X-ray Spectroscopic Study of Low-Mass X-ray Binaries: A Review of Recent Progress via the Example of GX 339-4

Abstract

Low-mass X-ray binaries (LMXB) serve as natural laboratories, where the predictions of general relativity can be tested in the strong field regime. The primary object of such sources can be a neutron star (NS) or a black hole (BH), and this object captures material from the secondary object through the inner Lagrange point via a process called Roche lobe overflow. Because of the angular momentum of the infalling matter, an accretion disk is formed, in which viscous effects transport the angular momentum radially outward. In the high/soft state of these sources, the accretion disk can extend all the way to the innermost stable circular orbit (ISCO); therefore, when the primary object is a BH, its X-ray spectrum contains information about the region very close to the event horizon. This paper aims to review the theoretical and observational works related to the X-ray spectroscopy of such sources via the example of GX 339-4, which is one of the most well-known and well-studied LMXBs.

1. Introduction

Black hole (BH) X-ray binaries (see review in [1,2]) have been the subject of intensive research during the last couple of decades. Their value is due to the fact that they serve as laboratories where general relativity can be tested. During the accretion of matter, the release of gravitational potential energy causes extreme temperatures to be reached, and the resulting radiation can provide information about the region close to the event horizon. Therefore, these objects turn up as X-ray sources in the sky. The first BH candidate, Cygnus X-1, was discovered in 1972 [3,4], and since that time, 70 BHs have been detected in X-ray binaries (https://www.astro.puc.cl/BlackCAT/transients.php -accessed on 28 August 2023) [5]. Most of these sources vary in their X-ray and optical luminosities, and there are time periods when the optical radiation is dominated by the accompanying star. This gives an opportunity for measuring its periodically changing radial velocity, which in turn provides information about the gravitational effect of the BH companion. Hence, the mass of the compact object and the inclination angle can be determined, and when the obtained mass is larger than 4M_ʘ, the accreting object being a NS can be ruled out (for detailed discussion about the upper mass limit for NSs, see [6] and the references therein). Since the accreted material usually has a considerable amount of angular momentum, it cannot directly fall into the BH or onto the surface of the NS. First, it has to lose some of its angular momentum, thus an accretion disk will form, in which matter is moving inward while the angular momentum is transported outward via viscous effects [7]. We distinguish three groups of these X-ray sources based on the mass of the companion star. If it is larger than 10 M_ʘ, we are talking about high-mass X-ray binaries (HMXB), if it is roughly between 1 and 10 M_ʘ, we refer to them as intermediate-mass X-ray binaries (IMXB), while in case of a star similar to our Sun or smaller, they are called low-mass X-ray binaries (LMXB). In the first case, the large star produces a strong wind, which supplies enough material for the compact object to be bright most of the time. In the second case, the wind-driven accretion contribution can still be significant, depending on the exact mass of the secondary. In the third case, however, accretion can only happen after the companion star fills up its Roche lobe, and the material is transferred through the inner Lagrangian point. These sources spend a significant portion of their time in the quiescent state, but due to instabilities in the accretion disk, they undergo occasional outbursts that last for a few days or even months. As the accretion rate changes, so does the geometry of the accretion disc and the emitted spectrum. At a few percent of the Eddington rate, it is thought that the disk is geometrically thin and optically thick. In some cases, this disk can extend all the way to the innermost stable circular orbit (R_ISCO), beyond which the material plunges into the BH without any considerable viscous interaction. Since the temperature of the disk is higher at smaller radii, the inner part will contribute more to the X-ray spectrum. This thin disk solution usually characterizes the accretion geometry close to the peak of the outburst, when the luminosity is high, and the spectrum is dominated by the relatively soft radiation. This is why this state is called the high/soft state. In this state, the X-ray spectrum peaks at around 1 keV energy, as opposed to the low/hard state, when the peak is at around 100 keV. In the first case, the contribution by the spherical hot flows close to the inner portion is negligible or quite weak; in the latter case, however, the emission is dominated by the radiatively inefficient, optically thin hot flows. Since R_ISCO strongly depends on the angular momentum of the BH, and whether the movement of the material in the accretion disk is prograde or retrograde (Figure 1), in principle, we can obtain information about the BH spin from the X-ray spectrum.

