Effects Of Frequency Redistribution Research Articles

Impact of the numerical conversion to optical depth on the transfer of polarized radiation

Making the conversion from the geometrical spatial scale to the optical depth spatial scale is useful in obtaining numerical solutions for the radiative transfer equation. This is because it allows for the use of exponential integrators, while enforcing numerical stability. Such a conversion involves the integration of the total opacity of the medium along the considered ray path. This is usually approximated by applying a piecewise quadrature in each spatial cell of the discretized medium. However, a rigorous analysis of this numerical step is lacking. This work is aimed at clearly assessing the performance of different optical depth conversion schemes with respect to the solution of the radiative transfer problem for polarized radiation, out of the local thermodynamic equilibrium. We analyzed different optical depth conversion schemes and their combinations with common formal solvers, both in terms of the rate of convergence as a function of the number of spatial points and the accuracy of the emergent Stokes profiles. The analysis was performed in a 1D semi-empirical model of the solar atmosphere, both in the absence and in the presence of a magnetic field. We solved the transfer problem of polarized radiation in different settings: the continuum, the photospheric Sr i AA modeled under the assumption of complete frequency redistribution, and the chromospheric Ca i AA taking the partial frequency redistribution effects into account during the modeling. High-order conversion schemes generally outperform low-order methods when a sufficiently high number of spatial grid points is considered. In the synthesis of the emergent Stokes profiles, the convergence rate, as a function of the number of spatial points, is impacted by both the conversion scheme and formal solver. The use of low-order conversion schemes significantly reduces the accuracy of high-order formal solvers. In practical applications, the use of low-order optical depth conversion schemes introduces large numerical errors in the formal solution. To fully exploit high-order formal solvers and obtain accurate synthetic emergent Stokes profiles, it is necessary to use high-order optical depth conversion schemes.

Scalable matrix-free solver for 3D transfer of polarized radiation in stellar atmospheres

We present an efficient and massively parallel solution strategy for the transfer problem of polarized radiation, for a 3D stationary medium out of local thermodynamic equilibrium. Scattering processes are included accounting for partial frequency redistribution effects. Such a setting is one of the most challenging ones in radiative transfer modeling. The problem is formulated for a two-level atomic model, which allows linearization. The discrete ordinate method alongside an exponential integrator are used for discretization. Efficient solution is obtained with a Krylov method equipped with a tailored physics-based preconditioner. A matrix-free approach results in a lightweight implementation, suited for tackling large problems. Near-optimal strong and weak scalability are obtained with two complementary decompositions of the computational domain. The presented approach made it possible to perform simulations for the Ca i line at 4227Å with more than 109 degrees of freedom in less than half an hour on massively parallel machines, always converging in a few iterations for the proposed tests.

The Circular Polarization of the Mn 1 Resonance Lines around 280 nm for Exploring Chromospheric Magnetism

We study the circular polarization of the Mn i resonance lines at 279.56, 279.91, and 280.19 nm (hereafter, UV multiplet) by means of radiative transfer modeling. In 2019, the CLASP2 mission obtained unprecedented spectropolarimetric data in a region of the solar ultraviolet including the Mg ii h and k resonance lines and two lines of a subordinate triplet, as well as two Mn i resonance lines. The first analysis of such data, in particular those corresponding to a plage region, allowed the inference of the longitudinal magnetic field from the photosphere to the upper chromosphere just below the transition region. This was achieved by applying the weak field approximation to the circular polarization profiles of the Mg ii and Mn i lines. While the applicability of this approximation to the Mg ii lines was already demonstrated in previous works, this is not the case for the Mn i UV multiplet. These lines are observed as absorptions between the Mg ii h and k lines, a region whose intensity is shaped by their partial frequency redistribution effects. Moreover, the only Mn i stable isotope has nuclear spin I = 5/2 and thus hyperfine structure must be, a priori, taken into account. Here we study the generation and transfer of the intensity and circular polarization of the Mn i resonance lines accounting for these physical ingredients. We analyze their sensitivity to the magnetic field by means of their response function, and we demonstrate the applicability of the weak field approximation to determine the longitudinal component of the magnetic field.

