Exploring Vertical Resonances in Earth-Moon Orbits

This paper explores vertical self-resonant (VSR) bifurcations from the distant retrograde orbit (DRO) family within the Earth-Moon circular restricted three-body problem (CR3BP). The research aims to investigate the vertical stability of DROs and identify vertical-critical DROs using a classical corrector-predictor algorithm. The study identifies fourteen such DROs, categorized into three groups based on their orbiting around the libration points L1, L2, L4, and L5. The analysis includes six VSR bifurcations of higher-order periods associated with DROs near the Moon, as well as six VSR bifurcations of multiplicity from five to ten for DROs near the Moon and around the L1 and L2 libration points. Additionally, two vertical single-turn branch points are identified within DROs orbiting around the L4 and L5 libration points.

In total, the research generates 25 bifurcated families of spatial symmetric periodic solutions and presents their orbital characteristics, including bridge families to Butterfly, prograde orbits, quasi DROs, and DROs. The study also identifies branches with long periods combining almost planar ecliptic motions with spatial excursions, resulting in Bumble Bee, Hoverfly, or Dragonfly-shaped trajectories. Furthermore, spatial orbits in resonance with the Earth and the Moon are discovered. To provide a structured overview of these bifurcation results, the authors determine Conley-Zehnder indices and construct bifurcation diagrams in view of symplectic invariants. This research contributes to a deeper understanding of the complex orbital dynamics within the Earth-Moon system, which is crucial for designing future lunar missions.

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