Regularization of the Hill four-body problem with oblate bodies, How many Keplerian arcs are there between two points of spacetime?

Abstract

Regularization of the Hill four-body problem with oblate bodies We consider the Hill four-body problem where three oblate, massive bodies form a relative equilibrium triangular configuration, and the 4th, infinitesimal body orbits in a neighborhood of the smallest of the three massive bodies. We regularize collisions between the infinitesimal body and the smallest massive body, via McGehee coordinate transformation. We describe the corresponding collision manifold and show that it undergoes a bifurcation when the oblateness coefficient of the smallest massive body passes through the zero value. How many Keplerian arcs are there between two points of spacetime? We consider the Keplerian arcs around a fixed Newtonian center joining two prescribed distinct positions in a prescribed flight time. We prove that putting aside the “opposition case” where infinitely many planes of motion are possible, there are at most two such arcs of each “type.” There is a bilinear quantity that we call b which is in all the cases a good parameter for the Keplerian arcs joining two distinct positions. The flight time satisfies a “variational” differential equation in b, and is a convex function of b.