= Critical Toomre parameter with exponential softening
{title2=$Q_c^2=2s_*e^{-\delta s_*}-s_*^2=s_*^2(1+\delta s_*)/(1-\delta s_*)$}
The <most unstable softened disk wavenumber> minimizes the squared mode frequency. Instability is $Q<Q_c$, while $Q\ge Q_c$ is spectrally stable for this compressive pair. Since $s_*<(1+\delta)^{-1}$, $Q_c^2<(1+2\delta)/(1+\delta)^2\le1$. At zero <gravitational softening> both bounds become equality. The physical path with fixed $\kappa\epsilon/c_s$ obeys $\delta=(\kappa\epsilon/c_s)/Q$.
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