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.     

The polarization of radiation reflected diffusely by an inhomogeneous plane medium

The problem of determining the intensity and the degree of polarization of radiation emerging from an inhomogeneous finite plane medium for the case of Rayleigh scattering with internal energy source is considered. A system of coupled integral equations are obtained and solved by the Galerkin method. The degree of polarization for homogeneous and inhomogeneous media are calculated for uniform and nonuniform sources.     

Application of the superconvergence properties of the Galerkin approximation to radiative transfer

The superconvergence Galerkin approximation is introduced to calculate some functionals arising in radiative transfer problems. Forward and backward radiation heat fluxes are calculated for comparison.     

Polarization of radiation from stellar atmospheres. The grey case

An accurate numerical method is presented for the solution of the transfer equations in a plane-parallel atmosphere in which scattering occurs according to Rayleigh's law. Some results are given for the polarization and limb darkening of both integrated and monochromatic radiation emerging from grey atmospheres with various ratios of scattering to absorption. The method is equally applicable to non-grey atmospheres.     

Luminosity profiles and the contraction of radiating spheres in general relativity

A semi-numeric method by Herrera, Jiménez and Ruggeri is used to study the evolution of general relativistic spheres in the free streaming radiation approximation when specific contraction schemes are imposed and for particular oscillatory radiation profiles. On one hand, the emerging radiation is studied for oscillating contracting spheres, on the other, the variation of the boundary surface is analyzed when a pulsating profile is provided. We considered three particular equations of state previously studied. It is found that the oscillatory frequency of the surface for the equations of state considered coincide with the frequency of the oscillation for the radiation profile.     

A new method for exact solution of transfer equations in finite media

In this paper we develop a new exact method combined with finite Laplace transform and theory of linear singular operators to obtain a solution of transport equation in finite plane-parallel steady-state scattering atmosphere both for angular distribution of radiation from the bounding faces of the atmosphere and for intensity of radiation at any depth of the atmosphere. The emergent intensity of radiation from the bounding faces are determined from simultaneous linear integral equations of the emergent intensity of radiation in terms ofX andY equations of Chandrasekhar. The intensity of radiation at any optical depth for a positive and negative direction parameter is derived by inversion of the Laplace transform in terms of intergrals of the emergent intensity of radiation. A new expression of theX andY equation is also derived for easy numerical computation. This is a new and exact method applicable to all problems in finite plane parallel steady scattering atmosphere.     

A method of computing the emergent radiation by the atmosphere in the region ranging from ultraviolet to infrared

A method of computing the diffuse reflection and transmission radiation by an inhomogeneous, plane-parallel planetary atmosphere with internal emission source is discussed by use of the adding method. If the atmosphere is simulated by a number of homogeneous sub-layers, the radiation diffusely reflected or transmitted by the atmosphere can be expressed in terms of the reflection and transmission matrices of the radiation of sub-layers. The diffusely transmitted radiation due to the internal emission source can be also easily computed in the same manner. These equations for the emergent radiation are in a quite general form and are applicable to radiative transfer in the atmosphere in the region from ultraviolet to infrared radiation. With this method, the tiresome treatment due to the polarity effect of radiation is overcome.     

Nonlinear diffuse reflection and transmission of radiative energy by a layer of finite thickness

The major results for the linear problem of diffuse reflection and transmission of radiation by a layer of finite thickness are carried over to the nonlinear case by successive application of Ambartsumyan’s approach for a one dimensional anisotropic medium. Formulas are given for nonlinear addition of layers which can be used to construct recurrence calculation procedures for uniform, periodic, and arbitrary stratified media. A complete set of differential equations for invariant imbedding is derived with the aid of these formulas. These equations are used to obtain a system of total invariance equations, which, in turn, offer the possibility of reducing the nonlinear problem of diffuse reflection and transmission during irradiation of a layer from both sides to the simpler problem of illuminating this medium from only one side, with the thickness of the layer remaining only as a fixed parameter. Finally, it is shown that the results obtained for the single frequency case (two-level atom) remain valid in the polychromatic case (multilevel atom), which is important for interpreting astrophysical data.     

A new technique for exact and unique solution of transfer equations in finite media

In this paper we develop a new method, combined with Laplace transformation and Wiener-Hopf technique, to obtain unique solutions of transport equations in finite media. For this purpose we consider the simple transfer equation for diffuse reflection by a plane-parallel finite atmosphere scattering radiation with moderate anisotropy. It is transformed, by Laplace transformation, into two coupled linear integral equations which are then reduced to two uncoupled Fredholm integral equations admitting of unique solutions by the method of iteration for values of the breadth of the atmosphere greater than that specified, depending on the scattering process.     

