# Constraining the Physical Parameters of Blazars Using the Seed Factor
Approach

Chang-Bin Deng  Department of Physics, Zhejiang Normal University, Jinhua
321004, People’s Republic of China  Yong-You Shi  Department of Physics,
Yunnan Normal University, Kunming 650500, People’s Republic of China  Yu-Jie
Song  Department of Physics, Yunnan Normal University, Kunming 650500,
People’s Republic of China  Rui Xue  Department of Physics, Zhejiang Normal
University, Jinhua 321004, People’s Republic of China  Lei-Ming Du  Department
of Physics, Yunnan Normal University, Kunming 650500, People’s Republic of
China  Ze-Rui Wang  College of Physics and Electronic Engineering, Qilu Normal
University, Jinan 250200, People’s Republic of China  Zhao-Hua Xie  Department
of Physics, Yunnan Normal University, Kunming 650500, People’s Republic of
China  [

###### Abstract

The discovery that blazars dominate the extra-galactic $\gamma$-ray sky is a
triumph in the Fermi era. However, the exact location of $\gamma$-ray emission
region still remains in debate. Low-synchrotron-peaked blazars (LSPs) are
estimated to produce high-energy radiation through the external Compton
process, thus their emission regions are closely related to the external
photon fields. We employed the seed factor approach proposed by Georganopoulos
et al. It directly matches the observed seed factor of each LSP with the
characteristic seed factors of external photon fields to locate the
$\gamma$-ray emission region. A sample of 1138 LSPs with peak frequencies and
peak luminosities was adopted to plot a histogram distribution of observed
seed factors. We also collected some spectral energy distributions (SEDs) of
historical flare states to investigate the variation of $\gamma$-ray emission
region. Those SEDs were fitted by both quadratic and cubic functions using the
Markov-chain Monte Carlo method. Furthermore, we derived some physical
parameters of blazars and compared them with the constraint of internal
$\gamma\gamma$-absorption. We find that dusty torus dominates the soft photon
fields of LSPs and most $\gamma$-ray emission regions of LSPs are located at
1-10 pc. The soft photon fields could also transition from dusty torus to
broad line region and cosmic microwave background in different flare states.
Our results suggest that the cubic function is better than the quadratic
function to fit the SEDs.

###### keywords:

radiation mechanisms: non-thermal – galaxies: jets – gamma-rays: general

These authors contributed equally to this work. These authors contributed
equally to this work. Rui Xue, Lei-Ming Du, Ze-Rui<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>
##  1 Introduction

Blazars are the most extreme subclass of active galactic nuclei (AGN) with a
relativistic jet pointing at the Earth Urry_Padovani95 (106). Due to the
beaming effect, they have high luminosity, fast variability and variable
polarization Urry_Padovani95 (106). According to the equivalent width (EW) of
the emission lines, blazars are divided into flat spectrum radio quasars
(FSRQs) with EW $\geq$ 5 Å and BL Lacertae objects (BL Lacs) with EW ¡ 5 Å,
respectively Urry_Padovani95 (106). In the $\log\nu$-$\log\nu F(\nu)$ diagram,
the non-thermal emission from jets dominates blazars’ spectral energy
distribution (SED), which usually shows a structure of double humps Macomb95
(72). Generally speaking, the low-energy hump is caused by synchrotron
radiation of relativistic electrons moving in the magnetic field
Marscher_Gear85 (76). Based on the peak frequency of the low-energy hump,
blazars are divided into low-synchrotron-peaked (LSP; i.e.,
$\nu^{\rm{S}}_{\rm{p}}<10^{14}$ Hz), intermediate-synchrotron-peaked (ISP;
i.e., $10^{14}<\nu^{\rm{S}}_{\rm{p}}<10^{15}$ Hz), high-synchrotron-peaked
(HSP; i.e., $\nu^{\rm{S}}_{\rm{p}}>10^{15}$ Hz), and extreme high-synchrotron-
peaked (EHSP; i.e., $\nu^{\rm{S}}_{\rm{p}}>10^{17}$ Hz) blazars Padovani95
(83, 31, 1). In the leptonic model, the high-energy hump is attributed to the
inverse Compton scattering (IC) from the same population of relativistic
electrons that emit the synchrotron emission. The seed photons for the IC
process could be from the synchrotron radiation (Maraschi92, 75, 105,
synchrotron self-Compton, SSC; e.g., ) or from external photon fields
(Dermer93, 39, 98, 21, external-Compton, EC; e.g., ). In addition, some
hadronic models have been proposed to explain the high-energy hump Aharonian00
(11, 23, 110).

Since the launch of the Fermi-Large Area Telescope (Fermi-LAT) in 2008, high-
energy astrophysics has undergone a transformative Fermi era marked by
profound discoveries Abdo10a (3, 4, 6). Though nearly 20% LSPs were out of
detection, it was found that the diffuse extra-galactic $\gamma$-ray
background is significantly dominated by emission from blazars Ajello15 (12,
8, 18). However, the exact location of $\gamma$-ray emission region is still
on debate Agudo12 (9, 62, 17, 101). Generally speaking, the $\gamma$-ray
emission of FSRQs is interpreted by the EC process, since strong ambient
photon fields are detected Madejski16 (73, 63). The LSP BL Lacs (LBLs) have
similar SEDs to those of FSRQs and occasionally show weak emission lines,
therefore, the $\gamma$-ray emission of LBLs can also be interpreted by the EC
process Madejski16 (73, 61). For LSPs whose high-energy emission originates
from the EC process, investigating the dominant soft photon fields could help
to locate the $\gamma$-ray emission region. If the $\gamma$-ray emission
region resides at the base of the jet, the soft photons should be dominated by
the accretion disc and the hot corona Dermer93 (39, 37, 109). When the
$\gamma$-ray emission region is positioned at sub-pc from the central
supermassive black hole (SMBH), the soft photons predominantly originate from
the broad line region (Sikora94, 98, 66, 20, 78, BLR; $R_{\rm{BLR}}\approx
0.1$ pc;). On the other hand, if the dissipation of the $\gamma$-ray emission
occurs at about 1-10 pc, the dominant soft photon source becomes the dusty
torus (Sikora08, 99, 117, DT; $R_{\rm{DT}}\approx 2.5$ pc;). In cases where
the $\gamma$-ray emission region is located in the extended jet, additional
external photon fields, such as the cosmic microwave background (CMB) and
starlight, play a significant role Bottcher08cmb (24, 88, 89, 90).

To pinpoint $\gamma$-ray emission regions of blazars, many methods have been
proposed: (i) variability: Tavecchio10_variability (103) studied the light
curves of 3C 454.3 and PKS 1510-089, and found significant short
variabilities, which indicates that the dissipation occurs in a very compact
region located in the BLR. Dotson12 (41) proposed that the variability
timescale of flares would not change in different bands within the BLR, but
should manifest faster variabilities at higher energies within the DT.
Applying this method to PKS 1510-089, they analyzed four prominent
$\gamma$-ray flares detected by Fermi in 2009 and concluded that $\gamma$-ray
emission regions are distributed over an extensive range of locations beyond
the BLR Dotson15 (42). (ii) radio core-shift: Based on radio core-shift
measurements, Yan18 (112) suggested that the distance between the SMBH and the
$\gamma$-ray emission region is less than 3.5 pc for PKS 1510-089 and less
than 0.02 pc for BL Lacertae in the framework of leptonic models. Wu18 (107)
determined the distance to be about ten times the typical size of the BLR for
23 LSPs. Jiang20 (65) used the time lags to derive the core size and inferred
that the $\gamma$-ray emission region of PMN J2345-1555 is probably inside the
BLR. (iii) model: Cao13 (26) reproduced the quasi-simultaneous SEDs of 21
FSRQs using the one-zone leptonic model and inferred that the locations of the
$\gamma$-ray emission regions are inside the BLR for 5 FSRQs and beyond the
BLR for 16 FSRQs. Tan20 (101) fitted the quasi-simultaneous SEDs of 60 FSRQs
with the same model and got similar results. Based on SED fitting, Arsioli18
(17) analysed the electron Lorentz factor and magnetic field strength for 104
LSPs, then found they are consistent with an EC model dominated by the DT.
However, SED fitting results are not always reliable due to coupling of
physical parameters, underscoring the importance of constraining some of them
through direct observations Yamada20 (111, 36).

In addition to the above three methods, Georganopoulos12 (49) proposed the
seed factor approach to study if the $\gamma$-ray emission region of blazars
is located near the BLR or DT. This method provides a convenient approach by
utilizing the peak frequencies and luminosities, which can be extracted from
the SEDs easily. Harvey20 (58) further applied this approach to a dataset
consisting of 62 FSRQs and demonstrated that the $\gamma$-ray emission regions
predominantly reside within the DT. This finding was subsequently confirmed by
Huang22 (63), who extended their analysis to a larger sample, including 619
FSRQs.

Recently, the SEDs of blazars in the Fourth Fermi-LAT 12-year Source catalog
(4FGL-DR3) have been fitted with the quadratic function by Yang22 (113, 114).
We apply the seed factor approach to this latest and largest sample of
$\gamma$-ray LSPs to study their dissipation region positions. Furthermore,
considering that blazars are highly variable objects Dotson15 (42, 18), some
flare states of various epochs are collected to investigate alterations in
their physical properties. This paper is organized as follows. In Section 2,
we present the methods, including the seed factor approach, SED fitting, and
comprehensive parameter analysis of the $\gamma$-ray emission regions. The
applications are presented in Section 3. In the end, we draw a conclusion in
Section 4. The cosmological parameters $H_{0}=69.6\rm{kms^{-1}Mpc^{-1}}$,
$\Omega_{0}=0.29$, and $\Omega_{\Lambda}=0.71$ are adopted in this work
Bennett14 (19).

