A Unified Hamiltonian Framework for Lunar Frozen Orbit Analysis Including Higher-Order Harmonics

Although higher-order zonal harmonics have been considered in early Earth frozen-orbit analyses, the lunar frozen-orbit problem differs fundamentally due to the comparable magnitudes of multiple lunar gravity harmonics. As a result, most analytic lunar frozen-orbit studies rely on low-order gravity models and simplified averaging assumptions. This research provides a new method for finding frozen orbits through a long-period canonical transformation with the inclusion of higher-order harmonics, which allows frozen orbit conditions to be derived from the structure of the doubly-averaged Hamiltonian rather than through the use of the Lagrange Planetary Equations. This paper compares the frozen orbit parameters of lunar orbits obtained with gravitational harmonics up to \((C_{6,6})\) to those from a simpler model and demonstrates the importance of higher-order harmonics for lower-altitude orbits.