# Testing afterglow models of FRB 200428 with early post-burst observations of
SGR 1935+2154

A. J. Cooper1,2 , A. Rowlinson1,2, R. A. M. J. Wijers1, C. Bassa2, K.
Gourdji3,7, J. Hessels1,2,A. J. van der Horst4, V. Kondratiev,2 D. Michilli5,
Z. Pleunis6, T. Shimwell2,8, S. ter Veen,2
1 Anton Pannekoek Institute, University of Amsterdam, Postbus 94249, 1090 GE
Amsterdam, The Netherlands
2 ASTRON, the Netherlands Institute for Radio Astronomy, Oude Hoogeveensedijk
4, 7991 PD, Dwingeloo, The Netherlands
3 Centre for Astrophysics and Supercomputing, Swinburne University of
Technology, Hawthorn VIC 3122, Australia
4 Department of Physics, George Washington University, 725 21st St NW,
Washington, DC 20052, U.S.A.
5 MIT Kavli Institute for Astrophysics and Space Research, 77 Massachusetts
Ave, Massachusetts Institute of Technology, Cambridge, MA 02139, U.S.A.
6 Dunlap Institute for Astronomy & Astrophysics, University of Toronto, 50 St.
George Street, Toronto, ON M5S 3H4, Canada
7 OzGrav: ARC Centre of Excellence for Gravitational Wave Discovery, Hawthorn,
VIC 3122, Australia
8 Leiden Observatory, Leiden University, P.O. Box 9513, 2300 RA Leiden, The
Netherlands E-mail<EMAIL_ADDRESS>
(Accepted XXX. Received YYY; in original form ZZZ)

###### Abstract

We present LOFAR imaging observations from the April/May 2020 active episode
of magnetar SGR 1935+2154. We place the earliest radio limits on persistent
emission following the low-luminosity fast radio burst FRB 200428 from the
magnetar. We also perform an image-plane search for transient emission and
find no radio flares during our observations. We examine post-FRB radio upper
limits in the literature and find that all are consistent with the multi-
wavelength afterglow predicted by the synchrotron maser shock model
interpretation of FRB 200428. However, early optical observations appear to
rule out the simple versions of the afterglow model with constant-density
circumburst media. We show that these constraints may be mitigated by adapting
the model for a wind-like environment, but only for a limited parameter range.
In addition, we suggest that late-time non-thermal particle acceleration
occurs within the afterglow model when the shock is no longer relativistic,
which may prove vital for detecting afterglows from other Galactic FRBs. We
also discuss future observing strategies for verifying either magnetospheric
or maser shock FRB models via rapid radio observations of Galactic magnetars
and nearby FRBs.

###### keywords:

stars: neutron – fast radio bursts – acceleration of particles – stars:
magnetars – stars: individual: SGR1935+2154

††pubyear: 2022††pagerange: Testing afterglow models of FRB 200428 with early
post-burst observations of SGR 1935+2154–Testing afterglow models of FRB
200428 with early post-burst observations of SGR 1935+2154

##  1 Introduction

Fast radio bursts (FRBs) are bright, millisecond duration flashes of coherent
radio emission (Lorimer et al., 2007; Thornton et al., 2013; Petroff et al.,
2016; The CHIME/FRB Collaboration et al., 2021). Over 20 FRBs have now been
localized111http://frbhosts.org to their host Galaxies (Chatterjee et al.,
2017; Ravi et al., 2019; Bannister et al., 2019; Marcote et al., 2020; Nimmo
et al., 2021; Heintz et al., 2020; Bhandari et al., 2020), confirming that
these bursts are (usually) of extragalactic origin, as was already implied by
large dispersion measures. Two varieties of FRB source have been identified,
categorized by whether the source is observed to repeat or not. Bursts
observed from these different categories of sources appear to have different
properties (Pleunis et al., 2021), which may be suggestive of different
progenitors or emission mechanisms.

Since the discovery of FRBs, many theoretical models have been presented to
explain the bursts involving various astrophysical sources and radiation
mechanisms (Platts et al., 2019). Many of these models focus on highly
magnetized neutron stars as prime progenitor candidates for a few reasons.
Firstly, the millisecond emission timescale implies compact emission regions
$R\approx ct_{\rm FRB}\Gamma^{2}\approx 10^{6}\,t_{\rm
FRB,-3}\,\Gamma^{2}_{0}\;{\rm cm}$, where $\Gamma$ is the bulk Lorentz factor
of the emission region towards the observer and we have used the convenient
notation $X_{\rm n}\equiv X/10^{n}$. In addition, the luminosity of FRBs
($\approx 8$ orders of magnitude larger than giant millisecond radio pulses in
the Milky Way) might imply they are powered by the large magnetic energy
reservoir in the magnetospheres of highly magnetized neutron stars known as
magnetars. For decades, magnetars within our Galaxy have been observed to emit
spontaneous X-ray bursts, which appear to follow a similar wait time
distribution and luminosity function to FRBs (e.g. SGR 1900+14 Göǧüş et al.
1999; SGR 1806-20 Göǧüş et al. 2000; FRB 121102 Gourdji et al. 2019), as well
as periods of activity and quiescence. Finally, other observed properties of
FRBs can be explained within a magnetized neutron star progenitor model: large
rotation measures (Michilli et al., 2018), downward drifting sub-bursts (e.g.
Hessels et al. 2019), and polarisation angle swings across bursts (Luo et al.,
2020).

The exact radiation mechanism within a magnetar progenitor framework is also
not understood, with the two primary classes of models distinguished by where
the FRB is emitted: close to the surface of the neutron star in magnetospheric
models (Katz, 2016; Kumar et al., 2017; Lu & Kumar, 2018; Kumar & Bošnjak,
2020; Wadiasingh & Timokhin, 2019; Lyutikov & Popov, 2020), and far away from
the surface in models which rely on maser emission in magnetized shocks
(Beloborodov, 2017; Metzger et al., 2019; Beloborodov, 2020). Both classes of
radiation model appear to explain the basic properties of FRBs.

Magnetospheric models of FRBs require the coherent radiation of accelerated
particles from close to the neutron star surface, through e.g. curvature
emission (Kumar et al., 2017). It has been shown that many of the observed
properties of FRBs discussed above can be reproduced within this model (e.g.
Lu & Kumar 2018; Wang et al. 2019; Cooper & Wijers 2021; Yang & Zhang 2021;
Wang et al. 2022). Coherent curvature radiation requires spatial
inhomogeneities in the particle distribution (‘clumps’ or ‘bunches’), which
may be created during the particle acceleration phase through the two-stream
instability (Cheng & Ruderman, 1977; Usov, 1987; Lu & Kumar, 2018; Kumar et
al., 2022). However, whether clumps form on the required timescales is less
clear (see Melrose et al. 2021 and references therein for critical discussion
of curvature radiation as a pulsar mechanism), and it has further been
suggested that the FRBs produced within the magnetosphere may fail to
propagate to an observer (Beloborodov, 2021).

The maser shock model appears to be a theoretically robust way in which
coherent emission can be produced, as the maser has been shown to emit narrow-
band radio emission in 1D & 3D particle-in-cell (PIC) simulations of
magnetized, relativistic shocks (Plotnikov & Sironi, 2019; Sironi et al.,
2021). Furthermore, such emission could be ubiquitous in nature wherever such
shocks occur, and non-magnetar progenitors have been invoked to explain
observed periodicity in repeating sources (Sridhar et al., 2021). Such
ubiquity could account for the high observed volumetric rate of FRBs (Ravi et
al., 2019; Lu & Kumar, 2019; Luo et al., 2020). However, the specific
properties of recently reported FRBs are difficult to explain in a maser shock
scenario. In particular, the microsecond variability of FRB 20180916B (Nimmo
et al., 2021) and the 30ms separation between distinct sub-bursts in Galactic
FRB 200428 (CHIME/FRB Collaboration et al., 2020) appear to be contradictory
to the large length scales associated with the synchrotron maser emission
region.

In magnetar models of FRBs, multi-wavelength or multi-messenger counterparts
to emission provide invaluable observing opportunities with which to
distinguish between theoretical models. Both magnetospheric and maser shock
models predict a high-energy counterpart to FRB emission (Metzger et al.,
2019; Cooper & Wijers, 2021; Yang & Zhang, 2021), however the large distances
to extragalactic FRBs mean predicted X-ray/gamma-ray fluxes are below the
detection threshold of current instrumentation. The first version of the
synchrotron maser shock model (Lyubarsky, 2014), in which relativistic
magnetar hyperflares ($U_{B}\approx 10^{48}\,{\rm erg}$) interact with pulsar
wind nebula, predicted a very high-energy TeV component to FRBs. In Metzger et
al. (2019) this model was adapted to consider interaction with a wind-like
environment, or a mildly relativistic baryonic shell emitted from a previous
flare. Using these models, Metzger et al. (2019) predict a multi-wavelength
afterglow due to thermal synchrotron emission, based on gamma-ray burst (GRB)
afterglow models (Sari et al., 1998). Notably, as the shocks required for the
maser are highly magnetized & relativistic, magnetic field lines are
compressed such that shocks are quasi-perpendicular. For this reason, shock
accelerated non-thermal electrons are not necessarily expected (Sironi &
Spitkovsky, 2009), in contrast to what is observed in other astrophysical
shocks. Observationally, identifying a multi-wavelength afterglow could
confirm the synchrotron maser model of FRBs, since magnetospheric models are
not predicted to produce them. Persistent radio counterparts have been
identified in two localized repeating FRB sources: FRB 121102 (Chatterjee et
al., 2017) and FRB 20190520B (Niu et al., 2021). The origin of the persistent
radio emission is not definitively known, the spectra and lack of variability
may imply the compact source is an AGN that resides relatively close to the
FRB source. Follow-up of all other localized FRBs have yielded only upper
limits to afterglow counterparts (Bhandari et al., 2018), although these are
generally not constraining for the maser shock model.

In this work, we present Low Frequency Array (LOFAR; van Haarlem et al. 2013)
imaging results of SGR 1935+2154, and discuss more broadly the multi-
wavelength afterglow predicted after FRB 200428 in the maser shock model. In
Section 2 we discuss the behaviour of magnetar SGR 1935+2154 before and during
the 2020 active phase. In Section 3, we present the LOFAR imaging results,
where we perform a search for bright radio bursts observable in the image-
plane (see Bailes et al. 2021 for the time-domain search, as well as other
radio observations) and provide limits for persistent low-frequency emission.
In Section 4 we interpret the LOFAR results and suggest observing strategies
for verification of FRB emission mechanisms. In Section 5, we discuss the
synchrotron maser shock model and its application to FRB 200428. In
particular, we find that early time upper limits appear to rule out simple
versions of the maser shock model as applied to the Galactic FRB. We further
discuss extensions to the model including non-thermal radiation at late times,
and make afterglow predictions for Galactic and nearby, extragalactic FRBs. We
conclude our findings in Section 6.

