# Accelerated protons produced by magnetic Penrose process in Sgr A*

Myeonghwan Oh<EMAIL_ADDRESS>Department of Astronomy and Atmospheric
Sciences, Kyungpook National University, Daegu, 41566, Republic of Korea
Myeong-Gu Park<EMAIL_ADDRESS>Department of Astronomy and Atmospheric
Sciences, Kyungpook National University, Daegu, 41566, Republic of Korea

###### Abstract

Typical mechanisms to extract energies from a rotating black hole are the
Blandford-Znajek process and the Penrose process. The Penrose process requires
a special condition that is difficult to occur in common astrophysical
situations. However, the magnetic Penrose process (MPP) does not require such
a special condition, and can produce ultra-high energy cosmic rays. When
neutrons decay near a rotating black hole, the MPP efficiency of the produced
proton is maximized. The supermassive black hole in Sagittarius A* (Sgr A*) is
likely to have a radiatively inefficient accretion flow that is hot enough to
produce neutrons by nuclear reactions, which can be subsequently accelerated
to high-energy by the MPP. We calculate the production rate of accelerated
protons from the Sgr A* to estimated the gamma-ray flux at Earth produced by
these accelerated protons and the flux of the accelerated protons themselves
transported from Sgr A* to Earth. We find that these very high-energy gamma
rays ($E_{\gamma}\gtrsim 10\,\mathrm{TeV}$) amount to a significant fraction
of the flux of the gamma-ray from the HESS J1745-290 and the central molecular
zone around $100\,\mathrm{TeV}$. The accelerated proton flux, when the
dimensionless spin parameter $a_{*}=0.5$ and the magnetic field strength in
the vicinity of the black hole $B_{0}=100\,\mathrm{G}$, is about $1.6-4.1\%$
of the cosmic ray proton flux from KASCADE experiment at about
$1\,\mathrm{PeV}$. Due to the finite decay time of neutrons which need to be
transported from the accretion flow to the acceleration zone, our acceleration
model can operate only around black holes with mass not much greater than
$\sim 10^{8}\,M_{\odot}$.

##  I Introduction

After the development of the black thermodynamics and the discovery of the
irreducible mass of a rotating black hole [1, 2, 3], many mechanisms of
extraction of the energy from a rotating black hole (Kerr black hole) have
been presented. Penrose process, one of such mechanisms, extracts the energy
by particle decay [4]. The observer at infinity sees one of the two fragments
fall into the black hole with a negative energy and also the other fragment
escapes to infinity with a positive energy. Through this process, the black
hole loses angular momentum and the rotational energy. However, the Penrose
process requires the relative velocity between the two fragments to be larger
than half the speed of light. Therefore, the normal Penrose process is not
generally expected in a typical astrophysical jet engine [5, 6].

About a decade later, Wagh, Dhurandhar & Dadhich [7] proposed that energy from
electromagnetic potential can alleviate the constraint on the relative
velocity. When particle 1 is the incident neutral particle and particle 2 one
of two charged fragments,

$\gamma\left(E_{1}-\left|\mathbf{v}\right|\sqrt{{E_{1}}^{2}+g_{00}}\right)-e_{2}\phi\leq
0,$ (1)

where $E_{1}$ is the rest mass energy of particle 1, $\mathbf{v}$ the relative
velocity in the rest frame of particle 1, $\gamma$ the Lorentz factor from the
relative velocity $\mathbf{v}$, $\phi$ an electric potential and $e_{2}$ an
electric charge of particle 2. Equation (1) is the Wald-Bardeen inequality
with an electromagnetic interaction and presents the minimum condition of an
energy extraction from the magnetic Penrose process (hereafter MPP). When the
relative velocity $\mathbf{v}=0$, the equation (1) becomes
$E_{1}\leq\left|e_{2}\phi\right|$. Therefore, even when $\mathbf{v}=0$,
through the condition of $E_{1}\leq\left|e_{2}\phi\right|$ energy can be
extracted through the MPP. Also, the inequalities showed that the MPP is
possible without any constraints on the relative velocity.

The efficiency of the Penrose process is defined as

$\eta\equiv\frac{E_{3}-E_{1}}{E_{1}},$ (2)

where $E_{1}$ is the incident particle energy and $E_{3}$ is the escape
particle energy. The maximum efficiency of the Penrose process is $20.7\%$,
while Parthasarathy et al. presented higher efficiency $\eta>1$ with the MPP
[8]. Afterwards, Narayan, McClintock & Tchekhovskoy showed that the efficiency
of the MPP can reach to few hundred percent with general relativistic
magnetohydrodynamics (GRMHD) calculations [9].

The MPP does not have a constraint on the relative velocity of fragments, and
nor a constraint on the magnetic field strength around the black hole.
Blandford-Znajek process has a threshold magnetic field strength to produce
electron-positron pair cascade for force-free condition [10] with

$B\approx 6.2\times
10^{4}\left(\frac{M}{a}\right)^{\frac{3}{4}}\left(\frac{10\,M_{\odot}}{M}\right)^{\frac{1}{2}}\,\mathrm{G}.$
(3)

For a $10\,M_{\odot}$ black hole and the Sagittarius A* (Sgr A*), the
supermassive black hole in the center of the Galaxy with mass of about
$4\times 10^{6}\,M_{\odot}$ [11], such threshold magnetic fields are about
$10^{4}\,\mathrm{G}$ and about few $10^{2}\,\mathrm{G}$, respectively. But, in
the MPP, the magnetic field strength required is only a few $\mathrm{mG}$ and
a few $10^{-8}\,\mathrm{mG}$ to reach one hundred percent efficiency for
$10\,M_{\odot}$ black hole and the Sgr A*, respectively [12].

To analyze the motion of diverged charged particles from MPP, one should know
the motion of a charged particle in the electromagnetic field near a black
hole. Since accretion flow tends to follow a Kerr black hole for axial
symmetry, one can use the Wald solution which is the solution for the Maxwell
equation when a Kerr black hole is immersed in an aligned test uniform
magnetic field to describe the electromagnetic field near the astrophysical
black hole for leading order [13, 14, 15, 16, 17, 18, 19, 20, 21, 22]. In such
electromagnetic field and special condition is met charged particles can
escape to infinity with ultrahigh energy. [14, 19].

Tursunov et al. suggested that charged particles produced through neutron
decay can be accelerated to ultra-high energy through MPP and calculated
trajectories of the escaping particle in the vicinity of SMBHs like Sgr A* and
M87* [19]. They showed that, unlike the Penrose process that is limited in the
ergosphere, MPP can occur in a broad region, if black hole mass and uniform
magnetic field strength are large enough. They also showed that the energy of
protons accelerated by the MPP around Sgr A* can reach $\simeq
10^{15.6}\,\mathrm{eV}$, which corresponds to the knee of cosmic rays
spectrum, and showed the compatibility with the HESS Collaboration analysis.
The HESS Collaboration argues that from the gamma-ray data, petaelectronvolt
protons are produced at the Galactic center (GC), and they estimated that the
Sgr A* is the pevatron [23].

However, Tursunov et al. did not concretely present the origin of the neutrons
and did not estimate the flux of the ultra-high energy proton at Earth. An
accretion disk that is hot enough to produce neutrons from thermo-nuclear
reactions can easily become the source of neutrons [24, 25, 26].

In 1976, Shapiro, Lightman & Eardley [27] presented the high ion temperature
accretion flow that is sufficiently hot for nuclear reactions to produce
neutrons. However, the model is thermally unstable [28, 29]. Narayan & Yi [30,
29] presented a thermally stable advection-dominated accretion flow (ADAF)
model in which ions are kept at a very high temperature. Ref. [30] found self-
similar solutions of such accretion flow that show the accreting gas
temperature is close to the virial temperature and the disk structure is
quasi-spherical. In the ADAF model, due to the large mass difference between
ions and electrons, ions dominate the heating rate $q^{+}$ while electrons
dominate the cooling rate $q^{-}$. Thus, the ion temperature is about $10^{3}$
times higher than the electron temperature. The ADAF was employed to explain
many X-ray sources, including Sgr A* [31, 32].

The accretion flow into the Sgr A* is considered to be a radiatively
inefficient accretion flow (RIAF), because the Sgr A* has a low luminosity.
Therefore, the accretion disk of the Sgr A* will be sufficiently hot to
produce neutrons from nuclear reactions. Also, ADAF parameters and the
accretion rate of the Sgr A* are reasonably well known. Thus, one can estimate
the neutron number that flows into the magnetic field near the Sgr A*.

The estimation of HESS collaboration, the high energy efficiency of the MPP,
and the accretion flow of Sgr A* show that the MPP acceleration model with
neutron decay is suitable for Sgr A*.

The MPP protons have a maximum energy of about $10^{15.6}\,\mathrm{eV}$. So
the gyroradius of the proton does not exceed the scale of the random magnetic
field in the Galaxy [33]. Thus, the MPP protons will be confined and diffuse
out in the Galaxy by scattering. Therefore, if we calculate the production
rate of the accelerated protons from the Sgr A* through the MPP, we can
estimate how much the accelerated protons from the Sgr A* will affect the
cosmic ray knee observed at Earth.

