Wakefield Acceleration in the Universe(Submitted to International Journal of Modern Physics D)
Toshikazu Ebisuzaki1, Toshiki Tajima2, and Barry C. Barish3
1

RIKEN, Cluster for Pioneering Research, 2-1 Hirosawa, Wako 351-1198, Japan2
Depart. of Physics and Astronomy, University of California, Irvine, CA 92679, USA
3
LIGO, California Institute of Technology, Pasadena, CA 91125, USA
Abstract
The important role of magnetic fields in the phenomena in and evolution of the Universe
is well appreciated. A salient example of this is to make (often episodic) large magnetic
fields in AGN accretion disks and their emanation of well collimated and longitudinallyextended astrophysical jets. Such typical cases or related astrophysical processes, we find,provide a fertile ground for exciting large amplitude oscillations in the magnetic fields
that constitute the spine of the jets. The energy sources of these oscillations can be traced
originally to the gravitational energy of the central object. During their long propagation
along the jet, because of the gradual changes of the density and magnetic fields, theselarge magnetic pulsations turn into relativistic amplitude electromagnetic (EM) pulses,
which in turn induce intense wakefields that are capable of acceleration of electrons,
positrons, and ions to high energies. In this review we survey a variety of astrophysicalobjects ranging from as large as the cosmic AGN accretion disks and their jets to as small
as microquasars, to find or predict that there exist common astrophysical processes ofemission of high energy particles and gamma (and other EM) emissions. A variety ofthese objects will be ideally observed and studied in the multi-messenger astrophysicalobservations. One example that already sticked out was the case of the simultaneous
observations of gravitational wave emission and gamma ray pulse from the collision of
the two neutron stars and their subsequent structure formation (such as a disk) aroundthem.
Keywords: blackholes, accretion disks, jets, particle acceleration, stellar core collapse
AGN, microquasars, multi-messenger astronomy

11.

Introduction
The importance of magnetic field and plasma in cosmic plasma has been pointed out
by H. Alfven.1 He showed and demonstrated that magnetic fields and plasma play animportance role in astrophysical structure and dynamics. Some of the large structures
observed (as shown in Figs. II.9 and II.10 of Alfven’s book1) in cosmos are essential to
be influenced or reflect the magnetic field influence. Lovelace2 and Blandford3, andUchida and Shibata4-7 showed that the presence of an axial magnetic field in accretion
disk can generate coherent, elongated, stable jets. As we will analyze below, the global
dynamics of the magnetic fields plays the essential role in this remarkable simulation.
Perhaps, in anticipation of such importance of the global magnetic field dynamics Kennel
and his group started efforts to develop a numerical global magnetohydrodynamics of
cosmical plasma.8,9 These works introduced and opened up the exploration of the essential
role of magnetic fields in plasma in their determining topological, global, and collective
influence on the structure and dynamics, while not denying magnetic field effects also in
microscopic properties of astrophysical plasmas. Alfven mentioned the filamentary
nature of cosmic plasma, whereas Uchida and Shibata showed the jets formation from anaccretion disk, while Leboeuf et al. observed a shock cone.8,92Figure 1. Astrophysical objects with gravitational center, its rotation, and magnetic fields (after TS97
p. 12 Fig. 1.5).10 While the gravitational system does not have a rotation, it evolves a simple sphericalcollapse (scenario (a)), with rotation the system can evolve into a disk, which resembles accretion
disk )scenario (b)). Finally if the primordial plasma in addition has magnetic fields embedded (scenario(c)), the system evolves into a accretion disk that emanates a pair of focused jets of plasma flow (andspiraling magnetic fields; after TS97).10

In the present review article, Uchida and Shibata’s jet formation from an accretion disklies in the heart of our understanding of the high energy astrophysical observations andcorresponding physical mechanisms. (Here we note that global magnetic effects are3

primarily represented by magnetohydrodynamic (MHD) effects. It is possible that kinetic
effects could give rise to the dynamo effects that spontaneously produce magnetic fields
out of non-magnetized plasma (see e.g., Mima et al.11). However, barring suchmicroscopic kinetic effects, the nonmagnetic dynamics in its equivalent gravitational
accretion disk setup would produce no jets. It shows the fundamental importance of thepresence of magnetic fields in the jet formation. We compare three cases of evolution ofastrophysical objects (Tajima and Shibata 199710 hereafter TS97): (a) a gravitational
object without rotation; (b) a gravitational object with rotation; (c) a gravitational object
with rotation and magnetic fields (see Figure 1. (out of TS97 Fig. 1.5). The rotation gives
rise to the formation of the central disk formation (the accretion disk). Only in case (c)we see the evolution of a large extended structure (jets formation) emanating out of thecentral region of the formed accretion disk that may appear from cases (b) or (c).

Figure 2. The magnetic evolution under the rotational motion of the astrophysical object, emanationof the torsional twist of magnetic fields and emission of the torsional Alfven wave. (after TS97 Fig.
4.17 p. 310).10 As shown in the scenario (c) in Figure 1, the plasma embedded with magnetic fields
rotates, its magnetohydrodynamic force f (shown in Eq. (2)) causes the uplifting force that resulting in
the emanating plasma flows, which are the origin of the jets from the central part of the disk (After
TS97).10

When there is a static constant axial magnetic field (call it 𝐵𝐵𝑧𝑧 ) originally penetrating
the accretion disk plasma, the rotational motion of the accretion disk twists the magnetic

field, which yields the azimuthal magnetic field (call it 𝐵𝐵θ ). See Figure 2 (from TS97 Fig.
4.17). Such bending/twisting of the originally uniform 𝐵𝐵𝑧𝑧 field gives rise to the radial4

inward current 𝐽𝐽𝑟𝑟 through the Ampere’s law:
4𝜋𝜋
𝑐𝑐

(1)

𝑱𝑱 = 𝛻𝛻 × 𝑩𝑩.

The plasma induces the magnetohydrodynamic (MHD) force density:
𝟏𝟏(2)

𝒇𝒇 = 𝑐𝑐 (𝑱𝑱 × 𝑩𝑩),

which points toward the axial (z) direction. This force is the very agent that generates thejet out of the edge of the accretion disk (Figure 3; TS97, Fig. 1.9), as already differentiatedin Figure 1. Toward the axial z direction, as shown in the first self-consistentmagnetohydrodynamic demonstration by Uchida-Shibata.4-7 Once such sheared magnetic
fields and flows are generated, these also shear the plasma flows and further magnetic
fields. The shear magnetic fields and shear plasma plows are interwoven. These
interrelations between the magnetic fields and plasma flows continuously generate
collimated spiral magnetic fields and flows that we now come to know as the(astrophysical) jets. See Shibata, Tajima and Matsumoto (1990)12 and Matsumoto et al.
(1996)13, also seen in the textbook (TS9710, pp. 346-387). This outcome is one of the mostbeautiful, largest-scaled, and most robust self-organized structure of nature in the
Universe. Such jet structure could stretch almost as far out as Mpc in the large AGN jets.
It is important to also notice that because of the differential rotation of the accretion diskthat is formed (see in Figure 1 (c), Figure 3, and also see Figure 4 (from Shibata and Uchida1986)5, the longitudinal 𝐵𝐵𝑧𝑧 and the induced toroidal field form a spiraling emanating
magnetic fields that wrap around the jets that we mention to be formed. Many subsequent
numerical computations have verified and reinforced the above pictures.13-15 Furthermore,
its wrapping pitch is greater in the more inside of the spiraling magnetic flux (Figure 5).
These intricate combination gives rise to a superbly well-collimated and long stretching
mammoth jet structure that we witness in the Universe often. Astrophysical jets (in termsof the linear dimension) must be the largest self-organized structures in the Universe. The
structure is mutually supported by the engine of the gravitational center, its rotating disk,
and its emanating and intricately woven fabric of large-scale magnetic fields that wrap
around the jets which int urn push the jets forward with collimation. A beautiful
photograph of Cygnus A jets is shown in Figure 6. The structure shows such a succinct
5highly organized form, only because the jet forming forces of the plasma and magnetic
field under the gravitational driven rotation mutually reinforce each other. The jets
eventually slow down its velocity from relativistic dynamics so that their interaction with
the intergalactic tenuous plasma eventually begins to kick in. This transition from ultrarelativistic jets to less than the relativistic dynamics induces the beam-plasma instabilities
that slows the jets considerably and also forms a couple of turbulent plasma lobes thatcan emit copious microwaves. Thus, we see the long straight and narrow jets ending withthe double humped lobes.

6Figure 3. The evolutionary sequences of an astrophysical object with gravitational center, its
rotation, and magnetic fields. (after TS97 Fig. 1.9, p. 17).10 The compact gravitational object(such as a black hole (or merged neutron starts) tends to evolve an accretion disk, due to theirrotational dynamics. With its presence of magnetic fields such dynamic evolution now
accompanies collimated jet formation with magnetic fields wrapping around these jets.
Meanwhile, the accretion disk embeds magnetic fields in it, which trigger the magneto-rotational
instability and its associated evolutions of the disk and its jets (After TS97).10

It is important to notice that the more the central gravitational object is strongly
localized, the more intensified toroidal rotation of the accretion disk is realized that has
stronger differential rotation. Thus, it is natural that we find that a large-scale jet formationemanating from the center of accretion disk is often from a compact gravitation object ofa black hole (e.g. Wheeler 199016). See a conceptual black hole and its emanating
magnetic fields in Figure 7 (b) (TS97 4.60 ). The recent observational revelation (Figure 7(a)) not only showed the resemblance to those shown by Matsumoto Figure 7 (b), but also
it exhibits a sure sign of the imprints of magnetic fields through its fibril stripes (Figure 7(b)).

Figure 4. The shear flow of the accretion disk plasma and its embedded magnetic field dynamics asshown in Figure 2 give rise to a helical winding and axially extended magnetic fields. This jet
magnetic fields accompany the plasma thrust emanating from the J x B force mentioned in Eq. (2) inIntroduction. After Shibata and Uchida 1986,7 also from TS97, p. 382-383).10 Jets are launched fromthe accretion disk by the shearing magnetic fields driven by the rotating accretion disk via the Uchida7

Shibata mechanism.

As we see above, only relatively recent when the confluence of the gravitational objectsof black hole (EHT)17 and neutron stars18,19 with the global structure formation that is
initiated or stimulated by the presence of magnetic fields has happened. However, we
reflect the brief history of research of these objects briefly here.

Figure 5. In the accretion disk the presence of magnetic fields is amplified magneto rotational
instability due to its differential rotation. It has its characteristics spatial structure and temporal growthtime scale, which impact the evolution of the disk and its jets. Schematic drawings inspired by the
figure of Matsumoto 1996.20

Additional magnetic-plasma interactions of importance are seen in the accretion disk
plasma. As has been suggested by Balbus and Hawley, it is known that even tiny seedmagnetic fields may be amplified to grow to a large twisted and sheared magnetic fields
inside of the accretion disk plasma.21,22 For example, a tiny magnetic field seed of 𝑩𝑩 in
the disk plasma encounter the sheared rotational flows in the accretion disk, as the matter(in this case plasma) differentially rotates due to the gravitational rotation (such as the
Keplerian motion). Balbus-Hawley21,22 and Matsumoto et al.20 showed that magnetic
fields in the rotationally sheared accretion disk give rise to the instability to amplify its
seed magnetic fields and flows in the disk (not just the Keplerian azimuthal systematic
gravitational motion, but also multidimensional flows of plasma). Matsumoto et al.20
showed a large magnetic field may be generated as a result of this instability, while there8remains a possible flux escape and/or saturation via a variety of mechanisms such as theParker instability23 to make the generated flux tubes to become buoyant out of theaccretion disk. There can be high 𝛽𝛽 (𝛽𝛽 being the ratio of plasma matter pressure to themagnetic pressure) disks and low β disks24,12. Both show some similarity of the growthof magnetic fields, in the low β disks tend to show more explosively large magnetic fields
that may become comparable to the plasma kinetic pressure. If the disk shows such highgrowth (or high level of) saturated magnetic fields in the disk, the plasma with such highmagnetic fields would give rise to significant deviation of the disk’s gravitationalrotational motion (such as the Keplerian motion). Such state of large magnetic fields inthe disk (see Matsumoto and Tajima 1995)24 could give rise to the formation of the plasma
clumps that may be no longer in the Keplerian motion in its orbital motion in the disk.
This would lead to the disruption of such clumps of matter to fall out of the equilibrium
Kepler motion in the disk. This leads to the disk disruption. (In some astrophysicalobjects such as microquasars shows such episodic transitions in its brightness).25 If andwhen such disk disruption occurs due to the Balbus-Hawley (B-H) instability, we expectthat a clump of mass falls toward the central gravitational object (such as the black hole).
This would trigger a large perturbation, otherwise a steady emanation of the above
described jet formation. This would be an episodic disruption of the jets whose root area
is severely bombarded by the infalling accreting matter disrupted by this outgrowth of the
B-H instability.21,22 The episodic disruptions tend to occur, because after the matter infall
due to the disk disruption, the magnetic fields have been erupted and released so that its
power and pressure are pumped out. Thus, this B-H instability has to start from all overagain from the minuscule stage. Because this above exponentiation of magnetic fields
would take some dozens of exponentiation time of the B-H instability growth time, theepisodic time would be on the order of several (or dozens) of times the B-H growth time.

9

Figure 6. Narrow and long gigantic structure of jets (emanating to upper right and lower left directions,in white nearly staright lines) from the core (a central white spot, showing the AGN), which ends up
in a pair of plasma cocoons (at the ends of the jets, shown in orange clouds) of the galaxy Cygnus A.
(Credit: NRAO/AUI). The astrophysical high energy particle creation have been incubated in thisstructure and dynamics, not only at jets’ bases but also along their long jet structures. After 26

In short, we have learned that the B-H instability of the accretion disk (that contains
magnetic fields) gives rise to the episodes of the fast magnetic field amplification in thedisk, accompanied by episodic disruption of the Keplerian (otherwise steady rotational)
motion of the disk, which accompanies a large-scale mass accretion of the non-Keplerianrotating matter in the disk toward its center. This also should accompany torsional shaking
disturbance of the emanating jet magnetic fields that we have discussed above. We may
call it a large disruptive shear Alfven shock. Once this release of a large flux of magnetic
flux that was pent up by the B-H instability happens, the disk mass rotation resumes themore normal Keplerian rotation by the gravitational force nearly now unhindered by themagnetic forces. This however can still build up toward the next episode of the B-H
instability buildup and repeats of the above disruptive emission of the Alfven shocks fromthe bottom of the jets’ magnetic flux tubes.

10Figure 7. Comparison of the observation of the structure surrounding a blackhole and its (theoretically
expected) evolution with the presence of surrounding magnetic fields. A) The shadow of the event
horizon of supermassive blackhole at the center of M87 (Credit: Event Horizon Telescope).27 The
stripes may reflect an influence of the magnetic fields. B) The numerical simulation of accretion disk
around a blackhole. The shear rotation and resultant magnetic shear (see Figure 2) give rise to the jetformation (After TS97).10

Ebisuzaki and Tajima28-29 suggested that such accretion disruption at the root of
the jet formation region would give rise to the strong excitation of the otherwise steadyemanation of the jet flow, which leads to the formation of the large collective plasma
wave excitations of the electromagnetic pulses (starting from the above-mentioned Alfven
shocks) and their further collective behavior in the jet plasma. As discussed above, as thejet flows accompany the woven fabric of twisted magnetic fields, such episodic
disruptions end up with episodic large excitation of Alfven waves (or shocks) starting at
the root of the jets, which would propagate along the jet flows, as their phase velocity
being the Alfven speed in that jet plasma. As we discovered, these Alfven shocks also
give rise to another spectacular collective and robust display of intense celestial displayof large intensity new structure formation, called wakefields. (We will discuss this in moredetail in Sec. 2.) As we will discuss in Sec. 2, these Alfven waves (or shocks) phasevelocity often is very close to the speed of light. Even if it is not, we will consider thatthese Alfven waves tend to mode convert into eventually high intensity electromagnetic
11

pulse (as the density and magnetic field that sustains the Alfven waves decrease as thejets propagate over the long distance). We will see the implications of this as aspectacularly self-organized nature’s development. The reason why the spectacularly selforganized motion and structure formation happen (we will find and show) is due largely
to the relativistically strong wave excitation and their high speed (close to speed of light).
We find that such relativistic entities tend to show robust structural and dynamical
stability and resilience. This is why it can propagate sometimes to the end of its jets.
In the above, we have learned that there is a fundamental principle of the large
structure formation by the presence of magnetic fields in astrophysical plasma such asnear a compact gravitational object such as blackhole (neutron star) etc. This is a part ofthe broader structure formation enabled by the presence of magnetic fields in plasma thatact as the agent of the coherent largely organized structure formation that Tajima andShibata (1997: TS97)10 found in their voluminous treatise of “Plasma Astrophysics”. In
that they state: “….The presence of magnetic fields and plasmas in the Universe .… (is)thus fundamental ……(in) enriching the astrophysical processes. …..(They) tend (i) toinitiate, enhance or maintain the structure formation of the Universe; (ii) to provide
efficient and sometimes violent paths to convert gravitational energies into flows,radiation, or kinetic energies; (iii) to facilitate or accelerate the evolution of the
Universe…..”
We will study the physical implications of these phenomena and theircharacteristics. Why they tend to be so robust and thus we can often be likely able toobserve their imparted implications of its phenomena as observable far away to our Earth.
This is the main reason why we believe and we try to show to the current extent that thephenomena we describe tend to show up in a multiple of observational channels such as
multi-messenger astronomy observations. These could include optical lights, X-rays,
Gamma-rays, microwaves, high energy cosmic rays, neutrinos, and even gravitational
waves.
In recent years we further accumulated evidence that in addition to the largescale structure formation such as the jets associated with an accretion disk around acompact gravitational object, there is an additional important principle in operation thatfacilitates the structural formation and evolution of the Universe in addition to the above
we discussed. That is, the principle of the robustness of the wave structure that has a large
phase velocity.30 Once the magnetic fields in the Universe could set up such a robust
structure as jets out of the accretion disk, when excited perturbations such as waves on
the jets have the high phase velocity (such as the speed of light), we find that such waves
stay robust and can propagate nearly un-disorganized over a huge propagation distance.
12

We call such a structure as wakefields. When wakefields excited by the large disturbancessuch as the B-H disruption by the large magnetic bulge, they remain as a coherent wavelike entity without decaying quickly in a turbulent mess. The reason they do not quickly
turn into turbulence is its high phase velocity 𝑣𝑣p of the wake waves. When the phasevelocity is far removed from the bulk plasma (as characterized by its thermal velocity 𝑣𝑣t ),i.e. 𝑣𝑣p ≫ 𝑣𝑣t , the wakefields remain to be highly organized and collectively interacting

robust waves. We will see details of this process and its emergence in astrophysicalplasmas such as in the astrophysical jets emanating from a compact gravitational object.
In astrophysics, this high phase velocity often is in fact the speed of light. This means thatwakefields are relativistic encasement of collectively organized plasma whose interaction
is characterized by collective rather than single particle dynamics.31 Here the single

particle interaction strength is linearly proportional to the charge 𝑁𝑁𝑁𝑁 , the collective
interaction goes up with a much faster rate such as (𝑁𝑁𝑁𝑁)2 . This means that even if thecosmic plasma density may be low, the reach of the collective fields such as the
wavelength of the wakefields is so huge that the collective fields can become quite
enormous. This is because the collective interaction reach is coherent and global in theplasma. We will see in the following that this combination of the magnetic field generated
large scale magnetic robust structure like jets combined with the wakefields that havebeen excited by the accretion disk magnetic disruptions leads to the highly efficientenergy converter of gravitational energy into kinetic and other energies over a truly
cosmic distance.
In summary, the wakefield acceleration (WFA) theory in the jets of the accreting
blackholes is considered as the backbone theory to explain signals through multimessengers, including cosmic rays (charged particles), gravitational waves, neutrinos,and high-energy gammas as well as conventional electro-magnetic waves (radio, infrared,
optical ultraviolet, and gammas).
In the rest of the paper, we will review how the observational results of multimessenger astronomy can be compared with the prediction of the WFA theory of accreting
blackhole/jet system. The paper is organized as follows. In section 2, we describe how
the WFA theory works in the accreting blackhole system. The production of the intense
Alfvenic puleses, their propagation and mode conversion in the jets, and acceleration of
the charged particles are described. In section 3, observational results are compared withthe prediction of the WFA theory in the individual astronomical objects, which harboraccreting blackhole/jet systems. Those are blasars, radio galaxies, Seyfert galaxies,
intermediate mass blackholes in the starburst galaxies, and galactic microquasars. In
section 4, we will discuss short and long gamma-ray bursts in terms of NDAF disk
13

produced in the NS-NS merging and the core-collapse of the rapidly rotating massive
stars. Section 5 is for the discussions. We will discuss the comparison of WFA theory withother theories such as the Fermi acceleration mechanism. We also will present theprospects and implications for the future missions/projects.
2.

