11institutetext: Ruhr-Universität Bochum, Fakultät für Physik und Astronomie,
Theoretische Physik IV, 44780 Bochum, Germany 22institutetext: Ruhr
Astroparticle and Plasma Physics Center (RAPP Center), 44780 Bochum, Germany
33institutetext: European Southern Observatory, Karl-Schwarzschild-Straße 2,
85748 Garching bei München, Germany 44institutetext: Ruhr-Universität Bochum,
Fakultät für Physik und Astronomie, Astronomisches Institut (AIRUB), 44780
Bochum, Germany 55institutetext: Department of Space Earth and Environment,
Chalmers University of Technology, 412 96 Gothenburg, Sweden

# On the possible jet contribution to the $\gamma$-ray luminosity in NGC 1068

S. Salvatore 11 2 2 B. Eichmann 11 2 2 X. Rodrigues 22 3 3 4 4 R.-J. Dettmar
2244 J. Becker Tjus 11 2 2 5 5

NGC 1068 is a nearby widely studied Seyfert II galaxy presenting radio,
infrared, X- and $\gamma$-ray emission as well as strong evidence for high-
energy neutrino emission. Recently, the evidence for neutrino emission could
be explained in a multimessenger model in which the neutrinos originate from
the corona of the active galactic nucleus (AGN). In this environment
$\gamma$-rays are strongly absorbed, so that an additional contribution from
e.g. the circumnuclear starburst ring is necessary. In this work, we discuss
whether the radio jet can be an alternative source of the $\gamma$-rays
between about $0.1$ and $100$ GeV as observed by Fermi-LAT. In particular, we
include both leptonic and hadronic processes, i.e. accounting for inverse
Compton emission and signatures from $pp$ as well as $p\gamma$ interactions.
In order to constrain our calculations, we use VLBA and ALMA observations of
the radio knot structures, which are spatially resolved at different distances
from the supermassive black hole. Our results show that the best leptonic
scenario for the prediction of the Fermi-LAT data is provided by the radio
knot closest to the central engine. For that a magnetic field strength $\sim
1\,\text{mG}$ is needed as well as a strong spectral softening of the
relativistic electron distribution at $(1-10)\,\text{GeV}$. However, we show
that neither such a weak magnetic field strength nor such a strong softening
is expected for that knot. A possible explanation for the $\sim$ 10 GeV
$\gamma$ rays can be provided by hadronic pion production in case of a gas
density $\gtrsim 10^{4}\,\text{cm}^{-3}$. Nonetheless, this process cannot
contribute significantly to the low energy end of the Fermi-LAT range. We
conclude that the emission sites in the jet are not able to explain the
$\gamma$-rays in the whole Fermi-LAT energy band.

###### Key Words.:

galaxies: active – galaxies: Seyfert – gamma rays: galaxies

##  1 Introduction

The nearby starburst-Seyfert II galaxy NGC 1068, located at a distance of
$D_{\text{L}}$ = 10.1 $\pm$ 1.8 Mpc (Tully et al., 2009), is not only one of
the first and brightest galaxies of its kind studied by Seyfert (1943), but
also one of the first sources to show strong evidence for neutrino emission,
as observed by the IceCube experiment (Abbasi et al., 2022). NGC 1068 shows
also clear non-thermal emission in the $\gamma$-ray band up to some tens of
GeV (Ackermann et al., 2012). The observation of a neutrino flux that is much
higher than the gamma-ray flux resulted in the expectation that neutrinos must
be produced in a $\gamma$-ray absorbed environment (Inoue et al., 2020; Murase
et al., 2020; Kheirandish et al., 2021; Eichmann et al., 2022; Inoue et al.,
2022). Since the AGN corona is optically thick to $\gamma$-rays, it seems to
be the most likely origin of the TeV neutrinos. The origin of the GeV
$\gamma$-rays is less clear, since it may either be located in the starburst
ring (Eichmann et al., 2022), the torus (Inoue et al., 2022), or the radio jet
(Lenain et al., 2010). In this work, we will be testing the hypothesis that
the jet of NGC 1068 contributes to the GeV $\gamma$-ray emission, based on
radio observations of the jet and a physical model of the multi-wavelength
non-thermal emission.

Radio observations (Gallimore et al., 2004; Michiyama et al., 2022) show
distinct emission sites at some tens of parsec as well as a few hundreds of
parsec from the supermassive black hole of NGC 1068. These are commonly
associated with its jet that is inclined to the line of sight by about 45∘.
Hereof, the most distant emission site shows spatial widening, including four
bright knot structures according to Michiyama et al. (2022), and the typical
lobe structure that marks the head of the jet. Here the whole structure seems
to have hit the interstellar medium (ISM) and therefore presents an overall
shock feature. Moreover, all of the observed radio structures can be inferred
by spherical symmetric knot structures with a radius $r_{\rm k}$ on the order
of about a few parsecs. According to Roy et al. (2000) these knots are
streaming non-relativistically away from the central engine with at most
$v\sim 0.075c$.

Without accounting for the spatial features of the radio emission sites,
Lenain et al. (2010) has provided a phenomenological explanation of the
observed $\gamma$-ray emission at GeV energies. They adopt an emission site at
a distance from the torus of $65\,\text{pc}$, with a radius of
$r_{\text{k}}=7\,\text{pc}$ and a magnetic field of $0.1\,\text{mG}$ that
however, is not observed in the radio band. Using relativistic electrons with
a broken power-law energy distribution they manage to explain observed GeV
$\gamma$-rays by inverse Compton (IC) scattering of IR photons from the torus.
Based on the given radio power, the expected jet power $P_{\text{jet}}$
according to Cavagnolo et al. (2010) is at the order of at most
$10^{43}\,\text{erg}\,\text{s}^{-1}$. Thus, the jet of NGC 1068 is comparably
weak to produce the observed $\gamma$-ray luminosity between about $0.1$ and
$100\,\text{GeV}$ of about $10^{41}\,\text{erg}\,\text{s}^{-1}$.

