# Solving Kepler's equation by fixed point iteration

The position of a planet in its elliptical orbit as a function of time can be found through solving Kepler's equation. Let \$t@ be time elapsed since perihelion (closest approach to the Sun), and \$T@ the period of the orbit - the amount of time it takes the planet to complete one revolution. Let

\$M = 2 pi (t/T)@

so that \$M@ varies from \$0@ to \$2 pi@ as an orbit is traversed. Then

\$M = E - e sin(E)@

where \$E@ is the angle of a point in theKepler circle constructed on the orbit. This is called Kepler's equation, and is derived from Kepler's Second Law, which states that the radial area swept out by a planet is proportional to time. The quantity \$M@ is called the mean anomaly in the literature since it measures the position of a fictitious planet moving uniformly with respect to time, and \$E@ is called the eccentric anomaly. They agree if the orbit is a circle. The position is given in terms of \$E@ as \$(a cos(E), b sin(E))@ where \$a@ and \$b@ are the semi-major and semi-minor axes of the orbit. The quantity \$b@ is related to \$a@ and \$e@ by the formula

\$b = a sqrt(1 - e^2) . @

Of course, Kepler's equation will tell us easily what \$t@ is if we are given \$E@, but going in the opposite direction involves solving a transcendental equation for \$E.@ We can rewrite it as

\$E = M + e sin(E)@

which means that \$E@ is a fixed point of the function \$E -> M + e sin(E).@ To find the fixed point of a function \$f(x)@ is simple if the root \$X@ we are looking for is stable - that is to say if \$|f'(X)|@ is less than \$1.@ This is always true for Kepler's equation if the condition \$e < 1@ is valid, which in fact always holds for elliptical orbit. The convergence rate will decresae, however, as \$e@ approaches \$1.@ The following very simple applet solves Kepler's equation for ellipses by fixed point iteration. Set \$e@ and \$M@ (press `carriage return' in a window to enter the data), and \$E@ will be set to \$M.@ `Iterate' changes \$E@ to \$M + e sin(E).@ Enough iterations will converge sooner or later.