Analytical Solution of the Perturbed Oribt-Attitude Motion of a Charged Spacecraft in the Geomagnetic Field

In this work we investigate the orbit-attitude perturbations of a rigid spacecraft due to the effects of several forces and torques. The spacecraft is assumed to be of a cylindrical shape and equipped with a charged screen with charge density sigma. Clearly the main force affecting the motion of the spacecraft is the gravitational force of the Earth with uniform spherical mass. The effect of oblate Earth up to J(2) is considered as perturbation on both the orbit and attitude of the spacecraft, where the attitude of the spacecraft is acted upon by what is called gravity gradient torque. Another source of perturbation on the attitude of the spacecraft comes from the motion of the charged spacecraft in the geomagnetic field. This motion generates a force known as the Lorentz force which is the source of the Lorentz force torque influencing the rotational motion of the spacecraft. In this work we give an analytical treatment of the orbital-rotational dynamics of the spacecraft. We first use the definitions of Delaunay and Andoyer variables in order to formulate the Hamiltonian of the orbit-attitude motion under the effects of forces and torques of interest. Since the Lorentz force is a non-conservative force, a potential like function is introduced and added to the Hamiltonian. We solve the canonical equations of the Hamiltonian system by successive transformations using a technique proposed by Lie and modified by Deprit and Kamel to solve the problem. In this technique we make two successive transformations to eliminate the short and long periodic terms from the Hamiltonian.