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

enter image description here

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

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

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  • $\begingroup$ That $T$ is a temperature and that $\Omega = 2\pi/P$? $\endgroup$
    – ProfRob
    Commented Mar 16, 2022 at 9:11
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    $\begingroup$ 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 $\endgroup$
    – WDUK
    Commented Mar 16, 2022 at 22:17

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$$ Q = \frac{\Omega c_s}{\pi G \Sigma} $$ $$ \Omega = 2\pi/P$$ 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 $$ Q = \frac{2\pi (kT)^{1/2}}{\pi P \mu^{1/2} G \Sigma} = \frac{2\sqrt{kT/\mu}}{P G\Sigma}$$

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  • $\begingroup$ Oh its angular velocity not frequency. That explains my confusion! Thanks ! :) $\endgroup$
    – WDUK
    Commented Mar 17, 2022 at 21:58
  • $\begingroup$ @WDUK Angular velocity is angular frequency. Both are measured in radians per second. $\endgroup$
    – ProfRob
    Commented Mar 17, 2022 at 22:40

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