exc6 - × 10-2 Mpc-3 and that in each galaxy there are 5 ×...

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Introduction to Astrophysics 0321.3108 Exercise 6 1. Measurements of the radial recession velocities of Fve galaxies in a cluster give velocities of 9700, 8600, 8200, 8500, and 10,000 km s - 1 . a. What is the distance to the cluster if the Hubble parameter is H 0 = 72 km s - 1 Mpc - 1 ? Hint: Use the Hubble law for the average velocity of the members in the cluster. b. Estimate, to an order of magnitude, the mass of the cluster if every galaxy is projected roughly half a degree from the cluster center. Hint: Use the virial theorem 2. The age of the Universe ±or a Hubble constant of H 0 = 72 km s - 1 Mpc - 1 , calculate the Hubble time t H = H - 1 0 . 3. The critical density a. Consider the ±riedmann equation: ± ˙ R R ² 2 = H 2 = 8 πG 3 ρ - kc 2 R 2 . What is the critical density ρ c that gives a marginally bound Universe. Assume: H 0 = 72 km s - 1 Mpc - 1 and ²at Universe. b. Estimate the stellar mass density ρ ? . Assume that the density of galaxies is 2
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Unformatted text preview: × 10-2 Mpc-3 and that in each galaxy there are 5 × 10 10 stars, and that each star has an average mass of 0 . 5M ¯ . What is the ratio ρ ? ρ c . 4. The proper distance to a source is rR where r is the comoving distance and R is the scale factor. a. Use the relation between redshift and the scale factor, i.e., 1 + z = R R ( t ) and show that 1 R ( t ) = 1 R-1 R H ( t-t ). Hint: Use Taylor series about the point t = t (the age of the Universe today) to the Frst order b. Consider the geodesic expression: cdt = R ( t ) dr √ 1-kr 2 , and assume a ²at Universe. Calculate to the Frst order the physical distance today, rR in terms of z and Hubble time t . Hint: Use the above approximation, and approximate to the Frst order....
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