Consider the case of axisymmetric perturbations.
Equation (3.32) becomes

(3.33) 
If the system is stable against the axisymmetric perturbation, while
if the system is unstable.
Defining

(3.34) 
if , for all wavenumbers .
On the other hand, if , becomes negative for some wavenumbers .
Therefore, the Toomre's number gives us a criterion whether the system is unstable or not
for the axisymmetric perturbation.
[Recommendation for a reference book of this section: Binney & Tremaine (1988).]
The condition is expressed as

(3.35) 
Kennicutt plotted
against the normalized radius as for various
galaxies, where represents the maximum distance of HII regions from the center
(Fig.3.8).
Since
, Figure 3.8 shows that HII regions are observed mainly in
the region with but those are seldom seen in the outer lowdensity region.
This seems the gravitational instability plays an important role.
Figure 3.8:
vs .
represents the maximum distance of HII regions from the center.
The sound speed is assumed constant
.
Taken from Fig.11 of
Kennicutt (1989).

Figure:
Numerical simulation of the swing amplification mechanism.
The number attached each panel shows the time sequence.
This is obtained by the timedependent linear analysis.
First, perturbation with leading spiral pattern is added to the Mestel disk with .
The leading spiral gradually unwinds and become a trailing spiral.
Loosely wound spiral pattern winds gradually and the last panel shows a tightly wound leading spiral pattern.
The final amplitude is times larger than that of the initial state.

The above discussion is for the gaseous disk.
The Toomre's value is also defined for stellar system as

(3.36) 
where represents the radial velocity dispersion.
For nonaxisymmetric waves, even if
the instability grows.
To explain this, the swing amplification mechanism is proposed (Toomre 1981).
If there is a leading spiral perturbation in the disk with
,
the wave unwinds and finally becomes a trailing spiral pattern.
At the same time, the amplitude of the wave (perturbations) is amplified (see Fig.3.9).
Kohji Tomisaka
20121003