Adiabatic invariance for tether deployment/retrieval problem

The idea of Tethered-Satellite-System (TSS) has been tested twice on the space shuttle in the past few years, however both failed. It is desirable to study the complicated dynamics of TSS more deeply. During the deployment phase or the retrieval phase of TSS operation, the length of the tether is varied slowly. This process may lead to a suitable application of the theory of adiabatic invariance. In short, adiabatic (or asymptotic) invariance is a quantity that is almost invariant when some parameter of the physical system is slowly changed. In this paper, we compute an adiabatic invariance for an approximate model of TSS during deployment/retrieval mode. It is observed that as the length of the tether becomes larger slowly (deployment mode), the amplitude of the oscillation becomes smaller. On the other hand, if the tether is shortened (retrieval mode), the oscillation becomes larger. This observation coincides with the assertion that the dynamics is stable during the deployment mode, but unstable during the retrieval mode.