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

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5. Implications and discussion

5.1. Calibrating [FORMULA] with evolutionary models of the Sun?

The RHD models predict almost no change of [FORMULA] during the solar main-sequence evolution and at most a very weak dependence of the convective efficiency on the helium abundance in the convective envelope (see Fig. 5 and Table 1). This is an important piece of information since it justifies the common practice of calibrating the solar [FORMULA] with evolutionary models. However, we think that the accurate knowledge about the solar envelope structure from helioseismology now provides a cleaner way to calibrate the efficiency of solar convection by the procedure outlined in Sect. 3. Calibrating [FORMULA] with the help of the present envelope structure would provide insight into the validity of parts of the physics put into the evolutionary model calculations.

For instance, calibrated solar models with and without element diffusion, respectively, provide different values for [FORMULA] (cf. e.g. Richard et al. 1996). Models including diffusion give a lower helium abundance at the surface for the present Sun and a somewhat higher [FORMULA] than models without diffusion. This might appear to contradict our findings, namely that [FORMULA] is almost independent of the helium content. The reason is of course that the evolutionary models without diffusion do not represent the Sun since their present surface helium abundance is not the actual one. This shortcoming of an evolutionary model without diffusion is compensated for by a change of [FORMULA]. As a price, no assessment of the quality of the model or the adequacy of the input physics is possible any more. If, however, one calibrated [FORMULA] at the present envelope structure, shortcomings in the input physics would not be hidden in the [FORMULA] calibration but would show up as a mismatch between the predicted and the actual solar radius.

5.2. How to make best use of the calibration data in evolutionary models?

In the case of the Sun our calibration underestimates the absolute value of the mixing-length parameter by about [FORMULA], a value which can be explained by a combined systematic effect due to the too small low-temperature opacities and the 2D approximation. The scaling behaviour of the [FORMULA] with effective temperature and surface gravity is a differential result and will be much less affected by these systematic shortcomings. From this perspective it seems that a scaling of the [FORMULA] values presented here by a constant factor (slightly larger than unity) is still admissible. In the context of stellar evolutionary models we suggest to calibrate [FORMULA] at the present Sun and use the ratio to [FORMULA] from our calibration as a scaling factor. We propose a constant scaling factor since it appears plausible to compensate for the systematic offset seen in the Sun in a simple, homogeneous fashion. For the time being this is the best one can do since the Sun is the only star where we know [FORMULA] with sufficient accuracy to critically test our results observationally. However, we emphasize that at the moment the scaling is a well-motivated modification to our results. We expect to eliminate the necessity of this step by improving on the identified systematic shortcomings of our approach in future models.

The simplified radiative transfer in the RHD models provides only an approximate atmospheric [FORMULA]-relation. We have seen by comparison of grey and non-grey models that the influence of the [FORMULA]-relation on the determination of [FORMULA] can be strongly reduced if one sticks to the same approximation in the envelope and RHD models. In this way we separated to some extend properties of convection from issues related to opacities and radiative transfer. If one now aims at a comparison with observations, the radiative properties should be modeled as accurately as possible. In evolutionary models intended to reproduce observations (e.g. colors) one should use as accurate [FORMULA]-relations as possible, preferentially derived from full-fledged model atmospheres. The [FORMULA] values from our calibration are not affected by such a change to a moderately differing [FORMULA]-relation.

5.3. Consistency with stellar stability

A variation of [FORMULA] and corresponding variation of [FORMULA] with the atmospheric parameters [FORMULA] and [FORMULA] can lead to a secular instability of a star on the Kelvin-Helmholtz timescale of its convective envelope. In order to see this, let's consider a small perturbation of [FORMULA] in a star's envelope. As known from stellar structure models such a disturbance alters its radius while the luminosity remains largely unchanged. The change in radius leads to a change of the surface parameters. Following Christensen-Dalsgaard (1997, his relations (10) and (12)) one approximately finds for the differential relation between [FORMULA] and [FORMULA] valid for stars similar to the Sun


where [FORMULA] is the typical specific heat at constant pressure in the convective envelope.

While relation (2) describes the response of the global stellar structure on changes of [FORMULA] the question is whether this is compatible with the surface conditions. Since from the surface conditions [FORMULA] is a function of [FORMULA] and [FORMULA], there is the possibility of a feedback on [FORMULA] which can amplify or damp an perturbation of [FORMULA]. One would obtain an amplification if [FORMULA] drops more strongly with [FORMULA] than given by relation (2). In case of an amplification, rapid changes of the stellar surface parameters would occur, leading to regions in the HRD devoid of stars.

