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A&A
Volume 674, June 2023
Article Number L2
Number of page(s) 4
Section Letters to the Editor
DOI https://doi.org/10.1051/0004-6361/202346265
Published online 31 May 2023

© The Authors 2023

Licence Creative CommonsOpen Access article, published by EDP Sciences, under the terms of the Creative Commons Attribution License (https://creativecommons.org/licenses/by/4.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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1. Introduction

The formation of the X-ray spectra of accreting highly magnetized (B ∼ 1012 G) neutron stars in high-mass X-ray binaries (HMXBs) is a complex process. The emitted continuum is mainly shaped by magnetic Comptonization in an accretion column or accretion mound. Resonant Compton scattering also results in the appearance of absorption line-like features in the spectra, or “cyclotron lines” (Staubert et al. 2019). Recent observations have revealed the complexity of continuum shapes, which often cannot be described only by a simple power law with high-energy cutoff (Müller et al. 2013), but require additional broad Gaussian features in absorption or emission at intermediate energies (the “10-keV” feature; see, e.g., Ferrigno et al. 2009; Diez et al. 2022). Alternatively, the continuum can be described as the sum of two Comptonization spectra from plasmas with different temperatures (see, e.g., Doroshenko et al. 2020). In contrast, the available angle- and polarization-averaged models for continuum formation (e.g., Becker & Wolff 2007; West et al. 2017a,b) yield a smooth spectral shape. The rich observational background therefore remains ahead of the spectral modeling, calling for new emission models that incorporate further physics, such as the effects of polarization and angular redistribution.

Polarization of radiation in the strong magnetic field introduces further complexity in the modeling. Apart from the magnetoactive plasma, photon polarization states can be affected by the quantum electrodynamic (QED) effect known as vacuum polarization. For the cold, absorption-dominated inhomogeneous atmospheres of highly magnetized isolated neutron stars (B ∼ 1014 G), radiative transfer solutions were obtained (e.g., Lai & Ho 2002; Ho & Lai 2003; Özel 2003), some of which also included partial mode conversion (Ho & Lai 2004; van Adelsberg & Lai 2006). For the lower fields that are characteristic of accreting neutron stars in HMXBs, however, the vacuum resonance has not been included in the radiative transfer discussions of the last three decades. Early studies of vacuum polarization effects by Pavlov & Shibanov (1979), Ventura et al. (1979), Kaminker et al. (1982), and Soffel et al. (1985) addressed the importance of this phenomenon for homogeneous media only, without Compton scattering, finding that in this case the vacuum resonance results in a narrow, low-energy absorption feature. Meszaros & Nagel (1985a,b) noted the importance of the combined vacuum and plasma polarization modes for the shape of the cyclotron line profile based on their homogeneous models including Comptonization, but no qualitative comparison with other cases was made. Later models for the accretion column and atmospheric emission considered only the polarization-averaged case (Becker & Wolff 2007; Farinelli et al. 2016; West et al. 2017b) or used pure magnetized plasma (Mushtukov et al. 2021; Caiazzo & Heyl 2021a,b) and pure vacuum normal modes (Sokolova-Lapa et al. 2021). Models for cyclotron lines on top of a predefined continuum also largely focused on the pure vacuum case (see, e.g., Araya & Harding 1999; Schwarm et al. 2017a,b).

Previous claims that for B ∼ 1012 G vacuum polarization does not alter the emission spectra (Ho & Lai 2004; van Adelsberg & Lai 2006) should be taken with a grain of salt, as they were based on computations for cool, absorption-dominated atmospheres. Vacuum polarization in the hot Comptonizing medium of accreting, highly magnetized neutron stars thus remains poorly studied. Here, we would like to highlight the effect of vacuum polarization in these environments and demonstrate its possible influence on the spectral shape and polarization signal. We also performed a comparison for different choices of photon normal modes. The problem of the mode definition is introduced in Sect. 2. Section 3 presents several models for emission from homogeneous and inhomogeneous media. Section 4 provides the discussion and conclusion of this study.

