Resumen de On the effect of the sun's gravity around the earth-moon l1 and l2 libration points

José Javier Rosales de Cáceres

• In this thesis we explored some aspects of the dynamics around the Earth-Moon L1 and L2 points in the context of two Restricted Four Body Problems: the Bicircular Problem (BCP) and the Quasi-bicircular Problem (QBCP). Both the BCP and QBCP model the dynamics of a massless particle moving under the influence of the Sun, Earth, and Moon. Although these two models focus on the same system, it is relevant to study both because their behavior around the L2 is qualitatively different.

  These two models can be written in the Hamiltonian formalism as periodic time-dependent perturbations of the RTBP. To study these Hamiltonians, we used numerical tools tailored to these type of models to get an insight on the phase space. These two techniques are the reduction to the center manifold, and the computation and continuation of 2D tori.

  For the BCP, the analysis focused around the L2 point. The results obtained showed that the reduction to the center manifold, and the non-autonomous normal form computed in this thesis do not provide useful information about the neutral motion around L2. The approach taken was to compute families of 2D tori, and explore any connections and their stability. As a summary of this effort we identified a total of six families of 2D tori: two Lyapunov-type planar quasi-periodic orbits, and four vertical. One of the vertical families was obtained by direct continuation of Halo orbits from the RTBP. This showed that the family of Halo orbits from the RTBP survive in the BCP, with the understanding that this new family is Cantorian. It was also shown that one of the other vertical families is Halo-like. Hence, members of this family may be potential candidates for future space missions. However, these tori are hyperbolic, as opposed the ones coming directly from the RTBP Halo obits, which are partially elliptic. It was also shown that this family of Halo-like tori comes from a family of quasi-periodic orbits in the RTBP that are resonant with the frequency of the Sun. Hence, these family of Halo-like orbits in the BCP have their counterparts in the RTBP.

  For the QBCP, the focus of the analyses was there Earth-Moon L1 and L2 points. In this model, the reduction to the center manifold provided relevant qualitative information about the dynamics around L1 and L2. The main takeaway was that L1 and L2 had a similar qualitative behavior. In both cases there were two families of quasi-periodic Lyapunov orbits, one planar and one vertical. It was also shown that the quasi-periodic planar Lyapunov family underwent a (quasi-periodic) pitchfork bifurcation, giving rise to two families of quasi-periodic orbits with an out-of-plane component. Between them, there was a family of Lissajous quasi-periodic orbits, with three basic frequencies. Qualitatively, the phase space of the center manifold, as constructed in this thesis, resembled the phase space of the center manifold of the RTBP around L1 and L2. In the QBCP we also continued families of invariant 2D tori, and for both L1 and L2. In these cases, the quasi-periodic planar and vertical families were continued. The bifurcations of the quasi-periodic planar Lyapunov were identified. A conclusion from this numerical experiment was that the family of out-of-plane orbits born from the bifurcation seemed not to be the RTBP Halo counterparts in the QBCP. The RTBP Halo orbits do survive in the QBCP, but do not seem to be connected to the quasi-periodic planar Lyapunov family.

  Finally, and also in the context of the BCP and the QBCP, numerical simulations to study transfers from a parking orbit around the Earth to a Halo orbit around the Earth-Moon L2 point were studied. The main conclusion is that the invariant manifolds of the target orbits studied intersect with potential parking orbits around the Earth. The relevance of this result is that it shows that there are one-maneuver transfers from a vicinity of the Earth to Earth-Moon L2 Halo orbits. This is not case when using the RTBP as reference model. Experiments were done for both the BCP and the QBCP, and in all cases is it was shown that the total cost in terms of ∆V and transfer time is comparable to other techniques requiring two or more maneuvers.