Solution Of Radiative Transfer Problems Research Articles

On the Development of a Recursive Method for the Solution of Radiative Transfer Problems

On the Development of a Recursive Method for the Solution of Radiative Transfer Problems

The solution of radiative transfer problems in molecular bands without the LTE assumption by accelerated lambda iteration methods

An iterative method based on the use of approximate transfer (or Λ) operators, which was designed initially to solve multilevel NLTE line formation problems in stellar atmospheres, is adapted and applied to the solution of the NLTE molecular band radiative transfer in planetary atmospheres. The matrices to be constructed and inverted are much smaller than those used in the traditional Curtis matrix technique, which makes possible the treatment of more realistic problems (including rotational NLTE, overlapping of lines in the bands and overlapping of bands with continuua) using relatively small computers. This technique converges much more rapidly than straightforward iteration between the transfer equation and the equations of statistical equilibrium (Λ-iteration). A test application of this new technique to the solution of NLTE radiative transfer problems for optically-thick and thin bands (the 4.3 μm CO 2 band in the Venusian atmosphere and the 4.7 and 2.3 μm CO bands in the Earth's atmosphere) is described. The necessity for careful treatment of rotational NLTE in weak lines of the 4.3 μm CO 2 fundamental band is shown. The possible influence of neglecting or treating approximately the CO 2 10 μm laser transitions and also effects of the overlap of the 4.3 μm CO 2 with the continuum are investigated. The rotational NLTE effects in the 4.7 μm CO band in the upper Earth's atmosphere are estimated.

An efficient method for the solution of 3-D radiative transfer problems

A novel efficient method for the solution of radiative transfer problems in media of two or three dimensions is presented. The approach is based on an implicit discretization of the transfer equation in Cartesian frames. The resulting linear system of equations is solved by a combination of recursion and iteration. Since the explicit handling of large matrices is avoided, the corresponding code is relatively simple and runs well on all types of modern computers. As examples of applications, we consider several spherical and non-spherical dust clouds heated by a central source and derive their temperature distributions and emergent spectra. Comparison of our spherical models with those of Rowan-Robinson shows good agreement.

An exact analytical solution of a radiative transfer problem in a binary mixture

We 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. Markov statistics are a special (degenerate) case of the class of statistics we are able to treat. Two different techniques, a singular eigenfunction analysis and an integral equation method, are applied to the problem.

Solution of radiative transfer problem in a plane layer for the model of complete frequency redistribution. I. One-dimensional medium

Solution of radiative transfer problem in a plane layer for the model of complete frequency redistribution. I. One-dimensional medium

Solution of radiative transfer problem in a plane layer for the model of complete frequency redistribution. II. Three-dimensional medium

In the first part of the study approximate analytic solutions were obtained for the problem of radiative transfer in the model of complete frequency redistribution in a one-dimensional medium of finite thickness. In the present paper analogous solutions are given for the case of a plane-parallel homogeneous layer for a spherical phase function (isotropic scattering). The accuracy of these solutions is higher than in the one-dimensional case and is a fraction of a percent at the line center. With increasing thickness the accuracy of the solutions rapidly increases. As an illustration, tables of Ambartsumyan's ~ and $ functions are given in the case of a Lorentzian profile. The asymptotic approximation for large layer thicknesses and in the neighborhood of the line center that can be extracted from the high-accuracy solutions is discussed. i. Introduction It appears to be in principle impossible to find analytic solutions to the problem of radiative transfer in a plane-parallel medium of finite thickness for the model of complete frequency redistribution within the line. Only asymptotic solutions are known, and these only for the X and Y functions for a layer of large optical thickness, and they are valid only in the immediate vicinity of the line center. Moreover, if these solutions are written down for the general formulation of the problem their error is too large (and increases rapidly with increasing distance from the line center), and such asymptotic solutions in the model of frequency redistribution are clearly unsatisfactory for applied purposes. In our previous [i] we found accurate analytic solutions to the problem of radiative transfer in the case of a one-dimensional medium of finite thickness in the general model of complete redistribution.

The solution of radiation transfer problem for a slab with space-dependent albedo and anisotropic scattering

The transport of thermal radiation has been considered within a finite slab which absorb and scatter anisotropically. The problem involves the space-dependent single-scattering albedow(x). Two approximations are taken forw(x). In the first it is represented in exponential form asw(x)=w0 exp(−x/s), wherew0 ands are given constants andx is the optical variable. The second approximation assumes the formw(x) = ∑r=0Rdr*pr(x/a), wheredr* are known expansion coefficients anda is the half optical thickness of the slab. Analytic expressions for the forward, backward radiation intensities and fluxes are given in each approximation. The solution of the linear transport equation is performed on the basis of integral Fourier transforms.

