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_{i}cosd_{i }L_{2i} = sina_{i}cosd_{i }L_{3i} = sind_{i} where i = the observation number (1, 2, and 3) AKA the column number of the Unit Vector matrix. L_{11}
(row 1, column 1) =
cosa_{1}cosd_{1}
= -0.3894713 L_{12} =
cosa_{2}cosd_{2}
= 0.7084021 L_{13} =
cosa_{3}cosd_{3}
= 0.6513514 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| = L_{11 }(L_{22}L_{33 }- L_{23}L_{32}) + L_{12 }(L_{23}L_{31 }- L_{21}L_{33}) + L_{13 }(L_{21}L_{32 }- L_{22}L_{31}) |L| = -0.0144496 The inverse of this matrix can now be calculated using the following formulae: [L^{-1}]_{11}
= [L_{22}L_{33} - L_{23}L_{32}] / |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} = I_{3} where
I_{3}
is a 3x3 Identity Matrix. |
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Step 3: Unit Vectors Was Last Modified On September 23, 2013 |