# Curvature of the spectral energy distribution, Compton dominance and
synchrotron peak frequency in jetted AGNs

Yongyun Chen∗ (陈永云)  College of Physics and Electronic Engineering, Qujing
Normal University, Qujing 655011, P.R. China  Key Laboratory of Modern
Astronomy and Astrophysics (Nanjing University), Ministry of Education,
Nanjing 210093, China  Qiusheng Gu∗ (顾秋生)  School of Astronomy and Space
Science, Nanjing University, Nanjing 210093, P. R. China  Key Laboratory of
Modern Astronomy and Astrophysics (Nanjing University), Ministry of Education,
Nanjing 210093, China  Junhui Fan (樊军辉)  Center for Astrophysics,Guang zhou
University,Guang zhou510006, China  Xiaoling Yu(俞效龄)  College of Physics and
Electronic Engineering, Qujing Normal University, Qujing 655011, P.R. China
Nan Ding (丁楠)  School of Physical Science and Technology, Kunming University
650214, P. R. China  Dingrong Xiong (熊定荣)  Yunnan Observatories, Chinese
Academy of Sciences, Kunming 650011,China  Xiaotong Guo (郭晓通)  Anqing Normal
University, 246133, P. R. China

###### Abstract

We collect a large sample with a reliable redshift detected by the Fermi
satellite after 10 years of data (4FGL-DR2), including blazars, $\gamma$-ray
Narrow-line Seyfert 1 galaxies ($\gamma$NLS1s), and radio galaxies. The
spectral energy distributions (SEDs) of these Fermi sources are fitted by
using a second-degree polynomial, and some important parameters including
spectral curvature, synchrotron peak frequency, and peak luminosity are
obtained. Based on those parameters, we discuss the Fermi blazar sequence and
the particle acceleration mechanism. Our main results are as follows:(i) By
studying the relationship between the synchrotron peak frequency and the
synchrotron peak frequency luminosity, jet kinetic power, and $\gamma$-ray
luminosity for jetted AGNs, we find an “L” shape in the Fermi blazar sequence.
(ii) There is a significant anti-correlation between Compton dominance, black
hole spin, and the synchrotron peak frequency for jetted AGNs, respectively.
These results support that the $\gamma$NLS1s and radio galaxies belong to the
Fermi blazar sequence. (iii) On the basis of previous work, statistical or
stochastic acceleration mechanisms can be used to explain the relationship
between synchrotron peak frequency and synchrotron curvature. For different
subclasses, the correlation slopes are different, which implies that the Fermi
sources of different subclasses have different acceleration mechanisms. (iv)
The FSRQs and $\gamma$NLS1s have a higher median spin of a black hole than BL
Lacs and radio galaxies.

Blazars (164); Radio loud quasars (1349); BL Lacerate objects (158); Seyfert 1
galaxies (1447); Active galactic nuclei (16); Gammay-rays (637)

##  1 Introduction

Blazar is a special subclass of active galactic nuclei (AGNs), whose jets
point to observers. According to the equivalent width (EW) of the emission
line, blazar is divided into two categories: flat spectrum radio quasars
(FSRQs) and BLLac objects (BLLac). FSRQ has $EW>5$Å, while the source of
$EW<5$Å is BLLacs (Urry & Padovani, 1995). Some authors proposed a physical
classification, Ghisellini et al. (2011) found that the ratio of broad-line
region luminosity ($L_{\rm BLR}$) to Eddington luminosity ($L_{\rm Edd}$) of
FSRQs is $L_{\rm BLR}/L_{\rm Edd}\geq 10^{-3}$, while BL Lacs are less than
this value. The result may reflect the difference in their accretion
mechanism. Foschini (2017) proposed another possible classification scheme by
studying the relation between jet power and the luminosity of accretion disk
for blazars and $\gamma$NLS1s. They found two branches, namely, electron
cooling in different environments and black hole mass, based on the main
factors driving the change of jet power. According to the mass of black holes,
FSRQs and BL Lacs are divided into one category, while the $\gamma$NLS1s is
another. In terms of electron cooling, FSRQs and $\gamma$NLS1s are divided
into one category, while BL Lacs are another.

Seyfert galaxies are low luminosity AGNs, which are used to probe the jet,
corona and disk connection (e.g., Wilkins et al., 2015). Seyfert galaxies are
classified into Seyfert 1 and Seyfert 2 according to whether there are broad
permitted emission lines in their optical spectra, respectively. Some Seyfert
1 galaxies show broad permitted lines with narrow widths, called narrow-line
Seyfert 1 galaxies (NLS1s). The NLS1s have a full width at half-maximum (FWHM)
of $H\beta<2000~{}\rm km~{}s^{-1}$, weak [O III] emssion line, and the
presence of strong Fe II multiplets (e.g., Osterbrock & Pogge, 1985). NLS1 is
usually found to reside in spiral galaxies with low black hole mass (e.g.,
Grupe & Mathur, 2004). Foschini (2011) found that the jet power of FSRQs and
$\gamma$NLS1s depends on the black hole mass, which suggests that the
accretion disk of FSRQs and $\gamma$NLS1s was dominated by radiation pressure.
Some authors found that the SED of $\gamma$NLS1s is Compton-dominated, similar
to that of FSRQs (e.g., Abdo et al., 2009a, b; Yang & Zhou, 2015). Foschini et
al. (2015) and Berton et al. (2016) already found that flat-spectrum radio-
loud narrow-line Seyfert 1 galaxies (F-RLNLS1s) might be the low-luminosity
and low-mass tail of FSRQs. According to the SED modeling, Paliya et al.
(2019) found that the Physical Properties of $\gamma$NLS1s are similar to
FSRQs.

Radio galaxies are considered to be the parent population of blazars: while
blazars are aligned to our line of sight, radio galaxies have their jets
oriented at larger viewing angles. According to radio morphology, radio
galaxies are historically divided into FR I and FR II radio galaxies (Fanaroff
& Riley, 1974). FR I radio galaxies have bright jets close to the nucleus,
while FR II radio galaxies show prominent hotspots far from it. Some authors
claim that the morphological separation of FR I and FR II radio galaxies may
be related to the formation of their jets, that is, through the Blandford-
Znajek (BZ: Blandford & Znajek (1977)) process of extracting the spin energy
of black holes, or the Blandford-Payne (BP, Blandford & Payne (1982)) process
of extracting the rotational energy through accretion disks. The observation
of EHT (Event Horizon Telescope, Event Horizon Telescope Collaboration et al.
(2019)) showed that M87 hosts an FR I jet, which appears consistent with the
BZ process. According to the unified model of AGNs, FSRQs and FRII radio
galaxies are unified, and BL Lacs and FR I radio galaxies are unified (Urry &
Padovani, 1995). Chen et al. (2015) found that the jet power of FSRQs and FR
II radio galaxies depends on the black hole mass, while the jet power of BL
Lacs and FR I radio galaxies depends on the accretion rates. These results
suggest that the accretion disk of FSRQs and FR II radio galaxies is dominated
by radiation pressure, and the accretion disk of BL Lacs and FR I radio
galaxies is dominated by gas pressure.

