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set2-06-sol - Astro 346 Spring Semester 2006 Homework 2nd...

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Astro 346, Spring Semester 2006 Homework, 2nd set, solutions. Problem 1: Radiation transport a) The optical depth is increased at the line frequency, because the line adds its emission and absorption coefficient to those of the continuum radiation processes. Any increase in the absorption coefficient implies an increase in the optical depth. b) We are dealing with a thermal gas, so LTE applies and the source function S ν , is a Planckian. Consider the general solution to the radiation transport equation I ν ( τ ) = I ν (0) exp( - τ ) + Z τ 0 0 S ν ( τ 0 ) exp( - τ + τ 0 ) (8 . 9) If the source function is a constant and LTE applies (thermal particle distribution!), then S ν = B ν ( T ) and I ν ( τ ) = I ν (0) exp( - τ ) + B ν ( T ) [1 - exp( - τ )] I ν (0) + j ν s for τ 1 B ν ( T ) for τ 1 (8 . 10) You can distinguish three cases: – The continuum optical depth is 1 . One observes a Planckian. – The continuum optical depth is 1 , but the line optical depth is 1 . The continuum intensity is always below that of the Planckian with the same temperature. The lines have an intensity equal to the value of the Planckian at the line frequency. We see only emission lines.
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