The purpose of this paper is to obtain a third-order expression, for the in-plane and out-of-plane amplitudes, of the solutions of the elliptic Hill–Clohessy–Wiltshire non-linear equations. The resulting third-order solution is explicit in terms of true anomaly. The coefficients of the expansions are given as functions of the eccentricity e of the orbit of the leader (i.e., are valid for all values of e). For e=0 we recover the solution given by Richardson and Mitchell for the circular case; for e≠0 the linear terms of the solution recover the solution found by Lawden for the linearised elliptic HCW equations, also known as the Tschauner–Hempel equations. In the last part of the paper we explain how a formal series solution of the elliptic HCW non-linear equations (in powers of the two amplitudes and the eccentricity) can be obtained, using the Lindstedt–Poincaré procedure.

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