Through the study of 126 blazars, it was found that there was a negative
correlation between the luminosity of the synchrotron peak ($L_{\rm pk}^{\rm
sy}$) and the synchrotron peak frequency ($\nu_{\rm pk}^{\rm sy}$) (Fossati et
al., 1998). Fossati et al. (1998) also noticed anti-correlations between the 5
GHz luminosity, synchrotron peak frequency ($\nu_{\rm pk}^{\rm sy}$), and the
Compton dominance (the ratio of Compton peak to synchrotron peak luminosity).
These correlations are claimed as evidence for a “blazar sequence”.
Subsequently, Ghisellini et al. (1998) proposed a physical explanation for the
blazar sequence. They fitted the broad-band spectra of blazars and suggested
that radiation cooling led to the emergence of the blazar sequence. The main
idea is that if the radiative cooling of emitted electrons is severe, then the
emitted electrons can not achieve high energies. According to this idea, in
FSRQs, the broad line region and a molecular torus are essential sources of
seed photons for the external Compton process, which means that the random
Lorentz factor of the electrons emitting at the peaks of the SED, $\gamma_{\rm
peak}$, is relatively small. Therefore, their synchrotron radiation peaks
appear in the submillimeter band, while the high-energy peaks appear in the
MeV band. Another important result is that the inverse Compton process becomes
dominant over the synchrotron process, which means a large Compton dominance.
In BL Lacs, the lack of an important thermal source implies the paucity of
seed photons for Compton scattering. Radiation cooling is less severe,
allowing electrons to achieve large $\gamma_{\rm peak}$. Therefore, the peak
of the synchrotron is in the UV-X-ray band, and the high-energy hump peaks are
in the GeV, sometimes even in the TeV band. The lack of ambient seed photons
implies that only the SSC process can contribute to high energy emissions.
Thus, high-energy components usually do not play a dominant role in BL Lacs.
This idea can be verified by applying a leptonic one-zone model, which allows
inferring the physical parameters of different types of sources. We also know
that the presence or absence of broad-emitting lines is related to the
luminosity of the accretion disk and its accretion regime. In addition, there
is a relationship between the jet power and the luminosity of accretion disk.
Therefore, the blazar sequence can also be considered as the sequence whose
main parameter is the jet power or equivalent to the disk luminosity. On the
other hand, we must also consider the mass of the central black hole, because
when the disk luminosity is about $10^{-3}-10^{-2}$ of the Eddington one, the
accretion regime will change. Some authors support the blazar sequence (e.g.,
Cavaliere & D’Elia, 2002; Maraschi & Tavecchio, 2003; Maraschi et al., 2008;
Celotti & Ghisellini, 2008; Ghisellini & Tavecchio, 2008; Ghisellini et al.,
2010; Chen & Bai, 2011; Finke, 2013; Ghisellini et al., 2010, 2017; Chen et
al., 2021). Alternatively, Potter & Cotter (2015) explained the sequence in
their inhomogeneous jet model, which is due to the different location of the
dissipation region: BL Lacs is closer to the central engine than FSRQs,
resulting in the stronger magnetic field, less Compton dominance, and larger
peak frequencies concerning FSRQs.

Since its birth in 1998, the blazar sequence has been an active debate. Some
authors believe that this sequence is not true, it is only the result of
observation biases (e.g., Giommi et al., 1999; Padovani et al., 2003; Nieppola
et al., 2006, 2008; Giommi et al., 2012; Keenan et al., 2021). Nieppola et al.
(2008) explained the observed blazar sequence as a sequence of Doppler
factors, that boosts the intrinsic luminosity and the intrinsic peak
frequencies. Considering that the method of deriving the beaming factor uses
the brightness temperature compared to the equipartition value of $5\times
10^{10}$K and the minimum variability timescale observed in the radio. This
implies that sources that are not varying during the observation period must
have very small Doppler factors, which in fact range from 1 to 30. Therefore,
some sources may be under-corrected. It is also considered that the beaming in
the radio band may be different from the beaming of more compact and innermost
regions producing the bulk of the emission (Prandini & Ghisellini, 2022). The
existence of blazar sequence was, and still is, a very controversial issue.
The main objection is that this may be due to the selection effect, which not
only worked when it was proposed but also works now, although there are more
sensitive instruments and more complete samples, making it reach a deeper
sensitivity limit (see also reviews Padovani, 2007; Ghisellini & Tavecchio,
2008). To prove this, someone recently proposed an alternative scheme, namely
”simplified blazar scenario” (Giommi et al., 2012). Giommi et al. (2012) put
forward detailed suggestions on the authenticity of this sequence. They put
forward what they called a ”simplified solution” for blazars. According to
these authors, the blazar sequence is the result of the selection effect, and
there is no physical connection between luminosity and the SEDs of blazars. We
have both red and blue blazars at high and low luminosities. In this case, it
is assumed that the distribution of the $\gamma_{\rm peak}$ reaches the peak
at $\sim 10^{3}$. Moreover, it is asymmetric, and the high-energy tail is
wider than the low-energy tail. In addition, the radio luminosity function is
$N(L)\propto L^{-3}$. It is assumed that the magnetic field of all blazar is
$B=0.15$ G, and the median Doppler factor is $<\delta>=15\pm 2$. As a direct
result, there are many low-power BL Lacs with “red” SED. These sources have
$\gamma_{\rm peak}\sim 10^{3}$ or smaller. Nearly 20 years after the
publication of the blazar sequence paper, about 1500 blazars were detected by
the Fermi-LAT Telescope, as reported by the 3LAC catalog Ackermann et al.
(2015), Ghisellini et al. (2017) proposed a revised sequence by using 747
sources, named Fermi blazar sequence. This time, the authors explored the
existence of SED sequences on a sample of gamma-ray selected blazars binned
according to the luminosity in the MeV-GeV energy range. Ghisellini et al.
(2017) found a significant inverse correlation between synchrotron peak and
gamma-ray luminosity by using the third catalog of AGN detected by Fermi/LAT
and supported the original blazar sequence. Although a large number of sources
have been considered, two types of sources may not have been studied. These
are MeV blazars and Extreme TeV blazars.

Keenan et al. (2021) mainly used 3FGL (Fermi Large Area Telescope Third Source
Catalogue), Low-frequency radio, and X-ray band data to select jetted AGNs,
and found that there was no blazar sequence. However, we have noticed that the
synchrotron peak frequency and synchrotron peak frequency luminosity are
estimated by using the radio core luminosity, the radio–optical spectral index
($\alpha_{r,o}$), and the optical–X-ray spectral index ($\alpha_{o,x}$) in the
sample of Keenan et al. (2021). The radio galaxies have a weaker beaming
effect than the blazar. Therefore, the estimation of synchrotron peak
frequency luminosity using radio core luminosity may lead to an underestimate
of synchrotron peak frequency luminosity, especially at low peak frequency. To
sum up, this may be the reason why they did not find the inverse correlation
between synchrotron peak frequency and synchrotron peak frequency luminosity.
At the same time, we find that most of the samples are mainly selected based
on Low-frequency radio and/or X-ray emssions. Some authors have found that the
Low-frequency radio radiation of some sources comes from star formation (e.g,
Caccianiga et al., 2015; Järvelä et al., 2022). Therefore, it may not be very
accurate to select the jetted AGNs by using the Low-frequency radio band. One
question, however, is how to choose a real jetted AGNs. Since the successful
launch of the Fermi-Large Area Telescope (Fermi-LAT; Atwood et al. (2009)),
many sources have been detected with gamma-ray radiation, including blazars,
radio galaxies, and NLS1s. This result shows that these Fermi sources have
strong relativistic jets, especially the NLS1s. Therefore, gamma-ray
luminosity is a useful tool for selecting real jetted AGNs.

