For an outward-going ray ( ) in a plane-parallel slab the transfer equation is

PHYSICS · Subjective · Fermi University Physics · Astrophysics & Cosmology · Radiation, radiative transfer and stellar atmospheres · Radiation, radiative transfer and stellar atmospheres

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

Problem

For an outward-going ray (μ>0\mu>0) in a plane-parallel slab the transfer equation is μ dI/dτ=I−S(τ)\mu\,\mathrm{d}I/\mathrm{d}\tau = I - S(\tau), with the boundary condition that I e−τ/μ→0I\,e^{-\tau/\mu}\to0 as τ→∞\tau\to\infty. (a) Using an integrating factor, derive the general formal solution I(τ,μ)=1μ∫τ∞S(t) e−(t−τ)/μ dtI(\tau,\mu) = \dfrac{1}{\mu}\displaystyle\int_\tau^\infty S(t)\,e^{-(t-\tau)/\mu}\,\mathrm{d}t for μ>0\mu>0. (b) A slab has a source function that decays with depth as S(τ)=S0 e−aτS(\tau) = S_0\,e^{-a\tau}, with S0=2.0×107 W m−2 sr−1S_0 = 2.0\times10^{7}\,\mathrm{W\,m^{-2}\,sr^{-1}} and a=2.0a = 2.0 (a dimensionless rate per unit optical depth). Use the formal solution to find I(0,μ)I(0,\mu) as a function of μ\mu, and evaluate it at μ=1\mu = 1 and μ=0.4\mu = 0.4.

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