Comet-type periodic motions and their out-of-plane bifurcations in the Earth-Moon CR3BP: a computational symplectic analysis

Abstract

Comet-type periodic orbits of the circular restricted three-body problem (CR3BP) are periodic solutions that are generated from very large retrograde and direct circular Keplerian motions around the common center of mass of the primaries. In this paper we first provide an analytical proof of the existence of comet-type periodic orbits by using the classical Poincaré continuation method. Within this analytical approach, we also determine their Conley-Zehnder index, defined as a Maslov index using a crossing form. Then, by applying a standard corrector-predictor technique, we explore numerically the two families of comet orbits within the Earth-Moon CR3BP. We compute their stability indices, identify vertical self-resonant orbits up to multiplicity six, investigate the vertically bifurcated periodic solutions and discuss their orbital characteristics. Our main results we illustrate in form of bifurcation graphs, based on symplectic invariants, which provide a topological overview of the connections of the bifurcated branches, including bridge families.

Figures (14)

• Figure 1: CR3BP in an inertial reference frame $(X,Y)$ and in a rotating reference frame $(x,y)$, with primaries $m_1$ and $m_2$, infinitesimal body $m_3$ and libration points $L_i$ ($i=1,2,3,4,5$).
• Figure 2: Conley--Zehnder index measures a twisting of the linearized flow along an orbit $\gamma$ with respect to a frame.
• Figure 3: The index jump. 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. Direction of crossing is determined by $\pm$ Krein signature.
• Figure 4: Family $\kappa_-$ of retrograde comet-type periodic orbits. Right top shows $\kappa_-$ orbits starting at low Jacobi constants; from dark to light (or from light to dark) indicates decreasing (or increasing) of energy values. Right bottom shows continuation of $\kappa_-$ orbits; grey dashed is an orbit of birth-death type. Left top shows planar and vertical stability diagrams, $s_p$ and $s_v$, continued left bottom where a logarithmic scale is used. In the elliptic range at left top, $d$:$k$ VSR orbits are denoted with respect to $s_v = \cos (2\pi d/k)$. $\pm$ sign indicates Krein signature.
• Figure 5: Family $\kappa_+$ of direct comet-type periodic orbits, continued up to approaching collision with the Earth. Right top shows $\kappa_+$ orbits starting at high Jacobi constants; from light to dark indicates increasing of energy values. Right bottom shows continuation of $\kappa_+$ orbits. Left top shows planar and vertical stability diagrams, $s_p$ and $s_v$. Left bottom shows zoomed regions, a) with respect to crossing of $-1$, and b) with respect to crossing of 1. In the elliptic range at left top, $d$:$k$ VSR orbits are denoted with respect to $s_v = \cos (2\pi d/k)$. $\pm$ sign indicates Krein signature.

Theorems & Definitions (9)

• Lemma 2.1
• proof
• Lemma 2.2
• proof
• Remark 2.3
• Theorem 4.1
• Theorem 4.2
• proof : Proof of Theorem \ref{['theorem2']}
• proof : Proof of Theorem \ref{['theorem1']}