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

Xu Zhang1, Dingrong Xiong2, Quangui Gao1,3, Guiqin Yang4
Fangwu Lu1,3, Weiwei Na1, Longhua Qin1
1 Department of Physics, Yuxi Normal University, Yuxi 653100, People’s
Republic of China
2 Yunnan Observatories, Chinese Academy of Sciences, 396 Yangfangwang, Guandu
District, Kunming, 650216, People s Republic of China
3 Key Laboratory of Astroparticle Physics of Yunnan Province, Kunming 650091,
People’s Republic of China
4 Faculty of Mechanical and Electrical Engineering, Kunming University of
Science and Technology,
Kunming 650031, People’s Republic of China
zx<EMAIL_ADDRESS>
###### Abstract

We examine the fundamental plane of 91 Blazars which include FSRQs and BL Lacs
with known X-ray luminosity ($L_{R}$), radio luminosity ($L_{X}$), and black
hole mass measurements ($M$) to reflect the relationship between jet and
accretion for blazars. The fundamental plane of Blazars are
log$L_{R}$=${0.273}_{+0.059}^{-0.059}$log$L_{X}$+${0.695}_{+0.191}^{-0.191}$log$M$+${25.457}_{+2.728}^{-2.728}$
and
log$L_{R}$=${0.190}_{+0.049}^{-0.049}$log$L_{X}$+${0.475}_{+0.157}^{-0.157}$log$M$+${28.568}_{+2.245}^{-2.245}$
after considering the effect of beam factor. Our results suggest that the jet
of blazars has connection with accretion. We set the black hole spin energy as
a new variable to correct the black hole mass and explore the effect of black
hole spin on the fundamental relationship. We find that the fundamental plane
of Blazars is effected by the black hole spin, which is similar to the
previous work for AGNs. We additionally examine a new fundamental plane which
is based on the black hole spin-mass energy ($M_{spin}$). The new fundamental
plane
(log$L_{R}$=${0.332}_{+0.081}^{-0.081}$log$L_{X}$+${0.502}_{+0.091}^{-0.091}$log$M_{spin}$+${22.606}_{+3.346}^{-3.346}$
with R-Square=0.575) shows that $M_{spin}$ has a better correlation
coefficient comparing to the $M$ for fundamental plane of Blazars. These
results support that the black hole spin should be considered as a important
factor for the study of fundamental plane for Blazars. And these may further
our understanding of the Blandford-Znajek process in blazars.

###### keywords:

accretion, accretion discs quasars: supermassive black holes black hole
physics

††pagerange: The fundamental plane of blazars based on the black hole spin-
mass energy–LABEL:lastpage††pubyear: 2023

##  1 INTRODUCTION

When studying the internal structure of binary star systems and galactic
cores, the existence of a black hole (BH) can be inferred through various
indirect indicators. These indicators show clear associations with black hole
activity. One such indicator is the synchrotron radiation emitted by
relativistic jets in the radio frequency range, while another common indicator
is the intense and compact power-law X-ray emission, often associated with the
inner regions of accretion disks (Begelman et al., 1984). It is widely
observed that the accretion of compact objects and the initiation of
relativistic outflows/jets are interconnected phenomena (Falcke et al., 1994).
This suggests an inherent correlation between the jet and disk accretion flux
to some extent. To discuss this correlation, the concept of a fundamental
plane relationship has been introduced. The fundamental plane for black hole
activity is commonly defined as an approximate power-law relationship among
the mass of BH, X-ray and radio luminosities. It usually applies to the hard
state BHs. The applicability of fundamental plane for inclusion of soft or
soft-intermediate state BHXBs, as well as Seyferts is still not sure. This
relationship applies to black holes that produce jets (Merloni et al., 2003;
Falcke et al., 2004; Kording et al., 2006a, b; Gultekin et al., 2009; Unal et
al., 2015). In black holes with outflows and an accretion disk, X-ray
luminosity is generally linked to accretion power, while radio luminosity
reflects the jet power in radiatively inefficient active galactic nuclei
(AGNs) which have sub-Eddington accretion rates (Heinz et al., 2003; Falcke et
al., 2004; Merloni et al., 2003; Saikia et al., 2015; Kording et al., 2006a;
Plotkin et al., 2012; Gultekin et al., 2009; Unal et al., 2015).

Blazars are the most extreme type of active galactic nuclei (AGNs), with their
jets pointing towards the observer (Urry et al., 1995) and they constitute the
brightest and most dominant population of AGNs in the gamma-ray band (Fichtel
et al., 1994; Abdo et al., 2010). Their extreme observational properties are
explained as a result of a beaming effect. Due to the relativistic beaming
effect, the emission in the jet is strongly enhanced in the observer’s frame
(Urry et al., 1995). Blazars are typically divided into two sub-classes: BL
Lacertae objects (BL Lacs) and flat spectrum radio quasars (FSRQs). FSRQs have
strong emission lines, while BL Lacs objects have very weak or no emission
lines. The classical division between FSRQs and BL Lacs is mainly based on the
equivalent width (EW) of the emission lines. In a rest frame, objects with EW
$>$ 5$\dot{A}$ are classified as FSRQs (e.g., Urry et al., 1995; Scarpa et
al., 1997). Blandford et al. (1978) originally proposed that the absence of
broad lines in BL Lacs is due to very bright, Doppler-enhanced continuous
synchrotron radiation.

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 et al., 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 et
al., 1991; Ghisellini et al., 2010; Gu et al., 2009; Cao et al., 1999;
Tavecchio et al., 2010; Sbarrato et al., 2012; Ghisellini et al., 2014; Xiong
et al., 2015; Begelman et al., 1984). Finally, A hybrid model for mixing BZ
and BP mechanisms has been proposed by Meier et al. (2001) and Garofalo et al.
(2010), which could provide an explanation for the observed differences in
AGNs with relativistic jets (Garofalo et al., 2010). The jet power of Blazars
often exceeds the luminosity of the accretion disk Ghisellini et al. (2014).
It is shown that besides accretion, there are other important physical
parameters that help to determine the jet power Meier et al. (2002), such as
the spin and black hole mass of supermassive black holes. In particular, the
spin of a black hole has often been speculated as the main parameter for
determining the jet power of BH (Blandford et al., 1977). Unal et al. (2015)
have examined the fundamental plane which confirmed that the jet energy scales
quadratically with BH spin of AGN. Considering the involvement of BH spin in
both accretion and jets, BH spin is crucial for the study of fundamental
plane(Unal et al., 2015).

As a special type of AGN, Blazars should theoretically conform to the
fundamental plane relation, which has been confirmed by previous studies
(Merloni et al., 2003; Falcke et al., 2004; Bariuan et al., 2022; Unal et al.,
2015; Saikia et al., 2015; Zhang et al., 2018). Merloni et al. (2003) and
Falcke et al. (2004) examined the fundamental plane with a large sample
including the blazars and confirmed that Blazars are also follow the
fundamental plane of AGN. Saikia et al. (2016) used the intrinsic radio
luminosity obtained through the Lorentz factor to calculate the fundamental
plane of Blazars. Zhang et al. (2018) selected the radiatively efficient
sources to get the fundamental plane of Blazars. In this paper, we conduct a
fundamental analysis of blazars, and compare them with normal radio-loud and
radio-quiet AGNs (Unal et al., 2015). Furthermore, we use a new calculation
method (Daly et al., 2022) to incorporate black hole spin energy as an new
variable to study the impact of black hole spin on the fundamental plane
relation. The structure of this article is as follows. The sample is described
in Section 2, and the analysis results are described in Section 3. Section 4
summarizes our conclusions.

##  2 THE SAMPLES

In order to do the fundamental plane of blazars, we tried to collect the
sample of blazars as much as possible with reliable redshift, 2-10 keV X-ray
luminosity, 5 GHz radio luminosity, BH mass, bolometric luminosity, beam
power. For this purpose, we select the large samples of blazars from Ackermann
et al. (2011) and Nolan et al. (2012) which can be detected by Very Large
Array (VLA) for radio flux density and Swift or XMM-Newton for X-ray flux
density. In order to collect as many sources as we need, we first obtain the
samples following broad-line data of blazars (Cao et al., 1999; Wang et al.,
2004; Liu et al., 2006; Sbarrato et al., 2012; Chai et al., 2012; Shen et al.,
2011; Shaw et al., 2012). Secondly, we cross-correlated these sample with
previous works (Woo et al., 2002; Xie et al., 2004; Liu et al., 2006; Zhou et
al., 2009; Zhang et al., 2012; Sbarrato et al., 2012; Chai et al., 2012; Leon-
Tavares et al., 2011; Shen et al., 2011; Shaw et al., 2012) to obtain the BH
mass. At last, we selected the sample of blazars with radio flux density and
X-ray flux density which are detected by VLA and Swift or XMM-Newton on
NASA/IPAC Extragalactic Database (NED). The beaming factor of sample in this
paper can be collected by cross-matching with the sources of Nemmen et al.
(2012). In total, we collected a sample containing 91 blazars which includes
70 FSRQs and 21 BL Lacs. All sample of 91 Blazars with redshift (z), ${\rm
log}M_{\rm dyn}$, ${\rm log}L_{\rm X}$, ${\rm log}L_{\rm R}$, ${\rm
log}L_{\rm{bol}}$, ${\rm log}f_{\rm b}$ and radio loudness (R) are listed in
Table 1. We use the collected data to calculate the black hole spin energy
($M_{\rm spin}$) and irreducible black hole mass ($M_{\rm irr}$) of sample for
further fundamental analysis. All sample of Blazars with ${\rm log}L_{\rm j}$,
${\rm log}L_{\rm{EDD}}$, ${\rm log}{\rm F}$, ${\rm log}M_{\rm irr}$, ${\rm
log}M_{\rm spin}$ are listed in Table 3. The sources are arranged in order of
increasing redshift. The source distribution of Blazars’ sample across several
different physical properties is shown in Figure 1. The z and ${\rm logR}$
distributions of the various classes are shown in part a and b of Figure 1.
The redshift distributions for all blazars are 0 $\leq z\leq$ 2.6 and mean
value is 0.93. Mean values for FSRQs and BL Lacs are 1.07 and 0.44. The R
distributions for all blazars are $10^{0.06}$ $\leq R\leq$ $10^{4.96}$ and
mean value is $10^{3.16}$. Mean values for FSRQs and BL Lacs are $10^{3.16}$
and $10^{2.70}$.

###  2.1 Black hole mass

The traditional virial BH mass also known as BH dynamical mass
($\rm{M_{dyn}}$) of AGN usually could be measured by the empirical
relationship between the BLR size and the ionization luminosity, as well as
the measured wide line width, assuming that the BLR cloud is gravitationally
bound by the central black hole (Xiong et al., 2014). In this paper we
collected the BH mass of FSRQs which are estimated by traditional virial
method (Woo et al., 2002; Wang et al., 2004; Liu et al., 2006; Shen et al.,
2011; Chai et al., 2012; Sbarrato et al., 2012; Shaw et al., 2012). If the BH
mass of sample can be attained from different lines, we take the average of BH
masses from different lines. The BH masses we got from Vestergaard et al.
(2006) attained by H $\beta$ and $\rm{C_{IV}}$ lines, and Vestergaard et al.
(2009) attained by Mg II lines. And for the BL Lac objects of sample, the
masses of black holes can be estimated by the $M_{BH}$-$\sigma$ or $M_{BH}$-L
relationship of their host galaxies, where $\sigma$ and L are the stellar
velocity dispersion and the bulge luminosity of the host galaxy (Xiong et al.,
2014; Woo et al., 2002; Zhou et al., 2009; Leon-Tavares et al., 2011; Chai et
al., 2012; Sbarrato et al., 2012; Zhang et al., 2012). There is a little
number of sources BH mass which are estimated by variation time-scale from Xie
et al. (1991, 2004). If there are more than one BH masses we got that belong
to the same sample, then we take the average of those data. The average
difference for BH mass between the method of traditional virial and the
properties of host galaxies from the same source is 0.023. Due to the small
number of black hole masses estimated by varying time-scales, the difference
between the black hole masses obtained from Xie et al. (1991, 2004) and other
methods has little effect on the results. The BH mass distributions of the
various classes are shown in part c of Figure 1. The BH mass distributions for
all blazars are $10^{7.5}$ $\sim$ $10^{10}$ $\rm{M_{\odot}}$ and mean value is
$10^{8.70\pm 0.30}$ $\rm{M_{\odot}}$. Mean values for FSRQs and BL Lacs are
$10^{8.78\pm 0.30}$ $\rm{M_{\odot}}$ and $10^{8.43\pm 0.30}$ $\rm{M_{\odot}}$
.

