Partial Frequency Redistribution Functions Research Articles

Full non-LTE spectral line formation

In this article we discuss a method for obtaining a numerical solution to the so-called full nonlocal thermodynamic equilibrium (non-LTE) radiation transfer problem. More specifically, the usual numerical iterative methods for non-LTE radiation transfer are coupled with that formalism and new numerical additions are explained in detail. We benchmarked the whole process with the standard non-LTE transfer problem for a two-level atom with Hummer’s RI − A partial frequency redistribution function. Finally, we present new quantities, such as the spatial distribution of the velocity distribution function of excited atoms, which can only be accessed by adopting a more general frame for non-LTE radiation transfer, such as the one we propose here.

Importance of Cross-redistribution in Scattering Polarization of Spectral Lines:The Cases of 3P−3S Triplets of Mg i and Ca i

Abstract Scattering on a multi-level atomic system has dominant contributions from resonance and Raman scattering. While initial and final levels are the same for resonance scattering, they are different for Raman scattering. The frequency redistribution for resonance scattering is described by the usual partial frequency redistribution functions of Hummer, while that for Raman scattering is described by cross-redistribution (XRD) function. In the present paper, we investigate the importance of XRD on linear polarization profiles of 3P−3S triplets of Mg i and Ca i formed in an isothermal one-dimensional atmosphere. We show that XRD produces significant effects on the linear polarization profiles when the wavelength separations between the line components of the multiplet are small, like in the cases of Mg i b and Ca i triplets.

An efficient decomposition technique to solve angle-dependent Hanle scattering problems

Hanle scattering is an important diagnostic tool to study weak solar magnetic fields. Partial frequency redistribution (PRD) is necessary to interpret the linear polarization observed in strong resonance lines. Usually angle-averaged PRD functions are used to analyze linear polarization. However, it is established that angle-dependent PRD functions are often necessary to interpret polarization profiles formed in the presence of weak magnetic fields. Our aim is to present an efficient decomposition technique, and the numerical method to solve the concerned angle-dependent line transfer problem. Together with the standard Stokes decomposition technique, we employ Fourier expansion over the outgoing azimuth angle to express in a more convenient form, the angle-dependent PRD function for the Hanle effect. It allows the use of angle-dependent frequency domains of Bommier to solve the Hanle transfer problem. Such an approach is self-consistent and accurate compared to a recent approach where angle-averaged frequency domains were used to solve the same problem. We show that it is necessary to incorporate angle-dependent frequency domains instead of angle-averaged frequency domains to solve the Hanle transfer problem accurately, especially for the Stokes U parameter. The importance of using angle-dependent domains has been highlighted by taking the example of Hanle effect in the case of line transfer with vertical magnetic fields in a slab atmosphere. We have also studied the case of polarized line formation when micro-turbulent magnetic fields are present. The difference between angle-averaged and angle-dependent solutions is enhanced by the presence of micro-turbulent fields.

The effect of electron scattering redistribution on atomic line polarization

The polarization of spectral lines is generated by the scattering of angularly anisotropic incident radiation field on the atoms in the stellar atmosphere. This atomic scattering polarization is modified by frequency non-coherent scattering of line photons on free electrons. With modern spectropolarimeters of high sensitivity, it is possible to detect such changes in the spectral line polarization caused by scattering on electrons. We present new and efficient numerical techniques to solve the problem of line radiative transfer with atomic and electron scattering frequency redistribution in planar media. The evaluation and use of angle-dependent partial frequency redistribution functions (both atomic and electron scattering type) in the transfer equation require a lot of computing effort. In this paper, we apply a decomposition technique to handle this numerically difficult problem. This recently developed technique is applied for the first time to the electron scattering partial redistribution. This decomposition technique allows us to devise fast iterative methods of solving the polarized line transfer equation. An approximate lambda iteration (ALI) method and a method based on Neumann series expansion of the polarized source vector are proposed. We show that these numerical methods can be used to obtain a solution of the problem, when both atomic and electron scattering partial frequency redistribution are considered together. This is in contrast with the classical numerical methods which require a great amount of computing time. We show the importance of electron scattering redistribution in the far wing line polarization, which has practical implications in the analysis of polarized stellar or solar spectra, where non-coherent electron scattering controls the line wing transfer.

