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.

Figures (23)

• Figure 1: CR3BP in an inertial reference frame $(X,Y)$ and in a rotating reference frame $(x,y)$, with primaries $P_1$ and $P_2$, infinitesimal body $P$, libration points $L_i$ ($i=1,2,3,4,5$), and distances $r_1$ and $r_2$ from $P$ to the primaries $P_1$ and $P_2$, respectively.
• Figure 2: The index jump (reproduced from aydin_czmoreno_aydin with minor modifications). Left: When eigenvalue 1 is crossed from above (or below), the index goes down (or up) by 1. Right: When Floquet multipliers after crossing eigenvalue 1 goes up (or down), the index goes up (or down) by 1. The direction of crossing is determined by the $\pm$ Krein signature.
• Figure 3: Top: Vertical stability diagram of the DRO family, continued at the bottom left (zoomed). Marked points in the form "$p\text{:}q$" indicate VSR orbits w.r.t. $s_v = \cos(2\pi p/q)$. The Six labeled VSR orbits to the right of the minimum of $s_v$ are plotted at the middle right. From light to dark corresponds to decreasing of Jacobi constant. The next six VSR orbits to the left of the minimum of $s_v$ are plotted at the middle left. Bottom right shows critical 1:1 self-resonant DROs that are marked at the bottom left.
• Figure 4: Left top: Bifurcation graph associated to 3-period DRO and P3DRO. The two critical 3-period DROs with horizontal stability index $\approx \cos(2\pi$$2/3)$, from which the P3DRO family arises, are plotted at the right top. Bottom plots show P3DROs, continued from left to right, that form a bridge (the blue branch) between the two critical 3-period DROs.
• Figure 5: Top left: $L_2$ planar Lyapunov orbits. From grey dashed orbit the $L_2$ halo branch is generated that is plotted at the top middle (pink orbits belong to southern $L_2$ halo family). Pink dashed orbit is of birth-death type, from which the near-rectilinear halo orbits (NRHOs) regime starts. Red orbits correspond to Butterfly orbits. These start at the top right, emerging from period-doubling bifurcation of last plotted NRHO, and are continued below from left to right.