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2004: Hopkins 2004). | 2004; Hopkins 2004). |
results on the evolution of the infrared (8—10007/m») and far-infrared luminosity functions reveal a degeneracy between luminosity and density evolution which is consistent with our adopted form of evolution (Le Floc'h et al. | results on the evolution of the infrared ) and far-infrared luminosity functions reveal a degeneracy between luminosity and density evolution which is consistent with our adopted form of evolution (Le Floc'h et al. |
2005: Huynh et al. | 2005; Huynh et al. |
2005). although constraints on the population with Lin<10 (equivalent to log [W/Hz/sr]) are limited at 2— 0.5. | 2005), although constraints on the population with $L_{\rm{IR}} < 10^{11}$ (equivalent to log $L_{\rm{1.4 GHz}}=22$ [W/Hz/sr]) are limited at $z > 0.5$ . |
More complex models of the evolution. of star-forming galaxies have been put forward to interpret mid- and far-infrared observations (e.g. Franceshini et al. | More complex models of the evolution of star-forming galaxies have been put forward to interpret mid- and far-infrared observations (e.g. Franceshini et al. |
2001: Lagache et al. | 2001; Lagache et al. |
2004). | 2004). |
((see SectionD)), in which the transverse field enhancement at the flaring PIL is unequivocally detected. | (see Section \ref{data}) ), in which the transverse field enhancement at the flaring PIL is unequivocally detected. |
We will discuss the rapid change of the 3D CMF associated with that of the PMF using nonlinear force-free field (NLFFF) extrapolations from the active region. | We will discuss the rapid change of the 3D CMF associated with that of the PMF using nonlinear force-free field (NLFFF) extrapolations from the active region. |
The HMI instrument obtains filtergrams in six polarization states at six wavelengths along the Fe 6173 sspectral line to compute Stokes parametersJQUV. | The HMI instrument obtains filtergrams in six polarization states at six wavelengths along the Fe 6173 spectral line to compute Stokes parameters. |
The inverted and disambiguated vector magnetic field data for the NOAA AR 11158 with a 12 minute cadence and ppixel size were recently released by the HMI et | The inverted and disambiguated vector magnetic field data for the NOAA AR 11158 with a 12 minute cadence and pixel size were recently released by the HMI \citep{hoeksema11}. |
We used the vector magnetograms to (Hoeksemaal.|2011).. coordinates to (1) the rapid remappedPMF evolution heliographicassociated with the M6.6 flare on investigate2011 February 13, which started at 17:28 UT, peaked at 17:38 UT, and ended at 17:47 UT in terms of 11-8 fflux, and (2) trace the surface flows with the differential affine velocity estimator for vector magnetograms (DAVE4VM; ⋅⋅ | We used the vector magnetograms remapped to heliographic coordinates to (1) investigate the rapid PMF evolution associated with the M6.6 flare on 2011 February 13, which started at 17:28 UT, peaked at 17:38 UT, and ended at 17:47 UT in terms of 1–8 flux, and (2) trace the surface flows with the differential affine velocity estimator for vector magnetograms \citep[DAVE4VM;][]{schuck08}. |
The Spectro-Polarimeter (SP) of the Solar Optical Telescope (SOT;[TsunetaetaL.|2008) on board oobtains polarization profiles of two magnetically sensitive Fe lines at 630.15 and 630.25 nm to generate Stokes images, which are then reduced with the Hinode//SP calibration/inversion pipeline to retrieve the vector PMF using the Milne-Eddington gRid Linear Inversion NetworkΠΝ)2. | The Spectro-Polarimeter (SP) of the Solar Optical Telescope \citep[SOT;][]{tsuneta08} on board obtains polarization profiles of two magnetically sensitive Fe lines at 630.15 and 630.25 nm to generate Stokes images, which are then reduced with the /SP calibration/inversion pipeline to retrieve the vector PMF using the Milne-Eddington gRid Linear Inversion Network. |
