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) \kappa_\nu at a particular frequency \nu of electromagnetic radiation. More specifically, if a beam of light with frequency \nu travels through a medium with opacity \kappa_\nu and mass density \rho, both constant, then the intensity will be reduced with distance
x according to the formula I(x) = I_0 e^{-\kappa_\nu \rho x} where •
x is the distance the light has traveled through the medium • I(x) is the intensity of light remaining at distance
x • I_0 is the initial intensity of light, at 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: \text{Opacity} = 100\% \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, B_{\nu}(T), as the weighting function, and averages \kappa_\nu directly: \kappa_{Pl}={\int_0^\infty \kappa_\nu B_\nu(T) d\nu \over \int_0^\infty B_\nu(T) d\nu }=\left( { \pi \over \sigma T^4}\right) \int_0^\infty \kappa_\nu B_\nu(T) d\nu , where \sigma is the
Stefan–Boltzmann constant.
Rosseland opacity (after
Svein Rosseland), on the other hand, uses a temperature derivative of the
Planck distribution, u(\nu, T)=\partial B_\nu(T)/\partial T, as the weighting function, and averages \kappa_\nu^{-1}, \frac{1}{\kappa} = \frac{\int_0^{\infty} \kappa_{\nu}^{-1} u(\nu, T) d\nu }{\int_0^{\infty} u(\nu,T) d\nu}. The photon
mean free path is \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: \kappa_{\rm es} = 0.20(1+X) \,\mathrm{cm^2 \, g^{-1}} where X is the hydrogen mass fraction. For
nonrelativistic thermal bremsstrahlung, or free-free transitions, assuming solar
metallicity, it is: \kappa_{\rm ff}(\rho, T) = 0.64 \times 10^{23} (\rho[ {\rm g}~ {\rm\, cm}^{-3}])(T[{\rm K}])^{-7/2} {\rm\, cm}^2 {\rm\, g}^{-1}. The Rosseland mean
attenuation coefficient is: \frac{1}{\kappa} = \frac{\int_0^{\infty} (\kappa_{\nu, {\rm es}} + \kappa_{\nu, {\rm ff}})^{-1} u(\nu, T) d\nu }{\int_0^{\infty} u(\nu,T) d\nu}. ==See also==