Calibration of mixing-length parameter α for MLT and FST models by matching with CO⁵BOLD models

¹ Astronomical Institute, Tohoku University, 6-3 Aramaki Aza-Aoba, Aoba-ku Sendai 980-8578, Japan
e-mail: sonoi@astr.tohoku.ac.jp
² LESIA, Observatoire de Paris, PSL University, CNRS, Université Pierre et Marie Curie, Université Denis Diderot, 92195 Meudon, France
³ Zentrum für Astronomie der Universität Heidelberg, Landessternwarte, Königstuhl 12, 69117 Heidelberg, Germany
⁴ GEPI, Observatoire de Paris, PSL University, CNRS, Place Jules Janssen, 92190 Meudon, France
⁵ Institut d’Astrophysique et de Géophysique, Université de Liège, Allée du 6 Août 17, 4000 Liège, Belgium
⁶ Dipartimento di Fisica e Astronomia, University of Padova, Vicolo dell’Osservatorio 3, 35122 Padova, Italy

Received: 25 May 2018
Accepted: 12 November 2018

Abstract

Context. Space observations by the CoRoT and Kepler missions have provided a wealth of high-quality seismic data for a large number of stars from the main sequence to the red giant phases. One main goal of these missions is to take advantage of the rich spectra of solar-like oscillations to perform precise determinations of stellar characteristic parameters. To make the best of such data, we need theoretical stellar models with a precise near-surface structure since a near-surface structure of a solar-like star has significant influence on solar-like oscillation frequencies. The mixing-length parameter is a key factor to determine the near-surface structure of stellar models. In current versions of the convection formulations used in stellar evolution codes, the mixing-length parameter is a free parameter that needs to be properly specified.

Aims. We aim at determining appropriate values of the mixing-length parameter, α, to be used consistently with the adopted convection formulation when computing stellar evolution models across the Hertzsprung–Russell diagram. This determination is based on 3D hydrodynamical simulation models.

Methods. We calibrated α values by matching entropy profiles of 1D envelope models with those of hydrodynamical 3D models of solar-like stars produced by the CO⁵BOLD code. For such calibration, previous works concentrated on the classical mixing-length theory (MLT). We also analyzed full spectrum turbulence (FST) models. To construct the atmosphere in the 1D models, we used the Eddington gray T(τ) relation and that with the solar-calibrated Hopf-like function.

Results. For both MLT and FST models with a mixing length l = αH_p, calibrated α values increase with increasing surface gravity or decreasing effective temperature. For the FST models, we carried out an additional calibration using an α^* value defined as l = r_top − r + α^*H_p, top, where α^* is found to increase with surface gravity and effective temperature. We provide tables of the calibrated α values across the T_eff–log g plane for solar metallicity. By computing stellar evolution with varying α based on our 3D α calibration, we find that the change from solar α to varying α shifts evolutionary tracks particularly for the FST model. As for the correspondence to the 3D models, the solar Hopf-like function generally gives a photospheric-minimum entropy closer to a 3D model than the Eddington T(τ). The structure below the photosphere depends on the adopted convection model. However, we cannot obtain a definitive conclusion about which convection model gives the best correspondence to the 3D models. This is because each 1D physical quantity is related via an equation of state (EoS), but it is not the case for the averaged 3D quantities. Although the FST models with l = r_top − r + α^*H_p, top are found to give the oscillation frequencies closest to the solar observed frequencies, their acoustic cavities are formed with compensatory effects between deviating density and temperature profiles near the top of the convective envelope. In future work, an appropriate treatment of the top part of the 1D convective envelope is necessary, for example, by considering turbulent pressure and overshooting.