Swift  Observations of SMC X-3 during Its 2016–2017 Super-Eddington Outburst

High-mass X-ray binaries contribute a large fraction of X-ray emission in normal galaxies, and they are believed to reflect the recent star formation activities in their host galaxies (e.g., Grimm et al. 2002; Mineo et al. 2012). According to the states of their optical companions, high-mass X-ray binaries can be subdivided into supergiant X-ray binaries and Be/X-ray binaries (BeXBs). A BeXB consists of a Be star and a compact object. Virtually all confirmed compact objects in BeXBs are neutron stars (NSs), and all these systems show X-ray pulsations (see Reig 2011 for reviews). As young systems, NSs in BeXBs have a high magnetic field ($B\gt {10}^{10}$ G); therefore, BeXBs provide unique natural laboratories for studying physics in extremely strong gravity and magnetic fields.

The direct measurement of an NS magnetic field strength can be achieved from the detection of a cyclotron scattering resonance feature (e.g., Coburn et al. 2002; Yan et al. 2012; Fürst et al. 2014; Walter et al. 2015, and references therein). Additionally, investigating the interaction between the magnetosphere and the accretion matter, we can acquire the information of NS magnetic field indirectly (e.g., Weng & Zhang 2011; Shi et al. 2015; Christodoulou et al. 2016). That is, the effect of the magnetic field strength manifests itself by the size of its magnetosphere corotating with the central NS. The boundary of the magnetosphere is determined where the ram pressure of in-falling flow is balanced by magnetic pressure; thus, it expands with field strength and decreases with the mass accretion rate (Lamb et al. 1973; Ghosh & Lamb 1979). As the accretion rate decreases below the critical value, the magnetospheric radius (${R}_{{\rm{m}}}$) grows beyond the corotation radius (${R}_{\mathrm{co}}=\sqrt[3]{\tfrac{{{GMP}}^{2}}{4{\pi }^{2}}}$), at which the Keplerian angular frequency is equal to that of the NS spin frequency, and the centrifugal barrier spins away accretion matter. If most of the material is prevented from accreting onto NS, X-ray flux and pulsation decay sharply in a few days, i.e., the "propeller" effect (e.g., Cui 1997; Campana et al. 2014). Alternatively, the magnetosphere is penetrated into the corotation radius at high luminosity, leading to the spin-up of an NS. If NSs are close to spin equilibrium, their magnetic fields can be estimated from long-term averaged spin parameters and X-ray luminosity (e.g., Klus et al. 2014; Shi et al. 2015). Meanwhile, the torque reversals between steady spin-up and spin-down are commonly shown in BeXBs (e.g., Bildsten et al. 1997).

Besides the long-term average spin evolution, the instantaneous torque measurements during episodic outbursts are essential to test accretion torque theories. Transient BeXBs experience periodic and less energetic (${L}_{{\rm{X}}}\lt {10}^{37}$ erg s⁻¹) outbursts or rare giant outbursts, which are referred to as type I and type II outbursts, respectively (Reig 2011). The tight relationships between the spin-up rate and the (pulsed) flux detected in the luminous outbursts of BeXBs (e.g., A0535+262 and 2S 1417−624) are interpreted as the sign of transient accretion disks around NSs, which are supported by the detection of simultaneous quasi-periodic oscillations (QPOs; Finger et al. 1996; Sartore et al. 2015). It is worth noting that a small number of sources (e.g., SMC X-1, LMC X-4, 4U 0115+63, and V0332+53) can reach a peak X-ray luminosity in excess of 10³⁸ erg s⁻¹ (e.g., Li et al. 2011; Mushtukov et al. 2015b, and references therein). Intriguingly, a growing number of ultraluminous X-ray sources (ULXs) in nearby galaxies have been found to exhibit coherent pulsations (Bachetti et al. 2014; Fürst et al. 2016; Israel et al. 2016, 2017), indicating a connection to X-ray pulsars (Mushtukov et al. 2015b; Shao & Li 2015; Kawashima et al. 2016; King & Lasota 2016; Mushtukov et al. 2017). Nowadays, super-Eddington accretions in magnetized NSs draws more attention (e.g., Ekşi et al. 2015; Pan et al. 2016; Tsygankov et al. 2016; Chen 2017). However, a detailed study on such dramatic phenomena is hampered by the lack of observations.

