The fundamental plane of FSRQs based on the black hole spin-mass energy

Previous studies have identified the Fundamental Plane (FP) of Active Galactic Nuclei (AGNs) in recent years. However, these findings relied on total black hole mass, also known as dynamical mass, (Mdyn). Instead the fundamental plane may be governed by BH spin. In this study, we take the black hole spin-mass energy (Mspin\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$M_{\mathrm{spin}}$\end{document}) as a new variable, which is closely related to the supermassive black hole (SMBH). We collected a sample of 62 flat-spectrum radio quasars (FSRQs) with gamma-ray luminosity (Lγ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$L_{\gamma }$\end{document}), X-ray luminosity (LX\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$L_{\mathrm{X}}$\end{document}) and spin-mass energy (Mspin\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$M_{ \mathrm{spin}}$\end{document}) to construct a new fundamental plane of blazars. Our analysis demonstrates that the fundamental plane (logLγ=0.662+0.193−0.193logLX+0.495+0.154−0.154logMspin+14.579+7.140−7.140\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\log L_{\gamma }={0.662}_{+0.193}^{-0.193}\log L_{ \mathrm{X}}+{0.495}_{+0.154}^{-0.154}\log M_{\mathrm{spin}} +{14.579}_{+7.140}^{-7.140}$\end{document} with R-Square = 0.783) has a stronger correlation with spin-mass energy (Mspin\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$M_{\mathrm{spin}}$\end{document}) compared to the black hole mass. Therefore, Mspin\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$M_{\mathrm{spin}}$\end{document} should be considered as an essential variable for the fundamental plane of blazars. Our findings may improve the understanding of the Blandford-Znajek process in FSRQs.


Introduction
The fundamental plane (FP) of black hole activity is generally described as an approximate power law relation between BH mass, X-ray and radio luminosity for jet producing BHs (Merloni et al. 2003;Falcke et al. 2004;Körding et al. 2006,b;Gultekin et al. 2009;Ünal and Loeb 2020).For black holes (BHs) with outflows and an accretion disk, the X-ray luminosity is usually associated with accretion power, while radio luminosity indicates jet power in radiatively inefficient active galactic nuclei (AGNs) with sub-Eddington accretion rates (Heinz and Sunyaev 2003;Körding et al. 2006,b;Falcke et al. 2004;Merloni et al. 2003;Saikia et al. 2015;Plotkin et al. 2012;Gultekin et al. 2009;Ünal and Loeb 2020).However, for blazars, which are an extreme subclass of active galactic nuclei (AGN) (Zhu et al. 2023), gamma rays can also be considered as an indicator of jet power.Blazars can be categorized into two types: flat-spectrum radio quasars (FSRQs) and BL Lacertae objects (BL Lac).FS-RQs are characterized by strong emission lines, while BL Lacs show featureless optical spectra or weak emission lines (Scarpa and Falomo 1997;Xiong et al. 2015a,b).Blazars have a typical two-hump structure in their spectral energy distribution (SED), with a wide emission range from radio to γ -ray bands (Abdo et al. 2010;Fan et al. 2016;Zhu et al. 2023).Previous research has identified many unique and iconic observational properties about the strong and variable γ -ray emissions of Blazars, as reported in various studies (Wills et al. 1992;Villata et al. 2006;Fan 2002; Urry and Padovani 1995;Abdollahi et al. 2020;Gupta et al. 2016;Xiao et al. 2015Xiao et al. , 2020;;Fan et al. 2014Fan et al. , 2021;;Xiong et al. 2020Xiong et al. , 2017Xiong et al. , 2016;;Zhu et al. 2023).The unique observational properties of γ -rays in Blazars can be attributed to the relativistic jets (e.g.Blandford and Levinson 1995), which relates γ -ray luminosity to jet power for the fundamental plane of Blazars.
There is no consensus on the formation mechanism of relativistic jets until now.Three main mechanisms have been proposed to explain their formation.The first one is the Blandford-Znajek (BZ) mechanism, which posits that the rotational energy of the black hole and accretion disc are the primary source of the jets (Blandford and Znajek 1977).Furthermore, the jet power in the BZ mechanism is dependent on the spin of the black hole.The second mechanism is the Blandford-Payne (BP) mechanism, which suggests that the jet extracts the rotational energy of the accretion disc, and the mass of the black hole can be ignored.This mechanism has been supported by previous studies (Rawlings and Saunders 1991;Ghisellini and Tavecchio 2010;Ghisellini et al. 2011;Sbarrato et al. 2012Sbarrato et al. , 2014;;Cao and Jiang 1999;Tavecchio et al. 2010).Finally, A hybrid model for mixing BZ and BP mechanisms has been proposed by Meier (2001) and Garofalo (2010), which could provide an explanation for the observed differences in AGNs with relativistic jets (Garofalo and Meier 2010).These models propose that the spin of the black hole plays a crucial role in the production of both emitted radiation and jets (McClintock et al. 2014).Therefore, deviations from the fundamental plane may result from the spin of the black hole.In this study, we utilized M spin to investigate the fundamental plane and establish a new fundamental plane for a sample of FSRQs.The sample is presented in Sect.2, and our findings are described in Sect.3. Section 4 outlines the conclusions of our study.

