X-Ray Emission Evolution of the Galactic Ultraluminous X-Ray Pulsar Swift J0243.6+6124 during the 2017–2018 Outburst Observed by the MAXI GSC

Figure 1 shows the background-subtracted X-ray light curves of Swift J0243.6 from 2017 September to 2018 October, obtained by the GSC in the 2–4, 4–10, and 10–20 keV bands in a 1 day time bin. Also plotted are the time variations of the soft color (SC), i.e., the 4–10 to 2–4 keV intensity ratio, and the hard color (HC), i.e., the 10–20 to 4–10 keV intensity ratio. These ratios have been calculated after the background subtraction. To visualize the quality of the degraded units (GSC_0, GSC_3, and GSC_6), we plot their data with different symbols. The statistical errors of these units are typically larger by a factor of 5–10 than those of the normal units.

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Figure 1 reveals that the present X-ray activity started at around MJD 58,025 (2017 September 29) and continued for over 1.5 yr. The first outburst developed into the largest one with the highest peak intensity (25 photons cm⁻² s⁻¹ ≃ 7 Crab in 2–20 keV) and the longest duration (∼150 days). After this, several outbursts with lower peaks (≲1 photons cm⁻² s⁻¹) and shorter durations (≲40 days) followed. Their recurrence cycles do not synchronize with the 27.3 day orbital period. This means that they are classified into the giant (type II) outbursts of Be XBPs (e.g., Reig 2011).

In Figure 2, we show the hardness-intensity diagrams (HIDs), i.e., SC or HC versus 2–20 keV photon flux ≡ I_2–20, using 2 days of bin data. As seen in Figure 1, the periods covered by the normal GSC units, MJD 58,062–58,108 and 58,135–58,165, are limited to the outburst decay phase, and they have a gap from MJD 58,108 to 58,135. We hence employed data taken by the degraded GSC units during the gap. To reduce their large statistical uncertainty, these data were averaged over a 5 day time bin. The obtained HIDs for the SC and HC are largely represented by a negative intensity-hardness correlation when the intensity is high (I_2–20 ≳ 0.8) and relatively constant hardness ratios when the intensity is low (I_2–20 ≲ 0.8). These features agree with those obtained from the NICER data (WMJ18).

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The two HIDs in Figure 2, though grossly similar, differ in details. In the very high intensity region of I_2–20 ≳ 4.5, which is just after the outburst peak, the SC changes little with I_2–20, but the HC changes significantly. During the intermediate region of 0.8 ≲ I_2–20 ≲ 4.5, the change of SC becomes larger, but that of HC becomes smaller than those at I_2–20 ≳ 4.5. In Figure 2, the boundaries of these regions at I_2–20 = 0.8 and 4.5 are marked by dashed lines.

To clarify the source evolution during the first outburst from MJD 58,025 to 58,175, we divided the time periods when Swift J0243.6 was observed by the normal GSC units into eight intervals and named them A through H, each covering 8–10 days, as illustrated in the top panel of Figure 1. These intervals have gaps from the outburst start to MJD 58,062 and from MJD 58,106 to 58,134, for which Swift J0243.6 was observed only by the degraded GSC units. We then decided to use the degraded units to fill in these two gaps and divided them into five intervals, U through Y, each of which has a length of 8–14 days. Table 2 summarizes the start and stop time (MJD), employed GSC units, exposure time (T_exp), and average detector area (A_eff) for the Swift J0243.6 direction in each interval. Below, we employ these interval definitions.

To study the time evolution of the pulsed X-ray emission, we performed pulse timing analysis. To begin with, every GSC event time was converted to that at the solar system barycenter. Then, these barycentric times were further corrected for the pulsar's orbital motion, using the binary orbital parameters shown in Table 1.

We examined the coherent pulsation, first with the GSC data. Considering the limited exposure and sparse time coverage, the epoch-folding period search was carried out for every 2 day interval. Figure 3(a) shows the obtained pulse frequencies of the 2 day intervals for which the pulsation was detected significantly, from MJD 58,038 to 58,172 during the first giant outburst. The pulse frequency of Swift J0243.6 has also been measured by the Fermi/GBM on an almost daily basis during the X-ray active periods. In Figure 3(a), the data from the Fermi/GBM are plotted together. We confirmed that the frequencies from the GSC data are all consistent with those of the Fermi/GBM within the errors quoted in the figure caption.

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We then investigated pulse-profile evolution. To derive phase-coherent pulse profiles considering the pulse-period changes, we calculated a sequential pulse phase ϕ(t) for the event time t as

$$\begin{eqnarray}&&\phi (t)={\int }_{{t}_{0}}^{t}\nu (\tau )d\tau ,\end{eqnarray} \tag{ 1 }$$

where ν(t) means the pulse-frequency time history, and t₀ is the phase-zero epoch, i.e., ϕ(t₀) = 0. As ν(t) represents the observations, we employed the daily frequencies taken by the Fermi/GBM at the measured time epochs, because they have better accuracies than those of the GSC. Also, t₀ was fixed at 58,027.499066 (MJD), which is the epoch of the first Fermi/GBM periodicity detection. The behavior of ν(t) between adjacent data points was estimated by a cubic spline-fit model. In Figure 3(a), the interpolated ν(t) model is drawn on the data.

Using Equation (1), we folded both the source and background light curves, normalized them to the average detector area for the source, and subtracted the latter from the former. In Figure 3(d), the pulse profiles obtained in this way every 2 day interval from MJD 58,038 to 58,172 are plotted in a two-dimensional color image. Figures 3(b) and (c) show the pulse phase–averaged X-ray flux and the rms pulsed fraction, f_pul (WMJ18), calculated from each pulse profile. These figures reconfirm the sequential pulse-profile change reported by WMJ18.

Figure 4(a) shows the pulse profiles averaged over the individual 8–14 day intervals of A through H and U through Y, defined in Table 2. The pulse profile changed from a double-peak shape in the brightest phase to a shallow single-peak one in the intermediate phase, and then to a dip-like feature developed in the fainter phase, as observed by NICER and Fermi/GBM (WMJ18).

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Figure 4(b) presents the I_2–20 dependence of f_pul, calculated from the pulse profiles in Figure 4(a). We also produced pulse profiles in the hard band of 10–20 keV with the same procedure. The I_2–20 dependence of f_pul in this band is plotted in Figure 4(b). These results from the two bands confirm the NICER results (WMJ18) that the pulsed fraction increases toward higher energies. The f_pul minimum at around I_2–20 ≃ 4.5, corresponding to the epoch of transition from the double peak to the single peak, agrees well with the boundary of the two regimes in the HC HID (right panel of Figure 2).

3.3.1. Pulse Phase–averaged Spectra

The source behavior on the HIDs, as seen in Figures 1 and 2, suggests that the energy spectrum changed with the X-ray luminosity. We thus analyzed X-ray spectra taken with the GSC and averaged over the pulse phase. The spectral model fits were carried out on the XSPEC software version 12.8 (Arnaud 1996), released as a part of the HEASOFT software package version 6.25.

