<p><strong>tl;dr</strong> The answer is opinion-based, but choosing an NFW profile probably won't offend anyone.</p>
<h3>Pseudo-isothermal profile</h3>
<p>There are many more or less physically motivated functional forms for the density profile of a dark matter halo.
The pseudo-isothermal sphere is a less unphysical profile than the simple, isothermal profile (which is unphysical in the center and has a diverging integrated mass for <span class="math-container">$r\rightarrow\infty$</span>). It gives good fit to the rotation curves in the optical disk of galaxies, but not to the light distribution.</p>
<h3>NFW profile</h3>
<p>Arguably, the most popular profile is the NFW profile (Navarro, Frenk, &amp; White <a href="https://ui.adsabs.harvard.edu/abs/1996ApJ...462..563N" rel="nofollow noreferrer">1996</a>;<a href="https://ui.adsabs.harvard.edu/abs/1997ApJ...490..493N" rel="nofollow noreferrer">1997</a>) — with a total of ~13,000 citations, these two papers are among the most cited papers in astrophysics ever. Its popularity is probably partly due to its simplicity. It isn't really physically motivated, but simply turned out to give good fits to numerical simulations.</p>
<h3>Einasto profile</h3>
<p>However, subsequent N-body simulations with higher resolution
(
<a href="https://ui.adsabs.harvard.edu/abs/2004MNRAS.349.1039N" rel="nofollow noreferrer">Navarro et al. 2004</a>;
<a href="https://ui.adsabs.harvard.edu/abs/2008MNRAS.388....2H" rel="nofollow noreferrer">Hayashi &amp; White 2008</a>;
<a href="https://ui.adsabs.harvard.edu/abs/2008MNRAS.387..536G" rel="nofollow noreferrer">Gao et al. 2008</a>;
<a href="https://ui.adsabs.harvard.edu/abs/2008MNRAS.391.1685S" rel="nofollow noreferrer">Springel et al. 2008</a>
)
showed small, but systematic, deviations from the NFW profile which are more accurately modeled with an <a href="https://ui.adsabs.harvard.edu/abs/1965TrAlm...5...87E" rel="nofollow noreferrer">Einasto (1965)</a> profile:
<span class="math-container">$$
\rho(r)=\rho_{-2} \exp \left[\frac{-2}{\alpha}\left\{\left(\frac{r}{r_{-2}}\right)^{\alpha}-1\right\}\right],
$$</span>
where <span class="math-container">$r_{-2}$</span> is the radius at which the slope of the profile is <span class="math-container">$-2$</span>, <span class="math-container">$\rho_{-2} = \rho(r_{-2})$</span>, and <span class="math-container">$\alpha$</span> is a fitting parameter with best-fit values that typically lie in the range <span class="math-container">$[0.12,0.25]$</span>, increasing with halo mass.</p>
<p>However, since there isn't much physics here either, and this form is more complicated, the NFW profile seems to remain the profile of choice for most people.</p>
<h3>Other profiles</h3>
<p>Many profiles can be written as double power-laws
<span class="math-container">$$
\rho(r)=\rho_{0}\left(\frac{r}{r_{0}}\right)^{-\gamma}\left[1+\left(\frac{r}{r_{0}}\right)^{\alpha}\right]^{(\gamma-\beta) / \alpha},
$$</span>
where the parameters <span class="math-container">$\alpha$</span>, <span class="math-container">$\beta$</span>, and <span class="math-container">$\gamma$</span> can take a variety of values (Tab. 5.1 from <a href="https://ui.adsabs.harvard.edu/abs/2010gfe..book.....M" rel="nofollow noreferrer">Mo et al. 2010</a>):
<span class="math-container">$$
\begin{array}{lll}
\hline
\hline (\alpha, \beta, \gamma)  &amp; \text{Name} &amp; \text{Reference} \\
\hline(1,3,1) &amp; \text{NFW profile} &amp; \text{Navarro et al. (1997)} \\
(1,4, \gamma) &amp; \text{Dehnen profile} &amp; \text{Dehnen (1993)} \\
(1,4,1) &amp; \text{Hernquist profile} &amp; \text{Hernquist (1990)} \\
(1,4,2) &amp; \text{Jaffe profile} &amp; \text{Jaffe (1983)} \\
(2,2,0) &amp; \text{Modified isothermal sphere} &amp; \text{Sackett &amp; Sparke (1990)} \\
(2,3,0) &amp; \text{Modified Hubble profile} &amp; \text{Binney &amp; Tremaine (1987) }\\
(2,4,0) &amp; \text{Perfect sphere} &amp; \text{de Zeeuw (1985)} \\
(2,5,0) &amp; \text{Plummer sphere} &amp; \text{Plummer (1911)} \\
\hline \hline
\end{array}
$$</span>
At small and large radii <span class="math-container">$\rho$</span> is proportional to <span class="math-container">$r-\gamma$</span> and <span class="math-container">$r-\beta$</span>, respectively, while <span class="math-container">$\alpha$</span> gives the sharpness of the break.</p>
<p>Some profiles are better at describing the inner structure, some are better farther out, some are better when you include gas, etc. What you choose depends on your particular problem, and your mood.</p>

