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1297_Physics ProblemsTechnical Physics

1297_Physics ProblemsTechnical Physics - 638*P46.47...

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638 Particle Physics and Cosmology *P46.47 (a) Consider a sphere around us of radius R large compared to the size of galaxy clusters. If the matter M inside the sphere has the critical density, then a galaxy of mass m at the surface of the sphere is moving just at escape speed v according to K U g + = 0 1 2 0 2 mv GMm R = . The energy of the galaxy-sphere system is conserved, so this equation is true throughout the history of the Universe after the Big Bang, where v dR dt = . Then dR d t GM R F H G I K J = 2 2 dR d t R GM = 1 2 2 / R dR GM d t R T 0 0 2 z z = R GM t R T 3 2 0 0 3 2 2 / = 2 3 2 3 2 R GM T / = T R GM R GM R = = 2 3 2 2 3 2 3 2 . From above, 2 GM R v = so T R v = 2 3 . Now Hubble’s law says v HR = . So T R HR H = = 2 3 2 3 . (b) T = × × F H G I K J = × = 2 3 17 10 3 10 1 1 18 10 11 8 3 8 10 m s ly m s ly yr yr billion years e j . . *P46.48 In our frame of reference, Hubble’s law is exemplified by v R 1 1 = H and v R 2 2 = H . From these we may form the equations
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