# The jet formation mechanism of Gamma-ray Narrow-line Seyfert 1 Galaxies

Yongyun Chen(陈永云)1, Qiusheng Gu(顾秋生)2, Junhui Fan(樊军辉)3, Xiaoling Yu (俞效龄)1
E-mail<EMAIL_ADDRESS><EMAIL_ADDRESS>Nan Ding(丁楠)4, Xiaotong Guo
(郭晓通)5, Dingrong Xiong(熊定荣)6
1College of Physics and Electronic Engineering, Qujing Normal University,
Qujing 655011, P.R. China
2School of Astronomy and Space Science, Nanjing University, Nanjing 210093, P.
R. China
3Center for Astrophysics,Guang zhou University,Guang zhou510006, China
4School of Physical Science and Technology, Kunming University 650214, P. R.
China
5School of mathematics and physics, Anqing Normal University 246011, P. R.
China
6Yunnan Observatories, Chinese Academy of Sciences, Kunming 650011, China

(Accepted XXX. Received YYY; in original form ZZZ)

###### Abstract

Under the coronal magnetic field, we estimate the maximal jet power of the
Blandford–Znajek (BZ) mechanism, Blandford–Payne (BP) mechanism, and hybrid
model. The jet power of the BZ and Hybrid model mechanisms depends on the spin
of a black hole, while the jet power of the BP mechanism does not depend on
the spin of a black hole. At high black hole spin, the jet power of the hybrid
model is greater than that of the BZ and BP mechanisms. We find that the jet
power of almost all gamma-ray narrow-line Seyfert 1 galaxies ($\gamma$NLS1s)
can be explained by the hybrid model. However, one source with jet power $\sim
0.1-1$ Eddington luminosity can not be explained by the hybrid model. We
suggest that the magnetic field dragged inward by the accretion disk with
magnetization-driven outflows may accelerate the jets in this $\gamma$NLS1.

###### keywords:

galaxies:active–quasars:general–galaxies:Seyfert–galaxies:jets

††pubyear: 2022††pagerange: The jet formation mechanism of Gamma-ray Narrow-
line Seyfert 1 Galaxies–The jet formation mechanism of Gamma-ray Narrow-line
Seyfert 1 Galaxies

##  1 Introduction

The narrow-line Seyfert 1 galaxy is a special subclass of active galactic
nuclei (AGNs) defined by Osterbrock & Pogge (1985). They show some extreme
characteristics, such as the full width at half maximum of their Balmer line
($FWHM(H\beta)<2000km~{}s^{-1}$), [O III]/H$\beta<3$, strong Fe II multiplets
emission, soft X-ray spectra and rapid X-ray variability (e.g. Boroson &
Green, 1992; Boller et al., 1996; Wang et al., 1996; Rani et al., 2017). These
narrow-line Seyfert 1 galaxies are generally considered to be radio quiet
(e.g. Ulvestad et al., 1995; Boroson, 2002). Therefore, it is surprising that
the radio-loud narrow-line Seyfert 1 galaxies (RLNLS1s) was discovered for the
first time (e.g. Remillard et al., 1986; Grupe et al., 2000; Oshlack et al.,
2001; Zhou et al., 2003; Yao & Komossa, 2021). In particular, since the launch
of Fermi Large Area Telescope (LAT), some RLNLS1 have been found to have
gamma-ray radiation (Abdo et al., 2009), which suggests that these RLNLS1s
have strong relativistic jets. These RLNLS1s have a compact core-jet
structure, flat radio spectra, and high brightness temperature (Komossa et
al., 2006; Doi et al., 2006). It is proposed that RLNLS1s exhibit the
characteristics of blazars. Yuan et al. (2008) studied a sample of 23 RLNLS1s
with radio loudness larger than 100 and found that these sources display a
blazar-like nature. Foschini (2011) used 7 $\gamma$-ray narrow-line Seyfert 1
galaxies ($\gamma$NLS1s) to study the relation between $\gamma$NLS1s and
blazars. These $\gamma$NLS1s and flat-spectrum radio quasars (FSRQs) are in
the region dominated by radiation pressure. Foschini et al. (2015) found that
the central engine of flat radio-loud Narrow-line Seyfert 1 galaxies
(F-RLNLS1s) is similar to blazars. Sun et al. (2015) suggested that the
radiation physics and the jet properties of the GeV NLS1 galaxies are similar
to that of FSRQs. Paliya et al. (2019) compared the broadband parameters
derived from the spectral energy distribution (SED) between 16 $\gamma$NLS1s
and Fermi blazars and found that the physical properties of $\gamma$NLS1s are
similar to Fermi blazars. Chen & Gu (2019) studied the relation between jet
and accretion for fermi blazars and F-RLNLS1s, respectively. They found that
the slope of the relation between jet and accretion in Fermi blazars is
similar to that of F-RLNLS1s. This result may imply that the jet formation
mechanism of Fermi blazars is similar to that of RLNLS1s. Chen et al. (2021a)
studied the relation between the fermi blazars sequence and $\gamma$NLS1s and
found that the $\gamma$NLS1s belong to the fermi blazars sequence.

