Planar retrograde periodic orbits of the asteroids trapped in two–body mean motion resonances with Jupiter

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We study planar resonant retrograde periodic orbits, using the model of the restricted three-body problem with the Sun and Jupiter as primaries. The position and the stability character of the periodic orbits are very useful for the study of the phase space structure and this will provide important piece of information on the stability and long term evolution of potential asteroids which are in retrograde motion with Jupiter. The starting point of the present study is the planar circular model. Families of periodic orbits are computed at the interior mean motion resonances of first, second and third order with Jupiter and these are used as a guide to select the energy levels for the computation of the Poincaré maps, so that the most important resonances are included in the study. Then we consider the second primary in an elliptic orbit. Families of retrograde symmetric periodic orbits of the planar elliptic restricted three-body problem are found. All these orbits bifurcate from the families of symmetric periodic orbits of the planar, circular, restricted three-body problem. The stability of all orbits is also examined. The resonant structure in the circular problem is similar for all resonances. For the elliptic model, the families of periodic orbits have been computed not only for Jupiter’s eccentricity but in the whole interval, 0<e’<1.