• DOI:10.1177/002182860003100404
• Corpus ID: 116599258

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, for E, which is what one always wants to do. In the Epitome of Copernican astronomy 5.2.4, he returned to the equation and set out an iterative solution, the first of many that have been discovered, and so simple and intuitive that it has doubtless been found independently many times since. There are literally hundreds of papers on the subject of Kepler's Equation, some of which must contain this solution, and I faintly remember programming on a calculator something identical or close to it many years ago. But even if it has been found before, it is Kepler's own solution and so is worth a report. Figure I is a standard diagram for the equations in Kepler's planetary model. On the apsidalline AB, in which A is aphelion and B perihelion, we describe an ellipse and its major auxiliary circle of radius R with centre M and the Sun S at a focus with eccentricity e. We let the planet be at P in the ellipse and on the same ordinate PN take the corresponding point P' in the major auxiliary circle. The mean anomaly M, proportional to time, is area ASP in the ellipse and area ASP' in the auxiliary circle, the eccentric anomaly E is the area of sector AMP', angle AMP', and arc AP' in the auxiliary circle, and the true anomaly u is angle ASP in the ellipse. There are two steps to finding the true anomaly u, the heliocentric direction of the planet. In the auxiliary circle, the mean anomaly, area ASP' = M, is the sum of the areas of sector AMP' = E, the eccentric anomaly, and of triangle SMP' = e sin E, the 'physical equation', that is, M = E + e sin E, which only applies to the auxiliary circle as the eccentric anomaly E does not appear in the ellipse. As noted earlier, M =E + e sin E is 'Kepler's Equation'. Given the eccentric anomaly E, one may easily find the mean anomaly M and the true anomaly u, thus,