The formation of jets has always been a hot issue in astrophysics research. At
present, there are three main theories of jet formation. First, the Blandford-
Znajek mechanism (BZ): the jet extracts the rotation energy of the black hole
and the accretion disk (Blandford & Znajek, 1977). The jet power depends on
the black hole mass and the spin of a black hole. Chen et al. (2021b) used a
large sample of Fermi blazars to study the relation between jet power and the
spin of black hole and found that the jet power depends on the spin of a black
hole. Second, the Blandford-Payne mechanism (BP): the jet extracts the
rotational energy of the accretion disk (Blandford & Payne, 1982). The jet
power depends on the luminosity of the accretion disk. This correlation has
been confirmed by many authors (e.g. Rawlings & Saunders, 1991; Celotti &
Ghisellini, 2008; Sbarrato et al., 2014; Ghisellini et al., 2010; Ghisellini
et al., 2014; Zamaninasab et al., 2014; Chen et al., 2015; Mukherjee et al.,
2019). Third, the hybrid model: a mixture of the BP and BZ mechanism (Meier,
2001; Garofalo et al., 2010). Zhou & Cao (2009) found that there is a strong
relation between Lorentz factors and black hole mass, which supports that the
BZ mechanism may dominate over the BP mechanism for blazars. The GeV-NLS1s
have high accretion rates, which may imply that the jet of GeV-NLS1s is
produced by the BP mechanism and/or BZ mechanism (Zhang et al., 2015). Some
AGNs with ultra-fast outflows may imply a hybrid mechanism in these AGNs, for
example, radio galaxies. Komossa et al. (2018) found that RLNLS1s have Extreme
gaseous outflows, which implies that RLNLS1s may be a hybrid mechanism.

It is generally believed that a large-scale magnetic field plays a vital role
in the acceleration and collimation of jet and/or outflows (e.g. Pudritz et
al., 2007). The structure of the magnetic field is a key ingredient in BZ, BP,
and hybrid jet models. The kinetic energy of the rotating black hole and the
accretion disk is connected to the jets with the corotating large-scale
magnetic field. The origin of the large-scale magnetic field passing through
the accretion disk has not been well understood. Some studies show that the
large-scale magnetic field of accelerating jet or outflow may be formed by the
advection of a weak external field (Bisnovatyi-Kogan & Ruzmaikin, 1974, 1976;
van Ballegooijen, 1989; Spruit & Uzdensky, 2005). However, Lubow et al. (1994)
found that the advection of the external field is rather inefficient in the
geometrically thin accretion disk ($H/R\ll 1$) due to its small radial
velocity. This may imply that the field in the inner region of the disk is not
much stronger than the external weak field (Lubow et al., 1994), which can not
accelerate strong jets in radio-loud quasars. A few mechanisms were suggested
to alleviate the difficulty of field advection in the thin disk (e.g., Spruit
& Uzdensky, 2005; Lovelace et al., 2009; Guilet & Ogilvie, 2012, 2013; Cao &
Spruit, 2013). It has been suggested that the external field can be
effectively dragged inward through the hot corona above the accretion disk,
i.e., the ‘coronal mechanism’ (Beckwith et al., 2009). The radial velocity of
the gas above the disk can be greater than the radial velocity of the middle
plane of the disk, which partially solves the problem of inefficient field
advection in the thin disk (Lovelace et al., 2009; Guilet & Ogilvie, 2012,
2013). Alternatively, Cao & Spruit (2013) suggested that if the magnetization-
driven outflows remove the most angular momentum of the gas in the thin disk,
the radial velocity of the disk increases significantly and, therefore, the
external field in the inner region of the thin disk with the magnetization-
driven outflows can be significantly enhanced.

Although there are many studies on RLNLS1s. However, there is a lack of
specific research on the jet formation mechanism of $\gamma$NLS1s. Some
authors have suggested that $\gamma$NLS1s (1H 0323+342) have shown the disk-
corona-jet connection (e.g., Paliya et al., 2014; Yao et al., 2015; Landt et
al., 2017; Ghosh et al., 2018). Thus, $\gamma$NLS1s is an ideal object for
exploring the jet–corona–disk connection (e.g. Wilkins et al., 2015). In this
work, we consider the strength of the magnetic field of the corona and
estimate the maximal jet power of BZ, BP, and hybrid jet models. We use a
sample of $\gamma$NLS1s to study their jet formation mechanisms.

