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Astron. Astrophys. 350, 587-597 (1999)

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4. Theoretical stellar models

4.1. Input physics

The stellar evolution calculations were computed with the CESAM code (Morel 1997) in which we have included appropriate input physics for the mass range under investigation.

The CEFF equation of state (Eggleton et al. 1973, Christensen-Dalsgaard 1991), was used. It includes the Coulomb corrections to the pressure and is appropriate when modeling low-mass stars of mass greater than about 0.6 [FORMULA] (Lebreton & Däppen 1988).

We used the nuclear reaction rates given by Caughlan & Fowler (1988).

We calculated the initial composition of the models either from the Grevesse & Noels (1993) solar mixture (GN93 mixture) or from a GN93 mixture where the [FORMULA]-elements O, Mg, Si, S, Ca, Ti are enriched relative to the Sun ([[FORMULA]/Fe]= +0.4 dex). Such an enrichment of [FORMULA]-elements is observed in metal deficient stars with metallicities [Fe/H] lower than -0.5 (Wheeler et al. 1989, Fuhrmann 1998) and its effect on models has to be taken into account.

We took the most recent OPAL opacities (Iglesias & Rogers 1996) provided for the two mixtures considered. They were complemented at low temperatures by atomic and molecular opacities from Alexander & Ferguson (1994) for the GN93 mixture or from Kurucz (1991) for the [FORMULA]-enriched mixture. Low and high temperatures tables were fitted carefully at a temperature of about 10 000 K, depending upon the chemical composition.

In most models the atmosphere was described with an Eddington [FORMULA] law which is easy to use. In order to estimate the uncertainties resulting from this choice we built a few models using [FORMULA] laws derived from the ATLAS9 atmosphere models (Kurucz 1991). This requires to calculate detailed atmosphere models with ATLAS9 for many values of the gravity, of the effective temperature and of the metallicity and then to derive the corresponding [FORMULA] laws (see Morel et al. 1994). Interpolation of the [FORMULA] laws is then performed to model the considered star. We checked on a test-case, that the use of those better atmospheres does not shift the position of the model in the HR diagram by an amount larger than the observational error bars.

Convection was treated according to the mixing-length theory of Böhm-Vitense (1958). The mixing-length parameter [FORMULA], ratio of mixing-length to pressure scale-height, is a free parameter of the models. As shown by Fernandes et al. (1998) the Sun and four visual binary systems spanning a wide range of masses and metallicities could be calibrated with very close values of [FORMULA]. Moreover the slope of the Hyades main-sequence is well-reproduced with a solar [FORMULA]-value (Perryman et al., 1998). On the other hand, Ludwig et al. (1999) made a calibration of the mixing-length based on 2-D radiation hydrodynamics simulations of solar-type surface convection and found that variations of [FORMULA] of about 10 per cent around the solar value are expected in the domain of effective temperature, gravity and metallicity studied here. We adopt the solar mixing-length value in our calculations. For unevolved stars a change of [FORMULA] of [FORMULA] 0.15 produces a change in effective temperature of the order of 50 K, smaller than the mean observational error on [FORMULA].

With the input physics described above the calibration in luminosity and radius of a solar model having [FORMULA] =0.0245, where X is the hydrogen abundance by mass (Grevesse and Noels 1993) requires a mixing-length [FORMULA], an initial helium abundance Y=0.266 and a metallicity [FORMULA]=0.0175.

4.2. Grids of stellar models and isochrones

In order to determine the ZAMS position, we calculated zero and terminal age main sequences (ZAMS and TAMS) for 10 mass values ranging from 0.5 to 1.4 [FORMULA], 5 helium values ranging from 0.18 to 0.43 and metallicities values Z= 0.004, 0.007, 0.01, 0.015, 0.02, 0.03, 0.04 and 0.06.

To discuss the global features of the HR diagram and the position of particular objects, we calculated detailed evolutionary sequences from the ZAMS to the beginning of the red-giant branch, for 14 masses ranging from 0.5 to 5 [FORMULA] and we derived isochrones using the Geneva isochrone program (Maeder 1974). We chose metallicities corresponding to the observational range (i.e. [Fe/H]=+0.3, 0.0, -0.5 and -1.0) and solar-scaled values of the helium abundance given by:

[EQUATION]

with [FORMULA] (Balbes et al. 1993). This implies a [FORMULA] value of 2.2.

Two distinct grids of models were calculated in the metal deficient cases ([Fe/H] = -0.5 and -1.0): a grid with normal solar mixture and a grid with an [FORMULA]-elements enhancement of 0.4 dex. For the solar mixture grid the metallicity Z in mass fraction is related to [Fe/H] by:

[EQUATION]

where [FORMULA] is the ratio of the solar mixture considered.

For the [FORMULA] -elements enhanced mixture the relation becomes:

[EQUATION]

where [FORMULA], difference between a solar [FORMULA]-elements enhanced mixture and a `normal' solar mixture.

The detailed results of all the stellar models computed for that work will be published separately (Lebreton et al., in preparation) and are available on request.

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© European Southern Observatory (ESO) 1999

Online publication: October 4, 1999
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