Solution of the equation of transfer for interlocked multiplets by the method of discrete ordinates with the planck function as a nonlinear function of optical depth

The equation of transfer for interlocked multiplets has been solved by the method of discrete ordinates, originally due to Chandrasekhar, considering nonlinear form of the Planck function to be

Solution of the equation of transfer for interlocked multiplets by the method of Laplace transform and Winer-Hopf technique with planck function as a nonlinear function of optical depth

The equation of transfer for interlocked multiplets has been solved by Laplace transformation and the Wiener-Hopf technique developed by Dasgupta (1978) considering two nonlinear forms of Planck function: i.e., (a) $$B{\text{ }}_{\text{v}} (T) = B(t) = b_0 + b_1 {\text{ }}e^{ - \alpha t} ,$$ (b) $$B{\text{ }}_{\text{v}} (T) = B(t) = b_0 + b_1 t + b_2 E_2 (t).$$ Solutions obtained by Dasgupta (1978) or by Chandrasekhar (1960) may be obtained from our solutions by dropping the nonlinear terms.     

Solution of the transfer equation for interlocked multiplets by probabilistic method

Sobolev's probabilistic method — The method of quantum exit from the medium — has been applied to solve the transfer equation for the case of interlocking without redistribution. The solution contains the function (x) which is same as theH-function involved in the solution given by Busbridge and Stibbs the method of principle of invariance.     

Solution of the equation of transfer for interlocked multiplets by the method of discrete ordinates

A method of discrete ordinates, originally due to Chandrasekhar, has been applied to solve the equation of transfer for the case of interlocked multiplet lines without redistribution. The solution thus deduced has been applied to find laws of darkening for the multiplets.     

An exact solution of transfer equations for interlocked multiplets

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.     

Solution of the equation of transfer with Rayleigh's phase function in a thin atmosphere

The equation of radiative transfer with scattering according to Rayleigh's phase function has been solved in a thin atmosphere by use of a modification of the spherical-harmonic method suggested by Wanet al. (1986).     

Solution of the equation of transfer with general phase function by a modified spherical-harmonic method

The equation of transfer with general phase function has been solved by a modified form of spherical-harmonic method. The solutions in case of certain particular phase functions are then derived from the general one.     

Solution of the equation of transfer for Rayleigh's phase function by the principle of invariance

In this paper we shall construct the solution of the equation of transfer in a semi-infinite atmosphere with no incident radiation for Rayleigh's phase function by the method of the Principles of Invariance and using the law of diffuse reflection. The solution will then be applied to find the laws of darkening for Rayleigh's phase function.     

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.     

Solution of a generalized equation of transfer in a two-region slab

The general equation of transfer in a two-region slab of unequal thickness with general boundary conditions has been solved by an analytical method developed by Menninget al. (1980). The scattering is regarded as isotropic and the source function is taken as a general one to accomodate different types of problems.     

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).     

Solution of a time-dependent equation of transfer by theF n -method

The time-dependent equation of transfer for a finite, plane-parallel, non-radiating, and isotropically-scattering atmosphere of arbitrary stratification is solved by using theF _n -method developed by Siewert.     

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.     

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.     

Mean optical depth and contribution function of spectral lines in a turbulent atmosphere

In this paper we present a new definition and its analytic expressions for the mean optical depth and the mean contribution function of spectral lines in a turbulent atmosphere. These mean values are based on the radiative transfer equation and thus satisfy the general properties of the radiation field. They can be used to study the line formation process in turbulent atmospheres.     

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).     

Solution of a transfer equation by a Modified Spherical Harmonic Method in a thin anisotropically scattering atmosphere

Wan, Wilson and Sen (1986) have examined the scope of Modified Spherical Harmonic Method in a plane medium scattering anisotropically. They have used the phase functionp(µ, µ) = 1 +aµµ. In this paper, the Transfer Equation has been solved by the Modified Spherical Harmonic Method using the phase functionp(µ, µ) = 1 + ₁ P ₁(µ)P ₁(µ) + ₂)P ₂(µ)P ₂(µ) and a few sets of numerical solution have been predicted for three different cases.