The resonant orbits, in the Earth-Moon system, generally refer to the orbits around the Earth whose orbital periods are approximately in integer ratios to the lunar sidereal period. This specific type of orbits exhibits favorable stability, diverse geometric shapes and a wide accessible range in cislunar space, leading to a growing interest for applications in long-term scientific observations [1], constellation design, space situational awareness [2] and orbital transfers [3].

Current research on resonant orbits is primarily conducted within the circular restricted three-body problem (CR3BP). To apply resonant orbits in practical missions, it is necessary to transition the trajectories into the ephemeris model, achieving long-term operation with minimal orbital maintenance consumption. The existing transition methods include the multiple-shooting method [3], the forward/backward multiple-shooting method [4], and the stacking corrections method [5], which have been widely utilized in the ephemeris transition process of different types of cislunar orbits. However, initial guesses from the CR3BP cannot guarantee a satisfying convergence of the algorithm for resonant orbits. One of the reasons is that the solar gravity, as a periodic perturbation, imposes obvious effects on the resonant orbits, especially when the orbits involve long-time segments in the outer cislunar space. Besides, the extended orbital periods of resonant orbits could result in significant cumulative errors.
 
In this study, an ephemeris transition method for cislunar resonant orbits is proposed in which the bicircular restricted four-body problem (BR4BP) serves as an intermediate dynamical model. Firstly, typical resonant orbits in the CR3BP are transitioned into the BR4BP through a homotopy process by considering the solar gravity and the synodic period. Next, by leveraging the multiple shooting method, initial guesses of resonant orbits in the BR4BP are adopted for multi-revolution ephemeris transition. Specifically, the initial ephemeris epoch must be matched to the initial solar phase. Several metrics are then proposed to evaluate the transition effectiveness of resonant orbits, including the fluctuation ranges of apogee altitude, perigee altitude and apse angle. These metrics could represent the extent to which the trajectory retains its geometric shape in the ephemeris model. Finally, by comparing the convergence and effectiveness of the ephemeris transitions directly from CR3BP versus via BR4BP, it could be confirmed that incorporating BR4BP as an intermediate dynamical model leads to a significant improvement in the quality of initial guesses.
 
The research in this paper improves the ephemeris transition process for cislunar resonant orbits, thereby promoting the application of resonant orbits in practical missions. Furthermore, the method of introducing an intermediate dynamical model holds great significance for high-fidelity ephemeris transition of various types of orbits.
 
References
[1] McComas, D. J., Carrico, J. P., Hautamaki, B., et al., A New Class of Long-Term Stable Lunar Resonance Orbits: Space Weather Applications and the Interstellar Boundary Explorer, Space Weather, Vol. 9, No. 11, 2011.
[2] Gupta, M., Howell, K., and Frueh, C., Constellation Design to Support Cislunar Surveillance Leveraging Sidereal Resonant Orbits, presented at the 33rd AAS/AIAA Space Flight Mechanics Meeting, Austin, Texas, 2023.
[3] Vaquero, M., and Howell, K. C., Leveraging Resonant-Orbit Manifolds to Design Transfers Between Libration-Point Orbits, Journal of Guidance, Control, and Dynamics, Vol. 37, No. 4, 2014, pp. 1143–1157.
[4] Guo, L., and Shang, H., Earth-moon resonant orbits design in the ephemeris model, Journal of Physics: Conference Series, Vol. 2977, No. 1, 2025, pp. 012023.
[5] Gupta, M., Howell, K., and Frueh, C., Long-Term Cislunar Surveillance via Multi-Body Resonant Trajectories, presented at the 2022 AAS/AIAA Astrodynamics Specialist Conference, Charlotte, North Carolina, 2023.