In this work, we will account for the observational details of the individual
radio knots and investigate if leptonic or hadronic emission in these knots
can actually explain the observed $\gamma$-ray signal. The paper is structured
as follows: in Sect. 2 we introduce the dominant emission scenarios for the
previously introduced emission sites, and in Sect. 3 we apply them to the case
of NGC 1068. In Sect. 4 we conclude and discuss whether or not the observed
radio knots can also be considered as origin of the observed $\gamma$-ray
signal.

##  2 Theoretical toolbox

###  2.1 Leptonic model

In the presence of a single relativistic electron (with $\beta\simeq 1$), its
emitted synchrotron power is given by
$P_{\text{syn}}=4\sigma_{\text{T}}c\gamma_{\text{e}}^{2}U_{\text{B}}/3$, where
$\sigma_{T}$ denotes the Thomson cross section and $U_{\text{B}}=B^{2}/8\pi$
yields the magnetic energy density. Moreover, the spectral energy distribution
of the emitted power shows a maximum at a frequency $\nu_{\rm
syn}\simeq\gamma_{\rm e}^{2}\,\nu_{\rm L}$, with the characteristic Larmor
frequency $\nu_{\rm L}=eB/2\pi m_{\text{e}}c$. In a similar manner also the
total emitted power by IC scattering of a monochromatic photon target
distribution with an energy density $U_{\rm ph}$ can be given in the Thomson
limit by $P_{\rm IC}=4\sigma_{\text{T}}c\gamma_{\rm e}^{2}\,U_{\rm ph}/3$. In
addition, it is well known in the Thomson regime that an initial photon with a
frequency $\nu_{0}$ obtains a mean frequency of $\nu_{\rm IC}=4\gamma_{\rm
e}^{2}\nu_{0}/3$.

In case of a differential energy distribution of electrons according to
$n_{\rm e}(\gamma_{\rm e})\propto\gamma_{\rm e}^{-q_{\rm e}}$, the emissivity
$\epsilon(\nu)$ at a given frequency $\nu$, within an interval
$\mathrm{d}\nu$, results for both emission processes from those electrons with
an appropriate energy $\gamma_{\rm e}$, within the interval
$\mathrm{d}\gamma_{\rm e}$, so that

$\epsilon_{\rm syn}(\nu_{\rm syn})\,\mathrm{d}\nu_{\rm
syn}\simeq\frac{1}{4\pi}\,P_{\rm syn}\left(\gamma_{\rm e}=\sqrt{\frac{\nu_{\rm
syn}}{\nu_{\rm L}}}\right)\,n_{\rm e}\left(\gamma_{\rm e}=\sqrt{\frac{\nu_{\rm
syn}}{\nu_{\rm L}}}\right)\,\mathrm{d}\gamma_{\rm e}\,,$ (1)

and

$\epsilon_{\rm IC}(\nu_{\rm IC})\,\mathrm{d}\nu_{\rm
IC}\simeq\frac{1}{4\pi}\,P_{\rm IC}\left(\gamma_{\rm e}=\sqrt{\frac{3\nu_{\rm
IC}}{4\nu_{0}}}\right)\,n_{\rm e}\left(\gamma_{\rm e}=\sqrt{\frac{3\nu_{\rm
IC}}{4\nu_{0}}}\right)\,\mathrm{d}\gamma_{\rm e}\,.$ (2)

Hereby, we suppose that all the power is emitted at the frequencies $\nu_{\rm
syn}$ and $\nu_{\rm IC}$, respectively, which is an appropriate estimate since
the synchrotron and IC emissions show a peak at these characteristic
frequencies. Note that these relations imply that all power is emitted at a
typical frequency. With

$\frac{\mathrm{d}\gamma_{\rm e}}{\mathrm{d}\nu_{\rm syn}}=\frac{\nu_{\rm
syn}^{-1/2}}{2\nu_{\rm
L}^{1/2}}\,,\quad\text{and}\quad\frac{\mathrm{d}\gamma_{\rm
e}}{\mathrm{d}\nu_{\rm IC}}=\frac{\nu_{\rm
IC}^{-1/2}}{2}\left(\frac{3}{4\nu_{0}}\right)^{1/2}\,$ (3)

the ratio of IC over synchrotron luminosity yields

$\frac{\nu_{\rm IC}L_{\nu_{\rm IC}}}{\nu_{\rm syn}L_{\nu_{\rm
syn}}}\equiv\frac{\nu_{\rm IC}\,\epsilon_{\rm IC}(\nu_{\rm IC})}{\nu_{\rm
syn}\,\epsilon_{\rm syn}(\nu_{\rm syn})}=\left[\frac{3}{4}\,\frac{\nu_{\rm
IC}/\nu_{0}}{\nu_{\rm syn}/\nu_{\rm L}}\right]^{\frac{3-q_{\rm
e}}{2}}\,\frac{U_{\rm ph}}{U_{\rm B}},$ (4)

where the equality only holds in the Thomson limit, where $\nu_{0}\nu_{\rm
IC}\ll(m_{\rm e}c^{2})^{2}/h^{2}$. Hence, for a given target photon field
(that is idealized to be monochromatic) and a known synchrotron flux spectrum
at $\nu_{\rm syn}$, the resulting $\gamma$-ray luminosity $\nu_{\rm
IC}L_{\nu_{\rm IC}}$ from IC scattering depends on the magnetic field strength
according to $\nu_{\rm IC}L_{\nu_{\rm IC}}\propto B^{-(1+q_{\rm e})/2}$.

