Nested spheroidal figures of equilibrium I. Approximate solutions for rigid rotations

TL;DR

This paper develops approximate analytical solutions for the equilibrium of a two-component spheroidal body in rigid or relative rotation, comparing them with numerical models and deriving practical formulas for slowly rotating celestial bodies.

Contribution

It introduces approximate solutions for nested spheroidal equilibrium configurations with two components, extending classical theory to include relative rotation and small ellipticities.

Findings

Analytical solutions match well with numerical models.

Configurations with small ellipticities are practically described by derived formulas.

Severe limitations exist for global rigid rotation configurations.

Abstract

We discuss the equilibrium conditions for a body made of two homogeneous components separated by oblate spheroidal surfaces and in relative motion. While exact solutions are not permitted for rigid rotation (unless a specific ambient pressure), approximations can be obtained for configurations involving a small confocal parameter. The problem then admits two families of solutions, depending on the pressure along the common interface (constant or quadratic with the cylindrical radius). We give in both cases the pressure and the rotation rates as a function of the fractional radius, ellipticities and mass-density jump. Various degrees of flattening are allowed but there are severe limitations for global rotation, as already known from classical theory (e.g. impossibility of confocal and coelliptical solutions, gradient of ellipticity outward). States of relative rotation are much less constrained, but these require a mass-density jump. This analytical approach compares successfully with the numerical solutions obtained from the Self-Consistent-Field method. Practical formula are derived in the limit of small ellipticities appropriate for slowly-rotating star/planet interiors.