Need help with determining the total mass using the NFW profile My review assignment has a question that asks us to use the Navarro–Frenk–White (NFW) profile to find total mass in the galaxy using 
$$\rho(R)=\frac{\rho_0}{1+\frac{R}{R_c}}$$ then taking a triple integral of it in spherical coordinates. 
I do not have a problem taking the integral or converting it, but I am struggling with what $R_c$ (the critical radius) should be. I must have missed it in class and cannot quite understand it by looking through other online resources. I feel like it would be a constant but I cannot find it online and am wondering if I need to calculate it somehow. Any help with explaining $R_c$ would be greatly appreciated.
 A: Well, first of all, that's not the NFW profile, instead you should have:
$$\rho(r) = \frac{\rho_0}{\frac{r}{r_s}(1+\frac{r}{r_s})^2}$$
The radius $r_s$ is usually called the scale radius, and is the place where the logarithmic derivative of the density is $-2$. This isn't especially physically meaningful, but is mathematically convenient. The integral is not particularly straightforward, but the result is well known:
$$M(<r) = 4\pi\rho_0r_s^3\left[\ln\left(\frac{r_s+r}{r_s}\right)-\frac{r}{r_s+r}\right]$$
Note that you need to pick a radius $r$ to integrate out to, because attempting to integrate to infinity will give an infinite mass. One common radius to pick is the "virial" radius, which can be defined in a few ways. One is the radius $r_{200{\rm c}}$ where the average enclosed density is 200 times the critical density of the Universe $\rho_{crit}=\frac{3H^2}{8\pi G}$. This radius is related to the scale radius by the concentration parameter $c$ as:
$$r_{200{\rm c}}=cr_s$$
The concentration parameter can be fixed for a halo of a given mass by using a mass concentration relation (e.g. Ludlow et al. 2014). This extra constraint reduces the NFW profile from a model with two free parameters to one with a single free parameter, which you could choose to call the mass of the halo.
