Spectral characteristics of the black hole binary 4U 1957+115: a multi mission perspective

ABSTRACT

We report spectral analysis of the persistent black hole X-ray binary, 4U 1957+115, using AstroSat, Swift, and NuSTAR observations carried out between 2016 and 2019. Modelling with a disc emission, thermal Comptonization, and blurred reflection components revealed that the source was in the high-soft state with the disc flux ∼87 per cent of the total and high-energy photon index ∼2.6. There is an evidence that either the inner disc radius varied by ∼25 per cent or the colour hardening factor changed by ∼12 per cent. The values of the inner disc radius imply that for a non-spinning black hole, the black hole mass is  < 7 M _⊙ and the source is located  > 30 kpc away. On the other hand, a rapidly spinning black hole would be consistent with the more plausible black hole mass of  < 10 M _⊙ and a source distance of ∼10 kpc. Fixing the distance to 10 kpc and using a relativistic accretion disc model, constrained the black hole mass to 6 M_⊙ and inclination angle to 72°. A positive correlation is detected between the accretion rate and inner radii or equivalently between the accretion rate and colour factor.

accretion; accretion discs, black hole physics – X-rays: low mass X-ray binaries, X-rays: individual: 4U 1957+115

1 INTRODUCTION

4U 1957+115 is a black hole low mass X-ray binary (BH-LMXB) that was discovered by Uhuru in 1973 (Giacconi et al. 1974) and its optical counterpart V1408 Aql was identified  four years later during the Aquila survey (Margon, Thorstensen & Bowyer 1978). Thorstensen (1987) estimated the orbital period of it to be 9.329 ± 0.011 hr by studying the photometric modulations in the optical lightcurves, probably due to the X-ray irradiation of the donor by the accretion disc and the accretor.

Hakala, Muhli & Dubus (1999) using the simultaneous data from UBVRI filters of the Nordic Optical Telescope at La Palma reported a high disc inclination angle (i = 70 – 75°). Furthermore, they suggested that optical modulations are due to the changes in accretion disc structure but not due to the changes in X-ray heated face of the donor. Bayless et al. ( 2011) and Mason et al. (2012) analysed the high-speed optical photometry data from Argos CCD photometer mounted on the 2.1 m Otto Struve Telescope at McDonald Observatory and provided a weak constraint on the orbital inclinations (20° <  i  < 70°) and mass ratios (0.025 – 0.3). Gomez, Mason & Robinson ( 2015) constrained the black hole mass (M_BH) and orbital inclination angle to be 3 M_⊙ < M_BH < 6.2 M_⊙ and 12.75°, respectively using Argos CCD photometer data. They found the source distance to lie between 20 and 40 kpc.

Yaqoob, Ebisawa & Mitsuda (1993) analysed seventeen GINGA observations ( 1.0 – 18.0 keV) of the source and estimated the |$\tt {powerlaw}$| index to be 2 – 3. They found the spectra to be dominated by a stable soft disc component and a varying hard |$\tt {powerlaw}$| component. Spectral analysis of ASCA ( 0.5 – 10.0 keV) and RXTE/PCA (0.8 – 10.0keV) data hinted the presence of a highly inclined accretion disc ( i > 65°). RXTE/ASM ( 2.0 – 20.0 keV) data revealed a 117 d period likely due to a warped precessing accretion disc (Nowak et al. 1999). However, a more detailed study of the entire RXTE/ASM data showed that the X-ray periodicity of 4U 1957+115 is variable and not exactly 117 d as predicted earlier and is probably due to the variations in the mass accretion rate on to the black hole (Wijnands, Miller & van der Klis 2002). Nowak et al. (2008) using data from RXTE/PCA ( 3.0 – 18.0 keV) tried to estimate the black hole spin ( a) by modelling the disc continuum with |$\tt {kerrbb}$| (Li et al. 2005), |$\tt {comptt}$| (Titarchuk 1994), and |$\tt {Gaussian}$| models. X-ray data analysis suggested a rapidly spinning accretor with spin varying in the range a = 0.83 – 1. Nowak et al. ( 2008) suggested that 4U 1957+115 is at a distance greater than 5 kpc using Chandra/HETGS (0.7 – 8.0keV) and XMM–Newton ( 0.3 – 12.0 keV) data. They probed the equivalent width of the Ne IX 13.45 Å line associated with warm/hot phase in the interstellar medium to arrive at this conclusion which was later confirmed by Yao et al. ( 2008).

