Magnetar Broadband X-Ray Spectra Correlated with Magnetic Fields: Suzaku Archive of SGRs and AXPs Combined with NuSTAR, Swift, and RXTE

2.1.1. Persistently Bright or Transient Sources

Table 1 summarizes all SGRs and AXPs that Suzaku has observed as of 2013 December. In this table, the "transient sources" exhibit prominent soft X-ray increases of two to three orders of magnitudes and subsequent decays on timescales of months to years, while the "persistently bright ones" are relatively stable with their X-ray luminosities around 10³⁵ erg s⁻¹. This is illustrated in Figure 3 by long-term X-ray flux records of representative sources. Since this "persistent" and "transient" classification is somehow phenomenological without a clear consensus (e.g., Pons & Rea 2012), we classified in this paper relatively variable sources as "transients."

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Table 1.  Log of the Suzaku SGR and AXP Observations

Namea,b	${B}_{{\rm{d}}}$ c	ObsID	Start Time	Epochd	Exp.e	Nominal	XIS Modeg	Process
	(1014 G)				(ks)	Pointingf	(XIS0, 1, 3)	Versionh
Persistently Bright Sources
SGR 1806−20	24	401092010	2006 Sep 09 22:13:43	AO1	48.9	HXD	(full, full, full)	2.0.6.13
	⋯	401021010	2007 Mar 30 15:08:00	AO1	19.6	HXD	(1/8, 1/8, 1/8)	2.0.6.13
	⋯	402094010	2007 Oct 14 05:35:49	AO2	42.7	HXD	(full, full, full)	2.1.6.15
1E 1841−045 (SNR Kes 73)	7.1	401100010	2006 Apr 19 10:51:40	PV	97.0	HXD	(1/8, 1/8, 1/8)	2.0.6.13
SGR 1900+14	7.0	401022010	2006 Apr 01 08:42:57	AO1	0.9	HXD	(1/4, 1/8, 1/4)	2.0.6.13
	⋯	404077010	2009 Apr 26 18:23:44	Key	40.6	HXD	(1/4, 1/4, 1/4)	2.3.12.25
CXOU J171405.7−381031	5.0	501007010	2006 Aug 27 01:27:07	AO1	75.6	XIS	(full, full, full )	2.0.6.13
1RXS J170849.0−400910	4.7	404080010	2009 Aug 23 16:25:08	Key	50.9	HXD	(1/4, 1/4, 1/4)	2.4.12.26
	⋯	405076010	2010 Sep 27 14:41:52	Key	47.0	HXD	(1/4, 1/4, 1/4)	2.5.16.28
1E 1048.1−5937	4.2	403005010	2008 Nov 30 23:02:01	AO3	85.0	HXD	(full, full, full)	2.2.11.22
4U 0142+61	1.3	402013010	2007 Aug 13 04:04:13	AO2	71.9	HXD	(1/4, 1/4, 1/4)	2.1.6.15
		404079010	2009 Aug 12 01:41:15	Key	82.7	HXD	(1/4, 1/4, 1/4)	2.4.12.26
		406031010	2011 Sep 07 15:43:32	ToO	36.7	XIS	(full, full, 1/4)	2.7.16.30
		408011010	2013 Jul 31 10:05:39	AO8	79.8	XIS	(1/8, 1/4, 1/4)	2.8.20.35
1E 2259+586	0.59	404076010	2009 May 25 20:00:17	Key	89.2	HXD	(1/4, 1/4, 1/4)	2.4.12.26
AX J1818.8−1559	⋯	406074010	2011 Oct 15 13:17:36	AO6	88.5	XIS	(1/8, 1/8, 1/8 )	2.7.16.30
Transient Sources
1E 1547.0−5408	3.2	903006010	2009 Jan 28 21:34:12	ToO	10.6	HXD	(psum, 1/4, 1/4)	2.3.12.25
	⋯	405024010	2010 Aug 07 03:52:07	AO5	34.1	HXD	(1/4, 1/4, psum)	2.5.16.28
SGR 0501+4516	1.9	903002010	2008 Aug 26 00:24:42	ToO	43.3	XIS	(1/4, 1/4, 1/4)	2.2.8.20
	⋯	404078010	2009 Aug 17 20:21:51	Key	29.5	HXD	(1/4, 1/4, 1/4)	2.4.12.26
	⋯	405075010	2010 Sep 20 17:27:15	Key	49.1	HXD	(1/4, 1/4, 1/4)	2.5.16.28
	⋯	408013010	2013 Aug 31 23:25:40	AO8	41.2	XIS	(1/8, 1/4, full)	2.8.20.35
SGR 1833−0832	1.8	904006010	2010 Mar 27 09:03:32	ToO	35.7	HXD	(1/8, full, full)	2.5.16.28
CXOU J164710.2−455216	1.6	901002010	2006 Sep 23 06:59:17	ToO	38.7	XIS	(1/8, 1/8, 1/8)	2.0.6.13
Swift J1834.9−0846	1.4	408015010	2013 Oct 17 07:17:57	AO8	30.4	XIS	(full, full, full)	2.8.20.35
Swift J1822.3−1606	0.14	906002010	2011 Sep 13 09:59:07	ToO	36.1	HXD	(1/8, full, full)	2.7.16.30

Notes.

^aAlthough there is accumulated evidence that SGRs and AXPs are intrinsically the same class of magnetars (Gavriil et al. 2002; Mereghetti 2008; Paper I), let us retain, in this paper, the historical terminology of "SGR" (usually discovered from burst activities) and "AXP" (identified as bright X-ray sources). Conventionally, some sources are labeled as both SGR and AXP; for example, 1E 1547.0−5408 is also called SGR 1550−5418 and PSR J1550−5418. For such an object, we employ either of the conventional names. ^bPreceding Suzaku studies: SGR 1806−20 (Esposito et al. 2007; Nakagawa et al. 2009), 1E 1841−45 (Morii et al. 2010), SGR 1900+14 (Nakagawa et al. 2009) 4U 0142+61 (Enoto et al. 2011; Makishima et al. 2014), AX J1818.8−586 (Mereghetti et al. 2012), 1E 1547.0−5408 (Enoto et al. 2010b; Iwahashi et al. 2013; Makishima et al. 2015), SGR 0501+4516 (Enoto et al. 2009, 2012, 2010c; Nakagawa et al. 2011) CXOU J164710.2−455216 (Naik et al. 2008), SGR 1833−0832 (Esposito et al. 2011), Swift J1822.3−1606 (Rea et al. 2012a), and Paper I. ^cObjects are sorted by the dipole field ${B}_{{\rm{d}}}$ calculated from the period P and its derivative $\dot{P}$, assuming magnetic dipole radiation. ^dType of observations: PV (Performance Verification phase, i.e., first nine months after the launch), AO (Announcement of Opportunity observations), ToO (Target of Opportunity observations), and Key (Key Project observations). ^eExposure time (ks) of one XIS instrument, the maximum value among the three cameras is shown. ^fNominal pointing position (XIS or HXD) at the observations; see Mitsuda et al. (2007). ^gXIS Observation modes: the full window reads out every 8 s, while the 1/4 and 1/8 window modes read out every 2 and 1 s, respectively. The timing P-sum mode (psum) provides a ∼7.8 ms time resolution together with one-dimensionally projected position information. ^hProcessing version of the data set.

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Our Suzaku sample in Table 1 includes 16 pointings of 9 persistently bright sources and 10 target of opportunity (ToO) observations to follow up 6 transient objects. This covers 15 of all the ∼29 sources or candidates known to date.¹¹ Due to operational constraints, we were unable to observe the recent transients SGR 0418+5729 and SGR 1745−29. In the following analyses, we reprocessed all the published data while adding newly observed objects, and performed comprehensive phase-averaged spectroscopic analyses.

2.1.2. Reduction of Broadband Suzaku Spectra

2.1.3. Detections of the Soft and Hard X-Rays

Figure 4 illustrates nine examples of the 1–10 keV XIS and 15–60 keV HXD-PIN spectra derived using the procedures in Section 2.1.2. The SXC below 10 keV was clearly detected with the XIS from all the observations in Table 1, except Swift J1834.9−0846, which had already faded so as to become undetectable. Hereafter, we analyze the other sources.

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After the NXB and CXB subtractions, we tabulate in Table 2 the 15–60 keV HXD-PIN source count rates ${R}_{\mathrm{pin}}$ together with the 1σ statistical and systematic uncertainties. The systematic uncertainty of the HXD-PIN background is a sum of the 1% level of the NXB (reproducibility of the LCFITDT model; Fukazawa et al. 2009) and 10% of the CXB (1σ sky-to-sky fluctuation). The associated 15–60 keV flux ${F}_{{\rm{x}}}$ was calculated via power-law fitting of the HXD data. If the HXC is undetectable, we set 3σ upper limits on the count rates, which were converted to those on ${F}_{{\rm{x}}}$ assuming spectral shapes of detected sources (caption of Table 2). We assigned 3σ upper limits on 1E 2259+586, 1E 1048.1−537, Swift J1822.3−1606, and latter observations of SGR 0501+4516. The upper limit on 1E 2259+586 is consistent with a recent detection by NuSTAR (Vogel et al. 2014).