Polarization effect on radiative transfer in planetary composite atmospheres with interacting interface

A procedure of computing the radiance and the polarization parameters of radiation diffusely reflected and transmitted by an inhomogeneous, plane-parallel terrestrial atmosphere bounded by a ruffled ocean surface is discussed with the aid of the adding method. If the atmosphere and the ocean are simulated by a number of homogeneous sublayers, the matrices of radiation reflected and transmitted diffusely by the atmosphere-ocean system can be expressed in terms of these matrices of sublayers by using only a couple of iterative equations in which the polarity effect of radiation is included. Furthermore, the upwelling radiance and the polarization degree of radiation at the top of the atmosphere can be calculated by using a single iterative equation without requiring the equation for the diffuse transmission matrix of radiation. The ruffled ocean surface can be treated as an interacting interface, where the transmitted radiation from beneath the ocean surface into the atmosphere is also taken into account into the derivation of equations. Finally, sample computations of the upwelling radiance and the polarization degree of radiation from the top of the atmosphere are carried out at the wavelength of 0.60 micron.     

Linear singular integral equations for polarized radiation in isotropic media

Linear singular integral equations are derived for polarized radiation fields in semi infinite and finite plane parallel atmospheres. An arbitrary phase matrix and any distribution of primary sources are assumed. The integral equations together with appropriate sets of linear constraints arise from functional relations derived by means of CASE 's eigenfunctions and their full range completeness and orthogonality. The emergent radiation is described by half range singular integral equations, whereas the STOKES vector of the inner radiation field obeys full range integral equations depending on the emergent radiation.     

The effects of mass on the radiation of a relativistically rotating neutron star

We investigate the effect of mass on the radiation of a relativistically rotating neutron star. The method of Haxton and Ruffini is used to find the radiation flux from a relativistically rotating neutron star. By extending the idea of a point charge orbiting a black hole, a pulsar is modeled by simulating a relativistically rotating magnetic dipole embedded within a neutron star. The resulting equations retain the mass of the neutron star, thereby introducing effects of general relativity on the radiation from the dipole. We present exact solutions to the modeling equation as well as plots of energy spectra at different rotational velocities and inclination angles. We also present plots of total energy versus mass and two tables containing a comparison of energy ratios. These demonstrate that, for realistic neutron star masses, the high speed enhancement of the radiation is always more than compensated by the frame dragging effect, leading to a net reduction of radiation from the star. It is found that the inclusion of mass not only reduced the special relativistic enhancement, but negates it entirely as the mass of the neutron star approaches the mass limit.     

Solution of linear radiative transport problems in plane-parallel atmospheres. I.

A new method for determining various quantities describing the radiation field in an inhomogeneous, plane-parallel atmosphere is proposed in this two-part paper. The essence of this method is the reduction of the boundary value problems which arise during the customary statement of various astrophysical problems associated with solving the radiative transfer equations to initial value problems. Compared to previous attempts in this area, the proposed method is universal and simple. The first part of this paper deals with one-dimensional media. Scalar, as well as vector–matrix problems relating to the diffusion of radiation in spectral lines with frequency redistribution are examined.     

Orbital Decay of a Binary System Consisting of Spin-Down Neutron Stars

{W}e consider the gravitational radiation from two time variable mass stars, orbiting around each other under the influence of gravity. The total rates of the variation of the energy, angular momentum, semimajor axis, eccentricity and orbital period are obtained. The results could be important for the understanding of general relativistic effects in the case of the variation of the gravitational mass due to spinning down of the compact stars, which sensitively depends on the equations of state. The cases of the binary systems PSR 1913+16 and PSR 1534+12 are analyzed in detail, and, for different equations of state of nuclear matter, the corrections to the orbital decay due to gravitational radiation and to the spinning down of the pulsars are calculated. The results show that a future significant improvement in the observational techniques could lead to the observation of the specific general relativistic effect of mass variation of pulsars due to spinning down, via the study of orbital decay, even in slowly rotating binary systems.     

