Insight-HXMT Study of the Inner Accretion Disk in the Black Hole Candidate EXO 1846–031

Following the MAXI/GSC discovery of a new outburst of EXO 1846–031, we triggered Insight-HXMT tartget-of-opportunity observations, which covered 85 days, from 2019 August 2 to October 25 (Table LABEL:tab:obsinf).

Insight-HXMT (Zhang et al., 2014, 2020b) is the first Chinese X-ray astronomy satellite, launched on 2017 June 15. It has a 550-km low-Earth-orbit with an inclination of $43^{\circ}$. It contains three slat-collimated instruments, sensitive to different energy ranges: the High Energy (HE) (Liu et al., 2020), Medium Energy (ME) (Cao et al., 2020), and Low Energy (LE) (Chen et al., 2020) X-ray Telescopes. HE, ME and LE are sensitive to the 20.0–250.0 keV, 5.0–30.0 keV and 1.0–15.0 keV bands, with detection areas of 5100 ${\rm cm}^{2}$, 952 ${\rm cm}^{2}$ and 384 ${\rm cm}^{2}$, respectively. The corresponding time resolutions are 4 ${\mu}$s, 240 ${\mu}$s and 1 ms.

We used the Insight-HXMT Data Analysis software (hxmtdas) v2.02¹¹1http://hxmt.org/software.jhtml to analyze all the data. The filtering criteria for the good time intervals were: (1) an offset angle from the pointing direction $<\ 0.04^{\circ}$; (2) a pointing direction above earth $>\ 10^{\circ}$; (3) a value of the geomagnetic cutoff rigidity $>$ 8 GeV; (4) a rejection of data within 300 s of the South Atlantic Anomaly passage. A detailed explanation of the Insight-HXMT data reduction is published on its website²²2http://hxmt.org/SoftDoc.jhtml. Backgrounds rates were estimated with the tools: lebkgmap, mebkgmap, hebkgmap (version 2.0.9), based on the standard Insight-HXMT background models (Liao et al., 2020a; Guo et al., 2020; Liao et al., 2020b). We obtained the response files with the tasks lerspgen, merspgen and herspgen.

We rebinned the spectra to a minimum of 30 counts per bin using the ftools (Blackburn, 1995) task grppha. We modelled them with xspec Version 12.10.1³³3https://heasarc.gsfc.nasa.gov/docs/xanadu/Xspec/ (Arnaud, 1996). We used the $\chi^{2}$ statistics, and added a systematic uncertainty of 1.5%. The energy bands adopted for our spectral analysis are: 1.0–10.0 keV (LE), 10.0–30.0 keV (ME) and 30.0–150.0 keV (HE). We jointly fitted the spectra of the three instruments and included a multiplicative constant to account for the relative flux calibration uncertainties (Li et al., 2020). All spectral models include a photoelectric absorption, modelled with TBabs and ‘wilm’ abundances (Wilms et al., 2000). The results of our spectral analysis are presented in the Section 3.2. We only report results from observations with an exposure time longer than 200 s. All uncertainties are quoted at the 90% confidence level.

Figure 1 shows the long-term light curves in different energy bands and the evolution of the LE hardness ratio (defined as the 4–10 keV count rate divided by the 2–4 keV rate). The HE and ME count rates show a similar evolution: first a decrease with time and then a relatively low, stable level (Figure  1). Instead, the LE light curve shows a more complex evolution, with two peaks. The LE count rate rapidly increased after MJD 58697.35, reached a peak value of $220.5\pm 0.4$ cts s^-1 on MJD 58705.53, and then decreased gradually. Then, it increased again during the SIMS and finally decreased to a relatively low value ($79.7\pm 0.3$ cts s^-1) during the HSS. We do not know the exact reason for the formation of these two peaks. From the LE hardness ratio we see that the spectrum gradually softened with time (bottom panel of Figure 1).

Liu et al. (2021) subdivided the outburst into four spectral states based on the relative changes in the hardness-intensity diagram and the fractional rms integrated over the $2^{-5}$–32 Hz band. In the following spectral analysis, we will adopt their spectral state classification, except for their definition of the LHS. Their preliminary classification was based on hardness ratios and included only the first three Insight-HXMT observations in the LHS. Here, we used spectral analysis to obtain a more accurate classification of this state. We adopted the canonical definition of LHS as the epochs when the photon index $\Gamma\sim$ 1.5–1.7 and the disk flux fraction $f_{\rm dbb}\leq$ 20% (Remillard & McClintock, 2006; Belloni, 2010). With this definition, the LHS interval includes the first eight Insight-HXMT observations (Section 3.2.1).

