Nonlinear librations of distant retrograde orbits: a perturbative approach—the Hill problem case

The non-integrability of the Hill problem makes its global dynamics be necessarily approached numerically. However, the analytical approach is feasible in the computation of relevant solutions. In particular, the nonlinear dynamics of the Hill problem has been thoroughly investigated by perturbation methods in two cases: when the motion is close to the origin, and in the case of motion about the libration points. Out of the Hill sphere, the analytical approach is also feasible, at least in the case of distant retrograde orbits. Previous analytical investigations of this last case succeeded in the qualitative description of the dynamics, but they commonly failed in providing accurate results. This is a consequence of the essential dependance of the dynamics on elliptic functions, a fact that makes progress in the perturbation approach beyond the lower orders of the solution really difficult. An alternative perturbation approach is proposed here that allows to provide a very simple low-order analytical solution in trigonometric functions, on the one hand, and, while still depending on special functions, to compute higher orders of the solution, on the other.