SPECTRO-TIMING STUDY OF GX 339-4 IN A HARD INTERMEDIATE STATE

We extracted mode 01 (SCIENCE, ∼10 ks per module) and mode 06 (SCIENCE_SC, ∼45 ks per module) data separately using a $90^{\prime\prime}$ radius extraction region centered at the J2000 coordinates of GX 339−4 for the mode 01 data and a larger $120^{\prime\prime}$ radius region for the mode 06 data. The larger region for mode 06 data was chosen to encompass the whole source at all times, as the region was defined in sky coordinates. We carefully compared the extracted spectra and found no significant differences in spectral shape or response. We then combined the data from the two modes using addascaspec, separately for focal plane modules (FPM) A and B.

For time-resolved spectroscopy we used manually created good-time interval files and supplied them to the pipeline and followed the same procedure as described above. The fastest timescales we extracted were of the length of one revolution of the NuSTAR satellite, corresponding to an average exposure of ∼2.7 ks per spectrum.

To measure the background we extracted spectra from a $120^{\prime\prime}$ region north of the source at the other end of the field of view. This places the region on a different detector with slightly different intrinsic background, but GX 339−4 dominates over the background by a factor of at least 10 up to 78 keV. Small background fluctuations therefore do not influence our results.

Data were extracted using nupipeline v.1.4.1 as distributed with HEASOFT v6.16 and CALDB files v20150316. At the time of writing, data below 4.5 keV suffer from slightly higher calibration uncertainties, due to a known shift in gain offset that has not been properly accounted for in the current response files. They have therefore been ignored in the spectral analysis but are still reliable for the light curves. As the relevant features for our conclusions (iron line, photon index) are located well above 4.5 keV, we do not expect this limitation to influence our results significantly. All analysis was performed using the Interactive Spectral Interpretation System (ISIS v1.6.2, Houck & Denicola 2000) and uncertainties are reported at the 90% confidence level unless otherwise noted.

We transferred the photon arrival times to the barycenter of the solar system using the standard FTOOL barycorr and the DE-200 solar system ephemeris (Standish et al. 1992). As the orbit of the binary system is unknown we could not correct for black hole's movement. We used the same source regions as for the spectral extraction and extract light curves with a time resolution of 10 s between 3–10 keV and 10–79 keV.

To measure the PSD we use the method described in Bachetti et al. (2015) to account for the variable dead-time of the NuSTAR detectors, which has typical values around 2.5 ms. In this method, the PSD is calculated using the cross-power spectral density (CPSD) between FPMA and FPMB, thus removing the contribution of the (independent) dead-time and therefore allowing us to accurately measure the PSD to frequencies above 1 Hz. We extracted the CPSD using MaLTPyNT.¹⁴

To calculate the CPSD we multiplied the Fourier transformation of the light curve of FPMA with the complex conjugated Fourier transformation of the light curve of FPMB. The real part of the resulting complex quantity is the CPSD, describing power in phase in both modules. As the deadtime is not correlated between FPMA and B and consequently not in phase, its power is removed from the CPSD. For details about the method see Bachetti et al. (2015).

We calculated the CPSD from light curves of 64 s length, extracted with a time resolution of ${2}^{-8}$ s in the energy range 3–78 keV based on the combined event files from mode 01 and 06 and used a $150^{\prime\prime}$ source region. The CPSD were averaged together for each satellite revolution, and geometrically rebinned with a factor 1.03. Uncertainties were propagated in quadrature from the unbinned data.

All CPSDs are presented in rms or Miyamoto normalization (Belloni & Hasinger 1990; Miyamoto et al. 1991).

To investigate the variability during the observation we first extracted light curves in the 3–10 keV (S) and the 10–78 keV (H) energy bands with 200 s time resolution. The light curves are shown in Figure 2 together with the hardness ratio (HR) calculated as HR = (H−S)/(H+S). The source softens steadily over the course of the observation, while the hard flux remains constant.

