# Fast radio bursts and their high-energy counterpart from magnetar
magnetospheres

Yuan-Pei Yang11affiliationmark:  and Bing Zhang22affiliationmark:  1 South-
Western Institute for Astronomy Research, Yunnan University, Kunming, Yunnan,
P.R.China<EMAIL_ADDRESS>
2 Department of Physics and Astronomy, University of Nevada, Las Vegas, NV
89154, USA<EMAIL_ADDRESS>
###### Abstract

The recent discovery of a Galactic fast radio burst (FRB) occurring
simultaneously with an X-ray burst (XRB) from the Galactic magnetar SGR
J1935+2154 implies that at least some FRBs arise from magnetar activities. We
propose that FRBs are triggered by crust fracturing of magnetars, with the
burst event rate depending on the magnetic field strength in the crust. Since
the crust fracturing rate is relatively higher in polar regions, FRBs are
preferred to be triggered near the directions of multipolar magnetic poles.
Crust fracturing produces Alfvén waves, forming a charge starved region in the
magnetosphere and leading to non-stationary pair plasma discharges. An FRB is
produced by coherent plasma radiation due to nonuniform pair production across
magnetic field lines. Meanwhile, the FRB-associated XRB is produced by the
rapid relaxation of the external magnetic field lines. In this picture, the
sharp-peak hard X-ray component in association with FRB 200428 is from a
region between adjacent trapped fireballs, and its spectrum with a high cutoff
energy is attributed to resonant Compton scattering. The persistent X-ray
emission is from a hot spot heated by the magnetospheric activities, and its
temperature evolution is dominated by magnetar surface cooling. Within this
picture, magnetars with stronger fields tend to produce brighter and more
frequent repeated bursts.

radio transient sources; magnetars; pulsars; X-ray bursts; non-thermal
radiation sources

##  1 Introduction

Fast radio bursts (FRBs) are millisecond-duration radio transients with
extremely high brightness temperatures (Lorimer et al., 2007; Petroff et al.,
2019; Cordes & Chatterjee, 2019; Zhang, 2020b). Hundreds of FRB sources have
been reported, dozens of which showed the repeating behavior (e.g., The
CHIME/FRB Collaboration et al. (2021), and see the catalog of published FRB
(http://frbcat.org, Petroff et al. 2016) and Transient Name Server system for
newly reported FRBs (https://www.wis-tns.org, Petroff & Yaron 2020)).
Recently, a Galactic FRB with fluence of $1.5~{}{\rm MJy-ms}$ was detected to
be associated with an X-ray burst (XRB) from the Galactic magnetar SGR
J1935+2154 (Bochenek et al., 2020; CHIME/FRB Collaboration et al., 2020a; Li
et al., 2020; Mereghetti et al., 2020; Ridnaia et al., 2020; Tavani et al.,
2020), which suggests that at least some FRBs originate from magnetars
(Lyubarsky, 2014; Katz, 2016; Murase et al., 2016; Beloborodov, 2017; Kumar et
al., 2017; Metzger et al., 2017; Yang & Zhang, 2018; Wadiasingh & Timokhin,
2019; Lu et al., 2020; Margalit et al., 2020; Wadiasingh et al., 2020; Wang,
2020; Wang et al., 2020; Yang et al., 2020; Ioka, 2020; Zhang, 2020b; Xiao et
al., 2021; Lyubarsky, 2021; Yu et al., 2021).

During the active phase of SGR J1935+2154 on 2020 April 28, over 200 XRBs were
detected, and a burst storm with a rate of $>0.2~{}{\rm s^{-1}}$ appeared
during the first 1120 seconds (Younes et al., 2020). The $T_{90}$ duration
distribution of the bursts peaks at $840~{}{\rm ms}$, and their spectra could
be fit with a blackbody spectrum with an average temperature of $1.7~{}{\rm
keV}$ and an emitting area of $53~{}{\rm km^{2}}$. Besides XRBs, the source
also shows persistent emission with a double-peaked pulse profile, with a
period defined by the magnetar spin period. It is interesting that the arrival
times of the two peaks of FRB 200428 aligned in phase with the brightest peaks
of the associated XRB (Li et al., 2020) and with the peak of the pulse profile
(Younes et al., 2020). The flux and blackbody temperature of the X-ray
persistent emission decreased rapidly in the early stages of the outburst with
a duration of $\sim(0.1-1)~{}{\rm day}$, while the size of the emitting area
remained unchanged (Younes et al., 2020).

Although a large number of XRBs were detected on 2020 April 28, deep
monitoring by Five-hundred-meter Aperture Spherical radio Telescope (FAST)
suggested that the associations between FRB and X-ray burst are very rare (Lin
et al., 2020). Three possibilities might be considered to explain the
association with low possibility (Lin et al., 2020): 1. the radiation beaming
angle of FRBs is much narrower than that of XRBs, so that there might exist
off-beam “slow” radio bursts (Zhang, 2021); 2. the peak frequency of FRBs has
a wide distribution than the band of radio telescopes; 3. and the associations
between FRBs and XRBs are intrinsically rare due to very special physical
conditions (e.g. Li et al., 2020; Younes et al., 2020; Yang et al., 2021).

For extragalactic FRBs, there is no confirmed multiwavelength counterpart so
far. If an XRB is associated with an extragalactic/cosmological FRB at a
distance $\sim 100~{}{\rm Mpc}$, the XRB flux would be much smaller than that
of the XRB associated with FRB 200428, which is almost undetectable with the
available high-energy detectors. Thus, identifying the magnetar origin of
extragalactic FRBs via multi-wavelength observations would be extremely
difficult. On the other hand, some extragalactic FRBs show special properties
that are not similar to the properties of Galactic magnetars. For example,
active repeating source FRB 180916B showed a period of $\sim 16~{}{\rm day}$
with a frequency-dependent active window of $\sim 5$ days (Marcote et al.,
2020; CHIME/FRB Collaboration et al., 2020b; Pastor-Marazuela et al., 2020).
The first repeater FRB 121102 has mysterious persistent radio emission, a
large, rapidly evolving rotation measure (Spitler et al., 2016; Chatterjee et
al., 2017; Michilli et al., 2018) and a possible 160-day period (Rajwade et
al., 2020). Most magnetars in the Milky Way galaxy have a typical age of
$\sim(1-10)~{}{\rm kyr}$. The extragalactic FRB sources may originate from
magnetars with a wider age distribution at cosmological distances. In
particular, those active repeaters may be related to magnetars with younger
ages and/or stronger magnetic fields.

Figure 1: Schematic configuration. (a) FRB triggered by the crust fracture: a
charge starved region is produced by Alfvén wave triggered by the crust
fracture, and further generates FRB via coherent plasma radiation. XRB is
produced by rapid relaxation of external magnetic field, and pair plasma
generate and form trapped fireballs. (b) sharp-peak X-ray component just after
FRB: At the bottom of trapped fireballs, E-mode X-ray photons escape to the
region between adjacent trapped fireballs, and are constrained in a beaming
angle due to the large optical depth of the trapped fireballs around it, which
corresponds to a sharp-peak component of XRB. Meanwhile, a part of XRB energy
inject to magnetar surface. The magnetar surface is heated and generate
persistent X-ray emission during a relatively long time. The dimensions in
this figure are distorted for clarity.

The high-energy emission from a magnetar has been suggested to be due to
magnetospheric activities (e.g., Thompson & Duncan, 1995, 2001; Beloborodov &
Thompson, 2007). When a magnetar magnetosphere is triggered by crustal
deformation, the magnetic energy in the crust would be released and be
converted to particle energy and to radiation. In this work, we conjecture
that FRBs are triggered by crust fracturing of magnetars (see a schematic
physical picture in Figure 1, see also Wadiasingh & Timokhin (2019) and Lu et
al. (2020)). In the crust, the geometric configuration of magnetic fields
evolves due to Hall drift. When the magnetic stress exceeds the shear modulus,
the crust would crack. Since the crust fracturing rate is relatively larger in
the polar regions with stronger magnetic fields, FRBs are preferably triggered
near the directions of multipolar magnetic poles at the surface. Crust
fracturing produces Alfvén waves, and a charge starved region is generated in
the magnetosphere (Kumar & Bošnjak, 2020; Lu et al., 2020), leading to non-
stationary pair plasma discharges. An FRB is produced by coherent plasma
radiation due to nonuniform pair production across magnetic field lines
(Philippov et al., 2020). The FRB-associated XRB is produced by rapid
relaxation of the external magnetic field. In the polar region between
adjacent trapped fireballs, a sharp-peak component of XRB emission is
produced. Its spectrum is modified by resonant Compton scattering, and the
radiation beam is constrained by the trapped fireballs surrounding it. After
the FRB and XRB, a persistent X-ray emission component would be generated by a
hot spot that is heated by the radiation near the magnetar surface. The paper
is organized as follows. We first discuss the physical conditions and the
event rate of crust fracturing in Section 2. We propose that FRBs are produced
by coherent plasma radiation in Section 3. The properties of FRB-associated
XRB and persistent X-ray emission are discussed in Section 4. The results are
discussed and summarized in Section 5. The convention $Q_{x}=Q/10^{x}$ is
adopted in cgs units, unless otherwise specified.

##  2 FRBs from crust fracturing and their burst rate

The association between FRB 200428 and an XRB from SGR J1935+2154 implies that
FRBs might be triggered by an event that triggered the XRB. We attribute this
trigger to crust cracking of the magnetar. According to this picture, FRBs
with a wide range of luminosities are caused by a wide range of crust
fracturing scales.

