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

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1. Introduction

The structure and evolution of late-type stars is intimately related to convective transport processes taking place in their outer envelopes. Although the underlying physical principles are well known, the non-linear and non-local character of the equations describing the convective motions in a radiating, partially ionized fluid has hampered the development of a closed analytical theory. Hitherto standard stellar structure models include the convective energy transport in the framework of mixing-length theory (MLT). Despite its heuristic nature MLT has proven to be rather successful and is still the working horse in stellar structure modeling. During recent years much effort has been put into the refinement of the theoretical description of microphysical stellar plasma properties - the opacity and the equation of state. To fully benefit from these improvements, an accompanying development of the description of hydrodynamical transport properties appears to be necessary, in the first place aiming at a better understanding of the convective energy transport. In this paper we report on such an effort relying on direct numerical simulations of convective flows in solar-type stars.

Primarily due to the heavy demands on computational resources, RHD simulations have concentrated very much on the Sun, and only rather few models have been constructed for other objects (Nordlund 1982, Steffen et al. 1989, Nordlund & Dravins 1990, Ludwig et al. 1994, Freytag et al. 1996). This sparseness has prevented a broader application of results of RHD simulations to the modeling of stellar atmospheres and evolution. We try to overcome this limitation in this work by computing a grid of RHD models , allowing a systematic study of the characteristics of convective flows in a larger region of the Hertzsprung-Russell diagram (HRD). We point out that the RHD simulations have now reached a level of sophistication far beyond idealized, simple numerical experiments. We consider it an important aspect of our work that it presents a quantitative study and - as such - puts emphasis on the estimation of the achieved accuracy. Aiming at an application in the field of stellar evolution, we concentrate on the evaluation of one structural quantity, namely the entropy of the adiabatically stratified layers deep in the convective stellar envelope. In the following we shall refer to this entropy as [FORMULA]. Its relevance for stellar structure stems from the fact that it strongly influences the radius of a star. To obtain handy numbers, all entropy values in this paper - unless stated otherwise - are given in units of [FORMULA]. To link our results more directly to standard stellar structure modeling we translate [FORMULA] into an equivalent mixing-length parameter.

Some words concerning our nomenclature: we use the term "solar-type" for stars with extended convective envelopes where the thickness of the superadiabatic layers at the top of this envelope is small in comparison to the stellar radius. "Grey" radiative transfer means that frequency-independent (mean) opacities were used in the computation of the radiation field. The opacities still depend on temperature and density and include contributions from spectral lines.

In the paper we proceed as follows: we start with methodical aspects describing our hydrodynamical models, the basic idea and the procedure to derive [FORMULA], and the translation of [FORMULA] into an equivalent mixing-length. We continue with the validation of our method by showing that we are able to predict the solar structure derived from helioseismic measurements within small uncertainties. We then present the calibrations of MLT and CMT. We discuss the application of our results to stellar modeling, point out consequences of stellar stability considerations, present a derivation of the gravity-darkening exponent, and contrast our approach with others. We conclude with future perspectives. In the appendix we provide some auxiliary data helping to utilize our findings in stellar structure models.

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© European Southern Observatory (ESO) 1999

Online publication: May 6, 1999