= Navarro--Frenk--White profile
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{title2=NFW profile}
{wiki=Navarro–Frenk–White_profile}
= NFW profile
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The <Navarro--Frenk--White profile> is an empirical model for the spherically averaged <mass density> of collisionless <dark matter halos>. <Julio F. Navarro>, <Carlos S. Frenk> and <Simon D. M. White> found that the same two-parameter shape fitted halos in their hierarchical-clustering <N-body simulations> over a broad resolved radial interval.<A Universal Density Profile from Hierarchical Clustering by Julio F. Navarro and others>[Abstract and density-profile analysis]{ref}
Writing $x=r/r_s$, the profile is
$$
\rho(r)=\frac{\rho_s}{x(1+x)^2},
$$
with positive scale radius $r_s$ and characteristic density $\rho_s$. Its inner <density> cusp scales as $r^{-1}$ and its outer tail as $r^{-3}$. Integrating $4\pi r^2\rho$ gives the enclosed <mass>
$$
M(<r)=4\pi\rho_sr_s^3\left[\log(1+x)-\frac{x}{1+x}\right].
$$
Although the central <density> diverges, the enclosed <mass> tends to zero as $r^2$. At large radius the total <mass> diverges logarithmically, so a quoted halo <mass> requires a finite boundary convention, such as a specified mean enclosed overdensity. The <halo concentration> compares that boundary radius with $r_s$. The simulation fit is useful for consistent halo descriptions and <gravitational lensing> calculations; its empirical success does not require every halo or baryon-modified inner region to follow it exactly.
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