Poncelet property of planar elliptic integrable Kepler billiards

Abstract We consider the integrable dynamics of a Kepler billiard in the plane bounded by a branch of a conic section with one focus at the Kepler center. We show that in this case, for negative energy orbits associated with trajectories being ellipses, the lines of consecutive second orbital foci along a billiard trajectory are all tangent to a fixed circle. Based on this observation, we analyse in detail the integrable dynamics of a planar Kepler billiard, both inside and outside an elliptic reflection wall, with the Kepler center at one of its foci. We identify the associated elliptic curve on which the dynamics are linearized, and the shift defined thereon. We also discuss explicit conditions on n -periodicity using Cayley’s criteria.