Kepler's Goat Herd: An Exact Solution to Kepler's Equation for Elliptical Orbits
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 rootfinding and series approaches by a factor greater than two.
 Publication:

Monthly Notices of the Royal Astronomical Society
 Pub Date:
 May 2021
 DOI:
 10.1093/mnras/stab1296
 arXiv:
 arXiv:2103.15829
 Bibcode:
 2021MNRAS.tmp.1324P
 Keywords:

 celestial mechanics;
 methods: analytical;
 numerical;
 Physics  Computational Physics;
 Astrophysics  Earth and Planetary Astrophysics;
 Astrophysics  Astrophysics of Galaxies;
 Astrophysics  Instrumentation and Methods for Astrophysics;
 Astrophysics  Solar and Stellar Astrophysics
 EPrint:
 7 pages, 4 figures, accepted by MNRAS. Code available at https://github.com/oliverphilcox/KeplersGoatHerd