The Evolution of GX 339-4 in the Low-hard State as Seen by NuSTAR and Swift

GX 339–4 is a low-mass X-ray binary (LMXB) and an archetypical black hole transient that shows a high level of activity in optical, infrared, radio and X-rays, with more than a dozen outburst cycles (typically every 2–3 years) of different strengths since its first discovery in 1973 (Markert et al. 1973). The high flux it can achieve in the hard state and the recurrent outburst activity make GX 339–4 an ideal source to study the evolution of the accretion disk in the low-hard state. A recent near-infrared study in Heida et al. (2017) has shown a mass function of 1.91 ± 0.08 M_⊙, much less than previously claimed (5.8 ± 0.5 M_⊙, Hynes et al. 2003); the inclination angle of the system is 37° < i < 78° from optical analysis, and the black hole mass can be as small as 2.3 M_⊙ with 95% confidence.

The evolution of the accretion disk properties is an observational foundation essential to understanding the physics governing the outbursts of LMXB systems. A body of evidence has shown that when a black hole binary is in the soft state, the accretion disk extends to the innermost stable circular orbit (ISCO, e.g., Gierliński & Done 2004; Steiner et al. 2010). The standard paradigm for the low-hard state is that the disk's truncation radius grows as luminosity decreases, leaving an interior hot advection-dominated accretion flow (ADAF, Narayan & McClintock 2008) or other coronal flow (e.g., Ferreira et al. 2006). There is good evidence that at very low luminosities the disk is largely truncated (see Narayan & McClintock 2008 for a review). However, for luminosities in a moderate range of 0.1%–10% of the Eddington limit, the values of the reported inner edge of the disk (R_in) vary significantly, making this a hotly debated topic. There are two widely adopted methods to estimate R_in: the continuum-fitting method, focusing on the thermal emission of the disk; and the reflection spectroscopy (commonly called the iron-line method), which models the reflection component coming from the reprocessing of the Comptonized photons, illuminating the optically thick disk. In this paper, we make use of the latter, since our observations are in the low-hard state, where the hard continuum and the reflected components dominate the spectra.

The reflection spectrum is a rich mixture of radiative recombination continua, absorption edges, fluorescent lines (most notably the Fe K complex in the 6–8 keV energy range), and a Compton hump at energies >10 keV. This reflected radiation leaves the disk carrying information on the physical composition and condition of the matter in the strong gravitational field near the black hole. The fluorescent lines are broadened and shaped by Doppler effects, light bending, and gravitational redshift. Under the assumption that astrophysical black holes are Kerr black holes, the method can be used to measure the spin parameter a_* = cJ/GM² (−1 ≤ a_* ≤ 1), where J is the black hole spin angular momentum and M is the black hole mass. By estimating the radius of the inner edge of the accretion disk, so long as the inner radius corresponds to the radius of the ISCO, R_ISCO, which simply and monotonically maps to a_* (Hughes & Blandford 2003), we can measure the black hole spin. For the three canonical values of the spin parameter, a_* = +1, 0, and −1, R_ISCO = 1 M, 6 M, and 9 M (c = G = 1). Alternatively, by fixing the spin parameter to its maximal value in relxill (a_* = 0.998), one can estimate the maximal truncation of the inner radius of the disk.

The most advanced reflection model to date is relxill (Dauser et al. 2014; García et al. 2014a), which is based on the reflection code xillver (García & Kallman 2010; García et al. 2013), and the relativistic line-emission code relline (Dauser et al. 2010, 2013). The relxill model family has different flavors.⁹ In two of these, the modeling of the incident spectrum is done by either the standard power law with a high-energy cutoff in the form of an exponential rollover, or by the continuum produced by a thermal Comptonization model (nthComp, Zdziarski et al. 1996). The results presented in this paper are derived using relxillCp to model the relativistically blurred reflection component from the inner disk and xillverCp to model unblurred reflection from a distant reflector, both adopting the continuum produced by the nthComp model.

In the past 10 years, great effort has been devoted to estimate the inner edge of the accretion disk of GX 339–4 in the low-hard state with reflection spectroscopy, analyzing data from eight outburst cycles of GX 339–4 (2002, 2004, 2007, 2008, 2009, 2010–2011, 2013, 2015) obtained from X-ray missions, including XMM-Newton (Miller et al. 2006; Reis et al. 2008; Kolehmainen et al. 2013; Plant et al. 2015; Basak & Zdziarski 2016), Swift (Tomsick et al. 2008), Suzaku (Tomsick et al. 2009; Shidatsu et al. 2011; Petrucci et al. 2014), Rossi X-ray Timing Explorer (RXTE, García et al. 2015), and the Nuclear Spectroscopic Telescope Array (NuSTAR, Fürst et al. 2015).

