On Resonant Motions of a Symmetric Satellite at Frequencies Equal or Close to Zero

The motion of a dynamically symmetric satellite (rigid body) relative to its center of mass in a central Newtonian gravitational field in a weakly elliptical orbit is considered. The motion occurs in the vicinity of stationary rotation of the satellite around the normal to the orbital plane (cylindrical precession). We study the cases where in the limiting case of a circular orbit the values of the problem parameters (inertial parameter, dimensionless angular velocity, and orbital eccentricity of the center of mass) belong to small neighborhoods of multiple resonance points in the parameter space, for which one of frequencies of small linear oscillations of the perturbed system is zero, and the other is an integer or half-integer number. We solve the problem of the existence, number, and stability of resonant periodic motions of the satellite, analytic in integer or fractional powers of a small parameter (orbital eccentricity of the center of mass). The study is based on previously obtained general theoretical results in the problem of nonlinear oscillations of a nearly autonomous, time-periodic Hamiltonian system with two degrees of freedom in the cases of multiple parametric resonances under consideration. Compared to the general theory, the problem of the stability of periodic satellite motions is studied more fully. For cases where the nonzero frequency is half-integer, a rigorous nonlinear stability analysis is performed. For cases where the nonzero frequency is an integer, a complete linear stability analysis is carried out, and the corresponding stability diagrams are obtained.