An improved algorithm of second order to construct consistent theories of equilibrium figures of rotating celestial bodies

Abstract

One of the main problems in celestial mechanics is the study of the figure adopted by large deformable bodies in slow rotation around an axis with a constant angular velocity ω when they reach their equilibrium configuration. This figure corresponds to the lowest equipotential surface containing the entire mass and, in order to determine it, to calculate its potential at an arbitrary point is required.Classical methods address this problem generally by performing a series development of the inverse-distance by using Clairaut's coordinates. These methods show convergence problems, already in first order in ω 2 , so that to avoid them they must assume no demonstrated hypotheses.The authors of this paper warned and proved this fact in a previous work, for which they used two methods:1. Taking into account the asymptotic properties of numeric quadrature formulas. 2. By a process similar to that used by Laplace to develop the inverse of the distance between two planets.