Gravity darkening in stars with surface differential rotation

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 Omega (theta) = Omega(o)[1 + alpha gamma(theta, k)] where gamma (theta, k) = cos(k) theta or sin(k) theta, and where (alpha, k) are free parameters. Results. The effective temperature varies with co-latitude theta, 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 (eta,alpha,k) (eta = ratio between centrifugal and gravitational accelerations at the equator) on the determination of the V sin i parameter and on the gravity-darkening exponent are studied. Depending on the values of (eta, alpha, k) the velocity V in the derived V sin i may strongly deviate from the equatorial rotational velocity. It is shown that the von Zeipel's-like gravity-darkening exponent beta(1) depends on all parameters (eta, alpha, 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,alpha). 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 (eta, beta(1)), it would be more useful to determine the quantities (eta, beta(1)) to characterize stellar rotation.