Table 2.  HXD-PIN Source Rate and Fluxes of the HXC

Name	ObsID	${T}_{\mathrm{pin}}$ a	${R}_{\mathrm{pin}}$ b	${F}_{{\rm{h}}}$ c
		(ks)
1806−20	401092010	51.9	$6.0\pm 0.3\pm 0.3$	33.7 ± 4.4
1806−20	401021010	15.6	$5.8\pm 0.5\pm 0.5$	21.1 ± 3.3
1806−20	402094010	46.6	$4.7\pm 0.3\pm 0.3$	27.1 ± 4.3
1841−04	401100010	59.8	$6.7\pm 0.3\pm 0.3$	48.9 ± 0.3
1900+14	401022010	13.5	$3.4\pm 0.6\pm 0.6$	20.6 ± 5.5
1900+14	404077010	39.1	$3.5\pm 0.3\pm 0.3$	16.5 ± 3.5
1708−40	404080010	47.9	$4.0\pm 0.3\pm 0.3$	24.4 ± 4.4
1708−40	405076010	55.4	$4.5\pm 0.3\pm 0.3$	24.4 ± 4.0
1048−59	403005010	63.3	$\lt 2.3$	$\lt 13.2$
0142+61	402013010	94.7	$4.1\pm 0.2\pm 0.2$	35.1 ± 6.4
0142+61	404079010	92.5	$3.1\pm 0.2\pm 0.2$	26.2 ± 5.8
0142+61	406031010	39.3	$4.7\pm 0.3\pm 0.3$	24.3 ± 3.7
0142+61	408011010	96.2	$3.8\pm 0.2\pm 0.2$	19.1 ± 2.8
2259+58	404076010	96.0	$\lt 1.8$	$\lt 10.1$
1547−54	903006010	31.0	$17.4\pm 0.4\pm 0.4$	110.2 ± 5.2
1547−54	405024010	39.6	$3.2\pm 0.3\pm 0.3$	13.5 ± 3.3
0501+45	903002010	50.7	$3.0\pm 0.3\pm 0.3$	28.1 ± 6.5
0501+45	404078010	25.5	$\lt 3.7$	$\lt 20.9$
0501+45	405075010	48.9	$\lt 2.9$	$\lt 16.2$
0501+45	408013010	33.4	$\lt 2.3$	$\lt 12.7$
1833−08	904006010	10.0	$3.9\pm 0.7\pm 0.7$	17.6 ± 4.5
1822−16	906002010	33.7	$\lt 0.9$	$\lt 4.9$

Notes.

^a ${T}_{\mathrm{pin}}$: The effective HXD exposure. ^bThe 15–60 keV HXD-PIN count rates with $1\sigma$ statistical and systematic errors. If not detected, the $3\sigma$ upper limits are shown. The Suzaku-detected HXC above were also reported with other satellites: 4U 0142+61, 1RXS J170849.0−400910 (Kuiper et al. 2006; den Hartog et al. 2008a, 2008b); 1E 1841−045 (An et al. 2013); SGR 1806−20 SGR 1900+14 (Götz et al. 2006); 1E 1547.0−5408 (Kuiper et al. 2012); SGR 0501+4516 (Rea et al. 2009). ^cThe 15–60 keV absorbed flux in units of 10⁻¹² erg s⁻¹ cm⁻² with 1σ statistical and systematic errors. If not detected, the $3\sigma$ upper limits are shown converted from the count rate ${R}_{\mathrm{pin}}$ with a conversion factor of ${F}_{{\rm{x}}}/{R}_{\mathrm{pin}}=6\,\times {10}^{-12}$ erg s⁻¹ cm⁻²/(0.01 counts s ${}^{-1})$.

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As reported in Paper I and references therein, we have detected the HXC from seven sources with >3σ significance (see the caption in Table 1). Out of 10 new observations of 6 sources added to Paper I, the HXC was reconfirmed from 4U 0142+61 in 2011 and 2013 and RXS J170849.0−400910 in 2010. A signature of the HXC was suggested in SGR 1833−0832, but ${F}_{{\rm{h}}}$ is rather weak compared with other sources, and the detection is marginal. Thus, we do not use this source in the correlation fittings in Section 3. Figure 5 illustrates a comparison of ${F}_{{\rm{s}}}$ and ${F}_{{\rm{h}}}$. Suzaku has detected the HXC down to $\sim {10}^{-11}$ erg cm⁻² s⁻¹ in the 15–60 keV band.

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The Nuclear Spectroscopic Telescope Array (NuSTAR; Harrison et al. 2013) provides a high spectral sensitivity in the 3–79 keV band and has already observed bright AXPs (An et al. 2013, 2014b, 2014a; Vogel et al. 2014; Tendulkar et al. 2015; Yang et al. 2015). In order to verify our results obtained from the non-imaging instrument HXD, here we analyze initial NuSTAR data sets available in the archive and listed in Table 3.

The data were processed and filtered with the standard nupipeline and nuproducts softwares of HEASOFT version 6.16 and the NuSTAR CALDB version 20141020. The on-source spectra were extracted from a circular region of 10 radius centered on the target objects within which nearly 90% photons are collected. Since some AXPs are faint hard X-ray sources, we used the background modeling software nuskybkg (Wik et al. 2014) for accurate background subtraction. This tool generates a simulated background spectrum expected from the selected source region by fitting blank-sky spectra from the same focal plane. We selected, for the background spectral modeling, three annular regions with radii 20–50, 50–80, and 80–123 centered on the target sources for each telescope, and adjusted model parameters to explain the actual blank-sky data. The background spectrum was simulated from the best-fit modeling parameters with a 100 times longer exposure to reduce statistical uncertainties. The accuracy of the simulated background spectrum was verified using blank-sky data as described in Kitaguchi et al. (2014).

If there was simultaneous Swift coverage during the NuSTAR observations, we also utilized the archived Swift/XRT data listed in Table 3, after the data processing as described in Section 2.3. If the observations were carried out in a sequential way over a month, we merged continuous data with different observation IDs into one spectrum and response for our long-term and phase-averaged analyses. Two series of observations of 1E 2259+586 were performed in 2013 April and March, so we derived two spectra (Table 3). As an example, we show the background-subtracted X-ray spectra of 4U 0142+61 in Figure 6. The derived NuSTAR spectra are consistent with previous works by An et al. (2013), Vogel et al. (2014), and Tendulkar et al. (2015).

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Transient magnetars are characterized by sporadic X-ray outbursts, namely sudden increases of persistent emission, which have recently provided a drastic increase of the number of this class (for early studies, see, e.g., Gavriil et al. 2002; Kaspi et al. 2003; Kouveliotou et al. 2003, and for a recent review, Rea & Esposito 2011). The onset of an outburst is usually noticed through a detection of short bursts by the Swift Burst Alert Telescope (BAT; Gehrels et al. 2004; Barthelmy et al. 2005), and then monitored with the Swift X-ray Telescope (XRT; Burrows et al. 2005) or with the Rossi X-Ray Timing Explorer (RXTE; Bradt et al. 1993), as shown in Figure 7.

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We conducted Suzaku ToO observations of some transients usually within a week after burst detection. The obtained Suzaku snapshots are also presented in Figure 7. We studied the characteristics of the eight latest known outbursts of seven sources which occurred after the Suzaku launch, as listed in Table 4 (see also Figure 3). The onset of an outburst is defined as the first short bursts reported by Swift/BAT or the Fermi/Gamma-ray Burst Monitor (GBM).

Table 4.  List of X-Ray Outbursts from Transient Sources

Name	${B}_{{\rm{d}}}$	Time Origina	References	Modelb	L0c	${\tau }_{0}$ c	pc	${E}_{\mathrm{tot}}$ c
	(1014 G)	(UT)			(1034 erg s−1)	(d)		(erg)
CXOUJ164710−455216	1.6	2006 Sep 21 01:34:52	1a-d	PL	8.2 ± 0.3	⋯	0.23 ± 0.02	$3.1\times {10}^{41}$
SGR 0501+4516	1.9	2008 Aug 22 12:41:59	2a-d	PD	5.9 ± 0.2	15.9 ± 2.9	0.76 ± 0.05	$1.5\times {10}^{43}$
1E 1547.0−5408	3.2	2008 Oct 03 09:28:08	3a-d	PL	9.1 ± 0.4	⋯	0.13 ± 0.02	$4.8\times {10}^{41}$
1E 1547.0−5408	3.2	2009 Jan 22 01:32:41	4a-e	PL	25.7 ± 0.6	⋯	0.29 ± 0.01	$7.9\times {10}^{41}$
SGR 0418+5729	0.061	2009 Jun 05 20:40:48	5a-e	PD	6.1 ± 0.4	42.9 ± 11.6	1.99 ± 0.30	$2.3\times {10}^{41}$
SGR 1833−0832	1.8	2010 Mar 19 18:34:50	6a-b	PL	8.8 ± 1.2	⋯	0.07 ± 0.06	$5.8\times {10}^{41}$
Swift J1822−16069	0.14	2011 Jul 14 12:47:47	7a-c	PD	7.2 ± 0.3	11.2 ± 0.9	1.25 ± 0.02	$2.9\times {10}^{41}$
Swift J1834.9−0846	1.4	2011 Aug 07 19:57:46	8a-c	PL	1.8 ± 0.4	⋯	0.28 ± 0.09	$5.7\times {10}^{40}$

Notes. X-ray outbursts after the Suzaku and Swift launches, i.e., outbursts from 1E 2259+586 in 2002, XTE J1810-197 in 2003, and 1E 1048.1-5937 in 2007 are not included. Further details of individual outbursts are listed in the references: (1a) Krimm et al. (2006), (1b) Israel et al. (2007), (1c) Naik et al. (2008), (1d) Clark et al. (2005), (2a) Holland et al. (2008), (2b) Enoto et al. (2009), (2c) Rea et al. (2009), (2d) Aptekar et al. (2009), (3a) Krimm et al. (2008), (3b) Israel et al. (2010), (3c) Tiengo et al. (2010), (4a) Gronwall et al. (2009), (4b) Enoto et al. (2010b), (4c) Ng et al. (2011), (5a) van der Horst et al. (2009), (5b) van der Horst et al. (2010), (5c) Rea et al. (2010), (5d) Esposito et al. (2010), (5e) Güver et al. (2011), (5d) Rea et al. (2013), (6a) Gelbord et al. (2010), (6b) Göǧüş et al. (2010), (6c) Esposito et al. (2011), (7a) Cummings et al. (2011), (7b) Rea et al. (2012a), (7c) Scholz et al. (2012), (8a) D'Elia et al. (2011), (8b) Kargaltsev et al. (2012), (8c) Esposito et al. (2013),

^aTime onset of the outbursts defined at the first short burst detected by Swift/BAT or Fermi/GBM. See the above references. ^bApplied fitting model used to unabsorbed 2–10 keV X-ray light curves; a single power-law model (PL; ${L}_{{\rm{x}}}{(t)={L}_{0}(t/1{\rm{d}})}^{-p}$) and a plateau decaying function (PD; ${L}_{{\rm{x}}}{(t)={L}_{0}(1+t/{\tau }_{0})}^{-p}$). ^cBest-fit parameters of unabsorbed 2–10 keV X-ray light curves (see details in Section 3.3). ${L}_{0}$ is an initial X-ray luminosity 1 day after the onset or during the plateau phase for the PL and PD models, respectively. ${\tau }_{0}$ and p are the duration of the plateau and the slope, respectively. The total emitted energy ${E}_{\mathrm{total}}$ is evaluated as an integration up to 100 days if $p\lt 1$.

