Time-dependentX- andY-functions for an isotropically scattering neutron transport problem in a finite slab

Expressions for time-dependentX- andY-functions for a one-speed neutron transport problem in a finite slab have been derived using a technique combining invariant imbedding method and eigenfunction expansion method. The atmosphere has been considered to scatter isotropically.     

Transfer of resonance radiation in an infinite isotropically expanding medium

Conclusions Besides the problems considered here, one can find comparatively easily asymptotic solutions in a problem with a spherically symmetric exponential distribution of the primary sources, and also in the case when the primary sources are distributed uniformly within a sphere of finite radius. However, we do not give these solutions because they are cumbersome. Note that some of these problems may have direct astrophysical applications. For example, the problem of a point source can arise in the determination of the source function in the problem of the emission of a planetary nebula.The results of the present work, and also of the majority of the other investigations have been obtained under the assumption of complete frequency redistribution for scattering in the comoving coordinate system. In reality, this assumption is not strictly satisfied. However, calculations made recently for moving media [13, 14] have shown that the results found for true and complete redistribution differ little from one another, these differences being less than in the case of a medium which is not in motion.A. A. Zhdanov Leningrad State University. Translated from Astrofizika, Vol. 14, No. 1, pp. 111–121, January–March, 1978.     

Average photon path-length in isotropically scattering finite atmospheres

The determination of the average path-length of photons in a finite isotropically scattering plane-parallel homogeneous atmosphere is discussed. To solve this problem we have used the kernel approximation method which easily allows us to find the derivatives of the intensity with respect to optical depth, optical thickness and albedo of single scattering.In order to check the results we have used another approach by exploiting the set of integrodifferential equations of Chandrasekhar for theX- andY-functions. This approach allows us to find the average path length only at the boundaries of the atmosphere but on the other hand it gives also the dispersion of the path-length distribution function, thus generating the input parameters for determining the approximate path-length distribution function. It occurred that the set so obtained is stable and the results are highly accurate.As a by-product we obtain the first two derivatives of theX- andY-functions with respect to the albedo of single scattering and optical thickness, and the mixed derivative.     

The complete solution for the scattering of polarized light in a Rayleigh and isotropically scattering atmosphere

TheF _N method is used to solve, in a concise manner, the complete problem concerning the diffusion of polarized light in a plane-parallel Rayleigh and isotropically scattering atmosphere.     

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)=w ₀ exp(–x/s), wherew ₀ ands are given constants andx is the optical variable. The second approximation assumes the formw(x) = _r=0 ^R d _r ^* p _r (x/a), whered _r ^* 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.     

Angular radiation transfer in inhomogeneous dispersive media

The equation of radiative transfer for an inhomogenous dispersive finite medium subject to general boundary conditions is solved. The Padé approximation technique is used to calculate the angular distribution of radiation. Numerical results for the [0/1] Padé approximant lead to numerical results that compared with the exact results.     

Angular fluctuations of relict radiation produced by cosmological turbulence

On the basis of a theory for the evolution of cosmological turbulence developed earlier by the present authors, the temperature fluctuations of the relict black-body radiation produced by turbulent motions are calculated. We take into consideration (i) the temperature fluctuations that appear at the moment of recombination (redshiftz10³) with the account of their subsequent diminishing as a result of the optical depth produced in the course of re-ionization of metagalactic gas by young galaxies at their bright phase, and (ii) the temperature fluctuations that emerge on the surface of the last scattering (1) atz10 from vortexes and from potential velocities generated by the vortexes. A comparison of these calculations with the available measurements (upper limits) for temperature fluctuations of relict radiation makes it possible to obtain important upper and lower bounds for the initial velocity of the vortex motions.     

Propagation of a burst of radiation in an expanding and recombining universe: Thomson scattering

The propagation of an instantaneous burst of nonpolarized isotropic radiation from the time of its onset at some redshift z ₀ to the time of its recording at the present epoch is considered within the framework of a flat cosmological model. Thomson (Rayleigh) scattering by free electrons is believed to be the only source of opacity. The spatial distributions of the mean (over the directions) radiation intensity as well as the angular distributions of the radiation intensity and polarization at various distances from the burst center have been constructed. The mean intensity profile normalized to the total number of photons emitted during the burst is shown to depend weakly on the initial conditions (the burst time z ₀, the width and shape of the initial radiation distribution) at fairly high z ₀ (≥1400). As regards the angular intensity and polarization distributions, they turn out to be rather narrow (3–10 arcmin), while the polarization can reach 70%. On average, the expected polarization can be about 15%.     

Compton scattering of electromagnetic radiation in pulsar magnetospheres

We have considered the spontaneous Compton scattering of radiation in the magnetic field by both a single ultra-relativistic electron and a system of electrons with the power-law energy distribution. The degree of radiation anisotropy was assumed arbitrary. Parameters of the scattering-generated radiation for the entire range of post-scattering photon energies are given in the paper for all possible scattering modes.     

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.     

A linear scattering problem in magnetohydrodynamics: Transmission resonances in a magnetic slab

The reduced linearized equations of ideal magnetohydrodynamics which are highly nonlinear in the eigenvalue parameter, are linearized about a prescribed value of that parameter, enabling the equation to be expressed as a Schrödinger equation with piecewise uniform coefficients. Reflection and transmission coefficients are obtained using standard techniques, and in addition to the possibility of total transmission of an incident wave occurring (together with complex-valued resonance energies), the magnetic field introduces other total transmission energy levels which have no counterpart in the absence of a magnetic field.     

Angular distribution of photoelectrons from atomic oxygen,nitrogen and carbon

The angular distributions of photoelectrons from atomic oxygen, nitrogen and carbon are calculated. Both Hartree-Fock and Hartree-Slater (Herman-Skillman) wave functions are used for oxygen and the agreement is excellent; thus only Hartree-Slater functions are used for carbon and nitrogen. The pitch angle distribution of photoelectrons is discussed and it is shown that previous approximations of energy independent isotropic or sin² θ distributions are at odds with our results, which vary with energy. This variation with energy is discussed as is the reliability of these calculations.     

Angular velocity distribution in convective regions and the origin of solar differential rotation

The observed differential rotation of the Sun is explained as a result of interaction between global rotation and convective eddies.