Observations of a GX 301–2 Apastron Flare with the X-Calibur Hard X-Ray Polarimeter Supported by NICER, the Swift XRT and BAT, and Fermi GBM

In this paper, we report on phase-resolved spectro-polarimetric observations of the accretion-powered, highly magnetized X-ray pulsar GX 301−2 with the X-Calibur balloon-borne mission (see Figure 1; Krawczynski et al. 2011; Guo et al. 2013; Beilicke et al. 2014, 2015; Kislat et al. 2017, 2018) in late 2018 December. The observations were accompanied by spectro-temporal observations in overlapping and adjacent periods by the Neil Gehrels Swift Observatory (Swift) Burst Alert Telescope (BAT; Barthelmy et al. 2005), the Swift X-ray Telescope (XRT; Burrows et al. 2007), the Neutron star Interior Composition Explorer Mission (NICER) X-ray telescope (Gendreau et al. 2012), and the Fermi Gamma-ray Burst Monitor (GBM; Meegan et al. 2009). The observations covered a particularly interesting epoch in which the pulsar exhibited rare flaring activity associated with a substantial pulsar spin up.

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The pulsar is in an orbit of period ∼41.5 days and eccentricity 0.462 about the star Wray 977, also known as BP Crucis (Koh et al. 1997; Sato et al. 1986; Doroshenko et al. 2010), an extremely bright B1 Ia hypergiant at a distance of kpc (Gaia Collaboration 2018). Wray 977 has an estimated mass of ∼39–63 M_⊙, a radius of au, and shines with a bolometric luminosity of ∼5 × 10⁵ L_⊙ (Kaper et al. 2006; Clark et al. 2012). The pulsar has a spin period of ∼680 s (White et al. 1976) and a 2–10 keV luminosity of 10³⁷–10³⁸ erg s⁻¹ (Liu et al. 2018). The pulsar displays bright flares prior to periastron at an orbital phase of ∼0.93 (Leahy & Kostka 2008). Although GX 301−2 is in a tight orbit with a hypergiant star (its semimajor axis is ∼19–30 au), the X-ray light curves do not show any evidence for eclipses. Parkes et al. (1980), Kaper et al. (2006), and Leahy & Kostka (2008) estimate the inclination (angle between the binary angular momentum vector and the observer) to lie between 44° and 78°.

GX 301−2 is believed to accrete from the wind of its companion, and possibly from a plasma stream (Leahy & Kostka 2008) or a temporary accretion disk (Koh et al. 1997; Nabizadeh et al. 2019). As the material sinks toward the neutron star, it latches onto the magnetic field lines at the magnetospheric radius r_m ∼ 2000–3000 km from the center of the neutron star (Lipunov 1992; Mészáros 1992, and references therein). Transferring its angular momentum to the neutron star, the plasma moves along the magnetic field lines until it dissipates its kinetic energy either in a radiative shock above the neutron star surface or in a hydrodynamic shock right at the neutron star surface (Basko & Sunyaev 1975, 1976; Becker et al. 2012; Mushtukov et al. 2015a). The X-ray emission is believed to form through the Comptonization of blackbody, bremstrahlung, and cyclotron seed photons emitted in and nearby the shocked plasma leading to a power law at low energies with an exponential cutoff in the 10–20 keV energy range (Becker & Wolff 2012; Farinelli et al. 2012; Postnov et al. 2015; West et al. 2017; Wolff et al. 2019).

The literature on accreting X-ray pulsars distinguishes between two idealized radiation patterns associated with the different locales for the energy dissipation, as illustrated in Figure 2. The dissipation in a radiative shock further up in the accretion column is believed to lead to a fan-shaped radiation pattern with most photons leaving the accretion column perpendicular to the flow direction (Davidson 1973). Emission associated with a hydrodynamic shock close to the neutron star surface is expected to lead to a more narrowly focused emission pattern resembling a pencil beam (Burnard et al. 1991; Nelson et al. 1993). Discriminating between these two scenarios is a prime goal of studies of X-ray binaries (XRBs), and X-ray polarimetry stands to play a decisive role.

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GX 301−2 observations with NuSTAR revealed two cyclotron resonant scattering features (CRSFs) with line centroids and Gaussian widths σ_CRSF of (E_CRSF, σ_CRSF) = (37 keV, 5 keV) and (50 keV, 8 keV) (Fürst et al. 2018; Nabizadeh et al. 2019). The CRSF energies and widths depend on time and on the pulsar phase (Kreykenbohm et al. 2004; Fürst et al. 2018; Nabizadeh et al. 2019). In XRBs, CRSFs are associated with electrons transitioning between quantized Landau levels, the transverse energy discretization relative to the magnetic field direction that emerges from the Dirac equation in quantum electrodynamics (QED). The observation of an electron  CRSF at energy constrains the magnetic field to be:

Here, the positive integer n is the harmonic number of the cyclotron transition. This relation applies to line features at energies significantly lower than m_ec², i.e., when B is much smaller than the quantum critical field , so that the harmonics are evenly spaced.

