Kepler Equation Newton Iteration

What you are seeing: the orbit of a planet in a Keplerian ellipse (left) and the Newton-iteration history for the Kepler equation $M = E - e \sin E$ (right). At each rAF frame the mean anomaly $M = 2\pi t / T$ ticks forward; the solver iterates on $E$ until $|E_{n+1} - E_n| \lt 10^{-12}$ and the planet is placed at $(a(\cos E - e), b \sin E)$ on the ellipse.

The right panel shows successive $|E_n - E_\infty|$ on a log axis. Newton's method has quadratic local convergence: each iteration roughly squares the remaining error. For moderate eccentricity ( $e \lesssim 0.9$ ) the residual drops to machine precision in 4-6 iterations; near $e = 0.99$ it takes 10-15.

Figure 1. Kepler-equation Newton iteration on an a-e ellipse. Method: closed-form residual + Newton step; quadratic convergence.

e (eccentricity)0.500

speed1.00

WHAT TO TRY

Raise the eccentricity toward 1: the planet races through perihelion and crawls through aphelion, and the transcendental Kepler equation M = E - e sin E grows harder to invert.
Watch the Newton iteration on the right: from a guess it converges on the eccentric anomaly in a few steps, doubling the correct digits each time.
At high eccentricity a naive starting guess can stall: this little root-find, solved billions of times a day, is the workhorse behind every ephemeris and spacecraft trajectory.