Milky Way Kinematics

1. our distance from the Galactic Center is R₀ = 8 kpc,
2. the proper motion of Sgr A* is Ω = 6.38 milliarcseconds/year, and
3. the measured radial and tangential velocity dispersions of local stars are related by σ_X² = 2.2 σ_Y²

What are the values of

1. Our orbital velocity around the center of the Milky Way?
2. The Oort A & B constants? Is the rotation curve rising or falling?
3. The period of our orbit around the Milky Way?
4. The period of one solar epicycle?
5. The mass interior to the solar circle? (You may assume a spherical mass distribution.)

ASTR 433 only: Consider the effects of solar motion on the above questions, given that the peculiar velocity of the sun exceeds that of the LSR in the Y direction by somewhere between 5 and 15 km/s. You may express your answer as a percentage change to each quantity for each of the two limits on V_⊙.
While this might sound like a minor effect, it is the second largest uncertainty in establishing the scale of our Galaxy (the first being our distance from the Galactic center).

Generalized Oort "Constants"

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

1. Derive expressions for A(R) and B(R).
   [Hint: what is μ in terms of V_f and R_f for the rotation curve to be continuous at R_f?]
   Comment on the identification of Oort's constant A with "shear" and the description of this rotation curve as "solid body" for R < R_f.
2. Use the result from (1) to find expressions for the orbital and epicyclic frequencies.
3. Consider an mode=2 pattern like a bar. Graph frequency as a function of radius to illustrate the locations of corotation and the Linblad resonances for illustrative pattern speeds.

Related reading: Section 6.3, Binney & Tremaine (2nd edition).

Mass Models

First, subtract off the rotation due to the luminous stars and gas. Whatever is left we call dark matter.

[Hint: Velocities add and subtract in quadrature. Real data are messy. Don't get hung up on tiny inconsistencies: deal with them.]

1. Fit the data with both a pseudo-isothermal halo and a NFW halo. Plot the results. Does either model provide a better fit?
2. In the NFW case, what is the escape velocity for a star at r = r_s?
   [Recall: V_esc² = 2 |Φ(r)|.]
3. Does the escape velocity seem well determined? Why did I not ask for the total mass of each halo? Or for the escape velocity of the pseudo-isothermal halo?

ASTR 433 only: A specific choice of stellar mass-to-light ratio is provided in the data file. Repeat the exercise assuming instead that M/L = 0.5 for both bulge and disk components.

[Hint: How does velocity scale with M/L?]

You may write your own code to fit the data, or use the fortran versions here: Pseduoisothermal halo code | NFW halo code. Note that getting fits with these codes to converge usually requires a reasonable initial guess for the parameters. The NFW code fits in terms of concentration and R₂₀₀ (see the NFW density profile).

Tully-Fisher

Derive an explanation for the Tully-Fisher relation.
Be sure to state explicitly any necessary assumptions.

Note: this problem is intentionally vague. This is the sort of thing we are faced with in science: an observed phenomenon that does not yet have a clean textbook explanation. Consider this an opportunity to be creative: why do you imagine TF happens? What must you assume? Can you justify your assumptions? What are the observable consequences? Be sure to derive these quantitatively: once you make a hypothesis, where does it lead?

ASTR 433 only: How should the rotation velocity vary with disk scale length at a given luminosity? What do you conclude from the observation that there are no residuals with scale length?