IXPE detection of polarized X-rays from magnetars and photon mode conversion at QED vacuum resonance

Significance

The recent detections of polarized X-rays from magnetars have opened up a new avenue to study magnetars, neutron stars endowed with superstrong magnetic fields. The detected polarization signals are intriguing, and can be explained by a Quantum electrodynamics (QED) vacuum resonance effect—an effect that is analogous to MSW neutrino oscillation that operates in the Sun and similar level-crossing phenomena in other areas of sciences.

Abstract

The recent observations of the anomalous X-ray pulsars 4U 0142+61 and 1RXS J170849.0-400910 by the Imaging X-ray Polarimetry Explorer (IXPE) opened up a new avenue to study magnetars, neutron stars endowed with superstrong magnetic fields (B ≳ 10¹⁴ G). The detected polarized X-rays from 4U 0142+61 exhibit a 90° linear polarization swing from low photon energies (E ≲ 4 keV) to high energies (E ≳ 5.5 keV). We show that this swing can be explained by photon polarization mode conversion at the vacuum resonance in the magnetar atmosphere; the resonance arises from the combined effects of plasma-induced birefringence and Quantum electrodynamics (QED)-induced vacuum birefringence in strong magnetic fields. This explanation suggests that the atmosphere of 4U 0142 is composed of partially ionized heavy elements and that the surface magnetic field is comparable to or less than 10¹⁴ G, consistent with the dipole field inferred from the measured spindown. It also implies that the spin axis of 4U 0142+61 is aligned with its velocity direction. The polarized X-rays from 1RXS J170849.0-400910 do not show such 90° swing, consistent with magnetar atmospheric emission with B ≳ 5 × 10¹⁴ G.

Magnetars are neutron stars (NSs) whose energy outputs (even in quiescence) are dominated by magnetic field dissipations (1, 2). Recently, the NASA/ASI Imaging X-ray Polarimetry Explorer (IXPE) (3) reported the detection of linearly polarized X-ray emission from the anomalous X-ray pulsar (AXP) 4U 0142+61, a magnetar with an inferred dipole magnetic field (based on the spindown rate) of ∼10¹⁴ G (4). This is the first time that polarized X-rays have been detected from any astrophysical point sources. The overall phase-averaged linear polarization degree is 12 ± 1% throughout the IXPE band (2 to 8 keV). Interestingly, there is a substantial variation of the polarization signal with energy: the polarization degree is 14 ± 1% at 2 to 4 keV and 41 ± 7% at 5.5 to 8 keV, while it drops below the detector sensitivity around 4 to 5 keV, where the polarization angle swings by ∼90°.

Taverna et al. (4) considered several possibilities to explain the observed polarization swing and suggested that the thermal X-rays from 4U 0142+61 are emitted from an extended region of the condensed neutron star (NS) surface. In this scenario, the 2 to 4 keV radiation is dominated by the O-mode (polarized in the plane spanned by the local magnetic field and the photon wave vector), while the 5.5 to 8 keV radiation by the X-mode (which is orthogonal to the O-mode) because of reprocession by resonant Compton scattering (RCS) in the magnetosphere.

While it is premature to draw any firm conclusion without detailed modeling, the “condensed surface + RCS” scenario may be problematic for several reasons: i) At the surface temperature of T_s ≃ 5 × 10⁶ K and B₁₄ ≡ B/(10¹⁴ G)≃1, as appropriate for 4U 0142, it is unlikely that the NS surface is in a condensed form, even if the surface composition is Fe (5, 6), simply because the cohesive energy of the Fe solid is not sufficiently large (7). ii) A stronger surface magnetic field is possible, but the emission from a condensed Fe surface (for typical magnetic field and photon emission directions at the surface) is dominated by the O-mode only for photon energies less than $E_c\simeq 0.1\eta Z^{2/5}{B}_{14}^{1/5}\mathrm{keV}$, where η ≲ 1 (8, 9). An unrealistically strong field (B₁₄ ≳ 10⁵ for Z = 26) would be required to make E_c ≳ 4 keV. iii) As acknowledged by Taverna et al. (4), the assumption that the phase-averaged low-energy photons are dominated by the O-mode would imply that the NS spin axis (projected in the sky plane) is orthogonal to the proper motion direction. This is in contradiction to the growing evidence of spin-kick alignment in pulsars (10–14).

In this paper, we show that the 90° linear polarization swing observed in 4U 0142 could be naturally explained by photon mode conversion associated with the “vacuum resonance” arising from QED and plasma birefringence in strong magnetic fields. The essential physics of this effect was already discussed in ref. 15, where it was shown that for neutron stars with H atmospheres, thermal photons with E ≲ 1 keV are polarized orthogonal to photons with E ≳ 4 keV, provided that the NS surface magnetic field is somewhat less than 10¹⁴ G. The purpose of this paper is to reexamine the mode conversion effect under more general conditions (particularly the atmosphere composition) and to present semianalytic calculations of the polarization signals for parameters relevant to 4U 0142. Most recently, IXPE detected polarized X-rays from another AXP 1RXS J170849.0-400910 and found that the polarization angle remains constant as a function of E (16). This result is expected for the magnetar atmospheric emission with B ≳ 5 × 10¹⁴ G—we comment on this at the end of the paper.