In the process of studying the SEDs of blazars, in addition to the synchrotron
peak frequency and synchrotron peak frequency luminosity, the energy spectrum
curvature is also an important physical parameter. If a log-parabolic law is
adapted to fit the peaked component, i.e., $\log\nu
f_{\rm\nu}=-b(\log\nu-\log\nu_{\rm p})^{2}+\log\nu_{\rm p}f_{\rm\nu_{\rm p}}$,
the b is the curvature of the SED peak (Chen, 2014). Some authors found a
relation between the synchrotron peak frequency and the curvature (e.g.,
Massaro et al., 2004, 2006; Tramacere et al., 2007, 2009). According to
previous theoretical studies, the relationship between peak frequency and
curvature of synchrotron can be explained within the framework of the
acceleration process of electron emission (see Massaro et al., 2006; Tramacere
et al., 2007; Paggi et al., 2009; Chen, 2014). Chen (2014) theoretically
analyzed the relationship between the peak frequency and the curvature of the
synchrotron, and proposed two electron acceleration mechanisms according to
the relationship coefficient between the peak frequency and the curvature of
the synchrotron. Chen (2014) proposed that for the stochastic acceleration $k$
($1/b_{\rm sy}=k\log\nu_{\rm p}+a$), the slope of the relationship between the
peak frequency and the curvature of the synchrotron is $k=2$, for statistical
acceleration with energy-dependent acceleration is 2.5, and for fluctuations
of fractional acceleration gain is 3.33. Chen (2014) found that the slope of
this relation was k = 2 by using 48 fermi blazars, which is consistent with
stochastic acceleration.

With the release of the fourth source catalog data of Fermi Large Area
Telescope (LAT), we can use a large sample of Fermi sources to re-examine
blazar phenomenology and in particular the evidence for Fermi blazar sequence
and the particle acceleration mechanism of jetted AGNs. One of the motives of
this work is to check the more sensitive instruments we now infer: Fermi/LAT
is 20 times more sensitive than EGRET, and patrols the whole sky in a more
effective way. Gamma-ray is usually the main part of the electromagnetic
output. Because of the abundance of gamma-ray sources, we use Fermi detection
and redshift knowledge as the only selection criteria. This is different from
the original blazar sequence, in which objects are selected based on their
radio and/or X-ray emission. Moreover, we can select the real jetted AGNs by
using $\gamma$-ray. At present, it is unclear whether the $\gamma$NLS1s and
radio galaxies belong to the Fermi blazar sequence. Therefore, we use a large
sample to study the relationship between the Fermi blazar sequence, radio
galaxies, and $\gamma$NLS1s. At the same time, we studied the particle
acceleration mechanism for these Fermi sources based on the
curvature–synchrotron peak frequency relation. The paper is structured as
follows. In Section 2 we present the sample; in Section 3 we present the
results and discussion; Section 4 describes the conclusions. A $\Lambda$CDM
cosmology with $H_{0}=70\rm km~{}s^{-1}Mpc^{-1}$, $\Omega_{\rm m}=0.27$,
$\Omega_{\Lambda}=0.73$ is adopted.

##  2 The sample

###  2.1 The fermi blazar and $\gamma$NLS1s sample

We try to collect large samples of Fermi sources with reliable redshift, black
hole mass, accretion disk luminosity, synchrotron peak frequency, and
synchrotron peak frequency luminosity. We consider the sample of Paliya et al.
(2021). Paliya et al. (2021) got 1077 sources with black hole mass and disk
luminosity. A detailed description of the calculation of black hole mass and
the luminosity of accretion disk is presented in the work of Paliya et al.
(2021). Paliya et al. (2021) used a second-degree polynomial to fit quasi-
simultaneous multiwavelength data of these fermi sources using the built-in
functionality provided in the SSDC SED builder tool 111
https://tools.ssdc.asi.it/SED/ and got the synchrotron and inverse Compton
peak frequencies and peak frequencies flux. We use the ratio of the high- and
low-energy peak luminosities to obtain the Compton Dominance (CD). We
carefully checked the sample of Paliya et al. (2021) and compared it with the
source classification of Abdollahi et al. (2020) and Foschini et al. (2021),
and found that 17 $\gamma$NLS1s were included in the sample of Paliya et al.
(2021). We only consider these sources with 1.4 GHz radio flux from the NED.
Finally, we get 504 FSRQs, 277 BL Lacs, and 17 $\gamma$NLS1s. The data is
listed in Table 1.

Komossa et al. (2018) used the following formula to estimate the jet kinetic
power of AGNs (Bîrzan et al., 2008),

$\displaystyle\log P_{\rm jet}=0.35(\pm 0.07)\log P_{1.4}+1.85(\pm 0.10)$ (1)

where $P_{\rm jet}$ is in units $10^{42}\rm erg~{}s^{-1}$, and $P_{1.4}$ in
units $10^{40}\rm erg~{}s^{-1}$. The $P_{1.4}$ is 1.4 GHz radio luminosity,
$P_{\rm 1.4}=4\pi d_{\rm L}^{2}\nu S_{\rm\nu}$, $S_{\rm\nu}=S_{\rm\nu}^{\rm
obs}(1+z)^{\alpha-1}$, where $\alpha$ is spectral index, $\alpha=0$ is assumed
(Donato et al., 2001; Abdo et al., 2010; Komossa et al., 2018). Komossa et al.
(2018) believes that if the relationship of Cavagnolo et al. (2010) is used
instead, the predicted value of $P_{\rm jet}$ can be up to an order of
magnitude. Thus, we estimate the jet kinetic power of these fermi sources by
using equation (1). Foschini et al. (2015) used the 15 GHz radio core
luminosity to estimate the jet kinetic power of F-RLNLS1s, $\log P_{\rm
jet,kinetic}=(0.90\pm 0.04)\log L_{\rm radio,core}+(6\pm 2)$. We compare the
jet kinetic power calculated using Equation (1) with that calculated using
Equation of Foschini et al. (2015). The jet kinetic powers calculated using
Equation (1) are $\log P_{\rm jet,kinetic}=44.32$ (1H 0323+342), $\log P_{\rm
jet,kinetic}=45.03$ (SBS 0846+513), $\log P_{\rm jet,kinetic}=44.83$ (PMN
J0948+0022), $\log P_{\rm jet,kinetic}=44.93$ (PKS 1502+036), $\log P_{\rm
jet,kinetic}=44.40$ (FBQS J1644+2619), and $\log P_{\rm jet,kinetic}=44.88$
(PKS 2004-447). The jet kinetic powers calculated using Equation of Foschini
et al. (2015) are $\log P_{\rm jet,kinetic}=43.40$ (1H 0323+342), $\log P_{\rm
jet,kinetic}=45.09$ (SBS 0846+513), $\log P_{\rm jet,kinetic}=45.27$ (PMN
J0948+0022), $\log P_{\rm jet,kinetic}=45.08$ (PKS 1502+036), $\log P_{\rm
jet,kinetic}=43.57$ (FBQS J1644+2619), and $\log P_{\rm jet}=44.36$ (PKS
2004-447). We do not consider the radiation part, but only the kinetic energy
part when calculating the jet power. We find that the largest difference
between them is less than one order of magnitude.

