Plane-parallel Medium Research Articles

Several Symbolic Augmented Chebyshev Expansions for Solving the Equation of Radiative Transfer

Three expansion methods are described using Chebyshev polynomials of the first kind for solving the integral form of the equation of radiative transfer in an isotropically scattering, absorbing, and emitting plane-parallel medium. With the aid of symbolic computation, the unknown expansion coefficients associated with this choice of basis functions are shown to permit analytic resolution. A unified and systematic solution treatment is offered using the projection methods of collocation, Ritz-Galerkin, and weighted Galerkin. Numerical results are presented contrasting the three expansion methods and comparing them with existing benchmark results. New theoretical results are presented illustrating rigorous error bounds, residual characteristics, accuracy, and convergence rates.

A nonstationary transfer problem with partial frequency redistribution

We analyze a system of equations for nonstationary transfer of resonant radiation taking into account nonlinear effects, and with partial frequency redistribution using a two-level model of the atom. We suggest a method for solving the corresponding problem in a three-dimensional plane-parallel medium.

Radiative transfer in participating media under conditions of radiative equilibrium

A rigorous integral theory is presented for the radiative transfer in a plane-parallel medium between two emitting and reflecting surfaces in radiative equilibrium. The medium is assumed to scatter radiation anisotropically. If the scattering phase function is expanded in Legendre polynomials the single scattering albedo modifies to a form which retains the medium inhomogeneity in radiative equilibrium. Expressions are given for the forward and backward radiation intensities, the heat fluxes, and the incident radiation. The case of isotropic scattering is considered in more detail and solved explicitly by a projection technique. Numerical results are reported for cases of physical interest and compared with the available data.

The solution of the transfer equation in a homogeneous isotropically scattering plane-parallel medium

The exact solution of the transfer equation (TE) in a homogeneous plane-parallel medium of any optical extent (i.e., infinite, semi-infinite, or finite) is derived in the case of isotropic local scattering. The sources are not necessarily isotropic. We start from our recent study of the Schwarzschild-Milne (SM) integral equation as derived from the TE with appropriate boundary conditions. Using the Laplace transform on the range of the optical depth variable, the solution of the TE is obtained as the transform, by an introduced ∧-operator, of the solution of the SM equation. The latter is then expressed in terms of the sources by means of the Green distribution of the SM equation. We are thus led to calculate the transform of this distribution by the ∧-operator, which is a fundamental auxiliary function of plane-parallel media. This transform, namely Γ, is thoroughly studied in the present article. The transform of the Γ-function by the ∧-operator, i.e., the bi-transform of the Green distribution, yields the solution of the TE in terms of the specified sources. It is thus the Green function of the problem. This function is studied in detail hereafter and written in terms of the Γ-function by means of simple algebraic formulae. Analytical and numerical studies of the auxiliary functions introduced here and in a former article are required before a numerical evaluation of our solution can be obtained.

On nonstationary radiative transfer in a spectral line in stellar atmospheres

We consider nonstationary radiative transfer in a line in stellar atmospheres simulated as a stationary semi-infinite plane-parallel medium. We assume complete frequency redistribution in the elementary act of scattering. We assume that the time a photon spends in the medium is determined only by the mean time spent in the absorbed state. We obtain an explicit expression for the resolvent of the nonstationary integral equation of transfer, which is a bilinear expansion with respect to the eigenfunctions found in [12] for the corresponding stationary transfer equation.

The solution of the Schwarzschild-Milne integral equation in an homogeneous isotropically scattering plane-parallel medium

The Schwarzschild-Milne integral equation−which is equivalent to the transfer equation with appropriate boundary conditions−is solved in an homogeneous plane-parallel medium of any optical extent (i.e. infinite, semi-infinite or finite). Local scattering of light is assumed isotropic. The three proposed methods use the Laplace transform on the I-interval explored by the optical depth variable. The first method is based on the Sobolev classical scheme yielding the resolvent kernel of the Schwarzschild-Milne equation in terms of the resolvent function. In the other two methods, the Laplace transform of the appropriate Green distribution is calculated from an integral equation of the Schwarzschild-Milne type (Ambartsumian method) or of the Cauchy type (a method from Leonard-Mullikin-Yanovitskii). Auxiliary functions defined in the complex plane are needed in the three methods. Their restrictions in the range [-1, +1] have already been introduced in the literature and have a clear physical meaning. The solution is provisionally written as an inverse Laplace transform of these auxiliary functions. Further developments based on the theorem of residues need a systematic study of the auxiliary functions in the complex plane (to be achieved in a forthcoming article).