Clearly, we do not observe such "gaps" in the region of the HRD studied here, and indeed one can verify with the aid of Fig. 3 that [FORMULA] as controlled by the surface conditions exhibits a stabilizing behaviour since


Similar relations hold for the atmospheric entropy minimum (not shown) and the entropy jump (see Fig. 4). Hence, all factors influencing [FORMULA] suggest that the convective instability described above will not be encountered. However, at least in principle, it cannot be excluded that under special circumstances a convectively driven runaway might occur.

5.4. The gravity-darkening 1 exponent

On the surface of a slowly rotating star the relation between local [FORMULA] and gravity g can be approximated by


Lucy (1967) argued that the so called gravity-darkening exponent [FORMULA] for stars with convective envelopes is given by


Fig. 3 represents lines [FORMULA], so the gravity-darkening exponent given by our RHD simulations can be readily deduced from this plot. We find an increase of [FORMULA] from 0.07 to 0.10 when going from the F- to the K-dwarfs, and a slight decrease of [FORMULA] with decreasing gravity. Basically, this confirms Lucy's result [FORMULA], with the novelty that our approach eliminates the weakest point of his analysis, namely that MLT provides a reasonable scaling relation [FORMULA] with constant [FORMULA]. Current observations of [FORMULA] (see e.g. Alencar & Vaz 1997) are consistent with our findings, but show a rather large scatter and do not allow a critical test of our results. We therefore refrain from a further discussion here.

5.5. Contrasting ours with other approaches

The long-standing lack of a reliable theory of convection has prompted numerous attempts to remedy the situation. In the following we comment on the work of two groups and highlight the major differences in the involved physics between their's and our approach, hoping to clarify the possible reasons for deviating results.

In a series of three papers Lydon et al. (1992, 1993a,b; hereafter LFS) presented a formulation of convective transport based on results of numerical experiments by Chan & Sofia (1989). The idealized numerical experiments were set up to describe the generic properties of almost adiabatic convection; radiative energy transport was included only as a diffusive flux computed with constant conductivity. Chan & Sofia extracted local statistical relations from their numerical data which were subsequently used by LFS to derive an expression for the convective energy flux suitable for calculating stellar structure models. LFS modeled the Sun as well as the A- and B-component of the [FORMULA] Centauri system. They found essentially the same [FORMULA] in all cases when they translated their formulation into an effective mixing-length parameter. This is consistent with our results, but in our opinion not a strong statement since the rather small differences in the surface parameters and chemical composition among the three stars makes it difficult to detect changes of [FORMULA]. Moreover, there is a principle problem with the LFS convection formulation since it relies on numerical experiments for adiabatic convection. In a follow-up project to the work of Chan & Sofia, Kim et al. (1995) found significantly different statistical relations among the fluctuating quantities in the superadiabatic regime, probably implying that larger uncertainties have to be attributed to the original results of LFS.

In CMT the main improvement with respect to MLT was the inclusion of a larger spectrum of eddies supposed to exist in a turbulent medium. Radiation was only included in a MLT-like fashion, and effects due to the compressibility of the medium were neglected. In our investigation we have regarded the characteristic length scale in CMT as a free parameter, adjusting it to fit the results of our simulations. If we compare the calibration of CMT shown in Fig. 6 with standard MLT shown in Fig. 5, we find a more extended plateau in the solar vicinity. The general tendency of CMT to make efficient convection more efficient and inefficient convection less efficient results in a smoother behaviour relative to MLT. But towards high and low effective temperatures, a clear over-compensation occurs, leading to a change of [FORMULA]. In MLT as well as in CMT, a significant variation of the free parameter is needed, and no simple scaling is apparent. We do not find a qualitative difference in their ability to reproduce [FORMULA] from the RHD simulations.

LFS as well as CMT essentially relate the convective flux to the local conditions in the flow. Such an approach is not well motivated from the simulation results where convection appears to be an extremely non-local phenomenon governed by the processes taking place in the thin cooling layer at the stellar surface (see Spruit 1997 for a recent discussion) while the bulk of the convection zone adapts to these layers. Even though we fit our simulation results to local theories, we do not assume that convection can be described in local terms since the fit is of purely formal nature. Furthermore, both LFS and CMT treat the effects of radiative transfer in a rudimentary fashion, despite the well-known fact that for a quantitative description of stellar convection a proper treatment of the radiative transfer is crucial. Indeed, much effort goes into the realistic modeling of radiative transfer in our RHD simulations. CMT describes convection within the picture of incompressible turbulence. However, it is just the compressibility of the stellar gas that gives turbulence in stars its special inhomogeneous character (see Nordlund et al. 1997 for a detailed discussion). In contrast, our RHD simulations fully include the effects of compressibility.

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© European Southern Observatory (ESO) 1999

Online publication: May 6, 1999