Astron. Astrophys. 331, 1011-1021 (1998)

3. Input physics

We computed solar evolutionary tracks starting from early pre-MS, prior to D-burning, since our aim was to have physically realistic and thermally relaxed models when starting investigation of Li-depletion during pre-MS phases. The initial Li-abundance has been set to ; we must however recall that the fractional amount of Li-depletion does not depend on the initial value since the reactions have no sensitive feedback on the structure of the star. The initial Li-abundance can be thus scaled up or down when making comparisons with given sets of observations (e.g. the open cluster data sets).

3.1. Microphysics update

The tracks have been computed by making use of the recently updated ATON 2.0 code; a complete description of it and of the input physics may be found in Mazzitelli et al. (1995) and in Ventura et al. (1998). We summarize here the most relevant features:

• 1 for the EOS we adopt Mihalas et al (MHD, 1988) in the range ; ; in the range the OPAL thermodynamics tables (Rogers et al. 1996) are instead used. In DM94 models only of the Mihalas et al. EOS was used;
• 2 we use Alexander & Ferguson (1994) opacities for K, and OPAL updated opacities (Rogers & Iglesias 1992a,b>/A>, Iglesias et al. 1992, Rogers & Iglesias 1993) at larger T: DM94 models were computed either with a preliminary release of Alexander opacities (1993, private communication), or with Kurucz (1993) opacities at low-T, and with the 1992 OPAL tables in the interior, which, around the Li-burning temperatures, displayed lower values of opacity than the following release, as discussed in Iglesias et al. 1992.

3.2. The convection model

The convective temperature gradient in the present models has been always evaluated via an FST model, with the updated turbulent fluxes by Canuto et al. (CGM 1996). In DM94 the FST computations had been performed with the previous Canuto & Mazzitelli (1991) fluxes. Only one test with the MLT has been performed for comparison. A detailed description of the use of the FST model (value of the Kolmogorov constant, convective scale length, a tiny amount of overshooting from the surface convective layer used as a fine tuning parameter , as opposed to the completely free tuning parameter in the MLT etc.) can be found in D'Antona et al. (1997, DCM), where also the main references to the successfull tests performed by various authors in various evolutionary phases can be found; in the following, unless otherwise specified, we will always imply that all the computations have been performed in an FST framework.

3.3. Treatment of overshooting

Convective overshooting should be in principle present in stellar structures, since it is an experimentally well known characteristics of any turbulent fluid. More difficult is to compute, starting from first principles, the amount of overshooting in stellar structures, and yet the information would be relevant when studying Li-depletion in pre-MS, since D'Antona and Mazzitelli (1984) have shown that it can play a crucial role by mixing the bottom of the shrinking convective envelope with deeper and hotter layers.

Overshooting is a non-local outcome of turbulence. Up today, all the attempts at getting simple recipes for stellar overshooting treating non-locality as a perturbation have failed (Canuto 1992). The few attempts at computing overshooting coupled to stellar structure in spite of the enormous numerical difficulties involved (Xiong 1985, Grossman 1996), have provided large "equivalent" overshooting thicknesses from a convective core (), not consistent with the much lower values (, Maeder & Meynet 1987>/A>, 1989; Stothers and Chin 1993) found when comparing observations of MS of young open clusters to results of parametric computations. Aa a matter of fact, the only semiempirical consideration which can help us in the present framework comes from Basu & Antia (1997). They have shown on helioseismological grounds that the overshooting from below the solar convective envelope cannot exceed . We will somewhat arbitrarily extend this latter conclusion to pre-MS stars of different masses, and perform test computations with instantaneous mixing below the formally convective envelope of 0.1 at most.

3.4. Diffusive mixing

Since the nuclear lifetimes at the bottom of the convective envelope can be of the same order of magnitude of the turbulent mixing times, the approximation of instantaneous mixing can be a very crude one, since even in the convective envelope can show a non-flat profile, with a minimum at the convective bottom. Instantaneous mixing is then likely to provide an overestimate of the actual amount of burnt in pre-MS. ATON 2.0 code also allows computations with diffusive mixing according to the diffusion equation (Cloutman & Eoll 1976) describing the local temporal variation of the element, but including (burdensome) full coupling between nuclear evolution and mixing (Sackmann et al. 1974) for all the chemical species considered:

The diffusion coefficient D should be computed according to: , where , the turbulent diffusion timescale, is a function of the one-point density-radial velocity correlation . Unfortunately, knowledge of the density-velocity momentum requires still unavailable solutions of the Navier-Stokes eqs. for a compressible stellar fluid in a huge variety of cases. We will then approximate D with where u is the average turbulent velocity and the FST convective scale length.

