Evolutionary sequences of stellar models with new radiative opacities. VI.

Astronomy and Astrophysics Supplement Series, Jul 2018

We present a large grid of stellar evolutionary models with the initial chemical composition . These tracks are conceived to extend the grid of stellar models described in the previous papers of this series, and are computed with the new radiative opacities by Iglesias et al. (1992) and convective overshoot. The tracks span the range of initial masses from to , and extend from the zero age main sequence (ZAMS) till very advanced evolutionary phases. Specifically, low- and intermediate-mass stars are followed till the beginning of the thermally pulsing regime of the asymptotic red giant branch phase (TP-AGB), while massive stars are followed till the core C-ignition. With respect to previous papers of this series, these models incorporate a number of small modifications in the input physics, particularly on the equation of state, which now incorporates the effect of Coulomb interactions between charged particles. The effect of these modifications is discussed. The corresponding theoretical isochrones are presented.

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Evolutionary sequences of stellar models with new radiative opacities. VI.

Astron. Astrophys. Suppl. Ser. Evolutionary sequences of stellar models with new radiative opacities. VI. Z = 0:0001 L. Girardi 1 2 A. Bressan 0 C. Chiosi 2 G. Bertelli 2 3 E. Nasi 0 0 Astronomical Observatory , Vicolo Osservatorio 5, 35122 Padua , Italy 1 Instituto de F sica, UFRGS , C.P. 15051, 91501-970 Porto Alegre RS , Brazil 2 Department of Astronomy , Vicolo Osservatorio 5, 35122 Padua , Italy 3 Fellow of the National Council of Research CNR | We present a large grid of stellar evolutionary models with the initial chemical composition [Z = 0:0001; Y = 0:23]. These tracks are conceived to extend the grid of stellar models described in the previous papers of this series, and are computed with the new radiative opacities by Iglesias et al. (1992) and convective overshoot. The tracks span the range of initial masses from 0:6 M to 100 M , and extend from the zero age main sequence (ZAMS) till very advanced evolutionary phases. Speci cally, low- and intermediate-mass stars are followed till the beginning of the thermally pulsing regime of the asymptotic red giant branch phase (TP-AGB), while massive stars are followed till the core C-ignition. With respect to previous papers of this series, these models incorporate a number of small modi cations in the input physics, particularly on the equation of state, which now incorporates the e ect of Coulomb interactions between charged particles. The e ect of these modi cations is discussed. The corresponding theoretical isochrones are presented. stars; evolution | stars; interiors | stars; Hertzsprung{Russel (HR) diagram | stars; abundances 1. Introduction This is a sequel of the series of papers aimed at providing a database of stellar models which has to be rstly complete in mass, chemical composition, and major evolutionary stages, and secondly homogeneous as well as updated in its physical ingredients. Let us briefly recall the main characteristics of the stellar tracks so far presented in this series. It started with the study by Alongi et al. (1993) presenting grids with chemical composition [Y = 0:250; Z = 0:008], adopting both the classical and the overshoot schemes for the treatment of convection in central cores and external envelopes. The opacity in use was the classical radiative opacity by Huebner et al. (1977 , thereinafter LAOL). Afterwards, LAOL opacities were superseded by the results of the Livermore group (Rogers & Iglesias 1992; Iglesias et al. 1992, thereinafter OPAL) , and tracks incorporating the new OPAL opacities were presented for the chemical compositions [Y = 0:280; Z = 0:020] (Bressan et al. 1993a) , [Y = 0:230; Z = 0:0004] and [Y = 0:352; Z = 0:05] (Fagotto et al. 1994a), and [Y = 0:240; Z = 0:004] and [Y = 0:250; Z = 0:008] (Fagotto et al. 1994b). An additional grid of stellar tracks with [Y = 0:475; Z = 0:1] and use of the LAOL opacities was calculated by Fagotto et al. (1994c). From all the above models, the grids presented by Bressan et al. (1993a) and Fagotto et al. (1994a, b) perfectly satisfy the request of homogeneity, physical accuracy and coverage of evolutionary stages that was outlined in the rst study by Alongi et al. (1993) and stressed in the subsequent papers. They gave origin to the large set of isochrones calculated by Bertelli et al. (1994), and so far were used for a series of purposes, from the study of