Astrophysical Wakefield Acceleration in the jets of the Accreting blackholes
The violent time variabilities are produced by the magnetic eruption of the disk as

described in section 1. In its strongly magnetized state, the plasma 𝛽𝛽, the ratio of thermalpressure to the magnetic pressure, decreases down to unity. In such a situation, magnetic
field cannot be confined by matter. Violent eruptions take place to emit strong pulses of
Alfvenic perturbations. This magnetic eruption can be considered as the cause of theviolent time variabilities, which are frequently observed accretion disk systems. The
pulses can propagate along the jets for long distance and accelerate charged particles(electrons, protons, and ions) by wakefiled acceleration up to high energies (above 1015
eV for electrons and above 1020 eV for protons and ions) to contribute various nonthermal phenomena.28-30
In turn, the pointing luminosity of the Alfvenic pulses, even if averaged over the time,is estimated as large as radiation luminosity and can sustain jets.28-30,32 In this sense, twoimportant features of accretion blackhole systems, those are 1) non-thermal emissions
produced via collective processes (rather than single particle effects such as the Fermi
acceleration) and 2) strong variabilities can be directly explained by a united collective
interaction theory of accretion disk instabilities and wakefield accelerations.
In the subsection 2.1, we first describe the structure of the steady state accretion disk,
according to Shakra and Sunyaev.33 The desk instability and amplification of magnetic
field are discussed in subsection 2.2 and the propagation of the Alfvenic pulse along thejets in subsection 2.3. In subsection 2.4, we present how charged particles are accelerated
in the wakefield excited in the jets.
2.1. Disk structure
As we discussed in Figure 1 (b), accreting gas forms a disk around a blackhole.33 In
the accretion disk, gas move slowly inward while orbiting in a circular orbit with the
Keplerian velocity around the blackhole without magnetic fields. If a viscosity works
among the material (plasma) orbiting different radius, they can dissipate (slowly) its
angular momentum resulting in the steady accretion of materials onto the central compact
objects, such a BH. Here, in the disk, gravitational (potential) energy by the material
14accretion is converted into rotational (kinetic) energy, which is further converted into
thermal energy by the friction and then eventually released in the form of various
radiations (such as optical, UV, X-rays and gamma radiation, and perhaps cosmic rays
depending on the situation).
Shakura and Sunyaev first obtained the structure of such an accretion disk arounda compact object.33 They estimated the physical quantities (densities, temperatures, andinternal energies) in the disk, assuming 1) the steady state and 2) the minor importance ofmagnetic field in the disk dynamics, using the three equations of mass, angularmomentum, and energy conservation. In this sense they are in the world of Figure 1(b).
In order to close the system of equations, they further assumed the proportionalityof the viscosity coefficient to the gas pressure and introduced the parameter, 𝛼𝛼, as the
proportionality constant. They, however, found that 𝛼𝛼 must be as large as 0.01-0.1 to becompatible to the observations of the compact objects and is by far (many orders ofmagnitude) large compared with the viscosity coefficient estimated by the atomic
diffusion of gas (plasma). They speculated that magnetic field may be responsible to thisanomalously large 𝛼𝛼. The work by Balbus-Hawley21,22 shows that the instability due tothe shearing plasma flow velocity of the accretion disk (such as the Keplerian velocity)
can give rise to exponential growth of the seed magnetic fields in the disk. Seed magnetic
fields may be generated, for example, by such a mechanism by another plasma motion.11
Matsumoto et al.24 showed that the B-H instability makes the saturated magnetic fields
either by the steady state formation of the enhanced magnetic viscosity or resistivity andthe break of the accretion disk equilibrium motion by the presence of large enough grownmagnetic fields. The self-consistently obtained magnetized plasma resistivity, 𝜂𝜂, may be
calculated using1𝑘𝑘 2
𝜋𝜋 2 1𝜂𝜂 = 4𝜋𝜋𝜎𝜎 = �2 � 4𝜋𝜋𝜋𝜋𝐶𝐶 ∑𝑘𝑘 ∫ 𝑑𝑑𝑑𝑑 𝑘𝑘J3 〈|𝑏𝑏 2 |(𝐤𝐤, 𝜔𝜔)〉,
e
s
𝑐𝑐 2

(3)

where 𝜎𝜎e is the electrical conductivity, 𝜌𝜌 the density of plasma, 𝐶𝐶s the sound velocity.
Here, 〈|𝑏𝑏 2 |(𝐤𝐤, 𝜔𝜔)〉 the time-averaged spectral intensity of magnetic fluctuations, and 𝑘𝑘J

the component of wavevector parallel to the mean electric current. By adopting themagnetic fluctuations that are produced in the disk by the BH instability, the magnitudeof such fields may be evaluated as〈𝛿𝛿𝐵𝐵2 〉4𝜋𝜋𝜋𝜋𝐶𝐶s2

𝜋𝜋𝑘𝑘 2 𝑘𝑘 2

𝑘𝑘 2

(4)

= 𝜒𝜒𝑒𝑒 �2 � � 𝑘𝑘 2z � � 𝑘𝑘∥2 � 𝑓𝑓(𝑞𝑞),J15where 〈𝛿𝛿𝐵𝐵 2 〉 is the time-averaged magnetic fluctuations and 𝜒𝜒𝑒𝑒 the ionization rate. Here,the factor 𝑘𝑘∥ 2 /𝑘𝑘 2 is unity for the case of purely poloidal case and 𝑘𝑘𝑦𝑦 2 /𝑘𝑘𝑧𝑧 2 for purely
toroidal case. By using this level of magnetic fields, we can estimate the enhancement
fact of the accretion viscosity alpha as〈𝛿𝛿𝐵𝐵2 〉𝛼𝛼𝐵𝐵 = 𝜒𝜒e 4𝜋𝜋𝜋𝜋𝐶𝐶 2−〈2𝛿𝛿𝐵𝐵𝑦𝑦𝛿𝛿𝐵𝐵𝑧𝑧

〉

,

(5)

2
s 〈�𝛿𝛿𝐵𝐵𝑦𝑦 � 〉+〈�𝛿𝛿𝐵𝐵𝑧𝑧 � 〉+1
𝛿𝛿𝐵𝐵𝑧𝑧𝛿𝛿𝐵𝐵𝑧𝑧−2Ω𝛾𝛾+�𝛾𝛾2 +𝜔𝜔𝐴𝐴 2 −4𝐴𝐴Ω��𝑘𝑘𝑦𝑦 /𝑘𝑘𝑥𝑥 �〈𝛿𝛿𝐵𝐵𝑦𝑦〉=〈𝛿𝛿𝐵𝐵𝑧𝑧〉 = −𝑞𝑞 2𝛿𝛿𝐵𝐵𝑥𝑥𝛿𝛿𝐵𝐵𝑥𝑥,𝛾𝛾2 +𝜔𝜔𝐴𝐴 2 +2Ωγ�𝑘𝑘𝑦𝑦 /𝑘𝑘𝑥𝑥 �(6)1 �𝛾𝛾 2 +𝜔𝜔 2 ��𝑘𝑘 /𝑘𝑘 +𝑘𝑘 /𝑘𝑘 +2𝐴𝐴/𝛾𝛾�
𝐴𝐴
𝑥𝑥 𝑦𝑦
𝑦𝑦 𝑥𝑥
𝛾𝛾2 +𝜔𝜔𝐴𝐴 2 +2Ωγ�𝑘𝑘𝑦𝑦 /𝑘𝑘𝑥𝑥 �.

(7)were 𝛿𝛿𝐵𝐵𝑥𝑥 , 𝛿𝛿𝐵𝐵𝑦𝑦 , and 𝛿𝛿𝐵𝐵𝑧𝑧 and 𝑘𝑘𝑥𝑥 , 𝑘𝑘𝑦𝑦 and 𝑘𝑘𝑧𝑧 are three components of turbulent magnetic

field, and wave vector, respectively, 𝛺𝛺 the rotational frequency, 𝐴𝐴 the Oort constant of thedisk, 𝜔𝜔𝐴𝐴 the Alfven frequency, and 𝛾𝛾 the growth rate of the magnetorotational instability.
Here 〈𝑥𝑥〉 denotes time average of the quantity 𝑥𝑥.
This leads to an evaluation that 𝛼𝛼 is a third of the ratio of the magnetic energy to thethermal energy,
1 〈𝛿𝛿𝐵𝐵2 〉1𝛼𝛼 = 3 4𝜋𝜋𝜋𝜋𝐶𝐶 2 = 3〈𝛽𝛽〉,s(8)where 〈𝛿𝛿𝐵𝐵 2 〉/4𝜋𝜋 is the turbulent magnetic energy averaged over the time and place, 𝜌𝜌 thegas density, 𝐶𝐶s the sound velocity, and 〈𝛽𝛽〉 the averaged ratio of the gas pressure to themagnetic pressure.
As can be seen in the next section, magnetic field can grow until their pressure iscomparable to the thermal pressure (in other words, 𝛽𝛽~1 ) in the accretion disk bymagnetorotational instability, so that 𝛼𝛼 could be as large as 0.1. In this sense, an accretion
disk actually evolves under the overall scenario of Figure 1 (c).
After we have shown its magnitude of 𝛼𝛼 may be in fact obtainable via selfconsistent calculation of the magnitude fields that evolves under the overall scenario of
Figure 1 (c).
16The orbital angular velocity is given by:
1/2

𝐺𝐺𝑀𝑀

𝛺𝛺 = � 𝑅𝑅3BH �

=

𝑐𝑐

1

√6𝑅𝑅0 𝑚𝑚𝑟𝑟 3/2.

(9)where 𝑐𝑐 is the light velocity, 𝑀𝑀BH the blackhole mass, 𝑅𝑅 the distance from the center ofthe blackhole, 𝐺𝐺 the gravitational constant, 𝑚𝑚 the blackhole mass normalized by solar
mass (𝑀𝑀⊙ ), and 𝑟𝑟 the distance from the center of the blackhole normalized by the radius,
𝑅𝑅ISCO , of the innermost stable circular orbit (ISCO):𝑅𝑅ISCO = 3𝑅𝑅𝑔𝑔 =6𝐺𝐺𝑚𝑚𝑚𝑚⊙
𝑐𝑐 2

(10)

= 𝑅𝑅0 𝑚𝑚,where,𝑅𝑅0 =6𝐺𝐺𝑀𝑀⊙
𝑐𝑐 2

,

(11)

In other words,(12)

𝑅𝑅 = 𝑅𝑅0 𝑚𝑚𝑚𝑚.

Inside the ISCO, the circular orbits are unstable due to the relativistic effects, and the gas
falls down at approximately the speed of light and are sucked into the blackholes. In otherwords, ISCO (𝑟𝑟 = 1) is the innermost radius of the gas disk.
According to Shakura and Sunyaev,33 the steady state solutions of the disk can
be calculated as follows. First, the disk is assumed to be axisymmetric and geometricallythin. The physical quantities in the disk are represented by the disk half thickness, 𝑧𝑧disk ,the density, 𝜌𝜌disk , and internal energy, 𝜀𝜀disk averaged over the vertical direction of thedisk (z-direction). Second, we assume the hydrostatic equilibrium in z-direction:

17

1

𝜀𝜀

(13)

𝑧𝑧disk = Ω �3𝜌𝜌disk �.
diskThird, the mass conservation equation turns out to be:(14)𝑀𝑀̇ = −4𝜋𝜋𝜋𝜋𝜌𝜌disk 𝑧𝑧disk 𝑣𝑣𝑟𝑟 = 𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐.,

in steady-state assumption, where 𝑀𝑀̇ is the mass accretion rate and 𝑣𝑣𝑟𝑟 the radial velocityof the gas. Fourth, the angular moment conservation equation:(15)

𝜀𝜀𝑀𝑀̇Ω = 4𝜋𝜋𝜋𝜋(𝜌𝜌disk 𝑧𝑧disk ) �3𝜌𝜌disk �,
diskwhere 𝛼𝛼 the phenomenologically introduced parameter reflecting on the nature of ofviscosity of the accretion disk. If all gravitational energy is released in the disk via theviscous heating of the disk plasma and that is released into the observable radiation
emissions, we can relate the viscous evolution of the disk with the kinetic energy andpotential energy of the disk, neglecting any magnetic energies (in the below shown
through the parameter 𝛼𝛼. ) Finally, when the gravitational energy release in the disk isassumed to be equal to the radiation flux from the disk, we obtain
3

8𝜋𝜋

4𝜀𝜀
𝑀𝑀̇Ω2 = 6𝜅𝜅 𝜌𝜌 disk𝑧𝑧

T disk disk

(16)

.

Now, we can obtain three quantities 𝑧𝑧disk , 𝜀𝜀disk , and 𝜌𝜌disk , solving three equations 13,
15, and 16 for a given parameter of 𝑀𝑀̇ as the function 𝑟𝑟 with the phenomenologicalparameter 𝛼𝛼.26-28 Expressing these three quantities, we obtain
𝑅𝑅

𝑧𝑧disk = 8𝜖𝜖0 𝑚𝑚̇𝑚𝑚,where 𝜖𝜖 = 0.06 the radiation efficiency of the accretion disk.
4𝑐𝑐𝑐𝑐

𝜀𝜀disk = 𝜅𝜅 𝛼𝛼 =
T

4𝑐𝑐 2

1

√6𝜅𝜅T 𝑅𝑅0 𝛼𝛼𝛼𝛼𝑟𝑟 3/2

,

(17)
(18)

18

and
1024𝜋𝜋 2 𝑐𝑐 3

256√6𝜖𝜖 2 𝑟𝑟 3/2

𝜌𝜌disk = 27𝜅𝜅 3𝛼𝛼Ω𝑀𝑀̇2 = 3𝜅𝜅 𝑅𝑅
T

T 0𝛼𝛼𝑚𝑚̇2 𝑚𝑚,(19)

where 𝜅𝜅T = 0.2 [cm2 g −1 ](1 + 𝑋𝑋) is the opacity for the Thomson electron scattering, 𝑋𝑋
the hydrogen concentration of gas, 𝑚𝑚̇ the mass accretion rate normalized by the criticalaccretion rate, 𝑀𝑀̇c ,4𝜋𝜋𝜋𝜋𝑀𝑀⊙
𝐿𝐿Edd
2𝜋𝜋𝜋𝜋𝑅𝑅
𝑀𝑀̇c = 𝜖𝜖𝑐𝑐
𝑚𝑚 = 3𝜖𝜖𝜅𝜅 0 𝑚𝑚.
2 = 𝑐𝑐𝑐𝑐𝜅𝜅
T

(20)

T

We note that the disk properties such as its thickness and diameter are expressed in termsof 𝛼𝛼.
In other words,
2𝜋𝜋𝜋𝜋𝑅𝑅𝑀𝑀̇ = 𝑚𝑚̇𝑀𝑀̇c = 3𝜖𝜖𝜅𝜅 0 𝑚𝑚̇𝑚𝑚,

(21)

T

The radiation luminosity, 𝐿𝐿rad , is calculated as:
𝐿𝐿rad =

4𝜋𝜋𝜋𝜋𝜋𝜋𝑀𝑀⊙𝜅𝜅T𝑚𝑚̇ 𝑚𝑚 =2𝜋𝜋𝑐𝑐 3 𝑅𝑅0
3𝜅𝜅T𝑚𝑚̇ 𝑚𝑚 = 𝜖𝜖𝑀𝑀̇𝑐𝑐 2 ,

19

(22)

Figure 8. An accretion disk shows transitions between the strongly magnetized states to the weakly
magnetized states, quasi-periodically, as suggested by Shibata et al.12 Its theoretical framework isshown schematically.

