Conflicting Disk Inclination Estimates for the Black Hole X-Ray Binary XTE J1550−564

Using the tool xselect, heasoft-v6.22.1, we extracted GIS 2 and 3 spectra from the available archival screened (using standard screening) ASCA event files. We first extracted a source spectrum from a circular region of 6' radius centered on the source. We then extracted an off-source spectrum of equal size to represent the ASCA background. Upon inspection of both the GIS 2 and GIS 3 light curves, we noticed dips in the count rate that appeared to be left after applying standard filtering (see Figure 2), likely explained by Earth occultation during satellite orbit. We manually excluded these portions of the light curve. We obtained the response matrix files for GIS 2 and 3, respectively, gis2v4_0.rmf and gis3v4_0.rmf, from the archive and used the FTOOL ascaarf to generate an ancillary response file for the selected region of the source spectrum. Finally, we corrected the GIS spectra for dead time using the FTOOL deadtime in accordance with Makishima et al. (1996). During extraction, we grouped the source spectrum by a factor of 4 such that there are 256 channels in both the GIS spectra. The GIS 2 and 3 spectra were then further combined into an average GIS spectrum, ancillary response, and background using the tool addascaspec. This combination is motivated by the fact that the GIS calibration against Crab spectra was performed using a coadded GIS 2 + GIS 3 spectrum (Y. Ueda 2019, private communication).

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During 1998 September 23–24, a simultaneous, roughly 3 ks observation of XTE J1550−564 was taken with RXTE, and archival PCA and HEXTE (cluster A and B) data are publicly available. We extracted data from all Proportional Counter Units (PCUs) of the PCA detector and both HEXTE clusters, discarding data within 10 minutes of the South Atlantic Anomaly (SAA). For the PCA, we focus on PCU 2 alone, given its superior calibration over the other four PCUs. The PCU 2 spectrum has been corrected using the tool pcacorr (García et al. 2014), and 0.1% systematics have been added to all channels accordingly. We then ignore the PCU 2 counts in channels 1–4 and above 45 keV. This limits the spectrum to roughly the 3–45 keV range. The HEXTE spectrum is also included, clusters A and B. The HEXTE B spectrum is corrected for instrumental effects using the tool hexBcorr (García et al. 2016b) analogously to the corrections made to the PCU 2 spectrum. We group both HEXTE spectra by factors of 2, 3, and 4 in the 20–30, 30–40, and 40–250 keV ranges, respectively, in order to achieve an oversampling of ∼3 times the instrumental resolution and group at a signal-to-noise ratio of 4. We constrain the HEXTE spectra to 25–200 keV after noticing that the 20–25 keV region shows a spectral turnover at odds with the PCA data.

Adopting a slope correction between the GIS and PCA instruments, we now fold in the RXTE-HEXTE data and apply our final model:

• 1.  
  Model 3:crabcorr∗TBabs(nthComp+diskbb+relxillCp+xillverCp+(powerlaw∗expabs)).

We fit model 3 to the full broadband 1–200 keV spectrum. Here we have replaced the Gaussian model, previously used to take into account the additional narrow-line residuals in the iron line region, with the xillverCp model. This represents a distant reflection component, undistorted by the general relativistic effects associated with a reflected component lying close to the black hole horizon (accounted for by relxillCp). We tie the iron abundance of relxillCp and xillverCp together, since both reflectors are components in the same accretion disk, and we fix the log ionization of the distant reflector (xillverCp) to log ξ = 0 (i.e., an almost neutral gas), since the irradiating flux impinging on the distant reflector is expected to be orders of magnitude lower than the one that strikes the inner regions of the disk. The ionization of the relativistic reflection component (relxillCp) is a free parameter. We continue to fix the black hole spin to a_⋆ = 0.5 (Steiner et al. 2011), given the lack of strong constraints on its value when allowed to vary freely (for example, see the second column of Table 1). As such, the disk inner radius is also kept fixed at the innermost stable circular orbit (ISCO). The temperature of the photon distribution impinging on the corona is set to the disk temperature of the diskbb component (kT_BB), and Γ, the photon index, is tied between the nthComp, relxillCp, and xillverCp components.

