Introduction

MAXI J1535-571 is a transient low-mass X-ray compact binary object with a black hole and a companion star1. The detection of MAXI J1535-571 in the constellation Norma, ~ 1° away from the Milky Way Galactic plane at Right Accession and Declination of 15 h 35 m 19.73 s, and − 57° 13′ 48″ respectively have been reported2,3,4. MAXI J1535-571 has a high angle of inclination of 67.4° and an orbital spin of about 0.995, and its distance of 4.1–6.5 kpc has been constrained6. The outburst of MAXI J1535-571 occurred on September 2, 2017, with dramatic episodes and ejection of the accretion flow. The accretion flow consists of Keplerian flow that is immersed inside the sub-Keplerian flow according to the Two-component Advective Flow (TCAF) model7,8. The sub-Keplerian and Keplerian flow contain optically thin- and optically thick-plasma8,9. These plasma explain the non-thermal (power-law) radiations and multi-color blackbody radiations respectively8,10. The Keplerian flow generates seed soft photons11and magnetic fields/activities mediate their motion12,13,14to the sub-Keplerian region. As a result, interception and thermally or inverse comptonization of seed soft photons by high-temperature electrons produce hard X-rays15,15,17. MAXI J1535-571 exhibited a hard spectral state from the onset till the peak of the outburst. The subsequent supply and the gradual manifestation of seed soft photons in the post-shock region dilute and cool the heating mechanism. This softened the hardness of the X-ray spectrum and caused MAXI J1535-571 to embark on spectral evolution on September 10, 2017, and was seen in the hard-intermediate state between September 12–18, 2017. It was later found in the soft-intermediate state on September 19 to October 11, 2017, and then in the soft state on January 19, 201818,19. This spectral evolution is accompanied by intermittent behavior and X-ray flux variability during the intermediate state2,20,21. Also, evidence of brightening and re-brightening events characterized by phenomenological soft-to-hard spectral states with different X-ray luminosity have been observed22,23. These phenomena, as seen in other black hole binaries/candidates, are tied to the evolution of the components of accretion flow24,25. That is, the X-ray flux variability, intermittent behavior, and geometry of the accretion flow in each spectral state are tied to the variations of components of accretion flow rates26,26,27,28,29,31. This implies that the isotropic advection of matter from the Keplerian region to the sub-Keplerian region determines the structure and geometry of the accretion flow32. Moreover, the evolution of ubiquitous humps; quasi-periodic oscillations (QPOs) in power density spectral is the most important temporal (timing) feature of outburst phases of transient BHCs33,34. The QPO exists in three forms namely, type A, B, and C respectively. Several models have been extensively discussed in the literature that QPO is produced from; (i) oscillation of post-shock waves in the Comptonizing regions35, (ii) perturbations/oscillations inside a Keplerian disk/flow36, (iii) oscillation of warped disk37, (v) accretion-ejection activities/ instabilities in Keplerian flow38, (iv) variations/fluctuations of mass accretion rates39, (vii) oscillations/perturbations in transition/slim layers separating Keplerian flow and sub-Keplerian flow40, (viii) relativistic effect of Lense-Thirring precession of accretion flow inner regions41,41,43, etc. Therefore, QPOs occur as a result of different physical processes. However, long duration and continuous evolution of QPOs require Propagating Oscillatory Shock (POS) model to explain the oscillation of post-shock/waves44,45. The variations/fluctuations of components of the accretion flow/rates create propagating oscillatory shock/waves (quasi-periodic oscillations29,39,46. The sub-Keplerian flow participates actively in oscillations of strong shock waves more than the Keplerian flow47. The frequency of the oscillating shock waves can be estimated via spectral analysis29,32. Strong oscillating shock is often observed during the hard spectral states where the resonance phenomenon is solely responsible for the generation of type–C QPO frequency of 0.1–10 Hz33,46,48,48,50. The frequency of type-C QPO is inversely proportional to the shock location/strength (rs) and its’ drift (or movement) is directly related to the spectral evolution51. Therefore, spectral and temporal (timing) accretion flow characteristics are related. Stiele and Kong52, Mereminskiy et al3., and Shang et al.53 carried out spectra-temporal (timing) analyses of MAXI J1535-571 during its hard-intermediate state (HIMS). These authors reported the photon index–QPO frequency (Γ–vQPO) relation, but the physical processes/mechanisms responsible for Γ–vQPO tight relation is ongoing research. This study provides the basis for investigating the origin of Γ–vQPO relation using spectral analysis approach. We adopted the TCAFmodel prediction32 which inferred that the QPO frequency of strong oscillating shock (during the hard spectral states) can be estimated empirically. In this paper, MAXI J1535-571 data observed by MAXI/GSC and SWIFT/BAT X-ray mission was analyzed to determine the photon index–QPO frequency (Γ–vQPO) characteristics and explain the physical processes/mechanisms responsible for their origin. This paper is organized as follows; data reduction and spectral analysis/fitting are presented in "Data acquisition/reduction and spectral analysis" section. Spectral analysis results and the discussion and conclusion is presented in presented in “Results” and “Discussion and conclusion” sections respectively.