Analyzing XMM-Newton data with reflection spectroscopy, Miller et al. (2006) presented for the first time strong evidence that the disk extended closely to the ISCO (R_in = 5 ± 0.5R_g) in the bright phase of the low-hard state (L/L_edd ∼ 5.4% assuming M_bh = 10 M_⊙ and D = 8 kpc), which was later confirmed by Reis et al. (2008) using the same XMM-Newton EPIC-MOS data taken in 2004. These results were challenged by Done & Diaz Trigo (2010), who reported that the iron-line profile appears much narrower in the XMM-Newton EPIC-pn data taken in timing mode (for the same observation), presumably because in this mode the pile-up is reduced. They obtained a much larger disk truncation (${R}_{\mathrm{in}}={60}_{-20}^{+40}{R}_{g}$). Other authors have also reported large disk truncation by analyzing the same EPIC-pn timing mode data: ${R}_{\mathrm{in}}\,={115}_{-35}^{+85}{R}_{g}$ (Kolehmainen et al. 2013), ${R}_{\mathrm{in}}={316}_{-74}^{+164}{R}_{g}$ (Plant et al. 2015), ${R}_{\mathrm{in}}={227}_{-84}^{+211}{R}_{g}$, and ${144}_{-96}^{+107}{R}_{g}$ separating the two revolutions (Basak & Zdziarski 2016). Nevertheless, Miller et al. (2010) argued that pile-up can still affect the timing mode and that if not corrected it can artificially make the continuum softer, which in turn will result in a narrower Fe K profile, leading to false estimates of large truncation. The discussion centered around pile-up effects suggest that it is a complicated instrumental issue for X-ray charge-coupled devices, for which we still do not have a complete model.

García et al. (2015) have independently analyzed the RXTE/PCA data tracking the evolution of GX 339–4 in the hard state with the luminosity ranging from 17% to 2% of the Eddington luminosity. Although the PCA data do not have problems with photon pile-up, and has archived extremely high signal-to-noise ratio and low systematic uncertainty by implementing the PCACORR tool (García et al. 2014b), it is limited by its relatively low spectral resolution to study the iron-line complex. With the most recently available data from NuSTAR (which is also free from pile-up), we can now extend the luminosity range down to 0.5% L_edd, to see the evolution of the accretion disk's truncation and other conditions in the system.

In this paper, we focus on Swift and NuSTAR to sidestep pile-up issues noting that there is some disagreement between the XMM-Newton and NuSTAR spectra. For example, in the recent analysis presented by Stiele & Kong (2017), the NuSTAR spectra can only be used down to 4 keV (see Figure 7 therein) due to this discrepancy. Thus, since the combination of XMM-Newton and NuSTAR observations seems to require special treatment, a detailed analysis of such data will be presented in a future publication.

This paper is organized as follows. Section 2 describes the observations and data reduction, Section 3 provides the details of our spectral fitting. We present our discussion in Section 4, and summarize the results in Section 5.

In 2015 August, X-ray monitoring detected the end of a new outburst of GX 339–4 and triggered observations with the NuSTAR (Harrison et al. 2013), Swift (Gehrels et al. 2004), and XMM-Newton (Jansen et al. 2001). We obtained six observations with NuSTAR at the end of the outburst, and for each a corresponding Swift snapshot within a day of the start time of NuSTAR (Figure 1).

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We also analyzed the data set from 2013, which was triggered by the detection of the onset of a new outburst. In this campaign, five observations were taken with NuSTAR, four during the rise and one during the decay of the outburst, and Swift observations every other day. However, the 2013 outburst was a failed outburst because the source did not follow the standard outburst pattern in the hardness-intensity diagram. The source remained in the low-hard state, and never switched to the high-soft state (Fürst et al. 2015). Table 1 provides a detailed observation log of NuSTAR and the matching Swift observations.

2.1. NuSTAR

2.2. Swift

3.1. The 2015 Data Set: During Decay in the Hard State

3.1.1. Model 1: The Standard Reflection Model

After fitting with an absorbed power law (i.e., tbabs∗powerlaw) in the 3–8 keV range with a fixed column density N_H = 5 × 10²¹ cm⁻², we can see from the the data-to-model ratio (Figure 2) that a disk component at $\lesssim 1\mbox{--}2\,\mathrm{keV}$ is present except for the last observation, and the iron line and Compton hump are clearly visible in all observations. Note that the total number of counts in Swift drops dramatically from ∼32,000 counts (observation 3) to ∼4900 counts (observation 4), ∼5300 counts (observation 5) and ∼1500 counts (observation 6), so the statistical precision for the last three observations is relatively poor.

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We perform a simultaneous fit on all six observations from 2015 using a more sophisticated model: const∗Tbabs∗(diskbb+nthComp+relxillCp+xillverCp) (2015-M1), where relxillCp models the relativistic reflection component and xillverCp represents the unblurred reflection coming from a distant reflection that could be wind or the outer region of a flared disk. The multi-color blackbody emission from the accretion disk is included via diskbb, and the Comptonization of the disk emission coming from the corona via nthComp. During the fit, we tie several global parameters that are expected to be unchanged during the time range for our observations (∼a month) including the column density N_H, the inclination angle i, and the iron abundance A_Fe. The spin parameter a_* is fixed at its maximal allowed value of 0.998, while the inner radius is left free to vary, so that R_in can be fully explored. The constants are introduced as cross-calibration factors, and thus are frozen at 1.0 for FPMA, tied together for all FPMB spectra but allowed to vary for XRT to account for the possible differences in the flux levels since these observations are not strictly simultaneous. The reflection fractions for the blurred and unblurred reflection components are frozen at R_f = −1, their iron abundances are tied, and the ionization parameter is fixed at $\mathrm{log}\xi =0$ in xillverCp as the gas in the distant reflector is expected to be cold and neutral (following García et al. 2015). The seed photon temperature kT_bb in nthcomp is tied with the temperature at inner disk radius kT_in in diskbb. If not specified, we use a canonical emissivity profile of ∝r⁻³ (i.e., emissivity index q = 3).

