Bifurcations of Highly Inclined Near Halo Orbits using Moser Regularization

TL;DR

This work develops a comprehensive, regularized Hamiltonian framework to study highly inclined near-halo polar orbits near the light primary in the CR3BP. By employing Moser regularization and a rescaled Hamiltonian bridging Hill's problem and CR3BP, the authors perform global continuation of vertical collision orbits and map their bifurcations, including pitchfork, period-doubling, and period-tripling (giving rise to halo, W4/W5, butterfly, and tri-fly families). They compute Floquet multipliers and Conley-Zehnder indices to classify the families and construct bifurcation surfaces across mass ratios, with representative cases Saturn-Enceladus, Earth-Moon, and the Copenhagen problem. The results yield a coherent global picture of polar orbit architecture near the light primary and provide a theoretical and computational foundation for mission design, including Enceladus plume sampling and NRHO-like trajectories.

Abstract

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.

Figures (22)

• Figure 1: Configuration of the Lagrange points and reflection symmetries in the CR3BP (left) and Hill's problem (right). In both models, the reflection symmetry $r_y$ across the plane $y=0$ is present, while the Hill's problem exhibits additional symmetry $r_x$ across the plane $x=0$. Only $L_1$ and $L_2$ Lagrange points remain in Hill's problem.
• Figure 2: Energy levels of the five Lagrange points in the standard CR3BP $H_\mu$ (left) and in the rescaled CR3BP $\hat{H}_\mu$ (right) as a function of the mass ratio $\mu\in [0,1]$. In the limit $\mu\to 0$, the rescaled Hamiltonian corresponds to the Hill's problem $\hat{H}_0$, in which only the two collinear Lagrange points $L_1$ and $L_2$ remain, both at the same energy level.
• Figure 3: Schematic picture of Moser regularization, where the momentum curve (also known as a hodograph) of an orbit is mapped to a unit sphere. The figure shows the case for the planar problem, where the unregularized momentum coordinates $(x_1, x_2)$ are mapped to $(\xi_0, \xi_1, \xi_2)$ on the sphere via inverse stereographic projection, with $\xi_0$ representing the vertical axis. Collision orbits correspond to trajectories passing through the north pole $\xi_0 = 1$, and the position coordinates correspond to tangent vectors at each point.
• Figure 4: Northern vertical collision orbit in Hill's problem, shown in configuration space (left), the $(q_3,p_3)$ plane (center), and Moser-regularized coordinates $(\xi_0,\eta_0)$ (right). Red marker denotes apoapsis (maximum $q_3$), and the cross indicates the periapsis corresponding to collision with the light primary.
• Figure 5: Bifurcation diagram of the vertical collision orbit (central line) and the $L_1$ and $L_2$ halo families (branches) in Hill's problem. The horizontal axis corresponds to the Hamiltonian energy $\hat{H}_0 = h$, and the vertical axis shows the $x$-coordinate at the apoapsis.

Theorems & Definitions (26)

• Proposition 2.1
• Remark 2.2
• Definition 2.3
• Lemma 2.4
• proof
• Definition 2.5
• Lemma 2.6
• proof
• Theorem 2.7
• proof