Note that within the MLT the physical reliability of the above choice for D can be disputable, since the velocity is the one for the largest (unique) eddy, whereas turbulent mixing is experimentally known to occur at the smallest (dissipative) scales; moreover, the parametrization grows meaningless when approaching the convective boundaries, where the turbulent scale length can be orders of magnitude lower. In an FST environment both the above problems find solution, since the velocity is the weighted average over all the scales, and is just the geometrical distance from the boundary.

In the following, we will always use the instantaneous mixing approximation since we are only interested in internal comparisons; one of these comparisons will be however performed also with a diffusive mixing pre-MS evolution. For more details on the diffusive scheme and on the numerical problems met with the inversion of the huge matrix required, see Ventura et al. (1998).

3.5. Influence of a magnetic field on convection

We finally describe how we dealt with the problem of including the effect of a magnetic field on convection. Stars in the pre-MS phase rotate relatively fast compared to the present Sun (, e.g. Bouvier et al. 1993), thus producing a magnetic field due to dynamo-effect. The rotational rate diminishes with age due to total angular momentum losses (e.g. Pinsonneault et al. 1989, Ghosh 1995). It is thus likely that the Sun itself experienced higher magnetic fields than the current one (surface G).

In the case of MLT, Gough & Tayler (1966) and Moss (1968) included the effect of a small magnetic field B finding that the new criterion to get convective instability becomes:

with where is the ratio of the specific heaths and P the local value of pressure. Including this latter term in the cubic equation for the flux, one then gets the convective (superadiabatic) temperature gradient.

In the FST case, the equation for the flux is not an algebraic one and an analytic solution as in the MLT case cannot be derived. However, Eq. (1) is a modified Schwarzschild cryterion for convective instability independent of the convective model. As for the superadiabatic gradient, we tested the MLT case for values of in the range and found that, as long as , the value of the gradient came out equal to the unperturbed one plus itself, the maximum discrepancy from this rule of thumb being of a very few percent of . Since the rule was found to hold for any flux (low or large), we assumed it to be valid independently of the convective model. We then limited our FST computations to values of , evaluating the FST temperature gradients as the gradients in the absence of magnetic field increased by . This confined us to values of B at the surface lower than G for all the masses considered here. Should it turn out from a more sophysticated analysis that the above rule of thumb is not completely correct, the main framework of the results will not be substantially modified.

The next problem was how to scale the magnetic field from the surface deep inside the star, down to the centre or to the bottom of the convective envelope. Since we are in the line of including in the ATON code rotation according to Endal and Sofia (1976) approximation (Sanctos Mendes et al. 1997), we postpone the problem of evaluating the dynamo-generated field from first principles, and simply assume that the same ratio between magnetic and gas energy density found at the surface was preserved throughout inside the star. This corresponds to assuming that scales like P and, thus, const.

We are aware that this is the most disputable of our choices. Note however the following three arguments.

-Semiempirical argument: in the present sun, the surface magnetic field is about 1 G, while the magnetic field at the bottom of the convective envelope is thought to be of the order of some in G (Spruit 1990), also on the ground of helioseismological results. If we consider the solar pressure stratification we find , almost constant through the envelope.

-Theoretical argument: a fossil magnetic fields at the centre of solar-type stars at the end of the Hayashi phase can be as large as G (Dudurov & Gorbenko 1991) while it is of the order of a few () G at the surface. Also in this case we have , constant throughout the star. The ratio between magnetic and gravitational energy is then nearly constant, and the same we expect from the virial theorem for the ratio magnetic to gas energy in contracting pre-MS stars with no large sources of nuclear energy.

-Epistemic argument: in any case, we restricted our computations to relatively weak magnetic fields; for pre-MS stars, an extreme field larger than G has been observed in TAP 35 (Basri et al. 1992). We can then say that, even if const. inside the star, we expect in real stars larger values of close to the surface than the ones here considered, and perhaps lower values in deep layers, at least partially compensating. Let us then explicitly state that we are, in the present paper, interested to study the qualitative and semi-quantitative effect of the magnetic field due to rotation on Li-depletion; the values of the surface magnetic fields we will quote here () are then not to be taken as the exact values of B at the surface of pre-MS stars, but are related to these latter in a way we will know only when detailed rotating models with dynamo will be available.

© European Southern Observatory (ESO) 1998

Online publication: March 3, 1998
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