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

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

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

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.