Argument of the Perihelion Projection

[Philosophaie]

In Celestial Mechanics the equation:

LP = w + N (Longitude of the Perihelion = Argument of the Perihelion + Longitude of the Ascending Node)

is confusing.
Both "LP" and "N" are on the Ecliptic Plane but "w" is not.
"w" is on the Elliptic Plane with a tilt of "i" Inclination from the Ecliptic Plane at "N".
Even though "i" is small then projection of "w" onto the x-y axis of the Ecliptic Plane is different than just plain "w".
How do you explain for this discrepancy?

 

[Simon Bridge]

It's not a discrepancy - that is how the LP is defined.
You are not the ony one to find the definition a bit odd, see also:
http://physics.stackexchange.com/questions/46649/the-validity-of-the-longitude-of-perihelion

 

[BobG]

In Celestial Mechanics the equation:

LP = w + N (Longitude of the Perihelion = Argument of the Perihelion + Longitude of the Ascending Node)

is confusing.
Both "LP" and "N" are on the Ecliptic Plane but "w" is not.
"w" is on the Elliptic Plane with a tilt of "i" Inclination from the Ecliptic Plane at "N".
Even though "i" is small then projection of "w" onto the x-y axis of the Ecliptic Plane is different than just plain "w".
How do you explain for this discrepancy?

If you rotate your orbit plane by the inclination prior to adding the Argument of the Perihelion + Longitude of the Ascending Node together, then they will be in the same plane. In other words, you're inventing a new coordinate system whose fundamental plane matches the orbit plane, and the angle between the fundamental plane of your new coordinate system is separated from the ecliptic plane by an angle equal to your inclination.

The new coordinate system avoids singularities that occur when the inclination is 0 and/or the eccentricity is 0. And what you're showing is roughly how to convert the elements of your new coordinate system into something more familiar.

There will be some quirks in the conversion, but the converted elements will still point to the location of the object you're interested in. For example, if you're talking about circular satellite orbits around the Earth, the conversion process from Equinoctial elements (with no singularities) to Keplerian elements (that can't be created when eccentricity and/or incination is 0) will probably yield incorrect values for the argument of perigee and the true anomaly. However, the sum of the two (the argument of latitude) will always be correct.