###  2.2 The radio galaxies sample

We consider the radio galaxies with 1.4 GHz radio flux from the 4FGL-DR2
catalog and select 38 samples. The 1.4 GHz radio flux comes from the NED. The
black hole mass of radio galaxies is estimated by using the following formula
(Marconi & Hunt, 2003; Sbarrato et al., 2014),

$\log\left(\frac{M}{M_{\rm\odot}}\right)=-2.77-0.464\times M_{\rm H}$ (2)

where $M_{\rm H}$ is the absolute magnitude in the $H$ band. We get $M_{\rm
H}$ from the NED and use equation (2) to estimate the black hole mass of radio
galaxies. Buttiglione et al. (2010) got the disk luminosity of 3CR sample. We
cross-match with their sample. We only get the disk luminosity of seven radio
galaxies. For the remaining 31 radio galaxies, we use the synchrotron peak
frequency luminosity to estimate the luminosity of the accretion disk, i.e,
$L_{\rm disk}\sim 2\nu L_{\rm\nu}$ (Calderone et al., 2013; Ghisellini &
Tavecchio, 2015). It should be noted that the luminosity of the accretion disk
obtained by this method is slightly larger than that obtained by optical
spectra (see Figure 1). We also use equation (1) to estimate the jet kinetic
power of radio galaxies. We also collect quasi-simultaneous multiwavelength
data of radio galaxies from the Space Science Data Center (SSDC) SED builder
tool. We use a second-degree polynomial to fit the quasi-simultaneous
multiwavelength data of these radio galaxies and get the synchrotron and
inverse Compton peak frequencies and corresponding flux values.

Table 1: The sample of jetted AGN Name | Type | Redshift | $log(M/M_{\odot})$ | $logL_{disk}$ | $logL_{\gamma}$ | $f_{1.4~{}GHz}$ | log$P_{jet}$ | $log\nu_{pk}^{sy}$ | $CD$ | $b_{sy}$ | $logL_{pk}^{sy}$
---|---|---|---|---|---|---|---|---|---|---|---
(1) | (2) | (3) | (4) | (5) | (6) | (7) | (8) | (9) | (10) | (11) | (12)
J0001.5+2113 | FSRQ | 0.439 | 7.539 | 44.65 | 46.54 | 0.217 | 44.87 | 13.81 | 30.9 | 0.117 | 45.17
J0003.2+2207 | BLL | 0.1 | 8.100 | 42.74 | 43.71 | 0.0087 | 43.84 | 15.15 | 0.21 | 0.102 | 43.23
J0004.4-4737 | FSRQ | 0.88 | 8.280 | 45.10 | 47.16 | 0.932 | 45.43 | 13.01 | 2.4 | 0.110 | 46.61
J0006.3-0620 | BLL | 0.347 | 8.924 | 44.52 | 45.01 | 2.051 | 45.12 | 12.92 | 0.11 | 0.180 | 45.74
J0010.6+2043 | FSRQ | 0.598 | 7.861 | 45.34 | 45.94 | 0.158 | 44.96 | 12.42 | 1.41 | 0.237 | 45.47
J0011.4+0057 | FSRQ | 1.491 | 8.664 | 45.71 | 48.2 | 0.167 | 45.51 | 12.81 | 3.63 | 0.196 | 47.30
J0013.6+4051 | FSRQ | 0.256 | 7.022 | 43.13 | 44.8 | 1.65 | 44.97 | 12.73 | 0.46 | 0.120 | 44.99
J0013.6-0424 | FSRQ | 1.076 | 7.816 | 45.03 | 47.07 | 0.304 | 45.38 | 12.3 | 1.55 | 0.220 | 46.42
J0013.9-1854 | BLL | 0.095 | 9.650 | 43.27 | 43.91 | 0.0295 | 44.01 | 17.43 | 0.08 | 0.058 | 44.02
J0014.1+1910 | BLL | 0.477 | 7.465 | 44.32 | 45.66 | 0.154 | 44.86 | 13.74 | 1.41 | 0.136 | 45.46
J0014.2+0854 | BLL | 0.163 | 8.850 | 43.37 | 44.42 | 0.326 | 44.55 | 15.64 | 0.43 | 0.051 | 43.75
J0014.3-0500 | FSRQ | 0.791 | 7.928 | 44.93 | 46.83 | 0.0318 | 44.86 | 12.6 | 2.75 | 0.277 | 45.98
J0015.6+5551 | BLL | 0.217 | 9.680 | 44.05 | 44.87 | 0.0849 | 44.45 | 17.11 | 0.35 | 0.052 | 44.67
J0016.2-0016 | FSRQ | 1.577 | 8.522 | 45.77 | 48.73 | 0.957 | 45.81 | 12.44 | 10.47 | 0.153 | 47.22
J0016.5+1702 | FSRQ | 1.721 | 8.874 | 45.74 | 48.41 | 0.135 | 45.58 | 12.79 | 4.79 | 0.184 | 47.37
J0017.5-0514 | FSRQ | 0.227 | 7.831 | 44.74 | 45.46 | 0.178 | 44.58 | 12.61 | 4.9 | 0.259 | 44.83
J0017.8+1455 | BLL | 0.303 | 8.270 | 44.16 | 45.13 | 0.0595 | 44.52 | 14.22 | 0.49 | 0.126 | 44.84
J0019.6+7327 | FSRQ | 1.781 | 9.306 | 46.62 | 49.32 | 1.25 | 45.95 | 12.29 | 14.12 | 0.182 | 47.77
J0021.6-0855 | BLL | 0.648 | 8.540 | 44.63 | 46.06 | 0.0472 | 44.82 | 14.22 | 1.26 | 0.109 | 45.33
J0022.0+0006 | BLL | 0.306 | 8.020 | 43.79 | 44.9 | 0.0042 | 44.13 | 16.67 | 0.31 | 0.073 | 44.57
J0023.7+4457 | FSRQ | 1.062 | 7.709 | 45.09 | 47.49 | 0.141 | 45.26 | 12.8 | 6.03 | 0.202 | 46.56
J0024.7+0349 | FSRQ | 0.546 | 7.114 | 44.62 | 45.91 | 0.022 | 44.62 | 13.47 | 2.69 | 0.172 | 45.15
J0025.2-2231 | FSRQ | 0.834 | 8.492 | 45.71 | 46.41 | 0.202 | 45.17 | 13.97 | 2.63 | 0.092 | 45.71
J0028.4+2001 | FSRQ | 1.553 | 8.426 | 45.57 | 48.37 | 0.287 | 45.62 | 12.29 | 2.57 | 0.239 | 47.36
J0032.4-2849 | BLL | 0.324 | 8.470 | 44.02 | 45.16 | 0.161 | 44.70 | 13.78 | 1.26 | 0.133 | 44.95
J0038.2-2459 | FSRQ | 0.498 | 8.139 | 44.97 | 45.92 | 0.413 | 45.03 | 12.43 | 1.86 | 0.233 | 45.59
J0039.0-0946 | FSRQ | 2.106 | 8.499 | 45.73 | 49.19 | 0.154 | 45.76 | 12.53 | 9.12 | 0.183 | 47.59
J0040.4-2340 | BLL | 0.213 | 8.680 | 43.75 | 44.62 | 0.0536 | 44.38 | 14.44 | 0.55 | 0.102 | 44.00
J0040.9+3203 | FSRQ | 0.632 | 7.173 | 44.47 | 46.09 | 0.414 | 45.14 | 13.14 | 1.91 | 0.109 | 45.17
J0042.2+2319 | FSRQ | 1.425 | 8.733 | 45.49 | 48.02 | 1.27 | 45.78 | 12.27 | 5.25 | 0.151 | 46.92
J0043.8+3425 | FSRQ | 0.969 | 7.828 | 44.76 | 47.81 | 0.0933 | 45.14 | 13.77 | 12.3 | 0.111 | 45.89