##  2 The Sample

We are trying to collect a large sample of $\gamma$NLS1s with reliable
redshift, black hole mass, disk luminosity, and X-ray luminosity. We consider
the sample of Paliya et al. (2019). Paliya et al. (2019) collected a sample of
16 $\gamma$NLS1s and studied their physical properties. The black hole mass
and disk luminosity of 16 $\gamma$NLS1s are from the work of Paliya et al.
(2019). The accretion disk luminosity and black hole mass are obtained by
using the SED modeling (see Paliya et al., 2019). Furthermore, whenever the
big blue bump is visible at optical–UV energies, it can be attributed to the
accretion disk radiation (e.g., Ghisellini & Tavecchio, 2009). By reproducing
this emission with a standard disk model (e.g., Shakura & Sunyaev, 1973), both
black hole mass and accretion disk luminosity can be calculated. The X-ray
luminosity in 0.3-10 keV comes from the work of Paliya et al. (2019).

Komossa et al. (2018) used the following formula to estimate the jet power of
RLNLS1s (Bîrzan et al., 2008),

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

where $P_{\rm cav}$ is in units $10^{42}erg~{}s^{-1}$, and $P_{1.4}$ in units
$10^{40}erg~{}s^{-1}$. The $P_{1.4}$ is radio luminosity at 1.4 GHz. The radio
luminosity is estimated by using the relation $L_{\rm\nu}=4\pi d_{\rm
L}^{2}S_{{\rm\nu}}$, $S_{(\rm\nu)}=S_{\rm\nu}^{obs}(1+z)^{\alpha-1}$, where
$\alpha$ is spectral index, $\alpha=0$ is assumed (Donato et al., 2001; Abdo
et al., 2010; Komossa et al., 2018). Following Komossa et al. (2018), we also
use equation (1) to estimate the jet power. The data is listed in Table 1.

##  3 Magnetic field of hot Corona

The detailed properties of the corona above the accretion disk have been
extensively studied by many authors (e.g. Shakura & Sunyaev, 1973; Galeev et
al., 1979; Haardt & Maraschi, 1993; Svensson & Zdziarski, 1994; Cao, 2009,
2018). Cao (2018) used relative thickness $\tilde{H}_{\rm c}=H_{\rm c}/R$ and
the optical depth $\tau_{\rm c}$ to describe the corona above the accretion
disk. The optical depth of the corona in the vertical direction is

$\tau_{\rm c}=\rho_{c}H_{\rm c}\kappa_{\rm T},$ (2)

where $\rho_{\rm c}$ is the density of the corona, $H_{\rm c}$ is the corona
thickness, and $\kappa_{\rm T}=0.4g^{-1}cm^{2}$ is the Compton scattering
opacity.

The gas pressure of the corona is

$p_{\rm c}=\frac{\rho_{\rm c}}{2}\left(\frac{H_{\rm
c}}{R}\right)^{2}\frac{L_{*}^{2}}{R^{2}},$ (3)

and the $L_{*}$ is as follows,

$L_{*}^{2}=L^{2}-j^{2}(E^{2}-1)$ (4)

where $j$ is the spin of a black hole, L is the conserved angular momentum of
the gas $L=u_{\rm\phi}$, $E$ is the conserved energy $E=-u_{\rm t}$. In order
to estimate the maximum jet power, $L_{*}=L_{\rm K}(r_{\rm ms})$ is used in
calculating the strength of the magnetic field (Cao, 2018).

Cao (2018) used a parameter $\beta$ to describe the magnetic field strength in
the corona,

$p_{\rm m}=\frac{B_{\rm z}^{2}}{8\pi}=\beta p_{\rm c},$ (5)

where $B_{\rm z}$ is the strength of the vertical component of the field, and
$\beta<1$ is required for the field advected by the corona. The magnetic field
strength of corona is as follows,

$B=4.37\times 10^{8}\beta^{1/2}\tau_{\rm c}^{1/2}\tilde{H}_{\rm
c}^{1/2}m^{-1/2}r^{-3/2}L_{*}^{2}~{}Gauss$ (6)

$\displaystyle r=\frac{Rc^{2}}{GM_{\rm bh}},m=\frac{M_{\rm
bh}}{M_{\odot}},\tilde{H}_{\rm c}=\frac{H_{\rm c}}{R}$ (7)

where $H_{\rm c}$ is the corona thickness (see Cao (2018), for the details).