There are two basic approaches for obtaining the BH spin and other information using the X-ray spectrum. One involves fitting the continuum emission of the disk using an appropriate model [8], while the other one is related to the study of broadened Fe Kα emission lines [9]. See Chapter 2 of [6] for more details on these methods. In order to obtain information about the BH spin from the X-ray spectrum, the accretion disk must extend all the way to R_ISCO, and it should dominate the observed emission. During the outburst, a LMXB goes through different spectral states depending on the mass accretion rate in terms of the Eddington limit. These states correspond to different accretion geometries (Figure 2), and as it was mentioned earlier, it is the high/soft state in which most of the emission is coming from the thin disk, and it is thought to extend all the way in to R_ISCO. Since the emission from the hot corona cannot be neglected even during the high/soft state, it is usually important to obtain high quality broadband spectra that include the higher energy regions as well, so that this component can be determined properly based on an energy interval that does not include disk emission anymore. Otherwise, this model component may result in unrealistic values for lower energies, and makes fitting the disk spectrum unreliable. Additional complication can be if the hard component of the spectrum is highly variable during the observation, as it was shown for the HMXB Cygnus X-1 [10]. In the case of LMXB sources, however, without the strong variable wind coming from the secondary star, this may not be such a serious problem. For active galactic nuclei (AGN), the excess in the 5–7 keV band can also be explained by a two-component warm absorber model [11,12], without the need for including a relativistically broadened Fe emission feature, but the required column densities do not seem realistic in the case of galactic X-ray binaries.

Another complication is that the local spectrum emitted by the disk is not exactly a blackbody spectrum, because it is “diluted” by electron scattering. The saturated comptonization causes the color temperature to differ from the effective temperature at a certain region of the disk: $T_{col}=f_{col}·T_{eff}$, where $f_{col}$ is the color correction factor [20].

It is also possible that the surface layer of the accretion disk deviates from the standard α-disk model [7] because of the heating by the hard X-ray source at the center [21,22]. Such a “warm” layer at the surface of the disk can add further complications in terms of the shape of the disk spectrum.

Obviously, an appropriate disk model should also take into account all the general relativistic (GR) effects that may arise so close to a spinning BH. Some of the photons can even cause self-irradiation in the disk because of the bending of the light in the strong gravity regime.

During the last couple of decades, significant progress was made in terms of the theoretical understanding and modeling of these processes, and at the same time, the technological development was also quite strong; several state-of-the-art satellites were launched, and provided high-quality data for the scientific community. This latter is also very important, because without broadband high-quality data, none of the different theories and models could be tested to an appropriate degree. In the following, we will review the recent theoretical and observational works in this field via the example of GX 339-4, which is a LMXB where the compact object is most likely a BH.

2. Review of the Recent Research on GX 339-4

Since its discovery in 1972 [23], GX 339-4 has undergone outbursts with different intensities and durations every 2–3 years, and demonstrated all of the spectral states depicted in Figure 2. Because of this, it has been the subject of intensive studies and frequent observations from radio to gamma ray frequencies and serves as one of the most important laboratories for research on LMXBs. Because of the relatively faint secondary star in this system, dynamical measurements in the optical or infrared band for the purpose of determining the parameters of the system turned out to be a difficult task. In 2003, Hynes et al. [24] performed outburst spectroscopy, and managed to determine the orbital period ($P_{orb}$) of the system and the radial velocity amplitude (K) using the light coming from the secondary star due to irradiation by the primary object. The orbital period of 1.7557 days gave a value of $(5.8\pm 0.5)$ M_ʘ for the mass function of the primary and $q\le 0.08$ for the mass ratio. The mass function can be considered as an absolute minimum for the mass of the primary object, according to the formula:

$$f\left(M_P\right)=\frac{K^3{P}_{orb}}{2\pi G}=\frac{{M_P}^3{\sin }^{3}i}{{\left(M_P+M_S\right)}^2},$$

