ABSTRACT
We present a broad-band spectral analysis of the atoll source 4U 1636−536 observed for ∼92 ks with NuSTAR. The source was found to be in a low-luminosity state during this observation with 3–79 keV X-ray luminosity of L3–79 keV = (1.03 ± 0.01) × 1037 erg s−1, assuming a distance of 6 kpc. We have identified and removed 12 type-I X-ray bursts during this observation to study the persistent emission. The continuum is well described by a thermal Comptonization model nthcomp with Γ ∼ 1.9, kTe ∼ 28 keV, and kTs ∼ 0.9 keV. The NuSTAR data reveal a clear signature of disc reflection, a significantly broad Fe–K emission line (around 5–8 keV), and the corresponding reflection hump (around 15–30 keV). We have modelled the data with two relativistically blurred reflection models. Both families of reflection models prefer truncated inner disc radii prior to the ISCO. We find that the inner disc is truncated with an inner radius of Rin = 3.2–4.7 RISCO (|$\simeq 16\!-\!24\, R_{\mathrm{ g}}\: \text{or}\: 36\!-\!54$| km). This inner disc radius suggests that the neutron star magnetic field strength is B ≤ 2.0 × 109 G.
1 INTRODUCTION
A neutron star low-mass X-ray binary system (NS LMXB) consists of a neutron star (NS) and a low-mass (≤1 M⊙) companion star. When the NS in an NS LMXB accretes matter from the companion star via Roche lobe overflow, a geometrically thin, optically thick disc-like structure is formed (Shakura & Sunyaev 1973). The radiation spectrum from the accretion disc, which is usually accompanied by a hot corona, is quasi-thermal in nature and is well known to be multicolour blackbody. The inverse Compton scattering of the thermal disc photon generates a power-law spectrum. Moreover, a hot single-temperature blackbody emission may arise from the boundary layer between the inner accretion disc and the NS surface. Hard X-rays (either a power-law continuum or a blackbody component) can illuminate the accretion disc and produce a reflection spectrum that consists of several emission lines and a broad hump-like shape.
The fluorescent Fe Kα line is the most prominent emission line due to its large cosmic abundance and high fluorescent yield (Bhattacharyya & Strohmayer 2007; Cackett et al. 2008; Pandel, Kaaret & Corbel 2008; Reis, Fabian & Young 2009; Degenaar et al. 2015). In the reflection spectrum, a broad hump-like shape is seen that is created by the high-energy photons that tend to Compton scatter back out of the disc (Ballantyne, Ross & Fabian 2001; Ross & Fabian 2007). Although the Fe Kα line is intrinsically narrow as expected, it becomes broad and asymmetric in the X-ray spectrum of the LMXBs due to the Doppler and the gravitational shift (Fabian et al. 2000). A profound study of this line profile is important due to its ability to provide information on the inner accretion flow in the NS LMXBs, which in turn provides constraints on the structure of the inner disc and inclination. In an NS binary system, the accretion disc may be truncated by a strong stellar magnetic field or by the boundary layer between the disc and the NS outer surface. The upper limit to the radius of the NS is related to the inner disc radius and may constrain the NS EOS (Piraino, Santangelo & Kaaret 2000; Cackett et al. 2008; Bhattacharyya 2011). Moreover, the Fe Kα line is also used to find out an upper limit to the strength of the magnetic field related to the NS (Degenaar et al. 2016; King et al. 2016; Ludlam et al. 2019).
4U 1636−536 is an atoll-type, bursting LMXB consisting of an NS and an 18th magnitude, 0.4 M⊙ companion star (van Paradijs et al. 1990). The source has been studied extensively in the literature. The source has exhibited type-I X-ray bursts, double-peaked X-ray bursts, superbursts, quasi-periodic oscillations (QPOs), and millisecond oscillations during thermonuclear bursts (Wijnands et al. 1997; Strohmayer & Markwardt 2002; Galloway et al. 2006). It has an orbital period of ∼3.8 h (van Paradijs et al. 1990). Its location has been estimated to be 6 ± 0.5 kpc by Galloway et al. (2006) who also studied its type-I X-ray bursts. Shih et al. (2005) have shown that the source regularly undergoes state transitions on ∼40-d time intervals. This source is known to show QPOs (Zhang et al. 1996; Wijnands et al. 1997; Barret, Olive & Miller 2007) at the domain of kHz. The inclination has been suggested to lie in the range ∼30–60° from optical observations (Casares et al. 2006). The source is well known to show burst oscillations at 581 Hz, which is remarkably coherent (Strohmayer & Markwardt 2002). This is possibly related to the rotation of the NS. The soft X-ray emission, modulated at the QPO frequency of kHz range, is known to have a phase lag behind the hard X-ray emission (Kaaret et al. 1999). This lag could be produced by the reprocessing of hard X-rays in a cooler Comptonizing corona with a size of at most a few kilometres.
Pandel et al. (2008) analysed three simultaneous XMM–Newton and RXTE observations of the source 4U 1636−536, once when the source was in the transitional state and twice when it was in the soft state. In all three spectra, they found clear evidence of a broad, asymmetric iron emission line extending over the energy range 4–9 keV. They reported that the line profile is consistent with the relativistically broadened Fe Kα emission line from the accretion disc. They found a lower limit of 64° on the disc inclination in 4U 1636−536 and reported that it is consistent with the 36–74° constraint on the orbital inclination by Casares et al. (2006). They reported an upper limit of the inner disc radius (Rin) that is larger than the ISCO while the source was in the transitional and soft states. Cackett et al. (2010) analysed the spectra of 10 NS LMXBs, including three XMM–Newton observations of 4U 1636−536, and found that fitting the spectrum with a self-consistent reflection model resulted in larger values for the inner disc radius than when the spectrum was fitted with the phenomenological Diskline model. They reported an inner disc radius of |$49\pm 24\,\,GM/c^2\,\,(\sim 8\pm 4\, R_{\mathrm{ ISCO}}$|, where RISCO = 6 GM/c2 for a spin parameter a = 0) from the disc blackbody component when the source was in the hard (low-flux) state, but the values were consistent with the ISCO when measured in the soft state. However, they found smaller inner disc radii (∼9 ± 2 GM/c2) when applying blurred reflection models to the low-flux state, and values larger than ISCO (∼13 ± 4 GM/c2) were measured in the soft state. Ng et al. (2010) analysed the same spectra as Pandel et al. (2008) and Cackett et al. (2010) considering pileup and background effects and suggested a symmetric line profile that could be well fitted with a GAUSSIAN model. They interpreted the line width as the result of broadening due to Compton scattering in the surface layers of the ion disc. Sanna et al. (2013) analysed six XMM–Newton observations of 4U 1636−536 with different phenomenological and reflection models. In all these observations, they found a broad Fe emission line at around 6.5 keV. They also found that Fe line profile exhibits a blue wing extending to high energies as observed by Pandel et al. (2008). Additionally, Sanna et al. (2013) measured the inner disc radius to be as large as 26.7 GM/c2 (= 4.45 RISCO) in the hard state. They reported that in four observations, the primary source of hard X-rays that reflect off the disc was the NS surface/boundary layer, whereas in the other two observations the reflected spectrum came from the corona. They also reported a high inclination of the source in most cases. Ludlam et al. (2017) analysed one NuSTAR observation of 4U 1636−536 in the hard state. They constrained the radius of the inner disc to 1.08 ± 0.06 ISCO for a = 0.3 through the disc reflection modelling. They found a high inclination of 77–80° for a = 0.0–0.3. Recently, Wang et al. (2017) analysed the other three NuSTAR observations of 4U 1636−536 while the source was in different states. In all the spectral states, they found prominent positive broad residuals around 5–10 keV. They applied different models to fit the reflected spectra and found a reasonable inclination of ∼56° with the RELXILLLP model.