We conduct a study on how many protons that are accelerated in Sgr A* by the
MPP model of Tursunov et al. (2020) impact the spectra of gamma rays from the
GC and the cosmic ray spectrum. In our model, first, neutrons are produced in
the accretion flow of the black hole by nuclear reactions. Subsequently, the
neutrons flow into the magnetic field of the vicinity of the black hole
(Chapter II). The neutrons that flow near the black hole decay into protons
and electrons. The electrons fall into the black hole and the protons are
accelerated by the MPP and escape to infinity from the black hole (Chapter
III). Finally, escaped protons diffuse out within the Galaxy and produce gamma
rays in the GC (Chapter IV) and cosmic rays flux (Chapter V).

##  II Neutron production for ADAF

We adopt the hot ADAF model to describe the accretion flow in Sgr A*. Narayan
& Yi [30, 29] presented self-similar solutions under Newtonian gravity. Radial
velocity $v_{r}$ (4), rotation angular velocity $\Omega$ (5), mass density
$\rho$ (6), and disk scale height $H$ (7) are described by simple power-laws
in radius.

$\displaystyle v_{r}$ $\displaystyle=$ $\displaystyle-2.12\times 10^{10}\alpha
c_{1}r^{-1/2}\,\mathrm{cm}\,\mathrm{s^{-1}},$ (4) $\displaystyle\Omega$
$\displaystyle=$ $\displaystyle 7.19\times
10^{4}c_{2}m^{-1}r^{3/2}\,\mathrm{s^{-1}},$ (5) $\displaystyle\rho$
$\displaystyle=$ $\displaystyle 3.79\times
10^{-5}\alpha^{-1}c^{-1}_{1}c^{-1/2}_{3}m^{-1}\dot{m}r^{-3/2}\,\mathrm{g\,cm^{-3}},$
(6) $\displaystyle H$ $\displaystyle=$ $\displaystyle(2.5c_{3})^{1/2}R,$ (7)

where $r$, $m$ and $\dot{m}$ are scaled disk radius, black hole mass, and mass
accretion rate with $r\equiv R/R_{s}$, $m\equiv M/M_{\odot}$ and
$\dot{m}\equiv\dot{M}/\dot{M}_{\mathrm{Edd}}$, respectively. The Schwarzschild
radius $R_{s}\equiv 2GM/c^{2}$ and the Eddington rate
$\dot{M}_{\mathrm{Edd}}\equiv L_{\mathrm{Edd}}/(0.1c^{2})$ where
$L_{\mathrm{Edd}}$ is the Eddington luminosity. Ion temperature $T_{i}$ and
electron temperature $T_{e}$ are also described by simple power-law in radius.

$T_{i}+1.08T_{e}=6.66\times 10^{12}\beta c_{3}r^{-1}\,\mathrm{K}$ (8)

The solutions depend on the parameters $\alpha$, $\beta$, and $f$ which
describe the physical processes in ADAF: $\alpha$ is the viscosity parameter,
$\beta$ the fraction of the gas pressure to the total pressure and $f$ the
fraction of viscously dissipated energy which is advected. Constants $c_{1}$
and $c_{3}$ are defined by parameters $\alpha$, $\beta$ and $f$ [30] with

$\displaystyle
c_{1}\equiv\frac{\left(5+2\epsilon^{\prime}\right)}{3\alpha^{2}}g\left(\alpha,\epsilon^{\prime}\right),\quad
c_{3}\equiv\frac{2\left(5+2\epsilon^{\prime}\right)}{9\alpha^{2}}g\left(\alpha,\epsilon^{\prime}\right),$
(9)

where

$\displaystyle g\left(\alpha,\epsilon^{\prime}\right)$ $\displaystyle\equiv$
$\displaystyle\left[1+\frac{18\alpha^{2}}{\left(5+2\epsilon^{\prime}\right)^{2}}\right]^{1/2}-1,$
$\displaystyle\epsilon^{\prime}$ $\displaystyle\equiv$
$\displaystyle\frac{\epsilon}{f}=\frac{1}{f}\left(\frac{5/3-\gamma}{\gamma-1}\right),$
$\displaystyle\gamma$ $\displaystyle=$
$\displaystyle\frac{32-24\beta-3\beta^{2}}{24-21\beta}$ (10)

Narayan, Kato & Honma (hereafter NKH) used the Paczyñsky-Wiita potential [34]
that mimics the relativistic effects of a Schwarzschild black hole [35].
Gammie & Popham (hearafter GP) presented fully relativistic ADAF solutions in
the Kerr geometry [36, 37]. These models are more relevant to our scenario.
However, one has to solve the two-point boundary value problems with
singularity while regulating a sonic point in the NKH model, and also one has
to solve the boundary value problems for a sonic point and viscous point in
the GP model whereas one can get analytic solutions from self-similar
solutions. In addition, NKH and GP models does not provide information about
the vertical structure of the flow, because these models are 1D solutions, and
these ADAF models do not reflect the complex magnetic field structure around
the black hole nor the inner region of the accretion flow, which will be
undoubtedly complex. Therefore, we adopt the self-similar solutions that
should be good enough approximations for our study, considering all the
uncertainties.

We choose ADAF parameters from previous studies based on observation. First,
we select the accretion rate. The accretion rate of the Sgr A* has been
estimated by Faraday rotation [38, 39, 40, 41, 42]. Refs. [40, 41] estimated
the mass accretion rate of the Sgr A* to be $(1.7-7.0)\times
10^{-8}\,M_{\odot}\,\mathrm{yr^{-1}}$ and $3.0\times
10^{-8}\,M_{\odot}\,\mathrm{yr^{-1}}$ using an MHD and a GRMHD calculation,
respectively. Using the model of Ref. [43], Ref. [38] found that the upper
limit of the accretion rate of the ADAF is about $3.0\times
10^{-6}\,M_{\odot}\,\mathrm{yr^{-1}}$. Faraday rotation measurements provide
upper limits of accretion rate of ADAF or a convection-dominated accretion
flow of the Sgr A* to be $2\times 10^{-7}\,M_{\odot}\,\mathrm{yr^{-1}}$ in
ordered, radial, equipartition magnetic field condition and lower limits to be
$2\times 10^{-9}\,M_{\odot}\,\mathrm{yr^{-1}}$ with sub-equipartition and
disordered magnetic field [39]. Bower et al. also estimated $\dot{M}\sim
10^{-8}\,M_{\odot}\,\mathrm{yr^{-1}}$ [42]. So in our model, we use the mass
accretion rate in $=1.0\times 10^{-8}\,M_{\odot}\,\mathrm{yr^{-1}}$ from Bower
et al., which is $\dot{m}=1.13\times 10^{-7}$. We choose the ADAF parameters
$\alpha=0.3$, $\beta=0.5$ (exact equipartition between the pressure of gas and
magnetic field) and $f=0.9994$. The parameters are referred from Ref. [32]
that applied the ADAF model to Sgr A*.

We calculate the creation rate of the neutron from nuclear reactions and the
decay of the neutrons within the ADAF with chosen parameters and determine the
number density of the produced neutrons.

###  II.1 Neutron creation rate

If neutrons are produced by a certain nuclear reaction, the number of
reactions per unit time per unit volume, i.e. the reaction rate
$R_{\mathrm{reaction}}$, becomes the creation rate of the neutrons. Even if
several neutrons are produced by a certain reaction, one just multiplies with
the number of produced neutrons. We use the non-relativistic reaction rate to
produce neutrons

$R_{\mathrm{reaction},AB}=n_{A}n_{B}{\left\langle\sigma v\right\rangle}_{AB},$
(11)

where

$\displaystyle{\left\langle\sigma v\right\rangle}_{AB}$ $\displaystyle=$
$\displaystyle\sqrt{\frac{8}{\pi m(k_{\mathrm{B}}T)^{3}}}$ (12)
$\displaystyle\times\int_{0}^{\infty}{\sigma_{AB}(E)}E\,\mathrm{exp}\left(-\frac{E}{k_{\mathrm{B}}T}\right)\,dE,$

and $A$, $B$, $\ldots$ are particle species. If $A=B$, one should divide
equation (11) by two. Thus, the creation rate of the neutron is

$\dot{n}_{C}=\sum_{k}^{N}(R_{\mathrm{reaction},AB})_{k},$ (13)

where $k$ corresponds to each nuclear reaction that creates neutrons. For
precise calculation, one should use the relativistic reaction rate [44].
However, in our ADAF model, the maximum ion temperature is about a few
$10^{12}\,\mathrm{K}$ ($k_{B}T/m_{p}c^{2}\lesssim 0.09$). There is little
difference between relativistic and non-relativistic reaction rates at this
temperature. Therefore, we use the non-relativistic Maxwell-Boltzmann
distribution as in equation (12).