The Hard X-ray Reflection on Cold Matter

We have simulated the reflection on cold matter in a variety of situations to determine which informations can actually be inferred from observations. We modelled a semi-infinite plane parallel medium of solar abundance matter, semi-isotropically illuminated by a X/γ ray source. The spectra are calculated from a Monte-Carlo method without any approximation in the cross-sections. Θ is the angle over which the reflecting matter is seen (90°=face-on), Θ=all means a spatially integrated spectrum. Fref is the ratio of the reflected over direct component.

An inverse source problem in radiative transfer

The spherical-harmonics method is used to develop a solution to an inverse source problem in radiative transfer. It is assumed that, with the exception of the inhomogeneous source term, all aspects of the radiation-transport problem are known, and we seek to determine the inhomogeneous source term from specified angular distributions of radiation exiting the two surfaces of a homogeneous plane-parallel medium. Anisotropic scattering is included in the monochromatic radiative-transfer model and general reflecting boundary conditions are considered.

A particular solution for polarization calculations in radiative transfer

A pair of biorthogonality relations, relevant to some elements of a generalized spherical-harmonics solution of the homogeneous equation of transfer, is derived and used, along with linear-algebra techniques, to develop a particular solution appropriate to the scattering of polarized light. The considered radiative-transfer model is based on a general treatment of polarization effects in a plane-parallel medium that contains a source that varies with position and both direction variables.

On intensity calculations in radiative transfer

A post-processing technique is used with the spherical-harmonics method to develop an accurate result for the radiation intensity in a homogeneous plane-parallel medium that contains a source that varies with position. Anisotropic scattering is included in the monochromatic radiative-transfer model, and general reflecting boundary conditions are considered.

Rain induced attenuation for 3mm wave band and its measurement system

In this paper, the effect of rain induced attenuation for millimeter wave is discussed. The theory of multiple scattering is used to obtain the solution for the plane wave propagation through a plane parallel medium of thickness L containing randomly distributed nonspherial particles. The coherent field and the total field are studied, respectively. The numerical results are good agreement with experimental data and the multiple scattering effects must be included. A 3mm wave propagation measurement system was made on a 0.8km terrestrial link.

Quantitative analysis in low-energy-electron transmission and reflection spectroscopy.

A recent model of particle transport in plane-parallel media, including both quasielastic and inelastic scattering, has been proposed [Goulet et al., Phys. Rev. A 41, 6006 (1990)] which claims to be fully three dimensional, to include the effects of multiple scattering, and to avoid the normal difficulties associated with such problems. The present paper demonstrates that a transport problem is being solved and that the method used is of comparable accuracy to elementary diffusion theory. It is shown that the model, applied here to the low-energy-electron transmission and reflection spectroscopy problem, is valid only under a fairly stringent assumption on the interface scattering at the boundaries of the layer. The model is rederived for this case in terms of standard transport theory. A comparison of the two models points to an ambiguity in the interface reflectivity in the previous model. In addition the method presented here leads to a more satisfactory application of the Snell-Descartes law and a full solution in terms of known functions.

On the accuracy of the Eddington approximation for radiative transfer in the microwave frequencies

Radiative transfer calculations in the presence of clouds and precipitation simplify as the wavelengths of the radiation approach the Rayleigh scattering regime. In particular, the transfer of microwave radiation is often simpler than the equivalent calculations at shorter wavelengths because the order of quadrature and moments of the phase function needed for the treatment of scattering decrease as the wavelength increases. The purpose of this paper is to examine how well an Eddington approximation can reproduce brightness temperatures obtained from a more complete, N‐stream discrete ordinate solution in the microwave regime. Radiation propagating through a plane parallel medium will be considered in this discussion. Although model discrepancies are complicated functions of the cloud constituents, the differences between an eight‐stream discrete ordinate solution and an analytical Eddington solution were generally small, ranging from 0 to 6°K when only one uniform layer of hydrometeors was considered. When realistic, multilayered cloud hydrometeor profiles were used, the differences between these two models never exceeded 3°K over the entire range of microwave frequencies considered (6.6–183 GHz). The models agreed to within 0.2°K in the absence of scattering constituents.

