Extreme values of relative distances for spacecraft in elliptic displaced orbits

This paper provides a framework to obtain a semi-analytical approximation of extreme values of relative distances between two spacecraft that cover elliptic displaced orbits. The relative motion is described in the rotating reference frame of the chief spacecraft and is parameterized with a new set of displaced orbital elements. The extreme values of the radial, along-track and cross-track distance are analytically evaluated (as roots of suitable algebraic equations) both for quasi-periodic orbits in the incommensurable case, and for periodic orbits in the 1:1 commensurable case. In particular, in the 1:1 commensurable case a Fourier series expansion is used to obtain a time-explicit expression of the relative motion. Finally, some illustrative examples are presented to validate the correctness of the proposed method.