Partial Frequency Redistribution Effects Research Articles

Modeling the scattering polarization in the solar Ca I 4227Å line with angle-dependent PRD effects and bulk velocities

Context. Modeling the scattering polarization signals of strong chromospheric lines requires solving the radiative transfer problem for polarized radiation, out of local thermodynamic equilibrium, taking partial frequency redistribution (PRD) effects into account. This problem is extremely challenging from a computational standpoint and, so far, most studies have been carried out by either modeling PRD effects under the angle-average approximation or by considering academic models of the solar atmosphere. Thanks to a new solution strategy, applicable to atomic systems that allow for a linearization of the problem, accurate solutions can now be routinely obtained in realistic 1D models, taking angle-dependent (AD) PRD effects into account. Aims. This work is aimed at assessing the suitability and performance of this new approach to handling dynamic scenarios. At the same time, it aims to explore the joint impact of magnetic fields and bulk velocities on the scattering polarization profiles of strong resonance lines, accounting for AD PRD effects and considering more realistic atmospheric models than in previous investigations. Methods. By using a two-level atomic model for neutral calcium, we synthesized the intensity and polarization profiles of the Ca I 4227 Å line. Our calculations were performed in 1D atmospheric models, both semi-empirical and extracted from 3D magnetohydrodynamic simulations, including vertical bulk velocities and magnetic fields of arbitrary strength and orientation, both constant and varying with height. Results. We obtained accurate solutions after only a few iterations across all considered scenarios. Even when formulating the problem in the observer’s reference frame, the frequency and angular grids required for accurate results were easily manageable. The calculated profiles showed the expected signatures of bulk velocities: wavelength shifts, enhancement of the line-core polarization amplitude, and prominent asymmetries in the wing signals. The results obtained in atmospheric models with complex thermal, dynamic, and magnetic structures unveiled the broad diversity of features in the emergent radiation that can be expected from realistic scenarios. Conclusions. The presented results assess the suitability of the proposed solution strategy and its parallel implementation, thus supporting its generalization to the 3D case. Our applications in increasingly realistic atmospheric models showed the difficulty related to precisely establishing the individual weight of bulk velocities and magnetic fields in the shape of the emergent profiles. This highlights the need to account for both these physical ingredients to perform reliable inversions of observed scattering polarization profiles.

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.

The Impact of Angle-dependent Partial Frequency Redistribution on the Scattering Polarization of the Solar Na i D Lines

The long-standing paradox of the linear polarization signal of the Na i D1 line was recently resolved by accounting for the atom’s hyperfine structure and the detailed spectral structure of the incident radiation field. That modeling relied on the simplifying angle-averaged (AA) approximation for partial frequency redistribution (PRD) in scattering, which potentially neglects important angle–frequency couplings. This work aims at evaluating the suitability of a PRD-AA modeling for the D1 and D2 lines through comparisons with general angle-dependent (AD) PRD calculations in both the absence and presence of magnetic fields. We solved the radiative transfer problem for polarized radiation in a 1D semiempirical atmospheric model with microturbulent and isotropic magnetic fields, accounting for PRD effects and comparing PRD-AA and PRD-AD modelings. The D1 and D2 lines are modeled separately as a two-level atomic system with hyperfine structure. The numerical results confirm that a spectrally structured radiation field induces linear polarization in the D1 line. However, the PRD-AA approximation greatly impacts the Q/I shape, producing an antisymmetric pattern instead of the more symmetric PRD-AD one while presenting a similar sensitivity to magnetic fields between 10 and 200 G. Under the PRD-AA approximation, the Q/I profile of the D2 line presents an artificial dip in its core, which is not found for the PRD-AD case. We conclude that accounting for PRD-AD effects is essential to suitably model the scattering polarization of the Na i D lines. These results bring us closer to exploiting the full diagnostic potential of these lines for the elusive chromospheric magnetic fields.

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.

The polarization signals of the solar K I D lines and their magnetic sensitivity

Aims. This work aims to identify the relevant physical processes in shaping the intensity and polarization patterns of the solar K I D lines through spectral syntheses, placing particular emphasis on the D2 line. Methods. The theoretical Stokes profiles were obtained by numerically solving the radiative transfer problem for polarized radiation considering one-dimensional semi-empirical models of the solar atmosphere. The calculations account for scattering polarization, partial frequency redistribution (PRD) effects, hyperfine structure (HFS), J- and F-state interference, multiple isotopes, and magnetic fields of arbitrary strength and orientation. Results. The intensity and circular polarization profiles of both D lines can be suitably modeled while neglecting both J-state interference and HFS. The magnetograph formula can be applied to both lines, without including HFS, to estimate weak longitudinal magnetic fields in the lower chromosphere. By contrast, modeling the scattering polarization signal of the D lines requires the inclusion of HFS. The amplitude of the D2 scattering polarization signal is strongly depolarized by HFS, but it remains measurable. A small yet appreciable error is incurred in the scattering polarization profile if PRD effects are not taken into account. Collisions during scattering processes have a clear depolarizing effect, although a quantitative analysis is left for a forthcoming publication. Finally, the D2 scattering polarization signal is particularly sensitive to magnetic fields with strengths around 10 G and it strongly depends on their orientation. Despite this, its center-to-limb variation relative to the amplitude at the limb is largely insensitive to the field strength and orientation. Conclusions. These findings highlight the value of the K I D2 line polarization for diagnostics of the solar magnetism, and show that the linear and circular polarization signals of this line are primarily sensitive to magnetic fields in the lower chromosphere and upper photosphere, respectively.