Supposing that the energy within the emission site is at equipartition, an
estimate of the corresponding magnetic field strength can be derived—similar
to the the procedure in Pacholczyk (1970)—as follows: the total electron
energy budget of those electrons that predominantly contribute to the
synchrotron luminosity $L_{\nu_{\rm syn}}$, i.e. those with
$\gamma_{\text{e}}=\sqrt{\nu_{\rm syn}/\nu_{\rm L}}$, can be estimated by

$E_{\text{e}}\simeq m_{\text{e}}c^{2}\,\frac{\nu_{\rm syn}}{\nu_{\rm
L}}\,n_{\text{e}}\left(\gamma_{\text{e}}=\sqrt{\frac{\nu_{\rm syn}}{\nu_{\rm
L}}}\right)\,V\,,$ (5)

where $V$ denotes the volume of the emission site. According to Eq. (1) the
corresponding synchrotron luminosity can be approximated by

$\nu_{\rm syn}L_{\nu_{\rm syn}}\simeq\frac{1}{2}\,\left(\frac{\nu_{\rm
syn}}{\nu_{\rm L}}\right)^{1/2}\,P_{\rm syn}\left(\gamma_{\rm
e}=\sqrt{\frac{\nu_{\rm syn}}{\nu_{\rm L}}}\right)\,n_{\rm e}\left(\gamma_{\rm
e}=\sqrt{\frac{\nu_{\rm syn}}{\nu_{\rm L}}}\right)\,\,V\,,$ (6)

so that the total electron energy dependent on the differential synchrotron
luminosity at $\nu_{\rm syn}$ yields

$E_{\text{e}}\simeq c_{\rm syn}\,B^{-3/2}\,\nu_{\rm syn}L_{\nu_{\rm
syn}}\,\quad\text{with}\quad c_{\rm syn}=\frac{12\pi
m_{\text{e}}c}{\sigma_{\text{T}}}\left(\frac{e}{2\pi m_{\text{e}}c\nu_{\rm
syn}}\right)^{1/2}\,.$ (7)

With a magnetic field energy of $E_{\rm B}=V\,U_{\rm B}$, the total energy
within the emission site is given by

$E_{\text{tot}}=E_{\text{e}}+E_{\text{p}}+E_{\text{B}}=(1+k)c_{\rm
syn}\,B^{-3/2}\,\nu_{\rm syn}L_{\nu_{\rm
syn}}+\frac{B^{2}\,r_{\text{k}}^{3}}{6}\,,$ (8)

where the heavy particle energy $E_{\rm p}$ is $k$ times the electron energy
of those electrons that predominantly contribute to $L_{\nu_{\rm syn}}$. Its
value depends on the acceleration mechanism of relativistic particles, but in
general it is expected that $k\gg 1$. Adopting that the total energy is at its
minimum, i.e. $\partial E_{\rm tot}/\partial B=0$, we obtain a magnetic field
strength of

$B_{\text{eq}}=(4.5)^{2/7}(1+k)^{2/7}c_{\rm
syn}^{2/7}r_{\text{k}}^{-6/7}\left(\nu_{\rm syn}L_{\nu_{\rm
syn}}\right)^{2/7}$ (9)

for which the magnetic field energy is about equal to the total particle
energy.

###  2.2 Hadronic model

In the case of an hadronic origin of the observed $\gamma$-ray luminosity,
there is no robust low-energy counterpart, such as in the leptonic scenario,
that can be used to constrain the energy distribution of the CR protons.
Moreover, $\gamma$-rays can in principle be produced by photomeson or hadronic
pion production, i.e. by inelastic interactions with some target gas and
target photon field, respectively.

In case of photomeson production, the corresponding photon emissivity is given
by (Dermer & Menon, 2009)

$\tilde{\epsilon}_{\pi\gamma}\left(E_{\gamma}\right)=\frac{c\,\zeta_{\gamma}\,\sigma_{\pi\gamma}^{s,m}}{64\pi^{2}\chi_{\gamma}m_{\text{p}}c^{2}}\,\frac{n_{\text{p}}\left(\bar{\gamma}_{\text{p}}\right)}{\bar{\gamma}_{\text{p}}^{2}}\int_{\frac{\varepsilon_{l}^{\prime}}{2\bar{\gamma}_{\text{p}}}}^{\infty}\mathrm{d}\varepsilon\frac{n_{\text{ph}}(\varepsilon)}{\varepsilon^{2}}f\left(\bar{\gamma}_{\text{p}},\varepsilon\right)\,,$
(10)

where $n_{\text{ph}}(\varepsilon)$ denotes the differential photon number
density at a dimensionless photon energy $\epsilon=h\nu/m_{\rm e}c^{2}$.
Moreover, the function
$f(\bar{\gamma}_{\text{p}},\varepsilon)\equiv\left\\{\big{[}\min\left(2\bar{\gamma}_{\text{p}}\varepsilon,\varepsilon_{u}^{\prime}\right)\big{]}^{2}-\varepsilon_{l}^{\prime
2}\right\\}\,$ accounts for single pion production, where
$\sigma^{s}_{\pi\gamma}$ = 340 $\mu$b, ${\varepsilon}_{l}^{\prime}$ = 390,
${\varepsilon}_{u}^{\prime}$ = 980, as well as multi pion production, where
$\sigma^{m}_{\pi\gamma}$ = 120 $\mu$b, ${\varepsilon}_{l}^{\prime}$ = 980,
${\varepsilon}_{u}^{\prime}\longrightarrow\infty$. Further,
$\zeta_{\gamma}^{s}=1$ and $\zeta_{\gamma}^{m}=2$, respectively, denote the
multiplicity in case of single and multi pion production respectively, and
$\chi_{\gamma}^{m,s}=0.1$ is the mean fractional energy of the produced
$\gamma$-ray compared to the incident primary proton. In addition,
$n_{\text{p}}(\bar{\gamma}_{\text{p}})$ is the differential intensity of CR
protons with a Lorentz factor
$\bar{\gamma}_{\text{p}}\equiv(E_{\gamma})/(\chi_{\gamma}m_{\text{p}}c^{2})$.