CMDs of star clusters and nearby galaxies to the study of the spectrophotometric properties of bulges and elliptical galaxies (e.g. Bressan et al. 1994b; Girardi et al. 1995a). In this paper, we present a grid of stellar models with initial chemical composition [Y = 0:230; Z = 0:0001]. They are primarily conceived to extend, into the lowermetallicity end, the set of stellar tracks available at Padua. Due to a number of improvements on the input physics, these models do not perfectly meet the requirements of homogeneity with the previous grids calculated by Bressan et al. (1993a) and Fagotto et al. (1994a, b). However, it will be shown that these changes in the input physics are not able to modify in a systematic or dramatic way the path described by the models on the HRD. Therefore, the set of isochrones that we calculate and present in this paper can be safely used together with those by Bertelli et al. (1994). The description of the models follows. We put emphasis in the description of the input physics which was improved with respect to the previous papers of this series, and on the particularities of the evolution which arise at this extremely low value for the initial metallicity. As usual the evolutionary tracks are presented sampled in three groups, in which the stars share similar properties. Speci cally, low-mass stars (M M HeF) are those which develop a degenerate helium core and ignite helium through a phase of violent burning episodes (HeliumFlash), intermediate-mass stars are those which avoid core He-flash but develop a highly electron-degenerate carbonoxygen core (M HeF M M up) in which carbon ignites violently, and massive stars (M M up) as those which are able to avoid degeneracy till core C-ignition. 2. Physical ingredients of the models As in the previous papers, the helium content at the limit of low metallicities is assumed to be Y = 0:23. As far as the opacity, the mixing scheme, and the massloss rates are concerned, the physical input is the same as that used by Bressan et al. (1993a) and Fagotto et al. (1994a, b) according to the request of homogeneity as previously claimed. Therefore, we will limit ourselves to briefly recall only the assumptions concerning these ingredients. The radiative opacity is from the Livermore Library (Iglesias et al. 1992) which supersedes the previous OPAL computations (Rogers & Iglesias 1992 and references) because of the inclusion of the spin-orbit interaction in the treatment of Fe atomic data and the adoption of recent measurements of the solar photospheric abundance of Fe. To be more precise, we use the latest (1993) release of OPAL, which di ers from the Iglesias et al. (1992) release of only 1−2% percent, due to the adoption of Grevesse & Noels (1993) solar composition instead of Grevesse (1991). For the outermost regions the radiative opacities are implemented with those of the LAOL and the molecular contribution according to the prescription by Bessell et al. (1989, 1991). We notice that at such low metallicities the contribution from metallic molecules to the opacities is very small. The full network of nuclear reactions which is followed is the same as in Bressan et al. (1993a). Reaction rates are from Caughlan & Fowler (1988), but for: (1) the reactions 17O(p; )14N and 17O(p; γ)18F in which we use the more recent determinations by Landre et al. (1990) ; and (2) 12C( ; γ)16O, which rate was set to be 1.7 times that given by Caughlan & Fowler (1988), as indicated by the study by Weaver & Woosley (1993) on the nucleosynthesis by massive stars. The equation of state (EOS) has been revised. For the interior, the EOS is that of a fully-ionized gas, including electron degeneracy in the way described by Kippenhahn et al. (1965) . We included the e ect of Coulomb interactions between the gas particles at high densities following mainly the formulation by Straniero (1988) . We however adapted this formulation to the general case of a multiplecomponent plasma, thus allowing us to explicitly calculate the Coulomb contribution to the thermodynamical variables at every mesh point of the models. Details are given in the Appendix. For the stellar envelope, thermodynamical quantities are derived from a H-He mixture with ionization equilibrium determined by a simple set of Saha equations (Baker & Kippenhahn 1962). Comparison of the thermodynamical quantities with those derived from the free-energy minimization technique of Mihalas et al. (1990 and references therein) reveals that it is a good approximation as long as the formation of the H2 molecule is not important. This condition is perfectly met in most of the stellar