. Hinode//SP scanned the entire isolated, near (ΜΕΕΙdisk-center (about S20?, E04?) AR nearly11158 at four time bins (12:00-12:32, 16:00-16:32, 19:00-19:32, and 21:30-22:02 UT) close to the M6.6 flare. | /SP scanned the nearly entire isolated, near disk-center (about $^{\circ}$, $^{\circ}$ ) AR 11158 at four time bins (12:00–12:32, 16:00–16:32, 19:00–19:32, and 21:30--22:02 UT) close to the M6.6 flare. |
We further processed the resulted vector magnetograms with a resolution of 0".32 pixel! to (1) resolve the 180° azimuthal ambiguity using the AZAM code (Lekaαἱ.2009) based on the “minimum energy" approachet (Metcalfetal]2006)., and (2) remove the projection effects by the observed fields to heliographic coordinatestransforming resolutionlyard|[1990). | We further processed the resulted vector magnetograms with a resolution of $''$ .32 $^{-1}$ to (1) resolve the $^{\circ}$ azimuthal ambiguity using the AZAM code \citep{leka09b} based on the “minimum energy” approach \citep{metcalf06}, and (2) remove the projection effects by transforming the observed fields to heliographic coordinates \citep{gary_hagyard90}. |
. Taking advantage of the high spatial and high polarization accuracy ofHinode,, we examine the CMF evolution by constructing NLFFF models using the “weighted optimization” method 2004) after preprocessing the photospheric boundary to best suit the force-free condition ⋅⋅ | Taking advantage of the high spatial resolution and high polarization accuracy of, we examine the CMF evolution by constructing NLFFF models using the “weighted optimization” method \citep{wiegelmann04} after preprocessing the photospheric boundary to best suit the force-free condition \citep{wiegelmann06}. |
The calculation was performed using 2 x 2 rebinned magnetograms within a box of 236 x 256 x 256 uniform grid points, which corresponds to ~1110 x 120 x 120 Mm? of balanced magnetic fluxes (with a ratio of opposite surface fluxes of ~999%—102%)). | The calculation was performed using 2 $\times$ 2 rebinned magnetograms within a box of 236 $\times$ 256 $\times$ 256 uniform grid points, which corresponds to 110 $\times$ 120 $\times$ 120 $^3$ of balanced magnetic fluxes (with a ratio of opposite surface fluxes of ). |
The temporal and spatial relationship between the PMF change and flare energy release can provide important clues concerning the mechanism. | The temporal and spatial relationship between the PMF change and flare energy release can provide important clues concerning the eruption mechanism. |
The evolution of the flare hard X-ray (HXR) eruptionemission was entirely registered by the citep[RHESST;][]lin02.. | The evolution of the flare hard X-ray (HXR) emission was entirely registered by the \\citep[\hsi;][]{lin02}. |
CLEAN images in the nonthermal energy range (25-100 keV) showing the flare footpoints were reconstructed using the front segments of detectors 2-8 with ~448 s integration time throughout the event. | CLEAN images \citep{hurford02} in the nonthermal energy range (25–100 keV) showing the flare footpoints were reconstructed using the front segments of detectors 2–8 with 48 s integration time throughout the event. |
To provide the observational context from the low chromosphere to the coronal, we also used 1700 ccontinuum (5000 K) and 94 ((FeXVII; 6 MK) images taken by the citep[AIA;][]leman!1 on boardSDO,, and Ca H images taken by Hinode//SOT. | To provide the observational context from the low chromosphere to the coronal, we also used 1700 continuum (5000 K) and 94 (Fe; 6 MK) images taken by the \\citep[AIA;][]{leman11} on board, and Ca H images taken by /SOT. |
The M6.6 flare is initiated at the center of the active region, where opposite magnetic flux concentrations undergo an overall counterclockwise, rotation-like motion as clearly tracked by DAVE4VM (Figure [If(a)), resulting in highly | The M6.6 flare is initiated at the center of the active region, where opposite magnetic flux concentrations undergo an overall counterclockwise, rotation-like motion as clearly tracked by DAVE4VM (Figure \ref{f1}( (a)), resulting in highly |
