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

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 generalizations 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, except for systems in which the orbits cross (this restriction also applies to the first expansion). Our generalizations make both formulations valid for arbitrary mass ratios. The classic versions of these appropriate to the restricted three-body problem are known as Kaula's expansion and the literal expansion, respectively. 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. © 2013 The Authors Published by Oxford University Press on behalf of the Royal Astronomical Society.