sol5-f7

# sol5-f7 - Astro 405/505 fall semester 2007 Homework 5th set...

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Astro 405/505, fall semester 2007 Homework, 5th set, solutions Problem 9: Time-dependent line profiles The problem is similar to that discussed in class except that now we would wish to calculate rates, i.e. also temporal probabilities. The ansatz would be P ( E γ , t obs ) dE γ dt obs = P ( v ) dv P (Ω) dφ d cos θ P ( t ) dt where we can use the Doppler shift formula E γ = E 0 1 + ~e k · ~ V c = E 0 1 + v cos θ c ! The relation between the observed time t obs and the standard time coordinate t is given by the changes in light travel time. Radiation from particles that are located on the far side of the supernova needs more time to arrive at the observer, and vice versa. After time t the particle has a separation x = vt from the supernova. If the supernova is located at a distance d to the observer, the particle will be at a distance D = q d 2 + x 2 - 2 x d cos θ = q d 2 + v 2 t 2 - 2 v t d cos θ . The observed time therefore is (we can add any constant) t obs = t + D c = t + 1 c q d 2 + v 2 t 2 - 2 v t d cos θ = t + d c s 1 + v 2 t 2 d 2 - 2 v t d

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