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According to the Keplerian system of orbital elements, a reference direction is given; the longitude of the ascending node is an angle with respect to that direction, and the argument of periapsis, specifying the orientation of the axes of the ellipse, is given relative to the LAN.

For a low-inclination, high-eccentricity orbit, the longitude of ascending node may be very uncertain (becoming undefined in the purely theoretical zero-inclination case) even though the orientation of the ellipse as projected to the reference plane is clear.

Why is the AOP not taken as the angle made with the reference direction when the periapsis is projected to the reference plane? This would make its measurement independent of the LAN, and the angle would be uncertain only in a very small region above the poles -- a much rarer situation than a near-zero-inclination orbit.

I see there's an element called longitude of periapsis (ϖ), which if I understand correctly is almost (but not quite) the same thing as what I'm proposing, that's occasionally used for orbital elements of the planets.

Why is AOP defined this way?

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The longitude of the ascending node $\Omega$, the inclination $i$, and the argument of periapsis $\omega$ are simply the classic Euler angles $\phi$, $\theta$, and $\psi$, in that order. Then the classic Euler transform applied to an orbit in the X-Y plane with the periapsis on the +X-axis gives you the actual orbit in X-Y-Z.

There is probably no reason other than the use of the conventional transform to specify the rotation of a rigid 3-D object.

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