Modeling relative motion of LEO satellites at different altitudes

This paper presents a new approach to model the relative motion of low Earth orbit satellites flying at different altitudes, with separations on the order of hundreds of kilometers. The main scope is describing the relative trajectory especially when the spacecraft are relatively close to each other and potentially able to use their payloads in a synergic way, as it happens in traditional formation flight. The derivation procedure accounts for the main orbit perturbing effects and assumes small differences between the orbit parameters. Final equations show that relative motion due to the altitude difference can be approximated by a circular in-plane trajectory. Then, when the along track separation is relatively small, in-plane dynamics results from the combination of several polynomial and oscillatory terms at orbital frequency and twice the orbital frequency. As for the out of plane motion, it consists of an oscillation with orbital frequency and slowly varying amplitude. A new mission concept is presented as an example of model application. It consists of a small sat-based distributed radar receiving the backscattered echoes of existing low Earth orbit transmitters. Hence, an orbital design is proposed to counteract the cross-track motion thus maximizing mission potential.