Regular Magnetic Field Research Articles

Cosmic Ray Anisotropy and Spectra as Probes for Nearby Sources

Cosmic ray (CR) spectra and anisotropy are closely related to the distribution of CR sources, making them valuable probes for studying nearby sources. There are 12 nearby sources located within 1 kpc of the solar system, and which ones are the optimal candidates? In this work, we have selected the Geminga, Monogem, Vela, Loop I, and Cygnus SNR sources as the focus of our research, aiming to identify the optimal candidate by investigating their contribution to the energy spectra and anisotropy using the Spatially Dependent Propagation (SDP) model. Additionally, the anisotropic diffusion effect of the local regular magnetic field (LRMF) on CR particles is also considered in the SDP model. Our previous work only provided 1D anisotropy along the right ascension; this current work will further present 2D anisotropy maps along the right ascension and declination. When the injection power of different nearby sources is roughly equal, the results show that the Geminga, Momogem, and Loop I SNR sources contribute significantly to the nuclear energy spectra. Under the isotropic diffusion without considering the LRMF, the 2D anisotropy maps indicate that the phase points to the nearby source below 100 TeV. We further adjust the injection power of the Monogem SNR source in accordance with the spin-down energy of the Geminga and Monogem pulsars, and find that the contribution of the corrected Monogem SNR can be disregarded. Because the Loop I SNR source is located in the direction of the Galactic Center (GC), it cannot contribute to the excess of CRs in the anti-GC direction. Under anisotorpic diffusion with the consideration of the LRMF, the 2D anisotropy maps show that only the Geminga SNR can match the anisotropy measurement, while the other sources cannot. Finally, we conclude that the Geminga SNR source is the optimal nearby source.

Large-scale Anisotropy of Galactic Cosmic Rays as a Probe of Local Cosmic-Ray Propagation

Recent studies have shown that the anisotropy is of great value to decipher cosmic rays’ origin and propagation. We have built a unified scenario to describe the observations of the energy spectra and the large-scale anisotropy and called attention to their synchronous evolution with energy. In this work, the impact of the local regular magnetic field (LRMF) and corresponding anisotropic diffusion on large-scale anisotropy have been investigated. When the perpendicular diffusion coefficient is much smaller than the parallel one, the dipole anisotropy points to the LRMF and the observational phase below 100 TeV could be reproduced. Moreover, we find that the dipole phase above 100 TeV strongly depends on the evolution of local diffusion. But the current measurements at that energy are still scarce. We suggest that more precise measurements at that energy could be carried out to unveil the local diffusion and further the local turbulence.

Disentangling Magnetic Fields in NGC 6946 with Wide-band Polarimetry

We present λ13 cm polarization observations of the nearby spiral galaxy NGC 6946 with the Westerbork Synthesis Radio Telescope (WSRT) to examine the nearside halo magnetic fields. Despite λ13 cm exhibiting similar two-dimensional morphology as observed at longer (λ18–22 cm) or shorter (λ3 and λ6 cm) wavelengths, more complete frequency coverage will be required to explain the gap in polarization in the southwest quadrant of the galaxy. We fit models of the turbulent and coherent line-of-sight magnetic fields to the fractional degree of linearly polarized emission at λ3, λ6, λ13, λ18, and λ22 cm from observations taken with the WSRT, Karl G. Jansky Very Large Array, and Effelsberg telescopes. The results favor a multilayer turbulent magneto-ionized medium consistent with current observations of edge-on galaxies. We constrain the physical properties of the synchrotron-emitting thin and thick disks (scale heights of 300 pc and 1.4 kpc, respectively) along with the thermal thick disk and halo (scale heights of 1 and 5 kpc, respectively). Our preferred model indicates a clumpy and highly turbulent medium within 1 kpc of the midplane, and a diffuse extraplanar layer with a substantially lower degree of Faraday depolarization. In the halo, we estimate a regular magnetic field strength of 0.4–2.2 μG and that turbulence and a total magnetic field strength of ∼6 μG result in a Faraday dispersion of σ RM = 4–48 rad m−2. This work is an example of how the advanced capabilities of modern radio telescopes are opening a new frontier for the study of cosmic magnetism.

