The energy balance of the MPP is determined by the local value of the electric part of the potential. We consider the MPP including the case of decaying charged particles or ionized Keplerian disks. For simplicity, we restricted attention to the equatorial motion, allowing for small impulsion to the vertical direction to enable the chaotic regime of the motion, allowing escape to infinity along the lines of the magnetic field. The energetic balance of the decay process was calculated in the equatorial plane where the highest efficiency of the MPP is obtained.
3.4.2. Extremely Efficient Regime of Mpp
The highly efficient regime of the MPP works for the ionization of neutral matter, and its efficiency is dominated by the electromagnetic component
In the extreme regime, the efficiency can be as large as ∼ for sufficiently large magnetic fields and sufficiently supermassive Kerr black holes.
It is very useful to demonstrate the differences in the efficiency of the moderate and extreme MPP, making comparisons in very similar situations. For these purposes, we considered two similar splittings near a magnetized Kerr black hole having
,
, and
G, due to an electron loss by a charged and uncharged Helium atom:
The estimate on the efficiency for the extreme MPP gave
and for the moderate MPP we obtained
We thus immediately see that for the split charged particle, we obtained efficiency of the order of 1, but, for the electrically neutral particle, the efficiency reached an order of
. We thus naturally expect that for supermassive black holes of mass
M∼
, in the field having
B∼
, the efficiency can reach values
∼
[
28], corresponding to protons accelerated up to the velocities with Lorentz factor
∼
. Of course, in the extreme regime of the MPP, the question of the energy gap to the negative energy states, important in the original Penrose process, is irrelevant, as the magnetic field present at the ionization point is the agent immediately acting to put the second particle into the state with negative energy relative to distant observers.
The crucial aspect of the MPP extreme regime is the neutrality of the first (incoming) particle that could reach the vicinity of the horizon, unavailable to charged particles, where the acceleration can be efficient—simultaneously, the space can be free of matter there, enabling thus the escape of the accelerated particle to infinity. Of course, the ionized Keplerian disks fulfill well these conditions. In the MPP related to ionized Keplerian disks, we can write
Assuming that the mass of the second particle is much smaller than the mass of the third particle,
we can put the restriction
In the ionized Keplerian disks, the splitting electrically neutral particle follows (nearly) circular geodesic orbits, so we can assume the third particle escaping with large canonical energy , while the second particle is captured with large negative energy .
Moreover, the chaotic scattering transmutes the original nearly circular motion of the ionized Keplerian disks to the linear motion of scattered particles along the magnetic field lines. The extreme MPP thus could model (in addition to the Blanford–Znajek model) the creation of strongly relativistic jets observed in active galactic nuclei. The external magnetic field plays the role of a catalyst of the acceleration of the charged particles generated by the ionization—extraction of the black hole rotational energy occurs due to captured negative-energy-charged particles. The magnetic field lines then collimate the motion of accelerated charged particles. Under the inner edge of the Keplerian disk, an under-dense region is located, where the charged particles acquire the highest possible energy due to acceleration by the strongest potential difference, and they could survive their travel to distant observers, if kept by the magnetic field close to the black hole rotation axis where the lowest density of the jets is expected.
In the vicinity of the horizon, the splitting process in the equatorial plane implies the efficiency of the extreme MPP taking the form (now in the standard units)
where
is the static limit radius (boundary of the ergosphere) at the equatorial plane, and
is the splitting point radius that can be potentially outside the ergosphere. The efficiency is governed by the electromagnetic acceleration—it exceeds the "annihilation" value of
for electrons accelerated around a stellar mass of black holes immersed in the field with
B∼
.
For a Keplerian disk ionized around a non-rotating black hole, the MPP generates winds not able to escape to infinity, as they can have only energy from the rotational energy of the orbiting matter extracted due to the chaotic scattering (similarly to the Payne–Blandford process [
71]).
3.4.3. Ultra-High Energy Cosmic Rays as Products of Mpp in the Extreme Regime
The cosmic rays are high-energy protons or ions, demonstrating an isotropic distribution that can be explained only by their extra-galactic origin. The ultra-high-energy cosmic rays (UHECRs) are particles with energy eV—particles exceeding eV are rarely observed and are of high interest as they overcome the GZK limit ( eV) caused by interactions with the cosmic microwave background.
