A, B, C...

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 = L₁₂R_ECI12 + L₂₂R_ECI22 + L₃₂R_ECI32

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 km·s²
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 + R²_obs ]
D = 0
E = 0
F = - 2md₂ (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