The US Air Force's ability to protect space assets is enhanced by a proficiency in satellite proximity operations and Space Situational Awareness (SSA). In pursuit of that proficiency, this research develops a key capability of interest to mission planners; the ability of a deputy satellite to "hover" within a defined volume fixed in the vicinity of a chief satellite for an extended period of time. Previous research has developed initial methodologies for maintaining restricted teardrop hover orbits that exist in a plane fixed within the chief's local reference frame. These methods use the natural drift of the deputy satellite in the relative frame and impulsive thrust to keep the deputy in a bounded volume relative to the chief, but do not address fuel-optimality. This research extends and enhances that work by finding optimal trajectories, produced with discrete-thrusts, that minimize fuel spent per unit time and stay within the user-defined volume, thus providing a practical hover capability in the vicinity of the chief. The work assumes the Clohessy-Wiltshire closeness assumption between the deputy and chief is valid, however, elliptical chief orbits are allowed. Using the new methodology developed in this work, feasible closed and non-closed relative orbits are found and evaluated based on a fuel criterion and compared to an easily calculated continuous-thrust baseline. It is shown that in certain scenarios the discrete-thrust solution provides the lowest overall fuel cost. These scenarios are generally constrained to a smaller total time-of-flight. A simple check is proposed that enables the mission planner to make the correct strategy choice.
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