Astron. Astrophys. 345, 156-162 (1999)

3. Physics of stellar models

Nuclear network and chemical mixture. The general nuclear network we used contains the following species: 1 H, 3 He , 4 He , 12 C, 13 C, 14 N, 15 N, 16 O, 17 O, and Ex. Ex is a fictitious mean non-CNO heavy element with atomic mass 28 and charge 13, which complements the mixture. With respect to time, due to the diffusion processes, the abundances of heavy elements are enhanced toward the center; Ex mimicks that enhancement for the non CNO metals which contribute to changes of Z, then to opacity variations. We have taken into account the most important nuclear reactions of PP+CNO cycles (Clayton, 1968) with the species ²H, ⁷Li, ⁷Be at equilibrium. The relevant nuclear reaction rates are taken from Caughlan and Fowler (1988); weak screening is assumed.

For helium the initial isotopic ratio is fixed at 3 He /4 He (Gautier & Morel 1997). In the mixture of heavy elements, the ratios between the CNO species are set to their -enriched Allard (1996) values (in number) C: 0.147909, N: 0.038904 and O: 0.616594, with the isotopic ratios: 13 C/12 C, 15 N/14 N, 17 O/16 O (Anders & Grevesse, 1989).

Diffusion. Different processes are participating to element separation or mixing. Whereas gravitational settling creates stratification, hydrodynamical instabilities can, at least in some phases of evolution, generate macroscopic motions which tend to reduce the chemical inhomogeneities. Presently, no general description exists of these processes, which could be easily incorporated in stellar evolution calculations. As the purpose of this paper is to evaluate the influence of elements segregation in comparing theory and observations of subdwarfs, we did not take into account any of these processes. Let us note that rotation, which has been identified as one major cause of partial mixing is almost absent in these old low mass objects.

Microscopic diffusion is described using the formalism of Michaud & Proffitt (1993) valid for main-sequence stars. The radiative forces are not taken into account. For Y the mass conservation equation gives: where the original equation has been slightly modified to take into account the heavy elements; and are respectively the diffusion velocities of hydrogen and helium. This description is parameter free, depending only on known coefficients. A more complete discussion of the method used to treat diffusion is given in Morel et al. (1997).

Equation of state and Opacities. We have used the EFF equation of state (Eggleton et al. 1973) sufficient for our purpose in this mass range.

We have chosen the Livermore Library (Iglesias & Rogers 1996) with the -enriched mixture of Allard (1996), complemented at low temperature opacities by the Kurucz's (1998) -enriched tables. Unfortunately there is no available common mixture in these libraries; we have retained the closest ones with a rough smoothing in the extreme low temperature atmospheric layers.

The opacities and equation of state are functions of the heavy elements content Z, through the number of free electrons and the abundances of efficient absorbers which do not necessarily belong to the nuclear network e.g. ⁵⁶Fe. Due to diffusion and nuclear reactions, Z changes as the star evolves, as well as the ratios between the abundances of chemicals. In the calculation of opacities and equation of state, Z is separated in two parts: the first one consists of the chemicals heavier than helium which, belonging to the CNO nuclear network, are both diffused and nuclearly processed; Ex, the second part, is only diffused. Hence in the estimate of Z, the changes of CNO abundances caused by diffusion, nuclear processes and the effects of the gravitational settling of the heaviest non-CNO species are taken into account.

Convection. In the convection zones the temperature gradient is computed according to the standard mixing-length theory, with the mixing-length defined as , where is the classical pressure scale height. The constancy of has been demonstrated for population I models close to the main-sequence (see e.g. Fernandes et al. 1998), and calibrations of from 2D simulations (Ludwig et al. 1998) lead to approximately the same result; so, we assume that is constant and equal to the value obtained for a solar model using the same physics, namely . This hypothesis has no influence on the results of this paper devoted to an estimate of the differential effect of microscopic diffusion.

In the models with diffusion the convection zones are mixed via a strong turbulent diffusion coefficient, which produces an homogeneous composition.

Atmosphere. An atmosphere is restored using the Hopf 's law: (Mihalas 1978) with as the Rosseland optical depth. The connection with the envelope is made at the optical depth , where the diffusion approximation for radiative transfer becomes valid. In the convective part of the atmosphere, a numerical trick (Henyey et al. 1965) is employed in connection with the purely radiative Hopf 's law in order to ensure the continuity of gradients at the limit between the atmosphere and the envelope. At each time step, the radius of the model is taken at the optical depth where ; the mass of the star , is the mass inside the sphere of radius . The external boundary is fixed at the optical depth , where a boundary condition on the pressure is expressed as: , with g as the gravity and as the Rosseland mean opacity.

Numerics. The models have been computed using the CESAM code (Morel 1997). Typically each evolutionary track needs of the order of 85 models of about 500 mass shells. The accuracy of the numerical scheme is one for the time and three for the space.

© European Southern Observatory (ESO) 1999

Online publication: April 12, 1999
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