Exploration of vertical self-resonant bifurcations from Distant Retrograde Orbits (DROs) in the Earth-Moon Circular Restricted Three-Body Problem (CR3BP)

Abstract

The purpose of this paper is to investigate vertical self-resonant (VSR) bifurcations from the distant retrograde orbit (DRO) family in the framework of the Earth-Moon circular restricted three-body problem (CR3BP). To this end, by using a classical corrector-predictor algorithm we compute the vertical stability of the DROs and identify fourteen vertical-critical DROs. We split them into three groups according to orbiting around the libration points $L_i$, $i=1,2,4,5$. (i) We first analyze six VSR bifurcations of higher order periods (of multiplicity from integer multiples of five to ten) associated with the DROs near the Moon. (ii) For the DROs that move near the Moon and additionally around the $L_1$ and $L_2$ libration points, we study six VSR bifurcations of multiplicity from five to ten as well. (iii) Within the DROs orbiting around the $L_4$ and $L_5$ libration points, two vertical single-turn branch points occur. In total, we generate 25 bifurcated families of spatial symmetric periodic solutions and present their orbital characteristics, including bridge families to the Butterfly, prograde orbits, quasi DROs and DROs. We also obtain branches whose members consist of long periods combining almost planar ecliptic motions with several spatial excursions, during which the trajectory repeatedly moves far from and then close to the Moon, being one of Bumble Bee, Hoverfly or Dragonfly shape. We also find spatial orbits that are in resonance with the Earth and the Moon. In order to provide a structured and systematic overview of such bifurcation results, we determine Conley-Zehnder indices and construct bifurcation diagrams in view of symplectic invariants.