(1)

where G is the universal gravitational constant, $M_P$ is the mass of the primary object, $M_S$ is the mass of the secondary star, and $i$ is the inclination of the system. A year later, Zdziarski et al. [25] used these parameters, and based on the X-ray variability of the source, they obtained $6.7\le d_{min}\left(i\right)\le 9.4$ kpc for the inclination dependent lower limit on the distance and constrained the inclination of the system to the interval between 45° and 80°. Muñoz-Darias et al. [26] applied K-correction to the velocity measurements of the secondary star using the stripped-giant evolutionary model. They assumed that the companion is more massive than about 0.17 M_ʘ, and that way obtained a lower limit of 6 M_ʘ for the BH mass. Taking into account the strong X-ray activity of the source, which suggests somewhat higher mass for the secondary star (≥0.3 M_ʘ), they estimated a lower limit of 7 M_ʘ for the mass of the BH. In 2017, Heida et al. [27] confirmed and refined the value for the orbital period reported in [24] by performing observations of the source in 2016 during its quiescent state and managed to determine the radial velocity curve. This was the first time that the radiation of the donor star was directly observed in the near-infrared (NIR). The absorption lines detected in the NIR band allowed them to determine the rotational velocity of the secondary star. They obtained $f\left(M_P\right)=(1.91\pm 0.08)$ M_ʘ for the mass function and $37°\le i\le 78°$ for the inclination, which gave a range of 2.3–9.5 M_ʘ for the possible mass of the BH. In 2019, Zdziarski et al. [28] performed a detailed evolutionary model for the secondary star and constrained its mass to the range of 0.5–1.4 M_ʘ. Based on model independent constraints and previously published values, they determined the following intervals for the BH mass, inclination, and distance: M = 4–11 M_ʘ, $40°\le i\le 60°$, and d = 8–12 kpc.

The LMXB GX 339-4 in its high/soft state was observed by Chandra in the 1–10 keV energy range during its 2002–2003 outburst. Miller et al. [29] reported on the analysis of the Chandra–HETGS spectrum. The model they used to fit the spectrum included a standard multi-color disk (MCD) blackbody [30] component (diskbb), and a pexriv [31] component, which is an exponentially cut-off power-law spectrum plus its reflection from the disk. The Fe Kα feature was fitted by the laor [32] component. They also used the CDID model [33], which is a constant-density ionized disk model, instead of the pexriv component. Modeling the shape of the Fe-line feature using the laor component resulted in a conclusion that the disk extends all the way to $R_{in}={{2.5}_{-0.3}^{+2.0}R}_g$, which would indicate a BH spin of about 0.9 or higher. For such a complex model, however, higher energy data would have been needed to better constrain the properties of the reflection component.

In their next work [34], Miller et al. analyzed simultaneous XMM-Newton and RXTE data from 2002 September, when the source was close to the peak of its outburst. However, they only used the higher energy data from RXTE as a “guide” to describe the higher energy component in the spectrum, and when they performed a joint fit on both the XMM-Newton and RXTE data using an MCD plus power-law model, they constrained the power-law index to lie within $\pm$0.1 from the value obtained from fitting the RXTE data alone. Using this simple model, they did not manage to obtain a good fit, and they claimed to have found significant excess in the 3–7 keV energy range, characteristic of a broadened Fe Kα emission line (Figure 3).

It should be mentioned that they ignored the 4–7 keV band when fitting the data. Attempts were made to describe the excess using a laor component, and they also included a smeared edge (smedge) absorption feature [35] in the model. Additionally, they also fit the spectrum using an MCD plus pexriv or MCD plus CDID model. The obtained fit parameters suggested a rapidly spinning BH (a* ≥ 0.8) for all three models; the laor component gave $R_{in}={{2.1}_{-0.1}^{+0.2}R}_g$, the CDID component gave $R_{in}={2.9\pm 0.1R}_g$, and the pexriv component gave $R_{in}={{2.1}_{-0.3}^{+0.5}R}_g$. However, the obtained best fit models did not describe the data too well, which is shown by the numbers calculated for the statistics. They were ${\chi }_{\nu }^2$/dof = 1.82/1894 for the MCD plus power-law model with the added laor component, ${\chi }_{\nu }^2$/dof = 2.05/1895 for the MCD plus CDID model, and ${\chi }_{\nu }^2$/dof = 1.84/1896 for the MCD plus pexriv model. The obtained low values for the inclination (11–12°) were at odds with the limits that were determined by independent methods (see earlier). Moreover, it does not seem to be consistent with the usual physical description of the system that the reflection component (CDID) gives significantly more contribution to the spectrum at around 1 keV energy than the MCD component (Figure 4). Two years later, Miller et al. [36] also applied the same models to simultaneous XMM-Newton and RXTE observations from 2004, which recorded the source in its low/hard state. They claimed to have detected broadened Fe Kα features in those datasets as well and came to the conclusion that the accretion disk extends all the way or close to R_ISCO even in the low/hard state. This could have provided evidence for the proposal that an inner thin disk may form even in the low/hard state by the recondensation of matter into cold clumps from the hot corona [16]. In the following years, however, further spectral and timing studies [37,38,39] of the low/hard state of GX 339-4 provided strong observational evidence that the disk is truncated in the low/hard state, and the inner radius changes according to the changing rate of mass accretion.

