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).

Dark Matter Halos

1. What do these density profiles imply for the total halo mass? (i.e., what do they integrate to as r ⇒ ∞?)

2. Derive the circular speed V(R) of a test particle orbiting in these mass distributions.
   [Hint: dark matter halos can be presumed to be spherical.]
   

3. How do the shapes of these halo models compare? Specifically, how do the density ρ(r) and circular velocity V(r) behave in the limits r → 0 and r → ∞?
   Express your answers as power laws: ρ ∝ r^x ↔ V ∝ r^y.

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

Use the data provided here to answer this question.

1. Construct the Tully-Fisher relation. Plot the data and fit it with the functional form
   log₁₀(L) = X log₁₀(Vf) + K
   where X is the slope and K the intercept of the Tully-Fisher relation.
   Report the fitted values of X and K along with their uncertainties and show the fitted line on the plot with the data. Do this for both the V-band (LV) and the 3.6 micron (L36) luminosities. [Note units.]

   When fitting, consider errors in both x and y. Measured uncertainties are given for the rotation velocity; those in luminosity are dominated by uncertainty in the distance to each galaxy. For the purposes of this problem, assume all galaxies have a distance uncertainty of 10%.
   [How does this propagate into luminosity?]
   [Recall that for base 10 logarithms, the uncertainty δu in variable u transforms to 0.43(δu/u) in log₁₀(u).]

   Are the resulting V-band and [3.6] Tully-Fisher relations the same? (indistinguishable within the uncertainties of the fits.) Should they be?

2. Construct the stellar mass Tully-Fisher relation,
   log₁₀(M*) = X log₁₀(Vf) + K
   using the B-V color (BV) to estimate the stellar mass from the luminosity. This is done by means of stellar population models that relate the mass-to-light ratio M*/L to color through formulas like
   log₁₀(M*/L) = b (B-V) − a.
   Two distinct stellar population models are given in the table:

   Model	log10(M*/LV)	log10(M*/L36)
   Zibetti	1.837 (B-V) − 1.075	1.176 (B-V) − 1.501
   Modified Bell	1.305 (B-V) − 0.628	−0.007 (B-V) − 0.322

   Compute stellar masses using both models for both luminosities, then plot and fit the resulting stellar mass Tully-Fisher relations as above. Note that there are now four distinct relations, two for each of the two bands. Are the fits consistent within their uncertainties? Should they be? Does one or the other population model perform "better" in this regard?
   [For the purposes of error propagation, pretend that the tabulated M*/L-color relations are perfect.]

3. Construct the baryonic Tully-Fisher relation,
   log₁₀(Mb) = X log₁₀(Vf) + K
   where the baryonic mass is the sum of stellar and gas mass: Mb = M*+Mg. Note that the gas masses given in the data table refer only to the mass of hydrogen; they should be corrected to include helium by multiplying by 1.33. Do this for all four realizations of the stellar mass determined in the previous part. Plot and fit the resulting relations. Are they consistent? Should they be?
   [The uncertainty in gas mass, like that in luminosity, is dominated by the uncertainty in distance, for which you should continue to assume a 10% error.]

ASTR 433 only: Compute the scatter about each of the best fit relations above. How should the scatter change as we go from the luminous Tully-Fisher relation to the stellar mass Tully-Fisher relation? Is your expectation reflected in the data?
How does the scatter in the baryonic Tully-Fisher relation compare to the scatter you expect from the uncertainties in measured velocities? What other sources of scatter might there be?