In this work, we present a broad-band NuSTAR observation of the source 4U 1636−536. We search for the presence of reflection features and place constraints on the position of the inner disc. In the presence of high-quality, pileup-free NuSTAR data and with the correct astrophysical model, X-ray reflection spectroscopy can be quite a powerful tool to probe the accretion geometry. Moreover, the good energy resolution of NuSTAR allows us to identify the presence of Comptonization and of a cut-off in the high-energy emission. The paper is structured in the following format: Section 2 presents the observations and the details of data reduction. Section 3 discusses spectral analysis and results, and Section 4 provides the discussion of the results.
2 OBSERVATION AND DATA REDUCTION
NuSTAR (Harrison et al. 2013) observed the source 4U 1636−536 on 2019 April 27 for a total exposure time of ∼92 ks (Obs. ID: 30401014002). The data were collected with the two co-aligned grazing incidence hard X-ray imaging focal plane module (FPM) A and B telescopes in the 3–79 keV energy band.
The data were reduced with the standard NuSTAR data analysis software (nustardasv1.7.1) and caldb (|$\mathrm{ v}$|20181030). We applied the standard routine nupipeline (v 0.4.6) to filter the event lists. Using the nuproducts tool, we created light curve, spectra, and response files for both the FPMA and FPMB telescopes. To produce a source spectrum for both the telescopes, we extracted a circular region with a radius of 100 arcsec centred around the source position. We extracted the background spectrum from a same-sized radial region away from the source. We grouped the FPMA and FPMB spectral data with a minimum of 100 counts per bin and fitted the two spectra simultaneously.
3 SPECTRAL FITTING
The source 4U 1636−536 is known to exhibit type-I X-ray bursts. The NuSTAR FPMA and FPMB light curves contain 12 type-I X-ray bursts that are shown in Fig. 1. We have also shown the light curve after removing the time interval when the type-I X-ray bursts occur. The source was detected at an average intensity of ∼38 counts s−1 during the non-burst period. After excluding all the type-I X-ray bursts, we have fitted both the NuSTAR FPMA and FPMB spectra simultaneously as the initial fits have showed a good agreement between these two spectra. An initial inspection of the FPMA and the FPMB spectra also suggests that the source is detected significantly in the entire energy bandpass of the NuSTAR. We have therefore performed the spectral analysis over the entire 3–79 keV energy band using xspecv 12.9 (Arnaud 1996). Due to flux variations between the detectors, we have added a multiplicative constant in each fit. We have fixed the constant for the FPMA spectrum to unity and allowed it to vary for the FPMB spectrum. A value of 1.002 has been measured for the FPMB spectrum. We have used the tbabs model to account for absorption along the line of sight to the source with the abundance set to wilm (Wilms, Allen & McCray 2000) and vern cross-sections (Verner et al. 1996). We have fixed the absorption column density to the Dickey & Lockman (1990) value of 4.1 × 1021 cm−2 as the NuSTAR low-energy bandpass cuts off at 3 keV and have found it difficult to constrain from the spectral fits. All quoted uncertainties in this paper are at |$90{{\ \rm per\ cent}}$| of the confidence level if not stated otherwise in particular.
Left: 3–79 keV NuSTAR/FPMA light curve of 4U 1636−536 with a binning of 100 s. It shows the presence of 12 brief type-I X-ray bursts. Right: Light curve after removing all the type-I X-ray bursts. In this observation, the source was detected with an average intensity of ∼38 counts s−1.
3.1 Continuum modelling
This NuSTAR observation has detected the source 4U 1636−536 with a luminosity of ∼1.03 × 1037 erg s−1 that corresponds to |$\sim 5{{\ \rm per\ cent}}$| of the Eddington luminosity (LEdd). So, the source is detected in a low-luminosity state with L/LEdd in the range of 0.05–0.1. The spectra of the low-luminosity state sources are typically characterized by a thermal Comptonization model with an electron temperature (kTe) around 25–30 keV (Barret et al. 2000). In the low-luminosity state, a soft component may be observed probably to represent the unscattered emission from an optically thick accretion disc (Barret et al. 2000). Additionally, Lin, Remillard & Homan (2007) also suggested that low/hard state spectra could be modelled with a cut-off power-law component and a single-temperature blackbody component when needed.
