Spectral and Timing Properties of IGR J17091–3624 in the Rising Hard State During Its 2016 Outburst

We processed the NuSTAR data using v.1.6.0 of the NuSTARDAS pipeline with NuSTAR CALDB v20170120. After the standard data cleaning and filtering, the total exposure time is 43.3, 20.2, and 20.7 ks for the three observations. For each observation, the source spectra and light curves were extracted at the position of IGR J17091–3624 within the radius of 100'' from the two NuSTAR focal plane modules (FPMA and FPMB). Corresponding background spectra were extracted from source-free areas on the same chip using polygonal regions. In the first NuSTAR observation (ObsID 80001041002), the source region was contaminated by the stray light from a nearby bright source GX 349+2. In this case, we also placed the background region in the part of the field illuminated by the stray light. We note that the influence from the stray light is minimal, as the background only contributes ∼2% to the total count rate. The NuSTAR spectra were grouped to have a signal-to-noise ratio (S/N) of at least 20 per bin after background subtraction.

For the NuSTAR timing analysis, we first applied barycenter correction to the event files to transfer the photon arrival times to the barycenter of the solar system using JPL Planetary Ephemeris DE-200. We then generated cross-power density spectra (CPDS) using MaLTPyNT (Bachetti 2015) from the cleaned event files. MaLTPyNT, developed for NuSTAR timing analysis, properly treats orbital gaps and uses the CPDS as a proxy for the power density spectrum. The CPDS uses the signals from two independent focal plane modules, FPMA and FPMB, and the real part of the CPDS provides a measure of the signal that is in phase between the two modules. Thus, this method helps to remove the spurious distortion to the power-density spectrum introduced by the instrumental dead time (for details, see Bachetti et al. 2015). We generated the CPDS with the binning time of 2⁻⁸ s and in 512 s long intervals. For all the power spectra, we used the root mean square (rms) normalization, and geometrically rebinned the power spectra with a factor of 1.03 to increase the S/N of the frequency bins and produce nearly equally spaced bins in the logarithmic scale.

The Swift/XRT data were processed using xrtpipeline v.0.13.2 with XRT CALDB v20160609 to produce clean event files. We extracted the source spectra with xselect from a circular region with a radius of 20 pixels centered on the position of IGR J17091–3624. The background was extracted from an annulus area with the inner and outer radii of 80 pixels and 120 pixels, respectively. The data were filtered for grade 0 events only and we used swxwt0s6_20131212v015.rmf for the response matrix. Ancillary response files were generated with xrtmkarf, incorporating exposure maps to correct for the bad columns. Finally, we rebinned the data to have a S/N greater than 5 for each energy bin.

We model the simultaneous NuSTAR and Swift/XRT spectra in XSPEC v12.9.0n (Arnaud 1996) using ${\chi }^{2}$ statistics. For spectral fitting, we adopt the cross-sections from Verner et al. (1996) and abundances from Wilms et al. (2000). All parameter uncertainties are reported at the 90% confidence level for one parameter of interest. Cross-normalization constants are allowed to vary freely for NuSTAR FPMB and Swift/XRT and assumed to be unity for FPMA. The values are within 5% of unity for Epoch 1 and Epoch 2, as expected from Madsen et al. (2015). In Epoch 3, the Swift normalization is ∼13% higher than NuSTAR. We note that it is probably caused by the exposure correction uncertainty when the source lies close to the CCD bad columns, which is sensitive to the exact source position and can be affected by the spacecraft pointing stability.¹¹

All three NuSTAR observations detect IGR J17091–3624 across the entire instrument bandpass. To highlight the reflection features, we fit the NuSTAR spectra with a simple absorbed power-law model, TBabs*powerlaw in XSPEC, using only data in the 3–5 keV and 8–12 keV energy range. As shown in Figure 2, the residuals display asymmetric broad iron lines around 4–7 keV and show evidence for changes in the cutoff energy as the source luminosity increases. From this simple model, we also detect a slight steepening in the power-law slope: Γ increases from ∼1.55 to ∼1.60 from Epoch 1 to Epoch 3. No parameter uncertainties are calculated here, as the fits are too poor to determine meaningful parameter uncertainties.

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3.1.1. Joint Fits with Disk Reflection Models

Given the disk reflection features observed (see Figure 2), we model the NuSTAR and Swift spectra simultaneously with the self-consistent reflection model relxill v0.5b (Dauser et al. 2014; García et al. 2014). We fit the NuSTAR and Swift spectra in the bands 3–79 keV and 1–10 keV, respectively. We ignore the Swift data below 1 keV to avoid possible low-energy residuals known to be present in the XRT Windowed Timing mode data of some absorbed sources.¹² We note that due to the relative weakness of the disk reflection component, as displayed in Figure 2(a), reflection features are not detected by Swift. Although measurements of the reflection spectra are dominated by the NuSTAR data, extending the spectral fitting to the soft energy band helps to constrain the absorption column density and the general continuum slope.

