## Low-luminosity AGNs

^{1}
PN Lebedev Physical Institute,
Leninsky Prospect 53,
119991
Moscow,
Russia

e-mail: istomin@lpi.ru

^{2}
LUTH, Observatoire de Paris, CNRS, Université Paris Diderot,
Place J. Janssen,
92195
Meudon,
France

e-mail: helene.sol@obspm.fr

Received:
1
July
2010

Accepted:
9
November
2010

*Context.* We propose that low-luminosity AGNs (LLAGNs), or some of them,
are sources extracting their energy from the black hole rotation by the Blandford-Znajek
mechanism.

*Aims.* It is shown that almost all energy of the black hole rotation is
converted to relativistic protons in a jet. Owing to the high magnetic-field magnitude
near the black hole, required for the Blandford-Znajek mechanism, electrons are not
strongly accelerated because of their high synchrotron losses. Conversely, protons gain
energies on the order of
(10^{4} − 10^{5})*m*_{p}*c*^{2}
when crossing the light cylinder surface. Protons are also accelerated in a disk by
2D turbulent motion of the disk matter.

*Methods.* We calculate the luminosity of the synchrotron radiation by
fast protons in the disk, the frequencies of this radiation being in the infrared band,
and the luminosities corresponding to LLAGNs. We measure the very high energy (VHE)
radiation luminosities from the disk and the jet, finding that VHE radiation is produced
by collisions of accelerated protons with surrounding matter.

*Results.* We predict a correlation between the infrared luminosity
*L*_{IR} and the VHE luminosity *L*_{VHE}
of the disk, ,
where *M* is the mass of a black hole. Two low-luminosity sources Sgr A*
and M 87, for which luminosities *L*_{IR} and
*L*_{VHE} are known, appear to follow this scheme.

*Conclusions.* The discovery of new bright VHE sources from LLAGNs could
confirm our hypotheses that they are energy sources powered by the Blandford-Znajek
mechanism.

Key words: galaxies: active / acceleration of particles / accretion, accretion disks / gamma rays: galaxies

*© ESO, 2011*

## 1. Introduction

The common point of view about active galactic nuclei (AGN) is that they are massive black
holes placed in the galactic center surrounded by an accretion disk. Energy-release
mechanisms that have been proposed up to now are the transformation of the gravitation
energy of accreting matter into heat and radiation, and the extraction of energy from the
black hole rotation. The latter is known as the Blandford-Znajek mechanism (1977). The former mechanism gives an AGN luminosity
*L* = *η*_{r}*Ṁ**c*^{2},
which is proportional to the mass rate accretion *Ṁ*,
*η*_{r} being the radiative efficiency (the standard value of which
is *η*_{r} ≃ 0.1 for bright AGNs). For the Eddington luminosity
*L*_{Edd} = 4*πGM**m*_{p}*c*^{2}/*σ*_{T} = 1.3 × 10^{38}(*M*/*M*_{⊙}) erg/s,
the accretion rate is ,
where *G* is the gravitational constant, *m*_{p} is
the proton mass, *c* is the speed of light, and
*σ*_{T} is the Thomson cross-section. The value *M*
is the black hole mass, and *M*_{⊙} is the solar mass. Accretion of
the interstellar gas into a black hole is defined by its gravitational radius
, at which the
gravitational energy of a particle in the field of a black hole equals the particle thermal
energy. The value of *c*_{s} is the gas sound velocity. In this case,
the accretion rate is equal to , where *ρ* is the
gas density. The gravitational radius *R*_{G} is much larger than the
radius of the horizon of a Schwarzchild black hole
*r*_{H} = 2*MG*/*c*^{2},
*R*_{G} = *r*_{H}(*c*/*c*_{s})^{2}.
Nevertheless, the accretion rate of interstellar gas into a black hole is small because of
the low gas density, ,
where *T* is the gas temperature and *n* is the gas
concentration
*n* = *ρ*/*m*_{p}.
If the accretion is fed by a dense molecular cloud or a disrupted star, then the accretion
flow has the form of an accretion disk, which possesses a large angular momentum. The
accretion rate *Ṁ* for disks can be as high as the Eddington rate
*Ṁ*_{Edd} and depends on the disk structure. The bolometric
luminosity of AGN *L*_{b} is often less than the Eddington
luminosity *L*_{Edd} by a factor 10^{-3} − 10^{-1}.
A noticeable fraction of nearby galaxies do not contain bright AGNs, but instead contain
low-luminosity AGNs (LLAGNs) for which
*L*_{b} ≈ (10^{-5} − 10^{-3})*L*_{Edd}.
The center of our Galaxy Sgr A* is dimmer, its bolometric luminosity being only
*L*_{b} ≈ 10^{36} erg/s ≈ 2 × 10^{-9} *L*_{Edd}(*M* ≈ 3.6 × 10^{6} *M*_{⊙}).
The reason for this low luminosity is the low value of the radiative
efficiency *η*_{r} or low value of the accretion mass
rate *Ṁ*. Low radiative efficiency is inherent to the advection-dominated
accretion flow (ADAF) (Narayan 1994). At low
accretion mass rates of *Ṁ* ≪ *Ṁ*_{Edd}, the radiative
cooling of the disk becomes inefficient. These accretion flows are generally called
radiatively inefficient accretion flows (RIAFs) (Narayan 2002). This class of disk flow also includes ADAFs. The accretion mechanism of the
energy release is ineffective for LLAGNs.

