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
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
They agree if the orbit is a circle.
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
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.