Robust resolution of Kepler’s equation in all eccentricity regimes

  • Davide FarnocchiaEmail author
  • Davide Bracali Cioci
  • Andrea Milani
Original Article


In this paper we discuss the resolution of Kepler’s equation in all eccentricity regimes. To avoid rounding off problems we find a suitable starting point for Newton’s method in the hyperbolic case. Then, we analytically prove that Kepler’s equation undergoes a smooth transition around parabolic orbits. This regularity allows us to fix known numerical issues in the near parabolic region and results in a non-singular iterative technique to solve Kepler’s equation for any kind of orbit. We measure the performance and the robustness of this technique by comprehensive numerical tests.


Two-body problem Kepler’s equation Newton’s method Starting points Universal variables Near parabolic motion 


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Copyright information

© Springer Science+Business Media Dordrecht 2013

Authors and Affiliations

  • Davide Farnocchia
    • 1
    Email author
  • Davide Bracali Cioci
    • 1
  • Andrea Milani
    • 2
  1. 1.SpaceDyS s.r.l.CascinaItaly
  2. 2.Department of MathematicsUniversity of PisaPisaItaly

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