r_i

In order to find the Keplerian orbit elements, we must first know the satellite's position and velocity. In order to determine those, we will first need to determine the satellite's topocentric ranges (distance from the observer to the satellite) for all three observations. This can be done by using the following equations:

c₁ = t₁ + mt₂ / (r₂)³
c₂ = -1
c₃ = t₃ + mt₄ / (r₂)³

c₁ = 0.5403414
c₂ = -1
c₃ = 0.5196976

Now we can determine the actual topocentric ranges of the satellite at all three observation times:

r₁ = (M₁₂ - M₁₁c₁ - M₁₃c₃) / c₁
r₂ = (M₂₂ - M₂₁c₁ - M₂₃c₃) / c₂
r₃ = (M₃₂ - M₃₁c₁ - M₃₃c₃) / c₃

r₁ = 2,373.1528 km
r₂ = 827.68474 km
r₃ = 2,419.6918 km

Note that the second topocentric range (828 km) is very near the estimated altitude of the satellite (816 km) determined in Step 7. This is encouraging, as the second observation of the satellite occurred when the satellite was at the location's zenith and therefore its topocentric range at that point would nearly be the orbit altitude of the satellite at that specific time. The other two topocentric ranges (r₁ and r₂) also look encouraging since the other two observations were made when the satellite was only 10 degrees above the local horizon.

BACK TO STEP #7

PROCEED TO STEP #9