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# 2.8: Stefan's Law (The Stefan-Boltzmann Law)

[ "article:topic", "Stefan\'s constant.", "authorname:tatumj", "Stefan-Boltzmann law" ]

The total exitance integrated over all wavelengths or frequencies can be found by integrating Equations 2.6.1 - 2.6.4. Integration of 2.6.1 over wavelengths or of 2.6.3 over frequencies each, of course, gives the same result:

$M= \sigma T^4 \label{Stefan}$

where

$\sigma = \dfrac{2\pi^5 k^4}{15h^3 c^2} = 5.6705 \times 10^{-8} \ \text{W m}^{-2} \text{K}^4$

Equation \ref{Stefan} is Stefan's Law, or the Stefan-Boltzmann law, and $$\sigma$$ is Stefan's constant.

Integration of Equation 2.6.2 over wavelengths or of 2.6.4 over frequencies each, of course, gives the same result:

$N = \rho T^3$

where

$\rho = \dfrac{4\pi \zeta (3) k^3}{h^3 c^2} = 1.5205 \times 10^{-8} \ \text{ph s}^{-1} \text{m}^{-2} \text{K}^{-3}$

Here $$\zeta(3)$$ is the Riemann zeta-function:

$\zeta(3) = 1 + \left(\dfrac{1}{2}\right)^3+ \left(\dfrac{1}{3} \right)^3 + \left(\dfrac{1}{4} \right)^3 + ... = 1.202057$