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We uniformly processed all the publicly available XRT data of the eight outbursts via the standard procedure FTOOLS xrtpipeline with default filtering criteria. We used the latest available RMF matrix in CALDB v20140610, while we generated the ARF files with the xrtmkarf tool. For the imaging Photon Counting (PC) mode, we extracted source photons from a circular region with a 48'' (20 pixels) radius centered on the target, while we collected background spectra from annular regions with inner and outer radii of 167'' (70 pixels) and 286'' (120 pixels), respectively. When the PC-mode count rates exceed ∼0.5 counts s⁻¹, we excluded a central 80 (3.4 pixels) region following a standard procedure.¹³ For the Windowed Timing (WT) mode with one-dimensional information and 1.76 ms time resolution, we extracted the source and background spectra from a strip of 94'' width around the source and surrounding regions 140'' away from the target, respectively. When the count rate exceeded 100 counts s⁻¹, we excluded the central 14'' strip to avoid pile-up. After the above standard pipelines, we discarded observations with poor photon statistics if the total source count is smaller than 100 counts per observation.

The non-imaging RXTE Proportional Counter Array (PCA; Jahoda et al. 2006) operates in the 2–60 keV energy band with a full width at half-maximum field of view of ∼1°. Due to nearby sources, we only used the PCA data for SGR 0501+4516 and SGR 0418+5729. The data were processed via the standard procedure using FTOOLS rex, pcarsp, and recofmi tasks.

We carried out unified spectral fitting of the Suzaku phase-average broadband spectra. Phase-resolved spectroscopy, available only for bright and slowly rotating objects (e.g., 1RXS J18049.0−400910), is beyond the scope of this paper. In order to avoid instrumental calibration uncertainties of the XIS, the 1.7–1.9 keV data were discarded, and a 2% systematic error was assigned to the XIS spectral bins. The XIS and HXD-PIN spectra were binned so that each bin has either >5σ significance or >30 counts. The cross-normalization factor of HXD-PIN was fixed at 1.164 and 1.181 relative to XIS-FI in the cases of the XIS- and HXD-nominal pointing positions (Maeda et al. 2008), respectively. The normalization of XIS-BI to XIS-FI was allowed to differ by up to 5%. For the simultaneous Swift and NuSTAR spectra (Section 2.2), the normalization was fixed to that of the Swift/XRT and the cross-normalization between the telescopes (FPMA and FPMB) was allowed by up to 5%.

A photo-absorption factor was multiplied to an intrinsic continuum spectral model of the SXC. We employed the phabs model (Balucinska-Church & McCammon 1992) in XSPEC with the solar metallicity abundance angr (Anders & Grevesse 1989) and cross-section bcmc; this combination has been widely used in the literature and allows us to compare our results with previous ones. When the HXC is detected in the 15–60 keV band, it was expressed by an additional single power law. We further added a plasma emission model when analyzing the 1E 1841−045 and CXU J171405.74−381031 data to describe the surrounding SNRs Kes 73 (Kumar et al. 2014) and CTB 37B (Sato et al. 2010), respectively, as described in Appendix A.

The intrinsic SXC spectrum is conventionally fitted by a model comprising two blackbody components (hereafter the 2BB model) or a combination of a blackbody plus an additional soft power law (BB+PL model). The second high-energy component of both models is considered to represent either a temperature anisotropy over the stellar surface, an upscattering of soft photons in the magnetosphere, or the effects of a magnetized neutron star atmosphere (e.g., Lyutikov & Gavriil 2006; Rea et al. 2008; Güver et al. 2011). These empirical 2BB and BB+PL models roughly explain the data, though only approximately sometimes. The low and high blackbody temperatures in the 2BB model, ${{kT}}_{{\rm{L}}}$ and ${{kT}}_{{\rm{H}}}$, are known to follow a relation of ${{kT}}_{{\rm{L}}}/{{kT}}_{{\rm{H}}}\sim 0.4$ (Nakagawa et al. 2009), and thus the number of free spectral parameters is expected to be three rather than four for the 2BB or BB+PL models.

Since the SXC spectral modeling has not reached a consensus, we utilize an empirical blackbody shape with a Comptonization-like power-law tail (hereafter CBB model, Tiengo et al. 2005; Halpern et al. 2008; Enoto et al. 2010b, and Paper I). This CBB model is mathematically described by three parameters: the temperature kT, the soft-tail power-law photon index ${{\rm{\Gamma }}}_{{\rm{s}}}$, and the normalization corresponding to the emission radius R. The model reproduces the soft tail at ≳5 keV of the BB+PL model without the large ${N}_{{\rm{H}}}$ that is needed by the 2BB.

We present the Suzaku CBB best-fit parameters in Table 5 and the corresponding $\nu {F}_{\nu }$ spectra in Figure 8. The NuSTAR fit results are given in Table 5 together with some $\nu {F}_{\nu }$ examples in Figure 9. The NuSTAR $\nu {F}_{\nu }$ shapes of the bright AXPs, 4U 0142+61 and 1E 1841−045, are consistent with those of Suzaku. The HXC of 1E 2259+586 was detected with NuSTAR (Vogel et al. 2014), but not with Suzaku.

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Table 5.  Best-fit Parameters of SGR/AXP Observations Using the CBB+PL Model with Suzaku and NuSTAR