The resulting ratio is shown in Figure 3 (left), the best-fit parameter values in Table 2 and the model components in Figure 4 (left). As we expect from the dramatic drop in count number for the last three observations, the Swift data cannot provide solid constraints on the intrinsic disk emission. However, we do obtain a decreasing trend in the disk temperature and the flux ratio between 2 and 20 keV of the disk component and the unabsorbed total one, except for the last observation, which has a physically unreasonable high disk temperature ${0.80}_{-0.10}^{+0.04}$ keV. The truncation of the inner disk and the decrease in R_in with increasing luminosity is a prediction of the standard paradigm for the faint hard state that a hot ADAF or other coronal flow appears when the inner edge of the disk recedes from the ISCO (Narayan & Yi 1994; Esin et al. 1997). In our best fit, we observe that during the decay, values of R_in are all between 3 and 15 R_g, with a tentative increase toward the end of the outburst. To test the statistical significance of this tentative variation, we perform another fit in which the inner radii, except for the last one, are tied together, and we find ${R}_{\mathrm{in},1-5}={1.6}_{-0.3}^{+0.4}$ R_g, and ${R}_{\mathrm{in},6}={12.2}_{-7.7}^{+8.4}$, with C-stat increasing by 8 and χ² increasing by 21 for 4 extra dof. This test suggests that the crucial value of R_in,6 determining the evolution with regard to the luminosity is not statistically significant. We also find the spectrum becomes harder with the photon index dropping from 1.72 to 1.62 when the luminosity decreases, while the ionization parameter in relxillCp is reduced from ξ ≃ 2200 to ξ ≃ 900 erg cm s⁻¹.

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Table 2.  Best-fit Parameter Values of Model const∗Tbabs∗(diskbb+nthComp+relxillCp+xillverCp) in a Simultaneous Fit for the 2015 Data Set (2015-M1)

Parameter	Obs.1	Obs.2	Obs.3	Obs.4	Obs.5	Obs.6
NH (1021 cm−2)	4.12+0.08−0.12
a*	0.998
i (deg)	${39.2}_{-1.8}^{+2.0}$
AFe	8.2 ± 1.0
CFPMA	1
CFPMB	1.015 ± 0.002
Γ	${1.724}_{-0.009}^{+0.011}$	${1.667}_{-0.009}^{+0.012}$	${1.628}_{-0.014}^{+0.013}$	${1.646}_{-0.019}^{+0.008}$	${1.605}_{-0.010}^{+0.009}$	${1.624}_{-0.013}^{+0.007}$
kTe (keV)	>195	>224	>95	>67	>152	46+35−8
kTin (keV)	${0.46}_{-0.01}^{+0.03}$	${0.30}_{-0.06}^{+0.03}$	${0.45}_{-0.05}^{+0.02}$	${0.36}_{-0.04}^{+0.08}$	${0.058}_{-0.006}^{+0.023}$	${0.80}_{-0.10}^{+0.04}$
Rin(RISCO)	2.5 ± 0.6	<2.3	<2.2	<2.4	${3.5}_{-0.9}^{+1.4}$	${12.4}_{-7.5}^{+8.4}$
logξ	${3.34}_{-0.02}^{+0.04}$	${3.11}_{-0.05}^{+0.06}$	${3.12}_{-0.07}^{+0.16}$	${3.02}_{-0.27}^{+0.02}$	${3.08}_{-0.06}^{+0.07}$	${2.95}_{-0.49}^{+0.15}$
Ndisk	${170}_{-36}^{+41}$	${58}_{-12}^{+32}$	${30}_{-8}^{+12}$	<31	>3 × 104	<1.1
NnthComp	${0.103}_{-0.003}^{+0.001}$	${0.107}_{-0.006}^{+0.003}$	${0.072}_{-0.003}^{+0.002}$	${0.066}_{-0.007}^{+0.001}$	0.049 ± 0.001	${0.017}_{-0.001}^{+0.002}$
NrelxillCp(10−3)	${1.15}_{-0.09}^{+0.16}$	${1.16}_{-0.14}^{+0.24}$	${0.84}_{-0.10}^{+0.18}$	${0.60}_{-0.16}^{+0.25}$	0.32 ± 0.07	${0.08}_{-0.02}^{+0.03}$
NxillverCp(10−5)	<12	<10	${7.2}_{-7.0}^{+7.6}$	${7.2}_{-5.1}^{+6.2}$	${9.1}_{-4.6}^{+5.0}$	${3.1}_{-2.2}^{+2.1}$
CXRT	1.017 ± 0.017	1.027 ± 0.012	1.086 ± 0.017	1.06 ± 0.03	1.044 ± 0.025	0.88 ± 0.04
L/Ledd (%)	2.0	1.8	1.7	1.2	1.0	0.5
Fdisk/Funabsorbed (%)	2.0	0.8	0.4	0.005	0	1.5
Rs	0.22	0.18	0.15	0.13	0.10	0.04
C-stat	10800
χ2/dof	12077/10730 = 1.126

Note. Luminosity calculated using unabsorbed flux between 0.1 and 300 keV, assuming a distance of 8 kpc and a black hole mass of 10 M_⊙. The flux ratio of disk emission and the total unabsorbed one is calculated in the 2–20 keV range. The reflection strength R_s is determined from the flux ratio between relxillCp and nthComp in the energy range of 20–40 keV.

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We also tried other emissivity profiles:

• 1.  
  Free emissivity index q₁ within the breaking radius R_br free, and a fixed outer emissivity index q₂ = 3. We find that q₁ is between the values of 3 and 4, R_br could not be constrained, and the other parameters were insignificantly affected, with the C-stat decreasing by only ∼23 for 12 fewer degrees of freedom.
• 2.  
  Free emissivity index q₁ = q₂ all over the disk. We again find that q falls between 3 and 4, the other parameters were insignificantly affected, with the C-stat decreasing by only ∼9 for 6 fewer degrees of freedom.
• 3.  
  Lamppost geometry. The fit is statistically worse by a ΔC-stat = 76 for 6 fewer degrees of freedom. The corona height was found to be fairly large (10–20R_g) and poorly constrained.

