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 the**Kepler
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.