We discuss the situation when the energy release due to accretion into the black hole of an AGN is small, and the energy activity is determined generally by the Blandford-Znajek mechanism. As we see below, the main energy output from a rotating black hole is in a relativistic jet. Thus, we mean LLAGNs with a jet. They are low-luminosity Seyfert galaxies and LINERs, the central source of which is a low luminosity AGN. A LINER is the low-ionization nuclear emission-line region at the center of a bright galaxy. LINERs are characterized by collisionally excited lines of neutral and singly ionized gas (Maoz 2007).

We first describe in Sect. 2 how the Blandford- Znajek mechanism operates. We describe
configurations of the magnetic field and electric currents in both a disk and the black hole
magnetosphere above a disk. We also show how the energy and angular momentum are transmitted
from the black hole rotation to a jet. In Sect. 3, we calculate the synchrotron radiation of
fast protons in a disk. The production of high energy *γ*-rays by
relativistic protons is presented in Sect. 4. The observational consequences of the
suggested scheme are described in the last section.

## 2. Blandford-Znajek mechanism

The Blandford-Znajek mechanism assumes that energy and angular momentum are extracted from
the black hole rotation. The rotation energy *E*_{rot} stored in the
black hole rotation is large, for slow rotation being
. Here we introduce the
dimensionless parameter of rotation,
*a* = *Jc*/*M*^{2}*G*,
where *J* is the angular momentum of the black hole. For a black hole,
*a* < 1. The value of Ω_{H} is the angular
velocity of black hole rotation,
Ω_{H} = *ac*/2*r*_{H}.
The extraction of the rotation energy is possible if there exists a poloidal magnetic
field *B* near the black hole horizon. The black hole in this case works as
a dynamo machine, creating the voltage *U*,
*U* = Ω_{H}*f*_{H}/2*πc*
(Landau & Lifshitz 1984; Thorne et al. 1986), where *f*_{H} is the flux
of the magnetic field reaching the horizon, . The voltage
generates the electric current
*I* = *U*/(*R* + *R*_{H}),
which on one side is closed at the black-hole horizon surface of resistivity
*R*_{H} = 4*π*/*c* ≈ 377 ohms
(Thorne et al. 1986). The resistivity of the outer
part of the system is *R*. Thus, the extracted power is
. The
power *L* reaches its maximum value *L*_{m} when
*R* = *R*_{H},
. This value is proportional to the
square of the black hole mass,
*L*_{m} ∝ *M*^{2}, and for large enough
magnetic fields can exceed the Eddington luminosity,
*B* > *B*_{Edd} = 5.5 × 10^{9}*a*^{-1}(*M*/*M*_{⊙})^{−1/2} Gauss.
For high masses of AGN black holes, the value of the magnetic field
*B*_{Edd} is quite moderate,
*B*_{Edd} ≃ 10^{5} Gauss (*a* ≃ 1).

We note that to ensure in general that the gravitation energy release is extremely
efficient, there must be a high mass accretion rate *Ṁ* on to the massive
black hole and a high radiative efficiency *η*_{r}, whereas the
Blandford-Znajek mechanism provides such efficiencies when there is black hole rotation,
Ω_{H} ≠ 0, and a strong enough magnetic field *B* near the black
hole horizon. Formally, the Blandford-Znajek mechanism does not need accretion. In addition,
the accumulation of a strong magnetic field requires some accretion process, but accretion
is not a source of energy.