Sample
To study the fundamental plane with FSRQs, the sample of FSRQs with γ -ray luminosity (L γ ), X-ray luminosity (L X ) in 2-10 keV and the BH spin-mass energy are necessary.In the redshift range of 0.2-2.5, we collected a sample of 62 FS-RQs with available black hole mass and available γ -ray luminosity (L γ ) from the Fermi Large Area Telescope (Abdo et al. 2010;Ackermann et al. 2011) (LAT).Because of the strong relativistic jets for FSRQs, we should consider the influence of anisotropic on our data of γ -ray luminosity.The intrinsic γ -ray luminosity for our sample can be measured by beaming factor f b with an average uncertainty, f b ≡ 1 − cos θ where θ is the jet opening angle (Nemmen et al. 2012;Zhang et al. 2017Zhang et al. , 2018)).
where the bulk Lorentz factor is collected by cross matching with previous work (Ghisellini et al. 2014) and L * γ is the intrinsic gamma luminosity respectively.The uncertainty of the Log of beaming factor we used is 0.02 (Xiong and Zhang 2014).
The X-ray luminosity in 2-10 keV for our sample of FS-RQs can not be directly collected from NASA/IPAC Extragalactic Database (NED) which includes 45 sample in 0.3-10 keV and 17 sample in 0.2-10 keV.The X-ray flux densities in 0.2-10 keV and 0.3-10 keV of FSRQs from NED can be converted into 2-10 keV by the power-law model with photon index τ = 1.87 for FSRQs (Zhang et al. 2018).
where F 2-10 keV is the X-ray flux density in 2-10 keV, F 2-10 keV is the X-ray flux density in 0.2-10 keV and F 0.3-10 keV is the X-ray flux density in 0.3-10 keV.ν is the energy dissipation of FSRQs (Zhang et al. 2018).
It is essential to calculate the M spin of BH for the fundamental plane.However, directly measuring the black hole spin for large samples by the X-ray reflection method is difficult due to observation limitations.Therefore, we estimated the BH spin with the beam power method, consistent with previous works such as the X-ray reflection method (Miller et al. 2009;Garcla et al. 2009;Reynolds 2014) and Daly's recent study (2019).To calculate the beam power of our sample, we used the theoretical model described by Willott et al. (1999), which established relationships between beam power and radio luminosity at 151 MHz.The TIFR Giant microwave radio telescope Sky Survey (Intema et al. 2017) detected the radio luminosity at 151 MHz for our sample.
where L 151 is the radio luminosity at 151 MHz in units of erg s −1 and L j is the beam power.The range of f is 1≤ f ≤ 20 (Cao 2003;Godfrey and Shabala 2013;Wu and Cao 2008;Fan andWu 2018, 2019).In this paper, we estimate the beam power with the low limit f = 1 (Cao 2003).The broad line region luminosity (L BLR ) which can be estimated by the standard templates (Francis et al. 1991;Celotti et al. 1997) is also needed.We can estimate the BH spin of our sample by the following formula (Daly 2019) where L bol is the bolometric luminosity (Netzer 1990), L bol = 10L BLR .We got the data of the bolometric luminosity with borad line region (Francis et al. 1991;Celotti et al. 1997).The uncertainty of logL bol in this paper is 0.32 (Calderone et al. 2013).L EDD is the Eddington luminosity, L Edd = 1.3 × 10 38 (M BH /M ).The uncertainty of logM dyn and the logL EDD we calculated are 0.04 (Xiong and Zhang 2014).g j = 0.1 and g bol = 1 (Daly 2019).A is a parameter defined by Daly et al. (2018), it can be measured by the coefficient of FP of BH (A = a/(a + b)), where the value of parameters a and b are from Merloni et al. (2003).In this paper, the value of A set as 0.43 (Daly et al. 2018).
In this paper, we use the method of Daly (2022) to estimate M spin for our sample using the relationships between the black hole spin function, F 2 , the total black hole mass, M (also referred to as M dyn ) and the mass-energy that can be extracted from the black hole, Mspin (Misner et al. 1973;Rees 1984;Blandford 1990;Thorne et al. 1986).Daly (2022) showed that where F is the spin function, F 2 ≡ f (j)/f max (Daly 2022), and M irr is the irreducible black hole mass.The uncertainty of the Log of bolometric luminosity we used is 0.15.We can estimate the M irr and M spin by Eq. ( 6) to do the fundamental palne.
All sample of 62 FSRQs with redshift, logM dyn , logL γ , logf b , logL x , logL j , logL bol and logL EDD are listed in Table 1.The sample of FSRQs with logF , logM irr , logM spin are listed in Table 2.The sources are arranged in order of increasing redshift.

