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

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A calibration of the mixing-length for solar-type stars based on hydrodynamical simulations

I. Methodical aspects and results for solar metallicity

Hans-Günter Ludwig 1, Bernd Freytag 1,2 and Matthias Steffen 3,2

1 Astronomical Observatory, Juliane Maries Vej 30, DK-2100 Copenhagen O, Denmark (hgl, bf@astro.ku.dk)
2 Institut für Theoretische Physik und Astrophysik der Universität Kiel, D-24098 Kiel, Germany
3 Astrophysikalisches Institut Potsdam, D-14482 Potsdam, Germany (MSteffen@aip.de)

Received 28 September 1998 / Accepted 22 March 1999

Abstract

Based on detailed 2D numerical radiation hydrodynamics (RHD) calculations of time-dependent compressible convection, we have studied the dynamics and thermal structure of the convective surface layers of solar-type stars. The RHD models provide information about the convective efficiency in the superadiabatic region at the top of convective envelopes and predict the asymptotic value of the entropy of the deep, adiabatically stratified layers (Fig. 3). This information is translated into an effective mixing-length parameter [FORMULA] suitable to construct standard stellar structure models. We validate the approach by a detailed comparison to helioseismic data.

The grid of RHD models for solar metallicity comprises 58 simulation runs with a helium abundance of [FORMULA] in the range of effective temperatures [FORMULA] and gravities [FORMULA]. We find a moderate, nevertheless significant variation of [FORMULA] between about 1.3 for F-dwarfs and 1.75 for K-subgiants with a dominant dependence on [FORMULA] (Fig. 5). In the close neighbourhood of the Sun we find a plateau where [FORMULA] remains almost constant. The internal accuracy of the calibration of [FORMULA] is estimated to be [FORMULA] with a possible systematic bias towards lower values. An analogous calibration of the convection theory of Canuto & Mazzitelli (1991, 1992; CMT) gives a different temperature dependence but a similar variation of the free parameter (Fig. 6).

For the first time, values for the gravity-darkening exponent [FORMULA] are derived independently of mixing-length theory: [FORMULA].

We show that our findings are consistent with constraints from stellar stability considerations and provide compact fitting formulae for the calibrations.

Key words: convection – hydrodynamics – stars: late-type – stars: evolution

Send offprint requests to: Hans-Günter Ludwig

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© European Southern Observatory (ESO) 1999

Online publication: May 6, 1999
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