Figure 9. Magneto-rotational instability is developed in in an accretion disk to build up its magnetic
fields in the disk and then leads to eruptions of magnetic field. And after its eruption, it relaxes backto a quiet stage, and repeat this episodic cycle. After the works by Tajima and Gilden34 and Haswell etal.35

2.2. Alfvenic pulses emitted by the magnetic eruptions of the disk
As discussed in the rest of the section, the accretion disks are by no means in the20steady state and rather unstable against magnet-rotational instability to become strongly
fluctuated, if we take into account of magnetic field. The physical quantities obtained here
must be understood as the average values integrated over a long period.
Many of astrophysical theoretical frameworks, as we reviewed above, are based on
the steady state and purely gravitational objects without temporal, explosive, andextensive behaviors. However, as more observational evidence accumulates, we begin tolearn the structured and behaviors that are beyond such models and descriptions. In these
phenomena additionally magnetic fields have been found to play an important role. An
example may be seen in Figure 6, in which central objects of AGN (seen as a tiny dot)
emanates a pair of long (many kpc in the longest cases) jets, which form a pair of large
lobes. Such objects, structure, and its emanating signals may not be explainable bygravitation alone. In fact, their strong active dynamics and high energy phenomena areconsidered to be mediated by magnetic activities.
The understanding of the production of the global magnetic field in the accretion disk
was advanced step-by-step, as follows. First, Uchida and Shibata4-7 pointed out that themagnetic field is amplified due to the winding by the differential rotation of the accretion
disk and a pair of jets (both sides of the accretion disk) is launched along the rotation axis
with the driving force of pinching and 𝒋𝒋 × 𝑩𝑩 force in the unwinding of the twisted
magnetic field. Secondly, Balbus and Hawley21,22 reveals that the accretion disk isunstable against magneto-rotational instability to develop magnetic turbulence in the disk.
Although one usually may think that the turbulence would destroy the global structure, itis actually not in the case for an accretion disk around a blackhole. Thirdly, Tout andPringle36, who solved a set of simplified dynamo equations, showed that magnetic field
is alternatively produced by the dynamo mechanism in the presence of the turbulence andthe differential rotation in the disk. The magnetic field is transported by buoyancy fromthe disk along the rotation axis. This is consistent that the Uchida-Shibata’s view thatmagnetic field twisted in the accretion disk drives the jets (Figure 8). In practice, Shibata
et al.12 performed a numerical simulation and revealed that the disk repeats transitions
between strongly magnetized and weekly magnetized states as shown in Figure 8.
When we introduce 𝛽𝛽 , the ratio of gas pressure to the magnetic pressure, themagnetic field is estimated as:
8𝜋𝜋

1/2

𝐵𝐵disk = �3𝛽𝛽 𝜀𝜀disk �

32𝜋𝜋𝑐𝑐 2121= �3√6𝜅𝜅 𝑅𝑅 � 𝛼𝛼1/2 𝛽𝛽1/2𝑚𝑚1/2 𝑟𝑟 3/4.
T 0

21

(23)

Figure 10. This episodic eruption of the disk and jets shown in Fig. 9 can lead to the generation oflarge perturbations of the jets and their magnetic fields. Such perturbations in the jets give rise to burstsof the Alfvenic perturbations, which are emitted in the transitions of the disk. An example is from
Canac, et al., 2020.37

At the quasi-periodical transitions from the strongly magnetized states to theweakly magnetized states are magnetic eruptions and emit pulses of Alfvenic
perturbations (Figure 9), which propagate along the jets perpendicular to the disk (along
the rotational axis; Figure 10). The wavelengths of the emitted electro-magnetic
disturbances are of the order of the size of the density clamps made in the disk. These areat the wavelength of the most unstable in magneto-rotational (B-H) instability;21,22 inother words𝑉𝑉𝜅𝜅 𝑀𝑀̇ΩT𝜆𝜆 = � A,disk� �𝐴𝐴 � 𝑧𝑧disk = 4𝜋𝜋𝜋𝜋=𝑣𝑣
s√2𝑅𝑅0 𝑚𝑚̇𝑚𝑚
,6𝜖𝜖 𝛽𝛽 1/2(24)where Ω/A = 4/3 for the Keplarian disk. Note that this value is a constant independent
to 𝑟𝑟. Here, 𝑉𝑉A,disk is the Alfven velocity in the disk:
𝑉𝑉A,disk =𝐵𝐵disk�4𝜋𝜋𝜌𝜌disk

𝑐𝑐

𝑚𝑚̇

= 8√3𝜖𝜖 𝛽𝛽1/2 𝑟𝑟 3/2 ,(25)

and 𝑣𝑣s is the sound velocity:
𝜀𝜀disk𝑣𝑣s = �3𝜌𝜌

disk

1
2

𝑐𝑐

� = 8√6𝜖𝜖,

(26)
22

The typical angular frequency the waves in the burst, which propagate in the light velocity
𝑐𝑐 is calculated as:
2𝜋𝜋𝜋𝜋𝜔𝜔 = 𝜆𝜆 =12𝜋𝜋𝜋𝜋𝜋𝜋 𝛽𝛽 1/2√2𝑅𝑅0 𝑚𝑚̇𝑚𝑚

.

(27)

The frequency of electro-magnetic wave bursts is given from the value of the ISCO (𝑟𝑟 =1):𝑉𝑉𝜈𝜈 = 𝑧𝑧A,disk = 𝑧𝑧
disk

𝐵𝐵diskdisk �4𝜋𝜋𝜌𝜌disk

=

𝑐𝑐

1

√3𝑅𝑅0 𝛽𝛽 1/2 𝑚𝑚𝑟𝑟 3/2

,

(28)

where 𝑉𝑉A,disk is the Alfven velocity of the quiet phase. Since the energy released in the

waves are dominated by the innermost region of the disk, the rising timescale, 𝜏𝜏rise andthe periodic recurrence time, 𝜏𝜏rec can be calculated from the value of the ISCO (𝑟𝑟 = 1)as:
2π

√2𝑅𝑅 𝑚𝑚̇ 𝑚𝑚

(29)

1

√3𝑅𝑅0 1/2
𝛽𝛽 𝑚𝑚,
𝑐𝑐(30)

𝜏𝜏rise = ω = 6𝜖𝜖𝜖𝜖0 𝛽𝛽1/2 ,and𝜏𝜏rec = ν =Therefore,𝜏𝜏rec𝜏𝜏rise=6√6𝜖𝜖𝛽𝛽 3/2
𝑚𝑚̇

,

(31)

The strongest waves are emitted around ISCO (𝑟𝑟 = 1) in the burst state, thismaximum flux of electromagnetic burst (Figure 11), is estimated as:𝛷𝛷w (𝑟𝑟 = 1) =𝑉𝑉A,disk 𝐵𝐵disk 2
4𝜋𝜋=

Ω2 𝑀𝑀̇
√2𝜋𝜋

𝑐𝑐 3

𝑚𝑚̇

= 9√2𝜖𝜖𝜅𝜅 𝑅𝑅 𝛼𝛼𝛽𝛽3/2 𝑚𝑚𝑟𝑟 3 .
T 0

23

(32)

The waves propagate along the perpendicular (jets) to the accretion disk. Wave flux𝛷𝛷w �𝐷𝐷 = 3𝑅𝑅g = 𝑅𝑅0 𝑚𝑚� is given by 𝛷𝛷w (𝑟𝑟 = 1). Since the Alfven velocity in the jets is

close to the speed of light, electric field 𝐸𝐸w of the wave is calculated as:𝛷𝛷w,jet (𝐷𝐷 = 𝑅𝑅0 𝑚𝑚) = 𝛷𝛷w (𝑟𝑟 = 1) =𝑐𝑐𝐸𝐸w 2
4𝜋𝜋

,

(33)

where 𝐷𝐷 is the distance from the blackhole along the jet. Therefore, we obtain
1/2

4𝜋𝜋

𝐸𝐸w = � 𝑐𝑐 𝛷𝛷w (𝑟𝑟 = 1)�

2𝑐𝑐

√2𝜋𝜋

1/2

𝑚𝑚̇1/2= 3 �𝜖𝜖𝜅𝜅 𝑅𝑅 �
T 0

1 3
𝛼𝛼 2 𝛽𝛽 4 𝑚𝑚1/2

.

(34)

The dimensionless vector potential 𝑎𝑎0 at the bottom of the jet is given by
𝑒𝑒𝐸𝐸

𝑒𝑒

√2𝑅𝑅

1/2

w
𝑎𝑎0 = 𝑚𝑚 𝜔𝜔𝜔𝜔
= 18𝑚𝑚 𝑐𝑐 �𝜋𝜋𝜖𝜖3 𝜅𝜅0 �eeT𝑚𝑚̇3/2 𝑚𝑚1/2
𝛼𝛼1/2 𝛽𝛽 5/4

.

(35)

Figure 11. A numerical simulation of the innermost part of the accretion disk around a blackhole.
Magnetic fields are amplified in the disk, while these turn into the jets and their episodic disruptive
dynamics. After Mizuta et al.14

On the other hand, the time-averaged luminosity, 𝐿𝐿w , integrated over the entire
disk is calculated as:
∞

𝐵𝐵

2

4𝜋𝜋𝑐𝑐 3 𝑅𝑅

𝑚𝑚̇𝑚𝑚

√2

𝐿𝐿w = ∫𝑚𝑚𝑚𝑚 2𝑧𝑧disk disk𝜈𝜈2𝜋𝜋𝜋𝜋𝜋𝜋𝜋𝜋 = 9√2𝜖𝜖𝜅𝜅 0 𝛼𝛼𝛽𝛽3/2 = �3𝛼𝛼𝛽𝛽3/2 � 𝑀𝑀̇c 𝑐𝑐 2 .
4𝜋𝜋
0

T

The ratio 𝐿𝐿w /𝐿𝐿rad of the wave luminosity to radiation luminosity,
24

(36)

𝐿𝐿w

𝐿𝐿rad

√2

= 3𝜖𝜖𝜖𝜖𝛽𝛽3/2,(37)

is larger than unity and as large as 10, in other words, the Alfvenic pulses emitted by therepeated magnetic eruptions of the disk are strong enough to maintain the relativistic flowof the jets emanating from the accreting blackhole system, as Shibata and Uchidapioneered.4-7 The jets, backboned by the relativistic flow and the longitudinal magnetic
field, extend long distances of Mega parsec ( ~1024 cm ) scales for the case ofcosmological quasars and parsec scales ~1018 cm for the case of galactic microquasars,
far beyond the size of their accretion disks.
2.3. Propagation of Alfvenic pulses along the jets
The Alfvenic pulses are propagates in the jets for long distance. Although their typicalfrequencies are lower than the plasma frequency at the bottom of the jets, they canpropagate with the restoring force of cyclotron resonance of electrons and ions. As
distance from the blackhole increases, the magnetic field decreases, so that the ratio of
the angular frequency of the wave to the cyclotron frequencies (both for electrons andions) increases. The waves are converted into the higher order of resonances keeping theirfrequencies constant through the channels near the light cone: the Alfven velocity is close
to the light velocity in the jets. As the plasma density also decreases, the frequencies
become eventually larger than the local plasma frequency. Afterwards, the waves canpropagate as plasma waves, which will be connected to the branch of the electro-magnetic
mode.
This subsection examines the dependence of physical parameters in the jet on
distance from the bottom and discusses how the waves propagate through it. First, themagnetic field 𝐵𝐵jet in the jet can be calculated assuming that the magnetic field flux isconserved in the jet.

𝐷𝐷

−1

𝐵𝐵jet = [𝐵𝐵disk (𝑟𝑟 = 1)](𝑏𝑏/𝑚𝑚𝑅𝑅0 )−2 = [𝐵𝐵disk (𝑟𝑟 = 1)] �𝑚𝑚𝑚𝑚 �
32𝜋𝜋𝑐𝑐 2

1/2

�3√6𝜅𝜅 𝑅𝑅 �T 01

𝛼𝛼1/2 𝛽𝛽 1/2 𝑚𝑚1/2

𝐷𝐷−1

�𝑚𝑚𝑚𝑚 � .
0

0

=

(38)

Next, we assume as𝛾𝛾 = 𝑎𝑎0 ,

(39)
25

within the jet, 𝑎𝑎0 can be calculated, assuming that the wave intensity within the jet isconserved, i.e., the flux 𝛷𝛷w,jet is inversely proportional to the cross-sectional area 𝜋𝜋𝑏𝑏 2 of

the jet.

𝑎𝑎0 (𝐷𝐷) = 𝑎𝑎0 (𝐷𝐷 = 𝑅𝑅0 ) �𝑏𝑏(𝐷𝐷) −1
𝑅𝑅0 𝑚𝑚� ,(40)

where 𝐷𝐷 is the distance from the bottom of the jet, and 𝑏𝑏(𝐷𝐷) is the radius of the jet, whichis assumed to 𝑏𝑏(0) = 3𝑅𝑅g = 𝑅𝑅0 𝑚𝑚 . In addition, Figure 13 shows the ratio of plasma
frequency 𝜔𝜔p′ /𝜔𝜔 are plotted against the distance 𝐷𝐷/(𝑅𝑅0 𝑚𝑚) from the bottom of the jet for
the typical cases (Γ = 10, 𝛼𝛼 = 0.1, 𝜉𝜉 = 10−2 , 𝑚𝑚̇ = 0.1 for 𝑚𝑚 = 1, 104 , 108 ). Here weassume that𝑏𝑏(𝐷𝐷) = 𝑅𝑅0 𝑚𝑚(𝐷𝐷/𝑅𝑅0 𝑚𝑚)1/2.

(41)

This relation is consistent with the observation of the jet of M87, the closest active
galactic nuclei M8738 and other AGN jet observation.39 Therefore, we get1

𝐷𝐷 −1/2√2𝑅𝑅 2 𝑚𝑚̇3/2 𝑚𝑚1/2
𝑎𝑎0 (𝐷𝐷) = 18𝑚𝑚 𝑐𝑐 �𝜋𝜋𝜖𝜖3 𝜅𝜅0 � 𝛼𝛼1/2 𝛽𝛽5/4 �𝑅𝑅 𝑚𝑚�
.
e
T
0
𝑒𝑒

(42)

The plasma frequency 𝜔𝜔p′ corrected for relativistic effects is given by
4𝜋𝜋𝑛𝑛jet 𝑒𝑒 2𝜔𝜔p′ = �𝑚𝑚𝑒𝑒 𝛾𝛾

1/2

�

.

(43)

The plasma density 𝑛𝑛jet in the jet can be calculated from as follows, if we assume thekinetic luminosity of the jet:

𝐿𝐿jet = 𝑛𝑛jet 𝜇𝜇𝑚𝑚H 𝑐𝑐 3 𝛤𝛤 2 𝜋𝜋𝑏𝑏 2 = 𝜉𝜉𝐿𝐿rad ,(44)

is conserved through the jet. Therefore,
2

𝜉𝜉𝑚𝑚̇

𝐷𝐷

−1

𝑛𝑛jet = 3𝜇𝜇𝑚𝑚 𝜅𝜅 𝑅𝑅 𝛤𝛤2 𝑚𝑚 �𝑅𝑅 𝑚𝑚� .
H T 0

0

(45)
26

Here, 𝜉𝜉2 is the ratio of the kinetic luminosity of the jet to the radiation luminosity, Γ thebulk Lorentz factor, and 𝜇𝜇 = 1.29 is the mean molecular weight of the accreting gas.
Substituting equations 44, 39, and 42 into equation 43, we get:
4𝜋𝜋𝑛𝑛jet 𝑒𝑒 2𝜔𝜔p′ = �𝑚𝑚𝑒𝑒 𝛾𝛾

1/2

�

= 4�

9𝜋𝜋 3 𝑒𝑒 2 𝑐𝑐 2 𝜖𝜖 3√2𝜇𝜇1/4 1/2 1/4 5/8𝜉𝜉 𝛼𝛼 𝛽𝛽�
2 𝑚𝑚 2 𝜅𝜅 𝑅𝑅 3
H

T 0

𝐷𝐷

−1/4

�
�
𝛤𝛤𝑚𝑚̇1/4 𝑚𝑚3/4 𝑅𝑅 𝑚𝑚
0

.

(46)

In Figure 13, we plot 𝜔𝜔c′ ,, 𝜔𝜔p′ and 𝜔𝜔 at the bottom of the jet (𝐷𝐷 = 𝑅𝑅0 ) against the blackhole

mass 𝑚𝑚 for the typical case (𝛤𝛤 = 10 , 𝛼𝛼 = 0.1 , 𝜉𝜉 = 10−2 , 𝑚𝑚̇ = 0.1 ). For most of theinteresting cases, the relationship of 𝜔𝜔p′ >𝜔𝜔 holds; In other words, at the bottom of the jets,
the plasma in the overdense state (𝜔𝜔p′ >𝜔𝜔 ), where plasma waves and electromagnetic
waves cannot propagate. On the other hand, Alfven wave or whistler wave can propagate,
the Alfven velocity 𝑉𝑉A,jet at the bottom of the jet are given by
𝑉𝑉A,jet =𝐵𝐵jet

�4𝜋𝜋𝑚𝑚H 𝑛𝑛jet

=

2

𝛤𝛤𝛤𝛤

√6 𝛼𝛼1/2 𝛽𝛽 1/2 𝑚𝑚̇1/2

.

(47)

In other words, the nominal values of the Alfven velocity
𝛤𝛤

𝜉𝜉

−1/2

𝑉𝑉A,jet ~1012 [cm s −1 ] �10� �10−2 �𝑚𝑚̇ −1/2

�0.1�

.

(48)

This can approach the speed of light, when the approximation breaks down. Then the
wave becomes that of EM waves in magnetized plasma. On the other hand, 𝜔𝜔p′ = 𝜔𝜔 atthe distance 𝐷𝐷0 given by:𝐷𝐷

4𝑒𝑒 2 𝑅𝑅�𝑅𝑅 0𝑚𝑚� = 9√2𝜋𝜋𝜋𝜋2𝑚𝑚 02𝑐𝑐 2 𝜅𝜅 𝜖𝜖0𝑛𝑛jet (𝐷𝐷0 ) =H

T

3√2𝜋𝜋𝑐𝑐 2 𝜇𝜇𝑚𝑚H 𝜖𝜖𝑒𝑒 2 𝑅𝑅02

𝜉𝜉 2 𝛼𝛼𝛽𝛽 1/2 𝑚𝑚̇3 𝑚𝑚
𝛤𝛤4

.

(49)

𝛤𝛤2

𝜉𝜉𝜉𝜉𝛽𝛽 1/2 𝑚𝑚̇2 𝑚𝑚2On the outside of the point 𝐷𝐷1 ( 𝐷𝐷 > 𝐷𝐷1 ), 𝜔𝜔 > 𝜔𝜔p′ so that the plasma wave

(electromagnetic wave) allows to propagate. The electromagnetic waves propagated as
Alfven wave and whistler wave are converted into plasma waves (electromagnetic waves)27

by nonlinear mode-conversion. This 𝐷𝐷 > 𝐷𝐷2 leads to the bow wakefield acceleration as
described in the next subsection.

Figure 12. The strong Alfvenic pulse (discussed in Figure 11) can mode-convert itself into an intenseelectromagnetic pulse (shown on the left of this picture) through the propagation in the jet plasma.
The acceleration structure produced by this intense electromagnetic pulse. An electron cloud is formedin front of the pulse due to the pondermotive force of the electromagnetic wave. The electrons areaccelerated in front of the slope of the electron cloud, while protons in the backside. This structure isstable and propagates at the velocity very close to the light velocity.28

28

Figure 13. The ratio of the eigenmode frequency of the jet plasma to the excited electromagtntic pulse
frequency 𝜔𝜔p′ /𝜔𝜔 is larger than unity at the bottom of the jets (𝐷𝐷/𝑅𝑅0 𝑚𝑚), which means the mode is in

overdense plasma.. The overdense disturbances propagate as the Alfven or whistler modes, which use
the magnetic field as restoring force, though they are highly relativistic and non-linear. The ratio𝜔𝜔p′ /𝜔𝜔 decreases as D increases and eventually become oscillations in underdense plasma 𝜔𝜔p′ /𝜔𝜔 <1, beyond where Alfvenic waves turn itself as a propagating electromagnetic wave. Alfven/whistler

waves are nonlinearly as well as linearly converted to electromagnetic waves to produce theacceleration structure shown in Figure 12.

