<p>In the <a href="https://en.wikipedia.org/wiki/Perifocal_coordinate_system" rel="nofollow noreferrer">Perifocal Coordinate system</a>, the unit vectors <span class="math-container">$\hat{p}$</span> and <span class="math-container">$\hat{q}$</span> define the plane of the orbit.</p>
<ul>
<li><span class="math-container">$\hat{p}$</span> indicates the direction from the orbital focus to the periapsis. It is in the same direction as the eccentricity vector <span class="math-container">$\vec{e}$</span></li>
<li><span class="math-container">$\hat{q}$</span> points at where the object would be at True Anomaly of 90°.</li>
<li><span class="math-container">$\hat{w} = \hat{p} \times \hat{q}$</span> is a unit vector that's perpendicular to the orbit, in the same direction as the specifc angular momentum vector.</li>
</ul>

CC BY-SA 4.0
2021-07-09T14:29:47.197Z
Many thanks for your reply. So if I understand correctly I just need to scale the unit vector ̂ by the magnitude of the eccentricity to find the components of  ⃗, don't I ?
mysterium
CC BY-SA 4.0
2021-07-09T15:30:08.050Z
@mysterium Yes, that should work.
notovny
CC BY-SA 4.0
2021-07-09T13:40:35.660Z
2021-07-09T13:40:35.660Z
notovny
<p>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 <a href="https://minorplanetcenter.net//iau/info/OrbElsExplanation.html" rel="nofollow noreferrer">Explanation Of Orbital Elements</a> page includes the following:</p>
<blockquote>
<p>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.</p>
</blockquote>
<p>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. What do they represent?</p>
<p>Eventually, I'm interested in computing the eccentricity vector (<span class="math-container">$\vec{e}$</span>) 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 <span class="math-container">$\vec{e}$</span> using <span class="math-container">$\vec{r}$</span> and <span class="math-container">$\vec{v}$</span> following <span class="math-container">$\vec e = \frac{\vec v \times \vec h}{\mu} - \frac{\vec r}{r}$</span>, where <span class="math-container">$\vec h$</span> is the specific angular momentum <span class="math-container">$\vec r \times \vec v$</span> , using JPL Horizons for example?</p>

<asteroids><orbital-elements><celestial-mechanics>
Querying MPC data : P and Q vectors of any use to compute the eccentricity vector?
mysterium
# Querying MPC data : P and Q vectors of any use to compute the eccentricity vector?

(Asked by: username_0 on 2021-07-08)

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](https://minorplanetcenter.net//iau/info/OrbElsExplanation.html) 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. What do 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?

**Tags:** asteroids, orbital-elements, celestial-mechanics

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## 1 answers

### Answer by username_1 on 2021-07-09. Score: 3 | Accepted answer

In the [Perifocal Coordinate system](https://en.wikipedia.org/wiki/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.

> **Comments:**
> **username_0**: Many thanks for your reply. So if I understand correctly I just need to scale the unit vector ̂ by the magnitude of the eccentricity to find the components of  ⃗, don't I ?  
> **username_1**: @mysterium Yes, that should work.  

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