= Minimum Toomre parameter in an exponential flat-rotation disk
{title2=$Q(x)=Q_0e^{x-1}/x$}
For $\Omega=\Omega_0r_0/r$, <surface density> $\Sigma=\Sigma_0e^{1-r/r_0}$ and constant <sound speed> $a_0$, the <Toomre parameter> is $Q(x)=Q_0e^{x-1}/x$, where $x=r/r_0$ and $Q_0=\sqrt2\Omega_0a_0/(\pi G\Sigma_0)$. 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.
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