###### Abstract

Fast radio bursts (FRBs) are recently discovered mysterious single pulses of
radio emission, mostly coming from cosmological distances ($\sim 1$ Gpc).
Their short duration, $\sim 1$ ms, and large luminosity evidence coherent
emission. I review the basic physics of coherent emission mechanisms proposed
for FRBs. In particular, I discuss the curvature emission of bunches, the
synchrotron maser, and the emission of radio waves by variable currents in the
course of magnetic reconnection. Special attention is paid to magnetar flares
as the most promising sources of FRBs. Non-linear effects are outlined that
could place bounds on the power of the outgoing radiation.

###### keywords:

non-thermal emission mechanisms; plasmas; shock waves; reconnection; neutron
stars

xx 1 5 Received: date; Accepted: date; Published: date Emission Mechanisms of
Fast Radio Bursts Yuri Lyubarsky Firstname Lastname, Firstname Lastname and
Firstname Lastname

##  1 Introduction

The discovery of fast radio bursts (FRBs) has inspired renewed interest of the
astrophysical community in coherent emission mechanisms. An extremely high
brightness temperature of the observed emission implies that the emission
mechanism is coherent, i.e., a large number of particles must emit the radio
waves in phase. Half a century ago, this sort of consideration led to
conclusion that the pulsar radio emission is coherent. However, a theory of
pulsar radio emission has not yet been developed. Therefore, it is not
surprising that a pessimistic view of the theoretical study of coherent
emission processes is common.

Under astrophysical conditions, coherent emissions are typically generated in
fully ionized plasma (a marked exception is the molecular line masers). The
plasma is a medium with a long-range interaction, which, in principle, enables
the coherent motions of large ensembles of particles. However, the same long-
range interaction strongly affects both the emission and the propagation of
electromagnetic waves. Therefore, coherent emission is a collective plasma
process that should be described in the language of plasma physics.
Unfortunately, this branch of physics is not popular among the astrophysical
community. Two views are prevalent. According to the first view, this is an
obscured and untrustworthy field, so there is no chance of any progress. The
second view is the opposite (but, in some sense, closely related): coherent
emission could be simply described by formulas from the Jackson’s textbook
(Jackson, 1962), into which the charge $Ne$ could be substituted with a large
enough $N$.

This review advocates for a more insightful approach. The theory of
collisionless plasmas supplemented by numerical simulations has already
provided spectacular progress in our understanding of astrophysical shock
waves, magnetic reconnection, and particle acceleration (Treumann and
Baumjohann, 2015; Sironi et al., 2015; Kagan et al., 2015; Pelletier et al.,
2017). The coherent emission from the Sun and planetary magnetospheres has
also become a mature field (Melrose, 2017). The situation with mechanisms of
pulsar radio emission looks disappointing but the reason for this is rather
specific: it is not that the processes in the magnetospheric plasma defy
analysis—the mechanisms of the plasma production and the very structure of the
plasma flows in pulsar magnetopsheres remain uncertain, which prevents the
confrontation of the theory and observations (Lyubarsky, 2008). Therefore,
there is no reason to halt efforts to resolve the enigma of FRBs.

This review is an attempt to present, at a basic level, the physics of
collective plasma emission mechanisms applied to FRBs. When dealing with
radiation processes in plasma, a sensible line of action is to address three
basic questions:

  1. 1.

What electromagnetic waves are supported by plasma at the assumed conditions?

  2. 2.

How could these waves be excited?

  3. 3.

Could they escape the system as radio waves?

Any reasonable model has to provide self-consistent answers to these
questions. I try to show how this could be achieved in particular cases
relevant to FRBs.

Only basic observational facts are used in this review. First, the theoretical
model should explain the $\sim 1$ ms burst duration and a wide range of
emitted isotropic energies, from $\sim 10^{35}$ to $\sim 10^{43}$ erg. Another
basic fact is that FRBs typically exhibit strong linear polarization,
sometimes reaching 100%. For a comprehensive analysis of observations, the
reader is referred to recent reviews (Cordes and Chatterjee, 2019; Petroff et
al., 2019). A concise summary of observational data, as well as the
description of theoretical models is given in ref. (Zhang, 2020).

The remainder of this paper is structured as follows: In Section 2, I describe
the coherent radiation mechanisms discussed in the context of FRBs. Magnetars
are considered the main candidate source of FRBs. In Section 3, I outline the
physics of magnetar magnetospheres and magnetar flares, and discuss how the
powerful coherent radiation may be produced by these flares. In Section 4, I
describe non-linear processes that could affect the propagation of high
intensity radio waves and even prevent the escape of radio emission from the
source.

##  2 Coherent Radiation Mechanisms

###  2.1 Brightness temperature

To estimate how many coherently emitting particles are minimally required to
produce the observed emission, let us estimate the typical brightness
temperature of FRBs. The brightness temperature of a radiation source,
$T_{b}$, is defined such that the radiation intensity of the source at the
frequency $\nu$ is presented according to the Rayleigh-Jeans law:

$I_{\nu}=\frac{2k_{B}T_{b}\nu^{2}}{c^{2}},$ (1)

where $k_{B}$ is the Boltzmann constant, and $c$ the speed of light. The
brightness temperature typically depends on the frequency.

If the source is at rest, the observed frequency and intensity are the same as
those in the source. For an isotropic source, the observed spectral flux is:

$F_{\nu}=I_{\nu}\Omega\sim\frac{a^{2}}{D^{2}}I_{\nu},$ (2)

where $\Omega$ is the solid angle subtended by the source, $a$ is the size of
the source, and $D$ is the distance. The characteristic spectral flux of FRBs
is of the order of 1 Jy; the duration of the bursts, $\tau$, is of the order
of 1 ms. The size of the source is limited by $a\leq\tau c$, which immediately
implies a low estimate of the brightness temperature:

$T_{b}\sim\frac{F_{\nu}D^{2}}{k_{B}\tau^{2}\nu^{2}}=7\cdot
10^{35}\frac{F_{\nu,\rm Jy}D^{2}_{\rm Gpc}}{\tau^{2}_{\rm ms}\nu^{2}_{\rm
GHz}}\,\rm K.$ (3)

The indexes like Jy and Gpc indicate the units in which the corresponding
quantities are measured. Below, I also employ the shorthand notation
$q_{x}=q/10^{x}$ in cgs units, e.g., $L_{44}=L/(10^{44}\rm erg/s)$.

In a moving source, the brightness temperature is related to the intensity in
the source frame, $I^{\prime}_{\nu^{\prime}}=2k_{B}T_{b}\nu^{\prime 2}/c^{2}$,
where the prime means that the corresponding quantities are measured in the
source frame. If the source moves toward the observer with the Lorentz factor
$\Gamma\gg 1$, the lower limit on the observed variability time significantly
decreases due to the transit time effect:

$\tau_{\rm obs}\geq\frac{a^{\prime}}{c\Gamma}.$ (4)

To find a relation between the measured spectral flux and the brightness
temperature of the source, take into account that the total luminosity of the
source,

$L^{\prime}\sim a^{\prime
2}I^{\prime}_{\nu^{\prime}}\Delta\nu^{\prime}\sim\frac{a^{\prime
2}k_{B}T_{b}\nu^{\prime 2}\Delta\nu^{\prime}}{c^{2}},$ (5)

is the relativistic invariant, $L=L^{\prime}$. Then, the observed spectral
flux is estimated, accounting for the relativistic beaming, as:

$F_{\nu}\sim\frac{L}{\Gamma^{2}D^{2}\Delta\nu}\sim\frac{a^{\prime
2}k_{B}T_{b}\nu^{\prime
2}}{c^{2}D^{2}\Gamma^{2}}\frac{\Delta\nu^{\prime}}{\Delta\nu}.$ (6)

Now, the Lorentz transform of the frequency, $\nu\sim\Gamma\nu^{\prime}$,
together with the upper limit on the source size (4), yields the low limit on
the brightness temperature:

$T_{b}\sim\frac{F_{\nu}D^{2}}{k_{B}\tau^{2}\nu^{2}\Gamma}=7\cdot
10^{35}\frac{F_{\nu,\rm Jy}D^{2}_{\rm Gpc}}{\tau^{2}_{\rm obs,ms}\nu^{2}_{\rm
GHz}\Gamma}\,\rm K.$ (7)

The brightness temperature for an ensemble of incoherently emitting particles
cannot exceed the temperature of the ensemble, understood as the
characteristic energy of particles expressed in degrees. This immediately
implies that the emission mechanism of FRBs should be coherent, i.e., a large
number of particles must emit radio waves in phase. Assuming that the emitting
particles are electrons, we estimate the minimal number of coherently
radiating particles as:

${\cal N}\sim\frac{k_{B}T_{b}}{m_{e}c^{2}}\sim 10^{26}\frac{F_{\nu,\rm
Jy}D^{2}_{\rm Gpc}}{\tau^{2}_{\rm obs,ms}\nu^{2}_{\rm GHz}\Gamma}.$ (8)

If the electrons in the source are relativistic, the required number of
particles is smaller by the characteristic Lorentz factor of electrons, but in
any reasonable case, $N$ remains tremendously large. This implies that the FRB
emission is definitely coherent. The question is what causes the concerted
motion of a macroscopic number of particles.

The simplest assumption is that bunches of charged particles are somehow
formed in the system, with each bunch emitting as a single entity. Then, two
questions immediately arise: what is the bunching mechanism and what are the
radiation properties of these bunches?

###  2.2 Curvature Emission of Bunches

At the very early stage of pulsar studies, particles were found to emit in the
radio band if they move with the Lorentz factors $\Gamma\sim 100$ along curved
magnetic field lines in the neutron star’s magnetosphere. Since then, the
popular view is that bunches of charged particles are somehow formed in the
magnetosphere, and that pulsar radio emission could be attributed to the
curvature emission of these bunches. Recently, the same model was applied to
FRBs (Falcke and Rezzolla, 2014; Cordes and Wasserman, 2016; Dai et al., 2016;
Kumar et al., 2017; Ghisellini and Locatelli, 2018; Katz, 2018; Yang and
Zhang, 2018; Lu and Kumar, 2018; Wang et al., 2019; Kumar and Bošnjak, 2020;
Lu et al., 2020; Wang et al., 2020; Yang et al., 2020).

The formation of bunches is typically attributed to the electrostatic plasma
waves excited by the two-stream instability. All the available estimates are
based on the theory developed at the assumption that the energy of
perturbations is small as compared with the energy of the plasma. However, the
electrostatic energy of the assumed bunches enormously exceeds the plasma
energy. Namely, for the emission to be coherent, the size of the bunch in the
comoving frame could not be larger than $c/\omega^{\prime}$, where
$\omega^{\prime}=2\pi\nu/\Gamma$ is the frequency in the frame of the bunch.
Using the estimate (8) for the minimal number of electrons in the bunch, we
find the electric potential in the bunch $V^{\prime}\sim{\cal
N}e\omega^{\prime}/c$. Then the ratio of the electrostatic to the plasma
energy is found as

$\epsilon\sim\frac{eV^{\prime}}{mc^{2}}\sim 10^{9}\frac{F_{\nu,\rm
Jy}D^{2}_{\rm Gpc}}{\tau^{2}_{\rm obs,ms}\nu_{\rm GHz}\Gamma_{2}^{2}}.$ (9)

In this case, the results of the linear theory are completely irrelevant. The
energy of plasma waves could not exceed the plasma energy because due to non-
linear effects, it efficiently goes to heat. One can speculate that the plasma
is so hot that $\epsilon$ does not exceed unity. However, the properties of
both the plasma waves and the two-stream instability in a relativistically
hot, strongly magnetized plasma differ drastically from those in the cold
plasma (Lominadze and Mikhailovskiǐ, 1979; Arons and Barnard, 1986). No
attempts were made to analyze the formation of the bunches at realistic
conditions.

Note that $\epsilon$ represents the repulsive electrostatic potential in units
of the electron rest energy. This enormous number clearly shows how difficult
is to form the bunches and to keep them against destruction. Unless physical
mechanisms capable of formation and maintenance of the bunches will be
elaborated, models assuming the existence of such bunches should be considered
as highly speculative.

In the papers quoted above, the emission of the bunches was estimated by
applying standard formulas of electrodynamics in a vacuum. The characteristic
frequency is estimated as:

$\omega_{c}\sim\frac{c\Gamma^{3}}{R_{c}},$ (10)

whereas the emission power of a bunch is assumed to be:

$P=\frac{2q^{2}c\Gamma^{4}}{3R_{c}^{2}},$ (11)

where $R_{c}\sim 10^{7}-10^{9}$ cm is the curvature radius of the magnetic
field line and $q={\cal N}e$ is the charge of the bunch. The demand that the
emitted frequency is in the radio band implies $\Gamma\sim 100$. Choosing an
appropriate $\cal N$, we find the required power.