Spectral line polarization with angle-dependent partial frequency redistribution

Context. The partial frequency redistribution (PRD) effects in line scattering are necessary ingredients for interpreting the linear polarization observed in strong resonance lines. It is a common practice to use angle-averaged PRD functions for simplicity (obtained by averaging over all scattering angles). It has been established that the use of angle-dependent PRD functions instead of angleaveraged functions is essential for weak fields. Aims. Here we present an efficient iterative method to solve the polarized line radiative transfer equation in weak magnetic fields using angle-dependent PRD functions. Methods. Based on the theory of Stokes vector decomposition for the Hanle effect combined with the Fourier azimuthal expansion of the angle-dependent PRD function, we try to formulate an efficient numerical method of solving the concerned transfer problem in one-dimensional media. This iterative method (referred to as the scattering expansion method, SEM) is based on a series expansion of the polarized source vector in mean number of scatterings (Neumann series expansion). We apply the SEM approach to handle both the exact and various approximate forms of the Hanle scattering redistribution matrix. Results. The SEM is shown to be an efficient method to solve angle-dependent PRD problems involving the Hanle effect. We show that compared to the earlier methods such as the perturbation methods, the SEM is stable and faster. We find that angle-dependent PRD significantly affects the Stokes U parameter.

POLARIZED LINE FORMATION IN MULTI-DIMENSIONAL MEDIA. IV. A FOURIER DECOMPOSITION TECHNIQUE TO FORMULATE THE TRANSFER EQUATION WITH ANGLE-DEPENDENT PARTIAL FREQUENCY REDISTRIBUTION

To explain the linear polarization observed in spatially resolved structures in the solar atmosphere, the solution of polarized radiative transfer (RT) equation in multi-dimensional (multi-D) geometries is essential. For strong resonance lines partial frequency redistribution (PRD) effects also become important. In a series of papers we have been investigating the nature of Stokes profiles formed in multi-D media including PRD in line scattering. For numerical simplicity so far we restricted our attention to the particular case of PRD functions which are averaged over all the incident and scattered directions. In this paper we formulate the polarized RT equation in multi-D media that takes into account Hanle effect with angle-dependent PRD functions. We generalize here to the multi-D case, the method of Fourier series expansion of angle-dependent PRD functions originally developed for RT in 1D geometry. We show that the Stokes source vector $\bm{S}=(S_I,S_Q,S_U)^T$ and the Stokes vector $\bm{I}=(I,Q,U)^T$ can be expanded in terms of infinite sets of components ${\tilde{\bm{\mathcal{S}}}}^{(k)}$, ${\tilde{\bm{\mathcal{I}}}}^{(k)}$ respectively, $k\in[0,+\infty)$. We show that the components ${\tilde{\bm{\mathcal{S}}}}^{(k)}$ become independent of the azimuthal angle ($\varphi$) of the scattered ray, whereas the components ${\tilde{\bm{\mathcal{I}}}}^{(k)}$ remain dependent on $\varphi$ due to the nature of RT in multi-D geometry. We also establish that ${\tilde{\bm{\mathcal{S}}}}^{(k)}$ and ${\tilde{\bm{\mathcal{I}}}}^{(k)}$ satisfy a simple transfer equation, which can be solved by any iterative method like an Approximate Lambda Iteration (ALI) or a Biconjugate-Gradient type projection method provided we truncate the Fourier series to have a finite number of terms.