2.4. Wakfefield acceleration
We have seen that the rotating accretion disk accompanied by the magnetic fields
emanates a well-collimated extended jets that are wrapped around by the netting magnetic
fields. These astrophysical jets are one of the largest (if not the largest) and long-standing
eminent structure of the Universe (see e.g. Figure 6). This structure, we have observed inmany astrophysical objects, also accompanies additional structured entities on it. That
was the formation of wakefields that are created upon the jets. First, the presence of themagnetic fields in the jets allow the disturbances from the accretion disk (such as the
Alfven shocks) to be propagating in the plasma due to the physical properties of theelectromagnetic waves in magnetized plasma (see. e.g. Ichimaru, 197331). This allows the
EM disturbances created at or near the accretion disk can propagate through the(otherwise evanescent (𝜔𝜔 < 𝜔𝜔p′ ) plasma to the propagating regime of plasma (𝜔𝜔 > 𝜔𝜔p′ ).
Thus the presence of the magnetic field in the jets acts as a conduit of these intense EM
disturbances that may be created by the accretion disk.
The created intense EM waves (due in part its highly relativistic dynamics) haveoften highly relativistic group velocity. As has been discussed above, the EM waves withhigh group velocity is capable of creating an intense longitudinal plasma wave structure(Figure 12), whose phase velocity is also relativistic (close to 𝑐𝑐) (called wakefields.40,30 It
has been shown and demonstrated (experimentally and theoretically) that thesewakefields generated with high phase velocity are robust. The robustness arises from thefact that (1) the phase velocity close to the speed of light, by far, different from the thermalvelocities of plasma bulk, so that plasma remains quite stable and the excited waves
remain robust. In other words, by the time the plasma electrons respond to the wakefield
electric fields, with the electron speed less than 𝑐𝑐, the waves have escaped away from theresponding electrons, thus keeping this high amplitude plasma waves intact.
It remains also robust due to the fact that (2) electrons in this regime, even if theelectrons respond to the high phase velocity, all the electrons can move at (or near) the29speed of light, which is in fact what the wakfields are moving. so that these electrons, infact, reinforce the movement of the wakefields. This is called the relativistic
coherence41,42 and is the reason why the wakefields themselves are stable and robust, in
addition to the robustness of the observed astrophysical jets.
We see that astrophysical jets only eventually begin to form lobes at their ends when
the particles that propagate through the jets are now expended their energies to become
low enough to interact with the bulk plasma strongly where a variety of plasma
instabilities with the bulk plasma particles may take place.
As discussed above, the conditions for the strong acceleration by wakefield are:32(a) the acceleration structure (wave) is very close to the relativistic propagation velocity(phase velocity), i.e. the speed of light; and(b) the wave has a relativistic amplitude (i.e. the particles in the wave have a relativistic
momentum 𝑒𝑒𝑗𝑗 𝐸𝐸/𝜔𝜔 > 𝑚𝑚𝑗𝑗 𝑐𝑐 in one photon cycle). Where 𝑒𝑒𝑗𝑗 and 𝑚𝑚𝑗𝑗 are the charge andthe mass of the particle 𝑗𝑗, and 𝐸𝐸 and 𝜔𝜔 are the electric field and angular frequencyof the wave.

The condition (b) came from the fact that the significant acceleration by a ponderomotive
force of the electromagnetic wave takes place only in the relativistic case: This term
becomes significant only if the amplitudes of the electromagnetic waves become
relativistic.32 If these two conditions are satisfied, as shown by a number of groundexperimental demonstrations and considerations,43-47 this acceleration mechanism is morerobust to Fermi acceleration,48 in which charged particles gradually gain energy as they
are scattered many times by magnetic clouds. Here, it is worth noting that the coherencein the driver is not necessary in the acceleration. What is important is the focusing energy
in a pulse to achieve the condition (a). In the case of photons, the coherence in photons isnecessary to concentrate energy, which automatically and naturally creates accelerating
structure.
As the pulse of the electromagnetic wave is highly relativistic in the sense thatthe magnitude, 𝑎𝑎0 = 𝑒𝑒𝐸𝐸w /(𝑚𝑚𝑒𝑒 𝜔𝜔𝜔𝜔) of the normalized vector potential is much larger thanunity and the propagation velocity is nearly equal to the light velocity, the two conditions(a) and (b) for the strong wakefield acceleration are well satisfied. Figure 12 shows theacceleration structure (wake) produced by an electromagnetic pulse. An electron cloud isformed in front of the pulse due to the ponderomotive force of electromagnetic wave. The
electrons are accelerated in the front slope of the electron cloud, while protons in the slopeof the backside (between two clouds). This acceleration structure is stable and propagatesat the velocity very close to the light velocity.
30

The pondermotive force, 𝐹𝐹pm , for the electrons of the electromagnetic wave is aforce generated from the Lorentz force, (𝑣𝑣/𝑐𝑐) × 𝐵𝐵, in the propagation direction of theelectromagnetic wave. If the motion of the electrons by the wave is not relativistic (𝑎𝑎0 <1), it can be calculated as the force resulting from the average of the profiles of the electro-

magnetic pulses. In the relative regime (𝑎𝑎0 > 1), this force is more simplified. Since theparticle velocity approaches to the light velocity and the plasma satisfies the underdense(𝜔𝜔 > 𝜔𝜔p′) condition as well.

Since to the particle velocity asymptotically approaches to the light velocity andthat the plasma satisfies the underdense (𝜔𝜔 > 𝜔𝜔p′ ) condition as well, 𝐵𝐵 is equal to 𝐸𝐸 (𝐵𝐵 =𝐸𝐸). In this case, 𝐹𝐹pm , is given by
𝐹𝐹pm = 𝛤𝛤 2 𝑚𝑚e 𝑐𝑐𝑎𝑎0 𝜔𝜔p′ .

(50)

The accelerated electrons emit high energy gamma rays with the energies
ranging GeV-PeV due to the collision with the magnetic perturbations. On the other hand,
protons and ions are released into interstellar or intergalactic space to be cosmic rays. The
highest energy charged particles with 1019 − 1020 eV are observed as Ultra High Energy
Cosmic Rays (UHECRs). The cosmic ray protons may lose their energy through the p-pinteraction in the interstellar medium in the galaxy, the resultant pions produce neutrinos
through their decays.

Figure 14. The emitted radiations from such collimated jets can be highly collimated in the directionof the jets as its inherent genealogical mechanism. Luminosity would be larger for on-axis observed
31

objects (blazars) and smaller for off-axis sources (other sources).

2.4.1. Electrons and gamma-ray emissions
The energies of electrons, which are accelerated in the front side of the bow wake
(Figure 12) are transferred to gamma-rays through the collisions of electrons withmagnetic fields. In other words,
𝐿𝐿γ = 𝐿𝐿TCR = 𝜎𝜎𝐿𝐿w .

(51)

This assumption in electron spectrum is consistent with the typical blazer SED (spectrum
energy distribution) and variability.35
The gamma-ray is most likely beamed along the jets, which is consistent with thecorrelation observed in gamma-ray flux and apparent opening angle of the jets of AGNs.43
When the emission is beamed, the isotropic luminosities are overestimated for on-axis
sources (blazars) but underestimated off-axis sources (other AGNs and microquasars;
Figure 14).
2.4.2. Protons and ionsProtons and ions are accelerated by an electric field (back side: Figure 12) generated
by bow wakefield (longitudinal polarization of electronic distributions). The acceleration
force 𝐹𝐹acc is given by2𝑒𝑒

9√2𝜋𝜋𝑒𝑒 2 𝑐𝑐 2

1/4

𝐹𝐹acc = 𝑧𝑧𝐹𝐹pm = 9 �𝜇𝜇2 𝑚𝑚 2 𝜅𝜅 3𝑅𝑅 𝜖𝜖3�HT0𝑧𝑧𝛤𝛤2 𝜉𝜉 1/2 𝑚𝑚̇4/5

𝛼𝛼1/4 𝛽𝛽 5/8 𝑚𝑚1/4

𝐷𝐷

−3/4

�𝑅𝑅 𝑚𝑚�0.

(52)Here, 𝑧𝑧 is the charge of the particle. Charged particles are repeatedly accelerated by thisacceleration structure. The acceleration distance. 𝑙𝑙acc , of one acceleration event isdetermined by dephasing of the accelerating particles and estimated as𝑐𝑐

𝑙𝑙acc = 2𝛾𝛾p 2 𝜔𝜔′ .

(53)

p

where 𝛾𝛾p = 𝜔𝜔/𝜔𝜔p′ .40 Please note that 𝑙𝑙acc can be larger than the wavelength (~𝑐𝑐/𝜔𝜔p′ ) ofelectromagnetic disturbances in the acceleration region, where 𝜔𝜔p′ < 𝜔𝜔. This makes thewakefiled acceleration much more efficient compared with the conventional Fermi
32acceleration, in which the acceleration distance is much smaller than the wavelength ofthe electro-magnetic disturbance. In such a strong acceleration case, the energy spectrum
𝑓𝑓(𝑊𝑊) of the charged particles has a power law function of the exponent -2,49,50 in otherwords, 𝑓𝑓(𝑊𝑊) = 𝐴𝐴(𝑊𝑊/𝑊𝑊min )−2.
The maximum energy, 𝑊𝑊max , of the spectrum of the accelerated particles isassumed to be determined by integrating 𝐹𝐹acc from 𝐷𝐷0 through 𝐷𝐷1 as:𝐷𝐷2𝑒𝑒

9√2𝜋𝜋𝑒𝑒 2 𝑐𝑐 2

1/4

𝑊𝑊max = ∫𝐷𝐷 1 𝐹𝐹acc 𝑑𝑑𝐷𝐷′ /𝛤𝛤 = 9 �𝜇𝜇2 𝑚𝑚 2 𝜅𝜅 3𝑅𝑅 𝜖𝜖3�0

1/4

8𝑒𝑒 9√2𝜋𝜋𝑒𝑒 2 𝑐𝑐 2 𝑅𝑅 3=

= 9 � 𝜇𝜇2𝑚𝑚 2𝜅𝜅 3𝜖𝜖03 �HT

8√2𝑒𝑒 2 𝑅𝑅0 �𝛿𝛿 1/4 −1� 𝑧𝑧𝑧𝑧𝑚𝑚̇2 𝑚𝑚
𝛤𝛤 𝛽𝛽 1/2

9𝜇𝜇𝑚𝑚H 𝜅𝜅T 𝜖𝜖

H

T

0

𝑧𝑧𝜉𝜉 1/2 𝑚𝑚̇4/5 𝑚𝑚3/4.

𝛼𝛼1/4 𝛽𝛽 5/8

𝐷𝐷𝑧𝑧𝑧𝑧𝜉𝜉 1/2 𝑚𝑚̇4/5

𝐷𝐷1

1/4

1/4

��𝑅𝑅 1𝑚𝑚�
0

𝐷𝐷 ′

−3/4

�
∫ �
𝛼𝛼1/4 𝛽𝛽 5/8 𝑚𝑚1/4 𝐷𝐷0 𝑅𝑅 𝑚𝑚
𝐷𝐷− �𝑅𝑅 0𝑚𝑚�
0

�

0

𝑑𝑑𝑑𝑑′
(54)

where, 𝐷𝐷1 = 𝛿𝛿𝐷𝐷0 is the distance at which acceleration is finished by the depletion of thedriving photon field. Here, we can eliminate 𝑚𝑚̇ using equation 22 as2√2𝑒𝑒 2 𝜅𝜅𝑊𝑊max = 𝜋𝜋2𝜇𝜇𝑚𝑚 𝑐𝑐 6𝑅𝑅T 𝜖𝜖H0𝑧𝑧𝑧𝑧�𝛿𝛿 1/4 −1�𝛤𝛤𝛽𝛽 1/2 𝑚𝑚

𝐿𝐿rad 2 .

(55)

Figure 15, the accreting blackholes of 1 − 1010 𝑀𝑀⊙ can produce UHECRs with theenergy above 5.7 × 1019 eV by the wakefield acceleration.

33

Figure 15. The maximum attainable energies of accelerated particles by the mechanism of wakefield
acceleration, 𝑊𝑊max are plotted against the mass of the central AGN 𝑚𝑚 for and its time derivative 𝑚𝑚̇ =10−1 , 10−3 , and 10−5 . Accreting blackholes can produce ultrahigh energy cosmic rays UHECRs(above 5.4 × 1019 eV), if accretion rates are high enough.

Given the energy efficiency, 𝜎𝜎 , of charged-particle acceleration, including theconversion of Alfven wave into electromagnetic waves, the total cosmic ray luminosity,
𝐿𝐿TCR , is given by𝑊𝑊

𝑊𝑊

𝐿𝐿TCR = 𝜎𝜎𝐿𝐿w = ∫𝑊𝑊 max 𝑊𝑊CR 𝑓𝑓(𝑊𝑊CR )𝑑𝑑𝑊𝑊CR = 𝐴𝐴𝑊𝑊min 2 ∫𝑊𝑊 max 𝑊𝑊CR −1 𝑑𝑑𝑊𝑊CR
min

min

= 𝐴𝐴𝑊𝑊min 2 l n(𝑊𝑊max /𝑊𝑊min ).

(56)

In other words,

𝐴𝐴 = 𝑊𝑊

𝜎𝜎
𝐿𝐿w .
ln(𝑊𝑊
max /𝑊𝑊min )
min

(57)

2

The cosmic-ray number spectrum is represented by:
𝑛𝑛CR = ln(𝑊𝑊𝜎𝜎𝐿𝐿wmax /𝑊𝑊min )𝑊𝑊CR −2,(58)

where 𝑊𝑊CR is the energy of a cosmic-ray particle. The UHECR luminosity is given by:
𝑊𝑊

𝐿𝐿UHECR,th = ∫𝑊𝑊 max 𝑊𝑊CR 𝑓𝑓(𝑊𝑊CR )𝑑𝑑𝑊𝑊CR =0=ln(𝑊𝑊max /𝑊𝑊0 )ln(𝑊𝑊max /𝑊𝑊min )𝜎𝜎𝐿𝐿w = 𝜎𝜎𝜎𝜎𝐿𝐿w .

𝜎𝜎𝑊𝑊ln� max �
𝑊𝑊min

𝑊𝑊

𝐿𝐿w ∫𝑊𝑊 max 𝑊𝑊CR −1 𝑑𝑑𝑊𝑊CR
0

(59)

Here,
𝜁𝜁 =

ln(𝑊𝑊max /𝑊𝑊0 )

.

(60)

ln(𝑊𝑊max /𝑊𝑊min )Since the hotspots in northern sky are seen for the events above 5.7 × 1019 eV,51 we set
𝑊𝑊0 = 0.57 × 1020 eV.
34

Cosmic-ray particles are originally beamed toward the direction of the jets. Afterreleased from the jets into the interstellar space, they may be deflected by the interstellar
magnetic field. The Larmor radius, 𝐷𝐷L , of a cosmic ray particle is calculated as:
𝑊𝑊

𝐵𝐵

−1

CR
𝐷𝐷𝐿𝐿 = 𝑍𝑍𝑍𝑍𝑍𝑍𝑍𝑍= 1.1 × 103 [pc] 𝑍𝑍 �100 μG�

𝑊𝑊

�1020 eV�.

(61)

Even UHECRs above 1020 eV be significantly lose their orientation toward the jets
because of the magnetic field of ~100 μG in the starburst regions of the galaxies.52,53 For
the objects with prominent jets, such as AGNs or microquasars, UHECRs are likely
beamed toward the direction of the jets. The isotropic luminosities are overestimated for
on-axis sources (blazars) but underestimated off-axis sources (other AGNs andmicroquasars; Figure 16a). However, the objects without prominent jets such as Seyfert
galaxies and intermediate mass black holes in the starburst galaxies, the UHECRs areejected in the starburst region, in which many supernova remnants and molecular locate,the directions of the UHECRs even above 1020 eV are likely to be randomized (Figure16b). UHECRs are deflected by magnetic field in the intergalactic/interstellar space
between their source and the Earth. The deflection angles are estimated as:
𝑑𝑑

𝐵𝐵

𝑊𝑊

−1

(62)

𝜃𝜃 = 0.5°𝑧𝑧 �Mpc� �nG� �1020 eV� .

Since the magnetic field in the intergalactic/interstellar space is turbulent, the spot of
UHECR from a point source is likely to be spread by the same magnitude.

35

Figure 16. The emission of the protons and ions is originally beamed as shown in (a). But could be
deflected by interstellar plasma and magnetic fields of the star burst region when the momentum of
the jet is decelerated and thereby strongly interacting with the interstellar plasma and generated plasma
turbulence, to form the jet lobes (b).

2.4.3. Neutrino emissions
The cosmic ray protons, accelerated by the wakefield, may lose their energy throughthe p-p interaction in the interstellar medium in the galaxy, the resultant pions produce
neutrinos and gamma-ray photons through their decays. The mean free path, 𝐷𝐷pp for thisprocess is calculated as:
1

𝐷𝐷pp = 𝑛𝑛 𝜎𝜎

0 pp

−1

𝜎𝜎pp

= 1.2 × 104 [pc] �2.7×10−26 cm2�

𝑛𝑛

−1

0
�103 cm
,
−3 �

(63)

where we use 𝜎𝜎pp is the cross section of the p-p interaction and 𝑛𝑛0 is the proton densityof interstellar space in the galaxy.

The resultant pions produce neutrinos through their decays. The number spectrum 𝑛𝑛ν
of the neutrinos is calculated as:
𝑑𝑑𝑊𝑊p

𝑓𝑓pp 𝜎𝜎𝜎𝜎𝜈𝜈 𝐿𝐿w

𝑛𝑛𝜈𝜈 (𝑊𝑊𝜈𝜈 ) = 𝑛𝑛𝐶𝐶𝐶𝐶 (𝑊𝑊) 𝑑𝑑𝑊𝑊 = ln(𝑊𝑊𝜈𝜈𝑊𝑊max /𝑊𝑊min )𝐷𝐷

where 𝜎𝜎𝜈𝜈 = 𝑊𝑊𝜈𝜈 ~0.05 and 𝑓𝑓pp = 𝐷𝐷P

pp𝑊𝑊𝜈𝜈 −2(64)

is the probability of the p-p interaction. Here, 𝑊𝑊ν ,36

and 𝑊𝑊p are the energies of the neutrino and the proton, respectively. The neutrino

luminosity can be calculated as:𝑊𝑊𝑓𝑓pp 𝜎𝜎𝜎𝜎𝜈𝜈

𝐿𝐿ν,th = ∫𝑊𝑊 max,ν 𝑛𝑛𝜈𝜈 𝑊𝑊𝜈𝜈 𝑑𝑑𝑊𝑊𝜈𝜈 = 𝑓𝑓pp 𝜎𝜎𝜎𝜎𝜈𝜈 𝐿𝐿w = 6𝜖𝜖𝜖𝜖𝛽𝛽3/2 𝐿𝐿rad .
min,ν

(65)

For the objects with prominent jets, such as AGNs or microquasars, neutrinos are likely
beamed toward the direction of the jets. The isotropic luminosities are overestimated for
on-axis sources (blazars) but underestimated off-axis sources (other AGNs andmicroquasars: Figure 17 (a)). On the other hand, the objects without prominent jets suchas Seyfert galaxies54 and intermediate mass black holes in the starburst galaxies,55,56 thedirection of the protons are significantly randomized (Figure 17 (b)). In such a case,neutrino emissions can be also isotropic rather than beamed.

Figure 17. The neutrinos (~1015 eV) are produced from the high energy protons (1016 − 1017 eV)through p-p interaction (a). They are beamed if the original protons are beamed but could be
isotropized, since protons with the energy of 1017 eV could be defused in the interstellar plasma andmagnetic fields of the star burst region (b). Such transition of the jets with its collimated narrow waist

into slowdown jets with strongly interacting jets with the interstellar medium corresponds to thetypical picture shown in Figure 6.

2.5. Summary
The scaling law for the times scales, maximum energy and luminosities are shown in
Table 1. We can summarize the results of the bow wake acceleration theory in the jets
emanating from an accreting blackhole as follows:
The bow wakefield accelerates protons and nuclei well above the 1020 eV with the wide37

range of parameters to produce UHECRs (and Table 1).
1) The electrons are also accelerated by the bow wakefield exactly the same way as
protons. They emit gamma-ray photons through the interaction with electromagnetic
perturbations (Table 1).
2) The accretion of charged particles and emission of gamma-rays show prominent
variabilities with the various time scales shown in Table 1 due to the accretion
instabilities.
3) The bow wake acceleration theory can provide the luminosity estimates in UHECRs,
gamma-rays, and neutrinos. The comparisons of them with the observed isotropic
luminosities are not straightforward, if the emissions are strongly beamed.

38

3.

