Comment on "Long-Term Evolution of Near-Geostationary Orbits"

Abstract

R 1 describes the development of a singleaveraging theory and its application to the prediction of a near-synchronous orbit. Single averaging usually denotes a formulation in which the high-frequency perturbations associated with the satellite period and the Earth's rotational period are removed from the equations of motion. Thus, step sizes on the order of one day are allowed in the integration of the averaged equations of motion. The purpose of this Comment is to connect the results in Ref. 1 with results that have already appeared in the literature. In many cases, the author of Ref. 1 gives a limited result when a more general result is available; these instances are noted. We also discuss the construction of initial conditions for a single-averaged orbit prediction. Finally, some aspects of the very long-term motion of desynchronized orbits (such as GEOS-2) are discussed. Fundamental to the development of this single-averaged theory are the differential equations for the motion of the equinoctial elements due to a disturbing potential. These are Eqs. (3) in Ref. 1. These same equations were given much earlier in Refs. 2 and 3. The disturbing potential due to third bodies (moon and sun) is developed in Ref. 1 with the aid of a Poisson series symbolic algebra program. The potentials include up to second order eccentricity terms. The author of Ref. 1 notes that lengthy computer-generated secular terms can be rearranged in extremely compact forms by introducing the C and S auxiliary parameters [Eq. (17) in Ref. 1 ]. The averaged potentials are obtained under the assumption that the third body positions are held constant during the averaging operation. Similar potentials were given much earlier in Refs. 3 and 5. In particular, Ref. 5 gave the following general form for the third body potential in terms of equinoctial elements: