Gravity darkening in late-type stars

I. The Coriolis effect

¹ School of Astronomy, Institute for Research in Fundamental Sciences (IPM), PO Box 19395-5531, Tehran, Iran
e-mail: raphael.raynaud@ipm.ir
² Université de Toulouse, UPS-OMP, IRAP, Toulouse, France
e-mail: Michel.Rieutord@irap.omp.eu
³ CNRS, IRAP, 14 avenue Édouard Belin, 31400 Toulouse, France
⁴ LERMA, Observatoire de Paris, PSL Research University, CNRS, Sorbonne Universités, UPMC Univ. Paris 06, École normale supérieure, 75005 Paris, France
e-mail: ludovic@lra.ens.fr
⁵ Institut de Physique du Globe de Paris, Sorbonne Paris Cité, Université Paris-Diderot, UMR 7154 CNRS, 1 rue Jussieu, 75005 Paris, France

Received: 7 August 2017
Accepted: 4 December 2017

Abstract

Context. Recent interferometric data have been used to constrain the brightness distribution at the surface of nearby stars, in particular the so-called gravity darkening that makes fast rotating stars brighter at their poles than at their equator. However, good models of gravity darkening are missing for stars that posses a convective envelope.

Aim. In order to better understand how rotation affects the heat transfer in stellar convective envelopes, we focus on the heat flux distribution in latitude at the outer surface of numerical models.

Methods. We carry out a systematic parameter study of three-dimensional, direct numerical simulations of anelastic convection in rotating spherical shells. As a first step, we neglect the centrifugal acceleration and retain only the Coriolis force. The fluid instability is driven by a fixed entropy drop between the inner and outer boundaries where stress-free boundary conditions are applied for the velocity field. Restricting our investigations to hydrodynamical models with a thermal Prandtl number fixed to unity, we consider both thick and thin (solar-like) shells, and vary the stratification over three orders of magnitude. We measure the heat transfer efficiency in terms of the Nusselt number, defined as the output luminosity normalised by the conductive state luminosity.

Results. We report diverse Nusselt number profiles in latitude, ranging from brighter (usually at the onset of convection) to darker equator and uniform profiles. We find that the variations of the surface brightness are mainly controlled by the surface value of the local Rossby number: when the Coriolis force dominates the dynamics, the heat flux is weakened in the equatorial region by the zonal wind and enhanced at the poles by convective motions inside the tangent cylinder. In the presence of a strong background density stratification however, as expected in real stars, the increase of the local Rossby number in the outer layers leads to uniformisation of the surface heat flux distribution.