Closed form perturbation theory in the restricted three-body problem without relegation

TL;DR

This paper introduces a novel closed-form normalization technique for analyzing the secular dynamics of small bodies influenced by a distant planet, avoiding convergence issues associated with traditional relegation methods.

Contribution

The authors develop a relegation-free, closed-form perturbation method using a book-keeping parameter, enhancing the analysis of small body dynamics in the restricted three-body problem.

Findings

Successfully applied to Jupiter's perturbations

Circumvents convergence issues of relegation techniques

Provides a new analytical tool for secular dynamics

Abstract

We propose a closed-form normalization method suitable for the study of the secular dynamics of small bodies in heliocentric orbits perturbed by the tidal potential of a planet with orbit external to the orbit of the small body. The method makes no use of relegation, thus, circumventing all convergence issues related to that technique. The method is based on a convenient use of a book-keeping parameter keeping simultaneously track of all the small quantities in the problem. The book-keeping affects both the Lie series and the Poisson structure employed in successive perturbative steps. In particular, it affects the definition of the normal form remainder at every normalization step. We show the results obtained by assuming Jupiter as perturbing planet and we discuss the validity and limits of the method.