Even leaving aside the question of how the bunches are formed, we cannot
ignore the fact that their radiation properties are strongly affected by the
plasma. The density of this plasma could not be much less than the particle
density in the bunches therefore the minimal comoving plasma density is:

$N^{\prime}\sim\frac{\omega^{\prime 3}\cal N}{c^{3}}\sim
10^{16}\frac{F_{\nu,\rm Jy}D^{2}_{\rm Gpc}\nu_{\rm GHz}}{\tau^{2}_{\rm
obs,ms}\Gamma^{4}_{2}}\,\rm cm^{-3}.$ (12)

In this case, the plasma frequency, $\omega^{\prime}_{p}=(4\pi
e^{2}N^{\prime}/m_{e})^{1/2}$, is well above the frequency of the emitted
waves:

$\frac{\omega_{p}^{\prime}}{\omega^{\prime}}\sim 10^{5}\frac{F_{\nu,\rm
Jy}^{1/2}D_{\rm Gpc}}{\tau_{\rm obs,ms}\nu_{\rm GHz}^{1/2}\Gamma_{2}}.$ (13)

Waves with frequencies below the plasma frequency could propagate in the
highly magnetized plasma of a neutron star’s magnetosphere , see, e.g., (Arons
and Barnard, 1986). However, their emission cannot be described by formulas of
vacuum electrodynamics.

The curvature emission of a point charge at the condition
$\omega^{\prime}_{c}\ll\omega^{\prime}_{p}$ was calculated in ref. (Gil et
al., 2004). It was found that the characteristic radiation frequency is the
same as that in the vacuum case, Eq. (10), but the emission power is strongly
suppressed:

$P=K_{0}\frac{q^{2}c\Gamma^{4}}{R_{c}^{2}}\left[\frac{\omega^{\prime}_{c}}{\omega^{\prime}_{p}}\left(1-\frac{\Gamma^{2}_{p}}{\Gamma^{2}}\right)\right]^{2},$
(14)

where $\Gamma$ is the Lorentz factor of the charge, $\Gamma_{p}$ is the
Lorentz factor of the plasma, $K_{0}\approx 0.1$, and the prime means that the
corresponding quantities are measured in the frame comoving with the plasma.
If the charge moves together with the plasma, $\Gamma=\Gamma_{p}$, it does not
radiate at all because it is completely shielded in this case. Note that in
this calculation, the plasma was assumed to be cold. Hot plasmas completely
shield charges moving with a velocity smaller than the electron thermal
velocity (Krall and Trivelpiece, 1973). Therefore, in a relativistically hot
plasma, the curvature emission could be expected to be completely suppressed
for a wide range of $\Gamma$. If the charge moves with respect to the plasma
such that its velocity is out of the range of the plasma thermal velocities,
it could emit but the emission rate would be suppressed by a factor
$(\omega^{\prime}/\omega^{\prime}_{p})^{2}\sim 10^{-10}$ compared with the
vacuum emission rate (11).

Of course, this highly idealized result could be modified by considering
additional effects, such as non-point-like charge distribution, hot plasma,
etc. However, this is a burden on the authors of the model to present a self-
consistent description of all relevant physical processes. The above example
just shows that the plasma effects could not be ingnored in the emission
process. It is worth noting that in pulsars, the plasma frequency is
comparable with the curvature frequency, and the electrostatic energy of the
assumed bunches is not too large in typical cases. Therefore, in this case,
the model of the curvature emission could not be certainly ruled out, even
though speculations of the origin of bunches are not convincing. In all the
available versions of the coherent curvature emission model for FRBs, both the
formation of bunches and their emission is described using theoretical results
obtained at the conditions by far violated in the model. Therefore for a
while, this model could not be considered physically consistent, so that the
discussion of any details of the model is premature.

###  2.3 Masers, Some Preliminaries

The standard explanation of maser action is based on the notion of induced
emission introduced by Einstein in his analysis of the quantum radiation
transitions. A photon with energy matching the energy difference between two
energy levels could not only be absorbed by the system, causing a transition
of an electron from the lower to the upper level, but could also cause an
electron in the upper level to jump downward, emitting one more photon with
the same energy and direction. Therefore, if the population of the upper level
exceeds that of the lower level, a seed radiation beam is exponentially
amplified.

The above quantum picture is well known and is, in principle, correct, even
though it could be sometimes misleading, as with any purely qualitative
picture. But it is worth stressing that we deal with classical systems, so
even though the classical view is paradoxically less vivid than the quantum
view, quantitative theory may be developed using only classical plasma
physics.

According to the classical picture, maser action occurs in resonantly unstable
systems. A seed wave satisfying the resonance conditions modulates the
unstable plasma, triggering currents that emit in phase with the seed wave,
thus amplifying it. In this case, the maser emission could also be attributed
to charged bunches formed in the course of the modulation process. However in
this approach, both the formation of the bunches and their emission are
considered self-consistently, accounting for the dispersion properties of the
medium.

###  2.4 Synchrotron Maser

Charged particles rotate in a magnetic field; therefore, an inverse population
in the magnetized plasma could be simply imagined as a ring in the particle
momentum space. Before considering how such a distribution could be formed in
nature, a couple of subtle points must be discussed.

First, maser action is impossible in an infinite system of equidistant energy
levels, like the system of non-relativistic Landau levels 111The cyclotron
maser works only due to relativistic corrections to Landau levels (Melrose,
2017). Even if the $n$th level is overpopulated, the resonant photon could
trigger both the transition $n\to n-1$ (induced emission) and $n\to n+1$
(absorption). The absorption rate is larger, so the resonant photons are
always absorbed in this case. In particular, maser action is impossible in a
system of linear oscillators. The reason why the induced emission was
discovered first in quantum physics is that we could not find a simple
classical system that exhibits this effect. In classical electrodynamics, the
amplified emission was found later, with the theory of the effect being quite
involved.

The system of relativistic Landau levels is not equidistant; therefore, at
first glance, an amplified synchrotron emission could be easily achieved
provided an inverse population is formed in a system of relativistic
electrons. However, there is another subtle point. By synchrotron, we
typically mean emission at very high harmonics of the rotational frequency,
such that the spectrum is continuous. In this case, waves emitted by an
electron could be absorbed by electrons with a different energy. The inverse
energy distribution can be formed only in a limited range of particle
energies, say, at $E<E_{0}$, whereas at $E>E_{0}$, the level population
decreases with the energy. Therefore, radiation from electrons with $E<E_{0}$
may be absorbed by electrons with $E>E_{0}$. For this reason, the synchrotron
maser is possible only in two cases (see (Melrose, 1980) and references
therein). The first is an extremely narrow energy distribution, which could
hardly form in realistic conditions. In the second case, the inversely
populated electrons radiate in the frequency range, where the dispersion of
the waves is modified by the plasma, $(\omega/ck)-1\geq\gamma^{-2}$, where
$\gamma$ is the characteristic Lorentz factor of electrons. Then, both
emission and absorption are suppressed by the Razin effect, with the
suppression being stronger for higher energy electrons. In ring-like
distributions, the inverse population is formed at smaller energies;
therefore, the induced emission of these electrons is less suppressed than the
absorption by higher energy electrons. Then, the total absorption coefficient
becomes negative, which implies maser emission.

It was noticed (Sazonov, 1970) that the dispersion of electromagnetic waves
may be sufficiently modified by the relativistic electrons themselves provided
their density is large enough. For this case, the theory of the synchrotron
maser instability has been developed both for spherically symmetric (Sagiv and
Waxman, 2002; Gruzinov and Waxman, 2019) and for ring-like electron
distributions (Lyubarsky, 2006). The growth rate of the maser instability is
estimated as

$\kappa\sim 0.1\frac{\Omega_{B}^{5/4}}{\Omega_{p}^{1/4}},$ (15)

where

$\Omega_{B}=\frac{eB}{m_{e}c\gamma};\qquad\Omega_{p}=\sqrt{\frac{4\pi
e^{2}N}{m_{e}\gamma}}$ (16)

are the relativistic Larmor and plasma frequencies, respectively. The
characteristic frequency of the amplified waves is:

$\omega\sim\frac{\Omega_{p}^{5/4}}{\Omega_{B}^{1/4}}.$ (17)

The above result was obtained under the condition $\Omega_{B}<\Omega_{p}$. In
relativistic sources, we can conveniently use the magnetization parameter,
defined as twice the ratio of the magnetic to the plasma energy density in the
source:

$\sigma=\frac{B^{2}}{4\pi m_{e}c^{2}\gamma
N}=\left(\frac{\Omega_{B}}{\Omega_{p}}\right)^{2}.$ (18)

Then, the characteristic frequency of amplified waves and the growth rate may
be respectively presented as:

$\kappa\sim
0.1\sigma^{1/4}\Omega_{B};\qquad\omega\sim\sigma^{-1/4}\Omega_{p};\qquad\sigma<1.$
(19)

Because of a very weak dependence on $\sigma$, the emission frequency of the
maser does not much exceed the plasma frequency.

Above, only radiation at high harmonics of the rotational frequency,
$\Omega_{B}$, was discussed. Note that the few first harmonics do not overlap
if the electron energy distribution is not too wide, $\Delta E<E$. Therefore,
the maser emission at the first harmonic is possible, even in a vacuum. In a
strongly magnetized plasma, $\sigma>1$, the Larmor frequency exceeds the
plasma frequency; therefore, waves with frequencies of the order of
$\Omega_{B}$ could propagate. Therefore, in this case, the maser emits at the
main and the few first harmonics, with the instability growth rate being
(e.g., (Aleksandrov et al., 1984)):

$\kappa\sim
0.3\sigma^{-1/3}\Omega_{B}\qquad\omega\sim\Omega_{B}=\sigma^{1/2}\Omega_{p};\qquad\sigma>1.$
(20)

One sees that the maser action is possible at any magnetization, provided an
inverse electron population is formed. As, in the systems of interest, the
magnetization is not too low and not too big, the characteristic emission
frequency may be generally presented as:

$\omega\sim\zeta\Omega_{p},$ (21)

where $\zeta\sim$ a few.

###  2.5 Coherent Emission from the Front of a Relativistic, Magnetized Shock

The reverse level population means that in some energy range, the particle
energy distribution function grows faster with the energy than the thermal
distribution. This distribution could be formed only in highly non-equilibrium
systems under rather non-trivial conditions.

A ring-like particle distribution is naturally formed at a front of the
collisionless shock in a magnetized flow because this shock is mediated by the
Larmor rotation: when the upstream flow enters the shock, the bulk velocity
sharply drops, and particles begin to rotate in the enhanced magnetic field.
It was assumed (Langdon et al., 1988) that at relativistic shocks, synchrotron
maser instability develops. Numerical simulations of relativistic, magnetized
shocks (Hoshino et al., 1992; Gallant et al., 1992; Sironi and Spitkovsky,
2011; Iwamoto et al., 2017, 2018, 2019; Plotnikov and Sironi, 2019; Babul and
Sironi, 2020) revealed strong electromagnetic precursors with the
characteristic frequencies compatible with the estimate (21).

Note that the frequencies (21) are in fact given in the downstream frame. In
the upstream frame, they are Lorentz-boosted so that the frequencies of the
emitted waves are well above the local plasma and Larmor frequencies.
Therefore, the precursor waves propagate away relatively easily.

The simulations show that a ring-like distribution is formed at the front of
the shock, provided the magnetization of the flow is high enough, $\sigma\geq
10^{-3}$. However, we cannot certainly attribute the precursor emission to the
maser mechanism. The ring is immediately destroyed just beyond the front, so
that the width of the unstable region in the shock frame is only $\sim
2c\Omega_{B}^{-1}$. As the growth rate of the instability is less than
$\Omega_{B}$, see Eqs.(19) and (20), a seed wave could not be amplified
considerably while crossing the shock. The waves propagating within the shock
plane perpendicular to the shock normal could be amplified, but then the
precursor should be attributed to the scattering of the amplified waves. It is
possible that the intrinsic unsteadiness of the shock front leads to strong
fluctuations in the charge density so that the thus-formed charged bunches
rotate in the magnetic field and emit coherent radiation. In any case, the
emitted frequency is of the order of a few $\Omega_{B}$ at $\sigma>1$ and a
few $\Omega_{p}$ at $\sigma<1$, which is compatible with our expectation from
the synchrotron maser emission. Despite this uncertainty, I refer to this
emission as the synchrotron maser emission, because this term is widely used
in the literature.

