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For $k_y=0$ and nonzero $k_x$, the wavenumber is constant. Continuity gives ${\dot{\tilde{\Sigma }}}^{\prime }=-i{k}_x{\Sigma }_0{\tilde{v}}_x$, and the conserved vortensity relation givesDifferentiate continuity and substitute radial momentum and this identity. The forced axisymmetric density mode with vortensity obeysThe constant forcing represents the stationary vortical or balanced component, which is lost if every mode is assumed to have nonzero frequency. Let $k_{\ast }=\pi G{\Sigma }_0/c_s^2=\kappa /(Q{c}_s)$. Completing the square givesFor $Q>1$, every nonzero radial wavenumber has positive ${\omega }_k^2$. The solution is a constant balanced offset $-2\Omega {\Sigma }_0^2{\tilde{q}}^{\prime }/{\omega }_k^2$ plus bounded sine and cosine oscillations. Pressure stabilizes short wavelengths, rotation stabilizes long wavelengths, and self-gravity is strongest at intermediate wavelengths.

For $Q<1$, the Toomre stability criterion fails in the bandThere ${\omega }_k^2<0$ and the homogeneous solutions grow or decay exponentially. The fastest rate is $s_{\max }=\kappa \sqrt{Q^{-2}-1}$ at $|k_x|=k_{\ast }$. Outside the band the modes oscillate; at its edges the frequency vanishes.

At $Q=1$, ${\omega }_k^2\ge 0$ and it vanishes only at $|k_x|=\kappa /c_s$. At that critical wavenumber,This Toomre marginal algebraic growth is spectrally marginal, not bounded for all data: a nonzero conserved vortensity gives quadratic secular growth, and even zero vortensity can allow the linear term. Other wavenumbers still oscillate about a balanced offset. $Q>1$ gives bounded axisymmetric modes, $Q<1$ allows exponential instability, and $Q=1$ has a zero-frequency marginal mode with possible algebraic growth. These statements are axisymmetric; nonaxisymmetric shearing disturbances can have transient swing behavior. For $k_x=k_y=0$, continuity instead fixes the uniform mass density shift, while the velocities execute epicyclic motion; no $1/k$ gravitational formula is used.

At the marginal radial wavenumber, the forced axisymmetric density mode with vortensity has zero squared frequency. Its mass density amplitude is $C_0+C_{1}t-\Omega {\Sigma }_0^2{q}^{\prime }{t}^2$. Thus there is no exponential instability but nonzero conserved vortensity can force quadratic growth, and a homogeneous generalized mode can grow linearly. The spectral boundary $Q=1$ of the Toomre stability criterion must not be mistaken for boundedness of every initial-value solution.