Retrograde resonance in the planar three-body problem

TL;DR

This paper investigates retrograde resonances in the planar three-body problem, deriving new methods to analyze their dynamics and constructing phase-space models to understand libration modes.

Contribution

It introduces a procedure to derive retrograde resonance terms and develops a semi-analytic model for the 1/-1 resonance using numerical averaging.

Findings

Surfaces of section reveal phase-space structures near main retrograde resonances.

The semi-analytic model accurately predicts libration modes in the 1/-1 resonance.

The approach extends understanding of retrograde orbital dynamics.

Abstract

We continue the investigation of the dynamics of retrograde resonances initiated in Morais & Giuppone (2012). After deriving a procedure to deduce the retrograde resonance terms from the standard expansion of the three-dimensional disturbing function, we concentrate on the planar problem and construct surfaces of section that explore phase-space in the vicinity of the main retrograde resonances (2/-1, 1/-1 and 1/-2). In the case of the 1/-1 resonance for which the standard expansion is not adequate to describe the dynamics, we develop a semi-analytic model based on numerical averaging of the unexpanded disturbing function, and show that the predicted libration modes are in agreement with the behavior seen in the surfaces of section.