Table 1: The sample of $\gamma$NLS1s

Name | Redshift | log$(M/M_{\odot})$ | log$L_{disk}$ | $f_{\nu}$ | log$P_{jet}$ | log$L_{X}$ | log$\lambda$ | Disk
---|---|---|---|---|---|---|---|---
(1) | (2) | (3) | (4) | (5) | (6) | (7) | (8) | (9)
1H 0323+342 | 0.061 | 7.3 | 45.3 | 613.5 | 44.163 | 44.432 | -1.114 | SS
SBS 0846+513 | 0.584 | 7.59 | 44.43 | 266.3 | 44.799 | 45.002 | -2.274 | ADAF+SS
CGRaBS J0932+5306 | 0.597 | 8 | 45.7 | 481.6 | 44.896 | 44.842 | -1.414 | SS
GB6 J0937+5008 | 0.275 | 7.56 | 43.71 | 166.6 | 44.461 | 44.892 | -2.964 | ADAF
PMN J0948+0022 | 0.585 | 8.18 | 45.7 | 69.5 | 44.594 | 45.792 | -1.594 | SS
TXS 0955+326 | 0.531 | 8.7 | 46.23 | 1247.1 | 44.998 | 44.852 | -1.584 | SS
FBQS J1102+2239 | 0.453 | 7.78 | 44 | 2.5 | 43.998 | 44.542 | -2.894 | ADAF
CGRaBS J1222+0413 | 0.966 | 8.85 | 46.18 | 800.3 | 45.155 | 46.132 | -1.784 | SS
SDSS J124634.65+023809.0 | 0.362 | 8.48 | 44.48 | 35.7 | 44.325 | 43.902 | -3.114 | ADAF
TXS 1419+391 | 0.49 | 8.48 | 45.34 | 85.7 | 44.563 | 44.762 | -2.254 | ADAF+SS
PKS 1502+036 | 0.407 | 7.6 | 44.78 | 394.8 | 44.733 | 44.402 | -1.934 | SS
TXS 1518+423 | 0.484 | 7.85 | 44.7 | 138.4 | 44.629 | 44.572 | -2.264 | ADAF+SS
RGB J1644+263 | 0.145 | 7.7 | 44.48 | 128.4 | 44.205 | 44.072 | -2.334 | ADAF+SS
PKS 2004-447 | 0.24 | 6.7 | 43 | 471 | 44.57 | 44.062 | -2.814 | ADAF
TXS 2116-077 | 0.26 | 7.2 | 44 | 96.1 | 44.357 | 43.892 | -2.314 | ADAF+SS
PMN J2118+0013 | 0.463 | 7.77 | 44.94 | 147.9 | 44.625 | 44.522 | -1.944 | SS

Columns (1) is the name of sources; Columns (2) is redshift; Columns (3) is
the black hole mass; Columns (4) is the disk luminosity; Columns (5) is the
1.4 GHz radio flux in units mjy; Columns (6) is the jet power in units
$erg~{}s^{-1}$; Columns (7) is the X-ray luminosity in 2-10 keV; Columns (8)
is the dimensionless accretion rates; Columns (9) is the disk.

##  4 Jet formation model

###  4.1 The BZ jet model

The jet power extracted from a rapidly rotating black hole is (e.g. MacDonald
& Thorne, 1982; Ghosh & Abramowicz, 1997)

$P_{\rm jet}^{\rm BZ}=\frac{1}{32}\omega_{\rm F}^{2}B_{\perp}^{2}R_{\rm
H}^{2}j^{2}c$ (8)

where $B_{\perp}$ is the magnetic field strength at the black hole horizon,
$B_{\perp}\simeq B$. $R_{\rm H}=[1+(1-j^{2})^{1/2}]GM_{\rm BH}/c^{2}$ is the
horizon radius. $\omega_{\rm F}\equiv\Omega_{\rm F}(\Omega_{\rm H}-\Omega_{\rm
F})/\Omega_{\rm H}^{2}$ is the angular velocity of the field lines
$\Omega_{\rm F}$ relative to that of the hole. $\omega_{\rm F}=1/2$ is used
when estimating the maximum BZ jet power (e.g. MacDonald & Thorne, 1982; Ghosh
& Abramowicz, 1997). Substituting equation (6) into equation (8), we can get
the the maximum BZ jet power.

###  4.2 The BP jet model

The jet power generated by the BP mechanism can be calculated using the
following formula (Livio et al., 1999; Cao, 2018),

$P_{\rm jet}^{\rm BP}\sim\frac{BB_{\rm\phi}^{s}}{2\pi}R_{\rm j}\Omega\pi
R_{\rm j}^{2}$ (9)

where $R_{\rm j}$ is the radius of the jet formation region in the corona,
$\Omega$ is the angular velocity of the gas in the corona. The
$B_{\rm\phi}^{\rm s}$ is the azimuthal component of magnetic field at the
corona surface, $B_{\rm\phi}^{\rm s}=\xi_{\phi}B$. The ratio $\xi_{\phi}\leq
1$ is required (Livio et al., 1999). Substituting equation (6) into equation
(9), the jet power of BP can be derived as follows

$P_{\rm jet}^{BP}\simeq 3.13\times 10^{37}\xi_{\phi}\tilde{\Omega}r_{\rm
j}^{-1/2}m\beta\tau_{\rm c}\tilde{H_{\rm c}}~{}erg~{}s^{-1}.$ (10)

Most of the gravitational force can be released in the inner region of the
accretion disk within the radius $\sim 2R_{\rm ms}$ (Shakura & Sunyaev, 1973).
$R_{\rm j}=2R_{\rm ms}$ is adopted in the estimates of BP jet power. The
$R_{ms}$ is defined by the following formula,