For a neutron star of mass M and an emission from radius r_em (measured from the center of the neutron star), the redshift z is approximately given by:

If the absorption features are interpreted as coming from one region, then the natural n = 3, 4 inference would yield Gauss, while an n = 2, 3 choice gives B ∼ 1.5 × 10¹² (1 + z) Gauss. In such a case, the absence of a prominent n = 1 fundamental at lower energies poses an issue. Thus, Fürst et al. (2018) interpret the two features as being fundamentals from distinct regions, in which case they possess higher fields, namely ∼3 × 10¹² G and ∼4.3 × 10¹² G (for z = 0), corresponding to cyclotron absorption radii differing by only around 12%. These fields are substantially above the values inferred from accretion torque models (see Table 1 of Staubert et al. 2019), the converse of what is usually obtained when comparing these two field estimates for X-ray binary pulsars. Some CRSFs are observed to depend on pulse phase, time, and luminosity (Staubert et al. 2019, and references therein). These variations are sometimes attributed to a movement of the radiative shock along the accretion column, or by changes in the magnetic field geometry (e.g., Becker et al. 2012; Mushtukov et al. 2015b).

The polarimetric capability of X-Calibur opens up a new degree of freedom in diagnosing the physical environment of GX 301−2. Observations of the linear polarization fraction and angle can provide qualitatively new information on the origin of X-rays in the accretion column or at its impact locale on the neutron star surface, on their birefringent propagation in the magnetosphere, and on the photon interaction cross sections.

The predictions of the polarization of the X-rays from highly magnetized neutron stars depend strongly on the strong-field QED predictions of the birefringence of the magnetized vacuum (Euler & Kockel 1935; Heisenberg & Euler 1936; Weisskopf 1936; Schwinger 1951; Toll 1952; Gnedin & Pavlov 1974; Chanan et al. 1979; Heyl & Shaviv 2000) and the mode dependence of the scattering cross sections and absorption coefficients (e.g., Adler et al. 1970; Adler 1971; Canuto et al. 1971; Mészáros & Ventura 1978; Ventura 1979; Arons et al. 1987; Mészáros 1992; Harding & Lai 2006). Kii et al. (1986), Kii (1987), and Mészáros et al. (1988) used polarization-dependent radiation transfer calculations to predict the polarization fractions of accreting X-ray pulsars. They found that the mode-dependent scattering cross sections lead to high polarization fractions in certain pulse intervals. Mészáros et al. (1988) determined that the models robustly predict that the phase-resolved flux and polarization fraction should be correlated (anti-correlated) in the fan-beam (pencil-beam) models. The detection of such a correlation can therefore discriminate between the fan-beam and pencil-beam models. This is a design driver for an upgraded version of X-Calibur, as described in Section 6.

The rest of the paper is structured as follows. The X-Calibur mission and experiment are described in Section 2. The X-Calibur, NICER, Swift, and Fermi observations, and data analysis methods are described in Sections 3 and 4, respectively. We present the results of the observations in Section 5 and conclude with a summary and an outlook for the scientific potential of follow-up flights in Section 6. The appendices include a description of the X-Calibur Stokes parameter analysis (Appendix A), our estimates of the systematic errors on the X-Calibur polarization parameters (Appendix B), and a summary of the spectral results (Appendix C).

All errors and uncertainties are quoted at 1σ-level (68.27% confidence level), unless noted otherwise.

X-Calibur combines an 8 m long X-ray telescope with arcseconds pointing and a scattering polarimeter (Figure 1). The telescope uses an aluminum-carbon fiber optical bench (Kislat et al. 2017), which is pointed with the Wallops Arc Second Pointer with a pointing stability of ∼1'' root mean square (rms) and a pointing knowledge of <15'' (3σ) (Stuchlik 2015). X-Calibur's energy range is limited to >15 keV by the absorption in the residual atmosphere at a float altitude of 125,000 feet, and to <60 keV by the mirror reflectivity. The mirror achieves an angular resolution of 25 half-power diameter and effective areas of 93 cm² at 20 keV and 46 cm² at 35 keV (Okajima et al. 2002; Berendse et al. 2003; Tueller et al. 2005; Ogasaka et al. 2008). Grazing incidence mirrors reduce the polarization of cosmic X-ray signals by less than 1% of the true polarization owing to the shallow scattering angles (Sanchez Almeida & Martinez Pillet 1993; Katsuta et al. 2009). The polarimeter is shown in Figure 3 and is made of a Be scattering element inside an assembly of Cadmium Zinc Telluride (CZT) detectors (each 2 mm thick, 2 × 2 cm² footprint, 64 pixels). Photons preferentially scatter perpendicular to the angle of the electric field of the beam with an azimuthal scattering angle distribution of:

with p₀ and ψ₀ being the true polarization fraction and angle, ψ the measured azimuthal scattering angle, and μ = 51.3% is X-Calibur's modulation factor. A rear CZT detector is positioned behind the scattering slab for monitoring the source location in the field of view. The timing resolution is ∼1 μs. The energy resolution increases from ∼3 keV FWHM at 15 keV to 5 keV FWHM at 35 keV. The detector assembly is shielded by a fully active CsI(Na) shield, and the polarimeter/shield assembly rotates at 1 rpm around the optical axis to minimize systematic errors. Detailed descriptions of the polarimeter and the in-flight performance of all components are given in Beilicke et al. (2014), Kislat et al. (2018), and Q. Abarr et al. (2020a, in preparation).

