Shapes and Gravitational Fields of Two-Layer Maclaurin Spheroids: Application to Planets and Satellites

The shapes and gravitational fields of rotationally and tidally distorted planets and satellites depend on their interior mass distributions. Measurements of these observable characteristics are therefore used to infer the internal structure of planetary bodies. Interpretations are based on approximate formulae such as the Radau-Darwin relation derivable from the theory of figures or more accurate evaluations of the theory. The exact solution for the shape and gravitational field of the rotationally distorted constant density Maclaurin spheroid has, until now, provided one of the only ways to assess the accuracy and range of validity of approximate theory of figure predictions. We generalize the Maclaurin spheroid solution to a 2-layer core-envelope body, a more realistic model of a real planet or moon. The exact 2-layer Maclaurin spheroid solution, e. g., the shapes of the surface and core-envelope interface, depend on 3 parameters, the core-envelope density ratio, the fractional volume of the core, and Ω2/2πGρ2, where Ω is the rotation rate, G is the gravitational constant, and ρ2 is the envelope density. For realistic parameter values, the flattening of the interface is smaller than that of the surface. Results of the exact solution are compared with predictions of the theory of figures up to order 3 in the small rotational parameter of the theory. The exact solution serves as a benchmark for numerical models that attempt to invert gravitational and shape data to infer internal planetary structure.