# Long term dynamics for the restricted N-body problem with mean motion resonances and crossing singularities

## TL;DR

This paper develops a mathematical framework to analyze the long-term dynamics of asteroids in mean motion resonance with planets, especially addressing crossing singularities, and demonstrates the dynamical protection mechanism against collisions.

## Contribution

It extends existing models by allowing the extension of the vector field across crossing singularities and analyzing the differentiability of the orbit distance near crossings.

## Key findings

- Extended the vector field to handle crossing singularities.
- Proved differentiability of the orbit distance near crossing times.
- Numerical validation with asteroid classes Alinda and Toro.

## Abstract

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 (Gronchi and Tommei 2007) 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 (Milani et al. 1989). This work extends the results in (Gronchi and Tardioli 2013) to the relevant case of asteroids in mean motion resonance with a planet.