the vertical shear that acts to diffuse the warp throughout the disk; and the third viscosity, ν 3, governs a torque component that tends to create the disk ring process if it is misaligned with its neighbors. The later two viscosities disappear in a flat disk. It is worth pointing out that Equation (1) remains valid for large amplitude warp (Ogilvie 1999). By analyzing the internal non-linear dynamics of warped accretion disks, Ogilvie (1999) formulated the effective viscosities and showed that they generally depend on the amplitude of the warp. For the sake of simplicity, we make use of the first-order approximations as follows ν 2 = ν 1 / 2 α 2 and ν 3 = 3 ν 1 / 8 α . As a result, the warp propagates in the disk on a timescale of an order of where t ν is the usual viscous timescale. In the weak field limit, the gravitomagnetic torque due to frame dragging has a form of (Bardeen & Petterson 1975) where J h is the angular momentum of the hole with its modulus J h = aGM 2 · / c , a is the spin parameter, M · is the mass of the black hole, G is the gravity constant, and c is the speed of light. Due to the mass accretion and the gravitomagnetic torque exerted by the warped accretion disk, the angular momentum of the black hole evolves with time as where ˙ M in is the mass accretion rate through the inner edge of the disk R in, which is determined by the marginal stable orbit, and j in is the specific angular momentum carried by matter at the inner edge. To study the roles of disk feeding, we consider two classes of accretion models to mimic the realistic accretion disks, as schematically shown in Figure 1. In the left panel, an outer clumpy region composite of clouds and/or stars continues to channel gas to the accretion disk from the outer environment (e.g., circumnuclear disk or torus). As stated above, the feedback from stellar radiation and supernova explosions are the potential mechanism responsible for exciting turbulence and viscosity, acting to transport the angular momentum of accreting matter (Wada & Norman 2002; Kumar & Johnson 2010; Wang et al. 2010). We denote this case as 'fixed flows' because the angular momentum of gas supply at the outer edge has a preferred direction. By contrast, the right panel of Figure 1 shows the other class of outer boundary conditions, in which there is no gas supply and the disk diffuses freely at its outer edge. We therefore denote this case as 'free flows'. Before numerically solving the above equations, we endeavor to explore the properties of warped accretion disks from a theoretical point of view in the next section.", "pages": [ 2 ] }, { "title": "3. WARPED ACCRETION DISKS", "content": "We start by specifying the accretion disk model used in this paper generally following Collin-Souffrin & Dumont (1990). To simplify the theoretical analysis and numerical calculations below, we parameterize accretion disks with a Keplerian rotation of Ω = √ GM · / R 3 , and a constant aspect ratio of h = H / R . The mass surface density of the disk obeys Σ = Σ 0 R -p , where Σ 0 is a coefficient determined by setting the total disk mass at a fraction of the black hole mass M d = fmM · , and p = 1 / 2 is an index to enforce the mass accretion rate of ˙ M ≈ 3 πν 1 Σ to a constant at a large radius as for a flat disk. The outer edge of the disk R out is located where self-gravitating becomes important, whose value depends on the mass accretion rate and black hole mass (e.g., Goodman 2003; King et al. 2008). In what follows, for illustration purposes, we adopt fiducial values of α =0 . 1, h =10 -2 , fm =10 -2 , R out = 10 4 R g, M · =10 8 M /circledot , and a =0 . 998, where R g = GM · / c 2 is the gravitational radius.", "pages": [ 2, 3 ] }, { "title": "3.1. Transporting Angular Momentum", "content": "In planar accretion disks, matter gradually falls inward the central black hole, whereas nearly all of the angular momentum is transported outward due to viscosities (Frank et al. 1992). This remains valid for warped accretion disks (Scheuer & Feiler 1996). The difference is that gravitomagnetic interactions of the hole and the disk induce extra transfers of angular momentum between them. This transfer rate depends on the Lense-Thirring precessing rate and warp propagation velocity. There exists a transition radius where the Lense-Thirring precessing timescale is comparable with the warp propagation timescale, which is usually denoted by a 'warp radius'. Within the warp radius, the Lense-Thirring precessions are dominant. Viscosities damp out the differential precessions rapidly, giving rise to an inner flat disk. It is easy to verify: if initially the angle of the disk and the hole is smaller than π/ 2, the inner flat disk is aligned with the hole. By contrast, if the angle is larger than π/ 2, the inner flat disk is anti-aligned. In the region around the warp radius, the incoming matter is inclined toward the inner flat disk, and the associated angular momentum is transferred to the hole through gravitomagnetic interaction. Further beyond the warp radius, warps propagate outward efficiently and the misaligned angular momentum is transported outward as well. Since the gravitomagnetic torque integrated over the whole disk in Equation (6) is always orthogonal to J h, it can be expressed in the form of (King et al. 2005) where a hat symbol on top of a vector stands for the corresponding direction vector, K 1 and K 2 are time-dependent coefficients, and is the total angular momentum of the disk. Here, the first term corresponds to the precessions and the second term corresponds to changes in the angle between the hole and the disk. Below we will numerically demonstrate that K 1 and K 2 are generally positive (see also Scheuer & Feiler 1996; King et al. 2005; Martin et al. 2007).", "pages": [ 3 ] }, { "title": "3.2. Magnitude Estimates", "content": "In in previous section, we demonstrate that the characteristic extension of warps is determined by the timescales for warp propagation given by Equation (4) and for the local Lense-Thirring precessing t LT ≈ | L | / | T LT | = c 2 R 3 / 2 GJ h (Scheuer & Feiler 1996). By equating these two timescales, one obtains the warp radius R w = 2 GJ h /ν 2 c 2 . Using the formulation of ν 2 and the fiducial values of parameters, R w is of the order of Considering that the Lense-Thirring torque in Equation (5) falls off rapidly with the radius in proportion to R -3 , the major contribution it has on the hole comes from the region of the disk around R w. As a result, the torque that the disk exerts on the hole approximates where the mass accretion rate ˙ M ∼ 3 πν 1 Σ is used for simplicity. This results in a timescale for the alignment between the hole and the whole disk (see also Perego et al. 2009) where t gr = M · / ˙ M is the growth timescale for a black hole accreting at a rate of ˙ M . Again, with the help of the approximation ˙ M ∼ 3 πν 1 Σ , one can estimate the mass accretion rate as and the growth timescale t gr ∼ 10 8 yr. We note that this estimated accretion rate is generally equal to the Eddington limit. If there is no gas supply to the accretion disk, the disk will be consumed on a timescale of an order of which is independent of fm and M · . As can be seen from Equation (11), once the disk is gradually depleted (without gas replenishment), the resulting decreases of surface density and the mass accretion rate will significantly prolong the alignment processes. The timescale for warp propagation throughout the disk in Equation (4) is The total angular momentum of the disk is This gives a ratio of J d over J h:", "pages": [ 3, 4 ] }, { "title": "3.3. Alignments and Anti-alignments", "content": "In this section, we qualitatively study what controls the final configuration of black hole-disk systems, i.e., alignments or anti-alignments, inspired by the arguments of King et al. (2005). As shown in the previous section, the (anti-)alignment timescale is generally orders of magnitude less than the growth timescale for black holes. Therefore, the accretion contribution to the changes of J h in Equation (6) can be safely neglected. By denoting θ inc as the angle between J h and J d and using Equation (7), we have In the meantime, above the time derivative can be alternatively written where we apply the fact that the gravitomagnetic interaction only modifies the direction of J d but keeps its modulus unchanged (i.e., d J h / d t = 0). A simple algebra manipulation yields Obviously, besides the gravitomagnetic interaction, the evolution of the inclination angle θ inc depends on how the angular momentum of the disk J d changes. According to the different types of disk feeding in the outer edges as illustrated in Figure 1, we distinguish the two following cases. 1) Fixed flows . If the disk is steadily fed and maintains a preferred angular momentum distribution, one shall expect J d, and therefore J d, to remain somewhat unchanged with time. As a result, Equation (19) is simplified into This means that, as found by previous intensive studies (Scheuer & Feiler 1996; Martin et al. 2007; Perego et al. 2009), a continuously fed disk always drives the hole to align with it regardless of the initial inclination. 2) Free flows . For disks without replenishment, the total angular momentum of the system ( J h + J d) is conserved. With the help of Equation (7), it is easy to derive the following equation (see also King et al. 2005) Note that here the coefficient K 2 is time-dependent. Because K 2 is positive, once initially cos θ inc /greaterorsimilar 0, i.e., θ inc /lessorsimilar π/ 2, the right hand side in the above equation is never smaller than zero. Therefore, θ inc will decrease continuously and the system ends up aligned. For θ inc /greaterorsimilar π/ 2, there exist two subcases: In real situations, J d generally decreases with time due to the internal viscous dissipation, and subsequently, the ratio J d / J h can transit from the former subcase to the latter. Therefore, the above conditions are indeed instantaneous for the behavior of the inclination. We next dig into exploring the detailed alignment/anti-alignment processes with help from numerical calculations.", "pages": [ 4 ] }, { "title": "4. NUMERICAL SCHEME", "content": "Numerically solving Equations (1) and (6) gives the selfconsistent evolution of the black hole accretion disk system. For this purpose, we implement a differencing scheme following Pringle (1992), but with some different treatment tricks. A detailed description of our differencing scheme and robust tests of the code are given in Appendices A and B, respectively. In order to produce the free-torque conditions in the inner boundary, we enforce L ( R in) = 0 such that the mass and angular momentum that reaches the inner boundary are removed, resulting in accretion onto the hole (see also Bregman & Alexander 2012; Nixon & King 2012). Accordingly, the initial surface density is setup by adding an extra factor to allow for the torque-free condition (Frank et al. 1992), namely, where R in is the inner edge of the disk. In consideration of the inner portion of the disk aligned/anti-aligned with the hole, we set ∂ /lscript /∂ R = 0 at R in. Given an initial inclination angle of θ inc , 0, we generate the initial distribution of the direction of disk rings as For this configuration, we presume that initially the black hole spin is oriented toward the z -axis. The outer boundary conditions at R out are treated differently for fixed flows and free flows. In our calculations, without specified otherwise, we adopt the fiducial values α = 0 . 1, p = 1 / 2, h = 10 -2 , R out = 10 4 R g, R in = 6 R g, and the initial black hole mass M · = 10 8 M /circledot and spin a = 0 . 998. In principle, R in is determined by the marginal stable orbit of the central black hole. However, since the inner portion of the disk is always aligned/anti-aligned with the hole, the location of R in does not affect the gravitomagnetic torque and hence the alignment rate. Moreover, a relatively larger R in helps to improve the time consumption of the computations in view of the timestep size limited by viscous timescales (see Equation (4)). To assess the processes of alignments and anti-alignments, we adjust the two free parameters: the mass fraction of the disk to the black hole fm , which determines the surface density of the disk in Equation (22) and hence the mass accretion; and the initial inclination angle between the disk and the hole θ inc , 0. We setup the grid with 202 points and evolve the equation set for a time of 1 . 5 t ν , where t ν = 2 × 10 6 yr is the viscous time at R out (see Appendix A for details).", "pages": [ 4, 5 ] }, { "title": "5.1. Evolution of Warped Accretion Disks", "content": "We define the inclination of disks at radius R with respect to black holes as In Figure 2, we illustrate how the shape of warped accretion disks evolves with times for free flows. The initial inclination angles are θ inc , 0 = 30 · in the upper panel and θ inc , 0 = 150 · in the bottom panel, respectively. We can find that an inner flat disk rapidly forms for all cases and is aligned to the black hole for θ inc , 0 = 30 · and anti-aligned for θ inc , 0 = 150 · . The size of the inner flat disk continuously grows until roughly 10 2 R g, corresponding to the warp radius R w defined in Equation (9). Warp propagation reaches the outer edge of the disk within a timescale of ∼ 0 . 01 t ν , consistent with the estimate from Equation (14). The disk for θ inc , 0 = 30 · gradually approaches full alignments; however, the alignment rate is prone to severe attenuation with time. The disk inclination is almost unchanged from 1 . 0 to 1 . 5 t ν . At the end of the numerical calculation 1 . 5 t ν , the disk maintains an inclination of θ inc ∼ 2 · at the outer edge. Henceforth, it hardly aligns with the black hole fully. This is because of a severe depletion of the surface density due to mass accretion (see Equation (13)). In Figure 3, it is verified that the mass accretion onto the black hole decreases rapidly with time by more than one order of magnitude as a result of disk depletion. This implies that if there is no gas supply, the luminosity of the accretion disk will fade out less than 10 6 yr. We further calculate the components of the gravitomagnetic torque that the whole disk exerts on the hole K 1 and K 2 as defined in Equation (7). As shown in Figure 4, it is confirmed that irrespective of the initial inclinations, both K 1 and K 2 are positive. Moreover, K 1 and K 2 are in the same order of magnitude (Scheuer & Feiler 1996; King et al. 2005; Martin et al. 2007). The system for θ inc , 0 = 150 · evolves toward anti-alignment in a similar fashion, as expected since initially J d / J h = 0 . 78 < cos θ inc , 0 = 0 . 87. However, the inclination θ inc at R out has changed only from 150 · to ∼ 160 · at the end of the calculation, which is very inefficient compared to the case of θ inc , 0 = 30 · . The corresponding reason is that, as shown in Equation (21), the change of θ inc from the gravitomagnetic torque is proportional to ∝ (1 + J h / J d cos θ inc). Its value for θ inc , 0 = 150 · is smaller by an order of magnitude than that for θ inc , 0 = 30 · . The cases for fixed flows are illustrated in Figure 5. By analogy to free flows, warps spread to the outer boundary within 0 . 01 t ν ; however, the subsequent evolution is somewhat different. The system with θ inc , 0 = 30 · quickly achieves full alignment after a time less than 1 . 0 t ν . For θ inc , 0 = 150 · , the disk shape shows complicated behaviors. The inner antialigned angular momentum is transported outward by warp propagation and confronts the angular momentum carried by the feeding matter. As a result, an abrupt discontinuity of the inclination appears and the disk breaks into two parts: an inner nearly anti-aligned portion and an outer misaligned portion (see also Lodato & Pringle 2006; Nixon & King 2012; Nixon et al. 2012). The location of the discontinuity transfers inward with time and the inner portion will be eventually swallowed by the hole due to mass accretion. In the meantime, the black hole is driven to align with the outer portion progressively. Around the time 0 . 34 -0 . 38 t ν , after the inner anti-aligned portion of the disk is consumed, the newly formed inner disk turns to be aligned with the black hole. At the end of the calculation 1 . 5 t ν , there is complete alignment between the disk and the black hole. The mass accretion rates onto the black holes for fixed flows are plotted in Figure 6. From Equation (2), the role of warping increases the inflow velocity and therefore enhances the mass accretion rate. As expected, for θ inc , 0 = 30 · , the accretion rate onto the hole ( ˙ M in) has a mildly decreasing trend by a factor of two at the beginning in response to the approaching alignment of the system. Then it maintains a steady value of ∼ 1 M /circledot yr -1 , equal to the gas supply rate at the outer boundary. Quite differently, for θ inc , 0 = 150 · there exist two peaks of the mass accretion rate at times ∼ 0 . 2 t ν and ∼ 0 . 5 t ν due to the large magnitude of warping developed at these moments (see the bottom panel of Figure 5). While the inner anti-aligned disk portion is being consumed, the accretion rate becomes significantly attenuated by orders of magnitude and reaches the minimum ( ˙ M in → 0) around 0 . 34 -0 . 38 t ν . Nevertheless, the mass accretion onto the hole eventually approaches a steady rate, the same as for θ inc , 0 = 30 · . It is worth pointing out that the black hole undergoes retrograde accretion for a time ∼ 0 . 35 t ν , after which it transits to prograde accretion.", "pages": [ 5, 6 ] }, { "title": "5.2. Alignment and Anti-alignment Timescales", "content": "where J d is the total angular momentum of the disk. The upper left panel shows θ inc /θ inc , 0 against time with θ inc , 0 = (10 · , 30 · , 40 · , 50 · , 60 · ) from bottom to top for free flows. The timescales that the inclination angle decreases to 1 / e of its initial value generally lie at several 10 5 yr, consistent with the magnitude estimate in Equation (11). However, the decaying rate of the inclination angle rapidly slows down with time because of the depletion of the disk (almost all of the disk angular momentum is transported to the large radius). After ∼ 10 6 yr, the inclinations for all the cases are prone to be frozen, indicating that the disk shall hardly approach complete alignment. This further confirms the results in the previous section. The upper right panel plots ( π -θ inc) / ( π -θ inc , 0) with θ inc , 0 = (120 · , 130 · , 140 · , 150 · , 170 · ) from top to bottom for free flows. Overall, although quite inefficient, the inclination angles have a tendency to approach θ inc → 180 · except for θ inc , 0 = 120 · , implying a trend of anti-alignment between the hole and the disk. The system for θ inc , 0 = 120 · tends toward alignment instead of anti-alignment because J d / J h > -cos θ inc is always retained throughout the calculation (see Section 3.3). Interestingly, there is an increasing trend at the beginning for θ inc , 0 = 130 · and 140 · , because J d / J h > -cos θ inc initially, which, however, transits to J d / J h < -cos θ inc later on. Again, the alignment rates are significantly slower compared with the counterparts in the upper left panel. The bottom panels of Figure 7 show the results for fixed flows. The timescales where θ inc decreases to 1 / e of its initial value are also generally several 10 5 yr for θ inc , 0 < 90 · (bottom left panel) and almost insensitive to θ inc , 0. Within a time of 1 . 0 t ν , the entire disks have been in complete alignment with the holes. In the bottom right panel, θ inc decreases with time even though θ inc , 0 > 90 · as was expected through the previous theoretical analysis. There are inflection points in the evolution curves of θ inc that correspond to when the disks break into two parts as illustrated in Figure 5. Generally, the timescales of the decaying rate for θ inc , 0 > 90 · (bottom right panel) are relatively longer compared to these for θ inc , 0 < 90 · (bottom left panel) due to the same reasons for free flows. However, because the disks are replenished and therefore the gravitomagnetic torques from the disks are not attenuated, the decreasing rates of θ inc are clearly more rapid than their counterparts (top right panel) for free flows. In summary, for fixed flows in which disks are continually fed, alignments always occur regardless of the initial inclination. The alignment timescale is of an order of 2 × 10 6 yr. For free flows in which there is no gas supply, both alignments and anti-alignments could occur. However, as a result of the notable depletion of the disk's surface density at a timescale", "pages": [ 6, 7 ] }, { "title": "5.3. Conditions for Anti-alignments", "content": "Previous sections show that anti-alignment between black holes and disks is possible only when the initial inclination angle is θ inc , 0 > 90 · for free flows. To assess how the final configuration of the system depends on the free parameters, in this section we evolve the equation set given different parameters fm and θ inc , 0 only for free flows. We note that the parameter fm determines the initial total angular momentum of the disks as in Equation (16). Figure 8 plots the evolution of inclination angle θ inc between J h and J d with fm / 10 -2 =0.1, 1, 1.5, 2, 3, 4, and 6, corresponding to the initial ratios J d / J h = 0 . 078, 0.78, 1.17, 1.56, 2.34, and 3.12, respectively. The initial inclination angles are θ inc , 0 = 160 · , 140 · , and 120 · from top to the bottom panel. Since gas diffuses outward through R out for free flows, there is an outflow of angular momentum at R out accordingly. Here J d includes these aspects of angular momentum. For comparison purposes, we also calculate the inclination angles in terms of the angular momentum of the disk just within R out, plotted by dotted lines in Figure 8. As can be seen, the evolution trends of θ inc depend both on the ratios J d / J h and the initial inclinations. Specifically, there are three prominent features as follows. 1) Consistent with the prediction in Section 3.3, the inclination angles θ inc generally go up when the initial ratios J d / J h < -cos θ inc , 0, but otherwise, drop off. We note that in the middle panel of Figure 8 the inclination of the system with an initial ratio J d / J h = 0 . 78, larger than -cos θ inc , 0 = 0 . 76, descends mildly at the very beginning and then increases to- d 180 · . This is because the ratio J d / J h rapidly decreases to J d / J h < -cos θ inc. 2) The inclination for θ inc , 0 = 160 · begins to enter θ inc < 90 · when the initial ratio J d / J h is larger than ∼ 3. This means that the system will approach alignment, and most importantly, the black holes undergo a transition from retrograde to prograde accretion. Similarly, the critical ratios for θ inc , 0 = 140 · and 120 · are ∼ 2 . 0 and 1 . 2, respectively. 3) Without replenishment, the disks are greatly depleted on a timescale of 10 6 yr. As a result, the gravitomagnetic torques are reduced with time and the inclination angles are almost frozen to values not equal to 0 · or 180 · at the end of calcu- ons. In other words, the systems hardly achieve full alignment or anti-alignment. On the basis of the angular momentum conservation of the system (free flows in the present study), King et al. (2005) proposed that anti-alignments will occur provided that the initial angle between the angular momenta of the hole and the disk satisfy J d / J h /lessorsimilar -2cos θ inc , 0. Applying this formula to the present cases, for θ inc , 0 = 160 · , anti-alignment occurs when the ratio obeys J d / J h /lessorsimilar 1 . 88 initially; for θ inc , 0 = 140 · and 120 · , the corresponding critical ratios are 1 . 53 and 1 . 0, respectively. Our results in Figure 8 are in considerable disagreement with these values. We ascribe such discrepancy to the neglect of the disk's depletion in the previous work. The ratio between the timescales for (anti-)alignments and the disk's depletion is of an order of unity for fiducial parameters from Figure 7. During accretion onto the black hole, the disk's mass is gradually consumed and the majority of the disk's angular momentum is transported to the large radius (Frank et al. 1992). Note that the gravitomagnetic torque is in proportion to R -3 , therefore the (anti-)alignment rate will be greatly reduced. This effect leads to some systems with ratios of J d / J h > -2cos θ inc , 0 still maintaining θ inc > 90 · at the end of the calculations in Figure 8.", "pages": [ 7, 8, 9 ] }, { "title": "6.1. Implications for Black Hole Spin", "content": "In addition to fueling black holes, accretion also adds angular momentum carried by matter to black holes and accordingly modifies their spin. How spin changes depends on the orbits at the inner edge of the accretion disks. Specifically, if the disk is aligned with the black hole, prograde accretion will spin up the hole; otherwise, retrograde accretion spin down the hole. The black hole will become maximally rotating ( a → 1) when its mass is doubled provided prograde or alternative retrograde accretion is preferably retained for a sufficient amount of time (Thorne 1974). In view of the episodic and random activities of black hole accretion (Wang et al. 2006; Li et al. 2010), it is crucial to model the spin evolution whether the accretion in each episode proceeds progradely or retrogradely. Our results obtained above imply that the configurations of the accretion disks depend on feeding them at the outer boundaries. If the disks are continually fed for a long enough lifetime (say, e.g., 10 7 yr), alignments always occur. In this case, prograde accretion is the dominated fashion. We expect that after a series of activities, black holes will be close to maximal rotation. On the other hand, if there is no gas feeding, both alignments and anti-alignments are possible, depending on the initial inclinations and the ratios J d / J h (see Section 5.3). One has to follow the probabilities of alignments and anti-alignments over episodes using the method presented here to determine spin evolution. An incorporation of the present calculations into semi-analytical models of galaxy formation and evolution will shed light on the existing studies in the field (e.g., Lagos et al. 2009; Fanidakis et al. 2011; Barausse 2012; Volonteri et al. 2012; Dotti et al. 2013), and provide more realistic information on SMBH spin. Note that black hole accretion only lasts a lifetime of ∼ 10 6 yr as a result of disk depletion without feeding. 3 Indeed, the lifetime of feeding disks directly determines the observable episodic lifetimes of AGNs. In this sense, observational measurements of the episodic lifetimes of AGNs can place useful constraints on the SMBH spin evolution. Unfortunately, thus far the episodic lifetime of AGNs remains elusive though some endeavor has been put forth (see a review of Martini 2004). The most compelling method-proximity effects estimated the episodic lifetimes in a broad range of ∼ 10 6 -10 8 yr, varying from study to study (Kirkman & Tytler 2008; Gonçalves et al. 2008; Furlanetto & Lidz 2011). Future observations from large sample quasar surveys with a variety of techniques would offer promising measurements on the 3 The lifetime can be generally estimated as follows: for black holes accreting at the Eddington limit ˙ M Edd ∼ 2 M 8 M /circledot yr -1 , where M 8 = M · / 10 8 M /circledot , the accretion disks with masses of ∼ ( H / R ) M · = 10 6 M 8 M /circledot will be consumed at ∼ 10 6 yr, where H / R ≈ 10 -2 . episodic lifetimes and help to understand the cosmological evolution of SMBH spin. On the other hand, we turn over the issue so as to constrain the episodic lifetimes of AGNs from the otherwise obtained SMBH spin evolution. Recently, quantifying the radiative efficiency of mass accretion through SMBH demography strongly suggests that SMBHs are spinning down with cosmic time since z ∼ 2 (Wang et al. 2009; Li et al. 2011, 2012; see also the discussion of Zhang et al. 2012). Combined with the present results, this indicates that the episodic lifetime of AGNs must not exceed several 10 6 yr and probably have cosmological evolution as well. More detailed modelings on the connections between SMBH spin evolution and the episodic lifetime of AGNs are highly deserved in future works. It is worth stressing that we presume black holes to be accreting at the Eddington limit in the calculations. The timescale for (anti-)alignments between the disks and the holes is in proportion to the reciprocal of the mass accretion rate (see Equation (11)). Therefore, if the presumed mass accretion rate is relaxed, the above proposed limit on the AGN lifetime changes in the same manner accordingly.", "pages": [ 9 ] }, { "title": "6.2. Uncertainties", "content": "In our calculations, the inner edge of the disk is fixed at R in = 6 R g. Considering that the inner portion of the disk within R w always remains aligned with the hole, this does not affect the gravitomagnetic torque on the hole and therefore the alignment rate. On the other hand, during the alignment course, the accreted mass and angular momentum are negligible with those of the black hole. Hence, the results presented in this paper are insensitive to the location of the inner boundary. For the sake of simplicity, the disk's outer edge (limited by the self-gravitating) is fixed at R out = 10 4 R g. In reality, the self-gravitating radius depends on the accretion rate, but generally lies at a range of ∼ 10 3 -10 4 R g for standard accretion disks (e.g., Goodman 2003; King et al. 2008). The location of R out just determines the ratio of J d / J h ∝ R 1 / 2 out as in Equation (16), but does not affect the gravitomagnetic torque and (anti-)alignment timescale as in Equations (5) and (11). The overall results remain unchanged. The viscosities for accretion disks are treated using the firstorder approximations given by Ogilvie (1999). The nonlinear fluid effects that appear for large amplitude warp would modify the expressions of the effective viscosities (Ogilvie 1999; Lodato & Price 2010). We expect this to become significant when the disks break as shown in the bottom panels of Figure 5. In this case, however, it is unclear whether the evolution equation set adopted here remains adequate. Dedicated investigations, including nonlinear fluid dynamics, are beyond the scope of the present study. We defer this to future sophisticated numerical simulations. Finally, we consider the accretion disks under the Newtonian timespace and treat the gravitomagnetic interaction in the weak-field limit (the Lense-Thirring precession). The general relativistic calculations are of somewhat less importance because the warp radius R w is far beyond the gravitational radius R g (see Equation (9)).", "pages": [ 9 ] }, { "title": "7. CONCLUSIONS", "content": "Wenumerically solve the evolution equation set of spinning black holes and their warped accretion disks. To mimic the realistic accretion processes, we consider two classes of accretion disks in terms of the outer boundary conditions: fixed flows for which the disk is continually fed with a preferred angular momentum through its outer edge; and free flows for which there is no gas supply and the disk diffuses freely at its outer edge (see Figure 1). Our main results are as follows. This research is supported by NSFC-11133006, 11173023, and 11233003, and a 973 project (2009CB824800). The numerical calculations in this work used the computer clusters at the Institute of High Energy Physics.", "pages": [ 9, 10 ] }, { "title": "A. DIFFERENCING THE EQUATION SET", "content": "We construct the differencing scheme of Equations (1) and (6) generally following Pringle (1992), but with some modifications. For the sake of completeness, we summarize our implementation here. The spatial domain is discretized into ( I + 2) logarithmic grid points: Ri = R in e i ∆ x , for i = 0 , ..., I + 1, with ∆ x the equal spacing of the logarithmic grid. The points R 0 and R I + 1 represent the inner and outer grid boundaries, respectively. If we define the advective velocity and use Equation (2), Equation (1) can be written as Notationally, we use a superscript n to denote the timestep and a subscript i to denote the spatial grid point. The differencing scheme of Equation (A2) is then built as follows (see also Bregman & Alexander 2012) where ∆ t is the timestep size, and the advective term is treated using upstream differencing: for V adv > 0, k = i -1; whereas for V adv < 0, k = i . The advective velocity is calculated by Similarly, the differencing scheme of Equation (6) is built as where the mass accretion rate onto the hole is calculated as ( ˙ M in) n = -2 π ( RVR Σ ) n 1 and the specific angular momentum at the inner edge is calculated as j n in = ( R 2 Ω ) n 1 /lscript n 1 . As described in Section 4, at the inner boundary, we take L n 0 = 0, and /lscript n 0 = /lscript n 1 ; at the outer boundary, for free flows , we take L n I + 1 = ( ν 1 L ) n I / ( ν 1) n I ; for fixed flows we take L n I + 1 = R 2 I + 1 Ω I + 1 Σ 0 R -p I + 1 (1 -√ R in / RI + 1) and fix /lscript n I + 1 = (sin θ inc , 0 , 0 , cos θ inc , 0). This enforces a coherent mass supply at a rate given roughly by Equation (12). The timestep size is adjusted for every step according to where", "pages": [ 10, 11 ] }, { "title": "B. VALIDITY OF THE NUMERICAL CODE", "content": "To verify the validity of our numerical code, we perform three tests. (1) We run the code upon a flat disk ( θ inc , 0 = 0 or π ) for a time of 1 . 5 t ν . The left panel of Figure 9 plots the surface density of the disk at the beginning and end of the calculations. There are tiny deviations less than 2% due to the dissipation of the numerical scheme. (2) For small warping, there exist analytical solutions if disks are continually fed at the outer edges, analogous to fixed flows defined in the present paper (Scheuer & Feiler 1996; Martin et al. 2007; Chen et al. 2009). We only evolve Equation (1) with a sufficient amount of time using the differencing scheme described above and obtain a steady shape of the disk. In the middle panel of Figure 9, we compare the numerically obtained inclination profile of the disk with the analytic solution given by Equation (24) in Martin et al. (2007). A complete match can be found. (3) Then initializing the disk with the above shape, we go on to evolve Equations (1) and (6) simultaneously to obtain the time-dependent inclination angle between the hole and the disk. The right panel of Figure 9 shows that our numerical calculations are in exact agreement with the analytical solutions given by Equation (52) in Martin et al. (2007). These three tests indicate a validity of our numerical code.", "pages": [ 11 ] }, { "title": "REFERENCES", "content": "Barausse, E. 2012, MNRAS, 423, 2533 Bardeen, J. M., & Petterson, J. A. 1975, ApJ, 195, L65 Benson, A. J., & Bower, R. 2010, MNRAS, 405, 1573 Bregman, M., & Alexander, T. 2012, ApJ, 748, 63 Chen, L., Wu, S., & Yuan, F. 2009, MNRAS, 398, 1900 Collin-Souffrin, S., & Dumont, A. M. 1990, A&A, 229, 292 Collin, S., & Zahn, J.-P. 1999, A&A, 344, 433 Dotti, M., Colpi, M., Pallini, S., Perego, A., & Volonteri, M. 2013, ApJ, 762, 68 Dotti, M., Volonteri, M., Perego, A., et al. 2010, MNRAS, 402, 682 Fanidakis, N., Baugh, C. M., Benson, A. J., et al. 2011, MNRAS, 410, 53 Frank, J., King, A., & Raine, D. 1992, Accretion Power in Astrophysics (Cambridge: Cambridge Univ. Press) Furlanetto, S. R., & Lidz, A. 2011, ApJ, 735, 117 Gallimore, J. F., Axon, D. J., O'Dea, C. P., Baum, S. A., & Pedlar, A. 2006, AJ, 132, 546 Gonçalves, T. S., Steidel, C. C., & Pettini, M. 2008, ApJ, 676, 816 Goodman, J. 2003, MNRAS, 339, 937 Hartle, J. B., 2003, Gravity: An Introduction to Einstein's General Relativity (San Francisco, CA: Addison-Wesley) Hopkins, P. F., Hernquist, L., Hayward, C. C., & Narayanan, D. 2012, MNRAS, 425, 1121 Kawakatu, N., & Wada, K. 2008, ApJ, 681, 73 King, A. R., Lubow, S. H., Ogilvie, G. I., & Pringle, J. E. 2005, MNRAS, 363, 49 King, A. R., & Pringle, J. E. 2006, MNRAS, 373, L90 King, A. R., Pringle, J. E., & Hofmann, J. A. 2008, MNRAS, 385, 1621 Kinney, A. L., Schmitt, H. R., Clarke, C. J., et al. 2000, ApJ, 537, 152 Kirkman, D., & Tytler, D. 2008, MNRAS, 391, 1457 Kumar, P., & Johnson, J. L. 2010, MNRAS, 404, 2170 Lagos, C. D. P., Padilla, N. D., & Cora, S. A. 2009, MNRAS, 395, 625 Larwood, J. D., Nelson, R. P., Papaloizou, J. C. B., & Terquem, C. 1996, MNRAS, 282, 597 Li, Y.-R., Ho, L. C., & Wang, J.-M. 2011, ApJ, 742, 33 Li, Y.-R., Wang, J.-M., & Ho, L. C. 2012, ApJ, 749, 187 Li, Y.-R., Wang, J.-M., Yuan, Y.-F., Hu, C., & Zhang, S. 2010, ApJ, 710, 878 Lodato, G., & Price, D. J. 2010, MNRAS, 405, 1212 Lodato, G., & Pringle, J. E. 2006, MNRAS, 368, 1196 Lubow, S. H., Ogilvie, G. I., & Pringle, J. E. 2002, MNRAS, 337, 706 Martin, R. G., Pringle, J. E., & Tout, C. A. 2007, MNRAS, 381, 1617 Martini, P. 2004, in Coevolution of Black Holes and Galaxies, ed. L. C. Ho (Carnegie Observatories Astrophysics Series, Vol. 1; Cambridge: Cambridge Univ. Press), 169 McKinney, J. C., Tchekhovskoy, A., & Blandford, R. D. 2013, Sci, 339, 49 Muñoz Marín, V. M., González Delgado, R. M., Schmitt, H. R., et al. 2007, AJ, 134, 648", "pages": [ 11 ] } ]

[ { "title": "ABSTRACT", "content": "Diffusion coefficients are usually used to describe the propagation of Cosmic Rays through the Universe. Whereas such transport parameters can be obtained from experiments in the Solar System, it is difficult to determine diffusion coefficients in the Milky Way or in external galaxies. Recently a value for the perpendicular diffusion coefficient in the nearby starburst halaxy NGC 253 has been proposed. In the present paper we reproduce this value theoretically by using an advanced analytical theory for perpendicular diffusion. Subject headings: cosmic rays - galaxies: magnetic fields - galaxies: individual (NGC 253) - galaxies: starburst", "pages": [ 1 ] }, { "title": "THEORETICAL EXPLANATION OF THE COSMIC RAY PERPENDICULAR DIFFUSION COEFFICIENT IN THE NEARBY STARBURST GALAXY NGC 253", "content": "K. Buffie 1 , V. Heesen 2 , and A. Shalchi 1 1 Department of Physics and Astronomy, University of Manitoba, Winnipeg, Manitoba R3T 2N2, Canada, [email protected] and 2 School for Physics and Astronomy, University of Southampton, Southampton SO17 1BJ, UK, [email protected] Draft version September 29, 2018", "pages": [ 1 ] }, { "title": "1. INTRODUCTION", "content": "Radio continuum observations of the nearby starburst galaxy NGC253 can be used to measure the distribution and transport of Cosmic Ray electrons (see Heesen et al. 2009a, 2011). The current understanding is that Cosmic Ray electrons are accelerated in the disk by shock waves induced by supernova explosions (Reynolds et al. 2012). The electrons are then transported away over their lifetime by either convection in a galactic wind or diffusion, where the two processes can be distinguished from their different typical transport length as a function of electron energy. The magnetic field in galaxies is normally dominated by the turbulent component, but averaged over spatial sizes larger than 1 kpc 1 (a typical resolution achieved with radio interferometers in external galaxies) ordered magnetic field components are detected by their linear polarisation from synchroton emission of Cosmic Ray electrons spiralling in the magnetic field. Observations suggest that galaxies in their disk plane are dominated by a magnetic field that is parallel to the disk and has a spiral pattern with a constant pitch angle of a few tens of degrees (Beck 2012). Further away from the disk plane at heights larger than 1 kpc the field becomes more vertical and the field lines open up further into the halo (Haverkorn & Heesen 2012). The motion of Cosmic Rays through the interplanetary or interstellar space is complicated due to their interaction with turbulent magnetic fields. Such fields lead to spatial diffusion of the energetic particles. In addition to the turbulent magnetic fields one can also observe a non-vanishing mean magnetic field which is in the case of the interstellar medium the Galactic magnetic field. This mean field breaks the symmetry of the physical system and one has to distinguish between diffusion along and across the Galactic magnetic field. The former process is also known as parallel diffusion whereas the latter process is called perpendicular transport. The theoretical investigation of Cosmic Ray transport has a long history (see, e.g., Schlickeiser 2002). Whereas the propagation of cosmic particles through the solar system seems to be well understood (see, e.g., Shalchi et al. 2006; Tautz & Shalchi 2012), it is unclear whether advanced transport theories can describe particle propagation with high accuracy in other systems like the Milky Way or external galaxies. At least parallel diffusion seems to be a process which can be described very well by more advanced diffusion theories. The measured decrease of the abundance ratio of secondary to primary Cosmic Ray nuclei as B/C and N/O at kinetic energies above 1 GeV/nucleon, implies a variation of the total column density as a function of rigidity R as (Swordy et al. 1990) λ ‖ ∝ R 0 . 6 where we have used the parallel mean free path λ ‖ . This behavior was explained by Shalchi & Schlickeiser (2005) who used a nonlinear diffusion theory. The behavior of low energy Cosmic Rays was also explained by using an extension of quasilinear theory (see Shalchi & Busching 2010). More complicated, however, is the process of perpendicular diffusion. In the recent years progress has been achieved. Shalchi (2010) developed an advanced theory for perpendicular diffusion which contains all known limits of diffusion theory. This theory is called the Unified Non-Linear Transport (UNLT) theory and provides the correct subdiffusive behavior for one-dimensional turbulence in agreement with the so-called theorem on reduced dimensionality (see Jokipii et al. 1993 and Jones et al. 1998) and computer simulations (see, e.g., Qin et al. 2002). Furthermore, the correct field line random walk limit can be derived in agreement with the nonlinear field line transport theory of Matthaeus et al. (1995). The theory discussed here was also tested numerically confirming its validity for solar wind parameters (see Tautz & Shalchi 2011). In analytical theories for energetic particle transport, the model of interstellar turbulence enters the corresponding nonlinear integral equation for the perpendicular diffusion coefficient. Shalchi et al. (2010) have therefore combined the UNLT theory with the turbulence correlation tensor proposed by Cho et al. (2002) based on Goldreich-Sridhar scaling (see Goldreich & Sridhar 1995). Later, however, Heesen et al. (2011) compared the analytical result obtained in the aforementioned paper with observations of NGC 253 and noticed a major difference. Using a multi-wavelength picture of the starburst nucleus they established that the nuclear outflow of hot X-ray emitting gas can be collimated by the magnetic field lines in the walls that surround the conical outflow. The magnetic field lines are aligned with the wall orientation as would be expected from compression by the hot gas in the outflow cone. An annotated figure of the multi-wavelength picture can be found in Figure 1, which shows the approximate location of the magnetic field filaments that are the walls of the outflow cone seen in projection. We have overlaid contours of the λ 3 cm radio continuum emission at a resolution of 150 pc that show the extent of the filaments extending along the northwestern outflow cone. The filaments are also observed at λλ 20 and 6 cm at the same spatial resolution, which offers to study the perpendicular diffusion coefficient as described in Section 3. These observations led Heesen et al. (2011) to the finding that the observed perpendicular diffusion coefficient is a factor of 10 too high in comparison to the predicted value by the theory. In the present paper we, therefore, revisit the problem of perpendicular transport of energetic particles in the nearby starbust galaxy NGC 253. We combine the approach of Shalchi et al. (2010) with more realistic particle and turbulence parameters to compute the perpendicular diffusion coefficient more accurately.", "pages": [ 1, 2 ] }, { "title": "2. THEORETICAL DESCRIPTION OF PERPENDICULAR DIFFUSION", "content": "The analytical description of diffusion across the mean magnetic field has a long history (see Shalchi 2009, for a review). Very recently an advanced theory for perpendicular transport has been developed (see Shalchi 2010) which shows good agreement with computer simulations performed for the parameters of the solar wind (see Tautz & Shalchi 2011). This theory is called the Unified Non-Linear Transport (UNLT) theory and provides a nonlinear integral equation for the perpendicular diffusion coefficient κ ⊥ . According to this equation κ ⊥ depends on magnetic turbulence described by the so-called magnetic correlation tensor. Therefore, the understanding of turbulence is crucial for the analytical description of cross field diffusion. A model for the magnetic correlation tensor in the interstellar medium was proposed by Cho et al. (2002) which is based on the GoldreichSridhar model (Goldreich & Sridhar 1995). For this specific turbulence model, Shalchi et al. (2010) derived the following equation where we have used the perpendicular mean free path which is related to the perpendicular diffusion coefficient via λ ⊥ = 3 κ ⊥ / 3. According to Eq. (1) the parameter λ ⊥ depends on the parallel mean free path λ ‖ = 3 κ ‖ /v ( v is the particle speed) and, therefore, there is a mutual influence of parallel and perpendicular transport. The parameter a 2 in Eq. (1) is an order one constant related to the probability that the particle is scattered away from the magnetic field lines (see, e.g., Dosch et al. 2009). In the present paper we set a 2 = 1 for simplicity. The perpendicular diffusion coefficient is also controlled by the characteristic length scale of the turbulence l , and the ratio E B ≡ δB/B 0 . The latter ratio is a measure for the turbulent magnetic field strength δB with respect to the mean field B 0 . The mean field is B 0 defined as the spatially averaged magnetic field vector on a scale corresponding to the length l , which is usually related to the correlation length of magnetic turbulence. The latter parameter is a characteristic scale for the correlation of magnetic fields at two different positions in space. Often this scale is directly proportional to the so-called bendover scale which denotes the turnover from the inertial range of the turbulence to the energy range. Furthermore, this scale is smaller than the largest scales on which turbulence can be observed. Shalchi et al. (2010) have solved Eq. (1) numerically to obtain values for the perpendicular diffusion coefficient in the interstellar medium. Heesen et al. (2011), however, compared the theoretical results with observations and found a disagreement. Below we, therefore, perform a parameter study to compute the perpendicular diffusion coefficient and compare it with observations. We will show that for the correct choice of input parameters, Eq. (1) is indeed able to reproduce the aforementioned observations.", "pages": [ 2 ] }, { "title": "3. OBSERVATIONS", "content": "The different parameters entering Eq. (1) as well is the perpendicular diffusion coefficient itself can be obtained from radio continuum observations of the nearby galaxy NGC 253 as described in Heesen et al. (2009a, 2011). In this section, we briefly summarize the techniques they have been using to quantify the Cosmic Ray transport. Radio continuum studies can be used to measure the length scale of Cosmic Ray diffusion. The time scale of Cosmic Ray electrons is determined by loss processes by which the electrons are losing their energy, such as synchrotron and inverse Compton radiation. The diffusion coefficient can be then calculated by where L diff is the diffusion length scale and τ is the electron lifetime. Depending on the magnetic field structure, the observed diffusion coefficient is either along ( κ ‖ ) the magnetic field or perpendicular to it ( κ ⊥ ). Ideally, one can measure the distance to the starformation sites, where the Cosmic Rays are accelerated and injected into the interstellar medium, to obtain the transport length scale. This is for instance the case when observing galaxies in so-called edge-on geometry, where the observer is located in or close to the disk plane of the galaxy (the inclination angle is close to 90 · ). Cosmic Ray acceleration in supernova remnants is confined to the relatively thin disc plane, where the formation of massive stars happens. The geometry in this case is thus simple: Cosmic Rays are transported away from the star formation sites over their lifetime and their transport length scale is equal to the vertical electron scaleheight. Typical scaleheights of galaxies are 1.8 kpc at observing wavelengths of λ 6 cm (Krause 2009), where longer wavelengths have larger scaleheights and shorter wavelengths smaller ones. Typical Cosmic Ray electron lifetimes in the interstellar medium are between 1 and 10 Myr. This basically determines the typical parallel diffusion coefficient in the ISM, where it is assumed that the halo magnetic field is opening up from the disk parallel to a vertical direction with increasing distance from the disk. Therefore, the vertical diffusion is predominantly along the magnetic field lines hence allowing us to measure κ ‖ . Heesen et al. (2009b) confirmed this for NGC 253 finding the characteristic X-shaped halo magnetic field with significant vertical components that is observed in a number of nearby galaxies (e.g., Soida et al. 2011). To measure the perpendicular diffusion coefficient is more difficult, because it requires to have variations of the Cosmic Ray distribution perpendicular to the magnetic field orientation. The observed radio continuum emission is mostly smooth in the disk-halo interface and filamentary structures are rare. Numerical MHD simulations suggest that the disk halo interface is dominated by filamentary magnetic fields (Breitschwerdt et al. 2012), but line-of-sight confusion and limited spatial resolution hampers their detection. An exception are starburst galaxies, where the spatially concentrated star formation activity results in exceptionally high radio continuum surface brightness, allowing us to employ high spatial resolution ( ≈ 100 pc). The spatially concentrated star formation can result into outflows of hot X-ray emitting gas in a galactic wind. The magnetic field is then concentrated and amplified by expansion of the hot gas until a pressure equilibrium is reached. This very specific geometry allowed Heesen et al. (2011) to measure the perpendicular diffusion across the magnetic field in the walls of the nuclear outflow cone in NGC 253. The latter authors found the width of radio continuum filaments weakly dependent on frequency and thus electron age, which can be interpreted as perpendicular Cosmic Ray diffusion. We note that they were able to measure the orientation of the ordered magnetic field along the filaments as it would be expected for a compression due to the hot X-ray emitting gas outflowing from the nuclear starburst. From the observations of particles with a magnetic rigidity of R = 3 × 10 12 Volt (equivalent to an electron energy of 3 GeV), they found a perpendicular diffusion coefficient of κ ⊥ = (2 . 6 ± 0 . 6) × 10 28 cm 2 s -1 . For the parallel diffusion coefficient one can assume κ ‖ = 1 . 0 × 10 29 cm 2 s -1 (again for an electron energy of 3GeV), as measured from the diffusion of Cosmic Ray electrons from the disk into the halo along the vertical halo magnetic field (Heesen et al. 2009a). It is important to measure the diffusion coefficients at roughly the same electron energy, because they are energy dependent. We have reduced the measured diffusion coefficient of Heesen et al. (2009a) to account for a possible contribution of convection. The southwestern halo is different from the northeastern one as indicated by the dependence of the electron scaleheight on the electron lifetime and by the different amount of extra-planar gas that is more abundant in the northeastern convective halo. However, we can not rule out the contribution of convection of Cosmic Rays by the disc wind and thus have lowered the diffusion coefficient assuming that both convection and diffusion contribute equally in the southwestern halo. Equation (1) depends on turbulence properties, namely the ratio of magnetic fields δB/B 0 and the scale l . According to Heesen et al. (2011) the equipartion estimate for the total field strength is B tot = 46 µ G. The ordered field strength B ord , ⊥ in the sky plane can be computed from the observed degree of polarization p of the synchrotron emission (e.g., Beck & Krause 2005): where p 0 is the intrinsic degree of polarization ( p 0 = (3 -3 α ) / (5 -3 α )) and q is the ratio of the isotropic turbulent field B turb to the ordered field B ord , ⊥ in the plane of the sky. For a non-thermal radio spectral index of α = -1 and a polarization degree of p = 0 . 21 we obtain B turb = 41 µ G and B ord , ⊥ = 21 µ G. For this particular case, the ordered magnetic field lies in the sky plane so that we find for the ordered magnetic field strength B 0 = 21 µ G. The fluctuations in the magnetic field strength are equal to the turbulent field field strength ( δB = B turb ) and thus we find for the the ratio δB/B 0 ≈ 2. We note that this is an upper limit as our spatial resolution of 150 pc that was available for the polarization measurements may be not be enough to resolve the filamentary magnetic fields. High resolution λ 20cm maps show a filament width of only 40 pc in radio continuum emission (see Figure 1), so that our measured ordered magnetic field strength B ord is only a lower limit. We have thus studied the implication of a lower value of δB/B 0 = 1 to indicate how it would change the theoretical expectation for the perpendicular diffusion coefficient. More difficult to estimate is the scale l . The width of the cone walls in which the magnetic fields are confined is equal or less than 40 pc (Heesen et al. 2011). This suggests that the upper value cannot be larger than about l ≈ 50 -100 pc. However the value for l is very uncertain and, therefore, we compute the perpendicular diffusion coefficient for a whole range of correlation lengths. Beck (2007) for instance suggested that the largest scales of turbulence are in the order of 10 -100 pc. Such largest scales, however, are not necessarily equal to the turbulence correlation scale. Actually they can be seen as the maximum of the scale l . Sometimes (see, e.g., Shalchi et al. 2009) it is assumed that the correlation length of interstellar turbulence is 1 pc but it could also be shorter. Therefore we compute the perpendicular diffusion coefficient for 1 pc ≤ l ≤ 1000 pc to explore the values of l which lead to agreement between theory and observations.", "pages": [ 2, 3, 4 ] }, { "title": "4. REPRODUCING THE OBSERVATIONS", "content": "In the following we solve Eq. (1) numerically for different turbulence scales l . To relate the mean free paths to the spatial diffusion coefficients we employ λ ‖ = 3 κ ‖ /v and λ ⊥ = 3 κ ⊥ /v . In the latter relations we replace the particle speed v by the speed of light c since we deal with relativistic particles. As described in the previous sections we set κ ‖ = 1 . 0 × 10 29 cm 2 s -1 and a 2 = 1 in our equation for the perpendicular diffusion coefficient. We have calculated the latter transport parameter for δB/B 0 = 1 and δB/B 0 = 2. Our theoretical results are shown together with the observations in Fig. 2. The observational value of the perpendicular diffusion coefficient is (2 . 6 ± 0 . 6) × 10 28 cm 2 s -1 for a correlation length of approximately 50 pc. The theoretical perpendicular diffusion coefficient is calculated for different values of the correlation length. For a correlation length of 50pc we found a κ ⊥ which is very close to the observations depending of the value of δB/B 0 . We note that the perpendicular diffusion coefficient is only weakly dependent on the turbulent correlation scale above a scale of approximately 10pc. The observed diffusion coefficient seems to rule out correlation lengths smaller than 1pc, but does not constrain very well the upper limit. It is reasonable to assume that the correlation length is smaller than the largest scales which are in the order of 10 -100 pc (see Beck 2007). We note that the observed value of the perpendicular diffusion coefficient is quite well reproduced by the measured value for δB/B 0 , which is in the range between 1 and 2 as argued in Section 3.", "pages": [ 4 ] }, { "title": "5. SUMMARY AND CONCLUSION", "content": "A fundamental process in astrophysics is the propagation of Cosmic Rays through the Universe. Whereas it seems that we understand the motion of energetic particles in the solar system very well (see, e.g., Shalchi et al. 2006; Tautz & Shalchi 2012), it is still not clear how the transport parameters look like in the interstellar space of our own or external galaxies. Shalchi et al. (2010) have calculated the perpendicular diffusion coefficient for interstellar turbulence parameters. Heesen et al. (2009a, 2011) obtained measurements for the parallel and perpendicular diffusion coefficient in the nearby starbust galaxy NGC253. They compared their results with the theoretical values presented in Shalchi (2010) and found one order of magnitude between the two results. However, the perpendicular diffusion coefficient obtained in Shalchi et al. (2010) was obtained for a different set of parameters. Especially the parallel diffusion coefficient used in the latter paper does not agree with the value obtained by the observers. Thus, in the current paper we combined the approach of Shalchi et al. (2010) with parameter sets tailored to the observations of NGC 253, such as the parallel diffusion coefficient and magnetic fields strengths. Furthermore, we computed the parameter κ ⊥ for different values of the correlation length l as this value is known only to a small degree of certainty. Our results now largely reconcile the observed values with the theoretical predicted ones as visualized in Fig. 2. For correlation lengths between 5 and 100 pc, the theoretical value and the observed one agree within a factor of 2. This agreement is quite good given the difficulties in the observational measurements of diffusion coefficients for Cosmic Ray electrons from radio continuum observations in galaxies. We can confidentially exclude correlation lengths smaller than 1 pc, which would result in too small perpendicular diffusion coefficients. We have done this study for only one galaxy, because so far measurements particularly of the vertical diffusion coefficient are very sparse in galaxies. However, the agreement of the observations with the theoretical prediction lends to some degree support both to the theoretical description of Cosmic Ray diffusion as described in this paper and to the observational attempts to measure it from the Cosmic Ray electron distribution. Support by the Natural Sciences and Engineering Research Council (NSERC) of Canada is acknowledged. V.H. is funded as postdoctoral research assistant by the UK's Science and Technology Facilities Council (STFC).", "pages": [ 4 ] } ]

