Radiative transfer in atmospheres with spherical symmetry

A new approximate method to solve the time-independent transfer equation in a system with spherical geometry is presented. No restriction is imposed on the radial variation of the opacity so that the method is applicable even when the opacity is tabulated. The approximation consists in solving the transfer equation only for the outward and inward radial directions and in using the corresponding intensities for the numerical computation of the mean intensity and flux. To obtain the integration weights, we make several hypotheses on the directional dependence of the intensity, taking into account the principal properties of radiation transfer in spherical symmetric systems. We point out the parallelism that exists between this approximation for spherical geometry and a similar one for the plane-parallel case. Results are presented for the conservative and non-conservative cases, and they are compared to those of other authors.     

Radiative decay times for thermal pertubations in a nongray medium

The radiative decay time of harmonic thermal perturbations in a nongray medium of infinite extent is obtained in closed form for two specific band absorption models. These models are the frequently used gray band and the exponential band, the latter being considered more realistic for molecular gases. It is found that the decay time at the boundary of a semi-infinite medium can be obtained in terms of that in an infinite medium. The decay time for combined thermal radiation and conduction is also discussed. The difference in radiative decay rates for a medium with gray bands and one with exponential-tailed bands is marked; in an infinite medium at large Bouguer number, the former falls to zero while the latter rises to a maximum.     

Radiative transfer in a two-dimensional rectangular medium exposed to diffuse radiation

The integral equation for radiative transfer in a two-dimensional rectangular scattering medium exposed to diffuse radiation is solved numerically by removing the singularity. This method yielded accurate results except at very large optical thicknesses. Graphical and tabular results for the source function, flux, and intensity are presented. The source function is also calculated using the first term of a Taylor series expansion. The Taylor series is fairly accurate for small optical thicknesses and columnar geometries. A method is presented for extending these results to the problem of a strongly anisotropic scattering phase function which is made up of a spike in the forward direction superimposed on an isotropic phase function.     

Radiative decay time of a nongray medium with arbitrary configurations

Based on the principle of conservation of energy, a simple analytical expression is presented for the radiative decay time of a nongray medium with arbitrary configurations. It is expressed explicity in terms of the mean beam length and the total band absorptance. In the high-pressure limit, where the decay time based on the harmonic thermal perturbation is available for two configurations, comparison is made with the present decay time. Good agreement is found.     

Radiative transfer model of a pyrotechnic flame

A two-line radiative transfer model for predicting the spectral radiant flux of pyrotechnic illuminating flares over a wide range of system variables such as formula, size, and ambient pressure, has been formulated and validated.To solve the transfer equation for observed radiant intensity, the flame is represented by a model whose main characteristics are (a) the flame is a homogeneous gaseous atmosphere with plane-parallel stratification, (b) the gas consists of inert molecules plus sodium atoms which can be excited to the ²P_{$\text{1}\text{2}$} or ²P_{$\text{3}\text{2}$} level, (c) there is local thermodynamic equilibrium governed by the local temperature, (d) the temperature gradient can be represented by a parabola whose vertex is at the center of the flame, (e) the dispersion profile and number density of sodium atoms have average values, inside the flame, that are independent of depth, and (f) the individual line dispersion profile is replaced with a two-line function to simultaneously describe the spectral distribution of both of the sodium D lines.The parameters of the radiative transfer theory were supplied from calculated thermodynamic properties of the flare. Optical thickness as a function of position in the flame was determined using computed sodium atom densities and physical flame size obtained photographically. A flame temperature gradient was constructed numerically as a function of temperature in the flame using the computed temperature at the flame center and the boundary. The two-line dispersion profile was constructed as a function of line broadening. The shape and intensity of the broadened flare spectrum was computed without introducing further assumptions.     

Radiative transfer in a layer of magnetized plasma with random irregularities

The problem of radio wave reflection from an optically thick plane uniform layer of magnetized plasma is considered in the present work. The plasma electron density irregularities are described by a spatial spectrum of arbitrary form. The small-angle scattering approximation in invariant ray coordinates is proposed as a technique for the analytical investigation of the radiation transfer equation. The approximate solution describing the spatial and angular distribution of radiation reflected from a plasma layer is obtained. The solution obtained is investigated numerically for the case of ionospheric radio wave propagation. Two effects occur as a consequence of multiple scattering: a change in the reflected signal intensity and an anomalous refraction.     

Radiative transfer in a spherical, emitting, absorbing and anisotropically scattering medium

The atmospheres of planets (including Earth) and the outer layers of stars have often been treated in radiative transfer as plane-parallel media, instead of spherical shells, which can lead to inaccuracy, e.g. limb darkening. We give an exact solution of the radiative transfer specific intensity at all points and directions in a finite spherical medium having arbitrary radial spectral distribution of: source (temperature), absorption, emission and anisotropic scattering. The power and efficiency of the method stems from the spherical numerical gridding used to discretize the transfer equations prior to matrix solution: the wanted ray and the rays which scatter into it both have the same physico-geometric structure. Very good agreement is found with an isotropic astrophysical benchmark [Avrett EH, Loeser R. Methods in radiative transfer. In: Kalkofen W, editor. Cambridge: Cambridge University Press; 1984. pp. 341-79]. We introduce a specimen arbitrary forward- side-back phase scattering function for future comparisons. Our method directly and exactly addresses spherical symmetry with anisotropic scattering, and could be used to study the Earth's climate, nuclear power (neutron diffusion) and the astrophysics of stars and planets.     

