Complete Frequency Redistribution Research Articles

Assessment of the CRD approximation for the observer’s frame RIII redistribution matrix

Context. Approximated forms of the RII and RIII redistribution matrices are frequently applied to simplify the numerical solution of the radiative transfer problem for polarized radiation, taking partial frequency redistribution (PRD) effects into account. A widely used approximation for RIII is to consider its expression under the assumption of complete frequency redistribution (CRD) in the observer’s frame (RIII−CRD). The adequacy of this approximation for modeling the intensity profiles has been firmly established. By contrast, its suitability for modeling scattering polarization signals has only been analyzed in a few studies, considering simplified settings. Aims. In this work, we aim to quantitatively assess the impact and the range of validity of the RIII−CRD approximation in the modeling of scattering polarization. Methods. We first present an analytic comparison between RIII and RIII−CRD. We then compare the results of radiative transfer calculations, out of local thermodynamic equilibrium, performed with RIII and RIII−CRD in realistic one-dimensional atmospheric models. We focus on the chromospheric Ca I line at 4227 Å and on the photospheric Sr I line at 4607 Å. Results. The RIII−CRD approximation provides accurate results for the Ca I 4227 Å line. Only when velocities are included can some appreciable discrepancies be found, especially for lines of sight close to the disk center. The approximation performs well also for the Sr I 4607 Å line, especially in the absence of magnetic fields or when a micro-turbulent field is included. However, some appreciable errors appear when deterministic magnetic fields or bulk velocities are considered. Conclusions. Our results show that the RIII−CRD approximation is suited for the PRD modeling of the scattering polarization signals of strong chromospheric lines, both in the core and wings. With a few minor exceptions, this approximation is also suitable for photospheric lines, although PRD effects generally play a minor role in their modeling.

Evolution of spectral lines under time-dependent illumination of an absorbing and scattering atmosphere

This is a continuation of the author’s research on the temporal variations in the spectral line profiles formed in absorbing and scattering atmospheres illuminated by non-stationary energy sources. The line radiation transfer in an atmosphere of finite optical thickness is considered under assumption of complete frequency redistribution. We treat two types of illumination of the medium: the sources of the form of Dirac δ(t) and Heaviside H(t) unit jump functions. The goal of the work is to identify the dependence of the dynamics of changes in the spectral line profiles on frequency redistribution and the values of various optical parameters such as the optical thickness of the medium and the value of scattering coefficient. It is shown that in some cases this dependence allows to use observational material on dynamics of the burst phenomena to estimate some of these characteristics.

Numerical solutions to linear transfer problems of polarized radiation

Context. The polarization signals produced by the scattering of anistropic radiation in strong resonance lines encode important information about the elusive magnetic fields in the outer layers of the solar atmosphere. An accurate modeling of these signals is a very challenging problem from the computational point of view, in particular when partial frequency redistribution (PRD) effects in scattering processes are accounted for with a general angle-dependent treatment. Aims. We aim at solving the radiative transfer problem for polarized radiation in nonlocal thermodynamic equilibrium conditions, taking angle-dependent PRD effects into account. The problem is formulated for a two-level atomic model in the presence of arbitrary magnetic and bulk velocity fields. The polarization produced by scattering processes and the Zeeman effect is considered. Methods. The proposed solution strategy is based on an algebraic formulation of the problem and relies on a convenient physical assumption, which allows its linearization. We applied a nested matrix-free GMRES iterative method. Effective preconditioning is obtained in a multifidelity framework by considering the light-weight description of scattering processes in the limit of complete frequency redistribution (CRD). Results. Numerical experiments for a one-dimensional (1D) atmospheric model show near optimal strong and weak scaling of the proposed CRD-preconditioned GMRES method, which converges in few iterations, independently of the discretization parameters. A suitable parallelization strategy and high-performance computing tools lead to competitive run times, providing accurate solutions in a few minutes. Conclusions. The proposed solution strategy allows the fast systematic modeling of the scattering polarization signals of strong resonance lines, taking angle-dependent PRD effects into account together with the impact of arbitrary magnetic and bulk velocity fields. Almost optimal strong and weak scaling results suggest that this strategy is applicable to realistic 3D models. Moreover, the proposed strategy is general, and applications to more complex atomic models are possible.

