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In our analysis. the central surface density (essentially the ML) will remain | In our analysis, the central surface density (essentially the M/L) will remain |
In (his section. we compare the Modified Polvivopic Cardassian Model of Eq. (2)) | In this section, we compare the Modified Polytropic Cardassian Model of Eq. \ref{eq:FRW}) ) |
with current supernovae and cosmüc microwave background (CMD) data. | with current supernovae and cosmic microwave background (CMB) data. |
We will see that the existing data can be well fit for several choices of the parameters » aud q. | We will see that the existing data can be well fit for several choices of the parameters $n$ and $q$. |
In a smooth Friedmann-BRobertson-Walker (FRW) universe. the metric is given by ds?=a(Ly(dr?/(A=kr?)+(dé?sin?0dc? )]. where a(1) is the cosmic scale factor. and / is the global curvature parameter. | In a smooth Friedmann-Robertson-Walker (FRW) universe, the metric is given by $ds^2=dt^2-a^2(t)[dr^2/(1-kr^2)+r^2 (d\theta^2
+\sin^2\theta \,d\phi^2)]$ , where $a(t)$ is the cosmic scale factor, and $k$ is the global curvature parameter. |
The comoving distance r is given by (Weinberg1972) Lem IQ,|E2. οι... pagο... Ete)TÉu-— CLες)MEAN+ TES7. where 0,—1—QuisOx. and S(x)esinh((r). 2emt,xo =x. Όση KG,og —sn(Cre). 2em&2,«04 | The comoving distance $r$ is given by \citep{Weinberg72}
1cm | _k, _X, _0^zdz' E(z') (1+z')^3+ _k(1+z')^2, where $\Omega_k = 1-\Omega_m^{obs}- \Omega_X$, and (x), 2cm _k>0 = x, 3cm _k=0 (x), 2cm _k<0. |
6 The angular diameter distance is given by d4(z)=r(z)/(1+2). and the luminosity clistance is given by dp(2)=(1+z)r(z). | The angular diameter distance is given by $d_A(z)=r(z)/(1+z)$, and the luminosity distance is given by $d_L(z)=(1+z) r(z)$. |
The distance modulus for a standard candle at recshilt z is slmmAb=Sslogle FH )+25. | The distance modulus for a standard candle at redshift $z$ is _p m-M= ( )+25, |
The distance modulus for a standard candle at recshilt z is slmmAb=Sslogle FH )+25.( | The distance modulus for a standard candle at redshift $z$ is _p m-M= ( )+25, |
The distance modulus for a standard candle at recshilt z is slmmAb=Sslogle FH )+25.(1 | The distance modulus for a standard candle at redshift $z$ is _p m-M= ( )+25, |
The distance modulus for a standard candle at recshilt z is slmmAb=Sslogle FH )+25.(17 | The distance modulus for a standard candle at redshift $z$ is _p m-M= ( )+25, |
The distance modulus for a standard candle at recshilt z is slmmAb=Sslogle FH )+25.(17) | The distance modulus for a standard candle at redshift $z$ is _p m-M= ( )+25, |
planetary migration scenario assumes the planetesimals to be initially closer to the planet. | planetary migration scenario assumes the planetesimals to be initially closer to the planet. |
If we started with planetesimals at a larger distance, we would have expected to see, as in the P-R drag scenario, more differences in the resonant structures for planets of different masses. | If we started with planetesimals at a larger distance, we would have expected to see, as in the P-R drag scenario, more differences in the resonant structures for planets of different masses. |
Our results show that, with the forced planetary migration scenario, it is easy to distinguish a planet on a circular orbit from another on a low-eccentricity orbit, except for very low mass planets or very massive planets, because the resonant structures are drastically different. | Our results show that, with the forced planetary migration scenario, it is easy to distinguish a planet on a circular orbit from another on a low-eccentricity orbit, except for very low mass planets or very massive planets, because the resonant structures are drastically different. |
Constraining the planetary mass is more difficult than in the P-R drag scenario and only an order of magnitude can be expected. | Constraining the planetary mass is more difficult than in the P-R drag scenario and only an order of magnitude can be expected. |
? used this scenario to reproduce Vega disk observations at submillimetric wavelengths (?).. | \citet{2003ApJ...598.1321W} used this scenario to reproduce Vega disk observations at submillimetric wavelengths \citep{1998Natur.392..788H}. |
We must take into account that, with the large SCUBA PSF, only the two major clumps can be observed. | We must take into account that, with the large SCUBA PSF, only the two major clumps can be observed. |
