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

We study the dynamics of the 3-D 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 semi-major 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 pericenter and/or longitudes of the node of both bodies. Short-period averaging and node reduction (Laplace plane) reduce the problem to two degrees of freedom. The four-dimensional dynamics is analyzed 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 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. (2011) and consisting of long-time stable and particularly low eccentric 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 in Libert & Henrard (2008), 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.