We have modelled the continuum above 3 keV using only a thermal Comptonization model nthcomp (Zdziarski, Johnson & Magdziarz 1996; Życki, Done & Smith 1999) that may arise from either a hot corona associated with accretion disc or a boundary layer between the disc and the NS surface (see e.g. King et al. 2016; Ludlam et al. 2019). This thermal Comptonization model has a power-law component with an index Γ, a low-energy cut-off determined by the temperature of the seed photons (kTs), and a high-energy roll-over determined by the electron temperature (kTe). Here, we have assumed that the seed spectrum is a multitemperature blackbody spectrum emitted from the disc. In our fits, we have allowed both these temperatures and the power-law index to vary. This model, (tbabs×nthcomp), describes the continuum very well but gives a poor fit with χ2/dof = 4412/1880. For the nthcomp component, we have obtained Γ ∼ 1.9, kTe ∼ 28 keV, and kTs ∼ 0.9 keV. This continuum model has left large positive residuals around ∼5–8 keV and ∼15–30 keV, which can be identified as a broad Fe–K emission line and the corresponding Compton hump (see Fig. 2). However, it may be noted that we have not detected any soft blackbody component in the spectrum. It may possibly be due to the inadequate low-energy coverage as in the case of NuSTAR.
NuSTAR (FPMA in black, FPMB in red) unfolded spectra. The data were fit with an absorbed, thermal Comptonization model nthcomp. There are prominent residuals at ∼5–8 and ∼10–20 keV. Those can be indentified as a broad Fe–K emission line and the corresponding Compton backscattering hump. The spectral data were rebinned for visual clarity.
3.2 Reflection modelling
As our continuum fit indicates the presence of reflection features (see Fig. 2), we therefore have proceeded by modelling our data with a physical reflection model. We have applied the standard reflionx (Ross & Fabian 2005) model that assumes a high-energy exponential cut-off power law irradiating the accretion disc. The model components of reflionx model are as follows: Γ is the photon index of the illuminating spectrum, ξ is the disc ionization parameter, AFe is the iron abundance relative to the solar value, Nnorm is the normalization of the reflected spectrum, and z is the redshift of the source.
We have modified this reflection model reflionx in such a way that it assumes nthcomp illuminating spectrum instead of a cut-off power law, as our broad-band fits prefer a Comptonized model to describe the continuum spectrum. The cut-off power law does not have a low-energy cut-off, while Comptonization spectra require a low-energy cut-off at the seed photon temperature (kTs). Moreover, the high-energy cut-off of the illuminating power law in the reflionx model is set to 300 keV. Therefore, we have modified reflionx in such a manner that it mimics the nthcomp continuum. In order to introduce low- and high-energy cut-off, we have multiplied reflionx by a high-energy cut-off, highecut, with the folding energy Efold set to ∼3 times of the electron temperature (kTe) and the cut-off energy (Ecut-off) tied to 0.1 keV. This is how we have introduced a cut-off in the reflection continuum. Moreover, we fixed the photon index (Γ) of the illuminating spectrum to that of the nthcomp component. Thus, we have modified the reflection model reflionx in order to reproduce the nthcomp continuum by introducing the model component highecut (see Matranga et al. 2017; Mondal, Dewangan & Raychaudhuri 2020, for details). To take relativistic blurring into account, we have convolved reflionx with relconv component (Dauser et al. 2010). Now the overall model becomes TBabs×(nthcomp + highecut*relconv*reflionx), hereafter Model 1. The parameters of the relconv include the inner and the outer disc emissivity indices (qin, qout), break radius (Rbreak), the inner and outer disc radii Rin and Rout, the disc inclination (i), and the dimensionless spin parameter (a).
We have imposed a few reasonable conditions when making fits with reflection models. We have assumed an unbroken emissivity profile with a fixed slope of q = 3, as the slope is not constrained by the data. We have also fixed the outer disc radius Rout to 1000 Rg. We set a redshift of z = 0 since 4U 1636−536 is a Galactic source. From previous measurements of the NS spin frequency 581 Hz, we have approximated the spin parameter a = 0.27 as a ≃ 0.47/Pms (Braje, Romani & Rauch 2000), where Pms is the spin period in ms. Furthermore, we have fixed the disc inclination, i, to 60° as it was poorly constrained when left free to vary (see also Pandel et al. 2008). Moreover, prior knowledge of disc inclination can significantly reduce the uncertainty of the measurement of the inner disc radius (Pandel et al. 2008).
Adding the relativistic reflection significantly improves our spectral fits with a χ2/dof = 2178/1876 = 1.16. The best-fitting parameters for the continuum and the reflection spectrum are listed in Table 1. The reflection component implies a large disc truncation prior to the ISCO at 3.2–4.7 RISCO (|$\simeq 16\!-\!24\, R_{\mathrm{ g}}\: \text{or}\: 36\!-\!54$| km). This is quite consistent with Cackett et al. (2010) and Sanna et al. (2013) when the source was in the hard state. This model yields an intermediate disc ionization of ξ ∼ 223 erg s−1 cm, which is consistent with log ξ ∼ (2–3) seen in other NS LMXBs (see e.g. Cackett et al. 2010). The Fe abundance obtained is consistent with solar composition (AFe = 1.45 ± 0.26). The fitted spectrum with relativistically blurred reflection model and the residuals are shown in the Fig. 3. We have computed the distribution of Δχ2 for the parameter inner disc radius (Rin) using steppar command in xspec to determine how the goodness-of-fit changed as a function of this parameter. Fig. 4 indicates that Rin is well constrained by the data. Moreover, it shows that Rin is inconsistent with the position of the ISCO, indicating that it is truncated far from the NS surface.
The NuSTAR (FPMA in black, FPMB in red) unfolded spectra of 4U 1636−536 with the best-fitting fitted model consisting of a thermal Comptonization model and a relativistically blurred reflection model (Model 1) i.e. TBabs×(nthcomp + highecut*relconv*reflionx). Lower panel shows residuals in units of σ.
Shows the variation of |$\Delta \chi ^{2}(=\chi ^{2}-\chi _{\mathrm{ min}}^{2})$| as a function of inner disc radius (in the unit of RISCO) obtained from the relativistic reflection model (Model 1). We varied the inner disc radius as a free parameter in between |$2\, R_{\mathrm{ ISCO}}$| and |$6\, R_{\mathrm{ ISCO}}$|. The parameter is clearly well constrained by the data. The value of the inner disc radius is inconsistent with the position of the ISCO. Horizontal lines are indicating 2σ and 3σ significance level.