We extracted X-ray spectra for the eight intervals A through H (Table 2), which were observed by the normal GSC units. Figure 5(a) shows the obtained 2–30 keV spectra, where the background has been subtracted as described in Section 2, but the instrumental responses are inclusive. To clarify the spectral evolution, we plot in Figure 5(b) their ratios to the spectra expected for a power-law function with a photon index Γ = 2, i.e., F(E) = E⁻² (photons cm⁻² s⁻¹ keV⁻¹). The ratios confirm the softening with the flux increase, as seen in the HIDs (Figure 2). In addition, the ratios are generally more convex than the Γ = 2 power law, with a mild bending at 6–8 keV. An enhancement at around 6.5 keV is considered to include the iron-K line emission.

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As inspired by Figure 5(b), we fitted these spectra with a model composed of a high-energy cutoff power law (HECut) and a Gaussian (Gaus) for the iron-K emission line. The HECut model is represented by a photon index Γ, cutoff energy E_cut, folded energy E_fold, and normalization factor A as a function of the photon energy E as

$$\begin{eqnarray}{F}_{\mathrm{HECut}}=\left\{\begin{array}{ll}{{AE}}^{-{\rm{\Gamma }}} & (E\leqslant {E}_{\mathrm{cut}})\\ {{AE}}^{-{\rm{\Gamma }}}\exp \left(-\tfrac{E-{E}_{\mathrm{cut}}}{{E}_{\mathrm{fold}}}\right) & ({E}_{\mathrm{cut}}\lt E).\end{array}\right.\end{eqnarray} \tag{ 2 }$$

The model has been successfully fitted to the spectra of major XBPs (e.g., White et al. 1983; Coburn et al. 2002). Because of the limited GSC energy resolution, we constrained the Gaussian centroid in a 6.4–7.0 keV range and fixed the width at σ = 0.3 keV, referring to the spectra of the typical XBPs. To account for the interstellar absorption, the continuum model was multiplied by a photoelectric absorption factor by a medium with the solar abundances (Wilms et al. 2000), with the equivalent hydrogen column density fixed at the Galactic H I density in the direction N_H = 0. 7 × 10²² cm⁻² (Kalberla et al. 2005). This N_H value is consistent with that determined by the NuSTAR spectrum in the outburst early phase (Jaisawal et al. 2018). The model is hence expressed as tbabs∗(powerlaw∗highecut+Gaussian) in the XSPEC terminology.

Figure 6(a) shows the unfolded νFν spectra of the A through H intervals, together with their best-fit HECut+Gaus models, and Figure 6(b) shows individual data-to-model ratios. Table 3 summarizes the best-fit model parameters, which include the absorption-corrected 0.5–60 keV flux, F_0.5–60, considered to approximate the bolometric flux. The value of F_0.5–60 = 38 erg cm⁻² s⁻¹ in interval A corresponds to the bolometric luminosity L_bol = 2. 2 × 10³⁹ erg s⁻¹ assuming an isotropic emission and D = 7 kpc. Although the HECut+Gaus model largely reproduced the data, the data-to-model ratios are not always consistent with 1. The discrepancies are evident in higher-luminosity intervals and energies ≳6 keV. The χ²_ν values indicate that the fits are not acceptable within the 95% confidence limits in the first half of the observation, intervals A through D, but those of the second half, E through H, are acceptable.

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Table 3.  The Best-fit Spectral Parameters with the HECut+Gaus and NPEX+Gaus Models

