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Astron. Astrophys. 318, 783-790 (1997)

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3. Theoretical models

The theoretical models were calculated with a multiple shooting point (Wilson 1981) one-dimensional hydrodynamic code developed by us at the Centro de Astrofísica da Universidade do Porto and the Astrophysical Institute of Vrije Universiteit Brussel. This code assumes a spherically symmetrical model with no rotation and no magnetic fields. In the following sections we summarise the more relevant inputs of the code (details in Figueiredo 1996).

3.1. Equation of state

The EOS is computed using the so-called "chemical picture", yielding the pressure and internal energy as a function of temperature, density and chemical composition. The Saha equation is iteratively solved for the following species: H, H [FORMULA], He, He [FORMULA] and He [FORMULA]. The formation of H [FORMULA] is also taken into account for temperatures below 10 000 K using the dissociation equilibrium constant of Sauval & Tatum (1984). The contribution of H [FORMULA] to the EOS is not included. However, neglecting H [FORMULA] should not affect our results in a significant way since the important role played by this ion as an opacity source is already included in the opacity tables we have used (see further).

The EOS applies to a semi-degenerated gas and includes ion, electron and radiation contributions, as well as corrections accounting for the formation of electron-positron pairs and the Coulomb interactions between charged particles. Coulomb corrections to the pressure and energy, as well as to the (laboratory) ionisation potentials, are considered. They are calculated using the numerical results obtained by Broyles et al. (1963) and Shaviv & Kovetz (1972). In the high-temperature low-density regime one recovers the results of the Debye-Hückel model (see e.g. Cox & Giuli 1968; Graboske et al. 1969). Pressure ionisation occurs each time the effective ionisation potential becomes negative.

3.2. Nuclear network

The nuclear network includes the p-p chains and the CNO tri-cycle (Clayton 1968). The thermonuclear reaction rates are computed using the analytical fitting formulae given by Caughlan & Fowler (1988) except for the 17 O(p, [FORMULA])18 F and 17 O(p, [FORMULA])14 N reaction rates which are taken from Landré et al. (1990). Screening factors accounting for weak, intermediate and strong screening effects are computed according to DeWitt et al. (1973) and Graboske et al. (1973). The evolution of all the fifteen chemical species considered is explicitly followed without imposing them to be in equilibrium. Furthermore, the evolution of species having a number fraction greater than 10-8 is followed on a time scale corresponding to abundance variations smaller than 20 [FORMULA].

3.3. Optical atmosphere

The atmosphere is computed using the gray atmosphere approximation. We assume the temperature and optical depth are related by [FORMULA]. The Hopf function [FORMULA] is computed using an analytical relation obtained by solving the equation of radiative transfer adopting the Wick-Chandrasekhar method (see e.g. Collins II 1989). The diffusion approximation is used to compute the temperature gradient and the radiative pressure only when [FORMULA], since at this optical depth this approximation is guaranteed within 1 [FORMULA] approximately. The stellar radius corresponds to an optical depth where [FORMULA].

3.4. Treatment of superadiabatic regions

The transport of energy in the convective envelopes is treated using the mixing-length theory (MLT) approximation (Vitense 1953; Böhm-Vitense 1958) or the Canuto and Mazzitelli (CM) model (Canuto & Mazzitelli 1991, 1992). Neither overshooting nor convective penetration (Zahn 1991) are considered in this work, since both effects are expected to be negligible in the mass range considered.

3.5. Opacities

For temperatures higher than 11 000 K we use the OPAL radiative opacity tables of Rogers & Iglesias (1994) computed taking the Grevesse & Noels (1993) relative metal abundances for the Sun. Whenever needed these opacities are completed with the opacities from the Los Alamos Opacity Library (LAOL, Huebner el al. 1977) computed with the Ross & Aller (1976) heavy elements mixture. For temperatures under 11 000 K we adopt the Neuforge (1993) opacity tables. If preferred it is possible to use the Kurucz (1992) opacity tables, which apply to hydrogen mass fractions close to 0.70. Both low-temperature sets were computed using the Anders & Grevesse (1989) mixture. The contribution of conductive opacity is also considered using the program of Hubbard & Lampe (1969).

3.6. Abundances

To compute the models we adopt an initial composition corresponding to the Grevesse & Noels (1993) mixture, except for the deuterium which is taken from Reeves (1994). The value adopted for [D/H], [FORMULA], agrees with the one obtained by the Hubble Space Telescope for the stellar gas in front of Capella (Linsky et al. 1992) and with the values obtained along other lines of sight (Vidal-Madjar 1991; McCullough 1992). The mass fractions of hydrogen ([FORMULA]) and helium ([FORMULA]) used in the initial composition of our models is always such that [FORMULA], where [FORMULA] is the metallicity, in agreement with the most recent estimated value for the solar photosphere (Grevesse & Noels 1993).

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© European Southern Observatory (ESO) 1997

Online publication: July 3, 1998