Name	ObsID	${F}_{1-10}$	${F}_{15-60}$	Unabsorbed ${F}_{{\rm{s}}}$, ${F}_{{\rm{h}}}$		${N}_{{\rm{H}}}$	kT	R	${{\rm{\Gamma }}}_{{\rm{s}}}$	${{\rm{\Gamma }}}_{{\rm{h}}}$	${\chi }_{\nu }^{2}$ (dof)
	Month			SXC	HXC	(1022 cm−2)	(keV)	(km)
Suzaku Observations
1806−20	401092010	${12.6}_{-0.1}^{+0.1}$	33.7(4.4)	${6.3}_{-0.8}^{+0.9}$	${58.4}_{-1.4}^{+1.4}$	6.7(3)	0.61(4)	1.8	⋯	1.62(5)	0.97 (377)
1806−20	401021010	${10.5}_{-0.2}^{+0.1}$	21.1(3.3)	${3.4}_{-0.8}^{+0.9}$	${50.1}_{-2.6}^{+2.9}$	$5.5(5)$	0.68(9)	1.1	⋯	1.51(9)	1.18 (171)
1806−20	402094010	${8.9}_{-0.2}^{+0.1}$	27.1(4.3)	${5.9}_{-0.8}^{+1.0}$	${42.2}_{-1.7}^{+1.8}$	$6.5(5)$	$0.65(6)$	1.5	⋯	1.50(7)	1.22 (193)
1841−04	401100010	${19.4}_{-0.1}^{+0.1}$	48.9(0.3)	${35.1}_{-1.4}^{+1.4}$	${50.9}_{-2.1}^{+2.2}$	$2.5(1)$	$0.27(1)$	21.1	3.41(10)	0.87(8)	1.22 (2087)
1900+14	401022010	${5.3}_{-0.5}^{+0.5}$	20.6(5.5)	${4.6}_{-0.6}^{+0.1}$	${25.0}_{-3.4}^{+3.2}$	$1.8(3)$	$0.57(2)$	2.5	⋯	0.96(14)	1.13 (44)
1900+14	404077010	${4.3}_{-0.1}^{+0.1}$	16.5(3.5)	${4.5}_{-0.3}^{+0.3}$	${26.3}_{-2.5}^{+2.9}$	$1.9(1)$	$0.52(2)$	3.0	⋯	0.78(9)	1.34 (57)
1714−38	501007010	${1.7}_{-0.1}^{+0.1}$	⋯	${5.0}_{-0.3}^{+0.5}$	⋯	$3.5(1)$	$0.24(4)$	15.7	3.25(8)	⋯	1.28 (179)
1708−40	404080010	${38.1}_{-0.2}^{+0.5}$	24.4(4.4)	${69.1}_{-3.3}^{+3.5}$	${28.4}_{-2.7}^{+2.8}$	$1.3(1)$	$0.26(2)$	14.2	3.48(9)	0.67(24)	0.98 (394)
1708−40	405076010	${35.8}_{-0.1}^{+0.4}$	24.4(4.0)	${55.0}_{-2.7}^{+4.0}$	${33.4}_{-2.2}^{+2.1}$	$1.2(1)$	$0.30(1)$	9.5	3.80(21)	1.14(16)	1.05 (540)
1048−59	403005010	${9.9}_{-0.1}^{+0.1}$	$\lt 13.2$	${12.3}_{-0.2}^{+0.2}$	$\lt 19.5$	$0.47(3)$	$0.45(1)$	4.7	4.88(20)	⋯	1.32 (78)
0142+61	402013010	${122.0}_{-0.1}^{+0.1}$	35.1(6.4)	${185.1}_{-0.7}^{+0.7}$	${38.3}_{-2.3}^{+2.3}$	$0.61(1)$	$0.28(1)$	19.0	4.68(2)	0.24(7)	1.43 (1725)
0142+61	404079010	${115.6}_{-0.2}^{+0.2}$	26.2(5.8)	${176.8}_{-1.5}^{+1.5}$	${27.1}_{-2.2}^{+2.3}$	$0.62(1)$	$0.28(1)$	18.6	4.71(4)	0.39(14)	1.15 (1401)
0142+61	406031010	${108.3}_{-0.9}^{+1.1}$	24.3(3.7)	${164.8}_{-4.4}^{+4.6}$	${41.3}_{-4.0}^{+4.1}$	$0.63(3)$	$0.28(1)$	18.0	4.65(10)	0.32(21)	1.08 (1062)
0142+61	408011010	${107.8}_{-0.4}^{+0.4}$	19.1(2.8)	${162.7}_{-1.4}^{+1.4}$	${32.9}_{-2.2}^{+2.3}$	$0.60(1)$	$0.28(1)$	17.9	4.81(4)	0.26(11)	1.37 (838)
2259+58	404076010	${30.4}_{-0.1}^{+0.1}$	$\lt 10.1$	${44.8}_{-0.5}^{+0.5}$	$\lt 14.9$	$0.55(1)$	$0.29(1)$	7.8	4.85(4)	⋯	1.20 (510)
1818−15	406074010	${1.0}_{-0.1}^{+0.1}$	⋯	${1.4}_{-0.1}^{+0.1}$	⋯	$2.1(21)$	$1.61(6)$	0.1	⋯	⋯	1.31 (83)
1547−54	903006010	${59.7}_{-0.8}^{+0.9}$	110.2(5.2)	${50.6}_{-1.7}^{+1.7}$	${158.7}_{-3.3}^{+3.3}$	$2.8(1)$	$0.67(2)$	1.9	⋯	1.53(4)	1.29 (140)
1547−54	405024010	${11.1}_{-0.2}^{+0.1}$	13.5(3.3)	${17.5}_{-0.6}^{+0.6}$	${23.4}_{-2.4}^{+2.5}$	$2.8(1)$	$0.62(1)$	1.3	⋯	1.15(12)	1.23 (69)
0501+45	903002010	${36.2}_{-0.2}^{+0.3}$	28.1(6.5)	${42.2}_{-0.8}^{+0.6}$	${30.4}_{-3.7}^{+4.0}$	$0.40(2)$	$0.49(1)$	2.7	4.35(17)	0.10(35)	1.23 (206)
0501+45	404078010	${2.9}_{-0.1}^{+0.1}$	$\lt 20.9$	${3.8}_{-0.2}^{+0.3}$	$\lt 30.8$	$0.44(10)$	$0.30(3)$	2.2	4.16(27)	⋯	0.92 (77)
0501+45	405075010	${1.7}_{-0.1}^{+0.1}$	$\lt 16.2$	${2.0}_{-0.2}^{+0.2}$	$\lt 23.9$	$0.24(13)$	$0.30(4)$	1.6	4.20(45)	⋯	0.80 (34)
0501+45	408013010	${1.7}_{-0.1}^{+0.1}$	$\lt 12.7$	${2.3}_{-0.1}^{+0.1}$	$\lt 18.8$	$0.41(7)$	$0.26(2)$	2.3	3.79(15)	⋯	0.96 (367)
1833−08	904006010	${3.8}_{-0.1}^{+0.1}$	17.6(4.5)	${7.9}_{-0.2}^{+0.3}$	${35.6}_{-8.8}^{+9.3}$	$9.6(5)$	$1.08(4)$	0.7	⋯	−0.38(40)	1.37 (167)
1647−45	901002010	${27.2}_{-0.1}^{+0.1}$	⋯	${45.9}_{-0.5}^{+0.5}$	⋯	$1.7(1)$	$0.49(1)$	3.4	4.39(6)	⋯	1.26 (248)
1822−16	906002010	${18.2}_{-0.3}^{+0.4}$	$\lt 4.9$	${18.5}_{-0.4}^{+0.5}$	$\lt 7.2$	$0.02(4)$	$0.54(2)$	0.7	5.86(71)	⋯	1.41 (59)
NuSTAR Observations
1841−04	30001025004	${23.6}_{-0.1}^{+0.1}$	35.2(0.6)	${29.9}_{-0.1}^{+0.1}$	${48.6}_{-0.5}^{+0.5}$	$2.3(1)$	$0.28(3)$	18.1	3.45(6)	1.13(2)	1.15 (1493)
0142+61	30001023002	${114.6}_{-0.1}^{+0.3}$	21.7(0.3)	${167.1}_{-0.1}^{+0.1}$	${27.4}_{-0.4}^{+0.4}$	$0.58(2)$	$0.30(1)$	15.8	4.84(2)	0.61(2)	1.11 (765)
2259+58	30001026002	${28.9}_{-0.9}^{+0.1}$	3.1(0.3)	${51.1}_{-2.3}^{+2.8}$	${3.7}_{-0.3}^{+0.3}$	$0.84(4)$	$0.28(1)$	8.9	4.95(6)	0.58(13)	1.07 (404)
2259+58	30001026007	${24.9}_{-2.3}^{+0.1}$	3.0(0.3)	${42.0}_{-3.7}^{+4.9}$	${3.4}_{-0.2}^{+0.2}$	$0.78(7)$	$0.30(1)$	7.0	5.06(8)	0.66(14)	0.94 (222)
1048−59	2013 Jul	∼8.9	$\lesssim 6.7$	∼18.3	N/A	from Yang et al. (2015)

Notes.

^a ${F}_{1-10}$ and ${F}_{15\mbox{--}60}$: absorbed 1–10 keV and 15–60 keV fluxes (10⁻¹² erg s⁻¹ cm⁻²). ^bUnabsorbed flux ${F}_{{\rm{s}}}$ and ${F}_{{\rm{h}}}$ of the SXC and HXC in the 1–60 keV band (10⁻¹² erg s⁻¹ cm⁻²). ^cRadius is evaluated from $R=0.09643\times {(d/\mathrm{kpc})(T/\mathrm{keV})}^{-2}$ ${({F}_{{\rm{s}}}/{10}^{-11}\mathrm{erg}{{\rm{s}}}^{-1}{\mathrm{cm}}^{-1})}^{0.5}$.

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3.2.1. Ratio of HXC to SXC versus Magnetic Field

In Paper I, we proposed a broadband spectral evolution of this class, i.e., (i) the hardness ratio of the HXC to SXC is positively (or negatively) correlated to their ${B}_{{\rm{d}}}$ (or ${\tau }_{{\rm{c}}}$), and (ii) the HXC photon index ${{\rm{\Gamma }}}_{{\rm{h}}}$ becomes harder toward the weaker ${B}_{{\rm{d}}}$ sources (see also Kaspi & Boydstun 2010).

Based on our updated sample shown in Figure 8, we revised these correlations. Figure 10, top-left panel, shows the ratio of absorbed fluxes, $\eta ={F}_{15-60}/{F}_{1-10}$, as a function of $\dot{P}$, which is derived from the pulsar timing information independently from the spectroscopy. The Spearman's rank-order test of this correlation gives a significantly high value, r_s = 0.94. The correlation is fitted as

$$\begin{eqnarray}\eta & = & {F}_{15-60}/{F}_{1-10}\\ & = & (0.59\pm 0.07)\times {(\dot{P}/{10}^{-11}{\rm{s}}{{\rm{s}}}^{-1})}^{0.51\pm 0.05},\end{eqnarray} \tag{ 1 }$$

using the Bayesian method in Kelly (2007; linmix package) to account for measurement errors and intrinsic scatter of the data. While the correlation is derived from spectral analyses below 70 keV, the potential HXC cutoff does not strongly affect the correlation (Paper I). Due to the limited photon statistics of several Suzaku sources, we can only study the total emission. Deconvolution into pulsed and unpulsed components (Kuiper et al. 2012) will be reported in future publications.

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As shown in the $\nu {F}_{\nu }$ plots (Figure 8), the HXC largely contributed to the SXC band below ≲10 keV in stronger ${B}_{{\rm{d}}}$ objects (e.g., SGR 1806−20, SGR 1900+14, and 1E 1547.0−5408). To remove this mixing and to eliminate the effects of photoabsorption, we also define the absorption-corrected luminosity ratio between the two components, $\xi ={L}_{{\rm{h}}}/{L}_{{\rm{s}}}$ (listed in Table 6), in the same way as in Paper I. The Spearman's rank-order significance becomes r_s = 0.97. The correlation slope of ξ becomes steeper than those of η as

$$\begin{eqnarray}\xi & = & {L}_{{\rm{h}}}/{L}_{{\rm{s}}}\\ & = & \,(0.62\pm 0.07)\times {(\dot{P}/{10}^{-11}{\rm{s}}{{\rm{s}}}^{-1})}^{0.72\pm 0.05}.\end{eqnarray} \tag{ 2 }$$

This is shown in the top-right panel of Figure 10. For these correlations, we revised the timing information (P and $\dot{P}$) from Paper I, referring to the McGill catalog (Olausen & Kaspi 2014) and further added NuSTAR observations. Data points of the canonical AXPs 4U 0142+61 and 1E 1841−045 are consistent with those from Suzaku, and the new HXC detection from 1E 2259+586 (Vogel et al. 2014) falls on the correlation. The Galactic center source SGR J1745−29, although not shown in Figure 10, is also expected to follow the relation since its wide-band spectrum resembles that of 1E 1547.0−5408 (Mori et al. 2013).

