11institutetext: Inter-University Institute for Data Intensive Astronomy &
Department of Astronomy, University of Cape Town, Private Bag X3, Rondebosch,
7701, South Africa 22institutetext: Department of Physics & Astronomy,
University of Utah, Salt Lake City, UT 84112, USA 33institutetext: Department
of Astrophysics/IMAPP, Radboud University, P.O. Box 9010, 6500 GL, Nijmegen,
The Netherlands 44institutetext: South African Astronomical Observatory, P.O.
Box 9, 7935, Observatory, South Africa 55institutetext: INAF-Osservatorio
Astrofisico di Catania, Via S. Sofia 78, I-95123 Catania, Italy
66institutetext: Thüringer Landessternwarte Tautenburg, Sternwarte 5, 07778
Tautenburg, Germany 77institutetext: Department of Physics, The George
Washington University, 725 21st Street NW, Washington, DC 20052, USA
88institutetext: DARK, Niels Bohr Institute, University of Copenhagen, Jagtvej
128, 2200 Copenhagen, Denmark 99institutetext: Cosmic Dawn Center (DAWN),
Denmark 1010institutetext: Niels Bohr Institute, University of Copenhagen,
Rådmandsgade 62-64, 2200, Copenhagen N, Denmark 1111institutetext: Department
of Astronomy, University of California, Berkeley, CA 94720-3411, USA
1212institutetext: Hessian Research Cluster ELEMENTS, Giersch Science Center,
Max-von-Laue-Strasse 12, Goethe University Frankfurt, Campus Riedberg, 60438
Frankfurt am Main, Germany 1313institutetext: Max-Planck-Institut für
Extraterrestrische Physik, Giessenbachstraße, 85748 Garching, Germany
1414institutetext: Department of Astronomy, School of Physics, Huazhong
University of Science and Technology, Wuhan, 430074, China

# The triple-peaked afterglow of GRB 210731A from X-ray to radio
frequencies††thanks: Table LABEL:tab:photometry is available in electronic
form at the CDS via anonymous ftp to cdsarc.cds.unistra.fr (130.79.128.5) or
via https://cdsarc.cds.unistra.fr/cgi-bin/qcat?J/A+A

S. de Wet Corresponding author 11<EMAIL_ADDRESS>T. Laskar 2233 P.J.
Groot 113344 F. Cavallaro 1155 A. Nicuesa Guelbenzu 66 S. Chastain 77 L. Izzo
88 A. Levan 33 D.B. Malesani 33991010 I.M. Monageng 1144 A.J. van der Horst 77
W. Zheng 1111 S. Bloemen 33 A.V. Filippenko 1111 D.A. Kann 1212 S. Klose 66
D.L.A. Pieterse 33 A. Rau 1313 P.M. Vreeswijk 33 P. Woudt 11 Z.-P. Zhu 1414

(Received ; accepted )

###### Abstract

Context. GRB 210731A was a long-duration ($T_{90}=22.5$ s) gamma-ray burst
discovered by the Burst Alert Telescope (BAT) aboard the Neil Gehrels Swift
observatory. Swift triggered the wide-field, robotic MeerLICHT optical
telescope in Sutherland; it began observing the BAT error circle 286 seconds
after the Swift trigger and discovered the optical afterglow of GRB 210731A in
its first 60-second $q$-band exposure. Multi-colour observations of the
afterglow with MeerLICHT revealed a light curve that showed three peaks of
similar brightness within the first four hours. The unusual optical evolution
prompted multi-wavelength follow-up observations that spanned from X-ray to
radio frequencies.

Aims. We present the results of our follow-up campaign and interpret our
observations in the framework of the synchrotron forward shock model.

Methods. We performed temporal and spectral fits to determine the spectral
regime and external medium density profile, and performed detailed multi-
wavelength theoretical modelling of the afterglow following the last optical
peak at ${\sim}0.2$ days to determine the intrinsic blast wave parameters.

Results. We find a preference for a stellar wind density profile consistent
with a massive star origin, while our theoretical modelling results in fairly
typical shock microphysics parameters. Based on the energy released in
$\gamma$ rays and the kinetic energy in the blast wave, we determine a low
radiative efficiency of $\eta\approx 0.02$. The first peak in the optical
light curve is likely the onset of the afterglow. We find that energy
injection into the forward shock offers the simplest explanation for the
subsequent light curve evolution, and that the blast wave kinetic energy
increasing by a factor of ${\sim}1000$ from the first peak to the last peak is
indicative of substantial energy injection. Our highest-likelihood theoretical
model over-predicts the 1.4 GHz flux by a factor of approximately three with
respect to our upper limits, possibly implying a population of thermal
electrons within the shocked region.

###### Key Words.:

Gamma-ray burst: individual: GRB 210731A

##  1 Introduction

Gamma-ray bursts (GRBs) are the most energetic explosions in the Universe,
with isotropic $\gamma$-ray energies of up to ${\sim}10^{55}$ erg (Kumar &
Zhang, 2015). Long-duration bursts are typically associated with the core
collapse of massive stars (Colgate, 1968; Woosley, 1993; van Paradijs et al.,
1997; Galama et al., 1998a; Woosley & Bloom, 2006), where the compact object
remnant acts as a central engine powering a collimated relativistic jet. In
the fireball shock model (Rees & Meszaros, 1992; Meszaros & Rees, 1993;
Mészáros & Rees, 1997; Sari et al., 1998), internal shocks within the
expanding ejecta are the source of the prompt $\gamma$-ray emission and the
interaction of the relativistic outflow with the circumburst medium (the
external forward shock) is the source of the afterglow emission.

Prior to the launch of the Neil Gehrels Swift Observatory (Gehrels et al.,
2004) in 2004, observations of afterglows showed broad agreement with the
basic external shock model (Galama et al., 1998b; Panaitescu & Kumar, 2001,
2002; Yost et al., 2003). Rapid X-ray and UV/optical follow-up of GRB triggers
with Swift have revealed rich features in early-time X-ray and optical light
curves that challenge the standard theory. Similar to the ‘canonical’ X-ray
light curve (Zhang et al., 2006; Nousek et al., 2006), optical light curves
have been decomposed into a number of distinct components that arise from
different emission sites and physical mechanisms (see the synthetic light
curve in Li et al., 2012). In addition to the normal and jet break decay
segments explained by the standard forward shock model, onset bumps, steep
decay segments, flares, and late-time re-brightenings have been observed in
optical afterglows. Early onset bumps with a smooth transition to the normal
decay segment are regarded as the onset of afterglow and were predicted within
the standard theory (Sari & Piran, 1999; Kobayashi & Zhang, 2007); steep decay
segments early on (some with an additional steep rise) have been attributed to
reverse shock emission (Mészáros & Rees, 1997; Sari & Piran, 1999; Zhang et
al., 2003; Yi et al., 2020); similar to X-ray light curves, shallow decay and
plateau segments and flares have been observed in a number of bursts (Mangano
et al., 2007; Greiner et al., 2009; Swenson et al., 2013); late-time re-
brightenings have also been observed in some optical afterglows (Nardini et
al., 2011; Liang et al., 2013), with a few having rarer simultaneous X-ray re-
brightenings (e.g. GRB 120326A; Melandri et al., 2014; Urata et al., 2014; Hou
et al., 2014; Laskar et al., 2015). Furthermore, small-scale bumps and wiggles
have been seen superposed over the larger-scale light curve features (e.g.
GRBs 021004 and 030329; Holland et al., 2003; Lipkin et al., 2004). Proposed
explanations for these additional features include inhomogeneities in the
circumburst medium, multiple-component jets, structured jets, varying
microphysical parameters, and energy injection into the forward shock (Lazzati
et al., 2002; Dai & Wu, 2003; Mundell et al., 2007; Racusin et al., 2008;
Filgas et al., 2011; Mészáros et al., 1998; Zhang & Mészáros, 2002a; Rossi et
al., 2002; Ioka et al., 2006; Fan & Piran, 2006; Granot et al., 2006; Sari &
Mészáros, 2000; Zhang & Mészáros, 2002b; Björnsson et al., 2004; Laskar et
al., 2015; Lin et al., 2018).

Here we report on multi-wavelength observations of the long-duration gamma-ray
burst GRB 210731A discovered by Swift (Gehrels et al., 2004). Our early-time
MeerLICHT optical observations show a complex light curve evolution with an
initial smooth bump followed by two further re-brightenings. We combine our
observations with X-ray and radio data that span 200 seconds to 118 days post-
trigger and interpret our observations in the traditional synchrotron forward
shock framework. We investigate the nature of the early optical light curve
evolution.

We adopt a $\Lambda$ cold dark matter cosmology with $\Omega_{m}=0.31$,
$\Omega_{\Lambda}=0.69$, and $H_{0}=68$ km s-1 Mpc-1 (Planck Collaboration et
al., 2016). All reported magnitudes are in the AB magnitude system unless
stated otherwise, and errors are reported at the $1\sigma$ level.

##  2 Observations

###  2.1 Prompt gamma-ray emission

The Swift Burst Alert Telescope (BAT; Barthelmy et al., 2005) was triggered by
GRB 210731A at 22:21:08 UT on 2021 July 31, with the mask-weighted 15–350 keV
light curve showing a single-pulse structure of duration $T_{90}=22.5\pm 2.8$
s (Stamatikos et al., 2021), making GRB 210731A a long-duration GRB under the
traditional $>2$ s duration limit. GRB 210731A triggered the Fermi Gamma-ray
Burst Monitor (GBM; Meegan et al., 2009; Lesage et al., 2021) one second
earlier than Swift/BAT, with the 10-1000 keV light curve showing a single
pulse with duration111Obtained from the online Fermi GBM Burst Catalog (von
Kienlin et al., 2020). $T_{90}=25.9\pm 5.3$ s, in agreement with the BAT
duration. The time-averaged spectra for both BAT and GBM were best-fitted with
a power law function and exponential high-energy cutoff with photon indices of
$-0.25\pm 0.59$ and $-0.1\pm 0.1$, and cutoff energies of $(107\pm 27)$ keV
and $(175\pm 11)$ keV, respectively (Stamatikos et al., 2021; Lesage et al.,
2021). The 10-1000 keV GBM fluence222See footnote 1. integrated over the burst
duration was $(3.05\pm 0.06)\times 10^{-6}$ erg cm-2. Using a measured
afterglow redshift of $z=1.2525$ obtained by X-Shooter on the Very Large
Telescope at 1.19 days post-trigger (Kann et al., 2021), this corresponds to
an isotropic-equivalent $\gamma$-ray energy of
$E_{\gamma,\mathrm{iso}}=(1.29\pm 0.03)\times 10^{52}$ erg.

We take the Swift/BAT trigger time as $T_{0}$ for this burst and reference all
future times with respect to this $T_{0}$.

###  2.2 X-ray observations

The Swift X-Ray Telescope (XRT; Burrows et al., 2005) started observing the
field of GRB 210731A 201 seconds post-trigger, finding a bright new X-ray
source consistent with the BAT position (Troja et al., 2021). The initial 62
seconds of data were obtained in windowed timing (WT) mode after which Swift
had to slew away. Data capture recommenced in photon counting (PC) mode at 3.3
hours post-trigger. We obtained the X-ray light curve and spectra from the
online Swift-XRT GRB Catalogue333The Burst Analyser for GRB 210731A is
available on the UK Swift Data Science Centre website. (Evans et al., 2007,
2009). The Burst Analyser (Evans et al., 2010) count-rate light curve showed
that the X-ray flux was decreasing rapidly during the WT-mode observations
with a spectrum that hardened from a photon index of $\Gamma_{\mathrm{X}}=3.2$
to 2.2 over 60 seconds. Once data capture resumed in PC mode at 10 ks, the
X-ray light curve was in a shallow decay phase before declining more steeply
at ${\sim}20$ ks post-trigger.

We fitted the PC-mode spectrum with a photoelectrically absorbed power-law
model (`tbabs*ztbabs*pow`) in Xspec version 12.12.0, fixing the source
redshift at 1.2525 and Galactic hydrogen column density at
$N_{\mathrm{H}}^{\mathrm{Gal}}=1.15\times 10^{21}$ cm-2 for consistency with
the online fit. The fitted spectrum was characterised by a photon index of
$\Gamma=2.00_{-0.11}^{+0.11}$ with a host galaxy column density of
$N_{\mathrm{H}}^{\mathrm{host}}=2.46_{-1.56}^{+1.99}\times 10^{21}$ cm-2 and
C-stat 182.1 for 213 degrees of freedom. There were insufficient photons in
the PC-mode light curve for time-resolved analysis and to test for spectral
evolution. We converted the PC-mode count-rate light curve to a 1 keV flux
density light curve using the spectral index from our PC-mode spectral fit of
$\beta_{\mathrm{X}}\equiv 1-\Gamma_{\mathrm{X}}\approx-1.00$ and the online
unabsorbed count-to-flux conversion factor of $4.36\times 10^{-11}$ erg cm-2
ct-1. We performed a similar procedure for the WT-mode data using the spectral
parameters in the automated fit on the Swift website, with a photon index of
3.07 and unabsorbed count-to-flux conversion factor of $4.69\times 10^{-11}$
erg cm-2 ct-1.

