Measuring Black Hole Spins through X-ray Reflection Spectroscopy and the Relativistic Precession Model: the case of XTE J1859+226

The development of techniques to measure accurately black hole spins is crucial to study the physics and astrophysics of these objects. X-ray reflection spectroscopy is currently the most popular method to estimate the spins of accreting black holes; so far it has provided a spin measurement of about 40 stellar-mass black holes in X-ray binaries and 40 supermassive black holes in active galactic nuclei. The relativistic precession model (RPM) is another method to measure the spins of stellar-mass black holes: it requires the measurement of the frequencies of three simultaneous quasi-periodic oscillations and can potentially provide precise estimates of the black hole mass and spin. However, the two methods do not seem to provide consistent results when applied to the same sources, which questions the reliability and accuracy of these measurements. Recently, the RPM has been applied to infer the spin of the black hole in XTE J1859+226. The authors found a ∗ = 0 . 149 ± 0 . 005 (68% CL). There are no other spin measurements of this source. We looked for archived RXTE observations of XTE J1859+226 with blurred reflection features and found 23 spectra suitable for measuring the spin. We employed two different models with relxill and relxillD and obtained a higher spin value from all these fits. From simultaneous fitting of 7 spectra of higher quality, we found a ∗ = 0 . 986 +0 . 001 − 0 . 004 and a ∗ = 0 . 987 ± 0 . 003 (90% CL, statistical) with relxill and relxillD , respectively. Our results confirm the discrepancy between the spin measurements inferred from the two techniques.


