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A&A
Volume 677, September 2023
Article Number A79
Number of page(s) 10
Section Interstellar and circumstellar matter
DOI https://doi.org/10.1051/0004-6361/202346511
Published online 08 September 2023

© The Authors 2023

Licence Creative CommonsOpen Access article, published by EDP Sciences, under the terms of the Creative Commons Attribution License (https://creativecommons.org/licenses/by/4.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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1 Introduction

Black holes (BHs) are fully defined by their mass, angular momentum, and charge, as stipulated by the no-hair theorem of general relativity (GR). However, in practice, a BH would inevitably undergo a discharge process to its environment, causing it to become neutral. Consequently, it is reasonable to assume that the space-time around BHs is well-represented by the Kerr Metric (Kerr 1963). The angular momentum of a black hole is characterized by the dimensionless spin parameter, a* ≡ cJ/GM2, where M is the BH mass, J is the angular momentum, G is Newton’s gravitational constant, c is the speed of light, and −1 ≤ a* ≤ 1.

There are two principal methods to extract the spin of a BH in accreting systems: the continuum fitting method (Zhang et al. 1997; Li et al. 2005), which requires accurate measurements of distance D, inclination i, and BH mass M, and the X-ray reflection method (Iwasawa et al. 1997; Miller et al. 2002), which is weakly dependent on inclination angle. More recently, various methods of spin extraction have been proposed, including the relativistic precession model (RPM; Stella & Vietri 1999), and the X-ray polarization method (Dovciak et al. 2008). Given sufficiently sensitive instruments, spin measurements in the nonaccreting regime are also possible via gravitational waves signals (see review by Reynolds 2021).

The thermal continuum method, or continuum fitting (CF), relies on the fundamental assumption that the accretion disk extends up to the innermost stable circular orbit (ISCO), and is truncated there. As the ISCO radius is a monotonic function of the spin, fitting the thermal continuum by identifying R¡n from the temperature maximum of soft X-rays would allow us to measure a.. Typically, the outburst is screened for the soft state dominated by thermal emission. Thin-disk solutions, which model a geometrically thin and optically thick disk, are typically realized when the disk luminosity lDisk ≤ 30% LEdd (Novikov & Thorne 1973; Shakura & Sunyaev 1973).

Among black hole transients (BHTs), 4U 1543–47 has been the subject of relatively extensive investigation due to its peculiar outburst and outflow history. The recurrent X-ray binary was discovered by the Uhuru satellite in 1971, and was observed in subsequent outbursts in 1983, 1992 and 2002 (Matilsky et al. 1972; Kitamoto et al. 1984; Harmon et al. 1992; Park et al. 2004). There have been several attempts to measure its spin, three of which used continuum fitting. Shafee et al. (2006) first measured a spin of 0.80 ± 0.05. Thereafter, Miller et al. (2009) and Morningstar & Miller (2014) measured spin values of 0.3 ± 0.1 and , respectively. All of these works use the fiducial values of M = 9.4 ± 2.0 M⊙ (Orosz 2003), i = 20.7° ± 1.5 (Orosz 2003), and D = 7.5 ± 0.5 kpc (Jonker & Nelemans 2004), with the exception of Miller et al. (2009) who used an inclination of obtained based on their constraints to the iron line. The latest spin measurement by Dong et al. (2020) found a moderately high black hole spin value of via X-ray reflection spectroscopy.

4U 1543–47 underwent a bright outburst in 2021, reaching a peak flux of ~ 9 Crab, as observed by the Monitor of All-sky X-ray Image (MAXI; see Fig. 1). The outburst was the brightest ever observed by NICER to date (Connors et al. 2022). Throughout the entirety of both Insight-HXMT and NICER observations, the source remains above 30% of its Eddington luminosity, and as such, no observations formally satisfy the traditional luminosity constraint employed when using continuum fitting. In addition to the thin-disk model, we therefore adopt the model slimbh in XSPEC (Sadowski 2009; Straub et al. 2011), which is a generalization of the Shakura-Sunyaev thin-disk model and can account for luminous, optically thick disks, or “slim” disks (Abramowicz et al. 1988) at high luminosities.