Vacuum Resonance and Mode Conversion

Quantum electrodynamics (QED) predicts that in a strong magnetic field, the vacuum becomes birefringent (17–21). This vacuum birefringence is significant for B ≫ B_Q = m_e²c³/(eℏ) = 4.414 × 10¹³ G, the critical QED field strength. However, when combined with the birefringence due to the magnetized plasma, vacuum polarization can greatly affect radiative transfer even when the field strength is modest. A “vacuum resonance” arises when the contributions from the plasma and vacuum polarization to the dielectric tensor “compensate” each other (15, 22–27).

Consider X-ray photons propagating in the magnetized NS atmospheric plasma. There are two polarization modes: The ordinary mode (O-mode) is mostly polarized parallel to the $\mathbf{k}$-$\mathbf{B}$ plane, while the extraordinary mode (X-mode) is perpendicular to it, where $\mathbf{k}$ is the photon wave vector and $\mathbf{B}$ is the external magnetic field (28). Throughout the paper, we assume that the photon energy E satisfies u_e = (E_Be/E)² ≫ 1 and E_Bi/E ≪ 1, where E_Be = ℏeB/(m_ec)=1158B₁₄ keV and E_Bi = 0.63(Z/A)B₁₄ keV are the electron and ion cyclotron energies, respectively. This distinction of photon modes is important since the two modes have very different absorption and scattering opacities in the atmosphere plasma (below): While the O-mode opacities are similar to the zero-field values, the X-mode opacities are significantly reduced because the electric field of the photon (EM wave), being orthogonal to $\mathbf{B}$, cannot effectively perturb the motion of the electron in the plasma when E ≪ E_Be. However, the above description of O-mode and X-mode breaks down near the “vacuum resonance.” To be concrete, let us set up the xyz coordinates with $\mathbf{k}$ along the z-axis and $\mathbf{B}$ in the x-z plane (such that $\widehat{\mathbf{B}}\times \widehat{\mathbf{k}}=sin{\theta }_{\mathrm{kB}}\widehat{\mathbf{y}}$, where θ_kB is the angle between $\mathbf{k}$ and $\mathbf{B}$). We write the transverse (xy) electric field of the photon mode as $\mathbf{E}\propto (iK,1)$. The mode ellipticity K is given by

$$\begin{array}{c}\hfill K_{\pm }=\beta \pm \sqrt{{\beta }^2+1},\end{array}$$ [1]

where

$$\begin{array}{c}\hfill \beta \simeq \frac{u_e^{1/2}{sin}^2{\theta }_{\mathrm{kB}}}{2cos{\theta }_{\mathrm{kB}}}\left(1-\frac{{\rho }_V}{\rho }\right).\end{array}$$ [2]

For a photon of energy E, the vacuum resonance density is given by

$$\begin{array}{c}\hfill {\rho }_V\simeq 0.964Y_e^{-1}{B}_{14}^2{E}_1^2{f}^{-2}{\mathrm{g}\mathrm{cm}}^{-3},\end{array}$$ [3]

where Y_e = ⟨Z/A⟩ is the electron fraction, E₁ = E/(1 keV), and f = f(B) is a slowly varying function of B, and is of order unity (f = 1 for B ≪ B_Q, f ≃ 0.991 at B₁₄ = 1 and f → (B/5B_Q)^1/2 for B ≫ B_Q; ref. 29 for a general fitting formula). For ρ >  ρ_V (where the plasma effect dominates the dielectric tensor) and ρ <  ρ_V (where vacuum polarization dominates), the photon modes (for typical θ_kB ≠ 0) are almost linearly polarized; near ρ = ρ_V, however, the normal modes become circularly polarized as a result of the “cancellation” of the plasma and vacuum effects (Fig. 1). The half width of the vacuum resonance (defined by |β|< 1) is

$$\begin{array}{c}\hfill ϵ\equiv \frac{\mathrm{\Delta }\rho }{{\rho }_V}=\frac{2cos{\theta }_{\mathrm{kB}}}{u_e^{1/2}{sin}^2{\theta }_{\mathrm{kB}}}.\end{array}$$ [4]

Fig. 1.

Polarization ellipticity of the photon mode as a function of density near the vacuum resonance. The two curves correspond to the (+) and (-) modes. In this example, the parameters are B = 10¹⁴G, E = 5 keV, and θ_kB = 30°. The ellipticity of a mode is specified by the ratio K = −iE_x/E_y, where E_x (E_y) is the photon’s electric field component along (perpendicular to) the $\mathbf{k}$-$\mathbf{B}$ plane. The O-mode is characterized by |K|≫1, and the X-mode |K|≪1.

