 Weak-drag secular gravitational growth rate

The mode originating at the neutral root grows for $0<{\omega }^2<{\Omega }^2$. At small nonzero $|k|$ this gives $s_{\sec }\sim 2\pi ϵG{\sigma }_0|k|/\Omega$. Gas drag transfers angular momentum to the fixed gas flow and permits secular gravitational instability even for a dust Toomre parameter above unity. If ${\omega }^2<0$, the small-root continuation is damped, while a separate dynamical root grows. The formula is singular at ${\omega }^2=0$.