In addition to the IC component and two reflection components, we also include an additional high-energy tail in the model, with a low-energy cutoff at 20 keV. This component represents the evidence for nonthermal electrons in a hybrid plasma, leading to a nonthermal IC tail, making our model similar to that adopted by Hjalmarsdotter et al. (2016). This high-energy component is required by the HEXTE spectra with a low statistical significance, and we include it mostly as a nuisance component, remaining indifferent to the physical implications. As such, we fix the index, Γ_pl = 1.7, in accordance with both standard diffusive shock acceleration theory (see, e.g., Drury 1983) and the slope of the injected nonthermal particle distribution in the modeling performed by Hjalmarsdotter et al. (2016). We cut the low-energy portion of the high-energy tail off at 20 keV to ensure limited degeneracy with the components modeling the iron line region, since this additional power law is unconstrained in the lower-energy regions. The choice of 20 keV is an appropriate one because it allows us to model out the high-energy tail, but it does not introduce confusion in the Fe K line region. Since the addition of this component clearly means that the cutoff energy of the thermal pool of electrons in the corona does not properly describe the limiting photon energy of the disk irradiator for reflection, we also fix the electron temperature in relxillCp and xillverCp to 200 keV. As discussed by García et al. (2015a), the cutoff energy for reflection can be constrained by, and thus has a strong impact on, the lower-energy portion of the reflection spectrum. It is therefore important to consider the high-energy radiation and make sure the reflection model knows about it. Our choice to fix the cutoff at higher energies than given by the temperature of the thermal IC component does have a drawback, since the high-energy tail has a harder spectral slope than the nthComp component; however, this choice is the most consistent of the two.

We first fit the full spectrum and run an error analysis, all using XSPEC v.12.10.0c (Arnaud 1996), and then make use of the EMCEE (Foreman-Mackey et al. 2013a) Markov chain Monte Carlo (MCMC) routine with a simple PYXSPEC wrapper to better explore the parameter space of the model. In this way, we can search for any possible multimodality to the fit, with particular attention paid to possible correlations between the disk inclination and other model parameters. We initialize 100 walkers per free parameter, run the MCMC exploration for 1 million steps (for every walker), and subsequently burn the first 30% of the run to ensure that the final distributions are fully converged with limited noise. We place flat log prior distributions on all normalization parameters in order to ensure sensible walker step sizes in the MCMC chains.

Figure 6 shows the final model fit to the full 1–200 keV ASCA and RXTE spectrum, based on our best fit using the standard Levenberg–Marquardt algorithm in XSPEC, and Table 2 shows the final parameters and their 90% confidence limits after running the full MCMC routine. Most strikingly, we find that, in accordance with the results of fitting only the PCA data with model 2, the disk inclination is constrained to a much lower value ($i={39.2}_{-0.9}^{\circ +0.9}$) than that given by the orbital inclination measurement of 75° ± 4°. Figure 7 shows 2D contours that demonstrate the correlations (or lack thereof) between the disk inclination i and other key parameters. There is evidence for a weak positive correlation between i and the coronal photon index, Γ, as well as anticorrelations between i and the disk ionization (log ξ) and iron abundance (A_Fe), but all of these key parameters are well constrained, such that we can safely conclude that the inclination of the disk must be low. Figure 9 in Appendix A shows the full 1D posterior distributions of every free parameter of the model, as well as all of the 2D contours that show potential parameter correlations. One can see that there are not many clear correlations between our parameters.

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Table 2.  Median Parameter Values for Fits of Model 3 to the ASCA and PCA/HEXTE 1998 September 23 Observations of XTE J1550−564, Calculated from the Final Posterior Probability Distributions Resulting from the MCMC Chain

