Ay 21 – Winter 2013 – Final Exam Distributed on March 13, due by 5 pm on Wednesday, March 20 (directly to the TA) 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 (attached). 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. The Honor System applies. 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, or the secretaries in rm. 211 Robinson. There are 7 problems, with a maximum total score of 150 points. The exam counts towards 30% of your grade. Please write legibly – it is in your own best interest. Good luck!Ay 21 Final – Winter ’13 Problem 1 [27 points, 3 for each item] Define or explain briefly (in a few sentences at most): a. The main differences between dwarf ellipticals and regular ellipticals; draw diagrams if needed. b. The Schechter luminosity function; draw a diagram; what are the typical values of its parameters? c. The reionization era and why is it important, and at what redshifts did it happen? d. Blazars (BL Lac’s) and how they differ from other types of AGN; why are they so interesting? e. Synchrotron emission and its origin; sketch its spectrum, label the axes. f. Kerr black hole vs. Schwarzschild black hole; which one can produce a more luminous AGN, for the same accretion rate? g. Unified models of AGN; do draw a picture. h. Types of absorbers in quasar spectra, and how do they relate to galaxies? i. The cosmic web, and how do we study it? Problem 2 [20 points] a. What is the Fundamental Plane of elliptical galaxies? [5 points] b. Use the Virial Theorem to derive it (or an analogous relation), and state clearly your assumptions. [5 points] c. What physical conditions must be satisfied in order to reproduce the observed slope and the small scatter? [5 points] d. What are its main cosmological uses? [5 points] Problem 3 [10 points] a. The Andromeda galaxy (M31) is about 700 kpc away. The velocity dispersion and the rotation speed are observed to increase in the center, so that the lower limit on the mean velocity of stars over and above the normal velocity dispersion in the bulge of M31 is 〈V〉 > 400 km/s, at projected radii < 0.5 arcsec. Estimate the lower limit to the mass of the central MBH in M31. [5 points] b. Consider a quasar with a SMBH of mass M bh ~ 108 M. The velocity half-width of the broad emission lines (BEL) is 5000 km/s in the quasar restframe. Estimate the physical radius of the BEL region. Can it be resolved with the HST (resolution ~ 0.1 arcsec) if it was in the Virgo cluster, 20 Mpc away? Can it be resolved with a VLBI experiment with a resolution of 1 microarcsec, if it is 100 Mpc away? [5 points]Ay 21 Final – Winter ’13 Problem 4 [18 points] Consider a quasar powered by an Eddington-limited accretion to a black hole, with an efficiency of 10%. Derive the function describing the evolution of the black hole’s mass and the quasar’s luminosity, and the numerical value of the characteristic time scale entering these formulas. Problem 5 [30 points] The observed energy density of the cosmic far-infrared background (CIRB) is u CIRB ≈ 710-15 erg cm-3. Assume that the background is generated by a population of obscured starbursts with a mean luminosity 〈L〉 ≈ 1012 L, with all energy emerging in the mid/far IR, with a mean redshift 〈z〉 ≈ 3, each lasting on average Δt ≈ 3107 yr. a. How many such starbursts should there be in the entire observable universe, in order to account for the CIRB? b. Estimate the comoving number density of their progeny at z ~ 0, and compare it with the estimated comoving number density of normal galaxies today. (Hint: estimate the total comoving volume of the observable universe first; an order-of-magnitude, reasoned estimate is OK.) c. Assuming the yield of nuclear reactions of 7 MeV/nucleon, how much helium was generated by these starbursts? How much metals, assuming the ratio of helium to metals production by mass ΔY/ΔZ = 5? If each of these starbursts was in a galaxy with a baryonic mass M ≈ 1011 M, what is the mean metallicity of the resulting stars? Problem 6 [20 points] The observed brightness of the cosmic X-ray background (CXRB) is νIν ≈ 310-10 W m-2 sr-1. a. Compute the corresponding volume energy density. Compare with the energy density of the CMB, assuming TCMB = 2.735°K. [4 points] b. It is estimated that on average there is a SMBH with M bh ~ 107 M per average (L ~ L*) galaxy. Assuming that the efficiency of accretion in converting the rest mass into energy was ~ 10% (i.e., 90% ends in the SMBH, 10% is radiated away), and that the mean redshift of emission was 〈z〉 ~ 3, compute the energy density today, generated by the making of these SMBHs. Compare it with the numbers for the CXRB and CMBR computed in (a). (Note: you will need to estimate the comoving density of L* galaxies today.) [16 points]Ay 21 Final – Winter ’13 Problem 7 [25 points] Consider a quasar with Lbol ≈ 51012 L, at z = 2. Assume an Einstein – de Sitter universe with H0 = 50 km/s/Mpc. a. Compute the luminosity distance to the quasar. [5 points] (Note: if you don’t know how to evaluate or derive this distance quickly, make a reasoned estimate for only 2 points credit, and move on.) b. Compute its absolute and apparent bolometric magnitudes, assuming Mbol = 4.8 mag. (Note: this is the absolute magnitude, not the mass!) [5 points] c. Assuming that 30% of the entire luminosity of this quasar is emitted in X-rays, compute the observed X-ray flux (in cgs or SI units). [5 points] d. How many such quasars would it take to generate all of the observed CXRB? (See the brightness in the previous problem). Compute their projected surface density on the sky (number per deg2), and compare with the numbers derived in problem 6. [10 points]Physical Constants and Astronomical DataNew! Try my Physical Calculator. It is a JavaScript calculator with all of the constants