When a photon propagates in an inhomogeneous medium, its polarization state will evolve adiabatically (i.e., following the K₊ or K₋ curve in Fig. 1) if the density variation is sufficiently gentle. Thus, an X-mode (O-mode) photon will be converted into the O-mode (X-mode) as it traverses the vacuum resonance, with its polarization ellipse rotated by 90° (Fig. 1). This resonant mode conversion is analogous to the Mikheyev–Smirnov–Wolfenstein neutrino oscillation that takes place in the Sun (30, 31) and similar level-crossing phenomena in other areas of sciences (e.g., Landau–Zener transition in atomic physics; EM wave propagation in inhomogeneous media and metamaterials; ref. 32). For this conversion to be effective, the adiabatic condition must be satisfied (25)

$$\begin{array}{c}\hfill E\gtrsim E_{\mathrm{ad}}=2.52(ftan{\theta }_{\mathrm{kB}}{)}^{2/3}{\left(\frac{1\mathrm{cm}}{H_{\rho }}\right)}^{1/3}\mathrm{keV},\end{array}$$ [5]

where H_ρ = |ds/dlnρ| is the density scale height (evaluated at ρ = ρ_V) along the ray. In general, the probability for nonadiabatic “jump” is given by

$$\begin{array}{c}\hfill P_{\mathrm{J}}=exp\left[-\frac{\pi }{2}(\frac{E}{E_{\mathrm{ad}}}{)}^3\right].\end{array}$$ [6]

The mode conversion probability is (1 − P_J).

Calculation of Polarized Emission

To quantitatively compute the observed polarized X-ray emission from a magnetic NS, it is necessary to add up emissions from all surface patches of the star, taking account of beaming/anisotropy due to magnetic fields and light bending due to general relativity (15, 33–37). While this is conceptually straightforward, it necessarily involves many uncertainties related to the unknown distributions of surface temperature T_s and magnetic field $\mathbf{B}$. In addition, the atmosphere composition is unknown, the opacity data for heavy atoms/ions for general magnetic field strengths are not available, and atmosphere models for many surface patches (each with different T_s and $\mathbf{B}$) are needed. Finally, to determine the phase-resolved lightcurve and polarization, the relative orientations of the line of sight, spin axis, and magnetic dipole axis are needed. Given all these complexities, we present a simplified, approximate calculation below. Our goal is to determine under what conditions (in terms of the magnetic field strength, surface composition, etc.) the polarization swing can be produced.

We consider an atmosphere plasma composed of a single ionic species (each with charge Ze and mass Am_p) and electrons. This is of course a simplification as, in reality, the atmosphere would consist of multiple ionic species with different ionizations. With the equation of state P = ρkT/(μm_p), hydrostatic balance implies that the column density y at density ρ is given by

$$\begin{array}{c}\hfill y=\frac{\rho kT}{\mu m_{p}g}=0.41\frac{{\rho }_1{T}_6}{\mu g_2}{\mathrm{g}/\mathrm{cm}}^2,\end{array}$$ [7]

where T = 10⁶ T₆ K is the temperature, μ = A/(1 + Z) is the “molecular” weight, ρ₁ is the density in 1 g/cm³, and g₂ is the surface gravity g = (GM/R²)(1 − 2GM/Rc²)⁻¹ in units of 2 × 10¹⁴ (For M = 1.4M_⊙, R = 12 km, we have g₂ ≃ 1.00). The density scale-height along a ray is

$$\begin{array}{c}\hfill H_{\rho }\simeq \frac{\mathit{kT}}{\mu m_{p}gcos\alpha }=0.41\frac{T_6}{\mu g_{2}cos\alpha }\mathrm{cm},\end{array}$$ [8]

where α is the angle between the ray and the NS surface normal.

To simplify our calculations, we shall neglect the electron scattering opacity and the bound–bound and bound–free opacities. The former is generally subdominant compared to the free–free opacity, while the latter are uncertain or unavailable. The free–free opacity of a photon mode (labeled by i) can be written as

$$\begin{array}{c}\hfill {\kappa }_i={\kappa }_0{\xi }_i,\end{array}$$ [9]

where the B = 0 opacity is (setting the Gaunt factor to unity)

$$\begin{array}{c}\hfill {\kappa }_0\simeq 9.3(Z^3/A^2){\rho }_1{T}_6^{-1/2}{E}_1^{-3}\mathcal{G},\end{array}$$ [10]

with $\mathcal{G}=1-exp(-E/kT)$. For the photon mode $\mathbf{E}\propto (iK,1)$, the dimensionless factor ξ is given by (26, 38)

$$\begin{array}{c}\hfill \xi =\frac{K^2{sin}^2{\theta }_{\mathrm{kB}}+(1+K^2{cos}^2{\theta }_{\mathrm{kB}})/u_e}{1+K^2}.\end{array}$$ [11]

Fig. 2 illustrates the behavior of ξ_X, ξ_O, ξ₊, and ξ₋ as a function of density. We see that for typical θ_kB’s, ξ_O ∼ 1 and ξ_X ∼ u_e⁻¹ ≪ 1 except near ρ = ρ_V.

Fig. 2.