Table 6.  X-Ray Luminosities of the Soft and Hard Components from Magnetars Measured with Suzaku

Name	ObsID	Month	P	${\dot{P}}_{11}$	B14	τ	${L}_{\mathrm{sd}}$	${L}_{{\rm{s}}}$	${L}_{{\rm{h}}}$	${L}_{{\rm{x}}}$	Abs. HR	HR
		yyyy mm	(s)			(kyr)				(total)	$\left(\tfrac{{F}_{15-60}}{{F}_{1-10}}\right)$	(ξ = ${L}_{{\rm{h}}}/{L}_{{\rm{s}}}$)
1806−20	401092010	2006 Sep	7.548	49.5	19.6	0.2	4.6	${5.7}_{-0.1}^{+0.8}$	${53.0}_{-0.1}^{+1.2}$	${58.8}_{1.5}^{+1.5}$	${2.7}_{-0.4}^{+0.4}$	${9.3}_{-1.2}^{+1.4}$
1806−20	401021010	2007 Mar	⋯	⋯	⋯	⋯	⋯	${3.1}_{-0.1}^{+0.9}$	${45.5}_{-0.1}^{+2.6}$	${48.6}_{2.5}^{+2.8}$	${2.0}_{-0.3}^{+0.3}$	${14.7}_{-3.7}^{+4.2}$
1806−20	402094010	2007 Oct	⋯	⋯	⋯	⋯	⋯	${5.3}_{-0.1}^{+0.9}$	${38.4}_{-0.1}^{+1.7}$	${43.7}_{1.7}^{+1.9}$	${3.0}_{-0.5}^{+0.5}$	${7.2}_{-1.0}^{+1.3}$
1841−04	401100010	2006 Apr	11.789	4.1	7.0	4.6	0.100	${30.5}_{-0.1}^{+1.2}$	${44.1}_{-0.1}^{+1.9}$	${74.6}_{2.2}^{+2.2}$	${2.5}_{-0.1}^{+0.1}$	${1.4}_{-0.1}^{+0.1}$
1900+14	401022010	2006 Apr	5.200	9.2	7.0	0.9	2.6	${8.7}_{-0.1}^{+0.2}$	${46.9}_{-0.1}^{+6.0}$	${55.5}_{6.5}^{+6.0}$	${3.9}_{-1.1}^{+1.1}$	${5.4}_{-1.0}^{+0.7}$
1900+14	404077010	2009 Apr	⋯	⋯	⋯	⋯	⋯	${8.4}_{-0.1}^{+0.5}$	${49.3}_{-0.1}^{+5.5}$	${57.7}_{4.8}^{+5.5}$	${3.8}_{-0.8}^{+0.8}$	${5.9}_{-0.7}^{+0.7}$
1714−38	501007010	2006 Aug	3.825	6.4	5.0	0.9	4.6	${10.5}_{-0.1}^{+0.9}$	⋯	(=${L}_{{\rm{s}}}$)	⋯	⋯
1708−40	404080010	2009 Aug	11.005	1.9	4.7	9.0	0.058	${12.0}_{-0.1}^{+0.6}$	${4.9}_{-0.1}^{+0.5}$	${16.9}_{0.7}^{+0.8}$	${0.64}_{-0.12}^{+0.12}$	${0.41}_{-0.04}^{+0.05}$
1708−40	405076010	2010 Sep	⋯	⋯	⋯	⋯	⋯	${9.5}_{-0.1}^{+0.7}$	${5.8}_{-0.1}^{+0.4}$	${15.3}_{0.6}^{+0.8}$	${0.68}_{-0.11}^{+0.11}$	${0.61}_{-0.05}^{+0.06}$
1048−59	403005010	2008 Nov	6.458	2.3	3.9	4.6	0.33	${11.9}_{-0.1}^{+0.2}$	<19.0	(=${L}_{{\rm{s}}}$)	<1.3	<2.0
0142+61	402013010	2007 Aug	8.689	0.20	1.3	68.1	0.012	${28.8}_{-0.1}^{+0.1}$	${6.0}_{-0.1}^{+0.4}$	${34.7}_{0.4}^{+0.4}$	${0.29}_{-0.05}^{+0.05}$	${0.21}_{-0.01}^{+0.01}$
0142+61	404079010	2009 Aug	⋯	⋯	⋯	⋯	⋯	${27.5}_{-0.1}^{+0.2}$	${4.2}_{-0.1}^{+0.3}$	${31.7}_{0.4}^{+0.4}$	${0.23}_{-0.05}^{+0.05}$	${0.15}_{-0.01}^{+0.01}$
0142+61	406031010	2011 Sep	⋯	⋯	⋯	⋯	⋯	${25.6}_{-0.1}^{+0.7}$	${6.4}_{-0.1}^{+0.6}$	${32.1}_{0.9}^{+1.0}$	${0.22}_{-0.03}^{+0.03}$	${0.25}_{-0.03}^{+0.03}$
0142+61	408011010	2013 Jul	⋯	⋯	⋯	⋯	⋯	${25.3}_{-0.1}^{+0.2}$	${5.1}_{-0.1}^{+0.4}$	${30.4}_{0.4}^{+0.4}$	${0.18}_{-0.03}^{+0.03}$	${0.20}_{-0.01}^{+0.01}$
2259+58	404076010	2009 May	6.979	0.048	0.59	229.0	0.0057	${5.5}_{-0.1}^{+0.1}$	<1.8	(=${L}_{{\rm{s}}}$)	<0.3	<0.5
1818−15	406074010	2011 Oct	2.482	0.80	1.4	4.9	2.1	${0.30}_{-0.01}^{+0.02}$	⋯	(=${L}_{{\rm{s}}}$)	⋯	⋯
1547−54	903006010	2009 Jan	2.072	4.8	3.2	0.7	21.4	${9.3}_{-0.1}^{+0.3}$	${29.1}_{-0.1}^{+0.6}$	${38.4}_{0.7}^{+0.7}$	${1.8}_{-0.1}^{+0.1}$	${3.1}_{-0.1}^{+0.1}$
1547−54	405024010	2010 Aug	⋯	⋯	⋯	⋯	⋯	${3.2}_{-0.1}^{+0.1}$	${4.3}_{-0.1}^{+0.5}$	${7.5}_{0.4}^{+0.5}$	${1.2}_{-0.3}^{+0.3}$	${1.3}_{-0.1}^{+0.2}$
0501+45	903002010	2008 Aug	5.762	0.59	1.9	15.4	0.12	${5.5}_{-0.1}^{+0.1}$	${4.0}_{-0.1}^{+0.5}$	${9.5}_{0.5}^{+0.5}$	${0.78}_{-0.18}^{+0.18}$	${0.72}_{-0.09}^{+0.10}$
0501+45	404078010	2009 Aug	⋯	⋯	⋯	⋯	⋯	${0.50}_{-0.01}^{+0.04}$	<4.0	(=${L}_{{\rm{s}}}$)	<7.3	<10.7
0501+45	405075010	2010 Sep	⋯	⋯	⋯	⋯	⋯	${0.26}_{-0.01}^{+0.03}$	<3.1	(=${L}_{{\rm{s}}}$)	<9.6	<14.2
0501+45	408013010	2013 Aug	⋯	⋯	⋯	⋯	⋯	${0.30}_{-0.01}^{+0.02}$	<2.5	(=${L}_{{\rm{s}}}$)	<7.4	<10.9
1833−08	904006010	2010 Mar	7.565	0.35	1.6	34.3	0.032	${7.6}_{-0.1}^{+0.3}$	${34.6}_{-0.1}^{+9.0}$	${42.3}_{8.6}^{+9.0}$	${4.7}_{-1.2}^{+1.2}$	${4.5}_{-1.1}^{+1.2}$
1647−45	901002010	2006 Sep	10.61	⋯	⋯	⋯	⋯	${8.4}_{-0.1}^{+0.1}$	⋯	(=${L}_{{\rm{s}}}$)	⋯	⋯
1822−16	906002010	2011 Sep	8.438	0.0021	0.14	6250.0	0.00014	${0.57}_{-0.01}^{+0.01}$	<0.2	(=${L}_{{\rm{s}}}$)	<0.3	<0.4
NuSTAR Observations
1841−04	30001025004	2013 Sep	11.789	4.1	7.0	4.6	0.100	${25.9}_{-0.1}^{+0.1}$	${42.1}_{-0.1}^{+0.4}$	${68.1}_{0.4}^{+0.4}$	${1.5}_{-0.1}^{+0.1}$	${1.6}_{-0.1}^{+0.1}$
0142+61	30001023002	2014 Jan	8.689	0.20	1.3	68.1	0.012	${26.0}_{-0.1}^{+0.1}$	${4.3}_{-0.1}^{+0.1}$	${30.2}_{0.1}^{+0.1}$	${0.19}_{-0.01}^{+0.91}$	${0.16}_{-0.01}^{+0.01}$
2259+58	30001026002	2013 Apr	6.979	0.048	0.59	229.0	0.0057	${6.3}_{-0.1}^{+0.3}$	${0.5}_{-0.1}^{+0.1}$	${6.7}_{0.3}^{+0.3}$	${0.11}_{-0.01}^{+0.01}$	${0.07}_{-0.01}^{+0.01}$
2259+58	30001026007	2013-05	⋯	⋯	⋯	⋯	⋯	${5.2}_{-0.1}^{+0.6}$	${0.4}_{-0.1}^{+0.1}$	${5.6}_{0.5}^{+0.6}$	${0.12}_{-0.02}^{+0.01}$	${0.08}_{-0.01}^{+0.01}$
1048−59	30001024002	2013 Jul	6.458	2.3	3.9	4.6	0.33	${17.8}_{-0.1}^{+0.1}$	$\lt {6.1}_{-0.1}^{+0.1}$	${23.9}_{0.0}^{+0.0}$	$\lt 0.74$	$\lt 0.34$

Note. Pulsar timing information: spin period P (s), period derivative ${\dot{P}}_{11}=\dot{P}/({10}^{-11}$ s s⁻¹), surface magnetic field ${B}_{14}=B/({10}^{14}$ G), and spin-down luminosity ${L}_{\mathrm{sd}}=4\pi {I}^{2}\dot{P}/{P}^{3}=3.2\times {10}^{33}{\dot{P}}_{11}{(P/5{\rm{s}})}^{-3}$ erg s⁻¹, where $I={10}^{45}$ g cm² is the neutron star momentum of inertia.

