Stefan–Boltzmann constant

The Stefan–Boltzmann constant (also Stefan's constant), a physical constant denoted by the Greek letter σ, is the constant of proportionality in the Stefan–Boltzmann law: the total energy radiated per unit surface area of a black body in unit time is proportional to the fourth power of the thermodynamic temperature.

The value of the Stefan–Boltzmann constant is given by

$\sigma =5.670400(40)\times 10^{-8}\ {\textrm {W}}\,{\textrm {m}}^{-2}\,{\textrm {K}}^{-4}$.

The value of the Stefan–Boltzmann constant is derivable as well as experimentally determinable; see Stefan–Boltzmann law for details. It can be defined in terms of the Boltzmann constant k_B as:

$\sigma ={\frac {2\pi ^{5}k_{\rm {B}}^{4}}{15h^{3}c_{0}^{2}}}={\frac {\pi ^{2}k_{\rm {B}}^{4}}{60\hbar ^{3}c_{0}^{2}}}$

where k_B is the Boltzmann constant, h is the Planck constant, ħ is the reduced Planck constant, and c₀ is the speed of light in a vacuum. The CODATA recommended value is calculated from the measured value of the gas constant R:

$\sigma ={\frac {2\pi ^{5}R^{4}}{15h^{3}c_{0}^{2}N_{\rm {A}}^{4}}}={\frac {32\pi ^{5}hR^{4}R_{\infty }^{4}}{15A_{\rm {r}}({\rm {e}})^{4}M_{\rm {u}}^{4}c_{0}^{6}\alpha ^{8}}}$

where N_A is the Avogadro constant, R_∞ is the Rydberg constant, A_r(e) is the "relative atomic mass" of the electron, M_u is the molar mass constant (1 g/mol by definition) and α is the fine structure constant.

A related constant is the radiation constant a which is given by:

$a={\frac {4\sigma }{c_{0}}}$.