# Ascending node with zero inclination

Long time ago, I was wondering which orbital element should I use to differentiate between the blue and red orientation of an otherwise same elliptical orbit on this image (all the three orbits are in an identical ecliptic plane; Sun in the center is, of course, at the foci.): After some studies I realized Euler angles are used for these purposes in the form of inclination and longitude of ascending node. However, longitude of a. n. is useful only when inclination is non-zero so that the nodes exist. For theoretical purposes this does not pose a problem since we are free to choose any other plane so the parameters can have definite values.

In practice there could be a problem for the used methods of simulation and orbit-element communication when inclination is near- or true-zero and we are bound to use the same plane of reference. As it is stated in this question:

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.

My questions are:

• Is there a recommended method for how to avoid non-ambiguous zero-inclination cases in computer simulation? If we solve the motion equations numerically we might like to know the present orbital parameters of an object without any possibility of errors. (You can answer this also with your own experience.)
• How is the longitude of a. n. communicated in practice through the two-line element sets in zero-inclination cases?
• The distinction between the red and blue orbits is the argument of periapsis, not the longitude of the ascending node. There's no requirement that the direction of the eccentricity be correlated with the LAN in an inclined eccentric orbit. en.wikipedia.org/wiki/Orbital_elements Sep 12 '17 at 16:23
• That is true. My question is rather technical. For example - we might want to measure the orbital parameters for every piece of space debris around Earth. We might have a fixed table for orbital parameters to be filled with numbers in the reference frame of Earth, Vernal point and the Ecliptic plane. But if some object has an inclination exactly 0 we cannot specify nor the AP nor the LAN. Sep 12 '17 at 16:37

In planar orbital mechanics, the argument of periapsis is measured directly from the reference direction used. To avoid confusion with the 3D usage it is sometimes called the orbital argument.

1. This is not actually an issue when implementing keplerian elements, as it should be very numerically robust against floating point artefacts. A relatively large change in longitude of ascending node means close to no change in the actual orbit when the inclination is close to zero. The opposite can be a problem though, like working with eccentricity values close to 1.

One issue you can encounter is that numerical comparison of orbits "are these orbits approximately the same?" is very touchy to corner cases with angular elements. A naive implementation here (+-margins or multiplicative error margins) will perform poorly.

You may consider using an internal representation using state vectors, and just use the keplerian elements as a front-end.

2. You can't make any assumptions about how an implementation of two-line elements handles this particular corner case. It is in principle allowed to give you any number between 0 and 360 depending on how it internally handles the numbers. You should however always expect the orbital argument to be accurate, given as a sum of line 2 field 4 and 6. (for cases with a noticeable inclination, you would have to project the argument of periapsis down on the reference plane first. This is however not useful for anything in practice as it only communicates the relationship with the reference plane, not their mutual angular separation. A single angular parameter is never sufficient.)

• You second point seems correct. I accept your answer. :) I was also wondering about the sufficiency of TLE precision in reality. I found the tables of modern ephemerides (link) and, if I am correct, these data are used instead of TLEs when higher accuracy is demanded. Sep 12 '17 at 17:41
• @Degauss I've been reading this to get the math, but your link is shorter also very helpful. Any idea what thing it is Chapter 8 of? Also, it's a good question how TLE propagators handle this, and you might consider asking it as a separate question. The details of standard SGP propagators are well documented and several people are familliar with them, so if you asked it as a short, concise question, you might get a good answer.
– uhoh
Sep 13 '17 at 14:01
• @Degauss Or look at pages 86-88 here for example: digitalcommons.calpoly.edu/cgi/…
– uhoh
Sep 13 '17 at 14:09