Dark Matter Halo + Virial theorem Q.

[neutralseer]

A year ago we had a HW problem about galactic rotation curves:

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

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

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

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

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?

 

[Orodruin]

The problem is looking for the galactic rotation curve, not a bunch of galaxies moving randomly as is the case for observations such as that of the Coma cluster. The orbital velocity at radius $r$ is related to the enclosed mass $M(r)$, which appears in the derivative of the potential, not in the potential itself.