Probabilistic model for radiative transfer problems in cylindrical shell media with complete redistribution in frequency

A probabilistic model for solving transfer problems in non-homogeneous, isotropic, and non-coherent scattering cylindrical shell media has been proposed. The source function is considered to be frequency independent. The scattering and transmission functions have been defined for the case of complete redistribution in frequency. A tractable integrodifferential equation for the scattering function has been derived.     

Combined-Operations method for diffuse reflection by an isotropic,non-coherent scattering homogeneous sphere

Combined-Operations method has been utilised to solve the problem of diffuse reflection by a homogeneous, isotropic, non-coherent scattering spherical medium. The source function is considered to be frequency independent. The auxiliary equation has been formulated, the scattering function defined, and the integro-differential equation for this function deduced. A method for obtaining the emergent intensity and the internal source function for non-zero internal source distribution has been suggested for a given line profile.     

On combined operations method for transfer problems in homogeneous,cylindrical media

The problem of diffuse reflection by a homogeneous, isotropically scattering, infinite cylindrical medium has been considered. The relevant auxiliary equation has been formulated, the scattering function defined and the integro-differential equation for such function deduced. For a medium having cylindrical distribution of source in addition to the incident flux at the outer surface, the integro-differential equation for the emergent intensity has been established.     

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.     

Non-coherent scattering in transfer problems in spherical shell media

The Combined Operations Method is utilised to solve Diffuse Reflection and Transmission Problems in inhomogeneous, isotropically and non-coherently scattering, spherical shell media. The source function is assumed to be frequency independent. TheN-solution of an auxiliary equation is sought, and tractable equations for the scattering and transmission functions are established. The solution of the problem for a scattering and emitting medium have been considered for a perfectly absorbing core.     

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.     

On the effect of the distribution function of the injected or formed relativistic particles on the frequency spectra of the radio sources

In the present paper a model of the cosmic radio source with plasma, relativistic electrons and chaotic magnetic fields is considered. It is assumed that the energy changes of relativistic electrons which are due to spontaneous scattering on the transverse and Langmuir plasmons take place in this radio source. For such a radio source the kinetic equation has been solved in the presence of the injected or formed secondary electrons and the frequency spectrum has been obtained.     

The effect of electron scattering on curves of growth

A new exact solution of the macroscopic line transfer equation including electron scattering terms has been obtained for a Milne-Eddington Model atmosphere, and curves of growth based on this solution have been calculated. The results indicate that for lines formed by scattering there is a systematic change in the appropriate theoretical curve of growth as electron scattering becomes an increasingly important source of continuous opacity. In a case where electron scattering is the dominant opacity source, the abundance necessary to produce a given line strength may be decreased by a factor of 2–5 and the derived velocity parameter decreased by 20–30% due to the shifts in the theoretical curves.Contributions from the Kitt Peak National Observatory No. 416.Operated by the Association of Universities for Research in Astronomy, Inc. under contract with the National Science Foundation.     

Time-dependentX- andY-functions by integral equation method

The time-dependent equation of radiative transfer for isotropic scattering has been solved by integral equation technique in terms ofX- andY-functions appropriate for the problem. It is seen thatX- andY-functions are reducible to the corresponding function for steady-state problems by simply changing the Laplace transform parameters-i.e., byS0.     

Numerical methods in polarized radiative transfer

This paper looks at three aspects of numerical methods for solving polarized radiative transfer problems associated with spectral line formation in the presence of a magnetic field. First we prove Murphy's law for Stokes evolution operators which is the basis of the efficient algorithm used in the SPSR software package to compute the Stokes line depression contribution functions. Then we use a two-stream model to explain the efficacy of the field-free method in which the non-LTE line source function in a uniform magnetic field is approximated by the source function neglecting the magnetic field. Finally we introduce a totally new and computationally efficient approach to solving non-LTE problems based on a method of sparsely representing integral operators using wavelets. As an illustration, the wavelet method is used to solve the source function integral equation for a two-level atomic model in a finite atmosphere with coherent scattering, ignoring polarization.     

Albedo-shifting method: Source function in a plane atmosphere

A classical problem in the theory of radiative transfer is considered: calculating the radiation field within a plane scattering atmosphere. The recently proposed albedo-shifting method is used to calculate the source function both in a semi-infinite atmosphere and in an atmophere of finite optical depth, illuminated by parallel rays. The method enables one to “suppress” scattering and obtain iterative solutions of the integral equation for the source function in only a few direct lambda iterations, even when the average number of photon scatterings in the atmosphere is very large. Translated from Astrofizika, Vol. 42, No. 4, pp. 485–500, October–December, 1999.     

Pitch-angle scattering of energetic particles in the current sheet of the magnetospheric tail and stationary distribution functions

An investigation of pitch-angle scattering of energetic particles in magnetic field configurations with a current sheet similar to that observed in the geomagnetotail has been performed. The magnetic field model is specified by two parameters which are the current sheet thickness in units of particle gyroradius and the angle between the magnetic field lines and the sheet plane. Computations of a considerable number of trajectories (about 20,000 for each model case) has provided the possibility of obtaining the matrix of pitch-angle scattering and the corresponding kernel function of the integral equation for the stationary particle distribution function. Solution of this equation shows that isotropic distributions are formed only in the case of a sufficiently thick current sheet. Particle scattering in a thin field reversal region leads to the formation of an anisotropic stationary distribution. The results can be used for interpretation of the data on the spatial distribution of energetic particle fluxes in the near part of the magnetospheric tail and in the vicinity of the outer boundary of the radiation belt.     

