Astro 346, Spring Semester 2006Homework, 2nd set, solutions.Problem 1: Radiation transporta) The optical depth is increased at the line frequency, because the line adds its emissionand absorption coefficient to those of the continuum radiation processes. Any increase in theabsorption coefficient implies an increase in the optical depth.b) We are dealing with a thermal gas, so LTE applies and the source functionSν, is a Planckian.Consider the general solution to the radiation transport equationIν(τ) =Iν(0) exp(-τ) +Zτ0dτ0Sν(τ0) exp(-τ+τ0)(8.9)If the source function is a constant and LTE applies (thermal particle distribution!), thenSν=Bν(T)andIν(τ) =Iν(0) exp(-τ) +Bν(T) [1-exp(-τ)]’Iν(0) +jνsforτ1Bν(T)forτ1(8.10)You can distinguish three cases:– The continuum optical depth is1. One observes a Planckian.– The continuum optical depth is1, but the line optical depth is1. The continuumintensity is always below that of the Planckian with the same temperature. The lines have anintensity equal to the value of the Planckian at the line frequency. We see only emission lines.
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