Applications of the two-variable expansion procedure to problems in celestial mechanics

Abstract

This work illustrates the application of the two-variable expansion procedure of reference 1 to the solution of two representative problems in celestial mechanics. The expansion procedure is applied first to the problem of aerodynamic perturbations of a satellite orbit. The case of planar motion is considered with both lift and drag perturbations acting on the satellite. A simplified model of the earth is used, but the motion is expected to exhibit a similar qualitative behavior in the more general case. It is found that the effect of drag causes the satellite to spiral toward the center of attraction while the orbit is tending to become circular. The effect of lift, to the order computed, is felt only by a slow advance of the apse. The second application of the expansion procedure is to the problem of third-body perturbations of a satellite orbit. A special case of the restricted three-body problem is used in which the plane of the satellite's orbit is coincident with the orbital plane of the two larger bodies. The two-variable expansion is applied to approximate equations which are valid for satellite orbits close to the smaller of the two large bodies. The results are in exact agreement with those of reference 2 and DePontecoulant's lunar theory. The solution of this problem is to serve as a preliminary step in establishing the choice of variables for the more general case in which the satellite's orbit has a high inclination to the orbital plane of the two larger bodies.