A second- and higher-order perturbation analysis of two-body trajectories

Abstract

An approach to the development of a second- and higher-order perturbation theory for two- body trajectories is presented. It is shown that the higher order analysis can be developed in a systematic manner in terms of series solutions to the two-body problem as functions of the time variable. Simple algebraic recursive formulas for the determination of the series coeffi- cients are derived and the radii of convergence for these series solutions are determined. Their accuracy and the rate of convergence are investigated in a number of numerical cases. ingly more precise mission objectives and with long flight times. It is the purpose of this paper to present an approach to the higher order perturbation analysis of two-body trajec- tories. A second-order theory, involving second partial derivatives of the six-dimensional state vector of a spacecraft at a given time with respect to the state vector at an initial epoch, is con- sidered in detail. It is shown that a second as well as higher order theory can be developed in a simple and systematic manner using power series solutions of the two-body problem as functions of the time variable, a technique suggested by the theory of Lie series.1 In this paper, the series representa- tions associated with the second-order theory are developed; they are uniformly valid for all two-body conies. It is shown that the coefficients of these series expansions can be generated by simple algebraic recursive formulas, and their radii of con- vergence can be determined analytically. The rate of convergence and truncation error associated with these series representations are investigated in a number of numerical cases. Results indicate that the convergence characteristics and accuracies can in general be estimated in terms of the eccentricity and the position vector at the initial epoch. Generally speaking, the results exhibit good con- vergence characteristics for elliptical orbits covering less than one orbital revolution. Results for hyperbolic orbits are less favorable.