	Model: HECut + Gaus
	A	Γ	Ecut	Efold	EFeb	EWFeb	I2–20c	F0.5–60d	fbole	${\chi }_{\nu }^{2}(\nu )$
Int.			(keV)	(keV)	(keV)	(eV)
A	32a	1.52a	5.2a	19a	6.4a	170a	25.3a	37.9a	1.50a	2.66 (28)
B	${20}_{-1}^{+1}$	$\,{1.50}_{-0.03}^{+0.03}$	$\,{4.8}_{-0.5}^{+0.5}$	${24}_{-2}^{+2}$	$\,{6.4}_{-0.00}^{+0.03}$	${210}_{-30}^{+40}$	${16.2}_{-0.1}^{+0.1}$	${26.0}_{-0.4}^{+0.5}$	$\,{1.61}_{-0.04}^{+0.04}$	1.95 (28)
C	$\,{9.1}_{-0.2}^{+0.4}$	$\,{1.41}_{-0.04}^{+0.04}$	$\,{4.2}_{-0.7}^{+0.8}$	${32}_{-4}^{+5}$	$\,{6.4}_{-0.00}^{+0.11}$	${120}_{-40}^{+40}$	$\,{8.74}_{-0.06}^{+0.06}$	${15.9}_{-0.4}^{+0.4}$	$\,{1.82}_{-0.06}^{+0.06}$	1.58 (28)
D	$\,{5.6}_{-0.3}^{+0.4}$	$\,{1.35}_{-0.05}^{+0.05}$	$\,{3.8}_{-1.3}^{+1.7}$	${40}_{-7}^{+11}$	$\,{6.4}_{-0.00}^{+0.14}$	${120}_{-30}^{+40}$	$\,{6.05}_{-0.04}^{+0.04}$	${12.1}_{-0.4}^{+0.4}$	$\,{2.01}_{-0.08}^{+0.08}$	1.82 (28)
E	$\,{4.5}_{-0.16}^{+0.17}$	$\,{1.46}_{-0.02}^{+0.02}$	${15}_{-4}^{+3}$	${28}_{-11}^{+27}$	$\,{6.4}_{-0.00}^{+0.12}$	${140}_{-40}^{+50}$	$\,{4.36}_{-0.05}^{+0.05}$	$\,{8.4}_{-0.7}^{+0.8}$	$\,{1.92}_{-0.17}^{+0.21}$	1.01 (28)
F	$\,{1.11}_{-0.09}^{+0.09}$	$\,{1.29}_{-0.05}^{+0.05}$	${12.7}_{-2.5}^{+2.4}$	${18}_{-8}^{+18}$	$\,{6.4}_{-0.0}^{+0.6}$	${40}_{-40}^{+90}$	$\,{1.40}_{-0.03}^{+0.03}$	$\,{2.5}_{-0.3}^{+0.5}$	$\,{1.82}_{-0.27}^{+0.39}$	1.44 (28)
G	$\,{0.57}_{-0.05}^{+0.05}$	$\,{1.12}_{-0.07}^{+0.06}$	$\,{9.0}_{-2.1}^{+2.0}$	${22}_{-5}^{+9}$	$\,{6.4}_{-0.0}^{+0.6}$	${40}_{-40}^{+120}$	$\,{0.93}_{-0.02}^{+0.02}$	$\,{1.90}_{-0.18}^{+0.22}$	$\,{2.03}_{-0.23}^{+0.27}$	1.15 (28)
H	$\,{0.29}_{-0.03}^{+0.04}$	$\,{1.09}_{-0.08}^{+0.08}$	$\,{8.9}_{-2.0}^{+2.4}$	${24}_{-8}^{+20}$	$\,{6.4}_{-0.0}^{+0.6}$	${210}_{-130}^{+140}$	$\,{0.51}_{-0.01}^{+0.01}$	$\,{1.09}_{-0.16}^{+0.21}$	$\,{2.13}_{-0.36}^{+0.47}$	0.80 (28)
	Model: NPEX + Gauss
	A1	${{\rm{\Gamma }}}_{1}$	${A}_{2}(\times {10}^{3})$	Efold	EFeb	EWFeb	${I}_{2-20}$ c	${F}_{0.5-60}$ d	fbole	${\chi }_{\nu }^{2}(\nu )$
Int.				(keV)	(keV)	(eV)
A	29a	1.04a	3.2a	7.2a	6.4a	180a	25.3a	37.4a	1.48a	2.25 (28)
B	${18}_{-0}^{+1}$	$\,{0.96}_{-0.04}^{+0.06}$	$\,{5.3}_{-2.3}^{+2.6}$	$\,{6.1}_{-0.6}^{+0.9}$	6.4a	${210}_{-30}^{+30}$	${16.2}_{-0.1}^{+0.1}$	${24.5}_{-0.6}^{+0.9}$	$\,{1.51}_{-0.05}^{+0.06}$	1.55 (29)
C	$\,{8.5}_{-0.3}^{+0.4}$	$\,{0.89}_{-0.03}^{+0.03}$	$\,{5.1}_{-1.5}^{+1.6}$	$\,{5.8}_{-0.4}^{+0.5}$	$\,{6.4}_{-0.00}^{+0.15}$	${130}_{-30}^{+40}$	$\,{8.78}_{-0.06}^{+0.06}$	${14.3}_{-0.4}^{+0.5}$	$\,{1.63}_{-0.05}^{+0.06}$	0.97 (28)
D	$\,{5.4}_{-0.2}^{+0.3}$	$\,{0.86}_{-0.03}^{+0.03}$	$\,{4.5}_{-1.2}^{+1.3}$	$\,{5.8}_{-0.4}^{+0.5}$	$\,{6.4}_{-0.00}^{+0.18}$	${140}_{-30}^{+40}$	$\,{6.10}_{-0.05}^{+0.05}$	${10.5}_{-0.3}^{+0.4}$	$\,{1.72}_{-0.06}^{+0.07}$	1.24 (28)
E	$\,{4.0}_{-0.3}^{+0.4}$	$\,{0.86}_{-0.05}^{+0.05}$	$\,{5.1}_{-1.7}^{+1.9}$	$\,{5.2}_{-0.4}^{+0.6}$	$\,{6.4}_{-0.00}^{+0.16}$	${140}_{-40}^{+70}$	$\,{4.37}_{-0.05}^{+0.05}$	$\,{7.4}_{-0.3}^{+0.4}$	$\,{1.68}_{-0.08}^{+0.11}$	1.09 (28)
F	$\,{1.32}_{-0.26}^{+0.32}$	$\,{0.84}_{-0.15}^{+0.17}$	$\,{4.7}_{-2.0}^{+2.5}$	$\,{4.1}_{-0.4}^{+0.6}$	$\,{6.4}_{-0.0}^{+0.6}$	${90}_{-90}^{+150}$	$\,{1.41}_{-0.03}^{+0.03}$	$\,{2.2}_{-0.1}^{+0.2}$	$\,{1.58}_{-0.13}^{+0.17}$	1.51 (28)
G	$\,{0.49}_{-0.05}^{+0.06}$	$\,{0.84}_{-0.15}^{+0.13}$	$\,{0.00}_{-0.00}^{+0.04}$	${17}_{-6}^{+7}$	$\,{6.4}_{-0.0}^{+0.6}$	${20}_{-20}^{+140}$	$\,{0.93}_{-0.02}^{+0.02}$	$\,{1.90}_{-0.18}^{+0.32}$	$\,{2.03}_{-0.22}^{+0.38}$	1.29 (28)
H	$\,{0.36}_{-0.10}^{+0.14}$	$\,{0.71}_{-0.22}^{+0.25}$	$\,{2.0}_{-2.0}^{+1.5}$	$\,{4.0}_{-0.5}^{+1.1}$	$\,{6.4}_{-0.0}^{+0.5}$	${280}_{-150}^{+160}$	$\,{0.52}_{-0.01}^{+0.01}$	$\,{0.83}_{-0.07}^{+0.12}$	$\,{1.61}_{-0.18}^{+0.28}$	0.93 (28)

Notes.

^aErrors are given with 90% limits of statistical uncertainty if the fits are within the acceptable level (${\chi }_{\nu }^{2}\lt 2$). ^bCentroid and equivalent width of iron-K line. ^cPhoton flux in 2–20 keV in photon cm⁻² s⁻¹. ^dAbsorption-corrected flux in 0.5–60 keV in 10⁻⁸ erg cm⁻² s⁻¹. ^eRatio of ${I}_{2\mbox{--}20}$ to ${F}_{0.5\mbox{--}60}$ in 10⁻⁸ erg photon⁻¹.

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We then examined another continuum model, the Negative and Positive power laws with a common EXponential cutoff (NPEX; Mihara et al. 1998), which has often been used in the study of XBPs more successfully than the HECut model. The NPEX model is represented by

$$\begin{eqnarray}&&{F}_{\mathrm{NPEX}}=\left({A}_{1}{E}^{-{{\rm{\Gamma }}}_{1}}+{A}_{2}{E}^{+{{\rm{\Gamma }}}_{2}}\right)\exp \left(-\displaystyle \frac{E}{{E}_{\mathrm{fold}}}\right),\end{eqnarray} \tag{ 3 }$$

with five parameters, ${{\rm{\Gamma }}}_{1}$, Γ₂, A₁, A₂, and E_fold. We fixed Γ₂(>0) at the typical value of 2.0 (Mihara et al. 1998). The best-fit NPEX+Gaus model parameters are listed in Table 3. The fits have been improved, particularly when the source is luminous. However, the ${\chi }_{\nu }^{2}$ values are still unacceptable in intervals A and B. In Figure 7, the data-to-model ratios are presented. Above 10 keV, they still exhibit a feature that is similar to those in the HECut+Gaus model.