[ { "title": "ABSTRACT", "content": "Massive stars at redshifts z ∼ > 6 are predicted to have played a pivotal role in cosmological reionization as luminous sources of ultra-violet (UV) photons. However, the remnants of these massive stars could be equally important as X-ray luminous ( L X ∼ 10 38 erg s -1 ) high-mass X-ray binaries (HMXBs). Because the absorption cross section of neutral hydrogen decreases sharply with photon energy ( σ ∝ E -3 ), X-rays can escape more freely than UV photons from the star-forming regions in which they are produced, allowing HMXBs to make a potentially significant contribution to the ionizing X-ray background during reionization. In this paper, we explore the ionizing power of HMXBs at redshifts z ∼ > 6 using a Monte Carlo model for a coeval stellar population of main sequence stars and HMXBs. Using the archetypal Galactic HMXB Cygnus X-1 as our template, we propose a composite HMXB spectral energy distribution consisting of black-body and power-law components, whose contributions depend on the accretion state of the system. We determine the time-dependent ionizing power of a combined population of UV-luminous stars and X-ray luminous HMXBs, and deduce fitting formulae for the boost in the population's ionizing power arising from HMXBs; these fits allow for simple implementation of HMXB feedback in numerical simulations. Based on this analysis, we estimate the contribution of high redshift HMXBs to the present-day soft X-ray background, and we show that it is a factor of ∼ 100 -1000 smaller than the observed limit. Finally, we discuss the implications of our results for the role of HMXBs in reionization and in high redshift galaxy formation. Subject headings: galaxies: formation - X-rays: binaries - cosmology:theory", "pages": [ 1 ] }, { "title": "FEEDBACK FROM HIGH-MASS X-RAY BINARIES ON THE HIGH REDSHIFT INTERGALACTIC MEDIUM : MODEL SPECTRA", "content": "Chris Power 1,4 , Gillian James 2 , Celine Combet 3 & Graham Wynn 2 2 Department of Physics & Astronomy, University of Leicester, Leicester, LE1 7RH, UK ARC Centre of Excellence for All-Sky Astrophysics (CAASTRO) ApJ, accepted 23/11/2012", "pages": [ 1 ] }, { "title": "1. INTRODUCTION", "content": "There is strong and compelling observational evidence that the Universe underwent an 'Epoch of Reionization' within the first ∼ 1 billion years after the Big Bang (e.g. Ouchi et al. 2010; Mesinger 2010; Shull et al. 2012; McGreer et al. 2011). During this period, the cosmic abundance of neutral hydrogen was 're-ionized' by a background of ionizing ultra-violet (UV) and X-ray radiation produced by the first generation of stars and galaxies (e.g. Barkana & Loeb 2007; Robertson et al. 2010). The precise nature of these sources remains an outstanding problem, but it can be argued reasonably that massive stars (M ∗ ∼ > 8 M /circledot ) must have been important (e.g. Wise & Abel 2008; Wise 2012). Massive stars have fleeting main sequence (MS) lives ( ∼ 10 Myrs), but as extremely luminous sources of hydrogen-ionizing UV photons (see, for example, Schaerer 2003) they are expected to play a crucial role in reionizing the Universe (e.g. Wyithe & Loeb 2003; Sokasian et al. 2004; Wise & Abel 2008). However, their predicted contribution to reionization depends on the ease with which UV photons can escape into the intergalactic medium (IGM), which depends on various factors, principally the clumpiness of the IGM (e.g. Pawlik et al. 2009). However, testing such predictions observationally is inherently challenging and a [email protected] clear consensus as to what the sources of reionization are is yet to emerge. For example, observations of star-forming galaxies at redshifts 7 ∼ < z ∼ < 10 hint that implied star formation rates are insufficient to produce enough massive stars to reionize the Universe by z /similarequal 6, without including a population of galaxies below the detection limits and relaxing assumptions about the escape fraction of ionizing photons and extrapolated star formation rates (e.g. Bunker et al. 2010; McLure et al. 2010; Finkelstein et al. 2010; Gonz'alez et al. 2010). In contrast, observations of high redshift galaxies also suggest that the star formation rate at z ∼ 9 could be as high as that at z =0 (cf. Ishida et al. 2011) and that the observable population of galaxies between 6 ∼ < z ∼ < 8 can sustain a fully reionized IGM at z =6 if the average escape fraction of ionizing photons is ∼ 30% (cf. Finkelstein et al. 2012; see also Bouwens et al. 2012). A similar conclusion is drawn by Kistler et al. (2009), who argue that a sufficient number of massive stars could have formed to reionize the Universe by using high-redshift gamma ray bursts as a proxy for the integrated star formation rate between 6 ∼ < z ∼ < 8. It is, however, worth noting that massive stars can continue to ionize during their post-MS lives, predominantly as sources of X-rays. Oh (2001) noted that Xray emission from supernovae in star forming regions in the high-redshift Universe should be large and compara- ble energetically to UV emission. The role of supernovae has been explored further by Johnson & Khochfar (2011) who examined how strong shocking of the ISM associated with supernovae leads to the production of ionizing photons with harder spectra and larger escape fractions than UV photons. They estimate that such X-rays can boost - briefly - the ionizing power of a massive star by ∼ 10%. Glover & Brand (2003) considered not only X-rays from supernovae but also from X-ray binaries in their examination of the contribution of star formation to the build up of the high redshift X- ray background. As Justham & Schawinski (2012) have demonstrated in their recent study, X-ray binaries are likely to be important sources of feedback across cosmic time (see also Fragos et al. 2012). In particular, we expect this to be the case at early times; in Power et al. (2009) (hereafter P09) we explored how high mass X-ray binaries (HMXBs) ∗ in primordial globular clusters at high redshifts could boost the UV ionizing power of the cluster. We found that harder ionizing spectra combined with enhanced escape fractions for X-rays implied that HMXBs could be just as efficient at ionizing the IGM as their MS progenitors. Similar ideas have been explored in Mirabel et al. (2011) and Wheeler & Johnson (2011), in which the primary is a stellar mass black hole, as well as by McQuinn (2012) in his census of potential sources of reionization. The analysis in P09 focused on the ionizing power of HMXBs in globular clusters because (i) the inferred ages of metal poor globular clusters imply that they formed at z ∼ > 6 (cf. Brodie & Strader 2006); (ii) the relationship between the initial mass functions (IMF) of stars and the dynamical evolution of clusters is well understood (cf. Vesperini & Heggie 1997); and (iii) the escape fraction for UV photons was likely to be large, assuming that globular clusters followed similar orbits then to the ones they follow now (cf. Ricotti 2002, who explored the ionizing power of massive stars in globular clusters). This last point is important because the large UV escape fraction allowed P09 to carry out a straightforward comparison of a globular cluster's UV and X-ray ionizing power. However, as noted in P09, HMXBs are likely to be a generic by-product of high mass star formation (cf. Helfand & Moran 2001), and should form as well in gas-rich galaxy discs as in young globular clusters. This is consistent with observations of star forming galaxies at z ∼ < 1 that show that luminous compact X-ray sources - with properties mirroring those of HMXBs in our Galaxy - are good indicators of recent star formation activity (see Figure 10 of Mineo et al. 2012a and Figure 2 of Mineo et al. 2012b). In this paper, we extend the analysis presented in P09 to study HMXBs as generic sources of ionizing radiation. Here our focus is on refining our treatment of their spectral energy distribution. Rather than assuming a simple power-law form for the HMXB spectrum (as we did in P09), we use the spectrum of Cygnus X-1, the archetypal Galactic HMXB (e.g. Remillard & McClintock 2006), as our template. We assess how this impacts on the ionizing power of a coeval stellar population over time and we quantify how the ionizing luminosity of the population is boosted by the presence of HMXBs. Finally, we estimate the possible contribution of HMXBs to the present-day soft X-ray background, assuming both our new template spectrum and power-law spectra of the kind that have been used in previous studies (e.g. P09). The structure of the remainder of this paper is as follows. In § 2 we describe our time-dependent Monte Carlo model for the spectral energy distribution of a coeval population of stars in which HMXBs are forming. In § 3 we present results for the time evolution of the ionizing power and spectral energy distribution of the population, and we quantify how HMXBs enhance the ionizing power of the stellar population. In § 4, we show that the X-ray luminosity produced in our model does not violate observed limits on the soft X-ray background. Finally, we summarize our results in § 5 and comment on their implications for cosmological reionization and high redshift galaxy formation.", "pages": [ 1, 2 ] }, { "title": "2.1. Modelling HMXBs in a Single Stellar Population", "content": "As in P09, we set up a Monte Carlo model of a stellar population, assumed to form in a single instantaneous burst, and follow the evolution of the massive stars over the first 250 million years, through their MS lives and into the HMXB phase. The main features of our model can be summarized as follows; THE MASSIVE STAR POPULATION: We assume the IMF of Kroupa (2001) as our fiducial case, with stellar masses spanning the range 0 . 01 ≤ M ∗ / M /circledot ≤ 100, but we also verify our results for the Salpeter (1955) and topheavy Chabrier (2001) IMFs. All stars with M ∗ ≥ 8M /circledot are assumed to form in binaries - these are the progenitor population from which the HMXBs are drawn. Initial binary parameters are assigned following Dray (2006) - that is, companion masses are drawn from a uniform distribution between 0 . 01 ≤ M ∗ / M /circledot ≤ 100 and orbital periods are distributed uniformly in logarithm between 1 and 10 4 days. Massive star lifetimes are estimated using the results of Marigo et al. (2001, 2003), Schaerer et al. (1993) and Meynet & Maeder (2000) for metallicities of Z = 0, 0.008 and 0.02 (i.e. solar metallicity) respectively; we explore the Z = 0 case in our results section. THE HMXB POPULATION: We assume that HMXBs form from binaries in which the initial MS mass of the primary exceeds M ∗ ∼ 8M /circledot , the threshold for neutron star formation (cf. Figure 1 of Heger et al. 2003a), and the donor (i.e. secondary) mass lies in the range M ∗ ≥ 3M /circledot † . Once the primary goes supernova, we estimate the remnant mass using Figure 3 of Heger et al. (2003a). The binary will be disrupted if it loses more than half its mass in the supernova - which, for our fiducial Kroupa IMF, implies that approximately 30% of bi- † This donor mass is lower than the usual definition of HMXBs - donor OB stars (cf. Table 1 of Fabbiano 2006) with typical masses ∼ > 10M /circledot (e.g. Justham & Schawinski 2012). However, it is reasonable to include these intermediate mass X-ray binaries because they are sufficiently luminous to contribute to the X-ray ionizing power of the stellar population and their MS lifetimes are of order ∼ 10 8 yrs. naries survive, thereby setting an upper limit to the total number of HMXBs that can potentially form. Following P09, we draw a survival fraction f sur of this ∼ 30% at random and consider them as HMXBs; f sur captures the various uncertainties that prevent massive binaries from evolving into HMXBs. We assume that HMXBs are active until the companion star evolves off the main sequence and goes supernova. We have noted already that the binaries that survive are more likely to host black holes than neutron stars, especially in low metallicity systems where the formation rate of HMXBs in which the primary is a black hole could be a factor of ∼ 10 higher than at Solar metallicity (cf. Linden et al. 2010; Justham & Schawinski 2012). There is also observational evidence that the black hole mass is likely to be larger in lower metallicity systems (cf. Crowther et al. 2010). This should, in principle, shape our HMXB luminosity function (cf. Dray 2006). However, we make the simplifying assumption to draw HMXB luminosities from a Weibull distribution with a peak luminosity of L X ∼ 10 38 erg s -1 but capped such that they do not accrete at superEddington rates; this sets an upper limit of approximately L X /similarequal 1 . 26 × 10 38 (M / M /circledot ) erg s -1 on the luminosity of an HMXB with primary mass M. This approach gives a distribution that is consistent with the luminosities of compact X-ray sources in nearby galaxies whose X-ray binary populations are dominated by HMXBs (cf. Figure 1 of Gilfanov et al. 2004); see P09 for further discussion of this point.", "pages": [ 2, 3 ] }, { "title": "2.2. Spectral Energy Distribution & Time Dependence", "content": "We split the spectral energy distribution of the stellar population into two components: MAIN SEQUENCE (MS) STARS: Each star is assumed to radiate as a black body with an effective temperature of where R ∗ /R /circledot = (M ∗ / M /circledot ) 0 . 8 is stellar radius and σ SB is the Stefan-Boltzmann constant. Stellar luminosity L ∗ is assumed to follow a mass-luminosity relation of the form where α , which governs the amplitude of the relation, and β , which governs its slope, are observed to depend on stellar mass (e.g. Henry 2004); their values and the stellar mass range in which they are applicable are summarized in Table 1. Empirically there is a wealth of evidence that the rate at which stellar luminosity varies with mass ( β = d log L/d log M ) decreases with increasing mass M ∗ . For example, Malkov (2007) find β ∼ 4 . 1 at M ∼ 1 M /circledot to β ∼ 3 . 2 at M ∼ 20 M /circledot (see their Table 6), while Vitrichenko et al. (2007) report that β ∼ 2 . 76 over the mass range 20 ∼ < M/ M /circledot ∼ < 50. Our values of α and β provide a good approximation to the functional form presented in Malkov (2007) for stellar masses M ∗ ∼ < 50 M /circledot . Above 50 M /circledot we must extrapolate because observationally inferred data are few; we assume that the relation is slightly shallower than it is below 50 M /circledot . This is an uncertainty, but it has negligible effect on our results because relatively few stars are formed in this mass range and their lifetimes are short. We find that the massluminosity relation can be well approximated by a 2 nd -order polynomial of the form, Here the coefficients have the values c 0 =-0.04172, c 1 =4.4954 and c 2 =-0.6041, and log indicates logarithm base 10. HIGH MASS X-RAY BINARIES: We model HMXB spectra using the archetypal Galactic HMXB Cygnus X1 as our template. Cygnus X-1 is the brightest HMXB in the Galaxy and it has been studied in exquisite detail (see, for example, the recent review by Remillard & McClintock 2006). It consists of a black hole of mass 8 . 7 ± 0 . 8M /circledot (Shaposhnikov & Titarchuk 2007) and a super-giant companion (HDE 226868). Its spectrum is observed to fluctuate between distinct lowhard and high-soft states, examples of which can be found in Gierli'nski et al. (1999). This suggests that we should adopt distinct spectral shapes for low-hard and high-soft states. However, we understand neither what sets the duration of these two states - for example, Cygnus X-1 is observed to be predominantly in its low hard state, but analogous systems such as LMC X-3 (Val-Baker et al. 2007) appear to spend more time in their high soft state - nor how the nature and duration of these states depend on factors such as metallicity. For this reason, we make the simplifying assumption that HMXB spectra do not vary in time and instead introduce a threshold in X-ray luminosity of 10 37 erg s -1 , above (below) which a source is in a soft (hard) state. In particular, we model the luminosities of the black body and power-law components matches that of Cygnus X-1, while the black body temperature is calculated by assuming ( L X /L CygX -1 ) = ( T/T CygX -1 ) 4 .", "pages": [ 3, 4 ] }, { "title": "3. RESULTS", "content": "In the following subsections, we show how the ionizing power of a stellar population formed in an instantaneous burst evolves over the first 150 Myrs after formation ( § 3.1). We compare and contrast results in the presence and absence of HMXBs and, where appropriate, we comment on the sensitivity of our results to the assumed value of the survival fraction ( f sur ) and our choice of IMF.", "pages": [ 4 ] }, { "title": "3.1. Ionizing Power over Time", "content": "In Figure 1 we show the spectral energy distribution of the fiducial stellar population (i.e. Kroupa IMF, f sur = 1 and N =10 6 ) as a function of time - a composite of MS stars, which we model individually as black bodies, and HMXBs, which we model as a combination of black bodies and power-law components. The amplitude of the black body corresponding to MS stars decreases with time, while its peak shifts to lower energies; this reflects the evolution of the most massive stars in the stellar population - which dominate the UV-luminosity - off the MS. Over the same period the amplitudes of both the black body and power-law components of the HMXBs decrease with time. The same qualitative trends can be seen if we vary the IMF from Kroupa to Chabrier or Salpeter - there is a systematic increase (decrease) in the amplitude for the Chabrier (Salpeter) IMF, which reflects an increase (decrease) in the proportion of massive stars that form. If we vary the size of population between N =10 4 to N =10 6 the amplitude varies linearly with the size of population. Varying the survival fraction f sur between 0.01 and 1 suppresses the amplitude of the HMXB black body and power- law contribution while leaving the black body contribution from the MS unchanged. In Figure 2 we estimate how much of this energy (instantaneous in the upper panel, cumulative in the lower panel) is available to ionize neutral atomic hydrogen (HI) in the IGM as a function of time. We evaluate this by integrating the spectra plotted in Figure 1 between a lower limit of 13.6 eV, the minimum photon energy required to ionize atomic hydrogen, and an upper limit of E lim , the energy above which atomic hydrogen becomes transparent to hard X-rays ‡ ; for the redshift range we consider ( z ∼ > 6), E lim ∼ > 1 keV. Provided f sur > 0 . 1, we find that HMXBs dominate the ionizing power of the stellar population after ∼ 20 Myrs. Interestingly we note that the X-ray ionizing power shows a pronounced bump between 20 ∼ < t ∼ < 40 Myrs. This is an imprint of our definition of an HMXB as a binary system in which the mass of the primary must exceed ∼ 8M /circledot at the end of its MS lifetime - the threshold for neutron star formation. Because the MS lifetime of an 8M /circledot is ∼ 25 Myrs (cf. Marigo et al. 2001, 2003) and because statistically we expect more HMXBs to be drawn from stars close to this 8M /circledot limit, we expect an excess in HMXBs to form around this time and to produce a bump in the ionizing power. After this time, HMXBs can no longer form and the ionizing power comes from a population that is in terminal decline. We quantify how HMXBs enhance the ionizing power of the stellar population by measuring the boost factor f HMXB = L HMXB /L MS , which compares the ionizing luminosity of the MS stars only ( L MS ) and of the combined MS stars and HMXBs ( L HMXB ). In Figure 3 we show how f HMXB varies with time for a stellar population of 10 6 stars and f sur =1 assuming Kroupa, Salpeter and Chabrier IMFs respectively (solid curves, left to right panels). Because the ionizing power of the MS component declines sharply after ∼ 10 -20 Myrs as the most massive stars end their lives, f HMXB grows steadily with ‡ As in P09, we estimate E lim by requiring that σ ( E lim ) = c -1 H ( z ) n H ( z ) -1 where σ ( E ) is the ionization cross section of neutral hydrogen, H ( z ) is the Hubble parameter at redshift z , n H ( z ) is the mean number density of hydrogen and c is the speed of light. time, peaking at f HMXB ∼ 1600 after t ∼ 180 Myrs before dropping off abruptly. The dashed curves in Figure 3 show fits to the numerical results that agree to better than 10%. We fit the results with 2 formulae, below and above the bump at 30 Myr. These are given by where t is population age in Myr, and are valid up to approximately 180 Myr, above which HMXB contribution falls rapidly to 0 (and f HMXB drops to 1). The parameters p , q and a to d are tabulated in Table 2.", "pages": [ 4, 5 ] }, { "title": "4. COSMIC SOFT X-RAY BACKGROUND", "content": "The Universe at z ∼ > 6 is effectively transparent to photons with energies ∼ > 1 keV. This means that hard photons remain unabsorbed by neutral hydrogen and so a hard X-ray background was built up in the early Universe that, through redshifting, contributes to the present day soft X-ray background (SXRB), the mean specific background intensity of soft Xrays. We wish to determine the contribution made to the SXRB by the HMXB population present at z ∼ > 6 and whether or not their contribution violates observed limits. Equivalent calculations have been performed by Dijkstra et al. (2004), Salvaterra et al. (2005) and Zaroubi et al. (2007) for mini-quasars in the early Universe. However, we adopt the approach of Dijkstra et al. (2012), who looked at the contribution of star-forming galaxies to both the soft and hard X-ray backgrounds from high redshift to the present day. In what follows, we assume values of H 0 = 100 h kms -1 Mpc -1 for the Hubble parameter with h =0.71, Ω M = 0 . 266 for the matter density parameter, and Ω Λ = 0 . 734 for the dark energy density parameter (cf. Komatsu et al. 2011), and we take as our observed limit on the SXRB the value of ∼ 3 . 4 ± 1 . 4 × 10 -13 erg s -1 cm -1 deg -2 obtained by Hickox & Markevitch (2007) for the excess flux between 1 -2 keV, which corresponds to hard X-rays at z ∼ > 6. We estimate the contribution made by high redshift HMXBs to the observed SXRB by evaluating where ∆Ω ∼ 3 × 10 -4 sr deg -2 ; c is the speed of light; z max is the earliest (unknown) redshift at which HMXB formation proceeds; E ( z ) = √ Ω M (1 + z ) 3 +Ω Λ ; ˙ ρ ∗ is the comoving star formation rate density in units of M /circledot yr -1 Mpc -3 ; and L X ( z ) is the 'K-corrected' X-ray luminosity per unit star formation rate in the observed energy range X = E 1 -E 2 . For ˙ ρ ∗ we explore four possibilities, spanning the range of potential star formation histories. We take (i) a fixed value of ˙ ρ ∗ ( z = 6) /similarequal 0 . 002 M /circledot yr -1 Mpc -1 , as measured at z =6 by Bunker et al. (2010); (ii) a fixed value of ˙ ρ ∗ ( z = 6) /similarequal 0 . 17 M /circledot yr -1 Mpc -1 , as measured at z =6 by Tanvir et al. (2012); (iii) a fixed value of ˙ ρ ∗ ( z = 6) /similarequal 0 . 02 M /circledot yr -1 Mpc -1 , estimated at z =6 using the functional form of Hopkins & Beacom (2006), with parameters u =0.017, v =0.13, w =3.3 and x =5.3 from Cole et al. (2001); and (iv) a redshift-dependent value of ˙ ρ ∗ , estimated using Equation 6. For L X ( z ) we follow Dijkstra et al. (2012) and recast it as L X ( z ) = c X K ( z ). Here we assume the range of values for 2 . 6 ≤ c X / 10 39 ergs -1 (M /circledot yr -1 ) -1 ≤ 3 . 7 measured by Mineo et al. (2012b) for compact resolved X-ray sources in galaxies (lower limit) and unresolved galaxies in the Chandra Deep Field North and ultra-luminous infra-red galaxies (upper limit); and we calculate where F ( E ) represents the spectrum of a single HMXB, E 1 =1 keV and E 2 =2 keV. Examples of the spectra we consider are shown in Figure 4 - the solid curve corresponds to the cumulative spectrum derived from HMXBs in the fiducial stellar population described in § 3, as measured at t =50 Myrs § , while the dashed, dotted and dotted-dashed curves correspond to power-law spectra ( ∝ E -α ) that have been normalized such that the total energy emitted between 1 and 2 keV is the same for both composite and power-law spectra. The units are arbitrary - it is the shape, not the amplitude, that is important when evaluating Equation 7 - and it is worth noting how poorly simple power-law spectra approximate the shape of the more realistic composite spectrum. In all cases we fix our lower bound in Equation 5 at z = 6 and allow our upper bound z max to vary between z = 20 and z =50. This corresponds to a range in age of the Universe of between 48 and ∼ 180 Myrs old ( z =50 and 20 respectively), which is likely to bracket the redshifts when the first generation of dark matter halos were sufficiently massive to support cooling and subsequent star formation formed (cf. Figure 1 of Glover 2005). However, we note that our results are insensitive to z max for the range of values that we consider. Evaluating Equation (5), we find that the contribution of HMXBs to the soft X-ray background is of order ∼ 5 × 10 -16 erg s -1 cm -2 deg -2 , depending on what precisely is assumed for the star formation rate density and spectrum - roughly a factor of ∼ 1000 smaller than the observed limit of ∼ 3 . 4 × 10 -13 erg s -1 cm -2 deg -2 measured by Hickox & Markevitch (2007). In general, the higher the star formation rate density, the larger the contribution that is possible, because of our assumption that HMXB formation tracks (massive) star formation. If power-law spectra are adopted, we find that SXRB ∼ 2 . 15 × 10 -16 erg s -1 cm -2 deg -2 for α =0 compared to ∼ 11 . 4 × 10 -16 erg s -1 cm -2 deg -2 for α =2; that is, the softer the spectrum, the larger the contribution HMXBs can make. Nevertheless, this contribution is still a factor of ∼ 100 smaller than the observed limit.", "pages": [ 5, 6 ] }, { "title": "5. SUMMARY", "content": "There are compelling astrophysical reasons to expect that HMXBs, which are observed to be a natural byproduct of massive star formation (e.g. Mineo et al. 2012a; Dijkstra et al. 2012), could be an important source of feedback in galaxy formation over cosmic time (e.g. Justham & Schawinski 2012). Because the cross section for photons to be absorbed by neutral hydrogen decreases strongly with energy as E -3 γ , X-rays from HMXBs can effectively diffuse through the IGM and deposit their energy at relatively large distances from the stellar population, in an act of non-local feedback (a similar effect is expected for mini-quasars; see, for example, Zaroubi & Silk 2005). This contrasts with lower energy UV photons, whose correspondingly larger absorption cross section means that they ionize the IGM in the vicinity of the stellar population, in an act of local feedback. In this paper, we build on the work of Power et al. (2009) in which we examined the X-ray luminosity and effective ionizing power ¶ of a coeval population of stars and HMXBs at z ∼ > 6. We do not compare the HMXB contribution to the ionizing X-ray background present during reionization to the contribution from other source populations, such as mini-quasars and accreting super-massive black holes, in this paper. Such a comparison has been carried out by McQuinn (2012) who concludes that HMXBs were likely to be a minor contributor to reionization. This may be the case - although the conclusion depends on the assumed spectral properties of the various sources considered, which in most cases are uncertain - but our work implies that HMXBs can have a significant impact on the ionization structure and heating rate of the IGM at z ∼ > 6, extending the volumes of partially ionized hydrogen by factors of ∼ > 1000. This will have a direct impact on the efficiency of galaxy formation by, for example, suppressing the collapse of gas onto low-mass dark matter halos (e.g. Machacek et al. 2003; Tanaka et al. 2012) and modifying the cooling rate of hot gas in galaxy halos (e.g. Cantalupo 2010). We will demonstrate this explicitly in a forthcoming paper (James et al., in preparation), in which we will use the more realistic composite spectrum in a radiative transfer calculation to assess the relative importance of such HMXB feedback in heating the high redshift IGM and the potential HMXB contribution to both cosmological reionization and galaxy formation in general.", "pages": [ 6, 7 ] }, { "title": "ACKNOWLEDGEMENTS", "content": "We thank the anonymous referee for their insightful report that has helped improve this paper. CP, GJ, CC and GW acknowledge the support of the theoretical astrophysics STFC rolling grant at the University of Leicester. Part of the research presented in this paper was undertaken as part of the Survey Simulation Pipeline (SSimPL; http://www.astronomy.swin.edu.au/SSimPL/ ). The Centre for All-Sky Astrophysics is an Australian Research Council Centre of Excellence, funded by grant CE11E0090.", "pages": [ 7 ] }, { "title": "REFERENCES", "content": "Dijkstra, M., Gilfanov, M., Loeb, A., & Sunyaev, R. 2012, MNRAS, 421, 213 Helfand, D. J., & Moran, E. C. 2001, ApJ, 554, 27 Hopkins, A. M., & Beacom, J. F. 2006, ApJ, 651, 142 Mineo, S., Gilfanov, M., & Sunyaev, R. 2012b, MNRAS, 426, 1870", "pages": [ 7 ] } ]

[ { "title": "ABSTRACT", "content": "We extend the one-dimensional, two-zone models of long-term protostellar disk evolution with infall of Zhu et al. to consider the potential effects of a finite viscosity in regions where the ionization is too low for the magnetorotational instability (MRI) to operate (the 'dead zone'). Wefindthat the presence of a small but finite dead zone viscosity, as suggested by simulations of stratified disks with MRI-active outer layers, can trigger inside-out bursts of accretion, starting at or near the inner edge of the disk, instead of the previously-found outside-in bursts with zero dead zone viscosity, which originate at a few AU in radius. These inside-out bursts of accretion bear a qualitative resemblance to the outburst behavior of one FU Ori object, V1515 Cyg, in contrast to the outside-in burst models which more closely resemble the accretion events in FU Ori and V1057 Cyg. Our results suggest that the type and frequency of outbursts are potentially a probe of transport efficiency in the dead zone. Simulations must treat the inner disk regions, R /lessorsimilar 0 . 5 AU, to show the detailed time evolution of accretion outbursts in general and to observe the inside-out bursts in particular. Subject headings: accretion disks, stars: formation, stars: pre-main sequence", "pages": [ 1 ] }, { "title": "Variable Accretion Outbursts in Protostellar Evolution", "content": "Jaehan Bae 1 , Lee Hartmann 1 , Zhaohuan Zhu 2 , Charles Gammie 3 , 4 [email protected], [email protected], [email protected], [email protected]", "pages": [ 1 ] }, { "title": "1. Introduction", "content": "The (re)discovery of the magnetorotational instability (MRI: e.g., Balbus & Hawley 1998, and references therein), appears to resolve the long-standing problem of the anomalous viscosity in sufficiently ionized accretion disks. To a crude approximation, this validates the use of Shakura & Sunyaev (1973) ' α viscosity\" disks, though there are differences in detail (e.g., Balbus & Papaloizou 1999; Gammie 1996). Constant α disks have been employed in many situations, including the evolution of pre-main sequence disks (e.g., Hartmann et al. 1998). However, as pointed out by Gammie (1996), thermal ionization levels in protostellar and protoplanetary disks generally are so low that it is unlikely that the MRI operates everywhere. Gammie suggested that transport in these regions might be limited to surface 'active' layers, in which non-thermal ionization (cosmic rays, stellar X-rays) could allow the MRI to operate, while in the central regions of the disk there could be a non-viscous 'dead zone'. A rich variety of phenomena are thus enabled beyond simple (quasi-steady) viscous disks (Vorobyov & Basu 2005, 2006, 2007, 2009; Zhu et al. 2010a,b; Martin & Lubow 2011; Martin et al. 2012a,b). Gammie (1996) noted that if the MRI-active layer has a roughly constant surface density, the result is a pileup of material in the inner disk which eventually could become gravitationally unstable and result in a rapid burst of accretion, perhaps producing an FU Ori outburst (Hartmann & Kenyon 1996). While early models of FU Ori accretion events relied on traditional thermal instability theory (Clarke et al. 1990; Bell & Lin 1994), Zhu et al. (2007) showed that the rapidly-accreting region of FU Ori extends much further in radius from the central star than can be achieved with this theory. The very large amount of mass accreted in one of FU Ori's outbursts ( ∼ 10 -2 M /circledot ) over such short timescales ( ∼ 10 2 yr) requires a large amount of material to be present in the disk at relatively small radii. This is a natural result of models in which gravitational instability (GI) triggers the outbursts (Armitage et al. 2001; Vorobyov & Basu 2006, 2007, 2009; Zhu et al. 2009c, 2010a,b; Martin & Lubow 2011; Martin et al. 2012a,b). Zhu et al. (2010b) presented one-dimensional, two-layer evolutionary disk models including infall from a rotating protostellar cloud, and showed that they could qualitatively reproduce the main features of the outbursts of FU Ori and V1057 Cyg, with rapid rise times and slowly-decaying accretion. The simulations included irradiation from the central star, but did not take into account the accretion luminosity, which during outbursts can be far larger than the stellar photospheric radiation. While the geometry of disk accretion may not favor self-irradiation (Bell 1999), during infall the dusty opaque envelope will act as a blanket, reradiating a significant portion of the accretion luminosity toward the disk. In addition the Zhu et al. (2010b) calculations assumed that the central layer - the dead zone - had no viscosity unless it became gravitationally unstable. However, detailed shearing box simulations of the MRI in disks with a stratified structure and a resistivity that increases toward the midplane indicate that MHD turbulence generated in the upper MRI-active layers produces some hydrodynamic turbulence in the inactive layers (Fleming & Stone 2003; Okuzumi & Hirose 2011; Gressel et al. 2012). Furthermore, these investigations suggest that this turbulence creates Maxwell stresses, which result in a small but non-zero viscosity that can transport angular momentum outward and thus mass inward. These considerations motivate further investigations of protostellar accretion using the two-layer onedimensional model. We find that the position and properties of the inner boundary, the inclusion of irradiation by the accretion luminosity generated in the inner disk, and any non-zero dead zone viscosity have significant effects on the resulting bursts of mass accretion. Our treatment is sufficiently limited to preclude detailed predictions, but the qualitative behavior is suggestive, given that continuing observations of protostars and pre-main sequence stars are increasingly found to exhibit a wide variety of accretion events, beyond the large FU Ori outbursts (Hartmann & Kenyon 1996; Herbig 2008; Muzerolle et al. 2005; Reipurth & Aspin 2010; Aspin et al. 2010; Covey et al. 2011; Lorenzetti et al. 2012). and", "pages": [ 1, 2, 3 ] }, { "title": "2. Methods", "content": "We use a modified version of the one-dimensional, two-zone disk model previously introduced in Zhu et al. (2010a,b), including changes to the infall model, enhanced disk heating, and viscosity in the dead zone. Here we review our scheme.", "pages": [ 3 ] }, { "title": "2.1. Surface density evolution", "content": "The surface density of a disk is evolved based on the mass and angular momentum conservation equations in cylindrical coordinates, and where Σ i is the surface density, Ω is the angular frequency, ˙ M i is the radial mass flux, W R φ , i = R Σ i ν i d Ω / dR , and ν i is the viscosity. The subscript i denotes either the active layer (' a \") or dead zone (' d \"). The terms 2 π g i ( R , t ) and 2 π Λ i ( R , t ) are the mass and angular momentum flux per unit distance of infall material from an envelope cloud (Cassen & Moosman 1981). Then, assuming instantaneous centrifugal balance (see Zhu et al. 2010b), equations (1) and (2) can be simplified to where M R is the sum of the mass of the central star and the disk mass within R . We then sequentially solve Equations (3) and (1) to evolve the disk. In Zhu et al. (2010b) mass and angular momentum were added to the disk using the infall model of Cassen & Moosman (1981), Here R c is the centrifugal radius, that is the outer radius at which mass is added to the disk at time t , and ˙ M in = 0 . 975 c 3 s / G is a constant total infall mass rate at a given cloud temperature (Shu 1977; Terebey et al. 1984). However, this model has a singularity at R = R c ; with finite grids this potentially causes non-convergent behavior at different grid resolutions. To avoid this we modified the infall model to eliminate the singularity by using a constant mass flux per unit distance. and The corresponding angular momentum added to the disk per unit distance is and A comparison of the radial mass infall profile of our model to that of the Cassen & Moosman (1981) model is presented in Figure 5. In total, our model adds 11 % less angular momentum to the disk per unit mass infall than the Cassen & Moosman (1981) model. These modifications result in better convergence with increasing grid resolution.", "pages": [ 3, 4 ] }, { "title": "2.2. Temperature evolution", "content": "The disk layer temperatures are determined by the balance between heating and radiative cooling. For the active layer, the energy equation is Another problem occurs at the inner boundary, which is difficult to make very small because for numerical reasons (short time steps, dust evaporation, etc.) as well as fundamental uncertainties; for example, is the disk truncated by a stellar magnetosphere, and if so where, and is there an outflow from the inner disk edge. In addition, mass infall right at the inner boundary produces different results depending upon on precisely which inner boundary radius we choose. To minimize these problems we take inner boundary radii which are relatively small but still comfortably outside the expected point of magnetospheric truncation. We further assume that envelope material does not fall onto the disk inside 0 . 2 R c , justified on the basis that the well-known emergence of jets and outflows seen in even the earliest protostellar phases should prevent the lowest-angular momentum material from reaching the disk or star (Reipurth & Bally 2001). By tying the inner radius of infall to R c we effectively assume that the same streamline denotes the boundary between outflow and inflow, such that the outflow cone retains the same opening angle, in this case, a half-angle of 26 . 6 · . This particular choice of opening angle is arbitrary and adopted mainly for numerical convenience. The mass infall rate of the modified model is where C Σ , a = Σ a c 2 s , a / T a is the heat capacity of the active layer. Here T ext characterizes the heating flux due to the irradiation of the disk by the stellar and (inner disk) accretion luminosity, and τ a and τ d are the optical depths of the active layer and the dead zone, respectively, and using the Rosseland mean opacity κ taken from Zhu et al. (2009a). In Equation (12), the first three terms are local heating of the active layer due to the viscosity, the infall, and the gravitational potential energy change, respectively. The fourth term consists of the external heating (see below) and the radiative heating from the underlying dead zone. The last term includes the radiative cooling toward each side of the active layer. The energy equation of the dead zone is similar to that of the active layer, where C Σ , d = Σ d c 2 s , d / T d is the heat capacity of the dead zone. If the active layer is optically thick, the energy equation is On the other hand, if the active layer is optically thin, the incident flux from the outside of the dead zone would be σ ( τ a T 4 a + T 4 ext ) so that the energy equation is Again, the first three terms in equations (16) and (17) represent local heating, while the last two terms account for radiative heating from the outside of the dead zone and the radiative cooling. In the energy equations, the viscous heating is where W R φ , i = (3 / 2) Σ i ν i Ω and ν i = α i c 2 s , i / Ω . The viscosity parameter α i is explained in detail in the next section. During infall the added material has smaller specific angular momentum than the disk material at the same radius. This results in a readjustment of the disk such a way that material moves inward. As we are assuming effectively instantaneous centrifugal balance, the increase in the gravitational potential energy driven by the readjustment process must be accompanied by the corresponding energy release. Here we assume that this heats the active layer only ( Q infall , d = 0), as this is the material directly impacted by the infalling matter. The heating by infalling material is then and As accretion proceeds, the central stellar mass increases and the disk gravitational potential energy will become more negative. In response, even in the absence of viscosity disk material will move inward, implying additional accretion luminosity. The heating by this effect is where ˙ M ∗ is change in the mass of the central star. The irradiation flux can be written as where L ∗ and L acc are the stellar luminosity and the accretion luminosity, respectively, and T env is the envelope cloud temperature. The coefficients f ∗ and f acc account for the non-normal irradiation of the disk surface. For the stellar irradiation we use f ∗ = 0 . 1 as in Zhu et al. (2010a) and assume the stellar luminosity follows the mass-luminosity relation which is an approximate power-law fit to pre-main sequence stars in the Taurus molecular cloud, using the luminosities and effective temperatures from Kenyon & Hartmann (1995), and adopting the Siess et al. (2000) evolutionary tracks to obtain the masses. The mass-luminosity relation is slightly modified from Zhu et al. (2010b). The inclusion of external heating by inner disk accretion is another new feature of our calculations. The accretion luminosity is calculated as where we assume a typical T Tauri stellar radius. In this case the appropriate value of f acc is quite uncertain. At low to moderate accretion rates, magnetospheric accretion onto the star can occur at high latitudes, so that adoption of f acc = 0 . 1, similar to that used for the stellar photospheric irradiation, seems reasonable. On the other hand, at high accretion rates, the spectra of FU Ori objects provide no indication of magnetospheric accretion (Hartmann & Kenyon 1996), and irradiation of the outer disk by a relatively flat inner disk should be much less effective (Bell 1999). However, if a substantial infalling envelope surrounds the disk, it can capture much of the accretion luminosity and reradiate a significant part toward the disk (Natta 1993). We therefore use both f acc = 0 . 1 and 0.01 to examine the importance of this heating. The last term in Equation (22) is the flux from the envelope cloud whose temperature is assumed to 20 K. Thus, the stellar luminosity irradiation Q ∗ and the accretion luminosity irradiation Q acc on the active layer become and We note that the accretion luminosity irradiation Q acc should be distinguished from the local viscous accretion heating Q vis . The relative importance of the individual heating terms, and their effects, during disk evolution will be discussed in §3.", "pages": [ 4, 5, 6, 7 ] }, { "title": "2.3. Disk Viscosity", "content": "The viscosity parameter α i is the sum of the MRI viscosity parameter α M , i and the GI viscosity parameter α Q , i . The MRI viscosity parameter is assumed to have a fixed value of α MRI only if a region can sustain the MRI. Thus, the active layer viscosity parameter is always set to α M , a = α MRI while the dead zone has MRI viscosity only if the midplane temperature is higher than a critical temperature T MRI to produce sufficient ionization levels. For the dead zone, we consider a residual viscosity as well as MRI viscosity and GI viscosity, α d = α M , d + α Q , d + α rd . The idea of the dead zone residual viscosity (DZRV) α rd is based on recent numerical magnetohydrodynamic simulations suggesting that magnetic turbulence in the active layers can drive hydrodynamic turbulence in the dead zone, implying a non-zero residual viscosity parameter ∼ 10 -3 -10 -5 (Bai & Stone 2011; Okuzumi & Hirose 2011; Gressel et al. 2012). Thus, for non-zero DZRV model we set where f rd is the efficiency of accretion in the dead zone whose value is chosen to be ≤ 1; this is intended to limit the effect of the active-layer induced turbulence such that the mass accretion rate of the dead zone (approximately) does not exceed that of the active layer ( ˙ M d ≤ ˙ M a ). This seems intuitively reasonable. We consider the upper limit f rd = 1 and consider a case with f rd = 0 . 1 as it is unlikely that the active layer can be that effective in driving accretion. Finally, the GI viscosity parameter is the same as in Zhu et al. (2010a), where Q is the Toomre parameter.", "pages": [ 7 ] }, { "title": "3.1. Initial conditions", "content": "We start with a 0 . 1 M /circledot central protostar surrounded by an M c = 1 M /circledot cloud. We parameterize the cloud rotation in terms of ω = Ω c / Ω b , where Ω c is the (constant) angular frequency of the initial cloud, and Ω b = 2 3 / 2 c 3 s / GM c is the breakup angular frequency at the outer cloud edge, and c s is the (uniform) cloud sound speed. Our fiducial models assume ω = 0 . 03, which results in ∼ 15 % larger cloud angular frequency than that used in the fiducial model of Zhu et al. (2010b) ( Ω c ∼ 1 . 15 × 10 -14 rad s -1 in our model). We set the maximum non-thermally ionized surface density Σ A to 100 g cm -2 for our fiducial choice and assume it is constant. We assume T MRI = 1500 K and α MRI = 0 . 01 for all calculations. We adopt a cloud envelope temperature of T env = 20 K, which yields a constant infall rate of ∼ 3 . 4 × 10 -6 M /circledot yr -1 . This is 20 % smaller than the infall rate for conventional singular isothermal collapse model (Shu 1977) because of our modified infall model (see §2.1). The infall lasts for ∼ 0 . 24 Myr, adding 0 . 8 M /circledot to the central star + disk in total.", "pages": [ 7, 8 ] }, { "title": "3.2. Zero dead zone viscosity model", "content": "Zhu et al. (2010a) found that GI moves matter from the outer disk to the inner disk, leading to a pileup at R ∼ 2 AU because GI is increasingly ineffective at small radii. Eventually enough material piles up to trap thermal energy that makes T d > T MRI , turns on the MRI thermally in the dead zone, and thus produces an outburst of accretion. Zhu et al. (2010b) investigated the long-term evolution of such disks and found that the evolution can be divided into three stages. The evolution starts with a quasi-steady disk accretion since the infall is to small radii where the inner disk can become hot enough to sustain the MRI thermally. Then, it turns into the outburst stage as the infall occurs at radii > 1 AU. After infall stops, the disk enters the T Tauri phase, having only a few GI-driven outside-in outbursts with a low mass accretion rate in between bursts. Figure 2 shows the mass accretion rate and the mass of the central star, the disk, and the central star + disk as a function of time. The modified infall model and additional heating sources discussed in §2 produce no qualitative difference in the overall evolution from Zhu et al. (2010b). The new heating sources for the first 0.3 Myr of the evolution at R = 1 and 10 AU are presented in Figures 3 and 4, together with the previously considered sources. In these figures, newly added terms in this paper are plotted in color while other terms have considered in Zhu et al. (2010a) and drawn in black. As one can see, heating by the change in gravitational energy is usually several orders of magnitude smaller than other terms so that it makes no change in the evolution. Infall heating provides comparable amount of heat to the active layer but is only limited to the region material falls onto, increasing the local disk temperature slightly. However, the large increase in the accretion luminosity during outburst produces enough irradiation to make the outer disk temperature increase dramatically. This is shown in Figure 5, where we show the mass accretion rate during a single outburst and the radial profiles of the disk surface density and the midplane temperature before outburst, at the maximum accretion rate, and at the end of the outburst. The temperature increase at the outer disk during outbursts does not affect the long-term evolution, because the viscous time of the outer regions ( ∼ 10 5 yr) is much longer than the outburst timescale of ∼ 10 3 yr. We also call attention to the jump in temperature at radii < 0 . 5 AU, which is due to thermal instability (Zhu et al. 2010a). This increase in temperature, which affects the behavior of the outburst of accreting material onto the central star, would not have been found if we had taken an inner radius of /greaterorsimilar 1 AU (see discussion in §4). The GI-driven outside-in bursts accrete ∼ 0 . 027 M /circledot of material onto the central star and last for ∼ 1340 years on average. Although the duration obtained in our calculation is longer than the typical outburst timescale seen in FU Ori, the timescale of an outburst can be scaled with a choice of α MRI since ∆ t burst ∼ R 2 /ν ∝ α MRI -1 (see Zhu et al. 2010a). The outside-in bursts are rare, because the disk needs a lot of material to trigger MRI through GI while it is difficult to do so with zero DZRV. The duty cycle of this model is thus pretty small, ∼ 0 . 06 during outburst stage and ∼ 0 . 005 during T Tauri phase. The details of disk properties at 0.24 and 1 Myr and of outside-in bursts are summarized in Table 1. All the outburst quantities in the table are time-averaged values after the initial quasi-accretion phase while those vary with time.", "pages": [ 8, 9 ] }, { "title": "3.3. Non-zero dead zone residual viscosity model", "content": "While the modified infall model and additional heating sources with zero DZRV make no qualitative change in the overall evolution, a finite DZRV makes a lot of difference in the long-term evolution and in the single outburst behavior as well. Figure 6 shows the mass accretion rate and the mass of the central star, the disk, and the central star + disk as a function of time. During infall the disk has a quasi-steady accretion phase at the beginning ( t /lessorsimilar 0 . 08 Myr) and the outburst stage follows, as in the zero DZRV case. However, the evolution after the accretion phase is different in that the non-zero DZRV model shows a lot of smaller outbursts instead of a few large outbursts. Figure 7 and 8 show the heating sources of active layer and dead zone of the non-zero DZRV model at R = 1 and 10 AU, respectively. The major difference between this model and the zero DZRV model is that dead zone viscous heating provides a significant amount of heat at the inner disk even after infall ends. We have run calculations with different T MRI (1300 K and 1800 K) and found that the overall features are not sensitive to the choice of T MRI . At the outer disk, the accretion luminosity irradiation is still important during bursts while the temperature increase is not as dramatic as in the zero DZRV model due to lower accretion peak. Infall and gravitational heating are less important than others. Figure 9(a) shows the mass accretion rate during a single outburst with non-zero DZRV. Radial surface density and midplane temperature profiles at the beginning, at the maximum accretion rate, and at the end of the burst are presented in Figure 9(b) and (c). The accretion behavior during a single outburst is remarkably different from that of the zero DZRV model. The outburst has a peak of ˙ M max ∼ 10 -5 M /circledot yr -1 , which is about two orders of magnitude smaller than that of the outside-in bursts. In addition, the accretion rate initially shows a rapid increase but has a slow rise time to its peak and a slow decrease after the peak as well. In this model, the dead zone is able to transport material with the help of the non-zero DZRV. Thus, the inner disk can be heated viscously and outbursts are initiated at the inner boundary of the disk before the GI piles up enough material at the middle of the disk ( R ∼ 2 AU) to initiate the MRI, which is the case for outside-in bursts. The ionization front propagates out to several AU from the inner boundary. Since the inside-out bursts have an MRI active inner boundary from their initiation, the disk continues to dump material from its innermost part during the whole bursts. Therefore, the system is not able to show a huge accretion rate as seen in outside-in bursts, but only generates moderate accretion rate. Note again that the outburst triggers first at small radii, inside of ∼ 0 . 5 AU (§4). On average, the mass accreted onto the central star during a single inside-out burst is ∼ 1 . 5 × 10 -3 M /circledot and it lasts ∼ 450 years. The inside-out bursts occur frequently enough to get a duty cycle of ∼ 0 . 16 during outburst stage and ∼ 0 . 06 during T Tauri phase. The disk properties and outburst details of the non-zero DZRV model are summarized in Table 1.", "pages": [ 9, 10 ] }, { "title": "3.4. Efficiency of accretion luminosity irradiation", "content": "As shown in the previous sections, irradiation by the inner disk plays an important role during outburst on the temperature profile at the outer disk. However, the efficiency of the accretion luminosity irradiation is uncertain as far as the non-normal irradiation of the disk is considered. We thus test the effect of changing f acc to 0.01, which is ten times smaller than the fiducial value. We found essentially no change in the overall evolution of both zero and non-zero DZRV cases, since the accretion luminosity irradiation is several orders of magnitude smaller than main heating sources - active layer viscous heating and stellar irradiation - during the quiescent phase. During outbursts, however, the accretion luminosity irradiation still dominates the heating even with a ten times smaller efficiency, making a significant difference to the outer disk temperature. Not surprisingly, the increase in outer disk temperature during bursts is smaller than the standard cases by a factor of ∼ 2.", "pages": [ 10 ] }, { "title": "3.5. Accretion Efficiency in Dead Zone", "content": "Intuitively, it seems unlikely that the turbulence generated by the active layers within the dead zone can transport as much mass as the active layer (e.g., Hartmann et al. 2006). In our models this happens when α rd /greaterorsimilar 10 -4 , where Σ d /greaterorsimilar 10 5 g cm -2 . This could be an overestimate of the efficiency with which the MRI turbulence in the active layers drives accretion in the dead zone. We therefore adjust the dead zone accretion efficiency f rd to 0.1 so that dead zone only has an accretion rate of ∼ 10 % of the active layer at most. Figure 10 shows the mass accretion rate and the mass of the central star, the disk, and the central star + disk as a function of time. Initially, the evolution resembles that of the standard zero DZRV model rather than the non-zero DZRV model; the system shows a distinct outburst phase during infall. This is because the mass that the dead zone can carry is now limited and thus generates less viscous heating at small radii than the standard non-zero DZRV case. Therefore, mass piles up at large radii through the GI before inner disk gets heated and triggers inside-out bursts. After infall ends, however, we still see inside-out bursts with much less frequency than the standard non-zero DZRV model, which is again due to less viscous heating at the inner disk. The duty cycle during T Tauri phase of this model is only 0.015, which is four times smaller than that of the standard non-zero DZRV model. First three outbursts after infall ends are outside-in bursts, since disk already collects enough material at outer disk during infall to make them. This emphasizes importance of understanding the effect of MRI turbulence on dead zones (§4).", "pages": [ 10 ] }, { "title": "3.6. Dependence on Σ A and α MRI", "content": "While we use Σ A = 100 g cm -2 as our fiducial value, several studies have pointed out that the active layer more likely has a lower surface density (e.g. Sano et al. 2000; Bai & Goodman 2009). We thus test Σ A = 20 g cm -2 and adjust the MRI viscosity parameter α MRI = 0 . 05 to maintain roughly the same mass accretion rate ( ˙ M ∝ α Σ ) during the quiescent phase as the standard cases (and also in agreement with typical T Tauri accretion rates). Figure 11 shows the mass accretion rates of both zero and non-zero DZRV models adopting the lower value of Σ A . The mass accretion rate during a single outburst and radial surface density and midplane temperature profiles at the beginning, at the maximum accretion rate, and at the end of a single outburst of the both models are presented in Figure 12. Since the outburst timescale depends on the MRI viscosity parameter ( ∆ t burst ∝ α MRI -1 ), the details of the outbursts, such as outburst duration and peak accretion rate, vary. However, the overall evolution as well as the initiation of outbursts remain the same. We see GIinduced MRI-driven outside-in bursts in the zero DZRV case and viscously triggered inside-out bursts in the non-zero DZRV case. This is because the overall evolution and the initiation of outbursts are governed by the mass accretion during the quiescent phase, which we manage to be unchanged. We note that the shorter timescales are in better agreement with FU Ori (see Zhu et al. 2007). The disk properties and outburst details are summarized in Table 1.", "pages": [ 11 ] }, { "title": "4. Discussion", "content": "Our simulations show that it is possible to obtain inside-out triggering of accretion outbursts as well as outside-in bursts (e.g., Zhu et al. 2010a,b), with the former enhanced if there is finite dead zone residual viscosity. The two types of outbursts were also obtained in the model developed by Bell & Lin (1994; BL) for FU Ori outbursts. In the BL model, the outbursts were due to thermal instability (TI), plus an assumed increase in α from a very low value to a much higher value. As Zhu et al. (2007, 2008) showed, the TI model is inconsistent with observations of FU Ori, because the high temperatures required limit the region of rapid accretion to smaller radii than inferred from modeling the spectral energy distribution including Spitzer Space Telescope data. Nevertheless, the finite dead zone residual viscosity models are qualitatively similar to the basic feature of the BL models which produce inside-out bursts; a small but finite viscosity allows material in the inner disk to produce enough trapping of viscously-generated heat to trigger a higher viscosity and eventually an outburst. As BL showed, such inner disk triggering leads to outbursts with slow rise times, qualitatively consistent with the observed outburst of V1515 Cyg (Herbig 1977; M. Ibrahimov, personal communication). BL showed that outbursts with rapid rise times, such as observed in FU Ori and V1057 Cyg (Herbig 1977), required outside-in accretion events. In the BL model, a large outer perturbation of the disk was required. In modern models, the event is triggered by GI, which piles up material at larger radii than possible in the TI model (Armitage et al. 2001; Vorobyov & Basu 2006, 2007, 2009, 2010; Zhu et al. 2010a,b; Martin et al. 2012a,b). Our current results build upon those of Zhu et al. (2010b) in that we clearly identify some inside-out bursts during the main phase of infall (they were actually present in the Zhu et al. simulation as well but were not emphasized). Our results also bear similarities to outbursts in models of cataclysmic variables (CV). Two different types of outbursts (outside-in and inside-out) in accretion disks around dwarf novae were first predicted by Smak (1984). In CV models, mass transfer from the secondary rises the effective temperature of a disk annulus to 5000 -8000 K, which corresponds to hydrogen recombination inside the disk and thus triggers the TI (Menou et al. 1999). While the dead zone residual viscosity parameter is the feature that changes outburst behavior in our model, CV models use the mass transfer rate to generate two different outbursts. If the mass transfer rate is high, the accumulation timescale of transferred material is shorter than the viscous timescale, allowing material piles up at outer disk. Thus, outside-in bursts are triggered. In contrast, if mass transfer rate is low, the accumulation timescale becomes longer than the viscous timescale so that the outbursts outside-in outbursts are replaced by inside-out ones. The resulting outburst behaviors of CV models are similar to ours; inside-out bursts have a smaller accretion peak and a slower rise time than outside-in ones (see Figures 3 and 4 of Hameury et al. 1998). It is worth emphasizing that the outburst behavior of systems, observed at optical and near-infrared wavelengths, is a result of accretion onto or near the star, i.e. at radial scales /lessorsimilar 0 . 1 AU. While our models do not reach magnetospheric or stellar radii, our inner boundary radius of 0 . 2 AU is small enough to capture behavior (thermal instability, inside-out bursts) which cannot be seen in simulations with inner boundaries > 1 AU. Thus, while our own treatment of non-steady accretion has its limitations, time histories of accretion in simulations with large inner disk radii must be treated with special caution. Using an inner boundary of a few AU, as in the series of papers by Vorobyov & Basu (2006, 2007, 2009) or Dunham & Vorobyov (2012), one would not find the outburst behavior characterized by our models or those of Armitage et al. (2001) or Martin et al. (2012a,b). Along these lines, we have found that the precise duration of the inside-out bursts during infall is sensitive function of the value of the inner radius. We do not explore this further here because there are other major uncertainties in our treatment, such as the assumption of a constant active layer surface density, the characterization of the GI via an α viscosity, and the way in which we implement a finite dead zone viscosity. In the following paragraphs we discuss these issues in turn. Advanced MHD treatments of the active layer are complex and involve a number of unknowns, such as whether low-energy cosmic rays can penetrate the accretion-driven winds, grain growth and settling, the presence of metal ions, etc. (e.g., Sano et al. 2000; Ilgner & Nelson 2006, 2008; Hirose & Turner 2011; Perez-Becker & Chiang 2011). Martin et al. (2012a,b) argue that a large critical Reynolds number is necessary for transport to occur, resulting 'active' regions very different than the constant Σ a we use. However, Martin et al. (2012b) predict essentially no accretion onto the central star in between outbursts, whereas the pre-outburst spectrum of V1057 Cyg shows emission lines typical of T Tauri stars accreting at ∼ 10 -8 M /circledot yr -1 (Herbig 1977). More recently, Miller et al. (2011) showed that the classical (accreting) T Tauri star LkHa 188-G4 underwent an FU Ori-type eruption in 2009. Finally, a pre-outburst spectrum of the FU Ori object V733 Cep (Reipurth et al. 2007) also showed characteristic accreting T Tauri emission lines (B. Reipurth, personal communication). Thus the pre-outburst state of at least some FU Ori objects is one of accretion at rates typical of T Tauri stars, which we obtain with our adopted value of α a Σ a , at least in the inner disk. Similarly, our treatment of the GI with an α viscosity is crude; as in the case of the MRI, threedimensional simulations are required to treat the GI properly (e.g., Rice et al. 2003; Boley et al. 2006; Durisen et al. 2007, and references therein). Two-dimensional simulations also do a better job of capturing the GI than our treatment (e.g., Vorobyov & Basu 2006, 2007, 2009, 2012). However, as pointed out by Zhu et al. (2009), and as shown in Zhu et al. (2010a,b) and the present simulations, the GI becomes harder and harder to sustain as one moves to smaller radii, whereas triggering of the MRI becomes easier. Finally, the presence and behavior of non-zero viscosity in dead zones is highly uncertain. While simulations of resistive, stratified disks in shearing boxes appear to show that the active layer produces a non-zero α in the central 'dead' layers (Fleming & Stone 2003; Okuzumi & Hirose 2011; Gressel, Nelson, &Turner 2012), the level to which this occurs, the amount of mass transport involved, and precisely where energy is dissipated is unclear. For example, our model assumes that the wave energy is dissipated near the midplane, and thus the heat generated can be trapped radiatively by the opacity of the disk at the midplane; however, it is possible that dissipation is concentrated at higher levels (N. Turner, personal communication). In all, these uncertainties show that our current results must be taken as suggestive rather than predictive for variations of accretion in young stellar objects. Nevertheless, these simple models, which can be evolved easily for significant evolutionary timescales, illustrate the potential information on transport processes in protostellar and protoplanetary disks that might ultimately be gleaned from the observed accretion outbursts. Due to the increasing monitoring of young stellar objects, it is becoming increasingly clear that a wide variety of accretion behavior is exhibited in young stars, emphasizing that further progress on challenging problems of globally simulating MRI and GI in protostellar disks may pay rich dividends.", "pages": [ 11, 12, 13 ] }, { "title": "5. Summary", "content": "In this paper, we have extended the one-dimensional, two-zone model of long-term protostellar disk evolution with infall, which is previously introduced in Zhu et al. (2010a,b). Our modified models include a revised treatment of infall, enhanced disk heating, and possible non-zero viscosity in the dead zone. While the former two changes produce no qualitative difference in the overall evolution from the Zhu et al. (2010b), we find that the presence of a small but finite dead zone viscosity can trigger inside-out bursts initiated at or near the inner edge of the disk through dead zone viscous heating, instead of GI-induced MRI-driven outside-in bursts with zero dead zone viscosity. These inside-out bursts not only bear a qualitative resemblance to the outburst behavior of one FU Ori objects, V1515 Cyg, but emphasize a careful treatment of the inner disk regions in simulations. Given the uncertainties, our results are rather suggestive than predictive. However, two types of outbursts seen in FU Ori objects can be successfully reproduced by the simple α treatment in the dead zone. This difference in accretion behavior could be a potential probe of transport efficiency in the dead zone. We acknowledge useful conversations with Bo Reipurth and Neal Turner. This work was supported in part by NASA grant NNX08A139G and by the University of Michigan.", "pages": [ 13, 14 ] }, { "title": "REFERENCES", "content": "Alexander, R. D., & Armitage, P. J. 2007, MNRAS, 375, 500 Armitage, P. J., Livio, M., & Pringle, J. E. 2001, MNRAS, 324, 705 Aspin, C. 2011, AJ, 142, 135 Aspin, C., Reipurth, B., Herczeg, G. J., & Capak, P. 2010, ApJ, 719, L50 (V1647) Bai, X.-N. & Goodman, J. 2009, ApJ, 701, 737 Bai, X.-N. & Stone, J. M. 2011, ApJ, 736, 144 Balbus, S. A., & Hawley, J. F. 1998, Reviews of Modern Physics, 70, 1 Balbus, S. A., & Papaloizou, J. C. B. 1999, ApJ, 521, 650 Bell, K. R. 1999, ApJ, 526, 411 Bell, K. R., & Lin, D. N. C. 1994, ApJ, 427, 987 Bell, K. R., Lin, D. N. C., Hartmann, L. W., & Kenyon, S. J. 1995, ApJ, 444, 376 Boley, A. C., Mejía, A. C., Durisen, R. H., et al. 2006, ApJ, 651, 517 Briceño, C., Vivas, A. K., Hernández, J., et al. 2004, ApJ, 606, L123 Cassen, A. & Moosman, A. 1981, Icarus, 48, 353 Clarke, C. J., Lin, D. N. C., & Pringle, J. E. 1990, MNRAS, 242, 439 Covey, K. R., Hillenbrand, L. A., Miller, A. A., et al. 2011, AJ, 141, 40 Dullemond, C. P., & Dominik, C. 2004, A&A, 421, 1075 Dunham, M. M., & Vorobyov, E. I. 2012, ApJ, 747, 52 Durisen, R. H., Boss, A. P., Mayer, L., et al. 2007, Protostars and Planets V, 607 Fleming, T., & Stone, J. M. 2003, ApJ, 585, 908 Gammie, C. F. 1996, ApJ, 457, 355 Gressel, O., Nelson, R. P., & Turner, N. J. 2012, MNRAS, 422, 1140", "pages": [ 14 ] } ]