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This softening indicates a significant variation in the observed X-ray spectrum and thereby a change in the underlying physical conditions. In order to avoid combining data with intrinsically different spectra as much as possible, we decided to split the data into segments each covering one revolution of NuSTAR around Earth, i.e., we split the data at the gaps due to eclipses in the light curve. Thanks to NuSTAR 's high sensitivity this still allows us to extract high signal-to-noise (S/N) energy spectra and PSD.

After splitting the data according to each revolution of NuSTAR, we obtained 21 energy spectra with about 2.7 ks exposure each. We rebinned each spectrum to about 300 bins, with a S/N  of 24 at the lowest energies decreasing to a S/N  of 8 by 50 keV. In the following analysis each spectrum was fitted individually, unless otherwise noted. Following typical procedure and for comparability with other analyses we start modeling the data with simpler models and increase the complexity of the model step-by-step.

We first describe the data with is an absorbed power-law with an exponential turn-over (cutoffpl) and an additional reflection component, described by the xillver model (García & Kallman 2010). The xillver model self-consistently describes the reflection spectrum of an optically thick, geometrically thin accretion disk, including the ${\rm{Fe}}\,{\rm{K}}\alpha$ line as well as the Compton-hump at high energies.

The continuum was modified at low energies with the absorption model phabs using abundances by Wilms et al. (2000) and cross-sections by Verner et al. (1996). We set the equivalent hydrogen column to $5\times {10}^{21}$ cm⁻², which is the expected Galactic absorption column along the line of sight toward GX 339−4 . The column could not be constrained with the NuSTAR data and we therefore kept it fixed. We also allowed for a cross-calibration constant between the two NuSTAR FPMs, setting it to 1 for FPMA.

This model (model 1) gives us seven free parameters per spectrum: the continuum flux, the photon index Γ, the cutoff energy ${E}_{\mathrm{cut}}$, the reflection flux ${A}_{\mathrm{refl}}$, the iron abundance ${A}_{\mathrm{Fe}}$, the ionization parameter ξ, and the cross-calibration constant. The abundances in the xillver model are relative   to the solar values found by Grevesse & Sauval (1998), i.e., an iron abundance of 1 corresponds to $2.5\times {10}^{-5}$ Fe-atoms per H-atom.

Model 1 results in an average ${\chi }_{\mathrm{red}}^{2}=1.12$ (${\chi }^{2}=13552$) for 578 degrees of freedom (dof) for each of the 21 spectra, i.e., 12,138 total dof. Note that we give the combined ${\chi }_{\mathrm{red}}^{2}$ as an indicator of the overall quality of the model, but that all spectra were fitted individually. The average X-ray luminosity (between 5 and 50 keV) over the complete observation is ${L}_{x}\approx 6.35\times {10}^{37}$ erg s⁻¹ for a distance of 8.4 kpc, i.e., $\sim \,5.5 \% \,{L}_{\mathrm{Edd}}$, assuming a $9\,{M}_{\odot }$ black hole. The average iron abundance is found to be ${A}_{\mathrm{Fe}}=1.9\pm 0.3$, where the uncertainty indicates the standard deviation over the 21 data sets.  

The average ${\chi }_{\mathrm{red}}^{2}$ value of model 1 is statistically not quite acceptable and an inspection of the combined residuals of all 21 spectra reveals strong structure around the ${\rm{Fe}}\,{\rm{K}}\alpha$ line as well as at the highest energies (Figure 3(b)). In each individual spectrum the model seems to describe the data well (hence the relatively good ${\chi }_{\mathrm{red}}^{2}$ value), but the combination of the residuals of all spectra reveals strong structure.

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To improve the model we added a relativistic convolution model (relconv, Dauser et al. 2010), broadening the reflection features as expected in the vicinity of the black hole (model 2). As in F15, we fixed the spin of the black hole to $a=0.93$, as found by Miller et al. (2008). Parker et al. (2016) found a slightly higher spin, $a={0.95}_{-0.08}^{+0.02}$, and we carefully checked that this increased value does not influence our results significantly. We kept the emissivity index of the accretion disk fixed at q = 3, as expected for a standard Shakura–Sunyaev accretion disk (Reynolds & Begelman 1997) with a large coronal height, and the outer disk radius fixed at ${r}_{\mathrm{out}}=400\,{r}_{g}$.