High-Energy Astronomy of blackholes with Multi-Messengers
Blackholes are first recognized as observable objects, when Greenstein and Schmidt
in 1964 reported the emission lines of radio stars 3C273 and 3C48 suffer from significant
red-shifts, which show that they locate rather far in cosmological distances and therefore
much brighter than normal stars, though they look compact, like stars.57 They named them
quasi-stellar objects, or “quasars”, in short. The following observations showed that they
are active galactic nuclei. Their high luminosities suggested that their central engines areblackholes swallowing (accreting) surrounding gas. The gravitational energy released bymass accretion onto a blackhole is assumed to be converted into the relativistic bulk
motion of the jets, though the details were unknown. The jets are found to emit nonthermal emission in almost all frequencies of photons, i.e., radio, infrared, optical, UV,
X-ray, and gamma-rays. They also exhibit violent variations in time scales in hours toyears. The direct imaging of the supermassive blackhole in the center of M87 was doneby the Event Horizon Telescope (Figure 7).17,58,59
The discovery of X-ray stars by Giacconi et al. in 196260 opened a new window to
study stellar-mass blackholes. In fact, a significant fraction of galactic X-ray sources are
considered to be blackholes with ~10𝑀𝑀⊙ in close binaries, where mass is accreting fromoptical components (normal stars) to the blackholes. They appeared to have jets with nonthermal emissions in addition to the thermal X-ray emission from the accretion disk
surrounding the blackhole. They also exhibit time variations in milli-seconds to years.61
Since these features suggest they are million times smaller version of the quasars,
astronomers are calling them now as microquasars.62
Now, we have a full set of accreting blackholes from stellar mass (~10𝑀𝑀⊙ : galactic
microquasars), intermediate (100-1000𝑀𝑀⊙ : ultra-luminous X-ray sources in the starburst
galaxies), and supermassive (106 − 1010 𝑀𝑀⊙ : active galactic nuclei). They share thecommon features, 1) relativistic jets, 2) non-thermal emissions of the full range of photon

spectra from radio to high energy gamma-rays ( > 1012 eV), and 3) violent time
variabilities. Although their natures were so far explained by phenomenological models
in the framework of the Fermi acceleration mechanism,48 they are a kind of patch works
with many parameters. A compact theory of the accreting blackhole/jet systems was longwaited for to unify three common features (relativistic jets, non-thermal emissions, andviolent time variabilities) of accreting blackhole systems. The acceleration mechanism
seems to be the key.
The accreting blackhole objects has been important targets of multi-messenger
astronomy, including cosmic rays (charged particles), high energy gammas, neutrinos,
39

and gravitational waves, as also mentioned above. First, there is growing evidence thatthe nearby starburst and Seyfert galaxies are the sources of ultra-high energy cosmic rays
(UHECRs). For the first place, the Telescope Array (TA) team suggested that there is a
hotspot in the northern sky where the directions of arrival of the UHECRs above 57 EeV.51
He et al. in 201663 pointed that starburst galaxy, M82, may contribute to the hot spot. In
addition, Pierre Auger Observatory (PAO) team reported that a hotspot in the southern
sky above 57 EeV, although it is less statistically significant compared with that in thenorthern sky.64 PAO team recently re-analyze the data in a bit lower energy and foundadditional enhancements of 4σ above 39 EeV toward two starburst galaxies (NGC253
and NGC4945) in the southern sky, too.65
These starburst (and Seyfert) galaxies with only less massive blackholes, 103 −106 𝑀𝑀⊙ but not supermassive (> 107 𝑀𝑀⊙ ), however, have a difficulty to accelerate up~1020 eV in the framework of Fermi acceleration. For example, Anchordoqui andSoriano66 found that acceleration is limited to at most 1019 eV for proton even in M82
even if they took into account of extensive shock and strong magnetic field in the nuclear
region of starburst galaxies. It is not enough for northern hot spot, in which proton
dominant composition was obtained by TA team.67
Second, the high energy phenomena associated with accreting blackholes has been
observed with gamma-ray emission. First, the Fermi Gamma-ray Space Telescope,formerly GLAST (Gamma-ray Large Area Space Telescope) launched in 200868 observed
accreting blackholes such as active galactic nuclei (AGN) and binary blackholes in the
GeV-100 GeV region.69-71 In addition, air Cherenkov telescopes, such as MAGIC,72
H.E.S.S.,73 or VERITAS74 and water Cherenkov detector such as High Altitude Water
Cherenkov (HAWC) observatory75 observed accreting blackholes in TeV and multi TeV
gamma rays.
In this section, we investigate various blackhole objects with the multi-messenger
approach, backboned by the wakefield acceleration (WFA) theory. We selected ten
astronomical objects, i.e., one blazars (BL: TXS0506+056), three radio galaxies (RG: CenA, M87, and For A), three Seyfert galaxies (SyG: NGC1068, NGC4995, NGC6814), two
starburst galaxies (SB: M82 and NGC0253), and one microquasars (MQ: SS433). The
loci of these ten objects are shown in 𝑚𝑚 − 𝐿𝐿rad diagram (Figure 18) and in skymap(Figure 19). The theoretical predictions of the WFA theory are compared with theobservations at source by source in the rest of the section 3.Although gamma-ray, proton,and neutrino emissions are likely to be beamed, as discussed in section 2, we will discussenergetics of the objects in terms of the isotropic luminosities because of the uncertaintyof the degree of the beaming of the emissions. Not that this assumption makes the40

observational luminosities overestimated in Blazers and underestimated in other sources,
as we will discuss for the individual objects.

Figure 18. Ten blackhole objects: one blazars (BL: TXS0506+056), three radio galaxies (RG: Cen A,
M87, and For A), three Seyfert galaxies (NGC1068, NGC4995, NGC6814), two starburst galaxies(SB: M82 and NGC0253), and one microquasars (MQ: SS433) are plotted in mass (𝑚𝑚)-radiation
luminosity (Lrad ) diagram. The solid line represents the locus of 𝑊𝑊max = 1020 eV. Seven objects arewell above the solid line, so that they can accelerate UHECRs. Dashed lines denote 𝑚𝑚̇ = 10−5, 10−3,
and 10−1, respectively.

Table 1. Comparison of the theoretical prediction by the WFA theory with the41

observations.

Figure 19. The skymap of the ten blackhole object examples that we consider in the present paper. BL:
blazar; RG: Radio Galaxy SyG: Seyfert Galaxy; SBG: Starbusrt Galaxy; MQ: Microquasar.

3.1. Blazars
3.1.1. TXS 0506+056
TXS 0506+056 in constellation Orion is a blazar, a quasar with a relativistic jet
pointing directly towards Earth, with a redshift of 0.3365 ± 0.0010,76 which corresponds
to about 1.8 Gpc from the Earth. TXS 0506+056 was first cataloged as a radio sourcein 1983,77 and then confirmed as a blazar.78 Gamma rays were detected by the EGRET
42

and Fermi-LAT missions.79-81 In addition, radio observations have shown apparent
superluminal motion in the jet.82
Furthermore, on 22 September 2017, the cubic-kilometer IceCube Neutrino
Observatory detected a high-energy neutrino emission from a direction consistent withthis flaring gamma-ray blazar TXS 0506+056.83 The most probable energy for theobserved neutrino is around 190 TeV with a 90% confidence level (CL) lower limit of
183 TeV, depending only weakly on the assumed astrophysical energy spectrum. Suchobservation may imply the existence of extremely high energy protons or nuclei with tensof PeV generated in the jet of the Blazar. The clear emission direction with high energy
particles may suggest a different acceleration mechanism for the ultrahigh energy cosmic
rays other than the Fermi's stochastic acceleration.48

Figure 20. The gamma ray blazar TXS 0506+056 shows quasi-periodic burst pattern in gamma-rays
with the energy between 0.1 GeV and 100 GeV.84 This quasi-periodic burst pattern may be explainedby the MRI instability of the accretion disk.37 After IceCube Collaboration 2018.84

During ±2 weeks of the neutrino observation, a peak flux of gamma ray emission
around 5.3 × 10−7 cm−2 s−1 is also reported by Fermi-LAT, with an energy range
between 0.1~10 GeV.81 The associated isotropic luminosity during the period reaches ashigh as 1.2 × 1047 erg s −1 .84 Analyzing the data prior to the event, a long-term isotropic
gamma ray luminosity between 0.1 GeV and 100 GeV is derived with an averaged valueof 0.28 × 1047 erg s −1 over 9.5 years of Fermi-LAT observations of TXS 0506+056.84
According to their study, the Gaussian-shaped time profile of the neutrino emission showsa quasi-periodic burst pattern as shown in Figure 20, which is similar to MRI instability
relaxation depicted by Canac et al.37
According to the WFA theory, the relativistic ponderomotive acceleration in the jet
can boost particles to an energy over ZeV.28-30,32 Also, because the particles areaccelerated linearly, the corresponding signal detected can be highly localized around thelocation of the blazar. In the rest of the subsection, we will estimate physical parameters
using the wakefield theory and compare them with their observational values.
Mass estimation is difficult in general for blazars. Padovani et al.85, however,estimated the central black hole mass to be 3 × 108 𝑀𝑀⊙ using the relations of black holes
43mass and R-band bulge magnitude 𝑀𝑀(𝑅𝑅)~ − 2.9,86 assuming the host galaxy to be a
giant elliptical. We adopt 1.7 × 1045 erg s −1 as the bolometric luminosity (≃ 𝐿𝐿rad ) ,taking into account of the overestimation due to jet-induced component using thebolometric luminosity derived from the the OII and OIII lines (LOII = 9 × 1045 erg s−1
and LOIII = 3 × 1045 erg s −1 . Substituting 𝑚𝑚 = 3 × 108 and 𝐿𝐿rad = 1.7 ×1045 erg s−1 into equations 29, 30, 51, and 54, we derived 𝜏𝜏rise,th = 3.0 × 102 , 𝜏𝜏rec,th =6.9 × 104 s, 𝐿𝐿γ,th = 1.5 × 1044 erg s−1, and 𝑊𝑊max = 4.6 × 1026 eV, as shown in Figure

15 and Table 1.

Although the maximum proton energy, 𝑊𝑊max for protons is well above 1020 eV,
UHECR protons propagate only 100 Mpc and cannot reach to the Earth because of the
GZK mechanism,87,88 in which protons undergo inelastic collisions with photons in thecosmic microwave background to their loose energy.
According to the MAGIC observation,89 very high energy (VHE) gamma rays,
above 90 GeV, from TXS 0506+056 varied, increasing by a factor of 6 within a day to
8.7 × 10−11 erg cm−2 s −1 , which corresponding to the isotropic luminosity of
𝐿𝐿γ,obs,TeV = 4.6 × 1045 erg s −1 . We may set 𝜏𝜏rise,obs = 4.8 × 104 s or shorter as an e-raising time. There are two periods (2017 October 3-4, and 2017 October 31) of enhanced
gamma-ray emission. We may set 𝜏𝜏rec,obs as 2.4 × 106 s or shorter, taking into accountthe incomplete observation in TeV gamma rays. Both 𝜏𝜏rise,obs and 𝜏𝜏rec,obs are consistent

with the theoretical prediction as shown in Table 1.
The theoretical gamma-ray luminosity is calculated as 𝐿𝐿γ,th = 1.5 × 1044 erg s−1(see equation 51), which is much less than the observed isotropic gamma-ray luminosity
𝐿𝐿γ,obs,GeV = 1.2 × 1047 erg s −1 observed by FERMI-LAT.84 This is probably due to theconcentration of the radiation being axially aligned with the jets. Since jets are close
parallel to the line of site in the case of blazars and the actual emission is strongly beamed
to the direction of the jets, such isotropic assumption will overestimate the luminosity byintegrating the whole sphere by a large factor. If we assume a diameter of 3° for the beam,
the corrected estimation of the Fermi-LAT luminosity will be 2 orders magnitudes lower
than their reported value. That is, Lγ,obs (3°) = ~3 × 1044 erg s−1, which is consistent
with the theoretical luminosity.
Substituting 𝑓𝑓pp = 1.0, 𝐿𝐿rad = 1.7 × 1045 erg s −1 , 𝑑𝑑 = 1.8 Gpc, σ = 0.1, α = 0.1,
ϵ = 0.06 , 𝜎𝜎𝜈𝜈 = 0.05 into equation 65, we obtain 𝐿𝐿ν,th = 6.7 × 1042 erg s−1 . On theother hand, the observation by IceCube suggested isotropic neutrino luminosity of
𝐿𝐿ν,obs = 1.5 × 1046 erg s −1 at 100 TeV,84 which is much higher than theoretical
expectation, probably due to the beaming, described above.

44

3.2. Nearby radio galaxies in the Local Super-cluster
Radio galaxies in the local supercluster, such as M87, For A, and Cen A can beconsidered possible source of UHECRs.
3.2.1. Cen A
Centaurus A (also known as NGC 5128) in constellation Centaurus, is a radio galaxy
with kilo-parsec size jets, and at a distance of 3.4 Mpc from the Earth.90-92 The mass ofthe central blackhole is well defined by stellar dynamics93 as 𝑀𝑀BH = 5.5 × 107 𝑀𝑀⊙ . Xray observations determined 𝐿𝐿rad = 2.3 × 1042 erg s −1 .94 As the distance of Cen A isestimated as 3.4 Mpc,95-97 Cen A, as well as NGC4945 can contribute to the excess flux.
In fact, the Pierre Auger Observatory reported that there may be excessive fluxes towards
the Centaurus region (Figure 21).64,65

Figure 21. Wakefield acceleration theory suggests three hot spots in the arrival direction of UHECRs.
They are consisted of the contributions from four galaxies: M82, NGC0253, Cen A, and NGC4945(a). The UHECRs from Cen A and NGC4945 cannot be distinguished. The arrival direction analysis
of UHECRs above 39 EeV suggested three hot spots (b). After Aab et al. 2018.65

Substituting 𝑚𝑚 = 5.5 × 107 and 𝐿𝐿rad = 2.3 × 1042 erg s −1 into equations 29, 30,
51, 55, and 59, we derived 𝜏𝜏rise,th = 4.0 × 10−1 , 𝜏𝜏rec,th = 1.3 × 104 s , and 𝐿𝐿γ,th =2.0 × 1041 erg s −1 , 𝑊𝑊max = 4.6 × 1021 eV, and 𝐿𝐿UHECR,th = 6.7 × 1039 erg s −1 ,The WFA theory predicts that Cen A is capable of accelerating protons to energies
45

above 1020 eV (see Table 1). The deflection angle of UHECRs due to intergalacticmagnetic field is estimated as small as 14 degrees, using equation 62 and shown in Table
1. The UHECR luminosity is predicted to 6.7 × 1039 erg s−1 . On the other hand, Aab etal.63 suggested UHECR flux of 0.016 event/(100 km2)/yr, which corresponds to theisotropic luminosity of 6.4 × 1037 erg s −1 (see Table 1). The difference between thetheoretical prediction and the observation may be explained by the beaming toward jets:in the radio galaxies, jets are not parallel to the line of sight unlike blasars, so that theisotropic assumption leads an underestimate by a large factor.
The episodic recurrence time, 𝜏𝜏rec,th,predicted by WFA theory is consistent with theobservation of Fukazawa et al.98 and Rothschild,99 who showed 50% time variability inthe time scale of 10-20 ks.
The theoretical gamma-ray luminosity for Cen A is calculated as 𝐿𝐿γ = 2.0 ×

1041 erg s −1 . On the other hand, Cen A has been known to emit gamma rays in the rangeof GeV and TeV energy, First, Fermi LAT detect gamma-ray flux of 3.0 ×
10−9 erg cm−2 s −1 and photon index of 2.763, which corresponds to the GeV luminosityof 1.7 × 1040 erg s −1.66 Second, the H.E.S.S. telescope determined the gamma-ray fluxof 0.45 ± 0.07 10−13 ph cm−2 s −1 TeV −1 at 1 TeV. The corresponding gamma-ray
luminosity calculated as 1.1 × 1038 erg s −1 .100 The difference between the theoreticaland observational values of 𝐿𝐿γ is most likely due to the fact that the axis of the jets is at alarge angle to our line of sight, and gamma-ray emissions are strongly beamed in theaxially direction of the jets.

3.2.2. M87
M87 is an elliptical galaxy in the constellation Virgo with the distance of 17 Mpc.101
It the nearest radio galaxy with the central blackhole of 6.2 × 109 102 has the coreluminosity of 7 × 1040 erg s −1 in normal phase103 and 1.5 × 1042 erg s −1 in flaring
phase.102 Substituting 𝑚𝑚 = 6.2 × 109 and 𝐿𝐿rad = 1.5 × 1042 erg s −1 into equations 29,
30, 51, and 55, we derived 𝜏𝜏rise,th = 0.27 , 𝜏𝜏rec,th = 1.4 × 106 s , 𝐿𝐿γ,th = 1.3 ×

1041 erg s−1 , and 𝑊𝑊max = 1.7 × 1019 eV as shown in Table 1.
The Event Horizon telescope17,58,59 first observed the ring structure of the radius of

21 μas, corresponds to the linear diameter of 5.3 × 1015 cm at the distance of 16.9 Mpc
by the VLBI observation at 230 GHz. Since the size of the ring is just consistent with theradius of the innermost circular stable orbit (5.5 × 1015 cm) of the accretion disk of a
non-rotating blackhole with the mass of 6.2 × 109 𝑀𝑀⊙ , this structure corresponds to theinner edge of the accretion disk.17,58,59 obtained the average electron density of𝑛𝑛e ~104−7 cm−3 , magnetic field strength of 1-30 G, and electron temperature46

of (1 − 12) × 1010 K . These values are consistent with the radiatively-inefficient andgeometrically-thick accretion flow inside the ISCO of the radiatively-efficient andgeometrically-thin accretion disk, which we described in section 2.
It is difficult of M87 to accelerate UHECRs, since 𝑊𝑊max is less than 1.9 × 1019 eVeven in the flaring phase (Table 1). Furthermore, a branch of the local-supercluster
filaments different from that toward M82 extends towards M87 from the Milky Way
Galaxy (Figure 22). Therefore, the path of UHECR from M87 to our galaxy is also
considered to be within the filament structure. In fact, galaxies in this direction, and ourgalaxy, is “falling” towards M87.105 The distance to M87 is as large as 17 Mpc,101 which
is about four times larger than that of M82, the deflection due to the magnetic field would
exceed 80 degrees and the spot would spread at least several tens of degrees (Table 1).
Thus, any excess flux from M 87, even if it exists, is difficult to detect with current
ground-based detectors.
According to Ackermann et al.,70 the GeV gamma-ray flux of 1.7 × 10−9 cm−2 s −1
and photon index of 2.17, which corresponds to the isotropic GeV gamma-ray luminosityof 6.5 × 1041 erg s −1 . On the other hand, the WFA theory predicts the gamma-ray

luminosity of 1.3 × 1041 erg s −1 , which is consistent within a factor of two (Table 1).
According to Aharonian et al.,106 M87 nucleus emit TeV gamma-ray radiation of

3 × 1040 erg s −1 and showed the variability of one day (~8 × 105 s) as well as one year(~3 × 107 s). We may set the upper limit of the times scales as 𝜏𝜏rise,obs < 8 × 105 s,
𝜏𝜏rec,obs < 3 × 107 s, taking into account of the incompleteness of the observations.

