An invariant region for the collisional dynamics of two bodies on Keplerian orbits

TL;DR

This paper proves the existence of a bounded invariant region for two bodies on Keplerian orbits that remain elliptic despite collisions or short-range interactions, based on explicit energy and angular momentum conditions.

Contribution

It establishes a new invariant region for collisional dynamics of two bodies on elliptic orbits, applicable to elastic, inelastic, and potential interactions.

Findings

Existence of a bounded invariant region for collisional two-body systems

Invariance holds for elastic and inelastic collisions

Invariant region persists with short-range potential interactions

Abstract

We study the dynamics of two bodies moving on elliptic Keplerian orbits around a fixed center of attraction and interacting only by means of elastic or inelastic collisions. We show that there exists a bounded invariant region: for suitable values of the total energy and the total angular momentum (explicitly computable) the orbits of the bodies remain elliptic, whatever are the number and the details of the collisions. We show that there exists a bounded invariant region even in the case of two bodies interacting by short range potential.