2. Polarization modes

The highly magnetized plasma in the vicinity of a neutron star is a complex medium, which can support various types of waves. In the medium, transverse electromagnetic waves, with electric vector, E, perpendicular to the wave vector, k, are allowed to propagate at frequencies higher than the electron plasma frequency. To model the X-ray emission from accreting neutron stars in HMXBs, the description of the radiation field in terms of two transverse waves (the two polarization normal modes) became a standard approach. We outline the classical definition of the two modes in Sect. 2.1 and discuss their ambiguity when both, the plasma and the QED vacuum alter the radiation propagation in Sect. 2.2.

2.1. Mode definition and ellipticity

For high-energy radiation in a magnetoactive plasma with B ∼ 1012 G, the electron response governs wave propagation. The solution of the wave equation with a linear response in the form of two independent modes is known for different temperature regimes (e.g., Ventura 1979; Pavlov et al. 1980; Nagel 1981). The polarization vectors, E, of the two normal modes have opposite directions of rotation. The modes are elliptically polarized, where the ellipticity K = Ex/Ey in a coordinate system in which the z-axis is parallel to k (Fig. 1, left). The “ordinary” mode is defined as the left hand polarized wave, with E oscillating mainly in the (B, k) plane (|K|≫1), where B is the external magnetic field vector. The polarization vector of the right hand polarized “extraordinary” mode oscillates mainly perpendicularly to this plane (|K|≪1). For the extraordinary mode, the co-directional rotation of E with electrons in the external B field allows the mode to resonate at the cyclotron frequency. In the absence of damping, the ellipses of the two modes are of equal shape, perpendicular to each other.

thumbnail Fig. 1.

Definition of the polarization modes. Left: coordinate system chosen to define ellipses of the polarization modes. Right: dependency of the ellipticity of the polarization modes on the electron number density. The vacuum resonance occurs in the center of the figure. Based on Fig. 1 from Lai & Ho (2003).

The effect of vacuum polarization in a strong B field alters this picture due to the birefringence introduced by the QED’s virtual electron-positron plasma, which allows both modes to exhibit cyclotron resonances. In the absence of a real plasma, the equality of the virtual charges makes the normal modes linear and strictly orthogonal to each other (except for k ∥ B). The combined effect of the virtual and the real plasma results in a more complex behavior of the modes, causing occasional departure from orthogonality.

2.2. Mode ambiguity

For a given B field, a given electron number density of the plasma, ne, and a photon propagation angle, θ, there are certain photon energies at which the effects of real and virtual particles compensate each other, since the QED vacuum and the plasma enforce linear polarization of the modes in a mutually orthogonal direction (Lai & Ho 2003). The two most relevant of these are at the vacuum resonance, EV, which depends on ne and B (see, e.g., Ho & Lai 2003) and the cyclotron resonance, Ecyc. The transition through these points results in a flip of the position angle of the polarization ellipse by ±90° or in the reversal of the direction of the rotation of the polarization vector due to the switch in the dominance of the influence of vacuum and plasma (Pavlov & Shibanov 1979). These effects make the classical definition of the modes introduced in Sect. 2.1 internally inconsistent. As discussed by Kirk (1980), a general definition of the modes can be made based on the continuity of the refractive indices of the waves. Figure 2 shows the cross-section for magnetic Compton scattering as a function of photon energy, illustrating the effect of the continuity of the refractive index across the spectrum. In the case of a discontinuous refractive index, corresponding to the change of the wave’s handedness, the cross sections of the modes acquire a sharp, resonance-like feature at EV (Ventura et al. 1979). The alternative choice of continuous refractive indices results in an abrupt change of opacity (see, e.g., Meszaros & Nagel 1985a).

thumbnail Fig. 2.

Cross-sections for magnetic Compton scattering for two different propagation angles, θ. The modes are shown for the cases of discontinuous (left) and continuous (right) refractive indices for B ≈ 2.6 × 1012 G.