CC BY-SA 4.0
2020-11-13T22:28:09.347Z
2020-11-13T23:01:28.210Z
2020-11-13T23:01:28.210Z
pela
<p>I was surfing the net for knowing the actual density of dark matter halo for a particular M<sub>h</sub>(Mass of the halo) and finally, I got two equations:</p>
<p><a href="https://i.sstatic.net/Oljjh.png" rel="nofollow noreferrer"><img src="https://i.sstatic.net/Oljjh.png" alt="enter image description here" /></a> for pseudo-isothermal halo</p>
<p>and</p>
<p><a href="https://i.sstatic.net/coBwq.png" rel="nofollow noreferrer"><img src="https://i.sstatic.net/coBwq.png" alt="enter image description here" /></a> by NFW-profile</p>
<p>So, with which density distribution of dark matter halo should I go corresponding to any M<sub>h</sub>?</p>

<galaxy><dark-matter><mass><density><matter>
density of dark matter halo
Nitin Kumar Chauhan
# density of dark matter halo

(Asked by: username_0 on 2020-11-13)

I was surfing the net for knowing the actual density of dark matter halo for a particular Mh(Mass of the halo) and finally, I got two equations:

[![enter image description here](https://i.sstatic.net/Oljjh.png)](https://i.sstatic.net/Oljjh.png) for pseudo-isothermal halo

and

[![enter image description here](https://i.sstatic.net/coBwq.png)](https://i.sstatic.net/coBwq.png) by NFW-profile

So, with which density distribution of dark matter halo should I go corresponding to any Mh?

**Tags:** galaxy, dark-matter, mass, density, matter

---

## 1 answers

### Answer by username_1 on 2020-11-13. Score: 2 | Accepted answer

**tl;dr** The answer is opinion-based, but choosing an NFW profile probably won't offend anyone.

### Pseudo-isothermal profile

There are many more or less physically motivated functional forms for the density profile of a dark matter halo.
The pseudo-isothermal sphere is a less unphysical profile than the simple, isothermal profile (which is unphysical in the center and has a diverging integrated mass for $r\rightarrow\infty$). It gives good fit to the rotation curves in the optical disk of galaxies, but not to the light distribution.

### NFW profile

Arguably, the most popular profile is the NFW profile (Navarro, Frenk, & White [1996](https://ui.adsabs.harvard.edu/abs/1996ApJ...462..563N);[1997](https://ui.adsabs.harvard.edu/abs/1997ApJ...490..493N)) — with a total of ~13,000 citations, these two papers are among the most cited papers in astrophysics ever. Its popularity is probably partly due to its simplicity. It isn't really physically motivated, but simply turned out to give good fits to numerical simulations.

### Einasto profile

However, subsequent N-body simulations with higher resolution
(
[Navarro et al. 2004](https://ui.adsabs.harvard.edu/abs/2004MNRAS.349.1039N);
[Hayashi & White 2008](https://ui.adsabs.harvard.edu/abs/2008MNRAS.388....2H);
[Gao et al. 2008](https://ui.adsabs.harvard.edu/abs/2008MNRAS.387..536G);
[Springel et al. 2008](https://ui.adsabs.harvard.edu/abs/2008MNRAS.391.1685S)
)
showed small, but systematic, deviations from the NFW profile which are more accurately modeled with an [Einasto (1965)](https://ui.adsabs.harvard.edu/abs/1965TrAlm...5...87E) profile:
$$
\rho(r)=\rho\_{-2} \exp \left[\frac{-2}{\alpha}\left\{\left(\frac{r}{r\_{-2}}\right)^{\alpha}-1\right\}\right],
$$
where $r\_{-2}$ is the radius at which the slope of the profile is $-2$, $\rho\_{-2} = \rho(r\_{-2})$, and $\alpha$ is a fitting parameter with best-fit values that typically lie in the range $[0.12,0.25]$, increasing with halo mass.

However, since there isn't much physics here either, and this form is more complicated, the NFW profile seems to remain the profile of choice for most people.

### Other profiles

Many profiles can be written as double power-laws
$$
\rho(r)=\rho\_{0}\left(\frac{r}{r\_{0}}\right)^{-\gamma}\left[1+\left(\frac{r}{r\_{0}}\right)^{\alpha}\right]^{(\gamma-\beta) / \alpha},
$$
where the parameters $\alpha$, $\beta$, and $\gamma$ can take a variety of values (Tab. 5.1 from [Mo et al. 2010](https://ui.adsabs.harvard.edu/abs/2010gfe..book.....M)):
$$
\begin{array}{lll}
\hline
\hline (\alpha, \beta, \gamma) & \text{Name} & \text{Reference} \\
\hline(1,3,1) & \text{NFW profile} & \text{Navarro et al. (1997)} \\
(1,4, \gamma) & \text{Dehnen profile} & \text{Dehnen (1993)} \\
(1,4,1) & \text{Hernquist profile} & \text{Hernquist (1990)} \\
(1,4,2) & \text{Jaffe profile} & \text{Jaffe (1983)} \\
(2,2,0) & \text{Modified isothermal sphere} & \text{Sackett & Sparke (1990)} \\
(2,3,0) & \text{Modified Hubble profile} & \text{Binney & Tremaine (1987) }\\
(2,4,0) & \text{Perfect sphere} & \text{de Zeeuw (1985)} \\
(2,5,0) & \text{Plummer sphere} & \text{Plummer (1911)} \\
\hline \hline
\end{array}
$$
At small and large radii $\rho$ is proportional to $r-\gamma$ and $r-\beta$, respectively, while $\alpha$ gives the sharpness of the break.

Some profiles are better at describing the inner structure, some are better farther out, some are better when you include gas, etc. What you choose depends on your particular problem, and your mood.


---