The Small Magellanic Cloud (SMC) is the second nearest galaxy (d = 62.1 kpc; Hilditch et al. 2005; Graczyk et al. 2014; Scowcroft et al. 2016) after the Large Magellanic Cloud, and it has high-mass X-ray binaries in abundance due to recent star-forming activities (Zaritsky et al. 2002; Sturm et al. 2013; Yang et al. 2017). SMC X-3 (also known as SXP 7.78) was discovered with SAS 3 X-ray observatory in 1978 (Clark et al. 1978), and was identified as an accreting pulsar with a detected pulsation of 7.78 s (Edge et al. 2004). The spectral type of the optical counterpart is identified as B1–B1.5 with V = 14.91 (McBride et al. 2008). An orbital period of ∼44.9 days was detected in both X-ray and optical bands (Corbet et al. 2003; Cowley & Schmidtke 2004; Galache et al. 2008; Bird et al. 2012). However, its eccentricity and other orbital parameters are still unknown. Recently, SMC X-3 underwent a giant type II outburst in 2016 with a peak X-ray luminosity of $\sim {10}^{39}$ erg s⁻¹, and it has been monitored in the Target of Opportunity mode by Swift since 2016 August 10. On 2016 November 8, we reported our preliminary results on the analysis of the Swift data (Weng et al. 2016), which are the basis of this work. During preparation of this paper, Townsend et al. (2017) investigated the optical and X-ray data (including the Swift data) of SMC X-3, and obtained similar orbital parameters of the binary as given in our paper. In this paper, we focus on the Swift data and carry out a comprehensive analysis on these data to investigate the physics of super-Eddington accretion around a magnetized NS. The data reduction is described in the next section. In Section 3, we perform the timing analysis and calculate the orbital elements of the binary. In Section 4, we discuss the physical implications of these results and present our main conclusions.

The MAXI/GSC was triggered by the brightening of SMC X-3 on 2016 August 8 (Negoro et al. 2016), which was confirmed by Swift during its survey of the SMC (Kennea et al. 2016). In this paper, we analyze all Swift pointing observations taken between 2016 August 10 and 2017 January 1. The Swift Gamma Ray Burst Explorer carries three scientific instruments covering a broad energy range of $\sim 0.002\mbox{--}150\,\,\mathrm{keV}$: the Burst Alter Telescope (BAT), the X-ray Telescope (XRT), and the UV/Optical Telescope (UVOT; Gehrels et al. 2004). The BAT daily light curve is adopted from Krimm et al. (2013).⁶ Meanwhile, both the XRT and the UVOT data are processed with the packages and tools available in heasoft 6.19.

When the source is bright, the observations are carried out in the windowed-timing (WT) mode. While after 2017 January 16, the count rate in 0.5–10 keV is less than 0.5 cts s⁻¹, and the observations are performed in the photon-counting (PC) mode without significant pile-up effects (Table 1). For the XRT data, the initial event cleaning is executed with the task xrtpipeline using standard quality cuts. The source and background events are extracted from a circle and an annulus region centered at the source position, respectively. The light curves are corrected for the telescope vignetting and point-spread-function losses with the task xrtlccorr, and then they are subtracted by the scaled background count rate to yield the net light curves (Figure 1). The hardness for each observation is calculated as the ratio of average count rates (CR) in (2.0–10.0 keV)/(0.5–2.0 keV) bands. The source becomes undetected in last three observations, and the upper limit of count rates is estimated by using the X-ray image package XIMAGE.

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Table 1.  Log of Swift/XRT Data

