Solution of radiative transfer equation with spherical-symmetry in partially scattering medium

We have solved the equation of radiative transfer in spherical symmetry with scattering and absorbing medium. We have set the albedo for single scattering to be equal to 0.5. We have set the Planck function constant throughout the medium in one case and in another case the Planck function has been set to vary asr ^–2. The geometrical extension of the spherical shell has been taken as large as one stellar radius. Two kinds of variations of the optical depth are employed (1) that remains constant with radius and (2) that varies asr ^–2. In all these cases the internal source vectors and specific intensities change depending upon the type of physics we have employed in each case.     

Solution of the transfer equation in a scattering atmosphere with spherical symmetry

An exact formal solution of then-approximation radiative transfer equations for the Compton scattering in a spherically symmetric atmosphere is obtained. In view of further applications, the simple case of a density ?(r)=?₀/r is fully developed and the 20 approximation equations have been studied with the computer.     

Bilinear integrals of the radiative transfer equation

It is shown that the group of problems in the theory of radiative transfer that are reducible to the sourcefree problem admits a class of integrals involving quadratic moments of the intensity of arbitrarily high orders. Based on a variational principle, it is found that these integrals, which include the R-integral, follow from the corresponding conservation laws. Some of the results are generalized to the case of anisotropic scattering.     

Exact solution of the transport equation for imperfect rayleigh scattering in a semi-infinite atmosphere

By performing the one-sided Laplace transform on the matrix integro-differential equation for a semi-infinite plane parallel imperfect Rayleigh scattering atmosphere we derive an integral equation for the emergent intensity matrix. Application of the Wiener-Hopf technique to this integral equation will give the emergent intensity matrix in terms of singularH-matrix and an unknown matrix. The unknown matrix has been determined considering the boundary condition at infinity to be identical with the asymptotic solution for the intensity matrix.     

Exact solution of equation for radiative transfer in semi-infinite Rayleigh scattering atmosphere using the laplace transform and the Wiener-Hopf technique

We have considered six scalar transport equations which are obtained from the vector transport equation to determine four Stokes's parameters to the problem of diffuse reflection in the semi-infinite plane parallel Rayleigh scattering atmosphere. By use of the Laplace transform and the Wiener-Hopf technique, these equations have been solved exactly to obtain the emergent intensity and the intensity at any optical depth and to reconstruct the Stokes's parameters. Solutions for emergent distribution so obtained are identical with the results of Chandrasekhar (1950).     

The comoving-frame equation of radiative transfer in relativistic flows

The comoving-frame equations of radiative transfer and moment equations to accurate terms of all orders inv/c are derived in the modified Lagrangian form. The equations exactly describe the interaction of radiation with matter in a relativistically moving medium in flat or curved spacetime. Two specialized sets of equations are presented: (1) the equation of radiative transfer and moment equations accurate to terms of second order (v ²/c ²), and (2) the transfer equation and moment equations for a radial flow in curved spacetime with the Schwarzschild-type metric.     

Solution of the equation of transfer for coherent scattering in an exponential atmosphere by Eddington's method

An approximate solution of the transfer equation for coherent scattering in stellar atmospheres with Planck's function as a nonlinear function of optical depth, viz., $$B_v \left( T \right) = b_0 + b_1 e^{ - \beta \tau } $$ is obtained by Eddington's method. is obtained by Eddington's method.     

Solution of the equation of transfer for coherent scattering in an exponential atmosphere by Busbridge's method

A solution of the transfer equation for coherent scattering in stellar atmosphere with Planck's function as a nonlinear function of optical depth, viz. $$B{\text{ }}_v (T) = b_0 + b_1 {\text{ }}e^{ - \beta \tau } $$ is obtained by the method developed by Busbridge (1953).     

