Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Demonstration of the jeans equation

  1. Feb 8, 2013 #1
    Hello,

    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 :

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

    [/tex]

    By integrating over all possible velocities, we can write :

    [tex]
    \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)
    [/tex]

    By defining the density and the mean stellar velocity :

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

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

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

    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 :

    [tex]
    \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)
    [/tex]

    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 :

    [tex]
    \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
    [/tex]

    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 : ?

    [tex]
    \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}
    [/tex]

    Any help would be appreciated
     
  2. jcsd
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook

Can you offer guidance or do you also need help?
Draft saved Draft deleted



Similar Discussions: Demonstration of the jeans equation
  1. Jeans mass (Replies: 7)

  2. Jeans instability (Replies: 1)

  3. Jeans mass - Formula (Replies: 2)

Loading...