##  3 Result and Discussion

###  3.1 The Fermi blazar sequence

Figure 2 shows the relation between synchrotron peak luminosity and peak
frequency. We find an “L” shape in this figure. Nieppola et al. (2006) found
an “V” shape by using BL Lacs, excluding FSRQ; If the FSRQ with high
luminosity and low peak is added, it may look more like an ”L” shape. Meyer et
al. (2011) also found an “L” shape by using 216 blazars and 41 radio galaxies.
We confirm their results. We also note that there is no inverse correlation
between synchrotron peak luminosity and peak frequency for most of the sources
with $\nu_{peak}>10^{15}$Hz. We carefully examine these sources with a peak
frequency greater than $10^{15}$Hz and found 176 BL Lacs. According to the
TeVCat catalogue222http://tevcat2.uchicago.edu/, we find that 19 of the 176
sources are TeV sources, about 10.8 percent. Extreme TeV blazars are expected
to have a steady but faint emissions in the GeV-TeV range. As a result, the
fainter or more distant objects might easily remain below the sensitivity
limit of Fermi. These subclasses represent the extremes of the sequence, and
it is very important to test the current understanding of the blazar sequence
through future experiments in the GeV-TeV range. Another key feature of this
class of source is its weak luminosity. For known extreme TeV blazars, the
measured luminosity is only slightly higher than the Fermi LAT sensitivity
limit in the gamma-ray range. In the context of the Fermi blazar sequence,
these extreme TeV blazars that are still missing would populate the lowest
luminosity range. At present, due to the small number of TeV samples, we can
not infer whether the TeV source causes the peak frequency and peak frequency
luminosity to have no anti-correlation when the peak frequency is greater than
$10^{15}$Hz. Current gamma-ray instruments like the Large High Altitude Air
Shower Observatory (LHAASO), future Cherenkov Telescope Array Observatory
(CTAO), and so on, will be the key to probing this expected trend.

From Figure 2, we also find that the synchrotron peak frequency of
$\gamma$NLS1s lies below $10^{14}$ Hz, which suggests that these $\gamma$NLS1s
are low-synchrotron-peaked (LSP) type beamed AGNs. The range of synchrotron
peak frequency of most radio galaxies is from $10^{14}$ to $10^{16}$ Hz, which
suggests that these radio galaxies are intermediate-synchrotron-peaked (ISP)
AGNs. The $\gamma$NLS1s and radio galaxies follow a fermi blazar sequence of
the “L” shape, which implies that $\gamma$NLS1s and radio galaxies belong to
the fermi blazar sequence. In the future, due to the improvement of the
sensitivity of the instrument, many sources may also be detected to have TeV
radiation, so the blazar sequence can be well tested by using large samples.

Figure 1: The accretion disk luminosity of Ghisellini & Tavecchio (2015) vs.
the disk luminosity of Buttiglione et al. (2010). The black dot is radio
galaxies.The y = x an equal line. The red dashed lines y = x$\pm$1 give an
idea of one order of magnitude difference with respect to the 1:1 ratio.
Figure 2: Relation between synchrotron peak luminosity ($L_{\rm pk}^{\rm sy}$)
and peak frequency ($\nu_{\rm pk}^{\rm sy}$) for the whole sample. FSRQs are
shown as red-filled circles, BL Lacs as black-filled circles, $\gamma$NLS1s as
stars, and radio galaxies as squares. The black line is the equal line.

In addition, it is found that there is a significant inverse correlation
between the 5 GHz luminosity and the peak frequency of synchrotron (Fossati et
al., 1998). The 5 GHz luminosity is an indication of jet power. Therefore, we
study the relationship between jet kinetic power, $\gamma$-ray luminosity, and
synchrotron peak frequency. In the top panel of Figure 3, we show the relation
between jet kinetic power, $\gamma$-ray luminosity, and synchrotron peak
frequency, respectively. We find that there is no anti-correlation between jet
kinetic power, $\gamma$-ray luminosity, and synchrotron peak frequency for
most of BL Lacs with high synchrotron peak frequency ($\nu>10^{15}$Hz). Keenan
et al. (2021) also found a similar tendency. They define these sources with
$\nu>10^{15}$Hz as sources of weak/type I jets. These sources with weak jets
are inefficient accretion. We also find that these sources with weak jets have
low $\log L_{disk}/L_{Edd}$, which suggests that these sources are inefficient
accretion. Our results are consistent with the work of Keenan et al. (2021).
The radio galaxies have lower $\gamma$-ray luminosity than other type AGNs,
which suggests that the radio galaxies have a weak beaming effect. Meyer et
al. (2011) found that Fermi radio galaxies have low Core dominance parameter
($R_{CE}$) than FSRQ and BL Lacs, which confirmed that these Fermi radio
galaxies have a weak beaming effect.

The $\gamma$-ray dominance was discussed as part of the original “blazar
sequence” by Fossati et al. (1998). Some authors have studied the relation
between $\gamma$-ray dominance or Compton dominance and synchrotron peak
frequency, in favor of testing the correlation between $L_{\rm pk}^{\rm sy}$
and $\nu_{\rm pk}^{\rm sy}$ (e.g., Padovani et al., 2003; Nieppola et al.,
2006; Giommi et al., 2012; Finke, 2013). We also test this correlation.
Pearson analysis shows that there is a significant inverse correlation between
Compton dominance and synchrotron peak frequency for the whole sample
($r=-0.69$, $P=5.66\times 10^{-121}$). Spearman and Kendall tau are used to
detect this correlation. The Spearman correlation coefficient and significance
level are $r=-0.67$ and $P=4.13\times 10^{-112}$. The Kendall tau correlation
coefficient and significance level are $r=-0.48$ and $P=1.75\times 10^{-94}$.
The fitted equation is as follows,

$\log CD=(-0.29\pm 0.01)\log\nu_{\rm pk}^{\rm sy}+(4.13\pm 0.14).$ (3)

