Exact Solution of a Class of Nongrey Milne Problems.
John C. Stewart · The Astronomical Journal · 1962
The method of discrete ordinates developed in Chandrasekhar's Radiative Transfer can be applied to a class of idealized nongrey problems in which the absorption coefficient is separable into a function of depth times a function of frequency such that the fraction of the Planck spectrum occupied by each of the values of a (=mean free path/Planck mean free path) is independent of temperature. Such problems have previously been treated by several authors (e.g., King, Astrophys. J. 121, 711, 1955) in low orders of approximation, employing separate quadrature formulas in a and ~ space (i --cos0). By using the single variable a~ and uniformly spaced quadrature points, the characteristic equation for the problem can be solved with the aid of an approximating function which becomes exact as N (the number of points) increases without limit. The limb darkening and the source function for the Milne problem, expressed in terms of the roots and constants of the characteristic equation, can then be evaluated exactly as explicit real integrals; in the grey case the results of Placzek and Mark are recovered. The present derivation does not employ complex variables. Because of the nature of the assumed absorption coefficient, the method is applicable to (e.g.) the picket-fence, Elsasser, and statistical (Mayer-Goody) models of line absorption, but not strictly to the Lyman-continuum blanketing effect. Numerical results for a number of cases are exhibited.