The problem of the mode definition becomes clear when one considers the propagation in an inhomogeneous medium, where a photon passes, for example, from a plasma-dominated region where E > EV, to a vacuum-dominated one where E < EV. After passing the “resonant density” where E = EV, modes with |K|≪1 and |K|≫1 will acquire left hand and right hand polarization, respectively. Figure 1 (right) illustrates the behavior of the modes’ ellipticities in an inhomogeneous medium. For a smooth density gradient, at energies above a few kiloelectronvolts it is expected that the modes evolve adiabatically (i.e., they keep their original helicity), while they behave non-adiabatically at lower energies (Özel 2003; Ho & Lai 2003). In general, however, the probability of a jump of mode characteristics across the resonance has to be considered (van Adelsberg & Lai 2006). The adiabatic and non-adiabatic cases correspond to the continuous and discontinuous behavior of the refractive index, respectively. In the following, we use the notations “mode 1” and “mode 2”, as denoted for these cases in Fig. 2 by σ1, 2.

3. Modeling and results

We used our polarization-dependent radiative transfer code FINRAD, which takes into account angular and energy redistribution of photons both in the continuum and in the cyclotron line to investigate the effect of vacuum polarization. The code is based on the Feautrier numerical scheme for two polarization modes (Nagel 1981; Meszaros & Nagel 1985a). More details on the implementation and verification of the code are given by Sokolova-Lapa et al. (2021). We performed each simulation for both choices of the cross-section behavior displayed in Fig. 2. For the inhomogeneous model, this choice corresponds to the adiabatic and non-adiabatic mode propagation. For some models we present a comparison with the modes for pure plasma (hot, but non-relativistic) and pure vacuum. All models assume a slab-like emission region, with the B-field parallel to the surface normal (same as in Nagel 1981). The slab is characterized by ne, the constant B field value, and the electron temperature kTe. The treatment for induced processes and boundary conditions are the same as in Sokolova-Lapa et al. (2021); that is, diffusion limit is assumed at the bottom of the slab and there is no illumination from the top. We describe the spectral shape with the photon flux , where the specific intensity I(E, θ) is obtained with FINRAD. The angular dependence of the emission will be discussed in future studies. Section 3.1 presents the spectra from several homogeneous models for various kTe, ne, and B. In Sect. 3.2, we then construct a simple model for an accretion mound with an inhomogeneous density profile and show the resulting spectra and polarization signal.

3.1. Homogeneous slab

Figure 3 illustrates the influence of vacuum polarization on the total spectra of the emission from the homogeneous slab and compares the spectra obtained for the pure plasma and the vacuum cases. The choice of modes for the low-temperature case (kTe = 0.5 keV, spectra with lower flux) has a very minor effect on the spectral shape. The Comptonized spectra for the hotter plasma, kTe = 6 keV, illustrate the strong influence of the cyclotron resonance near the cyclotron energy Ecyc = 60 keV for all cases except for the pure plasma spectra. The latter are dominated by mode 2 (in this case, the classically defined ordinary mode), whose weak resonance is introduced by the Doppler effect on the thermal electrons and then washed out by angle averaging. The spectra for the case of the discontinuous refractive indices closely follow the pure vacuum ones, except for a small region near the vacuum resonance (the polarization signal, however, is different, see Sect. 3.2). The spectral shape for the case of the continuous refractive indices differs significantly from the others. There is a noticeable depression of the continuum above the vacuum resonance energy; vacuum polarization effectively reduces the total optical depth of the emitting region. Lai & Ho (2002) noted similar behavior in their studies of photon propagation in magnetar-like fields with B ∼ 1014 G. When the vacuum resonance is located closer to the cyclotron line, it can mimic an additional line-like feature. For the cases presented in Fig. 3, EV < Ecyc. Figure 4 (left) illustrates the case when EV > Ecyc and shows contributions of the two normal modes. Here, the cyclotron line is significantly suppressed. The case of the continuous refractive indices is the most interesting one, as none of the modes exhibit a strong cyclotron resonance. The case of a higher B field shown in Fig. 4 (right) (EV ≪ Ecyc) yields a more complex continuum shape.

thumbnail Fig. 3.

