Helene Frisch

Université Côte d'Azur · Observatory of the Cote d'Azur

The construction of exact solutions for radiative transfer equations is always based on the transformation of these equations into new ones, for which exact methods of solutions have been developed, in other branches of physics or in mathematical physics. For infinite media, as shown in Sect. 3.1, the integral equation for the source function can b...

This chapter describes the method introduced by N. Wiener and E. Hopf in 1931 to construct an exact solution for the famous Milne integral equation, describing the radiative equilibrium of a stellar atmosphere. First method able to provide exact expressions for solutions of convolution integral equations on a half-line, it is still in use in many b...

In this Chapter, we use the integral equation for the source function to determine large scale properties of the radiation field. We use it, in particular, to distinguish between ordinary and anomalous diffusion processes, to introduce the thermalization length as a characteristic scale of variation of the radiation field and to introduce new equat...

The preceding chapter contains a presentation of several equations, differential and integral ones, describing multiple scatterings of photons. In this chapter we present, more or less in a chronological order, a brief summary of exact methods of solution. They are described in detail in the next chapters, following an inverse chronological order,...

In this chapter we generalize to spectral lines formed with partial frequency redistribution, asymptotic methods developed in preceding chapters, when the frequencies of the absorbed and reemitted photons are either unchanged (monochromatic scattering) or completely uncorrelated (complete frequency redistribution). For the elementary redistribution...

We show in this chapter how some exact results concerning radiative transfer in a semi-infinite medium can be derived from the radiative transfer equation and from the Wiener–Hopf integral equation for the source function itself, with elementary methods, to quote V.V. Ivanov. They are elementary in the sense that they do not make use of complex pla...

In Chap. 20, we derive the thermalization length from a large scale analysis of the integral equation for the source function and in Chap. 21 from the behavior of the random walk of photons that have undergone a large number of scatterings. In the present chapter, we derive the thermalization length and other properties of the propagation of the ph...

We show in this chapter how to construct an exact solution for the Milne integral equation and for its generalized complete frequency redistribution version. After a presentation of the properties of the conservative auxiliary functions \({\cal L}(z)\) and X(z), we use the exact expressions of the resolvent function obtained in Chap. 6 to derive an...

In this chapter, we present the scalar radiative transfer equations used in Part I to illustrate exact method of solutions for radiative transfer equations in semi-infinite media. We also present different types of integral equations that can be derived from the integro-differential equations. All the equations are time-independent and one-dimensio...

As shown by S. Chandrasekhar in 1946, the Milne problem for conservative Rayleigh scattering in a plane-parallel semi-infinite medium has an exact solution and the emergent polarized radiation field can be expressed in terms of two auxiliary functions H l(μ) and H r(μ), with explicit expressions. We show in this chapter how to construct these funct...

For problems with no exact solution, it is still possible to define a half-space auxiliary matrix X(z), as we have shown in the Chap. 17. We show in the present chapter, that it is also possible to solve vector or matrix Wiener–Hopf integral equations with the method based on the solution of singular integral equations, described for the polarized...

The book is organized in three parts: Part I, is devoted to the presentation of exact methods of solution for scalar radiative transfer problems, and Part II to their generalization to vector and matrix problems for linearly polarized radiation fields. In Part I we deal only with the radiation field intensity, whereas in Part II we also consider th...

In the preceding chapters, we have introduced the scattering kernel K(τ), the dispersion function, defined as V (k) or ℒ(z), and a half-space auxiliary function X(z), needed to solve radiative transfer equations in a semi-infinite medium. In this chapter we analyze in detail the properties of these functions, which all play a role in the build up o...

In this Chapter and in Chap. 18, we consider the following scattering mechanisms: the Rayleigh scattering with true absorption (non-conservative), the resonance polarization, and the Hanle effect. For these processes, there is no explicit solution for radiative transfer problems in semi-infinite plane parallel media. They have to be solved numerica...

We have shown in Sect. 2.4 that the surface Green function G(τ) = G(τ, 0) = G(0, τ 0) and its regular part the resolvent function Φ(τ) = G(τ) − δ(τ) can be considered as building blocks for solutions of radiative transfer problems. The convolution integral equations satisfied by G(τ) and Φ(τ) have fairly simple inhomogeneous terms, for G(τ) it is δ...

In this Chapter we show how some exact and asymptotic properties of the radiation field, derived in preceding chapters from the radiative transfer equation, can be recovered and given a new interpretation by considering the random motions of the photons. We use for the random walk a simple, popular, and yet very powerful discrete random walk model,...