##  2 Methods

###  2.1 Derivation of the Seed Factor

In this work, we adopt the seed factor approach to distinguish the location of
$\gamma$-ray emission region. Following Georganopoulos12 (49), we have peak
energies of synchrotron radiation and EC scattering in the observer’s frame,

$\epsilon^{\rm{obs}}_{\rm{syn}}=\frac{B}{B_{\rm{cr}}}\gamma^{2}_{\rm{b}}\delta/(1+z),$
(1)
$\epsilon^{\rm{obs}}_{\rm{EC}}=\frac{4}{3}\epsilon_{\rm{0,ext}}\gamma^{2}_{\rm{b}}\delta^{2}/(1+z),$
(2)

respectively Coppi_Blandford90 (30, 105, 50), where $B$ is the magnetic field
strength in units of Gauss; $\gamma_{\rm{b}}$ is the break Lorentz factor of
relativistic electrons responsible for the SED peaks; $\epsilon_{\rm{0,ext}}$
is the dimensionless energy of ambient soft photons in the AGN frame;
$B_{\rm{cr}}=m_{e}^{2}c^{3}/e\hbar$ is the critical magnetic field strength;
$\delta=[\Gamma(1-\beta\cos\theta_{\rm{obs}})]^{-1}$ is the Doppler factor,
where $\Gamma$ is the bulk Lorentz factor, $\beta c$ is the jet speed and
$\theta_{\rm{obs}}$ is the viewing angle. In this paper, by assuming
$\theta_{\rm{obs}}\lesssim 1/\Gamma$, we have $\delta\approx\Gamma$. It is
worth noting that equation (2) is only applicable within the Thomson regime.

Dividing equation (1) by equation (2), we obtain

$\frac{B}{\delta}=\frac{4B_{\rm{cr}}\epsilon_{\rm{0,ext}}\epsilon^{\rm{obs}}_{\rm{syn}}}{3\epsilon^{\rm{obs}}_{\rm{EC}}}.$
(3)

And the peak luminosities of synchrotron radiation and EC scattering in the
observer’s frame can be written as

$L^{\rm{obs}}_{\rm{syn,p}}=\frac{4}{3}\sigma_{\rm{T}}c\beta\gamma_{\rm{b}}^{2}n(\gamma_{\rm{b}})U_{\rm{B}}\delta^{4},$
(4)
$L^{\rm{obs}}_{\rm{EC,p}}=\frac{4}{3}\sigma_{\rm{T}}c\beta\gamma_{\rm{b}}^{2}n(\gamma_{\rm{b}})U_{\rm{ext}}\delta^{4},$
(5)

respectively Blumenthal_Gould70 (22, 93), where $\sigma_{\rm{T}}$ is the
Thomson cross section; $n(\gamma_{\rm{b}})$ is the electron density
distribution at $\gamma_{\rm{b}}$; $U_{\rm{B}}=B^{2}/8\pi$ is the energy
density of the magnetic field. Here, the energy density of the ambient photon
fields in the comoving frame can be calculated as

$U_{\rm{ext}}=\frac{17}{12}U_{\rm{0,ext}}\Gamma^{2},$ (6)

where $U_{\rm{0,ext}}$ is the energy density in the AGN frame
Ghisellini_Madau96 (55, 51).

Take the ratio of equation (4) and equation (5), we then get

$\frac{B^{2}}{\delta^{2}}=\frac{34\pi U_{\rm{0,ext}}}{3\it{CD}},$ (7)

where $\it{CD}=\it{L}^{\rm{obs}}_{\rm{EC,p}}/\it{L}^{\rm{obs}}_{\rm{syn,p}}$
is the Compton dominance.

Combining equation (3) and equation (7), we ultimately derive the seed factor
as

${\it{SF}}=\log(\frac{\sqrt{\it{U}_{\rm{0,ext}}}}{\epsilon_{\rm{0,ext}}})\approx\log(9863\times\frac{\nu^{\rm{obs}}_{\rm{syn,13}}}{\nu^{\rm{obs}}_{\rm{EC,22}}}\sqrt{\it{CD}}).$
(8)

Here, $\nu^{\rm{obs}}_{\rm{syn,13}}$ is the peak frequency of synchrotron
radiation in units of $\rm{10^{13}Hz}$ and $\nu^{\rm{obs}}_{\rm{EC,22}}$ is
the peak frequency of EC scattering in units of $\rm{10^{22}Hz}$.

###  2.2 Characteristic Values of the Seed Factor

As the ambient photon fields, the BLR and DT were discussed in the former seed
factor approach Georganopoulos12 (49, 58, 63). In addition, some studies
revealed the significance of CMB and starlight Bottcher08cmb (24, 88, 89, 90).
In this work, we comprehensively consider the seed factors of BLR, DT, CMB,
and starlight. The accretion disc is out of consideration, because it is not
suitable to this method.

To calculate the characteristic seed factor of BLR, the energy density
$U_{\rm{0,ext}}$ and the dimensionless energy $\epsilon_{\rm{0,ext}}$ of the
soft photons need to be determined. The typical size of BLR is
$R_{\rm{BLR}}\approx 1\times 10^{17}L_{\rm{d,45}}^{1/2}\ \rm{cm}$, where
$L_{\rm{d,45}}$ is the luminosity of the accretion disc in units of
$10^{45}\rm{erg\ s^{-1}}$ Kaspi07 (66, 20). The covering factor of BLR (the
fractions of the disk luminosity $L_{\rm{d}}$ reprocessed into the BLR
radiation) is $\xi_{\rm{BLR}}=0.1$ Ghisellini_Tavecchio09 (52). We then obtain
the energy density $U_{\rm{0,BLR}}=\xi_{\rm{BLR}}L_{\rm{d}}/4\pi
R_{\rm{BLR}}^{2}c=2.65\times 10^{-2}\rm{erg\ cm^{-3}}$ within the
characteristic distance. The BLR can be regarded as a grey body with peak
frequency of $1.5{\nu}_{{\rm{Ly}}_{\alpha}}$, then the dimensionless photon
energy is $\epsilon_{\rm{0,BLR}}=3\times 10^{-5}$ Tavecchio_Ghisellini081.5Lya
(104). Finally, with a $5\%$ uncertainty, we derive the characteristic seed
factor of BLR as ${\it{SF}}_{\rm{BLR}}=3.74\pm 0.19$.

The typical size of DT is found to be $R_{\rm{DT}}\approx 2.5\times
10^{18}L_{\rm{d,45}}^{1/2}\ \rm{cm}$ Ghisellini_Tavecchio09 (52, 87). In this
work, we set the covering factor of DT as $\xi_{\rm{DT}}=0.5$
Ghisellini_Tavecchio09 (52). Then the energy density of DT within the
characteristic distance is $U_{\rm{0,DT}}=\xi_{\rm{DT}}L_{\rm{d}}/4\pi
R_{\rm{DT}}^{2}c=2.12\times 10^{-4}\rm{erg\ cm^{-3}}$. In the studies, the DT,
which can also be approximated by a grey body, is endowed with three different
temperatures, e.g., 80K Lopez18_80K (70), 370K Ghisellini_Tavecchio09 (52),
1500K Almeyda17_1500K (14, 71). Then we obtain three dimensionless photon
energies for the DT, which are $\epsilon_{\rm{0,DT}}^{\rm{80K}}=5.30\times
10^{-8},\ \epsilon_{\rm{0,DT}}^{\rm{370K}}=2.45\times 10^{-7},\
\epsilon_{\rm{0,DT}}^{\rm{1500K}}=9.94\times 10^{-7}$. Considering the $5\%$
uncertainty, three distinct characteristic seed factors of DT can be described
as follows: ${\it{SF}}_{\rm{DT}}^{\rm{80K}}=5.44\pm 0.27,\
{\it{SF}}_{\rm{DT}}^{\rm{370K}}=4.77\pm 0.24,\
{\it{SF}}_{\rm{DT}}^{\rm{1500K}}=4.17\pm 0.21$.

For CMB, the energy density is $U_{\rm{CMB}}=4.02\times 10^{-13}\rm{erg\
cm^{-3}}$ and the typical temperature is $T_{\rm{CMB}}=2.72{\rm{K}}$ in the
observer’s frame, respectively Bottcher08cmb (24). In this case, the
characteristic seed factor of CMB with 5% uncertainty is
${\it{SF}}_{\rm{CMB}}=2.55\pm 0.13$. The energy density for starlight is
$U_{\rm{0,SL}}=8.01\times 10^{-13}\rm{erg\ cm^{-3}}$ and the typical
temperature is $T_{\rm{SL}}=30{\rm{K}}$. Then we get the characteristic seed
factor of starlight with 5% uncertainty ${\it{SF}}_{\rm{SL}}=1.65\pm 0.08$
HESS17_starlight (57).

When applying the seed factor approach, there are several caveats that should
be kept in mind. Firstly, the preceding derivation of the seed factor is
within the Thomson regime. As a result of $\gamma\epsilon<1/4$ Moderski05
(77), the corresponding peak frequency of the EC radiation belonging to BLR,
DT, CMB and starlight must be less than $1.03\times
10^{25}[\epsilon_{0}(1+z)/10^{-6}]^{-1}\rm{Hz}$. Since high-energy component
of LSP usually peaks around 1 GeV, the EC process associated to BLR occurs in
the Klein-Nishina regime, while others are cooling in the Thomson regime. Then
the available energy density of BLR could reduce and the actual characteristic
seed factor of BLR would be smaller than the above derived one. Secondly, the
energy density of the BLR and DT could be smaller at farther site as proposed
by Hayashida12 (59), i.e.,

$U_{\rm{0,BLR}}(r)=\frac{\xi_{\rm{BLR}}L_{\rm{d}}}{4\pi
R^{2}_{\rm{BLR}}c[1+(r/R_{\rm{BLR}})^{3}]},$ (9)

and

$U_{\rm{0,DT}}(r)=\frac{\xi_{\rm{DT}}L_{\rm{d}}}{4\pi
R^{2}_{\rm{DT}}c[1+(r/R_{\rm{DT}})^{4}]},$ (10)

where $r$ is the distance between the dissipation region and the central black
hole, both energy densities of the BLR and DT have been transformed into the
AGN frame (see also Figure 1). Then the actual characteristic seed factor
could also be smaller if the emission region is beyond the typical distance.
Since the above derived characteristic seed factor of DT is the largest one
among these four photon fields, the actual seed factor of DT covers that of
the others. For example, the actual seed factor of DT could decrease to about
3.5 and equal to the actual seed factor of BLR. Therefore, the above derived
seed factor of DT but not of BLR, CMB, or starlight is effective. If the
observed seed factor is approximated to the derived $\it{SF}_{\rm{DT}}$, it
can be ascertained that the DT dominates the soft photon fields.