Nowak et al. (2012) used |$\tt {diskbb}$| (Mitsuda et al. 1984; Makishima et al. 1986), |$\tt {powerlaw}$|⁠, |$\tt {eqpair}$| (Coppi 2000) and |$\tt {diskpn}$| (Gierliński et al. 1999) models to fit the Suzaku/XIS spectrum in the energy range 0.5 – 8.0keV. They found the source spectra to have limited hard photon excess (E > 8.0keV). For a set of mass and distance values (i.e. 3 M _⊙ and 10 kpc & 16 M_⊙ and 22 kpc), the black hole spin (a) was estimated to be greater than 0.9 for an assumed disc inclination angle of 75°. Maitra et al. (2014) analysed 26 data sets from Swift/XRT (0.3 – 8.0keV) and estimated the disc inclination angle to be ∼77.6° using Monte Carlo simulations and Chi-square fitting method. They modelled the spectra with |$\tt {kerrbb}$| using several combinations of mass, inclination angle, and distance triplets. Their results indicated a maximally spinning black hole (a∼0.98). The colour hardening factor was found to be slightly higher (1.9 – 2.1) than the typical value of 1.7 (Shimura & Takahara 1995). Sharma et al. (2021) analysed nine NuSTAR data sets in the energy range  3.0 – 40.0 keV and attempted to constrain the black hole mass and spin. The spectra was modelled primarily using |$\tt {kerrbb}$| along with |$\tt {thcomp}$| (Zdziarski et al. 2020) and |$\tt {Gaussian}$| models. Their results inferred a moderate spin of 0.85 for a black hole mass of  4 – 6 M _⊙ at a source distance of 7 kpc.

While a canonical black hole X-ray binary alternates between hard and soft spectral states, 4U 1957+115 has always been observed in the high-soft state with flux levels between 20 and 70 mCrab ( 2.0 – 12.0 keV) (Yaqoob et al. 1993; Nowak et al. 1999; Wijnands et al. 2002; Nowak et al. 2008; Dunn et al. 2010; Maitra et al. 2014). Despite being one of the most studied sources by various X-ray missions, precise values of the black hole spin, mass, and distance of 4U 1957+115 are yet to be determined. Most of the accepted estimates are obtained primarily from RXTE data which was sensitive to energy  > 3 keV while the characteristic temperature of the source is ∼1 keV (Nowak et al. 1999). Indeed, the spectral parameters of the BH-LMXB 4U 1957+115 in the soft state can be better constrained if it is observed by an instrument with broad-band spectral coverage with the sensitivity  < 1 keV. Data from Soft X-ray Telescope ( SXT) (Singh et al. 2016, 2017) and Large Area X-ray Proportional Counter (LAXPC) (Yadav et al. 2016; Agrawal 2017) aboard the Indian Space Observatory AstroSat (Agrawal 2006; Singh et al. 2014; Agrawal 2017) provide such an opportunity with simultaneous broad-band spectral coverage in the energy range  0.3 – 80.0 keV. Furthermore, there is also simultaneous data available from Swift/XRT ( 0.3 – 7.0 keV) (Burrows et al. 2005) and NuSTAR/FPMA and NuSTAR/FPMB ( 4.0 – 79.0 keV) (Harrison et al. 2013) to study the evolution of spectral parameters of 4U 1957+115.

In this work, we attempt to constrain the spectral parameters of 4U 1957+115 via comprehensive broad-band spectral analysis using simultaneous data from SXT and LAXPC of AstroSat, XRT of Swift, and FPMA and FPMB of NuSTAR. The paper is structured as follows. First, we discuss the data reduction and extraction procedure in Section 2. Next, we present the steps of broad-band spectral analysis in Section 3. Lastly, we discuss our results and summarize them in Section 4.

2 OBSERVATIONS AND DATA REDUCTION

2.1 AstroSat/SXT and LAXPC

The X-ray instruments SXT and LAXPC observed the source on several occasions between 2016 and 2018 (Fig. 1), which are referred to as Epochs (Table 1). Level 2 data of SXT was deduced from Level 1 data using AS1SXTLevel2-1.4b¹ version of SXT pipeline. xselectV2.4k tool of heasoft (v6.28) was used to extract individual SXT spectra from their respective processed Level 2 event files. Each of these spectra was extracted from a circular region of 16′ radius centred on the source. The net count rate in the energy range  0.3 – 7.0 keV was found to be 16 counts s ⁻¹. Therefore, an exclusion of the central source region of 1 – 3 ′ radius to account for the pile-up was deemed not necessary. Auxiliary response file (ARF) ‘corr|$\_$|crab|$\_$|long2018|$\_$|616A.arf’, response matrix file (RMF) ‘SXT|$\_$|pc|$\_$|mat|$\_$|g0to12.rmf’, and background spectrum ‘SkyBkg|$\_$|comb|$\_$|EL3p5|$\_$|Cl|$\_$|Rd16p0|$\_$|v01.pha’ provided by Payload Operations Centre (POC) were used for the analysis.

Figure 1.