The results
To conduct an analysis of the fundamental plane, this study investigates the correlation between the three key variables (L γ , L X , and M spin ) among FSRQs.The gamma ray luminosity of our samples requires beaming-corrected by beam factor f b as stated earlier due to the strong relativistic jets of FSRQs. Figure 1 presents the relationship between L γ and L X , which was best fitted with a straight line.The sample data show positive correlations, similar to previous work (Xie and Yuan 2017;Dong et al. 2014).This means that some part of the black hole's jet power is contributed by accretion power.
The relation between L γ and M dyn , M irr is presented in Fig. 2. We can see the different of correlations between M dyn and M irr .Because the contribution of M spin (M ≡ M dyn = M irr + E spin c −2 = M irr + M spin ), the differences on the mass is determined by the numerical value of BH spin.The difference is not obvious in the sample with small spin but the sample with large spin has a larger discrepancy.The positive correlation between black hole mass and jet energy is consistent with previous work, but the correlation analysis also shows that this relationship is affected by the black Fig. 1 The correlation between log L γ correcting with beam factor versus log L x for FSRQs.The red solid line is the best fitting Fig. 2 The correlation between log L γ correcting with beam factor versus M dyn and M irr for FSRQs.The red solid line is the best fitting for M dyn .The black dash line represents the best fitting for M irr hole spin.The relation between L γ and M spin is presented in Fig. 3.It means that the influence of the jet energy by black hole mass is partly due to the black hole spin.This provides a basis for us to use M spin as a variable to establish a new fundamental relationship.
In this paper, we use the multiple linear regression equation to obtain the fundamental relationship and verify the reliability of the fundamental relationship.
where L γ is the gamma-ray luminosity, L X is the X-ray luminosity in 2-10 keV, M spin is the BH spin-mass energy of FSRQs.a and b and c are fitting parameters (Zhang et al. 2018).The best fitting linear regression coefficient was obtained by multiple linear regression analysis (Merloni et al. 2003;Körding et al. 2006,b;Zhang et al. 2018).Three variables a, b and c with errors can be derived by linear regression analysis (Wang and Dai 2017).The fundamental plane of correcting anisotropism for our FSRQs' sample shows in Fig. 4. The slope of fitting for sam- Note: The first column is FGL name; the second column is redshift; the third column is the Logarithm of Black hole dynamical mass; the fourth column is the Logarithm of observational γ -ray luminosity; the fifth column is Logarithm of the beaming factor; the sixth column is Logarithm of observational x-ray luminosity in 2-10 keV (in units of erg s −1 ); the seventh column is Logarithm of the derived outflow beam power; the eighth column is Logarithm of the bolometric luminosity; the ninth column is Logarithm of the Eddington luminosity.with R-Square = 0.783.This shows that the jet energy of FSRQs is caused by accretion enegry and spin-mass energy of BH.It is consistent with the theory of BZ model.In order to verify that the black hole spin energy has better confidence than the black hole mass, we also conduct fundamental fitting for the M dyn and M irr as a comparison.For M dyn , the best-fitting result is L γ = 1.006 −0.180 +0.180 L X + 0.160 −0.150 +0.150 M dyn + 2.502 −6.601 +2.529 with R-Square = 0.744 Fig. 4 The fundamental plane of FSRQs with L γ , L X and M spin .The red solid line is the best fitting for whole sample which is similar to previous work of fundamental plane.
For M irr , the best-fitting result is L γ = 1.864 −0.6.540+6.540 L X + 0.133 −0.133 +0.133 M irr + 1.864 −6.54 +6.54 with R-Square = 0.73.This suggests that the proportion of accretion in the fundamental plane increases after the effect of the black hole spin is excluded.It means the previous correlations which is consistent with the BP mechanism may be fit for the lower spin AGNs.Both R-Square of latter two fundamental plane is lower comparing to the result we got in this paper.And the statistical significance of M dyn and M irr are 0.29 and 0.548.Both are above 0.05 which may should not be considered as a reliable variable for multiple linear regression for our data.
And the significance of M spin approaches 0.002.It means for our samples the spin-mass energy have a better confidence level comparing to the mass of black hole.

