Satgen319  In Orbit Pt3  Kepler Maths   by GM4IHJ               6 May 95

>From the Keplerian element parameters , we can read off the position of
the satellite at the Epoch time. To get its position at a latter time, we
calculate the number of Julian days which have elapsed between the Epoch
and the new date for which we want data. Then we use this calculation of
elapsed solar mean time , to calculate the Sidereal star time - because
satellites move in star time not earth time.

We can also use the elapsed time to calculate the number of orbits the
satellite has done in this time so that we can update the figure for
satellite position around its orbit = Mean anomaly.

Then with the updated mean anomally we can do the difficult bit of the
calculation , to find the Eccentric Anomaly , The arithmetic for this is
complex , More than 100 methods of working this out have been published
since Kepler first led the way. Modern computer programs tend to use
either the older Iterative method employed in the well known W3IWI
software, or, the more modern binary substitution method which has the
advantage of working over the full range of orbit eccentricity from 0 to
0.9 , ( useful when eccentricity changes as it did with Oscar 13 ). 

The next calculations solve the problem set by the non spherical nature of
the earth , and , the perturbation of the orbit right ascension. Then it
is just a case of getting the data into an X, Y, Z matrix  to define the
satellite and the observers positions . From which point satellite
Azimuth, Elevation, Slant Range, Sub Latitude and Longitude can be
calculated.

These answers must be used with caution. Several factors render the
situation much less accurate than the computer output would appear to
suggest , because :-
a. The radio signal from the satellite at the horizon is coming to the
receiver via a long path through the densest part of the atm sphere . The
signal path is bent around the horizon as it traverses this region,
usually causing the satellite to be heard earlier than expected. A
phenomenon which is best observed by watching the distortion of the Sun
when it is seen through this thick atmosphere near the horizon.
b. Equally important to any practical tracking considerations is the fact
that no land station has a clear horizon down to zero degrees elevation
,at all azimuth bearings . So in this case the actual point where the
satellite rises clear above your horizon is likely to be at some elevation
greater than zero. Thereby making signal reception later than the computer
predicted acquisition at zero elevation. Indeed it is good operating
practice to measure horizon elevation from your anhenna site , around the
full circle of azimuth bearings , so that you can anticipated changes in
acquisition times caused by none zero horizon elevation.

Please note that these factor affecting acquisition times , also affect
the time at which the satellite is lost as it descends below your horizon.