### 2.1. Magnetic field and electric current configurations

For disk accretion, the magnetic field inside the disk and nearby must have no component
that is perpendicular to the disk (Istomin & Sol 2009). In the axisymmetric stationary electromagnetic field, a charged particle
conserves the generalized angular momentum,
*ρ**p*_{φ} + *qf*/*c* = const*.*,
where *ρ* is the cylindrical distance from the center, *f*
is the flux of the poloidal magnetic field, and *q* is the charge of a
particle. However, two terms in this relation are practically not commensurable in the
case of disk accretion. The first one is proportional to the frequency of rotation of a
particle in the disk
*v*_{φ}/*ρ*,
the second one is proportional to the cyclotron frequency,
*ω*_{c} = *q**B*_{z}/*mc*,
of the particle rotation in the perpendicular magnetic field
*B*_{z}. A charged particle then can move in
the radial direction if it is not magnetized,
*ω*_{c} ≃ *v*_{φ}/*ρ*,
i.e. in a practically zero perpendicular magnetic field. The radial
*B*_{ρ} and the azimuthal
*B*_{φ} components can be arbitrary. We see
that the accretion of matter provides only the radial component of the poloidal magnetic
field towards the black hole vicinity. And the magnetic field in the expression for the
Blandford-Znajek luminosity is the radial magnetic field
*B* = *B*_{ρ} produced by the
conducting matter of the accretion disk. For there to be a zero component of the magnetic
field perpendicular to the disk, an electric current
*I*_{ρ} must flow through the disk, which is
just the current *I* = −*I*_{ρ}
generated by the voltage *U*. The accretion disk is not only the origin of
the flux of the matter on to the black hole, but also conducts the electric current. This
current then flows onto the black hole horizon and closes through a jet in outer space.
Owing to the spiral motion of charged particles in the disk, there exist not only a radial
current *j*_{ρ} but also an azimuthal current
*j*_{φ}. Only the azimuthal current creates the
radial magnetic field. The ratio of the radial current to the azimuthal one
*α*_{j} = *j*_{ρ}/*j*_{φ}
can be found by considering that from one side the current *I* is produced
by the black hole rotation,
*I* = *U*/(*R* + *R*_{H}) = (*a*/16*π*)*B*_{ρ}*r*_{H}*c* [ *R*_{H}/(*R* + *R*_{H}) ] ,
and from another side the Maxwell equation determines the radial magnetic field through
the azimuthal current,
*I* = *α*_{j}*r*_{H}*c**B*_{ρ}.
We obtain
*α*_{j} = (*a*/16*π*) [ *R*_{H}/(*R* + *R*_{H}) ] ≪ 1,
which shows that the toroidal magnetic
field *B*_{φ} is much less than the radial
magnetic field
*B*_{ρ}, *B*_{φ} = *α*_{j}*B*_{ρ}.

### 2.2. Jet power

The radial magnetic field at the black hole horizon allows the transfer of energy and
angular momentum from the black hole rotation to particles of the black hole magnetosphere
and to particles leaving the accretion disk. The lines of the radial magnetic field begin
to rotate with the angular velocity
Ω_{F} < Ω_{H}, Ω_{F} = Ω_{H}*R*/(*R* + *R*_{H})
(Thorne et al. 1986). For the optimal condition
*R* = *R*_{H},
Ω_{F} = Ω_{H}/2 (Blandford & Znajek 1977). Forced by the centrifugal acceleration towards
the light cylinder surface
(*r*_{L} = *c*/Ω_{F} = 2*a*^{-1}*r*_{H}*R*/(*R* + *R*_{H})),
particles achieve azimuthal velocities, which are close to the speed of light, and
significant energies. The main energy is in protons because of their low synchrotron
losses in the strong magnetic field. The Lorentz factor *γ* of protons on
the light cylinder surface is
*γ* = (*ω*_{cL}/Ω_{F})^{1/2}
(Istomin & Sol 2009), where
Ω_{cL} = *e**B*_{L}/*m*_{p}*c*
is the non – relativistic proton cyclotron frequency in the magnetic field
*B*_{L} on the light cylinder surface. Taking into account that
the radial magnetic field falls as *ρ*^{-1}, we obtain
*γ* = (*ω*_{cH}*r*_{H}/*c*)^{1/2},
where *ω*_{cH} is the proton cyclotron frequency near a black hole.
Almost all particle energy is in the azimuthal motion,
*p*_{φ} ≃ *m*_{p}*γc*,
and only a small part is in the radial one,
*p*_{ρ} ≃ *m*_{p}*γ*^{1/2}*c*
(Istomin & Sol 2009). Thus, rotating
energetic protons slowly flow outside the light cylinder surface,
*v*_{ρ} ≃ *c**γ*^{−1/2},
forming the relativistic jet. The energy density of particles on the light cylinder is
equal to the density of the electromagnetic energy (Istomin 2010). We can then calculate the jet luminosity
. The
total Blandford-Znajek luminosity *L* must, of course, be higher than the
jet luminosity *L*_{J}. This implies the definite condition for the
magnetic field stress near the black hole
(1)or for
*R* = *R*_{H}(2)For AGN black-hole
masses of *M* ≃ 10^{8} *M*_{⊙}, this
condition (*B* > 3 × 10^{3} Gauss) is not
onerous and AGN with such moderate magnetic fields can produce a relativistic jet. Less
massive black holes must have higher magnetic fields. At the center of our Galaxy, for
example, there must be *B* ≈ 10^{5} Gauss. However, a magnetic
field of the order of 10^{11} Gauss for micro-quasars seems problematic when
producing relativistic jets. However, Karitskaya et al. (2009) measured the disk magnetic field to be 600 Gauss at a distance
2 × 10^{5}*r*_{H} in Cygnus X-1. We see in Eqs. (1)
and (2) a strong dependence of the magnitude of the magnetic field near the black
hole *B* on the black hole rotation,
*B* ∝ *a*^{-8}. This increases the value of
magnetic fields for slowly rotating black holes. We note, however, that the rotation
cannot be too slow: the light cylinder surface must be inside the jet radius for producing
a relativistic jet,
*r*_{J} > *r*_{L}, *a* > 4*r*_{H}/*r*_{J}.