The hardening factor (also called color correction factor) $f_{\rm col}$ accounts for the deviation of the disk emission from a pure blackbody spectrum (Shimura & Takahara, 1995). The peak effective temperature $T_{\rm eff}$ is related to the fitted color temperature $T_{\rm in}$ by the relation $T_{\rm eff}=T_{\rm in}/f_{\rm col}$. Conversely, the relation between physical inner radius, $R_{\rm in}$, and apparent inner radius, $r_{\rm in}$, is:

(Kubota et al., 1998), where $\xi=0.412$ is the general relativity correction factor, $N_{\rm dbb}$ is diskbb normalization, $\theta$ is the inclination angle of the accretion disk and $D_{\rm 10}$ is the distance in units of 10 kpc. We can plausible assume that $R_{\rm in}\approx R_{\rm ISCO}$ and $f_{\rm col}\approx 1.7$ (”canonical” value from Shimura & Takahara 1995) in the SIMS and at least near the peak of the HSS, before the start of the decline (i.e., MJD 58717–58745). We adopt an inclination angle of $73^{\circ}$ (Draghis et al., 2020). Instead, the distance is completely unconstrained, in the absence of a detected optical counterpart⁵⁵5The value of 7 kpc introduced by Parmar et al. (1993) is only a guess based on a now out-of-date analogy with the luminosity of other BH transients.. The fitted value of $r_{\rm in}$ in the first epoch of the HSS is $59.0\pm 0.5$ $D_{\rm 10}$ km, and this value is indeed substantially unchanged throughout MJD 58717–58745. Then, from Eq.(1), $R_{\rm ISCO}=70.3\pm 0.6\,D_{\rm 10}$ km. In the LHS and HIMS, the physical inner radius $R_{\rm in}$ must be at least as large as $R_{\rm ISCO}$ (it may be equal if there is already a condensed thin disk in the innermost region). Thus, in our proposed scenario, we can determine the minimum value of $f_{\rm col}$ required to make $R_{\rm in}\geq R_{\rm ISCO}$ in the early epochs. We obtain $f_{\rm col}\gtrsim 2.7$ in the first few days of LHS observations. In the last part of the LHS and HIMS, $f_{\rm col}$ decreases to $\approx$1.7–2.4. Conversely, in the declining phase of the HSS, a canonical value $f_{\rm col}=1.7$ implies $R_{\rm in}\geq R_{\rm ISCO}$. This can be interpreted in two alternative ways: either the inner disk was starting to recede from ISCO after MJD 58745, or it was still at ISCO but the hardening factor declined to $f_{\rm col}\approx 1.6$. Finally, we derived the effective temperature $T_{\rm eff}=T_{\rm in}/f_{\rm col}$. We find that with our estimated hardening factors, $T_{\rm eff}$ increased monotonically during the LHS (as expected) instead of showing an unphysical decline in the first part of the outburst. We show the evolution of radii, temperatures and $f_{\rm col}$ in Figure 4.

The early stages of transient BH outbursts are characterized by the coupled evolution of a corona and of the underlying disk. For EXO 1846$-$031, the evolution of the coronal parameters follows the canonical expectations. For example, the scattering fraction $f_{\rm sc}$ follows the same trend seen in recent outbursts of well-known BH transients such as GX 339–4 (Sridhar et al., 2020), XTE J1550–564 (Connors et al., 2020) and 4U 1630–47 (Connors et al., 2021). Instead, the evolution of the disk parameters may have subtle differences.

Our spectral modelling shows (Section 3.2) that if the hardening factor $f_{\rm col}$ is kept constant from LHS to HSS, the inner radius of the disk is smaller in the LHS than in the HSS. We reject this possibility as unphysical: the inner disk radius in the LHS must be at least as large as in the HSS. If we assume that the inner disk radius is constant throughout the entire outburst, $f_{\rm col}$ must be decreasing from $\approx$2.7 at the beginning of the outburst, towards the canonical value of $\approx$1.6 in the HSS. With this assumption, we preserve the scaling law $F_{\rm dbb}~{}\propto T_{\rm eff}^{4}$ throughout the outburst, characteristic of a radiatively efficient disk with constant inner radius. Finally, if we also allow for the possibility that $R_{\rm in}>R_{\rm ISCO}$ in the LHS, the hardening factor must be even higher at the beginning of the outburst ($f_{\rm col}>2.7$).

