Contribution functions for polarized radiative transfer

The concepts of contribution functions (CF) and of mean depths of line formation of unpolarized light as well as of Stokes profiles will be critically discussed. After having outlined the historical development arguments are given in favour of the use of directly observable quantities such as the emergent line intensity or the polarized components seen through polarization optics only. The arguments are provided by a probability interpretation of the CF; the ambiguities of line depression CF as well as some physically strange features in Stokes profiles are avoided if the rules based on this interpretation are observed. Some problems of the interpretation of measurements in chromospheric lines will be discussed as well.     

Numerical methods in polarized radiative transfer

This paper looks at three aspects of numerical methods for solving polarized radiative transfer problems associated with spectral line formation in the presence of a magnetic field. First we prove Murphy's law for Stokes evolution operators which is the basis of the efficient algorithm used in the SPSR software package to compute the Stokes line depression contribution functions. Then we use a two-stream model to explain the efficacy of the field-free method in which the non-LTE line source function in a uniform magnetic field is approximated by the source function neglecting the magnetic field. Finally we introduce a totally new and computationally efficient approach to solving non-LTE problems based on a method of sparsely representing integral operators using wavelets. As an illustration, the wavelet method is used to solve the source function integral equation for a two-level atomic model in a finite atmosphere with coherent scattering, ignoring polarization.     

An iterative simultaneous solution of the equations of statistical equilibrium and radiative transfer in the comoving frame

A direct iteration has been performed to obtain a simultaneous solution of the equations of line transfer in an expanding spherically symmetric atmosphere in the comoving frame with statistical equilibrium for a non-LTE, two-level atom. The solution converges in three or four iterations to an accuracy of 1 per cent of the ratio of the population densities of the two levels. As initial values, the upper level population was set equal to zero or to the LTE densities. The final solution on convergence indicates enhanced population of these levels over the initial values assumed, Large velocity gradients enhance this effect whereas large geometrical sizes of the atmospheres tend to reduce it.     

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.     

Order-of-scattering and doubling method for the solution of transfer equations

A method to obtain scattering and transmission functions for computational purposes has been developed in this paper. This method can be easily extended to the scattering atmosphere involving internal energy and may be to the time-dependent case. Special cases such as the homogeneous process, the Helmholtz's principle of reciprocity and the self-adjoint are presented.     

A discrete spherical harmonics method for radiative transfer analysis in inhomogeneous polarized planar atmosphere

A discrete spherical harmonics method is developed for the radiative transfer problem in inhomogeneous polarized planar atmosphere illuminated at the top by a collimated sunlight while the bottom reflects the radiation. The method expands both the Stokes vector and the phase matrix in a finite series of generalized spherical functions and the resulting vector radiative transfer equation is expressed in a set of polar directions. Hence, the polarized characteristics of the radiance within the atmosphere at any polar direction and azimuthal angle can be determined without linearization and/or interpolations. The spatial dependent of the problem is solved using the spectral Chebyshev method. The emergent and transmitted radiative intensity and the degree of polarization are predicted for both Rayleigh and Mie scattering. The discrete spherical harmonics method predictions for optical thin atmosphere using 36 streams are found in good agreement with benchmark literature results. The maximum deviation between the proposed method and literature results and for polar directions \(\vert \mu \vert \geq0.1 \) is less than 0.5% and 0.9% for the Rayleigh and Mie scattering, respectively. These deviations for directions close to zero are about 3% and 10% for Rayleigh and Mie scattering, respectively.     

The solution of a time-dependent problem in radiative transfer

The time-dependent equation of radiative transfer for a finite, plane-parallel, non-radiating, and isotropically scattering atmosphere of arbitrary stratification is solved by using the integral equation method. The medium is taken to be inhomogeneous. The Laplace transform is used in the time domain. It is seen that the obtained solutions are reducible to the corresponding ones for steady-state problems by simply changing the Laplace transform parameter to zero.     

On multi-region problems in radiative transfer

TheF _N method is used to solve radiative transfer problems, based on the general anisotropically scattering model, in multi-layer atmospheres.     

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.     

