STEP 1: ORBIT ELEMENTS

A, B, C...



 
-14028.095 km -20859.718 km -28020.276 km
M = 11249.889 km 13626.673 km 16116.251 km
-13733.543 km -20437.266 km -27465.919 km

 

Variable Symbol Value
Time Coefficient 1 t1 0.5101194
Time Coefficient 2 t2 28233.955 s2
Time Coefficient 3 t3 0.4898806
Time Coefficient 4 t4 27855.560 s2
Earth Radius at Observer Robs 6367.8243 km
Gravitational Parameter m 398600.44 km3/s2

 

-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


Up to this point, we have been doing simple arithmetic and straightforward matrix algebra. From this point onward, things get tougher. This is because orbit determination is equally about intuition as it is about mathematical wizardry. If you are expecting the math to automatically give you the right answer, you have come to the wrong place.

This might sound counter-intuitive, but in order to perform accurate orbit determination, you must know something about the orbit beforehand so that you can "steer" the math in the right direction. This will become more apparent as we continue.

We now need to determine the coefficients of an 8th order equation that will be introduced in Step 7. The values and matrices that we have been calculating in the previous steps will now be used to determine these coefficients.

First. let us define some intermediate coefficients to make this process a little easier. These will be called d1, d2 and X:

d1 = M21t1 - M22 + M23t3
d2 = M21t2 + M23t4
X =
L12RECI12 + L22RECI22 + L32RECI32

Some of the more astute observers might have noticed that these equations only involve the second observation of the satellite and they would be right. This is because we are trying to estimate the second geocentric range using the change in angular rate between first and second and the second to third observations.

d1 = 7.1523363 km
d2 = 7.6655606x108 kms2
X = 6358.2789 km

The coefficients for the 8th order equation are A, B, C, D, E, F, G, H and K. The values for each are:

A = 1
B = 0
C = - [ d12 + 2Xd1 + R2obs ]
D = 0
E = 0
F = - 2
md2 (X + d1)
G = 0
H = 0
K = -
(md2)2

Don't forget the negative signs!

A = 1
B = 0
C = -4.0640191x107 km2
D = 0
E = 0
F = -3.8899000x1018 km5
G = 0
H = 0
K = -9.3360548x1028 km8

As can be seen, only the coefficients A, C, F and K are required. This will become clear in the next step.


BACK TO STEP #5

PROCEED TO STEP #7



CASTOR HOME

SITE MAP

GAUSS' METHOD

 

Step 6: Coefficients Was Last Modified On September 23, 2013