In this paper, we report the performance of both thin- and slim-disk models at high luminosities extending up to 1 LEdd, using Insight-HXMT observations of the source from its 2021 outburst. Given the performance of the models, we explore the possibility to constrain the spin of 4U 1543–47 with continuum fitting at these relatively high accretion rates, using both Insight-HXMT and NICER datasets. As in previous studies, we use fiducial values of M, i, and D for our source. The paper is organized as follows. In Sect. 2, we give an introduction to relativistic slim disks. In Sect. 3, we provide details on the data reduction and selection. In Sect. 4, we present our results, including the spectral evolution of the source, the behavior of slim- and thin-disk models, and provide our own spin measurement. Sections 5 and 6 present our discussion and conclusions, respectively. Unless otherwise stated, our errors are always reported within 1σ.

thumbnail Fig. 1

Temporal evolution of the light-curve and hardness ratio of 4U 1543–47 measured by all-sky monitor MAXI. The dashed black lines denote the start and end dates of Insight-HXMT monitoring.

2 Slim-disk versus standard thin-disk models

In the standard thin-disk model of the accretion disk (Shakura & Sunyaev 1973; Novikov & Thorne 1973), energy advection is neglected and all the gravitational energy release is in the form of radiation. The “slim-disk” describes a thicker accretion disk (h/r ≳ 0.03) at high luminosities, and takes into account three additional effects not assumed to be present in thin disk solutions (Abramowicz et al. 1988). In particular, the slim-disk model takes into account (1) heat advection, which becomes more dominant at higher luminosities and modifies the flux profiles of radiation emitted in the inner disk region; (2) at higher luminosities approaching the Eddington limit, radiation pressure begins to play a more significant role, causing a deviation from Keplerian angular momentum, especially in the inner disk region. The inner edge of the disk may also be much closer to the black hole than the ISCO radius due to heat advection. Straub et al. (2011) presents the flux profiles for three different accretion rates for a moderately rotating black hole. At high accretion rates, advection causes emission from within the ISCO to become more dominant. Compared to the Novikov-Thorne (NT) thin-disk model, a significant portion of the heat generated by disk α-viscosity is radiated closer to the BH horizon, and some heat may be advected into the horizon and thus never emitted. The effects of advection are clearly demonstrated in Fig. 1 of Straub et al. (2011); (3) the location of the photosphere, which may diverge from the equatorial plane at higher luminosities (Sadowski et al. 2009; Straub et al. 2011).

3 Observation and data reduction

We use Insight-HXMT observations (see Fig. 1 for Insight-HXMT’s observation epoch) to measure the spectral evolution of the source, and test the performance of the relativistic slim-and thin-disk models1. More specifically, we test the consistency of the spin estimates for both models over a luminosity range extending up to the Eddington limit. The performance and agreement between the models would serve as a nontrivial test of these models in attempting to measure spin at high accretion rates. This same procedure was employed by Straub et al. (2011), who find that both relativistic slim-disk and thin-disk models suffer from a spin drop-off at luminosities above 30% LEdd when measured for LMC X-3. Measuring spin at high luminosities is associated with more uncertainties, and so it is therefore imperative to test both models to see if they give relatively consistent and constant spin values over a wide range of luminosities before providing constraints on the spin.

In order to find a suitable epoch for spin extraction, that is where the edge of the accretion disk is truncated at the ISCO, we adopt the following criteria: (i) a low, stable inner disk radius epoch; and (ii) spectra dominated by thermal disk emission, where the proportion of thermal disk photons that scatter in the corona (the scattering fraction, ƒsc) does not exceed 10%. It is important that the suitable epoch chosen for spin extraction has the lowest values of the normalization or inner disk radius relative to the other observations, when measured by the multi-color disk blackbody model (diskbb in XSPEC). Moreover, results by Steiner et al. (2009) showed that when the disk emission accounted for over 75% of the total luminosity (ƒsc < 25%), the spin measurements remained relatively constant. These selection criteria for CF were successfully employed by Gou et al. (2009), Steiner et al. (2011), and Chen et al. (2016), and can also be extended to other spectral states besides the high soft state (HSS). Taken together, these two criteria will mean that the assumption of the disk being truncated at the ISCO is most justified. Any observations vigorously meeting these conditions are designated as “golden” spectra. Because all observations exceed the thin-disk luminosity limit, we use the slim-disk model slimbh to measure the spin for all “golden” observations and give a final spin result. Spectral fitting was carried out using XSPEC V. 12.12.1 and HEASoft version V6.30.

