# Solving Kepler's equation with high efficiency and accuracy

@article{Nijenhuis1991SolvingKE, title={Solving Kepler's equation with high efficiency and accuracy}, author={Albert Nijenhuis}, journal={Celestial Mechanics and Dynamical Astronomy}, year={1991}, volume={51}, pages={319-330} }

We present a method for solving Kepler's equation for elliptical orbits that represents a gain in efficiency and accuracy compared with those currently in use. The gain is obtained through a starter algorithm which uses Mikkola's ideas in a critical range, and less costly methods elsewhere. A higher-order Newton method is used thereafter. Our method requires two trigonometric evaluations.

## 24 Citations

On solving Kepler's equation for nearly parabolic orbits

- Physics
- 1996

We deal here with the efficient starting points for Kepler's equation in the special case of nearly parabolic orbits. Our approach provides with very simple formulas that allow calculating these…

An Improved Algorithm For The Solution of Kepler 's Equation For An Elliptical Orbit

- Physics
- 2010

In this paper, a root finding method due to iterative method is used first to the solution of Kepler's equation for an elliptical orbit. Then the extrapolation technique in the form of Aitken Δ 2…

Appropriate Starter for Solving the Kepler's Equation

- Physics
- 2014

This article focuses on the methods that have been used for solving the Kepler’s equation for thirty years, then Kepler's equation will be solved by Newton-Raphson's method, and one appropriate choice first guesses that increase the isotropy and decrease the time of Implementation of solving is introduced.

Numerical solution of the two-body problem for orbital motion is heavily dependent on efficient solution of Kepler's Equation

- Mathematics
- 1995

Kepler's Equation is solved over the entire range of elliptic motion by a fifth-order refinement of the solution of a cubic equation. This method is not iterative, and requires only four…

Kepler Equation solver

- Computer Science
- 1995

Kepler's Equation is solved over the entire range of elliptic motion by a fifth-order refinement of the solution of a cubic equation, and requires only four transcendental function evaluations.

Optimal starters for solving the elliptic Kepler’s equation

- Computer Science, Physics
- 2013

The highly accurate starter given by Markley is considered in proposing an improvement of it for low to medium eccentricities and new optimal starters with respect to these measures are derived.

QUASI CONSTANT-TIME ORBIT PROPAGATION WITHOUT SOLVING KEPLER'S EQUATION 1

- Physics
- 2008

Efficient methods for solving Kepler’s equation, a transcendental equation relating orbital position as a function of time, have been studied for centuries and generated a vast literature. This paper…

Sequential solution to Kepler’s equation

- Physics, Computer Science
- 2010

Seven sequential starter values for solving Kepler’s equation are proposed for fast orbit propagation and obtain improved accuracy at lower computational cost as compared to the best existing methods.

A method solving kepler's equation without transcendental function evaluations

- Physics
- 1996

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…

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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