Toomre's Stability Criterion

In astrophysics, the Toomre Stability Criterion (or Safronov-Toomre Criterion) refers to a simple relationship between parameters of a differentially rotating, gaseous disk which can be used to approximate 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^[1] for a disk to be stable can be expressed as,

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

(1)

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

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

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

(2)

The stability criterion can then simply be stated as, $Q>1$ 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 (e.g. the disk of a galaxy), yielding a kinematic Q parameter,^[1]

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

(3)

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

Background

Many astrophysical objects result from the gravitational collapse of gaseous objects (e.g. molecular clouds collapse to form stars), and thus the stability of gaseous systems is of critical interest. In general, a physical system is 'stable' if: 1) It is in equilibrium (there is a balance of forces such that the system is static), and 2) small 'perturbations' (deviations from equilibrium) will tend to damp out (i.e. the system tends to return to equilibrium).

The most basic gravitational stability analysis is the Jeans criteria, which addresses the balance between self-gravity and thermal pressure in a gas. In terms of the two above stability conditions, the system is stable if: i) thermal pressure balances the force of gravity, and ii) if the system is compressed slightly, the outward pressure force must become stronger than the inward gravitational force - to return the system to equilibrium. In the Jeans case, the stability criterion is 'wavelength' (i.e. size) dependent --- resulting in the concept of a Jeans Length and Jeans Mass.

The Toomre analysis, first studied by Viktor Safronov in the 1960s,^[2] considers not only gravity and pressure, but also shear forces from differential rotation. Conceptually, if a fluid is differentially rotating (i.e. in the keplerian motion of an astrophysical disk), gravity not only has to overcome the internal pressure of the gas, but also needs to halt the relative motion between two parcels of fluid --- allowing them to collapse together.

The analysis was expanded upon by Alar Toomre in 1964,^[1] and presented in a more general and comprehensive framework.

References

1. Toomre, Alar (1964). "On the gravitational stability of a disk of stars". Astrophysical Journal. 139: 1217–1238. Bibcode:1964ApJ...139.1217T. doi:10.1086/147861. Retrieved 18 July 2014.
2. Safronov, Viktor (1960). "On the gravitational instability in flattened systems with axial symmetry and non-uniform rotation". Annales d'Astrophysique. 23: 979. Bibcode:1960AnAp...23..979S. Retrieved 18 July 2014.