###  2.3 Optical/near-infrared observations

The fully robotic, 60 cm MeerLICHT optical telescope (Bloemen et al., 2016)
was automatically triggered by Swift/BAT and began observing the field of GRB
210731A 286 seconds after the BAT trigger, taking 60 second exposures in the
$u,g,r,i,z,$ and $q$ optical bands (where the $q$ band is roughly equivalent
to $g+r$), following the sequence $quqgqrqiqz$ in order to obtain high cadence
coverage in the wider and more sensitive $q$ band with quasi-simultaneous
multi-colour coverage of the evolving afterglow. Comparison of the first
$q$-band image with an existing MeerLICHT reference image revealed a new
transient candidate at
$\alpha=20^{\mathrm{h}}01^{\mathrm{m}}13.19^{\mathrm{s}}$,
$\delta=-28^{\mathrm{d}}03^{\mathrm{m}}40.10^{\mathrm{s}}$ (J2000). This
position was $0.3^{\prime\prime}$ away from the refined XRT position (D’Ai et
al., 2021), confirming the new source as the optical afterglow of GRB 210731A
(de Wet et al., 2021). These observations continued until the target set,
approximately 4.29 hours post-trigger. Four cycles of the same filter sequence
were obtained the following night in two time intervals separated by $\sim$2
hours. Since the afterglow was by this point in a declining phase and below
the 60 second single-exposure detection limit, repeated $q$-band exposures
were taken on the nights of 2021 August 2 and 3 in order to track the optical
light curve. The MeerLICHT pipeline (Vreeswijk et al., in prep) was used to
perform standard charge-coupled device (CCD) reduction tasks as well as
astrometry and point-spread function (PSF) photometry, producing a catalogue
file containing all 5$\sigma$ source detections for each image. For images
where the afterglow was fainter than 5$\sigma$ above the background we used
forced photometry to obtain magnitudes that were at least at the 3$\sigma$
level. Images from the night of 2021 August 1 onwards were co-added to produce
more significant detections or deeper upper limits.

The Swift UltraViolet and Optical Telescope (UVOT; Roming et al., 2005) took a
single 61.7 second exposure in the $white$ filter beginning 210.4 seconds
after the BAT trigger but did not continue observing the field of the GRB
until 3.27 hours after the trigger, whereafter it was observed with multiple
filters intermittently over the next five days (Troja et al., 2021; Kuin et
al., 2021). We performed aperture photometry on the Swift/UVOT data using
standard analysis tools from the HEASoft (Nasa High Energy Astrophysics
Science Archive Research Center (2014), Heasarc) Swift FTOOLS software package
(version 6.29c444Available at
https://heasarc.gsfc.nasa.gov/docs/software/lheasoft/). We extracted
magnitudes using the tool `uvotsource` with a $3.5^{\prime\prime}$ radius
aperture centred on the afterglow position, and a nearby background aperture
with a $10^{\prime\prime}$ radius. A total of 64 individual exposures in all
seven UVOT filters were taken over the course of the follow-up campaign. We
co-added exposures in the same filter with clear detections but taken close to
each other temporally using `uvotimsum` in order to produce more significant
detections, and once the afterglow became too faint to detect in individual
exposures we co-added images within wider time baselines in order to provide
the deepest limiting magnitudes.

The afterglow of GRB 210731A was observed simultaneously in the
$g^{\prime}r^{\prime}i^{\prime}z^{\prime}JHK$ bands with the Gamma-Ray Burst
Optical Near-Infrared Detector (GROND; Greiner et al., 2008) mounted at the
2.2 m MPG telescope at the European Southern Observatory (ESO) La Silla
Observatory in Chile. The afterglow was clearly detected in all bands in the
first epoch of observations taken 4.2 hours after the GRB trigger (Nicuesa
Guelbenzu et al., 2021a). A further three epochs were obtained at 1.225,
2.214, and 5.253 days post-trigger (Nicuesa Guelbenzu et al., 2021b). We also
obtained deep host-galaxy observations at 285 days that yielded detections in
the $g^{\prime}$ and $r^{\prime}$ bands, and which we regard as the host-
galaxy flux. The multi-colour GROND data were analysed through standard PSF
photometry using DAOPHOT (Stetson, 1987) and ALLSTAR tasks of IRAF (Tody,
1993), in a similar way to the procedure described in Krühler et al. (2008).
The optical data were calibrated against the Pan-STARRS catalogue555See
http://archive.stsci.edu/panstarrs/search.php. (Chambers et al., 2016), while
for the near-infrared (NIR) bands, photometric calibration was performed
against the 2MASS catalogue (Skrutskie et al., 2006), resulting in a typical
absolute accuracy of 0.04 mag in $g^{\prime}r^{\prime}i^{\prime}z^{\prime}$,
0.06 mag in $JH$ and 0.08 mag in $K$.

The 76 cm Katzman Automatic Imaging Telescope located at the Lick Observatory
(KAIT; Filippenko et al., 2001) obtained $20\times 60$ second exposures in the
clear band (similar to $R$; see Li et al., 2003), starting $\sim$9.04 hours
after the BAT trigger (Zheng et al., 2021). All images were reduced and co-
added using a custom pipeline (Ganeshalingam et al., 2010; Stahl et al.,
2019), whereafter PSF photometry was performed on the co-added image using
DAOPHOT. Several nearby stars were chosen from the Pan-STARRS1 catalogue for
flux calibration, with their magnitudes transformed into Landolt magnitudes
following Eq. 6 of Tonry et al. (2012). The optical afterglow of GRB 210731A
was clearly detected in the co-added image.

Images in the SDSS $g$, $r$, and $z$ filters were obtained at a single epoch
1.18 days post-trigger with the acquisition camera of the X-shooter
spectrograph, mounted on the ESO Very Large Telescope (VLT) UT3 (Melipal).
Reduction was carried out using standard procedures. For the $z$-band image, a
fringe correction was applied, using a template fringe pattern provided by the
observatory. We also observed the afterglow at two epochs in the $r$ and $z$
bands with the Nordic Optical Telescope (NOT) equipped with the ALFOSC imager.
The images were reduced following standard procedures including subtraction of
a master bias and correction with sky flats. Magnitudes were measured using
aperture photometry, and photometric calibration was carried out against the
Pan-STARRS catalogue.

We show all UV/optical/NIR photometry separated by instrument and filter in
Fig. 1.

$\displaystyle{10^{2}}$$\displaystyle{10^{3}}$$\displaystyle{10^{4}}$$\displaystyle{10^{5}}$Time
since trigger
(s)$\displaystyle{10^{-3}}$$\displaystyle{10^{-2}}$$\displaystyle{10^{-1}}$$\displaystyle{10^{0}}$$\displaystyle{10^{1}}$Time
since trigger
(days)$\displaystyle{10}$$\displaystyle{12}$$\displaystyle{14}$$\displaystyle{16}$$\displaystyle{18}$$\displaystyle{20}$$\displaystyle{22}$$\displaystyle{24}$$\displaystyle{26}$$\displaystyle{28}$$\displaystyle{30}$AB
magnitudeMeerLICHT/$\displaystyle u+2$MeerLICHT/$\displaystyle
g+1$MeerLICHT/$\displaystyle q$MeerLICHT/$\displaystyle
r-1$MeerLICHT/$\displaystyle i-2$MeerLICHT/$\displaystyle
z-3$UVOT/$\displaystyle uvw2+5$UVOT/$\displaystyle uvm2+4$UVOT/$\displaystyle
uvw1+3$UVOT/$\displaystyle u+2$UVOT/$\displaystyle b+1$UVOT/$\displaystyle
v+0.5$UVOT/$\displaystyle white+2$GROND/$\displaystyle
g^{\prime}+1$GROND/$\displaystyle r^{\prime}-1$GROND/$\displaystyle
i^{\prime}-2$GROND/$\displaystyle z^{\prime}-3$GROND/$\displaystyle
J-4$GROND/$\displaystyle H-5$GROND/$\displaystyle K-6$VLT/$\displaystyle
g+1$VLT/$\displaystyle r-1$VLT/$\displaystyle z-3$NOT/$\displaystyle
r-1$NOT/$\displaystyle z-3$KAIT/$\displaystyle clear-1$ Figure 1: Combined
UV/optical/NIR photometry of GRB 210731A. We only show detections where all
magnitudes are in the AB system and have not been corrected for Galactic
extinction. Times are relative to the Swift/BAT trigger time.

###  2.4 Radio observations

We obtained three epochs of radio continuum observations with the MeerKAT
radio telescope (Jonas & MeerKAT Team, 2016) in the L band (1.4 GHz) through
director’s discretionary time (DDT) proposal DDT-20120810-SD-01 (PI de Wet).
Each observation had a total integration time on source of 0.78 hours, using
J1939–6342 as the flux and bandpass calibrator and J1924–2914 as the gain
calibrator. All data were reduced using the oxkat pipeline666See
https://github.com/IanHeywood/oxkat/blob/master/README.md and references
therein.  (Heywood, 2020). No radio afterglow was detected at 1.4 GHz across
the three epochs at 10.8, 34.1 and 59.7 days post-trigger. The RMS noise was
$\approx 14$ $\mathrm{\SIUnitSymbolMicro}$Jy at the $1\sigma$-level in each
image. We take upper limits on the afterglow flux as three times the RMS
noise.

Radio continuum observations were also obtained with the Karl G. Jansky Very
Large Array (JVLA; Perley et al., 2011) in the C and X bands (centred on 6 and
10 GHz) through DDT proposals 21B-333 and 21B-342 (PI de Wet) at four epochs
spanning 18.2 to 118 days post-trigger. The total integration time per
observation was 0.44 hours in each band, with 3C286 used as the flux and
bandpass calibrator and J1924–2914 as the complex gain calibrator. We
performed preliminary imaging on the pipeline-calibrated measurements sets
using standard Common Astronomy Software Applications (CASA; McMullin et al.,
2007) procedures and detected the radio afterglow to GRB 210731A in all four
epochs at 10 GHz and in all but the first epoch at 6 GHz.

The first epoch National Radio Astronomy Observatory (NRAO) -calibrated
measurement set failed to pass internal quality thresholds for science
usability so we chose to calibrate manually from the raw data to obtain more
accurate flux measurements. We used CASA version 5.8.0 and performed imaging
with the task `tclean` and flux measurements with the task `imfit`. We
obtained satisfactory results with our X-band calibration, but the C-band
calibration contained persistent phase errors as a result of de-correlation
problems during observing, and the measured flux of all sources in the field
was substantially lower than in subsequent epochs. Despite these problems,
imaging on the re-calibrated data showed a radio source at the afterglow
position, in contrast with preliminary imaging. We therefore adopted the
following approach to calculate the flux of the afterglow. We identified six
point-like sources present at all four epochs in the C band and measured their
fluxes. For each source a flux correction factor could then be calculated
between the first epoch flux and the flux in each subsequent epoch. The
correcting factors ranged from 1.2 to 2.2, with a mean value of 1.77 and
standard deviation of 0.26. No obvious trends in the correcting factor were
found as a function of source brightness or offset from the image centre
across all epochs, so we took the mean correcting factor as the flux
correcting factor to apply to the measured afterglow flux in the first epoch
image. We incorporate the standard deviation of the correction factor as an
additional source of systematic uncertainty in the flux measurement from the
first epoch in the C band.

The third epoch X-band measurement set also had major phase issues, so we
performed the same procedure as for the C-band first epoch data, calibrating
the raw data and determining a mean flux correcting factor to apply to our
afterglow measurement. Only two sources were used in determining the mean
correcting factor as the X-band images had far fewer sources than the C-band
images. The correcting factors ranged from 1.3 to 1.6, with a mean value of
1.47 and standard deviation of 0.1.

All X-ray, optical, and radio flux measurements associated with GRB 210731A
are presented in Table LABEL:tab:photometry.

##  3 Afterglow temporal and spectral analysis

We interpret our combined multi-wavelength data in the framework of
synchrotron radiation emitted by electrons accelerated to a power-law
distribution in energy behind the forward shock, with
$N(\gamma)\propto\gamma^{-p}$ for $\gamma>\gamma_{\mathrm{min}}$, with
$\gamma_{\mathrm{min}}$ being the minimum Lorentz factor of electrons in the
distribution and $p$ being the electron energy index, which we assume to be
bounded between 2 and 3, though values of less than 2 have been suggested in
the literature (Dai & Cheng, 2001; Panaitescu & Kumar, 2002). The synchrotron
spectra are characterised by power-law segments that join at a number of break
frequencies, namely the synchrotron self-absorption frequency $\nu_{sa}$, the
characteristic synchrotron frequency $\nu_{m}$ corresponding to emission from
$\gamma_{\rm min}$ electrons, and the cooling frequency $\nu_{c}$. The
orderings of the spectral breaks depend on the hydrodynamic evolution of the
forward shock, which is described by the Blandford & McKee (1976) spherical
self-similar solution of an adiabatic relativistic blast wave expanding into a
cold medium with a circumburst density profile varying as a power-law with
radius: $n(r)=n_{0}r^{-k}$. We consider two density profiles: the constant
$k=0$ case corresponding to an interstellar-medium-like density profile; and
the $k=2$ case corresponding to a stellar wind from a massive star progenitor.
The synchrotron forward shock model is described in Sari et al. (1998),
Chevalier & Li (2000), and Granot & Sari (2002), and we follow the convention
$F_{\nu}(t)\propto t^{\alpha}\nu^{\beta}$ throughout.

