The Stefan-Boltzmann law

The total power radiated per unit area by a black-body at all frequencies is given by

$$P_{\rm tot}(T) = \int_0^\infty P(\omega) \, d\omega = \frac{... ...infty \frac{\omega^3\,d\omega} {\exp(\hbar\,\omega/k\,T) - 1},$$ (653)

or

$$P_{\rm tot}(T) = \frac{k^4\, T^4 }{4\pi^2\, c^2\,\hbar^3} \int_0^\infty \frac{\eta^3\,d\eta}{\exp\eta -1},$$ (654)

where $\eta = \hbar\,\omega/k \,T$. The above integral can easily be looked up in standard mathematical tables. In fact,

$$\int_0^\infty \frac{\eta^3\,d\eta}{\exp\eta -1} = \frac{\pi^4}{15}.$$ (655)

Thus, the total power radiated per unit area by a black-body is

$$P_{\rm tot}(T) = \frac{\pi^2}{60} \frac{k^4}{c^2\, \hbar^3} \,T^4 = \sigma \,T^4.$$ (656)

This $T^4$ dependence of the radiated power is called the Stefan-Boltzmann law, after Josef Stefan, who first obtained it experimentally, and Ludwig Boltzmann, who first derived it theoretically. The parameter

$$\sigma = \frac{\pi^2}{60} \frac{k^4}{c^2\, \hbar^3} = 5.67\times 10^{-8}\,\,{\rm W} \,{\rm m}^{-2} \,{\rm K}^{-4},$$ (657)

is called the Stefan-Boltzmann constant.

We can use the Stefan-Boltzmann law to estimate the temperature of the Earth from first principles. The Sun is a ball of glowing gas of radius $R_\odot\simeq 7\times 10^5$ km and surface temperature $T_\odot\simeq 5770^\circ$ K. Its luminosity is

$$L_\odot = 4\pi\, R_\odot^{~2} \, \sigma\, T_\odot^{~4},$$ (658)

according to the Stefan-Boltzmann law. The Earth is a globe of radius $R_\oplus\sim 6000$ km located an average distance $r_\oplus\simeq 1.5\times 10^8$ km from the Sun. The Earth intercepts an amount of energy

$$P_\oplus=L_\odot\,\frac{ \pi \,R_\oplus^{~2}/r_\oplus^{~2}}{4\pi}$$ (659)

per second from the Sun's radiative output: i.e., the power output of the Sun reduced by the ratio of the solid angle subtended by the Earth at the Sun to the total solid angle $4\pi$. The Earth absorbs this energy, and then re-radiates it at longer wavelengths. The luminosity of the Earth is

$$L_\oplus = 4\pi\, R_\oplus^{~2} \, \sigma\, T_\oplus^{~4},$$ (660)

according to the Stefan-Boltzmann law, where $T_\oplus$ is the average temperature of the Earth's surface. Here, we are ignoring any surface temperature variations between polar and equatorial regions, or between day and night. In steady-state, the luminosity of the Earth must balance the radiative power input from the Sun, so equating $L_\oplus$ and $P_\oplus$ we arrive at

$$T_\oplus = \left(\frac{R_\odot}{2\,r_\oplus}\right)^{1/2} T_\odot.$$ (661)

Remarkably, the ratio of the Earth's surface temperature to that of the Sun depends only on the Earth-Sun distance and the solar radius. The above expression yields $T_\oplus\sim 279^\circ$ K or $6^\circ$ C (or $43^\circ$ F). This is slightly on the cold side, by a few degrees, because of the greenhouse action of the Earth's atmosphere, which was neglected in our calculation. Nevertheless, it is quite encouraging that such a crude calculation comes so close to the correct answer.