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 the method of discrete ordinates

A solution of the transfer equation for coherent scattering in stellar atmosphere with Planck's function as a nonlinear function of optical depth, viz., $$B_v (T) = b_0 + b_1 {\text{ }}e^{ - \beta \tau } $$ is obtained by the method of discrete ordinates originally due to Chandrasekhar.     

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.     

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 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 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.     

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.     

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.,

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 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 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.     

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 equation for radiative transfer in semi-infinite Rayleigh scattering atmosphere using the laplace transform and the Wiener-Hopf technique

We have considered six scalar transport equations which are obtained from the vector transport equation to determine four Stokes's parameters to the problem of diffuse reflection in the semi-infinite plane parallel Rayleigh scattering atmosphere. By use of the Laplace transform and the Wiener-Hopf technique, these equations have been solved exactly to obtain the emergent intensity and the intensity at any optical depth and to reconstruct the Stokes's parameters. Solutions for emergent distribution so obtained are identical with the results of Chandrasekhar (1950).     

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.     

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.     

Light scattering calculations by the Fredholm integral equation method

The Fredholm integral equation method (FIM), originally introduced by Holtet al. to solve the light scattering problem for ellipsoidal particles, is reinvestigated by taking into account a recent great progress in numerical computers. A numerical code optimized for vector-processing computers is developed, and is applied to the light scattering by spherical and spheroidal particles. The results for these particles are compared with those by the Mie theory and by Asano and Yamamoto, respectively, and it is confirmed that the agreement with both of them is satisfactory. Sample calculations are also performed for the oblique incidence, in which the direction of incidence is not parallel nor perpendicular to the symmetry axis of the particle. No difficulties in the computation are found compared with the calculations for the parallel or perpendicular incidence. We study the efficiency factor for polarization (Q _pol) in general direction of incidence for spheroidal particles, and discuss the deviation from the Rayleigh approximation.     

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

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.     

Light scattering in a spherical,exponential atmosphere,with applications to Venus

The problem of the reflection of light from an optically thick, spherical atmosphere in which the scatterers are distributed exponentially with a scale height small compared to the radius of the planet is discussed. Exact formal solutions are obtained for the single scattered component. Useful approximate analytic solutions, which also include multiply scattered light, are given. The results are applied to the analysis of the Mariner 10 limb and terminator images of Venus. The altitude of the “detached” haze layer discovered by Mariner 10 is at 79–85 km, but in places the haze exists above 100 km. This layer apparently is a stable, planetwide feature which forms at the top of the Pioneer Venus upper haze layer. It was similar in location, scale height, and thickness at the times of the two missions, in contrast to the lower, high-altitude haze which changed dramatically. We discuss two possibilities for the nature of the limb hazes. (1) The lower haze is probably the sulfuric acid cloud and the “detached” layer may be a separate water-ice haze. (2)The “detached” haze layer may not be separate at all, but part of the sulfuric acid haze, and the apparent “gap” at 75–80 km may be the source region of a broadband absorber. The spatial distribution of the strong near-UV absorber, which may be elemental sulfur as first suggested by B. Hapke and R. Nelson (1975, J. Atmos. Sci.32, 1212–1218), is examined in light of our results. Several arguments indicate that there is no nonabsorbing, overlying haze and that the UV absorber extends to the top of the haze 8layer.     

Radiative transfer for incoherent scattering in a semiinfinite inhomogeneous atmosphere containing energy sources

Byurakanskaya Astrophysical Laboratory. Translated from Astrofizika, Vol. 36, No. 4, pp. 563–574, October–December, 1993.