# Comment on ”Case of thermodynamic failure in the Ginzburg-Landau approach to
fluctuation superconductivity”

A.V. Nikulov<EMAIL_ADDRESS>Institute of Microelectronics Technology and
High Purity Materials, Russian Academy of Sciences, 142432 Chernogolovka,
Moscow District, RUSSIA.

###### Abstract

Jorge Berger shows theoretically in the paper Phys. Rev. B 109, 024501 (2024)
that according to the Ginzburg-Landau theory the persistent current can create
the persistent voltage, i.e. a dc voltage at thermodynamic equilibrium, on
segments of nonuniform superconducting loop. A similar result was published
early and was collaborated experimentally. The persistent power estimated by
Berger is compared with the experimentally observed power. The attention of
experimenters is drawn to the possibility to observe the persistent voltage
thanks to its increase with the number of identical rings connected in series.

##  I Introduction

Jorge Berger has shown in Berger2024PRB  that the persistent current of Cooper
pairs $I_{p}$ can create a persistent voltage $V_{p}$, i.e. the dc voltage
$V_{dc}$ at thermodynamic equilibrium, on segments of nonuniform
superconducting loop. This result deserves special attention because of its
contradiction with the second law of thermodynamics. It is written in the
review article PhysRep1999 : ”The second law of thermodynamics is, without a
doubt, one of the most perfect laws in physics. Any reproducible violation of
it, however small, would bring the discoverer great riches as well as a trip
to Stockholm. The world’s energy problems would be solved at one stroke. It is
not possible to find any other law (except, perhaps, for super selection rules
such as charge conservation) for which a proposed violation would bring more
skepticism than this one. Not even Maxwell’s laws of electricity or Newton’s
law of gravitation are so sacrosanct”.

It is important to draw reader’s attention to the publications which confirm
experimentally and theoretically the result published in Berger2024PRB ,
because of this skepticism about violation of the sacrosanct law. Its
reproducible violation can solve world’s energy problems since we must used a
fuel in our heat engine because of the Carnot principle which we call ”as the
second law of thermodynamics since Clausius’s time” Smoluchowski . The
efficiency of any heat engine cannot exceed the maximum value

$\eta_{max}=1-\frac{T_{co}}{T_{he}}$ $None$

which is determined by the ratio $T_{co}/T_{he}$ of the cooler temperature
$T_{co}$ to the heater temperature $T_{he}$ according to the Carnot principle.
Therefore we must create and maintain a temperature difference $T_{he}-T_{co}$
with the help of a fuel in order to obtained a useful work from heat.

##  II A useful work from heat without any fuel

The theoretical result Berger2024PRB  allows to convert the heat energy
$\Delta Q$ to a useful work $W$ without any fuel. According to Berger’s
calculations the potential electric field $E(x)=-dV/dx$ at thermodynamic
equilibrium creates the persistent voltage on segments $A$
$\int_{A}dxE(x)=-V_{p}$ and $B$ $\int_{B}dxE(x)=V_{p}$ of the loop $L=A+B$.
The dc voltage are equal in value and opposite in sign since
$\int_{L}dxE(x)=\int_{L}dx(-dV/dx)\equiv 0$ Berger2024PRB . The sign of the
persistent voltage $V_{p}$ and the persistent power $V_{p}I_{p}$ is determined
relative to the direction of the persistent current $I_{p}$ circulating in the
loop $L=A+B$. The segment $A$ in which $V_{p}I_{p}<0$ is the power source and
the segment $B$ in which $V_{p}I_{p}>0$ is the load.

The persistent voltage $V_{p}$ can create a current $I_{load}$ in a useful
load connected to the segment $A$. The total power $-VI_{A}+VI_{B}<0$
delivered by the electric field in the whole loop will be negative in this
case since the $I_{A}$ current in the $A$ segment should be equal the sum
$I_{A}=I_{B}+I_{load}$ and $\int_{L}dxE(x)=\int_{L}dx(-dV/dx)\equiv 0$ in any
case. A useful work $W=\int_{t}dtV_{p}I_{load}$ can be obtained in the load
from heat without the need to create and to maintain a temperature difference
$T_{he}-T_{co}$. Thus, the question of the possibility of the persistent
voltage $V_{p}$ is of great fundamental and practical importance.

##  III A possibility of the persistent voltage

This possibility may be expected from experimental evidence of the persistent
current $I_{p}$ observed at a non-zero resistance $R_{n}>R>0$ in the
fluctuation region near superconducting transition $T\approx T_{c}$ LP1962 ;
Science2007 ; Letter2007 ; toKulik2010 . It is known that the conventional
electric current $I$ creates the potential difference

$V=0.5(R_{B}-R_{A})I$ $None$

on the halves of the ring with different resistance $R_{B}>R_{A}$. The value
and the direction (clockwise or counterclockwise) of the persistent current
change periodically with magnetic field $I_{p}(\Phi/\Phi_{0})$ with the period
$B_{0}=\Phi_{0}/\pi r^{2}$ corresponding to the flux quantum
$\Phi_{0}=2\pi\hbar/2e\approx 20.7\ Oe\ \mu m^{2}$ inside the ring with the
radius $r$, according to experimental results Science2007 ; Letter2007 ;
toKulik2010 .

Observations Letter2007 ; toKulik2010 ; PM2001 ; Letter2003 ; PLA2012PV ;
APL2016 ; PLA2017  on halves of asymmetric superconducting rings of the
quantum oscillations of the dc voltage $V_{dc}(\Phi/\Phi_{0})\propto
I_{p}(\Phi/\Phi_{0})$ give experimental evidence that persistent current can
also create the potential difference. The persistent current flows against the
total electric field $E=-\nabla V_{p}$ in the $A$ segment according to these
observations since the Faraday electric field $dA/dt=0$ at a magnetic flux
inside the ring $\Phi=\pi r^{2}B\neq n\Phi_{0}$ constant in time
$d\Phi/dt=2\pi rdA/dt=0$ Physica2019 .

##  IV The Ginzburg-Landau theory explains paradoxical phenomena

The quantum oscillations of the dc voltage $V_{dc}(\Phi/\Phi_{0})\propto
I_{p}(\Phi/\Phi_{0})$ were observed first in 1967 at the measurements of an
asymmetric dc SQUID Physica1967 . The dc power $P_{dc}=V_{dc}^{2}/R$ observed
in Physica1967  and also in Letter2007 ; toKulik2010 ; PM2001 ; Letter2003 ;
PLA2012PV ; APL2016 ; PLA2017 ; Physica2019  can be used for a useful work
$W=\int_{t}dtP_{dc}$. The authors Physica1967  concluded that the energy for
this work is taken from a non-equilibrium noise since they stopped observing
the dc voltage $V_{dc}(\Phi/\Phi_{0})\propto I_{p}(\Phi/\Phi_{0})$ after
careful shielding of all parts of their equipment. They did not taken into
account that the persistent current $I_{p}(\Phi/\Phi_{0})\propto
V_{dc}(\Phi/\Phi_{0})$ flowing against the total dc electric field $E=-\nabla
V_{dc}$ is a paradoxical phenomenon and that the persistent current observed
at a non-zero resistance $0<\overline{R}<R_{n}$ in the Little-Parks experiment
LP1962  implies the need for a DC power source $\overline{RI_{p}^{2}}$.

The GL theory GL1950  explains the both paradoxical phenomena as a consequence
of the quantization of the canonical momentum of
$\oint_{l}dlp=\oint_{l}dl(mv+qA)=n2\pi\hbar$ and with the help of postulate
that all $N_{s}=n_{s}V=n_{s}s2\pi r$ Cooper pairs have the same quantum number
$n$. The average persistent current
$\overline{I_{p}}=\int_{\Theta}dtI(t)/\Theta=q\overline{N_{s}}\overline{v}/2\pi
r$ does not damped at $0<\overline{R}<R_{n}$ since the velocity of Cooper
pairs should increase from $v=0$ to

$\oint_{l}dlv=\frac{2\pi\hbar}{m}(n-\frac{\Phi}{\Phi_{0}})$ $None$

each time when the ring returns to the superconducting state at its switching
by thermal fluctuations with a frequency $f_{sw}\gg 1/\Theta$. The total
change of the momentum $mv$ because of the quantization (3) during a time
unity $F_{q}=\hbar(\overline{n}-\Phi/\Phi_{0})f_{sw}/r$ was called ’quantum
force’ in PRB2001 . Thus, according to the GL theory the persistent current
$\overline{I_{p}}\neq 0$ is not damped at $0<\overline{R}<R_{n}$ LP1962 ;
Science2007 ; Letter2007 ; toKulik2010  due to the force balance
$\overline{F_{dis}}+F_{q}=0$: the average velocity of electrons should
decrease from $v=(\hbar/mr)(n-\Phi/\Phi_{0})$ to $v=0$ under influence of the
dissipation force $F_{dis}=-\eta\overline{v}$ because of electron scattering,
but the velocity of Cooper pairs should increase from $v=0$ to
$v=(\hbar/mr)(n-\Phi/\Phi_{0})$ because of the quantization (3).

A result similar to the one obtained in Berger2024PRB  was obtained in the
article LTP1998  for a ring whose halves have different critical temperatures
$T_{c,B}<T_{c,A}$. The GL theory predicts the persistent voltage at $T\approx
T_{c,B}<T_{c,A}$ since only the $B$ half should be switched by thermal
fluctuations LTP1998 . The persistent voltage
$V_{p}=0.5(R_{B}-R_{A})\overline{I_{p}}=0.5R_{B}\overline{I_{p}}$ may be
expected also according observations of the persistent current both at
$T<T_{c}$ where $R=0$ JETP07  and at $T\approx T_{c}$ where $0<R<R_{n}$ LP1962
; Science2007 ; Letter2007 ; toKulik2010 .

