Stability of periodic oscillations of almost axisymmetric satellite in plane of elliptical orbit

A solid satellite of almost axisymmetric form, but with an anomaly of its center of mass as independent variable, is considered moving in an elliptical Kepler orbit. Its periodic oscillations in the plane of that orbit, caused by the gravitational torque, are analyzed for stability. The corresponding system of equations of motion has the property that, if its solution satisfies the condition of plane motion at some initial anomaly of its center of mass, it will satisfy this condition at very other magnitude of the anomaly. The 2 pi - periodic solution to the corresponding boundary-value problem has been obtained by numerical methods. The stability region is established analytically as that for two independent linear systems, the two variational equations describing them reduced to an equivalent single one. After the dependence of the coefficients in the characteristic equation on e and mu (e- eccentricity of the ellipse, mu = 3(C - A)/B, A,B,C moments of inertia with respect to the central principle axes x,y,z) has been determined, existence and uniqueness of the 2 pi - periodic solution are proved on the basis of Poincare's theorem.