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

Jose Antonio Rebollo, Rafael Vazquez, Claudio Bombardelli

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.

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

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.

Paper Structure

This paper contains 10 sections, 1 theorem, 8 equations, 2 figures, 1 algorithm.

Table of Contents

  1. Introduction
  2. The Taylor Diffusion Framework
  3. Ansatz and Main Result
  4. Physical Interpretation
  5. Structural Preservation and Tail Modeling
  6. Application: Keplerian Orbit
  7. Problem Setup
  8. Implementation
  9. Validation and Results
  10. Conclusions and Outlook

Key Result

Theorem 1

Let the initial density be of the form eq:ansatz where $F(\cdot,t_0)$ is a diffeomorphism with isolated zero at $x_0(t_0)$, $Q(t_0)\succ 0$, and $\mu(x,t_0)$ a smooth scalar volume field. Under additive white Gaussian noise with diagonal covariance $\Sigma_0 = \sigma\sigma^\top$, this structural for where $J_F = \nabla_x F$ is the Jacobian of $F$; $\Lambda_i = \sqrt{\Sigma_{0,i}}\,\nabla F_i$ with

Figures (2)

  • Figure 1: Two-dimensional marginal PDFs at $t_f = 12\pi$ ($\approx$ 9.5 orbits) for the eccentric Keplerian orbit. Left column: Monte Carlo SDE reference (400000 samples). Right column: Taylor Diffusion (second-order map with evolved $Q$). Top row: position subspace $(\delta x, \delta y)$. Bottom row: velocity subspace $(\delta v_x, \delta v_y)$.
  • Figure 2: One-dimensional marginal PDFs for each state component. Blue histograms: Monte Carlo SDE. Red curves: Taylor Diffusion. The close agreement across all four components, including the asymmetric tails in $\delta x$ and $\delta v_y$, confirms that the method captures the non-Gaussian structure induced by the nonlinear dynamics and stochastic forcing.

Theorems & Definitions (1)

  • Theorem 1: Quadratic Form Conservation