INTRODUCTION
Einstein's theory of general relativity predicts the existence of black holes (BHs).These evolutionary remnants of massive stars can be fully described by three parameters: mass, spin (angular momentum), and electric charge (see, e.g., Chruściel et al. 2012).Since astrophysical BHs are believed to have negligible electric charge (Bambi 2017), they should be completely specified by the values of their mass and spin.Placing constraints on the BH masses and spins is crucial to study physics and the astrophysics of these objects (Reynolds 2014;Bambi et al. 2021).Spin measurements are challenging because they require observations that probe the immediate environment of the BH event horizon.
In the case of stellar-mass BHs, it is usually thought that the value of their spin is natal (King & Kolb 1999).If the BH is in a low-mass X-ray binary system (LMXB), the companion star is up to about one Solar mass, and even if the ⋆ Email: bambi@fudan.edu.cnBH swallows the whole star it cannot appreciably change the value of its spin because its mass is about ten times higher.If the BH is in a high-mass X-ray binary system, the companion star is a blue giant and its lifetime is too short to transfer a significant amount of mass to the BH and change its spin even if the BH accretes at its Eddington limit.If this picture is correct, the spins of stellar-mass BHs should be explained by the mechanism of gravitational collapse of the progenitor stars (but see Fragos & McClintock 2015, for different conclusions).
Today there are three leading techniques for measuring BH spins: the continuum-fitting method (CFM) (Zhang et al. 1997;McClintock et al. 2011McClintock et al. , 2014)), X-ray reflection spectroscopy (XRS) (Brenneman & Reynolds 2006;Reynolds 2014;Bambi et al. 2021), and the analysis of the gravitational waves (GWs) from the coalescence of binary BHs (Vitale et al. 2014).CFM and XRS involve X-ray observations of accreting BHs with geometrically thin and optically thick disks, while GWs can be used for compact binary systems BH-BH (or BH-neutron star) during the merger of the two bodies.
In the CFM, we fit the thermal spectrum of geometrically thin and optically thick accretion disks.The standard framework is the Novikov-Thorne disk model (Novikov & Thorne 1973;Page & Thorne 1974).Assuming the Kerr spacetime and that the inner edge of the accretion disk is located at the innermost stable circular orbit (ISCO), the model has five free parameters: the BH mass, the BH spin, the BH mass accretion rate, the BH distance, and the inclination angle of the disk with respect to the line of sight of the observer.If we can have independent estimates of the BH mass, distance, and inclination angle of the disk, for example from optical observations, we can fit the data and infer the BH spin and the BH mass accretion rate.Since the thermal spectrum of the disk is peaked in the soft X-ray band for stellar-mass BHs and in the UV band for supermassive BHs, this technique is normally used only to measure the spins of stellar-mass BHs.In the case of supermassive BHs, dust absorption in the UV band inhibits an accurate measurement of the thermal spectra.As of now, this technique has provided an estimate of the spin of about 20 stellar-mass BHs (Draghis et al. 2023).
XRS is the analysis of the disk's reflection spectrum, which is generated when an X-ray source illuminates the disk.The reflection spectrum in the rest-frame of the gas in the disk is normally characterized by narrow fluorescent emission lines in the soft X-ray band and by a Compton hump with a peak around 20-30 keV (Ross & Fabian 2005;García & Kallman 2010).The reflection spectrum of the whole disk as seen by a distant observer appears blurred because of relativistic effects in the strong gravity region (Fabian et al. 1989;Laor 1991;Bambi 2017).This technique can be used for both stellarmass BHs in XRBs and supermassive BHs in active galactic nuclei, and it is currently the most widely adopted method to estimate the spins of supermassive BHs.So far XRS has provided a spin measurement for about 40 stellar-mass BHs and for a similar number of supermassive BHs (Bambi et al. 2021;Draghis et al. 2023).
With current observational facilities, GWs can only be used to measure the spins of stellar-mass BHs in compact binary systems.GW signals are analysed to estimate the posterior probability distribution for the binary parameter values (Abbott & Abbott 2016).The spin parameter value that is well constrained using GW observations is the vector combination of the two spins, χ.The spins themselves are poorly constrained.The GW approach has an upper hand over electromagnetic observational techniques as the latter could be affected by the uncertainties of the astrophysical model but the former is limited due to the low signal-to-noise ratio (SNR) and systematics associated with the GW modeling.It is notable that X-ray techniques, and in particular XRS, often provide high values for the dimensionless spin parameters a * , close to the upper limit 1 (Bambi et al. 2021;Draghis et al. 2023;Liu et al. 2023Liu et al. , 2022)).On the contrary, with GWs, we normally find low values of the BH spin parameters (see, e.g., Tiwari et al. 2018).