###  2.2 Intrinsic 5 GHz radio luminosity

In this paper, 5 GHz radio luminosity was calculated by 5 GHz radio flux
density from NED. These radio fluxes were taken from 5 GHz high resolution VLA
observations directly. In order to ensure the accuracy of the collected data
and reduce errors caused by converting through other approximate frequency, we
only collect sources that have accurate 5 GHz radio flux density. At the same
time, we strive to make the observation time of the collected 5 GHz radio data
align as closely as possible with the observation time of other bands,
reducing errors caused by variations in observation time. If there is more
than one observation from the same source, we take the average value of all
observations. For different observations with the same wavelength from the
same source, the difference is usually controlled between 0.03 and 0.12. After
we averaged the difference, the error was basically controlled within 0.05.
Blazars usually have very powerful relativistic jets which includes gamma-ray
band and radio band. For this reason, We should consider the influence of the
beaming effect (Xiong et al., 2014; Zhang et al., 2023a). The intrinsic
luminosity of radio luminosity can be estimated by beaming factor
$f_{\rm{b}}$, $f_{\rm{b}}$$\equiv$1-cos$\theta$ where $\theta$ is the jet
opening angle (Nemmen et al., 2012; Zhang et al., 2017). The uncertainty of
$f_{\rm{b}}$ we took is 0.15 as reported in Nemmen et al. (2012).

$\emph{$f_{\rm{b}}$}=1-\cos{\theta_{\rm{j}}},$ (1)

$\theta_{\rm{j}}$ is the jet opening angle of Blazars. The beaming factor
$f_{\rm{b}}$ in this paper are obtained from Nemmen et al. (2012) and Xiong et
al. (2014). we can measure the intrinsic 5 GHz radio luminosity with beaming
factor.

$\begin{split}L_{\rm{R^{*}}}=L_{\rm{R}}\emph{$f_{\rm{b}}$},\end{split}$ (2)

$L_{\rm{R^{*}}}$ are the intrinsic luminosity of radio luminosity,
respectively. The 5 GHz radio luminosity and beam factor distributions of the
various classes are shown in part d and e of Figure 1. The 5 GHz radio
luminosity distributions for all blazars are $10^{40}$ $\sim$ $10^{46}$ erg
$\rm{s^{-1}}$ and mean value is $10^{43.95\pm 0.05}$ erg $\rm{s^{-1}}$. Mean
values for FSRQs and BL Lacs are $10^{44.24\pm 0.05}$ erg $\rm{s^{-1}}$ and
$10^{42.99\pm 0.05}$ erg $\rm{s^{-1}}$. The beam factor distributions for all
blazars are -4 $\sim$ -1.25 and mean value is -2.61. Mean values for FSRQs and
BL Lacs are -2.72 and -2.21.

###  2.3 2-10 keV X-ray luminosity

To do the fundamental analysis of Blazars, the X-ray luminosity in 2-10 keV is
required. In this paper, the X-ray luminosity in 2-10 keV was calculated by
the X-ray flux density from NASA/IPAC Extragalactic Database (NED). In order
to minimize errors caused by different observation devices, all data of X-ray
flux density we choose are come from the same X-ray observation satellite as
much as possible. Most of Blazars we collected can be observed by Swift with
reliable data. Only a small number of samples did not have Swift observations,
for these sources we collected the data of XMM-Netwon. Most data of flux
density in 2-10 keV can be obtained from the NED. But there are still some
data that do not have the direct observational data in the 2-10 keV. Because
of the limited number of X-ray flux densities with direct observational data
in the 2-10 keV range, we only collected another 0.3-10 keV flux density
directly observed by Swift, and used the power-law model to estimate the 2-10
keV flux density to expand the sample and reduce the error to a certain
extent. The X-ray flux densities in 0.3-10 keV of Blazars from NED can be
converted into 2-10 keV by the power-law model with photon index $\tau$=1.87
for FSRQs (Zhang et al., 2023b). The photon index we chose is the average
photon index of Blazars which is similar to Sahakyan et al. (2021). Due to the
small differences between two energy bands of X-rays, the uncertainty
generated by the spectral index has little impact on the results.

$\displaystyle F_{\rm 2-10keV}=F_{\rm 0.3-10keV}\frac{\int_{\rm 2keV}^{\rm
10keV}\nu^{1-\tau}d\nu}{\int_{\rm 0.3keV}^{\rm 10keV}\nu^{1-\tau}d\nu}$
$\displaystyle=F_{\rm 0.3-10keV}\times 0.5157,$ (3)

where $F_{\rm 2-10keV}$ and $F_{\rm 0.3-10keV}$ respectively are the X-ray
flux density in 2-10 keV and 0.3-10 keV. $\nu$ is the energy dissipation of
FSRQs. The 2-10keV X-ray luminosity distributions of the various classes are
shown in part f of Figure 1. The 2-10keV X-ray luminosity distributions for
all blazars are $10^{42.5}$ $\sim$ $10^{47.5}$ erg $\rm{s^{-1}}$ and mean
value is $10^{45.58\pm 0.35}$ erg $\rm{s^{-1}}$. Mean values for FSRQs and BL
Lacs are $10^{45.80\pm 0.35}$ erg $\rm{s^{-1}}$ and $10^{44.85\pm 0.38}$ erg
$\rm{s^{-1}}$.

###  2.4 Black hole spin energy

The black hole spin energy was also considered as a variable for fundamental
analysis. However, due to the limitations of observation and estimation
methods, it is difficult to measure the spin of large black holes directly by
conventional calculation methods. We measured the $M_{\rm spin}$ of BH with
the method of Daly et al. (2022). The accuracy of this estimation method has
been confirmed by previous work (Daly et al., 2019) which can consistent with
previous works such as the X-ray reflection method (Miller et al., 2009;
Reynolds et al., 2014). To obtain the key data (Spin function F) of this
method, the broad-line luminosity and the beam power of sample in this paper
are required.

####  2.4.1 Bolometric luminosity

In this paper, the bolometric luminosity $L_{\rm bol}$ of our sample were
collected from the previous work by cross matching. $L_{\rm BLR}$ which were
collected from the previous work (Xiong et al., 2014) can be converted into
$L_{\rm bol}$ ($L_{\rm bol}$= 10 $L_{\rm BLR}$) with an average uncertainty of
0.3dex (Calderone et al., 2013). According to Celotti et al. (1997), $L_{\rm
BLR}$ usually were obtained by scaling several strong emission lines to the
quasar template spectrum of Francis et al. (1991), with $L_{\rm y\alpha}$ as a
reference. The luminosity of blazars emission lines in the Sloan Digital Sky
Survey (SDSS) DR7 quasar sample (Xiong et al., 2014) can be converted to the
total luminosity of the broad lines (Sbarrato et al., 2012; Celotti et al.,
1997). A proportions is set to estimate $L_{\rm BLR}$ by these method that the
$L_{\rm y\alpha}$ flux contribution is set to 100 and the relative weights of
H$\alpha$, H$\beta$, $\rm Mg_{II}$ and $\rm C_{IV}$ are 77, 22, 34 and 63,
respectively. If there are more than one line was present, take the average of
$L_{\rm BLR}$ estimated from each line. The uncertainty of ${\rm
log}L_{\rm{bol}}$ we took is 0.32 which is consistent with Calderone et al.
(2013). The $L_{\rm bol}$ distributions of the various classes are shown in
part g of Figure 1. The $L_{\rm bol}$ distributions for all blazars are
$10^{42.5}$ $\sim$ $10^{49.5}$ erg $\rm{s^{-1}}$ and mean value is
$10^{45.79\pm 0.32}$ erg $\rm{s^{-1}}$. Mean values for FSRQs and BL Lacs are
$10^{46.07\pm 0.32}$ erg $\rm{s^{-1}}$ and $10^{44.85\pm 0.32}$ erg
$\rm{s^{-1}}$.

####  2.4.2 Beam power

We used the theoretical model described by Willott et al. (1999) to calculate
the beam power of our sample that replies on the relationships between beam
power and radio luminosity at 151 MHz. For our sample, most of the 151 MHz
radio luminosity directly comes from the Very Long Baseline Interferometry
(VLBI) by cross-matching on NED. Only 6 FSRQ sources 151 MHz flux density has
no direct observational data. Since the samples are all FSRQ, we collected the
radio flux density at 140-160MHz for this sample and set the spectral index
$\alpha=1$ to measure the 151 MHz radio flux density. Because 151 MHz radio
flux density are taken from extended radio flux density of samples. The
correction factor can be negligible because the frequencies (140-160 MHz) are
approximate.

$\displaystyle L_{\rm j}\approx 1.7\times
10^{45}f^{3/2}{(\frac{L_{151}}{10^{44}{\rm erg~{}s}^{-1}})}^{6/7}{\rm
erg~{}s}^{-1},$ (4)

where $L_{151}$ is the radio luminosity at 151 MHz in units of ${\rm
erg~{}s}^{-1}$ and $L_{\rm j}$ is the beam power. Willott et al. (1999)
contended that the normalization is highly uncertain, prompting the
introduction of the factor f to accommodate these uncertainties. The value
range of $f$ is 1$\leq f\leq$ 20 (Cao et al., 2003; Godfrey et al., 2013; Wu
et al., 2008; Fan et al., 2018, 2019). According to analysis result of Daly et
al. (2018) the uncertainty of beam power could be effected by the range of f.
To reduce the uncertainty of our result as much as possible, we used the the
lowest limit $f=1$ to measured the beam power just like Cao et al. (2003). The
$L_{151}$ distributions of the various classes are shown in part h of Figure
1. The $L_{151}$ distributions for all blazars are $10^{39.5}$ $\sim$
$10^{45}$ erg $\rm{s^{-1}}$ and mean value is $10^{42.85\pm 0.03}$ erg
$\rm{s^{-1}}$. Mean values for FSRQs and BL Lacs are $10^{43.17\pm 0.04}$ erg
$\rm{s^{-1}}$ and $10^{41.78\pm 0.03}$ erg $\rm{s^{-1}}$.

####  2.4.3 Spin function F

The spin function can be measured by the following formula (Daly et al., 2022;
Zhang et al., 2023a) with beam power, bolometric luminosity and Eddington
luminosity we got.