In previous studies of BH transients, a “canonical” value of $f_{\rm col}\approx 1.7$ (from Shimura & Takahara 1995) was usually adopted for the soft state, to account for a “diluted” disk spectrum. However, there are several observations of BH outbursts that show convincing evidence of a variable hardening factor, especially in the initial hard state. For example in GX 339-4, Salvesen et al. (2013) proposed that a variable hardening factor is an alternative to disk truncation, to explain changes in the disk spectrum. Other sources with possible evidence of variable hardening factor are 4U 1957$+$11 (Maitra et al., 2014), MAXI J1820$+$070 (Guan et al., 2021), and MAXI J1348$-$630 (Zhang et al., 2022). From simulations of disk spectra, Merloni et al. (2000) also suggested a variable hardening factor $f_{\rm col}\approx 1.7$–3, arguing that $f_{\rm col}$ increases when the disk emission is relatively less dominant. This result is supported by a global study of disk properties by Dunn et al. (2011). They found that for almost all BH transients, $f_{\rm col}$ is relatively stable in the disk-dominated state, but increase from 1.6 to 2.6 as the disk fraction decreases. The variation of $f_{\rm col}$ may be caused by changes in the accretion energy dissipation (Merloni et al., 2000), in the vertical disk structure (Davis et al., 2005), in the magnetic energy density (Blaes et al., 2006) and in the disk density and optical depth (Soria et al., 2008).

The presence or absence of a full disk has important implications also for the interpretation of QPOs. LFQPOs with a centroid frequency between 0.7 and 8 Hz have been detected in the HIMS of EXO 1846–031 by Liu et al. (2021) from a timing analysis of the same Insight-HXMT data used in this work. If there was already a thin disk at or very close to ISCO, as we have suggested here, this geometry disfavours QPO models (e.g., Stella & Vietri 1998; Ingram et al. 2009; Varniere & Vincent 2016) that require significant changes in the inner disk edge. Instead, jet precession would be a viable model. The presence of a jet in EXO 1846$-$031 was confirmed by VLA (Miller-Jones et al., 2019) and MeerKAT (Williams et al., 2019) detections in the HIMS. By comparison, the BH transient MAXI J1820$+$070 is another source in which LFQPOs have been detected in the LHS (e.g., Wang et al. 2020; Ma et al. 2021) and the disk is thought to be non-truncated based on the reflection-fitting method (Buisson et al., 2019). A likely interpretation for the origin of the QPOs is jet precession (Ma et al., 2021).

Done & Davis (2008) reported a detailed investigation of $f_{\rm col}$ in a disk-dominated state. Their work shows that $f_{\rm col}$ increases as the mass accretion rate $\dot{M}_{\rm acc}$ increases, which links the variation of $f_{\rm col}$ to the mass accretion rate. To examine if this relationship is still valid in a hard state, we plot $f_{\rm col}$ as a function of $\dot{M}_{\rm acc}$ in the LHS, HIMS and most of the HSS in Figure 5a. Data points between MJD 58717 and 58745 are excluded here since we have assumed a constant $f_{\rm col}$ in this period (we explained this in Section 3.3). It is clear that $f_{\rm col}$ and $\dot{M}_{\rm acc}$ indeed show a positive correlation in the HSS, which is in good agreement with the results of Done & Davis (2008). However, the values of $f_{\rm col}$ in the LHS and HIMS where $\dot{M}_{\rm acc}$ are relatively low, is significantly higher than that in the HSS. We thus suspect that there should be additional contributions to the $f_{\rm col}$ in the LHS and HIMS.

We show the relationship between $f_{\rm col}$ and power-law flux $F_{\rm pl}$ in Figure 5b, in which $f_{\rm col}$ increases as $F_{\rm pl}$ increases throughout the outburst. Apparently, $F_{\rm pl}$ gradually decreases as the source evolves from the LHS to the HIMS and becomes negligible in the HSS. This implies that the high value of $f_{\rm col}$ in the LHS and the HIMS is probably associated with the hard emission. Compared to HSS, the additional power-law emission in the hard state will heat up the disk surface, leading to an increase in the disk temperature, which results in a significant increase in $f_{\rm col}$. Similar results that $f_{\rm col}$ increases as the disk fraction decreases have been reported by Merloni et al. (2000) and Dunn et al. (2011). We plot $f_{\rm col}$ versus $F_{\rm pl}$ in Figure 5b, which is to emphasize the contribution of the power-law flux when the source was in the LHS and HIMS.

