sol1-f7 - at velocity ∞ to zero because otherwise the...

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Astro 405/505, fall semester 2007 Homework, 1st set, solutions Problem 1: Transformation of spectra We need to transform a spectrum that is diFerential in frequency, B ν ( T ) = dB = C ν 3 1 exp ± h ν k T ² - 1 into a spectrum that is diFerential in wavelength, B λ ( T ) = dB . We have B ν ( T ) = dB = dB = B λ ( T ) Consequently B λ ( T ) = B ν ( T ) ³ ³ ³ ³ ³ ³ ³ ³ = B ν ( T ) c λ 2 = C c 4 λ 5 1 exp ± h c k T λ ² - 1 Problem 2: Hydrodynamics I will use f ( t, ~x,~p ) as in the notes. We need to calculate the integral Z d 3 v m~v ∂f ∂t + ~v · ∂f ∂~x + ~ F m · ∂f ∂~v = 0 which splits up into three integrals, the ±rst of which is I 1 = Z d 3 v m~v ∂f ∂t = ∂t Z d 3 v m~v f = ∂t ( ρ ~ V ) because time and velocity are independent coordinates. ²or the second integral we ±nd I 2 = Z d 3 v m~v ~v · ∂f ∂~x ! = 3 X i Z d 3 v m~v v i ∂f ∂x i ! I 2 j = 3 X i ∂x i Z d 3 v m v j v i f = 3 X i ∂x i ( π ij + V i V j ρ )
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The last integral should be solved with partial integration, where we can set all absolute terms
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Unformatted text preview: at velocity ∞ to zero, because otherwise the number and fow velocity wouldn’t be Fnite. The third integral then is ±or each vector component I 3 j = Z d 3 v v j ~ F · ∂f ∂~v = 3 X i Z d 3 v v j F i ∂f ∂v i =-3 X i Z d 3 v f ∂v j F i ∂v i Because ±or all ±undamental interactions ∂F i ∂v i = 0 , we have ∂ ∂v i ( v j F i ) = δ ij F j ⇒ I 3 j = Z d 3 v f F j = ~ F j Pulling all three integrals together we Fnd ±or each component ∂ ∂t ( ρ V j ) + 3 X i ∂ ∂x i ( π ij + V i V j ρ )- F i = 0 Using the continuity equation ∂ρ ∂t + ~ ∇ · ± ~ V , ρ ² = 0 this reduces to the desired ρ ∂V j ∂t + V i ∂V j ∂x i ! + ∂ ∂x i π ij- F j = 0...
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This note was uploaded on 02/29/2012 for the course PHYS 227 taught by Professor Rabe during the Fall '08 term at Rutgers.

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sol1-f7 - at velocity ∞ to zero because otherwise the...

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