The effect of atmospheric drag on the motion of an artificial satellite in the gravitational field of an oblate earth is evaluated. A spherical exponential model atmosphere is assumed together with the velocity- square law in drag acceleration. Atmospheric rotation is neglected. The method of canonical transformation is shown to be powerful even in the problem of nonconservative systems like the drag problem. The principal effect of the coupling between the nonsphericity of the earth and air drag is shown to be mixed secular terms and long-period terms produced by short-period terms of the drag-free problem. The result is complete up to the second order in case of secular and mixed secular terms and the first order in case of periodic terms if the drag coefficient A * and the coefficient of the second harmonic of the earth potential k2 are considered small quantities of the first order. The only exception is with long-period terms in the mean anomaly, for which the result is complete to the order A*/k2. The result is complete to the fourth power of the eccentricity in all cases, and is valid for all inclination except the critical inclination case.
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Brouwer et al. (1961) studied this question.