We first fit the data with the unblurred version of the reflection model xillver (García & Kallman 2010) multiplied by a model that accounts for neutral Galactic absorption: TBabs*xillver. To maximize the parameter constraints, we jointly fit the spectra taken at the three epochs and link the values for the absorption column density ${N}_{{\rm{H}}}$, the iron abundance ${A}_{\mathrm{Fe}}$, and the disk inclination i, which are normally expected to remain constant. Other relevant parameters (the ionization parameter ξ, photon index Γ, high energy cutoff ${E}_{\mathrm{cut}}$, and reflection fraction ${R}_{\mathrm{ref}}$) are allowed to vary between epochs. The best-fit result yields the reduced-chi-square ${\chi }_{\nu }^{2}={\chi }^{2}/\nu =3851.1/3629=1.061$, where ν is the number of degrees of freedom. The model leaves obvious residuals in the Fe K band (see Figure 3), suggesting that relativistic blurring is required to provide an adequate fit for the data.

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3.1.2. Power-law Emissivity Index

In order to account for the relativistic effects and to better describe the changes in the high energy rollover as displayed in Figure 1, we fit the spectra with the relativistic reflection model relxillCp (Model 1a). It uses a thermally Comptonized input continuum nthcomp (Zdziarski et al. 1996; Życki et al. 1999) and parameterizes the high energy cutoff by the coronal electron temperature ${{kT}}_{{\rm{e}}}$. The disk emissivity profile in relxillCp is described by a broken power law with three parameters: the inner and the outer emissivity indices ${q}_{\mathrm{in},\mathrm{out}}$ and the break radius ${R}_{\mathrm{br}}$. We assume a canonical emissivity profile $\epsilon (r)\propto {r}^{-3}$ by fixing both ${q}_{\mathrm{in}}$ and ${q}_{\mathrm{out}}$ at 3, which is expected for the outer parts of a standard Shakura–Sunyaev disk (Dauser et al. 2013). If ${q}_{\mathrm{in}}$ and ${R}_{\mathrm{br}}$ are allowed to vary individually for each epoch, the fit would not be improved and the data cannot constrain the extra parameters. We fix the outer disk radius at 400 ${R}_{{\rm{g}}}$ and allow the inner disk radius ${R}_{\mathrm{in}}$ to vary to test the disk truncation hypothesis. There is no well-constrained black hole spin measurement for IGR J17091–3624; previous estimates range from a low or negative spin (Rao & Vadawale 2012) to a high spin (Reis et al. 2012). We first simply fix the dimensionless spin parameter a ($a\equiv {cJ}/{{GM}}^{2}$) at its maximum value of 0.998. As demonstrated below, the choice of spin does not significantly affect our results.

The broadband X-ray spectra can be well fitted by Model 1a with no obvious residuals (Figure 4). The reduced-chi-square is very close to unity (${\chi }_{\nu }^{2}=3627.0/3626=1.000$), indicating statistically a very good fit. Using the relativistic reflection model improves the fit significantly by ${\rm{\Delta }}{\chi }^{2}=224$ compared to the unblurred reflection model. The only excess noticeable is around 3.0–3.5 keV in the NuSTAR spectra at Epoch 2 (Figure 4(c)), which is possibly related to small calibration uncertainties near the edge of the instrument bandpass and is not statistically important. We include this part in the spectral modeling, as it helps to constrain the overall power-law slope. The soft disk component is either too cool to contribute much above 1 keV or intrinsically too faint to be detected by Swift/XRT, and a disk blackbody component is not required by our spectral modeling. From the best-fit parameters, we confirm the relatively high obscuration in IGR J17091–3624, with the equivalent hydrogen column density ${N}_{{\rm{H}}}=(1.58\pm 0.03)\,\times {10}^{22}$ cm⁻², consistent with previous measurements (e.g., Rodriguez et al. 2011; Capitanio et al. 2012). After including the reflection component in the spectral fitting, we measure a slightly steeper power-law index (${\rm{\Gamma }}\simeq 1.67\mbox{--}1.70$) compared to the simple absorbed power-law model. We confirm the existence of the high energy cutoff in the spectra, with the relatively low coronal temperature¹³ ${{kT}}_{{\rm{e}}}={64}_{-15}^{+46}$ keV, ${32}_{-4}^{+6}$ keV, and ${26}_{-2}^{+3}$ keV for Epochs 1, 2, and 3, respectively.