Using the reflection-fitting method, Draghis et al. (2020) obtained a spin parameter as high as $a_{*}=0.997^{+0.001}_{-0.002}$. They applied 12 reflection models to the NuSTAR spectra to test this value, with different assumptions on coronal geometry, disk density and emission spectrum, and obtained consistent spin values. However, Wang et al. (2021) studied the same NuSTAR data with two reflection models and argued that the spin cannot be constrained. Here, we try to constrain the spin parameter with the continuum-fitting method, and test whether the result is consistent with the extreme value claimed by Draghis et al. (2020).

As discussed in Section 3.3, the inner disk radius reached the ISCO in the HSS, with $R_{\rm ISCO}\approx 70\,D_{10}$ km (for $f_{\rm col}=1.7$ and $\theta=73^{\circ}$). We apply the relationship between the ISCO radius and the spin parameter (Zhang et al. 1997a; see also Bardeen et al. 1972):

where $r_{\rm g}=GM_{\rm BH}/c^{2}$, $A_{1}=1+(1-a_{*}^{2})^{1/3}[(1+a_{*})^{1/3}+(1-a_{*})^{1/3}]$ and $A_{2}=(3a_{*}^{2}+A_{1}^{2})^{1/2}$. Thus, from Equation 2, we can derive $a_{*}$ as a function of $M_{\rm BH}$ and distance. Since neither the BH mass nor the distance is known, we determined $a_{*}$ over a grid of values spanning a mass range of 5–15 $M_{\odot}$ (based on the distribution of kinematic masses in Galactic BH transients; Özel et al. 2010; Corral-Santana et al. 2016) and a distance $D$ of 2–12 kpc (based on the distribution of Galactic BH distances in Tetarenko et al. 2016).

The values of the spin parameter versus BH mass for different distances are shown in Figure 6a. The extreme value of $\emph{a}_{*}\approx 0.997$ suggested by Draghis et al. (2020) is consistent only with a small distance ($D\approx 2$–4 kpc) and relatively high mass ($M_{\rm BH}>7M_{\odot}$). However, this combination of mass and distance would imply a peak Eddington ratio $\lambda\equiv L_{\rm disk}/L_{\rm Edd}\approx 0.02$–0.04 in the HSS. This is much lower than the typical X-ray luminosities observed in the disk-dominated states of other Galactic BHs (Tetarenko et al., 2016). We also calculate the spin parameter when the inclination is assumed to be $40^{\circ}$ (Wang et al., 2021), as shown in Figure 6b. In this case, the extreme value of $\emph{a}_{*}\approx 0.997$ is consistent with the combination of distance ($D\approx 2$–6 kpc) and mass ($M_{\rm BH}\approx 5$–14 $M_{\odot}$). The corresponding peak Eddington ratio would then be $\approx$0.1 but is still lower than expected and observed in most other systems (Tang et al., 2011; Yan & Yu, 2015). For a higher Eddington ratio, the spin parameter has to be non-extremal.

Previous work has already reported the inconsistency of the spin measurements obtained for the same systems with the reflection-fitting and continuum-fitting methods. For example, for GRO J1655$-$40, the reflection-fitting method provides a higher spin than the continuum-fitting method (Shafee et al., 2006; Reis et al., 2009; Reynolds, 2014), similar to what we found here for EXO 1846$-$031. Allowing for uncertainties in $f_{\rm col}$ at a level of $\pm$0.2–0.3 can alleviate the discrepancy (Salvesen & Miller, 2021); this effect is more pronounced for non-extremal spins. For example, if we assume a smaller value of the hardening factor, $f_{\rm col}=1.4$, we can reach $a_{*}=0.997$ for $D\approx 5$–6 kpc and $M_{\rm BH}\gtrsim 12M_{\odot}$ at $\theta=73^{\circ}$, or $D\approx 4$–9 kpc and $M_{\rm BH}\approx 6$–14 $M_{\odot}$ at $\theta=40^{\circ}$. The corresponding peak Eddington ratio would then be $\approx$0.09 and $\approx$0.22, respectively.

In general, for the same continuum parameters, lower values of $f_{\rm col}$ imply higher Eddington ratios. We conclude that, in principle, the inconsistency of the spin parameter measured with the continuum-fitting and reflection-fitting methods can be somewhat mitigated by the choice of $f_{\rm col}$. However, without knowing its mass and distance, we cannot draw a further conclusion about the spin parameter of the BH in EXO 1846–031.