# The ‘blazar sequence’ in TeV band

Zhihao Ouyang  Shanghai Key Lab for Astrophysics, Shanghai Normal University,
Shanghai, 200234, China  Hubing Xiao Shanghai Key Lab for Astrophysics,
Shanghai Normal University, Shanghai, 200234, China  Jianzhen Chen  Shanghai
Key Lab for Astrophysics, Shanghai Normal University, Shanghai, 200234, China
Anton A. Strigachev  Institute of Astronomy and National Astronomical
Observatory, Bulgarian Academy of Sciences, 72 Tsarigradsko shosse Blvd., 1784
Sofia, Bulgaria  Rumen S. Bachev  Institute of Astronomy and National
Astronomical Observatory, Bulgarian Academy of Sciences, 72 Tsarigradsko
shosse Blvd., 1784 Sofia, Bulgaria  Xiangtao Zeng  Center for Astrophysics,
Guangzhou University, Guangzhou, 510006, China  Key Laboratory for
Astronomical Observation and Technology of Guangzhou, Guangzhou, 510006, China
Astronomy Science and Technology Research Laboratory of Department of
Education of Guangdong Province, Guangzhou, 510006, China  Marina Manganaro
Department of Physics, University of Rijeka, Rijeka, 51000, Croatia  Rui Xue
Department of Physics, Zhejiang Normal University, Jinhua, 321004, China
Zelin Li  College of Microelectronics, Feicuihu Campus of Hefei University of
Technology, Hefei, 230601, China  Junhui Fan Center for Astrophysics,
Guangzhou University, Guangzhou, 510006, China  Key Laboratory for
Astronomical Observation and Technology of Guangzhou, Guangzhou, 510006, China
Astronomy Science and Technology Research Laboratory of Department of
Education of Guangdong Province, Guangzhou, 510006, China Junhui Fan, Jianzhen
Chen, Hubing Xiao<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

The ‘blazar sequence’ has been proposed for more than 20 years, yet its nature
is still unclear. In this work, for the first time, we expand this topic to
the TeV band by using a sample of 58 TeV blazars including 48 blazars in the
quiescent state and 21 blazars in the flaring state. We investigate the
correlation between the TeV luminosity, which has been compensated for
attenuation from extragalactic background light, and the synchrotron peak
frequency. We note that there is no correlation between TeV luminosity and
peak frequency in the quiescent state and a strong anti-correlation in the
flaring state for the observed value. However, there is a strong positive
correlation in both the quiescent state and the flaring state for the
intrinsic value. This indicates that the blazar sequence is shown in the
flaring state rather than in the quiescent state for the observed value and
the blazar sequence is not present in both two states after removing the
beaming effect. In addition, to confirm whether the beaming effect results in
the blazar sequence, we compare the Fermi $\gamma$-ray luminosity between the
quiescent state and the flaring state. We find the Fermi $\gamma$-ray
luminosity in the flaring state is greater than that in the quiescent state
and the Doppler factor in the flaring state is greater. We suggest the blazar
sequence in the flaring state may be due to the stronger beaming effect.

TeV blazars; BL Lacertae objects; Flat-spectrum radio quasars

##  1 Introduction

Blazars are a subclass of active galactic nuclei (AGNs) with extreme
observational properties, including high polarization, fast rapid variability,
and strong high-energy $\gamma$-ray radiation, etc (Urry & Padovani, 1995; Fan
et al., 2004, 2014a, 2016; Lyutikov & Kravchenko, 2017; Ouyang et al., 2021;
Xiao et al., 2022, and references therein). They are divided into two
subclasses, flat spectrum radio quasars (FSRQs) and BL Lacertae objects (BL
Lacs), based on their optical spectrum. FSRQs have strong broad emission lines
(rest-frame equivalent width, EW $\geq$ 5 $\rm\mathring{A}$), while BL Lacs
have weak or no emission lines (EW $<$ 5 $\rm\mathring{A}$) (Stickel et al.,
1991; Urry & Padovani, 1995; Scarpa & Falomo, 1997). The blazar spectral
energy distribution (SED) is characterized by a two-hump structure. The low-
energy hump occurs at infrared to X-ray explained by the synchrotron emission
from electrons. The high-energy hump occurs at MeV to GeV which is produced
either by the inverse Compton process in a leptonic scenario (Tavecchio et
al., 1998; Ghisellini & Tavecchio, 2009), or the proton synchrotron emission
in a hadronic scenario (Mücke et al., 2003; Böttcher et al., 2009; Cerruti et
al., 2017) even the hybrid scenario (Cerruti et al., 2011; Böttcher et al.,
2013). According to the synchrotron peak frequency, blazars can be divided
into low-synchrotron-peaked sources (LSPs), intermediate-synchrotron-peaked
sources (ISPs), and high-synchrotron-peaked sources (HSPs) (Abdo et al., 2010;
Fan et al., 2016; Yang et al., 2022a).

An increasing number of extragalactic sources are detected in the TeV band,
which is also known as the very-high-energy band (VHE; $E\gtrsim\rm
100~{}GeV$). The radiation in the VHE band is mainly detected by ground-based
Cherenkov telescopes, e.g., the Major Atmospheric Gamma-ray Imaging Cherenkov
Telescopes (MAGIC), the High Energy Stereoscopic System (H.E.S.S.), the
Collaboration between Australia and Nippon for a Gamma Ray Observatory in the
Outback (CANGAROO), the Very Energetic Radiation Imaging Telescope Array
System (VERITAS), and the Large High Altitude Air Shower Observatory (LHAASO).
There are 251 TeV sources collected in TeVCat111http://tevcat2.uchicago.edu/
and the number is gradually increasing. The number of TeV blazars is still
small compared to GeV ones, but a lot of issues have been studied about the
TeV blazar. Massaro et al. (2013) suggested criteria to select TeV BL Lac
candidates based on infrared and X-ray observations; Lin & Fan (2016) compared
the observational properties of TeV-detected BL Lacs with non-TeV-detected BL
Lacs; Paiano et al. (2017) compiled 22 BL Lacs detected in TeV energy and
determined or constrained their redshifts, etc. The observations of TeV
blazars are usually performed during their flaring state, while some are
observed in their low energy state (quiescent state) owing to certain
circumstances, e.g., the source is very bright enough even in a quiescent
state, etc. However, observations of TeV blazar in VHE $\gamma$-rays are not
easy because of the strong absorption by the extragalactic background light
(EBL; Franceschini et al., 2008; Domínguez et al., 2011), the absorption would
steepen the TeV spectrum significantly.

Fossati et al. (1998) proposed a blazar sequence, from FSRQs to radio-selected
BL Lacs (RBLs) and to X-ray-selected BL Lacs (XBLs), whose synchrotron peak
luminosity decreases with synchrotron peak frequency increasing, which has
been explained as a consequence of the difference of electron cooling
efficiency (Ghisellini et al., 1998). The original blazar sequence was built
with a few sources and Ghisellini et al. (2017) used the 3LAC (The 3rd Catalog
of AGN Detected by the Fermi-LAT) sample and compiled all the archival data to
build their average SED, which agrees with the original blazar sequence and
the existence of the blazar sequence is proven. Many authors have performed
many studies on this issue, and there are basically two elucidations among
these scholars. One is that they acknowledged the validity of the blazar
sequence, but not the same as the original explanation. For instance, it is
found to be caused by a beaming effect (Nieppola et al., 2008; Fan et al.,
2017; Yang et al., 2022b) and the different location of the dissipation region
for BL Lacs and FSRQs (Potter & Cotter, 2015). The other one is that they
argued that the blazar sequence is caused by observational biases (Giommi et
al., 2012a, b).

The study of blazar sequence is still open, many of the previous works were
carried out using monochromatic luminosity (radio, optical, X-ray,
$\gamma$-ray, synchrotron peak luminosity) to study blazar sequence. However,
the blazar study in TeV band have not been performed yet. Using the TeV
$\gamma$-ray data to investigate the blazar sequence is of great significance
to understand the physical interpretation of the blazar sequence in the very-
high-energy band. In this work, for the purpose of exploring the blazar
sequence in TeV energy range, we will investigate the correlation between the
TeV luminosity and the synchrotron peak frequency for TeV blazars.

The sample acquisition and quantity calculation are presented in Section 2.
The results are given in Section 3 and Section 4. The discussions and
conclusions are given in Section 5 and Section 6. In this work, we use the
following flat $\Lambda$CDM cosmology, with $H_{0}\rm=71~{}km\cdot s^{-1}\cdot
Mpc^{-1}$, $\Omega_{\rm M}=0.27$ and $\Omega_{\Lambda}=0.73$.

##  2 Sample

There are 81 blazars, including 67 BL Lacs, 9 FSRQs, and 5 blazar candidates
of uncertain type (BCUs), confirmed as TeV emitters by associating TeVCat and
4FGL_DR3 (Abdollahi et al., 2022). We collect the TeV spectrum information in
literature and synchrotron peak frequency (${\rm log}\,\nu_{p}^{\rm ob}$) from
Fan et al. (2016) and Yang et al. (2022a) for these sources. In this case, we
compile a sample of 57 out of 81 TeV blazars with available the quiescent
state or (and) flaring state TeV spectra. The quiescent state refer to a low
flux density or being quiet and not active or being an average flux density
state, while flaring state designates conspicuous transient brightening and
shows an outburst in TeV band. Moreover, we also collect one source, 4FGL
J2042.1+2427 (RGB J2042.1+2426; Acciari et al., 2020), to our sample which was
observed by MAGIC, but not listed in TeVCaT. Finally, we get a sample of 58
sources (49 BL Lacs, 7 FSRQs and 2 BCUs), of which 48 sources (44 BL Lacs, 2
FSRQs and 2 BCUs) are in a quiescent state and 21 sources (14 BL Lacs and 7
FSRQs) are in a flaring state. These sources are listed in Table 1 and Table 2
respectively.

###  2.1 The EBL correction

The VHE spectrum of our sample is reduced by the EBL attenuation, making the
de-absorbed spectrum information can not obtain directly. For this reason, we
try to correct the absorbed spectrum.

Assuming the absorbed VHE spectrum of TeV blazar to form a power-law function:

$\frac{\mathrm{d}N}{\mathrm{d}E}_{\rm abs}=N_{0,\rm
abs}\times\left(\frac{E}{E_{0}}\right)^{-\Gamma_{\rm abs}},$ (1)

where $\Gamma_{\rm abs}$ is a photon spectral index in VHE spectrum, $E_{0}$
is a normalization energy and $N_{0,\rm abs}$ is differential photon flux at
$E_{0}$ in units of $\rm cm^{-2}\cdot s^{-1}\cdot TeV^{-1}$. Then the de-
absorbed spectrum can be obtained by the following function:

$\frac{\mathrm{d}N}{\mathrm{d}E}_{\rm de-
abs}=\frac{\mathrm{d}N}{\mathrm{d}E}_{\rm abs}e^{\tau(E,z)}=N_{0,\rm de-
abs}\times\left(\frac{E}{E_{0}}\right)^{-\Gamma_{\rm de-abs}},$ (2)

where $\tau(E,z)$, the optical depth, is a function of redshift and energy.
$N_{0,\rm de-abs}$ (hereafter $N_{0}$) is differential photon flux at $E_{0}$
and $\Gamma_{\rm de-abs}$ (hereafter $\Gamma$) is a photon spectral index with
EBL de-absorption. The model from Franceschini et al. (2008) is used to
correct the VHE spectrum and then the de-absorbed spectrum information can be
obtained. We consider the flux in TeV band in units of $\rm TeV\cdot
cm^{-2}\cdot s^{-1}$, which can be calculated by

$F=E^{2}\times\frac{\mathrm{d}N}{\mathrm{d}E}_{\rm de-abs},$ (3)

here $E$ is 1 TeV. Meanwhile, we show the absorbed (original) spectrum and the
de-absorbed spectrum in Figure 5 and Figure 6 in Appendix for the quiescent
state and the flaring state.

