Solving Kepler's Equation of Elliptical Motion

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Movie of elliptical motion
Plots of elliptical Kepler motion

The JavaScript form is preset to the values:

eccentricity e=0.5
mean anomaly M=27° or t/T=0.075.

The results are, as shown in the figure below:

true anomaly phi=75.84°
eccentric anomaly E=48.43°

A series expansion of the true anomaly φ can be found here or in Meeus (Astronomical Algorithms, Chapter 3):  

It is suitable only for small values of the eccentricity e.

eccentricity e

mean anomaly M (in °)

true anomaly phi (in °)
up to e3
up to e4
up to e5

true anomaly phi (in °)
8 decimal places


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(c) 2006 J. Giesen