Figure 1: Sketch of the environment, with the distances not in scale
($r_{\text{cor}}\sim 10^{-4}$ $\rm{pc}$ and $r_{\text{tor}}=3.6$ $\rm{pc}$).
The radii $r$ of the knots and respective distances $z$ from the core are
summarised in Table 1.  Figure 2: Spectral distribution of the energy density
within the torus and corona+disk environment, computed at a distance of
$10^{-4}$ pc. The torus field has its peak at a characteristic frequency of
$\nu_{\text{tor}}=1.7\times 10^{13}\text{Hz}$, the disk at a frequency of
$\nu_{\text{disk}}=4.5\times 10^{15}\text{Hz}$ and the corona at a frequency
of $\nu_{\text{cor}}=3.4\times 10^{19}\text{Hz}$.

In case of hadronic pion production, the corresponding photon emissivity is
given by (Schlickeiser, 2002)

$\begin{split}\tilde{\epsilon}_{\rm
pp}\left(E_{\gamma}\right)&=\frac{2c\,n_{\rm gas}}{3\pi
m_{\pi}c^{2}}\,\int_{E_{\gamma}+\frac{m_{\pi}^{2}c^{4}}{4E_{\gamma}}}^{m_{\pi}c^{2}\gamma_{\rm
p,max}^{3/4}}\mathrm{d}E_{\pi}\,\zeta\sigma_{\rm
pp}\left(\left(\frac{E_{\pi}}{m_{\pi}c^{2}}\right)^{4/3}\right)\\\
&\quad\times\left(\frac{E_{\pi}}{m_{\pi}c^{2}}\right)^{1/3}\,n_{\rm
p}\left(\left(\frac{E_{\pi}}{m_{\pi}c^{2}}\right)^{4/3}\right)\,\left[E_{\pi}^{2}-m_{\pi}^{2}c^{4}\right]^{-1/2},\end{split}$
(11)

Figure 3: The differential photon energy densities at their characteristic
frequencies. The data points for knot C and NE refer to the internal
background photon energy densities according to Gallimore et al. (2004) (at
$5\,\text{GHz}$) and the ones for P1-P4 refer to the data from Michiyama et
al. (2022) (at $92\,\text{GHz}$). For the conversion of the radio data of the
six knots knot radii of $r_{\text{k,1}}=0.2\pm 0.1\,\text{pc}$,
$r_{\text{k,2}}=0.3\pm 0.15\,\text{pc}$ and $r_{\text{k,3}}=3.5\pm
1.75\,\text{pc}$, respectively, are supposed. Figure 4: Optical thickness
$\tau_{\gamma\gamma}$ at 17 GeV dependent on the distance $z$ of the knot with
respect to the central engine. Here, the absorption coefficient
$\alpha_{\gamma\gamma}$ is calculated according to Dermer & Menon (2009) and
green shaded range corresponds to the uncertainty of the knot radius evolution
(see e.g. Zacharias et al. (2022) for more details).

where $n_{\rm gas}$ denotes the constant target gas density. Moreover, the
product of the total cross section $\sigma_{\rm pp}$ and the multiplicity
$\zeta$ can be approximated at $\gamma_{\rm p}\geq 1.75$ by $\zeta\sigma_{\rm
pp}(\gamma_{\rm p})=8.12\times 10^{-27}\,(\gamma_{\rm
p}-1)^{0.53}\,\text{cm}^{2}$. Note, that at $\gamma_{\rm p}<1.75$ the steep
decline of the inclusive cross section $\zeta\sigma_{\rm pp}$ yields a strong
suppression of the emissivity at sub-GeV energies.

Using a differential proton number density
$n_{\text{p}}(\gamma_{\text{p}})=n_{0}\left(\gamma_{\text{p}}/\gamma_{0,\text{p}}\right)^{-q_{\text{p}}}$
for
$\gamma_{\text{p},\text{min}}<\gamma_{\text{p}}<\gamma_{\text{p},\text{max}}$,
the associated total relativistic proton luminosity, $L_{\text{p}}\equiv 4\pi
r_{\rm k}^{2}v_{\rm b}\,m_{\rm p}c^{2}\,\int
d\gamma_{\text{p}}\,\gamma_{\text{p}}\,n_{\text{p}}(\gamma_{\text{p}})$,
cannot exceed the available jet power $P_{\text{jet}}$, i.e.
$L_{\text{p}}=f_{\rm jet}\,P_{\text{jet}}$ with $f_{\rm jet}<1$.

In this work, a homogeneous spherically symmetric emission site with a radius
$r_{\text{k}}$ is adopted, so that the $\gamma$-ray luminosity at a certain
energy $E_{\gamma}$ yields

$\nu_{\pi\gamma}L_{\nu_{\pi\gamma}}=A_{\pi\gamma}\,E_{\gamma}^{2}\,\frac{c}{v_{\rm
b}}\,r_{\rm k}\,f_{\rm
jet}P_{\text{jet}}\,\bar{\gamma}_{\text{p}}^{-q_{\text{p}}-2}\,\int_{\frac{\varepsilon_{l}^{\prime}}{2\bar{\gamma}_{\text{p}}}}^{\infty}{\mathrm{d}\varepsilon\frac{n_{\text{ph}}(\varepsilon)}{\varepsilon^{2}}f(\bar{\gamma}_{\text{p}},\varepsilon)}$
(12)

and

$\begin{split}\nu_{\rm pp}L_{\nu_{\rm pp}}&=A_{\rm
pp}\,E_{\gamma}^{2}\,\frac{c}{v_{\rm b}}\,\left(\frac{n_{\rm gas}\,r_{\rm
k}}{\text{cm}^{-2}}\right)\,f_{\rm jet}P_{\rm
jet}\,\int_{E_{\gamma}+\frac{m_{\pi}^{2}c^{4}}{4E_{\gamma}}}^{m_{\pi}c^{2}\gamma_{\rm
p,max}^{3/4}}\mathrm{d}E_{\pi}\\\
&\quad\times\left(\frac{E_{\pi}}{m_{\pi}c^{2}}\right)^{-\frac{4q_{\rm
p}-1}{3}}\,\left[\left(\frac{E_{\pi}}{m_{\pi}c^{2}}\right)^{\frac{4}{3}}-1\right]\,\left[E_{\pi}^{2}-m_{\pi}^{2}c^{4}\right]^{-\frac{1}{2}}\,,\end{split}$
(13)