$\displaystyle\frac{f(j)}{f_{\rm max}}=(\frac{L_{j}}{g_{j}L_{\rm
Edd}})(\frac{L_{\rm bol}}{g_{\rm bol}L_{\rm Edd}})^{-A},$ (5)

where j is BH spin, $L_{\rm bol}$ is the bolometric luminosity(Netzer et al.,
1990), $L_{\rm Edd}$ is the Eddington luminosity which can be measured by BH
mass, $g_{i}$ = 0.1 and $g_{bol}$ = 1 are directly obtained from Daly et al.
(2019), and A is a parameter defined by Daly et al. (2018), it can be measured
by the coefficient of fundamental plane of BH ($A=a/(a+b)$), where $a$ and $b$
are the value of parameters from Merloni et al. (2003). In this paper, we set
A as 0.43 (Daly et al., 2018). Based on the relationship among the black hole
spin function F, the total mass M of BH (also known as $M_{\rm dyn}$) and the
spin-mass energy ($M_{\rm spin}$) (Blandford et al., 1990; Thorne et al.,
1986; Rees et al., 1984; Misner et al., 1973; Zhang et al., 2023a), we
estimate $M_{\rm spin}$ by the black hole spin function $\rm{F^{2}}$.
M$\equiv$$M_{\rm dyn}$=$M_{\rm irr}$+$E_{\rm spin}$$c^{-2}$=$M_{\rm
irr}$+$M_{\rm spin}$. Daly et al. (2022) showed that

$\displaystyle\frac{M_{\rm irr}}{M_{\rm dyn}}=(F^{2}+1)^{-1/2},$
$\displaystyle\frac{M_{\rm spin}}{M_{\rm dyn}}=1-\frac{M_{\rm irr}}{M_{\rm
dyn}}=[1-(F^{2}+1)^{-1/2}],$ (6)

where F is the spin function, $F^{2}\equiv$f(j)/$f_{\rm max}$ (Daly et al.,
2022), $M_{\rm dyn}$ is the dynamic black hole mass and $M_{\rm irr}$ is the
irreducible black hole mass.

Table 1: The fundamental plane of Blazars

${\rm[1]}$ | ${\rm[2]}$ | ${\rm[3]}$ | ${\rm[4]}$ | ${\rm[5]}$ | ${\rm[6]}$ | ${\rm[7]}$ | ${\rm[8]}$
---|---|---|---|---|---|---|---
${\rm name}$ | type | z | ${\rm log}M_{\rm dyn}$ | ${\rm log}L_{\rm X}$ | ${\rm log}L_{\rm R}$ | ${\rm log}f_{\rm b}$ | ${\rm logR}$
0106+013/4C+01.02 | FSRQ | 2.099 | 8.830 $\pm$ 0.300 | 46.460 $\pm$ 0.415 | 45.234 $\pm$ 0.003 | -3.208 | 3.991
0110+495 | FSRQ | 0.389 | 8.340 $\pm$ 0.300 | 44.567 $\pm$ 0.370 | 42.966 $\pm$ 0.102 | -2.221 | 3.536
0133+476/OC 457 | FSRQ | 0.859 | 8.515 $\pm$ 0.300 | 45.623 $\pm$ 0.414 | 44.462 $\pm$ 0.017 | -2.618 | 3.752
0135-247/PKS0135-247 | FSRQ | 0.835 | 9.120 $\pm$ 0.300 | 45.695 $\pm$ 0.404 | 44.131 $\pm$ 0.022 | -2.623 | 4.200
0234+285/4C +28.07 | FSRQ | 1.213 | 9.220 $\pm$ 0.300 | 46.064 $\pm$ 0.409 | 44.642 $\pm$ 0.007 | -2.516 | 3.502
0235+164/PKS 0235+164 | Bllac | 0.940 | 9.073 $\pm$ 0.300 | 45.882 $\pm$ 0.426 | 44.005 $\pm$ 0.034 | -2.467 | 2.966
0336-019/PKS 0336-01 | FSRQ | 0.852 | 8.935 $\pm$ 0.300 | 45.503 $\pm$ 0.413 | 44.434 $\pm$ 0.005 | -3.032 | 3.584
0403-132/PKS 0403-13 | FSRQ | 0.571 | 9.075 $\pm$ 0.300 | 45.605 $\pm$ 0.320 | 44.108 $\pm$ 0.016 | -2.703 | 3.616
0420-014/PKS 0420-01 | FSRQ | 0.916 | 8.813 $\pm$ 0.300 | 45.827 $\pm$ 0.434 | 44.663 $\pm$ 0.003 | -2.400 | 3.196
0458-020/PKS 0458-020 | FSRQ | 2.291 | 8.965 $\pm$ 0.300 | 46.593 $\pm$ 0.408 | 45.284 $\pm$ 0.011 | -2.741 | 3.996
0528+134 | FSRQ | 2.060 | 9.800 $\pm$ 0.300 | 46.604 $\pm$ 0.423 | 45.031 $\pm$ 0.027 | -2.962 | 4.172
OG 050 | Bllac | 1.254 | 8.430 $\pm$ 0.350 | 45.989 $\pm$ 0.408 | 44.587 $\pm$ 0.026 | -2.886 | 3.739
0537-441/PKS 0537-441 | Bllac | 0.892 | 8.565 $\pm$ 0.300 | 46.173 $\pm$ 0.032 | 44.576 $\pm$ 0.022 | -2.829 | 3.086
0605-085/PKS 0605-085 | FSRQ | 0.872 | 8.628 $\pm$ 0.300 | 45.648 $\pm$ 0.406 | 44.357 $\pm$ 0.022 | -3.009 | 3.087
0637-752/PKS 0637-75 | FSRQ | 0.653 | 9.110 $\pm$ 0.300 | 45.782 $\pm$ 0.421 | 44.404 $\pm$ 0.022 | -2.782 | 3.482
B3 0650+453 | FSRQ | 0.928 | 8.170 $\pm$ 0.340 | 45.248 $\pm$ 0.377 | 43.369 $\pm$ 0.239 | -2.475 | 3.340
0716+714 | Bllac | 0.300 | 8.100 $\pm$ 0.300 | 45.269 $\pm$ 0.431 | 43.151 $\pm$ 0.087 | -2.319 | 2.316
0735+178 | Bllac | 0.424 | 8.300 $\pm$ 0.300 | 44.499 $\pm$ 0.399 | 43.145 $\pm$ 0.061 | -2.504 | 3.058
0736+017/PKS 0736+01 | FSRQ | 0.189 | 8.110 $\pm$ 0.300 | 44.468 $\pm$ 0.424 | 42.864 $\pm$ 0.052 | -2.736 | 3.111
0748+126/PKS 0748+126 | FSRQ | 0.889 | 8.150 $\pm$ 0.300 | 45.837 $\pm$ 0.409 | 44.524 $\pm$ 0.015 | -2.675 | 3.541
0754+100 | Bllac | 0.266 | 8.200 $\pm$ 0.300 | 44.731 $\pm$ 0.418 | 42.865 $\pm$ 0.042 | -2.974 | 2.514
PKS0808+019 | Bllac | 1.148 | 8.500 $\pm$ 0.300 | 45.265 $\pm$ 0.366 | 44.388 $\pm$ 0.035 | -2.615 | 3.583
4C +39.23 | FSRQ | 1.216 | 8.550 $\pm$ 0.300 | 45.544 $\pm$ 0.326 | 44.245 $\pm$ 0.068 | -2.779 | 3.046
OJ 535 | FSRQ | 1.418 | 9.260 $\pm$ 0.300 | 45.902 $\pm$ 0.378 | 44.375 $\pm$ 0.054 | -2.822 | 3.187
0823+033/PKS 0823+033 | Bllac | 0.506 | 8.830 $\pm$ 0.300 | 44.967 $\pm$ 0.407 | 43.699 $\pm$ 0.058 | -2.486 | 2.641
B2 0827+24 | FSRQ | 0.940 | 8.953 $\pm$ 0.300 | 46.004 $\pm$ 0.424 | 44.087 $\pm$ 0.039 | -3.104 | 3.024
PKS 0829+046 | Bllac | 0.174 | 8.630 $\pm$ 0.300 | 43.919 $\pm$ 0.415 | 42.403 $\pm$ 0.073 | -2.111 | 2.126
OJ 451 | FSRQ | 0.249 | 9.680 $\pm$ 0.090 | 43.805 $\pm$ 0.315 | 42.252 $\pm$ 0.087 | -1.972 | 2.367
0836+710/4C+71.07 | FSRQ | 2.172 | 9.360 $\pm$ 0.300 | 47.483 $\pm$ 0.012 | 44.950 $\pm$ 0.031 | -3.198 | 2.859
0851+202/OJ 287 | Bllac | 0.306 | 8.563 $\pm$ 0.300 | 45.009 $\pm$ 0.431 | 43.418 $\pm$ 0.022 | -2.229 | 2.792
S4 0859+470 | FSRQ | 1.466 | 9.250 $\pm$ 0.030 | 45.632 $\pm$ 0.320 | 44.529 $\pm$ 0.011 | -2.863 | 3.645
0906+015/PKS 0906+01 | FSRQ | 1.024 | 9.003 $\pm$ 0.300 | 45.871 $\pm$ 0.415 | 44.115 $\pm$ 0.056 | -2.617 | 2.889
S4 0913+39 | FSRQ | 1.267 | 8.620 $\pm$ 0.150 | 45.193 $\pm$ 0.320 | 43.968 $\pm$ 0.074 | -2.727 | 3.237
B3 0917+449 | FSRQ | 2.190 | 9.283 $\pm$ 0.300 | 46.758 $\pm$ 0.406 | 44.740 $\pm$ 0.051 | -2.893 | 2.792
OK 630 | FSRQ | 1.453 | 8.930 $\pm$ 0.160 | 45.819 $\pm$ 0.375 | 44.557 $\pm$ 0.037 | -2.760 | 3.678
4C +40.24 | FSRQ | 1.249 | 8.950 $\pm$ 0.060 | 45.717 $\pm$ 0.375 | 44.487 $\pm$ 0.074 | -3.275 | 3.471
0953+254/B2 0954+25A | FSRQ | 0.707 | 8.876 $\pm$ 0.300 | 45.256 $\pm$ 0.405 | 43.602 $\pm$ 0.099 | -2.555 | 3.252
4C+55.17 | FSRQ | 0.896 | 8.338 $\pm$ 0.300 | 45.159 $\pm$ 0.023 | 43.764 $\pm$ 0.020 | -2.800 | 2.848
4C+23.24 | FSRQ | 0.566 | 8.510 $\pm$ 0.098 | 45.098 $\pm$ 0.362 | 43.636 $\pm$ 0.079 | -2.483 | 3.117
1ES 1011+496 | Bllac | 0.212 | 8.300 $\pm$ 0.300 | 45.398 $\pm$ 0.429 | 42.015 $\pm$ 0.259 | -1.987 | 1.627
1055+018/PKS 1055+01 | Bllac | 0.888 | 8.410 $\pm$ 0.300 | 45.881 $\pm$ 0.422 | 44.605 $\pm$ 0.015 | -2.190 | 3.660
Mkn 421 | Bllac | 0.03 | 8.153 $\pm$ 0.300 | 44.990 $\pm$ 0.320 | 40.399 $\pm$ 0.066 | -1.459 | 0.060
1127-145 | FSRQ | 1.184 | 9.180 $\pm$ 0.300 | 46.657 $\pm$ 0.017 | 44.975 $\pm$ 0.039 | -2.984 | 3.365
4C+29.45 | FSRQ | 0.724 | 8.499 $\pm$ 0.300 | 45.458 $\pm$ 0.419 | 43.911 $\pm$ 0.045 | -3.100 | 3.323
1202-262/PKS 1203-26 | FSRQ | 0.789 | 8.795 $\pm$ 0.300 | 45.602 $\pm$ 0.320 | 43.937 $\pm$ 0.015 | -2.660 | 3.982
1219+285 | Bllac | 0.102 | 7.700 $\pm$ 0.300 | 43.606 $\pm$ 0.425 | 41.563 $\pm$ 0.163 | -1.850 | 1.514
4C+04.42 | FSRQ | 0.965 | 8.305 $\pm$ 0.300 | 45.956 $\pm$ 0.409 | 43.754 $\pm$ 0.078 | -2.607 | 1.702
4C+21.35 | FSRQ | 0.432 | 8.737 $\pm$ 0.300 | 45.263 $\pm$ 0.320 | 43.225 $\pm$ 0.116 | -3.609 | 2.580
1226+023/3C 273 | FSRQ | 0.158 | 8.510 $\pm$ 0.300 | 45.723 $\pm$ 0.430 | 44.087 $\pm$ 0.005 | -2.581 | 2.580
1253-055/3C 279 | FSRQ | 0.536 | 8.503 $\pm$ 0.300 | 45.922 $\pm$ 0.431 | 44.660 $\pm$ 0.005 | -2.937 | 3.451
1308+326/B2 1308+32 | FSRQ | 0.996 | 8.478 $\pm$ 0.300 | 45.823 $\pm$ 0.414 | 44.185 $\pm$ 0.018 | -2.986 | 3.982
B2 1315+34A | FSRQ | 1.056 | 9.215 $\pm$ 0.300 | 45.208 $\pm$ 0.308 | 43.556 $\pm$ 0.200 | -2.568 | 3.464
B2 1324+22 | FSRQ | 1.4 | 9.245 $\pm$ 0.300 | 46.011 $\pm$ 0.150 | 44.426 $\pm$ 0.047 | -2.376 | 3.033
1334-127/PKS 1335-127 | FSRQ | 0.539 | 7.980 $\pm$ 0.300 | 45.580 $\pm$ 0.416 | 44.195 $\pm$ 0.018 | -2.611 | 3.516
B3 1343+451 | FSRQ | 2.534 | 8.980 $\pm$ 0.990 | 46.398 $\pm$ 0.043 | 44.363 $\pm$ 0.157 | -2.733 | 3.543
S5 1357+76 | FSRQ | 1.585 | 8.255 $\pm$ 0.210 | 45.409 $\pm$ 0.339 | 44.106 $\pm$ 0.123 | -2.726 | 3.501
B3 1417+385 | FSRQ | 1.831 | 8.620 $\pm$ 0.250 | 45.799 $\pm$ 0.387 | 44.713 $\pm$ 0.039 | -2.759 | 3.639
PKS 1434+235 | FSRQ | 1.548 | 8.375 $\pm$ 0.300 | 45.776 $\pm$ 0.324 | 44.237 $\pm$ 0.087 | -2.756 | 3.739
PKS 1502+106 | FSRQ | 1.839 | 9.107 $\pm$ 0.300 | 46.221 $\pm$ 0.410 | 44.609 $\pm$ 0.051 | -2.665 | 3.355