Best-fitting spectral parameters of the both 2019 and 2015 NuSTAR observations of the source 4U 1636−536 using model: Model 1 TBabs×(nthcomp + highecut*relconv*reflionx).
| Component | Parameter (unit) | 2019 NuSTAR | 2015 NuSTAR |
|---|---|---|---|
| value | value | ||
| tbabs | NH (× 1021 cm−2) | 4.0(f) | 4.0(f) |
| nthcomp | Γ | 1.93 ± 0.01 | 2.28 ± 0.03 |
| kTe (keV) | |$28.8_{-2.8}^{+3.9}$| | |$18.9_{-2.2}^{+3.1}$| | |
| kTs (keV) | 0.99 ± 0.03 | |$1.43_{-0.15}^{+0.23}$| | |
| Norm | 0.13 ± 0.003 | 0.08 ± 0.02 | |
| highecut | Ecut (keV) | 0.1(f) | 0.1(f) |
| Efold (keV) | ≃ 3kTe | ≃ 3kTe | |
| relconv | i (deg) | 60(f) | |$86_{-4}^{a}$| |
| Rin (× RISCO) | |$3.8_{-0.6}^{+0.9}$| | 1.13 ± 0.05 | |
| reflionx | ξ (erg cm s−1) | |$223_{-6}^{+8}$| | |$2338_{-965}^{+1369}$| |
| Γ | 1.93 ± 0.01 | 2.28 ± 0.03 | |
| AFe (× solar) | 1.45 ± 0.25 | |$0.37_{-0.09}^{+0.05}$| | |
| Norm (×10−5) | 1.65 ± 0.21 | |$1.21_{-0.06}^{+0.14}$| | |
| |$F^{b}_{\mathrm{ total}}$| (×10−9 erg s−1 cm−2) | 2.4 ± 0.01 | 1.9 ± 0.02 | |
| Fnthcomp (×10−9 erg s−1 cm−2) | 1.9 ± 0.01 | 1.2 ± 0.01 | |
| Freflionx (× 10−9 erg s−1 cm−2) | 0.5 ± 0.01 | 0.7 ± 0.01 | |
| L3–79 keV (× 1037 erg s−1) | 1.03 ± 0.01 | 0.82 ± 0.02 | |
| χ2/dof | 2178/1876 | 998/1040 |
| Component | Parameter (unit) | 2019 NuSTAR | 2015 NuSTAR |
|---|---|---|---|
| value | value | ||
| tbabs | NH (× 1021 cm−2) | 4.0(f) | 4.0(f) |
| nthcomp | Γ | 1.93 ± 0.01 | 2.28 ± 0.03 |
| kTe (keV) | |$28.8_{-2.8}^{+3.9}$| | |$18.9_{-2.2}^{+3.1}$| | |
| kTs (keV) | 0.99 ± 0.03 | |$1.43_{-0.15}^{+0.23}$| | |
| Norm | 0.13 ± 0.003 | 0.08 ± 0.02 | |
| highecut | Ecut (keV) | 0.1(f) | 0.1(f) |
| Efold (keV) | ≃ 3kTe | ≃ 3kTe | |
| relconv | i (deg) | 60(f) | |$86_{-4}^{a}$| |
| Rin (× RISCO) | |$3.8_{-0.6}^{+0.9}$| | 1.13 ± 0.05 | |
| reflionx | ξ (erg cm s−1) | |$223_{-6}^{+8}$| | |$2338_{-965}^{+1369}$| |
| Γ | 1.93 ± 0.01 | 2.28 ± 0.03 | |
| AFe (× solar) | 1.45 ± 0.25 | |$0.37_{-0.09}^{+0.05}$| | |
| Norm (×10−5) | 1.65 ± 0.21 | |$1.21_{-0.06}^{+0.14}$| | |
| |$F^{b}_{\mathrm{ total}}$| (×10−9 erg s−1 cm−2) | 2.4 ± 0.01 | 1.9 ± 0.02 | |
| Fnthcomp (×10−9 erg s−1 cm−2) | 1.9 ± 0.01 | 1.2 ± 0.01 | |
| Freflionx (× 10−9 erg s−1 cm−2) | 0.5 ± 0.01 | 0.7 ± 0.01 | |
| L3–79 keV (× 1037 erg s−1) | 1.03 ± 0.01 | 0.82 ± 0.02 | |
| χ2/dof | 2178/1876 | 998/1040 |
Notes. Here, we have used the standard reflionx model that assumes a high-energy exponential cut-off power-law irradiating the accretion disc, modified in such a manner that it mimics the nthcomp continuum (see text). The outer radius of the relconv spectral component was fixed to 1000 Rg. We fixed emissivity index q = 3. The Efold parameter is fitted to be 3 times the kTe.
The parameter is at the hard-coded limit of the model.
All the unabsorbed fluxes are calculated in the energy band 3–79 keV.
Best-fitting spectral parameters of the both 2019 and 2015 NuSTAR observations of the source 4U 1636−536 using model: Model 1 TBabs×(nthcomp + highecut*relconv*reflionx).