We notice the large iron overabundance in our fits: 8.2 ± 1.0 in solar units. To show the data prefers the overabundance, we fix the iron abundance to be the solar value for this data set (2015-M1-AFe1), and find the C-stat increases by 791, for one additional degree of freedom. The disk becomes more truncated, especially for Obs.5, in which the value of R_in increases from ${4.3}_{-1.1}^{+1.7}{R}_{g}$ to >172R_g (see Table 7 for the best-fit parameters). To interpret this, and following the procedure in Section 6.1.4 in García et al. (2015), we plot the model components nthcomp+relxillCp for these two cases in Figure 5, which shows that it could be difficult to distinguish a case with a solar iron abundance and a disk truncated at hundreds of R_g from the case of iron overabundance and mild truncation, without good quality data covering the oxygen emission line below 0.7 keV and the Compton hump above 20 keV. Because of the low S/N of the Swift data, we cannot probe the oxygen line. However, with NuSTAR's wide energy coverage up to 79 keV, we can see evidence of discrepancy above ∼30 keV when the iron abundance is fixed at the solar value, as shown in Figure 3. This demonstrates the preference of these data to require large iron abundance.

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3.1.2. Model 2: Taking the Comptonization of Reflection into Account

The presence of a corona as the source of the hard photons in the continuum suggests the possibility for some of the reflected photons to intercept such a corona before they reach the observer. This will result in additional Compton scattering of some fraction of the reflection spectrum. As a first-order adjustment, we can convolve the reflection spectrum with a Compton-scattering kernel. For this, we use the model simplcut,¹⁰ which adopts a scattering kernel based upon nthComp (Zdziarski et al. 1996). It has four physical parameters: the scattered fraction f_sc, the spectral index Γ, the electron temperature kT_e, and the reflection fraction R_f. We follow the procedures in Steiner et al. (2017), but we do not implement any linking between the diskbb parameters in the hard and soft states. In xspec notation, the model we adopt is constant∗Tbabs∗[simplcut∗(diskbb+relxillCp)+xillverCp] (2015-M2).

Here, in applying simplcut in this way we are assuming that the fraction of disk photons that are up-scattered in the corona is the same as the fraction of reflected photons also intercepted by the corona, as they are governed by one single scattering fraction. The best-fit parameters are shown in Table 3. For the last observation with the lowest luminosity, the fit is consistent with the whole range of inner radii, 1.5–800 R_g, at the 90% confidence level. This might be due to the fact that the scattering fraction is so large (>0.97) that the reflection features, including the iron line, are heavily diluted, while the unblurred reflection component xillverCp can compensate for the iron emission seen in the spectrum with a small ionization parameter (logξ < 2.36). The iron-line profile becomes difficult to determine and thus, the inner edge of the disk is unconstrained. Also, the disk component is not evident in the data.

Table 3.  Best-fit Parameter Values of Model const∗Tbabs∗[simplcut∗(diskbb+relxillCp)+xillverCp] in the Simultaneous Fit Performed on the 2015 Outburst Data Set (2015-M2)

Parameter	Obs.1	Obs.2	Obs.3	Obs.4	Obs.5	Obs.6
NH (1021 cm−2)	${4.43}_{-0.06}^{+0.12}$
a*	0.998
i (deg)	${39.2}_{-1.5}^{+1.6}$
AFe	${7.7}_{-0.9}^{+1.0}$
CFPMA	1
CFPMB	1.0148 ± 0.0018
Γ	${1.781}_{-0.008}^{+0.009}$	1.717 ± 0.008	1.663 ± 0.007	${1.663}_{-0.007}^{+0.040}$	${1.635}_{-0.006}^{+0.004}$	${1.654}_{-0.032}^{+0.023}$
fsc	0.51 ± 0.02	0.64 ± 0.02	${0.68}_{-0.03}^{+0.04}$	${0.66}_{-0.08}^{+0.06}$	${0.45}_{-0.05}^{+0.08}$	>0.97
kTe (keV)	>196	>100	>66	${67}_{-18}^{+47}$	>117	${46}_{-12}^{+111}$
kTin (keV)	0.51 ± 0.03	${0.66}_{-0.12}^{+0.11}$	>0.57	<0.13	${0.110}_{-0.002}^{+0.018}$	>0.78
Ndisk	${227}_{-26}^{+74}$	${26}_{-8}^{+55}$	${8}_{-2}^{+11}$	$({4.8}_{-4.5}^{+1.8})\times {10}^{4}$	>7.1 × 104	${20}_{-1}^{+8}$
Rin(RISCO)	<1.9	${1.8}_{-0.6}^{+3.0}$	<1.9	<2.1	${5.0}_{-1.4}^{+2.7}$	⋯
logξ	${3.29}_{-0.06}^{+0.04}$	${3.07}_{-0.05}^{+0.07}$	${3.17}_{-0.07}^{+0.18}$	3.04 ± 0.05	${2.42}_{-0.29}^{+0.40}$	<2.36
NrelxillCp(10−3)	${2.7}_{-0.4}^{+0.2}$	${2.8}_{-0.5}^{+0.3}$	2.0 ± 0.4	${1.2}_{-0.5}^{+0.2}$	${0.59}_{-0.11}^{+0.15}$	<0.64
NxillverCp(10−5)	<8.3	<7.3	${6.5}_{-5.1}^{+4.2}$	${8.0}_{-4.3}^{+4.4}$	${8.5}_{-4.3}^{+4.4}$	6.8 ± 1.9
CXRT	${1.018}_{-0.015}^{+0.016}$	1.007 ± 0.016	${1.091}_{-0.007}^{+0.013}$	1.038 ± 0.025	${1.042}_{-0.025}^{+0.028}$	0.94 ± 0.04
L/Ledd (%)	2.0	1.8	1.7	1.2	1.0	0.5
Rs	1.87	4.16	4.13	0.83	0.21	0.18
C-stat	10822
χ2/dof	12067/10730 = 1.125

