In the diffusion limit the flux carried at frequency is

PHYSICS · Subjective · Fermi University Physics · Astrophysics & Cosmology · Radiation, radiative transfer and stellar atmospheres · Stellar structure, energy generation and evolution

Stars from the Inside, Ch Radiative transfer solved: the grey atmosphere, the diffusion limit and the curve of growth

Problem

In the diffusion limit the flux carried at frequency ν\nu is Fν=−4π3κνρ∂Bν∂rF_\nu = -\dfrac{4\pi}{3\kappa_\nu\rho}\dfrac{\partial B_\nu}{\partial r}. (a) Integrate FνF_\nu over frequency to obtain the total flux F=−16σT33κˉRρdTdrF = -\dfrac{16\sigma T^3}{3\bar\kappa_R\rho}\dfrac{\mathrm{d}T}{\mathrm{d}r}, defining the Rosseland mean opacity κˉR\bar\kappa_R by 1κˉR=∫0∞κν−1 ∂Bν∂T dν∫0∞∂Bν∂T dν.\frac{1}{\bar\kappa_R} = \frac{\displaystyle\int_0^\infty \kappa_\nu^{-1}\,\dfrac{\partial B_\nu}{\partial T}\,\mathrm{d}\nu}{\displaystyle\int_0^\infty \dfrac{\partial B_\nu}{\partial T}\,\mathrm{d}\nu}. State why the weighting makes κˉR\bar\kappa_R a harmonic, not an arithmetic, mean. (b) A layer has opacity κb=5.0 m2 kg−1\kappa_b = 5.0\,\mathrm{m^2\,kg^{-1}} everywhere except in a narrow spectral window, where the opacity drops to κw=0.05 m2 kg−1\kappa_w = 0.05\,\mathrm{m^2\,kg^{-1}}. At the local temperature, the window carries a fraction fw=0.20f_w = 0.20 of the frequency-integrated weight ∂Bν/∂T\partial B_\nu/\partial T. Compute κˉR\bar\kappa_R for this layer, and compare it with the (wrong) arithmetic mean fwκw+(1−fw)κbf_w\kappa_w + (1-f_w)\kappa_b.

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