Invariant manifolds near $$L_1$$ and $$L_2$$ in the quasi-bicircular problem

Abstract

The quasi-bicircular problem (QBCP) is a periodic time-dependent perturbation of the Earth–Moon restricted three-body problem (RTBP) that accounts for the effect of the Sun. It is based on using a periodic solution of the Earth–Moon–Sun three-body problem to write the equations of motion of the infinitesimal particle. The paper focuses on the dynamics near the $$L_1$$ L 1 and $$L_2$$ L 2 points of the Earth–Moon system in the QBCP. By means of a periodic time-dependent reduction to the center manifold, we show the existence of two families of quasi-periodic Lyapunov orbits around $$L_1$$ L 1 (resp. $$L_2$$ L 2 ) with two basic frequencies. The first of these two families is contained in the Earth–Moon plane and undergoes an out-of-plane (quasi-periodic) pitchfork bifurcation giving rise to a family of quasi-periodic Halo orbits. This analysis is complemented with the continuation of families of 2D tori. In particular, the planar and vertical Lyapunov families are continued, and their stability analyzed. Finally, examples of invariant manifolds associated with invariant 2D tori around the $$L_2$$ L 2 that pass close to the Earth are shown. This phenomenon is not observed in the RTBP and opens the room to direct transfers from the Earth to the Earth–Moon $$L_2$$ L 2 region.

A Frequency-Based Hierarchy of Dynamical Models in Cislunar Space: Leveraging Periodically and Quasi-Periodically Perturbed Models

“…3, it is obvious that the two models produce extremely similar behaviors across the time over one year, alluding to a global similarity across cislunar space. This observation complements the previous investigations that focus on specific domains to illustrate the similarity between the two models, for example, the vicinity of the L 2 Lagrange point as discussed in Andreu (1998); Rosales et al (2023); Peterson et al (2023); Henry et al (2023). The similarity is also apparent from the Earth-Moon distance plots in Fig.…”

“…(G2) Catalog the QBCP and HR4BP within the common formulation: The global capabilities and limitations of the two periodically perturbed models with coherent solar gravity, i.e., the Quasi Bi-Circular Problem (QBCP) and the Hill Restricted Four-Body Problem (HR4BP), are assessed within the common formulation. This effort complements the previous "bottom-up" approaches that focus on specific regions within cislunar space, e.g., the vicinity of Lagrange points (Rosales et al, 2023;Peterson et al, 2023;Henry et al, 2023). In the current analysis, additional insights are provided regarding the similarities and differences between the QBCP and the HR4BP.…”

“…For a case where ν C is in resonance with the model frequencies, examples are provided in Boudad et al (2020); Brown et al (2024); Peng and Xu (2015); Ferrari and Lavagna (2018); Sanaga and Howell (2024); Andreu (1998) that discuss resonant POs within periodically forced models including the ER3BP, BCR4BP, HR4BP, and QBCP. For a case where ν C is not commensurate with the model frequency, 2D-QPOs are determined within various models in Park and Howell (2024a); Rosales et al (2023); Henry et al (2023); Rosales et al (2021). The results within this section leverage the numerical processes investigated by these previous authors.…”

“…Within the pulsatingrotating frame, it is trivially demonstrated that the ER3BP also admits an equilibrium L 2 point (to observe this, refer to Park and Howell (2024b)). Within the QBCP, the L 2 point evolves to a PO with three distinct lobes as determined within the pulsatingrotating frame and discussed in Andreu (1998); Jorba-Cuscó et al (2018); Rosales et al (2023). The HR4BP's L 2 counterpart also displays a periodic motion, but represented within the uniform-rotating frame as discussed in Olikara et al (2016); Henry et al (2023); Peterson et al (2023).…”

Invariant manifolds near $$L_1$$ and $$L_2$$ in the Sun–Jupiter elliptic restricted three-body problem I

“…Extensions of this work were done in several directions, i.e. different models, such as the elliptic restricted three-body problem Guzzo (2021, 2022), the Hill restricted four-body problem Peterson et al (2023) and the quasi-bicircular problem Rosales et al (2023).…”