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 radiative transfer for semi-infinite plane-parallel isotropic scattering atmosphere with scattering albedo ω0 ≤ 1 by a method based on Laplace transform and Wiener-Hopf technique

By performing the one-sided Laplace transform on the scalar integro-differential equation for a semi-infinite plane-parallel isotropic scattering atmosphere with a scattering albedo ₀ 1, an integral equation for the emergent intensity has been derived. Application of the Wiener-Hopf technique to this integral equation will give the emergent intensity. The intensity at any optical depth for a positive scattering angle is also derived by inversion. The intensity at any optical depth for a negative scattering angle is also derived in terms of Cauchy's principal value using Plemelj's formulae.     

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.     

Polarization of resonance lines in the case of a Voigt absorption profile

Multiple resonance scattering of radiation in a spectral line is considered in the case of a Voigt absorption profile. The scattering is assumed to take place in a nonmagnetic semi-infinite atmosphere with uniformly distributed sources of unpolarized radiation. Polarization characteristics have been obtained for the emergent radiation by numerically solving the Ambartsumian-Chandrasekhar matrix integral equation.     

Linear singular integral equations for polarized radiation in isotropic media

Linear singular integral equations are derived for polarized radiation fields in semi infinite and finite plane parallel atmospheres. An arbitrary phase matrix and any distribution of primary sources are assumed. The integral equations together with appropriate sets of linear constraints arise from functional relations derived by means of CASE 's eigenfunctions and their full range completeness and orthogonality. The emergent radiation is described by half range singular integral equations, whereas the STOKES vector of the inner radiation field obeys full range integral equations depending on the emergent radiation.     

Asymptotic theory of the transfer of polarized radiation with resonance scattering in the doppler core of the line

In the first of the series of papers by Ivanov et al. it was shown that the model problem of the transfer of polarized radiation as a result of resonance scattering from two-level atoms in a homogeneous plane atmosphere in the absence of LTE comes down, in the approximation of complete frequency redistribution, to the solution of an integral matrix equation of the Wiener-Hopf type for a (2 × 2) matrix source function S(τ). In the second paper in this series, devoted to the vector Milne problem, complete asymptotic expansions of the matrix I(z) [which is essentially a Laplace transform of the matrix S(τ)] for the case of a Doppler profile of the coefficient of absorption, and the coefficients of asymptotic expansions of S(τ) (τ » 1) are expressed in terms of coefficients of the expansions of I(z). We show that asymptotic expansions of S(τ) can be found directly from an integral matrix equation of the Wiener-Hopf type for S(τ). We give new recursive equations for the coefficients of these expansions, as well as a new derivation of asymptotic expansions of the matrix I, including its second column, which was considered only briefly by Ivanov et al.     

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).     

Volterra equation for the matrix source function at resonance scattering

A matrix transfer equation for multiple resonance scattering of radiation in a spectral line in a semiinfinite atmosphere with a uniform distribution of primary radiation sources is examined. A nonlinear matrix integral is obtained for this equation as a generalization of the Rybicki two-point Q-integral. One special case of the matrix [^(Q)] {\mathbf{\hat{Q}}} -integral is the Volterra equation for the matrix source function of the problem discussed here. The Volterra equation is solved numerically for a Doppler profile of the absorption coefficient. Several polarization characteristics of the emerging radiation are obtained.     

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.     

Effect of continuum absorption on the polarization of resonance lines

Multiple resonance scattering of radiation in a spectrum line with absorption in the continuum is examined. It is assumed that the scattering atmosphere is semi-infinite and that there is no magnetic field or continuum emission at the frequencies of the spectrum line. The polarization characteristics of the emerging radiation are determined for unpolarized primary radiation sources distributed uniformly within an atmosphere in the case of a Voigt absorption profile. The calculations employ an iterative solution of the Ambartsumyan-Chandrasekhar matrix integral equation.     

Characteristics of the radiation emerging from the top of a Rayleigh atmosphere—II Total upward flux and albedo

The solution of the equation of radiative transfer in a medium exhibiting Rayleigh scattering, as developed by S. Chandrasekhar, has been used for an extensive series of computations(³) of the characteristics of the scattered and diffusely reflected radiation emerging from the top of an atmospheric model which corresponds in many respects to the sunlit portion of the earth's atmosphere. The first part of this two-part discussion dealt with the intensity, degree of polarization, plane of polarization and the neutral points of the emergent light as functions of sun elevation, direction in the downward hemisphere, optical thickness of the model atmosphere and reflectivity of the underlying surface. This second part is concerned with the upward flux obtained by an integration of the intensity over the entire hemisphere, for the incident radiation (a) being independent of wavelength or (b) having the spectral distribution of the extra-terrestrial solar radiation. Integration with respect to wavelength in the latter case, together with an approximation for the sphericity of the atmosphere, yields a value of 7.6 per cent for the earth's planetary albedo due to scattering by the clear atmosphere. An approximation for ozone absorption decreases the computed albedo to 6.9 per cent.     

