Addendum 11: Milky Way Galaxy: rotation curves, and dark matter (Chap 19)
First define some astronomical and physics constants and dimensional units that may be needed below
1. Finding the mass of a black hole using orbits. The mass of any central massive object, including a black hole, can be determined if the period and semi-major axis of a (much less massive) orbiting object can be observed, using Kepler's 3rd law. Since observations normally involve measuring angular separations, to obtain the semi-major axis involves the additional step of converting from angular to linear size using the distance via the small angle formula).
Example: The central region of the spiral galaxy NGC 4395 (distance 4.3 Mpc) has a massive black hole. Gas is seen orbiting around the central object with a velocity of 30 km/s at an angular diameter 0.3 arcsec. Estimate the mass of the black hole.
This mass is considered quite small for galactic black hole, and has been called the 'runt' of the galactic litter! See (click on):
2. Rotation curves of galaxies. The rotation curve of a galaxy is a plot of the circular speed versus distance from the galaxy's center. There are two characteristic portions of the curve: the inner 'solid body' part in which the mass interior to a given radius is growing rapidly, and the 'Keplerian' part in which most of the mass is contained.

For a uniform spherical distribution of matter the mass-radius relation is:
where Mgal is the effective galaxy mass, and ro is the characteristic radius beyond which there is very little mass, i.e. most of the mass is contained within ro. The velocity-radius relation is then:
For r > r0, the orbit is derived from Kepler's 3rd law:
Example: Plot the expected rotation curve for a galaxy with a radius 40 kpc, a total mass 109 solar masses, most of which is contained in the inner 2 kpc.
3. Dark Matter inferred from Rotation Curves. It turns out that almost no galaxies have rotation curves that resemble the one shown in the plot above. They do have the expected 'solid body' part, but the 'Keplerian' part is replaced by a nearly flat line (or slightly rising) out to very large distances, as shown below for the MW galaxy.
This implies that there is a large amount of 'unseen' material acting gravitationally, but which is undetected (visually, or at any other wavelength), hence the term 'dark matter'. The exact distribution of the dark matter is unknown, but a power-law like the following seems to fit the observed rotation curves:
Example: For the MW galaxy find approximate values of Mdm0 and rdm that will fit the observed rotation curve.

Solution: The total rotation curve is the sum of the 'normal' curve from known matter (section 2) and he dark matter contribution:
adjust these for best fit to observed rotation curve above
Note that:

1. There is 10x as much dark matter as 'normal' matter!
2. The mass distribution is very large (rdm = 25 kpc).

One of the most important question in all of astrophysics is:

What is the nature of this dark matter?
A Google search on dark matter will retrieve 1000's of web pages.
A couple of good web resources (clicking should work):