Long-term Spectral Variability of the Ultraluminous X-Ray Source Holmberg IX X–1

Ultraluminous X-ray sources (ULXs) are extragalactic, point-like, non-nuclear X-ray sources with observed X-ray luminosity, ${L}_{{\rm{X}}}\gt {10}^{39}\,\mathrm{erg}\ {{\rm{s}}}^{-1}$, exceeding the Eddington limit for a typical stellar-mass ($\sim 10\,{M}_{\odot }$) black hole (see Feng & Soria 2011, for a review on ULXs). Studies with Advanced Satellite for Cosmology and Astrophysics (ASCA) observations (Makishima et al. 2000) revealed the X-ray flux variability on different timescales, which suggests a binary accreting nature for ULXs. Early spectra of ULXs were described by a multi-color disk blackbody (MCD; Mitsuda et al. 1984)-plus-power law (PL) model, mimicking the popular spectral model for Galactic black hole X-ray binaries (BHXBs; see Remillard & McClintock 2006). Such modeling has provided evidence of cool thermal components (${{kT}}_{\mathrm{in}}\sim 0.20\,\mathrm{keV}$; Miller et al. 2003), which has been interpreted as the existence of intermediate-mass black holes (IMBHs) of mass $\sim {10}^{2}\mbox{--}{10}^{4}\,{M}_{\odot }$ in ULXs (Colbert & Mushotzky 1999; Coleman Miller & Colbert 2004). However, this interpretation has been challenged by the highest-quality XMM-Newton observations, where the observed spectra show a broad curvature (Stobbart et al. 2006) at high energies ($\gtrsim 3$ keV), which does not correspond to any of the known sub-Eddington accretion states in BHXBs (Remillard & McClintock 2006) and is hardly reconciled with the IMBH interpretation (Roberts 2007). The peculiar features (a soft component and the curvature at the high energies) in the highest-quality data suggest a new observational state for ULXs, which is referred as the ultraluminous state (Gladstone et al. 2009). Moreover, the observed properties of the ULXs are better explained on the basis of disk-plus-Comptonized-corona models in this state and such modeling suggests that the majority of ULXs are black holes of stellar origins accreting at near-Eddington and/or super-Eddington rates (Stobbart et al. 2006; Pintore & Zampieri 2012; Sutton et al. 2013). In addition, the recent discovery of pulsating neutron stars (NSs) in the three ULXs, M82 X–2 (Bachetti et al. 2014), NGC 5907 X–1, and NGC 7793 P13 (Israel et al. 2017a, 2017b), has proven the existence of NSs as the compact primaries in the ULX population. It is also noted that typical ULX spectra can be described equally well with phenomenological models adopted for Galactic X-ray pulsars (Pintore et al. 2017). Thus, a few non-pulsating ULXs may host NSs.

Spectral variability has been studied for individual as well as a sample of ULXs (Feng & Kaaret 2006b, 2009; Kajava & Poutanen 2009; Pintore & Zampieri 2012). In individual ULXs, the monitoring observations have been used to trace the long-term variability, which sometimes was interpreted as state transition (Kubota et al. 2001; Dewangan et al. 2004, 2010; Godet et al. 2009). The lower quality ASCA observations of two ULXs IC 342 X–1 and X–2 exhibited a canonical transition between low/hard and high/soft states (Kubota et al. 2001), where the two states were modeled with a PL and MCD component, respectively. However, there were flux variations not associated with any obvious state changes in some ULXs. For example, Swift monitoring observations of Holmberg II X–1, Holmberg IX X–1, NGC 5408 X–1, and NGC 4395 X–2 suggested that these ULXs remained in the same spectral states as their flux varied by an order of magnitude (Kaaret & Feng 2009; Grisé et al. 2010). Variability studies have also revealed several correlations between the spectral parameters when fitting relatively low counting statistics spectra with a MCD plus PL model, especially the disk-luminosity-versus-temperature and photon-index-versus-X-ray-luminosity (${\rm{\Gamma }}\mbox{--}{L}_{{\rm{X}}}$) correlations. Such correlations help compare the observed variations with the expectations from theoretical models. The disk-luminosity-versus-temperature correlation, ${L}_{\mathrm{disk}}\propto {T}^{4}$, has been reported for several ULXs (Feng & Kaaret 2006b, 2009; Kajava & Poutanen 2009), when modeled with absorbed MCD, which show consistency with the prediction from the standard accretion disk model. However, other ULXs appear to follow an anti-correlation between the parameters, ${L}_{\mathrm{disk}}\propto {T}_{\mathrm{in}}^{-3.5}$, when fitted with a cool-MCD-plus-PL model (Feng & Kaaret 2007; Soria 2007; Kajava & Poutanen 2009). In addition, ULXs exhibited a ${\rm{\Gamma }}-{L}_{{\rm{X}}}$ correlation when modeled with PL or MCD-plus-PL (Feng & Kaaret 2006a, 2006b, 2009; Kajava & Poutanen 2008, 2009, and references therein). Such a correlation has already been observed in BHXBs and active galactic nuclei (Magdziarz et al. 1998; Zdziarski et al. 2002, 2003), and suggests that the variability mechanism is related to the geometry of the disk corona in these sources (Haardt et al. 1997; Merloni & Fabian 2001).

Spectral variability studies of ULXs will thus help us understand the geometry and physical processes in these systems, which can provide constraints on the nature of ULXs. A large number of available observations provide a unique opportunity for such studies on the nearby ULXs. In this paper, we present the long-term spectral variability study of the ULX Holmberg IX X–1 (hereafter Ho IX X–1) using the archival Suzaku and XMM-Newton observations. The data from the observations cover nearly the same energy range, 0.3–10 keV, and are of good quality, allowing us to test different spectral models thoroughly and identify the model for describing the source emission. In our study, we used a two-component thermal model (disk-blackbody-plus-thermal-Comptonization), which has been found to be able to describe the spectral variability of ULXs quite well (Vierdayanti et al. 2010; Pintore & Zampieri 2012; Pintore et al. 2014; Luangtip et al. 2016). In Section 1.1, we briefly explain Ho IX X–1 and its previous variability studies. The observations and data reduction method are described in Section 2. The analysis and results are presented in Section 3. A discussion is presented in Section 4.

