2 Input physics

The basic physical ingredients used for the complete set of grids of the Geneva group are extensively described in previous papers (see Schaller et al. 1992, hereafter Paper I). With the exception of the equation of state, described in detail in the following subsection, we therefore just mention the main points.

2.1 Equation of state

We have already justified our choice of the MHD equation of state in the introduction. As mentioned there, MHD is one of the two recent equations of state that have been especially successful in modeling the Sun under the strong constraint of helioseismological data. Historically, the MHD equation of state was developed as part of the international "Opacity Project'' (OP, see Seaton 1987, 1992). It was realized in the so-called chemical picture, where plasma interactions are treated with modifications of atomic states, i.e. the quantum mechanical problem is solved before statistical mechanics is applied. It is based on the so-called free-energy minimization method. This method uses approximate statistical mechanical models (for example the nonrelativistic electron gas, Debye-Hückel theory for ionic species, hard-core atoms to simulate pressure ionization via configurational terms, quantum mechanical models of atoms in perturbed fields, etc.). From these models a macroscopic free energy is constructed as a function of temperature T, volume V, and the concentrations $N_1,\ \ldots, N_m$ of the m components of the plasma. The free energy is minimized subject to the stoichiometric constraint. The solution of this minimum problem then gives both the equilibrium concentrations and, if inserted in the free energy and its derivatives, the equation of state and the thermodynamic quantities.

The other of these two equations of state is the one underlying the OPAL opacity project (see Sect. 2.2). The OPAL equation of state is realized in the so-called physical picture. It starts out from the grand canonical ensemble of a system of the basic constituents (electrons and nuclei), interacting through the Coulomb potential. Configurations corresponding to bound combinations of electrons and nuclei, such as ions, atoms, and molecules, arise in this ensemble naturally as terms in cluster expansions. Any effects of the plasma environment on the internal states are obtained directly from the statistical-mechanical analysis, rather than by assertion as in the chemical picture.

More specifically, in the chemical picture, perturbed atoms must be introduced on a more-or-less ad-hoc basis to avoid the familiar divergence of internal partition functions (see e.g. Ebeling et al. 1976). In other words, the approximation of unperturbed atoms precludes the application of standard statistical mechanics, i.e. the attribution of a Boltzmann-factor to each atomic state. The conventional remedy of the chemical picture against this is a modification of the atomic states, e.g. by cutting off the highly excited states in function of density and temperature of the plasma. Such cut-offs, however, have in general dire consequences due to the discrete nature of the atomic spectrum, i.e. jumps in the number of excited states (and thus in the partition functions and in the free energy) despite smoothly varying external parameters (temperature and density). However, the occupation probability formalism employed by the MHD equation of state avoids these jumps and delivers very smooth thermodynamic quantities. Specifically, the essence of the MHD equation of state is the Hummer-Mihalas (1988) occupation probability formalism, which describes the reduced availability of bound systems immersed in a plasma. Perturbations by charged and neutral particles are taken into account. The neutral contribution is evaluated in a first-order approximation, which is good for stars in which most of the ionization in the interior is achieved by temperature (the aforementioned study (Trampedach & Däppen 1998) has verified the validity of this assumption down to the lowest mass of our calculation). For colder objects (brown dwarfs, giant planets), higher-order excluded-volume effects become very important (Saumon & Chabrier 1991, 1992; Saumon et al. 1995). In the common domain of application of the Saumon et al. (1995) and MHD equations of state, Chabrier & Baraffe (1997) showed that both developments yield very similar results, which strongly validates the use of the MHD equation of state for our mass range of 0.4 to 1.0 ${M_{\odot}}$.

Despite undeniable advantages of the physical picture, the chemical picture approach leads to smoother thermodynamic quantities, because they can be written as analytical (albeit complicated) expressions of temperature, density and particle abundances. In contrast, the physical picture is normally realized with the unwieldy chemical potential as independent variable, from which density and number abundance follow as dependent quantities. The physical-picture approach involves therefore a numerical inversion before the thermodynamic quantities can be expressed in their "natural'' variables temperature, density and particle numbers. This increases computing time greatly, and that is the reason why so far only a limited number of OPAL tables have been produced, only suitable for stars more massive than $\sim$0.8 ${M_{\odot}}$. Therefore we chose MHD for its smoothness, availability, and the possibility to customize it directly for our calculation, despite the - in principle - sounder conceptual foundation of OPAL.