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X-Calibur was launched at 20:45 on 2018 December 29 (all times and dates are UTC) and reached a float altitude of 39.6 km (130,000 feet) roughly 3 hr later. Following the checkout of the pointing system and the in-flight optimization of the anti-coincidence shield settings, X-Calibur observed the accreting X-ray pulsars GX 301–2 and Vela X-1 until the flight was aborted owing to a He leak of the balloon at 20:00 on 2019 January 1. The starting times and durations of the X-Calibur on-source observation windows are listed in Table 1. The high-balloon-altitude GX 301−2 data set comprises a total of 8.0 hr ON-source and 7.8 hr OFF-source (aiming 1° away from the source).

The NICER X-ray Timing Instrument (Gendreau et al. 2016) observed GX 301−2 for 2.0 ks on 2018 December 28 and for 0.2 ks on December 29. Observations on December 28 were split into five shorter observation windows and were analyzed independently. The Swift X-Ray Telescope (XRT) observed GX 301–2 from MJD 58,480 through MJD 58,488 in nine individual pointings between 0.5 and 1.1 ks for a total of 8.1 ks. Details about the NICER and Swift observations are summarized in Appendix C, Table 1. Observing windows are labeled X-I–X-XXXIV for X-Calibur, N−I to N-V for NICER, and S-I to S-IX for Swift XRT. The Swift BAT and Fermi GBM observe GX 301−2 on a regular basis. We use the following results obtained for individual orbits and results averaged over individual days.

4.1. X-Calibur Data Analysis

The X-Calibur data analysis uses single-pixel CZT events without shield veto. The energy deposited in the CZT is estimated based on the calibration of the polarimeter with a ¹⁵²Eu source with low-energy lines at 39.52 keV (K_α2), 40.12 keV (K_α1), 45.7 keV, and 121.78 keV.

An event consists of the pixel number i (located at position in the detector reference frame, with x and y being the coordinates in the focal plane and z pointing toward the source), the energy E deposited in the CZT detectors, and the GPS event time t. Consistent with the exponential cutoff of the energy spectrum (e.g., Fürst et al. 2018), X-Calibur does not detect a significant excess of photons with >35 keV energy deposits, and we thus only use E < 35 keV events. The events enter the analysis with weights that were optimized based on the detector response as inferred from Monte Carlo simulations (Appendix A). For light curves, we normalize the weights so that the weighted event rate equals the true source rate.

The polarization analysis uses the Stokes parameters I (total flux), Q (the linearly polarized flux along the north–south direction), and U (the linearly polarized flux along the direction rotated 45° counterclockwise from the north–south direction when looking at the source) which are the weighted sums of the corresponding Stokes parameters of individual events (Kislat et al. 2015; Strohmayer 2017). The main results are given in terms of the normalized Stokes parameters:

so that () equals 1 for a beam 100% linearly polarized along the north–south (northeast–southwest) direction. The reconstructed polarization fraction p_r is given by

and the reconstructed polarization angle ψ_r is given by

During the observations, we switch every 15 minutes between observations targeting GX 301−2 (ON observations) and observations of four fields each located in a cross-pattern 1° away from the source in pitch and in yaw (OFF observations). As the Stokes parameters are additive, we can infer the Stokes parameters of the source beam by calculating the Stokes parameters for the ON observations and OFF observations, and subtracting the OFF values from the ON values after scaling the OFF values according to the ON and OFF observation time ratio. Details of the Stokes parameter analysis and background subtraction procedure are given in Appendix A. The systematic error on a measured polarization fraction p_r is (Appendix B)

The error Δp_r is our best estimate of the maximum possible error.

We fit the X-Calibur energy spectrum with XSPEC (Arnaud 1996; Arnaud et al. 2017) using Response Matrix Files and Auxiliary Response Files derived from Monte Carlo simulations.

4.2. NICER, Swift, and Fermi Data Analysis

4.3. Orbital and Pulsar Phases

We compute the orbital phase with the parameters from Doroshenko et al. (2010), with the last recorded periastron passage on MJD 53,531.65 ± 0.01, an orbital period of P = 41.472 days, and a period derivative of  (−3.7 ± 0.5) × 10⁻⁶ s s⁻¹.