The typical FRB event rate density is $\sim 10^{4}~{}{\rm Gpc^{-3}yr^{-1}}$
with isotropic luminosity $L>10^{42}~{}{\rm erg~{}s^{-1}}$, and the power law
index of the cumulative luminosity function is $\alpha\sim-0.8$ for
$N(>L)\propto L^{\alpha}$ (Luo et al., 2020). This luminosity function seems
to extend to as low as $L\sim 10^{38}~{}{\rm erg~{}s^{-1}}$, the isotropic
luminosity of FRB 200428 associated with the XRB from SGR J1935+2154 (Lu et
al., 2020). A magnetar may also emit radio bursts in an even wider
luminosity/energy range. Indeed, in addition to FRB 200428, SGR J1935+2154
also emitted some radio bursts whose fluxes were several orders of magnitude
lower than FRB 200428 (Zhang et al., 2020; Kirsten et al., 2020; Good &
CHIME/FRB Collaboration, 2020; Zhu et al., 2020).

The evolution of magnetic fields in the crust triggers starquakes, further
initiates magnetosphere activities, and generates high-energy emission (e.g.,
Thompson & Duncan (2001); Beloborodov & Thompson (2007), see Wang et al.
(2018) for an FRB starquake model and Dehman et al. (2020) for a simulation of
crustal failures in young magnetars, although Levin & Lyutikov (2012) argued
that such fractures are unlikely to happen in magnetars.)111Besides internal
magnetic field triggers, accretion processes in a binary system might also
trigger crustal collapse and emit FRBs, which was recently considered in the
scenarios invoking ultra-luminous X-ray sources (Sridhar et al., 2021),
Be/X-ray binary systems (Li et al., 2021a) and strange stars (Geng et al.,
2021).. The magnetic energy inside the magnetar may be estimated as

$\displaystyle
E_{B}\sim\frac{B^{2}}{8\pi}\left(\frac{4\pi}{3}R^{3}\right)\simeq 1.7\times
10^{47}~{}{\rm erg}~{}B_{15}^{2},$ (1)

where $R$ is the magnetar radius. Assuming that the FRB emission makes use of
the dissipated magnetic energy with an efficiency $\eta$, the intrinsic energy
of one FRB should satisfy $E_{\rm FRB}\eta^{-1}<E_{B}$. For FRB 200428 from
SGR J1935+2154, the radio-to-X-ray flux ratio is about $10^{-5}$, suggesting
that the efficiency factor should be $\eta\lesssim 10^{-5}$ (e.g., Margalit et
al., 2020; Lu et al., 2020; Zhang, 2020b). Considering that the isotropic
energy of FRB 200428 is $\sim 10^{35}$ erg, the magnetic energy of the
magnetar is more than adequate to power this event, even if the efficiency is
much lower, e.g. $\eta\lesssim 10^{-5}$.

For repeating FRB sources, one should address how many radio bursts could be
produced within the energy budget of a magnetar. The first repeating source,
FRB 121102, has been bursting nearly continuously for over 8 years. Its
activity level significantly exceeds known Galactic magnetars. The FRB burst
event rate is about (Wang & Zhang (2019); Law et al. (2017); Gourdji et al.
(2019), also see Figure 3 of Margalit et al. (2020))

$\displaystyle R(>E)\simeq R_{0}\left(\frac{E}{E_{0}}\right)^{-\alpha},$ (2)

where $\alpha\sim 0.8$, $R_{0}\sim(10^{4}-10^{5})~{}{\rm burst~{}yr^{-1}}$ is
the average burst event rate with isotropic energy greater than
$E_{0}\sim(10^{37}-10^{38})~{}{\rm erg}$ for FRB 121102 (e.g., Li et al.,
2021b). In general, FRBs may be beamed with an average solid angle
$\Delta\Omega$. If the FRBs are on average emitted isotropically, the
intrinsic event rate of FRBs would be $(4\pi/\Delta\Omega)R(>E)$, and the
intrinsic energy of each burst would be $(\Delta\Omega/4\pi)E$. The two
beaming factors are canceled out (e.g. Zhang, 2020a). Thus, the total released
energy for a repeating source during the age $\tau$ would be

$\displaystyle E_{\rm tot}(\tau)$ $\displaystyle\simeq$
$\displaystyle\tau\int_{E_{0}}^{E_{\max}}dEE\left|\frac{dR}{dE}\right|\simeq\frac{\alpha}{1-\alpha}\left(\frac{E_{\max}}{E_{0}}\right)^{1-\alpha}E_{0}R_{0}\tau$
(3) $\displaystyle=$ $\displaystyle 1.0\times 10^{47}~{}{\rm
erg}\left(\frac{E_{\max}}{10^{40}~{}{\rm
erg}}\right)^{0.2}\left(\frac{E_{0}}{10^{38}~{}{\rm erg}}\right)^{0.8}$
$\displaystyle\times$ $\displaystyle\left(\frac{R_{0}}{10^{4}~{}{\rm
yr^{-1}}}\right)\left(\frac{\tau}{10~{}{\rm kyr}}\right),$

where $E_{\max}\sim 10^{40}~{}{\rm erg}$ is the observed maximum isotropic
energy. Such a burst energy is sustainable by a magnetar for a duration of

$\displaystyle\tau\lesssim 17~{}{\rm yr}~{}\eta_{-3}B_{15}^{2},$ (4)

even if the efficiency factor is higher, e.g. $\eta\sim 10^{-3}$. Such a time
is much shorter than the typical age of a magnetar. This suggests that within
the magnetar picture, the extremely high FRB event rate only lasts for a
relatively short period of time relative to the typical magnetar lifetime of
$\sim(1-10)~{}{\rm kyr}$ (e.g., Beniamini et al., 2019). Active FRB repeaters
in the universe therefore may originate from very young and highly magnetized
magnetars (e.g. Metzger et al., 2017; Lu et al., 2020).

We consider that FRBs are triggered by crust fracturing of a magnetar during
its active phase. Since the outer crust has a much lower density than that in
the inner crust, relatively frequent starquakes can occur in this region due
to the shear stress by magnetic fields. In the following discussion, we focus
on the region of outer crust. In the outer crust with a thickness of $L\sim
10^{4}~{}{\rm cm}$, the typical density is $\rho\sim 10^{-3}\rho_{\rm nuc}$,
where $\rho_{\rm nuc}=2.8\times 10^{14}~{}{\rm g~{}cm^{-3}}$ is the nuclear
density, and $Y_{e}\sim 0.3$ is the electron fraction (Cumming et al., 2004).
When the shear stress of the magnetic field $B\delta B/4\pi$ reaches a
critical value, the crust would crack, leading to starquakes and further
triggering bursting radiation (FRB or XRB) by releasing the magnetic energy.
The critical condition for magnetic shear stress reaching the threshold is
given by

$\displaystyle\frac{B\delta B}{4\pi}\simeq\sigma_{\rm
cr}\equiv\theta_{\max}\mu,$ (5)

where $\theta_{\max}$ is the maximum yield strain of the crust, the shear
modulus of the lattice is (e.g., Piro, 2005)

$\displaystyle\mu\simeq 1.0\times 10^{28}~{}{\rm erg~{}cm^{-3}}\rho_{\rm
nuc,-3}^{4/3},$ (6)

and $\rho_{\rm nuc,-3}=\rho/10^{-3}\rho_{\rm nuc}$, which is normalized to the
typical value of an NS outer crust (Douchin & Haensel, 2001). Here, we take
$Z=40$ as the number of protons per ion, $A=130$ as the number of nucleons per
ion, and $X_{n}=0$ as the fraction of neutrons outside nuclei (Douchin &
Haensel, 2001). We define the bent angle of magnetic field as
$\theta_{B}=\delta B/B$. According to Eq.(5), the critical value of the bent
angle is

$\displaystyle\theta_{B,c}=\left(\frac{4\pi\mu}{B^{2}}\right)\theta_{\max}=10^{-3}~{}{\rm
rad}~{}\mu_{28}B_{15}^{-2}\theta_{\max,-2}.$ (7)

During the evolution of the magnetic fields, once the magnetic bent angle
exceeds $\theta_{B,c}$, the outer crust would be fractured. In the following
discussion, we will analyze the magnetic field evolution in the outer crust.

The magnetic field evolution in plasma can be generally written as (e.g.,
Goldreich & Reisenegger, 1992; Thompson & Duncan, 1996; Cumming et al., 2004)

$\displaystyle\frac{\partial\bm{B}}{\partial
t}=\nabla\times(\bm{v}\times\bm{B})-\nabla\times(\eta\nabla\times\bm{B})-\nabla\times\left(\frac{\bm{J}\times\bm{B}}{n_{e}e}\right),$
(8)

where $\eta=c^{2}/4\pi\sigma$ is the magnetic diffusivity, $\sigma$ the
electrical conductivity, and $\bm{J}=(c/4\pi)(\nabla\times\bm{B})$ the current
density. In the outer crust, the advection and Ohmic terms (first two terms on
the right hand side of Eq.(8)) could be neglected since the plasma is almost
at rest and that the electrical conductivity is extremely large. Thus, the
magnetic field evolution is dominated by the Hall drift term (the third term
in the right hand side of Eq.(8)) in short term. The electron fluid drifts
with respect to ions, and the magnetic field is carried by the electron fluid.
Although Hall drift is non-dissipative, it can change the geometric
configuration of the magnetic field. The magnetic field evolution term due to
Hall drift is given by

$\displaystyle\left.\frac{\partial\bm{B}}{\partial t}\right|_{\rm
Hall}=-\nabla\times\left(\frac{\bm{J}\times\bm{B}}{en_{e}}\right),$ (9)

and the Hall timescale is given by

$\displaystyle\tau_{\rm Hall}=\frac{en_{e}L}{J}=\frac{4\pi en_{e}L^{2}}{cB},$
(10)

where $n_{e}=Y_{e}\rho/m_{p}$ is the electron density in the outer crust, and
$L\simeq P/\rho g$ is the typical length scale over which $B$ and $n_{e}$
vary, which is just of the order of magnitude of the outer crust thickness of
$\sim 10^{4}~{}{\rm cm}$. Taking the typical values of the outer crust, one
can estimate the Hall timescale as

$\displaystyle\tau_{\rm Hall}\simeq 11~{}{\rm yr}~{}Y_{e,-1}\rho_{\rm
nu,-3}L_{4}^{2}B_{15}^{-1},$ (11)

during which the magnetic field changes by $\delta B\sim B$. Since the
critical bent angle $\theta_{B,c}$ of magnetic fields is small (Eq.(7)), the
typical timescale triggering crustal cracking would be much shorter than the
Hall drift timescale, $\Delta t_{\rm cra}\sim\theta_{B,c}\tau_{\rm
Hall}\ll\tau_{\rm Hall}$. The burst rate is then estimated as

$\displaystyle\mathcal{R}=\Delta t_{\rm cra}^{-1}\simeq 0.26~{}{\rm
day}^{-1}~{}Y_{e,-1}^{-1}\rho_{\rm
nu,-3}^{-7/3}L_{4}^{-2}B_{15}^{3}\theta_{\max,-2}^{-1}.$ (12)

One can see that the burst rate satisfies $\mathcal{R}\propto B^{3}$,
suggesting that more frequent repeating FRBs originate from magnetars with
stronger magnetic fields within this picture.