ObsID	Date	Exposure	Mode	${R}_{\mathrm{Source}}$	${R}_{\mathrm{Background}}$	${L}_{0.6\mbox{--}10\mathrm{keV}}$
		(s)		(pixel)	(pixel/pixel)	(1037 erg s−1)
00034673001	2016 Aug 10	4671	WT	25	25/50	${35.3}_{-0.4}^{+0.4}$
00034673002	2016 Aug 12	1978	WT	25	25/50	${46.9}_{-0.7}^{+0.7}$
00034673003	2016 Aug 14	935	WT	25	25/50	${59.0}_{-1.1}^{+1.1}$
00034673004	2016 Aug 18	1997	WT	25	25/50	${68.3}_{-0.8}^{+0.8}$
00034673005	2016 Aug 20	2050	WT	25	25/50	${84.0}_{-1.0}^{+1.0}$
00034673006	2016 Aug 22	1958	WT	25	25/50	${90.1}_{-0.9}^{+0.9}$
00034673007	2016 Aug 24	1291	WT	25	25/50	${102.7}_{-1.3}^{+1.4}$
00034673008	2016 Aug 26	504	WT	25	25/50	${98.2}_{-1.9}^{+1.9}$
00034673009	2016 Aug 28	1583	WT	25	25/50	${98.2}_{-1.1}^{+1.1}$
00034673010	2016 Aug 30	953	WT	25	25/50	${88.1}_{-1.3}^{+1.3}$
00034673011	2016 Sep 01	793	WT	25	25/50	${92.9}_{-1.6}^{+1.6}$
00034673012	2016 Sep 03	1932	WT	25	25/50	${78.9}_{-1.0}^{+1.0}$
00034673013	2016 Sep 05	1345	WT	25	25/50	${56.6}_{-1.0}^{+1.0}$
00034673014	2016 Sep 06	2985	WT	25	25/50	${61.1}_{-0.6}^{+0.6}$
00034673015	2016 Sep 07	2977	WT	25	25/50	${61.3}_{-0.7}^{+0.7}$
00034673016	2016 Sep 08	2764	WT	25	25/50	${57.2}_{-0.7}^{+0.7}$
00034673017	2016 Sep 13	2973	WT	25	25/50	${45.1}_{-0.5}^{+0.5}$
00034673018	2016 Sep 14	2979	WT	25	25/50	${44.2}_{-0.5}^{+0.5}$
00034673019	2016 Sep 15	2981	WT	25	25/50	${42.8}_{-0.5}^{+0.5}$
00034673020	2016 Sep 17	3004	WT	25	25/50	${33.5}_{-0.4}^{+0.4}$
00034673021	2016 Sep 19	2918	WT	25	25/50	${30.8}_{-0.4}^{+0.4}$
00034673022	2016 Sep 21	2591	WT	25	25/50	${27.4}_{-0.4}^{+0.4}$
00034673023	2016 Sep 23	2992	WT	25	25/50	${26.0}_{-0.4}^{+0.4}$
00034673024	2016 Sep 25	2610	WT	25	25/50	${24.9}_{-0.4}^{+0.4}$
00034673025	2016 Sep 27	3289	WT	25	25/50	${23.6}_{-0.4}^{+0.4}$
00034673026	2016 Sep 29	2969	WT	25	25/50	${19.3}_{-0.4}^{+0.4}$
00034673027	2016 Oct 01	553	WT	25	25/50	${17.6}_{-0.7}^{+0.8}$
00034673028	2016 Oct 03	1718	WT	25	25/50	${16.2}_{-0.4}^{+0.4}$
00034673029	2016 Oct 06	726	WT	25	25/50	${17.5}_{-0.9}^{+1.0}$
00034673030	2016 Oct 07	2789	WT	25	25/50	${14.9}_{-0.3}^{+0.3}$
00034673031	2016 Oct 09	2839	WT	25	25/50	${13.2}_{-0.3}^{+0.3}$
00034673032	2016 Oct 11	1982	WT	25	25/50	${12.6}_{-0.3}^{+0.4}$
00034673033	2016 Oct 13	2028	WT	25	25/50	${12.2}_{-0.3}^{+0.3}$
00034673034	2016 Oct 14	1199	WT	25	25/50	${11.6}_{-0.4}^{+0.5}$
00034673035	2016 Oct 19	2493	WT	25	25/50	${10.2}_{-0.3}^{+0.3}$
00034673036	2016 Oct 20	2852	WT	25	25/50	${9.62}_{-0.26}^{+0.26}$
00034673037	2016 Oct 21	3435	WT	25	25/50	${9.31}_{-0.22}^{+0.23}$
00034673038	2016 Oct 23	3685	WT	25	25/50	${9.37}_{-0.22}^{+0.22}$
00034673039	2016 Oct 25	3626	WT	25	25/50	${8.51}_{-0.21}^{+0.22}$
00034673040	2016 Oct 27	2959	WT	25	25/50	${7.67}_{-0.25}^{+0.25}$
00034673041	2016 Oct 29	3984	WT	25	25/50	${9.04}_{-0.23}^{+0.23}$
00034673042	2016 Nov 06	5256	WT	25	25/50	${7.16}_{-0.17}^{+0.18}$
00034673043	2016 Nov 04	3973	WT	25	25/50	${7.59}_{-0.22}^{+0.22}$
00034673044	2016 Nov 08	4646	WT	25	25/50	${6.57}_{-0.18}^{+0.18}$
