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

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

View Full Document Right Arrow Icon
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
Image of page 1

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

View Full Document Right Arrow Icon
(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
Image of page 2
Image of page 3
This is the end of the preview. Sign up to access the rest of the document.

{[ snackBarMessage ]}

What students are saying

  • Left Quote Icon

    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.

    Student Picture

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

  • Left Quote Icon

    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.

    Student Picture

    Dana University of Pennsylvania ‘17, Course Hero Intern

  • Left Quote Icon

    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.

    Student Picture

    Jill Tulane University ‘16, Course Hero Intern