In the referenced simulations, the authors found that the emitted energy
reaches a few percent of the inflow power at $\sigma\sim 0.1$ and decreases
toward both large and small magnetizations. The strength parameter of the
electromagnetic waves is defined as the ratio of the cyclotron frequency in
the wave field to the wave frequency:

$a=\frac{eE}{m_{e}c\omega},$ (22)

where $E=B$ is the amplitude of the wave. The cases $a\ll 1$ and $a\gg 1$
correspond, respectively, to non-relativistic and highly relativistic
oscillations of electrons in the wave. This quantity is a relativistic
invariant. Let us define the efficiency of the maser emission, $\chi$, as the
ratio of the emitted power to the total power of the inflow, both measured in
the shock front reference frame. Using Eq. (21), we find that if the inflow
Lorentz factor is $\gamma$, the strength parameter of the emitted waves is
(Iwamoto et al., 2017; Plotnikov and Sironi, 2019):

$a\sim\frac{\chi^{1/2}}{\zeta}\gamma.$ (23)

One sees that a highly relativistic shock emits strong electromagnetic waves,
which could have profound influence on their interaction with the
surroundings.

The polarization of the radiation is an important issue. Most of the available
simulations of the maser emission from relativistic shocks were performed
either in 1D or 2D with the simulation plane perpendicular to the background
magnetic field. No information on polarization could be inferred from these
simulations. In (Gallant et al., 1992), 2D simulations were performed for the
magnetic field in the simulation plane; it was reported that the outgoing
radiation is 100% polarized perpendicular to the magnetic field. Conversely,
both polarization modes were found in other studies (Iwamoto et al., 2018;
Plotnikov and Sironi, 2019), where both 2D and 3D simulations were performed.
The last result seems to be more compatible with theoretical expectations. The
synchrotron emission is completely polarized only in the plane of the rotating
particles, but the out-of-plane emission has an electric field component
parallel to the magnetic field. Therefore, the total emission is polarized
predominantly perpendicular to the magnetic field, but the degree of
polarization never reaches 100%.

The dependence of the synchrotron maser emission on the pre-shock temperature
was numerically investigated (Babul and Sironi, 2020) for the strongly
magnetized case, $\sigma\geq 1$. It was found that the maser efficiency is
nearly independent of the temperature at $k_{B}T<0.03m_{e}c^{2}$, but sharply
drops at higher temperatures, becoming two orders of magnitude smaller at
$k_{B}T=0.1m_{e}c^{2}$. The reason is that even at such a relatively small
energy spread of the upcoming electrons, the ring at the shock front becomes
thick, $\Delta E\sim E$. As it was discussed in the previous subsection, the
system emits at the first harmonics of the Larmor frequency at $\sigma>1$,
only if these harmonics do not overlap, so that if $\Delta E<E$, which
explains the above result. How weakly magnetized shocks depend on the upstream
temperature remains unclear.

All the above refer to shocks in electron-positron plasma. The case of
electron-ion plasma is more complicated. In electron-ion flow, the electrons
transfer only a small fraction of the energy. However, the electrons lag
behind ions in the field of intense precursor waves. Then, a longitudinal
large-scale electric field is excited so that the electrons are accelerated,
taking the energy from the ion flow (Lyubarsky, 2006). Moreover the Raman
scattering of the precursor waves produces plasma waves; this process leads to
acceleration of the electrons and even to the formation of non-thermal
electron distributions (Hoshino, 2008). Therefore, the electrons could
eventually reach an energy equipartition with the ions before the flow reaches
the shock. In this case, the electron mass should be substituted by one-half
the ion mass in Eq. (21) for the frequency of the emitted radiation so that
the frequency of the precursor decreases. Numerical simulations of
relativistic shocks in magnetized electron-ion flows (Lyubarsky, 2006;
Hoshino, 2008; Iwamoto et al., 2018) reveal a large-scale longitudinal
electric field and the electron acceleration until the equipartition upstream
of the shock. The tendency of the emitted frequency to decrease was also found
(Hoshino, 2008), but the relatively small ion-to-electron mass ratio in
simulations (only 50) does not permit to firm conclusions.

The outlined energy equilibration mechanism assumes that the strong-enough
precursor is produced from the beginning, when the electrons are still light.
However, in this case, the shock is mediated by the Larmor rotation of ions;
it is not evident that the ring-like distribution of light electrons is
formed. The synchrotron radiation of ions could be suppressed at this stage
because the frequency is well below the electron plasma frequency; then, the
equilibration mechanism is not involved. However, if the electron
magnetization is not small, such that the electron Larmor frequency exceeds
the radiation frequency, the low-frequency radiation becomes possible. This
question has not been properly investigated theoretically. In numerical
simulations, the shock is initiated by hitting the flow upon a rigid
conductive wall. In this case, a strong electromagnetic perturbation is
initially formed, which is sufficient to trigger the acceleration process. It
is unclear how the process is triggered in reality.

Another problem is that the accelerated electrons have a wide quasithermal
energy distribution, in which case the synchrotron maser emission may be
suppressed. It was speculated (Iwamoto et al., 2018) that after the precursor
is quenched, cold undisturbed electrons enter the shock once again and the
whole positive feedback cycle is initiated. More simulations are necessary to
clarify the issue.

###  2.6 Radiation from Reconnecting Current Sheets

In this section, a sort of antenna mechanism is described. An antenna is a
linear conductor carrying a variable current. Variable linear currents are
naturally formed in the course of magnetic reconnection in a current sheet,
separating oppositely directed magnetic fields. In the reconnection process,
the current sheet breaks into a system of linear currents, which are called
magnetic islands because of their island-like appearance in 2D simulations.
The process is highly variable so that we can easily imagine that variable
currents are sources of electromagnetic waves. This analogy is not very useful
because antennas emit in a vacuum, whereas the reconnection process occurs in
plasma, the characteristic variability times being larger than the microscopic
plasma times. The frequency of the emitted waves is lower than both the plasma
and the Larmor frequency; therefore, the system should be described in terms
of magnetic hydrodynamics (MHD).

Inasmuch as the parallel currents are attracted one to another, the islands
continuously merge (see, e.g., (Kagan et al., 2015)). The merging of two
islands perturbs the magnetic field in the vicinity of the merging point, thus
exciting MHD waves around the reconnecting current sheet. There are generally
three types of MHD waves: the Alfv’en wave and two magnetosonic waves. In the
simplest case of cold plasma, only the Alfv’en and the fast magnetosonic (fms)
waves remain. In the Alfv’en waves, the magnetic field lines oscillate due to
the magnetic tension as stretched strings. These waves propagate along the
magnetic field lines; therefore, they do not transfer the energy away from the
current sheet. Fast magnetosonic waves propagate across the magnetic field
lines. These waves resemble sound waves, but instead of the gas pressure, the
restoring force is provided by the magnetic pressure. In cold plasma, the fms
velocity is (e.g., (Appl and Camenzind, 1988)):

$v_{\rm fms}=\sqrt{\frac{B^{2}}{4\pi\rho
c^{2}+B^{2}}}=\sqrt{\frac{\sigma}{1+\sigma}},$ (24)

where $\rho$ is the plasma density; all quantities are measured in the plasma
frame. The fms velocity is independent of the propagation direction;
therefore, any merging event produces a quasi-spherical fms pulse that
propagates away, with the duration of the pulse being $\sim a/c$, where $a$ is
the transverse size of the island. The size of magnetic islands scales with
the Larmor radius of particles in the current sheet, $a\sim{\rm
few}\,c/\Omega_{B}$. Therefore, the reconnection process produces an emission
with wavelength of the order of a dozen of Larmor radii of particles within
the sheet. The fms pulses are clearly observed in simulations of the
reconnection in the current sheet (Philippov et al., 2019); it was found that
they take away roughly 0.5% of the magnetic energy dissipated in the course of
the reconnection process.

In the fms wave, the plasma density oscillates along the propagation
direction, just as in sound waves. The magnetic field is frozen into the
plasma; therefore, plasma compression and rarefaction are accompanied by
oscillations in the field strength. The electric field vanishes locally in the
plasma frame; therefore, in the global frame where the plasma is on average at
rest, the electric field satisfies the relation
$\mathbf{E}+\frac{1}{c}\mathbf{v\times B}=0$, where $v$ is the oscillation
velocity. Thus, the electric field of the wave is perpendicular both to the
background magnetic field and to the propagation direction. The fms wave is
longitudinal as the hydrodynamic wave but it is a transverse electromagnetic
wave.

In the magnetically dominated plasma, $\sigma\gg 1$, the fms velocity is close
to the speed of light, and when the wave propagates toward smaller plasma
densities, such that $\sigma\to\infty$, it becomes a vacuum electromagnetic
wave polarized perpendicular to the background magnetic field. This could be
easily understood as follows: Consider a vacuum electromagnetic wave
superimposed on a homogeneous magnetic field such that the electric field of
the wave is perpendicular to the background field. In this system, a seed
charged particle oscillates due to drift in the crossed electric field of the
wave and the background magnetic field. The velocity of the electric drift is
independent of the particle charge; therefore, if a small amount of plasma is
added to the system, no electric current appears in the system and the wave
propagates as in vacuum.

The coherent radio emission could be produced in the course of reconnection in
magnetically dominated plasma, provided the Larmor radius of particles in the
current sheet is comparable with the radio wavelength. This mechanism was
recently proposed as a source of radio emission from the Crab and Crab-like
pulsars (Uzdensky and Spitkovsky, 2014; Lyubarsky, 2019; Philippov et al.,
2019). In ref. (Lyubarsky, 2020), the mechanism was applied to FRBs.

##  3 FRBs from Magnetar Flares

In the previous section, mechanisms of coherent radio emission were outlined,
which look promising in the FRB context. They all imply highly magnetized and
relativistic plasma, which could be found in many potential FRB progenitors.
Magnetars look to be the most promising sources of FRBs. Initially, the FRB-
magnetar connection was proposed on statistical ground (Popov and Postnov,
2007) because the rate of magnetar flares is comfortably above the FRB rate,
as distinct from, say, gamma-ray bursts. The recent discovery of weak FRBs
from the Galactic magnetar SGR 1935+2154 (CHIME/FRB Collaboration et al.,
2020; Bochenek et al., 2020) lends strong support to magnetar models. In this
section, I describe how the above-outlined coherent radiation mechanisms could
work in magnetars.

###  3.1 Magnetar Magnetosphere and Wind

Magnetars are neutron stars with surface magnetic field $\sim 10^{15}$ G. Even
a larger field, $\sim 10^{16}$ G, could be buried under the surface of the
star. The notion of magnetars was introduced to astrophysics by Duncan and
Thompson (Duncan and Thompson, 1992; Thompson and Duncan, 1995), who predicted
that ultrastrong magnetic fields could be generated in proto-neutron stars,
and demonstrated that the decay of this field could feed soft gamma repeaters
(for a recent review of magnetars, see (Kaspi and Beloborodov, 2017)).
Restructuring the magnetic field in a magnetar’s magnetosphere produces X-ray
flares with $10^{40}-10^{46}$ erg energies and with durations from a fraction
of a second to a few minutes. A rotating magnetar’s magnetosphere emits a
relativistic wind, similar to pulsar wind, which removes the rotational energy
of the neutron star. Therefore, the neutron star spins down; the magnetar
period increases to $\sim 1$ s for only $\sim 10$ years.

In the magnetosphere of an active magnetar, the magnetic field lines are
twisted; therefore, the field is non-potential. The twist currents in the
magnetosphere are carried by electron–positron pairs produced by the current
(Beloborodov and Thompson, 2007; Beloborodov, 2013): when the density of the
pairs drops below the limit sufficient to maintain the current, the induced
electric field (the displacement current) accelerates particles and triggers
an avalanche, producing new pairs. Nevertheless, the stability considerations
imply that in the persistent state, the magnetic field could hardly strongly
exceed the potential field; therefore, for rough estimates of the field
strength, we could use the dipole field.