Теоретическое исследование влияния рассеивающей среды, состоящей из атомов калия, на время затухания люминесцентного сигнала с использованием метода Монте-Карло

The study solves the problem of the luminescent photons propagation in a resonantly absorbing medium consisting of atoms of the studied substance and a buffer inert gas. The Monte Carlo method was used in numerical experiments carried out to simulate real processes that occur in a chamber designed to determine the lifetime of an individual atom in an excited state by the method of measuring the luminescence intensity decay time. Findings of the research show that when luminescent photons are repeatedly scattered (scattering means the process of absorption and re-emission of a photon by an atom) in a medium, the luminescence decay time noticeably increases, reaching a value greater than the average lifetime of an individual atom in an excited state. The reflection of photons from the walls that make up the chamber can lead to errors in measuring the lifetime. The process of luminescence decay is studied theoretically for various detunings of the laser frequency from the resonant transition frequency. A three-level model of the atom and a model of complete frequency redistribution were applied. The study describes an algorithm based on the Monte Carlo method, which is used to model the three-level population kinetics, laser radiation transfer, radiation trapping, and frequency redistribution

The Effects of Three-dimensional Radiative Transfer on the Resonance Polarization of the Ca i 4227 Å Line

Abstract The sizable linear polarization signals produced by the scattering of anisotropic radiation in the core of the Ca i 4227 Å line constitute an important observable for probing the inhomogeneous and dynamic plasma of the lower solar chromosphere. Here we show the results of a three-dimensional (3D) radiative transfer complete frequency redistribution investigation of the line’s scattering polarization in a magnetohydrodynamical 3D model of the solar atmosphere. We take into account not only the Hanle effect produced by the model’s magnetic field but also the symmetry breaking caused by the horizontal inhomogeneities and macroscopic velocity gradients. The spatial gradients of the horizontal components of the macroscopic velocities produce very significant forward scattering polarization signals without the need of magnetic fields, while the Hanle effect tends to depolarize them at the locations where the model’s magnetic field is stronger than about 5 G. The standard 1.5D approximation is found to be unsuitable for understanding the line’s scattering polarization, but we introduce a novel improvement to this approximation that produces results in qualitative agreement with the full 3D results. The instrumental degradation of the calculated polarization signals is also investigated, showing what we can expect to observe with the Visible Spectro-Polarimeter at the upcoming Daniel K. Inouye Solar Telescope.

Modeling Mg ii during Solar Flares. II. Nonequilibrium Effects

Abstract To extract the information that the Mg ii NUV spectra (observed by the Interface Region Imaging Spectrograph) carry about the chromosphere during solar flares, and to validate models of energy transport via model–data comparison, forward modeling is required. The assumption of statistical equilibrium (SE) is typically used to obtain the atomic level populations from snapshots of flare atmospheres, due to computational necessity. However, it is possible that relying on SE could lead to spurious results. We compare solving the atomic level populations via SE versus a nonequilibrium (NEQ) time-dependent approach. This was achieved using flare simulations from RADYN alongside the minority species version MS_RADYN from which the time-dependent Mg ii atomic level populations and radiation transfer were computed in complete frequency redistribution. The impacts on the emergent profiles, lightcurves, line ratios, and formation heights are discussed. In summary we note that NEQ effects during flares are typically important only in the initial stages and for a short period following the cessation of the energy injection. An analysis of the timescales of ionization equilibrium reveals that for most of the duration of the flare, when the temperatures and densities are sufficiently enhanced, the relaxation timescales are short (τ relax < 0.1 s), so that the equilibrium solution is an adequate approximation. These effects vary with the size of the flare, however. In weaker flares, effects can be more pronounced. We recommend that NEQ effects be considered when possible but that SE is sufficient at most stages of the flare.

