The attitude stability of the stationary motions of a dumbbell satellite in the case of air drag

This letter investigates the effect of the length of an Earth orbiting dumbbell system on the number and the Lyapunov stability of its stationary motions. This effect appears when a weak aerodynamical drag is taken into consideration and being important for low orbit satellites. The equation of motion is obtained as a Lagrange equation. Using analytical methods a critical length can be found causing essential changes in the structure of the set of stationary positions and in the stability properties, too. For longer systems the number of stationary positions is four, for shorter ones it is two. In the critical case the other two positions come together and disappear in a saddle-node bifurcation. The numerical values of the bifurcation point and of the critical length are also presented in a case described in the literature.