The Mean* Intensity, J_{ν 0}

This concerns the mean intensity used in conjunction with the Einstein coefficients, i.e., the numbers of absorptions and induced emissions between two energy levels of some particle per unit volume per unit time which are given by

n₁B₁₂J_ν0 and n₂B₂₁J_ν0.

Since the energy levels are not infinitely sharp, the actual probabilities per unit time per eligible particle of an absorption or an induced emission in frequency range ν to ν + dν are

B₁₂J_νφ(ν) dν   and   B₂₁J_νφ(ν) dν,

where φ(ν) is the normalized "broadening function" which determines the shape of the coefficient profiles.  The total probabilities of absorptions and induced emissions per eligible particle for this particular transition are therefore

 B₁₂ò₀ J_νφ(ν) dν = B₁₂J_ν0  and  B₂₁ò₀ J_νφ(ν) dν = B₂₁J_ν0 .

It follows that

J_ν0 º ò₀ J_νφ(ν) dν .

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*Actually this is the "mean, mean intensity" or "mean² intensity" since J_ν = (1/4π)ò_4π I_ν dΩ is already the
  "mean intensity" averaged over all directions.  This second averaging is over frequency, weighted by the
  normalized broadening function φ(ν).  Hence

J_ν0 º (1/4π)ò₀ ò_4π I_ν(θ) dΩ φ(ν) dν .