Note. Luminosity calculated using unabsorbed flux between 0.1 and 300 keV, assuming a distance of 8 kpc and a black hole mass of 10 M_⊙. The reflection strength R_s is determined from the flux ratio between relxillCp and nthComp in the energy range of 20–40 keV.

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In this framework of coronal Comptonization, there are several model components: the power-law continuum, the intrinsic disk emission, the relativistic reflection, and the Comptonized reflection. Besides the overall normalization, only two parameters determine the relative strength of each component: the scattered fraction f_sc, and the reflection fraction R_f. The former depends on the geometry of the disk-corona system and also the optical depth in the corona; while the latter is only associated with the geometry of the system. We find that f_sc increases when the luminosity decreases (see Table 3). This could be explained by changes in the corona structure. Figure 4 (right) shows how the model components change through observations. We calculated the reflection strength as defined in Dauser et al. (2016), and find that except for observation 1, the other five observations show a decreasing trend from ∼4 to ∼0.2, which is in line with the increasing inner radius of the accretion disk.

3.2. The 2013 Data Set: Rise and Decay in a Failed Outburst

The absorbed power-law fit on the 2013 data do not show any strong indication of the existence of a soft disk component, thus we started the fit by fixing the disk temperature to 0.05 keV. However, with free disk temperatures, the fit goes down in C-stat by 725 with 10 less d.o.f., which is a significant improvement. The flux ratio in the 2–20 keV range between the intrinsic disk emission and the unabsorbed total one is around 3%, which matches the expected faint disk in the low-hard state, but the determined disk temperatures are above 0.8 keV for the last three observations. In addition, the inner edge of the disk does not follow a one-way trend with luminosity. The best-fit parameters for this model (2013-M1) are shown in Table 4.

Table 4.  Best-fit Parameter Values of Model const∗Tbabs∗(diskbb+nthComp+relxillCp+xillverCp) in a Simultaneous Fit Performed on the 2013 Outburst Data Set (2013-M1)

Parameter	Obs.1	Obs.2	Obs.3	Obs.4	Obs.5
NH (1021 cm−2)	${4.12}_{-0.18}^{+0.06}$
a*	0.998
i (deg)	${40.7}_{-0.8}^{+0.7}$
AFe	3.83 ± 0.06
CFPMA	1
CFPMB	1.0219 ± 0.0009
Γ	1.56 ± 0.02	1.585 ± 0.001	1.606 ± 0.001	1.54 ± 0.02	1.616 ± 0.001
kTe (keV)	>473	${231}_{-21}^{+38}$	>540	>620	>497
kTin (keV)	0.422 ± 0.002	${0.53}_{-0.02}^{+0.08}$	0.892 ± 0.002	0.796 ± 0.001	0.80 ± 0.17
Rin(RISCO)	<1.5	3.9 ± 0.8	${14.0}_{-3.1}^{+3.5}$	${10.0}_{-1.5}^{+1.6}$	${32.3}_{-10.9}^{+17.2}$
logξ	${0.70}_{-0.06}^{+0.07}$	${1.01}_{-0.06}^{+0.03}$	${1.69}_{-0.49}^{+0.03}$	${1.54}_{-0.13}^{+0.04}$	${2.97}_{-0.08}^{+0.04}$
Ndisk	3.4 ± 0.2	⋯	7.4 ± 0.3	16.8 ± 0.5	1.05 ± 0.05
NnthComp	0.054 ± 0.019	${0.071}_{-0.012}^{+0.014}$	0.0920 ± 0.0001	0.1351 ± 0.0001	0.02039 ± 0.00003
NrelxillCp(10−3)	0.63 ± 0.03	1.5 ± 0.2	2.0 ± 0.4	2.7 ± 0.6	0.23 ± 0.03
NxillverCp(10−4)	2.7 ± 0.3	2.8 ± 0.4	5.0 ± 0.5	6.6 ± 0.4	1.1 ± 0.1
CXRT	1.057 ± 0.025	${1.174}_{-0.014}^{+0.015}$	0.982 ± 0.015	1.039 ± 0.010	1.074 ± 0.010
L/Ledd (%)	1.4	2.4	3.6	4.6	0.8
Fdisk/Funabsorbed (%)	2.7	0.1	3.4	2.8	2.3
Rs	0.12	0.21	0.15	0.19	0.09
C-stat	9556
χ2/dof	10253/9336 = 1.098

Note. Luminosity calculated using unabsorbed flux between 0.1 and 300 keV, assuming a distance of 8 kpc and a black hole mass of 10 M_⊙. The flux ratio of disk emission and the total unabsorbed one is calculated in the 2–20 keV range. The reflection strength R_s is determined from the flux ratio between relxillCp and nthComp in the energy range of 20–40 keV.