Some of the previously mentioned issues were addressed in our 2007 work [40], where XMM-Newton and RXTE data from the 2002–2003 outburst of GX 339-4 were jointly fitted, covering an energy range of 0.5–150 keV (2002 dataset) and 0.5–60 keV (2003 dataset). The source was in its very high state in 2002 and in the high/soft state in 2003. The high-quality data below 1 keV enable the reliable modeling of the accretion disk, while the high energy data help to constrain the parameters of the hard component. In order to determine the color correction factor, an observational approach proposed by Cui et al. [41] was used. Therefore, the disk component was modeled by a comptt component [42] in a disk geometry, which means saturated Compton scattering, and another comptt component was added to model the hard component of the spectra, but with a spherical geometry, which means unsaturated Compton scattering. The value of the color correction can be approximated from the $T_e$ and $T_0$ parameters of the comptt component according to the $f_{col}=T_e/\left(2.7{·T}_0\right)$ formula [41]. This method resulted in $f_{col}={1.48}_{-0.08}^{+0.09}$ for the 2002 data, and $f_{col}=1.35\pm 0.01$ for the 2003 data. Interestingly, the best fit for both datasets (${\chi }_{\nu }^2$/dof = 0.81/1201 and ${\chi }_{\nu }^2$/dof = 0.86/1076) was obtained using this comptt + comptt model (Figure 5).

In the 2003 dataset, significant excess could be seen in the 5–8 keV range (Figure 6), which was modeled using an additional laor component mentioned earlier, and a smedge component was also included for this dataset ($E_{edge}=8.5\pm 0.1$ keV, $W={2.7}_{-0.4}^{+0.5}$, and $\tau ={0.59}_{-0.05}^{+0.07}$).

The value of the inclination and $R_{in}$ were given by the laor component as $i={51}_{-1}^{+2}°$ and $R_{in}={{1.76}_{-0.06}^{+0.1}R}_g$. This result would imply a very high value for the dimensionless spin parameter: a* ≥ 0.97, while the measured inclination is consistent with the previously mentioned independent observations. An attempt was made to replace the comptt model describing the disk emission by a kerrbb [43] or a bhspec [44] component, but it was not possible to obtain formally acceptable fits for these broadband high-quality datasets (see Figure 7 for the attempt with kerrbb). These are both fully relativistic models for fitting the disk emission, and kerrbb contains the color correction factor as a free parameter, while bhspec incorporates the spectral hardening due to electron scattering.

The inclination was fixed at the 51° value obtained from the laor model, which was utilized for the best fit of the 2003 dataset, the mass of the BH was set to 10 M_ʘ, and the distance to the source to 8 kpc. This way, a value of about 0.7 was obtained for the BH spin using the kerrbb model, and about 0.5 when using the bhspec model. In both cases and for both datasets, significant features could be seen in the residuals. The ${\chi }_{\nu }^2$ values for the kerrbb fit turned out to be ${\chi }_{\nu }^2$/dof = 2.19/1203 (2002) and ${\chi }_{\nu }^2$/dof = 1.86/1079 (2003), and in case of the bhspec fit, they were ${\chi }_{\nu }^2$/dof = 1.87/1203 (2002) and ${\chi }_{\nu }^2$/dof = 2.32/1079 (2003).

