Cross-Track Motion of Satellite Formations in the Presence of J2 Disturbances

Abstract

Introduction T HIS Note augments and completes a previous work1 by the authors in which a set of constant coefficient linear differential equations were developed that describe the relative motion between two satellites in the presence of the Earth’s J2 geopotential. In the previous work, the motion was separated into in-plane and cross-track motion. The analysis of the in-plane motion was relatively straightforward and remains untouched in this Note. The cross-track motion was more difficult to derive; linearizations with respect to time were used to develop analytical solutions. In this Note, the equations are instead simplified using small angle approximations. These simplifications are valid due to the very small angles that describe the cross-track motion. Although no longer linear, the equations can still be solved analytically. The result is a set of differential equations that are more accurate and more intuitive.