Quasi-periodic oscillations (QPOs) are relatively narrow peaks at characteristic frequencies in the X-ray power density spectra of neutron stars and BHs (see, e.g., van der Klis 2004;Remillard & McClintock 2006).The exact physical mechanism responsible for the production of these QPOs is not yet well understood.The centroid frequency measurement of high frequency (∼ 100 Hz) QPOs can be highly precise and be associated with matter orbiting near the ISCO.In the relativistic precession model (RPM), the QPO frequencies are associated with the three fundamental frequencies of equatorial circular orbits of a test-particle (Stella & Vietri 1998;Stella et al. 1999;Stella & Vietri 1999).In the Kerr spacetime, these frequencies depend on three parameters: the BH mass, the BH spin parameter, and the radial coordinate of the oscillation.Motta et al. 2014a tested the RPM where Type-C QPOs (Casella et al. 2004) found in LMXBs are associated with the nodal precession frequency (or Lense-Thirring frequency), while the periastron precession frequency and the orbital frequency correspond, respectively, to the lower and upper high-frequency QPOs.If we observe these three frequencies simultaneously, we can infer the values of the BH mass, the BH spin, and the radial coordinate of the oscillation.Since it is relatively easy to get a very precise measurement of the QPO frequencies, if this interpretation is correct we can obtain very precise measurements of the BH mass and BH spin.
The RPM has so far provided a spin measurement for 5 stellar-mass BHs (see Table 1).The first spin measurement through the RPM was reported in Motta et al. (2014a) for the BH in GRO J1655-40.From the simultaneous observation of a type-C QPO and two high-frequencies QPOs, the authors inferred the BH spin parameter a * = 0.290±0.003(68% CL), which is significantly lower than the values inferred from the CFM and XRS.However, the BH spin measurement reported in Miller et al. (2009) from the analysis of the reflection spectrum was obtained in the early ages of XRS and GRO J1655-40 is a complicated source with strong absorption (see also Reis et al. 2009).The RPM was then used to infer the BH spin of XTE J1550-564 in Motta et al. (2014b); however, for this source, we do not have any observation with three simultaneous QPOs, so the authors determined the BH spin combining the detection of two simultaneous high-frequency QPOs with the dynamical measurement of the BH mass reported in Orosz et al. (2011).The reported spin measurement a * = 0.34 ± 0.01 (68% CL) is consistent with the spin measurement from the CFM and XRS, but the measurements from CFM and XRS have large uncertainties.The RPM was again used to measure the spin of the BH in MAXI J1820+070 in Bhargava et al. (2021).The authors found a * = 0.799 +0.016 −0.015 (68% CL), which is lower than the measurement obtained from XRS in Draghis et al. (2023) 1 .However, in this case, the authors did not observe a QPO triplet instead they observed a low-frequency QPO and broad noise features in the NICER observations.Motta & Belloni (2023) re-analysed RXTE observations of GRS 1915+105 and identified the 67 Hz QPO as the Lense-Thirring frequency at the ISCO yielding a value of 0.706 ± 0.034 (68% CL) for the black hole spin.The measurements from CFM and XRS provide a spin measurement very close to 1 (see McClintock et al. 2006 for the CFM andMiller et al. 2013 for XRS).Lastly, Motta et al. (2022) have recently used the RPM to measure the spin of the BH in XTE J1859+226.Their measurement is a * = 0.149 ± 0.005 (68% CL) and, as of now, there are no spin measurements of this source from CFM and XRS.It is worth noting that in the case of XTE J1859+226, the RPM did return the correct values for the mass of the BH which was consistent with the value obtained from optical measurements.In the case of GRO J1655-40, the value for the mass of the BH returned by RPM is consistent with one of the measurements obtained using the optical band (Beer & Podsiadlowski 2002), however, there are other measurements as well (e.g.Orosz & Bailyn 1997).
The goal of this work is to measure the spin of the BH in XTE J1859+226 through XRS in order to compare such a measurement with that inferred from the RPM in Motta et al. (2022).To do this, we looked for spectra with relativistic reflection features among the 138 RXTE pointings of XTE J1859+226.We selected 23 distinct observations with the strongest relativistic features and we analyzed these spectra with the reflection models relxill and relxillD (Dauser et al. 2013;García et al. 2014a).After performing the spectral analysis, we found the BH spin measurement a * to be 0.986 +0.001  −0.004 and 0.987 ± 0.003 (90% CL, statistical), respectively, which is clearly inconsistent with the RPM spin estimate.
The content of our paper is as follows.In Section 2, we briefly introduce our source, XTE J1859+226.In Section 3, we describe the data reduction process.In Section 4, we present our spectral analysis of the source.In Section 5, we discuss our results.
2 http://heasarc.nasa.gov/cgi-bin/Tools/w3nh/w3nh.pl(Jahoda et al. 2006) on-board the RXTE mission.We employed for our analysis only the spectra from PCU 2 as it is the most well-calibrated and has the most extensive coverage among all the units.We applied the publicly available tool pcacorr for improving the calibration to a ∼ 0.1% systematic precision (García et al. 2014b).The standard background model was applied and subtracted.We confine our spectral fitting in the 3-50 keV energy band.