The energy loss determined by the GZK-cutoff puts strong limits on the distance of sources of the cosmic rays with energy overcoming the GYK limit—the corresponding restricting distance is estimated as
l∼ 100 Mpc [
72,
73]. The observations give the correlation of the ultra-high energy particles with
eV to the active galactic nuclei at distances lower than 100 Mpc [
7].
The maximum of the energy of a charged particle generated in the extreme regime of the MPP is given (in physical units) as
This dependence is illustrated in
Figure 6. We can see that protons with energy
eV are generated by mildly spinning (
a∼
) supermassive black holes with mass
, in the magnetic field with
. The maximum energy of ions generated under the same conditions as protons is lowered by the factor corresponding to the specific charge of the considered particles.
The galaxy center SgrA* black hole, being the closest supermassive black hole with mass
∼
[
74], spin
∼
[
75], and the magnetic field intensity
B∼ 10 G [
76] should accelerate frequently observed particles due to its special position and shortest distance. The predicted maximal energy of protons generated near the horizon of SgrA* black hole
is very interesting from this point of view as it corresponds to the knee of the energy spectrum in the observed data, located at
∼
eV, where the observed particle flux is significantly suppressed, which is in agreement with assumed existence of a strong single source at short distance. Moreover, the maximal proton energy
∼
eV can be related to the M87 galaxy supermassive black hole with
and
.
If the maximal energy E∼ eV is related to protons, then in the same source we have to expect electrons accelerated up to energy eV because of the factor of ∼1820—however, nothing like that is observed. The explanation is hidden in the efficiency of the deceleration (energy damping) of the charged particles due to back-reaction onto their electromagnetic (synchrotron) radiation in the magnetic field near the black hole. To the energy E∼ eV, protons near the rotating magnetars are accelerated, as the magnetar mass decrease of 10 orders to M∼1 is just compensated by an increase in the magnetic strength to B∼ G. Again, nothing like that is observed—the reason is again the back-reaction on the radiation. The back-reaction due to the radiation self-force is thus extremely important in connection to the particle acceleration and their observations; so we discuss this point carefully.
3.4.4. Synchrotron Radiation of Accelerated Charged Particles
The charged particles (protons, ions, or electrons) accelerated to ultra-high energy can be detected by distant observers, if the role of the back-reaction force is small or negligible. Therefore, we discuss the charged particle motion considering both the standard Lorentz force and the radiation reaction force
. In the non-relativistic limit, the synchrotron radiation generates the back-reaction force
orthogonal to the four-velocity, satisfying thus the relation
. Its covariant form reads [
77]
In general, the Lorentz–Dirac equations take the form [
30]
The first term on the right-hand side of Equation (
83) is the Lorentz force that is determined by the electromagnetic tensor
of the external electromagnetic field, while the second term is the back-reaction self-force determined by the radiative field
. The vector potential of the self-electromagnetic field is governed by
where
is the Laplace operator,
is the covariant differentiation, and
is the Ricci tensor of the spacetime. The self-field is determined by the potential given by the retarded solution of Equation (
84)
is the retarded Green function. The integration in determining the self-field potential is considered for the particle worldline
, and the four-velocity
[
78].
The general relativistic form of the radiating charged particle dynamics takes the form [
79,
80]
the last term of Equation (
86) giving the tail integral reflecting the role of back-scattering of the radiation, which is determined by the Green function [
78,
80]
As the Ricci tensor vanishes in the vacuum Kerr spacetime, the related terms are irrelevant. Similarly, it can be demonstrated that the tail integral is irrelevant in our considerations [
14,
30,
81,
82]. Therefore, the radiation reaction force takes the form
and the Lorentz–Dirac equation reads
The Lorentz–Dirac equation has a weak point as it gives the runaway solutions [
30]—this disease can be cured by lowering the order of the differential equations [
14,
30]. Reducing the second derivative of the four-velocity, we arrive to the self-force expressed in the form
corresponding to the covariant form of the Landau–Lifshitz equations.
Detailed analysis of the motion of charged particles around a magnetized Schwarzschild black hole was presented in [
30]; the widening of circular orbits was discussed in [
31]. Examples of the role of the self-force on the motion around a magnetized Kerr black hole can be found in [
14] on page 56. The synchrotron radiation has been studied also in [
83,
84] using a covariant form of the flat space results and recently in [
85]—however, without inclusion of the radiation reaction force.