However, this is enough to distinguish between our three planet mass examples, at least for a migration rate of 0.5 AU Myr!: It is thus possible to obtain an estimate of the planetary mass. | However, this is enough to distinguish between our three planet mass examples, at least for a migration rate of $0.5$ AU $^{-1}$: It is thus possible to obtain an estimate of the planetary mass. |
The situation is well summarized by Figure 11 of ?.. | The situation is well summarized by Figure 11 of \citet{2003ApJ...598.1321W}. |
It defines several regions in the (planetary mass, migration rate) parameter space that can be observationally distinguished from each other, but inside each region a wide range of planetary masses is possible. | It defines several regions in the (planetary mass, migration rate) parameter space that can be observationally distinguished from each other, but inside each region a wide range of planetary masses is possible. |
Contradictions however appear between our simulation results and this previous study. | Contradictions however appear between our simulation results and this previous study. |
The asymmetry in the emission of the two observed clumps was interpreted as the migration of a Neptune mass planet by ?.. | The asymmetry in the emission of the two observed clumps was interpreted as the migration of a Neptune mass planet by \citet{2003ApJ...598.1321W}. |
In his model, a Neptune can trap planetesimals in the 3:2 and 2:1 resonances and generate two asymmetric clumps, like a Saturn mass planet in our simulations. | In his model, a Neptune can trap planetesimals in the $3$ $2$ and $2$ $1$ resonances and generate two asymmetric clumps, like a Saturn mass planet in our simulations. |
With our numerical model, we have found that a Neptune mass planet cannot trap planetesimals in the 2:1 resonance, but only in the 3:2 resonance: the two clumps are thus symmetric and cannot reproduce the Vega disk. | With our numerical model, we have found that a Neptune mass planet cannot trap planetesimals in the $2$ $1$ resonance, but only in the $3$ $2$ resonance: the two clumps are thus symmetric and cannot reproduce the Vega disk. |
A Neptune mass planet at a migration rate of about 0.5 AU Myr7! lies at the sharp transition between a 0 and a 100% trapping probability (?,Fig.4a).. | A Neptune mass planet at a migration rate of about $0.5$ AU $^{-1}$ lies at the sharp transition between a $0$ and a $100\% $ trapping probability \citep[][Fig. 4a]{2003ApJ...598.1321W}. |
A small change in the planetary mass or the migration rate in this configuration produces a large modification in the population of this resonance. | A small change in the planetary mass or the migration rate in this configuration produces a large modification in the population of this resonance. |
As ? uses a scaling law to predict the trapping probability, differences between our results may be explained by the approximation of this scaling law. | As \citet{2003ApJ...598.1321W} uses a scaling law to predict the trapping probability, differences between our results may be explained by the approximation of this scaling law. |
Nevertheless, the 2:1 resonance has an interesting behavior in the Neptune mass planet case: it perturbs all the | Nevertheless, the $2$ $1$ resonance has an interesting behavior in the Neptune mass planet case: it perturbs all the |
focusing on narrow regions where the spectral feature appears to be hiehlv suppressed. indicating a reduced concentration of micron-sized particles. | focusing on narrow regions where the spectral feature appears to be highly suppressed, indicating a reduced concentration of micron-sized particles. |
These features are likely a subset of (he compact optical depth enhancements identified in both Cassini images (Murray.e£af.2008;Beurleefa£.2010) and UVIS stellar occultations (Espositoefaf.2008:Meinkeefaf 2010). | These features are likely a subset of the compact optical depth enhancements identified in both Cassini images \citep{Murray08, Beurle10} and UVIS stellar occultations \citep{Esposito08, Meinke10}. |
. Finally. we discuss possible interpretations of the observed spectral variations in terms of spatially-varving particle densities aud velocity dispersions within these dusty rings. | Finally, we discuss possible interpretations of the observed spectral variations in terms of spatially-varying particle densities and velocity dispersions within these dusty rings. |
