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The Rosseland mean opacity (or simply Rosseland mean) is a weighted average across EMR frequencies of the opacity of a material, e.g., a gas or plasma, through which EMR is passing. Using it in modeling is a simplification allowing such models to carry out a single calculation covering all frequencies rather than varying the calculation by frequency. Given the complexity of the effect of a gas on a spectrum, it makes such models far more tractable. It is defined by and over frequency:
1 1 ∞ Iν ——— = — ∫ —— dν <κ> I 0 κν
where:
(A different-but-equivalent formula can be worked out that calculates the same value from specific intensities by and over wavelength). The above-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 approximation is termed a gray atmosphere.