Closed-Form Solutions to the Fokker-Planck Equation for Orbital Uncertainty Propagation

TL;DR

This paper introduces a closed-form, grid-free solution to the Fokker-Planck equation for orbital uncertainty propagation, efficiently capturing non-Gaussian features without Monte Carlo sampling.

Contribution

It proves that an exponential-of-quadratic ansatz is preserved under advection and diffusion, enabling a compact ODE system for probability density propagation.

Findings

Accurately captures non-Gaussian tails and asymmetric features.

Matches Monte Carlo benchmarks in orbit uncertainty propagation.

Significantly reduces computational cost compared to Monte Carlo methods.

Abstract

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.