An Introduction to Modern Astrophysics · Problem 9.4.1

Stellar Atmospheres — Carroll & Ostlie

Physics — Astrophysics · radiative-transfer-equation

Problem

To see how the intensity of a light ray tends to become equal to the local value of the source function, imagine a beam of light of initial intensity Iλ,0I_{\lambda,0} at s=0s = 0 entering a volume of gas of constant density, ρ\rho, that has a constant opacity, κλ\kappa_\lambda, and a constant source function, SλS_\lambda. Show that the transfer equation (Eq. 9.34) may be easily solved for the intensity of the light as a function of the distance ss traveled into the gas: Iλ(s)=Iλ,0 e−κλρs+Sλ(1−e−κλρs).I_\lambda(s) = I_{\lambda,0}\, e^{-\kappa_\lambda \rho s} + S_\lambda(1 - e^{-\kappa_\lambda \rho s}).

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