The isotropic TeV $\gamma$-ray luminosity in units of $\rm erg\cdot s^{-1}$ is
expressed as

$L=4\pi d_{L}^{2}(1+z)^{\Gamma-2}F{\rm,}$ (4)

here $(1+z)^{\Gamma-2}$ stands for a K-correction, $z$ is redshift, $\Gamma$
is the photon spectral index with EBL de-absorption, $F$ is the TeV
$\gamma$-ray flux and $d_{L}$ represents a luminosity distance expressed by
$d_{L}=\frac{(1+z)c}{H_{0}}\int_{z}^{1+z}\frac{1}{\sqrt{\Omega_{\rm
M}x^{3}+1-\Omega_{\rm M}}}\mathrm{d}x$. We calculate the de-absorbed TeV
$\gamma$-ray luminosity ($L_{\rm 1\,TeV,\,de-abs}^{\rm ob}$, hereafter $L_{\rm
1\,TeV}^{\rm ob}$ ) at 1 TeV for the sources in our sample.

##  3 Results

The blazar sequence is reflected in the diagrams as an anti-correlation
between luminosity in different bands and synchrotron peak frequency (Fossati
et al., 1998; Nieppola et al., 2006, 2008; Giommi et al., 2012a, b; Mao et
al., 2016; Fan et al., 2017; Yang et al., 2022b). Therefore, we can test the
blazar sequence in the TeV band by studying the correlation between the TeV
luminosity and synchrotron peak frequency.

In order to evaluate the correlation, the Pearson correlation analysis was
used in this work. This analysis presents the results in two parameters,
namely correlation coefficient ($r$) and chance probability ($p$). The
coefficient $r$, the most common way of measuring a linear correlation, is a
number between –1 and 1 that measures the strength and direction of the
relationship between two variables. If the $p$ value is smaller than the
critical value, which is usually set as 0.05, then the relationship is
statistically significant and allows you to reject the null hypothesis, in
which the two variables are not correlated. In addition, the least squares
linear regression was applied to find the linear correlation.

###  3.1 The observed correlation between TeV luminosity and synchrotron peak
frequency

Quiescent state: Based on the obtained TeV luminosity and synchrotron peak
frequency, we plot ${\rm log}\,L^{\rm ob}_{\rm 1\,TeV}$ vs ${\rm
log}\,\nu^{\rm ob}_{p}$ in the top panel of Figure 1. When the Pearson
correlation analysis is applied to test the correlation, we find that there is
no significant correlation between ${\rm log}\,L^{\rm ob}_{\rm 1\,TeV}$ and
${\rm log}\,\nu^{\rm ob}_{p}$ due to $r=-0.1$ and of $p=50.39\%$ for the case
of TeV blazars in the quiescent state. However, we can still find that there
is a trend of anti-correlation between the two parameters in this panel and
reveal the existence of a blazar sequence, because some ISP TeV sources are
absent in the circled region, which will be discussed in Section 5.

Figure 1: The observed correlation between ${\rm log}\,L^{\rm ob}_{\rm
1\,TeV}$ and ${\rm log}\,\nu_{p}^{\rm ob}$ in both quiescent state and flaring
state. (a) for the quiescent state, (b) for the flaring state. The blue circle
is for BL Lacs, the red one is for FSRQs, and the green one is for BCUs. The
solid line represents the corresponding linear regression for the whole
sample. The violet circle represents the particular region to be discussed in
Section 5 and the dashed line indicates a trend of the correlation.

Flaring state: Compared to the quiescent state, we find an anti-correlation
between ${\rm log}\,L^{\rm ob}_{\rm 1\,TeV}$ and ${\rm log}\,\nu^{\rm ob}_{p}$
in the flaring state and present a linear regression:

${\rm log}\,L^{\rm ob}_{\rm 1\,TeV}=-(0.61\pm 0.15)~{}{\rm log}\,\nu^{\rm
ob}_{p}+(54.23\pm 2.27)$ (5)

with the $r=-0.67$ and $p=7.95\times 10^{-4}$, as shown in the lower panel of
Figure 1. This result suggests a strong anti-correlation between TeV
luminosity and the synchrotron peak frequency in the flaring state and it
indicates that the blazar sequence is present in the flaring state rather than
in the quiescent state for TeV blazars.

###  3.2 The intrinsic correlation between TeV luminosity and synchrotron peak
frequency

In a relativistic beaming model, the observed flux density $f^{\rm ob}$ and
the intrinsic one $f^{\rm in}$ exist such correlation: $f^{\rm
ob}=\delta^{q}f^{\rm in}$ ($\delta=\left[\gamma_{\rm bulk}(1-\beta_{\rm
app}\cos\theta)\right]^{-1}$, where $\delta$ is a Doppler factor, $\beta_{\rm
app}$ is the apparent velocity, $\theta$ is the viewing angle and $\gamma_{\rm
bulk}$ is bulk Lorentz factor), then the intrinsic luminosity ($L^{\rm in}$)
and frequency ($\nu^{\rm in}$) can be calculated though the observed
luminosity ($L^{\rm ob}$) and frequency ($\nu^{\rm ob}$):

$L^{\rm in}=\frac{L^{\rm ob}}{\delta^{q+1}},~{}\nu^{\rm in}=\frac{\nu^{\rm
ob}}{\delta/(1+z)}{\rm,}$ (6)

where $q=2+\alpha$ for a continuous jet, $q=3+\alpha$ for a spherical jet and
$\alpha$ is the spectral index
($f_{\nu}\propto\nu^{-\alpha},\,\alpha=\Gamma-1$) (Lind & Blandford, 1985). We
compile the Doppler factors from Chen (2018) and the values range from 1.3 to
67.5 for the quiescent state and from 1.6 to 67.5 for the flaring state in our
sample. We calculate the intrinsic TeV luminosity $L_{\rm 1\,TeV}^{\rm in}$ at
1 TeV for the sources in our sample with available Doppler factor for both two
states. The remaining sources with unavailable Doppler factors are excluded in
the analysis of the intrinsic correlation.

Quiescent state: The top panel of Figure 2 shows a positive correlation
between the intrinsic TeV luminosity and the synchrotron peak frequency in the
quiescent state. We obtain

${\rm log}\,L^{\rm in}_{\rm 1\,TeV}=(1.20\pm 0.21)~{}{\rm log}\,\nu^{\rm
in}_{p}+(21.86\pm 3.23),$ (7)

with $r=0.67$ and $p\,\textless\,10^{-4}$ for the case of $q=\alpha+2$. And

${\rm log}\,L^{\rm in}_{\rm 1\,TeV}=(1.49\pm 0.24)~{}{\rm log}\,\nu^{\rm
in}_{p}+(16.54\pm 3.66),$ (8)

with $r=0.71$ and $p\,\textless\,10^{-4}$ for the case of $q=\alpha+3$.

Flaring state: The lower panel of Figure 2 also shows a positive correlation
between the intrinsic TeV luminosity and the synchrotron peak frequency in the
flaring state. We obtain

${\rm log}\,L^{\rm in}_{\rm 1\,TeV}=(0.86\pm 0.34)~{}{\rm log}\,\nu^{\rm
in}_{p}+(28.27\pm 4.72),$ (9)

with $r=0.53$ and $p=2.02\%$ for the case of $q=\alpha+2$. And

${\rm log}\,L^{\rm in}_{\rm 1\,TeV}=(1.13\pm 0.36)~{}{\rm log}\,\nu^{\rm
in}_{p}+(23.61\pm 5.05),$ (10)

with $r=0.60$ and $p=6.17\times 10^{-3}$ for the case of $q=\alpha+3$.

From the intrinsic relationship, we note that the intrinsic TeV luminosity and
synchrotron peak frequency show a very strong positive correlation in both
quiescent and flaring states. Thus, we suggest that the blazar sequence is not
present in the TeV band after removing the beaming effect. As above-mentioned,
we notice the absence (or only a trend) of the blazar sequence for the
quiescent state and the presence of a significant blazar sequence for the
flaring state in Figure 1. Moreover, the flatter slopes of intrinsic diagrams
for the flaring state than that for the quiescent state in Figure 2 indicate a
stronger beaming effect for the flaring state. And we shall discuss this issue
in Section 4.

Figure 2: The intrinsic correlation between ${\rm log}\,L^{\rm in}_{\rm
1\,TeV}$ and ${\rm log}\,\nu_{p}^{\rm in}$ in both quiescent state and flaring
state. (a) and (b) for the quiescent state, (c) and (d) for the flaring state;
(a) and (c) for the case of $q=2+\alpha$, (b) and (d) for the case of
$q=3+\alpha$. The blue circle is for BL Lacs and the red one is for FSRQs. The
solid lines represent the corresponding linear regression for the whole
sample. The dotted lines will be discussed in Section 5.

##  4 The comparison of Fermi $\gamma$-ray emission between the quiescent
state and flaring state

In order to investigate whether the stronger beaming effect results in (or
boosts) the presence of the blazar sequence in the flaring state, we try to
compare the $\gamma$-ray emission between the quiescent state and flaring
state for the sources in our sample. Previous studies have found that the bulk
Lorentz factor (or the Doppler factor, $\delta\approx\gamma_{\rm bulk}$ for
blazar) shows a strong correlation with $\gamma$-ray emission for the Fermi
detected AGNs (Fan et al., 2014b; Ghisellini et al., 2014; Blandford et al.,
2019; Zhang et al., 2020). Therefore, the $\gamma$-ray emission can be
considered to be an indicator of the beaming effect.