The dc voltage $V_{dc}\approx Lf_{sw}\overline{I_{p}}$ should be observed when
the $B$ half is switched with a low frequency $f_{sw}\ll 1/\tau_{RL}$ and
$V_{dc}\approx R_{B}\overline{I_{p}}$ with a high frequency $f_{sw}\gg
1/\tau_{RL}$ LTP1998 ; PRA2012QF  since the potential difference
$V(t)=R_{n,B}I(t)=R_{n,B}I_{p}\exp-t/\tau_{RL}$ should appear on the $B$ half
after each transition to the normal state with the resistance $R_{n,B}$, see
Fig.2 in Physica2021 . $\tau_{RL}=L/R_{B}$ is the relaxation time. It should
be noted that according to the GL theory, the amplitude of
$V_{dc}(\Phi/\Phi_{0})$ depends on the amplitude of $I_{p}(\Phi/\Phi_{0})$,
and not on the amplitude of the non-equilibrium noise, as the authors
Physica1967  thought.

The experimental observations $\overline{I_{p}}$ in the fluctuation region
LP1962 ; Science2007 ; Letter2007 ; toKulik2010  were explained in PRB2001  as
a directed (non-chaotic) Brownian motion. M. Smoluchowsk realized a hundred
and ten years ago that the impossibility of a directed Brownian motion is one
of the conditions for the validity of the second law of thermodynamics
Smoluchowski . He was right! Such well known chaotic Brownian motion Feynman
as the Nyquist Nyquist  (or Johnson Johnson ) noise cannot be used for a
useful work since its power $\overline{V_{Nyq}^{2}}/R=4k_{B}T\Delta f$ Feynman
is the same for all elements of electrical circuit in any frequency band
$\Delta f$. In contrast to this we obtain the power source $V_{p}I_{p}<0$ and
the load $V_{p}I_{p}>0$ according to Berger2024PRB ; LTP1998  in the case of
such directed Brownian motion as the persistent current LP1962 ; Science2007 ;
Letter2007 ; toKulik2010 .

Berger writes Berger2024PRB  that he considers a situation beyond the realms
considered in the articles Hirsch2020EPL  and Physica2021 . It is not quite
right. The well-known physicist W.H. Keesom wrote in 1934 that ”it is
essential that the persistent currents have been annihilated before the
material gets resistance, so that no Joule-heat is developed” Keesom1934
since he understood that the emergence of the persistent currents at the
Meissner effect Meissner1933  contradicts to the second law of thermodynamics
in the opposite case. The macroscopic persistent current $I_{p}=qN_{s}v/2\pi
r$ with the macroscopic kinetic energy $E_{p}=N_{s}mv^{2}/2\gg k_{B}T$ emerges
also at each transition to the superconducting state of a ring, for example
during measurements of the critical current PRB2014 . Jorge Hirsch draws
attention in Hirsch2020EPL  that the conventional theory of superconductivity
BCS1957  predicts that Joule heat is generated when the persistent currents
decrease. He states that this theory should be questioned because of its
contradiction with the laws of thermodynamics. But rather the laws of
thermodynamics than the theory GL1950 ; BCS1957  should be questioned
Physica2021 , because of numerous experimental evidences LP1962 ; Science2007
; Letter2007 ; toKulik2010  that the persistent currents are not annihilated
not only before but even after the material gets resistance.

##  V Observation of the persistent voltage

The DC power is summed $(NV_{dc})^{2}/NR=NV_{dc}^{2}/R$, as opposed to the
chaotic power of the Nyquist noise
$\overline{NV_{Nyq}^{2}}/NR=\overline{V_{Nyq}^{2}}/R=4k_{B}T\Delta f$. The
measurements Letter2007 ; PM2001 ; Physica2019  have corroborated the
$V_{dc}(\Phi/\Phi_{0})$ increase with the number $N$ of identical asymmetric
ring connected in series $V_{A,N}=NV_{A,1}$ Physica2019 . This advantage
allowed to conform Letter2007 ; toKulik2010 ; PM2001 ; PLA2012PV ; APL2016
that the dc voltage $V_{dc}(\Phi/\Phi_{0})$ do not disappear after careful
shielding of a non-equilibrium noise but their amplitude $V_{A}$ decreases
below the detection limit $V_{A}<V_{A,dl}$ on single ring. In accordance with
the GL theory the amplitude $V_{A}$ depends on the amplitude of the persistent
current $I_{p,A}$ which decreases with the temperature increase JETP07

$I_{p,A}(T)=I_{p,A}(T=0)(1-T/T_{c})$ $None$

The amplitude $V_{A}$ decreases after careful shielding Physica1967  since a
non-equilibrium noise Letter2007 ; toKulik2010 ; PM2001  or an external ac
current PLA2017 ; Physica2019 ; Letter2003 ; JETP07  cannot switch the ring in
the normal state at $T<T_{c}$ when their amplitude
$I_{A}=\overline{2I_{noise}^{2}}^{1/2}$ does not exceed the critical current

$I_{c}(T)=I_{c}(T=0)(1-T/T_{c})^{3/2}$ $None$

The persistent voltage $V_{p}(\Phi/\Phi_{0})$, i.e. the dc voltage at
thermodynamic equilibrium, can be observed only in a narrow temperature region
corresponding the resistive transition $R_{n}>R(T)>0$ where thermal
fluctuations decrease the critical current (5) down to zero $I_{c}(T)=0$ at
$T<T_{c}$ and makes the persistent current (4) non-zero $I_{p,A}(T)>0$ at
$T>T_{c}$ Skocpol1975 . No non-equilibrium noise is needed for ring switching
in this region. A system of 1080 asymmetric aluminum rings connected in series
was needed in order to observe the persistent voltage $V_{p}(\Phi/\Phi_{0})$
with the amplitude exceeding the detection limit $V_{A}<V_{A,dl}\approx 20\
nV=2\ 10^{-8}\ V$ in the temperature region $1.359\ K<T<1.373\ K$
corresponding to the lower part of the resistive transition
$0.01R_{n}<R(T)<0.32R_{n}$ PLA2012PV .

The observation of the persistent power in PLA2012PV  is a qualitative
confirmation of the theoretical result Berger2024PRB , despite the fact that
the maximum power $P\approx 5\ 10^{-12}\ W/m$ estimated on the base of
experimental results PLA2012PV  is less than the power $P\approx 10^{-10}\
W/m$ theoretically estimated by Berger Berger2024PRB . The maximum amplitude
$V_{A,1080}\approx 0.14\ \mu V=1.4\ 10^{-7}\ V$ observed on the 1080 rings
corresponds to $V_{A,1}\approx 1.4\ 10^{-10}\ V$ on one ring if each ring with
the length of semi-ring $L/2=\pi r\approx 2.8\ \mu m=2.8\ 10^{-6}\ m$
contributes to the observed power. The amplitude of the persistent
$I_{p}(\Phi/\Phi_{0})$ equals approximately $I_{p}\approx 0.1\ \mu A=10^{-7}\
A$ at the temperature corresponding to the resistance $R(T)\approx 0.2R_{n}$
for the rings used in PLA2012PV  according to the measurements of the Little-
Parks oscillations Letter2007 . The maximum persistent power
$V_{p}I_{p}\approx V_{A,1}I_{p,A}\approx 1.4\ 10^{-10}\ V\times 10^{-7}\
A\approx 1.4\ 10^{-17}\ W$ per one ring observed in PLA2012PV  is much less
than the dc power $P_{dc}\approx 2\ 10^{-10}\ W$ observed in Physica1967  and
$P_{dc}\approx 10^{-12}\ W$ observed in PM2001  since the persistent current
$I_{p}(\Phi/\Phi_{0})$ in the fluctuation region is much smaller than in the
superconducting state.

Non-equilibrium noise was suppressed by additional filtration with the help of
the help of low-temperature $\pi$-filters and coaxial resistive twisted pairs
in order to observe the persistent voltage $V_{p}(\Phi/\Phi_{0})$ in PLA2012PV
. This additional filtration allowed also to demonstrate that the system of a
big number of identical asymmetric superconducting rings can be used as a
detector of a noise: the oscillations $V_{dc}(\Phi/\Phi_{0})$ with the
amplitude up to $V_{A,667}\approx 1\ \mu V=1\ 10^{-6}\ V$ were observed on a
system of 667 asymmetric aluminum rings with $r\approx 0.5\ \mu m$ (the
amplitude $V_{A,1}\approx 1.5\ nV=1.5\ 10^{-9}\ V$ on each ring) when a source
of a noise with the amplitude $\overline{2I_{noise}^{2}}^{1/2}\approx 50\
nA=5\ 10^{-8}\ A$ was switched on APL2016 . The non-equilibrium noise
$\overline{2I_{noise}^{2}}^{1/2}\approx 5\ 10^{-8}\ A$ is weaker than the
noise $\overline{2I_{noise}^{2}}^{1/2}\approx 20\ \mu A=2\ 10^{-5}\ A$ induced
$V_{dc}\approx 1.5\ 10^{-5}\ V$ in Physica1967  and the noise
$\overline{2I_{noise}^{2}}^{1/2}\approx 3\ 10^{-6}\ A$ induced
$V_{dc}(\Phi/\Phi_{0})\approx 10^{-6}\ V$ in PM2001 .