Spectral line polarization with angle-dependent partial frequency redistribution

Context. The solar limb observations in spectral lines display evidence of linear polarization, caused by non-magnetic resonance scattering process. This polarization is modified by weak magnetic fields – the process of the Hanle effect. These two processes serve as diagnostic tools for weak solar magnetic field determination. In modeling the polarimetric observations the partial frequency redistribution (PRD) effects in line scattering have to be accounted for. For simplicity, it is common practice to use PRD functions averaged over all scattering angles. For weak fields, it has been established that the use of angle-dependent PRD functions instead of angle-averaged functions is essential.

Spectral line polarization with angle-dependent partial frequency redistribution

Context. The linear polarization of strong resonance lines observed in the solar spectrum is created by the scattering of the photospheric radiation field. This polarization is sensitive to the form of the partial frequency redistribution (PRD) function used in the line radiative transfer equation. Observations have been analyzed until now with angle-averaged PRD functions. With an increase in the polarimetric sensitivity and resolving power of the present-day telescopes, it will become possible to detect finer effects caused by the angle dependence of the PRD functions. Aims. We devise new efficient numerical methods to solve the polarized line transfer equation with angle-dependent PRD, in planeparallel cylindrically symmetrical media. We try to bring out the essential differences between the polarized spectra formed under angle-averaged and the more realistic case of angle-dependent PRD functions. Methods. We use a recently developed Stokes vector decomposition technique to formulate three different iterative methods tailored for angle-dependent PRD functions. Two of them are of the accelerated lambda iteration type, one is based on the core-wing approach, and the other one on the frequency by frequency approach suitably generalized to handle angle-dependent PRD. The third one is based on a series expansion in the mean number of scattering events (Neumann series expansion). Results. We show that all these methods work well on this difficult problem of polarized line formation with angle-dependent PRD. We present several benchmark solutions with isothermal atmospheres to show the performance of the three numerical methods and to analyze the role of the angle-dependent PRD effects. For weak lines, we find no significant effects when the angle-dependence of the PRD functions is taken into account. For strong lines, we find a significant decrease in the polarization, the largest effect occurring in the near wing maxima.

Spectral line polarization with angle-dependent partial frequency redistribution

Context. The linear polarization of a strong resonance lines observed near the solar limb is created by a multiple-scattering process. Partial frequency redistribution (PRD) effects must be accounted for to explain the polarization profiles. The redistribution matrix describing the scattering process is a sum of terms, each containing a PRD function multiplied by a Rayleigh type phase matrix. A standard approximation made in calculating the polarization is to average the PRD functions over all the scattering angles, because the numerical work needed to take the angle-dependence of the PRD functions into account is large and not always needed for reasonable evaluations of the polarization. Aims. This paper describes a Stokes parameters decomposition method, that is applicable in plane-parallel cylindrically symmetrical media, which aims at simplifying the numerical work needed to overcome the angle-average approximation. Methods. The decomposition method relies on an azimuthal Fourier expansion of the PRD functions associated to a decomposition of the phase matrices in terms of the Landi Degl’Innocenti irreducible spherical tensors for polarimetry T K Q (i, Ω )( i Stokes parameter index, Ω ray direction). The terms that depend on the azimuth of the scattering angle are retained in the phase matrices. Results. It is shown that the Stokes parameters I and Q, which have the same cylindrical symmetry as the medium, can be expressed in terms of four cylindrically symmetrical components I K (K = Q = 0, K = 2, Q = 0, 1, 2). The components with Q = 1, 2a re created by the angular dependence of the PRD functions. They go to zero at disk center, ensuring that Stokes Q also goes to zero. Each component I K is a solution to a standard radiative transfer equation. The source term S K are significantly simpler than the source terms corresponding to I and Q. They satisfy a set of integral equations that can be solved by an accelerated lambda iteration (ALI) method.

