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

$$\frac{\partial \delta \Sigma}{\partial t} + \frac{1}{R} \fr... ...c{\Sigma_0}{R} \frac{\partial \delta v}{\partial \phi} = 0,$$ (3.8)

where $\Omega=v_0/R$. Linearized equations of motion are

$$\left( \frac{\partial }{\partial t} + \Omega \frac{\partial... ... - \frac{\partial }{\partial R} (\delta \Phi + \delta h),$$ (3.9)

and

$$\left( \frac{\partial }{\partial t} + \Omega \frac{\partial... ...} \frac{\partial }{\partial \phi} (\delta \Phi + \delta h),$$ (3.10)

where $h$ is a specific enthalpy as $dh=dp/\Sigma$ and

$$\kappa=\left(4\Omega^2+R\frac{d \Omega^2}{d R}\right)^{1/2}$$ (3.11)

is the epicyclic frequency. We assume any solution of equations (3.8), (3.9) and(3.10) can be written as a sum of terms of the form

$$\delta u = u_a \exp[i(m\phi-\omega t)],$$ (3.12)

$$\delta v = v_a \exp[i(m\phi-\omega t)],$$ (3.13)

$$\delta \Sigma = \Sigma_a \exp[i(m\phi-\omega t)],$$ (3.14)

$$\delta h = h_a \exp[i(m\phi-\omega t)],$$ (3.15)

$$\delta \Phi = \Phi_a \exp[i(m\phi-\omega t)].$$ (3.16)

Using the equation of state of $p=K\Sigma^{\gamma}$,

$$h_a=c_s^2 \Sigma_a/\Sigma_0.$$ (3.17)

Using equations (3.12)-(3.16), equations (3.8), (3.9), and (3.10) are rewritten as

$$i(m\Omega - \omega) \Sigma_a + \frac{1}{R}\frac{\partial }{\partial R} (R\Sigma_0 u_a) + i m \frac{\Sigma_0 v_a}{R} = 0,$$ (3.18)

$$u_a [\kappa^2 -(m\Omega -\omega)^2] = -i\left[ (m\Omega - \... ...\Phi_a + h_a) + 2m\Omega \frac{(\Phi_a + h_a)}{R} \right],$$ (3.19)

and

$$v_a [\kappa^2 -(m\Omega -\omega)^2] = \left[ \frac{\kappa^2... ..._a) + m (m\Omega-\omega) \frac{(\Phi_a + h_a)}{R} \right],$$ (3.20)