##  VI Conclusion

The persistent power of one ring cannot exceed in any case the total power of
the Nyquist Nyquist  (or Johnson Johnson ) noise
$V_{p}^{2}/R<\overline{V_{Nyq}^{2}}/R=4k_{B}Tf_{ql}=4(k_{B}T)^{2}/h$ in the
whole frequency band from $f=0$ to the quantum limit $f=f{ql}\approx k_{B}T/h$
Feynman . More than $10^{13}$ of the rings with $T_{c}\approx 1\ K$ used in
PLA2012PV  are needed in order to get a power of only 1 W since
$4(k_{B}T)^{2}/h\approx 10^{-13}\ W$ at $T\approx 1\ K$. Fewer rings $>10^{9}$
will be required if high-temperature superconductors (HTSC) with $T_{c}\approx
100\ K$ can be used. Even the use of HTSC rings is unlikely to be able to
solve the world’s energy problems at one stroke. But the fact of violation of
the second law of thermodynamics is important in itself, since if the
violation is possible in one case, then it may be possible in other cases,
some of which may be more perspective for practical application.

One example of the violation may be enough to make scientists stop to consider
the second law of thermodynamics as sacrosanct law. In this case, the world’s
energy problems may be solved if not at one stroke, but rather quickly.
Therefore, it is important to draw the attention of experimenters to the
possibility to observe the persistent voltage not only in the lower part of
the resistive transition, but in the entire temperature region of thermal
fluctuations with the help of a system with a number of rings significantly
exceeding 1000.

This work was made in the framework of State Task No 075-00296-24-00.

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# The beaming effect for Fermi-LAT-detected FR-I radio galaxies

Xu-Hong Ye  Center for Astrophysics, Guangzhou University
Guangzhou, 510006, China  Astronomy Science and Technology Research Laboratory
of Department of Education of Guangdong Province,
Guangzhou 510006, China  Key Laboratory for Astronomical Observation and
Technology of Guangzhou
Guangzhou 510006, China  Xiang-Tao Zeng  Center for Astrophysics, Guangzhou
University
Guangzhou, 510006, China  Astronomy Science and Technology Research Laboratory
of Department of Education of Guangdong Province,
Guangzhou 510006, China  Key Laboratory for Astronomical Observation and
Technology of Guangzhou
Guangzhou 510006, China  Dan-Yi Huang  School of Physics and Material Science,
Guangzhou University
Guangzhou 510006, China  Zhuang Zhang  Center for Astrophysics, Guangzhou
University
Guangzhou, 510006, China  Astronomy Science and Technology Research Laboratory
of Department of Education of Guangdong Province,
Guangzhou 510006, China  Key Laboratory for Astronomical Observation and
Technology of Guangzhou
Guangzhou 510006, China  Zhi-Yuan Pei  Center for Astrophysics, Guangzhou
University
Guangzhou, 510006, China  Astronomy Science and Technology Research Laboratory
of Department of Education of Guangdong Province,
Guangzhou 510006, China  Key Laboratory for Astronomical Observation and
Technology of Guangzhou
Guangzhou 510006, China  Jun-Hui Fan  Center for Astrophysics, Guangzhou
University
Guangzhou, 510006, China  Astronomy Science and Technology Research Laboratory
of Department of Education of Guangdong Province,
Guangzhou 510006, China  Key Laboratory for Astronomical Observation and
Technology of Guangzhou
Guangzhou 510006, China

###### Abstract

Our knowledge of Giga-electron volt (GeV) radio galaxies has been
revolutionized by the Fermi-LAT Telescope, which provides an excellent
opportunity to study the physical properties of GeV radio galaxies. According
to the radio power and morphology, radio galaxies can be separated into
Fanaroff-Riley Type I radio galaxies (FR-Is) and Type II radio galaxies (FR-
IIs). In this paper, we consider the unification of FR-Is and BL Lacertae
objects (BL Lacs), and assume FR-Is to be a standard candle to discuss the
beaming effect for Fermi-LAT-detected FR-Is. Our main conclusions are as
follows: (1) The estimated Doppler factors for 30 Fermi-LAT-detected FR-Is are
in a range of $\delta_{\rm{I}}=0.88-7.49$. The average Doppler factor
($<\delta_{\rm{I}}>=2.56\pm 0.30$) of the 30 FR-Is is smaller than that
($<\delta_{\rm{BL}}>=10.28\pm 2.03$) of the 126 Fermi-LAT-detected BL Lacs,
supporting the unification model that FR-Is are regarded as the misaligned BL
Lacs with smaller Doppler factors; (2) We propose that different regions of
FR-Is in the plot of the $\gamma$-ray luminosity against the photon spectral
index $(\log L_{\gamma}-\alpha_{\rm{ph}})$ may indicate the different beaming
effects; (3) The average Doppler factor of the 6 TeV FR-Is is similar to that
of the 24 non-TeV FR-Is, which implies that the difference between the TeV and
GeV emissions is not driven by the beaming effect in the Fermi-LAT-detected
FR-I samples.

Active galaxies - Fanaroff-Riley radio galaxies - jets - Gamma-rays

##  1 Introduction

Active galactic nuclei (AGNs) are the most energetic objects with relativistic
jets in the Universe. Their physical structure, supermassive black hole (SMBH)
+ accretion disk + relativistic jet, is widely accepted for decades (Urry &
Padovani, 1995). Blazars are an extreme subclass of AGNs with the jet being
aligned with our line of sight, which can be separated into BL Lacertae
objects (BL Lacs) with weak or no emission lines (equivalent width,
EW$<5\mathring{A}$), or flat spectrum radio quasars (FSRQs) with broad
emission lines (EW$>5\mathring{A}$) (Stickel et al., 1991; Urry & Padovani,
1995). Their electromagnetic emission is strongly boosted due to the
relativistic jet pointing toward the observer, resulting that the observed
flux density ($f^{\rm{ob}}$) is enhanced by the Doppler factor ($\delta$),
namely $f^{\rm{ob}}=\delta^{q}f^{\rm{in}}$, where $f^{\rm{in}}$ denotes the
intrinsic flux density, $q=2+\alpha$ for a continuous jet and $3+\alpha$ for
the case of a moving sphere (Lind & Blandford, 1985; Ghisellini et al., 1993),
$\alpha$ is a spectral index ($f_{\nu}\propto\nu^{-\alpha}$). The Doppler
factor can be expressed by $\delta=[\Gamma(1-\beta\cos\theta)]^{-1}$, where
$\theta$ is an viewing angle, and $\Gamma$ is a Lorentz factor determined by
bulk velocity ($\beta=v/c$), $\Gamma=1/\sqrt{1-\beta^{2}}$.

Radio galaxies are also a subclass of AGNs. Based on the morphology and radio
power, Fanaroff & Riley (1974) classified the radio galaxies into Fanaroff-
Riley Type I radio galaxies (FR-Is) and Type II radio galaxies (FR-IIs). FR-Is
are lower radio power sources ($L_{\rm{178MHz}}<2.5\times 10^{33}$ erg s-1
Hz-1) with lobe dimmed morphology, while FR-IIs are all high radio power
sources ($L_{\rm{178MHz}}>2.5\times 10^{33}$ erg s-1 Hz-1) with one sided kpc-
scale jet and bright lobe (Fanaroff & Riley, 1974). According to the unified
schemes (Urry & Padovani, 1995), radio galaxies are regarded as edge-on
blazars with smaller Doppler factors and larger viewing angles. Thus radio
galaxies are expected to be the homologous population of blazars, namely, FR-
Is and FR-IIs are unified with BL Lacs and FSRQs, respectively.

Since the launch of the Large Area Telescope on board the Fermi Gamma-ray
Telescope (hereafter, Fermi-LAT) in 2008, blazars are the majority of Fermi-
LAT-detected sources (Abdollahi et al., 2020). Fermi-LAT-detected blazars show
stronger beaming effects than non-Fermi-LAT-detected blazars (Pushkarev et
al., 2009; Wu et al., 2014; Chen et al., 2015b; Fan et al., 2015; Xiao et al.,
2019; Pei et al., 2020a). Of the 3511 AGNs detected by the Fermi-LAT
incremental Data Release (4FGL-DR2), only 44 sources are identified as radio
galaxies (Abdollahi et al., 2020). Based on the unification of blazars and
radio galaxies, it is reasonable to consider that Fermi-LAT-detected radio
galaxies may be the transition sources that exhibit intermediate Doppler
beaming effects or special properties, due to these objects residing on the
boundary between blazars and radio galaxies.

Bassi et al. (2018) analyzed the multiband properties of a Fermi-LAT-detected
FR-I named IC 1531 (4FGL J0009.7-3217), and concluded that IC 1531 is a jet-
dominated source showing a moderate Doppler boosted flux amplification. This
source is potentially a good observational target residing on the boundary
between blazars and radio galaxies (Bassi et al., 2018).

