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Astron. Astrophys. 334, 953-968 (1998)

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5. Evolutionary tracks

In this section we will present and discuss evolutionary tracks for stars having chemical composition Y=0.274, Z=0.017 (the lower limit for the solar metal abundance) in the range [FORMULA], with and without diffusive overshooting. Tracks and isochrones for other compositions, mass ranges, [FORMULA] values etc. can be made available upon request, since both the ATON 2.0 and the isochrones codes are largely authomatized.

Let us first discuss the main differences in the evolution up to central [FORMULA] ignition for a star of [FORMULA], if we allow for either instantaneous mixing decoupled from nuclear evolution, or for coupled-diffusion (with and without overshooting), since - in the best of our knowledge - results with coupled diffusion have never been compared to results with nuclearly uncoupled diffusion or instantaneous mixing in MS.

5.1. Instantaneous mixing vs. coupled-diffusion

In the case of instantaneous mixing, all the chemical abundances profiles are obviously flat in the convective core. Not so for coupled-diffusion. Let us focus our attention on this latter case, since it is by far the more interesting and physically sound one.

Among the nine CNO reactions explicitly accounted for (Sect. 3.3), the fastest one is by far [FORMULA] ; all the others have cross sections at least two orders of magnitude lower. The average nuclear lifetime of [FORMULA] is [FORMULA] where [FORMULA] is the density, [FORMULA] the nuclear cross section of the reaction and [FORMULA] the hydrogen abundance. This lifetime is to be compared to the turbulent diffusion time which, for the core of the [FORMULA] star in MS, is of the order of [FORMULA] s. For the p-p reactions (always contributing for less than 1% to the total luminosity), [FORMULA] and [FORMULA] are the elements with shorter lifetimes.

At the beginning of MS, the central temperature ([FORMULA] MK) is such that the lifetime of [FORMULA] is still much larger than the mixing time; nearly complete mixing ensues both in the coupled-diffusive and in the instantaneous mixing case. [FORMULA] and [FORMULA] lifetimes are instead short, and their central abundances are only a fraction of the abundances at the boundary of the convective core, but [FORMULA] burning powers the star only for a minute fraction of the total. In the progress of evolution, however, [FORMULA] increases up to [FORMULA] MK; the lifetime of [FORMULA] decreases and also the [FORMULA] profile shows a minimum, of a few parts in ten thousands, at the centre of the star where the most of burning occurs.

Since H-burning is linear with each nuclear species involved (apart from the starting reaction of the p-p chain), central underabundances represent a bottleneck. The structure settles on a slightly lower luminosity than in the instantaneous mixing case, hardly detectable on Fig. 3 during the MS phase up to the overall contraction phase. The total duration on the MS phase is however somewhat increased. From Fig. 4, showing the behavior of [FORMULA] vs. time in several mixing and overshooting cases, one can see that the total duration of MS is, with coupled-diffusion, about 4% longer than with instantaneous mixing. This is not any more a negligible difference in modern stellar modeling, since it is of the same order of magnitude of the differences presently found when updating the main micro-physical inputs like radiative opacities etc. Coupled-diffusion can not be omitted any longer in numerical computations of stellar evolution, if sound quantitative results are required.

[FIGURE] Fig. 3. HR diagram for a [FORMULA] star, when treating mixing and overshooting according to different schemes

[FIGURE] Fig. 4. Evolution of [FORMULA] from the end of MS up to [FORMULA] burning for a [FORMULA] star, when treating mixing and overshooting according to different schemes

After the overall contraction, the behaviors of the tracks with and without diffusion remain similar but not identical, due to the large sensitivity of the surface conditions on the [FORMULA] ratio, the larger being the ratio, the bluer appearing the star (central [FORMULA] reactions ignite soon after central H -exhaustion). The delicate interplay among convective shells growing and merging around the convective core during MS and beyond (usually addressed to as semiconvection) is such that coupled-diffusion leads to slightly different chemical mixing and final chemistries in the region where the H -burning shell occurs, showing up amplified in the HR diagram.

This is not the place to address the long standing problem of the existence or not of the blue loop as a whole (see Chiosi 1997, and Salasnich et al. 1997 for a recent update). Suffice it to say the both our 15 [FORMULA] models, with instantaneous mixing and coupled-diffusion, show this feature for the chosen chemistry, and none of them reaches the red supergiant region before the end of central He -burning. From Fig. 4, one can see that C -burning takes over as soon as the star reaches the Hayashi track. Even if we can not follow the details of C -burning with ATON 2.0 code, we can in any case claim that the "red" phase for our 15 [FORMULA] star lasts for less than 1% of its total lifetime, leading to an expected deficiency of red supergiants of this mass. Different is the case if we include diffusive or instantaneous overshooting, as we will see right now.

5.2. Overshooting

From Figs. 3 and 4 one can see that, for the 15 [FORMULA] star, diffusive overshooting with [FORMULA] is nearly equivalent to instantaneous overshooting with [FORMULA] as long as morphology and duration of MS is concerned. We will come back on the tuning of [FORMULA] when discussing the general results for the whole range of masses considered and the comparisons to the observations.

After the MS, in both cases the star reaches the Hayashi track at the beginning of the core He -burning phase. The main qualitative difference with the no-overshooting case - apart from the duration and the amplitude in the HR diagram of the MS - is that overshooting is included not only for the core convective region, but also for the surrounding convective shell, allowing these latter to mix in external regions, where no H -burning had yet modified the original CNO abundances. As a result, at the end of MS, both C and N are now slightly overabundant in the H-burning shell than without overshooting, and the lower [FORMULA] ratio favors cooler surface conditions.

