Nested spheroidal figures of equilibrium II. Generalization to L layers

TL;DR

This paper develops a vectorial formalism to analyze multi-layered spheroidal bodies in equilibrium, generalizing previous methods to L layers, and provides analytical and numerical solutions for their rotation and shape configurations.

Contribution

It introduces a generalized analytical method for multi-layer spheroidal equilibrium figures with L layers, extending prior two-layer models and including new insights on rotation states.

Findings

Asynchronous layer rotation allows diverse equilibrium configurations.

Global rotation states are more constrained and exclude confocal and coelliptical shapes.

Analytical formulas for small ellipticities match numerical solutions successfully.

Abstract

We present a vectorial formalism to determine the approximate solutions to the problem of a composite body made of $L$ homogeneous, rigidly rotating layers bounded by spheroidal surfaces. The method is based on the 1st-order expansion of the gravitational potential over confocal parameters, thereby generalizing the method described in Paper I for $L=2$. For a given relative geometry of the ellipses and a given set of mass-density jumps at the interfaces, the sequence of rotation rates and interface pressures is obtained analytically by recursion. A wide range of equilibria result when layers rotate in an asynchronous manner, although configurations with a negative oblateness gradient are more favorable. In contrast, states of global rotation (all layers move at the same rate), found by solving a linear system of $L-1$ equations, are much more constrained. In this case, we mathematically demonstrate that confocal and coelliptical configurations are not permitted. Approximate formula for small ellipticities are derived. These results reinforce and prolongate known results and classical theorems restricted to small elliptiticities. Comparisons with the numerical solutions computed from the Self-Consistent-Field method are successful.