tracks presented in this paper; the only one which could be affected by H2 formation is the 0.6 M at the MS stage. We are presently working on the extension of the present stellar models to very-low masses (Girardi et al. 1995b), and anticipate that the e ect of the H2 molecule on the MS position of a 0.6 M star is already negligible. As already mentioned the models are calculated evaluating the extension of the convective regions (wherever necessary) by means of the overshoot scheme of mixing according to the prescriptions given by Bressan et al. (1981), Bertelli et al. (1985), Alongi et al. (1991, 1993), Bressan et al. (1993a), and Fagotto et al. (1993a). The e ciency of this process has been preliminarily calibrated on observational data for LMC clusters and Galactic old open clusters. Current estimates indicate a mild e ciency of core overshoot, or c 0:5, and of envelope overshoot, or e = 0:7 (see the discussion of this topic by Chiosi et al. 1992a) . Although signi cant mass loss by stellar wind may occur during the red giant (RGB) and asymptotic giant branch (AGB) phases of low- and intermediate-mass stars, the models are calculated at constant mass. This is possible because of the regular behaviour of the RGB and AGB phases for the adopted chemical compositions, so that mass loss by stellar wind can be included by means of the analytical method outlined long ago by Renzini (1977) and since then adopted in many studies (e.g. Bertelli et al. 1990, 1994, see also Sect. 5) when isochrones are constructed from evolutionary sequences. In brief, going from the tip of the RGB to the zero age horizontal branch (ZAHB) the mass of the star is simply scaled by removing a fraction of the envelope. Past the onset of the TP-AGB phase, the evolution can be followed analytically provided sets of low-mass stellar models: those by Bressan et al. that the relationship between the mass of the H-exhausted (1993a), and a limited set of tracks computed with initial core and the total luminosity, and the rate of mass loss as a composition [Z = 0:01886; Y = 0:273] and the same input function of basic stellar parameters (e.g. luminosity, mass, physics as described in this section. It is evident that both and radius) are assigned. sets almost perfectly superimpose in the HRD, all along On the contrary, massive stars with initial mass M from the main sequence to the red giant phases. Temper12 M are always evolved including the e ect of mass ature di erences between any two equivalent tracks are loss by stellar wind from the ZAMS stage because their always lower than 60 K. A similar test was performed on structure and evolution is entirely dominated by this phe- a limited set of tracks with [Y = 0:23; Z = 0:0004] and nomenon (Chiosi & Maeder 1986; Chiosi et al. 1992a, b) . [Y = 0:475; Z = 0:05], compared to the tracks presented The empirical mass-loss rates are the same as in Bressan by Fagotto et al. (1994a), with identical results. et al. (1994a) and Fagotto et al. (1994a). These are based upon observational data for supergiants, luminous blue variables (LBV), and Wolf Rayet (WR) stars. Precisely, for all evolutionary stages from the main sequence up to 3 the so-called de Jager limit in the Hertzsprung Russell Diagram (HRD), the mass-loss rates are derived from de Jager et al. (1988) however modulated on the metallicity according to the prescription by Kudritzki et al. (1989) . 2 The mass-loss rates are expressed as _ MZ = Z0:5 _ MZ=0:02: (1) Beyond the de Jager limit, the mass-loss rate M_ Z=0:02 is increased to 10−3 M yr−1 as suggested by the observational data for the LBV stars. 3. The mixing length calibration Since some physical ingredients of the models have been changed, we are forced to revise the calibration of the mixing length parameter = l=Hp in the outermost super-adiabatic convective region. With the introduction of Coulomb interactions in the EOS, models turned to be denser and brighter than their counterparts which did not consider these terms (see the Appendix). In consequence, the model which previously tted the luminosity and radius of the Sun at the solar age, now should have its increased and He content decreased in order to satisfy the same constraints. We nd that a 1:0 M model with Z = 0:01886 (the solar Z according to Anders & Grevesse 1989) reproduces the solar radius and luminosity at an age of 4.6 Gyr only if we assume = 1:56 and Y = 0:273. This value for replaces that used in the previous papers of this series. It is worth emphasizing that the use of a di erent value of does not constitute a serious source of heterogeneity among the di erent sets of tracks