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Manuscript Title: Numerical evaluation of the formal solution of radiative transfer problems in spherical geometries.
Authors: D.G. Hummer, C.V. Kunasz, P.B. Kunasz
Program title: TRANSPHERE
Catalogue identifier: AAAB_v1_0
Distribution format: gz
Journal reference: Comput. Phys. Commun. 6(1973)38
Programming language: Fortran.
Computer: CDC 6400.
Operating system: KRONOS.
Program overlaid: yes
RAM: 17K words
Word size: 60
Keywords: Astrophysics, Radiative transfer, Spherical geometry, Formal solution of Radiative transfer Equation, Stellar atmospheres, Spherical line formation.
Classification: 1.3, 21.2.

Nature of problem:
Given the opacity and source function on a discrete grid of radius and frequency points, this program evaluates the formal solution of the radiative transfer equation in a spherically symmetric atmosphere surrounding a central core, which may be hollow or may absorb and emit radiation in any manner specified by the user. In addition to the radiation intensity, the first three moments of the intensity (J, H and K) and the Eddington factors K/J and H/J are evaluated at radii and frequency of interest. This program is designed to be used in the variable-Eddington-factor procedure.

Solution method:
The transfer equation for each frequency of interest is integrated along parallel rays (lines of constant impact parameter parallel to the line of sight) by using a second-order difference operator to replace the second derivative, thus obtaining a system of linear algebraic equations that includes the boundary conditions at the inner and outer radii. The resulting tridiagonal system is solved by gaussian elimination. Cubic splines are used to obtain the second-order difference operator, to evaluate the optical depth from the given opacity, and to evaluate the angle-integrals for the moments of the intensity. The latter operation is performed in a way that does not require large arrays of intensity data to be stored.

Discountinuities in the input functions or in their first two deivatives will not be accurately represented by the splines, although by properly choosing the impact parameter-radius grid these effects can be minimized. If an opaque radiating core is specified, a discontinuity in the radiation field can be produced at the impact parameter tangent to the core. While in realistic stellar models the distinction between core and shell is artificial and the discontinuity will not appear, there will be problems in which it must be faced. We have found that by carefully choosing the spatial grid, such problems can be satisfactorily solved; alternative procedures are discussed in section 2.4. The source function and opacity are assumed to be isotropic. Very large (>10**3) ratios of outer to inner radii and/or very large ratios between the maximum and minimum opacities at any point (>10**5) may require a prohibitive number of mesh points and may give rise to cancellation problems.

Running time:
For a case with 6 frequencies, 100 radius points and 58 impact parameters, the total execution time (on the CDC 6400) was 26.4 s, with the actual formal solution requiring 10.4 s. Of this time, 9.1 s was used for computation and 1.3 s for disc reading. If optical depth increments and quadrature weights have been previously calculated, total execution time drops to 12.8 s.