Relegation-free closed-form perturbation theory and the domain of secular motions in the Restricted 3-Body Problem

TL;DR

This paper introduces a novel closed-form perturbation theory for the restricted three-body problem, enabling accurate secular motion analysis for exterior orbits without the need for relegation, demonstrated on the Sun-Jupiter system.

Contribution

It develops a relegation-free, closed-form perturbation method for exterior orbits in the R3BP using multipole expansion and Lie series normalization, extending previous interior orbit approaches.

Findings

Accurately reproduces orbital element variations far from resonances.

Provides a semi-analytical boundary estimate for the validity of the secular model.

Demonstrates effectiveness on Sun-Jupiter system parameters.

Abstract

We propose a closed-form (i.e. without expansion in the orbital eccentricities) scheme for computations in perturbation theory in the restricted three-body problem (R3BP) when the massless particle is in an orbit exterior to the one of the primary perturber. Starting with a multipole expansion of the barycentric (Jacobi-reduced) Hamiltonian, we carry out a sequence of normalizations in Delaunay variables by Lie series, leading to a secular Hamiltonian model without use of relegation. To this end, we introduce a book-keeping analogous to the one proposed in Cavallari and Efthymiopoulos (2022) for test particle orbits interior to the one of the primary perturber, but here adapted, instead, to the case of exterior orbits. We give numerical examples of the performance of the method in both the planar circular and the spatial elliptic restricted three-body problem, for parameters pertinent to the Sun-Jupiter system. In particular, we demonstrate the method's accuracy in terms of reproducibility of the orbital elements' variations far from mean-motion resonances. As a basic outcome of the method, we show how, using as criterion the size of the series' remainder, we reach to obtain an accurate semi-analytical estimate of the boundary (in the space of orbital elements) where the secular Hamiltonian model arrived at after eliminating the particle's fast degree of freedom provides a valid approximation of the true dynamics.