### Rosseland mean opacity

**(Rosseland mean)**
(a useful weighted average of opacities at all the wavelengths)

The **Rosseland mean opacity** (or simply **Rosseland mean**) is
a weighted average (across frequency) of the opacity of
a material, e.g., gas or plasma through which electromagnetic radiation is
passing. It is a simplification allowing models to carry out a
single calculation covering all wavelengths rather than
varying the calculation by wavelength. Given the complexity of the
effect of a gas on a spectrum, it makes such models far more
tractable.
It is:

1 1 ∞ I_{ν}
——— = — ∫ —— dν
<κ> I 0 κ_{ν}

where:

- <κ> - Rosseland mean opacity
- ν - wavelength
- I
_{ν} - Intensity at wavelength ν
- I - total intensity
- κ
_{ν} - opacity at wavelength ν

The calculation averages (with weighting) the mean free path
of photons, i.e., the reciprocal of opacity, taking the
reciprocal of the result as the Rosseland mean. The Rosseland
mean depends upon the function of intensity over frequency, and
for a substance at thermodynamic equilibrium (or local thermodynamic equilibrium), the
intensity of black-body radiation of the temperature is
often used, i.e., the Planck function.
A model stellar atmosphere using this short cut
is called a gray atmosphere.

(*optics,stars*)
**Further reading:**

https://en.wikipedia.org/wiki/Opacity_(optics)#Planck_and_Rosseland_opacities

http://scienceworld.wolfram.com/physics/RosselandMeanOpacity.html

**Referenced by pages:**

gray atmosphere

mean molecular weight (μ)

opacity (κ)

optical depth (τ)

equation of radiative transfer (RTE)

Index