###  3.1 Optical/X-ray temporal evolution

The most striking feature of our GRB 210731A dataset is the three peaks in our
early-time optical data (see Fig. 1). To characterise this light curve
further, we created a composite $R$-band light curve by combining our $q$-,
$r$-, and $R$-band data since they are the most well-sampled optical bands and
also have similar central wavelengths. We also included the KAIT clear flux
measurement since it is calibrated to the $R$ band. We used an optical
spectral index of $\beta_{\mathrm{opt}}=-0.81\pm 0.05$ derived from the first
GROND epoch (see Sect. 3.2) to transform the data to an $R$-band central
wavelength of 700 nm777For direct comparison with the sample in Li et al.
(2012) and Liang et al. (2013).. The composite $R$-band light curve (see Fig.
2) exhibits three distinct peaks occurring within the first 0.3 days of the
GRB trigger, each with rising and decaying segments of varying steepness and
smoothness. We investigate the nature of the optical peaks in Sect. 5. After
the last peak at ${\sim}0.2$ days, the light curve entered a final declining
phase until the last optical observation at 6.2 days post-trigger.

We follow two approaches to fit the data: first we fitted a single power-law
to each rising and decaying segment888We determine the boundary between each
segment by eye. These are shown as vertical dotted lines in Fig. 2. directly
in order to get an indication of the steepness of each segment (we use these
in Sect. 5.1.5); then we performed an empirical fit as in Li et al. (2012) by
decomposing the light curve into separate components, each of which may arise
from different emission sites or physical mechanisms. Since we have three
clear peaks in our light curve, we employed a model that comprises the sum of
three broken power-law (BPL) components (Beuermann et al., 1999; Price et al.,
2001; Zeh et al., 2004), each characterised by a normalising flux level,
$F_{0}$, rise and decay indices, $\alpha_{1}$ and $\alpha_{2}$, break time,
$t_{\mathrm{b}}$ , and break smoothness, $\omega$, according to

$F(t)=F_{0}\left[\left(\frac{t}{t_{\mathrm{b}}}\right)^{-\alpha_{1}\omega}+\left(\frac{t}{t_{\mathrm{b}}}\right)^{-\alpha_{2}\omega}\right]^{-1/\omega}.$
(1)

If $\alpha_{1}$ is positive and $\alpha_{2}$ is negative, the light curve
peaks at a time, $t_{\mathrm{p}}$, between rising and decaying segments;
$t_{\mathrm{b}}=t_{\mathrm{p}}$ in Eq. 1. We also include a constant term to
account for the host-galaxy $r^{\prime}$-band brightness of $24.7\pm 0.2$ mag
measured at 285 days by GROND. Considering values of 1, 3, 5 and 9 for
$\omega$, we find that a smoother break with a value of 1 produces a fit with
a $\chi_{\mathrm{r}}^{2}$ value closer to 1. Our fit allows us to compare the
temporal evolution in each optical band, which we do in Sect. 3.2.

Examining the X-ray light curve, the WT-mode data within the first 300 seconds
shows a steep decline with $\alpha_{\mathrm{X}}=-3.52\pm 0.36$ and a photon
index that hardened from 3.2 to 2.2, as taken from the online Burst Analyser.
The most likely explanation is high latitude prompt emission, as the temporal
and spectral indices agree broadly with the $\alpha=-2+\beta$ curvature effect
relation (Kumar & Panaitescu, 2000; Willingale et al., 2010). It is
unfortunate that we have no X-ray data during the time of the first two
optical peaks, rendering a direct comparison between the two bands unfeasible.
There is, however, X-ray data from ${\sim}0.13$ days onwards starting when the
optical light curve was rising to its final peak. The X-ray light curve
started in a shallow decay or plateau phase before steepening, which coincided
with the final decaying phase in the optical. We fit a BPL according to Eq. 1
to determine the break time and temporal slopes, fixing the break smoothness
at 1, 3, 5, or 9. Each fit had a similar reduced $\chi^{2}_{r}$ value
(${\sim}0.4$); therefore, we employed a break smoothness of $\omega=1$ to
match the value used in the optical fit, though this results in a pre-break
index that is poorly constrained. The $R$-band and 1 keV light curves are
presented in Fig. 2 along with their fits, and we present the results of the
fits in Table 1.

$\displaystyle{10^{3}}$$\displaystyle{10^{4}}$$\displaystyle{10^{5}}$Time since trigger (s)$\displaystyle{10^{0}}$$\displaystyle{10^{1}}$$\displaystyle{10^{2}}$$\displaystyle{10^{3}}$$\displaystyle F_{\nu}$ ($\mathrm{\SIUnitSymbolMicro}\mathrm{J}\mathrm{y}$)$\displaystyle t^{1.16}$$\displaystyle t^{-0.77}$$\displaystyle t^{0.64}$$\displaystyle t^{-0.11}$$\displaystyle t^{0.48}$$\displaystyle t^{-1.65}$$\displaystyle t^{-3.52}$Optical $\displaystyle R$-bandBPL componentCombined fitXRT 1 keV $\displaystyle\times 10^{1.5}$BPL fit$\displaystyle{10^{-2}}$$\displaystyle{10^{-1}}$$\displaystyle{10^{0}}$Time since trigger (days)$\displaystyle{0.5}$$\displaystyle{1.0}$$\displaystyle{1.5}$Ratio$\displaystyle{14}$$\displaystyle{15}$$\displaystyle{16}$$\displaystyle{17}$$\displaystyle{18}$$\displaystyle{19}$$\displaystyle{20}$$\displaystyle{21}$$\displaystyle{22}$$\displaystyle{23}$$\displaystyle{24}$$\displaystyle{25}$AB magnitude Figure 2: Composite $R$-band light curve and X-ray 1 keV light curve for GRB 210731A. For each rising and decaying segment of the optical light curve, we show the power-law slope ($t^{\alpha}$) as an indicator of the steepness of the light curve between each pair of adjacent vertical dotted lines. We also show the fit comprising the sum of three BPL components and a constant term equal to the $r^{\prime}$-band host galaxy flux measured by GROND at 285 days (solid red line), along with each individual BPL component (dashed blue lines). For the X-ray light curve, we indicate the steepness of the WT-mode segment and we show the BPL fit to the PC-mode data as a dashed-dotted red line. The results of the X-ray and optical fits are presented in Table 1. The ratio of observed flux to fitted flux is shown in the lower panel. Table 1: Parameters derived from fits to the composite $R$-band light curve and X-ray 1 keV light curve, as shown in Fig. 2. The $R$-band light curve was fitted with the sum of three BPLs, while the X-ray light curve was fitted with a single BPL.  | $\alpha_{1}$ | $\alpha_{2}$ |  $t_{\mathrm{p}}$ (days) |  $\chi^{2}$/dof
---|---|---|---|---
Optical BPL 1 | $1.39\pm 0.36$ | $-2.58\pm 0.75$ | $0.0088\pm 0.0013$ | 1.32
Optical BPL 2 | $1.19\pm 0.62$ | $-5.16\pm 2.58$ | $0.066\pm 0.007$ | -
Optical BPL 3 | $0.99\pm 0.16$ | $-1.84\pm 0.04$ | $0.27\pm 0.01$ | -
X-ray 1 keV BPL | $0.44\pm 0.62$ | $-1.69\pm 0.19$ | $0.26\pm 0.09$ | 0.39

###  3.2 Achromatic optical/X-ray spectral evolution

We now investigate if there is evidence for spectral evolution in the optical
data. In Fig. 3 we show the optical spectral energy distributions (SEDs)
formed using data from the first three of five GROND epochs corrected for a
Galactic extinction of $A_{\mathrm{V}}=0.24$ mag in the direction of the GRB
(Schlafly & Finkbeiner, 2011). We fitted the data with power-laws in
frequency, with the first epoch yielding a spectral index of
$\beta_{\mathrm{opt}}=-0.81\pm 0.05$. There does not appear to be substantial
spectral evolution between the first two epochs at 0.184 and 1.225 days,
particularly in the optical $g^{\prime}$, $r^{\prime}$, $i^{\prime}$, and
$z^{\prime}$ bands. It is unclear why there is excess emission in the near
infrared $J$, $H$, and $K$ bands during the second and third epochs. A
possible explanation is that there is contaminating emission from the host
galaxy, in which case we would expect the light curves to flatten towards a
constant value. The observed decline to below detection levels at 5.25 and 285
days appears to rule out this possibility. It could also be the case that
there is an additional unaccounted-for source of systematic photometric error.

4008001200160020002400Wavelength
(nm)$\displaystyle{10^{0}}$$\displaystyle{10^{1}}$$\displaystyle{10^{2}}$$\displaystyle{10^{3}}$$\displaystyle
F_{\nu}$ ($\mathrm{\SIUnitSymbolMicro}$Jy)0.18 days,
$\displaystyle\beta=-0.81\pm 0.05$1.23 days, $\displaystyle\beta=-0.79\pm
0.06$2.21 days, $\displaystyle\beta=-0.60\pm 0.14$5.25 days285 days, host
galaxy$\displaystyle{17}$$\displaystyle{18}$$\displaystyle{19}$$\displaystyle{20}$$\displaystyle{21}$$\displaystyle{22}$$\displaystyle{23}$$\displaystyle{24}$$\displaystyle{25}$AB
magnitude Figure 3: Power-law fits to the GROND optical/NIR data at three
epochs, corrected for a Galactic extinction of $A_{V}=0.24$ mag. The
uncorrected magnitudes are shown in a lighter shade below each corrected data
point. We also show the fourth epoch, which only had detections in the
$g^{\prime}$ and $r^{\prime}$ bands, as well as the detections (shown as
diamonds) and limits from deep observations of the host galaxy at 285 days.
Upper limits are shown as upside-down triangles.

For our early-time data, we took our composite $R$-band light curve fit (Fig.
2) and fitted the flux in each of the optical bands with this model, which
amounts to shifting the $R$-band fit light curve vertically until it fits the
data in a given band. The spectral slope in the blue and UV bands ($u$ through
$uvw2$) was steeper than in the optical owing to Galactic extinction and
damping by Ly$\alpha$ absorption at the redshift of the burst. We therefore
shifted the MeerLICHT $u$-band data to the UVOT $u$ band using an approximate
spectral index of $\beta_{\mathrm{UV}}\approx-4$ measured from the UVOT/$u$,
$uvw1$, $uvm1$, and $uvw2$ data. Figure 4 shows that the data in each of the
UV and optical bands is reasonably well fitted by the $R$-band light curve.
Deviations from the fit at earlier times ($<0.2$ days) are visible in the $u$,
$g$, $i$, and $z$ bands but they do not appear statistically significant -
only 7.5% of the UV/optical/NIR data points deviate by $2\sigma$ or more from
each model fit. Overall the optical evolution appears achromatic and
consistent with a constant optical spectrum, which points towards a
hydrodynamical rather than spectral origin to the complex early-time optical
light curve.

As mentioned in Sect. 2.2, insufficient X-ray photons were collected during
the PC-mode observations to create time-sliced spectra. The photon index from
the online Burst Analyser was fairly constant during PC mode, however, with a
mean value of $\Gamma_{\mathrm{X}}=1.84\pm 0.27$. This is indicative of
insubstantial spectral evolution in the X-ray band during PC mode. The X-ray
spectral index of $\beta_{\mathrm{X}}=-1.00\pm 0.11$ from our spectral fit in
Sect. 2.2 (with a mean photon arrival time of 0.42 days) is steeper than the
first GROND epoch (at 0.18 days) spectral index of
$\beta_{\mathrm{opt}}=-0.81\pm 0.05$, by $\Delta\beta=0.19\pm 0.12$. This
difference is suggestive of a spectral break lying between the X-ray and
optical bands. If the break was the cooling break, we would expect
$\Delta\beta=0.5$, however. This discrepancy could be explained by the fact
that the cooling break is always smooth (Uhm & Zhang, 2014) or that $\nu_{c}$
may lie near to either the optical or X-ray bands. We investigate this further
in Sect. 3.3.

$\displaystyle{10^{3}}$$\displaystyle{10^{4}}$$\displaystyle{10^{5}}$Time
since trigger
(s)$\displaystyle{10^{-2}}$$\displaystyle{10^{-1}}$$\displaystyle{10^{0}}$$\displaystyle{10^{1}}$$\displaystyle{10^{2}}$$\displaystyle{10^{3}}$$\displaystyle{10^{4}}$$\displaystyle
F_{\nu}$ ($\mathrm{\SIUnitSymbolMicro}$Jy)XRT 1 keV$\displaystyle uvw2\times
10^{-0.8}$$\displaystyle uvm2\times 10^{-0.7}$$\displaystyle uvw1\times
10^{-0.6}$$\displaystyle u\times 10^{-0.5}$$\displaystyle b\times
10^{-0.3}$$\displaystyle g\times 10^{-0.1}$$\displaystyle v\times
10^{0.1}$$\displaystyle q\times 10^{0.3}$$\displaystyle r\times
10^{0.5}$$\displaystyle i\times 10^{0.7}$$\displaystyle z\times
10^{0.9}$$\displaystyle J\times 10^{1.0}$$\displaystyle H\times
10^{1.1}$$\displaystyle K\times
10^{1.2}$$\displaystyle{10^{-2}}$$\displaystyle{10^{-1}}$$\displaystyle{10^{0}}$Time
since trigger
(days)$\displaystyle{0}$$\displaystyle{1}$$\displaystyle{2}$Ratio Figure 4:
Fits to the data in each optical band using the best-fit model to the
composite $R$-band light curve in Fig. 2. We also fitted this model to the
X-ray PC-mode data. The lower panel presents the ratio of measured flux to
model flux.