Semi-empirical model atmospheres for the chromosphere of the sunspot penumbra and umbral flashes

Context. The solar chromosphere and the lower transition region are believed to play a crucial role in the heating of the solar corona. Models that describe the chromosphere (and the lower transition region), accounting for its highly dynamic and structured character are, so far, found to be lacking. This is partly due to the breakdown of complete frequency redistribution (CRD) in the chromospheric layers and also because of the difficulty in obtaining complete sets of observations that adequately constrain the solar atmosphere at all relevant heights. Aims. We aim to obtain semi-empirical model atmospheres that reproduce the features of the Mg II h&k line profiles that sample the middle chromosphere with focus on a sunspot. Methods. We used spectropolarimetric observations of the Ca II 8542 Å spectra obtained with the Swedish 1 m Solar Telescope and used NICOLE inversions to obtain semi-empirical model atmospheres for different features in and around a sunspot. These were used to synthesize Mg II h&k spectra using the RH1.5D code, which we compared with observations taken with the Interface Region Imaging Spectrograph (IRIS). Results. Comparison of the synthetic profiles with IRIS observations reveals that there are several areas, especially in the penumbra of the sunspot, where most of the observed Mg II h&k profiles are very well reproduced. In addition, we find that supersonic hot down-flows, present in our collection of models in the umbra, lead to synthetic profiles that agree well with the IRIS Mg II h&k profiles, with the exception of the line core. Conclusions. We put forward and make available four semi-empirical model atmospheres. Two for the penumbra, reflecting the range of temperatures obtained for the chromosphere, one for umbral flashes, and a model representative of the quiet surroundings of a sunspot.

Fractional Differential Model of Resonant Transfer of Excitations in Plasma

The most crucial feature of excitation transfer in low-temperature plasma forming the physical basis for operation of contemporary lighting devices is participation of resonant photons in the process. The widely used approximation of complete frequency redistribution assumes independence of the emitted photon frequency from frequency of the absorbed one. Averaged over frequency distribution in a resonance line, the natural transfer equation with the exponential kernel becomes the equation with an inverse power kernel, which in case of an infinite homogeneous medium is asymptotically equivalent to the Laplace equation of non-integer (fractional) order, considered in this article.

Doppler Frequency Redistribution upon Coherent Photon Emission by Atoms in an Optically Dense Medium

The problem of transfer of line radiation in sodium vapor under photoexcitation by laser radiation of a resonance line with wavelength λ = 589.6 nm is numerically solved. The influence of Doppler frequency shifts on photon emission coherent in the atomic reference system is taken into account in the formation of an emission line profile using the model of partial frequency redistribution. In the case of a small optical thickness of a medium τ0 ≤ 1, the Doppler frequency redistribution in the laboratory reference system yields higher intensity values in the emission line profile core in comparison with the model of complete frequency redistribution. For an optically dense medium, thermal motion of atoms leads to an increase of reabsorption in the emission line core, and intensity increases in the line wings as compared to the complete redistribution model. The dependence of the Holstein trapping factor on optical thickness is closer to the analytic dependence than the complete redistribution model. This is explained by the fact that the frequencies of emitted photons in the laboratory reference system fall into the spectral profile core, which increases the effects of resonance radiation trapping in a dense medium.

Explicit Form of the Radiative and Collisional Branching Ratios in Polarized Radiation Transport with Coherent Scattering

Abstract We consider the vector emissivity of the polarized radiation transfer in a Λ-type atomic transition, which we recently proposed to account for both complete frequency redistribution (CRD) and partial redistribution (PRD) contributions to the scattered radiation. This expression can concisely be written as ε = ( ε ( 1 ) − ε f . s . ( 2 ) ) + ε ( 2 ) , where ε ( 1 ) and ε ( 2 ) are the emissivity terms describing, respectively, one-photon and two-photon processes in a Λ-type atom, and where “f.s.” means that the corresponding term must be evaluated assuming an appropriate “flat spectrum” average of the incident radiation across the spectral line. In this follow-up study, we explicitly consider the expressions of these various terms for the case of a polarized multi-term atom to derive the algebraic forms of the branching ratios between the CRD and PRD contributions to the emissivity. In the limit of a two-term atom with non-coherent lower term, our results are shown to be in full agreement with those recently derived by Bommier.