[ { "title": "ABSTRACT", "content": "After almost 4 years of operation, the two instruments onboard the Fermi Gammaray Space Telescope have shown that the number of gamma-ray bursts with high energy photon emission above 100 MeV cannot exceed roughly 9% of the total number of all such events, at least at the present detection limits. In a recent paper (Zheng et al. 2012c), we found that GRBs with photons detected in the Large Area Telescope (LAT) have a surprisingly broad distribution with respect to the observed event photon number. Extrapolation of our empirical fit to numbers of photons below our previous detection limit suggests that the overall rate of such low flux events could be estimated by standard image co-adding techniques. In this case, we have taken advantage of the excellent angular resolution of the Swift mission to provide accurate reference points for 79 GRB events which have eluded any previous correlations with high energy photons. We find a small but significant signal in the co-added field. Guided by the extrapolated power law fit previously obtained for the number distribution of GRBs with higher fluxes, the data suggests that only a small fraction of GRBs are sources of high energy photons. Subject headings: gamma-ray burst: general, gamma rays: stars", "pages": [ 1 ] }, { "title": "Limits to the fraction of high-energy photon emitting gamma-ray bursts", "content": "Carl W. Akerlof 1 [email protected] and WeiKang Zheng 1", "pages": [ 1 ] }, { "title": "1. Introduction", "content": "A most unexpected feature of gamma-ray bursts (GRBs) became apparent in 1994 with the Compton Gamma-Ray Observatory (CGRO) detection of an 18 GeV photon associated with GRB 940217 (Hurley et al. 1994). About a half-dozen high energy burst events were detected throughout the CGRO mission but the actual fraction of the total rate was poorly determined (Catelli et al. 1998, Dingus 2003). Prior to the launch of the Fermi Gamma-ray Space Telescope in 2008, it was possible to anticipate that the LAT instrument would detect more than 200 GRB events per year at energies above 100 MeV (Dingus 2003). Since the GRB spectral energy distribution at lower energies has been well characterized by a modified power law with peak fluxes at energies of the order of 200 KeV (Kaneko et al. 2006, Goldstein et al. 2012), the existence of photons at energies 10 4 times higher is a significant constraint on credible models of the GRB phenomenon (e.g. Band et al. 2009). Thus, there had been some anticipation that the Fermi mission might considerably enrich our knowledge of this aspect of GRB behavior. Such hopes were dampened by the Band paper which was drafted 7 months after the Fermi launch. With three LAT-detected events and an extrapolation of BATSE spectral data, Band et al. concluded that the high energy detection rate of the LAT would be about one per month. Almost four years of operation has demonstrated that the detection rate of energetic GRBs is closer to 9 per year. One interesting discovery that has emerged is that short as well as long bursts contribute to the population of > 100 MeV emitting GRBs. To date, six short events have been so identified, roughly comparable to their fraction within the total GRB population (GRB 080905A, 081024B, 090228A, 090510, 110529A and 111117A). This suggests that the high energy photon generation process is a characteristic of relativistic jets, independent of the details of their specific progenitors. Given the paucity of information about energetic bursts, we have embarked on a program to dig out as much information as possible from the available data. This has led to a string of papers describing the discovery of seven high energy GRBs with signatures too faint to be detected by more conventional statistical techniques (Akerlof et al. 2010, Akerlof et al. 2011, Zheng et al. 2012a, 2012b, 2012c, 2012d). The following work delves further into this realm to understand better the extent of the association of high energy events with the parent class of all GRBs.", "pages": [ 1, 2 ] }, { "title": "2. Data selection and analysis", "content": "In our first search of LAT data for faint GRBs (Akerlof et al. 2011), we recognized that the precise localizations of bursts ( σ PSF < 5 ' ) detected by Swift, INTEGRAL and AGILE were powerful constraints for identifying associated high energy photons in the Fermi LAT. More recently, we estimated the event intensity distribution and discovered that within the bounds set by limited statistics, the distribution was well fit by the number of LAT photons per unit area raised to a constant negative fixed power (Zheng et al. 2012c). To probe this empirical result at intensities below those required for reliable single event identification, we realized that the standard field coadding techniques of optical astronomy would work nicely with fields triggered by Swift and similar instruments. The GRB events were selected in the interval from June 11, 2008, ( Fermi launch date) through February 29, 2012 from the Swift catalog 1 for which the Fermi LAT boresight angle was computed from the Fermi spacecraft attitude data 2 . Since the publicly available LAT data stream was restricted to Pass 7 after 2011 August 6, we used Pass 6 prior to the switchover date and Pass 7 thereafter. Events were included only if they passed the maximum LAT acceptance angle of 74 · and no previous GRB correlations had been reported. Although the LAT acceptance drops significantly for boresight angles greater than 65 · , it is still large enough for burst detections such as reported by the LAT team for GRB 100414A (Takahashi et al. 2010) and GRB 120624B (Vianello & Kocevski 2012). Swift events were also ignored if the Fermi spacecraft was inside the South Atlantic Anomaly (SAA) or was otherwise inactive. A total of 79 events passed these cuts and are listed in Table 1. For reference later in this paper, this sample is called 'All Swift '. Since we were interested in investigating possible correlations with the intensity of bursts at lower energies, three further selections were made, based on the Fermi GBM fluence. The fluence data was obtained from either the GBM Fermitrig catalog 3 or GCN messages (Barthelmy 2000). The former took precedence whenever available. There were 46 events with such GBM data and this set is named the ' Swift GBM' sample. Third and fourth samples, ' Swift bright GBM' and ' Swift dim GBM' consists respectively of the 14 events with fluences greater than 10 µerg/cm 2 within ' Swift GBM' and the complementary set of 32 with fluences below this bound. Since ' Swift GBM' is a subset of 'All Swift ' and, in turn, ' Swift bright GBM' and ' Swift dim GBM' are disjoint subsets of ' Swift GBM', these four co-add sets are not statistically independent. Nevertheless, there are correlations to the low energy GRB fluxes that makes comparisons useful. It should be noted that very few Pass 7 GRB events are included in any of these data sets: All Swift - 5 out of 79, Swift GBM - 2 out of 46, Swift bright GBM - 1 out of 14 and Swift dim GBM - 1 out of 32. Thus, any systematic biases that might arise from somewhat different primary data processing are statistically insignificant.", "pages": [ 2, 3 ] }, { "title": "2.1. Matched filter weight computation and significance", "content": "The heart of our signal detection technique is the use of a matched filter to optimize the probability for correctly identifying a GRB in the presence of random backgrounds of gamma-rays. The matched filter technique is widely employed for time domain detection of radar and sonar signals but can be easily extended to more complex signal structures. In the simplest case, assume a uniform background noise, n ( t ), and a signal waveform, s ( t ). The detection significance for the signal is maximized by integrating a matched filter, f ( t ), over the effective duration of the signal. Variational methods show that f ( t ) = c · s ( t ) where c is a constant. For situations with non-constant backgrounds, the appropriate choice for f ( t ) is given by the ratio, c · s ( t ) /n ( t ). As described in Akerlof et al. 2011, the total weight for each photon was computed by the product of four quantities representing the estimated signal to background ratio for energy, position on the sky, time of arrival and LAT photon detection class. Finally, a somewhat ad hoc weight sharing factor, ζ , was applied to kill events in which the apparent total event weight was heavily dominated by only one or two photons, thus favoring the extreme tails of the background distribution. The photon data selection criteria for all events is the same as described in our previous work (Akerlof et al. 2010, 2011 and Zheng et al. 2012b, 2012c). Briefly stated, the photon energy is restricted to the range from 100 MeV to 300 GeV. A zenith angle cut of less than 105 · is applied as recommended by the LAT team. A detection time window is defined by the interval from T0 to T0+47.5s where T0 is the nominal trigger time. The photons passing these three criteria are then used for further analysis. The matched weight for each photon was calculated by the method described in Akerlof et al. 2011, equations 1 - 5. Note that all classes of photons are included together in computing the matched weight score using our previously described methodology. A weight-sharing factor, ζ , was computed via equation 8 and combined to yield the quantity, ζ ∑ w i . The final result is the sum of the matched weights for all single photons multiplied by the weightsharing factor, ζ . It is this value, ζ , that is propagated to all further analysis. Note that for each of the four co-add sets described above, the photons for every matching LAT data set were rotated to a common axis and the co-added ensemble was handled as if it were a single GRB event (see Figure 1 for a sky map of the Swift GBM co-add composite field). These various considerations determined how we treated two GRBs with previously reported LAT detections, GRB100728A and GRB110625A. GRB 100728A (Abdo et al., 2011) has been included in our analysis since although LAT emission was found to be associated with a late time X-ray flare, no LAT emission was reported during the GRB prompt phase from 0 to 167 s. In the case of GRB 110625A (Tam et al., 2012), the event was excluded on the basis of the LAT boresight angle of 88 · when the GRB occurred. The first question is whether there is evidence for any significant association of high energy photons with these GRB fields. To test this assumption, 1256 sets of random co-added fields were constructed with LAT data appropriate for each of the four samples with 79, 46, 14 and 32 GRB events respectively. For each of the 1256 random co-added sets, every single constituent field was selected at random time but with center position identical to the corresponding GRB field. These random fields also had boresight and zenith angles similar to the corresponding GRB data. The distribution of these weights is shown in Figure 2 for the ' Swift GBM' sample of 46. In this case, 7 random fields out of the 1256 yielded a weight greater than for the actual set associated with GRBs. Similar results were obtained for the other samples as well, setting an overall confidence level in the neighborhood of 99% that a modest excess of GRB-related photons has been detected. A synopsis of these estimates is provided in Table 2. The conclusion that a few GRB photons were associated with Swift-triggered events was probed in two different ways. It was noted that the fields associated with the 79 GRB events were not uniform in intensity as measured by the sum of matched filter weights over the 1256 observations of each field taken randomly in time. By comparison of one half of the random data observations with the other, a substantial pair-wise correlation was observed for each individual GRB direction. The noisiest 8 GRB fields were removed and the previous analysis re-run to see if background astrophysical sources were a major contamination. The significance levels for the GRB signals compared to the random co-added sets remained at the ∼ 99% level for all except the ' Swift dim GBM' sample. Secondly, it was noted that the matched filter weight method was moderately sensitive to infrequent statistical fluctuations with large weight values. To avoid this possible instability, a ranking procedure was invoked that involved two separate steps. First, the 80th percentile weight was determined for every set of 1256 random fields associated with each GRB event. Next, a score was computed for every random and GRB field based on simply summing the number of two highest photon weights above the 80th percentile cut for each field. For the two co-add samples with the greatest GBM fluences, ' Swift GBM' and ' Swift bright GBM', there appeared to be a significant correlation with the GRB co-add sets surpassing 93% and 97% of the equivalent random groups. In summary, we believe that this analysis has found reasonable evidence for a few GRB photons within a number of events but this number is remarkably small.", "pages": [ 6, 8 ] }, { "title": "2.2. Estimating the number of GRB high energy photons", "content": "Since the Swift sample is apparently associated with a non-zero number of GRB photons, the next step was to modify the 1256 random fields previously described by adding a fixed number of GRB photons taken randomly, with replacement, from the set of 851 photons described in Table 2 of Zheng et al. 2012c. Since the only parameter of interest is the GRB photon weight, these values were chosen by randomly selecting values from the appropriate 851-fold array. This procedure was iterated 1000 times for a total of 1256000 co-added fields for each integer number, n , of injected GRB photons. The integral distributions of the matched filter weights for the ensemble are shown in Figure 2 for pure random background fields (red line) and 1 to 20 injected GRB photons (blue lines). To find the most probable photon number for each data set, the integral distribution for each curve was estimated at the actual matched filter value. The quoted photon number was interpolated from the two values that bracketed 50%. A similar procedure was pursued for the ± 1σ ranges by finding the crossing points where the integral distributions took on the values of 0.158655 and 0.841345. The results for each of the four GRB sample sets are listed in Table 2. In order of increasingly more stringent GBM fluence requirements, the average number of photons per unit area is (0.23, 0.31, 0.53) m -2 , consistent with the high-energy/low-energy correlation shown in Figure 5 of Zheng et al. 2012c. (The ' Swift dim GBM' field has a corresponding flux of 0.30 photons m -2 .) This fluence trend lends additional support to the evidence of a small but finite number of GRB photons in the Swift -triggered co-added fields.", "pages": [ 8, 10 ] }, { "title": "3. Implications for the number distribution of high energy GRB photons", "content": "In a recently completed paper, we have found that the distribution of GRBs with photons > 100 MeV can be characterized by a distribution function that is proportional to the fixed power of the number of photons per unit area (Zheng et al. 2012c). This result is shown in Figure 3 which depicts the distribution of actual GRB events, the parent distribution and the distribution modified by the effective area of the LAT detector and analysis event discrimination efficiencies. As noted, the inferred number of possible GRB photons in the Swift co-added fields is relatively small. Thus for comparison with observation, we can try making the assumption that the GRB number distribution can be extrapolated below the detection threshold that originally defined the fit. We find that if the event distribution function observed at higher LAT fluxes can be extrapolated to lower rates, the inclusion of a fraction of all GRBs is sufficient to accommodate the numbers inferred from our analysis. Pursuing such questions will eventually offer a better insight as to whether high energy photon emission is a generic feature of all bursts or only a special subset.", "pages": [ 11 ] }, { "title": "3.1. Computing total photon number from a power law distribution", "content": "We need to find an expression for the total number of photons that can be expected from a set of bursts that are distributed in photon number according to a power law. Assume the number of GRB events emitting n photons per unit area is given by The exponent, p , is approximately -1.8, leading to the conclusion that this mathematical form is only valid above some minimum threshold value for n which will be designated as n min . Normalizing the integral of dN/dn to unity defines the maximum range over which the power law model can operate: For the problem at hand, we need to compute the total number of photons, ν Swift , that can be expected in the co-added field corresponding to all Swift triggers within the co-added set. Each GRB event will be characterized by a specific effective area determined by the boresight angle of the GRB with respect to the LAT z-axis. The probability of contributing m photons to the co-added field from the i 'th GRB event is determined by the Poisson distribution: where a i is the LAT effective area appropriate for the i 'th GRB event. The total number of photons is then given by where η ( m ) is the probability that an event with m photons would escape normal detection criteria. In practice, this limits the summation over to 8 photons or less. η ( m ) was computed by first estimating the probability that an event with m photons would have been previously detected by the matched filter weight method described in our earlier papers. This corresponds to a matched filter weight sum of approximately 15.0. An essentially pure sample of high energy GRB photons was obtained from 4 bright GRBs, 080916C, 090510, 090902B and 090926A with 125, 176, 164 and 177 photons respectively. For each of these burst events, a Monte Carlo program repeatedly computed the matched filter weight sum for m randomly selected photons to obtain an average probability of exceeding the threshold of 15.0. The independent results for the photon samples from the four bright GRBs were substantially similar and the average for each value of m estimates η ' ( m ), the probability of detection by previous searches. η ( m ) is computed immediately as 1 -η ' ( m ). Since dN/dn is simply a power of n , the evaluation of ν Swift devolves immediately to a summation of terms, each of which can be represented by an incomplete gamma function defined by Such functions can be found on familiar mathematical packages such as IDL and Mathematica. Taken together, equations 4 & 5 link assumptions about the GRB event photon number distribution to the actual number of photons that should be observed for a specified ensemble of exposures. In this way, the results given in Table 2 will lead to limits for these otherwise inaccessible regions of the event photon number distribution.", "pages": [ 11, 12 ] }, { "title": "3.2. Reconciling the Swift-triggered photons and the extrapolated number distribution", "content": "As noted previously, the lower limit of the range for a power-law distribution with negative exponent is bounded by the finite number of events it describes. That condition is satisfied by setting the lower bound, n min , by equation (2). Thus, the obvious first question is how many photons would be expected in the 'Swift GBM' co-add field if all 46 events obeyed this distribution over the range from n min to ∞ . With n min defined by equation 2, this can be obtained by application of the mathematical relations of equations 4 & 5. For the value of p which best fits the observed event distribution shown in Figure 3, the presumption that all GRBs contribute is clearly violated at the 97% confidence level as shown in Figure 2 by overpredicting the number of photons, 14, instead of the observed value of 6 given in Table 2. The obvious next step is to inquire where the lower bound should be set to match the observed photon number. To keep the notation unambiguous, this bound will be designated n thresh and n thresh > n min . For the most probable values for p and ν Swift , the value for n thresh is 1.7 photons/m 2 corresponding to 19% of the full GRB sample ( n min = 0 . 2 photons/m 2 ). Varying ν Swift over the ± 1σ range changes the percentage of the contributing GRBs from 11% to 40%. Keeping ν Swift at its central value but varying p over its ± 1σ range changes these percentages by a factor of roughly two. However if p and ν Swift are simultaneously set to their 1σ upper limits of -1.5 and 10, the fraction of GRBs rises to 100%. Admittedly, this extrapolation from present observations relies on extrapolation of the photon number distribution to low photon counts. Thus, the analysis should be viewed with circumspection but is intended to illustrate the twin constraints of an apparent powerlaw distribution and the limited total number of detected photons associated with a considerable number of bursts. The abrupt flux cutoff event distribution model described above is an unnatural extreme that favors a few events with fluxes near the current lower detection threshold. An alternative is to model the differential distribution in the region from zero to the detection threshold as a sum of two terms chosen to match the fitted distribution at threshold. With the constraint implied by the total of number of events inferred from the co-added analysis, the cumulative distribution plateaus at 24% of all GRBs (see dotted curve in Figure 3). The differential distribution for this segment is of the form, a + bn q , where a and b are constants, n is the GRB event flux and q is a fractional exponent chosen to match the co-added field photon count. This tends to favor a larger number of events at lower fluxes. Thus, despite wide differences in model assumptions, the GRB event population seems to divide between the ∼ 25% that emit high energy radiation and the 75% that don't. We find it exceedingly strange that nature has found a way to limit the number of events with high energy flux intensities in the 1 to 4 photons/m 2 range relative to what occurs at higher fluxes. Addressing this question might lead to a better understanding of the GRB high energy emission process. Finally, the stringent limits for total photon numbers in the various co-added trigger fields strongly suggest that there are very few modest intensity bursts that have gone undetected because of inadequate signal processing techniques.", "pages": [ 12, 14 ] }, { "title": "4. Summary", "content": "In this paper, we have found a small but statistically significant number of GRB-associated photons for co-added sets of events triggered by the Swift instrument. Although this number has large statistical errors, it points to a conclusion that high energy photon emission is relatively infrequent and can not be hidden in the somewhat fainter burst events. The necessary initial conditions that give rise to this phenomenon are not at all understood. With a larger event sample, it may be possible to better classify the population of high energy emitting GRBs and test models that explain their behavior. If nothing else, this analysis again demonstrates the importance of GRB detection with good angular resolution. We thank Timothy McKay for constructive suggestions for this manuscript. This research is supported by the NASA grant NNX08AV63G and the NSF grant PHY-0801007.", "pages": [ 14, 15 ] }, { "title": "REFERENCES", "content": "Abdo, A. A., et al., 2011, ApJ Letters, 734, L27 Akerlof, C., Zheng, W., Pandey, S. B., McKay, T. A., 2010, ApJ, 725, L15 Akerlof, C., Zheng, W., Pandey, S. B., McKay, T. A., 2011, ApJ, 726, 22 Atwood, W. B., et al., 2009, ApJ, 697, 1071 Band, D., et al., 2009, ApJ, 701, 1673 Barthelmy, S. D. et al. 2000 in Gamma-Ray Bursts , eds. R. M. Kippen, R. S. Mallozzi and G. J. Fishman (AIP, New York), p. 731 Catelli, J. R., Dingus, B. L. & Schneid, E. J., 1998, AIP Conf. Proc., 428, 309 Dingus, B. L., 2003, AIP Conf. Proc., 662, 240 Goldstein, A., et al., 2012, ApJS, 199, 19 Hurley, K., et al., 1994, Nature, 372, 652 Kaneko, Y., et al., 2006, ApJS, 166, 298 Takahashi, H., Ohno, M. & Omodei, N., 2010, GCN Circ., 10594 Tam, P. H. T., Kong, A. K. H., and Fan, Yi-Zhong , 2012, ApJ, 754, 117 Vianello, G. & Kocevski, D., 2012, GCN Circ., 13379 Zheng, W. & Akerlof, C., 2012a, GCN Circ., 12822 Zheng, W., Akerlof, C., Pandey, S. B., McKay, T. A., Zhang, B.-B., Zhang, B., 2012b, ApJ, 745, 72 (Z12) Zheng, W., et al., 2012c, ApJ, 756, 64 Zheng, W. & Akerlof, C., 2012d, GCN Circ., 13070 a Sum of the individual effective areas of all GRBs in the sample. The areas are determined from the LAT boresight angles at the burst trigger time. b Sum of the GBM fluences of all GRBs in the sample. These numbers are obtained from from the Fermi GBM Burst Catalog, FERMIGBRST (http://heasarc.gsfc.nasa.gov/W3Browse/all/fermigbrst.html), and the GCN catalog (searchable via GRBlog at http://grblog.org/grblog.php).", "pages": [ 15, 16 ] } ]

[ { "title": "ABSTRACT", "content": "We performed a series of hydrodynamical calculations of an ultra-relativistic jet propagating through a massive star and the circumstellar matter to investigate the interaction between the ejecta and the circumstellar matter. We succeed in distinguishing two qualitatively different cases in which the ejecta are shocked and adiabatically cool. To examine whether the cocoon expanding at subrelativistic speeds emits any observable signal, we calculate expected photospheric emission from the cocoon. It is found that the emission can explain early thermal X-ray emission recently found in some long gamma-ray bursts. The result implies that the difference of the circumstellar environment of long gamma-ray bursts can be probed by observing their early thermal X-ray emission. Subject headings: gamma-ray burst: general - radiation mechanisms: thermal - shock waves - supernovae: general", "pages": [ 1 ] }, { "title": "AKIHIRO SUZUKI 1 and TOSHIKAZU SHIGEYAMA 2", "content": "Draft version June 22, 2021", "pages": [ 1 ] }, { "title": "1. INTRODUCTION", "content": "Since the discovery of gamma-ray bursts (GRBs), numerous studies have been done to understand their progenitors, the mechanism to produce their highly energetic emission, and the central engine (see, e.g., Piran 1999; M'esz'aros 2006, for review). It is currently known that long GRBs are triggered by the gravitational collapse of massive stars. The spatial and temporal coincidence of GRB 980425 and SN 1998bw(Galama et al. 1998) has revealed the connection between long GRBs and a special class of type Ic supernovae (broad lined type Ic SNe), i.e., the firmly established SN-GRB connection (see, e.g., Woosley & Bloom 2006). For example, wellknown GRBs associated with SNe are GRB030329/SN 2003dh (Hjorth et al. 2003; Stanek et al. 2003), GRB 060218/SN 2006aj (Campana et al. 2006; Pian et al. 2006; Mazzali et al. 2006), GRB 100316D/SN 2010bh (Cano et al. 2011; Bufano et al. 2012; Olivares E. et al. 2012). The increasing number of detected samples of GRB-associated SNe has enabled us to investigate their circumstellar environments. Especially, whether the circumstellar matter (CSM) of the progenitor is dilute or dense is of particular interest, because it is expected that the CSM interacts with the ejecta and results in producing high-energy emission. The CSM may originate from the stellar material ejected prior to the explosion as a wind or the common envelope if the progenitor of the GRB was in a binary system (Podsiadlowski et al. 2004). Recently, it is reported that thermal components are found in X-ray spectra of some long GRBs, which are taken by Swift satellite 100-1000 seconds after the trigger (Campana et al. 2006; Starling et al. 2011; Page et al. 2011; Starling et al. 2012; Sparre & Starling 2012). The component is seen as an excess superposed on a powerlaw non-thermal component that is usually attributed to synchrotron emission from the forward shock, i.e., the af- terglow emission. Spectral analyses reveal that the component can be fitted by a single blackbody spectrum with temperature of k B T = 0 . 1-0 . 9 keV (see, Starling et al. 2012). The luminosity ranges from 10 45 to 10 49 erg s -1 . The contribution of the thermal emission to the total X-ray flux is typically a few % up to several 10 %. The emitting radii inferred from the fitting results are 10 12 -13 cm, which are much larger than the typical radius of the progenitor star /lessorsimilar 10 11 cm. Their durations are several 100 seconds, up to 1000 seconds for the longest case, GRB 060218, which is classified as a low luminosity GRB associated with a supernova SN 2006aj. The number of GRBs whose spectra exhibit the thermal component now reaches several dozens (see, Starling et al. 2012; Sparre & Starling 2012). Several models to explain this emission component have been presented. As an example, it is proposed that the supernova shock breakout can be responsible for the emission of some GRBs (e.g., Waxman et al. 2007; Li 2007). On the other hand, for GRB 060218/SN 2006aj, it is pointed out that the radiated energy and the inferred emitting radius are too large to ascribe the emission to the supernova shock breakout from the progenitor surface (e.g., Ghisellini et al. 2007). Therefore, some authors ascribe the large emitting radius to the presence of a stellar wind with a high mass-loss rate. In this model, the shock emerges from the photosphere located in the wind. Another proposed model is the cocoon emission. The cocoon is a hot plasma resulting from the interaction between the jet and the stellar material. It emerges from the star at the same time the collimated jet penetrates the stellar surface and then expands spherically at mildly relativistic speeds. Pe'er et al. (2006) investigated emission from the cocoon by combining a numerical radiative transfer calculation with an analytical treatment of the dynamical evolution of the cocoon. While their model is easy to treat, it is necessary to check whether some parameters used there, such as, the total energy of the cocoon, are realized in actual situations by using hydrodynamical calculations. In particular, by performing hydrodynamical calculations, one can estimate the amount of energy deposited into the cocoon out of the total injected energy in a self-consistent way. Furthermore, the large emitting radii inferred from spectral analyses indicate that the emission comes from the region where the CSM is expected to be present. If so, the ejecta-CSM interaction must give rise to thermal X-ray emission. This effect should also be investigated by hydrodynamical calculations. In addition, the cocoon emission might be important as a source of seed photons for inverse Compton to produce high-energy photons with energies of ∼ 100 MeV, as pointed out by Toma et al. (2009). In this Letter, to investigate the interaction between the ejected matter and the CSM, we perform special relativistic hydrodynamical calculations of the propagation of a relativistic jet emanating from a massive star in the CSM. In Section 2, we describe our method to calculate the evolution of the jet and the interaction with the CSM. Results of the hydrodynamical simulations and the expected light curves of the emission from the cocoon are presented in Section 3. Finally, in Section 4, we discuss implications from the results and conclude this Letter.", "pages": [ 1, 2 ] }, { "title": "2. METHOD", "content": "In this section, we briefly explain setups of the hydrodynamical calculations performed in this study. The detailed code description is found in Suzuki (2012).", "pages": [ 2 ] }, { "title": "2.1. Hydrodynamics", "content": "We perform hydrodynamical calculations of the propagation of an ultra-relativistic jet in a massive star and the subsequent interaction with the CSM by using the special relativistic hydrodynamics code in 2D spherical coordinates ( r, θ ) developed by one of the authors. In this code, we adopt a mapping procedure, in which the width of the radial zones is doubled as the jet head reaches a fraction ( ∼ 0 . 9) of the maximum of the radial coordinate, in order to calculate the propagation of the jet till t ∼ 1800 s. Thus, the radial resolution becomes coarser as the time elapses. At t = 0, the radial coordinate ranges from r = 10 9 cm to r = 10 11 cm. At the end of the calculations, the maximum of the radial coordinate reaches r ∼ 6 × 10 13 cm. The radial zone is divided into N r uniform cells and the number N r = 1024 is fixed. The angular coordinate θ ranges from θ = 0 to θ = π/ 2 and is composed of N θ = 256 uniform cells.", "pages": [ 2 ] }, { "title": "2.2. Simulation setup", "content": "As a presupernova model, we adopt 16TI model in Woosley & Heger (2006), which is commonly used in calculations of collapsar jets. In this study, we consider several models to clarify the effect of the ejecta-CSM interaction. Since the spatial distribution of the CSM is highly uncertain, we adopt the simplest steady wind model whose density profile is given by, The density profile is uniquely determined for a given ratio of the mass-loss rate ˙ M and the wind velocity v w . In this study, the wind velocity v w is fixed to be 1000 km s -1 . We performed calculations with the mass-loss rates of ˙ M = 10 -7 , 10 -6 , 10 -5 , 10 -4 , and 10 -3 M /circledot yr -1 . In the following, we especially focus on the two extreme cases, the models with ˙ M = 10 -7 and 10 -3 M /circledot yr -1 (hereafter they are referred to as the dense and dilute CSM models). The jet is injected from the inner boundary r = 10 9 cm from t = 0 to t = 60 s at a constant energy injection rate by using the same method as the previ- ous works (e.g., Zhang et al. 2003; Morsony et al. 2007; Mizuta et al. 2011). The parameters specifying the jet injection condition are as follows: the total energy E tot = 3 × 10 52 erg, the energy injection rate ˙ E = 5 × 10 50 erg/s, the opening angle θ j = 10 · , the initial Lorentz factor Γ 0 = 5, and the specific internal energy /epsilon1 0 /c 2 = 20.", "pages": [ 2, 3 ] }, { "title": "3.1. Jet dynamics", "content": "A lot of previous works on an ultra-relativistic jet emanating from the progenitor star have been carried out and unveiled the dynamical evolution of the jet, such as, the formation of the recollimation shock and the realization of the well-known fireball solution. (e.g., Zhang et al. 2003; Morsony et al. 2007; Mizuta et al. 2011). Our calculations successfully reproduce and confirm their findings. Some snapshots of the spatial distributions of the Lorentz factor and the density of the dilute CSM model are shown in Figure 1. The jet propagates in the interior of the progenitor star and then breaks out, and ejects stellar materials into the circumstellar space. As seen in the top right panel, the emergence of a hot material from the jet cavity follows the breakout of the collimated jet. The ejecta rapidly expand to form a spherical cocoon as seen in the bottom left panel of Figure 1. It is noteworthy that the cocoon expands at mildly relativistic speeds. The appearance and the subsequent expansion of the cocoon have also been reported and investigated by several previous works (see, e.g., Aloy et al. 2000; Ramirez-Ruiz et al. 2002; Zhang et al. 2003; Lazzati & Begelman 2005).", "pages": [ 3 ] }, { "title": "3.2. Effect of CSM interaction", "content": "Results of the dense and dilute CSM models are compared in Figures 2 and 3. Figure 2 represents the spatial distribution of the Lorentz factor (left) and the pressure (right) at t ∼ 200 s for the dense CSM model (lower panel) and the dilute CSM model (upper panel). Near the jet axis ( θ < 10 -20 · ), no difference between the two models is recognized. On the other hand, in the region with large inclination angles ( θ > 20 · ), we can see differences between the models. Denser CSM reduces the size of the cocoon in comparison with dilute CSM. In addition, the pressure distribution shows shelllike structure in the dense CSM. This difference can also be seen in the radial profiles of some physical variables of the cocoon, as illustrated in Figure 3. In both cases, the expansion velocities are mildly relativistic as seen in the top panel. From the bottom panel showing the pressure profiles, one can see that the reverse shock forms as a result of the cocoondense CSM interaction. On the other hand, in dilute CSM, the rarefaction wave propagates toward the center in the cocoon. This is due to the aspherical distribution of the energy deposited into the ejecta. Near the jet axis, the energy carried by the jet is too enormous for the CSM to affect the jet propagation. On the other hand, the energy deposited into the cocoon component is much smaller than that of the jet. The kinetic energy and mass of the cocoon component, which are now defined as those confined in the region outside the star and θ > 10 · , can be obtained from results of the simulation. They are found to be 3 × 10 50 erg and 2 × 10 -3 M /circledot at t = 10 s and 10 51 erg and 2 × 10 -2 M /circledot at t = 20 s. These values are almost independent of the mass-loss rate, because the dissipation of the kinetic energy of the jet to form the cocoon takes place in the star. The kinetic energy and mass of the cocoon increase due to the continuous energy and mass injection by the jet. In our calculations, the injection of the jet is terminated at t = 60 s , which means that the injection of the mass and kinetic energy into the cocoon lasts even after the cocoon begins to expand. The kinetic energy of the cocoon is up to a few % of the total injected energy and thus has a potential for producing thermal Xray photons with the observed luminosity ∼ 10 45 -48 erg s -1 for several hundreds seconds. On the other hand, the mass of the ultra-relativistic jet component, which is defined as the material with the Lorentz factor larger than 100, is 3 × 10 -6 M /circledot , while a substantial fraction ( ∼ 10 51 erg) of the injected energy is carried by this component. As the radial profiles along θ = 45 · at t = 200 s in Figure 3 shows, a reverse shock is formed in the dense CSM model. The other model with the mass-loss rates 10 -4 M /circledot yr -1 also form a reverse shock. The energy of the matter ejected immediately after the breakout of the jet from the surface results from the dissipation of a part of the kinetic energy of the jet for the initial several seconds. Denoting the dissipated kinetic energy by E dis and the fraction of the internal energy to the total by /epsilon1 , the pressure of the cocoon scales as where we have assumed that the cocoon is spherically expanding at the velocity v exp . On the other hand, the ram pressure of the CSM behind the forward shock is given by, where Γ exp = (1 -v 2 exp /c 2 ) -1 / 2 . The reverse shock forms when the pressure P c of the cocoon becomes comparable to the ram pressure ρ w Γ 2 c 2 of the shocked CSM. The balance between the pressure of the cocoon and the ram pressure yields the following expression for the time of the reverse shock formation, where we have derived the final expression by assuming v exp = 0 . 9 c . The value of the fraction /epsilon1 is found from the result of hydrodynamical calculations. This rough estimation is consistent with the fact that a reverse shock is observed for models with ˙ M = 10 -3 and 10 -4 M /circledot yr -1 at t = 200 s and no reverse shock for models with lower mass-loss rates.", "pages": [ 3, 4 ] }, { "title": "3.3. Photospheric emission", "content": "In the following, we investigate whether thermal Xray emission from GRBs can probe their circumstellar environments. We derive the expected light curve and the spectra of thermal X-ray emission from our models by calculating the photospheric emission. According to Starling et al. (2012), thermal emission with the isotropic luminosity of the order of 10 47 erg s -1 and the photon temperature ∼ 0 . 1 -0 . 9 keV is observed. Illuminated by the radiation, heavy atoms, such as oxygen and carbon, are rapidly photo-ionized. The recombination time scale much longer than the ionization time scale keeps those ions fully ionized and the dominant opacity source becomes electron scattering. Thus, we calculated the Thomson photosphere from a distant observer along the axis of the jet ( θ = 0). In deriving light curves and spectra, we have assumed that the matter and radiation are strongly coupled and the internal energy density on the photosphere is dominated by that of radiation. At first, we briefly consider the properties of the emission. Since the ejecta move at mildly relativistic velocities with the Lorentz factor of a few, the relativistic beaming effect strengthen the emission, especially, in the early phase. From the top and bottom panels of Figure 3, the radiation temperature of the shocked region for the dense CSM can be estimated to be, in the observer frame. This is consistent with observed values. The resultant light curves and time-integrated νF ν spectra of the photospheric emission for both models are presented in Figures 4 and 5. At first, for both models, the photospheric emission is very bright for the first ∼ 200 sec. This is because the cocoon is hot immediately after the emergence from the stellar or wind photosphere. After the early phase, the radial velocity at the photosphere decreases as the photosphere moves inward and the cocoon gradually cools. This corresponds to the decrease of the luminosity. The cocoon cools in different ways for the dense and the dilute CSM. As seen in Figure 3, the cocoon component is shocked in the dense CSMmodel. The shock converts the kinetic energy of the cocoon into the thermal energy and keeps the shocked region hot. As a result, the photospheric emission remains luminous even at t ∼ 1000 sec. In the dilute CSM model, on the other hand, the cocoon adiabatically cools. Figure 5 shows the time-integrated νF ν spectra for the dense (upper panel) and dilute (lower panel) CSM models. In each panel, νF ν spectra integrated over t = 0-200 s (solid line), t = 200-1500 s (dashed line), and t = 0-1500 s (dotted line) are plotted. If we fit a blackbody spectrum with a single temperature to each of these spectra, the temperature is found to be k B T = 0 . 16 , 0 . 077, and 0 . 13 keV for the 0-200 s, 200-1500 s, and 0-1500 s spectra of the dense CSM model and k B T = 0 . 061 , 0 . 038 , and 0 . 051 keV for the dilute CSM model. In high-energy part, however, a deviation from the planck function is prominent. This shows that each spectrum is actually superposition of blackbody spectra with different temperatures.", "pages": [ 4, 5 ] }, { "title": "4. DISCUSSION AND CONCLUSIONS", "content": "In this study, we performed hydrodynamical simulations of a jet emerging from a massive star surrounded by the CSM. Especially, we focus on the effect of the interaction between the ejecta and the CSM. The CSM is assumed to be a steady wind with the wind velocity v w = 1000 km s -1 and the mass-loss rates ranging from ˙ M = 10 -7 M /circledot yr -1 to 10 -3 M /circledot yr -1 . We found that the dynamical behavior of the cocoon, which expands at sub-relativistic speeds, is significantly affected by the ejecta-CSM interaction, while the collimated jet is so energetic that the CSM can not decelerate it even for the dense CSM model. In the dense CSM, the cocoon is shocked and thus remains hot even at ∼ 1000 s after the jet injection. On the other hand, in the dilute CSM, the cocoon cools adiabatically. Furthermore, calculating the photospheric emission from the cocoon, we found that the difference can be detected by observing their early thermal X-ray emission. Here we compare our results with the calculation by Pe'er et al. (2006). They consider emission from a freely expanding spherical cocoon with the initial internal energy of 3 × 10 51 , 10 52 , and 3 × 10 52 erg. On the other hand, the internal energy of the cocoon reproduced in the present calculations is less than these assumed values. As a result, for the dilute CSM model, where a freely expanding cocoon is realized, the photon temperature of the photospheric emission is lower than 0 . 1 keV. For the dense CSM model, a fraction of the kinetic energy of the cocoon can be converted into the internal one by the reverse shock, which leads to a higher photon temperature and brighter photospheric emission. This effect is not taken into account by Pe'er et al. (2006). From GRB 060218/SN 2006aj, a bright thermal X-ray emission with the temperature k B T /similarequal 0 . 1-0 . 2 keV is observed even in a few thousand seconds after the trigger. Campana et al. (2006) ascribed the long-lived thermal X-ray emission to a high mass-loss rate of ˙ M /similarequal 3 × 10 -4 M /circledot yr -1 . Our results show that the shocked cocoon is realized at this mass-loss rate. Interestingly, some fundamental features of the emission, such as the photon temperature and the long duration, are also reproduced by the photospheric emission from the shocked cocoon . Therefore, this event might occur in a dense circumstellar environment. However, we cannot reproduce the detailed temporal evolution of the observed photon temperature and the light curve by the present calculations in which the steady wind model is assumed. We attribute this discrepancy to the inhomogeneities of the CSM profile as described below. We should note that there are some uncertainties in this study. In particular, the properties of the photospheric emission calculated in this study are expected to strongly depend on the spatial distribution of the CSM. Although we adopted a steady wind model in this study, it may not be realized in actual circumstellar environments of progenitor systems of long GRBs. Of course, the slope of the density profile of the wind depends on its mass-loss history prior to the gravitational collapse. If the progenitor had been rapidly rotating, the wind could have angular dependence. Furthermore, some authors point out that long GRB progenitors have evolved in binary systems (e.g., Podsiadlowski et al. 2004), in which the structure of the material surrounding the system is expected to be much more complicated than single star progenitor cases. We regard the influence of the spatial distribution of the CSM on the properties of early thermal X-ray emission as one of future works. We appreciate A. Heger for kindly providing us the presupernova model used in this study. This work has been partly supported by Grant-in-Aid for JSPS Fellows (21 · 1726) of the Ministry of Education, Science, Culture, and Sports in Japan. Numerical computations were carried out in part on the Cray XT4 and the middle cluster at the Center for Computational Astrophysics, CfCA, of National Astronomical Observatory of Japan.", "pages": [ 5 ] }, { "title": "REFERENCES", "content": "Aloy, M. A., Muller, E., Ib'a˜nez, J. M., Mart'ı, J. M., & MacFadyen, A. 2000, ApJ, 531, L119 Bufano, F., Pian, E., Sollerman, J., et al. 2012, ApJ, 753, 67 Campana, S., Mangano, V., Blustin, A. J., et al. 2006, Nature, 442, 1008 Cano, Z., Bersier, D., Guidorzi, C., et al. 2011, ApJ, 740, 41 Chevalier, R. A. 2012, ApJ, 752, L2 Della Valle, M., Chincarini, G., Panagia, N., et al. 2006, Nature, 444, 1050 Galama, T. J., Vreeswijk, P. M., van Paradijs, J., et al. 1998, Nature, 395, 670 Ghisellini, G., Ghirlanda, G., & Tavecchio, F. 2007, MNRAS, 382, L77 Hjorth, J., Sollerman, J., Møller, P., et al. 2003, Nature, 423, 847 Lazzati, D., & Begelman, M. C. 2005, ApJ, 629, 903 Li, L.-X. 2007, MNRAS, 375, 240 Sparre, M., & Starling, R. L. C. 2012, MNRAS, 427, 296 Stanek, K. Z., Matheson, T., Garnavich, P. M., et al. 2003, ApJ, Mazzali, P. A., Deng, J., Nomoto, K., et al. 2006, Nature, 442, 1018 M'esz'aros, P. 2006, Reports on Progress in Physics, 69, 2259 Mizuta, A., Nagataki, S., & Aoi, J. 2011, ApJ, 732, 26 Morsony, B. J., Lazzati, D., & Begelman, M. C. 2007, ApJ, 665, 569 Olivares E., F., Greiner, J., Schady, P., et al. 2012, A&A, 539, A76 Page, K. L., Starling, R. L. C., Fitzpatrick, G., et al. 2011, MNRAS, 416, 2078 Pe'er, A., M'esz'aros, P., & Rees, M. J. 2006, ApJ, 652, 482 Pian, E., Mazzali, P. A., Masetti, N., et al. 2006, Nature, 442, 1011 Piran, T. 1999, Phys. Rep., 314, 575 Podsiadlowski, P., Mazzali, P. A., Nomoto, K., Lazzati, D., & Cappellaro, E. 2004, ApJ, 607, L17 Ramirez-Ruiz, E., Celotti, A., & Rees, M. J. 2002, MNRAS, 337, 1349", "pages": [ 5, 6 ] } ]