###  3.3 Closure relation analysis

Within the synchrotron forward shock model the ‘closure relations’ relate the
spectral index $\beta$ with the temporal index $\alpha$ in the convention
$F_{\nu}\propto t^{\alpha}\nu^{\beta}$ and depend on the physical regime,
spectral regime, and external medium density profile (Zhang & Mészáros, 2004;
Zhang et al., 2006). The closure relations can be adapted to describe a
variety of alternative scenarios to the standard self-similar deceleration
phase, including the post-jet break scenario, whether there is energy
injection involved, the reverse shock crossing phase, or the Newtonian/non-
relativistic phase (see the comprehensive review in Gao et al., 2013). It is
important to note that the spectral breaks are inherently smooth, so that a
transitioning spectral break or spectral break near to an observing band may
define a ‘grey zone’ where the $\alpha-\beta$ relations are not strictly
satisfied (Zhang et al., 2006; Uhm & Zhang, 2014).

If we assume that the final declining phase of our optical and X-ray light
curves arises from standard forward shock emission in the slow cooling regime
($\nu_{m}<\nu_{c}$), which is usually the case at later times, we can
determine which spectral regime and density profile best fits our data by
performing a closure relation analysis. Our X-ray and optical data both have
negative spectral slopes, with fluxes that decline with increasing frequency.
There are two spectral regimes that give rise to negative spectral slopes:
$\nu_{c}<\nu$ (Regime I) with $\beta=-p/2$, or $\nu_{m}<\nu<\nu_{c}$ (Regime
II) with $\beta=(1-p)/2$. For the optical spectral index of
$\beta_{\mathrm{opt}}=-0.81\pm 0.05$ derived from the first GROND epoch we
would have $p=1.62\pm 0.10$ or $p=2.62\pm 0.10$ in Regime I and II,
respectively. The decay index for both a wind or interstellar medium (ISM)
environment in Regime I is $\alpha=(2-3p)/4$ resulting in $\alpha=-0.72\pm
0.08$ with $p=1.62\pm 0.10$, too shallow for the observed decay rate of
$\alpha_{\mathrm{opt}}=-1.84\pm 0.04$ from our three BPL-component fit. In
Regime II, we have $\alpha=3(1-p)/4$ or $\alpha=(1-3p)/4$ for an ISM or wind
density profile, respectively. With $p=2.62\pm 0.10$ we get $\alpha=-1.22\pm
0.08$ or $\alpha=-1.72\pm 0.08$ for the ISM or wind profiles. Clearly, the
observed optical temporal index of $\alpha_{\mathrm{opt}}=-1.84\pm 0.04$ is
most consistent with the optical spectral index in Regime II for a wind
profile.

In a wind medium the cooling break moves to higher frequencies as $t^{1/2}$,
with light curves that are shallower by $\Delta\alpha=0.25$ above $\nu_{c}$.
With the cooling break between the optical and X-rays one could therefore
expect the X-ray light curve to decline more slowly than the optical light
curve. Our fits to the X-ray and optical light curves in Sect. 3.1 result in
$\Delta\alpha=0.15\pm 0.19$, which agrees within uncertainties with the
predicted difference of $\Delta\alpha=0.25$. We also see, however, that this
$\Delta\alpha$ value is consistent with a zero difference within $1\sigma$ and
is supported by the fact that the composite $R$-band light curve provides a
good fit to the X-ray light curve, as shown in Fig. 4. It is therefore not
possible to conclusively say whether $\nu_{c}$ lies between the optical and
X-ray bands from our data, though we note that the slightly different temporal
and spectral (see Sect. 3.2) indices between the two bands does hint at this
possibility. In summary, our optical and X-ray data can be accommodated within
the standard closure relations in a wind medium.

###  3.4 Broadband SED evolution

The optical and X-ray data alone can only place weak constraints on the
location of $\nu_{m}$ and the peak flux of the evolving synchrotron spectrum.
Our late-time radio observations, which probe the low-frequency end of the
synchrotron spectrum, can provide valuable constraints on the location of
$\nu_{m}$ and the peak flux, and can therefore lead to an estimation of the
intrinsic blast wave parameters (see Sect. 4).

Our three epochs of L-band (at 1.4 GHz) observations all yielded non-
detections, whereas our four epochs of C-band and X-band data yielded
detections spanning 18 to 118 days post-trigger. The flux across the first two
epochs at 18.2 and 34.2 days was fairly constant in both the C and X bands,
which is consistent with the predicted evolution of $t^{0}$ for the spectral
ordering $\nu_{sa}<\nu<\nu_{m}$ in a wind medium undergoing slow cooling. The
spectral slope between the first epoch C- and X-band detections of
$\beta_{6-10\leavevmode\nobreak\ \mathrm{GHz}}=0.57\pm 0.27$ is also close to
the predicted value of $\nu^{1/3}$ for the same spectral segment. The
subsequent decline in flux across the last two epochs can be interpreted as
the passage of $\nu_{m}$ through 6 and 10 GHz, whereafter both bands lie in
Regime II of the synchrotron spectrum in which the flux declines with time and
the spectrum declines with increasing frequency. This is seen in our last two
epochs where the C-band flux is in fact brighter than the X-band flux. At 34.2
days we have quasi-simultaneous flux measurements at 1.4, 6 and 10 GHz (see
Fig. 6). The spectral index between the C and X bands is
$\beta_{6-10\leavevmode\nobreak\ \mathrm{GHz}}=0.69\pm 0.23$, which is closer
to the optically thin spectral slope of $\nu^{1/3}$ than the synchrotron self-
absorbed slope of $\nu^{2}$. Based on the C-band detection and L-band upper
limit, we place a lower limit on the spectral slope of
$\beta_{1.4-6\leavevmode\nobreak\ \mathrm{GHz}}>0.84$. It therefore may be the
case that synchrotron self-absorption is responsible for the non-detections at
1.4 GHz, since our L-band limit is consistent with a $\nu^{2}$ spectrum. In
that case, the self-absorption frequency could lie between 1.4 and 10 GHz at
34.2 days.

In a wind medium, $\nu_{m}$ moves to lower frequencies as $t^{-3/2}$ with the
corresponding peak flux of the synchrotron spectrum declining as $t^{-1/2}$
for the spectral break ordering $\nu_{sa}<\nu_{m}<\nu_{c}$. From our GROND SED
at ${t=0.184}$ days (Fig. 3), we know that $\nu_{m}$ lies below the $K$ band
with a peak flux greater than the measured $K$-band flux of 555
$\mathrm{\SIUnitSymbolMicro}$Jy. If we assume that $\nu_{m}$ passes through
the radio X band (10 GHz) at 34.2 days with a peak flux of ${\sim}250$
$\mathrm{\SIUnitSymbolMicro}$Jy, we would have expected $\nu_{m}$ to be at
$1.48\times 10^{12}$ Hz with a peak flux of 1320
$\mathrm{\SIUnitSymbolMicro}$Jy at 1.22 days, the time of our second GROND
epoch. This frequency lies below the $K$-band frequency of $1.38\times
10^{14}$ Hz, as expected. The spectral index between this expected peak flux
value and the measured $K$-band flux value at 1.22 days results in
$\beta\approx-0.7$, which is in agreement with the measured optical spectral
index of $\beta_{\mathrm{opt}}=-0.79\pm 0.06$ at this time. We also note that
the X-ray to optical $R$-band spectral index at ${\sim}0.38$ days
($\beta_{\mathrm{opt,X}}\approx-0.95$) is between the X-ray-only (
$\beta_{\mathrm{X}}=-1.00\pm 0.11$) and optical-only index
($\beta_{\mathrm{opt}}=-0.81\pm 0.05$), demonstrating that both observing
bands can be accommodated via a forward shock model.

With these basic considerations, we attempted to find a first-guess set of
blast wave parameters that can explain our data. We have the following assumed
constraints: (i) $\nu_{m}$ passes through 10 GHz at 34.2 days; (ii) the
corresponding peak flux at this frequency and time is ${\sim}250$
$\mathrm{\SIUnitSymbolMicro}$Jy; (iii) $\nu_{c}$ lies between the optical and
X-ray bands at early times (i.e. $\nu_{c}\approx 10^{17}$ Hz at 0.3 days); and
(iv) $\nu_{sa}$ lies between 1.4 and 10 GHz at 34.2 days.

With these four constraints we can attempt to solve the system of four
equations describing the locations of the spectral breaks and their
corresponding flux densities in a wind medium, given in Table 2 of Granot &
Sari (2002). Our solution given the above constraints results in an unphysical
value of $\epsilon_{e}>1$, which is driven primarily by the requirement that
the self-absorption frequency lies between 1.4 and 10 GHz at 34.2 days.
Lowering $\nu_{sa}$ to a frequency of ${\sim}10^{7}$ Hz results in a physical
solution for all of the blast wave parameters. Our L-band limits therefore
pose a challenge to the interpretation of our multi-wavelength afterglow data.
We return to this point in Sect. 5.2.

###  3.5 Early jet-break scenario

Our optical light curve during the final declining phase had a temporal index
of $\alpha_{\mathrm{opt}}=-1.84\pm 0.04$ from the combined fit or
$\alpha_{\mathrm{opt}}=-1.65\pm 0.04$ from the direct fit to the late-time
data only, which is steep for normal pre-jet break evolution (see Fig. 4 in
Wang et al., 2018). An alternative scenario to explain the steep final
declining phase in the optical and X-ray light curves is post-jet break decay.
If the jet break is due to a purely geometric edge effect (Panaitescu et al.,
1998), the light curves within all spectral regimes should steepen by
$t^{-3/4}$ for the ISM case and $t^{-1/2}$ for the wind case once the ejecta
has slowed down such that the relativistic beaming angle $1/\Gamma$ is greater
than the jet half-opening angle $\theta_{j}$, assuming a top-hat jet. Sideways
expansion of a conical jet would result in a steeper jet break decay of
approximately $t^{-p}$ in Regimes I and II for an ISM (Rhoads, 1999; Sari et
al., 1999).

Considering the edge effect only, a jet break will not change the temporal
evolution of the synchrotron spectral break frequencies. If an early jet break
occurred we would expect our radio data to show declining light curves that
decay as $t^{-1/2}$ in a wind medium under the assumption that
$\nu_{sa}<\nu_{6,10\leavevmode\nobreak\ \mathrm{GHz}}<\nu_{m}$. Taking into
account sideways expansion, the evolution of the break frequencies is altered,
though we would still expect declining light curves at radio frequencies (Sari
et al., 1999). The rising radio light curves in the C and X bands until
${\sim}34$ days are therefore inconsistent with an early jet break. This
implies that the steep optical and X-ray decline is normal pre-jet break decay
in a wind medium, supporting our analysis in Sect. 3.3.

##  4 Theoretical modelling

We have shown in the previous sections that our X-ray, optical, and radio data
after the last optical peak can be reconciled within the synchrotron forward
shock model in a wind medium with $p\approx 2.6$ if we exclude our L-band
limits. We now proceed to find a set of blast wave parameters that can
describe our data by employing the smoothly connected power-law spectra
outlined in Granot & Sari (2002) and fitting for the forward shock parameters
$p$, $\epsilon_{e}$, $\epsilon_{B}$, $A_{\star}$, and $E_{\mathrm{K,iso}}$,
where $E_{\mathrm{K,iso}}$ is the total isotropic-equivalent kinetic energy in
the blast wave; $\epsilon_{e}$ and $\epsilon_{B}$ are the fractions of shock
internal energy given to the electrons and the magnetic fields, respectively;
and $A_{\star}=A/(5\times 10^{11}\mathrm{\leavevmode\nobreak\
g\leavevmode\nobreak\ cm}^{-1})$ is the wind density parameter as defined in
Chevalier & Li (2000). We correct the data for Galactic extinction with
$A_{V}=0.24$ mag.

We perform a Markov chain Monte Carlo (MCMC) analysis with emcee (Foreman-
Mackey et al., 2013) using 512 walkers and 2000 steps, discarding the initial
250 steps as burn-in. The details of our implementation are described in
Laskar et al. (2013, 2014). The host galaxy extinction, $A_{\mathrm{V,host}}$,
is a free parameter in our model. We include the effects of Klein-Nishina (KN)
corrections (G. McCarthy & T. Laskar in prep) using prescriptions from Nakar
et al. (2009) and Jacovich et al. (2021). We used uniform, uninformative
priors flat in log space, and restricted $\epsilon_{e}+\epsilon_{B}<1$,
although this limit is not reached. We did not include data before the
inferred time of the last optical peak at ${\sim}0.3$ days in the modelling,
and discuss these data further in Sect. 5. We also did not include the MeerKAT
$1.4$ GHz observations, as we do not expect these to be fit with this model
(Sect. 3.4). We present theoretical modelling including the L-band limits in
Appendix A. For completeness, we also include the possibility of a jet break
following Rhoads (1999). We set a lower limit on the jet break time of $t_{\rm
jet}\gtrsim 34$ days since there is no evidence for an earlier jet break in
the data, as discussed in Sect. 3.5.

