Constraints on black hole spins with a general relativistic accretion disk corona model

# Constraints on black hole spins with a general relativistic accretion disk corona model

Bei You 1Key Laboratory for Research in Galaxies and Cosmology, Shanghai Astronomical Observatory, Chinese Academy of Sciences, 80 Nandan Road, Shanghai, 200030, China; youbeiyb@gmail.com, cxw@shao.ac.cn
13University of Chinese Academy of Sciences, 19A Yuquanlu, Beijing 100049, China
3
Xinwu Cao 1Key Laboratory for Research in Galaxies and Cosmology, Shanghai Astronomical Observatory, Chinese Academy of Sciences, 80 Nandan Road, Shanghai, 200030, China; youbeiyb@gmail.com, cxw@shao.ac.cn
1
and Ye-Fei Yuan 2Department of Astronomy, University of Sciences and Technology of China, Hefei, Anhui 230026, China; yfyuan@ustc.edu.cn
2
###### Abstract

The peaks of the spectra of the accretion disks surrounding massive black holes in quasars are in the far-UV or soft X-ray band, which are usually not observed. However, in the disk corona model, the soft photons from the disk are Comptonized to high energy in the hot corona, and the hard X-ray spectra (luminosity and spectral shape) contain the information of the incident spectra from the disk. The values of black hole spin parameter are inferred from the spectral fitting, which spread over a large range, to . We find that the inclination angles and mass accretion rates are well determined by the spectral fitting, while the results are sensitive to the accuracy of black hole mass estimates. No tight constraints on the black hole spins are achieved, if the uncertainties of black hole mass measurements are a factor of four, which are typical for the single-epoch reverberation mapping method. Recently, the accuracy of black hole mass measurement has been significantly improved to  dex with velocity resolved reverberation mapping method(Pancoast et al., 2014). The black hole spin can be well constrained if the mass measurement accuracy is %. In the accretion disk corona scenario, a fraction of power dissipated in the disk is transported into the corona, and therefore the accretion disk is thinner than a bare disk for the same mass accretion rate, because the radiation pressure in the disk is reduced. We find that the thin disk approximation, , is still valid if , provided a half of the dissipated power is radiated in the corona above the disk.

Quasars: accretion disk; X-ray: corona; Black hole physics; Galaxies: active
\volnopage

Vol.0 (200x) No.0, 000–000

## 1 Introduction

It is believed that black holes reside in X-ray binaries and active galactic nuclei (AGN) with typical masses and respectively (Kormendy & Richstone, 1995; Ferrarese & Merritt, 2000; Remillard & McClintock, 2006; Hopkins et al., 2008; Alexander & Hickox, 2012). Massive black holes may grow up through gas accretion or/and mergers, which is also related with the galaxy evolution (e.g., Moderski & Sikora, 1996; Marconi et al., 2004; Volonteri et al., 2005; Berti & Volonteri, 2008; Dotti et al., 2013). The structure and radiation of accretion disks surrounding black holes have been extensively studied in the last several decades (Shakura & Sunyaev, 1973; Frank et al., 1992; Merloni, 2004). Relativistic jets have been observed in some X-ray binaries and AGNs, which is thought to be related to accretion disks or/and spinning black holes (Blandford & Znajek, 1977; Blandford & Payne, 1982).

A black hole can be described by two quantities: the mass and the dimensionless spin parameter , where is the angular momentum of the black hole. In the last decade, great progresses have been achieved on the mass measurement of black holes in either active or nonactive galaxies. The black hole mass in nearby galaxies can be estimated with the correlations between black hole mass and the stellar velocity dispersion or bulge luminosity of the host galaxy (Kormendy & Richstone, 1995; Ferrarese & Merritt, 2000; McConnell et al., 2011). For active galaxies, the size of the broad-line region (BLR) is measured with reverberation mapping method, and the black hole mass can be estimated with the broad-line width assuming the motion of BLR clouds to be virialized (Kaspi et al., 2000). There is a tight correlation between the BLR size and the optical continuum luminosity, which is widely used to estimate masses of black holes in AGNs, or called as single epoch reverberation mapping method (e.g., Kaspi et al., 2000; Woo & Urry, 2002; Shen et al., 2011; Tang et al., 2012).

Compared with the mass estimates, the measurements of the black hole spins are still far from satisfactory, though great efforts have been devoted on this issue. The observed broad Fe K emission lines provide a way to estimate the black hole spin parameters, which is applicable for black holes in both galactic X-ray binaries and AGNs. The observed broad Fe K emission lines are believed to be emitted from the inner regions of the accretion disks. The radii of the innermost stable circular orbits (ISCO) of the gas in the accretion disks can be inferred from the observed asymmetric Fe k line profiles (Martocchia et al., 2002; Miller et al., 2002; Miniutti et al., 2004; Brenneman & Reynolds, 2006; Lohfink et al., 2012), and the dimensionless spin can be determined with the radii of the ISCO (Barr et al., 1985; Reynolds & Nowak, 2003; Remillard & McClintock, 2006). The application of this method requires high quality X-ray spectroscopic data, and a few black hole spins in nearby Seyfert galaxies have been measured in this way (Goosmann et al., 2006; Brenneman & Reynolds, 2006; de La Calle Pérez et al., 2010; Patrick et al., 2011; Lohfink et al., 2012).

The black hole spin parameters in some X-ray binaries were measured by fitting the observed continuum spectra with general relativistic accretion disk models (so-called “continuum-fitting method”, Zhang et al., 1997; Shafee et al., 2006; Steiner et al., 2009a; Gou et al., 2011; McClintock et al., 2011; You et al., 2015). In principle, this method can be used to constrain the black hole spins for AGNs. In some early works, the accretion disk models were used to fit the observed multi-band spectra of quasars (e.g., Bechtold et al., 1987; Sun & Malkan, 1989), in which the general relativistic effects have been taken into account (Cunningham, 1975; Laor, 1991). Czerny et al. (2011) estimated the black hole spin to be by fitting the broad band (IR/optical/UV) continuum spectrum of the quasar SDSS J094533.99+100950.1. It was found that the model fitting suffers from the degeneracy of the disk parameters, namely, the black hole mass , the spin , and the mass accretion rate . At the time before the invention of the black hole mass measurements used nowadays, the derived disk parameters with the spectral fittings are quite uncertain. It is possible to constrain the the spins of the massive black holes in AGNs, when the black hole masses are measured. The degeneracy of the parameters in accretion disk spectral fittings can be eliminated if the peak frequency and luminosity of the accretion disk are well observed. However, the application of this method in AGNs meets a major difficulty, i.e., the spectral peaks of accretion disks surrounding massive black holes with are in the far-UV or soft X-ray band, which are unobservable for most AGNs.

