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

Rabindra Nath Das 《Astrophysics and Space Science》1981,76(2):441-463

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.  相似文献   

Exact solution of the problem of diffuse reflection and transmission by the method of Laplace transform and linear singular operators     

Rabindra Nath Das 《Astrophysics and Space Science》1978,58(2):419-425

The basic integro-differential equation is subjected to a one-sided finite Laplace transform to obtain linear integral equations of angular distribution of bounding faces. These linear integral equations have been transformed into linear singular integral equations which have been solved exactly to get the emergent distributions from the bounding faces by the theory of linear singular operators. Some solutions of linear singular integral equations have also been derived for future use in radiative transfer problems.  相似文献   

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     

Rabindra Nath Das 《Astrophysics and Space Science》2010,326(1):91-103

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.  相似文献   

Exact solution of the problem of diffuse reflection and transmission in Rayleigh scattering by use of the Laplace transform and the theory of linear singular operators     

Rabindra Nath Das 《Astrophysics and Space Science》1979,60(1):221-232

We have considered six scalar equations which are obtained from the vector transport equation for radiative transfer to the problem of diffuse reflection and transmission in finite plane-parallel Rayleigh scattering atmosphere. By use of the Laplace transform and the theory of linear singular operators these equations have been solved exactly to get the angular distribution of the intensity diffusely reflected from the surface and diffusely transmitted below the surface.  相似文献   

Solution of non-coherent spectral line transfer problem for finite atmosphere by theF n -method     

R. C. Raymandal  G. Biswas  S. Karanjai 《Astrophysics and Space Science》1988,146(2):311-319

By use of the orthogonality and normalization integrals developed by McCormick and Siewert (1970) a set of singular integral equations suitable forF _n -method is derived for non-coherent spectral line formation problem in finite media.F _n -equations for exit distributions are used to develop some algebraic equations with suitable recursion relations.  相似文献   

Exact solution of a key equation in a finite stellar atmosphere by the method of Laplace transform and linear singular operators     

Rabindra Nath Das 《Astrophysics and Space Science》1980,67(2):335-342

The key equation which commonly appears for radiative transfer in a finite stellar atmosphere having ground reflection according to Lambert's law is considered in this paper. The exact solution of this equation is obtained for surface quantities in terms of theX-Y equations of Chandrasekhar by the method of Laplace transform and linear singular operators. This exact method is widely applicable for obtaining the solution for surface quantities in a finite atmosphere.  相似文献   

Exact solution of a simple time-dependent integro-differential equation by the method of Laplace transform and the theory of linear singular operators     

Rabindra Nath Das 《Astrophysics and Space Science》1980,71(1):37-43

The simplest form of the equation of transfer for a time dependent radiation field in finite atmosphere is considered. This equation of transfer is an integro-differential equation, the solution of this equation is based on the theory of separation of variables, the Laplace transform and the theory of linear singular operators. The emergent intensities from the bounding faces of the finite atmosphere are determined in terms ofX-Y equations of Chandrasekhar.  相似文献   

Exact solution of a key equation in a finite planetary atmosphere by the method of laplace transform and linear singular operators     

Rabindra Nath Das 《Astrophysics and Space Science》1980,67(2):343-347

Considering the ground reflection according to Lambert's law, we establish a fundamental equation in finite planetary atmospheres. An exact form of the solution of this equation is obtained for the emergent quantities from the bounding faces in terms ofX-Y equations by the method of Laplace transform, in combination with the theory of linear singular operators.  相似文献   

An exact solution of transfer equations for interlocked multiplets     

S. R. Das Gupta 《Astrophysics and Space Science》1978,56(1):13-20

The transfer equations for non-coherent scattering arising from interlocking of principal lines without redistribution is exactly solved in a very simple way by Laplace tranform and Wiener-Hopf technique which are easily applied by the use of the new representation ofH-functions obtained recently by the author (1977). The emergent intensity in therth line is expressed in terms of anH-function and a Cauchy type integral admitting of closed form evaluation.  相似文献   

Average photon path-length in isotropically scattering finite atmospheres     

Tõnu Viik 《Astrophysics and Space Science》1995,225(2):205-219

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.  相似文献   

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

S. R. Das Gupta 《Astrophysics and Space Science》1978,56(1):21-49

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.  相似文献   

A new numerical method for solving radiation driven winds from hot stars     

M. Cur  D. F. Rial 《Astronomische Nachrichten》2007,328(6):513-520

We present a general method for solving the non‐linear differential equation of monotonically increasing steady‐state radiation driven winds. We graphically identify all the singular points before transforming the momentum equation to a system of differential equations with all the gradients explicitly given. This permits a topological classification of all singular points and to calculate the maximum and minimum mass‐loss of the wind. We use our method to analyse for the first time the topology of the non‐rotating frozen‐in ionisation m‐CAK wind, with the inclusion of the finite disk correction factor, and find up to 4 singular points, three of the x‐type and one attractor‐type. The only singular point (and solution passing through) that satisfies the boundary condition at the stellar surface is the standard m‐CAK singular point. (© 2007 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)  相似文献   