[ { "title": "ABSTRACT", "content": "We present Submillimeter Array observations of H 2 CO and N 2 H + emission in the disks around the T Tauri star TW Hya and the Herbig Ae star HD 163296 at 2 '' -6 '' resolution and discuss the distribution of these species with respect to CO freeze-out. The H 2 CO and N 2 H + emission toward HD 163296 does not peak at the continuum emission center that marks the stellar position but is instead significantly offset. Using a previously developed model for the physical structure of this disk, we show that the H 2 CO observations are reproduced if H 2 CO is present predominantly in the cold outer disk regions. A model where H 2 CO is present only beyond the CO snow line (estimated at a radius of 160 AU) matches the observations well. We also show that the average H 2 CO excitation temperature, calculated from two transitions of H 2 CO observed in these two disks and a larger sample of disks around T Tauri stars in the DISCS (the Disk Imaging Survey of Chemistry with SMA) program, is consistent with the CO freeze-out temperature of ∼ 20 K. In addition, we show that N 2 H + and H 2 CO line fluxes in disks are strongly correlated, indicative of co-formation of these species across the sample. Taken together, these results imply that H 2 CO and N 2 H + are generally present in disks only at low temperatures where CO depletes onto grains, consistent with fast destruction of N 2 H + by gas-phase CO, and in situ formation of H 2 CO through hydrogenation of CO ice. In this scenario H 2 CO, CH 3 OH and N 2 H + emission in disks should appear as rings with the inner edge at the CO midplane snow line. This prediction can be tested directly using observations from ALMA with higher resolution and better sensitivity. Subject headings: protoplanetary disks; astrochemistry; stars: formation; ISM: molecules; techniques: high angular resolution; radio lines: ISM", "pages": [ 2 ] }, { "title": "H 2 CO and N 2 H + in Protoplanetary Disks: Evidence for a CO-ice Regulated Chemistry", "content": "Chunhua Qi Harvard-Smithsonian Center for Astrophysics, 60 Garden Street, Cambridge, MA 02138, USA and Karin I. ¨ Oberg University of Virginia, Departments of Chemistry and Astronomy, Charlottesville, VA 22904, USA and David J. Wilner Harvard-Smithsonian Center for Astrophysics, 60 Garden Street, Cambridge, MA 02138, USA Received ; accepted", "pages": [ 1 ] }, { "title": "1. Introduction", "content": "Planetary systems are assembled from dust and gas in the disks surrounding pre-main sequence stars. The nature of the formed planets are intimately linked to the structure, composition and evolution of the parent circumstellar disk. Molecular emission lines serve as probes of disk characteristics, such as density, temperature and ionization fraction, that are not accessible by other observations. For example, N 2 H + along with the deuterated ions like H 2 D + and DCO + are believed to trace the ionization fraction near the midplane of the disks ( Oberg et al. 2011c). Molecular distributions in disks are also important to characterize because of their connection with the composition of forming planetesimals. This especially true of organic molecules, of which H 2 CO is an important representative. While most molecules are expected to be reprocessed in larger planetary bodies, the disk-chemical composition may survive quite intact in icy planetesimals, including comets (Mumma & Charnley 2011). Such planetesimals may have seeded the Earth with water and organics, connecting disk chemistry with the origins of life. Predicting the organic composition of these planetesimals depend on our understanding of the distribution of the organic composition of grains in the disks. The overall disk chemical structure is set by a combination of photochemistry at the disk surface and sequential freeze-out of molecules in the disk interior (e.g. Aikawa et al. 2002). The distributions of N 2 H + and H 2 CO and their relationship to CO present important test cases of the chemical models. The CO molecule is one of the last to freeze out and predicted to deplete quickly from the gas-phase at T < 20 -25 K for typical disk mid-plane densities. Abundant N 2 H + is expected only where CO is depleted because N 2 H + forms from protonation of N 2 , which remains in the gas-phase at temperatures a few degrees lower than CO ( Oberg et al. 2005), and is destroyed mainly by reactions with CO (Bergin et al. 2002). In dense cores in star forming regions, the abundance of CO is observed to show a strong anti-correlation with N 2 H + (e.g. Caselli et al. 1999; Bergin et al. 2002; Jørgensen 2004). In disk models, this effect is manifested as a jump in the N 2 H + column density at the CO 'snow line', one order of magnitude in a recent calculation (Walsh et al. 2012). The snow line is here defined as the disk radius where the midplane dust temperature is cold enough for volatiles to condense into ice grains. The H 2 CO chemistry is more complicated than N 2 H + because H 2 CO can form through several different pathways, both in the gas-phase and on grain-surfaces. Grain-surface formation of H 2 CO should depend directly on CO freeze-out; constrained by theory (Tielens & Hagen 1982; Cuppen et al. 2009) and experiments (e.g. Watanabe et al. 2003; Fuchs et al. 2009), H 2 CO (and CH 3 OH) form readily from CO ice hydrogenation. If this is the dominant formation pathway of H 2 CO in disks, and if ices are partially desorbed non-thermally (e.g. Garrod et al. 2007; Oberg et al. 2009b,a), then H 2 CO gas should coincide with N 2 H + exterior to the CO snow line. To date, emission from millimeter wavelength H 2 CO and N 2 H + lines has been detected toward 8 and 6 protoplanetary disks, respectively (Dutrey et al. 1997; Aikawa et al. 2003; Qi et al. 2003; Thi et al. 2004; Dutrey et al. 2007; Henning & Semenov 2008; Oberg et al. 2010, 2011b). Most detections are toward T Tauri stars with massive disks, and the detection fraction toward more luminous Herbig Ae stars is low ( Oberg et al. 2011b). Based on these observations, H 2 CO and N 2 H + are mainly abundant in disks with large reservoirs of cold dust and gas, where CO freeze-out is expected to occur. A direct connection between CO freeze-out and N 2 H + and H 2 CO in disks has yet to be observationally established, however. In this paper we present Submillimeter Array (SMA) observations of H 2 CO and N 2 H + toward the disks around HD 163296 and TW Hya, and we use these new observations along with H 2 CO and N 2 H + observations from DISCS (Disk Imaging Survey of Chemistry with SMA) to constrain the H 2 CO and N 2 H + distributions. The new data and their calibration are described in § 2. In § 3, we present the H 2 CO and N 2 H + images and spectra toward HD 163296 and TW Hya, models of the H 2 CO distribution toward HD 163296, H 2 CO excitation temperature calculations, and examine the relationship between H 2 CO and N 2 H + emission across the sample of disks. In § 4, we discuss the implications of these results, summarize the mounting evidence for CO-ice regulated chemistry and make predictions for future observations of H 2 CO and N 2 H + emission from disks with better sensitivity and resolution.", "pages": [ 3, 4, 5 ] }, { "title": "2. Observations", "content": "The observations of HD 163296 (R . A . = 17 h 56 m 21 . 279 s , decl . = -21 · 57 ' 22 . '' 38; J2000.0) were made between 2008 and 2012, and of TW Hydrae (R . A . = 11 h 01 m 51 . 875 s , decl . = -34 · 42 ' 17 . '' 155; J2000.0) between 2008 and 2012, using the eight-antenna Submillimeter Array (SMA) located atop Mauna Kea, Hawaii. Table 1 provides a summary of the observational parameters and results. For the 2007 and 2008 observations, the SMA receivers operated in a double-sideband mode with an intermediate frequency (IF) band of 4-6 GHz from the local oscillator frequency, sent over fiber optic transmission lines to 24 overlapping 'chunks' of the digital correlator. The 2012 observations were made after an upgrade that enabled a second IF band of 6-8 GHz, effectively doubling the bandwidth. The SMA observations of HD 163296 were carried out in the compact-north (COM-N), compact (COM) and subcompact (SUB) array configurations. The 2007 observations included the DCO + 3-2 line at 216.1126 GHz and the H 2 CO 3 1 , 2 -2 1 , 1 line at 225.698 GHz. The 2012 observations included the N 2 H + 3-2 at 279.512 GHz and the H 2 CO 4 1 , 4 -3 1 , 3 line at 281.527 GHz. The observing loops used J1733-130 as the main gain calibrator and observed J1744-312 every other cycle to check the phase calibration. Flux calibration was done using observations of Titan and Uranus. The derived fluxes of J1733-130 were 1.19 Jy (2007 Mar 20), 1.25 Jy (2012 Jun 10), 1.40 Jy (2012 Aug 12 and 14). The bandpass response was calibrated using observations of 3C279, Uranus and J1924-292. The SMA observations of TW Hya were carried out in the compact (COM) and subcompact (SUB) array configurations. The 2008 observations included the H 2 CO 5 1 , 5 -4 1 , 4 line at 351.769 GHz. The 2012 Jan 13 SUB observation included the H 2 CO 4 1 , 4 -3 1 , 3 and N 2 H + 3-2 lines, like HD 163296, but unfortunately the chunk containing N 2 H + was corrupted and unusable. The 2012 Jun 04 COM observation using a similar setting successfully included the N 2 H + line. The observing loops used J1037-295 as the gain calibrator. Flux calibration was done using observations of Titan and Callisto. The derived fluxes of J1037-295 were 0.73 Jy (2008 Feb 23), 0.73 Jy (2012 Jan 13) and 0.82 Jy (2012 Jun 4). The bandpass response was calibrated using observations of 3C279 and 3C273. Routine calibration tasks were performed using the MIR software package 1 , and imaging and deconvolution were accomplished in the MIRIAD software package.", "pages": [ 5, 6 ] }, { "title": "3. Results", "content": "In this section we present detections of H 2 CO and N 2 H + emission lines toward HD 163296 and TW Hya, display their respective distributions ( § 3.1), and compare the higher quality observations of H 2 CO in HD 163296 with models ( § 3.2). We then combine the new data with previously reported H 2 CO and N 2 H + detections in disks to examine trends with respect to H 2 CO excitation temperature ( § 3.3) and with each other ( § 3.4).", "pages": [ 6 ] }, { "title": "3.1. H 2 CO and N 2 H + towards HD 163296 and TW Hya", "content": "Figure 1 shows images of the spectrally integrated emission toward TW Hya and HD 163296 at the rest frequencies of two H 2 CO lines and the N 2 H + J = 3 -2 line. H 2 CO 4 1 , 4 -3 1 , 3 and 5 1 , 5 -4 1 , 4 and N 2 H + are detected toward TW Hya, and H 2 CO 3 1 , 2 -2 1 , 1 and H 2 CO 4 1 , 4 -3 1 , 3 and N 2 H + are detected toward HD 163296. The emission toward TW Hya appears to be centrally peaked at the size scale of the beams (FWHM > 2 '' ). Toward HD 163296, however, neither the H 2 CO lines nor the N 2 H + line emission peaks at the location of the continuum peak that marks the stellar position, but instead show significant offsets. This is most readily apparent for the H 2 CO 3 1 , 2 -2 1 , 1 line that was observed with a slightly smaller and more advantageously rotated beam, where the emission appears ring-like. This is the second reported observation of a ring-like H 2 CO distribution after DM Tau (Henning & Semenov 2008). Interpreting the H 2 CO emission toward DM Tau is however complicated by a large central cavity in dust emission (Andrews et al. 2011). Figure 2 shows the spatially integrated spectra. The line shapes and central velocities agree with what has been previously observed for other molecular lines toward these disks. The line fluxes are listed in Table 1, and the values are comparable to detections of these lines toward other large protoplanetary disks ( Oberg et al. 2010, 2011b).", "pages": [ 7 ] }, { "title": "3.2. HD 163296 H 2 CO Model Results", "content": "Figure 1 demonstrates that H 2 CO emission toward HD 163296 is spatially resolved and thus contains information on the radial distribution of H 2 CO in the disk. The relative excitation of the two H 2 CO transitions should probe primarily the vertical distribution and thus provide complementary constraints. We explore the H 2 CO distribution based on a previously developed accretion disk model with a well-defined temperature and density structure, constrained by the HD 163296 broadband spectral energy distribution, spatially resolved millimeter dust continuum, and multiple CO and CO isotopologue line observations (Qi et al. 2011). We adopt the same methods as Qi et al. (2008) for constraining the H 2 CO distribution, here fitting models that assume a radial power-law ( § 3.2.1) and a simple ring with inner boundary at the CO 'snow line' ( § 3.2.2).", "pages": [ 7, 8 ] }, { "title": "3.2.1. Power-law Model", "content": "For a first-order analysis of the distribution of H 2 CO, we model the radial variation in the column density as a power law N 100 × ( r/ 100) p between an inner radius R in and outer radius R out , where N 100 is the column density at 100 AU in cm -2 , r is the distance from the star in AU, and p is the power-law index. For the vertical distribution, we assume that H 2 CO is present with a constant abundance in a layer with boundaries toward the midplane and toward the surface of the disk (similar to Qi et al. (2008)). This assumption is motivated by chemical models (e.g. Aikawa & Nomura 2006) that predict a three-layered structure where most molecules are photodissociated in the surface layer, frozen out in the midplane, and have an abundance that peaks at intermediate disk heights. The surface ( σ s ) and midplane ( σ m ) boundaries are presented in terms of Σ 21 = Σ H / (1 . 59 × 10 21 cm -2 ), where Σ H is the hydrogen column density measured from the disk surface. This simple model approximates the vertical location where H 2 CO is most abundant. The excitation of multiple transitions can constrain both σ s and σ m , but in this case of very modest signal-to-noise, we fix σ s to 0.79, the surface boundary found for CO by Qi et al. (2011), and we fit σ m for the midplane boundary and the power-law parameters (N 100 , p , R in and R out ). Using the structure model, we compute a grid of synthetic H 2 CO visibility datasets over a range of R out , R in , p , σ m and N 100 values and compare with the observations. The best-fit model is obtained by minimizing χ 2 , the weighted difference between the real and imaginary part of the complex visibility measured in the ( u, v )-plane sampled by the SMA observations of both H 2 CO transitions. We use the two-dimensional Monte Carlo model RATRAN (Hogerheijde & van der Tak 2000) to calculate the radiative transfer and molecular excitation. The collisional rates are taken from the Leiden Atomic and Molecular Database (Schoier et al. 2005). Table 2 lists the best-fit parameters of the model. The power-law index of 2 implies an H 2 CO column density that strongly increases with radius. Figure 3 shows the best-fit radial distribution of the H 2 CO column density. Figures 4 and 5 present comparisons between the observed channel maps and the best-fit model. The model reproduces the main features of the observations remarkably well, in particular the flux ratio between the inner and outer channels, and the lack of emission at the location of the continuum peak. Figure 6 shows the χ 2 surfaces for the R in and R out versus the power law index p , which enables us to quantify the uncertainties associated with the inner and outer region sizes and the power-law index. We find that p is constrained between 0.5-3.0 (within 1 σ ) while R in is constrained to be < 200 AU. The χ 2 value does not change significantly for inner radii < 90 AU, as expected for the 2 '' beam size of the observations. The outer radius is better constrained, since the emission is very sensitive to the value of R out with a positive power-law index. Figure 7 shows the H 2 CO 3-2 line spectrum compared with the spectra derived from models with different radial column densities power-law indices. The spectra suggest a lack of high velocity line wings associated with emission originating in the inner regions of the disk, consistent with the results of the χ 2 analysis.", "pages": [ 8, 9 ] }, { "title": "3.2.2. Ring Model", "content": "The positive power-law index found for the H 2 CO radial distribution implies that H 2 CO is present mainly in the outer disk. This is expected if H 2 CO forms in situ from CO ice hydrogenation and is therefore present mainly beyond the CO snow-line, previously determined to be at 160 AU by Qi et al. (2011). Guided by this astrochemical ansatz , we have tried a second 'ring' model where the H 2 CO gas is only present where CO has frozen out. The vertical surface boundary is then defined by the CO freeze-out temperature of 19 K (Qi et al. 2011), while the midplane boundary can be constrained by the excitation of multiple H 2 CO transitions, as in the power-law model. Within this layer, the abundance of H 2 CO is assumed to follow that of H nuclei with a constant fractional abundance, which is also a parameter fit to the data. The best-fit abundance is 5 . 5 × 10 -11 and the midplane boundary σ m is consistent with what we find in the power-law model (Table 2). Figure 3 shows that the vertical surface boundary at 19 K effectively results in a ring-like radial structure, where the inner edge of the ring is at CO snow line. Figure 3 also shows that the best-fit power-law and ring models result in similar H 2 CO column densities beyond the CO snow line. The profile of the ring model is considerably flatter than the power-law model and even drops outside of 300 AU exponentially to the edge of CO emission. Figures 4-5 show that the power-law model and the ring model channel maps display some subtle differences. But both provide good fits to the data within the noise of the SMA observations. The same model also provides a good match to the N 2 H + flux, but the combination of low signal-to-noise and observations of just one transition preclude any independent modeling of the N 2 H + distribution.", "pages": [ 10 ] }, { "title": "3.3. H 2 CO Excitation Temperatures", "content": "HD 163296 is very favorable for H 2 CO and N 2 H + imaging since its relatively high luminosity and massive disk puts the CO snow line at a large angular distance compared to other disks, enabling us to resolve the H 2 CO and N 2 H + emission. Without such spatial information, however, we can still obtain a constraint on where H 2 CO emission originates in disks based on the average H 2 CO excitation temperatures. As a gross approximation, H 2 CO that coexists with CO ice should be cold, i.e. present at an excitation temperature comparable to the CO freeze-out temperature. To test the viability of using excitation temperatures to constrain the H 2 CO distribution we extracted spectra from three of the HD 163296 simulations presented in § 3.2, selecting models with the best outer radius, no inner hole and power-law indices of -2, 0, and 2. These distributions approximately correspond to a H 2 CO abundance that follows the H 2 column, that keeps constant with radius and that increases steeply, forming a ring. Assuming LTE, that both H 2 CO transitions trace the same underlying populations, and a single rotational excitation temperature, T rot , we calculate where E 0 and E 1 are the upper energy levels for the low and high H 2 CO transitions used in the calculation (H 2 CO 3 0 , 3 -2 0 , 2 and 4 1 , 4 -3 1 , 3 for most disks), ν and Sµ 2 the corresponding line frequencies and temperature independent transition strengths and dipole moments, ∫ Tdv the integrated line intensity, which is calculated from the integrated fluxes based on F/T = 13 . 6 λ 2 / ( a × b ), where F is the flux in Jy, T the intensity in K, λ the line wavelength in millimeters, and a and b the emission diameters in '' . Because of both vertical and radial temperature gradients in the disk, the size dependence of the emission regions from two transitions is complicated but the emission area is not expected to be very different. For simplicity we assume the extent of the emission is the same for both transitions on account of the model dependent effects of the temperature gradients. All of the line parameters were gathered from Splatalogue (a transition-resolved compilation of several spectroscopic databases), with the data originating from CDMS (Muller et al. 2005). LTE is a reasonable approximation if H 2 CO is mainly present at high densities. The critical densities for the observed H 2 CO transitions vary between 10 5 and 5 × 10 6 cm -3 (Troscomptet al. 2009), dependent on assumed kinetic temperatures. At radii < 300 AU, all gas colder than 25 K is at densities higher than 10 7 cm -3 (Qi et al. 2011), justifying this assumption. Using the simulated spectra we derive an excitation temperature of 27 K for the model with p = -2 and excitation temperatures close to or below 20 K for the other two models. Excitation temperatures thus provide some constraints on the H 2 CO distribution, but a temperature close to the expected CO freeze-out temperature only implies an increasing abundance with radius; it cannot be used to assess how steeply the abundance increases. By combining the new H 2 CO detections toward TW Hya and HD 163296 with H 2 CO data from DISCS ( Oberg et al. 2010, 2011b), we have a sample of 10 disks with two H 2 CO line detections or one H 2 CO line detection and one upper limit (Table 3). These data are sufficient to calculate excitation temperatures, albeit with substantial uncertainties. Figure 8 shows the calculated excitation temperatures for 9 of the 10 disks; the chemically peculiar Herbig Ae star HD 142527 is not included because its excitation temperature of 250 K is probably not due to thermal excitation. The H 2 CO excitation temperature, listed in Table 3, is consistent with, or lower than, the CO freeze-out temperature of ∼ 20 K for all of these disks. The average H 2 CO temperature in the sample (excluding HD 142527) is 18 ± 6 K. We note that the relatively high excitation temperature toward HD 163296 is most likely due to the fact that some H 2 CO 3-2 emission is resolved out by the SMA observations, as the compact-north antenna configuration has few short baselines.", "pages": [ 11, 12 ] }, { "title": "3.4. H 2 CO/N 2 H + Correlations", "content": "We examine the disk sources to test if N 2 H + and H 2 CO emission are correlated across the sample, as would be expected if 1) the two molecules form under the same physio-chemical conditions, i.e. only in the regions where CO has frozen out, 2) the line emission trace the total N 2 H + and H 2 CO column well, and 3) midplane ionization levels and CO hydrogenation efficiencies do not differ 'too much' across the sample (i.e. the size of the CO freeze-out region is the most important regulator of N 2 H + and H 2 CO column across the sample). It should be noted that there are other scenarios that could produce a correlation as well, and that the correlation analysis below should only be considered as a constraint on the H 2 CO distribution in combination with the results in the previous sections. In particular a constant H 2 CO/N 2 H + across a disk sample would be expected if the relative fractions of chemically characteristic disk regions is always similar in disks. To conclusively test this requires a larger sample than currently available, but the fact that we did not find that H 2 CO emission correlates with any other molecular emission than N 2 H + already challenges this scenario. Where possible, we base the comparison on the H 2 CO 3 0 , 3 -2 0 , 2 line that has E up = 21 K, similar to N 2 H + J = 3 -2 ( E up = 27 K), to minimize variations in fluxes caused by the different detailed temperature structures in different disks. Excluding HD 142527, this line has been observed toward 6/9 of the sample. For the remaining 3/9 disks, we calculate the expected H 2 CO 3 0 , 3 -2 0 , 2 line flux based on the H 2 CO excitation temperatures and fluxes of other H 2 CO lines toward each source. We then normalize the flux of each H 2 CO 3 0 , 3 -2 0 , 2 line to the (Taurus) distance of 140 pc, and we further normalize to a disk mass of 0.01 M /circledot to account for the fact that more nearby and more massive disks tend to have overall stronger line emission. Figure 9 shows that there is a strong correlation between the normalized H 2 CO and N 2 H + fluxes in the disk sample; the rank correlation is statistically significant at the 95% level. As expected, Figure 10 shows that this implies a nearly constant N 2 H + 3-2 / H 2 CO 3 0 , 3 -2 0 , 2 flux ratio across the sample.", "pages": [ 13, 14 ] }, { "title": "4. Discussion", "content": "Our modeling strategy in this study and in Qi et al. (2008) has been to first constrain the overall structure of molecular emission in disks using a parametric model with a minimum of free parameters, i.e. to determine whether the radial column density profile of a species decreases, increases or is flat as a function of disk radius. As demonstrated in Oberg et al. (2012), the slope of the radial column density profile already can place significant constraints on the formation pathway of a molecule. Here we find that H 2 CO toward HD 163296 belongs to the family of molecules that display an increasing column density with radius. This first-order constraint on the H 2 CO distribution motivated us to consider H 2 CO formation pathways that would result in an increase of H 2 CO with disk radius. H 2 CO formation through CO-ice hydrogenation results in a simple prediction that H 2 CO should be present in a ring, with the inner edge at the CO snow line. We therefore set up a second model, based on this prediction, to test if the observations are consistent with this hypothesis for H 2 CO formation. We propose that this combination of backward and forward modeling both provides a fair view of the constraints obtained by fitting the data, and challenges our basic understanding of disk chemistry.", "pages": [ 14 ] }, { "title": "4.1. H 2 CO Formation", "content": "H 2 CO can form through multiple chemical pathways. We have shown that the H 2 CO distribution towards HD 163296 and the sample statistics are consistent with formation through in situ CO ice hydrogenation. Here we consider the effects of additional pathways for H 2 CO formation, in particular (1) in situ gas phase formation, and (2) formation in the pre- and proto-stellar phases, followed by incorporation into the disk. H 2 CO can form in the gas-phase through ion-neutral reactions involving, e.g. CH + 3 or through neutral-neutral reactions between CH 3 and O (e.g. Aikawa & Herbst 1999). The neutral-neutral formation pathway is expected to result in a radially flat column density structure for a typical T Tauri disk (Aikawa & Herbst 1999), with most emission originating at temperatures of 20-40 K (Aikawa et al. 2003). It is not clear from existing disk models whether the structure will look substantially different if the neutral-ion reactions dominate. In a protostellar chemistry model (Bergin & Langer 1997), CH + 3 disappears when CO depletes (E. Bergin, private communication). This suggests that H 2 CO forming through this pathway should be anti-correlated with CO-freeze-out. Neither of these gas-phase pathways thus predicts excess H 2 CO column densities in the outer disk, or at low (T < 20 K) temperatures. The observed low excitation temperature could on its own be explained by efficient turbulent mixing of H 2 CO formed in the gas-phase and then cooled down in the midplane regions, similarly to what has been proposed to explain cold CO gas in disks (Aikawa 2007). Turbulent mixing would not, however, explain the observed deficiency of H 2 CO towards the inner disk in HD 163296. Gas-phase formation of H 2 CO then seems an unlikely dominant source of H 2 CO in disks in light of the new observations, but the case is unlikely to be conclusively settled until the exclusive grain-surface product CH 3 OH is observed to display a similar distribution. H 2 CO in disks could be a product of protostellar or molecular cloud chemistry, as H 2 CO is commonly observed in pre- and protostellar sources, and this molecular content may be preserved, at least in part, through the process of disk formation. Willacy (2007) has modeled this scenario, starting with a H 2 CO ice abundance of 10 -6 n H inherited from the cold cloud. The model also includes H 2 CO formation through gas-phase processes in the disk and results are presented with and without photodesorption. In the model without photodesorption, H 2 CO follows CO in the inner disk and has an additional outer disk component with an abundance that decreases with radius beyond the CO snow line. When photodesorption is included, the H 2 CO column is flat across the disk, corresponding to a power-law index of 0. Neither predicted abundance pattern is consistent with the new observations. In addition, H 2 CO is common towards protostellar sources of a range of luminosities (e.g. Schoier et al. 2004; Bisschop et al. 2007), while it is pre-dominantly detected towards disks around the low luminosity T Tauri stars. In short, H 2 CO formation through in situ CO-ice hydrogenation is not only consistent with the observations, but it is the only pathway proposed (so far) that naturally explains the observations. To conclusively demonstrate a CO-ice hydrogenation origin would, however, require the detection of co-spatial emission of CH 3 OH; CH 3 OH has no known efficient gas-phase formation pathway and is predicted to form together with H 2 CO whenever CO ice is hydrogenated (Cuppen et al. 2009).", "pages": [ 14, 15, 16 ] }, { "title": "4.2. Locating the CO 'Snow Line'", "content": "The 'snow line' is typically used to denote the midplane disk radius at which the temperature is low enough for water to condense out on dust grains. Outside of the snow line, grain accretion will be faster because of larger and stickier grains, which may substantially speed up the formation of planetesimals and eventually planets (e.g. Hayashi 1981; Ida & Lin 2004, 2008; Ciesla & Cuzzi 2006; Kretke & Lin 2007). In the Solar System, the dividing line between rocky planets and gas giants coincide with the H 2 O snow line (Lewis 1974). CO is another abundant volatile in disks and its snow line may boost planet formation in the outer disk by providing extra solid masses (Dodson-Robinson et al. 2009) and inducing planet traps (Masset et al. 2006; Hasegawa & Pudritz 2012), and could affect the elemental make-up of the forming gas-giants ( Oberg et al. 2011a). Because of its high volatility ( T freeze -out ∼ 20 K), the CO snow line is expected at disk radii of 10s-100s of AU. This should make it a far more accessible target than the H 2 O snow line for millimeter interferometry studies aimed at examining the general effects of snow lines on disk structures. Localizing the CO snow line directly from millimeter CO data is challenging, however. Disks have both a radial temperature gradient away from the central star, and a vertical one set by radiative heating at the disk surface (e.g. Aikawa & Herbst 1999). This results in a CO condensation front that is located at different radii at different disk heights (Fig. 11), and also that some CO is present at all radii in the upper disk layers. This fact, together with a complex radiative transfer (most CO lines are expected to be optically thick in the disk center and optically thin in the outer parts of the disk), means that the location of a CO snow line in a disk cannot be inferred from simply inspecting a CO disk image. The best constraint that exists to date on a CO snow line radius is toward the Herbig Ae star HD 163296 based on the analysis of multi-transition, multi-isotope, spatially resolved CO line data on a self-consistent physical disk model (Qi et al. 2011). The temperature structure of the model has been constrained by optically thick multiple CO lines and detailed analysis of the optically thinner 13 CO emission reveals a significant column density reduction at around 165 AU that cannot be explained by the overall disk column distribution as traced by the dust. This is interpreted as the result of CO freeze-out and the location as the CO snow line. Uncertainties in the temperature structure will mainly affect the determination of the CO freeze-out temperature at the location of the CO snow line, rather than the location itself , which is constrained between 135 and 175 AU in the disk of HD 163296. While fruitful, this is a time consuming approach that will be difficult to apply to larger samples of disks and will always include some degree of model dependency. Another approach to constrain the CO snow line location is to identify trace species that are only present where CO has begun to freeze out. Such molecules should display a ring-like structure with the inner edge corresponding to the midplane CO snow line. As we have described, N 2 H + and H 2 CO are good candidate probes of CO snow lines. In principle, these species can be used as powerful chemical imaging tools to constrain CO snow line locations in large samples of disks, rather than relying on complex analysis of the CO isotopologue observations.", "pages": [ 16, 17, 18 ] }, { "title": "4.2.1. Simulated ALMA Observations", "content": "It seems clear that both N 2 H + and H 2 CO are outer-disk species from the chemical perspective. To connect their formation to the onset of CO freeze-out conclusively requires a combination of higher sensitivity, and higher resolution imaging. This kind of imaging can be done with the newly available capabilities of the ALMA telescope in Chile, which is nearing completion of construction. To demonstrate the astrochemical predictions generated by our analysis of the SMA data, and the ease at which they can be tested with ALMA, we present a set of simulations of HD 163296 ALMA observations using the antenna configuration 5 (corresponding to 0.3 '' resolution) in Figure 12. The predicted H 2 CO and N 2 H + rings are readily detected at this resolution, and the power-law and ring models are clearly distinguished. For both models, the diameter of the emission maximum can be estimated directly from the images within a fraction of the beam size, enabling us to determine if the H 2 CO and N 2 H + emission truly trace the CO snow line. In addition, ALMA should have sufficient sensitivity to detect CH 3 OH if it is present with a similar abundance and distribution as H 2 CO, as expected if both of these species are formed through CO hydrogenation and then non-thermally desorbed.", "pages": [ 18 ] }, { "title": "5. Conclusions", "content": "We have presented three observational results that support the idea that CO freeze-out regulates the H 2 CO and N 2 H + chemistry in disks: These results suggest that both N 2 H + and H 2 CO should be present in rings, with the inner edge at the CO snow line. This may be used as a probe of CO snow line locations across samples of disk and is also important for predicting the organic content of comets forming at different disk radii. In general, the radial and vertical distributions of molecules constitute strong probes of the basic chemistry used into astrochemical models, while molecular abundances and column densities are probably best used to test our understanding of the structure and history of individual objects. Facilities: SMA The SMA is a joint project between the Smithsonian Astrophysical Observatory and the Academia Sinica Institute of Astronomy and Astrophysics and is funded by the Smithsonian Institution and the Academia Sinica. We thank Edwin Bergin and Paola D'Alessio for their helpful suggestions, and a referee for constructive comments on the paper. Support for K. I. O. is provided by NASA through a Hubble Fellowship grant awarded by the Space Telescope Science Institute, which is operated by the Association of Universities for Research in Astronomy, Inc., for NASA, under contract NAS 5-26555. We also acknowledge NASA Origins of Solar Systems grant No. NNX11AK63.", "pages": [ 19, 20 ] }, { "title": "REFERENCES", "content": "Aikawa, Y. & Herbst, E. 1999, A&A, 351, 233 Aikawa, Y., Momose, M., Thi, W., et al. 2003, PASJ, 55, 11 Aikawa, Y. & Nomura, H. 2006, ApJ, 642, 1152 Aikawa, Y. 2007, ApJ, 656, L93 Aikawa, Y., van Zadelhoff, G. J., van Dishoeck, E. F., & Herbst, E. 2002, A&A, 386, 622 Andrews, S., Wilner, D., Espaillat, C., et al. 2011, ArXiv e-prints Andrews, S. M. & Williams, J. P. 2005, ApJ, 631, 1134 Andrews, S. M., Wilner, D. J., Hughes, A. M., Qi, C., & Dullemond, C. P. 2009, ApJ, 700, 1502 Bergin, E. A. & Langer, W. D. 1997, ApJ, 486, 316 Bergin, E. A., Alves, J., Huard, T., & Lada, C. J. 2002, ApJ, 570, L101 Bisschop, S. E., Jørgensen, J. K., van Dishoeck, E. F., & de Wachter, E. B. M. 2007, A&A, 465, 913 Caselli, P., Walmsley, C. M., Tafalla, M., Dore, L., & Myers, P. C. 1999, ApJ, 523, L165 Ciesla, F. J. & Cuzzi, J. N. 2006, Icarus, 181, 178 Cuppen, H. M., van Dishoeck, E. F., Herbst, E., & Tielens, A. G. G. M. 2009, A&A, 508, 275 Dodson-Robinson, S. E., Willacy, K., Bodenheimer, P., Turner, N. J., & Beichman, C. A. 2009, Icarus, 200, 672 Dutrey, A., Guilloteau, S., & Guelin, M. 1997, A&A, 317, L55 Dutrey, A., Henning, T., Guilloteau, S., et al. 2007, A&A, 464, 615 Fuchs, G. W., Cuppen, H. M., Ioppolo, S., et al. 2009, A&A, 505, 629 Garrod, R. T., Wakelam, V., & Herbst, E. 2007, A&A, 467, 1103 Hasegawa, Y. & Pudritz, R. E. 2012, ArXiv e-prints Hayashi, C. 1981, in IAU Symposium, Vol. 93, Fundamental Problems in the Theory of Stellar Evolution, ed. D. Sugimoto, D. Q. Lamb, & D. N. Schramm, 113-126 Henning, T. & Semenov, D. 2008, in IAU Symposium, Vol. 251, IAU Symposium, ed. S. Kwok & S. Sandford, 89-98 Hogerheijde, M. R. & van der Tak, F. F. S. 2000, A&A, 362, 697 Ida, S. & Lin, D. N. C. 2004, ApJ, 616, 567 Ida, S. & Lin, D. N. C. 2008, ApJ, 685, 584 Jørgensen, J. K. 2004, A&A, 424, 589 Kretke, K. A. & Lin, D. N. C. 2007, ApJ, 664, L55 Lewis, J. S. 1974, Science, 186, 440 Lombardi, M., Lada, C. J., & Alves, J. 2008, A&A, 489, 143 Masset, F. S., Morbidelli, A., Crida, A., & Ferreira, J. 2006, ApJ, 642, 478 Muller, H. S. P., Schloder, F., Stutzki, J., & Winnewisser, G. 2005, Journal of Molecular Structure, 742, 215 Mumma, M. J. & Charnley, S. B. 2011, ARA&A, 49, 471 Thi, W., van Zadelhoff, G., & van Dishoeck, E. F. 2004, A&A, 425, 955 Tielens, A. G. G. M. & Hagen, W. 1982, A&A, 114, 245 Troscompt, N. and Faure, A. and Wiesenfeld, L. and Ceccarelli, C. and Valiron, P. 2009, A&A, 493, 687 Verhoeff, A. P., Min, M., Pantin, E., et al. 2011, A&A, 528, A91 Walsh, C., Nomura, H., Millar, T. J., & Aikawa, Y. 2012, ApJ, 747, 114 Watanabe, N., Shiraki, T., & Kouchi, A. 2003, ApJ, 588, L121 Willacy, K. 2007, ApJ, 660, 441 a Parameters in italics were fixed from CO modeling ( σ s for the power-law model and R out for the ring model). Star and disk data from: a Qi et al. (2004), b Qi et al. (2011), c Andrews & Williams (2005), d Lombardi et al. (2008),", "pages": [ 21, 22, 23, 24, 38, 39 ] } ]

[ { "title": "ABSTRACT", "content": "Recent studies have shown that baroclinic vortex amplification is strongly dependent on certain factors, namely, the global entropy gradient, the efficiency of thermal diffusion and/or relaxation as well as numerical resolution. We conduct a comprehensive study of a broad range and combination of various entropy gradients, thermal diffusion and thermal relaxation time-scales via local shearing sheet simulations covering the parameter space relevant for protoplanetary disks. We measure the Reynolds stresses as a function of our control parameters and see that there is angular momentum transport even for entropy gradients as low as b = -d ln s / d ln r = 1/2. Values we expect in protoplanetary disks are between b = 0.5 -2.0 The amplification-rate of the perturbations, G , appears to be proportional to b 2 and thus proportional to the square of the Brunt-Väisälä frequency( G GLYPH<181> b 2 GLYPH<181> N 2 ). The saturation level of Reynolds stresses on the other hand seems to be proportional to b 1/2 . This highlights the importance of baroclinic effects even for the low entropy gradients expected in protoplanetary disks. Subject headings: accretion, accretion disks, circumstellar matter, hydrodynamics, instabilities, turbulence, methods: numerical, solar system: formation, planetary systems", "pages": [ 1 ] }, { "title": "APARAMETER STUDY FOR BAROCLINIC VORTEX AMPLIFICATION", "content": "1,2 2,3,4 1 NATALIE RAETTIG WLADIMIR LYRA , AND HUBERT KLAHR Draft version March 2, 2022", "pages": [ 1 ] }, { "title": "1. INTRODUCTION", "content": "Angular-momentum transport and turbulence are important issues concerning protoplanetary disks. Magneto-hydrodynamic turbulence brought about by the magnetorotational instability (MRI, Balbus & Hawley 1991), is a reliable way to achieve a sufficient angular-momentum transport and with this also an accretion rate fitting observations (Andrews et al. 2009) and playing an important role in planet formation (Johansen et al. 2007; Lyra et al. 2008; Dzyurkevich et al. 2010; Flock et al. 2011; Uribe et al. 2011; Johansen et al. 2011). However, for MRI to be active the gas has to be sufficiently ionized. This is only the case in the outer regions, upper layers of the disk, and in regions close to the star. The other parts of the disk are too cold and dust-rich for sufficient ionization and thus the magnetic fields cannot couple to the gas. Because of this, the MRI cannot operate in this region, which is therefore called \"dead zone\" (Gammie 1996; Turner & Drake 2009). Since the precise ionization structure is still under debate (Turner & Drake 2009) as is the interplay between active and dead-zones (Lyra & Mac Low 2012) we want to assess the precise hydrodynamic behavior of dead zones, because accretion has to proceed through it somehow and it is where planets form. Therefore it is of interest to study purely hydrodynamic turbulence in circumstellar disks. Klahr & Bodenheimer (2003) found such a hydrodynamic instability creating vortices in three-dimensional radiation hydrodynamical simula- tions of baroclinic disks, e.g. with a radial entropy gradient and thus vertical shear, which they assumed to be a kind of baroclinic instability (BI) modified by the Keplerian shear profile. Observed protoplanetary disks have a non-zero radial entropy gradient b = -d ln s / d ln r , where s is the entropy and r the radial distance to the star. With b = q -( g 2D -1 ) p S , where q = -d ln T / d ln r and p S = -d ln S / d ln r are the temperature surface density gradient respectively and g 2D is the 2D adiabatic index, we see that disks that fulfill p S < q / ( g 2D -1 ) indeed have a negative entropy gradient with values from Andrews et al. (2009) of q ≈ 0.3 -0.5 and p S = 0.9. Therefore protoplanetary disks are not barotropic but rather baroclinic which means that planes of constant pressure and constant density are misaligned, creating a thermal wind, e.g. vertical shear. In a linear stability analysis that followed (Klahr 2004) it was shown that this instability can only be of non-linear nature (see also Cabot 1984; Knobloch & Spruit 1986). Thermal relaxation turned out to be crucial when Petersen et al. (2007a,b) studied baroclinic vortex amplification using an incompressible approximation. In fact thermal relaxation or diffusion, besides the entropy gradient, are key ingredient to establish baroclinic feedback that keeps the instability e.g. vortices in baroclinic disks growing. While both effects e.g. the baroclinic instability and baroclinic vortex amplification are a result of the superadiabatic radial stratification of a disk they are not to be confused. An operating linear baroclinic instability (compare Cabot 1984; Knobloch & Spruit 1986) would be able to create vortices in disks from infinitesimal perturbations, whereas the baroclinic vortex amplification deals with the growth of existing vortical perturbations, for which Lesur & Papaloizou (2010) used the term \"subcritical baroclinic instability\" (SBI). The occurrence of a classical BI in the disk in its geophysical definition is still under debate and shall be dis- cussed elsewhere. There are three possibilities: 1.) there is a classical BI working in protoplanetary disks creating the initial vortices, 2.) there is an other instability operating (see the discussion in Klahr 2004) for instance creating vortices via Kelvin-Helmholz instability from vorticity maxima in sheared waves of baroclinic disks or 3.) small vortical perturbations are triggered from other effects, e.g. waves from the MHD active region of the disks or maybe from the waves emitted by vortices at other radii. In any case the vortices are then growing as described by the BVA until they reach a sufficient size to influence the evolution of the disk, and this is the physics being subject of the present paper. Recently, Lyra & Klahr (2011) have examined the interplay of baroclinic vortex amplification and MHD. They found that as soon as magnetic fields are coupled to the gas, the MRI takes over and thus superseeds vortices which were previously amplified by vortex amplification. This is evidence that the vortex amplification is a phenomenon restricted to the dead-zone. All the above mentioned (lower resolution) studies had to apply entropy gradients 2-4 times stronger than to be expected in protoplanetary disks (Andrews et al. 2009, Klahr 2013 submitted) to drive BVA. We show in the current paper, through high resolution runs that realistic entropy gradients in protoplanetary disks are sufficient for BVA. Recently Paardekooper et al. (2010) have investigated the effect of radial vortex migration. They discovered that vortices migrate quickly radially inward once grown to their full size. While this effect will be of major importance to understand the life-cylce of a vortex, it plays a weaker role for the small/still growing vortices in the present paper. Of course migration will influence the effective angular momentum transport generated by the vortices via the emission of waves, but this is beyond the scope of 2D local simulations as in our study. Weshall return to vortex migration and have a better estimate for angular momentum transport once we return to global simulations. We carry out local, compressible shearing sheet simulations at various resolutions. We show that as we go to higher resolutions one can excite the nonlinear instability and achieve Reynolds stresses with the low entropy gradients deduced for observed accretion disks. We conduct an extensive parameter study for entropy gradients ( b ), resolution, thermal cooling ( t cool ) and diffusion times ( t diff ) respectively. Section 2 gives a brief overview of the physical background of the instability. In Section 3 we present the numerical setup of our simulations. In Section 4 we examine the amplification and decay-times of values such as enstrophy w 2 z =( ∇ × u ) 2 z and a -stresses. Here a = 〈 r uxuy ( qp 0 ) -1 〉 with r being the gas density, u the gas velocity, q = 1.5 the shear parameter, and p 0 the initial mean pressure. We also analyze the saturation values, e.g. how quantities like the entropy gradient, cooling processes in the disk or the size of the simulated domain influence the strength of angular momentum transport. Finally we summarize our results and give a conclusion in Section 5.", "pages": [ 1, 2 ] }, { "title": "2. PHYSICAL BACKGROUND", "content": "Vorticity is conserved in quasi-incompressible barotropic simulations, but in flows with density and pressure as independent quantities vorticity is produced via the so called baroclinic term Here r is the gas density, p the gas pressure, and b is the global radial entropy gradient. The ground state of a disk is geostrophic, e.g. all centrifugal forces and gravity are in balance with the strictly radial pressure gradient. If an entropy perturbation is introduced without perturbing the pressure, then this entropy perturbation will efficiently create vorticity in the presence of the global entropy and pressure gradients. This effect is basically radial buoyancy because of superadiabatic radial stratification 5 . Indeed the radial Brunt-Väisälä frequency (Tassoul 2000) is imaginary, which would lead to radial convection. However, shear stabilizes non-axisymmetric modes and for the dynamic stability of the axisymmetric system the Solberg-Høiland criterium (Tassoul 2000; Rüdiger et al. 2002) has to be considered. If one re-writes Eq. (3) for local approximation (see e.g. Balbus & Hawley 1998) the stabilizing action of the specific angular momentum shows up as the value of Oort's constant in the Coriolis term. If also the vertical stratification in velocity is taken under consideration, as it will occur in real three-dimensional accretion disks (Fromang et al. 2011), then the combined action of radial buoyancy and Coriolis forces lead to a thermal wind, e.g. a vertical shear in rotational velocity. This is precisely the initial state as baroclinic instability in rotating stars and planetary atmospheres. Yet, instability in these systems is not obstructed by radial shear, whereas in a Keplerian disk radial scales would have to be on the order of the vertical pressure scale-heigth ( H ) (Knobloch & Spruit 1986) to be linearly unstable with respect to baroclinic instability. Before we explain the motion of a gas parcel in a vortex we want to explain the cooling and heating processes in a disk as they proved to be crucial to maintain the baroclinic feedback (Petersen et al. 2007a,b). Dust particles absorb photons which heats them up. To cool they radiate photons in the infrared. This radiation can be absorbed by other particles. This happens on a typical length-scale. A convenient parametrization for the 2 diffusion time in our vortex system is t diff = a / K where a is the radius of the vortex and K the diffusion constant. The diffusion constant can be approached using a flux limited diffusion approach as in Kley et al. (2009). There K = l c 4 a R T 3 ( rk ) -1 where l is the flux limiter, c the speed of light, a R the radiation constant, T and r the gas temperature and density, respectively and k the opacity. Since K is constant and the vortex grows t diff will change over time. Thermal relaxation is the other process by which dust can deposit heat into the gas. When a dust particle has a certain temperature other than the equilibrium temperature it will exchange heat with the ambient medium until it reaches the background temperature again. t cool is the time needed to achieve this. This time-scale affects vortices of all sizes equally. The baroclinic feedback itself was explained in detail by Petersen et al. (2007b). A nice description of the mechanism can also be found in Lesur & Papaloizou (2010). In a baroclinic flow entropy is a function of pressure and density, s ( p , r ) . Pressure on the other hand is only a function of radius. The vortex interior transports high entropy material from small radii to large radii. After thermalization low entropy material is transported to small radii. Since the pressure variations, especially from weak vortices, are negligible in comparison to the global radial pressure gradient and much smaller than the azimuthal entropy gradient, pressure can be seen as approximately azimuthally constant (Klahr & Bodenheimer 2003; Klahr 2004; Petersen et al. 2007a). To keep the pressure constant an azimuthal density gradient is established, e.g. outflowing material has a lower density as inflowing material. Thus the vortex feels the effect of differential buoyancy which established the positive baroclinic feedback (Eq. (1)). If cooling is too fast (short time-scales) then the fluid parcel adapts the background temperature slope too quickly. The vortex becomes locally isothermal and no entropy transport is possible. Conversely, if cooling is too slow (long time-scales) then gas will not be thermalized fast enough. The vortex gas becomes adiabatic with constant entropy across the vortex. In both extreme cases, isothermal or adiabatic, the azimuthal entropy gradient across the vortex vanishes. As shown in Eq. (1) the vorticity source ceases to amplify the vortex, or at least stabilizes it against losses from numerical viscosity from radiating vorticity perturbations, e.g. Rossby waves. Therefore it is important that thermal cooling and diffusion times are in the right regime. We model both thermal relaxation and thermal diffusion separately because, dependent on the vortex size, either one or the other dominates thermalization. Always the process with the shorter time-scale sets the heat exchange between vortex and ambient gas.", "pages": [ 2, 3 ] }, { "title": "3. NUMERICAL SETUP", "content": "Our simulations were conducted with the PENCIL CODE 6 . We use a two-dimensional, local shearing sheet approach. We consider a sheet in the mid-plane that corotates with the co-rotational radius R 0 . This is a 2D version of the model used in Lyra & Klahr (2011). To include the baroclinic term they define a global entropy gradient b . Note that in our approximation the gradients for entropy ( s ) and pressure ( p ) are the same. Therefore we do not distinguish between them in our notation and call both b . However, in real disks both may easily differ. The total pressure p tot = ¯ p + p consist of a local fluctuation p and a time-independent part that follows a large scale radial pressure gradient b where r is the cylindrical radius. The full set of linearized equations used in our simulations is Here r is the gas density, u is the deviation of the gas velocity from the Keplerian value, T the temperature, cv the specific heat at constant volume and, K the heat conductivity. Tthermal diffusion time-scale is denoted by t cool . The symbol represents the Keplerian derivative where u ( 0 ) y = -3/2 W 0 x . For a more thorough derivation of these equations and the linearization of the global pressure gradient we refer to Lyra & Klahr (2011) and the appendix therein. In order to keep the numerical scheme stable we add sixth-order hyperdiffusion f D ( r ) , hyperviscosity f n ( u , r ) , and hyperconductivity f K ( s ) (Lyra et al. 2008, 2009; Oishi & Mac Low 2009). The radiation processes in the disk are implemented through the first (thermal diffusion as an approximation for flux limited diffusion of radiation energy density) and second (thermal relaxation to mimic heat exchange with the surface of the disk and thermal equilibration with the irradiation from the central object) terms on the right hand side of the entropy equation. As mentioned in the last chapter we keep the diffusion coefficient K , which is defined as in (Kley et al. 2009), constant and define its value via t diff = H 2 / K . So if the vortex has a radius of H , the pressure scale-hight of the disk, the diffusion time t diff has the value we quote in e.g. Table 1. If the vortex is smaller than H relaxation will be much faster. To clarify that it is indeed the global entropy gradient that produces the vorticity we take the curl of the Navier-Stokes Eq. (6) and assume an equilibrium state, ux = 0, and ∇ P = 0 so that Here we see that the negative azimuthal density gradient across the vortex is the source for vorticity production proportional to the global entropy gradient. Shearing sheet simulations with Zeus 7 like finite volume codes without explicit viscosity, e.g. the TRAMP code, have shown a weak amplification of kinetic energy for the pure adiabatic case, i.e. infinite cooling time (see Klahr 2013 ApJ submitted). This numerical artifact does not occur with simulations performed by the PENCIL CODE. See Appendix A for a 1D radial test/comparison simulation. Initially we apply a finite perturbation in the density so that with r 0 the constant background density and r ' the actual perturbation of the form where C describes the strength of the perturbation. We perturb the density in a way that r rms = 5% for b = 1.0,2.0 (runs A-I) and r rms = 10% for b = 0.5 (runs JP). To achieve a random perturbation we apply an arbitrary phase f ij between 0 and 1. The initial state is non-vortical. Again, this is the identical initial condition as used in Lyra & Klahr (2011) as well as the same amplitude, C , for simulations with b = 2.0, as was used in their simulations. Note that with this initial perturbation we do not perturb the pressure but the entropy. Thus it is really only the term in Eq. (9) that creates the development of non laminar flow structure. All our simulations are done in dimensionless codeunits. So that R 0 = W 0 = 1, g = 1.4, and cs = 0.1, which means that H = 0.1. All time-quantities are given in 2 p W -1 0 which is one local orbit at the co-rotational radius R 0 . The individual setups are given in Table 1. The thermal cooling times and thermal diffusion times are derived from standard disk models like in Bell et al. (1997), also see Klahr 2013 submitted. We explored different resolutions in our simulations, namely 288 2 , 576 2 and 1152 2 . The unusual non power of 2 resolution comes from our computational platform with 6 core processors. Typically we used up to 24 CPUs totaling 144 cores for our largest grids. Still we needed about 1200 hours per run. The grid covers ± 2 H around R 0 in the radial and [ 0 H , 16 H ] in azimuthal direction. This leads to an effective resolution of 72 (288 2 ), 144 (576 2 ) and 288 (1152 2 ) grid-points per scale hight in radial direction and 18 (288 2 ), 36 (576 2 ) and 72 (1152 2 ) grid-points per H in azimuthal direction. It is always necessary to compromise between resolution and computational time. Lower resolution simulations are computationally less expensive but might not resolve the necessary scales.", "pages": [ 3, 4 ] }, { "title": "4.1. Saturation Values and Convergence", "content": "We show the time-developement of a -stresses in Fig. 1. The green line shows the resolution of 288 2 , black of 576 2 and red 1152 2 for b = 2.0 (top), b = 1.0 (middle) and b = 0.5 (lower panel). In all simulations t diff = t cool = 10 local orbits. Wesee that for b = 1.0 and 0.5 and a resolution of 288 2 the perturbation decays right away. Higher resolution is required to increase the Reynolds-number of the system and have less dissipation on the smaller scales and thus excite the instability again. Wetake a stronger initial perturbation for b = 0.5 than for the higher b . The perturbation in entropy results in a perturbation in vorticity. This perturbation is proportional to b . For small b we have to apply a stronger perturbation to get the same effect on the vorticity. However, we expect that if we go to even higher resolution it is possible to keep the initial density perturbation at r rms = 5% (Petersen et al. 2007). If we compare the saturation values of runs with different resolution, we see that they differ by only 10 % from one another (see Table 1). It is important to note that the instability is excited and we measure a -values in the converged runs up to 4 × 10 -3 for entropy gradients as low as b = 0.5. In fact, in Section 4.5 we show that there is only a weak dependence of a on b as a GLYPH<181> b 0.5 . Fig. 1 shows that the saturation values of a do not depend strongly on b , but as we will see in the next section the amplification rates do.", "pages": [ 4, 5 ] }, { "title": "4.2. Amplification- and Decay Rates", "content": "We analyze the amplification timescales of the vortices, meaning how fast a vortex grows due to the baroclinic feedback. Thus it is independent of the precise shape of the initial condition as long as the amplitude is large enough for the given Reynolds number to have vortex growth. In fact, the initial strong kick needed to get the vortex going decays rather quickly as can be seen in e.g. Fig. 1. Here, the a -values start out in the order of 10 -5 then drop to around 10 -8 as the initial perturbation decays. As soon as the baroclinic feedback sets in, the values rise again. The timespan that follows is the one where we measure the amplification time. In analyzing the amplification-rates of the instability we find that the initial amplification-rate of the a -stress ( G ( a ) ), as can be seen in Fig. 2 for run C, can be fitted as exponential amplification a = a 0 exp ( t / t ) with t ≈ 70 b -2 . The proportionality to b -2 is not what one would naively expect from a linear convective or buoyancy driven turbulence. For a linear buoyancy driven turbulence one would expect an amplification rate proportional to the BruntVäisälä frequency, N which in our parameters looks like Here we explicitly wrote b p and b s to make clear that the Brunt-Väisälä frequency depends on the product of entropy and pressure gradient which can be different in global simulations. All quantities in Eq. (12) are positive. Thus the Brunt-Väisälä frequency is imaginary and therefore a linear buoyancy driven turbulence would have a amplification-rate G GLYPH<181> iN GLYPH<181> b . However, we found that G GLYPH<181> b 2 provides a better fit. This once again reflects that the baroclinic vortex amplification is a non-linear effect. In linear convective instability a displaced parcel of gas feels a buoyancy force and thus accelerates propotionally to b . But in the disk baroclinic instability first a vortex has to form with an azimuthal entropy gradient proportional to b (and t cool ) and in a second step this vortex feels a torque proportional to b . Therefore the amplification is proportional to b 2 . The b 2 and t cool dependance has also been derived by Lesur & Papaloizou (2010), see their Eq. (23) for an order of magnitude estimate of the growthrate. The amplification behavior in Fig. 1 already shows convergence for 576 grid cells resolution, e.g. 144/ H in radial direction. If we compare our amplification timescales for the lowest entropy gradients with the migration times obtained by Paardekooper et al. (2010) we see that they are of the same order of magnitude. Which means that the vortex could have drifted into the central star before it reaches strong a -values. However, Paardekooper et al. (2010) also state that their timescales refer to fully grown vortices of size H . Smaller vortices drift significantly slower. This gives them enough time to reach a size, with which they provide sufficient angular mo- mentum transport, before they drift inward. To study the numerical dissipation effects even further we now assess how the vortices decay if baroclinic driving is switched off (Fig. 3). To do this we first evolve runs C and D with b = 1.0 and the two resolutions of 576 2 and 1152 2 for 800 orbits and then turn off the entropy gradient so that b = 0.0. We observe that the vortices get smaller and that all relevant quantities like vorticity, w 2 z , or a -stresses decay with exponential behavior. Godon & Livio (1999) saw the same exponential decay of vorticity when they analyzed longevity of anticyclonic vortices in protoplanetary disks. Their dissipation was proportional to the effective viscosity applied in their numerical experiment. Here we find the same decay-rate for both resolutions, highlighting that the decay of vortices is no longer through numerical effects, but due to the radiation of waves as in Korotaev (1997).", "pages": [ 5, 6 ] }, { "title": "4.3. Saturation Values", "content": "We have established that even shallow entropy gradients lead to vortices but we still have to show that sufficient angular momentum transport can be reached with these shallow gradients. The saturation values of enstrophy, w 2 z , or u rms are of interest as well. Note that we talk about saturation values of our 2D local simulations, where certain restrictions apply, see a more detailed discussion in the conclusions. In the next sections we discuss the measured saturation values and analyze how the different controlling parameters influence amplification-phase and final values.", "pages": [ 6 ] }, { "title": "4.3.1. Influence of Entropy Gradient", "content": "In Fig. 4 we compare runs A, C and J (at a resolution of 576 2 and t diff = t cool = 10) which differ only regarding the value of b . There is an initial exponential amplification-phase of a , E kin and w 2 z that is shorter for high b , followed by a saturated state. We also see that for lower b the saturation values are lower. We want to stress that we did not reach saturation for simulations J and K (at a resolution of 576 2 and 1152 2 and t diff = t cool = 10). Even after 3000 local orbits vortex amplification was still ongoing. Here, t diff = 10 is much shorter than the amplification-rate