Conclusion and discussion
Although the standard fundamental plane provides a satisfactory description for the production of black hole jets (Ünal and Loeb 2020), it still has some deviations.To reduce these biases, we consider the effect of black hole spin.We assume that a greater fraction of the gravitational energy can be released as radiation as the spin increases, as the inner orbits of the accreting matter can approach closer to the horizon.This indicates that jet power is positively correlated with the spin-mass power of the black hole, as confirmed by the relevant analysis.Therefore, black hole spin should be considered in the fundamental plane for our samples.For greater accuracy, we want to include as many variables as possible.Thus we use M spin as the variable in our fundamental plane.This variable involves both the black hole spin and black hole mass.The new fundamental plane in this paper (log L γ = 0.662 −0.193  +0.193 log L X + 0.495 −0.154 +0.154 log M spin + 14.579 −7.140  +7.140 with R-Square = 0.783) shows that spin-mass energy has a stronger relation compared to black hole mass.Therefore, M spin should be considered an important variable for the fundamental plane of FSRQs.Our results emphasize the vital role of black hole spin in accreting and outflowing BHs to a certain extent (Ünal and Loeb 2020).This spin may also cause the differences seen in the three mechanisms of jet.It may help us understand the Blandford-Znajek process in FSRQs, and could provide a strong observational indication of the process's existence if the results are confirmed with larger samples.The deviations in our results may have resulted from various factors such as the thickness of the accretion disk, uncertainties in the data, the environment around our FSRQs, and uncertainties in the correlation between gamma-jet power and X-ray-accretion power.In order to improve the reliability of the fundamental plane, gamma luminosity and X-ray luminosity should be measured simultaneously (Zhang et al. 2018).However, observing data for our samples at the same time is rare.Miller et al. (2010) argued that on short time scales, an individual source may not be strictly controlled by the fundamental plane (Wang and Dai 2017).In this paper, the L γ and L x of FSRQs are nonsimultaneous, and the time interval between measurements can be more than a year.Because of the high variability of accreting black holes, the results of our fundamental plane can be affected by more than an order of magnitude (Körding et al. 2006,b;Zhang et al. 2018).Furthermore, the results of our study may also be influenced by selection bias (Zhang et al. 2018) due to the little number of sample.Therefore, further analysis is required with larger samples.

Fig.
Fig. The correlation between log L γ correcting with beam factor versus logM spin for FSRQs.The red solid line is the best fitting

Table 1
The inputs data of FSRQs

Table 2 The
The first column is FGL name; the second column is redshift; the third column is the Logarithm of the square root of the spin function; the fourth column is Logarithm of Black hole spin-mass energy; the fifth column is Logarithm of the black hole irreducible mass.