Question

A hypothesis once used to explain the Hubble relation is the "tired light hypothesis." The tired light hypothesis states that the universe is not expanding, but that photons simply lose energy as they move through space (by some unexplained means), with the energy loss per unit distance being given by the law ater ba some unecxplained macana), dE dr -kE where k is a constant. Show that this hypothesis gives a distance-redshift relation that is linear in the limit zく1. What must the value of k be in order to yield a Hubble constant of Ho = 68 kms-1 Mpc-1?

Answer 1

According to question, energy loss per unit distance dE/dr = - KE Rightarrow integral_E dE/E = integral^r_0 - k dr Rightarrow in epsilon Vertical-bar ^E_obs_E = - kr Rightarrow ln (epsilon_obs/E) = - kr Rightarrow E_obs/E = e^ - kr Rightarrow E_obs = E e^ - kr Therefore Red shift z = lambda_obs - lambda/lambda = since lambda = hc/lambda Therefore Z = - epsilon_obs + E/E_obs = E e^ - kr - epsilon/E_obs = E(- e^ - kr + 1/E e^ - kr = 1 - e^ - kr/e^ - kr Therefore z = e^kr - 1 = 1 - e^ - kr/e^ - kr
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