1.1. Holmberg IX X–1 and the Variability Studies

We used the observations of Ho IX X–1 obtained with Suzaku and XMM-Newton, which are publicly available, and analyzed the individual data in the 0.3–10 keV energy band. The list of the observations is given in Table 1.

2.1. Suzaku

2.2. XMM-Newton

3.1. Spectral Analysis

We performed detailed spectral modeling with simple models and complex physical models to understand the variability. The spectral modeling was performed with xspec version 12.8.1g (Arnaud 1996). We fitted the Suzaku and XMM-Newton (PN and MOS simultaneously) spectra in the 0.6–10 keV and 0.3–10 keV, respectively. In order to perform the spectral analysis in the same energy band, i.e., 0.3–10 keV, we extended the low-energy range of Suzaku data to 0.3 keV (using the energies command in xspec) and derived the spectral parameters based on the best-fit models. Due to the calibration uncertainties, the 1.7–2 keV energy range was excluded from the Suzaku XIS spectra. The uncertainties on the best-fit parameters were quoted at a 90% confidence level. Two multiplicative absorption components tbabs in xspec (Wilms et al. 2000) were used to incorporate the intervening absorption. The first component was fixed at the Galactic column ${N}_{{\rm{H}},\mathrm{Gal}}=5.54\times {10}^{20}\,{\mathrm{cm}}^{-2}$ (Kalberla et al. 2005) toward the direction of the source, while the second component, which was free to vary, represents the absorption local to the ULX.

Initially, we fitted the spectra with an absorbed PL model. The model provides a statistically acceptable fit for only six spectra, where the null hypothesis probability $\gt 0.05$, and fails to explain the spectra from XMM1 and XMM5, where the reduced ${\chi }^{2}\gt 2$ (${\chi }^{2}$/degrees of freedom (dof) = 651.6/323 and 1648.5/463, respectively; see Table 2). Thus, we added the MCD model (diskbb in xspec) to the PL. This combined model improved the fit significantly in the majority of the spectra compared to the absorbed PL, and the differences in the ${\chi }^{2}$ values of the two models ranges from 1106.9 (obs. XMM5) to 1.2 (obs. Suzaku 5) for the loss of two additional dof (see Table 2). In addition, improved fits were obtained for XMM1 and XMM5 spectra (${\chi }^{2}$/dof = 582.4/321 and 541.6/461, respectively). We then attempted with an exponentially cutoff PL (cutoffpl) by replacing the PL component, which further improved the fit, especially for the XMM-Newton observations, though the cutoff energies (${E}_{\mathrm{cutoff}}\gt 3$ keV) were not well constrained in some of the cases. As pointed out in Section 1, the high energy cutoff or curvature commonly found in the bright ULXs can be described by thermal Comptonization in a cool and optically thick corona. Using the thermal Comptonization models, comptt or nthcomp (in xspec), is appropriate in such circumstances. Thus, we tested these models by replacing the cutoffpl and both the models provide statistically acceptable fit to the spectra (see Table 2). While the former model is widely used to characterize the thermal Comptonization, here we proceeded with the nthcomp model to explain the observed spectral features.

Table 2.  The Obtained ${\chi }^{2}/\mathrm{dof}$ for Different Models

Data	Model 1	Model 2	Model 3	Model 4	Model 5
XMM1	651.6/323	582.4/321	399.5/320	388.6/320	393.2/320
XMM3	347.4/288	277.7/286	267.2/285	269.1/285	269.6/285
XMM4	373.4/323	318.4/321	316.4/320	315.5/320	316.2/320
XMM5	1648.5/463	541.6/461	467.2/460	463.4/460	463.4/460
XMM6	279.0/230	224.9/228	224.3/227	224.3/227	224.4/227
XMM8	384.5/321	377.9/319	376.8/318	373.6/318	373.8/318
XMM9	438.7/392	433.7/390	397.7/389	398.8/389	399.8/389
Suzaku1	417.4/346	403.2/344	359.2/343	356.7/343	357.1/343
Suzaku2	346.5/289	331.6/287	313.3/286	311.0/286	312.8/286
XMM10	388.1/303	317.3/301	316.5/300	315.7/300	316.0/300
Suzaku3	315.0/300	303.0/298	300.1/297	300.1/297	301.0/297
XMM11	388.6/331	319.0/329	314.5/328	313.1/328	313.8/328
Suzaku4	345.8/359	332.3/357	326.7/356	326.9/356	327.1/356
XMM12	354.1/301	334.3/299	320.5/298	319.3/298	318.8/298
XMM13	498.6/366	440.5/364	363.8/363	377.6/363	378.4/363
XMM14	572.9/381	511.2/379	380.2/378	379.7/378	380.1/378
XMM15	454.9/333	421.2/331	370.2/330	374.7/330	374.5/330
Suzaku5	294.4/271	293.2/269	288.3/268	288.2/268	287.2/268
Suzaku6	261.6/253	252.9/251	233.7/250	224.3/250	224.4/250
Suzaku7	295.4/248	279.9/246	275.0/245	276.0/245	275.9/245
Suzaku8	267.6/246	266.3/244	266.3/243	265.6/243	266.0/243

Note. Model 1: ${\mathtt{tbabs}}\times {\mathtt{tbabs}}\times {\mathtt{powerlaw}};$ Model 2: ${\mathtt{tbabs}}\times {\mathtt{tbabs}}\times ({\mathtt{diskbb}}+{\mathtt{powerlaw}});$ Model 3: ${\mathtt{tbabs}}\times {\mathtt{tbabs}}\times ({\mathtt{diskbb}}\,+{\mathtt{cutoffpl}});$ Model 4: ${\mathtt{tbabs}}\times {\mathtt{tbabs}}\times ({\mathtt{diskbb}}+{\mathtt{compTT}});$ Model 5: ${\mathtt{tbabs}}\times {\mathtt{tbabs}}\times ({\mathtt{diskbb}}+{\mathtt{nthcomp}})$.

