Gravitational Instability of Rotating Thin Disk

Here, we will derive the dispersion relation for the gravitational instability of a rotating thin disk. We will see the spatial variation of Toomre's $Q$ parameter, which determines the stability of the rotating disk, explains the nonlinearity of star formation rate, that is, there is a threshold density and no stars are formed in the low density region.

Use the cylindrical coordinate $(R,Z,\phi)$ and the basic equations for thin disk in $\S$2.6. In linear analysis, we assume $\Sigma(R,\phi)=\Sigma_0(R)+\delta \Sigma (R,\phi)$, $u(R,\phi)=0 + \delta u(R,\phi)$, $v(R,\phi)=v_0(R)+ \delta v(R,\phi)$, where $u$ and $v$ represent the radial and azimuthal components of the velocity. Linearized continuity equation is

\frac{\partial \delta \Sigma}{\partial t} + \frac{1}{R} \fr...
...c{\Sigma_0}{R} \frac{\partial \delta v}{\partial \phi} = 0,
\end{displaymath} (3.9)

where $\Omega=v_0/R$.

Linearized equations of motion are

\left( \frac{\partial }{\partial t} + \Omega \frac{\partial...
- \frac{\partial }{\partial R} (\delta \Phi + \delta h),
\end{displaymath} (3.10)

\left( \frac{\partial }{\partial t} + \Omega \frac{\partial...
...} \frac{\partial }{\partial \phi} (\delta \Phi + \delta h),
\end{displaymath} (3.11)

where $h$ is a specific enthalpy as $dh=dp/\Sigma$ and
\kappa=\left(4\Omega^2+R\frac{d \Omega^2}{d R}\right)^{1/2}
\end{displaymath} (3.12)

is the epicyclic frequency.

We assume any solution of equations (3.9), (3.10) and(3.11) can be written as a sum of terms of the form

$\displaystyle \delta u = u_a \exp[i(m\phi-\omega t)],$     (3.13)
$\displaystyle \delta v = v_a \exp[i(m\phi-\omega t)],$     (3.14)
$\displaystyle \delta \Sigma = \Sigma_a \exp[i(m\phi-\omega t)],$     (3.15)
$\displaystyle \delta h = h_a \exp[i(m\phi-\omega t)],$     (3.16)
$\displaystyle \delta \Phi = \Phi_a \exp[i(m\phi-\omega t)].$     (3.17)

Using the equation of state of $p=K\Sigma^{\gamma}$,
h_a=c_s^2 \Sigma_a/\Sigma_0.
\end{displaymath} (3.18)

Using equations (3.13)-(3.17), equations (3.9), (3.10), and (3.11) are rewritten as
i(m\Omega - \omega) \Sigma_a + \frac{1}{R}\frac{\partial }...
...tial R} (R\Sigma_0 u_a) + i m \frac{\Sigma_0 v_a}{R} = 0,
\end{displaymath} (3.19)

u_a [\kappa^2 -(m\Omega -\omega)^2] = -i\left[ (m\Omega - \...
...\Phi_a + h_a) + 2m\Omega \frac{(\Phi_a + h_a)}{R} \right],
\end{displaymath} (3.20)

v_a [\kappa^2 -(m\Omega -\omega)^2] = \left[ \frac{\kappa^2...
..._a) + m (m\Omega-\omega) \frac{(\Phi_a + h_a)}{R} \right],
\end{displaymath} (3.21)

Kohji Tomisaka 2009-12-10