On the averaging principle for one-frequency systems. An application to satellite motions

This paper is related to our previous works (Morosi and Pizzocchero in J. Phys. A, Math. Gen. 39:3673–3702, 2006; Nonlinear Dyn., 2008, DOI 10.1007/s11071-008-9444-1), on the error estimate of the averaging technique for systems with one fast angular variable. In the cited references, a general method (of mixed analytical and numerical type) has been introduced to obtain precise, fully quantitative estimates on the averaging error. Here, this procedure is applied to the motion of a satellite in a polar orbit around an oblate planet, retaining only the J2 term in the multipole expansion of the gravitational potential. To exemplify the method, the averaging errors are estimated for the data corresponding to two Earth satellites; for a very large number of orbits, computation of our estimators is much less expensive than the direct numerical solution of the equations of motion.