Qualitative behavior of ξ_i (the dimensionless mode opacity; Eq. 9) as a function of density for photon mode-i (with i = X, O, +, −). At the vacuum resonance (ρ = ρ_V), ξ_X has a spike, ξ_O has a dip, and ξ₊ and ξ₋ have discontinuities. Note that ξ₊ = ξ_X for ρ < ρ_V and ξ₊ = ξ_O for ρ > ρ_V, while ξ₋ = ξ_O for ρ < ρ_V and ξ₋ = ξ_X for ρ > ρ_V.

For a given photon energy E and wave vector $\mathbf{k}$ (which is inclined by angle α with respect to the surface normal), the transfer equation for mode i (with i = X, O or i = +, −) reads

$$\begin{array}{c}\hfill cos\alpha \frac{\mathrm{d}{I}_i}{\mathrm{d}y}={\kappa }_i\left(I_i-\frac{1}{2}{B}_{\nu }\right),\end{array}$$ [12]

where B_ν(T) is the Planck function and T = T(y) is the temperature profile (to be specified later). To calculate the emergent polarized radiation intensity from the atmosphere, taking account of partial mode conversion, we adopt the following procedure (33): i) We first integrate Eq. 12 for the X-mode and O-mode (i = X, O) from large y (=∞) to y_V, the column density at which ρ = ρ_V. This gives I_XV and I_OV, the X- and O-mode intensities just before resonance crossing. ii) We apply partial mode conversion

$$\begin{array}{cc}& I_{\mathrm{XV}}^{\prime }=I_{\mathrm{XV}}{P}_J+I_{\mathrm{OV}}(1-P_J),\hfill \end{array}$$ [13]

$$\begin{array}{cc}& I_{\mathrm{OV}}^{\prime }=I_{\mathrm{OV}}{P}_J+I_{\mathrm{XV}}(1-P_J),\hfill \end{array}$$ [14]

to obtain the mode intensities just after resonance crossing. This partial conversion treatment is valid since the resonance width is small (Δρ/ρ_V ≪ 1; Eq. 4). iii) We then integrate Eq. 12 for the X-mode and O-mode again from y = y_V (with the “initial” values I_XV′,I_OV′) to y ≪ 1. This then gives the mode intensities emergent from the atmosphere, I_{X, e}, I_{O, e}.

An alternative procedure is to integrate Eq. 12 for i = X, O, +, − from y ≫ 1 to y ≪ 1 (without applying partial mode conversion), which gives the intensities I_X(0),I_O(0),I₊(0), and I₋(0) at y ≃ 0. Then, apply the partial conversion

$$\begin{array}{cc}& I_{X,e}=I_X(0)P_J+I_{+}(0)(1-P_J),\hfill \end{array}$$ [15]

$$\begin{array}{cc}& I_{O,e}=I_O(0)P_J+I_{-}(0)(1-P_J),\hfill \end{array}$$ [16]

where P_J is evaluated at y = y_V.

It is straightforward to show that the above two procedures are equivalent. For example, after obtaining I_XV′, we can get the emergent X-mode intensity by

$$\begin{array}{c}\hfill I_{X,e}=I_{\mathrm{XV}}^{\prime }exp\left(-\frac{{\tau }_{\mathrm{XV}}}{cos\alpha }\right)+\mathrm{\Delta }{I}_{\mathrm{XV}},\end{array}$$ [17]

where τ_XV = ∫₀^y_Vκ_X dy is the optical depth of X-mode (measured along the surface normal) at y = y_V, and ΔI_XV is the contribution to I_{X, e} from the region 0 < y < y_V:

$$\begin{array}{c}\hfill \mathrm{\Delta }{I}_{\mathrm{XV}}={\int }_0^{{\tau }_{\mathrm{XV}}/cos\alpha }exp\left(-\frac{{\tau }_X}{cos\alpha }\right)\frac{1}{2}{B}_{\nu }(T)\frac{\mathrm{d}{\tau }_X}{cos\alpha }.\end{array}$$ [18]

On the other hand, when integrating Eq. 12 for i = X from y ≫ 1 to y ≪ 1, we obtain

$$\begin{array}{c}\hfill I_X(0)=I_{\mathrm{XV}}exp\left(-\frac{{\tau }_{\mathrm{XV}}}{cos\alpha }\right)+\mathrm{\Delta }{I}_{\mathrm{XV}}.\end{array}$$ [19]

Similarly,

$$\begin{array}{c}\hfill I_{+}(0)=I_{\mathrm{OV}}exp\left(-\frac{{\tau }_{\mathrm{XV}}}{cos\alpha }\right)+\mathrm{\Delta }{I}_{\mathrm{XV}}.\end{array}$$ [20]

It is easy to see that Eq. 17 together with Eq. 13 (the first procedure) and Eq. 15 with Eqs. 19–20 (the second procedure) yield the same emergent I_{X, e}.

Photosphere Densities and Critical Field.

Before presenting our sample results, it is useful to estimate the photosphere densities for different modes and the condition for polarization swing.

