Nested spheroidal figures of equilibrium III. Connection with the gravitational moments $J_{2n}$

Abstract

We establish, in the framework of the theory of nested figures, the expressions for the gravitational moments $J_{2n}$ of a systems made of ${\cal L}$ homogeneous layers separated by spheroidal surfaces and in relative rotational motion. We then discuss how to solve the inverse problem, which consists in finding the equilibrium configurations (i.e. internal structures) that reproduce ``exactly'' a set of observables, namely the equatorial radius, the total mass, the shape and the first gravitational moments. Two coefficients $J_{2n}$ being constrained per surface, ${\cal L}=1+\frac{n}{2}$ layers ($n$ even) are required to fix $J_2$ to $J_{2n}$. As shown, this problem already suffers from a severe degeneracy, inherent in the fact that two spheroidal surfaces in the system confocal with each other leave unchanged all the moments. The complexity, which increases with the number of layers involved, can be reduced by considering the rotation rate of each layer. Jupiter is used as a test-bed to illustrate the method, concretely for ${\cal L}=2,3$ and $4$. For this planet, the number of possible internal structures is infinite for ${\cal L} > 2$. Intermediate layers can have smaller or larger oblateness, and can rotate slower or faster than the surroundings. Configurations with large and massive cores are always present. Low-mass cores (of the order a few Earth masses) are predicted for ${\cal L} \ge 4$. The results are in good agreement with the numerical solutions obtained from the Self-Consistent-Field method.