Note: The first column is name; the second column is type of blazars; the
third column is redshift; the fourth column is the Logarithm of Black hole
dynamical mass; the fifth column is Logarithm of x-ray luminosity in 2-10 keV
(in units of ${\rm erg~{}s^{-1}}$); the sixth column is Logarithm of 5 GHz
radio luminosity (in units of ${\rm erg~{}s^{-1}}$); the seventh column is
Logarithm of the beaming factor; the eighth column is Logarithm of the radio
loudness.

Table 2: continued

${\rm[1]}$ | ${\rm[2]}$ | ${\rm[3]}$ | ${\rm[4]}$ | ${\rm[5]}$ | ${\rm[6]}$ | ${\rm[7]}$ | ${\rm[8]}$
---|---|---|---|---|---|---|---
${\rm name}$ | type | z | ${\rm log}M_{\rm dyn}$ | ${\rm log}L_{\rm X}$ | ${\rm log}L_{\rm R}$ | ${\rm log}f_{\rm b}$ | ${\rm logR}$
1508-055 | FSRQ | 1.185 | 8.970 $\pm$ 0.300 | 45.578 $\pm$ 0.035 | 44.606 $\pm$ 0.022 | -2.912 | 3.335
1510-089/PKS 1510-08 | FSRQ | 0.361 | 8.363 $\pm$ 0.300 | 45.426 $\pm$ 0.028 | 43.709 $\pm$ 0.031 | -2.929 | 3.259
1514-241 | Bllac | 0.049 | 7.650 $\pm$ 0.300 | 43.224 $\pm$ 0.419 | 41.912 $\pm$ 0.022 | -1.776 | 3.336
B2 1520+31 | FSRQ | 1.484 | 8.920 $\pm$ 0.310 | 45.506 $\pm$ 0.342 | 44.089 $\pm$ 0.113 | -2.606 | 1.344
1546+027/PKS 1546+027 | FSRQ | 0.414 | 8.618 $\pm$ 0.300 | 44.918 $\pm$ 0.422 | 43.490 $\pm$ 0.035 | -2.336 | 3.628
4C+05.64 | FSRQ | 1.422 | 9.180 $\pm$ 0.300 | 45.972 $\pm$ 0.386 | 44.908 $\pm$ 0.090 | -2.897 | 3.081
4C+10.45 | FSRQ | 1.226 | 8.970 $\pm$ 0.300 | 45.900 $\pm$ 0.434 | 44.316 $\pm$ 0.039 | -2.854 | 4.001
1611+343/B2 1611+34 | FSRQ | 1.397 | 9.343 $\pm$ 0.300 | 45.899 $\pm$ 0.405 | 44.847 $\pm$ 0.018 | -2.108 | 3.421
1622-297 | FSRQ | 0.815 | 9.100 $\pm$ 0.300 | 45.408 $\pm$ 0.320 | 44.228 $\pm$ 0.026 | -2.721 | 3.371
1633+382/4C+38.41 | FSRQ | 1.814 | 9.369 $\pm$ 0.300 | 46.718 $\pm$ 0.427 | 45.090 $\pm$ 0.024 | -3.287 | 2.910
4C +47.44 | FSRQ | 0.735 | 8.565 $\pm$ 0.475 | 45.330 $\pm$ 0.371 | 43.655 $\pm$ 0.077 | -2.564 | 3.455
Mkn 501 | Bllac | 0.034 | 8.837 $\pm$ 0.300 | 44.210 $\pm$ 0.431 | 41.236 $\pm$ 0.053 | -1.613 | 3.596
B3 1708+433 | FSRQ | 1.027 | 7.920 $\pm$ 0.350 | 45.110 $\pm$ 0.314 | 43.264 $\pm$ 0.372 | -2.350 | 0.819
1726+455 | FSRQ | 0.717 | 8.220 $\pm$ 0.300 | 45.054 $\pm$ 0.395 | 43.826 $\pm$ 0.050 | -2.532 | 2.870
1730-130 | FSRQ | 0.902 | 9.300 $\pm$ 0.300 | 45.615 $\pm$ 0.418 | 44.851 $\pm$ 0.007 | -3.934 | 3.013
1739+522 | FSRQ | 1.375 | 9.320 $\pm$ 0.300 | 46.028 $\pm$ 0.402 | 44.404 $\pm$ 0.086 | -2.718 | 4.961
4C+09.57/PKS1749+096 | Bllac | 0.322 | 8.413 $\pm$ 0.300 | 45.019 $\pm$ 0.422 | 43.449 $\pm$ 0.026 | -2.097 | 2.845
CGRaBS J1800+7828/S5 1803+78 | Bllac | 0.68 | 8.260 $\pm$ 0.300 | 45.498 $\pm$ 0.419 | 44.133 $\pm$ 0.022 | -2.248 | 3.179
3C 371 | Bllac | 0.051 | 8.605 $\pm$ 0.300 | 42.956 $\pm$ 0.423 | 41.524 $\pm$ 0.033 | -1.775 | 3.092
1823+568/4C+56.27 | Bllac | 0.664 | 9.260 $\pm$ 0.300 | 45.458 $\pm$ 0.407 | 43.763 $\pm$ 0.016 | -2.319 | 1.144
1849+670/CGRaBS J1849+6705 | FSRQ | 0.657 | 9.140 $\pm$ 0.300 | 45.270 $\pm$ 0.411 | 43.845 $\pm$ 0.040 | -2.412 | 3.047
1921-293/PKS B1921-293 | FSRQ | 0.353 | 8.695 $\pm$ 0.300 | 45.320 $\pm$ 0.414 | 44.341 $\pm$ 0.005 | -2.729 | 3.085
2145+067/4C+06.69 | FSRQ | 0.99 | 8.870 $\pm$ 0.300 | 46.225 $\pm$ 0.422 | 44.892 $\pm$ 0.008 | -2.097 | 4.042
2155-152/PKS 2155-152 | FSRQ | 0.672 | 7.590 $\pm$ 0.300 | 45.307 $\pm$ 0.414 | 44.237 $\pm$ 0.022 | -2.703 | 3.250
BL Lac | Bllac | 0.069 | 8.210 $\pm$ 0.300 | 43.828 $\pm$ 0.320 | 42.048 $\pm$ 0.030 | -1.767 | 3.861
PKS 2209+236 | FSRQ | 1.125 | 8.460 $\pm$ 0.450 | 45.217 $\pm$ 0.398 | 44.150 $\pm$ 0.058 | -2.597 | 2.265
2223-052/3C 446 | FSRQ | 1.404 | 8.417 $\pm$ 0.300 | 43.176 $\pm$ 0.320 | 45.186 $\pm$ 0.011 | -2.636 | 4.115
PKS 2227-08 | FSRQ | 1.56 | 8.785 $\pm$ 0.300 | 46.420 $\pm$ 0.402 | 44.845 $\pm$ 0.014 | -2.301 | 3.712
2230+114/CTA 102 | FSRQ | 1.037 | 8.780 $\pm$ 0.300 | 46.268 $\pm$ 0.320 | 44.937 $\pm$ 0.036 | -2.676 | 3.331
2251+158/3C 454.3 | FSRQ | 0.859 | 8.825 $\pm$ 0.300 | 46.884 $\pm$ 0.320 | 45.013 $\pm$ 0.013 | -2.881 | 3.677
TXS 2331+073 | FSRQ | 0.401 | 8.370 $\pm$ 0.500 | 44.780 $\pm$ 0.408 | 43.255 $\pm$ 0.023 | -2.281 | 3.495
2345-16/PKS 2345-16 | FSRQ | 0.576 | 8.595 $\pm$ 0.300 | 45.369 $\pm$ 0.413 | 44.007 $\pm$ 0.022 | -2.631 | 2.974

Note: The first column is name; the second column is type of blazars; the
third column is redshift; the fourth column is the Logarithm of Black hole
dynamical mass; the fifth column is Logarithm of x-ray luminosity in 2-10 keV
(in units of ${\rm erg~{}s^{-1}}$); the sixth column is Logarithm of 5 GHz
radio luminosity (in units of ${\rm erg~{}s^{-1}}$); the seventh column is
Logarithm of the beaming factor; the eighth column is Logarithm of the radio
loudness.

