Analytical theory of an artificial satellite of the moon

The dynamics of an artificial satellite of the moon is quite different from the dynamics of an artificial satellite of the Earth. Indeed, the C(22) term is only 1/10 of the J(2) term, and the effect of the Earth on the lunar satellite is much larger than the effect of the Moon on a terrestrial satellite. The method used here is the Lie method for averaging the Hamiltonian of the problem, in canonical variables. The solution is developed in powers of the small factors linked to J(2) and C(22). Short period terms (linked to l, the mean anomaly) are eliminated first, and then the long period terms (linked to g, the mean argument of the periaster, and to h, the longitude of the ascending node), which finally gives the secular motion. The results are obtained in a closed form, without any series developments in eccentricity or inclination. Thus, the solution applies for a wide range of values, except for few isolated critical values. The results are only very preliminary. As a side result, we were able to check the solution given by Kozai for the effect of the J(2) term on an artificial satellite.