BROADBAND X-RAY SPECTRAL ANALYSIS
We used XSPEC v12.9.1 (Arnaud 1996) for the spectral analysis.To examine whether this ensemble of spectra obtained from distinct observations is suitable for our study, we initiated by fitting each of them with an absorbed power law model.We fixed the hydrogen column density in tbabs, which calculates the cross section for X-ray absorption by the inter-stellar medium (Wilms et al. 2000), to NH = 0.2 • 10 22 atoms cm −2 (Radhika & Nandi 2014).Out of 138 spectra, 100 spectra had features resembling the relativistic reflection features.To quantify the strength of these features and select the best data for our analysis, we modeled every spectrum with a model consisting of a component to model the multi-color disc blackbody: diskbb (Mitsuda et al. 1984), a component to model the continuum from the corona: nthcomp and another component for the reflection features: relx-illCp (Dauser et al. 2013;García et al. 2014a).In XSPEC notation, the model becomes: tbabs*(diskbb + nthcomp + relxillCp).To compute the flux contribution of every component separately, we then convoluted each of them with the model cflux and calculated the total flux.The reflection component's strength was defined as the ratio of the flux of the reflection component to the total flux.For the selection pro-   2.
We continue the analysis of the selected observations.A self-consistent model that accounts for the impact of coronal Comptonization on the reflection spectrum and calculates the continuum from the corona directly from the thermal spectrum is simplcut (Steiner et al. 2017).This model has four parameters: the photon index Γ, the scattering fraction fSC, the high-energy cutoff Ecut, and the reflection fraction RF.The simplcut kernel employs a redistribution function to model the Comptonization of a general population of seed photons entering the corona (e.g.Wilkins & Gallo 2015) The scattering is not generically symmetric but includes both up-and down-scattering, for the fraction of photons that are scattered.In XSPEC notation, the total model can be written as tbabs*simplcut*(diskbb + relxill).We assumed that the emissivity profile of the reflection spectrum is described by a simple power law and therefore we have to fit only one parameter, the emissivity index q.The reflection fraction R f was frozen to −1, so the output of relxill is only a reflection spectrum and simplcut*diskbb describes both the thermal spectrum from the disk and the Comptonized spectrum from the corona.The spin parameter a * was left free in the fit and the inner edge of the accretion disk was set to the ISCO radius.The iron abundance AFe, the inclination angle of the disk with respect to the line of sight i, and the ionization parameter ξ were all free in the fit.This model provides a good fit for all observations.We also considered the scenario with higher electron density, allowing us to directly fit the density of the accretion disk in logarithmic terms logN , and we replaced relxill with relxillD.The model then read tbabs*simplcut*(diskbb + relxillD).On the whole, we found consistent measurements of the parameters between the fits obtained with relxill and relxillD.All the parameters, as they vary throughout the observations, are plotted in Figure 3 and Figure 4. Figure 5 shows the reduced χ 2 of these fits.Figure 7 is a representative spectral fit of one spectrum of the total set to illustrate the components.
After fitting spectral models to the data and treating all data sets independently, we ultimately produced a simultaneous fit linking several of the invariant parameters for a selected set from 23 observations.This small set of spectra (G1 Set) was selected taking into account various factors.The photon count rate of the 23 spectra was taken into account and only the spectra with more than 1000 counts/s were chosen.The spectrum whose hardness was above 1.3 and the reflection strength was above 16 were selected and thus a few more spectra were removed from the shortlisted set.Furthermore, we considered only the spectra in which the best-fit returned constrained parameters in the independent fit.The G1 Set eventually included 7 spectra to be used in the simultaneous fitting.Figure A1 shows the data to model ratios for this set of spectra.Using the models specified above, in this simultaneous fit, the spin parameter a * , the inclination angle i, and the iron abundance AFe are tied among all spectra, whereas the other parameters are allowed to vary among the observations.The best-fit parameter values for both model formulations, relxill and relxillD are listed in Tables A1 and A2, respectively, with the errors corresponding to the 90% confidence level.The unfolded spectra with residuals, for both the model formulations aforementioned,    Fe-Kα feature, we performed another simultaneous fit on another set of spectrum (G2 Set).For the G2 Set, we selected only those spectra, from the 23, whose relxill or relxillD fits returned a value of ionization parameter log ξ less than 4.3.We discarded the spectra whose exposure was less than 500 seconds and the ones whose individual spectral fitting did not return constrained parameters.We performed the simultaneous fitting as described above.The best-fit parameter values for both model formulations, relxill and relxillD are listed in Tables A3 and A4, respectively, with the errors corresponding to the 90% confidence level.The unfolded spectra with residuals, for both the model formulations aforementioned, are shown in Figures A3.We also explored a slightly different model formulation to exclude the impact of coro-nal comptonization on the reflection spectrum.In XSPEC notation, the model then was tbabs*((simplcut*diskbb) + relxill/relxillD).Only the scattering fraction fSC and the inner disk radius obtained from diskbb were inconsistent with the previous fits.Both these parameter values increased by 40%.However, every other parameter value remained consistent with the previous fits and the reduced χ 2 values also did not change significantly.Figure 8 is a representative spectral fit of one spectrum of the total set to illustrate the components.The results are discussed in the next section.

