Jeans Instability - Jeans Mass

Jeans Mass

The Jeans mass is named after the British physicist Sir James Jeans, who considered the process of gravitational collapse within a gaseous cloud. He was able to show that, under appropriate conditions, a cloud, or part of one, would become unstable and begin to collapse when it lacked sufficient gaseous pressure support to balance the force of gravity. Remarkably, the cloud is stable for sufficiently small mass (at a given temperature and radius), but once this critical mass is exceeded, it will begin a process of runaway contraction until some other force can impede the collapse. He derived a formula for calculating this critical mass as a function of its density and temperature. The greater the mass of the cloud, the smaller its size, and the colder its temperature, the less stable it will be against gravitational collapse.

The approximate value of the Jeans mass may be derived through a simple physical argument. One begins with a spherical gaseous region of radius, mass, and with a gaseous sound speed . Imagine that we compress the region slightly. It takes a time,

for sound waves to cross the region, and attempt to push back and re-establish the system in pressure balance. At the same time, gravity will attempt to contract the system even further, and will do so on a free-fall time,

where is the universal gravitational constant, is the gas density within the region, and is the gas number density for mean mass per particle g, appropriate for molecular hydrogen with 20% helium by number. Now, when the sound-crossing time is less than the free-fall time, pressure forces win, and the system bounces back to a stable equilibrium. However, when the free-fall time is less than the sound-crossing time, gravity wins, and the region undergoes gravitational collapse. The condition for gravitational collapse is therefore:

The resultant Jeans length is approximately:

This length scale is known as the Jeans length. All scales larger than the Jeans length are unstable to gravitational collapse, whereas smaller scales are stable. The Jeans mass is just the mass contained in a sphere of radius ( is half the Jeans length):

It was later pointed out by other astrophysicists that in fact, the original analysis used by Jeans was flawed, for the following reason. In his formal analysis, Jeans assumed that the collapsing region of the cloud was surrounded by an infinite, static medium. In fact, because all scales greater than the Jeans length are also unstable to collapse, any initially static medium surrounding a collapsing region will in fact also be collapsing. As a result, the growth rate of the gravitational instability relative to the density of the collapsing background is slower than that predicted by Jeans' original analysis. This flaw has come to be known as the "Jeans swindle". Later analysis by Hunter corrects for this effect.

The Jeans instability likely determines when star formation occurs in molecular clouds.

Michael K.-H. Kiessling believes that Jeans' results can be vindicated without engaging in a mathematical swindle on the premise that "as long as we obtain a sensible dynamics in some sensible limit, we should not worry too much if some potential ceases to exist in the same limit" concluding that in his treatment "All the ad hoc steps of the 'Jeans swindle' have materialized in a mathematically clean way." Mathematical Vindications of the "Jeans Swindle"