The Jeans Instability

Linear perturbative analyis on the self-gravitating fluid equations under the assumption of infinite, constant (v_0), static background density (\rho_0) results in the following 2nd order PDE in the density perturbation (\rho_1):

\frac{\partial^2\rho_1}{\partial t^2} - C_s^2 \triangledown^2 \rho_1 - 4\pi G \rho_0 \rho_1 = 0

where C_s is the sound speed.

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

\rho_1(\vec{x},t) = C e^{i(\vec{k} \cdot \vec{x} - \omega t)}

where the wave number k is given by k = 2 \pi /\lambda, with \lambda specifying the perturbation oscillation wave length, and \omega is the angular frequency of oscillation given by \omega = 2 \pi f. Plugging this into the wave equation above gives the following dispersion relation:

\omega ^2 = C_s^2 k^2 - 4 \pi G \rho_0

When

\omega^2 < 0

or

4 \pi G \rho_0 > C_s^2 k^2

we see that \omega is imaginary. This implies that there is an exponential growth or decay factor multiplying the oscillatory e^{i(\vec{k}\cdot \vec{x})} in the solution for \rho_1. That is,

\rho_1 = e^{i(\vec{k} \cdot \vec{x})} e^{-i(^+_- i \omega t)} =

e^{i(\vec{k} \cdot \vec{x})} e^{^+_- \omega t}

Taking the real part, we have:

\rho_1 = \cos(kx)e^{^+ _-\omega t}

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 \omega as a more intuitive form.