Explicit expansion of the three-body disturbing function for arbitrary eccentricities and inclinations

TL;DR

This paper presents a comprehensive analytical expansion of the three-body disturbing function valid for all eccentricities and inclinations, with explicit formulas and algorithms for higher orders, enhancing the modeling of planetary systems.

Contribution

It introduces explicit expressions and algorithms for the three-body disturbing function at any order, applicable to arbitrary eccentricities and inclinations, improving upon previous limited expansions.

Findings

Explicit expansions up to order ten in the planar case.

Explicit expansions up to order five in the spatial case.

Algorithms for extending expansions to higher orders and computing non-secular terms.

Abstract

Since the original work of Hansen and Tisserand in the XIXth century, there have been many variations in the analytical expansion of the three-body disturbing function in series of the semi-major axis ratio. With the increasing number of planetary systems of large eccentricity, these expansions are even more interesting as they allow us to obtain for the secular systems finite expressions that are valid for all eccentricities and inclinations. We revisited the derivation of the disturbing function in Legendre polynomial, with a special focus on the secular system. We provide here expressions of the disturbing function for the planar and spatial case at any order with respect to the ratio of the semi-major axes. Moreover, for orders in the ratio of semi-major axis up to ten in the planar case and five in the spatial case, we provide explicit expansions of the secular system, and simple algorithms with minimal computation to extend this to higher order, as well as the algorithms for the computation of non secular terms.