Another Fermi-LAT-detected FR-I, NGC 1275 (4FGL J0319.8+4130), also shows
particular observational properties that are similar to those of blazars:
rapid variability, strong polarization, and high $\gamma$-ray luminosity
(Martin et al., 1976; Abdo et al., 2010a; Sahakyan et al., 2018), which may be
caused by the strong beaming effect. 3C 120 (4FGL J0433.0+0522) is a broad-
line radio galaxy (BLRG) with a blazar-like jet, which has been discussed by
many authors due to its rapid variability in the Giga-electron volt (GeV)
band. The rapid variability suggests that the $\gamma$-ray emission of 3C 120
originates from a very compact region (Casadio et al., 2015; Rulten et al.,
2020). Casadio et al. (2015) analyzed the jet properties of 3C 120 based on
radio and $\gamma$-ray data from 2012 to 2014, and found that the apparent
superluminal motion is as high as 6.2$c$. When the viewing angle is very
small, according to the relationship $\theta\sim\arccos{\beta}$, a viewing
angle of $\sim 9.2\degree$ is obtained, which corresponds to a Doppler factor
of 6.2. Jones et al. (2001) comprehensively studied the jets and lobes of Cen
B (4FGL J1346.3-6026) using data from the Australia Telescope Compact Array
and Molonglo Observatory Synthesis Telescope. Both jet-counterjet ratio
brightness and polarization asymmetries imply the relativistic Doppler factor
involved (Jones et al., 2001; Katsuta et al., 2013). The Tera-electron volt
(TeV) emission of IC 310 (4FGL J0316.8+4120) was first detected by MAGIC
(Major Atmospheric Gamma Imaging Cherenkov Telescope) observations in
2009-2010 (Aleksić et al., 2010). The variability timescale of IC 310 is as
short as $\sim$ 5 mins, which is the fastest variability up to now (Aleksić et
al., 2014). Based on Very Long Baseline Interferometry (VLBI) data, Kadler et
al. (2012) found that IC 310 has a one-sided blazar-like jet. The jet-to-
counterjet ratio constrains the jet viewing angle to be $\leq 38\degree$,
implying a moderate beaming.

Xue et al. (2017) adopted the one-zone leptonic model to fit the spectral
energy distributions (SEDs) of 12 GeV radio galaxies, and ascertained that the
SED-derived Doppler factors are in a range of $\delta=1.2-9.8$, suggesting
that some GeV radio galaxies with larger Doppler factors may have smaller
viewing angles of jets like blazars.

Radio galaxies are regarded as the parent population of blazars, whose jet
direction is pointing away from the observer (Ghisellini et al., 1993; Urry &
Padovani, 1995). We consider that some radio galaxies keep small viewing
angles and moderate beaming effects, particularly for Fermi-LAT-detected radio
galaxies. Our aim in the present work is to estimate the Doppler factors and
discuss the beaming effect for these AGNs. This paper is arranged as follows:
In Section 2, we collected $\gamma$-ray properties and estimated the Doppler
factors for Fermi-LAT-detected FR-Is, and we discussed the beaming effect for
Fermi-LAT-detected FR-Is in Section 3. Finally, we draw our conclusions in
Section 4. For this work we assumed a $\Lambda$-CMD cosmology where
$\Omega_{\Lambda}\sim 0.68$, $\Omega_{\rm{M}}\sim 0.32$ and $H_{0}$ = 73 km
s-1 Mpc-1 (Planck Collaboration et al., 2016).

##  2 Samples and results

###  2.1  FR-I samples

Based on the catalog of 4FGL-DR2, 44 radio galaxies are detected by Fermi-LAT
(Abdollahi et al., 2020). We adopted the classifications of Angioni et al.
(2020), Harvey et al. (2020), and Rulten et al. (2020) to classify 44 radio
galaxies into FR Type I or Type II radio galaxies. If these references render
discrepant classifications for the same sources, we searched other references
to classify them. Finally, 44 sources are separated into 30 FR-Is, 10 FR-IIs,
and 3 uncertain type radio galaxies (4FGL J0958.3-2656, 4FGL J1236.9-7232,
4FGL J1724.2-6501). Subject to the FR-II samples, we only focused on the 30
Fermi-LAT-detected FR-Is and collected their redshifts from the NASA/IPAC
Extragalactic Database (NED) to estimate the Doppler factors and discuss their
beaming effect in the $\gamma$-ray band. The corresponding properties and
derived results of the 30 FR-Is are listed in Table 1. All the FR-Is are
nearby radio galaxies with redshift $z<$ 0.12. The $\gamma$-ray flux densities
are calculated from the integral photon fluxes and the $\gamma$-ray
luminosities are obtained from, $L_{\gamma}=4\pi
d_{\rm{L}}^{2}\nu_{\gamma}f_{\gamma}$, where $d_{\rm{L}}$ is the luminosity
distance determined by the redshift:

$d_{\mathrm{L}}=(1+z)\frac{c}{H_{0}}\int_{1}^{1+z}\frac{1}{\sqrt{\Omega_{\mathrm{M}}x^{3}+1-\Omega_{\mathrm{M}}}}dx$
(1)

The $\gamma$-ray flux densities and corresponding $\gamma$-ray luminosities
for our sample are listed in Col. (5) and (6) of Table. 1.

Table 1: The $\gamma$-ray properties and derived results of the 30 Fermi-LAT-detected FR-Is. 4FGL Name | Other Name | $z$ | $\alpha_{\rm{ph}}$ | $\log f_{\gamma}$ | $\log L_{\gamma}$ | $\delta_{\gamma}$ | $\delta_{\rm{max}}$ | $\delta_{\rm{min}}$
---|---|---|---|---|---|---|---|---
|  |  |  | ( erg/cm2/s) | (erg/s) |  |  |
(1) | (2) | (3) | (4) | (5) | (6) | (7) | (8) | (9)
J0009.7-3217 | IC 1531 | 0.026 | 2.15 | -12.01 | 42.10 | 1.63 | 2.20 | 1.21
J0057.7+3023 | NGC 315 | 0.016 | 2.37 | -11.87 | 41.82 | 1.31 | 1.73 | 0.99
J0153.4+7114 | TXS 0149+710 | 0.022 | 1.89 | -11.60 | 42.42 | 2.21 | 3.06 | 1.59
J0237.7+0206 | PKS 0235+017 | 0.022 | 2.15 | -12.08 | 41.88 | 1.39 | 1.88 | 1.03
J0308.4+0407 | NGC 1218 | 0.029 | 1.97 | -11.22 | 42.99 | 3.36 | 4.61 | 2.44
J0316.8+4120 | IC 0310 | 0.019 | 1.85 | -11.61 | 42.23 | 1.91 | 2.66 | 1.37
J0319.8+4130 | NGC 1275 | 0.018 | 2.11 | -9.64 | 44.14 | 7.49 | 10.15 | 5.53
J0322.6-3712e | Fornax A | 0.006 | 2.07 | -11.42 | 41.39 | 0.97 | 1.33 | 0.72
J0334.3+3920 | 4C +39.12 | 0.021 | 1.81 | -11.71 | 42.20 | 1.88 | 2.63 | 1.34
J0433.0+0522 | 3C 120 | 0.033 | 2.74 | -11.56 | 42.80 | 2.32 | 2.98 | 1.80
J0627.0-3529 | PKS 0625-35 | 0.055 | 1.92 | -10.96 | 43.86 | 6.85 | 9.47 | 4.95
J0708.9+4839 | NGC 2329 | 0.019 | 1.77 | -11.94 | 41.97 | 1.56 | 2.20 | 1.11
J0758.7+3746 | NGC 2484 | 0.043 | 2.20 | -12.09 | 42.47 | 2.12 | 2.85 | 1.58
J0931.9+6737 | NGC 2892 | 0.023 | 2.28 | -11.62 | 42.41 | 2.00 | 2.67 | 1.50
J1116.6+2915 | B2 1113+29 | 0.049 | 1.39 | -12.32 | 42.41 | 2.58 | 3.83 | 1.74
J1144.9+1937 | 3C 264 | 0.022 | 2.00 | -12.00 | 42.04 | 1.59 | 2.18 | 1.16
J1149.0+5924 | NGC 3894 | 0.011 | 2.18 | -11.89 | 41.53 | 1.07 | 1.44 | 0.80
J1219.6+0550 | NGC 4261 | 0.007 | 2.08 | -11.91 | 41.25 | 0.88 | 1.19 | 0.65
J1230.8+1223 | M 87 | 0.004 | 2.06 | -10.96 | 41.81 | 1.33 | 1.82 | 0.98
J1306.3+1113 | TXS 1303+114 | 0.086 | 1.98 | -11.98 | 43.26 | 4.11 | 5.63 | 2.99
J1325.5-4300 | Cen A | 0.002 | 2.64 | -10.86 | 41.30 | 0.92 | 1.20 | 0.71
J1346.3-6026 | Cen B | 0.013 | 2.40 | -10.92 | 42.66 | 2.30 | 3.04 | 1.74
J1449.5+2746 | B2.2 1447+27 | 0.031 | 1.54 | -12.12 | 42.19 | 1.99 | 2.88 | 1.37
J1518.6+0614 | TXS 1516+064 | 0.102 | 1.75 | -11.95 | 43.44 | 5.41 | 7.63 | 3.84
J1521.1+0421 | PKS B1518+045 | 0.052 | 2.04 | -12.02 | 42.76 | 2.75 | 3.75 | 2.01
J1630.6+8234 | NGC 6251 | 0.025 | 2.37 | -11.22 | 42.88 | 2.70 | 3.57 | 2.04
J1843.4-4835 | PKS 1839-48 | 0.111 | 2.03 | -11.98 | 43.49 | 4.77 | 6.52 | 3.50
J2227.9-3031 | ABELL 3880 | 0.058 | 1.98 | -12.08 | 42.78 | 2.85 | 3.91 | 2.07
J2329.7-2118 | PKS 2327-215 | 0.031 | 2.45 | -11.81 | 42.46 | 2.00 | 2.62 | 1.52
J2341.8-2917 | PKS 2338-295 | 0.052 | 2.24 | -12.02 | 42.73 | 2.53 | 3.39 | 1.89

###  2.2  Standard candles

From the radio-loud AGNs unification, the parent population of BL Lacs
consists of FR-Is (Ghisellini et al., 1993; Urry & Padovani, 1995; Bai & Lee,
2001; Xu et al., 2009; Chen et al., 2015a). Urry & Padovani (1995) proposed
that the mean host galaxy magnitude for nearby BL Lacs ($z<$ 0.2) is
$<M_{\rm{V}}>=-22.9\pm 0.3$, which is similar to the mean magnitude for nearby
FR-Is, $<M_{\rm{V}}>=-23.1\pm 0.1$.