Parameters	Model 3
	crabcorr*TBabs(nthComp+diskbb
	+relxillCp+xillverCp+(pl*expabs))
${N}_{\mathrm{crabcorr},\mathrm{GIS}}$	0.97a
${N}_{\mathrm{crabcorr},\mathrm{PCA}}$	${1.23}_{-0.01}^{+0.01}$
${N}_{\mathrm{crabcorr},\mathrm{HEXTEA}}$	${1.19}_{-0.02}^{+0.02}$
${N}_{\mathrm{crabcorr},\mathrm{HEXTEB}}$	${1.19}_{-0.02}^{+0.02}$
${\rm{\Delta }}{{\rm{\Gamma }}}_{\mathrm{crabcorr},\mathrm{GIS}}$	−0.01a
${\rm{\Delta }}{{\rm{\Gamma }}}_{\mathrm{crabcorr},{RXTE}}$	${0.090}_{-0.006}^{+0.006}$
NH (1022 cm−2)	${0.928}_{-0.009}^{+0.007}$
Γ	${2.13}_{-0.01}^{+0.01}$
kTe	${5.7}_{-0.2}^{+0.3}$
Nnth	${3.8}_{-0.2}^{+0.2}$
kTbb (keV)	${0.625}_{-0.007}^{+0.004}$
Ndbb	${7100}_{-200}^{+400}$
q	3a
${a}_{\star }$	0.5a
$i^\circ$	${39.2}_{-0.9}^{+0.9}$
AFe	${3.9}_{-0.6}^{+1.8}$
$\mathrm{log}\xi$	${4.44}_{-0.07}^{+0.21}$
Nrel	${0.027}_{-0.001}^{+0.002}$
Nxil	${0.021}_{-0.004}^{+0.003}$
${{\rm{\Gamma }}}_{\mathrm{pl}}$	1.7a
Npl	${0.22}_{-0.03}^{+0.04}$
Eexpabs	20a
${\chi }^{2}$	1075
ν	878
${\chi }_{\nu }^{2}$	1.22

Note. All limits are shown at 90% confidence. Parameter descriptions: N_crabcorr = Crab correction to normalization, shown for each detector; ΔΓ_crabcorr = Crab correction to slope, shown for each detector; N_H = galactic hydrogen column density; Γ = IC photon index; kT_e = coronal electron temperature; N_nth = normalization of nthComp; kT_BB = disk blackbody temperature; N_dbb = disk normalization; q = emissivity profile index; a_⋆ = black hole spin; i° = disk inclination; A_Fe = iron abundance; log ξ = disk ionization; N_rel = normalization of relxillCp; N_xil = normalization of xillverCp; Γ_pl = photon index of high-energy tail; N_pl = normalization of high-energy tail; E_expabs = low-energy cutoff of high-energy tail; χ² = total χ²; ν = degrees of freedom; ${\chi }_{\nu }^{2}={\chi }^{2}/\nu .$

^aFixed parameter.

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The reason for such a strict constraint on the inclination is geometrical in nature and a result of the relativistic Doppler broadening of the spectral reflection features, in particular the Fe K emission. Figure 8 demonstrates the effect of raising the inclination parameter in the model component relxillCp to 75°. The more the disk is inclined, the more the blue wing of the iron line increases in flux due to Doppler broadening, dramatically altering the shape of the spectrum in the ∼6–10 keV region. Given the high count rates of the source in this bright hard-intermediate state, the data plainly rule out a high-inclination relativistic reflection component.

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For completeness, we also perform the following tests. First, we fit model 3 with both the inner disk radius (R_in) and emissivity index (q) free to vary. We do this in order to test for additional degeneracies with inclination that may be associated with the relativistic effects both parameters generate in the Fe K line region. We present a brief description of these results in Appendix B, but to summarize, neither R_in nor q can be constrained; both are consistent with the fixed values previously chosen (R_in = R_ISCO, q = 3). We also find that the inclination is still entirely consistent with the value shown in Table 2. Second, we tested the effect of allowing both to vary freely with the inclination fixed at 75°, the orbital inclination. Again, we find that R_in and q are unconstrained at 3σ confidence, with R_in constrained to below a few times R_ISCO at 2σ confidence. We also obtain a best-fit χ² = 1113/877 = 1.27. We conclude from these tests that there is a strong preference, regardless of the freedom allowed in the fit of model 3, for the inclination of the disk of XTE J1550−564 to be low (i ∼ 40°).

