3 Stellar tracks

   
3.1 Evolutionary stages and mass ranges

Our models are evolved from the ZAMS, at constant mass. The evolution through the whole H- and He-burning phases is followed in detail. The tracks are stopped either during the TP-AGB phase in intermediate- and low-mass, or at the onset of carbon ignition in a helium-exhausted core in the case of our most massive models. In the case of stellar masses lower than 0.6 $M_{\odot }$, the main sequence evolution takes place on time scales much larger than a Hubble time. For them, we stopped the computations at an age of about 25 Gyr.

In low-mass stars with  $M_{\odot }$, the evolution is interrupted at the stage of He-flash in the electron degenerate hydrogen-exhausted core. This because the computation of the complete evolution through the He-flash requires too much CPU time. The evolution is then re-started from a ZAHB model with the same core mass and surface chemical composition as the last RGB model. The initial ZAHB model presents also a core in which 5 percent (in mass fraction) of the helium has been burned into carbon. This takes into account the approximate amount of nuclear fuel necessary to lift the core degeneracy during the He-flash. The evolution is then followed up to the thermally pulsing AGB phase.

Additional He-burning models with $0.5\le(M/\mbox{$M_{\odot}$ })<0.6$ have been computed, starting from a ZAHB model with the same core mass and surface chemical composition as the last 0.6 $M_{\odot }$ RGB model.

In intermediate-mass stars, the evolution goes from the ZAMS up to either the beginning of the TP-AGB phase, or to the carbon ignition in our most massive models (i.e. those with masses higher than about 5 $M_{\odot }$).

For the stellar masses in which the evolution goes through the TP-AGB, a small number of thermal pulses has been followed - typically, from 2 to 5 ones. In some few cases, however, the sequences contain only the first significant pulse, whereas few sequences present as much as 19 thermal pulse cycles.

Table 1 gives the values of the transition masses $M\mbox{$_{\rm Hef}$ }$ and $M\mbox{$_{\rm up}$ }$, as derived from the present tracks. $M\mbox{$_{\rm Hef}$ }$ is the maximum mass for a star to develop an electron-degenerate core after the main sequence, and sets the limit between low- and intermediate-mass stars (see e.g. Bertelli et al. 1986; Chiosi et al. 1992). For deriving the values of Table 1, we have selected the stellar track for which the core mass at He-ignition presents its minimum value (see Fig. 1 in Girardi 1999). It coincides, in most cases, with the least massive stellar track we were able to evolve through the He-flash. Given the low mass separation between the tracks we computed, the $M\mbox{$_{\rm Hef}$ }$ values here presented are uncertain by only 0.05 $M_{\odot }$.

$M\mbox{$_{\rm up}$ }$ is the maximum mass for a star to develop an electron-degenerate core after the CHeB phase, and sets the limit between intermediate- and high-mass stars (Chiosi et al. 1992). Stars with $M>\mbox{$M\mbox{$_{\rm up}$ }$ }$ should ignite carbon and avoid the thermally pulsing AGB phase. Since we do not follow the carbon burning in detail, as soon a small amount of carbon burning occurs in the stars with mass above $\sim4.5$ $M_{\odot }$, we are not able to determine with confidence whether such burning will increase or fade away with time (giving place to an AGB star). Therefore, in Table 1 we simply give a range of possible values to $M\mbox{$_{\rm up}$ }$, where the lower limit represents stars which probably enter in the double-shell thermally pulsing phase, and the upper limit that of stars which apparently burn carbon explosively.

 

 

Table 1: The transition masses $M\mbox{$_{\rm Hef}$ }$ and $M\mbox{$_{\rm up}$ }$

Z	Y	overshoot	$\mbox{$M\mbox{$_{\rm Hef}$ }$ }/\mbox{$M_{\odot}$ }$	$\mbox{$M\mbox{$_{\rm up}$ }$ }/\mbox{$M_{\odot}$ }$
0.0004	0.230	moderate	1.7	4.5-5.0
0.001	0.230	moderate	1.7	4.5-5.0
0.004	0.240	moderate	1.8	4.5-5.0
0.008	0.250	moderate	1.9	4.5-5.0
0.019	0.273	moderate	2.0	5.0-6.0
0.030	0.300	moderate	2.1	5.0-6.0
0.019	0.273	no	2.4	6.0-7.0

