The orbital dynamics of an artificial satellite in the Earth's atmosphere is considered. An analytic first-order atmospheric drag theory is developed using Lagrange's planetary equations. The short periodic perturbations due to the geopotential of all orbital elements are evaluated. And to construct a second-order analytical theory, the equations of motion become very complicated to be integrated analytically; thus we are forced to integrate them numerically using the method of Runge-Kutta of fourth order. The validity of the theory is checked on the already decayed Indian satellite ROHINI where its data are available.
S. E. Abd El-Bar. F. A. Abd El-Salam. "Analytical and Semi-Analytical Treatment of the Satellite Motion in a Resisting Medium." J. Appl. Math. 2012 1 - 18, 2012. https://doi.org/10.1155/2012/489849