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Density Wave Theory
Figure 3.7:
(Left:) A plot of the dispersion relation eq.(2.65).
 and axes are and .
Respective lines are for (straight lines), , , and .
The trailing part is only plotted.
Points , , and correspond respectively to Inner Lindbrad Resonance (ILR),
Corotation Resonance (CR), and Outer Lindbrad Resonance (OLR).
The relation is symmetric against the axis and the curve of represents the leading wave.
(Right:) Leading vs trailing spiral.

We have derived the dispersion relation of the gravitational instability in the rotating thin disk as

(3.36) 
where represents the number of spiral arms.
Although the stability of the stellar system is a little different,
we assume this is valid for the stellar system after is replaced to the velocity dispersion.
Since

(3.37) 
we obtain

(3.38) 
where
is a pattern speed,
is the normalized
frequency,
is the Toomre's critical wavenumber for a cold () system.
, which leads to , represents the Lindbrad resonance and is rewriten as

(3.39) 
Assuming ,
the resonance when
is called outer Lindbrad resonance while
that of
is called inner Lindbrad resonance.
means the corotation resonance
.
Plotting the wavenumber against the normalized frequency of
equation (3.38) as Figure 3.7(left),
it is shown that, in the case of , the wavenumber exists for all .
Since , 0, and +1 correspond to the points of ILR, CR, and OLR and these three resonance points
appear in accordance with the radial distance, the axis of Figure 3.7(left)
seems to correspond to the radial distance from the center.
In the case of , it is shown that a forbidden region appears around the corotation resonance point.
Waves cannot propagate into the region.
Figure 3.7(left) shows that
the has two possible wavenumbers in the permitted region.
The waves with larger and smaller are called short waves and long waves, respectively.
Consider a wave expressed by
.
If , moving from a point in the direction and
the phase difference between the two points
can be equal to zero.
That is, in the case of the wave is leading.
On the other hand, if , moving in the direction and the phase
will be unchanged.
In this case, the wave pattern is trailing.
Since the dispersion relation is symmetric for and , there are two waves, trailing waves and
leading waves.
Therefore there are four waves: a short trailing wave, a long trailing wave,
a short leading wave, and a long leading wave.
Subsections
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Up: Galactic Scale Star Formation
Previous: Spiral Structure
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Kohji Tomisaka
20070708