The observed ultraviolet (UV)/optical emission of AGNs is thought to be a thermal emission from the optically thick accretion flows (Shields, 1978; Malkan & Sargent, 1982; Sun & Malkan, 1989), while they are too cold to reproduce the observed power-law hard X-ray spectra of AGNs. The hard X-ray spectra of AGNs most likely originate from the inverse Compton process in which soft photons from the disks are scattered by the hot electrons in the coronas (Galeev et al., 1979; Haardt & Maraschi, 1991; Di Matteo, 1998). According to this disk-corona scenario, most gravitational energy is generated in the cold disk, while a fraction of it is carried into the corona likely by the magnetic loops in the disk (Di Matteo, 1998; Liu et al., 2002; Cao, 2009). Cao (2009) calculated the structure and spectrum of a geometrically thin accretion disk with a corona in AGNs, in which the corona is assumed to be powered by the reconnection of the magnetic loops originating from the cold accretion disk due to buoyancy instability. It is found that both the hard X-ray bolometric correction factor and the hard X-ray spectral index increase with the Eddington ratio, which are qualitatively consistent with the observations (e.g., Wang et al., 2004; Shemmer et al., 2006; Vasudevan & Fabian, 2007; Zhou & Zhao 2010; Fanali et al., 2013). This model was expanded in the general relativistic frame for a spinning Kerr black hole by You et al. (2012), in which the spectra of such a accretion disk-corona are recalculated taking into account general relativistic effect with the relativistic ray-tracing method.

As discussed above, the peaks of the spectra of the accretion disks surrounding massive black holes in AGNs are usually not observed (but also see Yuan et al., 2010, for the case of a relative small black hole in a narrow-line quasar). However, a portion of the soft photons originating from the disk undergoing Comptonization in the hot corona, and the scattered photons are dominantly in the hard X-ray band. This implies that the observed hard X-ray spectra (luminosity and spectral shape) contain the information of the incident spectra from the cold disk. It seems possible to constrain the values of black hole spin parameter in AGNs with a general relativistic accretion disk corona model, as a variant of the continuum-fitting method, even if the spectral peaks of the accretion disks are not observed.

In this paper, we use the general relativistic model of an accretion disk-corona surrounding a spinning black hole to fit the observed multi-band spectra of AGNs with measured black hole masses. The black hole spinning parameter are tentatively derived for a small sample of AGNs. We briefly summarize the accretion disk-corona model used for AGN spectral fitting in Section 2. The results and discussion are in Sections 3 and 4. Throughout this paper, a cosmology with , and is adopted.

## 2 Model

In this work, we adopt the Kerr black hole accretion disk-corona model developed by You et al. (2012). All the general relativistic effects are properly considered in both the calculations of the structure and emergent spectra of the accretion disks. We briefly summarize the model in the next two sub-sections (see You et al., 2012, for the details).

### 2.1 Structure of an accretion disk with corona

In the accretion disk corona model, most gravitational energy of the gas is dissipated in the cold disk, and a portion of it is carried into the corona by the magnetic loops. The particles in the corona are heated by the re-connection of the loops in the corona (Di Matteo, 1998). The electrons in the corona are hot, and the observed hard X-ray spectra mostly originate from the inverse Compton process in which soft photons from the thermal radiation of the cold disk are scattered to high energy by the hot electrons (e.g., Galeev et al., 1979; Haardt & Maraschi, 1991; Svensson & Zdziarski, 1994).

The gravitational power of the gas generated in the cold accretion disk surrounding a spining black hole is

 Q+dissi=3GM˙M8πR3LBC1/2, (1)

where , , are the general relativistic correction factors (Novikov & Thorne, 1973; Page & Thorne, 1974), and is the mass accretion rate of the disk.

The fraction of the dissipated power transported to the corona can be estimated provided the strength of the magnetic field is known. The magnetic pressure of the disk is usually assumed to scale with the gas or/and radiation pressure of the disk depending on the different magnetic stresses adopted (Sakimoto & Coroniti, 1981; Stella & Rosner, 1984; Taam & Lin, 1984; Cao, 2009; Hirose et al., 2009). The detailed physics of the field generation is still quite unclear, though the different magnetic stressed are tested against the observations in AGNs in the previous works (Wang et al., 2004; Cao, 2009; You et al., 2012). For simplicity, the ratio of the dissipated power in the corona to that in the disk is taken as a parameter in our model calculations,

 f=Q+corQ+dissi, (2)

which will be determined by the comparison with the observed spectra.

A portion of the soft photons from the disk undergo inverse Comptonization to high energy in the corona, which correspondingly cools the electrons in the corona. About half of the scattered photons are intercepted by the disk, some of which are reflected and the rest heats the disk and re-radiate as blackbody radiation (Zdziarski et al., 1999). Therefore the energy equation for the cold disk is

 Q+dissi−Q+cor+12(1−ε)Q+cor=4σT4d3τ=σT4s, (3)

where the reflection albedo is adopted in all our calculations (Zdziarski et al., 1999), is the optical depth in the vertical direction of the disk, and and are the temperature in the mid-plane of the disk and the temperature of the gas at the disk surface respectively.

### 2.2 Emergent spectrum of the accretion disk with corona

It is well known that most of the radiation of the disk comes from the inner region of the disk (Shakura & Sunyaev, 1973). The spectrum of an accretion disk-corona for a spinning black hole is influenced by the relativistic effects, i.e., the frame dragging, gravitational redshift, and photon trajectory bending, due to the strong gravity field of the black hole (Zhang et al., 1985; Laor & Netzer, 1989; Laor, 1991; Li et al., 2005, 2009; You et al., 2012), especially for the emission from the region near the black hole. The ray-tracing method is widely adopted in this kind of spectral calculations (Laor & Netzer, 1989; Laor, 1991; Zhang et al., 1997; Li et al., 2005; You et al., 2012). Considering the complexity of the radiation transfer of Comptonized photons in the geometrically thick corona, You et al. (2012) calculated the emergent Comptonized spectrum from the corona by dividing layers in the corona with the ray-tracing method. In this work, we follow the same routine of You et al. (2012) to calculate the spectrum of an accretion disk-corona for a rotating black hole, in which all the general relativistic effects are taken into account (see Section 3.4 in You et al., 2012, for the details)

The radiation from geometrically thin, optically thick accretion disk is anisotropic, and the specific intensity of the disk emission, , in the co-moving frame of the disk, where ( is the angle between the line of sight and the normal of the disk Chandrasekhar, 1960; Sunyaev & Titarchuk, 1985; Liu & Zhang, 2011).