respectively, with

$\begin{split}A_{\pi\gamma}&=\frac{\zeta_{\gamma}\,\sigma_{\pi\gamma}^{s,m}\,(2-q_{\text{p}})}{48\pi\,m_{\rm
p}^{2}c^{4}\,\chi_{\gamma}\,(\gamma_{\text{p},\text{max}}^{2-q_{\text{p}}}-\gamma_{\text{p},\text{min}}^{2-q_{\text{p}}})}\quad\text{and}\\\
A_{\rm pp}&=\frac{2.89\times 10^{-26}\,(2-q_{\rm
p})}{m_{\pi}m_{p}c^{4}\,(\gamma_{\text{p},\text{max}}^{2-q_{\text{p}}}-\gamma_{\text{p},\text{min}}^{2-q_{\text{p}}})}\,.\end{split}$
(14)

Dependent on the present target densities $n_{\rm gas}$ and $n_{\rm ph}$,
respectively, the hadronic $\gamma$-ray production is typically dominated by
one of these processes.

###  2.3 Photon target

As shown in Fig. 2 the most energetic target photon densities result from the
torus, which is located at a distance of about $(1-10)\,\text{pc}$ from the
BH, and the accretion disk (in the UV band) and the surrounding corona (where
these photons are comptonised into the X-ray band), whereof the latter expands
up to distances of about 100 Schwarzschild radii (Murase, 2022), which is
about $10^{-4}\,\text{pc}$ in the case of NGC 1068. Here, the torus with an
average temperature of $T_{\text{tor}}\sim 150$ K (Lopez-Rodriguez et al.,
2018) is modelled by a grey body spectrum with

$\nu
L_{\nu}^{\text{tor}}=\frac{L_{\text{bol}}}{6}\left(\frac{h\nu(1+z)}{k_{\text{B}}T_{\text{tor}}}\right)^{4}\,\exp\left(-\frac{h\nu(1+z)}{k_{\text{B}}T_{\text{tor}}}\right)$
(15)

where the prefactor of $1/6$ ensures that the integral over
$L_{\nu}^{\text{tor}}$ yields the bolometric luminosity of $L_{\rm
bol}=2.46\times 10^{44}\,\text{erg/s}$, as provided by the so-called CLUMPY
torus model by Lopez-Rodriguez et al. (2018). Here, an outer radius of $r_{\rm
tor}=3.6\,\text{pc}$ has been adopted111Under consideration of the different
assumptions on the distance of NGC 1068.. Note that the alternative scenario
of a smooth torus yields a photon density that is about a factor of 0.5 times
smaller. The term $T_{\text{tor}}$ denotes the temperature of the dusty torus,
and we consider it to be $T_{\text{tor}}=150\text{K}$.

The photon field of the accretion disk is modelled by an empirical model by Ho
(2008), where the UV spectrum is determined from the given Eddington ratio
$\lambda_{\text{Edd}}\equiv L_{\text{bol}}/L_{\text{Edd}}\sim 1.4$ of NGC
1068. The subsequent coronal X-ray spectrum is modelled by a power-law with an
exponential cutoff. Here, the photon index $\Gamma_{X}$ is correlated to
$\lambda_{\text{Edd}}$ through $\Gamma_{X}\approx
0.167\times\log(L_{\text{Edd}})$ \+ 2 and the cutoff energy is given by
$E_{\text{X,cut}}\sim[-74\log(\lambda_{\text{Edd}})+1.5\times 10^{2}]$ keV
(Murase et al., 2020). At low energies, we cut off the spectrum for energies
lower than 1 eV, since the infrared photons at these energies originate
predominantly from the previously described torus at significantly larger
distances from the BH. Note that this emission from the nuclear region of NGC
1068 gets reprocessed/ scattered by the clouds in the broad line region, so
that a fraction $f_{\rm diff}\ll 1$ of the central luminosity
$\nu_{0}L_{\nu_{0}}$ forms a diffuse radiation field at a distance from the BH
of $z\sim 0.1\,\text{pc}$. However, at the considered emission sites, where
$z\gg 1\,\text{pc}$, this diffuse photon field appears again as a point source
yielding a photon energy density in the reference frame of the knot of (Dermer
& Menon, 2009)

$U_{\rm trg}=\frac{\nu_{0}L_{\nu_{0}}}{4\pi z^{2}c}\,\left[\Gamma_{\rm
b}^{2}(1+\beta_{\rm b})^{2}\right]^{-1}\,,$ (16)

where $\beta_{\rm b}c$ denotes the velocity of the knot and $\Gamma_{\rm b}$
its corresponding Lorentz factor. Note that at small distances from the
isotropic photon target, the previous point source assumption is no longer
valid and details of the geometry of the isotropic radiation field have to be
taken into account to obtain the photon energy density in the reference frame
of the knot, see e.g. Sikora et al. (1994) for more details. But since all of
the considered emission sites show non-relativistic velocities with
$\beta_{\rm b}\equiv v_{\rm b}/c\lesssim 0.05$ (Roy et al., 2000), we can
neglect these relativistic beaming effects in the following. Note that Eq.
(16) can also be used for the isotropic photon field from the torus, with a
luminosity $\nu_{0}L_{\nu_{0}}$, if we adopt that $z\gg r_{\rm tor}$ and the
spatial extension of the torus is not orders of magnitudes smaller than
$r_{\rm tor}$. As shown in Fig. 3 the photon energy density from the
corona+disk environment is at all distances larger than the differential
photon density of the torus at its peak frequency of
$\nu_{\text{tor}}=1.7\times 10^{13}$ Hz. Moreover, it is shown in Fig. 3 for
the considered emission sites with a knot radius $r_{\rm k}$ as well as an
observed luminosity $\nu_{\rm obs}L_{\rm obs}$ in the radio band, as given in
Table 1, that the internal photon field222With internal we are referring to
the photon density that is produced within the considered knot structures, in
contrast to the previously discussed external fields. energy density at
$\nu_{\rm obs}$ yields

$U_{\text{trg}}^{\rm(int)}=\frac{\nu_{\rm obs}L_{\rm obs}}{4\pi r_{\rm
k}^{2}c}\ll U_{\rm trg}\,.$ (17)

This also holds for the considered emission sites if we account for the
observed spectral behaviour $\alpha$ and compare the energy densities at the
characteristic frequencies of the torus or the disk. Hence, in the following
only the external photon fields are taken into account, as the dominant photon
target to generate $\gamma$-rays up to some tens of GeV.