When the vacuum polarization effect is neglected (ρ_V = 0), |β|≫1 at all densities (for typical θ_kB’s not too close to 0), the ξ-factors for the O-mode (|K|≫1) and for the X-mode (with |K|≪1) are

$$\begin{array}{c}\hfill {\xi }_O\simeq {sin}^2{\theta }_{\mathrm{kB}},{\xi }_X\simeq \frac{1}{u_e{sin}^2{\theta }_{\mathrm{kB}}}.\end{array}$$ [21]

The photospheres of the O-mode and X-mode photons are determined by the condition

$$\begin{array}{c}\hfill {\int }_0^{y_{O,X}}{\kappa }_{O,X}\frac{\mathrm{d}y}{cos\alpha }=2/3.\end{array}$$ [22]

The corresponding photosphere densities can be estimated as

$$\begin{array}{cc}& {\rho }_O={\xi }_O^{-1/2}{\rho }_0,\hfill \end{array}$$ [23]

$$\begin{array}{cc}& {\rho }_X={\xi }_X^{-1/2}{\rho }_0,\hfill \end{array}$$ [24]

where the “zero-field” photosphere density is

$$\begin{array}{c}\hfill {\rho }_0\simeq 0.59{\left(\frac{\mu g_{2}cos\alpha }{\mathcal{G}}\right)}^{1/2}{\left(\frac{E_1}{Z}\right)}^{3/2}(\frac{A}{T_6^{1/4}}){\mathrm{g}\mathrm{cm}}^{-3}.\end{array}$$ [25]

The effect of the vacuum resonance on the radiative transfer depends qualitatively on the ratios ρ_V/ρ_O and ρ_V/ρ_X, given by

$$\begin{array}{c}\hfill \frac{{\rho }_V}{{\rho }_O}={\left(\frac{B}{B_{\mathrm{OV}}}\right)}^2,\frac{{\rho }_V}{{\rho }_X}=\frac{B}{B_{\mathrm{XV}}},\end{array}$$ [26]

where

$$\begin{array}{cc}& B_{\mathrm{OV}}=7.8\times {10}^{13}{\left(\frac{\mu g_{2}cos\alpha }{Z\mathcal{G}{E}_1{sin}^2{\theta }_{\mathrm{kB}}}\right)}^{1/4}\left(\frac{f}{T_6^{1/8}}\right)\mathrm{G},\hfill \end{array}$$ [27]

$$\begin{array}{cc}& B_{\mathrm{XV}}=7.1\times {10}^{16}{\left(\frac{\mu g_{2}cos\alpha }{Z\mathcal{G}{E}_1^3}\right)}^{1/2}\left(\frac{f^{2}sin{\theta }_{\mathrm{kB}}}{T_6^{1/4}}\right)\mathrm{G}.\hfill \end{array}$$ [28]

Clearly, the condition B ≪ B_XV or ρ_V ≪ ρ_X is satisfied for almost all relevant NS parameters of interest, while B_OV defines the critical field strength for the 90° polarization swing (Figs. 3 and 4): If B ≲ B_OV, the emergent radiation is dominated by the X-mode for E ≲ E_ad and by the O-mode for E ≳ E_ad; if B ≳ B_OV, the X-mode is dominant for all E’s.

Fig. 3.

A schematic diagram illustrating how vacuum resonance affects the polarization state of the emergent radiation from a magnetized NS atmosphere. This diagram applies to the B ≲ B_OV regime so that the vacuum resonance density ρ_V is less than the O-mode photosphere density ρ_O. For E ≲ E_ad, the photon evolves nonadiabatically across the vacuum resonance (for θ_kB not too close to 0), thus the emergent radiation is dominated by the X-mode. For E ≳ E_ad, the photon evolves adiabatically, with its plane of polarization rotating by 90° across the vacuum resonance, and thus, the emergent radiation is dominated by the O-mode. The plane of linear polarization at low energies is therefore perpendicular to that at high energies.

Fig. 4.

Same as Fig. 3, except for the B ≳ B_OV regime (ρ_V > ρ_O), in which the emergent radiation is always dominated by the X-mode for all E’s.

We can quantify the role of B_OV more precisely by estimating how vacuum resonance affects the photosphere densities. In the limit of no mode conversion (i.e., E ≪ E_ad), it is appropriate to consider the transport of the X-mode and O-mode, with the mode opacities modified around the vacuum resonance (Fig. 2). The O-mode opacity has a dip near ρ = ρ_V (where ξ ≃ sin²θ_kB/2), and since the resonance width Δρ/ρ_V ≪ 1, the photosphere density ρ_O′ is almost unchanged from the no-vacuum value, i.e., ρ_O′ ≃ ρ_O. On the other hand, the X-mode opacity has a large spike at ρ = ρ_V (where ξ = sin²θ_kB/2) compared to the off-resonance value (ξ ∼ u_e⁻¹). The X-mode optical depth across the resonance (from ρ_V − Δρ to ρ_V + Δρ) is of order Δτ_V ∼ ϵ(ρ_V/ρ_O)², where ϵ is given by Eq. 4. Thus, the modified X-mode photosphere density is ρ_X′ ≃ ρ_V for Δτ_V ≳ 1 and

$$\begin{array}{c}\hfill {\rho }_{X^{\prime }}\sim {\rho }_X{\left[1-ϵ{\left({\rho }_V/{\rho }_O\right)}^2\right]}^{1/2},\end{array}$$ [29]

for Δτ_V ≲ 1.

