An analytical theory for avoidance collision between space debris and operating satellites in LEO

Abstract

This paper provides a Hamiltonian formulation of equations of motion of an artificial satellite or space debris moving in Low Earth orbit (LEO). This Hamiltonian is the base to develop an analytical theory of order three, which the theory has been formulated using canonical transformations using Hori-Lie method. The theory accounts for the influence of the Earth gravity field up to degree and order of geopotential harmonics (50×50), luni-solar perturbations, solar radiation pressure, atmospheric drag, albedo forces, and the post Newtonian effects arising from Schwarzschild solutions. In this theory, we pay particular attention to the resonance and very long period perturbations, which are modeled with the use of semi-secular terms. The fourth order Runge–Kutta numerical integrator is used to integrate the equations of motion in order to compare with the analytical theory. Including all possible perturbations, in particularly Aledo force and post Newtonian effect in addition to the adequate section of the integrator found to have an importance on the improving the accuracy of the current theory compared with the previous solution. The orbital theory is precise and enables the short term predictions on a few centimeters level. Finally, we show some results concerning the short term dynamics of a different satellite and space debris in LEO under the influence of the considered perturbations.