Spin measurement of 4U 1543–47 with Insight-HXMT and NICER from its 2021 outburst

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 c$\mathrm{m^{2}}$), the medium-energy (ME) X-ray telescope (Cao et al., 2020), covering 8–35 keV (952 c$\mathrm{m^{2}}$), and the high-energy (HE) X-ray telescope (Liu et al., 2020), covering 20–250 keV (5100 c$\mathrm{m^{2}}$). We extracted spectra using the software HXMTDAS v2.05 ²²2http://hxmten.ihep.ac.cn/SoftDoc/501.jhtml 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 $\geq$ 10°; (2) a geomagnetic cut-off rigidity of $\geq$ 8 GeV; (3) a pointing position offset of $\leq$ 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 2).

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 c$\mathrm{m^{2}}$ 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).

We use a total of five different models, which are labeled M1–M5 (see table 3.2). 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, $N_{\text{H}}$, to 0.439$\times 10^{22}\mathrm{cm^{-2}}$ (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 $E_{\mathrm{Edge}}$ between 7 and 9 keV, optical depth $\tau_{\mathrm{max}}$, 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 $kT_{\mathrm{e}}$ and scattering fraction $f_{\mathrm{sc}}$. While not as comprehensive as ThComp, simpl is a semi-phenomenological model that nonetheless includes a parameter for the scattering fraction $f_{\mathrm{sc}}$, 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 reduced $\chi^{2}$ of 1.11 for M1 and 1.07 for M2.

We plot the evolution of the scattering fraction $f_{\mathrm{sc}}$, Comptonized photon index $\Gamma$, inner disk temperature $T_{\mathrm{in}}$, inner disk radius $R_{\mathrm{in}}$, and luminosity $L_{\mathrm{Edd}}$ obtained from both M1 and M2 in Figure 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 $\Gamma\geq 5$ and scattering fraction ($f_{\mathrm{sc}}$ ¿ 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 $2<\Gamma<3$ and $f_{\mathrm{sc}}$ ¡ 10 %, satisfying the criteria for spin extraction mentioned in section 3 (See green highlighted epoch, Figure 2). Here, the inner disk radius remains stable at around 4.6 $R_{\mathrm{g}}$. We therefore identify observations within these two epochs as those providing golden spectra from which to extract spin (See Table 4.1). Between MJD 59418.0–59450.0, the source also enters a period of a minimum, stable inner disk radius at around 4.5 $R_{\mathrm{g}}$. However, since $\Gamma$ and $f_{\mathrm{sc}}$ 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 $T_{\mathrm{in}}$ follow $L$ $\propto T^{\beta}_{\mathrm{in}}$), and fit the curve for observations before and after the state transition (see Figure 3) identified by Jin et al. (2023). We find a very steep curve before the state transition, with $\beta$ = 6.71 $\pm 0.25$, while after the state transition, the curve flattens with $\beta$ = 2.11 $\pm 0.24$.

For all observations ¡ 1 $L_{\mathrm{Edd}}$, 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 section 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, $f_{\mathrm{h}}$ (where $f_{\mathrm{h}}$ is the ratio of observed color temperature to effective temperature, $f_{\mathrm{h}}=T_{\rm{col}}/T_{\rm{Eff}}$), 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 (Figure 4(a), 4(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 $f_{\mathrm{val}}$ for kerrbb2 set to -1, allowing the spectral hardening factor $f_{\mathrm{h}}$ to be interpolated over the table, and $f$ for slimbh is set to -1, allowing $f_{\mathrm{h}}$ to be obtained from TLUSTY spectra. Above 0.6 $L_{\mathrm{Edd}}$ for $\alpha$-viscosity = 0.01, and 0.7 $L_{\mathrm{Edd}}$ for $\alpha=$ 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 $L_{\mathrm{Edd}}$ between the two models in the case of $\alpha$-viscosity = 0.01. We show the luminosity dependence of the hardening factor against the backdrop of the spin evolution obtained from slimbh in Figure 5. The hardening factor was obtained by fixing the previously obtained spins for each respective observation, and setting the $f_{\rm{h}}$ parameter of slimbh free ($f_{\rm{h}}$ ¿ 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 $f_{\rm{h}}$ increases with inner disk temperature ($T_{\rm{Eff}}$) and accretion rates.

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, $f(m)=\frac{M^{3}\sin{i}^{3}}{(M+M_{opt})^{2}}$ = 0.25 ± 0.01 (Park et al., 2004), where $M_{\mathrm{opt}}$ is the mass of the optical companion ($2.7\pm 1M_{\odot}$, Park et al. 2004). To this end, for each spectrum, we (1) generated 2000 parameter sets of $f(m)$, $i$, $D$, and $M_{\mathrm{opt}}$, (2) solved for the source mass M for a set of $f(m)$, $i$, $D$, and $M_{\mathrm{opt}}$, (3) generated a look-up table for each set, and (4) refitted the spectrum with models M4 (Inight-HXMT)and M5 (NICER) to obtain the distribution of $a_{*}$. This process was repeated for both viscosity parameters of 0.1 and 0.01.