Figure 22. Galaxies in the supergalactic plane. The filament structure of the local supergalaxy can be
seen.
47

3.2.3. For A (NGC1316)
Fornax A (NGC1316) is a radio galaxy locates in the constellation Fornax with the
distance of 19 Mpc107-109 and harbors the central blackhole of 1.3 × 108 𝑀𝑀⊙ 110 with theluminosity of 1.7 × 1040 erg s −1 .111 The maximum acceleration energy 𝑊𝑊max is as low
as 1.7 × 1017 eV.
Substituting 𝑚𝑚 = 1.3 × 108 and 𝐿𝐿rad = 1.7 × 1040 erg s −1 into equations 29, 30,
51, and 55, we derived 𝜏𝜏rise,th = 3.0 × 10−3 , 𝜏𝜏rec,th = 3.0 × 104 s, and 𝐿𝐿γ,th = 91.5 ×1038 erg s−1 , and 𝑊𝑊max = 1.1 × 1017 eV , as shown in Table 1. It does not have an
ability to produce UHECRs.
On the other hand, Ackerman et al.112 detected the extended emission of
5.38 × 10−12 erg cm−2 s −1 and photon index of 2.08, which corresponds to the isotropic
luminosity of 3.7 × 1040 erg s −1 . The WFA theory predicts the gamma-ray luminosity of
9.1 × 1038 erg s −1 , which is roughly consistent with the observation within factor offive.
3.3. Nearby Seyfert galaxies
Seyfert galaxies harbor profound nuclear activities with non-thermal emissions. They
are believed to be due to the jets emanated from the accretion disk around a withsupermassive blackhole with a mass of 106−7 𝑀𝑀⊙ (Table 1).54 The nuclear regions arecovered by Compton thick clouds, ionized by the interaction with the jets and excessive
star formation activities, including numerous numbers of supernovae; Many Seyfert
galaxies still exhibits the surplus starburst activities and are often difficult to distinguishfrom starburst galaxies.
3.3.1. NGC1068
NGC1068 is a Seyfert galaxy which has a bright nucleus with the central blackholeof 1.6 × 107 𝑀𝑀⊙ .111 It locates at the distance of 14 Mpc114 in the constellation Cetus. The

bolometric luminosity of the nucleus of 1.8 × 1045 erg s −1 was obtained from OIV
line.106 It is surrounded by a Compton thick cloud with 𝑁𝑁H > 1025 cm−2.115,116 It canaccelerate UHECR but the deflection by intergalactic magnetic field is too large (70degree) to form a hotspot.
Substituting 𝑚𝑚 = 1.6 × 106 and 𝐿𝐿rad = 1.8 × 1045 erg s −1 into equations 29, 30,
51, and 55 we derived 𝜏𝜏rise,th = 3.2 × 102 , 𝜏𝜏rec,th = 3.6 × 103 s , 𝐿𝐿γ,th = 1.6 ×
1044 erg s−1 , and 𝑊𝑊max = 1.0 × 1028 eV, as shown in Table 1. The nucleus of NGC
1068 cannot accelerate UHECRs.
48

There is no significant luminosity change in the intrinsic luminosity from theaccretion disk of the nucleus of NGC 1068. Although Zaino et al.117 reported the time
variability in the time scale of 1-6 months, the detailed spectral analysis revealed that thevariability is not due to the change in the intrinsic accretion rate but due to the change in
obscuring Compton thick cloud (NH > 1025 cm−2),114 which surrounds the nucleus. Thisview is consistent with the infrared and optical observations.118,119

Ackermann et al.71 observed isotropic gamma-ray luminosity of 1.5 × 1041 erg s−1 .
GeV gamma rays are generally believed to be from supernova remnants in the galaxy as
the result of intense starburst activity.
On the other hand, Acciari et al.120 set an upper limit of NGC 1068 in gamma-ray
above 200 GeV at 5.1 × 10−13 erg cm−2 s −1. It corresponds to the isotropic luminosityof 3.5 × 1045 erg s −1 , which is one order of magnitude large compared with thetheoretical prediction. On the other hand, the WFA theory predicts the gamma-ray

luminosity of 1.6 × 1044 erg s −1
The IceCube collaboration121 reported the positive detection of neutrinos at 1 TeV of

3 × 10−13 TeV −1 cm−2 s −1 from NGC 1068 (Figure 23). It corresponds to the isotropic
luminosity of 𝐿𝐿ν,obs = 1.8 × 1040 erg s −1 . Substituting 𝐿𝐿rad = 1.8 × 1045 erg s−1 ,𝑓𝑓pp = 1.0 , σ = 0.1 , 𝜎𝜎𝜈𝜈 = 0.05 , 𝛼𝛼 =0.1, and β = 20 into equation 65, we obtain theisotropic theoretical neutrino luminosity of 𝐿𝐿ν,th = 8.0 × 1042 erg s−1.

Figure 23. Seyfert galaxy example in the skymap. Local pre-trial p-value map around the Seyfert
galaxy NGC1068. The most significant point in the Northern hemisphere coincides with NGC1068(black cross). After IceCube Collaboration (2020).121

3.3.2. NGC4945
NGC4945 is a Seyfert galaxy with an enhanced star forming activity. It is located at
the distance of 3.6 Mpc96 from the Earth. It harbors the central supermassive blackholewith the mass of 1.1 × 106 𝑀𝑀⊙ .113 The X-ray emission from the core is heavily obscured

by Compton thick cloud with 𝑁𝑁𝐻𝐻 > 1024 cm−2 , like other type 2 Seyfert galaxies 115,122
and the unabsorbed luminosity is estimated as 3 × 1042 erg s −1 .122
49

Substituting 𝑚𝑚 = 1.1 × 106 and 𝐿𝐿rad = 3 × 1042 erg s−1 into equations 29, 30, 51,
55, and 59, we derived 𝜏𝜏rise,th = 5.3 × 10−1 , 𝜏𝜏rec,th = 2.5 × 102 s , 𝐿𝐿γ,th = 1.3 ×

1041 erg s−1 , 𝑊𝑊max = 4.0 × 1023 eV, and 𝐿𝐿UHECR,th = 8.9 × 1039 erg s−1 , as shown in
Table 1.
On the other hand, Aab et al.65 suggested UHECR flux of 0.016 event/(100 km2)/yr,
which corresponds to luminosity of 7.2 × 1037 erg s−1 assuming isotropic emission(Table 1). The difference between theoretical prediction and the observed isotropic
luminosity can be explained by the beaming of UHECRs. Note that the arrival direction
analysis of UHECRs cannot distinguish the contribution of NGC4945 from that of Cen A(Table 1 and Figure 21).
According to Ackerman et al.71 the GeV gamma-ray isotropic luminosity of

1.2 × 1040 erg s −1 . On the other hand, the WFA theory predicts the gamma-ray
luminosity of 2.6 × 1042 erg s −1

3.3.3. NGC6814
Seyfert galaxy NGC6814 locates at the distance of 23 Mpc114 and harbor the central
blackhole of ~3 × 106 𝑀𝑀⊙ .114,123 The luminosity of the nucleus source is as high as

2.7 × 1040 erg s −1 .124 Substituting 𝑚𝑚 = 3.0 × 106 and 𝐿𝐿rad = 2.7 × 1040 erg s −1 into
equations 29, 30, 51, and 55, we derived 𝜏𝜏rise,th = 4.7 × 10−3 , 𝜏𝜏rec,th = 6.9 × 102 s ,𝐿𝐿γ,th = 2.4 × 1039 erg s −1 , 𝑊𝑊max = 1.1 × 1039 eV . As shown in Table 1, it cannot

accelerate UHECRs. According to Ackermann et al.,70 the GeV gamma-ray flux of

6.8 × 10−10 cm−2 s −1 and photon index of 2.54, which corresponds to the GeV gammaray isotropic luminosity of 6.9 × 1040 erg s −1 . Tennant et al.125 observed the three foldvariability within 200 seconds. On the other hand, the WFA theory predicts the gammaray luminosity of 2.4 × 1039 erg s −1 .

3.4. Intermediate mass blackholes in starburst galaxies
Starburst galaxy has an enhanced star forming activity, which often caused by strong
disturbances due to the interaction with other galaxies. The starburst region sometimes
covers the entire galaxy (~10kpc ) or is constrained to a part of the galaxy acrosstypically~1kpc. Thick molecular clouds with the proton density higher than 100 cm−2
are pervaded in the starburst region. As a result of this starburst, many young supernova
remnants as well as compact young star clusters are embedded in the molecular cloudcomplex. Ultra-Luminous X-ray Sources (ULXs), which are believed to be intermediate
mass blackholes (IMBHs) with mass ranging 102 − 104 M⊙ . They have been discoveredin the starburst galaxies and are likely to be formed in young compact star clusters through50

runaway merging:126-128 The blackhholes heavier than 102 M⊙ are impossible to be

produced by stellar evolution, directly. Ebisuzaki et al.129 however suggested intermediate
mass blackholes with 100 − 1000 M⊙ can be formed through runaway merging ofmassive stars at the center of dense stellar cluster126-128 to be ultra-luminous X-ray (ULX)
sources in the starburst galaxies.
In a starburst galaxies, UHECR protons can be randomized significantly, because of

the enhanced magnetic field (~100 μG) due to high concentration of supernova remnants,
since the magnetic field in the starburst region is estimated as high as 100 − 1000 μG.52,53
Even UHECRs, which are originally beamed, can be significantly randomized indirection because of the magnetic field in the starburst regions.
3.4.1. M82
M82 is a starburst galaxy of the distance of 3.6 Mpc130 in the constellation Ursa
Major. The starburst activity takes place in a relatively small central region, radius of~200 pc131 from the dynamic center of the galaxy.
Ultra-luminous X-ray (ULX) sources of luminosity ≥ 1040 erg s −1 inside M82
have been observed (see Xu (2015)132 and the references therein). Among them, M82 X1 is the brightest ULX in M82, located about 200 pc away from the dynamic center of thegalaxy.133-138 In general, the maximum luminosity of a star bound by gravity is limited by
the Eddington luminosity, defined by:
𝐿𝐿Edd = 1.26 × 1038 [erg s −1 ]𝑚𝑚.

(66)

Thus, it can be concluded that M82 X-1 with an X-ray flux of 2 × 1041 erg/s or more 133135
must have a mass of at least around 103 𝑀𝑀⊙ . In addition, the mass of the M82 X-1

estimated from the frequency of the Quasi Periodic Oscillation (QPO) observed in the Xray luminous intensity as to be 100-1300 𝑀𝑀⊙ .139-141 Such a blackhole with mass of
102 − 105 𝑀𝑀⊙ is called an intermediate mass blackhole,129 and is considered to be an

important key for solving the mystery of formation of the central blackholes byconnecting a stellar mass blackhole ( ~103 𝑀𝑀⊙ ) and a supermassive blackhole(≫ 106 𝑀𝑀⊙ ) in galactic centers.131 Here, we adopted the mass of 400𝑀𝑀⊙ of Pasham etal., who used QPO frequency to a mass value.141

Substituting 𝑚𝑚 = 4 × 102 and 𝐿𝐿rad = 2 × 1041 erg s −1 into equations 29, 30, 51,
55, and 59, we derived 𝜏𝜏rise,th = 3.5 × 10−2 , 𝜏𝜏rec,th = 9.1 × 10−2 s , 𝐿𝐿γ,th = 1.8 ×1040 erg s−1 , 𝑊𝑊max = 4.8 × 1024 eV, and 𝐿𝐿UHECR,th = 5.8 × 1038 erg s −1 , as shown in
Table 1.

51

Applying WFA theory to the M82 X-1, it has been shown that acceleration to
10 eV is well feasible (Figure 15, Figure 21, and Table 1) in the accreting blackholesystem. On the other hand, the TA team acquired 72 cosmic rays of 57 EeV in 5 years,
and observed 19 events (4.5 events of which were expected from uniform arrival) within
the hotspot.51 Since the effective area of TA is 700 km2, the observed excess flux in the20

hot spot direction is about 0.4 UHECRs/(100km2/yr), which correspond to the isotropic
luminosity of 1.8 × 1038 erg s −1 , which is roughly consistent with the theoreticalprediction (Table 1).
If the hot spot in the northern sky comes from M82, the deflecting angle byintergalactic magnetic field is 17.4 degrees.63 Using equation 62, M 82 (𝑑𝑑 = 3.6 Mpc) is
the source of a proton (z=1), each requires a magnetic field of the order of 𝐵𝐵 = 9.7 nG.
In ordinal intergalactic space, the magnetic field strength is expected as low as 0.1 nG.142
Therefore, this large deflection angle given by observation could not be explained.
However, Ryu et al.143 carried out the simulation of the local large-scale structure of
the universe and found that the magnetic field of about 10 nG can be expected in thefilament structure. Figure 22 shows the distribution of galaxies close to the supergalactic
plain (within ± 1 Mpc). The distribution of galaxies represents the network (filament
structure) of the local supergalaxy to which our galaxies belong. It can be seen that our
Milky Way Galaxy and M82 are in the same filament structure. Therefore, the magnetic
field of the UHECR propagating path from M82 to the Milky Way Galaxy can be expected
to be about 10 nG, which is higher than that of the ordinary intergalactic space. It is
expected that the UHECR propagating through at distances of 3.6 Mpc is deflected bynearly about 18 degrees, as shown in Table 1.
He et al. 61 divided the events belonging to the northern hot spot into two by energy,
and found that there was a systematic deviation between them. Assuming that this is dueto the deflection by the magnetic field,144 the position of the true source was estimated.
While the estimated position, though extended to 10 degrees, included several highenergy celestial objects such as M82 and Mrk 180, only M82 was located within the GZKhorizon (~100 Mpc) that the UHECRs could reach.
The Telescope Array (TA) team detected 72 cosmic rays of 57 EeV in 5 years. Amongthem, 19 events are within the hot spot,51 while 4.5 events were expected from uniform
arrival. Since the effective area of TA is 700 km2 , the observed excess flux in the hot spot
direction is about 0.016 event/(100 km2)/yr, which is consistent with the expected
isotropic flux from equation 𝐿𝐿UHECR,obs = 1.8 × 1038 erg s−1 as shown in Table 1. The
direction of UHECRs may be randomized, due to the strong magnetic field inside M82.
The QPO period of M82 X-1 is observed in X-ray band to be 0.2 s.141 The theoretical52

recurrence time is 𝜏𝜏rec,obs = 9.1 × 10−2 s (Table 1). This predicted value is wellconsistent with the QPO period within a factor of 2.
In the WFA theory, electrons are also accelerated in the similar way with protons.
The high energy electrons, accelerated by the wakefield in the direction of the jet, emit
gamma rays by synchrotron process with the interaction with magnetic perturbations in
the jets. This gamma-ray luminosity of M82 X-1 can be also calculated as
1.7 × 1040 erg s −1 (Table 1). The energy spectrum is likely to be expressed by a single
power law from GeV to 100 TeV, with a constant index, which is close to 2 in the strongest
acceleration case, depending on the magnitudes of the acceleration field and the magnetic
field in the jets.37 Note that this gamma-ray emission is expected to be strongly
concentrated in the direction of the jets.
On the other hand, a bright and isolated gamma-ray excess, consistent with thelocation of the position of M82 of 100 MeV to 700 GeV gamma-rays that are isotropic inluminosity of 6.9 × 1040 erg s −1 with FERMI-LAT.71 This is consistent with thetheoretical prediction, though it might be just by chance, taking into account of the nonisotropic nature of WFA theory.
Gamma-ray emission of 3.7 × 10−13 cm−2 s−1 above 700 GeV were detected from
M82.145 The corresponding isotropic luminosity is about 2 × 1039 erg s −1 . The fitted
power-law spectrum suggests that a single physical emission mechanism, such as WFA
theory, dominates from GeV to TeV energies for M82, though a popular explanation isthose from numerous number of supernova remnants in the nucleus region of starburst
galaxy M82.
According to Ackerman et al.,71 time variabilities have not yet been reported from
Fermi-LAT observations. If it is the case in future observations, that will be an evidence
that gamma-ray emission comes from a compact object, such as M82 X-1, not from theextended sources, like supernova remnants.
3.4.2. NGC0253
NGC0253 is a nearly edge-on starburst galaxy located in the constellation Sculptor
at the distance of 3.9 Mpc146 from the Earth (Figure 22). It harbors at least three ULXs
with the luminosities ranging between (2.4 − 4.1) × 1039 erg s −1 .147 The sum of the

ULXs is 0.91 × 1039 erg s −1 . We adopt 600𝑀𝑀⨀ as the mass of an ULX of NGC0253

from the argument of the Eddington limit. Substituting 𝑚𝑚 = 6 × 102 and 𝐿𝐿rad =9.1 × 1039 erg s −1 into equations 29, 30, 51, 55, and 59, we derived 𝜏𝜏rise,th =1.8 × 10−3 , 𝜏𝜏rec,th = 1.4 × 10−1 s , and 𝐿𝐿γ,th = 8.8 × 1038 erg s −1 , 𝑊𝑊max = 7.9 ×
1021 eV, and 𝐿𝐿UHECR,th = 5.4 × 1037 erg s −1 , as shown in Table 1.
53

Gutierrez et al.148 proposed two candidates for the source of UHECRs, one is TH2149
and the other is NGC253 X-1. Although TH2 was presumed the brightest radio sourcenearly coincident to the center of the galaxy, the recent observation by ALMA revealed
that the position of TH2 exactly coincides to one of the knots in the central region of NGC
0253, which are most likely HII regions excited by young compact star clusters. The massof the clusters are estimated as ~106 𝑀𝑀⊙ and are not likely to have any blackholes, since

there are no X-ray emissions. Although one may still assume a hidden non-accreting blackhole in the cluster, any blackhole without accretion cannot emit any energy.

Gutierrez et al.148 assumed a strong magnetic field of 104 G around the blackhole toproduce jet luminosity through the Blandford-Znajek effect.148 However, this magnetic
field will decay rapidly if no accretion on the blackhole. The luminosity in equation 3 ofGutierrez et al.148 can not sustain without a certain amount of accretion. In conclusion,
TH2 is unlikely to be a source of UHECRs.
Ultra-Luminous X-ray Sources (ULXs), on the other hand, are promising as UHECR
sources, such as NGC 0253 X-1, if we take into account wakefield acceleration. NGC
0253 harbors at least three ULXs with the luminosity ranging between (2.4 − 4.1) ×

1039 erg s −1 .147 The sum of the luminosities of the ULXs reaches 9.1 × 1039 erg s−1 asshown in Table 1. They are considered to be intermediate black holes with masses that
range from 102 − 104 𝑀𝑀⊙ .

The maximum energy of protons are estimated to be 𝑊𝑊max = 1.2 × 1021 eV evenless massive black holes ( ~600 𝑀𝑀⊙ ) can generate UHECRs by WFA. Aab et al.65
reanalyzed the data of arrival direction observed by Pierre Auger Observatory (PAO) andfound that a significant (4σ level) enhancement in the arrival direction map of UHECRs
above 39 EeV with the search radius of 12.9 degree toward NGC0253. This corresponds
to the flux of 0.013 UHECRs/(100 km2 yr), which corresponds to 𝐿𝐿UHECR,obs =5.4 × 1037 rg s −1 . This is consistent with the theoretical isotropic luminosity of
2.7 × 1037 rg s −1 (Table 1 and Figure 21). Taking into account of rather large
uncertainty of theoretical prediction, discussed above. The result of Aab et al.65 is
consistent with the data of Telescope Array team, though statistically marginal.65,151
The episodic recurrence time is estimated by the WFA theory to be 𝜏𝜏rec,th =1.4 × 10−1 s. Barnard et al.147 reported significant variabilities can be seen in 100 second
bin, which are much longer when compared with the theoretical predictions. Since
observations for short time variabilities (less than seconds) have unfortunately not been
done for the ULXSs in NGC 253, the theory is not constrained by the observations.
The theoretical gamma-ray luminosity is estimated as 𝐿𝐿γ,th = 7.9 × 1038 erg s −1 .

