Orbital Velocity due to a Dark Matter Halo

[Dovahkiin]

Homework Statement

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.

Homework 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

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!

 

[turin]

ρ(r) = ρ₀r₀²/r²

M = ∫ρ dV = 4∏∫(from 0 to R) ρ(r) dr

... 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?

Shouldn't your integration measure include a r² (thus cancelling that pesky badly behaved explosion at r=0)?
(doh!)

 

[Dovahkiin]

Shouldn't your integration measure include a r² (thus cancelling that pesky badly behaved explosion at r=0)?
(doh!)

I knew I would have missed something simple ;) Thanks!