Optimal Minimum-Propellant Trajectories for De-Orbiting Satellites into a Northern Lunar Polar Graveyard Region

Recent developments in the global space exploration have put a spotlight on the Moon, attracting the attention of major space agencies and private companies. NASA's ambitious plans for sustained human presence in cislunar space, e.g. the Lunar Orbital Platform-Gateway program, has raised interest in cislunar orbits, looking for some desirable properties such as relatively low transfer costs from Earth, low orbit maintenance costs, and favourable communications opportunities with both Earth and the lunar south pole. Among libration-point orbits, Halo Orbit families, and particularly Near Rectilinear Halo Orbits, are of special interest due to their appealing characteristics from multiple perspectives. Perfectly periodic in the Circular Restricted Three Body Problem model, NRHOs comprise a subset of the halo orbit families in the Earth-Moon system, characterized by close lunar passages and nearly-stable behaviour, thus requiring low-cost maintenance. As the cislunar region is anticipated to become increasingly populated with spacecraft, including potential debris, it is important to highlight the necessity of strategic deorbiting planning and compliance with international laws governing space debris. This study focuses on optimizing lunar de-orbiting trajectories that use electric propulsion and depart from a NRHO, with a particular emphasis on minimizing propellant usage. The chosen reference orbit is the Gateway's southern L2 NRHO, with perilune and apolune radii of 3,300 km and 70,000 km and 9:2 synodic resonance with respect to the Moon's orbit around Earth. Its exact periodic motion is computed by single-shooting method via Differential Correction. The main objective of the proposed research is to optimize low-thrust lunar de-orbiting trajectories by employing an indirect method based on the Optimal Control Theory that transforms the propellant minimization problem into a Two-Point Boundary Value Problem. The single-shooting method shows bang-bang control derived from the Pontryagin's Maximum Principle to optimize the trajectories, ensuring that a specific region in the lunar north pole is targeted. The dynamic model considers 3-body gravitation (spacecraft subject to Earth and Moon gravity) within the Circular Restricted Three-Body Problem. Results identify a specific orbital arc in the NRHO, post-apolune, which is deemed ideal for de-orbiting the satellite via a two-burn trajectory that enables direct disposal towards the lunar north pole, significantly reducing propellant consumption. This reduction in propellant required for de-orbiting allows the mission to allocate more fuel for earlier operational phases, effectively extending the mission's operational lifespan.