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We study the long term evolution of the distance between two Keplerianconfocal trajectories in the framework of the averaged restricted 3-bodyproblem. The bodies may represent the Sun, a solar system planet and anasteroid. The secular evolution of the orbital elements of the asteroid iscomputed by averaging the equations of motion over the mean anomalies of theasteroid and the planet. When an orbit crossing with the planet occurs theaveraged equations become singular. However, it is possible to define piecewisedifferentiable solutions by extending the averaged vector field beyond thesingularity from both sides of the orbit crossing set. In this paper we improvethe previous results, concerning in particular the singularity extractiontechnique, and show that the extended vector fields are Lipschitz-continuous.Moreover, we consider the distance between the Keplerian trajectories of thesmall body and of the planet. Apart from exceptional cases, we can select asign for this distance so that it becomes an analytic map of the orbitalelements near to crossing configurations. We prove that the evolution of the'signed' distance along the averaged vector field is more regular than that ofthe elements in a neighborhood of crossing times. A comparison between averagedand non-averaged evolutions and an application of these results are shown usingorbits of near-Earth asteroids.

The evolution of the orbit distance in the double averaged restricted 3-body problem with crossing singularities