   
3.2 Tracks in the HR diagram

The complete set of tracks for very low-mass stars (M<0.6 $M_{\odot }$) are presented in the HR diagram of Fig. 1. The tracks start at a stage identified with the ZAMS, and end at the age of 25 Gyr. The ZAMS model is defined to be the stage of minimum $T\mbox{$_{\rm eff}$ }$ along the computed track; it follows a stage of much faster evolution in which the pp-cycle is out of equilibrium, and in which gravitation may provide a non negligible fraction of the radiated energy. It is evident from Fig. 1 that these stars evolve very little during the Hubble time.

The complete sets of evolutionary tracks with [Z=0.0004, Y=0.273]and [Z=0.03, Y=0.30] are presented in the HR diagrams of Figs. 2 and 3, respectively. In these figures, panel (a) presents the low-mass tracks from the ZAMS up to the RGB-tip, panel (b) the intermediate-mass ones from the ZAMS up to the last computed model (either on the TP-AGB phase or during C-ignition, depending on the mass), whereas panels (c) and (d) present the low-mass tracks from the ZAHB up to the TP-AGB phase. Figures 2 and 3 are aimed to illustrate the typical features and mass coverage of the present tracks, at the extreme values of metallicity for which they have been computed. The reader can notice, for instance, the extended Cepheid loops present in the Z=0.0004 intermediate-mass models, which are practically missing in the Z=0.03 ones.

We also computed an additional set of "canonical'' evolutionary tracks with solar composition, i.e. [Z=0.019, Y=0.273]. They differ from the models previously described only in the prescription for the mixing: they are computed assuming the classical Schwarzschild criterion for the convective boundaries (i.e. without overshooting). Semi-convection is assumed during the core-He burning phase. This set is presented only for $M\ge1.2$ $M_{\odot }$, since stars with  $M_{\odot }$ present a radiative core during the main sequence; therefore, the tracks of lower mass are not affected by the adoption of an overshooting scheme (see also Sect. 2.5), at least in the main sequence phase. The two sets of tracks for [Z=0.019, Y=0.273]provide a useful data-base for comparing the behaviour of canonical and overshooting models.

   
3.3 Description of the tables

The data tables for the present evolutionary tracks are available only in electronic format. They are stored at the CDS data center in Strasbourg, and are also available upon request to the authors. A WWW site with the complete data-base (including additional data and the future extensions) will be mantained at http://pleiadi.pd.astro.it

For each evolutionary track, the corresponding data file presents 21 columns with the following information:

1. age/yr: stellar age in yr;
2. logL: logarithm of surface luminosity (in solar units), $\log(L/L_{\odot})$;
3. logTef: logarithm of effective temperature (in K), $\log T\mbox{$_{\rm eff}$ }$;
4. grav: logarithm of surface gravity ( in cgs units);
5. logTc: logarithm of central temperature (in K);
6. logrho: logarithm of central density (in cgs units);
7. Xc,Yc: mass fraction of either hydrogen (up to the central H-exhaustion) or helium (later stages) in the stellar centre;
8. Xc_C: mass fraction of carbon in the stellar centre;
9. Xc_O: mass fraction of oxygen in the stellar centre;
10. Q_conv: fractionary mass of the convective core;
11. Q_disc: fractionary mass of the first mesh point where the chemical composition differs from the surface value;
12. logL_H: logarithm of the total luminosity (in solar units) provided by H-burning reactions;
13. Q1_H: fractionary mass of the inner border of the H-rich region;
14. Q2_H: fractionary mass of the outer border of the H-burning region;
15. logL_He: logarithm of the total luminosity (in solar units) provided by He-burning reactions; a null value indicates negligible energy generation by those reactions;
16. Q1_He: fractionary mass of the inner border of the He-burning region;
17. Q2_He: fractionary mass of the outer border of the He-burning region;
18. logL_C: logarithm of the total luminosity (in solar units) provided by C-burning reactions; a null value means that it is negligible;
19. logL_nu: logarithm of the total luminosity (in solar units) lost by neutrinos; a null value means that it is negligible;
20. Q_Tmax: fractionary mass of the point with the highest temperature inside the star;
21. stage: label indicating particular evolutionary stages.