The rotating magnetosphere becomes open at the light cylinder radius:

$R_{L}=cP/2\pi=4.8\times 10^{9}P\,\rm cm,$ (25)

where $P$ is the rotational period. The magnetic field at the light cylinder
is estimated as:

$B_{L}=\frac{\mu}{R_{L}^{3}}=9\cdot 10^{3}\frac{\mu_{33}}{P^{3}}\,\rm G,$ (26)

where $\mu$ is the magnetic moment of the star. In magnetars, the surface
magnetic field is $B_{*}\sim 10^{15}$ G therefore the magnetic moment is
$\mu\sim B_{*}R^{3}_{*}\sim 10^{33}$ G$\cdot$cm3, where $R_{*}\approx 10^{6}$
cm is the neutron star’s radius. The pairs are continuously ejected from the
magnetosphere, forming a magnetar wind. The particle flux in the wind from a
strongly twisted magnetar magnetosphere was estimated as (Beloborodov, 2020):

$\dot{\cal N}\sim{\cal M}\frac{c\mu}{eR_{L}R_{\pm}}=2.5\times
10^{39}\frac{{\cal M}_{3}\mu_{33}^{2/3}}{P}\,\rm s^{-1},$ (27)

where ${\cal M}$ is the pair multiplicity (should not be confused with the
multiplicity parameter used in the pulsar theory) and $R_{\pm}\sim 5\times
10^{6}\mu^{1/3}_{33}$ cm is the distance from the star where the magnetic
field falls to $10^{13}$ G, such that the pair production stops. The maximal
multiplicity was estimated to be of the order of ${\cal M}\sim 10^{3}$
(Beloborodov, 2020), but it should be stressed that the process of pair
production in magnetar’s magnetospheres is highly uncertain therefore one
cannot exclude that the pair loading of the magnetar wind is much larger or
smaller than that given by eq. (27).

The magnetar wind is strongly magnetized. Beyond the light cylinder, the field
is wound up by the rotating magnetosphere so that in the far zone, the
azimuthal field dominates:

$B_{\rm wind}=B_{L}\frac{R_{L}}{R}.$ (28)

The initial magnetization parameter of the magnetar wind, which is defined as
the ratio of the Poynting flux to the rest mass energy flux in the wind, is:

$\eta=\frac{B_{L}^{2}R_{L}^{2}}{\dot{\cal N}m_{e}c}=2.5\cdot
10^{4}\frac{\mu_{33}^{4/3}}{{\cal M}_{3}P^{3}}.$ (29)

The magnetization parameter is the maximal Lorentz factor achievable by the
wind if the magnetic energy is completely converted to kinetic energy.

Beyond the light cylinder, the strongly magnetized wind accelerates linearly
with distance $\gamma\sim R/R_{L}$, until it reaches the fast magnetosonic
point, $\gamma\sim\eta^{1/3}$. If there is no dissipation, the wind
accelerates very slowly beyond the fast magnetosonic point, $\propto(\ln
R)^{1/3}$ (Beskin et al., 1998). Therefore, we can take the Lorentz factor of
the wind in the far zone, $R\gg\eta^{1/3}R_{L}$, to be:

$\tilde{\gamma}_{\rm wind}=3\eta^{1/3}=90\frac{\mu_{33}^{4/9}}{{\cal
M}^{1/3}_{3}P}.$ (30)

The magnetic dissipation could lead to a gradual acceleration of the wind in
the equatorial belt, where the magnetic field changes sign every half of a
period (Lyubarsky and Kirk, 2001; Kirk and Skjæraasen, 2003); then, the
Lorentz factor of the wind exceeds $\tilde{\gamma}_{\rm wind}$ and may even
reach $\eta$. The efficiency of the dissipation is quite uncertain therefore
in all formulas below I retain $\gamma_{\rm wind}\geq\tilde{\gamma}_{\rm
wind}$ as a free parameter. The ratio of the Poynting to the plasma kinetic
flux in the wind may be generally presented as:

$\sigma_{\rm wind}=\frac{B^{2}}{4\pi mc^{2}\gamma_{\rm
wind}N}=\frac{\eta}{\gamma_{\rm wind}}=280\frac{\mu_{33}^{8/9}}{{\cal
M}^{2/3}_{3}P^{2}}\frac{\tilde{\gamma}_{\rm wind}}{\gamma_{\rm wind}},$ (31)

where $N$ is the pair density in the wind.

###  3.2 Magnetar Flares and FRBs

The energy of magnetar flares is sufficient to feed FRBs, even if a small
fraction is converted to radio waves. The flares are produced by a rapid
restructuring of the magnetar’s magnetic field. The duration of FRBs is
compatible with the Alfv’en crossing time of the inner magnetosphere, which is
the characteristic time of the development of MHD instabilities, which trigger
the flares. The duration of the flares significantly exceeds this time, which
could be attributed to the relatively long time necessary to radiate the
released energy away. It is also possible that during the flare, a few
magnetic explosions happen (see, e.g., (Yuan et al., 2020)). In this case, the
question remains why only one explosion typically produces an FRB.

In the magnetar magnetosphere, both the Larmor and the plasma frequencies are
far above the radio band. Therefore, if the source of FRBs was deep within the
magnetosphere, it should have emitted MHD waves. However, these waves are
unable to escape because of non-linear interactions, as will be shown in
Section 4.5. Therefore, FRBs could be produced only in the outer magnetosphere
or in the magnetar wind. Far from the neutron star, the magnetic energy
density is not sufficient in order to produce a short, powerful burst.
However, the necessary energy could be delivered to the far zone by a strong
magnetic pulse excited during a sudden rearrangement of the magnetosphere,
which gives rise to the magnetar flare. Let us consider the properties of such
a pulse.

A rapid restructuring of the magnetosphere produces a large-scale MHD
perturbation that propagates outward, sweeping the magnetic field lines into a
pulse of length $l=c\tau=3\times 10^{7}\tau_{\rm ms}$, where $\tau$ is the
duration of the magnetic restructuring process. The last is of the order of
the Alfv’en crossing time of the inner magnetosphere, i.e., of a few to a few
tens of stellar radii. The pulse opens the magnetosphere, and propagates
further out into the magnetar wind. As discussed in Section 2.6, only fms
waves can propagate across the magnetic field lines. Therefore, independently
of the initial magnetic configuration, we deal an fms pulse in the far zone.
The amplitude of the pulse may be conveniently presented as:

$B_{\rm pulse}=\sqrt{\frac{L_{\rm pulse}}{c}}\frac{1}{R}=3.8\times
10^{8}\frac{L_{\rm pulse,47}^{1/2}}{P}\frac{R_{L}}{R}\,\rm G,$ (32)

where $L_{\rm pulse}$ is the isotropic luminosity associated with the pulse.
The total energy in the pulse is:

${\cal E}=L_{\rm pulse}\tau=10^{44}L_{\rm pulse,47}\tau_{\rm ms}\,\rm erg.$
(33)

In the far zone, the amplitude of the pulse significantly exceeds the
background magnetic field.

Just beyond the light cylinder, the pulse enters the magnetar wind. Neglecting
plasma inertia, i.e., in the limit $\sigma\to\infty$, the pulse may be
considered purely electromagnetic. It propagates nearly at the speed of light;
the magnetic field is purely azimuthal, whereas the electric field is poloidal
and equal to the magnetic field $E_{\rm pulse}=B_{\rm pulse}$. The plasma is
squeezed in the pulse and pushed forward; the plasma velocity with respect to
the wind is estimated as the velocity of the zero electric field frame,
$v^{\prime}=E^{\prime}/B^{\prime}=B^{\prime}_{\rm pulse}/(B^{\prime}_{\rm
wind}+B^{\prime}_{\rm pulse})$, where the prime refers to quantities in the
wind frame. The corresponding Lorentz factor is:

$\Gamma^{\prime}=\sqrt{\frac{B^{\prime}_{\rm pulse}}{2B^{\prime}_{\rm
wind}}}=\frac{1}{2}\sqrt{\frac{B_{\rm pulse}}{B_{\rm wind}}}=100\frac{L_{{\rm
pulse,}47}^{1/4}P}{\mu_{33}^{1/2}}.$ (34)

In the lab frame, the Lorentz factor of the pulse is
$\Gamma=2\Gamma^{\prime}\gamma_{\rm wind}$.

The pulse itself, i.e., the waveform, moves with the fms velocity (24) with
respect to the local plasma velocity. For a large magnetization $\sigma\gg 1$
and a high amplitude of the pulse $B_{\rm pulse}\gg B_{\rm wind}$, the
difference between the velocity of the waveform and the speed of light is
estimated in the wind frame as:

$\frac{c-v^{\prime}_{\rm form}}{c}\sim(\sigma\Gamma^{\prime
2})^{-1}\sim\frac{B_{\rm wind}}{\sigma\gamma^{2}_{\rm wind}B_{\rm pulse}},$
(35)

so we can safely assume that the pulse moves with the speed of light. The
pulse moves through the magnetar wind as a propagating wave so that the plasma
enters the pulse through the front part and eventually leaves it through the
rear part. Within the pulse, the plasma moves with respect to the wind with
the large Lorentz factor (34). Therefore, the plasma is dragged within the
pulse to a large distance and is very slowly substituted by the wind plasma.

The dependence of the waveform velocity on the local density and magnetic
field, which vary across the pulse, could lead to the non-linear steepening of
the pulse. However, this velocity is very close to the speed of light (see Eq.
(35)); therefore, the nonlinearity is weak even if the amplitude of the pulse
is very large (Levinson and van Putten, 1997; Lyubarsky, 2003; Lyutikov,
2010). The physical reason for this is that the displacement current
significantly exceeds the conductivity current, so the pulse propagates nearly
as if in a vacuum. Moreover, the waveform propagates with the fms velocity in
the local plasma frame, which moves at relativistic speeds if the pulse
amplitude exceeds the background field, see Eq.(34). Substituting the
estimated above parameters of the wind and pulse in Eq. (35), the non-linear
steepening scale is estimated as (Lyubarsky, 2020):

$R_{\rm steep}=\frac{cl}{c-v_{\rm form}}=8\Gamma^{2}\sigma_{\rm
pulse}l=10^{13}\sigma_{\rm pulse}\gamma^{2}_{\rm wind}\frac{L_{\rm
pulse,47}^{1/2}P^{2}\tau_{\rm ms}}{\mu_{33}}\,\rm cm.$ (36)

The above considerations implicitly assume that the mean field in the wind
occurs in the same direction as the initial field of the pulse. However, we
can imagine a situation where the field in the wind is opposite to the field
in the pulse. In an ideal case, this does not affect the obtained results
because we can imagine a pulse within which the magnetic field changes sign
such that a current sheet separates the domains of the opposite polarity. The
electromagnetic stress is a quadratic function of the fields; therefore, a
solution of ideal MHD equations is not affected if we reverse fields in any
bundle of the magnetic field lines and insert an appropriate current sheet,
provided that the plasma in the sheet is light enough (so that the overall
inertia is not affected). In reality, the current sheet is unstable so that
the new flux could annihilate with the flux in the pulse. The flux in the
pulse, $B_{\rm pulse}l$, significantly exceeds the flux in the wind, $B_{\rm
wind}R=B_{L}R_{L}$; therefore, the pulse will not be destroyed. However, the
field annihilation heats the plasma and the heated plasma expands,
transforming heat into kinetic energy. A detailed analysis of this process has
not yet been performed.

###  3.3 FRBs Produced by Relativistic Shocks from Magnetar Flares

As shown in Section 2.5, relativistic magnetized shocks are efficient sources
of coherent emission. Let us discuss how such shocks could be produced by
outflows from magnetar flares and whether the properties of their emission are
compatible with the observed properties of FRBs.

It was proposed (Lyubarsky, 2014) that an FRB is produced when the
electromagnetic perturbation from the flare (magnetic pulse described in
Section 3.2) reaches the nebula inflated by the magnetar wind in the
surrounding gas. The inner boundary of the nebula is determined by the balance
of the pressure within the nebula and the dynamic pressure of the magnetar
wind:

$R_{s}=\frac{B_{L}R_{L}}{\sqrt{4\pi p}}=1.2\times
10^{15}\frac{\mu_{33}}{Pp_{-4}^{1/2}}\,\rm cm,$ (37)

where $p$ is the pressure within the nebula. The last quantity is highly
uncertain; normalization by $10^{-4}\,\rm dyne/cm^{2}$ was chosen because this
value is roughly compatible with the observational data for the nebula
surrounding FRB 121102 (Beloborodov, 2017).

When the electromagnetic pulse arrives at the wind termination shock, it
pushes the plasma outward like a magnetic piston. A forward shock propagates
through the magnetized and relativistically hot plasma of the nebula, whereas
a reverse shock enters the magnetic piston. Between the shocks, a contact
discontinuity separates the shocked plasma of the nebula from the magnetic
piston. At the contact discontinuity, the magnetic pressure of the pulse is
balanced by the bulk pressure of the relativistically hot plasma entering the
forward shock. The pressure balance condition yields the Lorentz factor of the
contact discontinuity:

$\Gamma_{\rm cd}=\left(\frac{B_{\rm pulse}^{2}}{32\pi
p}\right)^{1/4}=150\frac{L_{\rm pulse,47}^{1/4}P}{\mu^{1/2}_{33}}.$ (38)

The plasma in the nebula is mildly magnetized, $\sigma\leq 1$; therefore, the
forward shock moves relative to the contact discontinuity only mildly
relativistically; we can roughly take the Lorentz factor of the shock to be
$\Gamma_{\rm cd}$.

