The von Zeipel gravity-darkening law holds strictly only for stars with a conservative rotation law and radiative atmospheres in which the transfer of energy is approximated by a diffusion equation. If the transfer equation is used, but meridional circulation is neglected, a perturbation treatment shows that the emergent flux ℱ satisfies | ℱ | |. . ∝|g1/2 (where g is the effective gravity) for all conservative rotation laws. The inclusion of circulation, however, seems to restore von Zeipel's law, | ℱ | ∝ g . For non-conservative rotation laws, | ℱ | is no longer a simple power of g , even assuming the diffusion equation. Two examples are given for circulation-free rotation laws, which show that the dependence of emergent flux on latitude can be quite strong. This could have important effects on close binary light curves.
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Smith et al. (1974) studied this question.