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.

本文提出了一种近似求解开普勒方程的新方法。研究发现，通过级数逼近、角度恒等式、Sturm 定理的应用以及迭代修正方法，可以消除计算超越函数或查询查找表的需要。最终程序以 Mikkola 的方法为基础。最初，通过开普勒方程的级数近似导出十五阶多项式。 Sturm 定理用于证明对于给定的平均异常和偏心率范围，该多项式仅存在一个实根。使用三阶多项式找到该根的初始近似值。然后，应用单个广义牛顿-拉夫森校正以获得椭圆情况下的第十四位精度，该精度接近机器精度。本文将重点介绍椭圆情况的程序，尽管可以类似地开发通过类似方法应用于双曲轨道的程序。