The addition of relativistic blurring improved the fit drastically to an average ${\chi }_{\mathrm{red}}^{2}=1.03$ (${\chi }^{2}=12422$) for 576 dof per spectrum (the additional fit parameters being the inclination i and the inner accretion disk radius ${R}_{\mathrm{in}}$). This is an improvement of ${\rm{\Delta }}{\chi }^{2}=1130$ for 42 fewer dof, clearly preferring this model over model 1 according to the sample-corrected Akaike Information Criterion   (AIC, Akaike 1974), with ${\rm{\Delta }}{{\rm{AIC}}}_{\mathrm{1,2}}=1043$ (corresponding to a likelihood $\ll {10}^{-15}$) between model 1 and model 2. The average inclination is $i=33\pm 3^\circ$ and the average iron abundance ${A}_{\mathrm{Fe}}=5.3\pm 1.3$. Allowing for a free spin a resulted only in a marginal improvement of the statistical quality of the fit (${\rm{\Delta }}{\chi }^{2}=12.9$) and in consistently high values of the spin in all 21 data sets, validating our approach of fixing the spin to the literature value. Inspection of the combined residuals of model 2 again revealed strong deviations at high energies, as shown in Figure 3(c). 

In F15 we showed that there is evidence that the reflector sees a different continuum than the observed primary continuum during the hard state of GX 339−4 , and this was confirmed by Parker et al. (2016). This configuration can be crudely modeled by allowing the photon-indices of the continuum and the reflection model to be different (model 3a). With this approach, we obtain a very good fit with ${\chi }_{\mathrm{red}}^{2}=1.007$ $({\chi }^{2}=12158)$   for 575 dof per spectrum. We find ${\rm{\Delta }}{{\rm{AIC}}}_{\mathrm{2,3}}=220$ between models 2 and 3a, indicating that the latter is clearly preferred with a chance likelihood $\ll {10}^{-12}$, which is confirmed by an F-test.   The residuals to model 3a   are flat, as shown in Figure 3(d). Small deviations on the order of 2%–3% around 25 keV can be attributed to calibration uncertainties (Madsen et al. 2015). These can be reduced by adding an ad hoc Gaussian component at ∼22 keV, but this addition does not change any of the fit parameters significantly. As we already obtain a very good fit in terms of ${\chi }_{\mathrm{red}}^{2}$ and to keep the model as simple as possible, we chose not to include this component.  

Another way of approximating a complex coronal configuration is to allow the cutoff energy of the primary continuum and the input spectrum to the reflector to be different (model 3b). This setup might mimic different temperatures in an elongated corona. Using this approach, we find a slightly worse fit than with independent photon indices and obtain ${\chi }_{\mathrm{red}}^{2}=1.013\,({\chi }^{2}=12232)$ for 575 dof per spectrum. We obtain qualitatively similar results, with the input spectrum to the reflector being much harder (hotter) than the observed continuum. However, the cutoff energy of the reflector often pegs at the upper limit of 1 MeV. As this model also gives a statistically slightly worse fit than model 3a (${\rm{\Delta }}{\chi }^{2}=74$ for the same number of dof), which allowed for independent photon-indices, we will build on model 3a   for the remainder of this paper. The physical implications (e.g., the values of the inner radii) do not differ significantly between these two models.  

3.2.1. Simultaneous Fits

In the previous models, we kept all parameters independent between each spectrum. We can assume, however, that certain parameters do not change over the course of the observation, like the inclination i and the iron abundance ${A}_{\mathrm{Fe}}$ ¹⁵ , as long as the accretion disk is not strongly warped and precessing (as postulated for, e.g., Cyg X-1, Tomsick et al. 2014). To implement such constant parameters in our model we need to perform a simultaneous fit of all 21 spectra.  