Symmetric heat and mass transfer in a rotating spherical layer

The general theory of heat and mass transfer maintaining rotation with slightly different velocities under conditions typical for cores of planets in the solar system is developed for the first time. The analytic solution is obtained for thermal and diffusion equations without nonlinear terms responsible for the convective transfer. This spherically symmetric basic solution is applicable when the thermal flux from a planet core is weaker than or comparable to the adiabatic (radiative) flux. In the general case, by subtracting the basic solution, we simplified the inhomogeneous system of convective equations to obtain a completely homogeneous and dimensionless system. The latter system is controlled by two asymptotically small parameters: the Rossby number ε<10^?5, which characterizes the relative value of differential rotation, and the generalized Eckman number E<10^?12, which characterizes the relative role of viscosity-diffusion effects during rapid rotation. The principal order of the solution for ε →0 and then for \(\sqrt E \to 0\), for the transfer coefficients close to molecular coefficients, results in the basic flow, which is symmetric with respect to the rotation axis and directed predominantly along the azimuth. The basic-flow liquid ascends from a solid core along spirals inside an axial cylinder in contact with the equator of the solid core and descends in a narrow layer along the cylinder walls. The moment of viscous forces in the inner boundary Eckman layer provides a faster rotation of the inner solid core of terrestrial planets compared to a massive outer mantle due to the growth of the solid core at the expense of a low-density liquid core.     

Radiative transfer in three-dimensional rectangular enclosures containing inhomogeneous,anisotropically scattering media

Radiative transfer in a three-dimensional rectangular enclosure containing radiatively participating gases and particles is studied using the first- and third-order spherical harmonics approximations. Inhomogeneities in the radiative properties of the medium, as well as in the radiation characteristics of the boundaries, are allowed for. The scattering phase function is represented by the delta-Eddington approximation, and it is assumed to be a function of the location in order to account for density variation of the particles in the medium. Numerical solutions of the model equations are obtained using a finite difference scheme. For the purpose of validating the P₃-approximation, the results are compared with those based on Hottel's zonal method.     

Radiative equilibrium of a gray medium in a rectangular enclosure

Radiative equilibrium temperature distributions are calculated for an absorbing-emitting gray medium enclosed by rectangular walls of different temperatures. A “modified differential approximation” is used to obtain an approximate solution, which compares favorably with direct numerical computations carried out using a zonal method.     

Radiative transfer in atmospheres with spherical symmetry—IV. the non-conservative problem

Non-conservative transfer problems are conceptually more complicated than the conservative ones because the source function introduces a new scale height. This problem is specially important in spherical geometry, where, due to the curvature of the layers, one must also take into account the scale height of the opacity. With the introduction of the auxiliary variables μ_c and μ_r, it has been possible to obtain very simple linear closure relations for the μ moment system; μ_c is the cosine of the critical zenithal angle ?_c from which the intensity peak appears; μ_r is an intermediate value of μ between μ_c and 1 which permits an explanation of the geometrical dilution and leads easily to a peaking linear relation. With these closure relations, we have solved the moment system for different opacity laws and source functions which represent many cases of astrophysical interest. Likewise, we have compared the quality of our results for different orders of approximation.     

Radiative transfer equation for graded index medium in cylindrical and spherical coordinate systems

In graded index medium, the ray goes along a curved path determined by Fermat principle, and the curved ray-tracing is very difficult and complex. To avoid the complicated and time-consuming computation of curved ray trajectory, the methods not based on ray-tracing technique need to be developed for the solution of radiative transfer in graded index medium. For this purpose, in this paper the streaming operator along a curved ray trajectory in original radiative transfer equation for graded index medium is transformed and expressed in spatial and angular ordinates and the radiative transfer equation for graded index medium in cylindrical and spherical coordinate systems are derived. The conservative and the non-conservative forms of radiative transfer equation for three-dimensional graded index medium are given, which can be used as base equations to develop the numerical simulation methods, such as finite volume method, discrete ordinates method, and finite element method, for radiative transfer in graded index medium in cylindrical and spherical coordinate systems.     

Radiative transfer in atmospheres with spherical symmetry—III. The Eddington approximation

A new generalization of the Eddington approximation for systems with spherical geometry is presented. For this kind of geometry, we have deduced an analytical expression for the Eddington factor, where the opacity law, which we assume to be a known function of the radial distance r, is an explicit variable. This expression, which is easily computable, together with the two first moment equations, gives better results for spherical systems than the corresponding $\text{? =}\text{1}\text{3}$ relation in plane ones.     