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The nthcomp model (Zdziarski et al. 1996; Życki et al. 1999) describes the thermal Comptonization in a relatively cool and optically thick plasma. The electron plasma temperature (${{kT}}_{{\rm{e}}}$), photon index (Γ) and seed photon temperature (${{kT}}_{{\rm{S}}}$) are the free parameters of the model. The soft X-ray emission described by MCD is considered the principal component, providing seed photons for the Comptonization. Thus, the temperatures of the MCD component (${{kT}}_{\mathrm{in}}$) and the seed photons were kept the same and varied together. In the disk corona models, if the optically thick corona masks the underlying disk, the seed photon temperature is not always equal to the disk temperature. We also repeated the analysis by disconnecting the two temperatures. However, this assumption led to difficulties in finding a unique global minimum in the spectral fit (see also Feng & Kaaret 2009; Pintore & Zampieri 2012). Thus, we decided to keep the two temperatures the same. Because the ${\mathtt{tbabs}}\times {\mathtt{tbabs}}\times ({\mathtt{diskbb}}+{\mathtt{nthcomp}})$ model provides acceptable best-fits and reasonable physical explanations, we considered it as the baseline model. All the spectra were fitted with this baseline model and the best-fit parameters are listed in Table 3. The total unabsorbed luminosity (${L}_{{\rm{X}}}$) and the luminosity of the disk component (${L}_{\mathrm{disk}}$) were calculated using the convolution model cflux available in xspec.

Table 3.  Best-fit Parameters for the Baseline Model and the Derived Parameters from the nthcomp Model Function

Data	${N}_{{\rm{H}}}$	${{kT}}_{\mathrm{in}}$	Γ	${{kT}}_{{\rm{e}}}$	log ${L}_{{\rm{X}}}$	log ${L}_{\mathrm{disk}}$	${\chi }^{2}/\mathrm{dof}$	log ${L}_{\mathrm{Input}}$	A
XMM1	${0.19}_{-0.01}^{+0.01}$	${0.20}_{-0.01}^{+0.01}$	${1.72}_{-0.01}^{+0.01}$	$\gt 1.39$	${40.41}_{-0.01}^{+0.01}$	${39.41}_{-0.03}^{+0.03}$	$393.2/320$	39.83	3.90
XMM3	${0.12}_{-0.03}^{+0.03}$	${0.27}_{-0.04}^{+0.05}$	${1.66}_{-0.08}^{+0.06}$	${2.38}_{-0.38}^{+0.74}$	${40.10}_{-0.03}^{+0.03}$	${39.36}_{-0.11}^{+0.10}$	$269.6/285$	39.44	4.65
XMM4	${0.17}_{-0.04}^{+0.04}$	${0.21}_{-0.03}^{+0.05}$	${1.81}_{-0.06}^{+0.04}$	$\gt 2.56$	${40.21}_{-0.04}^{+0.05}$	${39.39}_{-0.18}^{+0.21}$	$316.2/320$	39.61	4.46
XMM5	${0.13}_{-0.01}^{+0.01}$	${0.24}_{-0.01}^{+0.01}$	${1.55}_{-0.02}^{+0.02}$	${2.45}_{-0.13}^{+0.16}$	${40.05}_{-0.01}^{+0.01}$	${39.38}_{-0.04}^{+0.04}$	$463.4/460$	39.26	6.25
XMM6	${0.15}_{-0.05}^{+0.06}$	${0.25}_{-0.04}^{+0.06}$	${1.61}_{-0.09}^{+0.06}$	$\gt 2.18$	${40.17}_{-0.05}^{+0.06}$	${39.52}_{-0.15}^{+0.18}$	$224.4/227$	39.43	6.05
XMM8	${0.13}_{-0.03}^{+0.05}$	${0.23}_{-0.08}^{+0.11}$	${1.83}_{-0.04}^{+0.05}$	$\gt 3.01$	${40.30}_{-0.03}^{+0.05}$	$\lt 39.20$	$373.8/318$	39.79	3.97
XMM9	${0.14}_{-0.03}^{+0.04}$	${0.25}_{-0.06}^{+0.08}$	${1.80}_{-0.04}^{+0.03}$	${2.22}_{-0.23}^{+0.32}$	${40.42}_{-0.02}^{+0.03}$	${39.02}_{-1.67}^{+0.31}$	$399.8/389$	39.91	3.68
Suzaku1	$\lt 0.09$	${0.30}_{-0.08}^{+0.10}$	${1.64}_{-0.05}^{+0.03}$	${2.57}_{-0.12}^{+0.25}$	${40.16}_{-0.02}^{+0.04}$	${38.91}_{-0.29}^{+0.28}$	$357.1/343$	39.55	4.76
Suzaku2	$\lt 0.19$	${0.21}_{-0.04}^{+0.12}$	${1.70}_{-0.04}^{+0.04}$	${2.76}_{-0.29}^{+0.45}$	${40.23}_{-0.06}^{+0.09}$	${39.18}_{-0.87}^{+0.47}$	$312.7/286$	39.59	5.05
XMM10	${0.14}_{-0.03}^{+0.04}$	${0.24}_{-0.04}^{+0.05}$	${1.68}_{-0.06}^{+0.04}$	$\gt 2.56$	${40.21}_{-0.03}^{+0.04}$	${39.40}_{-0.14}^{+0.16}$	$316.0/300$	39.54	5.43
Suzaku3	$\lt 0.17$	${0.23}_{-0.11}^{+0.16}$	${1.75}_{-0.04}^{+0.04}$	${3.03}_{-0.37}^{+0.75}$	${40.23}_{-0.03}^{+0.10}$	$\lt 39.55$	$301.0/297$	39.67	4.58
XMM11	${0.13}_{-0.03}^{+0.03}$	${0.28}_{-0.05}^{+0.06}$	${1.68}_{-0.07}^{+0.05}$	${2.94}_{-0.59}^{+1.78}$	${40.24}_{-0.02}^{+0.03}$	${39.43}_{-0.11}^{+0.10}$	$313.8/328$	39.60	4.80
Suzaku4	${0.17}_{-0.12}^{+0.09}$	${0.17}_{-0.05}^{+0.06}$	${1.80}_{-0.05}^{+0.04}$	${3.45}_{-0.59}^{+1.53}$	${40.33}_{-0.10}^{+0.11}$	${39.53}_{-0.90}^{+0.39}$	$327.1/356$	39.70	4.81
XMM12	${0.10}_{-0.03}^{+0.04}$	${0.31}_{-0.08}^{+0.10}$	${1.67}_{-0.15}^{+0.09}$	${2.16}_{-0.36}^{+0.53}$	${40.24}_{-0.03}^{+0.04}$	${39.39}_{-0.18}^{+0.17}$	$318.8/298$	39.63	4.13
XMM13	${0.22}_{-0.13}^{+0.09}$	${0.16}_{-0.03}^{+0.28}$	${1.73}_{-0.05}^{+0.06}$	${1.88}_{-0.13}^{+0.18}$	${40.52}_{-0.10}^{+0.12}$	$\lt 40.08$	$378.4/363$	39.89	4.59
XMM14	${0.17}_{-0.05}^{+0.06}$	${0.20}_{-0.03}^{+0.07}$	${1.62}_{-0.03}^{+0.04}$	${1.66}_{-0.09}^{+0.11}$	${40.51}_{-0.04}^{+0.06}$	${39.43}_{-0.51}^{+0.36}$	$380.0/378$	39.84	4.91
XMM15	${0.22}_{-0.09}^{+0.09}$	${0.17}_{-0.02}^{+0.07}$	${1.67}_{-0.06}^{+0.06}$	${1.72}_{-0.14}^{+0.20}$	${40.56}_{-0.08}^{+0.11}$	${39.68}_{-1.18}^{+0.46}$	$374.5/330$	39.87	4.86
Suzaku5	$\lt 0.15$	${0.39}_{-0.29}^{+0.12}$	${1.65}_{-0.17}^{+0.09}$	${2.63}_{-0.55}^{+1.01}$	${40.15}_{-0.01}^{+0.11}$	$\lt 39.46$	$287.2/268$	39.59	4.20
Suzaku6	$\lt 0.13$	${0.36}_{-0.10}^{+0.14}$	${1.44}_{-0.20}^{+0.10}$	${1.96}_{-0.29}^{+0.30}$	${40.05}_{-0.02}^{+0.05}$	${39.28}_{-0.19}^{+0.15}$	$224.4/250$	39.29	5.69
Suzaku7	${0.25}_{-0.19}^{+0.17}$	${0.19}_{-0.03}^{+0.08}$	${1.61}_{-0.08}^{+0.07}$	${2.62}_{-0.47}^{+1.18}$	${40.18}_{-0.15}^{+0.22}$	${39.70}_{-0.61}^{+0.45}$	$275.8/245$	39.32	6.27
Suzaku8	$\lt 0.35$	$\lt 0.48$	${1.81}_{-0.06}^{+0.06}$	$\gt 2.41$	${40.20}_{-0.14}^{+0.22}$	$\lt 40.07$	$266.0/243$	39.59	5.05