Apart from the difficulty fitting the spectrum using these models, there seems to be an inconsistency between the BH spin values obtained from the continuum fit and the iron-line fit. Since neither the mass, nor the distance were precisely known at the time, changing these values somewhat in the kerrbb model (13.5 M_ʘ and 7.5 kpc) resulted in BH spin values 0.93 and 0.96 for the 2002 and 2003 datasets, respectively. This latter result highlighted the need to determine the mass and distance from independent observations, so that the modeling of the continuum can really constrain the values for the BH spin and inclination. At the same time, however, the failure of these fully relativistic models to properly fit such high signal-to-noise spectra demonstrated the need for further theoretical development.

One year later, Reis et al. [45] analyzed another simultaneous XMM-Newton and RXTE dataset from the period of the very high state taken a month later (29 September 2002), as well as a 2004 dataset, during which the source was in its low/hard state. For the joint fit of the 2002 XMM-Newton and RXTE spectra within the 0.7–100 keV interval, instead of a relativistic accretion disk model, they used the refhiden model, which was developed by Ross and Fabian [46]. This is a self-consistent model of reflection from the hot inner portion of an accretion disk, which also includes the blackbody radiation of the disk, in case of a constant density atmosphere in hydrostatic equilibrium. The statistics (${\chi }_{\nu }^2$/dof = 1.48/1718) are still not perfect, but there is definitely a big improvement compared to the quality of our fit in [40], when the kerrbb or bhspec model was used for a similar dataset. The reflection model showed only a weak contribution due to the Fe Kα emission, which they explained with most of the iron being completely ionized. We also did not detect a strong Fe Kα feature during the very high state a month earlier. The obtained high value for the BH spin (0.935 ± 0.01) from the self-consistent reflection model is consistent with previous results.

In 2019, Aneesha et al. [47] studied the dynamical change in the behavior of the source by analyzing data from the outbursts between 2002 and 2011. During this work, they also analyzed the same 2003 dataset which was studied in [40]. They managed to obtain a formally acceptable fit (${\chi }_{\nu }^2$/dof = 1.07/1000), as seen in Figure 8. In their model, they used a standard MCD component (diskbb) plus a power-law component to model the high-energy emission, and they also included a smeared edge (smedge) component. They do not list the obtained parameters specifically for this observation, but they do mention values as high as 30 keV for the smearing with, and 7 for the maximum absorption factor, which seem quite high. They also list power-law indexes as high as 3, which are known to cause problems for the lower energy side of the spectrum. As the diskbb component does not include any relativistic effects, it is not suitable for extracting the most desired information about the system, such as the BH angular momentum and inclination.

The Suzaku satellite performed a long observation of GX 339-4 in 2007 February during its outburst and obtained high quality data in the 0.7–70 keV energy range. In their 2008 paper [48], Miller et al. reported on the analysis of this dataset using various models. First, they used a simple model including a diskbb and powerlaw component, and they omitted the 4–7 keV and 15–40 keV intervals while fitting the spectrum.

Their goal with this questionable practice was to highlight the excess due to the relativistically skewed Fe Kα emission and the Compton reflection hump in the omitted ranges, as seen in Figure 9. The latter is the result of electron downscattering of the high-energy photons on the disk. After this, they replaced the MCD component with the CDID reflection component but did not manage to obtain a formally acceptable fit (${\chi }_{\nu }^2$/dof = 2.34/5076) for this high-quality dataset. Nevertheless, the obtained BH spin (0.89 ± 0.04) seemed to be consistent with previous studies. Their result for the inclination (18 ± 1°), however, was still significantly lower than the constraints established by independent studies.

Garcia et al. [49] used XMM-Newton and INTEGRAL data to study GX 339-4 during its 2006–2007 outburst. The observations covered three different spectral states of the source, from the hard intermediate to the high/soft state. In one of the datasets, a broadened Fe-line emission feature was visible. They attempted to fit the broadband joint spectra as a combination of the pseudo-Newtonian accretion disk model diskpn, which considers a disk extending between 6R_g and 100R_g, and the EQPAIR model, which is a hybrid thermal/non-thermal Comptonization model [50].

This latter component was included to model the high-energy excess in the spectra. The diskpn model is an extension of the standard MCD diskbb model, which includes post-Newtonian corrections for the temperature distribution near the BH. Since the inner disk radius parameter in this model cannot be lower than 6R_g, it can only describe disk emission for a non-spinning BH, or if the disk does not extend all the way to R_ISCO. Moreover, no general relativistic effects are incorporated in the model, as in kerrbb for example. Apart from these issues with the applied model, they also obtained that the BH has a high a* value of about 0.95 by fitting the broadened emission feature by a laor component ($R_{in}=2.11\pm 0.11$).

