Relatively-periodic solutions of planetary systems with satellites and systems with slow and fast variables

TL;DR

This paper proves the existence of multiple symmetric periodic solutions in planetary systems with satellites, including resonances and stability conditions, using Hamiltonian system analysis with slow and fast variables.

Contribution

It introduces new existence results for symmetric periodic solutions with resonances in planetary systems with satellites, extending to Hamiltonian systems with slow and fast variables.

Findings

Existence of at least 2^{N-2} families of symmetric periodic solutions.

Identification of gaps corresponding to k:(k+1) resonances.

Conditions for orbital stability of some solutions.

Abstract

The partial case of the planar $N+1$ body problem, $N\ge2$, of the type of planetary system with satellites is studied. One of the bodies (the Sun) is assumed to be much heavier than the other bodies ("planets" and "satellites"), moreover the planets are much heavier than the satellites, and the "years" are much longer than the "months". The existence of at least $2^{N-2}$ smooth 2-parameter families of symmetric periodic solutions in a rotating coordinate system is proved, such that the distances between each planet and its satellites are much shorter than the distances between the Sun and the planets. The existence of "gaps" in these families of solutions is proved, corresponding to $k:(k+1)$ resonances of angular frequencies of planets' revolution around the Sun. Generating symmetric periodic solutions are described. Sufficient conditions for some periodic solutions to be orbitally stable in linear approximation are given. The results are extended to a class of Hamiltonian systems with slow and fast variables close to the systems of semidirect product type.