Semianalytic Satellite Theory Using the Method of Multiple Scales

The method of multiple scales is used to obtain a semianalytic solution to the equations of motion for earth orbiting satellites. First-order solutions are obtained for the centralbody gravitational potential, third-body gravitational potential, solar radiation pressure, and atmospheric drag. Second-order solutions are obtained for the second harmonic of the central-body gravitational potential, atmospheric drag, and the cross coupling between the second harmonic of the central-body gravitational potential and atmospheric drag. The second-order terms are accurate to any desired order in the eccentricity and use analytic expressions for the partial derivatives of the atmospheric density. Over a one day integration time span, for low earth orbit satellites, it is shown that the semianalytic solution is accurate to within 3km, and the orbit altitude, eccentricity, and inclination have a significant eect on the accuracy of the atmospheric drag calculations.