##  2 SGR 1935+2154

Galactic magnetar SGR 1935+2154 was discovered in July 2014 through a series
of short X-ray bursts (Cummings et al., 2014; Lien et al., 2014; Israel et
al., 2016), and past X-ray activity from 2011 was found in an archival search
(Campana et al., 2014). Further observations in 2015 with CHANDRA & XMM-Newton
found many more X-ray bursts (Israel et al., 2016), as well as the first
intermediate flares (Kozlova et al., 2016). This led to the first measurements
of a spin period and spin-down rate of $P=3.245$ seconds (Israel et al., 2016)
and $\dot{P}=1.43\times 10^{-11}$, relatively typical values for a Galactic
magnetar. Using these measured values, Israel et al. (2016) derive the
characteristic age and surface magnetic field of the magnetar as: $\tau_{\rm
c}=\dfrac{P}{2\dot{P}}\approx 3.6$ kyr and
$B_{s}=\sqrt{\dfrac{3c^{3}IP\dot{P}}{8\pi^{2}R_{\rm NS}^{6}}}\approx 2.2\times
10^{14}$ G. Interestingly, $\dot{P}$ shows a decreasing trend, with a measured
negative $\ddot{P}=-3.5(7)\times 10^{-19}\>{\rm s^{-1}}$ observed. The source
position was determined to be at ${\rm R.A.}=19:34:55.5978$, ${\rm
Dec.}=21:53:47.7864$, with an accuracy of 0.6” (90% confidence, Israel et al.
2016; see also the discovery of an infrared counterpart by Levan et al. 2018
at Galactic coordinates ${\rm b}=293.73128$, ${\rm l}=21.896608$). The
magnetar was associated with a supernova remnant SNR G57.2+0.8 by Gaensler
(2014), with a full multi-wavelength radio study of continuum and persistent
emission presented in Kothes et al. (2018). The authors find that the age of
the SNR is approximately $41$ kyr, i.e. much older than the characteristic age
of the magnetar derived from its spin properties (Israel et al., 2016). The
distance to SNR G57.2+0.8 was revised by Zhou et al. (2020), finding that the
SNR and thus the magnetar is likely closer than the $10$ kpc previously
assumed, at just $D=6.6\pm 0.7$ kpc.

In late 2019 (Wood et al., 2019; Ambrosi et al., 2019) and especially in early
2020 (Palmer, 2020; Younes et al., 2020), the magnetar entered a new phase of
extreme activity. On the 27th April 2020 at 14:34:24, an extremely bright
radio burst was emitted from SGR 1935+2154. The burst was observed by the
Survey for Transient Astronomical Radio Emission 2 (STARE-2; Bochenek et al.
2020b) an all-sky radio telescope (Bochenek et al., 2020a). It was also
detected in the side lobes of CHIME, the Canadian Hydrogen Intensity Mapping
Experiment telescope (CHIME/FRB Collaboration et al., 2020). STARE-2 detected
the burst with a $1.281-1.468$ GHz lower limit fluence of $1.5\pm 0.3\times
10^{6}$ Jy ms and a full-width half-maximum duration of $0.61$ ms. CHIME/FRB
observed two bursts separated by $28.9$ ms, with $400-800$ MHz durations of
$0.585\pm 0.014$ ms and $0.335\pm 0.007$ms respectively, and average fluences
of $0.48\pm^{0.48}_{0.24}\times 10^{6}$ Jy ms and $0.22\pm^{0.22}_{0.11}\times
10^{6}$ Jy ms. The lower observed fluence could be attributed to a steep
spectral rise with frequency such that the burst is brighter at the STARE-2
observing frequency. The implied luminosity was lower than extragalactic FRBs
by $\approx 4$ orders of magnitude, but was brighter than any other coherent
radio emission observed from within our Galaxy with the possible exception of
the brightest 2 MJy nanoshot observed from the Crab pulsar (Hankins & Eilek,
2007). On 30th April, a weak radio burst ($0.06$ Jy ms) was reported by Zhang
et al. (2020), and two further radio bursts were reported later by Kirsten et
al. (2021) with fluences of $112$ Jy ms and $24$ Jy ms.

FRB 200428 occurred during a very active period of high-energy bursts from SGR
1935+2154, and many hundreds of short X-ray bursts were observed by NICER,
Fermi, Swift, AGILE, INTEGRAL, Insight-HXMT, Konus-Wind and other X-ray
satellite observatories (Fig. 4; for catalogues of X-ray bursts see Younes et
al. 2020; Cai et al. 2022a; Li et al. 2022). Remarkably, an X-ray burst
temporally coincident with the FRB was observed by four X-ray instruments:
Integral (Mereghetti et al., 2020), Insight-HXMT (Li et al., 2021), AGILE
(Tavani et al., 2021) and Konus-Wind (Ridnaia et al., 2021). The X-ray/radio
energy ratio of the radio burst was $\dfrac{E_{\rm X}}{E_{\rm r}}\approx
10^{5}$. The radio-coincident X-ray burst had a harder spectrum than other
bursts seen from SGR 1935+2154 in the same window of activity, extending to
250 keV (Mereghetti et al., 2020; Li et al., 2021) despite a relatively
typical total fluence. The stark contrast is clear when the X-ray burst is
compared to the many Fermi-GBM and NICER observed bursts in Younes et al.
(2021). Within magnetospheric models of FRBs, this can plausibly be explained
by a non-thermal counterpart observable only when an FRB is observed (Cooper &
Wijers, 2021), in addition to a regular short magnetar burst; or due to rapid
relaxation of perturbed magnetic field lines (Yang et al., 2020). In the
synchrotron maser shock picture of FRBs, the hard X-ray burst represents the
initial peak of the afterglow emission that is observable at lower frequencies
at later times (Margalit et al., 2020b).

Figure 1: LOFAR image of SGR 1935+2154 (denoted by the green circle) and the
surrounding remnant SNR G57.2+0.8. This image makes use of the full datasets
from all epochs.

##  3 LOFAR observations

Observation ID | Start Date | Start Time | Calibrator ID
---|---|---|---
| (UTC) | (UTC) |
L780243 | April 29 2020 | 03:08 | L780239
L780251 | April 29 2020 | 05:20 | L780247
L780651 | May 11 2020 | 01:49 | L780655
L780659 | May 11 2020 | 04:01 | L780655
L780667 | May 11 2020 | 06:13 | L780671
L797090 | October 21 2020 | 14:30 | L797092
L797094 | October 21 2020 | 16:42 | L797096
Table 1: The observations attained of SGR 1935+2154. The ID numbers correspond
to the IDs for the preprocessed data in the LOFAR Long Term
Archive222https://lta.lofar.eu. The duration of each target observation was 2
hours, separated by 10 minute calibrator observations of 3C295.

SGR 1935+2154 entered an active phase in April 2020. We obtained Directors
Discretionary Time observations using LOFAR, comprising of 14 hours spread
over 3 dates in 2020. The observation details are provided in Table 2. Each
observation was obtained using the LOFAR High Band Antennas (HBA) and the full
Dutch array (24 core stations and 14 remote stations), covering a frequency
range of 120–168 MHz and a central frequency of 144 MHz. The frequency range
is divided into 244 sub-bands with bandwidths of 195.3 kHz. The recorded data
have a time resolution of 1s, and a frequency resolution of 64 channels per
sub-band. These data were pre-processed using the standard methods for LOFAR.

To calibrate these LOFAR data, we used the prefactor pipeline (version
3.2)333https://github.com/lofar-astron/prefactor developed by the LOFAR Radio
Observatory with the default parameters and version 3.1 of the LOFAR software.
The calibrator source chosen for these observations was 3C48. As the field of
SGR 1935+2154 is close to the A-Team source Cygnus A, we used the prefactor
demixing options to subtract the contribution of Cygnus A from the target
data. The data were averaged to a time step of 8 seconds and a frequency
resolution of 48.82 kHz (4 channels per subband).

###  3.1 Epoch imaging

All imaging of these data was conducted using WSClean version
2.7.0444http://wsclean.sourceforge.net (Offringa et al., 2014). The imaging
settings used for these observations are outlined in Table 2. Deep images of
the field were created for the 3 observation days separately, as well as a
fully combined image. Two versions of the deep images were created; one with
the full dataset and the second with a minimum baseline length of
1000$\lambda$ to remove the extended emission from the supernova remnant.
These images were used to search for faint persistent emission from the
location of SGR 1935+2154. No persistent emission was detected at the location
of SGR 1935+2154 and the limits obtained for persistent emission from SGR
1935+2154 are provided in Table 3. We provide a constrained flux density
measurement at the position of SGR 1935+2154 obtained using PySE (Carbone et
al., 2018) and assuming the restoring beam parameters. Additionally, we
measure the rms noise in a 30$\times$30 pixel box centred on the position of
SGR 1935+2154 and use the rms noise to calculate a 3$\sigma$ upper limit.

Paramter | Deep Images | Snapshot Images
---|---|---
Size (pixels) | 4096$\times$4096 | 2048$\times$2048
Pixel Scale (arcsec) | 5 | 5
-mgain | 10 | 10
-j | 10 | 10
Automatic masking of sources ($\sigma$) | 10 | 10
Maximum baseline length ($\lambda$) | 8000 | 8000
Minimum baseline length ($\lambda$) | 0/1000 | 700
Number of frequency channels | 6 | 1
Automatic threshold ($\sigma$) | 3 | 3
Briggs weighting robustness | -0.5 | -0.5
Number of iterations | 10000 | 10000
Primary beam applied | True | False
Weighting rank filter | 3 | 3
Clean border | 0 | 0
Reorder visibilities | True | False
Fit restoring beam | True | True
Number of time intervals | 1 | 180
Table 2: The parameters used for imaging each observation set in WSClean. Deep images are created using two different minimum baselines. Observation Date | Flux density | 3$\sigma$ upper limit
---|---|---
| (mJy) | (mJy/beam)
April 29 2020 | $2.2\pm 5.8$ | 9.5
May 11 2020 | $1.8\pm 4.3$ | 6.8
October 21 2020 | $0.7\pm 4.6$ | 7.9
All | $1.5\pm 3.8$ | 6.0
Table 3: The flux density measurements at the location of SGR 1935+2154 for
each observing epoch and a combined image. The flux density measurements were
obtained via a constrained flux density measurement using PySE. The 3$\sigma$
upper limits are obtained by measuring the rms noise in a region surrounding
SGR 1935+2154.