We calculate the pulsar phase with the following phase model derived from Fermi GBM data:

with t being the barycentered time. The model parameters are given in Table 2.

Table 2.  GX 301−2 Phase Model Parameters Used in this Paper

Parameter	Value
t0	58477.024509 (MJD)
	126.350509 day−1
	0.0769078 day−2
	−0.00868925 day−3
	0.001075897 day−4

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5.1. Timing Results

Figure 4 shows the 15–50 keV fluxes measured with the Swift BAT. The graph clearly shows the 41.5 day orbital period. The X-Calibur observations from MJD 58,482.1521–58,483.3912 (orbital phases 0.37–0.40) fall into a rare period of a flare close to apastron. Figure 5 compares the Swift BAT 15–50 keV count rate measured during the orbit covering the apastron flare with the average count rates measured during the previous eleven orbits. The activity was enhanced during the orbit of the apastron flare, with a pronounced peak at an orbital phase around 0.4.

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The Swift BAT data allows us to scrutinize the hard X-ray emission for spectral variability. Figure 6 presents the 14–20 keV, 20–24 keV, and 24–35 keV light curves and the 24–35 keV to 14–20 keV hardness ratios. The rms of the hardness ratios is 0.103 corresponding to an rms of the photon indices Γ (from ) of ΔΓ ≈ 1. We do not discern a clear pattern linking the hardness ratio excursions to the flux level or the flux history except for a pronounced hardening of the energy spectra at the end of the flaring periods at MJD 58,481.5539, MJD 58,483.1819, and MJD 58,484.9899.

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The spin frequencies measured with the Fermi GBM in the 12–50 keV band (Figure 7) show a spectacular spin up coinciding with the exceptionally bright orbit. During the orbit (41.5 days) covering the X-Calibur observations, the spin frequency (period) increased from 1.461 mHz (spin period 684 s) on MJD 58,471.2 to 1.482 mHz (spin period 675 s) on MJD 58,512.9 at a rate of 5.8 × 10⁻¹² Hz s⁻¹ (see also Nabizadeh et al. 2019). The next orbit saw a much slower spin up from 1.482 mHz on MJD 58,512.9 to 1.490 mHz on MJD 58,553.2 at a rate of 2.3 × 10⁻¹² Hz s⁻¹. The spin-up rate is clearly correlated with an enhanced X-ray flux (Figure 8), bolstering the hypothesis that a change of the accretion rate or accretion mode is causing the spin up. Interesting features include the simultaneous dip of the X-ray flux and spin-up rate at MJD 58,492, the decrease of the spin-up rate between MJD 58,502 and MJD 58,510 during a phase of rather constant elevated X-ray flux levels, and the factor two lower spin up during MJD 58,546 and MJD 58,548 when compared to the spin up one orbit earlier (MJD 58,503–MJD 58,510) at similar flux levels.

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Koh et al. (1997) and Bildsten et al. (1997) reported similar spin-up phases detected with the BATSE gamma-ray detectors. At the time, the spin frequency increased over 23 days (MJD 48,440–48,463) from 1.463 to 1.473 mHz at a rate of Hz s⁻¹ and over 15 days (MJD 49,245–49,230) from 1.474 to 1.478 mHz at a rate of 3.0 × 10⁻¹² Hz s⁻¹. All rapid spin-up periods were accompanied by heightened apastron activity.

Figure 9 shows the GX 301−2 detection in the X-Calibur rear CZT detector. The image allows us to verify and refine the X-ray mirror alignment calibration (see also Appendix B). Figure 10 presents the 15–35 keV ON and OFF light curves from the polarimeter section of the detector (without the rear CZT detector). Note that each data point corresponds to one 15 minute run covering slightly more than one pulsar period. X-Calibur detected the source with a mean 15–35 keV rate of 0.23 Hz. Figure 11 compares the X-ray light curves from X-Calibur, Swift BAT, Swift XRT, and NICER taken around the time of the X-Calibur campaign. The flux level increased as the observation campaign unfolded and peaked a day after the X-Calibur observations ended.

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Figure 12 reports the average pulse profiles measured with the Swift XRT (0.2–10 keV), NICER (0.2–12 keV), and X-Calibur (15–35 keV). All three pulse profiles show one peak strongly dominating over the other. The shape of the X-Calibur 15–35 keV pulse profile measured during the spin-up epoch deviates significantly from the shapes of the 18–30 keV pulse profiles recorded on 2014 October 29 (orbital phase 0.65), 2015 October 4 (orbital phase 0.85), and 2019 March 3 (orbital phase 0.89) with NuSTAR. Whereas the NuSTAR pulse profiles show two pulses with approximately equal fluences (flux integrated over time, see Figures 3 and 4 of Nabizadeh et al. 2019), the fluence of the main X-Calibur peak (phase 0.8–1.14) exceeds that of the secondary peak 1/2 period later by a factor of ≈2 with a statistical significance of more than five standard deviations. The NICER (not shown here) and Swift data sets have sufficiently high signal-to-noise ratios to reveal significant variations of the pulse profiles from pulse to pulse (Figure 13). Such pulse profile variations can be caused by alterations in the accretion rate and by changes of the accretion and emission geometries.

