Conceptual Question in Gibbs Orbit Determination

[enigma]

Hi all,

hopefully someone can help me out here:

I'm looking at the Gibbs' method for orbit determination, and I'm losing the conceptual picture in the vector calculations.

I have no problems seeing how D and N point in the W direction (PQW coordinate system), but I'll be damned if I can get a grip outside of simply swimming through the math why S: [(r2-r3)r1 + (r3-r1)r2 + (r1-r2)r3] points in the Q direction.

Anyone have any insights?

I'm reading Vallado Fundamentals of Astrodynamics and Applications section 7.5.1 if it helps.

 

[selfAdjoint]

Just a question. Are PQR supposed to be orthogonal? I ask because in celestial mechanics one often works with skew coordinate systems.

If they are orthogonal then could you show the inner product of the third one with each of the other two is zero? That would demonstrate it was in the right direction and working through the algebra might build some insight in you as to how that happens.

 

[enigma]

PQW are orthogonal and tied to the orbital plane

P points toward periapsis, W points in the angular momentum direction, and Q points toward the semilatus rectum (or semiparameter depending on the book)

 

[selfAdjoint]

Oh well sure then. P is aligned parallel to the major axis of the ellipse, Q is perpendicular to the plane of the ellipse (by the usual vector analysis representation of angular momentum) and R is parallel to the latus rectum which by the geometrical properties of the ellipse is perpendicular to the major axis. So there you are, orthogonal!

 

[enigma]

Sorry. I misread your first post.

the three r's are three measurements of position on the orbital ellipse at three different points in time. They are not necessarily perpendicular to any of the axes except for the W axis. The Q unit vector is perpendicular to both P and W. This is not a trivial cross-product direction problem.