A second order solution of the main problem of artificial satellites using multiple scales

A complete second order solution of the main problem in the theory of artificial satellites has been developed using the method of multiple scales. The basic equations of motion are expressed in terms of eight nonsingular noncanonical elements with two simple redundancies and an inertial angle as the independent variable. The obtained solution is expressed in the form of Fourier series whose trigonometric arguments are a linear combination of the unperturbed true anomaly and argument of perigee. The amplitudes of the Fourier series were found to be a nonlinear combination of the unperturbed inclination and eccentricity. The highest frequency of oscillation is found to occur in the eccentricity vector components along the radial and the transverse directions of the osculating orbit and it is produced by the seventh harmonic in the unperturbed true anomaly.