2.0 – 20.0 keV MAXI lightcurve of 4U 1957+115. The vertical lines green (solid), blue (dashed), and black (dotted) shows AstroSat, Swift, and NuSTAR observations of the source considered for the study, respectively.

Table 1.

AstroSat, Swift, and NuSTAR observations of 4U 1957+115.

Epoch	Obs ID	MJD	Start time (hh:mm:ss)	End time (hh:mm:ss)	Exposure time (ks)
			Date (yyyy-mm-dd)	Date (yyyy-mm-dd)	SXT LAXPC
1	9000000490	57549	22:31:52	01:00:19	12.93 5.10
			2016 June 10	2016 June 11
			2016 Sept 13	2016 Sept 13
2	9000001270	57879	14:09:38	18:17:38	6.44 9.33
			2017 June 05	2017 June 05
3	9000001362	57911	13:18:26	18:15:30	6.94 9.20
			2017 July 06	2017 July 06
4	9000001404	57961	08:08:13	09:13:42	7.74 18.40
			2017 July 27	2017 July 27
5	9000001658	58058	06:34:25	11:29:14	3.84 10.00
			2017 Nov 01	2017 Nov 01
6	9000002014	58213	06:18:08	06:32:38	1.08 7.85
			2018 Apr 05	2018 Apr 05
7	9000001380	57948	06:52:42	08:57:47	7.73 14.30
			2017 July 14	2017 July 14
	00030959007 *	57948	01:10:40	17:10:56	−−  − −
			2017 July 14	2017 July 15
8	000889750038 *	58776	14:28:26	14:56:56	−−  − −
			2019 Oct 20	2019 Oct 20
	30502007010 **	58776	12:51:09	23:46:09	−−  − −
			2019 Oct 20	2019 Oct 20

Note.* Swift; **NuSTAR, and Swift

Level 1 event mode data of LAXPC was converted to Level 2 data using the official laxpc pipeline.²laxpc command ‘laxpc|$\_$|make|$\_$|stdgti’ was used to generate good time interval file to remove Earth occultation of the source and South Atlantic Anomaly (SAA) passes of the satellite. Total spectrum along with RMF for LAXPC-10, LAXPC-20, and LAXPC-30 were obtained using ‘laxpc|$\_$|make|$\_$|spectra’ subroutine. Background spectrum was extracted by adopting the faint background model (Misra, Roy & Yadav 2021) of laxpc software as the source has very low countrate in the energy range greater than 30.0 keV. Since LAXPC-20 had lower background as compared to other two LAXPCs, we used LAXPC-20 data only for this study.

The LAXPCdata above 20.0 keV were not considered for the analysis as LAXPC spectrum in the higher energies was found to be background dominated. ftools subroutine ftgrouppha with group type set to ‘opt’ (Kaastra & Bleeker 2016) was used to bin the SXT spectrum for better statistics. SXT spectrum was considered only from 0.7 to 6.0 keV as the instrument has uncertainties in the effective area and response below  < 0.7 keV. A systematic error of three per cent was added as prescribed by the POC (Antia et al. 2017; Bhattacharya 2017). An offset gain correction of 25 keV determined with slope fixed at unity was applied to SXT data. Lower and higher energy range were extended to 0.1 keV and 100.0 keV respectively to supply an energy-binning array for the spectral fits.

2.2 Swift/XRT

It was found that both AstroSat and Swift have observed 4U 1957+115 simultaneously on 2017, July 07 (MJD 57948). Motivated by this, we proceeded with the joint spectral analysis of the data from Swift/XRT and SXT, LAXPC-20 of AstroSat. Swift data extraction and reduction were performed as per the steps outlined in Reynolds & Miller (2013) using heasoft software (v6.28). xrtpipeline command was used to reprocess the raw windowed timing mode data and then to apply the latest instrument calibrations and responses. The Swift spectra of the source were extracted from a circular region of 47″ centred around the source image. Adjoining source free region of the same size was used to extract the background spectrum. We used the latest RMF provided by the POC team and an ARF file created using xrtmkarf task for the analysis. Swift spectra were also binned on the similar lines of SXT spectra for better statistics. Spectral analysis was carried out in the energy range  0.3 – 6.0 keV. A three per cent systematic error, as mentioned in Swift/XRT CALDB Release Note4, was added while fitting.

2.3 NuSTAR/FPMA and FPMB

Of the many NuSTAR observations of the source, 4U 1957+115 was observed simultaneously by both Swift and NuSTAR (FPMA and FPMB telescopes) on 2019 October 20 (MJD 58776). The NuSTAR data were then processed using the latest nustardas software in the heasoft (v6.28) along with the latest calibration files (CALDB). The nupipeline (version 0.4.8) task was used for filtering event files. Total spectrum was extracted from a circular region of 10′ radius centred around the source whereas the background spectrum was extracted from a circular region of the same size farthest from the source. The nuproducts task was used to generate RMF and ARF for both telescopes- FPMA and FPMB. Total spectrum in the energy range  4.0 – 50.0 keV was considered for the study. No systematic error was added during the analysis.