Urry et al. (2000) reported Hubble Space Telescope (HST) images of 85 BL Lac
host galaxies and obtained their median K-corrected absolute magnitude
$<M_{\rm{V}}>=-23.7$ with a small dispersion of $\pm$0.6 mag. As mentioned in
Falomo et al. (2014), the host galaxies of BL Lacs can be regarded as the
standard candles, since the host galaxies of BL Lacs exhibited a relatively
narrow range of absolute magnitudes in the optical band.

The ranges of absolute magnitudes for radio galaxies are also narrow with a
small dispersion (Sandage, 1972; Hine & Longair, 1979; Magliocchetti et al.,
2002a, b). A study of the optical properties for 350 nearby FR-Is, shows their
absolute magnitudes ($M_{\rm{B}}$) are distributed within a narrow interval
around $M_{\rm{B}}\sim$ -21.3, suggesting that FR-Is are reliable standard
candles (Magliocchetti et al., 2002b). Based on the optical properties of FR-
Is, we consider FR-Is as a standard candle and assume that the luminosity for
FR-Is is a constant; therefore, the logarithm of the flux density ($\log f$)
will be expected to have a linear correlation with their logarithm of the
luminosity distance ($\log d_{\rm{L}}$), $\log f=-2\log d_{\rm{L}}+$const (Fan
& Xie, 1996; Ye et al., 2021). The correlation between the logarithm of the
$\gamma$-ray flux densities ($\log f_{\gamma}$) and the logarithm of the
luminosity distances ($\log d_{\rm{L}}$) for the 30 Fermi-LAT-detected FR-Is
are plotted in Fig. 1. The best-fitting ( black solid line) is shown in Fig. 1
[$\log f_{\gamma}=-(0.64\pm 0.25)\log d_{\rm{L}}-(10.36\pm 0.51)$ with
$r=-0.44$, $p=1.58\%$]. Although the flux densities are correlated with their
luminosity distances in the $\gamma$-ray band, the slope is not the same as
the expected value of $-2$. We proposed that this difference may be due to a
moderate beaming effect in Fermi-LAT-detected FR-Is.

Figure 1:  Plot of the correlation between the logarithm of the $\gamma$-ray
flux densities ($\log f_{\gamma}$) and the logarithm of the luminosity
distances ($\log d_{\rm{L}}$). The black square markers show the Fermi-LAT-
detected FR-Is. The black solid line is the best linear regression for the
FR-I samples [$\log f_{\gamma}=-(0.64\pm 0.25)\log d_{\rm{L}}-(10.36\pm
0.51)$]. The red dotted-dash line is obtained from the fixed slope of $-2$ and
a constant intrinsic luminosity [$\log L_{\gamma}=41.81\pm 0.41$ (erg/s)] from
the peak value of the Gaussian fitting of the 126 Fermi-LAT-detected BL Lacs.
The red dotted-dash line with red shaded area corresponds to $\log
f_{\gamma}=-2\log d_{\rm{L}}-(8.65\pm 0.41)$. The differences between the
$\gamma$-ray flux densities of FR-Is and the red dotted-dash line are
considered to be the beaming effect in the jet of FR-Is.

###  2.3  BL Lac samples

BL Lacs show extreme observational properties, which are due to the beaming
effect. The observed flux density ($f^{\rm{ob}}$) of BL Lacs is amplified by
the Doppler factor ($\delta$), $f^{\rm{ob}}=\delta^{q}f^{\rm{in}}$, and the
corresponding observed luminosity ($L^{\rm{ob}}$, in units of erg/s) is also
enhanced: $L^{\rm{ob}}=\delta^{q+1}L^{\rm{in}}$, where $L^{\rm{in}}$ is the
intrinsic luminosity in units of erg/s, $q$ for $2+\alpha$ (a continuous jet)
or $3+\alpha$ (the case of a moving sphere) (Ghisellini et al., 1993; Dondi &
Ghisellini, 1995; Fan et al., 2013a). If the Doppler factor is assumed to be
$\delta$=10 (average Doppler factor of BL Lacs in this work) in the case of a
continuous jet (Fan et al., 1997; Mondal & Mukhopadhyay, 2019; Pei et al.,
2020b) and a $\gamma$-ray spectral index of 1, the difference between the
observed luminosity and the intrinsic luminosity will be up to 4 orders of
magnitude. Therefore, the observed $\gamma$-ray luminosity of Fermi-LAT-
detected BL Lacs is strongly boosted on account of the beaming effect (Dondi &
Ghisellini, 1995; Fan et al., 2013a; Chen et al., 2015b; Yang et al., 2022).

The Doppler factor is important for understanding the physics of blazars, but
obtaining a reliable estimate is challenging because it is often difficult to
determine accurate viewing angles and bulk velocities from observations. As a
result, many indirect methods for estimating the Doppler factor have been
proposed (Ghisellini et al., 1993; Valtaoja et al., 1999; Zhang et al., 2002;
Fan et al., 2009; Hovatta et al., 2009; Fan et al., 2013b, 2014; Liodakis et
al., 2018; Pei et al., 2020b; Zhang et al., 2020). Such as the variability
Doppler factor estimations from the differences between the observed
brightness temperature and equipartition brightness temperature (Valtaoja et
al., 1999; Fan et al., 2009; Hovatta et al., 2009; Liodakis et al., 2018); the
SED-derived Doppler factors from the SED fitting (Chen, 2018); the Doppler
factor estimations from the combination between the X-ray data and the
synchrotron self-Compton (SSC) model (Ghisellini et al., 1993); the lower
limit of the $\gamma$-ray Doppler factor estimations from the pair-production
opacity in the $\gamma$-ray emissions (Dondi & Ghisellini, 1995; Fan et al.,
2013b, 2014; Pei et al., 2020b); the Doppler factor estimations from the
correlations between the broad-line luminosity and $\gamma$-ray luminosity
(Zhang et al., 2020). Zhang et al. (2002) assumed that the multiband radiation
of blazars originates from accelerated particles in the jet and used the
multiband data to obtain the Doppler factors of blazars at different wavebands
(radio, optical, X-ray, and $\gamma$-ray). The authors showed that, on
average, radio and $\gamma$-ray Doppler factors are comparable.

As mentioned in Liodakis et al. (2018), the variability Doppler factor method
is the best way to describe the blazar populations. Furthermore, the averaged
$\gamma$-ray Doppler factor is comparable to the averaged radio Doppler factor
(Zhang et al., 2002). Therefore, the Doppler factors given in Liodakis et al.
(2018) can be adopted to calculate the intrinsic $\gamma$-ray luminosity as in
Yang et al. (2022).

Zhang et al. (2020) obtained the Doppler factors from the correlations between
the $\gamma$-ray luminosity and broad-line luminosity, and concluded that
their Doppler factors are consistent with the results in Liodakis et al.
(2018). Therefore, The Doppler factors reported from Zhang et al. (2020) are
also available to obtain the intrinsic luminosity.

Following the method of Mattox et al. (1993), Pei et al. (2020b) assumed that
the X-ray and $\gamma$-ray emissions are from the same region and the
$\gamma$-ray variability timescale is around one day to estimate the
$\gamma$-ray Doppler factors. However, this method can only obtain the lower
limit of the $\gamma$-ray Doppler factors, which will cause the intrinsic
luminosity to be an upper limit value. Therefore, the Doppler factors of Pei
et al. (2020b) are not considered in this work.

By cross-checking the Doppler factors (Liodakis et al., 2018; Zhang et al.,
2020) and Fermi-LAT-detected BL Lacs (Abdollahi et al., 2020), we compiled a
sample of 126 Fermi-LAT-detected BL Lacs with available Doppler factors. We
took the Doppler factors from Zhang et al. (2020) for 22 sources in common. If
the Doppler factor $\delta<1.0$, $\delta=1$ is adopted to compute the
intrinsic luminosity, namely, $\log L^{\rm{ob}}_{\gamma}=\log
L^{\rm{in}}_{\gamma}$. The detailed parameters for BL Lacs are listed in Table
2.

The redshift distribution of the 126 Fermi-LAT-detected BL Lacs in this work
(BL LacsTW) is in a range from 0.030 to 2.017 with an average of 0.578, as
shown in Fig. 2 (red solid line). We also collected the redshifts from
NASA/NED for the whole Fermi-LAT-detected BL Lacs, and obtained a sample of
845 BL Lacs with available redshifts. The redshift distribution for the 845
Fermi-LAT-detected BL Lacs (BL LacsRef) is from 0.00001 $\sim 6.443$ and the
average is 0.448, which is also presented in Fig. 2 (blue dotted-dash line).
From this figure, the redshift distribution of BL LacsTW is similar to that of
BL LacsRef, and the difference between their average values is very small,
suggesting that the 126 Fermi-LAT-detected BL Lacs in this work are
representative of the much larger sample of Fermi-LAT-detected BL Lacs.

