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Astron. Astrophys. 346, 111-124 (1999)

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6. Concluding remarks

We presented a calibration of the asymptotic entropy [FORMULA] and the corresponding mixing-length parameter [FORMULA] for solar-type stars basing on radiation-hydrodynamics models. Despite the fact that [FORMULA] is all that is needed to construct a stellar structure model, it was helpful to translate [FORMULA] into a corresponding [FORMULA] since [FORMULA] proved to be less sensitive to the physical and numerical input to the models than [FORMULA] 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 [FORMULA] 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 [FORMULA] on the stellar parameters. Certainly, the simplicity of this dependence is a major reason for the relative success of MLT to predict [FORMULA], as evidenced by the moderate variation of [FORMULA] in Fig. 5. Looking at relative changes ignores the more fundamental problem of fixing the absolute value of [FORMULA], 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 [FORMULA]. Clearly, with an independent calibration of [FORMULA] 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 hydrodynamical model grids is computationally demanding. Nevertheless, the step towards 3D models is desirable, and first calibrations of [FORMULA] from 3D models are now available for smaller stellar samples: Trampedach et al. (1997) presented a calibration based on 6 models in the solar vicinity, Abbett et al. (1997) presented a calibration for the Sun. The latest version of the calibration of Trampedach et al. (Trampedach 1998, priv. comm.) is consistent with our calibration except for a systematic offset which is caused by differences in the employed low-temperature opacities; the scaling behaviour is similar as far as it is possible to judge from the small set of models. Abbett et al. do not provide [FORMULA] with sufficient precision to allow a critical comparison with our results. Own work is in progress to quantify systematic differences between 2D and 3D models. As already mentioned, first results for the Sun give a change of [FORMULA] by [FORMULA] with respect to the 2D models. We expect systematic changes of similar magnitude for other stars. Hence, it appears unlikely that conclusions of this paper will have to be altered substantially when more 3D results become available.

Last but not least we want to reiterate the warning that the presented calibration of [FORMULA] is only intended to reproduce [FORMULA] and the entropy jump. The detailed temperature profile of the superadiabatic layers is not necessarily represented adequately by an MLT model with our calibrated [FORMULA] (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 emergent radiation field of multidimensional simulations with 1D standard atmospheric models requires a quite different mixing-length parameter than is needed for matching the entropy jump. We do not consider this as a contradiction; it just indicates shortcomings of MLT and the 1D idealization.

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© European Southern Observatory (ESO) 1999

Online publication: May 6, 1999