A fourier transform solution of radiation transfer in a spherical-inhomogeneous medium

The radiative heat flux at the boundary of a sphere containing an internal energy source and subject to general boundary conditions (problem 1) is obtained in terms of the albedo of the corresponding source-free problem with isotropic boundary condition (problem 2). The solution of problem 2 is performed on the basis of the integral Fourier transforms method. Numerical results for the partial heat flux and emissivity for a given internal energy source and inhomogeneous medium, isotropic scattering are obtained.     

Radiative transfer in an infinite cylinder

The problem of radiation transfer in a cylinder with diffuse reflectivity and containing an energy source is connected with the source-free radiation transfer problem with isotropic boundary condition. Equation for the radiation heat flux is obtained for a polynomial source. In the special case of isotropic scattering, the radiation heat flux is given in terms of the albedo of the second problem. An expression is also given for the net radiation heat flux.     

The energy-dependent radiation transfer in a homogeneous finite cylinder

Radiation transfer problem in the slowing-down region for cylindrical geometry with diffuse reflectivity and internal source is connected with source-free problem with isotropic boundary condition. Modelled kernels are used to represent the slowing-down kernel. Calculations are performed for the radiation flux at the boundary of the cylinder. Numerical results are obtained using the bi-variation technique.     

Surface heat current in an inhomogeneous slab with reflecting walls

The problem of radiative transfer in an inhomogeneous finite medium with boundary surfaces which reflects both diffusely and specularly is connected with a source-free specular boundary condition radiation tramsfer. Numerical results are obtained for the partial heat flux using the bi-variational technique.     

Radiation transfer for linearly anisotropic phase functions

Radiative transfer equation in a plane-parallel medium with isotropic boundary conditions for linearly anisotropic scattering phase function is considered. Two coupled integral equations for total density of radiation and total radiation flux are obtained. The Galerkin method is used to solve these equations. Numerical results for the radiative fluxes at the boundaries show that the Galerkin method yields accurate results compared well with other exact methods.     

Angular distribution of radiation in an isotropically scattering slab

The equation of radiative transfer in an isotropically scattering slab subject to general boundary conditions is solved. The Padé approximation technique is used to calculate the reflected and transmitted angular distributions. Numerical results for angular distributions through and at the boundaries of a finite slab are calculated by the Padé approximation technique. The results for a Padé approximation of [0/1] are compared with results obtained by the Galerkin method.     

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.     

Solution of a generalized equation of transfer in a two-region slab

The general equation of transfer in a two-region slab of unequal thickness with general boundary conditions has been solved by an analytical method developed by Menninget al. (1980). The scattering is regarded as isotropic and the source function is taken as a general one to accomodate different types of problems.     

Radiation transfer in a diffuse and specular reflecting slab with Rayleigh scattering

A method of analysis is presented for solving the radiative transfer problem in an absorbing, emitting, inhomogeneous, and anisotropically scattering plane-parallel medium with specular and diffuse reflecting boundaries and internal source (problem 1). Exact relations for the radiation heat flux at the boundaries of problem 1 are obtained in terms of the radiation density and albedos of the corresponding source-free medium with specular reflecting boundaries (problem 2). Two coupled integral equations for the radiation density and the second moment of the radiation intensity for problem 2 with Rayleigh phase functions are obtained. The Galerkin method is used to solve these equations. Albedos of problem 2 are compared with theF _n method. Numerical results for radiation heat fluxes at the boundaries of problem 1 are tabulated for different forms of the internal source.     

Bianchi type-II,VIII, and IX cosmological models with matter and electromagnetic fields

We present analytic solutions of the Einstein-Maxwell equations for cosmological models of LRS Bianchi type-II, VIII, and IX. The solutions represent anisotropic universes with source-free electromagnetic fields and perfect fluids matter satisfying the equation of state that is a function of the cosmic-time. Some physical properties of the models have been discussed.     

Anisotropic radiation transfer in a sphere with internal source

In this work we connect the problem of anisotropic radiation transfer in a sphere with diffuse reflectivity and containing an energy source with the problem of source free amisotropic radiation transfer with isotropic boundary condition. Exact equation for radiation heat flux for the first problem is obtained in terms of the source, the flux and the albedo of the second problem. Modeled kernel is used to represent the anisotropy of the phase function. Numerical results are given.     

Modified spherical harmonic method in radiative transfer in linearly anisotropic scattering planar medium

Mengüç, and Viskanta (1983) examined the scope of some approximate methods for solving transfer problems in plane medium scattering anisotropically. His aim was to focus attention on methods capable of extension to multidimensional geometry. In the present paper, it is demonstrated that modified double-interval spherical harmonic method admirably suits that role. The transfer problem of Mengüç and Viskanta's model has been solved by this method. The results computed are found to be in good agreement with those obtained by other methods.     

A new approach to evaluation of the effect of the two half-Lambert surfaces composed of different albedoes on the emergent radiation at the top of the atmosphere

A new way is adopted for the evaluation of the upwelling radiation from atmosphere bounded by two half-Lambert surfaces. The atmosphere is assumed to be homogeneous, and is composed of aerosol, molecules, and absorbent gases, where the model aerosol is of the oceanic and water soluble types.In the computational procedure, an iterative doubling-adding equation is expanded into a series of the radiative interaction modes between atmosphere and surface. Next, a probability of radiation interacting with respective half surfaces is calculated based on the assumption of single-scattering in the atmosphere. On the basis of this probability, the emergent radiation at the top of the atmosphere is approximately calculated by considering the radiative intractions to be twice as large. The effect of the multiple-scattering is fully taken into account. A numerical simulation exhibits the extraordinary effect near the two half-surface boundary of different albedoes. The effect of the other half-surface on the radiance decreases monotonically with the distance from the boundary. The present new version enable us to quantitatively discuss radiative transfer near the boundary of two half-surfaces even if the optical thickness is large and (or) surface albedo is great.     