Figure 2:  The redshift distributions for the Fermi-LAT-detected BL Lacs. The
red solid line is the redshift distribution ($z=0.030\sim$ 2.017) for the 126
Fermi-LAT-detected BL Lacs in this work (BL LacsTW). The blue dotted-dash line
is the redshift distribution ($z=0.00001\sim$ 6.443) for the 845 Fermi-LAT-
detected BL Lacs from the Fermi-LAT reference (BL LacsRef) (Abdollahi et al.,
2020). The redshift distribution of 126 BL LacsTW is similar to that of 845 BL
LacsRef, and their average redshifts are comparable, suggesting that the 126
Fermi-LAT-detected BL Lacs in this work are representative of the much larger
sample of Fermi-LAT-detected BL Lacs.

By employing the Doppler factors of Liodakis et al. (2018) and Zhang et al.
(2020), we computed the intrinsic luminosities ($L^{\rm{in}}_{\gamma}$) for
the 126 Fermi-LAT-detected BL Lacs in the case of a continuous jet (Fan et
al., 1997; Mondal & Mukhopadhyay, 2019; Pei et al., 2020b), $\log
L^{\rm{in}}_{\gamma}=\log L^{\rm{ob}}_{\gamma}-(2+\alpha_{\gamma})\log\delta$,
in which $\alpha_{\gamma}$ is a $\gamma$-ray spectral index,
$\alpha_{\gamma}=\alpha_{\rm{ph}}-1$. The intrinsic luminosity for the 126 BL
Lacs ranges from $\log L^{\rm{in}}_{\gamma}=$38.87 to 45.98 with an average of
$<\log L^{\rm{in}}_{\gamma}>=42.07\pm 0.21$, and the median intrinsic
luminosity is $\log L^{\rm{in}}_{\gamma}=$41.91, as shown in the upper panel
of Fig. 3. The intrinsic luminosity distribution for the 126 BL Lacs is
compared with the Gaussian distribution by using the Kolmogorov-Smirnov (K-S)
test. The hypothesis for the K-S test is to discriminate between two
independent samples, and the probability $p$ value threshold that we are using
is 0.05. The $p$ value for the K-S test between our sample and the Gaussian
distribution is $p=0.606$, supporting that our intrinsic luminosity sample can
be regarded as the Gaussian distribution. The Quantile-quantile (QQ) plot is
also applied for the visual inspection (the lower panel of Fig. 3); most of
our data are located in the area of 95% confidence level interval of the
Gaussian distribution (purple shaded area), so we believed that our sample can
be analyzed by a Gaussian fitting. The peak value for the Gaussian fitting is
$\log L^{\rm{in}}_{\gamma}=41.81$ erg/s, which is regarded as the intrinsic
luminosity for the whole BL Lac sample. The 95% confidence level interval is
adopted to the error for the intrinsic luminosity, $\log
L^{\rm{in}}_{\gamma}=41.81\pm 0.41$ erg/s. Compared with the average of our
intrinsic luminosity sample, this intrinsic luminosity ($\log
L^{\rm{in}}_{\gamma}=41.81\pm 0.41$) obtained from the peak value of the
Gaussian fitting is less affected by outliers, which is more representative of
the intrinsic luminosity of 126 BL Lacs in this work.

Figure 3: The upper panel is the distribution of the logarithm of the
intrinsic luminosity for the 126 Fermi-LAT-detected BL Lacs. The lower panel
is a Quantile-quantile (QQ) plot for the comparison between the intrinsic
luminosity distribution of 126 BL Lacs (black circle) and the Gaussian
distribution (red solid line). The purple shaded area in the lower panel is
the 95% confidence level interval of the Gaussian distribution. From a simple
visual inspection, it is clear that most of the BL Lacs are located inside the
purple shaded area. Finally, the probability ($p=0.606$) resulting from a K-S
test also supports the hypothesis that the intrinsic luminosities for the 126
BL Lacs are normally distributed.

###  2.4 Doppler factors

From the unification of BL Lacs and FR-Is (Ghisellini et al., 1993; Urry &
Padovani, 1995; Bai & Lee, 2001; Xu et al., 2009; Chen et al., 2015a), the
intrinsic emissions (after removing the beaming effect) of BL Lacs should be
similar to those of FR-Is. Hence we adopted the intrinsic luminosity ($\log
L^{\rm{in}}_{\gamma}=41.81\pm 0.41$ erg/s) of the 126 BL Lacs as the intrinsic
luminosity of the 30 FR-Is. Therefore, the expected relation between the
$\gamma$-ray flux density and the luminosity distance can be obtained in the
case of the luminosity constant of $\log L_{\gamma}=41.81\pm 0.41$ erg/s,
$\log f_{\gamma}=-2\log d_{\rm{L}}-(8.65\pm 0.41)$. This linear relation with
its error is presented in Fig. 1 (red dotted-dash line with red shaded area).
The difference between the observed $\gamma$-ray flux densities of FR-Is and
the red shaded area may derive the Doppler factors,

$\log\delta=[\log f^{\rm{ob}}+2\log d_{\rm{L}}+(8.65\pm
0.41)]/1+\alpha_{\rm{ph}}$ (2)

where $f^{\rm{ob}}$ denotes the observed flux densities of FR-Is. The
corresponding $\gamma$-ray Doppler factors with their possible maximum and
minimum values are listed in Col. 7-9 of Table 1.

Table 2:  The $\gamma$-ray properties for the 126 Fermi-LAT-detected BL Lacs. 4FGL Name | $z$ | $\alpha_{\rm{ph}}$ | $\log f_{\gamma}$ | $\log L^{\rm{ob}}_{\gamma}$ | $\delta_{\gamma}$ | Ref | $\log L^{\rm{in}}_{\gamma}$
---|---|---|---|---|---|---|---
|  |  | ( erg/cm2/s) | (erg/s) |  |  | (erg/s)
(1) | (2) | (3) | (4) | (5) | (6) | (7) | (8)
J0006.3-0620 | 0.347 | 2.13 | -12.04 | 44.54 | 6.96 | L18 | 41.06
J0014.1+1910 | 0.473 | 2.28 | -11.87 | 45.02 | 5.99 | Z20 | 41.70
J0037.8+1239 | 0.090 | 2.22 | -11.37 | 43.90 | 2.51 | L18 | 42.21
J0047.0+5657 | 0.747 | 2.16 | -11.19 | 46.19 | 21.03 | L18 | 40.69
J0049.7+0237 | 1.474 | 2.21 | -11.04 | 47.07 | 12.83 | L18 | 42.41
J0050.7-0929 | 0.634 | 2.02 | -10.54 | 46.67 | 20.23 | L18 | 41.42
J0056.8+1626 | 0.206 | 2.20 | -11.99 | 44.06 | 5.89 | L18 | 40.83
J0105.1+3929 | 0.440 | 2.30 | -11.57 | 45.25 | 16.12 | L18 | 40.06
J0113.7+0225 | 0.047 | 2.48 | -11.98 | 42.71 | 0.97 | L18 | 42.71
J0127.2-0819 | 1.416 | 2.18 | -11.30 | 46.78 | 14.59 | L18 | 41.92
J0127.9+4857 | 0.067 | 2.59 | -11.98 | 43.02 | 7.95 | L18 | 38.89
J0141.4-0928 | 0.733 | 2.15 | -10.80 | 46.56 | 15.19 | L18 | 41.65
J0153.9+0823 | 0.681 | 1.95 | -10.90 | 46.38 | 7.63 | L18 | 42.89
J0203.7+3042 | 0.760 | 2.24 | -10.84 | 46.56 | 24.58 | Z20 | 40.66
J0209.9+7229 | 0.895 | 2.28 | -11.30 | 46.27 | 20.81 | L18 | 40.64
J0211.2+1051 | 0.200 | 2.12 | -10.46 | 45.56 | 8.41 | L18 | 41.76
J0217.2+0837 | 0.085 | 2.27 | -11.05 | 44.17 | 2.77 | L18 | 42.28
J0238.6+1637 | 0.940 | 2.16 | -10.13 | 47.50 | 33.52 | Z20 | 41.14
… | … | … | … | … | … | … | …

##  3 Discussions

Our understanding of the extragalactic $\gamma$-ray sky has been
revolutionized by Fermi-LAT, in particular, the $\gamma$-ray sky of AGNs, in
which more than 98% extragalactic sources are blazars, and only 44 sources are
identified as radio galaxies in 4FGL-DR2 (Abdollahi et al., 2020). Some Fermi-
LAT-detected radio galaxies show blazar-like observational properties, such as
rapid variability, strong polarization, superluminal motion or high
$\gamma$-ray luminosity (Martin et al., 1976; Bailey et al., 1986; Abdo et
al., 2010a; Casadio et al., 2015; Sahakyan et al., 2018; Rulten et al., 2020).
Like blazars, these special observational properties may be caused by the jet,
which prompts us to consider the beaming effect (modest or strong) for these
Fermi-LAT-detected radio galaxies. The $\gamma$-ray Doppler factor for the 30
Fermi-LAT-detected FR-Is is in a range from $\delta_{\rm{I}}=0.88$ to
$\delta_{\rm{I}}=7.49$. and their distribution is shown in Fig. 4.

