Analytic solution for COLA maneuver optimization for near-circular orbits

When a "Class Red" conjunction between a primary and a secondary object is identified as part of the collision avoidance (COLA) process, the collision risk is large enough to warrant consideration of a maneuver. Efficient numerical solutions to the maneuver problem have been developed and are implemented in The Aerospace Corporation program DVOPT that produces solutions quickly for 1- and 2-dimensional burns. However, achieving a full 3-dimensional solution (and even a 2-d solution under certain circumstances) can become time-inefficient in an operational environment. Therefore, it would be advantageous to have an analytical solution that can be generated quickly. The program DVOPT utilized backwards/forwards propagation in applying the burns during the search algorithm. These back-ward and forward propagations are conducted in either a 2-body sense, iterating in Kepler's Equation, or by numerical integration. The use of Kepler's Equation is faster than the numerical integration and is justified in that the burns to be performed are small and produce a similar change to the fully perturbed trajectory at the point of conjunction. Since the 2-body Keplerian solution is valid in determining the change in the state vector at conjunction for an arbitrary burn of low magnitude, then the backwards/forwards propagation can be eliminated in favor of an approximate direct solution. An approximation to the change in the primary's state vector at the time of conjunction due to an arbitrary burn is analytically derived. To avoid having to solve Kepler's Equation, the equivalent linearized equations were determined only for near-circular orbits. These changes are then used in existing probability models to find the optimal burn magnitude and direction. Favorable comparison is shown between the analytic approximation and both the numerical integration and Kepler's Equation solutions.