VIMS is most often. used to produce spatially-resolved reflectance spectra. οἱ planetary (targets. | VIMS is most often used to produce spatially-resolved reflectance spectra of planetary targets. |
ILowever. VIMS is a flexible instrument that can also operate in an occultation mode (Browne£al.2004). | However, VIMS is a flexible instrument that can also operate in an occultation mode \citep{Brown04}. |
. In (his mode. the imaging capabilities are disabled. the short-wavelength VIS channel of the instrument is turned off and the Ih channel obtains a series of spectra from a sinele pixel targeted. al a star. | In this mode, the imaging capabilities are disabled, the short-wavelength VIS channel of the instrument is turned off and the IR channel obtains a series of spectra from a single pixel targeted at a star. |
The raw spectra are composed of 248 measurements of the stellar brightness between 0.85 and 5.0 jan with a typical resolution of 0.016 yan Gn occultation mode. eight channels are used to encode timing data). | The raw spectra are composed of 248 measurements of the stellar brightness between 0.85 and 5.0 $\mu$ m with a typical resolution of 0.016 $\mu$ m (in occultation mode, eight channels are used to encode timing data). |
However. to save on data volume. these dala are usually co-acdded prior to transmission to earth. producing "summed" spectra consisting of 31 spectral measurements with a (vpical resolution of 0.1350n. The raw data used in this analvsis are the uncalibrated Data Numbers (DN) returned by the instrument. | However, to save on data volume, these data are usually co-added prior to transmission to earth, producing “summed” spectra consisting of 31 spectral measurements with a typical resolution of $\mu$ m. The raw data used in this analysis are the uncalibrated Data Numbers (DN) returned by the instrument. |
While these DN are linear measures of the photon flux 2004).. no attempt is made to convert these data to absolute fInxes here. although a nean insirumental thermal background spectrum has been subtracted Irom all the spectra for each occultation. | While these DN are linear measures of the photon flux \citep{Brown04}, no attempt is made to convert these data to absolute fluxes here, although a mean instrumental thermal background spectrum has been subtracted from all the spectra for each occultation. |
A precise time stamp is appended to every spectrum to acilitate reconstruction of the occultation geometry. | A precise time stamp is appended to every spectrum to facilitate reconstruction of the occultation geometry. |
Each occultation is geometrically navigated based on the positions of the star (obtained from the Hipparcos catalog. and adjusted (ο account for proper motion aud parallax al Saturn) and the position of the spacecralt derived from the appropriate SPICE kernels. | Each occultation is geometrically navigated based on the positions of the star (obtained from the Hipparcos catalog, and adjusted to account for proper motion and parallax at Saturn) and the position of the spacecraft derived from the appropriate SPICE kernels. |
This information was used to predict the apparent position (radius and inerGal longitude) of the star in Saturn's ring plane as a function of time in a planetocentric reference frame. taking into account stellar aberration. | This information was used to predict the apparent position (radius and inertial longitude) of the star in Saturn's ring plane as a function of time in a planetocentric reference frame, taking into account stellar aberration. |
In. nearly all cases. this estimate of the occultation geometry was confirmed to be accurate to within a few kilometers using (he known radii of nearly circular gap edges in the outer A Ring from (Frenchefal.1993). | In nearly all cases, this estimate of the occultation geometry was confirmed to be accurate to within a few kilometers using the known radii of nearly circular gap edges in the outer A Ring from \citep{French93}. |
. The exceptions were the low-inclination stars ο Ceti and ὁ Virginis. for which features could be tens of kilometers away [rom their nominal positions. | The exceptions were the low-inclination stars $o$ Ceti and $\delta$ Virginis, for which features could be tens of kilometers away from their nominal positions. |