The method of singular eigenfunctions expansion has been introduced by K. Case in 1960 for monokinetic neutron transport. It is modeled after the Fourier approach to partial differential equations. The basic idea is to construct a set of eigenfunctions, solutions of a homogeneous transport equation. The eigenfunctions are used to expand arbitrary s...

In Chap. 20, it was shown that monochromatic scattering, and more generally scattering processes with a finite second order moment, can be described, asymptotically, by an ordinary diffusion process, whenever the photons undergo a very large number of scatterings. The proof was based on the analysis of the one-dimensional integral equation for the...

For scalar scattering problems, we showed in Part I that many exact results for semi-infinite media can be derived with the so-called resolvent method, based on convolution-type integral equations satisfied by the Green and the resolvent functions. For a polarized radiation field, these functions become matrices or vectors. We show in this Chapter...

Spectral lines are in general embedded in a continuous background created by photoionizations and recombinations, and free-free emission. In cool stars such as the Sun the dominant source of continuous absorption is the negative hydrogen ion H−, an ion with a single bound state of low binding energy. In this Chapter we show how a continuous absorpt...

The equations of radiative transfer for a field polarized by a scattering process were formulated in the late forties by S. Chandrasekhar and V.V. Sobolev. In this chapter, we present a few linearly polarized radiative transfer equations describing monochromatic Rayleigh scattering, resonance polarization, and the Hanle effect, and then show how to...

In Chaps. 20, 21 and 22 we have presented some asymptotic results concerning the formation of spectral lines formed with the Doppler and Voigt complete frequency redistribution, in particular that the random walk of the photons has the structure of a Lévy walk and that the large scale behavior of the source function S(τ) can be described by an inte...

We have shown in Chap. 2, that for a plane parallel medium the source function S(τ), for monochromatic scattering as well as for complete frequency redistribution, satisfies a convolution type integral equation. We recall that monochromatic scattering is relevant to the formation of continuous spectra and complete frequency redistribution to the fo...

In Chap. 16, it is shown that the polarized radiative transfer equation for monochromatic Rayleigh scattering in a conservative semi-infinite medium can be solved exactly. We give in particular exact expressions for the emergent Stokes parameters I and Q in terms of the auxiliary functions H l(μ) and H r(μ). When the medium is non-conservative, the...

In this chapter we present exact results for the emergent intensity and the source function, for complete frequency redistribution, relevant to the formation of spectral lines, and for monochromatic scattering, relevant to the formation of continuous spectra. We make use of results obtained in the preceding chapters for the Green and resolvent func...

To give some physical content to the polarized radiative transfer equations that are considered in the following chapters, we give in this chapter a short survey of the main properties of a polarized radiation and of three different scattering processes: the Rayleigh scattering, the resonance scattering of spectral lines, and its associated Hanle e...

Context . The continuous spectrum of stellar and planetary atmospheres can be linearly polarized by Rayleigh or Thomson scattering. The polarization rate depends on the ratio κc /( κc + σc ), κc and σc being the absorption coefficients due to photo-ionizations and scattering processes, respectively. The scattering process is conservative if κc = 0,...

Context. The linear polarization of a strong resonance lines observed near the solar limb is created by a multiple-scattering process. Partial frequency redistribution (PRD) effects must be accounted for to explain the polarization profiles. The redistribution matrix describing the scattering process is a sum of terms, each containing a PRD functio...

We show that the spherical tensor decomposition of the Stokes parameters developed for angle-averaged frequency redistribution functions can be extended to angle-dependent frequency redistribution functions. The irreducible components of the Stokes parameters loose their cylindrical symmetry but one can still write an integral equation for the irre...

It is shown that the intensity and polarization of spectral lines formed with complete frequency redistribution in non-LTE conditions can be described by an integral equation for a source vector depending only on optical depth. The origin of the scalar integral equation for the non-polarized case is recalled and then it is shown show how a suitable...

A √∈-law was demonstrated by Landi Degl'Innocenti & Bommier (1994) for resonance polarization in a magnetic atmosphere where the primary source of photons is of thermal origin (isotropic and unpolarized). In this paper we propose a generalized form of this law by dropping the hypothesis on the primary source of photons. We restrict ourselves to the...