### 2.3. The resistivity R

The effective work done by the Blandford-Znajek mechanism depends on the value of
resistivity *R*. The maximum output is for
*R* = *R*_{H}. However, the real resistivity of
the system can differ from this value. We now estimate the resistivity
*R*_{c} of the current loop created by the unipolar inductor
voltage *U*. The current resistivity, of course, is determined by the
electron motion. We assume the conducting system to be a box with the
cross-section *S* and the length *l* along the direction
of the electric current. The resistivity of this system is then
*R*_{c} = *l*/*Sσ*,
where *σ* is the electron conductivity,
*σ* = *n**e*^{2}*τ*_{e}/*m*_{e},
*n* is the electron density, *m*_{e} and
*e* are its mass and charge, and *τ*_{e} is the
relaxation time of electrons. Coulomb collisions of electrons dominate even in a low
ionized plasma, and we can write the electron conductivity in the form
(3)We
introduce the coefficient
*η*_{σ} < 1, which takes
into account the possible abnormal electron conductivity due to turbulent or another
processes decreasing the electron conductivity. The other parameters
are *ε* the mean electron energy, and Λ the Coulomb logarithm,
Λ ≈ 15−20. Thus, the electron resistivity is
(4)The
electron resistivity strongly depends on the electron energy, decreasing as the energy
increases. The energy *ε* can be estimated from the bolometric luminosity
*L*_{b} in a continuous spectrum,
*L*_{b} = *S*_{1}*σ*_{S}*ε*^{4},
where *S*_{1} is the surface of the current system,
*S*_{1} ≈ 2*S* + 4*S*^{1/2}*l*.
The constant *σ*_{S} is the Stefan-Boltzmann constant in energetic
units,
*σ*_{S} = *π*^{2}/60ħ^{3}*c*^{2} = 1.6 × 10^{59} erg^{-3} cm^{-2} s^{-1}.
Substituting the expression
*ε* = (*L*_{b}/*S*_{1}*σ*_{S})^{1/4}
into Eq. (4), we get (5)(6)Characteristic
luminosities *L*_{1} and *L*_{2} are defined
as (7)(8)Equations
(7) and (8) show that for the real bolometric luminosities of AGN and LLAGN the
resistivity *R*_{c} is very small,
*R*_{c} ≪ *R*_{H}, and the current system
is far from the optimal condition
*R*_{c} = *R*_{H}. We note that for
relativistic electrons the characteristic luminosity *L*_{2} does
not depend on the size of the system, and the factor
*l*^{2}*S*_{1}/*S*^{2}
is only the geometric factor which is of the order of unity. In contrast, the
luminosity *L*_{1} for non-relativistic electrons is inversely
proportional to the system size *l*,
*L*_{1} ∝ *l*^{−2/3}.
According to this estimation, we can conclude that the Blandford-Znajek mechanism for
extracting energy from rotating black holes is ineffective for Ohmic heating of outer
space. The only possible way to extract power by means of the Blandford-Znajek mechanism
is to transform it into relativistic jet luminosity *L*_{J}. We can
attribute to the jet some value of the resistivity *R*_{J} using
the relation
*L*_{J} = *R*_{J}*I*^{2}.
Neglecting the electron resistivity, we have
*L* = *L*_{J}. This implies that the equation for
determining the jet resistivity is (9)Denoting the
quantity
*κ* = *a*^{2}(*ω*_{cH}*r*_{H}/*c*)^{1/4}/128*π,* (*κ* ≥ 1),
we find that (10)Only for
*κ* = 1 do we achieve the optimal efficiency of the central machine,
*R* = *R*_{H}. For
*κ* > 1, we can assign to the jet two values of
resistivity, one greater than *R*_{H} (the sign + in Eq. (10)),
another less than *R*_{H} (the sign – in Eq. (10)). For both
values, the jet luminosity is the same. However, the solution in Eq. (10) with the
negative sign is unstable because a fluctuation of the magnetic field *δB*
in this case results in a negative feedback with the power of the central machine,
*δL*/*δB* < 0,
while the value
*δ**L*_{J}/*δB*
is always positive since
*δ**L*_{J}/*δB* = 7*L*_{J}/4*B*.