In these cases. the fiducial position of Saturn's pole was adjusted slightly (bv al most 0.0157) to bring these cuts into alignment. with the other occultations. ( | In these cases, the fiducial position of Saturn's pole was adjusted slightly (by at most $^\circ$ ) to bring these cuts into alignment with the other occultations. ( |
Note that such corrections were not possible [or the Rev 12 ο Ceti occultation. which | Note that such corrections were not possible for the Rev 12 $o$ Ceti occultation, which |
electrons with energiesoO of 8 and 25 GeV traversinge thin egold and carbon targetse have been presented in Anthonyetal.(1995). | electrons with energies of $8$ and $25$ GeV traversing thin gold and carbon targets have been presented in \citet{An95}. |
. The suppression of bremsstralilung predicted by the LPAI theory is correct to within 5%. | The suppression of bremsstrahlung predicted by the LPM theory is correct to within $5\%$. |
The main parameters defining radiation processes in a medium are (he coherence length fo (he mean [ree path of a fast particle in matter /.. the radiation length Ny and the thickness of the target L (AkhiezerandShul'ga(1996) and references therein). | The main parameters defining radiation processes in a medium are the coherence length $l_c$, the mean free path of a fast particle in matter $l_{e}$, the radiation length $X_{0}$ and the thickness of the target $L$ \citet{AkSh96} and references therein). |
The coherence length is the distance along the particle momentum where interference effects during the radiation process are significant. | The coherence length is the distance along the particle momentum where interference effects during the radiation process are significant. |
/. rapidlv grows wilh an increase of the particle enerev and al hieh energies it can be macroscopic. | $l_c$ rapidly grows with an increase of the particle energy and at high energies it can be macroscopic. |
Η the energy w of the bremsstrahlung photon produced by an ultra-relativistic particle of energv. E satisfies the condition w<<e. then the average angle ϐ between the incident particle and the photon is small. &,2me?/e. | If the energy $\omega $ of the bremsstrahlung photon produced by an ultra-relativistic particle of energy $E$ satisfies the condition $\omega <<\epsilon
$, then the average angle $\theta $ between the incident particle and the photon is small, $\theta _c\approx mc^2/\epsilon $. |
The average angle between (he scattered. particles is smaller still. | The average angle between the scattered particles is smaller still. |
Neglecüng (hese angles. the longitudinal momentiun transfer between the interacting particles is pj~w/25?e. where 5=e/mc?. | Neglecting these angles, the longitudinal momentum transfer between the interacting particles is $p_{\left| {}\right| }\simeq
\omega /2\gamma ^{2}c$, where $\gamma =\epsilon /mc^{2}$. |
The uncertainty principle then requires (hat (he spatial position of the bremsstrahlung process has a longitudinal uncertainty of /.—h/pyzzολο. | The uncertainty principle then requires that the spatial position of the bremsstrahlung process has a longitudinal uncertainty of $l_{c}=\hbar /p_{\left| {}\right| }\approx 2\hbar c\gamma
^{2}/\omega $. |
The coherence length rapidly grows wilh an increase of the particle energy. and al high energies can be macroscopic. | The coherence length rapidly grows with an increase of the particle energy, and at high energies can be macroscopic. |
In alternate language. (he particle aud. photon slowly split apart over the distance /.. | In alternate language, the particle and photon slowly split apart over the distance $l_{c} $. |
In a sufficiently dense medium the mean [ree path of the incident particle is much smaller than /.. so a parücle will interact while (raversine the region /.. | In a sufficiently dense medium the mean free path of the incident particle is much smaller than $l_{c}$, so a particle will interact while traversing the region $l_{c}$ . |
Dremsstirahlung is suppressed when the mean square multiple scattering angle over the distance /.. is greater than or equal to 67 (AkhiezerandShul'ga(1996):lein(1999) and references (herein). | Bremsstrahlung is suppressed when the mean square multiple scattering angle over the distance $l_c$, is greater than or equal to $\theta _{k}^{2}$ \citet{AkSh96,Kl99}
and references therein). |
[ere is the radiation length and In Eq. (12)) | Here is the radiation length and In Eq. \ref{x0}) ) |
