= Solution
Use a local, homogeneous <razor-thin disk approximation> in a frame rotating with constant $\Omega$. The unperturbed planar <velocity> is zero in this frame, and the large-scale gravitational and centrifugal <forces> balance. Neglect viscosity, magnetic fields, thickness and background gradients across a <wavelength>. Take small planar disturbances, with <wavelength> short compared with the <galaxy>'s background scale but long enough for a fluid description. The <barotropic closure of a razor-thin disk> is $P=K\Sigma^\gamma$, with $K$ fixed and a positive derivative
$$
c^2=\left.\frac{dP}{d\Sigma}\right|_0=\frac{\gamma P_0}{\Sigma_0}.
$$
<Solid-body rotation> has no shear and has <radial epicyclic frequency> $\kappa=2|\Omega|$. By rotational symmetry of the local model choose a wavevector along $x$, and write perturbations proportional to $e^{i(kx-\omega t)}$.
Let $\Sigma_1,u,v,\Phi_1$ be the <surface density>, $x$-velocity, $y$-velocity and <Newtonian gravitational potential> amplitudes. The linearized <continuity equation> and <Euler equations> with the <Coriolis force> are
$$
-i\omega\Sigma_1+ik\Sigma_0u=0,
\qquad
-i\omega u-2\Omega v=-ik\left(c^2\frac{\Sigma_1}{\Sigma_0}+\Phi_1\right),
\qquad
-i\omega v+2\Omega u=0.
$$
The perturbing <Newtonian gravitational potential> solves the <Poisson equation for Newtonian gravity>
$$
(\partial_z^2-k^2)\Phi_1(z)=4\pi G\Sigma_1\delta(z).
$$
Its decaying solution is proportional to $e^{-|k||z|}$. The jump in its derivative is $-2|k|\Phi_1(0)=4\pi G\Sigma_1$, giving the <razor-thin disk Poisson kernel>
$$
\Phi_1(0)=-\frac{2\pi G\Sigma_1}{|k|}.
$$
Eliminating $v$ from the two momentum equations, and using $\Sigma_1=k\Sigma_0u/\omega$, gives the density-wave branch
$$
\boxed{\omega^2=c^2k^2-2\pi G\Sigma_0|k|+4\Omega^2.}
$$
The complete linear system also has a zero-frequency balanced mode; it is not the growing density-wave branch, and division by $\omega$ in this elimination excludes it. The result is the <uniformly rotating gas-sheet dispersion relation>: <pressure> opposes compression at large <wavenumber>, <self-gravity> promotes compression, and rotation provides epicyclic support.
A mode is exponentially unstable when $\omega^2<0$. With $\Omega=0$ and $c>0$, this occurs at
$$
\boxed{0<|k|<\frac{2\pi G\Sigma_0}{c^2}.}
$$
The uniform $k=0$ perturbation is marginal rather than growing in this local calculation. With $c=0$, rotation fails to stabilize sufficiently short waves:
$$
\boxed{|k|>\frac{2\Omega^2}{\pi G\Sigma_0}.}
$$
These limits show why both <pressure> and rotation are needed for stability at all <wavelengths>.
For nonzero <pressure> and rotation, put $q=|k|$. Complete the square:
$$
\omega^2=c^2\left(q-\frac{\pi G\Sigma_0}{c^2}\right)^2
+4\Omega^2-\frac{(\pi G\Sigma_0)^2}{c^2}.
$$
Its minimum lies at $q_*=\pi G\Sigma_0/c^2$. Thus a growing mode exists precisely when
$$
\boxed{\frac{|\Omega|c}{G\Sigma_0}<\frac\pi2.}
$$
\b[The printed inequality has the opposite physical meaning: it is the condition for no exponentially growing density wave.] Equality is marginal. In the usual gas <Toomre stability criterion>, $Q=\kappa c/(\pi G\Sigma_0)=2|\Omega|c/(\pi G\Sigma_0)$, so instability is $Q<1$ and stability is $Q\geq1$. The <unstable wavenumber band of a rotating gas sheet> is
$$
q_-<|k|<q_+,\qquad
q_\pm=\frac{\pi G\Sigma_0}{c^2}\left(1\pm\sqrt{1-Q^2}\right),
$$
when $Q<1$.
If $\Sigma_0$ and $\Omega\ne0$ remain fixed while $c^2$ decreases slowly, first reach marginality at
$$
c_{\mathrm{crit}}=\frac{\pi G\Sigma_0}{2|\Omega|},\qquad
q_{\mathrm{crit}}=\frac{4\Omega^2}{\pi G\Sigma_0}.
$$
The first <wavelength> to become unstable just below this threshold is the <marginal fragmentation wavelength of a rotating sheet>:
$$
\boxed{\ell_{\mathrm{crit}}=\frac{2\pi}{q_{\mathrm{crit}}}
=\frac{\pi^2G\Sigma_0}{2\Omega^2}
=\frac{2c_{\mathrm{crit}}^2}{G\Sigma_0}.}
$$
Density maxima are separated by approximately this <wavelength>. The expected fragment size is of this order; an overdense half-wave has width about $\ell_{\mathrm{crit}}/2$. Linear theory fixes a preferred <wavelength>, not an exact nonlinear clump radius or shape. Further cooling shifts the fastest-growing <wavelength> to $2c^2/(G\Sigma_0)$. A rough fragment <mass> is consequently of order $\Sigma_0\ell_{\mathrm{crit}}^2$, with a geometrical factor depending on the nonlinear fragmentation pattern.
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