We also find that the coronal electron temperature is very low (${{kT}}_{{\rm{e}}}={5.8}_{-0.4}^{+0.3}$), which agrees with the results of Hjalmarsdotter et al. (2016), who applied various flavors of hybrid plasma models to spectral data taken within the same period with both ASCA and RXTE. We note, however, that comparison of our modeling results with those of Gierliński & Done (2003) and Hjalmarsdotter et al. (2016)—despite the concurrence in the observational state of XTE J1550−564 and the instrumental data modeled—has its limitations due to the methods used to group data together. Both these works made use of coadded PCA spectra comprised of individual exposures taken within a roughly 2 week period between 1998 September 23 and October 6. The authors justified this grouping of the data by arguing that the X-ray spectral hardness of XTE J1550−564 does not vary significantly in this time frame. However, after careful inspection, we found that, at least for the purposes of reflection studies, the spectral hardening (ΔΓ ∼ 0.1) in this window must be accounted for, particularly for data with Crab-level count rates in the PCA detectors. In fact, the choice to group the spectra in this way actually hides the distinct disagreement in spectral slope between the ASCA-GIS and RXTE-PCA observations, which coincidentally happens to be around an offset of 0.1 in the photon index. Our modeling manages to account for the cross-calibration differences between the ASCA and RXTE data in a way that minimizes loss of interpretation of the physics.

The concept of misalignment between the inner and outer regions of an accretion flow goes back to the idea that viscous torques in the disk, along with Lense–Thirring precession (Lense & Thirring 1918) due to misalignment between the black hole spin and the disk plane, will cause the inner disk to line up with the spin axis (Bardeen & Petterson 1975). The term "warp" arises due to the transition between an outer disk region aligned with the orbit and an inner disk region aligned with the black hole spin axis. Thus, the extent of this warp depends on the extent of this misalignment between the spin axis and the orbital axis. While it is therefore possible to detect misalignment between the inner disk (the inner ∼10 r_g region principally responsible for the relativistic reflection component in the spectrum) and the outer disk, or orbit, the likelihood of this being true of XTE J1550−564, where we see a discrepancy of >30°, depends on two key factors: (i) the probability of the binary system forming with such a large misalignment between the black hole spin axis and the binary plane and (ii) the timescale for this misalignment to diminish due to accretion.

Factor (i), the probability of a binary system forming with high spin–orbit misalignment, can be roughly predicted through a population synthesis study, such as the one conducted by Fragos et al. (2010). In their work, the authors showed that since those binaries that survive the black hole–forming supernova explosion of one of the stellar components tend to be those with smaller "kicks," the resulting distribution of spin–orbit misalignments is highly skewed toward low angles, <10°. However, this still leaves a small proportion of all Galactic BHBs that will have very high spin–orbit misalignments, even up to 90°. Thus, although we should expect a small misalignment between the spin and orbital axes in most BHBs, we cannot rule out the seemingly atypical ∼30°–40° in XTE J1550−564 implied by our reflection modeling results.

Factor (ii), the prediction that the process of accretion onto the black hole should result in the alignment of the spin and orbital axes in BHBs, has been estimated in multiple theoretical studies (e.g., Natarajan & Pringle 1998; Maccarone 2002; Martin et al. 2008). Natarajan & Pringle (1998) calculated the timescale for the spin and accretion disk axes in active galactic nuclei (AGNs) to align, based on earlier work by Papaloizou & Pringle (1983) showing that warps travel in the disk in a way that is governed by the internal hydrodynamics of the disk. This timescale, t_align, is proportional to the black hole spin and disk viscosity and inversely proportional to both the source luminosity and black hole mass, though the mass dependence is very weak.

Maccarone (2002) showed that, based on this formalism, black holes formed with relatively low spin have alignment timescales of ∼10% of the binary lifetime, t_bin. We can tune this formula to the specific case of XTE J1550−564, giving

$$\begin{eqnarray}&&\displaystyle \frac{{t}_{\mathrm{align}}}{{t}_{\mathrm{bin}}}=0.003\,{a}_{\star ,0.5}^{11/16}{\alpha }_{0.03}^{13/8}{L}_{0.1{L}_{{\rm{E}}}}^{1/8}{M}_{9}^{15/16}{\epsilon }_{0.08}^{7/8}{M}_{2,\ 0.3}^{-1},\end{eqnarray} \tag{ 1 }$$