The validity of blazar sequence has been debated for a long time (Fossati et
al., 1998), i.e., whether such a sequence has a physical origin (Ghisellini et
al., 1998). From the right bottom of figure 3, we also find that FSRQs and
$\gamma$NLS1s have high accretion rates and low synchrotron peak frequency,
while BL Lacs and radio galaxies have low accretion rates and high synchrotron
peak frequency. Ghisellini & Tavecchio (2008) have suggested that the jet
power and SED of blazars are mainly related to two physical parameters of the
accretion process, namely the black hole mass and the accretion rate.
Ghisellini et al. (1998) suggested that radiative cooling leads to the
observational phenomenon of the blazar sequence. FSRQ and $\gamma$NLS1 have
high accretion rate, which leads to fast cooling of relativistic electrons
(see also Foschini, 2017). FSRQ and $\gamma$NLS1 have lower synchrotron peak
frequencies. However, the accretion rate of BL Lac and radio galaxies is very
low, which leads to the slow cooling of relativistic electrons. BL Lacs and
radio galaxies have high synchrotron peak frequencies. The Pearson analysis
shows a significant anti-correlation between accretion rate and synchrotron
peak frequency for the whole sample ($r=-0.74$, $P=3.8\times 10^{-144}$). The
Spearman correlation coefficient and significance level are $r=-0.70$ and
$P=3.13\times 10^{-122}$. The Kendall tau correlation coefficient and
significance level are $r=-0.49$ and $P=5.92\times 10^{-98}$. The fitted
equation is as follows,

$\log\frac{L_{\rm disk}}{L_{\rm Edd}}=(-0.57\pm 0.02)\log\nu_{\rm pk}^{\rm
sy}+(6.19\pm 0.25).$ (4)

Figure 3: Top: relation between the jet kinetic power (left), $\gamma$-ray
luminosity (right) and synchrotron peak frequency. Bottom: the Compton
dominance (left) and the accretion luminosity (right, in Eddington units)
versus synchrotron peak frequency. The solid line corresponds to the best
fitting linear model obtained by symmetric least square fitting. The dotted
line represents the confidence range of $3\sigma$.

In the physical scheme, a luminous disk implies an efficient accretion process
($L_{\rm disk}/L_{\rm Edd}\geq 1\times 10^{-2}$), which ionizes the broad-line
region clouds, thus detecting the broad optical emission lines. Therefore, the
jet electrons interact with the dense photon field in broad-line region
through the EC (External Compton) mechanism, and then reach high energy to
produce luminous gamma-ray emission, which makes the SED more dominated by
Compton and reaches its peak at low frequency. On the other hand, in the low-
accretion regime ($L_{\rm disk}/L_{\rm Edd}<1\times 10^{-2}$), the external
radiation field becomes weaker and the SED is less Compton dominated. We test
these hypotheses on our sample of jetted AGNs, and the results are shown in
Figure 4.

Figure 4: Top: the Compton dominance versus jet kinetic power (left) and
$\gamma$-ray luminosity (right). Bottom: the Compton dominance versus the
accretion disk luminosity (left) and the accretion disk luminosity (right, in
Eddington units). The solid and dashed lines are the same as in figure 3.

In the top panels of Figure 4, we show the relation between Compton dominance
and jet kinetic power and $\gamma$-ray luminosity, respectively. Strong
positive correlations are found in both cases. The Pearson analysis shows a
significant correlation between Compton dominance and jet kinetic power
($r=0.74$, $P=6.34\times 10^{-147}$) and $\gamma$-ray luminosity ($r=0.77$,
$P=1.81\times 10^{-170}$) for the whole sample, respectively. The Spearman
correlation coefficient and significance level for the relation between
Compton dominance and jet kinetic power are $r=0.73$ and $P=1.36\times
10^{-141}$, for the relation between Compton dominance and $\gamma$-ray
luminosity are $r=0.77$ and $P=5.05\times 10^{-169}$. The Kendall tau
correlation coefficient and significance level for the relation between
Compton dominance and jet kinetic power are $r=0.53$ and $P=3.51\times
10^{-115}$, for the relation between Compton dominance and $\gamma$-ray
luminosity are $r=0.57$ and $P=9.41\times 10^{-135}$. We find that FSRQs and
$\gamma$NLS1s with high $\gamma$-ray luminosity are Compton dominance, which
implies that the high-energy $\gamma$-ray emission is mainly provided by the
external Compton mechanism (e.g., Sikora et al., 1994; Ghisellini & Tavecchio,
2008; Ghisellini et al., 2010; Finke, 2013; Paliya et al., 2019). However, the
high-energy component of the BL Lacs and radio galaxies is dominated by the
synchrotron self-Compton (SSC) process (e.g., Ghisellini et al., 2010; Lister
et al., 2011; Ackermann et al., 2012; Zhang et al., 2013).

From figure 4, we get the average values of jet kinetic power and Compton
dominance for FSRQs are $\log P_{\rm jet}=45.48\pm 0.41$ and $\log CD=0.50\pm
0.43$, for BL Lacs are $\log P_{\rm jet}=44.44\pm 0.40$ and $\log CD=-0.41\pm
0.39$, for $\gamma$NLS1s are $\log P_{\rm jet}=44.96\pm 0.36$ and $\log
CD=0.38\pm 0.56$, and for radio galaxies are $\log P_{\rm jet}=44.14\pm 0.38$
and $\log CD=-0.51\pm 0.67$. Foschini et al. (2015) got that the average
values of jet power of F-RLNLS1s are $\log P_{\rm kin}=43.62$. We find that
there is no more than one order of magnitude difference between our results
and theirs within the error range. Our results are consistent with those of
Böttcher & Dermer (2002) and Cavaliere & D’Elia (2002), implying the evolution
of high-power, Compton-dominated sources to less-luminous, less Compton-
dominated ones. At the same time, the evolutionary sequence of blazar shows
that quasars are young AGNs. With their growth, they become BLLac objects
(Cavaliere & D’Elia, 2002; Böttcher & Dermer, 2002; Maraschi & Tavecchio,
2003). NLS1 is considered to be a low redshift analogue of early quasars
(Mathur, 2000). Berton et al. (2016) clearly shows that NLS1s might be the
young counterpart of FSRQs (see also Foschini et al., 2015). These results may
imply that the FSRQs, BL Lacs, $\gamma$NLS1s, and radio galaxies seem to form
an evolutionary sequence, $\gamma$NLS1s$\rightarrow$FSRQs$\rightarrow$BL
Lacs$\rightarrow$radio galaxies. However, we warn that strong claims cannot be
made for the following reasons. In our sample, there are very few sources
exceeding redshift 3. In our future work, we will try to address these
shortcomings by considering $\gamma$-ray detected high-redshift sources.

