hw3-sol - Ph 444 Solutions for Problem Set 3 1. The...

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Unformatted text preview: Ph 444 Solutions for Problem Set 3 1. The critical energy density is c = 3 H 2 / (8 Gc 2 ) and so the critical density is c = 3 H 2 / (8 G ). The uncertainty in the critical density is c = c H H = H parenleftBigg 3 H 2 8 G parenrightBigg H = parenleftbigg 6 H 8 G parenrightbigg H (1) c c = 2 H H (2) I find that fractional uncertainties are usually more meaningful than the uncertainty itself. The formulae are also often simpler. Plugging in the value of H gives c = 3(70 . 6 km s 1 Mpc 1 ) 2 (1 Mpc / 3 . 086 10 19 km) 2 8 (6 . 673 10 11 m 3 kg 1 s 1 ) = 9 . 362 10 27 kg m 3 . (3) For use in problem 2, convert the units to M Mpc 3 : c = (3 . 086 10 22 m / Mpc) 3 1 . 989 10 30 kg /M (9 . 362 10 27 kg m 3 ) = 1 . 38 10 11 M Mpc 3 . (4) The fractional uncertainty is c c = 2 parenleftBigg 1 . 8 km s 1 Mpc 1 70 . 6 km s 1 Mpc 1 parenrightBigg = 2(0 . 025) = 0 . 051 . (5) Thus, the critical density is c = (9 . 362 . 48) 10 27 kg m 3 = (1 . 38 . 07) 10 11 M Mpc 3 . (6) 2. If j is the luminosity density of galaxies, then the average M/L per galaxy needed for galaxies to produce the critical density is ( M/L ) c = c /j . The propagation of errors is slightly tricky in this case as both j and c depend on H and so errors in these quantities are not independent. Adding uncertainties in quadrature requires that the uncertainties be independent. The solution is to express (that the uncertainties be independent....
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This note was uploaded on 02/06/2012 for the course PHYSICS 444 taught by Professor Staff during the Fall '08 term at Rutgers.

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hw3-sol - Ph 444 Solutions for Problem Set 3 1. The...

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