The free-free absorption coefficient per unit volume derived from thermal…

PHYSICS · Multi-part · Fermi University Physics · Astrophysics & Cosmology · Radiation, radiative transfer and stellar atmospheres · Radiation, radiative transfer and stellar atmospheres

Stars from the Inside, Ch Hot and fast electrons: bremsstrahlung, Compton scattering and synchrotron emission

Problem

The free-free absorption coefficient per unit volume derived from thermal bremsstrahlung has the scaling ανff∝ρ2 T−1/2 ν−3(1−e−hν/kBT)\alpha_\nu^{ff} \propto \rho^2\,T^{-1/2}\,\nu^{-3}\left(1-e^{-h\nu/k_BT}\right) (density enters as ρ2\rho^2 because neni∝ρ2n_en_i\propto\rho^2 for gas of fixed composition). The opacity per unit mass is κνff=ανff/ρ\kappa_\nu^{ff}=\alpha_\nu^{ff}/\rho. ((a)) The Rosseland mean opacity is defined 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}. Substitute x=hν/kBTx=h\nu/k_BT everywhere (so ν∝xT\nu\propto xT and dν∝T dx\mathrm{d}\nu\propto T\,\mathrm{d}x), and show that every remaining integral over xx is a pure number independent of ρ\rho and TT, so that the full ρ\rho and TT dependence of κˉR\bar\kappa_R can be read off from the prefactors alone. Hence show κˉR∝ρ T−7/2\bar\kappa_R\propto \rho\,T^{-7/2}. ((b)) A layer of a red-giant envelope has ρ1=10 kg m−3\rho_1=10\,\mathrm{kg\,m^{-3}}, T1=2×106 KT_1=2\times10^{6}\,\mathrm{K} and Kramers opacity κˉ1=0.12 m2 kg−1\bar\kappa_1=0.12\,\mathrm{m^2\,kg^{-1}} there. Deeper in the envelope ρ2=50 kg m−3\rho_2=50\,\mathrm{kg\,m^{-3}} and T2=3×106 KT_2=3\times10^{6}\,\mathrm{K}. Using the scaling of part (a), find κˉ2\bar\kappa_2.

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