Magnetic Fields in Giant Filaments Probed by the Velocity Gradient Technique: Regular Magnetic Field Interrupted by Magnetization Gaps

We study the magnetic field structures in six giant filaments associated with the spiral arms of the Milky Way by applying the velocity gradient technique (VGT) to the 13CO spectroscopic data from the GRS, FUGIN, and SEDIGSM surveys. Unlike dust-polarized emission, the VGT allows us to separate the foreground and background using the velocity information, from which the orientation of the magnetic field can be reliably determined. We find that in most cases the magnetic fields stay aligned with the filament bodies, which are parallel to the disk midplane. Among these, G29, G47, and G51 exhibit smooth magnetic fields, and G24, G339, and G349 exhibit discontinuities. The fact that most filaments have magnetic fields that stay aligned with the Galactic disk midplane suggests that Galactic shear may be responsible for shaping the filaments. The fact that the magnetic field can stay regular at the resolution of our analysis (≲10 pc), where the turbulence crossing time is short compared to the shear time, suggests that turbulent motion cannot effectively disrupt the regular orientation of the magnetic field. The discontinuities found in some filaments can be caused by processes including filament reassembly, gravitational collapse, and stellar feedback.

Large-scale turbulence cascade in the spiral galaxy NGC 6946

ABSTRACT The generation mechanism of compressible fluid turbulence at kiloparsec scales in the interstellar medium is a long-lasting puzzle. In this work, we explore the nature of large-scale turbulence in the external spiral galaxy NGC 6946. We use the visibility moment estimator to measure the H i column density and line-of-sight turbulent velocity power spectra combining the new observations of A array configuration of Karl G. Jansky Very Large Array (VLA) with the VLA B, C, D array observations from The H i Nearby Galaxy Survey. The estimated power spectra are obeying a power law with a slope of −0.96 ± 0.05 in column density and −1.81 ± 0.07 in line-of-sight velocity in length-scales ranging from 6 kpc to 170 pc. This points towards a forward energy cascade in the plane of the disc with a driving scale at least as large as 6 kpc. The values of the power-law indices indicate a combination of solenoidal and compressive force responsible for driving the measured turbulence. The presence of strong regular magnetic fields from the magnetic spiral arms in the galaxy is possibly contributing to the solenoidal part, while self-gravity or gravitational instability can mostly be the input for the compressive part of the forcing in the driving mechanism.

Climatology of quiet time geomagnetic field variations at two locations in Antarctica

During geomagnetic quiet conditions, Indian Antarctic stations are considered to be located outside the auroral oval: Maitri (CGM coordinates: 63.3°S, 54.2°E) is equatorward and Bharati (CGM coordinates: 74.8°S, 98.4°E) is poleward of the auroral oval. Simultaneous observations of magnetic field variations at these two locations for 10-years (2013–2022) provide an opportunity to study quiet-time magnetic field patterns, if any. Geomagnetic quiet days with ΣKp ≤ 3 are selected, during which the lower values of solar wind and interplanetary parameters are also confirmed. Maitri station exhibits clear southern hemispheric solar quiet (Sq) type of magnetic field variation on geomagnetic quiet days in all seasons, indicating the influence of ionospheric dynamo due to thermospheric winds. Interestingly, Bharati station also displays regular and systematic magnetic field variations in all three components. The D-component at Bharati exhibits very strong variation at early morning hours (7–8 MLT), which is ∼2–4 times stronger than that of H-component, driving strong equatorward/northward currents during all seasons. Both stations show annual type of seasonal variation with peak amplitude during summer and least during winter. The schematic illustration of global and polar Sqs proposed here explains the results obtained through 10-years’ statistical study.

Nested invariant tori foliating a vector field and its curl: Toward MHD equilibria and steady Euler flows in toroidal domains without continuous Euclidean isometries

This paper studies the problem of finding a three-dimensional solenoidal vector field such that both the vector field and its curl are tangential to a given family of toroidal surfaces. We show that this question can be translated into the problem of determining a periodic solution with periodic derivatives of a two-dimensional linear elliptic second-order partial differential equation on each toroidal surface and prove the existence of smooth solutions. Examples of smooth solutions foliated by toroidal surfaces that are not invariant under continuous Euclidean isometries are also constructed explicitly, and they are identified as equilibria of anisotropic magnetohydrodynamics. The problem examined here represents a weaker version of a fundamental mathematical problem that arises in the context of magnetohydrodynamics and fluid mechanics concerning the existence of regular equilibrium magnetic fields and steady Euler flows in bounded domains without continuous Euclidean isometries. The existence of such configurations represents a key theoretical issue for the design of the confining magnetic field in nuclear fusion reactors known as stellarators.

