Radiative Transfer in Inhomogeneous Atmospheres. II

This paper is a continuation of a study of radiative transfer in one-dimensional inhomogeneous atmospheres. Two of the most important characteristics of multiple scattering in these media are calculated: the photon escape probability and the average number of scattering events. The latter is determined separately for photons leaving the medium and for photons that have undergone thermalization in the medium. The problem of finding the radiation field in an inhomogeneous atmosphere containing energy sources is also examined. It is assumed that the power of these sources, as well as the scattering coefficient, can vary arbitrarily with depth. It is shown that knowledge of the reflection and transmission coefficients of the atmosphere makes it possible to reduce all these problems to solving some first order linear differential equations with specified initial conditions. A series of new analytic results are obtained. Numerical calculations are done for two types of atmosphere with different depth dependences for the scattering coefficient. These are interpreted physically.     

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.     

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.     

Emergent radiation from the atmosphere bounded by the two half Lambert surfaces composed of different albedo,II

For the evaluation of the effect of the non-uniform surface albedo on the emergent radiation from the atmosphere, the emergent radiation from the atmosphere bounded by the two half Lambert surfaces composed of different albedo is computed. This paper is the improved version of the previous paper (Takashima and Masuda, 1991). The atmosphere is assumed to be homogeneous, which is composed of aerosol, molecules, and absorbent gases. Their optical thicknesses are (1) 0.25, 0.23, and 0.02, and (2) 0.75, 0.23, and 0.02, respectively. The model aerosol is of the oceanic and water soluble types.In the computational procedure, the emergent radiation is calculated approximately by the contributions due to the multiple scattering in the atmosphere, and due to the diffusely or directly transmitted radiation through the atmosphere which is reflected by the surfaces once (4 interactive radiative modes between atmosphere and surface). Furthermore, to perform the hemispherical integration processing the radiative interaction, the transmission function based on the single scattering in the atmosphere is introduced and then the transmission function is averaged over the hemisphere with weighting function. The numerical simulation exhibits the extraordinary effect near the two half surface boundary of different albedoes. The effect decreases exponentially with the distance from the boundary. The effect depends on the atmospheric aerosol type, optical thickness, and surface albedo. The present version enables us to quantitatively discuss the radiative transfer trend near the boundary of two half surfaces. The upward radiance would simply be evaluated using the present scattering approximation method if the surface albedo is less than 0.3. The present method is thought of as a first step extending the one-dimensional radiative transfer model to two-dimensional using the doubling-adding method.     

Order-of-scattering of partially polarized radiation in inhomogeneous,anisotropically scattering atmospheres

A complete set of transfer equations required for the order-of-scattering analysis of partially polarized radiation in inhomogeneous, anisotropically scattering atmospheres is provided. The equations have been derived for both a local study using the radiative transfer equation and its associated auxiliary equation for the source-matrix, and a global study in terms of the scattering and transmission matrices; they account for the polarity of the scattering medium. Their derivations for the finite order scattering and the finitely cumulative scattering, in particular, have yielded important new equations expressing the invariance principles and the integro-differential recurrences for the scattering and transmission matrices. These novel expressions contain as a special case Bellmanet al's (1972) equations for the simpler case of isotropic scattering of unpolarized light in homogeneous atmospheres.     

Average photon path-length of radiation emerging from finite atmospheres

The determination of the average path-length of photons emerging from a finite planeparallel atmosphere with molecular scattering is discussed. We examine the effects of polarisation on the average path-length of the emergent radiation by comparing the results with those obtained for the atmosphere where the scattering obeys the scalar Rayleigh function. Only the axial radiation field is considered for both cases.To solve this problem we have used the integro-differential equations of Chandrasekhar for the diffuse scattering and transmission functions (or matrices). By differentiation of these equations with respect to the albedo of single scattering we obtain new equations the solution of which gives us the derivatives of the intensities of the emergent radiation at the boundaries.As in the case of scalar transfer the principles of invariance by Chandrasekhar may be used to find an adding scheme to obtain both the scattering and transmission matrices and their derivatives with respect to the albedo of single scattering. These derivatives are crucial in determining the average path length.The numerical experiments have shown that the impact of the polarisation on the average pathlength of the emergent radiation is the largest in the atmospheres with optical thickness less than, or equal to, three, reaching 6.9% in the reflected radiation.     