$\displaystyle R_{\rm ms}=R_{\rm
G}\\{3+Z_{2}-\left[(3-Z_{1})(3+Z_{1}+2Z_{2})\right]^{1/2}\\},$ $\displaystyle
Z_{1}\equiv 1+(1-a^{2})^{1/3}\left[(1+a)^{1/3}+(1-a)^{1/3}\right],$
$\displaystyle Z_{2}\equiv(3a^{2}+Z_{1}^{2})^{1/2},$ $\displaystyle R_{\rm
G}=\frac{GM_{\rm bh}}{c^{2}}.$ (11)

Figure 1: The black hole spin versus jet power. The orange dashed line is jet
power $P_{\rm jet}^{\rm BP}$ extracted from a standard accretion disk. The
green dashed line is jet power $P_{\rm jet}^{\rm BZ}$ extracted from a rapidly
spinning black hole. The red dashed line is Hybrid jet model.

###  4.3 Hybrid model

The hybrid model is that jets are produced by the combined effort of the BZ
and BP mechanism. In the case of a thin accretion disk, the total jet power
for the hybrid model is given by (Garofalo, 2009; Garofalo et al., 2010)

$P_{\rm jet}^{\rm Hybrid}=2\times 10^{47}erg~{}s^{-1}\alpha
f^{2}\left(\frac{B_{\rm pd}}{10^{5}G}\right)^{2}m_{9}^{2}j^{2},$ (12)

$B_{\rm pd}\simeq B$, $m_{9}$ is the black hole mass in units of
$10^{9}M_{\odot}$, the functions $\alpha$ and $f$ capture the effects on BP
and BZ powers, respectively, from the numerical solution in the context of the
Reynolds conjecture (Garofalo, 2009)

$\displaystyle\alpha=\delta\left(\frac{3}{2}-j\right)$ (13)

and

$\begin{split}f=-\frac{3}{2}j^{3}+12j^{2}-10j+7-\frac{0.002}{(j-0.65)^{2}}+\frac{0.1}{(j+0.95)}\\\
+\frac{0.002}{(j-0.055)^{2}}.\end{split}$ (14)

where a conservative value for $\delta$ of about 2.5 is adopted but our
ignorance of how jets couple the BZ and BP components restrict our ability to
specify it and suggests that it might well be larger by an order of magnitude
or more (Garofalo et al., 2010). While $\alpha$ can be thought of as the
parameter that determines the effectiveness of the BP jet as a function of
spin, $f$ captures the enhancement of the disk thread field to the black hole,
both of which are within the Reynolds conjecture. The parameter $\delta$
determines the effective contribution of BP to overall jet power, with larger
values shifting jet power efficiency more towards the retrograde regime.

##  5 RESULTS AND DISCUSSION

The magnetic pressure is required to be lower than the gas pressure in the
corona when the magnetic field is dragged inward by the corona, i.e.
$\beta<1$. In order to calculate the maximum jet power, we use $\omega_{\rm
F}=1/2$ (Ghosh & Abramowicz, 1997), $\beta=1$, $\xi_{\phi}=1$ and
$\tilde{\Omega}=1$, while the typical values of the corona parameters,
$\tau_{\rm c}=0.5$ and $\tilde{H_{\rm c}}=0.5$ are used in the estimates (Cao,
2009, 2018). The relationship between the maximum jet power and the spin
parameters of the black hole is shown in Figure 1.

In Figure 1, the jet power of BZ and hybrid models changes similarly with the
spin of the black hole. The slight difference is that the jet power of the
hybrid model is higher than that of the BZ model. The sharp drop in the BZ
case reflects the drop in the rotational energy of the black hole, which can
be used to enhance the jet. In the case of hybrid, the decline is limited by
the contribution from the accretion disk. The most important point is that the
jet power of the BZ and hybrid model strongly depends on the spin of a black
hole. However, the jet power of BP does not depend on the spin of the black
hole. The jet power of BP is always higher than that of the BZ and hybrid
model when the spin of a black hole is less than 0.9. However, the jet power
of the hybrid model is greater than that of BP when the spin of the black hole
is greater than 0.9.

The right-hand axis of Figure 1 shows the jet efficiency factor, defined as
$\eta_{\rm jet}\equiv P_{\rm jet}/\dot{M}_{\rm ms}c^{2}$. Because the jet
power is related to $\dot{M}_{\rm ms}$, the corresponding jet efficiency only
depends on the spin of the black hole ($j$) and viscosity parameters. From
figure 1, we find that the hybrid models have high jet efficiency when the
spin of the black hole is greater than 0.9. The result suggests that the
magnetic field of hot corona may explain observations of AGNs with high jet
efficiency $\eta\approx few\times 100~{}per~{}cent$.

Figure 2: The black hole mass versus maximal jet power. The orange dashed line
is maximal jet power $P_{\rm jet}^{\rm BP}$ extracted from a standard
accretion disk. The green dashed line is maximal jet power $P_{\rm jet}^{\rm
BZ}$ extracted from a rapidly spinning black hole. The red dashed line is
Hybrid jet model. The black circle is $\gamma$NLS1s. The spin of black hole
$j=0.95$ are adopted. The cyan circle marks PKS 2004-447.

