Structure of Periodic Orbit Families in the Hill Restricted 4-Body Problem

TL;DR

This work advances the mapping of periodic orbit structure in the HR4BP by starting from CR3BP Earth–Moon orbits and continuing them under solar perturbation using a Melnikov-type function. The authors derive an HR4BP expansion around $m=0$, show that leading-order work vanishes due to time-scaling, and demonstrate that higher-order terms drive the continuation of resonant periodic orbits with period multiples of the forcing, $T_g= ext{π}$. They identify bifurcation points via a modified Jacobian and generate new HR4BP periodic orbit families, including resonant families around EM libration points, validating and extending prior results. The approach yields a flexible, general framework for locating and continuing resonant periodic orbits in periodically forced celestial-mechanics models, with potential implications for mission design in cislunar space.

Abstract

The Hill Restricted 4-Body Problem (HR4BP) is a coherent time-periodic model that can be used to represent motion in the Sun-Earth-Moon (SEM) system. Periodic orbits were computed in this model to better understand the periodic orbit family structures that exist in these types of systems. First, periodic orbits in the Circular Restricted 3-Body Problem (CR3BP) representation of the Earth-Moon (EM) system were identified. A Melnikov-type function was used to identify a set of candidate points on the EM CR3BP periodic orbits to start a continuation algorithm. A pseudo-arclength continuation scheme was then used to obtain the corresponding periodic orbit families in the HR4BP when including the effect of the Sun. Bifurcation points were identified in the computed families to obtain additional orbit families.

Figures (11)

• Figure 1: Coordinate Frames Used in cit:ScheeresHR4BP. Note $\bm{\rho} = \hat{\bm{\imath}}_m + \bar{\bm{\rho}}$.
• Figure 2: Continuing the CR3BP $L_4$ Planar Orbit with $T^* = 2 \pi$ into the HR4BP. Left: Melnikov function $\mathcal{M}(s,0)$ values. Right: HR4BP periodic orbit families hodograph.
• Figure 3: HR4BP Periodic Orbit Families Corresponding to the CR3BP $L_2$ N/S Halo Orbit with $T = \pi$.
• Figure 4: HR4BP Periodic Orbit Families Corresponding to the CR3BP $L_1$ Vertical Orbit with $T^* = \pi$.
• Figure 5: Hodographs of HR4BP Periodic Orbit Families Near $L_2$ with initial states at $\tau_0 = 0$.

Theorems & Definitions (6)

• Proposition 1
• Proposition 2
• Proposition 3
• Proof 1: Proof of \ref{['pro:P1']}
• Proof 2: Proof of \ref{['pro:P2']}
• Proof 3: Proof of \ref{['pro:P3']}