Table 3: The BH spin energy of Blazars

${\rm[1]}$ | ${\rm[2]}$ | ${\rm[3]}$ | ${\rm[4]}$ | ${\rm[5]}$ | ${\rm[6]}$ | ${\rm[7]}$ | ${\rm[8]}$
---|---|---|---|---|---|---|---
${\rm name}$ | ${\rm log}L_{\rm{151}}$ | ${\rm log}L_{\rm j}$ | ${\rm log}L_{\rm{Edd}}$ | ${\rm log}L_{\rm{bol}}$ | ${\rm log}{\rm F}$ | ${\rm log}M_{\rm spin}$ | ${\rm log}M_{\rm irr}$
0106+013/4C+01.02 | 44.378 $\pm$ 0.036 | 45.171 $\pm$ 0.031 | 46.944 $\pm$ 0.300 | 47.135 | -0.428 $\pm$ 0.129 | 8.767 $\pm$ 0.243 | 7.962 $\pm$ -0.613
0110+495 | 42.027 $\pm$ 0.036 | 43.540 $\pm$ 0.031 | 46.454 $\pm$ 0.300 | 45.780 | -0.812 $\pm$ 0.129 | 8.335 $\pm$ 0.285 | 6.407 $\pm$ -1.155
0133+476/OC 457 | 42.877 $\pm$ 0.022 | 44.268 $\pm$ 0.019 | 46.629 $\pm$ 0.300 | 45.444 | -0.426 $\pm$ 0.141 | 8.486 $\pm$ 0.263 | 7.320 $\pm$ -0.790
0135-247/PKS0135-247 | 43.415 $\pm$ 0.036 | 44.729 $\pm$ 0.031 | 47.234 $\pm$ 0.300 | 46.343 | -0.561 $\pm$ 0.129 | 9.104 $\pm$ 0.273 | 7.673 $\pm$ -0.912
0234+285/4C +28.07 | 43.451 $\pm$ 0.036 | 44.760 $\pm$ 0.031 | 47.334 $\pm$ 0.300 | 46.244 | -0.553 $\pm$ 0.129 | 9.204 $\pm$ 0.272 | 7.789 $\pm$ -0.905
0235+164/PKS 0235+164 | 42.839 $\pm$ 0.036 | 44.235 $\pm$ 0.031 | 47.187 $\pm$ 0.300 | 44.920 | -0.489 $\pm$ 0.129 | 9.052 $\pm$ 0.268 | 7.762 $\pm$ -0.844
0336-019/PKS 0336-01 | 42.903 $\pm$ 0.036 | 44.290 $\pm$ 0.031 | 47.049 $\pm$ 0.300 | 46.000 | -0.654 $\pm$ 0.129 | 8.925 $\pm$ 0.278 | 7.310 $\pm$ -1.001
0403-132/PKS 0403-13 | 43.272 $\pm$ 0.036 | 44.607 $\pm$ 0.031 | 47.189 $\pm$ 0.300 | 46.250 | -0.589 $\pm$ 0.129 | 9.061 $\pm$ 0.274 | 7.575 $\pm$ -0.939
0420-014/PKS 0420-01 | 43.049 $\pm$ 0.036 | 44.415 $\pm$ 0.031 | 46.927 $\pm$ 0.300 | 45.901 | -0.535 $\pm$ 0.129 | 8.796 $\pm$ 0.271 | 7.415 $\pm$ -0.888
0458-020/PKS 0458-020 | 44.385 $\pm$ 0.036 | 45.560 $\pm$ 0.031 | 47.079 $\pm$ 0.300 | 46.304 | -0.093 $\pm$ 0.129 | 8.856 $\pm$ 0.224 | 8.312 $\pm$ -0.492
0528+134 | 43.443 $\pm$ 0.036 | 44.753 $\pm$ 0.031 | 47.914 $\pm$ 0.300 | 49.650 | -1.454 $\pm$ 0.129 | 9.800 $\pm$ 0.296 | 6.591 $\pm$ -1.790
OG 050 | 43.407 $\pm$ 0.036 | 44.722 $\pm$ 0.031 | 46.544 $\pm$ 0.350 | 45.864 | -0.265 $\pm$ 0.129 | 8.374 $\pm$ 0.298 | 7.515 $\pm$ -0.385
0537-441/PKS 0537-441 | 43.297 $\pm$ 0.036 | 44.628 $\pm$ 0.031 | 46.679 $\pm$ 0.300 | 46.020 | -0.384 $\pm$ 0.129 | 8.531 $\pm$ 0.259 | 7.445 $\pm$ -0.746
0605-085/PKS 0605-085 | 43.257 $\pm$ 0.036 | 44.593 $\pm$ 0.031 | 46.741 $\pm$ 0.300 | 45.600 | -0.329 $\pm$ 0.129 | 8.584 $\pm$ 0.254 | 7.604 $\pm$ -0.695
0637-752/PKS 0637-75 | 43.737 $\pm$ 0.036 | 45.005 $\pm$ 0.031 | 47.224 $\pm$ 0.300 | 46.228 | -0.395 $\pm$ 0.129 | 9.077 $\pm$ 0.260 | 7.969 $\pm$ -0.757
B3 0650+453 | 42.552 $\pm$ 0.057 | 43.989 $\pm$ 0.049 | 46.284 $\pm$ 0.340 | 45.260 | -0.427 $\pm$ 0.111 | 8.142 $\pm$ 0.303 | 6.972 $\pm$ -0.655
0716+714 | 41.993 $\pm$ 0.007 | 43.510 $\pm$ 0.006 | 46.214 $\pm$ 0.300 | 45.766 | -0.756 $\pm$ 0.154 | 8.093 $\pm$ 0.283 | 6.278 $\pm$ -1.112
0735+178 | 42.386 $\pm$ 0.036 | 43.847 $\pm$ 0.031 | 46.414 $\pm$ 0.300 | 47.600 | -1.038 $\pm$ 0.129 | 8.298 $\pm$ 0.291 | 5.920 $\pm$ -1.378
0736+017/PKS 0736+01 | 41.635 $\pm$ 0.036 | 43.203 $\pm$ 0.031 | 46.224 $\pm$ 0.300 | 45.195 | -0.789 $\pm$ 0.129 | 8.104 $\pm$ 0.284 | 6.223 $\pm$ -1.133
0748+126/PKS 0748+126 | 43.010 $\pm$ 0.036 | 44.382 $\pm$ 0.031 | 46.264 $\pm$ 0.300 | 45.950 | -0.374 $\pm$ 0.129 | 8.114 $\pm$ 0.258 | 7.048 $\pm$ -0.736
0754+100 | 41.417 $\pm$ 0.036 | 43.017 $\pm$ 0.031 | 46.314 $\pm$ 0.300 | 47.500 | -1.404 $\pm$ 0.129 | 8.200 $\pm$ 0.296 | 5.091 $\pm$ -1.740
PKS0808+019 | 42.817 $\pm$ 0.036 | 44.216 $\pm$ 0.031 | 46.614 $\pm$ 0.300 | 44.623 | -0.271 $\pm$ 0.129 | 8.445 $\pm$ 0.247 | 7.574 $\pm$ -0.644
4C +39.23 | 43.760 $\pm$ 0.016 | 45.025 $\pm$ 0.013 | 46.664 $\pm$ 0.300 | 45.834 | -0.141 $\pm$ 0.147 | 8.459 $\pm$ 0.232 | 7.828 $\pm$ -0.540
OJ 535 | 43.615 $\pm$ 0.017 | 44.900 $\pm$ 0.015 | 47.374 $\pm$ 0.300 | 46.302 | -0.506 $\pm$ 0.146 | 9.240 $\pm$ 0.269 | 7.916 $\pm$ -0.868
0823+033/PKS 0823+033 | 42.039 $\pm$ 0.036 | 43.550 $\pm$ 0.031 | 46.944 $\pm$ 0.300 | 44.374 | -0.644 $\pm$ 0.129 | 8.819 $\pm$ 0.277 | 7.224 $\pm$ -0.992
B2 0827+24 | 42.679 $\pm$ 0.036 | 44.098 $\pm$ 0.031 | 47.066 $\pm$ 0.300 | 45.989 | -0.752 $\pm$ 0.129 | 8.946 $\pm$ 0.282 | 7.136 $\pm$ -1.097
PKS 0829+046 | 41.317 $\pm$ 0.036 | 42.931 $\pm$ 0.031 | 46.744 $\pm$ 0.300 | 43.568 | -0.724 $\pm$ 0.129 | 8.622 $\pm$ 0.281 | 6.870 $\pm$ -1.069
OJ 451 | 41.105 $\pm$ 0.069 | 42.749 $\pm$ 0.059 | 47.794 $\pm$ 0.090 | 44.070 | -1.222 $\pm$ 0.101 | 9.679 $\pm$ 0.083 | 6.934 $\pm$ -2.677
0836+710/4C+71.07 | 44.378 $\pm$ 0.008 | 45.555 $\pm$ 0.007 | 47.474 $\pm$ 0.300 | 47.430 | -0.450 $\pm$ 0.153 | 9.334 $\pm$ 0.266 | 8.120 $\pm$ -0.819
0851+202/OJ 287 | 41.489 $\pm$ 0.036 | 43.078 $\pm$ 0.031 | 46.677 $\pm$ 0.300 | 44.580 | -0.849 $\pm$ 0.129 | 8.559 $\pm$ 0.286 | 6.559 $\pm$ -1.191
S4 0859+470 | 43.945 $\pm$ 0.010 | 45.183 $\pm$ 0.009 | 47.364 $\pm$ 0.030 | 46.264 | -0.354 $\pm$ 0.151 | 9.211 $\pm$ -0.021 | 8.183 $\pm$ -0.389
0906+015/PKS 0906+01 | 42.962 $\pm$ 0.036 | 44.341 $\pm$ 0.031 | 47.117 $\pm$ 0.300 | 46.100 | -0.669 $\pm$ 0.129 | 8.994 $\pm$ 0.279 | 7.349 $\pm$ -1.017
S4 0913+39 | 43.498 $\pm$ 0.021 | 44.800 $\pm$ 0.018 | 46.734 $\pm$ 0.150 | 45.804 | -0.267 $\pm$ 0.143 | 8.564 $\pm$ 0.092 | 7.701 $\pm$ -0.339
B3 0917+449 | 43.867 $\pm$ 0.034 | 45.117 $\pm$ 0.029 | 47.397 $\pm$ 0.300 | 46.850 | -0.523 $\pm$ 0.132 | 9.265 $\pm$ 0.270 | 7.909 $\pm$ -0.877
OK 630 | 43.334 $\pm$ 0.015 | 44.660 $\pm$ 0.013 | 47.044 $\pm$ 0.160 | 46.055 | -0.479 $\pm$ 0.147 | 8.907 $\pm$ 0.125 | 7.636 $\pm$ -0.691
4C +40.24 | 43.615 $\pm$ 0.036 | 44.900 $\pm$ 0.031 | 47.064 $\pm$ 0.060 | 46.502 | -0.461 $\pm$ 0.129 | 8.925 $\pm$ 0.020 | 7.690 $\pm$ -0.459
0953+254/B2 0954+25A | 42.311 $\pm$ 0.061 | 43.783 $\pm$ 0.053 | 46.990 $\pm$ 0.300 | 45.950 | -0.880 $\pm$ 0.108 | 8.873 $\pm$ 0.286 | 6.810 $\pm$ -1.211
4C+55.17 | 43.535 $\pm$ 0.007 | 44.832 $\pm$ 0.006 | 46.451 $\pm$ 0.300 | 45.574 | -0.121 $\pm$ 0.155 | 8.239 $\pm$ 0.230 | 7.644 $\pm$ -0.527
4C+23.24 | 42.771 $\pm$ 0.020 | 44.177 $\pm$ 0.017 | 46.623 $\pm$ 0.098 | 45.893 | -0.566 $\pm$ 0.143 | 8.494 $\pm$ 0.067 | 7.053 $\pm$ -0.744
1ES 1011+496 | 41.067 $\pm$ 0.028 | 42.717 $\pm$ 0.024 | 46.414 $\pm$ 0.300 | 46.505 | -1.368 $\pm$ 0.136 | 8.300 $\pm$ 0.296 | 5.262 $\pm$ -1.708
1055+018/PKS 1055+01 | 43.494 $\pm$ 0.036 | 44.796 $\pm$ 0.031 | 46.524 $\pm$ 0.300 | 45.509 | -0.146 $\pm$ 0.129 | 8.320 $\pm$ 0.231 | 7.681 $\pm$ -0.535
Mkn 421 | 39.712 $\pm$ 0.028 | 41.555 $\pm$ 0.024 | 46.266 $\pm$ 0.300 | 42.699 | -1.089 $\pm$ 0.137 | 8.151 $\pm$ 0.292 | 5.672 $\pm$ -1.431
1127-145 | 43.664 $\pm$ 0.036 | 44.942 $\pm$ 0.031 | 47.294 $\pm$ 0.300 | 46.770 | -0.563 $\pm$ 0.129 | 9.164 $\pm$ 0.273 | 7.729 $\pm$ -0.914
4C+29.45 | 43.151 $\pm$ 0.020 | 44.502 $\pm$ 0.017 | 46.613 $\pm$ 0.300 | 45.710 | -0.361 $\pm$ 0.143 | 8.461 $\pm$ 0.258 | 7.419 $\pm$ -0.732
1202-262/PKS 1203-26 | 43.428 $\pm$ 0.036 | 44.740 $\pm$ 0.031 | 46.909 $\pm$ 0.300 | 45.070 | -0.189 $\pm$ 0.129 | 8.719 $\pm$ 0.237 | 8.001 $\pm$ -0.572
1219+285 | 40.336 $\pm$ 0.053 | 42.090 $\pm$ 0.045 | 45.814 $\pm$ 0.300 | 43.250 | -0.811 $\pm$ 0.115 | 7.695 $\pm$ 0.284 | 5.770 $\pm$ -1.147
4C+04.42 | 43.230 $\pm$ 0.036 | 44.571 $\pm$ 0.031 | 46.419 $\pm$ 0.300 | 45.857 | -0.303 $\pm$ 0.129 | 8.257 $\pm$ 0.251 | 7.325 $\pm$ -0.673
4C+21.35 | 42.783 $\pm$ 0.018 | 44.187 $\pm$ 0.015 | 46.851 $\pm$ 0.300 | 46.209 | -0.694 $\pm$ 0.145 | 8.728 $\pm$ 0.280 | 7.035 $\pm$ -1.048
1226+023/3C 273 | 42.983 $\pm$ 0.036 | 44.359 $\pm$ 0.031 | 46.624 $\pm$ 0.300 | 46.540 | -0.615 $\pm$ 0.129 | 8.498 $\pm$ 0.276 | 6.961 $\pm$ -0.964
1253-055/3C 279 | 43.557 $\pm$ 0.036 | 44.851 $\pm$ 0.031 | 46.616 $\pm$ 0.300 | 45.610 | -0.166 $\pm$ 0.129 | 8.420 $\pm$ 0.234 | 7.742 $\pm$ -0.553
1308+326/B2 1308+32 | 43.062 $\pm$ 0.019 | 44.426 $\pm$ 0.017 | 46.591 $\pm$ 0.300 | 46.090 | -0.475 $\pm$ 0.144 | 8.454 $\pm$ 0.267 | 7.192 $\pm$ -0.838
B2 1315+34A | 43.105 $\pm$ 0.036 | 44.463 $\pm$ 0.030 | 47.329 $\pm$ 0.300 | 46.070 | -0.662 $\pm$ 0.130 | 9.205 $\pm$ 0.278 | 7.574 $\pm$ -1.010
B2 1324+22 | 42.928 $\pm$ 0.036 | 44.311 $\pm$ 0.031 | 47.359 $\pm$ 0.300 | 45.896 | -0.709 $\pm$ 0.129 | 9.237 $\pm$ 0.280 | 7.513 $\pm$ -1.055
1334-127/PKS 1335-127 | 42.782 $\pm$ 0.036 | 44.187 $\pm$ 0.031 | 46.094 $\pm$ 0.300 | 45.180 | -0.257 $\pm$ 0.129 | 7.922 $\pm$ 0.246 | 7.077 $\pm$ -0.631
B3 1343+451 | 43.223 $\pm$ 0.089 | 44.565 $\pm$ 0.076 | 47.094 $\pm$ 0.990 | 46.120 | -0.555 $\pm$ 0.084 | 8.964 $\pm$ 0.971 | 7.544 $\pm$ -2.038
S5 1357+76 | 42.802 $\pm$ 0.064 | 44.204 $\pm$ 0.055 | 46.369 $\pm$ 0.210 | 45.200 | -0.331 $\pm$ 0.105 | 8.212 $\pm$ 0.160 | 7.227 $\pm$ -0.351
B3 1417+385 | 43.197 $\pm$ 0.036 | 44.542 $\pm$ 0.031 | 46.734 $\pm$ 0.250 | 46.104 | -0.461 $\pm$ 0.129 | 8.595 $\pm$ 0.214 | 7.361 $\pm$ -0.694
PKS 1434+235 | 43.078 $\pm$ 0.062 | 44.440 $\pm$ 0.053 | 46.489 $\pm$ 0.300 | 45.775 | -0.371 $\pm$ 0.107 | 8.339 $\pm$ 0.257 | 7.277 $\pm$ -0.724
PKS 1502+106 | 43.488 $\pm$ 0.036 | 44.792 $\pm$ 0.031 | 47.221 $\pm$ 0.300 | 46.297 | -0.516 $\pm$ 0.129 | 9.087 $\pm$ 0.270 | 7.745 $\pm$ -0.869

