While querying the Minor Planet Center orbital data for asteroids, among the returned data, I noticed 6 variables corresponding to x,y and z components of two 3-D vectors namely P and Q. The Explanation Of Orbital Elements page includes the following:

The vectors P and Q are an alternate form of representing the angular elements Peri., Node and Incl. For an explanation of how to convert between the two sets of quantities you are referred to standard celestial mechanics textbooks.

However, I'm not sure what these two vectors are. Quickly examining the components of P and Q, I realized that they are unit vectors. Could someone please give me a clue to what they represent ?

Eventually, I'm interested in computing the eccentricity vector ($\vec{e}$) of asteroid orbits. So I'm also keen on knowing whether these P and Q vectors are of any use for that purpose. Or would I be better off computing $\vec{e}$ using $\vec{r}$ and $\vec{v}$ following $\vec e = \frac{\vec v \times \vec h}{\mu} - \frac{\vec r}{r}$, where $\vec h$ is the specific angular momentum $\vec r \times \vec v$ , using JPL Horizons for example?

Thank you in advance !


In the Perifocal Coordinate system, the unit vectors $\hat{p}$ and $\hat{q}$ define the plane of the orbit.

  • $\hat{p}$ indicates the direction from the orbital focus to the periapsis. It is in the same direction as the eccentricity vector $\vec{e}$
  • $\hat{q}$ points at where the object would be at True Anomaly of 90°.
  • $\hat{w} = \hat{p} \times \hat{q}$ is a unit vector that's perpendicular to the orbit, in the same direction as the specifc angular momentum vector.
  • $\begingroup$ Many thanks for your reply. So if I understand correctly I just need to scale the unit vector 𝑝̂ by the magnitude of the eccentricity to find the components of 𝑒⃗, don't I ? $\endgroup$
    – mysterium
    Jul 9 '21 at 14:29
  • 2
    $\begingroup$ @mysterium Yes, that should work. $\endgroup$
    – notovny
    Jul 9 '21 at 15:30
  • $\begingroup$ Thanks @notovny ! $\endgroup$
    – mysterium
    Jul 9 '21 at 22:03

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