In the limit of complete mode conversion (i.e., E ≫ E_ad), it is appropriate to consider the transport of ( + )-mode and ( − )-mode, with the mode opacities exhibiting a discontinuity at ρ = ρ_V (Fig. 2). The ( + )-mode photosphere density ρ₊ is given by

$$\begin{array}{c}\hfill {\rho }_{+}^2\simeq {\rho }_O^2+\left(1+\frac{{\xi }_X}{{\xi }_O}\right){\rho }_V^2\simeq {\rho }_O^2+{\rho }_V^2.\end{array}$$ [30]

The ( − )-mode photosphere density ρ₋ is affected by the vacuum resonance only if ρ_O > ρ_V. Thus,

$$\begin{array}{c}\hfill {\rho }_{-}={\rho }_O\mathrm{for}{\rho }_O<{\rho }_V,\end{array}$$ [31]

and

$$\begin{array}{c}\hfill {\rho }_{-}^2={\rho }_V^2+\frac{{\xi }_O}{{\xi }_X}\left({\rho }_O^2-{\rho }_V^2\right)\mathrm{for}{\rho }_O>{\rho }_V.\end{array}$$ [32]

For general E’s with partial mode conversion, the emergent mode intensities are approximately given by

$$\begin{array}{cc}& I_{O,e}\simeq \frac{1}{2}{P}_J{B}_{\nu }({\rho }_O)+\frac{1}{2}(1-P_J)B_{\nu }({\rho }_{-}),\hfill \end{array}$$ [33]

$$\begin{array}{cc}& I_{X,e}\simeq \frac{1}{2}{P}_J{B}_{\nu }({\rho }_{X^{\prime }})P_J+\frac{1}{2}(1-P_J)B_{\nu }({\rho }_{+}),\hfill \end{array}$$ [34]

where (for example) B_ν(ρ_O) is the Planck function evaluated at ρ = ρ_O. These results are schematically depicted in Figs. 3 and 4.

Results.

To compute the polarized radiation spectrum emergent from a NS atmosphere patch using the method presented above (Eq. 12 with Eqs. 13–14 or with Eqs. 15–16), we need to know the atmosphere temperature profile T(y). This can be obtained only by self-consistent atmosphere modeling, which has been done for only a small number of cases (in terms of the local T_s, $\mathbf{B}$ and composition). Here, to explore the effect of different $\mathbf{B}$ and compositions, we consider two approximate models:

• Model (i): We use the profile T_H(y) for the T_s = 5 × 10⁶ K H atmosphere model with a vertical field B = 10¹⁴ G [Fig. 5 and equation 48 of (33); this model correctly treats the partial model conversion effect] and rescale it to take account of the modification of the free–free opacity for different Z, A (so that the rescaled profile yields the same effective surface temperature T_s):

  $$\begin{array}{c}\hfill T(y)={\left(2\mu Z^3/A^2\right)}^{1/8.5}{T}_{\mathrm{H}}(y).\end{array}$$ [35]
• Model (ii): We use a smooth (monotonic) fit to the T_H profile, given by

  $$\begin{array}{c}\hfill {log}_{10}{T}_{\mathrm{H}}(y)=0.11+0.147\left[{log}_{10}(0.4y)+3\right],\end{array}$$ [36]

  and then apply Eq. 35 for rescaling.

Fig. 5.

Polarization degree P_L (defined by Eq. 37) of the emergent radiation normal to the surface as a function of the photon energy E for H and He atmospheres with different magnetic field strengths and directions (θ_B, the angle between the surface $\mathbf{B}$ and the surface normal vector). All results are based on the temperature profile Model (ii).

Figs. 5 and 6 show a sample of our results for the polarization degree of the emergent radiation, defined by

$$\begin{array}{c}\hfill P_L\equiv \frac{I_{X,e}-I_{O,e}}{I_{X,e}+I_{O,e}}.\end{array}$$ [37]

Fig. 6.

Similar to Fig. 5, except for partially ionized Fe atmospheres with Z = 2, A = 56. In the upper panel, the solid/dashed lines are for Model (i)/(ii). In the lower panel, the results are shown for Model (i).

We see that for the H and He atmospheres (Fig. 5), P_L transitions from being positive at low E’s to negative at high E’s for B₁₄ ≲ 0.5, in agreement with the critical field estimate (Eq. 27). The transition energy (where P_L = 0) is approximately given by E_ad and has the scaling E_ad ∝ (μtan²θ_B)^1/3, where θ_B is the angle between the surface $\mathbf{B}$ and the surface normal (Eq. 5). To obtain a transition energy of 4 to 5 keV (as observed for 4U 0142) would require most of the emission to come from the surface region with θ_B ≳ 70°.

