# 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.

• I cleaned your question up a bit... please make sure I didn't change it in a bad way (I think you mean $\rho$ where you wrote $p$, and $\rho_0$ where you wrote $P(not)$) – Floris Mar 4 '16 at 0:48
• The density distribution section of your link talks of a "scaling" radius. Could that be what you need? It also seems to have a slightly different form of the density equation. – Floris Mar 4 '16 at 0:49
• Thank you very much for cleaning up my question, my laptop battery died when I was working on this assignment and so I was forced to use my phone. And I think the scaling radius may be the same, however I do not know, I cannot find any mention of scaling radius in my textbook and I cannot seem to find critical radius online. The different form is what makes me think they are different, it may be the virial radius, I am not sure as there seem to be hundreds of forms for this formula and every reasources I've looked at seems to use different notations. – user110240 Mar 4 '16 at 11:07
• Yes - I did notice there were various forms. But of course you could just pick a value and reach an answer expressed in terms of the ratio between the limits of integration and the scaling factor. That gives the answer in closed form "up to a term you can make up" - because I am pretty sure that value is something that is not well pinned down. Maybe gets you a B on the question... – Floris Mar 4 '16 at 12:47
• I'm a little confused. First, that's not the NFW profile. Next, the NFW profile is for dark matter, not the total mass of the matter in a galaxy. Lastly,I'm not sure what resources you've found, but this NED-IPAC document on the NFW profile seems to be pretty clear about what $R_c$ (though they've written it as $r_s$) is supposed to mean and do. – Kyle Kanos Mar 6 '16 at 20:00

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.