= Solution
The <secular gravitational instability> needs $|k|<k_{\mathrm{crit}}=2\pi G\sigma_0/c^2$. A finite radial extent $L$ removes arbitrarily small wavenumbers; the smallest available one is $k_{\min}\sim\alpha/L$, where $\alpha$ is a boundary-dependent constant of order unity. With $H=c/\Omega$ and the dust <Toomre parameter>,
$$
k_{\mathrm{crit}}=\frac{2}{QH},\qquad k_{\min}<k_{\mathrm{crit}}
\quad\Longrightarrow\quad Q<\frac2\alpha\frac LH.
$$
Thus the <finite-size secular gravitational criterion> is $\boxed{Q\lesssim L/H}$ at the rough accuracy requested. For example, a periodic radial interval has $k_{\min}=2\pi/L$ and gives $Q<L/(\pi H)$ in this idealized model. The numerical prefactor is not universal, but the scaling is. Finite size therefore restores a practical threshold for the otherwise long-wavelength instability. Growth must additionally occur within the disk lifetime, and a global wavelength comparable to disk radius lies beyond the strictly local <shearing sheet> approximation.
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