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5.2. Spectral Results

The large absorption column observable in the NICER and Swift energy spectra reduces the count rate dramatically below 2 keV. We select channels with energies between 2 and 10 keV for spectral analysis, and fit them with a power-law continuum going through a partially covered absorber, and an additional Gaussian line

where was fixed to the galactic equivalent column density of 1.7 × 10²² cm⁻², reported in (Kalberla et al. 2005).

The results are reported in Tables 3–4. Given the wide variation of the signal-to-noise ratios of the different data sets, some of the energy spectra do not constrain some of the parameters of the model from Equation (10). In those cases, the parameters without errors in Tables 3–4 were fixed to the reported values during the fitting process. For example, in observation (Figure 14, top, and Table 4) the data do not allow us to constrain the second absorption component, so we fit the spectrum using Equation (10) with c = 1.0 and . For the main absorbing component we find N_H values of between ∼33 × 10²² cm² and ∼90 × 10²² cm². The N_H of the main component decreases through the apastron flare until MJD 58,485.52 (observation S VI). Most of our values are higher than the pre-periastron column densities of between ∼15 × 10²² cm² and ∼40 × 10²² cm² from Suchy et al. (2012) and Fürst et al. (2018), and lower than the periastron values of between ∼115 × 10²² cm² and ∼175 × 10²² cm² of Fürst et al. (2011).

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Table 3.  Spectral Results from NICER Observations

Observation:	N-I	N-II	N-III	N-IV	N-V
	1.20 ± 0.06	0.78 ± 0.10	1.29 ± 0.08	1.21 ± 0.02	2.36 ± 0.12
[1022 cm−2]	78.1 ± 2.4	82.5 ± 4.5	80.3 ± 2.7	86.5 ± 1.8	52.2 ± 2.7
[1022 cm−2]	2.7 ± 1.0	4.4 ± 1.6	4.5 ± 1.8	2.6 ± 0.6	0.05 ± 0.83
Cov. Frac.	0.992 ± 0.001	0.986 ± 0.003	0.994 ± 0.001	0.990 ± 0.001	0.961 ± 0.005
PL [cm−2s−1keV−1]	0.53 ± 0.10	0.29 ± 0.09	0.60 ± 0.13	0.30 ± 0.04	0.09 ± 0.02
PL	1.21 ± 0.08	1.09 ± 0.14	1.21 ± 0.09	0.85 ± 0.06	0.16 ± 0.09
Fe Kα A s−1cm−2]	3.4 ± 0.3	2.6 ± 0.3	5.3 ± 0.5	4.3 ± 0.2	5.0 ± 0.7
Fe Kα E [keV]	6.41 ± 0.01	6.39 ± 0.01	6.40 ± 0.01	6.41 ± 0.01	6.41 ± 0.01
Fe Kα σ [keV]	0.02 ± 0.02	0.02	0.060 ± 0.011	0.039 ± 0.007	0.068 ± 0.018
χ2/NDF	0.97	1.00	0.93	1.14	1.06

Note. The errors are on 1σ confidence level.

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Table 4.  Spectral Results from the Swift XRT

Observation:	S-I	S-II	S-III	S-IV	S-V	S-VI	S-VII	S-VIII	S-IX
	1.15 ± 0.20	1.53 ± 0.12	2.27 ± 0.15	2.29 ± 0.11	3.69 ± 0.15	2.81 ± 0.12	1.91 ± 0.09	1.35 ± 0.20	1.16 ± 0.12
[1022 cm−2]	63.6 ± 5.8	76.5 ± 12.3	49.1 ± 2.9	67.6 ± 8.0	49.4 ± 3.1	32.8 ± 1.4	39.3 ± 2.0	42.4 ± 5.2	61.6 ± 19.3
[1022 cm−2]	0.0	21.1 ± 10.1	0.0	16.3 ± 9.8	0.0	0.0	0.0	0.0	19.8 ± 9.4
Cov. Frac.	1.0	0.945 ± 0.045	1.0	0.970 ± 0.031	0.988 ± 0.002	1.0	1.0	1.0	0.858 ± 0.134
PL [cm−2 s−1 keV−1]	0.32 ± 0.17	0.80 ± 0.41	0.23 ± 0.07	0.84 ± 0.34	0.33 ± 0.09	0.46 ± 0.09	0.25 ± 0.06	0.12 ± 0.07	0.38 ± 0.25
PLΓ	1.91 ± 0.36	1.33 ± 0.22	1.09 ± 0.24	1.19 ± 0.17	0.61 ± 0.12	1.01 ± 0.09	0.85 ± 0.12	0.67 ± 0.26	1.23 ± 0.27
Fe Kα A s−1 cm−2]	9.9 ± 2.4	5.0 ± 1.8	12.2 ± 3.6	7.6 ± 2.0	18.6 ± 3.4	8.9 ± 2.8	8.9 ± 1.4	9.5 ± 6.4	3.8 ± 1.1
Fe Kα E [keV]	6.56 ± 0.07	6.34 ± 0.05	6.37 ± 0.05	6.49 ± 0.04	6.52 ± 0.05	6.47 ± 0.04	6.43 ± 0.03	6.33 ± 0.06	6.33 ± 0.06
Fe Kα σ [keV]	0.317 ± 0.082	0.144 ± 0.085	0.316 ± 0.070	0.127 ± 0.073	0.296 ± 0.061	0.203 ± 0.058	0.145 ± 0.048	0.301 ± 0.078	0.200
χ2/NDF	1.58	1.53	1.07	0.90	1.28	1.13	0.75	0.66	1.20

