Toomre parameter ($Q=c\kappa /(\pi G\Sigma )$)

For a razor-thin fluid disk with sound speed or effective random speed $c$, surface density $\Sigma$ and radial epicyclic frequency $\kappa$, $Q=c\kappa /(\pi G\Sigma )$ compares pressure and rotational support with self-gravity. The local axisymmetric fluid Toomre stability criterion is $Q\ge 1$. For a Keplerian shearing sheet, $\kappa =\Omega$. Collisionless stellar disks use a different numerical normalization; gas drag can permit secular growth even when this fluid parameter exceeds unity.

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 razor-thin fluid disk dispersion relation, set $Q=\kappa a_0/(\pi G\Sigma )$ and $k_T={\kappa }^2/(2\pi G\Sigma )$. When $Q<1$, the unstable band isThe quadratic dispersion has its minimum at $|k{|}_{\ast }=\pi G\Sigma /a_0^2$, with fastest growth ${\gamma }_{\ast }=\kappa \sqrt{Q^{-2}-1}$. At $Q=1$ the two band edges coincide and the mode is marginal. These are local axisymmetric fluid results, not a global stability theorem.

In the razor-thin fluid disk dispersion relation, the cold limit is ${\omega }^2={\kappa }^2-2\pi G\Sigma |k|$. Rotation therefore suppresses wavelengths above ${\lambda }_T=2\pi /k_T=4{\pi }^{2}G\Sigma /{\kappa }^2$. With this convention, $k_T$ is the cold-disk boundary of instability, not the fastest-growing wave number when finite pressure is present. At marginal fluid stability $Q=1$, the minimum occurs at $|k|=2k_T$.

For constant circular speed $v_0$, angular velocity is $v_0/r$ and radial epicyclic frequency is $\sqrt{2}{v}_0/r$. Constant effective random speed and surface density then give $Q=r_Q/r$. The local axisymmetric fluid model is stable inside $r_Q$ and unstable outside it. This statement is conditional on the thin-disk local approximation, rather than a proof of global or collisionless stability.

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.