<p><span class="math-container">$$ Q = \frac{\Omega c_s}{\pi G \Sigma} $$</span>
<span class="math-container">$$ \Omega = 2\pi/P$$</span>
because <span class="math-container">$\Omega$</span>, the angular velocity/frequency in radians per unit time, is <span class="math-container">$2\pi$</span> radians divided by the time it takes to travel <span class="math-container">$2\pi$</span> radians, which is the orbital period <span class="math-container">$P$</span>.
<span class="math-container">$$ c_s = \left(\frac{kT}{\mu}\right)^{1/2}$$</span>
Leads to
<span class="math-container">$$ Q = \frac{2\pi (kT)^{1/2}}{\pi P \mu^{1/2} G \Sigma} = \frac{2\sqrt{kT/\mu}}{P G\Sigma}$$</span></p>

CC BY-SA 4.0
2022-03-17T21:58:10.500Z
Oh its angular velocity not frequency. That explains my confusion! Thanks ! :)
WDUK
CC BY-SA 4.0
2022-03-17T22:40:00.643Z
@WDUK  Angular velocity is angular frequency. Both are measured in radians per second.
ProfRob
CC BY-SA 4.0
2022-03-17T18:53:39.257Z
2022-03-17T22:38:45.380Z
2022-03-17T22:38:45.380Z
ProfRob
<p>I saw this equation in a literature review recently talking about the Toomre criterion for gravitational instability:</p>
<p><a href="https://i.sstatic.net/umN00.png" rel="nofollow noreferrer"><img src="https://i.sstatic.net/umN00.png" alt="enter image description here" /></a></p>
<p>Given here in section 2.1.1: <a href="https://arxiv.org/pdf/1801.06117.pdf" rel="nofollow noreferrer">https://arxiv.org/pdf/1801.06117.pdf</a>, viz.</p>
<p>But I am not seeing how they got the period to be in the denominator from my own workings i get it in the numerator:</p>
<p><a href="https://i.sstatic.net/EmVM5.png" rel="nofollow noreferrer"><img src="https://i.sstatic.net/EmVM5.png" alt="enter image description here" /></a></p>
<p>What am I misunderstanding here? How did they get the orbital period in the denominator ?</p>

CC BY-SA 4.0
2022-03-16T09:11:03.717Z
That $T$ is a temperature and that $\Omega = 2\pi/P$?
ProfRob
CC BY-SA 4.0
2022-03-16T22:17:33.000Z
Oh true i should've used lowercase ${t}$ for time to avoid confusion, but i am not seeing how you got P in the denominator there.. Where did you get angular frequency to be 2pi over the period ? I have always known it as ${2\pi/t}$ @ProfRob
WDUK
<gravity><mathematics><gas-giants>
How does this Toomre GI criteria have the period in the denominator?
WDUK
# How does this Toomre GI criteria have the period in the denominator?

(Asked by: username_0 on 2022-03-16)

I saw this equation in a literature review recently talking about the Toomre criterion for gravitational instability:

[![enter image description here](https://i.sstatic.net/umN00.png)](https://i.sstatic.net/umN00.png)

Given here in section 2.1.1: <https://arxiv.org/pdf/1801.06117.pdf>, viz.

But I am not seeing how they got the period to be in the denominator from my own workings i get it in the numerator:

[![enter image description here](https://i.sstatic.net/EmVM5.png)](https://i.sstatic.net/EmVM5.png)

What am I misunderstanding here? How did they get the orbital period in the denominator ?

**Tags:** gravity, mathematics, gas-giants
> **Comments:**
> **username_1**: That $T$ is a temperature and that $\Omega = 2\pi/P$?  
> **username_0**: Oh true i should've used lowercase ${t}$ for time to avoid confusion, but i am not seeing how you got P in the denominator there.. Where did you get angular frequency to be 2pi over the period ? I have always known it as ${2\pi/t}$ @ProfRob  

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## 1 answers

### Answer by username_1 on 2022-03-17. Score: 1 | Accepted answer

$$ Q = \frac{\Omega c\_s}{\pi G \Sigma} $$
$$ \Omega = 2\pi/P$$
because $\Omega$, the angular velocity/frequency in radians per unit time, is $2\pi$ radians divided by the time it takes to travel $2\pi$ radians, which is the orbital period $P$.
$$ c\_s = \left(\frac{kT}{\mu}\right)^{1/2}$$
Leads to
$$ Q = \frac{2\pi (kT)^{1/2}}{\pi P \mu^{1/2} G \Sigma} = \frac{2\sqrt{kT/\mu}}{P G\Sigma}$$

> **Comments:**
> **username_0**: Oh its angular velocity not frequency. That explains my confusion! Thanks ! :)  
> **username_1**: @WDUK  Angular velocity is angular frequency. Both are measured in radians per second.  

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