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.

Modeling the scattering polarization of the solar Ca I 4227 Å line with angle-dependent partial frequency redistribution

Context. The correct modeling of the scattering polarization signals observed in several strong resonance lines requires taking partial frequency redistribution (PRD) phenomena into account. Modeling scattering polarization with PRD effects is very computationally demanding and the simplifying angle-averaged (AA) approximation is therefore commonly applied. Aims. This work aims to assess the impact and the range of validity of the AA approximation with respect to the general angle-dependent (AD) treatment of PRD effects in the modeling of scattering polarization in strong resonance lines, with a focus on the solar Ca I 4227 Å line. Methods. Spectral line polarization was modeled by solving the radiative transfer problem for polarized radiation, under nonlocal thermodynamic equilibrium conditions, taking PRD effects into account in static one-dimensional semi-empirical atmospheric models presenting arbitrary magnetic fields. The problem was solved through a two-step approach. In step 1, the problem was solved for the intensity only, considering a multilevel atom. In step 2, the problem was solved including polarization, considering a two-level atom with an unpolarized and infinitely sharp lower level, and fixing the lower level population calculated at step 1. Results. The results for the Ca I 4227 Å line show a good agreement between the AA and AD calculations for the Q/I and U/I wings’ signals. However, AA calculations reveal an artificial trough in the line-core peak of the linear polarization profiles, whereas AD calculations show a sharper peak in agreement with the observations. Conclusions. An AD treatment of PRD effects is essential to correctly model the line-core peak of the scattering polarization signal of the Ca I 4227 Å line. By contrast, in the considered static case, the AA approximation seems to be suitable to model the wing scattering polarization lobes and their magnetic sensitivity through magneto-optical effects.

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.

Polarized Line Formation in Arbitrary Strength Magnetic Fields Angle-averaged and Angle-dependent Partial Frequency Redistribution

Abstract Magnetic fields in the solar atmosphere leave their fingerprints in the polarized spectrum of the Sun via the Hanle and Zeeman effects. While the Hanle and Zeeman effects dominate, respectively, in the weak and strong field regimes, both these effects jointly operate in the intermediate field strength regime. Therefore, it is necessary to solve the polarized line transfer equation, including the combined influence of Hanle and Zeeman effects. Furthermore, it is required to take into account the effects of partial frequency redistribution (PRD) in scattering when dealing with strong chromospheric lines with broad damping wings. In this paper, we present a numerical method to solve the problem of polarized PRD line formation in magnetic fields of arbitrary strength and orientation. This numerical method is based on the concept of operator perturbation. For our studies, we consider a two-level atom model without hyperfine structure and lower-level polarization. We compare the PRD idealization of angle-averaged Hanle–Zeeman redistribution matrices with the full treatment of angle-dependent PRD, to indicate when the idealized treatment is inadequate and what kind of polarization effects are specific to angle-dependent PRD. Because the angle-dependent treatment is presently computationally prohibitive when applied to realistic model atmospheres, we present the computed emergent Stokes profiles for a range of magnetic fields, with the assumption of an isothermal one-dimensional medium.

The Transfer of Resonance Line Polarization with Partial Frequency Redistribution in the General Hanle–Zeeman Regime

Abstract The spectral line polarization encodes a wealth of information about the thermal and magnetic properties of the solar atmosphere. Modeling the Stokes profiles of strong resonance lines is, however, a complex problem both from a theoretical and computational point of view, especially when partial frequency redistribution (PRD) effects need to be taken into account. In this work, we consider a two-level atom in the presence of magnetic fields of arbitrary intensity (Hanle–Zeeman regime) and orientation, both deterministic and micro-structured. Working within the framework of a rigorous PRD theoretical approach, we have developed a numerical code that solves the full non-LTE radiative transfer problem for polarized radiation, in one-dimensional models of the solar atmosphere, accounting for the combined action of the Hanle and Zeeman effects, as well as for PRD phenomena. After briefly discussing the relevant equations, we describe the iterative method of solution of the problem and the numerical tools that we have developed and implemented. We finally present some illustrative applications to two resonance lines that form at different heights in the solar atmosphere, and provide a detailed physical interpretation of the calculated Stokes profiles. We find that magneto-optical effects have a strong impact on the linear polarization signals that PRD effects produce in the wings of strong resonance lines. We also show that the weak-field approximation has to be used with caution when PRD effects are considered.

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.