The relations between Compton dominance and accretion disk luminosity (left)
and accretion disk luminosity in Eddington units (right) are shown in the
bottom panels of Figure 4. The Pearson analysis shows a significant
correlation between Compton dominance and disk luminosity ($r=0.69$,
$P=5.01\times 10^{-123}$) and disk luminosity in Eddington units ($r=0.70$,
$P=4.34\times 10^{-125}$) for the whole sample, respectively. The Spearman
correlation coefficient and significance level for the relation between
Compton dominance and disk luminosity are $r=0.69$ and $P=2.55\times
10^{-123}$, for the relation between Compton dominance and disk luminosity in
Eddington units are $r=0.70$ and $P=2.22\times 10^{-122}$. The Kendall tau
correlation coefficient and significance level for relation between Compton
dominance and disk luminosity are $r=0.49$ and $P=7.7\times 10^{-101}$, for
the relation between Compton dominance and disk luminosity in Eddington units
are $r=0.49$ and $P=3.39\times 10^{-100}$. The empirical relation between
Compton dominance and disk luminosity in Eddington units is as follows,

$\log CD=(0.37\pm 0.01)\log\frac{L_{\rm disk}}{L_{\rm Edd}}+(0.82\pm 0.03).$
(5)

From the bottom left panel of figure 4, we can see that FSRQs and
$\gamma$NLS1s with high disk luminosity tend to have high Compton dominance,
which hints that more Compton-dominated sources host more powerful disks. This
anti-correlation also indicates a physical connection between the luminosity
of accretion disk and the behavior of the SED in jetted AGNs. Stronger
accretion disk emission means a luminous broad-line region (BLR), whose strong
radiation field interacts with jet electrons, making them lose energy mainly
through EC-BLR process. The bottom right panel of figure 4 shows the obvious
difference of two groups in regions defined by $CD>1$, $L_{\rm disk}/L_{\rm
Edd}>0.01$ for both FSRQs and $\gamma$NLS1s and $CD<1$, $L_{\rm disk}/L_{\rm
Edd}<0.01$ for both BL Lacs and radio galaxies. Based on this finding, we
suggest that FSRQs and $\gamma$NLS1s are high-Compton-dominated (HCD) with
$CD>1$, and BL Lacs and radio galaxies are low-Compton-dominated (LCD) sources
for $CD<1$. At the same time, FSRQs have $CD>1$, $L_{\rm disk}/L_{\rm
Edd}>0.01$ and BL Lacs have $CD<1$, $L_{\rm disk}/L_{\rm Edd}<0.01$. This
classification scheme is analogous to that based on $L_{\rm BLR}/L_{\rm Edd}$
proposed by Ghisellini et al. (2011). Based on these findings and the results
discussed above, we can conclude that accretion rate plays an important role
in controlling the observational characteristics of powerful blazar, which
supports the statement that blazar sequence has a physical origin (e.g.,
Ghisellini et al., 2017).

###  3.2 Particle acceleration mechanisms

The statistical acceleration and the stochastical acceleration mechanisms can
be used to explain the correlation between $\log\nu_{\rm pk}^{\rm sy}$ and
$1/b_{\rm sy}$ (see details in Chen, 2014). Chen (2014) suggested that within
the framework of the statistical acceleration, it depends on the acceleration
probability of energy or the fluctuation of fractional acceleration gain. For
the case of energy-dependent acceleration probability, Chen (2014) got
$1/b_{\rm sy}\approx 5/2\log\nu_{\rm pk}^{\rm sy}+C$. For the case of
fluctuations of fractional acceleration gain, Chen (2014) got $1/b_{\rm
sy}\approx 10/3\log\nu_{\rm pk}^{\rm sy}+C$. The second scenario is in the
framework of stochastic acceleration, Chen (2014) got $1/b_{sy}\approx
2\log\nu_{\rm pk}^{\rm sy}+C$.

Figure 5 shows the relation between $\log\nu_{\rm pk}^{\rm sy}$ and $1/b_{\rm
sy}$ for Fermi blazars, $\gamma$NLS1s and radio galaxies. We derive the
correlation between $\log\nu_{\rm pk}^{\rm sy}$ and $1/b_{\rm sy}$ for the
whole sample ($r=0.89$, $P=1.66\times 10^{-287}$)

$1/b_{\rm sy}=(2.62\pm 0.05)\log\nu_{\rm pk}^{\rm sy}+(-27.78\pm 0.64)$ (6)

and for Fermi blazars ($r=0.92$, $P=1.75\times 10^{-310}$)

$1/b_{\rm sy}=(2.63\pm 0.04)\log\nu_{\rm pk}^{\rm sy}+(-27.85\pm 0.57)$ (7)

and for FSRQs ($r=0.63$, $P=6.48\times 10^{-57}$)

$1/b_{\rm sy}=(2.55\pm 0.14)\log\nu_{\rm pk}^{\rm sy}+(-26.68\pm 1.79)$ (8)

and for BL Lacs ($r=0.88$, $P=6.06\times 10^{-92}$)

$1/b_{\rm sy}=(2.96\pm 0.10)log\nu_{\rm pk}^{\rm sy}+(-33.12\pm 1.47)$ (9)

and for $\gamma$NLS1s ($r=0.61$, $P=0.009$)

$1/b_{\rm sy}=(2.33\pm 0.78)\log\nu_{\rm pk}^{\rm sy}+(-24.18\pm 10.36)$ (10)

and for radio galaxies ($r=0.37$, $P=0.02$)

$1/b_{\rm sy}=(2.23\pm 1.38)\log\nu_{\rm pk}^{\rm sy}+(-22.54\pm 19.77).$ (11)

The slopes of the relation between synchrotron peak frequency and synchrotron
curvature for the whole sample are $k_{\rm whole}=2.62\pm 0.05$, for Fermi
blazars are $k_{\rm Fermi~{}blazars}=2.63\pm 0.04$, for FSRQs are $k_{\rm
FSRQs}=2.55\pm 0.14$, and for $\gamma$NLS1s are $k_{\rm\gamma NLS1s}=2.33\pm
0.78$. We find that these slopes are close to 2.5, which implies that these
AGNs can be explained by statistical acceleration for the case of energy-
dependent acceleration probability. Chen (2014) found that the slope of the
relation was $2.04\pm 0.03$ by using 48 blazars, which is consistent with the
stochastic acceleration. Our results are different from theirs, which may be
due to the number of samples.

The slope of the relation between synchrotron peak frequency and synchrotron
curvature for BL Lacs $k_{\rm BLLacs}=2.96\pm 0.10$ is close to 10/3, which
can be explained by statistical particle acceleration for the case of
fluctuation of the fractional acceleration gain. The slope of the relation for
radio galaxies $k_{\rm radio,galaxies}=2.23\pm 1.38$ is close to 2, which can
also be explained by stochastic particle acceleration.

We find that the slope of the relationship between the synchrotron peak
frequency and curvature of different AGNs is different. Tramacere et al.
(2011) suggested that the acceleration and cooling processes may explain the
difference of the slope. During acceleration, the energy gain has significant
dispersion, leading to a momentum diffusion term, i.e., reducing the curvature
(namely increasing of $1/b_{\rm sy}$) causes the peak frequency to shift
towards higher peak frequencies. When cooling dominates the acceleration
process, the slope of this correlation may change (Tramacere et al., 2011;
Kalita et al., 2019). Tramacere et al. (2011) thought that magnetic field
plays an important role in the evolution of spectral parameters. They proposed
that when the magnetic field is weak, the evolution of the particles around
the peak is mainly accelerated, while the strong magnetic field is driven by
cooling. The FSRQs have a slightly larger magnetic field $B$ (1-10G) than in
BL Lacs (0.1-1G) (Ghisellini et al., 2010). The peak of the magnetic field of
radio galaxies (M87) is $B=1.61~{}G$ (Fraija & Marinelli, 2016). The average
magnetic field derived for $\gamma$NLS1s is $B=0.91$G (Paliya et al., 2019).
The FSRQs and radio galaxies have a slightly larger magnetic field than BL
Lacs and $\gamma$NLS1s. These results may indicate that for FSRQs and radio
galaxies, cooling dominates the acceleration process, while for BL Lacs and
$\gamma$NLS1s, acceleration dominates the cooling process.

