On periodic motions of a gyrostat satellite with a large inner (gyrostatic) angular momentum

Rotational motion of an axially symmetric gyrostat satellite under the action of its gravitational torque in a circular orbit is considered. Periodic motions of the symmetry axis of the satellite relative to the orbital coordinate system are investigated. In absolute space, these motions appear as a slow precession about the normal to the orbital plane. Such motions are described by an autonomous system of fourth-order differential equations. The gyrostatic angular momentum is assumed to be large, which allows us to introduce a large parameter into the equations of motion. Solutions that are a rest in absolute space, with the symmetry axis forming a nonzero angle with the orbital plane, serve as the generating solutions. The period of the found solutions depends on this angle. Earlier, the limit case of such periodic motions when the symmetry axis of the satellite lies in the orbital plane in the generating solutions was investigated. The limit solutions describe small oscillations of the symmetry axis of the satellite in absolute space, and their period is equal to half the orbital period. To prove the existence of the new motions, we reduce the boundary value problem configuring the periodic solutions to a system of integral equations, which is solved by the method of successive approximations. This reduction is carried out according to the same scheme as in the degenerate case, but the necessary solutions of the integral equations are constructed differently. The result obtained explains the appearance of the limit solutions although the latter cannot be constructed within the framework of the considered general case.