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As for the key reflection parameters: the ionization parameter ξ¹⁴ increases with source flux from Epoch 1 to Epoch 3; an abundance of ${A}_{\mathrm{Fe}}={0.78}_{-0.13}^{+0.10}$ (in units of solar value) is derived for the accretion disk and the disk is estimated to be viewed at a low inclination angle $i={37}_{-4}^{^\circ +3};$ the measured inner disk radius is ${R}_{\mathrm{in},\mathrm{Epoch}1}={20}_{-8}^{+33}$ ${r}_{{\rm{g}}}$, ${R}_{\mathrm{in},\mathrm{Epoch}2}={17}_{-7}^{+14}$ ${r}_{{\rm{g}}}$, and ${R}_{\mathrm{in},\mathrm{Epoch}3}={40}_{-22}^{+47}$ ${r}_{{\rm{g}}}$ (see Table 2 for a list of the best-fit parameters).

Table 2.  Best-fit Parameters of the Disk Reflection Models

Epoch ${N}_{{\rm{H}}}$	${q}_{\mathrm{in}}$	${q}_{\mathrm{out}}$	h	i	${R}_{\mathrm{in}}$	a	Γ	log $\xi$	${A}_{\mathrm{Fe}}$	${k}_{\mathrm{Te}}$	${R}_{\mathrm{ref}}$
	($\times {10}^{22}$ cm2)			(${r}_{{\rm{g}}}$)	(°)	(${r}_{{\rm{g}}}$)			(log[erg cm s−1])		(keV)
Model 1a: TBabs*relxillCp
1	${(1.59\pm 0.03)}^{l}$	3*	3*	⋯	${({37}_{-5}^{+8})}^{l}$	${20}_{-8}^{+33}$	0.998*	${1.674}_{-0.009}^{+0.008}$	${1.95}_{-0.18}^{+0.41}$	${({0.74}_{-0.15}^{+0.10})}^{l}$	${64}_{-15}^{+46}$	$0.26\pm 0.03$
2	⋯	3*	3*	⋯	⋯	${17}_{-7}^{+14}$	0.998*	${1.680}_{-0.010}^{+0.012}$	${2.73}_{-0.21}^{+0.08}$	⋯	${32}_{-4}^{+6}$	${0.19}_{-0.04}^{+0.06}$
3	⋯	3*	3*	⋯	⋯	${40}_{-22}^{+47}$	0.998*	${1.698}_{-0.008}^{+0.009}$	$2.75\pm 0.09$	⋯	${26}_{-2}^{+3}$	${0.15}_{-0.03}^{+0.04}$
${\chi }^{2}$/ν							3627.0/3626(1.000)
Model 1b: TBabs*relxillCp (i Allowed to Vary Between Epochs)
1	${(1.56\pm 0.03)}^{l}$	3*	3*	⋯	${34}_{-6}^{+8}$	${17}_{-6}^{+24}$	0.998*	$1.672\pm 0.009$	${2.07}_{-0.26}^{+0.34}$	${({0.74}_{-0.14}^{+0.16})}^{l}$	${69}_{-7}^{+11}$	${0.24}_{-0.03}^{+0.04}$
2	⋯	3*	3*	⋯	${42}_{-7}^{+11}$	${21}_{-10}^{+20}$	0.998*	${1.683}_{-0.010}^{+0.012}$	${2.72}_{-0.30}^{+0.07}$	⋯	${33}_{-4}^{+6}$	${0.21}_{-0.05}^{+0.07}$
3	⋯	3*	3*	⋯	${29}_{-8}^{+21}$	${21}_{-9}^{+77}$	0.998*	${1.696}_{-0.009}^{+0.010}$	${2.79}_{-0.08}^{+0.16}$	⋯	${26}_{-2}^{+3}$	${0.13}_{-0.03}^{+0.04}$
${\chi }^{2}$/ν							3623.7/3624(1.000)
Model 2a: TBabs*relxilllpCp
1	${(1.59\pm 0.03)}^{l}$	⋯	⋯	${9.4}_{-4.6}^{+29.3}$	${({36}_{-2}^{+5})}^{l}$	${19}_{-8}^{+21}$	0.998*	$1.673\pm 0.008$	${2.04}_{-0.29}^{+0.31}$	${(0.73\pm 0.13)}^{l}$	${67}_{-17}^{+39}$	0.51
2	⋯	⋯	⋯	${5.7}_{-3.0}^{+6.5}$	⋯	${18}_{-7}^{+8}$	0.998*	${1.680}_{-0.009}^{+0.010}$	${2.74}_{-0.20}^{+0.07}$	⋯	${33}_{-4}^{+6}$	0.40
3	⋯	⋯	⋯	${11.9}_{-7.8}^{+20.2}$	⋯	${39}_{-20}^{+21}$	0.998*	$1.698\pm 0.008$	${2.75}_{-0.08}^{+0.09}$	⋯	${26}_{-2}^{+3}$	0.31
${\chi }^{2}$/ν							3626.9/3626(1.000)
Model 2b: TBabs*relxilllpCp (Rin Linked for All Epochs)
1	${(1.59\pm 0.03)}^{l}$	⋯	⋯	${10.3}_{-2.5}^{+3.3}$	${(35\pm 4)}^{l}$	${({21}_{-6}^{+11})}^{l}$	0.998*	${1.673}_{-0.008}^{+0.006}$	${2.04}_{-0.29}^{+0.30}$	${({0.72}_{-0.13}^{+0.12})}^{l}$	${67}_{-17}^{+35}$	0.50
2	⋯	⋯	⋯	${6.6}_{-3.0}^{+7.3}$	⋯	⋯	0.998*	$1.678\pm 0.009$	${2.74}_{-0.19}^{+0.08}$	⋯	${33}_{-4}^{+5}$	0.36
3	⋯	⋯	⋯	${5.5}_{-2.4}^{+7.1}$	⋯	⋯	0.998*	${1.700}_{-0.008}^{+0.009}$	${2.79}_{-0.06}^{+0.10}$	⋯	${26}_{-2}^{+3}$	0.31
${\chi }^{2}$/ν							3629.3/3628(1.000)

