Spherical Formulation Research Articles

A Semianalytic Method for the Solution of the Neutron Transport Equation in Plane and Spherical Geometries

The source-free monoenergetic transport equation can be written as where for plane and for spherical geometries, respectively; Ω is the direction of motion with θ as its angle to the axis, An analytic expression for ψ(Z, cosθ) of the form is considered where Pn(cosθ) is the Legendre polynomial of order n; are combinations of exponential functions for the plane case and Bessel functions in the spherical formulation. A simple algorithm is developed that enables solutions to be found for any N using a small computer program.

A Rigorous Treatment of Transverse Buckling Effects in Two-Dimensional Neutron Transport Computations

The variational formulation of the even-parity form of the within-group neutron transport equation is generalized to include complex trial functions. The introduction of transverse leakage effects through the buckling term exp(iB·r) leads, in general, to a coupled set of Euler equations for the real and imaginary even-parity flux components. The coupling between real and imaginary flux components is retained in both discrete-ordinates and finite element angular approximations. Employment of the spherical harmonics approximations in angle, however, leads to an uncoupled set of Euler equations if an appropriate choice of axes is made. Hence, a rigorous buckling treatment of third-dimensional leakage can be incorporated into two-dimensional transport computations without solving for the imaginary flux component. The foregoing spherical harmonic formulation is combined with finite element discretization in space in the multigroup criticality code FESH. One- and multigroup results are presented to demonstrate the elimination of ray effects and to examine the errors introduced by the DB2 leakage correction used in conventional transport calculations.

The circumstellar shells and mass loss rates of four M supergiants

view Abstract Citations (93) References (34) Co-Reads Similar Papers Volume Content Graphics Metrics Export Citation NASA/ADS The circumstellar shells and mass loss rates of four M supergiants. Bernat, A. P. Abstract A reanalysis of the physical structure of the circumstellar gas shells of four bright M super- giants, Betelgeuse, Antares, a Herculis, and Cephei, has been undertaken. The observational data include old Hale Observatories plates, recent McDonald Struve telescope plates, and McDonald 2.7 m photoelectric scans. These data are analyzed in the full expanding spherical geometry formulation of the radiative transfer equation. The results of the present analysis indicate that column densities in the gas shells must be revised downward compared with the previous plane-parallel results. However, the physical extents of the shells are considerably larger than previously assumed. These extents are inferred through ionization modeling, Weymann's Ca ii technique, and direct observation. Also inferred are schematic wavelength-dependent chromospheric color temperatures. These results lead to much larger mass loss rates (in the range 6.7 x l0- to 4.2 x 10- M0 yr-2) than previously inferred. The influence of these large rates of mass loss on the evolution of both stars and the Galaxy is briefly discussed. Subject headings: stars: circumstellar shells - stars: mass loss - stars: supergiants Publication: The Astrophysical Journal Pub Date: May 1977 DOI: 10.1086/155205 Bibcode: 1977ApJ...213..756B Keywords: M Stars; Mass Transfer; Radiative Transfer; Stellar Envelopes; Stellar Models; Supergiant Stars; Abundance; Chromosphere; Gas Ionization; Photoionization; Stellar Atmospheres; Stellar Evolution; Stellar Spectrophotometry; Astrophysics full text sources ADS |

Remedies for Ray Effects

Remedies are considered for the ray effect, a flux distortion producing defect of the discrete ordinates approximation to the transport equation. The partially effective remedies of increasing the number of directions, selecting quadrature sets invariant under discrete rotations, and introducing coupling terms into the representation of the transport divergence operator are considered. Numerical results are presented showing the effectiveness of these remedies in a standard test problem. Remedies that eliminate the ray effect are also considered. We show how to formulate multidimensional discrete ordinate equations equivalent to spherical harmonic equations and relate the order of the equivalent spherical harmonic equations to the degree of precision of the numerical quadrature. We give numerical results for this kind of remedy in three test problems, including one with material discontinuities. We show how the use of orthonormal polynomials permits the formulation of discrete ordinates approximations which have properties “like” those of the highest order spherical harmonic equations consistent with the number of directions in the discrete ordinates approximation. We find that these formulations also eliminate ray effects. We conclude that the partially effective remedy of using more, specially chosen, directions is the most practical remedy for most applications, but that for especially difficult situations or for reference calculations, the defect-eliminating spherical harmonic-like formulations should be available for use.

Dipole-quadrupole polarization correlation in photo-selection

The intensity and polarization correlation formulas for an electric dipole-electric quadrupole tandem two-photon process are explicitly developed using a general spherical-tensor formulation for multipole-radiation of any order. This is applicable to relative polarization studies in photoselection or photoluminescence studies. The molecules are oriented randomly with respect to the laboratory coordinates defined by the radiation field, and the intensity is written in terms of transition matrix elements taken over coordinates of the parent molecule (assumed to interact with one of the photons through the quadrupole mode) and the daughter molecule (assumed to interact with the other photon through the dipole mode). Some bipolar harmonics through the third order needed in the correlation are computed. As an alternative to the above spherical tensor formulation, we expressed the individual one-photon intensities for electric dipole and electric quadrupole transitions in terms of direction cosines between laboratory and molecular coordinates. Then, the two-photon intensities in terms of sixth power products of direction cosines are averaged over all molecular orientations in space. The two formulations yield identical results in terms of matrix elements over cartesian coordinates. This dipole-quadrupole correlation is evaluated for molecules of a few common point group symmetries.

A Numerical Investigation of Classical Approximations Used in VLF Propagation

A computer program based on formalism developed by Budden will be used to evaluate the adequacy of some approximations that have been used in VLF waveguide propagation studies. The program makes allowance for the inhomogeneity and anisotropy of the lowest part of the ionosphere, allows for earth curvature, and may be used for arbitrary direction of propagation at any latitude.The results of this program will be compared with calculations of Wait and Spies who used a model ionosphere consisting of an exponential variation for both electron density and collision frequency. This comparison will allow us to demonstrate a close numerical equivalence between the modified refractive index formulation and the spherical coordinate formulation as implemented by Wait and Spies. Also, it will be shown that an approximation introduced by Wait and Spies during the course of applying the effective surface impedance concept, though quite valid for first‐order mode calculations, is inadequate for daytime second‐order mode calculations.