Figure 1: Energy density distribution of the broad line region (BLR) and the
dusty torus (DT). $L_{\rm{d}}=1\times 10^{45}{\rm{erg\ s^{-1}}}$ is adopted.

###  2.3 The Fitting of Spectral Energy Distribution

As shown in equation (8), the observed seed factor can be determined by
extracting the peak frequencies and luminosities associated with the two
humps. Then we fit each SED by the quadratic and cubic functions,
respectively. Namely,

$\left\\{\begin{aligned} &\log(\nu
F_{\nu})=a_{2}(\log\nu)^{2}+b_{2}\log\nu+c_{2},\\\ &\log(\nu
F_{\nu})=a_{3}(\log\nu)^{3}+b_{3}(\log\nu)^{2}+c_{3}\log\nu+d_{3}.\end{aligned}\right.$
(11)

We use two kinds of functions because some SEDs possess high symmetry but
others do not, which causes difference on the parameters Xue16_Curvature
(108). Markov-chain Monte Carlo (MCMC) analysis is employed since it returns
robust uncertainties on the parameters Speagle19 (100). It works by randomly
sampling from the posterior distribution, which are derived from the product
of prior distribution and likelihood function. We use an uninformative uniform
prior distribution because we have little knowledge about the parameters of
quadratic and cubic functions in advance. Conservatively, it is expressed by

$p(m)=\left\\{\begin{aligned}
&\frac{1}{1000},&{\rm{if}}m_{0}-500<m<m_{0}+500\\\
&0,&\rm{otherwise}\end{aligned}\right.$ (12)

where $m$ denotes the paremeters in both quadratic and cubic functions, such
as $a_{2},b_{2}...$ And $m_{0}$ is the preprocessed $m$ derived by
numpy.polyfit111https://numpy.org/doc/stable/reference/generated/numpy.polyfit.html.
The likelihood function is written as Yamada20 (111)

${\it{L}}=\prod_{i=1}^{n}\frac{1}{\sqrt{2\pi{\sigma_{i}}^{2}}}\exp{\left(-\frac{({\nu}F_{\nu,i}-\nu
F_{\nu}(\nu_{i}))^{2}}{2\sigma^{2}_{i}}\right)}.$ (13)

Here, $\sigma_{i}$ is the Gaussian error of data point $i$ and $n$ is the
number of data points in each energy hump. The emcee Python
package222https://emcee.readthedocs.io/en/stable/ (Foreman13_emcee, 46,
version 3.1.2,) is utilized to perform the MCMC algorithm. This package
employs an affine invariant MCMC ensemble sampler with interdependent chains
Goodman_Weare10 (56). While there is no fixed number of samples needed to make
the convergence, we evaluate the convergence by inspecting the corner plot of
parameters. The autocorrelation time, also the time that the chain ”forgets”
where it started, range from 35 to 250 in our Python program. Conservatively,
we adopt 32 walkers (chains) initialized by the above preprocessed values with
a $10^{-10}$ Gaussian error, run 17000 steps, burn 2000 steps and thin by 25.
The results of posterior distribution are presented in Figure 2.

Figure 2: Posterior distribution from the Markov-chain Monte Carlo analysis of
OD 166’s low-energy (synchrotron) hump with quadratic function. We present
only one corner plot here, those of other SEDs are available in machine-
readable form.

While fitting the SEDs, we also compare the goodness of the two different
function models with modified Akaike Information Criterion (Akaike74, 13, 25,
AICc;), written as

${\rm{AICc}}=-2\ln(\hat{L})+2k+\frac{2k^{2}+2k}{n-k-1}.$ (14)

In this formula, $\hat{L}$ is the maximum likelihood, which corresponds to the
maximum posterior since we used an uniform prior, and $k$ is the number of
free parameters. This criterion is chosen because the sample size of data
points is small. AICc evaluates the loss of information during the fitting.
The smaller AICc means the better model. Actually, the additional parameters
in a model should improve the fitting to the data points, but we should also
consider the increase in model complexity which causes overfitting. In this
case, the AICc includes a penalty.

###  2.4 Parameters of the Emission Region

By fitting the SEDs of blazars, the observed peak frequencies and peak
luminosities are determined. Then we select the blazars whose seed factors
fall within the $\it{SF}_{\rm{DT}}$. With the gathered information on
variability timescales and Doppler factors, we can deduce the other physical
parameters of the $\gamma$-ray emission region. For blazars, we have such a
formula Tavecchio98 (105, 78):

$\frac{L^{\rm{obs}}_{\rm{syn}}}{U_{\rm{B}}}=\frac{L^{\rm{obs}}_{\rm{EC}}}{U_{\rm{ext}}}=\frac{L^{\rm{obs}}_{\rm{SSC}}}{U_{\rm{syn}}}.$
(15)

Here, $L^{\rm{obs}}_{\rm{syn}}$, $L^{\rm{obs}}_{\rm{EC}}$,
$L^{\rm{obs}}_{\rm{SSC}}$ are the luminosities of the synchrotron, EC, and SSC
radiations, respectively. $L^{\rm{obs}}_{\rm{syn}}$ and
$L^{\rm{obs}}_{\rm{EC}}$ are determined by the integral of the low-energy and
high-energy humps, respectively. For simplicity, we boldly assume that
$L^{\rm{obs}}_{\rm{SSC}}=10\times L^{\rm{obs}}_{\rm{X}}$, where
$L^{\rm{obs}}_{\rm{X}}$ is the maximum luminosity in X-ray band.
$U_{\rm{syn}}=L^{\rm{obs}}_{\rm{syn}}/(4\pi R^{2}c\delta^{4})$ is the energy
density of the synchrotron radiation. The radius of the $\gamma$-ray emission
region in the comoving frame can be estimated as

$R\approx ct_{\rm{var}}\frac{\delta}{1+z},$ (16)

where $t_{\rm{var}}$ is the variablility timescale.

If we have measured the variability timescale, we can derive the following
parameters by combining the above formulas:

$\left\\{\begin{aligned}
&\delta=6.39\left(\frac{L^{\rm{obs}}_{\rm{syn}}L^{\rm{obs}}_{\rm{EC}}}{L^{\rm{obs}}_{\rm{SSC}}\cdot
10^{45}\rm{erg\ s^{-1}}}\frac{\rm{10^{-4}erg\
cm^{-3}}}{U_{\rm{0,ext}}}\right)^{1/8}\left(\frac{t_{\rm{var}}}{1\rm{d}(1+z)}\right)^{-1/4},\\\
&R=1.66\times
10^{16}\left(\frac{L^{\rm{obs}}_{\rm{syn}}L^{\rm{obs}}_{\rm{EC}}}{L^{\rm{obs}}_{\rm{SSC}}\cdot
10^{45}\rm{erg\ s^{-1}}}\frac{\rm{10^{-4}erg\
cm^{-3}}}{U_{\rm{0,ext}}}\right)^{1/8}\left(\frac{t_{\rm{var}}}{1\rm{d}(1+z)}\right)^{3/4},\\\
&B=0.382\left(\frac{L^{\rm{obs}\ 5}_{\rm{syn}}}{L^{\rm{obs}\
3}_{\rm{EC}}L^{\rm{obs}}_{\rm{SSC}}\cdot 10^{45}\rm{erg\
s^{-1}}}\right)^{1/8}\left(\frac{U_{\rm{0,ext}}}{\rm{10^{-4}erg\
cm^{-3}}}\right)^{3/8}\left(\frac{t_{\rm{var}}}{\rm{1d}(1+z)}\right)^{-1/4}.\end{aligned}\right.$
(17)

When the Doppler factor is measured, we obtain:

$\left\\{\begin{aligned} &t_{\rm{var}}=1.44\times
10^{4}\left(\frac{L^{\rm{obs}}_{\rm{syn}}L^{\rm{obs}}_{\rm{EC}}}{L^{\rm{obs}}_{\rm{SSC}}\cdot
10^{45}\rm{erg\ s^{-s}}}\frac{\rm{10^{-4}erg\
cm^{-3}}}{U_{\rm{0,ext}}}\right)^{1/2}\left(\frac{\delta}{10}\right)^{-4}(1+z),\\\
&R=4.33\times
10^{15}\left(\frac{L^{\rm{obs}}_{\rm{syn}}L^{\rm{obs}}_{\rm{EC}}}{L^{\rm{obs}}_{\rm{SSC}}\cdot\rm{10^{45}erg\
s^{-1}}}\frac{\rm{10^{-4}erg\
cm^{-3}}}{U_{\rm{{0,ext}}}}\right)^{1/2}\left(\frac{\delta}{10}\right)^{-3},\\\
&B=0.597\left(\frac{L^{\rm{obs}}_{\rm{syn}}}{L^{\rm{obs}}_{\rm{EC}}}\frac{U_{\rm{0,ext}}}{\rm{10^{-4}erg\
cm^{-3}}}\right)^{1/2}\frac{\delta}{10}.\end{aligned}\right.$ (18)

Besides, $\gamma_{\rm{b}}$ can be calculated by Tavecchio98 (105)

$\gamma_{\rm{b}}=5.2\times
10^{-4}\left(\frac{\nu^{\rm{obs}}_{\rm{syn}}(1+z)}{B\delta}\right)^{1/2}.$
(19)

###  2.5 Constraint of the Internal $\gamma\gamma$-Absorption

To make a comprison with the above derivation of physical parameters, we
further make a constraint on $\delta$ through $\gamma\gamma$-absorption
(Dondi_Ghisellini95, 40, see also). Due to the $\gamma\gamma$ annihilation,
electron-positron pairs are generated. Applying Delta-approximation, the
corresponding optical depth can be calculated as Foffano22 (45):

$\tau_{\gamma\gamma}=\sigma_{\gamma\gamma}n_{\rm{soft}}R,$ (20)

where $n_{\rm{soft}}=U_{\rm{soft}}/h\nu_{\rm{soft}}$ is the number density of
the soft photons and $\sigma_{\gamma\gamma}=1.68\times 10^{-25}\rm{cm^{2}}$ is
the $\gamma\gamma$-absorption cross section, which is assumed to be a constant
when such a condition is fulfilled:

$\epsilon_{\rm{soft}}\epsilon_{\gamma}=2,$ (21)

where $\epsilon_{\rm{soft}}$ and $\epsilon_{\gamma}$ are the dimensionless
energies of the soft photons and $\gamma$-ray photons (comoving frame),
respectively Dermer_Menon09 (38).