Data acquisition/reduction and spectral analysis

The MAXI J1535-571 data observed by SWIFT/BAT (Swift/Burst Alert Telescope) and MAXI/GSC (Monitor of All-sky X-ray Image/Gas slit camera) from UT 21:57:38 to UT 22:13:58 and UT 21:18:46 to UT 22:32:26 on September 13, 2017, were obtained from the HEASARC (High Energy Astrophysics Science Archive Research Center). The standard pipeline products; batsurvey product and CALDB version 20170131 (SWIFT/BAT) and mxproduct (MAXI/GSC), were downloaded and incorporated into the HEASoft (High Energy Astrophysical Software)/Flexible Image Transport System Tools (FTOOLS) version 6.28 and its software packages. The batsurvey alongside the Swift/BAT CALDB 01/31/2017 version was used for data reduction, and the time-averaged spectra and the response files were created using make_survey_pha. For the MAXI/GSC detector, the spectrum events were extracted using 80 arc-minute circular region/radius which covers about 90% of the MAXI J1535-571’s flux. The extraction of the background spectrum was made in a blank-sky area near the source region using 160 arc-minute circular region/radius. The source.pha file was generated after subtracting the background spectrum from the source spectrum. Thereafter, the two X-ray mission data were simultaneously fitted/modelled in XSPEC version 12.10.1f. The data with a low signal-to-noise ratio was removed using the XSPEC ignore command. The data within 2–135 keV energy bands were utilized while fitting the data. The phenomenological and physical models that explain the components of the accretion flow were used in modelling the data. The multiplicative tbabsmodel54) that accounts for the interstellar absorption alongside the power-law radiation model and multi-temperature accretion disk (diskbb55) were first used. Hence, tbabs*(pow + diskbb; M1). The diskbbmodel explains the thermal-component radiations originating from the optically thick plasma whereas the power-law explains the non-thermal component radiations originating from the optically thin plasms9,10. The hydrogen column density was “steppar” between 2.00 and 4.00 and this improved the fit statistic. A best-fit photon index of 2.34 and a reduced Chi-squared value of 1.27 (173.19/136) was obtained. This value is above the acceptable reduced Chi-squared limit ≤ 1.2. This prompted us to replace the power-law model in M1 with a two-component thermal comptonization model (nthcomp56). Hence, tbabs*(nthcomp + diskbb; M2). The inner and seed photon temperatures of the diskbb and nthcomp models were “steppar” such that their value is equivalent and represents the temperature of the thermal component of the accretion flow. This improved the spectral fit statistics with an acceptable reduced Chi-squared value of 1.20 and best-fit photon index of 2.30. Thereafter, we included the TCAFmodel in XSPEC as a local additive model51, and explored the option of using this physical model alongside the tbabs on the data; tbabs*(TCAF v0.3.2: M3). The compression ratio and shock location were “steppar” in the range of 3.85–4.00 and ≥ 38 respectively. This gives a good fit-statistics with a reduced Chi-squared value of 1.23 (165.46/134). The compression ratio of 4.0 indicates the presence of a strong propagating oscillating shock in the accretion flow48,49. Also, the Keplerian and sub-Keplerian mass accretion rates were linear “steppar” to run 10 iterations, and their contour plot were obtained. Moreover, the best-fit photon index of 2.30 and 2.34 obtained in “M1” and “M2” indicates that MAXI J1535-571 was in the hard-intermediate state on September 13, 201718,19,53. This gives us clues on the feasibility of estimating the QPO frequency based on the TCAFmodel prediction. Therefore, the existence of QPO in the accretion flow can be ascertained via resonance phenomenon. Resonance occurs due to the oscillation of components of the accretion flow when their timescales are comparable29,46,48,57,

$$\tau_r=\frac{t_{Kep}}{t_{suKep}}=3.5\times10^{-4}\bigg(\frac{1+A_{r}}{f_{o}\lambda}\bigg)\bigg(1-\frac{1}{R^{2}}\bigg)=0.5<\tau_r<1.5$$
(1)

where tKep and tsuKep is the Keplerian flow timescale and sub-Keplerian flow timescale respectively.