we estimated in the previous section ( t ≈ 300). As we will see in the next section the amplification-phase is shortest if those timescales are comparable, because t diff also defines how fast pressure perturbations are damped. Although we expect the saturation values of simulation J and K to be higher than what they are right now, it is possible that they will still stay below the saturation values obtained in simulations with higher b . The vorticity can be seen as a measure of the strength of the vortex. The higher the absolute value of the vorticity the stronger the vortex. The only stable vortices in disks are anticyclonic 8 and therefore the vorticity has negative values. So the minimum value of vorticity ( w z ,min ) shows how strong a vortex is. To explain the behavior of w z ,min (3rd panel in Fig. 4), cooling processes have to be taken into account. During the early phases thermalization is dominated by thermal diffusion (Petersen et al. 2007b). As mentioned before this time-scale is shorter for smaller vortices. Therefore heat exchange between the vortex gas and the ambient gas is more efficient than in later stages. Once the vortex has grown to its final size, thermal relaxation takes over. However heat exchange in the center of the vortex is less efficient than in the earlier stages. The baroclinic feedback, e.g. the azimuthal entropy gradient across the vortex, is less efficient, the vortex grows weaker, and w z ,min rises again, creating a flat yet extended vortex.", "pages": [ 6, 7 ] }, { "title": "4.3.2. Influence of Thermal Diffusion and Cooling Times", "content": "We take a closer look at simulations with b = 1 and different combinations of K and t cool to see how thermal diffusion and relaxation influence the saturation values and the amplification-phases. As long as t diff ( l ) = l 2 / K < t cool , t diff ( l ) will dominate the heat exchange from the inside of the vortex to the ambient disk. As the vortex grows t diff ( l ) will increase and with that only contribute to the heat exchange at the outskirts of the vortex. t cool will then dominate the interior of the vortex. For the simulations where we set t diff = t cool , t cool will take over when the vortex has reached a size of H . In radial extend this happens once the vortex has grown to its final size. This is consistent with what we see in Fig. 5. During the early amplification-phase simulations with equal t diff behave exactly the same. Eventually t cool takes over so that the saturation values are determined by t cool . For longer t cool saturation values are lower than for shorter t cool .", "pages": [ 7 ] }, { "title": "4.3.3. Influence of Physical Domain", "content": "A problem with local shearing sheet simulations is that eventually vortices grow to box-size. We cannot say whether they have reached their final size or just do not have any more room to grow. Another problem that arises with the periodic boundary conditions is that the vortices potentially interact with themselves and thus forcing (shaking) them to shed more waves and therefore increase the a -values. To deal with that, we re-did y y simulations A, C and J with a doubled physical domain (simulations A2, C2, J2 in Table 1). The resolution is the same. Instead of x =[ -0.2,0.2 ] and y =[ 0.0,1.6 ] we switch to x = [ -0.4,0.4 ] and y = [ 0.0,3.2 ] . We did not adjust the initial perturbation in any way. Therefore the initial state is perturbed at smaller wave numbers than in the smaller domain. If we go to even larger boxes the initial condition has to be adjusted so the the effective perturbation in the density is of the same strength as in the smaller physical domain. If we compare the time development of runs with a different physical domain (see Fig. 6), we see that vortices in fact do not merge as fast in the large domain because there now is more space between them in radial direction, and they thus pass each other less frequently due to the extended azimuthal domain. Eventually they can merge as Godon & Livio (1999) saw, but the larger the box the longer it takes. We do not want to discuss the mechanism of how the process of vortex merging happens exactly. This has been explained extensively in the field of fluid dynamics (see e.g. Cerretelli & Williamson 2003). The merging process itself is not the focus of our study, because a) the vortex merging is strongly influenced by the box dimensions in a shearing sheet simulation and b) 2D flat vortices merge differently than full scale 3D vortices. The important thing is that vortices do indeed merge if the are sufficiently close to one another, but conserve w in the process. Another unphysical process that can occur in local periodic simulations is that when the vortex approaches the integral scale it interacts with itself, the outer edges of the one side of the vortex almost touches the other side of the same vortex. We do not see this for the runs with the larger physical domain. Since the vortices in the larger domain do not interact with themselves, the saturation values are lower. However, they are still in the same order of magnitude (see Table 1). In Fig. 6 we show snapshots of the vorticity for b = 0.5 (simulations J and J2). Initially there are several vortices. The larger ones sweep up the smaller vortices and thus grow further. At 1500 local orbits there is only one vortex left for the small physical domain, whereas in the larger physical domain there are still three vortices. If we look at the a -value and enstrophy for these two simulations (see Fig. 7) we see that the value seems to decay in the larger box at the end of the run. However this does not mean that the vortices die out. It y more so reflects fluctuations in the vortex interaction, modulating a , as also can be seen in the small domain case at high frequency. We calculate the values as a mean over the entire box but especially the angular momentum transport is a very localized process as can be seen in Fig. 8 (this time for b = 1.0 after 1000 orbits). Here we show the product uxuy at each location in the box. Most areas of the box have an uxuy -value close to zero. However, one can clearly see bands excited by the vortex with positive uxuy -values. These bands are inertia-acoustic waves which are responsible for the angular momentum transport (Klahr & Bodenheimer 2003; Mamatsashvili & Chagelishvili 2007; Heinemann & Papaloizou 2009; Tevzadze et al. 2010). If we had an ideal vortex with a smooth surface we would expect that uxuy sums up to zero within the vortex. However the vortex has a more complex structure as can be seen in the lower right plot of Fig. 8. This leads to an negative net a -value across the vortex. To properly compare the values of a for both physical domains, the box average has to be taken. If the average over an equal physical size centered around a vortex, as indicated by the white dashed lines in Fig. 6, is taken, then the a -values agree again. The a -values are generated only in the vicinity of vortices.", "pages": [ 7, 8, 9 ] }, { "title": "4.4. Correlations", "content": "It is a feature of baroclinic instability that the saturation values of u rms, w 2 z , r rms seem to correlate with each other. In Fig. 9 we plot the different quantities as a function of a . Figure 9 shows the dependencies on a for all our simulations. The colors represent the different entropy gradients: b = 2.0 (black), b = 1.0 (red) and b = 0.5 (green). The different combinations of diffusion and cooling times are represented through the different symbols. We find that the following relations are good fits to our simulation results We can derive the typical length-scale of angular momentum transport L , of the system if Eq. (13) is inserted into the general a formalisms (Shakura & Sunyaev 1973) n = a csH = u rms L so that indicating smaller structures than the vortices in our simulations and also smaller than the vorticity in standard a -models where w GLYPH<181> √ a with a different coefficient (Cuzzi et al. 1994). We do not perform a more exact analysis of these dependencies (varying initial conditions) before we do three-dimensional simulations.", "pages": [ 9, 10 ] }, { "title": "4.5. Dependence on b", "content": "In Section 4.2 we showed that amplification of vortices for low entropy gradients is computationally demanding in terms of evolution time. Thus it is difficult to extract saturation values for entropy gradients even shallower than b = 0.5 with the computational resources at hand. In Fig. 10 we plot the a -stresses as a function of the entropy gradient. Note that we choose a different colorcoding than in Fig. 9. Here symbols represent the thermal cooling times whereas colors represent thermal diffusion times. The dashed black line illustrates a slope GLYPH<181> b 0.5 which is a reasonable fit for the set of points with t cool = 30, t diff = 10 (black triangles) and t cool = 100, t diff = 30 (orange x). We cannot predict a -values for specific entropy gradients and thermal cooling and relaxation times. The key issue is less a strong correlation between a and b but rather the lack thereof. The strength of the a -stresses reflects the size and the amplitude of the largest vortex. Its size is defined by H only and not by any of the other t and b parameters. As long as t and b are sufficient to replenish vorticity at the loss-rate, the a -stresses should be independent of t and b . The loss time-scale via generation of waves and Reynolds stresses is rather long, see Section 4.2 and Fig. 3. Thus as long as the amplification-rates are faster than decayrates one should always obtain roughly the same a -values.", "pages": [ 10 ] }, { "title": "5. SUMMARY AND CONCLUSION", "content": "In this paper we have conducted an extensive parameter analysis for the baroclinic vortex amplification. In particular we analyzed the influence of the global entropy gradient, thermal relaxation and cooling as well as numerical parameters such as resolution, box size, and amplification-rates for vortices and saturation values of a . The most important result of our study is that we find vortex growth even for entropy gradients as low as b = 0.5. However the amplification rate is of the order of several 100 local orbits which makes it difficult to extract reliable saturation values for the efficiency of angular momentum transport. Recently Paardekooper et al. (2010) studied the migration behavior of vortices in global accretion disks. They found significant radial drift for fully grown vortices with drift times shorter than the vortex amplification times we measure in this paper. Nevertheless, this is not a contradiction, because as also shown in Paardekooper et al. (2010) drift rates strongly depend on vortex size. Thus the typical life cycle of a growing vortex might be starting as a growing small vortex without relevant radial drift, which starts drifting as soon as it reaches its saturated state. Therefore radial drift does not affect the study of vortex amplification discussed here. However, it will affect the time a single vortex can partake in angular momentum transport. Future work will have to investigate radial drift of growing vortices in global simulations. Note here that Paardekooper et al. (2010) studied the migration in barotropic disks, in which no vortex amplification occurs. The amplification-phase of the vorticies can be measured in the strength of the overall velocity fluctuation which seem to be growing exponentially on a certain time-scale t GLYPH<181> b -2 . Therefore amplification for steeper entropy gradients is faster, i.e. t = 16 for b = 2.0 and t = 70 for b = 1.0. With these short amplification-times we do reach saturation. Whereas the b = 0.5 was still growing after 3000 orbital periods, when we stopped the simulation. Other parameters that influence the evolution of a -stresses are the thermal cooling and relaxation times. The diffusion times define the amplification phase of the vortices because diffusion dominates small scales, e.g. small vortices. We see faster amplification for longer diffusion times. Cooling time on the other hand determines the saturation values. Here, longer time-scales produce lower saturation values. For the angular momentum transport we get a -values up to 10 -2 for b = 2.0 and 10 -3 for b = 1.0 and b = 0.5. These values are not so different to the ones found with MRI in active layers (Flock et al. 2011) and stronger than the 10 -4 found in dead zones (Dzyurkevich et al. 2010), which shows that entropy gradients can be an important mechanism to transport angular momentum in a dead-zone. Realistic entropy gradients in protoplanetary disks are around b = 0.5 and b = 1.0 which can be derived out of the data obtained by Andrews et al. (2009) as discussed in Klahr (2013 submitted to ApJ). Although we could not reach saturation in all our simulations for these entropy gradients we do see reasonable a -stresses of the order of 10 -3 to 10 -2 . We expect the final values to be in this range which still provides sufficient angular momentum transport in a disk. Yet, we have to consider certain cavities: 1.) Our simulations are 2D simulations and lack the 3 dimensional structure of the vortices. This might very well affect the strength of the a -values. 2.) We do not consider migration of vortices, but rather have periodic boundary conditions. It is not clear for how long vortices can play a role in angular momentum transport before they migrate into the central star. Thus we cannot say how many vortices are in a disk at any given time. The higher the number of vortices, the higher the a -values will be. The interplay between migration and Reynolds stresses definitely has to be analyzed in future models. 3.) The formation process for vortices is still not clear. It is unknown how long the initial formation of a vortex takes, by which process they are formed and if there are processes which can destroy them before the reach full growth. Therefore, our saturation values have to be viewed with caution and cannot be seen as face values for protoplanetary accretion disks. As relation between entropy gradient and strength of angular momentum transport we only find a weak dependence of a GLYPH<181> b 1/2 . Since local simulations are always limited by the box size we also conduct simulations in larger boxes. We do not see a difference in the initial amplification-phase. At later stages the amplification last longer for larger boxes and also is slower. Since part of the vortex evolution happens through merging of smaller vortices, growth takes longer in larger boxes simply because there the radial distance between vortices is bigger and thus mergers are less likely. The saturation values of velocity fluctuations reached for the larger box sizes are slightly lower than for the smaller box sizes. This is due to two reasons. One is that we see some artificial enhancement in vortex strength in the smaller box. Once the vortex has reached boxsize it can no longer grow. It is forced to interact with itself thus emitting more waves. This does not happen in larger boxes. The other reason is that the number of vortices per radial distance is independent of box size because their typical maximum size is in the order of a pressure scaleheight. In the azimuthal direction the number of vortices is limited to 1 per radius, because otherwise merging will occur on short time-scales. Therefore the overall density of vortices per simulation volume (area) is lower in simulations with the larger azimuthal extend. Here we want to note that our larger boxes with H / r = 0.1 and Ly = 32 are only a factor of about two shy of the equivalent 2 p global simulation. Overall, we conclude that the baroclinic vortex amplification works reasonably well for entropy gradients as low as b = 0.5. This b corresponds to a Richardsonnumber of Ri = -1.5 × 10 -3 . This makes BVA a relevant mechanism for angular momentum transport in the dead-zone. An exploration of lower entropy values will have to be postponed due to the long evolution time required. In the future we will study stratified 3D boxes and the interaction of dust with the vortices. Our simulations were conducted partly on the MPIA cluster THEO in Garching, and on the JUGENE machine of the JSC using the grand HHD19. This work was partially supported by the National Institute for Computational Sciences (NICS) under TG-MCA99S024 and utilized the NICS Kraken system. This collaboration was made possible through the support of the Annette Kade Graduate Student Fellowship Program at the American Museum of Natural History. NR also wants to thank IMPRS-HD. !h", "pages": [ 10, 11, 12 ] }, { "title": "NUMERICAL ARTEFACTS", "content": "Shearing sheet simulations with the TRAMP code have displayed unreliable behavior for the extreme cases of cooling times, either isothermal ( t cool = 0) or adiabatic ( t cool = ¥ ). In the first case, a global pressure gradient in a locally isothermal disk leads to the amplification of radially propagating sound waves, which is a physically realistic case (see the derivation in Klahr 2013 ApJ submitted), but only shows up in local radially periodic simulations because the sound wave can propagate through the the box for an unlimited amount of time, which of course is not possible in a global disk. This physical instability can thus be found both in 1D radial TRAMP as well as in PENCIL CODE simulations with remarkably identical growth behavior. This means, having a too short cooling time artifacts from these radially propagating sound waves could ruin our models. Nevertheless, as pointed out by Klahr (2013 ApJ submitted) already a cooling time of t cool = 0.01 will suppress these sound wave instability completely. On the other hand the adiabatic simulations using the TRAMP code were showing a weak amplification of kinetic energy over very long time scales which is the accumulation of numerical error in the quasi dissipation free TRAMP scheme. This behavior is independent of the chosen entropy gradient and results from the conservative treatment of Coriolis forces. Again the PENCIL CODE with its explicit dissipation does not allow for this accumulation of this numerical error, even in the presence of a radial entropy gradient (see solid and dashed-dotted line in Fig. 11).", "pages": [ 12 ] }, { "title": "REFERENCES", "content": "Andrews, S. M., Wilner, D. J., Hughes, A. M., Qi, C., & Dullemond, C. P. 2009, ApJ, 700, 1502 Balbus, S. A., & Hawley, J. F. 1991, ApJ, 376, 214 -. 1998, Reviews of Modern Physics, 70, 1 Bell, K. R., Cassen, P. M., Klahr, H. H., & Henning, T. 1997, ApJ, 486, 372 Cabot, W. 1984, ApJ, 277, 806 Cerretelli, C., & Williamson, C. H. K. 2003, Journal of Fluid Mechanics, 475, 41 Cuzzi, J. N., Dobrovolskis, A. R., & Hogan, R. C. 1994, LPI Contributions, 844, 6 Dzyurkevich, N., Flock, M., Turner, N. J., Klahr, H., & Henning, T. 2010, A&A, 515, A70 Flock, M., Dzyurkevich, N., Klahr, H., Turner, N. J., & Henning, T. 2011, ApJ, 735, 122 Fromang, S., Lyra, W., & Masset, F. 2011, A&A, 534, A107 Gammie, C. F. 1996, ApJ, 457, 355 Godon, P., & Livio, M. 1999, ApJ, 523, 350 Heinemann, T., & Papaloizou, J. C. B. 2009, MNRAS, 397, 64 Johansen, A., Klahr, H., & Henning, T. 2011, A&A, 529, A62 Johansen, A., Oishi, J. S., Mac Low, M.-M., Klahr, H., Henning, T., & Youdin, A. 2007, Nature, 448, 1022 Klahr, H. 2004, ApJ, 606, 1070 Klahr, H. H., & Bodenheimer, P. 2003, ApJ, 582, 869 Kley, W., Bitsch, B., & Klahr, H. 2009, A&A, 506, 971 Knobloch, E., & Spruit, H. C. 1986, A&A, 166, 359 Korotaev, G. K. 1997, Surveys in Geophysics, 18, 567 Lesur, G., & Papaloizou, J. C. B. 2010, A&A, 513, A60 Lyra, W., Johansen, A., Klahr, H., & Piskunov, N. 2008, A&A, 479, 883 Lyra, W., Johansen, A., Zsom, A., Klahr, H., & Piskunov, N. 2009, A&A, 497, 869 Lyra, W., & Klahr, H. 2011, A&A, 527, A138 Lyra, W., & Mac Low, M.-M. 2012, ApJ, 756, 62 Mamatsashvili, G. R., & Chagelishvili, G. D. 2007, MNRAS, 381, 809 Oishi, J. S., & Mac Low, M.-M. 2009, ApJ, 704, 1239 Paardekooper, S.-J., Lesur, G., & Papaloizou, J. C. B. 2010, ApJ, 725, 146 Petersen, M. R., Julien, K., & Stewart, G. R. 2007a, ApJ, 658, 1236 Petersen, M. R., Stewart, G. R., & Julien, K. 2007b, ApJ, 658, 1252 Rüdiger, G., Arlt, R., & Shalybkov, D. 2002, A&A, 391, 781 Shakura, N. I., & Sunyaev, R. A. 1973, A&A, 24, 337 Tassoul, J.-L. 2000, Stellar Rotation (Cambridge University Press) Tevzadze, A. G., Chagelishvili, G. D., Bodo, G., & Rossi, P. 2010, MNRAS, 401, 901 Turner, N. J., & Drake, J. F. 2009, ApJ, 703, 2152 Uribe, A. L., Klahr, H., Flock, M., & Henning, T. 2011, ApJ, 736, 85", "pages": [ 12, 13 ] } ]

[ { "title": "ABSTRACT", "content": "We have tested some relations for star formation rates used in extra-galactic studies for regions within the Galaxy. In nearby molecular clouds, where the IMF is not fully-sampled, the dust emission at 24 µ m greatly underestimates star formation rates (by a factor of 100 on average) when compared to star formation rates determined from counting YSOs. The total infrared emission does no better. In contrast, the total far-infrared method agrees within a factor of 2 on average with star formation rates based on radio continuum emission for massive, dense clumps that are forming enough massive stars to have L TIR exceed 10 4 . 5 L /circledot . The total infrared and 24 µ m also agree well with each other for both nearby, low-mass star forming regions and the massive, dense clumps regions. Subject headings: galaxies: ISM - infrared: ISM - ISM: clouds - ISM: dust - star:formation", "pages": [ 1 ] }, { "title": "TESTING 24 µ m AND INFRARED LUMINOSITY AS STAR FORMATION TRACERS FOR GALACTIC STAR FORMING REGIONS", "content": "Nalin Vutisalchavakul, Neal J. Evans II The University of Texas at Austin, Department of Astronomy, 2515 Speedway, Stop C1400 Austin, TX 78712-1205, USA Draft version October 21, 2018", "pages": [ 1 ] }, { "title": "1. INTRODUCTION", "content": "Star formation is a fundamental process in the formation and evolution of galaxies (Kennicutt 1998b, Hopkins 2004, Bigiel et al. 2008, Gao & Solomon 2004). A unified picture of star formation across different scales and types of regions would benefit from unified measures of star formation rates (Krumholz et al. 2011a, 2011b; Schruba et al. 2011; Shi et al. 2011; Kennicutt 1998a). The most direct way to measure the rate of star formation is to count stars of a known age and mass. Because most galaxies are too far away for individual star forming regions to be resolved, alternative measures of star formation rates have been developed. Many different methods have been used to estimate the star formation rate (SFR) in galaxies (Kennicutt 1998b, hereafter K98). Commonly used tracers include continuum UV emission, recombination lines of hydrogen and other atomic species, total infrared luminosity ( L TIR ), monochromatic infrared emission, and radio emission (Kennicutt 1998b; Kennicutt et al. 2003, 2009; Calzetti et al. 2007, 2010; Perez-Gonzalez et al. 2006; Murphy et al. 2011; Kinney et al. 1993; Condon 1992). Each of these indicators traces star formation in somewhat different ways, averaging over different timescales (e.g., Kennicutt & Evans 2012). UV continuum emission in the wavelength range of 125-250 nm directly measures radiation from high mass stars, with peak contributions from stars of several M /circledot ; consequently, it can average SFR over 10-200 Myr. Hydrogen recombination lines, such as H α , or free-free radio continuum emission trace H II regions surrounding high mass stars ( M > 15 M /circledot ), with a peak contribution from M = 30 to 40 M /circledot ; thus they average SFR over only 3-10 Myr (Kennicutt & Evans 2012 and references therein). Most studies of star formation in galaxies use UV continuum or optical lines (e.g., Bigiel 2008, Kinney 1993, Salim 2007, Hao et al. 2011). However, optical emission can be strongly affected by dust-extinction, and the UV continuum is even more sensitive to extinction (Calzetti 1994, Hao et al. 2011, Buat et al. 2005, Burgarella et al. 2005). The recombination lines trace only very massive stars, so they are sensitive to assumptions about the IMF (see Figure 1 in Chomiuk & Povich 2011). As supplements to UV and optical tracers, IR fluxes have been used to study SFR in regions that are obscured by dust (Calzetti et al. 2007, 2010; Perez-Gonzalez et al. 2006; Kennicutt et al. 2009). Infrared dust emission traces the stellar luminosity that has been absorbed by dust and reemitted in the infrared (K98, Calzetti et al. 2007). It is less biased towards the highest mass stars and hence less sensitive to the IMF. If all the photons inside star forming regions get absorbed by dust, then the total infrared emission from dust ( L TIR ) should trace the total luminosity of the stars. One problem with using L TIR to trace star formation is that sources other than young stars, such as older stars or AGNs, can contribute to heating the dust. For galaxies less active in star formation, a significant amount of dust heating can come from the general interstellar radiation field, arising from older stellar populations (K98, Draine et al. 2007). In that case, L TIR would trace emission that is not relevant to the current star formation. Monochromatic IR emission has also been widely used. One particularly widely used tracer is the 24 µ m continuum emission (Calzetti et al. 2007, Wu et al. 2005a, Rieke et al. 2009, Alonso-Herrero et al. 2006, Helou et al. 2004). In principle, 24 µ memission has the advantage over L TIR that it requires quite warm dust. In active star forming regions, the warm dust is more intimately associated with the forming stars. The diffuse part of the interstellar medium that has been heated by the average interstellar radiation field should be at a comparatively low temperature and should not emit much in the 24 µ m wavelength band compared to the emission from high mass star forming regions. Stronger radiation fields from high mass stars can heat the dust to higher temperatures over a larger region; therefore, 24 µ m emission should be a good tracer for high mass star forming regions with less contamination from non-star-forming sources. There are several studies of how emission from nonstar-forming sources compares to emission relevant to star formation in the 24 µ m wavelength (Rahman et al. 2011, Verley et al. 2008, Draine et al. 2007). Draine et al. showed from fitting dust models to numbers of galaxies that for galaxies with high star formation rates (starburst galaxies), the main contribution to the 24 µ m emission comes from photodissociation regions associated with high mass stars. For high mass star forming regions, 24 µ m emission should be a good tracer of SFR. Observations of nearby galaxies show strong concentrations of 24 µ m emission toward H II regions, but with a diffuse component. Unifying studies of star formation in other galaxies with studies within the Milky Way can be mutually illuminating. Chomiuk and Povich (2011) have compared tracers of SFR on global scales and found a potential discrepancy of a factor of two between extragalactic relations applied to the Milky Way as a whole and more direct measures of the Milky Way star formation rate. Our goal is to test extragalactic relations on still smaller scales of individual clouds and dense clumps. Images of the Galactic Plane at 24 µ m are available from MIPS on Spitzer from the infrared survey of the plane of the Milky Way (MIPSGAL) (Carey et al. 2009) and at 25 µ m from IRAS. If these could be used to measure star formation rates in regions of our Galaxy, it would be very useful. The goal of this paper is to test the limits of applicability of the extragalactic relations for regions within our Galaxy. Since we can observe star forming regions in the Milky Way in more detail, testing extragalactic SFR relations on nearby regions can also provide some perspective on the use of such relations in other galaxies. In order to test how well 24 µ memission can trace SFR, another method for tracing SFR is needed for comparison. We tie our measurements to those in nearby clouds, where we can count YSOs of a certain age. These provide a completely independent and reasonably accurate measure of the SFR. These nearby clouds are not forming high mass stars, which means that the IMF is not fullysampled in these regions. Since one of the assumptions in deriving SFR from IR emission is that the IMF is fullysampled in the regions, studying the use of IR tracers in these nearby clouds can tell us about the effect of undersampling the IMF on SFR calibration. We then extend the study to regions forming massive stars. These regions are at larger distances than the nearby clouds, and counting individual YSOs in these regions as a measure of SFR is not applicable. With the lack of a direct method of measuring SFR, we instead compared SFR measured from 24 µ m, L TIR and radio continuum emission. In section 2 we describe the sample of star forming regions used in the study. In section 3 we describe how the SFR was calculated for a sample of nearby molecular clouds. In section 4, we consider high mass star forming regions using samples of massive, dense, clumps from Wu et al. (2010). The resulting comparison of all the SFRs in this study is described in section 5, and we summarize the results in section 6.", "pages": [ 1, 2 ] }, { "title": "2. THE SAMPLE", "content": "Two groups of sources were included in this study. The first group consists of nearby molecular clouds with evidence of low-mass star formation. This group has the advantage of having an independent estimate of the SFR from counting YSOs. The second group consists of massive dense clumps with evidence of high mass star forma- tion. This group does not have SFRs from YSO counting, but it is more representative of the star formation regions that might be seen in other galaxies. The first group consists of 20 clouds within 1 kpc of the Sun, in the structure known as the Gould Belt (GB). They have data from Spitzer Legacy programs and ancillary data (Evans et al. 2003, core to disk (c2d); and Allen et al. in prep., GB). The clouds are listed in Table 1, along with their distances. All the clouds have been observed in all IRAC (3.6, 4.5, 5.6, 8.0 µm ) and MIPS bands (24, 70, 160 µm ), using the same procedures and data reduction methods. Young Stellar Objects (YSOs) were identified and categorized into their SED classes (Class I, Flat, Class II, and Class III) using the spectral index following the criteria from Green et al. (1994). The details on identifying YSOs and calculating SFR in these clouds can be found in Evans et al. (2009) and Heiderman et al. (2010). We also make use of data from the IRAS data archive for assessing the large scale emission from the clouds. The second group contains massive dense clumps with evidence of high mass star formation, selected from Wu et al. (2010). This sample is a subsample of a large survey by Plume et al. (1997) of regions associated with water masers, which are indicators of an early phase of massive star formation, most of which contain compact or ultracompact H II regions. These clumps have characteristic densities from CS excitation of about 10 6 cm -3 (Plume et al. 1997). The mean and median virial masses are 5300 and 2700 M /circledot , respectively. Most of these clumps have been observed in many molecular line transitions, such as CS lines (Plume et al. 1992, 1997; Shirley et al. 2003), HCN J = 1 → 0 and J = 3 → 2 (Wu et al. 2010), HCO + and several others (Reiter et al. 2011). Some of the clumps have also been observed in 350 µ m dust continuum emission by Mueller et al. (2002), who also tabulated IRAS data.", "pages": [ 2 ] }, { "title": "3. ANALYSIS OF THE REGIONS FORMING LOW-MASS STARS", "content": "Emission at 24 µ mhas been used in many extragalactic studies as a star formation tracer. A number of studies have derived an expression for the SFR as a function of the 24 µ m emission [SFR(24 µ m)] (Calzetti et al. 2007, Alonso-Herrero et al. 2006, Rieke et al. 2009, Wu et al. 2005, Zhu et al. 2008, Relano et al. 2009, Perez-Gonzalez et al. 2006). Various calibrations of SFR(24 µ m) are compared in Calzetti (2010). Our goal is to test these relations by comparing the SFR using 24 µ m emission with the SFR using YSO counting (Evans et al. 2009, Heiderman et al. 2010). The YSO counting method uses the following equation. Assuming an average stellar mass of 〈 M ∗ 〉 = 0 . 5 M /circledot and an average time for YSOs to have an infrared excess of t excess = 2 Myr, the SFRs were calculated by Evans et al. (2009) and Heiderman et al. (2010). The average mass was chosen to be consistent with IMF studies (Chabrier 2003, Kroupa 2002) and consistent with an average mass for some clouds although there may be variations between clouds (Evans et al. 2009). They are collected in Table 1. The largest source of uncertainty is the lifetime of the infrared excess (perhaps ± 1 Myr).", "pages": [ 2 ] }, { "title": "3.1. 24 µ m emission from YSOs", "content": "We now compare the SFRs calculated from counting YSOs [SFR(YSO count)] to the SFRs calculated using SFR(24 µ m). Since 24 µ m emission comes from dust that has been heated by stellar radiation and does not require high energy photons, it may be able to pick up the star formation rate of even low-mass YSOs. The first step was to calculate the total 24 µ memission coming from all the YSOs in each cloud. The flux densities at 24 µ m for individual YSOs were extracted from data bases and summed over all the YSOs in individual clouds. The resulting total YSO flux for each cloud is shown in Table 1. Using the distances to the clouds (Heiderman et al. 2010, updated distances can be found in Dunham et al. 2012 in prep), the 24 µ m luminosity can be calculated from the total 24 µ m flux density. From the total 24 µ m emission from YSOs, we computed SFR(YSO, 24 µ m). The relation for SFR(24 µ m) that we used in this study came from the work of Calzetti et al. (2007), who adopted the starburst99 stellar synthesis model and Kroupa's IMF (Kroupa et al. 2001) in the calibration. Kroupa's IMF has been used in many studies for calibrating SFR; it has the form and stellar mass range described by (Chomiuk et al. 2011, Kennicutt et al. 2009, Murphy el al. 2011): Calzetti et al. (2007) uses Kroupa's IMF but with an upper mass limit of 120 M /circledot . The SFR(24 µ m) is where L 24 µm is the total 24 µ m luminosity per unit frequency times the frequency ( νL ν ). The calculated SFRs for each cloud are as shown in Table 1. It is clear that SFR(YSO, 24 µ m) vastly underestimates SFR(YSO count). The mean ratio of SFR(YSO count) to SFR(YSO, 24 µ m) is 1867 ± 1335.", "pages": [ 3 ] }, { "title": "3.2. Total 24 µ m Emission", "content": "Since the relation in equation 2 was derived for extragalactic star formation, where individual YSOs are not resolved, we should expect the detected flux to be contributed from diffuse emission as well as from point sources. In this section, we consider the total emission, which includes diffuse as well as point source emission in SFR(24 µ m). To compare SFR from the total 24 µ m emission with the SFR from YSO counting, the calculations have to come from the same area of the clouds. Boundaries for each cloud used for identifying YSOs were chosen using contours from extinction maps. Therefore, we chose the same boundaries for calculating diffuse emission. All clouds' boundaries were chosen to be extinction contours of A V = 2 . The exceptions are Serpens and Ophiuchus for which the c2d survey extended down to A V = 6 and A V = 3 respectively (Evans et al. 2009). The total flux used to calculate the SFR should also be emission only from the clouds themselves. Images that cover the area inside the cloud's boundary can still contain foreground and background emission not associated with the clouds. To include only emission from the clouds, we subtracted background emission. To do this, we needed large scale images that cover not only the area of the cloud defined by extinction contours, but also the area surrounding the contour boundaries. MIPS images from the Spitzer survey have good spatial resolution but lack the area coverage needed for background estimations. Therefore, we chose to use IRAS images for our diffuse emission analysis. The Infrared Astronomical Satellite (IRAS) observed 96 % of the sky in four bands (12, 25, 60, 100 µ m). We used 25 µ m IRAS images from the the Improved Reprocessing of the IRAS Survey (IRIS) obtained from the Infrared Processing and Analysis Center (IPAC) as a substitute for 24 µ mdata. First the total flux densities inside contour boundaries were calculated for each cloud. We then chose a 'sky annulus' for each cloud separately by choosing an area surrounding the cloud's boundary while avoiding any extended emission that seemed to be connected to the cloud. The background level was estimated by summing over the flux inside the sky annulus divided by the total number of pixels to estimate the background value per pixel (Jy/pix). The total flux inside contour boundaries minus the background flux (background flux = average background level per pixel × number of pixels inside the boundary) gave the actual flux from the clouds. The 25 µ m emission coming from the clouds themselves turns out to be very small compared to the foreground/background emission. The 25 µ m luminosities calculated from the background subtracted flux for all the c2d and Gould's Belt clouds are shown in Table 2. For clouds with background emission comparable to the total emission inside the boundaries, namely Lupus IV and Auriga North, we set the 25 µ m luminosities and SFR(24 µ m) to zero. With the 25 µ m luminosities, the SFR for each cloud was obtained using Equation 2. The differences between luminosities measured at 24 µ m and 25 µ m should be quite small. Table 2 compares the SFR(24 µ m), which is calculated from the total emission including point sources and diffuse emission, with SFR(YSO count). It is clear from the table that SFR(24 µ m) greatly underestimates SFR(YSO count). The average ratio of SFR(YSO count) to SFR(24 µ m) is 107 ± 109 , with a median of 61.6. Figure 1(a) shows a plot of SFR(24 µ m) over SFR(YSO count), and Figure 1(b) shows a ratio of SFR(24 µ m)/SFR(YSO count) over SFR(YSO count).", "pages": [ 3 ] }, { "title": "3.3. Contributions from Stellar Continuum Emission", "content": "Calzetti et al. (2007) developed relations between SFR and emission at two MIR wavelengths of 8 and 24 µ m. Since only the dust emission should measure SFR, stellar continuum emission needed to be subtracted from the flux. The stellar continuum subtraction was performed for the 8 µ m emission, but contributions to the 24 µ m flux from stars was considered to be negligible. We used c2d clouds as sample regions to see how much stellar continuum contributes to the total flux. The c2d project identified all point sources, which include background and foreground stars, for all clouds. These background/foreground stars in fact dominate the source counts in each cloud. With the available data, we can compare the contributions from point sources, which can be separated into YSO and non-YSO, to the total 24 µ m flux. First, we calculated the flux from all identified objects in the 24 µ m MIPS images. Then the flux from YSOs was subtracted from the all-object flux to get the non-YSO object flux. In extragalactic studies, when looking at star forming regions the flux is the total flux emitted from the projected area. To see how much stellar emission contribute to total flux, we compare the non-YSO flux to the total flux (before background subtraction). The results show that stellar continuum contributes very little to the total flux. The contribution is larger for some clouds, specifically clouds with little diffuse emission, but stellar contributions to the total flux are less than 10 percent for all clouds (Table 3).", "pages": [ 3, 4 ] }, { "title": "3.4. L TIR", "content": "Another tracer of star formation often used in extragalactic studies is the total infrared luminosity. While 24 µ m emission arises from warm dust grains or from small, transiently heated dust grains, most of the emission from dust in molecular clouds peaks at a longer wavelength, in the far-infrared. The total infrared luminosity should then trace the bulk of the dust emission. With the available IRAS data, the total infrared luminosity ( L TIR ) for all the c2d and GB clouds can be estimated from: where f i is the flux in each IRAS band in units of Jy, D is the distance in kpc, and L TIR (8-1000 µ m) is in units of L /circledot (Wu et al. 2010). Each of the IRAS bands have a slightly different angular resolution: 3.8 ' , 3.8 ' , 4.0 ' , and 4.3 ' for IRIS plate of 12, 25, 60, and 100 µ m respectively (Miville-Deschenes & Lagache 2005). However, the angular size of our objects are in the order of a few degrees. We therefore did not take into account the differences in the resolutions. The flux in each band was computed with the same technique used for the flux at 25 µ m, including background subtraction. To calculate SFR( L TIR ), we used the extragalactic relation for starburst galaxies from K98. However, the SFR( L TIR ) from K98 assumed a Salpeter form of the IMF. For consistency, all our calculations should be based on the same IMF model. A Salpeter IMF gives a Lyman continuum photon rate of 1.44 times higher than Kroupa IMF (from 0.1-100 M /circledot ) for the same SFR (Chomiuk et al. 2011, Kennicutt et al. 2009). Assuming that L TIR scales with Lyman continuum photon rates, we then divided SFR( L TIR ) from K98 by 1.44 to obtain where L is the total infrared luminosity (8-1000 µm ). TIR The results (Table 2) show that L TIR underestimates SFR(YSO count) for all the clouds, with the mean ratio of SFR(YSO count) to SFR( L TIR ) of 969 ± 1870 and median of 480. Figure 2(a) shows SFR( L TIR ) over SFR(YSO count), and Figure 2(b) shows the ratio of SFR( L TIR )/SFR(YSO count) versus SFR(YSO count). With both the 24 µ mand L TIR available, we also compared SFR(24 µ m) with SFR( L TIR ). Figure 3 shows SFR(24 µ m) over SFR( L TIR ) with the low mass star forming clouds data represented by orange circles. The two SFRs agree well with each other with average ratio of SFR( L TIR )/SFR(24 µ m) of 0 . 22 ± 0 . 08 and a median of 0.33. A curved fit was performed using MPFITEXY routine (Williams et al. 2010; Markwardt 2009) with adopted uncertainties of 50% for both SFRs. The solid black line represents a line of SFR(24 µ m) = SFR( L TIR ) while the dot-dashed, orange line represents a leastsquare fit for the nearby clouds of", "pages": [ 4 ] }, { "title": "4. ANALYSIS OF REGIONS FORMING HIGH-MASS STARS", "content": "So far we have found that the extragalactic relations between SFR and 24 µ m or total infrared badly underestimate the SFR in nearby molecular clouds, which are not forming stars of high mass. Here we address the issue for regions forming massive stars, using the dense clump sample discussed in §2. These clumps have an average distance of 3 . 9 ± 2 . 4 kpc and a median of 3.5 kpc.", "pages": [ 4 ] }, { "title": "4.1. IRAS 25 µ m emission and total infrared luminosity L TIR", "content": "The fluxes for the IRAS bands for these clumps are available from the IRAS point source catalog (PSC) and tabulated by Mueller et al (2002). However, most of the massive dense clump sources are extended sources. Examining the images of these sources showed that the IRAS point source catalog could underestimate the flux because the average source size is larger than the IRAS beam size. To obtain more accurate values of the flux, we performed photometry on the massive dense clump sample instead of adopting the flux from PSC. IRAS IRIS images in all four bands were used for photometry. Aperture photometry was performed on each source with the use of IDL routine APER and by setting the aperture radius to be equal to the FWHM of a 1D gaussian fit. Most of the sources are in a crowded field, which complicated the photometry. Sky subtraction was done by choosing a sky region for each source by eye and averaging the flux within the region to obtain sky level. The result gives a flux in all four IRAS bands for a total of 56 sources. The total infrared luminosity and the SFR( L TIR ) was calculated from the same equation used in the last section (Equation 3 and 4). Note that L TIR from our photometry is higher than L TIR from the PSC by a factor of 2 on average. The SFR(24 µ m) was also calculated in the same way by using the relation in Equation 2. Ideally, we would now compare the SFRs from infrared emission to SFR(YSO count)as we did for low-mass regions. However, because of the greater distance and the presence of diffuse emission, counting YSOs is not practical in these regions. Without the YSO count, we cannot test the IR SFR tracers against a direct measure of SFR. With more than one method of tracing star formation, we can test to see if different tracers give consistent measures of SFRs. As shown in Table 4, the two IR SFRs are comparable to each other with the average ratio of SFR( L TIR ) to SFR(24 µ m) = 0 . 41 ± 0 . 19 . The median is 0.37. Figure 3 shows the comparison between SFR(24 µ m) and SFR( L TIR ) for the clumps, which is represented by blue diamonds. The dashed, blue line represents a least- are fit for the massive dense clump data of", "pages": [ 4, 5 ] }, { "title": "4.2. Radio Continuum Emission", "content": "In addition to infrared emission, radio continuum emission is also used as a SFR tracer for galaxies in several studies (Condon et al. 1992, Yun el al. 2001, Jogee et al. 2005, Murphy et al. 2011). For normal and starburst galaxies, most of the radio emission is free-free emission from ionized gas and synchrotron emission from relativistic electrons (Yun et al. 2001). Free-free emission traces ionized gas inside H II regions, along with some more diffuse emission from extended ionized gas, while synchrotron emission traces relativistic electrons accelerated by supernova remnants, which are much more widely distributed. Both of the sources of the radio emission are related to high mass star formation because high mass stars produce H II regions while stars with M ≥ 8 M /circledot produce core-collapse supernova (Yun et al. 2001). However, the quantitative relation between synchrotron emission and star formation is less direct, being derived from a correlation between the synchrotron and far-infrared emission (de Jong et al. 1985; Helou et al. 1985; Condon 1992). For this study, we used radio continuum as another independent source of SFR tracer for comparison with L TIR since both radio continuum and L TIR should trace the presence of high mass stars. In a spectrum of a whole galaxy, synchrotron emission dominates emission at ν ≤ 30 GHz (Condon et al. 1992). However, our samples are on much smaller scales than for extragalactic studies. In the absence of nearby supernova remnants, radio emission from high mass star forming regions is dominated by thermal free-free emission. To use radio continuum as a SFR tracer for the massive dense clump samples we need to connect free-free emission to a total number of massive stars. Thermal (free-free) luminosity is related to the rate of photoionizing photons (Lyman continuum photons) by where N UV is the production rate of Lyman continuum photons per second, T e is the electron temperature, ν is the frequency, and L T is the thermal emission luminosity, assuming it is optically thin in this part of the spectrum (Condon et al. 1992). Using Kroupa's IMF and stellar spectral model from Starburst99 (Leitherer et al. 1999), the rate of photoionizing photons is related to SFR by (Chomiuk et al. 2011) We get For an electron temperature of T e ∼ 10 4 K, the thermal radio SFR relation is For the radio continuum data, we used radio surveys that cover the regions of the Galactic plane that coincide with the massive dense clump sample. The radio data in this study was obtained from two surveys. The first set of data came from a survey of the Galactic plane at 4.875 GHz by Altenhoff et al. (1979; hereafter A79). The radio data were obtained with the 100-m Effelsberg with a half-power beamwidth of 2.6 ' over the galactic longitude range of l = 357 . 5 · to 60 · and galactic latitude of b = ± 2 · . The second set of radio data were obtained from an earlier survey by Altenhoff et al. (1970; hereafter A70). The survey of the Galactic plane at 1.414, 2.695, and 5.000 GHz covered a range of l = 335 · to 75 · and b = ± 4 · with a half-power beamwidth of approximately 11 ' . The observations for the three wavelength bands were made with the 300-ft transit paraboloid antenna at the NRAO, the 140-ft antenna at NRAO, and the 85ft parabolic antenna at Fort Davis for 1.414, 2.695, and 5.000 GHz respectively (Altenhoff et al. 1970). Using the 4.875 GHz (A79) survey has the advantage of having a comparable resolution to the infrared data from IRAS ( 2 . 6 ' for A79 and ∼ 2 ' for IRAS 100 µ m), making it suitable for comparison between radio and infrared data. We first matched objects from the radio surveys to the massive dense clump objects by matching their coordinates. The matching objects have center coordinates within a few arcminutes of each other. Lockman (1989) provides radio recombination line data for these radio sources from his survey of radio H II regions in the northern sky. We compared radio recombination line velocities of matched objects to line velocities (HCN J = 1 → 0 , J = 3 → 2 and CS J = 2 → 1 , J = 7 → 6 ) from Wu et al. (2010). We kept the objects with velocities approximately within ± 5 km s -1 between the two data sets. Our matching resulted in a total of 22 objects with available radio continuum flux, radio recombination line velocity, and infrared luminosity. A79 provides a peak intensity for each radio source along with a FHWM. The integrated flux for each object was calculated for a total of 18 objects by assuming a Gaussian profile for both the source flux distribution and the beam profile. A70 provides integrated flux and FWHM data for an additional 4 objects. Then SFR(radio) was calculated from Equation 7. After obtaining SFR(radio), our next step was to compare them to IR SFR. However in order to compare radio data to infrared data, the two sets of data should come from equal areas of the objects. Aperture photometry was performed on IRAS IRIS images with a chosen aperture radius equal to the radio FWHM size of each object. The aperture size was chosen to capture most of the infrared flux of the objects without contamination from other nearby sources and to make the observed areas comparable to those of the radio data. The resulting SFR(radio), L TIR , SFR( L TIR ) and SFR(24 µ m) are included in Table 5. SFR(radio) and SFR( L TIR ) are well correlated, with an average ratio of SFR(radio)/SFR( L TIR ) of 1 . 8 ± 0 . 8 , a median of 1.9, and a linear correlation coefficient of 0.90. There are many sources of uncertainties in our calculations of SFR, which makes it difficult to estimate realistic errors for each source. We instead adopted a 50 % uncertainties for both SFRs and performed a curve fit using MPFITEXY routine (Williams et al. 2010; Markwardt 2009). Figure 4(a) shows SFR( L TIR ) versus SFR(radio) with a solid line representing SFR ratio of one and a dashed line representing a best fit to the data of SFR(radio) and SFR(24 µ m) are also well correlated with an average ratio of SFR(radio)/SFR(24 µ m) of 0 . 76 ± 0 . 42 , a median of 0.79, and a linear correlation coefficient of 0.98. Figure 4(b) shows SFR(24 µ m) versus SFR(radio) with a dashed line representing a best fit of", "pages": [ 5, 6 ] }, { "title": "5.1. Low mass SF", "content": "From the results for c2d and Gould's Belt survey, it is clear that the SFRs from 24 µ m do not agree well with SFRs from YSO counting. First of all, 24 µ m emission from YSO point sources contributes very little to the total emission of the clouds. Even when we included the diffuse emission into our calculation of SFR(24 µ m), the resulting values are still much lower (by a factor of about 100 than SFR(YSO count)). Nonetheless, we can ask whether there is any relation at all between SFR(24 µ m) and SFR(YSO count). Figure 1(a) shows a plot of SFR(24 µ m) versus SFR(YSO count). The solid black line represents a ratio of 100. The figure shows that there is a general correlation between the two with the Pearson linear correlation coefficient of 0.83. Perhaps the 24 µ m emission might provide a rough guide to the SFR, but with a different conversion factor. However, the scatter is large. Figure 1(b) shows the ratio of SFR(24,diffuse)/SFR(YSO count). The discrepancies and scatter between the two SFRs persists throughout the range of SFRs. A similar result was obtained for the comparison of SFR( L TIR ) with SFR(YSO count), as shown in Figure 2. There is again a weak correlation with a correlation coefficient of 0.77, but the underestimate of SFR(YSO count) is even greater. The solid black line represents the same line of SFR(YSO count) = 100 × SFR( L TIR ), as shown in Figure 2(a). The disagreement between SFR(IR) and SFR(YSO count) is not surprising since these clouds are not forming very massive stars, which would dominate the luminosity if the IMF is fully sampled. The undersampling of the IMF along with other possible causes behind the discrepancy in SFRs are discussed below.", "pages": [ 6 ] }, { "title": "5.1.1. External Heating", "content": "As discussed earlier, the total fluxes from the actual clouds are generally small fractions of the total emission toward the regions, which means that a lot of the emission is background emission. Furthermore, much of the diffuse emission that is associated with the cloud does not correspond to regions of high extinction or intense ongoing star formation. As examples, Figure 5 and Figure 6 show the images for Lupus I and Ophiuchus, with extinction contour levels overlaid. In Lupus I, the diffuse emission at 24 µ m is located away from the regions of current star formation. In contrast, in Ophiuchus, most of the diffuse emission is associated with the cluster of forming stars spatially, and the excitation peaks on embedded early-type stars (Padgett et al. 2008, see, Fig. 2). In the case of the Perseus cloud, much of the diffuse 24 µ m emission comes from regions heated by a star lying behind the cloud (unrelated to current star formation) or from the IC348 cluster (related to recent star formation) (Rebull et al. 2007). Such differences from cloud to cloud will introduce large scatter into the relations. In the absence of high mass stars in these clouds, external sources of heating could dominate the infrared emission. The IRAS 100 µ m images show more correlation with the extinction contours than the 25 µ m images. The contribution to the L TIR is also larger from the 100 µ m, which is closer to the peak of the general dust emission from molecular clouds. The resulting L TIR may then trace the amount of dust inside the clouds as opposed to star formation in the clouds. Then the correlation in Figure 2 could be a secondary effect of the correlation of SFR with amount of dust for the cloud as a whole.", "pages": [ 6 ] }, { "title": "5.1.2. Undersampled IMF", "content": "Since these clouds are not forming very massive stars, clearly there are no stars to populate the high-end of the IMF. The lack of high-mass stars means that it requires more mass in the form of lower-mass stars to produce a certain luminosity than if the IMF is fully-sampled. Using SFR relations derived by assuming the full IMF will then underestimate the SFR in these regions. To see how much this affects the discrepancies in the SFRs, we looked at the details of the SFR calibrations. Calzetti et al. (2007) calibrated the SFR-24 µ m relation by empirically fitting L(24 µ m) to H α . H α was then connected to SFR through a stellar population model assuming Kroupa's IMF, solar metallicity, and a constant SFR over a timescale of 100 Myr. Any differences in the IMF would have an effect on the two steps: SFR-H α (or directly related, N UV ) relation and H α - 24 µ mratio. We performed a test by running starburst99 with the same IMF but with a different upper limit on the stellar mass (M upper ). We also assume that a constant fraction of the bolometric luminosity (L bol ) is being re-emitted in the 24 µ m band. Taking the Perseus molecular cloud as an example, the highest mass star in the cloud is an early B star (Rebull et al. 2007). We set M upper = 15 M /circledot and a constant SFR over 100 Myr. The results showed an underestimation of SFR(24 µ m) by a factor of 2.1 when assuming a full IMF. For SFR( L TIR ), the relation in equation 4 was derived from assuming that all of L bol is re-emitted in the infrared so that L bol = L TIR . L bol was connected to SFR directly from the stellar synthesis model. This would result in the same underestimation of SFR( L TIR ) by a factor of 2.1. A factor of 2 difference from the cut-off IMF is still much less than the observed discrepancies in SFR(YSO count)/SFR(24 µ m) of a factor of 43 and SFR(YSO count)/SFR( L TIR ) of 210 in Perseus. The effect of under-sampling the IMF on underestimating the SFR will be greater for clouds with lower M upper . For many clouds M upper is even lower than 15 M /circledot . We tested the model with M upper = 5 M /circledot , which showed an underestimation of SFR by a factor of 10. Even with the lower M upper , undersampled IMF still cannot account for the large discrepancies in the whole sample. We tested the the effect of under sampling IMF by changing M upper , but in regions of low SFR stochastic sampling of the IMF could also be important, especially in contributing to the scatter in the sample (da Silva et al. 2011, Eldridge 2012).", "pages": [ 6, 7 ] }, { "title": "5.1.3. Star Formation Timescale", "content": "The time scale of constant star formation assumed in the SFR relations is 100 Myr, much longer than a lifetime of an average molecular cloud (few × 10 7 Yr; McKee & Ostriker 2007, Murray 2011) or the time scale over which YSO counting is relevant ( ≈ 5 Myr). On a longer time scale the contribution of high mass stars to the total luminosity will get smaller since low mass stars will outlast the short-lived high mass stars. On the time scale of molecular clouds, not accounting for the lack of massive stars will lead to even greater underestimations of SFR than on a longer time scale. Taking an average age of the clouds to be 10 Myr, the model results from combining the cut-off IMF (M upper =15 M /circledot ) and the change in time scale showed a higher SFR by a factor of 9.9, still lower than the observed differences in Perseus. Combining the change in time scale to 10Myr and a cut-off IMF of M upper =5 M /circledot gave a higher SFR by a factor of 110, close to the average discrepancy in our data. Additionally, the assumption that all of the bolometric luminosity is being re-emitted in the infrared might not be valid in these regions. If the fraction of energy emitted in the infrared or 24 µ m band over L bol is not constant or is lower in regions with low SFR than in the regions used in the SFR calibration, then this would be another cause for underestimation of the SFR.", "pages": [ 7 ] }, { "title": "5.2.1. L TIR and 24 µ m", "content": "Limited resolution, extinction, and the confusing effects of diffuse emission prevent accurate star counts for the massive dense clumps. Instead, we calculated the SFR from both 25 µ m and total infrared emission. There is a good correlation between SFR(24 µ m) and SFR( L TIR ). Ideally, this would mean that both 24 µ m and L TIR can trace SFR well in high mass star forming regions. However without an absolute SFR for comparison, we cannot tell if the SFR from both tracers are accurate or if the calibration is off by some factor. Moreover, the correlation could also result if all the clumps have similar SEDs. One way to distinguish these explanations is to compare SFR(24 µ m) and SFR( L TIR ) in low mass star forming clouds. If they show a strong correlation even when both fail to represent accurate SFR, the explanation of similar SEDs is likely. SFR(24 µ m) is plotted versus SFR( L TIR ) for both the massive dense clump sample and the nearby cloud sample in Figure 3. The solid black line represents a line of SFR(24 µ m)/SFR( L TIR ) = 1. For both data sets, SFR(24 µ m) is higher than SFR( L TIR ) on average with the average ratio of SFR(24 µ m)/SFR( L TIR ) higher for the nearby cloud sample than for high mass sample. The dashed red line represents a fit for the nearby clouds while the dashdot, green line represent a fit for the massive dense clump data. The fact that both fit similar relationships, even though we know that neither SFR( L TIR ) nor SFR(24 µ m) is accurately tracing SFR in the nearby clouds suggests that the correlation is mostly driven by the similarity of the SEDs. The nearby cloud sample shows a smaller scatter in the data than the high mass sample. The smaller scatter in the low mass sample suggests that the SED for low mass star forming clouds are more uniform that those of massive dense clumps. If the diffuse dust continuum emission is dominated by grains responding to the generally interstellar radiation field, the SED would be fairly uniform. In regions forming massive stars, the dust energetics could instead be dominated by luminous sources internal to the cloud, and the SED would depend more on the distribution of luminosities of the sources and the geometry.", "pages": [ 7 ] }, { "title": "5.2.2. IR and radio continuum", "content": "After comparing SFR(24 µ m) to SFR( L TIR ) , we then compared them to SFR(radio). The thermal radio emission comes from a different mechanism than the infrared emission. While infrared emission mostly traces dust surrounding H II regions, thermal radio traces ionized gas inside H II regions. Radio data then provides a more independent tracer of SFR in a different part of the spectrum. The result shows that SFR(radio) also correlates very well with L TIR with a correlation coefficient of 0.90. Radio data gives a slightly larger SFR than does L TIR , as shown in Figure 4(a), where a solid line represents a SFR ratio of one and the dashed line represents a best fit. Similarly, 24 µ malso correlates well with radio data as shown in Figure !4(b). In the are of 24 µ m the SFR(radio) is slightly lower than SFR(24 µ m) on average. The fact that SFR(radio) and SFR( L TIR ) are comparable to each other could indicate that both radio and infrared emission originate from the same source of heating, namely photons from high mass stars. The radio and infrared data also imply a good correlation between L TIR and radio luminosity. As seen from many previous studies, FIR-radio correlation have been well observed among galaxies with a wide luminosity range and spatial scales (Murphy et al. 2006, Dumas et al. 2011, Hughes et al. 2006, Tabatabaei et al. 2007, Zhang et al. 2010). It is interesting that even though radio continuum emission in galaxies is dominated by synchrotron emission instead of free-free emission, our results still show that the correlation between TIR and radio emission extends down to parsec scales in high mass star forming regions.", "pages": [ 7 ] }, { "title": "5.3. Combining Both Samples", "content": "Our results indicate that L TIR underestimates SFR by a large factor for low mass regions while L TIR gives consistent (within a factor of 2) SFR with SFR(radio) for high mass regions. Figure 7 shows the ratio of SFR( L TIR )/SFR(best) for both low mass and high mass regions. SFR(best) refers to SFR(YSO count) for low mass regions and SFR(radio) for high mass regions. We note that SFR(YSO count) is a more direct measurement of current SFR than SFR(radio), which depends on certain assumptions that went into the calibration. With the lack of SFR(YSO count) for high mass regions, we use SFR(radio) as a comparison. The blue stars, which represents low mass clouds, show a general trend between the SFR ratio and L TIR . SFR( L TIR ) is closer to the SFR(YSO count) at higher L TIR . L TIR traces SFR better for L TIR closer to ≈ 10 4 . 