Table 1

Insight-HXMT ‘golden’ observation log.

3.1 Insight-HXMT

Insight-HXMT is China’s first X-ray Astronomy satellite, and was launched on June 15, 2017. Its scientific payload consists of the low-energy (LE) X-ray telescope (Chen et al. 2020), covering 1–15 keV (384 cm2), the medium-energy (ME) X-ray telescope (Cao et al. 2020), covering 8–35 keV (952 cm2), and the high-energy (HE) X-ray telescope (Liu et al. 2020), covering 20–250 keV (5100 cm2). We extracted spectra using the software HXMTDAS V2.052 and the pipeline prescribed by the official user guide. The latest calibration database was also used (CALDB V2.06). The following criteria were used for extraction: (1) An elevation angle ≥10°; (2) a geomagnetic cut-off rigidity of ≥8 GeV; (3) a pointing position offset of ≤0.05°; and (4) at least 300 s away from the South Atlantic Anomaly (SAA). Out of 151 sub exposures, 108 survived this screening for Good Time Interval (GTI), all of which were used to measure the spectral evolution of the source. We only used data from the LE and ME telescopes, as they provide sufficient coverage of the relevant energy range for continuum fitting, and because of the very low photon count in HE. The backgrounds are estimated with the tools provided by the Insight-HXMT team: LEBKGMAP (Liao et al. 2020) and MEBKGMAP (Guo et al. 2020), version 2.0.9 based on the current standard Insight-HXMT background models, for LE and ME, respectively. We select a total of 13 exposures for the final spin analysis, which we refer to as the golden observations (see Table 1).

Table 2

NICER ‘golden’ observation log.

3.2 NICER

The Neutron Star Interior Composition Explorer (NICER), a payload on the International Space Station (ISS), provides coverage of the soft X-ray range (0.2–12 keV). The X-ray timing instrument (XTI) of NICER is composed of 56 identical and co-aligned cameras, each of which contains an X-ray concentrator (XRC, Okajima et al. 2016) and silicon drift detector (SDD) pairs. With a peak effective area of 1900 cm2 at 1.5 keV, it is well suited to thermal fitting, and hence spin measurements of BHs. We obtained cleaned event files by applying the standard calibration and filtering tool nicerl2, and obtained the response files with NICERRMF and NICERARF tools with HEASOFT v 6.30. The background spectrum was computed using nibackgen3C50 tool. We select a total of six observations observed during the golden epoch to perform the spin extraction (see Table 2).

4 Analysis and results

4.1 Accretion state evolution

We use a total of five different models, which are labeled M1–M5 (see Table 3). All used models account for interstellar absorption with the TBabs component in XSPEC, with Wilms et al. (2000) abundances and Verner et al. (1996) cross sections. We fix the column density, NH, to 0.439×1022cm−2 (Connors et al. 2021). In addition, due to the presence of an excess at energies of between 6–8 keV, which is likely due to disk reflection, we use the model Smedge in XSPEC (Ebisawa et al. 1994) component in all models. The SMEDGE parameters are the absorption edge EEdge between 7 and 9 keV, optical depth τmax, photoelectric cross section (fixed to −2.67), and width (fixed at 7 keV). We study the spectral evolution of the source with the first two models, M1 and M2, both of which contain the multicolored absorbed blackbody component diskbb (Mitsuda et al. 1984; Makishima et al. 1986), accounting for the bulk of emission, but with differing Componization components: Simpl in M1 (Steiner et al. 2009), and ThComp in M2 (Zdziarski et al. 2020). ThComp is a much more accurate representation of thermal comptonization compared to the simple phenomenological model powerlaw, taking into account the electron temperature kTe and scattering fraction ƒsc. While not as comprehensive as ThComp, simpl is a semi-phenomenological model that nonetheless includes a parameter for the scattering fraction ƒsc, which is important for finding a suitable epoch for spin extraction. Both ThComp and simpl are convolution models that take in a thermal component as their seed spectrum. These two models are well fitted, with an average reducedχ2 of 1.11 for M1 and 1.07 for M2.

