Dynamical regimes of small bodies perturbed by an eccentric giant planet

The dynamics of small bodies perturbed by an eccentric planet was done mostly under the assumption of well separated orbits using analytical approximations appropriate for the hierarchical case. In this work we study the dynamics of small bodies in a wide range of eccentricities and inclinations perturbed by a giant planet with $e_p=0.4$, in the non-hierarchical case. We consider small bodies both interior and exterior to the planet. We apply semi-analytical models for the study of resonances and the properties of the secular disturbing function. We perform a frequency analysis of numerical integration of the exact equations of motion to obtain the proper frequencies and corresponding dynamical secular paths. We study the dependence of proper frequencies with the initial mutual inclination and we find a critical inclination around 30 degrees for which the pericenter proper frequency vanishes giving rise to the increase of small bodies eccentricities followed by unstable dynamics. This happens for both interior and exterior small bodies and constitutes a stability barrier in the inclination. For greater inclinations the ZLK mechanism dominates both populations. By means of numerical integration of thousands of small bodies we reproduce the well known pericenter shepherding, but for the exterior populations with low inclinations we also find concentrations of the longitude of the ascending node in the direction of the planetary line of apsides.