The relation between jet power and black hole mass is shown in Figure 2. The
red dashed line is a hybrid jet model. The orange dashed line is the BP jet
model. The green dashed line is the BZ jet model. The black circle dot is
$\gamma$NLS1s. From figure 2, we find that the hybrid jet model has high jet
power than that of both BP and BZ jet power. About 55% sources are above both
the orange and green lines, which implies that BP and BZ mechanisms cannot
explain the jet power of these sources. However, we can see that almost all
sources are located below the red line, which shows that the jet power of
these sources can be explained by the hybrid jet model. Here, the jet power of
only one source (PKS 2004-447) is above the red line. There are two possible
explanations for one source. One explanation is that the one source has a
strong beaming effect, and it has high jet power (PKS 2004-447). Schulz et al.
(2016) found a compact core by using the Very-Long-Baseline Interferometry
(VLBI) observation in PKS 2004-447. The PKS 2004-447 is a core-dominated, one-
sided parsec-scale jet with a strong beaming effect. Orienti et al. (2015)
also found that the PKS 2004-447 have a high Doppler factor ($D_{factor}=30$).
These results suggest that the PKS 2004-447 have a strong beaming effect.
Thus, it has high jet power. Another explanation is that the one source may
require other jet models, such as the accretion disk with magnetization-driven
outflows models.

Figure 3: The black hole mass versus jet power. The dashed lines are $P_{\rm
jet}/L_{\rm Edd}=0.01,0.1,1$, respectively. The cyan circle marks PKS
2004-447.

In Figure 3, we present the results as in Figure 2, but the dash lines are
$P_{\rm jet}/L_{\rm Edd}=0.01,0.1,1$, respectively. All sources have jet power
less than 1$L_{\rm Edd}$. There are about two sources located at $\sim
0.1-1L_{\rm Edd}$. Some authors suggested that the magnetic field dragged
inwards by the accretion disk with magnetization-driven outflows may
accelerate the jets in AGNs (e.g. Cao, 2014; Li, 2014; Cao, 2016; Li & Cao,
2019; Cao et al., 2021). In this scenario, the hot corona/gas above the
accretion disk may help to enhance the jet/outflow (Li & Cao, 2021). Most of
the angular momentum of the accretion disk is removed by magnetization-driven
outflows, and the magnetic field will be enhanced than the thin disk without
outflow (Cao & Spruit, 2013). Cao (2018) suggested that the magnetic field
dragged inward by the accretion disk with magnetization-driven outflows may
help explain the sources with jet power $\sim 0.1-1L_{\rm Edd}$ (or even
higher). The outflow can be enhanced by the radiation pressure of the
accretion disk (e.g. Bisnovatyi-Kogan & Blinnikov, 1977; Shlosman et al.,
1985; Murray et al., 1995). This scenario is very effective for the jet of a
source with a high mass accretion rate, for example, RLNLS1s and radio
galaxies. Radiation pressure may play an important role in magnetization-
driven outflows (Cao, 2014). Some authors have found that the accretion disk
of $\gamma$NLS1s is dominated by radiation-pressure (e.g., Foschini, 2011;
Chen & Gu, 2019). Our results may imply that the jet power of the one source
may be explained by the magnetization-driven outflows model.

We also note that the NLS1 has a high accretion rate, which may imply
different accretion disk models. Wang et al. (2002) defined the line
dimensionless accretion rate form as

$\displaystyle\lambda=\frac{L_{\rm lines}}{L_{\rm Edd}},L_{\rm lines}=\xi
L_{\rm disk}.$ (15)

where $\xi\approx 0.1$ (Netzer, 1990). Wang et al. (2002) got the relation
between $\lambda$ and the dimensionless accretion rate ($\dot{m}$) for an
optically thin ADAF,

$\dot{m}=2.17\times 10^{-2}\alpha_{0.3}\xi_{-1}^{-1/2}\lambda_{-4}^{1/2}.$
(16)

where $\alpha_{0.3}=\alpha/0.3$, $\xi_{-1}=\xi/0.1$, and
$\lambda_{-4}=\lambda/10^{-4}$. A necessary condition for the presence of an
optically thin ADAF is $\dot{m}\leq\alpha^{2}$ (Narayan et al., 1998).
Equation (16) can then be rewritten as

$\lambda_{1}=1.72\times 10^{-3}\xi_{-1}\alpha_{0.3}^{2}.$ (17)

Optically thin ADAFs require $\lambda<\lambda_{1}$. The optically thick,
geometrically thin disk obey

$\dot{m}=\frac{L_{\rm line}}{\xi L_{\rm Edd}}=10\xi_{-1}^{-1}\lambda.$ (18)

