Linearization of the Fluid Equations

The system of equations to describe a fluid with pressure and self-gravity are the following:

\frac{\partial \rho}{\partial t} + \nabla \cdot (\rho \vec{v} ) = 0 ~(continuity)

\frac{\partial \vec{v}}{\partial t} + \vec{v} \cdot \nabla \vec{v} = - \frac{\nabla P}{\rho} - \nabla \phi ~(Euler)

\nabla ^ 2 \phi = 4 \pi G \rho ~(Poisson)

A equilibrium solution to this non-linear system of PDE's is easily given by:

\rho = \rho_0 = ~constant

v = 0

P = P(\rho) ~(constant ~entropy ~S)

However, note that this is the source of the infamous 'Jeans Swindle', as plugging this equilibrium solution into the Euler equation implies that phi is a constant (\phi = \phi_0), in contradiction with Poisson's equation. Ignoring this though, we can do a standard linear analysis on these equations to check for stability of small perturbations. As usual, a linear analysis 1) transforms the original non-linear system into a solvable linear system, and 2) provides valuable insight into the basic physics within the fluid.

Inducing a small linear perturbation to the equilibrium solution gives the following:

\rho = \rho_0 + \rho_1

P = P_0 + P_1

v = v_1

\phi = \phi_0

Plugging these solutions into the above system of equations leads to the Jeans Instability as described as follows.

The Jeans Instability

Linear perturbative analyis on the self-gravitating fluid equations under the assumption of infinite, constant density (\rho = \rho_0), static (v = 0) background 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

\omega is real, and the density perturbation has the form:

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

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

\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.

The Jeans Length

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

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

Substituting 2 \pi / \lambda for k and solving for \lambda we have

\lambda < \frac{2 \pi C_s}{(4 \pi G \rho_0)^{1/2}}

That is, for wavelengths smaller than 2 \pi C_s / (4 \pi G \rho_0), the perturbation is stable to collapse. You can think of this wavelength as length scales over which the average \rho = \rho_0. The critical length is known as the Jeans length and is given by:

\lambda_J = Cs(\frac{\pi}{G \rho_0})^{1/2}

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 \omega into a more intuitive form.

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

\omega = ^+_- [-4 \pi G \rho_0(1-\frac{C_s^2 k^2}{4 \pi G \rho_0})]^{1/2} \Rightarrow

\omega = ^+ _- i[4 \pi G \rho_0 (1 - \frac{C_s^2 k^2}{4 \pi G \rho_0})]^{1/2} \Rightarrow

\omega = ^+ _- i[4 \pi G \rho_0 (1 - \frac{\lambda_J ^2}{\lambda^2})]^{1/2}

Taking the positive root and plugging into \rho_1, we see

\rho_1 \propto e^{-i i [4 \pi G \rho_0 (1-\frac{\lambda_J^2}{\lambda^2})]^{1/2}t}

which gives

\rho_1 = C \cos(kx) e^{[4 \pi G \rho_0 (1-\frac{\lambda_J^2}{\lambda^2})]^{1/2}t}

Thus we have found the explicit form of the density perturbation. For \lambda > \lambda_J , the density perturbation will be unstable and grow exponentially with time. For \lambda < \lambda_J, the density perturbation is oscillatory, and will travel through the medium as a sound wave. (Given this analytical expression, one can compute the growth rate explicitly with AstroBEAR. That is, by seeding the grid with this density perturbation with self gravity turned off, one can visualize this perturbation function over time. To test the code's self-gravity, one could seed the grid with the density perturbation without the exponential factor (i.e. only sinusoidal density perturbation), turn self-gravity on, and see if the density grows with a rate that matches that of the analytical solution.)

Now, the growth rate of these modes is given by

\Gamma = [4 \pi G \rho_0 (1-\frac{\lambda_J^2}{\lambda^2})]^{1/2}

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

\tau = \frac{1}{\Gamma}

High lambda limit

From

\rho_1 = C \cos(kx) e^{[4 \pi G \rho_0 (1-\frac{\lambda_J^2}{\lambda^2})]^{1/2}t}

it is easy to see that in the limit

\lambda \rightarrow \infty

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

\tau = [\frac{1}{4 \pi G \rho0}]^{1/2} \sim t_{ff}

where t_{ff} is the freefall time for a spherical region of uniform gas. It is interesting that if you calculated the proportionality constant, you'd see that \tau \sim \frac{1}{2} t_{ff}.

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.

References

​Drexel Course Notes

​Cosmology Derivation