Fully analytical propagator for lunar satellite orbits in closed form

Abstract

We present a fully analytical propagator for the orbits of lunar artificial satellites in a lunar gravity and third-body model sufficiently precise for a wide range of practical applications. The gravity model includes the twelve most important lunar gravity harmonics as well as the Earth's quadrupole tidal terms with a precise representation of the Earth's lunicentric ephemeris, and it gives an accuracy comparable to the way more extended semi-analytical propagator SELENA [6] for satellite orbits at altitudes from 300 to 3000 km. Extra terms of a more complete gravity model are straightforward to include using the formulas of the presently discussed analytical theory. The theory is based on deriving an approximate analytical solution of the secular part of the equations of motion using a Hamiltonian normal form in closed form. In total, we have two types of element transformations: from osculating to mean elements (as in [6]), and from mean to proper elements. The solution of the problem in proper elements is trivial, and, through the inverses of the above transformations, it allows to recover the position and velocity of a satellite analytically at any time t given initial conditions of the osculating elements at time $t_0$ without any intermediate numerical propagation. The propagator model is valid in time spans of several decades, and for every initial condition leading to no-fall on the Moon's surface, except for identified thin zones around a set of secular resonances corresponding to commensurabilities between the satellite's secular frequencies and the secular frequencies of the lunicentric Earth's orbit. Open software python and symbolic routines implementing our propagator are provided in the repository [14]. Precision tests with respect to fully numerical orbital propagation in Cartesian coordinates are reported.