A study of the moderate altitude frozen orbits around the Moon

Lunar frozen orbits, characterized by fixed eccentricity and argument of perigee on average, have been previously studied using different dynamical models. In this work, frozen orbits about the Moon are investigated on the basis of an averaged Hamiltonian. The gravitational field of the Moon is considered up to the seven zonal harmonic plus the third body perturbation (Earth). The third body is assumed to move in an elliptic inclined orbit. The averaging procedure is performed through Lie transformation method. We used the eccentricity-inclination diagrams to obtain a deep insight about the evolution of frozen orbits. For the reduced system, we found two frozen solutions which correspond to argument of perigee ω=π2,3π2. Moreover, we studied the evolution of the eccentricity and inclination as a function of time. The results showed that, for moderate altitude orbits, the eccentricity oscillates with small amplitudes around its initial value, while the inclination almost remains constant. In higher altitudes case, we observed that variations in eccentricity and inclination are larger than the moderate ones. The present dynamical model gives acceptable results for low initial eccentricity and inclination.