A sqrt epsilon -law was demonstrated by Landi Degl'Innocenti & Bommier (1994) for resonance polarization in a magnetic atmosphere where the primary source of photons is of thermal origin (isotropic and unpolarized). In this paper we propose a generalized form of this law by dropping the hypothesis on the primary source of photons. We restrict ourse...

Resonance polarization, which is created by the scattering of an anisotropic radiation field in regions of zero or weak magnetic fields, is strongly dependent on the frequency redistribution taking place during the scatterings. Here we discuss the frequency redistribution matrix relevant to resonance lines, concentrating on linear polarization. Fir...

Singular integral equations with Cauchy type kernels are considered on a real interval, and a new approach to the numerical construction of their solutions is proposed. In particular, analytical solutions are given, so that a direct numerical evaluation becomes possible. The case when the known term is expressed by a power series is analyzed in det...

This paper considers monochromatic radiative transfer in a diffusive three dimensional random binary mixture. The absorption coefficient, along any line-of-sight is a homogeneous Markov process, which is described by a three-dimensional Kubo-Anderson process. The transfer equation is solved numerically by Monte-Carlo simulations on a massively para...

(1) Introduction. (2) Formation ETL des raies: (2.1) Intensité du rayonnement et équation de transfert. (2.2) Approximation de diffusion. (2.3) Equation de transfert ETL pour les raies. (3) Formation non-ETL des raies: (3.1) Fonction source d'un atome à deux niveaux. (3.2) Fonction de redistribution. (3.3) Equation de transfert non-ETL. (3.4) Analy...

It is proved that certain singular equations, which have no classical solutions because of singularities of the coefficients on the interval of integration, still have distributional solutions. The explicit form of these distributional solutions is presented.

The authors give an exact analytical solution of a statistical radiative transfer problem. This problem consists of time-independent, non-scattering transport in a binary statistical mixture. The mixing statistics are taken as homogeneous. The chord-length distribution of each constituent is assumed to be representable as a Laplace transform. Marko...

Polarized transfer asymptotic and first order escape probability methods developed for the nonpolarized case are generalized to include linear polarization produced by the scattering of anisotropic radiation in the absence of magnetic fields. The analyses are based on a coupled integral equation for two-angle-dependent source functions. Some genera...

This paper is devoted to a method of Cauchy integral equation for the solution of half-space convolution equations. It was introduced by Frisch and Frisch to solve Wiener-Hopf integral equations with algebraically decreasing kernels, arising in non-coherent transfer with complete frequency redistribution. The standard Wiener-Hopf technique, which r...

Some one-dimensional boundary value problems of the kinetic theory of gases can be solved analytically in closed form when the Boltzmann collision operator is replaced a simple kinetic model such as the linear BGK (Bhatnagar-Gross-Krook) model. Such are the slip-flow and the diffusion slip problems. Analytical solutions were obtained by Wiener-Hopf...

The propagation of EM radiation in gaseous media is characterized in chapters based on lectures presented at Leysin, Switzerland, on March 6-12, 1988. The emphases are on the fundamental astrophysics and the experimental and observational methods employed. Topics addressed include hot-star atmosphere models and their application to observations, ra...

General Introduction Asymptotic methods. Some general ideas Outline of the lectures Diffusive and Non-Diffusive Large Scale Behaviour Introduction Asymptotic analysis of the integral equation for the source function Specific intensity in the interior Transfer with Partial Frequency Redistribution Introduction The ingredients of a partial redistribu...

The problem is examined of constructing an approximation of the radiation field valid everywhere in a medium, by matching asymptotically the interior and the boundary-layer solution. Various frequency redistribution mechanisms are considered: complete redistribution and the four basic types of partial redistribution introduced by Hummer (1962), RI...

What follows is a summary and a review by the editors of the contents of the September 4th pannel discussion, which followed the first two sessions of the Workshop, devoted to general aspects of partial redistribution and its astrophysical implications. In presenting here our version of the main outlines of a long and articulate discussion, we hope...

A mean escape-probability approximation for resonance lines, which encompasses both the effectively thin and the effectively thick limits, is described. Global conservation of photons and the large scale diffusive behavior of the radiation field are reasonably well preserved in this approximation. This approximation is tested on a two-level atom by...

Resonance lines of ions with a large Z are broadened essentially by natural damping, and therefore photons with frequencies larger than a few Doppler widths are scattered almost coherently. This "coherent" scattering of resonance-line photons can be described by a space and frequency diffusion process when the mean number of scatterings undergone b...

