Long-term evolution of mid-altitude quasi-satellite orbits

Quasi-satellite orbits are of great interest for the exploration of planetary moons because of their dynamical features and close proximity with respect to the surface of scientifically relevant objects like Phobos and Deimos. This paper explores the equations of the elliptical Hill problem, offering a new analytical insight into the long-term evolution of mid-altitude quasi-satellite orbits. Our developments are based on the Yamanaka-Ankersen solution of the Tschauner-Hempel equations and capture the effects of the secondary's gravity and orbital eccentricity on the shape and orientation of near-equatorial retrograde relative trajectories. The analytical solution of the in-plane and out-of-plane components of the secular motion is achieved by averaging over the relative longitude of a spacecraft as seen from the co-rotating frame of the two primaries. Developments are validated against the numerical integration of quasi-periodic trajectories that densely cover the surface of three-dimensional invariant tori. This analysis confirms the stable nature of quasi-satellite orbits and provides new tools for future spacecraft missions such as the Martian Moons eXploration envisaged by JAXA.