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Stefan-Boltzman Law Proof with Regards to Solar Radiation

  1. Sep 18, 2012 #1
    1. The problem statement, all variables and given/known data

    The unit area intensity of radiation from the Sun at the photosphere is 6.33*107 W/m2.

    a) Check this value using the calculus of the Stefan-Boltzman Law, assuming the Sun is a blackbody emitter ([itex]\epsilon[/itex] = 1) with a surface temperature of 5777K.

    2. Relevant equations

    Stefan-Boltzmann Law:
    [tex]E_b = \int_0^∞ \! E_{\lambda,b} \, \mathrm{d} \lambda = σT^4[/tex]
    Planck's Law:
    [tex]E_{\lambda, b} = {\frac{2\pi hc^2}{\lambda^5[e^{(hc/\lambda kT)}-1]}} [/tex]

    where:

    [itex]c = 2.998*10^{14} \ \mu m \ s^{-1}[/itex]
    [itex]h = 6.626 * 10^{-34} \ J \ s[/itex]
    [itex]k = 1.381 * 10^{-23} \ J/K [/itex]
    3. The attempt at a solution

    I'm not sure how to solve the integral of the Stefan-Boltzmann Law. I know I can substitute [itex]E_{\lambda, b}[/itex] from Planck's law into the Stefan-Boltzmann law, but I have no idea how to integrate it then. Integration by substitution fails here and I have to prove using calculus, that the sun's [itex]E_b[/itex] is equal to 6.33*107 W/m2. Thanks.
     
  2. jcsd
  3. Sep 18, 2012 #2

    Dick

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    It's not an elementary integral, as you've probably guessed. It involves zeta functions and stuff like that. You can reduce it to a dimensionless integral which contains the hard part. See http://en.wikipedia.org/wiki/Stefan–Boltzmann_law But I'm not sure you need to do that. Doesn't it just ask you to use the Stefan-Boltzann constant? You can look that up. Do they really mean you should derive it from Planck's Law?
     
    Last edited: Sep 18, 2012
  4. Sep 18, 2012 #3
    Thanks for the reply. It's certainly not. The Stefan-Boltzmann constant comes from the integration, right? No, it doesn't say I need to derive it from Planck's law. However, I don't see how to integrate it without the substitution.
     
  5. Sep 18, 2012 #4

    Dick

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    I don't think you need to do any integration. The Stefan-Boltzmann constant already includes the integral. Just put numbers in. It's a hard integral. Involves functions that aren't elementary. Stuff beyond just substitution and integration by parts.
     
  6. Sep 18, 2012 #5
    Alright thanks. I'll give it a shot and see if I can come up with anything.
     
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