00034673045	2016 Nov 10	4453	WT	25	25/50	${6.30}_{-0.18}^{+0.18}$
00088012001	2016 Nov 13	1803	WT	25	25/50	${5.56}_{-0.26}^{+0.26}$
00034673046	2016 Nov 14	4452	WT	25	25/50	${5.78}_{-0.16}^{+0.16}$
00034673047	2016 Nov 16	4328	WT	25	25/50	${5.91}_{-0.19}^{+0.19}$
00034673048	2016 Nov 18	4628	WT	25	25/50	${5.04}_{-0.15}^{+0.16}$
00034673049	2016 Nov 20	4881	WT	25	25/50	${5.07}_{-0.16}^{+0.16}$
00034673050	2016 Nov 22	3287	WT	25	25/50	${4.21}_{-0.17}^{+0.17}$
00034673051	2016 Nov 24	4484	WT	25	25/50	${3.89}_{-0.14}^{+0.14}$
00034673053	2016 Nov 26	4430	WT	25	25/50	${3.82}_{-0.15}^{+0.15}$
00034673054	2016 Nov 28	4589	WT	25	25/50	${3.37}_{-0.14}^{+0.15}$
00034673055	2016 Nov 30	5026	WT	25	25/50	${3.35}_{-0.12}^{+0.13}$
00034673056	2016 Dec 02	4563	WT	25	25/50	${3.01}_{-0.13}^{+0.13}$
00034673057	2016 Dec 04	4915	WT	25	25/50	${2.57}_{-0.12}^{+0.12}$
00034673058	2016 Dec 06	4917	WT	25	25/50	${2.24}_{-0.11}^{+0.11}$
00034673059	2016 Dec 08	4526	WT	25	25/50	${2.68}_{-0.12}^{+0.13}$
00034673060	2016 Dec 10	4711	WT	25	25/50	${2.70}_{-0.11}^{+0.11}$
00034673061	2016 Dec 12	3073	WT	25	25/50	${2.92}_{-0.16}^{+0.16}$
00034673062	2016 Dec 14	3538	WT	25	25/50	${2.83}_{-0.13}^{+0.13}$
00034673063	2016 Dec 16	4380	WT	25	25/50	${3.08}_{-0.12}^{+0.13}$
00034673064	2016 Dec 28	4189	WT	25	25/50	${2.25}_{-0.11}^{+0.11}$
00034673065	2016 Dec 30	4288	WT	25	25/50	${2.36}_{-0.12}^{+0.12}$
00034673066	2017 Jan 01	2026	WT	25	25/50	${2.30}_{-0.18}^{+0.19}$
00034673067	2017 Jan 10	1179	WT	25	25/50	${1.37}_{-0.23}^{+0.26}$
00034673069	2017 Jan 13	1666	WT	25	25/50	${0.75}_{-0.11}^{+0.12}$
00034673071	2017 Jan 16	768	PC	15	15/30	${0.68}_{-0.08}^{+0.09}$
00034673073	2017 Jan 18	82	PC	15	15/30	⋯a
00034673074	2017 Jan 19	445	PC	15	15/30	${0.77}_{-0.11}^{+0.12}$
00034673075	2017 Jan 20	382	PC	15	15/30	${0.83}_{-0.11}^{+0.12}$
00034673076	2017 Jan 21	347	PC	15	15/30	${1.05}_{-0.14}^{+0.16}$
00034673077	2017 Jan 22	329	PC	15	15/30	${1.06}_{-0.15}^{+0.17}$
00034673078	2017 Jan 23	355	PC	15	15/30	${1.38}_{-0.15}^{+0.17}$
00034673079	2017 Jan 24	336	PC	15	15/30	${1.67}_{-0.17}^{+0.19}$
00034673080	2017 Jan 25	235	PC	15	15/30	${1.65}_{-0.19}^{+0.22}$
00034673081	2017 Jan 27	2295	WT	15	15/30	${1.98}_{-0.16}^{+0.16}$
00034673082	2017 Jan 29	237	PC	15	15/30	${1.75}_{-0.20}^{+0.23}$
00034673083	2017 Jan 31	394	PC	15	15/30	${1.86}_{-0.16}^{+0.17}$
00034673084	2017 Feb 02	552	PC	15	15/30	${1.76}_{-0.14}^{+0.15}$
00034673087	2017 Feb 20	138	PC	15	15/30	⋯a
00034673088	2017 Feb 24	385	PC	15	15/30	⋯a
00034673089	2017 Feb 26	411	PC	15	15/30	⋯a
00034673091	2017 Mar 07	198	PC	15	15/30	⋯a
00034673092	2017 Mar 08	57	PC	15	15/30	⋯a
00034673093	2017 Mar 09	345	PC	15	15/30	${0.46}_{-0.10}^{+0.13}$
00034673094	2017 Mar 10	384	PC	15	15/30	${0.46}_{-0.10}^{+0.12}$
00034673095	2017 Mar 11	350	PC	15	15/30	${0.42}_{-0.08}^{+0.09}$
00034673096	2017 Mar 23	229	PC	15	15/30	undetected
00034673097	2017 Mar 24	104	PC	15	15/30	undetected
00034673098	2017 Mar 26	192	PC	15	15/30	undetected

