Toomre's stability criterion

In astrophysics, Toomre's stability criterion (also known as the Safronov–Toomre criterion) is a relationship between parameters of a differentially rotating, gaseous accretion disc which can be used to determine approximately whether the system is stable. In the case of a stationary gas, the Jeans stability criterion can be used to compare the strength of gravity with that of thermal pressure. In the case of a differentially rotating disk, the shear force can provide an additional stabilizing force.

The Toomre criterion for a disk to be stable can be expressed as,

${\displaystyle {\frac {c_{s}\kappa }{\pi G\Sigma }}>1,}$

(1)

where ${\displaystyle c_{s}}$ is the speed of sound (and measure of the thermal pressure), ${\displaystyle \kappa }$ is the epicyclic frequency, G is Newton's gravitational constant, and ${\displaystyle \Sigma }$ is the surface density of the disk.[1]

The Toomre Q parameter is often defined as the left-hand side of Eq.1,

${\displaystyle Q_{\mathrm {gas} }\equiv {\frac {c_{s}\kappa }{\pi G\Sigma }}.}$

(2)

The stability criterion can then simply be stated as, ${\displaystyle Q>1}$ for a disk to be stable against collapse.

The previous discussion was for a gaseous disk, but a similar analysis can be applied to a disk of stars (for example, the disk of a galaxy), yielding a kinematic Q parameter,[1]

${\displaystyle Q_{\mathrm {star} }\equiv {\frac {\sigma _{R}\kappa }{3.36G\Sigma }},}$

(3)

where ${\displaystyle \sigma _{R}}$ is the radial velocity dispersion, and ${\displaystyle \kappa }$ is the local epicyclic frequency.