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Dark Matter Halo + Virial theorem Q.

  1. Jul 16, 2008 #1
    A year ago we had a HW problem about galactic rotation curves:

    If the dark matter density is
    [tex] \rho= \frac{\rho_0}{1+\left(\frac{r}{r_0}\right)^2}[/tex],
    Then how does velocity depend radius at large r ([itex] r>>r_o[/itex])?

    You want to use the virial theorem here, so you calculate M(r) and then finally calculate the potential:
    [tex] \phi(r)=\int_{-\infty}^{r} \frac{k}{s}ds=\infty[/tex]
    with k = constant.

    Oops, phi(r) diverges logarithmically. Oh well, you just have to set the zero point somewhere else besides infinity, say at [itex] a > r_0 [/itex], so we get,
    [tex] \phi(r)=k\ln{\frac{r}{a}}[/tex]

    Now, since the force between any two particles is an inverse square law, the virial theorem says:
    [itex] 2KE+PE=0 [/itex]. Thus we can write:
    [tex] v=\sqrt{-\phi(r)} =\sqrt{ -k\ln{\frac{r}{a}}} [/tex].

    Obviously, something is wrong with this answer. An observable quantity like the velocity, ‘v’, cannot depend on an arbitrarily chosen constant, ‘a’. Where did my logic go south?
     
  2. jcsd
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