Linear stability and shape analysis of spinning three-craft Coulomb formations

This paper describes the discovery of families of multiple invariant shape solutions for collinear three-craft Coulomb formations with set charges, as well as the results of linear stability analysis on such formations. The charged spacecraft are assumed to be spinning about each other in deep space without relevant gravitational forces present. Up to three invariant shape solutions are possible for a single set of craft charges. This behavior, only speculated in previous work, is confirmed through analysis and numerical simulation examples. In fact, distinct regions are analytically described where two or three invariant shape solutions exist for a single charge set. These regions are analyzed to determine what range of trajectories are possible. Linear stability analysis for circular trajectories yields the first examples of marginally stable three-craft invariant shape formations. Linearly stable behavior is only observed when two invariant shape solutions result for one set of charges, where one shape will be unstable and the other marginally stable. Numerical simulation illustrates stability for ten orbital periods when perturbations are confined to the orbital plane. When out of plane motion is considered the shapes are found to be weakly unstable, though the out of plane motion appears to be decoupled from in plane motion to first order.