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A method is described for solving the integral transfer equation in an infinite medium and in a homogeneous sphere with a point source at its center. Isotropic, coherent scattering is assumed. The Ambartsumian function is used for a semi-infinite medium. A nonintegrable singularity of the primary source function is eliminated.

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 to make hypotheses on the solution itself, namely the specific intensity I(tau;n) of the radiation field. On the contrary, integral methods require to make hypotheses on the source function S(tau). It looks of course more reasonable to make hypotheses on the latter because one can expect that the run of S(tau) with depth be smoother than that of I(tau;n). In previous works we assumed a piece-wise parabolic approximation for the source function, which warrants the continuity of S(tau) and its first derivative at each depth point. Here we impose the continuity of the second derivative S"(tau). In other words, we adopt a cubic spline representation to the source function, which highly stabilize the numerical processes., Accepted for publication in Astrophysics (2012, N.1)

A matrix transfer equation for multiple resonance scattering of radiation in a spectral line in a semiinfinite atmosphere with a uniform distribution of primary radiation sources is examined. A nonlinear matrix integral is obtained for this equation as a generalization of the Rybicki two-point Q-integral. One special case of the matrix $ {\mathbf{\hat{Q}}} $ -integral is the Volterra equation for the matrix source function of the problem discussed here. The Volterra equation is solved numerically for a Doppler profile of the absorption coefficient. Several polarization characteristics of the emerging radiation are obtained.

This is a discussion of V. A. Ambartsumyan’s studies of the mean number of scatterings for photons in scattering media and of further work and development in this area, especially at Ambartsumyan’s St. Petersburg school. The following questions are discussed briefly: (a) the traditional method for calculating the number of scatterings from the source function and critiques of this method. (b) The equation for the number N(τ; τ0 ) of scatterings for a photon born at optical depth τ in a plane layer of optical thickness τ0 and its use for calculating the number $\bar{N}$ of scatterings, averaged over the entire ensemble of photons for a medium with arbitrary internal sources. These questions are first considered for the case of monochromatic scattering, and then for scattering in a spectral line with complete frequency redistribution (CFR). (c) The mean path length $\bar{T}$ for a resonance line photon in a scattering medium with CFR and continuum absorption: the basic equations and asymptotic behavior of an optically thick layer. (d) A review of calculations of $\bar{N}$ and $\bar{T}$ in media that are so thick that the CFR approximation breaks down and the effects of partial frequency redistribution (PFR) become dominant. The presentation is at a semiquantitative level in many parts of this paper, with stress on physical significance rather than the mathematics, through the use of approximate and asymptotic solutions.

A new method is proposed to account for multiple scattering by electrons in calculations of the correlation functions describing the angular fluctuations in the cosmic microwave background radiation (CMBR). The apparatus of the theory of radiative transport with Rayleigh scattering is used. The problem is reduced to solving an integral equation for the vector source function (dependent only on time), along with differential equations for the other quantities (scalar potentials, baryon velocities, etc.) which show up in the problem. The quantities which describe the angular fluctuations in the CMBR (in the temperature and in the polarization) are then calculated by integrating the vector source function along the line of sight. As an illustration, the correlation functions and power spectra are calculated for the case where the fluctuations are produced by some initial gaussian perturbations of the CMBR.

The albedo-shifting method is used to solve the problem of radiative transfer at line frequencies. Radiative transfer with complete frequency redistribution in a plane-parallel, semi-infinite atmosphere is considered. It is shown that the method works well in this case and enables one to considerably improve the convergence in an iterative solution of the equation for the source function.

In the first of the series of papers by Ivanov et al. it was shown that the model problem of the transfer of polarized radiation as a result of resonance scattering from two-level atoms in a homogeneous plane atmosphere in the absence of LTE comes down, in the approximation of complete frequency redistribution, to the solution of an integral matrix equation of the Wiener-Hopf type for a (2 × 2) matrix source function S(τ). In the second paper in this series, devoted to the vector Milne problem, complete asymptotic expansions of the matrix I(z) [which is essentially a Laplace transform of the matrix S(τ)] for the case of a Doppler profile of the coefficient of absorption, and the coefficients of asymptotic expansions of S(τ) (τ » 1) are expressed in terms of coefficients of the expansions of I(z). We show that asymptotic expansions of S(τ) can be found directly from an integral matrix equation of the Wiener-Hopf type for S(τ). We give new recursive equations for the coefficients of these expansions, as well as a new derivation of asymptotic expansions of the matrix I, including its second column, which was considered only briefly by Ivanov et al.

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 » 1, 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).

A classical problem in the theory of radiative transfer is considered: calculating the radiation field within a plane scattering atmosphere. The recently proposed albedo-shifting method is used to calculate the source function both in a semi-infinite atmosphere and in an atmophere of finite optical depth, illuminated by parallel rays. The method enables one to “suppress” scattering and obtain iterative solutions of the integral equation for the source function in only a few direct lambda iterations, even when the average number of photon scatterings in the atmosphere is very large.

Questions relating to the formation of multicomponent blends in supernova spectra are discussed. An algorithm for calculating such blends for an arbitrary number of lines with allowance for the effects of nonlocal radiative coupling is described. The source functions, radiation pressure, and profiles of the multicomponent blends are investigated. In special cases, the results for the source function and the radiation pressure force are obtained analytically. The application of the method to the calculation of synthetic spectra of supernovas is discussed.

The above examples show that the method of calculating profiles of resonance lines based on the replacement of the exact equation for the source function (1) by an equation of diffusion type enables one to avoid the very laborious calculations but still gives a reasonable accuracy. In contrast to the small-parameter method, it can be generalized very simply to the case of more complicated geometries of the velocity field, and this makes it possible to consider more realistic models. The solution of some of these problems (the cases of spherical and cylindrical geometry) will be given in a following paper.

The problem of radiative transfer in an infinite three-dimensional stochastic medium is considered in the case when the probability I of a photon's surviving an elementary scattering event is a statistically homogeneous isotropic Gaussian field with exponential correlation function. A diagram technique is used to solve the problem. The first moment of the source function and the mean number of photon scatterings are found in the Bourret approximation. The case of a strongly correlated field is considered as a special case.

Besides the problems considered here, one can find comparatively easily asymptotic solutions in a problem with a spherically symmetric exponential distribution of the primary sources, and also in the case when the primary sources are distributed uniformly within a sphere of finite radius. However, we do not give these solutions because they are cumbersome. Note that some of these problems may have direct astrophysical applications. For example, the problem of a point source can arise in the determination of the source function in the problem of the emission of a planetary nebula.

The matrices A sup (-1) and T (tau sub O = 0.1 - 1.0) are tabulated, permitting a source function and emission line profile to be easily calculated. It is assumed that the radiation diffuses with complete redistribution in frequency.