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Aims. In this paper we investigate the deviations from von Zeipel’s theorem at the upper layers of a distorted star. In addition, we derive an analytic expression for the gravity-darkening exponent β1. Methods. We introduce two different methods to derive the theoretical gravity-darkening exponents. In the first one we use a perturbation theory to derive an analytical expression for the gravity-darkening exponent for slow rotating stars as a function of the rotation law, colatitude, and opacity derivatives. In the second procedure we explore the validity of the mentioned theorem in the upper layers of a distorted star by adapting grey and non-grey atmosphere models to our numerical method, designed originally for stellar envelopes. Results. We have found significant deviations from von Zeipel’s theorem when we compute gravity-darkening exponents at the upper layers of a distorted star using our modified numerical method. This is a consequence of using a transfer equation that is more elaborated than the diffusion approach, therefore such a theorem is not strictly valid at lower optical depths. The shifts depend on the optical depth, on the effective temperature, and on the adopted atmosphere models. For large depths, we restore the classical value of β1, say, 1.0 for hotter stars. Taking such deviations into account, it may be possible to explain the scattering around the theoretical predictions for double-lined eclipsing binaries, as well as the low value of β1 (0.75), recently inferred for the very fast rotating star α Leo for which the classical von Zeipel’s theorem predicts β1 = 1.0.

On the deviations of the classical von Zeipel’s theorem at the upper layers of rotating stars