Accurate Kepler equation solver without transcendental function evaluations
Pimienta-Penalver, Adonis Reinier
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The goal for the solution of Kepler's Equation is to determine the eccentric anomaly accurately, given the mean anomaly and eccentricity. This study presents a new approach to solve this very well documented problem. It is found that by means of a series approximation, an angle identity, the application of Sturm's theorem and an iterative correction method, the need to evaluate transcendental functions or store tables is eliminated. A 15 th -order polynomial is developed through a series approximation of Kepler's Equation. Sturm's theorem is used to prove that only one real roots exists for this polynomial for the given range of mean anomalies and eccentricities. An initial approximation for this root is found using a 3 rd -order polynomial. Then a one-time generalized Newton-Raphson correction is applied to obtain accuracies to the level of around 10 -15 rad for the elliptical case and 10 -14 rad for the hyperbolic case, which is near machine precision. The procedure is demonstrated for both the elliptical and hyperbolic cases.