Efficient Solutions of Kepler’s Equation via Hybrid and Digital Approaches

Abstract

The solution of Kepler’s equation is accomplished via families of hybrid and table lookup techniques. A new arithmetic operation timing approach with general application to scientific programming is used to accurately estimate performance of each Kepler’s equation solution technique. The hybrid approaches couple a power series expansion starting approximation with nine different types of higher-order corrective step methods. The resulting computationally efficient non-iterative methods avoid “if” statements and directly yield in-plane Euler rotation angles necessary to map orbit elements to orbital position. The best-performing of the nine hybrid methods are up to two times faster than the original efficient Laguerre iterative method and achieve worst-case resultant true anomaly accuracies down to machine precision at 3 × 10−11° for eccentricities up to 0.999999. This matches or exceeds the performance of iterative methods and translates to less than three micro-meters at GEO altitude. Meanwhile, six digital approaches were explored, with the best table lookup approach boasting a ten-fold speed increase with a worst-case accuracy of 1 × 10−6° (154 mm) for an eccentricity of 0.999999. These combinations of accuracy and speed performance make both the hybrid and table lookup approaches well-suited for in-line incorporation into a wide range of low- to high-fidelity semi-analytic orbit propagators and multi-threaded or vector programming languages and computing hardware (GPUs, etc.). And finally, a new operator-based computational timing technique is designed and employed to estimate the performance of math-laden computer programs, with sample application to all of the aforementioned Kepler’s equation solution techniques.