h

When observed at zenith, the satellite range from the observer is approximately the same as the satellite altitude above the Earth's surface.

The orbit range (AKA height or altitude at zenith) is found by using the equation below:

h = (r_E / 3) { 2cos [ cos^-1 ( 5381105.9 / r³_E w²_app  - 1 ) / 3 ] - 1 }

This equation can only be used if the following condition is true:

w_app > [ 2690553 / r_E³ ]^1/2

For the observer's latitude, [ 2690553 / r_E³ ]^1/2 = 0.003228 rad/s.

Since w_app for this example is indeed greater than 0.003228 rad/s, it is alright to proceed with the calculation.

h = 614.1 km
 

r_orbit ~ a = h + r_E

a ~ 6981.9 km
 

The satellite's orbit period can therefore be found from the determined semi-major axis by using Kepler's Third Law.

T = 2p [ a³ / 398600.44 ]^1/2

T = 5806 s = 96.8 m = 1.613 h

From the orbit period, the mean motion (n) of the satellite's orbit can be found:

n = 1 / T

n = 14.881157 orbits/day

These results look adequate for a typical LEO satellite.

We have determined the first of the six Keplerian orbit elements (semi-major axis) for this satellite using a single image! The satellite streak endpoint coordinates will be used again to determine the satellite's inclination, right ascension of ascending node (RAAN) and the mean anomaly.

PROCEED TO STEP #5