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

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.

Automated summary

This article discusses the equilibrium conditions for a body made of two homogeneous components separated by oblate spheroidal surfaces and in relative motion. Exact solutions are not allowed for rigid rotation unless a specific ambient pressure is present, but approximations can be obtained for configurations involving a small confocal parameter. The problem has two families of solutions, depending on the pressure along the common interface. The analytical approach compares well with numerical solutions obtained from the self-consistent-field method and practical formulas are derived for slowly rotating star/planet interiors.