3 SPECTRAL ANALYSIS

We carried out the combined spectral analysis of the data from AstroSat/SXT−LAXPC-20, Swift, and NuSTAR observations using xspec version−12.11.1d (Arnaud 1996). The errors in the values of all the best fit parameters were estimated at 90 per cent confidence level. The combined spectra were well represented by a thermal disc emission component and a Comptonization component. Disc emission was modelled using multi colour blackbody emission (⁠|$\tt {diskbb}$|⁠), whereas the Comptonization component was fitted with Comptonization model (⁠|$\tt {simpl}$|⁠) (Steiner et al. 2009). Interstellar absorption model: Tuebingen−Boulder Inter Stellar Medium absorption model (⁠|$\tt {tbabs}$|⁠) (Wilms, Allen & McCray 2000) was used to compensate for the photon loss due to the intervening medium between the source and observer. Model |$\tt {relxill}$| (García et al. 2014) was included to account for the relativistic reflection component from the accretion disc in the vicinity of the black hole by freezing the disc inclination angle and black hole spin, respectively, at 77.6° (Maitra et al. 2014) and 0.998. We discuss the variations of these parameters in the next section. Iron abundance (A_Fe = 1.0), log ionization (logX_i = 1.0), and high energy cut-off (E_cut = 100.0 keV) parameters were fixed during the fitting. For |$\tt {relxill}$|⁠, the inner disc radius was taken to be the last stable orbit while the outer one was kept at 400 R_g, where R_g  = (GM_BH)/c² is the gravitational radius, G is the universal gravitational constant (cm³ g⁻¹ s⁻²) and c is the speed of light (cm s⁻¹). The power-law index parameters for |$\tt {simpl}$| and |$\tt {relxill}$| were together treated as a single variable. We introduced a constant factor for all spectral fits to account for any overall calibration uncertainty between different instruments. While the constant was fixed to unity for LAXPC, it was allowed to vary for SXT in the AstroSat data analysis. The SXT constant was allowed to vary while those of LAXPC andXRT were fixed to unity in AstroSat/Swift analysis. For Nustar/Swift analysis, the variable constant was applied to XRT spectra. The best fit representative spectra and the residuals are shown in the top left (Epoch 7) and bottom left (Epoch 8) of Fig. 3 and the best fit parameters are presented in Table 2.

Table 2.

Best fit spectral parameters for the model combination |$\tt {constant} \times \tt {tbabs} \times \tt {(relxill + simpl} \otimes \tt {diskbb)}$| for (i) AstroSat data ( 0.7 – 20.0 keV), (ii) AstroSat and Swift data ( 0.3 – 20.0 keV), and (iii) Swift and NuSTAR data ( 0.3 – 50.0 keV). The best fit constant factor for AstroSat data and AsroSat/Swift data were applied to the SXT spectra, while for NuSTAR/Swift, it was applied to the XRT data.