| Component | Parameter (unit) | 2019 NuSTAR | 2015 NuSTAR |
|---|---|---|---|
| value | value | ||
| tbabs | NH (× 1021 cm−2) | 4.0(f) | 4.0(f) |
| nthcomp | Γ | 1.93 ± 0.01 | 2.28 ± 0.03 |
| kTe (keV) | |$28.8_{-2.8}^{+3.9}$| | |$18.9_{-2.2}^{+3.1}$| | |
| kTs (keV) | 0.99 ± 0.03 | |$1.43_{-0.15}^{+0.23}$| | |
| Norm | 0.13 ± 0.003 | 0.08 ± 0.02 | |
| highecut | Ecut (keV) | 0.1(f) | 0.1(f) |
| Efold (keV) | ≃ 3kTe | ≃ 3kTe | |
| relconv | i (deg) | 60(f) | |$86_{-4}^{a}$| |
| Rin (× RISCO) | |$3.8_{-0.6}^{+0.9}$| | 1.13 ± 0.05 | |
| reflionx | ξ (erg cm s−1) | |$223_{-6}^{+8}$| | |$2338_{-965}^{+1369}$| |
| Γ | 1.93 ± 0.01 | 2.28 ± 0.03 | |
| AFe (× solar) | 1.45 ± 0.25 | |$0.37_{-0.09}^{+0.05}$| | |
| Norm (×10−5) | 1.65 ± 0.21 | |$1.21_{-0.06}^{+0.14}$| | |
| |$F^{b}_{\mathrm{ total}}$| (×10−9 erg s−1 cm−2) | 2.4 ± 0.01 | 1.9 ± 0.02 | |
| Fnthcomp (×10−9 erg s−1 cm−2) | 1.9 ± 0.01 | 1.2 ± 0.01 | |
| Freflionx (× 10−9 erg s−1 cm−2) | 0.5 ± 0.01 | 0.7 ± 0.01 | |
| L3–79 keV (× 1037 erg s−1) | 1.03 ± 0.01 | 0.82 ± 0.02 | |
| χ2/dof | 2178/1876 | 998/1040 |
| Component | Parameter (unit) | 2019 NuSTAR | 2015 NuSTAR |
|---|---|---|---|
| value | value | ||
| tbabs | NH (× 1021 cm−2) | 4.0(f) | 4.0(f) |
| nthcomp | Γ | 1.93 ± 0.01 | 2.28 ± 0.03 |
| kTe (keV) | |$28.8_{-2.8}^{+3.9}$| | |$18.9_{-2.2}^{+3.1}$| | |
| kTs (keV) | 0.99 ± 0.03 | |$1.43_{-0.15}^{+0.23}$| | |
| Norm | 0.13 ± 0.003 | 0.08 ± 0.02 | |
| highecut | Ecut (keV) | 0.1(f) | 0.1(f) |
| Efold (keV) | ≃ 3kTe | ≃ 3kTe | |
| relconv | i (deg) | 60(f) | |$86_{-4}^{a}$| |
| Rin (× RISCO) | |$3.8_{-0.6}^{+0.9}$| | 1.13 ± 0.05 | |
| reflionx | ξ (erg cm s−1) | |$223_{-6}^{+8}$| | |$2338_{-965}^{+1369}$| |
| Γ | 1.93 ± 0.01 | 2.28 ± 0.03 | |
| AFe (× solar) | 1.45 ± 0.25 | |$0.37_{-0.09}^{+0.05}$| | |
| Norm (×10−5) | 1.65 ± 0.21 | |$1.21_{-0.06}^{+0.14}$| | |
| |$F^{b}_{\mathrm{ total}}$| (×10−9 erg s−1 cm−2) | 2.4 ± 0.01 | 1.9 ± 0.02 | |
| Fnthcomp (×10−9 erg s−1 cm−2) | 1.9 ± 0.01 | 1.2 ± 0.01 | |
| Freflionx (× 10−9 erg s−1 cm−2) | 0.5 ± 0.01 | 0.7 ± 0.01 | |
| L3–79 keV (× 1037 erg s−1) | 1.03 ± 0.01 | 0.82 ± 0.02 | |
| χ2/dof | 2178/1876 | 998/1040 |
Notes. Here, we have used the standard reflionx model that assumes a high-energy exponential cut-off power-law irradiating the accretion disc, modified in such a manner that it mimics the nthcomp continuum (see text). The outer radius of the relconv spectral component was fixed to 1000 Rg. We fixed emissivity index q = 3. The Efold parameter is fitted to be 3 times the kTe.
The parameter is at the hard-coded limit of the model.
All the unabsorbed fluxes are calculated in the energy band 3–79 keV.
It may be noted that the reflection features are better explained by the utilization of the self-consistent reflection model RELXILL. Some new flavours of the RELXILL model are available today. A flavour of the RELXILL model, RELXILLCP, uses nthcomp as an illuminating continuum. In this model, the reflection spectrum is self-consistently calculated using the more physical illumination continuum calculated with nthcomp. The downside of RELXILLCP is the fixed seed photon temperature of 0.05 keV. We detected a higher seed photon temperature of ∼0.9 keV in our continuum fit with nthcomp. Still, we attempted to fit the spectrum with RELXILLCP model. The overall model now becomes TBabs×(nthcomp + RELXILLCP), hereafter Model 2. It would be helpful to check how the best-fitting parameter values depend on the choice of the model since neither of them is ideal. The parameters of RELXILLCP are as follows: the inner and the outer disc emissivity indices (qin, qout), break radius (Rbreak), the inner and outer disc radii Rin and Rout, respectively, the disc inclination (i), the dimensionless spin parameter (a), the redshift to the source (z), which we fixed to 0, the photon index of the power law (Γ), the ionization parameter (log ξ), the iron abundance (AFe), the coronal electron temperature (kTe), the reflection fraction (frefl), and the normalization. For this fit, the emissivity profile is assumed to follow a power law with an index fixed at the canonical value of q = 3. We have tied the emissivity indices to create one emissivity index and fixed the outer disc radius Rout to 1000 Rg. We have fixed the spin parameter a = 0.27. To get a consistent fit, we tied the RELXILLCP photon index (Γ) and electron temperature (kTe) to that of the nthcomp component.
The addition of this model improved the overall fit significantly to χ2/dof = 2108/1874 = 1.13. Most of the parameters of Model 2 follow the same trend as those of Model 1. The inner disc radius Rin = 2.1–6.7 RISCO is consistent with the value obtained with the previous modelling of Model 1. This is also consistent with Cackett et al. (2010) and Sanna et al. (2013) when the source was in the hard state. This demonstrates that the measurement of the inner disc radius is robust. Moreover, the photon index (Γ), iron abundance (AFe), and the electron temperature (kTe) values are also consistent with the values of the parameters of Model 1. Importantly, Model 2 gives a reasonable disc inclination of ∼44° which is consistent with Casares et al. (2006) and Wang et al. (2017) who reported an inclination of ∼56° with the RELXILLLP model. Also, different authors reported it larger than 64° in all cases (Pandel et al. 2008; Sanna et al. 2013; Ludlam et al. 2017). However, with this model, we found a higher disc ionization parameter (log ξ) compared to that in Model 1 but comparable with other similar kinds of accreting sources. At the same time, we are unable to constrain the lower limit of the reflection fraction frefl value. The best-fitting parameter values are given in Table 2. The best-fitting spectrum, individual components, and residuals are shown in Fig. 5. Moreover, the distribution of Δχ2 is calculated using the command steppar in xspec over 20 steps in the inner disc radius (Rin) as well as in the inclination angle (i). The left-hand and right-hand panels of Fig. 6 show the Δχ2 of the fit versus the Rin and i, respectively, for Model 2.
The NuSTAR (FPMA in black, FPMB in red) unfolded spectra of 4U 1636−536 with the best-fitting fitted model consisting of a thermal Comptonization model and a self-consistent reflection model (Model 2) i.e. TBabs×(nthcomp + RELXILLCP). Lower panel shows residuals in units of σ.
