= Solution
Setting $\nu=\nu_b=0$, the <dispersion relation> reduces to
$$
\lambda\left(\lambda^2+\Omega^2-2\pi G\Sigma_0k+v_s^2k^2\right)=0.
$$
An axisymmetric density mode grows when
$$
2\pi G\Sigma_0k-v_s^2k^2>\Omega^2.
$$
The left-hand side is a concave <quadratic function> of $k$, with maximum $(\pi G\Sigma_0)^2/v_s^2$ at $k=\pi G\Sigma_0/v_s^2$. Growth is therefore possible precisely when the <Toomre stability criterion> is violated:
$$
\boxed{Q=\frac{v_s\Omega}{\pi G\Sigma_0}<1}.
$$
Back to article page
