# NFW Dark Matter Halos and Virial Radius

1. Sep 5, 2014

### zephyr5050

I've been working with NFW Dark Matter Halos recently. This is a particular density model for the halo developed by Navarro, Frenk, & White (NFW). The density structure has the form
$$\rho (r) = \frac{\delta_c \rho_c}{(r/r_s)(1+r/r_s)^2}$$
where
$$\delta_c = \frac{200}{3} \frac{c^3}{ln(1+c)-c/(1+c)}$$
$$r_s = r_{200}/c$$
and $\rho_c$ is the critical density of the universe (as a function of redshift). The parameter $r_{200}$ is the virial radius which is defined as the radius at which the mass density of the halo is $200\rho_c$.

Now we can't really talk about the mass of this halo because the integral from 0 to $\infty$ diverges. Instead, we use the fiducial radius $r_{200}$ and define the quantity $M_{200}$ to be the mass inside the radius $r_{200}$. It can be shown that
$$M_{200} = \frac{800\pi}{3}\rho_c r_{200}^3$$
While all this makes sense to me, there's one thing that I don't understand here. Where does this 200 come from? Why say $r_{200} \equiv 200 \rho_c$? Is there any logic to this, is it historical, arbitrary? What's going on here?

2. Sep 5, 2014

### phyzguy

I think it's basically historical and arbitrary. You have to draw a line somewhere, and the accepted way to do it is when the average density of the cluster falls to 200X the critical density. It's not universal, however. You will find papers referring to R500, R100, R150, etc., all defined in the same way, but with different multipliers.

3. Sep 5, 2014

### zephyr5050

Do you happen to know the paper which proposed this commonly accepted line? If any?

4. Sep 5, 2014