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Plummer Sphere, Density from Potential

  1. Feb 26, 2013 #1
    1. The problem statement, all variables and given/known data
    It wants me to get the density function of a Plummer sphere from its gravitational potential.


    2. Relevant equations
    Plummer sphere potential:

    [itex]\Phi (r) = -\frac{GM}{\sqrt{r^{2}+a^{2}}}[/itex]

    where phi is the potential as a function of radius from the mass, M. And a is a scale factor of the model. I think I'm just supposed to take M as a constant here.

    I am supposed to end up with

    [itex]\rho = \frac{3a^{2}}{4\pi}\frac{M}{(r^{2} + a^{2})^{5/2}}[/itex]


    3. The attempt at a solution

    So according to Poisson's equation

    [itex]\nabla ^{2} \Phi = 4\pi G \rho (x)[/itex]

    So to solve for ρ I just took the derivative of Phi twice with respect to r twice and then divided by 4πG

    first derivative got me

    [itex]2rGM(r^{2} + a^{2})^{-1/2}[/itex]

    and then the second derivative got me

    [itex]\frac{GM}{2}\left[2(r^{2} + a^{2})^{-3/2} - (r^{2} + a^{2})^{-1/2}\right][/itex]


    then after a bit of rearranging I have ended up with

    [itex]\frac{M}{8\pi}(r^{2} + a^{2})^{-1/2}(2 - \frac{1}{r^{2} + a^{2}})[/itex]

    I'm not really sure if I'm on the right track...
     
  2. jcsd
  3. Feb 26, 2013 #2
    The Laplacian is not "taking the derivative with respect to r twice". Look up its expression in spherical coordinates (or derive it, if you are not supposed to know).
     
  4. Feb 26, 2013 #3
    ohhh yes you are right, thank you. That explains a lot :redface:
     
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