r,=ες> is the classical radius of the particle (the electron). | $r_{e}=e^{2}/mc^{2}$ is the classical radius of the particle (the electron). |
The effect of multiple suppression on the total radiation emission of the charged particle. d1/duw. can be obtained. for iw<<ee/e?Xy. in the form (Ixlein(1999). and references therein) Thus. multiple scattering results in a decrease of the coherence length. leading. in (urn. to radiation suppression. | The effect of multiple suppression on the total radiation emission of the charged particle, $dI/d\omega
$, can be obtained, for $\omega <<\epsilon _{s}^{2}c/\epsilon
^{2}X_{0}$, in the form \citet{Kl99} and references therein) Thus, multiple scattering results in a decrease of the coherence length, leading, in turn, to radiation suppression. |
The effect appears in (lie case when (he mean-square angle ofthe | The effect appears in the case when the mean-square angle ofthe |
favours rates significantly smaller. particularly at the beginning of the AGB phase. with cillerences up to ~1.5 dex compared to the others. | favours rates significantly smaller, particularly at the beginning of the AGB phase, with differences up to $\sim~1.5$ dex compared to the others. |
A smaller. dillerence exists between the Blocker(1995) and the Weiss&Fergu-son(2009) recipes. the latter being higher by ~0.20.3 dex during most of the evolution. | A smaller difference exists between the \citet{blo} and the \citet{achim} recipes, the latter being higher by $\sim~0.2-0.3$ dex during most of the evolution. |
In the early AGB the Weiss&Ferguson(2009) treatment predicts rates of the order of 10"10.9AL. fve. which are. however. not sullicient to diminish the tota number of LPs so as to recover their original result. (6). | In the early AGB the \citet{achim} treatment predicts rates of the order of $10^{-7}-10^{-6}~M_{\odot}$ /yr, which are, however, not sufficient to diminish the total number of TPs so as to recover their original result $6$ ). |
Dhis suggests that the main reasons for the smaller number of TPs in the Weiss&Ferguson(2009) investigation compare to our models is an earlier transition to the C-star stage. | This suggests that the main reasons for the smaller number of TPs in the \citet{achim} investigation compared to our models is an earlier transition to the C-star stage. |
‘This dillerence is due to two reasons: a) The mocdelling of convective overshoot [rom the bottom of the envelope (see their equation. 2). based on an exponential decay of the dilfusive coefficient. with the same e-folding clistance usec for main-sequences width fitting. | This difference is due to two reasons: a) The modelling of convective overshoot from the bottom of the envelope (see their equation 2), based on an exponential decay of the diffusive coefficient, with the same e-folding distance used for main-sequences width fitting. |
This approach determines a much deeper extension of EDU when compared with our models (where no overshoot is assumed). and with those by Cristalloetal.(2009).. who adopt an exponential decay. of the convective velocity [rom the inner border of the external mantle. with an e-folding distance tuned in order to allow the synthesis of a given amount of £C available for neutron-captures. | This approach determines a much deeper extension of TDU when compared with our models (where no overshoot is assumed), and with those by \citet{sergio2}, who adopt an exponential decay of the convective velocity from the inner border of the external mantle, with an e-folding distance tuned in order to allow the synthesis of a given amount of $^{13}$ C available for neutron-captures. |
This amount of extra-mixing is also the plausible reason for the smaller number of TIs experienced. by. the Cristalloetal.(2009). model compared to our 806€. and to the different final core-masses (0.67AL... to be compared to τοAM. b) the convective overshoot from the He-burning shell. adopted by. Weiss&Ferguson(2009).. and. neglected here and in Cristalloetal.(2009)... which. as described bv Llerwig&Austin(2004).. renders the pulses stronger. | This amount of extra-mixing is also the plausible reason for the smaller number of TPs experienced by the \citet{sergio2}
model compared to our S06C, and to the different final core-masses $0.67~M_{\odot}$, to be compared to $0.70~M_{\odot}$ ); b) the convective overshoot from the He-burning shell, adopted by \citet{achim}, and neglected here and in \citet{sergio2}, which, as described by \citet{falk3}, renders the pulses stronger. |
