Lagrangian dynamics and the discovery of cislunar periodic orbits

Through the art of applying assumptions which restrict the orbital shape and relative inclination of celestial bodies such as the Earth and Moon, trajectory prediction of small bodies of comparatively negligiblemass (i.e., a satellite) within amulti-body gravitational system is possible. In this research, Lagrangian analytical methods are applied to formulate succinct derivations of the circular restricted three-body problem (CR3BP), the elliptical restricted three-body problem (ER3BP), and the bicircular restricted four-body problem (BCR4BP). The presence of, or lack thereof, equilibrium points within each dynamical model is discussed and presented in both graphical and tabular form. In terms of application, a form of periodic trajectories within the Earth-Moon system, identified herein as cislunar periodic orbits, is propagated using each of the presented dynamical models. The dynamical variations in these cislunar periodic orbits when transitioning between dynamical models are analyzed and discussed. The methodology behind cislunar periodic orbit generation is also discussed with 33 cislunar periodic orbits presented. Finally, through means of differential correction, it is shown how much error in Delta V (Delta eV), the BCR4BP dynamics introduce on the CR3BP solutions for a given number of patchpoints. Results of this analysis show the ER3BP to have a significantly higher perturbative effects than the BCR4BP on cislunar periodic orbits which are closed in the CR3BP. Based on the 33 orbits analyzed, correlation was also observed between the Jacobi constant and the dynamical variations present during the transition of cislunar periodic orbits to higher fidelity models, with larger Jacobi constants being associated with more dynamical variations as an orbit transitions from the CR3BP to both the ER3BP and BCR4BP.