The frequency of the maser radiation in the shock frame is estimated by Eq.
(21). The plasma pressure beyond the forward shock is of the order of the
magnetic pressure in the pulse. Therefore, the relativistic plasma frequency
(16) at the shock front is roughly equal to the Larmor frequency of particles
rotating with the Lorentz factor $\Gamma_{\rm cd}$ in the field
$B^{\prime}_{\rm pulse}=B_{\rm pulse}/\Gamma_{\rm cd}$. Making the Lorentz
transform, we estimate the observed frequency as:

$\nu\sim\frac{\zeta eB_{\rm pulse}}{2\pi m_{e}c\Gamma_{\rm
cd}}=28\zeta\frac{L_{\rm pulse,47}^{1/4}p_{-4}^{1/2}P}{\mu_{33}}\,\rm MHz.$
(39)

The observed duration of the pulse is determined by the time spread of the
arrival of radiation emitted by the fraction of the forward shock, subtending
the angle $\Gamma^{-1}_{\rm cd}$. This yields:

$\tau_{\rm obs}\sim\frac{R_{s}}{2c\Gamma_{\rm
cd}^{2}}=0.9\frac{\mu_{33}}{L^{1/2}_{\rm pulse,47}P^{4}p_{-4}^{1/2}}\,\rm s.$
(40)

The observed duration of the burst, $\tau\sim 1$ ms, and the frequency,
$\nu\sim 1$ GHz, may be reproduced only by assuming a long-period magnetar,
$P\sim 5$ s, and the power, $L_{\rm pulse}\sim 10^{38}$ erg/s. However, the
nebula around such a long-period magnetar could hardly have large energy
density and pressure, whereas decreasing $p$ demands a further increase in the
burst power. In the original paper (Lyubarsky, 2014), FRBs were attributed to
very rare superflares with the total energy of $\sim 10^{48}$ erg, which could
occur when the neutron star eventually becomes unstable to a dynamic
overturning instability that destroys most of its dipole moment in a single
event (Eichler, 2002). Later-discovered repeating FRBs, as well as the
ubiquity of single FRBs, indicated that the energy beyond a typical FRB could
hardly ever be that large. Moreover, the discussion in Section 2.5 of
simulations casts doubt of the possibility of maser emission from shocks in a
relativistically hot medium.

To resolve these difficulties, it was assumed that the shock arises in the
magnetar wind at distances of the order of $10^{14}$ cm (Beloborodov, 2017,
2020; Yuan et al., 2020). In this case, the pulse acts as a piston driving a
blast wave into the cold pre-explosion wind. The plasma in the pulse moves
with the Lorentz factor (34) with respect to the wind. In a highly magnetized
medium, the shock propagates with respect to the downstream plasma with the
fast magnetosonic Lorentz factor $\sqrt{\sigma}$, Eq. (24). Therefore, in the
lab frame, the shock Lorentz factor is $\Gamma_{\rm shock}=2\Gamma\sigma_{\rm
wind}^{1/2}$. In the downstream frame, the particles rotate with the Lorentz
factor $\Gamma^{\prime}$ in the magnetic field $B^{\prime}_{\rm pulse}=B_{\rm
pulse}/\Gamma$. The frequency of the observed maser emission is obtained by
the Lorentz transform:

$\nu=\zeta\frac{eB^{\prime}_{\rm pulse}}{2\pi
m_{e}c\Gamma^{\prime}}\Gamma=0.5\zeta\frac{\mu_{33}^{1/2}L^{1/2}_{\rm
pulse,47}}{PR_{14}}\,\rm GHz,$ (41)

which is compatible with observations (recall that $\zeta\sim$ a few, see Eq.
21), if the radiation occurs at the distance $R\sim 10^{14}$ cm. The observed
duration of the burst is:

$\tau_{\rm obs}=\frac{R}{2c\Gamma^{2}_{\rm
shock}}=0.01\frac{\mu_{33}R_{14}}{\sigma_{\rm wind}\gamma^{2}_{\rm
wind}L^{1/2}_{\rm pulse,47}P^{2}}\,\rm s.$ (42)

The observed duration, $\sim 1$ ms, may be obtained only if the magnetar wind
is mildly relativistic, $\gamma_{\rm wind}\sim$ a few, and moderately
magnetized, $\sigma_{\rm wind}\sim 1$. Moderate Lorentz factor and
magnetization of the wind are also required by the condition that the
deceleration scale of the piston, $\sim\Gamma^{2}l$, does not significantly
exceed $\sim 10^{14}$ cm because in the opposite case, too little energy is
emitted in the GHz band.

This model is based on the assumption that the pulse may be considered as a
piston, i.e., the front part of the pulse is a flat wall pushing the plasma of
the wind outward. However, initially, the pulse has a smooth shape, with the
magnetic field gradually growing to the maximum at the scale of $\sim
l=c\tau$. The shock could only arise due to the non-linear steepening
discussed in Section 3.2. The larger the local magnetic field, the closer the
velocity of the waveform to the speed of light, see Eq. (35). Fractions of the
pulse with a larger field overtake the foot of the pulse, where the magnetic
field approaches the background field; therefore, initially, a weak shock
arises at the foot of the pulse. The field jump at the shock grows gradually
as the fractions of the pulse with a larger field reach the shock. The shock
velocity increases when the jump increases; therefore, the steepening process
slows. The shock matches the amplitude of the pulse when the point of the
maximum field arrives at the shock. This occurs at the distance given by Eq.
(36). Note that the magnetization $\sigma_{\rm pulse}$ in this formula is
determined by the magnetization just downstream of the shock because these
parts of the pulse overtake the shock. There, the matter is already
substituted by the matter of the wind; therefore, we can substitute
$\sigma_{\rm pulse}=\sigma_{\rm wind}$ into this expression. Substituting
$R_{\rm steep}$ into Eq. (41), we obtain:

$\nu=5\frac{\zeta}{\sigma_{\rm wind}\gamma_{\rm
wind}^{2}}\frac{\mu^{3/2}_{33}}{L^{1/4}_{\rm pulse,47}P^{3}\tau_{\rm ms}}\,\rm
GHz.$ (43)

We again see that we obtain a reasonable observed frequency if $\gamma_{\rm
wind}\sim$ a few and $\sigma_{\rm wind}\sim 1$. Substituting $R_{\rm steep}$
into Eq. (42), we find that $\tau_{\rm obs}=\tau\sim 1$ ms, independent of
other parameters of the pulse.

At the distance (36), the shock passes across a significant fraction of the
pulse, thus releasing a significant fraction of the pulse energy. Therefore,
the isotropic energy of the FRB is only proportional to the isotropic energy
of the pulse:

${\cal E}_{\rm FRB}=\chi{\cal E}=10^{41}\chi_{-3}L_{\rm pulse,47}\tau_{\rm
ms}\,\rm erg.$ (44)

The model easily explains even bright FRBs. As the radiation frequency very
weakly depends on the pulse energy, whereas the burst duration does not depend
on it at all, the model can explain the large scatter of the observed bursts
in energy. For example, taking small pulse energy and luminosity, say, $L_{\rm
pulse}\sim 10^{42}-10^{41}$ erg/s, and the magnetar period of a few seconds,
we obtain the same radiation frequency of about GHz, whereas the energy of the
burst is only $\sim 10^{35}$ erg, which is compatible with the data for an FRB
from galactic magnetar SGR 1935+2154. Some repeaters exhibit bursts over a
broad range of frequencies from 400 MHz to 8 GHz. This could be attributed to
the variations in $\sigma_{\rm wind}$ and $\gamma_{\rm wind}$.

The model of the shock in the magnetar wind successfully confronted to
observations, provided the magnetar wind is only mildly relativistic and
moderately magnetized. This implies that the initial magnetization of the
wind, $\eta$, is of the order of a few so that the wind is heavily loaded by
the plasma. According to Eq. (29), the required plasma multiplicity is ${\cal
M}\sim 10^{7}$, which is about four orders of magnitude larger than the
available magnetar models could provide. It was claimed (Beloborodov, 2017,
2020) that in a highly twisted pre-flare magnetosphere, the effective magnetic
moment increases; then, the observed parameters of the burst may be achieved
at $\gamma_{\rm wind}\sim 10$ and $\sigma_{\rm wind}\sim$ a few, so that
$\eta$ may reach a few dozen. Still, this demands an extremely large
multiplicity. What mechanism could load the wind by the plasma so efficiently
is still unclear.

It was assumed (Yuan et al., 2020) that during the magnetar flare, repeating
magnetic explosions eject a chain of plasmoids. The magnetic field lines
gradually reconnect behind each plasmoid, generating a wind far stronger than
the normal spindown. Repeating ejections drive blast waves in the amplified
wind, producing FRBs. This model successfully reproduces the observed
properties of an FRB from the Galactic magnetar SGR 1935+2154 (CHIME/FRB
Collaboration et al., 2020; Bochenek et al., 2020), and specifically two
subburst components separated by 30 ms. However, the model is based on
axisymmetric simulations. In 3D, these plasmoids become unstable and dissipate
for about 10 Alfven times (Mehta et al., 2020). Therefore, they could hardly
escape the magnetosphere. However, copious pair production in the course of
the magnetar flare could load the wind, at least temporarily. The model
deserves further development.

To resolve the problem of a highly mass loaded magnetar wind, it was suggested
that the shock arises when the pulse collides with the mildly relativistic
baryon cloud ejected by a previous flare (Beloborodov, 2017; Metzger et al.,
2019; Beloborodov, 2020; Margalit et al., 2020a, b; Xiao and Dai, 2020). It is
an observational fact that a significant amount of baryonic matter is released
in giant magnetar flares (Gaensler et al., 2005; Gelfand et al., 2005; Granot
et al., 2006). The model assumes that not earlier than 1 day before the FRB, a
strong flare ejected the required amount of material. Then, the magnetic pulse
drives a shock in the cloud. However, the magnetization rapidly decreases
within an expanded cloud, which means that the conditions for the synchrotron
maser could be violated in this shock. Even if the magnetization of the cloud
is not too small, the maser emission from a shock in the baryonic plasma is
determined by proton plasma and cyclotron frequencies, which are very low.
Therefore, the observed FRBs could hardly be produced by shocks in such
clouds. After the cloud is ejected, the magnetar wind flows around it and
forms a bow shock from behind. Therefore, before the pulse enters the cloud,
it passes through a magnetized electron-positron plasma of the shocked
magnetar wind. The pulse drives the shock in this plasma; therefore, we can
speculate that the emission is produced by this shock. However, this plasma is
relativistically hot; therefore, the synchrotron maser may be disabled in this
case, see (Babul and Sironi, 2020) and the discussion in Section 3.3.
Therefore, for a while, there is no satisfactory solution to the problem of a
mildly relativistic, heavily-plasma-loaded medium required for the synchrotron
maser model.

In all the above models, the shock propagates outward; therefore, the emission
frequency, which scales with the local Larmor frequency, decreases. So, the
sometimes-observed downward frequency drift in FRBs (CHIME/FRB Collaboration
et al., 2019; Hessels et al., 2019) could be naturally explained by the
synchrotron maser models (Metzger et al., 2019; Beloborodov, 2020; Margalit et
al., 2020a).

The synchrotron maser emission is polarized perpendicular to the background
magnetic field. This agrees with the typically observed high linear
polarization. However, this implies that in repeaters, the polarization
position angle remains the same, namely, along the rotation axis of the
magnetar. This is incompatible with the diverse polarization angle swings
observed in FRB 180301 (Luo et al., 2020). As discussed in Section 2.5, the
synchrotron radiation could not be fully polarized. Therefore, the observed
100% degree of polarization in a few FRBs (Gajjar et al., 2018; Michilli et
al., 2018; Osłowski et al., 2019) could hardly be explained within the scope
of the synchrotron maser model.

###  3.4 FRBs from Magnetic Reconnection in the Upper Magnetar Magnetosphere

As discussed in Section 2.6, in the course of magnetic reconnection, fast
magnetosonic (fms) waves are efficiently generated, and they are converted to
electromagnetic waves when propagating toward a decreasing plasma density. The
wavelength of this radiation is of the order of the size of magnetic islands
in the current sheet, which, in turn, are scaled with the Larmor radius of the
particles in the sheet. The violent reconnection occurs during the magnetar
flare when the unstable magnetic configuration is disrupted (e.g., (Parfrey et
al., 2013; Carrasco et al., 2019)). However, all microscopic plasma parameters
are very small in the inner magnetosphere, so it is unclear why the waves with
wavelengths $\sim 10$ cm could be generated with a sufficiently large power.
In any case, Section 4.5 shows that these waves could not escape because of
non-linear interactions.