The models presented here. though including no overshoot. are much more similar to those by Cristalloetal.(2009) in terms of duration of the whole AGB phase and of the chemistry of the ejecta. | The models presented here, though including no overshoot, are much more similar to those by \citet{sergio2} in terms of duration of the whole AGB phase and of the chemistry of the ejecta. |
A ctailecl investigation of the ΕΙ phenomenology. and the possible effects. of assuming some convective overshoot from the bottom of the convective envelope. is bevond the scope ofthe present paper: the interested reader may find in Llerwig(2000).. Llerwig (2005).. Mowlavi(1999) and Stranieroctal.(1997) a Cull discussion on the kev issues relevant to address this problem. | A detailed investigation of the TDU phenomenology, and the possible effects of assuming some convective overshoot from the bottom of the convective envelope, is beyond the scope of the present paper; the interested reader may find in \citet{falk}, \citet{falk2}, \citet{mowlavi} and \citet{oscar} a full discussion on the key issues relevant to address this problem. |
Anvhow. the present study. clearly illustrates that the AGB evolution is the product. of a complex. interplay between various processes: the treatment of the extra-mixing. for instance. has a remarkable impact not only on the chemical but also on the physical. properties of these stars. | Anyhow, the present study clearly illustrates that the AGB evolution is the product of a complex interplay between various processes: the treatment of the extra-mixing, for instance, has a remarkable impact not only on the chemical but also on the physical properties of these stars. |
In view of improving the robustness of the results provided by AGB modelling (or at least to quantify the uncertainty range). we have investigated the sensitivity of the main physical ancl chemical properties to the choices inherent the input macro- and micro-physies. | In view of improving the robustness of the results provided by AGB modelling (or at least to quantify the uncertainty range), we have investigated the sensitivity of the main physical and chemical properties to the choices inherent the input macro- and micro-physics. |
We [ind a threshold. mass. that is AJ~3.5M. for Z-—0.001 in the present analysis. (note that it also depends on the assumed overshooting from the central regions during the core L- and Le-burnine). above which the results are little sensitive to the opacity treatment. because LBB prevents the surface C/O ratio to exceed unity. | We find a threshold mass, that is $M\simeq 3.5~M_{\odot}$ for $Z=0.001$ in the present analysis, (note that it also depends on the assumed overshooting from the central regions during the core H- and He-burning), above which the results are little sensitive to the opacity treatment, because HBB prevents the surface C/O ratio to exceed unity. |
In this range of masses mass loss modelling plavs a crucial role. | In this range of masses mass loss modelling plays a crucial role. |
Models calculated with a treatment of the mass loss only mildly. dependent on the luminosity. such as the classic Vassiliacis&Wood(1993). or the Stranieroetal.(2006) recipe used here. precict smaller rates during mos of the AGB phase. thus more “PPs are experienced. | Models calculated with a treatment of the mass loss only mildly dependent on the luminosity, such as the classic \citet{VW93}
or the \citet{oscar2} recipe used here, predict smaller rates during most of the AGB phase, thus more TPs are experienced. |
This also favours the growth of the luminosity ancl temperature at the bottom of the convective zone. and a more acvancec LBB nucleosynthesis: the vields will contain the signature of proton-capture nucleosvnthesis. c.g. lower carbon ane oxvecn abuncanees. and higher nitrogen content. | This also favours the growth of the luminosity and temperature at the bottom of the convective zone, and a more advanced HBB nucleosynthesis: the yields will contain the signature of proton-capture nucleosynthesis, e.g. lower carbon and oxygen abundances, and higher nitrogen content. |
