Critical Toomre parameter with exponential softening ($Q_c^2=2s_{\ast }{e}^{-\delta s_{\ast }}-s_{\ast }^2=s_{\ast }^2(1+\delta s_{\ast })/(1-\delta s_{\ast })$)

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 $s_{\ast }<(1+\delta {)}^{-1}$, $Q_c^2<(1+2\delta )/(1+\delta {)}^2\le 1$. At zero gravitational softening both bounds become equality. The physical path with fixed $\kappa ϵ/c_s$ obeys $\delta =(\kappa ϵ/c_s)/Q$.