# 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

- Mathematics
- 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

- Computer Science
- 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

- 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…

Optimal starters for solving the elliptic Kepler’s equation

- Mathematics
- 2013

In this paper starting algorithms for the iterative solution of elliptic Kepler’s equation are considered. New global efficiency measures to compare the quality of starters are introduced and several…

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
- 2010

Seven sequential starter values for solving Kepler’s equation are proposed for fast orbit propagation. The proposed methods have constant complexity (not iterative), do not require pre-computed data,…

A method solving kepler's equation without transcendental function evaluations

- Mathematics
- 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

- Mathematics
- 2006

Kepler’s equation needs to be solved many times for a variety of problems in Celestial Mechanics. Therefore, computing the solution to Kepler’s equation in an efficient manner is of great importance…

## References

SHOWING 1-10 OF 10 REFERENCES

Procedures for solving Kepler's equation

- Mathematics
- 1986

We review starting formulae and iteration processes for the solution of Kepler's equation, and give details of two complete procedures. The first has been in use for a number of years, but the second…

The solution of Kepler's equation, I

- Mathematics
- 1983

Methods of iteration are discussed in relation to Kepler's equation, and various initial ‘guesses’ are considered, with possible strategies for choosing them. Several of these are compared; the…

A cubic approximation for Kepler's equation

- Mathematics
- 1987

We derive a new method to obtain an approximate solution for Kepler's equation. By means of an auxiliary variable it is possible to obtain a starting approximation correct to about three figures. A…

On solving Kepler's equation

- Mathematics, Physics
- 1989

Intrigued by the recent advances in research on solving Kepler's equation, we have attacked the problem too. Our contributions emphasize the unified derivation of all known bounds and several…

A general algorithm for the solution of Kepler's equation for elliptic orbits

- Mathematics
- 1979

An efficient algorithm is presented for the solution of Kepler's equationf(E)=E−M−e sinE=0, wheree is the eccentricity,M the mean anomaly andE the eccentric anomaly. This algorithm is based on simple…

An improved algorithm due to laguerre for the solution of Kepler's equation

- Physics
- 1986

A root-finding method due to Laguerre (1834–1886) is applied to the solution of the Kepler problem. The speed of convergence of this method is compared with that of Newton's method and several…

The hyperbolic Kepler equation (and the elliptic equation revisited)

- Mathematics
- 1988

A procedure is developed that, in two iterations, solves the hyperbolic Kepler's equation in a very efficient manner, and to an accuracy that proves to be always better than 10−20 (relative…

The solution of Kepler's equation, II

- Mathematics, Physics
- 1983

Starting values for the iterative solution of Kepler's equation are considered for hyperbolic orbits, and for generalized versions of the equation, including the use of universal variables.

Iterative Methods for the Solution of Equations

- Mathematics
- 1982

General Preliminaries: 1.1 Introduction 1.2 Basic concepts and notations General Theorems on Iteration Functions: 2.1 The solution of a fixed-point problem 2.2 Linear and superlinear convergence 2.3…