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We then try the model taking the Comptonization of reflection into account in this data set (2013-M2), following the same procedures as in Section 3.1.2. Compared to 2013-M1, C-stat increases by 224 with the same dof, which is statistically worse; but we also notice that M2 reduced χ² by 7. Additionally, this model provides a more reasonable combination of disk and power-law components. As shown in Table 5, the disk temperatures fall into a range of values closer to the expectation for this source (${{kT}}_{\mathrm{in}}\lesssim 0.2$ keV). In Figure 6 (right), the intrinsic disk flux becomes much smaller which is more typical for the low-hard state.

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Table 5.  Best-fit Parameter Values of Model const∗Tbabs∗[simplcut∗(diskbb+relxillCp)+xillverCp] in a Simultaneous Fit Performed on the 2013 Outburst Data Set (2013-M2)

Parameter	Obs.1	Obs.2	Obs.3	Obs.4	Obs.5
NH (1021 cm−2)	${6.85}_{-0.09}^{+0.10}$
a*	0.998
i (deg)	${39.7}_{-2.0}^{+2.6}$
AFe	${2.82}_{-0.15}^{+0.17}$
CFPMA	1
CFPMB	1.0219 ± 0.0012
Γ	${1.640}_{-0.010}^{+0.011}$	${1.635}_{-0.009}^{+0.010}$	1.676 ± 0.006	1.705 ± 0.007	${1.626}_{-0.004}^{+0.005}$
fsc	${0.78}_{-0.07}^{+0.06}$	0.69 ± 0.06	0.79 ± 0.03	${0.78}_{-0.03}^{+0.02}$	${0.31}_{-0.01}^{+0.05}$
kTe (keV)	>148	>159	>272	>142	>210
kTin (keV)	${0.130}_{-0.024}^{+0.011}$	${0.130}_{-0.009}^{+0.006}$	${0.204}_{-0.020}^{+0.008}$	${0.156}_{-0.028}^{+0.006}$	0.116 ± 0.002
Ndisk(104)	<2.0	${8.8}_{-1.0}^{+0.3}$	${1.6}_{-1.3}^{+3.2}$	${8.1}_{-0.7}^{+0.6}$	>9.4
Rin(RISCO)	>11.4	${4.4}_{-1.0}^{+1.7}$	${14.3}_{-6.1}^{+7.4}$	${12.6}_{-3.6}^{+4.7}$	${15.6}_{-5.9}^{+14.7}$
logξ	2.69 ± 0.02	<1.81	1.76+0.25−0.34	${2.00}_{-0.12}^{+0.02}$	$\lt 1.78$
NrelxillCp(10−3)	${2.5}_{-0.9}^{+0.2}$	${3.0}_{-0.6}^{+0.3}$	${6.1}_{-0.6}^{+0.7}$	${9.2}_{-0.5}^{+0.4}$	${0.33}_{-0.09}^{+0.27}$
NxillverCp(10−4)	<2.2	${3.0}_{-1.6}^{+1.3}$	${3.9}_{-1.8}^{+1.7}$	${5.3}_{-1.5}^{+1.4}$	1.5 ± 0.5
CXRT	1.025 ± 0.024	1.140 ± 0.015	${0.947}_{-0.015}^{+0.016}$	${1.012}_{-0.011}^{+0.012}$	${1.028}_{-0.012}^{+0.011}$
L/Ledd (%)	1.4	2.4	3.6	4.6	0.8
Rs	1.99	0.50	0.74	0.99	0.14
C-stat	9780
χ2/dof	10246/9336 = 1.097

Note. Luminosity calculated using unabsorbed flux between 0.1 and 300 keV, assuming a distance of 8 kpc and a black hole mass of 10 M_⊙. The reflection strength R_s is determined from the flux ratio between relxillCp and nthComp in the energy range of 20–40 keV.

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The parameters that are global to all observations are the Galactic hydrogen column density N_H, the spin parameter a_*, the inclination angle i and the iron abundance A_Fe. Table 6 shows a summary of these intrinsic parameter values found in different simultaneous fits performed in this paper. The inclination is consistent with i = 40° ± 2° through all fits except for 2015-M1-AFe1. Assuming that the inclination of the inner disk is equal to the binary orbit inclination, with the latest measurement of the mass function $f(M)=\tfrac{{M}_{\mathrm{bh}}{\sin }^{3}i}{{(1+q)}^{2}}=1.91\pm 0.08$ M_⊙ and $q\,=\tfrac{{M}_{c}}{{M}_{\mathrm{bh}}}=0.18\pm 0.05$ (Heida et al. 2017), we estimate the mass of the black hole to be M_bh = 10.0 ± 0.6 M_⊙.

Table 6.  The Intrinsic Parameters of the System Found in Different Simultaneous Fits in This Paper

Fit	Model Desciption	C-stat	χ2/dof	NH	i	AFe
				(1021 cm−2)	(deg)
2015-M1	Standard reflection model	10800	12077/10730	${4.12}_{-0.12}^{+0.08}$	${39.2}_{-1.8}^{+2.0}$	8.2 ± 1.0
	(diskbb+nthcomp+relxillCp+xillverCp)		=1.126
2015-M1-AFe1	Standard reflection model,	11591	12584/10731	${4.53}_{-0.05}^{+0.04}$	75 ± 5	1.0
	AFe = 1.0		=1.173
2015-M2	Model considering the coronal Comptonization	10822	12067/10730	${4.43}_{-0.06}^{+0.12}$	${39.7}_{-2.0}^{+2.6}$	${7.7}_{-0.9}^{+1.0}$
	[simplcut∗(diskbb+relxillCp)+xillverCp]		1.125
2013-M1	Standard reflection model	9556	10253/9336	${4.12}_{-0.18}^{+0.06}$	${40.7}_{-0.8}^{+0.7}$	3.83 ± 0.06
			=1.098
2013-M2	Model considering the coronal Comptonization	9780	10246/9336	${6.85}_{-0.09}^{+0.10}$	${39.7}_{-2.0}^{+2.6}$	${2.82}_{-0.15}^{+0.17}$
			=1.097

Note. Parameters include hydrogen column density (N_H), the dimensionless spin parameter a_* = 0.998, which is frozen in all, the inclination of the inner disk i, the iron abundance with respect to the solar value A_Fe. The model description, C-Stat, and χ² values are also provided.

