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Unformatted text preview: AST 3722C  Spring 2008 Homework #8 – Due before class April 1 no foolin’ Instructions: Solve each part of each problem below. Where math is involved, and unless otherwise indicated, show your work. • 1. (2 points.) It is actually possible to integrate the Planck function, it’s just not easy. Start with B ν = 2 hν 3 c 2 × 1 e hν/kT − 1 . You want to find an expression for S : S = π integraldisplay ∞ B ν dν, i.e., the integral of the Planck function over all frequencies (times π steradians). (a) What are the units of B ν ? (b) What are the units of S ? (c) One way to do the integral is to do a substitution. Let u = hν/kT . Rewrite S by inserting the Planck function and making it an integral over u instead of over ν . (d) Now for a trick. Start with the identity 1 = 1. Next we take advantage of the fact that y − y = 0 for any y , and we can always add 0 as many times as we want. So: 1 = 1 = 1 + 0 + 0 + 0 + ... = 1 + ( e − x − e − x ) + ( e − 2 x − e − 2 x ) + ( e − 3 x − e − 3 x ) + ... = 1 + e − x + e − 2 x + e − 3 x + ... − e − x − e − 2 x − e − 3 x + ... = ∞ summationdisplay n =0 e − nx − ∞ summationdisplay n =1 e − nx = ∞ summationdisplay n =1 e − ( n − 1) x − ∞ summationdisplay n =1 e − nx = ∞ summationdisplay n =1 ( e − ( n − 1) x − e − nx ) = ∞ summationdisplay n =1 (( e x − 1) e − nx ) = ( e x − 1) ∞ summationdisplay n =1 e − nx . So we get that 1 e x − 1 = ∞ summationdisplay n =1 e − nx . Use this to rewrite the integrand as an infinite sum of integrals. 1 (e) Next do integrationbyparts. Don’t evaluate the summation, integratebyparts on a generic integrand of the form u 3 e − nu . Evaluate the result at infinity and zero (i.e., actually do out the definite integral)....
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This note was uploaded on 11/09/2009 for the course AST 4700 taught by Professor Fernandez during the Spring '09 term at University of Central Florida.
 Spring '09
 Fernandez
 Astronomy

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