Dark matter from speed of stars in outer galaxy

My understanding is that the first evidence of dark matter came from measurements of the orbital velocity of stars in the galaxy. In theory, the further out from the centre, the slower the stars should move (following KeplerIII?)

But measurements found that no matter how far from the centre the stars were, they all had around the same orbital speed (~200 km/sec).

...which meant there was a bunch of missing mass, dark matter.

Question 1: I assume that if dark matter was distributed evenly throughout the galaxy, then orbital speed would still follow Kepler. Is that right? So, do we assume there's more dark matter in the outer galaxy than in the inner galaxy? (Or vice versa, pardon my maths.)

Question 2: If there's precisely the right amount and distribution of dark matter to cause every part of the total mass to be rotating at the same speed, wouldn't that be a mighty coincidence?

• What research have you done on this topic? – sammy gerbil Oct 22 '16 at 2:23

Kepler's laws hold only for bodies orbiting what can be approximated as point masses. On scales as large as planetary motion, even stars can be approximated as point masses. Galaxies, however, can not, and so Kepler's laws don't hold.

In fact, there isn't a uniform density distribution of dark matter; generally, data is fitted using a Navarro-Frenk-White density profile1,2: $$\rho(r)=\frac{\rho_0}{\frac{r}{R_s}\left(1+\frac{r}{R_s}\right)^2}$$ for some density parameter $\rho_0$ (not the central density) and a scale length $R_s$; this works well in most areas, although it fails at the galactic center, where $\rho\to\infty$ as $r\to0$. Also, note that this is the wrong density distribution to yield Keplerian behavior. Furthermore, this profile shows that for most $r\ll R_s$, $\rho(r)\sim r^{-1}$, and in all cases, the density decreases as you get further out from the center. The dark matter halo is not uniformly distributed.

Regarding your statement about constant velocity, I'd recommend looking at some rotation curves extrapolated for many galaxies. It's true that after a certain radius, the curve seems to be relatively flat, but there are actually plenty of irregularities and oscillations, and in some cases, the velocities even tail off a little at the outer reaches of the galaxy. There's enough variation - and certainly not an artefact of experimental error - to cast aside any doubts that there's a giant coincidence here; I challenge you to find a curve which is perfectly flat after a certain peak.

You also may be interested in answers by Kyle Oman and Rob Jeffries.

1. For more information, start reading the original papers, e.g. Navarro et al. (1996).
2. The Einasto profile is another popular choice; it uses an exponentially decreasing radial density model with finite central density.
• That's really interesting, thanks HDE. I'd assumed that Kepler's laws would hold because the net gravitational pull on any particular star would only depend on the mass 'inside' its own orbit, and that would be the same as if it was a point mass. But interesting to learn that's no so. Every day is a day at aschool! I have seen those charts of orbital speed vs distance, and seen the little wiggles and irregularities. I just thought it was still a bit of a coincidence that the chart was so flat. I'd have thought that of all the different dark-matter distributions, the one that kept things so – Errol Hunt Oct 19 '16 at 22:15
• Contd... I have seen those charts of orbital speed vs distance, and seen the little wiggles and irregularities. I just thought it was still a bit of a coincidence that the chart was so flat. I'd have thought that of all the different dark-matter distributions, the one that kept things so flat would be a rare thing. Thanks or the note on dark-matter profiles. That's fascinating. I followed the link but wasn't sure how the NFW profile was actually derived. (I assume it's not just a fit to measured speeds.)But I did learn it was a sphere, not a disk. And, not rotating? – Errol Hunt Oct 19 '16 at 22:26
• @ErrolHunt It's correct that it's a sphere; a galaxy's dark matter halo is spherical, and so so is the NFW profile. As for the derivations, I'd recommend looking at some of the original papers; start perhaps with this one. – HDE 226868 Oct 19 '16 at 22:30
• @ErrolHunt The interesting thing about rotation is that the halo doesn't rotate with the rest of the galaxy, which is why non-rotating models should work decently. – HDE 226868 Oct 20 '16 at 19:15
• Thanks HDE. Actually yes I remember now someone explaining that the DM halo doesn't rotate. And that's why some scientists think we might be able to measure a difference 6-months apart between when the Earth's orbital speed adds to Sun's movement around galaxy, and 6M later when they counter act. – Errol Hunt Oct 24 '16 at 2:53