Kepler's Iterative Solution to Kepler's Equation

@article{Swerdlow2000KeplersIS,
  title={Kepler's Iterative Solution to Kepler's Equation},
  author={Noel M. Swerdlow},
  journal={Journal for the History of Astronomy},
  year={2000},
  volume={31},
  pages={339 - 341}
}
  • N. Swerdlow
  • Published 1 November 2000
  • Physics, Geology
  • Journal for the History of Astronomy
The subject of this note is quite simple. In Astronomia nova 60, Kepler set out demonstrations and equations for finding the position of a planet moving in either an ellipse or a circle such that a line from the Sun to the planet describes areas proportional to time.' One of them, written as M = E + e sin E, known as 'Kepler's Equation', has a well known difficulty, which Kepler himself was the first to recognize, in that it can easily be solved in one direction, for M, but not in the other… 

Figures from this paper

Early Numerical Analysis in Kepler's New Astronomy
Argument Johannes Kepler published his Astronomia nova in 1609, based upon a huge amount of computations. The aim of this paper is to show that Kepler's new astronomy was grounded on methods from
Padé Approximation to the Solution of the Elliptical Kepler Equation
In orbital mechanics, the elliptical Kepler equation is a basic nonlinear equation which determines the eccentric anomaly of a planet orbiting the Sun. In this paper, Kepler’s equation has been
Properties of Bessel Function Solution to Kepler’s Equation with Application to Opposition and Conjunction of Earth–Mars
Abstract In this article, a simple approach is suggested to calculate the approximate dates of opposition and conjunction of Earth and Mars since their opposition on August 28, 2003 (at perihelion of
From Keplerian Orbits to Precise Planetary Predictions : the Transits of the 1630 s
The first transits of Mercury and Venus ever observed were important for quite different reasons than were the transit of Venus observed in the eighteenth century. Good data of planetary orbits are
The Homotopy Perturbation Method for Accurate Orbits of the Planets in the Solar System: The Elliptical Kepler Equation
Abstract Accurate trajectories for the orbits of the planets in our solar system depends on obtaining an accurate solution for the elliptical Kepler equation. This equation is solved in this article
Procedural generation of multiple stable, small-scale solar systems using 3D N-Body simulation.
TLDR
A galaxy generator is developed that generates a small galaxy with a user-specified number of solar systems and internal bodies, and an approximate, specialized NBody algorithm and integration method for both CPU and GPU that is able to calculate the movement of the bodies fast by using the center of mass as an approximation.
On the precision of the successive approximations method tested on Kepler's equation from astronomy.
In this paper we point out a method to ensure the same precision for the exact solution as the precision obtained by the last two iterations in the method of successive approximations. The method of