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Stefan-Boltzman Law Proof with Regards to Solar Radiation

  • #1

Homework Statement



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.

Homework 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]

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.
 

Answers and Replies

  • #2
Dick
Science Advisor
Homework Helper
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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:
  • #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.
 
  • #4
Dick
Science Advisor
Homework Helper
26,258
618
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.
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.
 
  • #5
Alright thanks. I'll give it a shot and see if I can come up with anything.
 

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