Carson Hunsberger, Pennsylvania State University; Roshan Eapen, Pennsylvania State University; Puneet Singla, Pennsylvania State University
https://doi.org/10.64861/GLCX4148
Keywords: Characterization, Hamiltonian, normal form, libration point, center manifold, action-angle
Abstract:
In the neighborhoods of space near the Earth and the Moon, satellite motion is well-approximated by the completely integrable two-body problem, with its orbital element sets. Even when additional perturbations come into play, such as Earth’s atmosphere or its irregular gravitational field, these orbital element sets are still utilized. Even when the orbital elements are fluctuating with time along a real trajectory, they provide useful intuition about the satellite’s dynamical behavior. Unfortunately, as the distance from either the Earth or the Moon grows, trajectories become very non-Keplerian, and the two-body orbital elements are no longer useful for trajectory characterization even in their osculating form. It is a well-known result that in a lower fidelity model known as the circular restricted three-body problem (CR3BP), there is only a single constant of motion. Once the fidelity of the model is increased further, to the elliptic restricted three-body problem (ER3BP), for example, this constant of motion is lost. One is then forced to work with the full six-dimensional state, which must be numerically propagated in order to determine which family the trajectory belongs to.
Near the collinear libration points, the equilibrium points along the Earth-Moon axis at which the gravitational attraction of both primaries cancel out, additional constants of motion can be obtained by leveraging Hamiltonian normal forms. The Birkhoff normal form completely isolates the saddle and center subspaces, providing three constants of motion referred to as actions, along with their three corresponding linearly varying angles. These actions provide intuition about the behavior of a trajectory without the need to propagate the initial state. However, it is generally accepted that the Birkhoff normal form has a small radius of validity — so much so that only a small section of the family of halo trajectories is accurately characterized. Thus, the resonant normal form is often used instead when one requires a larger radius of validity, along with characterization of the halo and quasihalo families. Furthermore, states in the normal form space can be projected onto the saddle subspace, where the origin represents the center manifold, the axes correspond to the stable and unstable manifolds, and the four quadrants separate transit trajectories from trajectories that return in the direction from which they came.
In previous work, the accuracy of the normal forms has been evaluated by propagating states simultaneously in the synodic frame and the action-angle space, and then comparing the Cartesian error at the final state, upon transformation from action-angle variables to the synodic frame. This work calls into question whether this metric is overly strict. The dynamics of the CR3BP and ER3BP are notoriously sensitive to the initial condition. The error between states propagated in the synodic frame and action-angle space for a halo trajectory, for example, will diverge quickly even if the action-angle state maps very close to a state along the halo trajectory. What is perhaps more important in terms of the intuition provided by the normal forms is the ability to provide qualitative information about the behavior of a trajectory given just a single initial state. As long as this capability exists, the normal forms can be used to define osculating action-angle variables that can be used to characterize a much longer trajectory.
The normal forms depend on a polynomial expansion of the potentials of the two primaries about a given libration point. It is generally accepted that the normal forms will provide more accurate characterization of trajectories within a neighborhood of the libration when the degree of the expansion is increased. This work will quantify the region of validity of various degree normal forms of both the CR3BP and ER3BP by taking a large grid of sampled initial states within the synodic frame, propagating them according to the dynamics of a higher fidelity ephemeris model (HFEM), and determining at what point the short-arc characterization capabilities of the normal forms begin to break down. As mentioned, the saddle subspace should separate distinct types of trajectories, and so the accuracy of the normal forms can be tested by checking an initial state’s location within the saddle subspace and seeing if the trajectory behaves as expected when propagated numerically with the HFEM. After the ideal degree of the normal form has been determined, with optimality being evaluated as a combination of accuracy and computational expense, the normal forms of the CR3BP and ER3BP can be used to obtain osculating action-angle representations of members of various center manifold trajectories from multiple families obtained numerically in the HFEM. The earlier analysis will provide a limit on the size of the quasi-/periodic trajectories that are to be considered.
Date of Conference: September 15-18, 2026
Track: Cislunar SDA