In the case of the accretion disk around the solar mass black hole or neutron
star, we should consider the secondary reactions of particles that are
produced by the first nuclear reactions. But in the Sgr A* case, the produced
particle number density is much lower than the ion number density of the disk.
So we only consider the first nuclear reactions.

In our model, we consider that protons and $\alpha$ particles
(${}^{4}\mathrm{He}$) dominate the accretion fow and consider reactions
$p(\alpha,pn)\mathrm{{}^{3}He}$, $p(\alpha,2pn)d$,
$\alpha(\alpha,pn)\mathrm{{}^{6}Li}$, $\alpha(\alpha,n)\mathrm{{}^{7}Be}$ and
$p(p,n\pi^{+})p$. Also, we choose the mass fractions $X_{p}=0.75$ and
$X_{\alpha}=0.25$ [29]. Other reactions are excluded because their reaction
cross-section is much smaller than the above reactions. We use the cross-
section of $p-\alpha$ reactions from Refs. [45, 46], $\alpha-\alpha$ reactions
from Refs. [47, 48, 49, 50, 51, 52] and $p(p,n\pi^{+})p$ reaction from Ref.
[53]. Fig. 1 shows ${\left\langle\sigma v\right\rangle}$ of these reactions in
our model as a function of $r$.

Figure 1: The reaction rate ${\left\langle\sigma v\right\rangle}$ of
$p(\alpha,pn)\mathrm{{}^{3}He}$ (red solid), $p(\alpha,2pn)d$ (orange dashed),
$\alpha(\alpha,pn)\mathrm{{}^{6}Li}$ (green dotted),
$\alpha(\alpha,n)\mathrm{{}^{7}Be}$ (blue dash-dotted) and $p(p,n\pi^{+})p$
(black long dash-dotted) are shown, respectively. The radial range is
$1<r<100$. We use the ADAF parameters with $\alpha=0.3$, $\beta=0.5$ and
$f=0.9994$.

###  II.2 Neutron inflow

In general, ions and electrons can not escape from the accretion disk and are
accreted to the central object because of the electromagnetic interaction. But
neutrons are free from the electromagnetic force and easily escape from the
disk. Thus, it is important to estimate how many neutrons do not escape to
infinity and flow into the acceleration zone close to the black hole.

Neutrons can be confined by the gravitational binding and the scattering with
ions [54]. Neutrons are scattered with ions with the mean free path.

$\lambda_{n}=\frac{1}{n_{i}\sigma},$ (14)

where $n_{i}$ is an ion number density and $\sigma$ is a cross-section with
ions. In the ADAF disk of the Sgr A*, the mean free path of the neutrons
produced within $r\sim 10$ is longer than $r\sim 10^{4}$. therefore, neutrons
are not affected by the plasma in the disk and can be treated as
collisionless.

Aharonian & Sunyaev [24] presented an escape fraction of the neutrons produced
in an accretion disk. They used the condition that if the sum of the thermal
energy of neutrons and the energy from their flow velocity exceeds the
gravitational binding energy, neutrons can freely escape to infinity:

$\frac{1}{2}m_{\mathrm{n}}(v_{\mathrm{flow}}+v_{\mathrm{th}})^{2}\geq\frac{GMm_{\mathrm{n}}}{R}.$
(15)

From this condition, the escape fraction of neutrons is

$\displaystyle f_{esc}=\left(\frac{m}{2\pi
k_{B}T_{i}}\right)^{3/2}\int_{0}^{2\pi}\,d\phi\int_{0}^{\pi}\mathrm{sin}\theta\,d\theta$
$\displaystyle\times\int_{v_{min}}^{\infty}v_{th}^{2}e^{-mv_{th}^{2}/2k_{B}T}\,dv_{th}.$
(16)

using the Maxwell-Boltzmann distribution.

But, our purpose is to estimate the inflow rate of neutrons into the magnetic
field near the black hole. So we integrate fractions that flow into the
magnetic field using geodesic integration. To be exact, we need to know the
real neutron velocity distribution which may be different from the Maxwell-
Boltzmann distribution. However, there is no experimental data on the neutron
energy spectrum of nuclear reactions that are used in our calculation, and
there are more than three products in the reactions. Therefore, we assume the
neutron energy spectrum to be the Maxwell-Boltzmann distribution of ion
temperature $T_{i}$ with the following assumptions.

When a neutron and a $X$ particle are produced by a nuclear reaction, momentum
conservation at the center of mass is

$m_{n}\vec{u}_{n}+m_{X}\vec{u}_{X}=0,$ (17)

where $\vec{u}$ is the velocity with respect to the center of mass. Energy
conservation requires

$Q+K=\frac{1}{2}m_{n}{u_{n}}^{2}+\frac{1}{2}m_{X}{u_{X}}^{2},$ (18)

where $Q$ is the mass difference before and after the reaction and $K$ is the
kinetic energy of reactants at the center of mass. If there are multiple $N$
particles, one needs the relative velocity of each of them. However, if $N$ is
greater than two, this is unknown. So, we apply following assumptions to
equation (17) and (18):

$\displaystyle
m_{n}\vec{u}_{n}+\sum^{N}_{i=1}(m_{i})\vec{u}_{\mathrm{CM}}=0,\qquad\ $
$\displaystyle
Q+K\approx\frac{1}{2}m_{n}{u_{n}}^{2}+\frac{1}{2}\sum^{N}_{i=1}(m_{i}){u_{\mathrm{CM}}}^{2},$
(19)

where $\vec{u}_{\mathrm{CM}}$ is the center of mass velocity of $N$ particles.
Under this condition, the mean energy of neutrons [55] is

$\left\langle E_{n}\right\rangle=\frac{1}{2}m_{n}\left\langle
V^{2}\right\rangle+\frac{\sum^{N}_{i=1}(m_{i})}{m_{n}+\sum^{N}_{i=1}(m_{i})}\left(Q+\left\langle
K\right\rangle\right),$ (20)

where $V$ is the total center of mass velocity and

$\displaystyle\left\langle
V^{2}\right\rangle=\frac{3k_{B}T_{i}}{m_{n}+\sum^{N}_{i=1}(m_{i})},\quad\quad\
$ $\displaystyle\left\langle
K\right\rangle=\frac{\int_{0}^{\infty}K^{2}dK\sigma(K)\mathrm{exp}(-K/k_{B}T_{i})}{\int_{0}^{\infty}KdK\sigma(K)\mathrm{exp}(-K/k_{B}T_{i})}.$
(21)

For the dominant nuclear reaction $p(\alpha,pn)\mathrm{{}^{3}He}$ within
$r\lesssim 10$, the mean neutron energy $\left\langle E_{n}\right\rangle$ and
the mean energy of Maxwell-Boltzmann distribution with $T_{i}$ are almost
identical. Therefore, we assume $T_{i}$ to be the neutron temperature.

Neutrons decay with a lifetime. So, the fraction of neutrons reaching a
certain radius $r$ decreases as the travel time increases. We calculate the
neutron travel time $\tau$ from the geodesic equation.

$\frac{du^{\beta}}{d\tau}=-\Gamma^{\beta}_{\mu\nu}u^{\mu}u^{\nu},$ (22)

where the Christoffel symbol is

$\Gamma^{\beta}_{\mu\nu}=\frac{1}{2}g^{\beta\alpha}(\partial_{\nu}g_{\alpha\mu}+\partial_{\mu}g_{\alpha\nu}-\partial_{\alpha}g_{\mu\nu}).$
(23)

Since self-similar ADAF solutions, used in our model, are based on Newtonian
gravity and Sgr A* has a relatively small spin parameter, we calculate the
motions of neutrons with Schwarzschild metric as a leading order. The
Schwarzschild metric for a black hole of mass M is

$\displaystyle ds^{2}$ $\displaystyle=$ $\displaystyle-c^{2}d\tau^{2}$ (24)
$\displaystyle=$
$\displaystyle-\left(1-\frac{R_{s}}{R}\right)c^{2}dt^{2}+\left(1-\frac{R_{s}}{R}\right)^{-1}dR^{2}$
$\displaystyle+R^{2}\left(d\theta^{2}+\mathrm{sin}^{2}\theta
d\phi^{2}\right).$

Self-similar ADAF solutions provide only radial velocity and angular frequency
profiles of the accretion flow without the vertical flow structure information
[29]. Therefore, we vertically average the number of neutrons. From the
assumption, the total neutron number at initial radius $r_{i}$ is

$\displaystyle N_{n,i}(r_{i})$ $\displaystyle=$ $\displaystyle\int
n_{n,i}\,dV\sim n_{n,i}(r_{i})\Delta V$ (25) $\displaystyle=$ $\displaystyle
4\pi H{R_{s}}^{2}n_{n,i}(r_{i})r_{i}\Delta r,$

where $2\pi$ is from the axial symmetry of the self-similar ADAF model.

