L, L-1

 Variable Symbol Value R.A. 1 a1 232.70667 Dec. 1 d1 +49.998889 R.A. 2 a2 7.9400000 Dec. 2 d2 +44.335556 R.A. 3 a3 39.669583 Dec. 3 d3 -32.199444

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:

L1i = cosaicosdi
L2i = sin
aicosdi
L3i = sin
di

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) = sin
a1cosd1 = -0.5113776
L31 (row 3, column 1) = sin
d1 = 0.7660319

L12 = cosa2cosd2 = 0.7084021
L22 = sin
a2cosd2 = 0.0988030
L32 = sin
d2 = 0.6988592

L13 = cosa3cosd3 = 0.6513514
L23 = sin
a3cosd3 = 0.5401785
L33 = sin
d3 = -0.5328680

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

 -0.389471 0.708402 0.651351 L = -0.511378 0.098803 0.540179 0.766032 0.698859 -0.532868

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|

 29.7698 -57.627 -22.0289 L-1 = -9.77862 20.168 8.49176 29.9709 -56.3922 -22.4075

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

 SITE MAP Step 3: Unit Vectors Was Last Modified On September 23, 2013