Exact solution of a problem in radiative transfer by the laplace transform and the Wiener-Hopf technique when the incoming intensity at the surface is known

We consider the problem of determining the emergent intensity from the bounding face of a semi-infinite atmosphere having conservative scattering and the intensity at any optical depth by use of the Laplace transform in combination with the Wiener-Hopf technique when the incoming intensity at the bounding face of the atmosphere is known. The solution is exact.     

Exact solution of a problem in a stellar atmosphere using the Laplace transform and the Wiener-Hopf technique

The general equation for radiative transfer of line scattering intensity — including the effects of scattering, absorption and thermal emission — in the Milne-Eddington model is considered here. The scattering function is assumed to be quadratically anisotropic in the cosine of the scattering angle, and Planck's intensity function is assumed for thermal emission. The exact solutions for emergent intensity from the bounding face and the intensity at any optical depth are obtained by the method of the Laplace transform in combination with the Wiener-Hopf technique.     

Exact solution of the transport equation for imperfect rayleigh scattering in a semi-infinite atmosphere

By performing the one-sided Laplace transform on the matrix integro-differential equation for a semi-infinite plane parallel imperfect Rayleigh scattering atmosphere we derive an integral equation for the emergent intensity matrix. Application of the Wiener-Hopf technique to this integral equation will give the emergent intensity matrix in terms of singularH-matrix and an unknown matrix. The unknown matrix has been determined considering the boundary condition at infinity to be identical with the asymptotic solution for the intensity matrix.     

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

An approximate solution of the transfer equation for coherent scattering in stellar atmospheres with Planck's function as a nonlinear function of optical depth, viz.,

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.     

Exact solution of a radiative transfer problem by the Laplace Transform and Wiener-Hopf Technique

In this paper the authors have extended Bayin's (1978) work to the case of charged fluid spheres. These solutions are matched at the boundary with the Reissner-Nordström solution.     

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

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.     

An exact solution of the equation of transfer for coherent scattering in an exponential atmosphere

An exact solution of the transfer equation for coherent scattering in stellar atmospheres with Planck's function as a nonlinear function of optical depth, of the form $$B_v (T) = b_0 + b_1 {\text{ }}e^{ - \beta \tau } $$ is obtained by the method of the Laplace transform and Wiener-Hopf technique.     

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.     

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

A finite atmosphere having distribution of intensity at both surfaces with definite form of scattering function and source function is considered here. The basic integro-differential equation for the intensity distribution at any optical depth is subjected to the finite Laplace transform to have linear integral equations for the surface quantities under interest. These linear integral equations are transformed into linear singular integral equations by use of the Plemelj's formulae. The solution of these linear singular integral equations are obtained in terms of theX-Y equations of Chandrasekhar by use of the theory of linear singular operators which is applied in Das (1978a).     

An exact solution of the equation of transfer with three-term scattering indicatrix in an exponential atmosphere

The general equation for radiative transfer in the Milne-Eddington model is considered here. The scattering function is assumed to be quadratically anisotropic in the cosine of the scattering angle and Planck's intensity function is assumed for thermal emission. Here we have taken Planck's function as a nonlinear function of optical depth, viz.,B _v(T)=b _o+b ₁ e ^–. The exact solution for emergent intensity from the bounding face is obtained by the method of the Laplace transform in combination with the Wiener-Hopf technique.     

Solution of the radiative transfer equation in combination with rayleigh and isotropic scattering

A theory is constructed for solving half-space, boundary-value problems for the Chandrasekhar equations, describing the propagation of polarized light, for a combination of Rayleigh and isotropic scattering, with an arbitrary probability of photon survival in an elementary act of scattering. A theorem on resolving a solution into eigenvectors of the discrete and continuous spectra is proven. The proof comes down to solving a vector, Riemann—Hilbert, boundary-value problem with a matrix coefficient, the diagonalizing matrix of which has eight branching points in the complex plane. Isolation of the analytical branch of the diagonalizing matrix enables one to reduce the Riemann—Hilbert problem to two scalar problems based on a [0, 1] cut and two vector problems based on an auxiliary cut. The solution of the Riemann—Hilbert problem is given in the class of meromorphic vectors. The conditions of solvability enable one to uniquely determine the unknown expansion coefficients and free parameters of the solution of the boundary-value problem. Translated from Astrofizika, Vol. 41, No. 2, pp. 263–276, April-June, 1998.     

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

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.     

Solution of radiative transfer equation with spherical-symmetry in partially scattering medium

We have solved the equation of radiative transfer in spherical symmetry with scattering and absorbing medium. We have set the albedo for single scattering to be equal to 0.5. We have set the Planck function constant throughout the medium in one case and in another case the Planck function has been set to vary asr ^–2. The geometrical extension of the spherical shell has been taken as large as one stellar radius. Two kinds of variations of the optical depth are employed (1) that remains constant with radius and (2) that varies asr ^–2. In all these cases the internal source vectors and specific intensities change depending upon the type of physics we have employed in each case.     

Exact solution of the problem of diffuse reflection for a combination of Rayleigh and isotropic scattering

We consider the basic vector equation of transfer for radiation in a semi-infinite atmosphere for diffuse reflection which scatters radiation in accordance with the phase matrix obtained from a combination of Rayleight and isotropic scattering. This equation will give an integral equation for emergent intensity while subjected to the Laplace transform. The integral equation will give rise to the emergent intensity matrix on application of the Wiener-Hopf technique. This is an exact method.     

Solution of the transfer equation in a scattering atmosphere with spherical symmetry

An exact formal solution of then-approximation radiative transfer equations for the Compton scattering in a spherically symmetric atmosphere is obtained. In view of further applications, the simple case of a density ?(r)=?₀/r is fully developed and the 20 approximation equations have been studied with the computer.     

Exact solution of the equation of transfer with planetary phase function

We have considered the transport equation for radiative transfer to a problem in semi-infinite atmosphere with no incident radiation and scattering according to planetary phase function w(1 + xcos ). Using Laplace transform and the Wiener-Hopf technique, we have determined the emergent intensity and the intensity at any optical depth. The emergent intensity is in agreement with that of Chandrasekhar (1960).     

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

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.     

Solution of the equation of transfer for coherent scattering in an exponential atmosphere by Eddington's method

An approximate solution of the transfer equation for coherent scattering in stellar atmospheres with Planck's function as a nonlinear function of optical depth, viz., $$B_v \left( T \right) = b_0 + b_1 e^{ - \beta \tau } $$ is obtained by Eddington's method. is obtained by Eddington's method.     

Solution of the equation of transfer for coherent scattering in an exponential atmosphere by Busbridge's method

A solution of the transfer equation for coherent scattering in stellar atmosphere with Planck's function as a nonlinear function of optical depth, viz. $$B{\text{ }}_v (T) = b_0 + b_1 {\text{ }}e^{ - \beta \tau } $$ is obtained by the method developed by Busbridge (1953).