Abstract
The Bicircular Problem (BCP) is a periodic time dependent perturbation of the Earth-Moon Restricted Three-Body Problem that includes the direct gravitational effect of the Sun. In this paper we use the BCP to study the existence of Halo-like orbits around \(L_2\) in the Earth-Moon system taking into account the perturbation of the Sun. By means of computing families of 2D invariant tori, we show that there are at least two different families of Halo-like quasi-periodic orbits around \(L_2\).
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This article is part of the topical collection on Toward the Moon and Beyond.
Guest Editors: Terry Alfriend, Pini Gurfil and Ryan P. Russell.
This work has been supported by the Spanish grant PGC2018-100699-B-I00 (MCIU/AEI/FEDER, UE) and the Catalan Grant 2017 SGR 1374. The project leading to this application has received funding from the European Union’s Horizon 2020 research and innovation programme under the Marie Skłodowska-Curie Grant Agreement #734557. The authors thank the comments by the reviwers that helped to improve this manuscript.
Appendix
Appendix
In this section we provide additional information concerning the paper. In Table 3 we provide the initial conditions (identified by the energy) of the Halo family of the RTBP that have been used to produce Fig. 5.
We also provide some examples of tori from the other families found are given (see Fig. 5). They are provided here to illustrate the richness of the Sun-Earth-Moon BCP, and to evidence that the vertical families V1 and V2 are not Halo-like. A complete study of their stability properties and how they transition from the RTBP to the BCP is under work, and no details are provided here.
The planar tori from the families H1 and H2 are very similar, and two examples of each one are shown in Fig. 16. The representative of the family H1 (left) has rotation number \(\rho = {0.522687812628674}\). The rotation number of the representative of the family H2 (right) is \(\rho = {0.258684108104417}\).
Projections of a H1 orbit (left) and a H2 orbit (right). (Identified with an empty triangle and a full pentagon respectively in Fig. 6c)
More interesting are the families V1 and V2 with a vertical component. Different projections of a representative of the family V1 with rotation number \(\rho = {0.651014628070470}\) are shown in Fig. 17. The projection onto the plane \({x=0}\) (bottom-left image) shows that this orbit falls behind the Moon.
Different projections of a V1 orbit. (Identified with a full circle in Fig. 6d)
Finally, and example of the family V2 is illustrated in Fig. 18. This torus has rotation number \(\rho = {0.585297052915989}\). It also falls behind the Moon. However, the projection onto the plane \({x=0}\) (bottom-left image) show that has different symmetry than the representative of the V1 family.
Different projections of a V2 orbit. (Identified with a full square in Fig. 6d)
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Rosales, J.J., Jorba, À. & Jorba-Cuscó, M. Families of Halo-like invariant tori around \(L_2\) in the Earth-Moon Bicircular Problem. Celest Mech Dyn Astr 133, 16 (2021). https://doi.org/10.1007/s10569-021-10012-0
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DOI: https://doi.org/10.1007/s10569-021-10012-0