In long term, the magnetic field in the crust would decay via Ohmic
dissipation. Two mechanisms dominate this process: 1. the nonlinear Hall term
gives rise to a turbulent cascade to smaller scales; 2. the Ohmic dissipation
rate (the second term of Eq.(8)) is enhanced by Hall cascades. The long-term
magnetic field evolution by Hall cascades can be approximated by $dB/dt\simeq-
AB^{2}$ with $A=10^{-18}~{}{\rm G^{-1}yr^{-1}}$ (Colpi et al., 2000) so that
the magnetic decay may be quantified as

$\displaystyle B=\frac{B_{0}}{1+AB_{0}t},$ (13)

where $B_{0}$ is the initial magnetic field in the crust. The typical
timescale for magnetic field decay is

$\displaystyle\tau_{B}=\frac{1}{AB_{0}}=100~{}{\rm yr}~{}B_{0,16}^{-1}$ (14)

If $t\gg\tau_{B}$, the field decay satisfies $B\simeq B_{0}(t/\tau_{B})^{-1}$.
According to Eq.(12) and Eq.(13), for a given magnetar, the event rate
satisfies

$\displaystyle\mathcal{R}\propto\left\\{\begin{aligned} &t^{0},~{}&{\rm
for}~{}t\ll\tau_{B}\\\ &t^{-3},~{}&{\rm
for}~{}t\gg\tau_{B}.\end{aligned}\right.$ (15)

Therefore, active repeating FRBs (i.e., FRBs with high bursting rates) are
proposed to originate from young magnetars with strong magnetic fields when
$t<\tau_{B}$. Galactic magnetars typically have $t\gtrsim\tau_{B}$, so that
their burst rate drops significantly, consistent with their low FRB rate as
indicated by the only detection from SGR J1935+2154 in several years since
CHIME became online. Since the number of young magnetars in the universe is
relatively small, we expect that active repeating FRBs are rarer compared with
apparent one-off FRBs, which are also repeaters but with a much lower
repetition rate within this picture. Such a result may also offer an
interpretation to the periodic activities of FRB 180916B (CHIME/FRB
Collaboration et al. (2020b); Pastor-Marazuela et al. (2020) and the possible
periodic activity of XRBs of SGR 1806-20 (Zhang et al., 2021)), since the
internal strong magnetic field of a young magnetar can deform the star,
leading to possible free precession with a long period (Zanazzi & Lai, 2020;
Levin et al., 2020; Li & Zanazzi, 2021) (see, however, Ioka & Zhang (2020) and
Lyutikov et al. (2020) for an alternative interpretation of the periodicity
due to the binary motion of the FRB-generating magnetar).

On the other hand, the repeating behavior may also depend on the line of sight
if the FRB emission is beamed, which is naturally expected in a magnetar
magnetosphere. Since the polar regions have much stronger fields, crust
fracturing may more easily occur in these regions. The near-surface magnetic
field configuration is not well known, and it is possible that there exist
multipolar fields near the magnetar crust. We generally delineate the
multipolar magnetic field configuration by spherical harmonics $(m,l)$. For a
multipole field, the magnetic field vector
$\bm{B}^{lm}(r,\theta,\varphi)=[B^{lm}_{r}(r,\theta,\varphi),B^{lm}_{\theta}(r,\theta,\varphi),B^{lm}_{\varphi}(r,\theta,\varphi)]$
satisfies (e.g., Jackson, 1975)

$\displaystyle B^{lm}_{r}(r,\theta,\varphi)$ $\displaystyle=$ $\displaystyle
4\pi\frac{l+1}{2l+1}\frac{q_{lm}}{r^{l+2}}Y_{lm}(\theta,\varphi),$
$\displaystyle B^{lm}_{\theta}(r,\theta,\varphi)$ $\displaystyle=$
$\displaystyle-\frac{4\pi}{2l+1}\frac{q_{lm}}{r^{l+2}}\frac{\partial}{\partial\theta}Y_{lm}(\theta,\varphi),$
$\displaystyle B^{lm}_{\varphi}(r,\theta,\varphi)$ $\displaystyle=$
$\displaystyle-\frac{4\pi}{2l+1}\frac{q_{lm}}{r^{l+2}}\frac{im}{\sin\theta}Y_{lm}(\theta,\varphi).$
(16)

For simplicity, we assume an axisymmetric multipolar magnetic field with $m=0$
and $l={\rm arbitrary~{}number}$, leading to $B^{lm}_{\varphi}=0$, and
$Y_{l0}=\sqrt{(2l+1)/4\pi}P_{l}(\cos\theta)$, where $P_{l}(x)$ is Legendre
polynomial. At the crust, the strength of magnetic field at the polar angle
$\theta$ is

$\displaystyle
B=\sqrt{\frac{4\pi}{2l+1}}\frac{q_{l0}}{R^{l+2}}\sqrt{(l+1)^{2}P_{l}(\cos\theta)^{2}+P^{\prime}_{l}(\cos\theta)^{2}}$

where the prime symbol represents the derivative with respect to $\theta$.
According to Eq.(12), the fracturing rate satisfies $\mathcal{R}\propto
B^{3}$. Assuming that the scale $L$ is independent of $\theta$ in the crust,
one may estimate the fracturing rate as

$\displaystyle\mathcal{R}(\theta)\propto\left[(l+1)^{2}P_{l}(\cos\theta)^{2}+P^{\prime}_{l}(\cos\theta)^{2}\right]^{3/2}.$
(18)

As shown in Figure 2, the fracturing rate is relatively larger at the main
poles of a multipolar field, and the $\theta$-range of a crust fracturing
region is smaller for a multipole field with a larger $l$. We call the region
with a large fracturing rate as a “fragile region”. A real multipole field may
be composed of sub components with different $(m,l)$ so that there are
multiple fragile regions on the magnetar crust, as shown in the schematic
picture in Figure 3. If the fragile regions sweep across the line of sight as
the magnetar rotate, one would detect FRBs with a high repetition rate.
Recently, Li et al. (2021b) found that the bursts of FRB 121102 appears to
have two components in the energy distribution function. According to this
picture, this might correspond to two fragile regions sweeping the line of
sight. Due to the different magnetic field strengths at these fragile regions,
they show different typical energies and burst rate distributions. On the
other hand, if all fragile regions are always outside the line of sight, the
burst rate would be much lower. In the latter case, if some bursts from
fragile regions are slightly off-axis, their observed fluxes would be
suppressed by the beaming structure or the Doppler effect (Zhang, 2021), which
might correspond to the case of SGR J1935+2154 with a lower luminosity and a
lower burst rate (based on the observation time of FRB 200428).

Figure 2: The fracturing rate as a function of polar angle $\theta$ for a
multipole field with $m=0$ and $l={\rm arbitrary}$. The black, red, blue lines
corresponds to $l=1,2,3$, respectively. The unit of rate is arbitrary. Figure
3: A schematic view of a magnetar with a multipolar magnetic field
configuration near surface. At polar regions, FRBs are more easily triggered
by crust fracturing. When the fragile regions sweep across the line of sight
during magnetar rotation, the source would appear to have a high repetition
rate. When the fragile regions are always outside the line of sight, the burst
rate would be much lower.

##  3 Coherent plasma radiation of FRBs

The extremely high brightness temperature of FRBs, typically $T_{B}\sim
10^{35}~{}{\rm K}$, implies that the radiation mechanism of FRBs must be
coherent. Since FRBs have some properties similar to radio emission of normal
pulsars, several pulsar-like coherent radiation mechanisms have been proposed
(Zhang (2020b) and references therein). The connections between FRBs and radio
pulsars are manifested by the following facts:

  * •

FRBs and radio pulsars are the most coherent sources in all kinds of
astrophysical phenomena. In addition to normal radio pulses, some pulsars show
giant pulses with the brightness temperature comparable to FRBs (e.g., Hankins
& Eilek, 2007), although the luminosity of giant pulses is much smaller than
that of FRBs.

  * •

Both periodic radio pulses and FRB 200428 were emitted from Galactic magnetar
SGR J1935+2154 (Zhu et al., 2020). Meanwhile, there have been some bright
radio bursts with fluxes between those of pulsed emission and of FRB emission
detected from this magnetar (Zhang et al., 2020; Kirsten et al., 2020; Good &
CHIME/FRB Collaboration, 2020).

Based on the observed flux and frequency of an FRB, one can estimate the
electric field strength $E_{w}$ of the electromagnetic wave at a distance $r$
from the center of the magnetar as

$\displaystyle E_{w}$ $\displaystyle\simeq$ $\displaystyle\left(\frac{4\pi\nu
F_{\nu}}{c}\right)^{1/2}\frac{d}{r}$ (19) $\displaystyle\simeq$ $\displaystyle
6\times 10^{8}~{}{\rm V~{}cm^{-1}}\nu_{\rm GHz}^{1/2}F_{\nu,{\rm
Jy}}^{1/2}d_{\rm Gpc}r_{7}^{-1},$

where $d$ is the distance between the FRB source and the observer. We consider
the generation of FRB radiation in the magnetosphere of a magnetar. According
to Kumar & Bošnjak (2020) and Lu et al. (2020), Alfvén waves are induced by
the crust cracking in the polar region and propagate along the magnetic field
lines. Since the plasma density decreases with the radius, a strong parallel
electric field may be induced as Alfvén waves reach a critical radius of
$r_{\rm cr}\sim 10R$ where charge starvation occurs (Kumar & Bošnjak, 2020; Lu
et al., 2020), as shown in panel (a) of Figure 1. Different from Kumar &
Bošnjak (2020) and Lu et al. (2020) who invoked the accelerating electric
field to continuously accelerate bunches to emit coherent radiation, in the
following we consider a pair cascade in the charge starved region, which
further generates coherent plasma radiation.

