## 6. Concluding remarksWe presented a calibration of the asymptotic entropy and the corresponding mixing-length parameter for solar-type stars basing on radiation-hydrodynamics models. Despite the fact that is all that is needed to construct a stellar structure model, it was helpful to translate into a corresponding since proved to be less sensitive to the physical and numerical input to the models than itself. We gave only a description of the numerical results without providing an explanation in physical terms - not even on a qualitative level - of the scaling behaviour of which we observe. We shall come back to this interesting issue in the next paper of this series where we can look at it from a broader perspective by including models of sub-solar metallicity. Fig. 3 shows a remarkably simple structure if one considers the complex interplay of fluid flow and radiation which governs the dependence of on the stellar parameters. Certainly, the simplicity of this dependence is a major reason for the relative success of MLT to predict , as evidenced by the moderate variation of in Fig. 5. Looking at relative changes ignores the more fundamental problem of fixing the absolute value of , which in practice is done by taking recourse to empirical constraints. Our RHD models provide a determination of the zero point from first principles. Since in MLT important pieces of the physics of convection are missing - at least within the present physical interpretation of the MLT formulae - our work should not be considered as validation of MLT, even though MLT is capable of matching some of our simulation results quite well. Work is under way to check and apply the hydrodynamical convection models beyond the comparison with the Sun. There are classical procedures which allow the determination of the convective efficiency at various locations in the HRD (position of the red giant branch, shape of the main sequence, evolution of binary stars). A lot of work has already been dedicated to empirical determinations, but conclusions are sometimes conflicting and no clear picture has emerged yet. We suspect that systematic uncertainties are actually often larger than estimated. E.g. Castellani et al. (1999) emphasize discrepancies in fitting the main sequence of open clusters which are related to the temperature-color transformation and the uncertainties in . Clearly, with an independent calibration of at hand one can disentangle both effects. For the case of globular clusters Freytag & Salaris (1999) have studied the effects which are expected from our calibration on the shape of the turn-off and the position of the red giant branch. By using our calibration the effective temperatures of their evolutionary models become essentially unaffected by the uncertainties inherent to MLT. The uncertainties related to the temperature-color transformation remain present but one can at least judge the internal accuracy of the transformation. The precise HIPPARCOS data - in particular for some open clusters - might allow a detailed investigation of these issues. Moreover, helioseismology proved to be an invaluable tool in the case of the Sun, and asteroseismic measurements of the internal stellar structure appear to be a promising way to gain further insight. Already in 2D, the construction of Last but not least we want to reiterate the warning that the
presented calibration of is only
intended to reproduce and the entropy
jump. The detailed temperature profile of the superadiabatic layers is
not necessarily represented adequately by an MLT model with our
calibrated (see Fig. 2). Moreover,
the calibration is not suitable for providing the optimum
mixing-length parameter for convective stellar atmospheres.
Preliminary results for the Sun (Steffen & Ludwig 1999) show
that matching the © European Southern Observatory (ESO) 1999 Online publication: May 6, 1999 |