5 L /circledot , which is a transition between regions forming low-mass and regions forming high mass stars. If SFR(radio) gives an accurate measure of SFR, then the results would mean that SFR( L TIR ) is a good tracer above 10 4 . 5 L /circledot . This result would be consistent with the suggestion by Wu et al. (2005b) that the L TIR traces star formation above that luminosity. Resolving YSOs in regions forming high mass stars is a next important step in further understanding of the use of these tracers. The failure of SFR(24 µ m) and SFR( L TIR ) to accurately trace SFR in nearly all the nearby clouds has some interesting implications. An observer in another galaxy using H α or radio continuum emission would miss all star formation in a 300 pc radius of the Sun; we find that using 24 µ m emission would underestimate the local star formation by a factor of about 100. If the local volume were representative of most star formation in galaxies, the SFRs would be vastly underestimated. The fact that the same extragalactic observers would get the global SFR in the Milky Way right to a factor of about 2 (Chomiuk and Povich 2011) indicates that most star formation in the Milky Way occurs in regions forming massive stars, but this might not be the case in the outer parts of the galaxies. Finally, we note that the apparently good correlation of two purported tracers of star formation, even in regimes where neither is accurate, serves as a warning about ac- cepting 'consistency' as evidence of accuracy.", "pages": [ 7, 8 ] }, { "title": "6. SUMMARY", "content": "We studied two groups of star forming clouds in the Milky Way: 20 nearby clouds from Spitzer c2d and Gould Belt Legacy surveys; and 32 massive dense clumps that are forming massive stars. We determined the total diffuse 24 µ m emission for each cloud and calculated the corresponding SFR using the relation from Calzetti et al. (2007). Comparing 24 µ m images with extinction maps shows that a significant portion of 24 µ m emission does not come from star-forming regions in some clouds. We calculated the total infrared emission from the IRAS data and the corresponding SFR. For massive dense clumps, we also obtained radio continuum data and calculated SFR(radio) for a total of 22 clumps. Then the resulting SFRs were compared with SFRs calculated using the method of counting number of YSOs for the nearby clouds. We compared SFR( L TIR ) with SFR(24 µ m) and SFR(radio) for massive dense clumps. The comparison shows quite a good correlation between the three SFR tracers for the massive dense clumps, which are high-mass star forming regions, with the average ratio of SFR( L TIR )/SFR(24 µ m) = 0.6 ± 0 . 6 and SFR(radio)/SFR( L TIR ) = 1.8 ± 0 . 9 . Neither SFR(24 µ m) nor SFR( L TIR ) trace the SFR(YSO count) accurately in the nearby clouds, where we can calibrate with an independent method. There is a weak correlation between both tracers and SFR(YSO count), but a very different calibration value would be needed, and the scatter is large. Both 24 µ m and L TIR severely underestimate SFR for the nearby clouds. SFR( L TIR ) shows better agreement to SFR(YSO count) for clouds with higher luminosity. We would like to thank G. Helou for suggesting this study and the referee for suggestions that improved the work. We would also like to thank Mike Dunham and Amanda Heiderman for helpful discussions. we acknowledge support from NSF Grant AST-1109116 to the University of Texas at Austin.", "pages": [ 8 ] }, { "title": "REFERENCES", "content": "Krumholz, M. R., Dekel, A., & McKee, C. F. 2011, arXiv:1109.4150 Krumholz, M. R., Leroy, A. K., & McKee, C. F. 2011, ApJ, 731, 25 Kroupa, P. 2001, MNRAS, 322, 231 Kroupa, P. 2002, Science, 295, 82 Leitherer, C., Schaerer, D., Goldader, J. D., et al. 1999, ApJS, 123, 3 Lockman, F. J. 1989, ApJS, 71, 469 Markwardt, C. B. 2009, Astronomical Data Analysis Software and Systems XVIII, 411, 251 McKee, C. F., & Ostriker, E. C. 2007, ARA&A, 45, 565 Miville-Deschênes, M.-A., & Lagache, G. 2005, ApJS, 157, 302 Mueller, K. E., Shirley, Y. L., Evans, N. J., II, & Jacobson, H. R. 2002, ApJS, 143, 469 Murphy, E. J., Braun, R., Helou, G., et al. 2006, ApJ, 638, 157 Murphy, E. J., Condon, J. J., Schinnerer, E., et al. 2011, ApJ, 737, 67 Murray, N. 2011, ApJ, 729, 133 Reiter, M., Shirley, Y. L., Wu, J., et al. 2011, ApJS, 195, 1 Relaño, M., Lisenfeld, U., Pérez-González, P. G., Vílchez, J. M., & Battaner, E. 2007, ApJ, 667, L141 Rieke, G. H., Alonso-Herrero, A., Weiner, B. J., et al. 2009, ApJ, 692, 556 Salim, S., Rich, R. M., Charlot, S., et al. 2007, ApJS, 173, 267 Schruba, A., Leroy, A. K., Walter, F., et al. 2011, AJ, 142, 37 Shi, Y., Helou, G., Yan, L., et al. 2011, ApJ, 733, 87 Shirley, Y. L., Evans, N. J., II, Young, K. E., Knez, C., & Jaffe, D. T. 2003, ApJS, 149, 375 Tabatabaei, F. S., Beck, R., Krause, M., et al. 2007, A&A, 466, 509 Verley, S., Corbelli, E., Giovanardi, C., & Hunt, L. K. 2009, A&A, 493, 453 Williams, M. J., Bureau, M., & Cappellari, M. 2010, MNRAS, 409, 1330 Wu, H., Cao, C., Hao, C.-N., et al. 2005a, ApJ, 632, L79 Wu, J., Evans, N. J., II, Gao, Y., et al. 2005b, ApJ, 635, L173 Wu, J., Evans, N. J., II, Shirley, Y. L., & Knez, C. 2010, ApJS, 188, 313 Yun, M. S., Reddy, N. A., & Condon, J. J. 2001, ApJ, 554, 803 Zhang, J., Hopkins, A., Barnes, P. J., et al. 2010, Publications of the Astronomical Society of Australia, 27, 340 Zhu, Y.-N., Wu, H., Cao, C., & Li, H.-N. 2008, ApJ, 686, 155", "pages": [ 9 ] }, { "title": "SFRs for the c2d and Gould's Belt clouds", "content": "Comparison of different sources of 24 µ", "pages": [ 10 ] } ]

[ { "title": "ABSTRACT", "content": "Observations of accreting black holes often provoke suggestions that their jets precess. The precession is usually supposed to result from a combination of the Lense-Thirring effect and accretion disc viscosity. We show that this is unlikely for any type of black hole system, as the disc generally has too little angular momentumcompared with a spinning hole to cause any significant movement of the jet direction across the sky on short timescales. Uncorrelated accretion events, as in the chaotic accretion picture of active galactic nuclei, change AGN jet directions only on timescales /greaterorsimilar 10 7 yr. In this picture AGN jet directions are stable on shorter timescales, but uncorrelated with any structure of the host galaxy, as observed. We argue that observations of black-hole jets precessing on timescales short compared to the accretion time would be a strong indication that the accretion disc, and not the standard Blandford-Znajek mechanism, is responsible for driving the jet. This would be particularly convincing in a tidal disruption event. We suggest that additional disc physics is needed to explain any jet precession on timescales short compared with the accretion time. Possibilities include the radiation warping instability, or disc tearing. Subject headings: accretion, accretion disks - black hole physics - galaxies: active - galaxies: evolution -galaxies: jets", "pages": [ 1 ] }, { "title": "DO JETS PRECESS... OR EVEN MOVE AT ALL?", "content": "CHRIS NIXON 1,2,3 & ANDREW KING 2 Draft version August 1, 2018", "pages": [ 1 ] }, { "title": "1. INTRODUCTION", "content": "Jets appear in all accreting systems, from protostars (e.g. Davis et al. 1994) to AGN (e.g. Nagar & Wilson 1999; Kinney et al. 2000). In all cases the terminal speed of the jet is /greaterorsimilar the escape speed from the surface of the accreting object. Studies of protostellar jets usually assume that the ultimate power source is the accretion energy of the gas disc forming the star, mediated by strong magnetic fields (e.g. Price et al. 2012, and references therein). To tap the maximum accretion energy, a jet produced in this way must come from the innermost part of the disc near the stellar surface, and so naturally gives a terminal velocity of order the escape speed. For black holes there is debate as to whether the jet driver is again the disc accretion energy (e.g. Blandford & Payne 1982; Livio et al. 1999) or instead the black hole spin (Blandford & Znajek 1977). Observations of jets from AGN often encourage suggestions that the jets precess (e.g. Falceta-Gonc¸alves et al. 2010; Kharb et al. 2010; Gong et al. 2011; Mart'ı-Vidal et al. 2011). For the two suggested types of black hole jet-driving, this requires precession either of the disc plane close to the central accretor (where the jet is launched), or instead, of the blackhole spin axis. In this Letter we consider these processes, and show that precessing jets are not easy to obtain via any of the mechanisms usually invoked. The reasons are simply: (1) the angular momentum of any single realistic accretion event is always smaller than the angular momentum of the hole; and (2) the inner disc settles rapidly into a steady shape. This is aligned to the spin if α > H/R , and a steady warp if α < H/R . Here α is the Shakura-Sunyaev viscosity parameter and H/R is the disc angular semithickness, and the two cases correspond to diffusive and wavelike warp propagation respectively.", "pages": [ 1 ] }, { "title": "2. LENSE-THIRRING EFFECT IN DISCS WITH α > H/R", "content": "We briefly describe the evolution of a misaligned disc around a spinning black hole in the regime where warps propagate diffusively - i.e. α > H/R (Papaloizou & Pringle 1983). We discuss the wavelike case ( α < H/R ) in Section 3. The diffusive case is considered at length in the literature (e.g. Bardeen & Petterson 1975; Pringle 1992; Scheuer & Feiler 1996; Lodato & Pringle 2006 and Nixon & King 2012). The Lense-Thirring effect of a spinning black hole makes tilted disc orbits precess around its angular momentum vector at a frequency Ω LT = a ( R/R g ) -3 Ω K ( R g ) (where a is the Kerr spin parameter, R g = GM/c 2 is the black hole's gravitational radius, and Ω K ( R g ) is the Kepler frequency at disc radius R g ) which decreases strongly with radius (Thirring 1918; Lense & Thirring 1918). This differential precession is communicated through the disc by its viscosity, which acts to co- or counter-align the disc with the plane of the hole. The inner parts of the disc quickly settle in the equatorial plane of the black hole and the outer parts remain misaligned, with the two parts joined by a warped region. This is the Bardeen-Petterson effect (Bardeen & Petterson 1975) (but note that the equations of that paper do not conserve angular momentum; see Papaloizou & Pringle 1983). If an external torque (e.g. from a misaligned binary companion) maintains the tilt at the outer edge of the disc the warp can remain stationary, but otherwise the warp propagates outwards until the entire disc lies in the equatorial plane. The hole-disc system thus ends up aligned (or counter-aligned) along its original total angular momentum (the vector sum of the original spin and disc angular momenta; King et al. 2005). We note that so far all calculations of the Bardeen-Petterson effect have used Shakura & Sunyaev (1973) α discs; a demonstration of the effect for discs explicitly driven by the magnetorotational instability (MRI) has not yet been attempted. The Bardeen-Petterson evolution assumes that the disc viscosity is strong enough to communicate the differential precession efficiently through the disc. Recently Nixon & King (2012) and Nixon et al. (2012a) have shown that for realistic parameters this often does not hold. Instead the disc is torn into many distinct planes which precess almost independently of each other (Nixon et al. 2012a). If the disc inclination to the black hole spin is high enough this generates significantly counter-rotating disc orbits and these lead to rapid accretion (cf. Nixon et al. 2012b). These results markedly alter the picture of how black holes accrete, and may allow for strong precession of the inner disc plane. We return to this possibility in Section 4, but for the moment consider the usual BardeenPetterson evolution. To discuss possible jet precessions we let J d , J h and J t = J d + J h be the disc, hole and total angular momentum vectors respectively, with magnitudes J d , J h and J t . During the alignment process J h precesses around J t with an initial amplitude θ i defined by This angle is small (i.e. J t and J h are in a similar direction) either when the disc is oriented in a similar direction to the hole, or when J d /lessmuch J h (and so J h /similarequal J t ). It is clear that if J d /lessmuch J h alignment cannot move the hole spin vector very far. The inner disc must quickly become anchored to the spin plane of the hole (e.g. King et al. 2005), so alignment cannot move the inner disc very far either. So if J d /lessmuch J h the Lense-Thirring effect cannot drive a precessing jet. Thus if we have the usual Bardeen-Petterson evolution, precessions are confined at best to cases where J d /greaterorsimilar J h . However this still does not generate repeated jet precession. The initial amplitude of the precession can be large, since J t /greatermuch J h . But the alignment and precession timescales for the disc are similar (Scheuer & Feiler 1996): after only one precession time the hole is significantly aligned with the disc. This is shown explicitly in Lodato & Pringle (2006), who get at most a single precession of the jet (see their Figs 6 & 11) with significant amplitude. We conclude that in a tilted disc propagating warps in the diffusive regime ( α > H/R ) , the Lense-Thirring effect alone cannot drive repeated jet precession, unless the disc is torn into many distinct planes (Nixon et al. 2012a).", "pages": [ 1, 2 ] }, { "title": "2.1. Do jets move?", "content": "We have argued above that sustained Lense-Thirring precessions are inhibited by the dynamics of the disc-hole system. We now ask how much angular momentum can be transferred from an accretion event on to a black hole. In particular, can this change its direction significantly? We derive a simple expression for J d /J h and use it to consider realistic parameters for various astrophysical systems. The disc angular momentum is where M d is the disc mass, M is the black hole mass, R d a characteristic radius for the disc, V K the Keplerian velocity and G is the gravitational constant. The spin angular momentum of a black hole with dimensionless spin parameter a is (Kumar & Pringle 1985) where c is the speed of light. Combining (2) and (3) gives us or equivalently where R g = GM/c 2 ∼ 10 13 M 8 cmis the gravitational radius (with M 8 = M/ 10 8 M /circledot ). It is clear that this ratio can take very different values for various astrophysical systems, as we now consider.", "pages": [ 2 ] }, { "title": "2.1.1. Tidal Disruption Events", "content": "In a tidal disruption event, a star on a near-parabolic orbit around a supermassive black hole fills its tidal lobe near pericenter and is torn apart. This condition implies a pericenter separation p given by where the star has mass and radius M ∗ , R ∗ . Since R d < p and M d < M ∗ we find Even in the most favorable case of a giant star ( R ∼ 10 13 cm), (7) implies a tiny ratio This makes it obvious that any observational evidence for the movement (let alone precession) of a jet in a tidal disruption event is incompatible with jet driving by the hole spin, as is central to the standard axisymmetric Blandford-Znajek mechanism. If instead it is assumed that the jet is driven by the inner accretion disc, this must involve physics more complex than a standard thin disc warped by the Lense-Thirring effect. Tidal disruption events may produce geometrically thick discs and therefore could propagate warps as waves (see Section 3), but this requires α to be unusually small (cf. King et al. 2007).", "pages": [ 2 ] }, { "title": "2.1.2. Black hole binaries", "content": "This case appears slightly more promising than a tidal disruption as the black hole and the donor star have comparable masses M 1 , M 2 , with 0 . 1 /lessorsimilar M 2 /M 1 /lessorsimilar 10 . However at any one instant only a small fraction of the donor star feeds the black hole and thus again we have M d /lessmuch M . As favorable parameters we take R g ≈ 3 × 10 6 cm (i.e. a 10M /circledot black hole), and a large disc radius R d /lessorsimilar 10 13 cm. The largest realistic disc mass is M d /lessorsimilar 10 -5 M /circledot (e.g. Eq. 5.51 of Frank et al. 2002, with viscosity parameter α = 0 . 1 and an accretion rate ˙ M = 10 19 g s -1 corresponding to the Eddington limit for a 10M /circledot black hole). This gives Thus the disc has far too little instantaneous angular momentum to cause the hole spin axis to move on a directly observable timescale. We again conclude that jet movement would imply that the jet is not driven by the black hole spin, or by the alignment of a standard thin disc warped by the LenseThirring effect. We note that if the disc is geometrically thick it could propagate warps as waves (see Section 3), but this requires α to be unusually small (cf. King et al. 2007).", "pages": [ 2, 3 ] }, { "title": "2.1.3. Active Galactic Nuclei", "content": "This case has been considered by King et al. (2008). The main constraint on J d is the fact that discs which are too large tend to fragment into stars under self-gravity. King et al. (2008) show that a maximal disc of this type has J d /J h /lessorsimilar few × 10 -2 a -1 and has an instantaneous mass ∼ 10 -3 M , where M is the SMBH mass. Thus a mass ∼ 0 . 01 aM must pass through this kind of disc, with constant orientation, to move the direction of a centrally-produced jet by ∼ 0 . 1 radian. This would take at least 10 -2 a Salpeter times, i.e. /lessorsimilar 4 × 10 5 a yrs, even with continuous accretion at the Eddington rate, and typically /greaterorsimilar 10 7 a yrs if accretion is slower and slightly intermittent. If the orientation of successive accretion disc events changes randomly, as envisaged in the chaotic accretion picture of AGN (King & Pringle 2006, 2007) the spin direction would perform a random walk and so deviate less from its original direction. We again conclude that detectable jet precession is unlikely in AGN. In the chaotic accretion picture jets generally move very little for timescales /lessorsimilar a few × 10 6 yr. However a sequence of significant but random accretion events can move AGN jets across the sky on longer timescales ( /greaterorsimilar 10 7 yrs). These conclusions agree with the facts that jets with relatively stable or closely correlated directions are seen (e.g. Kharb et al. 2006), but jet directions do not correlate at all with any features of the host galaxy (Kinney et al. 2000).", "pages": [ 3 ] }, { "title": "3. LENSE-THIRRING EFFECT IN DISCS WITH α < H/R", "content": "We have argued above that Lense-Thirring precession in standard thin discs cannot be responsible for repeated precessions of jets. However it is unlikely that the innermost regions of black hole accretion discs remain thin. In this section we discuss the possibility of precession in discs with H/R > α . We again find that repeated precession of the jet is generally unlikely, but this time not impossible. In Section 2 we assumed α > H/R , so that warps propagate diffusively. But if α < H/R , warps instead propagate efficiently as waves with near-sonic velocities, and are not locally damped by viscosity. It is therefore possible that the transmission of such waves in the inner disc region could produce a precession. However this requires quite specific initial conditions - i.e. that the accreting material be arranged into a radially narrow ring, and α must be small. If instead the radial extent of the disc is large, the wave induced by the LenseThirring effect propagates outwards, and either never returns (on timescales of interest) or significantly damps before returning (the wave has to reach the outer disc edge before reflecting back inwards). Lubow et al. (2002) give an example where the disc has R out /R in = 90 with H/R = 0 . 1 and α = 0 . 05 . In this case the inner disc effectively settles into a steady shape while the wave slowly propagates to the outer disc. As Lubow et al. (2002) remark (last paragraph of their Section 4), 'the steady-state shape of the disc close to the hole is essentially established'. The disc quickly sets up a shape in which the internal disc torques balance the Lense-Thirring precession torque. Thus for any precession to occur and move the jet, the inner regions must wait for the outward propagating wave to reach a boundary and reflect back inwards. The reflection timescale is ∼ 2 R out /c s (e.g. Nixon & Pringle 2010) where R out is the distance the wave must travel and c s / 2 is the wave speed (Papaloizou & Lin 1995). This reasoning is not inconsistent with the simulations of Fragile et al. (2007) which suggest repeated precession of a tilted disc around a black hole. Here the authors do not assume an α viscosity, but instead simulate the MRI in an inclined thick disc ( H/R ∼ 0 . 2 ). As is known to happen in such cases (e.g. King et al. 2007), this implies an effective viscosity parameter ( α ≈ 0 . 01 ) rather lower than implied by observations ( α ≈ 0 . 1 -0 . 3 ). Fig. 13 of Fragile et al. (2007) shows the value of alpha in their computation, ranging from α ≈ 0 . 5 near the innermost stable circular orbit (ISCO) to α ≈ 2 × 10 -3 in the centre of their disc ( R = 25 R g ) to α ≈ a few × 10 -4 in the outer parts ( R ≈ 50 R g ). Away from the ISCO these values are far from those inferred from observations or those predicted by shearing box simulations (e.g. Simon et al. 2012). This may well be because the simulation run time is not long enough to allow the MRI to develop fully; for example the run time is ∼ 10 orbits at R = 25 R g , and only ∼ 3 orbits at 50 R g . We note that the disc precession (Fig. 16 of Fragile et al. 2007) is averaged over the disc region 20 R g < R < 50 R g . We also note that the timescale on which the disc is expected to reach a steady (not precessing) shape is ∼ 1 / ( α Ω) (see Eq. 4 of Lubow et al. 2002). This timescale is much longer than the runtime of the simulations showing precession. Longer runs are needed to check whether for realistic viscosities and disc sizes the repeated precession observed in Fragile et al. (2007) remains, rather than damping away after only a few orbits of the disc. A thick ( H/R /greaterorsimilar α ) small ( R /lessmuch c s t damp ) disc can in principle precess. If one can arrange a disc like this to make a sharp transition (on a scale length less than the warp wavelength) to a thin disc outside it, the wave could see this as a hard boundary and efficiently reflect back inwards. The dynamics of such a setup is largely unexplored, but since the thick region is fed by the thin region, a minimum condition is that the tilt in the thin region must be maintained. This requires extra physics, as we advocate below. The disc geometry needed for repeated precession in the wave-like regime is feasible for a tidal disruption event, where the gas circularizes very close to the accreting black hole, and the instantaneous accretion rate can be superEddington. However again this is problematic: for a thick disc with H/R ∼ 0 . 1 and α ∼ 0 . 1 the inner disc ( R < 10 R g ) aligns after at most a few precessions (Eq. 35 of Bate et al. 2000).", "pages": [ 3 ] }, { "title": "4. DISCUSSION", "content": "We have argued that the physics of standard warped discs (diffusive or wave-like) strongly suggests that the LenseThirring effect alone is not a promising mechanism for explaining jet precessions, except possibly in rather rare cases (see Section 3). The essential reason for this is that the accretion disc generally has total angular momentum small compared with that of the spinning black hole, strongly restricting the motion of any jet across the sky. Two alternative mechanisms, so far largely unexplored, may offer more promising ways of moving jets. First Pringle (1996) shows that an accretion disc can be unstable to warping driven by irradiation from a central source. If there is initially a small tilt in the disc, this can grow to provide a substantial global tilt in the disc with the angle between inner and outer parts differing by up to ∆ θ ∼ π . The inner regions of the disc precess with a quasi-periodic change in inclination (Pringle 1997). This mechanism uses the angular momentum induced by anisotropic scattering of the central accretion luminosity, so could potentially be more powerful than the Lense-Thirring effect. A second possibility for large precessions of the disc plane close to the black hole is that for large disc tilts it may break into distinct planes, with only tenuous viscous communication between them. This happens when the Lense-Thirring torque is strong enough to overcome the viscous torques holding the disc together (Nixon & King 2012; Nixon et al. 2012a). Nixon et al. (2012a) show that rapid precessions can occur here. We shall explore these ideas in future papers. Finally we note that interaction of the jet with superEddington winds coming from the disc can also generate precession of the jet as suggested for SS433 (Begelman et al. 2006). Here the jet collides with a precessing gas mass and is deflected (and slowed). The jet precession here is purely a consequence of the deflection. We thank Phil Armitage for useful discussions. Support for this work was provided by NASA through the Einstein Fellowship Program, grant PF2-130098. Research in theoretical astrophysics at Leicester is supported by an STFC Rolling Grant.", "pages": [ 3, 4 ] }, { "title": "REFERENCES", "content": "Kumar, S., & Pringle, J. E. 1985, MNRAS, 213, 435 Bardeen, J. M., & Petterson, J. A. 1975, ApJ, 195, L65+ Bate, M. R., Bonnell, I. A., Clarke, C. J., et al. 2000, MNRAS, 317, 773 Begelman, M. C., King, A. R., & Pringle, J. E. 2006, MNRAS, 370, 399 Blandford, R. D., & Payne, D. G. 1982, MNRAS, 199, 883 Blandford, R. D., & Znajek, R. L. 1977, MNRAS, 179, 433 Davis, C. J., Mundt, R., & Eisloeffel, J. 1994, ApJ, 437, L55 Falceta-Gonc¸alves, D., Caproni, A., Abraham, Z., Teixeira, D. M., & de Gouveia Dal Pino, E. M. 2010, ApJ, 713, L74 Fragile, P. C., Blaes, O. M., Anninos, P., & Salmonson, J. D. 2007, ApJ, 668, 417 Gong, B. P., Li, Y. P., & Zhang, H. C. 2011, ApJ, 734, L32 Kharb, P., Hota, A., Croston, J. H., et al. 2010, ApJ, 723, 580 Kharb, P., O'Dea, C. P., Baum, S. A., Colbert, E. J. M., & Xu, C. 2006, ApJ, 652, 177 King, A. R., & Pringle, J. E. 2006, MNRAS, 373, L90 -. 2007, MNRAS, 377, L25 King, A. R., Pringle, J. E., & Livio, M. 2007, MNRAS, 376, 1740 Kinney, A. L., Schmitt, H. R., Clarke, C. J., et al. 2000, ApJ, 537, 152 Lense, J., & Thirring, H. 1918, Phys. Z., 19, 156 Livio, M., Ogilvie, G. I., & Pringle, J. E. 1999, ApJ, 512, 100 Lodato, G., & Pringle, J. E. 2006, MNRAS, 368, 1196 Lubow, S. H., Ogilvie, G. I., & Pringle, J. E. 2002, MNRAS, 337, 706 Mart'ı-Vidal, I., Marcaide, J. M., Alberdi, A., et al. 2011, A&A, 533, A111 Nagar, N. M., & Wilson, A. S. 1999, ApJ, 516, 97 Nixon, C., King, A., Price, D., & Frank, J. 2012a, ApJ, 757, L24 Nixon, C. J., & King, A. R. 2012, MNRAS, 421, 1201 Nixon, C. J., King, A. R., & Price, D. J. 2012b, MNRAS, 2780 Nixon, C. J., & Pringle, J. E. 2010, MNRAS, 403, 1887 Papaloizou, J. C. B., & Lin, D. N. C. 1995, ApJ, 438, 841 Papaloizou, J. C. B., & Pringle, J. E. 1983, MNRAS, 202, 1181 Price, D. J., Tricco, T. S., & Bate, M. R. 2012, MNRAS, 423, L45 Pringle, J. E. 1992, MNRAS, 258, 811 -. 1997, MNRAS, 292, 136 Scheuer, P. A. G., & Feiler, R. 1996, MNRAS, 282, 291 Shakura, N. I., & Sunyaev, R. A. 1973, A&A, 24, 337 Simon, J. B., Beckwith, K., & Armitage, P. J. 2012, MNRAS, 422, 2685 Thirring, H. 1918, Phys. Z., 19, 33", "pages": [ 4 ] } ]

[ { "title": "ABSTRACT", "content": "We report interferometric imaging of CO( J =3 → 2) emission toward the z =2.846 submillimeterselected galaxy SMM J04135+10277, using the Combined Array for Research in Millimeter-wave Astronomy (CARMA). SMMJ04135+10277 was previously thought to be a gas-rich, submillimeterselected quasar, with the highest molecular gas mass among highz quasars reported in the literature. Our maps at ∼ 6 × improved linear resolution relative to earlier observations spatially resolve the emission on ∼ 1.7 '' scales, corresponding to a (lensing-corrected) source radius of ∼ 5.2 kpc. They also reveal that the molecular gas reservoir, and thus, likely the submillimeter emission, is not associated with the host galaxy of the quasar, but with an optically faint gas-rich galaxy at 5.2 '' , or 41.5 kpc projected distance from the active galactic nucleus (AGN). The obscured gas-rich galaxy has a dynamical mass of M dyn sin 2 i =5.6 × 10 11 M /circledot , corresponding to a gas mass fraction of /similarequal 21%. Assuming a typical M BH / M ∗ ratio for z /greaterorsimilar 2 quasars, the two galaxies in this system have an approximate mass ratio of ∼ 1.9. Our findings suggest that this quasar-starburst galaxy pair could represent an early stage of a rare major, gas-rich/gas-poor ('wet-dry') merger of two massive galaxies at z =2.8, rather than a single, gas-rich AGN host galaxy. Such systems could play an important role in the early buildup of present-day massive galaxies through a submillimeter-luminous starburst phase, and may remain hidden in larger numbers among rest-frame far-infrared-selected quasar samples at low and high redshift. Subject headings: galaxies: active - galaxies: starburst - galaxies: formation - galaxies: high- redshift - cosmology: observations - radio lines: galaxies", "pages": [ 1 ] }, { "title": "SMM J04135+10277: A CANDIDATE EARLY-STAGE 'WET-DRY' MERGER OF TWO MASSIVE GALAXIES AT Z =2.8", "content": "Dominik A. Riechers 1,2 draft version July 11, 2021, accepted for publication in the Astrophysical Journal Letters", "pages": [ 1 ] }, { "title": "1. INTRODUCTION", "content": "Studies of gas- and dust-rich, starbursting AGN host galaxies out to cosmological distances are important to better understand the connection between supermassive black hole and stellar bulge growth in galaxies that gives rise to the present-day M BH -M bulge relation (Magorrian et al. 1998; Haring & Rix 2004). A particularly important cosmic epoch for these studies is the redshift range 2 /lessorsimilar z /lessorsimilar 3 where most of the growth of stellar and black hole mass in galaxies occurs, i.e., where the volume densities of both cosmic star formation and AGN activity peak (e.g., Magnelli et al. 2009; Richards et al. 2006). It has recently been found that the dynamical masses of some of the most distant quasars at z> 4 appear to be too small to host stellar components as expected from the local M BH -M bulge relation, and that the available gas masses are too small to produce a sufficient amount of stars to approach this relation (e.g., Walter et al. 2004; Riechers et al. 2008a, 2008b). Thus, these galaxies appear to require a source of external gas supply (or stars) to assemble sufficient stellar mass by z =0 to approach the local M BH -M bulge relation. This gas supply could either be due to accretion of gas through cold streams (e.g., Dekel et al. 2009), or due to gas-rich, gas-rich ('wet-wet') or gas-rich, gas-poor ('wet-dry') mergers with massive and/or gas-rich galaxies (e.g., Springel et al. 2005). 1 Astronomy Department, California Institute of Technology, MC 249-17, 1200 East California Boulevard, Pasadena, CA 91125, USA 2 Astronomy Department, Cornell University, 220 Space Sciences Building, Ithaca, NY 14853, USA; [email protected] Examples of the latter may be found among highredshift, submillimeter-selected quasars. A strong submillimeter detection is suggestive of a large amount of warm dust heated by young stars formed at high rates (e.g., Isaak et al. 2002). Follow-up observations of the molecular interstellar medium (ISM) in these galaxies, typically through the detection of CO lines, are important to measure the mass of the ISM that constitutes the reservoir for star formation, and to confirm that the starburst and gas are at the same redshift as the AGN (e.g., Coppin et al. 2008). A particularly interesting submillimeter-selected quasar was found in the field of the z =0.088 galaxy cluster Abell 478, SMMJ04135+10277 at z =2.837 ± 0.003 (Knudsen et al. 2003). The source was identified in 450 and 850 µ m observations with the JCMT/SCUBA instrument, revealing high submillimeter fluxes of 25 ± 2.8 and 55 ± 17mJy, respectively, which suggest a total infrared luminosity of (2.9 ± 0.5) × 10 13 L /circledot . Subsequent interferometric CO( J =3 → 2) observations at 15 '' × 11 '' resolution and single-dish CO( J =1 → 0) observations revealed a massive molecular gas reservoir at z =2.846 ± 0.002, consistent with both the redshift and position of the quasar within the relative uncertainties (Hainline et al. 2004; Riechers et al. 2011a). None of these past studies offered sufficient spatial resolution to spatially resolve and/or precisely locate the CO or submillimeter continuum emission. We here report higher spatial resolution CO( J =3 → 2) observations with CARMA to determine the size and dynamical mass of the molecular gas reservoir. We use a concordance, flat ΛCDM cosmology throughout, with H 0 =71 kms -1 Mpc -1 , Ω M =0.27, and Ω Λ =0.73 (Spergel et al. 2003, 2007).", "pages": [ 1, 2 ] }, { "title": "2. OBSERVATIONS", "content": "We observed the CO( J =3 → 2) transition line ( ν rest = 345 . 7959899 GHz, redshifted to 89.911 GHz, or 3.33 mm) towards SMMJ04135+10277, using CARMA. A total bandwidth of 3.7 GHz ( ∼ 12,400 km s -1 ; at 5.208MHz resolution) was used to cover the CO( J =3 → 2) line and the underlying 3.33 mm (rest-frame 870 µ m) continuum emission. Observations were carried out under good 3mm weather conditions for 3 tracks in C configuration (18-367m baselines, which corresponds to probing spatial scales of 1.5 '' -31 '' , or 11-250kpc) on 2012 February 25, 29, and March 12. This resulted in 6.3 hr of 15 antenna-equivalent on-source time after discarding unusable visibility data. The nearby source 3C120 was observed every 15 minutes for pointing, amplitude and phase calibration. Fluxes were bootstrapped relative to Mars. The bright nearby calibrators 3C84 and J0423-013 were observed for bandpass calibration, yielding ∼ 15% calibration accuracy. The MIRIAD package was used for data reduction and analysis. All data were mapped using the CLEAN algorithm with 'natural' weighting, resulting in a synthesized beam size of 2.5 '' × 1.9 '' . The final rms is 0.55 mJy beam -1 over 229.2 MHz (corresponding to 765 km s -1 ), and 1.8 mJy beam -1 over 20.8 MHz (69 kms -1 ).", "pages": [ 2 ] }, { "title": "3. RESULTS", "content": "We have detected and spatially resolved strong CO( J =3 → 2) emission toward SMMJ04135+10277 (Fig. 1). CO emission is detected in each of the tracks individually at the same position. By fitting an elliptical, two-dimensional Gaussian to the u -v data, we find a source size of 1.66 '' ± 0.40 '' along its major axis, corresponding to 13.2 ± 3.2kpc at z =2.846. The source remains unresolved down to /lessorsimilar 1.2 '' ( /lessorsimilar 9.5 kpc) along its minor axis. From fitting a four-parameter Gaussian to the spectrum of the CO( J =3 → 2) line emission (Fig. 2), we measure a line peak flux of 6.6 ± 0.9mJy at a FWHM velocity width of 679 ± 120 kms -1 . Within the relative uncertainties, the line width is consistent with that measured in the CO( J =1 → 0) line (505 ± 75 kms -1 ; Riechers et al. 2011a), and marginally consistent with a previous measurement of the CO( J =3 → 2) line at lower significance (340 ± 120 kms -1 ; Hainline et al. 2004). Our measurements correspond to an integrated CO( J =3 → 2) line flux of 4.78 ± 0.67Jy kms -1 , and a CO( J =3 → 2)/CO( J =1 → 0) brightness temperature ratio of r 31 =0.82 ± 0.15. The Gaussian peaks at a redshift of z CO =2.8458 ± 0.0006, consistent with previous estimates within the errors. We marginally detect the underlying 3.33 mm continuum emission at a level of 0.31 ± 0.17mJy. The CO( J =3 → 2) emission peaks at a position of α =04 h 13 m 26 s . 989 ± 0 '' . 11, δ =+10 · 27 ' 37 '' . 89 ± 0 '' . 12. The i -band position of the quasar as determined from a Hubble Space Telescope WFPC2 F814W image obtained from the Hubble Legacy Archive 3 is α =04 h 13 m 27 s . 28, δ =+10 · 27 ' 40 '' . 77. Thus, the molecular gas reservoir is spatially offset by 5.2 '' , or 41.5 kpc, from the AGN position (Fig. 3). There is no evidence for any restframe ∼ 210nm emission at the position of the CO emission. Spitzer Space Telescope IRAC 3.6-8.0 µ m images obtained from the Spitzer Heritage Archive 4 reveal a faint counterpart to the CO-emitting galaxy at restframe near-infrared wavelengths (0.9-2.1 µ m; Fig. 4). Its rest-frame near-infrared spectrum appears to be flatter than that of the quasar, consistent with a dust-obscured star-forming galaxy and the lack of a strong active galactic nucleus component. There is no evidence for any CO or continuum emission at the position of the quasar. Assuming a line FWHM of 400 kms -1 , we derive a 3 σ upper limit of 0.9 Jy km s -1 for the CO( J =3 → 2) line flux of the quasar host galaxy. This corresponds to < 20% of the CO( J =3 → 2) line flux of the submillimeter source. We also derive a 3 σ upper limit of 0.4 mJy for the 3.33 mm continuum emission at the position of the quasar.", "pages": [ 2 ] }, { "title": "4. ANALYSIS", "content": "To account for a small amount of gravitational magnification by the foreground galaxy cluster, we will adopt a lensing magnification factor of µ QSO L =1.3 for the ac- ive galactic nucleus (as determined by Knudsen et al. 2003), and of µ CO L =1.6 ± 0.5 for the molecular gas and far-infrared continuum emission (as determined using the CO J =1 → 0 line luminosity and FWHM from Riechers et al. 2011a and a µ L -L ' CO(1 -0) -d v FWHM scaling relation for submillimeter-selected galaxies; Harris et al. 2012) in the following. We further adopt the gas mass determined by Riechers et al. (2011a) based on the CO( J =1 → 0) line luminosity, but scaled to our updated lensing magnification factor. This suggests M (H 2 ) = 1 . 2 × 10 11 (1.6/ µ L ) -1 M /circledot (assuming a L ' CO(1 -0) -M (H 2 ) conversion factor of α CO =0.8 M /circledot (K kms -1 pc 2 ) -1 for ultra-luminous infrared galaxies (ULIRGs); Downes & Solomon 1998, but also see recent discussion by Papadopoulos et al. 2012). The size estimate (5.2 kpc radius; corrected by a factor of µ 1 / 2 L ; e.g., Riechers et al. 2009) and width of the CO line suggest a dynamical mass of M dyn sin 2 i =5.6 × 10 11 M /circledot . 5 This suggests a gas mass fraction of f gas = M (H 2 )/( M dyn sin 2 i ) /similarequal 21%. To determine the black hole mass of the quasar, we adopt the λL 5100 ˚ A luminosity determined by Knudsen et al. (2003) and the M BH -L 5100 ˚ A relationship determined 5 Due to the flattening of the baryonic mass distribution in a disk, possible biases due to clumpiness of the gas, and non-circular motions of the gas, virial estimates for clumpy, disk-like galaxies may underpredict the dynamical mass by typically ∼ 30% (e.g., Daddi et al. 2010). Estimates for more complex dynamical systems likely exhibit at least comparable uncertainties, and thus, require model-based correction factors. For submillimeter-bright dusty starburst galaxies, an isotropic virial estimator is commonly adopted, which gives by a factor of ∼ 1.5 larger values than standard estimates for rotating disk galaxies at an average inclination (Engel et al. 2010). Adopting the isotropic virial estimator instead of the inclined disk model used here would result in a ∼ 20% larger dynamical mass relative to an edge-on configuration. Both estimators result in the same value when assuming an inclination of i /similarequal 66 · . The uncertainties on the dynamical mass estimate, and thus, f gas , amount to at least 20%-30%. by Peterson et al. (2004). This suggests a black hole mass of M BH =1.7 × 10 9 M /circledot . If the source were to follow the M BH -M bulge relation for nearby galaxies (Haring & Rix 2004), this would suggest M bulge /similarequal 8.2 × 10 11 M /circledot . However, Peng et al. (2006) suggest that this ratio is likely typically ∼ 4 × lower at z ∼ 2. In the following, we thus assume a stellar mass of M /star /similarequal 2 × 10 11 M /circledot for the quasar host galaxy. Approximating the total mass of the system as M tot = M BH + M /star + M (H 2 )+ M dust + M DM /similarequal 3 × 10 11 M /circledot (adopting a gas-to-dust ratio of 100, and assuming a contribution from dark matter (DM) of 25%), we find a gas fraction of f t gas = M (H 2 )/ M tot < 8% (assuming the same gas excitation as for the CO-detected galaxy).", "pages": [ 2, 3 ] }, { "title": "5. DISCUSSION AND CONCLUSIONS", "content": "We have imaged CO( J =3 → 2) emission towards the high-redshift galaxy SMMJ04135+10277, using CARMA. Our observations suggest that the molecular gas reservoir previously detected towards SMMJ04135+10277 is not associated with the host galaxy of the optically-detected quasar at z =2.837, but with an optically faint, gas-rich galaxy at z =2.846, separated by /greaterorsimilar 40kpc from the active galactic nucleus. The spectral properties of the CO-emitting source are consistent with those of an optically obscured star-forming galaxy. Its CO( J =3 → 2)/CO( J =1 → 0) brightness temperature ratio of r 31 =0.82 ± 0.15 is within the range of values observed for z> 2 submillimeter-selected starburst galaxies (e.g., Riechers et al. 2011b, 2011c; Ivison et al. 2011), but higher than the values typically observed in highz disk galaxies (e.g., Dannerbauer et al. 2009; Aravena et al. 2010). Assuming no difference in CO excitation, this gas-rich companion carries at least 5 × the gas mass of the quasar host galaxy. 6 The gas mass and gas mass fraction of the companion are comparable to those of other z> 2 submillimeter galaxies (e.g., Tacconi et al. 2006; Riechers et al. 2011d). We estimate the total mass of the SMMJ04135+10277 system to be /greaterorsimilar 8.6 × 10 11 M /circledot (with considerable uncertainty), which may be dominated by the gas-rich companion galaxy (mass ratio of ∼ 1.9). Gas-rich companions have been detected in other highz quasars as well, such as the z =4.4 and 4.7 systems BRI1335-0417 (which is already actively merging with its companion on 5 kpc scales; Riechers et al. 2008a) and BR1202-0725 (which has a companion at 26kpc projection; e.g., Carilli et al. 2002, 2013). However, in all examples known so far at high z , both the quasar host and companion are gas-rich. Thus, the quasarstarburst galaxy pair SMM J04135+10277 could be the first high redshift example of an early-stage gas-rich, gaspoor merger, in which the optically faint submillimeter galaxy provides the gas supply to further build up the stellar component of the quasar host. Models of hierarchical structure formation (e.g., Springel et al. 2005) lend support to the idea that a close massive galaxy pair like SMM J04135+10277 will likely result in a major merger in which the gas-rich companion may replenish the gas supply in the quasar host (which perhaps will yield configurations similar to those 6 High redshift quasar host galaxies commonly show high r 31 of > 0.9, which may suggest an even higher ratio in gas mass (e.g., Riechers et al. 2006, 2011a). observed in nearby infrared-luminous galaxies in intermediate or late merger stages that contain both AGN and gas-rich galaxy components; e.g., Evans et al. 2002). The relatively large projected separation of /greaterorsimilar 40kpc suggests that the two massive galaxies are likely physically related and gravitationally interacting, but cannot (yet) be considered part of a common gravitational potential. This is consistent with what is expected for a merging system in an early stage. Given the early phase in the merging process implied by this scenario, it is plausible but not unambiguous to assume that the ongoing black hole accretion in the quasar and star formation in the gas-rich companion could have been triggered by interaction. Submillimeter-selected high redshift quasars are good candidates for transition objects from hyper-luminous infrared galaxies to optically bright quasars, linking the most intense starbursts in the universe the the most actively accreting black holes (e.g., Coppin et al. 2008; Simpson et al. 2012). This scenario would be consistent with a high redshift analogue of the ULIRG-quasar transition scenario proposed by Sanders et al. (1988). However, despite being a good candidate for a transition object initially, SMM J04135+10277 is not an example of such sources. Instead, our observations suggest that it is a good candidate for an early-stage gasrich, gas-poor ('wet-dry') merger 7 of two massive galaxies at z =2.8, and thus, a possibly more extreme highredshift analogue to the z =0.3 quasars HE0450-2958 and J1821+643 (which were identified to not be transition objects through similar observational strategies; e.g., Papadopoulos et al. 2008; Aravena et al. 2011). Gas-rich, gas-poor ('wet-dry') mergers at high redshift are predicted by cosmological simulations of hierarchical structure formation (e.g., Springel et al. 2005), but successful observations of such systems are still scarce. Based on the discovery of SMMJ04135+10277 alone, it remains unclear what the incidence of such systems within submillimeter-selected quasar samples is. It however clearly motivates observations of larger samples with high-quality optical/infrared data in CO and submillimeter continuum emission at high spatial resolution. Such studies will become feasible in the near future, with the completion of both the Karl G. Jansky Very Large Array (VLA) and the Atacama Large (sub-) Millimeter Array (ALMA). We thank the anonymous referee for a helpful report. DR acknowledges support from the National Aeronautics and Space Administration (NASA) through a Spitzer Space Telescope grant. Support for CARMA construction was derived from the G. and B. Moore Foundation, the K. T. and E. L. Norris Foundation, the Associates of the California Institute of Technology, the states of California, Illinois, and Maryland, and the NSF. Ongoing CARMA development and operations are supported by the NSF under a cooperative agreement, and by the CARMA partner universities. Based in part on observations made with the NASA/ESA Hubble Space Telescope, and obtained from the Hubble Legacy Archive, which is a collaboration between the Space Telescope Science Institute (STScI/NASA), the Space Telescope European Coordinating Facility (ST-ECF/ESA), and the Canadian Astronomy Data Centre (CADC/NRC/CSA). Based in part on observations made with the Spitzer Space Telescope, and obtained from the Spitzer Heritage Archive through the NASA/IPAC Infrared Science Archive, which is operated by the Jet Propulsion Laboratory, California Institute of Technology, under contract with NASA.", "pages": [ 3, 4 ] }, { "title": "REFERENCES", "content": "Aravena, M., et al. 2010, ApJ, 718, 177 Aravena, M., Wagg, J., Papadopoulos, P. P., & Feain, I. J. 2011, ApJ, 737, 64 Carilli, C. L., et al. 2002, AJ, 123, 1838 Carilli, C. L., Riechers, D., Walter, F., Maiolino, R., Wagg, J., Lentati, L., McMahon, R., & Wolfe, A. 2013, ApJ, 763, 120 Coppin, K., et al. 2008, MNRAS, 389, 45 Daddi, E., et al. 2010, ApJ, 713, 686 7 We note that 'dry' here refers to the fact that the companion galaxy far dominates the gas content of this system. It does not imply that there is no gas whatsoever in the host of the quasar. Dannerbauer, H., Daddi, E., Riechers, D. A., Walter, F., Carilli, C. L., Dickinson, M., Elbaz, D., & Morrison, G. E. 2009, ApJ, 698, L178 Dekel, A., et al. 2009, Nature, 457, 451 Downes, D., & Solomon, P. M. 1998, ApJ, 507, 615 Engel, H., et al. 2010, ApJ, 724, 233 Evans, A. S., Mazzarella, J. M., Surace, J. A., & Sanders, D. B. 2002, ApJ, 580, 749 Haring, N., & Rix, H.-W. 2004, ApJ, 604, L89 Hainline, L. J., Scoville, N. Z., Yun, M. S., Hawkins, D. W., Frayer, D. T., & Isaak, K. G. 2004, ApJ, 609, 61 Harris, A. I., et al. 2012, ApJ, 752, 152 Isaak, K. G., et al. 2002, MNRAS, 329, 149", "pages": [ 4 ] } ]