2.2 Opacity tables, treatment of convection, atmosphere and mass loss

• The OPAL radiative opacities from Iglesias & Rogers (1996) including the spin-orbit interactions in Fe and relative metal abundances based on Grevesse & Noels (1993) are used. These tables are completed at low temperatures below 10000 K with the atomic and molecular opacities by Alexander & Fergusson (1994).
• We use a value of 1.6 for the mixing length parameter $\alpha$. Various observational comparison support this choice. $\alpha = 1.6 \pm 0.1$ leads to the best fit of the red giant branch for a wide range of clusters (see Paper I). It is also the value we get for the calibration of solar models including the same input physics (Richard et al. 1996).
• A grey atmosphere in the Eddington approximation is adopted as boundary condition. Below $\tau = 2/3$, full integration of the structure equations is performed. We discuss the implications of such an approximation in Sect. 4.
• Evolution on the pre-main sequence and on the main sequence are calculated at constant mass. On the red giant branch, we take mass loss into account by using the expression by Reimers (1975): $\dot{M} = 4 \ 10^{-13} \eta L$ R/M (in ${M_{\odot}}$ yr^-1) where L, M and R are the stellar luminosity, mass and radius respectively (in solar units). At solar metallicity, $\eta=0.5$ is chosen (see Maeder & Meynet 1989). At Z=0.001, the mass loss is lowered by a factor (0.001/0.020)^0.5 with respect to the models at Z=0.020 for the same stellar parameters.

2.3 Nuclear reactions

Nuclear reaction rates are due to Caughlan & Fowler (1988). The screening factors are included according to the prescription of Graboske et al. (1973).

Deuterium is destroyed on the pre-main sequence at temperatures higher than 10⁶ K by $D(p,\gamma)^3$He and, to a lower extent, by $D(D,p)^3{\rm H}(\rm e^- \nu)^3$He and D(D,n)³He. We take into account these three reactions. In order to avoid the follow-up of tritium, we consider the last two reactions as a single process, D(D, nucl)³He. The $\beta$ desintegration is considered as instantaneous, which is justified in view of the lifetime of tritium ($\tau _{1/2}=12.26$ yr) compared with the evolutionary timescale. The rate of the D(D, nucl)³He reaction is written as

$$<DD\gt _{\rm nucl} = (1 + {{<DD\gt _p}\over{<DD\gt _n}} ) <DD\gt _n$$

where we take for ${{<DD\gt _p}\over{<DD\gt _n}}$ a mean value of 1.065, in agreement with the rates given by Caughlan & Fowler (1988). The corresponding mean branch ratios are I_p = 0.5157 and I_n = 0.4843.

2.4 Initial abundances

• The initial helium content is determined by $Y=0.24+(\Delta Y/\Delta Z)Z$, where 0.24 corresponds to the current value of the cosmological helium (Audouze 1987). We use the value of 3 for the average relative ratio of helium to metal enrichment ($\Delta Y/\Delta Z$) during galactic evolution. This leads to (Y, Z) = (0.300, 0.020) and (0.243, 0.001). In addition, computations were also performed with (Y, Z) = (0.280, 0.020).
• The relative ratios for the heavy elements correspond to the mixture by Grevesse & Noels (1993) used in the opacity computations by Iglesias & Rogers (1996).
• Choosing initial abundance values for D and ³He is more complex. Pre-solar abundances for both elements have been reviewed in Geiss (1993), however galactic chemical models face serious problems to describe their evolution (see e.g. Tosi 1996), and no reliable prescription exists to extrapolate their values in time. Deuterium is only destroyed by stellar processing since the Big Bang Nucleosynthesis, so that its abundance decreases with time (i.e. with increasing metallicity). On the other hand, the actual contribution of stars of different masses is still subject to a large debate (Hogan 1995; Charbonnel 1995; Charbonnel & Dias 1998), and observations of (³He/H) in the proto-solar nebulae (Geiss 1993) and in galactic HII regions present a large dispersion. We adopt the same (D/H) and (³He/H) initial values for both metallicities, namely $2.4\ 10^{-5}$ and $2.0 \ 10^{-5}$ respectively.

2.5 Initial models

At the low mass range considered in this paper, the observed quasi-static contraction begins rather close to the theoretical deuterium main sequence, once the stars emerge from their parental dense gas and dust. For the present purposes then, we take as starting models polytropic configurations on the Hayashi boundary, neglecting the corrections brought to isochrones and upper tracks by the modern accretion paradigm of star formation (Palla & Stahler 1993; Bernasconi & Maeder 1996). For the mass range considered here, these corrections are likely not to exceed 3$\%$ of the Kelvin-Helmholtz timescale for the pre-main sequence contraction times (Bernasconi 1996). We note, however, that the predicted upper locus for the optical appearance of T Tauri stars in the HR diagram can be as much as half less luminous than the deuterium ignition luminosity on the convective tracks ($\Delta \log L \approx$ 0.3).