###  4.1 Introduction to Fermi-LAT and data analysis

The Fermi-LAT222https://fermi.gsfc.nasa.gov detects $\gamma$-ray in the energy
from 20 MeV to beyond 300 GeV and surveys the entire sky every day (Atwood et
al., 2009). In order to obtain the GeV $\gamma$-ray emission in the quiescent
state and flaring state, we analyse Fermi data in the same observation periods
when the VHE observation is performing. We use MJD 57863 as the starting time
of the data for analysis if the VHE observation period encompasses the
campaign before the launch of the Fermi-LAT. For TeV J1443+120, for instance,
the observation period spans the months of May 2008 and June 2010, totaling 53
hours and then we choose MJD 57863 as its start time. We perform the binned
likelihood analysis of the data using the latest LAT
Fermitools333https://fermi.gsfc.nasa.gov/ssc/data/analysis/ 2.2.0 and the
instrument response functions
P8R3_SOURCE_V3444https://fermi.gsfc.nasa.gov/ssc/data/analysis/documentation/Cicerone/Cicerone_LAT_IRFs/IRF_overview.html.
The energy of the data ranges from 0.1 GeV to 300 GeV and we select the
maximum zenith angle value as 90° to reduce the photons coming from the Earth
limb. The condition ‘evclass = 128, evtype = 3’ is used to filter events with
high probability of being photons. For each source, the region of interest
(ROI) considered is the circle of 10° radius surrounding the catalog source
position. The condition ‘(DATA_QUAL $\geq$ 0)&&(LAT_CONFIG==1)’ is used to
select the good time intervals. The background model consists of the Galactic
(gll_iem_v07.fits) and isotropic extragalactic diffuse emission models
(iso_P8R3_SOURCE_V3_v1.txt555https://fermi.gsfc.nasa.gov/ssc/data/access/lat/BackgroundModels.html).
The normalizations of the two diffuse emission components are set as free
parameters in the analysis. When performing the likelihood fitting, we fix the
parameters of the least significant sources until convergence is reached.

The spectrum of each source is described with a log-parabola function

$\frac{{\rm d}N}{{\rm
d}E}=N_{0}\times(\frac{E}{E_{0}})^{-\Gamma-\beta\log(\frac{E}{E_{0}})}{\rm,}$
(11)

where $N_{0}$ is the normalization in units of $\rm cm^{-2}\cdot s^{-1}\cdot
MeV^{-1}$, $E_{0}$ is the pivot energy and $\Gamma$ is photon index at $E_{0}$
and $\beta$ is the curvature index, or a power-law function

$\frac{{\rm d}N}{{\rm d}E}=N_{0}\times(\frac{E}{E_{0}})^{-\Gamma}{\rm,}$ (12)

where $N_{0}$ is the normalization in units of $\rm cm^{-2}\cdot s^{-1}\cdot
MeV^{-1}$, $E_{0}$ is the pivot energy and $\Gamma$ is photon index. After we
obtain the normalisation and the parameters, the integral flux ($F$) in units
of $\rm GeV\cdot cm^{-2}\cdot s^{-1}$ can be expressed in the form

$F=\int_{E_{\rm L}}^{E_{\rm U}}E\times\frac{{\rm d}N}{{\rm d}E}\,{\rm d}E$
(13)

for the log-parabola or the power-law function. Here $E_{\rm L}$ and $E_{\rm
U}$ are corresponding to 1 GeV and 100 GeV, respectively. Then we calculate
the $\gamma$-ray luminosity ($L_{\gamma}$) in units of $\rm erg\cdot s^{-1}$
through the Equation (4).

###  4.2 The results and conclusions obtained from the Fermi-LAT data analysis

Finally, we obtain the sample of Fermi $\gamma$-ray luminosity in the
quiescent state and flaring state, with sample size 25 (22 BL Lacs, 2 FSRQs
and 1 BCU) and 15 (9 BL Lacs and 6 FSRQs), respectively. We exclude the
sources which were not significantly detected with a test statistic TS $<$ 9
and/or the model-predicted photons $N_{\rm pred}<$ 8 (Kapanadze & Gurchumelia,
2022). The fitting normalizations and parameters of each source are listed in
Table 3 and Table 4. The Fermi $\gamma$-ray luminosity distribution of both
quiescent state and flaring state is shown in Figure 3. We compute the mean
value of $\gamma$-ray luminosity of both two states and get the mean value of
$\log L_{\gamma,\,\rm flaring}^{\rm mean}=47.37\pm 0.30{\,\rm erg\cdot
s^{-1}}$ and $\log L_{\gamma,\,\rm quiescent}^{\rm mean}=45.51\pm 0.24{\,\rm
erg\cdot s^{-1}}$, respectively. A Mann-Whitney U test is employed to compare
the Fermi $\gamma$-ray luminosity distribution of these two states, the test
gives a statistical value of 327.0 and a $p$-value smaller than $10^{-4}$,
which suggests that their mean values are different and the Fermi $\gamma$-ray
luminosity of the flaring state is greater than that of the quiescent state.
This result also reveals that blazars have a stronger beaming effect during
the flaring state than that during the quiescent state, therefore, we suggest
the blazar sequence may be caused by the stronger beaming effect for those
sources observed in the flaring state.

Figure 3: The distribution of Fermi $\gamma$-ray luminosity in both quiescent
state and flaring state. The blue histogram denotes the quiescent state and
red histogram denotes the flaring state.

##  5 Discussions

Fossati et al. (1998) calculated the SEDs for 126 blazars, investigated the
correlation between radio (5 GHz) luminosity and synchrotron peak luminosity,
between $\gamma$-ray luminosity and synchrotron peak frequency, and proposed
the ‘blazar sequence’. This sequence was later confirmed with a larger sample
of Fermi sources (Ghisellini et al., 2017).

The synchrotron radiation of blazars is produced by the relativistic electrons
moving in the magnetic field. The $\gamma$-ray emission is attributed to the
inverse Compton process, but different seed photons supply for FSRQs and for
BL Lacs. FSRQs are considered to contain a soft-photon-rich environment, these
soft photons could come from the accretion disk, the broad emission line
region (BLR), the dusty torus (DT) (Ghisellini et al., 1998; Prandini &
Ghisellini, 2022). Hence, the electron cooling for FSRQs is more efficient
than that for BL Lacs and the energy of relativistic electrons in BL Lac jets
is more likely powerful than those in FSRQ jets. As a result, the energy of
soft photons produced through synchrotron radiation in BL Lac jets is higher
than that in FSRQ jets, making BL Lac have a higher synchrotron peak frequency
(Ghisellini et al., 1998). These seed photons could be scattered into an
extremely high energy, e.g., TeV range, mainly by the population of
relativistic electrons through the synchrotron-self-Compton (SSC) process for
BL Lacs, and through the external Compton (EC) process for FSRQs. The more
dramatic electron cooling efficiency makes the lower energy of relativistic
electrons and the subsequent lower synchrotron peak frequency. However, BL
Lacs have a less efficient electron cooling process, the relativistic electron
could preserve energy and result in higher energy of synchrotron peak and
produce SSC TeV emission.

In this work, we expand the study of the ‘blazar sequence’ to the TeV band. We
notice that there is an anti-correlation between the observed TeV luminosity
and the observed synchrotron peaked frequency during the flaring state, and
there is no significant anti-correlation between the observed TeV luminosity
and the observed synchrotron peaked frequency for the sources in the quiescent
state, as seen in Figure 1. In the top panel of Figure 1, we consider and
analyse the correlation for FSRQs, BL Lacs and BCUs together as a whole
sample, because we have an extremely limited sample size of FSRQs and BCUs.
Besides, the empty circled region, synchrotron peak frequency ranging from
$10^{14}$ up to $10^{16}$, in this panel is intriguing. Those sources located
in this region should be mostly considered as ISPs with intermediate TeV
luminosity compare to TeV J0222+430 (3C 66A) and those sources below the
circle. These sources, somehow, in this region are not detected. Therefore,
the anti-correlation is diluted in the quiescent state. In this case, our
results indicate a trend of blazar sequence at the TeV band in the quiescent
state of blazars and the trend is plotted in a dashed line in the top panel of
Figure 1. However, one should bear in mind that it is possible that there are
no more other sources in this empty circle. Because these potential sources
are detectable to those TeV telescopes. In this case, the anti-correlation
trend for the quiescent state does not exist, and the TeV blazar sequence
appears only in the flaring state. To sum up, the existence of the TeV blazar
sequence in the quiescent state is uncertain.

Fan et al. (2017) studied the blazar sequence in radio, optical, X-ray and
$\gamma$-ray bands using a sample of 86 flaring blazars. They noticed that the
blazar sequence exists in the diagram of observed quantities and the sequence
disappears when the intrinsic quantities are considered, suggesting that the
blazar sequence is a consequence of a beaming effect. Yang et al. (2022b), an
extended work of Fan et al. (2017) with a larger sample, also found the blazar
sequence exists for the observed quantities and disappears for the intrinsic
quantities. Our results in Figure 2 are consistent with the conclusion of Fan
et al. (2017) and Yang et al. (2022b). In the top panel of Figure 2, we notice
that there are two FSRQs mixed with those BL Lacs. We reanalysed the
correlation after removing the two FSRQs, the linear regressions are plotted
in the top panel of Figure 2 with dotted lines and give the formulas

${\rm log}\,L^{\rm in}_{\rm 1\,TeV}=(1.45\pm 0.20)~{}{\rm log}\,\nu^{\rm
in}_{p}+(17.90\pm 3.03)$ (14)

with $r=0.77$ and $p\,\textless\,10^{-4}$ for the case of $q=2+\alpha$;

${\rm log}\,L^{\rm in}_{\rm 1\,TeV}=(1.78\pm 0.22)~{}{\rm log}\,\nu^{\rm
in}_{p}+(11.86\pm 3.38)$ (15)

with $r=0.80$ and $p\,\textless\,10^{-4}$ for the case of $q=3+\alpha$. For
the lower panel of Figure 2, we notice TeV 1159+292 (Ton 599), shows an
extremely powerful TeV emission, apart from the flaring sample population,
where linearly distributed along the regression line. Ton 599 was occurring in
an extraordinary $\gamma$-ray flaring state. This flare was detected by AGILE
(the Astro-rivelatore Gamma a Immagini Leggero) from 16 to 18 December 2017
(MJD 58103.24 - 58105.24) (Bulgarelli et al., 2017) and VERITAS from 15 to 16
December 2017 (MJD 58102 - 58104) (Mukherjee & VERITAS Collaboration, 2017).
The average flux on the two nights is $F(E\geq 100\,\rm GeV)=(1.0\pm
0.1)\times 10^{-10}\,\rm ph\cdot cm^{-2}\cdot s^{-1}$ or 16% of the Crab
Nebula flux above the same threshold (Mukherjee & VERITAS Collaboration,
2017). According to our fitting results for the VHE spectrum, we find the
photon spectral index of Ton 599 is $\Gamma=0.16$ (see Figure 6) and the
spectrum is so hard that it has an extremely high luminosity in TeV emission.

Thus, we consider reanalysing the linear regression without Ton 599, the
results are plotted in the lower panel of Figure 2 in dotted lines and
expressed as

${\rm log}\,L^{\rm in}_{\rm 1\,TeV}=(1.11\pm 0.26)~{}{\rm log}\,\nu^{\rm
in}_{p}+(24.51\pm 3.65)$ (16)

with $r=0.73$ and $p=5.80\times 10^{-4}$ for the case of $q=2+\alpha$;

${\rm log}\,L^{\rm in}_{\rm 1\,TeV}=(1.37\pm 0.29)~{}{\rm log}\,\nu^{\rm
in}_{p}+(19.87\pm 4.15)$ (17)

with $r=0.76$ and $p=2.65\times 10^{-4}$ for the case of $q=3+\alpha$.

It is clear that the correlations become more significant when we removed
these particular sources. And, our results of Figure 2 indicate the TeV blazar
sequence is caused by the beaming effect which is consistent with the results
in Fan et al. (2017) and Yang et al. (2022b). Moreover, we find these TeV
sources have larger average Fermi GeV $\gamma$-ray luminosity during the TeV
flaring state than during the quiescent state, see in Figure 3. These results
suggest blazars in flaring state show a stronger beaming effect and show a
more significant blazar sequence than those blazars in quiescent state.

