New developments for modern celestial mechanics. I. General coplanar three-body systems. Application to exoplanets

TL;DR

This paper introduces simplified, general expansions of the three-body disturbing function for coplanar systems, applicable to all eccentricities and mass ratios, enhancing the analysis of exoplanetary and stellar dynamics.

Contribution

It provides new, accessible formulations of classic three-body expansions, valid for arbitrary parameters, and demonstrates their equivalence and physical significance.

Findings

Valid for all eccentricities and mass ratios.

Introduction of principal resonances concept.

Guidance for choosing the appropriate expansion.

Abstract

Modern applications of celestial mechanics include the study of closely packed systems of exoplanets, circumbinary planetary systems, binary-binary interactions in star clusters, and the dynamics of stars near the galactic centre. While developments have historically been guided by the architecture of the Solar System, the need for more general formulations with as few restrictions on the parameters as possible is obvious. Here we present clear and concise generalisations of two classic expansions of the three-body disturbing function, simplifying considerably their original form and making them accessible to the non-specialist. Governing the interaction between the inner and outer orbits of a hierarchical triple, the disturbing function in its general form is the conduit for energy and angular momentum exchange and as such, governs the secular and resonant evolution of the system and its stability characteristics. Focusing here on coplanar systems, the first expansion is one in the ratio of inner to outer semimajor axes and is valid for all eccentricities, while the second is an expansion in eccentricity and is valid for all semimajor axis ratios [...]. Our generalizations make both formulations valid for arbitrary mass ratios. [...]. We demonstrate the equivalence of the new expansions, identifying the role of the spherical harmonic order m in both and its physical significance in the three-body problem, and introducing the concept of principal resonances. Several examples of the accessibility of both expansions are given including resonance widths and the secular rates of change of the elements. Results in their final form are gathered together at the end of the paper for the reader mainly interested in their application, including a guide for the choice of expansion.