ANALYTIC SOLUTION OF PERTURBED RELATIVE MOTION WITH ZONAL AND TESSERAL HARMONICS

A state transition matrix for satellite relative motion including the perturbation effects due to zonal and tesseral harmonics is presented. Recently, a new solution for short-period variations of the orbital elements of a satellite due to tesseral and sectorial harmonics was found by the authors without resorting to eccentricity expansions or the method of relegation. This novel approach makes use of non-conservative canonical transformations to normalize the disturbing potential and produce generalized analytic formulae, closed-form in eccentricity, for second-order tesseral short-period effects. Using these analytic formulae, a state transition matrix solution for the perturbed relative motion that includes the effects of zonals and tesserals up to second order is presented in this work. Since the tesseral harmonics have a significant effect on the mean longitude of a satellite, their inclusion in the relative motion solution improves the prediction of along-track motion between two satellites in a formation. The proposed solution for the perturbed relative motion is valid for any reference elliptic orbit with an arbitrary value of the eccentricity. Accuracy verification of the proposed analytic solution is carried out by comparing the results with numerical propagation of a formation of two satellites using GMAT simulation package.