Theoretical study of partial frequency redistribution function in irradiated, moving atmospheres of close binary components

We have studied the effects of partial frequency redistribution function with angle-averaged R II-A in irradiated and moving atmospheres of close binary components. We have considered the atmospheric extension of the primary component to be twice the radius of the primary component in a close binary system. We have considered two cases: (i) when the atmosphere is at rest and (ii) when the atmosphere is moving. In both the cases, we have computed the line profiles along the line of sight for a given optical depth. The irradiation from the secondary component is assumed to be one, five and 10 times the self-radiation. The line fluxes in the line of sight are calculated by using the total source functions due to self-radiation of the primary component and due to the irradiation from the secondary component. We have noted double-peaked emission lines in the case of a static medium and a reduction of emission peaks in the case of velocity field.

Experimental and theoretical study of the imprisonment of the [formula omitted] 123.58 nm resonance radiation

The apparent decay of the Kr 3P 1 level, populated by means of a 3-photon excitation process, is studied in the [0.006, 10] Torr pressure range for two different geometry's of the imprisonment cell. The reproducibility of the experimentally obtained fundamental mode radiative decay rates β is better than 3%. For the theoretical description of the imprisonment process effects of partial frequency redistribution must be accounted for. It is found that the analytical expression of Post for the fundamental mode radiative decay rate underestimates β in the pressure region around 0.03 Torr, due to the Jefferies–White approximation, which is used in this formula. Decay rates, deduced from Monte Carlo simulations of the imprisonment process, in which no approximations of the partial frequency redistribution functions are made, show excellent agreement with the experimental results for pressures p<0.6 Torr. A natural lifetime τ n =3.5 ns is obtained for the 3P 1 resonance state in Kr. The value of the FWHM of the pressure broadened line profile in the impact approximation is derived and compared with previous results in literature. For p≥0.6 Torr the present theory, based on the impact approximation, predicts a nearly constant β. The good accuracy of our results allows us to observe a distinct decrease of β with increasing pressure in the [0.6, 4] Torr range. In the corresponding optical depth range the impact approximation is invalid and the duration of collision effects has to be taken into account. A generalised form of the formula of Post, valid regardless the duration of the collisions, is proposed. This formula yields an accurate description of the observed decrease of the decay rate β.

Effects of redistribution with dipole scattering on line source functions

The partial frequency redistribution function for zero natural line width with dipole scattering (RI) has been considered in obtaining the simultaneous solution of the statistical equilibrium and line transfer equations in the comoving frame of the expanding gas. We have considered a non-LTE two level atom in an expanding spherical medium whose outer radii are 3, 10 and 20 times the stellar radius with a total optical depthT ≃ 2 × 103. In all the cases, we have calculated the population ratio of the two levels N2/N1 and compared these results with those obtained by using different expansion velocities and geometrical extensions. Initially, the upper level population (N2) is set equal to zero. The converged simultaneous solution shows that the upper level population is enhanced considerably from the initial value. Variation in velocity gradients seem to have little effect on the ratio N2/N1 when the geometrical thickness of the medium is 3 or 10 times the stellar radius. However, when the thickness is increased to 20 times the central radius, the velocity gradients change the ratio N2/N1 considerably in the region where log T ≤ 2. The effect of variation of geometrical thickness is to reduce the N2/N1 ratio atτ = 0.

Comoving frame calculations of spectral lines formed in rapidly expanding media with the partial frequency redistribution function for zero natural line width

Comoving frame calculations have been used to compute the spectral lines formed in rapidly expanding spherical media. We have employed the angle-averaged partial frequency redistribution functionR I with a two-level atom model in non-LTE atom approximation. A linear velocity law increasing with radius has been employed with maximum velocity at Τ=0 being set equal to 30 mean thermal units. It is found that one obtains almost symmetric emission line profiles at large velocities similar to those found in quasars.

Monte Carlo Simulation of Emission Frequencies from Partial Frequency Redistribution Functions

Abstract : Simple and efficient algorithms are presented for randomly selecting photon emission frequencies from a class of partial frequency redistribution functions. These formulas are developed by applying some basic rules of probability theory, in particular, Bayes' rule. Theoretical proofs of these formulas are given, and numerical values of the generated frequency distributions are presented to validate the algorithms. These algorithms are intended for Monte Carlo studies of photon diffusion problems. (Author)