Note: The first column is name; the second column is Logarithm of 151 MHz
radio luminosity; the third column is Logarithm of beam power; the fourth
column is the Logarithm of Eddington luminosity; the fifth column is Logarithm
of the beaming factor; the sixth column is the Logarithm of the spin function;
the seventh column is the Logarithm of black hole spin-mass energy; the eighth
column is Logarithm of the black hole irreducible mass.

Table 4: continued

${\rm[1]}$ | ${\rm[2]}$ | ${\rm[3]}$ | ${\rm[4]}$ | ${\rm[5]}$ | ${\rm[6]}$ | ${\rm[7]}$ | ${\rm[8]}$
---|---|---|---|---|---|---|---
${\rm name}$ | ${\rm log}L_{\rm{151}}$ | ${\rm log}L_{\rm j}$ | ${\rm log}L_{\rm{Edd}}$ | ${\rm log}L_{\rm{bol}}$ | ${\rm log}{\rm F}$ | ${\rm log}M_{\rm spin}$ | ${\rm log}M_{\rm irr}$
1508-055 | 44.229 $\pm$ 0.036 | 45.426 $\pm$ 0.031 | 47.084 $\pm$ 0.300 | 46.520 | -0.208 $\pm$ 0.129 | 8.899 $\pm$ 0.240 | 8.147 $\pm$ -0.588
1510-089/PKS 1510-08 | 42.587 $\pm$ 0.036 | 44.019 $\pm$ 0.031 | 46.476 $\pm$ 0.300 | 45.750 | -0.573 $\pm$ 0.129 | 8.347 $\pm$ 0.273 | 6.894 $\pm$ -0.923
1514-241 | 40.254 $\pm$ 0.036 | 42.020 $\pm$ 0.031 | 45.764 $\pm$ 0.300 | 44.500 | -1.100 $\pm$ 0.129 | 7.649 $\pm$ 0.292 | 5.146 $\pm$ -1.439
B2 1520+31 | 43.048 $\pm$ 0.053 | 44.415 $\pm$ 0.045 | 47.034 $\pm$ 0.310 | 45.900 | -0.566 $\pm$ 0.115 | 8.905 $\pm$ 0.283 | 7.464 $\pm$ -0.998
1546+027/PKS 1546+027 | 41.986 $\pm$ 0.036 | 43.504 $\pm$ 0.031 | 46.731 $\pm$ 0.300 | 45.670 | -0.885 $\pm$ 0.129 | 8.614 $\pm$ 0.287 | 6.540 $\pm$ -1.227
4C+05.64 | 43.754 $\pm$ 0.036 | 45.020 $\pm$ 0.031 | 47.294 $\pm$ 0.300 | 46.057 | -0.371 $\pm$ 0.129 | 9.144 $\pm$ 0.258 | 8.082 $\pm$ -0.734
4C+10.45 | 43.635 $\pm$ 0.036 | 44.918 $\pm$ 0.031 | 47.084 $\pm$ 0.300 | 46.006 | -0.351 $\pm$ 0.129 | 8.931 $\pm$ 0.256 | 7.907 $\pm$ -0.716
1611+343/B2 1611+34 | 43.635 $\pm$ 0.019 | 44.917 $\pm$ 0.016 | 47.456 $\pm$ 0.300 | 46.870 | -0.643 $\pm$ 0.144 | 9.332 $\pm$ 0.278 | 7.738 $\pm$ -0.999
1622-297 | 43.201 $\pm$ 0.036 | 44.545 $\pm$ 0.031 | 47.214 $\pm$ 0.300 | 48.780 | -1.171 $\pm$ 0.129 | 9.099 $\pm$ 0.293 | 6.456 $\pm$ -1.509
1633+382/4C+38.41 | 43.833 $\pm$ 0.023 | 45.087 $\pm$ 0.020 | 47.483 $\pm$ 0.300 | 46.820 | -0.555 $\pm$ 0.141 | 9.353 $\pm$ 0.273 | 7.933 $\pm$ -0.912
4C +47.44 | 43.102 $\pm$ 0.013 | 44.460 $\pm$ 0.011 | 46.679 $\pm$ 0.475 | 45.581 | -0.373 $\pm$ 0.149 | 8.529 $\pm$ 0.438 | 7.463 $\pm$ -0.768
Mkn 501 | 39.879 $\pm$ 0.024 | 41.698 $\pm$ 0.021 | 46.951 $\pm$ 0.300 | 43.204 | -1.321 $\pm$ 0.140 | 8.836 $\pm$ 0.295 | 5.894 $\pm$ -1.663
B3 1708+433 | 42.410 $\pm$ 0.068 | 43.867 $\pm$ 0.058 | 46.034 $\pm$ 0.350 | 45.032 | -0.368 $\pm$ 0.102 | 7.883 $\pm$ 0.308 | 6.828 $\pm$ -0.504
1726+455 | 42.192 $\pm$ 0.036 | 43.681 $\pm$ 0.031 | 46.334 $\pm$ 0.300 | 45.850 | -0.722 $\pm$ 0.129 | 8.212 $\pm$ 0.281 | 6.463 $\pm$ -1.068
1730-130 | 43.602 $\pm$ 0.036 | 44.890 $\pm$ 0.031 | 47.414 $\pm$ 0.300 | 45.830 | -0.422 $\pm$ 0.129 | 9.271 $\pm$ 0.262 | 8.112 $\pm$ -0.781
1739+522 | 43.223 $\pm$ 0.039 | 44.564 $\pm$ 0.033 | 47.434 $\pm$ 0.300 | 46.160 | -0.661 $\pm$ 0.127 | 9.310 $\pm$ 0.278 | 7.682 $\pm$ -1.007
4C+09.57/PKS1749+096 | 42.570 $\pm$ 0.036 | 44.005 $\pm$ 0.031 | 46.527 $\pm$ 0.300 | 44.699 | -0.368 $\pm$ 0.129 | 8.377 $\pm$ 0.258 | 7.321 $\pm$ -0.731
CGRaBS J1800+7828 | 42.784 $\pm$ 0.008 | 44.188 $\pm$ 0.007 | 46.374 $\pm$ 0.300 | 45.851 | -0.480 $\pm$ 0.153 | 8.237 $\pm$ 0.268 | 6.964 $\pm$ -0.848
3C 371 | 40.722 $\pm$ 0.007 | 42.421 $\pm$ 0.006 | 46.719 $\pm$ 0.300 | 43.000 | -0.849 $\pm$ 0.155 | 8.601 $\pm$ 0.286 | 6.599 $\pm$ -1.204
1823+568/4C+56.27 | 43.017 $\pm$ 0.015 | 44.388 $\pm$ 0.013 | 47.374 $\pm$ 0.300 | 44.299 | -0.332 $\pm$ 0.148 | 9.217 $\pm$ 0.255 | 8.231 $\pm$ -0.707
1849+670 | 42.492 $\pm$ 0.036 | 43.938 $\pm$ 0.031 | 47.254 $\pm$ 0.300 | 45.418 | -0.763 $\pm$ 0.129 | 9.134 $\pm$ 0.283 | 7.303 $\pm$ -1.108
1921-293/PKS B1921-293 | 42.273 $\pm$ 0.036 | 43.751 $\pm$ 0.031 | 46.809 $\pm$ 0.300 | 45.020 | -0.645 $\pm$ 0.129 | 8.684 $\pm$ 0.277 | 7.088 $\pm$ -0.992
2145+067/4C+06.69 | 43.409 $\pm$ 0.036 | 44.724 $\pm$ 0.031 | 46.984 $\pm$ 0.300 | 46.766 | -0.583 $\pm$ 0.129 | 8.856 $\pm$ 0.274 | 7.381 $\pm$ -0.933
2155-152/PKS 2155-152 | 42.948 $\pm$ 0.036 | 44.329 $\pm$ 0.031 | 45.704 $\pm$ 0.300 | 44.682 | 0.032 $\pm$ 0.129 | 7.423 $\pm$ 0.203 | 7.095 $\pm$ -0.398
BL Lac | 40.470 $\pm$ 0.036 | 42.204 $\pm$ 0.031 | 46.324 $\pm$ 0.300 | 43.519 | -0.957 $\pm$ 0.129 | 8.207 $\pm$ 0.289 | 5.992 $\pm$ -1.297
PKS 2209+236 | 42.615 $\pm$ 0.036 | 44.044 $\pm$ 0.031 | 46.574 $\pm$ 0.450 | 45.786 | -0.596 $\pm$ 0.129 | 8.446 $\pm$ 0.427 | 6.947 $\pm$ -1.458
2223-052/3C 446 | 44.516 $\pm$ 0.036 | 45.672 $\pm$ 0.031 | 46.531 $\pm$ 0.300 | 46.601 | 0.056 $\pm$ 0.129 | 8.237 $\pm$ 0.199 | 7.948 $\pm$ -0.381
PKS 2227-08 | 43.788 $\pm$ 0.036 | 45.049 $\pm$ 0.031 | 46.899 $\pm$ 0.300 | 46.664 | -0.375 $\pm$ 0.129 | 8.749 $\pm$ 0.258 | 7.681 $\pm$ -0.737
2230+114/CTA 102 | 43.668 $\pm$ 0.036 | 44.946 $\pm$ 0.031 | 46.894 $\pm$ 0.300 | 46.870 | -0.469 $\pm$ 0.129 | 8.756 $\pm$ 0.266 | 7.505 $\pm$ -0.825
2251+158/3C 454.3 | 43.860 $\pm$ 0.036 | 45.110 $\pm$ 0.031 | 46.939 $\pm$ 0.300 | 46.650 | -0.352 $\pm$ 0.129 | 8.786 $\pm$ 0.256 | 7.761 $\pm$ -0.717
TXS 2331+073 | 41.628 $\pm$ 0.036 | 43.198 $\pm$ 0.031 | 46.484 $\pm$ 0.300 | 45.932 | -1.025 $\pm$ 0.129 | 8.368 $\pm$ 0.490 | 6.017 $\pm$ -1.164
2345-16/PKS 2345-16 | 42.811 $\pm$ 0.036 | 44.211 $\pm$ 0.031 | 46.709 $\pm$ 0.300 | 45.359 | -0.459 $\pm$ 0.129 | 8.570 $\pm$ 0.265 | 7.339 $\pm$ -0.815

