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,
\end{displaymath} (2.66)

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

\frac{\partial^2 \delta \phi}{\partial z^2}-k^2 \delta \phi =4\pi G \delta\sigma\delta(z),
\end{displaymath} (2.68)

where the third term of r.h.s. of equation (2.67) represents the Colioris force. Equation (2.68) gives a solution identical to equation (2.64) as
\delta \phi=-\frac{2\pi G \delta \sigma}{k}
\exp\left[i(\omega t-kx)-k\vert z\vert\right].
\end{displaymath} (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 \sigma_0 \delta u_x =0,
\end{displaymath} (2.70)

i\omega \delta u_x = -c_{is}^2 (-ik) \frac{\delta \sigma}{\sigma_0}
-2\pi i G \delta \sigma+2\Omega \delta u_y,
\end{displaymath} (2.71)

i\omega \delta u_y = -2 \Omega \delta u_x.
\end{displaymath} (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 k +4\Omega^2.
\end{displaymath} (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].
\end{displaymath} (2.74)

Q\equiv 2\Omega c_{is}/\pi G \sigma_0,
\end{displaymath} (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
\end{displaymath} (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}.
\end{displaymath} (2.77)

See section 3.5 for the galactic disk.

Kohji Tomisaka 2009-12-10