Fast and accurate approximation of the angle-averaged redistribution function for polarized radiation

Context. Modeling spectral line profiles taking frequency redistribution effects into account is a notoriously challenging problem from the computational point of view, especially when polarization phenomena (atomic polarization and polarized radiation) are taken into account. Frequency redistribution effects are conveniently described through the redistribution function formalism, and the angle-averaged approximation is often introduced to simplify the problem. Even in this case, the evaluation of the emission coefficient for polarized radiation remains computationally costly, especially when magnetic fields are present or complex atomic models are considered. Aims. We aim to develop an efficient algorithm to numerically evaluate the angle-averaged redistribution function for polarized radiation. Methods. The proposed approach is based on a low-rank approximation via trivariate polynomials whose univariate components are represented in the Chebyshev basis. Results. The resulting algorithm is significantly faster than standard quadrature-based schemes for any target accuracy in the range [10−6, 10−2].

Temperature constraints from inversions of synthetic solar optical, UV, and radio spectra

Context. High-resolution observations of the solar chromosphere at millimeter wavelengths are now possible with the Atacama Large Millimeter Array (ALMA), bringing with them the promise of tackling many open problems in solar physics. Observations from other ground and space-based telescopes will greatly benefit from coordinated endeavors with ALMA, yet the diagnostic potential of combined optical, ultraviolet and mm observations has remained mostly unassessed. Aims. In this paper we investigate whether mm-wavelengths could aid current inversion schemes to retrieve a more accurate representation of the temperature structure of the solar atmosphere. Methods. We performed several non-LTE inversion experiments of the emergent spectra from a snapshot of 3D radiation-MHD simulation. We included common line diagnostics such as Ca II H, K, 8542 Å and Mg II h and k, taking into account partial frequency redistribution effects, along with the continuum around 1.2 mm and 3 mm. Results. We find that including the mm-continuum in inversions allows a more accurate inference of temperature as function of optical depth. The addition of ALMA bands to other diagnostics should improve the accuracy of the inferred chromospheric temperatures between log τ ∼ [−6, −4.5] where the Ca II and Mg II lines are weakly coupled to the local conditions. However, we find that simultaneous multiatom, non-LTE inversions of optical and UV lines present equally strong constraints in the lower chromosphere and thus are not greatly improved by the 1.2 mm band. Nonetheless, the 3 mm band is still needed to better constrain the mid-upper chromosphere.

The transfer of resonance line polarization with partial frequency redistribution andJ-state interference

The linear polarization signals produced by scattering processes in strong resonance lines are rich in information on the magnetic and thermal structure of the chromosphere and transition region of the Sun and of other stars. A correct modeling of these signals requires accounting for partial frequency redistribution effects, as well as for the impact of quantum interference between different fine structure levels (J-state interference). In this paper, we present a theoretical approach suitable for modeling the transfer of resonance line polarization when taking these effects into account, along with an accurate numerical method of solution of the problem's equations. We consider a two-term atom with unpolarized lower term and infinitely sharp lower levels, in the absence of magnetic fields. We show that by making simple formal substitutions on the quantum numbers, the theoretical approach derived here for a two-term atom can also be applied to describe a two-level atom with hyperfine structure. An illustrative application to the MgII doublet around 2800A is presented.

J-state interference signatures in the second solar spectrum

The scattering polarization in the solar spectrum is traditionally modeled with each spectral line treated separately, but this is generally inadequate for multiplets where J-state interference plays a significant role. Through simultaneous observations of all the 3 lines of a Cr I triplet, combined with realistic radiative transfer modeling of the data, we show that it is necessary to include J-state interference consistently when modeling lines with partially interacting fine structure components. Polarized line formation theory that includes J-state interference effects together with partial frequency redistribution for a two-term atom is used to model the observations. Collisional frequency redistribution is also accounted for. We show that the resonance polarization in the Cr I triplet is strongly affected by the partial frequency redistribution effects in the line core and near wing peaks. The Cr I triplet is quite sensitive to the temperature structure of the photospheric layers. Our complete frequency redistribution calculations in semi-empirical models of the solar atmosphere cannot reproduce the observed near wing polarization or the cross-over of the Stokes Q/I line polarization about the continuum polarization level that is due to the J-state interference. When however partial frequency redistribution is included, a good fit to these features can be achieved. Further, to obtain a good fit to the far wings, a small temperature enhancement of the FALF model in the photospheric layers is necessary.

The influence of frequency redistribution on the transfer of gyroresonant photons in the atmospheres of compact stars: Monte-Carlo analysis

We numerically model the frequency redistribution of gyroresonant photons in a plane-parallel semi-infinite plasma atmosphere of a compact star by means of the Monte-Carlo method. We calculate the mean frequency shift and the mean number of scatterings for photons leaving the atmosphere as functions of the optical depth at emission, plasma temperature, and magnetic field strength. The probability of the escape of cyclotron-radiation photons from the atmosphere as a function of their optical depth at birth is calculated. The influence of the frequency redistribution effects on the spectrum of emergent radiation is studied.

