Orbit of a lunar artificial satellite: Analytical theory of perturbations

Abstract

We are currently developing an analytical theory of an artificial satellite of the Moon. It is an interesting problem because the dynamics of a lunar orbiter is quite different from that of an artificial satellite of the Earth, by at least two aspects: the $J_2$ lunar gravity term is only $1/10$ of the $C_{22}$ term and the third body effect of the Earth on the lunar satellite is much larger than the effect of the Moon on a terrestrial satellite. So we have to account at least for these larger perturbations. We use here the method of the Lie Transform as perturbation method. The Hamiltonian of the problem is first averaged over the fast angle, in canonical variables. The solution is developed in powers of the small factors linked to $ n_{\Moon}$, $J_2$, $C_{22}$ and to the Earth's position. The Earth location is determined by the lunar theory ELP2000 (Chapront-Touzé & Chapront 1991) from which we take the leading terms. Series developments are made with our home-made Algebraic Manipulator, the MM (standing for “Moon's series Manipulator”). The results are obtained in a closed form, without any series developments in eccentricity or inclination. So the solution applies for a wide range of values, except for few isolated critical values. We Achieved, among others, second order results for the combined effect of $J_2$ and $C_{22}$. As a side result, we were able to check the second order generator $\mathcal{W}_{2}$ given by Kozai for the effect of the $J_2$ term on an artificial satellite.To search for other articles by the author(s) go to: http://adsabs.harvard.edu/abstract_service.html