Analytical formulations to the method of variation of parameters in terms of universal Y's functions

Abstract

The method of variation of parameters still has a great interest and wide applications in mathematics, physics and astrodynamics. In this paper, universal functions (the Y's functions) based on Goodyear's time transformation formula were used to establish a variation of parameters method which is useful in slightly perturbed two-body initial value problem. Moreover due to its universality, the method avoids the switching among different conic orbits which are commonly occurring in space missions. The position and velocity vectors are written in terms of f and g series. The method is developed analytically and computationally. For the analytical developments, exact literal formulations for the differential system of variation of the epoch state vector are established. Symbolical series solution of the universal Kepler's equation was also established, and the literal analytical expressions of the coefficients of the series are listed in Horner form for efficient and stable evaluation. For computational developments of the method, an efficient algorithm was given using continued fraction theory. Finally, a short note on the method of solution was given just for the reader guidance.