Bounds on Relative Distance in the J2 Problem

What is it about?

This paper proposes an analytic method to determine the bounds of the relative distance between two satellites that includes the $J_2$ perturbation. As the dynamics of the $J_2$-perturbed relative motion are nonlinear and complicated, averaged dynamics are used to simplify the analysis model. The relative distance is calculated analytically as the mean orbital elements, and the Fourier--Bessel series is utilized to separate the secular and periodic variations. The secular change and the amplitude of the periodic oscillations are quadratic functions of time. Then, an algorithm to determine the bounds is proposed by solving a quartic equation. The numerical results validate the accuracy of the proposed method, which is a promising approach to predicting collision accidents.

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