The Keplerian flow timescale is the ratio of heat energy to the cooling rate;

$$t_{K_{ep}}=\frac{3\pi k_{B}r^{2}_{s}H_{s}T_{e}n_{e}}{[\frac{2\pi}{r_{s}}}\cdot \frac{5\times10^{9}M_{d}}{m^{2}}(\frac{GM^{3}}{c_{2}})]\cdot f_{o} \lambda$$
(2)

The number density of electron (ne) comptonizing the soft photons is;

$$n_{e}=\frac{M_{d}+M_{h}}{4\pi V+r_{s}H_{s}m_{p}}$$
(3)

The height (Hs) of the post-shock is;

$$H_{s}\Bigg[\frac{\gamma(R-1)r^{2}_{s}}{R^{2}}\Bigg]^{\frac{1}{2}}$$
(4)

The speed of the waves in the post-shock is;

$$V_{+}=(r_{s}-1)^{\frac{-1}{2}}$$
(5)

The timescale of the sub-Keplerian flow is;

$$t_{suKep}=\frac{r_{s}}{V_{+}}\sim Rr_{S}(r_{s}-1)^{\frac{1}{2}}$$
(6)

Where rs, Te, KB, mp, R, G, M, c, \((m=M/M_\odot)\), and γ is the shock location, electron temperature, Boltzmann constant, the mass of the proton, compression ratio, the gravitational constant, mass of the black hole, the speed of light, the ratio of the mass of the black hole to that of the Solar mass, and adiabatic index [(4/3) for optically thick and optically thin flow] respectively. \(A_{r}=\dot{M}_h + \dot{M}_d/\dot{M}_d\) is the ratio of the accretion flow rate to the Keplerian mass accretion rate in kilograms/second, \(M_{h}=m_{h} \times M_{E}, M_{d}=m_{d} \times M_{E}\). mh and md is the sub-Keplerian mass accretion rate and Keplerian mass accretion rate (in the Eddington limit) respectively. \(\dot{M}_E\) is the Eddington mass accretion rate. The fraction of the intercepted and comptonized soft photons in the post-shock region is “fo”. “λ” accounts for the intercepted soft photon’s energy. The value of λ and foin the hard states is 10–40, and 0.01–0.05 respectively29,44,48,51,57,56,59. The frequency of the strong oscillating shock is32,49

$$v_{qpo}=\frac{c}{2\pi Rr_{g}r_{s}(r_{s}-1)^\frac{1}{2}}$$
(7)

where\(r_{g}= 2 {G} {M}_bh/{c}^2\), G, Mbh, c, rs, R, is the radial distance (unit of Schwarzchild radius), the gravitational constant, the mass of the black hole, the speed of light, shock location (unit of rg), and compression ratio respectively. Given this, we re-normalized the spectral fitting and modelled again using a cross-calibration energy–independent constant (const) factor alongside nthcomp and TCAF models. Hence; const*tbabs*(nthcomp + TCAF v0.3.2; M4). The energy–independent factor (const) accounts for the instrumental artifact. The “M4” reproduced the data well with a reduced Chi-squared value of 1.09 (137.54 /126). A best-fit photon index of 2.010 and a compression ratio of 3.985 were obtained. Also, the Keplerian mass accretion rate (mdotd), and sub-Keplerian mass accretion rate (mdoth) were obtained. The accretion flow rate {mdot(d + h); the sum of mdotd and mdoth} were calculated. The model-fitted parameters of “M4” alongside the adopted values of λ = 40, and fo= 0.0348,57 were utilized in MATLAB written codes of Eqs. (1–7). The resonance condition, Keplerian flow timescale, sub-Keplerian time scale, and QPO frequency were estimated respectively. Standard errors associated with each model’s–fitted parameter were estimated, and Pearson correlations and plots of these parameters were obtained. It is worth noting that nthcomp and TCAF are two-component accretion flow models and each model (in combination with “const” and “tbabs”) fitted the data very well in a similar manner with a reduced Chi-squared value of 1.07 (140.77/131) and 1.24 (161.13/128) respectively. However, the sole aim of using both models in “M4” was to simultaneously constrain model-fitted parameters required to estimate other accretion flow characteristics. We chose to report the best-fit parameters of “M4” (see Table 1) because this model prescribed the data very well when compared to other models used. The errors in each best-fit parameter of “M4” were estimated using the XSPEC error command.