Modelling artificial night-sky brightness with a polarized multiple scattering radiative transfer computer code

As part of an ongoing investigation of radiative effects produced by hazy atmospheres, computational procedures have been developed for use in determining the brightening of the night sky as a result of urban illumination. The downwardly and upwardly directed radiances of multiply scattered light from an offending metropolitan source are computed by a straightforward Gauss–Seidel (G–S) iterative technique applied directly to the integrated form of Chandrasekhar's vectorized radiative transfer equation. Initial benchmark night-sky brightness tests of the present G–S model using fully consistent optical emission and extinction input parameters yield very encouraging results when compared with the double scattering treatment of Garstang, the only full-fledged previously available model.     

Non-LTE polarized radiative transfer in intermediate magnetic fields: Numerical problems and results

This paper presents some numerical results relative to a solution, based on the density matrix formalism, of the non-LTE, polarized radiative transfer problem for a two-level atom. The results concern the atomic upper level population and alignment, and the emergent radiation Stokes profiles, for a plane-parallel, static, isothermal atmosphere embedded in a magnetic field of intermediate strength, such that the Zeeman splitting has to be taken into account in the line profile. Zeeman coherences are neglected, whereas magneto-optical effects are taken into account, resulting in a full 4×4 absorption matrix. Induced emission is neglected and complete frequency redistribution, in the rest and laboratory frames, is assumed. Pure Doppler absorption profile (gaussian shape) has also been assumed. The presentation of the results is preceded by a brief discussion of their accuracy and of the numerical difficulties that were met in the solution of the problem.On leave from the Dipartimento di Astronomia e Scienza dello Spazio, Università di Firenze, Largo E. Fermi 5, I-50125 Firenze, Italia     

Hyperbolic character of the angular moment equations of radiative transfer and numerical methods

We study the mathematical character of the angular moment equations of radiative transfer in spherical symmetry and conclude that the system is hyperbolic for general forms of the closure relation found in the literature. Hyperbolicity and causality preservation lead to mathematical conditions allowing us to establish a useful characterization of the closure relations. We apply numerical methods specifically designed to solve hyperbolic systems of conservation laws (the so-called Godunov-type methods) to calculate numerical solutions of the radiation transport equations in a static background. The feasibility of the method in all regimes, from diffusion to free-streaming, is demonstrated by a number of numerical tests, and the effect of the choice of the closure relation on the results is discussed.     

Linear hydrodynamical equations coupled with radiative transfer in a non-isothermal atmosphere

In a previous paper (Schmieder, 1977), we solved simultaneously the hydrodynamical and radiative transfer equations, so we do not have to assume any relaxation time of the atmosphere. In this paper, we use that theory to interpret photospheric observations of the Mg i line at 5172 Å.For periods between 400 and 140 s, the phase-shifts observed between velocities and the phase shifts between intensity and velocity fluctuations are explained by the existence of radiative dissipation coupled with evanescent waves or upward propagating waves, according to the frequency.For smaller periods partial or total reflections must be considered.The results relative to radiative dissipation are expressed in terms of the variation of a relaxation time with frequency through the atmosphere (10^–3<₅₀₀₀<1).     

Linear hydrodynamical equations coupled with radiative transfer in a non-isothermal atmosphere

A method coupling the hydrodynamical equations and radiative transfer in a realistic solar model atmosphere is described. The influence of the temperature gradient of the model and the radiative dissipation is pointed out.The effect of the large temperature gradient is important in the layers where the optical depth ₅₀₀₀ is greater than 0.5; the ratio between the amplitude of the temperature and the velocity fluctuations decreases with the altitude by a factor 2 between = 1 and = 0.5 and in the case of the acoustic waves, the phase shift between these fluctuations is small.The radiative energy loss in the thick layers ( ₅₀₀₀ = 1) leads to a decrease of the vertical phase velocity of the waves and to a damping of their amplitudes in the layers of intermediate optical depth (10^-2 < ₅₀₀₀ < 0.5). The effect of the dissipation is negligible in the thin layers (₅₀₀₀ < 10^-2).     

Material coefficients and transfer of polarized radio radiation in a plasma

By a perturbation and diagram resummation method, a transport equation for the transverse field polarization matrix is established. This equation is then transformed into an equation for the Stokes parameters of the radiation. The equation takes the usual form of a transfer equation; the absorption and emission coefficients are matrix, the elements of which are given as a function of the dissipative part of the microcurrent correlation tensor and conductivity tensor. Finally this equation is expressed as a system for the intensities of the proper modes. The equations of the system are usually coupled.