Rotating Disk

In the case of a rotating disk with an angular rotation speed $\Omega$, basic equations are

$$\frac{\partial \delta \sigma}{\partial t}+\sigma_0\frac{\partial \delta \mbox{\boldmath${u}$}}{\partial x}=0,$$ (2.66)

$$\sigma_0\frac{\partial \delta \mbox{\boldmath${u}$}}{\partia... ...2\mbox{\boldmath${\Omega}$}\times\delta \mbox{\boldmath${u}$},$$ (2.67)

$$\frac{\partial^2 \delta \phi}{\partial z^2}-k^2 \delta \phi =0.$$ (2.68)

Equation (2.68) gives a solution

$$\delta \phi=-\frac{2\pi G \delta \sigma}{k} \exp\left[i(\omega t-kx)-k\vert z\vert\right].$$ (2.69)

Choosing a direction in which the wavenumber vector can be expressed as $\mbox{\boldmath${k}$}=(k,0,0)$, we can reduce equations (2.66) and (2.67) to

$$i\omega \delta \sigma -ik \delta u_x =0,$$ (2.70)

$$i\omega \delta u_x = -c_{is}^2 (-ik) \delta \sigma -ik\frac{2\pi G \delta \sigma}{\vert k\vert}+2\Omega \delta u_y,$$ (2.71)

$$i\omega \delta u_y = -2 \Omega \delta u_x.$$ (2.72)

These three equations together with equation (2.69) bring us a dispersion relation as

$$\omega^2=c_{is}^2k^2-2\pi G \sigma_0 \vert k\vert +4\Omega^2.$$ (2.73)

Comparing with equation(2.65), this indicates rotation works to stabilize the system.

Equation (2.73) is rewritten as

$$\omega^2=c_{is}^2\left(k-\frac{\pi G \sigma_0}{c_{is}^2}\rig... ... \left[\frac{4\Omega^2c_{is}^2}{(\pi G \sigma_0)^2}-1\right].$$ (2.74)

Defining

$$Q\equiv 2\Omega c_{is}/\pi G \sigma_0,$$ (2.75)

we can see that $\omega^2>0$ for all wavenumbers if $Q>1$ and that if $Q<1$ for some range of wavenumber $\omega^2$ becomes negative. A rotating disk with $Q<1$ is unstable for some range of wavenumber. This number is called Toomre's $Q$. This is useful to see whether a galactic disk is stable or not. For the galactic disk, Toomre's $Q$ must be modified as

$$Q\equiv \kappa c_{is}/\pi G \sigma_0$$ (2.76)

where $\kappa$ represents the epicyclic frequency as

$$\kappa\equiv \left(R \frac{d \Omega^2}{d R}+4\Omega^2\right)^{1/2}.$$ (2.77)

See section 3.4 for the galactic disk.