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

摘要

We study the dynamics of the 3-D three-body problem of a small body movingunder the attractions of a star and a giant planet which orbits the star on amuch wider and elliptic orbit. In particular, we focus on the influence of aneccentric orbit of the outer perturber on the dynamics of a small highlyinclined inner body. Our analytical study of the secular perturbations relieson the classical octupole hamiltonian expansion (third-order theory in theratio of the semi-major axes), as third-order terms are needed to consider thesecular variations of the outer perturber and potential secular resonancesbetween the arguments of the pericenter and/or longitudes of the node of bothbodies. Short-period averaging and node reduction (Laplace plane) reduce theproblem to two degrees of freedom. The four-dimensional dynamics is analyzedthrough representative planes which identify the main equilibria of theproblem. 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 innerbody on quasi-circular orbit: its eccentricity variations are very limited formutual inclination between the orbital planes smaller than ~40^{\deg}, whilethey become large and chaotic for higher mutual inclination. Particularattention is also given to a region around 35^{\deg} of mutual inclination,detected numerically by Funk et al. (2011) and consisting of long-time stableand particularly low eccentric orbits of the small body. Using a 12th-orderHamiltonian expansion in eccentricities and inclinations, in particular itsaction-angle formulation obtained by Lie transforms in Libert & Henrard (2008),we show that this region presents an equality of two fundamental frequenciesand can be regarded as a secular resonance. Our results also apply to binarystar systems where a planet is revolving around one of the two stars.