When the crust cracks, an magnetic field disturbance is produced by the
wiggling of the magnetic field foot points by the crust fracturing motion,
i.e.

$\displaystyle\delta B$ $\displaystyle\sim$
$\displaystyle\frac{v}{c}B_{p}\sim\frac{1}{c}\left(\frac{\mu}{\rho}\right)^{1/2}B_{p}$
(20) $\displaystyle\simeq$ $\displaystyle 6\times 10^{12}~{}{\rm
G}~{}B_{p,15}\rho_{\rm nu,-3}^{1/6},$

where $v$ is comparable to the shear wave velocity in the outer crust,

$\displaystyle v\sim v_{\mu}=\left(\frac{\mu}{\rho}\right)^{1/2}=6\times
10^{-3}c~{}\rho_{\rm nu,-3}^{1/6}.$ (21)

The Alfvén wave packet travels along magnetic field lines and its amplitude
$\delta B$ decreases with distance, $\delta B(r)\propto r^{-3/2}$. At the
charge starved region $r_{\rm cr}$, the parallel electric field is (Kumar &
Bošnjak, 2020)

$\displaystyle E_{\rm gap}$ $\displaystyle=$
$\displaystyle\frac{k_{\perp}}{k_{\parallel}}\left(\frac{r_{\rm
rc}}{R}\right)^{-3/2}\delta B$ $\displaystyle=$ $\displaystyle 2\times
10^{10}~{}{\rm V~{}cm^{-1}}~{}\eta_{k,-1}B_{p,15}\rho_{\rm nu,-3}^{1/6}r_{\rm
cr,7}^{-3/2}$

where $k_{\perp}$ and $k_{\parallel}$ are the components of the wavevector of
Alfvén waves perpendicular and parallel to the background magnetic field,
respectively, and

$\displaystyle\eta_{k}\equiv\frac{k_{\perp}}{k_{\parallel}}\sim 0.1$ (23)

at $r_{\rm cr}\sim 10R$ (Kumar & Bošnjak, 2020).

In the charge starved region near at $r_{\rm cr}\sim 10R$, due to the
existence of $E_{\rm gap}$, electrons and positrons would be accelerated and
emit gamma-ray photons due to curvature radiation or inverse Compton
scattering off the thermal photons (Zhang & Harding, 2000). The gamma-ray
photons produce new pairs via the QED processes222Photon splitting may
suppress pair production when $B\gtrsim B_{Q}$ (Baring & Harding, 2001).
However, if only the E$\rightarrow$OO mode is operating, then there is no
suppression of pair yields with photon splitting alone (Shabad & Usov, 1986;
Harding & Lai, 2006). $\gamma+B\rightarrow e^{+}+e^{-}$ (Daugherty & Harding,
1996) or $\gamma+\gamma\rightarrow e^{+}+e^{-}$ with thermal photons near the
magnetar surface (Zhang, 2001)333There should be $\gamma$-ray emission
accompanied from this processs. However, the luminosity of this emission
component is comparable to that of the FRB, which is much dimmer than the
other XRB-emission mechanism as discussed in Sect.4.. Then the cascade process
is interrupted when the fresh pairs generated by the cascade process is able
to screen the accelerating electric field $E_{\rm gap}$. When $E_{\rm gap}$ is
screened, the plasma still carries a finite current that reverses the initial
electric field. This process could be repeated, driving a inductive
oscillations as proposed in the pulsar scenario444In the active magnetar
scenario, the oscillation of Alfvén wave in the magnetosphere might also cause
an oscillating electric field at the same time. (Levinson et al., 2005;
Timokhin & Arons, 2013; Cruz et al., 2021).

Philippov et al. (2020) performed a simulation and claimed that coherent
electromagnetic radiation can be directly generated in non-stationary pair
plasma discharges. The mechanism was applied to interpret pulsar radio
emission. Consider that the accelerating electric field in the gap region is
$E_{\rm gap}$, then the amplitude of electrostatic wave is
$E_{x}\sim\xi_{L}E_{\rm gap}$ with the parameter $\xi_{L}\sim(0.1-1)$ for
large-amplitude oscillations (e.g., Levinson et al. (2005); Timokhin & Arons
(2013); Cruz et al. (2021), also see the simulation of Philippov et al.
(2020)). For this coherent radiation mechanism, the most important condition
to generate electromagnetic waves is nonuniform pair creation across magnetic
field lines. Define $\Theta$ as the angle between the normal to plasma
injection front and the background magnetic field. The amplitude of
electromagnetic wave is then about $\sim E_{x}\sin\Theta\sim\xi_{L}E_{\rm
gap}\sin\Theta$. For coherent radiation from pulsar polar gap, nonuniform pair
creation across magnetic field lines is due to the different degrees of
acceleration at different positions in the polar cap, finally leading to
$\Theta\sim 0.1$ (Philippov et al., 2020). Therefore, the amplitude of
electromagnetic waves is about $\sim 0.01E_{\rm gap}$ as shown by the
simulation results presented in Figure 1 of Philippov et al. (2020).

In the following discussion, we will give a semi-quantitative analysis for
this mechanism and apply it to FRBs. First, we consider that the discharge
occurs in the background magnetic field along $x$-axis, similar to Philippov
et al. (2020). Particles move along the field line and the electric current is
$j_{x}$. The magnetic field is assumed to be uniform in the $z$-direction,
leading to $\partial_{z}=0$ for all quantities, as shown in Figure 4. For an
E-mode wave, since its field does not couple with the plasma, it cannot be
directly emitted in the discharging process. For an O-mode wave, the Maxwell’s
equations give

$\displaystyle\partial_{y}B_{z}$ $\displaystyle=$
$\displaystyle\frac{4\pi}{c}j_{x}+\frac{1}{c}\partial_{t}E_{x},$
$\displaystyle-\partial_{x}B_{z}$ $\displaystyle=$
$\displaystyle\frac{1}{c}\partial_{t}E_{y},$
$\displaystyle\partial_{x}E_{y}-\partial_{y}E_{x}$ $\displaystyle=$
$\displaystyle-\frac{1}{c}\partial_{t}B_{z},$ (24)

which suggests that O-mode waves are coupled with the plasma because of the
$j_{x}$ term. Meanwhile, transverse modes can be only excited when the system
is non-uniform in the $y$-direction. Therefore, the non-uniform discharge in
the $y$-direction is the necessary condition to generate electromagnetic
waves. Based on Eq.(24), one obtains

$\displaystyle\partial_{t}\left(\frac{1}{c^{2}}\partial_{t}E_{x}+\frac{4\pi}{c^{2}}j_{x}\right)+\partial_{y}\left(\partial_{x}E_{y}-\partial_{y}E_{x}\right)=0.$
(25)

Injection of particles due to pair formation gives rise to the electric
current $j_{x}$ driven by the accelerating electric field in the gap, and
$j_{x}$ induces a fluctuating electric field $E_{x}$. In the charge starved
region, the parallel electric field $E_{x}$ still dominates plasma processes,
leading to

$\displaystyle\partial_{t}E_{x}+4\pi j_{x}\simeq 0.$ (26)

For an electron in a one-dimensional relativistic plasma, its equation of
motion is

$\displaystyle\frac{d(\gamma
m_{e}v)}{dt}\sim(\gamma^{3}m_{e})\frac{v}{\tau_{p}}\simeq eE_{x},$ (27)

where $\gamma^{3}m_{e}$ corresponds to the “longitudinal mass” of the electron
in one-dimensional plasma due to $d(\gamma v)=\gamma^{3}dv$. Using Eq.(26),
Eq.(27) and the definition of the current $j_{x}=n_{e}ev$, the plasma
oscillation frequency is estimated as

$\displaystyle\omega_{p}=\frac{1}{\tau_{p}}=\left(\frac{4\pi
e^{2}n_{e}}{\gamma^{3}m_{e}}\right)^{1/2}=5.6~{}{\rm
GHz}~{}n_{e,16}\gamma_{2}^{-3/2}$ (28)

where $n_{e}$ is the electron density in the emission region555This density of
$n_{e}\sim 10^{16}~{}{\rm cm^{-3}}$ is consistent with the density of pair
flow entailed by XRB at $r\sim 10R$, see Ioka (2020) that focuses on the
discussion of chocked scenario of FRB in active magnetosphere. Notice that
here $\gamma$ is the local Lorentz factor of electron by accelerated electric
field.. If $\partial_{y}=0$, according to Eq.(25) and Eq.(26), there would be
no electromagnetic wave generated. However, for $\partial_{y}\neq 0$ (implying
an inevitable nonuniformity of pair formation across the magnetic field
lines), the non-uniformity of $E_{x}$ across magnetic field lines would give
rise to the fluctuating magnetic field $B_{z}$, and further induce fluctuating
perpendicular electric field $E_{y}$. The electric field $E_{y}$ directly
reflects the amplitude of the transverse wave, and the typical frequency of
the coherent radiation of the transverse wave is $\nu\sim\omega_{p}/2\pi$
(Philippov et al., 2020). According to Eq.(25) and Eq.(26), for
$\partial_{y}\neq 0$, one obtains the relation between $E_{x}$ and $E_{y}$,
i.e.

$\displaystyle\partial_{x}E_{y}-\partial_{y}E_{x}\simeq 0.$ (29)

Figure 4: The transversal displacements of plasma and magnetic field in the
Alfvén wave. The dashed line denote the wave front of Alfvén wave.