Note. (1) Data; (2) neutral hydrogen column density in units of ${10}^{22}\,{\mathrm{cm}}^{-2};$ (3) inner disk temperature in keV; (4) photon index; (5) electron temperature in keV; (6)–(7) logarithmic total unabsorbed 0.3–10 keV X-ray luminosity and the luminosity of the disk component in $\,\mathrm{erg}\ {{\rm{s}}}^{-1}$, calculated by assuming the distance of 3.4 Mpc (Hill et al. 1993); (8) ${\chi }^{2}$ statistics and degrees of freedom; (9) logarithmic luminosity of input seed photons in $\,\mathrm{erg}\ {{\rm{s}}}^{-1};$ (10) amplification factor. Seed photon temperature, ${{kT}}_{{\rm{S}}}={{kT}}_{\mathrm{in}}$ and input type = 1.

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3.2. Spectral Variability

We first investigated how the spectral parameters, ${N}_{{\rm{H}}}$, ${{kT}}_{\mathrm{in}}$, ${{kT}}_{{\rm{e}}}$, ${L}_{\mathrm{disk}}$, and Γ, changed with X-ray luminosity ${L}_{{\rm{X}}}$. Their variations as a function of ${L}_{{\rm{X}}}$ are shown in Figures 1 and 2. From the figures, it is clear that the parameters exhibit an approximate trend with ${L}_{{\rm{X}}}$. Among them, the plasma temperature (${{kT}}_{{\rm{e}}}\sim 1\mbox{--}4$ keV) has large uncertainties and was poorly constrained in some of the observations, although it first increases, and then decreases as ${L}_{{\rm{X}}}$ increases. The disk luminosity ${L}_{\mathrm{disk}}$ is in the range of a few times ${10}^{39}\,\mathrm{erg}\ {{\rm{s}}}^{-1}$, with large uncertainties and poor constraints in some of the observations. Thus, we can consider ${L}_{\mathrm{disk}}$ not strongly variable or relatively stable in these observations. The photon index Γ varies significantly with ${L}_{{\rm{X}}}$ (Figure 2). It first has a strong correlation with ${L}_{{\rm{X}}}$, but turns to be lower at the highest ${L}_{{\rm{X}}}$.

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We quantified the correlations between the X-ray luminosity and the best-fit parameters by obtaining the Spearman rank correlation coefficient (${r}_{{\rm{s}}}$) and the probability (p-value) for the null hypothesis. If the p-value is 0.05 or less, the correlation is considered to be significant. The parameters ${N}_{{\rm{H}}}$ and Γ are positively correlated with ${L}_{{\rm{X}}}$, where ${r}_{{\rm{s}}}=0.53$ ($p=1.43\,\times {10}^{-2}$) and ${r}_{{\rm{s}}}=0.52$ ($p=1.51\times {10}^{-2}$), respectively. The parameter ${{kT}}_{\mathrm{in}}$ is negatively correlated with both ${L}_{{\rm{X}}}$ and ${L}_{\mathrm{disk}}$, where ${r}_{{\rm{s}}}=-0.58$ ($p=5.57\times {10}^{-3}$) and ${r}_{{\rm{s}}}=-0.61$ ($p=3.41\times {10}^{-3}$), respectively. However, the parameters ${L}_{\mathrm{disk}}$ and ${{kT}}_{{\rm{e}}}$ do not show any significant correlation with ${L}_{{\rm{X}}}$ and the coefficients are ${r}_{{\rm{s}}}=0.31$ (p = 0.17) and ${r}_{{\rm{s}}}=-0.30$ (p = 0.18), respectively.