[ { "title": "ABSTRACT", "content": "Linear analysis of the formation of protostellar cores in planar magnetic interstellar clouds yields information about length scales involved in star formation. Combining these length scales with various distributions of other environmental variables, (i.e., column density and mass-to-flux ratio) and applying Monte Carlo methods allow us to produce synthetic core mass functions (CMFs) for different environmental conditions. Our analysis shows that the shape of the CMF is directly dependent on the physical conditions of the cloud. Specifically, magnetic fields act to broaden the mass function and develop a high-mass tail while ambipolar diffusion will truncate this high-mass tail. In addition, we analyze the effect of small number statistics on the shape and high-mass slope of the synthetic CMFs. We find that observed core mass functions are severely statistically limited, which has a profound effect on the derived slope for the high-mass tail. Subject headings: diffusion - ISM: clouds - stars: formation - stars: luminosity function, mass function - ISM: magnetic fields - ISM: structure", "pages": [ 1 ] }, { "title": "THE EFFECT OF MAGNETIC FIELDS AND AMBIPOLAR DIFFUSION ON CORE MASS FUNCTIONS", "content": "Nicole D. Bailey and Shantanu Basu Department of Physics and Astronomy, University of Western Ontario 1151 Richmond Street, London, Ontario, N6A 3K7 Draft version April 24, 2018", "pages": [ 1 ] }, { "title": "1. INTRODUCTION", "content": "Observations of the stellar initial mass function (IMF) and the core mass function (CMF) show similarities in the shape and high mass slope of these two functions (Motte et al. 1998; Testi & Sargent 1998; Johnstone et al. 2000; Alv'es et al. 2007; Nutter & Ward-Thompson 2007; Simpson et al. 2008; Enoch et al. 2008; Sadavoy et al. 2010, among others). As such, much theoretical effort has been invested in order to explain these similarities. Various different approaches to this problem have been explored, including analytic and numerical studies which invoke gravitational fragmentation or accretion (Silk 1995; Inutsuka 2001; Basu & Jones 2004), turbulence (Padoan et al. 1997; Padoan & Nordlund 2002; Ballesteros-Paredes et al. 2006; Hennebelle & Chabrier 2008, 2009), independent stochastic processes (Larson 1973; Elmegreen 1997) and magnetic fields (Dib et al. 2008), among others. Results of these studies vary from those which seem to agree with the fiducial Salpeter form, dN/d log M ∝ M -α where α = 1 . 35 is the value of the Salpeter slope, to those that do not. The high mass slope of the IMF was initially derived by Salpeter (1955) and later improved upon by Kroupa (2002) and Chabrier (2003a,b, 2005). Despite variations in observed and theoretically derived IMF slope values, it is often assumed that the shape and high mass slope of the IMF and CMF are identical and universal. From a theoretical view, such a one-to-one correspondence between these two functions implies that high-mass cores beget high-mass stars and likewise for low-mass cores. The need for extensive simulations of how a complex of cores turns into a cluster of stars is simplified tremendously if it is assumed that each core will collapse into a single star with some mass loss to account for the mass shift between the CMF and IMF. The underlying tenet of universality is that all star- [email protected] (NDB); [email protected] (SB) forming regions are the same and undergo the same process to form stars, however recent observations and simulations have started to reveal cracks in this assumption. In a study of the effect of turbulence on the formation of the CMF, Hennebelle & Chabrier (2008, 2009) find that comparisons between their IMF and observations for different cloud conditions suggest that star formation should predominantly occur in clouds five times denser than characterized by Larson (1981). This led them to question the universality of the IMF since, as they say, choosing different cloud parameters would lead to a different CMF/IMF. Several recent studies of the IMF also tend to disagree with the assumed universality. Observations of different star clusters in both the Milky Way and the Large Magellanic Cloud (LMC) show a wide scatter of slopes: α = 0 . 5 -2 . 0 (Elmegreen 1999). A survey of high mass slope values for different stars (i.e., cluster stars versus association stars versus field stars) yields a wide range of values; α = 2 . 0 -4 . 0 for extreme field stars to α = 1 . 5 -2 . 0 for cluster stars (Elmegreen 1997). Further to this, Elmegreen (1999) shows that through stochastic fractal sampling of a cloud, the derived IMF slopes can vary from α as low as 1.0 to as high as 1.7. Clark et al. (2007) note that if the lifetime of a more massive core is longer than a less massive one, the slope of the CMF should be shallower in order to obtain the IMF. Finally, Zaritsky et al. (2012) show that there may be evidence for two distinct stellar IMFs that depend on the age and metallicity of the cluster in question. Based on the above evidence and arguments, it is not clear why one should insist on using α = 1 . 35 as the universal slope for both the CMF and IMF. The majority of the work in this area has focused on the effects of turbulence within the molecular clouds on the formation and shape of the CMF. Research which considers the effect of magnetic fields and ambipolar diffusion on the CMF is sparse. Kunz & Mouschovias (2009) used the results of a non-ideal MHD linear analysis of a partially ionized sheet (Morton 1991; Ciolek & Basu 2006) to generate a broad CMF, assuming ambipolar-diffusion initiated core formation. Their model assumed subcritical to critical initial conditions with a uniform distribution of mass-to-flux ratios between 0.1 and 1.0 times the critical value for gravitational instability (see Section 5 for more discussion of their model). In this study, we use the results of the linear analysis of a partially ionized sheet along with a lognormal distribution of initial column density and various distributions of mass-to-flux ratio. We explore both subcritical and supercritical initial conditions. Mildly supercritical initial conditions are the most likely to lead to massive core formation, as seen in e.g., Figure 2 of Ciolek & Basu (2006). Furthermore, we use a lognormal distribution of column densities, as expected in molecular clouds on both theoretical grounds for a turbulent medium (Padoan et al. 1997) and from observations (Kainulainen et al. 2009). The aim of this paper is two fold. In the first part we show the effects of a magnetic field on the shape of the CMF. Starting from an assumption of lognormal column density probability we show the broadening effect of neutral-ion drift via ambipolar diffusion and differing mass-to-flux ratio distributions. In the second part, we address the inherent limitations of observed core mass functions, i.e. sample size and bin size. Specifically, we aim to compare small sample synthetic CMFs to large sample synthetic CMFs to show effect of small number statistics on the observed features of the CMF. In Section 2 we outline our model and methods for constructing our synthetic CMFs. Section 3 shows the results for the different distribution models considered. Section 4 shows the effect of small number statistics and the variance in derived analytic slopes. Finally Sections 5 and 6 give our discussion and conclusions.", "pages": [ 1, 2 ] }, { "title": "2. SYNTHETIC CORE MASS FUNCTIONS", "content": "To better understand the effects of the environment on the shape and peak of the core mass function, we produce synthetic CMFs (synCMFs) based upon varying physics and properties of molecular clouds. These include the column density ( σ n, 0 ), ionization fraction ( χ i = log[ n e /n H ]), mass-to-flux ratio ( µ 0 ), and neutral ion-collision time ( τ ni ). The synCMFs are produced by randomly sampling predefined column density and massto-flux ratio distributions (where applicable) and using a preferred fragmentation length scale to calculate the core mass. We choose to use such methods due to the random nature of molecular cloud properties. This allows us to statistically determine the shape of the CMF for a wide range of randomly chosen σ n -µ 0 pairs.", "pages": [ 2 ] }, { "title": "2.1. Physical Model", "content": "We consider the formation of cores and the resulting CMF within ionized, isothermal, interstellar molecular clouds. These clouds are modelled as planar sheets with infinite extent in the x - and y - directions and a local vertical half thickness Z . The nonaxisymmetric equations and formulations of our assumed model have been described in detail in several papers (Ciolek & Basu 2006; Basu et al. 2009a,b; Bailey & Basu 2012). For this work we consider three models: nonmagnetic, flux-frozen magnetic field and a magnetic field with ambipolar diffusion. The key ingredient to this analysis is the assumed length scale for the core. This length scale for collapse can be derived through linear analysis. The nonaxisymmetric equations of Ciolek & Basu (2006) and Basu et al. (2009a,b) include the effect of ambipolar diffusion. This is quantified by the timescale for collisions between ions bound to the magnetic field and free neutral particles. This timescale is Here, m i is the ion mass, n i is the number density of ions and 〈 σw 〉 iH 2 is the neutral-ion collision rate. The typical atomic and molecular species within a molecular cloud are singly ionized Na, Mg and HCO which have a mass of 25 amu. Assuming collisions between H 2 and HCO + , the neutral-ion collision rate is 1 . 69 × 10 -9 cm 3 s -1 (McDaniel & Mason 1973). Collisions between neutrals and ions transfer information about the magnetic field to the neutral particles. The threshold for whether a region of a molecular cloud is stable or unstable to collapse is given by the mass-to-flux ratio of the background reference state where (2 πG 1 / 2 ) -1 is the critical mass-to-flux ratio for gravitational collapse in the adopted model and B ref is the magnetic field strength of the reference state. Regions with µ 0 < 1 are defined as subcritical, regions with µ 0 > 1 are defined to be supercritical and regions with µ 0 ∼ 1 are transcritical. A dispersion relation for the governing magnetohydrodynamic equations can be found via linear analysis (Ciolek & Basu 2006; Basu et al. 2009b; Bailey & Basu 2012) . Here we follow the analysis as described in Bailey & Basu (2012). For a model with ambipolar diffusion, the resulting dispersion relation is where Here, ω is the angular frequency of the perturbations, τ ni, 0 is the initial neutral-ion collision time, k is the wavenumber in the z -direction, V A, 0 is the Alfv'en speed, where Z 0 is the initial half-thickness of the sheet, and C eff , 0 is the local effective sound speed, such that Here, c s = ( k B T/m n ) 1 / 2 is the isothermal sound speed, k B is the Boltzmann constant, T is the temperature in Kelvins and m n is the mean mass of a neutral particle ( m n = 2 . 33 amu). For this analysis, we assume a temperature T = 10 K and a normalized external pressure ˜ P ext ≡ 2 P ext /πGσ 2 n, 0 = 0 . 1. In the limit of flux freezing, τ ni, 0 → 0, which gives the reduced dispersion relation The gravitationally unstable mode corresponds to one of the roots of ω 2 < 0 and occurs for µ 0 > 1. The growth time for this mode can be written as for λ ≥ λ MS , where The length scale corresponding to the minimum growth time is λ g,m = 2 λ MS . This is the length scale used to produce our synCMFs for models with flux freezing. The variation of this length scale as a function of µ 0 is shown by the solid line in Figure 1. For the case with no magnetic field, Equation 9 reduces down to the thin disk equivalent of the Jeans length, Again, the length scale corresponding to the minimum growth time is λ g,m,J = 2 λ J , which is the scale used in our nonmagnetic model. The addition of ambipolar diffusion complicates the process somewhat. In these cases, the gravitationally unstable mode corresponds to one of the roots of the full dispersion relation (Equation 3). However since it is a cubic function, there is no simple expression to describe these roots. Therefore, each length scale is computed numerically. The value of this length scale is related to the degree of ambipolar diffusion i.e., the degree of ionization within the cloud, and the mass-to-flux ratio of the region. Previous studies show that the ionization fraction within a molecular cloud resembles a step function (Ruffle et al. 1998; Bailey & Basu 2012) such that the outer layers are highly ionized due to UV photoionization while ionization of denser inner regions is primarily due to cosmic rays. For this study, we choose to fix the neutral-ion collision time to the dimensionless value τ ni, 0 /t 0 = 2 πGσ n, 0 τ ni, 0 /c s = 0 . 2 ; a value typical of the denser inner regions where most cores are likely to form (Basu et al. 2009a, and references within). This corresponds to an ionization fraction χ i = 5 . 2 × 10 -8 at a neutral column density σ n, 0 = 0 . 023 g cm -2 . Figure 1 (dotted line) shows the relation between the collapse length scale and the mass-to-flux-ratio for this neutral-ion collision time. By fixing the neutral-ion collision time, our ambipolar diffusion models have only two free parameters, the column density and mass-to-flux ratio distributions. Our choices for these two parameters are discussed in the following sections.", "pages": [ 2, 3 ] }, { "title": "2.2. Column Density Distribution", "content": "A survey of column density σ n distributions within various molecular clouds shows that they generally exhibit log-normal distributions either with or without a high density tail (Kainulainen et al. 2009). Correlation of these different shapes with the conditions within the clouds suggest that regions with a pure lognormal distribution tend to be quiescent while those with high density tails show signs of active star formation. Since the aim of this paper is to investigate the shape of the core mass function as an initial condition for star formation, we choose a simple lognormal distribution as shown in Figure 2. This plot shows the distribution as a function of both the column density ( σ n , lower axis) and the visual extinction ( A v , upper axis). Following the prescription of Pineda et al. (2010), the conversion from visual extinction to column density is achieved by combining the ratio of H 2 column density to color excess (Bohlin et al. 1978) with the total selective extinction (Whittet 2003) to yield a conversion factor N ( H 2 ) = 9 . 35 × 10 20 A v cm -2 mag -1 . Although this conversion is specifically for H 2 , the abundance ratio of CO to H 2 is ∼ 10 -4 and other molecular contributions are even smaller, so they do not add significantly to the number density of H 2 . Therefore we assume this number density is representative of all species. Assuming a mean molecular weight of 2.33 amu, this translates into a mass column density conversion of the form The variance and mean ( σ 2 and µ ) of this distribution were chosen based upon observational information. Previous studies of molecular clouds show visual extinction thresholds for core and star formation to be on the order of A v = 5 mag (Johnstone et al. 2004; Kirk et al. 2006) and A v = 8 mag (see Johnstone et al. 2004; Froebrich & Rowles 2010, among others) respectively. As such, we adopted a mean visual extinction value of 8 magnitudes for our lognormal density distribution. The variance reflects the typical width of the lognormal fits to cloud density functions presented by Kainulainen et al. (2009).", "pages": [ 4 ] }, { "title": "2.3. Mass-to-Flux Ratio Distributions", "content": "Although density/visual extinction maps are fairly commonplace, measurements of magnetic field strengths within molecular clouds are difficult to obtain. Due to limitations in techniques and resolution, studies of magnetic fields within clouds are generally on a more global scale (see Crutcher 1999; Heiles & Troland 2004; Troland & Crutcher 2008; Falgarone et al. 2008; Crutcher et al. 2010; Chapman et al. 2011, among others) which does not give much insight into the exact nature of µ 0 within denser small scale regions. Therefore, the mass-to-flux ratio of specific regions are not generally known, let alone a distribution over an entire cloud. Recent simulations of cloud formation with magnetic fields (V'azquez-Semadeni et al. 2011) show that the mass-toflux ratio distribution seems to exhibit a lognormal shape. On the other hand, analysis of the likelihood of different magnetic field distributions (Crutcher et al. 2010) show that the magnetic field strengths for various regions (HI diffuse clouds, OH dark clouds, etc) exhibit a uniform distribution ranging from very small values up to a maximum value. This seems to disagree with the simulations of V'azquez-Semadeni et al. (2011). With these results in mind, we choose to explore both options (i.e., uniform and lognormal distributions). As shown by the linear analysis results presented in Bailey & Basu (2012) and Figure 1, the length scale for collapse is dependent on the value of the mass-to-flux ratio. The value of µ 0 is selected from a predefined distribution that is independent of the distribution of σ n . This implies that the magnetic field strength is not constant and varies according to the choices of σ n and µ 0 . The independent sampling of values of σ n and µ 0 does not then allow for any systematic dependence of one quantity on the other. We believe this is an acceptable first approximation since the initial conditions of the mass-to-flux ratio distribution in a molecular cloud are poorly constrained. We test several possible µ 0 distributions in an attempt to determine if the shape of an observed CMF could reveal information about the underlying mass-to-flux ratio distribution. We consider both uniform and lognormal distributions. Figures 3 & 4 show the adopted lognormal mass-to-flux ratio distributions for the flux freezing and ambipolar diffusion models respectively. Specifically, all distributions sample the transcritical peak in fragmentation scale, λ g,m (see Figure 1). The properties of all µ 0 distributions considered are given in Table 1.", "pages": [ 4 ] }, { "title": "2.4. Producing Synthetic Core Mass Functions", "content": "To produce a synthetic CMF, we randomly sample the column density distribution for the nonmagnetic case and both the column density and mass-to-flux ratio distributions for the magnetic cases. These values are then used to find the preferred length scale for collapse from the linear analysis. Finally, the mass is determined by multiplying the column density by the square of the corresponding length scale. By randomly sampling each model distribution 10 6 times, a synthetic CMF is produced.", "pages": [ 5 ] }, { "title": "3. MODELS AND RESULTS", "content": "Our analysis covers several different mass-to-flux ratio distributions and assumptions about the neutral-ion collision time and column density distribution. As stated earlier, the column density distribution is the same for all models (see Figure 2) and the neutral-ion collision time for the ambipolar diffusion models is set to a normalized value, τ ni, 0 /t 0 = 0 . 2. In addition to the models listed in Table 1, we also present a nonmagnetic (NM) fiducial case. The following subsections present the results for each model individually. An in depth comparison between all the models and implications regarding observed CMFs will be discussed in Sections 3.4 & 4 respectively.", "pages": [ 5 ] }, { "title": "3.1. Non-Magnetic Model", "content": "The nonmagnetic model serves as a baseline for our investigation. The left panel of Figure 5 shows the resulting core mass function from this technique. As discussed in Section 2.2, we choose the peak of our density distribu- tion to correspond to the apparent visual extinction threshold for the creation of star forming cores; A v ∼ 8 magnitudes. The right hand panel of Figure 5 shows the contributions from high density gas ( A v > 8 mag, dotted line) and low density gas ( A v < 8 mag, dashed line). As expected from the Jeans theory, the core mass distribution mimics the column density distribution, with high mass cores formed from low density gas and low mass cores formed from high density gas. The distribution of masses for this model peaks at a value of log( M/M /circledot ) = 0 . 4 or M /similarequal 2 . 5M /circledot which is consistent with observations (Nutter & Ward-Thompson 2007).", "pages": [ 5 ] }, { "title": "3.2. Flux Frozen Magnetic Model", "content": "A main aim of this paper is to show the effect of a magnetic field on the CMF. A flux frozen field represents the simplest case. Such a scenario arises in highly ionized regions where frequent collisions between ions and neutral particles would ensure perfect coupling to the magnetic field. Figures 6-8 show the resulting synthetic core mass function for the three models FF1, FF2, and FF3 respectively. Under the assumption of a uniform mass-to-flux ratio distribution (FF1), the resultant CMF (Figure 6, left) exhibits a narrow peak with a distinct high mass tail. The right hand panel of Figure 6 again shows the contributions to the CMF from the two column density regimes ( A v < 8 mag (dashed line) and A v > 8 mag (dotted line)). This composite plot shows that like the NM case, and in line with the Jeans theory, the low density gas forms high mass cores, while high density gas forms low mass cores. However, unlike the Jeans theory and nonmagnetic case, we see that with the addition of a magnetic field, the high density gas also contributes to the formation of high mass cores , albeit to a lesser extent. Compared to the NM case, the peak of this core mass function is shifted to M /similarequal 10 0 . 7 M /circledot /similarequal 5 . 0 M /circledot . On the right hand side of this peak, the trend can be described by two distinct slopes. For 0 . 7 < log( M/M /circledot ) < 1 . 2, α = 0 . 8 while for log( M/M /circledot ) > 1 . 2 the slope becomes shallower; α ∼ 0 . 6. Neither of these values correspond to the typical Salpeter and observational values. This discrepancy will be discussed further in Section 4. The formation of the high mass tail is due to the relationship between µ 0 and λ as defined by Equation 9. For µ 0 -σ n pairs which have mass-to-flux ratios closer to the critical value ( µ 0 = 1), the corresponding length is up to 23 times larger than the thermal Jeans length for the same column density (see Figure 1). This increase in length scale has a direct effect on the mass of the core that is formed. Conversely, the low mass distribution is formed by µ 0 -σ n pairs that have low column density and mass-to-flux values that are closer to the other limit ( µ 0 = 3), where λ is only about 1.5 times larger than the thermal length scale. Figure 7 shows the resulting synCMF for a broad lognormal µ 0 distribution (FF2). The two panels again show the total and composite CMFs as described above. This distribution results in a CMF that is similar to that of model FF1 (Figure 6), with a few minor differences. First, the high mass tail exhibits a steeper slope that results in a more pronounced peak region. Second, the peak of the mass function has shifted to a slightly smaller value of M /similarequal 10 0 . 5 M /circledot = 3 . 16 M /circledot . As before, the trend of the high mass side can be described by two distinct slopes. For 0 . 5 < log( M/M /circledot ) < 1 . 0, α = 1 . 31 while for log( M/M /circledot ) > 1 . 0 the slope becomes shallower; α = 0 . 63. models, the effect of adding a flux-frozen field is the appearance of a broad shallow tail at the high mass end of the core mass function.", "pages": [ 5, 6, 7 ] }, { "title": "3.3. Ambipolar Diffusion Magnetic Model", "content": "In the previous section we looked at the effect of a simple flux-frozen field on the shapes of the resulting CMF(s). Here we look at how the addition of neutralion slip via ambipolar diffusion affects the shape of the CMF. As discussed above, we have fixed the normalized neutral-ion collision time to τ ni, 0 /t 0 = 0 . 2. This implies a high degree of ambipolar diffusion and therefore less frequent collisions between the neutrals and ions. Such a situation would occur in the inner regions of a molecular cloud where the main ionization mechanism is cosmic rays. Figures 9-13 show the resulting synCMFs for all five mass-to-flux ratio distributions respectively. To establish how the sub- and supercritical regions of the massto-flux ratio affect the shape of the CMF, we start our analysis by presenting two cases that isolate each regime. Figures 9 & 10 show the resulting synCMFs for the subcritical and supercritical uniform mass-to-flux ratio distributions (AD1 and AD2) respectively. The two panels show the total and constituent core mass functions as described in the previous section. Focusing on model AD1, Figure 9, the left panel shows that the core mass function is very similar to the nonmagnetic model (see Figure 5, left). This is due to the fact that the curve on the subcritical side of Figure 1 converges to the nonmagnetic limit faster than in the transand supercritical regions. Upon closer comparison, AD1 peaks at the approximately the same value as NM, however the density composite CMF (Figure 9, right) reveals differences between these two models. Unlike the nonmagnetic model, AD1 shows evidence that a portion of the high column density gas goes toward forming high mass cores (Figure 9, right). Figure 10 shows the resulting synCMF under the assumption of a uniform supercritical distribution (AD2). The left panel shows that the total CMF is a hybrid between the nonmagnetic and flux-frozen models presented above. Specifically, this CMF shows the same peaked nature with high mass tail as the flux frozen model, however this tail abruptly declines at about 100 M /circledot . This truncation makes the over all shape of the CMF resemble the nonmagnetic case, albeit broader, with the beginnings of a 'shoulder' feature between 10 and 100 M /circledot . Looking at the composite column density CMF (Figure 10, right), we see that the lowest and highest mass cores are formed by the highest and lowest density gas respectively, while the middle has contributions from both density regimes. The peak of the mass function for this model occurs at about log( M/M /circledot ) = 0 . 7. Model AD3 assumes a uniform mass-to-flux ratio distribution that samples the peak of the λ versus µ 0 graph (see Figure 1). The resulting CMF (Figure 11, left) is very similar to the one produced by AD2. Looking at the contributions from the low and high column density gas, we again see that the lowest and highest mass cores are formed by the highest and lowest density gas respectively while the middle range has contributions from both density regimes. The right panel of Figure 11 shows the contributions from the subcritical ( µ 0 ≤ 1, dashed line) and supercritical ( µ 0 > 1, dotted line) gas. We see that the total synCMF for AD3 (Figure 11, left) is a combination of models AD1 and AD2. Specifically, we see that the majority of the cores are formed from supercritical gas, while the subcritical gas yields a minor contribution to the population of low mass cores. By mentally combining the middle and right hand plots in Figure 11, one can determine that the highest mass cores are formed by supercritical gas and fall into the non-star-forming regime while low-mass cores are formed by both supercritical and subcritical gas, and fall into both the star-forming and nonstar-forming regimes. The peak of the mass function for this model occurs at about log( M/M /circledot ) = 0 . 7 and the average slope of the high mass 'tail' is α = 1 . 42. Finally, Figures 12 & 13 show the resulting synCMFs for the two lognormal µ 0 distributions, AD4 and AD5, respectively. The broad lognormal distribution (AD4) is similar to models AD2 and AD3, however this model shows a more distinct 'peak' and 'shoulder' region as compared to the other two. Looking at the composite mass-to-flux ratio plot (Figure 12, right) we see that the peak region is mainly formed by subcritical gas while the shoulder region is formed mainly by contri- utions from supercritical gas. This model peaks at M = 10 0 . 7 M/M /circledot /similarequal 5 . 0 M /circledot , and the average slope of the high mass tail is α = 1 . 18. Switching to the narrow lognormal distribution (Figure 13), we see that this model results in a double peaked function. Examination of the composite plots show that the low mass peak is formed by the subcritical material while the second peak is formed by supercritical material. These peaks occur at log( M/M /circledot ) ∼ 0 . 7 and log( M/M /circledot ) ∼ 1 . 5 respectively. The formation of the high mass peak is due to the extremely narrow mass-to-flux ratio distribution. It picks out only large length scales from the peak of the λ -µ 0 curve (with τ ni, 0 /t 0 = 0 . 2) in Figure 1.", "pages": [ 7, 8 ] }, { "title": "3.4. Assessment of Synthetic Core Mass Functions", "content": "The previous subsections presented the overall results and features of each of the models. Within these results we found three main features that changed between the different models. These are the overall shape of the core mass function, the location of the peak(s) and the slope of the high mass tail (if it exists). Here we discuss these three features across all models.", "pages": [ 8 ] }, { "title": "3.4.1. Shape", "content": "Within the nine models presented, there were three distinct recurring shapes; pure lognormal as represented by the NM and AD1 models, lognormal peak with a shoulder as represented by AD2, AD3, AD4 and AD5, and the lognormal peak with high mass tail as represented by FF1, FF2, and FF3. The appearance of these shapes are directly connected to the state of the magnetic field in the region. In the absence of a magnetic field, the CMF is a pure lognormal function. This shape is also observed in model AD1. As mentioned earlier, the reason that this AD model shows such a shape while the other ones do not is due to the shape of the λ -µ 0 curve on the subcritical side of Figure 1; the curve asymptotes to the nonmagnetic limit faster on that side than on the supercritical side. Therefore one would expect a model with only subcritical mass-to-flux values to look similar to the nonmagnetic model, but with a slight broadening due to a narrow region of mass-to-flux ratios with λ larger than the non-magnetic limit. For models with an increasing supercritical regime, the broadening becomes more pronounced as a shoulder develops. This shoulder is due to an increase in higher mass cores that are the product of the larger length scales picked out by the supercritical mass-to-flux ratios. The extent of the shoulder depends on the mass-to-flux ratio distributions. For uniform distributions, the CMF is narrower with a less defined shoulder region, while for a broad lognormal distribution, the shoulder region is much broader and distinct. Finally, the appearance of the double peaked CMF in AD5 is an example of an extreme shoulder. This second peak is due solely to the extremely narrow mass-to-flux ratio range used in this model. This preferentially picks out only mass-to-flux ratios with length scales much larger than the nonmagnetic model. The appearance of the pure high mass tail is entirely a product of flux-freezing. This is due to the asymptotic nature of the flux-frozen curve as it nears the critical mass-to-flux ratio (see Figure 1). This allows for transcritical mass-to-flux ratio values to produce much larger masses for the same column density.", "pages": [ 8, 9 ] }, { "title": "3.4.2. Peak Location", "content": "The location of the CMF peak depends on the distribution of the mass-to-flux ratio. The location of the peak in the nonmagnetic case, which occurs at log( M/M /circledot ) = 0 . 4 ( M ∼ 2 . 5 M /circledot ) serves as the comparison point. For magnetic models, the location of the peak was generally larger than this value as long as the mass-to-flux ratio distribution was uniform with some contribution from the supercritical regime (see models FF1, AD2, and AD3). Model AD1, although also assuming a uniform mass-to-flux ratio distribution, exhibits a similar peak value to NM due to the exclusion of supercritical massto-flux ratio values. When considering the lognormal mass-to-flux ratio distributions, we find that the peak location is dependent on the width of the distribution. Specifically, broader distributions exhibit values closer to the NM peak value, while narrower distributions exhibit peak values that are higher than the nonmagnetic case. Model AD5 is an anomaly and does not fit within these trends given that it exhibits two peaks.", "pages": [ 9, 10 ] }, { "title": "3.4.3. High Mass Slope", "content": "As alluded to earlier, the shape and extent of the high mass slope was found to be variable and connected to the influence of the magnetic field. Specifically, the appearance of the 'shoulder' feature is directly connected to the presence of ambipolar diffusion. The degree of the shoulder in the ambipolar diffusion models was found to be dependent on the range of allowed mass-to-flux ratio values. Overall, these differences in shapes result in a wide range of slopes. For the flux-frozen models, the slopes were as steep as α = 1 . 31 in the case of FF2, and as shallow as α = 0 . 44 in the case of FF3. For the ambipolar diffusion models, the average high mass slope ranges between α = 1 . 18 and α = 1 . 42. Although some of these slopes are consistent with the Salpeter value, α = 1 . 35 (Salpeter 1955), others are significantly different. Further analysis of this discrepancy is given in the following section.", "pages": [ 10 ] }, { "title": "4. SCALING TO OBSERVATIONS", "content": "Unlike our synCMFs, typical observational CMFs usually contain on the order of 200 cores, not 10 6 . Therefore, to make our analysis relevant for typical observed CMFs, we must scale back our sample sizes to those typically observed. The following two sections explore the effect of two observational constraints, sample size and bin size, on the shape and slope of observed CMFs.", "pages": [ 10 ] }, { "title": "4.1. Effect of Sample Size", "content": "To test the effect of the sample size on the resultant CMF, we scaled three synCMFs (NM, FF1, and AD3) down to plausible observational sample sizes (100, 200, 300, 400, and 500 cores). Figure 14 shows the resulting synCMFs for each of the fifteen cases. In addition to scaling the sample size, we have also truncated the mass range considered to one more typically found in observed CMFs ( -1 . 0 < log( M/M /circledot ) < 1 . 3). Under these scaled conditions, we see that the nonmagnetic CMFs still maintain the overall shape exhibited by the full sample curve (Figure 5), however the two magnetic cases are fairly different. The high mass tail and truncated shoulder features present in the full sample curves for FF1 and AD3 respectively are no longer quite as distinct at these sample sizes. For a definitive difference between the ambipolar diffusion and flux-frozen cases, observations would have to extend up to objects with masses between 10 2 and 10 3 solar masses. Therefore, on typical observational scales, conclusions about the nature of the magnetic field from the shape of the CMF are possible, but highly uncertain.", "pages": [ 10 ] }, { "title": "4.2. Effect of Bin Size", "content": "Constructing histograms for the purposes of determining a CMF requires binning data into predetermined mass bins. For the above synCMFs, we used ∆log( M/M /circledot ) = 0 . 1 size bins. Variations in the bin size acts to change the resolution of the resulting curve; smaller bins yield more detail while larger bins show only the broad strokes. To determine the effect of the bin size on the resulting CMF, we re-binned the histograms for AD3 in Figure 14 (bottom row) using ∆ log( M/M /circledot ) = 0 . 25 bins. Figure 15 shows the comparison of the original bin size (∆ log( M/M /circledot ) = 0 . 1) to the new bin size (top row). As expected, with the larger bin size, the detail becomes smeared out, resulting in an average curve.", "pages": [ 10 ] }, { "title": "4.3. Effect of sample size and bin size on CMF slopes", "content": "The main piece of data generally extracted from a CMF is the slope of the high mass tail. This information is then used to compare different regions to each other, and to the initial mass function (IMF) in an attempt to determine the true nature of star formation and the possible relation between the CMF and IMF. However, as discussed above, the sample size and bin size have a profound effect on the shape of the curve. This effect translates over to the derived slopes. To determine the extent of this effect, we generate multiple small sample CMFs (2000+) for each sample size and compute the average slope. Figure 16 shows the results of this analysis for models FF1 and AD3 for both mass bin sizes. The filled symbols show the average slope for each of the models while the open symbols depict the slope of the full sample (10 6 ). Tests with larger numbers of samples for each sample size showed differences in the average slope of up to 0.01, which is encompassed in the size of the symbols. As shown in Figure 16, the size of the bin clearly affects the average slope. The larger bin size yields slopes that are steeper than the Salpeter slope, while the smaller bin size shows an overall shallower average slope. The size of the sample also effects the slope. Smaller samples generally result in steeper slopes than those derived using the full sample. Furthering this analysis we look at both the minimum and maximum slopes calculated for each filled point in Figure 16, as well as the distribution of slopes. Figure 17 shows the distribution of slopes for four of the points on Figure 16 as indicated (Top row: AD3, Bottom Row, FF1. Left column: 100 cores, Right column: 500 cores) assuming a CMF constructed with ∆log( M/M /circledot ) = 0 . 1 bins. All four cases show that the preferred slope value is close to the average slope value. The maximum and minimum computed slopes exhibit a very wide range for the small sample sizes (i.e., α = -0 . 1 to α = 11 for AD3, 0.1, 100 cores) while the larger sample sizes exhibit a smaller maximum-minimum range (i.e., α = 0 . 34 to α = 1 . 79 for AD3, 0.1, 500 cores). This decrease in the slope variance is evident when comparing the left column to those in the right column in Figure 17. From these plots, we conclude that although there can be a wide variance in possible slope values, the preferred slope value is in general smaller than the typical Salpeter value, α = 1 . 35, and the range of slopes decreases as the number of samples increases. Figure 18 shows the distribution of slopes for two CMFs (left: AD3, right: FF1) assuming a 100 core sample size and ∆ log( M/M /circledot ) = 0 . 25 bins. Comparing to the left hand plots in Figure 17, we see that the larger bin size results in a bimodal distribution with the peaks occurring at α ≈ 2 . 0 and α ≈ 1 . 5 for AD3 and α ≈ 1 . 9 and α ≈ 1 . 45 for FF1. The result of this bimodal distribution is to shift the average slope values to smaller values than the dominant peak. This is particularly evident in Figure 16 in the trend of slopes for the smallest sample sizes of the blue triangles (FF, 0.25). Further analysis of the effect of the original column density distribution on the variance and mean of the resulting slope histogram showed that a larger variance in the column density distribution shifts the mean in the slopes to smaller values ( < 1) while a smaller variance results in a larger mean value, α ∼ 1 . 35.", "pages": [ 10, 11 ] }, { "title": "5. DISCUSSION", "content": "Our analysis shows that the shape of the CMF is highly dependent on the magnetic field strength and neutralion coupling within the cloud. Specifically, a flux-frozen magnetic field broadens the nonmagnetic lognormal distribution to have a significant power-law high mass tail, though it is much shallower than the Salpeter value. When ambipolar diffusion is taken into account, there is an intermediate mass tail and a high mass cutoff. The extent of all these features are dependent on the range of mass-to-flux ratio values in the initial cloud. Kunz & Mouschovias (2009) carried out a more focused study of the effect of magnetic fields and ambipolar diffusion in creating a broad CMF. Their model explored only the subcritical portion of the fragmentation scales seen in Figure 1. Furthermore, they assumed a uniform distribution of subcritical mass-to-flux ratios and effectively a fixed Jeans mass in order to generate their mass distribution. The low-mass tail in their distribution originates in the assumption that the subcritical clouds ultimately form dense cores with masses that are scaled by µ 0 for subcritical values of µ 0 . This is because numerical simulations of Basu & Mouschovias (1995) show that only an inner region where the mass-to-flux ratio exceeds the critical value undergoes rapid collapse. We do not make that assumption in this study, since cores that form by ambipolar drift have an appearance that is similar to those that are forming by a more rapid gravitationally-dominated process (see Basu et al. 2009b). Since the resultant CMF in our model is generated from an underlying lognormal function, it has an intrinsic peak even when binned in linear mass bins. An advantage of the KM09 model is that they do not need to assume an underlying lognormal distribution to obtain a lognormal-like CMF, however that CMF is peaked only when binned in log mass. Upon scaling our models down to observational sample sizes and ranges, we found that the distinction between the different models is lost within typical observational mass ranges and therefore no information regarding the magnetic field can be reliably gleaned from the shape of the observed CMFs. Further to this, analysis of the slopes for each of the sample sizes showed that the smaller samples sizes result in slopes that are 1 . 1 -1 . 4 times larger than the slope derived from the full sample, while the derived slopes for the larger binsize are ∼ 1 . 3 -2 . 0 times larger than the corresponding smaller binsize slope measurements. Although we have taken care to scale our analysis down to those typically used in observations, the question still remains as to how well our results and conclusions correspond to actual observations. A recent study of the CMF for five separate star forming regions (Ophiuchus, Taurus, Perseus, Serpens and Orion) performed by Sadavoy et al. (2010) provides the perfect platform for comparison. Looking at the core mass distributions for these regions, as expected, it is hard to definitively discern any characteristic features that are indicative of a particular magnetic field model. With limited data, it is plausible that the CMFs for Ophiuchus, Taurus and Perseus could exhibit the indicative shoulder of the ambipolar diffusion models, while the full Orion CMF could show evidence of a flux frozen field. Looking at the slopes of the CMFs for these regions, Sadavoy et al. (2010) showed each region gave slope values that are close to the α = 1 . 35 Salpeter slope, within their adopted errors. Comparing their slope values to those in Figure 16, most of them would fall somewhere in the lower half of the graph in amongst the diamonds and squares while the Orion with OMC slope would fall in amongst the triangles and circles. However looking at the binsize of the observations, all of the slopes should be within the triangle/circle regime of the graph. Comparing these values to the corresponding slope histograms (see Figure 18) we see that these values all fall within the regime of possible slopes. On the surface, this seems to be a huge discrepancy between our results and observations, however each of these five observational slopes represents a single slope within our 2000+ values used to derive an average slope. However, looking at the range of slopes derived from our analysis, these observed slopes fall within this range. As shown in Figure 17, the only way to produce a narrower range of slope values is to increase the sample size, which is not always possible observationally since the number of objects detected depends entirely upon the number of objects actually present and the sensitivity of the instrument. Based on our analysis and the above comparison to the work by Sadavoy et al. (2010), we argue that the observed CMFs are extremely statistically limited, both in the size of the sample and the number of samples over which the slope of the CMF is averaged. Through our analysis, we have shown that with larger number statistics, not only is the measured slope of the CMF much different than the typical Salpeter value α = 1 . 35, but highly dependent on the size of the mass bin. In addition, the range of individual slope values within the set size decreases as the number of cores in the sample size increases. This is analogous to the results found by Elmegreen (1999), where although it was determined that the most probable value for the IMF slope is the Salpeter value, α = 1 . 35, it is a highly reduced average of all possible outcomes. Subsequently, we argue that based on our analysis and the results of Elmegreen (1999), there seems to be no clear cut correlation between the slope of the CMF and the IMF and that the shape and slope of the CMF are entirely controlled by the conditions within the cloud itself. Since it is unfeasible to claim that all clouds exhibit identical conditions, it is therefore unrealistic to expect a universal shape and slope value for all star forming regions.", "pages": [ 11, 12 ] }, { "title": "6. SUMMARY", "content": "We have studied the effect of magnetic fields on the formation and properties of the core mass function using a combination of the results from linear analysis and Monte Carlo methods. In addition, we have studied the effects of low number statistics on the slope of the high mass tail. Here we summarize the main results of our analysis.", "pages": [ 12 ] }, { "title": "ACKNOWLEDGMENTS", "content": "NDB was supported by a scholarship from the Natural Science and Engineering Research Council (NSERC) of Canada. SB was supported by a Discovery Grant from NSERC.", "pages": [ 12 ] }, { "title": "REFERENCES", "content": "Alv'es, J., Lombardi, M., & Lada, C. J. 2007, A&A, 462, L17 Bailey, N. D. & Basu, S. 2012, ApJ, 761, 67 Ballesteros-Paredes, J., Gazol, A., Kim, J., et al. 2006, ApJ, 637, 384 Basu, S., Ciolek, G. E., Dapp, W. B., & Wurster, J. 2009a, NewA, 14, 483 Basu, S., Ciolek, G. E., & Wurster, J. 2009b, NewA, 14, 221 Basu, S. & Jones, C. E. 2004, MNRAS, 347, L47 Basu, S. & Mouschovias, T. C. 1995, ApJ, 453, 271 Bohlin, R. C., Savage, B. D., & Drake, J. F. 1978, ApJ, 224, 132 Chabrier, G. 2003a, PASP, 115, 763 -. 2003b, ApJ, 586, L133 Chabrier, G. 2005, in Astrophysics and Space Science Library, Vol. 327, The Initial Mass Function 50 Years Later, ed. E. Corbelli, F. Palla, & H. Zinnecker, 41 Troland, T. H. 2010, ApJ, 725, 466 -. 1999, ApJ, 515, 323 -. 2009, ApJ, 702, 1428 Johnstone, D., Wilson, C. D., Moriarty-Schieven, G., et al.2000, ApJ, 545, 327", "pages": [ 12, 13 ] } ]

[ { "title": "ABSTRACT", "content": "Neumayer et al. established the existence of a blue-shifted cloud in the core of Centaurus A, within a few parsecs of the nucleus and close to the radio jet. We propose that the cloud has been impacted by the jet, and that it is in the foreground of the jet, accounting for its blue-shifted emission on the Southern side of the nucleus. We consider both shock excitation and photoionization models for the excitation of the cloud. Shock models do not account for the [SiVI] and [CaVIII] emission line fluxes. However, X-ray observations indicate a source of ionizing photons in the core of Centaurus A; photoionization by the inferred flux incident on the cloud can account for the fluxes in these lines relative to Brackettγ . The power-law slope of the ionizing continuum matches that inferred from synchrotron models of the X-rays. The logarithm of the ionization parameter is -1.9, typical of that in Seyfert galaxies and consistent with the value proposed for dusty ionized plasmas (Dopita et al. 2002). The model cloud density depends upon the Lorentz factor of the blazar and the inclination of our line of sight to the jet axis. For acute inclinations, the inferred density is consistent with expected cloud densities. However, for moderate inclinations of the jet to the line of sight, high Lorentz factors imply cloud densities in excess of 10 5 cm -3 and very low filling factors, suggesting that models of the gamma ray emission should incorporate jet Lorentz factors glyph[lessorsimilar] 5. Subject headings: black hole physics - galaxies: active - galaxies: individual (Centaurus A) - galaxies: jets - line: formation - relativistic processes", "pages": [ 1, 2 ] }, { "title": "The Kinematics and Ionization of Nuclear Gas Clouds in Centaurus A", "content": "Geoffrey V. Bicknell 1 , Ralph S. Sutherland 2 Research School of Astronomy & Astrophysics Australian National University Mt. Stromlo Observatory Cotter Rd., Weston ACT, Australia 2611 and Nadine Neumayer 3 European Southern Observatory Karl-Schwarzschild Strasse 2 85748 Garching - Germany", "pages": [ 1 ] }, { "title": "1. Introduction", "content": "Neumayer et al. (2007) presented high resolution (0 . 12 '' ) infrared spectral images of the central parsecs of Centaurus A, which they utilized to derive important dynamical constraints on gas motions in this region, as well as the mass of the central black hole (see Neumayer (2010) for a review of the black hole mass measurements). A feature of the data, which has not yet been exploited is the existence of emission line gas clouds approximately aligned with the parsec-scale radio jet. Questions which naturally arise in this context are: What is the excitation mechanism for this gas and what is the cause of the ∼ 100 km s -1 velocities of this gas relative to the nucleus? Two candidates for the excitation mechanism are: (1) Shock waves driven by the jet into clouds adjacent to the jet and (2) Photoionization by either emission from an accretion disk or from the jet close to the back hole. Shock excitation could arise naturally in the process of entrainment of clouds into the jet or simply by jet deflection off these clouds. Considering the possibility of photoionization, Lenain et al. (2008) modeled the radio through X-ray and gamma-ray emission from the nucleus of Centaurus A in terms of either synchrotron plus synchrotron self-Compton (SSC) emission or two component synchrotron emission from a relativistic jet. In a subsequent paper they incorporated the newly discovered very high energy (VHE) gamma-ray emission from Centaurus A into a revised model in which the 1 -10 5 eV emission is due to synchrotron radiation. The significance of a synchrotron explanation for the 1 -10 5 eV emission is that it may have a high enough flux (depending upon beaming effects) to be an important source of ionizing photons. If an SSC model is used for the 1 -10 5 eV flux, then the flux density of ionizing photons is much lower at around 10 eV (cf Lenain et al. (2008)). If one or both of the jets in Centaurus A are indeed responsible for the photoionization of the nuclear gas clouds then modelling of the emission may lead to interesting constraints on the Lorentz factor, which strongly affects the number density of ionizing photons through relativistic beaming. In this paper, we summarize the observational data in § 2, consider both shock and photoionization models in § 3 and discuss our results in § 4. In an appendix, we derive expressions for the ionizing photon density incident on a nuclear cloud in terms of the observed flux density and the jet parameters relating to the beaming in both the observer and cloud directions.", "pages": [ 2, 3 ] }, { "title": "2. Observational data", "content": "as us and that it is pushed to one side by the jet - hence the blueshift. The geometrical configuration that we envisage is represented in Figure 2. Such an interpretation for the cloud velocity raises the question of why the cloud would not acquire a substantial forward momentum along the jet direction, leading to a redshift. Simulations of jets interacting with inhomogeneous media both in the form of disks Sutherland & Bicknell (2007) and spherically distributed clouds Wagner & Bicknell (2011a,b) show that jet-cloud interactions are complex and are often affected by the backflow of radio plasma, either from the head of the radio source or from the downstream interaction of the jet with other clouds. The back flow impedes forward motion of the cloud so that its resultant motion is primarily perpendicular to the jet.", "pages": [ 4 ] }, { "title": "3. EMISSION LINE MODELS", "content": "In this section we discuss the two principal possible excitation mechanisms of gas in the nucleus of Centaurus A: Shock excitation and photoionization. In doing so, we concentrate on the blue-shifted cloud (see Figure 1) since this is separated in velocity from the rest of the gas in the nuclear region and offers the prospect of being described by a well-defined set of parameters of number density and photon density. However, we also consider the integrated emission from the nucleus. The K-band infrared spectra do not offer a large number of emission lines for comparison with the models. The principal lines that are evident in the spectra are those of [SiVI] at 1 . 9602 µ m, Br γ at 2 . 1656 µ m and [CaVIII] at 2 . 322 µ m so that the models are not uniquely constrained by the data. However, [CaVIII] is an important discriminant of shock-excitation and photoionization and it is possible to derive reasonable conclusions on the relative merits of these two mechanisms and also to estimate the cloud densities.", "pages": [ 4 ] }, { "title": "3.1. Shock Models", "content": "Since we are suggesting that the velocity of the blue cloud is the result of being pushed to one side by the passage of the jet it is natural to investigate whether the emission from the cloud could be the result of shock excitation. The emission from an astrophysical shock wave comprises contributions from the shocked gas and the precursor region photoionized by radiation from the shock. Using the MAPPINGS III emission line code, we have computed a comprehensive grid of shock + precursor models for velocities between 160 and 1000 km s -1 . The results for [CaVIII] and [SiVI] are shown in Figure 3 wherein the shock and precursor regions are shown both separately and combined. The effect of varying the pre-shock magnetic field is also shown in Figure 3. The compelling message from the shock models is that there is no combination of shock velocity and magnetic field, which adequately reproduces both the observed [CaVIII]/Br γ and [SiVI]/Br γ line ratios. The adjacency of gas to the jet combined with its velocity offsets with respect to the core indicates that its kinematics may be affected by the jet. However, it does not appear that shock waves associated with a jet-cloud interaction dominate the emission.", "pages": [ 4, 5, 6, 7 ] }, { "title": "3.2. Photoionization Models", "content": "We now consider photoionization models for the emission from the nuclear region of Centaurus A. The point of view that we take here, is inspired by that proposed by Pedlar et al. (1989) and Taylor et al. (1992) for Seyfert galaxies. Clouds of moderate density are impacted by the radio jets producing higher density radiatively shocked regions. The shocked gas is photoionized by the nuclear continuum, producing the observed emission line spectrum.", "pages": [ 7 ] }, { "title": "3.2.1. Photoionizing Flux", "content": "One possible source of the ionizing continuum is the nuclear sub-parsec-scale jet. In order to assess the implications of such a source, which may be relativistically beamed, we need to determine the flux of ionizing radiation incident on the cloud given the radiation that we observe along a line of sight, which is inclined at a different angle to the velocity of the radiating plasma. The high energy emission of the Northern jet from hard X-rays through to very high energy (Tev) gamma rays has been modeled by Lenain et al. (2009) in terms of a SynchrotronSelf-Compton (SSC) model. This was an update of an earlier model (Lenain et al. 2008) in which both synchrotron and inverse Compton models were considered for the X-ray emission alone. However, the detection of VHE γ -ray emission from Centaurus A (Aharonian et al. 2009) has made the synchrotron interpretation more appealing. This is important since the inverse Compton model has a much lower ionizing flux. In the model that we develop here, we assume that the base of the Southern jet has a similar spectrum to the Northern jet and that this provides the flux of photoionizing radiation. Figure 1 in Lenain et al. (2009) shows that the continuum region that is of interest here, namely the ionizing continuum with energies in excess of approximately 10 eV in the rest frame of the irradiated gas, is only constrained by the observational data in the range ∼ 1 -100 keV. Hence we rely on the modeled spectrum in the region ∼ 1 -10 5 eV. The model spectral energy distribution (SED) between approximately 1 eV and 100 keV (the power-law synchrotron component indicated on their Figure 1) has a flux spectral index of α ≈ 0 . 39 ( F ν ∝ ν -α ). This is slightly flatter than usual (say ∼ 0 . 6 -0 . 7) for a nonthermal spectrum. Nevertheless, values of 0.4 and 0.3 have been inferred for the synchrotron component of the blazar emission in MKN 501 by Bicknell et al. (2001) and Konopelko et al. (2003) respectively. Lenain et al. (2009) present three different models. In the following we use the model which most accurately reproduces the X-ray data, the thin line in their Figure 1. Lenain et al. (2009) infer a Lorentz factor of 15 for the Centaurus A jet. Lorentz factors of this magnitude are controversial in view of the fact that Lorentz factors estimated on the parsec scale from radio observations are always lower than this (e.g. Piner et al. 2008) and Centaurus A is no exception: The pattern speeds of knots in the Centaurus A jets are of order 0.1 c (Tingay et al. 1998) - much less than the superluminal pattern speeds seen in many quasars, although Tingay et al. note that there is evidence for a faster underlying flow. Estimated high Lorentz factors in blazars may point to significant deceleration between the sub-parsec and parsec scales or to localized fast-moving regions of the flow and these issues are currently unresolved. However, the inference of high Lorentz factors and the observation of superluminal motions in AGN in general, means that we need to take account of relativistic beaming. As noted above, in Centaurus A the radiation field observed by us should be different from the radiation field intercepted by the blue-shifted cloud and it is important to allow for this in the photoionization models. The various relationships required to relate the observed flux density to the ionizing photon density at the cloud are derived in the appendix. We summarize the main results here: Our model is summarized in Figure 2. We envisage the inner jet, within say about 100 gravitational radii ( ∼ 20 mpc) as the source of ionizing photons (the inner jet); let D cl be the distance of the cloud from the source; let D A be the angular diameter distance of Centaurus A; let the Doppler factors of the radiation received by observer and cloud respectively be δ obs and δ cl and let z ≈ 0 . 0018 (Graham 1978) be the redshift of Centaurus A. If ν is the frequency of a photon intercepted by the cloud, it is emitted by the jet with a rest frame frequency δ -1 cl ν and a photon emitted with this frequency reaches the observer with a frequency ν obs = δ obs (1 + z ) -1 δ -1 cl ν . Let F obs ( ν obs ) be the observed flux density of the inner jet as a function of the observing frequency ν obs . With these definitions, the photon density per unit frequency at the cloud (derived in the Appendix) is Let ν 0 . = 3 . 28 × 10 15 Hz be the frequency corresponding to the Rydberg limit. The total number density of ionizing photons at the cloud (also derived in the appendix) is Between about 10 14 . 5 and an upper frequency ν u ≈ 10 19 . 3 Hz the modeled spectrum may be described as a power-law F ( ν obs ) ≈ F ( ν 0 )( ν obs /ν 0 ) -α with F ( ν 0 ) ≈ 1 . 35 × 10 -27 ergss -1 Hz -1 and α ≈ 0 . 39. Hence, Since ν u glyph[greatermuch] ν 0 , the contribution of the second term in brackets is minor. Let the inclination of our line of sight to the jet be θ obs , the angle between the direction of a ray from the core through the center of the cloud and the jet be ψ cl and the (projected) angular displacement of the cloud from the core be ξ cl , the geometry of the model (see Figure 2) implies that the ratio of angular diameter distance to cloud distance is given by (This equation is also derived in the Appendix.) Furthermore, the ratio of Doppler factors is given by: As expected, the expression for the ionizing photon density has a strong dependence on the ratio of Doppler factors, when ψ cl is small. We estimate ψ cl assuming that the cloud is adjacent to the jet and that is depth along the line of sight is the same as its transverse diameter ≈ 0 . 35 '' . The expression for ψ cl is derived in the Appendix (see equation A18).", "pages": [ 7, 8, 9 ] }, { "title": "3.2.2. Isochoric Photoionization Models", "content": "In framing models for cloud photoionization one question to resolve is whether an isobaric or isochoric model is more appropriate. Here, an isobaric model is relevant if the expansion time scale for the gas heated by photoionization is less than the time scale for γ cloud shredding resulting from the jet-cloud interaction. The photoionized cloud would expand at approximately the sound speed corresponding to a temperature of 10 4 K, that is, about 12kms -1 . If we take the anomalous velocity of the blue cloud ∼ 100kms -1 as indicative of shocks driven into it by the jet, then the shock-shredding time scale ∼ 18pc / 100kms -1 ≈ 1 . 8 × 10 5 yrs is approximately a factor of eight shorter than the expansion time scale. Hence, we mainly consider isochoric, constant density, models in the following. We have calculated a grid of models for solar metallicity using the solar abundance scale of Grevesse et al. (2010). The models are parameterized by the ionization parameter, U , the ratio of the ionizing photon density, n ph to the particle density n ( U = n ph /n ) and the power-law index, α of the photoionizing flux density ( F ν ∝ ν -α ). The models are ionization bounded and are for a density n = 10 4 cm -3 . For this fiducial density, the extent of the ionization bounded region ≈ 10 19 . 5 cm and fits comfortably within the projected major axis ≈ 10 20 . 7 cm of the blue cloud. Below, we consider lower filling factors, involving higher densities. However, the model line ratios remain independent of density for all the densities considered here. Figure 4 shows the results of the grid of models for the line ratios [SiVI]/Br γ and [CaVIII]/Br γ . The best-fit model is represented by (log U, α ) ≈ ( -1 . 9 , 0 . 42). The bestfit value of α is in practice indistinguishable from the value inferred from the synchrotron emission fit to the X-ray data, strongly supporting the notion of photoionization by jet emission. We note that the best fit ionization parameter of log U ∼ -2 . 0 is in rough agreement with the self-limiting radiation pressure dominated dusty AGN nebula models of Dopita et al. (2002); Groves et al. (2004a,b,c). In future work it may be fruitful to explore the possibility of such dusty models in the present context. However, with a scarcity of diagnostic lines such as [OIII] λ 5007, and a lack of constraints on the properties of the dust, and any depletion factors that may be present in the blue cloud (affecting the abundances of both Silicon and Calcium), additional model parameters are not justified by the present spectral data. If more data on the cloud composition, dust properties and more spectral line measurements were available, then radiation pressure dominated dusty models may improve the models and place better constraints on the Br γ emission for example. This would affect the filling factor of the cloud (see below). In our isochoric models, the ratios of [SiVI] and [CaVIII] to Br γ are primarily determined by the slope of the ionizing continuum and the level of the emission is determined by the density. Hence, we can estimate the cloud number density, n from the ionization parameter, U , and the ionizing photon density, n ph (using n = U -1 n ph ). Since the ionizing photon density depends upon the observed continuum, the angle of inclination of the radio jet and its Lorentz factor, through equation (3), we can assess the effect of these quantities on the inferred density of the cloud. Note that our estimates of the number density would not be substantially revised by dusty models since such models predict a very similar ionization parameter to what we have inferred here.", "pages": [ 9, 10, 11, 12 ] }, { "title": "3.2.3. Filling Factor.", "content": "The filling factor of the Hydrogen-line emitting region of the cloud can be estimated from the Br γ luminosity, L(Br γ ). Let α (Br γ ) be the effective Case B recombination coefficient for Br γ , f the volume filling factor, n e and n H the electron and Hydrogen number densities and V cl the cloud volume, then We estimate a cloud volume where θ maj ≈ 1 . 1 '' and θ min ≈ 0 . 7 '' are the angular sizes of the major and minor axes of the cloud respectively; we assume that the depth of the cloud is similar to the length of the minor axis. In using Equations (6) and (7) to estimate the filling factor, we use the measured value of the Br γ flux of 2 . 42 × 10 -16 ergs s -1 cm -2 and adopt a value of 2 . 67 × 10 -27 for the Br γ emission coefficient, appropriate for the average temperature of 12 , 500 K obtained in the photoionization models. Note that the estimate for the filling factor depends upon the assumption that the cloud depth is similar to its width. If the depth were an order of magnitude smaller, for example as a result of ablation by the jet, then the filling factor would increase by an order of magnitude. Our estimates of the density of the photoionized gas and the related filling factor both as a function of Lorentz factor and θ obs are presented in the left and right panels of Figure 5. For all θ obs , the density increases quite rapidly with Lorentz factor. For θ obs = 40 · , 60 · and 80 · , the density increases to values in excess of 10 4 cm -3 . In keeping with this, the filling factor decreases quite rapidly with Lorentz factor for all inclination angles but especially for θ obs ≥ 40 · . The strong dependence of the density and filling factor on Lorentz factor results from the effects of beaming and the high inferred photon density at the cloud when the line of sight to the observer is greater than approximately 30 · to the jet axis (see equation (3)). That is, for moderate values of the angle of the observer to the line of sight, the observer sees a considerably lower intensity than that which is incident on the cloud. Various inclination angles have been proposed for the Centaurus A jet but most are within the range 50 · ≤ θ obs ≤ 80 · proposed by Tingay et al. (1998). The exception is Hardcastle et al. (2003) who estimate an angle of inclination of approximately 15 · if the inner Very Large Array (VLA) jets are intrinsically symmetric . However, they note the inconsistency of this estimate with other estimates of inclination, leaving open the question of the asymmetry of the inner VLA structure. What is the evidence for gas with densities in excess of 10 3 cm -3 ? Using a spherical free-free absorption model for the very long baseline interferometry (VLBI) jets, Tingay & Murphy (2001) derived a number density ∼ 9 × 10 4 T 1 . 