FRBs could be produced when the magnetic perturbation from the magnetar flare
(magnetic pulse) reaches the current sheet separating, just beyond the light
cylinder, the oppositely directed magnetic fields (Lyubarsky, 2020). When the
magnetic pulse arrives at the sheet, the sharp acceleration and compression
cause violent reconnection. We can speculate that the current sheet is
destroyed by the Kruskal–Schwarzschild instability (Lyubarsky, 2010; Gill et
al., 2018), which is the magnetic counterpart of the Rayleigh–Taylor
instability, so that the field line tubes with the oppositely-directed fields
fall into the magnetic pulse, forming multiple small current sheets scattered
over the body of the pulse. Within each of the small current sheets, the
reconnection process occurs via formation and merging of magnetic islands,
which procudes an fms noise. This noise is converted, as was outlined in
Section 2.6, into radio emission. Since the sources of the noise are
distributed in the body of the magnetic pulse but not concentrated at the
front part, the duration of the observed radiation burst is of the order of
the duration of the pulse, $\tau_{\rm obs}\sim l/c=\tau$.

Assuming the above picture, let us estimate the parameters of the outgoing
radiation. The magnetic pulse from the flare enters the magnetar wind just
beyond the light cylinder. The plasma is squeezed in the pulse and pushed
forward with the Lorentz factor (34). In the magnetar wind, a current sheet
separates two magnetic hemispheres, with the shape of the sheet having been
likened to a ballerina’s skirt. The total reconnecting magnetic flux may be
estimated as $B_{L}R_{L}^{2}$. Within the pulse, the stripe with the
oppositely-directed fields is compressed $B_{\rm pulse}/B_{L}$ times. Then,
the total energy of annihilated fields is roughly:

${\cal E}\sim\left(B_{\rm pulse}/B_{\rm
wind}\right)B^{2}_{L}R_{L}^{3}=\left(L_{\rm
pulse}/c\right)^{1/2}B_{L}R_{L}^{2}.$ (45)

According to the simulations (Philippov et al., 2019), the fraction $f\sim
0.005$ of the reconnecting magnetic energy is emitted in the form of fms
waves. Making use of Eqs. (25) and (26), we estimate the isotropic energy of
the radio burst as:

${\cal E}_{\rm FRB}=f{\cal E}=3.8\cdot 10^{39}\frac{f_{-2}\mu_{33}L_{{\rm
pulse,}47}^{1/2}}{P}\,\rm erg.$ (46)

The wavelength of the emitted waves is of the order of the size of magnetic
islands within the current sheet, which are 10–100 times larger than the width
of the current sheet (Philippov et al., 2019). Therefore, we can estimate the
emitted frequency in the comoving frame as:

$\omega^{\prime}=\frac{c}{\xi a^{\prime}},$ (47)

where $a^{\prime}$ is the width of the sheet in the comoving frame, $\xi\sim
10-100$. The reconnection occurs via the collisionless tearing instability;
therefore, the width of the sheet is of the order of a few Larmor radii:

$a^{\prime}=\zeta\frac{\varepsilon_{T}}{eB^{\prime}_{\rm pulse}},$ (48)

where $\varepsilon_{T}$ is the characteristic thermal energy of the pairs
within the sheet, $\zeta\sim$ a few. To estimate $\varepsilon_{T}$, let us
consider the energy and the pressure balance within the sheet.

Within the sheet, the pressure of the external magnetic field is balanced by
the pressure of hot pairs:

$\frac{1}{3}N^{\prime}\varepsilon_{T}=\frac{B^{\prime 2}_{\rm pulse}}{8\pi},$
(49)

where $N^{\prime}$ is the pair density within the sheet. The synchrotron
cooling time is very short at the inferred parameters. Therefore, within the
sheet, the reconnection energy release is balanced by the synchrotron cooling.
The energy release per unit square of the sheet is determined by the Poynting
flux into the sheet, $cE^{\prime}B^{\prime}_{\rm pulse}/4\pi$. Introducing the
reconnection rate, $\epsilon=E^{\prime}/B^{\prime}_{\rm pulse}\sim 0.1$, the
energy balance may be written as:

$\epsilon\frac{B^{\prime 2}_{\rm
pulse}}{4\pi}c=N^{\prime}\sigma_{T}\frac{B^{\prime 2}_{\rm
pulse}}{4\pi}c\left(\frac{\varepsilon_{T}}{m_{e}c^{2}}\right)^{2}a^{\prime},$
(50)

where $\sigma_{T}$ is the Thomson cross-section. Eliminating from the last
three equations $\varepsilon_{T}$ and $n$ in favor of $a$, we obtain:

$a^{\prime}=\left(\frac{\epsilon\zeta}{r_{e}}\right)^{1/2}\left(\frac{c}{\omega^{\prime}_{B}}\right)^{3/2},$
(51)

where $r_{e}$ is the classical electron radius and
$\omega^{\prime}_{B}=eB^{\prime}_{\rm pulse}/m_{e}c$ is the cyclotron
frequency.

The emitted frequency in the observer’s frame may be estimated by substituting
the obtained width of the sheet into Eq. (47) and making a Lorentz transform.
The Lorentz factor of the plasma within the pulse with respect to the wind is
given by Eq. (34). Just beyond the light cylinder, the wind is only mildly
relativistic; therefore, we can use this estimate as the Lorentz factor in the
observer’s frame. Then, we obtain:

$\nu=\Gamma^{\prime}\frac{\omega^{\prime}}{2\pi}=\frac{1}{2\pi\xi}\left(\frac{r_{e}}{\epsilon\zeta
c\Gamma^{\prime}}\right)^{1/2}\omega_{B}^{3/2}=3\frac{\mu_{33}^{1/4}L_{\rm
pulse,47}^{5/8}}{\xi_{1}\zeta_{1}^{1/2}\epsilon_{-1}^{1/2}P^{2}}\,\rm GHz.$
(52)

One sees that the basic parameters of the outgoing radiation are compatible
with the observed properties of typical FRBs.

The emitted fms waves are formed within the magnetic pulse, which moves with
the relativistic velocity. Therefore, they propagate within the pulse and
escape from it far away from the magnetar, when the pulse is decelerated,
colliding with the surrounding plasma. Importantly, these waves are polarized
perpendicular to the background magnetic field. The magnetic pulse is formed
well within the magnetosphere, and the magnetic flux from this region is
transferred by the pulse outward. Therefore, the direction of the magnetic
field in the pulse is determined by the rotational phase of the magnetar. When
the pulse travels in the magnetar wind, it picks up the azimuthal magnetic
field, which is accumulated at the front of the pulse. The radiation passes
through this layer on the way out. If the field varies gradually, the
polarization is adiabatically adjusted to the local magnetic field, so that
the outgoing radiation eventually becomes completely polarized perpendicular
to the field in the wind, i.e., along the rotational axis of the magnetar. If
the field varies sharply or the accumulated flux is too small, the waves are
split into two normal modes corresponding to the local magnetic field so that
the radiation is depolarized, whereas the position angle depends on the
details of the transition zone.

In contrast to the synchrotron maser, the reconnection emission model could
explain the observed (Gajjar et al., 2018; Michilli et al., 2018; Osłowski et
al., 2019) 100% degree of linear polarization in a few FRBs. The position
angle in this model may be determined by both the field in the magnetar wind
and by the magnetosheric field, depending on how the magnetic pulse propagates
through the wind. This is compatible with the diverse polarization patterns
observed in FRBs (Luo et al., 2020). However, the model could not be applied
to weak FRBs because, according to Eq. (52), the emitted frequency becomes too
low for small luminosities. This is because the magnetic field in weak pulses
is too low already at light cylinder distances. It was suggested (Yuan et al.,
2020) that in the course of the magnetar flare, the reconnection emission
could be generated in the upper magnetosphere inside the light cylinder. The
ejected plasmoids push out the magnetospheric field lines, so current sheets
are formed behind the plasmoid separating oppositely-directed fields. The
reconnection in these current sheets could produce the radio emission. A more
detailed analysis of this idea is necessary.

##  4 Non-Linear Effects and Escape of Radio Emission

The high brightness temperature of FRBs implies that the non-linear processes
could significantly affect the properties of the outgoing radiation and even
prevent escape of radio waves. Here, I briefly outline the relevant processes.

###  4.1 Electron in a Strong Electromagnetic Wave

Let us first consider non-linear effects in the non-magnetized plasma. The
strength of the electromagnetic wave is measured by the parameter $a$
introduced in Section 2.5 in Eq. (23). This parameter represents the four-
velocity of electron oscillations in the field of the wave (see, e.g.,
(Melrose, 1980)). In FRBs, the strength parameter exceeds unity at the
distance from the source (Luan and Goldreich, 2014):

$R<2\times 10^{13}\frac{F^{1/2}_{\nu,\rm Jy}D_{\rm Gpc}}{\nu^{1/2}_{\rm
GHz}}\,\rm cm.$ (53)

The standard linear theory of electromagnetic waves in a plasma assumes that
$a\ll 1$. Even under this condition, the non-linear effects could
significantly affect the propagation of the wave because even small non-linear
corrections accumulate considerably at a long enough path. At $a>1$, the
electron oscillates relativistically so that the Lorentz force,
$(1/c)\mathbf{v\times B}$, becomes comparable with the electric force. In a
linearly-polarized strong wave, the electrons oscillate both in the transverse
and in the longitudinal directions, producing a figure-of-eight trajectory in
the oscillation-center frame.

If an electron at rest is illuminated by a strong wave, it is pushed forward
with the Lorentz factor $\sim a$. Together with oscillations with amplitude
$\sim a$ in the oscillation-center frame, this produces the total energy $\sim
m_{e}c^{2}a^{2}$. Therefore, the electron-positron plasma is boosted forward
by a strong wave. Conversely, electrons in an electron-ion plasma are tied
electrostatically to ions. Therefore, they remain, on average, at rest and
only experience oscillations with the amplitude of $\sim a$ (Waltz and Manley,
1978; Sprangle et al., 1990).

In many cases, an electron in a high frequency electromagnetic wave may be
considered a particle with the effective mass:

$m_{\rm eff}=m_{e}\sqrt{1+\frac{1}{2}a^{2}},$ (54)

moving with the velocity of the electron oscillation center, $\mathbf{v}_{\rm
d}$, and having the energy and momentum $\varepsilon=m_{\rm
eff}c^{2}\gamma_{\rm d}$ and $\mathbf{P}=m_{\rm eff}\mathbf{v}_{\rm
d}\gamma_{\rm d}$, respectively, where $\gamma_{\rm d}=(1-v_{\rm
d}^{2}/c^{2})^{-1/2}$ is the Lorentz factor of the oscillation center (see,
e.g., (Melrose, 1980)). In particular, the conservation laws for the Compton
scattering of a strong wave look like

$\varepsilon+s\hbar\omega=\varepsilon^{\prime}+\hbar\omega^{\prime};\qquad\mathbf{P}+s\hbar\mathbf{k}=\mathbf{P^{\prime}}+\hbar\mathbf{k^{\prime}},$
(55)

where $s$ is an integer. The scattering of a strong wave may be described as
the absorption of $s$ photons and the reemission of a single photon.
Neglecting recoil, we recover the classical picture: a relativistically
oscillating electron emits at harmonics of the oscillation frequency. In the
regime $a\gg 1$, the power of the scattered radiation is dominated by the
higher harmonics of the incident wave. As with synchrotron emission, the
maximal power is achieved at $s\sim a^{3}$, and the scattering cross-section
exceeds the Thomson cross-section $a^{2}$ times.

The amplitude-dependent relativistic mass of electrons has many implications.
The plasma refraction index,

$n=\sqrt{1-\frac{\omega_{p}^{2}}{\omega^{2}}},$ (56)

depends on the plasma frequency,

$\omega_{p}=\sqrt{\frac{4\pi e^{2}N}{m_{e}}}.$ (57)

Increasing the effective electron mass decreases the plasma frequency.
Therefore, the cutoff frequency, below which the wave could not propagate,
decreases from $\omega_{\rm cutoff}=\omega_{p}$ to $\omega_{\rm
cutoff}\sim\omega_{p}/\sqrt{a}$. The dispersion measure decreases
appropriately, which could have implications for FRBs (Lu and Phinney, 2020;
Yang and Zhang, 2020). An important point is that these relations are written
in the plasma frame, which is the oscillation center frame of electrons.