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This characteristic excess feature has already been noticed in the NuSTAR and NICER data (Jaisawal et al. 2019; Tao et al. 2019). There, it was considered a "broad iron line" and thus fitted with a Gaussian with σ ∼ 1.5 keV. We hence attempted to fit the GSC spectra with a model consisting of an NPEX continuum plus three Gaussians representing three lines at fixed energies of 6.4, 6.7, and 7.0 keV. The 6.4 keV line was allowed to take a free width, whereas the other two were assumed to be narrow. The fit was acceptable with ${\chi }_{\nu }^{2}=1.07$ (26 degrees of freedom). The spectrum in the interval A (=outburst peak) gave a 6.4 keV width of $\sigma ={1.27}_{-0.29}^{+0.27}$ keV and an equivalent width of ${{\rm{EW}}}_{\mathrm{Fe}}={0.54}_{-0.17}^{+0.20}$ keV, which are consistent with those measured with NICER and NuSTAR spectra (Jaisawal et al. 2019; Tao et al. 2019).

Although the excess feature in the GSC spectra can thus be interpreted as a broad iron line, its origin is not necessarily clear (Jaisawal et al. 2019; also see later discussion). Therefore, other interpretations should be explored. The characteristic excess also reminds us of the "10 keV feature" that has been observed in several XBPs (e.g., Coburn et al. 2002) and interpreted as either a bump or an absorption on the cutoff power-law continuum (Klochkov et al. 2008). In the bump case, it can be fitted with a broad Gaussian (e.g., Müller et al. 2013; Reig & Nespoli 2013) or a blackbody (BB; Reig & Coe 1999). In the absorption case, it can look like a cyclotron-resonance absorption (CYAB; Mihara et al. 1990). We hence repeated the model fits by incorporating either a BB (bump case) or a CYAB model (absorption case) to the HECut or NPEX continuum.

Table 4 summarizes the best-fit parameters of these models for the A, B, C, and D spectra. Because E_cut in HECut or A₂ in NPEX was consistent with zero, the continuum in both models can be replaced by a simple cutoff power law (Cutoffpl) as ${F}_{\mathrm{Cutoffpl}}\,=A\exp (-E/{E}_{\mathrm{fold}})$. Therefore, the results are given in simple model forms as Cutoffppl+BB+Gaus and Cutoffpl*CYAB+Gaus. Figure 7 compares data-to-model ratios of intervals A and B when using the modeling of (1) HECut+Gaus, (2) NPEX+Gaus, (3) Cutoffpl+BB+Gaus, and (4) Cutoffpl*CYAB+Gaus. The fits are significantly improved by adding the BB or CYAB component. In the first two models of HECut+Gaus and NPEX+Gaus, the ratios show a dip-like structure at 6.4 keV because the broad excess feature was fitted with a narrow Gaussian line. It was reduced in the latter two models. Figure 8 shows the implied Cutoffpl+BB+Gaus and Cutoffpl*CYAB+Gaus models that give the best fits to the interval-A spectrum.

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Table 4.  The Best-fit Spectral Parameters with the Cutoffpl+BB+Gaus and Cutoffpl*CYAB+Gaus Models

	Model: Cutoffpl + BB + Gaus
	A	Γ	Efold	kTBBa	RBBa	—	EFe	EWFe	${F}_{0.5-60}$	fbol	${\chi }_{\nu }^{2}(\nu )$
Int.			(keV)	(keV)	(km)		(keV)	(eV)
A	${32}_{-2}^{+2}$	$\,{1.63}_{-0.10}^{+0.10}$	${49}_{-18}^{+70}$	$\,{1.61}_{-0.10}^{+0.09}$	${17}_{-2}^{+2}$	—	$\,{6.4}_{-0.00}^{+0.12}$	${120}_{-40}^{+50}$	${41.3}_{-1.6}^{+1.7}$	${1.64}_{-0.07}^{+0.08}$	1.29 (27)
B	${18}_{-1}^{+1}$	$\,{1.43}_{-0.06}^{+0.07}$	${27}_{-5}^{+8}$	$\,{1.34}_{-0.14}^{+0.14}$	${14.0}_{-2.5}^{+3.2}$	—	$\,{6.4}_{-0.00}^{+0.04}$	${190}_{-30}^{+50}$	${26.4}_{-0.7}^{+0.8}$	${1.63}_{-0.05}^{+0.06}$	1.48 (27)
C	$\,{7.8}_{-1.1}^{+0.8}$	$\,{1.28}_{-0.10}^{+0.08}$	${28}_{-6}^{+8}$	$\,{1.08}_{-0.16}^{+0.21}$	${13.6}_{-4.6}^{+7.1}$	—	$\,{6.4}_{-0.00}^{+0.18}$	${120}_{-50}^{+30}$	${15.8}_{-0.6}^{+0.6}$	${1.81}_{-0.08}^{+0.08}$	1.27 (27)
D	$\,{4.9}_{-1.0}^{+0.8}$	$\,{1.23}_{-0.15}^{+0.10}$	${31}_{-8}^{+12}$	$\,{0.94}_{-0.16}^{+0.36}$	${12.3}_{-7.4}^{+10.1}$	—	$\,{6.4}_{-0.00}^{+0.17}$	${130}_{-40}^{+60}$	${11.9}_{-0.6}^{+0.5}$	${1.96}_{-0.11}^{+0.10}$	1.74 (27)
	Model: Cutoffpl∗CYAB + Gaus
	A	Γ	Efold	Eab	Wab	Dab	EFe	EWFe	${F}_{0.5\mbox{--}60}$	fbol	${\chi }_{\nu }^{2}(\nu )$
Int.			(keV)	(keV)	(keV)		(keV)	(eV)
A	${28}_{-1}^{+1}$	$\,{1.22}_{-0.05}^{+0.05}$	${15}_{-1}^{+2}$	${10.9}_{-0.7}^{+0.5}$	$\,{3.7}_{-1.1}^{+1.6}$	$\,{0.22}_{-0.04}^{+0.04}$	$\,{6.4}_{-0.00}^{+0.09}$	${110}_{-30}^{+40}$	${37.5}_{-0.3}^{+0.3}$	${1.48}_{-0.02}^{+0.02}$	1.08 (26)
B	${18}_{-1}^{+1}$	$\,{1.24}_{-0.05}^{+0.05}$	${16}_{-1}^{+1}$	$\,{9.5}_{-1.0}^{+0.6}$	$\,{3.2}_{-1.1}^{+1.5}$	$\,{0.14}_{-0.03}^{+0.03}$	$\,{6.4}_{-0.00}^{+0.04}$	${160}_{-30}^{+30}$	${25.1}_{-0.2}^{+0.2}$	${1.55}_{-0.02}^{+0.02}$	1.13 (26)
C	$\,{8.4}_{-0.6}^{+0.5}$	$\,{1.20}_{-0.09}^{+0.06}$	${21}_{-3}^{+3}$	$\,{8.5}_{-2.2}^{+0.9}$	$\,{3.2}_{-1.4}^{+2.1}$	$\,{0.12}_{-0.03}^{+0.04}$	$\,{6.4}_{-0.0}^{+0.3}$	${80}_{-40}^{+40}$	${15.3}_{-0.2}^{+0.2}$	${1.75}_{-0.03}^{+0.04}$	0.98 (26)
D	$\,{5.4}_{-0.9}^{+0.3}$	$\,{1.21}_{-0.36}^{+0.07}$	${27}_{-9}^{+6}$	$\,{8.2}_{-8.2}^{+1.7}$	$\,{3.2}_{-1.5}^{+2.8}$	$\,{0.08}_{-0.04}^{+0.06}$	$\,{6.4}_{-0.0}^{+0.3}$	${100}_{-50}^{+80}$	${11.8}_{-0.2}^{+0.2}$	${1.94}_{-0.04}^{+0.05}$	1.63 (26)

Notes. Errors are given with 90% limits of statistical uncertainty if the fits are within the acceptable level (${\chi }_{\nu }^{2}\lt 2$).