We plot the evolution of the scattering fraction ƒsc, Comptonized photon index Γ, inner disk temperature Tin, inner disk radius Rin, and luminosity LEdd obtained from both M1 and M2 in Fig. 2. The inner disk temperature and luminosity show a steep decrease in the initial phase of the outburst (MJD 59379.0– 59400.0), followed by a shallower decrease. The source starts with a high power-law index with Γ ≥ 5 and scattering fraction (ƒSc > 50 % for M2); between MJD 59410.0–59440.0, the power-law index and scattering fraction cannot be constrained well due to the high background in ME. The source enters into a stable configuration between MJD 59460.0-59469.0 and MJD 59472.0–59480.0 with 2 < Γ < 3 and ƒSc < 10 %, satisfying the criteria for spin extraction mentioned in Sect. 3 (see green highlighted epoch, Fig. 2). Here, the inner disk radius remains stable at around 4.6 Rg. We therefore identify observations within these two epochs as those providing golden spectra from which to extract spin (see Table 4). Between MJD 59418.0–59450.0, the source also enters a period of a minimum, stable inner disk radius at around 4.5 Rg. However, since Γ and ƒsc cannot be constrained well here due to the low photon count above 15 keV, these observations have been excluded.

BHB sources are expected to undergo several phases of exponential decay during transitions between different disk states within outbursts (Eckersall et al. 2015). Jin et al. (2023) fit the light curve and hardness-intensity diagram (HID) of the source with exponential curves, and identify two branches of the decay phase, pointing to a disk state transition from a thicker slim-disk to a thin-disk between MJD 59395.0 and MJD 59405.0. The source luminosity during this transition period incidentally crosses one Eddington. Motivated by this finding, we plot the disk luminosity and the inner-disk temperature (where L and Tin follow L ∝ ), and fit the curve for observations before and after the state transition (see Fig. 3) identified by Jin et al. (2023). We find a very steep curve before the state transition, with ß = 6.71 ± 0.25, while after the state transition, the curve flattens with ß = 2.11 ± 0.24.

For all observations <1 LEdd, we compare the luminosity dependence of the spin from two relativistic accretion disk models by replacing diskbb with kerrbb2 (M3; McClintock et al. 2006) and slimbh (M4; Sadowski 2011). kerrbb2 is based on the NT thin-disk solutions, and is a hybrid of kerrbb (Li et al. 2005) and BHSPEC (Davis & Hubeny 2006). The relativistic slim-disk model, slimbh, is a generalization of the standard thin-disk model, kerrbb2, taking into account the aforementioned deviations from thin-disk solutions presented in Sect. 2, in particular, ray tracing which can be done from the disk photosphere rather than the equatorial plane. As is the case for kerrbb2, slimbh can account for the color correction factor, ƒh (where ƒh is the ratio of observed color temperature to effective temperature, ƒh = Tcol/TEff), with the aid of TLUSTY stellar atmospheres code spectra, which computes the vertical structure and radiation transfer in accretion disks, and can model the X-ray continuum up to the Eddington limit (Hubeny & Lanz 1995). The comparison of the spin between the two models was performed twice for two values of viscosity: 0.1 and 0.01 (Figs. 4a,b). By doing so, we are able to better gauge (i) the effect of viscosity on spin measurements, and (ii) the disk structure by comparing the NT model and slim-disk models. M, i, and D are fixed to their fiducial values. As mentioned, we use M3-M4, with ƒval for kerrbb2 set to -1, allowing the spectral hardening factor ƒh to be interpolated over the table, and ƒ for slimbh is set to -1, allowing ƒh to be obtained from TLUSTY spectra. Above 0.6 LEdd for α-viscosity = 0.01, and 0.7 LEdd for α = 0.1, kerrbb2 consistently estimates a higher spin value than slimbh, and the effect is more pronounced at higher luminosities. This is expected, because emission from within the ISCO is predicted by slim-disk models at higher accretion rates (Straub et al. 2011). There is excellent agreement below 0.6 LEdd between the two models in the case of α-viscosity = 0.01. We show the luminosity dependence of the hardening factor against the backdrop of the spin evolution obtained from slimbh in Fig. 5. The hardening factor was obtained by fixing the previously obtained spins for each respective observation, and setting the ƒh parameter of slimbh free (ƒh > 1). The hardening factor starts around 1.7 at the state transition and gradually decreases to around 1.6 toward the end of Insight-HXMT’s observation epoch of the source. This is indeed consistent with previous findings; through a series of fits to synthetic spectra produced via TLUSTY code, Davis & El-Abd (2019) found that ƒh increases with inner disk temperature (TEff) and accretion rates.

