Nonlinear dynamical model of relative motion for the orbiting debrisproblem

Abstract

A nonlinear dynamical model of relative motion is derived for application to the orbital breakup problem. This model is designed for analysis of debris density beginning at the time of breakup and continuing until after the cloud rings the Earth. A linear relative motion model is derived, which generalizes the Clohessy-Wiltshire equations to include all values of eccentricity and the secular term Ji. Then, selected higher-order terms are added, yielding generalized equations that are useful at large displacements from the reference ephemeris. A significant improvement over linear relative motion techniques is demonstrated for the expanding debris cloud. The inverse of the relative motion transformation is then shown to be an approximate solution to Lambert's problem for multiple orbits in a perturbed force field. This model enables an analysis of debris density differentially at a point in space as a function of initial breakup spread velocity statistics. Analysis is provided showing that particle density can vary by four orders of magnitude within the instantaneous cloud. Density extrema always occur at the pinched points of the cloud and, in a perturbed force field, also occur at discrete locations within the envelope of the expanding cloud.