Low-energy, three-dimensional, symmetric jumping Trojan trajectories for the exploration of the Earth-Moon triangular libration point regions

Technical Summary: Low-energy, three-dimensional, symmetric jumping Trojan trajectories for the exploration of the Earth-Moon triangular libration point regions

Problem Statement
The triangular libration point (TLP) regions ($L_4$ and $L_5$) in the Earth-Moon system offer unique opportunities for scientific missions, particularly due to the potential existence of Kordylewski dust clouds (KDCs). While previous missions like HITEN visited these regions, they typically passed each TLP only once, limiting the ability to conduct comprehensive surveys. An ideal trajectory for exploring these regions would allow a spacecraft to perform multiple revolutions around both $L_4$ and $L_5$ while ballistically transitioning between them to minimize fuel consumption. This behavior mimics "jumping" Trojan asteroids, which naturally transfer between tadpole orbits around the two triangular points. However, a comprehensive understanding of which specific low-energy, symmetric jumping Trojan trajectories are available for mission design in the Earth-Moon system remains an open question.

Methodology
The study investigates these trajectories within the framework of the spatial Circular Restricted Three-Body Problem (CR3BP) using the Earth-Moon mass parameter $\mu = 0.0121506683$. To address the computational complexity of searching a six-dimensional phase space, the authors leveraged the system's symmetries to reduce the search space dimensionality:

• Symmetry Utilization: Time-reversal symmetries ($s_1$ and $s_2$) were used to construct axi-symmetric ($L_5$-to-$L_4$) and $xz$-planar symmetric trajectories. This reduced the search space to three dimensions and eliminated the need for backward propagation during the initial search. A third symmetry ($s_3$) regarding the $xy$-plane further halved the search space by restricting variables to non-negative values.
• Grid Search: A comprehensive grid search was conducted over initial conditions ($x_0$, Jacobi constant $C$, and out-of-plane velocity/position variables). The search focused on "low-energy" conditions defined as $C \geq 3$ (higher energy than the triangular points but below the $L_3$ energy level) to ensure compatibility with low-energy lunar transfer scenarios.
• Novel Criteria for Identification: A critical methodological contribution was the development of new criteria to distinguish tadpole orbits from other trajectory types during propagation. The authors utilized the unique energetic switching behavior of tadpole orbits induced by the secondary body's gravitational perturbation. Specifically, a trajectory was identified as a tadpole orbit around $L_4$ if it experienced the angle $\theta = 60^\circ$ at least three times, coupled with a specific oscillation in Keplerian energy $K$ between $-0.6 < K < -0.4$ (switching between $K > -0.5$ and $K < -0.5$). This energetic constraint prevented overcounting caused by orbital loops.

Key Results
The grid search successfully identified a vast variety of low-energy, symmetric jumping Trojan trajectories.

• Orbital Characteristics: The discovered trajectories exhibit complex dynamics, including multiple loops near the $L_3$ point and irregular fluctuations in energy, attributed to chaotic dynamics in that region. The solutions generally possess low eccentricity ($e < 0.2$) and low inclination relative to the lunar orbital plane ($i < 10^\circ$).
• Solution Distribution: The solutions were found to reside well within the search bounds, avoiding the boundaries of $x_0$ and the upper energy limit $C_{L3}$.
• Mission Compatibility: A specific strategy was developed to extract candidate trajectories that connect with low-energy lunar transfer missions. By analyzing the backward legs of the trajectories, the authors identified groups of solutions that reach low perilune altitudes after the second effective event.
  • The lowest perilune altitude found for axi-symmetric trajectories was approximately 2,764 km.
  • The lowest perilune altitude for $xz$-planar symmetric trajectories was approximately 1,483 km.
  • These "Moon-grazing" trajectories provide promising initial guesses for mission design, allowing a spacecraft to explore the TLP regions before or after being inserted into a Moon-centric orbit.

Significance and Claims
The paper claims to provide the first comprehensive search for low-energy, three-dimensional, symmetric jumping Trojan trajectories in the Earth-Moon CR3BP. The primary significance lies in the development of robust criteria for identifying tadpole orbits, which enables the systematic discovery of these complex trajectories via grid search. The authors posit that these trajectories offer a fuel-efficient mechanism for exploring both $L_4$ and $L_5$ regions, potentially enabling single-spacecraft missions or mother-daughter spacecraft configurations. The identified Moon-grazing solutions are presented as practical starting points for mission designers to optimize trajectories for specific engineering constraints and scientific goals, such as the detection of Kordylewski dust clouds. The study acknowledges that the results are based on a low-fidelity CR3BP model and suggests that future work should assess these trajectories using higher-fidelity models.