Three-dimensional retrograde periodic orbits of asteroids moving in mean motion resonances with Jupiter

We study the dynamics of interior mean motion resonances of first, second and third order with Jupiter using the model of the restricted three-body problem with the Sun and Jupiter as primaries and we focus on asteroids which are in retrograde motion with the perturbing planet. The basis of our study is the computation of three-dimensional symmetric periodic orbits and their linear stability. The position and stability of periodic orbits determine critically the phase space topology in the resonant regimes. The periodic orbits are continued analytically forming families which show particular structures. They may continue from the planar model to the three-dimensional one forming “direct” families, which start from inclination i = 0°, or “retrograde” families, which start from i = 180°. Also there exist families that start from inclined circular orbits and do not contain any planar orbits. Families that contain a polar orbit (i = 90°) consist of both direct and retrograde parts and are defined in the whole inclination domain. Instead, pure direct or retrograde families tend to high eccentricity values, i.e., they terminate to a collision with Sun. The resonance 3/1 is studied further in order to understand the dynamics of the asteroid (343158) Marsyas, which is in retrograde motion.