The physics of rotational flattening and the point core model

Abstract

A point mass at the center of an ellipsoidal homogeneous fluid is used as a simple model to study the effect of rotation on the shape and external gravitational field of planets and stars. Maclaurin's analytical result for a homogenous body is generalized to this model. The absence of a third order term in the Taylor expansion of the Maclaurin function leads to further simple but very accurate analytical results connecting the three observables: oblateness ($\epsilon$), gravitational quadrupole ($J_{2}$), and angular velocity parameter ($q$). These are compared to observational data for the planets. The moments of inertia of the planets are calculated and compared to published values. The oblateness of the Sun is estimated. Oscillations near equilibrium are studied within the model.