Gravity darkening in stars with surface differential rotation
1 Sorbonne Universités, UPMC Université Paris 6 et CNRS, UMR 7095 Institut d’Astrophysique de Paris, 75014 Paris, France
2 CNRS, UMR 7095, Institut d’Astrophysique de Paris, 98bis Bd. Arago, 75014 Paris, France
3 Université de Toulouse, UPS-OMP, IRAP, 31062 Toulouse, France
4 CNRS, IRAP, 14 avenue Edouard Belin, 31400 Toulouse, France
5 University of Alcalá, 28871, Alcalá de Henares, Spain
6 Royal Observatory of Belgium, 3 Av. Circulaire, 1180 Bruxelles, Belgium
7 Université Côte d’Azur, Observatoire de la Côte d’Azur, CNRS UMR 7293, Lagrange, 28 avenue Valrose, 06108 Nice Cedex 2, France
8 GEPI, Observatoire de Paris, PSL Research University, CNRS UMR 8111, 5 place Jules Janssen, 92190 Meudon, France
Received: 18 March 2017
Accepted: 10 May 2017
Context. The interpretation of stellar apparent fundamental parameters (viewing-angle dependent) requires that they be treated consistently with the characteristics of their surface rotation law.
Aims. We aim to develop a model to determine the distribution of the effective temperature and gravity, which explicitly depend on the surface differential rotation law and on the concomitant stellar external geometry.
Methods. The basic assumptions in this model are: a) the external stellar layers are in radiative equilibrium; b) the emergent bolometric flux is anti-parallel with the effective gravity; c) the angular velocity in the surface obeys relations like Ω(θ) = Ωo [ 1 + αΥ(θ,k) ] where Υ(θ,k) = coskθ or sinkθ, and where (α,k) are free parameters.
Results. The effective temperature varies with co-latitude θ, with amplitudes that depend on the differential-rotation law through the surface effective gravity and the gravity-darkening function (GDF). Although the derived expressions can be treated numerically, for some low integer values of k, analytical forms of the integral of characteristic curves, on which the determination of the GDF relies, are obtained. The effects of the quantities (η,α,k) (η = ratio between centrifugal and gravitational accelerations at the equator) on the determination of the Vsini parameter and on the gravity-darkening exponent are studied. Depending on the values of (η,α,k) the velocity V in the derived Vsini may strongly deviate from the equatorial rotational velocity. It is shown that the von Zeipel’s-like gravity-darkening exponent β1 depends on all parameters (η,α,k) and that its value also depends on the viewing-angle i. Hence, there no unique interpretation of this exponent determined empirically in terms of (i,α).
Conclusions. We stress that the data on rotating stars should be analyzed by taking into account the rotational effects through the GDF, by assuming k = 2 as a first approximation. Instead of the classic pair (η,β1), it would be more useful to determine the quantities (η,α,i) to characterize stellar rotation.
Key words: stars: rotation
© ESO, 2017