Jeans instability

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The Jeans instability occurs when internal pressure is no longer strong enough to prevent gravitational collapse of a region filled with matter.

$\frac{dp}{dr}=-\frac{G\varrho M_{(<r)}}{r^2}$

$M ( < r)$ is the enclosed mass, p is the pressure, G is the gravitational constant, r is the radius of the region

[edit] Jeans mass

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 more unstable it will be to 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 $R$ , mass $M$ , and with a gaseous sound speed $c s$ . Imagine that we compress the region slightly. It takes a time,

t s o u n d = R / c s

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,

t f f = 1 / (G ρ) 1 / 2

where $G$ is the universal gravitational constant, and $ρ$ 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:

t f f < t s o u n d

With a little bit of algebra, one can show that the resultant Jeans mass $M J$ is approximately:

$M_J = c_s^3 / (G^{3/2} \rho^{1/2} )$

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 $R J$ is approximately:

R J = c s / (G ρ) 1 / 2

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.

[edit] When it occurs

Jeans instability occurs once the enclosed matter exceeds the Jeans mass or the region grows beyond the Jeans length. If the gravitational instability is governed by waves of type $\Delta=\Delta_0 e^{\Gamma t + i \mathbf{k\cdot r}}$ , a value of gamma

$\Gamma=\left[4\pi G \varrho_0\left(1-\frac{\lambda^2_J}{\lambda^2}\right)\right]^{1/2}$

represents an exponentially growing instability. $λ J$ is the Jeans length and $\varrho_0$ is the mass density. The time scale for collapse is given by