| Jeans Mass |
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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, and is the gas density within the region. 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: : With a little bit of algebra, one can show that the resultant Jeans mass is approximately: : The stability criterion can also be equivalently expressed in terms of a length instead of a mass.This length scale is known as the Jeans Length . All scales less than the Jeans length are stable to gravitational collapse, whereas larger scales are unstable. One can use the same derivation above to demonstrate that the Jeans length is approximately: : 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. |
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