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ch44_p43 - 43(a During the time interval t the light...

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43. (a) During the time interval t , the light emitted from galaxy A has traveled a distance c t . Meanwhile, the distance between Earth and the galaxy has expanded from r to r' = r + r α t . Let c t r r ra t = ′ = + , which leads to t r c r = α . (b) The detected wavelength λ ' is longer than λ by λ α t due to the expansion of the universe: λ ' = λ + λ α t . Thus, ∆λ λ λ λ λ = ′ − = = α α α t r c r . (c) We use the binomial expansion formula (see Appendix E): ( ) ( ) ( ) 2 2 1 1 1 1 1! 2! n n n x nx x x ± = ± + + < " to obtain ( )( ) 1 2 2 3 1 2 1 1 1 1! 2! . r r r r r r c r c c c c c r r r c c c α α α α α α α α α α ª º ∆λ § · § · § · = = = + + + « » ¨ ¸ ¨ ¸ ¨ ¸ λ © ¹ © ¹ © ¹ « » ¬ ¼ § · § · + + ¨ ¸ ¨ ¸ © ¹ © ¹ " (d) When only the first term in the expansion for ∆λ λ is retained we have ∆λ λ α r c . (e) We set ∆λ λ = = v c Hr c and compare with the result of part (d) to obtain α = H . (f) We use the formula ∆λ λ = α α r c r b g to solve for r :
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( ) ( ) ( ) ( ) ( )( ) 8 8 8 2.998 10 m s 0.050 6.548 10 ly 6.5 10 ly. 1 0.0218 m s ly 1 0.050 c r α × ∆λ λ = = = × × + ∆λ λ + (g) From the result of part (a), ( ) ( ) ( ) ( ) 8 15 16 8 8 6.5 10 ly 9.46 10 m ly 2.17 10 s, 2.998 10 m s 0.0218 m s ly 6.5 10 ly r t c r α × × ∆ =
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