Polarized X-Rays Detected from the Anomalous X-Ray Pulsar 1E 2259+586

Anomalous X-ray pulsars (AXPs) and soft gamma-ray repeaters (SGRs) are believed to be associated with magnetars, which have extremely strong magnetic fields. Recently, with the operation of the Imaging X-ray Polarimetry Explorer (IXPE), the polarization information of two AXPs and one SGR have been investigated. In this work, we report the observational results of the fourth magnetar, 1E 2259+586, with IXPE, and perform a joint analysis with observations from Neutron Star Interior Composition Explorer. We find that the emission from 1E 2259+586 is linearly polarized, with a polarization degree (5.3% ± 1.3%) and a polarization angle −77° ± 7° in the 2–8 keV energy range. Additionally, both the polarization degree and polarization angle exhibit variability with the pulse phase, and there is a hint of anticorrelation between the polarization degree and the flux, which is similar to AXP 1RXS J170849.0-400910. The phase-dependent polarization angle displays a sinusoidal profile and can be well fitted with the rotating vector model, indicating that the magnetic dipole field dominated the magnetic structure of the pulsar, and the variation in the polarization angle was modulated by the pulsar’s rotation.


Introduction
Magnetars are a small class of young neutron stars (NSs) characterized by bright X-ray emissions (L X ∼ 10 31 -10 36 erg s −1 ) and various activities, including short bursts, intermediate flares, giant flares, and long-term outbursts, which are believed to involve their extremely strong magnetic fields (see Kaspi & Beloborodov 2017, for reviews).The current magnetar catalog contains 16 soft gamma repeaters (SGRs), and 14 anomalous X-ray pulsars (AXPs),5 which have long spin periods (P ∼ 1-12 s) and large spin-down rates ( - ~--P 10 10 15 10 s s −1 , Olausen & Kaspi 2014).Typically, the inferred dipolar magnetic field is of 10 13 -10 15 G, which is much greater than those of normal radio rotation-powered pulsars (RPPs).However, two low-magnetic-field magnetars (B 10 12 G) have been discovered due to the short bursts and the transient behavior (Rea et al. 2010;Zhou et al. 2014).Meanwhile, the derived spin-down powers of AXPs and SGRs are significantly smaller than their observed X-ray luminosity; hence, both of them are referred to as magnetars, i.e., powered by ultrastrong magnetic fields.
In contrast to RPPs, magnetars are X-ray bright, and their magnetospheres are expected to be twisted due to the deformation of the NS crust (Thompson et al. 2002;Beloborodov & Thompson 2007;Lander 2023).The twisted magnetosphere can support dense mildly relativistic charges, which further supply a significant optical depth to resonant Compton scattering (RCS) in the X-rays.The soft X-ray phaseaveraged spectra of magnetars can be empirically fitted by the two-component model, a blackbody (BB) emission from the NS surface and a steep power-law (PL) component due to the reprocessing of soft photons through RCS in the twisted magnetosphere (Lyutikov & Gavriil 2006;Fernández & Thompson 2007).Besides, an additional hard PL component has been detected above 10 keV from a few magnetars by INTEGRAL, RXTE, SUZAKU, and NuSTAR (e.g., Kuiper et al. 2006;Vogel et al. 2014;Weng & Göğüs 2015), which, however, cannot be accounted by the RCS.Alternatively, this component could result from the resonant inverse Compton scattering of soft photons by another population of ultra-relativistic electrons (Baring & Harding 2007;Beloborodov 2013;Wadiasingh et al. 2018Wadiasingh et al. , 2019)).In this work, we focus on the soft X-rays from magnetars.
