Changes between Version 17 and Version 18 of u/erica/JeansTest
- Timestamp:
- 06/30/13 20:20:20 (12 years ago)
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u/erica/JeansTest
v17 v18 23 23 = Determining the form of the perturbation function = 24 24 25 Recall, 25 Recall, in our linear perturbative analysis on the gravito-hydrodynamic equations, we had: 26 27 [[latex($\rho = \rho_0 + \rho_1$)]] 28 29 [[latex($v = v_1$)]] 30 31 [[latex($P = P_0 + P_1$)]] 32 33 [[latex($\phi = \phi_0 + \phi_1$)]] 34 35 This led to a 2nd order ODE in the density perturbation which had the following solution: 26 36 27 37 [[latex($\rho_1 = C \cos(kx) e^{[4 \pi G \rho_0 (1-\frac{\lambda_J^2}{\lambda^2})]^{1/2}t}$)]] 28 38 29 so in addition to specifying the sound speed and ambient density, we need to specify the wave length of the perturbation. With the desire to keep the ratio [[latex($\lambda_J / \lambda $)]] small, I chose39 In addition to specifying the sound speed and ambient density for this perturbation (implicit to the Jeans length), we need to specify the wave length of the perturbation. With the desire to keep the ratio [[latex($\lambda_J / \lambda $)]] small, I chose 30 40 31 41 [[latex($\lambda = 10 \lambda_J = 350 ~ pc$)]] 42 43 Using this, we can set the initial (t=0) density inside of the mesh as: 44 45 [[latex($\rho = \rho_0 + \rho_1 = \rho_0 + cos(kx)$)]] 46 47 To get the form of the corresponding velocity perturbation, we use the linearized continuity equation: 48 49 [[latex($\frac{\partial \rho_1}{\partial t} + \rho_0 \nabla \cdot \vec{v_1} = 0$)]] 50 51 Rearranging, 52 53 [[latex($\frac{d}{dx} v_1 = -\frac{1}{\rho_0} \frac{\partial \rho_1}{\partial t} = - \frac{1}{\rho_0} e^{ikx}\Gamma e^{\Gamma t}$)]] 54 55 [[latex($\frac{d}{dx} v_1 = - \frac{\Gamma e^{\Gamma t}}{\rho_0} cos(kx)$)]] 56 57 and so, 58 59 [[latex($v_1 = (\frac{-\Gamma e^{\Gamma t}}{\rho_0})(\frac{1}{k})sin(kx)$)]] 60 61 Thus, we have the form of the initial (t=0) velocity perturbation: 62 63 [[latex($v = v_1 = \frac{- \Gamma}{\rho_0 k} sin(kx)$)]] 32 64 33 65 = Setting up the problem domain =