Padé approximant calculations for dispersive,isotropic scattering,homogeneous media

Exact relations for radiation heat flux at the boundaries of a slab with diffusely reflecting boundary conditions and internal source are obtained in terms of the reflection and transmission coefficients of a source free slab with isotropic boundary conditions. The integral equation defining the radiation heat flux contains explicitly the internal source. So, the particular solution for radiative transfer equation is not required. Available exact values for albedos give exact values of radiation heat flux. Padé approximant technique is used to obtain numerical values for homogenous media.     

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

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.     

The transfer of polarized radiation in spectral lines: Formalism and solutions in simple cases

A general treatment of the transfer of polarized radiation in spectral lines assuming a Rayleigh phase function and a general law of frequency redistribution is derived. It is shown how nine families of coupled integral equations for the moments of the radiation field arise which are necessary to fully describe the state of polarization of the emergent radiation from a plane-parallel, semi-infinite atmosphere. The special case of angle independent redistribution functions is derived from the general formalism, and it is shown how the nine families of integral equations reduce to the six linearly independent integral equations derived by Collins (1972). To serve as a test of the formulation, solutions for isothermal atmospheres are given.     

Rayleigh scattering: Volterra equation for the matrix source function

Multiple Rayleigh scattering is examined in a semi-infinite atmosphere with uniformly distributed primary sources of partially polarized radiation. The resulting linear polarization is described by a 2×2 matrix transfer equation. A matrix generalization of Rybicki's two point Q-integral is obtained for this case. It is shown that the Volterra equation for the matrix source function for this problem is a particular case of our Q integral. Applying the Laplace transform to it yields the matrix form of the Ambartsumyan-Chandrasekhar H-equation. The Volterra equation for Sobolev's matrix resolvent function is another simple consequence of this equation. Translated from Astrofizika, Vol. 52, No. 2, pp. 301–310 (May 2009).     

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

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.     

Laser radiation in active amplifying media treated as a transport problem: Transfer equation derived and exactly solved

The flow of laser radiation in a plane-parallel cylindrical slab of active amplifying medium with axial symmetry is treated as a problem in radiative transfer. The appropriate one-dimensional transfer equation describing the transfer of laser radiation has been derived by an appeal to Einstein'sA, B coefficients (describing the processes of stimulated line absorption, spontaneous line emission, and stimulated line emission sustained by population inversion in the medium) and considering the rate equations to completely establish the rational of the transfer equation obtained. The equation is then exactly solved and the angular distribution of the emergent laser beam intensity is obtained; its numerically computed values are given in tables and plotted in graphs showing the nature of peaks of the emerging laser beam intensity about the axis of the laser cylinder.     

Line Formation in a Purely Scattering, Optically Thick Atmosphere

A model problem in the theory of line formation in an optically thick, purely scattering, stellar atmosphere is considered. The integral equation of radiation transfer at line frequencies is solved numerically for a two-level atom in the approximation of complete frequency redistribution in scattering. The numerical results are compared with those calculated from equations of the asymptotic theory. On the basis of the asymptotic theory, the positions of intensity maxima in a line are found for different absorption profiles.     

Curvature effects in extended stellar atmospheres — Pure absorption

The effects of curvature in an atmosphere with pure absorption are investigated. Numerical solution of the transfer equation has been obtained in the framework of the Discrete Space Theory of Radiative Transfer. Two cases have been considered: (a) the atmosphere is irradiated at the bottom and there is no incident radiation at the top of the atmosphere; and (b) no radiation is incident on either side of the atmosphere. It is found that the thermal sources inside the atmosphere dominantly influence the emergent radiation and this is very much so, in the spherical case and for large optical thickness. The emergent luminosities increase with the geometrical thickness although the emergent specific intensities are reduced and the former seems to be because of the larger surface area and later seems to be because of the effects of curvature.     

Radiative transfer in spectral lines by noncoherent scattering

The functional analytic method of solution is applied to investigation of the radiative transfer equation in spectral lines. A problem of scattering in the spectral line with the frequency redistribution in anisotropic-scattering infinite and semi-infinite media is considered. Continuum absorption in the line is also taken into account.The solution is presented as the exponential function of the operatorA and the functional calculus is developed. The eigenfunction and the expansion coefficients, in terms of which the explicit solution is expressed, have been found. The nonlinear equation and the explicit expressions for theX-function are derived. The albedo problem with the determined expansion coefficients and the intensity of the emergent radiation is given as an example.