(i) AstroSat
Epoch	Constant	Hydro-	Relxill	Asympto	Scattered	Tempera-	Disc	Total	Disc	Flux	Reduc-
	relative	gen	norma-	tic power-	fraction of	ture at	norma-	unabs-	flux	ratio	ed chi
	norma-	column	lization	law	the seed	inner disc	lization	orbed			square
	lization	density		index	photons	radius		flux
		NH	Nrelxill	(Γ)	FractSctr	kTin	Ndisc	Ftotal	Fdisc		χ2/dof
		× 1022	× 10−4		× 10−2			× 10−9	× 10−9
		(cm−2)				(keV)		(erg cm−2 s−1)	(erg cm−2 s−1)	(%)
1	1.23|$^{+0.04}_{-0.04}$|	0.11|$^{+0.02}_{-0.02}$|	2.08|$^{+1.98}_{-1.45}$|	2.75|$^{+0.36}_{-0.25}$|	<1.41	1.58|$^{+0.02}_{-0.02}$|	9.03|$^{+0.77}_{-0.71}$|	1.111|$^{+0.035}_{-0.033}$|	1.084|$^{+0.029}_{-0.027}$|	97.57	62/99
2	1.18|$^{+0.05}_{-0.05}$|	0.12|$^{+0.01}_{-0.01}$|	<1.06	2.36|$^{+0.24}_{-0.22}$|	10.93|$^{+3.94}_{-2.90}$|	1.42|$^{+0.04}_{-0.04}$|	11.45|$^{+1.58}_{-1.34}$|	1.017|$^{+0.033}_{-0.032}$|	0.894|$^{+0.029}_{-0.030}$|	87.91	71/96
3	1.13|$^{+0.04}_{-0.04}$|	0.13|$^{+0.03}_{-0.03}$|	2.28|$^{+2.28}_{-1.55}$|	2.29|$^{+0.32}_{-0.42}$|	3.49|$^{+2.64}_{-1.90}$|	1.54|$^{+0.04}_{-0.04}$|	10.55|$^{+1.50}_{-1.28}$|	1.235|$^{+0.049}_{-0.046}$|	1.152|$^{+0.034}_{-0.032}$|	93.28	63/96
4	1.31|$^{+0.04}_{-0.04}$|	0.16|$^{+0.02}_{-0.02}$|	2.57|$^{+1.75}_{-1.19}$|	2.39|$^{+0.16}_{-0.20}$|	<0.38	1.59|$^{+0.02}_{-0.02}$|	8.42|$^{+0.65}_{-0.59}$|	1.101|$^{+0.035}_{-0.031}$|	1.057|$^{+0.027}_{-0.025}$|	96.00	101/100
5	1.21|$^{+0.04}_{-0.04}$|	0.17|$^{+0.04}_{-0.04}$|	4.79|$^{+3.36}_{-2.41}$|	2.52|$^{+0.13}_{-0.20}$|	<0.86	1.53|$^{+0.02}_{-0.02}$|	10.50|$^{+1.04}_{-0.92}$|	1.182|$^{+0.047}_{-0.043}$|	1.113|$^{+0.035}_{-0.032}$|	94.16	83/94
6	1.12|$^{+0.05}_{-0.05}$|	0.16|$^{+0.05}_{-0.05}$|	4.22|$^{+6.52}_{-3.60}$|	2.83|$^{+0.22}_{-0.38}$|	<1.08	1.34|$^{+0.03}_{-0.03}$|	13.58|$^{+1.74}_{-1.52}$|	0.881|$^{+0.048}_{-0.044}$|	0.842|$^{+0.036}_{-0.032}$|	95.57	77/83
(ii) Swift +
AstroSat
7	0.81|$^{+0.03}_{-0.03}$|	0.15|$^{+0.02}_{-0.02}$|	2.79|$^{+1.92}_{-1.40}$|	2.51 |$^{+0.26}_{-0.25}$|	7.51|$^{+3.07}_{-2.37}$|	1.56|$^{+0.04}_{-0.04}$|	11.12 |$^{+1.28}_{-1.09}$|	1.531|$^{+0.060}_{-0.053}$|	1.378|$^{+0.038}_{-0.036}$|	90.01	106/148
(iii) Swift +
NuSTAR
8	0.95|$^{+0.02}_{-0.02}$|	0.23|$^{+0.02}_{-0.02}$|	13.03|$^{+3.39}_{-3.89}$|	3.01 |$^{+0.11}_{-0.09}$|	1.52|$^{+0.59}_{-0.63}$|	1.61|$^{+0.01}_{-0.01}$|	11.72 |$^{+0.31}_{-0.29}$|	1.307|$^{+0.007}_{-0.007}$|	1.268|$^{+0.007}_{-0.006}$|	97.02	572/567

The disc normalization, N_disc varies from |$8.42^{+0.65}_{-0.59}$| for Epoch 4 to |$13.58^{+1.74}_{-1.52}$| for Epoch 6 (Table 2). The variation of N_disc may be due to the change in the apparent inner disc radius, r_in (km),

$$\begin{eqnarray} \begin{aligned} N_{disc} = \biggl (\frac{r_{in}}{D_{10}}\biggl)^2 cos(i) \end{aligned}, \end{eqnarray}$$

(1)

where D₁₀ is the distance to the source (in units of 10 kpc) and i is the disc inclination angle (°) of the source. The absolute or true or colour corrected inner disc radius (R_in) is related to r_in as |$R_{\rm in} \simeq f_c^2\times r_{in}$|⁠, where f_c is the colour hardening factor. Thus the variation of N_disc implies that either the inner disc radius has changed by ∼25 per cent or the colour hardening factor is not constant and has a variation of ∼12 per cent. Fig. 2 shows the absolute inner disc radii inferred by considering the standard colour hardening factor value f_c = 1.7 (Shimura & Takahara 1995) and inclination angle of ∼78° (Maitra et al. 2014) at four different distances D₁₀  = 5 kpc, 10 kpc, 20 kpc, and 30 kpc for various epochs. The inner most stable circular orbit radii (R_{ISCO_sw}) for a non-spinning black hole of mass 5 M_⊙ and 10 M_⊙ are marked with the black solid line in Fig.   2. Since the inner disc radii should be larger than the R_ISCO values for a non-spinning black hole and for a colour factor close to the standard value, the distance must be larger than 30 kpc for a black hole of mass > 5M _⊙. The values of inner most stable circular orbit radii (R_{ISCO_kerr}) for rapidly spinning black hole of mass 5 M_⊙, 10 M_⊙, and 15 M_⊙ are also plotted (yellow solid line) in the Fig.   2. Thus, for a rapidly spinning black hole, more plausible values such as distance of ∼10 kpc and black hole mass of  < 10 M _⊙ are allowed. We note that the inferred luminosity of the source is around ∼5 × 10³⁶ erg cm⁻² s⁻¹and ∼5 × 10³⁸ erg cm⁻² s⁻¹for an assumed distance of 5 and 30 kpc, and hence the source is sub-Eddington if the black hole mass is > 5M _⊙.