Table 3

Utilized models.

thumbnail Fig. 2

Spectral parameter evolution of 4U 1543–47 as observed by Insight-HXMT, where L/LEdd is the total disk luminosity (0.01 – 30 keV) in Eddington units, with LEdd = 1.2 × 1039 ergs s−1 and systematic uncertainties from the distance and BH mass accounted for; ƒsc is the scattering fraction; Γ is the photon power-law index of the Comptonization component; Tin is the inner disk temperature in units of keV; and Rin is the inner disk radii in units of gravitational radii Rg. The shaded red regions denotes the epoch for state transition (Jin et al. 2023). The shaded green regions denote ideal epochs for spin extraction. Due to the similar luminosity values between M1 and M2 which appear indistinguishable, we present only the M2 luminosity values. The Γ factors and ƒsc between 59 400 and 59 440 cannot be constrained well due to the low photon count in the ME band for observations in this epoch.

4.2 Spin error analysis

The bulk of the uncertainty on continuum fitting derives from the uncertainties in the BH mass, distance, and inclination angle. For our selected “golden” spectra, we perform monte carlo simulations of the mass, inclination angle, and distance following the prescriptions of Gou et al. (2011). M and i can be decoupled with the aid of the mass function, (Park et al. 2004), where Mopt is the mass of the optical companion (2.7 ± 1 M⊙, Park et al. 2004). To this end, for each spectrum, we (1) generated 2000 parameter sets of ƒ(m), i, D, and Mopt, (2) solved for the source mass M for a set of ƒ(m), i, D, and Mopt, (3) generated a look-up table for each set, and (4) refitted the spectrum with models M4 (Insight-HXMT) and M5 (NICER) to obtain the distribution of a*. This process was repeated for both viscosity parameters of 0.1 and 0.01.

4.3 Insight-HXMT

We identify 13 sub-exposures observed during the golden epoch identified in Sect. 4.1. We fitted the selected Insight-HXMT datasets with M4 in the 2-30 keV band, with a systematic error of 1% for LE, again with Wilms et al. (2000) abundances and Verner et al. (1996) cross sections. The switch for limb darkening is off (lflag = −1), while (rflag = 1) allowing raytracing to be done from the photosphere and taking into account vertical disk thickness. The normalization is fixed to 1. ƒh was set to −1 allowing the spectral hardening factor to be interpolated from TLUSTY. The summed spin histogram distributions for our chosen golden Insight-HXMT spectra are shown in Figs. 6a,b. Combined, we estimate a spin value of and for α = 0.01 and 0.1, respectively. The spectra are well fitted with an average reducedχ2 statistic of 0.99 and 1.04 for α-viscosity = 0.01, 0.1, respectively (see Table 5).

4.4 NICER

We identify six NICER observations performed between MJD 59460.0 and 59469.0, and between 59472.0 and 59480.0. No GTIs survived between MJD 59473.0 and 59480.0 because of abnormal background conditions, and so six NICER observations were excluded3. We consider the energy range between 2 and 9 keV, this time using M5, which does not have a Comptonized component because higher Comptonized energies are not used. We applied the same abundances, column density, and systematic as those used for Insight-HXMT. The smeared edge width was fixed to 7 keV, and an additional edge component was added (edge) and fixed to 2.0 keV to account for the Au edge. Combined, we estimate a spin value of for α = 0.1, and for α = 0.01 (see Figs. 6c and d). The spectra are well fitted with an average reduced χ2 statistic of 0.99 and 1.03 for α-viscosity = 0.01 and 0.1, respectively (see Table 5).

Table 4

Fitting results with M2 on selected ‘golden’ Insight-HXMT data.

thumbnail Fig. 3

Luminosity-temperature (L-T) relation (L ∝ Tß) for super-Eddington observations with M2 values, which yields a best-fit power value of ß = 6·71 ± 0·25. The shaded red region denotes the disk state transition epoch found by Jin et al. (2023). For observations after MJD 59405.0, the best-fit power value is ß = 2·11 ± 0·24.

