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 stellar mass Tully-Fisher and Baryonic Tully-Fisher relations. Comment on the difference, contrasting with the situation for the brighter, star dominated galaxies that are usually the subject of Tully-Fisher studies.

• A Baryonic Tully-Fisher relation with a slope of 4 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 has a mass of 0.02 M_☉, and recalling the result for the local density of dark matter from the first homework (0.015 M_☉pc^-3), 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?

• The optical depth τ of gravitational lensing for MACHOs in the halo is

  

• What is the optical depth τ to gravitational lensing from a halo of brown dwarfs in the direction of the LMC?
  Hint: We are 8 kpc from the Galactic Center; the LMC is 50 kpc away. Where is the typical lens? What is its typical speed (σ)?

• The observed limit found by EROS is τ < 3.6 x 10^-8. How does this compare with the optical depth you computed? What does this imply?

• The EROS limit only applies for objects of mass > 10^-8 M_☉. What famous solar system object has about this mass? What local number density and typical separation would such objects have if they were to be the dark matter? Would we notice them?

  ASTR 433 only:

  What would the kinetic energy of such a halo object should it happen to collide with the Earth? How does this compare with the gravitational binding energy of the Earth?

WIMPs in you

• Assuming the dark matter is made of WIMP mass of 100 GeV, how many are passing through your body at any given moment?

Cosmic Microwave Background Radiation

• What is the number density of photons?

• If the mean mass density of baryons in the universe is ρ₀ = 2 x 10^-31 g cm^-3, what is the number density of baryons?

• What is the baryon-to-photon ratio?
  

Ω_m from Cluster Baryon Fractions