Changes between Version 1 and Version 2 of u/erica/JeansInstability
 Timestamp:
 06/20/13 19:53:44 (11 years ago)
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u/erica/JeansInstability
v1 v2 5 5 [[latex($\frac{\partial^2\rho_1}{\partial t^2}  C_s^2 \triangledown^2 \rho_1  4\pi G \rho_0 \rho_1 = 0$)]] 6 6 7 where 8 9 [[latex($\rho_1 \equiv density ~perturbation$)]] 10 [[latex($C_s \equiv sound ~speed$)]] 7 where [[latex($C_s$)]] is the sound speed. 11 8 12 9 This is a wave equation, so we propose a solution of the form: … … 14 11 [[latex($ \rho_1(\vec{x},t) = C e^{i(\vec{k} \cdot \vec{x}  \omega t)}$)]] 15 12 16 Plugging this into the wave equation above gives the following dispersion relation:13 where the wave number k is given by [[latex($k = 2 \pi /\lambda$)]], with [[latex($\lambda$)]] specifying the perturbation oscillation wave length, and [[latex($\omega$)]] is the angular frequency of oscillation given by [[latex($\omega = 2 \pi f$)]]. Plugging this into the wave equation above gives the following dispersion relation: 17 14 18 15 [[latex($\omega ^2 = C_s^2 k^2  4 \pi G \rho_0$)]]