Toomre's stability criterion ($Q$)

For a razor-thin isothermal gas disk,Axisymmetric disturbances are stable when $Q\ge 1$: pressure stabilizes short wavelengths, epicyclic motion stabilizes long wavelengths, and self-gravity drives intermediate wavelengths.

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.

Linear axisymmetric compressive modes of a barotropic closure of a razor-thin disk combine radial epicyclic restoration, softened self-gravity and pressure. The off-plane razor-thin Poisson kernel supplies the exponential gravity reduction. Here $k$ is the nonnegative radial wavenumber magnitude. The full linear system also has an axisymmetric geostrophic mode.

The most unstable softened disk wavenumber minimizes the squared mode frequency. Instability is $Q<Q_c$, while $Q\ge Q_c$ is spectrally stable for this compressive pair. Since $s_{\ast }<(1+\delta {)}^{-1}$, $Q_c^2<(1+2\delta )/(1+\delta {)}^2\le 1$. At zero gravitational softening both bounds become equality. The physical path with fixed $\kappa ϵ/c_s$ obeys $\delta =(\kappa ϵ/c_s)/Q$.

With $s=Q{c}_{s}k/\kappa$ and $\delta =\kappa ϵ/(Q{c}_s)$, maximize $F(s)=2s{e}^{-\delta s}-s^2$. Its unique positive critical point satisfies the displayed equation. For positive gravitational softening, $e^{-\delta s_{\ast }}<1$ and $1-\delta s_{\ast }>0$, giving the strict bound. Unsoftened gravity has $s_{\ast }=1$.

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.

In a star-dominated Keplerian disk with $H\simeq c_s/\Omega$ and characteristic disk mass $M_D\sim \pi r^2\Sigma$, the Toomre stability criterion becomes $Q\sim (H/r)(M_{\ast }/M_D)$. Thus instability can occur at a disk-to-star mass ratio comparable to the small disk aspect ratio. The numerical coefficient depends on whether $M_D$ means a local annular estimate or integrated enclosed mass.

In a cooling self-gravitating disk, $Q<1$ excites gravitational turbulence whose dissipation raises the sound speed, while $Q>1$ suppresses that heating and lets cooling lower the sound speed. A sustained state can therefore regulate itself near $Q\simeq 1$.

A mixing-length estimate for gravitational turbulence takes the largest unstable wavelength $\ell$ and orbital growth rate $\Omega$, giving turbulent velocity $u_{\mathrm{turb}}\sim \Omega \ell$ and kinematic viscosity $\nu \sim u_{\mathrm{turb}}\ell \sim \Omega {\ell }^2$.

For $P=K{\rho }^{1+1/n}$, dimensionless pseudo-enthalpy $w=n(\rho /{\rho }_0{)}^{1/n}$, and $\zeta =\Omega z/c_0$, vertical hydrostatic balance and the Poisson equation reduce toThe midplane conditions are $w(0)=n$ and $w^{\prime }(0)=0$, and the first zero gives the disk surface.

The self-gravitating radius of an accretion disk is the radius at which its Toomre stability criterion reaches $Q=1$. Beyond it, a smooth thin disk is vulnerable to spiral structure, gravitational transport, or fragmentation.