A hybrid iterative–series method for solving kepler’s equation with enhanced convergence

Abstract

Kepler’s equation, a fundamental relation in celestial mechanics, poses significant numerical challenges due to its transcendental nature and the nonlinearity introduced by eccentricity. Existing iterative and series-based methods, while efficient in specific domains, often fail to deliver uniformly fast and stable convergence across the full eccentricity range. This paper introduces a hybrid solution method that dynamically combines a truncated Fourier–Bessel series for low-eccentricity scenarios with an optimized Newton–Halley iterative scheme for moderate-to-high eccentricities. A switching method in line with Laplace’s limit is used for robust convergence around the threshold eccentricity. The numerical experiments demonstrate that the method proposed, in comparison to classical methods, achieves better results for convergence speed, numerical stability, and better results for accuracy, particularly in the vicinity of critical points where classical methods demonstrate divergence. The method offers a new tool for orbital prediction and mission planning in astrodynamics and planetary science, enabling high-precision solutions with reduced computational cost.