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Homework 5 University Astronomy University Astronomy 1017-301 Homework Assignment 5 Instructor: Dr Andrew Robinson Due Date: 9 am Friday, 14 May...


Homework 5 University Astronomy 5/4/10 1 University Astronomy 1017-301 Homework Assignment 5 Instructor: Dr Andrew Robinson Due Date: 9 am Friday, 14 May 2010. To tackle some of the following problems you will need to do some research, using either the textbook or other resources, to find various pieces astronomical data and the values of certain physical constants. For calculations, show all your workings in order to receive full credit . 1. a. Estimate the total number of orbits around the galactic center that the sun will make during its entire main sequence lifetime. b. How far would an O star, 5 × 10 5 times more luminous than the sun, travel along its orbit during its lifetime, if it formed at the same distance from the galactic center as the sun? 2. The quasar PG1700+518 has a redshift z = 0.292 and a flux in the visible band of F V = 1.0 × 10 -14 W/m 2 . a. What is the distance to this quasar in Mpc? (Note : use the formula for relativistic redshift given in Lecture 21) b. What is its luminosity in the visual band, L V ? c. Assuming that its bolometric (i.e., total) luminosity is 9 L V , calculate the mass of the smallest black hole that is required to power this quasar. 3. The Galactic Center black hole is a relatively weak X-ray source, which suggests that it is currently dormant through lack of “fuel” in the form of interstellar gas. What if, for example, the tidal disruption of a passing star (as in HW 4) was to suddenly provide a fuel supply? a. Calculate the luminosity of the black hole if it were accreting gas at the maximum possible rate (think Eddington). b. Calculate the rate (in solar masses per year) at which the black hole would have to accrete gas in order to sustain this luminosity. 4. The Coma cluster consists of about 1000 bright galaxies and has a linear diameter of about 6 Mpc. Its visual luminosity is 5 × 10 12 L . You can assume that the cluster is spherical. a. The rms velocity of the galaxies in the cluster is 1700 km s -1 . What are the total mass (think “virial theorum”) and the mass-to-light ratio of the cluster? Express your answers in units of L and M . b. X-ray observations show that the cluster is filled with hot gas. Assuming that the gas is in hydrostatic equilibrium, estimate its temperature at the core of the cluster.
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Homework 5 University Astronomy 5/4/10 2 (Hint: you may find it useful to review the slides in lecture 15, on hydrostatic equilibrium in stars.) 5. The inner radius of the circum-nuclear torus around an AGN is determined by the temperature ( T evap ) at which dust grains evaporate due to heating by absorption of UV radiation from the accretion disk. a. Assuming that the dust grains are perfect backbodies, show that the torus inner radius is given by r torus = 1 4 T evap 2 L UV "# $ % & ( ) 1 2 , where L UV is the ultra-violet luminosity of the AGN. b. What is the torus inner radius if T evap = 1500 K and L UV = 10 39 W? At what wavelength will the blackbody radiation from the dust grains peak? 6. The distance measurements that Hubble used in his original work were poorly calibrated leading him to greatly over estimate the value of the Hubble constant. a. The currently accepted value for H 0 is 73 km s -1 Mpc -1 . Estimate the age of the universe, assuming that it has continued to expand at a constant rate since the Big Bang. b. Hubble’s original estimate of H 0 was 500 km s -1 Mpc -1 . What is the age of the universe implied by this value? Comment on how this age for the universe compares with what you know other about astronomical timescales (e.g., age of the solar system, stellar lifetimes etc.) 7. It is thought that protogalactic clouds began to collapse soon after the recombination era when the density and temperature of the primordial gas were ρ ~10 -18 kg m -3 and T ~3000 K, respectively. Assuming that this gas is pure Hydrogen and that the cloud is spherical, estimate the characteristic mass and size of a proto-galactic fragment. What, if any, components of galaxies in the local universe have masses comparable to this characteristic fragment mass? (Hint: you may find useful to review Lecture 13, on star formation.) 8. Sketch a diagram of the pair production process that results in a proton–anti-proton pair. Label all particles or photons involved. a. Estimate the temperature of the universe at the time when this reaction ceased . b. Estimate the redshift when proton–anti-proton pair production ceased. What was the scale factor of the universe at this time? BONUS Can cosmic neutrinos close the universe? As well as a microwave background, the universe has a cosmic neutrino background. The number density of these neutrinos is calculated to be n =3.4 × 10 8 m -3 . So what average neutrino mass would be required to make the matter density equal to the current critical density of the universe? What can you find out about the current experimental limits on neutrino mass? Based on your research, is this idea plausible, or busted?
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