 

A number of evolutionary stages are indicated along the tracks (Col. 21). They correspond either to: the initial evolutionary stages (ZAMS or ZAHB), local maxima and minima of L and $T\mbox{$_{\rm eff}$ }$ (Te-M, Te-m, L-M, and L-m), the exhaustion of central hydrogen (Xc=0) and helium (Yc=0), the lowest L and highest $T\mbox{$_{\rm eff}$ }$ during the He-burning of intermediate-mass stars (Bhe and LpT, respectively), the base and tip of the first ascent of the red giant branch (Brg and Tip, respectively), the maximum L immediately preceding a thermal pulse (1tp), and the onset of C-burning (Cb). These stages delimit characteristic evolutionary phases, and can be useful for the derivation of physical quantities (as e.g. typical lifetimes) as a function of either mass or metallicity. Notice that some of these evolutionary stages may be absent from particular tracks, depending on the precise value of stellar mass and metallicity.

For the sake of conciseness and homogeneity, the evolutionary tracks in the data-base do not include the fraction of the TP-AGB evolution which was actually computed, and which is also presented in Figs. 2 and 3. Detailed data about the initial TP-AGB evolution, for any metallicity, can be obtained upon request to the authors. However, we remark that the complete TP-AGB evolution is included, in a synthetic way, in the isochrones to be described below.

Figure 4: Comparison between the empirical and theoretical initial-final mass relations. The dots are data for WDs in open clusters, according to Herwig (1996; full circles) and Jeffries (1997, full squares). The smaller dots represent data points of lower quality (cf. Herwig 1996). The open circle instead represents the mean masses of field white dwarfs and their progenitors. The mean initial-final mass relation from Herwig is also shown (dashed line); it is based only in the most reliable mass determinations for white dwarfs. The continuous line, instead, is the initial-final mass relation as derived from our Z=0.019 models (see text for details)

Figure 5: Theoretical isochrones in the HR diagram, for the compositions [Z=0.0004, Y=0.230] (panel a), and [Z=0.030, Y=0.300] (panel b). The age range goes from $\mbox{$\log(t/{\rm yr})$ }=7.8$ to 10.2, at equally spaced intervals of $\Delta\log t=0.3$. In both cases, the main sequence is complete down to 0.15 $M_{\odot }$

   
3.4 Changes in surface chemical composition

The surface chemical composition of the stellar models change on two well-defined dredge-up events. The first one occurs at the first ascent of the RGB for all stellar models (except for the very-low mass ones which are not evolved out of the main sequence). The second dredge-up is found after the core He-exhaustion, being remarkable only in models with  $M_{\odot }$. We provide tables with the surface chemical composition of H, ³He, ⁴He, and main CNO isotopes, before and after these events (when present in our data). Table 2 shows, as an example, the table for the solar composition.

 

 

Table 2: Surface chemical composition (by mass fraction) of [Z=0.019, Y=0.273] models