##  6 Conclusions

In this work, for the purpose of studying the blazar sequence in the TeV
energy range, We have collected a sample of 48 TeV blazars in the quiescent
state and 21 TeV blazars in the flaring state with available VHE spectrum
information, synchrotron peak frequency, and Doppler factor from the
literature. We have investigated the correlation, which is representative of
the blazar sequence, between the observed/intrinsic TeV luminosity and the
observed/intrinsic synchrotron peak frequency.

Our main conclusions are as follows:

  * (1)

There is no correlation between TeV luminosity and synchrotron peak frequency
for observed value in the quiescent state, however, there is a strong anti-
correlation between these two parameters in the flaring state. Therefore, we
suggest that the TeV blazar sequence is present in the flaring state.

  * (2)

The correlation for intrinsic value shows a strong positive correlation in
both the quiescent state and flaring state. This result demonstrates that the
blazar sequence is not present for the intrinsic value and the anti-
correlation for the observed quantities derives from the beaming effect.

  * (3)

Through the comparison of Fermi $\gamma$-ray luminosity between the quiescent
state and flaring state, we find that the Fermi $\gamma$-ray luminosity of the
flaring state is greater than that of the quiescent state, which means the
Doppler factor in the flaring state is greater than that in the quiescent
state. We suggest the presence of the blazar sequence or the anti-correlation
between TeV luminosity and synchrotron peak frequency may be due to the
stronger beaming effect in the flaring state.

We thank the support of the key laboratory for astrophysics of Shanghai. H. B,
Xiao acknowledges the support from the National Natural Science Foundation of
China (NSFC 12203034) and from the Shanghai Science and Technology Fund
(22YF1431500); J. H, Fan acknowledges the support from the NSFC (NSFC
U2031201, NSFC 11733001, U2031112), Scientific and Technological Cooperation
Projects (2020-2023) between the People’s Republic of China and the Republic
of Bulgaria, Guangdong Major Project of Basic and Applied Basic Research
(Grant No. 2019B030302001), the science research grants from the China Manned
Space Project with NO. CMS-CSST-2021-A06, and the support for Astrophysics Key
Subjects of Guangdong Province and Guangzhou City. This research was partially
supported by the Bulgarian National Science Fund of the Ministry of Education
and Science under grants KP-06-H28/3 (2018), KP-06-H38/4 (2019), KP-06-KITAJ/2
(2020) and KP-06-PN68/1(2022).