POLARIZED LINE FORMATION WITH LOWER-LEVEL POLARIZATION AND PARTIAL FREQUENCY REDISTRIBUTION

ABSTRACT In the well-established theories of polarized line formation with partial frequency redistribution (PRD) for a two-level and two-term atom, it is generally assumed that the lower level of the scattering transition is unpolarized. However, the existence of unexplained spectral features in some lines of the Second Solar Spectrum points toward a need to relax this assumption. There exists a density matrix theory that accounts for the polarization of all the atomic levels, but it is based on the flat-spectrum approximation (corresponding to complete frequency redistribution). In the present paper we propose a numerical algorithm to solve the problem of polarized line formation in magnetized media, which includes both the effects of PRD and the lower level polarization (LLP) for a two-level atom. First we derive a collisionless redistribution matrix that includes the combined effects of the PRD and the LLP. We then solve the relevant transfer equation using a two-stage approach. For illustration purposes, we consider two case studies in the non-magnetic regime, namely, the J a = 1, J b = 0 and J a = J b = 1, where J a and J b represent the total angular momentum quantum numbers of the lower and upper states, respectively. Our studies show that the effects of LLP are significant only in the line core. This leads us to propose a simplified numerical approach to solve the concerned radiative transfer problem.

Polarization of Resonance Lines in the Case of Partially Polarized Primary Radiation Sources

The transfer of polarized radiation in a resonance line in a semi-infinite plane-parallel nonmagnetic atmosphere is examined. It is assumed that scattering takes place with complete frequency redistribution within the line and continuum absorption is taken into account. The primary radiation sources in the atmosphere are assumed to be partially polarized; here the function describing these sources is given by the product of a polynomial and an exponential of the optical depth. The Stokes vector of the emerging radiation is found using a general analytic theory of I ˆ -matrices (the I -matrix is a generalization of the scalar Chandrasekhar H-function). It is shown that the Stokes vector of the radiation emerging from an atmosphere in which primary sources of this form are distributed can be expressed in terms of the solution of a single, so-called standard problem.

Unified numerical model of collisional depolarization and broadening rates that are due to hydrogen atom collisions

Interpretation of solar polarization spectra accounting for partial or complete frequency redistribution requires data on various collisional processes. Data for depolarization and polarization transfer are needed but often missing, while data for collisional broadening are usually more readily available. Recent work by Sahal-Br\'echot and Bommier concluded that despite underlying similarities in the physics of collisional broadening and depolarization processes, relationships between them are not possible to derive purely analytically. We aim to derive accurate numerical relationships between the collisional broadening rates and the collisional depolarization and polarization transfer rates due to hydrogen atom collisions. Such relationships would enable accurate and efficient estimation of collisional data for solar applications. Using earlier results for broadening and depolarization processes based on general (i.e. not specific to a given atom), semi-classical calculations employing interaction potentials from perturbation theory, genetic programming (GP) has been used to fit the available data and generate analytical functions describing the relationships between them. The predicted relationships from the GP-based model are compared with the original data to estimate the accuracy of the method.

THE FORMATION OFIRISDIAGNOSTICS. V. A QUINTESSENTIAL MODEL ATOM OF C ii AND GENERAL FORMATION PROPERTIES OF THE C ii LINES AT 133.5 nm