5 Discussion

5.1 Disk evolution

Our study of the spectral evolution of the source shows that the source undergoes a disk transition from a slim disk to a standard thin-disk around 1 LEdd, and more detailed studies are ongoing on this specific subject. This is suggested by the evolution of the L-T relationship (Fig. 3), where the source starts out with a steep power law with L ∝ before MJD 59395.0, and shallows to L ∝ after MJD 59405.0. It is generally expected that in the case of a fixed emitting area, the luminosity would scale with temperature with L ∝ . A shallower relationship may indicate an unstable inner disk radius, and this is indeed the case, because the power-law fitting includes epochs where the inner disk recedes from the ISCO (MJD 59445.0-59460.0 and MJD 59469.0-59472.0; see epochs shown in grey in Fig. 2). Moreover, as ƒh decreases with decreasing temperature, and Tin = ƒh TEff, ƒh > 1, the modified temperature would further shallow the L–T relation. In addition, advection may still play a role in these moderate luminosities below 1 LEdd. Work by Watarai et al. (2000) showed that in the slim-disk regime, Tin ∝ r−1/2, suggesting that slim-disk models should render ß ~ 2. Therefore, ß ~ 2 is not surprising. More intriguing is the very steep relation before the disk state transition (L/LEdd > 1). There are two possible explanations for this at these super-Eddington observations: either the disk temperature profile deviates from the prediction of Shakura and Sunyaev that T(r) ∝ r−3/4 in unforeseen ways, or a model decomposition issue is the culprit, given that many of these observations have a larger fraction of emission in the harder component.

The “evolution” of the spin over time also seems to indicate a state transition. Although Straub et al. (2011) find a clear luminosity dependence of the spin measured by both slim and NT models, we find both models to be relatively stable for both values of α viscosity. This gives further credence to the state transition hypothesis around 1 LEdd. Furthermore, the fact that ƒh decreases with decreasing Tin while obtaining a constant spin suggests a softening of the spectrum with decreasing luminosity, which is consistent with model predictions (Davis & El-Abd 2019).

In all previous CF spin measurements, the thin-disk criterion of <0.3 LEdd was strictly employed when using continuum fitting for spin measurements of BHBs. Our results suggest the possibility of recovering thin-disk solutions at even higher luminosities. However, we have chosen to remain conservative and use the slim-disk model in our spin measurements, as all our selected ‘golden’ spectra have luminosities greater than >0.45 LEdd· Between 0.52 LEdd–0.58 LEdd, and 0.48 LEdd– 0.49 LEdd, there is a sharp drop in spin. The time of these observations corresponds to the rise in the inner disk radius between MJD 59445.0–59460.0 and MJD 59469.0–59472.0, a rise in the scattering fraction ƒsc, along with a dip in the inner disk temperature Tin (see epochs marked in gray in Figs. 2 and 5). A sharp hardening in the former epoch is found in the HR in Fig. 1, and hard flares have been detected by ME, HE, and AstroSAT (Prabhakar et al. 2023), suggesting that the source enters an intermediate hard state before transitioning back to a soft state, and this would manifest itself as lower spin estimates. This epoch may also correspond to the presence of jet emissions, because jets driven by magnetic fields around accretion disks will use up part of the accretion power, which will result in a reduction of the disk temperature (Blandford & Payne 1982; Li & Begelman 2014; Li & Cao 2009). In the meantime, a receding disk is also observed, suggesting that a jet base may occur in the inner part of the accretion disk pushing material out, hence the increased inferred inner disk radius (Ferreira et al. 2006; Marcel et al. 2022). We exclude these epochs from our final spin estimate. Between MJD 59460.0 and 59469.0, and between MJD 59472.0 and 59480.0 (green epochs, Fig. 2), the constant, low inner disk radius coupled with photon index in the typical soft range 2 < Γ < 3 and a low scattering fraction ƒsc < 10% make our implicit assumption that rin = rISCO most robust.

thumbnail Fig. 4

Evolution of both slimbh and kerrbb2 spin estimates vs. luminosity and time for all Insight-HXMT observations below 1 LEdd. The dips around 0.55 LEdd and 0.49 LEdd correspond to the rise in inner disk radius and ƒsc between MID 59445–59460 and 59469–59472. Both kerrbb2 and slimbh give stable spin values up to ~0.9 LEdd.
thumbnail Fig. 5

Luminosity dependence of the hardening factor ƒh (blue) and spin a* (red) measured with slimbh (α = 0.01).
thumbnail Fig. 6

Summed spin distributions for Insight-HXMT and NICER from MC analysis, with α viscosity = 0.01 (panels a and c) and α viscosity = 0.1 (panels b and d). Dashed lines denote 1σ errors.