We note that ${N}_{{\rm{H}}}$ is positively correlated with ${L}_{{\rm{X}}}$. If the correlation has a physical origin, the increasing luminosity is a signature of the denser environment, while a non-physical origin is due to the degeneracy between ${N}_{{\rm{H}}}$ and Γ in model-fitting, artificially boosting the unabsorbed luminosity (see more details in Kajava & Poutanen 2009). For some of the ULXs, ${N}_{{\rm{H}}}$ is also correlated with Γ, when fitted with a PL or a disk-plus-PL model (Feng & Kaaret 2009; Kajava & Poutanen 2009). However, we did not find an ${N}_{{\rm{H}}}$–Γ correlation for Ho IX X–1 with the baseline model. This result is consistent with that in Kajava & Poutanen (2009), where the ${N}_{{\rm{H}}}\mbox{--}{\rm{\Gamma }}$ correlation no longer exists when fitted with the PL-plus-cool-MCD model. Since ${N}_{{\rm{H}}}$ varies marginally with ${L}_{{\rm{X}}}$, we tried to fix the absorption column to their mean value (${N}_{{\rm{H}}}=1.34\,\times {10}^{21}\,{\mathrm{cm}}^{-2}$) and fitted the spectra again. The spectral fits were worse for the cases where the best-fit values of ${N}_{{\rm{H}}}$ differ from the mean value, but we found that the ${\rm{\Gamma }}\mbox{--}{L}_{{\rm{X}}}$ correlation still appears. Thus, although we considered the local absorption column as a free parameter in the baseline model, this choice did not affect the correlation (for a given spectrum, higher ${N}_{{\rm{H}}}$ can cause higher Γ and higher ${L}_{{\rm{X}}}$).

Examining the correlation between Γ and ${L}_{{\rm{X}}}$ (Figure 2), Γ reaches ∼1.8 around the luminosity of $2\times {10}^{40}\,\,\mathrm{erg}\ {{\rm{s}}}^{-1}$ and actually turns out to be lower with values of ∼1.6–1.7 at higher luminosities. We investigated this plausible turnover feature by fitting a line to all data points first. A poor fit with a reduced ${\chi }^{2}$ of 3.6 (19 dof) was obtained. Excluding the data points above the luminosity of $2\times {10}^{40}\,\,\mathrm{erg}\ {{\rm{s}}}^{-1}$, then the fitting was improved to have a reduced ${\chi }^{2}$ of 1.0 (13 dof). This improvement is significant at a $99.95 \%$ confidence level compared to the first fit, indicating the presence of the turnover. We also attempted a broken PL fit for the Γ and ${L}_{{\rm{X}}}$ correlation. The fitting identified a break at $(1.99\pm 0.10)\,\times {10}^{40}\,\,\mathrm{erg}\ {{\rm{s}}}^{-1}$ with a reduced ${\chi }^{2}$ of ∼1.7 for 17 dof, and the slopes were 0.27 ± 0.04 and −0.17 ± 0.05 below and above the break, respectively. The fit results further support a turnover around $\sim 2\times {10}^{40}\,\,\mathrm{erg}\ {{\rm{s}}}^{-1}$.

Many ULXs, when their spectra were fitted with a PL-plus-cool-MCD model, exhibited an anti-correlation between the disk luminosity and temperature, ${L}_{\mathrm{disk}}\propto {T}_{\mathrm{in}}^{-3.5}$ (Soria 2007; Kajava & Poutanen 2009). This anti-correlation suggests that the ULXs considered in those work were in a state with a high accretion rate and low disk temperature (ultraluminous branch; Soria 2007). We checked the correlation between ${L}_{\mathrm{disk}}$ and ${{kT}}_{\mathrm{in}}$ derived from the baseline model, and the anti-correlation appears to be consistent with the previous reported results (Soria 2007; Kajava & Poutanen 2009). The X-ray luminosity of Ho IX X–1 has changed by a factor of ∼3–4 during these observations and among them, XMM5 and Suzaku 6 data have the lowest flux. During the observations conducted in 2012 November (XMM13–XMM15), the source was observed with the highest X-ray luminosity (${L}_{{\rm{X}}}\sim 3.6\times {10}^{40}\,\mathrm{erg}\ {{\rm{s}}}^{-1}$) compared to the other observations. The MCD component appears relatively less variable and the flux contribution from it is always less (≲33%) than that from the Comptonized component in all observations. Thus, the observed variability is dominated by the Comptonization process.

Our baseline model with a black hole suggests that the compact object is surrounded by the corona and multi-color accretion disk, where the inner region of the disk may be covered by the corona. The input seed photons from the accretion disk are injected into the corona and the corona Comptonizes these seed photons. Since the source exhibits variability, it is important to know how the changes occur in the radiation mechanism and geometry of the source. Using the xspec model function (nthcomp.f) for the nthcomp model, we estimated the amplification factor, $A={L}_{{\rm{C}}}/{L}_{\mathrm{Input}}$, where ${L}_{{\rm{C}}}$ and ${L}_{\mathrm{Input}}$ are the luminosities of the Comptonizing cloud and the input seed photons, respectively. The parameter ${L}_{\mathrm{Input}}$ depends on the luminosity of the disk as well as the fraction of input seed photons seen by the corona. Moreover, the fraction of input seed photons entering the corona depends on the accretion geometry of the system. We derived A, ${L}_{{\rm{C}}}$, and ${L}_{\mathrm{Input}}$ from the model function for all the spectra using the best-fit parameters. The variations of the derived parameters with ${L}_{{\rm{X}}}$ are shown in Figure 3.

