<p>You are missing one last ingredient: the correlation between the virial radius $r_{\mathrm{vir}}$ and the virial mass $M_{\mathrm{vir}}$. The idea is that the orbital time scale of a particle orbiting in the halo becomes roughly equal to the age of the universe at $r_{\mathrm{vir}}$. This implies a relation between the (mean) density of the halo and of the universe:
$$
    \bar{\rho}_{\mathrm{vir}} \equiv \frac{3M_{\mathrm{vir}}}{4\pi r_{\mathrm{vir}}^3} = \Delta \rho_{\mathrm{crit}}(z)
$$
where $\bar{\rho}_{\mathrm{vir}}$ is the halo's mean density interior of $r_{\mathrm{vir}}$, $\rho_{\mathrm{crit}}$ is the critical density of the universe (as function of redshift $z$), and $\Delta$ a parameter that depends on the cosmology. Often $\Delta=200$ is used and the corresponding quantities labeled $r_{200}$ and $M_{200}$ instead. See also <a href="https://en.wikipedia.org/wiki/Virial_mass" rel="nofollow noreferrer">this Wikipedia page</a>.</p>

<p>So the way to construct a halo is as follows:</p>

<ol>
<li>Pick $M_{\mathrm{vir}}$ (or $M_{200}$) and redshift $z$ (and your favorite cosmology)</li>
<li>Use above relation to find $r_{\mathrm{vir}}$ (or $r_{200}$)</li>
<li>Use the relation from Napolitano et al. (2005), or equivalent, to find $c$ and hence $r_s$</li>
<li>Obtain $\rho_{\mathrm{DM}}$ (if you still need it)</li>
</ol>

CC BY-SA 3.0
2017-05-27T11:12:05.150Z
2017-05-27T13:24:41.320Z
2017-05-27T13:24:41.320Z
Walter
<p>This should be a straightforward exercise, but I'm struggling. Help is much appreciated</p>

<p>I would like to parameterize a Navarro-Frenk-White profile with one parameter, using the correlation between total mass and the concentration. </p>

<p>As a starting point, I have the density in terms of two parameters: the total dark matter mass inside the virial radius $r_{vir}$, and the concentration parameter $c=r_{vir}/r_{s}$:</p>

<p>$\rho_{DM}(r) = \frac{M_{DM}}{4\pi A(c)}\frac{1}{r(r_{s}+r)^{2}}$</p>

<p>where the function $A(c)$ is given by</p>

<p>$A(c) = \ln{(1+c)} - \frac{c}{1+c} $</p>

<p>Using the correlation between $M_{DM}$ and $c$ (from e.g. Napolitano et al. 2005)</p>

<p>$c(M_{DM}) \approx 9.195 \left(\frac{M_{DM}}{h^{-1}10^{12} M_{\odot}} \right)^{-0.094}$</p>

<p>it should then in principle be possible to parameterize $\rho_{DM}$ as a function of $M_{DM}$ only. This is how they do it in e.g. Williams et al. 2009.</p>

<p>However, I'm not managing to reproduce that. Sure, given the correlation above, $M_{DM}$ determines $A(c)$, but won't $\rho_{DM}$ still depend on $r_{s}$? </p>

CC BY-SA 3.0
2017-05-26T14:38:11.153Z
Can you provide the equation you end up getting and the equation from Williams et al. 2009 you're trying to match?
zephyr
CC BY-SA 3.0
2017-05-26T22:55:59.397Z
What is $r_s$? Is it small enough to be neglected?
probably_someone
CC BY-SA 3.0
2017-05-27T16:27:04.283Z
@zephyr, Williams et al. mention the equations that I state above allow them to rewrite it, but they don't give the rewritten expression. 
I've solved it by now actually, doing essentially what Walter below also says. Thanks!
user1991
CC BY-SA 3.0
2017-05-27T16:28:23.830Z
@probably_someone $r_{s}$ is the scale radius of the NFW profile, the radius at which the functional behaviour changes significantly. It is not negligible,  no. Thanks!
user1991
<dark-matter>
1-parameter NFW dark matter profile
user1991
# 1-parameter NFW dark matter profile

(Asked by: username_0 on 2017-05-26)

This should be a straightforward exercise, but I'm struggling. Help is much appreciated

I would like to parameterize a Navarro-Frenk-White profile with one parameter, using the correlation between total mass and the concentration.

As a starting point, I have the density in terms of two parameters: the total dark matter mass inside the virial radius $r\_{vir}$, and the concentration parameter $c=r\_{vir}/r\_{s}$:

$\rho\_{DM}(r) = \frac{M\_{DM}}{4\pi A(c)}\frac{1}{r(r\_{s}+r)^{2}}$

where the function $A(c)$ is given by

$A(c) = \ln{(1+c)} - \frac{c}{1+c} $

Using the correlation between $M\_{DM}$ and $c$ (from e.g. Napolitano et al. 2005)

$c(M\_{DM}) \approx 9.195 \left(\frac{M\_{DM}}{h^{-1}10^{12} M\_{\odot}} \right)^{-0.094}$

it should then in principle be possible to parameterize $\rho\_{DM}$ as a function of $M\_{DM}$ only. This is how they do it in e.g. Williams et al. 2009.

However, I'm not managing to reproduce that. Sure, given the correlation above, $M\_{DM}$ determines $A(c)$, but won't $\rho\_{DM}$ still depend on $r\_{s}$?

**Tags:** dark-matter
> **Comments:**
> **username_1**: Can you provide the equation you end up getting and the equation from Williams et al. 2009 you're trying to match?  
> **username_2**: What is $r_s$? Is it small enough to be neglected?  
> **username_0**: @zephyr, Williams et al. mention the equations that I state above allow them to rewrite it, but they don't give the rewritten expression.  I've solved it by now actually, doing essentially what Walter below also says. Thanks!  
> **username_0**: @probably_someone $r_{s}$ is the scale radius of the NFW profile, the radius at which the functional behaviour changes significantly. It is not negligible,  no. Thanks!  

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## 1 answers

### Answer by username_3 on 2017-05-27. Score: 4 | Accepted answer

You are missing one last ingredient: the correlation between the virial radius $r\_{\mathrm{vir}}$ and the virial mass $M\_{\mathrm{vir}}$. The idea is that the orbital time scale of a particle orbiting in the halo becomes roughly equal to the age of the universe at $r\_{\mathrm{vir}}$. This implies a relation between the (mean) density of the halo and of the universe:
$$
\bar{\rho}\_{\mathrm{vir}} \equiv \frac{3M\_{\mathrm{vir}}}{4\pi r\_{\mathrm{vir}}^3} = \Delta \rho\_{\mathrm{crit}}(z)
$$
where $\bar{\rho}\_{\mathrm{vir}}$ is the halo's mean density interior of $r\_{\mathrm{vir}}$, $\rho\_{\mathrm{crit}}$ is the critical density of the universe (as function of redshift $z$), and $\Delta$ a parameter that depends on the cosmology. Often $\Delta=200$ is used and the corresponding quantities labeled $r\_{200}$ and $M\_{200}$ instead. See also [this Wikipedia page](https://en.wikipedia.org/wiki/Virial_mass).

So the way to construct a halo is as follows:

1. Pick $M\_{\mathrm{vir}}$ (or $M\_{200}$) and redshift $z$ (and your favorite cosmology)
2. Use above relation to find $r\_{\mathrm{vir}}$ (or $r\_{200}$)
3. Use the relation from Napolitano et al. (2005), or equivalent, to find $c$ and hence $r\_s$
4. Obtain $\rho\_{\mathrm{DM}}$ (if you still need it)


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