Computation of eigenfrequencies for equilibrium models including turbulent pressure

¹ LESIA, Observatoire de Paris, PSL Research University, CNRS, Université Pierre et Marie Curie, Université Denis Diderot, 92195 Meudon, France
e-mail: takafumi.sonoi@obspm.fr
² Institut d’Astrophysique et de Géophysique, Université de Liège, Allée du 6 Août 17, 4000 Liège, Belgium
³ Zentrum für Astronomie der Universität Heidelberg, Landessternwarte, Königstuhl 12, 69117 Heidelberg, Germany
⁴ GEPI, Observatoire de Paris, PSL Research University, CNRS, Université Denis Diderot, Sorbonne Paris Cité, 5 place Jules Janssen, 92195 Meudon, France

Received: 8 August 2016
Accepted: 16 January 2017

Abstract

Context. The space-borne missions CoRoT and Kepler have provided a wealth of highly accurate data. However, our inability to properly model the upper-most region of solar-like stars prevents us from making the best of these observations. This problem is called “surface effect” and a key ingredient to solve it is turbulent pressure for the computation of both the equilibrium models and the oscillations. While 3D hydrodynamic simulations help to include properly the turbulent pressure in the equilibrium models, the way this surface effect is included in the computation of stellar oscillations is still subject to uncertainties.

Aims. We aim at determining how to properly include the effect of turbulent pressure and its Lagrangian perturbation in the adiabatic computation of the oscillations. We also discuss the validity of the gas-gamma model and reduced gamma model approximations, which have been used to compute adiabatic oscillations of equilibrium models including turbulent pressure.

Methods. We use a patched model of the Sun with an inner part constructed by a 1D stellar evolution code (CESTAM) and an outer part by the 3D hydrodynamical code (CO⁵BOLD). Then, the adiabatic oscillations are computed using the ADIPLS code for the gas-gamma and reduced gamma model approximations and with the MAD code imposing the adiabatic condition on an existing time-dependent convection formalism. Finally, all those results are compared to the observed solar frequencies.

Results. We show that the computation of the oscillations using the time-dependent convection formalism in the adiabatic limit improves significantly the agreement with the observed frequencies compared to the gas-gamma and reduced gamma model approximations. Of the components of the perturbation of the turbulent pressure, the perturbation of the density and advection term is found to contribute most to the frequency shift.

Conclusions. The turbulent pressure is certainly the dominant factor responsible for the surface effects. Its inclusion into the equilibrium models is thus necessary but not sufficient. Indeed, the perturbation of the turbulent pressure must be properly taken into account for computing adiabatic oscillation frequencies. We propose a formalism to evaluate the frequency shift due to the inclusion of the term with the turbulent pressure perturbation in the variational principle in order to extrapolate our result to other stars at various evolutionary stages. Although this work is limited to adiabatic oscillations and the inclusion of the turbulent pressure, future works will have to account for the nonadiabatic effect and convective backwarming.