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

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.