= Total hemispherical emissivity
{title2=$\epsilon(T)=M(T)/(\sigma T^4)$}
<Total hemispherical emissivity> integrates over all emitted <wavelengths> and the outward <hemisphere>. If $\epsilon_{\lambda,\Omega}$ is <spectral emissivity> resolved by direction, and $B_\lambda(T)$ is <blackbody> <spectral radiance> from <Planck's law>, then
$$
\epsilon(T)=\frac{\displaystyle\int_0^\infty\!\int_{\Omega_{\rm outward}}
\epsilon_{\lambda,\Omega}(T)B_\lambda(T)\cos\theta\,d\Omega\,d\lambda}{\sigma T^4}.
$$
It is a weighted average, not an unweighted mean of surface properties. Even if the wavelength-resolved material response stays unchanged, the <Planck's law> weighting changes with <temperature>, so the total ratio can change. Likewise, total <radiometric absorptance> averaged over radiation from hotter surroundings generally differs from this emission-weighted ratio. <Kirchhoff's law of thermal radiation> equates matched spectral and directional quantities; it does not equate arbitrary broadband averages.
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