Figure 2.

Absolute inner disc radius (R_in) as a function of Epochs calculated at 5 kpc, 10 kpc, 20 kpc, and 30 kpc. Each of the horizontal lines represents R_ISCO values for maximally spinning (ISCO_kerr) black hole of mass 5 M_⊙, 10 M_⊙, and 15 M_⊙ and non-spinning (ISCO_sw) black hole of mass 5 M_⊙ and 10 M_⊙.

The above inferences are based on the model |$\tt {diskbb}$|⁠, which is an approximation to the spectra arising from a relativistic disc, extending nearly to the R_ISCO value. In order to study the more detailed variation of physical parameters such as the accretion rate, we fit the spectra by replacing |$\tt {diskbb}$| with the relativistic disc model |$\tt {kerrd}$| (Laor 1991; Ebisawa et al. 2003). The xspec model |$\tt {kerrd}$| has been chosen instead of |$\tt {kerrbb}$|⁠, since the latter allows for variation of the inner disc radius as a parameter and the analysis using |$\tt {diskbb}$| suggests that the inner disc radius varies for the different Epochs (Fig. 2). Based on the analysis mentioned above we fix the spin of the black hole to a = 0.998 and the source distance 10 kpc. The inner disc radii in models |$\tt {relxill}$| and |$\tt {kerrd}$| were tied together and treated as one variable. Outer disc radii in the models |$\tt {relxill}$| and |$\tt {kerrd}$| were assigned to 400 R_g and 1 × 10⁵ R_g, respectively. Mass accretion rate and photon index parameters were allowed to vary during the fitting. Remaining parameters were treated in the same way as described in Section 3.

We first analysed the Epoch 4 data, which has the smallest inferred inner disc radius from the |$\tt {diskbb}$| model and assume that for this observation the inner disc radius is equal to the R_ISCO value of R_in = 1.235 R_g. For this spectral fitting, owing to the complexity of the model and the low value of scattering fraction, the photon index (Γ) was not constrained and hence was fixed to ∼2.4 (i.e. the value obtained before (Table 2)). For this combination the black hole mass was constrained to be |$6.10^{+0.82}_{-0.70}$| M_⊙ and the inclination angle of |$i = 72.5^{+2.10}_{-2.04}$| degree. We then fitted the other observations with the mass fixed at 6.10 M_⊙ and inclination angle of i = 72.5°, but now allowing for the inner disc radius to vary. The results of the spectral fits are tabulated in Table 3. Alternatively, we can also fix the inner disc radii for all the other observations at R_ISCO and leave the colour hardening factor to vary. This led to the variation of the colour hardening factor from 1.7 (fixed for Epoch 4) to 1.46.

Table 3.

Best fit spectral parameters for the model combination |$\tt {constant} \times \tt {tbabs} \times \tt {(relxill + simpl} \otimes \tt {kerrd)}$| for (i) AstroSat data ( 0.7 – 20.0 keV), (ii) AstroSat and Swift data ( 0.3 – 20.0 keV), and (iii) Swift and NuSTAR data (0.3 – 50.0keV), with fixed disc inclination (72.5°), black hole mass (6.10 M _⊙), black hole spin (0.998), and source distance (10 kpc). The best fit constant factor for AstroSat and AsroSat/Swift data was applied to the SXT spectra, while for NuSTAR/Swift it was applied to the XRT data.

