= Solution
The branch originating at $s=0$ has $s=\epsilon s_1+O(\epsilon^2)$. Keeping first-order terms in the cubic <dispersion relation> gives $\omega^2s_1+\Omega(\omega^2-\Omega^2)=0$. Hence the <weak-drag secular gravitational growth rate> is
$$
\boxed{s_{\mathrm{sec}}=\epsilon\Omega\frac{\Omega^2-\omega^2}{\omega^2}+O(\epsilon^2).}
$$
Carrying forward the positive-$\omega^2$ regime of the preceding part, this mode grows when $\boxed{0<\omega^2<\Omega^2}$. This specifies the scope of the source's abbreviated condition. If $\omega^2<0$, the small root written here is instead damped, while the separate dynamical branch already grows; the dust is still unstable, but growth is not identified with this particular small-root continuation. The expansion also fails at $\omega^2=0$.
At sufficiently small nonzero $|k|$, $\omega^2$ is positive and just below $\Omega^2$, since
$$
\Omega^2-\omega^2=2\pi G\sigma_0|k|-c^2k^2>0
\quad\text{for}\quad0<|k|<\frac{2\pi G\sigma_0}{c^2}.
$$
Therefore an infinite dust layer with nonzero self-gravity and nonzero drag has a long-wavelength <secular gravitational instability of an astrophysical disk>, even when $Q>1$. Its small-$k$ growth rate tends to $2\pi\epsilon G\sigma_0|k|/\Omega$, and vanishes at $k=0$; the uniform-density mode itself is neutral. Drag transfers perturbation angular momentum to the gas, weakening rotational support while self-gravity exceeds pressure on long wavelengths.
The instability is not limited to weak drag as an existence statement. For any $\epsilon>0$, the cubic has $P(0)=\epsilon\Omega(\omega^2-\Omega^2)<0$ and $P(s)\to+\infty$ on the positive real axis. Thus it has a positive real root whenever $\omega^2<\Omega^2$. This exact <positive-root criterion for dust self-gravity with drag> distinguishes the growth criterion from the approximation used to compute its slow rate.
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