2/1 resonant periodic orbits in three dimensional planetary systems

TL;DR

This paper investigates the dynamics of three-dimensional planetary systems near 2/1 mean motion resonance, identifying stable and unstable periodic orbits that influence long-term system evolution.

Contribution

It introduces methods to compute families of symmetric periodic orbits in 3D planetary systems, including inclined orbits, using analytical continuation from known resonant solutions.

Findings

Many stable periodic orbits found with mutual inclinations up to 60 degrees.

Unstable orbits dominate but stable ones may explain observed planetary configurations.

Stable orbits could support long-term regular planetary system evolution.

Abstract

We consider the general spatial three body problem and study the dynamics of planetary systems consisting of a star and two planets which evolve into 2/1 mean motion resonance and into inclined orbits. Our study is focused on the periodic orbits of the system given in a suitable rotating frame. The stability of periodic orbits characterize the evolution of any planetary system with initial conditions in their vicinity. Stable periodic orbits are associated with long term regular evolution, while unstable periodic orbits are surrounded by regions of chaotic motion. We compute many families of symmetric periodic orbits by applying two schemes of analytical continuation. In the first scheme, we start from the 2/1 (or 1/2) resonant periodic orbits of the restricted problem and in the second scheme, we start from vertical critical periodic orbits of the general planar problem. Most of the periodic orbits are unstable, but many stable periodic orbits have been, also, found with mutual inclination up to $50^\circ$ - $60^\circ$, which may be related with the existence of real planetary systems.