On the other hand, for a partially ionized heavy-element atmosphere (such that μ/Z is much larger than unity), the critical field B_OV can be increased (Eq. 27). Fig. 6 shows that at B₁₄ = 1, a Z = 2, A = 56 atmosphere can have a sign change in P_L around E ∼ 3 − 5 keV depending on the θ_B value.

To determine the observed polarization signal, we must consider the propagation of polarized radiation in the NS magnetosphere, whose dielectric property in the X-ray band is dominated by vacuum polarization (39). As a photon propagates from the NS surface through the magnetosphere, its polarization state evolves following the varying magnetic field it experiences, up to the “polarization-limiting radius” r_pl, beyond which the polarization state is frozen. It is convenient to set up a fixed coordinate system XYZ, where the Z-axis is along the line of sight, and the X-axis lies in the plane spanned by the Z-axis and Ω (the NS spin angular velocity vector). The polarization-limiting radius r_pl is determined by the condition Δk = 2|dϕ_B/ds|, where Δk = |k_X − k_O| is the difference in the wavenumbers of the two photon modes, and ϕ_B(s) is the azimuthal angle of the magnetic field along the ray (s measures the distance along the ray). For a NS with surface dipole field B_d and spin frequency ν = Ω/(2π), we have (33) r_pl/R ∼ 150 (E₁B_{d, 14}²/ν₁)^1/6, where B_{d, 14} = B_d/(10¹⁴ G) and ν₁ = ν/Hz. Note that since R ≪ r_pl ≪ r_l (with r_l = c/Ω the light-cylinder radius), only the dipole field determines r_pl. Regardless of the surface magnetic field structure, the radiation emerging from most atmosphere patches with mode intensities I_{X, e} and I_{O, e} evolves adiabatically in the magnetosphere such that the radiation at r > r_pl consists of approximately the same I_{X, e} and I_{O, e}, with a small mixture of circular polarization generated around r_pl (33). The exception occurs for those rays that encounter the quasi-tangential point (where the photon momentum is nearly aligned with the local magnetic field) during their travel from the surface to r_pl (40). Since only a small fraction of the NS surface radiation is affected by the quasi-tangential propagation, we can neglect its effect if the observed radiation comes from a large area of the NS surface. Let the azimuthal angle of the $\mathbf{B}$ field at r_pl be ϕ_B(r_pl) in the XYZ coordinate system. The observed Stokes parameters (normalized to the total intensity I) are then given by

$$\begin{array}{cc}& Q/I=-P_{L}cos2{\varphi }_B(r_{\mathrm{pl}}),\hfill \end{array}$$ [38]

$$\begin{array}{cc}& U/I=-P_{L}sin2{\varphi }_B(r_{\mathrm{pl}}).\hfill \end{array}$$ [39]

Note that when r_pl ≪ r_l, the transverse (XY) component of $\mathbf{B}(r_{\mathrm{pl}})$ is opposite to the transverse component of the magnetic dipole moment $\mathit{\mu }$; thus, ϕ_B(r_pl)≃π + ϕ_μ, where ϕ_μ is the azimuthal angle of $\mathit{\mu }$.

To determine the emission from the whole NS surface, we need to add up contributions from different patches (area element dS), including the effect of general relativity (41, 42). For example, the observed spectral fluxes F_I, F_Q (associated with the intensities I, Q) are

$$\begin{array}{cc}& F_I=g_r^3\int \frac{\mathrm{d}Scos\alpha }{D^2}(I_{X,e}+I_{O,e}),\hfill \end{array}$$ [40]

$$\begin{array}{cc}& F_Q=g_r^3\int \frac{\mathrm{d}Scos\alpha }{D^2}(I_{O,e}-I_{X,e})cos2{\varphi }_B(r_{\mathrm{pl}}),\hfill \end{array}$$ [41]

where g_r ≡ (1 − 2GM/Rc²)^1/2 and α is the angle between the ray and the surface normal at the emission point). Clearly, to compute F_I, F_Q and F_U requires the knowledge of the distributions of NS surface temperature and magnetic field, as well as various angles (the relative orientations between the line of sight, the spin axis, and the dipole axis). This is beyond the scope of the paper. [Note that since ϕ_B(r_pl)≃π + ϕ_μ, the phase variation of F_Q, F_U follows the rotation of the magnetic dipole, as in the rotating vector model.] Nevertheless, our results depicted in Figs. 5 and 6 (with different values of local field strengths and orientations) show that the 90° linear polarization swing observed in AXP 4U 0142 can be explained by emission from a partially ionized heavy-element atmosphere with surface field strength about 10¹⁴ G or a H/He atmosphere with B₁₄ ≲ 0.5 and a more restricted surface field geometry (i.e., most of the radiation comes from the regions with θ_B ≳ 70°).

