Density Wave Theory

Figure 3.11: (Left:) A plot of the dispersion relation eq.(2.65). $x$- and $y$-axes are $\nu$ and $k/k_{\rm T}$. Respective lines are for $Q=1$ (straight lines), $Q=1.2$, $Q=1.5$, and $Q=2$. The trailing part $k>0$ is only plotted. Points $\nu =-1$, $\nu =0$, and $\nu =1$ correspond respectively to Inner Lindbrad Resonance (ILR), Corotation Resonance (CR), and Outer Lindbrad Resonance (OLR). The relation is symmetric against the $x$-axis and the curve of $k<0$ 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

$$\left(m\Omega -\omega\right)^2=k^2c_s^2 - 2\pi G \Sigma_0 \vert k\vert + \kappa^2,$$ (3.37)

where $m$ 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 $c_s$ is replaced to the velocity dispersion. Since

$$\left(m\Omega -\omega\right)^2=k^2c_s^2 - 2\pi G \Sigma_0 \... ...ight)^2+\kappa^2-\left(\frac{\pi G \Sigma_0}{c_s}\right)^2,$$ (3.38)

we obtain

$$\vert k\vert= k_{\rm T}\frac{2}{Q^2}\left[1 \pm \sqrt{1 -Q^{2}(1 - \nu^2)}\right],$$ (3.39)

where $\Omega_{\rm P}=\omega/m$ is a pattern speed, $\nu=m(\Omega_{\rm P}-\Omega)/\kappa$ is the normalized frequency, $k_{\rm T}=\kappa^2/2\pi G \Sigma_0$ is the Toomre's critical wavenumber for a cold ($c_s=0$) system. $\nu=\pm 1$, which leads to $\vert k\vert=0$, represents the Lindbrad resonance and is rewriten as

$$\Omega_{\rm P}=\frac{\omega}{m}=\Omega \pm \frac{\kappa}{2}.$$ (3.40)

Assuming $m=2$, the resonance when $\Omega_{\rm P}=\Omega + \kappa/2$ is called outer Lindbrad resonance while that of $\Omega_{\rm P}=\Omega - \kappa/2$ is called inner Lindbrad resonance. $\nu =0$ means the co-rotation resonance $\Omega_{\rm P}=\Omega$.

Plotting the wavenumber $k$ against the normalized frequency $\nu$ of equation (3.39) as Figure 3.11(left), it is shown that, in the case of $Q=1$, the wavenumber exists for all $\nu$. Since $\nu =-1$, 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 $x$-axis of Figure 3.11(left) seems to correspond to the radial distance from the center. In the case of $Q>1$, it is shown that a forbidden region appears around the co-rotation resonance point. Waves cannot propagate into the region. Figure 3.11(left) shows that the $k/k_{\rm T}$ has two possible wavenumbers in the permitted region. The waves with larger $k$ and smaller $k$ are called short waves and long waves, respectively.

Consider a wave expressed by $\Sigma \propto \exp[im\phi+ikr]$. If $k<0$, moving from a point $(R_0,\phi_0)$ in the direction $\Delta \phi>0$ and $\Delta r>0$ the phase difference between the two points $[m(\phi_0+ \Delta \phi) + k (R_0+\Delta r)] - [m\phi_0+k R_0]$ can be equal to zero. That is, in the case of $k<0$ the wave is leading. On the other hand, if $k>0$, moving in the direction $\Delta \phi<0$ and $\Delta r>0$ the phase will be unchanged. In this case, the wave pattern is trailing. Since the dispersion relation is symmetric for $k>0$ and $k<0$, 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.