Celestial Mechanics and Dynamical Astronomy

, Volume 118, Issue 1, pp 1–11 | Cite as

Solving Kepler’s equation using implicit functions

Original Article


A new approach to solve Kepler’s equation based on the use of implicit functions is proposed here. First, new upper and lower bounds are derived for two ranges of mean anomaly. These upper and lower bounds initialize a two-step procedure involving the solution of two implicit functions. These two implicit functions, which are non-rational (polynomial) Bézier functions, can be linear or quadratic, depending on the derivatives of the initial bound values. These are new initial bounds that have been compared and proven more accurate than Serafin’s bounds. The procedure reaches machine error accuracy with no more that one quadratic and one linear iterations, experienced in the “tough range”, where the eccentricity is close to one and the mean anomaly to zero. The proposed method is particularly suitable for space-based applications with limited computational capability.


Kepler equation Optimal starter Bézier functions Root finder 



The authors would like to thank Rose Sauser for checking and improving the use of English in the manuscript.


  1. Battin, R.H.: An Introduction to the Mathematics and Methods of Astrodynamics, Revised Edition. AIAA Education Series, Reston (1999)CrossRefMATHGoogle Scholar
  2. Bézier, P.E.: Procédé de Définition Numérique des Courbes et Surfâces non Mathématiques. Automatisme XIII(5), 189–196 (1968)Google Scholar
  3. Calvo, M., Elipe, A., Montijano, J.I., Rández, L.: Optimal starters for solving the elliptical Kepler’s equation. Celest. Mech. Dyn. Astron. 115, 143–160 (2013)Google Scholar
  4. Colwell, P.: Solving Kepler’s Equation Over Three Centuries. Willmann-Bell, Richmond (1993)MATHGoogle Scholar
  5. Feinstein, S.A., McLaughlin, C.A.: Dynamic discretization method for solving Kepler’s equation. Celest. Mech. Dyn. Astron. 96, 49–62 (2006)Google Scholar
  6. Fukushima, T.: A method solving Kepler’s equation without transcendental function evaluations. Celest. Mech. Dyn. Astron. 66, 309–319 (1996)Google Scholar
  7. Markley, F.L.: Kepler equation solver. Celest. Mech. Dyn. Astron. 63, 101–111 (1996)Google Scholar
  8. Meeus, J.: Astronomical Algorithms, 2nd edn. Willmann-Bell Inc., Richmond (1999)Google Scholar
  9. Mortari, D.: Root Finder using Implicit Functions (in progress) (2013)Google Scholar
  10. Mortari, D., Clochiatti, A.: Solving Kepler’s equation using Bézier curves. Celest. Mech. Dyn. Astron. 99, 45–57 (2007)Google Scholar
  11. Nijenhuis, A.: Solving Kepler’s equation with high efficiency and accuracy. Celest. Mech. Dyn. Astron. 51, 319–330 (1991)Google Scholar
  12. Serafin, R.A.: Bounds on the solution to Kepler’s equation. Celest. Mech. 38, 111–121 (1986)Google Scholar
  13. Vallado, D.A.: Fundamentals of Astrodynamics and Applications, Vol. 2. McGraw-Hill, New York (2001)Google Scholar

Copyright information

© Springer Science+Business Media Dordrecht 2013

Authors and Affiliations

  1. 1.Department of Aerospace EngineeringTexas A&M UniversityCollege StationUSA
  2. 2.Centro Universitario de la DefensaZaragozaSpain
  3. 3.Dpto. Matemática Aplicada—IUMAUniversidad de ZaragozaZaragozaSpain

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