^aTemperature (kT_BB) and radius (R_BB) of BB emission assuming the distance D = 7 kpc. ^bCyclotron-resonance energy (E_a), width (W_a), and depth (D_a) in the CYAB model.

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While the latter two models are at the acceptable levels, their data-to-model ratios in Figure 7 still seem to have a small (≲3%) structure at around 5 keV. This is considered partly due to the systematic errors on the GSC response function, associated with the Xe-L edge at 4.8 keV (Mihara et al. 2011). We confirmed that the model-fit results did not change significantly even if their energy range (4.5–5.5 keV) was masked.

To visualize the spectral-parameter evolution, Figure 9 summarizes these best-fit parameters against the X-ray luminosity, where we plot the results with the HECut+Gaus, Cutoffpl+BB+Gaus, and Cutoffpl*CYAB+Gaus fits that are acceptable within the 90% confidence limits. The power-law index Γ increased with the luminosity, as expected from the negative correlation in the HIDs (Figure 2). The Gaussian centroid for the iron line remained at E_Fe = 6.4 keV throughout the period. This appears inconsistent with the NuSTAR and NICER results that the narrow (σ ≲ 300 eV) iron-line centroid shifted from 6.4 to 6.7 keV in the luminous regime over the Eddington limit (Jaisawal et al. 2019; Tao et al. 2019), but this discrepancy is because the GSC spectrum with the resolution ΔE ∼ 0.8 keV (at 6 keV) was dominated by the broad structure with a peak at ∼6.4 keV. The equivalent width is almost constant at EW_Fe ∼ 100 eV, in agreement with the NICER result that the iron-line flux was approximately proportional to the luminosity (Jaisawal et al. 2019), as well as with the behavior of the typical XBPs (Reig & Nespoli 2013). When the 10 keV feature was fitted with a BB model, the BB temperature increased from kT_BB ∼ 1 to 1.4 keV, but the BB radius did not change significantly from R_BB ∼ 10 km. When it was fitted with the CYAB absorption model, the CYAB energy and width remained at E_a ∼ 10 and W_a ∼ 3 keV, respectively, but the depth increased from D_a ∼ 0.1 to 0.2 with the luminosity.

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3.3.2. Pulse Phase–resolved Spectra

Pulse profiles obtained by NICER in 0.2–12 keV were not very dependent on the energy bands during the luminous (≳2 × 10³⁸ erg s⁻¹) period, but their pulsed fractions increased toward higher energies (WMJ18). As seen in Section 3.2 (Figure 4), the same trend was observed in the GSC 2–20 keV data. This suggests that the X-ray spectrum gets harder around the pulse peaks.

We hence extracted pulse phase–resolved spectra for four pulse phases as illustrated in Figure 4, which we hereafter call the minimum (PP1), intermediate high (PP2), intermediate low (PP3), and maximum (PP4), respectively, in the double-peak profile. Figure 10 shows the ratios of each phase-resolved spectrum to the entire phase average during the luminous period of intervals A, B, C, and D. It confirms that the pulsed fraction indeed increases toward higher energies. We also performed the model fit to the individual pulse-phase spectra but were not able to find significant phase-dependent parameter changes, except for the power-law index Γ and the emission normalization.

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As seen in Figure 3, the spin-frequency increase, i.e., the pulsar spin-up, is closely correlated with the X-ray intensity. Although correlation was already reported by Doroshenko et al. (2018) and Zhang et al. (2019), we here refine the analysis by jointly using the MAXI GSC light curve and the Fermi GBM pulse period. These data have the advantage that both are available almost with a daily sampling.

For the above purpose, we need to convert I_2–20 to the bolometric luminosity L_bol. The bolometric correction factor ${f}_{\mathrm{bol}}={F}_{0.5-60}/{I}_{2\mbox{--}20}$ used in this conversion depends on the energy spectrum. Figure 11 shows the relation between I_2–20 and F_0.5–60 calculated from the best-fit spectral models in Tables 3 and 4. Although the values of F_0.5–60 depend to some extent on the fitting models, the effect is within the statistical uncertainties (≲10%). The factor f_bol slightly decreases toward the higher I_2–20, according to the spectral softening as observed in the HID (Figure 2). Based on the HID behavior, we assumed that the f_bol–I_2–20 relation can be expressed as

$$\begin{eqnarray}{f}_{\mathrm{bol}}=\left\{\begin{array}{ll}{f}_{0} & ({I}_{2-20}\lt 4.5)\\ {f}_{0}{\left({I}_{2-20}/4.5\right)}^{-\gamma } & ({I}_{2-20}\gt 4.5),\end{array}\right.\end{eqnarray} \tag{ 4 }$$

which is constant at f₀ in I_2–20 < 4.5 where HC is constant and decreases by a power law in I_2–20 > 4.5. We fitted Equation (4) to the f_bol–I_2–20 data obtained from the NPEX spectral parameters and determined the best-fit values of f₀ = 1.74 × 10⁻⁸ erg photon⁻¹ and γ = 0.10. The scale of L_bol in Figures 2, 3, and 4(a) and (b) associated with I_2–20 has been calculated by ${L}_{\mathrm{bol}}=4\pi {D}^{2}{I}_{2\mbox{--}20}{f}_{\mathrm{bol}}$ and D = 7 kpc.

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Figure 12 shows the obtained ${\dot{\nu }}_{{\rm{s}}}$–L_bol relation, where we calculated the spin-frequency derivative ${\dot{\nu }}_{{\rm{s}}}$ from the Fermi/GBM pulsar data with the same procedure as in Sugizaki et al. (2017). It clearly reveals a positive correlation close to the proportionality. We fitted the data points with a power law, ${\dot{\nu }}_{{\rm{s}}}\propto {L}_{\mathrm{bol}}^{\alpha }$, and obtained the best-fit power-law index α = 1.0 (±0.02), where the fitting error is estimated by adding appropriate systematic errors so as to make the fit formally acceptable. The best-fit α value is somewhat higher than those of the theoretical predictions, 6/7 in Ghosh & Lamb (1979, hereafter GL79), 0.85 in Lovelace et al. (1995), and 0.9 in Kluźniak & Rappaport (2007), but agrees with the empirical relations determined from the observed data of major Be XBPs (Bildsten et al. 1997; Sugizaki et al. 2017).

