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.
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Battin, R.H.: An Introduction to the Mathematics and Methods of Astrodynamics. American Institute of Aeronautics and Astronautics, Reston, VA (1987)
Burkardt, T.M., Danby, J.M.A.: The solutions of Kepler’s equation. II. Celest. Mech. 31, 317–328 (1983)
Calvo, M., Elipe, A., Montijano, J.I., Rández, L.: Optimal starters for solving the elliptic Kepler’s equation. Celest. Mech. Dyn. Astron. 115 (2013). doi:10.1007/s10569-012-9456-5
Colwell, P.: Solving Kepler’s Equation over Three Centuries. Willmann-Bell, Richmond (1993)
Conway, B.A.: An improved algorithm due to Laguerre for the solution of Kepler’s equation. Celest. Mech. 39, 199–211 (1986)
Danby, J.M.A.: The solution of Kepler’s equations—Part three. Celest. Mech. 40, 303–312 (1987)
Danby, J.M.A., Burkardt, T.M.: The solution of Kepler’s equation. I. Celest. Mech. 31, 95–107 (1983)
Davis, J.J., Mortari, D., Bruccoleri, C.: Sequential solution to Kepler’s equation. Celest. Mech. Dyn. Astron. 108, 59–72 (2010)
Everhart, E., Pitkin, E.T.: Universal variables in the two-body problem. Am. J. Phys. 51, 712–717 (1983)
Feinstein, S.A., McLaughlin, C.A.: Dynamic discretization method for solving Kepler’s equation. Celest. Mech. Dyn. Astron 96, 49–62 (2006)
Fukushima, T.: A fast procedure solving Kepler’s equation for elliptic case. Astron. J. 112, 2858 (1996)
Markley, F.L.: Kepler equation solver. Celest. Mech. Dyn. Astron. 63, 101–111 (1995)
Mikkola, S.: A cubic approximation for Kepler’s equation. Celest. Mech. 40, 329–334 (1987)
Milani, A., Gronchi, G.F., Farnocchia, D., Knežević, Z., Jedicke, R., Denneau, L., Pierfederici, F.: Topocentric orbit determination: algorithms for the next generation surveys. Icarus 195, 474–492 (2008)
Mortari, D., Clocchiatti, A.: Solving Kepler’s equation using Bézier curves. Celest. Mech. Dyn. Astron. 99, 45–57 (2007)
Odell, A.W., Gooding, R.H.: Procedures for solving Kepler’s equation. Celest. Mech. 38, 307–334 (1986)
Palacios, M.: Kepler equation and accelerated Newton method. J. Comput. Appl. Math. 138, 335–346 (2002)
Schaub, H., Junkins, J.L., Schetz, J.A.: Analytical Mechanics of Space Systems, 2nd edn. American Institute of Aeronautics and Astronautics, Reston, VA (2009)
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Farnocchia, D., Cioci, D.B. & Milani, A. Robust resolution of Kepler’s equation in all eccentricity regimes. Celest Mech Dyn Astr 116, 21–34 (2013). https://doi.org/10.1007/s10569-013-9476-9
- Two-body problem
- Kepler’s equation
- Newton’s method
- Starting points
- Universal variables
- Near parabolic motion