$\displaystyle{10^{14}}$$\displaystyle{10^{15}}$$\displaystyle{10^{16}}$$\displaystyle{10^{17}}$$\displaystyle{10^{18}}$Frequency
(Hz)$\displaystyle{10^{-1}}$$\displaystyle{10^{0}}$$\displaystyle{10^{1}}$$\displaystyle{10^{2}}$$\displaystyle{10^{3}}$$\displaystyle
F_{\nu}$ ($\mathrm{\SIUnitSymbolMicro}$Jy)Unabsorbed model SEDAbsorbed model
SEDXRT PC-mode spectrumUV/Optical/NIR photometry Figure 5: Optical to X-ray
SED at 0.3 days along with the highest-likelihood theoretical model SED, both
absorbed (dashed line) and unabsorbed (solid line). The optical photometric
data points were derived from the light curve fits to each observing band (see
Fig. 4) through interpolating each fit to 0.3 days. The X-ray PC-mode spectrum
had a mean photon arrival time of 0.43 days, so we used the X-ray BPL light
curve fit in Fig. 2 to determine a correcting factor to shift the spectrum to
the expected flux level at 0.3 days.
$\displaystyle{10^{8}}$$\displaystyle{10^{9}}$$\displaystyle{10^{10}}$$\displaystyle{10^{11}}$$\displaystyle{10^{1}}$$\displaystyle{10^{2}}$$\displaystyle{10^{3}}$$\displaystyle
F_{\nu}$ ($\mathrm{\SIUnitSymbolMicro}$Jy)$\displaystyle t=10.8$
days$\displaystyle{10^{8}}$$\displaystyle{10^{9}}$$\displaystyle{10^{10}}$$\displaystyle{10^{11}}$Frequency
(Hz)$\displaystyle t=34.2$
days$\displaystyle{10^{8}}$$\displaystyle{10^{9}}$$\displaystyle{10^{10}}$$\displaystyle{10^{11}}$$\displaystyle
t=59.7$ days Figure 6: Radio SEDs at the times of the three MeerKAT epochs. We
show the highest-likelihood unabsorbed model SED along with the effects of
Galactic scintillation at the $1\sigma$ level as derived from the NE2001 model
(Cordes & Lazio, 2002) for the GRB line of sight through the Milky Way. For
the second epoch we show the C- and X-band detections obtained quasi-
simultaneously at 34.2 days, while for the third epoch we show the C- and
X-band detections obtained 7.4 days after the third MeerKAT epoch.

The physical parameters for the highest-likelihood model and those derived
from the MCMC analysis are presented in Table 2, while the corresponding light
curves are presented in Fig. 7. For these parameters, both inverse Compton and
KN effects are important at early times ($\lesssim 1$ day), with Compton
$Y\approx 4$ at ${\sim}0.3$ days. At this time, the relevant spectral break
frequencies are located at $\nu_{m}\approx 4\times 10^{13}$ Hz,
$\nu_{c}\approx\hat{\nu}_{c}\approx 10^{16}$ Hz, and $\hat{\nu}_{m}\approx
10^{21}$ Hz, resulting in the spectral ordering $\nu_{\rm
opt}<\nu_{c}\approx\hat{\nu}_{c}\lesssim\nu_{\rm X}$, where $\hat{\nu}_{c}$
and $\hat{\nu}_{m}$ are KN spectral breaks as outlined in Nakar et al. (2009).
The cooling frequency passes through the X-ray band between $\approx 0.6$ to
12 days, consistent with the discussion in Sects. 3.2 and 3.3. In this regime,
the spectral index in the X-rays is expected to be intermediate between
$(1-p)/2\approx-0.88$ and $-p/2\approx-1.34$, which is consistent with the
observed X-ray spectral index of $\beta_{\rm X}=-1.00\pm 0.11$. Taking into
account the $1\sigma$ confidence intervals from the MCMC analysis, the derived
value of $p=2.75\pm 0.03$ is consistent with the value of $p=2.62\pm 0.10$
inferred from our closure relation analysis in Sect. 3.3. This model also
requires an intrinsic extinction of $A_{\mathrm{V,host}}\approx 0.2$ mag,
consistent with the observed UV suppression (Fig. 5). The corner plot in Fig.
8 shows that there are strong correlations between some pairs of parameters,
especially those involving $\epsilon_{e}$, $\epsilon_{B}$, $A_{\star}$, and
$t_{\mathrm{jet}}$. The model over-predicts the $1.4$ GHz flux by a factor of
$\approx 3$ with respect to our MeerKAT upper limits (Fig. 6); even when
taking scintillation into account, the upper limits are all more than
$4\sigma$ below the model flux. We return to this point in Sect. 5.2.

Our shock microphysics parameters are fairly typical. Our value of $p=2.75\pm
0.03$ is within the $1\sigma$ range of the sample in Wang et al. (2015), for
which they find $p=2.33\pm 0.48$. Santana et al. (2014) collect $\epsilon_{e}$
and $\epsilon_{B}$ values in the literature and find that $\epsilon_{e}$ is
narrowly distributed across one order of magnitude between ${\sim}0.02$ to 0.6
with a median of value 0.22. Our value of $\approx 0.1$ is a normal value
within their sample. For the magnetic field equipartition factor, they find a
much wider distribution varying across almost 5 orders of magnitude from
${\sim}3.5\times 10^{-5}$ to 0.33. Our value of $\epsilon_{B}\approx 0.2\times
10^{-2}$ is close to their median value of $1.0\times 10^{-2}$. Additionally,
Beniamini & van der Horst (2017) use radio light curve peaks to determine the
distribution of $\epsilon_{e}$ and find a value of
$\mathrm{log}_{10}\epsilon_{e}=-0.88\pm 0.26$ for a wind medium. Again, our
derived valued is consistent with their sample value.

The model requires a jet break at $t_{\rm jet}\approx 64$ days, which is
driven by the declining radio light curves after this time. For the highest-
likelihood parameters, this corresponds to a jet opening angle of $\theta_{\rm
jet}\approx 5^{\circ}$. In the absence of a jet break, the model over-predicts
the final X-band detection by $\approx 3.5\sigma$. However, we note that the
evidence in support of a jet break is fairly weak and this inferred opening
angle should be interpreted with caution. For a limit of $t_{\rm jet}\gtrsim
118$ days (the last radio detection), the highest-likelihood model yields a
lower limit on the opening angle of $\theta_{\rm jet}\gtrsim 6^{\circ}$. The
corresponding beaming correction of ${f_{b}=(1-\cos\theta_{\rm jet})\gtrsim
5.48\times 10^{-3}}$ implies constraints on the true $\gamma$-ray and kinetic
energy of $E_{\gamma}\gtrsim 7.07\times 10^{49}$ erg and
$E_{\mathrm{K}}\gtrsim 3.45\times 10^{51}$ erg, respectively, where we have
used the maximum-likelihood (ML) estimate from Table 2 for
$E_{\mathrm{K,iso}}$.

$\displaystyle{10^{3}}$$\displaystyle{10^{4}}$$\displaystyle{10^{5}}$$\displaystyle{10^{6}}$$\displaystyle{10^{7}}$Time since trigger (s)$\displaystyle{10^{-2}}$$\displaystyle{10^{-1}}$$\displaystyle{10^{0}}$$\displaystyle{10^{1}}$$\displaystyle{10^{2}}$Time since trigger (days)$\displaystyle{10^{-2}}$$\displaystyle{10^{2}}$$\displaystyle{10^{6}}$$\displaystyle{10^{10}}$$\displaystyle{10^{14}}$$\displaystyle{10^{18}}$$\displaystyle{10^{22}}$$\displaystyle{10^{26}}$$\displaystyle F_{\nu}$ ($\mathrm{\SIUnitSymbolMicro}$Jy)1 keV $\displaystyle\times 10^{23}$$\displaystyle uvw2\times 10^{20}$$\displaystyle uvm2\times 10^{19}$$\displaystyle uvw1\times 10^{18}$$\displaystyle white\times 10^{17}$$\displaystyle u\times 10^{16}$$\displaystyle b\times 10^{15}$$\displaystyle g\times 10^{14}$$\displaystyle v\times 10^{13}$$\displaystyle q\times 10^{12}$$\displaystyle r\times 10^{11}$$\displaystyle i\times 10^{10}$$\displaystyle z\times 10^{9}$$\displaystyle J\times 10^{8}$$\displaystyle H\times 10^{7}$$\displaystyle K\times 10^{6}$X $\displaystyle\times 10^{2}$C $\displaystyle\times 10^{1}$L $\displaystyle\times 10^{0}$ Figure 7: Light curves from our highest-likelihood theoretical model shown for each observing band, spanning X-ray, optical, and radio frequencies. Only data points after 0.3 days were used in the modelling, and we show these as filled-in data points, in contrast to the earlier time data points shown as empty circles. Upper limits are shown as upside-down triangles. The shaded regions surrounding the three radio bands (X, C, and L) represent the effects of Galactic scintillation at the $1\sigma$ level. The optical $g$\- and $r$-band model fits plateau towards the measured host-galaxy flux levels at 285 days, shown as diamonds.  $\displaystyle p$ = $\displaystyle{2.75}_{-0.03}^{+0.03}$$\displaystyle{-1.50}$$\displaystyle{-1.25}$$\displaystyle{-1.00}$$\displaystyle{-0.75}$log($\displaystyle\epsilon_{e}$)log($\displaystyle\epsilon_{e}$) = $\displaystyle{-1.02}_{-0.05}^{+0.06}$$\displaystyle{-4}$$\displaystyle{-3}$$\displaystyle{-2}$$\displaystyle{-1}$log($\displaystyle\epsilon_{B}$)log($\displaystyle\epsilon_{B}$) = $\displaystyle{-2.66}_{-0.23}^{+0.26}$$\displaystyle{-2.5}$$\displaystyle{-2.0}$$\displaystyle{-1.5}$$\displaystyle{-1.0}$$\displaystyle{-0.5}$log($\displaystyle A_{\star}$)log($\displaystyle A_{\star}$) = $\displaystyle{-1.19}_{-0.11}^{+0.09}$$\displaystyle{1.2}$$\displaystyle{1.5}$$\displaystyle{1.8}$$\displaystyle{2.1}$$\displaystyle{2.4}$log($\displaystyle E_{\mathrm{K,iso,52}}$)log($\displaystyle E_{\mathrm{K,iso,52}}$) = $\displaystyle{1.74}_{-0.13}^{+0.15}$$\displaystyle{0.20}$$\displaystyle{0.22}$$\displaystyle{0.24}$$\displaystyle{0.26}$$\displaystyle{0.28}$$\displaystyle A_{\mathrm{V,host}}$$\displaystyle A_{\mathrm{V,host}}$ = $\displaystyle{0.23}_{-0.01}^{+0.01}$$\displaystyle{2.58}$$\displaystyle{2.64}$$\displaystyle{2.70}$$\displaystyle{2.76}$$\displaystyle{2.82}$$\displaystyle p$$\displaystyle{1.6}$$\displaystyle{2.4}$$\displaystyle{3.2}$$\displaystyle{4.0}$log($\displaystyle t_{\mathrm{jet}}$)$\displaystyle{-1.50}$$\displaystyle{-1.25}$$\displaystyle{-1.00}$$\displaystyle{-0.75}$log($\displaystyle\epsilon_{e}$)$\displaystyle{-4}$$\displaystyle{-3}$$\displaystyle{-2}$$\displaystyle{-1}$log($\displaystyle\epsilon_{B}$)$\displaystyle{-2.5}$$\displaystyle{-2.0}$$\displaystyle{-1.5}$$\displaystyle{-1.0}$$\displaystyle{-0.5}$log($\displaystyle A_{\star}$)$\displaystyle{1.2}$$\displaystyle{1.5}$$\displaystyle{1.8}$$\displaystyle{2.1}$$\displaystyle{2.4}$log($\displaystyle E_{\mathrm{K,iso,52}}$)$\displaystyle{0.20}$$\displaystyle{0.22}$$\displaystyle{0.24}$$\displaystyle{0.26}$$\displaystyle{0.28}$$\displaystyle A_{\mathrm{V,host}}$$\displaystyle{1.6}$$\displaystyle{2.4}$$\displaystyle{3.2}$$\displaystyle{4.0}$log($\displaystyle t_{\mathrm{jet}}$)log($\displaystyle t_{\mathrm{jet}}$) = $\displaystyle{1.84}_{-0.14}^{+0.16}$ Figure 8: Corner plot showing the marginalised posterior distributions for each model parameter along with the 2D marginalised posterior distributions for each pair of model parameters, from our MCMC analysis. Contours are at the $1\sigma$, $2\sigma$, and $3\sigma$ levels, and the red lines denote the median values derived from the marginalised posterior distributions for each model parameter. Table 2: Parameters derived from our multi-wavelength theoretical modelling of the afterglow data after 0.3 days. We show both the highest-likelihood model parameters from a maximum-likelihood (ML) estimation and the median values along with their corresponding $1\sigma$ confidence intervals from the MCMC marginalised posterior distributions presented in Fig. 8. The beaming-corrected prompt $\gamma$-ray and kinetic energies are given in the lower panel of the table, where we have placed a lower limit on the opening angle of the jet based on a limit of $t_{\mathrm{jet}}\gtrsim 118$ days.  Parameter | ML estimate | MCMC results
---|---|---
$p$ | 2.75 | $2.75\pm 0.03$
$\epsilon_{e}$ | $9.7\times 10^{-2}$ | $9.6^{+1.3}_{-1.0}\times 10^{-2}$
$\epsilon_{B}$ | $1.7\times 10^{-3}$ | $2.2^{+1.9}_{-0.9}\times 10^{-3}$
$A_{\star}$ | $7.1\times 10^{-2}$ | $6.4^{+1.6}_{-1.5}\times 10^{-2}$
$E_{\mathrm{K,iso}}$ ($10^{52}$ erg) | 63 | $69^{+32}_{-19}$
$A_{\mathrm{V,host}}$ (mag) | 0.22 | $0.23\pm 0.01$
$t_{\mathrm{jet}}$ (days) | 64 | $55^{+23}_{-14}$
$\theta_{\mathrm{jet}}$ (deg) | $\gtrsim 6$ | -
$E_{\gamma}$ (erg) | $\gtrsim 7.07\times 10^{49}$ | -
$E_{\mathrm{K}}$ (erg) | $\gtrsim 3.45\times 10^{51}$ | -

##  5 Discussion

The optical light curve of GRB 210731A is unusual for showing three distinct
peaks of similar brightness within the first five hours of the GRB trigger.
Multiple peaks in optical light curves have been observed before (e.g. GRBs
060904B, 080928, 100621A, 100814A, 100901A; Klotz et al., 2008; Rossi et al.,
2011; Greiner et al., 2013; Nardini et al., 2014; Laskar et al., 2015). We
investigate a number of explanations proposed in the literature for peaks and
re-brightenings in afterglow light curves, including the passage of a spectral
break, flaring behaviour, secondary jets, an off-axis viewing angle, energy
injection into the forward shock, and the onset of afterglow. We also consider
the implications of our L-band upper limits.