Both the spectral profile and the polarization of soft X-rays from magnetars are modified by not only the twisted magnetosphere but also the thin atmosphere due to the vacuum polarization and the proton cyclotron resonances (e.g., Lai & Ho 2002;Özel 2003;van Adelsberg & Lai 2006;Barchas et al. 2021;Hu et al. 2022).As such, we can explore the nature of the magnetar surface and the surrounding magnetosphere by investigating the properties of soft X-ray emissions (Harding & Lai 2006;Fernández & Davis 2011;Lai 2023).The twisted/ untwisted magnetosphere scenario is well consistent with the observational data of some outbursts of magnetars (Beloborodov 2009;Weng et al. 2015;Hu et al. 2020).But, in details, the axisymmetric magnetosphere assumed in the standard theories is oversimplification (Thompson et al. 2002;Fernández & Davis 2011), and it was inconsistent with some observations (e.g., the crustal motion found in Original content from this work may be used under the terms of the Creative Commons Attribution 4.0 licence.Any further distribution of this work must maintain attribution to the author(s) and the title of the work, journal citation and DOI.Younes et al. 2022).Meanwhile, it is expected that X-ray emissions from magnetars exhibit strong polarization (Taverna et al. 2020;Caiazzo et al. 2022).Therefore, more precise information about the geometry of twisted magnetic fields could be achieved by the X-ray polarimetry observations (e.g., Fernández & Davis 2011;Tong et al. 2021).
The Imaging X-ray Polarimetry Explorer (IXPE; Weisskopf et al. 2022) is the first telescope entirely devoted to X-ray imaging polarimetry.Since its launch on 2021 December 9, IXPE has conducted precise polarimetric measurement for tens of diverse X-ray sources, including four magnetars.The IXPE observation of the first magnetar, the AXP 4U 0142+61, revealed an averaged linear polarization degree (PD) of ∼13.5% in the 2-8 keV.Both the PD and the polarization angle (PA) vary with the photon energies, which can be explained by the RCS model (Taverna et al. 2022).In contrast, the AXP 1RXS J170849.0-400910(1RXS J1708 hereafter) presents a much stronger polarization signal (PD ∼ 35%), and the PD monotonically increases from 20% at 2-3 keV up to 80% at 6-8 keV with a constant PA (Zane et al. 2023).Alternatively, no significant polarization was detected from SGR J1806-20 (Turolla et al. 2023).In this work, we analyze the IXPE observation of the fourth magnetar, 1E 2259+586, to expand the sample.
1E 2259+586 is located in the supernova remnant CTB 109 (Fahlman & Gregory 1981) at a distance of 3.2 ± 0.2 kpc (Kothes & Foster 2012).The rotation power derived from its spin period (P ∼ 6.98 s and  ~´- P 5 10 13 s s −1 ) is orders of magnitude lower than the observed X-ray luminosity; therefore, 1E 2259+586 is considered as the AXP prototype (Gavriil & Kaspi 2002).1E 2259+586 entered into an outburst in 2002 with over 80 SGR-like short bursts detected, implying the connection between SGRs and AXPs (Gavriil et al. 2004).Dramatically, applying the three-dimensional surface thermal emission and magnetospheric scattering model to the XMM-Newton spectra of 1E 2259+586 during its 2002 outburst, we found that an increased twist angle was coincident with the outburst and then decreased following the flux decayed (Weng et al. 2015).The evolution trend was well consistent with the twisting/untwisting magnetosphere scenario (Thompson et al. 2002).In 2012, 1E 2259+586 entered into another episode, where it exhibited an unexpected antiglitch behavior (i.e., a sudden decrease in spin frequency; Archibald et al. 2013;Younes et al. 2020).
As an important source of study on the evolution of magnetospheric geometry, 1E 2259+586 has been regular monitored by Neutron Star Interior Composition Explorer (NICER) and was observed by IXPE recently.In this work, we report on the timing and polarimetric analyses of about 7 months of NICER data and the IXPE observation.The paper is organized as follows.The data analysis is described in Section 2, the results are interpreted with the rotating vector model (RVM; Radhakrishnan & Cooke 1969;Levin et al. 2012;Tong et al. 2021) in Section 3, and the conclusion follows in Section 4.

