Radiative Transfer in Inhomogeneous Atmospheres. III

The approach proposed in the previous parts of this series of papers is used to solve the radiative transfer problem in scattering and absorbing multicomponent atmospheres. Linear recurrence relations are obtained for both the reflectance and transmittance of these kinds of atmospheres, as well as for the emerging intensities when the atmosphere contains energy sources. Spectral line formation in a one-dimensional inhomogeneous atmosphere is examined as an illustration of the possibility of generalizing our approach to the matrix case. It is shown that, in this case as well, the question reduces to solving an initial value problem for linear differential equations. Some numerical calculations are presented.     

Group-theoretical description of radiative transfer in one-dimensional media

Group theory is used to describe a procedure for adding inhomogeneous absorbing and scattering atmospheres in a one-dimensional approximation. The inhomogeneity originates in the variation of the scattering coefficient with depth. Group representations are derived for the composition of media in three different cases: inhomogeneous atmospheres in which the scattering coefficient varies continuously with depth, composite or multicomponent atmospheres, and the special case of homogeneous atmospheres. We extend an earlier proposal to solve problems in radiative transfer theory by first finding global characteristics of a medium (reflection and transmission coefficients) and then determining the internal radiation field for an entire family of media without solving any new equations. Semi-infinite atmospheres are examined separately. For some special depth dependences of the scattering coefficients it is possible to obtain simple analytic solutions expressed in terms of elementary functions. An algorithm for numerical solution of radiative transfer problems in inhomogeneous atmospheres is described.     

Diffuse reflection of line radiation from a semi-infinite turbulent atmosphere

The diffuse reflection of line radiation from a one dimensional semi-infinite turbulent atmosphere is examined in two limiting regimes of micro-and macroturbulence. Ambartsumyan’s invariance principle is used to solve this problem. In addition to the observed spectral line profile, statistical averages describing the diffusion process in the atmosphere (mean number of scattering events, average time spent by a photon in the medium) are determined. The dependence of these quantities on the average hydrodynamic velocity and scattering coefficient is studied. It is shown that in the microturbulent regime the intensity at the line center depends only slightly on the mean nonthermal velocity. In both regimes, photons in the far wings undergo scattering more frequently than in a static atmosphere, although they spend, on average, less time in the medium. __________ Translated from Astrofizika, Vol. 50, No. 3, pp. 391–403 (August 2007).     

Radiative Transfer in Inhomogeneous Atmospheres. I

This series of papers is devoted to multiple scattering of light in plane parallel, inhomogeneous atmospheres. The approach proposed here is based on Ambartsumyan's method of adding layers. The main purpose is to show that one can avoid difficulties with solving various boundary value problems in the theory of radiative transfer, including some standard problems, by reducing them to initial value problems. In this paper the simplest one dimensional problem of diffuse reflection and transmission of radiation in inhomogeneous atmospheres with finite optical thicknesses is considered as an example. This approach essentially involves first determining the reflection and transmission coefficients of the atmosphere, which, as is known, are a solution of the Cauchy problem for a system of nonlinear differential equations. In particular, it is shown that this system can be replaced with a system of linear equations by introducing auxiliary functions P and S. After the reflectivity and transmissivity of the atmosphere are determined, the radiation field in it is found directly without solving any new equations. We note that this approach can be used to obtain the required intensities simultaneously for a family of atmospheres with different optical thicknesses. Two special cases of the functional dependence of the scattering coefficient on the optical thickness, for which the solutions of the corresponding equations can be expressed in terms of elementary functions, are examined in detail. Some numerical calculations are presented and interpreted physically to illustrate specific features of radiative transport in inhomogeneous atmospheres.     

Radiative transfer in a semi-infinite plane parallel atmosphere with an inhomogeneous source function

We investigate the multiple scattering of radiation in semi-infinite homogeneous atmosphere when the sources of the radiation are distributed inhomogeneous, for example, are created by restricted beams penetrating into the medium. The case of isotropic scattering is considered. It is shown that the density of radiation and the intensity of outgoing radiation for any forms of the sources can be represented as some integrals with the real and imaginary parts of the universal H-function, which satisfies the nonlinear integral equation. We calculated the intensity of radiation emerging from the surface after multiple scattering for the case when a beam with a finite radius incident perpendicular on the medium surface. The results allowed us to estimate quantitatively when the intensity of outgoing radiation in the center of a beam coincides with that for the classical case of unbounded flux (the case considered by Chandrasekhar et al.). We compared our exact solutions with those in the diffusion approximation. For conservative medium the difference is ?20–30%, depending on the particular forms of the radiation sources. For absorbing medium the difference is much larger. Our exact semi-analytical solution can be generalized for the cases of multiple anisotropic scattering of the polarized beams. The presented simple theory can be used at the consideration of close binary systems, flare stars etc.     

