Shapes of two-layer models of rotating planets

To first approximation the interiors of many planetary bodies consist of a core and mantle with significantly different densities. The shapes of the surface and interface between the core and the mantle are basic properties reflecting planetary structure and rotation. In addition, interface shape is an important parameter controlling the dynamics of a fluid core. We present a theory for the rotational distortion of a two-layer model of a planet (two-layer Maclaurin spheroid) that determines the shapes of both the interface and the outer free surface without treating departure from sphericity as a small perturbation. Since the interface and the outer free surface, in general, have different shapes, two different spheroidal coordinates are required in the mathematical analysis, and the transformation between them is at the heart of the complexity of the theory. Furthermore, two different cases have to be considered. In the first case, the core is sufficiently large, or the rate of rotation is sufficiently small, that the foci of the outer free surface are located within the core. In the second case, the core is sufficiently small, or the rate of rotation is sufficiently fast, that the foci of the free surface are located within the outer layer. In comparison to the classical Maclaurin solution which is explicitly analytical, the relevant multiple integrals for the equilibrium solution of a two-layer Maclaurin spheroid have to be evaluated numerically. The shape of a two-layer rotating planet is characterized by three dimensionless parameters that are explored systematically in the present study.