Discrete-ordinates finite-element method for atmospheric radiative transfer and remote sensing

Advantages and disadvantages of modern discrete-ordinates finite-element methods for the solution of radiative transfer problems in meteorology, climatology, and remote sensing applications are evaluated. After the common basis of the formulation of radiative transfer problems in the fields of neutron transport and atmospheric optics is established, the essential features of the discrete-ordinates finite-element method are described including the limitations of the method and their remedies. Numerical results are presented for 1-D and 2-D atmospheric radiative transfer problems where integral as well as angular dependent quantities are compared with published results from other calculations and with measured data. These comparisons provide a verification of the discrete-ordinates results for a wide spectrum of cases with varying degrees of absorption, scattering, and anisotropic phase functions. Accuracy and computational speed are also discussed. Since practically all discrete-ordinates codes offer a builtin adjoint capability, the general concept of the adjoint method is described and illustrated by sample problems. Our general conclusion is that the strengths of the discrete-ordinates finite-element method outweight its weaknesses. We demonstrate that existing general-purpose discrete-ordinates codes can provide a powerful tool to analyze radiative transfer problems through the atmosphere, especially when 2-D geometries must be considered.

Numerical evaluation of the formal solution of radiative transfer problems in spherical geometries

We present a new and general method for the solution of complex multi-level non-LTE radiative transfer problems both in static and moving media. The method is based on the approximate Λ-iteration method and is a direct extension of the scheme devised by Rybicki and Hummer (1991). We demonstrate that this method is able to treat both active continua and overlapping lines with relatively small amount of computer resources. Some simplified test cases are presented and discussed.

Difference-equation methods for the solution of radiative transfer problems in media with discontinuities

Difference-equation methods are developed for solving the equation of transfer in media with discontinuities in their physical properties. These should prove useful in calculating the radiation field in dynamical atmospheres having shocks. Two examples with thermal or scattering source functions, for which exact solutions can be obtained, are used to evaluate the accuracy of the techniques, which prove quite satisfactory.

An Hermitian method for the solution of radiative transfer problems

A new method for the solution of the radiative transfer equation is developed. The resulting system, like the ordinary difference equations is tridiagonal in form and easy to use. Numerical examples are given which show the superiority of the new technique to previous differential and integral equation methods.

Numerical evaluation of the formal solution of radiative transfer problems in spherical geometries

Numerical evaluation of the formal solution of radiative transfer problems in spherical geometries

Limitation of the variance of “local” estimates in the solution of radiative transfer problems by the Monte Carlo method

USING the concept of attributes of trajectories introduced previously, a new modification of the method of the “local” estimate with finite variance is described. The algorithms take into account the combination of solutions corresponding to the direct and conjugate transfer equations.

Solution of radiative transfer problems in planetary atmospheres

The numerical solution of radiative transfer problems in inhomogeneous, anisotropically scattering, plane-parallel planetary atmospheres with internal source distributions is discussed. We propose a simple recursive method to derive the internal distribution of radiation in such an atmosphere in the manner of Grant and Hunt (1968a). We also describe a second algorithm to compute only the radiation reflected by the atmosphere. Specimen numerical results for a homogeneous planetary atmosphere whose lower surface reflects radiation according to Lambert's Law are given to illustrate these methods.

On the scattering of sunlight into planetary shadow cones

Refraction and multiple scattering of solar radiation in a planetary atmosphere cause the propagation of appreciable amounts of radiative energy into the planet's geometric shadow cone. The implications of this atmospheric phenomenon to research in various fields are discussed. Examples are studies of lunar eclipses, the presence of the Venus ring near inferior conjunction, the astronomical problem of the appearance of the earth as a planet, and the flights of spacecraft in earth orbit and in cislunar space. It is shown that the results of such studies can provide data in disciplines involving geophysics, planetary atmospheres, solar physics, astronomy, and astronautics. Rigorous solutions of radiative transfer problems in a real atmosphere, however, still are beyond the capabilities of present theory. Some preliminary quantitative results in the visible range of the electromagnetic spectrum have been obtained through the use of a semiempirical method. This technique promises to become a useful tool in the fields of study discussed.

Radiation from plasmas

While radiative transfer problems in laboratory plasma are usually of lesser importance than in stellar plasmas (except for resonance lines), measurement and theoretical analysis of radiation from laboratory plasmas does serve to enhance our knowledge of basic data required for actual solutions of radiative transfer problems. Examples are atomic transition probabilities, spectral line shape parameters and collisional rate coefficients. Recent progress in the measurement and interpretation of these quantities will be reviewed.