We calculate neutron inflow using the geodesic of neutrons. The neutron
density at initial position $\vec{r}_{i}=(r_{i},\pi/2)$ with an initial
velocity on the local rest frame of the accretion flow
$\vec{v}_{i}=({v_{i}}^{(r)},{v_{i}}^{(\theta)},{v_{i}}^{(\phi)})=({v_{i}}\mathrm{cos}\theta^{\prime},{v_{i}}\mathrm{sin}\theta^{\prime}\mathrm{cos}\phi^{\prime},{v_{i}}\mathrm{sin}\theta^{\prime}\mathrm{sin}\phi^{\prime})$
is

$\displaystyle n_{n,i}(\vec{v}_{i},r_{i})$ $\displaystyle\sim$
$\displaystyle\dot{n}_{c}(r_{i})\frac{R_{s}\Delta
r}{\left|{v_{r}}(r_{i})\right|}\left(\frac{m_{n}}{2\pi
k_{B}T_{i}}\right)^{3/2}$ (26)
$\displaystyle\times{v_{i}}^{2}e^{-m{v_{i}}^{2}/2k_{B}T_{i}}\mathrm{sin}\theta^{\prime}\Delta\theta^{\prime}\Delta\phi^{\prime}\Delta
v_{i},$

where $\theta^{\prime}$ and $\phi^{\prime}$ are from the coordinate of the
local rest frame, and ${v_{r}}(r_{i})$ is the proper radial velocity of the
accretion flow at an initial position. We choose the ADAF radial velocity of
the equation (4) to be the proper radial velocity of an accretion flow at an
initial position.

The neutron number of the $k\mathrm{th}$ step at
$\vec{r}_{k}=(r_{k},\theta_{k})$ is

$N_{n,k}(\vec{v}_{i},\vec{r}_{k},r_{i})=N_{n,i}(\vec{v}_{i},r_{i})\exp\left[-\frac{\tau(\vec{v}_{i},\vec{r}_{k},r_{i})}{\tau_{n}}\right],$
(27)

where $\tau$ the neutron travel time from $r_{i}$ to $\vec{r}_{k+1}$, the
neutron life time $\tau_{n}$ is about 879.4 seconds [56].

Therefore, the total neutron number at the final position $\vec{r}_{f}$ is

$\Delta
N_{n}(\vec{r}_{f})=\sum_{r_{i}}\sum_{\vec{v}_{i}}N_{n,f}(\vec{v}_{i},\vec{r}_{f},r_{i})$
(28)

Fig. 1 shows that the neutron production reactions become efficient only at
$r\lesssim 10$. Thus, for our calculation, we set $1<r<10$ with $\Delta
r=5\times 10^{-3}$. We also set that $\Delta\theta^{\prime}=\pi/30$,
$\Delta\phi^{\prime}=2\pi/30$ and $\Delta v_{i}=3\sqrt{k_{B}T_{i}/m_{n}}/100$
with $0<\theta^{\prime}<\pi$, $0\leq\phi^{\prime}<2\pi$ and
$0<v<3\sqrt{k_{B}T_{i}/m_{n}}$, respectively.

We compare the escape fraction of our model with that of Aharonian & Sunyaev
[24]. For $T_{i}\approx 1.3\times 10^{11}\,\mathrm{K}$, our model shows that
about $18\%$ of neutrons at $R\sim(7-9)R_{s}$ escape to infinity when neutron
decay is not considered, which is almost same as the estimate of Aharonian &
Sunyaev. However, the difference between the two models will be larger much
closer to the black hole by general relativistic effect.

##  III Proton acceleration through the MPP

In this chapter, we present the accelerated proton energy spectrum from the
MPP in the Kerr space-time. The Kerr metric for a black hole of mass $M$ and
an angular momentum of $J$ is

$\displaystyle ds^{2}$ $\displaystyle=$ $\displaystyle-c^{2}d\tau^{2}$ (29)
$\displaystyle=$
$\displaystyle-\left(1-\frac{R_{s}R}{\Sigma}\right)c^{2}dt^{2}+\frac{\Sigma}{\Delta}dR^{2}+\Sigma
d\theta^{2}$
$\displaystyle+\left(R^{2}+a^{2}+\frac{R_{s}Ra^{2}}{\Sigma}\mathrm{sin}^{2}\theta\right)\mathrm{sin}^{2}\theta
d\phi^{2}$
$\displaystyle-\frac{2R_{s}Ra\mathrm{sin}^{2}\theta}{\Sigma}cdtd\phi$

in the Boyer–Lindquist coordinates, where $\Sigma$, $\Delta$ and $a$ are

$\displaystyle\Sigma=R^{2}+a^{2}\mathrm{cos}^{2}\theta,\qquad\quad$
$\displaystyle\Delta=R^{2}-R_{s}R+a^{2},\quad a=\frac{J}{Mc},$ (30)

respectively.

In the MPP model of Tursunov et al., protons that are produced by neutron
decay are accelerated through electric potential difference [19]. If protons
are produced by neutron decay in an accretion disk, it is challenging to
accelerate protons because plasma in the accretion disk can screen the induced
electric field induced by the rotation of a black hole and an external uniform
magnetic field. In such case, if the charged particle density of plasma is
lower than the Goldreich-Julian density, acceleration of charged particles can
operate [21, 57, 58]. However, the charged particle density of the accretion
disk of the Sgr A* from equation (6) is $n_{SgrA*}\sim
10^{6}-10^{5}\,\mathrm{cm}^{-3}$ at $r\lesssim 10$ and the Goldreich-Julian
density of the Sgr A* is $n_{GJ}\sim a_{*}B_{0}c^{2}/\left(4\pi eGM\right)\sim
10^{-2}a_{*}\left[B_{0}/\left(30\,\mathrm{G}\right)\right]\,\mathrm{cm}^{-3}$
[58], where $a_{*}=2a/R_{s}$ is a dimensionless spin parameter. So the proton
acceleration after neutron decay by the induced electric field is difficult in
the accretion flow of Sgr A* and does not operate inside the accretion flow.
We assume that there is no plasma in the polar region of the accretion flow
close to Sgr A* from estimate to accretion flow more likely due to the outflow
[59, 60, 61]. The electron-positron pair density is estimated to be
$n_{\pm}\approx 10^{-8}\,\mathrm{cm^{-3}}$ which is much lower than the
Goldreich-Julian density in the GRMHD simulations [62, 63] and will not screen
the electric field. Therefore, we set the conditions of the acceleration zone
to operate MPP are
$0<\theta<\theta_{cut,1}=\pi/2-\mathrm{tan}^{-1}\left(H/R\right)$,
$\theta_{cut,2}=\pi/2+\mathrm{tan}^{-1}\left(H/R\right)<\theta<\pi$ and
$1<r<5$. We assume that protons from neutrons decay are accelerated and escape
without any hindrance of the plasma in the acceleration zone. We express the
schematic view of our model in Fig. 2.

Figure 2: The schematic view of our model at the vicinity of a black hole is
described by cylindrical coordinates. Neutrons are produced by nuclear
reactions in the accretion disk (the region of grey color) and flow into the
acceleration zone (the region of beige color).

Tursunov et al. (2020) calculated the maximum efficiency
$\eta_{\mathrm{ultra}}$ of a proton that is produced in the vicinity of the
black hole by a neutron decay with Wald solution.

$\eta_{\mathrm{ultra}}=\eta_{\mathrm{PP}}+\frac{q}{m}A_{t},\quad\eta_{\mathrm{PP}}\simeq
0.21,$ (31)

where $q$ is a charge of the proton, $m$ is a mass of the neutron, and $A_{t}$
is a covariant electric potential. At this efficiency, the energy of the
accelerated proton is

$E_{\mathrm{p}}=1.7\times
10^{20}\,\mathrm{eV}\left(\frac{B}{10^{4}\,\mathrm{G}}\right)\left(\frac{M}{10^{9}M_{\odot}}\right)\left(\frac{a_{*}}{0.8}\right).$
(32)

In the Sgr A* case, this energy is $E_{p}\simeq 5\times 10^{15}\,\mathrm{eV}$.