Note. Frozen parameters are marked with asterisks. Superscripts l indicate the parameters whose values are linked at the three epochs. For all models, the outer disk radius ${R}_{\mathrm{out}}$ is fixed at 400 ${r}_{{\rm{g}}}$ and the black hole spin a is fixed at 0.998. As discussed in the text, altering the spin parameter a causes no significant change to the fitting result.

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The accretion disk is considered truncated if the inner edge of the disk is located outside the ISCO. The radius of the ISCO is a function of the black hole spin, and its value decreases monotonically from 9 ${r}_{{\rm{g}}}$ (${r}_{{\rm{g}}}\equiv {GM}/{C}^{2}$ is the gravitational radius) for an extreme retrograde spinning black hole to 1.235 ${r}_{{\rm{g}}}$ for a black hole with a maximum positive spin. The lower limits of the inner disk radius from Model 1a are all in excess of 10 ${r}_{{\rm{g}}}$ at the 90% confidence level and radii smaller than 6 ${r}_{{\rm{g}}}$ can be ruled out at 3σ (see Figure 5), which points to the truncated disk scenario.

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The reflection fraction ${R}_{\mathrm{ref}}$ is defined to be the ratio of the coronal intensity illuminating the disk to that reaching the observer (Dauser et al. 2016). The low reflection fraction measured (${R}_{\mathrm{ref}}\sim 0.15\mbox{--}0.26$) suggests the photon reprocessing is free from the strong light-bending effects in the vicinity of the black hole, also consistent with a truncated accretion disk. To investigate the effect of the black hole spin on the fitting results, we fix a at −0.998, 0, and 0.998 respectively, and obtain basically identical fits for all other parameters. Therefore, we note that the reflection modeling is simply not sensitive to the spin parameter, which is to be expected when the disk is truncated at this level. By fixing ${R}_{\mathrm{in}}$ at the ISCO and in turn fitting for the black hole spin a would yield a worse fit by ${\rm{\Delta }}{\chi }^{2}\simeq 20$ with a pegged at −0.998. This indicates that the fitting is attempting to find a larger inner disk radius than the maximum value allowed for the ISCO, which is also evidence for a truncated disk.

In Model 1a, we assume the iron abundance, inclination, and absorption column density remain constant through the outburst, which can be an oversimplification in some cases. For instance, a strongly warped disk could cause the apparent disk inclination to vary with radius. Evidence for disk warping has been found in, e.g., Cyg X–1, for which the inner disk and the orbital plane inclination are found to disagree by ∼10°–15° (Tomsick et al. 2014; Walton et al. 2016). We explore this possibility by fitting independently the disk inclination parameter i at the three epochs (Model 1b). This only improves the fitting marginally with two extra free parameters ${\rm{\Delta }}{\chi }^{2}/{\rm{\Delta }}\nu =-4.3/-2$. The best-fit inner disk radius at Epoch 3 is a bit lower, about the same value at Epoch 2 (although still consistent with Model 1a results within errors). This is more reasonable, as an obvious increase of the inner radius returned by Model 1a is not expected given the short time interval between the later two observations. However, since there is no clear indication for a change in the inner disk radius between epochs, disk warping is not explicitly required to describe the data.