35 4 where the temperature of the absorbing gas is 10 4 T 4 K and the radius of the free-free absorbing sphere is 0.016 pc. Wild et al. (1997) determined gas densities ≈ 1 -3 × 10 4 cm -3 from CO observations of the Centaurus A dust lane, at a spatial resolution of 44 '' . Subsequent work (Wild & Eckart 2000) using HCN and CS observations at a resolution of 54 '' showed that approximately 10% of the molecular gas in the nuclear region has densities in excess of 10 4 cm -3 . Thus, there is good evidence for dense gas clouds in the nuclear region of Centaurus A, associated with the dust lane. Future ALMA observations should provide more information on the spatial distribution of these clouds. However, we do not expect the cloud densities at a few parsecs to be higher than the 10 5 cm -3 inferred by Tingay & Murphy (2001) near the core so that we adopt a conservative upper estimate on the density of 10 5 cm -3 . It is only for inclinations glyph[lessorsimilar] 50 · that the constraint n < 10 5 cm -3 is realized for all Lorentz factors up to the estimate of Γ = 15 by Lenain et al. (2009). This density constraint is consistent with the Lenain et al. (2009) models in which a low inclination angle, θ obs = 25 · , is assumed but not with the range of inclinations discussed above. Let us now consider the dependence of filling factor on Lorentz factor. While describing the density structure of a photoionized region using this parameter is not ideal, we adopt if here for the purpose of approximate comparisons. In the narrow line region of active galaxies the value inferred for the filling factor is typically ∼ 10 -4 -10 -3 . Pedlar et al. (1989) and Taylor et al. (1992) explained this low value in terms of the dynamics of radio plasma driven radiative shocks, which produce high density, but low volume filling factor, shocked regions. The high density regions are photoionized by the nuclear radiation and the Pedlar et al. and Dyson et al. models explain why photoionization models require a low filling factor. Pedlar et al. and Dyson et al. formulated their models to explain the observed properties of Seyfert galaxies, and the physical situation envisaged here is similar to their Seyfert model: Gas shocked by the jet is photoionized by the central source producing regions of photoionized gas with a low filling factor. Hence a comparison of the filling factors in the narrow line regions of Seyfert galaxies and those deduced here for Centaurus A is useful. We therefore adopt a conservative lower limit on the filling factor of approximately 10 -4 . When we examine the dependence of filling factor on Lorentz factor (right panel of Figure 5) we see that the inferred filling factor decreases rapidly with Lorentz factor. This results from the increase of the beamed photon density and atomic density with Lorentz factor. The effect is highlighted for moderate inclinations. For acute inclinations the value of the filling factor is acceptable up to Γ ≈ 10. However, for moderate inclinations, filling factors glyph[lessorsimilar] 10 -4 are only possible for Lorentz factors less than a few.", "pages": [ 12, 13, 14 ] }, { "title": "4. Conclusions and discussion", "content": "The discovery of blueshifted line-emitting gas on the side of the nucleus that we would normally associate with redshifted outflow has led us to consider detailed models for the location, kinematics and excitation of this gas. The features of the model that we have investigated in detail include the following: the jet and between the jet and us. As we have seen in § 3.2 precise conclusions concerning the Lorentz factor of the jet in Centaurus A (and consequently the viability of models for the high energy emission) depend strongly on our viewing angle and the interpretation of the milliarcsecond and arcsecond radio data. Broadly speaking, if the inclination of our line of sight to the jet axis is as acute as the interpretation of the arcsecond radio data may suggest (Hardcastle et al. 2003) then our photoionization models are consistent with a reasonably high Lorentz factor (for both jets). If, on the other hand, the inclination is in the range of 50 · -80 · inferred from the millarcsecond radio data (Tingay et al. 1998), then low to moderate Lorentz factors (Γ glyph[lessorsimilar] 5) are implied . The is consistent with the lack of evidence for very high Lorentz factors in the form of superluminal proper motions (Tingay et al. 1998) such as are found in quasar jets for example (Kellermann et al. 2004). The physical distance, D cl of the blue-shifted cloud from the nucleus is determined by the model parameters, specifically θ obs , the angle between the observer and the Northern jet direction, and ψ cl , the angle between the direction of the southern jet and the direction of the cloud from the nucleus, together with the projected distance from the nucleus ≈ 16 . 5 pc corresponding to an angular separation of approximately 1 arcsec. (See Figure 2 and Equation A15.) For the most acute angle of inclination we have considered, θ obs = 20 · , D cl ≈ 36 pc. For the range of angles referred to in the above paragraph 50 · < θ obs < 80 · the distance of the cloud from the core varies very little: 18 pc > D cl > 17 pc. Note however, it is only in the case of the photoionization models that these parameters are relevant. The main parameter in the shock models is the shock velocity. Our conclusion that the blue cloud is photoionized by beamed emission from the core invites the question as to the relevance of photoionization for other clouds on the kiloparsec scales in the interstellar medium of Centaurus A (Blanco et al. 1975; Peterson et al. 1975; Graham & Price 1981; Morganti et al. 1991). In particular Morganti et al. (1991) argued that the region referred to as the inner filaments are photoionized by beamed radiation from the core. On the other hand, Sutherland et al. (1993), presented a model in which the filaments are locally shock-excited through interaction with the radio plasma. More recently Sharp & Bland-Hawthorn (2010) have found an ionization cone in the inner galaxy of Centaurus A aligned with the radio jet as well as other nearby emission line regions, which are excited by star formation. In the light of the calculations presented in this paper, we can consider some examples based on typical parameters which indicate the conditions under which photoionization may be feasible for the different regions well outside the core. Note that these regions are all to the North of the nucleus so that we do not need to assume the equivalence of Southern and Northern jet ionizing fluxes. In Table 1 we present some indicative calculations of ionization parameters for these different regions for various values of jet Lorentz factor, nominal values of number density and a jet inclination of 60 · . The angle between jet and cloud directions were estimated as follows. For the ionization cone Sharp & Bland-Hawthorn (2010) estimated a half-opening angle of 20 · ; deprojected this is 17 . 5 · ; our estimate of ionization parameter refers to the edge of the cone. The inner and outer filaments have position angle differences with respect to the milliarcsecond jet of 4 · and 12 · respectively. Since we are only determining indicative numbers, we assume that the inclination of the cloud direction to the line of sight is the same as the inclination of the jet to the line of sight (i.e. θ cl = θ obs ) and we calculate the angle between the jet and the cloud direction, ψ cl , using equation (A18). The cloud densities assumed for the inner filaments are the same as the estimates of Morganti et al. (1991); the other densities are nominal but the resulting ionization parameters can easily be scaled for other densities. For the spectra that are observed in Centaurus A the required ionization parameters are of order a few × 10 -3 to 10 -2 . Indeed Morganti et al. (1991) estimated a reference value from their variable density models for the inner filaments of 5 × 10 -3 . Our estimates for the ionization cone and the inner filaments are in this range for jet Lorentz factors of four and five but not for the lower Lorentz factor of two. If the outer filaments are to fall in this range the density would need to be an order of magnitude lower. Hence the estimates of jet parameters derived from models of the blue cloud are consistent with photoionization of the Sharp & Bland-Hawthorn (2010) ionization cone. However, the situation is not as clear for the outer filaments. We thank Dr. Rob Sharp for informative discussions on Centaurus A. This work was supported by the Australian Research Council Discovery Project DP0664434. NN acknowledges the support by the DFG cluster of excellence Origin and Structure of the Universe . GVB thanks the European Southern Observatory, where this work was initiated, for its hospitality.", "pages": [ 14, 15, 16, 17 ] }, { "title": "REFERENCES", "content": "Aharonian, F., et al. 2009, ApJ, 695, L40-L44 Begelman, M. C., Blandford, R. D., & Rees, M. J. 1984, Rev. Mod. Phys., 56, 255 Bicknell, G. V., Wagner, S. J., & Groves, B. A. 2001, in Particles and Fields in Radio Galaxies, ed. R. A. Laing & K. M. Blundell, Volume 250 of Astronomical Society of the Pacific Conference Series 80 Blanco, V. M., Graham, J. A., Lasker, B. M., & Osmer, P. S. 1975, ApJ, 198, L63 Dopita, M. A., Groves, B. A., Sutherland, R. S., Binette, L., & Cecil, G. 2002, ApJ, 572, 753-761 Graham, J. A. 1978, PASP, 90, 237-240 Graham, J. A. & Price, R. M. 1981, ApJ, 247, 813 Grevesse, N., Asplund, M., Sauval, A. J., & Scott, P. 2010, Ap&SS, 328, 179-183 Groves, B., Dopita, M., & Sutherland, R. 2004a, in The Interplay Among Black Holes, Stars and ISM in Galactic Nuclei, ed. T. Storchi-Bergmann, L. C. Ho, & H. R. Schmitt, Volume 222 of IAU Symposium 263-266 Groves, B. A., Dopita, M. A., & Sutherland, R. S. 2004b, ApJS, 153, 9-73 Groves, B. A., Dopita, M. A., & Sutherland, R. S. 2004c, ApJS, 153, 75-91 Hardcastle, M. J. 2003, New Astronomy Reviews, 47, 649-652 Hardcastle, M. J., Worrall, D. M., Kraft, R. P., Forman, W. R., Jones, C., & Murray, S. S. 2003, ApJ, 593, 169-183 Kellermann, K. I., et al. 2004, ApJ, 609, 539-563 Konopelko, A., Mastichiadis, A., Kirk, J., de Jager, O. C., & Stecker, F. W. 2003, ApJ, 597, 851-859 Lenain, J.-P., Boisson, C., Sol, H., & Katarzy'nski, K. 2008, A&A, 478, 111-120 Lenain, J.-P., Medina, M. C., Boisson, C., Romero, G. E., Sol, H., & Zech, A.ArXiv e-prints: 0907.2258, Lind, K. R. & Blandford, R. D. 1985, ApJ, 295, 358 Morganti, R., Robinson, A., Fosbury, R. A. E., Serego Aligheri, di S., Tadhunter, C., & Malin, D. F. 1991, MNRAS, 249, 91 Neumayer, N. 2010, PASA, 27, 449-456 Neumayer, N., Cappellari, M., Reunanen, J., Rix, H.-W., van der Werf, P. P., de Zeeuw, P. T., & Davies, R. I. 2007, ApJ, 671, 1329-1344 Pedlar, A., Dyson, J., & Unger, S. W. 1989, MNRAS, 214, 463 - 473", "pages": [ 17, 18, 19 ] }, { "title": "A. RELATIONSHIP BETWEEN OBSERVED EMISSION AND BEAMED PHOTOIONIZING FLUX", "content": "We often see high energy X-ray and γ -ray emission from the cores of active galaxies, which is interpreted as emission from a relativistic jet. Potentially, this emission can ionize clouds along the direction of the jet where it is more highly beamed. However, as a result of the beaming pattern of the emission, such clouds see a different intensity to that at the observer. In this Appendix we provide a ready way of estimating the ionizing photon density given parameters such as the Lorentz factor of the jet, the directions of the observer and cloud and the respective distances of cloud and observer.", "pages": [ 19 ] }, { "title": "A.1. Ionizing Photon Density", "content": "We use the following symbols and definitions: (xvi) δ obs = Γ -1 (1 -β cos θ obs ) -1 is the Doppler factor of the source along a ray to the observer; θ obs is the direction of the observer with respect to the blob's velocity. (xvii) The frequency ν ' of a photon emitted in the jet rest frame and the observed frequency ν obs are related by ν obs = δ obs (1 + z ) -1 ν ' , where z is the redshift of the galaxy. The number density of ionizing photons (per unit frequency and total) are, respectively: We relate these quantities to the parameters of the relativistically moving emission region as follows. The intensity in the cloud frame and jet rest frame are related by: where and the integral is along a ray through the relativistically moving emitting region R . That is, Hence, the number density of photons per unit frequency is The number density of ionizing photons in the cloud frame may be determined by integration over the volume of the relativistic emitting region R . Since d Ω = dA/D 2 cl = dA ' /D 2 cl , then dx ' d Ω = dx ' dA ' /D 2 cl = dV ' /D 2 cl and The integral over solid angle and path length has been replaced by an integral over comoving volume. We integrate over frequency to obtain the total number of ionizing photons. Thus,", "pages": [ 20, 21 ] }, { "title": "A.2. Flux Density at Observer", "content": "For completeness, and in order to relate observed flux density to ionizing photon density we repeat the analysis in Lind & Blandford (1985) in the following expression for the flux density. This enables us to estimate the volume integral of the emissivity which appears in the expressions for the ionizing photon density, equations (A7) and (A8). The flux density of the blob, at frequency ν obs , as measured by the observer is: As in previous analyses (e.g. Lind & Blandford 1985; Begelman et al. 1984) we determine the observed flux density in two stages, from blob to a point well outside the blob in the galaxy rest frame and from galaxy rest frame to observer. The first stage involves relativistic effects; the second stage cosmological effects. The result for the observed flux density is: where j ' ( ν ' ) is the emissivity in the jet rest frame.", "pages": [ 22 ] }, { "title": "A.3. Using the Observed Flux Density to Estimate the Ionizing Photon Density", "content": "We now use the observed flux to determine the volume integral of the emissivity and then the ionizing photon density. The combination δ obs (1+ z ) -1 δ -1 cl ν which appears in the following reflects the fact that a photon which intersects the cloud with frequency ν , originates from the blob rest frame with frequency δ -1 cl ν . Photons emitted with this frequency reach the observer with frequency δ obs (1 + z ) -1 δ -1 cl ν . From equation (A10) for the flux density, we have: When we insert this expression into equation (A7) for the number density of ionizing photons per unit frequency we obtain: and the number density of ionizing photons is: We now change the integration frequency to the observer's frequency ν obs = δ obs (1+ z ) -1 δ -1 cl ν to obtain: We now estimate the factor D A /D cl appearing in this equation. Let the angular separation of the cloud from the nucleus in the plane of the sky be ξ cl and the projected linear separation be D cl , p . Then, referring to Figure 2 for the source geometry, the polar angle of the ray intersecting the center of the cloud is θ cl = π -θ obs -ψ cl and the projected cloud distance from the core is Hence, Thus, the number density of ionizing photons at the cloud is: This form makes it straightforward to use the observed ionizing spectrum to evaluate the photon density at the cloud as a function of the jet Lorentz factor and the angular direction of the rays intersecting the cloud. The angular parameter ψ cl depends upon the angle of inclination θ obs . Consider a cloud which is adjacent to the jet and assume that it is approximately spherical. The angle between a ray through the cloud center and the jet axis is determined by the sum of the jet and cloud radii and the distance of the cloud from the core. The jet and cloud radii are independent of the angle of inclination but the distance from the core is affected by projection. Let Φ cl ≈ 0 . 35 '' and Φ jet ≈ 0 . 35 '' be the observed angular diameter of the cloud and jet respectively. Then the angle, ψ cl between the ray through the center of the cloud and the jet axis is given by where, as above, ξ cl is the projected angular separation of the cloud from the core. In our calculations we round up the value of ψ cl to the nearest degree.", "pages": [ 22, 23 ] } ]

[ { "title": "ABSTRACT", "content": "The measured emission-weighted metal abundance of the hot gas in early-type galaxies has been known to be lower than theoretical expectations for 20 years. In addition, both X-ray luminosity and metal abundance vary significantly among galaxies of similar optical luminosities. This suggests some missing factors in the galaxy evolution process, especially the metal enrichment process. With Chandra and XMM-Newton , we studied 32 early-type galaxies (kT glyph[lessorsimilar] 1 keV) covering a span of two orders of L X, gas /L K to investigate these missing factors. Contrary to previous studies that X-ray faint galaxies show extremely low Fe abundance ( ∼ 0 . 1 Z glyph[circledot] ), nearly all galaxies in our sample show an Fe abundance at least 0.3 Z glyph[circledot] , although the measured Fe abundance difference between X-ray faint and X-ray bright galaxies remains remarkable. We investigated whether this dichotomy of hot gas Fe abundances can be related to the dilution of hot gas by mixing with cold gas. With a subset of 24 galaxies in this sample, we find that there is virtually no correlation between hot gas Fe abundances and their atomic gas content, which disproves the scenario that the low metal abundance of X-ray faint galaxies might be a result of the dilution of the remaining hot gas by pristine atomic gas. In contrast, we demonstrate a negative correlation between the measured hot gas Fe abundance and the ratio of molecular gas mass to hot gas mass, although it is unclear what is responsible for this apparent anti-correlation. We discuss several possibilities including that externally originated molecular gas might be able to dilute the hot gas metal content. Alternatively, the measured hot gas Fe abundance may be underestimated due to more complex temperature and abundance structures and even a two-temperature model might be insufficient to reflect the true value of the emission weighted mean Fe abundance. Subject headings: galaxies: luminosities and abundance - galaxies: elliptical and lenticular - galaxies: ISM - X-rays: galaxies", "pages": [ 1 ] }, { "title": "Investigating the Potential Dilution of the Metal Content of Hot Gas in Early-Type Galaxies by Accreted Cold Gas", "content": "Yuanyuan Su 1 and Jimmy A. Irwin 1 [email protected]", "pages": [ 1 ] }, { "title": "1. Introduction", "content": "Most early-type galaxies (elliptical and lenticular galaxies) are composed of old, low-mass stars with little on going star formation activity. Over several gigayears, gas lost from aging stars serves as a substantial source of hot metal-rich interstellar medium (ISM). Such enriched hot gas radiates mainly in X-rays at a temperature of 10 6 . 4 -7 . 0 K via thermal bremsstrahlung with metal line emission. In addition to hot gas, unresolved low mass X-ray binaries (LMXBs) and other stellar components contribute to the diffuse X-ray emission. Early-type galaxies with similar optical luminosities ( L opt ) might be expected to have similar X-ray luminosities ( L X ), since a comparable amount of stellar mass from their similar stellar populations should release similar amounts of material into the ISM over a given time period. The surprisingly large dispersion in the L X /L opt relation has been reported since Einstein Observatory (e.g. Canizares et al. 1987; Fabbiano et al. 1992), with a variation in X-ray luminosity among early-type galaxies of similar optical luminosities up to two orders of magnitude (O'Sullivan et al. 2001). Such a discrepancy has been a well-known puzzle in X-ray astronomy for the last three decades. Another puzzle involves the metallicity of the hot ISM in early-type galaxies. The ISM metal abundance traces the history of the star formation and galaxy evolution. The ultimate sources of enriched hot gas are red giant winds, planetary nebulae and supernovae ejecta. Given that the stars of early-type galaxies have measured abundances near solar, and Type Ia supernovae (SNIa) contribute even higher metallicities, the hot ISM should show an abundance well above solar. However, contrary to theoretical derivation, ASCA detected an Fe abundance approaching ∼ 0 . 1 Z glyph[circledot] for most elliptical galaxies (Arimoto et al. 1997). Such anomalously low abundance values likely resulted from the inability of ASCA to spatially resolve the LMXBs components or temperature gradients (the 'Fe bias' 1 ). To some extent, the low metallicity issue has been partially resolved at least for some X-ray bright galaxies. Current missions such as Chandra , XMM-Newton and Suzaku observe that the metal abundance of the ISM in hot gas-rich galaxies tend to be approximately solar or slightly super-solar (e.g., Humphrey & Buote 2006; Xu et al. 2002; Loewenstein & Davis 2010). However, this is still at odds with our classical understanding of the enrichment processes of such systems. The origin of the metal discrepancy for X-ray bright galaxies has been reviewed in Pipino & Matteucci (2011, and references therein). One possibility is that a large fraction of SNIa ejecta may condense into dust rather than staying in the X-ray emitting hot phase. Fe-enriched gas cools faster than metal-poor gas because of its larger radiative emissivity. In fact, giant ellipticals may contain up to 10 7 M glyph[circledot] of dust (Temi et al. 2004), which makes dust-assisting cooling efficient. However, simulations have revealed that it is unlikely for Fe ejecta to cool and drop out of the hot gas phase (Tang & Wang 2010). Alternatively, a variation in SNIa yields or a large uncertainty in star formation history may also help explain the disagreement. Moreover, Pipino et al. (2005) also investigated whether the dilution from freshly accreted cold gas could reduce hot gas metal abundance. For X-ray faint galaxies (with only small amounts of hot ISM), the metallicity discrepancy is much worse, where very sub-solar ( ∼ 10% solar) abundances have been reported even with current missions (e.g., NGC 1291, Irwin et al. 2002; NGC 4697, Sarazin et al. 2001; NGC 3585, NGC 4494, NGC 5322, O'Sullivan & Ponman 2004). Early-type galaxies of various masses may have intrinsically different stellar Fe abundance which may be a factor in driving abundance differences. However, even elliptical galaxies with the same stellar metallicity show up to a factor of ten variation in ISM metal abundance (Humphrey & Buote 2006), making such an explanation unlikely. The inconsistency with theory and the fact that abundances vary widely among early-type galaxies of similar optical luminosities predict a crucial missing (possibly external) factor in the enrichment process. It has been suggested that galactic winds, SNIa and AGN could drive gaseous outflow (Mulchaey et al. 2010; Alatalo et al. 2011). There is also observational evidence for gas being removed from galaxies through ram pressure in dense environments (Owen et al. 2006; Sun et al. 2007). Through such processes, enriched gas of early-type galaxies can be transported into their environments, eventually enriching intragroup medium (IGrM) or intracluster medium (ICM). It also helps to explain the large scatter in the L X /L opt relation, in the sense that outflows were able to more efficiently remove gas from X-ray faint galaxies. However, such a process cannot explain the heavy element deficiency of the remaining gas, since it should work equivalently for both metals and hydrogen. Another explanation associates the dilution by the accretion of the relatively pristine local intergalactic medium. However, this explanation faces difficulties when it comes to isolated galaxies not to mention field galaxies tend to have a lower ISM metallicity than cluster galaxies. The role played by dilution via the accretion of surrounding cold gas in the enrichment processes has not been observationally investigated thoroughly. While the classical view of early-type galaxies is that they contain little or no cold (atomic or molecular) ISM, an increasing number of early-type galaxies have been found to contain significant amount of cold gas in the phase of H i (up to 10 9 M glyph[circledot] ; Oosterloo et al. 2010) and H 2 (up to 10 9 M glyph[circledot] ; Young et al. 2011). The fraction of early-type galaxies that are reported to have strong H i absorption is around 50%, which is as significant as the H i detection fraction for star formation galaxies (Thom et al. 2012). Such H i gas is also bound to its host galaxy, with the velocities of detected H i below the escape velocity (Thom et al. 2012). Additional studies show that the H i structures in early-type galaxies can reach out to many tens of kpc from the stellar body (Helmboldt 2007). Due to their relatively large orbit, when cold gas is compressed toward the galaxy center, their large potential energy difference would be eventually transformed into kinetic energy. Such a process may heat up the cold gas into the X-ray emitting phase. The overall detection rate of H 2 gas is 22% according to the largest volume-limited CO survey to date (Atlas 3 D collaboration 2 ). H 2 gas is more bound to their host galaxies than H i , and their distributions are more concentrated (Davis et al. 2009). Molecular gas could be shock heated to 10 6 K or higher via relative stellar velocities and interacting with hot gas (Young et al. 2011). Moreover, field galaxies tend to contain larger H i halo than galaxies in groups and clusters which may have had their H i halo destroyed by stripping or ram pressure (Oosterloo et al. 2010), while H 2 gas mass of galaxies does not seem to depend on environments (Young et al. 2011). The origin of such cold gas remains ambiguous. One explanation is that they may be the leftovers from the epoch of galaxy formation, or high angular momentum tidal gas that survived the merging process when early-type galaxies were transformed from spirals and settled into stable orbits around the newly-formed galaxies (Barnes 2002). They also have been proposed to originate from recent accretion from filaments, recent mergers and internal star formation (Davis et al. 2009). Li et al. (2011) provide evidence for the interaction between cold gas and hot gas through SNIa feedback. It is therefore desirable to explore the part played by cold gas in galaxy enrichment processes. This paper focuses on the metal abundance of X-ray faint early-type galaxies and the metal abundance difference between X-ray faint and the much better studied X-ray bright galaxies. As mentioned above, one explanation for this discrepancy is that accreted cold gas may have played a crucial role in the enrichment process of the X-ray hot gas in early-type galaxies by diluting the metal content of hot gas-poor (hot gas mass ∼ 1 × 10 7 M glyph[circledot] ) galaxies from their original approximately solar value to ∼ 20-30% solar. For galaxies with such H i structures destroyed (such as cluster galaxies which may have experienced stripping or ram pressure in a more dense ICM environment), the metal abundance of the hot gas should be much closer to solar. In this scenario X-ray bright galaxies would be relatively unaffected by dilution with cold gas due to their copious amount of existing hot metal-enriched gas ( ∼ 10 10 M glyph[circledot] ), making any cold gas dilution inconsequential. Many observations of X-ray faint galaxies do not have a sufficiently high X-ray count rate to constrain model parameters well, due to their low X-ray brightness. Their unresolved LMXBs and background also contribute a large fraction to the diffuse emission, leading to a poor signal-tonoise ratio for the hot gas component. It is natural to suspect that the low measured abundance of X-ray faint early-type galaxies is an outcome of low S/N or some other artificial bias. Unlike hot gas in the ICM, temperatures of hot gas in the ISM are usually ≈ 0.3-1 keV. Fe abundances in this temperature range can only be derived with complicated, incompletely ionized Fe-L lines. This makes the Fe abundance measurement sensitive to instruments, choice of spectral models, background subtraction, etc. We address such artificial factors in this work. We assume H 0 = 70 km s -1 Mpc -1 , Ω Λ = 0 . 7, and Ω M = 0 . 3. Throughout this paper, all uncertainties are given at the 90% confidence level unless otherwise stated. We adopt the solar abundance standard of Asplund (2009), which considered the deviations from local thermodynamic equilibrium. Sample selection is presented in Section 2. Section 3 and 4 are dedicated to observa- and data reduction. In Section 5 we report several relationships between galaxy properties. In Section 6 we examined potential biases. Implications of our results are discussed in Section 7. Finally, we summarize our main conclusions in Section 8.", "pages": [ 2, 3, 4, 5 ] }, { "title": "2.1. X-ray observations", "content": "Our ultimate goal is to study the X-ray emitting hot gas properties (e.g., metal abundance) of nearby early-type galaxies as completely as possible. We select our sample from a volume-limited sample of 260 nearby early-type galaxies from ATLAS 3 D plus 61 galaxies studied with ROSAT from Irwin et al. (1998). We only considered galaxies that have Chandra or XMM-Newton observations with a total exposure time of at least 25 ksec. We did not consider any galaxies residing in cluster centers since it is extremely challenging to disentangle the strong ICM emission from relatively weak ISM emission. We ultimately selected 32 galaxies with sufficient X-ray counts to have their metal abundance constrained with our model. Our sample included galaxies both in groups/clusters and in the field. Classification of galaxy environments is based on Faber (1989). Eight of our galaxies are at group centers (NGC 507, NGC 1399, NGC 4472, NGC 4636, NGC 1332, NGC 4261, NGC 1407, NGC 5846). We kept such relatively bright group center galaxies to form the bright end of this relatively continuous sample. The archived data itself does not form a statistically complete sample. Still, our sample of 32 early-type galaxies covers a span of L X /L K of more than two orders of magnitudes (0 . 03-3 . 00 × 10 30 ergs s -1 L K, glyph[circledot] -1 ), containing a sufficient number of X-ray faint, X-ray bright and intermediate-brightness galaxies. For the majority of the galaxies in this sample we adopted the distance estimation derived from surface brightness fluctuations of Tonry (2001). If not available, we used the distance determined from redshift as given in NED 3 and our assumed cosmology. Galaxy properties and observation logs are summarized in Table 1.", "pages": [ 5 ] }, { "title": "2.2. Assumed H i and H 2 gas mass values", "content": "We use the atomic neutral gas (M(H i )) mass and molecular gas (M(H 2 )) mass to represent the amount of cold gas mass in each galaxy obtained from the literature. Only 24 galaxies in our sample have published masses of atomic and molecular gas as listed in Table 2. Most of the atomic neutral gas masses were obtained with the Westerbork Synthesis Radio Telescope (WSRT). Molecular gas masses were derived with the measurements of 12 CO(1-0) and (2-1) emission lines obtained with IRAM 30-m Telescope.", "pages": [ 5 ] }, { "title": "3.1.1. Chandra", "content": "We used CIAO4.3 to reduce ACIS-I or ACIS-S data (Table 3). All data were reprocessed from level 1 events, which guarantees the latest and consistent calibrations. Only the events with grades 0, 2, 3, 4, and 6 are included. We also removed bad pixels, bad columns, and node boundaries. We filtered background flares with the light curve filtering script lc clean . The effective exposure times are shown in Table 3. Bright point sources including nuclei resolved with wavdetect were removed. In our spectral analysis, each spectrum contains at least 25 counts per energy bin.", "pages": [ 6 ] }, { "title": "3.1.2. XMM-Newton", "content": "Only data from the European Photon Imaging Camera (EPIC) are reported in this paper (Table 3). Both MOS and PN detectors were processed. The standard Science Analysis System (SAS 11.0.0) pipeline tools were used throughout this analysis. Tasks emchain and epchain were used to generate calibrated event files from raw data. PATTERN ≤ 12 was used to select MOS data sets, while PATTERN ≤ 4 was used for PN data sets. The removal of bright pixels and hot columns was done by applying the expression (FLAG==0). Point sources resolved with edetect chain and verified by eye were removed. The remaining exposure time after filtering for background flares is shown in Table 3. The minimum counts for each energy bin is 25 for MOS and 50 for PN.", "pages": [ 6 ] }, { "title": "3.1.3. Regions and Background", "content": "We adopted the effective radii for each galaxy from the Third Reference Catalogue of Bright Galaxies (RC3, de Vaucouleurs et al. 1991). The extracted aperture for L X,gas , M X,gas as well as metal abundance and temperature determinations was chosen to be exactly two effective radii for each galaxy. Local background, extracted from a region away from the source region on the same ccd chip, is used as background for spectral analysis for both Chandra and XMM-Newton . The area of the local background was chosen to be at least twice the area of the source region to ensure a sufficient S/N ratio for background subtraction. For X-ray bright galaxies observed with Chandra , the emission sometimes filled the entire chip due to the relatively small area of the S3 chip. Since X-ray bright galaxies are usually in clusters or at group centers, the adoption of local background enables us to subtract the surrounding ICM or IGrM, assuming the surface brightnesses of ICM/IGrM are uniform on a scale of ∼ 20 kpc. The variation of background emission is also relatively insignificant for X-ray bright galaxy studies. For X-ray faint galaxies, the emission usually does not fill the entire chip, which makes it ideal to use local background for spectral studies, unless they are very nearby. One counter example is the X-ray faint galaxy NGC 4697, which is at a distance of only 11 Mpc and has a very extended X-ray emission distribution that extends beyond the S3 chip boundaries. For this galaxy, we also tried to use 'stowed background' 4 for the spectral fitting. We fit a spectrum extracted from the S1 chip with a stowed background 5 of the same region on the S1 chip to determine the surface brightness of cosmic X-ray and Galactic emission background since the S1 chip is more offset and less contaminated by source emission. Then, we fit the spectrum of NGC 4697 extracted from the S3 chip with a corresponding stowed background by adding scaled X-ray background components obtained with the S1 chip to the fitting. The determinations for Fe metal abundance (see model fitting procedure in § 4 . 2) with these two different methods are 0 . 42 +0 . 22 -0 . 14 Z glyph[circledot] with χ 2 ν = 1 . 01 (local background) and 0 . 35 +0 . 17 -0 . 13 Z glyph[circledot] with χ 2 ν = 1 . 18 (stowed background), consistent within the uncertainties. Therefore, it is reliable to use local background for galaxies in our sample, even in extreme cases such as NGC 4697.", "pages": [ 6, 7 ] }, { "title": "3.2. 2MASS", "content": "To characterize the optical brightness of each galaxy, we use the K-band luminosity, which is more representative of relatively old stellar populations in early-type galaxies, instead of the historically-used B-band luminosity. L K of these galaxies are derived from Two Micron All Sky Survey (2MASS) archived images. The K-band photometry region is the same as that used in the Chandra and XMM-Newton analyses. Bright nuclear and foreground sources (detected by eye) were excluded and refilled with a local surface brightness component using dmfilth in CIAO4.3. We obtained the counts from the source region after subtracting the local background component. We converted it to the corresponding magnitudes and corrected for Galactic extinction. K-band infrared solar luminosity is assumed to be L K, glyph[circledot] = 5 . 67 × 10 31 ergs s -1 (Mannucci et al. 2005). L K of galaxies in this sample are listed in Table 4.", "pages": [ 7 ] }, { "title": "4.1. Spectral analysis", "content": "4.1.1. Spectral modelling For those galaxies with both Chandra and XMM-Newton observations, we conducted joint fits for the measurements of temperature and metal abundance. We performed spectral anal- ysis with Xspec 12.7.0. The model we adopted to fit the diffuse emission for each galaxy is phabs ∗ ( vapec + vapec + powerlaw + mekal + powerlaw ). The absorbing column density ( N H ) was fixed at the Galactic value (Dickey & Lockman 1990). vapec + vapec represent two temperature components of bremsstrahlung emission from the hot gas, with their elemental abundances tied to each other. We set Mg=Al, Si=S and He=C=N=Ar=Ca=1 (Nagino & Matsushita 2010; Hayashi et al. 2009). We use this two temperature component model to reduce the Fe bias, brought about by multi-temperature gas. The first Powerlaw with an index of 1.6 represents the contribution from unresolved LMXBs (Irwin et al. 2003). In addition to hot gas and unresolved LMXBs, faint stellar X-ray sources such as cataclysmic variables (CVs) and coronally active binaries (ABs) also contribute to L X . Revnivtsev et al. (2007, 2008, 2009) calibrated the X-ray emission from such old stellar populations in several extremely gas-poor galaxies. We estimated such stellar contributions from their L K based on a L X /L K relation averaged over these gas-poor early-type galaxies given by Revnivtsev et al. (2008): L 0 . 5-2 . 0 keV /L K = 5 . 9 × 10 27 ergs s -1 L K, glyph[circledot] -1 . The mekal + powerlaw component represents CV/ABs. The temperature of mekal is fixed at 0.5 keV, and the index of powerlaw is fixed at 1.9 (Revnivtsev et al. 2008). In our spectral analysis, we fixed such CV/ABs components at the estimated flux based on the L K of each galaxy. The ratio of the fluxes of the mekal component and the powerlaw was set to 2.03 (Revnivtsev et al. 2008). glyph[negationslash] glyph[negationslash] We tested the effects of our spectral model and assumptions on NGC 4459, which is the Xray faintest galaxy in this sample. The two temperature thermal components model gives a Fe abundance of 0 . 22 +0 . 09 -0 . 06 Z glyph[circledot] . After setting Al = Mg and S = Si, we obtained a Fe abundance of 0 . 23 +0 . 07 -0 . 07 . The calibration of the X-ray emission of CV/ABs is not well determined, and varies between L X /L K = 4 . 1 -6 . 9 × 10 27 for M32, N3379, and M31 (Revnivtsev et al. 2007; 2008). We varied the CV/ABs component by 100% by completely ignoring it and by doubling the contribution of such component, which gives a Fe abundance of 0 . 23 +0 . 07 -0 . 06 Z glyph[circledot] and 0 . 22 +0 . 07 -0 . 07 Z glyph[circledot] , respectively, indicating that uncertainties in the CV/AB normalization are not relevant to our metal abundance determinations.", "pages": [ 7, 8 ] }, { "title": "4.1.2. Joint fitting Chandra and XMM", "content": "In order to minimize statistical uncertainty, if available, Chandra and XMM-Newton observations were jointly fit for each galaxy, with all normalizations varied independently, but only the flux and normalization of Chandra are used to determine L X and M X,gas 6 , since we left all normalizations of each data set untied. We use 0.5-8.0 keV for ACIS-I, ACIS-S and PN, 0.3-8.0 keV for MOS to fit the spectra. It is important to justify that the cross calibrations are sufficient to provide reliable abundances. We examined all galaxies in our sample by determining their hot gas Fe abundance within two effective radii separately with Chandra and XMM-EPIC . Among them, 15 galaxies contain sufficient data to determine temperatures and metallicities separately from either data set. All of these 15 galaxies show consistency within the uncertainties between Chandra and XMM-EPIC results. Overall, this cross check gives us confidence in our joint fitting strategy.", "pages": [ 8, 9 ] }, { "title": "4.2. Determination of L X, gas and M X, gas", "content": "L X, gas estimated in this paper is contributed only by hot gas from 0 . 1 -2 . 0 keV, excluding Xray emission contributed by CV/ABs, and unresolved LMXBs, which have been removed spectrally. Assuming a spherical distribution of hot gas, we obtained their volume from the size of extraction region. Based on the sum of the best fit normalizations of the thermal emission model vapec + vapec , we derived the average hot gas density. With hot gas density and volume, we obtained hot gas mass ( M X, gas ) for galaxies in this sample. We assume that the hot gas density is a single value in the given volume for each galaxy. To test how a density gradient affect the result, we divide NGC 720 into 10 spatial bins within two effective radii. We analyzed each bin separately and obtained a sum of gas masses of the 10 bins that is within 10% of the gas mass obtained by analyzing a single integrated bin of a size of two effective radii.", "pages": [ 9 ] }, { "title": "5. Results", "content": "In Figure 1, we plot L X, gas for each galaxy as a function of L K , which shows the scatter of L X, gas /L K is more than a factor of 50. This factor is in line with but somewhat smaller than previously found (e.g. Boroson et al. 2011). Since our work aims at studying hot gas Fe abundance only in galaxies with sufficient X-ray counts, we eliminated galaxies that are extremely X-ray faint, leading to a smaller scatter in the L X, gas -L K relation. The span of L X, gas is from 7 × 10 38 ergs s -1 to 1 . 7 × 10 42 ergs s -1 ; the span of L K is from 1 . 5 × 10 10 L glyph[circledot] to 5 . 6 × 10 11 L glyph[circledot] (Table 4). Galaxies that reside at group centers are among the brightest. Galaxies in the field are on average fainter than galaxies in groups and clusters. Hot gas Fe abundance ranges from 0.22 Z glyph[circledot] (NGC 4459) to 1.9 Z glyph[circledot] (NGC 1407) (Table 5; Figure 2). The Fe abundance generally increases with L X,gas /L K for most galaxies (Figure 2), which gives a Spearman correlation coefficient of ρ = 0 . 449 with a null hypothesis probability of 1 . 24% for the L X,gas /L K -Fe relation using the ASURV software package 7 . The slope becomes flatter for X-ray bright galaxies such as those at group centers. Excluding group centers, we obtained a Spearman correlation coefficient of ρ = -0 . 673 with a null hypothesis probability of 0 . 12%. We also break galaxies in our sample into various environments as galaxies in the field, galaxies in groups and clusters but not at centers, and galaxies that reside at group centers, although only two galaxies in our sample are in field. A gradient in luminosity, temperature and hot gas Fe abundance can be seen as galaxies in denser environments are brighter, hotter and Fe richer. Only 22 galaxies in this sample have published Lick/IDS index measurements (listed in Table 1) from which we obtain the stellar metallicity [Fe/H] (Thomas et al. 2011; Maraston et al. 2011; Thomas et al. 2010; Johansson et al. 2011). The relation between the hot gas Fe and stellar metallicity is indeed very random (Figure 3 (a)), although NGC 1407, the galaxy with the highest hot gas Fe, does have the highest stellar metallicity. For galaxies of similar stellar abundance, the scatter of their ISM Fe is up to a factor of five. Therefore, the discrepancy in ISM abundance among galaxies is unlikely to be a result of the variation in stellar metallicity. We also tried to associate hot gas Fe abundance with stellar ages which we obtained from the literature (listed in Table 1). There is virtually no correlation between these two variables as shown in Figure 3 (b). Therefore, the variation of hot gas Fe abundance is unlikely to be caused by differing galaxy ages either. To test the cold gas dilution scenario, we studied a sub sample of 24 galaxies with cold gas data available. We related the hot gas Fe abundance to the ratio of their atomic gas mass M(H i ) to hot gas mass M X, gas and to the ratio of molecular gas mass M ( H 2 ) to hot gas mass M X, gas , respectively, (M(H i ) /M X, gas and M( H 2 ) /M X, gas ). Such a ratio is larger for X-ray faint galaxies at a given cold gas mass. We find that there is virtually no correlation between Fe and M(H i ) /M X, gas as shown in Figure 4(a); using ASURV which takes into account upper limit of variables, we obtained a Spearman correlation coefficient of ρ = -0 . 089 with a null hypothesis probability of 67 . 1%, illustrating the limited effect of atomic gas on the hot gas metal content. In contrast, we find that Fe abundance generally decreases with M ( H 2 ) /M X, gas as shown in Figure 4(b); we obtained a Spearman correlation coefficient of ρ = -0 . 459 with a null hypothesis probability of 2 . 76%. We summarized the correlations for each relation in Table 6.", "pages": [ 9, 10 ] }, { "title": "6. Possible abundance determination biases", "content": "Previously determined extremely low Fe abundances for NGC 1291 (Irwin et al. 2002) and NGC 4697 (Sarazin et al. 2001) are not confirmed because in our analysis we free individual elements, take account of CV/ABs, allow multi-temperature components and adopt a revised abundance table. Only by eliminating such improvements, can we recover the ∼ 10% Fe abundances for these galaxies found by previous studies. Here, we discuss some additional potential biases to abundance determinations.", "pages": [ 10 ] }, { "title": "6.1. 'Frankenstein' test", "content": "One potential bias that may affect abundance determination may originate from the different ratios of X-ray emitting components for X-ray faint and X-ray bright galaxies. The fraction of X-ray emission produced by hot gas is smaller for X-ray faint galaxies than for X-ray bright galaxies. For example, the ratio of hot gas to unresolved LMXBs to CV/ABs to background of a typical X-ray bright galaxy is 83:5:2:10 while that of a typical X-ray faint galaxy is 20:20:10:50. To explore this effect, we built a spectrum of a typical X-ray faint galaxy out of the X-ray components of a typical X-ray bright galaxy by separating and rearranging the contributions from the hot gas, LMXBs, CV/ABs, and background: a so-called 'Frankenstein' galaxy. We chose NGC 720 as a representative X-ray bright galaxy. The X-ray emission from this galaxy is partitioned as follows: 5.7% unresolved LMXB, 62.5% gas, 1.8% CV/AB and 30% background. NGC 720, as a galaxy in the field, is not influenced by any environmental effects such as ram pressure, tidal disruption, or galaxy harassment. The morphology of NGC 720 also appears very symmetric. The temperature radial profile of this galaxy shows little gradience (Humphrey et al. 2011), which minimizes any biasing in the measurement of Fe abundance due to multi-temperature components. All these factors assure a relatively robust metal abundance measurement. There are five main Chandra ACIS-S observations of NGC 720 with comparable exposure times. We combined the spectra of resolved point sources 8 from all five observations for the LMXBs spectrum. The spectrum of the hot gas was extracted from only one observation (the extracted region is the same as § 4 . 1 . 2). The background component was extracted from various source-free off-center regions of three observations. We then simulated the spectrum of CV/ABs ( mekal + powerlaw ). Combining these spectra produced a typical X-ray faint galaxy spectrum that contains 23% hot gas, 20% LMXBs, 10% CV/ABs and 47% background. We fit this Frankenstein spectrum with a separate local background spectrum and compared it with the results of the original X-ray bright galaxy NGC 720. The best fit Fe for this Frankenstein galaxy is 0 . 79 +1 . 31 -0 . 31 Z glyph[circledot] , while that of the original X-ray bright galaxy NGC 720 is 0 . 91 +0 . 22 -0 . 16 Z glyph[circledot] , consistent within the uncertainties. Therefore, the ratio of spectral components is unlikely to cause the measured low abundance of X-ray faint galaxies.", "pages": [ 10, 11 ] }, { "title": "6.2. Complex temperature structure", "content": "When fitting intrinsically multi-temperature systems with a single temperature model, the measured Fe abundance is systematically low, and is sometimes referred to as the Fe bias (Buote 2002). To address how significant this effect is, we tried to fit the hot gas component with just a single vapec model for each galaxy. We compared the Fe abundance as measured with a single temperature model (1T) to the Fe abundance measured with a two temperature model (2T). The measured Fe abundances are plotted in Figure 5. The Fe abundance measured with 2T is on average 20% greater than with 1T. In Figure 6 (a), we show the ratio of Fe abundance determined with 2T model and that with 1T model as a function of L X, gas /L K . 'Fe bias' does bias low the measured hot gas Fe abundance but seems equally effective for both X-ray faint and X-ray bright galaxies, therefore, unlikely to cause the Fe abundance discrepancy between X-ray faint and X-ray bright galaxies. We also simulated a spectrum with a model of phabs ∗ ( cevmkl + powerlaw + mekal + powerlaw ), where the Xspec model cevmkl represents a continuously varying multi-temperature hot gas emission. The Fe abundance of cevmkl was set to 1.0 Z glyph[circledot] . The simulated spectrum was built to have the same counts rate, exposure times, CV/ABs and LMXBs as NGC 720. We fit this simulated spectrum with the 2T model: phabs ∗ ( vapec + vapec + powerlaw + mekal + powerlaw ) as described earlier. We still obtained a Fe abundance of vapec + vapec to be around 1.0 Z glyph[circledot] , even when we vary the power-law index for the temperature distribution and the maximum temperature in cevmkl by at least 50%. This test demonstrates that the 2T model we applied is sufficient to describe continuous varying multi-temperature structures. However, the hot gas Fe abundance may indeed be underestimated due to more complex temperature and abundance structures, such that even a 2T model might be insufficient to reflect the real value of the Fe abundance. It is impractical to fit galaxies in our sample with 2T/2Fe model with the limited S/N of out data set. Therefore, we simulated a spectrum based on NGC 720 with a 2T model. We let each vapec component have different temperatures and Fe abundances but the same flux (2T/2Fe model: T 1 =0.2, Fe 1 =1.4 Z glyph[circledot] and T 2 =0.6, Fe 2 =0.4 Z glyph[circledot] ; higher metallicity gas is supposed to cool faster). We fit this spectrum with a 2T model but tied the Fe abundance of each vapec to each other. We obtained a Fe abundance of only 0.5 Z glyph[circledot] , smaller than the emission weighted average abundance of the two vapec but still between the bookend values. The measured Fe abundance is further reduced by 25% if we fit this spectrum with a 1T model. We did the same test based for NGC 4697 (2T/2Fe model: T 1 =0.2, Fe 1 =0.8 Z glyph[circledot] and T 2 =0.4, Fe 2 =0.2 Z glyph[circledot] ), and obtained a best fit Fe abundance of 0.22 Z glyph[circledot] with 2T model for this simulated spectrum. The Fe abundance was underestimated to the same degree for NGC 720 and NGC 4697. In our work, we focus on the relative difference of Fe abundance between X-ray faint and X-ray bright galaxies instead of their absolute values. Such a uniform underestimation for Fe abundance should not seriously affect our conclusions. Exploring a larger Fe/T parameter space modeling and a more realistic simulation are needed in future work to address this point more fully.", "pages": [ 11, 12 ] }, { "title": "7. Discussion", "content": "In this paper we have demonstrated a discrepancy in the hot gas metal abundance between X-ray faint and X-ray bright galaxies. As shown in Figure 2, it is evident that abundance increases with L X, gas /L K for X-ray faint galaxies but less so for X-ray brightest galaxies. This indicates that X-ray faint galaxies are more affected by whatever the origin of this variation is. In this work, we found that there is essentially no correlation between hot gas Fe abundance and the ratio of atomic gas mass to hot gas mass (M(H i ) / M X, gas ). Hot gas Fe abundance does not appear to be related to M(H i ) either as shown in Table 6. Thus, this would seem to rule out the scenario that the accretion of pristine atomic gas could affect the metal content of hot gas of early-type galaxies. Atomic gas is usually distributed in large orbits and in some cases they are in a shape of a ring surrounding the galaxy (e.g. NGC 1291). The accretion of such outskirt gas may not be efficient. Moreover, such accretion may be suppressed during the process of gaseous outflows. In contrast, we found a significant anti-correlation between the hot gas Fe abundance as measured in X-ray and the ratio of molecular gas mass to hot gas mass (M( H 2 ) / M X, gas ); galaxies with a larger molecular gas fraction tend to have lower Fe abundance while galaxies with a smaller cold gas fraction show a higher Fe abundance (Figure 4). The Spearman correlation coefficient is ρ = -0 . 459 with a null hypothesis probability of 2 . 76%. Hot gas Fe is also correlated with M( H 2 ) as shown in Table 6. Unlike atomic gas, molecular gas is usually located in the inner regions of galaxies which makes it easier for it to interact with hot gas. This result seemingly suggests a scenario where the molecular gas has been shock heated to the X-ray emitting phase by relative velocities between stellar wind and ambient hot gas. However, unlike atomic gas, it is unjustified to assume that molecular gas is pristine. In fact, molecular gas is usually associated with new star formation, which makes its metallicity no less than that of hot gas, putting this dilution scenario in doubt. To search for an alternative explanation, we first examined the reliability of the molecular gas mass estimation. M( H 2 ) adopted in this paper were derived from CO emission lines (e.g, Young et al. 2011). Those authors assume a constant conversion between M( H 2 ) and CO emission. In fact, M( H 2 )-F CO is a function of stellar metallicity as lower metallicity systems have a higher M( H 2 ) / F CO ratio (Genzel et al. 2012). However, as illustrated in § 5, stellar metallicity and hot gas metallicity are not directly related. Therefore, this bias should not be responsible for this apparent anti-correlation between hot gas Fe and M( H 2 ) / M X, gas . Moreover, even if higher stellar metallicity galaxies tend to have higher hot gas Fe abundance, the true anti-correlation between hot gas Fe - M( H 2 ) / M( X, gas ) would have been even stronger because assuming a constant conversion for all galaxies could only underestimate the M( H 2 ) for lower metallicity systems. Second, this apparent anti-correlation may be a result of complex temperature structures in a sense that a larger molecular gas fraction may lead to a more non-uniform temperature distribution of hot gas through cooling and therefore bias low the measurement of hot gas Fe abundance. This scenario predicts that the ratio of Fe abundance determined with a 2T model to that with a 1T model should depend on molecular gas fraction. However, as shown in Figure 6(b), these two factors are not related. Therefore, we do not think complex temperature structure drives this hot gas Fe - M( H 2 ) / M( X, gas ) anti-correlation. Still, it could be a result of complex abundance structure as proposed in § 6 . 2. The interaction with molecular gas may cause multi-phase metal abundance distributions which would bias low the measured hot gas Fe abundance. The test of this scenario is beyond the scope of this paper. We expect more simulation works in the future to cast light on this issue. Third, this apparent anti-correlation may stem from a third factor that is linked to both molecular gas and hot gas Fe abundance. Whatever this third factor is, the question why there is a discrepancy of the hot gas Fe metallicity between X-ray faint and X-ray bright galaxies remains to be answered. Using 3D simulations, Tang & Wang (2010) found that SN ejecta has a tendency to move outward substantially faster than the ambient medium via buoyancy force and effectively reduces the average Fe abundance of hot gas. If this process is plausible, we can conclude that systems with smaller angular momentum and larger potential wells are more likely to retain their Fe abundance since it is easier for them to resist such buoyancy force. X-ray bright galaxies are usually massive galaxies which also tend to be slow rotating galaxies (Emsellem et al. 2011). Consequently, X-ray bright galaxies would have a larger Fe abundance than X-ray faint galaxies in this scenario. It has been speculated that slow rotating galaxies as well as massive galaxies contain less molecular gas (Young et al. 2011). Hence, X-ray bright galaxies tend to lack molecular gas. As a result, molecular gas and hot gas Fe abundance appear to be anti-correlated. Yet, it is unclear how dynamic mass, angular momentum and molecular gas content of galaxies are physically related. Finally, we still try to explore the possibility that molecular gas may have diluted the metal content of hot gas. In addition to internal stellar mass loss, molecular gas in early-type galaxies have also been proposed to originate externally such as accretion through filaments and mergers with late-type galaxies. Davis et al. (2009) show that quite a few galaxies have their molecular gas kinematically misaligned with respect to the stars, suggesting external origin. Combes et al. (2007) inferred that CO-rich galaxies may be more metal and α -element poor owing to a slow star formation fuelled by relatively pristine gas. Davis et al. (2009) also found that the molecular and atomic gas are always kinematically aligned. Therefore, it is possible that molecular gas in early-type galaxies may originate from the condensation of surrounding atomic gas. In that case, the molecular gas might be relatively less contaminated and potentially dilute the hot gas metal content. There are many other factors that may contribute to the discrepancy between X-ray faint and X-ray bright galaxies as well as the apparent anti-correlation between hot gas Fe abundance and molecular gas content. In fact, early-type galaxies may be quite heterogeneous with various assembly and enrichment histories. Both a case-to-case study and a larger and more complete sample are required to address such issues in the future. Nevertheless, various explanations are not exclusive to each other. A combined scenario may eventually solve the discrepancy.", "pages": [ 12, 13, 14 ] }, { "title": "8. Summary", "content": "We studied a sample of 32 early-type galaxies with quality Chandra and XMM-Newton data covering a large span of X-ray luminosities. We derive a number of their properties including L X , L K , temperature, and ISM Fe metallicity. We attempt to relate these properties to their stellar metallicity, stellar age, and cold gas masses, to investigate the causes of the low metallicity of hot gas in X-ray faint galaxies and the metal abundance difference between X-ray faint and X-ray bright galaxies. We summarize our main results as follows: Deeper observations of X-ray as well as radio observations of more early-type galaxies, especially for galaxies in the field, are needed to further test the role played by neutral gas more thoroughly.", "pages": [ 14, 15 ] }, { "title": "9. Acknowledgments", "content": "We are grateful to Daniel Thomas and Jonas Johansson for calculating stellar metallicities. We thank Dong-Woo Kim and Raymond White for useful discussions and suggestions. We thank Evan Million, Ka-Wah Wong, Milhoko Yukita and Zhiyuan Li for reading an early draft and helpful comments.", "pages": [ 15 ] }, { "title": "REFERENCES", "content": "Beuing, J., Bender, R., Mendes de Oliveira, C., Thomas, D., & Maraston, C. 2002, A&A, 395, 431 Boroson, B., Kim, D. & Fabbiano, G. 2011, ApJ, 729, 12 Buote, D. 2002, ApJ, 574L, 135 Zhang, Y., Gu, Q.-S., & Ho, L.C. 2008, A&A, 487, 177 Note. -∗ 0: Galaxies in the field. 1: Galaxies in groups and clusters but not at centers. 2: Galaxies at group center (Faber et al. 1989) References. - a. Serra & Oosterloo (2010), b. Welch (2010), c. Haynes (1988), d. Taniguchi (1994), e. Serra (2011), f. Beuing (2002), g. Young (2011), h. http://goldmine.mib.infn.it/, i. Oosterloo (2010), j. Li (2011), k. Lees (1991), l. Emsellem (2011), l. Denicolo (2005), m. Serra (2008), n. Trager (2000), o. Li (2006), p. Terlevich (2002), q. Noll (2009), r. Humphrey (2008), s. Annibali (2001), t. Sil'chenko (2006), u. Zhang (2008), v. Kuntschner (2010), w. Shapiro (2009), x. Howell (2005), y. McDermid (2006), z. Gallagher (2008) Note. - All quantities were measured within two effective radii given in Table 1. Note. - T 1 (L 1 ) and T 2 (L 2 ) represent the best fit temperatures (luminosities) of each vapec component in the two temperature model. All quantities were measured within two effective radii given in Table 1. Note. -∗ group center galaxies excluded. ∗∗ galaxies with cold gas data. X, gas", "pages": [ 15, 16, 17, 18, 19, 21, 22, 23, 26 ] } ]