Ay 21 – Winter 2013 – Midterm Exam Distributed on Wed. Feb. 6, due by 12 noon on Wednesday, Feb. 13 The Rules: Closed book, closed notes, no web access, closed everything (except your minds) … but you can use tables of physical and astronomical constants or units. You can use a pocket calculator, but not if it has display of formulas and such. You cannot discuss the problems with anyone until after everyone turns in their exams. You have a maximum of 5 hours (it should take less) from the moment you start until the moment you finish. Please mark your exam with the start and stop times. You have to turn it in in person, either to the Prof or the TA; do not just leave it in a mailbox. Please write legibly – it is in your own best interest. Good luck! Problem 1 [32 points, 4 for each item] Define or explain briefly (in a few sentences at most): a. The distinction between commoving and proper coordinates? b. You and an alien astronomer in a galaxy which for you is at z = 1 observe a quasar which for you is at z = 2 (along the same line of sight); what is the redshift of the quasar from the viewpoint of your alien colleague? c. List at least 3 distinct methods to measure the H0, and their principal advantages and disadvantages. d. What is the K-correction, and what does it depend on? e. WIMPs and MACHOs? f. What is the Einstein radius, and the difference between strong and weak lensing? g. Cosmological constant vs. quintessence, and what is that parameter w? h. The recombination epoch, its redshift, approx. age of the universe at the time? Problem 2 [45 points] Assume a spatially flat universe, with Ωtotal = Ωmatter = 1, and that H0 = 70 km/s/Mpc. a. Derive the formula for the angular diameter distance DA(z) in this cosmological model, and for the age of the universe t(z). [10 points] b. Compute the proper size in Mpc of the particle horizon diameter at the time of recombination, at z = 1100 (hint: from how far would light have traveled at that time in the history of the universe?) [5 Points] c. Compute the corresponding angular size as observed now, if Ωtotal = Ωmatter = 1 [5 points] d. Same as (a), but for the empty universe with Ωtotal = 0. [10 Points] e. Same as (b), for the empty universe model. [5 Points] f. Same as (c), for the empty universe model. [5 Points] g. Comment on these results and the actual observed angular scale (~ 0.9°) for the first Doppler peak of the CMBR fluctuations. [5 Points]Ay 21 Midterm – Winter ’08 Problem 3 [18 points total] The Cosmic Neutrino Background is expected to have the number density (in neutrinos per cm3) nearly equal to (actually 9/11 of) the number density of photons in CMBR. a. Estimate the CMBR photon number density today, assuming TCMBR = 2.7º K, and from that the relict neutrino number density today [5 points]. b. Estimate the number of these relict neutrinos passing through your body every second; state your assumptions [5 points]. c. How massive would these neutrinos have to be (in eV) in order to account for all of the dark matter (state your choices of the relevant cosmological parameters) [8 points]. Problem 4 [40 points total]: Consider a typical disk galaxy like the Milky Way, with a flat rotation curve with Vcirc = 220 km s-1, and a halo extending out to Rmax = 100 kpc. Assume that we live in an Einstein – de Sitter universe with Ωm = Ω0 = 1, and h = 0.5. a. Derive the formula for the free-fall time (of the collapsing object itself) as a function of the object’s mass M, and the initial radius R [10 points] b. What is the total mass of this galaxy? [5 points] c. If it formed via dissipationless collapse, what was the free-fall time? How does it compare with the orbital period at the galaxy’s edge today? (hint: what is the ratio of the radius today to the initial radius?) [5 points] d. What are the present density and the age in this universe? [10 points] e. Assuming that the halo is virialized today, at what redshift did it start collapsing? [10 points] f. How old was the universe then? [5 points] Problem 5 [15 points total]: What would be the form of the galaxy 2-point correlation function if: a. Galaxies were distributed uniformly in space? [5 points] b. All galaxies were on sheets/walls? [5 points] c. All galaxies were in filaments? [5 points] (Maximum total score for this exam = 150 points. The exam counts towards 20% of your
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