###  3.2 Snapshot Imaging

To search for bright pulses from SGR 1935+2154, we also imaged the
observations on shorter snapshot timescales. We integrated across the observed
frequencies to create a single image. As SGR 1935+2154 has a high dispersion
measure ($332.7{\rm pc~{}cm^{-3}}$; CHIME/FRB Collaboration et al. 2020), it
takes time for a dispersed pulse to traverse the bandwidth of the HBA
observation. This time is calculated using:

$\displaystyle\tau_{\rm DM}=8.3\Delta\nu DM\nu^{-3}\mu{\rm s},$ (1)

where $\tau_{\rm DM}$ is the dispersion delay in seconds, $\Delta\nu$ is the
bandwidth in MHz, $\nu$ is the observing frequency in GHz, and $DM$ is the
dispersion measure in pc cm-3 (Taylor & Cordes, 1993). For these LOFAR
observations, we find that the dispersion delay corresponds to 44 seconds.
Therefore, we create snapshot images of 40 seconds in duration (as the data
are integrated to 8 second intervals) and obtained 1081 images in total. We
note that we could conduct de-dispersion imaging (e.g. Anderson et al. 2021;
Tian et al. 2022) to obtain deeper constraints on very short duration
dispersed pulses such as FRBs. However, beam-formed observations were also
attained for these observations and very short duration dispersed pulses would
be optimally detected in those observations (Bailes et al., 2021). Our
snapshot images targeted longer duration flares that may not have been
detectable in the beam-formed searches. The snapshot images were created using
the settings given in Table 2. A minimum baseline length of 700$\lambda$ was
used so that the extended emission from the supernova remnant was not visible
in the snapshot images as this leads to a better constraint on emission from a
point source at the location of SGR 1935+2154.

Following the method outlined in Rowlinson et al. (2022), we correct for any
systematic flux density variations between the snapshot images produced by
assuming that the majority of sources are not variable. As SGR 1935+2154 is in
the centre of the field, we consider sources out to a radius of 1 degree
(covering the inner 50% of the full width half maximum of the LOFAR HBA beam
at 150 MHz555http://old.astron.nl/radio-observatory/astronomers/lofar-imaging-
capabilities-sensitivity/lofar-imaging-capabilities/lofa). We extract all
sources in the first 40 second snapshot image within that radius using PySE
with a detection threshold of 8$\sigma$ and confirm they are point sources by
visual inspection (one source was rejected as it consisted of 2 blended point
sources). These selection criteria resulted in 10 sources that were input into
the LOFAR Transients Pipeline (TraP; Swinbank et al., 2015) using the
monitoring list capability. A flux density correction factor is then
calculated and applied to each individual image666using the script
https://github.com/transientskp/TraP_tools/blob/master/exampleScripts/correctSystematicFluxOffset.py.
The average flux density correction is $\sim$6%.

Following the correction for systematic flux density variations, we conduct
image quality control by measuring the rms noise in the inner eighth of each
image and plot a histogram of these data in Figure 2. We fit the typical rms
noise distribution for the images with a Gaussian distribution to give an
average rms value of 17.4${}^{+2.6}_{-2.3}$ mJy/beam. Any image with an rms
noise that is more than 3$\sigma$ deviant from the average rms value is
rejected. This analysis rejected 110 images, corresponding to 10% of the
sample.

Figure 2: Histogram of the rms values of the inner eighth of all the 40 second
snapshot images used. The values are fit with a Gaussian distribution shown by
the solid red curve. All images that are $>3\sigma$ deviant from the average
rms are rejected and the thresholds are shown with the black dashed lines.

To search for emission at the location of SGR 1935+2154, we processed the
images using TraP and monitor the position of SGR 1935+2154 to provide a flux
measurement in each epoch. We used the standard TraP settings apart from using
a 4$\sigma$ detection threshold and setting the force beam parameter to True.
As we are focusing on the location of SGR 1935+2154, which lies at the centre
of the field, we use a small extraction radius of 120 pixels (corresponding to
10 arcmin). We detect no emission at the location of SGR 1935+2154. In Figure
3, we plot a histogram of the flux density values measured at the location of
SGR 1935+2154 and fit the distribution with a Gaussian distribution. The
fitted Gaussian distribution gives an average flux density measurement per
epoch of 13$\pm$0.7 mJy with a standard deviation of 23 mJy, corresponding to
a 3$\sigma$ upper limit for persistent point source emission of 82 mJy. This
suggests the flux at the location of SGR 1935+2154 is significantly non-zero,
which by inspection of Fig. 1, can be attributed to extended emission from the
surrounding SNR.

Figure 3: A histogram of the flux density values for SGR 1935+2154 obtained by
conducting a constrained fit at the location in each snapshot image. Figure 4:
We show the times of LOFAR observations (denoted by orange background hue) and
AARTFAAC (Prasad et al., 2016) observations (denoted by purple background hue)
of SGR 1935+2154 during the April/May 2020 active episode. We note that
observations on the 30th April were beamformed only, and therefore do not
appear in our imaging analysis Section 3. The time of the FRB as observed by
CHIME & STARE-2 is denoted by the vertical dashed orange line. We also show
counts/fluence and times of X-ray bursts as observed by Insight-HXMT (blue
dots; Cai et al. 2022a), NICER (red dots; Younes et al. 2020) and Fermi-GBM
(green dots; Lin et al. 2020b). Insight-HXMT data points refer to the total
number of photon counts in the burst across all detectors (left y-axis),
whereas NICER and Fermi-GBM data points refer to the total fluence of the
burst in the 0.5-10 keV and 8-200 keV bands respectively. The second row of
plots show timeline zoomed in on LOFAR observations, where 10 minute
calibrator scans are visible, as are the two X-ray bursts for which we were on
source. Interpretation of these observations is discussed in Section 4.

##  4 Interpretation of LOFAR observations & discussion

###  4.1 Simultaneous radio & X-ray limits on transient flares

Low-frequency image-plane searches for radio transients have recently yielded
detections of new types of coherent radio transient with long periods (Hurley-
Walker et al., 2022; Caleb et al., 2022). These new sources have inferred
surface magnetic fields consistent with magnetars. We have performed a search
for transient low-frequency emission in snapshot images on timescales of $40$
seconds (i.e. the dispersion delay across the LOFAR band of a burst from SGR
1935+2154) and find no transient emission at the location of the magnetar.
Given the non-detections in the LOFAR time-domain search presented in Bailes
et al. (2021), coupled with the fact that second-hour timescale transient
radio emission has not been yet observed from Galactic magnetars, we did not
necessarily expect a detection from this search.

One of the primary motivators of radio observations of magnetars in active
phases, other than detecting period-modulated radio pulses from magnetars (see
Sect. 4.2), is to obtain constraints on the X-ray/radio luminosity ratio
during X-ray magnetar bursts. In Fig. 4 we show the times of LOFAR and
AARTFAAC observations, overplotted with high-energy bursts observed by various
X-ray observatories. One high-energy burst observed by Insight-HXMT (burst
number 56 in Cai et al. 2022a) occurred during our LOFAR observations on
11/5/2022. The burst had a reported 1-250 keV flux of
$2.29^{+0.18}_{-0.17}\times 10^{-7}\>{\rm erg\,s^{-1}\,cm^{-2}}$ (Cai et al.,
2022b). The duration of this burst was $0.093$, $0.06$ and $0.076$ seconds in
the high-energy (HE), medium-energy (ME) and low-energy (LE) detectors
respectively, resulting in a X-ray fluence of approximately $2\times
10^{-8}\>{\rm erg\,cm^{-2}}$ across 1-250 keV. The time-domain search for
LOFAR bursts in this observation provided upper limits to the spectral fluence
of a 1ms width radio burst of $0.075$ Jy ms (Bailes et al., 2021). Assuming
the bandwidth of a typical radio burst is $\delta\nu=10^{9}\,{\rm Hz}$ (i.e.
similar to the observed bandwidth of FRB 200428), we find a fluence limit of
$7.5\times 10^{-18}\>{\rm erg\,cm^{-2}}$, resulting in a minimum X-ray/radio
fluence ratio of $\dfrac{\mathcal{F}_{\rm X}}{\mathcal{F}_{\rm
r,144MHz}}>3\times 10^{9}$. This ratio is very constraining, as shock models
of FRBs typically predict luminosity ratios of approximately $10^{5}$
(Margalit et al., 2020b).

The Amsterdam-ASTRON Radio Transient Facility And Analysis Centre (AARTFAAC),
is an all-sky radio transient monitor operating in parallel with LOFAR (Prasad
et al., 2016), sensitive to bright transient and variable emission at low-
frequencies. AARTFAAC was operating with 16 subbands spanning $60.15-64.84$
MHz during an X-ray burst observed from SGR 1935+2154 by Insight-HXMT (burst
19 in Cai et al. 2022a). The dispersion delay from SGR 1935+2154 from the
Insight-HXMT X-ray band to the AARTFAAC band is $\tau_{\rm X-ray}=324$
seconds, and the in-band dispersion delay across $60.15-64.84$ MHz is
$\tau_{\rm in-band}=54$ seconds. No radio transient was detected at the time
of the X-ray burst. We can derive a meaningful fluence sensitivity by
comparing the weakest bursts observed in Kuiack et al. (2020) from PSR
B0950+08 ($4\times 10^{4}$ Jy ms), and rescaling by a factor proportional to
the square root of the ratio of in-band dispersion delay from the different
sources to account for a loss of sensitivity due to pulse smearing:
$\bigg{(}\dfrac{\tau_{\rm PSR\,B0950+08}}{\tau_{\rm
SGR\,1935+2154}}\bigg{)}^{1/2}\approx 11$. We therefore find a conservative
fluence sensitivity of approximately $10^{6}$ Jy ms for the dispersed burst
from SGR 1935+2154, and we can constrain the fluence ratio of the X-ray burst
observed by Insight-HXMT to $\dfrac{\mathcal{F}_{\rm X}}{\mathcal{F}_{\rm
r}}\gtrsim 10^{2}$.

Some magnetospheric models of FRBs predict that FRBs are always produced with
short magnetar X-ray bursts, but are only beamed towards the observer in a
small number of cases $\approx\Gamma^{-1}$ (Lu et al., 2020). Here, $\Gamma$
is the Lorentz factor of the emitting set of particles, and is on the order of
$50-1000$ (Kumar et al., 2017). In the context of these models, coordinated
radio and X-ray observations of active magnetars will either lead to the
detection of another Galactic FRB, or build up a sample of radio-quiet
magnetar bursts. If such a sample is significantly large (i.e. $N\gg\Gamma$)
it can be used to statistically constrain $\Gamma$, and thus magnetospheric
models of FRBs more broadly. Using single-dish 25m class telescopes, Kirsten
et al. (2021) were able to place strong limits on radio counterparts to 59
high-energy bursts from SGR 1935+2154, effectively ruling out values of
$\Gamma\ll 50$. Furthermore, Lin et al. (2020a) found that 29 Fermi-GBM bursts
were co-observed during one observing epoch with the FAST radio telescope,
providing stringent upper limits on the radio/X-ray ratio of these bursts. The
caveats with pursuing these kinds of simultaneous observation are two-fold.
Firstly, we know that (repeating) FRBs generally have a narrow bandwidth (e.g.
The CHIME/FRB Collaboration et al. 2021) and therefore observing at a one
frequency does not necessarily preclude the emission of a radio burst at
another. Second, magnetar X-ray bursts are stochastic and difficult to
predict. Building such a sample will require either a pipeline from X-ray
detections to radio observatories that enable rapid target-of-opportunity
observations777A possible observation strategy would be to use low-frequency
radio telescopes and utilize the dispersion delay of radio emission to probe
radio bursts associated with rapidly reported X-ray bursts. Alternatively, one
may attempt to identify X-ray burst storms to trigger observations when more
X-ray bursts are expected within the observation duration.; or long-term
observations of magnetars with underutilized radio telescopes. For the latter,
we note that achieving a constraint of $\dfrac{L_{\rm X}}{L_{\rm r}}>10^{5}$
for a typical Galactic magnetar bursts only requires fluence limits on the
order of $100-1000$ Jy ms, and therefore does not require time on extremely
sensitive radio instruments.