where a_⋆,0.5 is the black hole spin, assumed to be 0.5; α_0.03 is the disk viscosity parameter; L is the luminosity in units of 10% L_E, the Eddington luminosity; M₉ is the black hole mass tuned to 9 M_⊙; _0.08 is the accretion efficiency, assumed to be roughly 8% in the case of XTE J1550−564; and M_2,0.3 is the mass of the donor star, determined to be ∼0.3 M_⊙ (Orosz et al. 2011). Adopting the more up-to-date analytical calculation of Martin et al. (2008), we find that this fraction increases by roughly a factor of 2–3. Nonetheless, the spin axis should, in principle, align with the orbit within the binary lifetime, and one should not expect to see such a large misalignment. Indeed, Steiner & McClintock (2012) used similar principles to support their finding that the spin axis (and thus jet launching axis) is aligned with the binary orbit in XTE J1550−564.

Regardless of the apparent likelihood that the inner disk and jet of XTE J1550−564 are aligned with the binary orbit, the comparatively low measurement we have obtained from our reflection modeling naturally leads us to consider a geometrical scenario that could explain the disparity. The current breeding ground for physical estimates of the behavior of misaligned disks lies in computationally expensive general relativistic magnetohydrodyamic (GRMHD) simulations (e.g., Fragile et al. 2007; Liska et al. 2018, 2019) that can capture the fluid dynamics of a plasma with magnetic fields and strong gravity. One such code adopted to simulate black hole accretion, H-AMR, is now being applied to the spin–orbit misalignment problem, with the goal of characterizing the entire jet–corona–disk geometry (Liska et al. 2018, 2019). The results thus far indicate that in the very inner regions (<10 r_g), the jet, disk, and corona all align with the black hole spin axis. However, what is rather more noteworthy is that Liska et al. (2018) found that the disk remains at least partially aligned with spin axis out to further distances than both the jet and corona, not approaching full alignment with the plane of the feeding torus until tens of r_g. An inner disk that is misaligned in this way would naturally lead to a lower inclination in the reflection modeling, since the strongest portions of the reflection spectrum occur in those inner regions.

Liska et al. (2018) first investigated the effects of applying a relatively minor (10°) spin–orbit misalignment, which is difficult to relate to the apparent >30° misalignment we find here for XTE J1550−564, whereas Liska et al. (2019) extended this work to the exploration of such large misalignments, as high as ∼65°, and a much thinner disk (H/R < 0.03). At such high misalignments and low disk thickness, the inner disk is seen to tear away from the broader accretion flow. The inner disk and jet both precess rapidly, with both the outer disk and jet aligning with the binary orbit.

Strong caveats arise when drawing any connection between our reflection modeling results and the more complex/involved simulations results of Liska et al. (2018, 2019). Such simulations tell us little about how radiative losses and a full ray-tracing treatment may impact the resultant geometry and thus also what the reflection spectrum may look like. In addition, GRMHD simulations are giving us insight into the physics of accretion in the presence of strong, dynamically important magnetic fields. Furthermore, the misalignment of the jet/corona/disk system depends strongly on the assumed disk thickness, which is another element of the geometrical setup we do not consider in any detail in our analysis (the disk is approximated as a slab in the reflection model relxill). Thus, the details of how dynamically important magnetic fields may impact the geometry and thus what the reflector sees, as well as the observer, and the role of disk thickness on both the degree of misalignment and the resultant shape of the Fe K line in the reflection spectrum are still entirely open questions.

Observational constraints on the jet inclination over time also seem to rule out a warped inner jet, at least. The extensive years-long radio observations of the transient jet formed by XTE J1550−564 show no signs of precession (Corbel et al. 2002). The 7 Crab flare that preceded the detection of the transient jet, and was thus presumed to coincide with the launching window of the jet, lasted roughly 1 day, significantly longer than the timescale of the predicted precession of the inner ∼10 r_g of the accretion flow if it were misaligned with the orbit (basic size scale estimates give precession timescales orders of magnitude shorter than the daylong launching window; see, e.g., Fragile et al. 2007; Ingram et al. 2009). The precession of these inner regions should be observable as a significant angular variation in the outer jet, yet this is not observed. A good test case to compare to is the recent result concerning BHB V404 Cygni, in which high-resolution radio observations of its transient jet during a bright 2015 outburst reveal precession on minute timescales (Miller-Jones et al. 2019).