Note: The first column is name; the second column is Logarithm of 151 MHz
radio luminosity; the third column is Logarithm of beam power; the fourth
column is the Logarithm of Eddington luminosity; the fifth column is Logarithm
of the beaming factor; the sixth column is the Logarithm of the spin function;
the seventh column is the Logarithm of black hole spin-mass energy; the eighth
column is Logarithm of the black hole irreducible mass.

Figure 1: Distribution of the FSRQ and BL Lacs samples used in our study of
fundamental plane for blazars. Histograms illustrate the distribution of
several different physical properties, including:(a) redshift z, (b) radio
loudness logR, (c) black hole mass log$M$, (d) 5 GHz radio luminosity
log${L_{R}}$, (e) beam factor log$f_{b}$, (f) 2-10 keV X-ray luminosity
log${L_{X}}$, (g) bolometric luminosity log${L_{bol}}$, (h) 151 MHz radio
luminosity log${L_{151}}$.

##  3 The RESULTS

###  3.1 fundamental plane of blazars

To examine the fundamental plane of Blazars, we should first explore the
correlation between radio luminosity ($L_{\rm{R}}$), X-ray luminosity
($L_{\rm{X}}$), and black hole mass (M) which also can be converted to
Eddington luminosity $L_{\rm{Edd}}$. We use the method of Merloni et al.
(2003) to determine the exist of fundamental plane. The results of the best
fit is showed in Table 5. We found that the fundamental plane exists for
Blazars, just as previous research has shown for AGN (Merloni et al., 2003;
Unal et al., 2015; Bariuan et al., 2022). This means that there is a
correlation between $L_{\rm{X}}$/$L_{\rm{Edd}}$ and $L_{\rm{R}}$ (Merloni et
al., 2003; Zhang et al., 2018). And we found the best fit of blazars is very
similar to radio-loud quasars (RL). The data points are distributed along the
direction of the extended line of fit. This confirms the current theoretical
classification of Blazars as a subtype of radio-loud active galactic nuclei.
After correcting for the beam effect, the radio data points of FSRQs change
from near the radio loud quasar region ($logL_{\rm{R}}$=42.5-45) with average
value 45.26 to near the radio quiet quasar region after correcting for the
beam effect ($logL_{\rm{R^{*}}}$=39.5-43) with average value 42.65. It
suggests that the differences for Blazars between left plane and right plane
of Figure 2 may be caused by the beaming effect induced by relativistic jets.
This further underscores the importance of accounting for beaming effects when
conducting fundamental plane analysis of Blazars.

Figure 2: Left panel: The correlation between $L_{\rm{X}}$/$L_{\rm{Edd}}$ and
$L_{\rm{R}}$. Right panel: The correlation between $L_{\rm{X}}$/$L_{\rm{Edd}}$
and $L_{\rm{R}}$ with beaming corrected. Table 5: The correlation analysis for
$L_{R}$ and $\frac{L_{\rm{X}}}{L_{\rm{Edd}}}$

Type | Best fit | R
---|---|---
RL | log$L_{R}$=${0.677}\pm{0.074}$ log($\frac{L_{\rm{X}}}{L_{\rm{Edd}}}$)+${44.396}\pm{0.147}$ | 0.272
RQ | log$L_{R}$=${0.206}\pm{0.077}$ log($\frac{L_{\rm{X}}}{L_{\rm{Edd}}}$)+${41.518}\pm{0.193}$ | 0.054
FSRQ | log$L_{R}$=${0.625}\pm{0.103}$ log($\frac{L_{\rm{X}}}{L_{\rm{Edd}}}$)+${44.977}\pm{0.139}$ | 0.355
Bllac | log$L_{R}$=${0.909}\pm{0.115}$ log($\frac{L_{\rm{X}}}{L_{\rm{Edd}}}$)+${45.138}\pm{0.172}$ | 0.391
Blazar | log$L_{R}$=${0.909}\pm{0.115}$ log($\frac{L_{\rm{X}}}{L_{\rm{Edd}}}$)+${45.138}\pm{0.172}$ | 0.416
${\rm{FSRQ}}^{*}$ | log$L_{R}$=${0.478}\pm{0.107}$ log($\frac{L_{\rm{X}}}{L_{\rm{Edd}}}$)+${42.071}\pm{0.145}$ | 0.229
${\rm{BLLac}}^{*}$ | log$L_{R}$=${0.651}\pm{0.205}$ log($\frac{L_{\rm{X}}}{L_{\rm{Edd}}}$)+${41.883}\pm{0.386}$ | 0.347
${\rm{Blazar}}^{*}$ | log$L_{R}$=${0.634}\pm{0.095}$ log($\frac{L_{\rm{X}}}{L_{\rm{Edd}}}$)+${42.166}\pm{0.142}$ | 0.338

Note: The R is R-Square as correlation coefficient; p is significance level; *
indicates that the corresponding data has been corrected with beaming factor.

In this paper, the fundamental plane of Blazars we defined is consistent with
Merloni et al. (2003). We use a multiple linear regression equation to obtain
the fundamental relationship and verify its reliability.

$\displaystyle{\rm log}L_{\rm R}=a{\rm log}L_{\rm X}+b{\rm log}M+c,$ (7)

where $L_{\rm R}$ is 5 GHz luminosity, $L_{\rm X}$ is the X-ray luminosity in
2-10 keV, M is the BH mass of Blazars. $a$ and $b$ and $c$ are fitting
parameters (Zhang et al., 2018). Multiple linear regression analysis was
performed to obtain the best fitting linear regression coefficient (Merloni et
al., 2003; Kording et al., 2006a; Zhang et al., 2018, 2023a). Linear
regression analysis enables the derivation of three variables, namely a, b,
and c, each with associated errors (Wang et al., 2017). The fundamental plane
for Blazars,
log$L_{R}$=${0.273}_{+0.059}^{-0.059}$log$L_{X}$+${0.695}_{+0.191}^{-0.191}$log$M$+${25.457}_{+2.728}^{-2.728}$
with R-Square=0.355, is showed in left plane of Figure 3. We examined the
fundamental plane relation for Blazars. It shows that the jet energy of
Blazars has a correlation with accretion and spin of BH. It is consistent with
the theory of BZ model (Blandford et al., 1977). Due to the strong
relativistic jets of Blazars, 5 GHz radio luminosity originating from jet can
be affected by the beaming effect. It may influence the analysis results of
the fundamental plane. Therefore, we measured the intrinsic 5 GHz radio
luminosity for conducting the fundamental plane analysis of Blazars. The
fundamental plane of Blazars after correcting anisotropism,
log$L_{R}$=${0.190}_{+0.049}^{-0.049}$log$L_{X}$+${0.475}_{+0.157}^{-0.157}$log$M$+${28.568}_{+2.245}^{-2.245}$
with R-Square=0.279, is showed in right plane of Figure 3. The existing FP
relations from previous work (Saikia et al., 2018; Gultekin et al., 2019) are
also included in Figure 3 for visualizing the results. The difference between
our result and previous work has decreased after correcting anisotropism. This
suggests that the fundamental plane of FSRQs are more similar to the
fundamental plane of AGNs (Saikia et al., 2018) with beaming corrected. Both
fundamental plane relations of Blazars are positive correlation and generally
similar. However, there is a slight decrease in the R-Square of the
fundamental plane, comparing to the original version, indicating that the
beaming effect can have some impact on the fundamental plane analysis. The
beaming effect enhances the role of jets in the fundamental plane
relationship. The analysis results are showed in Table 6. The R-values (0.595
and 0.528) are above 0.5 and the significance of each variable is below the
threshold of 0.05. The fitting results indicate that the correlation of
fundamental plane for Blazars is strong. It supports the validity of the
fundamental relationship. We also separately fitted FSRQs and BL Lacs by the
same fundamental plane which is calculated by total sample. It can be observed
that there are differences between the best-fitting lines of FSRQs and BL
Lacs, and the best fit for the overall sample lies between these two.
Furthermore, this characteristic remains even after the elimination of the
beaming effect. The fundamental plane relationship still exists when Blazars
and BL Lacs are discussed separately.

Figure 3: Left panel: The fundamental plane of Blazars. Right panel: The
fundamental plane of Blazars with beaming corrected. R is the radio loudness.
FP of Saikia is from Saikia et al. 2018, FP of Gultekin is from Gultekin et
al. 2019.  Table 6: The correlation analysis for the fundamental plane of
Blazars

Type | R-Square | p($L_{\rm{X}}$) | p(M) | DW
---|---|---|---|---
Blazars | 0.355 | 0.001 | 0.001 | 2.011
$\rm{{Blazars}^{*}}$ | 0.279 | 0.001 | 0.003 | 2.024

Note:R-Square is correlation coefficient; p($L_{\rm{X}}$) is significance
level of log$L_{\rm{X}}$; p(M) is significance level of logM ; * indicates
that the corresponding data has been corrected with beaming factor. DW is the
Durbin-Watson statistic which is a statistical method used to assess the
presence of autocorrelation in time series data.

We also used redshift as third variable to conduct a partial correlation
analysis of the correlation and discussed the effect of distance on result.
The R-square value decreases from 0.279 to 0.232 and the significance level of
both variables increased to 0.029 (still less than 0.05) after excluding the
effect of redshift (distance) on the fundamental plane which has been
corrected with beaming factor. It indicates that the result of fundamental
plane is partly effected by distance. The correlation of fundamental plane
still exists after removing the effect of distance from both the variables.

