# How does this Toomre GI criteria have the period in the denominator?

I saw this equation in a literature review recently talking about the Toomre criterion for gravitational instability: 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: What am I misunderstanding here? How did they get the orbital period in the denominator ?

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

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