The accretion disk is standard thin disk (SS) when accretion rate satisfy
$1>\dot{m}\geq\alpha^{2}$ (e.g Shakura & Sunyaev, 1973). Equation (18) can
then be rewritten as

$\lambda_{2}=9.0\times 10^{-3}\xi_{-1}\alpha_{0.3}^{2}.$ (19)

A standard thin disk can exist when gives $\lambda\geq\lambda_{2}$. It is a
super Eddington accretion (SEA) when the accretion rate reaches $\dot{m}\geq
1$, we have

$\lambda_{3}=0.1\xi_{-1}.$ (20)

A slim disk requires $\lambda\geq\lambda_{3}$ (Wang et al., 2002, 2003).
Interestingly, in the transition region between $\lambda_{1}$ and
$\lambda_{2}$, the accretion flow may be in a mixed state where the standard
disk and ADAF coexist. Several authors have discussed the possibility of mixed
states in AGN accretion disks (e.g., Quataert et al., 1999; RóżaŃska & Czerny,
2000; Ho et al., 2000). A transition from an SS disk to an ADAF is possible
(e.g., Gu & Lu, 2000), perhaps via evaporation (Liu et al., 1999). The
transition radius depends on the accretion rate, the black hole mass, and
viscosity. The disk structure, however, is complicated in such a regime.

Figure 4: The distribution of $\lambda$. The distribution of $\lambda$ is
divided into four regions, corresponding to different states of the accretion
disks:(1) $\lambda<\lambda_{1}$ (pure ADAF); (2)
$\lambda_{1}\leq\lambda\leq\lambda_{2}$ (ADAF+SS); (3)
$\lambda_{2}\leq\lambda\leq\lambda_{3}$ (SS); (4) $\lambda\geq\lambda_{3}$
(SEA).

The distribution of the three critical values of $\lambda$ define four regimes
in accretion states are shown in Figure 4. The average values of $\lambda$ for
$\gamma NLS1s$ are $\langle\log\lambda\rangle=-2.16\pm 0.56$. From figure 4,
we find that the distribution of $\log\lambda$ is from -3.2 to -1.1. These
$\gamma$NLS1s do not have super Eddington accretion and do not need slim disk
interpretation. From Figure 2, almost all $\gamma$NLS1s can be explained by
the standard disk. We also note that values of $\lambda_{1}$ and $\lambda_{2}$
are sensitive to $\alpha$. The most likely value of $\alpha$ is 0.3 (Narayan
et al., 1998). However, if $\alpha=0.1$, it will follow that
$\lambda_{1}=1.91\times 10^{-4}\xi_{-1}\alpha_{0.1}^{2}$ and
$\lambda_{2}=1.0\times 10^{-3}\xi_{-1}\alpha_{0.1}^{2}$. All the disks listed
as hybrid models of SS and ADAF will be SS disks in Table 1. In any case, this
does not affect the conclusion that most of the objects have the standard
accretion disks.

It is well known that the luminosities of typical NLSys are around the
Eddington value. However, we find that this sample has a lower Eddington
ratio. The possible reason is due to the black hole mass. Previously, the
black hole mass derived from the optical spectroscopy is argued to be
underestimated, due to the radiation pressure effect in highly accreting
systems and also possibly due to a flat BLR geometry (Decarli et al., 2008;
Marconi et al., 2008). The latter effect is proposed to be more pronounced in
the sources with a pole-on view, typically blazars and $\gamma$NLS1s. On the
other hand, the disk modeling approach is free from these issues and relies
mainly on the visibility of the big blue bump. Calderone et al. (2013) modeled
the SED of a sample of radio-loud NLSy1 galaxies with a standard Shakura &
Sunyaev (1973) accretion disk spectrum and estimated a consistently higher
black hole mass than that derived from the optical spectroscopy. Because the
black hole mass of our sample comes from the SED model, these gamma-ray
Narrow-line Seyfert 1 galaxies may have a low Eddington ratio.

Meier (2001) considered the SS disk and ADAF extracting energy from the spin
of the black hole and the disk itself. His model uses an intermediate region
solution for SS disks and a self-similar solution for ADAF; it takes into
account Schwarzschild and Kerr’s black holes. In this model, if the Kerr black
holes with the same mass and accretion rate are $M_{\rm BH}=10^{9}M_{\odot}$
and $\dot{M}=0.1$, the jet power of the Schwarzschild black hole is only is
$P_{\rm jet}=10^{41.7}\rm erg~{}s^{-1}$, and a Kerr black hole with the same
mass and accretion rate is $P_{\rm jet}=10^{42.7}\rm ergs^{-1}$. The current
data does not support this model, as many sources have SS disks that support
higher luminosity. Wang et al. (2003) considered the jet power of slim disk
with hot coronae, $P_{\rm jet}\approx 8.0\times 10^{37}M_{\rm
BH}ln((1-f)\dot{m}/50)\rm erg~{}s^{-1}$ ($2.5\ll\dot{m}\leq 100$). If the
black hole mass, accretion rate and free parameter $f$ are
$M_{BH}=10^{9}M_{\odot}$, $\dot{m}=100$, and $f=0$, the maximal jet power is
$P_{\rm jet}=10^{46.74}\rm erg~{}s^{-1}$. All $\gamma$NLS1s can be explained
by the slim disk with hot coronae.

