Origins and Application of the Translation Theorem in Extremely Low Lunar Orbits

Extremely low lunar orbits (eLLOs) (altitudes [Formula: see text]) are highly perturbed by the nonspherical lunar gravitational field, presenting unique challenges to orbit maintenance. At these altitudes, stable “frozen” orbits do not exist, and conventional station-keeping methods such as circularization tend to be ineffective. However, a distinctive dynamic feature has been observed in such orbits, dubbed the “translation theorem,” wherein the eccentricity vector follows a predictable monthly evolution that is largely independent of initial conditions. In this work, the dynamic origins of this effect are identified; it results from a combination of low orbital eccentricity with the dominant effects of a particular subset of sectoral and tesseral harmonics. Based on this result, automated station-keeping strategies for eLLOs are developed, which leverage eccentricity vector control. Several constraints for station-keeping within the eccentricity vector plane are explored, including circular, annular, and elevation-model-derived regions, each forming distinct strategies for varying orbital configurations. The optimal control profiles for these strategies in the ephemeris model are then obtained using sequential convex programming (SCP). These station-keeping strategies offer computational simplicity while achieving [Formula: see text] costs that are comparable to full trajectory optimization.