Spectral information: The 1–60 keV luminosity of the SXC and HXC, ${L}_{{\rm{s}}}$, ${L}_{{\rm{h}}}$, and their total ${L}_{{\rm{x}}}={L}_{{\rm{s}}}+{L}_{{\rm{h}}}$. All the luminosities, ${L}_{\mathrm{sd}}$, ${L}_{{\rm{s}}}$, ${L}_{{\rm{h}}}$, and ${L}_{{\rm{x}}}$, are shown in units of 10³⁴ erg s⁻¹. Hardness ratio (HR): Evaluated from absorbed fluxes of 15–60 keV and 1–10 keV ($\eta ={F}_{15-60}/{F}_{1-10}$) or luminosities ($\xi ={L}_{{\rm{h}}}/{L}_{{\rm{s}}}$) after correcting for absorption.

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The above correlation to the directly measured quantity $\dot{P}$ can be converted to correlations to ${B}_{{\rm{d}}}$ and ${\tau }_{{\rm{d}}}$, although the two additional relations are not independent of Equation (2), since ${B}_{{\rm{d}}}$ and ${\tau }_{{\rm{c}}}$ are estimated using combinations of the same P and $\dot{P}$, i.e., ${B}_{{\rm{d}}}\propto {P}^{1/2}{\dot{P}}^{1/2}$ and ${\tau }_{{\rm{c}}}\propto P{\dot{P}}^{-1}$. Furthermore, considering the clustering of rotational periods of magnetars in a narrow range (P = 2–11 s), correlating with Equation (2), $\xi \propto {\dot{P}}^{-k}$ (k ∼ 0.72), gives $\xi \propto {B}_{{\rm{d}}}^{2k}\sim {B}_{{\rm{d}}}^{1.4}$ and $\xi \propto {\tau }_{{\rm{c}}}^{-k}\sim {\tau }_{{\rm{c}}}^{-0.72}$. These relations are shown in the middle and bottom panels in Figure 10, and the same fitting procedures gives

$$\begin{eqnarray}\eta & = & {F}_{15-60}/{F}_{1-10}\\ & = & (0.097\pm 0.036)\times {({B}_{{\rm{d}}}/{B}_{\mathrm{QED}})}^{1.00\pm 0.14}\end{eqnarray} \tag{ 3 }$$

$$\begin{eqnarray}&&=\,(1.91\pm 0.33)\times {({\tau }_{{\rm{c}}}/1\mathrm{kyr})}^{-0.48\pm 0.05}\end{eqnarray} \tag{ 4 }$$

and

$$\begin{eqnarray}\xi & = & {L}_{{\rm{h}}}/{L}_{{\rm{s}}}\\ & = & \,(0.050\pm 0.022)\times {({B}_{{\rm{d}}}/{B}_{\mathrm{QED}})}^{1.41\pm 0.16}\end{eqnarray} \tag{ 5 }$$

$$\begin{eqnarray}&&=\,(3.25\pm 0.50)\times {({\tau }_{{\rm{c}}}/1\mathrm{kyr})}^{-0.68\pm 0.05}.\end{eqnarray} \tag{ 6 }$$

The slopes of Equations (5) and (6) are consistent with those from Paper I within error bars. The Suzaku upper limit for the second lowest ${B}_{{\rm{d}}}$-field source Swift J1822.3−1606 (${B}_{{\rm{d}}}\,=1.4\times {10}^{13}$ G, i.e., ${B}_{{\rm{d}}}/{B}_{\mathrm{QED}}\sim 0.32$) is also consistent with this picture. Thus, we reconfirm, and reinforce, the evolution in η and ξ as reported in Paper I.

3.2.2. Photon Index Γ_h of HXC versus Magnetic Field

The second prediction of Paper I is the HXC spectral hardening toward weaker ${B}_{{\rm{d}}}$ objects, as seen in the representative $\nu {F}_{\nu }$ spectra in Figure 11 (top). Figure 11 (bottom) also shows the HXC photon index ${{\rm{\Gamma }}}_{{\rm{h}}}$ as a function of ${B}_{{\rm{d}}}$, which is fitted as

$$\begin{eqnarray}&&{{\rm{\Gamma }}}_{{\rm{h}}}=(0.40\pm 0.11)\times {({B}_{{\rm{d}}}/{B}_{\mathrm{QED}})}^{0.35\pm 0.09}.\end{eqnarray} \tag{ 7 }$$

The ${{\rm{\Gamma }}}_{{\rm{h}}}$ values are stable on a long timescale for persistently bright sources such as 4U 0142+61, SGR 1806−20, and 1E 1841−045, while some transients show slope changes during the outbursts. This was also reported during the ∼400 days of INTEGRAL monitoring of 1E 1547.0−5408, changing from ${\rm{\Gamma }}\sim 1.4$ to ∼0.9 (Kuiper et al. 2012). This figure clearly indicates a peculiar trend that relatively weaker HXC intensity sources show a harder HXC spectral slope.

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3.2.3. Surface Temperature kT of SXC versus ${B}_{d}$

The surface temperature ${T}_{{\rm{s}}}$ of the SXC is plotted as a function of ${B}_{{\rm{d}}}$ in Figure 12 where we also added the ${T}_{{\rm{s}}}$ of other isolated neutron stars from previous studies. This plot indicates (1) higher ${T}_{{\rm{s}}}$ of magnetars than that of other isolated neutron stars, (2) a tendency toward a positive correlation between ${T}_{{\rm{s}}}$ and ${B}_{{\rm{d}}}$ in the quiescent neutron star sample, and (3) increase of ${T}_{{\rm{s}}}$ during transient magnetar outbursts. All these properties suggest that the values of ${T}_{{\rm{s}}}$ reflect the effects of magnetic energy dissipation, which is an implicit but direct consequence of the magnetar hypothesis.

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The magnetic field decay in the high field regime ${B}_{{\rm{d}}}\gtrsim {10}^{13}$ G (Goldreich & Reisenegger 1992), postulated in the magnetar hypothesis, may be formulated as (Colpi et al. 2000; Dall'Osso et al. 2012)

$$\begin{eqnarray}&&\displaystyle \frac{{dB}}{{dt}}=-{{aB}}^{1+\alpha },\end{eqnarray} \tag{ 8 }$$

where a and α are the normalization and decay index, respectively. The solution is obtained as

$$\begin{eqnarray}&&B(t)=\displaystyle \frac{{B}_{0}}{{(1+\alpha t/{\tau }_{B})}^{1/\alpha }},\end{eqnarray} \tag{ 9 }$$

with ${\tau }_{B}={({{aB}}_{0}^{\alpha })}^{-1}$. This formula successfully explains the clustering of the rotational periods of SGRs and AXPs around 2–12 s (Colpi et al. 2000), and resolves the overestimation of ${\tau }_{\mathrm{ch}}$ in 1E 2259+586 when compared with the age derived from the plasma diagnostics of the surrounding SNR CTB 109 (Nakano et al. 2015; but see also Suwa & Enoto 2014). Then, let us here assume that the surface temperature ${T}_{{\rm{s}}}$ is determined by a balance between the radiative cooling and heating by the magnetic energy dissipation of $d({B}^{2}/8\pi )/{dt}$ in the crust. Following Pons et al. (2007), this is described as

$$\begin{eqnarray}&&S\sigma {T}^{4}=-S\bigtriangleup R\displaystyle \frac{d}{{dt}}\left(\displaystyle \frac{{B}^{2}}{8\pi }\right),\end{eqnarray} \tag{ 10 }$$

where S, $\bigtriangleup R$, and σ are the surface area, crust thickness, and the Stefan–Boltzmann constant, respectively. Combining this with Equation (8), we derive

$$\begin{eqnarray}&&4\pi \sigma {T}^{4}=a\bigtriangleup {{RB}}^{2+\alpha },\end{eqnarray} \tag{ 11 }$$

or the relation $T\propto {B}^{1/2+\alpha /4}$. The slope of Figure 12 is close to a range, α ∼ 1–2, estimated from comparisons between pulsar and SNR ages (Nakano et al. 2015). This is another support for the magnetar hypothesis.

In our sample, transients were observed basically during X-ray outbursts. The decay of the SXC is often composed of two distinct components (e.g., Woods et al. 2004): an initial quickly fading emission within ∼1 day, and a longer timescale one decaying in about a month. The former component, possibly related to burst activities, is beyond our present analyses. Below, we focus on the longer-decay component.

The absorbed SXC fluxes were already shown in Figure 7 and are in agreement with the previous studies listed in Table 4. The light curves were derived in the following way over individual outbursts from the data described in Section 2.3. Individual Swift and RXTE spectra were at first fitted by a single blackbody model (Section 2.3), while the CBB model (Section 2.1) was further employed to explain residuals in the higher energy band if the resultant reduced chi-square is not accepted (${\chi }_{\nu }^{2}\gt 1.4$). All the spectral parameters were allowed to vary in each observation, except the absorption column density ${N}_{{\rm{H}}}$ which is fixed at the value determined by high statistics observations from early phase observations with Suzaku/XIS or Swift/XRT. Below, we use the flux measurements of the acceptable fits, and detailed spectral characterization will be discussed elsewhere.

Figure 13 shows the 0.1–20 keV luminosities of these outbursts after correcting for the photoabsorption (Section 2.3) and distances d (Appendix B). Such fading light curves in Figure 7 from different objects have so far been fitted by some empirical formulae: exponential (Rea et al. 2009), double-exponential (Scholz et al. 2012), power-law (Lyubarsky et al. 2002; Kouveliotou et al. 2003), or broken power-law functions. Since there is no commonly established function applicable to all the outbursts nor consensus on the physical understanding of the decay, it is still meaningful to search for an empirical formula to uniformly investigate the outbursts.