FAST Polarization Mapping of the Supernova Remnant VRO 42.05.01

We have obtained the polarization data cube of the supernova remnant (SNR) VRO 42.05.01 at 1240 MHz using the Five-hundred-meter Aperture Spherical Radio Telescope (FAST). Three-dimensional Faraday synthesis is applied to the FAST data to derive the Faraday depth spectrum. The peak Faraday depth map shows a large area of enhanced foreground rotation measure (RM) of ∼60 rad m−2 extending along the remnant’s “wing” section, which coincides with a large-scale H i shell at −20 km s−1. The two depolarization patches within the “wing” region with RMs of 97 rad m−2 and 55 rad m−2 coincide with two H i structures in the H i shell. Faraday screen model fitting on the Canadian Galactic Plane Survey 1420 MHz full-scale polarization data reveals a distance of (0.7–0.8)d SNR in front of the SNR with enhanced regular magnetic field there. The highly piled-up magnetic field indicates that the H i shell at −20 km s−1 could originate from an old evolved SNR.

Peculiarities of medium parameter dynamics and cosmic ray density in strong Forbush decreases associated with magnetic clouds

Diffusion and electromagnetic mechanisms determine the formation of sporadic Forbush decreases. The diffusion mechanism affects the Forbush decrease amplitude in the turbulent layer, and the part of the coronal mass ejection preceding the magnetic cloud, and its efficiency depends on the level of magnetic field turbulence. The electromagnetic mechanism works in a magnetic cloud, and its efficiency depends on the intensity of regular magnetic and electric fields. We analyze solar wind parameters and cosmic ray density, using the superposed epoch analysis. In 1996–2006, 23 strong Forbush decreases (amplitude >5 %) were detected. The average amplitude of 7 % is equally formed by both mechanisms. The events can be divided into 2 groups depending on the contribution of the mechanisms to Forbush decrease amplitude. Group 1 includes the strongest Forbush decreases (amplitude=8.5 %), formed by both diffusion and electromagnetic mechanisms. The diffusion mechanism forms 0.26 amplitude, and the electromagnetic mechanism is responsible for 0.74 one. In group 2, the averege amplitude Forbush decrease =5.7 %, the diffusion mechanism forms 0.79 of amplitude; and the electromagnetic one, 0.21. The spatial distributions of the mean values of the medium parameters in the region of disturbances in the groups differ. This difference can be explained by the fact that Forbush decrease amplitude in groups 1 and 2 are formed in the central and peripheral parts of coronal mass ejection respectively.

Особенности динамики параметров среды и плотности космических лучей в сильных форбуш-понижениях, связанных с магнитными облаками

Diffusion and electromagnetic mechanisms determine the formation of sporadic Forbush decreases. The diffusion mechanism affects the Forbush decrease amplitude in the turbulent layer, and the part of the coronal mass ejection preceding the magnetic cloud, and its efficiency depends on the level of magnetic field turbulence. The electromagnetic mechanism works in a magnetic cloud, and its efficiency depends on the intensity of regular magnetic and electric fields. We analyze solar wind parameters and cosmic ray density, using the superposed epoch analysis. In 1996–2006, 23 strong Forbush decreases (amplitude >5 %) were detected. The average amplitude of 7 % is equally formed by both mechanisms. The events can be divided into 2 groups depending on the contribution of the mechanisms to Forbush decrease amplitude. Group 1 includes the strongest Forbush decreases (amplitude=8.5 %), formed by both diffusion and electromagnetic mechanisms. The diffusion mechanism forms 0.26 amplitude, and the electromagnetic mechanism is responsible for 0.74 one. In group 2, the averege amplitude Forbush decrease =5.7 %, the diffusion mechanism forms 0.79 of amplitude; and the electromagnetic one, 0.21. The spatial distributions of the mean values of the medium parameters in the region of disturbances in the groups differ. This difference can be explained by the fact that Forbush decrease amplitude in groups 1 and 2 are formed in the central and peripheral parts of coronal mass ejection respectively.

