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.