We know that the fraction of the corona to the disk emission increases with
the increasing Eddington ratio. The properties of the coronae may be related
to their hard X-ray emission. If the coronal jet model is indeed at work in
$\gamma$NLS1s, one may expect a correlation between radio and hard X-ray
emission. We test this correlation. In order to get consistent hard X-ray
luminosity in 2-10 keV, we roughly convert X-ray luminosity in 0.3-10 keV,
$L_{X,0.3-10keV}$, to the hard X-ray luminosity with a mean energy spectral
index of $\alpha_{\rm X}=1(f_{x}\propto E^{-\alpha_{X}})$,
$L_{X,2-10keV}=0.46L_{X,0.3-10keV}$. This is consistent with the general
features of the theoretical calculations for the X-ray spectra from the disk
corona (e.g., Nakamura & Osaki, 1993). Figure 5 shows the relation between
radio luminosity and hard X-ray luminosity. The Pearson analysis shows that
there is a significant correlation between radio luminosity and hard X-ray
luminosity ($r=0.61,P=0.01$, significant correlation P$<$0.05 confidence
level). The Spearman correlation coefficient and significance level are
$r=0.58$ and $P=0.02$. The Kendalltau correlation coefficient and significance
level are $r=0.38$ and $P=0.04$. These two tests also show a significant
correlation.

Figure 5: Relation between radio luminosity and hard X-ray luminosity for
$\gamma$NLS1s. The solid lines correspond to the best-fit linear models
obtained with the symmetric least-squares fit. The dashed lines indicate
2$\sigma$ confidence band. The cyan circle marks PKS 2004-447. Figure 6:
Relation between jet power and accretion disk luminosity for $\gamma$NLS1s.
The solid lines correspond to the best-fit linear models obtained with the
symmetric least-squares fit. The dashed lines indicate 2$\sigma$ confidence
band. The cyan circle marks PKS 2004-447.

It is generally believed that there is a close connection between jet power
and accretion disk luminosity for radio galaxies/AGNs. We test this
correlation for $\gamma$NLS1s. The relation between jet power and accretion
disk luminosity for $\gamma$NLS1s is shown in Figure 6. The Pearson analysis
shows that there is a significant correlation between jet power and accretion
disk luminosity ($r=0.56,P=0.02$). The Spearman correlation coefficient and
significance level are $r=0.55$ and $P=0.03$. The Kendalltau correlation
coefficient and significance level are $r=0.36$ and $P=0.04$. These two tests
also show a significant correlation. These results show that there is a close
connection between jet and accretion.

##  6 SUMMARY

We consider the magnetic field in the corona and calculate the jet power of
the BZ mechanism, BP mechanism, and hybrid model. The jet power of the BZ
mechanism and hybrid model depends on the spin of a black hole, while the jet
power of the BP mechanism has no obvious relationship with the spin of a black
hole. At high a spin of the black hole, the jet power of the hybrid model is
higher than that of the BZ mechanism and BP mechanism.

We find that the jet power of 99% $\gamma$NLS1s is lower than that of the
hybrid model. This result shows that the hybrid model can explain almost all
the jet power of $\gamma$NLS1s. However, the jet power of only one source is
located at $\sim 0.1-1L_{\rm Edd}$. Cao & Spruit (2013) suggested that most of
the angular momentum of the disk can be removed by the magnetization-driven
outflows, and therefore the radial velocity of the disk increases
significantly. Because the most angular momentum of the disk is carried away
by the outflows, the gas in the disk rapidly accretes onto the black hole.
Thus, the magnetic field dragged inward by the accretion disk with
magnetization-driven outflows may accelerate the jet in one source.

## Acknowledgements

We thank the referee and Editor for the helpful comments and suggestions that
improved the presentation of the manuscript substantially. Yongyun Chen thanks
for the financial support from the National Natural Science Foundation of
China (No. 12203028). This work was support from the research project of
Qujing Normal University (2105098001/094). This work is supported by the youth
project of Yunnan Provincial Science and Technology Department
(202101AU070146, 2103010006). This work is supported by the National Natural
Science Foundation of China (NSFC 11733001). This work is supported by the
National Key Research and Development Program of China (No. 2017YFA0402703)
and the National Natural Science Foundation of China (Grant Nos. 11733002 and
11773013).Nan Ding thanks for the financial support from the National Natural
Science Foundation of China (No. 12103022) and the Special Basic Cooperative
Research Programs of Yunnan Provincial Undergraduate Universities’Association
(No. 202101BA070001-043).

## Data Availability

All the data used here are available upon reasonable request. All datas are in
Table 1.

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