Figure 4:  The distribution of Doppler factors derived in this work for the 30
Fermi-LAT-detected FR-Is. The Doppler factor estimations for 30 Fermi-LAT-
detected FR-Is is in a range from $\delta_{\rm{I}}=0.88$ to
$\delta_{\rm{I}}=7.49$ with an average of $<\delta_{\rm{I}}>=2.56\pm 0.30$,
and the medium Doppler factor $\delta_{\gamma}=2.06$.

The average Doppler factor of the 30 Fermi-LAT-detected FR-Is,
$<\delta_{\rm{I}}>=2.56\pm 0.30$, is significantly smaller than that
($<\delta_{\rm{BL}}>=10.28\pm 2.03$) of the 126 Fermi-LAT-detected BL Lacs in
this work. This is consistent with the unified scheme that FR-Is are the
parent population of BL Lacs, which have larger viewing angles and smaller
Doppler factors.

###  3.1  Weak beaming

6 of 30 Fermi-LAT-detected FR-Is residing in the red shaded area [$\log
f_{\gamma}=-2\log d_{\rm{L}}-(8.65\pm 0.41)$] of Fig. 1 may be lobe-dominated
FR-Is, of which the dominant emission is located further down the parsec scale
jet as opposed to the core. M87 is the first detected TeV radio galaxy showing
the shortest variability as short as $\sim$2 days (Aharonian et al., 2006).
The jet viewing angle is constrained at $\sim 30\degree$ (Bicknell & Begelman,
1996; Acciari et al., 2011). During the monitoring of VLBA, its average
apparent superluminal motion in the core is 1.1$c$. Adopting the jet
orientation angle $\sim 30\degree$, one can obtain the average Doppler factor
$\approx 1.5$ for M87. Ghisellini et al. (1993) combined the X-ray emission
and SSC model to estimate the Doppler factor of M87 ($\delta_{\rm{SSC}}=1$).
Liodakis et al. (2018) calculated the difference between the observed
brightness temperature and equipartition brightness temperature to estimate
the Doppler factor of M87 ($\delta_{\rm{var}}=0.2$). Our Doppler factor
estimation for M87, $\delta_{\gamma}=1.33$, is consistent with previous
results, also indicating that the total emission of M87 is not significantly
boosted by its jet.

Cen A, the first FR-I whose core and lobe emissions can be separated by Fermi-
LAT (Abdo et al., 2010b), had been found that the $\gamma$-ray jet morphology
is similar to the radio one, implying the $\gamma$-ray emission is from the
core and lobe components. Assuming a power law for the $\gamma$-ray spectra of
Cen A, Abdo et al. (2010b) found a large fraction ($>1/2$) of the $\gamma$-ray
emission of Cen A to originate from the lobe. Its radio 5GHz core flux density
(6.98 Jy) is significantly smaller than its 5GHz extended flux density (55.85
Jy) (Di Mauro et al., 2014). The Doppler factors for the core of Cen A were
estimated by the single-zone SSC model ($\delta_{\rm{D}}$=1), the SSC model
excluding X-rays ($\delta_{\rm{D}}$=3.9), the SSC model for High Energy
Stereoscopic System data only ($\delta_{\rm{D}}$=3.1) (Abdo et al., 2010c) or
the decelerating jet model ($\delta_{\rm{D}}=1.79\rightarrow 1.01$ for the
case of $\Gamma=5\rightarrow 2$) (Georganopoulos & Kazanas, 2003; Abdo et al.,
2010c). Our derived Doppler factor for Cen A ($\delta{\gamma}=0.92$) is
consistent with the estimation in the case of the decelerating jet model.

Fornax A is the second FR-I whose $\gamma$-ray emission can be observed from
the lobe by Fermi-LAT (Ackermann et al., 2016). Ackermann et al. (2016)
reported its GeV band behavior using 6.1 years of Fermi-LAT data, and
concluded that its $\gamma$-ray emission originates in the lobes and the core
emission contributes the total emission no more than 14%. Its radio 5GHz core
flux density (0.05 Jy) also accounts for a small fraction of the total flux
density (72 Jy) (Di Mauro et al., 2014). Our Doppler factor estimation for
Fornax A ($\delta_{\gamma}=0.97$) is also small, supporting that the core
emission is not boosted by its jet. The small $\delta_{\gamma}$ estimations
for Cen A and Fornax A may account for the core and lobe emissions separated
by Fermi-LAT. Some Doppler factors $\delta<1$ in the FR-I samples may be due
to the systematic error or the limitation of this method.

NGC 315 was first identified as a radio galaxy in 4FGL (Abdollahi et al.,
2020). Park et al. (2021) employed VLBA observation to investigate the jet
collimation and acceleration of NGC 315. They found an apparent outward motion
($\beta=1.85\pm 0.44c$) of the core of NGC 315, then combined the jet-to-
counterjet ratio to constrain the jet viewing angle of $\approx 50\degree$.
They suggested that the flux density of NGC 315 from the Doppler boosting
effect is at most by a factor of about two. Our Doppler factor estimation for
NGC 315 ($\delta_{\gamma}=1.31$) is suggesting that the flux density of NGC
315 is not strongly enhanced by a Doppler factor, in accordance with Park et
al. (2021).

Besides the four FR-Is mentioned above, the other two FR-Is (NGC 3894, NGC
4261) are also possessing the Doppler factors $\sim 1$, first reported in
4FGL. There is one FR-I, NGC 4261, whose $\gamma$-ray properties ($\log
L_{\gamma}=41.25$ erg/s, $\alpha_{\rm{ph}}=2.08$, $\delta=0.88$) are similar
to those ($\log L_{\gamma}=41.39$ erg/s, $\alpha_{\rm{ph}}=2.07$,
$\delta=0.97$) of Fornax A, and the redshifts between NGC 4261 (0.007) and
Fornax A (0.006) are similar. We hoped that NGC 4261 is a potential target
whose dominant $\gamma$-ray emissions in the lobe component can be detected by
Fermi-LAT in the future just like the observations of Fornax A.

###  3.2 Modest or strong beaming

24 of 30 Fermi-LAT-detected FR-Is residing outside the red shaded area [$\log
f_{\gamma}=-2\log d_{\rm{L}}-(8.65\pm 0.41)$] of Fig. 1 may show modest or
strong Doppler-boosted flux amplification. IC 1531 is classified as a radio
galaxy in Abdollahi et al. (2020), but as a blazar of uncertain type in the
third source catalog (3FGL) (Acero et al., 2015). Bassi et al. (2018) used the
multiband data to investigate its nature and jet properties, and concluded
that IC 1531 is seen at moderate angles ($\theta=10\degree-20\degree$) and
presented a moderate Doppler beaming effect in the flux emission. They also
assembled the multiband data to fit the SED of the core of IC 1531, in which
the $\gamma$-ray flux of IC 1531 can be explained by the SSC model with
$\Gamma\sim 4$. When a viewing angle of $\theta=15\degree$ is adopted (Bassi
et al., 2018), the SED-derived Doppler factor corresponds to $\delta\approx
3.86$. For IC 1531, our Doppler factor estimation ($\delta_{\gamma}=1.63$) is
smaller than the SED-derived Doppler factor (3.86) from Bassi et al. (2018).

3C 120 shows a blazar-like jet with a Doppler factor of 6.2 (Casadio et al.,
2015). Hovatta et al. (2009) compared the observed radio brightness
temperature and radio equipartition brightness temperature to obtain the
Doppler factor ($\delta_{\rm{var}}=5.9$) of 3C 120. Our Doppler factor
estimation for 3C 120, $\delta_{\gamma}=2.32$, is derived from the average
flux density during the Fermi-LAT observations. The rapid short-timescale
$\gamma$-ray flux variability of 3C 120 can be observed during the Fermi-LAT
observations. If the $\gamma$-ray flux density is adopted from the prominent
flare, our estimated Doppler factor for 3C 120 will increase to be comparable
to the Doppler factors of Casadio et al. (2015) and Hovatta et al. (2009).

IC 310 presented a one-sided blazar-like jet during VLBI observations,
implying a Doppler factor involved (Kadler et al., 2012). Ghisellini et al.
(1993) estimated the Doppler factor of IC 310, $\delta_{\rm{SSC}}=1.6$, based
on the X-ray emission and SSC model; Liodakis et al. (2018) obtained the radio
variability Doppler factor, $\delta_{\rm{var}}=2.11$, from the difference
between the observed brightness temperature and equipartition brightness
temperature. Our Doppler factor estimation for IC 310 ($\delta_{\gamma}=1.91$)
is consistent with the previous results.

PKS 0625-35, a TeV-detected source, displays both FR-I and BL Lac properties
(Wills et al., 2004; Rani, 2019). Based on the Tracking Active Nuclei with
Austral Milliarcsecond Interferometry (TANAMI) observations, Mueller et al.
(2012) found the presence of the superluminal motion of 3$\pm 0.5c$ in the jet
of PKS 0625-35. The superluminal motion of 3$c$ is also later confirmed by
Angioni et al. (2019). The jet viewing angle for PKS 0625-35 is constrained to
$\theta\approx 15\degree$ from the X-ray variability (HESS Collaboration et
al., 2018), or $\theta<53\degree$ from the VLBI observations (Angioni et al.,
2019). Our Doppler factor estimation for PKS 0625-35, $\delta=6.85$, is
similar to the median Doppler factor ($\delta_{\rm{BL}}=7.26$) of the 126 BL
Lacs in this work, suggesting that this FR-I is a misaligned blazar producing
a strong beaming effect.

