Satgen 629   Stuck in a Loop by GM4IHJ  (BID SGEN629) 2001-04-14 

The computation of a satellite orbit is a high speed, simple affair ,
in this age of microcomputers, provided all things human and nonhuman
behave. But we should not forget that in the 370 or so years since
Kepler produced the essential equation  E = M + e*sin(E)  , that
mathematicians too numerous to mention , have strived to find new
ways of solving it. Noting that simple arithmetic does not work.

You have to use trial and error . You know M the mean anomaly of the
orbit and you know e the orbital eccentricity. But you need to use
ITERATION to solve for E. You start by guessing E and see if the two
sides of the equation match. If not you alter your guess slightly ,
say increasing it. If that gets closer to balancing the equation you
proceed to alter E a little more , till step by step it balances. If
however the steps take you further from a balanced solution , you
change round and start to decrease the value of E step by step till
it balances. Doing this by computer you must use careful step sizes
and set limits, so the process cannot overshoot. 

Normally this works . The equation runs into a balanced solution .
But lets say the orbit the satellite adopts is not quite the one you
intended. Your carefully selected sizes and limits , and , or your
original guess for E at the start of the iterative process will fail
to produce a smooth run into a solution and your software may crash
or simply oscillate between unsatisfactory solutions, never achieving
balance.

The problem is at its worst . When the software is up in the
satellite . Suddenly the software is asked to solve Keplers equation
in order to check whether the satellites location requires a change
of state. The program cannot revert to normal and will never work
until the control steps and or limits on the calculation are changed.
So the satellite is inoperable until new software is uploaded.

All of which is quite unnecessary. The following BINARY SEARCH
procedure first published by Roger SINNOTT in 1985 outlines a
software fix that can be substituted for the iterative algoriths if
the orbit eccentricity is anywhere between 0 and 0.95 .
E0 =eccentricity : M = Mean anomaly
P1=3.14159265 : E=P1/2 : D=P1/4
For I1 = 1 to 23    ( range may be increased >23 if required )
M1=E - E0*SIN(E)
E= E + SGN(M-M1)*D : D=D/2
Next I1

This works for all eccentricities from 0 to 0.95  . But as
eccentricity approaches 1 , the transfer from mean anomaly to
eccentric anomaly and thereafter to true anomaly , becomes
increasingly inaccurate and by eccentricity =1 the answers are
meaningless. A method due to Karl Gauss is therefore preferred for
orbits with eccentricity > 0.95. Orbits with this sort of
eccentricity are of no interest to satellite communicators , so the
Gaussian method should not be needed by radio amateurs.