If the inner regions of the accretion flow, as implied by the wealth of evidence cited thus far, are aligned with the binary orbit, then we must invoke another scenario to explain the low inclination of the reflector. Another possible explanation is simply that the inner regions of the disk are not geometrically thin, as prescribed in detail by Shakura & Sunyaev (1973). The luminosity of XTE J1550−564 during 1998 September 23–24, the window of these observations, reached ∼0.1 L_Edd. At these luminosities, we should not expect the disk to increase its scale height due to either radiative or thermal pressures (see, e.g., Shakura & Sunyaev 1973; Paczyńsky & Wiita 1980), since such effects are only expected to occur at close to Eddington luminosities. However, it could be that the radial profile of the disk has some structure within the inner regions that is explained by neither high radiative flux nor thermal pressures. Recent work by Jiang et al. (2019b) has shown that simulations are now being used to explore the vertical structure of disks in more magnetically dominated states, for example. The inclination determined by the reflection modeling may be somewhat of an artifact of a complex inner accretion flow, rather than a true representation of the orientation of the disk itself.

An additional possibility may be that at high disk inclinations (assuming there is no warp in the disk), there is some obscuration of the blueshifted line emission on the front-facing side of the disk. As shown in Figure 8, at higher inclinations, the iron emission line is Doppler-broadened, and this emission originates in gas traveling toward the observer in front of the black hole. If this portion of the disk were to be obscured due to the scale-height profile, we should expect a reduction in the blueward flux of the line, much akin to an iron line predicted by reflection at lower disk inclinations. Taylor & Reynolds (2018) explored this scenario and found that when the coronal height in a lamppost model is comparable to the inner disk thickness, one indeed expects modifications to the line profile due to obscuration effects, an inclination-dependent effect. At this stage, we make this suggestion speculatively, given the limited work on the effects of disk geometries on reflection, and leave proper investigation to future work, since such a test requires substantial modifications to the relxill model to include these geometrical dependencies, and indeed no other model currently has such capabilities.

The location of the irradiating source for disk reflection determines the fraction of photons that strike the disk, as well as the location of the disk at which they strike, due to the relativistic effects in the presence of the strong gravity of the black hole. In addition, as shown by Dauser et al. (2013), if the irradiating source is in motion at speeds comparable to the speed of light, similar implications arise. As shown in multiple papers over the past few decades, there is an inherent degeneracy between a static corona, which produces the observable hard X-ray spectrum in BHBs, and a jet/outflow in relativistic motion (Beloborodov 1999; Markoff et al. 2005; Connors et al. 2017, 2019). Furthermore, the recent simulations of Liska et al. (2018, 2019) showed that the jet/corona/disk trinity is a connected physical system in which the individual components dynamically impact one another. For example, the jet may actually be collimated by the pressure provided by the enveloping corona.

This leads to a natural question: how would the broad Fe K line profile appear if the irradiating source were a region traveling at mildly relativistic velocities perpendicular to the disk? Dauser et al. (2013) addressed this to some degree and showed that indeed, the height and velocity of the irradiating source (considered in this case to be a lamppost) strongly affects the broad-line profile. However, these effects primarily occur within the red portion of the line, since the primary changes are in the gravitational redshift of the photons. Nonetheless, it is possible, if the source height is beyond the inner ∼10 r_g region, that the peak blueward emission due to Doppler effects may be reduced, since a greater source height leads to more photons striking the outer regions of the disk. Thus, this is an additional effect that could lead one to derive a low disk inclination when performing reflection modeling of the kind we have performed in this paper, though it is difficult to generate such a large (>30°) discrepancy with respect to the binary inclination with this one effect.

As well as the location and velocity of the irradiating source, the lateral structure of the jet/corona may have an impact on the inclination estimate. Liska et al. (2019) investigated Bardeen–Petterson misalignment in accreting black holes with very thin accretion disks (H/R < 0.03) and showed that in such cases, the jet/corona are very broad geometrically. One would need to extend current reflection models to include a scenario in which the irradiating source is not only vertically extended but has lateral structure, such that the origin of much of the irradiating photons is not at x = 0.