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We also compared the coefficient of proportionality between ${\dot{\nu }}_{{\rm{s}}}$ and L_bol with those of the theoretical models. Specifically, the data in Figure 12 are compared with the relations predicted by the representative GL79 model, assuming the canonical neutron-star mass 1.4 M_⊙, radius 10 km, moment of inertia 10⁴⁵ g cm², and typical surface magnetic fields B_s = 1 × 10¹² and 1 × 10¹³ G. Although the data and models slightly disagree in α, the data are mostly distributed between the two model curves. This means that the data prefer B_s between these two values, i.e., a few ×10¹². The best-fit GL79 model suggests B_s = 3.4 × 10¹² G.

The simultaneous changes in the X-ray spectrum and the pulse profile of Swift J0243.6 have been noticed in the NICER and Fermi/GBM data (WMJ18). However, possible relations between the two attributes have not necessarily been clear because of their uneven time coverage. Here we study this issue by using the MAXI GSC results.

As shown in Figure 2, during the remarkable X-ray active period of I_2–20 ≳ 0.8, the two hardness ratios, the SC and HC, both showed a negative correlation against I_2–20. According to the simple HID classification (Reig 2008), the period of I_2–20 > 0.8 is classified into the diagonal branch (DB), and the part of I_2–20 ≲ 0.8 is thought to be the horizontal branch (HB) from the result of NICER (WMJ18). However, the two HIDs employing SC and HC show characteristic differences in the DB. We hence divide the DB region into the following two states: (i) the intermediate DB state of 0.8 ≲ I_2–20 ≲ 4.5, where the SC changed more than the HC, and (ii) the extreme DB state of I_2–20 ≳ 4.5, where the HC changed more than the SC. Using f_bol in Equation (4), these characteristic intensities of I_2–20 = 0.8 and 4.5 correspond to the luminosities of L_bol = 0.9 × 10³⁸ and 5 × 10³⁸ erg s⁻¹, respectively.

The spectral analysis clarified how the 2–30 keV spectrum changed between the two DB states. Generally, X-ray spectra of Be XBPs are represented with a Cutoffpl continuum (Makishima et al. 1999; Coburn et al. 2002), where their luminosity-dependent changes in the DB are characterized by a correlation between L_bol and Γ (Reig & Nespoli 2013). As shown in Figure 9, the best-fit parameters obtained from Swift J0243.6 exhibit this general behavior. In the extreme DB state (intervals A, B, C, and D), the increased 6 keV excess on top of the Cutoffpl continuum further enhanced the change in the HC but reduced the change in the SC.

The pulse-profile evolution in Figures 3 and 4 also suggests that it is related to the two DB states because a transition between the single and double peak occurred at I_2–20 ≃ 4.5, just at the boundary of the two DB states. These correlated changes in the spectrum and pulse profile are considered to reflect luminosity-related changes in the physical condition of the X-ray emission region. Table 5 summarizes how the spectral and temporal properties depend on the X-ray intensity.

Table 5.  Luminosity-dependent Changes in the X-Ray Properties

HID Branch	HB		DB
Substate in DB			Intermed.		Extreme
${I}_{2\mbox{--}20}$ a		0.8		4.5		30
Lbol (1038)b		0.9		5.0		26
Intervalc	H	G F (Y X)		E D C B A
SC-${I}_{2\mbox{--}20}$ sloped	+(↗)		−(↘)		∼0 (→)
HC-${I}_{-20}$ sloped	+(↗)		∼0 (→)		−(↘)
Spec. profile	Cutoffpl + iron-K line				+ $\gtrsim 6\,\mathrm{keV}$ excess
Pulse profile	Single-peak				Double-peak

Notes.

^a2–20 keV photon flux (photons cm⁻² s⁻¹). ^bBolometric luminosity (erg s⁻¹). ^cGSC data intervals defined in Table 2. ^d+(↗) means positive correlation, (↘) means negative correlation, and ∼0(→) means little dependence.

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As discussed above, the X-ray spectrum of Swift J0243.6 in the extreme DB state is characterized by the excess at ≳6 keV. Because the feature can be represented by a Gaussian function with a centroid of ∼6.4 keV and width σ ∼ 1.2 keV, Jaisawal et al. (2019) interpreted it as a broad iron-K line. However, a question about what causes such a broad iron line has not been answered. The broad Gaussian model also needs to have a large equivalent width of ∼1 keV (Jaisawal et al. 2019; Tao et al. 2019), which would be realized only when the direct X-ray component is suppressed by source obscuration. However, such an obscuration feature has not been observed. Meanwhile, to explain the power spectrum obtained from the Insight-HXMT data during the DB period, Doroshenko et al. (2020) proposed a scenario that the major X-ray emission came from an accretion disk, which made the transition from a state dominated by Coulomb collisions to that by radiation. However, the picture is also considered difficult from the pulsed fraction evolution, which increased up to ≳40% (rms amplitude) in proportion to the luminosity. Furthermore, an accretion disk in an XBP must be truncated at the magnetospheric radius, or so-called Alfvén radius (Ghosh & Lamb 1979b), ${R}_{{\rm{A}}}=1400{L}_{38}^{-2/7}{M}_{1.4}^{1/7}{R}_{6}^{10/7}{B}_{12}^{4/7}\,\mathrm{km}$, where L₃₈, M_1.4, R₆, and B₁₂ are the source luminosity in 10³⁸ erg s⁻¹, neutron-star mass 1.4 M_⊙, radius 10⁶ cm, and surface magnetic field 10¹² G. Therefore, the specific gravitational energy that the accreting matter acquires throughout the disk would be two orders of magnitude smaller than is available by the time it reaches the neutron-star surface. In other words, the disk would not provide a major source for the observed pulsed hard X-rays. The absorption line detected with the Chandra HETGS can be explained without invoking an X-ray-emitting disk because the strong radiation pressure would produce outflows from the cool disk outside R_A or the accretion stream inside R_A. Hence, we consider another interpretation for these spectral and pulse-profile behaviors.

As shown in Figure 7, the MAXI GSC spectra with the excess feature can be fitted if either a bump represented by a BB or an absorption represented by a CYAB model is incorporated into the HECut or NPEX continuum. These model parameters are consistent with those for the "10 keV feature" that has been reported previously in several XBPs (e.g., Coburn et al. 2002; Klochkov et al. 2008). Also, similar spectral and pulse-profile changes have been observed in several Be XBPs, 4U 0115+63 (Ferrigno et al. 2009), X 0331+53 (Tsygankov et al. 2010), EXO 2030+375 (Epili et al. 2017), and SMC X-3 (Weng et al. 2017), when close to the Eddington limit. These facts imply that the behavior is not unique to Swift J0243.6 but common to the other XBPs.

