Long Term Dynamics for the Restricted N-Body Problem with Mean Motion Resonances and Crossing Singularities

We consider the long term dynamics of the restricted N-body problem, modeling in a statistical sense the motion of an asteroid in the gravitational field of the Sun and the solar system planets. We deal with the case of a mean motion resonance with one planet and assume that the osculating trajectory of the asteroid crosses the one of some planet, possibly different from the resonant one, during the evolution. Such crossings produce singularities in the differential equations for the motion of the asteroid, obtained by standard perturbation theory. In this work we prove that the vector field of these equations can be extended to two locally Lipschitz-continuous vector fields on both sides of a set of crossing conditions. This allows us to define generalized solutions, continuous but not differentiable, going beyond these singularities. Moreover, we prove that the long term evolution of the "signed" orbit distance [G. F. Gronchi and G. Tommei, Discrete Contin. Dyn. Syst. Ser. B, 7 (2007), pp. 755-778] between the asteroid and the planet is differentiable in a neighborhood of the crossing times. In case of crossings with the resonant planet we recover the known dynamical protection mechanism against collisions. We conclude with a numerical comparison between the long term and the full evolutions in the case of asteroids belonging to the "Alinda" and "Toro" classes [A. Milani et al., Icarus, 78 (1989), pp. 212-269]. This work extends the results in [G. F. Gronchi and C. Tardioli, Discrete Contin. Dyn. Syst. Ser. B, 18 (2013), pp. 1323-1344] to the relevant case of asteroids in mean motion resonance with a planet.