# Dark matter density as a function of radius

1. Jul 5, 2012

### dillont

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
$F=\frac{GMm}{R^{2}}$

$a=\frac{v^{2}}{R}$

3. The attempt at a solution
$\frac{F}{m}=\frac{GM}{R^{2}}$

$4\pi \frac{F}{m}=\frac{4\pi GM}{R^{2}}$

$\frac{v^{2}}{R}=\frac{GM}{R^{2}}$

$dM=4\pi\rho (R)R^{2}dR$

$dM=\frac{v^{2}}{G}dR$

$4\pi\rho (R)R^{2}dR=\frac{v^{2}}{G}dR$

$\rho (R)=\frac{v^{2}}{4\pi GR^{2}}$

Does this look correct to you guys? Thanks.