Interstellar Polarization Survey. III. Relation between Optical Polarization and Reddening in the General Interstellar Medium

Optical starlight can be partially polarized while propagating through the dusty, magnetized interstellar medium (ISM). The polarization efficiency describes the polarization intensity fraction per reddening unit, P V /E(B − V), related to the interstellar dust grains and magnetic field properties. The maximum value observed, [P V /E(B − V)]max, is thus achieved under optimal polarizing conditions of the ISM. Therefore, the analysis of polarization efficiency observations across the Galaxy contributes to the study of magnetic field topology, small-scale magnetic fluctuations, grain-alignment efficiency, and composition. Infrared observations from Planck satellite have set [P V /E(B − V)]max to 13% mag−1. However, recent optical polarization observations in Planck's highly polarized regions showed polarization efficiency values between 13.6% mag−1 and 18.2% mag−1 (depending on the extinction map used), indicating that [P V /E(B − V)]max is not well constrained yet. We used V-band polarimetry of the Interstellar Polarization Survey (consisting of ∼10,500 high-quality observations distributed in 34 fields of 0.°3 × 0.°3) to accurately estimate the polarization efficiency in the ISM. We estimated the upper limit of P V /E(B − V) with the weighted 99th percentile of the field. In five regions, the polarization efficiency upper limit is above 13% mag−1. Furthermore, we found [P V /E(B − V)]% mag−1 using diffuse intermediate-latitude (∣b∣ > 7.°5) regions with apparently strong regular Galactic magnetic field in the plane-of-sky. We studied the variations of P V /E(B − V) across the sky and tested toy models of polarization efficiency with Galactic longitude that showed some correspondence with a uniform spiral magnetic field.

Influence of Magnetic Fields on the Gas Rotation in the Galaxy NGC 6946

Magnetic fields can play an important role in the energy balance and formation of gas structures in galaxies. However, their dynamical effect on the rotation curve of galaxies is immensely unexplored. We investigate the dynamical effect of the known magnetic arms of NGC 6946 on its circular gas rotation traced in H i, considering two dark-matter mass-density models, ISO, and the universal NFW profile. We used a three-dimensional model for the magnetic field structure to fit the modeled rotation curve to the observed data via a χ 2 minimization method. The shape of the H i gas rotation curve is reproduced better including the effect of the magnetic field, especially in the outer part, where the dynamical effect of the magnetic field could become important. The typical amplitude of the regular magnetic field contribution in the rotation curve is about 6–14 km s−1 in the outer gaseous disk of the galaxy NGC 6946. The contribution ratio of the regular magnetic field to the observed circular velocity and to dark matter increases with the galactocentric radius. Its ratio to the observed rotational velocity is about 5% and, to dark matter, is about 10% in the outer regions of the galaxy NGC 6946. Therefore, the large-scale magnetic fields cannot be completely ignored in the large-scale dynamics of spiral galaxies, especially in the outer parts of galaxies.

Geminga SNR: Possible Candidate of Local Cosmic-Ray Factory (II)

Accurate measurements of the energy spectrum and anisotropy can help us discover local cosmic-ray accelerators. Our recent works have shown that spectral hardening above 200 GeV in the energy spectra and transition of large-scale anisotropy at ∼100 TeV are of an unifying origin. Less than 100 TeV, both spectral hardening and anisotropy explicitly indicate the dominant contribution from nearby sources. Recent observations of CR anisotropy suggest that this phase is consistent with the locally regular magnetic field (LRMF) of the interstellar boundary explorer (IBEX) below 100 TeV. In this work, we further investigate the parameter space of sources allowed by the observational energy spectra and amplitude and phase of dipole anisotropy. To obtain the best-fit source parameters, a numerical algorithm is to compute the parameter posterior distributions based on Bayesian inference. We found that by combining the observations of the energy spectrum and anisotropy, the parameters of the model can be well constrained. The LRMF and the effect of the corresponding anisotropic diffusion are considered in this work. Finally, the phase results’ right ascension (R.A.)=3.2 h below 100 TeV was obtained by fitting, which is in general agreement with the experimental observations. Since the Geminga SNR is very close to the mean of the fitted parameters, it could be a candidate for a local cosmic-ray accelerator.

Magnetic Field in some Selected Directions of the Galaxy: Sagittarius Spiral Arm

Faraday rotation data on 180 pulsars are used in a detail study of the magnetic field in the direction of galactic longitude 40◦ < l < 70◦, including the Sagittarius spiral arm region. The highly regular magnetic field in northern hemisphere of galaxy is directed to the Sun, when the magnetic field of southern hemisphere is directed from the Sun. We propose that the Sagittarius spiral arm lies mainly to the north of the Galactic plane, while the magnetic field with opposite direction below this plane is the field of the halo of the southern hemisphere of the Galaxy.

