# Laplace Method for Orbit Estimation

I have a question about Laplace angle-only method for orbit determination where the line of sight vectors are being interpolated. I read somewhere that the method fails (due to a matrix being singular) when the observation site is in the motion plane of the observed body, but failed to see proof for that. Could anyone explain to me why is that?

The Laplace method could be found here. In pages 8-12 it explains the method, and in page 12 it references to Escobal's book that should explain that, however I do not have access to that book.

edit: Schaeperkoetter, Andrew Vernon (2011) Masters Thesis A Comprehensive Comparison Between Angles-Only Initial Orbit Determination Techniques

From page 12:

Escobal (1) determines that this will only occur when the observation site lies on the great circle of the satellite’s orbit.

(1) ESCOBAL, P. R., Methods of Orbit Determination, John Wiley and Sons, Inc., New York, 1965.

The determinants in Vallado's notation are equal to vector triple products ($$(A \times B) \cdot C$$) in Battin's notation, which are zero when all three vectors are coplanar. The three vectors in this case are the middle observation, and its first and second derivatives. Because those derivatives are estimated from the other two observations, if all three observation directions are coplanar, then the velocity and acceleration measured from the first to the last also lie in that plane, so the triple product is zero. In any other case, at least one of those three is not in the same plane, so the determinant (the volume of the parallelepiped with those vectors as edges) is nonzero.