Analytic Solution of Satellite Orbit Disturbed by Atmospheric Drag

The analytic solution of a satellite orbit disturbed by the atmospheric drag is derived in this paper. The disturbing force vector is first transformed and rotated to the orbital frame so that it can be used in the simplified Gaussian equations of satellite motion. Then the force vector is expanded to triangle functions of the Keplerian angular elements and the disturbances are separated into three parts: short periodic terms with triangle functions of M, long periodic terms with triangle functions of (ω, i), and secular terms (non-periodic functions of (a, e)) with a program using mathematic symbolic operation software. The integrations are then carried out with respect to M, (ω, i), and t, respectively, to obtain the analytic solutions of satellite orbits disturbed by the atmospheric drag. Some interesting conclusions are obtained theoretically. The atmospheric disturbing force is not a function of Ω. The semi-major axis a of the orbital ellipse will be reduced in a constant and strong manner by the air disturbance; the shape of the ellipse (eccentricity e) will change towards a more circular orbit in a linear and weak manner. The right ascension of ascending node Ω and the mean anomaly M are influenced by the disturbance only short-periodically.