DISCUSSION AND CONCLUSIONS
We carried out broadband spectral analysis of 23 distinct observations of XTE J1859+226 as observed by RXTE to determine the spin parameter of the compact object of this source via XRS.To obtain this result we fit the reflection features of the spectra with relxill/relxillD, diskbb, and simplcut.We note that both the reflection models provided a good fit for all the spectra.The best-fit parameter values obtained from the independent fits are plotted in Figures 3 and 4 at a 90% CL (statistical).We note that the obtained values of spin parameter a * are close to the maximum possible value of the parameter.The values of the spin parameter plotted in Figure 4 seem to be systematically increasing with time.This is a topic of further investigation whether this could be an indication of an evolving disk geometry through the observations.However, this is beyond the scope of this work.The spin parameter a * is 0.986 +0.001 −0.004 and 0.987 ± 0.003, as inferred from the simultaneous fitting of the G1 Set of spectra, for relxill and relxillD models, respectively.The value obtained for the inclination angle of the accretion disk, i, is 71.2 +0.4  −0.8 degrees and 71.7 +0.9 −0.3 degrees, respectively.The iron abundance AFe obtained is 10.0−0.4 and 7.12 +0.28 −0.18 , respectively.For the G2 Set, the values of a * from two models are 0.981 +0.006  −0.007 and 0.982 +0.006 −0.007 ; i is 68.4 +1.9 −1.4 degrees and 69.1 +0.5 −0.8 degrees; AFe is 10.0−1.7 and 5.23 +0.5  −0.25 .This measurement of spin is consistent with the spin distribution in X-ray binaries (Fishbach & Kalogera 2022).The high value of the inclination angle of the disk is found consistent with suggested values of the same in previous studies (Corral-Santana et al. 2011;Yanes-Rizo et al. 2022).The reduced χ 2 is < 1 for some of the individual fits as reported in the aforementioned tables.It is possible that the systematic uncertainties included have led to this overfitting.However, this is important for instrumental and calibration uncertainty and thus can not be reduced.Figure 6 presents the histogram of the reduced χ 2 values obtained throughout all the selected observations.For comparison, we have plotted the theoretical chi-square distribution curve for 66 degrees of freedom and it is consistent with the histogram.
Our reported value of a * is clearly inconsistent with the value reported by Motta et al. (2022) (see Section 1).It is worth mentioning that the respective authors have reported that they found one clear QPO triplet in the observation ID 40124-01-07-00 (Spectrum 1, Table 2) which is also one of the observations we have made use of in our study (Table 2).Such discrepancy between the spin measurements obtained via XRS and RPM has also been noted in the case of another LMXB source, GRO J1655-40.The spin measurements via RPM rely strongly on two parameters: the mass of the BH and the radial coordinate of the oscillation.A stringent constraint on these parameters can be obtained using a simultaneous observation of a low-frequency QPO and twin high-frequency QPOs, with the assumption that these QPOs arise at a given radius.Stuchlík & Kološ (2016) demonstrated for the case of GRO J1655-40 that if this assumption that the twin high-frequency QPOs and the simultaneously observed low-frequency QPO being observed at a common radius is abandoned, then the a * prediction made by this model is in good agreement with the values that are obtained via XRS.
In our fits, the inner edge of the accretion disk in relxill/relxillD is set to the ISCO radius, and therefore it can range from 1.23 Rg (a * = 0.998) to 9 Rg (a * = −1), where Rg = GNM/c 2 is the gravitational radius of the BH.In the RPM, the oscillation can be at the ISCO or at a larger radius.If the disk is truncated at a radial coordinate larger than 9 Rg, this is a problem for the XRS measurement.One of the observations considered in our final sample is the observation with the QPO triplet considered by Motta et al. (2022): for that observation, Motta et al. (2022) inferred a characteristic radius of 6.85 Rg, which is in the range of our reflection model.Fig. 4 in Motta et al. (2022) reports the centroid frequencies of the Type-C QPOs of the source and they are in the range ∼1-10 Hz.A few QPOs have a characteristic frequency centered around 1 Hz, which would correspond to a nodal precession radius exceeding 9 Rg according to the RPM.However, none of those observations have been used in our work.Ingram et al. (2009) and Ingram & Done (2011) presented a physical model for the QPOs where a hot, geometrically thick accretion flow is undergoing the Lense-Thirring precession and is misaligned to the BH spin axis.To account for the effects of a geometrically thin disk surrounding this hot, geometrically thick accretion flow, Bollimpalli et al. (2022) performed general relativistic magnetohydrodynamics simulations.They concluded that the outer thin disk affects the precession of the inner thick disk by significantly reducing its rate, which, in turn, has direct implications for the truncation radius estimation and the observed QPO frequency range.Another possible way to obtain a truncated disk is by fitting the spectra with two Comptonisation and reflection components, mimicking the effect of a stratified extended corona.The extra curvature this creates in the spectrum has a similar effect to the red wing of a very broad iron line and this would impact the estimated spin parameter (e.g.Frontera et al. 2001;Shidatsu et al. 2011;Zdziarski et al. 2021).
Indeed these timing-based estimates are close to the spin distribution emerging from GW data analysis.On the other hand, in the case of MAXI J1820+070, the various values obtained in the literature are not fully consistent with each other.However, Bhargava et al. (2021) does mention that the RPM tends to introduce a bias.This happens due to the use of broad features instead of QPO peaks and it underestimates the value of the spin parameter, thus their reported value should be regarded as a lower limit.The reliability of the RPM model depends on the interpretation of QPOs.The measured BH spins, assuming the QPOs are the dynamic frequencies, are consistent with values obtained from reflection spectroscopy (Misra et al. 2020;Liu et al. 2021;Zhang et al. 2023).Draghis et al. (2023) discuss in detail how the overall distribution of spin measurements conducted via XRS is stacked near high values.All the available reflection models present a number of simplifications that can, at different levels, have an impact on the precision measurements of BHs.It is important to remember for both these methods of measurement that the discrepancy probably boils down to the assumptions inherent in the two models and these models can give an estimate of the spin assuming only that framework.
Table A1.Summary of the best-fit values from the analysis of the G1 Set of spectra of XTE J1859+226 when the emissivity profile of the disk is described by a power law.* indicates that the parameter is frozen in the fit.The reported uncertainties correspond to the 90% confidence level for one relevant parameter (∆χ 2 = 2.71).