Shows the variation of |$\Delta \chi ^{2}(=\chi ^{2}-\chi _{\mathrm{ min}}^{2})$| as a function of inner disc radius (in the unit of RISCO) and inclination obtained from the self-consistent reflection model RELXILLCP in Model 2. Both the parameters are well constrained by the data. Horizontal lines are indicating 1σ and 2σ significance level.
Best-fitting spectral parameters of the 2019 NuSTAR observation of the source 4U 1636−536 using model: Model 2 TBabs×(nthcomp + RELXILLCP).
| Component | Parameter (unit) | Value |
|---|---|---|
| tbabs | NH (× 1021 cm−2) | 4.0(f) |
| nthcomp | Γ | 1.83 ± 0.01 |
| kTe (keV) | |$22.3_{-1.1}^{+1.5}$| | |
| kTs (keV) | |$0.89_{-0.04}^{+0.12}$| | |
| Norm | 0.11 ± 0.01 | |
| relxillcp | i (deg) | |$44_{-6}^{+8}$| |
| Rin (× RISCO) | |$4.8_{-2.7}^{+1.9}$| | |
| logξ (erg cm s−1) | 3.58 ± 0.12 | |
| Γ | 1.83 ± 0.01 | |
| AFe (× solar) | 2.06 ± 0.50 | |
| kTe (keV) | |$22.3_{-1.1}^{+1.5}$| | |
| frefl | |$2.57^{+1.1}$|a | |
| Norm (× 10−4) | |$1.44_{-0.81}^{+1.71}$| | |
| χ2/dof | 2108/1874 |
| Component | Parameter (unit) | Value |
|---|---|---|
| tbabs | NH (× 1021 cm−2) | 4.0(f) |
| nthcomp | Γ | 1.83 ± 0.01 |
| kTe (keV) | |$22.3_{-1.1}^{+1.5}$| | |
| kTs (keV) | |$0.89_{-0.04}^{+0.12}$| | |
| Norm | 0.11 ± 0.01 | |
| relxillcp | i (deg) | |$44_{-6}^{+8}$| |
| Rin (× RISCO) | |$4.8_{-2.7}^{+1.9}$| | |
| logξ (erg cm s−1) | 3.58 ± 0.12 | |
| Γ | 1.83 ± 0.01 | |
| AFe (× solar) | 2.06 ± 0.50 | |
| kTe (keV) | |$22.3_{-1.1}^{+1.5}$| | |
| frefl | |$2.57^{+1.1}$|a | |
| Norm (× 10−4) | |$1.44_{-0.81}^{+1.71}$| | |
| χ2/dof | 2108/1874 |
Note. Here, we have used a flavour of the RELXILL model, RELXILLCP, uses nthcomp as a illuminating continuum. The outer radius of the RELXILLCP spectral component was fixed to 1000 Rg. We fixed emissivity index q = 3.
The lower bound of the parameter is not well constrained by the data.
Best-fitting spectral parameters of the 2019 NuSTAR observation of the source 4U 1636−536 using model: Model 2 TBabs×(nthcomp + RELXILLCP).
| Component | Parameter (unit) | Value |
|---|---|---|
| tbabs | NH (× 1021 cm−2) | 4.0(f) |
| nthcomp | Γ | 1.83 ± 0.01 |
| kTe (keV) | |$22.3_{-1.1}^{+1.5}$| | |
| kTs (keV) | |$0.89_{-0.04}^{+0.12}$| | |
| Norm | 0.11 ± 0.01 | |
| relxillcp | i (deg) | |$44_{-6}^{+8}$| |
| Rin (× RISCO) | |$4.8_{-2.7}^{+1.9}$| | |
| logξ (erg cm s−1) | 3.58 ± 0.12 | |
| Γ | 1.83 ± 0.01 | |
| AFe (× solar) | 2.06 ± 0.50 | |
| kTe (keV) | |$22.3_{-1.1}^{+1.5}$| | |
| frefl | |$2.57^{+1.1}$|a | |
| Norm (× 10−4) | |$1.44_{-0.81}^{+1.71}$| | |
| χ2/dof | 2108/1874 |
| Component | Parameter (unit) | Value |
|---|---|---|
| tbabs | NH (× 1021 cm−2) | 4.0(f) |
| nthcomp | Γ | 1.83 ± 0.01 |
| kTe (keV) | |$22.3_{-1.1}^{+1.5}$| | |
| kTs (keV) | |$0.89_{-0.04}^{+0.12}$| | |
| Norm | 0.11 ± 0.01 | |
| relxillcp | i (deg) | |$44_{-6}^{+8}$| |
| Rin (× RISCO) | |$4.8_{-2.7}^{+1.9}$| | |
| logξ (erg cm s−1) | 3.58 ± 0.12 | |
| Γ | 1.83 ± 0.01 | |
| AFe (× solar) | 2.06 ± 0.50 | |
| kTe (keV) | |$22.3_{-1.1}^{+1.5}$| | |
| frefl | |$2.57^{+1.1}$|a | |
| Norm (× 10−4) | |$1.44_{-0.81}^{+1.71}$| | |
| χ2/dof | 2108/1874 |
Note. Here, we have used a flavour of the RELXILL model, RELXILLCP, uses nthcomp as a illuminating continuum. The outer radius of the RELXILLCP spectral component was fixed to 1000 Rg. We fixed emissivity index q = 3.
The lower bound of the parameter is not well constrained by the data.
4 2015 NuSTAR OBSERVATION
Apart from the recent 2019 NuSTAR observation stated in the last section, it observed the source on 2015 June 6 (Obs ID: 30101024002) for a total exposure of ∼20 ks with a net count rate of 34 counts s−1. This coordinated NuSTAR observation was previously analysed by Ludlam et al. (2017). Here, we have re-analysed the 2015 NuSTAR observation in the very similar approach as done above for the recent 2019 NuSTAR observation to see if there is any change between the observations. In this connection, it may be noted that the other three NuSTAR observations (between 2015 August 25 and September 18) that are subsequent to the observation in Ludlam et al. (2017) was analysed by Wang et al. (2017). In between these three observations, the source evolves from the soft to the transitional, and the hard state (Wang et al. 2017). In this observation, the source was in the hard state as reported by Ludlam et al. (2017).