## References

  * Abdalla et al. (2020) Abdalla, H., Adam, R., Aharonian, F., et al. 2020, MNRAS, 494, 5590, doi: 10.1093/mnras/staa999
  * Abdo et al. (2010) Abdo, A. A., Ackermann, M., Agudo, I., et al. 2010, ApJ, 716, 30, doi: 10.1088/0004-637X/716/1/30
  * Abdollahi et al. (2022) Abdollahi, S., Acero, F., Baldini, L., et al. 2022, ApJS, 260, 53, doi: 10.3847/1538-4365/ac6751
  * Abeysekara et al. (2016) Abeysekara, A. U., Archambault, S., Archer, A., et al. 2016, MNRAS, 459, 2550, doi: 10.1093/mnras/stw664
  * Abeysekara et al. (2017) —. 2017, ApJS, 233, 7, doi: 10.3847/1538-4365/aa8d76
  * Abeysekara et al. (2018) Abeysekara, A. U., Archer, A., Benbow, W., et al. 2018, ApJ, 861, L20, doi: 10.3847/2041-8213/aad053
  * Acciari et al. (2008) Acciari, V. A., Aliu, E., Beilicke, M., et al. 2008, ApJ, 684, L73, doi: 10.1086/592244
  * Acciari et al. (2009a) Acciari, V. A., Aliu, E., Arlen, T., et al. 2009a, ApJ, 695, 1370, doi: 10.1088/0004-637X/695/2/1370
  * Acciari et al. (2009b) Acciari, V. A., Aliu, E., Aune, T., et al. 2009b, ApJ, 703, 169, doi: 10.1088/0004-637X/703/1/169
  * Acciari et al. (2010a) Acciari, V. A., Aliu, E., Arlen, T., et al. 2010a, ApJ, 715, L49, doi: 10.1088/2041-8205/715/1/L49
  * Acciari et al. (2010b) Acciari, V. A., Aliu, E., Beilicke, M., et al. 2010b, ApJ, 709, L163, doi: 10.1088/2041-8205/709/2/L163
  * Acciari et al. (2010c) Acciari, V. A., Aliu, E., Arlen, T., et al. 2010c, ApJ, 708, L100, doi: 10.1088/2041-8205/708/2/L100
  * Acciari et al. (2011) —. 2011, ApJ, 738, 169, doi: 10.1088/0004-637X/738/2/169
  * Acciari et al. (2020) Acciari, V. A., Ansoldi, S., Antonelli, L. A., et al. 2020, ApJS, 247, 16, doi: 10.3847/1538-4365/ab5b98
  * Adams et al. (2022a) Adams, C. B., Batista, P., Benbow, W., et al. 2022a, ApJ, 932, 129, doi: 10.3847/1538-4357/ac6dd9
  * Adams et al. (2022b) Adams, C. B., Batshoun, J., Benbow, W., et al. 2022b, ApJ, 924, 95, doi: 10.3847/1538-4357/ac32bd
  * Aharonian et al. (2007a) Aharonian, F., Akhperjanian, A. G., Barres de Almeida, U., et al. 2007a, A&A, 475, L9, doi: 10.1051/0004-6361:20078462
  * Aharonian et al. (2007b) —. 2007b, A&A, 473, L25, doi: 10.1051/0004-6361:20078412
  * Aharonian et al. (2007c) Aharonian, F., Akhperjanian, A. G., Bazer-Bachi, A. R., et al. 2007c, A&A, 470, 475, doi: 10.1051/0004-6361:20077057
  * Aharonian et al. (2008) Aharonian, F., Akhperjanian, A. G., Barres de Almeida, U., et al. 2008, A&A, 481, L103, doi: 10.1051/0004-6361:200809603
  * Ahnen et al. (2015) Ahnen, M. L., Ansoldi, S., Antonelli, L. A., et al. 2015, ApJ, 815, L23, doi: 10.1088/2041-8205/815/2/L23
  * Ahnen et al. (2016a) —. 2016a, MNRAS, 459, 3271, doi: 10.1093/mnras/stw689
  * Ahnen et al. (2016b) —. 2016b, A&A, 595, A98, doi: 10.1051/0004-6361/201629461
  * Ahnen et al. (2017) —. 2017, A&A, 603, A31, doi: 10.1051/0004-6361/201629540
  * Albert et al. (2006) Albert, J., Aliu, E., Anderhub, H., et al. 2006, ApJ, 648, L105, doi: 10.1086/508020
  * Albert et al. (2007) —. 2007, ApJ, 667, L21, doi: 10.1086/521982
  * Aleksić et al. (2011) Aleksić, J., Antonelli, L. A., Antoranz, P., et al. 2011, ApJ, 730, L8, doi: 10.1088/2041-8205/730/1/L8
  * Aleksić et al. (2012) Aleksić, J., Alvarez, E. A., Antonelli, L. A., et al. 2012, A&A, 539, A118, doi: 10.1051/0004-6361/201117967
  * Aleksić et al. (2015) Aleksić, J., Ansoldi, S., Antonelli, L. A., et al. 2015, MNRAS, 451, 739, doi: 10.1093/mnras/stv895
  * Aliu et al. (2009) Aliu, E., Anderhub, H., Antonelli, L. A., et al. 2009, ApJ, 692, L29, doi: 10.1088/0004-637X/692/1/L29
  * Aliu et al. (2011) Aliu, E., Aune, T., Beilicke, M., et al. 2011, ApJ, 742, 127, doi: 10.1088/0004-637X/742/2/127
  * Aliu et al. (2012) Aliu, E., Archambault, S., Arlen, T., et al. 2012, ApJ, 750, 94, doi: 10.1088/0004-637X/750/2/94
  * Aliu et al. (2015) Aliu, E., Archer, A., Aune, T., et al. 2015, ApJ, 799, 7, doi: 10.1088/0004-637X/799/1/7
  * Allen et al. (2017) Allen, C., Archambault, S., Archer, A., et al. 2017, MNRAS, 471, 2117, doi: 10.1093/mnras/stx1756
  * Archambault et al. (2013) Archambault, S., Arlen, T., Aune, T., et al. 2013, ApJ, 776, 69, doi: 10.1088/0004-637X/776/2/69
  * Archambault et al. (2015) Archambault, S., Archer, A., Beilicke, M., et al. 2015, ApJ, 808, 110, doi: 10.1088/0004-637X/808/2/110
  * Archambault et al. (2016) Archambault, S., Archer, A., Barnacka, A., et al. 2016, MNRAS, 461, 202, doi: 10.1093/mnras/stw1319
  * Archer et al. (2018) Archer, A., Benbow, W., Bird, R., et al. 2018, ApJ, 862, 41, doi: 10.3847/1538-4357/aacbd0
  * Arlen et al. (2013) Arlen, T., Aune, T., Beilicke, M., et al. 2013, ApJ, 762, 92, doi: 10.1088/0004-637X/762/2/92
  * Atwood et al. (2009) Atwood, W. B., Abdo, A. A., Ackermann, M., et al. 2009, ApJ, 697, 1071, doi: 10.1088/0004-637X/697/2/1071
  * Baloković et al. (2016) Baloković, M., Paneque, D., Madejski, G., et al. 2016, ApJ, 819, 156, doi: 10.3847/0004-637X/819/2/156
  * Blandford et al. (2019) Blandford, R., Meier, D., & Readhead, A. 2019, ARA&A, 57, 467, doi: 10.1146/annurev-astro-081817-051948
  * Böttcher et al. (2009) Böttcher, M., Reimer, A., & Marscher, A. P. 2009, ApJ, 703, 1168, doi: 10.1088/0004-637X/703/1/1168
  * Böttcher et al. (2013) Böttcher, M., Reimer, A., Sweeney, K., & Prakash, A. 2013, ApJ, 768, 54, doi: 10.1088/0004-637X/768/1/54
  * Bulgarelli et al. (2017) Bulgarelli, A., Parmiggiani, N., Lucarelli, F., et al. 2017, The Astronomer’s Telegram, 11073, 1
  * Cerruti et al. (2011) Cerruti, M., Zech, A., Boisson, C., & Inoue, S. 2011, in SF2A-2011: Proceedings of the Annual meeting of the French Society of Astronomy and Astrophysics, ed. G. Alecian, K. Belkacem, R. Samadi, & D. Valls-Gabaud, 555–558, doi: 10.48550/arXiv.1111.0557
  * Cerruti et al. (2017) Cerruti, M., Zech, A., Emery, G., & Guarin, D. 2017, in American Institute of Physics Conference Series, Vol. 1792, 6th International Symposium on High Energy Gamma-Ray Astronomy, 050027, doi: 10.1063/1.4968973
  * Chen (2018) Chen, L. 2018, ApJS, 235, 39, doi: 10.3847/1538-4365/aab8fb
  * Domínguez et al. (2011) Domínguez, A., Primack, J. R., Rosario, D. J., et al. 2011, MNRAS, 410, 2556, doi: 10.1111/j.1365-2966.2010.17631.x
  * Fan et al. (2014a) Fan, J.-H., Bastieri, D., Yang, J.-H., et al. 2014a, Research in Astronomy and Astrophysics, 14, 1135, doi: 10.1088/1674-4527/14/9/004
  * Fan et al. (2014b) Fan, J. H., Bastieri, D., Yang, J. H., et al. 2014b, Journal of Astrophysics and Astronomy, 35, 231, doi: 10.1007/s12036-014-9201-4
  * Fan et al. (2004) Fan, J.-H., Wang, Y.-J., Yang, J.-H., & Su, C.-Y. 2004, Chinese J. Astron. Astrophys., 4, 533, doi: 10.1088/1009-9271/4/6/533
  * Fan et al. (2016) Fan, J. H., Yang, J. H., Liu, Y., et al. 2016, ApJS, 226, 20, doi: 10.3847/0067-0049/226/2/20
  * Fan et al. (2017) Fan, J. H., Yang, J. H., Xiao, H. B., et al. 2017, ApJ, 835, L38, doi: 10.3847/2041-8213/835/2/L38
  * Fossati et al. (1998) Fossati, G., Maraschi, L., Celotti, A., Comastri, A., & Ghisellini, G. 1998, MNRAS, 299, 433, doi: 10.1046/j.1365-8711.1998.01828.x
  * Franceschini et al. (2008) Franceschini, A., Rodighiero, G., & Vaccari, M. 2008, A&A, 487, 837, doi: 10.1051/0004-6361:200809691
  * Ghisellini et al. (1998) Ghisellini, G., Celotti, A., Fossati, G., Maraschi, L., & Comastri, A. 1998, MNRAS, 301, 451, doi: 10.1046/j.1365-8711.1998.02032.x
  * Ghisellini et al. (2017) Ghisellini, G., Righi, C., Costamante, L., & Tavecchio, F. 2017, MNRAS, 469, 255, doi: 10.1093/mnras/stx806
  * Ghisellini & Tavecchio (2009) Ghisellini, G., & Tavecchio, F. 2009, MNRAS, 397, 985, doi: 10.1111/j.1365-2966.2009.15007.x
  * Ghisellini et al. (2014) Ghisellini, G., Tavecchio, F., Maraschi, L., Celotti, A., & Sbarrato, T. 2014, Nature, 515, 376, doi: 10.1038/nature13856
  * Giommi et al. (2012a) Giommi, P., Padovani, P., Polenta, G., et al. 2012a, MNRAS, 420, 2899, doi: 10.1111/j.1365-2966.2011.20044.x
  * Giommi et al. (2012b) Giommi, P., Polenta, G., Lähteenmäki, A., et al. 2012b, A&A, 541, A160, doi: 10.1051/0004-6361/201117825
  * H. E. S. S. Collaboration et al. (2010a) H. E. S. S. Collaboration, Acero, F., Aharonian, F., et al. 2010a, A&A, 511, A52, doi: 10.1051/0004-6361/200913073
  * H. E. S. S. Collaboration et al. (2010b) H. E. S. S. Collaboration, Abramowski, A., Acero, F., et al. 2010b, A&A, 520, A83, doi: 10.1051/0004-6361/201014484
  * H. E. S. S. Collaboration et al. (2010c) —. 2010c, A&A, 516, A56, doi: 10.1051/0004-6361/201014321
  * H. E. S. S. Collaboration et al. (2012a) —. 2012a, A&A, 538, A103, doi: 10.1051/0004-6361/201118406
  * H. E. S. S. Collaboration et al. (2012b) —. 2012b, A&A, 542, A94, doi: 10.1051/0004-6361/201218910
  * H. E. S. S. Collaboration et al. (2013a) —. 2013a, A&A, 554, A72, doi: 10.1051/0004-6361/201220996
  * H. E. S. S. Collaboration et al. (2013b) —. 2013b, A&A, 559, A136, doi: 10.1051/0004-6361/201321639
  * H. E. S. S. Collaboration et al. (2013c) —. 2013c, A&A, 552, A118, doi: 10.1051/0004-6361/201321108
  * H. E. S. S. Collaboration et al. (2015) H. E. S. S. Collaboration, Abramowski, A., Aharonian, F., et al. 2015, A&A, 573, A31, doi: 10.1051/0004-6361/201321436
  * H. E. S. S. Collaboration et al. (2021) H. E. S. S. Collaboration, Abdalla, H., Adam, R., et al. 2021, A&A, 648, A23, doi: 10.1051/0004-6361/202038949
  * HESS Collaboration et al. (2013) HESS Collaboration, Abramowski, A., Acero, F., et al. 2013, MNRAS, 434, 1889, doi: 10.1093/mnras/stt1081
  * Kapanadze & Gurchumelia (2022) Kapanadze, B., & Gurchumelia, A. 2022, A&A, 668, A75, doi: 10.1051/0004-6361/202244748
  * Lin & Fan (2016) Lin, C., & Fan, J.-H. 2016, Research in Astronomy and Astrophysics, 16, 103, doi: 10.1088/1674-4527/16/7/103
  * Lind & Blandford (1985) Lind, K. R., & Blandford, R. D. 1985, ApJ, 295, 358, doi: 10.1086/163380
  * Lyutikov & Kravchenko (2017) Lyutikov, M., & Kravchenko, E. V. 2017, MNRAS, 467, 3876, doi: 10.1093/mnras/stx359
  * MAGIC Collaboration et al. (2008) MAGIC Collaboration, Albert, J., Aliu, E., et al. 2008, Science, 320, 1752, doi: 10.1126/science.1157087
  * MAGIC Collaboration et al. (2018a) MAGIC Collaboration, Ansoldi, S., Antonelli, L. A., et al. 2018a, MNRAS, 480, 879, doi: 10.1093/mnras/sty1753
  * MAGIC Collaboration et al. (2018b) MAGIC Collaboration, Acciari, V. A., Ansoldi, S., et al. 2018b, A&A, 619, A159, doi: 10.1051/0004-6361/201833618
  * MAGIC Collaboration et al. (2018c) MAGIC Collaboration, Ahnen, M. L., Ansoldi, S., et al. 2018c, A&A, 619, A45, doi: 10.1051/0004-6361/201832677
  * MAGIC Collaboration et al. (2018d) —. 2018d, A&A, 617, A30, doi: 10.1051/0004-6361/201832624
  * MAGIC Collaboration et al. (2021) MAGIC Collaboration, Acciari, V. A., Ansoldi, S., et al. 2021, A&A, 647, A163, doi: 10.1051/0004-6361/202039687
  * Mao et al. (2016) Mao, P., Urry, C. M., Massaro, F., et al. 2016, ApJS, 224, 26, doi: 10.3847/0067-0049/224/2/26
  * Massaro et al. (2013) Massaro, F., Paggi, A., Errando, M., et al. 2013, ApJS, 207, 16, doi: 10.1088/0067-0049/207/1/16
  * Mücke et al. (2003) Mücke, A., Protheroe, R. J., Engel, R., Rachen, J. P., & Stanev, T. 2003, Astroparticle Physics, 18, 593, doi: 10.1016/S0927-6505(02)00185-8
  * Mukherjee & VERITAS Collaboration (2017) Mukherjee, R., & VERITAS Collaboration. 2017, The Astronomer’s Telegram, 11075, 1
  * Nieppola et al. (2006) Nieppola, E., Tornikoski, M., & Valtaoja, E. 2006, A&A, 445, 441, doi: 10.1051/0004-6361:20053316
  * Nieppola et al. (2008) Nieppola, E., Valtaoja, E., Tornikoski, M., Hovatta, T., & Kotiranta, M. 2008, A&A, 488, 867, doi: 10.1051/0004-6361:200809716
  * Ouyang et al. (2021) Ouyang, Z., Xiao, H., Zheng, Y., Xu, P., & Fan, J. 2021, Ap&SS, 366, 12, doi: 10.1007/s10509-020-03915-1
  * Paiano et al. (2017) Paiano, S., Landoni, M., Falomo, R., et al. 2017, ApJ, 837, 144, doi: 10.3847/1538-4357/837/2/144
  * Potter & Cotter (2015) Potter, W. J., & Cotter, G. 2015, MNRAS, 453, 4070, doi: 10.1093/mnras/stv1657
  * Prandini & Ghisellini (2022) Prandini, E., & Ghisellini, G. 2022, Galaxies, 10, 35, doi: 10.3390/galaxies10010035
  * Scarpa & Falomo (1997) Scarpa, R., & Falomo, R. 1997, A&A, 325, 109
  * Seta et al. (2009) Seta, H., Isobe, N., Tashiro, M. S., et al. 2009, PASJ, 61, 1011, doi: 10.1093/pasj/61.5.1011
  * Stickel et al. (1991) Stickel, M., Padovani, P., Urry, C. M., Fried, J. W., & Kuehr, H. 1991, ApJ, 374, 431, doi: 10.1086/170133
  * Tagliaferri et al. (2008) Tagliaferri, G., Foschini, L., Ghisellini, G., et al. 2008, ApJ, 679, 1029, doi: 10.1086/586731
  * Tavecchio et al. (1998) Tavecchio, F., Maraschi, L., & Ghisellini, G. 1998, ApJ, 509, 608, doi: 10.1086/306526
  * Urry & Padovani (1995) Urry, C. M., & Padovani, P. 1995, PASP, 107, 803, doi: 10.1086/133630
  * Valverde et al. (2020) Valverde, J., Horan, D., Bernard, D., et al. 2020, ApJ, 891, 170, doi: 10.3847/1538-4357/ab765d
  * Xiao et al. (2022) Xiao, H., Ouyang, Z., Zhang, L., et al. 2022, ApJ, 925, 40, doi: 10.3847/1538-4357/ac36da
  * Yang et al. (2022a) Yang, J. H., Fan, J. H., Liu, Y., et al. 2022a, ApJS, 262, 18, doi: 10.3847/1538-4365/ac7deb
  * Yang et al. (2022b) Yang, W. X., Wang, H. G., Liu, Y., et al. 2022b, ApJ, 925, 120, doi: 10.3847/1538-4357/ac3a09
  * Zhang et al. (2020) Zhang, L., Chen, S., Xiao, H., Cai, J., & Fan, J. 2020, ApJ, 897, 10, doi: 10.3847/1538-4357/ab9180

Table 1: The sample of 48 TeV blazars in the quiescent state.

