Opacity

Quantitative definition

The words "opacity" and "opaque" are often used as colloquial terms for objects or media with the properties described above. However, there is also a specific, quantitative definition of "opacity", used in astronomy, plasma physics, and other fields, given here.

In this use, "opacity" is another term for the mass attenuation coefficient (or, depending on context, mass absorption coefficient, the difference is described here) ${\displaystyle {\kappa }_{\nu }}$ at a particular frequency ${\displaystyle \nu }$ of electromagnetic radiation.

More specifically, if a beam of light with frequency ${\displaystyle \nu }$ travels through a medium with opacity ${\displaystyle {\kappa }_{\nu }}$ and mass density ${\displaystyle \rho }$, both constant, then the intensity will be reduced with distance x according to the formula $${\displaystyle I(x)=I_0{e}^{-{\kappa }_{\nu }\rho x}}$$ where

• x is the distance the light has traveled through the medium
• ${\displaystyle I(x)}$ is the intensity of light remaining at distance x
• ${\displaystyle I_0}$ is the initial intensity of light, at ${\displaystyle x=0}$

For a given medium at a given frequency, the opacity has a numerical value that may range between 0 and infinity, with units of length²/mass.

Opacity in air pollution work refers to the percentage of light blocked instead of the attenuation coefficient (aka extinction coefficient) and varies from 0% light blocked to 100% light blocked:

$${\displaystyle \text{Opacity}=100\mathrm{\%}\left(1-\frac{I(x)}{I_0}\right)}$$

Planck and Rosseland opacities

It is customary to define the average opacity, calculated using a certain weighting scheme. Planck opacity (also known as Planck-Mean-Absorption-Coefficient) uses the normalized Planck black-body radiation energy density distribution, ${\displaystyle B_{\nu }(T)}$, as the weighting function, and averages ${\displaystyle {\kappa }_{\nu }}$ directly: $${\displaystyle {\kappa }_{Pl}=\frac{{\int }_0^{\mathrm{\infty }}{\kappa }_{\nu }{B}_{\nu }(T)d\nu }{{\int }_0^{\mathrm{\infty }}{B}_{\nu }(T)d\nu }=\left(\frac{\pi }{\sigma T^4}\right){\int }_0^{\mathrm{\infty }}{\kappa }_{\nu }{B}_{\nu }(T)d\nu ,}$$ where ${\displaystyle \sigma }$ is the Stefan–Boltzmann constant.

Rosseland opacity (after Svein Rosseland), on the other hand, uses a temperature derivative of the Planck distribution, ${\displaystyle u(\nu ,T)=\mathrm{\partial }{B}_{\nu }(T)/\mathrm{\partial }T}$, as the weighting function, and averages ${\displaystyle {\kappa }_{\nu }^{-1}}$, $${\displaystyle \frac{1}{\kappa }=\frac{{\int }_0^{\mathrm{\infty }}{\kappa }_{\nu }^{-1}u(\nu ,T)d\nu }{{\int }_0^{\mathrm{\infty }}u(\nu ,T)d\nu }.}$$ The photon mean free path is ${\displaystyle {\lambda }_{\nu }=({\kappa }_{\nu }\rho {)}^{-1}}$. The Rosseland opacity is derived in the diffusion approximation to the radiative transport equation. It is valid whenever the radiation field is isotropic over distances comparable to or less than a radiation mean free path, such as in local thermal equilibrium. In practice, the mean opacity for Thomson electron scattering is: $${\displaystyle {\kappa }_{\mathrm{e}\mathrm{s}}=0.20(1+X)\mathrm{c}{\mathrm{m}}^2{\mathrm{g}}^{-1}}$$ where ${\displaystyle X}$ is the hydrogen mass fraction. For nonrelativistic thermal bremsstrahlung, or free-free transitions, assuming solar metallicity, it is: $${\displaystyle {\kappa }_{\mathrm{f}\mathrm{f}}(\rho ,T)=0.64\times {10}^{23}(\rho [\mathrm{g}{\mathrm{c}\mathrm{m}}^{-3}])(T[\mathrm{K}]{)}^{-7/2}{\mathrm{c}\mathrm{m}}^2{\mathrm{g}}^{-1}.}$$ The Rosseland mean attenuation coefficient is: $${\displaystyle \frac{1}{\kappa }=\frac{{\int }_0^{\mathrm{\infty }}({\kappa }_{\nu ,\mathrm{e}\mathrm{s}}+{\kappa }_{\nu ,\mathrm{f}\mathrm{f}}{)}^{-1}u(\nu ,T)d\nu }{{\int }_0^{\mathrm{\infty }}u(\nu ,T)d\nu }.}$$