Note. The errors are on 1σ confidence level.

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The NICER energy spectra show clear Fe Kα lines, and some marginally significant deviations of the data from the best-fit model between 2 and 3 keV (Figure 14, top). The Swift XRT spectra also show the presence of the Fe Kα line (Figure 14, middle) throughout the whole observation period.

The X-Calibur 15–35 keV energy spectrum is fitted with a power-law model. We obtain a 15–35 keV flux of erg cm⁻² s⁻¹ and a power-law index of 4.2 ± 0.6 (1σ errors). The photon index agrees within statistical errors with the energy spectrum measured with NuSTAR on 2019 March 3, which exhibits a rollover from a photon index of Γ = 2 at 20 keV to Γ = 4 at 30 keV (Figure 6 of Nabizadeh et al. 2019).

We study the broadband 2–35 keV energy spectrum by simultaneously fitting the Swift XRT (observations S III and S IV) and X-Calibur data (Figure 14, bottom) with a power-law model with an exponential cutoff, a partially covered absorber, and an additional Fe–Kα fluorescence line:

A model with Γ = 0.04 ± 0.21, E_fold = 7.95 ± 0.78 keV, and cm² gives a good fit to the broadband data, with . The values of the spectral parameters are similar to those obtained by Fürst et al. (2018) using NuSTAR observations, with the exception of the softer photon index of Γ ∼ 0.8 obtained by Fürst et al. (2018).

5.3. X-Calibur Polarization Analysis

All polarization results are given in the 15–35 keV band for three data sets (see lower panel in Figure 12): (i) the entire data set, (ii) the main pulse (pulsar phase 0.8–1.14), and for (iii) the bridge and secondary pulse emission (pulsar phase 0.14–0.8). Figure 15 presents the modulation curves (azimuthal scattering angle distributions) for the ON and OFF observations. Neither the ON nor the OFF distributions show obvious modulations.

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Figure 16 presents the results in the – plane for all three data sets. The statistical significance for a polarization detection can be calculated with Q and U, which have slightly smaller relative errors than and . The overall results deviate by 1.41 (entire emission), 1.47 (main pulse), and 0.78 (bridge and secondary pulse) standard deviations from zero polarization (Q = 0 and U = 0). The X-Calibur observations thus did not lead to a significant detection of a non-zero polarization.

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For the pulse-integrated emission, Figure 17 shows the and parameters for the background-subtracted ON data and the OFF background data as a function of time. It can be seen that the and parameters of the ON and OFF observations are consistent with zero polarization for all time intervals. The same applies to the and parameters of the entire OFF data set.

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Figure 18 presents the observational constraints on the polarization fraction p₀ and angle ψ₀. We use a Bayesian analysis with a flat prior of the polarization fraction p₀ between 0% and 100% and the polarization angle ψ₀ between 0 and π (Quinn 2012; Kislat et al. 2015):

The most likely true parameter combination p₀ and ψ₀ is shown by a cross mark, and the confidence regions are shown by contours and the color scales. Table 5 lists the most likely values of p₀ and ψ₀ together with the confidence intervals derived from the distributions in Figure 18. The table includes the 90% confidence interval upper limits on the polarization fraction p₀ calculated by marginalizing the probability density function over ψ₀.

Table 5.  X-Calibur 15–35 keV Polarization Results

Phase Interval			Deviation from p = 0 [σ]	p [%]	ψ ]	Upp. Lim. p (90% CL) [%]
All (0-1)	18.4 ± 19.4	20.2 ± 19.4	1.41		21 ± 43	46.9
Main Pulse (0.8–1.14)	26.6 ± 21.2	16.1 ± 21.1	1.47		30 ± 40	52.3
Bridge and Secondary Pulse (0.14–0.8)	8.3 ± 33.5	24.6 ± 33.6	0.78		10	62.2

Note. Errors on p and ψ are on 90% confidence level. The polarization angle of the third data set is unconstrained on the 90% confidence level.