Ludlam et al. [51] revisited the 2002 XMM-Newton data on the very high state of GX 339-4 and fitted those data jointly with some 2007 Suzaku data within the 1–60 keV energy band. Their model included a standard multicolor disk component (diskbb) and a relxill component, which is a self-consistent reflection model [52], combining a relativistic ray tracing kernel (relline) and an X-ray reflection code (xillver). They managed to obtain a very good fit (${\chi }_{\nu }^2$/dof = 1.23/1472), and their results seemed to be consistent with previous works (a* > 0.97, $i=36\pm 4°$). However, since the source was in the very high state in 2002, and it was in the intermediate state in 2007, it is questionable whether the joint fitting of the two datasets can be warranted. We should also note that the diskbb component used to model the disk emission does not include any relativistic effects, nor does it account for the spectral hardening in any way.

Simultaneous Swift and NuSTAR spectra recorded during the 2015 outburst of GX 339-4 were fitted by Parker et al. [53] in 2016 within the 1–60 keV energy band. The source was in the very high state, showing signs of both a strong disk emission and a high-energy power-law component. In order to fit the broadband spectrum, they employed a model consisting of a kerrbb, a comptt, and a relxilllp component. The latter is a variation of the self-consistent relxill reflection model, assuming a lamp post geometry for the source that illuminates the disk. They managed to obtain a good fit (${\chi }_{\nu }^2$/dof = 1.32/407) using this advanced model (Figure 10), and it is very important that both the kerrbb and the relxilllp component include a parameter related to the BH spin (a* and R_in, respectively). These parameters were tied together during the fit, forcing the two components to give consistent results, but the mass and distance parameters of the kerrbb component were allowed to vary, since they were poorly determined at the time anyway. The obtained results (a* $={0.95}_{-0.08}^{+0.02}$ and $i=30\pm 1°$) were consistent with previous studies, and they even managed to constrain the BH mass and distance quite well ($M={9.0}_{-1.2}^{+1.6}$∙M_ʘ and $D=8.4\pm 0.9$ kpc).

Based on these studies and extensive simulations, in their 2019 paper [54], they showed that multiple physical parameters can be obtained from the disk spectrum, if the high energy region is measured precisely. With the 2012 launch of the NuSTAR satellite, high-sensitivity measurements became possible within the 3–79 keV interval. They also emphasize, however, that high-quality low energy data are also essential for this exercise, especially for low-temperature accretion disks which have emission peaks below 1 keV. According to their simulations, the largest source of systematic error is the unknown value of the color correction factor.

Since the value of f_col can be as small as 1.4 and as large as 2, this results in an uncertainty of $\pm 0.1$ for the spin and $\pm$10° for the inclination. They also concluded that they significantly underestimated the errors for the BH mass and distance in their previous work [53], and that X-ray spectroscopy alone cannot be used to simultaneously constrain the values of the mass, distance, and mass accretion rate. At least one of these parameters must be determined precisely from independent methods, for example the distance to the source using parallax.

In their 2023 paper [55], Liu et al. reported on the hard to soft state transition of GX 339-4 during its 2021 outburst. The source was observed by Insight-HXMT, and the recorded spectra were analyzed in the 2–100 keV energy range for 7 different epochs, as the contribution of the disk gradually increased. First, they applied a simple model for all seven datasets, consisting of an MCD and a cut-off power-law component. Although they did not mention excluding any part of the energy range during the fitting of the spectra, they reported on large excesses at around 6–7 keV and 30 keV for all 7 epochs, similarly to Miller et al. in [48], as seen in Figure 9. Subsequently, they added a relativistic reflection component to their model. They experimented with different members of the relxill model family (relxilllp, relxillD, relxillcp), xillver, and refhiden (see earlier). In addition, they also included the simplcut kernel [56] in order to take the coronal Comptonization of the reflection spectrum into account. They carried out the fitting of the seven spectra simultaneously, and linked those parameters that were not supposed to change between observations, such as the BH spin, the inclination angle, the iron abundance, and the galactic absorption. All of the advanced models gave a good fit to the seven spectra, and both for the BH spin (a* $>0.86$) and for the inclination ($35°\le i\le 43°$), they gave results that were consistent with previous measurements and independent studies. What is contradictory to the commonly accepted picture, however, is that irrespective of the spectral states throughout the seven epochs, they always obtained very low values for the inner radius of the disk, with the exception of epoch 1. This would mean that the disk is not significantly truncated during the low/hard state, and hardly (or abruptly) changes during the transition between spectral states. One would naively assume that if the disk always extends to R_ISCO, it would dominate the X-ray emission all the time. The root of the discrepancy could be the use of the standard MCD component, as opposed to a fully relativistic disk model, and the missing low energy data below 2 keV may also cause the unreliable modeling of the disk contribution, since its peak is expected to be at around 1 keV, or even lower during the low/hard state. The high values for the cut-off energies in case of the cut-off power-law model also seem odd, since they are well above the range of available data.

