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# The Jeans Instability

Linear perturbative analyis on the self-gravitating fluid equations under the assumption of infinite, constant (

), static background density ( ) results in the following 2nd order PDE in the density perturbation ( ):

where

is the sound speed.This is a wave equation, so we propose a solution of the form:

where the wave number k is given by

, with specifying the perturbation oscillation wave length, and is the angular frequency of oscillation given by . Plugging this into the wave equation above gives the following dispersion relation:

When

or

is real, and the density perturbation has the form:

which is oscillatory. Thus the density perturbation propagates through the medium as a sound wave, i.e. pressure gradients are strong enough to support the medium against self-gravitational collapse. On the other hand, when

or

we see that

is imaginary. This implies that there is an exponential growth or decay factor multiplying the oscillatory in the solution for . That is,

Taking the real part, we have:

Thus, the perturbation takes on a form that is oscillatory in x and exponential in time.

# The Jeans Length

To get the critical length scale (aka Jeans length) for the oscillations, we write the oscillatory condition

Substituting

for k and solving for we have

That is, for wavelengths smaller than

, the perturbation is stable to collapse. You can think of this wavelength as length scales over which the average . The critical length is known as the Jeans length and is given by:

# Unstable Modes

What we'd like to do now is determine, for which frequencies is the perturbation *unstable* (exponentially growing). In order to determine this we can rewrite into a more intuitive form.

Taking the positive root and plugging into

, we see

which gives

Thus we have found the condition on

which gives unstable perturbation modes. Now, the growth rate of these modes is given by

and so the characteristic timescale for exponential growth, that is the time in which the perturbation increases by a factor of e, is given by

# High lambda limit, i.e. pressure-less collapse, or 'uniform collapse'

From

it is easy to see that in the limit

for finite

(i.e. pressure gradients are null/gravity dominates), the characteristic collapse time becomes

where

is the freefall time for a spherical region of uniform gas.# Writing a module to test self-gravity using the Jeans Instability

I go through the procedure of setting up the problem on this page here.

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