Flux from a homogeneous slab summed over the two polarization modes for the higher (left) and lower (right) electron number densities, for B ≈ 5.2 × 1012 G. Different types of polarization modes are shown: pure-plasma (“pl”, dotted), pure-vacuum (“v”, thick dashes) modes, and modes with continuous and discontinuous (“pl+v: cri” and “pl+v: dri”, dashed and solid, respectively) refractive indices.

thumbnail Fig. 4.

Emission from a highly magnetized optically thick slab for continuous and discontinuous behavior of the refractive indices, for two B-field values. Left: B ∼ 1.7 × 1012 G (Ecyc = 20 keV and EV ∼ 24 keV); right: B ∼ 5.2 × 1012 G (Ecyc = 60 keV and EV ∼ 8 keV).

3.2. Inhomogeneous accretion mound

In an inhomogeneous medium, the radiation field encounters the vacuum resonance at a continuum of energies. To understand how this effect contributes to the total spectrum, we created a simplified model of an accretion mound confined to the polar cap of a neutron star. The mound is represented by a slab of magnetized isothermal plasma with kTe = 5 keV and Ecyc = 40 keV, a height of 500 m and a radius of 300 m. We assume that the plasma is accreted at a mass-accretion rate of Ṁ = 1017 g s−1 and that the accretion flow is mainly decelerated higher up, above the emission region. The velocity on the top of the mound is set to the 5% of the free-fall velocity near the surface, vff, and decreases linearly within the mound to 10−3 vff at the bottom, such that the density distribution within the mound increases from ∼1022 cm−3 at the top to ∼1024 cm−3 at the bottom.

Figure 5 shows the total flux for the previously discussed types of polarization modes. Here, the adiabatic and non-adiabatic cases (left panel) are similar, both exhibiting an imprint of the vacuum resonance at ∼6.5 keV from the most tenuous top layer of the slab. For adiabatic propagation the total flux is slightly suppressed above EV, with the cyclotron line core becoming noticeably more shallow. Due to the absence of the vacuum resonance, the pure plasma and vacuum cases (right panel) naturally lack the additional complexity of the spectra.

thumbnail Fig. 5.

Emission from an inhomogeneous mound with kTe = 5 keV and B ∼ 3.6 × 1012 G (Ecyc = 40 keV). Top: flux for the cases including the plasma and vacuum polarization effects for non-adiabatic,“non-ad” and adiabatic, “ad”, mode propagation (left) and the pure plasma and vacuum cases (right). The left top panel shows the corresponding blackbody spectrum (“BB”). Bottom: corresponding degree of polarization. The energy range accessible by the Imaging X-ray Polarimetry Explorer (IXPE) is shown by the light blue transparent region.

For both cases, the polarization degree (Fig. 5, bottom) reaches a maximum of ∼50%1. It is overall lower than predicted by the accretion column model of Caiazzo & Heyl (2021b) for the emission in the neutron star rest frame. This difference is probably due to radiative propagation effects that were omitted by these authors, such as redistribution during scattering, which are enhanced by the presence of the hot plasma. The degree of polarization in the pure vacuum case is higher at intermediate energies; for the plasma case it reaches almost 100% near the cyclotron line.

4. Discussion and conclusions

In this Letter, we present the results of radiative transfer calculations in the hot Comptonizing medium in the vicinity of accreting neutron stars with B ∼ 1012 G, including the effect of vacuum polarization. We found that the choice of polarization modes and the vacuum resonance can potentially strongly influence the emitted spectrum. Together with angle- and energy-dependent magnetic Comptonization, those polarization effects result in a complex continuum shape and can alter the cyclotron line. The natural switch in the dominance of the two polarization modes in the total flux across the spectrum produces excesses and dips on top of the power-law-like continuum. While this phenomenon is not specific to vacuum polarization, it is enhanced by it due to the additional interplay between the modes. The vacuum resonance itself can also introduce a weak dip in the spectra, mimicking the cyclotron line, as shown for the inhomogeneous model. For the conditions studied here, we find that pure vacuum normal modes provide a closer spectral shape to the mixed (magnetized plasma and polarized vacuum) case than pure plasma modes.

