# Navarro–Frenk–White profile

The Navarro–Frenk–White (NFW) profile is a spatial mass distribution of dark matter fitted to dark matter haloes identified in N-body simulations by Julio Navarro, Carlos Frenk and Simon White.[1] The NFW profile is one of the most commonly used model profiles for dark matter halos.[2]

## Density distribution

In the NFW profile, the density of dark matter as a function of radius is given by:

${\displaystyle \rho (r)={\frac {\rho _{0}}{{\frac {r}{R_{s}}}\left(1~+~{\frac {r}{R_{s}}}\right)^{2}}}}$

where ρ0 and the "scale radius", Rs, are parameters which vary from halo to halo.

The integrated mass within some radius Rmax is

${\displaystyle M=\int _{0}^{R_{\max }}4\pi r^{2}\rho (r)\,dr=4\pi \rho _{0}R_{s}^{3}\left[\ln \left({\frac {R_{s}+R_{\max }}{R_{s}}}\right)-{\frac {R_{\max }}{R_{s}+R_{\max }}}\right]}$

The total mass is divergent, but it is often useful to take the edge of the halo to be the virial radius, Rvir, which is related to the "concentration parameter", c, and scale radius via

${\displaystyle R_{\mathrm {vir} }=cR_{s}}$

The virial radius is often referred to as ${\displaystyle R_{200}}$, and is defined as the radius at which the average density within this radius is 200 times the critical density. In this case, the total mass in the halo is

${\displaystyle M=\int _{0}^{R_{\mathrm {vir} }}4\pi r^{2}\rho (r)\,dr=4\pi \rho _{0}R_{s}^{3}\left[\ln(1+c)-{\frac {c}{1+c}}\right]}$

The value of c is roughly 10 or 15 for the Milky Way, and may range from 4 to 40 for halos of various sizes.

The integral of the squared density is

${\displaystyle \int _{0}^{R_{\max }}4\pi r^{2}\rho (r)^{2}\,dr={\frac {4\pi }{3}}R_{s}^{3}\rho _{0}^{2}\left[1-{\frac {R_{s}^{3}}{(R_{s}+R_{\max })^{3}}}\right]}$

so that the mean squared density inside of Rmax is

${\displaystyle \langle \rho ^{2}\rangle _{R_{\max }}={\frac {R_{s}^{3}\rho _{0}^{2}}{R_{\max }^{3}}}\left[1-{\frac {R_{s}^{3}}{(R_{s}+R_{\max })^{3}}}\right]}$

which for the virial radius simplifies to

${\displaystyle \langle \rho ^{2}\rangle _{R_{\mathrm {vir} }}={\frac {\rho _{0}^{2}}{c^{3}}}\left[1-{\frac {1}{(1+c)^{3}}}\right]\approx {\frac {\rho _{0}^{2}}{c^{3}}}}$

and the mean squared density inside the scale radius is simply

${\displaystyle \langle \rho ^{2}\rangle _{R_{s}}={\frac {7}{8}}\rho _{0}^{2}}$

### Gravitational potential

Solving Poisson's equation gives the gravitational potential

${\displaystyle \Phi (r)=-{\frac {4\pi G\rho _{0}R_{s}^{3}}{r}}\ln \left(1+{\frac {r}{R_{s}}}\right)}$

with the limits ${\displaystyle \lim _{r\to \infty }\Phi =0}$ and ${\displaystyle \lim _{r\to 0}\Phi =-4\pi G\rho _{0}R_{s}^{2}}$ .

### Radius of the maximum circular velocity

The radius of the maximum circular velocity (confusingly sometimes also referred to as ${\displaystyle R_{\max }}$) can be found from the maximum of ${\displaystyle M(r)/r}$ as

${\displaystyle R_{\mathrm {circ} }^{\max }=\alpha R_{s}}$

where ${\displaystyle \alpha \approx 2.16258}$ is the positive root of

${\displaystyle \ln \left(1+\alpha \right)={\frac {\alpha (1+2\alpha )}{(1+\alpha )^{2}}}}$.

## Dark matter simulations

The NFW profile is an approximation to the equilibrium configuration of dark matter produced in simulations of collisionless dark matter particles by numerous groups of scientists.[3] Before the dark matter virializes, the distribution of dark matter deviates from an NFW profile, and significant substructure is observed in simulations both during and after the collapse of the halos.

Alternative models, in particular the Einasto profile, have been shown to represent the dark matter profiles of simulated halos as well as or better than the NFW profile.[4][5] The Einasto profile has a finite (zero) central slope, unlike the NFW profile which has a divergent (infinite) central density. Because of the limited resolution of N-body simulations, it is not yet known which model provides the best description of the central densities of simulated dark-matter halos.

## Observations of halos

The observations of bright galaxies like the Milky Way and M31 may be compatible with the NFW profile,[6] but this is open to debate. The NFW dark matter profile is not consistent with observations of low surface brightness galaxies,[7][8] which have less central mass than predicted. This is known as the cusp-core or cuspy halo problem.