General integral operator for radiative transfer problems

General integral operator for radiative transfer problems is considered and will be denoted asK ^k {g(t)}.General theorems for theK-operator valid to any smooth functiong(t) will be established. The effect of the operator to the functions oftenly occur in transfer problems will also be established analytically and computationally.     

Generalised eddington approximation for radiative transfer problems in spherically-symmetric moving media

A moment method with three stream division of the radiation field was suggested by Wilson, Wan and Sen (1980) for solving radiative transfer problems in stationary, non-grey extended shells surrounding a central star. Use was made of the generalised Eddington relations as the closure conditions of the moment equations. In the present paper the same method has been utilised to study the radiative transfer problems in a non-grey, expanding gaseous spherical shells surrounding a central star. The transfer equation has been set in comoving frame in spherical geometry. The radiation and material quantities, angles and frequencies have been expressed in comoving frame. The mean intensity, flux and K-integrals have been calculated for extensive atmospheres in the presence of different velocity fields.     

Advances in radiative transfer

This review describes advances in radiative transfer theory since about 1985. We stress fundamental aspects and emphasize modern methods for the numerical solution of the transfer equation for spatially multidimensional problems, for both unpolarized and polarized radiation. We restrict the discussion to two-level atoms with noninverted populations for given temperature, density and velocity fields. Unfortunately this article was originally published with typesetter's errors: The correct publication date was 25 February 2006, not 3 January 2006. The content was not in the final form. The publishers wish to apologize for this mistake. The online version of the original version can be found at /10.1007/s00159-005-0025-8.     

Non-LTE polarized radiative transfer in intermediate magnetic fields: Numerical problems and results

This paper presents some numerical results relative to a solution, based on the density matrix formalism, of the non-LTE, polarized radiative transfer problem for a two-level atom. The results concern the atomic upper level population and alignment, and the emergent radiation Stokes profiles, for a plane-parallel, static, isothermal atmosphere embedded in a magnetic field of intermediate strength, such that the Zeeman splitting has to be taken into account in the line profile. Zeeman coherences are neglected, whereas magneto-optical effects are taken into account, resulting in a full 4×4 absorption matrix. Induced emission is neglected and complete frequency redistribution, in the rest and laboratory frames, is assumed. Pure Doppler absorption profile (gaussian shape) has also been assumed. The presentation of the results is preceded by a brief discussion of their accuracy and of the numerical difficulties that were met in the solution of the problem.On leave from the Dipartimento di Astronomia e Scienza dello Spazio, Università di Firenze, Largo E. Fermi 5, I-50125 Firenze, Italia     

Estimate of the accuracy of depth asymptotics in some problems of radiative transfer

Principal Astronomical Observatory, Ukrainian Academy of Sciences. Translated from Astrofizika, Vol. 34, No. 3, pp. 403–418, May–June, 1991.     

On nonstationary radiative transfer in a spectral line in stellar atmospheres

We consider nonstationary radiative transfer in a line in stellar atmospheres simulated as a stationary semi-infinite plane-parallel medium. We assume complete frequency redistribution in the elementary act of scattering. We assume that the time a photon spends in the medium is determined only by the mean time spent in the absorbed state. We obtain an explicit expression for the resolvent of the nonstationary integral equation of transfer, which is a bilinear expansion with respect to the eigenfunctions found in [12] for the corresponding stationary transfer equation.Translated fromAstrofizika, Vol. 37, No. 3, 1994.I am grateful to the American Astronomical Society for financial support of this work.     

Unified structure of analytical solutions for radiative transfer problems in plane-parallel,homogeneous media

The basic concepts for developing a system of analytic solutions for the standard problems of radiative transfer theory are discussed. These solutions, which are found using Ambartsumyan’s layer addition method in Sobolev’s probabilistic interpretation for radiative diffusion problems, are maximally compact and easily used in numerical computations. New expressions are obtained for the resolvents and the resolvent functions, as well as a unified structure for the form of an integral representation for solving different radiative transfer problems in semi-infinite media and in finite layers. Block diagrams of the sequence of stages for solving these problems are provided, where the Ambartsumyan function φ(η) (more precisely, 1/φ(η)) plays a fundamental role in the case of semi-infinite media while the functions a(η, τ₀ ) and b(η, τ₀) play an analogous role for finite layers.     

OnH-functions of radiative transfer

Chandrasekhar'sH-functionH(z) corresponding to the dispersion functionT(z)=| _rs –frs(z)|, where [f _rs (z)] is of rank 1, is obtained in terms of a Cauchy integral whose density functionQ(x, ₁, ₂,...) can be approximated by approximating polynomials (uniformly converging toQ(x)) having their coefficients expressed as known functions of the parameters _r 's. A closed form approximation ofH(z) to a sufficiently high degree of accuracy is then readily available by term by term integration.     

Probabilistic model for radiative transfer problems in cylindrical shell media with complete redistribution in frequency

A probabilistic model for solving transfer problems in non-homogeneous, isotropic, and non-coherent scattering cylindrical shell media has been proposed. The source function is considered to be frequency independent. The scattering and transmission functions have been defined for the case of complete redistribution in frequency. A tractable integrodifferential equation for the scattering function has been derived.     

An improved version of the implicit integral method to solving radiative transfer problems

Radiative transfer (RT) problems in which the source function includes a scattering-like integral are typical two-points boundary problems. Their solution via differential equations implies making hypotheses on the solution itself, namely the specific intensity I (τ; n) of the radiation field. On the contrary, integral methods require making hypotheses on the source function S(τ). It seems of course more reasonable to make hypotheses on the latter because one can expect that the run of S(τ) with depth is smoother than that of I (τ; n). In previous works we assumed a piecewise parabolic approximation for the source function, which warrants the continuity of S(τ) and its first derivative at each depth point. Here we impose the continuity of the second derivative S′′(τ). In other words, we adopt a cubic spline representation to the source function, which highly stabilizes the numerical processes.     

Factorization methods in radiative transfer theory

Published in Astrofizika, Vol. 38, No. 4, pp. 706–708, October–December, 1995.     

Numerical methods in polarized radiative transfer

This paper looks at three aspects of numerical methods for solving polarized radiative transfer problems associated with spectral line formation in the presence of a magnetic field. First we prove Murphy's law for Stokes evolution operators which is the basis of the efficient algorithm used in the SPSR software package to compute the Stokes line depression contribution functions. Then we use a two-stream model to explain the efficacy of the field-free method in which the non-LTE line source function in a uniform magnetic field is approximated by the source function neglecting the magnetic field. Finally we introduce a totally new and computationally efficient approach to solving non-LTE problems based on a method of sparsely representing integral operators using wavelets. As an illustration, the wavelet method is used to solve the source function integral equation for a two-level atomic model in a finite atmosphere with coherent scattering, ignoring polarization.     

The interaction principle in radiative transfer

We describe the interaction principle which is of fundamental importance to the theory of radiative transfer in one-, two-, and three-dimensional geometry. We also describe the practical difficulties associated with this principle in these geometries.     

A note on modified spherical-harmonic method for radiative transfer problems with reflective boundary conditions

The modified double-interval spherical-harmonic method is used to compute the radiative flux in a linearly anisotropically scattering plane-parallel medium with specularly and diffusely reflecting boundaries.