Kepler's Goat Herd: An exact solution to Kepler's equation for elliptical orbits

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.

Automated summary

Kepler's equation is a fundamental relation in celestial mechanics that links mean anomaly, eccentric anomaly, and eccentricity. Conventional methods cannot directly solve for eccentric anomaly, so approximate methods have been developed. In this study, an explicit integral solution is presented, which has been found to be highly accurate and efficient in numerical computations for arbitrary eccentricities. The C++ implementation of this method outperforms traditional root-finding and series approaches by more than two times.