Notes. ${R}_{\mathrm{Source}}$: radius of source region; ${R}_{\mathrm{Background}}$: inner and outer radius of background region; ${L}_{0.6\mbox{--}10\mathrm{keV}}$: unabsorbed luminosities are calculated with the distance of $d=$ 62.1 kpc.

^aSpectra are unavailable due to short exposure times and low flux.

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The ancillary response files are created using the task xrtmkarf, and the latest response files are taken from the CALDB database for the spectral analyses. The spectral fitting is restricted to the 0.6–10 keV energy range due to calibration residuals below 0.6 keV for the WT mode data.⁷ For the PC mode data, we employ the C-statistic (Cash 1979) instead of the common ${\chi }^{2}$ for spectral fitting in 0.3–10 keV because of low count rates and short exposure times. Unfortunately, spectra are unavailable for some observations that have very limited photons (Table 1). The absorption column density along the line-of-sight to SMC X-3 is difficult to constrain and could yield an extremely small value (${nH}\lt {10}^{14}$ cm⁻²) for some of the observations; therefore, we fix it to the Galactic absorption toward the direction of the source ($6.57\,\times {10}^{20}$ cm⁻²; Dickey & Lockman 1990). At the early stage of the outburst, the phase-averaged spectra can be well fitted by an absorbed power-law (PL) model with a photon index of $\sim 0.7\mbox{--}1.2$. These results are consistent with those seen in NuSTAR data, which can be fitted by a cutoff PL model with a photon index of $\sim 0.5\mbox{--}0.7$ and a folding energy of $\sim 11\mbox{--}15$ keV (Pottschmidt et al. 2016; Tsygankov et al. 2017). During 2016 September 25 and December 12, the soft excess emerges below 1 keV, which can be described by a blackbody (BB) emission with ${kT}\sim 0.1\mbox{--}0.2\,\,\mathrm{keV}$ and ${R}_{\mathrm{BB}}\sim {10}^{2}\mbox{--}{10}^{3}$ km. The soft thermal component is variable and contributes a small fraction ($\sim 1 \% \mbox{--}3 \%$) of the total flux in 0.6–10 keV; however, because of the low S/N of band-limited data, we cannot put a tight constraint on this component.

In order to rule out the instrumental influence, we also analyze the XMM-Newton EPIC-pn spectrum observed on 2016 October 14–15 and confirm the existence of the thermal component. The EPIC-MOS data were taken in imaging mode and suffered from the pile-up effect; therefore, we only use the EPIC-pn data that was in timing mode. The data in the first 4.5 ks are excluded because of background flares, and the spectrum is extracted from the rest data with an exposure time of 28 ks. The EPIC-pn spectrum cannot be well fitted by a single PL model with a reduced ${\chi }^{2}$ larger than 2.9. When a cool BB component is added, the reduced ${\chi }^{2}$ decreases to 1.07, and it further reduces to 0.75 with the inclusion of an additional Gaussian line (phabs*(bbodyrad+power law+gauss) in XSPEC; Figure 2). The best-fit parameters are: ${nH}={1.4}_{-0.2}^{+0.2}\times {10}^{21}$ cm⁻², ${kT}={0.19}_{-0.01}^{+0.01}$ keV, ${\mathrm{Norm}}_{\mathrm{BB}}\,={1136}_{-392}^{+560}$, ${\rm{\Gamma }}={0.99}_{-0.01}^{+0.01}$, ${\mathrm{Norm}}_{\mathrm{PL}}={1.54}_{-0.03}^{+0.03}\times {10}^{-2}$, ${E}_{{\rm{l}}}\,={6.65}_{-0.10}^{+0.10}$ keV, $\sigma ={0.35}_{-0.10}^{+0.14}$ keV, ${\mathrm{Norm}}_{\mathrm{Gau}}={1.64}_{-0.45}^{+0.53}\times {10}^{-5}$, and ${\chi }^{2}/\mathrm{dof}=123.8/165$. A more detailed analysis on the XMM-Newton data will be presented elsewhere.