The synchrotron absorption of strong waves occurs as if the absorbing
particles have mass (54) and rotate in the magnetic field with the Lorentz
factor $\gamma_{\rm d}$ (Lyubarsky, 2018).

In some cases, the results of linear theory are strongly modified even at
$a<1$. For example, the free-free absorption is determined by the velocity of
electrons. Therefore, if the velocity of oscillations in the wave exceeds the
thermal electron velocity, $a>v_{T}/c$, the absorption coefficient is
suppressed by the factor $(ac/v_{T})^{3}=a^{3}(m_{e}c^{2}/k_{B}T)^{3/2}$ (Lu
and Phinney, 2020).

All the above effects are based on strong oscillations of electrons in the
field of the wave. If the plasma is magnetized such that the Larmor frequency
exceeds the wave frequency, these effects are suppressed for the waves
polarized perpendicularly to the background magnetic field (Lyutikov, 2020);
these waves represent the so called X-mode. In the O-mode, where the
polarization vector has a component along the background magnetic field, the
electron oscillates strongly. Therefore the refraction indexes for the two
modes are different, which has important implications for the polarization
transfer. When the radiation propagates through a slowly varying magnetic
field, the polarization of each mode is adjusted to the local direction of the
field if the wavelength of beating between the two modes is small as compared
with the inhomogeneity scale. The polarization of the outgoing radiation is
fixed at the so called limiting polarization radius, where this condition is
violated (e.g., (Ginzburg, 1970; Cheng and Ruderman, 1979)). For strong waves,
these effects were studied in ref. (Lu et al., 2019).

###  4.2 Induced Compton Scattering

The induced Compton scattering may be described in terms of radiation transfer
theory. The scattering process could be considered as absorption followed by
emission. Therefore, to consider the induced scattering, we must only
multiply, according to the general quantum prescription, the scattering rate
by $1+n_{\nu,\mathbf{l}}$, where $n_{\nu,\mathbf{l}}$ is the occupation number
of the scattered photons with frequency $\nu$ propagating along the unit
vector $\mathbf{l}$. The photon occupation number is simply related to the
brightness temperature, $n_{\nu,\mathbf{l}}=k_{B}T_{b}/h\nu$. The scattering
rate from state $(\nu^{\prime},\mathbf{l^{\prime}})$ to state
$(\nu,\mathbf{l})$ is proportional to
$n_{\nu^{\prime},\mathbf{l^{\prime}}}(1+n_{\nu,\mathbf{l}})$. At first glance,
this means that the induced scattering rate dominates when the occupation
number exceeds unity. However, the net scattering rate is determined by the
difference between the scattering $\mathbf{l}\to\mathbf{l^{\prime}}$ and the
reverse process $\mathbf{l^{\prime}}\to\mathbf{l}$. Then, the product terms
$n_{\nu\mathbf{l}}n_{\nu^{\prime}\mathbf{l^{\prime}}}$ cancel out unless we
take into account that, due to a small recoil, the frequency of the scattered
photon decreases in the electron rest frame:

$\delta\nu=\frac{h\nu^{2}}{m_{e}c^{2}}(1-\mathbf{l\cdot l^{\prime}}).$ (58)

Let us consider photons with the frequency $\nu$ propagating into the
direction $\mathbf{l}$ and assume for a while that they are scattered only
into or from the direction $\mathbf{l}^{\prime}$. The photon scattered from
the direction $\mathbf{l}^{\prime}$ to the direction $\mathbf{l}$ has the
frequency $\nu+\delta\nu$ whereas when a photon is scattered from the
direction $\mathbf{l}$ into the direction $\mathbf{l}^{\prime}$, the frequency
of the scattered photon is $\nu-\delta\nu$. Therefore the net rate of
scattering into and out of the state $(\nu,\mathbf{l})$ is proportional to

$n_{\nu+\delta\nu,\mathbf{l^{\prime}}}\left(1+n_{\nu\mathbf{l}}\right)\nu^{2}-n_{\nu\mathbf{l}}\left(1+n_{\nu-\delta\nu,\mathbf{l^{\prime}}}\right)(\nu-\delta\nu)^{2}=(n_{\nu\mathbf{l^{\prime}}}-n_{\nu\mathbf{l}})\nu^{2}+2n_{\nu\mathbf{l}}\frac{\partial\nu^{2}n_{\nu\mathbf{l^{\prime}}}}{\partial\nu}\delta\nu,$
(59)

where the factor $\nu^{2}$ takes into account the phase volume. The induced
scattering rate exceeds the spontaneous scattering rate only when the second
term in the rhs of this equation exceeds the first term, i.e., if
$n_{\nu\mathbf{l}}\delta\nu>\nu$. This requires a very high brightness
temperature, $k_{B}T_{b}(1-\mathbf{l\cdot l^{\prime}})\gg
m_{e}c^{2}=k_{B}\times 6\cdot 10^{9}$ K.

The total scattering rate is found by multiplying the above expression by the
scattering probability and integrating over the directions
$\mathbf{l}^{\prime}$. When the plasma is nonrelativistic, $v_{T}\ll c$, and
the radiation spectrum is wide, $\Delta\nu/\nu\gg v_{T}/c$, the resulting
radiation transfer equation for the induced scattering is written as (e.g.,
(Wilson, 1982)):

$\frac{dI_{\nu,\mathbf{l}}}{dt}=\frac{r_{e}^{2}Nc}{m_{e}}I_{\nu,\mathbf{l}}\int(\mathbf{e\cdot
e}^{\prime})^{2}(1-\mathbf{l\cdot
l}^{\prime})\frac{\partial}{\partial\nu}\left(\frac{I_{\nu,\mathbf{l^{\prime}}}}{\nu}\right)d\Omega^{\prime},$
(60)

where $I_{\nu,\mathbf{l}}=h\nu^{3}n_{\nu,\mathbf{l}}/c^{2}$ is the radiation
intensity, $N$ is the electron number density, and $\mathbf{e}$ is the
polarization unit vector. The time derivative is along the ray. The induced
scattering does not affect the escape time of photons from the source, but
redistributes them toward lower frequencies, thus heating the plasma
(Syunyaev, 1971). The total number of photons,
$\int\omega^{2}n_{\nu,\mathbf{k}}d\omega d\Omega$, is conserved. If the
initial spectrum has a maximum, such that $I_{\nu,\mathbf{l}}/\nu$ increases
with $\nu$ at $\nu<\nu_{0}$ and decreases at $\nu>\nu_{0}$, the induced
scattering increases the intensity at $\nu<\nu_{0}$ and decreases it at
$\nu>\nu_{0}$, the maximum intensity shifting toward larger frequencies.

Note that the equation does not contain the Planck constant, even though the
above simple picture is based on the quantum approach. The reason is that the
effect is purely classical: the non-linear interaction of two waves yields a
beating wave, which exerts a constant force on electrons moving with the
velocity equal to the beating phase velocity,
$\omega-\omega^{\prime}=\mathbf{v\cdot(k-k^{\prime})}$. This transfers the
energy from the wave to the plasma, which means that the wave with the higher
frequency decays. The classical derivation of the induced Compton scattering
of electro-magnetic waves is given in (Galeev and Syunyaev, 1973; Drake et
al., 1974).

The probability of the induced scattering into a state is proportional to the
number of photons already available in the state. The radiation transfer
equation (60) has the form:

$\frac{dI_{\nu,\mathbf{l}}}{dt}=GI_{\nu,\mathbf{l}},\qquad
G==\frac{r_{e}^{2}Nc}{m_{e}}\int(\mathbf{e\cdot
e}^{\prime})^{2}(1-\mathbf{l\cdot
l}^{\prime})\frac{\partial}{\partial\nu}\left(\frac{I_{\nu,\mathbf{l^{\prime}}}}{\nu}\right)d\Omega^{\prime},$
(61)

so that the quantity $G$, which depends on the radiation intensity, is the
induced scattering rate. Within the source, where the radiation is nearly
isotropic, the recoil factor $(1-\mathbf{l\cdot l}^{\prime})$ is of the order
of unity. Then, we can estimate the scattering rate as
$G\sim(k_{B}T_{b}/m_{e}c^{2})\sigma_{T}nc$, where $\sigma_{T}$ is the Thomson
cross-section. If the plasma in the source is relativistically hot, the
induced scattering rate decreases by $\gamma_{T}$, which is the characteristic
thermal Lorentz factor of electrons in the source (Lyubarsky, 2008). The
escape time from a source of size $L$ is $L/c$. Then, the optical depth to the
induced scattering is conveniently defined as the ratio of the escape time to
the frequency redistribution time:

$\tau_{\rm ind}=GL/c\sim\frac{k_{B}T_{b}}{m_{e}c^{2}}\sigma_{T}NL.$ (62)

The source is transparent to the induced scattering if $\tau_{\rm ind}<1$. In
the opposite case, the radiation is redistributed toward smaller frequencies
before escaping.

If the radiation is highly beamed, as is the case at large distances from the
source, the recoil factor $(1-\mathbf{l\cdot l^{\prime}})$ makes the
scattering within the beam inefficient. Then, the scattering outside the beam
dominates (Coppi et al., 1993; Lyubarsky, 2008) because, according to Equation
(61), even weak isotropic background radiation (created, e.g., by spontaneous
scattering) grows exponentially. Let us choose in Eq. (61)
$\mathbf{l^{\prime}}$ in the direction of the beam and $\mathbf{l}$ in the
direction well outside the beam, where only a weak background emission is
initially present. Then, the intensity in the direction $\mathbf{l}$ grows
exponentially with the rate:

$G\sim\sigma_{T}Nc\frac{F_{\nu}}{m_{e}\nu^{2}},$ (63)

where $F_{\nu}=I_{\nu,\mathbf{l}}\Delta\Omega$ is the spectral flux at the
scattering point and $\Delta\Omega$ is the solid angle subtended by radiation
at this point. Eventually, the scattered radiation takes whole the energy of
the primary beam. Inasmuch as the intensity in the direction $\mathbf{l}$ is
initially very low, many e-folding times are necessary to remove significant
energy from the beam. Therefore, in this case, the transparency condition may
be written as:

$\tau_{\rm ind}=\frac{1}{c}\int GdR\sim GR/c<10.$ (64)

Here, the flux, and therefore the rate $G$, decreases as $R^{-2}$; therefore,
only the region $\Delta R\sim R$ contributes into the integral.

The above assumes that the emission is steady. In the case of a short pulse
with duration $\tau<R/c$, the induced scattering occurs only while the
scattered ray remains within the zone illuminated by the primary radiation.
Therefore, the optical depth is determined not by the size of the medium but
by the duration of the pulse, so that we must substitute $R$ by $c\tau$ in Eq.
(64) (Lyubarsky, 2008).

The above considerations also assume that the electrons oscillate non-
relativistically in the field of the wave. This means that the wave strength
parameter (Eq. (23) is smaller than unity. With this parameter, the induced
scattering rate (63) may be presented as:

$G\sim\frac{\omega_{p}^{2}a^{2}}{\omega},$ (65)

where $\omega_{p}$ is the plasma frequency (57). When $a>1$, the scattering
rate decreases because the effective mass of a relativistically oscillating
electron increases. A larger effective mass reduces recoil, which is crucial
for the induced scattering. In this case, the scattering rate is estimated as
(Lyubarsky, 2019):

$G\sim\frac{\omega_{p}^{2}}{\omega a}.$ (66)

Even though the spontaneous scattering is dominated by high harmonics at $a\gg
1$, the induced scattering occurs predominantly in the first harmonic at any
$a$.

In FRBs, the induced scattering could affect the wave propagation close enough
to the source. In synchrotron maser models (see Section 3.3), the transparency
condition (64) is marginally satisfied for $\sim 1$ GHz and violated for
smaller frequencies (Metzger et al., 2019; Beloborodov, 2020), which could
explain the non-detection of low frequency FRBs (Karastergiou et al., 2015).
The effect of the induced scattering was also used in order to show that FRBs
could not be produced in stellar coronae (Lyubarsky and Ostrovska, 2016).

###  4.3 Induced Raman Scattering

Raman scattering is the scattering of electromagnetic waves on plasma
oscillations. The process may be interpreted as a decay of an electromagnetic
wave into another electromagnetic wave and a plasma wave or/and a merging of
an electromagnetic and a plasma waves into another electromagnetic wave. The
energy and momentum conservations imply relations between the frequencies and
the wave vectors of three waves:

$\omega=\omega^{\prime}+\omega_{p};\qquad\mathbf{k=k^{\prime}+q},$ (67)

where $\omega$, $\mathbf{k}$, and $\omega^{\prime}$ and $\mathbf{k^{\prime}}$
are the frequencies and wave vectors of the electromagnetic waves,
respectively; $\omega_{p}$ is the plasma frequency (57); and $\mathbf{q}$ is
the wave vector of the plasma wave. If a powerful beam propagates through a
medium with no preexisting plasma turbulence, the induced Raman scattering
results in the exponential growth of both the intensity of the scattered wave
and the level of plasma turbulence (parametric instability (Drake et al.,
1974)). With the classical language, the non-linear interaction between two
waves produces a beating wave, which resonantly excites the third wave.