A method of computing the emergent radiation by the atmosphere in the region ranging from ultraviolet to infrared

A method of computing the diffuse reflection and transmission radiation by an inhomogeneous, plane-parallel planetary atmosphere with internal emission source is discussed by use of the adding method. If the atmosphere is simulated by a number of homogeneous sub-layers, the radiation diffusely reflected or transmitted by the atmosphere can be expressed in terms of the reflection and transmission matrices of the radiation of sub-layers. The diffusely transmitted radiation due to the internal emission source can be also easily computed in the same manner. These equations for the emergent radiation are in a quite general form and are applicable to radiative transfer in the atmosphere in the region from ultraviolet to infrared radiation. With this method, the tiresome treatment due to the polarity effect of radiation is overcome.     

Albedo perturbation models: General formalism and applications to LAGEOS

The force due to radiation pressure on a satellite of arbitrary shape is written in a general form within a formalism similar to that used in the theory of radiative transfer in atmospheres. Then the corresponding integrals are evaluated for the simple case of a spherically symmetric satellite, and applied to model the perturbation due to the Earth-reflected radiation flux on LAGEOS. For this purpose, the optical behaviour of the Earth's surface and atmosphere is described as a combination of Lambertian diffusion (continents), partial specular reflection consistent with Fresnel law (oceans) and anisotropic diffusion according to Chandrasekhar's radiative transfer theory (clouds). The in-plane Gauss componentsT andS vs. mean anomaly are computed for a simple orbital geometry and for different models of the Earth's optical properties. A sensitive dependence is found on the assumed cloud distribution, with significant perturbations possibly arising from oceanic specular reflection when the satellite is close to the Earth's shadow boundaries.On leave from Astronomical Institute, Charles University, védská 8, 15000 Prague 5, Czechoslovakia     

Scattering and transmission functions of radiation by finite atmospheres with reflecting surfaces

In the present paper, with the aid of invariance principles in connection with the scattering matrix, we get the exact solution of diffuse reflection and transmission problems by finite inhomogeneous, anisotropically scattering atmospheres bounded by reflecting sufaces. On making use of the reflection and transmission integral operators, we show how to obtain the non-linear integro-differential equations for these operators, which do not depend on the initial condition. Then, we have a system of the required integro-differential equations for the scattering and transmission functions. The obtained result is new, so far as we know. Finally, using the scattering matrix, we reduce the diffuse reflection and transmission problems for planetary atmospheres with reflecting surfaces to the standard diffuse reflection and transmission problems.Supported by the National Science Foundation under Grant No. GP29049 and the Atomic Energy Comminission, Division of Research under Contract No. AT(40-3)-133, Project 19.     

Theory of optical identification of submicron particles in high atmosphere

Interaction between planetary atmospheres and small bodies is connected with radiation effects. Submicron particles in the Earth's upper atmosphere strongly influence the scattering of the shortwave solar radiation. Based on the mutual connection between the environmental and radiation field structures it is possible to determine the physical characteristics of the particle set in this environment. Generaly, the diffused radiation field in the real atmosphere is given by a sum of elementary and multiple scattering components. Solving the inverse problems always leads to complicated integral equations. A major part of the diffused radiation field in the upper atmosphere is due to the first order scattering. The paper presents a new method for determination of the effective complex refractive index and size distribution of the particles based on the radiance data. The solution of integral equations is to be found in the space of quadratically integrable and continuous functionsf L ₂.     

Spectral line formation in a mesoturbulent atmosphere

The formation of spectral lines in a turbulent atmosphere with a spatially correlated velocity field is examined. A new approach for solving this problem is proposed which is not based on the Fokker-Planck formalism. The invariant imbedding method makes it possible to reduce the problem of finding the mean radiant intensity observed in a line to solving a system of differential equations. This possibility is based on determining the mean intensity of the radiation emerging from the medium for a fixed value of the turbulent velocity at its boundary. A separate integral equation is derived for this quantity. The dependence of the line profile, integrated intensity, and width on the mean correlation length and the average value of the hydrodynamic velocity is studied. It is shown that the transition from a microturbulent regime to a macroturbulent regime occurs within a comparatively narrow range of variation in the correlation length. The proposed method yields a solution to the problem for a family of inhomogeneous atmospheres with different optical thicknesses, which makes it easy to determine the radiation field inside the turbulent medium. This approach can be generalized in various ways, in particular, it can be applied without significant changes to the case where the correlation length depends on position within the atmosphere. __________ Translated from Astrofizika, Vol. 50, No. 2, pp. 219–232 (May 2007).     