Paschen-Back effect involving atomic fine and hyperfine structure states

AbstractThe linear polarization in spectral lines produced by coherent scattering is significantly modified by the quantum interference between the atomic states in the presence of a magnetic field. When magnetic fields produce a splitting which is of the order of or greater than the fine or hyperfine structure splittings, we enter the Paschen-Back effect (PBE) regime, in which the magnetic field dependence of the Zeeman splittings and transition amplitudes becomes non-linear. In general, PBE occurs for sufficiently strong fields when the fine structure states are involved and for weak fields in the case of hyperfine structure states. In this work, we apply the recently developed theory of PBE in the atomic fine and hyperfine structure states including the effects of partial frequency redistribution to the case of Li i 6708 Å doublet. We explore the signatures of PBE in a single scattering event and their applicability to the solar magnetic field diagnostics.

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.

THE QUANTUM INTERFERENCE EFFECTS IN THE SC II 4247 Å LINE OF THE SECOND SOLAR SPECTRUM

The Sc II 4247 A line formed in the chromosphere is one of the lines well known, like the Na I D_2 and Ba II D_2, for its prominent triple peak structure in Q/I and the underlying quantum interference effects governing it. In this paper, we try to study the nature of this triple peak structure using the theory of F-state interference including the effects of partial frequency redistribution (PRD) and radiative transfer (RT). We compare our results with the observations taken in a quiet region near the solar limb. In spite of accounting for PRD and RT effects it has not been possible to reproduce the observed triple peak structure in Q/I. While the two wing PRD peaks (on either side of central peak) and the near wing continuum can be reproduced, the central peak is completely suppressed by the enhanced depolarization resulting from the hyperfine structure splitting. This suppression remains for all the tested widely different 1D model atmospheres or for any multi-component combinations of them. While multidimensional radiative transfer effects may improve the fit to the intensity profiles, they do not appear capable of explaining the enigmatic central Q/I peak. This leads us to suspect that some aspect of quantum physics is missing.

POLARIZED LIGHT SCATTERING WITH THE PASCHEN-BACK EFFECT, LEVEL-CROSSING OF FINE STRUCTURE STATES, AND PARTIAL FREQUENCY REDISTRIBUTION

The quantum interference between the fine structure states of an atom modifies the shapes of the emergent Stokes profiles in the Second Solar Spectrum. This phenomenon has been studied in great detail both in the presence and absence of magnetic fields. By assuming a flat-spectrum for the incident radiation, the signatures of this effect have been explored for arbitrary field strengths. Even though the theory which takes into account the frequency dependence of the incident radiation is well developed, it is restricted to the regime in which the magnetic splitting is much smaller than the fine structure splitting. In the present paper, we carry out a generalization of our scattering matrix formalism including the effects of partial frequency redistribution (PRD) for arbitrary magnetic fields. We test the formalism using available benchmarks for special cases. In particular we apply it to the Li\,{\sc i} 6708\,\AA\ D$_1$ and D$_2$ line system, for which observable effects from the Paschen-Back regime are expected in the Sun's spectrum.

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.

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.

Quantum interference with angle-dependent partial frequency redistribution: solution of the polarized line transfer in the non-magnetic case

Angle-dependent partial frequency redistribution (PRD) matrices represent the physical redistribution in the process of light scattering on atoms. For the purpose of numerical simplicity, it is a common practice in astrophysical literature to use the angle-averaged versions of these matrices, in the line transfer computations. The aim of this paper is to study the combined effects of angle-dependent PRD and the quantum interference phenomena arising either between the fine structure (J) states of a two-term atom or between the hyperfine structure (F) states of a two-level atom. We restrict our attention to the case of non-magnetic and collisionless line scattering on atoms. A rapid method of solution based on Neumann series expansion is developed to solve the angle-dependent PRD problem including quantum interference in an atomic system. We discuss the differences that occur in the Stokes profiles when angle-dependent PRD mechanism is taken into account.

POLARIZED PARTIAL FREQUENCY REDISTRIBUTION IN SUBORDINATE LINES. II. SOLUTION OF THE TRANSFER EQUATION WITH RAYLEIGH SCATTERING

It is quite common in line formation theory to treat scattering in subordinate lines under the assumption of complete frequency redistribution (CRD). The partial frequency redistribution (PRD) in subordinate lines cannot always be approximated by CRD, especially when the polarization state of the line radiation is taken into account. Here we investigate the PRD effects in subordinate lines including scattering polarization. The line formation is described by a polarized non-LTE line transfer equation based on a two-level atom model. We use the recently derived subordinate line redistribution matrix. We devise polarized approximate lambda iteration methods to solve the concerned transfer problem. The linear polarization profiles of subordinate lines formed in non-magnetic (Rayleigh) scattering atmospheres are discussed. We consider one-dimensional isothermal planar model atmospheres. We show that in the polarized line transfer calculations of subordinate lines, PRD plays as important of a role as it does in the case of resonance lines. We also study the effect of collisions on linear polarization profiles of subordinate lines.