Since the $\gamma$-ray is detected, optical depth must be less than 1. In
order to solve the optical depth, energy density need to be determined. For
internal soft photon fields such as synchrotron and IC radiation, we employ

$U_{\rm{soft}}=\frac{\nu L^{\rm{obs}}_{\nu,{\rm{soft}}}}{4\pi
R^{2}c\delta^{4}},$ (22)

where $\nu L^{\rm{obs}}_{\nu,{\rm{soft}}}=4\pi d_{\rm{L}}^{2}\nu
F^{\rm{obs}}_{\nu,{\rm{soft}}}$ is the observed luminosity of the soft
photons, $d_{\rm{L}}$ is the luminosity distance, and $\nu
F^{\rm{obs}}_{\nu,{\rm{soft}}}$ is the observed flux. Then we derive

$\tau_{\gamma\gamma}=\frac{\sigma_{\gamma\gamma}d^{2}_{\rm{L}}\nu
F^{\rm{obs}}_{\nu,{\rm{soft}}(1+z)}}{hc^{2}t_{\rm{var}}\delta^{5}\nu_{\rm{soft}}}<1.$
(23)

Here, $\nu_{\rm{soft}}$ could be derived from equation (21) and expressed by

$\nu_{\rm{soft}}=\frac{2(m_{\rm{e}}c^{2})^{2}}{h^{2}\nu_{\gamma}},$ (24)

where $\nu_{\gamma}=\nu_{\gamma}^{\rm{obs}}(1+z)/\delta$ is the frequency of
$\gamma$-ray in the comoving frame. Then we obtain the lower limit of
$\delta$:

$\delta>\left(\frac{h\sigma_{\gamma\gamma}d_{\rm{L}}^{2}\nu
F_{\nu,{\rm{soft}}}^{\rm{obs}}\nu_{\gamma}^{\rm{obs}}(1+z)^{2}}{2m_{\rm{e}}^{2}c^{6}t_{\rm{var}}}\right)^{1/6}.$
(25)

Not only do internal photon fields constrain the physical parameter, but
external photon fields also give an additional constraint on $r$. The
absorption of DT could be omitted according to equation (21), since the
corresponding $\gamma$-ray
($\nu_{\gamma}^{\rm{obs}}=10^{27}(\nu_{\rm{DT}}/10^{13}\rm{Hz})^{-1}\rm{Hz}$)
is beyond detection in our collected SEDs. However, the $\gamma$-ray up to
$10^{25}(\nu_{\rm{BLR}}/10^{15}\rm{Hz})^{-1}\rm{Hz}$ that is detectable could
be absorbed by BLR. Therefore, we inspect the $\gamma\gamma$-absorption of BLR
by unfolding its frequency spectrum. Given that the BLR is a grey body, we
have

$\frac{\rm{d}\it{U}}{{\rm{d}}\nu}=\frac{8\pi
h\nu^{3}}{c^{3}}(e^{h\nu/k_{\rm{B}}T}-1)^{-1},$ (26)

where $T=h\nu_{\rm{BLR}}/3.93k_{\rm{B}}$ is the characteristic temperature of
BLR and $k_{\rm{B}}$ is the Boltzmann constant Ghisellini_Tavecchio09 (52).
From the combination of equation (9) and (26), we derive the energy density of
BLR as a function of both $r$ and $\nu$:

$U_{\rm{BLR}}(r,\nu)=\Gamma^{2}U_{\rm{0,BLR}}(r)\nu\frac{\rm{d}\it{U}}{{\rm{d}}\nu}/\int\frac{\rm{d}\it{U}}{{\rm{d}}\nu}{\rm{d}}\nu.$
(27)

With the same condition about $\tau<1$ and lower limit of $\delta$ derived
from equation (25), we could get the lower limit of $r$:

$r>R_{\rm{BLR}}\left(\frac{\sigma_{\gamma\gamma}R}{h}\frac{\xi_{\rm{BLR}}\Gamma^{2}L_{\rm{d}}}{3\pi
R^{2}_{\rm{BLR}}c}\frac{\rm{d}\it{U}}{{\rm{d}}\nu}\bigg{|}_{\nu=\nu_{\rm{soft}}}/\int\frac{\rm{d}\it{U}}{{\rm{d}}\nu}\rm{d}\nu-1\right)^{1/3}.$
(28)

##  3 Application

###  3.1 Low-Synchrotron-Peaked Blazars

Table 1: Observed quantities and seed factors of 1138 low-synchrotron-peaked
blazars.

Fermi Name | Classification | $z$ | $\log\nu_{\rm{syn}}^{\rm{obs}}$ | $\log L_{\rm{syn}}^{\rm{obs}}$ | $\log\nu_{\rm{IC}}^{\rm{obs}}$ | $\log L^{\rm{obs}}_{\rm{IC}}$ | $\it{SF}$ | $\delta$
---|---|---|---|---|---|---|---|---
(1) | (2) | (3) | (4) | (5) | (6) | (7) | (8) | (9)
4FGL J0001.5+2113 | FSRQ | 1.106 | 13.2 | 46.17 | 20.6 | 47.21 | 6.10 |
4FGL J0003.3-1928 | BCU | 2.000 | 13.3 | 46.21 | 22.5 | 46.42 | 3.90 |
4FGL J0003.9-1149 | BLL | 0.860 | 13.2 | 45.9 | 23.1 | 45.28 | 2.81 |
4FGL J0004.3+4614 | FSRQ | 1.810 | 13.1 | 45.93 | 21.2 | 46.84 | 5.35 | 7.75
4FGL J0004.4-4737 | FSRQ | 0.880 | 13 | 45.96 | 21.7 | 45.89 | 4.30 |
··· | ··· | ··· | ··· | ··· | ··· | ··· | ··· | ···

  * •

NOTE: Column (1) gives the Fermi name. Column (2) represents the spectral
classification. Column (3) gives the redshift. Column (4) and (5) are the
synchrotron peak frequencies and luminosities from Yang22 (113), respectively.
Column (6) and (7) are the IC peak frequencies and luminosities from Yang23
(114), respectively. Column (8) gives the observed seed factors. Column (9)
gives the Doppler factors of 383 blazars from Liodakis18 (69). We present only
5 items here, full table is available in machine-readable form.

Figure 3: Histogram of observed seed factors. FSRQs, BL Lacs and blazar
candidates of uncertain type (BCU) in our sample are distinguished by three
kinds of grids. Three red areas represent the scales of characteristic seed
factors belonging to the dusty torus (DT) in three different temperatures.
Blue, green and yellow areas represent the scales of characteristic seed
factors belonging to the broad line region (BLR), cosmic microwave background
(CMB) and starlight (SL).

Locating the $\gamma$-ray emission region of blazars has been a significant
issue in the Fermi era. Debates continuously happen because of the limited
accuracy of instruments and complicated radiation mechanism of blazars. In
this work, we adopted the seed factor approach proposed by Georganopoulos12
(49). Based on the derivation in Section 2.1, some observed quantities need to
be determined. We collected a sample of 1138 LSPs with synchrotron peak
frequencies and luminosities from Yang22 (113), and with IC peak frequencies
and luminosities from Yang23 (114). There are 630 FSRQs, 132 BL Lacs and 376
blazar candidates of uncertain type (BCUs) in this sample (see also Table
LABEL:table1). The Doppler factors of 383 blazars are also recorded from
Liodakis18 (69) for further calculation. Figure 3 displays the histogram of
observed seed factors belonging to the LSPs, which is obtained using equation
(8). The observed seed factors of FSRQs and BCUs converge around the areas of
DT, which demonstrates that DT dominates the soft photon fields of FSRQs and
BCUs. This can be attributed to the strong radiation from DT, which is
reproduced by the strong radiation from accretion disc of FSRQs and BCUs
Madejski16 (73, 63). The observed seed factors of BL Lacs converge around the
area of BLR. This suggests that the soft photons either originate from BLR or
from DT, as the actual seed factor of DT could be smaller than the one
depicted in Figure 3. The areas of CMB and starlight appear on the left edge
of the histogram, indicating that their contributions to the soft photons in
the EC process are relatively small. In general, observed seed factors of 552
in 1138 LSPs directly fall into the areas of DT, which locates the
$\gamma$-ray emission region at 1-10 pc.

This result is consistent with the former analysis using the seed factor
approach. Harvey20 (58) calculated the seed factors of 62 FSRQs and found the
distribution peaking at a value corresponding to DT. Huang22 (63) used a
sample of 619 sources and also found the distribution is located at DT. In our
work, rather than setting the temperature of DT to 1200K, we considered three
different temperatures of DT because it is a relatively thick gas cloud with
its inner temperature varying from the outer one Lyu18_1500K (71). Figure 3
shows 370K dominates the distribution of observed seed factors, indicating
most $\gamma$-ray emission regions are located inside the DT.

###  3.2 LSPs Dominated by the Dusty Torus

Figure 4: Scatterplots of observed seed factors belonging to low-synchrotron-
peaked blazars dominated by the dusty torus. These data correspond to the
first 23 flare states in Table 2, ordered from top to bottom. The upper and
lower panels are derived from SEDs fitted by quadratic and cubic functions,
respectively. Red, blue and green areas depict the characteristic seed factors
of dusty torus, broad line region and cosmic microwave background,
respectively.

