Angle-dependent Redistribution Research Articles

Fast approximation of angle-dependent partial redistribution in moving atmospheres

Radiative transfer modeling of spectral lines including partial redistribution (PRD) effects requires the evaluation of the ratio of the emission to the absorption profile. This quantity requires a large amount of computational work if one employs the angle-dependent redistribution function, which prohibits its use in 3D radiative transfer computations with model atmospheres containing velocity fields. We aim to provide a method to compute the emission to absorption profile ratio that requires less computational work but retains the effect of angle-dependent scattering in the resulting line profiles. We present a method to compute the profile ratio that employs the angle-averaged redistribution function and wavelength transforms to and from the rest frame of the scattering particles. We compare the emergent line profiles of the \MgII\,k and \Lyalpha\ lines computed with angle-dependent PRD, angle-averaged PRD and our new method in two representative test atmospheres. The new method yields a good approximation of true angle-dependent profile ratio and the resulting emergent line profiles while keeping the computational speed and simplicity of angle-averaged PRD theory.

Hanle effect with angle-dependent partial redistribution

The polarized line transfer equation for the Hanle effect is solved in the framework of an exact partial frequency redistribution (PRD) theory developed by Bommier (1997a,b). In that theory the effect of collisions on the Hanle effect is considered self-consistently. We follow that approach in the line transfer computations presented here. The theory formulated by Bommier clearly recognizes two levels of approximations for exact PRD, in order to facilitate the solution of the line transfer equation. The second level employs angle-dependent redistribution functions, and numerically represents a more difficult problem compared to the third level, which involves only the use of angle-averaged frequency redistribution functions. We present a method which can solve the problem in both the levels of approximation. The method is based on a perturbative approach to line polarization. Although computationally expensive, it offers the only practical means of solving the angle-dependent Hanle PRD problem. We discuss the numerical aspects of assembling the so called “frequency domain dependent redistribution matrices”, and also an efficient way of computing the scattering integral. Some examples are presented to illustrate the interesting aspects of the Hanle-PRD problem with angle-dependent frequency redistribution. A comparison of the emergent profiles computed under angle-averaged and angle-dependent redistribution is carried out, and the effect of collisions is investigated. We show that it is necessary to incorporate an angle-dependent redistribution mechanism especially in the computation of the Stokes U parameter. We demonstrate that the use of simple frequency domains is good enough in practical applications of the Hanle PRD theory.

The effect of angle-dependent non-coherent electron scattering and partial redistribution on the polarization of resonance lines

The polarized radiative transfer equation is solved when angle-dependent partial redistribution and non-coherent electron scattering are included as line-scattering mechanisms. A static atmosphere with plane parallel symmetry is assumed. Test calculations are used to illustrate the effects of the electron-scattering coefficient, the thermalization parameter and the continuous absorption coefficient on the line polarization. Results of angle-averaged and angle-dependent redistribution functions are compared and it is shown that angle-dependent functions should be used to model the wing polarization of optically thin lines. The lower the continuous absorption compared with the electron scattering, the higher the wing polarization.

2-D Radiative Transfer Simulations with Angle-Dependent Partial Frequency Redistribution

AbstractWe demonstrate how the angle-dependent redistribution function can be incorporated into the 2-D transfer modelling of solar prominences. Some preliminary numerical simulations have been performed and we present their results by comparing the emergent hydrogen Lα line profiles computed with the angle-averaged and angle-dependent redistributions.

Comments on ‘the angle-dependent redistribution functionsR III andR IV’ by S. J. McKenna

It is shown that the complicated form of the line absorption probability function for scattering in subordinate lines, derived by McKenna (1984), is an artifact of adopting a mathematically inconvenient expression for the corresponding redistribution function. It is demonstrated that the absorption probability must be given by the Voigt function.

Redistribution effects on line formation in moving stellar atmospheres—II. The non-isotropic redistribution functions [formula omitted

The effects of photon frequency and angular redistribution arising from pure Doppler ( R I ), Doppler with natural line broadening R II and from the combined Doppler, natural and collisional line broadening ( R III ) are examined for line formation in plane-parallel semi-infinite and slab differentially expanding atmospheres. Line source functions and their corresponding emergent line intensities are obtained using the full angle-dependent redistribution functions. The results are compared with those obtained assuming complete redistribution and also with those using two types of angle-averaged redistribution functions; those angle-averaged in the observer's rest frame and those angle-averaged in the fluid frame. It is found that the redistribution functions angle-averaged in the observer's rest frame can produce spurious effects on line formation in moving media, as pointed out by Magnan. 1 On the other hand, emergent line intensities obtained using the redistribution functions angle-averaged in the fluid frame are found to differ by little from those obtained using the full angle-dependent redistribution functions, for the velocity fields considered here.

Non-isotropic redistribution effects on spectral line formation

The combined effects of frequency and angular redistribution are examined in slab and semi-infinite static isothermal atmospheres for the cases of pure Doppler damping and Doppler with natural damping using both the dipole and isotropic phase function. The main effect of angle-dependent redistribution is that photons which scatter more coherently in direction are re-emitted more coherently in frequency and this effect is enhanced by the dipole phase function. For the slab atmosphere it was generally found that the source functions, and hence the emergent intensities, for the angle-dependent redistribution were greater in the line core and smaller in the line wings than those obtained for angle-averaged or complete redistribution. For the semi-infinite atmosphere the behaviour of the results was the same as in the slab atmosphere except that in the line core the reverse was true for the emergent intensities and for the source functions at small optical depths.

A simple quadrature method for the evaluation of the redistribution functions R III,IV(x′, n′; x, n)

A simple and last method for computing the frequency and angle-dependent redistribution functions R III(x′, n ′x, n ) and R IV(x′;x n ) is given along with graphs of these functions for various values of the relevant parameters which control their behaviour. These redistribution functions describe photon redistribution when there are deviations from strict frequency coherency in the atom's frame. The redistribution function R III(x′, n ′;x, n) assumes collisional broadening of the upper transition states resulting in complete redistribution in the atom's frame, while R IV(x′, n ′;x, n ) assumes both transition states to be radiation broadened only and thus describes resonance scattering resulting in partial frequency coherency in the atom's frame. A comparison between these angle-dependent redistribution functions and their angle-averaged counterparts is also given.

Resonance-line transfer with partial redistribution. VII Angle-dependent redistribution

A method is presented for treating radiative transfer in resonance lines, allowing for the full angle and frequency dependence of redistribution in the scattering process, as seen in the laboratory frame. The case of an equivalent-two-level-atom source function is considered; the problem to be treated is then linear in the radiation field. We apply this method to the Ca II lines in the solar atmosphere, using a redistribution function which takes into account a mixture of coherence in the atom's frame, with Doppler redistribution in the laboratory frame (for atoms which have not suffered an elastic collision), and of complete redistribution in the laboratory frame (for atoms that are collisionally perturbed during the emission process). Both the angle-averaged approximation and the full angle-dependent solution were obtained, and were compared to assess, differentially, the effects of angular redistribution upon the computed line profile and its center-to-limb behavior. For the Ca II line in a homogeneous solar chromosphere the angle-dependent effects are found to be negligible, indicating that one may use angle-averaged redistribution functions when studying partial redistribution effects in line profiles.