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In addition to the pointing data, we include the 27 survey observations (on the SMC) with an average exposure of ∼60 s for the following photometry. For each observation, we first sum the sky images with uvotimsum, and then perform aperture photometry with the summed images by using uvotsource. A source aperture of radius 5 arcsec and a larger neighboring source-free region for background are used. The AB magnitudes for all filters are shown in Figure 1.

The spin evolution during giant outbursts is crucial for investigating the accretion processes in BeXBs and for determining orbital parameters using the orbital Doppler effect (e.g., Li et al. 2011). Only WT mode data, which have a high time resolution and relatively long exposure time for individual observations (Table 1), are used for the following timing analysis. Before computing the spin frequency, we apply the barycentric correction with the ftool barycorr to the source event files (in the range of 0.5–10 keV). The source position is adopted from the 2MASS all sky Catalog of point sources (Cutri et al. 2003). The spin frequency is obtained by folding the observed counts to reach the maximum Pearson ${\chi }^{2}$. The derived values of the observed spin frequency are the same as those presented by the Fermi/GBM Pulsar Project⁸ ; however, the source becomes faint and is below the detection limit of GBM after MJD 57671 (Figure 3).

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The background-subtracted light curves with a time bin size of 0.05 s are extracted from the barycentric corrected event files and are folded over the best frequency. For the purpose of comparing the profiles between the observations, the pulse profiles are aligned by the cross correlation function (Figure 4). The energy-dependent pulse profiles clarify that the spectra harden during the peak. The pulse fraction is defined as $\mathrm{PF}=(M-N)/(M+N)$, where M and N are the maximum and minimum flux of the profile, respectively. To reduce the statistical error, we calculate the mean pulse fraction every 15 days, which are shown in the bottom panel of Figure 1.

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3.1. Orbital Elements

To study the frequency evolution, the Doppler effects of the binary should be considered here. However, the frequency evolution is very fast, so we utilize seven frequency derivatives to describe the spin evolution. The detailed processes are described as follows: (1) search the spin frequencies without considering the spin evolution; (2) fit the spin parameters and orbital elements; (3) search the spin frequencies considering the spin evolution and orbital modulation; (4) repeat steps (2) and (3) several times to get the best spin parameters and orbital elements. The fitting in step (2) is based on the Levenberg–Marquardt algorithm for the nonlinear least square method, weighted by the error of each point, which is similar to the fitting process described in Li et al. (2011, and references therein). The spin-frequency errors of GBM are multiplied by a factor of 5 to balance the weights between XRT and GBM in the fitting process. The best-fit parameters listed in Table 2 are quite consistent with those reported in Townsend et al. (2017), and the uncertainties of parameters are 1σ.

Table 2.  Timing Results of SMC X-3 from the X-Ray Observations

Parameters	Value
R.A.	${00}^{{\rm{h}}}{52}^{{\rm{m}}}05\buildrel{\rm{s}}\over{.} 64$
Decl.	−72°26'042
Epoch(MJD)	57606
ν(mHz)	128.005(2)
$\dot{\nu }$(10−5 Hz d−1)	−0.10(5)
$\ddot{\nu }$(10−6 Hz d−2)	2.08(7)
${\nu }_{3}$(10−7 Hz d−3)	−1.88(7)
${\nu }_{4}$(10−8 Hz d−4)	1.1(4)
${\nu }_{5}$(10−10 Hz d−5)	−3.9(2)
${\nu }_{6}$(10−12 Hz d−6)	8.9(5)
${\nu }_{7}$(10−13 Hz d−7)	−0.96(7)
Porb(day)	44.52(9)
asini(lt-s)	194(1)
e	0.259(3)
ω°	202(2)
${T}_{\omega }$(MJD)	57632.1(3)
${\chi }_{{re}}^{2}$(MJD)	1.1
d.o.f.	89

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It is difficult to obtain the derivative of the spin frequency with each individual observation alone. Alternatively, we calculate the spin-up rates from the fitting results and the X-ray fluxes for each two successive observations, and plot them in Figure 5.

