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

Abstract

The non-integrability of the Hill problem makes that its global dynamics must 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 close to the origin, and the libration point dynamics have been thoroughly investigated by perturbation methods. 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 to progress in the perturbation approach beyond the lower orders of the solution really difficult. We propose an alternative perturbation approach that allows us 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.

Long-term evolution of mid-altitude quasi-satellite orbits

“…In order to provide a detailed mapping between mean and osculating orbit elements, different methods of perturbation theories can be considered (see for instance [36], [22] and references therein). We choose to introduce the near-identity transformation T : [0, 2 π ] × M → M so as to reconstruct the unbiased oscillations of motion of the original system from the averaged trajectories.…”

“…Lidov and Vashkov'yak extended this approach to the elliptical three-body problem, but failed to come up with an analytical solution that was valid in the orbital regime of the Martian moon [21]. Owing to small mass ratios and QSO altitudes, Lara more recently considered Lie-Deprit transformations within the framework of the planar circular Hill problem [22]. Differently from the CRTBP, the Hill problem does not depend on any external parameter and is the ideal laboratory for dynamical investigations of many planetary systems.…”

Design of distant retrograde orbits based on a higher order analytical solutiont

“…Then, Ω is obtained using Eqs. (20) and (16). Initial conditions of the deferential motion are then solved from Eq.…”

“…However, purely analytical efforts can be done without the need of relying on numerical averaging. In particular, focusing on the approximation to the dynamics provided by the Hill problem, both a low and a high order perturbation solutions have been recently reported without limitation to the evolutionary equation [16]. Indeed, further than the usual averaging, the short-period corrections that provide the transformation from osculating to mean elements and vice-versa, have also been obtained analytically, yet only to some order of the perturbation approach due to the special functions that arise in the procedure when the coupling of the different disturbing effects takes place in the perturbation approach.…”

“…Thus, after formulating the relative motion described by the Hill problem in epicyclic variables, the phase of the orbiter in its "epicycle" is removed by perturbation methods in [16]. As a result of the elimination of this fast angle, and up to certain truncation order of the perturbation approach, the planar Hill problem is decoupled into a reduced system of one degree of freedom for the motion of the "deferent", which is integrable, and a quadrature that provides the epicycle.…”