TeV name | Other name | $z$ | Class | $\log\,\nu_{p}^{\rm ob}$ | $\delta$ | $N_{0,\rm abs}$ | $\Delta N_{0,\rm abs}$ | $\Gamma_{\rm abs}$ | $\Delta\Gamma_{\rm abs}$ | $E_{0}$ | $N_{0}$ | $\Delta N_{0}$ | $\Gamma$ | $\Delta\Gamma$ | $E_{0}$ | Ref.
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---
TeV J0013-188 | SHBL J001355.9-185406 | 0.094 | B | 14.96 | 29 | 1.20E-12 | 2.09E-14 | 3.43 | 0.05 | 0.51 | 1.94E-12 | 5.98E-14 | 2.78 | 0.08 | 0.51 | H. E. S. S. Collaboration et al. (2013a)
TeV J0033-193 | KUV 00311-1938 | 0.61 | B | 16.81 | 13.2 | 3.14E-11 | 3.00E-27 | 4.39 | 2.60E-16 | 0.17 | 8.37E-11 | 1.47E-26 | 1.92 | 4.77E-16 | 0.17 | Abdalla et al. (2020)
TeV J0112+227 | S2 0109+22 | 0.265 | B | 13.90 | 53.4 | 1.56E-10 | 1.86E-11 | 3.39 | 0.22 | 0.16 | 4.84E-10 | 7.47E-11 | 3.03 | 0.35 | 0.12 | MAGIC Collaboration et al. (2018a)
TeV J0152+017 | RGB J0152+017 | 0.08 | B | 15.73 | 24.8 | 6.24E-13 | 5.81E-14 | 2.95 | 0.12 | 1 | 1.34E-12 | 1.14E-13 | 2.43 | 0.11 | 1 | Aharonian et al. (2008)
TeV J0214+517 | TXS 0210+515 | 0.049 | B | 15.92 | 4.7 | 1.32E-13 | 3.28E-15 | 1.95 | 0.03 | 1.574 | 2.38E-13 | 1.47E-15 | 1.65 | 0.01 | 1.574 | Acciari et al. (2020)
TeV J0222+430 | 3C 66A | 0.444 | B | 14.96 | 36 | 1.67E-11 | 2.24E-12 | 3.08 | 0.18 | 0.3 | 2.48E-10 | 2.49E-11 | 0.67 | 0.10 | 0.3 | Aliu et al. (2009)
TeV J0232+202 | 1ES 0229+200 | 0.139 | B | 16.26 | 14.2 | 6.74E-13 | 9.44E-14 | 2.51 | 0.15 | 1 | 2.70E-12 | 4.08E-13 | 1.44 | 0.20 | 1 | Aharonian et al. (2007a)
TeV J0303-241 | PKS 0301-243 | 0.2657 | B | 16.10 | 16.4 | 1.09E-11 | 1.85E-12 | 4.48 | 0.62 | 0.29 | 2.27E-11 | 2.83E-12 | 2.95 | 0.47 | 0.29 | H. E. S. S. Collaboration et al. (2013b)
TeV J0319+187 | RBS 0413 | 0.19 | B | 16.98 | 3.7 | 1.37E-11 | 1.11E-12 | 3.18 | 0.15 | 0.3 | 2.10E-11 | 8.82E-13 | 1.89 | 0.08 | 0.3 | Aliu et al. (2012)
TeV J0349-119 | 1ES 0347-121 | 0.188 | B | 17.47 | 14.3 | 4.52E-13 | 6.97E-14 | 3.10 | 0.18 | 1 | 3.27E-12 | 4.29E-13 | 1.81 | 0.16 | 1 | Aharonian et al. (2007b)
TeV J0416+010 | 1ES 0414+009 | 0.287 | B | 17.61 | 2 | 5.54E-12 | 6.33E-13 | 3.29 | 0.15 | 0.305 | 1.59E-11 | 2.91E-12 | 1.50 | 0.25 | 0.305 | H. E. S. S. Collaboration et al. (2012a)
TeV J0449-438 | PKS 0447-439 | 0.205 | U | 13.08 | – | 2.81E-13 | 1.13E-13 | 3.99 | 0.44 | 1 | 2.53E-12 | 9.81E-13 | 2.57 | 0.43 | 1 | H. E. S. S. Collaboration et al. (2013c)
TeV J0509+056 | TXS 0506+056 | 0.3365 | B | 14.43 | – | 6.42E-11 | 1.16E-11 | 4.81 | 0.91 | 0.15 | 8.20E-11 | 1.68E-11 | 4.05 | 1.05 | 0.15 | Abeysekara et al. (2018)
TeV J0521+211 | VER J0521+211 | 0.108 | B | 15.24 | 14.3 | 1.99E-11 | 1.27E-12 | 3.46 | 0.14 | 0.4 | 3.08E-11 | 1.70E-12 | 2.83 | 0.13 | 0.4 | Archambault et al. (2013)
TeV J0648+152 | RX J0648.7+1516 | 0.179 | B | 17.07 | 6.7 | 2.32E-11 | 2.71E-12 | 4.37 | 0.43 | 0.3 | 3.74E-11 | 4.30E-12 | 3.48 | 0.43 | 0.3 | Aliu et al. (2011)
TeV J0710+591 | RGB J0710+591 | 0.125 | B | 17.25 | 1.5 | 5.58E-12 | 1.03E-12 | 2.70 | 0.20 | 0.5 | 1.08E-11 | 2.16E-12 | 1.88 | 0.22 | 0.5 | Acciari et al. (2010a)
TeV J0809+523 | 1ES 0806+524 | 0.138 | B | 16.26 | 3.7 | 3.90E-12 | 7.56E-13 | 2.77 | 0.25 | 0.5 | 7.77E-12 | 2.33E-12 | 2.18 | 0.41 | 0.5 | Aleksić et al. (2015)
TeV J0847+115 | RBS 723 | 0.198 | B | 16.37∗ | 1.9 | 1.10E-11 | 2.00E-12 | 3.76 | 0.71 | 0.3 | 1.84E-11 | 3.96E-12 | 2.93 | 0.86 | 0.3 | Acciari et al. (2020)
TeV J0854+201 | OJ 287 | 0.3056 | B | 13.62 | 67.5 | 3.20E-11 | 6.45E-12 | 2.24 | 0.29 | 0.1 | 3.53E-11 | 1.23E-11 | 1.56 | 0.50 | 0.1 | Seta et al. (2009)
TeV J1010-313 | 1RXS J101015.9-311909 | 0.142639 | B | 16.26 | 45.8 | 2.04E-13 | 8.68E-14 | 3.00 | 0.46 | 1 | 8.71E-13 | 3.71E-13 | 2.06 | 0.47 | 1 | H. E. S. S. Collaboration et al. (2012b)
TeV J1015+494 | 1ES 1011+496 | 0.212 | B | 16.68 | 1.3 | 1.97E-10 | 3.47E-11 | 4.11 | 0.51 | 0.2 | 2.60E-10 | 5.50E-11 | 3.56 | 0.67 | 0.2 | Albert et al. (2007)
TeV J1103-234 | 1ES 1101-232 | 0.186 | B | 17.07 | 1.4 | 5.52E-13 | 7.95E-14 | 2.94 | 0.19 | 1 | 3.92E-12 | 4.90E-13 | 1.68 | 0.16 | 1 | Aharonian et al. (2007c)
TeV J1104+382 | Mkn 421 | 0.031 | B | 16.61 | 1.5 | 1.58E-10 | 1.48E-11 | 2.89 | 0.19 | 0.3 | 1.70E-10 | 1.58E-11 | 2.74 | 0.19 | 0.3 | Baloković et al. (2016)
TeV J1136+701 | Mkn 180 | 0.045 | B | 16.72 | 1.4 | 4.27E-11 | 2.58E-12 | 3.16 | 0.08 | 0.3 | 4.75E-11 | 3.90E-12 | 2.91 | 0.11 | 0.3 | Albert et al. (2006)
TeV J1217+301 | 1ES 1215+303 | 0.131 | B | 15.35 | 15.1 | 1.53E-11 | 1.28E-12 | 3.29 | 0.16 | 0.4 | 2.66E-11 | 2.27E-12 | 2.61 | 0.17 | 0.4 | Valverde et al. (2020)
TeV J1221+282 | W Comae | 0.103 | B | 15.54 | 13.8 | 7.35E-12 | 7.10E-13 | 3.06 | 0.21 | 0.5 | 1.26E-11 | 1.23E-12 | 2.38 | 0.22 | 0.5 | Acciari et al. (2009a)
TeV J1221+301 | 1ES 1218+304 | 0.182 | B | 17.37 | – | 1.15E-11 | 7.81E-13 | 3.07 | 0.10 | 0.5 | 3.39E-11 | 1.76E-12 | 1.98 | 0.08 | 0.5 | Acciari et al. (2010b)
TeV J1315-426 | 1ES 1312-423 | 0.105 | B | 16.44 | – | 2.40E-13 | 2.51E-14 | 2.68 | 0.15 | 1 | 6.68E-13 | 7.73E-14 | 2.00 | 0.17 | 1 | HESS Collaboration et al. (2013)
TeV J1422+323 | B2 1420+32 | 0.682 | F | 13.63 | 4.3 | 1.04E-09 | 4.23E-11 | 4.19 | 0.09 | 0.1 | 1.70E-09 | 2.52E-10 | 3.31 | 0.36 | 0.1 | MAGIC Collaboration et al. (2021)
TeV J1427+238 | PKS 1424+240 | 0.16 | B | 16.26 | 34.8 | 5.01E-11 | 4.67E-12 | 3.85 | 0.27 | 0.2 | 6.05E-11 | 6.62E-12 | 3.32 | 0.34 | 0.2 | Acciari et al. (2010c)
TeV J1428+426 | 1ES 1426+428 | 0.129 | B | 17.35 | 2.5 | 2.47E-11 | 3.69E-12 | 2.50 | 0.16 | 0.242 | 3.29E-11 | 3.62E-12 | 1.84 | 0.13 | 0.242 | Acciari et al. (2020)
TeV J1443+120 | 1ES 1440+122 | 0.16306 | B | 16.97 | 1.5 | 1.50E-12 | 3.24E-13 | 3.09 | 0.44 | 0.5 | 3.75E-12 | 8.12E-13 | 2.10 | 0.46 | 0.5 | Archambault et al. (2016)
TeV J1443-391 | PKS 1440-389 | 0.065 | B | 16.33 | – | 3.10E-11 | 4.75E-12 | 3.38 | 0.27 | 0.274 | 3.63E-11 | 4.97E-12 | 3.05 | 0.25 | 0.274 | Abdalla et al. (2020)
TeV J1512-091 | PKS 1510-089 | 0.361 | F | 13.93 | 10.5 | 7.73E-11 | 4.17E-12 | 3.05 | 0.10 | 0.175 | 1.22E-10 | 2.07E-11 | 2.42 | 0.35 | 0.175 | MAGIC Collaboration et al. (2018b)
TeV J1517-243 | AP Librae | 0.048 | B | 14.12 | 20.9 | 4.00E-12 | 4.96E-13 | 2.73 | 0.18 | 0.45 | 5.02E-12 | 6.07E-13 | 2.47 | 0.18 | 0.45 | H. E. S. S. Collaboration et al. (2015)
TeV J1555+111 | PG 1553+113 | 0.36 | B | 16.33 | 11.4 | 5.19E-11 | 6.38E-12 | 4.22 | 0.30 | 0.3 | 1.77E-10 | 1.60E-11 | 2.55 | 0.23 | 0.3 | Aliu et al. (2015)
TeV J1653+397 | Mkn 501 | 0.034 | B | 16.81 | 2.3 | 1.74E-10 | 1.04E-11 | 2.48 | 0.06 | 0.3 | 1.86E-10 | 1.09E-11 | 2.28 | 0.06 | 0.3 | Ahnen et al. (2017)
TeV J1725+118 | H 1722+119 | 0.4 | B | 16.35 | 14.3 | 4.21E-11 | 8.10E-12 | 3.56 | 0.24 | 0.2 | 8.24E-11 | 2.96E-11 | 2.83 | 0.51 | 0.2 | Ahnen et al. (2016a)
TeV J1743+196 | 1ES 1741+196 | 0.084 | B | 15.54 | 17.3 | 9.28E-12 | 2.39E-13 | 2.70 | 0.10 | 0.3 | 1.12E-11 | 4.43E-13 | 2.36 | 0.15 | 0.3 | Abeysekara et al. (2016)
TeV J1943+213 | HESS J1943+213 | 0.16 | U | 16.66 | – | 9.61E-12 | 5.81E-13 | 2.81 | 0.10 | 0.4 | 3.24E-11 | 2.82E-12 | 1.81 | 0.13 | 0.3 | Archer et al. (2018)
TeV J1959+651 | 1ES 1959+650 | 0.047 | B | 17.32 | – | 2.67E-12 | 2.28E-13 | 2.54 | 0.11 | 1 | 4.13E-12 | 3.80E-13 | 2.25 | 0.12 | 1 | Tagliaferri et al. (2008)
TeV J2009-488 | PKS 2005-489 | 0.071 | B | 16.63 | 6.7 | 1.34E-11 | 1.78E-12 | 3.20 | 0.17 | 0.4 | 1.70E-11 | 2.14E-12 | 2.71 | 0.16 | 0.4 | H. E. S. S. Collaboration et al. (2010a)
TeV J2039+523 | 1ES 2037+521 | 0.053 | B | 15.82 | 53 | 6.40E-12 | 1.13E-13 | 2.27 | 0.02 | 0.4 | 8.06E-12 | 1.37E-13 | 1.99 | 0.02 | 0.4 | Acciari et al. (2020)
TeV J2158-302 | PKS 2155-304 | 0.116 | B | 16.25 | 11.1 | 1.83E-12 | 1.05E-13 | 3.53 | 0.05 | 1 | 5.30E-12 | 3.86E-13 | 2.88 | 0.06 | 1 | H. E. S. S. Collaboration et al. (2010b)
TeV J2250+384 | B3 2247+381 | 0.1187 | B | 16.55 | 25.7 | 1.36E-11 | 1.17E-12 | 3.31 | 0.13 | 0.3 | 1.94E-11 | 1.79E-12 | 2.69 | 0.14 | 0.3 | Aleksić et al. (2012)
TeV J2347+517 | 1ES 2344+514 | 0.044 | B | 16.62 | 2.7 | 2.64E-12 | 1.77E-13 | 2.49 | 0.08 | 0.91 | 3.88E-12 | 2.40E-13 | 2.20 | 0.07 | 0.91 | Allen et al. (2017)
TeV J2359-306 | H 2356-309 | 0.165 | B | 16.67 | 14.3 | 3.50E-13 | 4.55E-14 | 2.97 | 0.15 | 1 | 1.89E-12 | 2.03E-13 | 1.89 | 0.12 | 1 | H. E. S. S. Collaboration et al. (2010c)
4FGL J2042.1+2427 | RGB J2042.1+2426 | 0.104 | B | 16.74 | 15.2 | 2.61E-12 | 3.00E-13 | 2.31 | 0.13 | 0.379 | 4.11E-12 | 6.31E-13 | 1.76 | 0.18 | 0.379 | Acciari et al. (2020)

