Calculation of Precession of Space Objects Using an Approximate Theory of a Gyroscope with Error Estimation

A space object is considered as a dynamically symmetric rigid body with a fixed point at the center of mass under the action of a periodic moment of force. Two small parameters are introduced: the first characterizes the smallness of the amplitude of the moment of force, and the second characterizes the smallness of the component of the kinetic moment perpendicular to the axis of symmetry. The smallness of the second parameter is usually the basis for using the approximate theory of the gyroscope. Using this approximation, one can quite easily find the speed of precession of the top under the action of a small periodic torque. It is shown that the relative error of the precession period calculated in this way is very small: it is proportional to the product of two small parameters. In this way, a simple formula is found for the precession of the Earth’s satellite under the influence of the Earth’s gravitational field. The resulting formula for the speed of the lunar–solar precession of the Earth agrees well with astronomical observations.