###  III.1 Wald solution

Wald solution is the vacuum Maxwell equation solution when a Kerr black hole
is immersed in a uniform magnetic field. Although Wald solution is a vacuum
solution, for the condition that the density of plasma near black hole $n\ll
n_{GJ}$ and the large-scale magnetic field shares the axis-symmetry and
stationary of space-time geometry, Wald solution has been employed to
demonstrate the motion and the acceleration of charged particles in the
vicinity of a black hole to leading order in many studies [13, 14, 15, 16, 17,
18, 19, 20, 21, 22]. The electromagnetic vector field presented by the Wald
solution ($G=c=1$) is

$A_{\mu}=\frac{1}{2}B_{0}\left(\psi_{\mu}+\frac{2J}{M}\eta_{\mu}\right)-\frac{Q}{2M}\eta_{\mu},$
(33)

where $B_{0}$, $J$, and $Q$ are a uniform magnetic field, an angular momentum
of the black hole, and a charge of the black hole, and $\psi_{\mu}$ and
$\eta_{\mu}$ Killing vectors that respond to $\partial/\partial\phi$ and
$\partial/\partial t$, respectively. Wald solution, expressed by metric
tensor, is

$A_{\mu}=\frac{1}{2}B_{0}\left(g_{\mu\phi}+2ag_{\mu
t}\right)-\frac{Q}{2M}g_{\mu t}.$ (34)

When the charge $Q=0$, a contravariant electric vector field $A^{t}$ is non-
zero. Thus, the vector field raises the selective charge accretion in a very
short time in astrophysical black holes [16, 19]. When the black hole charge
$Q=2aMB_{0}$ which is a Wald charge, the black hole finishes the selective
accretion. Thus, we can write the vector potential as

$A_{\mu}=\frac{1}{2}B_{0}g_{\mu\phi}.$ (35)

From the short timescale of selective charge accretion of an astrophysical
black hole, a scenario that a black hole is induced with a Wald charge is
applied to describe the motion of charged particles. For gamma-ray burst cases
where a black hole has just been born, the acceleration of charged particles
is estimated from Wald solution with charge $Q=0$, and electrons escape to
infinity and protons fall into a black hole at particular direction [17, 20,
21, 22]. When a black hole has a Wald charge and decaying neutrons nearby,
proton escape scenario is preferred because it is highly likely that the black
hole will be charged with a positive Wald charge [13, 64, 65]. Thus, we adopt
the scenario that electrons fall into the black hole and protons escape to
infinity after neutrons decay. When protons escape from the vicinity of the
black hole to infinity, they have two components of energy, which are the
vertical escape energy $E_{z}$ and oscillatory (Larmor) energy in the
equatorial plane $E_{L}$.

$E^{2}_{\infty}=E^{2}_{z}+E^{2}_{L}.$ (36)

If a black hole is induced with a Wald charge, the vertical energy of charged
particles is maximized, and the charged particles escape to infinity in a
vertical direction. [19, 14].

###  III.2 The production rate of the accelerated protons

First, we assume that the Galactic center black hole, Sgr A*, is induced with
the Wald charge and protons just escape to infinity without falling into the
black hole with maximized vertical escape energy. Since the Kerr metric has
two Killing vectors $\psi_{\mu}$ and $\eta_{\mu}$, the proton energy and
angular momentum are conserved. The conserved total energy of the proton is

$E=-\left(mu_{t}\right)_{\mathrm{decay}}-qA_{t,\mathrm{decay}}=-\left(mu_{t}\right)_{\infty}-qA_{t,\infty},$
(37)

where $-\left(mu_{t}\right)_{\mathrm{decay}}$ and
$-\left(mu_{t}\right)_{\infty}$ are proton energies at a decay point and
infinity, respectively, and $-qA_{t}$ is electric potential energies at
corresponding locations. Since protons escape vertically, and we can ignore
electric field $A_{t,\infty}$ ($A_{t,\infty}\ll A_{t,\mathrm{decay}}$), the
proton energy at infinity is

$E_{\mathrm{p}}=-\left(mu_{t}\right)_{\infty}\simeq-\left(mu_{t}\right)_{\mathrm{decay}}-qA_{t,\mathrm{decay}}.$
(38)

Since the proton is accelerated in a vertical direction regardless of the
proton’s motion at the decay point and
$|-\left(mu_{t}\right)_{\mathrm{decay}}|\ll\left|-qA_{t,\mathrm{decay}}\right|$,
one can write the proton energy at infinity as

$E_{\mathrm{p}}\simeq-qA_{t,\mathrm{decay}}.$ (39)

Thus, when we adopt the equation (35) for $A_{t,\mathrm{decay}}$ ($c\neq G\neq
1$),

$E_{p}=-\frac{1}{2}qB_{0}cg_{t\phi}=\frac{qB_{0}c}{2}\frac{R_{s}ra\,\mathrm{sin}^{2}\theta}{r^{2}+a^{2}\mathrm{cos}^{2}\theta}.$
(40)

The proton production rate $Q_{MPP}\left(E_{p}\right)$ is now

$Q_{MPP}=\frac{d\dot{N}_{p}}{dE_{p}}\sim\frac{\Delta\dot{N}_{p}}{\Delta
E_{p}}=\frac{\Delta N_{n}}{\tau_{n}\Delta E_{p}},$ (41)

where $\Delta E_{p}$ is the total differential of the accelerated proton,

$\Delta E_{p}=\frac{\partial E_{p}}{\partial r}\Delta r+\frac{\partial
E_{p}}{\partial\theta}\Delta\theta.$ (42)

However, $\Delta N_{n}$ and $\Delta E_{p}$ depend on $r$ and $\theta$. Thus,
we calculate the total proton production rate after dividing the acceleration
zone with radius $r$ and calculating each production rate. For a given $r$,

$\displaystyle E_{p}$ $\displaystyle=$ $\displaystyle
E_{p,r}\left(\theta\right),\quad\Delta N_{n}=\Delta
N_{n,r}\left(\theta\right),$ (43) $\displaystyle\Delta
E_{p,r}\left(\theta\right)=\left.\frac{\partial
E_{p}}{\partial\theta}\right|_{r}\left(\theta\right)\Delta\theta.$

Proton production rate at $r\sim r+\Delta r$ is

$Q_{MPP,r}\left(E_{p,r}\right)\sim\frac{\Delta
N_{n,r}\left(\theta\right)}{\tau_{n}\Delta E_{p,r}\left(\theta\right)}.$ (44)

Therefore, the total production rate is

$Q_{MPP}\left(E_{p}\right)=\sum_{r}Q_{MPP,r}\left(E_{p}\right).$ (45)

We set $\Delta r=0.01$ and $\Delta\theta=\pi/100$.

We select the dimensionless spin parameter and uniform magnetic field strength
in the vicinity of the black hole as $a_{*}=0.5$ and
$B_{0}=30,\,100\,\mathrm{G}$ [19]. The spin parameter value corresponds to
orbiting spot models with variable emission [66]. The magnetic field strength
coincides with the emission model of Sgr A* that requires the ordered magnetic
field with $B_{0}=30-100\,\mathrm{G}$ and most accretion models in
equipartition state have the magnetic field with few hundred gauss [67]. For
example, self-similar solutions that are used in our model estimate the
magnetic field of a few hundred gauss [29], and RIAF based on Keplerian shell
model with radio emission proposed the magnetic field strength to about one
hundred gauss at $r\sim 1$ [68].

Figure 3: The production rates of protons that accelerated by MPP from
equation (45). The solid line and dashed line show the proton production rates
for $100\,\mathrm{G}$ and $30\,\mathrm{G}$.

From equation (45), we get the production rate of protons accelerated by MPP
as shown in Fig. 3. At the neighborhood of the polar axis ($\lesssim
10^{14}\,\mathrm{eV}$ for $B_{0}=100\,\mathrm{G}$ and $\lesssim
10^{13}\,\mathrm{eV}$ for $B_{0}=30\,\mathrm{G}$), $\Delta N_{n}$ and
$\left.\partial E_{p}/\partial\theta\right|_{r}$ increase similarly as
$\theta$ approaches $\theta_{cut}$. At the neighborhood of the $\theta_{cut}$
($\gtrsim 10^{14}\,\mathrm{eV}$ for $B_{0}=100\,\mathrm{G}$ and $\gtrsim
10^{13}\,\mathrm{eV}$ for $B_{0}=30\,\mathrm{G}$), $\Delta N_{n}$ increase
faster than $\left.\partial E_{p}/\partial\theta\right|_{r}$ as $\theta$
approaches $\theta_{cut}$. From those feature of the neutron number and the
partial derivative of the energy of protons, the production rate of protons
increases from about $10^{14}\,\mathrm{eV}$ for $B_{0}=100\,\mathrm{G}$ (about
$10^{13}\,\mathrm{eV}$ for $B_{0}=30\,\mathrm{G}$) to the peak energy
$E_{peak}\simeq 4.8\times 10^{14}\,\mathrm{eV}$ ($E_{peak}\simeq 1.5\times
10^{14}\,\mathrm{eV}$ for $B_{0}=30\,\mathrm{G}$). The production rate of
protons at $E_{peak}$ is $5.5\times 10^{18}\,\mathrm{eV^{-1}\,s^{-1}}$ for
$B_{0}=100\,\mathrm{G}$ and $1.8\times 10^{19}\,\mathrm{eV^{-1}\,s^{-1}}$ for
$B_{0}=30\,\mathrm{G}$. $E_{peak}$ is the energy of protons accelerated at
$r=5$ and $\theta=\theta_{cut}$. As the radius where the proton is accelerated
decreases, protons are accelerated to an energy higher than $E_{peak}$ on
$r<5$. Therefore production rate of protons steeply decreases from $E_{peak}$.

##  IV Emission from accelerated protons

###  IV.1 Synchrotron

The main radiation processes of high-energy charged particles are synchrotron,
inverse Compton scattering and bremsstrahlung.