In general, the $\gamma$-ray luminosity, according to Eq. (4), (12) and (13),
scales linearly with the present target photon density, if absorption via
$\gamma\gamma$ pair production (with an absorption coefficient
$\alpha_{\gamma\gamma}$) can be neglected. Fig. 4 shows that this is actually
a proper presumption as even at the highest considered $\gamma$-ray energies
the corresponding optical thickness
$\tau_{\gamma\gamma}\equiv\alpha_{\gamma\gamma}\,r_{\rm k}\ll 1$ for the
considered radio knots. Moreover, it illustrates that dependent on the
supposed knot radius evolution the observed $\gamma$-rays at a few $\times
10\,\text{GeV}$ cannot be explained by alternative emission sites that are
closer than about a few $\times 0.1\,\text{pc}$ to the central nucleus.

##  3 Results

###  3.1 Leptonic Scenario

Figure 5: Produced luminosity via IC process, according to Eq. (4). The dotted
lines indicate the values of the magnetic field according to Eq. (9). The grey
shaded band excludes all values higher than $P_{\rm jet}$.

In the following, we determine the $\gamma$-ray luminosity at the low and
high-energy end of the recently updated Fermi-LAT flux (Abdollahi et al.,
2022), i.e. at $\nu_{\rm IC,1}=0.18\,\text{GeV}/h$ (luminosity of $1.7\times
10^{40}$ erg/s) and $\nu_{\rm IC,2}=17\,\text{GeV}/h$ (luminosity of
$8.2\times 10^{39}$ erg/s). Hereby, we use Eq. (4) to determine the
$\gamma$-ray luminosity from IC scattering in the Thomson limit, so that

$\nu_{\rm IC}L_{\nu_{\rm IC}}=2\left[\frac{3}{8\pi}\,\frac{\nu_{\rm
IC}/\nu_{0}}{\nu_{\rm syn}}\,\frac{e}{m_{\rm e}c}\right]^{\frac{3-q_{\rm
e}}{2}}\,\frac{\nu_{0}L_{\nu_{0}}}{z^{2}c}\,B^{-\frac{1+q_{\rm
e}}{2}}\,\nu_{\rm syn}L_{\nu_{\rm syn}},$ (18)

where $\nu_{\rm syn}L_{\nu_{\rm syn}}=\nu_{\rm obs}L_{\nu_{\rm obs}}$ at
$\nu_{\rm syn}=\nu_{\rm obs}$ as given in Table 1. A maximal $\gamma$-ray
luminosity is obtained for a dense target energy distribution at $\nu_{0}$ as
well as a low value of $\nu_{0}$ in case of $q_{\rm e}<3$. In case of
optically thin synchrotron emission at $\nu_{\rm syn}$ according to
$L_{\nu_{\rm syn}}\propto\nu_{\rm syn}^{-\alpha}$ the spectral behavior of the
relativistic electron distribution can be determined by
$q_{\text{e}}=2\alpha+1$. Note that this spectral behavior can in principle
become softer towards higher energies due to the impact of synchrotron or IC
losses. But without these softening effects the spectral behavior as given in
Table 1 indicates that the torus with a spectral luminosity of
$\nu_{0}L_{\nu_{0}}=1.72\times 10^{44}\rm{erg/s}$ at
$\nu_{0}=\nu_{\text{tor}}=1.7\times 10^{13}\,\text{Hz}$ is the dominant photon
target.

As shown in Fig. 5 the emission sites C is at $\nu_{\rm
IC,2}=17\,\text{GeV}/h$ able to exceed the observed $\gamma$-ray luminosity
for the whole range of reasonable magnetic field strength values, such as its
$B_{\rm eq}$ value (for a proton-to-electron energy ratio of $k=100$) that is
indicated by the red dotted line. Note that for all emission sites the values
of $B_{\rm eq}$ are rather large. Comparing the results at $0.18$ and
$17\,\text{GeV}$, it is clear that the adopted spectral behavior of the
relativistic electrons, that has been derived from the radio emission, is too
hard with respect to the observed spectral behavior in the $\gamma$-ray band.
Thus, this emission site has in principle enough total energy to produce the
observed $\gamma$-rays above about a few GeV, if electrons can be accelerated
up to $\gamma_{\rm e}\simeq\sqrt{3\nu_{\gamma}/4\nu_{\rm tor}}\simeq 4\times
10^{5}$. But to explain also the sub-GeV $\gamma$-ray emission, a magnetic
field strength of about $0.5\,\text{mG}$ is needed, which is about an order of
magnitude smaller than what is expected from the minimal total energy
estimate. Moreover, a significant softening of the spectral behavior of the CR
electrons to an index of $q_{\rm e}\sim 3.3$ at $\gamma_{\rm
e}\simeq\sqrt{3\nu_{\rm IC,1}/4\nu_{\rm tor}}=4\times 10^{4}$ is needed to
agree with the spectral behavior of the observed $\gamma$-ray data.

