Closed-Form Solutions to the Fokker-Planck Equation for Orbital Uncertainty Propagation
This provides a computationally efficient method for orbital uncertainty propagation in conjunction assessment, addressing a domain-specific bottleneck.
The paper tackles the challenge of capturing non-Gaussian tails in orbital collision probability estimates without Monte Carlo sampling by presenting a closed-form, grid-free solution to the Fokker-Planck equation, which propagates probability density via a compact ODE system and matches Monte Carlo benchmarks.
Non-Gaussian tails dominate collision probability estimates in conjunction assessment, yet capturing them without Monte Carlo sampling is challenging, especially when process noise is included. We present a closed-form, grid-free solution to the Fokker-Planck equation by proving that an exponential-of-quadratic-form ansatz is structurally preserved under advection and diffusion. The probability density function propagates via a compact ODE system, significantly cheaper than Monte Carlo and without spatial discretization. As an application, the method performs orbit uncertainty propagation under stochastic forcing representative of atmospheric drag. Results demonstrate the method faithfully captures non-Gaussian features, asymmetric tails, and stochastic broadening, matching a Monte Carlo benchmark.