Observation and Data Reduction
IXPE is NASAʼs dedicated X-ray polarimetry mission, which has on board three identical units of gas iixel detectors, DU1, DU2, and DU3 (Baldini et al. 2022;Weisskopf et al. 2022).Three units independently record the spatial, time, and energy-resolved polarimetric information (Di Marco et al. 2022).IXPE has observed 1E 2259+586 in 2023 June.We perform the polarimetric analysis using the software suite ixpeobssim 29.2.01 (Baldini et al. 2022) for phase-resolved results, which is equipped with various tools to process IXPE level 2 event files and produce scientific results.The data process is described as follows: (1) A circular region (radius of 60″) and an annular region (inner and outer radii of 180″ and 240″) are selected for the extraction of source and background events, respectively.(2) The polarization cubes of the source including count I and Stokes Q and U are generated with tool XPBIN for energy range 2-8 keV for specific phase ranges (Kislat et al. 2015), which are calculated with the ephemeris in Table 2. (3) The polarization cubes of the background are obtained similarly but for all phase.(4) The version of response matrices for spectral fitting is v012 of ixpeobssim.
(5) The phase-resolved PA and PD are estimated from polarization cubes according to the method discussed in Kislat et al. (2015).
NICER is an International Space Station payload dedicated to the study of NSs through soft X-ray (0.2-12 keV) timing.1E 2259+586 has been regularly monitored by NICER since  its launch on 2017 June 3.In this work, we focus on the 13 pointed observations made in the first half year of 2023 (Table 1), in order to coordinate the data analysis on the IXPE observations.We use HEASOFT v6.31.1 to process the NICER data, employing the default standard procedure of nicerl2 to generate cleaned event data.Subsequently, the event data in 0.5-5 keV are extracted with xselect, and are applied for barycentric corrections with the task barycorr using the source coordinates R.A. = 345°.284563,decl.= 58°.879014 (Hulleman et al. 2001).Meanwhile, the default standard  procedure of using nicerl3-spect is employed to generate the spectral data.

Timing Process
We utilize a fully phase-coherent timing (FPCT) analysis to obtain the timing solution of 1E 2259+586 using TEMPO2 (Hobbs et al. 2006).To perform the FPCT analysis, we require spin frequencies and time of arrivals (ToAs) at different epochs.They are obtained through χ 2 searching, where the frequency causing the folded profile to deviate the most from a uniform distribution, as indicated by the highest χ 2 value, is considered as the spin frequency at each observation time.Additionally, the phase of the maximum point in each profile is taken as the ToA for that specific observation (Ge et al. 2012).Finally, the FPCT analysis is performed for ToAs in about 196 days from MJD 59952 to 60148 with Equation (1) in TEMPO2 where f 0 , ν 0 ,  n 0 are phase, frequency, and frequency derivative at the reference time t 0 .With the fitting result, the spin parameters are given in Table 2.

RVM Fitting
We utilize the RVM, which is based on the assumption of a dipole magnetic field, to model the variation in PA with respect to the pulse phase (Radhakrishnan & Cooke 1969;Poutanen 2020;Doroshenko et al. 2022).The PA is determined by the orientation of the pulsar's magnetic axis and is modulated on the sky plane due to the pulsar's rotation.In the framework of the RVM, if radiation escapes in the O-mode of plasma (Lai & Ho 2002), the PA can be described by the following formula: where i p represents the angle between the pulsar spin vector and the line of sight, θ is the magnetic obliquity, χ p is the position angle of the pulsar spin axis, f corresponds to the pulse phase, and f 0 indicates the phase when the magnetic pole is closest to the observer.The fitting process is carried out employing the EMCEE package, a Python implementation of an affine-invariant Markov Chain Monte Carlo ensemble sampler (Foreman-Mackey et al. 2013).The errors of parameters could be estimated from the Markov Chain.Here, all errors of parameters are reported with 1σ confidence (68%) level.