Note. — Col. (1): TeV name; Col. (2): other name; Col. (3): redshift; Col.
(4): classification, B for BLL, F for FSRQ and U for BCU; Col. (5):
synchrotron peak frequency, TeV J0847+115 is from Fan et al. (2016) and the
others are from Yang et al. (2022a); Col. (6): Doppler factor from Chen
(2018); Col. (7): the absorbed normalization in units of $\rm cm^{-2}\cdot
s^{-1}\cdot TeV^{-1}$; Col. (8): the error of the absorbed normalization in
units of $\rm cm^{-2}\cdot s^{-1}\cdot TeV^{-1}$; Col. (9): the absorbed
photon index; Col. (10): the error of the absorbed photon index; Col. (11):
the normalization energy in units of TeV; Col. (12): the de-absorbed
normalization in units of $\rm cm^{-2}\cdot s^{-1}\cdot TeV^{-1}$; Col. (13):
the error of the de-absorbed normalization in units of $\rm cm^{-2}\cdot
s^{-1}\cdot TeV^{-1}$; Col. (14): the de-absorbed photon index; Col. (15): the
error of the de-absorbed photon index; Col. (16): the normalization energy in
units of TeV; Col. (17): the reference of TeV spectra. The symbol ‘–’
indicates a null value.

Table 2: The sample of 21 TeV blazars in the flaring state.

TeV name | Other name | z | Class | $\log\,\nu_{p}^{\rm ob}$ | $\delta$ | $N_{0,\rm abs}$ | $\Delta N_{0,\rm abs}$ | $\Gamma_{\rm abs}$ | $\Delta\Gamma_{\rm abs}$ | $E_{0}$ | $N_{0}$ | $\Delta N_{0}$ | $\Gamma$ | $\Delta\Gamma$ | $E_{0}$ | Ref.
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---
TeV J0112+227 | S2 0109+22 | 0.265 | B | 13.90 | 53.4 | 1.62E-09 | 5.92E-11 | 3.57 | 0.09 | 0.11 | 1.38E-09 | 7.38E-12 | 3.28 | 0.01 | 0.12 | MAGIC Collaboration et al. (2018a)
TeV J0218+359 | S3 0218+35 | 0.944 | F | 12.99 | – | 2.18E-09 | 2.17E-10 | 3.91 | 0.45 | 0.1 | 4.28E-09 | 7.75E-10 | 2.45 | 0.89 | 0.1 | Ahnen et al. (2016b)
TeV J0521+211 | VER J0521+211 | 0.108 | B | 15.24 | 14.3 | 2.35E-10 | 1.96E-11 | 3.16 | 0.12 | 0.3 | 3.32E-10 | 1.44E-11 | 2.72 | 0.07 | 0.3 | Adams et al. (2022a)
TeV J0721+713 | S5 0716+714 | 0.31 | B | 14.42 | 20.3 | 1.23E-09 | 7.35E-11 | 4.65 | 0.16 | 0.15 | 1.47E-09 | 1.27E-10 | 3.65 | 0.24 | 0.15 | MAGIC Collaboration et al. (2018c)
TeV J0809+523 | 1ES 0806+524 | 0.138 | B | 16.26 | 3.7 | 1.27E-11 | 1.65E-12 | 2.92 | 0.19 | 0.5 | 2.75E-11 | 2.93E-12 | 2.19 | 0.16 | 0.5 | Aleksić et al. (2015)
TeV J0854+201 | OJ 287 | 0.3056 | B | 13.62 | 67.5 | 1.48E-11 | 1.15E-12 | 2.18 | 0.11 | 0.1 | 1.59E-11 | 3.76E-12 | 1.42 | 0.33 | 0.1 | Seta et al. (2009)
TeV J0958+655 | S4 0954+65 | 0.367 | B | 14.04 | 26.5 | 1.01E-09 | 5.63E-11 | 4.64 | 0.24 | 0.15 | 1.38E-09 | 2.86E-11 | 3.98 | 0.09 | 0.15 | MAGIC Collaboration et al. (2018d)
TeV J1104+382 | Mkn 421 | 0.031 | B | 16.61 | 1.5 | 2.49E-11 | 1.24E-12 | 2.28 | 0.06 | 1 | 3.31E-11 | 1.71E-12 | 2.09 | 0.06 | 1 | Acciari et al. (2009b)
TeV J1159+292 | Ton 599 | 0.7247 | F | 13.54 | 29.3 | 3.08E-10 | 1.76E-11 | 4.37 | 0.13 | 0.2 | 1.56E-09 | 3.15E-10 | 0.16 | 0.61 | 0.2 | Adams et al. (2022b)
TeV J1217+301 | 1ES 1215+303 | 0.131 | B | 15.35 | 15.1 | 9.42E-11 | 6.82E-12 | 3.59 | 0.14 | 0.4 | 8.49E-11 | 6.77E-12 | 2.93 | 0.12 | 0.5 | Valverde et al. (2020)
TeV J1221+282 | W Comae | 0.103 | B | 15.54 | 13.8 | 2.11E-11 | 3.23E-12 | 3.72 | 0.37 | 0.4 | 3.19E-11 | 4.39E-12 | 3.11 | 0.35 | 0.4 | Acciari et al. (2008)
TeV J1224+213 | 4C +21.35 | 0.435 | F | 14.06 | 11.3 | 7.85E-10 | 9.37E-11 | 3.73 | 0.21 | 0.2 | 1.85E-09 | 5.69E-11 | 2.63 | 0.06 | 0.2 | Aleksić et al. (2011)
TeV J1256-057 | 3C 279 | 0.5362 | F | 12.79 | 27.7 | 4.45E-10 | 1.08E-10 | 4.33 | 0.44 | 0.2 | 8.66E-10 | 5.20E-10 | 3.75 | 1.03 | 0.2 | MAGIC Collaboration et al. (2008)
TeV J1422+323 | B2 1420+32 | 0.682 | F | 13.63 | 4.3 | 2.41E-09 | 2.35E-10 | 4.12 | 0.18 | 0.1 | 3.75E-09 | 5.15E-10 | 3.49 | 0.27 | 0.1 | MAGIC Collaboration et al. (2021)
TeV J1443+250 | PKS 1441+25 | 0.93978 | F | 13.99 | – | 1.13E-09 | 1.14E-10 | 4.63 | 0.24 | 0.1 | 2.59E-09 | 1.25E-10 | 3.16 | 0.14 | 0.1 | Ahnen et al. (2015)
TeV J1512-091 | PKS 1510-089 | 0.361 | F | 13.93 | 10.5 | 3.27E-10 | 5.10E-11 | 4.22 | 0.18 | 0.265 | 8.46E-10 | 8.63E-11 | 3.34 | 0.13 | 0.265 | H. E. S. S. Collaboration et al. (2021)
TeV J1653+397 | Mn 501 | 0.034 | B | 16.81 | 2.3 | 6.32E-10 | 4.56E-11 | 2.26 | 0.07 | 0.3 | 6.76E-10 | 4.95E-11 | 2.06 | 0.07 | 0.3 | Ahnen et al. (2017)
TeV J1728+502 | 1ES 1727+502 | 0.055 | B | 16.52 | 2 | 7.71E-12 | 9.10E-13 | 2.19 | 0.24 | 0.62 | 1.09E-11 | 1.30E-12 | 1.82 | 0.24 | 0.62 | Archambault et al. (2015)
TeV J2202+422 | BL Lacertae | 0.069 | B | 13.93 | 3.8 | 5.55E-10 | 7.96E-11 | 3.69 | 0.48 | 0.3 | 6.50E-10 | 9.07E-11 | 3.38 | 0.47 | 0.3 | Arlen et al. (2013)
TeV J2243+203 | RGB J2243+203 | 0.39 | B | 15.14 | 14.3 | 6.33E-10 | 4.79E-11 | 4.62 | 0.24 | 0.15 | 8.69E-10 | 1.05E-10 | 3.44 | 0.44 | 0.15 | Abeysekara et al. (2017)
TeV J2347+517 | 1ES 2344+514 | 0.044 | B | 16.62 | 2.7 | 1.73E-11 | 1.27E-12 | 2.44 | 0.14 | 1 | 2.61E-11 | 1.83E-12 | 2.14 | 0.13 | 1 | Acciari et al. (2011)