However. rw general patterns of the C-N and O-Na relations. anc vw very small increase in the overall CNO abundance. are almost independent of the opacity anc mass-loss treatment: we expect convection modelling plavs a relevant role here. in agreement with the analysis by Ventura&D'Anton | However, the general patterns of the C-N and O-Na relations, and the very small increase in the overall CNO abundance, are almost independent of the opacity and mass-loss treatment: we expect convection modelling plays a relevant role here, in agreement with the analysis by \citet{paolo4}. . |
a For masses smaller than 3.5A. the interplay between opacity ancl mass-loss treatment is more tricky. because 1e lack of UBB favours the increase in surface carbon. | For masses smaller than $3.5~M_{\odot}$ the interplay between opacity and mass-loss treatment is more tricky, because the lack of HBB favours the increase in surface carbon. |
Generally speaking. we confirm the results by 2007).. Cristalloetal.(2008.2009). and Weiss&Ferguson(2009):: the use of the correct opacities in the Iow-T regime specds-up the loss of the stellar envelope. via the strong winds associated to the general expansion of the structure when the surface C/O ratio exceeds unity. | Generally speaking, we confirm the results by \citet{paola02, paola07}, \citet{sergio, sergio2} and \citet{achim}: the use of the correct opacities in the low-T regime speeds-up the loss of the stellar envelope, via the strong winds associated to the general expansion of the structure when the surface C/O ratio exceeds unity. |
We expect in this case a smaller. number of PPPs. and a less advanced nucleosvnthesis (i£ any) at the bottom of the convective envelope. | We expect in this case a smaller number of TPs, and a less advanced nucleosynthesis (if any) at the bottom of the convective envelope. |
Alocdels caleulated with a miass-loss rate steeply dependent on the luminosity show a more homogeneous behaviour. whereas when a milder dependeney of AL with luminosity. c.g. a relationship that relates A. to the stellar period. is adopted. many physical properties. c.g. core-mass. temperature at the bottom of the convective zone. total number of thermal pulses experienced. follow a completely different behaviour. | Models calculated with a mass-loss rate steeply dependent on the luminosity show a more homogeneous behaviour, whereas when a milder dependency of $\dot M$ with luminosity, e.g. a relationship that relates $\dot M$ to the stellar period, is adopted, many physical properties, e.g. core-mass, temperature at the bottom of the convective zone, total number of thermal pulses experienced, follow a completely different behaviour. |
In this case the quenching of HB determined. by the use of the C-rich opacities. predicted. by Alavigo(2007).. is confirmed. being a common feature of all he models with LS3.0M... | In this case the quenching of HBB determined by the use of the C-rich opacities, predicted by \citet{paola07}, is confirmed, being a common feature of all the models with $1.8-3.0~M_{\odot}$. |
A narrower range of masses (around ~3 AL.) is otherwise allected by this uncertainty when an ellicient. mass-loss prescription. like the Dlócker(1995) formula. is adopted. | A narrower range of masses (around $\sim~3~M_{\odot}$ ) is otherwise affected by this uncertainty when an efficient mass-loss prescription, like the \citet{blo}
formula, is adopted. |
Contrary to the models experiencing LBB. the vields rom lower mass stars are highly uncertain and moclel-dependent: the average C/Fe] in the ejecta shows up an overall uncertainty of 0.7 dex. though this dillerence reduces Oo 0.3.0.4 dex if mocels calculated with the same opacities are compared. | Contrary to the models experiencing HBB, the yields from lower mass stars are highly uncertain and model-dependent: the average [C/Fe] in the ejecta shows up an overall uncertainty of $0.7$ dex, though this difference reduces to $\sim 0.3-0.4$ dex if models calculated with the same opacities are compared. |
The situation for nitrogen. and sodium is more extreme. as they are particularly sensitive to he ignition of LIBB. | The situation for nitrogen and sodium is more extreme, as they are particularly sensitive to the ignition of HBB. |
Ehe dilferences in ΑΟ and. NaEc] can reach ~2 dex. and even when comparing mocoels with the same assumptions for the opacity treatment. discrepancies of one order of magnitude still persist. | The differences in [N/Fe] and [Na/Fe] can reach $\sim 2$ dex, and even when comparing models with the same assumptions for the opacity treatment, discrepancies of one order of magnitude still persist. |