presented in this series of papers. Apart from the fact that can only affect models within a limited range of e ective temperatures, the modi cations in the stellar structure were small enough such that the small changes in e ective temperature and luminosity of the models were almost completely compensated by the changes made in order to recalibrate the solar model. Therefore, the new models turn to follow almost the same track on the HRD as before. It is illustrated in Fig. 1, in which we superimpose two di erent 1 0 -1 4 2.0 1.8 1.6 1.4 1.2 1.0 0.8 3.8 3.6 3.4 The main e ect of the introduction of Coulomb interactions in the EOS is on the lifetimes of low-mass stars. Figure 2 illustrates the change in lifetimes from the ZAMS to the beginning of core He-burning, for tracks of two different chemical compositions, with respect to the tracks presented by Bressan et al. (1993a) and Fagotto et al. (1994a). As expected, the change in lifetime is almost null for a 1 M star with solar composition, and di erences of a few percent arise for the remaining stars. For the models with near-solar composition, lifetimes di er by at most 5%, an e ect that could be safely neglected in the study of young and intermediate-age stellar populations. For Z = 0:0004, however, a more dramatic change occurs 0.05 0 -0.05 -0.1 0.7 0.8 0.9 1 2 3 4 5 at the low-mass end: a 0.8 M star decreases its lifetime of 8%, without any appreciable e ect in the turn-o luminosity. It is a small but not negligible e ect, since it would imply a decrease of 1 Gyr in the absolute age estimates of globular clusters made with our isochrones. 4. Evolutionary results In the following we make a short discussion on some global features of the HRDs, lifetimes, and lifetime ratios of the major evolutionary phases, and the variation of the surface abundance induced by dredge-up and mass loss. We inform the reader that complete tables containing all the relevant information on the models (the same described in previous papers of this series), can be obtained upon request to the authors. The data is also available in a public directory at Padua, and is accessible through the World Wide Web (WWW) at the node http://www.pd.astro.it. Figures 3, 4 and 5 show the corresponding HRDs for the low mass range (stars undergoing core He-flash), from the ZAMS to the tip of the RGB, and from the ZAHB to the start of the TP-AGB phase. For the sake of clarity, the set of tracks from the ZAHB to later phases is presented in two di erent gures: Fig. 4 with ZAHB mass in the range 0:5 − 0:8 M , and Fig. 5 displaying the same but for ZAHB masses in the range 0:9 − 1:7 M . Looking at Fig. 3, the reader can notice the main features of the low-metallicity grid of low-mass evolutionary tracks. Particularly interesting are the narrowness of the main sequence band, the large temperature range covered by the SGBs, and the almost vertical track of the RGB. The transition mass from low to intermediate mass stars is M HeF = 1:7 M . During the core He-burning and later phases of low mass stars, the evolutionary path in the HRD develops at the high temperatures characteristic of low metallicity models. The ZAHB is shifted to higher and higher temperatures at both decreasing and increasing ZAHB mass (Figs. 4 and 5, respectively). Following core He-exhaustion the models evolve toward the Hayashi line to start the AGB phase. The 0:51 M model, however, settles on very hot temperatures and proceeds exhausting the central fuel growing its luminosity and maintaining its high temperature. A pronounced He-shell flash takes place when the star has not yet settled on the Hayashi line. 3 2 1 0 1.7 1.6 1.5 1.4 1.3 1.2 1.1 1.0 0.9 -1 4.1 4 3.9 0.8 3.8 0.7 0.6 3.7 Figure 6 presents the HRD for intermediate mass stars and those massive stars of lower mass in which the e ect of 1 4.6 9 7 6 5 4 3 2.5 mass loss by stellar wind can be neglected. The models go from the ZAMS to the latest evolutionary phase, i.e. TPAGB for stars up to M up and central C-ignition above. The value of M up falls between 4:0 M and 5:0 M . The more impressive features exhibited by these tracks are the extended loops on the HRD of the He-burning sequences. Figure 7 shows the models of massive stars in which the e ect of mass loss all over their evolution is important. Each sequence goes from the ZAMS to the stage of central C-ignition. Relevant is the dependence of massloss rate on metallicity which strongly a ects both the morphology of the HRD and the lifetimes of the various evolutionary phases. In particular these metal-poor models of any initial mass start burning helium early in their way to the