Solid-solid interaction in the two body problem

TL;DR

This paper extends the analysis of solid-solid interactions in the two-body problem by expanding the gravitational potential to the fourth order, revealing integrable secular dynamics and explicit analytical solutions, with implications for n-body rigid body systems.

Contribution

It introduces a higher-order potential expansion and demonstrates the integrability of the secular problem, providing explicit analytical approximations and a decomposition of solutions into precession and periodic motions.

Findings

The secular system is integrable after averaging.

Explicit analytical solutions for the motions are derived.

General n-body solutions decompose into precession and quasiperiodic motion.

Abstract

We consider the solid-solid interactions in the two body problem. The relative equilibria have been previously studied analytically and general motions were numerically analyzed using some expansion of the gravitational potential up to the second order, but only when there are no direct interactions between the orientation of the bodies. Here we expand the potential up to the fourth order and we show that the secular problem obtained after averaging over fast angles, as for the precession model of Boue and Laskar [Boue, G., Laskar, J., 2006. Icarus 185, 312-330], is integrable, but not trivially. We describe the general features of the motions and we provide explicit analytical approximations for the solutions. We demonstrate that the general solution of the secular system can be decomposed as a uniform precession around the total angular momentum and a periodic symmetric orbit in the precessing frame. More generally, we show that for a general n-body system of rigid bodies in gravitational interaction, the regular quasiperiodic solutions can be decomposed into a uniform precession around the total angular momentum, and a quasiperiodic motion with one frequency less in the precessing frame.