The combination of these effects may also explain several complexities observed in the spectra of accreting neutron stars in HMXBs, such as the “10-keV” feature or the two-component-like continua. Standalone absorption-line-like features at intermediate energies (see, e.g., Reig & Milonaki 2016; Doroshenko et al. 2020 for 4U 1901+03 and KS 1947+300) are potential candidates for the mode interchange regions. Vacuum polarization can also result in significant suppression of the cyclotron line when EV > Ecyc, potentially explaining the missing cyclotron lines in the spectra of some sources.

Concerning the polarization degree, our results for a simplified model of the accretion mound show a maximum of ∼50% in the continuum. The vacuum resonance results in a broad region of depolarization. A detailed understanding of the angular redistribution is required in order to draw any further conclusions, which will be addressed in a future work, along with the study of emission beaming and formation of pulse profiles under the influence of light bending. We expect that angle-dependent beaming and the propagation of the X-rays in the magnetosphere will only lower the observable degree of polarization, bringing the total value closer to the low values observed recently by IXPE (see, e.g., Tsygankov et al. 2022; Doroshenko et al. 2022). The presented model is a first step toward a more realistic description of the radiation field in the accretion channel, which would ideally include partial mode conversion, higher relativistic corrections, and consistent simulation of the structure of the accretion mound, as well as the influence of the falling flow.

1

For the angle-resolved spectra of our calculations (not presented here) the polarization degree reaches a maximum of ∼65% in the continuum at θ ∼ 70°.

Acknowledgments

This research has been funded by DFG grants WI1860/11-2 and WI1860/19-1. The work originates in part from the text of the thesis draft by ESL. We are grateful to Aafia Zainab, Philipp Thalhammer, and Katrin Berger for their valuable comments. The authors thank the ESA Space Science Faculty for funding of the workshop “X-ray Tracking of Magnetic Field Geometries in Accreting X-ray Pulsars” and all participants, who supported fascinating discussions and raised new inspiring questions for this research. The authors are also grateful to the XMAG Collaboration.

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All Figures

thumbnail Fig. 1.

Definition of the polarization modes. Left: coordinate system chosen to define ellipses of the polarization modes. Right: dependency of the ellipticity of the polarization modes on the electron number density. The vacuum resonance occurs in the center of the figure. Based on Fig. 1 from Lai & Ho (2003).
In the text
thumbnail Fig. 2.

Cross-sections for magnetic Compton scattering for two different propagation angles, θ. The modes are shown for the cases of discontinuous (left) and continuous (right) refractive indices for B ≈ 2.6 × 1012 G.
In the text
thumbnail Fig. 3.

Flux from a homogeneous slab summed over the two polarization modes for the higher (left) and lower (right) electron number densities, for B ≈ 5.2 × 1012 G. Different types of polarization modes are shown: pure-plasma (“pl”, dotted), pure-vacuum (“v”, thick dashes) modes, and modes with continuous and discontinuous (“pl+v: cri” and “pl+v: dri”, dashed and solid, respectively) refractive indices.
In the text
thumbnail Fig. 4.

Emission from a highly magnetized optically thick slab for continuous and discontinuous behavior of the refractive indices, for two B-field values. Left: B ∼ 1.7 × 1012 G (Ecyc = 20 keV and EV ∼ 24 keV); right: B ∼ 5.2 × 1012 G (Ecyc = 60 keV and EV ∼ 8 keV).
In the text
thumbnail Fig. 5.

Emission from an inhomogeneous mound with kTe = 5 keV and B ∼ 3.6 × 1012 G (Ecyc = 40 keV). Top: flux for the cases including the plasma and vacuum polarization effects for non-adiabatic,“non-ad” and adiabatic, “ad”, mode propagation (left) and the pure plasma and vacuum cases (right). The left top panel shows the corresponding blackbody spectrum (“BB”). Bottom: corresponding degree of polarization. The energy range accessible by the Imaging X-ray Polarimetry Explorer (IXPE) is shown by the light blue transparent region.
In the text

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