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During the 2016–2017 giant outburst, the X-ray luminosity of SMC X-3 reached a peak value of $\sim {10}^{39}$ erg s⁻¹ (in 0.6–10 keV) around August 24, which is a few times that of NS Eddington luminosity and close to the low-luminosity tail of ULXs. Since 2017 January, the source flux dropped sharply with two humps separated by about one orbital period (∼42 days), and became lower than the detection limit of Swift/XRT (Figure 1). Its X-ray spectra became harder as flux decays. The BAT daily light curve shows an indication of flat plateau during MJD $\sim 57618\mbox{--}57640$. The magnitudes in the near-ultraviolet bands increased by $\sim {0.2}^{m}\mbox{--}{0.3}^{m}$ during outburst, and returned to a constant level after two months. Alternatively, the U-band flux changes with the XRT count rates with the Spearman's rank correlation coefficients of $\rho /P=0.86/3.6\times {10}^{-10}$.

According to the Doppler motion of the binary, we fit the orbital modulation of the observed pulse period, obtain an orbital period of $P=44.52\pm 0.09$ days that is consistent with the previous works to within 3σ uncertainty (Galache et al. 2008; Bird et al. 2012), and determine the projected semimajor axis ${asini}=194\pm 1$ lt-s and an eccentricity of $e=0.259\,\pm 0.003$. After correcting for the orbital motion, we find that the spin frequency increases steadily, indicating the formation of a transient accretion disk surrounding the central NS during the outburst. According to the accretion torque theory, we would expect the spin-up rate to increase with the accretion rate ($\dot{\nu }\propto {\dot{M}}^{6/7}$; Ghosh & Lamb 1979; Wang 1981). The power-law relationships between $\dot{\nu }$ and X-ray luminosity have been confirmed in some BeXBs during their type II giant outbursts and are consistent with the results of QPOs evolution (e.g., Finger et al. 1996; Içdem et al. 2011; Sartore et al. 2015). However, the fitted power-law indices are generally greater than the prediction of 6/7. Alternatively, fitting the spin-up rate and the 0.6–10 keV flux of SMC X-3 with a power law, we obtain an index of $0.84\pm 0.02$ that is in agreement with 6/7, but the relation deviates from the power law at peak and low luminosity (Figure 5).

Klus et al. (2014) investigated all RXTE/PCA data to determine the long-term average spin period ($P=7.7836\,\pm 0.0001$ s), spin-down rate ($\dot{P}=0.00262\pm 0.00003\,{\rm{s}}$ yr⁻¹), and the average X-ray luminosity (${L}_{{\rm{X}}}\sim 3.7\times {10}^{36}$ erg s⁻¹) for SMC X-3. These results point to a torque switch from spin-down to spin-up at the beginning of the giant outburst (${L}_{{\rm{X}}}\lt {10}^{38}$ erg s⁻¹). By assuming spin equilibrium, they derived a magnetic field of $\sim 2.9\times {10}^{12}$ G. Here, we report the instantaneous spin-period measurements during the 2016 giant outburst. The spin frequency with the orbital correction is evidently near the torque equilibrium with the spin-frequency derivative close to zero at an unabsorbed luminosity of $\sim {L}_{{\rm{X}}}\sim 2\times {10}^{37}$ erg s⁻¹, assuming the distance of 62.1 kpc (Figures 3 and 5). Thus, we can estimate a magnetic field of SMC X-3 with the assumption of ${R}_{\mathrm{co}}={R}_{{\rm{m}}}$, or $B\,=\phantom{\rule{}{3.25ex}}[4.8\times {10}^{10}{P}^{7/6}$ ${\left(\tfrac{\mathrm{flux}}{{10}^{-9}\mathrm{erg}{\mathrm{cm}}^{-2}{{\rm{s}}}^{-1}}\right)}^{1/2}$ $\times \left(\tfrac{d}{1\ \mathrm{kpc}}\right)\times {\left(\tfrac{M}{1.4{M}_{\odot }}\right)}^{1/3}$ × ${\left(\tfrac{R}{{10}^{6}\mathrm{cm}}\right)}^{-5/2}\phantom{\rule{}{3.25ex}}]$ G for the simplified model (Cui 1997), yielding $B\sim 6.8\times {10}^{12}$ G with the canonical value of NS mass and radius, i.e., 1.4 M_⊙ and 10 km. That is, the obtained value is significantly higher than that reported by Klus et al. (2014).

