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

摘要

Modern applications of celestial mechanics include the study of closelypacked systems of exoplanets, circumbinary planetary systems, binary-binaryinteractions in star clusters, and the dynamics of stars near the galacticcentre. While developments have historically been guided by the architecture ofthe Solar System, the need for more general formulations with as fewrestrictions on the parameters as possible is obvious. Here we present clearand concise generalisations of two classic expansions of the three-bodydisturbing function, simplifying considerably their original form and makingthem accessible to the non-specialist. Governing the interaction between theinner and outer orbits of a hierarchical triple, the disturbing function in itsgeneral form is the conduit for energy and angular momentum exchange and assuch, governs the secular and resonant evolution of the system and itsstability characteristics. Focusing here on coplanar systems, the firstexpansion is one in the ratio of inner to outer semimajor axes and is valid forall eccentricities, while the second is an expansion in eccentricity and isvalid for all semimajor axis ratios [...]. Our generalizations make bothformulations valid for arbitrary mass ratios. [...]. We demonstrate theequivalence of the new expansions, identifying the role of the sphericalharmonic order m in both and its physical significance in the three-bodyproblem, and introducing the concept of principal resonances. Several examplesof the accessibility of both expansions are given including resonance widthsand the secular rates of change of the elements. Results in their final formare gathered together at the end of the paper for the reader mainly interestedin their application, including a guide for the choice of expansion.