1. For this question assume (somewhat inaccurately) that the universe has a flat spatial geometry, zero cosmological con

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1 For This Question Assume Somewhat Inaccurately That The Universe Has A Flat Spatial Geometry Zero Cosmological Con 1 (83.42 KiB) Viewed 9 times

1. For this question assume (somewhat inaccurately) that the universe has a flat spatial geometry, zero cosmological constant, and is always dominated by matter, i.e. with scale factor given by a(i) = ($)". If He = 67 kms Mpc!, show that the corresponding age of the universe is to 100 years. Calculate the comoving distance that light could have travelled in the time between the hot Big Bang and the present day (express your answer in Mpc). [1 year = 3.156 x 10's, 1 pc = 3.086 x 101m, c = 3.076 x 10-7 Mpc yr ') 2. Write down the expression for the free-fall time and define what this time scale describes. Consider a cloud of primordial gas in a freshly virialized, spherical dark matter halo of mass 10" M and radius 13 kpc. Will this gas cloud be able to contract into a protogalaxy if its cooling time is 8x10 yr? Justify your answer. 3. The initial mass function of stars in a recently formed star cluster can be described by a Salpeter IMF o(m) = Dom 235, with limiting masses mi = 0.1M, and m = 100M. Calculate the fraction of the total luminosity of the star cluster that is contributed by stars more massive than the sun. Assume that for main-sequence stars the luminosity scales roughly as LL. = (M/M)'. How do you expect this fraction to evolve over time as the stellar population in the cluster ages? 4. A supermassive black hole with mass M. = 2 x 10% M, is accreting matter at the Eddington rate, resulting in a luminosity 476cm M. Lead 07 where mm = 1.672x10-27 kg is the proton mass and or = 6.65x10-29 m² is the Thom- son scattering cross section. 0) Determine the black hole mass accretion rate assuming an accretion effi- ciency n=0.1. (ii) This black hole is located at the centre of a massive elliptical galaxy with gas fraction ſgas = Mgas/(M. +Mgas) = 0.015, where M. is stellar mass. The elliptical galaxy and its black hole follow the Magorrian relation. Assuming a constant accretion rate and that all of the gas can be used for feeding the black hole, calculate for how long it will remain observable as an active galactic nucleus.