Stefan–Boltzmann law

Remarks

Proof idea: by Lambert's cosine law, the exitance is $\pi$ times the radiance, so $M = \pi \int_0^\infty B_\nu \, d\nu$ with $B_\nu$ from Planck's law. The substitution $x = h\nu / (k_B T)$ gives $M = \frac{2\pi k_B^4 T^4}{c^2 h^3} \int_0^\infty \frac{x^3}{e^x - 1} \, dx$. The integral equals $\Gamma(4) \zeta(4) = 6 \cdot \pi^4/90 = \pi^4/15$, by the Riemann zeta function. Josef Stefan inferred the $T^4$ law from measurements in 1879, and Ludwig Boltzmann derived it from thermodynamics in 1884.

Example: the surface of the Sun at about $5772$ K emits about $\sigma \cdot 5772^4 \approx 6.3 \times 10^7$ W per square metre. For a real surface with emissivity $\varepsilon$, the emitted power is $\varepsilon \sigma T^4$. The law needs absolute temperature: in degrees Celsius the formula is wrong.

Going deeper

For a student

The net radiation between a small grey body at $T_1$ and large surroundings at $T_2$ is $\varepsilon \sigma A (T_1^4 - T_2^4)$, a basic formula of radiative heat transfer. Linearising it gives a radiation heat transfer coefficient $4 \varepsilon \sigma T^3$ for small temperature differences. The law is used to estimate the temperatures of stars and planets from their power output.