Based on the canonical models of X-ray emission from XBPs (e.g., Basko & Sunyaev 1976; Becker et al. 2012), these X-rays are considered to originate from accretion columns that are formed on the neutron-star surface through the magnetic field lines. In this scenario, the two HID branches, HB and DB, are thought to represent two accretion regimes where accreting matter flows are decelerated by Coulomb collisions (subcritical accretion regime) and radiation pressure (supercritical accretion regime), respectively. The spectral softening in the DB is interpreted by a development of Comptonized emission in the accretion columns. As the luminosity increases, the region responsible for the Comptonization extends farther from the neutron-star surface, and then the temperature of the Comptonizing plasma decreases. The scenario also explains the pulsed emission evolution (Basko & Sunyaev 1975; Becker et al. 2012). Theoretically (Becker et al. 2012), the emission column height h_s is expected to be proportional to L_bol in the supercritical regime until it reaches a few km at the Eddington luminosity. When h_s becomes larger than the column radius r_c (∼1 km), the pulsed emission geometry changes from pencil beam to fan beam, which results in the transition from the single-peak to the double-peak pulse profile. Furthermore, if h_s ≫ r_c, the pulsed fraction tends to be approximately proportional to h_s and thus to L_bol. The observed correlation between f_pul and I_2–20 in Figure 4(b) agrees with this prediction.

Then, what produces the 6 keV excess in the extreme DB state? When the BB bump model is employed, the change of the feature with L_bol is represented by the BB temperature, which increased from kT_BB = 1.0 to 1.6 keV. On the other hand, the BB radius was almost constant at R_BB ∼ 10 km. Assuming that the BB emission came from the accretion column of r_c ∼ 1 km, its height needs to be h_s ∼ 100 km to attain the BB area = πR_BB² ∼ 100 km². The estimated h_s seems too high compared with the theoretical prediction of a few km. This difficulty would not be solved even if we considered a significant temperature gradient in the emission region.

Alternatively, assuming the CYAB interpretation, we obtained the best-fit parameters as E_a ≃ 10 keV, W_a ≃ 3 keV, and D_a ≃ 0.1–0.2. Compared with other XBPs (e.g., Makishima et al. 1999; Coburn et al. 2002), the values of E_a and D_a are at the lower ends of their distributions but still within their observed ranges. The value of W_a is typical. Therefore, the CYAB parameters are not so unusual.

In this scenario, E_a = 10 keV means ${B}_{{\rm{s}}}=0.86(1+{z}_{{\rm{g}}})\,\times {10}^{12}\simeq 1.1\times {10}^{12}$ G, where z_g represents the gravitational redshift. This estimate is consistent with the implication of Figure 12. On the other hand, the L_bol dependence of the parameters, including an increase of D_a from 0.1 to 0.2 and relatively constant values of E_a and W_a, is not necessarily typical of the cyclotron-resonance effects in other XBPs, where D_a is relatively constant and E_a often decrease toward high L_bol (e.g., Mihara et al. 2004). Therefore, we retain this interpretation as a possible candidate.

The surface magnetic field B_s is one of the key parameters of the accretion process. In the section above, we arrived at a possibility of B_s ≃ 1.1 × 10¹² G, assuming that the 6 keV excess feature in the spectrum of the extreme DB state is a result of a CYAB at ∼10 keV. Meanwhile, several other attempts to constrain B_s have been performed so far. Tsygankov et al. (2018) derived B_s < 1 × 10¹³ G from the upper limit on the propeller luminosity. An estimate of 0.1–2 × 10¹³ G was derived by WMJ18 from the HID transition luminosity and quasiperiodic oscillation (QPO) frequency. From the correlated X-ray flux and spin-up evolution observed by the Insight-HXMT, Zhang et al. (2019) estimated B_s ∼ 1 × 10¹³ G. While all of these constraints are consistent, they have large uncertainties that stem from those in the theoretical relations employed to interpret the observed data. As a result, these published reports do not enable us to either assess the reality of our cyclotron hypothesis or examine whether the B_s of Swift J0243.6 is different from those of typical XBPs.

We also studied this subject using the ${\dot{\nu }}_{{\rm{s}}}-{L}_{\mathrm{bol}}$ relation from the MAXI/GSC and Fermi/GBM data (Section 3.4) and found that the positive correlation between the two quantities smoothly extends up to the maximum luminosity, L_bol ≳ 2 × 10³⁹ erg s⁻¹ (Figure 12). Assuming a neutron-star mass of 1.4 M_⊙, a radius of 10 km, and the GL79 disk–magnetosphere interaction model, the data are best explained with B_s ≃ 3.4 × 10¹² G. Although the model largely reproduces the data, the fit is not as good as being acceptable. The discrepancy is considered mainly on the assumed physical conditions in GL79, which are estimated to affect the coefficient of proportionality between ${\dot{\nu }}_{{\rm{s}}}$ and L_bol by a factor of ∼2 (e.g., Bozzo et al. 2009). In fact, Sugizaki et al. (2017) confirmed that the GL79 model reproduced the observed ${\dot{\nu }}_{{\rm{s}}}$–L_bol relations of 12 Be XBPs with an accuracy of a factor of ≲3.

To avoid these model uncertainties, in Figure 13, we compare the observed ${\dot{\nu }}_{{\rm{s}}}$–L_bol relation of Swift J0243.6 with those of other Be XBPs in which B_s is determined by the cyclotron-resonance feature. These are the nine Be XBPs in Sugizaki et al. (2017): 4U 0115+63, X 0331+53, RX J0520.5−6932, H 1553−542, XTE J1946+274, KS 1947+300, GRO J1008−57, A 0535+262, and GX 304−1. The results for these XBPs have been derived from the MAXI/GSC and Fermi/GBM data in the same way as for Swift J0243.6. The values of L_bol for four objects, 4U 0115+63, X 0331+53, A 0535+262, and GX 304−1, have been revised using the updated D in the Gaia DR2. (These changes in D from the values employed by Sugizaki et al. 2017 are <15%.) Except for one outlier, X 0331+53, the ${\dot{\nu }}_{{\rm{s}}}$–L_bol relations of these objects all line up within a factor of ∼3. The data of Swift J0243.6 are located almost at the bottom of them, in agreement with the fact that the best-fit GL79 model implies the lowest B_s among the known XBPs. The result suggests that the B_s of Swift J0243.6 is not much different from the B_s range of XBPs and tends to be relatively low. The timing analysis hence reinforces the cyclotron-absorption interpretation of the ∼6 keV excess feature.

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In our Galaxy, Swift J0243.6 is the first example of a ULXP, as well as a ULX. Therefore, the MAXI GSC results should give important hints about their unknown origins. Table 6 compares the basic parameters of Swift J0243.6 with those of the five known ULXPs, M82 X-2 (Bachetti et al. 2014), NGC 300 ULX-1 (Carpano et al. 2018), NGC 7793 P13 (Fürst et al. 2016), NGC 5907 ULX-1 (Israel et al. 2017), and SMC X-3 (Tsygankov et al. 2017), which have all been securely identified as ULXPs with maximum luminosities >2.5 × 10³⁹ erg s⁻¹.

