Lunar satellite analytic theory with complete gravity and third-body perturbations Part-I: Gravity harmonics and their coupling effects

Abstract Artificial satellite theories for the Moon differ in important aspects from the similar theories for Earth because of more uneven nature of lunar gravity, strong third-body perturbations, and no atmospheric drag. The existing lunar satellite theories include only a few gravity spherical harmonics along with Hill’s approximation of the third-body disturbing function. Despite such simplifications, they provide semi-analytical solutions and require numerical propagation, thus limiting their applications. In this work, a fully analytic theory for lunar satellites is developed using generalized formulae that enable inclusion of spherical harmonics up to an arbitrary degree and order. The complete gravity and third-body disturbing functions are together considered as the dominant perturbation to the Keplerian Hamiltonian, which is then fully normalized up to second order by constructing three canonical transformations for averaging out the short-, medium-, and long-period terms. The gravity harmonics are fully treated in Part-I of this paper with the short- and medium-period terms developed up to first order and the long-period and secular terms up to second order. These second-order terms capture the coupling effects of the gravity harmonics. For the third-body perturbation, the secular terms are developed up to first order here. The periodic effects (including those originating from the coupling of gravity and third-body terms) will be explicitly treated in Part-II of this paper. The significance of the coupling effects from the gravity harmonics is assessed by comparing the analytically-propagated ephemerides of low-altitude lunar satellites with the numerically-propagated results.