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Homework Help: Orbital Velocity due to a Dark Matter Halo

  1. Mar 9, 2012 #1
    1. The problem statement, all variables and given/known data

    I am struggling to calculate the gravitational force from a dark matter halo given that its density is given by ρ(r) = ρ0r02/r2. Once i have found the force i plan to equate it to the centripetal force, F = mv2/r, to find the velocity, v.

    2. Relevant equations

    Density of DM halo: ρ(r) = ρ0r02/r2

    Centripetal Force: F = mv2/r

    Gravitational force: F = GMm/r2

    Mass of DM: M = ∫ρ dV = 4∏∫(from 0 to R) ρ(r) dr

    3. The attempt at a solution

    I (think) i can see what i need to do but using a density ρ(r) = ρ0r02/r2 and trying to integrate from 0 (centre of halo) to R (radial position of the orbiting object) obviously gives and infinite mass?

    However the answer is given to be:

    v2 = 4∏Gρ0r02

    I can't see how you could get to this answer without modifying the density, or am I incorrectly assuming that the force of a mass distribution is the same as the total mass centred at the "origin"?

    Any help will be much appreciated... this question seems like it should be pretty trivial but i'm stumped!
  2. jcsd
  3. Mar 9, 2012 #2


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    Homework Helper

    Shouldn't your integration measure include a r2 (thus cancelling that pesky badly behaved explosion at r=0)?
  4. Mar 12, 2012 #3
    I knew I would have missed something simple ;) Thanks!
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