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

TL;DR

This paper analyzes the long-term evolution of the distance between two celestial bodies in a restricted 3-body problem, especially near orbit crossings, by extending and improving the mathematical framework to handle singularities.

Contribution

It introduces an improved technique for singularity extraction, proving Lipschitz continuity of extended vector fields and the analyticity of the signed distance near crossings.

Findings

Extended vector fields are Lipschitz-continuous.

Signed distance is analytic near crossing configurations.

The evolution of signed distance is more regular than orbital elements.

Abstract

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.