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It is clear from the figures that ${L}_{\mathrm{Input}}$ and ${L}_{{\rm{C}}}$ increase with ${L}_{{\rm{X}}}$. Moreover, ${L}_{\mathrm{Input}}$ varies more rapidly (by a factor of ∼5) than ${L}_{{\rm{C}}}$ (a factor of ∼3) and it appears to be flat ($\sim 7\times {10}^{39}\,\,\mathrm{erg}\ {{\rm{s}}}^{-1}$) at the highest ${L}_{{\rm{X}}}$. The changes in ${L}_{\mathrm{Input}}$ and ${L}_{{\rm{C}}}$ lead to the variations of A , which decreases from 6.3 to 3.7 as ${L}_{{\rm{X}}}$ increases, but then turns out to be ∼4.5–5 at the highest luminosity (Figure 3). In the Comptonization context, Γ is inversely proportional to A. We did observe such variability in the case of Ho IX X–1, where the source evolves from hard (${\rm{\Gamma }}\,\sim$ 1.4) to soft (${\rm{\Gamma }}\,\sim$ 1.8) as A decreases. We also estimated the optical depth τ (Zdziarski et al. 1998) as

$$\begin{eqnarray}&&\tau ={\left(\displaystyle \frac{{{kT}}_{e}}{{m}_{e}{c}^{2}}\right)}^{-1/2}{\left[{\left({\rm{\Gamma }}+\displaystyle \frac{1}{2}\right)}^{2}-\displaystyle \frac{9}{4}\right]}^{-1/2}.\ \ \ \end{eqnarray} \tag{ 1 }$$

The derived τ values are consistent with having optically thick corona ($\tau \sim 6-13$) and showed marginal variations with ${L}_{{\rm{X}}}$, first dropping and then going up as ${L}_{{\rm{X}}}$ increases.

We analyzed the Suzaku and XMM-Newton observations of the bright ULX Ho IX X–1 conducted over a period of ∼14 years, to study its spectral variability. We systematically studied the spectra with different models. The data were better represented by a MCD-plus-thermal-Comptonization (nthcomp) model. The best-fit spectral parameters derived from this model were consistent with the results of broadband spectral studies in the 0.3–30 keV energy band using the disk-corona-plus-PL-like-tail model (${N}_{{\rm{H}}}\sim 1.4\times {10}^{21}\,{\mathrm{cm}}^{-2}$, ${{kT}}_{\mathrm{in}}\sim 0.3\,\mathrm{keV}$, ${{kT}}_{{\rm{e}}}\,\sim 2.4\,\mathrm{keV}$; Walton et al. 2014). Using the MCD-plus-thermal-Comptonization model, we studied the variability behavior of the accretion disk plus the corona as a function of X-ray luminosity. Our analysis revealed that the best-fit model parameters showed strong trends with X-ray luminosity. The parameters ${N}_{{\rm{H}}}$ and Γ were found to be positively correlated with X-ray luminosity, while ${{kT}}_{\mathrm{in}}$ was negatively correlated. The plasma temperature derived from this model description ranges from ∼1 to 4 keV, consistent with the low-temperature, optically thick corona seen in other ULXs (Stobbart et al. 2006; Gladstone et al. 2009). Moreover, ${{kT}}_{{\rm{e}}}$ did not show a statistically significant correlation with ${L}_{{\rm{X}}}$, although it marginally varied with ${L}_{{\rm{X}}}$. The parameter Γ significantly varied with ${L}_{{\rm{X}}}$ and exhibited a statistically significant positive correlation. It evolved from hard (${\rm{\Gamma }}\sim 1.4$) to soft (${\rm{\Gamma }}\sim 1.8$) as ${L}_{{\rm{X}}}$ increases, while at higher X-ray luminosities Γ turned out to be slightly lower (${\rm{\Gamma }}\sim 1.7$). In these observations, the X-ray luminosity of Ho IX X–1 varied by a factor of ∼3–4 and the flux contribution from the Comptonized component was much higher than that from the disk component.

In the MCD-plus-thermal-Comptonization model, one can assume a geometry in which a standard cold accretion disk is truncated at a radius ${R}_{\mathrm{tr}}$ and the inner region contains hot plasma. The plasma Comptonizes the seed photons from the outer disk, and the fraction of input seed photons that enters the plasma region is related to a solid angle subtending between the plasma region and the outer disk. For this geometry, the input seed photon luminosity must be smaller than the disk luminosity, ${L}_{\mathrm{Input}}/{L}_{\mathrm{disk}}\lesssim 1$, but if ${L}_{\mathrm{Input}}\ll {L}_{\mathrm{disk}}$, it poses an unphysically small solid angle. This geometry has been proposed for the ULX NGC 1313 X–1, where the source was at a low flux state (Dewangan et al. 2010). For Ho IX X–1, such geometry is consistent with the spectra of XMM5, XMM6, and Suzaku 7, where ${L}_{\mathrm{Input}}/{L}_{\mathrm{disk}}\sim 0.4\mbox{--}0.8$. The values suggest the solid angle ${\rm{\Delta }}{\rm{\Omega }}\sim (0.4\mbox{--}0.8)\times 2\pi$. The estimated truncation radius from the normalization of the disk component for these three spectra ranges between $8.7\times {10}^{8}\,\mathrm{and}\,2.1\,\times {10}^{9}\,\mathrm{cm}$, which is corrected for color factor $\kappa =1.7$ (Shimura & Takahara 1995; the inclination angle of the binary is assumed to be 60^◦). The proposed scenario is consistent with the existence of a massive black hole of mass 50–200 M_☉. If we assume a mass of 200 M_☉ for Ho IX X–1, the Schwarzschild radius would be ${r}_{{\rm{s}}}=2{GM}/{c}^{2}\sim 6\times {10}^{7}\,\mathrm{cm}$, making the transition radius ${R}_{\mathrm{tr}}\sim 15\mbox{--}35\,{r}_{{\rm{s}}}$. Thus, the three spectra were consistent with a model where a massive black hole surrounded by a standard accretion disk truncated at a radius of $\sim 15\mbox{--}35\,{r}_{{\rm{s}}}$.