Table 6.  Basic Parameters of Known ULX Pulsars

Source Name	Pspin (s)	Porb (days)	Lmax (erg s−1)	D (Mpc)	Opt.	P/T	${B}_{{\rm{s}}}^{\mathrm{su}}$ (G)	${B}_{{\rm{s}}}^{\mathrm{pr}}$ (G)
NGC 5907 ULX-1${}^{* 1}$	1.14	5?	$6.0\times {10}^{40}$	17	⋯	P	⋯	⋯
M82 X-2${}^{* 2}$	1.37	2.5	$2.0\times {10}^{40}$	3.5	B9I	P	⋯	$\sim 1\times {10}^{14}$
NGC 7793 P13${}^{* 3}$	0.42	64	$1.0\times {10}^{40}$	3.9	⋯	P	$(1.5\times {10}^{12}$)	–
NGC 300 ULX-1${}^{* 4}$	31.6	⋯	$5.0\times {10}^{39}$	1.9	Be	T	$3\times {10}^{12}$	–
SMC X-3${}^{* 5}$	7.8	45.1	$2.5\times {10}^{39}$	0.062	Be	T	$2.6\times {10}^{12}$	$(1\mbox{--}5)\times {10}^{12}$
Swift J0243.6${}^{* 6}$	9.7	27.6	$2.5\times {10}^{39}$	0.007	Be	T	$2.5\times {10}^{12}$	$\lt 6.2\times {10}^{12}$

Note. P_spin—spin period; P_orb—orbital period; D—source distance; L_max—observed maximum luminosity; Opt.—optical counterpart; P/T—persistent or transient; ${B}_{{\rm{s}}}^{\mathrm{su}}$—B_s from luminosity–spin-up relation; ${B}_{{\rm{s}}}^{\mathrm{pr}}$—B_s from propeller effect.

References. ${}^{* 1}$Israel et al. (2017), ${}^{* 2}$Bachetti et al. (2014) and Tsygankov et al. (2016), ${}^{* 3}$Fürst et al. (2016), ${}^{* 4}$Carpano et al. (2018), ${}^{* 5}$Tsygankov et al. (2017), ${}^{* 6}$Tsygankov et al. (2018) and WMJ18.

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The X-ray properties of XBPs depend considerably on the type of their mass-donating companions. The XBPs known in our Galaxy are mostly classified into those accompanied by supergiant primaries, i.e., Sg XBPs, and Be XBPs (e.g., Reig 2011; Walter et al. 2015), which have been a major focus of the present paper. While Sg XBPs show persistent X-ray activities often involving flare-like time variations, Be XBPs show mostly periodical outbursts lasting for a week to months (e.g., Bildsten et al. 1997). Out of the six ULXPs in the sample, four have allowed optical identification and hence the classification of one Sg XBP and three Be XBPs. From the type of their X-ray activity, the remaining two are naturally considered to be Sg XBPs. Therefore, regardless of its optical companion type, any XBP may become, under certain conditions, a ULXP. In Table 6, the ULXPs with Be companions are generally found to have longer P_s, as well as longer P_orb, than the objects of Sg companions, in agreement with those of the XBPs in our Galaxy (Corbet 1986). This suggests that the binary evolution of ULXPs is not much different from that of standard XBPs.

As the origin of the super-Eddington luminosity in ULXPs, a strong B_s reaching ∼10¹⁴ G has been proposed with a theoretical model (Mushtukov et al. 2015). However, it would be natural to presume that a stronger dipole field would enlarge the Alfvén radius and make it closer to the Bondi radius for gravitational capture of the accreting gas, thus suppressing the accretion. Actually, Yatabe et al. (2018) found, through the ${\dot{\nu }}_{{\rm{s}}}$–L_bol technique, that the very low L_bol XBP X Persei has B_s ∼ 10¹⁴ G. Then, how about the values of B_s of the six ULXPs? In any of them, B_s has not been determined by the cyclotron feature. Instead, its likely range has been estimated empirically and indirectly, employing either (a) the simultaneous luminosity–spin-up evolution (e.g., this work; Doroshenko et al. 2018; Zhang et al. 2019), (b) the propeller effect (e.g., Tsygankov et al. 2016, 2017, 2018); (c) the assumption of a torque equilibrium between the accreting matter and the pulsar magnetosphere (Carpano et al. 2018), (d) the HID and/or pulse-profile transitions (Tsygankov et al. 2017; WMJ18), or (e) the QPO frequency (WMJ18). Table 6 refers to the results obtained by (a) or (b) because they are based on relatively simple theoretical models and have been better calibrated against observation data. Among these values, ${B}_{{\rm{s}}}^{\mathrm{pr}}\sim {10}^{14}$ G in M82 X-2, which was derived by Tsygankov et al. (2016), looks extraordinarily higher than the others (∼10¹² G). However, M82 X-2 is consider to be an Sg XBP from the persistent X-ray activity, so its rapid flaring episodes could mimic the propeller effect. All of the other estimates agree with those of the standard XBPs, (1–8) × 10¹² G (e.g., Makishima et al. 1999; Yamamoto et al. 2014). This suggests that the B_s values of the ULXPs are not different from those of the standard XBPs.

As discussed in Sections 4.1 and 4.2, the X-ray behavior of Swift J0243.6 during the extreme DB state is represented by the spectral softening due to the broad 6 keV enhancement and the transition from the single-peak to the double-peak pulse profiles. Similar spectral and pulse-profile changes at the luminosity close to the Eddington limit have already been reported in several Be XBPs, even if they are not identified as UXLPs (Ferrigno et al. 2009; Tsygankov et al. 2010; Epili et al. 2017; Weng et al. 2017). On the other hand, an absorption-like profile that can be fitted with a cyclotron-resonance model was observed from another ULXP, NGC 300 ULX-1 (Walton et al. 2018). These results suggest that the observed properties of Swift J0243.6 during the extreme DB state are common to ULXPs and smoothly extrapolated from those of normal XBPs.

In summary, we find neither clear differences between normal XBPs and ULXPs in the basic parameters listed in Table 6 nor discontinuity in their luminosity-dependent X-ray behavior. Therefore, the question of what causes the extraordinarily high luminosity in ULXPs still remains unanswered. The key parameter might be in those that have not been discussed above. One possible candidate would be the angle θ_m of the magnetic dipole moment to the neutron-star spin axis. If θ_m gets close to 90°, the accretion path from the inner edge of the disk onto the neutron-star surface through the field lines becomes shorter and straighter. In the θ_m ≃ 90° case, radiation pressure in the fan-beam geometry, which is expected under supercritical accretion (Section 4.2), gets maximum in the direction perpendicular to the accretion plane; thus, it does not work effectively to decelerate the matter flow. This mechanism will increase the maximum luminosity.