For our purposes, the calculation of the energy loss is crucial. For the equatorial motion, the energy loss is given by the relation [
30]
For the ultra-high-energy particles (
), the most significant contribution to the energy loss is given by the first term in square brackets of (
91). The energy loss is related to the relaxation time
required for decay of the radial oscillatory motion of a charged particle. The rate of the energy loss is related to the relaxation time as
where
(
) denote the initial (final) energy of the particle. For ultrarelativistic particles, the energy loss reads
giving the solution
with
denoting the initial energy. The relaxation time
can be expressed as [
30]
For large values of
, we arrive to the simple form [
14]
enabling a fast estimation of the relevance of the self-force effects in connection to realistic astrophysical scenarios. We thus have to relate the particle and background parameters to the relaxation time.
For the characteristic values of the magnetic fields near the stellar mass (
M∼10
,
B∼
) and supermassive black holes (
,
B∼
) [
86,
87], we find for electrons
For protons, the values of
in (
97) and (
98) decrease by the factor
. The extremely large values of
imply a strong role of magnetic fields in charged particles dynamics in realistic astrophysical scenarios.
The influence of the radiation reaction force on the energy damping, represented by the relaxation time
, depends strongly on the parameter combining the particle and the black hole characteristics—the parameter
k is expressed in dimensionless form as
The parameter
k governs strongly the realistic astrophysical scenarios, although it is very small, much lower than
. For example, we find for electrons orbiting stellar mass and supermassive black holes
For protons orbiting the same object as electrons, k decreases by the factor , as for .
The parameter
k is very low in relation to the parameter
, but the particle energy damping can be very strong, as the relaxation time depends quadratically o a
that is large for realistic magnetized black holes. In
Table 2, the relaxation time for electrons and protons is given for the same conditions around magnetized black holes. The relaxation times have to be confronted with the orbital timescales
; particles orbiting at the ISCO imply
For Sgr A* supermassive black holes, we find the electron decay time ∼ s, while the ISCO orbital time is ∼ s, being by one order smaller that the decay time.
The relaxation time due to the charged particle oscillatory motion can be estimated by the relation [
14]
depending cubically on the particle mass and quadratically on the magnetic field intensity. Typical relaxation decay times of electrons and protons are given in
Table 2.
Since , the ratio of relaxation times of proton to electron, at fixed conditions, is very large, ∼, in correspondence with the factor of ∼. For this reason, the energy decay of electrons is relevant around magnetized black holes with plausible magnetic fields giving ultra-high energetic particles, so that electrons are significantly slowed and can not be observed as UHECR. The energy decay of protons (and ions) is irrelevant around magnetized black holes accelerating ultra-high energetic particles, and such energetic protons can also keep their energy on the distances ∼100 Mpc comparable to the GZK limiting distance—we thus can observe them as UHECR. Simply saying, under fixed conditions, electrons are accelerated with efficiency ∼ larger than protons, but efficiency of their energy decay is ∼ larger than for protons. On the other hand, the energy due to acceleration by a given electromagnetic field depends linearly on B, but energy decay caused by the radiative reaction force depends on ; for protons, the energy decay is relevant exclusively around magnetars.
Charged particles (e.g., protons) can be accelerated to the same energy around magnetized supermassive black holes with M∼, B∼, and magnetars with M∼, B∼ G, but around magnetars, the particle energy decays with efficiency higher than around the magnetized supermassive black hole. Therefore, there are no extremely energetic particles coming from magnetars, but we can see protons (ions) coming from magnetized supermassive black holes.
The play of the MPP acceleration and related energy decays at fixed conditions around a magnetized black hole, along with the energy decay related to the intergalactic travel of the ultra-high energy protons and ions, could help in localization of the active galatic nuclei emitting such particles.
For example, the calculations of energy decay of particles with
eV, traveling across very weak magnetic field of
B∼
G representing the intergalactic magnetic field, demonstrate that particles with energy
eV can survive the distance
l∼100 Mpc comparable to the GZK limit, but particles with energy
E∼
eV can survive at the distance
l∼10 Mpc [
28].