5.2 α-viscosity

Although the underlying mechanism behind angular momentum transport within accretion disks remain unknown, it is generally accepted that the turbulence arising from magneto-rotational instability is the main driver of accretion. Typically, continuum fitting is performed with the dimensionless viscosity parameter fixed at 0.1. Indeed, strong observational evidence points toward a value of between 0.1 and 0.4 to describe fully ionized, thin-disks (see, e.g., Dubus et al. 2001, Cannizzo 2001, Schreiber et al. 2003). However, numerical simulations tend to estimate α an order of magnitude lower than what is suggested by observations (α ≤ 0.02, see King et al. 2007 and references therein). Although measurements of the α viscosity tend to be less reliable in the case of partially ionized disks, estimates are consistently finding values of a an order of magnitude smaller (α ~ 0.01) than those for fully ionized disks (Martin et al. 2019).

The lower spin measurements from assuming a higher α-viscosity of 0.1 is not surprising; higher values of α-viscosity yield lower disk densities, which would in turn yield harder spectra (higher ƒh), and therefore lower spin. This is because the ratio of absorption opacity to scattering opacity decreases with decreasing disk density (Davis & El-Abd 2019). Although all former thermal spin measurements of 4U 1543–57 fixed α to 0.1, we take the α viscosity of 0.01 to be a more accurate dimensionless representation of angular momentum transport in our selected spectra for several reasons. Firstly, although iron emission lines for the source are prominent and wide during the start of the outburst, they are largely diminished after the state transition between MJD 59395.0 and 59405.0; the spectral analysis by Jin et al. (2023) finds somewhat diminished ionization ξ and a diminished contribution from the reflection component after MJD 59400.0. Hence, we take the disk to be partially ionized for the ‘golden’ observations. In addition, there is excellent agreement between kerrbb2 and slimbh for lDisk < 0.6 LEdd in the case of α = 0.01, while the difference between the models is around ≈0.07 for α = 0.01.

5.3 System parameter accuracy

Indeed, the bulk of uncertainty on the spin measurements arises from uncertainties in M, i, and D (Gou et al. 2011). Accurate measurements of the spin therefore entail accurate and precise measurements of these orbital parameters. The reflection studies of 4U 1543–47 performed by Miller et al. (2009) and Dong et al. (2020) both constrained the inclination to ~10° higher than the value found by Park et al. (2004) and adopted by us. It is unlikely that this is an indication of a misalignment between the orbital inclination angle (~21°) and the inclination angle of the inner disk, because accretion would have already torqued the system into alignment, given that the timescale for such an alignment is on the order of 106–108 yr (Martin et al. 2008). Regarding distance, the second data release (DR2) by Gaia gives a distance of kpc inferred from a Bayesian estimation of the parallax, which is consistent with the fiducial distance of 7.5 ± 1.0 kpc found from photometry (Gandhi et al. 2019). When fixing the distance to 8.6 kpc, the final spin estimate from MC analysis decreases to for α = 0.01, which is consistent with the error range in the case of D = 7.5 ± 1.0 kpc. However, considering that this Gaia distance was obtained from a Bayesian inference of the parallax - which may not be accurate at long distances (Astraatmadja & Bailer-Jones 2016) – as well the fact that we explore the possibility of measuring the spin at high luminosities to compare our results to previous spin measurements, we still use the same distance of D = 7.5 ± 0.5 kpc by Jonker & Nelemans (2004) in reporting a final spin result.

Table 5

Spin estimates of all ‘golden’ observations.

