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For a dust layer of extent $L$, its smallest admissible wavenumber is of order $1/L$. The secular gravitational instability band ends at $k_{\mathrm{crit}}=2\pi G{\sigma }_0/c^2=2/(QH)$, with Toomre parameter $Q=c\Omega /(\pi G{\sigma }_0)$ and $H=c/\Omega$. Requiring an admissible mode in the band gives $Q\lesssim L/H$. Boundary conditions fix the order-unity prefactor; a finite disk lifetime also limits practical growth.

At fluid Toomre parameter $Q=1$, the local spiral-wave dispersion relation becomes $n^2({\Omega }_P-\Omega {)}^2=(a_0|k|-\kappa {)}^2$. The minus-sign, longer-wavelength root is nonnegative only between the two Lindblad resonance conditions. The plus-sign, shorter-wavelength root remains positive outside that interval. Thus this dispersion relation alone does not prove that every density wave is confined between Lindblad resonances; an additional physical restriction is needed.

For $\Omega ={\Omega }_0{r}_0/r$, surface density $\Sigma ={\Sigma }_0{e}^{1-r/r_0}$ and constant sound speed $a_0$, the Toomre parameter is $Q(x)=Q_0{e}^{x-1}/x$, where $x=r/r_0$ and $Q_0=\sqrt{2}{\Omega }_0{a}_0/(\pi G{\Sigma }_0)$. Its logarithmic derivative is $1-1/x$, so the minimum is $Q_0$ at $r_0$. The formal inner and outer limits both diverge; if $Q_0<1$, local axisymmetric instability occupies a bounded intermediate annulus. A central departure from the assumed flat rotation law can change the inner conclusion.

For the radial epicyclic frequency, introduce the effective potential at fixed specific angular momentum $l$: ${\Phi }_{\mathrm{eff}}=\Phi +l^2/(2r^2)$. A circular orbit at $r_c$ satisfies ${\Phi }^{\prime }(r_c)=l^2/r_c^3=r_c{\Omega }^2(r_c)$. A small radial displacement $\xi$ obeys $\ddot{\xi }=-{\Phi }_{\mathrm{eff}}^{\prime \prime }(r_c)\xi$. Differentiate at fixed $l$ before inserting the circular-orbit relation:For a flat rotation curve, $\Omega \propto r^{-1}$, so $\kappa =\sqrt{2}\Omega$.

In a perturbation proportional to $\exp [i(\int kdr+n\varphi -\omega t)]$, $n$ is the integer azimuthal mode number, giving $|n|$ arms for a nonaxisymmetric pattern. Its angular pattern speed is ${\Omega }_p=\omega /n$ for a real frequency and $n\ne 0$; $a_0$ is the unperturbed gas sound speed, or the specified effective fluid pressure-response speed. For growing patterns the angular phase speed uses $\mathrm{Re}\omega /n$. Axisymmetric modes have $n=0$: their pattern speed is undefined, and one must use $(\omega -n\Omega {)}^2\to {\omega }^2$, not set the pattern-speed expression to zero.

Define the Toomre parameter and Toomre wavenumber byThen $q=|k|/k_T$ givesFor $a_0>0$, the minimum is negative precisely when $Q<1$. Thus this condition is necessary and sufficient for an exponentially growing axisymmetric mode within the given local razor-thin fluid model; $Q=1$ is marginal. It is not a guarantee against global or nonaxisymmetric instabilities, and the collisionless stellar-disk criterion has a different pressure-response normalization.

The Toomre unstable band and fastest growth follow by minimizing ${\omega }^2$:Here the growing branch has $\omega =i{\gamma }_{\ast }$ under the stated Fourier convention. Both signs of the radial wave number have the same growth rate. The unstable band is $q_{-}<q<q_{+}$, where $q_{\pm }=2(1\pm \sqrt{1-Q^2})/Q^2$.

For the minimum Toomre parameter in an exponential flat-rotation disk with constant $a_0$, put $x=r/r_0$:The logarithmic derivative is $d\log Q/dx=1-1/x$, so the unique minimum occurs at $r=r_0$. Both $Q\to \infty$ as $r\downarrow 0$ and $Q\to \infty$ as $r\to \infty$. Hence inner and outer regions are stable within this idealized rotation law, even when an intermediate annulus is unstable. The whole disk is strictly stable in this local axisymmetric test ifEquality makes the ring at $r_0$ marginal. The central conclusion uses the assumed flat rotation curve all the way inward; real central rotation laws need not retain that limit.

Assume $\kappa ,c_s,{\Sigma }_0>0$, and use the positive Toomre parameter. The dimensionless dispersion relation isAt fixed dimensionless gravitational softening $\delta$, maximum growth corresponds to maximizing $F(s)=2s{e}^{-\delta s}-s^2$. Differentiating gives the most unstable softened disk wavenumberFor $\delta >0$, 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 $\delta s_{\ast }<1$ and $s_{\ast }<1-\delta s_{\ast }$, soInstability occurs precisely when $Q^2<F(s_{\ast })$. Thus the critical Toomre parameter with exponential softening isThe 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 $s_{\ast }<1/(1+\delta )\le 1$ and $2s-s^2$ increases on $[0,1]$,Both strict comparisons become equality in the unsoftened limit $\delta =0$: then $s_{\ast }=1$ 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.

The plot holds $Q$ fixed and changes gravitational softening. If $Q$ is varied physically at fixed $\kappa ϵ/c_s$, remember $\delta =(\kappa ϵ/c_s)/Q$: the marginal curve must be evaluated at its corresponding $\delta$, rather than treating these two dimensionless parameters as independently fixed along that physical variation.

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,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.

Turbulence usually makes the onset of dust gravitational instability of an astrophysical disk harder in three complementary ways. It increases the dust random velocity dispersion, raising $c$ and the Toomre parameter; it stirs dust vertically, reducing the self-gravity enhancement of a razor-thin disk; and it mixes density enhancements through turbulent diffusion. The last effect is particularly important for a slowly growing secular gravitational instability, which can be erased before it amplifies appreciably.

A simple diffusion model adds $D{\nabla }^2\sigma$ to the dust continuity equation, giving a damping scale of order $D{k}^2$. Growth then has to compete with mixing as well as pressure and rotational support. In the long-wavelength weak-drag regime where ${\omega }^2\simeq {\Omega }^2$, the indicative competition iswith further pressure and finite-thickness corrections. This is a model-dependent long-wave estimate, not a substitution into the original cubic without changing its continuity equation. In an infinite domain, diffusion proportional to $k^2$ need not remove growth proportional to $|k|$ at every arbitrarily long wavelength; in a finite disk, the remaining wavelengths and their growth times may be inadequate. The conclusion is therefore a higher practical collapse threshold and slower growth, rather than guaranteed stability for all turbulent flows.

Turbulence can also concentrate dust through coherent structures or pressure maxima, increasing local surface density and encouraging collapse. Its net quantitative effect requires a model for stirring, diffusion, thickness and concentration. The deterministic fixed gas flow used in the preceding parts does not specify those statistics. This competition is turbulent mixing of a secularly unstable dust layer.

The mode originating at the neutral root grows for $0<{\omega }^2<{\Omega }^2$. At small nonzero $|k|$ this gives $s_{\sec }\sim 2\pi ϵG{\sigma }_0|k|/\Omega$. Gas drag transfers angular momentum to the fixed gas flow and permits secular gravitational instability even for a dust Toomre parameter above unity. If ${\omega }^2<0$, the small-root continuation is damped, while a separate dynamical root grows. The formula is singular at ${\omega }^2=0$.