Inversions for axisymmetric galactic disks     

N. Hiotelis  P. A. Patsis 《Astrophysics and Space Science》1993,206(1):21-35

We use two models for the distribution function to solve an inverse problem for axisymmetric disks. These systems may be considered - under certain assumptions - as galactic disks. In some cases the solutions of the resulting integral equations are simple, which allows the determination of the kinematic properties of self-consistent models for these systems. These properties for then = 1 Toomre disk are presented in this study.  相似文献   

Unrestricted second-order tensor virial equations for linear oscillations of magnetic configurations     

M. Goossens 《Astrophysics and Space Science》1979,60(2):401-421

This paper deals with the second-order tensor virial equations for the linear oscillations of a gaseous mass in the presence of a magnetic field. It is shown that the commonly used linearized versions of the tensor virial equations are restricted integral equations that incorporate the linearized equation of motion but not the boundary condition. These restricted equations only allow trial functions that fulfil the boundary condition and are of limited practical value.The unrestricted variational principle for the linear oscillations of a magnetic configuration is used to derive a more general formulation of the second-order tensor virial equations so that the linear trial function _i =X _ij x _j can be used to study the oscillations of a configuration with a magnetic field that extends in the exterior vacuum. The unrestricted virial equations have been applied to Ferraro's model and approximate results for the eigenfrequencies and eigenfunctions have been obtained for nine oscillation modes.  相似文献   

The polarization of radiation reflected diffusely by an inhomogeneous plane medium     

M. T. Attia  E. M. Abulwafa 《Astrophysics and Space Science》1991,184(2):247-259

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.  相似文献   

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

Rabindra Nath Das 《Astrophysics and Space Science》1979,61(1):169-173

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.  相似文献   

An improved version of the implicit integral method to solving radiative transfer problems     

E. Simonneau  O. Cardona  L. Crivellari 《Astrophysics》2012,55(1):110-126

Radiative transfer (RT) problems in which the source function includes a scattering-like integral are typical two-points boundary problems. Their solution via differential equations implies making hypotheses on the solution itself, namely the specific intensity I (τ; n) of the radiation field. On the contrary, integral methods require making hypotheses on the source function S(τ). It seems of course more reasonable to make hypotheses on the latter because one can expect that the run of S(τ) with depth is smoother than that of I (τ; n). In previous works we assumed a piecewise parabolic approximation for the source function, which warrants the continuity of S(τ) and its first derivative at each depth point. Here we impose the continuity of the second derivative S′′(τ). In other words, we adopt a cubic spline representation to the source function, which highly stabilizes the numerical processes.  相似文献   

Solution of a Riemann–Hilbert problem for a new expression of Chandrasekhar’s H-function in radiative transfer     

Rabindra Nath Das 《Astrophysics and Space Science》2008,317(1-2):119-126

The linear singular integral equation derived from the nonlinear integral equation of Chandrasekhar’s H-function in radiative transfer is considered here to develop a new form of H-function as a solution of a Riemann–Hilbert problem using Plemelj and Cauchy integral formulae for complex domain. This new form of H-function is a simple integral of known functions. Forms of H-function both for conservative and nonconservative cases are obtained. Their numerical evaluations are made by Simpson’s one-third rule to arrive at an accuracy to ninth places of decimals.  相似文献   

Radiation transfer in a diffuse and specular reflecting slab with Rayleigh scattering     

M. S. Abdel Krim  E. M. Abulwafa  S. M. Shouman 《Astrophysics and Space Science》1992,189(2):279-287

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.  相似文献   

The critical layers and other singular regions in ideal hydrodynamics and magnetohydrodynamics     

J. A. Adam 《Astrophysics and Space Science》1984,105(2):401-412

The governing singular differential equations are stated for certain systems in ideal hydrodynamics and magnetohydrodynamics. The solutions of these equations are studied in the neighborhood of the singular regions, and can be conveniently characterised by the roots of the corresponding indicial equation. The nature of the solution determines the nature of the dispersion relation (relating frequency and wavenumberk) and of the Green's function for the problem. The differences between the various classes of problems are discussed, and exemplified by considering the initial value problem for three cases: (a) unstratified shear flow, (b) stratified shear flow, and (c) a static magnetofluid. The latter case is typical of a number of problems of astrophysical interest, and possesses a rich mathematical and physical structure.  相似文献