The flattenings of the layers of rotating planets and satellites deformed by a tidal potential

TL;DR

This paper extends Clairaut's theory to model the equilibrium shapes of layered, non-homogeneous rotating bodies under tidal forces, providing analytical solutions and applying them to various internal structure models.

Contribution

It introduces a generalized solution for the flattenings of layered bodies under tidal deformation, incorporating non-homogeneous density distributions and deriving boundary conditions via the Radau transformation.

Findings

Derived expressions for external polar flattenings and layer radii.

Provided solutions for bodies with polynomial and polytropic density profiles.

Applied the theory to specific layered models demonstrating its versatility.

Abstract

We consider the Clairaut theory of the equilibrium ellipsoidal figures for differentiated non-homogeneous bodies in non-synchronous rotation adding to it a tidal deformation due to the presence of an external gravitational force. We assume that the body is a fluid formed by $n$ homogeneous layers of ellipsoidal shape and we calculate the external polar flattenings and the mean radius of each layer, or, equivalently, their semiaxes. To first order in the flattenings, the general solution can be written as $\epsilon_k={\cal H}_k*\epsilon_h$ and $\mu_k={\cal H}_k*\mu_h$, where $\cal{H}_k$ is a characteristic coefficient for each layer which only depends on the internal structure of the body and $\epsilon_h, \mu_h$ are the flattenings of the equivalent homogeneous problem. For the continuous case, we study the Clairaut differential equation for the flattening profile, using the Radau transformation to find the boundary conditions when the tidal potential is added. Finally, the theory is applied to several examples: i) a body composed of two homogeneous layers; ii) bodies with simple polynomial density distribution laws and iii) bodies following a polytropic pressure-density law.