3.1.3. Lamp-post Geometry

Low-frequency QPOs observed in black hole binaries are classified into different types, namely type-A, type-B, and type-C, based on their characteristics in the power spectra and are related to different spectral states (e.g., see Casella et al. 2004, 2005). In the hard state, type-C QPOs are most commonly found, which are variable and highly coherent peaks on top of strong flat-topped noise. A type-C QPO is evident in the CPDS generated in the full energy band (3–79 keV) from our NuSTAR observations of IGR J17091–3624 (see Figure 6).

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We model the NuSTAR CPDS of the three epochs in XSPEC jointly with a unit response file. We use a multi-Lorentizan model, which is commonly used to fit the power spectra of black hole binaries (e.g., Nowak 2000; Pottschmidt et al. 2003). The model consists of two zero-centered Lorentzians for the underlying broad component, plus one Lorentzian for the fundamental QPO and two for possible sub- and super-harmonics. This provides an adequate fit for the power spectra, with ${\chi }_{\nu }^{2}=829.9/798=1.04$, leaving no obvious excess (see Figure 6). The Lorentzian used to fit for the possible sub-harmonic is less significant compared to other components, and the centroid frequency and the line width cannot be very well constrained. Therefore, we focus our discussion on the two more significant peaks in the CPDS. The type-C QPO frequency ${\nu }_{0}$ is well measured to be $0.131\pm 0.002$ Hz, $0.219\pm 0.002$ Hz, and $0.327\pm 0.002$ Hz, with the rms amplitude of 6.1% ± 0.6%, 7.3% ± 0.5%, and 7.3% ± 0.4% for Epochs 1–3, respectively. No significant deviation in the QPO frequency is observed during any of the three individual NuSTAR observations. The relatively low type-C QPO frequencies are consistent with the values found during the onset of the 2011 outburst of IGR J17091–3624 (Rodriguez et al. 2011; Iyer et al. 2015). However, by adopting the disk reflection model, we measure a slightly higher photon index of ${\rm{\Delta }}{\rm{\Gamma }}\sim 0.2$ from the energy spectra when about the same QPO frequencies are detected, compared to Iyer et al. (2015) who used different spectral models.

As displayed in Figure 6, there is a secondary peak at higher frequencies than the type-C QPO in all three data sets. Such a feature is commonly associated with the first QPO harmonic, which occurs at twice the fundamental QPO frequency. However, this secondary frequency ${\nu }_{1}$ is measured to be centered at 0.31 ± 0.01 Hz, 0.50 ± 0.01 Hz, and ${0.76}_{-0.04}^{+0.05}$ Hz at Epochs 1, 2, and 3, respectively. ${\nu }_{1}$ is about 2.3 times the type-C QPO frequency, making them unlikely to be harmonically related. Although some small residuals can been seen in the fit of the power spectra (e.g., around 0.3 Hz in Figure 6(b)), these are probably due to the fact that the underlying continuum cannot be perfectly described by two Lorenzians and would not affect the measurement of the secondary frequency. When modeling the CPDS, the Lorentzian width of the secondary peak cannot always be constrained within a reasonable value. In order to obtain a reasonable fit, we set an upper limit of 0.2 Hz for the Lorentzian FWHM. This feature is detected at a $\gtrsim 4\sigma$ confidence level in all epochs, and is most significant at Epoch 2 with the detection at $7.6\sigma$.¹⁵ Although its origin is unclear, we note that this is not caused by instrumental effects, as its characteristic frequency is increasing between epochs.

We also investigate the energy dependence of the CPDS. In order to allow for enough statistics in each energy band, we only calculate the CPDS in two energy intervals: 3–6 keV (low) and 6–79 keV (high). The type-C QPO frequency basically remains constant in the two energy bands. The only noteworthy difference is evidence for an apparent shift of the secondary frequency, which is present at Epochs 1 and 2 (see Figure 7). In the high energy band, this secondary frequency is detected at $3.9\sigma$ and $5.1\sigma$ in Epoch 1 and Epoch 2, respectively; while it becomes considerably weaker at lower energies, the confidence levels are $2.0\sigma$ in Epoch 1 and $3.6\sigma$ in Epoch 2. The fitting finds different values for the secondary frequency ${\nu }_{1}$ in the two energy bands: at lower energies, we measure ${\nu }_{1,\mathrm{Epoch}1}={0.270}_{-0.004}^{+0.006}$ Hz and ${\nu }_{1,\mathrm{Epoch}2}={0.45}_{-0.02}^{+0.03}$ Hz; whereas ${\nu }_{1}$ derived from the high energy band agrees with the full-band results (see Table 3 for a list of frequencies and significances). We note ${\nu }_{1}$ in the 3–6 keV band is roughly two times the type-C QPO frequency, consistent with that of a conventional first harmonic. If the secondary frequency ${\nu }_{1}$ is required to be the same in the two energy bands, the overall fit would worsen by ${\rm{\Delta }}{\chi }_{\mathrm{Epoch}1}^{2}\simeq 8$ and ${\rm{\Delta }}{\chi }_{\mathrm{Epoch}2}^{2}\simeq 5$. In the Epoch 3 data, the secondary peak is the weakest and cannot be detected above the 95% confidence level in the low-energy band. The absence of the same pattern in Epoch 3 could imply it is transient, or it just lacks the statistics to be detected as the power spectrum becomes noisy close to the high-frequency end. In addition, we calculate the lag-frequency spectra between the two energy bands. The time lags are consistent with zero around the QPO frequencies.