For the Alfvén waves in the magnetosphere, the components of wavevector are
$k_{\perp}$ and $k_{\parallel}$, respectively. Thus, the angle $\Theta$
between the background magnetic field direction and the normal direction to
the plasma injection front satisfies
$\sin\Theta=k_{\perp}/k_{\parallel}=\eta_{k}$. Based on Eq.(29), the amplitude
of transverse wave is

$\displaystyle E_{w}\sim
E_{y}\simeq\frac{k_{\perp}}{k_{\parallel}}\xi_{L}E_{\rm gap}$ (30)

Since the parallel electric field with large-oscillation amplitude is dominant
in the $x$-direction, one has $E_{x}\sim\xi_{L}E_{\rm gap}$ with parameter
$\xi_{L}\sim(0.1-1)$ (see the simulation of (Philippov et al., 2020)). This
suggests that an ordinary mode is excited in the charge starved region, and
the amplitude of the electromagnetic wave is $E_{w}\sim E_{y}$. For an FRB
with typical parameters, according to Eq.(19), Eq.(LABEL:Egap2), and Eq.(30),
$\xi_{L}\sim 0.3$ is required, which is consistent with the simulation result.
Therefore, the predicted isotropic energy of FRBs is estimated as

$\displaystyle E_{\rm FRB,iso}$ $\displaystyle=$ $\displaystyle
f_{b}\left(\frac{cE_{w}^{2}}{4\pi}\right)\Delta r^{2}\Delta t$
$\displaystyle=$ $\displaystyle 9\times 10^{37}~{}{\rm
erg}~{}f_{b,1}\eta_{k,-1}^{4}\xi_{L,-1}^{2}\beta_{-2}^{2}B_{p,15}^{2}r_{\rm
cr,7}^{-3}\Delta r_{7}^{3},$

where $f_{b}=4\pi/\Delta\Omega$ which is normalized tp 10, $\Delta\Omega$ is
the radiation beaming angle, $\beta=v/c$, $v$ is the shear wave velocity given
by Eq.(21), $\Delta t\sim\Delta r/c$ is the duration, $\Delta r$ is the scale
of the FRB emission region. The energy given by Eq.(3) is consistent with the
energy distribution of extragalactic FRBs (Luo et al., 2020; Lu et al., 2020).

On the other hand, as proposed by Thompson & Duncan (2001), the high-energy
radiation, i.e. X-ray or gamma-ray emission, is driven by internal magnetic
stresses in the crust. The creation of fireballs by Alfvén waves outside the
star involves the rapid relaxation of the external magnetic stresses. Assuming
that the fireball scale $\Delta R$ (notice that it is different from the FRB
emission scale $\Delta r$) is comparable to the neutron star radius, i.e.
$\Delta R\sim R$, one may estimate the energy powering the XRB as

$\displaystyle E_{\rm XRB}\sim\frac{\delta B^{2}}{8\pi}\Delta R^{3}=4.0\times
10^{42}~{}{\rm erg}~{}\delta B_{13}^{2}\Delta R_{6}^{3}.$

Taking $\delta B\sim\beta B$, then the energy ratio between the FRB and the
XRB is

$\displaystyle\eta=\frac{E_{\rm FRB,iso}}{E_{\rm XRB}}\sim 2\times
10^{-5}\eta_{k,-1}^{4}\xi_{L,-1}^{2}r_{\rm cr,7}^{-3}\Delta r_{7}^{3}\Delta
R_{6}^{-3}$

for the above other typical parameters. The ratio depends on the properties of
Alfvén waves, pair cascade processes, and size of the trapped fireball. In
conclusion, magnetars with a stronger field tends to generate FRBs with a
larger burst energy and a higher event rate, which implies that extragalactic
active repeating FRB sources are from younger magnetars with stronger fields
than Galactic magnetars.

##  4 X-ray emission from an FRB source

###  4.1 X-ray emission properties associated with FRB 200428

FRB 200428 with two peaks separated by $\sim$ 30 ms (Bochenek et al., 2020;
CHIME/FRB Collaboration et al., 2020a) was detected from SGR J1935+2154 during
its active phase in association with a hard X-ray burst (Li et al., 2020;
Mereghetti et al., 2020; Ridnaia et al., 2020; Tavani et al., 2020). The FRB-
associated XRB showed two hard X-ray peaks whose arrival times are consistent
with the two FRB peaks after de-dispersion (Li et al., 2020). The delay time
of the X-ray peak from the FRB pulse was $\sim 6.5$ ms, but the onset of the
X-ray burst started $\sim 30$ ms before the FRB (Li et al., 2020; Mereghetti
et al., 2020). Meanwhile, the associated XRB appears to have some different
properties from normal XRBs, including a sharp peak with millisecond duration
and a harder energy cutoff (Li et al., 2020; Mereghetti et al., 2020; Younes
et al., 2020).

Based on the estimated distance of SGR J1935+2154 (Pavlović et al., 2013;
Kothes et al., 2018; Zhou et al., 2020; Zhong et al., 2020), the isotropic
energy of FRB 200428 is $E_{\rm FRB,iso}\sim 10^{35}~{}{\rm erg}$, and the
energy of the associated X-ray burst is $E_{\rm XRB}\sim 10^{40}~{}{\rm erg}$
(XRBs from magnetars are widely considered to be nearly isotropic). Thus, the
ratio between the radio and X-ray fluxes is $\eta=E_{\rm FRB,iso}/E_{\rm
XRB}\sim 10^{-5}$.

The FRB-associated XRB is considered to be generated from trapped fireballs
and/or relativistic outflows from an active magnetar magnetosphere (e.g.,
Ioka, 2020; Yamasaki et al., 2020a). Since the onset of the XRB started 30 ms
before the FRB, whether an FRB can break out from the magnetosphere becomes an
important issue. Ioka (2020) argued that an electron pair outflow from a
trapped-expanding fireball would be entailed by the associated XRB, which
would further pollute the magnetosphere before the FRB emission. Although the
pair flow is opaque to induced Compton scattering of FRB photons due to the
extremely high brightness temperature of the FRB, the FRB can break out of the
pair outflow by a radiation force if the FRB emission radius is larger than a
few tens of magnetar radii. The critical condition is given by $L_{\rm
FRB}>\tau_{\rm T}L_{\rm XRB}$ (Ioka, 2020), where $L_{\rm FRB}$ and $L_{\rm
XRB}$ are the luminosities of the FRB and the XRB, respectively, $\tau_{\rm
T}$ is the Thomson optical depth in the FRB emission region. Based on the
above critical condition, a breakout radius of a few tens of $R$ is obtained
for FRB 200428 (Ioka, 2020), which is consistent with the radius of charge
starved region of FRB as discussed in Section 3. On the other hand, for a
given FRB emission radius, if the luminosity ratio between an FRB and its
associated XRB becomes smaller, the FRB would be chocked in the magnetosphere,
which might explain the low association rate by deep monitoring of FAST
observations (Lin et al., 2020).

In the following discussion, we focus on the properties of the sharp-peak
component of the associated XRB of FRB 200428, including the sharp peaks and a
harder spectral cutoff. We take a typical value $B_{p}=10^{14}~{}{\rm G}$,
which is more consistent with that of SGR J1935+2154.

###  4.2 Resonant Compton scattering process during X-ray emission

Resonant Compton scattering has been studied to explain the spectra of
magnetar emission (e.g., Thompson et al., 2002; Baring & Harding, 2007;
Beloborodov & Thompson, 2007; Fernández & Thompson, 2007; Baring et al., 2020;
Yamasaki et al., 2020b). In the following, we focus on the resonant Compton
scattering process of the escaping photons from the bottom of a trapped
fireball. Under the strong magnetic field of a magnetar, X-ray photons have
two polarization modes: The O-mode has the electric vector in the plane of the
wavevector and the background magnetic field, and its cross section is about
$\sigma_{\rm O}\sim\sigma_{\rm T}$; and the E-mode has an electric vector
perpendicular to this plane and its cross section is B-dependent, $\sigma_{\rm
E}\simeq\left(\omega/\omega_{B}\right)^{2}\sigma_{\rm T}$, where
$\omega_{B}=eB/m_{e}c$ is the cyclotron frequency. The stronger the magnetic
field, the more transparent the magnetosphere to the E-mode. Therefore, when a
trapped fireball forms, the E-mode X-ray photons would escape from the bottom
of trapped fireball near the magnetar surface because $\sigma_{\rm E}\propto
B^{-2}\propto r^{6}$, as shown in panel (b) of Figure 1.

