The Stationary Points of the Hierarchical Three Body Problem

TL;DR

This paper analyzes stationary points in the hierarchical three body problem, revealing new stable equilibria at octupole order and connecting these to chaotic dynamics and orbital flips.

Contribution

It extends the analysis of stationary points to octupole order, identifying new stable equilibria and their relation to chaos and orbital flips.

Findings

New stable fixed points at octupole order.

Resilience of equilibria to relativistic precession.

Connection between stationary points and chaotic orbital behavior.

Abstract

We study the stationary points of the hierarchical three body problem in the planetary limit (m_2, m_3 << m_1) at both the quadrupole and octupole orders. We demonstrate that the extension to octupole order preserves the principal stationary points of the quadrupole solution in the limit of small outer eccentricity e_2 but that new families of stable fixed points occur in both prograde and retrograde cases. The most important new equilibria are those that branch off from the quadrupolar solutions and extend to large e_2. The apsidal alignment of these families is a function of mass and inner planet eccentricity, and is determined by the relative directions of precession of omega_1 and omega_2 at the quadrupole level. These new equilibria are also the most resilient to the destabilizing effects of relativistic precession. We find additional equilibria that enable libration of the inner planet argument of pericentre in the limit of radial orbits and recover the non-linear analogue of the Laplace-Lagrange solutions in the coplanar limit. Finally, we show that the chaotic diffusion and orbital flips identified with the Eccentric Kozai Lidov mechanism and its variants can be understood in terms of the stationary points discussed here.