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A Method Solving Kepler’s Equation for Hyperbolic Case

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Abstract

We developed a method to solve Kepler’s equation for the hyperboliccase. The solution interval is separated into three regions; F ≪ 1, F≈ 1, F ≫ 1. For the region F is large, we transformed the variablefrom F to exp F and applied the Newton method to the transformed equation.For the region F is small, we expanded the equation with respect to F andapplied the Newton method to the approximated equations. For the middleregion, we adopted a discretization of the Newton method and used the Taylorseries expansion for the evaluation of transcendental functions (Fukushima,1997). Numerical measurements showed that, in the case of Intel Pentiumprocessor, the new method is more than 3.7 times as fast as the combinationof a fourth order correction formula (Fukushima, 1997) and Roy’sstarting procedure (Roy, 1988) and others.

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References

  • Burkardt T.M. and Danby J.M.A.: 1983, 'The Solution of Kepler's Equation, II',Celest. Mech. 31, 317–

    Google Scholar 

  • Colwell P.: 1993, Solving Kepler's Equation Over Three Centuries, Willmann-Bell, Richmond, VA.

    Google Scholar 

  • Danby J.M.A.: 1988, Fundamentals of Celestial Mechanics, 2nd Ed. MacMillan, New York.

    Google Scholar 

  • Fukushima T.: 1982, 'Stable Starters for Newton Iteration and its Application to Kepler's Equation' in G. Hori and M. Yuasa (eds), Proc. of 15th Symp. on Celest. Mech., held in Tokyo, Feb. 22-23, 1982, 33–36.

  • Fukushima T.: 1997, 'A Method Solving Kepler's Equation without Transcendental Function Evaluations', Celest. Mech. Dynam. Astron. 66, 309–319.

    Google Scholar 

  • Gooding R.H. and Odell A.W.: 1988, 'The Hyperbolic Kepler's Equation (and the Elliptic Equation Revisited)', Celest. Mech. 44, 267–282.

    Google Scholar 

  • Prussing: 1977, J. Astronaut. Sci., 25, 123.

    Google Scholar 

  • Roy A.E.: 1988, Orbital Motion,3rd Ed., Adam Hilger, Bristol, U.K.

    Google Scholar 

  • Serafin R.A.: 1986, 'Bounds on the Solution to Kepler's Equation', Celest. Mech. 38, 111–121.

    Google Scholar 

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Authors and Affiliations

  1. National Astronomical Observatory, 2-21-1, Ohsawa, Mitaka, Tokyo, 181, Japan

    Toshio Fukushima

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  1. Toshio Fukushima
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Fukushima, T. A Method Solving Kepler’s Equation for Hyperbolic Case. Celestial Mechanics and Dynamical Astronomy 68, 121–137 (1997). https://doi.org/10.1023/A:1008254717126

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  • DOI: https://doi.org/10.1023/A:1008254717126

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