4.1 Spectral fitting
We processed the 2015 NuSTAR data in the similar fashion as described before for the 2019 NuSTAR observation. There were six type I X-ray bursts in this observation. We created good time intervals to eliminate the bursts from the spectrum of the persistent emission. We tried to fit the continuum spectrum from 3 to 70 keV with the absorbed thermal Comptonization component nthcomp assuming that the seed spectrum is a multitemperature blackbody spectrum emitted from the disc. This model, tbabs×nthcomp, describes the continuum well but left large positive residuals around 5–10 keV and comparatively less positive residuals around 20–30 keV (see left-hand panel of Fig. 7). This time the reflection hump at high energies is not as clear as detected for the 2019 NuSTAR observation. The other three NuSTAR observation, analysed by Wang et al. (2017), also did not show the presence of a clear reflection hump. For this observation, the features can be seen in the left-hand panel of Fig. 7. To describe the reflection features and relativistic effects properly, we employed our Model 1 (described earlier) to the data with very similar conditions imposed on some parameters. This model describes the spectrum successfully with χ2/dof = 998/1040 = 0.96. It predicts a small inner disc radius of Rin = (1.08–1.18) RISCO that is consistent with Ludlam et al. (2017). This fit returns a higher disc inclination of ∼86° also consistent with Ludlam et al. (2017). The ionization parameter is comparatively high but comparable with other similar kinds of accreting sources. The iron abundance is less than 1. The best-fitting parameters of Model 1 for this observation are presented in Table 1. The best-fitting spectrum with individual components can be seen in the right-hand panel of Fig. 7. The variation of Δχ2 of the fit versus the inner disc radius (Rin) is shown in Fig. 8.
Left: The 2015 NuSTAR observation shows the strong reflection features when continuum is fitted with TBabs× nthcomp. Right: The NuSTAR (FPMA in black, FPMB in red) unfolded spectra of 4U 1636−536 with the previous best-fitting model (Model 1). Lower panel shows residuals in units of σ. In both the cases, the spectra have been rebinned for plotting purposes.
The 2015 NuSTAR data: This plot shows the variation of |$\Delta \chi ^{2}(=\chi ^{2}-\chi _{\mathrm{ min}}^{2})$| as a function of inner disc radius (in the unit of RISCO) obtained from the relativistic reflection model (Model 1). We varied the inner disc radius as a free parameter in between |$1\, R_{\mathrm{ ISCO}}$| to |$4\, R_{\mathrm{ ISCO}}$|. The parameter is clearly well constrained by the data. The value of the inner disc radius is consistent with the position of the ISCO. Horizontal line is indicating 3σ significance level.
5 DISCUSSION
We have presented here a broad-band spectral analysis of the NuSTAR observation of the source 4U 1636−536, aimed to study the reflection spectrum and to constrain the accretion geometry. The continuum spectrum is hard and well described by a thermal Comptonization model nthcomp with Γ ∼ 1.9, kTe ∼ 28 keV, and kTs ∼ 0.9 keV. For the first time, we have detected a clear signature of strong disc reflection in its spectrum. The most prominent disc reflection features, a broad Fe–K emission line around 5–8 keV and the corresponding Compton hump around 15–30 keV are clearly visible in the spectrum. It must be noted that previous NuSTAR observations analysed by Wang et al. (2017) and Ludlam et al. (2017) did not show the presence of a clear Compton hump in the X-ray spectra. A correct choice of self-consistent relativistically blurred disc reflection model helps us to determine the position of the inner disc along with some other important NS parameters. The 3–79 keV persistent spectrum is well described by a combination of the Comptonization model nthcomp and a relativistic reflection of this Comptonized emission. We have modelled the data with two relativistically blurred reflection models, relconv*reflionx and RELXILLCP. Both the models assume that a thermal Comptonization spectrum is irradiating the disc. It may be physically interpreted as the region of main energy release, where hard X-rays produced would be either an optically thin boundary layer between the disc and the NS surface or a hot corona associated with the disc. A part of this hard X-ray emission may illuminate the accretion disc and produce the reflection spectrum.
The source was detected at an average intensity of ∼38 counts s−1 during the non-burst state. Our best-fitting model yields an unabsorbed flux in the 3–79 keV band of F3–79 ∼ 2.4 × 10−9 erg s−1 cm−2. The source was observed in a low-luminosity and hard spectral state (low/hard) during this observation and we measured a 3–79 keV luminosity of LX ∼ 1.03 × 1037 erg s−1, which corresponds to |$\sim 5{{\ \rm per\ cent}}$| of the Eddington luminosity assuming a distance of 6 kpc. It suggests that the rate of any outflow in 4U 1636−536 is significantly below the Eddington mass accretion rate. It also allows us to study the disc reflection of NS LMXBs in a relatively low accretion regime.
From the reflection spectrum (with Model 1), we have measured an inner disc radius of Rin = 3.2–4.7 RISCO, given that RISCO = 5.1 GM/c2 for an NS spinning at a ≃ 0.3. This would correspond to |$R_{\mathrm{ in}}=16\!-\!24\, R_{\mathrm{ g}}$| or 36–54 km for a 1.5 M⊙ NS. Another self-consistent reflection model (our Model 2) prefers an inner disc radius of Rin = 2.1–6.7 RISCO, which is consistent with the value obtained by using Model 1. So, both families of reflection models prefer an inner disc radius larger than ISCO. Moreover, the values are consistent with Cackett et al. (2010) and Sanna et al. (2013) when the source was in the hard spectral state. It indicates that the disc is truncated at a large distance away from the NS surface. The disc has a relatively low ionization (ξ ∼ 223 erg s−1 cm) and iron abundance is comparable to the solar abundance (AFe ∼ 1.4). However, the RELXILLCP model in Model 2 predicts a higher disc ionization of log ξ ∼ 3.5 erg s−1 cm. In our modelling of the reflection spectrum (using Model 1), we fixed the inclination angle to 60° as it prefers a value close to 90° when we left it free. A very high inclination (∼90°) of the accretion disc is at odds with the fact that no dips or eclipses have been observed in this source. Previously, high inclination angles have been reported by different authors (Pandel et al. 2008; Sanna et al. 2013; Ludlam et al. 2017). In our case, the reflection modelling with RELXILLCP in Model 2 gives a reasonable inclination of ∼44°, which is quite similar (∼56°) to that reported by Wang et al. (2017) using the RELXILLLP model. During this observation, the source was in the hard state and the corresponding electron temperature (kTe) of the corona was ∼28 keV. This amount of electron temperature indicates that the corresponding cut-off energy (Ecut) is also very high, which is ≃2–3 kTe. The high value of Ecut indicates that the illuminating source that produces the reflection component is essentially the corona.
