Modeling the position and velocity distribution of space objects by maximizing entropy with energy constraint

TL;DR

This paper develops a maximum entropy-based 6D probability density function for space objects in LEO, incorporating energy constraints to accurately model their position and velocity distributions for uncertainty analysis.

Contribution

It introduces a novel MaxEnt approach with energy constraints to model joint position-velocity distributions of space objects in LEO, ensuring physically consistent uncertainty representation.

Findings

Provides a PDF for space object states adhering to energy conservation.

Enables sampling and propagation of orbital uncertainties without prior assumptions.

Facilitates uncertainty analysis via Monte Carlo or Fokker-Planck methods.

Abstract

In this work, we have developed a 6-dimensional joint probability density function for the 3-dimensional position and 3-dimensional velocity vectors of space objects in the Low Earth Orbit (LEO) based on the Principle of Maximum Entropy (MaxEnt), adhering to the principle of energy conservation. For the problem under consideration, maximizing entropy subject to energy conservation ensures that the derived probability density function (PDF) is the best representation of the uncertainty of a space object while the sampled position and velocity vectors from the PDF adhere to the orbital dynamics. We approach the entropy maximization by constructing a Lagrangian functional incorporating the energy conservation constraint and the normalization constraint of the PDF using Lagrange multipliers, setting the functional derivative of the Lagrangian to zero. This PDF can be used to generate position and velocity samples for space objects without any prior assumption and can further be utilized for orbital uncertainty propagation either using the Monte-Carlo method or by direct propagation of the PDF through the Fokker-Planck Equation.