Since the 1-vear period ofthe solar eycle was discovered x Sclavabe (1811). the long-term activity of the solar evele lias become a hot issue in the feld of solar ouwesies. | Since the 11-year period of the solar cycle was discovered by Schwabe (1844), the long-term activity of the solar cycle has become a hot issue in the field of solar physics. |
Sunspots are taken as the most famous and vpical indicators of solar activity. | Sunspots are taken as the most famous and typical indicators of solar activity. |
Suuspot activity las Couples spatial aud temporal behavior. | Sunspot activity has complex spatial and temporal behavior. |
Carrington (15858. 1859) investigate a dift latitude of suuspot notion towards the equator and a variation of the rotation rate of the Sun. | Carrington (1858, 1859) investigated a drift latitude of sunspot motion towards the equator and a variation of the rotation rate of the Sun. |
Hathaway et al. ( | Hathaway et al. ( |
2003) examined the drift of the centroid of the sunspot area oward the equator in cach hemisphere from 1871 to 2002 and found that the dift rate slows as the centroid approaches the equator. | 2003) examined the drift of the centroid of the sunspot area toward the equator in each hemisphere from 1874 to 2002 and found that the drift rate slows as the centroid approaches the equator. |
The distribution of sunspots and flares i a solar evcle exhibits a "butterfly diagram” (Carrington. i858. Maunder 1901. 1913: Carcia 1990: Li ct al. | The distribution of sunspots and flares in a solar cycle exhibits a “butterfly diagram" (Carrington, 1858, Maunder 1904, 1913; Garcia 1990; Li et al. |
2003) anc he evcle appears uniforuly in both hemispheres on average (Newton Milsou 1955: White Trotter 1977). | 2003) and the cycle appears uniformly in both hemispheres on average (Newton Milson 1955; White Trotter 1977). |
Suuspot activity is described by sunspot nunubers and sunspot areas in ecucral. | Sunspot activity is described by sunspot numbers and sunspot areas in general. |
Consequently. the sunspot uunubers and sunspot areas are used to investigate long-term solar activity (Li ct al. | Consequently, the sunspot numbers and sunspot areas are used to investigate long-term solar activity (Li et al. |
2009). | 2009). |
Because of the periodicity of sunspot activity (sunspot nunboers and sunspot areas). the associated cruptions (flares. CNIES) appear to have the similar periodicity to that of sunspot activity (Storini IHofer 1999). | Because of the periodicity of sunspot activity (sunspot numbers and sunspot areas), the associated eruptions (flares, CMEs) appear to have the similar periodicity to that of sunspot activity (Storini Hofer 1999). |
In addition. the soft N-rav flares were significautlv delaved with | In addition, the soft X-ray flares were significantly delayed with |
the decline in both radio and X-ray luminosity as a function of spectral type. | the decline in both radio and X-ray luminosity as a function of spectral type. |
The best-fit linear trend ts: which in the range of X-ray luminosities for MO-M6 dwarfs. Ly~1075ο corresponds Συ Hz'. | The best-fit linear trend is: which in the range of X-ray luminosities for M0-M6 dwarfs, $L_X\sim
10^{28}-10^{29.5}$, corresponds to $L_{\rm \nu,rad}/L_X\approx
10^{-15.5}$ $^{-1}$. |
Beyond spectral— type M6. however. we find a clear trend of increasing ratio of Li4/Ly. from about 1074 for Μ7Τ-Μδ to Z107 beyond M9. | Beyond spectral type M6, however, we find a clear trend of increasing ratio of $L_{\rm \nu,rad}/L_X$, from about $10^{-14}$ for M7-M8 to $\gtrsim 10^{-12}$ beyond M9. |
We note that not all objects necessarily violate the correlation. since some are detected in the X-rays with no corresponding radio emission. | We note that not all objects necessarily violate the correlation, since some are detected in the X-rays with no corresponding radio emission. |
However. with the exception of the marginal detection of Kelu-1. all of these objects have spectral types of M7-M9. | However, with the exception of the marginal detection of Kelu-1, all of these objects have spectral types of M7-M9. |
