We study the long term evolution of the distance between two Keplerian
confocal trajectories in the framework of the averaged restricted 3-body
problem. The bodies may represent the Sun, a solar system planet and an
asteroid. The secular evolution of the orbital elements of the asteroid is
computed by averaging the equations of motion over the mean anomalies of the
asteroid and the planet. When an orbit crossing with the planet occurs the
averaged equations become singular. However, it is possible to define piecewise
differentiable solutions by extending the averaged vector field beyond the
singularity from both sides of the orbit crossing set. In this paper we improve
the previous results, concerning in particular the singularity extraction
technique, and show that the extended vector fields are Lipschitz-continuous.
Moreover, we consider the distance between the Keplerian trajectories of the
small body and of the planet. Apart from exceptional cases, we can select a
sign for this distance so that it becomes an analytic map of the orbital
elements near to crossing configurations. We prove that the evolution of the
'signed' distance along the averaged vector field is more regular than that of
the elements in a neighborhood of crossing times. A comparison between averaged
and non-averaged evolutions and an application of these results are shown using
orbits of near-Earth asteroids.Comment: 29 pages, 8 figure