= Stefan-Boltzmann constant
{c}
{title2=$\sigma$}
The <Stefan-Boltzmann constant> converts the fourth power of absolute <temperature> into <blackbody> <radiant exitance>:
$$
\sigma=\frac{2\pi^5k_B^4}{15h^3c^2}
\approx5.670374419\times10^{-8}\,\mathrm{W\,m^{-2}\,K^{-4}}.
$$
Here $k_B$ is the <Boltzmann constant>, $h$ the <Planck constant> and $c$ the <speed of light>. In the current <International System of Units>, these three constants have exact defining values, so $\sigma$ is exact as an expression in them and $\pi$. The displayed finite decimal is an approximation, not an exact decimal or a measured uncertainty. The 2022 recommended-constants table lists an ellipsis and an exact uncertainty for $\sigma$.<Fundamental Physical Constants — Complete Listing by National Institute of Standards and Technology and others>[2022 table, Stefan-Boltzmann constant row; Boltzmann constant, Planck constant and speed of light in vacuum rows]{ref}
A dimensional check uses $[k_B]=\mathrm{J\,K^{-1}}$, $[h]=\mathrm{J\,s}$ and $[c]=\mathrm{m\,s^{-1}}$. Consequently $[k_B^4/(h^3c^2)]=\mathrm{J\,s^{-1}\,m^{-2}\,K^{-4}}=\mathrm{W\,m^{-2}\,K^{-4}}$, exactly the units needed in the <Stefan–Boltzmann law>.
For an equilibrium <photon gas>, the <energy density> is $u=a_{\rm rad}T^4$ with <radiation constant> $a_{\rm rad}=4\sigma/c$. The <radiation pressure> is $p_{\rm rad}=u/3=4\sigma T^4/(3c)$. These describe the radiation filling an equilibrium cavity; $u$ has units $\mathrm{J\,m^{-3}}$, whereas the outward <radiant exitance> through a small opening is $cu/4$. The factor $1/4$ comes from averaging propagation directions and their <projected area> over a radiation field with <isotropy>, not from reducing the opening's <emissivity>.
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