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Fürst et al. (2015) found the disk to be truncated at tens of R_g, based on the same NuSTAR and Swift data set of the 2013 outburst, using the model constant∗tbabs∗[powerlaw+relconv(reflionx)+ Gaussian], which includes the older reflection model reflionx (Ross & Fabian 2005), convolved with the relativistic kernel relconv (Dauser et al. 2010). By comparing the simplcut∗relxillCp with the relxillCp models shown in Figures 4 and 6 (right), the slope of the reflection component is reduced as a pure consequence of coronal scattering. This could potentially explain the results found in Fürst et al. (2015). After allowing a difference between the photon index feeding the reflection (∼1.3) and the one in the power-law continuum (∼1.6) to account for a possible physically extended corona with a nonuniform temperature profile, they found that the iron abundance was also reduced (from ∼5 to ∼1.5), and thus, forces the disk to be much more truncated to minimize the relativistic effects that blur the line profile. Nevertheless, in our case, M2 only provides a significant reduction in the iron abundance compared to M1 for the 2013 data.

We do not observe a clear evolution for a decrease of disk temperature with decreasing luminosity, as another prediction in the truncation disk scenario. The reasons for this are threefold. First, the Swift data have the total numbers of counts much smaller than NuSTAR (10–100 times smaller), which makes the determination of disk temperatures governed by the low energy range very difficult. Second, the large disk temperatures we find with M1 could be artificially produced by the complexity of the Comptonization model (Kolehmainen et al. 2013). In the frequency resolved spectra, the most rapidly variable part of the flow has harder spectra and less reflection than the slowly variable emission (Axelsson et al. 2013). This feature would give rise to spectral curvature in broadband data (as seen in, e.g., Makishima et al. 2008), and thus, requires an additional soft component when such a continuum is fitted with a single Comptonization component. Lastly, as we do not observe a strong evolution pattern of the disk's inner radius with luminosity, it is understandable that the disk temperature does not evolve as expected either.

The evolution of the inner disk radius changing with respect to the luminosity we find in different models, and those reported by García et al. (2015) and Petrucci et al. (2014) are shown in Figure 7. For a detailed summary of estimations of R_in in previous literature for GX 339–4 between a luminosity range of 0.1%–20% L_edd in low-hard state obtained from the reflection spectroscopy, see Table 5 in García et al. (2015).

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Among all the fits we performed, 2015-M1 shows the most promising decreasing trend of R_in with increasing luminosity. However, this result is not statistically significant, as we suggested in Section 3.1.1. By comparing the trends M1 and M2 give for the 2015 data set (see the upper panel in Figure 7), except for the one missing data point in M2 where R_in is unconstrained, the other five values agree well with each other, suggesting a consistent and model-robust conclusion.

Another interesting aspect to notice is that in the luminosity range covered by the two data sets, the values of R_in found for the 2013 observations is slightly larger. This could be due to the fact that the 2013 observations were taken in the rising phase (obs.1–4), and at the end of a failed outburst (obs.5), while the 2015 data was taken during the decay of a successful one. The hysterisis pattern typically observed in the hardness-intensity diagram of this source suggests that the evolution during the rising and decay phases displays a different phenomenology, which is likely to affect the evolution of the inner radius.

The evolution of R_in with luminosity in the low-hard state is a matter of central importance for the study of black hole binaries. As our results are limited by the relatively small luminosity range we explore, we plot the reported results in previous literature and our preferred ones (2015-M1 and 2013-M2) of inner radius versus Eddington-scaled luminosity in Figure 8, sorted and colored with regard to satellites, instruments, and observation mode. At luminosities larger than 1% L_edd, there are two groups of results: an upper group with inner radii between 20R_g and 800R_g comprised by values from XMM-Newton pn timing mode and two imaging mode data; and a bottom group with R_in < 20R_g aligned with NuSTAR, RXTE, Suzaku, Swift, XMM-Newton MOS, and one set of XMM-Newton pn imaging mode data. These results indicate the possibility of calibration issues with XMM-Newton pn timing mode data as the main factor responsible for the very extreme truncation.

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We have analyzed 11 observations of GX 339–4 in the low-hard state seen by NuSTAR and Swift; 5 were taken in a failed outburst in 2013 and the other 6 during the decay of the 2015 outburst. The luminosity covers the range of 0.5% to 5% L_edd, which only covers a fraction of the usual luminosity range typically observed during the outburst for this source (up to 20%–30% L_edd). Each spectrum spans the energy range 3–79 keV from NuSTAR, and 0.5–8 keV from Swift. The data have in total 10.7 million counts, and a composed exposure time of 790 ks.

Both data sets are fitted with two models: a standard reflection model including intrinsic disk emission, power-law continuum, and both the relativistic and unblurred reflection components const∗Tbabs∗(diskbb+nthComp+relxillCp+xillverCp) (M1); and a model in which the reflection component is Comptonized by the corona constant∗Tbabs∗[simplcut∗(diskbb+relxillCp) +xillverCp].