Charged particles have chaotic motion when gravity and electromagnetic force
are strong. One can check which force more strongly affects charged particles
through the parameter

$\mathcal{B}=\frac{qGMB}{2mc^{4}}.$ (46)

When $\mathcal{B}<1$, charged particles fall freely into the black hole, while
$\mathcal{B}\sim 1$, the motion of charged particles is chaotic. For
$\mathcal{B}\gg 1$, the electromagnetic effect on charged particles is greater
than gravity, and the charged particles escape to the direction parallel to
the rotational axis of the black hole and magnetic field. For larger
$\mathcal{B}$, charged particles are accelerated faster and the pitch angle
goes to zero [19, 14, 15, 16, 69]. Since the pitch angle is close to zero,
synchrotron radiation of accelerated protons by MPP in the vicinity of Sgr A*
will be minimized. We can simply compare the synchrotron radiation and
luminosity of Sgr A*. Without gravitational redshift, the power spectrum of
synchrotron radiation of a charged particle is [70]

$\displaystyle P_{\nu}$ $\displaystyle=$
$\displaystyle\frac{\sqrt{3}q^{3}B}{mc^{2}}F(\nu/\nu_{c})\mathrm{sin}\psi,$
$\displaystyle F(x)$ $\displaystyle=$ $\displaystyle
x\int_{x}^{\infty}K_{5/3}(y)\,dy,$ (47)

where $\nu_{c}$ is the characteristic frequency of synchrotron radiation,

$\nu_{c}=\frac{3\gamma^{2}qB}{4\pi mc}\mathrm{sin}\psi.$ (48)

For $B=100\,\mathrm{G}$ and proton with $E=0.48\,\mathrm{PeV}$, the
characteristic frequency is $6.24\times 10^{18}\mathrm{sin}\psi\,\mathrm{Hz}$,
and the characteristic energy of the photon is
$258\mathrm{sin}\psi\,\mathrm{eV}$. If one assumes that protons escape the
vicinity of Sgr A* with velocity $v\sim c$ and parallel direction to the
uniform magnetic field and the rotational axis of Sgr A*, from equation (45)
and (IV.1), at the peak frequency $\sim 2\times 10^{15}\,\mathrm{Hz}$,
$\left(\nu L_{\nu}\right)_{peak}\sim 10^{26}\,\mathrm{erg}\,\mathrm{s^{-1}}$
on the scale of the Schwarzschild radius of Sgr A*. We estimated $(\nu
L_{\nu})_{peak}$ only on the scale of the Schwarzschild radius, taking into
account the decrease in the magnetic field with distance from Sgr A*. However,
the best-fitting power law from the mean NIR to X-ray spectra of the very
bright flare VB3 showed that $\nu L_{\nu}\sim 10^{35}\,\mathrm{erg\,s^{-1}}$
at $\nu\sim 10^{15}\,\mathrm{Hz}$ [71]. Also RIAF models show $\nu L_{\nu}\sim
10^{33}\,\mathrm{erg\,s^{-1}}$ at $\nu\sim 10^{15}\,\mathrm{Hz}$ [32, 72]. The
small pitch angle will further decrease the synchrotron emission, and
synchrotron radiation from accelerated proton does not affect the spectrum of
Sgr A*.

###  IV.2 Inverse Compton scattering and bremsstrahlung

The effect of inverse Compton scattering from the Compton $y$ factor. For
mono-energetic proton, the Compton $y$ factor is

$y=\frac{4}{3}\gamma^{2}\beta^{2}\times\mathrm{max}(\tau,\tau^{2}),\quad\tau=n_{p}\sigma_{T}\left(\frac{m_{e}}{m_{p}}\right)^{2}L,$
(49)

where $\tau$ is optical depth and $L$ is the length scale. On the scale of the
Schwarzschild radius with $B_{0}=100\,\mathrm{G}$, the number density of
accelerated protons

$\displaystyle n_{p}$ $\displaystyle\sim$
$\displaystyle\frac{1}{{R_{s}}^{3}}\cdot\frac{2R_{s}}{c}\cdot\int
Q_{MPP}(E_{p})\,dE_{p}$ (50) $\displaystyle\sim$
$\displaystyle\frac{2}{{R_{s}}^{2}c}\cdot
Q_{MPP}\left(E_{peak}\right)E_{peak}\sim 0.123\,\mathrm{cm^{-3}}$

Assuming $L=R_{s}$, $\tau\sim 2.9\times 10^{-20}$. Therefore,

$y\sim\frac{4}{3}\cdot(4.8\times 10^{5})^{2}\cdot 2.9\times 10^{-20}=8.9\times
10^{-9}.$ (51)

For high-energy photons like X-rays, the Klein-Nishina scattering cross-
section $\sigma_{KN}$ should be used, and $\sigma_{KN}$ will make the Compton
$y$ factor even smaller. With this small value of the Compton $y$ factor,
inverse Compton scattering of accelerated protons will not affect the spectrum
of Sgr A*.

The emission of photons by high-energy ions interact with electrons is called
the inverse bremsstrahlung [73]. The inverse bremsstrahlung against the
electrons in the accretion flow is very low for cross-section of inverse
bremsstrahlung [73] and the inverse bremsstrahlung neither will affect the
spectrum of Sgr A*.

###  IV.3 Proton-proton interaction

Many observations found that the central molecular zone (CMZ) in the Galactic
center (GC) produces very high energy (VHE) gamma-rays (which is called HESS
J1745-290 or VER J1745-290) [23, 74, 75, 76, 77, 78, 79, 80, 81]. Inverse
Compton scattering of high-energy electrons and p-p interaction of high energy
protons can produce the VHE gamma-rays. HESS collaboration argues that the p-p
interaction is more favorable than the inverse Compton scattering of high-
energy electrons, in the case of the diffuse VHE emission of the CMZ based on
the propagation scale of high-energy protons and electrons [23]. They also
argue that one or more multi-TeV accelerators are in the CMZ based on VHE
gamma-ray observation data and gas distribution of the CMZ.

If the accelerators inject high-energy particles at a continuous rate in the
GC, the radial distribution of the particles will be

$\frac{dn}{dE}=\frac{Q(E)}{4\pi D(E)r_{CMZ}},$ (52)

where $Q(E)$ is a injection rate of the particles and $D(E)$ is a diffusion
coefficient. Therefore, using the energy spectrum of the protons which are
accelerated from the MPP at the equation (45), one can calculate the MPP’s
contribution to the gamma-ray flux from the GC.

The production rate of gamma-rays and neutrinos from p-p interaction is

$\displaystyle\frac{d\dot{n}_{\gamma,\nu}}{dE_{\gamma,\nu}}=\qquad\qquad\qquad\qquad\qquad\qquad$
$\displaystyle
cn_{\mathrm{H}}\int_{E_{\gamma,\nu}}^{\infty}\sigma_{\mathrm{inel}}(E_{p})\frac{dn_{p}}{dE_{p}}F_{\gamma,\nu}\left(\frac{E_{\gamma,\nu}}{E_{p}},E_{p}\right)\frac{dE_{p}}{E_{p}},$
(53)

where $n_{\mathrm{H}}$ is the density of the hydrogen gas,
$\sigma_{\mathrm{inel}}(E_{p})$ the inelastic cross-section of the p-p
interaction and $F_{\gamma,\nu}$ functions of the energy spectra of the gamma-
rays and neutrinos [53]. Thus, the fluxes of the diffuse VHE gamma-rays and
neutrinos from the p-p interaction of protons which are accelerated by the MPP
in the vicinity of the Sgr A* is

$F_{\gamma,\nu}\left(E_{\gamma,\nu}\right)=\frac{1}{4\pi\left(8\,\mathrm{kpc}\right)^{2}}\int\frac{d\dot{n}_{\gamma,\nu}}{dE_{\gamma,\nu}}\left(E_{\gamma,\nu}\right)\,dV,$
(54)

where $8\,\mathrm{kpc}$ is the distance from Sgr A* to Earth [11]. If one
assumes that the scales of the CMZ are $-50\,\mathrm{pc}<z<50\,\mathrm{pc}$
and $r<200\,\mathrm{pc}$, the fluxes of the gamma-rays and neutrinos are

$\displaystyle F_{\gamma,\nu}$ $\displaystyle\sim$
$\displaystyle\frac{cn_{\mathrm{H}}}{16\pi^{2}(8\,\mathrm{kpc})^{2}}Y_{\gamma,\nu}\left(E_{\gamma,\nu}\right)$
(55)
$\displaystyle\times\int_{0}^{2\pi}\int_{-50\,\mathrm{pc}}^{50\,\mathrm{pc}}\int_{0}^{200\,\mathrm{pc}}\frac{1}{r}\,rdrdzd\theta$
$\displaystyle=$
$\displaystyle\frac{cn_{\mathrm{H}}}{25600\pi}Y_{\gamma,\nu}\left(E_{\gamma,\nu}\right),$

where

$\displaystyle
Y_{\gamma,\nu}\left(E_{\gamma,\nu}\right)=\qquad\qquad\qquad\qquad\qquad\qquad$
$\displaystyle\int_{E_{\gamma,\nu}}^{\infty}\sigma_{\mathrm{inel}}\left(E_{p}\right)\frac{Q\left(E_{p}\right)}{D\left(E_{p}\right)}F_{\gamma,\nu}\left(\frac{E_{\gamma,\nu}}{E_{p}},E_{p}\right)\frac{dE_{p}}{E_{p}}.$
(56)

