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1 Introduction

In stellar evolution computations, and in particular in the case of stars more massive than the Sun, it is generally sufficient to use a simple equation of state. The plasma of the stellar interior is treated as a mixture of perfect gases of all species (atoms, ions, nuclei and electrons), and the Saha equation is solved to yield the degrees of ionization or molecular formation. In the case of low mass stars however, non ideal effects, such as Coulomb interactions become important. It is then necessary to use a more adequate equation of state than the one employed in the Geneva code for more massive stars. This simple equation of state essentially contains a mixture of ideal gases, ionization of the chemicals is dealt with by the Saha equation, excited states and molecules are neglected, complete pressure ionization is artificially imposed above certain temperatures and pressures, and no Coulomb-pressure correction is included (see Schaller et al. 1992).

For the present grids of models of 0.4 to 1.0 ${M_{\odot}}$ stars these assumptions are obviously inadequate. For such stars, the most useful equations of state, as far as their smooth realization and versatility are concerned, are (i) the so-called Mihalas-Hummer-Däppen (MHD) equation of state (Hummer & Mihalas 1988; Mihalas et al. 1988; Däppen et al. 1988), and (ii) the OPAL equation of state, the major alternative approach developed at Livermore (Rogers 1986, and references therein; Rogers et al. 1996). A brief description of these two equations of state is given in the next section.

Here, we chose the MHD equation of state. First, we were able to compute very smooth tables specifically for our cases of chemical composition, instead of relying on pre-computed, relatively coarse tables that would require interpolation in the chemical composition. Second, our choice was forced by the fact that the currently available OPAL equation of state tables do not allow to go below stars less massive than $\sim$0.8 ${M_{\odot}}$. Third, we validated our choice by a comparative calculation with OPAL at its low-mass end. We found results that are virtually indistinguishable from MHD. Fourth, we examined in a parallel theoretical study (Trampedach & Däppen 1998) the arguments about the validity of the MHD equation of state down to the limit of our calculation of 0.4 ${M_{\odot}}$ (see below). Therefore we do not have to include a harder excluded-volume term such as the one included in the Saumon-Chabrier (SC) equation of state (Saumon & Chabrier 1991, 1992).

Although the MHD equation of state was originally designed to provide the level populations for opacity calculations of stellar envelopes, the associated thermodynamic quantities of MHD can none the less be reliably used also for stellar cores. This is due to the fact that in the deeper interior the plasma becomes virtually fully ionized. Therefore, in practice, it does not matter that the condition to apply the detailed Hummer-Mihalas (1988) occupation formalism for bound species is not fulfilled, because essentially there are no bound species. Other than that, the MHD equation of state includes the usual Coulomb pressure and electron degeneracy, and can therefore be used for low-mass stars and, in principle, even for envelopes of white dwarfs (W. Stolzmann, private communication). The present paper, with its MHD-OPAL comparison (see Sect. 3) corroborates this assertion.

This broad applicability of the MHD equation of state for entire stars was specifically demonstrated by its successes in solar modeling and helioseismology (Christensen-Dalsgaard et al. 1988; Charbonnel & Lebreton 1993; Richard et al. 1996; Christensen-Dalsgaard et al. 1996). A solar model that is based on the MHD equation of state from the surface to the center is in all respects very similar to one based on the OPAL equation of state, the major alternative approach developed at Livermore (Rogers 1986, and references therein; Rogers et al. 1996). This similarity even pertains to the theoretical oscillation frequencies that are used in comparisons with the observed helioseismic data.

Although the difference between the MHD and OPAL equations of state is of helioseismological relevance, it has no importance for the lower-mass stellar modeling of the present analysis. This is explicitly validated in the present paper. For the much finer helioseismological analyses, it turned out that in some respect the OPAL model seems to be closer than the MHD model to the one inferred from helioseismological observations (Christensen-Dalsgaard et al. 1996; Basu & Christensen-Dalsgaard 1997). However, we stress that for the present stellar modelling these subtle differences are no compelling reason to abandon the convenience of our ability to compute MHD equation of state tables ourselves, and to go below the range of the available OPAL tables ($\sim$0.8 ${M_{\odot}}$).

Not only helioseismology, but also fine features in the Hertzsprung-Russell diagram of low- and very low-mass stars impose strong constraints on stellar models (Lebreton & Däppen 1988; D'Antona & Mazzitelli 1994, 1996; Baraffe et al. 1995; Saumon et al. 1995). They have all confirmed the validity of the principal equation-of-state ingredients employed in MHD (Coulomb pressure, partial degeneracy of electrons, pressure ionization). Finally, we have checked that even at the low-mass end of our calculations the physical mechanism for pressure ionization in the MHD equation of state is still achieved by the primary pressure ionization effect of MHD (the reduction of bound-state occupation probabilities due to the electrical microfield; see Hummer & Mihalas 1988). Such a verification was necessary to be sure that our results are not contaminated by the secondary, artificial pressure-ionization device included in MHD for the very low-temperature high-density regime (the so-called $\Psi$ term of Mihalas et al. 1988). A parallel calculation has confirmed that in our models a contamination by this $\Psi$ term can be ruled out (Trampedach & Däppen 1998).

In the present paper, we expand the current mass range of the Geneva evolution models from 0.8 down to 0.4 ${M_{\odot}}$, by using a specifically calculated set of tables of the MHD equation of state. This work aims to complete the base of extensive grids of stellar models computed by the Geneva group with up-to-date input physics (Z=0.020 and 0.001, Schaller et al. (1992); Bernasconi (1996), and Charbonnel et al. (1996); Z=0.008, Schaerer et al. (1992); Z=0.004, Charbonnel et al. (1993); Z=0.040, Schaerer et al. (1993); Z=0.10, Mowlavi et al. (1998); enhanced mass loss rate evolutionary tracks, Meynet et al. (1994)). In Sect. 2, we present the characteristics of our equation of state and recall the physical ingredients used in our computations. In Sect. 3, we summarize the main characteristics of the present models and discuss the influence of the equation of state on the properties of low mass stars. Finally, we compare our solar-metallicity models with recent models computed by other groups and with observations in Sect. 4.

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