###  4.2 Radio-quiet magnetars as FRB sources

Pulsed, persistent emission is observed from a sub-population of magnetars
which are radio-loud (Camilo et al., 2006, 2007; Kramer et al., 2007; Levin et
al., 2010; Anderson et al., 2012; Lower et al., 2020; Esposito et al., 2020).
In Rea et al. (2012), the authors describe the fundamental plane of radio
magnetars, suggesting that in general radio-loud magnetars have quiescent
$2-10$ keV X-ray luminosities below their rotational energy loss rate $L_{\rm
rot}=4\pi^{2}IP\dot{P}^{-3}$, where I is the neutron star moment of inertia.
The conjecture that enhanced X-ray emission precludes radio emission (or vice
versa) is supported by observations of PSR J1119-6127, a radio pulsar with a
high magnetic field where radio pulses shut off when X-ray bursts are observed
(Archibald et al., 2017). Similar X-ray and radio mode switches have also been
observed in older (but non-recycled) pulsars with modest inferred magnetic
fields, including PSR B0943+10 (Hermsen et al., 2013) and PSR B0823+26
(Hermsen et al., 2018).

SGR 1935+2154 is thus far the only Galactic magnetar to have produced a bright
radio burst, and the radio limits presented here and in Bailes et al. (2021)
suggest SGR 1935+2154 is not a radio-loud magnetar. This is consistent with
the fundamental plane of Rea et al. (2012) given the observed persistent X-ray
luminosity of SGR 1935+2154. Three individual radio bursts/pulses were
reported after the FRB (Zhang et al., 2020; Kirsten et al., 2021), albeit at
lower luminosities. The one-off radio bursts and pulses strongly suggest that
high X-ray bursting activity does not disrupt a magnetar’s ability to produce
transient radio bursts, and possibly that persistent pulsed radio emission
precludes transient bursts. The association between radio-quiet magnetars and
FRB-emitting magnetars is more intriguing when considered in the context of
the recent discovery of highly magnetized neutron stars which exist beyond the
canonical pulsar death lines (e.g. Ruderman & Sutherland 1975; Chen & Ruderman
1993) in $\dot{P}-P$ phase space (Caleb et al. 2022; see also Hurley-Walker et
al. 2022). The implication that a neutron star’s inability to produce typical
pulsar-like emission could be a prerequisite for a FRB production may hint
that pulsed radio emission observed from (ultra) long period magnetars is
powered by a different emission mechanism than pulsars, possibly more similar
to the radiation mechanism that powers FRBs. Understanding the particle
acceleration and coherent radiation mechanism operating in these long period
sources may advance our understanding of how FRBs are generated.

##  5 Maser shock model & afterglow constraints

In the synchrotron maser shock model of FRBs, coherent radiation is generated
by the gyration of particles in relativistic, magnetized shocks (Lyubarsky,
2014; Beloborodov, 2017; Metzger et al., 2019; Beloborodov, 2020; Sironi et
al., 2021), as has been previously discussed in the literature before the
discovery of FRBs (Langdon et al., 1988; Hoshino et al., 1992; Gallant et al.,
1992; Usov & Katz, 2000). The model suggests that flaring magnetars can
produce relativistic ejecta, which provide conditions conducive to synchrotron
maser emission upon interaction with external material in a surrounding nebula
(Lyubarsky, 2014), or from a previous flare (Metzger et al., 2019). As the
shock propagates relativistically, a multi-wavelength afterglow is expected
(Metzger et al., 2019), with emission peaking at successively lower
frequencies due to absorption and a decreasing bulk Lorentz factor $\Gamma$.

###  5.1 Afterglow model

We present relevant dynamical & radiation models, following the afterglow
prescription detailed in Metzger et al. (2019) & Margalit et al. (2020b) in
this subsection. We extend the model of Metzger et al. (2019) into the non-
relativistic expansion phase relevant for late-time radio observations,
remaining agnostic about the nature of the circumburst medium. Much of the
early afterglow model is similar to models developed for GRB afterglows (e.g.
Mészáros et al. 1998; Sari et al. 1998). The late-time dynamics of interest
here after a deceleration time $t_{\rm dec}$ rely on the self-similar model of
blast-waves described in Blandford & McKee (1976) and non-relativistic Sedov-
Taylor model of an expanding blast-wave (Sedov, 1959; Taylor, 1950). Using
these models we can make detailed predictions of the evolution of the shock
and multi-wavelength afterglow.

####  5.1.1 Deceleration phase

As shown in Metzger et al. (2019), the shock reaches the deceleration radius
$r_{\rm dec}$ in a time $t_{\rm dec}$, which is equal to the central engine
activity time, i.e. $t_{\rm dec}\approx t_{\rm FRB}\approx 1$ ms, such that we
use these subscripts interchangeably. All parameters denoted with these
subscripts refer to the value at the deceleration time. All timescales of
interest for emission below the X-ray band occur at $t\gg t_{\rm dec}$,
therefore we are only interested in solutions to the dynamics and radiation in
the deceleration and non-relativistic phases. The bulk Lorentz factor for an
adiabatically evolving shock at times $t\gg t_{\rm FRB}$ is:

$\begin{split}\Gamma(r>r_{\rm FRB})&=\bigg{(}\frac{E_{\rm flare}(17-4k)}{16\pi
m_{p}n_{\rm ext}r^{3}c^{2}}\bigg{)}^{1/2}\\\ &=\Gamma_{\rm
FRB}\bigg{(}\frac{r}{r_{\rm FRB}}\bigg{)}^{\frac{k-3}{2}}\end{split}$ (2)

Where $r$ and $n_{\rm ext}$ are the radius of the shock from the central
engine and the external number density of the local environment that the shock
ploughs into. $n_{\rm ext}$ depends on the type of environment considered and
in general has the form $n_{\rm ext}\propto r^{-k}$ where $0\leq k<3$. Wind-
like medium ($k=2$) and constant medium ($k=0$) are commonly used, and either
may be applicable to the environment surrounding a magnetar (Metzger et al.,
2019). Assuming an external density at $n_{\rm ext}(r_{\rm FRB})$ at $r=r_{\rm
FRB}$:

$n_{\rm ext}(r)=n_{\rm ext}(r_{\rm FRB})\bigg{(}\frac{r}{r_{\rm
FRB}}\bigg{)}^{-k}$ (3)

In the observer frame, the distance of the adiabatic shock front from the
central engine as a function of time is:

$\begin{split}r(t>t_{\rm FRB})&=2c\Gamma^{2}t=2ct\Gamma_{\rm
FRB}^{2}\bigg{(}\frac{r}{r_{\rm FRB}}\bigg{)}^{k-3}\\\ &\propto
t^{\frac{1}{4-k}}\end{split}$ (4)

Therefore, by Eqs. 2 & 4 the shock Lorentz factor varies as a function of
observer time as:

$\begin{split}\Gamma(r>r_{\rm FRB})&=\Gamma_{\rm FRB}\bigg{(}\frac{1}{r_{\rm
FRB}}\bigg{)}^{\frac{k-3}{2}}\bigg{(}\big{(}2ct\Gamma_{\rm
FRB}^{2}\big{)}^{\frac{1}{4-k}}r_{\rm
FRB}^{\frac{3-k}{4-k}}\bigg{)}^{\frac{k-3}{2}}\\\ &\propto
t^{\frac{k-3}{8-2k}}\end{split}$ (5)

Given this, Metzger et al. (2019) compute the afterglow emission during this
deceleration phase. Electrons in the shock are assumed to be in energetic
equipartition with the ions such that their mean thermal Lorentz factor is:
$\gamma_{\rm therm}=\dfrac{m_{p}}{2m_{e}}\Gamma$. Particles are assumed not to
undergo non-thermal acceleration by the shock (see Sect. 5.5). The magnetic
field is parameterized in terms of the magnetization parameter $\sigma$ as a
fraction of the thermal energy density of the shock such that:

$B(t>t_{\rm FRB})=\sqrt{64\pi\sigma\Gamma^{2}m_{p}c^{2}n_{\rm ext}}\propto
t^{\frac{-3}{8-2k}},$ (6)

where we assume throughout $0.1\leq\sigma\leq 1$, as required for coherent
maser emission. As in GRB afterglows, the critical synchrotron frequency
$\nu_{\rm sync}$ and the cooling frequency $\nu_{\rm cool}$ (Sari et al.,
1998) in the observer frame is:

$\begin{split}\nu_{\rm sync}&=\frac{qB\gamma_{\rm therm}^{2}\Gamma}{2\pi
m_{e}c}\propto t^{\frac{3k-12}{8-2k}}\\\ \nu_{\rm
cool}&=\frac{qB}{m_{e}c}\gamma_{\rm c,cool}^{2}\Gamma\propto
t^{\frac{3k-10}{8-2k}}\end{split}$ (7)

Where $\gamma_{\rm c,cool}=\dfrac{6\pi m_{e}c}{\sigma_{\rm T}\Gamma B^{2}t}$.
Using the parameter values of Margalit et al. (2020b), we find that for the
flare associated with FRB 200428 from SGR 1935+2154, $\nu_{\rm sync}<\nu_{\rm
cool}$ after a time of just $t=0.99\;\sigma_{-1}^{2}$ seconds. This means that
for even the earliest optical data discussed in Section 5.3, the slow cooling
regime holds. We note that for larger magnetization values (i.e. $\sigma=1$)
the earliest time predictions within a minute of the burst (e.g. Figs. 6 & 7)
will be affected.

