The theory of secondary resonances in the spin-orbit problem

TL;DR

This paper develops a novel analytical method using higher order normal forms and a canonical normalization procedure to study the stability and bifurcation of primary and secondary resonances in the spin-orbit problem, with applications to satellite dynamics.

Contribution

It introduces a new normalization technique combining detuning expansion and book-keeping to analyze secondary resonances in spin-orbit interactions.

Findings

Analytical bifurcation curves match numerical results.

Stability regions for satellites are identified.

Secondary resonances influence satellite stability.

Abstract

We study the resonant dynamics in a simple one degree of freedom, time dependent Hamiltonian model describing spin-orbit interactions. The equations of motion admit periodic solutions associated with resonant motions, the most important being the synchronous one in which most evolved satellites of the Solar system, including the Moon, are observed. Such primary resonances can be surrounded by a chain of smaller islands which one refers to as secondary resonances. Here, we propose a novel canonical normalization procedure allowing to obtain a higher order normal form, by which we obtain analytical results on the stability of the primary resonances as well as on the bifurcation thresholds of the secondary resonances. The procedure makes use of the expansion in a parameter, called the detuning, measuring the shift from the exact secondary resonance. Also, we implement the so-called `book-keeping' method, i.e., the introduction of a suitable separation of the terms in orders of smallness in the normal form construction, which deals simultaneously with all the small parameters of the problem. Our analytical computation of the bifurcation curves is in excellent agreement with the results obtained by a numerical integration of the equations of motion, thus providing relevant information on the parameter regions where satellites can be found in a stable configuration.