Abstract
In this paper, new set of universal functions, Y (n) (chi;alpha); based on Goodyear's time transformation formula will be developed analytically and computationally for the two body-initial value problem, where n is non negative integer, chi, new independent variable, a kind of generalized anomaly, and alpha is the reciprocal of the orbit's semi-major axis.
For the analytical developments, the proofs of the linear independence of the Y's functions, differential and recurrence formulae satisfied by them and their relations with the elementary function are given. Full, set of the identities for the Y's functions are listed to serve as a ready reference when need. Exact analytical expressions of the general universal Kepler's equation for the time ta[t (a"") ,t (s) ], together with some orbital parameters are developed as power series of chi; also some recurrence formulae are given to facilitate their computations. Finally, symbolical series solution of the general Kepler's equation is established, and the literal analytical expressions of the coefficients of the series are listed in Horner form for efficient and stable evaluation.
For the computational developments, algorithm for the evaluation of the Y's functions is constructed. Computational initial value problem in terms of the Y's functions is developed for which an efficient iterative method of arbitrary positive integer order of convergence a parts per thousand yen2 is established for the universal Kepler's equation. The method is of dynamic natural in the sense that, on going from one iterative scheme to the subsequent one, only additional instruction is needed. The applications of the method are illustrated by numerical examples of some test orbits of different eccentricities. The numerical results are accurate and efficient.