Emissivity

Definition

Let a surface be at absolute temperature $T$. Fix a wavelength $\lambda$ and a direction $(\theta, \phi)$, with $\theta$ the angle from the surface normal. Let $I_{\lambda}(\lambda, \theta, \phi, T)$ be the spectral radiance of the thermal radiation that the surface emits. Let $I_{\lambda, b}(\lambda, T)$ be the spectral radiance of a black body at the same temperature $T$.

1. The spectral directional emissivity is $\varepsilon_{\lambda, \theta}(\lambda, \theta, \phi, T) = I_{\lambda}(\lambda, \theta, \phi, T) / I_{\lambda, b}(\lambda, T)$.
2. The total hemispherical emissivity is $\varepsilon(T) = E(T) / E_b(T)$. Here $E(T)$ is the power emitted by the surface per unit area over all wavelengths and directions. And $E_b(T)$ is the same quantity for a black body at $T$.