ASTR 333/433 - Dark Matter
Homework 2 - Due at the beginning of class October 15

Late homework suffers 15% depreciation per day

  1. Plummer Potential

    Derive the density ρ(r) associated with the spherically symmetric Plummer potential Φ(r) = -GM/(r2+a2)1/2.

  2. Milky Way Kinematics

    Use three pieces of information to quantify some basic properties of our Milky Way Galaxy:
    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 ‹VX›2 = 2.2 ‹VY›2
    [You may ignore the peculiar motion of the sun here.]

    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?
    For comparison, the baryonic mass of the Milky Way is about 6 x 1010 M⊙. Most but certainly not all of this is interior to the solar radius.

    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).

  3. 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

    V = mRforR < R0
    V = VfforR > R0

    1. Derive expressions for A(R) and B(R).
      [Hint: what is m in terms of Vf and R0?]
      Comment on the identification of Oort's constant A with "shear" and the description of this rotation curve as "solid body".
    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.

    Note: the slope m is the slope of the rotation curve and has nothing to do with the mode number of a bar: do not set the slope equal to 2; solve for it in terms of Vf and R0.

    Related reading: Section 6.3, Binney & Tremaine (2nd edition) [especially 6.3.3]

  4. Dark Matter Halos

    The asymptotically flat rotation curves of galaxies are usually interpreted to require extended halos of dark matter. Two forms of halo are generally considered: empirically motivated pseudo-isothermal halos and "NFW" halos motivated by numerical simulations of structure formation (Binney & Merrifield eqns 8.57 & 8.58, respectively).

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

    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.]
    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 ∝ rx ↔ V ∝ ry.

  5. Mass Models

    Get the rotation curve data for NGC 2998.
    A detailed mass model of the bulge and disk (both stellar and gaseous) is included.

    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.]
    Fit the data with a pseudo-isothermal halo and plot the results. ASTR 333 only: a fit by eye suffices.
    What constraint might seem an obvious one to impose given the flatness of observed rotation curves?
    [Hint: Can you express the characteristic velocity of the halo in terms of the parameters ρ0 and a?]

    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?]

    In addition to pseudo-isothermal halos, also fit NFW halos to the dark matter for this galaxy, for both choices of M/L. You may write your own code, 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, which provide a more common parameterization than that given by Binney & Merrifield. See Navarro, Frenk, & White (1997).

    Plot the results for both halo types for both assumptions about M/L. Plot the original data together with lines for each of the mass components and the sum of all mass components. How good are these fits? Can you distinguish between
    • halo types?
    • M/L choices?