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Table 3.  Fitting Results of the NuSTAR CPDS in Different Energy Bands

Epoch	Energy (keV)	${\nu }_{0}$ (Hz)	${\nu }_{1}$ (Hz)	$P({\nu }_{1})$
1	3.0–79.0	$0.131\pm 0.002$	0.31 ± 0.01	$5.3\sigma$
	3.0–6.0	${0.134}_{-0.003}^{+0.004}$	${0.270}_{-0.004}^{+0.006}$	$2.0\sigma \,(2.4\sigma )$
	6.0–79.0	$0.131\pm 0.002$	${0.30}_{-0.01}^{+0.02}$	$3.9\sigma$
2	3.0–79.0	$0.219\pm 0.002$	0.50 ± 0.01	$7.6\sigma$
	3.0–6.0	0.218 ± 0.004	${0.45}_{-0.02}^{+0.03}$	$3.6\sigma$
	6.0–79.0	0.220 ± 0.002	0.50 ± 0.01	$5.1\sigma$
3	3.0–79.0	$0.327\pm 0.002$	${0.76}_{-0.04}^{+0.05}$	$4.0\sigma$
	3.0–6.0	$0.327\pm 0.004$	⋯	$\lt 95 \%$ (79%)
	6.0–79.0	$0.327\pm 0.003$	${0.70}_{-0.04}^{+0.05}$	$2.7\sigma \,(2.6\sigma )$

Note. ${\nu }_{0}$ and ${\nu }_{1}$ denote the type-C QPO and the secondary frequency. $P({\nu }_{1})$ is the significance of the secondary frequency determined by the F-test. For comparison, significances calculated by simulations (1000 trials) are listed in parentheses.

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Our reflection modeling finds the accretion disk to be truncated. The lower limits of the inner disk radius are all in excess of $10\,{r}_{{\rm{g}}}$, which is larger than the maximum ISCO radius corresponding to a retrograde spin. We have also demonstrated that the choice of black hole spin has negligible effects on the measurement of disk truncation in our case. The best-fit values for the inner disk radius are around $17\mbox{--}40\,{r}_{{\rm{g}}}$. Considering the parameter uncertainties, the truncation radius is at the level of a few tens of gravitational radii. Due to controversies over the source distance and the black hole mass, the Eddington ratio of the source during our observations is uncertain. It can be estimated if we assume IGR J17091–3624 emits at the Eddington limit during its "hearbeat" state. From Altamirano et al. (2011a), the 2–50 keV source flux was ∼4 × 10⁻⁹ erg cm⁻² s⁻¹ when GRS 1915+105 like variabilities were detected. For a crude estimation, taking a bolometric correction factor of 3, we find an Eddington fraction $\lambda ={L}_{1\mbox{--}100\mathrm{keV}}/{L}_{\mathrm{Edd}}\sim 20 \% \mbox{--}30 \%$ at the time of the NuSTAR observations (the source flux only increases by ∼50% from Epoch 1 to 3 as shown in Figure 8). Therefore, our results are currently consistent with the disk still being truncated at a considerable level during the bright phase of the outburst ($L/{L}_{\mathrm{Edd}}\gg 1 \%$), although we stress that the exact value for the Eddington ratio is highly uncertain.

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We obtain an abundance of ${A}_{\mathrm{Fe}}\simeq 0.7$ for the accretion disk from the spectral fitting. Super-solar abundances of ${A}_{\mathrm{Fe}}\sim 3\mbox{--}5$ have been frequently required in the reflection modeling of Galactic black hole binaries: e.g., Cyg X–1 (Fürst et al. 2015; Parker et al. 2015; Fürst et al. 2016; Walton et al. 2016), GX 339–4 (Tomsick et al. 2014; García et al. 2015), and GRS 1739–278 (Miller et al. 2015). The lower elemental abundance for IGR J17091–3624 could result from the weak Fe K line/edge relative to the Compton hump.

