## Toomre's Value

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 low-density region. This seems the gravitational instability plays an important role.

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 non-axisymmetric 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 2012-10-03