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# solution_4 - ASTRONOMY 294Z The History of the Universe...

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ASTRONOMY 294Z: The History of the Universe Professor Barbara Ryden SOLUTIONS TO PROBLEM SET # 4 1) [20 points] Today, the average density of matter in the universe is ρ = 3 × 10 - 27 kg / m 3 . If the matter consisted entirely of hydrogen atoms, how many hydrogen atoms (on average) would be contained in a cubic meter of the universe? If the matter consisted entirely of regulation baseballs (of mass M = 0 . 145 kg apiece), how many baseballs (on average) would be contained in a cubic astronomical unit of the universe? The mass of a single hydrogen atom is m = 1 . 7 × 10 - 27 kg. In order for the mass density to be ρ = 3 × 10 - 27 kg / m 3 , the number of hydrogen atoms per cubic meter would have to be n = ρ m = 3 × 10 - 27 kg / m 3 1 . 7 × 10 - 27 kg = 1 . 76 / m 3 . (1) Fewer than two atoms, on average, per cubic meter of space – that’s a very low density. The mass of a single baseball is M = 0 . 145 kg. In order for the mass density to be ρ = 3 × 10 - 27 kg / m 3 , the number of baseballs per cubic AU would have to be n = ρ M = ˆ 3 × 10 - 27 kg / m 3 0 . 145 kg ! ˆ 1 . 5 × 10 11 m 1 AU ! 3 = 69 , 800 , 000 / AU 3 . (2)

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2) [20 points] The temperature of the cosmic background light today is T 3 K . At the time the universe became transparent, the temperature of the cosmic background light was T 3000 K . This means that the universe has
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