We have detected a significantly broader Fe Kα emission line (σ ∼ 0.98 keV) from this source. The broadness of the line requires it to be located deep within the potential well where the orbital velocities are mildly relativistic (see also King et al. 2016). Moreover, the observed symmetric nature (a broad blue wing in addition to the broad red wing) of the Fe Kα line profile requires it to be located far enough away as to not suffer severe relativistic Doppler beaming (see also King et al. 2016). Pandel et al. (2008) suggest that such a broader Fe Kα line needs a higher disc inclination of around 60–70°.
To place this recent 2019 NuSTAR observation in context, we have re-analysed another NuSTAR observation (carried out in 2015), which was previously analysed by Ludlam et al. (2017). Like the recent 2019 NuSTAR observation, this particular observation was also in the hard spectral state as reported by Ludlam et al. (2017). In this analysis, we have adopted the same models and approach as previously done to investigate whether there is any change between the observations. This also allows us to probe how the choice of reflection and continuum models can affect the conclusions about the disc parameters inferred from the reflection component. We found that our results (obtained with Model 1) are more in line with those already found by Ludlam et al. (2017). In particular, we found an inner disc radius to be ∼1.13 RISCO with an inclination angle of ∼85°. Additionally, the photon index, high-energy cut-off, and ionization parameters are largely consistent with the values found by Ludlam et al. (2017). However, Ludlam et al. (2017) reported an iron overabundance of 4.5–5.0 relative to solar value. According to them, it is unlikely that the abundance of iron is that high in the low-mass companion star in 4U 1636−536. Our model estimates an iron abundance of ∼0.5 relative to the solar value, which is quite reasonable for the low-mass companion star. Further, we found that the major difference between these two observations lies in the fact that the recent 2019 NuSTAR observation predicts a truncated inner disc radius, whereas 2015 NuSTAR observation predicts a small inner disc radius consistent with ISCO. The standard accretion disc model suggests that the spectral state may be associated with the level of disc truncation (Done, Gierliński & Kubota 2007). Here, both the NuSTAR observations under consideration were in the hard spectral state, but a different level of disc truncation was observed. This behaviour is quite puzzling and further critical analysis in this direction is needed to find how similar kinds of spectral states give rise to the different levels of disc truncation.
Disc truncation is likely the result of the presence of a boundary layer that lies between the disc and the NS surface or the associated magnetic field of the NS. Here, we estimate the mass accretion rate (|$\dot{m}$|) per unit area at the NS surface using equation (2) of Galloway et al. (2008). The estimated value of |$\dot{m}$| during this observation is ∼1.6 × 10−9 M⊙ yr−1 using the persistent flux Fp = 2.4 × 10−9 erg s−1 cm−2 and assuming the bolometric correction cbol is ∼1.38 for the non-pulsing sources (Galloway et al. 2008) and considering 1 + z = 1.31 for an NS with mass 1.5 M⊙ and radius 10 km, where z is the surface redshift. At this mass accretion rate, we estimate the maximum radial extent (Rmax) of the boundary layer region using equation (2) of Popham & Sunyaev (2001). It estimates a maximum radial extent of ∼5.4 Rg for the boundary layer (assuming MNS = 1.5 M⊙ and RNS = 10 km). The extent of the boundary layer region is small to account for the disc position. It may be because it does not account for a spin and viscous effects in this layer.
The inferred radial extension of the boundary layer is small compared to the disc truncation radius and the magnetic field of the NS would be responsible for the disc truncation. We can use our measured inner disc radius to estimate an upper limit for the magnetic field strength of the NS. We use equation (1) of Cackett et al. (2009) to calculate the magnetic dipole moment (μ). We estimate a bolometric flux of Fbol ≃ 2.94 × 10−9 erg cm−2 s−1 by extrapolating the best fit over the 0.01–100 keV range. We assume an NS of mass 1.5 M⊙ and radius 10 km, and a distance of 6 kpc. We keep similar assumptions regarding the geometrical and efficiency parameters as in Cackett et al. (2009): kA = 1, which is a factor depending on the geometry, spherical or disc-like, of the accretion flow; fang = 1, which is known as the anisotropy correction factor; and accretion efficiency in the Schwarzschild metric η = 0.1. The constraint that Rin ≤ 24 Rg from the best-fitting model then yields B ≤ 2.0 × 109 G at the magnetic poles.
In this observation, we have observed 12 brief (10–100 s) X-ray bursts. Accretion on to NS in LMXBs provides the fuel for thermonuclear burning that powers X-ray bursts. Therefore, the detection of X-ray bursts indicates that accreted material is still reaching the surface of 4U 1636−536, even when the disc is truncated at a larger distance. This behaviour can be explained with the model proposed by Kluzniak & Wilson (1991). They suggest that in this scenario accreted material can free-fall crossing the ‘gap’ between the disc and the NS surface and then strike the NS surface, creating a hot accretion belt with a temperature inversion. According to Lamb et al. (1977), the accreted material can reach the NS via a magnetic gate, but it requires a relatively low magnetic field. If the accreted material is channelled along the magnetic field lines, it would cause a hotspot on the magnetic pole (Lamb et al. 2009). However, any further possible mechanisms for accretion on to the NS are out of the scope of this work. Moreover, these studies can further be extended to investigate how the accretion disc responds to an X-ray burst and the impact of bursts on the accretion disc dynamics.
ACKNOWLEDGEMENTS
We thank the referee for many invaluable comments and suggestions. This research has made use of data and/or software provided by the High Energy Astrophysics Science Archive Research Centre (HEASARC). This research also has made use of the NuSTAR data analysis software (nustardas) jointly developed by the ASI Science Center (ASDC, Italy) and the California Institute of Technology (Caltech, USA). ASM and BR would like to thank Inter-University Centre for Astronomy and Astrophysics (IUCAA) for their hospitality and facilities extended to them under their Visiting Associateship Programme.
DATA AVAILABILITY
This data set with Obs. ID: 30401014002 dated 2019 April 27 is in public domain put by NASA at their website (https://heasarc.gsfc.nasa.gov). The public date is 2020 May 8.
