ASTR 333/433 - Dark Matter
Homework 3 - Due at the beginning of class November 19

Late homework suffers 15% depreciation per day
  1. Milky Way baryons

    Suppose the rotation curve of the Milky Way stays flat at 220 km/s out to the "edge" of the dark matter halo at 180 kpc.

  2. Elliptical galaxies: big & bright; small & dim

    M89 is a giant elliptical galaxy with luminosity L = 1011 L, effective radius Re = 500 pc, and velocity dispersion σ = 252 km/s.

    Draco is a dwarf Spheroidal satellite of the Milky Way. It has luminosity L = 2 x 105 L, effective radius Re = 220 pc, and velocity dispersion σ = 10 km/s.

    You may assume isotropy; i.e., the usual virial theorem applies.

  3. 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?

  4. Timing Argument

    Andromeda is 770 kpc away from the Milky Way and approaching us at 120 km/s. Lets assume that these two most massive galaxies of the Local Group (having a total mass m+M) started out expanding away form each other with the Hubble flow but are now falling together on a perfectly radial orbit (e = 1).


    Eine kleine orbit theory:
    The position of an object r(t) along an orbit can be described by the development parameter η such that

    r = a[1-e cos(η)]

    t = {a3/[G(m+M)]}1/2 [η-e sin(η)]

    where e is the eccentricity of the orbit and a is the semimajor axis.
    η measures the progress from one pericenter (η = 0 at t = 0) to the next (at η = 2π).
    The above expressions can be combined to find the speed

    dr/dt = (dr/dη)/(dt/dη) = {[G(m+M)]/a}1/2 e sin(η)/[1-e cos(η)] = (r/t0){e sin(η)[η-e sin(η)]}/[1-e cos(η)]2

    (see the end of chapter 4 of Sparke & Gallagher).


    Traditionally one assumes an age for the universe (t0) in order to solve for η. That's a tad tedious, so instead accept that η=4.2 and determine

    ASTR 433 only: Now suppose the orbit is not perfectly radial, but still pretty eccentric. The development parameter doesn't change much from the e = 1 case. Adopting e = 0.8 and η = 4.3 for an age of 13.2 Gyr, determine the semi-major axis a of the orbit.