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Demonstration of the jeans equation

  1. Feb 8, 2013 #1

    I have two problems on an article about the demonstration of the jeans equation. I have 2 problems :

    1*) Firstly, I begin with the collisionless Boltzmann equation :

    \dfrac{\partial\,f}{\partial t}+{\bf v}\,\cdot\,\nabla\,f-\nabla\,\Phi\,\cdot\,\dfrac{\partial\,f}{\partial {\bf v}}=0\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,(1)


    By integrating over all possible velocities, we can write :

    \int\,\dfrac{\partial\,f}{\partial t}\,d^{3}{\bf v}+\int\,v_{i}\,\dfrac{\partial\,f}{\partial x_{i}}\,d^{3}{\bf v}-\dfrac{\partial\,\Phi}{\partial x_{i}}\,\int\,\dfrac{\partial\,f}{\partial v_{i}}\,d^{3}{\bf v}=0\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,(2)

    By defining the density and the mean stellar velocity :

    \nu=\int\,f\,d^{3}{\bf v}\,\,\,\,\,\,\,\,\,\,\,,\,\,\,\,\,\,\,\,\,\bar{v_{i}}=\dfrac{1}{\nu}\,\int\,f\,v_{i}\,d^{3}{\bf v}

    I don't understand how we can get from eq(2) the continuity equation :

    \dfrac{\partial\,\nu}{\partial t}+\dfrac{\partial\,(\nu\,\bar{v_{i}})}{\partial x_{i}}\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,(3)

    How can we make vanish the third term of eq(2) in order to get eq(3) ?

    2*) My second problem : get the jeans equation.

    If I multiply the collisionless Boltzmann equation (1) by [tex]v_{j}[/tex] and integrate over all velocities, we get :

    \dfrac{\partial}{\partial t}\int\,f\,v_{j}\,d^{3}{\bf v}+\int\,v_{i}\,v_{j}\,\dfrac{\partial\,f} {\partial x_{i}}\,d^{3}{\bf v}-\dfrac{\partial\,\Phi}{\partial x_{i}}\,\int\,v_{j}\,\dfrac{\partial\,f}{\partial v_{i}}\,d^{3}{\bf v}=0\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,(4)

    In the article, they say that : "using the fact that [tex]f\rightarrow 0[/tex] for large [tex]v[/tex] and applying the divergence theorem, eq(4) can be written as :

    \dfrac{\partial\,\nu\,\bar{v_{j}}}{\partial t}+
    \dfrac{\partial\,(\nu\,\overline{v_{i}\,v_{j}})}{\partial x_{i}}+\nu\dfrac{\partial\,\Phi}{\partial x_{j}}=0

    How can I simplify the eq(4), especially the third term of eq(4) with the two above assumptions ?

    From the divergence theorem, can I write : ?

    \dfrac{\partial\,\Phi}{\partial x_{i}}\,\int\,v_{j}\,\dfrac{\partial\,f}{\partial v_{i}}\,d^{3} {\bf v}=\dfrac{\partial\,\Phi}{\partial x_{i}}\,\int\,v_{j}\,f\,d^{2}{\bf v}

    Any help would be appreciated
  2. jcsd
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