A calibration of the mixing-length for solar-type stars based on hydrodynamical simulations I. Methodical aspects and results for solar metallicity

Based on detailed 2D numerical radiation hydrodynamics (RHD) calculations o f 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 supe radiabatic region at the top of convective envelopes and predict the asympt otic value of the entropy of the deep, adiabatically stratified layers (Fig . 3). This information is translated into an effective mixing-length parame ter alpha(MLT) 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 w ith a helium abundance of Y = 0.28 in the range of effective temperatures 4 300 K less than or equal to T-eff less than or equal to 7100 K and gravitie s 2.54 less than or equal to log g less than or equal to 4.74. We find a mo derate, nevertheless significant variation of alpha(MLT) between about 1.3 for F-dwarfs and 1.75 for K-subgiants with a dominant dependence on T-eff ( Fig. 5). In the close neighbourhood of the Sun we find a plateau where alph a(MLT) remains almost constant. The internal accuracy of the calibration of alpha(MLT) is estimated to be +/-0.05 with a possible systematic bias towa rds lower values. An analogous calibration of the convection theory of Canu to & 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 beta are deri ved independently of mixing-length theory: beta = 0.07...0.10. We show that our findings are consistent with constraints from stellar stab ility considerations and provide compact fitting formulae for the calibrati ons.