Although our result suggests that DT dominates the soft photon fields of LSPs,
previous study demonstrated that blazars might have various $\gamma$-ray
emission regions in different flare epochs Dotson15 (42). We investigated this
property by collecting some historical flare states. LSPs whose observed seed
factors directly fall into the red areas in Figure 3 (i.e., LSPs dominated by
the DT) were selected, since only their soft photon fields had been determined
effectively. The SEDs that we collected fulfilled these conditions: Possessing
multi-wavelength quasi-simultaneous data except for the radio band. The
observation times of each band intersect within two months. In total, we
collected 23 SEDs. The related references are given in Table 2. These SEDs
were fitted by both quadratic and cubic functions using the MCMC method (see
also Figures 7 & 8). Then we extracted the peak frequencies and luminosities
of two humps, and calculated the observed seed factors. The results are
displayed as scatterplots in Figure 4. The distributions of data scatters vary
under two different function fits. This demonstrates that the values of seed
factors are significantly influenced by the choice of the function. On the
other hand, both fitting results of two different functions show that some
scatters, i.e., 7 scatters of quadratic function and 9 scatters of cubic
function, move outside DT areas. Although the actual seed factor of DT could
be smaller and cover them, they have already moved to the areas of BLR or CMB.
This indicates that the location of $\gamma$-ray emission region changed in
historical flare states, and the soft photon fields could transition from DT
to BLR and CMB.

###  3.3 Four Typical Low-Synchrotron-Peaked Blazars

Figure 5: From top to bottom, the scatterplots represent the observed seed
factor distributions of historical states for CTA 102, 3C 279, TXS 0506+056,
and OJ 287, respectively. The left and right panels are derived from SEDs
fitted by quadratic and cubic functions, respectively. Red, blue and green
areas depict the characteristic seed factors of dusty torus, broad line region
and cosmic microwave background, respectively.

In order to further verify the alteration of $\gamma$-ray emission region in
different flare epochs, we collected SEDs of four typical LSPs, including 7
SEDs of CTA 102, 7 SEDs of 3C 279, 3 SEDs of TXS 0506+056 and 5 SEDs of OJ
287. The references of these SEDs are presented in Table 2. These SEDs
fulfilled the same conditions as above. Similarly, we fitted these SEDs with
both quadratic and cubic functions (see also Figures 10 to 13), extracted the
peak frequencies and luminosities, and created the scatterplots of observed
seed factors (see also Figure 5). The scatter distributions still vary
significantly under two different function fits. Figure 5 also shows that the
observed seed factors of the same LSPs are variable in different flare states.
Some scatters are positioned in red areas, while others are not, indicating
multiple locations of $\gamma$-ray emission regions in the same blazar. Some
previous studies also support our results. Pati18 (86) collected multi-
wavelength light curves for 3C 279 over 6 years and divided them into three
flaring periods. They analyzed the time delays and $\gamma$-ray spectral
index, then found that the dominant radiation mechanism and $\gamma$-ray
emission regions varied in different periods. Similar to the above LSPs,
Deng23 (35) located the $\gamma$-ray emission region of OT 081 at the edge of
BLR during the 2016 multiwavelength flare and at about 1-10 pc away from the
black hole during the 2009-2012 orphan X-ray flare. Given to the variability
of blazars, broadtime analysis has become a typical approach of Fermi-LAT to
reduce the impact of some short-lived flares Arsioli18b (18).

On September 22 2017, the IceCube Observatory detected a $\sim$290 TeV
neutrino from the direction of TXS 0506+056 IceCube18 (64, 82). The neutrino
was produced in the photopion process, in which the $\triangle^{+}(1232)$
resonance contributes the main cross section. Using Delta-approximation, we
could derive the external soft photon energy in the AGN frame,

$E_{\rm{0,soft}}\simeq\frac{50{\rm{eV}}}{1+z}\left(\frac{\delta}{\Gamma}\right)\left(\frac{290{\rm{TeV}}}{E^{\rm{obs}}_{\nu}}\right).$
(29)

It shows that the dissipation region was in the BLR. This neutrino event was
closely followed by two very high energy $\gamma$-ray flares of TXS 0506+056
Ansoldi18 (16, 96). These flares were denoted as Flare 2 and Flare 3 in Figure
12. The seed factors are presented in Figure 5, corresponding to serial number
2 and 3. Result of quadratic function depects that the dissipation region was
in the DT, contradicting the above calculation. But that of cubic function
approximately supports the BLR. This manifests the difference between two
functions and possible superiority of cubic function.

###  3.4 Model Comparison and Parameter Analysis

Figure 6: Parameter distribution of OD 166 when Doppler factor is known.
Possible value counts of each parameter is 19200 (=32$\times$(17000-2000)/25)
in total. The number of bins is 100. We present only one sample here, others
are available in machine-readable form.

We found that different authors adopted various functions to fit the SEDs.
Abdo10SEDfitting (2) employed the cubic function, while Chen14 (28, 113, 114)
used a simpler quadratic function. Xue16_Curvature (108) fitted the
Synchrotron hump with both functions and found the derived Synchrotron
luminosity with cubic function is lower. In this work, we tried both of the
two to further illuminate the difference. The AICc of each hump derived from
two functions are presented on the figures in Appendix. For the low-energy
humps, 33 (45 in total, one is null because $n=k+1$) AICc of cubic function
are smaller. As for high-energy humps, 35 (45 in total) AICc of cubic function
are smaller. This demonstrates a dominance of cubic function over quadratic
function both in two kinds of humps. We found that the humps with perfect
symmetry could be fitted well (i.e., smaller AICc) with quadratic function,
such as low-energy humps in PKS 2123-463, OJ 287 Flare 3, OJ 287 Flare 4,
etc., and high-energy humps in PKS 0208-512 Flare 1, PKS 0402-362 Flare 2, PKS
0420+022, etc. The asymmetrical SEDs, which take the most part of our samples,
could be explained well with cubic function. On the other hand, we found some
quadratic function curves deviate badly from the GeV data points, such as PKS
0208-512 Flare 2, TXS 0506+056 Flare 1, and NARO 512. It is worth noting that
GeV data is rarer than other wavebands’ in practical, given that the
$\gamma$-ray telescopes are relatively scarce. Thus the goodness of fit of
data points in GeV band has superiority. For 3C 279 Flare 4 and CTA 102 Flare
3, the quadratic function curves in high-energy hump even continuedly
increase, which breaks the physical reality. Therefore, we recommend the cubic
function to fit the SEDs.

However, cubic function is not suitable to all the SEDs, such as high-energy
humps in TXS 0536+145 and CTA 102 Flare 3 (see Appendix). The hard
$\gamma$-ray spectra and lack of higher-energy data make it difficult to form
a full peak. We also noticed that the conventional one-zone leptonic or
hadronic model could hardly explain the SEDs of bright TeV blazars, which
results in extreme values Abdo11 (5, 27, 68). In this case, some models of
inhomogeneous jet were constructed, such as spine-layer model Ghisellini05
(53, 102), two-zone leptonic model Shukla15 (97) and two-zone leptohadronic
model Aguilar22 (10).

Besides collecting the quasi-simultaneous data and fitting the SEDs, we
collected the corresponding Doppler factor from Liodakis18 (69) or observed
variability timescales from the original work, and obtained the distribution
of some related parameters using formulas in Section 2.4. For example, Figure
6 presents the derived parameter distribution of OD 166 when the corresponding
Doppler factor from Liodakis18 (69) is known. There are 19200 possible values
for each parameter from the MCMC fitting. The 16-50-84 percentile rule is
employed to characterize parameter uncertainty. The detailed results of
parameter analysis are listed in Table 2. The average value for variability
timescale, Doppler factor, magnetic field strength, radius of emission region,
and break Lorentz factor of electrons are 36.58 days, 23.40, 0.674 G,
$3.540\times 10^{16}$ cm, and 845.28, respectively. Table 2 shows that the
physical parameters of a specific blazar changed in different flaring epochs.
Feng22 (44) reproduced the SEDs of various flares belonging to 3C 454.3 under
a one-zone leptonic scenario and found similar variations in physical
parameters. Interestingly, Table 2 demonstrates that parameters under two
different function fits are analogous, which is unlike the scatterplots. This
could be attributed to the effect of symmetry in two humps. It causes more
influences in peak frequencies which are mainly used to calculate the observed
seed factors, but less in the integral of two humps which are used to
calculate the other parameters. We also constrained the Doppler factors and
$r$ for the collected LSPs whose observed timescales are known. Table 2
depicts that all the former derived Doppler factors are consistent with their
corresponding lower limit. We noticed that some lower limits of $r$ equal to
$0$. Because the results of derivation in Section 2.5 are negative numbers,
but $r$ must be non-negative numbers in real world. This indicates that the
constraint of internal $\gamma\gamma$-absorption on $r$ is relatively weak.

Table 2: Results of parameter analysis.