The spectral luminosity at the critical synchrotron frequency is:

$L_{\nu,\rm sync}\propto N_{\rm therm}\Gamma B,$ (8)

where $N_{\rm therm}\propto n_{\rm ext}R_{\rm sh}^{3}$ is the number of
radiating thermal electrons. In the slow-cooling regime (Sari et al., 1998;
Margalit et al., 2020b), the spectral luminosity is:

$L_{\nu}=\begin{cases}L_{\nu,\rm sync}\big{(}\frac{\nu}{\nu_{\rm
sync}}\big{)}^{1/3}\>\>\>\>\>\>\>\>\>\>\>\>\>\>\>\>\>\>\>\>\>\>\>\>\>\nu<\nu_{\rm
sync}\\\ L_{\nu,\rm sync}\exp\bigg{(}\big{(}-\frac{\nu}{\nu_{\rm
sync}}\big{)}^{1/3}\bigg{)}\>\>\>\>\>\>\>\>\nu>\nu_{\rm sync},\end{cases}$ (9)

where the $1/3$ exponent in the second line reflects recent theoretical work
by Margalit & Quataert (2021). In practice, for FRB 200428 we compute the
predicted afterglow by fitting the initial X-ray data point and using the
temporal scaling relations developed for GRB afterglow models described below
(Mészáros et al., 1998; Sari et al., 1998; Metzger et al., 2019). To
approximate the absorption of lower frequency emission, we follow the method
of Margalit et al. (2020b) used in their Fig. 9, by ensuring the spectral
luminosity is not larger than the expected synchrotron self-absorption value
of:

$L_{\nu,\rm SSA}\approx\frac{8}{3}\pi^{2}m_{e}R_{\rm sh}^{2}\nu^{2}\gamma_{\rm
therm}$ (10)

####  5.1.2 Non-relativistic phase

The low luminosity of FRB 200428 means it enters a non-relativistic expansion
phase much earlier than a bright FRB (Margalit et al., 2020b). We can model
this phase by comparison to the dynamics of supernova remnants, i.e. Sedov-
Taylor expansion (Taylor, 1950; Sedov, 1959). We can find the time at which
the shock enters this phase by considering when $\Gamma\approx 1$. Inspection
of Eqs. 4 & 5 tell us $\Gamma(t>t_{\rm FRB})\propto t^{\frac{k-3}{8-2k}}$ such
that the approximate time and radius at which the shock becomes non-
relativistic as:

$\begin{split}t_{\rm non-rel}&\approx t_{\rm FRB}\bigg{(}\frac{1}{\Gamma_{\rm
FRB}}\bigg{)}^{\frac{8-2k}{k-3}}\\\ r_{\rm non-rel}&\approx r_{\rm
FRB}\bigg{(}\frac{t_{\rm non-rel}}{t_{\rm
FRB}}\bigg{)}^{\frac{1}{4-k}}\end{split}$ (11)

For FRB 200428, we find that $t_{\rm non-rel}\approx 45$ seconds. We can find
the synchrotron frequency and spectral luminosity of the thermal electrons at
the time the shock is non-relativistic by Eq. 7:

$\nu_{\rm sync,non-rel}=\nu_{\rm sync,dec}\bigg{(}\frac{t_{\rm non-
rel}}{t_{\rm FRB}}\bigg{)}^{\frac{3k-12}{8-2k}}$ (12)

In the slow-cooling regime, $L_{\nu,\rm sync}\approx$ constant in the
deceleration phase (Sari et al., 1998) such that:

$L_{\nu,\rm sync,non-rel}=L_{\nu,\rm sync,dec}$ (13)

After defining these quantities at the non-relativistic transition time, we
can make predictions for this phase. Using the well-known Sedov-Taylor
expansion solution in a constant medium, the shock radius in this regime is:

$r(r>r_{\rm non-rel})=r_{\rm non-rel}\bigg{(}\frac{t}{t_{\rm non-
rel}}\bigg{)}^{2/5}$ (14)

In the slow-cooling non relativistic regime, the critical thermal synchrotron
frequency and synchrotron luminosity are:

$\begin{split}\nu_{\rm sync}(t>t_{\rm non-rel})&=\nu_{\rm sync,non-
rel}\bigg{(}\frac{t}{t_{\rm non-rel}}\bigg{)}^{-3}\\\ L_{\nu,\rm
sync}(t>t_{\rm non-rel})&=L_{\nu,\rm sync,non-rel}\bigg{(}\frac{t}{t_{\rm non-
rel}}\bigg{)}^{\frac{3}{5}}\end{split}$ (15)

The above description assumes the thermal particles are relativistic, and
therefore is valid while $\gamma_{\rm therm}\gg 1$. For FRB 200428, in the
constant medium case the thermal electrons become non-relativistic
approximately 1 day post-burst due to the low initial Lorentz factor. By this
time the synchrotron cut-off frequency has dropped below the observing
frequency for radio wavelengths of interest (see Fig. 5). However,
extrapolating afterglow lightcurves beyond this time will require additional
consideration of the deep-Newtonian regime similarly to GRB afterglows (e.g.
Barniol Duran & Giannios 2015). Finally, we note that similar to SNR
evolution, the shock is expected to enter the radiative snow-plough phase at a
threshold velocity $v_{\rm sh}\approx\sqrt{\dfrac{kT}{m_{e}}}\approx
10^{7}\,{\rm cm\,s^{-1}}$. For the constant medium case, this occurs for the
FRB 200428 at approximately 1000 days post-burst affecting the predicted shock
dynamics and lightcurves at very late times, which are not relevant for the
observed upper limits.

Using the above, we can make lightcurve predictions through the deceleration
and non-relativistic phases for a variety of FRB bursts, as in Margalit et al.
(2020a). Following the prescription of Margalit et al. (2020b), we use the
observed properties of FRB 200428 & and the coincident X-ray burst to
normalize the initial values $r_{\rm FRB}$, $\Gamma_{\rm FRB}$, $t_{\rm FRB}$
given the requirements of the maser and the high-energy burst. Crucially, the
peak of the X-ray afterglow at $t_{\rm FRB}$ is normalized to the fluence and
peak frequency observed by Insight-HXMT ($\mathcal{F}\approx 7\times
10^{-7}\,{\rm erg\,cm^{-2}}$; $\nu_{\rm peak}\approx 50$ keV; Li et al. 2021).
These parameters are outlined for FRB 200428 from SGR 1935+2154 in Margalit et
al. (2020b) (Eqs. 10-13; used in Figs. 5 & 9).

###  5.2 Radio afterglow

In Fig. 5, we show predicted afterglow lightcurves for four frequencies for
which radio limits were obtained within a short time after the initial burst
(Bailes et al., 2021), assuming a constant medium $(k=0)$. We find that that
model-constraining observations could have been attained with the Karl G.
Jansky Very Large Array (VLA; Thompson et al. 1980) & European VLBI Network
(EVN888https://www.evlbi.org/) at 6 and 1.67 GHz respectively if these
instruments were on source before the synchrotron frequency of thermal
electrons falls below the observing frequency. However, this would necessitate
observations beginning just hours after the initial burst, which is
incompatible with typical target-of-opportunity delay for large-scale radio
facilities. In this case the obtained LOFAR limits presented in Section 3 are
not very constraining, primarily attributable to the brightness of the
extended emission from SNR G57.2+0.8 at $144$ MHz.

Figure 5: Radio afterglow lightcurves following the FRB from SGR 1935+2154,
following the prescription of Margalit et al. (2020b) in their Fig. 9 as
discussed in the text. We show those frequencies with the most stringent and
earliest radio observations (Bailes et al., 2021), including the LOFAR limits
we present in this work. The lack of non-thermal particles leads to a steep
decline in all lightcurves after just hours as the synchrotron frequency drops
below the observing frequency.

###  5.3 Optical afterglow

In Lin et al. (2020a), the authors report on FAST observations of SGR
1935+2154, as well as a multi-wavelength campaign spanning X-ray, optical and
radio observations taken after FRB 200428. Of particular note is a minute-long
z-band BOOTES-3 (Castro-Tirado et al., 1999) observation that occurred
simultaneously with the FRB 200428. The observation began on the 28th April
2020 at 14:34:24.03 (Extended Data Table 1; Lin et al. 2020a), concluding a
minute later and setting an upper limit of 17.9 mag. The authors revise this
to just 11.7 mag when corrected for dust extinction of 6.2 mag, corresponding
to a flux density of approximately 75 mJy. The peak of the X-ray burst
coincident with FRB 200428 occurred after the optical observations began
(Mereghetti et al., 2020; Ridnaia et al., 2021), meaning the optical limit is
constraining for the very early time afterglow of the FRB. Lin et al. (2020a)
discuss the limit with respect to fast optical bursts (FOBs) predicted by Yang
et al. (2019) to be produced coincident with FRBs. In Fig. 6, we show z-band
lightcurves using the above method (see also Fig. 9 in Margalit et al. 2020b),
and note that the BOOTES-3 upper limit significantly constrains the FRB
afterglow. We assume that the flux limit scales as $F_{\rm lim}\propto(T_{\rm
obs})^{-1/2}$ and that limits could be placed 1 second after the FRB and the
start of the observations. We note that there are uncertainties with
extrapolating the upper limit to the start time of the observation in this
manner. Even without such extrapolation, the reported data strongly suggests
that the predicted afterglow of the maser shock model in a uniform medium
presented in Margalit et al. (2020b) is not compatible with the observed upper
limit.

Figure 6: We show in red the z-band lightcurve following the FRB from SGR
1935+2154, following the prescription of Margalit et al. (2020b). In orange,
the span of the observation taken by BOOTES-3 telescope (after 1 second on
source), and in black we include additional optical upper limits by BOOTES-2
and LCOGT presented in Lin et al. (2020a). The peak of the predicted flux is
approximately one order of magnitude above the extinction-corrected limit set
by the BOOTES-3 observation.

####  5.3.1 Wind-like case

One way in which the maser shock model could circumvent the optical upper
limit is to invoke a more complex, non-uniform environment. In this
subsection, we consider the FRB 200428 afterglow shock propagating into a
wind-like ($k=2$) medium at early times. To correct the initial values of
$\Gamma_{\rm FRB}$ & $r_{\rm FRB}$ we refer to the wind-like case in Section
2.2.3 of Metzger et al. (2019).

$\Gamma_{\rm FRB}=2.8\>E_{\rm flare}^{1/4}\,\dot{M_{19}}^{-1/4}\,\beta_{\rm
W}^{1/2}\,\delta t^{1/2}$ (16) $r_{\rm FRB}=1.5\times 10^{9}\>{\rm cm}\;E_{\rm
flare}^{1/2}\,\dot{M_{19}}^{-1/2}\,\beta_{\rm W}^{1/2}\,\delta t^{1/2}$ (17)

Where $E_{\rm flare}$ is the total energy of the flare, $\dot{M}$ is the mass
injection rate in units of grams per second, and $\beta_{\rm W}=v_{\rm w}/c$
is the time-averaged magnetar wind velocity divided by the speed of light. We
note that the Lorentz factor required to explain the low-luminosity X-ray
flare is relatively low, challenging the implicit assumption required for
maser emission that the flare is initially ultra-relativistic. As a lower
limit to the flare energy $E_{\rm flare}$, we adopt the X-ray luminosity of
the coincident flare so that $E_{\rm flare}=7\times 10^{39}$ erg. We also
consider a range of values for the unknown mass injection rate for this
source: $\dot{M}=10^{17-21}$ g s-1. Values of $\dot{M}=10^{19-21}$ g s-1 have
been shown to be consistent with the persistent radio nebula and rotation
measure of FRBs from FRB 121102 (Margalit & Metzger, 2018). However, we also
consider values $\dot{M}<10^{19}$ g s-1 due to the lack of any persistent
radio nebula at the location of SGR 1935+2154 in Section 3, and the fact that
this source does not appear to be a prolific FRB emitter. In the non-
relativistic regime, we use the Sedov-Taylor expansion solution corrected for
wind-like media such that $R\propto t^{2/3}$.

