The Oort constants are commonly defined and employed in the solar neighborhood, but of course their definition could be applied to any location in the galaxy, or for that matter, to any galaxy. In addition to flat rotation curves, many galaxies show a region of "solid body" rotation. Consider a galaxy whose rotation curve is approximated by

A. Derive expressions for A(R) and B(R).

[Hint: what is m in terms of V_f and R₀?]

D. Use the result from (A) to find the orbital and epicyclic frequencies. Sketch a graph of these.

C. Consider a pattern like a bar or spiral arms. Can these be composed of the same stars? (It helps to think of the spiral arm as a solid piece of string winding out from the center.) What happens to such a structure in each regime?

Comment on the identification of Oort's constant A with "shear" and the description of this rotation curve as "solid body".

A. Binney & Merrifield 4.12

B. The current record holder for the most distant quasar is CFHQS J2329-0301 at z = 6.43. Its apparent brightness is J = 21.56. What is the absolute magnitude of this object? (The solar absolute magnitude in J is 3.66. You may ignore the K-correction, but not cosmology: See BM eqn 7.7. Assume and H_o = 72 km/s/Mpc and q_o = 0.) How does the luminosity of this object compare to that of the Milky Way? By how much does this change if instead we assume q_o = 1/2? How far off would we be if we ignored cosmology entirely and just used the usual Euclidean distance modulus?

C. Using the results of (A) & (B), what is the minimum mass of the black hole required to power this object?

Binney & Tremaine 10-8
It may help to know that Ω_o = 0.25.

Build your own model galaxy by combining an exponential disk with an NFW halo.
Plot the rotation curve showing the disk, halo, and total rotation curve.
What parameters do you have to specify to do this?
Tweak it until your model looks realistic...
How easy is it to get a rotation curve in that is flat the outer parts of the disk?