Fluorescence weakening due to radiation trapping in a high density cold-atom cloud

We have measured the change of fluorescence from cold 87Rb atoms after magnetic field of the magneto-optic trap is turn off. A rise in fluorescence within several tens of milliseconds was observed. Analyzing this phenomenon in connection with the experimental conditions, we found that incoherent pumping field due to radiation trapping in the high density cold-atom cloud was the reason of fluorescence weakening under tight confinement by the magneto-optic trap. We have also performed a simple numerical simulation, which indicated that the calculations agree with experimental results only when the effect of frequency redistribution of scattered light is taken into account.

Hanle effect in the solar Ba II D2 line: a diagnostic tool for chromospheric weak magnetic fields

Context. The physics of the solar chromosphere depends in a crucial way on its magnetic structure. However there are presently very few direct magnetic field diagnostics available for this region. Aims. Here we investigate the diagnostic potential of the Hanle effect on the Ba ii D2 line resonance polarization for the determination of weak chromospheric turbulent magnetic fields. Methods. The line formation is described with a non-LTE polarized radiative transfer model taking into account partial frequency redistribution with an equivalent two-level atom approximation, in the presence of depolarizing collisions and the Hanle effect. We investigate the line sensitivity to temperature variations in the low chromosphere and to elastic collision with hydrogen atoms. We compare center-to-limb variations of the intensity and linear polarization profiles observed at THEMIS in 2007 to our numerical results. Results. We show that the line resonance polarization is very strongly affected by partial frequency redistribution effects both in the line central peak and in the wings. Complete frequency redistribution cannot reproduce the polarization observed in the line wings. The line is weakly sensitive to temperature differences between warm and cold components of the chromosphere. The effects of elastic collisions with hydrogen atoms and of alignment transfer due to multi-level coupling with the metastable 2 D5/2 levels have been studied in a recent paper showing that they depolarize the 2 P3/2 level of the line. In the conditions where the line is formed we estimate the amount of depolarization due to this mechanism as a factor of 0.7 to 0.65. If we first neglect this effect and determine the turbulent magnetic field strength required to account for the observed line polarization, we obtain values between 20 G and 30 G. We show that this overestimates the magnetic strength by a factor between 1.7 and 2. Applying these correction factors to our previous estimates, we find that the turbulent magnetic field strength is between 10 G and 18 G. Conclusions. Because of its low sensitivity to temperature variations, the solar Ba ii D2 line appears as a very good candidate for the diagnosis of weak magnetic fields in the low chromosphere (z ≥ 900 km) by means of the Hanle effect.

Photonic superdiffusive motion in resonance line radiation trapping Partial frequency redistribution effects

The relation between the jump length probability distribution function and the spectral line profile in resonance atomic radiation trapping is considered for partial frequency redistribution (PFR) between absorbed and reemitted radiation. The single line opacity distribution function [M. N. Berberan-Santos et al., J. Chem. Phys. 125, 174308 (2006)] is generalized for PFR and used to discuss several possible redistribution mechanisms (pure Doppler broadening; combined natural and Doppler broadening; and combined Doppler, natural, and collisional broadening). It is shown that there are two coexisting scales with a different behavior: the small scale is controlled by the intricate PFR details while the large scale is essentially given by the atom rest frame redistribution asymptotic. The pure Doppler and combined natural, Doppler, and collisional broadening are characterized by both small- and large-scale superdiffusive Levy flight behaviors while the combined natural and Doppler case has an anomalous small-scale behavior but a diffusive large-scale asymptotic. The common practice of assuming complete redistribution in core radiation and frequency coherence in the wings of the spectral distribution is incompatible with the breakdown of superdiffusion in combined natural and Doppler broadening conditions.

Effect of frequency redistribution in X-ray laser modeling

Calculations are presented of the spectral emissivity corrected for higher-order processes such as photon scattering. Usually, photon scattering is taken into account by postulating a complete frequency redistribution approximation, i.e., same absorption and emission spectra. However, cases exist in hot and dense plasmas for which partial frequency redistribution and angular redistribution play an important role for the radiative transfer mechanisms. Examples of X-ray lasers based on neon-like germanium ion emitters and hydrogen-like carbon ion emitters are considered.