Investigation of the magnetron balancing effect on the ionized flux fraction and deposition rate of sputtered titanium species for the high-power impulse magnetron sputtering pulses of different lengths

The study is focused on the impact of different magnetic field configurations of a high-power impulse magnetron sputtering (HiPIMS) in a nonreactive mode on the film precursors. Ionized flux fraction and total flux deposited onto the substrate were measured with the magnetic quartz crystal microbalance probe placed in front of the target racetrack. Particularly, we investigated the degree of magnetron balancing and the geometry of the magnetic field above the Ti target surface (4 in. diameter), as crucial factors influencing the thin film deposition, for different HiPIMS pulse lengths. Three unbalanced (II type) magnetron configurations have been chosen for this study: two symmetric geometries—with a regular magnetic field (B parallel to the target about 80 and 35 mT) and one asymmetric (highly unbalanced) magnetron configuration with an intermediate magnetic field (B parallel to the target about 48 mT). The HiPIMS was operated keeping constant the peak current at 43 A for C0-E0 and C10-E0 B-field configurations and a lower value, 33 A, when operating in C10-E10 configuration. In addition to the peak current, the pulse frequency was kept constant at 100 Hz but the pulse length (power on-time, Ton) was varied from 50 up to 100 μs. Obviously, the pulse power and the average power continuously increase with the length of the pulse. The results reveal a significant difference in the trends of the deposition rate and ionized flux fraction reaching the substrate with respect to the degree of balancing of the magnetron. It was found that the ionized fraction of metal arriving at the substrate reaches its maximum for the pulse length between Ton ≈ 70–80 μs in both symmetric cases, with strong and weak magnetic fields. The ionized fraction of Ti atoms in the asymmetric configuration increased in all measured range with the pulse length and the growth rate has a smooth increase.

Monte Carlo Simulations on Possible Collimation Effects of Outflows to Fan Beam Emission of Ultraluminous Accreting X-Ray Pulsars

Pulsating ultraluminous X-ray sources (PULXs) are accreting pulsars with apparent X-ray luminosity exceeding 1039 erg s−1. We perform Monte Carlo simulations to investigate whether a high collimation effect (or strong beaming effect) is dominant in the presence of accretion outflows, for the fan beam emission of the accretion column of the neutron stars in PULXs. We show that the three nearby PULXs (RX J0209.6−7427, Swift J0243.6+6124, and SMC X-3), namely, the Three Musketeers here, have their main pulsed emission not strongly collimated even if strong outflows exist. This conclusion can be extended to the current sample of extragalactic PULXs, if accretion outflows are commonly produced from them. This means that the observed high luminosity of PULXs is indeed intrinsic, which can be used to infer the existence of very strong surface magnetic fields of ∼1013–14 G, possibly multipole fields. However, if strong outflows are launched from the accretion disks in PULXs as a consequence of disk spherization by radiation pressure, regular dipole magnetic fields of ∼1012 G may be required, comparable to that of the Three Musketeers, which have experienced large luminosity changes from well below their Eddington limit (2 × 1038 erg s−1 for an NS) to super-Eddington and whose maximum luminosity fills the luminosity gap between Galactic pulsars and extragalactic PULXs.

The effect of magnetic field on the inner Galactic rotation curve

ABSTRACT In the past few decades, some studies pointed out that magnetic field might affect the rotation curves in galaxies. However, the impact is relatively small compared with the effects of dark matter and the baryonic components. In this letter, we revisit the impact of magnetic field on the rotation curve of our Galaxy. We show that the inner Galactic rotation curve could be affected significantly by the magnetic field. The addition of the inner bulge component, which has been proposed previously to account for the inner rotation curve data, is not necessary. The magnetic field contribution can fully account for the excess of the inner rotation velocity between 5 to 50 pc from the Galactic Centre. Our analysis can also constrain the azimuthal component of the central regular magnetic field strength to $B_0 \sim 50-60\, \mu$G, which is consistent with the observed range.

