A Differential Rotation Law for Stars and Fluid Planets

We derive, supported on a generalization of Bernoulli’s equation, a law of rotation for any axial-symmetric, self-gravitating fluid mass. For a homogeneous mass, the law depends solely on the derivative of the potential with respect to the distance to the rotation axis, implying generally differential rotation, the Maclaurin spheroids representing the only case of solid-body rotation. We turn then to a heterogeneous mass consisting of any number l of concentric layers, each of constant density, finding that the angular velocity profile of a given layer depends on that of the layer immediately above it. Finally, we let l tend to infinity to convert our model into continuous mass distribution, the result being a certain rotation profile for the surface, and law of differential rotation change at its interior. To support the fundamentals of our approach, we write the potential integrals for the three mass distributions. The aim of a continuous distribution is that it may facilitate a comparison---to be carried out in a future paper---between our results and those of other researchers who employ structure equations. We point out that the distribution of angular velocity is a consequence of the equilibrium, rather than being imposed ad initio. The law was used in a past paper to construct a Jupiter multi-layer model adopting the spheroidal (a distorted spheroid) shape for each of the layers, taking as reference the gravitational data surveyed by the Juno mission. The procedure used here is not restricted to axial-symmetric cases.