Figure 4 shows the evolution of all fit parameters as a function of time in model 3a. While we know from the change in hardness that a significant evolution in the spectrum is present, the relatively large uncertainties on all parameters in model 3a render them insensitive to these changes. To quantify the spectral changes, we therefore have to choose parameters to which we ascribe the observed change in hardness, while we require others not to change over the course of the observation.

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We select the photon index of the continuum ${{\rm{\Gamma }}}_{\mathrm{cont}}$ as the main driver of the hardness changes, and tie the cutoff energy ${E}_{\mathrm{cut}}$ and the photon index of the reflector ${{\rm{\Gamma }}}_{\mathrm{refl}}$ between all spectra in the simultaneous fits. This selection is somewhat arbitrary, but results in a very good description of the data.   We note that if we instead allow the photon index of the reflector, ${{\rm{\Gamma }}}_{\mathrm{refl}}$, to vary, and tie the photon index of the continuum, ${{\rm{\Gamma }}}_{\mathrm{cont}}$, across all spectra, we obtain very similar correlations and implications. We   will base our discussion on a variable photon index of the continuum with the understanding that this is not necessarily the correct physical parameter, but just a convenient parameterization of the hardness. 

For the simultaneous fit, we have the parameters $i,{A}_{\mathrm{Fe}},{E}_{\mathrm{cut}},{{\rm{\Gamma }}}_{\mathrm{refl}}$, and the cross-calibration constant for FPMB, ${{CC}}_{\mathrm{FPMB}}$ tied across all spectra. These are "global" parameters as defined by Kühnel et al. (2016). This simultaneous fit provided a statistically very good description of the data, with ${\chi }_{\mathrm{red}}^{2}=1.009\,({\chi }^{2}$ = 12,290 for 12,175 dof).

  However, as this fit is highly complex and a single evaluation of the spectrum takes almost 2 s on a typical workstation, it was not feasible to calculate uncertainties for all 110 free parameters. Instead, we only calculated uncertainties for the parameters tied across all data sets (Table 1). In a second step, we go back to modeling the 21 spectra individually, but constrain the global parameters to their 90% uncertainty interval found in the simultaneous fit (model 4). This approach ensures that the values of the global parameters are similar across all data sets, while still capturing their statistical variability and allows us to obtain a realistic estimate for uncertainties of the local parameters.  

Table 1.  Parameters and Uncertainties of the Best-fit Model by Fitting all 21 Spectra Simultaneously

Parameter	Value
${E}_{\mathrm{cut}}$ (keV)	${136}_{-9}^{+6}$
${{\rm{\Gamma }}}_{\mathrm{refl}}$	${1.37}_{-0.04}^{+0.05}$
${A}_{\mathrm{Fe}}$	${4.8}_{-0.4}^{+0.6}$
i (deg)	${29.7}_{-1.5}^{+2.2}$
${{CC}}_{\mathrm{FPMB}}$	${1.0299}_{-0.0012}^{+0.0013}$

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Model 4 provided a   good fit to the data, with ${\chi }_{\mathrm{red}}^{2}=1.012\,({\chi }^{2}$  = 12,221 with 575   dof per spectrum). The results of the local   parameters are shown in Figure 5. The global parameters do not show a trend as function of time, and their uncertainties encompass the complete allowed range in each individual spectrum, i.e., they follow the blue bars shown in Figure 4. As is obvious in a comparison of Figures 4 and 5, the local parameters are better constrained in model 4, in particular the photon index ${{\rm{\Gamma }}}_{\mathrm{cont}}$ and the inner radius of the accretion disk, ${R}_{\mathrm{in}}$. As expected from our model setup, the only parameter that shows a significant evolution as a function of time is ${{\rm{\Gamma }}}_{\mathrm{cont}};$ all other parameters are constant within their uncertainties. The photon index ${{\rm{\Gamma }}}_{\mathrm{cont}}$, however, shows a strong steepening over the course of the observation, in step with the observed softening shown in Figure 2.

