Changes between Version 17 and Version 18 of u/erica/JeansTest


Ignore:
Timestamp:
06/30/13 20:20:20 (12 years ago)
Author:
Erica Kaminski
Comment:

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  • u/erica/JeansTest

    v17 v18  
    2323= Determining the form of the perturbation function =
    2424
    25 Recall,
     25Recall, 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
     35This led to a 2nd order ODE in the density perturbation which had the following solution:
    2636
    2737[[latex($\rho_1 = C \cos(kx) e^{[4 \pi G \rho_0 (1-\frac{\lambda_J^2}{\lambda^2})]^{1/2}t}$)]]
    2838
    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 $)]] small, I chose
     39In addition to specifying the sound speed and ambient density for this perturbation (implicit to the Jeans length), we need to specify the wave length of the perturbation. With the desire to keep the ratio [[latex($\lambda_J / \lambda $)]] small, I chose
    3040
    3141[[latex($\lambda = 10 \lambda_J = 350 ~ pc$)]]
     42
     43Using this, we can set the initial (t=0) density inside of the mesh as:
     44
     45[[latex($\rho = \rho_0 + \rho_1 = \rho_0 + cos(kx)$)]]
     46
     47To get the form of the corresponding velocity perturbation, we use the linearized continuity equation:
     48
     49[[latex($\frac{\partial \rho_1}{\partial t} + \rho_0 \nabla \cdot \vec{v_1} = 0$)]]
     50
     51Rearranging,
     52
     53[[latex($\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)$)]]
     56
     57and so,
     58
     59[[latex($v_1 = (\frac{-\Gamma e^{\Gamma t}}{\rho_0})(\frac{1}{k})sin(kx)$)]]
     60
     61Thus, we have the form of the initial (t=0) velocity perturbation:
     62
     63[[latex($v = v_1 = \frac{- \Gamma}{\rho_0 k} sin(kx)$)]]
    3264
    3365= Setting up the problem domain =