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Poisson Problem - Gravitation

  1. Feb 11, 2005 #1
    If I have an infinite slab of incompressible self-gravitating fluid of density rho within the region |z|<a, and I am asked to find the potential both inside and outside the slab, where should I start?
     
  2. jcsd
  3. Feb 11, 2005 #2

    dextercioby

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    With writing the (differential) equations which account for the dynamics...??

    Daniel.
     
  4. Feb 11, 2005 #3
    Do you mean by "potential" the gravitational potential and by "rho" the mass density?
     
  5. Feb 11, 2005 #4

    dextercioby

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    Sure he does.

    Daniel.
     
  6. Feb 11, 2005 #5
    A very fast solution to this problem can be obtained using a Gauss-like law for the gravitational field (it can be demostrated by direct integration of the Poisson equation and using the divergence theorem). The "gravitational flux" through a closed surface must equal the total mass inside the surface times gravitational constant.
    [tex]\oint_S \vec{\Gamma}\cdot d\vec{S}=-\gamma \int \rho dv [/tex]
    Then, if you know [tex]\Gamma(z)[/tex], the potential is just
    [tex]V=-\int \Gamma dz[/tex]
    (for the integration constant you can impose V(0)=0)
    You can choose a cylindrical gaussian surface with its axis parallel to Oz and play with this theorem. For this cylinder, the total flux is [tex]2\Gamma S[/tex] (S is basis area)

    I think [tex]\Gamma[/tex] will vary linearly from z=0 to z=a and would be uniform for z>a. So the potential will be quadratic and linear respectively.....but you must verify that.....
     
    Last edited: Feb 11, 2005
  7. Feb 11, 2005 #6

    dextercioby

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    Your analysis would be okay,if the "infinite slab of incompressible self-gravitating fluid of density rho" would not mean what i think it does:namely a fluid to which u have to apply not only the Poisson equation (for a gravitostatic field),but also Euler's equations and the continuity of mass (for an incompressible fluid).You'd have then 5 equations with 5 unknowns:the gravity potential,the velocity field and the density field...
    It would be really nasty,indeed.

    Daniel.
     
  8. Feb 14, 2005 #7
    Thanks Clive. But wouldn't using Gauss Law introduce a factor of 4*Pi? The answers have no 4*Pi in it. And do I have to consider also the pressure and the boundary condition?

    By the way, here's the answers provided:
    Code (Text):
    V = (1/2)G rho (z)^2             |z|<a
                 = G rho a (|z| - (1/2)a)     |z|>a
     
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