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

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.

💡 Research Summary

The paper addresses a class of periodic solutions in the circular restricted three‑body problem (CR3BP) that originate from extremely large retrograde and direct Keplerian circles about the barycenter of the two primaries. These “comet‑type” orbits are first proved to exist analytically by applying the classical Poincaré continuation method. In this analytical framework the authors introduce a small continuation parameter that deforms the infinite‑radius circular motion into a finite‑energy periodic orbit while preserving non‑degeneracy and avoiding resonances. The proof guarantees smooth dependence of the solution on the parameter and establishes the global continuation of the family.

A central contribution of the analytical part is the computation of the Conley‑Zehnder (CZ) index for each comet‑type orbit. By interpreting the CZ index as a Maslov index, the authors evaluate a crossing form associated with the linearized Hamiltonian flow. This index captures the winding of the symplectic eigenvectors and directly reflects the linear stability of the orbit, especially in the vertical (z‑direction) subspace. The CZ index thus serves as a symplectic invariant that can predict where vertical bifurcations are likely to occur.

The second part of the study turns to the Earth‑Moon CR3BP and implements a standard predictor‑corrector scheme to continue numerically the two families of comet‑type orbits: one retrograde and one direct. For each computed periodic solution the monodromy matrix is obtained, from which the Floquet multipliers, the horizontal stability index, and the vertical stability index are extracted. By scanning the parameter space the authors locate vertical self‑resonances satisfying νz = k·νt for integers k up to six. At each resonance a vertical bifurcation is observed, giving rise to new periodic branches that inherit the same period and Jacobi constant but possess distinct inclination and out‑of‑plane amplitude.

The bifurcation structure is visualized using graphs whose axes are the CZ index and the vertical stability index. These graphs, unlike traditional Poincaré‑section diagrams, encode symplectic invariants and therefore provide a topological overview of how the various branches interconnect, including the appearance of “bridge families” that link otherwise separate families. The authors analyze the geometry of the bifurcated orbits, noting that some retain high‑order vertical resonances (multiplicity six) and exhibit pronounced vertical oscillations, while others form smoother, low‑inclination families.

From an application standpoint, the vertically bifurcated comet‑type orbits are of interest for low‑altitude lunar missions, station‑keeping strategies, and long‑duration spacecraft trajectories, because the vertical stability enhancement can increase resilience against small perturbations. The paper concludes by emphasizing that the combination of rigorous existence proofs, symplectic index calculations, and comprehensive numerical continuation offers a unified methodology for exploring high‑energy periodic motions in the CR3BP. It also suggests future extensions to the elliptic restricted three‑body problem, inclusion of additional perturbations (solar radiation pressure, non‑spherical gravity), and the use of the presented symplectic invariants in other celestial‑mechanics contexts.