Rotationally and tidally distorted compact stars. A theoretical approach to the gravity-darkening exponents for white dwarfs

Context. To the best of our knowledge, there are no specific calculations of gravity-darkening exponents for white dwarfs in the literature. On the other hand, the number of known eclipsing binaries whose components are tidally and/or rotationally distorted white dwarfs is increasing year on year. <BR /> Aims: Our main objective is to present the first theoretical approaches to the problem of the distribution of temperatures on the surfaces of compact stars distorted by rotation and/or tides in order to compare with relevant observational data. <BR /> Methods: We used two methods to calculate the gravity-darkening exponents: (a) a variant of our numerical method based on the triangles strategy and (b) an analytical approach consisting in a generalisation of the von Zeipel theorem for hot white dwarfs. <BR /> Results: We find discrepancies between the gravity-darkening exponents calculated with our methods and the predictions of the von Zeipel theorem, particularly in the cases of cold white dwarfs; although the discrepancy also applies to higher effective temperatures under determined physical conditions. We find physical connections between the gravity-darkening exponents calculated using our modified method of triangles strategy with the convective efficiency (defined here as the ratio of the convective to the total flux). A connection between the entropy and the gravity-darkening coefficients is also found: variations of the former cause changes in the way the temperature is distributed on distorted stellar surfaces. On the other hand, we have generalised the von Zeipel theorem for the case of hot white dwarfs. Such a generalisation allows us to predict that, under certain circumstances, the value of the gravity-darkening exponent may be smaller than 1.0, even in the case of high effective temperatures. <BR /> Conclusions: To constrain the gravity-darkening exponent values observationally it would be necessary to find and investigate eclipsing binaries constituted by white dwarfs showing tidal and/or rotational distortions that were double-lined and that were bright enough to obtain good radial-velocity semi-amplitudes for both components. It would be very interesting and useful if observer were to focus their attention on this kind of system to check our theoretical results.