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 in SI units by

σ = 5.670373(21)×10⁻⁸ W m⁻² K⁻⁴.

In cgs units the Stefan-Boltzmann constant is:

$\sigma \approx 5.6704 \times 10^{-5}\ \textrm{erg}\,\textrm{cm}^{-2}\,\textrm{s}^{-1}\,\textrm{K}^{-4}$.

In US customary units the Stefan-Boltzmann constant is :

$\sigma = 0.1714 \times 10^{-8}\ \textrm{BTU}\,\textrm{hr}^{-1}\,\textrm{ft}^{-2}\,\textrm{R}^{-4}$.

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 as:

$\sigma = \frac{2\pi^5k_{\rm B}^4}{15h^3c^2} = \frac{\pi^2k_{\rm B}^4}{60\hbar^3c^2} = 5.670373(21) \, \cdot 10^{-8}\ \textrm{J}\,\textrm{m}^{-2}\,\textrm{s}^{-1}\,\textrm{K}^{-4}$

where:

• k_B is the Boltzmann constant;
• h is the Planck constant;
• ħ is the reduced Planck constant;
• c is the speed of light in vacuum.

The CODATA recommended value is calculated from the measured value of the gas constant:

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

where:

• R is the Universal gas constant;
• 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);
• α is the fine structure constant.

A related constant is the radiation constant (or radiation density constant) a which is given by:

$a = \frac{4\sigma}{c} = 7.5657 \times 10^{-15} \textrm{erg}\,\textrm{cm}^{-3}\,\textrm{K}^{-4} = 7.5657 \times 10^{-16} \textrm{J}\,\textrm{m}^{-3}\,\textrm{K}^{-4} .$