UV/soft X-ray photons are emitted from the inner region of the accretion disk, where a hot plasma layer is present above the disk surface caused by the illumination of the external hard X-ray emission from the corona. The electron scattering opacity dominates over absorption in this hot layer, and the emergent spectrum of the disk departures from the black body emission due to radiative transfer. Hubeny et al. (2001) calculated the vertical structure and emergent spectra of the disks with a hot plasma layer by including a self-consistent treatment of Compton scattering. Chiang (2002) found that the derived emergent spectrum could be well reproduced by blackbody emission with a higher color temperature, . The empirical color correction factor is given by

 fcol(Td)=f∞−(f∞−1)[1+exp(−νb/Δν)]1+exp[(νp−νb)/Δν] (4)

where , and . The local spectrum emitted from the disk can be calculated with

 Fν∝1f4colπIν(fcolTd), (5)

where is the local blackbody emission.

As described in Section 2.1, the structure of the accretion disk-corona can be calculated when the values of the black hole mass , the dimensionless spin , the mass accretion rate , and the ratio of the dissipated power in the corona to that in the disk, are specified. With the derived structure of the accretion disk, we use the ray tracing method to calculate the emergent spectrum of the disk corona system observed in infinity at a viewing angle . We plot the representive emergent spectra with different values of the ratio in Fig. 1 to show the effect of on the overall shape of the spectra. At a given radius, both the energy dissipated in the disk and the disk temperature decrease with energy ratio according to Eq. (2). Therefore, the specific intensity of the soft photons from the disk will also decrease with the ratio . In this case, the electron temperature will increase with , since the cooling of the corona is dominated by the inverse Compton scattering of the soft photons from the disk by the hot electrons in the corona (Cao, 2009; You et al., 2012). This means that the resulting spectra in the hard X-ray band become much harder and brighter with the increase in the ratio , given that more soft photons from the disk will be Compton scattered to higher energy by the hot electrons in the corona (see Fig. 1).

## 3 Results

Using the accretion disk model described in Section 2, we fit the observed multi-band spectral energy distributions (SEDs) of five quasars which exhibit obvious feature of so-called “Big Blue Bump” (BBB) with measured black hole masses and good X-ray measurements. The peak of BBB at  Hz and a power-law hard X-ray spectrum are the typical features of the spectrum of an accretion disk with corona (e.g., Cao, 2009; You et al., 2012). As discussed in Section 1, the peaks of the BBBs are beyond the waveband coverage of the instruments for most AGNs due to their massive black holes. However, the left tail of the BBBs can be observed in infared/optical/UV wavebands, which provide useful information of the BBBs (e.g., Calderone et al., 2013). The SED data of five quasars in this work is taken from Shang et al. (2011) and summarized in Tables 1 and 2.

The black hole masses of these five quasars are given in Tang et al. (2012), which are estimated with the broad-line width and the scaling relationships (Vestergaard & Peterson, 2006; Vestergaard & Osmer, 2009). The infrared emission from these quasars may probably be dominated by the radiation from the dust tori, while the origin of the soft X-ray excess observed in AGNs is still controversial (see, e.g., Done et al., 2012, and the references therein). Therefore, the infrared and soft X-ray spectral data are not used in our model fitting on the SEDs.

We use the general relativistic accretion disk corona model to fit the observed optical/UV spectra and the hard X-ray continuum spectra in  keV. There are four free parameters in our model calculations, i.e., the mass accretion rate , the spin parameter , the power fraction dissipated in the corona, and the viewing angle with respect to disk axis. In the spectral fitting, we find that the fraction is mainly determined by the X-ray spectral index, which is almost independent of the values of other parameters. For the broad-line type I quasars, the values of the inclination angle are tuned in the range of .

We plot the spectral fitting results of the five quasars in Figures 2-6. In the optical/UV wavebands, the data points (marked as blue filled circles) which are used in our spectral fitting are extracted from the observed SED in the line-free continuum windows in order to avoid the contamination by emission lines, as done in Zheng et al. (1997) and Vanden Berk et al. (2001).

In this work, the measured hard X-ray luminosity and the spectral index are used for our spectral fitting. In the spectral fitting, the errors of the fitting in the hard X-ray band (both the luminosity and spectral index) are minimized by tuning the model parameters, meanwhile the observed optical/UV spectrum is taken as upper limits.

The best-fitting results of the disk corona parameters are listed in Table 3. The error of the parameter is determined according to the error of the hard X-ray spectral index in Table 2, given that strongly depends on . The fitting error is calculated in the parameter plane, and the distributions of the error contour for the spin , the mass accretion rate and the inclination angle are plotted in Fig. 7. The uncertainties on the three parameters correspond to 90% confidence level. . The 2D confidence contours of and for PG 1322659 with the measured black hole mass and fraction are plotted in Fig. 8. We find that, the value of black hole spin parameter is constrained in a narrow range with best-fitting value . The best-fitting accretion rate , and the inclination angle of the disk . We know that uncertainties of the black hole spin constraints are closely related to the errors of the black hole mass estimates. In Fig. 9, we plot the fitting results for PG 1322+659 with the different values of reported in the literature, and find that the constraints on the black hole spin parameter are sensitive to the black hole mass estimate (see Table 4). Within the uncertainties of a factor of four on the mass measurements (Vestergaard & Peterson, 2006), using single epoch reverberation mapping method, from to with the step of , the best-fitting values of black hole spin can vary from -0.94 to 0.998 (see Fig. 10). In Fig. 11, we plot the fitting results for PG 1322+659 with a small shift on the observed X-ray luminosity to assess how the black hole spin is affected by X-ray variability (see Table 4).

We fit the emission line properties of PG 1322+659 by subtracting the best spectral fitting continuum in Fig. 12, to compare with the typical broad line region spectrum. The derived EW and FWHM for Mg ii, H, and H are consistent with those of the typical BLR spectrum given in Shen et al. (2011). In Fig. 13, we plot the relative disk thickness varying with the mass accretion rate . It is found that the thin disk approximation can be achieved even if . Note that when the mass accretion rate is as high as , is only slightly larger than 0.1 for a typical value of .

