# A cubic approximation for Kepler's equation

@article{Mikkola1987ACA, title={A cubic approximation for Kepler's equation}, author={Seppo Mikkola}, journal={Celestial mechanics}, year={1987}, volume={40}, pages={329-334} }

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 high order iteration formula then corrects the solution to high precision at once. The method can be used for all orbit types, including hyperbolic. To obtain this solution the trigonometric or hyperbolic functions must be evaluated only once.

## 38 Citations

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…

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- 1991

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…

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- 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…

Robust resolution of Kepler’s equation in all eccentricity regimes

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- 2013

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.…

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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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- 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…

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.

Improved First Estimates to the Solution of Kepler's Equation

- Physics
- 2017

The manuscripts provides a novel starting guess for the solution of Kepler's equation for unknown eccentric anomaly E given the eccentricity e and the mean anomaly M of an elliptical orbit.

Dynamic discretization method for solving Kepler’s equation

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- 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…

Optimal starters for solving the elliptic Kepler’s equation

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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…

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