The measured system parameters for GX 339-4 are summarized in

Table 1. Summary of the measured system parameters of GX 339-4.

Method	Dimensionless BH Spin (a*)	BH Mass (Mʘ)	Inclination (°)	Distance (kpc)
Dynamical measurement	-	4–11	40–60	8–12
X-ray spectral fitting	${0.95}_{-0.08}^{+0.02}$	${9.0}_{-1.2}^{+1.6}$	$30 \pm 1$	$8.4 \pm 0.9$

below, both for the most recent dynamical measurement of the secondary star [28] and for the most physically consistent X-ray spectral fitting [53] according to our opinion.

3. Future Prospects

Based on the highlighted works that were described in the previous section, one can see that significant progress has been made in this field during the last few decades. The combination of the developments in theoretical work and technological advancements will continue to fuel an accelerating march toward better understanding the most extreme locations in the universe. As computational capacity continues to increase, more and more complicated simulations will be possible, and more and more sophisticated models can be fitted to the high-quality data. The very fast development of AI and deep learning will most likely make data analysis easier and more efficient. Currently, we are almost at the point where Einstein’s theory of gravity itself can be put to the test, and possible deviations from the Kerr metric can be detected in the strong field regime. Such a study was reported by Tripathy et al. [57], for example, where they used combined NuSTAR and Swift data on GX-339-4 in the 1–60 keV energy interval to measure the value of the Johannsen deformation parameter ${\alpha }_{13}$. For this, they used modified versions of both the accretion disk model and the reflection model, which are calculated in non-Kerr spacetimes. So far, the results are consistent with the Kerr metric within measurement error, which is obtained for the ${\alpha }_{13}=0$ case. Possible deviations may be detected in the future, when higher quality data become available, and the testing of other modifications, such as rotating bumblebee black holes [58] or the predictions of pseudo-complex general relativity [59]. The parameters of BHs, such as mass and spin, and possible deviations from general relativity can also be tested in the future by studying quasi-periodic oscillations [60] and by creating even more detailed images of supermassive BHs [61] using the planned extended version of the event horizon telescope. For the near future, the dominant driving force behind the progress in this field could be the ongoing construction and development of high-sensitivity, high-precision telescopes. If we can detect the faint secondary star in the optical or IR band, we will get a chance to better constrain some of the system parameters, such as distance, mass, and inclination, the uncertainty of which is usually responsible for our inability to constrain other properties of the system or the metric. If we focus on the X-ray spectroscopic study of LMXBs, future satellites should have good temporal resolution and a large effective area in the 0.1–1000 keV energy band, so that both the faint disk emission at low energies and the non-thermal Comptonization up to hundreds of keV energies in the very high state can be studied in detail. If this very high energy radiation without any sign of a spectral break is really caused by the bulk motion Comptonization by free-falling electrons into the BH, then it would be impossible to observe such a high-energy tail in systems containing NS primaries, and that would give a tool to differentiate between the two sources based on the high-energy X-ray data alone during the very high state. Good timing properties and large effective area in the high-energy region will also allow more precise time-resolved spectroscopy to better study the variable hard X-ray component [10], which may be weak in the high/soft state, but it still has to be precisely determined, and it can still show some level of fast variability, even in LMXBs. The different rate of variability in the thin disk and the hot corona can help better disentangle these two components in future studies.