= Solution
Assume $\kappa,c_s,\Sigma_0>0$, and use the positive <Toomre parameter>. The dimensionless dispersion relation is
$$
\frac{\omega^2}{\kappa^2}=1+\frac{s^2-2se^{-\delta s}}{Q^2}.
$$
At fixed dimensionless <gravitational softening> $\delta$, maximum growth corresponds to maximizing $F(s)=2se^{-\delta s}-s^2$. Differentiating gives the <most unstable softened disk wavenumber>
$$
\boxed{s_*=e^{-\delta s_*}(1-\delta s_*).}
$$
For $\delta>0$, the right side is strictly decreasing on $0<s<1/\delta$, while the left side increases; there is exactly one positive solution. The equation implies $\delta s_*<1$ and $s_*<1-\delta s_*$, so
$$
\boxed{s_*<\frac1{1+\delta}.}
$$
Instability occurs precisely when $Q^2<F(s_*)$. Thus the <critical Toomre parameter with exponential softening> is
$$
\boxed{Q_c^2=2s_*e^{-\delta s_*}-s_*^2=s_*^2\frac{1+\delta s_*}{1-\delta s_*},\qquad Q<Q_c\ \text{is unstable}.}
$$
The printed description of a minimum $Q$ for instability reverses the threshold: $Q_c$ is the upper boundary of unstable values and the lower boundary of stable ones. Because $s_*<1/(1+\delta)\le1$ and $2s-s^2$ increases on $[0,1]$,
$$
Q_c^2<2s_*-s_*^2<\frac2{1+\delta}-\frac1{(1+\delta)^2}
=\frac{1+2\delta}{(1+\delta)^2}\le1.
$$
Both strict comparisons become equality in the unsoftened limit $\delta=0$: then $s_*=1$ and $Q_c=1$. <Gravitational softening> suppresses short-wave gravity, shifts the most dangerous mode to a longer <wavelength> and requires stronger <self-gravity>, or smaller $Q$, for instability.
\Image[/past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2013/iii/paper-54-softened-dispersion.png]
{title=Exponential gravitational softening narrows and can remove the unstable density-wave band}
{height=460}
The plot holds $Q$ fixed and changes <gravitational softening>. If $Q$ is varied physically at fixed $\kappa\epsilon/c_s$, remember $\delta=(\kappa\epsilon/c_s)/Q$: the marginal curve must be evaluated at its corresponding $\delta$, rather than treating these two dimensionless parameters as independently fixed along that physical variation.
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