ASTR 333/433 - Dark Matter
Homework 3 - Due at the beginning of class March 24

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Further depreciation occurs for every additional day late
  1. Overdensities

    Estimate the overdensity Δ for

    What volume of the universe would each object correspond to now?
    i.e., what is the radius of a sphere with equal mass but density equal to the cosmic mean?
    (Since the universe began with uniform density, this tells you how big a chunk of the current universe was scooped up to form each object). [Hint: what is the appropriate mean density to compare to for each object?]

    You will need to refer to scholarly resources to obtain the necessary data. The first hit on a google search rarely qualifies. NED is a useful resource.

    It helps to think first about what data you need.

  2. 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.

  3. 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.

  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.