Table 1 Best-fit spectral parameters of “M4”.
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Results

Table 1 shows the best-fit parameters of “M4” that reproduced MAXI J1535-571 data very well in comparison to other models. The errors in each best-fit parameter are at 90% confidence. Table 2 shows the Pearson product-moment correlation of the parameters.

Table 2 Pearson correlation coefficient of model’s-fitted parameters.
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The best-fit photon index of 2.010 and 2.340 was obtained using different phenomenological models (M1–M4). A compression ratio and oscillating shock location of 3.985 and 43.570 rg were respectively obtained from the physical (TCAF) model. Also, the QPO frequency of 0.840–4.961 Hz was obtained. The resonance condition in the range of (0.507–1.248) ± 0.080 was obtained. Figure 1 is the X-ray spectrum of MAXI J1535-571. The fitted MAXI/GSC and SWIFT/BAT data in the 2–40 keV (red) and 15–135 keV (black) are shown by crosses whereas the TCAF and nthcomp model is the solid thick and dotted lines respectively in the upper panel. The ratio of the fitted data to the models is shown in the lower panel. Figure 2 is the contour plot of the Keplerian and sub-Keplerian mass accretion rates at different confidence levels. The plus sign “+” is the minimum fit statistics (2.051 × 102) of the first confidence level. The three confidence fit statistic is 2.074 × 102, 2.097 × 102, and 2.143 × 102 respectively. The color bar on the right hand is the delta-fit statistic. The right-hand color bar is the delta-fit statistic. The “zig-zag” (though not elaborate) pattern in Fig. 2 suggests a correlation between the two parameters. It is worth noting that “stepparing” the Keplerian and sub-Keplerian mass accretion rates to run more iterations while fitting did not produce robust contour plot when compared to the one presented. Figure 3 shows the anti-correlation of the Keplerian and sub-Keplerian mass accretion rates. While the sub-Keplerian mass accretion rate is decreasing, the Keplerian mass accretion rate correspondingly increases. Figure 4 shows the correlation of the Keplerian and sub-Keplerian mass accretion rates with the accretion flow rate (mdot(d + h)). When the Keplerian mass accretion rate decreases from 0.0030–0.0025 MEdd, the sub-Keplerian mass accretion rate and mdot(d + h) are relatively constant. The Keplerian mass accretion rate decreases and later saturates at ~ 0.00171 MEdd while the sub-Keplerian mass accretion rate and mdot(d + h) increase. Figure 5 shows the correlation of the Keplerian and sub-Keplerian mass accretion rates with vQPO. As vQPO increases in unison with the sub-Keplerian mass accretion rate, the Keplerian mass accretion rate decreases. Therefore, vQPO and sub-Keplerian mass accretion rate are positively correlated while Keplerian mass accretion rate is anti-correlated with them. Figure 6 shows the Keplerian and sub-Keplerian mass accretion rates with photon index. As the photon index increases from 1.830 to 1.859, the Keplerian mass accretion rate decreases from 0.0030–0.0253 MEdd while the sub-Keplerian mass accretion rate saturates at ~ 0.0510 MEdd. Thereafter, the Keplerian mass accretion rate further decreases to 0.0018 MEdd and saturates at ~ 0.0017 MEdd while the sub-Keplerian mass accretion rate increases from 0.0510 to 0.0730 MEdd. The photon index increases in unison with an increase in the sub-Keplerian mass accretion rate but saturates at ~ 2.010. Figure 7 shows the anti-correlation of the Keplerian (Kep_t) and sub-Keplerian (sub-Kep_t) timescales with the accretion flow rate (mdot(d + h)). Ab initio, the Kep_t, and sub-Kep_t were decreasing when mdot(d + h) was relatively constant (0.5169 MEdd). While the accretion flow rate, mdot(d + h)), is increasing, the Kep_t and sub-Kep_t timescales correspondingly decrease to 1.770 × 1012 s and 1.689 × 1012 s respectively. The Kep_t timescale increases from (1.820–1.870) × 1012 s while the sub-Kep_t timescale decreases from (1.540–1.500) × 1012 s with a further increase in mdot(d + h). Figure 8 shows the strong positive correlation of the power-law photon index (Γ) and the QPO frequency (vQPO). The photon index fluctuates between 1.830 and 1.859 while vQPO increases from 0.840 Hz to 1.424 Hz. The photon index increases in unison with the increase in vQPO but saturates at ~ 2.000 with further increases in vQPO.