Intriguingly, if taking the color-correction factor into account, the emission size of the thermal component (${R}_{\mathrm{BB}}\sim 1000$ km) detected during the source returning to the quiescence state is significantly larger than that of the NS radius. Alternatively, the spin periodic modulation indicates that the BB emission is not homogeneous, but comes from a place even farther away from the central NS. The location where the thermal component is produced could be close to the corotation radius of SMC X-3 (${R}_{\mathrm{co}}\sim 6000$ km), supporting the torque equilibrium hypothesis. The emission line detected in XMM-Newton data is consistent with that from the highly ionized Fe XXV, which could originate from the illumination of cold material (i.e., the cool BB component) by central hard X-rays. The relatively large line widths ($\sigma ={0.35}_{-0.10}^{+0.14}$ keV) can be interpreted as the results of Keplerian rotation at a radius of ∼1000 km, which agrees with the size of the BB component discussed above. We, however, caution that the broad emission line with the central energy of 6.65 keV could be due to the blending of lines from different ionization states of Fe, i.e., Fe Kα and Fe Kβ, which have been detected in other accreting pulsars (e.g., Reynolds & Miller 2010).

After 2017 January, the flux drops abruptly, which hints at the "propeller" effect as the result of the further decrease in the accretion rate and that the centrifugal force prevents material from entering the magnetosphere. It is worth noting that there are two X-ray flux jumps exhibited around MJD 57780 and 57822 (corresponding to the orbital phase of ∼0.3) in Figure 1, and that they can be interpreted as the increase in the mass accretion rate when the NS travels through its periastron, i.e., Type I outbursts. Since the typical X-ray luminosity of its Type I outburst is above the Swift/XRT detection limit, we expect that the periodic outbursts can be observed by future Swift monitoring data. However, these outbursts are beyond the scope of this paper, and we suggest that the source has ended its 2016–2017 giant outburst in the end of 2017 March.

At low luminosity, the single-peak pulse profiles were observed in the RXTE (Galache et al. 2008), Chandra, and XMM-Newton archival data (Haberl et al. 2008) more than 10 years ago. As shown explicitly in Figure 4, the pulse profiles of SMC X-3 during the 2016 outburst are variable and have double peaks at the beginning of the outburst, and then converge to a single peak as the flux decays below ${L}_{{\rm{X}}}\sim 4\times {10}^{37}$ erg s⁻¹. The similar evolution pattern of pulse profiles has been reported in other outbursts but with lower transient luminosity, e.g., the 1994 giant outburst of A0535+262 (Bildsten et al. 1997). These results can be interpreted as different geometries of the accretion column beyond and below the critical luminosity ($\sim {10}^{37}$ erg s⁻¹; e.g., Basko & Sunyaev 1976; Becker et al. 2012; Mushtukov et al. 2015a; Sartore et al. 2015), respectively. At lower luminosity, the in-falling gas may be decelerated via Coulomb interactions and a pencil-beam of emission is formed. At the supercritical state, the accretion flow is decelerated via the radiative shock and the photons can only escape from accretion column walls, resulting in a double-peak structure (i.e., fan beam mode; e.g., Becker et al. 2012). As can be seen in Figure 1, the pulse fraction generally increases with the hardness during the outburst, which is consistent with that of those recently found in ULXs. Investigating a sample of ULX broadband X-ray spectra, Pintore et al. (2017) suggest that the ULX pulsars have harder spectra than that of the majority of other ULXs and the spectra becomes softer when no pulsations are detected. It is worth noting that the luminosity of all three accreting NSs can be two orders of magnitude higher than that of SMC X-3, but all have single-peak pulse profiles (Bachetti et al. 2014; Fürst et al. 2016; Israel et al. 2017a, 2017b). More extreme surface magnetic fields are required to account for the observed characteristics of ULX pulsars.

We thank the anonymous referee for their helpful comments. S.S.W. thanks Dr. Kim Page from the Swift team for discussions on data analysis. We thank Jian Li, Xiao-Chuan Jiang, and Fang-Jun Lu for many valuable suggestions. This work is supported by the National Natural Science Foundation of China under grants 11303022, 11133002, 11233001, 11573023, 11173016, 11673013, 11373024, 11233003, 11503027, 11622326, and 11433005, the National Program on Key Research and Development Project (grant No. 2016YFA0400802 and 2016YFA0400803), and by the Special Research Fund for the Doctoral Program of Higher Education (grant No. 20133207110006).