Diffusion of radiation in planetary atmospheres

The problem of interaction of the solar radiation with the turbid Earth atmosphere, containing complicated polydispersive aerosol systems, is discussed in this paper. Equations for computing the angular functions ofn-th order scattering are derived. On the basis of these functions the spectral radiance, radiation flows and radiation balance of the atmosphere in the short-wave spectral range are calculated. The relations obtained can be used to calculate the complex index of refraction, distribution function and other characteristics of the submicron aerosol fraction, by solving the inverse problems.     

The H N method for Milne problem with polarization

The H _N method, employed for studies in neutron transport theory, is used to establish numerical results basic to the vector equation describing the transfer of polarized light in a Rayleigh scattering atmosphere with true absorption. The method has been applied to the classical Milne problem. The exit distribution is defined as a series in powers of the zenith observation angle. The numerical results are computed and compared with exact values obtained using the exit distribution in terms of the H-matrix. The numerical results are in good agreement with previously published findings.     

Inverse radiation modeling of Titan's atmosphere to assimilate solar aureole imager data of the Huygens probe

During the descent of the Huygens probe through Titan's atmosphere in January 2005, the Descent Imager/Spectral Radiometer (DISR) will perform upward and downward looking measurements at various spectral ranges and spatial resolutions. This internal radiation density could be estimated by radiative transfer calculations for Titan's atmosphere. However, to do this, the optical properties—i.e. volume extinction coefficient, single scattering albedo and scattering phase function—have to be prescribed at every altitude, and these are apriori not known. Herein, an inverse approach is investigated, which retrieves the single scattering albedo and the phase function of the aerosols from DISR observations. The method uses data from a DISR subinstrument, the Solar Aureole imager (SA), to estimate the optical properties of the atmospheric layer between two successive observation altitudes. A unique solution for one layer can in principle be calculated directly from a linear system of equations, but due to the sparseness of the data and the unavoidable noise in the measurements, the inverse problem is ill-posed. The problem is stabilized by the regularization method requiring smoothness of the resultant solution. A consistent set of solutions for all layers is obtained by iterating several times downward and upward through the layers. The method is tested in a simulated radiation density scenario for Titan, which is based on a microphysical aerosol model for the haze layer. Within this scenario, the expected coverage of SA data allows a reconstruction of the angular dependence of the scattering phase function with an explained variance better than 90%.     

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.     

Exact solution of a problem in a stellar atmosphere using the Laplace transform and the Wiener-Hopf technique

The general equation for radiative transfer of line scattering intensity — including the effects of scattering, absorption and thermal emission — in the Milne-Eddington model is considered here. The scattering function is assumed to be quadratically anisotropic in the cosine of the scattering angle, and Planck's intensity function is assumed for thermal emission. The exact solutions for emergent intensity from the bounding face and the intensity at any optical depth are obtained by the method of the Laplace transform in combination with the Wiener-Hopf technique.     

International conference on planetary satellites

Further analysis of visual (V) wavelength photometric function data for Saturn's ring is presented. Evidence indicating both that primary scattering dominates, and that mutual shadowing is an irrelevant concept, is reviewed. Simple anisotropic scattering radiative transfer models are used to define the probable ranges in the single scattering albedo, and in the general shape of the scattering phase function of the individual particles. Limitations on the mean perpendicular optical thickness of the ring are also obtained. Results indicate that the ring particles are highly efficient back-scatterers of visual radiation. Macroscopic particles account for the basic shape of the scattering phase function. Based on an infinite optical thickness for the ring, a minimum single scattering albedo ≈0.75 is found. Use of conservative scattering leads to a minimum optical thickness ≈0.7. The analysis is consistent with the ring particles being centimeter-size pieces of ice.     

Stochastic transfer of polarized radiation in finite cloudy atmospheric media with reflective boundaries

The problem of monoenergetic radiative transfer in a finite planar stochastic atmospheric medium with polarized (vector) Rayleigh scattering is proposed. The solution is presented for an arbitrary absorption and scattering cross sections. The extinction function of the medium is assumed to be a continuous random function of position, with fluctuations about the mean taken as Gaussian distributed. The joint probability distribution function of these Gaussian random variables is used to calculate the ensemble-averaged quantities, such as reflectivity and transmissivity, for an arbitrary correlation function. A modified Gaussian probability distribution function is also used to average the solution in order to exclude the probable negative values of the optical variable. Pomraning-Eddington approximation is used, at first, to obtain the deterministic analytical solution for both the total intensity and the difference function used to describe the polarized radiation. The problem is treated with specular reflecting boundaries and angular-dependent externally incident flux upon the medium from one side and with no flux from the other side. For the sake of comparison, two different forms of the weight function, which introduced to force the boundary conditions to be fulfilled, are used. Numerical results of the average reflectivity and average transmissivity are obtained for both Gaussian and modified Gaussian probability density functions at the different degrees of polarization.