The observed gamma ray luminosity (isotropic) is 6.0 × 1039 erg s−1 in 1-100 GeV71 and54

2.8 × 1039 erg s −1 above 220 GeV, which is one order of magnitude higher than the
expected gamma-ray flux of ULXs by the wakefield acceleration theory. Directcomparison of 𝐿𝐿γ luminosity with theory is difficult, however, due to other contributions
from other supernova remnants].152

3.5. Galactic microquasars
Galactic blackhole binaries, such as SS433, Cyg X-1, Cyg X-3, Sco X-1 exhibit
relativistic jets, violent variabilities in time scales ranging from milliseconds to years andemit radiation from radio to high energy gamma rays (~ TeV; Figure 24).153,154 Becauseof such non-thermal phenomena, they are considered counterparts of quasars(~106−9 𝑀𝑀⊙ ) in million times smaller scales with masses of ~10 𝑀𝑀⨀ , in other words,
micorquasars, and yet we find they are capable of generating high energy gamma rays
and UHECRs.
Our Milky Way Galaxy has a number of maicroqasars, which are X-ray binaries
containing stellar mass blackholes (~10 𝑀𝑀⊙ ),155 Many of them are brightest sources in

the X-ray sky with the luminosity of 1037 − 1038 erg s-1,156,157 such as Sco X-1, Cyg X1, Cyg X-3, GX5-1, and GX339-4 (Figure 24), exhibiting strong time variabilities all thetime scales from milli-second to months.158,61,159 Most of them have well collimated jets
just like those of AGN but million times smaller in their scale and emit gamma-rays andeven high energy gamma-rays.160,161 Those observational facts shows that an efficientmechanism must work at these stellar size blackholes. One of the candidates would thewakefield acceleration, which accelerates charged particles to 1018-1020 eV in their jets,
as shown in
Figure 15. When they are in dense molecular cloud, the accelerated charged particles
may interact with protons in the cloud to produce neutrinos. Figure 24 shows the sky mapof such galactic microquasars. They might contribute to the enhancement along thegalactic plane and galactic center in the neutrino arrival directions.
In the rest of the section, we will discuss SS433 in detail.

55

Figure 24 Skymap of microquasas..

3.5.1. SS 433
SS 433 is a galactic binary system consisting of a supergiant star 𝑀𝑀 = 10 − 30 𝑀𝑀⨀
and a compact object of 𝑀𝑀 = 2 − 3 𝑀𝑀⨀ (commonly considered to be a black hole) in theconstellation Aquarius. The distance to the S433 system was estimated as 3.5 kpc162 andis located inside of the supernova remnant W50, which exploded 20-40 thousands years
ago.163

Figure 25. Galactic microquasar SS433. Since the jets are precessing, a pair of cork screw shapes areformed (Credit: B. Saxton from data provided by R.M. Hjellming, K.J. Johnston, NRAO/AUI/NSF).164

SS433 emits jets that have an approximate length of 40 pc, and a bulk velocity of
0.26c.165-166 The two precessing jets model is well established (Figure 25).167-171
Kubota et al.172 determined the mass of the compact object from orbital analyses tobe 2.5𝑀𝑀⨀ . According to Cherepashchuk et al.173 and Abeysekara174, the jet luminosity is
as high as 1040 erg s −1 , because of super-critical accretion, in spite o low luminosity(1035−36 erg s −1) in X-ray band.175
Substituting 𝑚𝑚 = 2.5 and 𝐿𝐿rad = 1040 erg s −1 into equations 29, 30, 51, 55, and 59,
we derived 𝜏𝜏rise,th = 1.8 × 10−3 , 𝜏𝜏rec,th = 5.8 × 10−4 s , 𝐿𝐿γ,th = 8.8 × 1038 erg s−1 ,𝑊𝑊max = 1.9 × 1024 eV, and 𝐿𝐿UHECR,th = 2.9 × 1037 erg s−1 as shown in Table 1.

According to the WFA theory, SS433 is capable of accelerating protons, and thus
UHECRs. The UHECRs produced in SS433, may not be much localized unfortunately,
since it is located near the galactic center, where the magnetic field is higher comparedwith the outer region. It may produce a broad (more than several ten degrees)
concentration toward the galactic center, together with other microquasars in the galactic
center region.
56

We can also estimate the theoretical recurrence time as 𝜏𝜏rec,th = 5.8 × 10−4 s

according to wakefield theory (Table 1). Although a significant variation in flux at thetime scale of 10 s was observed,176 there is no information in the millisecond range.
The theoretical gamma-ray luminosity is calculated as 𝐿𝐿γ,th = 8.7 × 1039 erg s −1.
Since gamma rays are strongly beamed in the direction of the jets, they are not necessarily
seen from Earth; our line of site is not aligned with the jets. The angle between our lineof site and the axis of the jet precession is about 74 degree177 and the precession angle isabout 20 degree.
SS 433 has been observed to emit gamma rays. First, careful analysis of data fromthe Fermi gamma-ray observatory Large Area Telescope reveals that SS433 system emits

extended gamma rays of 2.8 × 10−11 erg cm−2 s−1 in 500 MeV-10 GeV by Fermi-

LAT.178 The corresponding isotropic luminosity is about 1.0 × 1035 erg s−1 . It also
showed a modulation of ~10−10 erg s −1 correlation with the precession period.179 The
corresponding isotropic luminosity is 3.6 × 1037 erg s −1 . This component may be related
to the gamma rays emitted from the electrons accelerated by wakefield in the jets, assuggested by Tajima et al.30 in 2020, though the observed flux is much less comparedwith the theoretical prediction for the case of isotropic emission.
At 20 TeV, the HAWC detector in Abeysekara et al.174 reported the emission is
spatially localized in the three lobes (e1, e2, and w1), 40 pc away from the SS433
system.174 The corresponding isotropic luminosity is 1.1 × 1030 erg s −1 . Since the lobes
are located where jets interact with the nebula gas, the gamma rays can be explained by
the synchrotron emission from high energy electrons accelerated in the wakefiled in thejets, colliding with the magnetic field of ~10μGauss produced by the interaction of jetswith nebula clouds.
3.6. Background (isotropic) fluxes of UHECR and neutrinos
The cosmic ray spectrum summing-up the contributions of such numerous numbersof UHECR sources (starburst galaxies and AGNs, distant than 10 Mpc, like NGC1068)
over the entire sky is estimated as follows. First, the cosmic-ray flux 𝐽𝐽CR per unit energy,
area, time, and solid angle of cosmic-ray particles is calculated as:
𝐽𝐽CR =

𝑐𝑐𝑙𝑙𝛾𝛾 𝜏𝜏CR
4𝜋𝜋

= 1.8 ×

𝑊𝑊 −2
𝑊𝑊

−2

10−28 [particles⁄(GeV cm2 s sr)] �1019 eV�

57

𝑙𝑙

𝜏𝜏

𝛾𝛾
CR
�1038 erg s−1
� �3.4×10
9 yr�,(Mpc)−3(67)where 𝑙𝑙𝛾𝛾 is the local gamma-ray luminosity function of blazars, estimated as 1037 −

1038 erg s−1 (Mpc)−3 , taking into account of the beaming effect of the relativistic
jets180,181 and 𝜏𝜏CR the lifetime of the cosmic-ray particles.182 The expected spectrum of
cosmic-rays is consistent with the observational flux of UHECR.

If a fraction 𝑓𝑓 of the cosmic rays lose their energies through p-p interaction in thegalaxies, resultant pions produce neutrinos through their decays. According to Waxman
and Bahcall,183 the total background accumulated flux of neutrinos through the entire

history of the universe can be calculated as:
𝑓𝑓̅

𝑙𝑙

𝜏𝜏

𝛾𝛾
𝑊𝑊𝜈𝜈2 𝐽𝐽ν = 1.1 × 10−8 [ GeV ⁄(cm2 s sr)] �0.02� �1038 erg s−1� �1.5×10𝜈𝜈10 yr�
(Mpc)−3(68)

where 𝜏𝜏𝜈𝜈 is the neutrino accumulation time, which is the order of Hubble time (~1010 yr).
Here, we assumed neutrinos are produced with the average energy of 𝑊𝑊𝜈𝜈 = 0.05𝑊𝑊p by

one p-p collision. The best-fit single power-law parametarization for the astrophysicalenergy spectrum results in a normalization of 1.44 × 10−18 [GeV cm−1 s−1 sr −1 ] at 100

TeV and the spectral index of 2.37 from 15 TeV to 5 PeV,184 which is consistent withequation 68 for 𝑊𝑊𝜈𝜈 = 100 TeV = 105 GeV.
3.7. Summary

1) The time scales (𝜏𝜏rise and 𝜏𝜏rec ) of variabilities are clues to clarify how the WFA
theory works in the astronomical objects, though the observation intervals are too
sparse to constrain the theory in the current stage.
2) The theoretical luminosities are consistent with the observed isotropic luminosities
in the sense that the former is lower than the later in on-axis source (blazars), while
the former is higher than the later in off-axis sources (other than blazars), taking into
account the beaming effects.
3) UHECR and neutrino emissions could be partially isotropic from the sources withthe enhanced star formation (Seyfert and starburst galaxies), because of therandomization of the proton beam be interstellar magnetic field.
4) The observed background components of UHECRs and neutrinos are consistent of
those predicted by WFA theory with AGN luminosity functions.

585) Gamm-ray emissions from the jets by the WFA theory likely show time variabilities,
by which one can distinguish them from those from the supernova remnants
pervading in the star forming regions.
4.

Neutron star mergers and core collapse of massive stars
Both neutron star mergers and core collapse of massive stars harbor neutrino
dominated accretion flow (NDAF) disk, which emit neutrinos rather than photons.185 An
NDAF disk is present for about 1-3 seconds, until all the mass is sucked into the central
black hole of the 2-3 solar masses. If the mechanism similar to bow wake acceleration
takes place in NDAF disk, which has 𝐿𝐿w ~1051−52 erg s-1, or about 6-7 orders ofmagnitude higher than that of photon-cooled accretion disk, the UHECR can be expected
to be formed by acceleration from a far stronger wakefield than that of radiation dominantaccretion disk.15 We will discuss wakefield acceleration in NDAF disks in the rest of thesection.
4.1. Neutron star mergers (Short gamma-ray bursts)
GW170817 and GRB 170817A mark the discovery of a binary neutron star (BNS)
merger detected both as a gravitational wave (GW)18 and a short-duration gamma-ray
burst (SGRB).186 Detecting GW radiation from the coalescence of BNS and neutron star(NS)– black hole (BH) binary systems has been a major goal of the LIGO187 and Virgo188
experiments.
The astrophysical implication of a joint detection of a SGRB and GWs from a BNS
merger is the confirmation that binary neutron star mergers are the progenitors of at least
some SGRBs. GW170817 has been identified as a GW signal from the in spiral of two
low-mass compact objects, interpreted as the first GW observation of a BNS
coalescence.18,189 It was first observed by a low-latency search190,191 on 2017 August 17
at 12:41:04 UTC by a single-detector trigger in the LIGO-Hanford detector.189
GRB 170817A was independently and simultaneously detected in-orbit by the GBM
light software known as “triggering.”192 the signal exceeded 5𝜎𝜎 in three (of twelve) GBM
NaI detectors. The GBM detection showed two distinct components, the triggering pulse,that lasts about half a second and fell consistent with expectations for SGRBs, followed
by a subsequent weaker emission that lasted a few seconds.192 The GBM light-curves
show the two distinct components, as can be seen in the top two panels in Figure 26.
The GBM time-tagged event data is binned to match the SPI-ACS temporal resolution(100 msec) and phase (matching bin edges) to allow for an easier comparison between
the gamma-ray instruments.
59

Figure 26. The joint, multi-messenger signal of gravitational wave (GW170817) and gamma-ray burst(GRB 170817A) is shown. GW170817 was followed by GRB 170817A by 1.74 seconds. Top: thesummed GBM lightcurve for sodium iodide (NaI) detectors 1, 2, and 5 for GRB 170817A between 10
and 50 keV, matching the 100 msec time bins of the SPI-ACS data. The background estimate fromGoldstein et al.192 is overlaid in red. The middle panel shows the signals in the 50–300 keV energy
range. The SPI-ACS lightcurve with energy range beginning at 100 keV and with a high energy limitof least 80 MeV.

The bottom frame shows the time-frequency map of GW170817 coherently

combining LIGO-Hanford and LIGO- Livingston gravitational wave data. All times here arereferenced to the GW170817 trigger time 𝑇𝑇0 GW.186

In one of the early theoretical works on the formation of the binary neutron60

start merger fireball and subsequent formation of an accretion disk and its associated jet
formation may be seen in Figure 27 of Takahashi et al.193
We can have a strongly differentially rotating accretion disk from another kindof compact gravitational object such as the merged neutron star and another neutronstar.191 We show the latter case in Figure 27. In this work, Takahashi et al.193 clearly
predicted simultaneous emissions of gamma-ray burst and gravitational waves. In one of
the latest multiples of astrophysical observational signals (in this case, in addition to theoptical signal), the NS-NS collision was observed by the arrival of the gravitational
wave18 and simultaneous (delayed only by 1.74 second) arrival of gamma burst.186

Figure 27. A schematic picture of the model of neutron star coalescence and associated high energy
phenomena including gamma-ray burst after Takahashi’s model.193 The NS-NS collision created anaccretion disk and its associated jets. While the collision of the compact massed neutron starts may
emit gravitational waves, subsequent formation of the accretion disk and jets may give rise to the61electron acceleration and emission of gamma bursts, generating high energy phenomena including
gamma-ray burst. After Takahashi et al. (2000).193

The neutron stars gradually approach each other with the emission of gravitational
waves, and when the lighter star reaches its Roche limit, it is destroyed by the tidal forceof the heavier star, to form an accretion disk. This accretion disk becomes a neutrinocooled accretion disk, NDAF (Neutrino Dominated Accretion Flow),185 rather thanelectromagnetic wave, because of the high temperature (~109−10 K) and density(1013−15 protons cm-3). Such an accretion disk is present for about 1-3 seconds until all
the mass are sucked into the central blackhole of the 2-3 solar mass. If a mechanism

similar to the bow wake acceleration does work in 𝐿𝐿w ~1044−45 erg s −1, which is 6-7
orders of magnitude higher than that of radiation-cooled accretion disk, the UHECR canbe expected to be formed by acceleration by a far stronger wakefield than that of radiation
dominant accretion disk, which we discussed mainly in the present paper. On the other
hand, the coalescence of neutron star should be accompanied by the dispersal of a large
amount of materials processed in the star interior. When a beam of accelerated particles
enters the materials, ultra-high energy (1020−22 eV ) neutrinos can be emitted by theinteraction of the beam and the materials. Since most of the energy used to accelerate
electrons is emitted as gamma rays, there is also a possibility that a burst of gamma rays

is produced. The delay of the gamma rays to the emission of gravitational waves is
consistent with the time required for the NDAF disk and jets to form. The formation of
the NDAF disk and the formation of the acceleration field due to the coalescence of theneutron stars will be discussed in a separate paper.
Kato et al.15 investigated such a NDAF disk around a merged neutron star/blackholewith the extremely high accretion rate of 0.01-10 𝑀𝑀⊙ /s. They concluded that collimatedrelativistic jets emanated from the NDAF disk. The total expansion energy reaches

1051 − 1051 ergs, depending on the total accretion mass. The expansion energies arecomparable to those required for the gamma-ray bursts as well as core-collapsed

supernova (Figure 28). The wakefield acceleration in the jets produces ultra-high enegy

protons up to 1022−24 eV. When a beam of accelerated particles enters the materials,

ultra-high energy (1020−22 eV) neutrinos can be emitted by the interaction of the beam
62

and the materials. Since most of the energy used to accelerate electrons is emitted as
gamma rays, there is also a possibility that a burst of gamma rays is produced. The delayof the gamma rays to the emission of gravitational waves is consistent with the timerequired for the NDAF disk and jets to form. They also obtained the neutrino spectrum
thermally emitted from the NDAF disk (Figure 29).

Figure 28. The explosion energy of the jets as a function of the accreted mass, supplied byelectromagnetic waves from a neutrino driven accretion flow (NDAF) disk. After Kato, et al., 2022.15

63

Figure 29. Neutrino luminosity spectrum as a function of the neutrino energy thermally emitted froma neutrino driven accretion flow (NDAF) disk. After Kato et al., 2022.15

4.2. Core collapse of massive stars (Long Gamma-ray bursts and Super-/Hyper-novae)
A stellar-mass blackhole is formed in the final stage of massive star’s life, when astar implodes on itself.194 Type Ib/c and Type II supernovae,195 most hypernovae196 andlong-duration gamma-ray burst197,198 are believed to be produced by the gravitationalenergy released when the core of a massive (~8 − 10𝑀𝑀⊙ or greater) star collapses down

to a neutron star or blackhole. If the specific angular momentum of the core is higher than
4.6 × 1016 [cm2 s −1 ]�𝑀𝑀BH /3𝑀𝑀⊙ � . According to MacFadyen and Woosley,198 thespecific angular momentum is less than this critical value in the most part of the coremade by the stellar evolution of a single star. However, the stellar number density in the
nucleus of a young compact stellar is so large that massive stars (several ten solar) aresinking down by the dynamical friction among stars to form a binary. The binary
continues to shrink through three-body interaction of lighter stars and eventually merge
into a massive star before the supernova explosion.126-129 Suzuki et al.199 performed thesimulation of the merging and found the core of the merging remnants rapidly rotates in
the almost break-up rotation: the specific angular momentum of the core is as high as641020 cm2 s −1 . When such a rapidly rotating core of the massive star undergoes the corecollapse, the most of the materials cannot fall into the blackhole but to form accretion
disk around it.
The connection between long-duration Gamma-Ray Burst (GRN) and a particularclass of core-collapse Supernovae (SNe) has been established200 with the discovery ofbright SNe (so called “Hypernovae”) in positional and temporal coincidence with the timenearest GRBs.201,202 The spectra of the SNe are similar to one another, and characterizedby P-Cygni lines with broad absorption components, indicative of the presence ofmaterial expelled at high velocities. These SNe are thought to be the result of the
explosion of the carbon-oxygen core of a massive star, which had lost its outer hydrogenand helium envelopes prior to the collapse of the core. Analysis of the spectra and thelight curves of these GRB-SNe suggests that they are all energetic explosions, whichejected large quantities of matter: typically the explosion energy is 𝐸𝐸~5 × 1052 erg,
which is about 50 times larger than in normal core-collapse SNe, and ejected mass is~10 𝑀𝑀⊙ . Because of the high energy, these have been also called “Hypernovae”.
5.

