Stefan-Boltzman Law Proof with Regards to Solar Radiation

[walterwhite]

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-Boltzmann Law, assuming the Sun is a blackbody emitter ($\epsilon$ = 1) with a surface temperature of 5777K.

Homework Equations

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

where:

$c = 2.998*10^{14} \ \mu m \ s^{-1}$
$h = 6.626 * 10^{-34} \ J \ s$
$k = 1.381 * 10^{-23} \ J/K$
​

The Attempt at a Solution

I'm not sure how to solve the integral of the Stefan-Boltzmann Law. I know I can substitute $E_{\lambda, b}$ 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 $E_b$ is equal to 6.33*10⁷ W/m². Thanks.

 

[Dick]

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?

 

[walterwhite]

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.

 

[Dick]

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.

 

[walterwhite]

Alright thanks. I'll give it a shot and see if I can come up with anything.