Asymptotic calculation of the influence of the relative motion of a satellite on the motion of its centre of mass

Abstract

The numerical integration of the complete system of equations of motion of a satellite encounters great difficulties in those cases where the satellite has a high relative motion, while the position of its centre of mass changes insignificantly. High-frequency vibrations or rotations of the satellite lead to the necessity to decrease the integration step, which considerably increases the time of numerical calculations and leads to reduction in accuracy. On the other hand the presence of high relative motion enables us to introduce into the system of equations a small parameter and then to apply to it standard methods of asymptotic analysis. The possibility of such an approach to the problems of calculating the motion of a satellite was indicated in [1]. In [2], [3] by means of the theory of adiabatic invariants, the influence of aerodynamic moments on the relative motion of a satellite was considered, but such an approach made it possible to consider the phenomenon only to a first approximation. The method of averaging used in [1] (see [4], [5]) enables us to construct higher approximations. The present paper is a further development of [1]; here the following two problems of the motion of a satellite are solved to a second approximation: the influence of a rapid rotation of the satellite during its latest revolutions round the earth on the motion of its centre of mass, and the problem of the relative motion of a guided satellite in dense atmospheric layers. In both problems it is assumed that the relative motion takes place in the plane of motion of the centre of mass and that the aerodynamic forces cause no lateral forces and moments.