Once the E-mode X-ray photons escape from the base of a trapped fireball, they
will enter a region between two adjacent trapped fireballs, and further be
resonantly scattered by the electrons in this region. The frequency-dependent
cross section of resonant Compton scattering is

$\displaystyle\sigma_{\rm res}(\omega^{\prime})\simeq
2\pi^{2}r_{e}c\delta(\omega^{\prime}-\omega_{B})=\sigma_{\rm
res,0}\omega_{B}\delta(\omega^{\prime}-\omega_{B}),$ (34)

where $\omega^{\prime}$ is the electromagnetic wave frequency in the electron
rest frame, $r_{e}=e^{2}/m_{e}c^{2}$ is the classical electron radius, and
$\sigma_{\rm res,0}=2\pi^{2}e/B$. At the resonant frequency, the cross section
of resonant Compton scattering is much larger than that of Compton scattering
in the magnetosphere, $\sigma_{\rm res,0}/\sigma_{\rm
T}\sim(1/\alpha)\left(B/B_{Q}\right)^{-1}$, where $\alpha\simeq 1/137$ is the
fine structure constant. We consider that the strength of the dipole magnetic
field depends on radius $r$, i.e. $B\simeq B_{p}(r/R)^{-3}$. Thus the critical
radius is $r_{\rm res}\sim\left(\alpha B_{p}/B_{Q}\right)^{1/3}R$. The
resonant Compton scattering would be dominant at $r\gg r_{\rm res}$. For a
given incident photon energy, the typical resonant Lorentz factor for resonant
Compton scattering is (e.g., Beloborodov & Thompson, 2007)

$\displaystyle\gamma_{\rm res}$ $\displaystyle=$
$\displaystyle\frac{1}{1-\cos\theta_{kB}}\frac{B}{B_{Q}}\left(\frac{\epsilon}{m_{e}c^{2}}\right)^{-1}$
(35) $\displaystyle=$ $\displaystyle 512f(\theta_{kB})(B/B_{Q})\epsilon_{\rm
keV}^{-1},$

where $\epsilon=\hbar\omega=1~{}{\rm keV}\epsilon_{\rm keV}$ is the energy of
the incident photon, $\theta_{kB}$ is the angle between the incident photon
and the magnetic field, $f(\theta_{kB})=1/(1-\cos\theta_{kB})$. The radiative
force exerted on an electron near the magnetar surface is estimated by

$\displaystyle F_{r}$ $\displaystyle\simeq$ $\displaystyle\int\sigma_{\rm
res}(\omega)\frac{L_{\omega}}{4\pi r^{2}c}d\omega=\frac{\pi
eL_{X}}{2cB_{p}R^{2}}$ (36) $\displaystyle=$ $\displaystyle 2.5\times
10^{-5}~{}{\rm dyne}~{}L_{X,41}B_{p,14}^{-1},$

where $\sigma_{\rm res}(\omega)$ is the cross section of resonant Compton
scattering given by Eq.(34),
$\omega^{\prime}=\gamma(1-\cos\theta_{kB})\omega$, and $L_{\omega}\sim
L_{X}/\omega$. After the FRB is emitted (with a delay time of 6 ms), the
radiative force would change the electron momentum distribution. The
relaxation timescale for electrons to acquire the equilibrium momentum
distribution can be estimated by $\gamma m_{e}c/\tau_{\rm rel}\sim F_{r}$,
which gives

$\displaystyle\tau_{\rm rel}$ $\displaystyle\sim$ $\displaystyle\frac{\gamma
m_{e}c}{F_{r}}\simeq\frac{2\gamma m_{e}c^{2}B_{p}R^{2}}{\pi eL_{X}}$ (37)
$\displaystyle\simeq$ $\displaystyle 1.1\times 10^{-12}~{}{\rm s}~{}\gamma
L_{X,41}^{-1}B_{p,14}.$

This means that the momentum distribution of electrons changes in an extremely
short period of time. Yamasaki et al. (2020b) proposed that resonant Compton
scattering of the balanced relativistic electrons would change the X-ray
spectrum, leading to a higher cut-off energy. In the electron rest frame, the
incident photon angle with respect to the local magnetic field is

$\displaystyle\cos\theta_{kB}^{\prime}=\frac{\cos\theta_{kB}-\beta}{1-\beta\cos\theta_{kB}}.$
(38)

Consider an electron that initially moves in parallel with the magnetic field
with $\beta\sim 0$. It would be accelerated by the radiation force. Once the
electron has a relativistic velocity $\beta\sim 1$, it would be decelerated by
the radiation force, because the incident photon is from
$\cos\theta_{kB}^{\prime}\sim-1$ in the electron rest frame. Therefore, the
electron under radiative force would be immediately accelerated or decelerated
to reach an equilibrium state. In the equilibrium state, the radiation force
is directed perpendicularly to the magnetic field so that
$\theta_{kB}^{\prime}=\pi/2$. The balancing velocity of the electron under
radiative force is

$\displaystyle\beta_{\rm res}=\cos\theta_{kB};~{}~{}~{}~{}~{}\gamma_{\rm
res}=\frac{1}{\sin\theta_{kB}}.$ (39)

Using Eq.(35) and Eq.(39). The resonant Compton energy of incident photon is

$\displaystyle\epsilon=\frac{\sin\theta_{kB}}{1-\cos\theta_{kB}}\frac{B}{B_{Q}}m_{e}c^{2}.$
(40)

For a magnetar with $B_{p}\sim 10B_{Q}$, at a distance $r\sim 5R$, the energy
of the incident photon is required to be $\epsilon\simeq 40~{}{\rm
keV}g(\theta_{kB})$, where
$g(\theta_{kB})=\sin\theta_{kB}/(1-\cos\theta_{kB})$. The energy of the
scattered photon is about

$\displaystyle\epsilon_{s}\sim\gamma_{\rm res}^{2}\epsilon$ (41)

for $\theta_{kB}\gtrsim 1$. Since $\gamma_{\rm res}\sim$ a few, according to
Eq.(39), the cutoff energy of X-ray emission would be increased by several
times compared with those without resonant Compton scattering, which might
explain the high-energy cutoff of the associated XRB of FRB 200428. A detailed
discussion about the spectral modification could be found in (Yamasaki et al.,
2020b), which has been applied to model the spectra of giant flares and even
intermediate flares of magnetars.

Another important observation is that the light curve of the FRB-associated
XRB showed sharp peaks aligned with the FRB pulses (Li et al., 2020). We
propose that the sharp-peak component is from the region between two adjacent
trapped fireballs, as shown in panel (b) of Figure 1. For E-mode X-ray
photons, the radiative flux across the trapped fireball surface is
concentrated close to the bottom of the trapped fireball near the magnetar
surface, and further escapes along more extended magnetic field lines in the
region between two adjacent trapped fireballs. A large flux is emitted in this
region and the radiation beam is confined by the trapped fireballs, leading to
the sharp-peak component in light curve. The effect of the confined beam is
discussed as follow.

As discussed in Section 2, during the magnetar active phase, the magnetic
field near the magnetar surface may have a general multipolar configuration
described by $(m,l)$. For simplicity, we still assume an axisymmetric
multipolar magnetic field with $m=0$ and $l={\rm arbitrary}$. The first-order
approximation for the field line equation is (e.g., Asseo & Khechinashvili,
2002)

$\displaystyle r\sim R_{\max}\sin^{2/l}\theta$ (42)

for $\theta\lesssim 1$, where $\theta$ is the angle from the multipolar pole.
Then the opening angle of a field line at one pole is estimated as

$\displaystyle\theta_{\rm open}\sim\left(\frac{R}{R_{\max}}\right)^{l/2}.$
(43)

For example, for a closed field line with $R_{\max}\sim 10^{4}R$, a dipole
field ($l=1$) has an opening angle of $\theta_{\rm open}\sim 0.01~{}{\rm
rad}$, but a tripole field ($l=2$) has $\theta\sim 10^{-4}~{}{\rm rad}$. Thus,
multipolar fields have much narrower opening angles than dipolar fields.

When the X-ray photons enter the region between two adjacent trapped
fireballs, they will be confined in a beaming angle due to the large optical
depth $\tau\gg 1$ at the top region of the trapped fireball around it, as
shown in panel (b) of Figure 1. For two adjacent trapped fireballs with a size
of $\Delta R$, one has $R_{\max}\sim R+\Delta R$. According to Eq.(43), the
scale of the region between two adjacent trapped fireballs is estimated by

$\displaystyle l_{\rm beam}=R\theta_{\rm open}\sim R\left(\frac{R}{R+\Delta
R}\right)^{l/2}.$ (44)

Consider that X-ray photons escape from the base of a trapped fireball. Since
$\sigma_{\rm E}\propto r^{6}$, the upper regions of the adjacent trapped
fireballs are opaque to X-rays. Thus, the beaming angle is about

$\displaystyle\theta_{\rm beam}\sim\frac{l_{\rm beam}}{\Delta
R}\simeq\frac{R}{\Delta R}\left(\frac{R}{R+\Delta R}\right)^{l/2}.$ (45)

We assume that the energy of a trapped fireball is $\Delta E\sim E_{\rm
XRB}\simeq 10^{40}~{}{\rm erg}$ and the temperature is about $kT\sim 10~{}{\rm
keV}$, then the size of the trapped fireball is about $\Delta R\sim R$. For a
trapped fireball with $\Delta R\sim R$ and $l=4$, the beaming angle of X-ray
emission is $\theta_{\rm beam}=0.25~{}{\rm rad}$. Therefore, the radiation
from this region will be confined in a narrow beam, especially for multipolar
fields near the magnetar surface.

###  4.3 Temperature evolution of persistent X-ray emission

In addition to XRBs, SGR J1935+2154 also showed persistent emission with a
double-peaked pulse profile with the rotation period of the magnetar during
the magnetar active phase. The arrival time of FRB 200428 aligned in phase
with the brightest peak of the pulse profile (Younes et al., 2020). The
blackbody temperature of the X-ray persistent emission decreases rapidly in
the early stage of the outburst, while the size of the emitting area remains
the same with a radius of a few km. In this section, we propose that the X-ray
persistent emission is from a hot spot on the magnetar surface and calculate
its temperature evolution. See panel (b) of Figure 1.

