L, L^-1

The individual coordinates of the three observations can be represented as three unit vectors by using the following equations for each column. This particular step is very similar to Step 10 of the Orbit Propagation section of this website:

L_1i = cosa_icosd_i
L_2i = sina_icosd_i
L_3i = sind_i

where i = the observation number (1, 2, and 3) AKA the column number of the Unit Vector matrix.

L₁₁ (row 1, column 1) = cosa₁cosd₁ = -0.3894713
L₂₁ (row 2, column 1) = sina₁cosd₁ = -0.5113776
L₃₁ (row 3, column 1) = sind₁ = 0.7660319

L₁₂ = cosa₂cosd₂ = 0.7084021
L₂₂ = sina₂cosd₂ = 0.0988030
L₃₂ = sind₂ = 0.6988592

L₁₃ = cosa₃cosd₃ = 0.6513514
L₂₃ = sina₃cosd₃ = 0.5401785
L₃₃ = sind₃ = -0.5328680

The entire unit vector matrix should look like (or very similar to) the following:

The inverse of this unit matrix will also be required to proceed with this initial orbit determination. If done manually, this can involve a large amount of calculations. The individual steps are shown here with respect to those who had originally derived them.

First, we calculate the determinant of the unit matrix L. The full calculation is offered in the following equation. Be extremely careful if doing this manually! One wrong move will ruin the entire orbit determination (no pressure)!:

|L| = L₁₁ (L₂₂L₃₃ - L₂₃L₃₂) + L₁₂ (L₂₃L₃₁ - L₂₁L₃₃) + L₁₃ (L₂₁L₃₂ - L₂₂L₃₁)

|L| = -0.0144496

The inverse of this matrix can now be calculated using the following formulae:

[L^-1]₁₁ = [L₂₂L₃₃ - L₂₃L₃₂] / |L|
[L^-1]₁₂ = [L₁₃L₃₂ - L₁₂L₃₃] / |L|
[L^-1]₁₃ = [L₁₂L₂₃ - L₁₃L₂₂] / |L|
[L^-1]₂₁ = [L₂₃L₃₁ - L₂₁L₃₃] / |L|
[L^-1]₂₂ = [L₁₁L₃₃ - L₁₃L₃₁] / |L|
[L^-1]₂₃ = [L₁₃L₂₁ - L₁₁L₂₃] / |L|
[L^-1]₃₁ = [L₂₁L₃₂ - L₂₂L₃₁] / |L|
[L^-1]₃₂ = [L₁₂L₃₁ - L₁₁L₃₂] / |L|
[L^-1]₃₃ = [L₁₁L₂₂ - L₁₂L₂₁] / |L|

The best method to determine if the L and L^-1 are correct is to multiply them. If all of the terms are correct:

L L^-1 = I₃

where I₃ is a 3x3 Identity Matrix.

BACK TO STEP #2

PROCEED TO STEP #4