q, g

At this stage, it will be very beneficial to determine if the orbit that is slowly materializing is indeed an orbit. In other words, the three determined positions for the satellite have to form an orbit plane. The angles between the first and second and the second and third observations need to be separated by a suitable angle so that Gauss' Method will work properly.

The three observations form an orbit plane if the following is true:

q = sin^-1 { (Z₁₂r₁₁+Z₂₂r₂₁+Z₃₂r₃₁) / [ r₁ [(Z₁₂)²+(Z₂₂)²+(Z₃₂)²]^1/2] } ~ 0

Since no measurements are perfect, this angle is seldom 0, however it is acceptable if this angle is under 5 degrees. If this angle is above 5 degrees, you might want to check your calculations and/or your tracking data.

q = 3^°.2008249

The angular separation between the first and second observation is acceptable when the following is true:

g₁₂ = cos^-1 [ (r₁₁r₁₂+r₂₁r₂₂+r₃₁r₃₂) / r₁r₂ ] > 5°

The angular separation between the second and third observation is acceptable when the following is true:

g₃₂ = cos^-1 [ (r₁₂r₁₃+r₂₂r₂₃+r₃₂r₃₃) / r₂r₃ ] > 5°

g₁₂ = 19°.121367
g₃₂ = 19°.931927

You might have expected that these angles would be closer to 80 degrees, since the first and third observations were taken when the satellite was observed at 10 degrees above the local horizon and the second observation was taken when the satellite was observed near the local zenith. What must be remembered here is that you are comparing the angles that are measured from the center of the Earth (geocentric) rather than the topocentric angles.

if you are very confident that your tracking data and calculations are correct, you can skip this step altogether; however it is strongly recommended that your method initially use this plane and angle check in order to confirm its reliability.

Since this example has passed the plane and angle tests, we can confidently move on to the next step.

BACK TO STEP #10

PROCEED TO STEP #12