Fully analytical propagator for lunar satellite orbits in closed form]{Fully analytical propagator for lunar satellite orbits in closed form

We present a fully analytical propagator for the long-term motion of lunar artificial satellites, based on a lunar gravity and third-body model sufficiently accurate for many practical applications. The model includes the twelve most important lunar gravity harmonics together with the Earth's quadrupole tidal perturbation, computed using a precise representation of the Earth's lunicentric ephemeris. Numerical tests indicate a satisfactory precision with respect to high-order (GRAIL) gravity models for all trajectories at altitudes ranging from approximately 300 km to 3000 km above the lunar surface, or frozen orbits at lower altitudes. Additional gravitational terms can be incorporated in a straightforward manner using the analytical framework here developed. Our theory yields an analytical solution of the secular equations of motion through a Hamiltonian normal-form approach in 'closed form'. Two successive transformations are employed together with their inverses: from osculating to mean orbital elements, and from mean to proper elements. Hence, the method allows to recover analytically the satellite's cartesian position and velocity in the lunicentric frame (PALRF) for any future or past time $t$ without any intermediate numerical propagation of the initial conditions. The propagator is valid over timescales of several decades for all non-impacting orbits, except within narrow regions associated with specific secular resonances. Numerical comparisons with full Cartesian propagation are used to assess the accuracy of the method.