On the radiative equilibrium of a stellar atmosphere

Abstract

In this paper the equation of transfer μdI/dτ=I-I₀; I₀=ƒ⁺¹_-1Idμ is solved by expanding I in terms of spherical harmonics in the form I(τ,μ)=Σ^∞_l=0I_l(τ)P_l(μ). It is shown how successively higher approximations to I can be obtained by retaining more and more terms in the foregoing expansion; thus, the first, second, and third approximations are obtained by retaining the first three, five, and seven terms, respectively, in the expansion. The first three approximations to I₀(τ) are explicitly obtained, and the corresponding laws of darkening derived. The exact function describing the law of darkening and given by Hopf's. theory is also numerically evaluated. A comparison of this exact law with those derived on our second and third approximations indicates that the method described in this paper is sufficiently rapidly convergent.