How many Keplerian arcs are there between two points of spacetime?

TL;DR

This paper investigates the number of Keplerian arcs connecting two points in spacetime within a fixed flight time, establishing that typically there are at most two such arcs per type, with a focus on a bilinear parameter b.

Contribution

The authors prove a maximum of two Keplerian arcs of each type connect two points, introducing the bilinear quantity b as a key parameter and analyzing the flight time as a convex function of b.

Findings

At most two Keplerian arcs of each type connect two points.

The bilinear quantity b effectively parameterizes Keplerian arcs.

Flight time is a convex function of b, satisfying a variational differential equation.

Abstract

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.