(i) AstroSat
Epoch	Constant	Hydrogen	Relxill	Asymptotic	Scattered	Accretion	Inner disc	Eddington	Reduced
	relative	column	norma-	power-law	fraction of	rate	radius	ratio	chi-square
	norma-	density	lization	index	the seed
	lization				photons
		NH	Nrelxill	Γ	FractSctr	|$\dot{M}$|	Rin	L(a)	χ2/dof
		× 1022	× 10−4		× 10−2	× 1018
		(cm−2)				(g s−1)	(Rg)
1	1.21|$^{+0.02}_{-0.03}$|	0.08|$^{+0.03}_{-0.03}$|	1.74|$^{+1.83}_{-1.36}$|	2.75(f)	<0.38	0.084 |$^{+0.009}_{-0.006}$|	1.869|$^{+0.231}_{-0.288}$|	0.054|$^{+0.006}_{-0.004}$|	79/100
2	1.16|$^{+0.04}_{-0.04}$|	0.12|$^{+0.01}_{-0.01}$|	<1.65	2.32|$^{+0.20}_{-0.17}$|	9.50 |$^{+3.24}_{-2.32}$|	0.091 |$^{+0.009}_{-0.008}$|	2.511|$^{+0.176}_{-0.172}$|	0.058|$^{+0.006}_{-0.005}$|	77/96
3	1.12|$^{+0.04}_{-0.03}$|	0.13|$^{+0.02}_{-0.03}$|	2.89|$^{+1.49}_{-1.47}$|	2.29(f)	2.78 |$^{+1.30}_{-1.25}$|	0.106 |$^{+0.011}_{-0.009}$|	2.320|$^{+0.160}_{-0.168}$|	0.068|$^{+0.006}_{-0.007}$|	77/97
4	1.31|$^{+0.04}_{-0.04}$|	0.15|$^{+0.01}_{-0.01}$|	3.05|$^{+0.42}_{-0.42}$|	2.39(f)	<0.38	0.076 |$^{+0.004}_{-0.003}$|	1.235(f)	0.049|$^{+0.002}_{-0.003}$|	52/82
5	1.21|$^{+0.04}_{-0.04}$|	0.15|$^{+0.01}_{-0.01}$|	<4.90	2.57|$^{+0.03}_{-0.03}$|	<0.86	0.096 |$^{+0.009}_{-0.009}$|	2.208|$^{+0.158}_{-0.167}$|	0.062|$^{+0.006}_{-0.006}$|	101/95
6	1.21|$^{+0.05}_{-0.05}$|	0.19|$^{+0.06}_{-0.06}$|	9.20|$^{+12.60}_{-6.28}$|	3.05|$^{+0.19}_{-0.27}$|	<1.08	0.109 |$^{+0.019}_{-0.016}$|	3.029|$^{+0.297}_{-0.272}$|	0.070|$^{+0.010}_{-0.012}$|	87/84
(ii) AstroSat +
Swift
7	0.80|$^{+0.02}_{-0.02}$|	0.19|$^{+0.02}_{-0.02}$|	5.09|$^{+2.09}_{-2.70}$|	2.54|$^{+0.19}_{-0.20}$|	6.62 |$^{+2.45}_{-2.15}$|	0.133 |$^{+0.011}_{-0.010}$|	2.553|$^{+0.155}_{-0.157}$|	0.085|$^{+0.006}_{-0.007}$|	117/148
(iii) Swift +
NuSTAR
8	0.94|$^{+0.01}_{-0.01}$|	0.28|$^{+0.02}_{-0.02}$|	25.65|$^{+8.89}_{-7.13}$|	3.09|$^{+0.10}_{-0.11}$|	0.66|$^{+0.74}_{-0.64}$|	0.174 |$^{+0.005}_{-0.005}$|	2.729|$^{+0.051}_{-0.059}$|	0.112|$^{+0.003}_{-0.003}$|	565/567

Note.^(a)L =  L_acc/L_Edd, where |$L_{\rm acc} = \eta \dot{M}c^{2}$| with η ∼ 0.3 for a = 0.998 and L_Edd = 1.3 × 10³⁸ × (M_BH/M_⊙) erg s⁻¹, M_BH =  6.10 M_⊙

The spectra and the residuals for Epoch 7 (which includes AstroSat and Swift data) and for Epoch 8 (Swift and NuSTAR data) are shown in the bottom left and right panel of the Fig. 3. The primary component is the disc emission which dominates below 20.0 keV. However, note that the blurred reflection has a significant contribution below 1.0 keV. There is also a correlation between the inner disc radius and the mass accretion rate as shown in Fig. 4. The Spearman Rank correlation tests confirmed the correlation with a coefficient r = 0.929 and a null hypothesis probability of p = 0.001.

Figure 3.

The combined spectra for Epoch 7 (Top panels) and Epoch 8 (Bottom panels). The left panels show the best fitted model using the non-relativistic disc model (⁠|$\tt {constant}\times \tt {tbabs}\times \tt {(relxill+simpl}\otimes \tt {diskbb)}$|⁠), while the right panels show the fitting using the relativistic disc model (⁠|$\tt {constant}\times \tt {tbabs}\times \tt {(relxill+simpl}\otimes \tt {kerrd)}$|⁠).

Figure 4.

Inner disc radius in the units of R_g of the source as function of mass accretion rate for all the eight Epochs.