A model of dust-driven wind relevant to red giant stars is investigated, in which the usual hypothesis of 'momentum coupling' (amounting to neglect of grain particle mass) is relaxed. When the momentum coupling approximation is abandoned, the sonic point is shifted outwards, and the gas and grain expansion velocities are reduced. In the supersonic...

A numerical method for investigating the possibility of blow-up after a finite time is introduced for a large class of nonlinear evolution problems. With initial data analytic in the space variable(s), the solutions have for any t > 0 complex-space singularities at the edge of an analyticity strip of width δ(t) Loss of regularity corresponds to the...

Scaling laws for conservative scattering in a finite slab are extracted from an asymptotic analysis of the integral equation for the source function. The solution is separated into an interior and a surface boundary layer part. The matching between the two parts provides a scaling law for the surface value of the source function. When expressed in...

An examination of the multiple scattering of resonance-line photons in nonconservative media, where the photons have a small probability of destruction at each scattering, yields an approximation for the mean number of scatterings on the basis of scaling arguments which takes into account both the destruction of photons and escape through boundarie...

Commission 36 acts as a cosponsor of the following Symposia: (1) IAU Symposium No. 102 “Solar and Stellar Magnetic Fields: Origin and Coronal Effects” Zurich, Switzerland (2-6 August 1982) and (2) IAU Symposium No. 103 “Planetary Nebulae” London, UK (10-1U August 1982). The commission participates jointly with Commissions 29, 35, and 45 in the orga...

Approximation procedures frequently used in handling self-absorption effects in the hydrogen emission lines of quasars are discussed and compared with an exact numerical treatment of line transfer. The model here is a finite slab with prescribed density and temperature. It is noted that if a medium has finite thickness and is such that subordinate...

Resonance-line scattering in static low density media with large optical thickness has a diffusive behavior in both space and frequency because photons belonging to the Lorentzian wings of the line may be scattered almost monochromatically a very large number of times. This diffusive behavior holds on frequency scales and spatial scales, χc and τc,...

Profiles of non-LTE lines broadened by a turbulent velocity field with a finite correlation length are computed by the 'effective source function' technique. This method can reformulate stochastic transfer as a standard non-LTE problem with an effective escape probability determined by averaging all realizations of velocity fields. Effective source...

Resonance line scattering in the presence of a source of continuous absorption is studied for very small values of the ratio of the continuous to line opacity, beta. Scaling laws for the thermalization length, the thermalization frequency, the mean number of scatterings and the mean path length are extracted from an asymptotic analysis of the equat...

The large-scale behavior of noncoherent radiative transfer with partial frequency redistribution is examined. Scaling laws for four basic scattering processes are extracted from an asymptotic analysis of the integral equation for the source function in the limit, epsilon approaches zero (where epsilon is the probability of photon destruction) to de...

Turbulence dominated by eddies of a finite size produces effects on a line profile which are similar to both macro- and micro-turbulence but which are at the same time neither. It is suggested that one of these effects in the Fourier-transform domain, namely the narrowing of the first natural side lobe relative to the width of the main lobe, can be...

An attempt is presented to explain the center-to-limb increase in width of solar lines on the basis of the theory of line formation in a turbulent medium where finite eddy-size is taken into account (Auvergne et al., 1973). Hydrodynamic velocities along the line of sight are represented by a step-wise constant stochastic process (Kubo-Anderson Proc...

A singnlar perturbation expansion method commonly used in boundary layer ; analysis is applied to the study of the coupling between thermal conduction and ; radiative transfer in a moving atmosphere. The expansion parameter is the ; conduction coefficiert assumed to be small in suitable dimensionless units. The ; model consists of a plane slab of f...

Our present knowledge on the average physical properties of the chromosphere and of the transition region between chromosphere and corona is reviewed. It is recalled that shock wave dissipation is responsible for the high temperatures observed in the chromosphere and corona and that, due to the non-linear character of the dissipation mechanism, no...

Context.It has been shown for the weak-field Hanle effect that the Stokes parameters $I$, $Q$, and $U$ can be represented by a set of six cylindrically symmetrical functions. The proof relies on azimuthal Fourier expansions of the radiation field and of the Hanle phase matrix. It holds for a plane-parallel atmosphere and scattering processes that c...

We study the influence of the line absoprtion on the cooling of the upper layers of the solar photosphere in the non-LTE approximation. We find that only the H and K lines of Ca II, the resonance lines of Mg II, and the Ca II triplet at 8500 Å may contribute to the cooling. We give 4400°K as a lower limit of the surface temperature.