# A method solving kepler's equation without transcendental function evaluations

@article{Fukushima1996AMS, title={A method solving kepler's equation without transcendental function evaluations}, author={Toshio Fukushima}, journal={Celestial Mechanics and Dynamical Astronomy}, year={1996}, volume={66}, pages={309-319} }

We developed two approximations of the Newton-Raphson method. The one is a sort of discretization, namely to search an approximate solution on pre-specified grid points. The other is a Taylor series expansion. A combination of these was applied to solving Kepler's equation for the elliptic case. The resulting method requires no evaluation of transcendental functions. Numerical measurements showed that, in the case of Intel Pentium processor, the new method is three times as fast as the original…

## 36 Citations

A Method Solving Kepler’s Equation for Hyperbolic Case

- Physics
- 1997

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…

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The goal for the solution of Kepler’s equation is to determine the eccentric anomaly accurately, given the mean anomaly and eccentricity. This paper presents a new approach to solve this very well…

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This work presents an algorithm to compute the eccentric anomaly and even its cosine and sine terms without usage of other transcendental functions at run-time with slight modifications for the hyperbolic case.

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- Computer Science, PhysicsAstronomy & Astrophysics
- 2018

An algorithm to compute the eccentric anomaly and even its cosine and sine terms without usage of other transcendental functions at run-time without using trigonometric or root functions is presented.

Dynamic discretization method for solving Kepler’s equation

- Computer Science
- 2006

This paper defines Kepler’s equation for the elliptical case and describes existing solution methods, and presents the dynamic discretization method and shows the results of a comparative analysis, demonstrating that, for the conditions of the tests, dynamicDiscretization performs the best.

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- Computer Science
- 2014

New upper and lower bounds are derived for two ranges of mean anomaly that have been compared and proven more accurate than Serafin’s bounds and are particularly suitable for space-based applications with limited computational capability.

An efficient code to solve the Kepler equation. Elliptic case

- Physics
- 2017

A new approach for solving Kepler equation for elliptical orbits is developed in this paper. This new approach takes advantage of the very good behaviour of the modified Newton?Raphson method when…

Solving Kepler’s Equation using Bézier curves

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- 2007

This paper presents a non-iterative approach to solve Kepler’s Equation, M = E − e sin E, based on non-rational cubic and rational quadratic Bézier curves. Optimal control point coordinates are first…

Fast Procedure Solving Universal Kepler's Equation

- Physics
- 1999

We developed a procedure to solve a modification of the standard form of the universal Kepler’s equation, which is expressed as a nondimensional equation with respect to a nondimensional variable.…

A Fast Procedure Solving Gauss' Form of Kepler's Equation

- Physics
- 1998

We developed a procedure solving Gauss' form of Kepler's equation, which is suitable for determining position in the nearly parabolic orbits. The procedure is based on the combination of asymptotic…

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