red supergiant phase, and are found to never enter the Wolf-Rayet stages. Table 1 contains the lifetimes of the central H- and Heburning phases and early AGB (E-AGB) of all the models. The three lifetimes are indicated by H, He, and E−AGB, respectively. Table 2 summarizes the changes in the surface abundances (by mass) induced by the rst dredge-up (the pollution of the original surface composition by the products of H-burning) as function of the stellar mass. For parameters for core and envelope overshoot are c = 0:5 and e = 0:7, respectively 4 the elements 1H, 3He, 4He, 12C, 13C, 14N, 15N, 16O, 17O, and 18O, this table displays both their initial abundance and their surface abundance after completion of the rst dredge-up. The third column gives the fractionary mass (Qconv) of the layer reached by the external convection at its maximum penetration, while the last two columns show the ratios 12C/13C and (14N)/(14N)i. This latter ratio is the enhancement factor of 14N with respect to the initial value (14N)i. For each star Table 3 indicates the age at which the second dredge-up occurs (when the external convection reaches the deep He-rich layers between the inert carbonoxygen core and the H-burning shell temporarily extinguished), the mass Qconv of the layer reached by the external convection at its maximum penetration, and the surface abundances (by mass) of 1H, 4He, 12C, 13C, 14N and 16O. Remarkably, models with mass higher than 3 M do not become red giants before the start of He-burning, and consequently they miss any dredge-up event before entering on the E-AGB. Moreover, when the second dredgeup occurs on these stars, C-burning starts on the interior when the external convection has not yet attained its maximum extension. Consequently, the surface abundance continuously varies during their evolution along the Hayashi line. In massive stars, the variation of the surface abun- mass loss on the internal structure of models at the tip of dances in the course of evolution is not as simple as in the the RGB are negligible. lower mass range, but it is complicated by the occurrence Mass loss is the key parameter during the evolution in of mass loss and depends also on the e ciency of central the TP-AGB, while it can be neglected during the early mixing in the sense that the larger the convective core, AGB phase. From the beginning of the TP-AGB phase up the easier is the exhibition of nuclearly processed material to the stage of envelope ejection, the evolutionary tracks at the surface. are evolved analytically (see Renzini 1977; Iben & Renzini 1983; Bertelli et al. 1990, 1994; Gr¨onewegen & de Jong 5. Theoretical isochrones 1993 for details) . Like in Bertelli et al. (1994), we adopt the core mass-luminosity relationship from the models by Theoretical isochrones were constructed from the tracks Boothroyd & Sackmann (1988), Reimers' mass loss rate presented in this paper. A detailed description of the with the parameter increasing with the stellar mass, methods is presented in Bertelli et al. (1990, 1994). Below and a luminosity-e ective temperature relation obtained we just recall some aspects of the construction of these extrapolating the slope of the early AGB phase of our isochrones. models to higher luminosities and lower e ective temper To include in the isochrones the e ects of mass loss by atures. The tracks are ended when the entire mass of the stellar wind towards the tip of the RGB for low mass stars envelope is removed. and during the AGB evolution of low and intermediate Therefore, the terminal stage of the isochrones is either mass stars, the empirical formulation by Reimers (1975) a white dwarf for M i 4 M , or core C-ignition for the is used. The mass loss parameter varies from 0.35 for remaining stars. low mass stars increasing gradually to = 1 for interme- When the star evolves toward the white dwarf stage, diate mass stars. In passing from the tip of the RGB to the isochrones include also the evolutionary phase across the ZAHB, we integrate the mass loss rate along the RGB the HRD as the central star of a planetary nebula (CSPN). to estimate the total amount of mass that has to be re- In this case we derive a suitable relation between the inimoved. Then the mass of the evolutionary models (that tial mass and the mass of the CSPN. The evolutionary sewere computed at constant mass) is simply scaled down quences for the CSPNs are either taken from Scho¨nberner to the value suited to the ZAHB stars, as the e ects of (1983) and Bl¨ocker & Sch¨onberner (1990) for M cs = 0:546, 8.5 7.5 9 8 7 6.5 14 20 9 of log t = 0:1 (ages are in years). The characteristic