Invariant imbedding and Chandrasekhar's planetary problem of radiative transfer

In connection with Chandrasekhar's planetary problem of radiative transfer the total scattering and the diffuse transmission functions have been discussed by several authors (cf. Chandrasekhar, 1950; van de Hulst, 1948; Sobolev, 1948; Bellman,et al., 1967; Kagiwada and Kalaba, 1971). With the aid of the Bellman-Krein formula for the resolvent kernel of the auxiliary equation governing the source function, we show how the invariant imbedding equations governing the diffuse scattering and transmission functions can readily be obtained. So far as we know, the Cauchy system of the functional equations for the scattering and transmission functions is new and is well-suited for the numerical computation.Supported by the National Science Foundation under Grant No. GP 29049, and by the Atomic Energy Commission under Grant No. AT (40-3)-113 Project 19.     

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).     

3-Dimensional Energetic Particle Distribution Past an Instantaneous Uni-Directional Injection

The exact analytic expression for the density of energetic charged particles, which were injected by an instantaneous point source at a particular pitch angle into the interplanetary medium, has been derived. We start from the Boltzmann kinetic equation with the collision integral describing the isotropic particle scattering by "massive" magnetic clouds. The solution has been obtained without any expansion parameters in the 3-dimensional vector form, then it was projected into the cylindrical coordinate system. The space-time particle distribution is disscussed. This revised version was published online in July 2006 with corrections to the Cover Date.     

Angular fluctuations in the cosmic microwave background owing to initial perturbations in the radiation temperature

A new method is proposed to account for multiple scattering by electrons in calculations of the correlation functions describing the angular fluctuations in the cosmic microwave background radiation (CMBR). The apparatus of the theory of radiative transport with Rayleigh scattering is used. The problem is reduced to solving an integral equation for the vector source function (dependent only on time), along with differential equations for the other quantities (scalar potentials, baryon velocities, etc.) which show up in the problem. The quantities which describe the angular fluctuations in the CMBR (in the temperature and in the polarization) are then calculated by integrating the vector source function along the line of sight. As an illustration, the correlation functions and power spectra are calculated for the case where the fluctuations are produced by some initial gaussian perturbations of the CMBR. __________ Translated from Astrofizika, Vol. 50, No. 4, pp. 621–631 (November 2007).     

A relaxation method for the computation of model stellar atmospheres

A general Monte Carlo relaxation method has been formulated for the computation of physically self-consistent model stellar atmospheres. The local physical state is obtained by solving simultaneously the equations of statistical equilibrium for the atomic and ionic level populations, the kinetic energy balance equation for the electron gas to obtain the electron temperature, and the equation of radiative transfer. Anisotropic Thomson scattering is included in the equation of transfer and radiation pressure effects are included in the hydrostatic equation. The constraints of hydrostatic and radiative equilibrium are enforced. Local thermodynamic equilibrium (L.T.E.) is assumed as a boundary condition deep in the atmosphere. Elsewhere in the atmosphere L.T.E. is not assumed.The statistical equilibrium equations are solved with no assumptions made concerning detailed balance for the bound-bound radiative processes. The source function is formulated in microscopic detail. All atomic processes contributing to the absorption and emission of radiation are included. The kinetic energy balance equation for the electron gas is formulated in detail. All atomic processes by which kinetic energy is gained and lost by the electron gas are included.The method has been applied to the computation of a model atmosphere for a pure hydrogen early-type star. An idealized model of the hydrogen atom with five bound levels and the continuum was adopted. The results of the trial calculation are discussed with reference to stability, accuracy, and convergence of the solution.Contribution No. 385 from the Kitt Peak National Observatory.Operated by the Association of Universities for Research in Astronomy, Inc., under contract with the National Science Foundation.     

Radiation transfer through atmospheric aerosol media with anisotropic scattering

Radiation transfer in atmospheric aerosol media with general boundary conditions has been studied for anisotropic scattering. The considered aerosol medium assumed to have specular and diffused reflecting boundary surfaces and in the presence of internal source. The radiation transfer scattering parameters as single scattering albedo, asymmetry factor, scattering, absorption, extinction efficiencies and anisotropic scattering coefficient have been calculated using the Mie theory. The problem with general boundary conditions is solved in terms of the solution of source-free problem with simply boundary conditions. Pomraning-Eddington approximation is used to solve the source-free problem. For the sake of comparison, a weight function is introduced and used in two special forms. The calculated partial heat fluxes with the two methods are compared and showed good agreement. Some of our results are found in a good agreement with published data.     

Volterra equation for the matrix source function at resonance scattering

A matrix transfer equation for multiple resonance scattering of radiation in a spectral line in a semiinfinite atmosphere with a uniform distribution of primary radiation sources is examined. A nonlinear matrix integral is obtained for this equation as a generalization of the Rybicki two-point Q-integral. One special case of the matrix [^(Q)] {\mathbf{\hat{Q}}} -integral is the Volterra equation for the matrix source function of the problem discussed here. The Volterra equation is solved numerically for a Doppler profile of the absorption coefficient. Several polarization characteristics of the emerging radiation are obtained.