## 3. Acceleration of protons in accretion disk: proton synchrotron radiation

Producing a relativistic jet, a rotating black hole transmits an electric
current *I* of high magnitude through an accretion disk. This current
creates the magnetic field not only outside the disk, but also inside. Internal magnetic
fields *B*_{ρ} and
*B*_{φ}, such that
*B*_{φ} ≪ *B*_{ρ},
are of the same order as that above the disk, except that they are equal to zero at the disk
equator. Fields are frozen to the disk plasma motion. Therefore, a turbulent motion in the
disk induces a turbulent electric field , where
is the turbulent plasma velocity. This stochastic electric field accelerates disk particles.
This scenario of particle acceleration by large-scale 2D turbulence in a disk was discussed
by Istomin & Sol (2009). They found that
protons are indeed accelerated by this mechanism up to high energies. The maximum value of
the proton Lorentz factor *γ*_{m} ≫ 1. Electrons are almost not
accelerated at all because of large synchrotron losses (Istomin & Sol 2009). Thus, for the pure Blandford-Znajek mechanism
almost no non-thermal high-energy radiation is produced by disk electrons because of the
strong magnetic field and, connected with this, large synchrotron losses in any acceleration
process.

In contrast, fast disk protons can radiate the synchrotron emission in strong magnetic
fields. The distribution function of fast protons follows a power-law function
*f*_{p} = *b**γ*^{−β}.
The index *β* is the ratio of the loss energy rate to the rate of
acceleration by stochastic electric field (Istomin & Sol 2009) (11)where *n* is
the proton density, *σ*_{E} is the cross-section of the proton-proton
collisions in the disk *σ*_{E} ≈ 10^{-26} cm^{2},
and the time *τ*_{c} is the correlation time of the turbulence. We
consider that relativistic protons have energies in the range
10 < *γ* < 10^{8} eV/*T*_{d},
where the pricipal means of energy loss for fast protons is in their collisions with disk
protons and *T*_{d} is the disk temperature in eV units. The
correlation time *τ*_{c} is of the order of the time of the azimuthal
gyration of the matter in the disk,
*τ*_{c} ≃ *ρ*/*u*_{φ} ∝ *ρ*^{3/2}.
We consider the velocity *u*_{φ} to be
*u*_{φ} ∝ *ρ*^{−1/2},
as in the Keplerian disk. The same law applies to the turbulent motion,
, and the disk density is
*n* ∝ *ρ*^{-2}. We see that the power-law
index *β* increases with the radius *ρ* as
*ρ*^{3/2}, *β* = *β*_{H}(*ρ*/*r*_{H})^{3/2},
where *β*_{H} is the index value near the black hole. The density of
energetic protons is determined by the condition that their energy density is of the order
of the energy density of the stochastic electric field
. Therefore, the
distribution function of fast protons is
(12)We see that the fast
proton density decreases as ∝ *ρ*^{-3}. Substituting this function
into the well known expression for synchrotron radiation of one particle
*P*_{1} = 2(*e*^{2}/*m*_{p}*c*^{2})^{2}*c**B*^{2}*γ*^{2}/3
and integrating over *γ* up to *γ*_{m}, we find that
the density of the synchrotron power *W* is
(13)The total synchrotron
luminosity *L*_{s} is the integral of Eq. (13) over the disk volume
(14)Here
we introduce the dimensionless disk width near the black hole,
*h* = *H*/*r*_{H}.
Because the index *β* increases with distance from the black hole, i.e.,
*β* = *β*_{H}*x*^{3/2},
the main contribution to the total synchrotron luminosity comes from the inner part of the
disk,
*x* < *x*_{1} = (3/*β*_{H})^{2/3}
if *β*_{H} < 3. For
*β*_{H} > 3, the luminosity is lower. For
ln*γ*_{m} ≫ 1, the result of the integration in Eq. (14) with
logarithmic accuracy is (15)(16)To generate a
relativistic proton jet, an AGN must have a strong magnetic field near the central black
hole. This condition is given by Eq. (1),
*ω*_{cH}*r*_{H}/*c* ≥ (128*π*)^{4}*a*^{-8}.
Using that, we obtain (17)(18)For
our Galaxy, the estimated synchrotron luminosity of the disk is
*L*_{s} ≃ 10^{35} erg/s, which is
close to its bolometric luminosity
*L*_{b} ≃ 10^{36} erg/s. In any case,
the synchrotron luminosity from a turbulent disk given by Eqs. ((15), (16)) is always much
less than the total power extracted from a AGN rotating black hole.

