STEP 1: ORBIT ELEMENTS

r

 
Variable Symbol Value
Topocentric Range 1 r1 2,373.1528 km
Topocentric Range 2 r2 827.68474 km
Topocentric Range 3 r3 2,419.6918 km

 
-0.3894713 0.7084021 0.6513514
L = -0.5113776 0.0988030 0.5401785
0.7660319 0.6988592 -0.5328680

 

4488.1674 km 4466.1251 km 4440.4533 km
RECI = 867.33155 km 974.49353 km 1085.4885 km
4433.2015 km 4433.2015 km 4433.2015 km

Although we have found the possible topocentric ranges of the satellite for all three observation times, we still have not found the Keplerian orbit elements. This will be done by first determining the "state vector", which comprises of the satellite's geocentric distance matrix (which will be determined in this step) and the velocity vector at the second observation time (which will be determined in the following steps).

The geocentric matrix is determined by using the following equations:

r11 = r1L11 + R11
r12 = r2L12 + R12
r13 = r3L13 + R13
r21 = r1L21 + R21
r22 = r2L22 + R22
r23 = r3L23 + R23
r31 = r1L31 + R31
r32 = r2L32 + R32
r33 = r3L33 + R33
 

3563.8925 km 5052.4587 km 6016.5229 km
r = -346.24563 km 1056.2713 km 2392.5540 km
6251.1122 km 5011.6366 km 3143.8252 km

If you determine the sum of the squares of each column, you will find the estimated geocentric distances of the satellite at each observation time:

r1 = [ (r11)2 + (r21)2 + (r31)2 ] 1/2
r2 = [ (r12)2 + (r22)2 + (r32)2 ] 1/2
r3 = [ (r13)2 + (r23)2 + (r33)2 ] 1/2

r1 = 7,204.0002 km
r2 = 7,194.4110 km
r3 = 7,197.6732 km

Note that the second of these ranges (r2) is the same as that determined from the 8th order equation in Step 7. This is correct. In fact, in order to check your algebra, the r2 value here should be very similar to that value determined in Step 7.

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Step 9: Geocentric Matrix Was Last Modified On September 23, 2013