Timing Results
We combine the ToAs of NICER and IXPE, and the timing residuals are shown in Figure 1 with the spin parameters listed in Table 2.The radiation from 1E 2259+586 remains relatively stable, with a count rate of ∼16.4 cnts s −1 for NICER and 0.35 cnts s −1 for IXPE.It can be seen that the timing results of NICER and IXPE are consistent with each other, and the overall timing results are satisfactory.The obtained results (ν ∼ 0.143 Hz and  n ~-´-9.5 10 15 Hz s −1 ) are in broad agreement with previous measurements (Dib & Kaspi 2014;Vogel et al. 2014;Younes et al. 2020).Due to the consistent results of joint timing with NICER and IXPE, we utilize the same spin parameters to fold the pulse profiles, and both profiles are binned into 50 intervals.To show a clear comparison of the profiles, both profiles are normalized with minimum 0 and maximum 1, as shown in Figure 1.Pulse profiles obtained from NICER and IXPE are consistent with each other, exhibiting a double-peaked, nearly sinusoidal profile.The two peaks are not the same, they can be seen as a main peak and a secondary peak, and the main peak has larger maximum and minimum values compared to the secondary peak.These features are also consistent with the previous reports (Pizzocaro et al. 2019).Meanwhile, the pulse profile of 1E 2259 +586 is also similar to the those observed before and after 2002 outburst (Woods et al. 2004).The time span for IXPE is 27 days,  while NICER covers a span of 196 days, and no glitches, bursts, or other activities were observed within the time span of this work.

Polarization Results
We apply the two BB components combined with a constant polarization model (Tbabs * POLCONST * (BB+BB) in XSPEC) to fit the IXPE +NICER spectra and polarization results of 1E 2259+586.An additional multiplicative factor (constant) is added to account for difference in normalization of IXPE and NICER.In addition, we also fit the spectra and polarization results with BB and PL (Tbabs * POLCONST * (BB+POW)) as described in Heyl et al. (2023).The fitting result is shown in Figure 2, and the fitting parameters are listed in Table 3.However, we cannot obtain more details about energy-dependent polarization from the low signal-to-noise ratio data.So the polarization results only in 2-4 and 4-8 keV are analyzed as listed in Table 4.
We found the emission is linearly polarized with PD = (5.3%± 1.3%), and PA = −77°± 7°(east of north) for both models in the 2-8 keV range, which is consistent with the result from Heyl et al. (2023).In this energy range, compared to the polarimetric results reported for the three magnetars prior to IXPE, 1E 2259+586 exhibits a lower degree of polarization, with 4U 0142+61 having a PD of ∼13.5%, while 1RXS J1708 achieves the highest PD of 35%.Although significant polarization from SGR J1806-20 was detected only in the 4-5 keV energy range, it exhibits approximately 31% within this range.
We analyze the variation of PA and PD with pulsar phase and both of them are binned into eight intervals, in the energy range of 2-8 keV.Additionally, We provide the corresponding values of minimum detectable polarization (MDP) for comparison, and also plot the pulse profile together.The results are shown in Figures 3 and 4. We find that both the PA and PD exhibit variability with pulse phase.In terms of the PD, the variation profile also shows a roughly double-peak structure, and there is a hint of anticorrelation between the PD and the flux.Similar to 1RXS J1708 and 4U 0142+61, there is also some similarity between the variation profile of the PD and the pulse profile.In case 4U 0142+61, both the PD and the pulse profile exhibit a double-peak profile, and there is a slight correlation between them, in which the main and secondary peak of the light curve exhibit a higher polarization fraction compared to the valley, in the energy range of 2-4 keV (Taverna et al. 2022).In the case of 1RXS J1708, both exhibit a single-peak structure but showing a broadly anticorrelated with them, in the energy ranges of 2-4 and 2-8 keV (Zane et al. 2023).For the variation of the PA, we fit the PA data using RVM, and the fitting results are consistent with the observed data, also shown in Figure 4. Unlike the complex variability of PD, both 1RXS J1708 and 4U 0142 +61 exhibit a simple sinusoidal modulation shape similar to 1E 2259+586, and the results fitted with the RVM are all satisfactory.Additionally, no correlation has been found between the variation of PA with the changes in the PD or flux.These results indicate that the variations of PA still align with the theoretical prediction of RVM framework.

