Kepler's Goat Herd: An Exact Solution to Kepler's Equation for Elliptical Orbits

TL;DR

This paper presents an explicit integral solution to Kepler's equation for elliptical orbits, enabling efficient and accurate computation of eccentric anomalies using numerical integration, surpassing traditional methods in speed.

Contribution

The authors introduce a novel integral-based method for solving Kepler's equation exactly, improving computational efficiency and accuracy over existing approximate techniques.

Findings

The integral solution is highly accurate across eccentricities.

The method outperforms conventional root-finding algorithms.

Implementation in C++ is more than twice as fast as traditional methods.

Abstract

A fundamental relation in celestial mechanics is Kepler's equation, linking an orbit's mean anomaly to its eccentric anomaly and eccentricity. Being transcendental, the equation cannot be directly solved for eccentric anomaly by conventional treatments; much work has been devoted to approximate methods. Here, we give an explicit integral solution, utilizing methods recently applied to the "geometric goat problem" and to the dynamics of spherical collapse. The solution is given as a ratio of contour integrals; these can be efficiently computed via numerical integration for arbitrary eccentricities. The method is found to be highly accurate in practice, with our C++ implementation outperforming conventional root-finding and series approaches by a factor greater than two.