Frequency redistribution effect on emissivity and intensity of Ge22+ and C5+ X-ray lasers

This work concerns the effect of frequency redistribution on X-ray laser beams. The redistribution function, calculated by extending the frequency fluctuation model to 2-photon processes, shows coherent scattering and a redistributed contribution. The spectral emissivity, corrected in order to account for this effect, is calculated for X-ray lasers, the amplifier plasma being composed either of neon-like germanium ions or hydrogen-like carbon ions. The coupled radiative transfer equation and population equations are solved consistently, and the intensity of the X-ray laser output is given for two 3p-3s lines (19.6 and 23.6 nm) of neon-like germanium. We also study the variation with plasma length (i.e. line intensity) of the emissivity of the C 5+ Balmer-α lasing line. The physical conditions of the calculations are obtained from a modelling of experiments conducted at the Rutherford Appleton Laboratory.

Atomic coherences and level-crossings physics

A synthesis work about the interaction of matter with polarized radiation, applied to solar magnetic field diagnostics, has recently been done by Stenflo (1994). This synthesis uses the classical theory of matter-radiation interaction - supplemented by the theory of partial redistribution of Omont, Smith, and Cooper (1972), on the one hand, and full quantum matter-radiation interaction theory, unable to take into account the partial frequency redistribution effects, on the other hand. The need of a full quantum approach taking into account the partial frequency redistribution effects appears as a unifying purpose; the present work, using the density matrix formalism, is a first attempt in this direction.

Partial frequency redistribution of polarized radiation

Resonance polarization, which is created by the scattering of an anisotropic radiation field in regions of zero or weak magnetic fields, is strongly dependent on the frequency redistribution taking place during the scatterings. Here we discuss the frequency redistribution matrix relevant to resonance lines, concentrating on linear polarization. First we analyze in detail the redistribution matrix in a zero magnetic field given by the theory of Omont, Smith and Cooper (1972), revisited by Domke and Hubeny (1988). We explain that the linear polarization maxima which may appear in the wings of the Stokes Q profiles of strong resonances lines such as the Ca I 4227 A line are coherent frequency redistribution effects. Various approximate forms of the frequency redistribution matrix are also examined. For resonance polarization in a weak magnetic field, we suggest a new expression for the redistribution matrix which can be used at all line frequencies, and is consistent with the condition that the Hanle effect acts only in the line core.

Partially coherent scattering in stellar chromospheres. II - The first-order escape probability method. III - A second-order probability method

Approximate analytic expressions are derived for resonance-line wing diagnostics, accounting for frequency redistribution effects, for homogeneous slabs, and slabs with a constant Planck function gradient. Resonance-line emission profiles from a simplified conceptual standpoint are described in order to elucidate the basic physical parameters of the line-forming layers prior to the performance of detailed numerical calculations. An approximate analytic expression is derived for the dependence on stellar surface gravity of the location of the Ca II and Mg II resonance-line profile peaks. An approximate radiative transfer equation using generalized second-order escape probabilities, applicable even in the presence of nearly coherent scattering in the damping wings of resonance lines, is derived. Approximate analytic solutions that can be applied in special regimes and achieve good agreement with accurate numerical results are found.

Radiative transfer of resonance lines with internal sources

The modeling of resonance lines in the aurorae and dayglow usually requires a large computer code to solve the radiative transfer problem. We present here a simple theory that gives reasonable estimates of the emergent intensities, including the angular distribution, for very optically thick lines with internal sources. The method uses an extension of the H-functions (implying a semi-infinite atmosphere) and assumes that the source is monochromatic and may be reasonably represented by a Chapman profile. The theory is compared with detailed numerical calculations for resonance lines with Voigt emission line profiles, including the cases of both partial and complete frequency redistribution. The effects of frequency redistribution on the angular distribution of the emergent radiation and on the line profiles are also shown.

Frequency Redistribution Effects in the Formation of Lyman α in Prominences and Their Influence on the Ratio of Hα to Lα

The effect of partial frequency redistribution in the formation of Lyman α in the chromosphere has been discussed by Milkey and Mihalas (1973) and others, and it has been shown that in this case the coherency of scattering in the wings of the line substantially influences the line profile. Although there are non-negligible sources for La photons within a prominence, a large fraction of the emergent line photons are due to scattering of photons incident on the surface of the prominence so that one expects that in a prominence the frequency redistribution processes will play an important role in determining the emergent intensity.