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Astron. Astrophys. 317, 90-98 (1997)

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3. Theoretical VLM stellar structures

Selected sequences of metal poor stellar models have been computed to cover the range of H burning stars below [FORMULA], assuming Y=0.23 everywhere. In this section we will discuss some structural evolutionary features of the computed models which represent a common feature for all VLM structures independently of the assumed amount of metal and/or the treatment of stellar atmospheres. According to the calibration of solar models, all computations assume a mixing length parameter as given by [FORMULA] =2.2. However, numerical experiments confirmed that below, about, M=0.5 M [FORMULA] varying the assumption on the mixing length within reasonable limit (1.5 [FORMULA] 2.5) or varying Y within [FORMULA] Y [FORMULA] 0.02 have quite a negligible influence on the results. All models have been followed from an early pre-main sequence phase till, at least, an age of 20 Gyr. Selected models have been followed till the last phases of exhaustion of central H, often for a time much larger than the Hubble time. Of course, in this way stellar structures that will populate the Universe only in an extremely far future, have been computed. However, similar unrealistic models have been used to learn some interesting features of the VLM evolutionary scenario that we will discuss here in the following.


Table 1. Selected evolutionary quantities (see text) for VLM stellar structures.

As well known, when reaching the range of VLM structures, the common notion of Zero Age Main Sequence, as given by structures where secondary elements in H burning already attained their equilibrium values, becomes more and more meaningless since [FORMULA] behaves as a pseudo primary element, with a lifetime becoming comparable to the lifetime for central H burning, so that models start depleting H with [FORMULA] still well below its equilibrium value. This is shown in Table 1 where for selected values of stellar masses the typical time spent by the structures burning H ( [FORMULA] ), the age at which the contribution of gravitational energy vanishes ( [FORMULA] ), the age of the models attaining [FORMULA] equilibrium ( [FORMULA] ), the ratio between the amount of H at the center of this model and the initial H abundance, and the abundance by mass of [FORMULA] at the equilibrium, are reported. However, the same table shows that after 1 Gyr the structures are in all cases already supported by nuclear burning only, placed on what we can regard as the initial Main Sequence location. Last column in Table 1 reports the convective structure of these models. As already known, below about [FORMULA] only fully convective stellar structures, with a progressive amount of electronic degeneracy which, eventually, succeeds in inhibiting the release of gravitational heating and the ignition of central H nuclear burning, are found. Top and bottom panels of Fig. 1 disclose the time behavior of selected structural parameters for two models with 0.4 and [FORMULA], respectively. The [FORMULA] model behaves like the well known models with moderately larger masses populating the upper portion of the cluster main sequence. The increase of [FORMULA] toward its equilibrium value increases the efficiency of the burning and the structure reacts decreasing both central temperature and density, which start increasing again only when the equilibrium value for [FORMULA] has been attained. However, fully convective structures behave quite differently, and central density keeps decreasing all along the major phase of H burning. A similar steady decrease of central density is a well known feature but not yet discussed of the He burning ignition in the center of more massive stars.

[FIGURE]Fig. 1. The variation with time of stellar luminosity (L) in solar unit, contribution of gravitation to the total luminosity ( [FORMULA] ), central density ( [FORMULA] ), central temperature ( [FORMULA] ), size of the convective core ( [FORMULA] ) as fraction of the total mass, mass location of the bottom of the convective envelope ( [FORMULA] ), and central abundances by mass of H and [FORMULA] for models with [FORMULA] (top panel) and [FORMULA] (lower panel) with [FORMULA].

According to plain mathematical elaborations, one finds that the curious behavior of VLM fully convective structures is just the one expected in homological models with increasing molecular weight. As a matter of the fact, as shown in the same Fig. 1, one finds that the central density of the [FORMULA] model starts suddenly increasing again in the very last phases of central H burning, when the increased abundance of He and the corresponding decrease of radiative opacities induces a radiative shell which rapidly grows to eventually form a radiative core. Further details on the argument, as well as a detailed description of VLM evolutionary features, can be found in Ciacio (1994).

[FIGURE]Fig. 2. The evolution with time of the HR diagram location of models with Z=0.0003 for the labeled assumptions on the cluster age.

Fig. 2 shows the effect of age on the HR diagram distribution of a typical set of models. Data in this figure are presented as an example of evolutionary effects, showing that in "not-too-young" VLM stellar systems, the age plays a negligible role on the HR diagram location of stars below, about [FORMULA]. Interesting enough, the [FORMULA] model appears the less affected by age, less massive models shoving a progressively increasing sensitivity to the adopted ages. The reason for such a behavior is disclosed by Fig. 3, where the variation with time of the amount of central [FORMULA] for selected choices about the stellar mass, has been reported. It appears that in the [FORMULA] case, at t=10 Gyr central [FORMULA] has already approached its equilibrium value, so that the following evolution is governed by the depletion of central H only. On the contrary, less massive models keep increasing central [FORMULA], thus increasing the efficiency of proton-proton burning and readjusting the structure according to such an occurrence. A readjustment that in the less massive models is amplified by the decreasing electronic degeneracy induced by the decrease of central density.

[FIGURE]Fig. 3. The behavior with time of the abundance by mass of central [FORMULA] for the labeled assumptions about the stellar masses and for Z=0.0003 - Y=0.23.

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