In the crust of a magnetar, the Coulomb lattice of heavy nuclei coexists in
$\beta$-equilibrium with a degenerate electron gas. The Fermi energy of
degenerate electrons in a strong field is (e.g., Ruderman, 1971; Flowers &
Ruderman, 1977; Lai, 2001)

$\displaystyle\epsilon_{F,e}=\frac{2\pi^{4}\hbar^{4}c^{2}Y_{e}^{2}\rho^{2}}{e^{2}m_{p}^{2}m_{e}B^{2}}=180~{}{\rm
GeV}~{}Y_{e,-1}^{2}\rho_{\rm nuc,-3}^{2}B_{15}^{-2},$

where $B$ is the magnetic field in magnetar crust. Since the fraction of the
effective thermal electrons is only $\sim kT/\epsilon_{F,e}$ for a Fermi
electron gas, the internal energy by relativistic electrons is

$\displaystyle U_{e}$ $\displaystyle\simeq$
$\displaystyle\frac{1}{2}N_{e}kT\left(\frac{kT}{\epsilon_{F,e}}\right)$
$\displaystyle=$ $\displaystyle 7.3\times 10^{30}~{}{\rm
erg}~{}Y_{e,-1}^{-1}\rho_{\rm nuc,-3}^{-1}T_{\rm keV}^{2}L_{4}\Delta R_{\rm
hs,5}^{2}B_{15}^{2}$

where $N_{e}\simeq n_{e}L\Delta R_{\rm hs}^{2}$ is the total electron number
in the hot spot with size $\Delta R_{\rm hs}$ and depth $L$, $\rho\sim
10^{-3}\rho_{\rm nuc}$ is the density of the outer crust, $T=1~{}{\rm keV}\
T_{\rm keV}/k$ is the temperature of the hot spot, and the factor of $1/2$ is
due to the one-dimensional freedom of electrons in strong fields. On the other
hand, the ions in the outer crust are nondegenerate, and each ion has thermal
energy of $(3/2)kT$. Thus, the internal energy stored by ions in the outer
crust is

$\displaystyle U_{i}$ $\displaystyle=$ $\displaystyle\frac{3}{2}N_{i}kT$ (48)
$\displaystyle=$ $\displaystyle 3.1\times 10^{38}~{}{\rm erg}~{}\rho_{\rm
nuc,-3}T_{\rm keV}L_{4}\Delta R_{\rm hs,5}^{2},$

where $N_{i}\simeq(\rho/Am_{p})L\Delta R_{\rm hs}^{2}$ is the total ion number
with $A=130$.

Therefore, the thermal energy of the hot spot is dominated by ions, $U\simeq
U_{i}$, and such an energy could be produced by $\sim 1\%$ of an XRB. The
temperature evolution is given by $dU_{i}/dt=-\Delta R_{\rm hs}^{2}\sigma
T^{4}$. The typical cooling time is defined as

$\displaystyle t_{\rm cool}=\frac{U_{i}}{\Delta R_{\rm hs}^{2}\sigma
T^{4}}=0.3~{}{\rm day}~{}\rho_{\rm nuc,-3}^{2/3}T_{\rm keV}^{-3}L_{4},$ (49)

which is consistent with the rapid temperature evolution of the X-ray
persistent emission, as shown in Figure 8 of Younes et al. (2020). Meanwhile,
the cooling time from temperature $T_{0}$ to $T$ is given by

$\displaystyle\Delta t=t-t_{0}=\frac{k\rho
L}{2Am_{p}\sigma}\left(\frac{1}{T^{3}}-\frac{1}{T_{0}^{3}}\right),$ (50)

where $T_{0}$ is the initial temperature at time $t_{0}$.

##  5 Conclusion and Discussion

Prompted by the observed association between FRB 200428 and its X-ray
counterpart from Galactic magnetar SGR J1935+2154 (Bochenek et al., 2020;
CHIME/FRB Collaboration et al., 2020a; Li et al., 2020; Mereghetti et al.,
2020; Ridnaia et al., 2020; Tavani et al., 2020), we propose that FRBs and
XRBs are triggered by crust fracturing of magnetars. Due to Hall drift, the
geometric configuration of the magnetic fields would evolve in the crust. When
the shear stress reaches a critical value depending on the shear modulus, the
outer crust would crack and further trigger magnetosphere activities. We find
that the event rate of crust fracturing depends on the magnetic field
strength, i.e. $\mathcal{R}\propto B^{3}$. Since the magnetic field in the
crust decays by Ohmic dissipation and Hall cascades in long term, our results
suggest that the active repeating FRB sources are younger magnetars with
stronger magnetic fields. On the other hand, the observed repeating behavior
also depends on the line of sight if the FRB emission is beamed. If the
fragile regions sweep across the line of sight as the magnetar rotates, one
would see FRBs with a high repetition rate. However, if the fragile regions
are always the line of sight, the burst rate would be much lower.

According to Kumar & Bošnjak (2020) and Lu et al. (2020), a charge starved
region could be generated by Alfvén waves triggered by crust fracturing after
propagating to a distance of $\sim 10R$. Meanwhile, the association between an
FRB and an XRB also implies that the emission region is higher than $\sim 10R$
in order for an FRB to break out from the magnetosphere (Ioka, 2020). Non-
stationary pair plasma discharges would be produced in this charge starved
region, similar to inner gap sparking theorized in radio pulsars (Ruderman &
Sutherland, 1975) but with a higher emission region at $\sim 10R$. Inspired by
the simulations of Philippov et al. (2020), we consider coherent plasma
radiation as the radiation mechanism of FRBs due to the similarity of the
coherent radio emission between FRBs and normal radio pulsars. Near the charge
starved region, nonuniform pair production across magnetic field lines would
lead to electromagnetic waves whose amplitude is proportional to the strength
of the induced electric field. This electromagnetic wave would be strong
enough to be an FRB. According to this mechanism, magnetars with stronger
fields tend to generate FRBs with larger burst energies and a higher event
rate.

On the other hand, the sharp-peak component of the FRB-associated XRB is
proposed to be generated in the region between adjacent trapped fireballs. The
E-mode X-ray photons escape from the base of the trapped fireballs and get
resonantly scattered to higher energies, leading to the observed high cutoff
energy of the FRB-associated XRB (e.g., Li et al., 2020; Younes et al., 2021).
Meanwhile, when X-ray photons escape from the bottom of the trapped fireballs
and enter the region between two adjacent trapped fireballs, they are confined
in a beaming angle due to the large optical depth $\tau\gg 1$ of the trapped
fireballs around it. Therefore, a large flux is emitted in this region and the
radiation beam is collimated by the trapped fireballs, leading to the sharp-
peak component in the light curve.

After an XRB, a part of energy (about $1\%$) of the magnetosphere activity is
injected to the magnetar surface and produces a hot spot. The persistent X-ray
emission is suggested to be the thermal emission from this hot spot. Since
cooling of the hot spot is dominated by thermal ions on the magnetar surface,
the typical cooling timescale is about $\sim 0.3~{}{\rm day}$ for a hot spot
with size of a few km and temperature of $\sim 1~{}{\rm keV}$. This cooling
timescale is consistent with the observations of SGR J1935+2154 (Younes et
al., 2020).

We thank the anonymous referee for helpful comments and suggestions. We also
thank Clara Dehman, Jinjun Geng, Ersin Gogus, Qiaochu Li, Zenan Liu, Kohta
Murase, Yuanhong Qu, Fayin Wang, Weiyang Wang, Zhaoyang Xia, Shotaro Yamasaki
and Jinping Zhu for helpful discussions. Y.P.Y is supported by National
Natural Science Foundation of China grant No. 12003028.