$M/\mbox{$M_{\odot}$ }$	H	3He	4He	12C	13C	14N	15N	16O	17O	18O
Initial:
all	0.708	2.92$\:10^{-5}$	0.273	3.26$\:10^{-3}$	3.92$\:10^{-5}$	1.01$\:10^{-3}$	3.97$\:10^{-6}$	9.15$\:10^{-3}$	3.70$\:10^{-6}$	2.06$\:10^{-5}$
After the first dredge-up:
0.60	0.697	3.03$\:10^{-3}$	0.281	3.25$\:10^{-3}$	4.10$\:10^{-5}$	1.01$\:10^{-3}$	3.93$\:10^{-6}$	9.14$\:10^{-3}$	3.71$\:10^{-6}$	2.06$\:10^{-5}$
0.70	0.693	2.35$\:10^{-3}$	0.285	3.24$\:10^{-3}$	5.62$\:10^{-5}$	1.01$\:10^{-3}$	3.71$\:10^{-6}$	9.15$\:10^{-3}$	3.71$\:10^{-6}$	2.06$\:10^{-5}$
0.80	0.691	1.86$\:10^{-3}$	0.288	3.16$\:10^{-3}$	8.49$\:10^{-5}$	1.07$\:10^{-3}$	3.47$\:10^{-6}$	9.14$\:10^{-3}$	3.71$\:10^{-6}$	2.05$\:10^{-5}$
0.90	0.690	1.54$\:10^{-3}$	0.290	3.04$\:10^{-3}$	9.52$\:10^{-5}$	1.20$\:10^{-3}$	3.27$\:10^{-6}$	9.15$\:10^{-3}$	3.72$\:10^{-6}$	2.03$\:10^{-5}$
1.00	0.690	1.20$\:10^{-3}$	0.290	2.94$\:10^{-3}$	9.89$\:10^{-5}$	1.32$\:10^{-3}$	3.12$\:10^{-6}$	9.15$\:10^{-3}$	3.75$\:10^{-6}$	1.99$\:10^{-5}$
1.10	0.690	9.83$\:10^{-4}$	0.290	2.83$\:10^{-3}$	1.00$\:10^{-4}$	1.44$\:10^{-3}$	2.98$\:10^{-6}$	9.15$\:10^{-3}$	3.89$\:10^{-6}$	1.94$\:10^{-5}$
1.20	0.693	8.32$\:10^{-4}$	0.287	2.77$\:10^{-3}$	1.02$\:10^{-4}$	1.51$\:10^{-3}$	2.89$\:10^{-6}$	9.14$\:10^{-3}$	4.06$\:10^{-6}$	1.91$\:10^{-5}$
1.30	0.695	7.40$\:10^{-4}$	0.286	2.72$\:10^{-3}$	9.63$\:10^{-5}$	1.57$\:10^{-3}$	2.88$\:10^{-6}$	9.14$\:10^{-3}$	4.31$\:10^{-6}$	1.87$\:10^{-5}$
1.40	0.695	6.78$\:10^{-4}$	0.286	2.61$\:10^{-3}$	1.01$\:10^{-4}$	1.70$\:10^{-3}$	2.68$\:10^{-6}$	9.14$\:10^{-3}$	4.86$\:10^{-6}$	1.82$\:10^{-5}$
1.50	0.693	5.92$\:10^{-4}$	0.288	2.39$\:10^{-3}$	1.07$\:10^{-4}$	1.95$\:10^{-3}$	2.38$\:10^{-6}$	9.14$\:10^{-3}$	6.47$\:10^{-6}$	1.72$\:10^{-5}$
1.60	0.693	5.22$\:10^{-4}$	0.287	2.33$\:10^{-3}$	1.06$\:10^{-4}$	2.03$\:10^{-3}$	2.31$\:10^{-6}$	9.12$\:10^{-3}$	2.00$\:10^{-5}$	1.68$\:10^{-5}$
1.70	0.693	4.57$\:10^{-4}$	0.288	2.28$\:10^{-3}$	1.05$\:10^{-4}$	2.12$\:10^{-3}$	2.26$\:10^{-6}$	9.06$\:10^{-3}$	3.80$\:10^{-5}$	1.65$\:10^{-5}$
1.80	0.692	4.08$\:10^{-4}$	0.288	2.29$\:10^{-3}$	1.08$\:10^{-4}$	2.16$\:10^{-3}$	2.24$\:10^{-6}$	9.00$\:10^{-3}$	4.28$\:10^{-5}$	1.64$\:10^{-5}$
1.90	0.691	3.65$\:10^{-4}$	0.289	2.25$\:10^{-3}$	1.07$\:10^{-4}$	2.25$\:10^{-3}$	2.20$\:10^{-6}$	8.93$\:10^{-3}$	5.13$\:10^{-5}$	1.62$\:10^{-5}$
