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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, its just not easy. Start with B = 2 h 3 c 2 1 e h/kT 1 . You want to find an expression for S : S = integraldisplay B d, i.e., the integral of the Planck function over all frequencies (times steradians). (a) What are the units of B ? (b) What are the units of S ? (c) One way to do the integral is to do a substitution. Let u = h/kT . Rewrite S by inserting the Planck function and making it an integral over u instead of over . (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 integration-by-parts. Dont evaluate the summation, integrate-by-parts 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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