## 3. Input physicsWe 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 updateThe 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 modelThe 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 ## 3.3. Treatment of overshootingConvective 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 mixingSince 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 Note that within the MLT the physical reliability of the above
choice for 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 convectionWe 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 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 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 We are aware that this is the most disputable of our choices. Note however the following three arguments. - - - © European Southern Observatory (ESO) 1998 Online publication: March 3, 1998 |