In the case of a rotating disk with an angular rotation speed
,
basic equations are
![\begin{displaymath}
\frac{\partial \delta \sigma}{\partial t}+\sigma_0\frac{\partial \delta \mbox{\boldmath${u}$}}{\partial x}=0,
\end{displaymath}](img524.png) |
(2.66) |
![\begin{displaymath}
\sigma_0\frac{\partial \delta \mbox{\boldmath${u}$}}{\partia...
...0\mbox{\boldmath${\Omega}$}\times\delta \mbox{\boldmath${u}$},
\end{displaymath}](img525.png) |
(2.67) |
![\begin{displaymath}
\frac{\partial^2 \delta \phi}{\partial z^2}-k^2 \delta \phi =4\pi G \delta\sigma\delta(z),
\end{displaymath}](img526.png) |
(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
![\begin{displaymath}
\delta \phi=-\frac{2\pi G \delta \sigma}{k}
\exp\left[i(\omega t-kx)-k\vert z\vert\right].
\end{displaymath}](img527.png) |
(2.69) |
Choosing a direction in which the wavenumber vector can be expressed as
,
we can reduce equations (2.66)
and (2.67) to
![\begin{displaymath}
i\omega \delta \sigma -ik \sigma_0 \delta u_x =0,
\end{displaymath}](img529.png) |
(2.70) |
![\begin{displaymath}
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}](img530.png) |
(2.71) |
![\begin{displaymath}
i\omega \delta u_y = -2 \Omega \delta u_x.
\end{displaymath}](img531.png) |
(2.72) |
These three equations together with equation (2.69)
bring us a dispersion relation as
![\begin{displaymath}
\omega^2=c_{is}^2k^2-2\pi G \sigma_0 k +4\Omega^2.
\end{displaymath}](img532.png) |
(2.73) |
Comparing with equation(2.65),
this indicates rotation works to stabilize the system.
Equation (2.73) is rewritten as
![\begin{displaymath}
\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}](img533.png) |
(2.74) |
Defining
![\begin{displaymath}
Q\equiv 2\Omega c_{is}/\pi G \sigma_0,
\end{displaymath}](img534.png) |
(2.75) |
we can see that
for all wavenumbers if
and
that if
for some range of wavenumber
becomes negative.
A rotating disk with
is unstable for some range of wavenumber.
This number is called Toomre's
.
This is useful to see whether a galactic disk is stable or not.
For the galactic disk, Toomre's
must be modified as
![\begin{displaymath}
Q\equiv \kappa c_{is}/\pi G \sigma_0
\end{displaymath}](img537.png) |
(2.76) |
where
represents the epicyclic frequency as
![\begin{displaymath}
\kappa\equiv \left(R \frac{d \Omega^2}{d R}+4\Omega^2\right)^{1/2}.
\end{displaymath}](img539.png) |
(2.77) |
See section 3.5 for the galactic disk.
Kohji Tomisaka
2009-12-10