Distribution of dark matter in galactic halos

Often dark matter around galaxies is referred to as a 'halo'. I've seen the galactic rotation curves, but I'm having trouble visualizing how the dark matter is distributed for a typical rotating galaxy.

I'm familiar with the expected relation $v \approx \frac{1}{\sqrt{r}}$ for the orbital speeds at distance $r$ from the center.

I simply cannot imagine what the distribution of additional gravitational sources should look like to make $v$ a constant (even ignoring the center).

Is there a function with respect to $r$ that can describe the distribution of dark matter in galactic halos?

Intuitively it seems at odds that adding more gravitational sources (anywhere) would increase the speeds of outer objects more so than inner objects; the reasoning for this is not clear to me.

Why / how does adding in additional sources of gravity allow for faster orbital velocities farther from the center of a galaxy?

• Just to clarify where I see the problem, don't really want to change the question - if we're not seeing a keplerian orbital speed distribution, why would adding mass change the expected curve? – Xeren Narcy Feb 27 '15 at 5:59
• >> Why / how does adding in additional sources of gravity allow for faster orbital velocities farther from the center of a galaxy? << This is explained by the en.wikipedia.org/wiki/Shell_theorem - only masses inside the shpere contribute to gravitational force, all masses outside the shell cancel out on average. – Yukterez Mar 2 '15 at 10:44
• I see... Thought this might be the case. However it then raises more questions about whether DM orbits the galactic center or not; if it doesn't I'd find it odd that it doesn't get sucked in, and if it does (being more abundant than regular matter) it ought to create orbiting 'clumps' provided it is orbiting in the same direction as regular matter. – Xeren Narcy Mar 2 '15 at 23:44
• Of course it orbits the center, otherwise it would indeed get sucked in. – Yukterez Mar 3 '15 at 0:12

Is there a function with respect to r that can describe the distribution of dark matter in galactic halos?

Yes, it is called the NFW-profile and it looks like this:

$$\rho_{(r)}=\frac{\rho_0}{\frac{r}{r_s}\left(1+\frac{r}{r_s}\right)^2}$$

where $\rho_{(r)}$ is the dark matter density inside the radius $r$, and $\rho_0$ and the scale-radius, $r_s$ are different for different galaxy types and sizes.

To integrate the mass inside the radius, $M_{(r)}$, you get

$$M_{(r)}=\int_0^{r} \{4\cdot \pi\cdot R^2\cdot \rho_{(R)}\} \, \text{d}R$$

The function for whole clusters is approximated by the function

$$M_{(r)}=4 \cdot \pi \cdot \delta \cdot \rho_c \cdot r_s^3 \int_0^{\frac{r}{r_s}} \frac{u^{2-\mu }}{\left(1+x^v\right)^{\lambda }} \, \text{d}x$$

where $\delta$ is the concentration parameter, $\mu$, $v$ and $\lambda$ some numerical values which may vary from cluster to cluster (for examples see this link) and $\rho_c$ is the critical density of the universe given by the equation

$$\rho_c = \frac{3 \cdot H_0^2}{8 \cdot \pi \cdot G} = 8.47\cdot 10^{-27} \, \text{kg}/\text{m}^3$$

with $H_0$ beeing the Hubble-constant and $G$ Newton's constant.

• Ok, after some reading on this I think I'm understanding this a bit better. So the NFW profile basically works backwards from the newtonian / keplerian orbital speed in order to fix it to a constant value; as evidence the DM distribution falls off with r^-2, which also explains the lack of convergence towards r = 0. Is this right? – Xeren Narcy Mar 2 '15 at 5:26
• Also, I realize this may have already been considered (and rejected) but I'm highly suspicious of a r^-2 falloff as having some other explanation, it seems too convenient. Would it not be possible for the central mass of a galaxy to be underestimated combined with Shapiro-like time dilation effects due to the same? That is, the stars seen on the distant side of a galaxy would appear to be delayed giving a relative perceived speed increase to the near side (since gravity too approximately falls off with r^-2)? – Xeren Narcy Mar 2 '15 at 5:33

For a spherically symmetric mass distribution, you can go further than saying $v \propto 1/r^{1/2}$. It is in fact $v \propto (M(r)/r)^{1/2}$, where $M(r)$ is the mass enclosed within an approximately circular orbit.

If $M(r)$ increases as $r$ or faster, then the rotation curve will be flat or increase with radius. The strongly increasing mass within a given radius gives sufficient gravitational force to centripetally accelerate orbiting objects to increasing speeds at larger radii.

In detail:

The most common hypothesised dark matter profile is the Navarro-Frenk-White formulation. $$\rho(r) = \frac{\rho_0 R_s}{r(1 + r/R_s)^2},$$ where $\rho_0$ is normalisation and $R_s$ is a scalelength parameter.

This can be integrated in spherical shells thus: $$M(r) = \int_{0}^{r} 4\pi r^2 \rho(r)\ dr = 4\pi\rho_0 R_s^{3}\left[ \ln (1 + r/R_s) - \frac{r}{r+R_s}\right]$$

It's a bit tricky to immediately see how this complicated function behaves, so I plot it below, using logarithmc scales showing a normalised $\log M(r)$ versus $\log (r/R_s)$. You can perhaps see from this plot that $M(r) \propto r^{\alpha}$, where $\alpha \sim 1.5$ when $r/R_s < 1$, but flattens to $\alpha \sim 1$ for $r \sim 3R_s$ and gets shallower at larger radii.

Thus the rotation curve is flattened or even rising out to about $3R_s$ (depending on the proportion of dark matter, which controlled by $\rho_0$).

For the Milky Way, $R_s \sim 15$ kpc gives a reasonable fit to a flat/slowly rising rotation curve out to 30-40 kpc from the Galactic centre.

We can use orbital mechanics to solve this problem. For all bodies in orbit, the inward force is always balanced by the outward force. Galaxies has mostly flat rotation curves as all stars orbit the galactic center with almost the same orbital speed as shown here:

Dark matter provides almost all the inward force that balances out the centrifugal force. We might calculate the net force of dark matter as a galactic center force like this:

This net force of dark matter results in a velocity that is not dependent on distance to the center, and all stars get the same orbital speed around the galactic center, which is what we observe.

So the net force of dark matter can be described in relation to the galactic center, but this is no normal force as it does not follow inverse square law, and has 1/r instead of 1/r^2. So if it is an actual force it might be some strange flat black hole force acting along the galactic disk.