Another significant result of our reflection modeling of XTE J1550−564 is the high iron abundance (A_Fe = 3. 9^+1.7_−0.5). High iron abundances have been measured for multiple BHBs and AGNs alike (García et al. 2018) in reflection modeling (Fürst et al. 2015; García et al. 2015b; Parker et al. 2015), and the current interpretation of these estimates is that the disk density in reflection modeling has been underestimated until recently (Tomsick et al. 2018; Jiang et al. 2019a). Thus, given the very high iron abundance resulting from our modeling, we suggest that future modeling of the reflection spectrum of XTE J1550−564 ought to be tuned to higher disk densities. An increase in disk electron number density to beyond ∼10²⁰ cm⁻³ results in excess free–free heating via bremsstrahlung, which leads to higher soft X-ray flux. This increased soft X-ray flux can subsume the blackbody disk emission to a certain extent and has even been invoked to explain the soft excess in AGNs (García et al. 2019).

As yet, it is not entirely clear what the effects of introducing higher disk densities may be on the inclination constraint we have presented in this work. Given that the broadening of the blue wing of the iron line is largely inclination-dependent due to relativistic effects that are linked directly to the geometry of the system, it seems unlikely that higher disk densities will somehow alter the resultant geometrical interpretation of the system. Indeed, recent applications of high-density reflection models to Cygnus X-1 and GX 339−4 spectra have shown that the iron abundance is the only parameter that changes with respect to low-density modeling (Tomsick et al. 2018; Jiang et al. 2019a); most notably, the inclination remains unaffected. We have chosen not to explore high-density models in our fits because the models are currently limited in terms of the range of values of key physical parameters, such as the electron temperature (fixed at a single value in the most recent high-density models; García et al. 2016a).

Steiner et al. (2011) provided two independent constraints on black hole spin via modeling of the soft-state thermal continuum of XTE J1550−564 and through reflection modeling. Both methods yielded a low spin for the source, thus giving an estimate of ${a}_{\star }\sim 0.5$ as an average of the two measurements. However, this estimate was guided by the assumption that the disk is inclined to the same degree as the binary orbit (i ∼ 75°). Steiner et al. (2011) showed that the black hole spin is quite strongly anticorrelated with the disk inclination in thermal continuum modeling. This leads to the obvious implication of our much lower inclination estimate: that adopting our inclination would produce a much higher value of spin from continuum fitting.

Much of our discussion has been focused thus far on the inclination constraints of our reflection modeling. There are, however, a few other interesting parameter constraints to discuss. Table 2 and Figure 9 show the best-fit parameters with 90% limits and the associated MCMC parameter correlations. One can see that the coronal electron temperature is very low (kT_e ∼ 6 keV), a feature of hard-intermediate-state BHBs (see, e.g., Remillard & McClintock 2006) and in line with the results of modeling XTE J1550−564 with a hybrid coronal plasma (Hjalmarsdotter et al. 2016). We also found very high disk ionization (>10⁴ erg cm s⁻¹), and though its value is tightly correlated with the iron abundance in the disk, it is well constrained to such high values. In addition, though the ionization may appear surprisingly high, it can be expected for softer states in which the Fe K emission is stronger than in hard states in which we typically see lower values (García et al. 2013). The constraints on the coronal power-law photon index (Γ), hydrogen column density (N_H), and disk blackbody temperature (kT_BB) all agree well with previous estimates (Miller et al. 2003; Tomsick et al. 2003) of the spectral parameters of XTE J1550−564 in its hard-intermediate state. In addition, we checked that the N_H value is consistent with simple modeling of Chandra grating spectra during an observation in 2000 May when the source was in a similar hard-intermediate state, ObsID 680 and 681, finding only a difference of ∼0.2 between the derived value and our results from fitting model 3 (a minor difference, given the slight difference in model approach). As already discussed in Section 4, there are some differences in our spectral fitting results when compared to the modeling by Gierliński & Done (2003) and Hjalmarsdotter et al. (2016). These differences are due to the differences in treatment of the data, with both of the respective previous works combining PCA spectra across a small range in X-ray hardness. Selecting the data in this way, due to the higher PCA count rates being skewed toward harder spectra, actually serendipitously mitigates the offset in spectral slope between the ASCA-GIS and RXTE-PCA data, a feature that we have shown is important to account for in the 1998 September 23–24 observation.