Darin Koblick, Raytheon
Keywords: Tulip Orbits, Cislunar, SSA, SDA, CR3B, Multi-Level Shooting, Three Body, N body
Abstract:
Several new families of quasi-periodic orbits, termed tulip orbits were recently discovered from forking a near rectilinear halo orbit (NRHO) around the second Earth-Moon Lagrange point. Characterized by their nearly symmetric lobes, periods spanning 12 to 26 earth-days, and low perilune altitudes suitable for near-surface fly-bys, these orbits are similar to NRHOs but distinguishable by their quasi-periodic nature and variable apolune and perilune altitudes. This variability makes them attractive candidates for tracking, surveillance, communication, and navigation missions requiring geometric measurement diversity. Their apolune can occur high above the northern or southern hemisphere while perilune occurs on the opposite side. Tulip orbits are symmetric about the x-z plane and their orbital period can be adjusted to achieve a specific resonance.
The quasi-periodic nature of these tulip orbits persisted after transitioning from the circular restricted three-body (CR3B) problem to a higher-fidelity force model. For this analysis, NASAs General Mission Analysis Tool (GMAT) engine was used to model third-body perturbation effects. The DE 421 model, a comprehensive and accurate representation of the moons orbit, used by many organizations for lunar navigation and planning, was used to determine the position and velocity of the moon within submeter accuracy.
The orbit ephemeris transition process from the CR3B model to the GMAT force model, consisted of generating several dozen patch points per orbital period and then correcting them with a multi-level shooting technique. The multi-level shooting technique, as outlined in An Improved Corrections Process for Constrained Trajectory Design in the n-body Problem employs a two-level differential correction process:
Level One – differential correction process operates on the velocity components of each patch point. The resulting output is a continuous ballistic trajectory in the higher fidelity force model (no discontinuities in position after a single pass). This process is embarrassingly parallel as the corrector can operate on each segment separately. The velocity components at the start of each segment can be corrected simultaneously.
Level Two – differential correction process operates on the position and epoch of all patch points concurrently. The resulting output is a smooth trajectory which has minimal discontinuities in velocities between segments. This process operates on all segments at once using a minimum-norm solution of least-square system to drive discontinuities in velocity (delta-v) down to zero.
A smooth continuous trajectory was obtained after several iterations of the multi-level shooting method. This smoothed trajectory was propagated for several orbital periods using a numerical ordinary differential equation solver on the higher-fidelity force model. The resulting properties of each trajectory were compared against those of the CR3B orbit to verify their periodicity and stability.
The family of tulip orbits was compared to more traditional three-body colinear orbit families such as NRHOs and butterfly orbits for lunar surface monitoring and space domain awareness (SDA) missions. Surface ISR and SDA performance was evaluated by standard catalog coverage metrics such as gap times. A comparative analysis was performed with a cislunar target catalog, consisting of 15 targets across eight orbit families, and 64 surface target locations from the NASA catalog of man-made material on the moon. Findings demonstrate that average minimum and maximum gap times decrease with an increase in petals.
Date of Conference: September 19-22, 2023
Track: Cislunar SDA