Stability Analysis of All Possible Equilibria for Gyrostat Satellities under Gravitational Torques

Introducing a source of angular momentum into a gravitational!) stabilized satellite affects both the equilibria and the stability properties. A previous paper found that there is a two parameter family of orientations which can be made into equilibria, relative to gravitational torques, by using a source of internal angular momentum. Here a stability analysis is made of all possible equilibria for such gyrostat satellites. The parameters involved are the three inertias of the satellite, two parameters identifying the equilibrium, and the component of angular momentum perpendicular to the orbital plane. The boundaries in this parameter space along which the qualitative nature of the stability behavior changes, are identified using the theory of bifurcation of Poincare with the quadratic approximation to the dynamic potential. The region of Liapunov stable satellites (using the Hamiltonian as a Liapunov function) is identified. It is shown that the case 4a solutions previously described (having the axis of intermediate moment of inertia any place in the orbital plane) forms a boundary dividing those satellites which can be made Liapunov stable from those which cannot be made Liapunov stable by choosing a sufficiently large component of internal angular momentum along the perpendicular to the orbital plane. This boundary can be easily visualized. Contour plots show the magnitude of this angular momentum component required to obtain stability.