Fermi Name | Source Name | Time | $t_{\rm{var}}$ | $\delta$ | $B$ | $\log R$ | $\log\gamma_{\rm{b}}$ | $\delta_{\rm{low}}$ | $r_{\rm{low}}$ | Ref
---|---|---|---|---|---|---|---|---|---|---
|  |  | (days) |  | (G) | (cm) |  |  | (pc) |
(1) | (2) | (3) | (4) | (5) | (6) | (7) | (8) | (9) | (10) | (11)
4FGL J0242.3+1102∗ | OD 166 | 2013.04.18-2013.07.27 | $504.12^{+32.78}_{-27.82}$ | $4.37$ | $0.272^{+0.015}_{-0.017}$ | $18.191^{+0.027}_{-0.025}$ | $3.271^{+0.015}_{-0.014}$ |  |  | 1
|  |  | $627.67^{+15.68}_{-15.72}$ |  | $0.270^{+0.007}_{-0.007}$ | $18.286^{+0.011}_{-0.011}$ | $3.006^{+0.014}_{-0.014}$ |  |  |
4FGL J1354.8-1041∗ | PKS 1352-104 | 2008.08.11-2008.09.11 | $7.45^{+0.20}_{-0.20}$ | $6.85$ | $0.229^{+0.006}_{-0.006}$ | $16.997^{+0.012}_{-0.012}$ | $3.657^{+0.027}_{-0.027}$ |  |  | 2
|  |  | $6.84^{+0.23}_{-0.22}$ |  | $0.283^{+0.009}_{-0.009}$ | $16.959^{+0.014}_{-0.014}$ | $3.280^{+0.044}_{-0.043}$ |  |  |
4FGL J1549.5+0236∗ | PKS 1546+027 | 2010.01.18-2010.03.18 | $3.80^{+0.09}_{-0.09}$ | $10.75$ | $0.221^{+0.005}_{-0.006}$ | $16.874^{+0.011}_{-0.011}$ | $3.021^{+0.010}_{-0.010}$ |  |  | 3
|  |  | $3.39^{+0.09}_{-0.08}$ |  | $0.247^{+0.006}_{-0.006}$ | $16.825^{+0.011}_{-0.010}$ | $2.991^{+0.012}_{-0.011}$ |  |  |
4FGL J1635.2+3808∗ | 4C +38.41 | 2010.02.07-2010.04.07 | $0.58^{+0.02}_{-0.02}$ | $26.96$ | $1.274^{+0.057}_{-0.050}$ | $16.160^{+0.017}_{-0.019}$ | $2.376^{+0.010}_{-0.011}$ |  |  | 3
|  |  | $0.65^{+0.05}_{-0.04}$ |  | $1.222^{+0.077}_{-0.080}$ | $16.208^{+0.029}_{-0.027}$ | $2.412^{+0.016}_{-0.014}$ |  |  |
4FGL J1640.4+3945∗ | NRAO 512 | 2010.07.07-2010.09.07 | $0.31^{+0.02}_{-0.02}$ | $31.29$ | $0.960^{+0.070}_{-0.059}$ | $15.973^{+0.029}_{-0.032}$ | $2.518^{+0.017}_{-0.018}$ |  |  | 3
|  |  | $0.32^{+0.02}_{-0.01}$ |  | $0.974^{+0.048}_{-0.044}$ | $15.980^{0.020}_{-0.021}$ | $2.515^{+0.012}_{-0.012}$ |  |  |
4FGL J0438.4-1254∗ | PKS 0436-129 | 2010.06.01-2010.08.01 | $0.05^{+0.00}_{-0.00}$ | $46.93$ | $0.549^{+0.017}_{-0.018}$ | $15.423^{+0.014}_{-0.014}$ | $2.579^{+0.017}_{-0.016}$ |  |  | 3
|  |  | $0.04^{+0.00}_{0.00}$ |  | $0.623^{+0.030}_{-0.032}$ | $15.335^{+0.023}_{-0.021}$ | $2.494^{+0.020}_{-0.020}$ |  |  |
4FGL J2110.2-1021c∗ | PKS 2107-105 | unknown | $337.97_{-13.39}^{+14.27}$ | $4.03$ | $0.143^{+0.006}_{-0.006}$ | $18.004_{-0.018}^{+0.018}$ | $3.482^{+0.024}_{-0.022}$ |  |  | 4
|  |  | $359.10_{-13.47}^{+13.19}$ |  | $0.142_{-0.006}^{+0.006}$ | $18.030_{-0.016}^{0.016}$ | $3.478_{-0.026}^{+0.025}$ |  |  |
4FGL J1345.5+4453∗ | B3 1343+451 | 2013.07.27 | $0.10^{+0.00}_{-0.00}$ | $45.61$ | $0.647^{+0.014}_{-0.013}$ | $15.542^{+0.009}_{-0.009}$ | $2.624^{+0.010}_{-0.011}$ |  |  | 1
|  |  | $0.10^{+0.00}_{0.00}$ |  | $0.628^{+0.014}_{-0.014}$ | $15.517^{+0.009}_{-0.009}$ | $2.538^{+0.017}_{-0.018}$ |  |  |
4FGL J0210.7-5101 | PKS 0208-512 | 2016.10.20-2016.11.10 | $8.45$ | $10.67^{+0.19}_{-0.22}$ | $0.414^{+0.026}_{-0.021}$ | $17.066^{+0.008}_{-0.009}$ | $2.993^{+0.008}_{-0.013}$ | $3.06$ | $0$ | 5
|  |  |  | $10.81^{+0.35}_{-0.25}$ | $0.398^{+0.030}_{-0.036}$ | $17.072^{+0.014}_{-0.010}$ | $2.990^{+0.021}_{-0.016}$ | $3.16$ | $0$ |
|  | 2019.12.14-2019.12.25 | $5.21$ | $12.74^{+0.07}_{-0.06}$ | $0.197^{+0.003}_{-0.003}$ | $16.934^{+0.002}_{-0.002}$ | $3.160^{+0.002}_{-0.009}$ | $4.30$ | $0.064$ | 5
|  |  |  | $12.26^{+0.07}_{-0.07}$ | $0.220^{+0.004}_{-0.004}$ | $16.917^{+0.003}_{-0.002}$ | $3.130^{+0.011}_{-0.010}$ | $4.75$ | $0.069$ |
4FGL J0403.9-3605 | PKS 0402-362 | 2010.01.20-2010.03.01 | $2.20$ | $17.92^{+0.08}_{-0.08}$ | $0.480^{+0.008}_{-0.008}$ | $16.625^{+0.002}_{-0.002}$ | $2.771^{+0.008}_{-0.007}$ | $7.23$ | $0.043$ | 6
|  |  |  | $17.79^{+0.08}_{-0.08}$ | $0.435^{+0.008}_{-0.007}$ | $16.622^{+0.002}_{-0.002}$ | $2.735^{+0.012}_{-0.012}$ | $7.23$ | $0.043$ |
|  | 2010.03.01-2010.03.21 | $2.20$ | $18.58^{+0.11}_{-0.11}$ | $0.455^{+0.010}_{-0.010}$ | $16.640^{+0.003}_{-0.002}$ | $2.948^{+0.012}_{-0.012}$ | $7.34$ | $0.059$ | 6
|  |  |  | $18.63^{+0.11}_{-0.11}$ | $0.460^{+0.010}_{-0.010}$ | $16.641^{+0.002}_{-0.003}$ | $2.804^{+0.021}_{-0.020}$ | $7.59$ | $0.06$ |
|  | 2011.09.20-2011.10.04 | $2.75$ | $23.22^{+0.11}_{-0.11}$ | $0.371^{+0.008}_{-0.008}$ | $16.834^{+0.002}_{-0.002}$ | $2.765^{+0.011}_{-0.010}$ | $5.70$ | $0.044$ | 6
|  |  |  | $23.17^{+0.10}_{-0.10}$ | $0.384^{+0.007}_{-0.006}$ | $16.642^{+0.003}_{-0.003}$ | $2.804^{+0.021}_{-0.020}$ | $5.88$ | $0.045$ |
|  | 2014.07.31-2014.08.17 | $3.57$ | $19.64^{+0.08}_{-0.08}$ | $0.562^{+0.009}_{-0.009}$ | $16.875^{+0.002}_{-0.002}$ | $2.756^{+0.011}_{-0.011}$ | $5.01$ | $0.023$ | 6
|  |  |  | $19.61^{+0.09}_{-0.09}$ | $0.558^{+0.009}_{-0.010}$ | $16.873^{+0.002}_{-0.002}$ | $2.688^{+0.010}_{-0.011}$ | $5.01$ | $0.023$ |
4FGL J0530.9+1332 | PKS 0528+134 | 2009.09.08 | $0.94$ | $27.06^{+0.27}_{-0.28}$ | $0.648^{+0.025}_{-0.024}$ | $16.335^{+0.004}_{-0.005}$ | $2.976^{+0.031}_{-0.029}$ | $6.48$ | $0.039$ | 7
|  |  |  | $28.21^{+0.65}_{-0.62}$ | $1.290^{+0.153}_{-0.136}$ | $16.353^{+0.010}_{-0.010}$ | $2.829^{+0.029}_{-0.029}$ | $6.73$ | $0.041$ |
4FGL J0539.6+1432 | TXS 0536+145 | 2012.03.04-2012.04.04 | $1.00$ | $30.33^{+0.46}_{-0.45}$ | $0.907^{+0.054}_{-0.054}$ | $16.328^{+0.007}_{-0.006}$ | $2.928^{+0.035}_{-0.033}$ | $7.71$ | $0.061$ | 8
|  |  |  | $29.90^{+0.62}_{-0.59}$ | $0.721^{+0.049}_{-0.045}$ | $16.322^{+0.009}_{-0.009}$ | $3.051^{+0.036}_{-0.036}$ | $9.45$ | $0.071$ |
4FGL J2253.9+1609 | 3C 454.3 | 2011.01.27-2011.02.08 | $0.78$ | $24.39^{+0.11}_{-0.11}$ | $0.758^{+0.015}_{-0.015}$ | $16.425^{+0.002}_{-0.002}$ | $2.715^{+0.007}_{-0.007}$ | $7.47$ | $0$ | 9
|  |  |  | $25.27^{+0.22}_{-0.21}$ | $0.933^{+0.038}_{-0.036}$ | $16.440^{+0.004}_{-0.004}$ | $2.739^{+0.008}_{-0.007}$ | $7.85$ | $0$ |
|  | 2010.11.11-2010.12.06 | $0.78$ | $26.98^{+0.14}_{-0.14}$ | $0.387^{+0.009}_{-0.009}$ | $16.468^{+0.002}_{-0.002}$ | $3.007^{+0.018}_{-0.016}$ | $8.32$ | $0.066$ | 9
|  |  |  | $27.02^{+0.13}_{-0.13}$ | $0.386^{+0.009}_{-0.009}$ | $16.469^{+0.002}_{-0.002}$ | $3.037^{+0.018}_{-0.018}$ | $7.77$ | $0.063$ |
|  | 2011.05.19-2012.09.30 | $0.56$ | $19.20^{+0.14}_{-0.14}$ | $0.698^{+0.017}_{-0.017}$ | $16.172^{+0.003}_{-0.003}$ | $2.809^{+0.008}_{-0.008}$ | $5.21$ | $0$ | 10
|  |  |  | $20.15^{+0.14}_{-0.14}$ | $0.711^{+0.016}_{-0.016}$ | $16.193^{+0.003}_{-0.003}$ | $2.583^{+0.015}_{-0.015}$ | $5.38$ | $0$ |
4FGL J2329.3-4955 | PKS 2326-502 | 2010.07.31-2010.09.29 | $0.53$ | $17.05^{+0.23}_{-0.24}$ | $0.173^{+0.012}_{-0.012}$ | $16.190^{+0.006}_{-0.006}$ | $2.995^{+0.025}_{-0.024}$ | $2.86$ | $0$ | 11
|  |  |  | $20.03^{+1.16}_{-1.18}$ | $0.456^{+0.148}_{-0.119}$ | $16.260^{+0.024}_{-0.026}$ | $2.945^{+0.050}_{-0.058}$ | $3.87$ | $0$ |
|  | 2012.06.25-2012.07.05 | $1.50$ | $18.39^{+0.41}_{-0.37}$ | $0.220^{+0.024}_{-0.020}$ | $16.674^{+0.010}_{-0.009}$ | $2.934^{+0.026}_{-0.027}$ | $1.88$ | $0$ | 11
|  |  |  | $20.10^{+0.38}_{-0.33}$ | $0.437^{+0.024}_{-0.025}$ | $16.713^{+0.008}_{-0.007}$ | $3.014^{+0.017}_{-0.016}$ | $1.98$ | $0$ |
4FGL J2126.3-4605 | PKS 2123-463 | 2011.12.10-2011.12.19 | $1.97$ | $19.57^{+0.26}_{-0.26}$ | $0.280^{+0.013}_{-0.012}$ | $16.572^{+0.006}_{-0.006}$ | $3.456^{+0.057}_{-0.051}$ | $4.20$ | $0$ | 12
|  |  |  | $19.82^{+0.30}_{-0.28}$ | $0.269^{+0.014}_{-0.013}$ | $16.578^{+0.007}_{-0.006}$ | $3.467^{+0.062}_{-0.062}$ | $4.35$ | $0$ |
4FGL J0108.6+0134 | 4C +01.02 | 2015.11.23-2015.12.15 | $0.66$ | $31.85^{+0.11}_{-0.11}$ | $0.271^{+0.003}_{-0.003}$ | $16.245^{+0.002}_{-0.002}$ | $2.695^{+0.003}_{-0.003}$ | $6.84$ | $0.051$ | 13
|  |  |  | $33.00^{+0.12}_{-0.12}$ | $0.343^{+0.004}_{-0.004}$ | $16.260^{+0.0062}_{-0.002}$ | $2.803^{+0.001}_{-0.003}$ | $8.43$ | $0.061$ |