## 4 Discussion

The main features of quasar SED with a big blue bump in optical/UV wavebands and a power-law hard X-ray continuum spectrum can be reproduced by the accretion disk corona scenario (Galeev et al., 1979; Qiao & Liu 2015). There are various models which are developed in the exploring the radiation from the accretion disk and corona. For instance, KERRBB is a relativistic multi-temperature blackbody model for the thin accretion disk around a Kerr black hole, which is usually used to fit the disk emission from disk-dominated X-ray binaries and quasars (Li et al., 2005), while SIMPL is a Comptonization model for modeling the power-law hard X-ray spectra of X-ray binaries and AGNs (Steiner et al., 2009b). The convolution between KERRBB and SIMPL in XSPEC (Arnaud, 1996) could simulate the broad band SEDs of quasars, namely, the optical BBB and power-law X-ray spectra, for the purpose of our paper here. In the resulting spectra of SIMPL*KERRBB, the hard X-ray power-law index depends on the parameter , while the flux is determined by the parameters which is the fraction of the soft photons from the disk being scattered to high energy (see Fig. 1 in Steiner et al., 2009b). As an alternative Comptonization model, a general relativistic model for an accretion disk corona surrounding a Kerr black hole was developed by You et al. (2012), in which the energy fraction dissipated into the corona was assumed in order to solve the pattern of the disk corona system. The emergent spectrum of the disk corona observed at infinity can be calculated when the values of the disk parameters are specified.

In this work, we try to fit the observed SEDs of quasars with the general relativistic accretion disk corona model, in which the values of the disk parameters for five quasars, especially the black hole spin parameter , are derived. The black hole masses of these quasars are estimated with single epoch reverberation mapping method (see Tang et al., 2012, for the details).

The red tail of the BBB in optical wavebands is mostly emitted from the outer region of the accretion disk, which can be used to constrain the mass accretion rate almost independent of the black hole spin parameter if the black hole mass and the inclination angle are known (see Davis & Laor, 2011). In our accretion disk corona model, the mass accretion rate can be inferred similar to their work. The peaks of the BBBs in most quasars are in the far-UV/soft X-ray wavebands, which are usually unobserved. This is sensitive to the black hole spin , which is the main obstacle for measuring the spin parameters of massive black holes with the spectral fitting method. In the accretion disk corona model considered in this work, the photons emitted from the disk, including the unobserved far-UV/soft X-ray photons are inverse Compton scattered in the hot corona, and the hard X-ray spectra (luminosity and spectral shape) contain the information of the incident spectra from the cold disk.

The fraction of the power dissipated in the corona to the total power is a key parameter in the accretion disk-corona model. The detailed physical processes for the energy transported from the disk to the corona were explored by many different authors (e.g., Di Matteo, 1998; Meyer et al., 2000; Liu et al., 2002; Wang et al., 2004; Cao, 2009). It is found that both the hard X-ray spectral index and the hard X-ray bolometric correction factor are correlated with the Eddington ratio (e.g., Wang et al., 2004; Shemmer et al., 2006; Vasudevan & Fabian, 2007; Zhou & Zhao 2010; Fanali et al., 2013), which can be reproduced roughly by the accretion disk corona model on the assumption that the corona is heated by the reconnection of the magnetic fields generated by buoyancy instability in the cold accretion disk (Cao, 2009). However, the precise value of is still undetermined in the theoretical model calculations. In this work, we adopt the fraction as a free parameter in the spectral fittings.

In the spectral fitting of a quasar SED with measured black hole mass, the value of is dominantly determined by the hard X-ray photon index , which is almost insensitive to all other model parameters (Cao, 2009; You et al., 2012). With the derived value of from , the mass accretion rate and the inclination angle are constrained with the luminosity and spectral shape in optical waveband (the red tail of the BBB). These two parameters are almost independent of the black hole spin parameter , because the emission in this waveband is dominantly from the outer region of the disk (Davis & Laor, 2011) . Finally, the black hole spin parameter is constrained mainly with the observed hard X-ray luminosity.

In Figures 2-6, we plot the fitting results of five quasars. The values of black hole spin parameter are well constrained with small uncertainties, which are in the range of to (see Table 3). The estimated accretion rate is moderate, approximately 0.4 except OS 562 with , and these quasars are possibly viewed with small inclination angles , which support the classification of the sources as type 1 quasars. The fractions of the power radiated in the corona to the total dissipated power in the disk are (see Table 3).

In the spectral fitting on the SEDs of these sources in this work, the electron temperatures of the coronas are in the range of  keV, which are roughly consistent with the previous works (e.g., Zdziarski et al., 1996; Liu et al., 2002). The maximal temperatures of the electrons in the coronas for the sources are listed in Table 3. The corresponding cutoff energy of the X-ray spectra is around several hundred keV (see You et al., 2012). No direct observations in this energy band are available for these five sources. The cutoff energy in the power law X-ray spectra is observed only in a small fraction of AGNs, which is typically in the range of  keV (Molina et al., 2013). This is consistent with our results.

We plot the distribution of the fitting error for the spin , the mass accretion rate and the inclination angle in Fig. 7, in which the uncertainties of the fitted parameters are given by confidence levels. We also plot 2D confidence contours in Fig. 8, and find that the black hole spin is indeed well constrained at fairly good accuracy if the black hole mass is accurately measured.

The observed hard X-ray spectra may be affected by the reflection components (Gou et al., 2011, 2014). However, there are observational evidences that the reflection components are absent in many bright quasars (high luminosities) (Nandra et al., 1997; Page et al., 2005; Gilli et al., 2007). The XMM-Newton observations of a sample of high redshift quasars show that the hard X-ray spectra ( keV in the rest frame of the sources) can be well fitted for each source by a simple power law, and no Compton reflection humps are present (Page et al., 2005). The five sources in this work are all luminous quasars with , and therefore we believe that their hard X-ray continuum spectra may probably not be contaminated by the reflection components. Further X-ray observations in high energy with/without Compton reflection hump for these sources may help resolve this issue.

