A note on the first-order theories of equilibrium figures of celestial bodies

Abstract

One of the main problems in celestial mechanics is the determination of the shape of the equilibrium configuration of celestial bodies. In this paper, a model of a fluid mass rotating in space like a rigid body will be developed. To this aim, the equipotential surfaces are developed by using the Neumann series with respect to the Clairaut coordinates, and from these developments, the equilibrium equations and the boundary conditions can be obtained. Classical methods involve convergence problems, and in this paper two methods are developed to solve this problem, one based on numerical quadrature methods and the other one based on an analytical development.