Dynamics on the Laplace surface, revisited

Abstract

We are living through revolutionary times in exo-planetary exploration, both observational and theoretical, with nearly four thousand exo-planetary systems confirmed, and most presenting us with fundamental challenges to cherished theories of planet formation and evolution. Of those, more than a hundred reside in wide binaries, where the stellar binary companion evolves on a fairly eccentric, and possibly inclined orbit. Such a binary companion can strongly perturb both classical processes of planet formation, as well as the resulting planetary system, in the course of its dynamical evolution. Those perturbations can be explored with the help of N-Body simulations, or through semi-analytic, perturbative treatment of the dynamics or both. In this thesis, we extend the semi-analytic tradition as we study the orbital architecture of systems of self-gravitating particles around a central body (star or planet) which are further affected by a wide, eccentric, hierarchical perturber (stellar binary companion, super-Jupiter, an eccentric debris disk, or a combination of what preceded). To explore equilibrium architecture, we revisit long term dynamical evolution within a classical regime in celestial mechanics, associated with the so-called Laplace Surface. Here equilibrium configurations of a test particle’s orbit arise from the combined action of an inner quadrupole (usually from a non-spherical mass distribution associated with an oblate planet, or a planet around the central star), and an outer, inclined perturber, again modelled as a quadrupole. The combined effect allows for equilibrium planes known as the Laplace Planes. Our contribution is in generalizing the classical scenario by allowing for higher order multipoles to account for the breaking of axisymmetry which is introduced by an eccentric perturber. The theory allows for multipoles of arbitrary order and exploits the geometric structure of the dynamics to provide a complete characterization of equilibrium configurations, their stability and bifurca