Emissivity

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The emissivity of a material (usually written $ε$ ) is the ratio of energy radiated by the material to energy radiated by a black body at the same temperature. It is a measure of a material's ability to absorb and radiate energy. A true black body would have an $ε = 1$ while any real object would have $ε < 1$ .

This emissivity depends on factors such as temperature, emission angle, and wavelength. However, a typical engineering assumption is to assume that a surface's spectral emissivity and absorptivity do not depend on wavelength, so that the emissivity is a constant. This is known as the grey body assumption. When dealing with non-black surfaces, the deviations from ideal black body behavior are determined by both the geometrical structure and the chemical composition, and follow Kirchhoff's law of thermal radiation: emissivity equals absorptivity (for an object in thermal equilibrium), so that an object that does not absorb all incident light will also emit less radiation than an ideal black body.

[edit] Astrophysical Greybody

The monochromatic flux density radiated by a greybody at frequency $ν$ through solid angle $d Ω$ is given by $F ν = B ν (T) Q ν d Ω$ where $B ν$ is the Plank function for a blackbody at temperature T and emissivity $Q ν$ .

For a uniform medium of optical depth $τ ν$ radiative transfer means that the radiation will be reduced by a factor $e - τ$ giving . The optical depth is often approximated by the ratio of the emitting frequency to the frequency where τ=1 all raised to an exponent β. For cold dust clouds in the interstellar medium β is approximately two. Therefore Q becomes,

$Q_{\nu} = 1 - e^{-\tau_{\nu}} = 1 - e^{-(\nu / \nu_{\tau=1} )^{\beta}}$