Asymmetry of compton scattering by relativistic electrons with an isotropic velocity distribution: Astrophysical implications

The angular distribution of low-frequency radiation after a single scattering by relativistic electrons with an isotropic velocity distribution differs markedly from the Rayleigh angular function. In particular, the scattering by an ensemble of ultrarelativistic electrons is described by the law p=1?cosα, where α is the scattering angle. Thus, photons are mostly scattered backward. We discuss some consequences of this fact for astrophysical problems. We show that a hot atmosphere of scattering electrons is more reflective than a cold one: the fraction of incident photons reflected after a single scattering can be larger than that in the former case by up to 50%. This must affect the photon exchange between cold accretion disks and hot coronae (or advective flows) near relativistic compact objects, as well as the rate of cooling (through multiple inverse-Compton scattering of the photons supplied from outside) of optically thick clouds of relativistic electrons in compact radio sources. Scattering asymmetry also causes the spatial diffusion of photons to proceed more slowly in a hot plasma than in a cold one, which affects the shapes of Comptonization spectra and the time delay in the detection of soft and hard radiation from variable X-ray sources.     

The solution of a time-dependent problem in radiative transfer

The time-dependent equation of radiative transfer for a finite, plane-parallel, non-radiating, and isotropically scattering atmosphere of arbitrary stratification is solved by using the integral equation method. The medium is taken to be inhomogeneous. The Laplace transform is used in the time domain. It is seen that the obtained solutions are reducible to the corresponding ones for steady-state problems by simply changing the Laplace transform parameter to zero.     

Lyman α radiative transfer in a multiphase medium

Hydrogen Lyman α (Lyα) is our primary emission-line window into high-redshift galaxies. Despite an extensive literature, Lyα radiative transfer in the most realistic case of a dusty, multiphase medium has received surprisingly little detailed theoretical attention. We investigate Lyα resonant scattering through an ensemble of dusty, moving, optically thick gas clumps. We treat each clump as a scattering particle and use Monte Carlo simulations of surface scattering to quantify continuum and Lyα surface scattering angles, absorption probabilities, and frequency redistribution, as a function of the gas dust content. This atomistic approach speeds up the simulations by many orders of magnitude, making possible calculations which are otherwise intractable. Our fitting formulae can be readily adapted for fast radiative transfer in numerical simulations. With these surface scattering results, we develop an analytic framework for estimating escape fractions and line widths as a function of gas geometry, motion, and dust content. Our simple analytic model shows good agreement with full Monte Carlo simulations. We show that the key geometric parameter is the average number of surface scatters for escape in the absence of absorption,     , and we provide fitting formulae for several geometries of astrophysical interest. We consider the following two interesting applications. (i) Equivalent widths ( EWs ). Lyα can preferentially escape from a dusty multiphase interstellar medium if most of the dust lies in cold neutral clouds, which Lyα photons cannot penetrate. This might explain the anomalously high EWs sometimes seen in high-redshift/submillimetre sources. (ii) Multiphase galactic outflows . We show the characteristic profile is asymmetric with a broad red tail, and relate the profile features to the outflow speed and gas geometry. Many future applications are envisaged.     

The polarization of radiation reflected diffusely by an inhomogeneous plane medium

The problem of determining the intensity and the degree of polarization of radiation emerging from an inhomogeneous finite plane medium for the case of Rayleigh scattering with internal energy source is considered. A system of coupled integral equations are obtained and solved by the Galerkin method. The degree of polarization for homogeneous and inhomogeneous media are calculated for uniform and nonuniform sources.     

Average photon path-length in isotropically scattering finite atmospheres

The determination of the average path-length of photons in a finite isotropically scattering plane-parallel homogeneous atmosphere is discussed. To solve this problem we have used the kernel approximation method which easily allows us to find the derivatives of the intensity with respect to optical depth, optical thickness and albedo of single scattering.In order to check the results we have used another approach by exploiting the set of integrodifferential equations of Chandrasekhar for theX- andY-functions. This approach allows us to find the average path length only at the boundaries of the atmosphere but on the other hand it gives also the dispersion of the path-length distribution function, thus generating the input parameters for determining the approximate path-length distribution function. It occurred that the set so obtained is stable and the results are highly accurate.As a by-product we obtain the first two derivatives of theX- andY-functions with respect to the albedo of single scattering and optical thickness, and the mixed derivative.     