The frequencies of radiation are in the range . According to Eq. (1)
*ω*_{cH} ≥ (128*π*)^{4}*a*^{-8}*c*/*r*_{H},
we estimate that .
For *γ*_{m} ≃ 10^{3} − 10^{4}, frequencies are in the
infrared band for AGNs
(*M* ≃ 10^{8} *M*_{⊙}). Observed LLAGNs
with radiatively inefficient accretion flow indeed show a peak in infrared emission (Maoz
2007). The center also radiates infrared light, and
according to observations (Genzel et al. 2003;
Nishiyama et al. 2009), this emission comes from the
rotating accretion disk. We suggest that this infrared emission is due to the proton
synchrotron radiation from the disk. The emitted spectrum of radiation is locally a power
law
*F*(*ν*) ∝ *ν*^{−(β−1)/2}
because the fast proton distribution function is a power law with index *β*.
However, *β* changes in the disk since
*β* = *β*_{H}(*ρ*/*r*_{H})^{3/2}.
The integration over the disk gives the following dependence
*F*(*ν*) ∝ *ν*^{−(βH−1)/2}/ln(*ν*),
which is almost a power law, but corrected by the logarithmic function. We note that the
observed power-law index of the infrared radiation from the Galactic center is −0.6 (Meyer
et al. 2009), implying that index of the proton
distribution is *β*_{H} ≃ 2.2.

## 4. Very high energy radiation

In the Blandford-Znajek mechanism, almost all energy is transformed into protons, a jet, or
a disk. Thus, it appears to be a barionic scenario for the very high energy (VHE) photon
production. VHE photons are measured by Cherenkov telescopes and have energy in the TeV
band. Energetic protons collide with the ambient matter and produce pions and then gamma
quanta. Sources of the VHE radiation can be in the disk and the jet. We first calculate the
VHE radiation from a disk. The spectrum of photons reproduces the spectrum of fast protons
in Eq. (12), *γ* ≫ 1, and is equal to
(19)where *n*
is the disk proton density, *γ*_{ph} is the photon energy
*E*_{ph} in units of the proton rest energy,
*γ*_{ph} = *E*_{ph}/*m*_{p}*c*^{2}.
The integration in Eq. (19) is over the disk volume. The quantities
,
and *β* depend on the radial distance *ρ* as we have
discussed. The result of the integration with logarithmic accuracy,
ln*γ*_{ph} > 1, is
(20)We obtain the total
luminosity of VHE radiation from the disk by integrating the spectrum given by Eq. (19) over
photon energies
*m*_{p}*c*^{2}*γ*_{ph}(21)We see that the VHE
luminosity is proportional to the energy of the magnetic field near the black hole,
*L*_{VHE} ∝ *B*^{2}*M*^{3},
and increases with the black hole mass. Substituting the condition in Eq. (2) in to the
expression (21), we get (22)which implies that
for
*M* ≃ 10^{8} *M*_{⊙}. We note that the
luminosity of VHE photons from the disk increases with the black hole mass in Eq. (22),
while the bolometric luminosity of the proton synchrotron radiation in the disk represented
by Eqs. (17) and (18) decreases with mass because of the different dependences of
luminosities on the magnetic field *B*,
*L*_{s} ∝ *B*^{4},
*L*_{VHE} ∝ *B*^{2}. The magnetic field must
be stronger for low-mass black holes (see Eq. (2)).

Fast protons of the jet can also produce VHE photons. Their energy density on the light
cylinder surface,
*ρ* = *r*_{L} = *c*/Ω_{F} ≃ 4*a*^{-1}*r*_{H},
is equal to the energy density of the electromagnetic field near this surface
(Istomin 2010). Thus, the jet proton density on the light cylinder surface is
*n*_{L} = *a*^{2}*B*^{2}/64*π**m*_{p}*c*^{2}*γ*.
Moving further away the light surface, protons diminish in density
*n*_{J} in line with the jet poloidal magnetic field to which they
are frozen,
*n*_{J} = *n*_{L}(*R*_{L}/*ρ*)^{2}.
The total luminosity of the jet in *γ*-rays is

where
*n* is the density of the interstellar gas inside the jet
and *V* is the jet volume. The integration provides a simple formula
analogous to Eq. (21) (23)where *l* is
the jet length and *r*_{J} is the outer radius of the jet at its
base. Comparing Eqs. (21) and (23), we conclude that they are similar and that both contain
the column density of the matter, which for the disk is
*n*_{H}*H* and the jet is *nl*. We
assume an angular resolution Δ*φ*. In the field of the central source, there
is a contribution of the jet emission along its length
*l* = *D*Δ*φ*, where *D* is
the distance to the source. For a resolution
Δ*φ* < (*r*_{H}/*D*)(*n*_{H}/*n*),
this means that the VHE luminosity of the disk dominates over the observed flux of the
central source. In contrast, for a resolution
Δ*φ* > (*r*_{H}/*D*)(*n*_{H}/*n*)
we will observe only the radiation of the jet. Using the condition given by Eq. (2), the
expression in Eq. (23) becomes (24)If the jet length
*l* does not depend on the black-hole scale
length *r*_{H}, as it occurs over the accretion disk
width *H*, then the VHE luminosity of the jet does not depend on the black
hole mass and is defined only by the column density *nl*,
.