Polarization power spectra and dust cloud morphology

Context In the framework of studying cosmic microwave background polarization and characterizing its Galactic foregrounds, the angular power spectrum analysis of the thermal dust polarization map has led to intriguing evidence of an E/B asymmetry and a positive TE correlation. The interpretation of these observations is the subject of theoretical and simulation-driven studies in which the correlation between the density structure of the interstellar medium (ISM) and the magnetic field appears to be a key aspect. In this context, and when the magnetized ISM structures are modeled in three dimensions, dust clouds are generally considered to be filamentary structures only, but both filamentary and sheet-like shapes are supported by observational and theoretical evidence. Aims. We aim to study the influence of the cloud shape and its connection to the local magnetic field, as well as the influence from the viewing angle, on the angular power spectra measured on thermal dust polarization maps; we specifically focus on the dependence of the E/B power asymmetry and TE correlation. Methods. To this end, we simulated realistic interstellar clouds with both filament-like and sheet-like shapes using the software Asterion, which also allowed us to generate synthetic maps of thermal dust polarized emission with an area of 400 square degrees. Then, we computed their polarization power spectra in the multipole range ℓ ϵ [100, 500] and focused on the E/B power asymmetry, quantified through the ℛEB ratio, and the correlation coefficient rTE between Τ and Ε modes. We quantified the dependence of ℛEB and rTE values on the offset angle (between the longest cloud axis and local magnetic field lines) and inclination angle (between the line of sight and the magnetic field) for both types of cloud shapes, either embedded in a regular magnetic field or coupled to a nonregular field to mimic turbulence. Results. We find that both types of cloud shapes cover the same regions of the (ℛEB, rTE) parameter space. The dependence on the inclination and offset angles is similar for both shapes, although sheet-like structures generally show larger scatter than filamentary structures. In addition to the known dependence on the offset angle, we find a strong dependence of ℛEB and rTE on the inclination angle. Conclusions. The very fact that filament-like and sheet-like structures may lead to polarization power spectra with similar (ℛEB,rTE) values complicates their interpretation. We argue that interpreting them solely in terms of filament characteristics is risky, and in future analyses, this degeneracy should be accounted for, as should the connection to the magnetic field geometry. Our results based on maps of 400 square degrees clarify that the overall geometrical arrangement of the magnetized ISM surrounding the observer leaves its marks on polarization power spectra.

On the potential of faraday tomography to identify shock structures in supernova remnants

ABSTRACT Knowledge about the magnetic fields in supernova remnants (SNRs) is of paramount importance for constraining Galactic cosmic ray acceleration models. It could also indirectly provide information on the interstellar magnetic fields. In this paper, we predict the Faraday dispersion functions (FDFs) of SNRs for the first time. For this study, we use the results of three dimensional (3D) ideal magnetohydrodynamic (MHD) simulations of SNRs expanding into a weak, regular magnetic field. We present the intrinsic FDFs of the shocked region of SNRs for different viewing angles. We find that the FDFs are generally Faraday complex, which implies that conventional rotation measure study is not sufficient to obtain the information on the magnetic fields in the shocked region and Faraday tomography is necessary. We also show that the FDF allows to derive the physical-depth distribution of polarization intensity when the line of sight is parallel to the initial magnetic field orientation. Furthermore, we demonstrate that the location of contact discontinuity can be identified from the radial profile of the width of the FDF with the accuracy of 0.1–0.2 pc.

The Milne problem with chaotic magnetic field

ABSTRACT We consider the semi-infinite plane-parallel electron atmosphere with chaotic magnetic field B′ ≤ 1010 G, when the parameter $x^{\prime 2}=(\omega _{B^{\prime }}/\omega)^2 \simeq 0.87 \times 10^{-16}\lambda ^2(\mu \mathrm{ m}) B^{\prime 2}(G)$ can be both ≪1 and ≫1. Regular magnetic field $\boldsymbol{B}_0$ is absent. All magnetic fluctuations are assumed as Gaussian type and isotropic. The isotropic magnetic fluctuations $\boldsymbol{B}^{\prime }$ give rise to the same extinction for all Stokes parameters and the additional extinction factor h for parameters Q and U, which arises due to chaotic Faraday rotations. We consider the Milne problem, which is described by the system of transfer equations for intensity I and parameter Q. It is shown that the phase matrix depends on parameter G(B′), which is equal to 3 for x′2 ≪ 1 and to 27 for x′2 ≫ 1. The calculations demonstrate that the polarization of outgoing radiation strongly depends on parameter h and degree of absorption ε.