Kepler equation solution without transcendental functions or lookup tables

Abstract

This paper presents a new approach to approximate the solution of Kepler’s equation. 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 query lookup tables is eliminated. The final procedure builds upon Mikkola’s approach. Initially, a fifteenth-order polynomial is derived through a series approximation of Kepler’s equation. Sturm’s theorem is used to prove that only one real root exists for this polynomial for the given range of mean anomaly and eccentricity. An initial approximation for this root is found using a third-order polynomial. Then, a single generalized Newton–Raphson correction is applied to obtain fourteenth-place accuracies in the elliptical case, which is near machine precision. This paper will focus on demonstrating the procedure for the elliptical case, though an application to hyperbolic orbits through a similar methodology may be similarly developed.