We identify 13 sub-exposures observed during the golden epoch identified in Section 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. $f_{\rm h}$ 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 Figures 6(a) and 6(b). Combined, we estimate a spin value of $\alpha_{*}=0.64^{+0.14}_{-0.24}$ and $\alpha_{*}=0.56^{+0.17}_{-0.27}$ for $\alpha=$ 0.01 and 0.1, respectively. The spectra are well fitted with an average reduced $\chi^{2}$ statistic of 0.99 and 1.04 for $\alpha$-viscosity = 0.01, 0.1, respectively (see table 4.4).

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 excluded ³³3https://heasarc.gsfc.nasa.gov/lheasoft/ftools/headas/nibackgen3C50.html. 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 $\alpha_{*}=0.55^{+0.18}_{-0.27}$ for $\alpha=$ 0.1, and $\alpha_{*}=0.65^{+0.14}_{-0.24}$ for $\alpha=$ 0.01 (See Figure 6(c) and 6(d)). The spectra are well fitted with an average reduced $\chi^{2}$ statistic of 0.99 and 1.03 for $\alpha$-viscosity = 0.01 and 0.1, respectively (see table 4.4).

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 $L_{\mathrm{Edd}}$, and more detailed studies are ongoing on this specific subject. This is suggested by the evolution of the L-T relationship (Figure 3), where the source starts out with a steep power law with $L\propto T_{\mathrm{in}}^{6.71}$ before MJD 59395.0, and shallows to $L\propto T_{\mathrm{in}}^{2.11}$ after MJD 59405.0. It is generally expected that in the case of a fixed emitting area, the luminosity would scale with temperature with $L\propto T^{4}_{\mathrm{Eff}}$. 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 Figure 2). Moreover, as $f_{\rm{h}}$ decreases with decreasing temperature, and $T_{\mathrm{in}}=f_{\rm{h}}T_{\rm{Eff}}$, $f_{\rm{h}}$ ¿ 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 $L_{\mathrm{Edd}}$. Work by Watarai et al. (2000) showed that in the slim-disk regime, $T_{\rm{in}}\propto r^{-1/2}$, suggesting that slim-disk models should render $\beta\sim 2$. Therefore, $\beta\sim 2$ is not surprising. More intriguing is the very steep relation before the disk state transition ($L/L_{\rm{Edd}}$ ¿ 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 $T(r)\propto r^{-3/4}$ 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 $\alpha$ viscosity. This gives further credence to the state transition hypothesis around 1 $L_{\mathrm{Edd}}$. Furthermore, the fact that $f_{\rm{h}}$ decreases with decreasing $T_{\mathrm{in}}$ 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 $L_{\mathrm{Edd}}$ 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 $L_{\mathrm{Edd}}$. Between 0.52 $L_{\mathrm{Edd}}$ – 0.58 $L_{\mathrm{Edd}}$, and 0.48 $L_{\mathrm{Edd}}$ – 0.49 $L_{\mathrm{Edd}}$, 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 & MJD 59469.0 – 59472.0, a rise in the scattering fraction $f_{\mathrm{sc}}$, along with a dip in the inner disk temperature $T_{\mathrm{in}}$ (see epochs marked in gray in Figures 2, 5). A sharp hardening in the former epoch is found in the HR in Figure 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, Figure 2), the constant, low inner disk radius coupled with photon index in the typical soft range 2 ¡ $\Gamma$ ¡ 3 and a low scattering fraction $f_{\mathrm{sc}}$ ¡ 10% make our implicit assumption that $r_{\mathrm{in}}=r_{\mathrm{ISCO}}$ most robust.

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 $\alpha$ an order of magnitude lower than what is suggested by observations ($\alpha\leq 0.02$, see King et al. (2007) and references therein). Although measurements of the $\alpha$ viscosity tend to be less reliable in the case of partially ionized disks, estimates are consistently finding values of $\alpha$ an order of magnitude smaller ($\alpha\sim 0.01$) than those for fully ionized disks (Martin et al., 2019).

The lower spin measurements from assuming a higher $\alpha$-viscosity of 0.1 is not surprising; higher values of $\alpha$-viscosity yield lower disk densities, which would in turn yield harder spectra (higher $f_{\rm{h}}$), 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 $\alpha$ to 0.1, we take the $\alpha$ 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 $\xi$ 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 $l_{\mathrm{Disk}}<0.6L_{\mathrm{Edd}}$ in the case of $\alpha$ = 0.01, while the difference between the models is around $\approx$ 0.07 for $\alpha$ = 0.01.

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 $\sim$ 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 ($\sim$ 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 $10^{6}$ – $10^{8}$ years (Martin et al., 2008). Regarding distance, the second data release (DR2) by Gaia gives a distance of $8.60^{+3.84}_{-2.46}$ kpc inferred from a Bayesian estimation of the parallax, which is consistent with the fiducial distance of $7.5\pm 1.0$ 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 $0.47^{+0.16}_{-0.22}$ for $\alpha$ = 0.01, which is consistent with the error range in the case of $D$ = $7.5\pm 1.0$ 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 $D$ = $7.5\pm 0.5$ kpc by Jonker & Nelemans (2004) in reporting a final spin result.