Figure 5: Relation between $\log\nu_{\rm pk}^{\rm sy}$ and $1/b_{\rm sy}$ for
the whole sample. The solid lines and the dashed lines is the same as figure
3.

###  3.3 Black hole spins

We have searched the literature for the most advanced model that can explain
the emission of relativistic jets from thin accretion disks, which is
applicable to jetted AGNs. The most promising model is the the General
relativistic magnetohydrodynamic (GRMHD) simulations of thin the magnetically
arrested disks by Avara et al. (2016). Avara et al. (2016) derived an
empirical expression of the jet production efficiency based on the thin MAD
model,

Figure 6: Distribution of spins. The red solid histogram is FSRQs. The green
dashed histogram is BL Lacs. The orange dashed line is $\gamma$NLS1s. The cyan
dashed histogram is radio galaxies.

$\eta_{\rm model}\approx 4\omega_{\rm H}^{2}\left(1+\frac{0.3\omega_{\rm
H}}{1+2h^{4}}\right)^{2}h^{2},$ (12)

among $\omega_{\rm H}\equiv a_{*}/r_{\rm H}$ is the rotation frequency of the
black hole, $r_{\rm H}=1+\sqrt{1-a_{*}^{2}}$ is the event horizon radius, and
$h\approx H/R$ is disk thickness (McKinney et al., 2012, 2013; Avara et al.,
2016). Soares & Nemmen (2020) solved equation (12) to obtain the spin of a
black hole by assuming $\eta=\eta_{\rm model}$, $\eta=P_{\rm
jet}/(\dot{Mc^{2}})$, $\dot{M}=0.1\dot{M_{\rm Edd}}=L_{\rm Edd}/c^{2}$, and
$h=0.13$. The observed jet efficiency should satisfy the following
requirements,

$\eta<max(\eta_{\rm model})=\eta_{\rm model}(a_{*}=0.998)=0.098,$ (13)

where the maximum allowed spin is $max(a)=0.998$ (Thorne, 1974). Following the
work of Soares & Nemmen (2020), we also use equation (12) to get the black
hole spins for our sample with $\eta<max(\eta_{\rm model})$.

Figure 6 shows the distributions of black hole spin for FSRQs, $\gamma$NLS1s,
BL Lacs, and radio galaxies. The average values of black hole spin for FSRQs
are $a_{*}=0.93\pm 0.11$, for BL Lacs are $a_{*}=0.44\pm 0.26$, for
$\gamma$NLS1s are $a_{*}=0.91\pm 0.22$, and for radio galaxies are
$a_{*}=0.23\pm 0.17$. Soares & Nemmen (2020) got that the average values of
black hole spin for 154 FSRQs are $a_{*}=0.84_{-0.25}^{+0.11}$. Our results
are slightly different from theirs. There are two possible reasons. One is the
number of objects, our number of objects is larger than theirs. Another reason
is that the calculation method of jet power is different. They mainly use
gamma-ray luminosity to get jet power, and we use the 1.4 GHz radio
luminosity. We find that the FSRQs and $\gamma$NLS1s have high black hole spin
than BL Lacs and radio galaxies. The anti-correlation between jet kinetic
power and the synchrotron peak frequency reflects the existence of the blazar
sequence. Thus, the relation between the spin of black hole and the
synchrotron peak frequency also can be used to test the Fermi blazar sequence.

The relation between the spin of black and the synchrotron peak frequency is
shown in Figure 7. The Pearson analysis shows a significant anti-correlation
between black hole spin and the synchrotron peak frequency for the whole
sample ($r=-0.71$, $P=1.35\times 10^{-118}$). The Spearman correlation
coefficient and significance level for this relation are $r=-0.68$ and
$P=3.21\times 10^{-102}$. The Kendall tau correlation coefficient and
significance level for this relation are $r=-0.45$ and $P=1.36\times
10^{-77}$. The correlation between black hole spin and the synchrotron peak
frequency is significant even after the effects of jet kinetic power are
removed ($r_{\rm XY,Pjet}=-0.35$, $P=1.09\times 10^{-22}$). The fitted
equation is as follows,

$a_{*}=(-0.129\pm 0.005)\log\nu_{\rm pk}^{\rm sy}+(2.493\pm 0.065).$ (14)

To sum up, all the results and discussions support the Fermi blazar sequence.
The $\gamma$NLS1s and radio galaxies belong to the Fermi blazar sequence.
Paliya et al. (2013) suggested that the physical properties of the two gamma
ray emitting NLSy1 galaxies (PKS1502+036 and PKS2004-447) are similar to
blazar, between FSRQs and BLLac objects, so these sources may belong to the
traditional blazar sequence.

Figure 7: Relation between black hole spin and the synchrotron peak frequency
for the whole sample. The solid lines and the dashed lines is the same as
figure 3.

##  4 Conclusions

We use the largest Fermi samples, including Fermi blazars, $\gamma$NLS1s, and
radio galaxies, to study the relationship between them. Our main results are
as follows,

(1) In the plane of the synchrotron peak frequency luminosity, jet kinetic
power, $\gamma$-ray luminosity, and the synchrotron peak frequency, the Fermi
blazar sequence is “L” shaped.

(2) There is a significant inverse correlation between Compton dominance,
black hole spin and the synchrotron peak frequency for the whole sample. These
results support the Fermi blazar sequence, and $\gamma$NLS1s and radio
galaxies belong to the Fermi blazar sequence.

(3) There is a significant correlation between synchrotron peak frequency and
synchrotron curvature for the whole sample, Fermi blazars, FSRQs, and
$\gamma$NLS1s, respectively. The slopes of such a correlation are close to
2.5, which implies that these AGNs can be explained by statistical
acceleration for the case of energy-dependent acceleration probability.
However, the slope of such a correlation for BL Lacs is close to 10/3, which
implies that BL Lacs can be explained by statistical particle acceleration in
the case of fluctuation of the fractional acceleration gain. The slope of this
correlation for radio galaxies is close to 2, which implies that radio
galaxies can be explained by stochastic particle acceleration.

(4) The average value of black hole spin for FSRQs is $a_{*}=0.93\pm 0.11$,
for BL Lacs is $a_{*}=0.44\pm 0.26$, for $\gamma$NLS1s is $a_{*}=0.91\pm
0.22$, for radio galaxies is $a_{*}=0.23\pm 0.17$.

We are very grateful to the referee and Editor for the very helpful comments
and suggestions that improved the presentation of the manuscript
substantially. Yongyun Chen is gratefulfor financial support from the National
Natural Science Foundation of China (No. 12203028). This work was support from
the research project of Qujing Normal University (2105098001/094). This work
is supported by the youth project of Yunnan Provincial Science and Technology
Department (202101AU070146, 2103010006). Yongyun Chen is grateful for funding
for the training Program for talents in Xingdian, Yunnan Province. QSGU is
supported by the National Natural Science Foundation of China (No. 11733002,
12121003, 12192220, and 12192222). We also acknowledge the science research
grants from the China Manned Space Project with NO. CMS-CSST-2021-A05. This
work is supported by the National Natural Science Foundation of China
(11733001 and U2031201).

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