Figure 6: Interaction timescales for the relevant radiative processes in C
knot. The horizontal black line represents the size of the region based on
radio observations. We can see that inverse Compton cooling (blue curve)
becomes efficient for electron energies
$E_{\mathrm{e}}>1.1\leavevmode\nobreak\ \mathrm{GeV}$, as denoted by the
vertical line, leading to a cooling break in the electron spectrum.

Hence, in addition to the low magnetic field strength, the data also shows the
need for a strong cooling break in the electron spectrum at about 1 GeV. But
as shown in Fig. 6 a break at about these energies can only occur if the
escape of CR electrons is delayed by at least an order of magnitude with
respect to the free streaming condition which is determined by the light-
crossing time of the knot, suggested to be about 0.2 pc by VLBA observations
(cf. Tab. 1). This can in principle be realized by diffusive escape from the
knot, which however, already softens the initial spectral behavior so that
only a minor softening by $\Delta q_{\rm e}\simeq 0.6$ emerges at the
transition from diffusive escape to IC losses. Moreover, we can see that
synchrotron emission (green line) is comparatively inefficient, since the
energy density of the magnetic field is much lower than that of the external
thermal fields. That also applies to relativistic bremsstrahlung losses
(purple line) even for the case of a dense target plasma. For completeness, we
also show the optical thickness to photon-photon pair production (dashed
orange curve), which is inefficient up to photon energies of about 30 GeV, as
shown previously in Fig. 4. Thus, all potential interaction processes in knot
C are not able to explain the necessary softening of the spectrum. A
significant $\gamma$-ray contribution from the NE knot is only obtained for a
magnetic field strength $\ll 0.1\,\text{mG}$. Further, it can be excluded that
the observed $\gamma$-rays originate from the head of the jet (i.e. P1-4).

Table 1: Properties for the different emission sites that have been observed
in the radio band by Gallimore et al. (2004) (for C and NE) and Michiyama et
al. (2022) (for P1-4) as well as the expected magnetic field strength
according to Eq. (9).

| $z$ | $r_{\text{k}}$ | $\nu_{\text{obs}}$ | $\nu_{\text{obs}}L_{\nu_{\text{obs}}}$ | $\alpha$ | $B_{\rm eq}(k=100)$
---|---|---|---|---|---|---
| [pc] | [pc] | [GHz] | [$10^{36}$ erg/s] |  | [mG]
C | 15 | 0.2 | 5 | $6.4$ | 0.23 | 15.4
NE | 30 | 0.3 | 5 | $9.5$ | 0.90 | 10.9
P1 | 484 | 3.5 | 92 | $7.6$ | 0.50 | 1.40
P2 | 477 | 3.5 | 92 | $8.6$ | 0.59 | 1.40
P3 | 468 | 3.5 | 92 | $8.8$ | 0.65 | 1.40
P4 | 468 | 3.5 | 92 | $7.5$ | 0.50 | 1.40

  * •

Notes. Length scales are determined supposing a source distance of
$10.1\,\text{Mpc}$, so that $1^{\prime\prime}=49\,\text{pc}$. Here, $z$ is the
distance of the knot with respect to the central engine, $r_{\text{k}}$ is the
knot radius, $\nu_{\text{obs}}L_{\nu_{\text{obs}}}$ the observed radio
luminosity at a frequency $\nu_{\text{obs}}$ and $\alpha$ the spectral index
at this frequency.

###  3.2 Hadronic Scenario

Accounting for the external photon targets by the torus, disk and corona with
a differential photon number density

$n_{\text{ph}}(\varepsilon)=\frac{1}{4\pi}\,\frac{m_{\rm
e}c^{2}}{(h\nu_{0})^{2}}\,U_{\rm trg}$ (19)

at a dimensionless photon energy $\epsilon=h\nu_{0}/m_{\rm e}c^{2}$, we
determine in the following the resulting $\gamma$-rays according to photomeson
production as given by Eq. (12). Since the torus only provides photons at
energies $\ll 100\,\text{keV}$, these target photons are not energetic enough
to provide $\gamma$-rays up to some tens of GeV. However, the X-ray photons
from the corona with energies up to few $\times 100\,\text{keV}$ are in
principle energetic enough to yield $\gamma$-rays at these energies, but not
at sub-GeV energies. Moreover, $\nu_{\pi\gamma}L_{\nu_{\pi\gamma}}$ is—even
for the most optimistic scenarios—also at $17\,\text{GeV}$ multiple order of
magnitudes below the observations. Hence, we can clearly rule out that the jet
produces the observed $\gamma$-rays by a photohadronic scenario.

In the case of hadronic pion production, the resulting luminosity $\nu_{\rm
pp}L_{\rm pp}$ is at a few GeV much closer to the observations than the
photohadronic scenario. But due to the fundamental kinetics of the inelastic
scattering the observed sub-GeV $\gamma$-rays can still not be explained. In
order to match the observed $\gamma$-ray luminosity at $17\,\text{GeV}$, the
Fig. 7 shows the necessary product of target gas density ($n_{\rm gas}$) and
the fraction ($f_{\rm jet}$) of the jet power that goes into CR protons
dependent on their spectral index ($q_{\rm p}$). Even in the most optimistic
scenario of a jet that is completely dominated by the CR protons, i.e. $f_{\rm
jet}\simeq 1$, the target gas density needs to be multiple orders of magnitude
higher than the average ISM density of about $1\,\text{cm}^{-3}$. But as shown
by García-Burillo et al. (2014) dense molecular gas with $n(H_{2})\geq
10^{5-6}\,\text{cm}^{-3}$ is present within the circumnuclear disk of NGC
1068. In particular the knot C seems to impact a giant molecular cloud
(Gallimore et al., 2004) not least due to the need of free-free absorption at
$1.4\,\text{GHz}$. Thus, if the target gas in the considered knot structures
also shows an increased density, the slight excess of the observed
$\gamma$-ray flux at about $17\,\text{GeV}$ could actually be a result of the
contribution from $\gamma$-rays by hadronic pion production. But in any case
there is an additional $\gamma$-ray emitter needed to also explain the
observed $\gamma$-rays at sub-GeV energies.

