Lunar missions below ∼ 50 km over the lunar surface (as NASA’s Lunar Reconnaissance Orbiter-LRO, the Australian Binary mission, or the European Space Agency’s SER3NE, for instance) challenge traditional orbit design and maintenance procedures due to the highly inhomogeneous gravitational field of Moon. The orbital motion is dominated in the long-term by the effects of the zonal harmonics, which need to consider high-degree truncations of the lunar potential. However, the modulation imposed on the mean evolution of the eccentricity vector by monthly effects induced by the sectorial and tesseral harmonics—and, in particular, those of odd degree—cannot be ignored due to their notable amplitude.

Due to the small eccentricity of non-impact low lunar orbits, these tesseral effects admit a remarkably simple analytical formulation. When superimposed to an analogously simple approximate analytical solutions of the long-term motion, which is obtained after the linearization of the mean variations, the combined solution generally provides quite accurate predictions of the evolution of the eccentricity vector, thus allowing for the implementation of highly efficient automatic stationkeeping algorithms. However, this is not always the case, and significant changes of the analytical predictions with respect to the actual eccentricity-vector dynamics are eventually observed.

We show that the observed differences between the available analytical solutions and the actual long-term behavior stem from the incomplete modeling of the mean dynamics, on the one hand, and for the insufficient accuracy in the initialization of the constants of the analytical solution, on the other hand. Beyond the long-term dynamics imposed by the zonal harmonics alone, the former requires the consideration of long-term effects due to the coupling of the tesseral harmonics. Regarding the latter, the monthly modulation of low lunar orbits  needs to be refined with  the inclusion of higher order terms. We will prove that both needed refinements, the additional long-term effects and the improved monthly modulation, can be computed analytically. Our demonstrations rely on a degree and order 4 truncation of the lunar potential, thus including the main tesseral and sectoral disturbances of the lunar potential. General analytical formulas for arbitraries degree and order truncations of the gravitational potential, which will be definitely useful in the implementation of operational tools for the preliminary analysis of low lunar orbits, are under development.

Related bibliography:
1.- G. Metris, P. Exertier, Y. Boudon, F. Barlier, Long period variations of the motion of a satellite due to non-resonant tesseral harmonics of a gravity potential. Celest. Mech. Dyn. Astron. 57, 175188 (1993)
2.- H. Holt, J. Yarndley, R. Armellin, C. Bombardelli, M. Lara, R. Howie, P. Bland, Extremely low- altitude lunar station keeping using eccentricity vector control, 29th ISSFD, 22–26 April 2024, ESOC, Darmstadt, Germany.
3.- J. Yarndley, M. Lara, H. Holt, R. Armellin, Origins and application of the translation theorem in extremely low lunar orbits, J. Guid. Control Dyn., in press (2025), arXiv:2504.19559 [astro-ph.IM]
4.- M. Lara, Nonresonant tesseral coupling: shape matters. Celest. Mech. Dyn. Astron., in press (2025), doi: 10.1007/s10569-025-10271-1