1.95	0.691	3.45$\:10^{-4}$	0.290	2.25$\:10^{-3}$	1.07$\:10^{-4}$	2.28$\:10^{-3}$	2.20$\:10^{-6}$	8.91$\:10^{-3}$	4.50$\:10^{-5}$	1.62$\:10^{-5}$
2.00	0.690	3.25$\:10^{-4}$	0.291	2.22$\:10^{-3}$	1.03$\:10^{-4}$	2.35$\:10^{-3}$	2.19$\:10^{-6}$	8.86$\:10^{-3}$	5.27$\:10^{-5}$	1.61$\:10^{-5}$
2.20	0.687	2.61$\:10^{-4}$	0.294	2.20$\:10^{-3}$	1.06$\:10^{-4}$	2.48$\:10^{-3}$	2.17$\:10^{-6}$	8.74$\:10^{-3}$	5.13$\:10^{-5}$	1.59$\:10^{-5}$
2.50	0.683	2.09$\:10^{-4}$	0.298	2.21$\:10^{-3}$	1.08$\:10^{-4}$	2.62$\:10^{-3}$	2.13$\:10^{-6}$	8.58$\:10^{-3}$	4.93$\:10^{-5}$	1.59$\:10^{-5}$
3.00	0.680	1.46$\:10^{-4}$	0.301	2.19$\:10^{-3}$	1.08$\:10^{-4}$	2.76$\:10^{-3}$	2.09$\:10^{-6}$	8.47$\:10^{-3}$	3.08$\:10^{-5}$	1.58$\:10^{-5}$
3.50	0.682	1.11$\:10^{-4}$	0.299	2.23$\:10^{-3}$	1.08$\:10^{-4}$	2.71$\:10^{-3}$	2.12$\:10^{-6}$	8.48$\:10^{-3}$	2.13$\:10^{-5}$	1.60$\:10^{-5}$
4.00	0.682	9.04$\:10^{-5}$	0.299	2.26$\:10^{-3}$	1.15$\:10^{-4}$	2.67$\:10^{-3}$	2.11$\:10^{-6}$	8.48$\:10^{-3}$	1.49$\:10^{-5}$	1.63$\:10^{-5}$
4.50	0.681	7.52$\:10^{-5}$	0.299	2.27$\:10^{-3}$	1.17$\:10^{-4}$	2.69$\:10^{-3}$	2.10$\:10^{-6}$	8.44$\:10^{-3}$	1.19$\:10^{-5}$	1.64$\:10^{-5}$
5.00	0.680	6.37$\:10^{-5}$	0.300	2.30$\:10^{-3}$	1.16$\:10^{-4}$	2.70$\:10^{-3}$	2.12$\:10^{-6}$	8.39$\:10^{-3}$	1.16$\:10^{-5}$	1.65$\:10^{-5}$
6.00	0.681	4.93$\:10^{-5}$	0.300	2.35$\:10^{-3}$	1.21$\:10^{-4}$	2.66$\:10^{-3}$	2.14$\:10^{-6}$	8.36$\:10^{-3}$	8.92$\:10^{-6}$	1.67$\:10^{-5}$
7.00	0.679	4.03$\:10^{-5}$	0.302	2.33$\:10^{-3}$	1.20$\:10^{-4}$	2.76$\:10^{-3}$	2.13$\:10^{-6}$	8.28$\:10^{-3}$	7.84$\:10^{-6}$	1.66$\:10^{-5}$
After the second dredge-up:
4.50	0.653	7.06$\:10^{-5}$	0.328	2.14$\:10^{-3}$	1.14$\:10^{-4}$	3.15$\:10^{-3}$	1.98$\:10^{-6}$	8.09$\:10^{-3}$	1.30$\:10^{-5}$	1.55$\:10^{-5}$
5.00	0.635	5.80$\:10^{-5}$	0.346	2.11$\:10^{-3}$	1.13$\:10^{-4}$	3.42$\:10^{-3}$	1.94$\:10^{-6}$	7.82$\:10^{-3}$	1.23$\:10^{-5}$	1.51$\:10^{-5}$
6.00	0.610	4.29$\:10^{-5}$	0.371	2.07$\:10^{-3}$	1.16$\:10^{-4}$	3.74$\:10^{-3}$	1.89$\:10^{-6}$	7.51$\:10^{-3}$	9.47$\:10^{-6}$	1.49$\:10^{-5}$
7.00	0.622	3.56$\:10^{-5}$	0.359	2.09$\:10^{-3}$	1.16$\:10^{-4}$	3.65$\:10^{-3}$	1.90$\:10^{-6}$	7.58$\:10^{-3}$	8.57$\:10^{-6}$	1.48$\:10^{-5}$