Collision Probability of Debris Clouds Based on Higher-Order Boundary Value Problems

Abstract

Debris clouds stemming from the breakup of objects pose a threat to spacecraft. Here, a method based on higher-order boundary value problems (BVPs) is presented to analyze the hazard of debris clouds. The higher-order transfer map from initial velocity space to final position space is obtained with the use of differential algebra. An adaptive homotopic scheme is proposed to solve the perturbed BVP, where the transfer map is used to correct the residuals. Then, higher-order expansions of the BVP solution and the Jacobian with respect to the final position are derived, and the debris density inside a spacecraft envelope is calculated with the expansions. The simulation results show that the proposed higher-order method contributes to accelerating the convergence of perturbed BVP solvers. The density transformation method based on the map is validated by Monte Carlo simulations. The effects of perturbations on the collision risk are shown.