# Dark matter density calculation

I find that the general dark matter density of the Milky Way is $$6.87 \times 10^9 \: \rm GeV/m^3$$ or $$1.225 \times 10^{-17} \: \rm kg/m^3$$ (by taking the size of the Milky Way and dividing it to the dark matter mass in the Milky Way).

Is this the true value? I couldn't find an online source to prove that its indeed the case.

I want to also ask that the dark matter density is lower in the middle and higher around the edges or other way around. Is there any source that I can read about it?

• Interesting that you’re speaking of dark matter as if it is real. Very heavy confirmation bias on this topic. – Lambda Nov 16 '18 at 16:26

## 2 Answers

The local density of the dark matter halo is believed to be 0.4 GeV/cm^3, or 0.01 solar masses/pc^3. This comes from global fits to the Galactic rotation curve and is consistent with kinematics of local Milky Way stars.

The density should be highest in the center of the galaxy and lower in the outskirts.

There is a review here: http://pdg.lbl.gov/2018/reviews/rpp2018-rev-dark-matter.pdf and you can also look at the references there. The book "Galactic Dynamics" by Binney & Tremaine is also very good.

The number you calculated seems to be 3 orders of magnitude too high (convert meters to cm). If you take a mass of the Galaxy of 10^12 solar masses, and a halo radius of 30 kiloparsecs, it gives exactly 0.01 $$M_\odot/{\rm pc^3}$$.

• Thanks for your info. I take the dark matter mass as $10^{12}$ Solar massess as you said. For the the size of the milky way I used $V=\pi r^2h$ ( I thought the milky way as a cylinder). I take $r=53.5kpc$ and $h=0.6 kpc$. To derive correct density we should use Volume of a sphere formula or something else ?. – Reign Nov 15 '18 at 14:52
• Yes the visible matter is a disk like you said but the dark matter halo is roughly like a sphere. – Eric David Kramer Nov 17 '18 at 18:47

Arthur Morgan said: "I want to also ask that the dark matter density is lower in the middle and higher around the edges or other way around. Is there any source that I can read about it?"

On that subject, the better known and useful dark matter density distribution in a Galaxy , is the Navarro-Frenk-White profile: The Structure of Cold Dark Matter Halos, where you can find:

$$\rho (r) = \frac{\rho_0}{\dfrac{r}{R_s}\left (1+\dfrac{r}{R_s}\right)^2}$$

$$M=\int_0^{R_\max} 4\pi r^2 \rho (r) \, dr=4\pi \rho_0 R_s^3 \left[ \ln\left(\frac{R_s+R_\max}{R_s}\right)-\frac{R_\max}{R_s+R_\max}\right]$$

Best regards