Effective stability around the Cassini state in the spin-orbit problem

TL;DR

This paper analyzes the long-term stability of celestial bodies near the Cassini state using Hamiltonian normal forms, applying the method to Titan and finding stability times exceeding the Universe's age.

Contribution

It develops a high-order Birkhoff normal form approach to estimate effective stability times in the spin-orbit problem, with explicit application to Titan's rotation.

Findings

Titan's librations are stable over timescales exceeding the Universe's age.

Stability time depends on orbital inclination, precession rate, and polar moment of inertia.

Method provides physical bounds on librations in the spin-orbit problem.

Abstract

We investigate the long-time stability in the neighborhood of the Cassini state in the conservative spin-orbit problem. Starting with an expansion of the Hamiltonian in the canonical Andoyer-Delaunay variables, we construct a high-order Birkhoff normal form and give an estimate of the effective stability time in the Nekhoroshev sense. By extensively using algebraic manipulations on a computer, we explicitly apply our method to the rotation of Titan. We obtain physical bounds of Titan's latitudinal and longitudinal librations, finding a stability time greatly exceeding the estimated age of the Universe. In addition, we study the dependence of the effective stability time on three relevant physical parameters: the orbital inclination, $i$, the mean precession of the ascending node of Titan orbit, $\dot\Omega$, and the polar moment of inertia, $C$.