Radiative transfer in a moving medium

We illustrate first, for a one-dimensional medium, how to generalize the concept of the probability of quantum exit to the case of a moving atmosphere. The transfer of radiation takes place in a two-level atom with total redistribution in frequency at each scattering. The values of the reflection coefficient and of the source function at the surface of a semi-infinite medium with constant properties (especially concerning the velocity gradient) are given. The results are then extended to the three-dimensional case in the Appendix.     

Radiative transfer in atmospheres with spherical symmetry—II. The conservative Milne problem

A new method of successive approximations is developed to solve transfer problems in spherical coordinates. It is equivalent to methods involving the introduction of a closure relation for the μm-moment system. From a practical point of view, it is a two-direction method with variable quadrature weights which are easily computable for any depth variation of the opacity. These quadrature weights are taken as new variables; this allows us, even in the lowest orders, to dispose of closure relations that take into account the principal properties of the radiation field in spherical systems: quasi-isotropic radiation at great optical depths and unidirectional beam “peaking effect” at large values of the radial distance.In the first approximation, we find simple properties like those of the first order in the discretization method for plane-parallel systems. Therefore, it is formally possible to identify these two methods which have a similar practical application. We find for the mean intensity J(r), in the Milne problem, an analytical expression which corresponds to J = 3H(τ + cte) in the plane-parallel geometry.This work is a generalization of several important aspects of another study published in 1976 from which we have only kept the formalism. The new closure relation for the moment system is physically more complete and permits the development of new important results and allows the possibility of an easy solution in every order. From the first results obtained here, we show and analyze the differences between this study and the preceding one.     

Radiative equilibrium in a rectangular enclosure bounded by gray walls

Two-dimensional temperature and heat flux distributions are calculated for an absorbing-emitting gray medium at radiative equilibrium in a rectangular enclosure. The bounding walls are gray and diffuse with arbitrary surface temperature distributions, and heat generation may take place inside the medium. As a first approximation, the problem is solved for optically thick systems (differential approximation). These results are subsequently improved by the introduction of a number of geometrical parameters to yield good accuracy for all optical thicknesses. As examples, two cases are discussed in detail: (1) uniform heat generation in a black enclosure and (2) an enclosure with one gray surface at constant temperature. Comparison with some numerical solutions generated by Hottel's zonal method shows excellent agreement.     

On “Radiative transfer in three-dimensional rectangular enclosures containing inhomogeneous, anisotropically scattering media”

Our 1985 paper (JQSRT 1985; 33: 533-549) reported the result of the research we conducted back then to better understand heat transfer processes in large-scale combustion chambers, especially in pulverized coal-fired furnaces. It was one of the first works exploring radiative transfer in three-dimensional enclosures where absorption and scattering coefficients due to combustion particles and gases were allowed to vary within the medium. This flexibility of the mathematical model made it useful for applications to realistic furnaces and different types of high-temperature systems. This note briefly discusses the motivation behind the paper and the immediate extension of the idea to different systems.     

Radiative corrections in a vector-tensor model

In a recently proposed model in which a vector non-Abelian gauge field interacts with an antisymmetric tensor field, it has been shown that the tensor field possesses no physical degrees of freedom. This formal demonstration is tested by computing the one-loop contributions of the tensor field to the self-energy of the vector field. It is shown that despite the large number of Feynman diagrams in which the tensor field contributes, the sum of these diagrams vanishes, confirming that it is not physical. Furthermore, if the tensor field were to couple with a spinor field, it is shown at one-loop order that the spinor self-energy is not renormalizable, and hence this coupling must be excluded. In principle though, this tensor field does couple to the gravitational field.     

Effect of band or line shape on radiative transfer in a nongray two-dimensional medium

An exact formulation is presented for a nongray two-dimensional, finite, planar, absorbing-emitting medium in radiative equilibrium. The absorption coefficient consists of an array of equal intensity, nonoverlapping bands or lines. Rectangular, triangular, exponential, Doppler and Lorentz shapes are specifically considered. Exact expressions are obtained for a medium subjected to collimated and diffuse radiation. The integral equations are linearized by the narrow-band approximation. The solution for the cosine-varying, collimated, monochromatic radiation model is used to construct the solutions for other boundary conditions. The two-dimensional equations are reduced to one-dimensional equations by the method of separation of variables. Results for the diffuse case are presented for several spatial variations.     

Radiative energy transfer in a randomly fluctuating medium

The basic laws of the phenomenological theory of radiative energy transfer are derived, under certain conditions, within the framework of the stochastic scalar wave theory. An equation of radiative energy transfer is derived for wave propagation in a statistically quasihomogeneous medium. Our results relate the extinction and scattering coefficients (which are introduced heuristically in the conventional theory of radiative energy transfer) to the stochastic characteristics of the medium.