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The ${\chi }_{\mathrm{red}}^{2}$-values of this model vary around 1, in a manner fully consistent with a ${\chi }^{2}$-distribution with 580 dof according to the KS-test.

To test the dependence of the values of the inner radius on the assumed geometry we also tested the lamppost geometry using the relxilllp model (Dauser et al. 2014),   which assumes a point-like corona on the spin-axis above the black hole. We find a statistically comparable good fit, using the same approach of allowing the   global parameters to only vary within their 90% uncertainties found in a simultaneous fit.   All parameters, in particular the photon index and the inner radius, follow the results from model 4 closely. The similarity between the two models indicates   that in our data the exact geometry and emissivity profile of the accretion disk do not significantly influence the results.  

In the same way as for the time-resolved energy spectra, we analyzed the CPSD on a revolution-by-revolution basis. All 21 CPSD show a very prominent type-C QPO around 0.8 Hz (for a definition of QPO types, see, e.g., Casella et al. 2005). We modeled each CPSD with two zero-centered Lorentzians to describe the band-limited noise, a Lorentzian for the type-C QPO and one Lorentzian for the sub- and super-harmonic (at $\tfrac{1}{2}$ and $\tfrac{3}{2}$ the frequency of the QPO) each. This model provides a very good fit to the data, with an average ${\chi }_{\mathrm{red}}^{2}=1.01$ for 183 dof Figure 6 shows the average CPSD and best-fit model in red. The QPO around 0.8 Hz appears broad, as it is a superposition of individual QPOs in each individual CPSD.

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By looking at the QPO frequency as function of time, we find that it is significantly increasing, rising from ∼0.65 Hz at the beginning of the observation to ∼1.0 Hz at the end (Figure 6, in blue and green respectively). This change is in step with the softening of the photon index ${{\rm{\Gamma }}}_{\mathrm{cont}}$, as shown in Figure 7, where we plot the QPO frequency versus ${{\rm{\Gamma }}}_{\mathrm{cont}}$ together with a linear correlation with a slope of $b=0.22\pm 0.11.$  This correlation is often observed for type-C QPOs, in GX 339−4 as well as in other sources (e.g., Vignarca et al. 2003; Motta et al. 2011; Stiele et al. 2013).

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We do not find a significant correlation between the inner accretion disk radius and the QPO frequency, with a Pearson correlation coefficient of 0.16 and a linear slope of ${b}_{\mathrm{rin}}=6\pm 12$. The uncertainty is dominated by the larger uncertainty on the inner radius, which cannot be constrained better than $\approx 3.5\,{r}_{g}$.

To investigate the dependence of the QPO frequency on energy we extracted CPSDs in the energy bands 3–5 keV, 5–7.5 keV, 7.5–12 keV, and 12–79 keV. We do not see any significant change in frequency as a function of energy, and all four energy bands give the same time dependence. While type-C QPOs show a strong energy dependence in some sources if they are located at higher frequencies ($\gtrsim 4\,{\rm{Hz}}$ for H1743−322, Li et al. 2013), at lower frequencies they are energy independent (e.g., in XTE J1550−564, Cui et al. 1999), in agreement with our results.

We carefully searched for evidence of QPOs at higher frequencies, but did not find a significant signal. For that search we also analyzed the time-averaged CPSD to obtain a better signal at high frequencies. Here, we find a slight improvement of the statistical quality of the fit when including a QPO at around 55 Hz. However, by using a Monte Carlo approach to test its significance and simulating 500 CPSDs from the best-fit model, scattered around within the uncertainties of the measured CPSD, we find that a similar improvement in ${\chi }^{2}$ can occur by chance with $\gt 5 \%$ probability. We therefore conclude that this feature is not significantly detected.

Using relativistic reflection models, we find that the inner radius of the accretion disk is truncated significantly outside the ISCO, at around 9 r_g. This result is independent of the assumed geometry, either a standard disk emissivity profile with an index of q = 3, or a lamppost geometry. This amount of truncation is similar to the one found by Tamura et al. (2012), assuming a $9\,{M}_{\odot }$ black hole.