In Fig. 7 we show the z-band predictions for the wind-like medium case,
considering a range of values for $\dot{M}$. We also show the flux limits
obtained by BOOTES-3, again assuming that the flux limit sensitivity scales as
$T_{\rm obs}^{-1/2}$ from one second after the start of the observation. We
find that the optical limits are consistent with scenarios in which
$\dot{M}>10^{19}$ g s-1, but caution that a more detailed analysis of the
earliest second of the optical dataset may rule out all values of
$\dot{M}<10^{21}$ g s-1. Finally, we note that $n\propto R^{-k}$ where $k>2$
might be expected during the early-time deceleration phase at the time of the
BOOTES-3 observation. This is because the charge density of the magnetar’s
magnetosphere (which has a radial dependence of $k\approx 3$) likely
contributes to the local density at $r\approx 10^{9}$ cm. Estimated in terms
of the Goldreich-Julian density (Goldreich & Julian, 1969), the contribution
is:

$n_{\rm GJ}(r_{\rm FRB})\approx\frac{2B_{s}}{cqP}\bigg{(}\frac{R_{\rm
NS}}{R}\bigg{)}^{-3}\approx 10^{4}\;{\rm cm^{-3}}\;B_{\rm s,14}\,P_{0}^{-1}$
(18)

This is comparable to the required value of $n_{\rm ext}=4\times 10^{4}\,{\rm
cm^{-3}}$ in the maser shock model of FRB 2000428 in Eq. 12 of Margalit et al.
(2020b), therefore considering variable values of $k$ as a function of $r$ may
be required for a more accurate description of the afterglow.

Figure 7: Afterglow z-band lightcurves the wind-like environment case for
various values of the mass injection rate $\dot{M}$. The flux sensitivity of
the BOOTES-3 limit is assumed to scale as $T_{\rm obs}^{-1/2}$ and we show
only limits from one second after the start of the observation. This appears
to rule out the most values of the mass injection rate, however extrapolation
of the observations to the time of the FRB (where they are reported to begin)
may completely rule out the maser shock model of FRB 200428.

###  5.4 Prospects of radio afterglow detection for the next Galactic FRB

As aforementioned, the lower limit to the radio luminosity of the Galactic FRB
200428 from SGR 1935+2154 was approximately 4-5 orders of magnitude less than
typical FRB luminosities. In this subsection, we discuss strategies with which
to detect an afterglow for a future Galactic FRB.

A bright FRB occurring within the Galaxy could be detected by all-sky radio
telescopes such as STARE-2999STARE-2 is now decommissioned, but a successor
instrument to the project is planned: Connor et al. (2021) or AARTFAAC (Prasad
et al., 2016), or in the side-lobes of smaller field-of-view radio telescopes.
Furthermore, the prompt X-ray component of the afterglow (or coincident
magnetar X-ray burst) will be very bright and possibly detected by wide-field
gamma-ray instruments such as Fermi-GBM, or other X-ray instruments as in the
case of the burst from SGR 1935+2154.

In Fig. 8, we present $6$ GHz, $1.67$ GHz and $144$ MHz afterglow lightcurves
for a ‘typical’ luminosity ($E_{\rm flare}=10^{43}$ erg) FRB from within our
Galaxy, along with obtained radio limits for the SGR 1934+2154. The afterglow
in this Figure is calculated using the reference values of $r_{\rm FRB}$,
$\Gamma_{\rm FRB}$, $t_{\rm FRB}$ & $\nu_{\rm sync,dec}$ for a typical FRB
described in Eqs. 30-32, 57 in Metzger et al. (2019). We scale the early X-ray
afterglow fluence appropriately such that the $\mathcal{F}_{\nu,\rm
sync,max}\approx E_{\rm flare}/(4\pi D^{2})$. We find that of the radio limits
attained after the 2020 burst, only the earliest LOFAR & VLA limits can
constrain the model in its current form. We note that for this brighter burst,
the condition that $\gamma_{\rm therm}>1$ as required by the afterglow model
holds until 1000 hrs post-burst, i.e. the entire time of interest for the
radio afterglow.

Figure 8: Radio afterglow predictions for a typical luminosity FRB from SGR
1935+2154, at frequencies for which early upper limits were obtained for FRB
200428. We include the expected sensitivity from triggered observations with
low-frequency radio telescopes that have rapid response capabilities, such as
LOFAR or the Murchison Wide Array (MWA; Tingay et al. 2013). We assume a
conservative $5$ mJy sensitivity for a 4 hr integration at $144$ MHz, and that
sensitivity increases as $\sqrt{T}$ where T is the observation time (dashed
orange line). The observations are presumed to begin 6 minutes after the
burst, longer than LOFAR’s current rapid response capability. Inverted
triangles correspond to radio upper limits at different frequencies from
Bailes et al. (2021) as in Fig 5.

We also show in Fig. 8 plausible flux limits that could have been attained if
radio telescopes capable of rapid automated response such as LOFAR (van
Haarlem et al., 2013; Rowlinson et al., 2022) and MWA (Tremblay et al., 2015;
Hancock et al., 2019; Anderson et al., 2021) were triggered on the initial
burst and were on source within 10 minutes. We have assumed a $5$ mJy flux
limit after $4$ hr observation, where the flux limit scales as $F_{\rm
lim}\propto(T_{\rm obs})^{-1/2}$ as generally expected, unless the flux
threshold is limited by poor u-v coverage for very short integration times. If
we are fortunate enough to observe a bonafide FRB from within our Galaxy,
rapid optical and radio observations on minute to day timescales will be
crucial to observe the afterglow and verify or falsify the maser shock model
of FRBs. Such observations are only possible if the FRB event is reported in a
timely manner on networks that distribute astronomical alerts rapidly (such as
GCN Barthelmy et al. 1998 or VOEvent Williams & Seaman 2006 networks), and
radio telescopes have programs in which observations can be interrupted for
rapid or automatic repointing.

###  5.5 Non-thermal radio afterglows

Shock acceleration of particles resulting in non-thermal distributions and
radiation occurs almost ubiquitously in high energy astrophysical transients
such as supernovae remnants (Yuan et al., 2011; Ackermann et al., 2013),
gamma-ray bursts (Waxman, 1997), active galactic nuclei (Blandford & Königl,
1979) and X-ray binary jets (Markoff et al., 2001). However, particle-in-cell
(PIC) simulations of relativistic, magnetized shocks suggest that particle
acceleration in shocks does not occur very efficiently (Sironi & Spitkovsky,
2009; Sironi et al., 2015). As relativistic shocks sweep up organised magnetic
field lines, they are compressed into the downstream shocked medium. As
compression occurs, the angle between the magnetic field lines and the shock
velocity $\theta$ increases, such that they are quasi-perpendicular
$\theta\approx 90\,\deg$. PIC simulations show no significant self-generated
turbulence or magnetic field from particles in relativistic magnetized shocks
(Sironi & Spitkovsky, 2011), and as such the particles are forced to slide
along the background magnetic field lines. As these field lines run
perpendicular to the shock velocity, particles do not escape the shock and
therefore do not undergo repeated shock crossings required for Fermi-like
shock acceleration. This is quantified in Sironi & Spitkovsky (2009), where
the authors define a critical angle $\theta_{\rm crit}$, such that if
$\theta>\theta_{\rm crit}$, particles would have to propagate at greater than
the speed of light in order to outrun the shock and undergo multiple shock
crossings as required for non-thermal acceleration. For this reason, the FRB
afterglow model presented in Metzger et al. (2019); Margalit et al. (2020b)
assumes purely thermal radiation, nor are non-thermal particles included in
Figs. 5, 6, 7 or 8.

As the FRB-initiating shock wave progresses it decelerates to smaller bulk
Lorentz factors and propagates into regions of lower magnetic field strength
(Eqs. 5 & 6). The typical magnetization and Lorentz factor values assumed in
simulations of relativistic, magnetized shocks are $\sigma>0.1$ and
$\Gamma>10$. For the SGR 1935+2154 burst, the afterglow model in Margalit et
al. (2020b) suggests the shock enters a non-relativistic Sedov-Taylor phase at
$t_{\rm non-rel}\approx 45$ seconds via Eq. 11. This is exceptionally early
due to the low-luminosity of this FRB: $t_{\rm non-rel}$ will be larger by a
factor of approximately $100$ for a bright FRB, using the shock values from
Margalit et al. (2020a), corresponding to a time $t_{\rm non-rel}\approx 1.5$
hours. We suggest that in this non-relativistic phase, particles are able to
undergo multiple shock crossing as has been observed in magnetized shocks of
supernovae remnants.

To model the non-thermal radiation in the Sedov-Taylor phase, we assume 10% of
the total (constant) energy of the shock $E_{\rm sh}=\dfrac{4}{3}\pi R_{\rm
sh}^{3}n_{\rm ext}m_{\rm p}v_{\rm sh}^{2}\propto t^{0}$ is available for non-
thermal particle acceleration, as is canonically expected for cosmic ray
acceleration in SNRs (e.g. Strong et al. 2007). We further assume an
equipartition in energy between hadronic and leptonic acceleration. For this
calculation we assume a uniform density medium throughout such that $n_{\rm
ext}=n_{\rm FRB}$, noting that if the constant density does not extend to such
radii, the non-thermal radiation would be lower than predicted here. Non-
thermal radiation strongly depends on the magnetic field at the shock.
However, in the non-relativistic regime the shock’s magnetic field strength is
uncertain due to the unknown magnetization of the magnetosphere-ISM transition
medium, and possible amplification of the compressed magnetic field due to
non-resonant Bell instability (Vink & Laming, 2003; Bell, 2004). We
parameterize the magnetic field in terms of a fraction of the thermal energy
density (i.e. Eq. 6) and the shock-amplified ISM value, such that:

$B={\rm max}\bigg{(}\sqrt{64\pi\sigma\beta^{2}m_{p}c^{2}n_{\rm
ext}}\>,\>\>\chi_{\rm sh}B_{\rm ISM}\bigg{)}$ (19)

Where $\chi_{\rm sh}\approx 4$ is the shock compression ratio, the
magnetization $\sigma=0.1-1$ (Metzger et al., 2019), $\beta$ is the velocity
of the blast-wave in units of the speed of light, and we assume $B_{\rm
ISM}\approx 3\;\mu$G. The minimum electron energy can be expressed in terms of
the thermal particle energy:

$E_{\rm min}=\epsilon_{\rm e}\frac{(p-2)}{2(p-1)}\frac{m_{\rm
p}\beta^{2}c^{2}}{m_{\rm e}}+m_{e}c^{2}$ (20)

Where $\epsilon_{\rm e}=0.05$ is the fraction of shock energy (i.e. half of
10% of the total energy due to equipartition) that goes into accelerating
electrons and $p$ is the slope of the power law distribution of shock-
acceleration electrons which we assume to be $p=2.2$ (Sironi & Giannios,
2013). The spectral luminosity in the slow cooling regime relevant for the
radio afterglow is therefore:

$L_{\nu}\propto\frac{K}{p+1}\nu^{-(p-1)/2}B^{(p+1)/2}$ (21)

where $K=E_{\rm sh}\epsilon_{\rm e}(p-2)E_{\rm min}^{p-2}$ is the
normalization factor of the non-thermal electron distribution. We note that
exponential suppression of non-thermal particle radiation is assumed for times
$t<t_{\rm non-rel}$, since we do not expect shock acceleration at these times
(see above). We treat synchrotron self-absorption using Eq. 10, replacing
$\gamma_{\rm therm}$ with the Lorentz factor for which the critical
synchrotron frequency is the observing frequency $\nu$.

