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Thinking about evanescent solutions to self-gravity...

So we start with the linearized differential equations

[ \left [ \begin{array}{c} \partial_t{\rho_1} \\ \partial_t{v_1} \\ \partial_{xx}{\phi_1} \\ \end{array} \right ] =\left [ \begin{array}{c} -\rho_0\partial_x v_1 \\ -\frac{c_s^2}{\rho_0}\partial_x \rho_1 -\partial_x \phi_1 \\ 4 \pi G \rho_1 \end{array} \right ] = \left [ \begin{array}{ccc} 0 & -\rho_0\partial_x & 0 \\ -\frac{c_s^2}{\rho_0}\partial_x & 0 & -\partial_x \\ 4 \pi G & 0 & 0 \end{array} \right ] \left [ \begin{array}{c} \rho_1\\ v_1 \\ \phi_1 \\ \end{array} \right ] \]

which we Fourier transform in physical space and solve the third equation for \phi and substitute it into the second.

[ \partial_t \left [ \begin{array}{c} \rho_1 \\ v_1 \\ \end{array} \right ] = \left [ \begin{array}{cc} 0 & -ik\rho_0 \\ -ik\frac{c_s^2}{\rho_0} + \frac{4 \pi i G}{k} & 0 \\ \end{array} \right ] \left [ \begin{array}{c} \rho_1\\ v_1 \\ \end{array} \right ] \]

which gives us a matrix equation for the time evolution operator. Now we just need to find eigenvalues and eigenvectors.

The characteristic equation is

\lambda^2 + k^2 c_s^2 -4 \pi G \rho_0 = 0

and we have

\lambda = \pm \sqrt{4 \pi G \rho_0-k^2 c_s^2}

with eigen vectors

[ \left [ \begin{array}{c} k \rho_0 \\ i \lambda \end{array} \right ] \]

However given the presence of self-gravity we see that \lambda can be real as well as imaginary.

So for stable waves we have \lambda = \pm i\omega where \omega is real and positive. And there are two solutions… (a left and right going gravity-modified sound wave. Gravo-accoustic modes)

[ \left [ \begin{array}{c} \rho_1 \\ v_1 \\ \end{array} \right ] = d\rho \left [ \begin{array}{c} e^{\pm i \omega t} e^{i k x} \\ \mp \frac{\omega}{k} e^{\pm i \omega t} e^{i k x} \\ \end{array} \right ] \]

And if \lambda=\pm\omega where \omega is real - then we have two solutions as well.

[ \left [ \begin{array}{c} \rho_1 \\ v_1 \\ \end{array} \right ] = d\rho \left [ \begin{array}{c} e^{\pm \omega t} e^{i k x} \\ \mp \frac{\omega}{k} e^{\pm \omega t} e^{i k x + i \pi} \\ \end{array} \right ] \]

these correspond to an exponentially increasing collapse and an exponentially decreasing decay (which is just the time reversed version of the 1st) Note the velocity field is phase shifted.

astrobear

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Thinking about evanescent solutions to self-gravity...

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