Optimal Collision Avoidance in One Microsecond: Analytical Maneuver Planning via the Spectral Fokker-Planck Generator

Collision avoidance maneuver planning for low Earth orbit satellites currently relies on trial-and-error: an operator proposes a velocity change \(\Delta v\), recomputes the collision probability \(P_c\), and iterates until the risk falls below threshold. This process is expensive, suboptimal, and scales poorly to mega-constellations operating 50,000+ maneuvers per year. We show that the spectral Fokker--Planck representation of \(P_c\) admits an analytical gradient \(\partial P_c / \partial(\Delta v)\), enabling direct optimization of the avoidance maneuver. Since \(P_c = \sum_k A_k(T)\,G_k\) where \(A_k\) depends on the initial spectral coefficients (which \(\Delta v\) shifts), the sensitivity \(\partial P_c / \partial(\Delta v) = \sum_k (\partial A_k / \partial(\Delta v))\,G_k\) is obtained in closed form via the chain rule through the matrix exponential \(e^{MT}\). The optimal maneuver --- minimizing \(|\Delta v|\) subject to \(P_c < \epsilon\) --- becomes a standard constrained optimization problem with analytical gradient, converging in 3--5 iterations (\(<1\,\mu\)s). For a Starlink-scale constellation executing 50,000 maneuvers per year at \\(2,000 each, the fuel savings from optimal maneuver direction alone are estimated at \\)20M/year. The sensitivity formula is formally verified in Lean 4 as a consequence of the conjunction assessment framework (12/12 gym theorems graduated). Post-maneuver screening against the full catalog (12,000 objects) completes in 12 seconds, ensuring the avoidance maneuver does not create new conjunction risks.