stages are: a) The turno (TO), i.e. the bluest point during core Hburning. b) The reddest point before the overall contraction phase at the end of the core H-burning (indicated as stage B). c) The stage of core H-exhaustion (indicated as stage C). d) The base of the RGB (BRGB). e) The tip of the RGB (TRGB). f) The mean locus of the core-He burning phase (MHeb). This is evaluated graphically, plotting the luminosity (or MV ) versus the current mass along the isochrone. g) The bluest stage during core He-burning in presence of a loop (BHeb). h) The reddest stage during core He-burning in presence of a loop (RHeb). i) The tip of the AGB (TAGB). j) Finally, the last computed model (LM) for all cases in which the AGB phase does not occur. The layout of Table 4 is as follows: Column 1 (Log(age)): the logarithm of the age in yr; Column 2 (Phase): the characteristic stage; Column 3 (M=M ): the current mass in solar units; Column 4 (log Te ): the logarithm of the e ective temperature; Column 5 (log L=L ) the logarithm of the luminosity in solar units; Column 6 (MV ): the absolute visual magnitude; Columns 7-10: the colors (B −V ), (V −I), (V −J ), and (V −K), respectively. Complete tables with the isochrones are available on request, and can also be obtained through the WWW (see Sect. 4). These tables are specially useful in the study of very metal poor populations as those found on halo globular clusters and some dwarf spheroidal galaxies. Fig. 9. In the plane (P; T ), we limit the regions in which our EOS change its characteristics. Going to higher pressures at constant temperatures: = 0:5 limits the region in which gas pressure begins to dominate over radiation pressure. The lines labelled with O, C, He and H limit the areas in which Pc becomes 1% of the gas pressure for, respectively, pure O, C, He and H compositions. We use Eq. (A12) to determine this limit for any chemical mixture. The continuous straight line limits the region in which we start considering electron degeneracy. Superimposed, we see the run of central temperature and pressure for some stars of di erent masses (this paper) 0.565, 0.605 and 0.836 M or explicitly calculated for 6. Summary and conclusions M cs = 0:5167 and 0.646 M (see Bertelli et al. 1994). All these models possess the same initial chemical compo- We have presented a new grid of evolutionary tracks and sition, namely [Z = 0:020; Y = 0:28] and have the same isochrones with extremely low metallicity. They cover a initial stage taken at log T e = 3:70 (the short-lived part wide range of evolutionary stages and masses and are suitof the sequence between the tip of the AGB and this ini- able for population synthesis purposes. They complement tial stage is neglected). The age of the stellar models in those by Bertelli et al. (1994a and references therein) at the CSPN stage is inclusive of the lifetime elapsed from the low-metallicity end. the zero age main sequence up to the tip of the AGB. Results and peculiar features of the models are briefly Theoretical luminosities and e ective temperatures commented. We discuss the main di erences that arise along the isochrones are translated to magnitudes and col- on these models due to the adoption of some di erent ors using extensive tabulations of bolometric corrections physical ingredients. Particularly, we evaluate the change and colors obtained from properly convolving the spec- in the lifetimes of the models as a function of stellar mass, tral energy distributions contained in the library of stellar due to the introduction of Coulomb interactions on the spectra kindly made available by Kurucz (1992) . The re- EOS for the stellar interior. The tracks described by the sponse functions for the various pass-bands in which mag- models on the HRD do not change signi cantly due to the nitudes and colors are generated are from the following introduction of these di erent ingredients. sources: Buser & Kurucz (1978) for the U BV passbands, These grids of models belong to a wider data base of Bessell (1990) for the R and I Cousins pass-bands, and stellar models aimed at studying the spectro-photometric nally Bessell & Brett (1988) for the J HK pass-bands. evolution of stellar populations of di erent complexity go In Fig. 8 we present some of the isochrones on the ing from star clusters to galaxies. Most of this data base HRD, while Table 4 presents a summary of the most sig- has already been published in the previous papers of this ni cant stages of the isochrones, sampled at age intervals series. Acknowledgements. We thank Dr. O. Straniero for his help in clarifying some of the aspects concerning the Coulomb corrections. This work has been nancially supported by the Italian Ministry of University, Scienti c Research and Technology (MURST) and the Italian Space Agency (ASI). L. Girardi acknowledges a fellowship from Brazilian funding agency CNPq. A. Appendix: The interior EOS In a dense plasma the EOS deviates from that of an ideal tghaes dchuaertgoedthpearetiecclets.ofDtuhee tCootuhleo mtebn dinetnecryacotfioenlescbtreotnwseetno Γz = Γ0Zz5=3 = Zhr01i=k3Te2 cluster around ions, the net e ect is that a negative term In the latter equation, is added to the pressure and internal energy of the stellar material at these conditions. 