Another uncertainty of the black hole spin constraints is caused by the errors of the black hole mass estimates, because the constraints on the spin parameter is sensitive to the black hole mass (see Table 4). The black hole spin of quasar SDSS J094533.99+100950.1 is derived by fitting its multi-band (IR/optical/UV) continuum spectrum. The black hole spin varies from to if the black hole mass is taken as a free parameter, which means that the black hole mass is crucial in constraining the black hole spin (Czerny et al., 2011). In order to evaluate how the constraint on the spin parameter is affected by the black hole mass, we re-fit the SED of PG 1322+659 with different values of black hole mass given in different works (see Table 4). For the black hole mass estimated with C iv line (Vestergaard & Peterson, 2006), the best fitting of black-hole spin appears as , while if is adopted. For the black hole mass estimated by Kawakatu et al. (2007), no satisfactory spectral fitting is available. In most previous works, the uncertainty of the black hole mass estimates with single epoch reverberation mapping approach is within a factor of four (e.g., Wu et al., 2004; Peterson et al., 2004; Vestergaard & Peterson, 2006). Increasing the mass by a factor of two and four, i.e., , we find that the black hole spin parameters are constrained as respectively (see Fig. 9). With typical accuracy of the black hole mass measurements, the best fitted black hole spins are in a large range from -0.94 to 0.998 (see Fig. 10), which indicates that no tight constraint on the black hole spin is achieved with the uncertainty of a factor of four in mass estimates. However, it is not unique. Actually this issue is already discovered by previous works when the continuum-fitting method is applied to both AGNs and X-ray binaries. For instance, as for AGNs, the best-fitting values of the spin a for different black hole masses and the inclination angles can be found in Fig 5 of Done et al. (2013), which showed that the spin could vary from -1 to 1 if the mass is in a large range; as for X-ray binaries, even the system parameters of LMC X-3 are fairly well determined, the spin can spread over a large range with different combinations of the blacho hole mass and inclination angle (see Table 2 of Kubota et al., 2010). The improved estimate of the spin of LMC X-3 (Steiner et al., 2014), with the dynamical parameters to date (Orosz et al., 2014). Although the spin covers almost all the possible range for the spin parameter due to large errors of the black hole mass, fitting the disk-corona model to the optical/UV and X-ray SED in our paper is meriting exploration.

The main uncertainty of traditional reverberation mapping analysis originates from the virial coefficient or normalizing factor. Recently, Pancoast et al. (2011) proposed a method to analyze reverberation mapping data, with which the geometry and kinematics of the BLR can be well constrained by modeling the continuum light curve and broad line profiles directly with the velocity resolved broad line data. This allows to make a measurement of the black hole mass that does not depend on the virial coefficient (Pancoast et al., 2011, 2012). The accuracy of the black hole mass estimates with this method has been significantly improved (Barth et al., 2011; Pancoast et al., 2012; Li et al., 2013). The preliminary results show that five black hole masses have been measured at an uncertainty of dex depending upon data quality with this method (Pancoast et al., 2014). We search the literature, and find that the X-ray spectral data are available only for the two sources. Unfortunately, the X-ray photon spectral index for these two sources, which are unsuitable for our present investigation. The black hole spins can be well constrained with the method in this work, when a variety of black hole masses are measured at an uncertainty of dex with the velocity resolved reverberation mapping method in the near future.

In our model calculations, the hard X-ray emission of these quasars is assumed to originate predominantly from the corona above the disk, which is true for radio quiet sources. However, it should be cautious for radio loud quasars, of which the observed hard X-ray emission may be contaminated by the beamed jet emission (e.g., Chen et al., 2012). We find that OS 562 is a Flat Spectrum Radio Quasar (FSRQ), which is also characterized as low-spectral peaked source with the synchrotron peak frequency (Lister et al., 2009). The Very Long Baseline Array (VLBA) observations indicate that this source is observed with a small inclination angle with respect to the jet axis (Lister, 2001), which is consistent with the viewing angle derived in our spectral fitting (see Table 3). Donato et al. (2001) analyzed the X-ray spectral of 268 blazars including BL Lacs and FSRQs, and found that the X-ray spectra of FSRQs are always hard, i.e., the average X-ray spectral index (). The X-ray spectral index of OS 562 is , much softer than a typical FSRQ, which may imply that the X-ray emission is predominantly from the corona. This is the case also for the other two radio quasars. The radio core dominance (defined as ) is usually taken as a radio orientation indicator (Ghisellini et al., 1993; Wills & Brotherton, 1995). The radio core dominance parameters of the other two radio loud quasars in this work are, (4C 10.06) and (4C 39.25) (Runnoe et al., 2013), which imply that the jets are inclined at large angles with respect to the line of sight. This is consistent with our fitting results of for these two sources. The beaming effects for the jets viewed at such large angles are always unimportant (e.g., Wu et al., 2013). This strengthens the conclusion that the beamed X-ray emission from the jets can be neglected compared with that from the coronas in these two sources.

Although there are no additional information of X-ray variability for 4C 10.06 and OS 562, some previous observations indicated that the other three sources have low X-ray variability, for PG 1322+659 and PG 1115+407 (Grupe et al., 2001), variation for a sample of flat-spectrum radio quasars including 4C 39.25. In Fig. 11, we show that the constraint on the black hole spin is affected by the X-ray variability. It is found that the spin parameter becomes for , while for . This implies that the sources with violent variability are not suitable for present investigation on black hole spin. Our model is applicable for steady accretion disk corona systems. For those slightly variable sources, the mean observed continuum spectrum can still be used to constrain the black hole spin parameters.

Although the soft X-ray spectral data are not used in our model fitting on the SEDs, given the origin of the soft X-ray excess observed in AGNs is still controversial (Done et al., 2012), it is still necessary to investgate the influence of the soft X-ray excess on determining black hole spin. It was done for MCG -06-30-15 which showed a soft excess below 0.7 keV (Brenneman & Reynolds, 2006). Fitting physical models to the broadband spectrum at 2-10 keV with broad iron line, the spin of the black hole in MCG -06-30-15 is well constrained to be . Moreover, if including the observed features at the lower energy bands 0.6-2 keV and introducing additional components as the possible origin of soft X-ray excess, the best fits to the overall spectrum estimated the black hole spin of or 0.989, which indicated the weak influence of the soft X-ray excess on determining the spin of the black hole. Given that we don’t have the observation of soft X-ray excess for the five sources in our work, we try to discuss the influence of the soft X-ray excess in terms of power dissipation. We assume that 20% of the dissipated disk power is radiated as soft X-ray excess component (Marinucci et al., 2014), then the parameter is defined as the ratio of the dissipated power in the corona to the remaining power in the disk, i.e., . We re-fit the SED of PG 1322+659 with this definition of , and find that the best-fitting value of black hole spin , showing small difference in the spin parameter. The remaining model parameters are , , and , respectively.

