Solving Kepler’s equation using implicit functions
- 392 Downloads
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.
KeywordsKepler 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.
- Bézier, P.E.: Procédé de Définition Numérique des Courbes et Surfâces non Mathématiques. Automatisme XIII(5), 189–196 (1968)Google Scholar
- 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
- Feinstein, S.A., McLaughlin, C.A.: Dynamic discretization method for solving Kepler’s equation. Celest. Mech. Dyn. Astron. 96, 49–62 (2006)Google Scholar
- Fukushima, T.: A method solving Kepler’s equation without transcendental function evaluations. Celest. Mech. Dyn. Astron. 66, 309–319 (1996)Google Scholar
- Markley, F.L.: Kepler equation solver. Celest. Mech. Dyn. Astron. 63, 101–111 (1996)Google Scholar
- Meeus, J.: Astronomical Algorithms, 2nd edn. Willmann-Bell Inc., Richmond (1999)Google Scholar
- Mortari, D.: Root Finder using Implicit Functions (in progress) (2013)Google Scholar
- Mortari, D., Clochiatti, A.: Solving Kepler’s equation using Bézier curves. Celest. Mech. Dyn. Astron. 99, 45–57 (2007)Google Scholar
- Nijenhuis, A.: Solving Kepler’s equation with high efficiency and accuracy. Celest. Mech. Dyn. Astron. 51, 319–330 (1991)Google Scholar
- Serafin, R.A.: Bounds on the solution to Kepler’s equation. Celest. Mech. 38, 111–121 (1986)Google Scholar
- Vallado, D.A.: Fundamentals of Astrodynamics and Applications, Vol. 2. McGraw-Hill, New York (2001)Google Scholar