Secular orbit variation due to solar radiation effects: a detailed model for BYORP

A detailed derivation of the effect of solar radiation pressure on the orbit of a body about a primary orbiting the Sun is given. The result is a set of secular equations that can be used for long-term predictions of changes in the orbit. Solar radiation pressure is modeled as a Fourier series in the body’s rotation state, where the coefficients are based on the shape and radiation properties of the body as parameters. In this work, the assumption is made that the body is in a synchronous orbit about the primary and rotates at a constant rate. This model is used to write explicit variational equations of the energy, eccentricity vector, and angular momentum vector for an orbiting body. Given that the effect of the solar radiation pressure and the orbit are periodic functions, they are readily averaged over an orbit. Furthermore, the equations can be averaged again over the orbit of the primary about the Sun to give secular equations for long-term prediction. This methodology is applied to both circular and elliptical orbits, and the full equations for secular changes to the orbit in both cases are presented. These results can be applied to natural systems, such as the binary asteroid system 1999 KW4, to predict their evolution due to the Binary YORP effect, or to artificial Earth orbiting, nadir-pointing satellites to enable more precise models for their orbital evolution.     

Average photon path-length of radiation emerging from finite atmospheres

The determination of the average path-length of photons emerging from a finite planeparallel atmosphere with molecular scattering is discussed. We examine the effects of polarisation on the average path-length of the emergent radiation by comparing the results with those obtained for the atmosphere where the scattering obeys the scalar Rayleigh function. Only the axial radiation field is considered for both cases.To solve this problem we have used the integro-differential equations of Chandrasekhar for the diffuse scattering and transmission functions (or matrices). By differentiation of these equations with respect to the albedo of single scattering we obtain new equations the solution of which gives us the derivatives of the intensities of the emergent radiation at the boundaries.As in the case of scalar transfer the principles of invariance by Chandrasekhar may be used to find an adding scheme to obtain both the scattering and transmission matrices and their derivatives with respect to the albedo of single scattering. These derivatives are crucial in determining the average path length.The numerical experiments have shown that the impact of the polarisation on the average pathlength of the emergent radiation is the largest in the atmospheres with optical thickness less than, or equal to, three, reaching 6.9% in the reflected radiation.     

Radiant heat transfer in isotropically-scattering media bounded by diffusely-reflecting plates

The partial heat fluxes of thermal radiation in an emitting, absorbing, and scattering finite medium bounded by diffusely-reflecting plates is studied, assuming the medium to scatter isotropically. Two algebraic equations connecting the partial heat fluxes relevant to this problem and the corresponding one with simple boundary conditions are obtained. The intergral equation is solved in a semi-analytical from by collection method, using Legendre polynomials as basis functions. Numerical results are presented and compared.     

Photon path-length distribution function (II)

The determination of the photon path-length distribution function (PLDF) in a semi-infinite plane-parallel homogeneous atmosphere is discussed while the atmosphere scatters radiation according to the 2 × 2 Rayleigh-Cabannes phase matrix. The Piessens-Huysmans method of numerically inverting the Laplace transform which proved to be successful for the non-polarized radiation works in this special case as well. To employ this method we had to define the complex H-matrix and to find a fast method to determine its numerical values. For determining the average path-lengths and the dispersion we set up a system of integral equations the solution of which gave us the H-matrix and its first two derivatives with respect to the albedo of single scattering.The influence of different parameters characterizing the interaction of the polarized radiation with the atmosphere on the PLDF and the average path-length is studied in detail and a sample of average path-lengths is given in Table I.     

Equations of general relativistic radiation hydrodynamics from a tensor formalism

Radiation interacts with matter via exchange of energy and momentum. When matter is moving with a relativistic velocity or when the background space–time is strongly curved, rigorous relativistic treatment of hydrodynamics and radiative transfer is required. Here, we derive fully general relativistic radiation hydrodynamic equations from a covariant tensor formalism. The equations can be applied to any three-dimensional problems and are rather straightforward to understand compared to the comoving frame-based equations. The current approach is applicable to any space–time or coordinates, but in this work we specifically choose the Schwarzschild space–time to show explicitly how the hydrodynamic and the radiation moment equations are derived. Some important aspects of relativistic radiation hydrodynamics and the difficulty with the radiation moment formalism are discussed as well.     

Angular distribution of radiation in an isotropically scattering slab

The equation of radiative transfer in an isotropically scattering slab subject to general boundary conditions is solved. The Padé approximation technique is used to calculate the reflected and transmitted angular distributions. Numerical results for angular distributions through and at the boundaries of a finite slab are calculated by the Padé approximation technique. The results for a Padé approximation of [0/1] are compared with results obtained by the Galerkin method.