Importantly, this process is impossible in electron-positron plasma. The
reason is that the beating wave exerts the same force both on electrons and on
positrons, so that the plasma waves in which electrons and positrons oscillate
in antiphase are not excited.

In electron-ion plasma, the rate of the induced Raman scattering could be
represented as (Thompson et al., 1994; Lyubarsky, 2008):

$G=\sigma_{T}Nc\frac{F_{\nu}}{m_{e}\nu_{p}\nu}(1-\cos\theta),$ (68)

where $\theta$ is the scattering angle. If $\theta$ is not small, the Raman
scattering rate exceeds that of Compton scattering (63). However, due to
Landau damping, the phase velocity of the plasma wave could not be smaller
than a few electron thermal velocities, which may be written as
$q<\omega_{p}/4v_{T}$. Therefore, the scattering angle is limited by the
resonance conditions (67), so that the Raman scattering rate exceeds the
Compton scattering only if:

$\frac{\omega}{\omega_{p}}<\frac{m_{e}c^{2}}{8k_{B}T}.$ (69)

The condition of efficient scattering is still given by Eq. (64) because, just
as for the case of the induced Compton scattering, many e-folding times are
necessary for the scattered waves to grow significantly.

Importantly, the scattering by a small angle does not prevent escape of
radiation, it only makes the beam wider, which could smear a short pulse.
However, the situation with short pulses is more complicated so that we cannot
only substitute $R$ by $c\tau$ as with induced Compton scattering. As for the
case of induced Compton scattering, the process occurs only in the illuminated
region. However, if the scattering angle is small, the scattered wave remains
within the illuminated region along a large distance,
$L=c\tau/(1-\cos\theta)$. Convesely, the group velocity of the plasma waves is
small; therefore, they are amplified only for the time $\tau$. The Raman
scattering of short pulses was addressed considering these effects in
(Lyubarsky, 2008).

Notably, the scattering rate (68) was obtained by neglecting the decay of the
plasma waves, i.e., assuming that the decay rate is less than the scattering
rate. If this condition is violated, the Raman scattering is suppressed. The
collisional decay of plasma waves could be easily taken into account by
comparing the decay rate with the scattering rate (Thompson et al., 1994;
Lyubarsky, 2008). Since the latter is proportional to the beam intensity, the
collisional decay of plasma waves may be neglected for powerful enough
radiation beams. However, the Raman scattering of powerful beams could be
affected by the non-linear interactions of plasma waves, such as the induced
scattering of plasma waves (non-linear Landau damping) and the modulation
instability (see, e.g., (Shapiro and Shevchenko, 1984; Brejzman, 1990)). This
could prevent the growth of plasma turbulence beyond some level, thus
suppressing the rate of Raman scattering. To determine the role of the Raman
scattering in any particular case, one must consider all the above effects.

###  4.4 Modulation and Filamentation Instabilities

Due to the non-linear interactions of radiation and matter, the refraction
index depends on the radiation power, which could lead, under some conditions,
to self-modulation or/and self-focusing of the radiation beam (see, e.g.,
(Karpman and Cap, 1975)). Let the refraction index increase with the radiation
intensity. Then, any transverse intensity gradient is amplified because the
rays are focused toward the intensity maximum. In wide beams, self-focusing
leads to filamentation, i.e., to modulation in the direction perpendicular to
the propagation direction. Self-modulation in the longitudinal direction
occurs if the wave group velocity decreases with intensity. Then, the waves
accumulate near the intensity maximum.

In plasma, the refraction index is given by Eq. (56). Three effects provide
the dependence of the refraction index on the radiation intensity and,
therefore, could potentially lead to modulation and/or filamentation
instabilities:

The first effect is based on the ponderomotive force, which expels plasma from
the regions of enhanced radiation intensity. Therefore, if this fluctuation
forms, the plasma frequency in the region decreases so that the refraction
index increases. Due to refraction, the radiation is focused at the region,
thus repelling more plasma and amplifying the focusing effect. This effect
leads to filamentation of the beam; however, it could not produce self-
modulation (Shearer and Eddleman, 1973; Kaw et al., 1973; Drake et al., 1974).
The effect works efficiently in electron-positron plasma, whereas in electron-
ion plasma, it is suppressed by the large inertia of ions, and therefore it is
only involved at very large intensities.

The second effect reduces the plasma frequency in the regions of enhanced
radiation because, as discussed in Section 4.1, the effective mass of the
oscillating electron increases with oscillation amplitude. This results in
increasing the refraction index.

The third effect arises because, in the field of the wave, the longitudinal
$\mathbf{v\times B}$ force produces an electron density perturbation, $\Delta
n$, at the double wave frequency. The beating between this perturbation and
the electron velocity oscillations in the pumping wave yields current
$\mathbf{j}=\delta N\mathbf{v}$ at the wave frequency, which affects the
propagation of the pumping wave. The last two effects are not suppressed by
the ion inertia because they do not affect the average plasma density. They
are of the same order, and therefore should be considered together. It is
shown (Max et al., 1974) that they lead to both filamentation and modulation
of the radiation beam. The scale of filamentation is significantly larger than
the longitudinal modulation scale, so the beam breaks into pancakes transverse
to the radial direction.

Thus, in the electron-positron plasma, where the effect of the ponderomotive
force dominates, only the filamentation instability could develop. In
particular, simulations of the maser emission from relativistic shocks reveal
filamentation of the upstream flow (Iwamoto et al., 2017). In electron-ion
plasma, the effect of the ponderomotive force is suppressed by ion inertia;
therefore, in a wide range of parameters, both filamentation and modulation
are possible. The comprehensive analysis of different regimes in electron-ion
plasma was given in (Sobacchi et al., 2020).

Neither filamentation nor modulation prevent escape of radiation from the
source. However, they could affect the properties of the outgoing radiation.
It was suggested (Yang and Zhang, 2020; Lyutikov, 2020) that self-focusing
could affect the collimation angle and the true event rate of FRBs. However,
when the beam is macroscopically wide, it is not focused as a whole. Instead,
the filamentation instability develops faster on small scales; therefore, the
beam only breaks into narrow subbeams, each of which may be also modulated in
the longitudinal direction. This could produce other observable effects
(Sobacchi et al., 2020). Namely, when the burst exits the plasma slab, the
subbeams are strongly diffracted, which could lead to (1) smearing of the
burst in time if the diffraction angle is large enough and (2) frequency
modulation of the observed intensity due to interference between the subbeams.
The longitudinal modulation could, in turn, imprint a microsecond structure on
the light curve of the burst.

###  4.5 Non-Linear Interaction of Waves in a highly magnetized plasma

Let us consider the most relevant for the magnetar magnetospheres case when
the wave frequency is small compared with both the plasma and the Larmor
frequencies and when the magnetic energy considerably exceeds the plasma
energy. Then, we can work in the scope of the force-free MHD. Only two types
of waves exist in this system: the fast magnetosonic (fms) and Alfven waves.
The fms waves were briefly discussed in Section 4. Within the limit
$\sigma\to\infty$, their dispersion law is reduced to $\omega=ck$. The
dispersion law of the Alfven waves in the same limit is
$\omega=ck|\cos\theta|$, where $\theta$ is the angle between the background
magnetic field and the propagation direction of the wave. The fms waves are a
low frequency $\omega\ll\omega_{p}$ limit of the so called X-mode of
electromagnetic wave (e.g., (Arons and Barnard, 1986)). When propagating
towards decreasing plasma density, the fms wave is smoothly converted into the
X-mode and could, in principle, eventually escape as outgoing electromagnetic
waves. The Alfven waves propagate only along the magnetic field lines and
therefore if they are excited on closed field lines, they remain trapped.

The non-linear interactions of force-free MHD waves were studied in refs.
(Thompson and Blaes, 1998; Lyubarsky, 2020). The strongest are the resonant
three-wave interactions, i.e., the decay of a wave into two waves and the
merging of two waves into one. In this case, the resonant conditions (the
conservation laws in the quantum language) looks like

$\omega=\omega_{1}+\omega_{2};\qquad\mathbf{k=k_{1}+k_{2}}.$ (70)

Substituting the dispersion laws into the conservation laws, one sees that
only interactions involving both types of waves are possible:

${\rm fms}\longleftrightarrow{\rm fms}+{\rm Alfven}\quad{\rm and}\quad{\rm
fms}\longleftrightarrow{\rm Alfven}+{\rm Alfven}.$ (71)

The rate of the interaction may be roughly estimated as:

$G\sim\left(\frac{\delta B}{B}\right)^{2}\omega,$ (72)

where $\delta B$ is the amplitude of the waves and $B$ is the background
field.

Assume that there is a powerful source of fms waves well within the magnetar
magnetosphere. As they propagate outwards, their amplitude decreases with the
distance as $1/R$, whereas the background field in the magnetosphere decreases
as $1/R^{3}$, so that:

$\frac{\delta B}{B}=\sqrt{\frac{L_{\rm FRB}}{c}}\frac{R^{2}}{\mu}.$ (73)

Therefore, the characteristic time of the non-linear interaction, $1/G$,
rapidly decreases with distance. For waves in the GHz band, it becomes smaller
than the propagation time, $R/c$, already at distance (Lyubarsky, 2020):

$R=10^{7}\left(\frac{\mu_{33}^{2}}{L_{\rm FRB,43}\nu_{9}}\right)^{1/5}\,\rm
cm,$ (74)

which is well within the magnetosphere. Beyond this distance, the non-linear
interaction yields a cascade redistribution of the waves toward larger and
smaller frequencies. High-frequency waves eventually decay. The amplitude of
low-frequency waves increases with decreasing frequency because of energy
conservation. Therefore, eventually, the non-linear steepening is involved and
they decay too. As no other waves in the GHz band can propagate across
magnetic field lines, even if a source of GHz waves with the required power
existed inside the magnetosphere, no radiation would escape.

In the outer magnetosphere, the situation is different. The FRB is triggered
by the magnetar flare, which produces a magnetic perturbation (the magnetic
pulse) described in Section 3.2. In the outer magnetosphere, the field of the
pulse is larger than the local magnetospheric field, and the pulse propagates
with relativistic velocity. When the pulse excites high-frequency waves, as
described in Section 3.4, they propagate on the top of the pulse to very large
distances. They are not strongly affected by non-linear interactions because
the ratio of the wave amplitude to the field in the pulse is small and because
of relativistic time dilation within the pulse (Lyubarsky, 2020).

##  5 Conclusions

In this review I outlined the radiation mechanisms proposed for FRBs. The aim
was to demystify, at least somewhat, the field of the coherent emission
mechanisms. I tried to describe them at the very basic physics level, avoiding
technical details. However, one has to stress again, that these mechanisms may
be understood only as collective plasma processes; an over-simplistic approach
could not provide reasonable results.

For a while, we have two workable mechanisms: the synchrotron maser at the
front of a relativistic magnetized shock and the radiation from variable
currents in a reconnecting current sheet. Both mechanisms assume relativistic
magnetized outflows, which could be found in different astrophysical
environments. Observational evidence as well as theoretical considerations
point to magnetar flares as the most promising progenitor. Therefore, both
types of FRB models were developed within the scope of the magnetar paradigm.
Both models have their pros and cons. The synchrotron maser produces radio
emission in a wide range of luminosities; however, this emission could hardly
be 100% polarized as is sometimes observed. Moreover, shock waves with the
required parameters may be obtained only with an assumption that a moderately
relativistic and mildly magnetized medium is present relatively close to the
magnetar. The origin of such a medium is unclear. The emission from a
reconnecting current sheet may have a 100% degree of polarization; however,
this model could not explain weak FRBs because the emission frequency becomes
too low for small luminosities. This emission has been studied only for a
quiet, steady reconnection. It is unclear how the process develops in a highly
unsteady and turbulent reconnection when the sheet is destroyed by a strong
electromagnetic perturbation caused by the magnetar flare. Further development
of both models is necessary. It is possible that both mechanisms work in
different cases. And of course we could not exclude that new models will be
presented that will successfully agree with observations.

###### Acknowledgements.

I gratefully acknowledge grant I-1362-303.7/2016 from the German-Israeli
Foundation for Scientific Research and Development and grant 2067/19 from the
Israeli Science Foundation. References yes

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