TY - JOUR

T1 - Line formation in moving media

T2 - Asymptotic expansions of some special functions

AU - Grachev, S. I.

N1 - Funding Information: I thank V. V. Ivanov for numerous comments aimed at improving this paper. This work was partially supported by a small grant from the American Astronomical Society and grant No. 96-15-96622 from the Russian Fund for Fundamental Research. Copyright: Copyright 2020 Elsevier B.V., All rights reserved.

PY - 1999

Y1 - 1999

N2 - Line formation in the spectrum of a moving medium with a spherical geometry is considered. In the Sobolev approximation there are some special functions that determine the source function and the force of radiation pressure in the line. The most important case is that of a small dimensionless velocity gradient (i.e., a large dimensionless Sobolev length τ) and a small ratio β of the opacity in the continuum to the opacity in the line. Until now there has been no detailed analytical information about the asymptotic behavior of these functions. For the case of a Doppler profile of the absorption coefficient, we clarify the nontrivial structure of their total asymptotic expansions for τ ≫ l, β ≪ 1, and arbitrary βτ. We give an algorithm for obtaining all the coefficients of these expansions and give explicit expressions for the first few coefficients. We also compare the asymptotic expansions with the numerical calculations of these functions available in the literature. We also briefly consider the case of a power-law decrease in the absorption coefficient in the line wing (and, in more detail, the case of Lorentz wings of the Voigt profile).

AB - Line formation in the spectrum of a moving medium with a spherical geometry is considered. In the Sobolev approximation there are some special functions that determine the source function and the force of radiation pressure in the line. The most important case is that of a small dimensionless velocity gradient (i.e., a large dimensionless Sobolev length τ) and a small ratio β of the opacity in the continuum to the opacity in the line. Until now there has been no detailed analytical information about the asymptotic behavior of these functions. For the case of a Doppler profile of the absorption coefficient, we clarify the nontrivial structure of their total asymptotic expansions for τ ≫ l, β ≪ 1, and arbitrary βτ. We give an algorithm for obtaining all the coefficients of these expansions and give explicit expressions for the first few coefficients. We also compare the asymptotic expansions with the numerical calculations of these functions available in the literature. We also briefly consider the case of a power-law decrease in the absorption coefficient in the line wing (and, in more detail, the case of Lorentz wings of the Voigt profile).

UR - http://www.scopus.com/inward/record.url?scp=53149114517&partnerID=8YFLogxK

U2 - 10.1007/BF02700944

DO - 10.1007/BF02700944

M3 - Article

AN - SCOPUS:53149114517

VL - 42

SP - 376

EP - 390

JO - Astrophysics

JF - Astrophysics

SN - 0571-7256

IS - 4

ER -