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