Caption: An animation of galaxy rotation with inset plots showing galaxy rotation curves.
To see the animation---which is cool--click on the image and the next one.
Features:
v=sqrt[GM(r)/r] ,where gravitational constant G = 6.67430(15)*10**(-11) (MKS units), r is the orbital radius, and M(r) is enclosed within a sphere of radius r and distributed with spherically symmetry.
If the mass is centrally concentrated, then v will fall of as 1/sqrt(r). This behavior is, in fact, what one expects for galaxy rotation curves outside of the central regions if there we only the observed baryonic matter.
However, M(r) ∝ r gives v = constant. Since we observe v ≅ constant outside of the central regions, we must have have M(r) ∝∼ r for galaxies. In fact, the rotation velocity plateaus extend well beyond almost all observable baryonic matter (typically to tens of kiloparsecs), and so invisible matter extends that far at least. Note we do, of course, see some baryonic matter (isolated stars, globular clusters, cold neutral atomic hydrogen gas) out to tens of kiloparsecs in order to the measure the galaxy rotation curves that far.
Big Bang nucleosynthesis (BBN) tells us most of the invisible matter we detect from its gravitational effect CANNOT be baryonic matter. We conclude it is exotic matter: an exotic particle or, in a currently less-favored theory, primordial black holes (PBHs). We call this invisible, exotic matter dark matter---which term we have already been using.
In fact, almost all large galaxies are embedded in dark matter halos.
For orbital radii ∼ 10 kpc, the orbital periods will typically be ∼ 200 Myr.
This is shorter than the main-sequence lifetimes of stars of ≤∼ 3 M_☉ (lifetime ≥∼ 370 Myr: Wikipedia: Stellar evolution).
Thus, such stars at orbital radius ∼ 10 kpc (e.g., the Sun) move far from their star formation regions (which typically break up and disperse on time scales of order tens of megayears) in an orbital period. Usually, all traces of their particular star formation regions are erased.