ASTR 333/433 - Dark Matter
Homework 1 - Due at the beginning of class September 17

Late homework suffers 15% depreciation per day

  1. Local Dark Matter Density

    Assume that most of the mass of the Milky Way interrior to the solar radius Ro is in a spherical dark matter halo. If the rotation curve is flat with V = 230 km/s and Ro = 8 kpc, what is the local density of dark matter?
    [Hint: what must the density profile be to obtain a flat rotation curve?]
    Use your result to compute the mass of the dark matter within the solar system (out to 40 AU).
    Should the dark matter have a noticeable impact on planetary dynamics?
    [See the the useful numbers page for Newton's constant in convenient units.]

  2. Oort limit

    This dataset contains radial velocities, and apparent and absolute magnitudes for a sample of K giant stars at high galactic latitude (looking "down" out of the plane of the Milky Way's disk). We are going to use this dataset to measure the mass density of the Milky Way's disk in the solar neighborhood using the Oort limit. To do this, we need the velocity dispersion and scale height of the stars.

    Scale Height

    Calculate the distance to each star from its apparent and absolute magnitude. [If it isn't immediately obvious to you how to do this, see "distance modulus" on the useful numbers page.] Because these stars are at high galactic latitude, their Z coordinate is essentially the same as their distance, once we correct for the Z distance of the Sun from the disk plane (Z☉ = +30pc). Then find the number of stars (N) as a function of Z by counting the number of stars in bins of Z. [Hint: 400 pc is a good bin size.] Make a plot of ln(N) (y axis) vs Z (x axis) and find the scale height of the stars.

    Velocity Dispersion

    Just as the distance is roughly the Z coordinate (once we corrected for the Sun's Z coordinate), the radial velocity is roughly the Z velocity once we correct for the Sun's Z velocity. From the distribution of velocities, calculate the Sun's Z velocity. Then subtract that from all the velocities to get the Z velocity of each star. Then calculate the velocity dispersion.
    ASTR 433 only: Compute the velocity dispersion in each bin for the same bins you used to measure the scale height. How does the velocity dispersion vary with height above the plane? Does this make sense? [i.e., what is σ(Z) responding to?]

    Surface Density

    Given the scale height and velocity dispersion, estimate the mass density of the disk. Consider sources of error and how they may have affected your result.

  3. The IMF of stars

    Salpeter IMF

    Consider a Salpeter IMF with a number of stars
    ξ(M) = dN/dM = ξ0 M-2.35
    for a mass range from a lower limit Ml to Mu.
    Find expressions in terms of ξ0, Ml, and Mu for For the last item, assume a main sequence mass-luminosity relation L=M3.5

    The Pleiades

    For a star cluster like the Pleiades with a mass of 800 M☉,

  4. Spiral Galaxies

    Exponential Disks

    Spiral and irregular galaxies have azimuthally averaged radial light profiles that are tolerably well approximated as "exponential disks":
    Σ(r) = Σ0 e-r/Rd,
    where Σ0 is the central surface brightness (in solar luminosities per square parsec) and Rd is the scale length (in kpc) of a galaxy fit by this description.

    Integrate Σ(r) from 0 to 2π and r = 0 to x scale lengths (r = xRd) to obtain an expression for the luminosity enclosed by x scale lengths.
    • Is the total luminosity finite as x ⇒ ∞ infinity? If not, what is it?
    • How many scale lengths contain half the total light? (This is known as the half-light radius, Re.)
    • Plot the cumulative enclosed luminosity L(< x).

    Disk rotation curve

    • Assuming M/L is constant with radius and making the fudge of a "spherical" disk, use the expression for L(x) to compute and plot Vd(r) for an exponential disk.
    • ASTR 433 only: The actual rotation curve of a thin exponential disk is given by Binney & Tremaine eqn. 2-169 (1st edition) or 2.165 (2nd edition). Solve this numerically for a reasonable sampling or radii and plot it on the same diagram as the spherical approximation.
      Why are the two different? Which rotates faster?
    • At what radius (in scale lengths) does the rotation curve of an exponential disk peak?
    • ASTR 433 only: do this for the thin disk as well as the spherical approximation.

    Disk Stability

    Consider a galaxy with a flat rotation curve V(r) = Vc. The condition of maximum disk is Vd,max ≤ Vc (the contribution of the disk cannot exceed the total mass budget).
    • What is the central mass density (in solar masses per square parsec) of a maximum disk with Rd = 5 kpc and Vc = 200 km/s?
    • ASTR 433 only: do this for the thin disk as well as the spherical approximation.

      That bars and 2-armed spirals are observed to exist implies a minimum disk mass in order for the m=2 mode to grow.

    • Use the stability criteria X2 < 3 to find a limit on Vd,min in terms of Vc. Make use of your result above for the radius of peak disk rotation. How does Vd compare to Vc at this radius?
    • ASTR 433 only: do this for the thin disk as well as the spherical approximation.