Bifurcations of highly inclined near halo orbits using Moser regularization

We study the bifurcation structure of highly inclined near halo orbits with close approaches to the light primary, in the circular restricted three-body problem (CR3BP). Using a Hamiltonian formulation together with Moser regularization, we develop a numerical framework for the continuation of periodic orbits and the computation of their Floquet multipliers which remains effective near collision. We describe vertical collision orbits and families emerging from its pitchfork, period-doubling, and period-tripling bifurcations in the limiting Hill’s problem, including the halo and butterfly families. We continue these into the CR3BP using a perturbative framework via a symplectic scaling, and construct bifurcation graphs for representative systems (Saturn-Enceladus, Earth-Moon, Copenhagen) to identify common dynamical features. Conley-Zehnder indices are computed to classify the resulting families. Together, these results provide a coherent global picture of polar orbit architecture near the light primary, offering groundwork for future mission design, such as Enceladus plume sampling missions.

💡 Research Summary

The paper investigates the global bifurcation architecture of highly inclined, near‑collision periodic orbits in the circular restricted three‑body problem (CR3BP) when the mass ratio μ of the primaries is small. The authors focus on orbits that approach the lighter primary (the “light body”) closely, a regime of particular interest for mission concepts such as nearly rectilinear halo orbits (NRHOs) and Enceladus plume‑sampling trajectories.

The study proceeds in several stages. First, the classical Hamiltonian of the CR3BP is written in a uniformly rotating frame and then rescaled by a symplectic transformation that uses μ¹ᐟ³ as a small parameter. In the limit μ → 0 the rescaled Hamiltonian converges to the Hill problem, a well‑known approximation that retains only the two collinear Lagrange points L₁ and L₂. The Hill problem possesses additional reversing symmetries (rₓ, r_y, r_yz) that are exploited throughout the analysis.

A central methodological contribution is the adoption of Moser regularization instead of the more common Kustaanheimo‑Stiefel (KS) regularization. Moser’s technique works in arbitrary dimensions, preserves the periodicity of orbits, and provides a direct correspondence between Floquet multipliers of the regularized and original systems (Lemma 8). After regularization the phase space becomes a six‑dimensional constrained submanifold embedded in ℝ⁸; the authors develop a differential‑correction scheme that solves the constrained Hamiltonian equations using Newton–Raphson iterations with Lagrange multipliers. This framework yields high‑precision energy conservation (≈10⁻¹⁴) and reliable computation of monodromy matrices even arbitrarily close to collision.

In the Hill problem the authors identify a “vertical collision orbit” that moves strictly along the q₃‑axis, reaches a maximal height, and then collides with the primary. After regularization it becomes a genuine periodic orbit and serves as a generating orbit for a whole family of polar trajectories. By tracking its Floquet spectrum they locate three fundamental bifurcations:

1. Pitchfork bifurcation – produces two symmetric families, the classic halo family and a “butterfly” family, both doubly symmetric with respect to the y = 0 plane. Their Conley‑Zehnder (CZ) indices drop from (4, 3, 2) to (3, 2).

2. Period‑doubling bifurcation – yields 2‑fold periodic orbits associated with L₁ and L₂ halo families that retain a collisional character.

3. Period‑tripling bifurcation – gives rise to three‑lobed “tri‑fly” orbits. In the Hill limit these appear as left/right (x‑symmetric) or up/down (y‑symmetric) variants, with CZ indices ranging from 10 to 12.

The authors then perturb these families back to the full CR3BP using the symplectic scaling. Because the scaling is analytic in ν = μ¹ᐟ³, non‑degenerate periodic orbits persist for sufficiently small μ (Proposition 1). The continuation is performed for three representative systems: Saturn‑Enceladus (μ≈1.9×10⁻⁷), Earth‑Moon (μ≈1.2×10⁻²), and the Copenhagen problem (μ≈0.5).

In the Saturn‑Enceladus case the tri‑fly families emerge through a combination of “touch‑and‑go” and pitchfork bifurcations near the triple‑cover of the L₂ halo orbit. These orbits stay at very low altitude over Enceladus, making them attractive candidates for repeated plume‑sampling passes. In the Earth‑Moon system the same families are realized via period‑doubling and non‑symmetric pitchfork bifurcations, reflecting the larger μ and resulting in higher CZ indices (10–12) and more pronounced asymmetry. For the Copenhagen problem the additional symmetry rₓ is exact, so halo and butterfly families coincide more closely and the bifurcation graph is highly symmetric.

A comprehensive table (Table 1) lists every family studied, its symmetry properties, the μ interval where it exists, and its CZ index. The CZ index, a symplectic invariant, remains constant along a non‑degenerate family and changes by ±1 at each bifurcation, providing both a diagnostic for detecting bifurcations numerically and a topological classification of the families. The authors detail the computation of CZ indices in Appendix D, showing consistency with the observed Floquet multiplier crossings.

The paper concludes that Moser regularization combined with symplectic scaling offers a robust, high‑precision tool for exploring near‑collision polar dynamics in the CR3BP. The global bifurcation graphs assembled for the three systems reveal universal features (e.g., the persistence of the vertical‑collision generating orbit, the sequence of pitchfork → period‑doubling → period‑tripling) while also highlighting system‑specific differences driven by the mass ratio. These insights lay a solid foundation for future mission design that exploits highly inclined, low‑altitude orbits—particularly for icy‑moon exploration where repeated close passes are essential. The methodology can be extended to other three‑body configurations, to higher‑order resonances, and to the inclusion of additional perturbations (e.g., solar radiation pressure), thereby opening a broad avenue for both theoretical celestial mechanics and practical astrodynamics.