We show in the Fig. 4 the fluxes of the gamma-rays and the neutrinos if the
gas in the CMZ has the typical density of $n_{\mathrm{H}}\sim
100\,\mathrm{cm}^{-3}$ [82] and diffusion coefficient $D\left(E_{p}\right)\sim
3\times{10}^{28}\left(E_{p}/\mathrm{GeV}\right)^{1/3}\,\mathrm{cm}^{2}\,\mathrm{s}^{-1}$
along with the flux of the gamma-ray data from HESS and MAGIC collaboration
[23, 80]. The flux ${E_{\gamma}}^{2}F_{\gamma}$ has a peak value $1.1\times
10^{-13}\,\mathrm{TeV\,cm^{-2}\,s^{-1}}$ at $61\,\mathrm{TeV}$ for
$B_{0}=100\,\mathrm{G}$ and $4.3\times
10^{-14}\,\mathrm{TeV\,cm^{-2}\,s^{-1}}$ at $19\,\mathrm{TeV}$ for
$B_{0}=30\,\mathrm{G}$.

Figure 4: Observation data of diffuse VHE gamma-ray from CMZ (black open
circles and from HESS with the pac-man region [23] and blue crosses and from
MAGIC [80]), and the fluxes of gamma-rays and neutrinos from protons
accelerated by the MPP in Sgr A*. Black lines and red lines represent fluxes
of gamma-ray and neutrinos respectively. Solid lines and dashed lines show
gamma-ray and neutrino fluxes in the cases of the magnetic field strength of
the vicinity of the Sgr A* $B_{0}=100\,\mathrm{G}$ and $B_{0}=30\,\mathrm{G}$,
respectively.

From the information on the gas density at 0.1° circular region (inner $\sim
10\,\mathrm{pc}$ region), one can deduce the contribution of the MPP on HESS
J1745-290. At a few parsec scales around Sgr A*, the gas density shows a
distribution of $\sim 10^{3}\,\mathrm{cm}^{-3}$ on average, and there is a
dense region at $1-3\,\mathrm{pc}$ [83, 84]. The gas density averaged over
solid angle is about $8\times 10^{4}\,\mathrm{cm}^{-3}$ at about
$1-3\,\mathrm{pc}$ and about $2\times 10^{3}\,\mathrm{cm}^{-3}$ at about
$0-1\,\mathrm{pc}$ and $3-10\,\mathrm{pc}$ [85]. Using the gas density and the
equation (54), we determine the gamma-ray flux from accelerated protons by the
MPP, ${E_{\gamma}}^{2}F_{\gamma}$. It has peak values $4.4\times
10^{-14}\,\mathrm{TeV\,cm^{-2}\,s^{-1}}$ at $61\,\mathrm{TeV}$ for
$B_{0}=100\,\mathrm{G}$ and $1.8\times
10^{-14}\,\mathrm{TeV\,cm^{-2}\,s^{-1}}$ at $19\,\mathrm{TeV}$ for
$B_{0}=30\,\mathrm{G}$ (Fig. 5).

Figure 5: Observation data of HESS J1745-290 (black open circles from HESS
[23], blue crosses from MAGIC [80] and red rectangles from VERITAS [81]), and
the fluxes of gamma rays and neutrinos from protons which are accelerated by
the MPP in Sgr A*.

Fig. 4 and Fig. 5 show that the flux from MPP can have a significant effect on
the gamma-ray flux of CMZ and HESS J1745-290 at $E_{\gamma}\gtrsim
10\,\mathrm{TeV}$. The flux from MPP in the case of $100\,\mathrm{G}$
($30\,\mathrm{G}$) is about $3.5\%$ ($2.3\%$) of the observed gamma-ray flux
of CMZ at about $10\,\mathrm{TeV}$. The flux from MPP in the case of
$100\,\mathrm{G}$ ($30\,\mathrm{G}$) is about $1.0-2.4\%$ ($0.7-1.6\%$) of
observed gamma-ray flux of HESS J1745-290 at about $10\,\mathrm{TeV}$.

The flux from MPP surpasses the broken power-law based on gamma-ray flux of
CMZ from MAGIC at about $63\,\mathrm{TeV}$ in the case of $100\,\mathrm{G}$
and surpasses the broken power-law (blue dot-dashed line) at about
$95\,\mathrm{TeV}$ in the case of $30\,\mathrm{G}$. The flux from MPP is lower
than the power-law (black dot-dashed line) based on gamma-ray flux of CMZ from
HESS at every energy region. The flux from MPP surpasses the broken power-law
based on gamma-ray flux of HESS J1745-290 from HESS (black dot-dashed line),
MAGIC (blue dot-dashed line) and VERITAS (red dot-dashed line) data at about
$49\,\mathrm{TeV}$ in the case of $100\,\mathrm{G}$ and surpasses the broken
power-law at about $67\,\mathrm{TeV}$ in the case of $30\,\mathrm{G}$. These
results show that detailed future observation of gamma-ray flux of CMZ and
HESS J1745-290 at $\lesssim 100\,\mathrm{TeV}$ may provide the existence of
MPP or limit of parameters for MPP.

##  V Accelerated proton cosmic rays flux

In the range of $E\gtrsim 10^{10}\,\mathrm{eV}$, cosmic rays are not affected
by solar modulation and the cosmic ray spectrum shows a power-law spectrum
with knee structures at few $10^{15}\,\mathrm{eV}$ and at
$10^{17}-10^{18}\,\mathrm{eV}$. Random Galactic magnetic field effectively
confines cosmic rays in the Galaxy and makes an anisotropic flux depending on
the location of the source to an isotropic flux by scattering. The diffusion
approximates such transport of cosmic rays. The mean magnetic field strength
of the Galaxy is about $3\,\mathrm{\mu G}$ and the scale of the random
magnetic field is about $100\,\mathrm{pc}$ [33]. So one can use the diffusion
equation to estimate the transport of the accelerated protons from MPP at Sgr
A* to Earth if the gyroradius of the proton is less than $100\,\mathrm{pc}$.
Gyroradius is given by

$r_{g}=\frac{p}{|q|B}=\frac{E}{|q|Bc}\simeq
0.3\left(\frac{E}{10^{15}\,\mathrm{eV}}\right)\,\mathrm{pc}.$ (57)

The peak energy of the accelerated proton from Sgr A* in our model is about
$0.5\times 10^{15}\,\mathrm{eV}$ for $100\,\mathrm{G}$ magnetic field around
Sgr A*, much greater than the proton mass of $\sim 10^{9}\,\mathrm{eV}$, and
one can use $p=E/c$ in the equation (57). The gyroradius of the proton with
$0.5\times 10^{15}\,\mathrm{eV}$ is about $0.1\,\mathrm{pc}$, and diffusion
approximation is valid.

When the number of cosmic rays in the Galaxy is stationary and there are no
energy loss and reacceleration, the diffusion equation is

$\bm{\bigtriangledown}\cdot\bm{j}=q(\bm{x}),\quad
j_{i}=-D_{ij}(\bm{x})\triangledown_{j}n,$ (58)

where $q(\bm{x})$ is the time independent source term, $\bm{j}$ the cosmic ray
current and $D_{ij}$ the diffusive coefficient tensor [86, 87]. We assume that
the diffusive coefficient tensor only depends on the energy of the cosmic ray
to simplify the equation (58). The diffusion coefficient can be scaled by the
cosmic ray rigidity.

$D=D_{0}\beta R^{\delta},\quad\beta=\frac{v}{c},\quad R=\frac{pc}{ze},$ (59)

where $R$, $z$, and $e$ are rigidity, atomic number, and elementary charge,
respectively. Most models estimate $\delta=1/3$ and this is consistent with
the Kolmogorov spectrum [33, 88]. Thus, we use

$D_{K}=3\times
10^{28}\beta\left(R/\mathrm{GV}\right)^{1/3}\,\mathrm{cm^{2}s^{-1}},$ (60)

where $\mathrm{GV}$ is gigavolt. Sgr A* is effectively a delta function source
in the Galaxy and equation (58) has the same form as the central force.
Therefore,

$\frac{dn}{dE_{p}}=\frac{Q_{MPP}}{4\pi Dr_{s}},\quad
q(\bm{x})=Q_{MPP}\delta(\bm{x}),$ (61)

where $r_{s}$ is the distance from the source to Earth. The main structure of
the Galaxy is a disk-like, not spherical. Ref. [89] use equation (58) with to
obtain top-down boundaries with diffusive volume. The particle distribution
model:

$\frac{dn}{dE_{p}}=\frac{Q_{MPP}}{4\pi Dr_{s}}\times
2\sqrt{\frac{r_{s}}{L}}e^{-\pi r_{s}/2L},$ (62)

where $L$ is the height of the Galactic halo (diffusive volume). Equation (62)
does not consider the side boundary of the Galaxy. However, the equation fits
well in $L\lesssim 10\,\mathrm{kpc}$ with the model that considers all
boundaries of the Galaxy. Analyse of various cosmic rays observed on Earth,
Refs. [88, 90, 91] estimate $L=4\,\mathrm{kpc}$, $L\simeq 6\,\mathrm{kpc}$ and
$L=10\,\mathrm{kpc}$, respectively. These results enable that we just use the
equation (62). We use $L=4\,\mathrm{kpc}$ with equation (62) from Ptuskin
(2006) that presented $L=4\,\mathrm{kpc}$ with static diffusion model without
the Galactic wind [33]. We assume the diffused protons flow to be isotropic,
and the accelerated proton flux to Earth from Sgr A* is

$F_{MPP}(E_{p})=\frac{Q_{MPP}c}{16\pi Dr_{s}}\times
2\sqrt{\frac{r_{s}}{L}}e^{-\pi r_{s}/2L},$ (63)

where $r_{s}$ is the distance from Sgr A* to Earth, chosen to be
$8\,\mathrm{kpc}$ [11].