However, the majority of the spectra, 18 out of 21, appear to be inconsistent with this geometry as ${L}_{\mathrm{Input}}\gt {L}_{\mathrm{disk}}$. The observed properties of the source instead can be explained by an alternative geometry, the "sandwich model" (Liang & Price 1977; Haardt & Maraschi 1993; Svensson & Zdziarski 1994), which has been successful at describing the observed properties of Galactic BHXBs (Kubota & Done 2004; Done & Kubota 2006, and references therein). In this model, the corona covers the standard accretion disk and takes some fraction of the total gravitational power, while the remaining fraction is dissipated in the disk. The corona Comptonizes the seed photons from the underlying disk and a fraction of Comptonized photons (ξ) impinge on the disk and get absorbed. This geometry is valid only when the fraction ξ is less than the maximum value ($\xi \lt {\xi }_{\max }=1/A$). Such geometry can explain the observed properties of NGC 1313 X–1 in the high flux state (Dewangan et al. 2010), where the corona covers the entire disk. For Ho IX X–1, the MCD component is clearly evident in all the spectra, suggesting that the corona covers only the inner part of the disk. This can be confirmed by estimating the size of the corona region ${R}_{{\rm{c}}}$. Assuming the input seed photon luminosity as a blackbody, $\sigma {T}_{{\rm{s}}}^{4}2\pi {R}_{{\rm{c}}}^{2}={L}_{\mathrm{Input}}$ (see also Dewangan et al. 2010, for more details), where ${T}_{{\rm{s}}}$ is the temperature of the disk. The derived values of ${R}_{{\rm{c}}}$ exhibit an increasing trend with L_X (see Figure 3) and ${R}_{{\rm{c}}}\sim (1.4\mbox{--}13.6)\,\times {10}^{8}\ \mathrm{cm}$, which can definitely mask the inner part of the disk. The negative correlation between ${{kT}}_{\mathrm{in}}$ and ${L}_{{\rm{X}}}$ (see Figure 1) can thus be because of the larger part of the inner disk covered by the corona.

From our studies, we revealed the existence of a more complex ${\rm{\Gamma }}\mbox{--}{L}_{{\rm{X}}}$ relation: Γ evolves from hard to soft (∼1.4–1.8) as ${L}_{{\rm{X}}}$ increases, while at higher ${L}_{{\rm{X}}}$, Γ turns out to be slightly harder (∼1.7). The positive ${\rm{\Gamma }}\mbox{--}{L}_{{\rm{X}}}$ correlation has been reported for Ho IX X–1, based on an absorbed PL fit, with a limited sample of observations (Kajava & Poutanen 2009). The observed values of photon index (${\rm{\Gamma }}\sim 1.5\mbox{--}2.0$) from the simple PL are also consistent with that from our baseline model. We further note that the ${\rm{\Gamma }}\mbox{--}{L}_{{\rm{X}}}$ correlation has been reported for several other ULXs, for example, NGC 1313 X–1 (Feng & Kaaret 2006b), Antennae X–11 (Feng & Kaaret 2006a), NGC 253 X–4, IC 342 X–6, Holmberg II X–1, NGC 5204 X–1, and NGC 5408 X–1 (Feng & Kaaret 2009; Kajava & Poutanen 2009), in which their spectra were fitted with a PL or a MCD-plus-PL model. This correlation phase of ULXs was explained as an intermediate state with hybrid properties from the thermal and steep PL states (Feng & Kaaret 2009).

However in the disk corona model that we used, the ${\rm{\Gamma }}\mbox{--}{L}_{{\rm{X}}}$ correlation can be explained as being due to the process of the thermal Comptonization of the seed photons from the accretion disk by the hot corona (Zdziarski et al. 2003, and references therein). In this process, the seed photons are variable, and when the seed photon flux increases, the X-ray emission becomes softer and stronger (Zdziarski & Grandi 2001; Zdziarski et al. 2002). Therefore, in this model the variability of ${L}_{\mathrm{Input}}$ should be stronger than that of ${L}_{{\rm{C}}}$. Ho IX X–1 is likely such a case. We found that ${L}_{\mathrm{Input}}$ is more variable (by a factor of ∼5) than ${L}_{{\rm{C}}}$ and the amplification factor decreases with ${L}_{{\rm{X}}}$. Moreover, the input seed photon flux for Ho IX X–1 shows signs of saturation at high luminosities. Thus, at high luminosities although the radiation flux from the corona has increased, the seed photon flux remains more or less the same, which should lead to hardening of the spectra, as is marginally observed. The saturation of the seed photon flux implies that while the coronal radiative power has increased, the disk flux remained nearly constant or did not increase proportionally. This suggests that at high luminosities a larger fraction of the accretion energy is dissipated in the corona compared to the disk, perhaps because the corona has become larger, covering a greater fraction of the accretion disk. The variable seed photon flux also affects the coronal parameters. Our modeling showed a marginal increase in ${{kT}}_{{\rm{e}}}$ as ${L}_{{\rm{X}}}$ (or ${L}_{\mathrm{Input}}$) increases, while it drops at high luminosity. The former behavior has been observed in other ULXs, NGC 5204 X–1 and Holmberg II X–1 (Roberts et al. 2006; Feng & Kaaret 2009), while the latter is consistent with earlier studies of Ho IX X–1 (Vierdayanti et al. 2010) and IC 342 X–1 (Feng & Kaaret 2009). The change in ${{kT}}_{{\rm{e}}}$ is also reflected in the optical depth, which first decreases (from $\tau \,\sim$ 13–6) and then becomes very thick ($\tau \sim 12$) at high luminosity. The variability indicates that Ho IX X–1 exhibits a cooler (${{kT}}_{{\rm{e}}}\sim 2$ keV) and very thick corona ($\tau \sim 12$) at different luminosity values. We note that in NGC 1313 X–2, the different corona states ("very thick" and "thick") correlate with luminosity and the source becomes more luminous in the "very thick" state (Pintore & Zampieri 2012). The comparison indicates the more complex variability behavior in Ho IX X–1.