Source vector in Rayleigh-Cabannes scattering atmosphere: the nonconservative milne problem

We consider the radiative transfer in a nonconservative homogeneous plane-parallel semi-infinite planetary atmosphere where the scattering processes are described by the Rayleigh-Cabannes phase matrix and where the primary sources are in infinitely deep layers. If we use the superposition principle we derive the Cauchy problem for the source vector.As a by-product the external field of radiation for the problem described is obtained using the principle of invariance by Chandrasekhar. The respective formulae for the radiation field in the deep layers and for the extrapolation distance are given. It is shown that the Rubenson degree of polarization even in the case of near-conservative atmospheres reaches the asymptotic regime at rather small values of the optical depth. The-plane reliefs of the characteristic equation, extrapolation distance and the normalized components of the source vector at the boundary are given along with a sample of zeros of the characteristic equation.     

A discrete spherical harmonics method for radiative transfer analysis in inhomogeneous polarized planar atmosphere

A discrete spherical harmonics method is developed for the radiative transfer problem in inhomogeneous polarized planar atmosphere illuminated at the top by a collimated sunlight while the bottom reflects the radiation. The method expands both the Stokes vector and the phase matrix in a finite series of generalized spherical functions and the resulting vector radiative transfer equation is expressed in a set of polar directions. Hence, the polarized characteristics of the radiance within the atmosphere at any polar direction and azimuthal angle can be determined without linearization and/or interpolations. The spatial dependent of the problem is solved using the spectral Chebyshev method. The emergent and transmitted radiative intensity and the degree of polarization are predicted for both Rayleigh and Mie scattering. The discrete spherical harmonics method predictions for optical thin atmosphere using 36 streams are found in good agreement with benchmark literature results. The maximum deviation between the proposed method and literature results and for polar directions \(\vert \mu \vert \geq0.1 \) is less than 0.5% and 0.9% for the Rayleigh and Mie scattering, respectively. These deviations for directions close to zero are about 3% and 10% for Rayleigh and Mie scattering, respectively.     

Solution of linear radiative transport problems in plane-parallel atmospheres. I.

A new method for determining various quantities describing the radiation field in an inhomogeneous, plane-parallel atmosphere is proposed in this two-part paper. The essence of this method is the reduction of the boundary value problems which arise during the customary statement of various astrophysical problems associated with solving the radiative transfer equations to initial value problems. Compared to previous attempts in this area, the proposed method is universal and simple. The first part of this paper deals with one-dimensional media. Scalar, as well as vector–matrix problems relating to the diffusion of radiation in spectral lines with frequency redistribution are examined.     

Three-dimensional numerical cosmological radiative transfer in an inhomogeneous medium

A numerical scheme is proposed for the solution of the three-dimensional (3D) radiative transfer equation with variable optical depth. We show that time-dependent ray tracing is an attractive choice for simulations of astrophysical ionization fronts, particularly when one is interested in covering a wide range of optical depths within a three-dimensional clumpy medium. Our approach combines the explicit advection of radiation variables with the implicit solution of local rate equations given the radiation field at each point. Our scheme is well suited to the solution of problems for which line transfer is not important, and could, in principle, be extended to those situations also. This scheme allows us to calculate the propagation of supersonic ionization fronts into an inhomogeneous medium. The approach can be easily implemented on a single workstation and should also be fully parallelizable.     

Exact solution of the equation of transfer with planetary phase function

We have considered the transport equation for radiative transfer to a problem in semi-infinite atmosphere with no incident radiation and scattering according to planetary phase function w(1 + xcos ). Using Laplace transform and the Wiener-Hopf technique, we have determined the emergent intensity and the intensity at any optical depth. The emergent intensity is in agreement with that of Chandrasekhar (1960).     

Exact solution of the transport equation for radiative transfer with scattering albedo ωO < 1 using the Laplace transform and the Wiener-Hopf technique and an expression ofH-function

We have considered the transport equation for radiative transfer to a problem in semi-infinite non-conservative atmosphere with no incident radiation and scattering albedo ₀ < 1. Usint the Laplace transform and the Wiener-Hopf technique, we have determined the emergent intensity and the intensity at any optical depth. We have obtained theH-function of Dasgupta (1977) by equating the emergent intensity with the intensity at zero optical depth.     

Curvature effects in extended stellar atmospheres-absorption and scattering

The numerical solution of radiative transfer equation including curvature with both absorption and scattering has been developed in the frame work of Discrete Space Theory. Two cases have been considered: (A) irradiation of the atmosphere at =T and (B) no irradiation on either side of the atmosphere. Isotropic scattering has been assumed. It is found that the emergent luminosities (defined by r ² I(r, ) ) from scattering dominated atmospheres are smaller than those from absorption dominated atmospheres.     

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.