Periodic oscillations of the satellite symmetry axis under the action of gravitational and aerodynamic torques

We consider the satellite, which is an axially symmetric rigid body. The motion of the satellite symmetry axis relative to the orbital coordinate system under the action of gravitational and aerodynamic torques is described by the fourth order differential system. We suppose the satellite orbit is circular and the atmosphere drag is constant along the orbit. Then the differential system is autonomous and admits the first integral of the energy type as well as three families of stationary solutions. We investigate periodic solutions of the system close to the stationary solutions in which the satellite symmetry axis coincides with the normal to the orbital plane. There are two types of such periodic solutions, when the satellite angular rate around the symmetry axis is large. They differ by their periods. Short-period solutions describe motions of the satellite similar to Euler’s regular precession. Long-period solutions describe satellite rotations around the symmetry axis, which moves slowly in absolute space. At some values of parameters the existence of the periodic solutions is proved analitically. Then the solutions are continued numerically in the domain of arbitrary parameter values. Orbital stability in the first approximation of the solutions is investigated. Some solutions are used for explaining uncontroled attitude motion of the satellite Foton-12.