Averaging on Earth-crossing orbits

The orbits of planet-crossing asteroids (and comets) can undergo close approaches and collisions with some major planet. This introduces a singularity in the hi-body Hamiltonian, and the averaging of the equations of motion, traditionally used to compute secular perturbations, is undefined. We show that it is possible to define in a rigorous way some generalised averaged equations of motion, in such a way that the generalised solutions are unique and piecewise smooth. This is obtained, both in the planar and in the three-dimensional case, by means of the method of extraction of the singularities by Kantorovich. The modified distance used to approximate the singularity is the one used by Wetherill in his method to compute probability of collision. Some examples of averaged dynamics have been computed; a systematic exploration of the averaged phase space to locate the secular resonances should be the nest step.