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This paper presents the results of the observations of the accretion-powered X-ray pulsar GX 301−2 with X-Calibur, NICER, the Swift XRT and BAT, and Fermi GBM. The observations reveal a rare flaring period between the periastron flares associated with a spin up of the pulsar similar to earlier events (Bildsten et al. 1997; Koh et al. 1997). Historically, the spin of GX 301−2 exhibited values around 1.4 mHz (pulsar period: 715 s) between 1975 and 1985 and values around the current value of 1.47 mHz (pulsar period: 680 s) between 1993 and now (White et al. 1976; Nagase 1989; Lutovinov et al. 1994; Bildsten et al. 1997; Koh et al. 1997), indicating an approximate equilibrium between spin-up and spin-down torques during these two long epochs (Lipunov 1992; Doroshenko et al. 2010). The spin-up epochs typically last about one orbit, in which the pulsar frequency changes linearly. The spin-up period, starting at the time of the X-Calibur observations, lasted for two orbits with a marked decline of the spin-up rate after the first orbit. The spin-up events start briefly after periastron (Figure 7 in this paper, and Figure 11 of Koh et al. 1997).

A possible interpretation of these signatures is that the neutron star acquires a temporary accretion disk (Koh et al. 1997) shortly after periastron passage. The temporary disk provides fuel for one orbit during which the pulsar spins up continuously, and is destroyed during the next periastron passage. The disk may form, for example, when the neutron star crosses the plasma stream from Wray 977 at the orbital phase of ∼0.25 (Leahy & Kostka 2008). We note that during the spin-up periods, the crossing always results in a large increase in X-ray flux at the orbital phase of ≈0.4. Independent of what exactly triggers the X-ray flares, it is an open question why only some flares spin up the neutron star.

We report here on the first constraints on the hard X-ray polarization of an accreting neutron star at energies fairly close to the cyclotron line energy. Owing to the short balloon flight time, the X-Calibur observations did not yield a definitive polarization detection, but did offer constraints on the polarization fraction and the polarization angle plane. The results can be compared to the predictions from Mészáros et al. (1988). The authors find that the propagation of the radiation in the ordinary and extraordinary mode and the strongly mode-dependent scattering cross sections can lead to very high (∼80%) polarization fractions for certain pulse phases close to the cyclotron resonant energy. Interestingly, they find that fan-beam models predict, rather robustly, a positive correlation of the peak intensity and the polarization fraction. In contrast, pencil-beam models predict the opposite: a minimum (maximum) of the polarization fraction during the peak (valley) of the pulsed emission.

The X-Calibur observations constrain the polarization fraction in the 15–35 keV band, somewhat below the centroids of the CRSFs at 35 and 50 keV. The calculations of Mészáros et al. (1988) were carried out for a cyclotron resonance at 35 keV. At 25 keV the pencil-beam (fan-beam) model predicts polarization fractions of ∼20% (<5%). The X-Calibur GX 301−2 result of % cannot distinguish between the two models. Doing so with high statistical certainty will require future observations with a one-sigma error of <4%.

Driven by this requirement, we are now working on an X-Calibur follow-up mission called XL-Calibur (Q. Abarr et al. 2020b, in preparation) which promises hard X-ray polarimetric observations with one to two orders of magnitude improved signal-to-background ratio. The mission uses the 12 m focal length mirror fabricated for the Formation Flight Astronomical Survey Telescope (Tsunemi et al. 2014) which offers more than three times larger effective areas than the current mirror (Awaki et al. 2014; Matsumoto et al. 2018). We furthermore expect more than one order of magnitude lower background rates owing to the use of thinner (0.8 mm thick) CZT detectors, improved shielding, and flights closer to solar maximum rather than solar minimum (see Shaw et al. 2003; Potgieter 2008). Simulated XL-Calibur observations of GX 301−2 (Figure 19) show that the improved mission could clearly distinguish between the fan-beam and the pencil-beam model. Joint observations with the Imaging X-Ray Polarimetry Explorer (2–8 keV, launching in 2021) (Weisskopf et al. 2016) and XL-Calibur (launching in 2022, 2023, and 2025), will enable detailed comparisons of predicted and observed signatures.

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We thank A. Awaki (Ehime Uniersity), K. Hayashida (Osaka University, Project Research Center for Fundamental Sciences, ISAS), Y. Maeda (ISAS), H. Matsumoto (Osaka University, Project Research Center for Fundamental Sciences), T. Tamagawa (RIKEN), and K. Tamura (Nagoya University) for fruitful discussions and for their comments on this paper. We thank V. Mikhalev for contributing the code for barycentring the X-Calibur event times and Rakhee Kushwah (KTH, Oskar Klein Centre) for contributing to the flight monitoring shifts.