For the traditional thin accretion disk model, the mass accretion rate is required in order to satisfy the geometrically thin approximation, i.e., (e.g., Laor & Netzer, 1989; Esin et al., 1997; King, 2012; Abramowicz & Fragile, 2013). We find that is required for the spectral fitting in five sources in this work. Our calculations show that the situation is different for the accretion disk with corona, because a fraction of gravitational power is transported into the corona, which reduces the radiation pressure of the disk for a given mass accretion rate . In Fig. 13, we find that the geometrically thin disk assumption is still valid even if when a typical value of is adopted. This implies that the thin accretion disk corona model is applicable for all quasars in this work. For more luminous quasars with , a general relativistic slim accretion disk corona model is required, which will be reported in our future work.

In this work, we tentatively propose a variant of the continuum-fitting method to constrain the black hole spin parameter in AGNs with a general relativistic accretion corona model. A reliable way to test the feasibility of this proposed method is to apply it to a few stellar-mass black hole X-ray binaries with available estimates of the black hole spin from both continuum-fitting and broad Fe K line fitting methods, to see whether the similar results can be obtained. In this paper, it is found that the black hole spin parameter can be well constrained, if the hard X-ray continuum spectrum and the black hole mass are accurately measured. We have not studied the statistics of the black hole spin parameter in this work, due to the small number of the quasars, which can provide useful information when a large sample of quasars with measured black hole spins .

###### Acknowledgements.
We thank Zhaohui Shang for the SED data, careful reading the manuscript, and helpful comments/suggestions. Weimin Yuan, Tinggui Wang, Minfeng Gu, Shiying Shen, and Jianmin Wang are thanked for the helpful discussion. Hengxiao Guo is thanked for fitting the emission lines in Fig. 12. This work is supported by the NSFC (grants 11173043, 11121062, 11233006, 11073020, 11373056, and 11473054), the Fundamental Research Funds for the Central Universities (WK2030220004), the CAS/SAFEA International Partnership Program for Creative Research Teams (KJCX2-YW-T23), and Shanghai Municipality.