The 133.5 nm lines are important observables for the NASA/SMEX mission Interface Region Imaging Spectrograph (IRIS). To make 3D non-LTE radiative transfer computationally feasible it is crucial to have a model atom with as few levels as possible while retaining the main physical processes. We here develop such a model atom and we study the general formation properties of the C II lines. We find that a nine-level model atom of C I-C III with the transitions treated assuming complete frequency redistribution (CRD) suffices to describe the 133.5 nm lines. 3D scattering effects are important for the intensity in the core of the line. The lines are formed in the optically thick regime. The core intensity is formed in layers where the temperature is about 10kK at the base of the transition region. The lines are 1.2-4 times wider than the atomic absorption profile due to the formation in the optically thick regime. The smaller opacity broadening happens for single peak intensity profiles where the chromospheric temperature is low with a steep source function increase into the transition region, the larger broadening happens when there is a temperature increase from the photosphere to the low chromosphere leading to a local source function maximum and a double peak intensity profile with a central reversal. Assuming optically thin formation with the standard coronal approximation leads to several errors: Neglecting photoionization severly underestimates the amount of C II at temperatures below 16kK, erroneously shifts the formation from 10kK to 25kK and leads to too low intensities.

Redistribution of light frequency by multiple scattering in a resonant atomic vapor

The propagation of light in a resonant atomic vapor can a priori be thought of as a multiple scattering process, in which each scattering event redistributes both the direction and the frequency of the photons. Particularly, the frequency redistribution may result in L\'evy flights of photons, directly affecting the transport properties of light in a resonant atomic vapor and turning this propagation into a superdiffusion process. Here, we report on a Monte Carlo simulation developed to study the evolution of the spectrum of the light in a resonant thermal vapor. We observe the gradual change of the spectrum and its convergence towards a regime of complete frequency redistribution as the number of scattering events increases. We also analyze the probability density function of the step length of photons between emissions and reabsorptions in the vapor, which governs the statistics of the light diffusion. We observe two different regimes in the light transport: superdiffusion when the vapor is excited near the line center and normal diffusion for excitation far from the line center. The regime of complete frequency redistribution is not reached for excitation far from resonance even after many absorption and reemission cycles due to correlations between emitted and absorbed frequencies.

Partial Redistribution Effects on Polarized Lines Formed in Moving Media in the Presence of a Weak Magnetic Field

AbstractMacroscopic velocity fields in stellar atmospheres significantly affect the shapes of the emergent Stokes profiles. The inextricable coupling between the angle and frequency variables becomes more complex in a moving medium when compared to a static medium. In this paper we consider both complete frequency redistribution (CRD) and partial frequency redistribution (PRD) in the line scattering of a two-level atom in the presence of an external weak magnetic field. For simplicity we consider empirical velocity laws to represent motion of the atmospheric layers. We present emergent Stokes profiles computed with CRD, angle-averaged PRD, and angle-dependent PRD. We show that angle-dependent PRD effects are important both in non-magnetic and magnetized scattering when vertical velocity gradients are present in the atmosphere. The results are presented for simple atmospheric models. They are expected to be of relevance to polarized line formation in slowly expanding chromospheric layers.

Accelerating NLTE radiative transfer by means of the Forth-and-Back Implicit Lambda Iteration: A two-level atom line formation in 2D Cartesian coordinates

State-of-the-art methods in multidimensional NLTE radiative transfer are based on the use of local approximate lambda operator within either Jacobi or Gauss–Seidel iterative schemes. Here we propose another approach to the solution of 2D NLTE RT problems, Forth-and-Back Implicit Lambda Iteration (FBILI), developed earlier for 1D geometry. In order to present the method and examine its convergence properties we use the well-known instance of the two-level atom line formation with complete frequency redistribution. In the formal solution of the RT equation we employ short characteristics with two-point algorithm. Using an implicit representation of the source function in the computation of the specific intensities, we compute and store the coefficients of the linear relations J=a+bS between the mean intensity J and the corresponding source function S. The use of iteration factors in the ‘local’ coefficients of these implicit relations in two ‘inward’ sweeps of 2D grid, along with the update of the source function in other two ‘outward’ sweeps leads to four times faster solution than the Jacobi’s one. Moreover, the update made in all four consecutive sweeps of the grid leads to an acceleration by a factor of 6–7 compared to the Jacobi iterative scheme.

