Approximate gyroscope theory and its applications to the motion of space objects

The motion of an axisymmetric rigid body with a fixed point under the action of a periodic torque is considered. Two small parameters are introduced: one characterizes the smallness of the torque amplitude, and the other characterizes the smallness of the angular momentum component perpendicular to the axis of symmetry. The smallness of the latter parameter usually underlies the approximate theory of gyroscopes. Using this approximation, one can quite simply find the precession velocity of a top under the action of a small periodic torque. It is shown that the relative accuracy of the velocity calculated in this way is nearly independent of the latter small parameter, which does not exceed a value of the order of unity. In this way, a simple formula is found for the precession of an Earth satellite under the action of the Earth’s gravity field. Additionally, a simple formula for the lunisolar precession rate of the Earth’s axis is derived, which agrees well with astronomical observations.