X-Calibur is funded by the NASA APRA program under contract number 80NSSC18K0264. We thank the McDonnell Center for the Space Sciences at Washington University in St. Louis for funding of an early polarimeter prototype, as well as for funds for the development of the ASIC readout. H.K. acknowledges NASA support under grants 80NSSC18K0264 and NNX16AC42G. KTH authors acknowledge support from the Swedish National Space Agency (grant No. 199/18). M.P. also acknowledges support from the Swedish Research Council (grant No. 2016-04929). H.K. acknowledges support from the National Science Foundation under the Independent Research and Development program.

The analysis of the X-Calibur events starts with the de-rotation of the x and y coordinates of the energy deposition in the detector reference frame into the reference frame of the telescope truss. Subsequently, we correct for the offset of the focal spot of the X-ray mirror from the center of the scattering element as determined from the excess recorded in the rear CZT detector (Figure 9). Finally, the coordinates are referenced to the celestial north pole based on the truss orientation measured by the pointing system. Choosing a coordinate system with the y-coordinate pointing north and the x-coordinate pointing east, the azimuthal scattering angle is given by:

so that ψ = 0 corresponds to scatterings along the north–south direction, and 0 < ψ < π/2 corresponds to scatterings along the northeast direction. We calculate a set of Stokes parameters (Kislat et al. 2015) for the kth event:

The factor μ is the modulation factor (see Equation (3)). The minus signs in the expressions of q_k and u_k account for the 90° offset between the electric field vector of the photons and the preferred scattering direction.The factor 2/μ normalizes q_k (u_k) so that its average is 1 for a beam 100% linearly polarized along the north–south direction (looking into the sky, 45° anti-clockwise from the north–south direction).

The kth event enters the analysis with weight w_k that is proportional to the expected signal-to-background ratio, and is the product of two functions (spectral analysis) or three functions (light curves) optimized with Monte Carlo simulations of the detector. The first function depends on the position of the energy deposition along the optical axis (the z coordinate) and accounts for the approximately exponential distribution of the depths of the Compton scattering in the scattering element. As a consequence, most source photons are detected near the front of the polarimeter. The second function depends on the position of the triggered pixel relative to the scattering element and is proportional to the azimuthal scattering angle interval Δψ that the pixel covers as seen from the axis of the scattering element. The function weighs events close to the middle of the side walls of the rectangular detector assembly more heavily than those close to the edges, as those pixels achieve a better signal-to-background ratio. The third function f₃(E) (only for light curves) weighs events according to the energy E deposited in the CZT detectors and is proportional to the expected source detection rate as a function of energy accounting for the source spectrum, atmospheric absorption, and the mirror effective area.

With t_ON and t_OFF being the ON and OFF observation times and α = t_ON/t_OFF, we define the total background-subtracted Stokes parameters as

where the sums run over the ON and OFF events.

Compared to the unweighted analysis, the weighted analysis improves the signal-to-background ratio of the GX 301−2 results by ∼20%. Further sensitivity improvements might be achieved with a maximum likelihood analysis (see Krawczynski 2011; Lowell 2017; Lowell et al. 2017).

We calculate statistical errors on I, Q, U, , and from error propagation. Each event contributes with the following rms values to the analysis (Kislat et al. 2015):

The estimates of and are conservatively chosen for p₀ = 0. For p₀ > 0 the errors are smaller. When calculating the error on (), we assume that the errors on I and Q (I and U) are statistically independent. A toy simulation shows that this is indeed an excellent assumption.

We calibrated the polarimeter at the Cornell High Energy Synchrotron Source using a 40 keV beam with a ∼90% polarization (Beilicke et al. 2014). The measurements were carried out with different polarimeter orientations allowing us to simulate an unpolarized beam by combining data taken at orientations differing by 90°. It is important to note that the rotation of the detector and shield assembly removes systematic errors due to detector non-uniformities (e.g., dead pixels, noisy pixels) and geometrical effects (including uncertainties in the distances between the center of the scattering element and the CZT detectors and gaps between the detectors). Based on the calibration data, we estimate that we know the modulation factor μ within an uncertainty of ±2%. The uncertainty on μ introduces a relative systematic error on the measured polarization fraction p_r of Δp_r = 2%p_r.

The misalignment of the center of the mirror point-spread function and the rotation axis of the polarimeter can lead to a spurious polarization that is independent of the true polarization fraction (Beilicke et al. 2014). Based on the image of GX 301−2 in the rear CZT detector (Figure 9), we estimate that the center of the point-spread function and the rotation axis of the polarimeter were offset by ( mm). Correcting for d, the uncertainty in d leads to a residual systematic polarization fraction error of <0.25%.

We performed the full Stokes analysis for the background data runs, and obtain Stokes parameters that are consistent with 0. For example, for the entire 15–35 keV background, we get

where the errors are given for a 1σ confidence interval (see also Figure 17). The fact that the background looks unpolarized implies that an under- or over-subtraction of the background (owing for example to a time variable background) does not create a spurious polarization detection. We estimate that the background subtraction introduces a relative 5% error on measured polarization fractions.

Adding all systematic errors linearly, we get a total systematic error on the polarization fraction quoted in Equation (8).