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.

The acceleration scale

The mass discrepancy appears at a particular physical scale, a_†. One way to see this is from the Tully-Fisher relation. Download the data for gas rich dwarf galaxies. These are rotationally supported systems with more gas than stars. Use these data to

• Plot the tradition (stellar mass) Tully-Fisher and Baryonic Tully-Fisher relations.
• A line fit to the Baryonic Tully-Fisher relation is a line of constant acceleration a_†. Calculate this characteristic accelertion scale a_† = χ V_c⁴/(GM_b) and the corresponding surface density Σ_† = a_†/G. (Fun fact - Σ_† appears to impose an upper limit for disk stability.)
  Here χ = 0.8 is a factor that corrects for the fact that disks rotate faster than the equivalent sphere (recall #4 on homework 1).
• Compute the scatter (i.e., what is σ about the mean a_†?)
  How does this compare to the scatter expected from the uncertainties?
  Remember: uncertainties add in quadrature. For well behaved random errors, the intrinsic scatter of a relation can be obtained from σ_intr² = σ_obs² - σ_err² - i.e., the observed scatter less that expected from the errors.

  ASTR 433 only:

• The given value of χ has been determined empirically. Estimate χ theoretically by comparing a thin exponential disk with the spherical equivalent in the outer parts where V_c is measured (at 4 scale lengths). (Again, recall #4 on homework 1.)

Brown dwarf MACHOs

• Early on, brown dwarfs were a popular dark matter candidate. Presuming that the typical brown dwarf weighs 0.02 M_☉, and recalling the result for the local density of dark matter from the first problem of the first homework, what local number density would be required for brown dwarfs to be the dark matter in the Milky Way? What is the corresponding separation between these brown dwarfs?

• Now suppose we only wish to explain the missing baryons from the first problem of homework 3. What is the local density now?
  You may assume the same shape for the mass distribution; only the amount of mass is different.

• Another idea for the missing baryons is that they reside in a tenuous, diffuse ionized gas (of ~10⁵ K). Presuming this gas is composed of 3/4 hydrogen and 1/4 helium by mass, what is its density? How does this compare to the typical density of the interstellar medium in the plane of the Milky Way of 1 atom per cubic cm?
  [Bear in mind that we are presuming this gas, and the putative halo of brown dwarfs, to be in a quasi-spherical rather than planar distribution.]

Donkey Dark Matter

Professor Mihos once jokingly speculated that the dark matter could be composed of free floating space donkeys (FFSDs).
Can we constrain this possibility?

Live Donkeys

FFSDs would not radiate in the optical, but would emit in the infrared (presuming they have space suits to keep them at a comfortable body temperature of 37 C). If the halo mass of a typical L* (≈ 2 x 10¹⁰ L_☉) galaxy is 10¹² M_☉, what is its infrared luminosity? How does this compare to its optical luminosity? Could we detect this? (Bear in mind that while IR detector technology lags behind optical detector technology, it has nevertheless gotten pretty good).

It may help to recall the Wien and Stefan-Boltzmann Laws. To estimate plausible donkey parameters you may assume a spherical donkey.

ASTR 433 (required); ASTR 333 (extra credit):

Dead Donkeys

FFDDs (Free Floating Dead Donkeys - those lacking space suits) would quickly come into thermal equilibrium with the CMB and any other stray energy sources. Can you think of a way to detect them now that they do not emit in the IR?