Discontinuous Behavior of Time-of-Flight Distribution for Bi-impulsive Earth-Moon Transfers in the Three-Body Model

TL;DR

This paper analyzes bi-impulsive Earth-Moon transfers within the planar circular restricted three-body problem (PCR3BP) by constructing a large set of transfers through grid search and trajectory correction. It introduces a construction-parameter framework $\mathbf y=[α_i,β_i,TOF]^T$ and maps the resulting solutions across multiple planes, uncovering a novel discontinuous behavior in the time of flight $TOF$ as a function of the departure phase angle $α_i$, with roughly six distinct branches separated by about one month in $TOF$. The work provides practical guidance for selecting initial guesses and highlights a structural feature of the solution space that deepens understanding of transfer construction. The findings offer a pathway to more efficient design of Earth-Moon transfers and can be extended to higher-fidelity models, informing mission planning and optimization strategies.

Abstract

As interest in the Earth-Moon transfers renewed around the world, understanding the solution space of transfer trajectories facilitates the construction of transfers. This paper is devoted to reporting a novel or less-reported phenomenon about the solution space of bi-impulsive Earth-Moon transfers in the Earth-Moon planar circular restricted three-body problem. Differing from the previous works focusing on the transfer characteristics of the solution space, we focus on the distribution of the construction parameters, i.e., departure phase angle at the Earth parking orbit, initial-to-circular velocity ratio, and time of flight. Firstly, the construction method of bi-impulsive transfers is described, and the solutions satisfying the given constraints are obtained from the grid search method and trajectory correction. Then, the distribution of the obtained solutions is analyzed, and an interesting phenomenon about the discontinuous behavior of the time-of-flight distribution for each departure phase angle is observed and briefly reported. This phenomenon can further provide useful insight into the construction of bi-impulsive transfers, deepening the understanding of the corresponding solution space.

Figures (1)

• Figure 1: The distribution of obtained bi-impulsive Earth-Moon transfers in the Earth-Moon PCR3BP. (a) $\left( {{\text{TOF}},{\text{ }}\Delta v} \right)$ Map; (b) $\left( {{\text{TOF}},{\text{ }}{\alpha _i}} \right)$ Map; (c) the $\left( {{\alpha _i},{\text{ }}{\beta _i}} \right)$ Map; (d) $\left( {{\text{TOF}},{\text{ }}{\beta _i}} \right)$ Map.