# homework8 - AST 3722C Spring 2008 Homework#8 Due before...

This preview shows pages 1–3. Sign up to view the full content.

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 3 c 2 × 1 e hν/kT 1 . You want to find an expression for S : S = π integraldisplay 0 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

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
(e) Next do integration-by-parts. Don’t 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). (f) You should wind up with something that depends on summationdisplay n =1 1 n 4 . It turns out that this expression equals π 4 / 90. Replace the summation with this expres- sion. At this point you should have an equation of the form S = CT 4 , where C is a constant that depends on a bunch of universal constants. Evaluate
This is the end of the preview. Sign up to access the rest of the document.

{[ snackBarMessage ]}

### What students are saying

• As a current student on this bumpy collegiate pathway, I stumbled upon Course Hero, where I can find study resources for nearly all my courses, get online help from tutors 24/7, and even share my old projects, papers, and lecture notes with other students.

Kiran Temple University Fox School of Business ‘17, Course Hero Intern

• I cannot even describe how much Course Hero helped me this summer. It’s truly become something I can always rely on and help me. In the end, I was not only able to survive summer classes, but I was able to thrive thanks to Course Hero.

Dana University of Pennsylvania ‘17, Course Hero Intern

• The ability to access any university’s resources through Course Hero proved invaluable in my case. I was behind on Tulane coursework and actually used UCLA’s materials to help me move forward and get everything together on time.

Jill Tulane University ‘16, Course Hero Intern