## 5. Discussion

It seems that LLAGNs are good candidates in which to observe the Blandford-Znajek
mechanism. Accreting matter onto a black hole in LLAGNs is a weak source of energy because
of either the low accretion mass rate or low radiative efficiency. It may then be possible
to observe a black hole operating like a dynamo machine. The accretion disk would play the
role of a conductor through which the electric current would flow and the electric current
would then follow a jet. For jet creation, a strong magnetic field near the black hole is
needed. For AGNs of mass 10^{8} *M*_{⊙}, this field
should not be too high, *B* ≥ 3 × 10^{3} Gauss. The magnetic field
could be accumulated during previous epochs of high accretion rate. The rotating black hole
loses its rotation energy and angular momentum, which are both transmitted to the jet.
Rotating with the black hole, the radial magnetic field transfers its rotation to the
surrounding matter, which leeds to relativistic energies being attained on the light
surface. The energy is mainly in protons, which form the relativistic jet. The accretion
disk around the black hole can be observed in millimetre and infrared bands. To accrete
matter, the disk must be turbulent (abnormal transport coefficients). The turbulent motion
in the strong magnetic field generates a turbulent electric field, which accelerates disk
ions. Electrons are not accelerated to relativistic energies because of their large
synchrotron losses. Disk fast protons radiate synchrotron emission in the infrared range.
The high proton energies should correspond to a high disk luminosity in the very high energy
(VHE) photon range, and that a correlation exists between the infrared luminosity
*L*_{s} of the disk and its VHE luminosity
. Using Eqs. (15)
and (21), and excluding the unknown value of the magnetic field, we find the correlation
(25)We can check the
validity of this relation for low-luminosity AGNs with known luminosities
*L*_{VHE} and *L*_{s}, namely the Galactic
center (Sgr A*), M 87, and Centaurus A. The nucleus of Centaurus A has a high bolometric
luminosity *L*_{b} ≃ 1.3 × 10^{41} erg/s
(Meisenheimer et al. 2009). At this luminosity, a
high density of infrared photons of energy *ε*_{ph} ≃ 0.1 eV in the
central source prevents the free escape of VHE photons due to photon-photon collisions and
the production of electron-positron pairs. The simple estimate
*L*_{b} < *ε*_{ph}*cd*/*σ*_{T},
where *σ*_{T} is the Thomson cross-section and *d* is
the length scale of the central engine
*d* ≃ 10^{2}*r*_{H} = 1.5 × 10^{15}cm
for Centaurus A for which
*M* = 5 × 10^{7} *M*_{⊙}, gives the
condition
*L*_{b} < 1.1 × 10^{37} erg/s
at which the photon-photon annihilation is ineffective. The absorption of VHE quanta and the
generation of e^{ + }e^{ − } pairs result in the re-radiation of VHE
emission, as described by Stawarz et al. (2006), for
the interaction of VHE radiation with the starlight radiation from stars of the host galaxy.
Thus, for Centaurus A the observed luminosity *L*_{VHE} does not
reflect the direct VHE radiation from the black hole vicinity.

For Sgr A*, we have
*L*_{VHE} = 3 × 10^{34} erg/s
(Aharonian et al. 2009a) and
*L*_{s} = *L*_{IR} = 10^{36} erg/s
(Yuan et al. 2003),
*M* = 3.6 × 10^{6} *M*_{⊙}, and for M 87
*L*_{VHE} = 3 × 10^{40} erg/s
(Aharonian et al. 2006) and
*L*_{s} = *L*_{IR} = 10^{39} erg/s
(Perlman et al. 2007),
*M* = 3 × 10^{9} *M*_{⊙}. Substituting
these values in to Eq. (25), we find

for M 87. These data do
not contradict our model. To reach this conclusion, we must be sure that the observed VHE
luminosity *L*_{VHE} comes from the disk, and not the jet. For the
angular resolution of current VHE instruments, this requires that
*n*_{H}/*n* > 10^{-3}(*D*/*r*_{H}).
For M 87, the argument in favor of a disk origin of the VHE radiation is its short time
variability, which excludes the large scale jet of 2 kpc length (Aharonian et al. 2006). On the basis of our estimate of the VHE luminosity
of the total jet given by Eq. (21), , the jet is less
luminous than the luminosity observed, while from Eq. (22). For the
region of Sgr A*, the origin of the VHE emission is not quite clear (HESS collaboration
2010). One argument in favor of a disk origin of the
VHE radiation is that the observed spectral index and the cut-off energy of VHE radiation
(Aharonian et al. 2009a), *β* ≃ −2.1
and *E*_{c} ≃ 16 TeV, are close to the values that follow from the
observation of IR radiation from the disk, *β* ≃ −2.2,
*γ*_{m} ≃ 10^{4}, and
*E*_{m} ≃ 10 TeV. The argument of Aharonian et al. (2008) that VHE radiation from Sgr A* is likely from the
jet rather than the disk is based on the assumptions that the X-ray emission originates in
the disk and there is no time variability in the VHE flux during the X-ray flare. However,
we discuss below another possible origin of X-ray radiation in the Blandford-Znajek
mechanism, which is not in the disk.