Sobolev's Φ-function of radiative transfer in planar and spherical scattering media

In recent years Sobolev's -function of radiative transfer has been discussed in connection with the resolvent of Milne's integral equation such that it plays an important role in the determination of the radiation field in semi-infinite (or finite) atmospheres with internal sources (cf. Sobolev, 1963). In the present paper, the part of Sobolev's -function in plane-parallel and spherical, isotropically scattering, atmospheres with internal source distribution is investigated from analytical and numerical aspects. With the aid of invariant imbedding (cf. Bellmanet al., 1968), we computed Sobolev's -function of Milne's integral equation for the planar case by solving the Cauchy system for the auxiliary function and Chandrasekhar'sX- andY-functions. The corresponding -function for the spherical case is readily obtained from the for the planar case.Investigation supported by the National Science Foundation under Grant No. GP 29049, the Atomic Energy Commission, Division of Research under Contract No. AT(04-3)-113, Project 19, and the National Institutes of Health under Grant No. 16197-05.     

The comoving-frame equation of radiative transfer in a curvilinear coordinate system

The comoving-frame equation of radiative transfer and moment equations are derived in orthogonal, curvilinear coordinates, inclusive of terms of orderv/c. The equation of radiative transfer, which contains the terms due to the effect of curvature of coordinate lines explicitly as well as those of Doppler shift and aberration, is the generalization of Castor's equation for spherical symmetry and of Buchler's equation for Cartesian coordinates. The moment equations agree with Buchler's.     

Solution of the equation of transfer for coherent scattering in an exponential atmosphere by the method of discrete ordinates

A solution of the transfer equation for coherent scattering in stellar atmosphere with Planck's function as a nonlinear function of optical depth, viz., $$B_v (T) = b_0 + b_1 {\text{ }}e^{ - \beta \tau } $$ is obtained by the method of discrete ordinates originally due to Chandrasekhar.     

Solution of a transfer equation by a Modified Spherical Harmonic Method in a thin anisotropically scattering atmosphere

Wan, Wilson and Sen (1986) have examined the scope of Modified Spherical Harmonic Method in a plane medium scattering anisotropically. They have used the phase functionp(µ, µ) = 1 +aµµ. In this paper, the Transfer Equation has been solved by the Modified Spherical Harmonic Method using the phase functionp(µ, µ) = 1 + ₁ P ₁(µ)P ₁(µ) + ₂)P ₂(µ)P ₂(µ) and a few sets of numerical solution have been predicted for three different cases.     

Modelling artificial night-sky brightness with a polarized multiple scattering radiative transfer computer code

As part of an ongoing investigation of radiative effects produced by hazy atmospheres, computational procedures have been developed for use in determining the brightening of the night sky as a result of urban illumination. The downwardly and upwardly directed radiances of multiply scattered light from an offending metropolitan source are computed by a straightforward Gauss–Seidel (G–S) iterative technique applied directly to the integrated form of Chandrasekhar's vectorized radiative transfer equation. Initial benchmark night-sky brightness tests of the present G–S model using fully consistent optical emission and extinction input parameters yield very encouraging results when compared with the double scattering treatment of Garstang, the only full-fledged previously available model.     

The effects of scattering and conduction upon radiative transfer in lunar and mercurian surfaces

The linearized analysis of Cess and Srinivasan (1971) for thermal emission from lunar and Mercurian surfaces following a sudden eclipse has been extended to include two additional factors. One is the separate influence of scattering upon the radiative transport process within the surface material. The second is the effect of thermal conduction as an additional energy transport mechainsm.     

Solution of the equation of transfer with Rayleigh's phase function in a thin atmosphere

The equation of radiative transfer with scattering according to Rayleigh's phase function has been solved in a thin atmosphere by use of a modification of the spherical-harmonic method suggested by Wanet al. (1986).     

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.     

Exact solution of the problem of diffuse reflection for a combination of Rayleigh and isotropic scattering

We consider the basic vector equation of transfer for radiation in a semi-infinite atmosphere for diffuse reflection which scatters radiation in accordance with the phase matrix obtained from a combination of Rayleight and isotropic scattering. This equation will give an integral equation for emergent intensity while subjected to the Laplace transform. The integral equation will give rise to the emergent intensity matrix on application of the Wiener-Hopf technique. This is an exact method.