###  5.1 The nature of the optical re-brightenings

####  5.1.1 Passage of a spectral break

The passage of the spectral break associated with the peak of the synchrotron
spectrum ($\nu_{m}$) through the optical bands could in principle give rise to
a peak in the optical light curves. Since $\nu_{m}$ moves to lower frequencies
with increasing time, we would expect the spectral index to transition from a
positive to negative value over the rise and fall of the light curve. We have
shown in Sect. 3.2 and in Fig. 4, however, that our optical evolution is
achromatic, ruling out a spectral break origin to any of the peaks.

####  5.1.2 Flaring behaviour

The peaks in the early light curve are too smooth and long-lived to be due to
flares. Hence, the observed structure in the optical light curve cannot be due
to any of the mechanisms that are typically invoked to explain flares, such as
late-time central engine activity, density fluctuations, or reverse shock
emission.

####  5.1.3 Off-axis viewing angle

It is possible to obtain a rising light curve if the GRB jet is viewed from an
angle outside the cone of the jet (i.e.
$\theta_{\mathrm{obs}}\gtrsim\theta_{\mathrm{jet}}$; Granot et al. 2002). The
peak of the light curve corresponds to the time when the Lorentz factor of the
jet is $\sim 1/\theta_{\mathrm{obs}}$, whereafter the light curve evolves in a
post-jet break manner. Our radio data and theoretical modelling does not
support an early jet break however, as discussed in Sects. 3.5 and 4. Although
it would be possible to explain a single peak in the light curve through
viewing angle effects, it is difficult to interpret all three light curve
peaks within such a scenario. Due to the relativistic beaming effect, one
would also expect to observe negligible prompt $\gamma$-ray emission when the
viewing angle is outside the jet cone, resulting in an orphan afterglow
(Rhoads, 1999; Granot et al., 2002; Zou et al., 2007). The prompt $\gamma$-ray
observations of GRB 210731A therefore do not support an off-axis viewing angle
interpretation.

####  5.1.4 Two-component jet model

The two-component jet model (Peng et al., 2005) has been invoked to explain
chromatic behaviour and late-time re-brightenings observed in a number of GRB
afterglows (Berger et al., 2003; Racusin et al., 2008; Filgas et al., 2011;
Nicuesa Guelbenzu et al., 2011; Kann et al., 2018). In this model a fast,
narrow inner jet powers the prompt emission and early afterglow emission,
while a slower, wider jet powers the late-time afterglow evolution, with both
jets viewed on-axis. This model was preferred by Liang et al. (2013) to
explain the re-brightenings observed in their sample of optical afterglows, in
which the deceleration of the slow jet explains the re-brightening peaks,
analogous to the onset peaks. They claim that the similar rising and decaying
indices of the onset and re-brightening peaks supports this interpretation.
Furthermore, they find that the properties of the re-brightenings are not
correlated with the prompt emission properties (contrary to the onset peaks),
and so they are likely independent emission components. The fact that the GRB
210731A optical light curve shows three distinct peaks appears to rule out the
two-component jet model. A three-component jet might be able to explain the
three light curve peaks, but we consider it beyond the scope of this work.

####  5.1.5 Energy injection

The most straightforward explanation for the optical light curve evolution is
energy injection into the forward shock. Energy injection, or a refreshed
shock, has been invoked to explain the plateaus and shallow decay segments
seen in X-ray (Campana et al., 2005; Vaughan et al., 2006; Nousek et al.,
2006; Zhang et al., 2006; Liang et al., 2007; Evans et al., 2009) and optical
light curves (Mangano et al., 2007; Greiner et al., 2009; Swenson et al.,
2013). In this framework, the blast wave energy increases with time during the
energy injection period, rather than remaining constant. Two physical
mechanisms have been proposed: the first is a long-lasting central engine that
continuously injects a Poynting flux into the blast wave as in the case of a
spin-down millisecond magnetar (Dai & Lu, 1998; Zhang & Mészáros, 2001), where
the central engine luminosity is described as a power law in time with
$L(t)=L_{0}(t/t_{0})^{-q}$. The injected energy $E_{\mathrm{inj}}$ is
essentially constant when $q\geq 1$, while the total energy in the blast wave
can only increase significantly with time when $q<1$. Once the injected energy
begins to exceed the original energy in the blast wave, the total energy will
scale as $E_{\mathrm{tot}}\propto t^{1-q}$. The second case could arise from
an impulsive central engine injection episode, producing a stratified ejecta
distribution where the ejecta mass above a certain Lorentz factor $\Gamma$ is
described as a power law, for example $M(>\Gamma)\propto\Gamma^{-s}$. The
energy in a shell with Lorentz factor $\Gamma$ is added to the blast wave when
the blast wave bulk Lorentz factor has slowed down to $\Gamma$, so that the
energy in the blast wave scales as $E(>\Gamma)\propto\Gamma^{1-s}$ (Rees &
Mészáros, 1998; Sari & Mészáros, 2000). For both cases the blast wave scaling
laws can be derived and applied to the synchrotron spectra to obtain closure
relations that depend on either $q$ or $s$. Both forms of energy injection can
be cast in an equivalent form, and simple relations between $q$ and $s$ can be
derived (Zhang et al., 2006). From a closure relation analysis with energy
injection alone, it is therefore impossible to distinguish energy injection
from a long-lasting central engine or from injection of a stratified ejecta.

The $R$-band light curve segments between the first and final peaks have
temporal indices of $\alpha=[-0.77,0.64,-0.11,0.48]$ (from the direct fit to
each segment in Fig. 2). Adopting the $q$-formalism, we can determine the
energy injection index $q$ for each segment by making use of the closure
relations for a wind medium in the slow cooling regime (Zhang et al., 2006).
We make the assumption that the $R$ band remained in the spectral regime
satisfying $\nu_{m}<\nu_{R}<\nu_{c}$ during all four segments of energy
injection, which is valid since our data supports achromatic optical
evolution. In this regime, $\alpha=\frac{(2-2p)-(p+1)q}{4}$, so employing
$p=2.75$ from our theoretical modelling we derive values of
$q=[-0.11,-1.62,-0.82,-1.45]$ for each light curve segment after the first
peak. The energy increase during a time period from $t_{0}$ to $t_{1}$ is
calculated as

$E_{\mathrm{K,iso,1}}=E_{\mathrm{K,iso,0}}\left(\frac{t_{1}}{t_{0}}\right)^{1-q}.$
(2)

Assuming that the blast wave kinetic energy evolves according to Eq. 2 during
each of the four power law segments, we can determine the energy at the time
of the first peak, $E_{\mathrm{K,iso,0}}$, using

$E_{\mathrm{K,iso,f}}=E_{\mathrm{K,iso,0}}\left(\frac{t_{1}}{t_{0}}\right)^{1-q_{1}}\left(\frac{t_{2}}{t_{1}}\right)^{1-q_{2}}\left(\frac{t_{3}}{t_{2}}\right)^{1-q_{3}}\left(\frac{t_{4}}{t_{3}}\right)^{1-q_{4}}$
(3)

along with the start and end times of each segment999We use the vertical
dashed lines in Fig. 2 as the start and end times of each segment. These have
values of $t_{0}=0.0077$, $t_{1}=0.021$, $t_{2}=0.057$, $t_{3}=0.09$, and
$t_{4}=0.22$ days., our values calculated for $q$ above, and a final blast
wave energy of $E_{\mathrm{K,iso,f}}=6.3\times 10^{53}$ erg from our
theoretical modelling. We find that the blast wave energy at the time of the
first optical peak is equal to $7.3\times 10^{50}$ erg, smaller by a factor of
${\sim}1000$ compared to the final kinetic energy, indicative of substantial
energy injection.

Laskar et al. (2015) argue that the significant X-ray and optical re-
brightenings seen in a sample of GRB afterglows are best explained by the
stratified ejecta model, since energy injection from a spinning-down
millisecond magnetar should not lead to a significant increase in the blast
wave energy (i.e. $q\geq 1$). They also exclude fall-back accretion onto a
black hole as the theoretically-predicted accretion rate is insufficient to
power plateaus or re-brightenings. In the stratified ejecta formalism, there
is a gap between the initial blast wave shell and the fast outer shell of the
stratified ejecta that is moving with some maximum Lorentz factor,
$\Gamma_{\mathrm{max}}$. As the initial shell slows down, the stratified
ejecta deposits energy into the blast wave until the slowest shell moving with
Lorentz factor $\Gamma_{\mathrm{min}}$ has deposited its energy, whereafter
the afterglow evolves following the standard framework. From their study of a
sample of afterglows exhibiting later-time re-brightenings, Laskar et al.
(2015) showed that a large amount of the kinetic energy deposited into the
blast wave comes from the slowest-moving ejecta. They also find that the GRBs
with significant energy injection have low radiative efficiencies, consistent
with the prompt $\gamma$-ray emission being produced by the fastest-moving
ejecta and a large amount of kinetic energy being carried by the slower
ejecta. With our value of $E_{\gamma,\mathrm{iso}}=1.29\times 10^{52}$ erg and
$E_{\mathrm{K,iso}}=63\times 10^{52}$ erg, we calculate a radiative efficiency
using $\eta\equiv
E_{\gamma,\mathrm{iso}}/(E_{\gamma,\mathrm{iso}}+E_{\mathrm{K,iso}})$ of
$\eta\approx 0.02$, a low radiative efficiency consistent with significant
energy injection. We also note that the energy released in $\gamma$ rays is
$\approx 18$ times greater than the kinetic energy in the blast wave derived
at the time of the first optical peak. This is unphysical if the kinetic
energy at that time is the true energy reservoir from which the prompt
emission is drawn, and adds support to the final kinetic energy being the true
energy reservoir.

####  5.1.6 Afterglow onset and initial Lorentz factor, $\Gamma_{0}$

The synchrotron forward shock model predicts a smooth onset peak in optical
light curves as the expanding blast wave is slowed down by the circumburst
medium (Sari & Piran, 1999; Kobayashi & Zhang, 2007). Such onset peaks are not
uncommon in early optical afterglows and have been studied by a number of
authors (Zhang et al., 2003; Molinari et al., 2007; Xue et al., 2009; Liang et
al., 2010, 2013). For an ISM density profile, the deceleration time (or onset
peak time) is most sensitive to the bulk Lorentz factor of the ejecta and
depends weakly on other parameters. Onset peaks can therefore provide a
valuable way of constraining the initial Lorentz factor $\Gamma_{0}$ of the
GRB blast wave. For a non-ISM density profile, however, the dependence of the
deceleration time on other parameters becomes stronger.

The first optical detection associated with GRB 210731A was made in the UVOT
$white$ filter starting 210 seconds post-trigger, with subsequent detections
showing a steady rise to a smooth peak around 700 seconds. Thereafter, the
light curve entered a declining phase before starting to rise steadily again
at ${\sim}$1700 seconds. Li et al. (2012) and Liang et al. (2013) define an
onset peak as a smooth hump peaking within one hour post-trigger that is
followed by a normal power-law decay component. By comparing our onset peak
with the sample of 38 onset peaks in Liang et al. (2013), we can assess how
likely it is that our first peak is indeed the onset of afterglow. From our
combined R-band fit (Fig. 2 and Table 1) we derive a number of additional
properties from each BPL component. These include the peak flux
($F_{\mathrm{p}}$), peak time ($t_{\mathrm{p}}$), full width at half maximum
($w$), peak $R$-band luminosity ($L_{\mathrm{R,p}}$), and the isotropic energy
release in the interval $[t_{\mathrm{p}}/5,5t_{\mathrm{p}}]$, as shown in
Table 3.

Table 3: Parameters derived from each BPL component of our $R$-band light curve fit. $F_{\mathrm{p}}$ ($10^{-12}$ erg cm-2 s-1) |  $t_{\mathrm{p}}$ (s) |  $w$ (s) |  $L_{\mathrm{R,p}}$ ($10^{45}$ erg s-1) |  $E_{\mathrm{R,iso}}$ ($10^{48}$ erg)
---|---|---|---|---
$1.92\pm 0.27$ | $760\pm 111$ | $956$ | $15.69\pm 2.22$ | $8.28$
$0.75\pm 0.27$ | $5704\pm 645$ | $5385$ | $6.11\pm 2.21$ | $16.96$
$2.62\pm 0.10$ | $23161\pm 1179$ | $39824$ | $21.41\pm 0.78$ | $464.53$

Liang et al. (2013) find that the onset peak times of their sample span a
range of 30-3000 seconds, with typical rising and decaying indices of 1.5 and
$-1.15$ in ranges of [0.3,4] and [$-1.8$,$-0.6$], respectively. Our peak time
of $760\pm 111$ seconds and rising index of $1.39\pm 0.36$ are typical values
within this sample, but our decaying index of $-2.58\pm 0.75$ from BPL 1 in
Table 1 is steeper than average, though it does fall within the sample range
within uncertainty. It should be noted that the decaying slope of our first
peak is not well determined owing to the uncertain contribution of the second
BPL component at this time. Furthermore, the authors derive a number of
empirical relations between pairs of properties of onset peaks (see their
Figs. 7 and 9): the width of a peak is strongly correlated with the peak time;
the R-band peak luminosity is anti-correlated with the rest-frame time; and
the peak luminosity and energy are correlated with the isotropic $\gamma$-ray
energy. We find that our measured values in Table 3 agree closely with their
empirical relations, lending support to the onset peak claim.

