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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

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

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