A comparison of the trends in Figures A2 and A6 suggests that the breakdown in the radio/X-ray correlation beyond ~M7 1s largely due to the rapid decline in quiescent X-ray luminosity. | A comparison of the trends in Figures \ref{fig:lx} and \ref{fig:lr}
suggests that the breakdown in the radio/X-ray correlation beyond $\sim {\rm M7}$ is largely due to the rapid decline in quiescent X-ray luminosity. |
This is a surprising result since the strong correlation for objects earlier than M6 suggests that heating (X-ray emission) and particle acceleration (radio emission) are either directly correlated. or share a common origin. presumably through dissipation of the magnetic fields (e.g.. Guedel&Benz 1993)). | This is a surprising result since the strong correlation for objects earlier than M6 suggests that heating (X-ray emission) and particle acceleration (radio emission) are either directly correlated, or share a common origin, presumably through dissipation of the magnetic fields (e.g., \citealt{gb93}) ). |
Since the radio luminosities remain relatively unchanged from early-M to mid-L. dwarfs. it appears that the fraction of magnetic energy that goes into accelerating electrons is roughly constant. and that the process of field dissipation remains uniformly efficient despite the increasing neutrality of the stellar atmospheres. | Since the radio luminosities remain relatively unchanged from early-M to mid-L dwarfs, it appears that the fraction of magnetic energy that goes into accelerating electrons is roughly constant, and that the process of field dissipation remains uniformly efficient despite the increasing neutrality of the stellar atmospheres. |
Atmospheric neutrality has been argued to cause the decline in heating (1.e.. Ha emission) due to the resulting decrease in magnetic stresses and field dissipation (Mohantyetal.2002). | Atmospheric neutrality has been argued to cause the decline in heating (i.e., $\alpha$ emission) due to the resulting decrease in magnetic stresses and field dissipation \citep{mbs+02}. |
. Since. the field dissipation and electron. acceleration efficiency remain relatively unchanged. we are forced to conclude that the breakdown of the radio/X-ray correlation ts due to a decrease in the plasma heating efficiency. | Since the field dissipation and electron acceleration efficiency remain relatively unchanged, we are forced to conclude that the breakdown of the radio/X-ray correlation is due to a decrease in the plasma heating efficiency. |
This may be due to enhanced trapping of the radio-emitting electrons. if these electrons are directly responsible for plasma heating in higher mass stars. | This may be due to enhanced trapping of the radio-emitting electrons, if these electrons are directly responsible for plasma heating in higher mass stars. |
Another possibility is a decline in the bulk coronal density leading to a strong suppression in X-ray heating. but with a minor impaet on the radio emission. which requires a smaller population of relativistic electrons. | Another possibility is a decline in the bulk coronal density leading to a strong suppression in X-ray heating, but with a minor impact on the radio emission, which requires a smaller population of relativistic electrons. |
Alternatively. the geometry of the radio-emitting regions may evolve to smaller sizes such that their impact on large-scale coronal plasma heating decreases. | Alternatively, the geometry of the radio-emitting regions may evolve to smaller sizes such that their impact on large-scale coronal plasma heating decreases. |
This latter explanation appears less likely since rotationally-stable quiescent radio emission and periodic Ha emission from several ultracool dwarfs in our sample point to large magnetic field covering fractions (e.g.. Bergeretal.2008a. 2009)). | This latter explanation appears less likely since rotationally-stable quiescent radio emission and periodic $\alpha$ emission from several ultracool dwarfs in our sample point to large magnetic field covering fractions (e.g., \citealt{bbg+08,brp+09}) ). |