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The light curves imply that some transients exhibit a plateau-like constant period at the earliest phase, followed by a power-law decay, such as in three transients: SGR 0501+4516, SGR 0418+5729, and Swift J1822.3-1606. In order to represent this behavior, we employed the mathematical form

$$\begin{eqnarray}&&L(t)=\displaystyle \frac{{L}_{0}}{{(1+t/{\tau }_{0})}^{p}}.\end{eqnarray} \tag{ 12 }$$

Hereafter, we call Equation (12) the plateau-decaying (PD) function since it becomes constant, L₀, at $t\ll {\tau }_{0}$, while it becomes a power law with an index of p at $t\gg {\tau }_{0}$. For sources in which the plateau is not clear, we just used the power-law (PL) model, ${L}_{{\rm{x}}}={L}_{0}{(t/{\tau }_{0})}^{-p}$, which is normalized at ${\tau }_{0}=1$ day. The resultant parameters are summarized in Table 4, and further comparison among different fittings are described in Appendix C. As described there, the plateau is not an artifact of the definition of the time origin. The plateau periods are ${\tau }_{0}$ ∼ 11–43 days for the three sources above, and subsequent decay indices are p ∼ 0.7–2. Other sources (e.g., 1E 1547.0−5408 and Swift J1834.9−0846) faded with a flatter index p ∼ 0.1–0.3 without such a clear plateau duration.

The total energy ${E}_{\mathrm{total}}$ released during a single outburst, an integration of Equation (12), becomes ${E}_{\mathrm{total}}={\tau }_{0}{L}_{0}/(p-1)$ in a steep decay case ($p\gt 1$), while that in the flatter decay is calculated by assuming an integration upper bound at 100 days as a typical decaying duration. Even under present uncertainties of distance measurements, as listed in Table 7, a typical ${E}_{\mathrm{tot}}$ is a few 10⁴¹ erg.

Table 7.  Distances d to Known SGRs and AXPs Used in This Paper

Source Name	d	Assumption or Method (Reference)
	(kpc)
SGR 1806−20	${8.7}_{-1.5}^{+1.8}$	Host cluster 1806−20 (G10.0−0.3) (Bibby et al. 2008)
1E 1841−045	${8.5}_{-1.0}^{+1.3}$	Association with SNR Kes 73 (Tian & Leahy 2008)
SGR 1900+14	${12.5}_{-1.7}^{+1.7}$	Host cluster C1 1900+14 (Davies et al. 2009)
1RXS J170849.0−400910	${3.8}_{-0.5}^{+0.5}$	Red clump star method (Durant & van Kerkwijk 2006)
1E 1048.1−5937	${9.0}_{-1.7}^{+1.7}$	Red clump star method (Durant & van Kerkwijk 2006)
4U 0142+61	${3.6}_{-0.4}^{+0.4}$	Red clump star method (Durant & van Kerkwijk 2006)
1E 2259+586	${3.2}_{-0.2}^{+0.2}$	Radio observation to SNR CTB 109 (Kothes et al. 2002)
1E 1547.0−5408	${3.91}_{-0.07}^{+0.07}$	Dust-scattering halo (Tiengo et al. 2010)
SGR 0501+4516	3.3	Assumed to be on the Perseus arm
SGR 1833−0832	9.0	Assumed to be on the Scutum–Crux arm
CXOU J164710.2−455216	${3.9}_{-0.7}^{+0.7}$	Assocation with Westerlund I (Kothes & Dougherty 2007)
Swift J1834.9−0846	${4.2}_{-0.3}^{+0.3}$	Association with SNR W41 (Leahy & Tian 2008)
Swift J1822.3−1606	${1.6}_{-0.3}^{+0.3}$	Association with ${{\rm{H}}}_{\mathrm{II}}$ region M17 (Nielbock et al. 2001)

Note. Referring to up-to-date data from Olausen & Kaspi (2014) with previous distance studies: 1E 1841−045, >5 kpc (red clump star method; Durant & van Kerkwijk 2006); 1RXS J170849.0−400910, 3.2–4.0 kpc (dust-scattering halo; Rivera-Ingraham & van Kerkwijk 2010); 1E 1048.1−5937, 5.7–6.2 kpc (dust-scattering halo; Rivera-Ingraham & van Kerkwijk 2010); 4U 0142+61, 3.5–6.8 kpc (dust-scattering halo; Rivera-Ingraham & van Kerkwijk 2010); 1E 2259+586, 3.0 ± 0.5 kpc (radio observation to SNR CTB 109; Kothes et al. 2002), 7.5 ± 1.0 (red clump star method; Durant & van Kerkwijk 2006), 4.0 ± 0.8 (radio observation to SNR CTB 109; Tian et al. 2010); 1E 1547.0−5408, ∼9 kpc (dipersion measure; Camilo et al. 2007), ∼4 kpc (possible association with SNR G327.24−0.13; Gelfand & Gaensler 2007); SGR 0501+4516, 0.8 ± 0.4 kpc (possible association with SNR HB9,; Leahy & Tian 2007), ∼1.5 kpc (possible association with SNR HB9; Gaensler & Chatterjee 2008), ∼2 kpc (Lin et al. 2011); Swift J1834.9−0846, ∼5.4 kpc (dust-scattering halo; Esposito et al. 2013).

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The characteristic spectral shapes, composed of the SXC ($\lesssim 10$ keV) and HXC ($\gtrsim 10$ keV), have been revealed to be a common feature of this class. Figure 14, an expanded P–$\dot{P}$ plot, summarizes the currently available HXC information of all AXPs and SGRs. Combining INTEGRAL, RXTE, Suzaku (with seven detections), and NuSTAR, which provided the HXC detections for 1E 2259+586 (Vogel et al. 2014) and SGR J1745−29 near Sgr A* (Mori et al. 2013; Kaspi et al. 2014), the HXC has been confirmed from nine objects among ∼23 confirmed sources, and the detected 15–60 keV HXC flux level is in the range of ∼0.3–$11\times {10}^{-11}$ erg s⁻¹ cm⁻².

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One important property of the HXC is that it exhibits clear scalings in terms of its luminosity ratio ξ (Section 3.1; Figure 10; Equations (2), (5), and (6)) and its spectral slope ${{\rm{\Gamma }}}_{{\rm{h}}}$ (Section 3.2.2; Figure 11(c); Equation (7)). Since the scalings apply both to AXPs and SGRs, they can be collectively called magnetars at least from the view point of the wide-band spectrum. These relations also allow us to predict the HXC intensity of a source when its P, $\dot{P}$, and the SXC intensity are given. In Section 3.2.1, we pointed out the narrow P clustering of magnetars, and showed the η and ξ correlations to $\dot{P}$. In the future, if young and fast rotating magnetars (e.g., $P\sim 0.1\,{\rm{s}}$ at ∼10–100 years, as the origin of fast radio bursts; e.g., see Beloborodov 2017), or hypothetical slowly rotating magnetars (e.g., $P\sim 100\,{\rm{s}}$, related to gamma-ray bursts; Rea et al. 2015), would be discovered, a wider range of P will provide another hint to judge a main control parameter of the spectral shape.

Do we observe such two-component spectra from other classes of magnetized neutron stars? Some rotation-powered pulsars, such as the Vela pulsar or Geminga, exhibit a two-component spectrum, consisting of thermal and non-thermal components. However, they have lower ${{kT}}_{{\rm{s}}}$ and steeper ${{\rm{\Gamma }}}_{{\rm{h}}}\sim 1.5\mbox{--}2.5$ than those of the SGRs and AXPs, and show no clear scaling of ξ–${B}_{{\rm{d}}}$. Accretion-powered X-ray pulsars emit predominantly in a single hard component (at least in >2 keV), regardless of their luminosity (e.g., Terada et al. 2006). Thus, the two-component spectral composition is considered to be specific to magnetars.

Different from the magnetar model, an alternative scenario to explain SGRs and AXPs assumes that ${B}_{{\rm{d}}}$ calculated from P and $\dot{P}$ neither represents the true dipole field strength near a stellar surface nor exceeds the critical field ${B}_{\mathrm{QED}}$. In this case, X-ray radiation is not powered by magnetic energy but by the accretion of a fallback disk that is left over from a supernova explosion of the progenitor (Chatterjee et al. 2000; Alpar 2001; Benli et al. 2013). The bulk-motion Comptonization of the accretion column is attributed to the HXC radiation (Trümper et al. 2010, 2013; Kylafis et al. 2014).

The accretion scenario still has difficulties explaining the broadband X-ray observations. (1) First, as already stated above, the accretion-powered pulsars do not show two distinct spectral components even in the low X-ray luminosity. (2) The power-law HXC extend up to ∼100 keV with the hard ${{\rm{\Gamma }}}_{{\rm{h}}}$ without any absorption nor cutoff features which are usually expected from the electron cyclotron resonance of the ordinary accretion-powered neutron stars. (3) Finally, the spectral scalings implies that the ${B}_{{\rm{d}}}$ values play an important role in the emission mechanism, especially in the HXC. The nominal value ${B}_{{\rm{d}}}$ derived from P and $\dot{P}$ is, therefore, considered to be a true poloidal component near the stellar surface.

The present work suggested that ${B}_{{\rm{d}}}$ is one main control parameter of the HXC radiation. Some theoretically motivated emission models have already been developed so far in the magnetar scheme; these include thermal bremsstrahlung (Thompson & Beloborodov 2005), synchrotron radiation (Heyl & Hernquist 2005; Thompson & Beloborodov 2005), resonant scattering (Baring & Harding 2007; Nobili et al. 2008; Viganò et al. 2012; Beloborodov 2013), and down-cascade due to photon splitting (Paper I). Although the mechanism is still observationally poorly understood, the appropriate radiation scenario would include the physics in the strong magnetic field to explain the scaling. For example, the hard photon index ${{\rm{\Gamma }}}_{{\rm{h}}}$ in our sample is correlated with ${B}_{{\rm{d}}}$. This cannot be explained only by the difference of viewing angles, and should be further compared with models (e.g., Beloborodov 2013).

