The Fn Method for Spectral-line Formation by Completely Noncoherent Scattering

ÐThe FN method is used to develop a solution to a class of nongrey problems in the theory of radiative transfer. The model considered allows for scattering with complete frequency redistribution (completely noncoherent scattering) and continuum absorption. In addition to a general formulation, a speci®c solution is developed for an inhomogeneous source term (Planck function) that varies linearly with optical depth in a semi-in®nite medium. Test problems based on Doppler and Lorentz pro®les of the line-scattering coecient are considered, and numerical results (thought to be correct to ®ve signi®cant ®gures) are given for the frequency-dependent intensity exiting the medium and for the source function within the medium. For comparison purposes, a previously reported solution that is expressed in terms of Chandrasekhar's H function is evaluated numerically. # 1998 Elsevier Science Ltd. All rights reserved 1. INTRODUCTION Some 25 years or so ago, McCormick and Siewert used an expansion in terms of singular eigenfunctions to solve analytically a class of nongrey problems in radiative transfer that was based on the equation of transfer written, after Hummer, as m @ @t Ix…t, m† ˆ ‰f…x† ‡ bŠ Sx…t† ÿ Ix…t, m† …1† where Sx(t) is the source function, ‰f…x† ‡ bŠSx…t† ˆ 1 2 $f…x† …1 ÿ1 f…x 0† …1 ÿ1 Ix 0 …t, m 0†dm 0dx 0 ‡ ‰rb‡ …1ÿ$†f…x†ŠB…t†, …2† and B(t) is the Planck function. We note that tr0 is the optical variable and m $ [ÿ1, 1] is the cosine of the polar angle (as measured from the positive t axis) that describes the direction of propagation of the radiation. In addition, $ $ [0, 1] is the albedo for single scattering, b>0 is the ratio of the continuum absorption coecient to the average line coecient, r is the ratio of the continuum source function to the Planck function and f(x) is the line-scattering pro®le. We note that while some of the quantities we compute here remain valid in the limit b 4 0, other quantities, as we shall see, do not. In fact the case b = 0 requires special attention since for some speci®c line-scattering pro®les certain integrals we use fail to exist for this case. For a speci®ed Planck function B(t) we seek a solution of Eq. (1) subject to the boundary condition that no radiation is entering the medium, i.e. Ix…0, m† ˆ 0 …3† for m $ (0,1]. In addition to providing a formulation to the problem we consider here, Hummer developed some asymptotic results, reported a solution based on a discrete-ordinates method and provided some of the ®rst numerical results for this challenging class of problems. We note also that Ivanov and colleagues have reported numerous works devoted to analytical and computational aspects of this problem; Ref. 4 in particular provides an excellent entry into this body of work. J. Quant. Spectrosc. Radiat. Transfer Vol. 60, No. 2, pp. 261±276, 1998 # 1998 Elsevier Science Ltd. All rights reserved Printed in Great Britain 0022-4073/98 $19.00+0.00 PII: S0022-4073(97)00171-4