Table 2: -$continued$.

Fermi Name | Source Name | Time | $t_{\rm{var}}$ | $\delta$ | $B$ | $\log R$ | $\log\gamma_{\rm{b}}$ | $\delta_{\rm{low}}$ | $r_{\rm{low}}$ | Ref
---|---|---|---|---|---|---|---|---|---|---
|  |  | (days) |  | (G) | (cm) |  |  | (pc) |
(1) | (2) | (3) | (4) | (5) | (6) | (7) | (8) | (9) | (10) | (11)
4FGL J1256.1-0547 | 3C 279 | 2014.04.03-2014.04.07 | $0.09$ | $41.50^{+0.22}_{-0.23}$ | $1.354^{+0.030}_{-0.030}$ | $15.799^{+0.002}_{-0.002}$ | $2.445^{+0.013}_{-0.014}$ | $7.96$ | $0.005$ | 14
|  |  |  | $42.00^{+0.24}_{-0.24}$ | $1.181^{+0.030}_{-0.029}$ | $15.804^{+0.002}_{-0.002}$ | $2.422^{+0.002}_{-0.012}$ | $7.96$ | $0.005$ |
|  | 2012.04.03 | $0.08$ | $44.36^{+0.30}_{-0.29}$ | $0.562^{+0.012}_{-0.013}$ | $15.795^{+0.003}_{-0.002}$ | $2.695^{+0.013}_{-0.013}$ | $8.79$ | $0$ | 15
|  |  |  | $43.58^{+0.27}_{-0.27}$ | $0.593^{+0.014}_{-0.014}$ | $15.787^{+0.003}_{-0.003}$ | $2.684^{+0.002}_{-0.011}$ | $9.63$ | $0$ |
|  | 2011.02.08-2011.04.12 | $3.49$ | $13.49^{+0.12}_{-0.13}$ | $1.096^{+0.044}_{-0.043}$ | $16.900^{+0.009}_{-0.009}$ | $2.526^{+0.028}_{-0.028}$ | $2.87$ | $0$ | 16
|  |  |  | $14.33^{+0.35}_{-0.34}$ | $1.514^{+0.181}_{-0.167}$ | $16.926^{+0.010}_{-0.011}$ | $2.530^{+0.010}_{-0.031}$ | $2.87$ | $0$ |
|  | 2011.06.01-2011.06.08 | $3.49$ | $15.01^{+0.46}_{-0.39}$ | $0.456^{+0.038}_{-0.039}$ | $16.945^{+0.012}_{-0.013}$ | $3.389^{+0.020}_{-0.021}$ | $3.30$ | $0$ | 16
|  |  |  | $14.88^{+0.15}_{-0.16}$ | $0.931^{+0.034}_{-0.032}$ | $16.942^{+0.004}_{-0.005}$ | $2.523^{+0.004}_{-0.009}$ | $3.30$ | $0$ |
|  | 2014.03.25-2014.04.02 | $11.61$ | $11.73^{+0.06}_{-0.06}$ | $0.358^{+0.007}_{-0.006}$ | $17.361^{+0.002}_{-0.002}$ | $3.046^{+0.009}_{-0.010}$ | $2.76$ | $0$ | 16
|  |  |  | $11.23^{+0.05}_{-0.05}$ | $0.402^{+0.007}_{-0.007}$ | $17.342^{+0.002}_{-0.002}$ | $3.011^{0.002}_{-0.008}$ | $3.07$ | $0$ |
|  | 2015.06.16 | $3.17$ | $18.02^{+0.13}_{-0.11}$ | $0.106^{+0.003}_{-0.003}$ | $16.973^{+0.003}_{-0.003}$ | $3.260^{+0.016}_{-0.017}$ | $4.31$ | $0$ | 16
|  |  |  | $16.39^{+0.11}_{-0.10}$ | $0.131^{+0.003}_{-0.003}$ | $16.943^{+0.003}_{-0.003}$ | $3.244^{+0.003}_{-0.015}$ | $4.44$ | $0$ |
|  | 2010.01.14-2010.06.28 | $2.89$ | $12.24^{+0.08}_{-0.08}$ | $0.745^{+0.022}_{-0.021}$ | $16.776^{+0.004}_{-0.004}$ | $2.552^{+0.011}_{-0.011}$ | $3.48$ | $0$ | 10
|  |  |  | $12.75^{+0.09}_{-0.10}$ | $0.708^{+0.025}_{-0.022}$ | $16.794^{+0.003}_{-0.003}$ | $2.416^{+0.003}_{-0.007}$ | $3.48$ | $0$ |
4FGL J2232+1143 | CTA 102 | 2016.12.23 | $0.56$ | $27.80^{+0.12}_{-0.12}$ | $0.771^{+0.013}_{-0.013}$ | $16.298^{+0.002}_{-0.002}$ | $2.940^{+0.002}_{-0.009}$ | $6.97$ | $0$ | 17
|  |  |  | $27.25^{+0.32}_{-0.25}$ | $0.702^{+0.025}_{-0.025}$ | $16.289^{+0.005}_{-0.005}$ | $2.962^{+0.013}_{-0.013}$ | $6.97$ | $0$ |
|  | 2012.09.18-2012.10.03 | $3.93$ | $21.20^{+0.15}_{-0.15}$ | $0.225^{+0.005}_{-0.005}$ | $17.022^{+0.003}_{-0.003}$ | $3.247^{+0.003}_{-0.009}$ | $6.06$ | $0.105$ | 18
|  |  |  | $20.83^{+0.27}_{-0.24}$ | $0.224^{+0.008}_{-0.009}$ | $17.018^{+0.005}_{-0.006}$ | $3.247^{+0.011}_{-0.011}$ | $5.98$ | $0.104$ |
|  | 2017.04.19 | $0.17$ | $68.81^{+1.79}_{-1.81}$ | $0.170^{+0.013}_{-0.014}$ | $16.172^{+0.012}_{-0.011}$ | $3.135^{+0.014}_{-0.014}$ | $11.33$ | $0$ | 19
|  |  |  | $57.05^{+2.91}_{-2.53}$ | $0.243^{+0.036}_{-0.035}$ | $16.091^{+0.020}_{-0.021}$ | $3.117^{+0.029}_{-0.027}$ | $11.70$ | 0 |
|  | 2017.01.08 | $0.05$ | $69.69^{+0.25}_{-0.25}$ | $1.487^{+0.027}_{-0.028}$ | $15.601^{+0.002}_{-0.002}$ | $3.061^{+0.006}_{-0.006}$ | $16.60$ | $0.040$ | 20
|  |  |  | $76.81^{+0.31}_{-0.30}$ | $1.964^{+0.030}_{-0.030}$ | $15.643^{+0.002}_{-0.002}$ | $3.120^{+0.006}_{-0.006}$ | $15.95$ | $0.039$ |
|  | 2016.12.30 | $0.50$ | $38.33^{+0.18}_{-0.18}$ | $0.573^{+0.012}_{-0.012}$ | $16.387^{+0.002}_{-0.002}$ | $3.371^{+0.009}_{-0.009}$ | $9.96$ | $0.072$ | 21
|  |  |  | $31.29^{+0.67}_{-0.66}$ | $0.217^{+0.024}_{-0.022}$ | $16.299^{+0.009}_{-0.009}$ | $3.073^{+0.065}_{-0.066}$ | $10.45$ | $0.075$ |
|  | 2016.12.26-2016.12.31 | $0.21$ | $49.42^{+0.17}_{-0.17}$ | $1.421^{+0.021}_{-0.022}$ | $16.121^{+0.001}_{-0.002}$ | $3.073^{+0.001}_{-0.006}$ | $16.15$ | $0.064$ | 22
|  |  |  | $52.23^{+0.38}_{-0.37}$ | $1.954^{+0.069}_{-0.065}$ | $16.150^{+0.003}_{-0.003}$ | $2.663^{+0.026}_{-0.027}$ | $13.39$ | $0.555$ |
|  | 2011.09.04-2011.10.18 | $2.49$ | $14.77^{+0.11}_{-0.11}$ | $0.421^{+0.012}_{-0.012}$ | $16.669^{+0.003}_{-0.003}$ | $3.179^{+0.011}_{-0.011}$ | $3.46$ | $0$ | 10
|  |  |  | $16.08^{+0.17}_{-0.16}$ | $0.433^{+0.019}_{-0.019}$ | $16.706^{+0.004}_{-0.004}$ | $2.848^{+0.036}_{-0.038}$ | $3.63$ | $0$ |
4FGL J0509.4+0542 | TXS 0506+056 | 2018.10.06 | $0.14$ | $25.07^{+0.14}_{-0.13}$ | $1.646^{+0.033}_{-0.033}$ | $15.833^{+0.002}_{-0.002}$ | $3.079^{+0.008}_{-0.008}$ | $9.13$ | $0$ | 23
|  |  |  | $23.86^{+0.11}_{-0.12}$ | $1.529^{+0.029}_{-0.029}$ | $15.816^{+0.002}_{-0.002}$ | $3.359^{+0.014}_{-0.014}$ | $7.34$ | $0$ |
|  | 2017.10.03-2017.10.04 | $1.16$ | $13.30^{+0.10}_{-0.10}$ | $1.040^{+0.025}_{-0.024}$ | $16.493^{+0.003}_{-0.003}$ | $3.819^{+0.046}_{-0.042}$ | $10.76$ | $0.028$ | 24
|  |  |  | $14.02^{+0.06}_{-0.06}$ | $0.962^{+0.017}_{-0.017}$ | $16.498^{+0.002}_{-0.002}$ | $3.551^{+0.034}_{-0.038}$ | $9.83$ | $0.023$ |
|  | 2017.10.31 | $1.16$ | $13.87^{+0.27}_{-0.31}$ | $0.661^{+0.038}_{-0.047}$ | $16.493^{+0.008}_{-0.010}$ | $3.648^{+0.016}_{-0.016}$ | $9.97$ | $0.029$ | 24
|  |  |  | $13.30^{+0.31}_{-0.35}$ | $0.765^{+0.059}_{-0.057}$ | $16.475^{+0.010}_{-0.011}$ | $3.669^{+0.019}_{-0.020}$ | $9.97$ | $0.028$ |
4FGL J0854.8+2006 | OJ 287 | 2015.12.03 | $1.00$ | $14.92^{+0.07}_{-0.07}$ | $0.899^{+0.017}_{-0.017}$ | $16.471^{+0.002}_{-0.002}$ | $3.107^{+0.029}_{-0.031}$ | $3.32$ | $0$ | 25
|  |  |  | $14.87^{+0.14}_{-0.09}$ | $0.910^{+0.021}_{-0.028}$ | $16.470^{+0.004}_{-0.003}$ | $3.115^{+0.040}_{-0.036}$ | $3.07$ | $0$ |
|  | 2008.08.11-2008.11.11 | $10.70$ | $7.40^{+0.03}_{-0.03}$ | $0.763^{+0.012}_{-0.011}$ | $17.196^{+0.002}_{-0.002}$ | $3.210^{+0.008}_{-0.008}$ | $2.17$ | $0$ | 26
|  |  |  | $7.78^{+0.03}_{-0.03}$ | $0.961^{+0.017}_{-0.017}$ | $17.218^{+0.002}_{-0.002}$ | $3.157^{+0.007}_{-0.007}$ | $1.83$ | $0$ |
|  | 2009.10.20-2009.10.27 | $2.50$ | $10.16^{+0.12}_{-0.12}$ | $0.492^{+0.026}_{-0.024}$ | $16.702^{+0.005}_{-0.005}$ | $3.261^{+0.059}_{-0.053}$ | $4.25$ | $0$ | 27
|  |  |  | $10.30^{+0.14}_{-0.14}$ | $0.501^{+0.029}_{-0.028}$ | $16.708^{+0.006}_{-0.005}$ | $3.267^{+0.046}_{-0.043}$ | $4.02$ | $0$ |
|  | 2009.10.27-2009.11.17 | $2.50$ | $11.57^{+0.17}_{-0.17}$ | $1.344^{+0.076}_{-0.070}$ | $16.759^{+0.006}_{-0.006}$ | $2.846^{+0.037}_{-0.034}$ | $3.04$ | $0$ | 27
|  |  |  | $12.81^{+1.66}_{-1.12}$ | $1.925^{+1.594}_{-0.712}$ | $16.803^{+0.053}_{-0.040}$ | $2.375^{+0.534}_{-0.352}$ | $3.04$ | $0$ |
|  | 2009.11.17-2009.12.19 | $2.50$ | $11.11^{+0.19}_{-0.20}$ | $1.821^{+0.111}_{-0.101}$ | $16.741^{+0.007}_{-0.008}$ | $2.832^{+0.035}_{-0.031}$ | $3.04$ | $0$ | 27
|  |  |  | $11.04^{+0.20}_{-0.19}$ | $1.812^{+0.113}_{-0.110}$ | $16.738^{+0.008}_{-0.008}$ | $2.874^{+0.040}_{-0.041}$ | $3.04$ | $0$ |

