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

Oliver H.E. Philcox, Jeremy Goodman, Zachary Slepian

Research output: Contribution to journalArticlepeer-review

11 Scopus citations


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.

Original languageEnglish (US)
Pages (from-to)6111-6116
Number of pages6
JournalMonthly Notices of the Royal Astronomical Society
Issue number4
StatePublished - Oct 1 2021

All Science Journal Classification (ASJC) codes

  • Astronomy and Astrophysics
  • Space and Planetary Science


  • Celestial mechanics
  • Methods: Analytical
  • Methods: numerical


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