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| The '''Navarro–Frenk–White profile''' or '''NFW profile''' is a spatial mass distribution of [[dark matter]] fitted to dark matter haloes identified in [[N-body]] simulations by [[Julio Navarro (astrophysicist)|Julio Navarro]], [[Carlos Frenk]] and [[Simon White]].<ref name="NAVARROETAL1996">{{cite journal|author=Navarro, Julio F.; Frenk, Carlos S.; White, Simon D. M.|title=The Structure of Cold Dark Matter Halos|journal=The Astrophysical Journal|volume=463|pages=563|date=May 10, 1996|bibcode=1996ApJ...462..563N|doi=10.1086/177173|arxiv=astro-ph/9508025}}</ref> The NFW profile is one of the most commonly used model profiles for dark matter halos.<ref name="BERTONE2005">{{Cite book|last=Bertone|first=Gianfranco|title=[[Particle Dark Matter|Particle Dark Matter: Observations, Models and Searches]]|publisher=Cambridge University Press|year=2010|pages=762|isbn=978-0-521-76368-4}}</ref>
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| ==Density distribution==
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| <!-- Deleted image removed: [[File:rho nfw.svg|thumb|right|The NFW density profile on a log radius versus density plot.]] -->
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| In the NFW profile, the density of dark matter as a function of radius is given by:
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| <math>
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| \rho (r)=\frac{\rho_0}{\frac{r}{R_s}\left(1~+~\frac{r}{R_s}\right)^2}
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| </math>
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| where ρ<sub>0</sub> and the "scale radius", ''R<sub>s</sub>'', are parameters which vary from halo to halo.
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| The integrated mass inside of some radius ''R<sub>max</sub>'' is
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| <math>
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| M=\int_0^{R_{max}} 4\pi r^2 \rho (r) dr=4\pi \rho_0 R_s^3 \left[
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| \ln\left(\frac{R_s+R_{max}}{R_s}\right)-\frac{R_{max}}{R_s+R_{max}}\right]
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| </math>
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| The total mass is divergent, but it is often useful to take the edge of the halo to
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| be the virial radius, ''R<sub>vir</sub> '',
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| which is related to the "concentration parameter", ''c'', and scale radius via
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| <math>
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| R_{vir}=cR_s
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| </math> | |
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| The virial radius is often referred to as <math> R_{200} </math>, 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
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| <math>
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| M=\int_0^{R_{vir}} 4\pi r^2 \rho (r) dr=4\pi \rho_0 R_s^3 \left[\ln(1+c) - \frac{c}{1+c}\right]
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| </math>
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| The value of ''c'' is roughly 10 or 15 for the Milky Way, and may range from 4 to 40
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| for halos of various sizes.
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| The integral of the ''squared density'' is
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| <math>
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| \int_0^{R_{max}} 4\pi r^2 \rho (r)^2 dr=\frac{4\pi}{3} R_s^3 \rho_0^2
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| \left[1-\frac{R_s^3}{(R_s+R_{max})^3}\right]
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| </math>
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| so that the mean squared density inside of ''R<sub>max</sub>'' is
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| <math>
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| \langle \rho^2 \rangle_{R_{max}}=\frac{R_s^3\rho_0^2}{R_{max}^3}
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| \left[1-\frac{R_s^3}{(R_s+R_{max})^3}\right]
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| </math>
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| which for the virial radius simplifies to
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| <math>
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| \langle \rho^2 \rangle_{R_{vir}}=\frac{\rho_0^2}{c^3}
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| \left[1-\frac{1}{(1+c)^3}\right]
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| \approx \frac{\rho_0^2}{c^3}
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| </math>
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| and the mean squared density inside the scale radius is simply
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| <math>
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| \langle \rho^2 \rangle_{R_s}=\frac{7}{8}\rho_0^2
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| </math>
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| ==Dark matter simulations==
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| The NFW profile is an approximation to the [[Thermodynamic equilibrium|equilibrium]] configuration of dark matter produced in simulations of [[collision]]less dark matter particles by numerous groups of scientists.<ref>{{cite journal|author=Y. P. Jing|title=The Density Profile of Equilibrium and Nonequilibrium Dark Matter Halos|journal=The Astrophysical Journal|date=20 May 2000|volume=535|issue=1|pages=30–36|bibcode=2000ApJ...535...30J|doi=10.1086/308809|arxiv=astro-ph/9901340}}</ref> Before the dark matter [[virial theorem|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.
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| 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.<ref>{{cite journal|author=[[David Merritt|Merritt, David]]; Graham, Alister; Moore, Benjamin; Diemand, Jurg; Terzić, Balsa|title=Empirical Models for Dark Matter Halos|journal=The Astronomical Journal|date=20 December 2006|volume=132|issue=6|pages=2685–2700|bibcode=2006AJ....132.2685M|doi=10.1086/508988|arxiv=astro-ph/0509417|url=http://adsabs.harvard.edu/abs/2006AJ....132.2685M}}</ref><ref>{{cite journal|last1=Merritt|first1=David|authorlink1=David Merritt|last2=et al.|title=A Universal Density Profile for Dark and Luminous Matter?|journal=The Astrophysical Journal|date=May 2005|volume=624|issue=2|pages=L85–L88|bibcode=2005ApJ...624L..85M|doi=10.1086/430636|arxiv=astro-ph/0502515|url=http://adsabs.harvard.edu/abs/2005ApJ...624L..85M}}</ref> 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.
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| ==Observations of halos==
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| The observations of both the [[Milky Way]] and [[Andromeda Galaxy|M31]] may be compatible with the NFW profile for the dark matter halo.<ref name="KLYPIN2002">{{cite journal|author=Klypin, Anatoly; Zhao, HongSheng; Somerville, Rachel S.|title=ΛCDM-based Models for the Milky Way and M31. I. Dynamical Models|date=10 July 2002|journal=The Astrophysical Journal|volume=573|issue=2|pages=597–613|bibcode=2002ApJ...573..597K|doi=10.1086/340656|arxiv=astro-ph/0110390}}</ref> The dark matter profile of smaller galaxies tend to have flatter distributions of dark matter in the central region, known as the [[Cuspy halo problem]].
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| ==References==
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| {{reflist}}
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| {{Dark matter}}
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| {{DEFAULTSORT:Navarro-Frenk-White profile}}
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| [[Category:Dark matter]]
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