Nonlinear Stability Analysis of Low Lunar Frozen Orbits for Design Considerations of Space Architectures

Abstract

Lunar orbits have become a topic of interest for future space mission architectures. However, pursuing long-term missions in low-altitude lunar orbit is particularly difficult due to the Moon's highly irregular gravitational field. Previous research identified candidate Lunar Frozen Orbits (LFOs) through analytical or numerical methods, but without a systematic understanding of where more frozen orbits may occur. This work attempts to fill this gap by using nonlinear analysis techniques to analyze the stability of current candidate Lunar Frozen Orbits through a Python computational framework which applies Lagrangian Coherent Structures (LCS) and the State Transition Matrix (STM). By mapping Lagrangian coherent structures for a grid of initial states, we were able to visualize local stretching and contracting, showing how small perturbations in nominal frozen orbits affect their long-term trajectories. Experiments were conducted on candidate Lunar Frozen Orbits identified by Folta and Quinn (2006) and Miceli (2023), where 90×90 grids of initial conditions varying eccentricity and inclination were propagated for 1, 5, and 10 periods through NASA GMAT using the GRAIL gravity model at spherical harmonic order and degree 100×100. The STM for each grid point was reconstructed via central finite differencing, and its Singular Values (SVs) were plotted as contour fields to reveal the stretching and contracting manifolds of the local phase space. Results show highly organized, patterned singular value fields that evolve over time and reveal regions of differing sensitivity to initial conditions. A six-month long-term propagation of select representative points further validates that trajectories embedded within similar short-term Lagrangian coherent structure environments exhibit comparable long-term behavior, suggesting that LCS-based analysis may provide a more systematic and efficient method for identifying families of Lunar Frozen Orbits than existing large-scale numerical sampling approaches.