New State Transition Matrix for Formation Flying in J2-Perturbed Elliptic Orbits

Abstract

A state transition matrix for formation flying in eccentric orbits is considered in this paper. The relative motion of the formation flying obeys the Tschauner–Hempel equations, which are extended for the presence of a J2 perturbation. Furthermore, the state transitionmatrix of the Tschauner–Hempel equations is extended for J2, and it is demonstrated that the extended state transition matrix becomes the solution of the extended Tschauner–Hempel equations. Because the extended state transition matrix only uses osculating orbital elements in the initial state and does not use mean orbital elements, an analytical treatment of the state transition matrix is possible. When the state transition matrix uses only the first-order term of J2, the errors sometimes become large compared with the relative distancemultiplied by J2 in a time range that is longer thanone orbit period.Amethod to suppress errors numerically is also proposed.