Fig. 1
figure 1

MAXI J1535-571 X-ray spectrum. In the upper panel, the fitted MAXI/GSC (red) and SWIFT/BAT (black) data are shown by crosses whereas the TCAF and nthcomp model are shown by solid thick and dotted lines respectively. The ratio of the fitted data to the models is shown in the lower panel.

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Fig. 2
figure 2

Contour plot of Keplerian and sub-Keplerian mass accretion rates. The plus sign “+” is the minimum fit statistic (2.051 × 102). The three confidence fit statistic is 2.074 × 102, 2.097 × 102, and 2.143 × 102 respectively. The color bar on the right hand is the delta-fit statistic.

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Fig. 3
figure 3

Correlation of Keplerian and sub-Keplerian mass accretion rates. The plotted data is a red hexagram while the error bars are black crosses with caps.

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Fig. 4
figure 4

Correlation of Keplerian and sub-Keplerian mass accretion rates vs accretion flow rate. The plotted data is a blue and green hexagram while the error bars are black crosses with caps.

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Fig. 5
figure 5

Correlation of Keplerian and sub-Keplerian mass accretion rates vs QPO frequency. The plotted data is blue and green circles while the error bars are black and red crosses with caps respectively.

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Fig. 6
figure 6

Correlation of Keplerian and sub-Keplerian mass accretion rates vs photon index. The plotted data is blue and green circles while the error bars is the black cross with a cap.

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Fig. 7
figure 7

Correlation of Keplerian and sub-Keplerian flow timescales vs accretion flow rate. The Keplerian and sub-Keplerian flow timescale is blue and green circle respectively while the error bars are black crosses with caps.

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Fig. 8
figure 8

Correlation of photon index and QPO frequency. The plotted data is circled magenta and the error bars are the dark cross.

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Discussion and conclusion

The spectral results show that the adopted phenomenological and physical models8,9,51,55,56,60 explained the thermal and non-thermal components of accretion flow X-radiations of MAXI J1535-571 with an acceptable statistical fit of a reduced Chi-squared value of ~ 1.1 (see Table 1 and Fig. 1). The spectral results shows that the accretion flow consists of optically thick/Keplerian and optically thin/sub-Keplerian regions. There is a strong correlation between the Keplerian and sub-Keplerian mass accretion rates (see Fig. 2). The Keplerian and sub-Keplerian mass accretion rates are anti-correlated61 to one another with a coefficient of − 0.920. Also, the Keplerian and sub-Keplerian mass accretion rates fluctuate and saturate at different points with an increase in accretion flow rates, QPO frequency, and power-law photon index respectively (see Figs. 3, 4, 5, 6). This suggests that MAXI J1535-571 accretion flow and its components are dynamics47,61. The sub-Keplerian flow timescale and Keplerian flow timescale are comparable or roughly match (see Fig. 7). This suggests propagating oscillatory shock/waves (QPO) in the accretion flow when resonance phenomenon in the range of 0.50–1.24 occurs35,36,39,48,62. The QPO frequency of 0.840–4.961 Hz indicates that the oscillating shock/wave in the accretion flow is type-C QPO. Type-C QPO appears in the hard spectral states and a best-fit photon index of 2.010 indicates that MAXI J1535-571 is in the HIMS18,19,29,33,50. This affirms the TCAF model prediction. The Γ–vQPO relation shows strong positive correlation values of 0.973 (see Table 2 & Fig. 8). The track of Γ–vQPO relation obtained is tight and similar to results of previous studies on MAXI J1535-571 and other BHCs41,63,62,65(references therein). The origin of Γ–vQPO characteristics seems to be tied to the accretion flow intrinsic parameters; components of accretion flow rates (mass accretion rates27,28) that determine the spectral states of BHCs (e.g9,27,28,66,65,68.). This indicates that the components of accretion flow interact at varying distances influencing the variations of optical depth, and plasma temperature, and causing the distribution of power-law energy spectral indices in the post-shock region8,57,60,69,68,69,70,73. Hence, the origin of Γ–vQPO relation is tied to the variations/fluctuations of components of accretion flow rates. This is consistent with the work of Eze et al.29. The investigation of the origin of Γ–vQPO relation in the hard states via spectral analysis is ongoing research. Further spectral analysis using multi-X-ray mission (AstroSat, HXMT, NICER, and NuSTAR) data of MAXI J1535-571 and other Galactic black hole candidates at different epochs of hard states are paramount to elucidate our view on the origin of Γ–vQPO relation and this will be reported elsewhere.