During the decay in 2015, with fit M1, we find that the inner disk recedes from the ISCO, values of R_in are all between 3 and 15 R_g, with a tentative increase toward the end of the outburst, although we do notice that the largest truncation radius here is not statistical significant. Fit M2 provides similar results, except for the last observation whose inner radius is unconstrained. As for the 2013 data set, the disk temperatures determined from M1 are unphysically large for these luminosities in the low-hard state, while M2 can effectively reconcile these values (${{kT}}_{\mathrm{in}}\lesssim 0.2$ keV) and provide more physical trends. The evolution of R_in with luminosity for the 2013 data is somewhat less monotonic than for the 2015 data, and while the inner radius is larger in the former, we find the largest disk truncation is constrained to be less than 37R_g when the source is at 0.8% L_edd.

Part of this work was carried out by JW during attendance to the Summer Undergraduate Research Fellowship (SURF) at California Institute of Technology in 2017. Warm hospitality and tutelage are kindly acknowledged, in particular from the members of the NuSTAR Science Operations Team (K. Foster, B. Grenfestette, K. Madsen, M. Heida, M. Brightman, and D. Stern). We would like to thank the referee for useful comments toward the improvement of this paper. We also thank B. de Marco and G. Ponti for useful discussions. J.A.G. acknowledges support from NASA grants NNX17AJ65G and 80NSSC177K0515, and from the Alexander von Humboldt Foundation. J.F.S. has been supported by NASA Hubble Fellowship grant HST-HF-51315.01. S.C. is supported by the SERB National Postdoctoral Fellowship (No. PDF/2017/000841). We thank the NuSTAR Operations, Software, and Calibration teams for support with the execution and analysis of these observations.

Table 7 shows the best-fit parameters when we fix the iron abundance to be the solar value for the 2015 observations (2015-M1-AFe1). The disk becomes more truncated, especially for Obs.5, in which the value of R_in increases from ${4.3}_{-1.1}^{+1.7}{R}_{g}$ to >172R_g. However, the fit is significantly worse in statistics with regard to 2015-M1, with C-stat increasing by 791 for one additional degree of freedom. In addition, with NuSTAR's wide energy coverage of up to 79 keV, we can see evidence of discrepancy above ∼30 keV, as shown in Figure 3. This demonstrates the preference of these data to require large iron abundances, and a systematic discussion about the iron overabundance found by reflection spectroscopy will be presented in a future publication.

Table 7.  Best-fit Parameter Values of Model const∗Tbabs∗(diskbb+nthComp+relxillCp+xillverCp) with a Frozen Iron Abundance at the Solar Value in the Simultaneous Fit for the 2015 Data Set (2015-M1-AFe1)

Parameter	Obs.1	Obs.2	Obs.3	Obs.4	Obs.5	Obs.6
NH (1021 cm−2)	${4.53}_{-0.05}^{+0.04}$
a*	0.998
i (deg)	75.0 ± 5.0
AFe	1.0
CFPMA	1
CFPMB	${1.015}_{-0.002}^{+0.004}$
Γ	1.767 ± 0.002	${1.70}_{-0.05}^{+0.05}$	${1.665}_{-0.010}^{+0.006}$	${1.665}_{-0.010}^{+0.007}$	1.637 ± 0.001	1.653 ± 0.002
kTe (keV)	>388	>381	>308	>250	>241	${182}_{-31}^{+42}$
kTin (keV)	<0.06	<0.13	${0.31}_{-0.11}^{+0.08}$	${0.34}_{-0.01}^{+0.23}$	${0.059}_{-0.006}^{+0.003}$	0.752 ± 0.003
Ndisk	⋯	⋯	<86	<38	>3.2 × 106	<1.08
Rin(Rg)	<1.4	${25.2}_{-8.8}^{+4.8}$	>18.7	${36.1}_{-6.3}^{+9.5}$	>172	${55.5}_{-17.6}^{+68.2}$
logξ	3.321 ± 0.001	${3.22}_{-0.07}^{+0.05}$	${3.20}_{-0.07}^{+0.10}$	${3.002}_{-0.053}^{+0.006}$	3.027 ± 0.006	${2.75}_{-0.08}^{+0.02}$
NnthComp(10−3)	75.2 ± 0.3	${108.8}_{-6.4}^{+0.2}$	${77}_{-9}^{+5}$	${65.7}_{-14.1}^{+0.2}$	49.1 ± 0.1	17.4 ± 0.1
NrelxillCp(10−3)	${3.509}_{-0.009}^{+0.013}$	${1.9}_{-0.9}^{+1.9}$	${1.9}_{-0.2}^{+0.6}$	1.10 ± 0.03	0.88 ± 0.02	0.36 ± 0.02
NxillverCp(10−4)	<0.54	12 ± 3	${12}_{-3}^{+4}$	9.7 ± 0.6	4.8 ± 0.4	2.0 ± 0.2
CXRT	${1.064}_{-0.011}^{+0.008}$	${1.013}_{-0.009}^{+0.015}$	${1.082}_{-0.015}^{+0.014}$	${1.052}_{-0.024}^{+0.025}$	${1.042}_{-0.023}^{+0.024}$	0.87 ± 0.04
L/Ledd (%)	2.0	1.8	1.7	1.2	1.0	0.5
C-stat	11591
χ2/dof	12584/10731 = 1.173

Note. Luminosity calculated using unabsorbed flux between 0.1 and 300 keV, assuming a distance of 8 kpc and a black hole mass of 10 M_⊙.

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