The oxygen vields appear. instead. more stable. | The oxygen yields appear, instead, more stable. |
These trends should. be considered: representative of AGB models with a low metal content (Z 0.001) while they need to be further exploredat larger mictallicities i.c. including those characteristic of the LAIC (Ventura et al.. | These trends should be considered representative of AGB models with a low metal content $Z \sim 0.001$ ) while they need to be further exploredat larger metallicities i.e. including those characteristic of the LMC (Ventura et al., |
in preparation). | in preparation). |
Moreover. in the present. unresolved. scenario | Moreover, in the present unresolved scenario |
the quark core appears. the star only spins-up with angular momentum loss. for the rest of its lifetime. | the quark core appears, the star only spins-up with angular momentum loss, for the rest of its lifetime. |
To summarize. for the particular EOS that we have studied in this paper. normal pulsars do not feature a spin-up era during phase transition, while supramassive pulsars only spin up after the phase transition. | To summarize, for the particular EOS that we have studied in this paper, normal pulsars do not feature a spin-up era during phase transition, while supramassive pulsars only spin up after the phase transition. |
The limiting sequence that divides normal from supramassive pulsars shows a very small era of spin-up and appears to be the dividing line between stars that can spin up and stars that only spin down. | The limiting sequence that divides normal from supramassive pulsars shows a very small era of spin-up and appears to be the dividing line between stars that can spin up and stars that only spin down. |
In the previous sections we demonstrated that the refined tabulated/analytic EOS produces a constant baryonic mass sequence with equilibrium properties that are in sharp contrast to the equilibrium properties computed on the basis of the original tabulated EOS. | In the previous sections we demonstrated that the refined tabulated/analytic EOS produces a constant baryonic mass sequence with equilibrium properties that are in sharp contrast to the equilibrium properties computed on the basis of the original tabulated EOS. |
In this section. we will use an independent check to verify that the equilibrium properties obtained with the refined EOS are indeed the physically acceptable ones. while the equilibrium sequence obtained with the tabulated EOS suffers from numerical errors. | In this section we will use an independent check to verify that the equilibrium properties obtained with the refined EOS are indeed the physically acceptable ones, while the equilibrium sequence obtained with the tabulated EOS suffers from numerical errors. |
The independent check of the accuracy of the computed equilibrium properties is provided by the use of a relation that was first proved by Ostriker and Gunn (1969) in Newtonian theory and then derived in general relativity by Bardeen (1970). | The independent check of the accuracy of the computed equilibrium properties is provided by the use of a relation that was first proved by Ostriker and Gunn (1969) in Newtonian theory and then derived in general relativity by Bardeen (1970). |
Along a sequence of uniformly rotating models of constant baryonic mass. changes in M and J are related by which can be regarded as an expression of the first law of thermodynamies for such sequences. | Along a sequence of uniformly rotating models of constant baryonic mass, changes in $M$ and $J$ are related by which can be regarded as an expression of the first law of thermodynamics for such sequences. |
We have evaluated the above relation numerically along the obtained sequences of equilibrium models (using sufficiently many individual models to ensure an accurate evaluation of the numerical derivative). | We have evaluated the above relation numerically along the obtained sequences of equilibrium models (using sufficiently many individual models to ensure an accurate evaluation of the numerical derivative). |
We define an error indicator. A. as The evaluation of X along the evolutionary sequence of My=1.551M. (Fig. 7). | We define an error indicator, $\lambda$, as The evaluation of $\lambda$ along the evolutionary sequence of $M_0=1.551
M_\odot$ (Fig. \ref{lambda}) ), |
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