In the progress of time, however, the star with instantaneous overshooting goes back to the blue until core He -exhaustion and C -ignition. The star with diffusive overshooting, at variance with the above behavior, stays on the Hayashi track until C -ignition. The reason for this difference must be traced back in the presence of a convective region around the [FORMULA] interface. The H -burning shell is thick enough that about [FORMULA] of the H -luminosity is born in the surrounding convective region.

With overshooting, the convective region is enriched in CNO due to leakadge into more external regions, previously untouched by H -burning and, as above elucidated, the H -burning shell turns out more powered. However, with instantaneous overshooting, overabundance of CN is present only once, at the beginning of the H -shell phase, and it is not refurbished by further and further overshooting; when CN equilibrium is reached in the convective region, the shell is not overpowered any more, and the [FORMULA] ratio increases leading the star to the blue.

On the contrary, diffusive overshooting goes on penetrating the external layers slower than instantaneous overshooting, but for a larger extension - even if mixing there is only partial. Continuous enrichment of fresh C and N in the convective layer follows, and the efficiency of H -burning shell is kept larger (and almost constant) than with instantaneous overshooting. The ratio [FORMULA] always remains low enough to maintain the star in the red, and a blue loop never arises.

This different behavior leads to the prediction of a larger fraction of red supergiants ([FORMULA]) in the diffusive case than in the instantaneous mixing one ([FORMULA]). Careful comparisons (with adequate chemistry and statistics) between theoretical and observed star countings in MS and red supergiants for young open clusters, should be then a more powerful tool in tuning the value of [FORMULA] than thickness in the MS alone.

5.3. Tracks for smaller masses

Let us now turn to the whole grid of models, from 0.6 to 15 [FORMULA] in which diffusive overshooting has been included. For [FORMULA], as already seen (Fig. 1), [FORMULA] leads to almost negligible effects. Figure 7 shows that the chosen amount of diffusive overshooting has almost no effect also upon the [FORMULA] star, while the [FORMULA] model begins to display a small remnant convective core at the TO, which it would not have shown without overshooting. For the chosen chemical composition, in fact, only the [FORMULA] star would maintain a convective core at the TO even in the absence of overshooting. The chosen treatment and tuning of [FORMULA], then, does not substantially modify the occurrences at the TO for solar-type stars. Different would have been the case with instantaneous overshooting [FORMULA] (Fig. 7, dashed line), since the TO morphology in the whole range [FORMULA] would have been substantially modified.

At the upper extreme of masses considered here, we see instead (Figs. 3 and 4) that diffusive overshooting with [FORMULA] is, for a [FORMULA] star, nearly equivalent to instantaneous overshooting with [FORMULA], both for the HR diagram morphology, and for its effect of prolongating the MS lifetime (by about 15%). More in general, comparing Figs. 5 and 6 one can see that the effect of diffusive overshooting increases with the stellar mass in the range 1.5-2.0 [FORMULA], and finally becomes comparable with that of instantaneous overshooting with [FORMULA]. So, working with diffusive mixing, a single value of [FORMULA] seems to be consistent with all the range of masses considered. In fact, it gives zero "equivalent" instantaneous overshooting for the solar mass, a slowly increasing effect for larger masses up to [FORMULA], after which it mimicks [FORMULA] of instantaneous overshooting.

[FIGURE] Fig. 5. HR diagram with no overshooting. The masses considered (in [FORMULA]) are: 0.6; 0.65; 0.7; 0.8; 0.9; 1.0; 1.1; 1.2; 1.3; 1.4; 1.5; 1.6; 1.7; 1.8; 1.9; 2.0; 2.1; 2.2; 2.5; 3; 4; 5; 6; 7; 8; 9; 10; 12 and 15

[FIGURE] Fig. 6. HR diagram with diffusive overshooting, [FORMULA]

[FIGURE] Fig. 7. Blow-up of the theoretical HR diagram for stars in the range [FORMULA], with and without diffusive overshooting. The difference between the two cases is that, with [FORMULA], also the star of [FORMULA] maintains a small convective core at the turn-off, while in the former case the same occurrence was found for [FORMULA]

The reason for this smooth growth of "equivalent" overshooting is mainly in the non-local flavor introduced in the models thanks to [FORMULA] in Eq. (4). As already quoted in Sect. 4.2, for low mass stars like the Sun the size of the initial convective core is quite small, much less than the value of [FORMULA] at the convective boundary, and also the value of [FORMULA] is correspondingly low. This gives rise to a stiff velocity profile in the overshooting region, which in turn leads to very little mixing and fast fading of the core. For larger masses, the size of the core grows and so also [FORMULA], until saturation or quite so. Including some non-locality, even in a rough approximation, in the modeling of turbulent convection always gives more straightforward and physically realistic results, as already found for the superadiabatic temperature gradient thanks to the introduction of z as convective scale length (Sect. 2.1). In the present case, we do not have to select different mass ranges with different tunings of the instantaneous overshooting parameter [FORMULA] ; we can simply apply the same tuning for [FORMULA] to any mass range as a first, reasonable approximation.

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© European Southern Observatory (ESO) 1998

Online publication: June 2, 1998