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.     

Time-dependentH-,X- andY-functions for an anisotropically-scattering atmosphere

We have considered a homogeneous atmosphere scattering anisotropically with Dirac -function type time-dependent incidence. We used the method of integral operator developed by Ambartsumian and the theory ofN-solutions developed by Busbridge to find the correspondingH-function (in semi-infinite atmosphere) andX- andY-functions (in finite atmosphere).     

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.     

Time-dependentX- andY-functions for anisotropically scattering inhomogeneous atmosphere by principle of invariance

The nonlinear integral equations forX- andY-functions have been developed for an inhomogeneous atmosphere scattering anisotropically using the principle of invariance. The anisotropy is represented by means of a phase function expressed in terms of finite-order Legendre polynomials.     

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.     

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.     

Solution of Riemann–Hilbert problems for determination of new decoupled expressions of Chandrasekhar’s <Emphasis Type="Italic">X</Emphasis>- and <Emphasis Type="Italic">Y</Emphasis>-functions for slab geometry in radiative transfer

In radiative transfer, the intensities of radiation from the bounding faces of a scattering atmosphere of finite optical thickness can be expressed in terms of Chandrasekhar’s X- and Y-functions. The nonlinear nonhomogeneous coupled integral equations which the X- and Y-functions satisfy in the real plane are meromorphically extended to the complex plane to frame linear nonhomogeneous coupled singular integral equations. These singular integral equations are then transformed into nonhomogeneous Riemann–Hilbert problems using Plemelj’s formulae. Solutions of those Riemann–Hilbert problems are obtained using the theory of linear singular integral equations. New forms of linear nonhomogeneous decoupled expressions are derived for X- and Y-functions in the complex plane and real plane. Solutions of these two expressions are obtained in terms of one known N-function and two new unknown functions N ₁- and N ₂- in the complex plane for both nonconservative and conservative cases. The N ₁- and N ₂-functions are expressed in terms of the known N-function using the theory of contour integration. The unknown constants are derived from the solutions of Fredholm integral equations of the second kind uniquely using the new linear decoupled constraints. The expressions for the H-function for a semi-infinite atmosphere are obtained as a limiting case.     

External radiation field in Rayleigh-Cabannes atmospheres with constant and linear sources

The external field of radiation in Rayleigh-Cabannes atmospheres with constant and linear sources is found using the resolvent matrix approach. If the internal sources are constant the external field may be described by theX-, Y-, andH-matrices. For the case with linear sources we need the derivatives of these matrices with respect to angular variable. The respective scheme for their determination is given.A set of integro-differential equations for theX- andY-matrices is derived and solved numerically. Some relations between the moments of theH-matrix are given and a sample of results for external fields are provided.     

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.     

The solution of resonance radiative transfer equation for diffusion by the method of laplace transform and linear singular operators

The equation for radiative transfer in the case of resonance radiation for isotropic scattering has been solved by the method of the Laplace transformation and linear singular operators. The solution for emergent intensities have come out in terms ofX- andY-functions.     

Derivation of frequency-dependentX- andY-functions for a non-coherent scattering problem

The equation of transfer for the case of non-coherent scattering (Hummer, 1968; Ivanov, 1973; McCormick and Siewert, 1970) has been considered. The correspondingX- andY-functions have been derived by a combination of eigenfunction method developed by Case, and from the principle of invariance as developed by Chandrasekhar (1960).     

Derivatives of theH-function

The first two derivatives and the mixed derivative of theH-function with respect to the angular variable and the albedo of single scattering are studied in the case of isotropic scattering using the kernel approximation method. This allows to obtain simple formulae the accuracy of which is estimated to be very good in rather low orders of approximation. Some samples of numerical results are given with eight decimal accuracy.     

The X- Y-, and H-matrices for molecular scattering

A method is described to calculate theX-,Y-, andH-matrices for molecular scattering in a homogeneous plane-parallel atmosphere. This method is based on the method of discrete ordinates by Chandrasekhar. A sample of results in 7 decimals is given and the accuracy of the method presented is discussed.     

5D space-time-mass gravity theory with off-diagonal components in the metric tensor

The bi-variational technique is used to calculated Chandrasekhar'sX- andY-functions and their first two moments for an isotropic homogeneous finite slab. Numerical results obtained are compared with published results.     

An exact linearization and decoupling of the integral equations satisfied by time-dependent X- and Y-functions

We discuss a simple method of linearization and decoupling of the integral equations satisfied by time-dependentX - andY -functions which play an important rôle in the study of non-stationary radiative transfer problems.     

Exact solution of the transport equation for radiative transfer with scattering albedo ωO < 1 using the Laplace transform and the Wiener-Hopf technique and an expression ofH-function

We have considered the transport equation for radiative transfer to a problem in semi-infinite non-conservative atmosphere with no incident radiation and scattering albedo ₀ < 1. Usint the Laplace transform and the Wiener-Hopf technique, we have determined the emergent intensity and the intensity at any optical depth. We have obtained theH-function of Dasgupta (1977) by equating the emergent intensity with the intensity at zero optical depth.     

The complexH-function

The properties of theH -function for the complex albedo of single scattering are studied. It is shown that basically all the formulae derived for the real albedo of single scattering can be transferred into the complex-plane without alteration.An iterational procedure to find the numerical values of the complexH - function is set up. Some examples of the numerical results are given in the figures.     

Application of the Wiener-Hopf technique for a new representation of X- and Y-equations

The application of the Wiener-Hopf technique to the coupled linear integral equation ofX- andY-equations gives rise to the Fredholm equations with simpler kernels.X-equation is expressed in terms ofY-equation and vice-versa. These are unique in representation with respect to coupled linear constraints.     

Calculation of effective optical depth of absorption line formation in homogeneous semi-infinite planetary atmosphere during anisotropic scattering

The algorithm for determining effective optical thickness of absorption line formation in a plane-parallel homogeneous planetary atmosphere is presented. The case of anisotropic scattering is considered. The results of numerical calculations of τ _e (μ₀) at the scattering angle γ = π for some values of the single scattering albedo λ and the parameter of the Heyney-Greenstein scattering indicatrix g are given. The refined equation for the function T ^m (−μ, μ₀) is presented.