In Fig. 9, we show the predicted thermal (dashed) and non-thermal (solid)
afterglow lightcurves for two observing frequencies. For FRB 200428, the weak
burst means the afterglow is relatively dim, and does not challenge the radio
upper limits previously discussed. However, we note that particularly at low
frequencies the non-thermal radiation could contribute to the overall flux
shortly before the thermal synchrotron cut-off. In Fig. 10, we show a similar
plot for a bright FRB for two distances, corresponding to Galactic or nearby
extragalactic FRBs. For a bright Galactic FRB, non-thermal radio emission
could be detectable up to years after the initial burst. Long-term monitoring
of such a source would provide opportunities for detailed modelling of the
afterglow, contributing to our understanding of shocks more generally.
However, non-thermal radio emission from even a nearby extragalactic FRB is
not likely to be bright enough to probe.

The closest repeating FRB resides at a distance of $D\approx 3.6$ Mpc in M81
(Bhardwaj et al., 2021; Kirsten et al., 2022). Kirsten et al. (2022) obtained
deep persistent flux constraints ($6.5\mu$Jy at 1.5 GHz) of the source before
their reported FRBs, but 1-6 months after previously reported bursts by
Bhardwaj et al. (2021). Unfortunately, the low luminosity of the observed FRBs
mean that predicted afterglow emission is not observable.

Figure 9: Dotted and solid lines denote the thermal and non-thermal components
respectively to the afterglow associated with the FRB from SGR 1935+2154, as
described in the text. The major break denotes the time at which the emission
becomes optically thin. The non-thermal afterglow contributes most
significantly for lower frequencies and at later times, near the thermal
synchrotron cut-off. For all times of interest, the shock magnetic field
strength is always greater than the amplified ISM value. Black and orange
inverted triangles correspond radio upper limits taken after FRB 200428 by VLA
and LOFAR at 6 GHz and 144 MHz respectively. Figure 10: Dotted and solid lines
denote the thermal and non-thermal components respectively to the afterglow
after a typical luminosity FRB, where $\nu_{\rm obs}=1.4\,{\rm GHz}$. We show
two distances representative of a bright Galactic burst (black) and a nearby
extragalactic FRB. The non-thermal emission is relatively weak, but allows for
late-times monitoring of the lightcurve in the Galactic FRB case. The thermal
radio afterglow is tentatively observable ($>\mu$Jy) in the hours following a
close extragalactic FRB. We assume exponential suppression of non-thermal
acceleration before the shock becomes non-relativistic.

##  6 Conclusions

In this work we have presented results from LOFAR imaging observations of SGR
1935+2154, following the Galactic magnetar burst FRB 200428. We discuss
interpretations of the LOFAR results, stressing the importance of simultaneous
X-ray/radio observations of active magnetars to constrain both magnetospheric
and flare/shock models of FRBs. We also make recommendations of rapid
observations on minute-day timescales following Galactic or nearby
extragalactic FRBs. We have analysed optical and radio early time upper limits
in the context of models of FRB 200428 that predict a multi-wavelength
afterglow, namely the synchrotron maser shock model. We have found that early
BOOTES-3 optical observations appear to rule out simple versions of this
model, but invoking a wind-like environment close to the FRB emission zone may
mitigate the constraints. We also suggest that shock accelerated particle
populations should be considered at late times when the shock is non-
relativistic, however we find that such non-thermal emission is too faint to
be observable, except for future Galactic FRBs.

## Acknowledgements

AC would like to thank Ben Margalit for a thorough explanation of the shock-
maser afterglow model for the specific case of SGR 1935+2154, and for
describing the calculation underlying the afterglow light curve predictions in
the non-relativistic regime in their work. We would also like to thank Jeremy
Harwood, Oliver Boersma & Rhaana Starling for useful discussions and
suggestions, and the anonymous referee whose comments improved this work.

Sections of this paper are based on data obtained with the International LOFAR
Telescope (ILT) under project codes DDT13_004, DDT13_006, DDT14_007. LOFAR is
the Low Frequency Array designed and constructed by ASTRON. It has observing,
data processing, and data storage facilities in several countries, that are
owned by various parties (each with their own funding sources), and that are
collectively operated by the ILT foundation under a joint scientific policy.
The ILT resources have benefited from the following recent major funding
sources: CNRS-INSU, Observatoire de Paris and Université d’Orléans, France;
BMBF, MIWF-NRW, MPG, Germany; Science Foundation Ireland (SFI), Department of
Business, Enterprise and Innovation (DBEI), Ireland; NWO, The Netherlands; The
Science and Technology Facilities Council, UK; Ministry of Science and Higher
Education, Poland. K.G. acknowledges Australian Research Council (ARC) grant
DP200102243.

## Data Availability

The LOFAR visibility data are publicly available at the LOFAR Long Term
Archive (https://lta.lofar.eu/) under the relevant project codes. A basic
reproduction package providing the scripts and data required to reproduce the
figures and tables of this paper will be available upon publication.