Finally. the combination of rapid rotation and the shrinking co-rotation radius of lower mass stars may lead to a decline in X-ray emission through centrifugal stripping of the corona. | Finally, the combination of rapid rotation and the shrinking co-rotation radius of lower mass stars may lead to a decline in X-ray emission through centrifugal stripping of the corona. |
In Figure ΑΘ. we plot the ratio Lx{πρ as a function of rotation period. | In Figure \ref{fig:lxlbol_rot} we plot the ratio $L_X/L_{\rm bol}$ as a function of rotation period. |
We find a decline in Ly/Lpo as a function of decreasing period for objects later than M7. with a median value of Ly(Lio107 for P>0.3 d. and Ly/Liac107 for P«0.3 d (for typical parameters. Pz0.3 d corresponds tov~15-20 km s! ). | We find a decline in $L_X/L_{\rm bol}$ as a function of decreasing period for objects later than M7, with a median value of $L_X/L_{\rm bol}\approx 10^{-4}$ for $P>0.3$ d, and $L_X/L_{\rm bol}\approx 10^{-5}$ for $P<0.3$ d (for typical parameters, $P\approx 0.3$ d corresponds to $v\sim 15-20$ km $^{-1}$ ). |
For this decline to be due to coronal stripping. the magnetic field scale height should be a few stellar radi. | For this decline to be due to coronal stripping, the magnetic field scale height should be a few stellar radii. |
We presented simultaneous X-ray. radio. and Πα observations of three late-M and L dwarfs. which along with our previous published results double the number ultracool dwarfs and triple the number of L dwarfs observed in X-rays. | We presented simultaneous X-ray, radio, and $\alpha$ observations of three late-M and L dwarfs, which along with our previous published results double the number ultracool dwarfs and triple the number of L dwarfs observed in X-rays. |
The overall X-ray detection fraction m our survey. to a typical limit of Ly/2107. is about 20%. | The overall X-ray detection fraction in our survey, to a typical limit of $L_X/L_{\rm bol}\approx 10^{-5}$, is about $20\%$ . |
Combining our sources with all objects later than M7 from the literature leads to a detection fraction of about 50%. | Combining our sources with all objects later than M7 from the literature leads to a detection fraction of about $50\%$. |
This fraction is dominated by late-M dwarfs. with at most ~15% of L dwarfs detected to date (this is based on the marginal detection of Kelu-1). | This fraction is dominated by late-M dwarfs, with at most $\sim 15\%$ of L dwarfs detected to date (this is based on the marginal detection of Kelu-1). |
We further find a significant drop in X-ray activity beyond spectral type ~M7. to Ly/Lpaz107 for M7-M9 and Lx/LyX10? for L dwarfs. | We further find a significant drop in X-ray activity beyond spectral type $\sim {\rm M7}$, to $L_X/L_{\rm bol} \approx
10^{-4}$ for M7-M9 and $L_X/L_{\rm bol}\lesssim 10^{-5}$ for L dwarfs. |
The decline inX-ray activity. and the observed level of emission. are similar to those measured for low mass stars and brown dwarfs in young star forming regions (Feigelsonetal.2002;Mokler&Stelzer2002;Preibisch&Zinnecker2002:Grossoetal. 2007). | The decline inX-ray activity, and the observed level of emission, are similar to those measured for low mass stars and brown dwarfs in young star forming regions \citep{fbg+02,ms02,pz02,gbg+07}. |
. A similar decline in activity is observed in Ho. although 75% of our targets are detected. | A similar decline in activity is observed in $\alpha$, although $75\%$ of our targets are detected. |
In the radio band. however. we find that {μμ remains relatively unchanged in the range MO-L4. and hence ζωή[μοι Increases by about two orders of magnitudes over the same spectral type range. | In the radio band, however, we find that $L_{\rm rad}$ remains relatively unchanged in the range M0-L4, and hence $L_{\rm rad}/L_{\rm bol}$ increases by about two orders of magnitudes over the same spectral type range. |
The rapid decline in. X-ray emission and the uniform level of radio emission lead to a breakdown in the radio/X-ray correlation. | The rapid decline in X-ray emission and the uniform level of radio emission lead to a breakdown in the radio/X-ray correlation. |