4 DISCUSSION AND SUMMARY

Spectral analysis of seven AstroSat observations of 4U 1957+115 (one of them having simultaneous Swift Observation) along with a joint Swift/NuSTAR observation in the energy range  0.5 – 50.0 keV was conducted. The spectra are well described by a disc emission ( |$\tt {diskbb}$|⁠), thermal Comptonization (⁠|$\tt {simpl}$|⁠), and a blurred reflection component (⁠|$\tt {relxill}$|⁠). The normalization of the disc emission was found to vary for the different observations, implying that the inner disc radii may have changed. Thus, a model where the disc extends to the last innermost stable circular orbit (ISCO) (i.e. |$\tt {kerrbb}$|⁠) cannot be applied to all the observations. On the other hand, it is also possible that the variation in the disc emission normalization is caused by changes in the colour factor. From the inferred values of the inner disc radii, a non-spinning black hole should have mass  > 5 M _⊙ and at distance larger than 30 kpc, while for a rapidly spinning black hole, these estimates turn out be more reasonable values with a black hole mass of  < 10 M _⊙ and distance ∼10 kpc.

The spectral model used in the initial analysis was an approximate one, since the relativistic nature of the disc emission was not taken into account. Epoch 4 had the least estimate of inner disc radius and hence we identified that data as being represented by a disc extending to the last stable orbit. Assuming a distance of 10 kpc and a black hole spin of 0.998, we fitted Epoch 4 with a relativistic disc model and were able to constrain the disc inclination angle to ∼72° and a black hole mass of ∼6.1 M_⊙. If instead we assume some other data set such as Epoch 8 (Swift/NuSTAR data) to be extending to the last stable orbit we get estimates of the black hole mass and inclination angle to be ∼4.2 M_⊙ and ∼61°. However, we emphasize that these estimates are based on the assumption that the distance to the source is 10 kpc and the black hole spin is 0.998. After fitting Epoch 4, with a disc extending to the last stable orbit, we fitted the other epochs allowing for the radii to vary, but fixing the inclination and black hole mass. Significant variation of inner disc radius was found with the largest value to be ∼3 R_g, which is nearly twice the last stable orbit (1.235 R_g) assumed for Epoch 4. The relativistic disc component fitting allows to constrain the accretion rate for the different observations and we find a positive correlation between the accretion rate and inner disc radii. This is in contrast to the behaviour of black hole X-ray transients where during the beginning of the transition, the system is in the low hard state with a low accretion rate and large inner disc radius. The system then evolves over time-scales of days, with increasing accretion rate to a soft state where the inner disc radius is small. Hence for such systems, the inner disc radius is inversely correlated with the accretion rate. On the other hand, time resolved spectral analysis of rapid (∼minutes) variability of black hole systems such as GRS 1915+105 (Rawat et al. 2022), reveal that for such systems and time-scales, the inner disc radius correlates with the accretion rate. Our results suggest that this correlation maybe extended to longer time variations of persistent black hole systems in the disc dominated (i.e. soft) state. Long and short term monitoring observations of such persistent sources using sensitive detectors would provide more clues regarding the drivers of the spectral variability of black hole systems.

The results presented here are from several observations which span  three years, each covering a wide energy range. One of the principle results of this analysis is that the inner disc radius varies in time and seems to be correlated to the accretion rate. Hence for a given observation it may not be identified with the last stable orbit, as has been done in previous studies. However, identifying the observation having the smallest inner disc radius with the last stable one, we find qualitatively similar results to what has been reported earlier which is that the source has an highly spinning black hole of mass ∼6 M _⊙ and inclination angle of ∼65°. The uncertainty in the distance to the source does not allow for more stringent constraints.

ACKNOWLEDGEMENTS

The authors thank the SXT and LAXPC POC teams at Tata Institute of Fundamental Research (TIFR), Mumbai, India for verifying and releasing the data via the Indian Space Science Data Centre (ISSDC) and providing the necessary software tools. This publication uses data from the AstroSat mission of the Indian Space Research Organization (ISRO). This work has made use of data and/or software provided by National Aeronautics and Space Administration (NASA)’s HEASARC. SBG thanks the Inter-University Centre for Astronomy and Astrophysics (IUCAA), Pune, India for the Visiting Associateship. SPM thanks IUCAA, Pune, India for providing facilities to complete part of this work. We thank the anonymous referee for the valuable suggestions and comments, which improved the content of the manuscript further.

DATA AVAILABILITY

The data utilized in this article are available at AstroSat−ISSDC website (http://astrobrowse.issdc.gov.in/astro_archive/archive/Home.jsp), NuSTAR Archive (https://heasarc.gsfc.nasa.gov/docs/nustar/nustar_archive.html), and Swift Archive Download Portal (https://www.swift.ac.uk/swift_portal/). The software used for data analysis is available at HEASARC website (https://heasarc.gsfc.nasa.gov/lheasoft/download.html).

Footnotes

1

http://www.tifr.res.in/~astrosat_SXT/SXTpipeline.html

2

http://astrosat-ssc.iucaa.in/?q = laxpcData