This result indicates that the accretion disk does not move all the way to the ISCO before the source has entered the soft state. This seems to be at odds with results finding the accretion disk to be at the ISCO in bright hard states (e.g., Miller et al. 2015). On the other hand, it might simply show that different outbursts behave differently. It is in theory also possible that the truncation is not a strict function of luminosity, and truncation appears again during the HIMS, given its slightly different radiation balance at the accretion disk/corona boundary (e.g., a corona with a steeper photon index and therefore likely higher optical thickness).

Finding a truncated accretion disk in the HIMS nonetheless indicates that the geometry and physical conditions in this state are very different than in the soft state, e.g., that the inner parts of the accretion disk are still replaced with an optically thin flow, like an ADAF. During the switch to the soft state, which happens on a very short timescale, the accretion disk has then to move all the way to the ISCO and significantly heat up. As can be seen in Figure 1, the source is in the proper soft state only 3–4 days after the NuSTAR observation.

Despite being an ubiquitous feature of black hole binaries, the physical origins of QPOs are still highly uncertain. While QPOs come in different shapes (see, e.g., Casella et al. 2004), the most often observed one is the type-C QPO, i.e., a strong, narrow feature in the PSD on top of flat-top noise dominated continuum (Remillard et al. 2002). Different theories have been put forward as to their origin, with one of the most prevalent being that the QPO is produced by Lense–Thirring precession of the innermost parts of the accretion flow (e.g., Stella & Vietri 1998; Stella et al. 1999; Ingram et al. 2009).

This model, known as the RPM, predicts three QPOs to be observed, all of which are produced by the same accretion flow. The lowest frequency QPO is identified with the nodal precession of the accretion flow due to Lense–Thirring precession and observable in black hole binaries as a type-C QPO around 0.5–5 Hz (Stella et al. 1999). If all three QPOs are observed (as in, e.g., GRO J1655−40, Motta et al. 2014), the RPM allows for a measurement of the mass and spin of the black hole, as well as the inner accretion disk radius, independent of results or assumptions from fitting the energy spectra.

In GX 339−4 , we only observe the low-frequency QPO, which can be identified with the nodal precession frequency ${\nu }_{\mathrm{nod}}$. If we include measurements of the spin and inner radius from reflection modeling, we can investigate possible black hole masses. We follow the calculations presented in Ingram & Motta (2014), and assume a spin of $a=0.95$ (Parker et al. 2016). Using the measurements of the inner radius of model 4, Figure 8(a) shows the correlation between ${\nu }_{\mathrm{nod}}$ and ${r}_{\mathrm{in}}$ for different masses of the black hole, as predicted by the RPM. As can be seen, our measured values are incompatible with a black hole mass of ∼$9\,{M}_{\odot }$ and a fast spinning black hole. Instead they seem to indicate a very massive black hole around $75\mbox{--}100\,{M}_{\odot }$, which is too large for typical X-ray binaries.  Within the RPM the sampled range of QPO frequencies results only in a very small variation of the inner radius, clearly below the uncertainties of the measured values.

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As shown in Figure 8(a) we can consolidate the measurements of the inner radius and QPO frequency with a black hole mass of $9\,{M}_{\odot }$ when assuming a very low spin of $a=0.1$. Such a low spin is in stark contrast to all relativistic reflection modeling where a broad ${\rm{Fe}}\,{\rm{K}}\alpha$ line is observed (Miller et al. 2008; Reis et al. 2008; García et al. 2015; Ludlam et al. 2015; Parker et al. 2016) and also ruled out by disk continuum fitting for inclinations $i\gt 20^\circ$ (Kolehmainen & Done 2010).

