Interesting dynamics at high mutual inclination in the framework of the Kozai problem with an eccentric perturber

Abstract

We study the dynamics of the 3D three-body problem of a small body moving under the attractions of a star and a giant planet which orbits the star on a much wider and elliptic orbit. In particular, we focus on the influence of an eccentric orbit of the outer perturber on the dynamics of a small highly inclined inner body. Our analytical study of the secular perturbations relies on the classical octupole Hamiltonian expansion (third-order theory in the ratio of the semimajor axes), as third-order terms are needed to consider the secular variations of the outer perturber and potential secular resonances between the arguments of the pericentre and/or longitudes of the node of both bodies. Short-period averaging and node reduction (by adoption of the Laplace plane reference frame) reduce the problem to two degrees of freedom. The 4D dynamics is analysed through representative planes which identify the main equilibria of the problem. As in the circular problem (i.e. perturber on a circular orbit), the ‘Kozai-bifurcated’ equilibria play a major role in the dynamics of an inner body on a quasi-circular orbit: its eccentricity variations are very limited for mutual inclination between the orbital planes smaller than ∼40°, while they become large and chaotic for higher mutual inclination. Particular attention is also given to a region around 35° of mutual inclination, detected numerically by Funk et al. and consisting of long-time stable and particularly low-eccentricity orbits of the small body. Using a 12th-order Hamiltonian expansion in eccentricities and inclinations, in particular its action-angle formulation obtained by Lie transforms from Libert & Henrard, we show that this region presents an equality of two fundamental frequencies and can be regarded as a secular resonance. Our results also apply to binary star systems where a planet is revolving around one of the two stars.