Discussion

We have shown that the observed X-ray polarization signal from AXP 4U 0142, particularly the 90° swing around 4 to 5 keV, can be naturally explained by the mode conversion effect–associated vacuum resonance in the NS atmosphere. In this scenario, the 2 to 4 keV emission is dominated by the X-mode, while the 5 to 8 keV emission by the O-mode as a result of the adiabatic mode conversion from the X-mode to the O-mode. This interpretation of the polarization swing would imply that the NS’s kick velocity is aligned with its spin axis, in agreement with the spin–kick correlation observed in other NS systems.

It is important to note that in our scenario, the existence of the X-ray polarization swing depends sensitively on the actual value of the magnetic field on the NS surface (Eq. 27). To explain the observation of AXP 4U 0142, the magnetic fields in most region of the NS surface must be less than about 10¹⁴ G, and a lower field strength would be preferred in terms of producing the polarization swing robustly (for a wide range of geometrical parameters). Using the pulsar spindown power derived from force-free electrodynamics simulations (43), L_sd = (μ²Ω⁴/c³)(1 + sin²θ_sμ) (where θ_sμ is the angle between the magnetic dipole $\mathit{\mu }$ and the spin axis), we find that for the observed $P,\dot{P}$ of 4U 0142, the dipole field at the magnetic equator is $B_d=1.1\times {10}^{14}{I}_{45}^{1/2}{R}_6^{-3}{(1+{sin}^2{\theta }_{s\mu })}^{-1/2}$ G, where R₆ is the NS radius in units of 10⁶ cm, and I₄₅ is the moment of inertia in units of 10⁴⁵ g cm². Note that with R₆ ≃ 1.3 (44, 45), the above estimate is reduced by a factor of 2. In addition, if the magnetar possesses a relativistic wind with luminosity L_w ≳ L_sd, the wind can open up field lines at r_open ∼ (B_d²R⁶c/L_w)^1/4 ≲ c/Ω and significantly enhance the spindown torque (46–48). This would imply that a smaller B_d is needed to produce the observed $\dot{P}$ in 4U 0142. Thus, the observation of X-ray polarization swing is consistent with the indirectly “measured” dipole field and requires that the high-order multipole field components be not much stronger than the dipole field.

We reiterate some of the caveats of our work: We have not attempted to calculate the synthetic polarized radiation from the whole NS and to compare with the X-ray data from AXP 4U 0142 in detail; our treatment of partially ionized heavy-element magnetic NS atmospheres is also approximate in several aspects (e.g., bound–free opacities are neglected; the vertical temperature profiles are assumed based on limited H atmosphere models). At present, these caveats are unavoidable, given the uncertainties (and large parameter space needed to do a proper survey) in the surface temperature and magnetic field distributions on the NS and the fact that self-consistent heavy-element atmospheres models for general magnetic field strengths and orientations have not been constructed, especially in the regime where the vacuum resonance effects are important.

In this work, we have interpreted the 2 to 8 keV polarization signal from AXP 4U 0142 in terms of thermal emission from the NS surface. It is well established that for quiescent magnetars, the spectrum turns up above 10 keV, such that the bulk of their energy comes out as nonthermal hard X-rays (2, 49). The 0.5 to 10 keV spectrum can be parameterized either by an absorbed blackbody plus a power-law component (of a photon index between −4 and −2) or by the sum of several blackbodies. So it is possible that the 5 to 8 keV emission from AXP 4U 0142 has a significant nonthermal contribution. Thompson and Kostenko (50) have studied a model in which the hard X-ray emission of quiescent magnetars comes from the “annihilation bremsstrahlung” of an electron–positron magnetospheric plasma. Such emission is mainly polarized in the O-mode. If this emission dominates the 5 to 8 keV spectrum of APX 4U 0142 while negligible at lower energies, it could provide an alternative explanation for the polarization swing for a wide range of surface field strengths (recall that for B ≳ B_OV, the atmosphere thermal emission is dominated by the X-mode for all E’s (Eq. 27).

Very recently, while this paper was under review, IXPE reported the detection of polarized X-rays from another magnetar, 1RXS J170849.0-400910 (16): The phase-averaged polarization signal exhibits an increase with energy, from ∼20% at 2 to 3 keV to ∼80% at 6 to 8 keV, while the polarization angle is independent of the photon energy. This constant polarization angle is consistent with atmospheric emission dominated by the X-mode at all energies and is expected since the measured dipole field (based on spindown) for this AXP, B_d ≃ 5 × 10¹⁴ G, is significantly larger than B_OV (Eq. 27). The increase of linear polarization with E can be a result of the magnetic field geometry (relative to the spin axis and line-of-sight) and the surface temperature distribution [ref. 33 for examples].

Overall, our work demonstrates the important role played by the vacuum resonance in producing the observed X-ray polarization signature from magnetars (and NSs with weaker magnetic fields). The observations of AXP 4U 0142 and 1RXS J170849.0-400910 by IXPE have now opened up a new window in studying the surface environment of NSs. Future X-ray polarization mission [such as eXTP; ref. 51] will provide more detailed observational data. Comprehensive theoretical modelings of magnetic NS surface radiation and magnetosphere emission will be needed to confront these observations.

Data, Materials, and Software Availability

All study data are included in the main text.