The Stability of Tidal Equilibrium for Hierarchical Star-Planet-Moon Systems

TL;DR

This paper investigates the stability of tidal equilibrium in hierarchical star-planet-moon systems, finding that such systems lack stable long-term states due to orbital and energy constraints, especially outside the Hill radius.

Contribution

It introduces an approximate analysis of tidal stability in three-body systems, revealing the absence of stable equilibrium states similar to two-body systems.

Findings

Hierarchical systems have no stable tidal equilibrium in the Keplerian limit.

A shallow minimum exists when using a time-averaged three-body approximation.

Critical lunar orbits are outside the Hill radius, indicating instability.

Abstract

Motivated by the current search for exomoons, this paper considers the stability of tidal equilibrium for hierarchical three-body systems containing a star, a planet, and a moon. In this treatment, the energy and angular momentum budgets include contributions from the planetary orbit, lunar orbit, stellar spin, planetary spin, and lunar spin. The goal is to determine the optimized energy state of the system subject to the constraint of constant angular momentum. Due to the lack of a closed form solution for the full three-body problem, however, we must use use an approximate description of the orbits. We first consider the Keplerian limit and find that the critical energy states are saddle points, rather than minima, so that these hierarchical systems have no stable tidal equilibrium states. We then generalize the calculation so that the lunar orbit is described by a time-averaged version of the circular restricted three-body problem. In this latter case, the critical energy state is a shallow minimum, so that a tidal equilibrium state exists. In both cases, however, the lunar orbit for the critical point lies outside the boundary (roughly half the Hill radius) where (previous) numerical simulations indicate dynamical instability. These results suggest that star-planet-moon systems have no viable long-term stable states analogous to those found for two-body systems.