Note. — Col. (1): TeV name; Col. (2): other name; Col. (3): redshift; Col.
(4): classification, B for BLL and F for FSRQ; Col. (5): synchrotron peak
frequency from Yang et al. (2022a); Col. (6): Doppler factor from Chen (2018);
Col. (7): the absorbed normalization in units of $\rm cm^{-2}\cdot s^{-1}\cdot
TeV^{-1}$; Col. (8): the error of the absorbed normalization in units of $\rm
cm^{-2}\cdot s^{-1}\cdot TeV^{-1}$; Col. (9): the absorbed photon index; Col.
(10): the error of the absorbed photon index; Col. (11): the normalization
energy in units of TeV; Col. (12): the de-absorbed normalization in units of
$\rm cm^{-2}\cdot s^{-1}\cdot TeV^{-1}$; Col. (13): the error of the de-
absorbed normalization in units of $\rm cm^{-2}\cdot s^{-1}\cdot TeV^{-1}$;
Col. (14): the de-absorbed photon index; Col. (15): the error of the de-
absorbed photon index; Col. (16): the normalization energy in units of TeV;
Col. (17): the reference of TeV spectra. The symbol ‘–’ indicates a null
value.

Table 3: The normalizations and parameters of spectrum in the quiescent state.

TeV name | $N_{0}$ | $\Delta N_{0}$ | $\Gamma$ | $\Delta\Gamma$ | $\beta$ | $\Delta\beta$ | $E_{0}$ | Observation periods
---|---|---|---|---|---|---|---|---
TeV J0013-188† | 7.6E-14 | 3.18E-14 | 1.71 | 0.26 | – | – | 2498.28394 | MJD 54683 - 55912
TeV J0033-193 | 1.36E-12 | 1.77E-13 | 1.74 | 0.11 | 0.10 | 0.06 | 1571.1472 | MJD 55145 - 56954
TeV J0112+227 | 3.29E-11 | 7.19E-12 | 1.71 | 0.26 | 0.06 | 0.10 | 755.25354 | MJD 57225 - 57231
TeV J0214+517† | 3.71E-14 | 1.75E-14 | 1.66 | 0.32 | – | – | 4042.61401 | MJD 57370 - 58042
TeV J0303-241 | 4.35E-12 | 7.08E-13 | 2.06 | 0.13 | 0.01 | 0.06 | 945.1168 | MJD 55044 - 55895
TeV J0319+187† | 1.51E-14 | 8.52E-15 | 1.43 | 0.31 | – | – | 6216.29443 | MJD 54732 - 55485
TeV J0416+010† | 4.16E-14 | 5.19E-15 | 1.67 | 0.07 | – | – | 3730.71851 | MJD 54683 - 55158
TeV J0449-438 | 4.52E-12 | 4.03E-13 | 1.87 | 0.08 | 0.03 | 0.04 | 1581.183 | MJD 55136 - 55227
TeV J0509+056 | 2.04E-11 | 1.46E-12 | 2.09 | 0.05 | 0.06 | 0.03 | 1074.2167 | MJD 58019.39 - 58155.18
TeV J0521+211 | 5.43E-12 | 9.49E-15 | 1.81 | 1.23E-03 | 0.01 | 4.33E-04 | 1542.1824 | MJD 55126 - 55212
TeV J0648+152† | 1.54E-14 | 3.87E-15 | 2.92 | 0.09 | – | – | 4960.08594 | MJD 55259 - 55301
TeV J0710+591† | 3.68E-14 | 1.4E-14 | 1.50 | 0.23 | – | – | 4958.52441 | MJD 54801 - 54921
TeV J0809+523 | 1.88E-12 | 3.46E-13 | 2.27 | 0.14 | 2.13E-06 | 1.16E-03 | 1345.8887 | MJD 55568 - 55622
TeV J1217+301 | 6.85E-12 | 6.52E-14 | 1.82 | 0.01 | 0.06 | 3.28E-03 | 1093.8949 | MJD 57202 - 57562
TeV J1221+301† | 2.09E-13 | 3.53E-14 | 1.54 | 0.11 | – | – | 4501.46387 | MJD 54801 - 54952
TeV J1422+323 | 3.96E-10 | 1.24E-12 | 1.87 | 2.63E-03 | 0.04 | 1.08E-03 | 579.01245 | MJD 58873.5 - 58880.5
TeV J1427+238 | 7.55E-12 | 5.36E-13 | 1.76 | 0.06 | 0.04 | 0.02 | 1204.567 | MJD 54881 - 55003
TeV J1443+120† | 6.28E-14 | 1.68E-14 | 1.79 | 0.20 | – | – | 3828.12671 | MJD 54683 - 55362
TeV J1443-391 | 7.18E-13 | 1.68E-13 | 1.25 | 0.15 | 0.20 | 0.07 | 1978.045 | MJD 55985 - 56074
TeV J1512-091 | 7.26E-11 | 2.27E-12 | 2.45 | 0.02 | 0.02 | 0.01 | 723.6382 | MJD 56000 - 58000
TeV J1517-243 | 8.55E-12 | 1.15E-12 | 2.32 | 0.12 | 6.77E-09 | 3.46E-05 | 811.1857 | MJD 55326 - 55689
TeV J1555+111 | 4.86E-12 | 3.36E-13 | 1.66 | 0.05 | 0.07 | 0.03 | 1802.4828 | MJD 55317 - 56108
TeV J1653+397 | 4.73E-12 | 4.49E-14 | 1.71 | 0.01 | 2.00E-10 | 5.10E-07 | 1486.4363 | MJD 54907 - 55004
TeV J1743+196† | 2.7E-14 | 2.54E-14 | 1.25 | 0.37 | – | – | 2791.80957 | MJD 54940 - 56834
TeV J2347+517 | 2.65E-13 | 2.21E-14 | 1.70 | 0.04 | 0.01 | 0.01 | 1934.7299 | MJD 54683 - 57023

Note. — ${\dagger}$: power-law function. The other sources are log-parabola
function.

Col. (1): TeV name; Col. (2):the normalization in units of $\rm cm^{-2}\cdot
s^{-1}\cdot MeV^{-1}$; Col. (3): the error of normalization in units of $\rm
cm^{-2}\cdot s^{-1}\cdot MeV^{-1}$; Col. (4): the photon index; Col. (5): the
error of the photon index; Col. (6): the curvature index of log-parabola
function; Col. (7): the error of the curvature index; Col. (8): the pivot
energy in units of MeV; Col. (9): the same observation periods as the VHE
observation. The symbol ‘–’ indicates a null value.

Table 4: The normalizations and parameters of spectrum in the flaring state.

TeV name | $N_{0}$ | $\Delta N_{0}$ | $\Gamma$ | $\Delta\Gamma$ | $\beta$ | $\Delta\beta$ | $E_{0}$ | Observation periods
---|---|---|---|---|---|---|---|---
TeV J0112+227 | 6.27E-11 | 2.67E-11 | 1.73 | 0.38 | 0.16 | 0.21 | 755.25354 | MJD 57228
TeV J0218+359 | 5.86E-11 | 1.49E-11 | 1.30 | 0.23 | 0.30 | 0.13 | 764.40204 | MJD 56863
TeV J0521+211 | 3.05E-11 | 7.51E-12 | 1.90 | 0.22 | 0.31 | 0.20 | 1542.1824 | MJD 56628.5 - 56632.5
TeV J0721+713 | 1.34E-10 | 9.44E-12 | 1.86 | 0.05 | 0.03 | 0.03 | 734.5862 | MJD 57040 - 57050
TeV J0809+523 | 1.39E-11 | 8.02E-12 | 2.14 | 0.49 | 0.14 | 0.30 | 1345.8887 | MJD 55616
TeV J0958+655 | 3.94E-10 | 5.94E-11 | 1.83 | 0.14 | 0.11 | 0.08 | 651.3296 | MJD 57067
TeV J1159+292 | 6.2E-10 | 6.11E-11 | 1.65 | 0.11 | 0.05 | 0.04 | 523.50714 | MJD 58102 - 58104
TeV J1217+301 | 5.74E-11 | 2.13E-11 | 1.09 | 0.48 | 0.17 | 0.15 | 1093.8949 | MJD 57844
TeV J1224+213 | 2.93E-09 | 2.49E-10 | 1.70 | 0.09 | 0.08 | 0.03 | 392.393 | MJD 55364
TeV J1422+323† | 3.28E-10 | 2.66E-11 | 1.90 | 0.06 | – | – | 749.958679 | MJD 58868.3 - 58870.3
TeV J1443+250 | 8.28E-11 | 8.22E-12 | 1.60 | 0.10 | 0.11 | 0.04 | 850.17804 | MJD 57130 - 57139.5
TeV J1512-091 | 2.57E-10 | 4.75E-11 | 1.74 | 0.16 | 0.01 | 0.06 | 723.6382 | MJD 57538
TeV J1653+397 | 9.38E-12 | 3.96E-12 | 1.48 | 0.27 | 0.05 | 0.10 | 1486.4363 | MJD 54952.41 - 54955
TeV J2202+422 | 1.63E-10 | 4.12E-11 | 1.92 | 0.24 | 0.33 | 0.21 | 796.1543 | MJD 55739.5 - 55740.5
TeV J2243+203 | 1.01E-11 | 5.48E-12 | 1.74 | 0.37 | 9.64E-09 | 3.98E-05 | 1517.4087 | MJD 57012

Note. — ${\dagger}$: power-law function. The other sources are log-parabola
function.

Col. (1): TeV name; Col. (2):the normalization in units of $\rm cm^{-2}\cdot
s^{-1}\cdot MeV^{-1}$; Col. (3): the error of normalization in units of $\rm
cm^{-2}\cdot s^{-1}\cdot MeV^{-1}$; Col. (4): the photon index; Col. (5): the
error of the photon index; Col. (6): the curvature index of log-parabola
function; Col. (7): the error of the curvature index; Col. (8): the pivot
energy in units of MeV; Col. (9): the same observation periods as the VHE
observation. The symbol ‘–’ indicates a null value.

##  Appendix A The fitting result of VHE spectrum

(a) (a) Figure 5: The fitting VHE spectrum of the quiescent state for each
source. The red circle for the original spectrum and the blue one for the de-
absorbed spectrum corrected by the EBL model from Franceschini et al. (2008).
Figure 6: The fitting VHE spectrum of the flaring state for each source. The
red circle for the original spectrum and the blue one for the de-absorbed
spectrum corrected by the EBL model from Franceschini et al. (2008).