Figure 7: The necessary parameter condition for the different knot
environments C, NE and P1 to P4 to explain the observed $\gamma$-ray flux of
$8.2\times 10^{39}\,\text{erg}\,\text{s}^{-1}$ at $17\,\text{GeV}$ by hadronic
pion production. Here a knot velocity of $v_{\rm b}=0.075\,c$ is used as well
as $\gamma_{\text{p},\text{min}}=1.$ and $\gamma_{\text{p},\text{max}}=2\times
10^{3}$ to obtain a cut-off in the resulting $\gamma$-ray flux at about a few
$\times 10\,\text{GeV}$ as indicated by the observations.

##  4 Discussion and conclusions

In this work, we discuss the possibility that the different radio emission
sites, so-called knots, within the jet-like structure of NGC 1068 can be
associated as the origin of the observed $\gamma$-ray flux between about $0.1$
and $100\,\text{GeV}$. Hereby, we account for the observed spatial and
spectral details of these radio structures and discuss the potential leptonic
and hadronic $\gamma$-ray emission scenarios.

For relativistic protons within these radio knots, hadronic pion production is
the most promising scenario to explain the mild excess of the observed
$\gamma$-ray flux at about $17\,\text{GeV}$. For this purpose, it is needed
that

$\left(\frac{v_{\rm b}/c}{0.075}\right)^{-1}\,\left(\frac{n_{\rm
gas}}{1\,\text{cm}^{-3}}\right)\,\left(\frac{r_{\rm
k}}{5\,\text{pc}}\right)\,\left(\frac{f_{\rm jet}P_{\rm
jet}}{10^{43}\,\text{erg}\,\text{s}^{-1}}\right)\sim 10^{2-3}\,,$

hence the target gas needs to be significantly denser than in the average ISM.
In particular the knot C shows indications of the impact of a dense molecular
cloud (Gallimore et al., 2004) that would be sufficient to explain the GeV
$\gamma$-rays. But in any case, an additional leptonic $\gamma$-ray
contribution is needed to account for the observed sub-GeV $\gamma$-rays.

For relativistic electrons, IC scattering with the IR photons from the torus
yields the highest flux of $\gamma$-rays based on the expected spectral
behavior of the electrons from radio observations. Depending on the supposed
magnetic field strength within these radio knots we could constrain that the
knot C, which is the closest with respect to the central engine, represents
the most likely emission site for the production of the observed
$\gamma$-rays. But for this purpose it is necessary that: (i) the magnetic
field strength is $\lesssim 1\,\text{mG}$; (ii) the electron spectrum needs to
show a break with a strong softening at about an energy of
$(1-10)\,\text{GeV}$ to agree with the observed spectral behavior of the
$\gamma$-ray flux. However, we also show that such a strong softening (by
$\Delta q_{\rm e}\simeq 1.8$ if at low energies $q_{\rm e}\simeq 1.5$ as
indicated from VLBA observations) cannot emerge from the potential interaction
processes in knot C. In general, these findings are in good agreement with the
results from Lenain et al. (2010), where however, the previously mentioned
conditions (i) and (ii) have been supposed, to explain the $\gamma$-ray data
by IC scattering with IR photons from the torus. Nonetheless, these conditions
do not hold, because the radius and magnetic field needed to achieve the right
$\gamma$-ray luminosity are $\sim$ 0.1 pc and $0.1$ mG. These values are,
respectively, an order of magnitude lower and higher than what is observed and
expected. Moreover, we do not naturally find a spectral break at an energy of
$(1-10)\,\text{GeV}$. Hence, by accounting for the observational constraints
by Gallimore et al. (2004) it is not possible to explain the observed
$\gamma$-rays by the radio knot C or any other observed knot structure. But if
one of these sites impacts a dense target gas, they could still provide some
hadronic $\gamma$-ray contribution at about $10\,\text{GeV}$.

One of the key parameters of these results is the strength of the magnetic
field: based the minimal total energy condition, as given in Eq. (9), a value
of about $(1+k)^{2/7}\,\text{mG}$ is obtained, which however still includes
the uncertain proton-to-electron energy ratio $k$. Moreover, this classical
estimate can be revised, because it does not account for inhomogeneous
magnetic fields, the synchrotron spectrum over the real energy range and the
real ratio $k$ of total energies. Considering these factors, it has been shown
by Beck et al. (2005) that the resulting magnetic field could become slightly
smaller by a factor of about $0.86-0.99$. Hence, even in the most extreme
scenario of a vanishing hadronic particle energy, the field strength $B_{\rm
eq}$ that are given in Table 1 would only become smaller by a factor of about
$0.2$.

Further, knot C only manages to produce sufficient $\gamma$-rays due to its
non-relativistic bulk motion. In the case of a mildly relativistic motion,
i.e. $\Gamma_{b}\gg 1$, the target photon energy density in the rest frame of
the knot would be reduced according to Eq. (16) and the target photon
frequency $\nu_{0}$ would suffer from relativistic Doppler de-boosting which
would in total lower the expected IC $\gamma$-ray luminosity by a factor
$\Gamma_{\rm b}^{-(1+q_{\rm e})/2}$.

Apart from the previously discussed spatially resolved radio knots, there
could be the chance that the observed $\gamma$-rays are produced much closer
to the foot of jet, where radio telescopes cannot resolve individual emission
sites. But as shown in Fig. 4, the emission sites already become opaque for
$\sim 10\,\text{GeV}$ $\gamma$-rays at distances of about $0.1\,\text{pc}$ due
to $\gamma\gamma$ pair production. Therefore, such alternative emission sites
could not explain the observed $\gamma$-rays at the highest energies.

Acknowledgements. We acknowledge funding from the German Science Foundation
DFG, within the Collaborative Research Center SFB 1491 “Cosmic Interacting
Matters - From Source to Signal”.

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