A face-on geometry is favored for IGR J17091–3624 with a low inclination of $i\simeq 30^\circ \mbox{--}40^\circ$. A high inclination ($\gt 65^\circ$) can be excluded by all models at $\gt 90 \%$ confidence level (see Table 2), and in the best constrained case (Model 2b), this can be ruled out at $\gt 5\sigma$. A low inclination is in contrast with several previous estimations. Capitanio et al. (2012) proposed IGR J17091–3624 could be a highly inclined system, as the faintness of the source could be ascribed to the spectral deformation effects due to a high inclination angle. Rao & Vadawale (2012) performed a phase-resolved spectral fitting of the heartbeat oscillations of IGR J17091–3624 observed with RXTE and XMM-Newton with a multi-temperature disk blackbody and a power-law component, and their results favor a high inclination of ∼70°. In addition, high velocity outflows have been observed in IGR J17091–3624 by Chandra (King et al. 2012), that are usually expected from nearly edge-on systems. Suggestions of a high inclination for IGR J17091–3624 also come from its general similarities with GRS 1915+105, that has been measured to have an inclination angle of $\sim 65^\circ \mbox{--}75^\circ$ (e.g., Middleton et al. 2006; Miller et al. 2013).

A warped accretion disk could be one possible explanation for this discrepancy. All previous arguments for the high inclination are based on the soft state properties of IGR J17091–3624, while our measurements indicating a low inclination are from the hard state. From the standard diagram for black hole binaries, the inner disk radius is believed to move inward toward the ISCO during the transition from the hard state to the soft state, the inner edge of the accretion disk can appear to be viewed at different angles if the disk is strongly warped: the inner part is aligned with the black hole spin while the outer part is tilted, aligned with the binary orbital axis. This misalignment is believed to be driven by the precessional torque from the Lense–Thirring effect (Bardeen & Petterson 1975). Location of the transition of orientations can reach a steady state and is at ∼10–20 ${r}_{{\rm{g}}}$ as shown by recent simulations (Nealon et al. 2015), which is relevant to the scale of truncation radius discussed here. Therefore, the viewing angle 30°–40° we measured might be the orbital inclination or it might be somewhere in between the orbital and inner disk inclinations. Unfortunately, as we have not detected significant changes in the inner disk radius considering the error range, it is not possible to test this hypothesis via spectral fitting within the timespan of our observations.

Regarding the nature of the peak around 2.3 times the fundamental frequency, the basic question is whether it is harmonically related to the type-C QPO frequency or not. Harmonics of low-frequency QPOs have been discovered from a number of Galactic black hole binaries, and have been used as an additional probe to investigate the physical origins of QPOs (e.g., Axelsson et al. 2014; Ingram & van der Klis 2015; Axelsson & Done 2016). While the fundamental QPO and its harmonics have been found to display different behaviors (e.g., the energy dependence of phase lags and amplitudes), the ratio of the fundamental and first harmonic frequency has always been found at 1:2 (e.g., Rodriguez et al. 2002; Casella et al. 2005; Ingram & van der Klis 2015). However, based on the measurements from the full-band CPDS of IGR J17091–3624, the frequency ratio of the secondary peak and the type-C QPO ${\nu }_{1}/{\nu }_{0}$ is 2.37 ± 0.08, 2.28 ± 0.05, and 2.32 ± 0.02 for Epoch 1–3, respectively, making it inappropriate to associate the secondary frequency with the first QPO harmonic.

In the low-energy band (3–6 keV) CPDS, the secondary peak is found at the conventional value of two times the QPO frequency. We note that instead of being an apparent shift of the secondary frequency detected in the full band, it is possible that this is the real first harmonic. It is not unusual for the super-harmonic to be more prominent in the lower energy band, while it is undetected at higher energies, as the harmonic energy spectrum has been found to be systematically softer than that of the fundamental QPO. This is likely a result of the inhomogeneity of the Comptonizing region generating the QPOs (Axelsson et al. 2014; Axelsson & Done 2016). Also, the harmonic features are usually transient, which could explain why it is not detected in the soft band CPDS at Epoch 3.

