Minimum Toomre parameter in an exponential flat-rotation disk ($Q(x)=Q_0{e}^{x-1}/x$)

For $\Omega ={\Omega }_0{r}_0/r$, surface density $\Sigma ={\Sigma }_0{e}^{1-r/r_0}$ and constant sound speed $a_0$, the Toomre parameter is $Q(x)=Q_0{e}^{x-1}/x$, where $x=r/r_0$ and $Q_0=\sqrt{2}{\Omega }_0{a}_0/(\pi G{\Sigma }_0)$. Its logarithmic derivative is $1-1/x$, so the minimum is $Q_0$ at $r_0$. The formal inner and outer limits both diverge; if $Q_0<1$, local axisymmetric instability occupies a bounded intermediate annulus. A central departure from the assumed flat rotation law can change the inner conclusion.