Transfer of Radiation. II. Radiative Transfer in Absorption Lines.
Abstract
The equation of the formation of absorption lines in the MilneEddington model of a stellar atmosphere has been solved by an extension of the method previously described (A p. J. 110, 1,1949). The equation of transfer in the standard case is a 1i 'ir)di#X(a$br) MI (T,I#) =I (T, ) where r is the optical depth in the total (line and continuous) absorption, X is the ratio of the continuous to the total absorption coefficient, and a and b are the first two coefficients of the Taylor exparision of the Planck function in powers of r. The authors assume an expansion of the "source function," J(r) of the form J(r) =a+b $ce + AjKj+ (T), where m is the positive root of the equation m = (t  X) tanli3 m, and K (r) is the exponential integral of the nth order, defined by Kn(r) =f1 dx. The A `s and c in J(r) are constants which are found to be the solutions of a set of n simultaneous linear equations in the nth order. Finally, the quantities r and R, defined as the ratios of the emergent intensities and fluxes in the line and continuum respectively, are obtained as functions of , p (frequency), and the A `s. The equations have been solved for a fourterm expansion of J(r), and the values of R and r obtained have been tabulated for the standard value of X = 0.2 and various values of the parameter x, which is a function of the frequency p They are more accurate than the values obtained by Chandrasekhar (Ap. J., 100, 355, 19 ) in the third approximation b Gaussian quadrature and comparable in accuracy to the values obtained by him (Ap. J.,106,145, 194 ) from the H(j#) function and its moments. A special feature of the method is that it is not restricted to the atmospheric surface but enables one to evaluate the "source function" J(r) for any r.
 Publication:

The Astrophysical Journal
 Pub Date:
 May 1951
 DOI:
 10.1086/145418
 Bibcode:
 1951ApJ...113..482M