Figure 1 .
Figure 1.The observation log of XTE J1859+229 by ASM on-board RXTE.Each red point represents the respective epoch number's average ASM unit count rate.The blue asterisk denotes the observations considered in this work.

Figure 2 .
Figure 2. The Hardness-Intensity Diagram of the 1999 outburst of XTE J1859+226 as observed by RXTE.Each point represents data from one day of observation.The rainbow colour gradient represents the reflection strength.The observation points in the red box are used in this work.The green and blue stars represent the observations that were analysed in the sample sets G1 and G2, respectively.

Figure 3 .
Figure3.Evolution of different physical parameters throughout all the selected observations.The plotted uncertainties correspond to the 90% confidence level for one relevant parameter (∆χ 2 = 2.71).The black and red data points and error bars represent the parameter values obtained from fitting relxill and relxillD models respectively.The green and blue stars represent the observations that were analysed in the sample sets G1 and G2, respectively.

Figure 4 .Figure 5 .
Figure4.Evolution of different physical parameters throughout all the selected observations.The plotted uncertainties correspond to the 90% confidence level for one relevant parameter (∆χ 2 = 2.71).The black and red data points and error bars represent the parameter values obtained from fitting relxill and relxillD models respectively.The orange cross represents those parameter values that remained unconstrained.The green and blue stars represent the observations that were analysed in the sample sets G1 and G2, respectively.

Figure 6 .Figure 7 .Figure 8 .
Figure6.Histogram of the reduced χ 2 throughout all the selected observations.The black and red bars represent the number of observations in different bins of the reduced χ 2 values obtained from fitting the relxill model with 66 degrees of freedom and relxillD model with 65 degrees of freedom, respectively.The blue line represents the theoretical chi-square distribution curve with 66 degrees of freedom.

Figure A1 .Figure A2 .
Figure A1.Data to model ratio of the 7 spectra in the G1, Set when fitting with a simple absorbed continuum model: tbabs*(diskbb+powerlaw). Data are rebinned for visual clarity.

Figure A3 .
FigureA3.The top panel shows the unfolded spectra obtained from the joint fitting analysis of the G2 Set of spectra obtained from the relxill model.The middle and bottom panels show the ∆χ when the relxill and relxillD model formulations were employed to fit the data set, respectively.

Table 1 .
Motta et al. (2014b)urement from the relativistic precession model reported in the literature.The table also shows the existing spin measurements from the continuum-fitting method and X-ray reflection spectroscopy of these objects.†InMottaet al. (2014b), the BH spin is obtained within the relativistic precession model from the detection of two simultaneous QPOs and the dynamical measurement of the BH mass.

Table 2 .
Summary of the observations analyzed in the present work.