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sol11-f7 - Astro 405/505 fall semester 2007 Homework 11th...

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Astro 405/505, fall semester 2007 Homework, 11th set, solutions Problem 21: Optically thick bremsstrahlung First determine the source function, which is the Planckian under LTE. S ν = j ν α ν = B ν ( T ) = C 1 C 2 T ν 2 In addition we have B ν (4 T 0 ) = 4 B ν ( T 0 ) and from the density scaling the optical depth of the cloud in front (cloud 2) as τ f = τ b / 128 , where τ b is the optical depth of the cloud in the background (cloud 1). At the front of the background cloud the intensity would then be I ν ( τ b ) = B ν ( T 0 ) [1 - exp( - τ b )] This intensity must pass through the foreground cloud, so added to its emission the observer would see I ν ( τ b + τ f ) = B ν ( T 0 ) exp( - τ f ) [1 - exp( - τ b )] + B ν (4 T 0 ) h 1 - exp( - τ f ) i I ν ( τ b + τ f ) = B ν ( T 0 ) 4 - exp - τ b 129 128 - 3 exp - τ b 128 ) At high frequencies the system is optically thin, τ b 1 , so we see cloud 1. I ν ( τ b + τ f ) = B ν ( T 0 ) τ b 33 32 τ b 1 At intermediate frequencies τ b 1 , but τ b 128 , and we still see cloud 1.
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