Under the assumption that GRB 210731A occurred in a stellar wind medium
($k=2$), we calculate the initial Lorentz factor following Zhang (2018) as

$\Gamma_{0}\simeq 1.3^{1/4}\left(\frac{3E_{\mathrm{K,iso}}(1+z)}{8\pi
Ac^{3}t_{\mathrm{dec}}}\right)^{1/4}\simeq
120t_{\mathrm{dec}}^{-1/4}\left(\frac{1+z}{2}\right)^{1/4}E_{52}^{1/4}A_{\star}^{-1/4},$
(4)

where after the last equality, $t_{\mathrm{dec}}$, is measured in seconds,
$E_{52}$ in units of $10^{52}$ erg, and $A_{\star}$ is the wind density
parameter as described previously. Equation 4 depends on the assumption of an
impulsive fireball, that is, the thin shell regime. The isotropic-equivalent
kinetic energy of the blast wave $E_{\mathrm{K,iso}}$ can be inferred from
theoretical modelling of the afterglow, which we do in Sect. 4. However, the
value calculated in Sect. 4 was only applicable to the late-time light curve,
so we employ the value for $E_{\mathrm{K,iso}}$ of $7.3\times 10^{50}$ erg at
the time of the first optical peak calculated in Sect. 5.1.5. Using the
$A_{\star}$ value from our theoretical modelling and the peak time of our
first peak, we calculate an initial Lorentz factor of $\Gamma_{0}\approx 24$.
As mentioned previously, the dependence of the deceleration time on other
parameters is stronger for a non-ISM density profile. This is also the case if
there is energy injection during deceleration, which may be the case during
our early optical observations (Sect. 5.1.5). We therefore caution that our
calculation of $\Gamma_{0}$ is an estimate.

###  5.2 Suppressed L-band flux

Our highest-likelihood theoretical model can provide an adequate fit to all of
our late-time data except for the L band, where the model over-predicts the
flux by a factor of $\approx 3$ compared to our MeerKAT upper limits. We did
not expect our model to fit the L-band data since we found that requiring the
synchrotron self-absorption frequency to lie above the L band at 34.2 days
resulted in an unphysical value of $\epsilon_{e}>1$. We therefore need to
consider an additional source of opacity at these observing frequencies to
explain our L-band limits.

One possible source of additional opacity is a thermal electron population
within the GRB shock front. Ressler & Laskar (2017) modelled afterglow spectra
and light curves while considering the effect of such a population and find
that it has two effects on the spectra: an excess of flux near the peak
synchrotron frequency of the thermal electrons that fades with time as the
electrons cool; and additional opacity in the optically thick portion of the
spectrum compared to the case with only non-thermal electrons. The latter
effect is consistent with a higher self-absorption frequency by a factor of
10-100. It could therefore be the case that our suppressed L-band flux points
towards a population of thermal electrons. We leave the detailed modelling
including a thermal population of electrons to a future work.

##  6 Conclusion

GRB 210731A was a long-duration burst discovered by Swift/BAT. Observations
with the optical telescope MeerLICHT starting 286 seconds post-trigger found
an unusual optical light curve evolution with three peaks of similar
brightness within the first 4.3 hours; afterwards, the burst entered a
declining phase. We find that the early optical evolution is consistent with a
constant optical spectrum, pointing towards a hydrodynamical origin. A closure
relation analysis based on the optical SED and temporal decay after the last
peak showed a preference for a stellar wind environment, consistent with the
long GRB duration and therefore a massive star origin. We find that the first
optical peak can be explained as the onset of afterglow, while energy
injection into the forward shock from a stratified ejecta is a natural
explanation for the two subsequent re-brightenings. We estimate that the blast
wave kinetic energy increased by a factor of ${\sim}1000$ from the first
optical peak to the last peak. Detailed theoretical modelling of the optical,
X-ray, and radio data after the last optical peak at ${\sim}0.3$ days resulted
in typical blast wave and shock microphysics parameters. Our MeerKAT L-band
upper limits could not be reconciled with our model, however, possibly
implying a thermal electron population within the shocked region that provided
an additional source of opacity. Future multi-wavelength modelling of GRB
afterglows, especially at millimetre and radio frequencies, will shed light on
the electron distribution in GRB shocks.

###### Acknowledgements.

The MeerLICHT consortium is a partnership between Radboud University, the
University of Cape Town, the Netherlands Organisation for Scientific Research
(NWO), the South African Astronomical Observatory (SAAO), the University of
Oxford, the University of Manchester and the University of Amsterdam, in
association with and, partly supported by, the South African Radio Astronomy
Observatory (SARAO), the European Research Council and the Netherlands
Research School for Astronomy (NOVA). We acknowledge the use of the Inter-
University Institute for Data Intensive Astronomy (IDIA) data intensive
research cloud for data processing. IDIA is a South African university
partnership involving the University of Cape Town, the University of Pretoria
and the University of the Western Cape. SdW and PJG are supported by NRF
SARChI Grant 111692. Part of the funding for GROND (both hardware and
personnel) was generously granted from the Leibniz-Prize to G. Hasinger (DFG
grant HA 1850/28-1) and by the Thüringer Landessternwarte Tautenburg. AVF is
grateful for financial assistance from the Christopher R. Redlich Fund and
numerous individual donors. KAIT and its ongoing operation were made possible
by donations from Sun Microsystems, Inc., the Hewlett-Packard Company,
AutoScope Corporation, Lick Observatory, the U.S. NSF, the University of
California, the Sylvia & Jim Katzman Foundation, and the TABASGO Foundation.
Research at Lick Observatory is partially supported by a generous gift from
Google. This work made use of data supplied by the UK Swift Science Data
Centre at the University of Leicester. This publication made use of the python
package `corner.py` (Foreman-Mackey, 2016).

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##  Appendix A Theoretical modelling with MeerKAT L-band limits

We repeated the theoretical modelling following the procedure outlined in
Sect. 4, but including the MeerKAT L-band limits. The radio light curves from
the highest-likelihood model are shown in Fig. 9, and the MCMC parameter
distributions are presented in Fig. 10 and Table 4.

Including the MeerKAT L-band limits leads to a poorer fit. This is
demonstrated by the highest-likelihood light curve under-predicting the C- and
X-band fluxes while still over-predicting the L-band flux, and the actual
likelihood value of this fit being lower than for the fit excluding the
MeerKAT limits. We measured a maximum log likelihood of 236.9 for the fit
without the limits vs 225.1 for the fit including the MeerKAT limits. As we
expected, including the MeerKAT data does not lead to a model with a steep
enough spectral index to accommodate the non-detections.

Table 4: Same as Table 2, but for the fit including the L-band upper limits. Parameter | ML estimate | MCMC results
---|---|---
$p$ | 2.76 | $2.76^{+0.03}_{-0.04}$
$\epsilon_{e}$ | $1.2\times 10^{-2}$ | $1.2^{+0.1}_{-0.2}\times 10^{-1}$
$\epsilon_{B}$ | $3.1\times 10^{-3}$ | $5.7^{+17.7}_{-2.5}\times 10^{-3}$
$A_{\star}$ | $4.1\times 10^{-2}$ | $3.1^{+0.8}_{-1.3}\times 10^{-2}$
$E_{\mathrm{K,iso}}$ ($10^{52}$ erg) | 39 | $30^{+11}_{-10}$
$A_{\mathrm{V,host}}$ (mag) | 0.22 | $0.23\pm 0.01$
$t_{\mathrm{jet}}$ (days) | 125 | $186^{+37}_{-87}$
$\displaystyle{10^{6}}$$\displaystyle{10^{7}}$Time since trigger
(s)$\displaystyle{10^{1}}$$\displaystyle{10^{2}}$Time since trigger
(days)$\displaystyle{10^{1}}$$\displaystyle{10^{2}}$$\displaystyle{10^{3}}$$\displaystyle{10^{4}}$$\displaystyle{10^{5}}$$\displaystyle
F_{\nu}$ ($\mathrm{\SIUnitSymbolMicro}$Jy)X $\displaystyle\times 10^{2}$C
$\displaystyle\times 10^{1}$L $\displaystyle\times 10^{0}$ Figure 9: Highest-
likelihood model for the fit that includes the L-band data (dashed lines) and
the fit that excludes the L-band data (solid lines). The fit to the optical
and X-ray light curves is similar in both models.  $\displaystyle p$ =
$\displaystyle{2.76}_{-0.04}^{+0.03}$$\displaystyle{-1.35}$$\displaystyle{-1.20}$$\displaystyle{-1.05}$$\displaystyle{-0.90}$$\displaystyle{-0.75}$log($\displaystyle\epsilon_{e}$)log($\displaystyle\epsilon_{e}$)
=
$\displaystyle{-0.93}_{-0.08}^{+0.04}$$\displaystyle{-4}$$\displaystyle{-3}$$\displaystyle{-2}$$\displaystyle{-1}$log($\displaystyle\epsilon_{B}$)log($\displaystyle\epsilon_{B}$)
=
$\displaystyle{-2.25}_{-0.25}^{+0.61}$$\displaystyle{-2.4}$$\displaystyle{-2.0}$$\displaystyle{-1.6}$$\displaystyle{-1.2}$$\displaystyle{-0.8}$log($\displaystyle
A_{\star}$)log($\displaystyle A_{\star}$) =
$\displaystyle{-1.50}_{-0.23}^{+0.10}$$\displaystyle{1.2}$$\displaystyle{1.6}$$\displaystyle{2.0}$$\displaystyle{2.4}$log($\displaystyle
E_{\mathrm{K,iso,52}}$)log($\displaystyle E_{\mathrm{K,iso,52}}$) =
$\displaystyle{1.48}_{-0.18}^{+0.13}$$\displaystyle{0.200}$$\displaystyle{0.225}$$\displaystyle{0.250}$$\displaystyle{0.275}$$\displaystyle
A_{\mathrm{V,host}}$$\displaystyle A_{\mathrm{V,host}}$ =
$\displaystyle{0.23}_{-0.01}^{+0.01}$$\displaystyle{2.64}$$\displaystyle{2.70}$$\displaystyle{2.76}$$\displaystyle{2.82}$$\displaystyle{2.88}$$\displaystyle
p$$\displaystyle{1.6}$$\displaystyle{2.4}$$\displaystyle{3.2}$$\displaystyle{4.0}$$\displaystyle{4.8}$log($\displaystyle
t_{\mathrm{jet}}$)$\displaystyle{-1.35}$$\displaystyle{-1.20}$$\displaystyle{-1.05}$$\displaystyle{-0.90}$$\displaystyle{-0.75}$log($\displaystyle\epsilon_{e}$)$\displaystyle{-4}$$\displaystyle{-3}$$\displaystyle{-2}$$\displaystyle{-1}$log($\displaystyle\epsilon_{B}$)$\displaystyle{-2.4}$$\displaystyle{-2.0}$$\displaystyle{-1.6}$$\displaystyle{-1.2}$$\displaystyle{-0.8}$log($\displaystyle
A_{\star}$)$\displaystyle{1.2}$$\displaystyle{1.6}$$\displaystyle{2.0}$$\displaystyle{2.4}$log($\displaystyle
E_{\mathrm{K,iso,52}}$)$\displaystyle{0.200}$$\displaystyle{0.225}$$\displaystyle{0.250}$$\displaystyle{0.275}$$\displaystyle
A_{\mathrm{V,host}}$$\displaystyle{1.6}$$\displaystyle{2.4}$$\displaystyle{3.2}$$\displaystyle{4.0}$$\displaystyle{4.8}$log($\displaystyle
t_{\mathrm{jet}}$)log($\displaystyle t_{\mathrm{jet}}$) =
$\displaystyle{2.26}_{-0.27}^{+0.46}$ Figure 10: Same as Fig. 8, but for the
fit including the L-band upper limits.

##  Appendix B Flux measurements

Table LABEL:tab:photometry contains all the X-ray, optical, and radio
observations used in this work.