Discussion and Summary
Magnetars are highly magnetized NSs and are considered one of the most promising targets for detecting X-ray polarization.In the recent detection results of IXPE, the polarization results of 4U 0142+61, 1RXS J1708, and SGR J1806-20 have been reported (Taverna et al. 2022;Turolla et al. 2023;Zane et al. 2023).Here, we report the polarization information of the emission from 1E 2259+586, as well as the results of PD and PA variation with phase.The detected PD is (5.3% ± 1.3%), which is evidently low, even compared to previous magnetar detection results by IXPE, which results are also lower than expected.Additionally, the using of two BB components fit to the spectrum provides a better fit.However, it is unclear whether this can explain the double-peak structure of the pulsar and whether there is no component of RCS.This is because, based solely on the insignificant value of PD, the change in PA reaches approximately 120°, which theoretically can partially cancel out the PD and also display a low polarization result.Therefore, further theoretical research on magnetar polarization requires more samples and comprehensive analysis.
Meanwhile, we find that the variation of PD in 1E 2259+586 may also exhibit a double-peak structure and seems to have some slight anticorrelation with the pulse profile.In 4U 0142+61, the variation of PD and the pulse profile may also have a similar structure (double-peak) and show some correlation.In 1RXS J1708, both of them exhibit a single-peak structure but show a broadly anticorrelation.Due to the low significance of the data results and the scarcity of similar correlated/anticorrelated samples, it is not possible to draw a definitive conclusion about the true situation.The operation of IXPE and the enhanced X-ray Timing and Polarimetry mission (Zhang et al. 2016) in the near future will enlarge the sample and promote the studies on magnetar polarization.Compared to the complex nature of PD, the change in PA is relatively simple and conforms to expectations.The observed sinusoidal dependence of PA with respect to pulse phase can also be explained by the basic assumptions of X-ray pulsar modeling.The results of the RVM fitting indicate that the variation of PA with pulse phase is solely due to the modulation caused by the pulsar's rotation, which is theoretically expected to be purely geometric and not related to changes in the PD or flux.In fact, the results confirm this expectation.We used the RVM with four free parameters: χ p , θ, i p , and f 0 , of which the range and form of the priors are −90 χ p 90, 0 θ 90, -  i 1 cos 1 p , and  i sin 0 p , −0.5 f 0 0.5.Meanwhile, the number of walkers, burn-in, and steps taken are 100, 2000, 12,000, respectively.Finally, the resulting posterior distributions are shown in Figure 5, and the results of the four parameters are also shown in Table 5 for clarity.
We note that Heyl et al. (2023) have released their analysis results on arXiv when our work was submitted.We clarify that our work is finished independently with the work done by Heyl et al. (2023), and the main results are consistent with each other.

Figure 1 .
Figure 1.Panel (A): the pulse profiles of 1E 2259+586 with NICER and IXPE.For better comparison, both profiles have been normalized with minimum 0 and maximum 1.Two cycles are shown for clarity.Panel (B): the timing residuals of 1E 2259+586.In all panels, red and blue data represent IXPE and NICER, respectively.

Figure 2 .
Figure 2. The red and blue data in panel (A) represent the spectrum of IXPE and NICER for 1E 2259+586, respectively.The blue, green, and gray data in panels (B), (C), and (D) represent the results of three detector units for the Stokes parameters I, Q, and U.The corresponding residual results for each panel are shown below their respective plots.

Figure 3 .
Figure 3.The normalized Stokes parameters of 1E 2259+586 for 2-8 keV.The red square represents phase-averaged parameters while the green circles represent phase-resolved parameters.The blue circle means MDP for the integrated data.

Figure 4 .
Figure 4.The polarization results of 1E 2259+586.Panel (A): the blue curve represents the variation of polarization degree as function of phase, and the magenta curve represents the corresponding minimum detectable polarization at 99% confidence level.Panel (B): the green curve represents the variation of polarization angle with phase, and the red dashed line shows the best-fitting approximation for the PA with the rotating vector model.In both panels, the gray curve represents the pulse profile obtained from IXPE.

Table 1
The Observations of NICER and IXPE

Table 4
The Polarization Results for the Two Energy Bands and Phase-resolved ResultsNote.The phase-resolved results in the middle and lower parts of the table are for the energy ranges of 4-8 keV and 2-8 keV, respectively.