Satellite Motion about an Oblate Earth

A general theory of the method of averaging is used to study the effect of the Earth's oblateness on the motion of an artificial satellite. The first-order harmonic J2 and the second-order harmonics J3 and J4 are included in the analysis. This paper illustrates a thorough and straightforward application of a second-order method of averaging to the orbital motion of a satellite about an oblate Earth with the well-known conventional orbital elements as the time-dependent variables. The first-order periodic variations together with the second-order averaged and secular variations are derived for all the six orbital elements. Essential transformations and initialization procedures are also outlined. T is well known that for a close-Earth satellite, the inclusion of the perturbations due to the Earth's oblateness and the atmospheric drag is of vital importance to predict the lifetime and the trajectory of the satellite accurately. Except for the pioneer work of Brouwer,1 Lane,2 and Cranford,3 relatively little analytical work has been done in studying the joint effects of the Earth's oblateness and atmospheric drag on an orbiting satellite. Although this study will not develop a joint treatment, it does present a second-order theory of an artificial satellite about an oblate Earth. This theory establishes a foundation for the joint treatment of the Earth's oblateness, atmospheric drag, and other perturbations (e.g., extra-terrestrial gravitation) through application of the general theory of the method of averaging. This approach is feasible not only because the transformations involved need not be canonical, but also because it provides a rigorous, systematic and straightforward procedure for studying a dynamical system perturbed by conservative and/or nonconservative forces. Further, this approach is also general in the sense that both the Von Zeipel method and the two-variable asymptotic expansion procedures are particular cases of the method of averaging.4'5 Specifically, the present task is to develop a second-order theory which includes the effects contributed by the second, third, and fourth harmonics of the Earth's gravitational potential by applying the general theory of the method of averaging with the conventional orbital elements as its variables. General perturbation theories applied to a close-Earth satellite under the influence of the oblateness of the Earth have been published by a number of authors to predict satellite motions. To name a few, second-order treatments have been given by Kozai,6 Brouwer,7 Blitzer,8 Kyner,9 Lorell et al.,10 and Arsenault et al. 11 These and a large number of other theories and their results have been compiled and compared in Ref. 11. Kozai6 used an averaging technique to study the second-order solutions by introducing his mean elements. Brouwer7 used the Von Zeipel method to study the same problem. Blitzer8 introduced dimensionless variables and adopted the method of Bogoliuboff and Mitropolsky12 to obtain a second-order solution. Kyner9 used the method of averaging to study the problem with the use of spherical coordinates and angular momentum. In Ref. 10, the authors also applied the method of averaging to obtain the averaged variational equations to the second order. In their analysis, however, the second harmonic of the Earth's gravitational potential was the only perturbation under consideration and the mean anomaly, rather than time t, was treated as the