As mentioned above, the source turns out to be marginally hard at high luminosities, i.e., ${L}_{{\rm{X}}}\gt 2\times {10}^{40}\,\,\mathrm{erg}\ {{\rm{s}}}^{-1}$. This turnover of Γ in the ${\rm{\Gamma }}\mbox{--}{L}_{{\rm{X}}}$ plane is significant, and results in a change in the slope of the correlation. Interestingly, the optical depth also exhibits a similar variability pattern, as the source appears to be optically very thick ($\tau \sim 12$) above the luminosity $2\times {10}^{40}\,\,\mathrm{erg}\ {{\rm{s}}}^{-1}$. In Pintore et al. (2014), the softening behavior was observed for Ho IX X–1 in the "thick" state as the intensity increases in the hardness-intensity diagram (left panel of Figure 6 in Pintore et al. 2014). We also observe such softening variability behavior below the luminosity $2\times {10}^{40}\,\,\mathrm{erg}\ {{\rm{s}}}^{-1}$ (mostly in the "thick" state) in the ${\rm{\Gamma }}\mbox{--}{L}_{{\rm{X}}}$ plane. However, the behavior of low-luminosity sources (NGC 253 X–1 and NGC 1313 X–2 in the "thick" state; Pintore et al. 2014) is different, where these sources become marginally hard as the intensity increases to a level consistent with the "very thick" state seen in a few ULXs. Then, in the "very thick" state, NGC 1313 X–2 turns out to have softening behavior as the intensity increases. Thus, comparing Ho IX X–1 to NGC 1313 X–2, these sources have different behavior below and above a certain luminosity threshold. The threshold is also different in the two sources; as for the former, its value is $\sim 2\times {10}^{40}\,\,\mathrm{erg}\ {{\rm{s}}}^{-1}$, while it can be approximated to be $5\times {10}^{39}\,\,\mathrm{erg}\ {{\rm{s}}}^{-1}$ (see Figure 3 in Pintore & Zampieri 2012) for the latter.

In the models with a corona, the energy balance is achieved by adjusting the coronal parameters: varying either the electron temperature alone or both ${{kT}}_{{\rm{e}}}$ and τ. For example, if the optical depth is constant, the corona adjusts the electron temperature itself to the variable seed photon flux. Thus, the energy balance is satisfied by an increased cooling of the plasma (${{kT}}_{{\rm{e}}}$ decreases), which results in the increase of Γ and a softer spectrum (Zdziarski & Grandi 2001; Zdziarski et al. 2002, 2003). However, it is also possible that the variation in the photon index is due to the optical depth variation (Svensson 1994; Haardt et al. 1997, and references therein). In such cases, the energy balance is satisfied by adjusting τ and the softening of the spectrum is associated with an increase of ${{kT}}_{{\rm{e}}}$ (Nicastro et al. 2000; Chiang 2002; Zdziarski et al. 2003). The latter explanation seems to be consistent with the observed behavior of the coronal parameters of Ho IX X–1 below a luminosity of $2\times {10}^{40}\,\,\mathrm{erg}\ {{\rm{s}}}^{-1}$. At a luminosity above $2\times {10}^{40}\,\,\mathrm{erg}\ {{\rm{s}}}^{-1}$, the reason for the change in the slope of the ${\rm{\Gamma }}\mbox{--}{L}_{{\rm{X}}}$ relation appears to be the saturation of the input seed photon luminosity. Considering this and the energy balance of the system at high luminosity, one would expect that at constant optical depth the coronal electron temperature increases as ${L}_{{\rm{X}}}$ increases, while we observed an opposite behavior. Thus, this behavior can be explained as when ${L}_{{\rm{X}}}$ increases above $2\times {10}^{40}\,\,\mathrm{erg}\ {{\rm{s}}}^{-1}$, the corona becomes not only more energetic but also more mass-loaded, which increases the coronal density and the optical depth of the source. In such a case, the decrease in the mean energy per scattering is largely compensated for by an increase of the average number of scatterings, which produces the observed hardening of the spectrum.

While in this work, we have interpreted Ho IX X–1 as a massive black hole accreting at a near-Eddington rate, the source has also been interpreted as a stellar-mass black hole accreting at super-critical rates leading to a strong, radiation-driven winds (Sutton et al. 2013; Pintore et al. 2014; Luangtip et al. 2016). In this scenario, the wind may be sufficiently dense and cover a large fraction of the outer disk. When a ULX system is observed face-on, the inner region around a black hole emits a hard spectrum, and at high inclination angles, the line of sight passes through the outer, cold region of the wind and a softer spectrum is observed (see e.g., Middleton et al. 2011; Sutton et al. 2013; Middleton et al. 2014). Sutton et al. (2013) studied the spectral variability of a large sample of ULXs, using a doubly absorbed MCD-plus-PL model, and empirically classified the ULX population into three spectral regimes, referred as broadened-disk, hard ultraluminous, and soft ultraluminous. These spectral states can be explained as being due to different viewing angles to the sources. However, Ho IX X–1 shows spectral characteristics of the hard ultraluminous and broadened-disk class at different times (Sutton et al. 2013). This was confirmed by another detailed study of a large sample of ULXs, where color–color and hardness-intensity diagrams were used (Pintore et al. 2014). Luangtip et al. (2016) also characterized the spectral evolution of Ho IX X–1 below 10 keV. They used a two-thermal component model, diskbb+comptt, similar to what has been used in this work, and our results are consistent with theirs. For example, the flux variations primarily came from the hard component and at the high-luminosity end, the spectra of the hard component is harder. The difference between theirs and this work is in the interpretation of the spectral variability, where they consider the wind model of a stellar-mass black hole, while here we consider the system to be a massive black hole accreting at near-Eddington rates. We note that no direct evidence for a strong disk wind has been found in Ho IX X–1 (Walton et al. 2012, 2013), although some residuals in the source's soft spectra have been suggested to be line features from the wind (Middleton et al. 2015). In any case, our study has revealed interesting variability features of the source, in particular the ${\rm{\Gamma }}\mbox{--}{L}_{X}$ correlation. These features can be considered as significant constraints, and remain to be explained by any scenario proposed for this ULX.

We thank the anonymous referee for the constructive comments and suggestions that improved this manuscript. This research has made use of data obtained from the High Energy Astrophysics Science Archive Research Center (HEASARC), provided by NASA's Goddard Space Flight Center. This research was supported by the National Program on Key Research and Development Project (Grant No. 2016YFA0400804) and the National Natural Science Foundation of China (11373055, 11633007). V.J. acknowledges the IUCAA Visitor's Program and the financial support from the Chinese Academy of Sciences through the President's International Fellowship Initiative (CAS PIFI, Grant No. 2015PM059). Z.W. acknowledges the support by the CAS/SAFEA International Partnership Program for Creative Research Teams.

Software: heasoft (v 6.15.1), sas (v 14.0), xspec (v 12.8.1g; Arnaud 1996).