## References

• Abramowicz & Fragile (2013) Abramowicz, M. A., & Fragile, P. C. 2013, Living Reviews in Relativity, 16, 1
• Alexander & Hickox (2012) Alexander, D. M., & Hickox, R. C. 2012, Nature, 56, 93
• Antonucci (1999) Antonucci, R. 1999, High Energy Processes in Accreting Black Holes, 161, 193
• Arnaud (1996) Arnaud, K. A. 1996, Astronomical Data Analysis Software and Systems V, 101, 17
• Barr et al. (1985) Barr, P., White, N. E., & Page, C. G. 1985, \mnras, 216, 65P
• Barth et al. (2011) Barth, A. J., Pancoast, A., Thorman, S. J., et al. 2011, \apjl, 743, L4
• Bechtold et al. (1987) Bechtold, J., Czerny, B., Elvis, M., Fabbiano, G., & Green, R. F. 1987, \apj, 314, 699
• Beloborodov (1999) Beloborodov, A. M. 1999, \apjl, 510, L123
• Bentz et al. (2009) Bentz, M. C., Walsh, J. L., Barth, A. J., et al. 2009, \apj, 705, 199
• Blaes (2013) Blaes, O. 2013, \ssr, 58
• Blandford & Znajek (1977) Blandford, R. D., & Znajek, R. L. 1977, \mnras, 179, 433
• Blandford & Payne (1982) Blandford, R. D., & Payne, D. G. 1982, \mnras, 199, 883
• Berti & Volonteri (2008) Berti, E., & Volonteri, M. 2008, \apj, 684, 822
• Brenneman & Reynolds (2006) Brenneman, L. W., & Reynolds, C. S. 2006, \apj, 652, 1028
• Brinkmann et al. (1997) Brinkmann, W., Yuan, W., & Siebert, J. 1997, \aap, 319, 413
• Calderone et al. (2013) Calderone, G., Ghisellini, G., Colpi, M., & Dotti, M. 2013, \mnras, 431, 210
• Cao (2009) Cao, X. 2009, \mnras, 394, 207
• Chandrasekhar (1960) Chandrasekhar, S. 1960, New York: Dover, 1960,
• Chen et al. (2012) Chen, L., Cao, X., & Bai, J. M. 2012, \apj, 748, 119
• Chiang (2002) Chiang, J. 2002, \apj, 572, 79
• Cunningham (1975) Cunningham, C. T. 1975, \apj, 202, 788
• Czerny et al. (2011) Czerny, B., Hryniewicz, K., Nikołajuk, M., & Sa̧dowski, A. 2011, \mnras, 415, 2942
• Davis & Laor (2011) Davis, S. W., & Laor, A. 2011, \apj, 728, 98
• de La Calle Pérez et al. (2010) de La Calle Pérez, I., Longinotti, A. L., Guainazzi, M., et al. 2010, \aap, 524, A50
• Di Matteo (1998) Di Matteo, T. 1998, \mnras, 299, L15
• Donato et al. (2001) Donato, D., Ghisellini, G., Tagliaferri, G., & Fossati, G. 2001, \aap, 375, 739
• Done et al. (2012) Done, C., Davis, S. W., Jin, C., Blaes, O., & Ward, M. 2012, \mnras, 420, 1848
• Done et al. (2013) Done, C., Jin, C., Middleton, M., & Ward, M. 2013, \mnras, 434, 1955
• Dotti et al. (2013) Dotti, M., Colpi, M., Pallini, S., Perego, A., & Volonteri, M. 2013, \apj, 762, 68
• Esin et al. (1997) Esin, A. A., McClintock, J. E., & Narayan, R. 1997, \apj, 489, 865
• Fanali et al. (2013) Fanali, R., Caccianiga, A., Severgnini, P., et al. 2013, \mnras, 433, 648
• Ferrarese & Merritt (2000) Ferrarese, L., & Merritt, D. 2000, \apjl, 539, L9
• Frank et al. (1992) Frank, J., King, A., & Raine, D. 1992, Camb. Astrophys. Ser., Vol. 21,,
• Galeev et al. (1979) Galeev, A. A., Rosner, R., & Vaiana, G. S. 1979, \apj, 229, 318
• Ghisellini et al. (1993) Ghisellini, G., Padovani, P., Celotti, A., & Maraschi, L. 1993, \apj, 407, 65
• Gilli et al. (2007) Gilli, R., Comastri, A., & Hasinger, G. 2007, \aap, 463, 79
• Goosmann et al. (2006) Goosmann, R. W., Czerny, B., Mouchet, M., et al. 2006, \aap, 454, 741
• Gou et al. (2011) Gou, L., McClintock, J. E., Reid, M. J., et al. 2011, \apj, 742, 85
• Gou et al. (2014) Gou, L., McClintock, J. E., Remillard, R. A., et al. 2014, \apj, 790, 29
• Grupe et al. (2001) Grupe, D., Thomas, H.-C., & Beuermann, K. 2001, \aap, 367, 470
• Haardt & Maraschi (1991) Haardt, F., & Maraschi, L. 1991, \apjl, 380, L51
• Haardt et al. (1994) Haardt, F., Maraschi, L., & Ghisellini, G. 1994, \apjl, 432, L95
• Hirose et al. (2009) Hirose, S., Blaes, O., & Krolik, J. H. 2009, \apj, 704, 781
• Hopkins et al. (2008) Hopkins, P. F., Hernquist, L., Cox, T. J., & Kereš, D. 2008, \apjs, 175, 356
• Hubeny et al. (2001) Hubeny, I., Blaes, O., Krolik, J. H., & Agol, E. 2001, \apj, 559, 680
• Kaspi et al. (2000) Kaspi, S., Smith, P. S., Netzer, H., et al. 2000, \apj, 533, 631
• Kawakatu et al. (2007) Kawakatu, N., Imanishi, M., & Nagao, T. 2007, \apj, 661, 660
• King (2012) King, A. 2012, \memsai, 83, 466
• Kishimoto et al. (2008) Kishimoto, M., Antonucci, R., Blaes, O., et al. 2008, \nat, 454, 492
• Koratkar & Blaes (1999) Koratkar, A., & Blaes, O. 1999, \pasp, 111, 1
• Kormendy & Richstone (1995) Kormendy, J., & Richstone, D. 1995, \araa, 33, 581
• Kubota et al. (2010) Kubota, A., Done, C., Davis, S. W., et al. 2010, \apj, 714, 860
• Laor & Netzer (1989) Laor, A., & Netzer, H. 1989, \mnras, 238, 897
• Laor (1991) Laor, A. 1991, \apj, 376, 90
• Laor & Davis (2014) Laor, A., & Davis, S. W. 2014, \mnras, 60
• Li et al. (2005) Li, L.-X., Zimmerman, E. R., Narayan, R., & McClintock, J. E. 2005, \apjs, 157, 335
• Li et al. (2013) Li, Y.-R., Wang, J.-M., Ho, L. C., Du, P., & Bai, J.-M. 2013, \apj, 779, 110
• Li et al. (2009) Li, Y.-R., Yuan, Y.-F., Wang, J.-M., Wang, J.-C., & Zhang, S. 2009, \apj, 699, 513
• Lister (2001) Lister, M. L. 2001, \apj, 562, 208
• Lister et al. (2009) Lister, M. L., Aller, H. D., Aller, M. F., et al. 2009, \aj, 137, 3718
• Liu et al. (2002) Liu, B. F., Mineshige, S., & Shibata, K. 2002, \apjl, 572, L173
• Liu & Zhang (2011) Liu, Y., & Zhang, S. N. 2011, \apjl, 728, L44
• Lohfink et al. (2012) Lohfink, A. M., Reynolds, C. S., Miller, J. M., et al. 2012, \apj, 758, 67
• Malkan & Sargent (1982) Malkan, M. A., & Sargent, W. L. W. 1982, \apj, 254, 22
• Marconi et al. (2004) Marconi, A., Risaliti, G., Gilli, R., et al. 2004, \mnras, 351, 169
• Marinucci et al. (2014) Marinucci, A., Matt, G., Miniutti, G., et al. 2014, \apj, 787, 83
• Martocchia et al. (2002) Martocchia, A., Matt, G., Karas, V., Belloni, T., & Feroci, M. 2002, \aap, 387, 215
• McClintock et al. (2011) McClintock, J. E., Narayan, R., Davis, S. W., et al. 2011, Classical and Quantum Gravity, 28, 114009
• McConnell et al. (2011) McConnell, N. J., Ma, C.-P., Gebhardt, K., et al. 2011, \nat, 480, 215
