Asymptotic behavior of an elastic satellite with internal friction

TL;DR

This paper analyzes the long-term behavior of an elastic satellite under gravitational forces, showing that energy dissipation leads to three possible outcomes: unbounded orbit, collision, or capture in synchronous resonance.

Contribution

It establishes the asymptotic outcomes of an elastic satellite with internal friction, using LaSalle's invariance principle and Hamiltonian elastodynamics, which is a novel analysis in this context.

Findings

Satellite orbits are either unbounded, lead to collision, or become synchronous.

Energy dissipation ceases only on synchronous orbits.

The analysis leverages Hamiltonian structure and symmetry properties.

Abstract

We study the dynamics of an elastic body whose shape and position evolve due to the gravitational forces exerted by a pointlike planet. The main result is that, if all the deformations of the satellite dissipate some energy, then under a suitable nondegeneracy condition there are only three possible outcomes for the dynamics: (i) the orbit of the satellite is unbounded, (ii) the satellite falls on the planet, (iii) the satellite is captured in synchronous resonance i.e. its orbit is asymptotic to a motion in which the barycenter moves on a circular orbit, and the satellite moves rigidly, always showing the same face to the planet. The result is obtained by making use of LaSalle's invariance principle and by a careful kinematic analysis showing that energy stops dissipating only on synchronous orbits. We also use in quite an extensive way the fact that conservative elastodynamics is a Hamiltonian system invariant under the action of the rotation group.