Discussions

5.1. WFA in comparison with Fermi mechanism and shock acceleration
The recognition that the magnetic fields play an important role in astrophysics isgaining strength. It is no longer gravitational fields alone that are the engine of the
Universe organization and its morphology. The work such as Uchida and Shibata (1984)4
showed clearly that the magnetic fields are essential that drive the formation of jets
emanating from an accretion disk and push the plasma along the jets over the cosmic
distance way far greater than the size of the thickness or even the diameter of the accretion
disk. Another work that made such recognition stronger is by Balbus-Hawley (1991).21,22
Yes, there have been classical works preceding these such as Alfven (1981)1 andChandrasekhar (1961).203 The structure forming ability of magnetic fields (in addition to
the gravitational fields) and thus associated evolution have been summarized by Tajima
and Shibata (1997; TS97).10 In this review we are making one step beyond the structureformation to recognize that magnetic fields and their active dynamics can serve togenerate high energy phenomena of the Universe, such as high energy cosmic rays, burstsof high energy gamma rays and neutrinos.
The magnetic fields emanating from the accretion disk is spiraling out along the axisof the jets (or rather more precisely, the disk-driven spiral magnetic fields and their shear
structure drive the emanation of plasma out of the accretion disk to form the jets). Thus
65the magnetic fields are primarily stretched along the axis of the jets (with some associatedpoloidal components) by this jet forming mechanism due to the magnetic stress thatpropels the jet plasma (Figs. 1, 2, 5, and 7). Such a structure may be regarded as shear
Alfven waves itself, or a structural foundation of the jets’ magnetic fields. This overallmagnetic shear structure is in fact the reason why the jet plasma has been emanated fromthe root of the accretion disk and accelerated along the jest axis. Sometimes these jet
motions give rise to even relativistic and highly relativistic jet flows (we codified such by
the symbol of Γ in such places as in Eq. 44 in the above). [Sometimes, also, such a disevolution may happen if and when a couple of compact objects such as neutron star andneutron star collide (as have been considered in Sec. 4.1).]
What we have studied and summarized above in terms of high energy acceleration
processes is the additional dynamics on top of this approximately “steady state” structural
evolution of the accretion disk-jet systems. As Balbus and Hawley21,22 demonstrated andfurthered by Matsumoto and Tajima in 1995,24 the accretion disk with magnetic fields can
exhibit growing magnetic fields within the disk due to its differential azimuthal motionof the disk, which amplifies the toroidal magnetic fields in a relatively short time (in thescale of the Universe’s time). When such outbursts of magnetic fields growth happen in
the disk, the originally gravitationally equilibrated disk motion (Keplerian or its
relativistic cousin) becomes no long a sustainable dynamically equilibrium of the disk
plasma. This outburst can give rise to an additional accretion not shown by the BlandfordZnajek150 (steady accretion disk rate of plasma accretion) across the radial infall. Thisphenomenon and quantitative enhanced infall of a large chunk of plasma mass were
quantitatively analyzed by Matsumoto and Tajima.24 This causes, according to Ebisuzaki
et al. in 2014,28,29 large disturbances on the jet forming magnetic fields that emanate fromthe inner edge of the accretion disk and make huge disturbances on the jet magnetic fields.
Since the jets magnetic fields are primarily the axial component (let us call Bz ), the infall
large accretion of the plasma chunk would cause radial disturbances on the jet magnetic
fields primarily. It is important to recognize that this is a large motion of shear magnetic
field perturbation, characterized as shear Alfven motions. (This is unlike thecompressional squeeze of magnetic flux, which would cause compressional Alfven
waves.) Thus we conclude that the evolution of the disk’s magnetic fields trigger thelarge shear bending (and its propagation) along the axial direction of the jets. In otherwords, these are huge shear Alfvenic pulse propagation along the jets. The shear Alfven
pulse propagates along the axial magnetic fields of the jets. Depending on the accretion
conditions, disk conditions, and the nonlinear outcome of the Balbus-Hawley disturbance,
the generated shear Alfven pulse parameter would vary, but in general it propagates with66

the group velocity of the Alfven speed 𝑣𝑣𝐴𝐴 (which under certain condition tends to close
to be 𝑐𝑐 ). As the jet stretches from the root of the disk to away from the disk, the density
and magnetic fields etc. decrease and in general the Alfven speed increases, experiencing
the mode conversion from shear Alfven waves into ordinary electromagnetic pulse.28,29
The shear Alfven wave and its derivative of EM wave pulses are thus generated andpropagate along the jet plasma. This shear (or transverse) electromagnetic (EM) fields(that propagates close to the speed of light c) are capable of causing huge wakefields thatare primarily composed of the longitudinal electric field visa ponderomotive forces of the
EM fields.40,28,29 Thus, our mainline accelerating process is vis the shear perturbation ofmagnetic fields and their derived transverse EM fields. The wakfields are their outcome.
Although details vary depending upon the parameters and precise astrophysicalconditions, the generic dynamics of this mechanism is the transverse EM wave generation
with the group velocity close to 𝑐𝑐. If and when the group velocity of the EM waves isclose to 𝑐𝑐, the wakefields and the accelerated high energy particles whose velocities areclose to c) accelerated particles can resonate and ride on the phase with the wakefield forlong time, as particles and waves are synchronized (this may be seen by the energy
amplification factor of 2γ2, where γ is the Lorentz factor of the group velocity of the EM
pulse.40 Though probably minority, some particles (such as high energy component ofprions (and ions) can stick along the jet for over cosmic distance. Most of other particles
may drop out of this synchronicity with the waves, but still enjoy a directed ride over the
wakefields. This implies that the generated electrons (and gamma rays subsequently) andions (such as protons) and their derived neutrinos are directly largely along the jet
direction upon their generation and temporarily also synchronized.
E. Fermi48 in 1951 considered a multitude of particles (such as ions) entering a regionof magnetic fields. He showed that the particle bending caused by each encounter of theparticle with the magnetic field bends the ion transverse to the magnetic field andsuperposition of such magnetic bending of the particles stochastically accumulated cangain its overall energies. As we see, the fundamental acceleration of Fermi mechanism isto the transverse to the magnetic field direction. A subsequent derivative of the Fermi
stochastic acceleration has been the magnetized shock acceleration. Particle encountering
a bursting magnetic shock (with the Mach number 𝑀𝑀 ) may see the shock with its

propagating speed of 𝑀𝑀�𝑣𝑣𝐴𝐴 2 + 𝑐𝑐𝑠𝑠 2 , which is the propagation speed of shock in amagnetized plasma with s being the sound speed. Fundamentally, the magnetize shock (acompressional Alfven mode) is basically a transverse to the magnetic field lines and acompressional mode.

67

5.2. Implications for future missions/projects
The current mechanism as described above is inherently coherent process between
the excited waves (EM-like waves and their wakefields) and the particles that are to beaccelerated (electrons and their derived gammas, positrons, protons and ions, and theirderived particles such as neutrinos). Though the presence of intergalactic magnetic fields
could deflect ions (and protons) orbits before arriving to the Earth, at least these particles
were emanated from a point source (as compared to the size of their traveling distance).
Thus one of the derived conclusive characteristics is the point like emission of various
species of emissions of particles. Not only spatially correlated, but also can be temporarily
correlated.
On the other hand, there are also possibilities that some less coherent process ofacceleration along the propagation of the jets, but from the immediate vicinity of theaccretion disk via wakefields is possible.204 Emitted accelerated particles from such a
process are less energetic and less aligned in the direction than the that in the jets. Thus,
we surmise that these particles comprise lower energy portions of the spectrum.
Following the discussion in Sec. 5.1, we can view that the future observations canbegin to separate the mechanisms of emission of radiation of the currently discussed one
and the others. We will begin to learn the degree of coherence and lack of it by studying
the future missions carefully with the following points of view. First, without thepresence of plasma dynamics (i.e. only gravitational dynamics), radiations are mainly
stemming from the bright thermonuclear start burning surface and its ensembles (such as
galaxies). If and when we have both gravitational objects as well as the presence ofmagnetic fields in the plasma, while the gravitational interactions make stars and globallycondensed objects such as galaxies and their clusters, with plasma the galactic accretion
disks evolve in far more dynamically rich and enhanced modes. First, the presence ofplasma allows emission of various additional radiations (even in electromagnetic
radiations include Cherenkov radiation, synchrotron radiation and Bremsstrahlung under
the non-collective (linear) mechanisms.205 Each of these radiations (not nonlinear ones)is traceable to the emission by a single particle (such an electron). Thus the amount ofradiation is proportional to the simple sum of these particles and thus is proportional to
the number N of such particles. However, as we have seen in our review, the presence ofplasma and magnetic fields in that now open up farm richer behavior of the plasma
dynamics and thus the observationally apt phenomena (such as specific and enhanced
radiations). If the plasma creates a dynamics that coherently amass a lump of 𝑁𝑁 electron
excess (charge of −𝑁𝑁𝑁𝑁 ) in the plasma, which would force another clump of charge (nowpositively) excess with +𝑁𝑁𝑁𝑁 in the other area. The collective interaction in such a plasma
68amounts to the strength in proportion to 𝑁𝑁 2 . This squared intensity of the interaction is
the origin of the superior intensity of the plasma collective phenomena and their radiation
signatures from the collectively interacting phenomena. The wakefields wield is anintense collective phenomena, as they are the relativistic dynamics and this is further
strengthened by the relativistic coherence.41
On the other hand, consider a chaotic or disorganized ensemble of random N charges.
We understand that their average fluctuations of charges are proportional to √𝑁𝑁. Thus
the strength of mutual interactions of such random charge fluctuation is square of the sqrt

of 𝑁𝑁 , which is proportional to 𝑁𝑁 . In other words, the random chaotic plasma (noncoherent plasma) exhibit interaction strength (and thus its signature to the emissivity) is

proportional linearly to 𝑁𝑁. This is similar to the simple (non-interactive) linear addition
due to the single particle dynamics (such as Cherenkov emission, synchrotron emission)
we considered before we considered collective phenomena.
We now understand that plasma’s ability of collectively interactive clumps(wakefields included) can emit far stronger radiative signatures than those by linear
interactions. This is one of the most important characteristics of collective phenomena ofplasma in the Universe, as that means we can either farther objects that contain and emit
collective radiation or more intense and detailed patterns if not far away. By employing
different bandwidths of radiation, we may thus capture a broader spectrum of plasma
nonlinear evolution and dynamics intimately. We also note that since the emitted signals
from collective phenomena are brighter as 𝑁𝑁 2 , we can receive those signals far away indeeper places than the linear phenomena. In addition, it reveals the interplay of theastrophysical object’s gravitational interaction (or structure) and plasma dynamics. Thisway, we shall learn more details of the Universe’s dynamics and evolution.
One example is the case of a neutron star- neutron star collision, as discussed in Sec.
4.1. If NS and NS did not collide, such a dense compact gravitational object would emit
neither gravitational radiation, but nor intense gamma ray emission. However, when NS
and NS collide and erupt plasma and its disrupted dynamics (such as jets, disk and otherstructural features, as discussed in Sec. 4.1) commences, intense plasma waves andwakefields are excited, which are capable of making intense and coinciding gamma
emission. We can well imagine in the future that various combinations of emissions fromplasma’s strong collective interactions, particularly those of wakefields.
6.

Conclusions
We have reviewed the astrophysical basis of the wakefield acceleration in the Universe.
This mechanism may arise in a variety of circumstances and astrophysical objects.
69However, a standard case may be instructive to find the basic understanding, so that wefirst focus on the typical astrophysical accretion disk and its emanating jets. The major
astrophysical energy that drives such an object is the gravitational energy. The central
object, the accretion disk, is driven by its central gravitational object (be it a black hole,or other heavy central gravitational object(s) (AGN (active galactic nuclei))) thatmaintains the rapidly rotating accretion disk (made up with plasma and embedded
magnetic fields in it). These combinations are known to form: (1) emanating jets by its
magnetic fields (such as by Uchida-Shibata mechanism4-7), (2) created and enhanced
shearing magnetic fields driven by the magneto-rotational instabilities (such as the
Balbus-Hawley instability21,22 and the Parker instability23), which sometimes episodic
large inflow of accretion from the disk plasma toward the root of the jets. In this review
we have described and clarified how such processes can make up fairly predictive
dynamical outlook (not chaotic behaviors, but rather dynamically predictive
consequences, as sometimes called the structure forming universe; TS9710). These areinteresting and fascinating aspects of the predictions of this model of acceleration
processes in astrophysical plasma. This is partially due to the fact that our process’
definitive drive is directly related to and orchestrated by the well-organized central
gravitational object.
We are suggesting this as a potentially relevant physical process and mechanism,
which may be an emerging latest theoretical model as a possible alternative to the existingmodels such as the Fermi acceleration48 (and its derived or related shock acceleration204),
magnetic reconnection-driven acceleration,6,10,205-216 etc. that have been considered for along period preceding our current scheme. In the Fermi acceleration the stochastic
cumulative accumulation of the magnetic field kicks that are randomly moving plasma
containing embedded magnetic fields. In a related shock acceleration (again
accompanied with magnetic fields embedded in the shocks) the magnetic kicks may bedone via the moving plasma shocks (a fast moving supersonic plasma dynamics). In amagnetic reconnection related acceleration the accelerating mechanism may be arisingfrom the snapping plasma flow motion accompanied by the topologically reconnected
flux causes fast reconnected flows (again related to the mass motion as its fundamental
velocity). These can give rise either cumulatively large energies or some violent motions
that may arise. All of this involves the energy conversion from the plasma bulk motion,i.e. ions are moving, which may be called acoustic type (both ions and electrons partakein their motions fundamentally. Not electronic type. On the other hand, the wakefieldprocess (though it may be superposed on the supersonically moving on the jet motion
itself) fundamentally is the electronic only process, where magnetic flux is not any more70

tied with the plasma, but electromagnetic pulses may be of a freely propagating photon
nature (This a kind that is freed from the magnetic flux frozen to the plasma (i.e. ionicmotion’s dictation to the central dynamics)). That is, the photons are driving at the levelof the speed of light so that only electrons can keep up with the moving electromagnetic
pulse motions. (Though this mechanism is more tightly bound by the original motherhoodof the accretion disk and its jets, it also has nature’s stochasticity in such aspects as its
phase of the wakefields,49 which determines the power law nature of spectrum accelerated
by the wakfield generated in Nature.) This point makes some important manifestations
and unique areas of applications in astrophysics as well as unique phenomenology that
may not be arising in the past models. We will list some of the interesting aspects.
In contrast to these conventional mechanisms, the wakefield acceleration is based on
more organized motions in the Universe (though it may accompany some chaotic natures
such as the wakefield phases). The most typical is from the primordial galaxy (i.e.
accretion disk and its plasma rotation-driven magnetic fields) assisted organized (often)
large scale motion in the Universe.6,10 Characteristics are such as the well-extended jets
(over many kps or even Mps) and their accompanying magneto-rotational propulsion of
the jet plasma,4 accompanied by large magnetic field pulses driven by the magnetorotational episodic disruption,20 of the accreting matter onto the jet’s magnetic flux. This
causes huge and relativistically propagating cosmic electromagnetic wave pulsations,2830
which drives the wakefield generation. Even though this is among the largest size
phenomena, similar mechanisms can be at work in smaller scales.
As an important special case of such, Takahashi et al.191 conjectured that a collisionof two compact neutron stars that form a rapidly rotating accretion disk and emanating
jets from the collision can form a tiny version of wakefield acceleration along its jets andhigh energy particles (electrons ending up with gamma emissions). This in fact forms abasis of the simultaneous emissions of gravitational waves from the two colliding neutronstars and associated gamma emission.186 This was discussed in Sec. 4.1. There have been
also suggestions and in fact partial realization of observatories in space orbits to use
earth’s atmosphere as a gigantic scintillation detector such as proposed MASS
(Maximum-energy Auger (Air)-Shower Satellite)211 and EUSO (Extreme Universe Space
Observatory).212-216
A version smaller than the AGN cosmic scale is the microquasars from the center of
our Galaxy. And anything in between, such as starburst galaxies, Seyfert galaxies, etc.
are discussed. The characteristics of the conjectured WFA phenomenology includes:(1) Simultaneous emissions of gammas and (ultra-)high energy cosmic rays
(UHECR), and neutrinos.
71

(2) Near point-like spatial co-incidences of these (UHECR may be bentslightly off by the intervening cosmic magnetic fields).
(3) Possible identification of optical or microwave identification of its sources(unlike the Fermi acceleration).
(4) Multi-messenger astrophysics (including LIGO, James Webb Space
telescope, etc.) will tell untold details and nature of the emitting stellar tocosmic objects and their processes.
(5) The time scales and patterns of emitted gamma, optics, microwaves,
UHECR, etc. may tell microscopic (relatively speaking as to the global
cosmic scales) of the inside of the accelerating process.
Thus the introduction of the wakefield acceleration process to the cosmic acceleration
genre will open up not only a rich addition of the physical mechanism to the conventional
set of those, but also a new vista and details of the object that emits and its details physicalprocesses in real time in a variety of astrophysical objects as mentioned in this paper. It
will enrich astrophysics so much more.
Acknowledgements:
Three of us (TE, TT, and BB) have followed their own path of research to encounter
with each other and three of us, which amounted to a joint work of this review. TE andTT ran into a cross point that the late Professor Yoshiyuki Takahashi brought us together
by his vision of very active and high energy astrophysical phenomena such as collisionsof compact stars and associated emissions of radiation, extreme high energy cosmic rays
and their possible origin, and pioneering detection concept of them by space-based
observatory of EUSO (Extreme Universe of Space Observatory) in early 2000s. TE joined
EUSO projects, led by Professors Yoshi Takahashi, Livio Scarsi, Piergiorgio Picozza, andTE, and got interested in the acceleration of charged particles in the astrophysical jets.
Stimulating discussions between TT and TE and other collaborators, such as Professor
Etienne Parizot gradually grew into the idea of wakefield acceleration in the universe. TT
and BB encountered in the preparation of SSCL (Superconducting Super Collider
Laboratory) to try to detect extremely high energy phenomena. After the demise of the
SSCL, BB went on to utilize laser to detect never observed Einstein’s gravitational waves
and their astrophysical implications, while TT continued to make realization of his earlier
wakefield accelerators on Earth. BB and TT again met in the committee of ELI (Extreme
Light Infrastructure) SAC (2008), which tried to realize high intensity light phenomena
including high energy particle acceleration with laboratory lasers. These encounters andcollaborations founded a basis for three of us to investigate and review the Nature’s
72

acceleration mechanism to extreme high energies, i.e. via wakefield acceleration. BB
went on successfully building LIGO and discovered gravitational waves, for which he
was recognized for the 2017 Nobel. BB wishes to thank the LIGO Laboratory for theirsupport, the LIGO Scientific Collaboration for their contributions to LIGO data analysis
and science, and lastly the National Science Foundation for their long-term support fromLIGO conception to the present day.
Along these meanderings and paths, we encountered a huge number of colleagues
whose enlightening, collaboration, and encouragements were invaluable. They include:Drs. K. Abazajian, J. Adams, F. Albert, H. Alfven, R. Assmann, S. Barwick, F. Brunel, S.
Bulanov, S. Cable, A. Caldwell, P. Chen, S. Corde, J. Dawson, M. Downer, E. Esarey, D.
Gilden, C. Haswell, K. Holcomb, W. Horton, G. Huxtable, S. Ichimaru, S. Karsch, Y.
Kato, Y. Kishimoto, M. Kivelson, M. Kole, R. Kulsrud, T. Kurki-Suonio, C. Lau, R. Li,
J. Leboeuf, R. Lovelace, K. Makishima, R. Matsumoto, K. Mima, S. Mineshige, S.
Miyamoto, A. Mizuta, G. Mourou, K. Nakajima, S. Nicks, E. Parizot, F. Perkins, N.
Petviashvili, P. Picozza, M. Rosenbluth, N. Rostoker, J.I, Sakai, H. Sato, L. Scarsi, C.
Schroeder, K. Shibata, V. Shiltsev, C. Siders, H. Sobel, D. Sugimoto, Y. Takahashi, K.
Thorne, S. Weinberg, J. Wheeler, M. Yamada, X. Yan, H. Yu, F. Zimmermann, E. Zweibel.
The Rostoker Fund is also acknowledged for its support.

73

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