4 N a −1=3 We refer to Straniero (1988) for a general description hri = 3 0 of the subject. Here we limit to a description of the formulas adopted by ourselves, and a summary analysis of is the mean distance between ions, and the e ects of the corrections in our models. We consider the total pressure P , and the energy by unit mass of stellar material u given as a function of density and temperature T by Z0 = Pz Zz X0z =Az = 0e Zz5=3: P = u = N ak 0 3R + ue + 2 0 aT 4 aT 4 3 + uc( ; T ) T + P e + + P c( ; T ) is the mean charge of the ions. The above formulas are identical to those given by Hansen & Vieillefosse (1976) for two-component plasmas, (A1) and are expected to be good approximations for 3 or 4 element mixtures. Therefore, are fully adequate to de(A2) scribe the Coulomb energy of ions in the interior of stars at H and He-burning stages. The Coulomb energy of the electrons is, however, much more uncertain. For a one-component plasma in the limit of low-Γ, Straniero's (1988) Eq. (45) gives us an energy by unit mass which is a factor uc = Z + 1 uci Z3=2 Z + F1=2( ) !1=2 F1=2( ) mixture of ions labelled z, of electric charge Zz, atomic mass Az , at mass fractions Xz , we have: f(Γ) = 0 X Xz f(Γz ) Az z with f(Γz ) = Γz3=2 The right-hand side terms refer respectively to the contributions of kinetic energy, radiation, and the Coulomb interactions between all gas particles. Symbols have their usual meaning, namely: 0 is the mean atomic weight, N a is Avogadro's number, k is Boltzmann's constant, a is the radiation pressure constant. The subscript e stands for the electrons. uc is related to P c by uc = 3P c= . As usual, the parameter which describes the magnitude of the Coulomb energy is Γ, which is a ratio between the mean kinetic and potential energy of ions. For a onecomponent neutral plasma, Γ = (Ze)2=hrikT , with Z and A being respectively the charge and atomic mass of ions, and hri = (4 N a=3A)−1=3 the mean distance between ions. The Coulomb energy for the ions, uci, is then a simple function of Γ, being expressed by a fraction f(Γ) of the kinetic energy of ions: uci = N akT f(Γ)= 0 (see Straniero 1988) . There are analytical formulas for f(Γ) which t both the results of Debye-Hu¨ckel theory at low Γ and the results of Monte Carlo simulations at high Γ (e.g. Hansen 1973; Hansen & Vieillefosse 1976). To deal with the Coulomb correction for arbitrary chemical mixtures, we generalize the expressions given by Hansen & Vieillefosse (1976) for a two-component plasma. Accordingly, for an arbitrary greater than that would be obtained simply by the ions (being = kT =mec2, F1=2( ) the known Fermi-Dirac integral, and F1=2( ) its derivative with respect to the degeneracy parameter ). It is attractive to use this result in order to obtain the total (ions+electrons) Coulomb energy once the ionic contribution is obtained from Eq. (A3), for all values of Γ. Using the same simple reasoning that led us to generalize Hansen & Vieillefosse (1976) formulas to Eqs. (A4 { A8), with the aid of Eq. (A8) we transformed Eq. (A9) (A3) into uc = 1 + uci e 0 1 + e F1=2( ) !1=2 0 F1=2( ) : (A10) This latter was actually used for the multiple-component plasmas dealt with by our stellar evolution code. The Coulomb terms to the pressure and internal energy were included in the derivation of all thermodynamical quantities (and their derivatives) of our stellar code. Calculation of these terms is accomplished only when the Coulomb pressure overcomes a threshold fraction (set to 1%) of the total gas pressure. This condition can be shown to translate into: log P > 4 log T − 2 log higher. This experiment indicates that the main e ect of Coulomb interactions comes from the large region of the star with intermediate-Γ. 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L. Girardi, A. Bressan, C. Chiosi, G. Bertelli, E. Nasi. Evolutionary sequences of stellar models with new radiative opacities. VI., Astronomy and Astrophysics Supplement Series, 113-125, DOI: 10.1051/aas:1996144