A Monte Carlo study of polarization structures in the Thomson-scattered line radiation

Thomson scattering is often invoked to explain broad wing features that are seen in various objects including active galactic nuclei and symbiotic stars. Despite the wavelength-independent scattering cross-section of Thomson scattering, the line flux may exhibit wavelength-dependent linear degree of polarization, because various parts of emission wings are contributed by photons with different scattering numbers. Specifically, more scattered and hence more weakly polarized photons tend to fill the farther wing parts from the line centre, while the neighbourhood of the line centre is dominated by less-scattered photons with higher degree of polarization. Using a Monte Carlo technique, we investigate the polarization structure of Thomson-scattered line radiation. A detailed analysis of polarization structure formation is conducted by investigating the dependence of the polarization and profile width on the scattering number for various finite electron scattering slabs. Significantly varying degree of polarization is obtained when the scattering medium has Thomson optical depth  τ_Th≥ 1  . We present our high-resolution spectrum of the symbiotic star V1016 Cyg obtained with the Bohyunsan Optical Echelle Spectrograph (BOES) in order to fit the broad profile around Hα by electron scattering wings adopting an oblate spheroidal geometry with Thomson optical depth  τ_Th= 0.5  and electron temperature   T _e= 6.2 × 10⁴  K  . Local maxima in the linear degree of polarization of Thomson-scattered line radiation are expected to appear in the spectral regions characterized by the average scattering number ≃1.     

The radiation field in a semi-infinite atmosphere containing energy sources

The problem of determining the radiation field, in a medium in which the energy sources are distributed in depth according to the power law, under broad assumptions concerning the elementary scattering process, has been reduced to a solution of similar problem in an isothermal atmosphere. In addition, the radiation intensity at a specified point in an isothermal atmosphere is related with that in an atmosphere, illuminated by continuum isotropic radiation. This fact, in particular, enables one to express the intensity of the outgoing radiation for an arbitrary distribution of the internal energy sources in terms of Ambartsumian's -function, or in terms of -functions obtained in more complicated situations, characterized by anisotropic scattering, the general case of frequency redistribution etc. For illustration, the classical Milne-Eddington problem of spectral-line formation is considered.     

Rayleigh-Cabannes scattering in planetary atmospheres II. Constant internal sources in nonconservative atmospheres

The vector equation of radiative transfer is solved for non-conservative homogeneous plane-parallel atmosphere using the method of discrete ordinates. The scattering processes in the atmosphere bounded by a Lambert bottom are described by the Rayleigh-Cabannes phase matrix. The primary radiation field is generated by constant internal sources. A package of FORTRAN subroutines is compiled to find the axial radiation field for such an atmosphere at arbitrary optical depth.     

Reflection and transmission by a vertically inhomogeneous planetary atmosphere

By appealing to the reciprocity principle simple expressions are derived for the plane albedo and the transmissivity of a vertically inhomogeneous, plane parallel atmosphere. The plane albedo is shown to equal the angular distribution of the reflected intensity for isotropie Illumination of unit intensity incident at the top of the atmosphere, while the transmissivity equals the angular distribution of the transmitted intensity for isotropie illumination of unit Intensity incident at the bottom of the atmosphere. Chandrasekhar's solution of the planetary problem (including a Lambert reflecting lower boundary) in terms of the solution to the standard problem (no reflecting ground) is extended to apply to an inhomogeneous atmosphere resting on a surface that reflects radiation anisotropically but with no dependence on the direction of incidence (anisotropic Lambert reflector). The computational aspects are discussed and a procedure for computing the planetary albedo and transmissivity Is outlined for a vertically inhomogeneous, anisotropically scattering atmosphere overlying a partially reflecting surface. Numerical verification and illustration are also provided and it is shown that the assumed vertical variation of the single scattering albedo strongly affects the plane albedo but only weakly the transmissivity.     

Line formation in spherical media with partial frequency redistribution

The effects of partial redistribution of frequency on the formation of spectral lines in a static and spherically symmetric media have been investigated. The partial redistribution functionsR _I andR _II (Hummer, 1962) have been employed to calculate the lines for a two-level atom in non-LTE in a spherically symmetric medium with homogenous physical characteristics whose ratiosB/A (of outer to inner radii) are equal to 2 and 10. These results are compared with those formed in a plane-parallel medium withB/A=1. Two types of atmosphere are treated: (1) a pure scattering medium with =0 and =0, and (2) an atmosphere with a constant source of emission =10^–4 and =0, where is the probability per scatter that a photon will be destroyed by collisional de-excitation and is the ratioK _c/K _l of opacity due to continuous absorption per unit interval of frequency to that in the line. Lines formed in complete redistribution also have been calculated for the sake of comparison, and the total optical depth in all cases has been taken to be 10³ at the line centre.Vast differences have been found between the lines formed by complete and partial redistribution functions (which, for the sake of simplicity, we shall hereafter refer to as CRD and PRD, respectively). In the case of a purely scattering medium, a small amount of emission is observed in the wings for all cases of scattering functions in the spherical medium as a result of the combined effects of curvature and physical scattering. In the scattering medium, more photons are scattered into the cores of the lines by PRD than in the case of CRD. The lines formed in the medium with internal sources show emission in all cases with small absorption in the cores, except those lines formed by the angle-dependent PRD functions which again depend on the geometrical extension of the medium.     

