2015A&A...577A..87C

Rotation affects various aspects of the stellar structure and evolution. For example, it distorts the star and causes the energy flow to be dependent on the local gravity (von Zeipel effect). Recent advances in the semi-empirical derivation of the gravity-darkening exponents in eclipsing binaries and very fast rotators require new theoretical calculations so that the results can be compared with these observations. Using an analytical alternative method, we studied how the temperature is distributed over distorted neutron star (NS) surfaces through the gravity-darkening exponent (GDE). We also extended these investigations to non-relativistic stars. The envelopes of NS, whose properties are necessary to derive the GDE, were computed using routines of the code mesa. The non-relativistic stellar models were computed following the code granada. We use a perturbation theory to derive an equation for the GDE for neutron and non-relativistic stars as a function of the rotation law, of the colatitude, and of the logarithmic derivatives of the opacity. Significant deviations from the von Zeipel's theorem were found for differentially rotating NS as well as for non-relativistic stars. This equation is also capable of predicting the transition zone in the GDE around log T_eff=3.9 for non-relativistic stars, in good agreement with observational data. We use this equation to explore the effects of differential rotation to explain the anomalous values of semi-empirical GDE found in some early-type eclipsing binaries. We also analysed the role of convection and of the change of the main thermonuclear energy source in the GDE calculations for late-type stars.