Our Doppler factor estimation ($\delta_{\gamma}=7.49$) for NGC 1275 is the
largest value in the FR-I samples, which is also similar to the median Doppler
factor ($\delta_{\rm{BL}}=7.26$) of the 126 BL Lacs. From the radio
observations, Hovatta et al. (2009) obtained the observed brightness
temperature of NGC 1275 and assumed the equipartition temperature as $5\times
10^{10}$ K to calculate the variability Doppler factor
($\delta_{\rm{var}}=0.3$) for NGC 1275. The significant discrepancy between
the variability Doppler factor ($\delta_{\rm{var}}=0.3$) and our Doppler
factor ($\delta_{\gamma}=7.49$) may be due to the possible bending in the jet
of NGC 1275 (Pei et al., 2020b). If the jet of NGC 1275 is aligned to the
observer in the $\gamma$-ray emission region but is bent to a large viewing
angle in the radio emission region, it may lead to this discrepancy.

The observational properties of the 24 FR-Is residing outside the red shaded
area of Fig. 1, are likely to exhibit modest or strong Doppler beaming
effects. As discussed in Abdo et al. (2010a), the $\gamma$-ray luminosities
for some Fermi-LAT-detected radio galaxies are similar to those of blazars,
implying a strong beaming effect for the $\gamma$-ray emission. These 24 FR-Is
are likely effective transition sources between the lobe-dominated FR-Is and
blazars.

###  3.3 Intrinsic luminosity

If we consider the unification of BL Lacs and FR-Is (Ghisellini et al., 1993;
Urry & Padovani, 1995), it is expected that the observed $\gamma$-ray
luminosity of FR-Is should be similar to the intrinsic luminosity of BL Lacs;
however, as shown in the Fig. 5, most FR-Is have larger observed $\gamma$-ray
luminosities than the intrinsic luminosity ($\log L^{in}_{\gamma}=41.81\pm
0.41$ erg/s) of BL Lacs. Figure 5 shows that the $\gamma$-ray luminosities of
some FR-Is fall between the intrinsic luminosity band (pink shaded area, $\log
L^{in}_{\gamma}=41.81\pm 0.41$ erg/s) and the cluster of BL Lacs (blue-filled
circles), which resemble a bridge linking both AGN classes. The observed
$\gamma$-ray luminosities for the lobe-dominated FR-Is are comparable with the
intrinsic luminosity. Those FR-Is with high $\gamma$-ray luminosity showing
special observational properties are thought to be caused by the strong
beaming effect. The position of distribution ($\log
L_{\gamma}-\alpha_{\rm{ph}}$) may be an effective indicator for the beaming
effect of FR-Is.

Figure 5:  Plot of the correlation between the logarithm of $\gamma$-ray
luminosities ($\log L_{\gamma}$) and photon spectral indexes
($\alpha_{\rm{ph}}$) for 30 FR-Is and 126 BL Lacs. The red-filled square
markers show the FR-Is, and the blue-filled circle markers show the BL Lacs.
The black solid line with pink shaded area is the intrinsic luminosity with
error ($\log L_{\gamma}=41.81\pm 0.41$ erg/s) obtained from the peak value of
the Gaussian fitting of the 126 BL Lacs. The position of distribution ($\log
L_{\gamma}-\alpha_{\rm{ph}}$) may be an effective indicator for the beaming
effect of FR-Is.

###  3.4 Comparison with SED-derived Doppler factors

Xue et al. (2017) compiled a sample of SEDs for 12 GeV radio galaxies (8 FR-Is
and 4 FR-IIs) with one-zone leptonic model and obtained their Doppler factors.
We compare our Doppler factor estimations with those from Xue et al. (2017)
for 8 GeV FR-Is. The result is shown in Fig. 6. The probability of linear
correlation is 0.053; therefore, our estimated $\gamma$-ray Doppler factors
are not correlated with the SED-derived Doppler factors from Xue et al.
(2017). The radiation in SED fitting is mainly dominated by the behavior of
electrons; however, the radiation for Fermi-LAT-detected FR-Is in the GeV band
is dominated by photons. The different particle behaviors may be the reason
why our Doppler factors are not correlated with the SED-derived Doppler
factors.

Figure 6:  The comparison between our Doppler factors derived in this paper
($\delta_{\rm{TW}}$) and the SED-derived Doppler factors ($\delta_{\rm{X17}}$)
from Xue et al. (2017) for the 8 FR-Is in common. The object names are
annotated next to each point. The probability of linear correlation between
$\delta_{\rm{TW}}$ and $\delta_{\rm{X17}}$ is 0.053, suggesting that our
estimated $\gamma$-ray Doppler factors are not correlated with the SED-derived
Doppler factors from Xue et al. (2017).

###  3.5 TeV-detected radio galaxies

In the Fermi-LAT-detected 30 FR-I samples, 6 radio galaxies (Cen A, M87, 3C
84, IC 310, 3C 264, PKS 0625-35) have been detected at TeV energies (Wakely &
Horan, 2008; Rieger & Levinson, 2018; Rani, 2019; Rulten, 2022). Our Doppler
factor estimations for the 6 TeV FR-Is are in a range from $\delta=0.92$ to
$\delta=7.49$ with an average of $<\delta_{\rm{TeV}}>=3.35\pm 1.22$, while the
Doppler factors for the 24 remaining GeV FR-Is range from $\delta=0.88$ to
$\delta=5.41$ with $<\delta_{\rm{GeV}}>=2.36\pm 0.23$. We perform a Student’s
T-test (T-test) between these two samples and ascertain that the probability
is $p=0.46$, suggesting that they are from the same distribution, and the
difference between the TeV and GeV emissions for radio galaxies is not driven
by the Doppler beaming effect.

###  3.6 Core dominance

During the first 15 months of Fermi-LAT sky survey, 11 radio galaxies had been
detected, including 7 FR-Is and 4 FR-IIs (Abdo et al., 2010a). 11 $\gamma$-ray
radio galaxies from Third Cambridge catalog of radio sources (3CRR) showed
higher core dominance than that of non-Fermi-LAT-detected radio galaxies.
However, the $\gamma$-ray luminosities for a larger sample of Fermi-LAT-
detected radio galaxies do not show any correlation on the core dominance
parameter ($R=f_{\rm{core}}/f_{\rm{ext}}$), suggesting that high-energy
emission in Fermi-LAT-detected radio galaxies is not driven by the Doppler
beaming effect (Angioni et al., 2019). The incompatible result between Abdo et
al. (2010a) and Angioni et al. (2019) may be due to the selection effect. From
the first 15 months of observation, Fermi-LAT preferentially detected the
strong Doppler factor sources or nearby sources, such as NGC 1218
($\delta_{\gamma}=3.36$), NGC 1275 ($\delta_{\gamma}=7.49$), PKS 0625-354
($\delta_{\gamma}=6.85$), 3C 120 ($\delta_{\gamma}=2.32$) with moderate or
large Doppler factors or Cen A ($z=0.002$) and M87 ($z=0.004$), the two
closest FR-Is. During the 2008-2018 observations, more FR-Is can be detected
by Fermi-LAT, in which more and more lobe-dominated FR-Is will dilute the
correlation between $\gamma$-ray luminosities and core dominance parameters.

##  4 Conclusions

Based on the unification of BL Lacs and FR-Is (Urry & Padovani, 1995;
Ghisellini et al., 1993) and the assumption of standard candles for FR-Is, we
adopt the relation between the flux density and the luminosity distance, $\log
f_{\gamma}=-2\log d_{\rm{L}}+$const, to estimate the Doppler factors for the
30 Fermi-LAT-detected FR Is and discuss their beaming effect. Our conclusions
are as follows:

  1. 1.

Our Doppler factor estimations for the 30 FR-Is are in a range of
$\delta=0.88-7.49$ with average $<\delta_{\rm{I}}>=2.56\pm 0.30$ for the case
of a continuous jet, which is smaller than that of the 126 BL Lacs
($<\delta_{\rm{BL}}>=10.28\pm 2.03$), in accordance with the unification of BL
Lacs and FR-Is (Urry & Padovani, 1995).

  2. 2.

In the plot of the $\gamma$-ray luminosity versus the photon spectral index
($\log L_{\gamma}-\alpha_{\rm{ph}}$), we found that FR-Is residing close to
the intrinsic luminosity of the 126 BL Lacs are lobe-dominated FR-Is, whose
observed emission is regarded as the intrinsic emission, and those FR-Is with
high luminosities should be considered about their emission which is strong
Doppler-boosted. The position of $\log L_{\gamma}-\alpha_{\rm{ph}}$ may be an
effective indicator for the beaming effect of Fermi-LAT-detected FR-Is.

  3. 3.

A statistical test (T-test) of average Doppler factors between the TeV-
detected FR-Is and GeV-detected FR-Is results in a probability $p$ = 0.46.
This suggests that the Doppler beaming effect may not be a key differentiating
parameter between the GeV and TeV emissions.

We thank the anonymous referee for the constructive comments that made us
improve our manuscript. We also thank the NASA Fermi-LAT collaboration for the
Fermi-LAT data. This research has made use of the NASA/IPAC Extragalactic
Database (NED) which is operated by the Jet Propulsion Laboratory, California
Institute of Technology, under contract with the National Aeronautics and
Space Administration. The work is partially supported by the National Natural
Science Foundation of China (NSFC U2031201, NSFC 11733001, U2031112),
Guangdong Major Project of Basic and Applied Basic Research (grant No.
2019B030302001). Z. Y. Pei acknowledges support from the National Science
Foundation for Young Scientists of China (grant 12103012). We also acknowledge
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 University (grant No. YM2020001)

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