Sources of error in the photoclinometric determination of planetary topography: A reappraisal

The technique of photoclinometry has frequently been used to determine planetary topography without proper consideration of possible sources of error. Previous studies of error sources have been limited in extent and have overlooked the importance of factors such as atmospheric scattering and the choice of a surface photometric function. This paper adopts a thorough and more direct approach to error analysis, whereby known topography is compared with photoclinometric profiles derived from synthetic quantised reflectance scans.Instrumental and geometric sources of error are found to exert a minimal influence on profiles in practice, provided that sufficient care is taken in the selection of images and the extraction of scans from those images. Environmental factors — relating to the scattering properties of the surface and, if present, atmosphere — are far more important. It is found that a simple Lommel-Seeliger law is unlikely to be appropriate to the majority of planetary terrains, given its inability to model the effects of multiple scattering or unresolved macroscopic roughness. It is further demonstrated that a Minnaert function or combination of Lommel-Seeliger and Lambert laws may empirically compensate for the first of these phenomena but not the second; in this respect, Hapke's equation is a far superior model of surface optical properties. In the case of an atmosphere, the need to correct for scattering by aerosols or suspended dust becomes more acute as atmospheric opacity increases and as particle scattering becomes more forward-biased. To perform this correction, a model for the combined reflectance of surface and atmosphere must be used when deriving profiles.Two case studies — of a small impact crater on Triton and a dust-mantled basaltic lava flow on Mars - are presented here. Regarding the latter, the implications that errors in photoclinometric flow thickness measurements have for inferred lava rheology are examined. Conservative estimates of errors in yield strength and apparent viscosity easily exceed 100% when one of the simplest photometric models possible — a Lommel-Seeliger law — is used to derive a profile.In the light of these findings, strategies are suggested for improving the results obtained from photoclinometry in the future.     

Intensity Fluctuations of Radiation Escaping from a Multicomponent Stochastic Atmosphere. I

This is the first part of a paper devoted to a theoretical investigation of intensity fluctuations of radiation at the frequencies of a spectral line formed in a multicomponent stochastic atmosphere. It is assumed that the optical depth of structural elements and the power of the energy sources contained in them undergo random variations. The frequency dependence of the relative mean-square deviation of the intensity of radiation escaping from the atmosphere is determined. Two special cases are considered and it is shown that the behavior of this quantity is different, depending on which of the indicated characteristics of the medium undergoes random variations. The results make it possible to judge the character of random variations in the fine structure of a radiating medium from observations of it in the cores and wings of spectral lines. Recent observations of prominences made using the SUMER spectrometer in the SOHO international project served as the specific motivation for the work.     

The effects of radiative transfer on the reionization of an inhomogeneous universe

Assuming simple dynamics for the growth of density fluctuations, we implement six-dimensional (6D) radiative transfer calculations to elucidate the effects of photon propagation during the reionization of an inhomogeneous universe. The ionizing sources are postulated to be AGN-like in this paper. The present simulations reveal that radiative transfer effects are still prominent considerably after the percolation epoch, in which patchy ionized regions connect with each other. In other words, owing to the collective opacity, the Universe does not become perfectly transparent against ionizing radiation even though strongly self-shielded regions disappear. It turns out that the inhomogeneity of the medium enhances the opacity effects and delays the end of reionization. Owing to such radiative transfer effects, the reionization in an inhomogeneous universe proceeds fairly slowly, in contrast to the prompt reionization in a homogeneous universe, and as a result the surface of reionization is not so sharply edged, but highly uneven. As a signature of the uneven surface of reionization, the cosmic IR background (CIB) radiation, which is produced by Ly photons resulting from radiative recombination, could exhibit strong anisotropies, reflecting the amplitude of density fluctuations at the reionization era. The predicted CIB intensity lies on a level of possible detection by forthcoming IR space telescope facilities.     

Statistics of the frequency redistribution for gyroresonance radiation in the atmospheres of compact stars

We analyze in detail the statistics of the frequency redistribution of photons during the transfer of gyroresonance radiation under conditions typical of compact stars. The probabilities of photon escape from a scattering atmosphere of arbitrary optical thickness in a single scattering have been found. The effects of gyroresonance photon diffusion in space and in frequency have been simulated. We show that when photons escape from a semi-infinite atmosphere with weak absorption, the frequency redistribution effects lead to a considerable increase in the probability of photon escape from large optical depths and, consequently, modify significantly the dynamics of gyroresonance photon transfer for neutron stars and white dwarfs.