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Astron. Astrophys. 347, 272-276 (1999)

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3. Calculations and results

We have added Eq. (1) into the diffusion part of our SSM program, which already takes into consideration particle diffusion (coefficients calculated according to Thoul et al. 1994) of hydrogen, helium and 7 heavier elements, among them 7Li. Turbulent convection is treated as a diffusive process as well, with the diffusion velocity calculated within our convection theory approach.

The solar models we calculate correspond to those described in Schlattl et al. (1997), with some modifications: we now use the nuclear reaction cross sections recommended by Adelberger et al. (1998) and treat nuclear burning and diffusion simultaneously as one system of equations. In addition, the T-P-stratification of the stellar envelopes throughout the whole evolution is taken from 2d-hydro-models provided by H.-G. Ludwig (private communication). These are compatible with those of Freytag et al. (1996). They are extending down to an optical depth of [FORMULA] and are continued by convective layers with convection treated according to Canuto & Mazzitelli (1991; 1992), where the usual mixing-length parameter [FORMULA] is used. All calculations are full solar model calculations, implying the inclusion of the PMS phase and a complete calibration of the free parameters. Fig. 1 shows that our SSM is fully competitive with the best ones published so far.

We have performed calculations of standard solar models including overshooting (SSMO) for the cases listed in Table 1. For overshooting from the convective envelope (downwards) and the core (upwards) we have applied different values of f. The final solar models were investigated for their agreement with the seismic model of Basu et al. (1996). Helium and metal diffusion is always present. Except for case 1a, all calculations were started on the PMS.


Table 1. Parameter f (Eq. (1)) for overshooting from envelope and core, final 7Li abundance and calibrated values of initial helium content [FORMULA] and mixing length parameter [FORMULA] for the SSMO calculations; for comparison, the standard case (G4=GARSOM4) is listed as well

We first applied [FORMULA] to reproduce the result by Blöcker et al. (1998), with the difference that no overshooting during the PMS evolution was included. This is because Blöcker et al. (1998) did not specify the value for f they used but only quoted the change in lithium abundance (-0.3 dex) until the ZAMS. Fig. 3 demonstrates that we obtain a final depletion of 200 in our SSMO. The seismic properties, however, change only partially as intended (Fig. 2, short-dashed line). Although the deviation just beneath the convective boundary is removed, a new region of similar deviation but larger extent exists below, such that the total deviation has grown somewhat. The core remains almost unaffected, although the neutrino flux increases slightly (Table 2) due to a temperature increase.

[FIGURE] Fig. 2. Comparison of solar models with overshooting (see text) with the seismic model by Basu et al. (1996). Our standard model (GARSOM4, solid line) is shown for comparison

[FIGURE] Fig. 3. 7Li-abundance as a function of time for the different values of the (envelope) overshooting parameter


Table 2. Predicted event rates of the Gallex ("Ga"), Homestake ("Cl") and Super-Kamiokande ("SK") neutrino experiments for the SSMO models and the standard case (G4=GARSOM4)

The assumption that the overshooting parameter might be the same also for the PMS evolution results, for [FORMULA], in a complete destruction of lithium (case 1a). Therefore, a smaller (but still constant) value of [FORMULA] was chosen for case 2 (this value is the one Herwig et al. 1997 employed for envelope overshooting in asymptotic giant branch stars). In this case, almost the same final 7Li abundance is reached already on arrival on the ZAMS, with only minor further depletion thereafter due to sedimentation. This constancy during the main-sequence evolution is in contradiction to the observed correlation between the lithium abundance in open cluster stars (of solar mass) and time, which indicates a gradual depletion during the main-sequence phase (Chaboyer 1998; Jones et al. 1997; Jones et al. 1999). The sound speed profile and the neutrino rates lie in between the standard case and case 1, with hardly any noticable change (Fig. 2, dotted line, and Table 2). Brun et al. (1998b) found a rather similar effect on the sound speed profile, when introducing turbulent diffusion in the same region, while Richard et al. (1996) noticed a small improvement for their implementation of rotation-induced mixing.

Next, we tested the effect of overshooting on the small convective core, which exists for a few 10 million years during the PMS evolution. Assuming [FORMULA] for the core (case 3) leads to a completely non-standard solar evolution (Fig. 4) with a persistent convective core of [FORMULA] and strongly reduced neutrino rates (Table 2). We do not show the comparison with the seismic model, because even in cases 4 and 5 ([FORMULA] and 0.005), where the convective core vanishes shortly after respectively before the solar age (Fig. 4), the SSMO deviate strongly from the seismic models (Fig. 2, long-dashed and dash-dotted lines). While the effect on the core structure is well known (see, e.g., Richard & Vauclair 1997), this experiment demonstrates that for overshooting from the core one can allow only for values of f much smaller than those required for lithium depletion, which in all these three cases amounts to either the same as in case 1 (for [FORMULA]) or to that of the standard case GARSOM4 (for [FORMULA]).

[FIGURE] Fig. 4. Evolution of the convective core for different assumptions about the overshooting parameters

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© European Southern Observatory (ESO) 1999

Online publication: June 18, 1999