Overdensities

Estimate the overdensity Δ for

• the Earth
• the globular cluster 47 Tuc
• M31 (Andromeda)
• the Virgo cluster of galaxies

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.

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.

• What is the dynamical mass of the Milky Way within this edge?
• What is the detected baryon fraction of the Milky Way?
  The tally of known stars and gas is approximately 5 x 10¹⁰ M_☉.
• If the cosmic baryon fraction f_b = Ω_b/Ω_m = 0.15, what mass of baryons is missing?
• Speculate on the current location & state of the missing baryons.
  i.e., where are they now? what form are they in?

Elliptical galaxies: big & bright; small & dim

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

• What is the mass-to-light ratio of M89?
• Does this system require dark matter?

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

• What is the mass-to-light ratio of Draco?
• Does this system require dark matter?

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

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).

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).

• the semimajor axis a
• the age t₀
• the total mass m+M
• the time to when Andromeda crashes into us.
• the mass-to-light ratio of the Local Group.
  The luminosity of Andromeda is 2.7 x 10¹⁰ L_☉ and that of the Milky Way is 1.5 x 10¹⁰ L_☉. You may neglect the rest of the Local Group (see Table 4.1 of Sparke & Gallagher).
• is dark matter required?

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.

• Will M31 collide with the Milky Way now?
  Assume their centers require a clearance of 100 kpc for safe passage. That's twice the distance to the Magellanic Clouds. Recall that the pericenter = a(1-e).
• Does the implied mass increase or decrease?
• Do either of these changes (to a and m+M) seem so dramatic that they would be readily obvious?