Although Centaurus A is unsuitable for a comparison of the discussed model with observations, substituting its data into Eq. (25) provides values of the VHE luminosity not far from previous estimates, i.e.,

where
*L*_{VHE} = 2.6 × 10^{39} erg/s
(Aharonian 2009b),
*L*_{IR} = 1.3 × 10^{41} erg/s, and
*M* = 5 × 10^{7} *M*_{⊙}. This implies
that the re-radiation of VHE emission does not strongly affect its power.

We should also consider whether the observed infrared radiation from Sgr A* and M 87 might
originate in the disk. For Sgr A*, the short time variability of the NIR emission, which has
a period of the order of 20 min, is strong evidence that it originates in the disk (Genzel
et al. 2003; Nishiyama et al. 2009). The measured period corresponds to the rotation of a hot spot on
the disk around the black hole at distances close to the black hole horizon. We recall that
M 87 is a LINER. Apart from the thermal component of its mid-infrared emission, we also
observe its power law synchrotron-like emission of similar intensity
≃10^{39} erg/s (Perlman et al. 2007). The thermal component of temperature ≃50 K is the radiation of
the dust around the central energy source. The power-law component is thought to be the
radiation from the disk.

LLAGNs also radiate significant power in the X-ray band (Maoz 2007). Because the Thomson cross-section of the scattering of the
electromagnetic radiation for protons is
(*m*_{e}/*m*_{p})^{2}
times less than that for electrons the process of the inverse Compton scattering, which is
important to models of standard AGN radiation, can not be applied to explain the X-ray
radiation of LLAGNs in the scheme suggested here. However, a second component of fast
protons exists in the jet formed by the Blandford-Znajek mechanism. These are protons
accelerated in the disk, then ejected into the black hole magnetosphere that obtain
additional energy while crossing the light cylinder surface. This two-step mechanism of
proton acceleration up to very high energies was suggested by Istomin & Sol (2009). The Lorentz factor of these particles is
*γ* = (*γ*_{d}*ω*_{cL}/Ω_{F})^{1/2}.
We recall that *ω*_{cL} is the non-relativistic proton cyclotron
frequency at the light surface, Ω_{F} is the angular frequency of rotation of
magnetic field lines, Ω_{F} ≃ Ω_{H}/2, and
*γ*_{d} is the Lorentz factor of disk fast protons. The ratio
*ω*_{cL}/Ω_{F} is a very large number,
≫ *γ*_{d}, and the energy of these particles is much greater than
the energy of fast particles in the disk. They radiate synchrotron emission in the region
behind the light surface above the outer part of the disk near the base of the jet. The
ratio of frequencies of the synchrotron radiation in both this region and the inner disk is
, where
is the proton cyclotron frequency at the jet base averaged over the volume from
*r*_{L} to *r*_{J}. In the jet behind the
light surface, the poloidal magnetic field weakens like ∝ *ρ*^{-2},
the toroidal field decreases more slowly, ∝ *ρ*^{-1}, but initially
at
*ρ* = *r*_{L}*B*_{φ}
is small,
*B*_{φ} ≪ *B*_{ρ}.
Because of this we can consider
*B*(*ρ*) = *B*_{L}(*r*_{L}/*ρ*)^{2}
out to the outer jet radius *r*_{J} and
.
As a result, we obtain

Using the
expression in Eq. (1) for the ratio
*ω*_{cH}*r*_{H}/*c* ≃ 3 × 10^{10}
and the estimates *γ*_{d} ≃ 10^{4} and
*r*_{J}/*r*_{L} ≃ 10−10^{2},
we obtain
*ν*_{J}/*ν*_{d} ≃ 10^{4}−10^{5},
which corresponds to *ν*_{J} frequencies in the X-ray band when
*ν*_{d} is in the IR band. Unfortunately, we cannot estimate the
X-ray luminosity from the jet base because we do not know the fraction of fast protons
escaping the disk and being collected by the jet, which depends on the disk model. The
variability of this X-ray emission is expected from plasma instabilities in the jet (Istomin
2010).

In conclusion, we can say that LLAGNs, or at least some of them, could be be extracting the
energy from the black hole rotation by means of the Blandford-Znajek mechanism. The black
hole spends almost all its energy on the jet production and proton acceleration. These
LLAGNs are probably sources of high-energy cosmic rays. Their VHE luminosity should reflect
the intensive process of proton acceleration. Our scenario predicts that the value of
*L*_{VHE} increases with the black hole mass as
*M*^{3/2} and with the infrared luminosity of
the disk as
(see Eq. (25)). The discovery of new bright VHE sources from LLAGNs could confirm our
hypotheses.

## Acknowledgments

We acknowledge support from the Observatoire de Paris and the LEA ELGA. This work also was partially supported by the Russian Foundation for Basic Research (grant No. 08-02-00749) and the State Agency for Science and Innovation (state contract No. 02.740.11.0250).

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