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Dark matter density as a function of radius

  1. Jul 5, 2012 #1
    1. The problem statement, all variables and given/known data
    Evidence for dark matter comes from “flat” rotation curves of galaxies. Assume
    that the observed matter moves in circular orbits about the center of the galaxy
    and that the velocity of the matter as a function of the radius v(r) is a constant.
    Also assume the motion of the observed matter is purely due to the gravity of
    the dark matter (mass of luminous matter is negligible) and the dark matter is
    distributed with spherical symmetry about the center of the galaxy. What is the
    density ρ(r) of the dark matter as a function of radius?​

    2. Relevant equations
    [itex]F=\frac{GMm}{R^{2}}[/itex]

    [itex]a=\frac{v^{2}}{R}[/itex]

    3. The attempt at a solution
    [itex]\frac{F}{m}=\frac{GM}{R^{2}}[/itex]

    [itex]4\pi \frac{F}{m}=\frac{4\pi GM}{R^{2}}[/itex]

    [itex]\frac{v^{2}}{R}=\frac{GM}{R^{2}}[/itex]

    [itex]dM=4\pi\rho (R)R^{2}dR[/itex]

    [itex]dM=\frac{v^{2}}{G}dR[/itex]

    [itex]4\pi\rho (R)R^{2}dR=\frac{v^{2}}{G}dR[/itex]

    [itex]\rho (R)=\frac{v^{2}}{4\pi GR^{2}}[/itex]

    Does this look correct to you guys? Thanks.
     
  2. jcsd
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