FIRST ORDER ANALYTICAL SOLUTION FOR DISTANT RETROGRADE ORBITS IN THE CIRCULAR RESRICTED THREE-BODY PROBLEM

This paper proposes an analytical solution of the first order for Distant Retrograde Orbits (DROs) in the Circular Restricted Three-Body Problem (CR3BP). Starting from the Hamiltonian formulation of the problem, we apply the theory of canonical perturbations, by addressing the non-perturbed and the perturbed terms separately. The first step is to simplify the problem through a canonical invertible transformation for the non-perturbing part, the second step is to apply the Lie transformation to the perturbed part. This procedure allows us to obtain first a mean Hamiltonian that can be analytically solved, then to obtain short-periodic corrections, that take into consideration the short-term fluctuations neglected during the averaging process. The solutions obtained are compared with the numerical solution simulated with the CR3BP, both in terms of maximum error and in terms of computational speed. Even if the solution shows to be computationally efficient and accurate, the improvement over the solution of Hill on the same problem is not noticeable unless one considers very low mass ratios. This is because the potential of the primary is considered, in part, as a perturbation and the closer we get to the primary the less this assumption is valid . In particular, the modulation of the geometric assumptions, allows to better manage the potential of the primary and to increase the range of validity of the analytical solution