The secondary frequency at about 2.3 times the type-C frequency is consistently detected in the 6–79 keV and full energy band, indicating that it is dominated by hard X-ray photons. One possibility is that higher energy photons are generated at a smaller distance from the black hole where the timescales are shorter, corresponding to a higher QPO frequency. Pairs of non-harmonically related QPOs have been simultaneously discovered during the 2005 outburst of GRO J1655–40, which were identified as a type-C QPO and a type-B QPO (Motta et al. 2012). However, the secondary frequency detected in IGR J17091–3624 is unlikely a type-B QPO, since type-B QPOs are normally detected at higher frequencies around ∼5–6 Hz and when the source is at a higher hardness ratio (e.g., Casella et al. 2004, 2005). Although GRO J1655–40 is not an exact analog to that discussed here, it indicates that there are multiple mechanisms generating the low-frequency QPOs in black hole binaries. Currently, there is no consensus about the nature of low-frequency QPOs. If the secondary frequency discussed here is indeed not harmonically related with the type-C QPO, its origin is rather uncertain but it should be an independent frequency that increases with the type-C QPO frequency.

The 2–10 keV flux of IGR J17091–3624 increased by ∼50% during the timespan of our three NuSTAR observations. We observe several parameters from the spectral and timing analysis to change systematically with the source flux (see Figure 8).

The evolution in the spectral fitting parameters is broadly similar to what has been found in the systematic study of GX 339-4 recently with RXTE (García et al. 2015). The ionization parameter increases with the source flux, which is naturally expected given the definition of the ionization parameter ξ.¹⁴ Regarding the two parameters governing the general curvature of the continuum, the photon index Γ increases whereas the electron temperature ${{kT}}_{{\rm{e}}}$ decreases with rising source flux. The slope of the thermal Comptonized continuum is described by an asymptotic power law with photon index Γ in the nthcomp model, which depends on the electron scattering optical depth ${\tau }_{{\rm{e}}}$ and the electron temperature ${{kT}}_{{\rm{e}}}$ (Lightman & Zdziarski 1987). Therefore, optical depth ${\tau }_{{\rm{e}}}$ can be estimated from Γ and ${{kT}}_{{\rm{e}}}$ ¹⁶ and we find ${\tau }_{{\rm{e}},\mathrm{Epoch}1}\simeq 1.95$, ${\tau }_{{\rm{e}},\mathrm{Epoch}2}\simeq 3.13$, ${\tau }_{{\rm{e}},\mathrm{Epoch}3}\simeq 3.51$. The measurements suggest the corona is cooling more efficiently by Compton up-scattering more photons as the source flux increases, which is also physically consistent with the increase in the electron opacity.

The type-C QPO frequency is known to be well correlated with the general changes in the energy spectra, e.g., photon index, strength of the thermal disk component (Remillard & McClintock 2006). One promising explanation for the low-frequency QPO is the Lense–Thirring precession of an inner hot flow of a truncated disk, which correlates the increase of the QPO frequency with the inward motion of the inner edge of the accretion disk (e.g., Ingram et al. 2009; Ingram & Done 2011). The model assigns the nodal precession frequency to the origin of type-C QPOs. In principle, it is possible to test the consistency of this QPO model using spectral-timing methods. We make a simple comparison of the QPO frequency ${\nu }_{0}$ and the inner disk radius ${r}_{\mathrm{in}}$ to the theoretical calculation of the nodal precession frequency ${\nu }_{\mathrm{nod}}$ from Ingram & Motta (2014). Since neither the black hole spin a nor mass m is known for the system, we calculate the theoretical curve for several combinations of these parameters (see Figure 9). However, the inner disk radius measured at the three epochs overlaps with each other within errors and no clear trend can be inferred from the three data points. It is clear from Figure 9 that in order to detect possible tension between the spectral and timing results in the frame of the QPO model, e.g., whether the inner disk radius changes in step with the QPO frequency, the inner disk radius must be constrained to be better than ∼5 ${r}_{{\rm{g}}}$.

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The anti-correlation of the reflection fraction with the source flux has also been observed and discussed in García et al. (2015), although we note that the reflection fraction is defined differently here. From the lamp-post geometry, the decrease of the reflection fraction in IGR J17091–3624 can be self-consistently explained by a reduction of the lamp-post height assuming the inner disk radius remains constant (as the case with Model 2b), which can still be valid if the inner disk is moving inward as predicted by the QPO model. This only works if the corona has a compact size compared to the level of disk truncation. In this scenario, because there is no inner disk to reflect the emission, a lower source height means that a larger fraction of the photons that would have been reflected by an inner disk reaching the ISCO are now lost in the gap between the disk and the horizon, reducing the reflection fraction. We note that the decrease of the reflection fraction could also be caused by the beaming effect of an outflowing corona (Beloborodov 1999; Malzac et al. 2001). However, it is difficult to explore this scenario with our limited S/N data, since additional parameters such as the corona velocity and geometric extent should be considered.