The steady motions of a satellite with a two-degree-of-freedom powered gyroscope in a central gravitational field and their stability

The positions of relative equilibrium of a satellite carrying a two-degree-of-freedom powered gyroscope, in the axes of the framework of which only dissipative forces can act, are investigated within the limits of a restricted circular problem. For the case when the “satellite - gyroscope” system possesses the property of a gyrostat and the axis of the gyroscope frame is directed parallel to one of the principal central axes of inertia of the satellite, all the equilibrium positions are found as a function of the magnitude of the angular momentum of the rotor. It is established that the minimum number of equilibrium positions is equal to 32 and, in certain ranges of values of the system parameters, it can reach 80. All the positions satisfying the sufficient conditions for stability are also determined. The number of them is either equal to 4 or 8 depending on the values of the system parameters.