6 Conclusions

We carefully measured the spectral evolution of 4U 1543–47 throughout its 2021 outburst with data provided by Insight-HXMT, and compared the performance between two accretion disk models, slimbh and kerrbb2. The consistent and constant spin measurements over luminosities extending up to the Eddington limit, coupled with the two different regimes of the L–T relationship point to a disk state transition occurring around ~1 L/LEdd. Our results indicate that thin-disk solutions can be recovered at luminosities higher than the typical thin-disk selection criterion of 0.3 LEdd. We have rigorously filtered observations for epochs when the source is most likely truncated at the ISCO, and measured the spin with both Insight-HXMT and NICER. Although all past measurements of spin for this source have assumed a viscosity parameter of 0.1, we adopt a lower a-viscosity parameter of 0.01 given evidence of a partially ionized disk and the consistency between the NT and slim-disk model, and report a final spin of when combining Insight-HXMT and NICER, which is in good agreement with the reflection results obtained by Dong et al. (2020), and is also broadly consistent with the results of Shafee et al. (2006).

Acknowledgements

This work is supported by the National Key R&D Program of China (2021YFA0718500). This work made use of the data from the Insight-HXMT mission, a project funded by China National Space Administration (CNSA) and the Chinese Academy of Sciences (CAS), as well as data obtained through the High Energy Astrophysics Science Archive Research Center Online Service, provided by the NASA/Goddard Space Flight Center. Q.B. acknowledges support by the National Natural Science Foundation of China (NSFC) under grants U1938102 and U1938107. A.V. acknowledges support from the Bundesministerium für Wirtschaft und Energie through Deutsches Zentrum für Luft-und Raumfahrt (DLR) under the grant number 50 OR 1917.

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All Tables

Table 1

Insight-HXMT ‘golden’ observation log.

In the text
Table 2

NICER ‘golden’ observation log.

In the text
Table 3

Utilized models.

In the text
Table 4

Fitting results with M2 on selected ‘golden’ Insight-HXMT data.

In the text
Table 5

Spin estimates of all ‘golden’ observations.

In the text

All Figures

thumbnail Fig. 1

Temporal evolution of the light-curve and hardness ratio of 4U 1543–47 measured by all-sky monitor MAXI. The dashed black lines denote the start and end dates of Insight-HXMT monitoring.
In the text
thumbnail Fig. 2

Spectral parameter evolution of 4U 1543–47 as observed by Insight-HXMT, where L/LEdd is the total disk luminosity (0.01 – 30 keV) in Eddington units, with LEdd = 1.2 × 1039 ergs s−1 and systematic uncertainties from the distance and BH mass accounted for; ƒsc is the scattering fraction; Γ is the photon power-law index of the Comptonization component; Tin is the inner disk temperature in units of keV; and Rin is the inner disk radii in units of gravitational radii Rg. The shaded red regions denotes the epoch for state transition (Jin et al. 2023). The shaded green regions denote ideal epochs for spin extraction. Due to the similar luminosity values between M1 and M2 which appear indistinguishable, we present only the M2 luminosity values. The Γ factors and ƒsc between 59 400 and 59 440 cannot be constrained well due to the low photon count in the ME band for observations in this epoch.
In the text
thumbnail Fig. 3

Luminosity-temperature (L-T) relation (L ∝ Tß) for super-Eddington observations with M2 values, which yields a best-fit power value of ß = 6·71 ± 0·25. The shaded red region denotes the disk state transition epoch found by Jin et al. (2023). For observations after MJD 59405.0, the best-fit power value is ß = 2·11 ± 0·24.
In the text
thumbnail Fig. 4

Evolution of both slimbh and kerrbb2 spin estimates vs. luminosity and time for all Insight-HXMT observations below 1 LEdd. The dips around 0.55 LEdd and 0.49 LEdd correspond to the rise in inner disk radius and ƒsc between MID 59445–59460 and 59469–59472. Both kerrbb2 and slimbh give stable spin values up to ~0.9 LEdd.
In the text
thumbnail Fig. 5

Luminosity dependence of the hardening factor ƒh (blue) and spin a* (red) measured with slimbh (α = 0.01).
In the text
thumbnail Fig. 6

Summed spin distributions for Insight-HXMT and NICER from MC analysis, with α viscosity = 0.01 (panels a and c) and α viscosity = 0.1 (panels b and d). Dashed lines denote 1σ errors.
In the text

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