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Astron. Astrophys. 363, 675-691 (2000)

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7. Evolutionary models

Basically the physics employed for the calculation of models is the same as in Morel et al. (1997b). The ordinary assumptions of stellar modeling are made, i.e. spherical symmetry, no rotation, no magnetic field, no mass loss. It has been already claimed, see Morel et al. (2000a), that stellar models of about one solar mass, computed with the same physics but initialized either on pre main-sequence or at zero-age homogeneous main-sequence are almost identical after a few [FORMULA] years, a small quantity with respect to the expected age of the [FORMULA] Cen system. To save computations, along the search of a solution with the [FORMULA] minimization, we have initialized each evolution with a homogeneous zero-age main-sequence model. The models presented in Table 5 include the pre main-sequence evolution. They are initialized with homogeneous zero-age stellar model in quasi-static contraction with a central temperature close to the onset of the deuteron burning, i.e. [FORMULA] MK.

7.1. Nuclear and diffusion network

We have taken into account the most important nuclear reactions of PP+CNO cycles (Clayton 1968). The relevant nuclear reaction rates are taken from the NACRE compilation (Angulo et al. 1999) with the reaction [FORMULA] taken from the compilation of Adelberger et al. (1998). Weak screening (Salpeter 1954) is assumed. We have used the meteoritic value (Grevesse & Sauval 1998) for the initial lithium abundance, [FORMULA]. For the calculations of the depletion, lithium is assumed to be in its most abundant isotope [FORMULA] form. The initial abundance of each isotope is derived from isotopic fractions and initial values of [FORMULA] and Z in order to fulfill the basic relationship [FORMULA] with [FORMULA]. For the models computed for the [FORMULA] fitting, the isotopes [FORMULA], [FORMULA], [FORMULA] are set at equilibrium.

Microscopic diffusion is described by the simplified formalism of Michaud & Proffitt (1993) with heavy elements as trace elements. We have neglected the radiative accelerations as they amount only to a tiny fraction of gravity in the radiative part for stars with masses close to the solar one (Turcotte et al. 1998). We assume that changes of Z, as a whole, describe the changes of metals and we use the approximation:


with [FORMULA] is the iron mass fraction within Z.

7.2. Equation of state, opacities, convection and atmosphere

We have used the CEFF equation of state (Christensen-Dalsgaard & Däppen 1992) and the opacities of Iglesias & Rogers (1996) complemented at low temperatures by Alexander & Ferguson (1994) opacities for the solar mixture of Grevesse & Noels (1993). We have not taken into account the changes of abundance ratios between the metals within Z due to diffusion; for stellar masses close to the solar one they do not really affect the structure of models (Turcotte et al. 1998).

In the convection zones the temperature gradient is computed according to either MLTBV or MLTCM convection theories. The mixing-length is defined as [FORMULA], where [FORMULA] is the pressure scale height. The convection zones are mixed via a strong full mixing turbulent diffusion coefficient [FORMULA] which produces a homogeneous composition (Morel 1997a).

At the end of the pre main-sequence both components, have for a few million years, a temporary convective core. For [FORMULA] Cen A, slightly more massive than the Sun, a second convective core is formed during the main-sequence due to the onset of the CNO burning (e.g. Guenther & Demarque 2000). Following the prescriptions of Schaller et al. (1992) we have calibrated models with overshooting of convective cores over the distance [FORMULA] where [FORMULA] is the core radius.

The atmosphere is restored using a grid of [FORMULA] laws, provided by Cayrel (2000), ([FORMULA] is the Rosseland optical depth) derived from atmosphere models with the solar mixture of Grevesse & Noels (1993) and metallicity [FORMULA]. The atmosphere models were computed with the Kurucz (1991) ATLAS12 package. The connection with the envelope is made at the optical depth [FORMULA] where the diffusion approximation for radiative transfer becomes valid (Morel et al. 1994). A smooth connection of the gradients is insured between the uppermost layers of the envelope and the optically thick convective bottom of the atmosphere. It is an important requirement for the calculation of eigenmode frequencies. The radius [FORMULA] of any model is taken at the optical depth [FORMULA] where [FORMULA]. Typically, [FORMULA] increases from [FORMULA] in the initial pre main-sequence model, until [FORMULA] at the present time. The mass of the star [FORMULA] is defined as the mass enclosed in the sphere of radius [FORMULA]. The external boundary is located at the optical depth [FORMULA], where the density is fixed to its value in the atmosphere model [FORMULA] g cm-3. To simplify, the chemical composition within the atmosphere models is assumed to be unaffected by the diffusion.

7.3. Numerics

Models have been computed using the CESAM code (Morel 1997a). The numerical schemes are fully implicit and their accuracy is of the first order for the time and third order for the space. Each model is described by about 600 mass shells, this number increases up to 2100 for the models used in seismological analysis. Evolutions are described by about 80 models. About half of them concerns the pre main-sequence evolution.

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© European Southern Observatory (ESO) 2000

Online publication: December 11, 2000