One potential clue is how ξ or ${{\rm{\Gamma }}}_{{\rm{h}}}$ behave on the scaling plots during the outburst. For example, there is a signature of a faster decay of the HXC than the SXC in some sources, e.g., SGR 0501+4516, (Rea et al. 2009; Enoto et al. 2010c), and 1E 1547.0−5408 (T. Enoto et al. 2017, in preparation). This question has not yet been clearly answered in the present Suzaku data. In addition, our current observations of the HXC are limited up to ∼60 keV. The HXC cutoff is expected at ∼400 keV from the fossil disk model (Kylafis et al. 2014), while the annihilation line is suggested in some magnetar models (Beloborodov 2013). Thus, the soft gamma-ray observation is expected to provide a smoking gun.

In Figure 15 and Table 6, we compare the total X-ray luminosity ${L}_{{\rm{x}}}={L}_{{\rm{h}}}+{L}_{{\rm{s}}}$ (Section 3) with the spin-down powers ${L}_{\mathrm{sd}}$. The 2–10 keV luminosities of 41 ordinary rotation-powered pulsars follow a linear empirical relation, $\mathrm{log}{L}_{{\rm{x}}}=1.34\mathrm{log}{L}_{\mathrm{sd}}-15.34$ (Possenti et al. 2002). They are well below the critical luminosity $\mathrm{log}{L}_{\mathrm{crit}}\,=1.48\mathrm{log}{L}_{\mathrm{sd}}-18.5$, which is thought to be the maximum efficiency of the conversion of the spin-down luminosity to X-ray emission. Our inclusion of ${L}_{{\rm{h}}}$ improved ${L}_{{\rm{x}}}$ measurements, and more clearly show the luminosity excess (${L}_{{\rm{x}}}\gg {L}_{\mathrm{sd}}$). Furthermore, the ${L}_{{\rm{x}}}$ values exhibit little dependence on ${L}_{\mathrm{sd}}$. These properties have indeed been providing strong support to the magnetar hypothesis.

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Highly variable transients mostly reach $\sim {10}^{35}$ erg s⁻¹ at an early stage of their outbursts and gradually decay back to quiescence below $\lesssim {10}^{33}$ erg s⁻¹. Some of them (e.g., 1E 1547.0−5408 and Swift J1834.9−0846) decay back to ${L}_{{\rm{x}}}\lt {L}_{\mathrm{sd}}$, to a region on Figure 14 that is adjacent to radio-loud normal rotation-powered pulsars (Rea et al. 2012b). However, it is still unclear whether AXPs and SGRs in their deep quiescence can become dimmer than ${L}_{\mathrm{crit}}$.

On the ${L}_{{\rm{x}}}$–${L}_{\mathrm{sd}}$ diagram in Figure 15, the persistent X-ray emission does not exceed ∼${10}^{36}\,$erg s⁻¹ either in persistent sources or in transients during an early phase of outbursts. In the light curves (Section 3.3), transients sometimes show an initial plateau phase for ∼10–40 days, keeping ${L}_{{\rm{s}}}\sim {10}^{35}$–10³⁶ erg s⁻¹, in the same ${L}_{{\rm{s}}}$ range as the persistent sources. If the emission is a pure blackbody radiation, this typically corresponds to ${L}_{\max }=4\pi {R}_{\mathrm{spot}}^{2}$σT⁴ = 1.3 × 10³⁵(R_spot/1 km)²(kT/1 keV)⁴ erg s⁻¹. Such a ceiling X-ray luminosity could be interpreted as regulation via temperature-sensitive neutrino cooling, if the magnetic energy is converted to thermal energy in the stellar crust (Potekhin et al. 2007; Pons & Rea 2012; Esposito et al. 2013; Rea et al. 2013).

As shown in Figure 16, outbursts of higher-${B}_{{\rm{d}}}$ magnetars show more prolonged decay than those of lower-${B}_{{\rm{d}}}$ sources. This supports the basic concept that outbursts from magnetars are powered by a sudden dissipation of magnetic energy, although we do not yet know whether the release occurs in the stellar interior (Pons & Rea 2012; Li & Beloborodov 2015) or in the magnetosphere (Beloborodov & Thompson 2007; Kojima & Kato 2014; Benli et al. 2015), or both.

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The total emitted energy during a single X-ray outburst is typically $\sim {10}^{41}$–10⁴² erg. If this outburst can be attributed to the magnetic energy stored in the magnetar crust with a thickness of $\bigtriangleup R$ and a hot spot radius of ${R}_{\mathrm{spot}}\sim 1$ km, i.e., ${E}_{\mathrm{mag}}=\pi {R}_{\mathrm{spot}}^{2}\bigtriangleup R\,\cdot \,$B²/8π = 2.4 × 10⁴¹ (B/B_QED)² (R_spot/1 km)² (△R/1 km) erg, the required $\bigtriangleup R\sim 1$ km is comparable to the crust thickness. Since low-${B}_{{\rm{d}}}$ sources below ${B}_{\mathrm{QED}}$ (SGR 0418+5729 and Swift J1822.3−1606) also radiate an amount of energy similar to other high-${B}_{{\rm{d}}}$ sources, the actual surface field of low-${B}_{{\rm{d}}}$ transients would be much stronger than estimated from P–$\dot{P}$ method.

Incidentally, the fall-back disk model tries to explain the outbursts in terms of sudden mass accretion (Alpar 2001; Benli et al. 2013, 2015). However, in such cases, the luminosity would hit a ceiling at the Eddington limit ($\sim {10}^{38}$ erg s⁻¹) rather than at $\sim {10}^{36}$ erg s⁻¹; the outburst would decay exponentially (Tanaka & Shibazaki 1996) or sometimes fall abruptly via the propeller effect (Asai et al. 2013) rather than in power-law form; and the spectrum around the outburst peak would exhibit a clear fluorescence Fe-K line (e.g., EW > 100 eV), if the fast initial rise of the X-ray outbursts indicates a mass reservoir close to the neutron star. Therefore, the scenario fails to account for several important properties of the actual observed outburst of magnetars.

In the magnetar scenario, the fading X-ray radiation of outbursts is usually theoretically modeled as a transient thermal response to a sudden energy release inside the stars. Originally proposed for post-glitch afterglows of ordinary neutron stars (Eichler & Cheng 1989; Hirano et al. 1997), this idea has recently been investigated in the magnetar context (Lyubarsky et al. 2002) and applied to X-ray observations of transient sources (Rea et al. 2012a; Scholz et al. 2012).

Instead of a large delta-function energy release assumed in previous models, we may alternatively assume that many small subsequent energy deposits take place during the X-ray afterglow as small starquakes or reconnections. Such a scheme remind us of seismology, where the occurrence frequency of the aftershock sequence n(t) after a large earthquake is known to follow an empirical formula, called the modified Omori's law, as $n{(t)\propto (t+c)}^{-p}$ (Utsu 1961). In seismology, t is the elapsed time after the mainshock, p ∼ 0.7–1.5 is the slope index of the decay, and $c\lesssim 1$ day is the empirical plateau phase duration (Utsu et al. 1995). This formula, thought to represent an underlying relaxation process after a mainshock, has the same mathematical form as the plateau-decaying function we used for X-ray outbursts (Equation (9) in Section 2.3).

We already know another statistical similarity of seismology to magnetars, i.e., the Gutenberg–Richter law (Ishimoto & Iida 1938; Gutenberg & Richter 1941) seen in SGR short bursts: the number–intensity relations of their short burst fluence follows a power-law distribution (Götz et al. 2006; Nakagawa et al. 2007) as commonly seen in earthquakes and solar flares. This statistical relation is believed to represent the self-organized criticality of the system (Aschwanden 2011). If, furthermore, the magnetar persistent emission (or a part of it) is composed of small energy releases, the plateau-decaying light curve is understood as changes in the normalization of the Gutenberg–Richter law during the outbursts (Nakagawa et al. 2009), providing another similarity of the underlying physical process to seismology, where subsequent energy dissipation is triggered by a previous event.

On 2010 March 19, SGR 1833−0832 was discovered from short bursts. A Suzaku ToO was performed on March 27 for an effective 40 ks exposure, ∼8.4 days after the discovery. The 0.5–10 keV XIS light curve is stable without any clear short burst detections. The pulsation was detected with XIS0 (1/8 window mode) at $P=7.565\pm 0.001\,{\rm{s}}$ at the MJD epoch of 55272. The pulsed fraction is $\mathrm{PF}=({F}_{\max }-{F}_{\min })/({F}_{\max }\,+{F}_{\min })\,\sim$ 60%–80% (2–10 keV), where ${F}_{\max }$ and ${F}_{\min }$ are the background-subtracted maximum and minimum count rates of the pulse profile. When fitting the 1–10 keV XIS data alone, the SXC spectrum is fitted by a single blackbody of kT = 1.22 keV with ${N}_{{\rm{H}}}=9.0\times {10}^{22}$ cm⁻² or by a single power law of Γ = 3.1 with ${N}_{{\rm{H}}}=1.6\times {10}^{23}$ cm⁻². The HXC detection is marginal (Table 2). Comparing with Earth occultation data, the simulated HXD-PIN NXB was found to be slightly underestimated by ∼2% than other observations, and if we assigned this additional 2% to the NXB uncertainty, the HXC detection significance reduced to 2.4σ level. Thus, we just regard the HXC of SGR 1833−0832 as a marginal signal.

A.1.1. 4U 0142+61

A.1.2. AX J1818.8−1559

A.1.3. 1E 1841−045 (Kes 73)

A.1.4. Swift J1822.3−1606

A.1.5. CXOU J171405.7−381031 (CTB37B)

A.1.6. 1E 1048.1–5937