With our increased sample of objects we find that the ratio transitions sharply. but smoothly. from {ναωνzz107072 to ~1071? Hz! over the spectral type range M7-M9. | With our increased sample of objects we find that the ratio transitions sharply, but smoothly, from $L_{\rm \nu,rad}/ L_X\approx 10^{-15.5}$ to $\sim
10^{-11.5}$ $^{-1}$ over the spectral type range M7-M9. |
The radio/X-ray correlation in objects earlier than M7 is likely due to a correlated or common origin for particle acceleratiot and plasma heating. | The radio/X-ray correlation in objects earlier than M7 is likely due to a correlated or common origin for particle acceleration and plasma heating. |
We conclude that its breakdown is due to a decline in the plasma heating efficiency. as a result of efficient electron trapping. increased neutrality. decrease i coronal density. a decline in the size of the radio-emitting regions, or coronal stripping. | We conclude that its breakdown is due to a decline in the plasma heating efficiency, as a result of efficient electron trapping, increased neutrality, decrease in coronal density, a decline in the size of the radio-emitting regions, or coronal stripping. |
In light of our results. we suggest that continuec observational progress in the study of X-ray emissior from ultracool dwarfs can be achieved with two distinct approaches: (1) deeper observations of a small number of nearby objects: and (11) a shallower survey of a significantly larger sample of objects than is currently available. | In light of our results, we suggest that continued observational progress in the study of X-ray emission from ultracool dwarfs can be achieved with two distinct approaches: (i) deeper observations of a small number of nearby objects; and (ii) a shallower survey of a significantly larger sample of objects than is currently available. |
Inthe context of the former approach. our existing survey indicates that most L dwarts do not produce quiescent X-ray emissionat alevel of Ly/Lyo2107? based on 30 ks observations. | Inthe context of the former approach, our existing survey indicates that most L dwarfs do not produce quiescent X-ray emissionat a level of $L_X/L_{\rm bol}\gtrsim 10^{-5}$ based on 30 ks observations. |
Since our targets were located within about 15 pe. deeper limits can only be achieved with ~100 ks observations of targets within ~5 pe. | Since our targets were located within about 15 pc, deeper limits can only be achieved with $\sim 100$ ks observations of targets within $\sim 5$ pc. |
However. the only early-L dwarfs that come close to matching this criterion (2M0036+18 and 2M1507-16) were already observed as part of our survey. | However, the only early-L dwarfs that come close to matching this criterion (2M0036+18 and $-$ 16) were already observed as part of our survey. |
Thus. it is unlikely that we can observe any ultracool dwarfs to limits that are more than a factor of 2—3 times deeper than our current survey. and we therefore conclude that long X-ray observations do not provide the best way forward. | Thus, it is unlikely that we can observe any ultracool dwarfs to limits that are more than a factor of $2-3$ times deeper than our current survey, and we therefore conclude that long X-ray observations do not provide the best way forward. |
Instead. such observations should be targeted at objects with strong radio and/or Ha emission. for which the likelihood of an X-ray detection may be higher. | Instead, such observations should be targeted at objects with strong radio and/or $\alpha$ emission, for which the likelihood of an X-ray detection may be higher. |
In addition. a large shallow survey may be highly fruitful in detecting X-ray emission from ultracool dwarfs. | In addition, a large shallow survey may be highly fruitful in detecting X-ray emission from ultracool dwarfs. |
The existing observations indicate that late-M dwarfs are capable of producing flares at a level similar to the saturated X-ray emissionfrom early-M dwarfs. Lyων 107. | The existing observations indicate that late-M dwarfs are capable of producing flares at a level similar to the saturated X-ray emissionfrom early-M dwarfs, $L_X/L_{\rm bol}\sim 10^{-3}$ . |