###  3.2 fundamental plane of blazars base on BH spin energy

Before using BH spin energy as a variable for fundamental analysis, we need to
discuss its effect on the black hole mass and its correlation with other
variables. The relation between z and $M_{\rm dyn}$, $M_{\rm irr}$ is
presented in upper left panel of Figure 4. The differences between $M_{\rm
dyn}$ and $M_{\rm irr}$ are evident. This is due to the contribution of
$M_{\rm spin}$ (where M$\equiv$$M_{\rm dyn}$=$M_{\rm irr}$+$E_{\rm
spin}$$c^{-2}$=$M_{\rm irr}$+$M_{\rm spin}$), where the differences in mass
are determined by the numerical value of the black hole spin. While the
difference is negligible in samples with low spin, samples characterized by
high spin exhibit a more significant discrepancy (Daly et al., 2022; Zhang et
al., 2023a). The 3D correlation diagram: upper right panel, lower left panel
and lower right panel in Figure 4, show that there are obvious correlations
among ${\rm log}L_{\rm X}$, ${\rm log}L_{\rm R}$ , ${\rm log}M_{\rm spin}$ and
${\rm log}M_{\rm irr}$. They are positively correlated with each other. The
results of R-Square are listed in Table 7. This correlations existed after the
effects of beam factors were taken into account. The analysis results indicate
that the correlation observed in the fundamental plane relation of $M_{\rm
dyn}$ is partly contributed by the $M_{\rm spin}$. The analysis results also
indicate that the correlation of black hole spin is more strongly correlated
with the fundamental plane relation than irreducible black hole mass. It is
important to analyze the role of black hole spin energy in the fundamental
plane relation.

Figure 4:  Upper left panel: The differences between ${\rm log}M_{\rm spin}$ and ${\rm log}M_{\rm irr}$. Upper right panel: The correlation between ${\rm log}L_{\rm X}$, ${\rm log}M_{\rm spin}$ and ${\rm log}M_{\rm irr}$. Lower left panel: The correlation between ${\rm log}L_{\rm R}$, ${\rm log}M_{\rm spin}$ and ${\rm log}M_{\rm irr}$. Lower right panel: The correlation between ${\rm log}L_{\rm R^{*}}$, ${\rm log}M_{\rm spin}$ and ${\rm log}M_{\rm irr}$ Table 7: Correlation coefficient between $L_{\rm X}$, $L_{\rm R}$ and $M_{\rm irr}$, $M_{\rm spin}$ Type | ${\rm log}L_{\rm X}$ | ${\rm log}L_{\rm R}$ | ${\rm log}L_{\rm R^{*}}$
---|---|---|---
| R-Square | R-Square | R-Square
$M_{\rm irr}$ | 0.211 | 0.158 | 0.121
$M_{\rm spin}$. | 0.362 | 0.604 | 0.501

Consistent with Merloni et al. (2003), we examined the fundamental plane of
Blazars based on $M_{\rm spin}$. We replace the $M_{\rm dyn}$ variable with
$M_{\rm irr}$ and $M_{\rm spin}$, and discuss them separately in this paper.
The results for new fundamental plane is showed in Table 8. After separately
analyzing $M_{\rm spin}$ and $M_{\rm irr}$, as shown in Figure 5. All the
correlations are positive, and there are differences between the fundamental
plane of $M_{\rm irr}$ and $M_{\rm spin}$ for Blazars. This may indicate that
both $M_{\rm irr}$ and $M_{\rm spin}$ contribute to the correlation foundation
for fundamental plane of Blazars. According to the results for new fundamental
plane, $M_{\rm spin}$ contributes more correlation than $M_{\rm irr}$. It
means that the effect of BH spin on fundamental plane is relatively large.
Since XRBs applie to fundamental relationships and its jet mechanism is also
similar to AGNs. This effect is similar to the radio loud and radio quiet
sources in the $L_{r}-L_{x}$ correlation of XRBs (Fender et al., 2003; Gallo
et al., 2003) for the sources of XRBs which have black hole spins. Therefore,
it is necessary to discuss the effect of $M_{\rm spin}$ on the traditional
fundamental plane and consider the $M_{\rm spin}$ as a important factor for
the study of fundamental plane.

Figure 5: Left panel: The fundamental plane of Blazars based on ${\rm
log}M_{\rm irr}$ after beam-correction. Right panel: The fundamental plane of
Blazars based on ${\rm log}M_{\rm spin}$ after beam-correction. Table 8: The
correlation analysis for the fundamental plane based on $M_{\rm{irr}}$ and
$M_{\rm{spin}}$

Type(left plane) | R-Square | p($L_{\rm{X}}$) | p($M_{\rm{irr}}$) | DW
---|---|---|---|---
$\rm{{Blazars}^{*}}$ | 0.429 | 0.001 | 0.010 | 1.9
Type(right plane) | R-Square | p($L_{\rm{X}}$) | p($M_{\rm{spin}}$) | DW
$\rm{{Blazars}^{*}}$ | 0.575 | 0.001 | 0.001 | 1.7

Note: R-Square is correlation coefficient; p($L_{\rm{X}}$) is significance of
log$L_{\rm{X}}$; p($M_{\rm{irr}}$) is significance of log$M_{\rm{irr}}$ ;
p($M_{\rm{spin}}$) is significance level of log$M_{\rm{spin}}$ ; * indicates
that the corresponding data has been corrected with beaming factor. DW is the
Durbin-Watson statistic which is a statistical method used to assess the
presence of autocorrelation in time series data.

##  4 CONCLUSIONS

In this paper, we repeated the fundamental plane of previous work (Merloni et
al., 2003) with Blazars. A total of 91 samples were obtained through sample
collection. We examined a traditional fundamental plane of black hole activity
for Blazars. The fundamental plane is
log$L_{R}$=${0.273}_{+0.059}^{-0.059}$log$L_{X}$+${0.695}_{+0.191}^{-0.191}$log$M$+${25.457}_{+2.728}^{-2.728}$
with R-Square=0.355. Although this fundamental plane is similar to the
previous work of AGN in general (Merloni et al., 2003; Unal et al., 2015;
Bariuan et al., 2022), there are quite differences in the coefficients. The
fundamental plane for our Blazars after considering the effect of beam factor
is
log$L_{R}$=${0.190}_{+0.049}^{-0.049}$log$L_{X}$+${0.475}_{+0.157}^{-0.157}$log$M$+${28.568}_{+2.245}^{-2.245}$
with R-Square=0.279 . The fundamental plane of Blazars with beaming effect
corrected is more similar to the radio-quiet quasars of previous work (Bariuan
et al., 2022). This could be attributed to the jet opening angles of Blazars,
which results in a significant influence of beaming effect on the jet
emission. The work of Bariuan et al. (2022) has indicated that radio loudness
has an impact on the fundamental plane within quasar samples. To accurately
model the dichotomy between radio-loud and radio-quiet quasars, two different
fundamental plane models are required. It has been previously concluded that
radio-loud and radio-quiet quasars may be governed by distinct fundamental
plane models (Bariuan et al., 2022). Typically, the emission mechanism for
radio-quiet samples is dominated by accretion flows, while radio-loud samples
are likely a combination of jet and disk. However, in this study, as shown in
Figure 2, our Blazars tend to exhibit characteristics of radio-loud samples
before correcting anisotropism, while their characteristics lean towards
radio-quiet samples after considering the effect of beam factor. We speculate
that the difference in jet opening angles leads to the manifestation of both
jet-dominated and accretion disk-dominated fundamental plane models for
Blazars. Due to the fact that all samples are radio-loud objects, and the
distribution of radio loudness data points in Figure 3 wasn’t affected by the
beaming effect. We found that Blazars are more suitable for the jet-dominated
fundamental plane models.

Standard fundamental models can accurately describe the production of black
hole jets (Unal et al., 2015), but they still have some deviations. In this
paper, we consider the influence of black hole spin to reduce these deviations
in the fundamental models. As the spin increases, a larger portion of the
gravitational energy can be converted into radiation due to the closer
proximity of accreted matter to the event horizon. The correlation indicates a
positive relationship between the jet power and the black hole spin-mass
energy, which is validated by correlation analysis. Therefore, the effect of
black hole spin should be considered for the study of fundamental plane. To
improve accuracy of conclusion, our goal is to include as many variables as
possible. Thus, we introduce $M_{\rm spin}$ as a variable in the fundamental
model, which is calculated by the spin and mass of BH. The new fundamental
plane in this paper (log$L_{\rm R}$=${0.332}_{+0.081}^{-0.081}$log$L_{\rm
X}$+${0.502}_{+0.091}^{-0.091}$log$M_{\rm spin}$+${22.606}_{+3.346}^{-3.346}$
with R-Square=0.575) shows that the fundamental plane based on $M_{\rm spin}$
has a stronger relation compared to the fundamental plane based on the mass of
BH also know as $M_{\rm dyn}$ (log$L_{\rm
R}$=${0.190}_{+0.049}^{-0.049}$log$L_{\rm
X}$+${0.475}_{+0.157}^{-0.157}$log$M$+${28.568}_{+2.245}^{-2.245}$ with
R-Square=0.279). Hence, the parameter $M_{\rm spin}$ holds significant
importance in the fundamental plane of Blazars (Zhang et al., 2023a). Our
findings underscore the crucial role of black hole spin in the accretion and
outflow processes (Unal et al., 2015). We confirmed the correlation among jet,
accretion, and spin of BH which indicates the applicability of BZ model for
Blazars. It is consistent with the conclusions of previous work (Blandford et
al., 1977, 2019; Unal et al., 2015) and could provide a observational
indication for the existence of the Blandford-Znajek process.

Due to the flux variation of Blazars, the observed value of the sample will
change with time, especially for radio luminosity, if the observation time
interval is large, it may produce errors, which will affect the fundamental
plane analysis results. This change may cause the correlation of R-Square to
decrease. Agns and blazars have different bands of radiation that are
correlated with each other and there is a significant time lag between these
radiations (Xie et al., 2017). In order to improve the accuracy of the
analysis results, we limited the observation time of the collected data
sources in 2-5 years. However, the time interval between radio and X-ray
emissions in Blazars generally ranges from months to years. AGNs exhibit
variability, with the potential for amplitude reaching approximately 100$\%$
on timescales that are shorter than the observation time interval (Ho et al.,
2001). This uncertainty may govern the actual observed uncertainty of flux
(Xie et al., 2017). Therefore, to investigate the intrinsic and physically
related Fundamental Plane, it is important to account for the radio/X-ray time
interval. But implementing such a correction for Blazars is exceedingly
difficult due to the requirement of long-term, coordinated monitoring of
individual sources in both radio and X-ray bands (Xie et al., 2017).
Additionally, this monitoring must capture at least one outburst in each
Blazars to obtain a dependable timelag measurement. Consequently, only a
limited number of AGNs have undergone such intensive monitoring to date, as it
poses significant challenges (Bell et al., 2011; King et al., 2013). In this
paper the uncertainty because of non-simultaneous (radio and X-ray)
observations we taken is from Ho et al. (2001) which adopted a slightly
smaller scatter in radio luminosities (0.2 dex).

## Acknowledgments

We gratefully thank the anonymous referee for the very helpful comments, which
helped us to greatly improve this paper. This work is partially supported by
Natural Science Foundation of Yunnan(202101AU070006), National Natural Science
Foundation of China (12063006), Young and middle-aged academic and technical
reserve talents of Yunnan Province (202205AC160087) and Yunnan Provincial
Department of Education Research Fund (2021J0671). The authors gratefully
acknowledge the computing support provided by the JRT Science Data Center at
Yuxi Normal University. QLH gratefully acknowledges the financial support from
the Hundred Talents Program of Yuxi (grant 2019).

## Data availability

The data underlying this article are available in the article and in its
online supplementary material.

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