On the formation of Mg ii h and k lines in solar prominences

Aims. With the recent launch of the IRIS mission, it has become urgent to develop the spectral diagnostics using the Mg ii resonance h and k lines. In this paper, we aim to demonstrate the behavior of these lines under various prominence conditions. Our results serve as a basis for analysis of new IRIS data and for more sophisticated prominence modeling.Methods. For this exploratory work, we use a canonical 1D prominence-slab model, which is isobaric and may have three different temperature structures: isothermal, PCTR-like (prominence-corona transition region), and consistent with the radiative equilibrium. The slabs are illuminated by a realistic incident solar radiation obtained from the UV observations. A five-level plus continuum Mg ii model atom is used to solve the full NLTE problem of the radiative transfer. We use the numerical code based on the ALI techniques and apply the partial frequency redistribution for both Mg ii resonance lines. We also use the velocity-dependent boundary conditions to study the effect of Doppler dimming in the case of moving prominences. Finally, the relaxation technique is used to compute a grid of models in radiative equilibrium.Results. We computed the Mg ii h and k line profiles that are emergent from prominence-slab models and show their dependence on kinetic temperature, gas pressure, geometrical extension, and microturbulent velocity. By increasing the line opacity, significant departures from the complete frequency redistribution take place in the line wings. Models with a PCTR temperature structure show that Mg ii becomes ionized to Mg iii in the temperature range between roughly 15 000 and 30 000 K. Doppler dimming is significant for Mg ii resonance lines. At the velocity 300 km s-1 , the line intensity decreases to about 20% of the value for static prominences. Finally, we demonstrate the role of Mg ii h and k radiation losses on the prominence energy balance. Their dominant role is at lower pressures, while the losses due to hydrogen and Ca ii dominate at higher pressures.

LINE-INTERLOCKING EFFECTS ON POLARIZATION IN SPECTRAL LINES BY RAYLEIGH AND RAMAN SCATTERING

The polarized spectrum of the Sun and stars is formed from the scattering of anisotropic radiation on atoms. Interpretation of this spectrum requires the solution of polarized line transfer in multilevel atomic systems. While sophisticated quantum theories of polarized line formation in multilevel atomic systems exist, they are limited by the approximation of complete frequency redistribution in scattering. The partial frequency redistribution (PRD) in line scattering is a necessary component in modeling the polarized spectra of strong lines. The polarized PRD line scattering theories developed so far confine themselves to a two-level or a two-term atom model. In this paper, we present a heuristic approach to the problem of polarized line formation in multilevel atoms taking into account the effects of PRD and a weak magnetic field. Starting from the unpolarized PRD multilevel atom approach of Hubeny et al., we incorporate the polarization state of the radiation field. However, the lower level polarization is neglected. Two iterative methods of solving the polarized PRD line transfer in multilevel atoms are also presented. Taking the example of a five-level Ca II atom model, we present illustrative results for an isothermal one-dimensional model atmosphere.

MODELING THE QUANTUM INTERFERENCE SIGNATURES OF THE Ba II D24554 Å LINE IN THE SECOND SOLAR SPECTRUM

Quantum interference effects play a vital role in shaping the linear polarization profiles of solar spectral lines. The Ba II D2 line at 4554 A is a prominent example, where the F-state interference effects due to the odd isotopes produce polarization profiles, which are very different from those of the even isotopes that have no F-state interference. It is therefore necessary to account for the contributions from the different isotopes to understand the observed linear polarization profiles of this line. Here we do radiative transfer modeling with partial frequency redistribution (PRD) of such observations while accounting for the interference effects and isotope composition. The Ba II D2 polarization profile is found to be strongly governed by the PRD mechanism. We show how a full PRD treatment succeeds in reproducing the observations, while complete frequency redistribution (CRD) alone fails to produce polarization profiles that have any resemblance with the observed ones. However, we also find that the line center polarization is sensitive to the temperature structure of the model atmosphere. To obtain a good fit to the line center peak of the observed Stokes Q/I profile, a small modification of the FALX model atmosphere is needed, by lowering the temperature in the line-forming layers. Because of the pronounced temperature sensitivity of the Ba II D2 line it may not be a suitable tool for Hanle magnetic-field diagnostics of the solar chromosphere, because there is currently no straightforward way to separate the temperature and magnetic-field effects from each other.