Fig. 6 shows that proton cosmic ray flux from JACEE (black, open circle) [92]
and RUNJOB (black, filled circle) [93] balloon experiment data, KASCADE (blue
rectangle and cross) [94] experiment data and the accelerated proton flux from
Sgr A* (solid lines) estimated by equation (45) and (63). Compared to JACEE
data, the accelerated proton flux is $0.2\%$ (${E_{p}}^{2}F_{MPP}=2.9\times
10^{-3}\,\mathrm{m^{-2}\,s^{-1}\,sr^{-1}\,GeV}$) in the case of
$B_{0}=100\,\mathrm{G}$ (black solid line) and the accelerated proton flux is
$0.6\%$ (${E_{p}}^{2}F_{MPP}=8.1\times
10^{-3}\,\mathrm{m^{-2}\,s^{-1}\,sr^{-1}\,GeV}$) in the case of
$B_{0}=30\,\mathrm{G}$ (blue solid line) at $0.25\,\mathrm{PeV}$. The
accelerated proton flux is about 1.6%, 2.9% and 4.1% of cosmic ray proton flux
with KASCADE EPOS1.99 (blue, open rectangle), QGSJET01 (black, filled
rectangle) and QGSJETII (orange, open triangle), respectively, at
$E_{p}=1.1\,\mathrm{PeV}$ (${E_{p}}^{2}F_{MPP}=9.1\times
10^{-2}\,\mathrm{m^{-2}\,s^{-1}\,sr^{-1}\,GeV}$).

Figure 6: The accelerated proton flux (black solid and blue solid line) from
Sgr A* estimated by equation (45) and (63) and proton cosmic ray flux from
JACEE (black, open circle) [92] and RUNJOB (black, filled circle) [93] balloon
experiment and KASCADE experiment with QGSJET01 (blue, filled rectangle),
QGSJETII (blue, open triangle) and EPOS1.99 (blue, cross) [94].

##  VI Conclusion and discussion

Magnetic Penrose process (MPP) considers the electromagnetic interaction,
unlike the normal Penrose process. These feature not only removes the limit of
the normal Penrose process but can also accelerate charged particle to high-
energy. We calculate the production rate of the high-energy protons
accelerated by MPP at Sgr A*. We also estimate the flux of gamma-rays and the
flux of the accelerated protons diffused from Sgr A* to Earth.

We model the accretion flow onto Sgr A* with the simple ADAF model that is hot
enough to produce neutrons. We then estimate how many neutrons are produced
within the ADAF and transported into the accelerated zone. Neutrons decay into
protons, that are subsequently accelerated along with rotational axis of the
black hole in MPP by electric potential difference described by Wald solution.
The peak production rate of high-energy accelerated protons is $5.5\times
10^{18}\,\mathrm{eV^{-1}\,s^{-1}}$ at $0.48\,\mathrm{TeV}$ in the case of
uniform magnetic field strength around Sgr A* of $B_{0}=100\,\mathrm{G}$ and
$1.8\times 10^{19}\,\mathrm{eV^{-1}\,s^{-1}}$ at $0.15\,\mathrm{TeV}$ in the
case of $B_{0}=30\,\mathrm{G}$.

Radiation fluxes of synchrotron, inverse Compton and bremsstrahlung of the
high-energy protons do not affect the current radiation flux of Sgr A*. On the
other hand, the flux of VHE gamma-rays from p-p interaction of the high-energy
protons against Galactic center hydrogen significantly affects the flux of VHE
gamma-rays of CMZ and HESS J1745-290 at $\geq 10\,\mathrm{TeV}$. The peak flux
${E_{\gamma}}^{2}F_{\gamma}$ from MPP on CMZ scale are $1.3\times
10^{-13}\,\mathrm{TeV\,cm^{-2}\,s^{-1}}$ at $61\,\mathrm{TeV}$ for
$B_{0}=100\,\mathrm{G}$ and $4.2\times
10^{-14}\,\mathrm{TeV\,cm^{-2}\,s^{-1}}$ at $19\,\mathrm{TeV}$ for
$B_{0}=30\,\mathrm{G}$. The peak flux ${E_{\gamma}}^{2}F_{\gamma}$ from MPP on
HESS J1745-290 scale are $4.4\times 10^{-14}\,\mathrm{TeV\,cm^{-2}\,s^{-1}}$
at $61\,\mathrm{TeV}$ for $B_{0}=100\,\mathrm{G}$ and $1.8\times
10^{-14}\,\mathrm{TeV\,cm^{-2}\,s^{-1}}$ at $19\,\mathrm{TeV}$ for
$B_{0}=30\,\mathrm{G}$. At $\geq 50-100\,\mathrm{TeV}$, The flux
${E_{\gamma}}^{2}F_{\gamma}$ from MPP surpasses the expected flux from the
broken power-law fit based on observed data. This result implies that precise
future observation at $\leq 100\,\mathrm{TeV}$ will provide the existence of
MPP or a limit of parameters of MPP.

The gyroradius of high-energy protons is much smaller than the scale of the
Galaxy. We estimate flux at Earth with the diffusion equation that is
considered for stationary high-energy proton production. MPP proton flux is
about $1.6-4.1\%$ of cosmic ray proton flux from KASCADE experiment at about
$1\,\mathrm{PeV}$.

The essential conditions for our model to operate are whether the electric
field from Wald solution is not screened, the ability to produce neutrons and
the timescale of the neutron inflow. We can ask if the same process can
operate in other SMBHs, such as M87. Equation (3) is the threshold magnetic
field strength to produce the electron-positron pair cascade under the force-
free condition. The magnetic field strength in the vicinity of the black hole
in M87 is $1-30\,\mathrm{G}$ [95] which is comparable to the threshold
magnetic field strength. Previous GRMHD simulations studies of the pair
production in M87 also present that the pair density at the jet spine in the
vicinity of the black hole is much higher than the Goldreich-Julian density
[63, 96]. This pair density makes our model difficult to operate in M87.

The neutron production rate scales with the mass accretion rate and the mass
of the black hole. From the self-similar ADAF solutions, the physical gas
density of the accretion flow is $\rho\propto\dot{m}m^{-1}$. The reaction rate
of neutron production nuclear reactions is
$R_{reaction}\propto{\dot{m}}^{2}m^{-2}$ by equation (11). Since the total
number of produced neutrons per unit time $\dot{N}$ is proportional to the
volume, $\dot{N}\propto{\dot{m}}^{2}m$, which shows that heavy SMBH is more
efficient in neutron production for the same value of $\dot{m}$.

However, heavy SMBHs like M87 are critically disadvantaged due to the longer
timescale of the neutron transport from the accretion flow to the acceleration
zone. Assuming that the transport distance is approximately of the order of
Schwarzschild radius and the speed of the neutron is approximately the speed
of light, the number of neutrons that arrive into the acceleration zone per
unit time is
$\dot{N}_{in}\propto{\dot{m}}^{2}m\exp(-2GM_{\odot}m/c^{3}\tau_{n})$, where
$\tau_{n}$ is the mean lifetime of the neutron.

Since this exponential factor will be dominant for large $m$, even when the
acceleration zone and the accretion flow are favorably configured, the
accretion flow can supply neutrons only when $m$ is not much greater than
$9\times 10^{7}$. Therefore, the acceleration model considered in this work
will operate only in SMBHs with mass not much greater than $\sim
10^{8}\,M_{\odot}$, and M87 is much heavier than this critical mass.

###### Acknowledgements.

The authors thank the anonymous referee for many useful and insightful
comments that have improved this paper. This work was supported by the Basic
Science Research Program through the National Research Foundation of Korea
(NRF No-2019R1I1A3A02062242) and the National Research Foundation of Korea
(NRF No-2018R1A6A1A06024970) funded by the Ministry of Education, Korea.

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