  * •

Note: Column (1) and (2) give the Fermi name and source name, respectively.
The sources marked with ∗ are the DT dominated LSPs with observed Doppler
factors, and the others possess observed variability timescales. Column (3)
gives the observed time period. Column (4) gives the observed or derived
variability timescales. Column (5) gives the observed or derived Doppler
factors. Column (6), (7) and (8) are the derived magnetic field strength, the
derived radius of emission region and the derived break Lorentz factor of
relativistic electrons, respectively. Column (9) and (10) give the lower
limits of Doppler factors and distance between the black hole and emission
region, which are derived from internal $\gamma\gamma$ absorption. Column (11)
is the related reference. The table contains some rows with double sub-rows.
The upper one is the parameter obtained by fitting SEDs with quadratic
function, and the lower one is the parameter obtained by fitting SEDs with
cubic function.

  * •

References: (1) Sahakyan20 (95); (2) Ghisellini10_SED (54); (3) Tan20 (101);
(4) Sahakyan20 (95); (5) Ammenadka22 (15); (6) Das23 (33); (7) Palma11 (84);
(8) Orienti14 (80); (9) Das20 (34); (10) Roy21 (92); (11) Dutka17 (43); (12)
DAmmando12 (32); (13) Malik22 (74); (14) Patel21 (85); (15) Hayashida15 (60);
(16) Fraija19 (47); (17) Zacharias17 (115); (18) Pacciani14 (81); (19)
Gasparyan18 (48); (20) Prince18 (91); (21) Zacharias19 (116); (22)
Sahakyan20_CTA102 (94); (23) Acciari22 (7); (24) Sahakyan18 (96); (25)
Oikonomou19 (79); (26) Chen10 (29); (27) Kushwaha13 (67).

##  4 Conclusions

In this work, we calculated the observed seed factors of 1138 LSPs and the
characteristic seed factors of four external photon fields, then plotted the
histogram distribution to locate the $\gamma$-ray emission region. SEDs
related to historical flare states were collected to investigate the variable
locations. These SEDs were fitted by both quadratic and cubic functions using
the MCMC method. Furthermore, we derived some parameters of emission region
and employed a constraint of internal $\gamma\gamma$-absorption to verify the
derivation. Our main results are as follows:

  * 1.

We find that DT dominates the soft photon fields of LSPs and $\gamma$-ray
emission regions of LSPs are mainly located at 1-10 pc. Histogram shows that
the corresponding distribution of BL Lacs peaks at the area of BLR, but this
area could also be covered by the actual value of DT. CMB and starlight make
little contribution to the $\gamma$-ray emission of LSPs.

  * 2.

The locations of $\gamma$-ray emission region of LSPs are variable in
different flare epochs. Most $\gamma$-ray emission regions are within the DT,
but the soft photon fields could also transition to BLR and CMB.

  * 3.

The cubic function is better than the quadratic function to fit the SEDs of
blazars. We find that some high-energy humps of blazars cannot be fitted well
by quadratic function due to the symmetry of the SEDs.

We thank the anonymous referee for valuable comments and constructive
suggestions. This work is supported by the National Natural Science Foundation
of China (NSFC) under Grant No. 12203043, 12203024 and 12263009.

##  5 Data Availability

The data derived in this paper are available on Zenodo with a DOI
10.5281/zenodo.11119117 333https://zenodo.org/records/11119118.

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##  Appendix A SED Fitting Results

Figures 7 to 13 present all the 45 SEDs fitted by both quadratic and cubic
functions. A and B denote the SEDs fitted by quadratic and cubic functions,
respectively. Blue and light blue scatters represent the simultaneous and
archival data, respectively. Black lines are plotted using the maximum
posterior values. Yellow areas denote the 1-$\sigma$ uncertainties under
16-50-84 rule. The AICc of each hump and observed seed factors are displayed
on the figures.

Figure 7: SED fitting results of the blazars dominated by dusty torus with
Doppler factors.

Figure 8: SED fitting results of the blazars dominated by dusty torus with
variability timescales.

Figure 9: SED fitting results of the blazars dominated by dusty torus with
variability timescales.

Figure 10: SED fitting results of CTA 102.

Figure 11: SED fitting results of 3C 279.

Figure 12: SED fitting results of TXS 0506+056.

Figure 13: SED fitting results of OJ 287.