Another model to explain the origin of low-frequency QPOs was presented by Titarchuk & Osherovich (2000). These authors assume that the QPO is produced by global disk mode (GDM) oscillations, i.e., by the oscillations from a disk off-set from the equatorial plane around a compact object without the need to invoke relativistic effects. In this theory the total mass of the disk plays an important role, which is strongly influenced by our choice of the outer radius when assuming a standard Shakura–Sunyaev disk. Using an outer radius of 400 r_g, and the measured inner radii and QPO frequency, we find that the data are best described with a $100\,{M}_{\odot }$ black hole (Figure 8(b)).

However, as the outer radius is not known and cannot be constrained in our spectral fits, we can also find an acceptable solution for a $9\,{M}_{\odot }$ black hole with an outer accretion disk radius of ${10}^{4}\,{r}_{g}$. We note, however, that the predicted correlation between ${r}_{\mathrm{in}}$ and ${\nu }_{0}$ in that case seems somewhat flatter than what we measure, as shown in Figure 8(b).

Motta et al. (2011) and Stiele et al. (2013) present a detailed study of the correlation between QPO frequency and spectral parameters in GX 339−4 using RXTE data of outbursts between 2002 and 2010. They find a clear correlation between QPO frequency and photon index which, for hard spectra and frequencies around 1 Hz, is linear. While their data sample a large range in both Γ and ${\nu }_{\mathrm{QPO}}$, the sampling during the transitional state around ${\rm{\Gamma }}\approx 1.7$ is very sparse. This makes a comparison to our data not feasible, but places the unique state we sampled with NuSTAR in context.

Shaposhnikov & Titarchuk (2009) use a similar selection of RXTE data to compare the correlation of GX 339−4 to the one observed in other sources, e.g., XTE J1650−500 and Cyg X-1. By following the idea that the differences in the observed QPO-Γ correlations between different sources can be reduced to differences in the mass of the compact object (Titarchuk & Fiorito 2004), they estimate a black hole mass for GX 339−4 of $12.3\pm 1.4\,{M}_{\odot }$. This method requires the measurement of the correlation over a large range of QPO frequencies, in particular covering the transition regime, at which the correlation flattens.

The NuSTAR data cover only a very small range of QPO frequencies, but assuming that the measured range is well below the transition frequency (which is, e.g., at 6.64 ± 0.48 Hz for GRO J1655−40, Shaposhnikov & Titarchuk 2007), we can fit a linear function to the correlation. This gives a slope of $b=0.22\pm 0.11$   (see Figure 7). Comparing this to values for other sources presented by Shaposhnikov & Titarchuk (2007), we estimate a mass of $\sim 10\,{M}_{\odot }$ for GX 339−4 from the scaling relation, in agreement with the mass measured by Parker et al. (2016). This value is dominated by systematic uncertainties given the limited QPO frequency range covered during the NuSTAR observation and therefore compatible with the mass found by Shaposhnikov & Titarchuk (2009).

By combining precise spectral modeling with recent theories concerning the formation of QPOs in accreting black holes, we have shown that there are still many questions left to answer. Our data show that in both the RPM as well as the GDM oscillation model, a black hole mass on the order of $100\,{M}_{\odot }$ seems to be necessary to explain the observed combination of QPO frequency and inner accretion disk radius. As such a massive black hole is highly unlikely to exist in GX 339−4 (Parker et al. 2016), it shows that further improvement of the measurements and QPO theory is necessary. We find a significantly truncated accretion disk (${r}_{\mathrm{in}}\approx 4.4\,{r}_{\mathrm{ISCO}}$ for $a=0.95$) but the measurement of its inner radius through reflection fitting is strongly dependent on emission geometry and inclination of the accretion disk. While we have shown that the lamppost geometry gives similar inner radii as a standard emissivity profile, more complicated coronal geometries cannot be ruled out. The measurement of the photon index ${{\rm{\Gamma }}}_{\mathrm{cont}}$ is more reliable, for which we find a tight correlation to the observed QPO frequency. However, theoretical predictions based on physical models are currently lacking, due to the fact that the physics of the corona is still poorly understood. Further high-quality observations similar to the ones presented here will allow us to increase our knowledge about the corona and its physics in the future.