Emissivity

The emissivity of a material (usually written ε or e) is the ratio of energy radiated by a particular material to energy radiated by a black body at the same temperature. It is a measure of a material's ability to radiate absorbed energy. A true black body would have an $$\varepsilon=1$$ while any real object would have $$\varepsilon<1$$. Emissivity is a dimensionless quantity (does not have units).

In general, the duller and blacker a material is, the closer its emissivity is to 1. The more reflective a material is, the lower its emissivity. Highly polished silver has an emissivity of about 0.02.

Explanation
Emissivity depends on factors such as temperature, emission angle, and wavelength. 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.

Emissivity of earth's atmosphere
The emissivity of Earth's atmosphere varies according to cloud cover and the concentration of gases that absorb and emit energy in the thermal infrared (i.e., wavelengths around 8 to 14 micrometres). These gases are often called greenhouse gases, from their role in the greenhouse effect. The main naturally-occurring greenhouse gases are water vapor, carbon dioxide, methane, and ozone. The major constituents of the atmosphere, N2 and O2, do not absorb or emit in the thermal infrared.

Astrophysical graybody
The monochromatic flux density radiated by a greybody at frequency $$\nu$$ through solid angle $$d\Omega$$ is given by $$F_{\nu} = B_{\nu}(T) Q_{\nu} d\Omega $$ where $$B_{\nu}$$ is the Planck function for a blackbody at temperature T and emissivity $$Q_{\nu}$$.

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

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

Emissivity between 2 walls

$${\varepsilon}_{1,2}=\frac{1}{{\frac{1}{\varepsilon_1}}+{\frac{1}{\varepsilon_2}}-1}$$