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

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2. The physical inputs

Since the pioneering work by Limber (1958) investigations given by several authors have already brought to light a series of happy features characterizing the structure of very low mass (VLM) stars. VLM stars appear rather insensitive to two evolutionary parameters which affect the structures of more massive stars, namely the mixing length parameter and the amount of original He (see e.g. Vanderberg et al. 1983). Unfortunately, when entering the range of VLM cool stars the theoretical evaluations about these structures are critically dependent on the difficult evaluation of both the opacity and the equation of state for a cool gas, where molecules and grains play a relevant role. Interested readers can find a detailed discussion about this last point in Dorman et al. (1989).

As for low temperature opacities, tabulations available in the literature including molecules and grains, as presented by Alexander et al. (1983, 1989) and Alexander & Ferguson (1994) were found not to cover the whole range of pressure experienced by a VLM star, thus requiring more or less tentative extrapolations. A new extended tabulation has been computed for this project, allowing a full coverage of VLM structures for selected choices about the star metallicity. We expect that such new evaluations should maintain validity for densities lower or of the order of 0.01 gr/cm 3, a range which appears fully adequate for our purposes.

Low temperature opacities were implemented at T > 10000 [FORMULA] K with radiative opacities by Roger & Iglesias (1992) (OPAL), as made available at the anonymous ftp account. When necessary, this last tabulation was further integrated with opacity values from the Los Alamos Opacity Library (Huebner et al. 1977), computed for the same mixture (Grevesse 1991) of the OPAL and Alexander & Ferguson opacity tables. In all cases, this occurs at large values of temperature and density, where Los Alamos opacities nicely overlap OPAL predictions.

As already indicated, the second critical ingredient for the theory of VLM structures is given by the appropriate evaluation of the EOS for similar dense and cool objects. Courtesy of dr. Chabrier, we were enabled to insert in our evolutionary code the most recent EOS evaluated adopting a free-energy minimization technique (Saumon & Chabrier 1992, Saumon et al. 1995) for a zero metal mixture. According to a well established scenario, the presence of metals should play little or no role in the EOS up to metal contents of the order of the solar metallicity (see, e.g. Hubbard 1994). However, we will mainly refer to computations in the range Z= 0.0001-0.001 which appear suitable for the large majority of galactic globular clusters and, in the same time, should allow a safe use of the zero metal EOS.

With the above quoted assumptions, present computations appear based on the most updated physics presently available. However, VLM stars present another difficulty as given by the treatment of stellar atmospheres. As it has been widely debated (see, e.g., Baraffe et al. 1995 and reference therein) in solar metallicity models below about [FORMULA] (and down to about 2500K ) the Eddington approximation and, thus, the [FORMULA] relation is expected to become unrealistic. According to Burrows et al. (1993) this limits increases above [FORMULA] ( [FORMULA] ) decreasing the star metallicity down to the extreme case Z=0. Because of the lack of suitable model atmospheres, we were forced to neglect such an occurrence. Atmospheric integrations where thus performed adopting the Krishna-Swamy (1966) solar scaled [FORMULA] formula until reaching [FORMULA] or, alternatively, until the onset of convection, where mixing length has been used to evaluate the degree of superadiabaticity. Comparison of models in Burrows et al. (1989, 1993: but see also Fig. 2 in Baraffe et al. (1995)) discloses that in quoted range of temperature our models should tend to be a bit hotter and more luminous than expected on the basis of a proper treatment of the atmosphere.

According to such an evidence, present results can be regarded as an investigation on the effect of updating the physical input (EOS and opacity) in previous results appeared in the literature, as based on similar assumptions about the treatment of the atmospheric layers (as in Dorman et al. 1989, D'Antona & Mazzitelli 1994, 1996). However, as it will be discussed later on, even restricting the investigation to the range of validity of [FORMULA] one can significatively extend previous computations into the range of very low stellar masses. Moreover, it appears that the HR diagram location of the sequence of VLM models is not dramatically affected by the release of Eddington approximation, so that the use of a [FORMULA] relation can be regarded at least as a first order approximation to the expected models behavior.

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