• Merloni (2004) Merloni, A. 2004, \mnras, 353, 1035
• Meyer et al. (2000) Meyer, F., Liu, B. F., & Meyer-Hofmeister, E. 2000, \aap, 361, 175
• Miller et al. (2002) Miller, J. M., Fabian, A. C., in’t Zand, J. J. M., et al. 2002, \apjl, 577, L15
• Miniutti et al. (2004) Miniutti, G., Fabian, A. C., & Miller, J. M. 2004, \mnras, 351, 466
• Moderski & Sikora (1996) Moderski, R., & Sikora, M. 1996, \mnras, 283, 854
• Molina et al. (2013) Molina, M., Bassani, L., Malizia, A., et al. 2013, \mnras, 433, 1687
• Nandra et al. (1997) Nandra, K., George, I. M., Mushotzky, R. F., Turner, T. J., & Yaqoob, T. 1997, \apjl, 488, L91
• Narayan & Yi (1994) Narayan, R., & Yi, I. 1994, \apjl, 428, L13
• Novikov & Thorne (1973) Novikov, I. D., & Thorne, K. S. 1973, Black Holes (Les Astres Occlus), 343
• Orosz et al. (2014) Orosz, J. A., Steiner, J. F., McClintock, J. E., et al. 2014, \apj, 794, 154
• Page & Thorne (1974) Page, D. N., & Thorne, K. S. 1974, \apj, 191, 499
• Page et al. (2005) Page, K. L., Reeves, J. N., O’Brien, P. T., & Turner, M. J. L. 2005, \mnras, 364, 195
• Pancoast et al. (2011) Pancoast, A., Brewer, B. J., & Treu, T. 2011, \apj, 730, 139
• Pancoast et al. (2012) Pancoast, A., Brewer, B. J., Treu, T., et al. 2012, \apj, 754, 49
• Pancoast et al. (2014) Pancoast, A., Brewer, B. J., Treu, T., et al. 2014, \mnras, 445, 3073
• Patrick et al. (2011) Patrick, A. R., Reeves, J. N., Porquet, D., et al. 2011, \mnras, 411, 2353
• Peterson et al. (2004) Peterson, B. M., Ferrarese, L., Gilbert, K. M., et al. 2004, \apj, 613, 682
• Porquet et al. (2004) Porquet, D., Reeves, J. N., O’Brien, P., & Brinkmann, W. 2004, \aap, 422, 85
• Qiao & Liu (2015) Qiao, E., & Liu, B. F. 2015, \mnras, 448, 1099
• Remillard & McClintock (2006) Remillard, R. A., & McClintock, J. E. 2006, \araa, 44, 49
• Reynolds & Nowak (2003) Reynolds, C. S., & Nowak, M. A. 2003, \physrep, 377, 389
• Runnoe et al. (2013) Runnoe, J. C., Shang, Z., & Brotherton, M. S. 2013, \mnras, 435, 3251
• Sakimoto & Coroniti (1981) Sakimoto, P. J., & Coroniti, F. V. 1981, \apj, 247, 19
• Sambruna (1997) Sambruna, R. M. 1997, \apj, 487, 536
• Schlegel et al. (1998) Schlegel, D. J., Finkbeiner, D. P., & Davis, M. 1998, \apj, 500, 525
• Shafee et al. (2006) Shafee, R., McClintock, J. E., Narayan, R., et al. 2006, \apjl, 636, L113
• Shakura & Sunyaev (1973) Shakura, N. I., & Sunyaev, R. A. 1973, \aap, 24, 337
• Shang et al. (2011) Shang, Z., Brotherton, M. S., Wills, B. J., et al. 2011, \apjs, 196, 2
• Shemmer et al. (2006) Shemmer, O., Brandt, W. N., Netzer, H., Maiolino, R., & Kaspi, S. 2006, \apjl, 646, L29
• Shen et al. (2011) Shen, Y., Richards, G. T., Strauss, M. A., et al. 2011, \apjs, 194, 45
• Shields (1978) Shields, G. A. 1978, \nat, 272, 706
• Steiner et al. (2009a) Steiner, J. F., McClintock, J. E., Remillard, R. A., Narayan, R., & Gou, L. 2009a, \apjl, 701, L83
• Steiner et al. (2009b) Steiner, J. F., Narayan, R., McClintock, J. E., & Ebisawa, K. 2009b, \pasp, 121, 1279
• Steiner et al. (2014) Steiner, J. F., McClintock, J. E., Orosz, J. A., et al. 2014, \apjl, 793, L29
• Stella & Rosner (1984) Stella, L., & Rosner, R. 1984, \apj, 277, 312
• Sun & Malkan (1989) Sun, W.-H., & Malkan, M. A. 1989, \apj, 346, 68
• Sunyaev & Titarchuk (1985) Sunyaev, R. A., & Titarchuk, L. G. 1985, \aap, 143, 374
• Svensson & Zdziarski (1994) Svensson, R., & Zdziarski, A. A. 1994, \apj, 436, 599
• Taam & Lin (1984) Taam, R. E., & Lin, D. N. C. 1984, \apj, 287, 761
• Tang et al. (2012) Tang, B., Shang, Z., Gu, Q., Brotherton, M. S., & Runnoe, J. C. 2012, \apjs, 201, 38
• Vanden Berk et al. (2001) Vanden Berk, D. E., Richards, G. T., Bauer, A., et al. 2001, \aj, 122, 549
• Vasudevan & Fabian (2007) Vasudevan, R. V., & Fabian, A. C. 2007, \mnras, 381, 1235
• Vestergaard & Osmer (2009) Vestergaard, M., & Osmer, P. S. 2009, \apj, 699, 800
• Vestergaard & Peterson (2006) Vestergaard, M., & Peterson, B. M. 2006, \apj, 641, 689
• Volonteri et al. (2005) Volonteri, M., Madau, P., Quataert, E., & Rees, M. J. 2005, \apj, 620, 69
• Wang et al. (2004) Wang, J.-M., Watarai, K.-Y., & Mineshige, S. 2004, \apjl, 607, L107
• Wills & Brotherton (1995) Wills, B. J., & Brotherton, M. S. 1995, \apjl, 448, L81
• Woo & Urry (2002) Woo, J.-H., & Urry, C. M. 2002, \apjl, 581, L5
• Wu et al. (2013) Wu, Q., Cao, X., Ho, L. C., & Wang, D.-X. 2013, \apj, 770, 31
• Wu et al. (2004) Wu, X.-B., Wang, R., Kong, M. Z., Liu, F. K., & Han, J. L. 2004, \aap, 424, 793
• You et al. (2012) You, B., Cao, X., & Yuan, Y.-F. 2012, \apj, 761, 109
• You et al. (2015) You, B., Czerny, B., Sobolewska, M. et al.: 2015, arXiv:1506.03959
• Yuan et al. (2010) Yuan, W., Liu, B. F., Zhou, H., & Wang, T. G. 2010, \apj, 723, 508
• Zdziarski et al. (1996) Zdziarski, A. A., Gierlinski, M., Gondek, D., & Magdziarz, P. 1996, \aaps, 120, 553
• Zdziarski et al. (1999) Zdziarski, A. A., Lubiński, P., & Smith, D. A. 1999, \mnras, 303, L11
• Zhang et al. (1985) Zhang, J. L., Xiang, S. P., & Lu, J. F. 1985, \apss, 113, 181
• Zhang et al. (1997) Zhang, S. N., Cui, W., & Chen, W. 1997, \apjl, 482, L155
• Zheng et al. (1997) Zheng, W., Kriss, G. A., Telfer, R. C., Grimes, J. P., & Davidsen, A. F. 1997, \apj, 475, 469
• Zhou & Zhao (2010) Zhou, X.-L., & Zhao, Y.-H. 2010, \apjl, 720, L206
You are adding the first comment!
How to quickly get a good reply:
• Give credit where it’s due by listing out the positive aspects of a paper before getting into which changes should be made.
• Be specific in your critique, and provide supporting evidence with appropriate references to substantiate general statements.
• Your comment should inspire ideas to flow and help the author improves the paper.

The better we are at sharing our knowledge with each other, the faster we move forward.
The feedback must be of minimum 40 characters and the title a minimum of 5 characters