Celestial Mechanics and Dynamical Astronomy

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

Solving Kepler’s equation using implicit functions

Original Article

Abstract

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.

Keywords

Kepler equation Optimal starter Bézier functions Root finder 

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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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