### Stefan-boltzmann constant

Not to be confused with 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

Template:Physconst

In cgs units the Stefan–Boltzmann constant is:

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

In US customary units the Stefan–Boltzmann constant is:

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

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:

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

where:

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

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

where:

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

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

A simple rule to remember the Stefan–Boltzmann constant is to think "5-6-7-8;" and try not to forget the negative sign before the final eight.