## References

  * Ackermann et al. (2013) Ackermann M., et al., 2013, Science, 339, 807
  * Ambrosi et al. (2019) Ambrosi E., Barthelmy S. D., D’Ai A., D’Elia V., Lien A. Y., Page K. L., Neil Gehrels Swift Observatory Team 2019, GRB Coordinates Network, 26153, 1
  * Anderson et al. (2012) Anderson G. E., et al., 2012, ApJ, 751, 53
  * Anderson et al. (2021) Anderson G. E., et al., 2021, Publ. Astron. Soc. Australia, 38, e026
  * Archibald et al. (2017) Archibald R. F., et al., 2017, ApJ, 849, L20
  * Bailes et al. (2021) Bailes M., et al., 2021, MNRAS, 503, 5367
  * Bannister et al. (2019) Bannister K. W., et al., 2019, Science, 365, 565
  * Barniol Duran & Giannios (2015) Barniol Duran R., Giannios D., 2015, MNRAS, 454, 1711
  * Barthelmy et al. (1998) Barthelmy S. D., et al., 1998, in Meegan C. A., Preece R. D., Koshut T. M., eds, American Institute of Physics Conference Series Vol. 428, Gamma-Ray Bursts, 4th Hunstville Symposium. pp 99–103, doi:10.1063/1.55426
  * Bell (2004) Bell A. R., 2004, MNRAS, 353, 550
  * Beloborodov (2017) Beloborodov A. M., 2017, ApJ, 843, L26
  * Beloborodov (2020) Beloborodov A. M., 2020, ApJ, 896, 142
  * Beloborodov (2021) Beloborodov A. M., 2021, ApJ, 922, L7
  * Bhandari et al. (2018) Bhandari S., et al., 2018, MNRAS, 475, 1427
  * Bhandari et al. (2020) Bhandari S., et al., 2020, ApJ, 895, L37
  * Bhardwaj et al. (2021) Bhardwaj M., et al., 2021, ApJ, 910, L18
  * Blandford & Königl (1979) Blandford R. D., Königl A., 1979, ApJ, 232, 34
  * Blandford & McKee (1976) Blandford R. D., McKee C. F., 1976, Physics of Fluids, 19, 1130
  * Bochenek et al. (2020a) Bochenek C. D., McKenna D. L., Belov K. V., Kocz J., Kulkarni S. R., Lamb J., Ravi V., Woody D., 2020a, PASP, 132, 034202
  * Bochenek et al. (2020b) Bochenek C. D., Ravi V., Belov K. V., Hallinan G., Kocz J., Kulkarni S. R., McKenna D. L., 2020b, Nature, 587, 59
  * CHIME/FRB Collaboration et al. (2020) CHIME/FRB Collaboration et al., 2020, Nature, 587, 54
  * Cai et al. (2022a) Cai C., et al., 2022a, arXiv e-prints, p. arXiv:2203.16855
  * Cai et al. (2022b) Cai C., et al., 2022b, arXiv e-prints, p. arXiv:2204.07369
  * Caleb et al. (2022) Caleb M., et al., 2022, Nature Astronomy,
  * Camilo et al. (2006) Camilo F., Ransom S. M., Halpern J. P., Reynolds J., Helfand D. J., Zimmerman N., Sarkissian J., 2006, Nature, 442, 892
  * Camilo et al. (2007) Camilo F., Reynolds J., Johnston S., Halpern J. P., Ransom S. M., van Straten W., 2007, ApJ, 659, L37
  * Campana et al. (2014) Campana S., de Ugarte Postigo A., Thoene C. C., Gorosabel J., Rea N., Coti Zelati F., 2014, GRB Coordinates Network, 16535, 1
  * Carbone et al. (2018) Carbone D., et al., 2018, Astronomy and Computing, 23, 92
  * Castro-Tirado et al. (1999) Castro-Tirado A. J., et al., 1999, A&AS, 138, 583
  * Chatterjee et al. (2017) Chatterjee S., et al., 2017, Nature, 541, 58
  * Chen & Ruderman (1993) Chen K., Ruderman M., 1993, ApJ, 402, 264
  * Cheng & Ruderman (1977) Cheng A. F., Ruderman M. A., 1977, ApJ, 212, 800
  * Connor et al. (2021) Connor L., et al., 2021, PASP, 133, 075001
  * Cooper & Wijers (2021) Cooper A. J., Wijers R. A. M. J., 2021, MNRAS, 508, L32
  * Cummings et al. (2014) Cummings J. R., Barthelmy S. D., Chester M. M., Page K. L., 2014, The Astronomer’s Telegram, 6294, 1
  * Esposito et al. (2020) Esposito P., et al., 2020, ApJ, 896, L30
  * Gaensler (2014) Gaensler B. M., 2014, GRB Coordinates Network, 16533, 1
  * Gallant et al. (1992) Gallant Y. A., Hoshino M., Langdon A. B., Arons J., Max C. E., 1992, ApJ, 391, 73
  * Goldreich & Julian (1969) Goldreich P., Julian W. H., 1969, ApJ, 157, 869
  * Gourdji et al. (2019) Gourdji K., Michilli D., Spitler L. G., Hessels J. W. T., Seymour A., Cordes J. M., Chatterjee S., 2019, ApJ, 877, L19
  * Göǧüş et al. (1999) Göǧüş E., Woods P. M., Kouveliotou C., van Paradijs J., Briggs M. S., Duncan R. C., Thompson C., 1999, ApJ, 526, L93
  * Göǧüş et al. (2000) Göǧüş E., Woods P. M., Kouveliotou C., van Paradijs J., Briggs M. S., Duncan R. C., Thompson C., 2000, ApJ, 532, L121
  * Hancock et al. (2019) Hancock P. J., et al., 2019, Publ. Astron. Soc. Australia, 36, e046
  * Hankins & Eilek (2007) Hankins T. H., Eilek J. A., 2007, ApJ, 670, 693
  * Heintz et al. (2020) Heintz K. E., et al., 2020, ApJ, 903, 152
  * Hermsen et al. (2013) Hermsen W., et al., 2013, Science, 339, 436
  * Hermsen et al. (2018) Hermsen W., et al., 2018, MNRAS, 480, 3655
  * Hessels et al. (2019) Hessels J. W. T., et al., 2019, ApJ, 876, L23
  * Hoshino et al. (1992) Hoshino M., Arons J., Gallant Y. A., Langdon A. B., 1992, ApJ, 390, 454
  * Hurley-Walker et al. (2022) Hurley-Walker N., et al., 2022, Nature, 601, 526
  * Israel et al. (2016) Israel G. L., et al., 2016, MNRAS, 457, 3448
  * Katz (2016) Katz J. I., 2016, ApJ, 826, 226
  * Kirsten et al. (2021) Kirsten F., Snelders M. P., Jenkins M., Nimmo K., van den Eijnden J., Hessels J. W. T., Gawroński M. P., Yang J., 2021, Nature Astronomy, 5, 414
  * Kirsten et al. (2022) Kirsten F., et al., 2022, Nature, 602, 585
  * Kothes et al. (2018) Kothes R., Sun X., Gaensler B., Reich W., 2018, ApJ, 852, 54
  * Kozlova et al. (2016) Kozlova A. V., et al., 2016, MNRAS, 460, 2008
  * Kramer et al. (2007) Kramer M., Stappers B. W., Jessner A., Lyne A. G., Jordan C. A., 2007, MNRAS, 377, 107
  * Kuiack et al. (2020) Kuiack M., Wijers R. A. M. J., Rowlinson A., Shulevski A., Huizinga F., Molenaar G., Prasad P., 2020, MNRAS, 497, 846
  * 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
  * Kumar et al. (2022) Kumar P., Gill R., Lu W., 2022, MNRAS,
  * Langdon et al. (1988) Langdon A. B., Arons J., Max C. E., 1988, Phys. Rev. Lett., 61, 779
  * Levan et al. (2018) Levan A., Kouveliotou C., Fruchter A., 2018, ApJ, 854, 161
  * Levin et al. (2010) Levin L., et al., 2010, ApJ, 721, L33
  * Li et al. (2021) Li C. K., et al., 2021, Nature Astronomy,
  * Li et al. (2022) Li X., et al., 2022, arXiv e-prints, p. arXiv:2204.03253
  * Lien et al. (2014) Lien A. Y., et al., 2014, GRB Coordinates Network, 16522, 1
  * Lin et al. (2020a) Lin L., et al., 2020a, Nature, 587, 63
  * Lin et al. (2020b) Lin L., Göğüş E., Roberts O. J., Baring M. G., Kouveliotou C., Kaneko Y., van der Horst A. J., Younes G., 2020b, ApJ, 902, L43
  * Lorimer et al. (2007) Lorimer D. R., Bailes M., McLaughlin M. A., Narkevic D. J., Crawford F., 2007, Science, 318, 777
  * Lower et al. (2020) Lower M. E., Shannon R. M., Johnston S., Bailes M., 2020, ApJ, 896, L37
  * Lu & Kumar (2018) Lu W., Kumar P., 2018, MNRAS, 477, 2470
  * Lu & Kumar (2019) Lu W., Kumar P., 2019, MNRAS, 483, L93
  * Lu et al. (2020) Lu W., Kumar P., Zhang B., 2020, MNRAS, 498, 1397
  * Luo et al. (2020) Luo R., et al., 2020, Nature, 586, 693
  * Lyubarsky (2014) Lyubarsky Y., 2014, MNRAS, 442, L9
  * Lyutikov & Popov (2020) Lyutikov M., Popov S., 2020, arXiv e-prints, p. arXiv:2005.05093
  * Marcote et al. (2020) Marcote B., et al., 2020, Nature, 577, 190
  * Margalit & Metzger (2018) Margalit B., Metzger B. D., 2018, ApJ, 868, L4
  * Margalit & Quataert (2021) Margalit B., Quataert E., 2021, ApJ, 923, L14
  * Margalit et al. (2020a) Margalit B., Metzger B. D., Sironi L., 2020a, MNRAS, 494, 4627
  * Margalit et al. (2020b) Margalit B., Beniamini P., Sridhar N., Metzger B. D., 2020b, ApJ, 899, L27
  * Markoff et al. (2001) Markoff S., Falcke H., Fender R., 2001, A&A, 372, L25
  * Melrose et al. (2021) Melrose D. B., Rafat M. Z., Mastrano A., 2021, MNRAS, 500, 4530
  * Mereghetti et al. (2020) Mereghetti S., et al., 2020, ApJ, 898, L29
  * Mészáros et al. (1998) Mészáros P., Rees M. J., Wijers R. A. M. J., 1998, ApJ, 499, 301
  * Metzger et al. (2019) Metzger B. D., Margalit B., Sironi L., 2019, MNRAS, 485, 4091
  * Michilli et al. (2018) Michilli D., et al., 2018, Nature, 553, 182
  * Nimmo et al. (2021) Nimmo K., et al., 2021, arXiv e-prints, p. arXiv:2105.11446
  * Niu et al. (2021) Niu C. H., et al., 2021, arXiv e-prints, p. arXiv:2110.07418
  * Offringa et al. (2014) Offringa A. R., et al., 2014, MNRAS, 444, 606
  * Palmer (2020) Palmer D. M., 2020, The Astronomer’s Telegram, 13675, 1
  * Petroff et al. (2016) Petroff E., et al., 2016, Publ. Astron. Soc. Australia, 33, e045
  * Platts et al. (2019) Platts E., Weltman A., Walters A., Tendulkar S. P., Gordin J. E. B., Kandhai S., 2019, Phys. Rep., 821, 1
  * Pleunis et al. (2021) Pleunis Z., et al., 2021, ApJ, 923, 1
  * Plotnikov & Sironi (2019) Plotnikov I., Sironi L., 2019, MNRAS, 485, 3816
  * Prasad et al. (2016) Prasad P., et al., 2016, Journal of Astronomical Instrumentation, 5, 1641008
  * Ravi et al. (2019) Ravi V., et al., 2019, Nature, 572, 352
  * Rea et al. (2012) Rea N., Pons J. A., Torres D. F., Turolla R., 2012, ApJ, 748, L12
  * Ridnaia et al. (2021) Ridnaia A., et al., 2021, Nature Astronomy, 5, 372
  * Rowlinson et al. (2022) Rowlinson A., et al., 2022, arXiv e-prints, p. arXiv:2203.16918
  * Ruderman & Sutherland (1975) Ruderman M. A., Sutherland P. G., 1975, ApJ, 196, 51
  * Sari et al. (1998) Sari R., Piran T., Narayan R., 1998, ApJ, 497, L17
  * Sedov (1959) Sedov L. I., 1959, Similarity and Dimensional Methods in Mechanics
  * Sironi & Giannios (2013) Sironi L., Giannios D., 2013, ApJ, 778, 107
  * Sironi & Spitkovsky (2009) Sironi L., Spitkovsky A., 2009, ApJ, 698, 1523
  * Sironi & Spitkovsky (2011) Sironi L., Spitkovsky A., 2011, ApJ, 726, 75
  * Sironi et al. (2015) Sironi L., Keshet U., Lemoine M., 2015, Space Sci. Rev., 191, 519
  * Sironi et al. (2021) Sironi L., Plotnikov I., Nättilä J., Beloborodov A. M., 2021, Phys. Rev. Lett., 127, 035101
  * Sridhar et al. (2021) Sridhar N., Metzger B. D., Beniamini P., Margalit B., Renzo M., Sironi L., Kovlakas K., 2021, ApJ, 917, 13
  * Strong et al. (2007) Strong A. W., Moskalenko I. V., Ptuskin V. S., 2007, Annual Review of Nuclear and Particle Science, 57, 285
  * Swinbank et al. (2015) Swinbank J. D., et al., 2015, Astronomy and Computing, 11, 25
  * Tavani et al. (2021) Tavani M., et al., 2021, Nature Astronomy, 5, 401
  * Taylor (1950) Taylor G., 1950, Proceedings of the Royal Society of London Series A, 201, 159
  * Taylor & Cordes (1993) Taylor J. H., Cordes J. M., 1993, ApJ, 411, 674
  * The CHIME/FRB Collaboration et al. (2021) The CHIME/FRB Collaboration et al., 2021, arXiv e-prints, p. arXiv:2106.04352
  * Thompson et al. (1980) Thompson A. R., Clark B. G., Wade C. M., Napier P. J., 1980, ApJS, 44, 151
  * Thornton et al. (2013) Thornton D., et al., 2013, Science, 341, 53
  * Tian et al. (2022) Tian J., et al., 2022, Publ. Astron. Soc. Australia, 39, e003
  * Tingay et al. (2013) Tingay S. J., et al., 2013, Publ. Astron. Soc. Australia, 30, e007
  * Tremblay et al. (2015) Tremblay S. E., et al., 2015, Publ. Astron. Soc. Australia, 32, e005
  * Usov (1987) Usov V. V., 1987, ApJ, 320, 333
  * Usov & Katz (2000) Usov V. V., Katz J. I., 2000, A&A, 364, 655
  * Vink & Laming (2003) Vink J., Laming J. M., 2003, ApJ, 584, 758
  * Wadiasingh & Timokhin (2019) Wadiasingh Z., Timokhin A., 2019, ApJ, 879, 4
  * Wang et al. (2019) Wang W., Zhang B., Chen X., Xu R., 2019, ApJ, 876, L15
  * Wang et al. (2022) Wang W.-Y., Yang Y.-P., Niu C.-H., Xu R., Zhang B., 2022, ApJ, 927, 105
  * Waxman (1997) Waxman E., 1997, ApJ, 485, L5
  * Williams & Seaman (2006) Williams R. D., Seaman R., 2006, in Gabriel C., Arviset C., Ponz D., Enrique S., eds, Astronomical Society of the Pacific Conference Series Vol. 351, Astronomical Data Analysis Software and Systems XV. p. 637
  * Wood et al. (2019) Wood J., Bissaldi E., Fermi GBM Team 2019, GRB Coordinates Network, 25975, 1
  * Yang & Zhang (2021) Yang Y.-P., Zhang B., 2021, ApJ, 919, 89
  * Yang et al. (2019) Yang Y.-P., Zhang B., Wei J.-Y., 2019, ApJ, 878, 89
  * 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., et al., 2020, ApJ, 904, L21
  * Younes et al. (2021) Younes G., et al., 2021, Nature Astronomy,
  * Yuan et al. (2011) Yuan Q., Liu S., Fan Z., Bi X., Fryer C. L., 2011, ApJ, 735, 120
  * Zhang et al. (2020) Zhang C. F., et al., 2020, The Astronomer’s Telegram, 13699, 1
  * Zhou et al. (2020) Zhou P., Zhou X., Chen Y., Wang J.-S., Vink J., Wang Y., 2020, ApJ, 905, 99
  * van Haarlem et al. (2013) van Haarlem M. P., et al., 2013, A&A, 556, A2