Graphical generation of periodic orbits of tschauner-hempel equations

Graphical generation of periodic orbits of Tschauner-Hempel (TH) equations is presented. The general solution of the relative dynamics based on the orbital elements is obtained, and the relative motion invariant manifold, where the solution lies, is determined. The region where all periodic solutions lie is determined by two ellipses, whose size depends on the eccentricity of the leader orbit. Size and phase parameters for periodic orbits are introduced for the TH equations, and a simple way to draw a periodic orbit is proposed for a given pair of size and phase parameters and a given initial true anomaly of the elliptic orbit. Results show that the design of a relative orbit with size and position specifications is as simple as in the Hill-Clohessy-Wiltshire (HCW) case.