Angle-dependent Partial Frequency Redistribution 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.

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.

Polarized resonance line transfer in a spherically symmetric medium with angle-dependent partial frequency redistribution

ABSTRACT In a stellar atmosphere, the resonance line polarization arises from scattering of limb-darkened radiation field by atoms. This spectral line polarization gets affected particularly in the wings, when the line photons suffer scattering on electrons in thermal motion. Scattering of line photons by atoms and electrons are, respectively, described by the atomic and Thomson electron scattering redistribution functions, which in general depend on both the frequencies and directions of incident and scattered photons. In this paper, we consider the polarized spectral line formation in spherically symmetric extended and expanding media accounting for the angle-dependent partial frequency redistribution (AD-PRD) in scattering on both atoms and electrons. We solve this computationally demanding polarized transfer problem using an accelerated lambda iteration method and a method based on orders of scattering approach. In the case of expanding spherical medium, the concerned transfer problem is solved in the comoving frame. Because of the computational limitations, we consider optically thin isothermal spherically symmetric media of different extensions for the static case as well as when the velocity fields are present. For the considered model, we show that the AD-PRD effects on the linear polarization profiles are significant and have to be accounted for.

Hanle Effect with Angle-dependent Partial Frequency Redistribution in Three-dimensional Media

There has been a constant improvement in the observational measurement of linear polarization in chromospheric spectral lines in the last three decades. However, modeling polarized profiles of these lines still remains incomplete, due to the lack of inclusion of fundamental physics in modeling efforts. To model the observed solar spectrum and its linear polarization, we need a solution to the polarized radiative transfer equation. The polarization in strong resonance lines originates from the scattering mechanism known as the partial frequency redistribution (PFR) of an anisotropic radiation field. The understanding of the linear polarization in spatially resolved structures needs radiative transfer solutions in multidimensional geometries. In this paper, we explore the effects of angle-dependent PFR on scattering polarization profiles formed in three-dimensional (3D) media. We find that the 3D geometry combined with angle-dependent PFR produces more scattering polarization than an angle-averaged one.

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.

Importance of Angle-dependent Partial Frequency Redistribution in Hyperfine Structure Transitions Under the Incomplete Paschen–Back Effect Regime

Abstract Angle-frequency coupling in scattering of polarized light on atoms is represented by the angle-dependent (AD) partial frequency redistribution (PRD) matrices. There are several lines in the linearly polarized solar spectrum, for which PRD combined with quantum interference between hyperfine structure states play a significant role. Here we present the solution of the polarized line transfer equation including the AD-PRD matrix for scattering on a two-level atom with hyperfine structure splitting and an unpolarized lower level. We account for the effects of arbitrary magnetic fields (including the incomplete Paschen–Back effect regime) and elastic collisions. For exploratory purposes we consider a self-emitting isothermal planar atmosphere and use atomic parameters that represent an isolated Na i D2 line. For this case we show that the AD-PRD effects are significant for field strengths below about 30 G, but that the computationally much less demanding approximation of angle-averaged PRD may be used for stronger fields.

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.

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.

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.

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.

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.

POLARIZED LINE FORMATION IN MULTI-DIMENSIONAL MEDIA. V. EFFECTS OF ANGLE-DEPENDENT PARTIAL FREQUENCY REDISTRIBUTION

The solution of polarized radiative transfer equation with angle-dependent (AD) partial frequency redistribution (PRD) is a challenging problem. Modeling the observed, linearly polarized strong resonance lines in the solar spectrum often requires the solution of the AD line transfer problems in one-dimensional (1D) or multi-dimensional (multi-D) geometries. The purpose of this paper is to develop an understanding of the relative importance of the AD PRD effects and the multi-D transfer effects and particularly their combined influence on the line polarization. This would help in a quantitative analysis of the second solar spectrum (the linearly polarized spectrum of the Sun). We consider both non-magnetic and magnetic media. In this paper we reduce the Stokes vector transfer equation to a simpler form using a Fourier decomposition technique for multi-D media. A fast numerical method is also devised to solve the concerned multi-D transfer problem. The numerical results are presented for a two-dimensional medium with a moderate optical thickness (effectively thin), and are computed for a collisionless frequency redistribution. We show that the AD PRD effects are significant, and can not be ignored in a quantitative fine analysis of the line polarization. These effects are accentuated by the finite dimensionality of the medium (multi-D transfer). The presence of magnetic fields (Hanle effect) modifies the impact of these two effects to a considerable extent.

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.

Effects of the heliospheric interface on the interplanetary Lyman α glow seen at 1 AU from the Sun

This work presents multiple scattering computations of interplanetary background line profiles using Angle Dependent Partial Frequency Redistribution to model the scattering process as explained by Quémerais ([CITE]). The density and velocity distribution of hydrogen atoms in the heliosphere was computed using a self-consistent model of the solar wind interaction with the two-component (H atoms and plasma) interstellar plasma (Izmodenov et al. [CITE]). We show that the main difference of the heliospheric interface model as compared to the classical hot model lies in the existence of three populations of H atoms in the heliosphere. These populations are primary interstellar atoms, secondary interstellar atoms created by charge exchange in the outer heliosheath and a hot component (around 150 000 K) created in the inner heliosheath. The two interstellar components provide additional asymmetry in the backscattered Lyα profile at 1 AU, which is not expected from the classical hot model. At the Earth orbit in the upwind direction, the hot component of the backscattered intensity represents less than 5% of the total intensity. It reaches 15% in the downwind direction. Although the heliospheric interface has some effect on the upwind to downwind ratio of total backscattered Lyα intensity, on the apparent velocity and apparent temperature of interplanetary line profiles, they are difficult to discriminate from the results of the variations of the solar parameters. Therefore our following models should include time-dependence effects on the interplanetary hydrogen distribution. Finally, it is concluded that measurements of interplanetary Lyα profiles in the inner heliosphere may serve as a good diagnostic of the heliospheric interface. The existence of a faint hot component in the Lyman α backscattered profiles would serve as a new tool to study the heliospheric interface.

Hanle effect with angle-dependent partial redistribution

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