Propagation of Space Objects

Abstract

A ctual, important problems of space dynamics are related to the protection of the Earth and its close space environment. This includes the study of disposal strategies for artificial satellites, the real-time propagation of space catalogues of thousands of objects, the fast prediction of orbit evolution of fragmentation models, among others. Accurate predictions of orbit dynamics require the integration of complete force models, including both gravitational and non-gravitational effects. The most accurate integration is expected from numerical methods. However, in many cases, it is important to have available tools for the instant propagation of the orbit evolution, as opposite to precise ephemerides computation. These tools can be implemented with the help of modern perturbation methods, which, in addition to the mean element propagation, allow for the recovery of short-period effects to provide osculating elements.Within a perturbative approach, the highest frequencies of the motion, which normally have small amplitudes, are analytically filtered via averaging procedures. From the mathematical point of view, the averaging is the result of a transformation from old to new variables, which, in reference to perturbed Keplerian motion, are called osculating and mean elements, respectively. This transformation is rarely attained in closed form, and one feels satisfied with computing the first terms of the Taylor series expansion of the transformation. The averaged equations depend only on long period angles and, therefore, are integrated numerically with very long step sizes. The short-period effects, if required, can be recovered at any step of the integration by the simple evaluation of the analytical expressions of the averaging transformation.The averaging transformation can be performed either directly over the variation of parameters equations, using the generalized method of averaging (Bogoliuvov and Mitropolski, 1961), or in the Hamiltonian function, using canonical perturbation theory. For the latter, the Lie transforms method (Hori 1966, Deprit 1969) has the advantage -over Poincaré's nouvelle method (Poincaré, 1893) or its von Zeipel-Brouwer variant (Ferraz-Mello, 2007) -of providing the canonical transformation of the averaging in explicit form, which makes the computation of higher orders in the perturbation approach simpler.Analytical and semi-analytical theories based on averaging have been used in the propagation of catalogs of space objects since the beginning of the space era. Different algorithms based on Brouwer's theory (Brouwer, 1959) still serve for this task, and the generation of the so-called two-line elements is based on them (Hoot and Roehrich 1980;Vallado et al., 2006). More sophisticated theories were later implemented, as Draper semi-analytic satellite theory (DSST; McClain, 1977) or analytic orbit propagation program (AOPP; Coffey et al., 1995) in the United States, the numeric-analytic theory THEONA, developed at the Keldysh Institute of Applied Mathematics (Golikov, 1990), or the h...