Such flares can be detected with or to a distance of ~50 pe in about 10 ks. | Such flares can be detected with or to a distance of $\sim 50$ pc in about 10 ks. |
Thus. a survey of about 50 ultracool | Thus, a survey of about $50$ ultracool |
ol this non-gaussian distribution is shown in Figure {,, | of this non-gaussian distribution is shown in Figure \ref{BAO_pdf}. |
An important component of the full posterior is that it must be correctly normalized: fP(@|d)dé=1. | An important component of the full posterior is that it must be correctly normalized: $\int P(\theta | d) d\theta = 1$. |
Assuming that the prior is much wider than the measurement. near (he peak of the likelihood. Figure (2)) shows how the errorbars grow as one decreases the detection probability D,,;,,. | Assuming that the prior is much wider than the measurement, $P(\theta | d, detected) \gg P(\theta)$ near the peak of the likelihood, Figure \ref{BAO_sigma}) ) shows how the errorbars grow as one decreases the detection probability $ P_{detect}$. |
This is done by finding the 68.3%. 95.57. and 99.7% regions. using the normalization D,«l1 for a gaussian distribution. | This is done by finding the $68.3\%$ , $95.5\%$, and $99.7\%$ regions, using the normalization $P_{detect} < 1$ for a gaussian distribution. |
Let us look at the example of 2,4,=0.99 (Figure 1)). | Let us look at the example of $P_{detect} = 0.99$ (Figure \ref{BAO_pdf}) ). |
Although the 1 and 2—6 error bars are essentially unaffected. these are of little interest in parameter estimation since adv real conclusions must be supported at the 99.7% (30) level or better. | Although the $1$ and $2-\sigma$ error bars are essentially unaffected, these are of little interest in parameter estimation since any real conclusions must be supported at the $\%$ $3\sigma$ ) level or better. |
At this level. Figure (2)) shows that there is a dramatic change: (here are constraints on the parameter al all at this significance. despite the likelihood (0|d.detected) possibly claiming wonderful constraints. | At this level, Figure \ref{BAO_sigma}) ) shows that there is a dramatic change: there are constraints on the parameter at all at this significance, despite the likelihood $P(\theta | d, detected)$ possibly claiming wonderful constraints. |
What is happening is (hat one is transitioning from the likelihood to the prior when one moves sufficiently. far [rom the maximum likelihood value such that: since (he prior should vary will 9 much less rapidly than the likelihood. there is a point al which one's constraints are actually driven by the prior. not the data. | What is happening is that one is transitioning from the likelihood to the prior when one moves sufficiently far from the maximum likelihood value such that: Since the prior should vary with $\theta$ much less rapidly than the likelihood, there is a point at which one's constraints are actually driven by the prior, not the data. |
The basic concept is simple: one cannot extrapolate data bevond its realm of validity. as illustrated by (he figures. | The basic concept is simple: one cannot extrapolate data beyond its realm of validity, as illustrated by the figures. |
We can go [further and show how the non-gaussini sting can radically change even 1—c errors when combining constraints [rom different measurements. | We can go further and show how the non-gaussian sting can radically change even $1-\sigma$ errors when combining constraints from different measurements. |
In Fig. (3)) | In Fig. \ref{BAO_comb}) ) |
we show a tov model with (wo independent measurements of (wo parameters (D4.Ds)=(0.3.0.3)+(0.1.0.15) and (Dy.Do)=(0.7.0.7)E(0.1.0.15). whieh are cliflerent due to some unaccounted for svstematic error. | we show a toy model with two independent measurements of two parameters $(D_1,D_2) = (0.3,0.3) \pm (0.1,0.15)$ and $(D_1,D_2)=(0.7,0.7)\pm (0.1,0.15)$, which are different due to some unaccounted for systematic error. |
Furthermore. each measurement is based on a 3.66 detection. which causes a slight expansion of 1 and 26 constraints (blue solid contours. corresponding {ο AA?= 2.3 and 6) with respect to the gaussian constraints (black dotted contours). as described in thisfe/fer (see Equation —below). | Furthermore, each measurement is based on a $\sigma$ detection, which causes a slight expansion of 1 and $\sigma$ constraints (blue solid contours, corresponding to $\Delta\chi^2=$ 2.3 and 6) with respect to the gaussian constraints (black dotted contours), as described in this (see Equation \ref{dchi} below). |
However. as we see on the second. panel. combining the two constraints leads to radically different posteriors. with and without the assumption of gaussianitv. even the non-gaussian lo combined errors are much bigger than (he gaussian ones. illustrating that the effects need not only limited be limited to >30 results. | However, as we see on the second panel, combining the two constraints leads to radically different posteriors, with and without the assumption of gaussianity, even the non-gaussian $\sigma$ combined errors are much bigger than the gaussian ones, illustrating that the effects need not only limited be limited to $>3\sigma$ results. |
The frequenüst analog of this statement is (hat. for a finite detection confidence level. the function A?(9|d) should asvmptote to a plateau far away [rom its mininmm. rather than growing indefinitely, | The frequentist analog of this statement is that, for a finite detection confidence level, the function $\chi^2(\theta | d)$ should asymptote to a plateau far away from its minimum, rather than growing indefinitely. |
The maximun dillerence between the plateau and theminimum is roughly the square of the signal-to-noise of the detection. S/N. | The maximum difference between the plateau and theminimum is roughly the square of the signal-to-noise of the detection, $S/N$ . |
Therefore. (he gaussian (quadratic) approximation to the likelihood (4?) breaks down when νοBallas)ονο (S/N). | Therefore, the gaussian (quadratic) approximation to the likelihood $\chi^2$ ) breaks down when $\Delta\chi^2_{\rm gauss} \gtrsim (S/N)^2$ . |
2004). essentially packing the entire molecular gas supply of two eas-rich spirals within a few hundred parsecs. | 2004), essentially packing the entire molecular gas supply of two gas-rich spirals within a few hundred parsecs. |
Their emergent Ut luminosities will then be less than proportional to their dust. mass reservoirs. and can be further absorbed by outer dust distributions that are optically thick at UR/far-LR wavelengths (Condon et al. | Their emergent IR luminosities will then be less than proportional to their dust mass reservoirs, and can be further absorbed by outer dust distributions that are optically thick at IR/far-IR wavelengths (Condon et al. |
1991: Solomon οἱ al. | 1991; Solomon et al. |
1997). | 1997). |
For such verv dense gas disks (he dust mass estimates using submnm fluxes and teniperatures obtained from global SED fits (e.g. Dunne at al. | For such very dense gas disks the dust mass estimates using submm fluxes and temperatures obtained from global SED fits (e.g. Dunne at al. |
2000) can be significant. since an outer cooler dust distribution. while not containing the bulk of the mass. dominates most of the observed SED (see also 4.2). | 2000) can be significant since an outer cooler dust distribution, while not containing the bulk of the mass, dominates most of the observed SED (see also 4.2). |
This may explain why subnumn interferometric imagine can recover a Galactic value for. M(II3)/Mau4 in ULIBRGs (e.g. Wilson et al. | This may explain why submm interferometric imaging can recover a Galactic value for $_2$ $_{\rm dust}$ in ULIRGs (e.g. Wilson et al. |
2008) while global SED fits can vield significantly lower ones (Table 3: Dunne Eales20011. | 2008) while global SED fits can yield significantly lower ones (Table 3; Dunne Eales. |
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