A spherical-harmonics method for multi-group or non-gray radiation transport

The spherical-harmonics method, also called the P N method, is used to develop solutions to a class of multi-group or non-gray radiation transport problems. The multi-group model considered allows an anisotropic scattering law and transfer from any group to any group. In addition to a spherical-harmonics solution for the case of a homogeneous radiative-transfer equation, a particular solution for the P N method is derived for the case of multi-group radiative transfer in a homogeneous plane-parallel medium that contains group sources that vary with position and direction. Computational aspects of the developed solutions are discussed, and numerical results for a test case are reported.

Asymptotic theory for optically thick layers: application to the discrete ordinates method

Asymptotic expressions for the reflected, transmitted, and internal scattered radiation field in optically thick, vertically homogeneous, plane-parallel media are derived from first principles by using the discrete ordinates method of radiative transfer. Compact matrix equations are derived for computing the escape function, diffusion pattern, diffusion exponent, and the reflection function of a semi-infinite atmosphere in terms of the matrices, eigenvectors, and eigenvalues that occur in the discrete ordinates method. These matrix equations are suitable for numerical computations and are valid throughout the full range of single scattering albedos. The present formulations are validated by comparing them with established methods of radiative transfer.

Alternative to the Pomraning-Eddington approach to radiative transfer.

The modified Eddington approximation, introduced in 1969 by Pomraning [J. Quant. Spectrosc. Radiat. Transfer 9, 407 (1969)], is used for solving the radiative-transfer equation for plane-parallel media. The specific intensity I(x,\ensuremath{\mu}) is expressed as the sum of an even and an odd function of the angular variable \ensuremath{\mu}, i.e., I(x,\ensuremath{\mu})=E(x)\ensuremath{\epsilon}(x,\ensuremath{\mu})+F(x)o(x,\ensuremath{\mu}); the integro-differential transport equation is thereby transformed into two coupled first-order differential equations involving the energy density E(x), the radiative flux F(x), and D(x)\ensuremath{\equiv}${\mathcal{F}}_{\mathrm{\ensuremath{-}}1}^{1}$${\mathrm{\ensuremath{\mu}}}^{2}$\ensuremath{\epsilon}(x,\ensuremath{\mu})d\ensuremath{\mu}. In Pomraning's treatment, one of these two exact equations is replaced by an approximate version derived by neglecting the spatial dependence of D(x); despite this simplification, the coupled equations usually defy, if one is treating inhomogeneous media, attempts at analytic solutions; the boundary conditions for the two differential equations are fabricated by multiplying the condition satisfied by I(x,\ensuremath{\mu}) at each boundary by a prescribed weight function and integrating with respect to \ensuremath{\mu}. We also outline an alternative strategy wherein the integral equations satisfied by D(x), E(x), and F(x) are deduced and solved by means of the variational method. The advantages of the latter approach are twofold. First, since the pertinent boundary conditions are automatically incorporated in the integral equations, the problem of inferring suitable weight functions is obviated; second, the spatial dependence of D(x) is taken into account explicitly. Numerical results are presented to illustrate the performance of the adapted Pomraning-Eddington approach.

Calculation of radiative transfer in plane media

Combined radiation and heat conduction in a plane parallel medium is investigated using a special integro-differential form of the radiative transfer equation as a starting point. Opaque reflecting boundaries are assumed; steady and time-dependent cases are considered. The solution technique used is efficient in computer time and can be extended to treat much more complicated problems.

Ionization and excitation in cool giant stars. I - Hydrogen and helium

The influence that non-LTE radiative transfer has on the electron density, ionization equilibrium, and excitation equilibrium in model atmospheres representative of both oxygen-rich and carbon-rich red giant stars is demonstrated. The radiative transfer and statistical equilibrium equations are solved self-consistently for H, H(-), H2, He I, C I, C II, Na I, Mg I, Mg II, Ca I, and Ca II in a plane-parallel static medium. Calculations are made for both radiative-equilibrium model photospheres alone and model photospheres with attached chromospheric models as determined semiempirically with IUE spectra of g Her (M6 III) and TX Psc (C6, 2). The excitation and ionization results for hydrogen and helium are reported.

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.

Particular solutions for the radiative transfer equation

Particular solutions for both the (formally exact) method of elementary solutions and the P N method are derived for the case of monochromatic radiative transfer in a homogeneous plane-parallel medium which contains a source that varies with position and direction.