## References

  * Asseo & Khechinashvili (2002) Asseo, E., & Khechinashvili, D. 2002, MNRAS, 334, 743
  * Baring & Harding (2001) Baring, M. G., & Harding, A. K. 2001, ApJ, 547, 929
  * Baring & Harding (2007) —. 2007, Ap&SS, 308, 109
  * Baring et al. (2020) Baring, M. G., Wadiasingh, Z., Gonthier, P. L., Harding, A. K., & Hu, K. 2020, arXiv e-prints, arXiv:2012.10815
  * Beloborodov (2017) Beloborodov, A. M. 2017, ApJ, 843, L26
  * Beloborodov & Thompson (2007) Beloborodov, A. M., & Thompson, C. 2007, ApJ, 657, 967
  * Beniamini et al. (2019) Beniamini, P., Hotokezaka, K., van der Horst, A., & Kouveliotou, C. 2019, MNRAS, 487, 1426
  * Bochenek et al. (2020) Bochenek, C. D., Ravi, V., Belov, K. V., et al. 2020, Nature, 587, 59
  * Chatterjee et al. (2017) Chatterjee, S., Law, C. J., Wharton, R. S., et al. 2017, Nature, 541, 58
  * CHIME/FRB Collaboration et al. (2020a) CHIME/FRB Collaboration, Andersen, B. C., Bandura, K. M., et al. 2020a, Nature, 587, 54
  * CHIME/FRB Collaboration et al. (2020b) CHIME/FRB Collaboration, Amiri, M., Andersen, B. C., et al. 2020b, Nature, 582, 351
  * The CHIME/FRB Collaboration et al. (2021) The CHIME/FRB Collaboration, :, Amiri, M., et al. 2021, arXiv:2106.04352
  * Colpi et al. (2000) Colpi, M., Geppert, U., & Page, D. 2000, ApJ, 529, L29
  * Cordes & Chatterjee (2019) Cordes, J. M., & Chatterjee, S. 2019, ARA&A, 57, 417
  * Cruz et al. (2021) Cruz, F., Grismayer, T., & Silva, L. O. 2021, ApJ, 908, 149
  * Cumming et al. (2004) Cumming, A., Arras, P., & Zweibel, E. 2004, ApJ, 609, 999
  * Daugherty & Harding (1996) Daugherty, J. K., & Harding, A. K. 1996, ApJ, 458, 278
  * Dehman et al. (2020) Dehman, C., Viganò, D., Rea, N., et al. 2020, ApJ, 902, L32. doi:10.3847/2041-8213/abbda9
  * Douchin & Haensel (2001) Douchin, F., & Haensel, P. 2001, A&A, 380, 151
  * Fernández & Thompson (2007) Fernández, R., & Thompson, C. 2007, ApJ, 660, 615
  * Flowers & Ruderman (1977) Flowers, E., & Ruderman, M. A. 1977, ApJ, 215, 302
  * Geng et al. (2021) Geng, J.-J., Li, B., & Huang, Y.-F. 2021, arXiv e-prints, arXiv:2103.04165
  * Goldreich & Reisenegger (1992) Goldreich, P., & Reisenegger, A. 1992, ApJ, 395, 250
  * Good & CHIME/FRB Collaboration (2020) Good, D., & CHIME/FRB Collaboration. 2020, The Astronomer’s Telegram, 14074, 1
  * Gourdji et al. (2019) Gourdji, K., Michilli, D., Spitler, L. G., et al. 2019, ApJ, 877, L19
  * Hankins & Eilek (2007) Hankins, T. H., & Eilek, J. A. 2007, ApJ, 670, 693
  * Harding & Lai (2006) Harding, A. K., & Lai, D. 2006, Reports on Progress in Physics, 69, 2631
  * Ioka (2020) Ioka, K. 2020, ApJ, 904, L15
  * Ioka & Zhang (2020) Ioka, K., & Zhang, B. 2020, ApJ, 893, L26
  * Jackson (1975) Jackson, J. D. 1975, Classical electrodynamics
  * Katz (2016) Katz, J. I. 2016, ApJ, 826, 226
  * Kirsten et al. (2020) Kirsten, F., Snelders, M. P., Jenkins, M., et al. 2020, Nature Astronomy, arXiv:2007.05101
  * Kothes et al. (2018) Kothes, R., Sun, X., Gaensler, B., & Reich, W. 2018, ApJ, 852, 54
  * Kumar & Bošnjak (2020) Kumar, P., & Bošnjak, Ž. 2020, MNRAS, 494, 2385
  * Kumar et al. (2017) Kumar, P., Lu, W., & Bhattacharya, M. 2017, MNRAS, 468, 2726
  * Lai (2001) Lai, D. 2001, Reviews of Modern Physics, 73, 629
  * Law et al. (2017) Law, C. J., Abruzzo, M. W., Bassa, C. G., et al. 2017, ApJ, 850, 76
  * Levin & Lyutikov (2012) Levin, Y. & Lyutikov, M. 2012, MNRAS, 427, 1574. doi:10.1111/j.1365-2966.2012.22016.x
  * Levin et al. (2020) Levin, Y., Beloborodov, A. M., & Bransgrove, A. 2020, ApJ, 895, L30
  * Levinson et al. (2005) Levinson, A., Melrose, D., Judge, A., & Luo, Q. 2005, ApJ, 631, 456
  * Li et al. (2020) Li, C. K., Lin, L., Xiong, S. L., et al. 2020, arXiv e-prints, arXiv:2005.11071
  * Li et al. (2021a) Li, Q.-C., Yang, Y.-P., Wang, F.Y., Xu, K, Shao, Y., Liu, Z.-N., Dai, Z.-G., 2021, submit
  * Li et al. (2021b) Li, D., Wang, P., Zhu, W. W., et al. 2021, Nature, accepted.
  * Li & Zanazzi (2021) Li, D., & Zanazzi, J. J. 2021, arXiv e-prints, arXiv:2101.05836
  * Lin et al. (2020) Lin, L., Zhang, C. F., Wang, P., et al. 2020, Nature, 587, 63
  * Lorimer et al. (2007) Lorimer, D. R., Bailes, M., McLaughlin, M. A., Narkevic, D. J., & Crawford, F. 2007, Science, 318, 777
  * Lu et al. (2020) Lu, W., Kumar, P., & Zhang, B. 2020, MNRAS, 498, 1397
  * Luo et al. (2020) Luo, R., Men, Y., Lee, K., et al. 2020, MNRAS, 494, 665
  * Lyubarsky (2014) Lyubarsky, Y. 2014, MNRAS, 442, L9
  * Lyubarsky (2021) —. 2021, arXiv e-prints, arXiv:2103.00470
  * Lyutikov et al. (2020) Lyutikov, M., Barkov, M. V., & Giannios, D. 2020, ApJ, 893, L39
  * Marcote et al. (2020) Marcote, B., Nimmo, K., Hessels, J. W. T., et al. 2020, Nature, 577, 190
  * Margalit et al. (2020) Margalit, B., Beniamini, P., Sridhar, N., & Metzger, B. D. 2020, ApJ, 899, L27
  * Mereghetti et al. (2020) Mereghetti, S., Savchenko, V., Ferrigno, C., et al. 2020, ApJ, 898, L29
  * Metzger et al. (2017) Metzger, B. D., Berger, E., & Margalit, B. 2017, ApJ, 841, 14
  * Michilli et al. (2018) Michilli, D., Seymour, A., Hessels, J. W. T., et al. 2018, Nature, 553, 182
  * Murase et al. (2016) Murase, K., Kashiyama, K., & Mészáros, P. 2016, MNRAS, 461, 1498. doi:10.1093/mnras/stw1328
  * Pastor-Marazuela et al. (2020) Pastor-Marazuela, I., Connor, L., van Leeuwen, J., et al. 2020, arXiv e-prints, arXiv:2012.08348
  * Pavlović et al. (2013) Pavlović, M. Z., Urošević, D., Vukotić, B., Arbutina, B., & Göker, Ü. D. 2013, ApJS, 204, 4
  * Petroff et al. (2019) Petroff, E., Hessels, J. W. T., & Lorimer, D. R. 2019, A&A Rev., 27, 4
  * Petroff & Yaron (2020) Petroff, E., & Yaron, O. 2020, Transient Name Server AstroNote, 160, 1
  * Petroff et al. (2016) Petroff, E., Barr, E. D., Jameson, A., et al. 2016, PASA, 33, e045
  * Philippov et al. (2020) Philippov, A., Timokhin, A., & Spitkovsky, A. 2020, Phys. Rev. Lett., 124, 245101
  * Piro (2005) Piro, A. L. 2005, ApJ, 634, L153
  * Rajwade et al. (2020) Rajwade, K. M., Mickaliger, M. B., Stappers, B. W., et al. 2020, MNRAS, 495, 3551
  * Ridnaia et al. (2020) Ridnaia, A., Svinkin, D., Frederiks, D., et al. 2020, arXiv e-prints, arXiv:2005.11178
  * Ruderman (1971) Ruderman, M. 1971, Phys. Rev. Lett., 27, 1306
  * Ruderman & Sutherland (1975) Ruderman, M. A., & Sutherland, P. G. 1975, ApJ, 196, 51
  * Shabad & Usov (1986) Shabad, A. E., & Usov, V. V. 1986, Ap&SS, 128, 377
  * Spitler et al. (2016) Spitler, L. G., Scholz, P., Hessels, J. W. T., et al. 2016, Nature, 531, 202
  * Sridhar et al. (2021) Sridhar, N., Metzger, B. D., Beniamini, P., et al. 2021, arXiv e-prints, arXiv:2102.06138
  * Tavani et al. (2020) Tavani, M., Casentini, C., Ursi, A., et al. 2020, arXiv e-prints, arXiv:2005.12164
  * Thompson & Duncan (1995) Thompson, C., & Duncan, R. C. 1995, MNRAS, 275, 255
  * Thompson & Duncan (1996) —. 1996, ApJ, 473, 322
  * Thompson & Duncan (2001) —. 2001, ApJ, 561, 980
  * Thompson et al. (2002) Thompson, C., Lyutikov, M., & Kulkarni, S. R. 2002, ApJ, 574, 332
  * Timokhin & Arons (2013) Timokhin, A. N., & Arons, J. 2013, MNRAS, 429, 20
  * Wadiasingh et al. (2020) Wadiasingh, Z., Beniamini, P., Timokhin, A., et al. 2020, ApJ, 891, 82
  * Wadiasingh & Timokhin (2019) Wadiasingh, Z., & Timokhin, A. 2019, ApJ, 879, 4
  * Wang & Zhang (2019) Wang, F. Y. & Zhang, G. Q. 2019, ApJ, 882, 108
  * Wang (2020) Wang, J.-S. 2020, ApJ, 900, 172
  * Wang et al. (2018) Wang, W., Luo, R., Yue, H., et al. 2018, ApJ, 852, 140
  * Wang et al. (2020) Wang, W.-Y., Xu, R., & Chen, X. 2020, ApJ, 899, 109
  * Xiao et al. (2021) Xiao, D., Wang, F., & Dai, Z. 2021, arXiv e-prints, arXiv:2101.04907
  * Yamasaki et al. (2020a) Yamasaki, S., Kashiyama, K., & Murase, K. 2020, arXiv:2008.03634
  * Yamasaki et al. (2020b) Yamasaki, S., Lyubarsky, Y., Granot, J., & Göğüş, E. 2020, MNRAS, 498, 484
  * Yang et al. (2021) Yang, Y.-H., Zhang, B.-B., Lin, L., et al. 2021, ApJ, 906, L12
  * Yang & Zhang (2018) Yang, Y.-P., & Zhang, B. 2018, ApJ, 868, 31
  * Yang et al. (2020) Yang, Y.-P., Zhu, J.-P., Zhang, B., & Wu, X.-F. 2020, ApJ, 901, L13
  * Younes et al. (2020) Younes, G., Güver, T., Kouveliotou, C., et al. 2020, ApJ, 904, L21
  * Younes et al. (2021) Younes, G., Baring, M. G., Kouveliotou, C., et al. 2021, Nature Astronomy, arXiv:2006.11358
  * Yu et al. (2021) Yu, Y.-W., Zou, Y.-C., Dai, Z.-G., & Yu, W.-F. 2021, MNRAS, 500, 2704
  * Zanazzi & Lai (2020) Zanazzi, J. J., & Lai, D. 2020, ApJ, 892, L15
  * Zhang (2001) Zhang, B. 2001, ApJ, 562, L59
  * Zhang (2020a) —. 2020a, ApJ, 890, L24
  * Zhang (2020b) —. 2020b, Nature, 587, 45
  * Zhang (2021) —. 2021, ApJ, 907, L17
  * Zhang & Harding (2000) Zhang, B., & Harding, A. K. 2000, ApJ, 532, 1150
  * Zhang et al. (2020) Zhang, C. F., Jiang, J. C., Men, Y. P., et al. 2020, The Astronomer’s Telegram, 13699, 1
  * Zhang et al. (2021) Zhang, G. Q., Tu, Z.-L., & Wang, F. Y. 2021, arXiv e-prints, arXiv:2101.07923
  * Zhong et al. (2020) Zhong, S.-Q., Dai, Z.-G., Zhang, H.-M., & Deng, C.-M. 2020, ApJ, 898, L5
  * Zhou et al. (2020) Zhou, P., Zhou, X., Chen, Y., et al. 2020, ApJ, 905, 99
  * Zhu et al. (2020) Zhu, W., Wang, B., Zhou, D., et al. 2020, The Astronomer’s Telegram, 14084, 1