Results of true-anomaly regularization in orbital mechanics

Presented herein are some analytical results available from regularization of the differential equations of satellite motion. True-anomaly regularization is developed as a special case of a more general Sundman-type transformation of the independent variable (time) in the equations of motion. Constants of the unperturbed motion are introduced as extra state variables, and regularization with several types of coordinates is considered. Because analytical results are sought, those regularizing transformations which produce rigorously linear governing equations are of main interest. When solutions of the linear regular equations in the true-anomaly domain are examined, it is found that the initial value and boundary value problems of unperturbed motion, typically requiring iterative solutions of the time equation, can be solved with only a single transcendental function evaluation per iteration cycle. Various means are described which can accelerate the evaluation of this function. The time equation developed in this study is a new universal relation between time of flight and true anomaly, and applies uniformly to all types of orbits, including rectilinear ones. It is a well-behaved function, the zero of which can be found reliably by Newton's method or other typical iteration methods. Once this time equation has been solved, the initial and final state vector on the transfer arc can be related to each other by rational algebraic formulae; no other transcendental function is needed. When the two problems are generalized by variation of parameters to the case of oblate-gravity perturbed motion, it is found that, to first order, the corrections of the unperturbed solution can be obtained by direct, noniterative formulae valid for all types of orbits. Moreover, it is possible to compute these corrections with only a single extra evaluation of the same transcendental function used in the unperturbed problem. Additional results are also presented, including exact solutions of the first-order averaged differential equations governing secular variations of the regular orbital elements in the true-anomaly domain. Complete universal expressions are given for the Keplerian state transition matrix in terms of the orbital transfer angle, and a simple midcourse guidance scheme is rederived in terms of universal variables valid for all non-rectilinear transfer orbits.