Table 5: X-ray/optical/radio flux measurements of GRB 210731A. $\Delta t$ (days) | Telescope | Band/Filter |  Frequency (Hz) |  Flux ($\mathrm{\SIUnitSymbolMicro}$Jy) |  Uncertainty ($\mathrm{\SIUnitSymbolMicro}$Jy) |  Detection? ($1=$ yes)
---|---|---|---|---|---|---
0.00241 |  Swift/XRT | 1 keV | 2.42e+17 | 177.589 | 27.077 | 1
0.00243 |  Swift/XRT | 1 keV | 2.42e+17 | 144.408 | 21.527 | 1
0.00245 |  Swift/XRT | 1 keV | 2.42e+17 | 173.501 | 26.453 | 1
0.00247 |  Swift/XRT | 1 keV | 2.42e+17 | 173.423 | 27.678 | 1
0.00249 |  Swift/XRT | 1 keV | 2.42e+17 | 135.870 | 20.254 | 1
0.00251 |  Swift/XRT | 1 keV | 2.42e+17 | 166.523 | 25.390 | 1
0.00254 |  Swift/XRT | 1 keV | 2.42e+17 | 104.861 | 16.356 | 1
0.00257 |  Swift/XRT | 1 keV | 2.42e+17 | 122.493 | 18.260 | 1
0.00259 |  Swift/XRT | 1 keV | 2.42e+17 | 118.424 | 17.654 | 1
0.00262 |  Swift/XRT | 1 keV | 2.42e+17 | 132.423 | 20.190 | 1
0.00264 |  Swift/XRT | 1 keV | 2.42e+17 | 115.389 | 17.201 | 1
0.00267 |  Swift/XRT | 1 keV | 2.42e+17 | 119.856 | 19.577 | 1
0.00270 |  Swift/XRT | 1 keV | 2.42e+17 | 102.343 | 17.113 | 1
0.00273 |  Swift/XRT | 1 keV | 2.42e+17 | 87.334 | 13.938 | 1
0.00276 |  Swift/XRT | 1 keV | 2.42e+17 | 106.299 | 16.965 | 1
0.00279 |  Swift/XRT | 1 keV | 2.42e+17 | 94.213 | 15.036 | 1
0.00283 |  Swift/XRT | 1 keV | 2.42e+17 | 93.477 | 13.935 | 1
0.00287 |  Swift/XRT | 1 keV | 2.42e+17 | 72.925 | 13.765 | 1
0.00290 |  Swift/XRT | 1 keV | 2.42e+17 | 89.665 | 14.310 | 1
0.00294 |  Swift/XRT | 1 keV | 2.42e+17 | 85.993 | 14.046 | 1
0.00298 |  Swift/XRT | 1 keV | 2.42e+17 | 64.979 | 11.965 | 1
0.00302 |  Swift/XRT | 1 keV | 2.42e+17 | 96.119 | 15.340 | 1
0.00305 |  Swift/XRT | 1 keV | 2.42e+17 | 70.237 | 13.596 | 1
0.00309 |  Swift/XRT | 1 keV | 2.42e+17 | 64.675 | 13.529 | 1
0.13866 |  Swift/XRT | 1 keV | 2.42e+17 | 0.507 | 0.116 | 1
0.20459 |  Swift/XRT | 1 keV | 2.42e+17 | 0.438 | 0.114 | 1
0.20839 |  Swift/XRT | 1 keV | 2.42e+17 | 0.538 | 0.093 | 1
0.27115 |  Swift/XRT | 1 keV | 2.42e+17 | 0.415 | 0.109 | 1
0.27459 |  Swift/XRT | 1 keV | 2.42e+17 | 0.455 | 0.103 | 1
0.33652 |  Swift/XRT | 1 keV | 2.42e+17 | 0.441 | 0.115 | 1
0.33905 |  Swift/XRT | 1 keV | 2.42e+17 | 0.378 | 0.100 | 1
0.34243 |  Swift/XRT | 1 keV | 2.42e+17 | 0.285 | 0.075 | 1
0.34705 |  Swift/XRT | 1 keV | 2.42e+17 | 0.287 | 0.076 | 1
0.35107 |  Swift/XRT | 1 keV | 2.42e+17 | 0.427 | 0.087 | 1
0.40418 |  Swift/XRT | 1 keV | 2.42e+17 | 0.280 | 0.073 | 1
0.40918 |  Swift/XRT | 1 keV | 2.42e+17 | 0.283 | 0.074 | 1
0.41349 |  Swift/XRT | 1 keV | 2.42e+17 | 0.233 | 0.061 | 1
0.42004 |  Swift/XRT | 1 keV | 2.42e+17 | 0.282 | 0.060 | 1
0.56853 |  Swift/XRT | 1 keV | 2.42e+17 | 0.234 | 0.061 | 1
0.60579 |  Swift/XRT | 1 keV | 2.42e+17 | 0.165 | 0.043 | 1
0.61160 |  Swift/XRT | 1 keV | 2.42e+17 | 0.207 | 0.046 | 1
0.68902 |  Swift/XRT | 1 keV | 2.42e+17 | 0.125 | 0.027 | 1
0.74695 |  Swift/XRT | 1 keV | 2.42e+17 | 0.133 | 0.035 | 1
0.77010 |  Swift/XRT | 1 keV | 2.42e+17 | 0.150 | 0.032 | 1
2.50026 |  Swift/XRT | 1 keV | 2.42e+17 | 0.018 | 0.005 | 1
0.00368 | MeerLICHT | $q$ | 5.169e+14 | 59.71 | 4.40 | 1
0.00485 | MeerLICHT | $u$ | 7.889e+14 | 74.48 | 13.03 | 1
0.00607 | MeerLICHT | $q$ | 5.169e+14 | 106.67 | 3.93 | 1
0.00719 | MeerLICHT | $g$ | 6.246e+14 | 97.28 | 5.38 | 1
0.00844 | MeerLICHT | $q$ | 5.169e+14 | 106.67 | 3.93 | 1
0.00962 | MeerLICHT | $r$ | 4.789e+14 | 128.24 | 7.09 | 1
0.01086 | MeerLICHT | $q$ | 5.169e+14 | 97.28 | 3.58 | 1
0.01210 | MeerLICHT | $i$ | 3.919e+14 | 167.50 | 13.88 | 1
0.01320 | MeerLICHT | $q$ | 5.169e+14 | 80.91 | 3.73 | 1
0.01447 | MeerLICHT | $z$ | 3.276e+14 | 124.75 | 28.72 | 1
0.01575 | MeerLICHT | $q$ | 5.169e+14 | 74.48 | 4.12 | 1
0.01823 | MeerLICHT | $q$ | 5.169e+14 | 57.55 | 3.71 | 1
0.01950 | MeerLICHT | $g$ | 6.246e+14 | 50.59 | 5.59 | 1
0.02076 | MeerLICHT | $q$ | 5.169e+14 | 56.50 | 3.64 | 1
0.02193 | MeerLICHT | $r$ | 4.789e+14 | 68.55 | 7.58 | 1
0.02311 | MeerLICHT | $q$ | 5.169e+14 | 58.62 | 3.78 | 1
0.02438 | MeerLICHT | $i$ | 3.919e+14 | 77.27 | 14.23 | 1
0.02564 | MeerLICHT | $q$ | 5.169e+14 | 61.38 | 3.96 | 1
0.02691 | MeerLICHT | $z$ | 3.276e+14 | 100.01 | 32.24 | 1
0.02818 | MeerLICHT | $q$ | 5.169e+14 | 69.19 | 5.10 | 1
0.02936 | MeerLICHT | $u$ | 7.889e+14 | 47.87 | 15.87 | 1
0.03061 | MeerLICHT | $q$ | 5.169e+14 | 67.30 | 4.34 | 1
0.03189 | MeerLICHT | $g$ | 6.246e+14 | 67.30 | 7.44 | 1
0.03318 | MeerLICHT | $q$ | 5.169e+14 | 64.87 | 3.58 | 1
0.03438 | MeerLICHT | $r$ | 4.789e+14 | 78.71 | 7.97 | 1
0.03559 | MeerLICHT | $q$ | 5.169e+14 | 84.73 | 4.68 | 1
0.03684 | MeerLICHT | $i$ | 3.919e+14 | 86.30 | 12.72 | 1
0.03811 | MeerLICHT | $q$ | 5.169e+14 | 82.42 | 4.55 | 1
0.03942 | MeerLICHT | $z$ | 3.276e+14 | 128.24 | 35.43 | 1
0.04071 | MeerLICHT | $q$ | 5.169e+14 | 87.91 | 4.05 | 1
0.04199 | MeerLICHT | $u$ | 7.889e+14 | 55.47 | 17.37 | 1
0.04321 | MeerLICHT | $q$ | 5.169e+14 | 86.30 | 3.97 | 1
0.04450 | MeerLICHT | $g$ | 6.246e+14 | 77.99 | 6.46 | 1
0.04577 | MeerLICHT | $q$ | 5.169e+14 | 90.37 | 4.16 | 1
0.04696 | MeerLICHT | $r$ | 4.789e+14 | 88.72 | 8.17 | 1
0.04815 | MeerLICHT | $q$ | 5.169e+14 | 93.76 | 5.18 | 1
0.04939 | MeerLICHT | $i$ | 3.919e+14 | 154.18 | 12.78 | 1
0.05064 | MeerLICHT | $q$ | 5.169e+14 | 92.90 | 4.28 | 1
0.05189 | MeerLICHT | $z$ | 3.276e+14 | 197.71 | 38.24 | 1
0.05312 | MeerLICHT | $q$ | 5.169e+14 | 97.28 | 3.58 | 1
0.05430 | MeerLICHT | $u$ | 7.889e+14 | 76.56 | 18.33 | 1
0.05548 | MeerLICHT | $q$ | 5.169e+14 | 106.67 | 4.91 | 1
0.05675 | MeerLICHT | $g$ | 6.246e+14 | 89.54 | 6.60 | 1
0.05801 | MeerLICHT | $q$ | 5.169e+14 | 103.76 | 4.78 | 1
0.05919 | MeerLICHT | $r$ | 4.789e+14 | 125.90 | 9.28 | 1
0.06037 | MeerLICHT | $q$ | 5.169e+14 | 101.87 | 4.69 | 1
0.06149 | MeerLICHT | $i$ | 3.919e+14 | 124.75 | 12.64 | 1
0.06275 | MeerLICHT | $q$ | 5.169e+14 | 101.87 | 5.63 | 1
0.06398 | MeerLICHT | $z$ | 3.276e+14 | 164.45 | 46.95 | 1
0.06522 | MeerLICHT | $q$ | 5.169e+14 | 100.01 | 5.53 | 1
0.06766 | MeerLICHT | $q$ | 5.169e+14 | 100.93 | 6.51 | 1
0.06893 | MeerLICHT | $g$ | 6.246e+14 | 75.86 | 7.69 | 1
0.07020 | MeerLICHT | $q$ | 5.169e+14 | 98.18 | 5.43 | 1
0.07143 | MeerLICHT | $r$ | 4.789e+14 | 115.88 | 11.74 | 1
0.07275 | MeerLICHT | $q$ | 5.169e+14 | 99.09 | 5.48 | 1
0.07403 | MeerLICHT | $i$ | 3.919e+14 | 151.37 | 16.73 | 1
0.07527 | MeerLICHT | $q$ | 5.169e+14 | 100.93 | 5.58 | 1
0.07651 | MeerLICHT | $z$ | 3.276e+14 | 210.88 | 40.79 | 1
0.08293 | MeerLICHT | $q$ | 5.169e+14 | 97.28 | 6.27 | 1
0.08413 | MeerLICHT | $r$ | 4.789e+14 | 114.82 | 14.81 | 1
0.08532 | MeerLICHT | $q$ | 5.169e+14 | 103.76 | 6.69 | 1
0.08659 | MeerLICHT | $i$ | 3.919e+14 | 129.43 | 23.84 | 1
0.08914 | MeerLICHT | $z$ | 3.276e+14 | 307.63 | 62.33 | 1
0.09540 | MeerLICHT | $q$ | 5.169e+14 | 120.23 | 7.75 | 1
0.09662 | MeerLICHT | $r$ | 4.789e+14 | 121.35 | 14.53 | 1
0.10654 | MeerLICHT | $g$ | 6.246e+14 | 60.82 | 9.52 | 1
0.10781 | MeerLICHT | $q$ | 5.169e+14 | 92.90 | 6.85 | 1
0.10899 | MeerLICHT | $r$ | 4.789e+14 | 114.82 | 10.58 | 1
0.11016 | MeerLICHT | $q$ | 5.169e+14 | 93.76 | 6.05 | 1
0.11138 | MeerLICHT | $i$ | 3.919e+14 | 110.67 | 17.33 | 1
0.11252 | MeerLICHT | $q$ | 5.169e+14 | 98.18 | 6.33 | 1
0.11377 | MeerLICHT | $z$ | 3.276e+14 | 157.05 | 40.50 | 1
0.11503 | MeerLICHT | $q$ | 5.169e+14 | 99.09 | 6.39 | 1
0.11754 | MeerLICHT | $q$ | 5.169e+14 | 104.72 | 6.75 | 1
0.11877 | MeerLICHT | $g$ | 6.246e+14 | 69.83 | 10.29 | 1
0.12003 | MeerLICHT | $q$ | 5.169e+14 | 125.90 | 6.96 | 1
0.12121 | MeerLICHT | $r$ | 4.789e+14 | 140.61 | 14.25 | 1
0.12248 | MeerLICHT | $q$ | 5.169e+14 | 108.65 | 10.01 | 1
0.13362 | MeerLICHT | $r$ | 4.789e+14 | 131.83 | 14.57 | 1
0.13482 | MeerLICHT | $q$ | 5.169e+14 | 114.82 | 9.52 | 1
0.13604 | MeerLICHT | $i$ | 3.919e+14 | 134.28 | 25.97 | 1
0.13725 | MeerLICHT | $q$ | 5.169e+14 | 113.77 | 11.53 | 1
0.13958 | MeerLICHT | $q$ | 5.169e+14 | 100.01 | 11.05 | 1
0.14195 | MeerLICHT | $q$ | 5.169e+14 | 127.07 | 8.19 | 1
0.14305 | MeerLICHT | $g$ | 6.246e+14 | 97.28 | 11.65 | 1
0.14430 | MeerLICHT | $q$ | 5.169e+14 | 125.90 | 6.96 | 1
0.14550 | MeerLICHT | $r$ | 4.789e+14 | 159.97 | 14.73 | 1
0.14670 | MeerLICHT | $q$ | 5.169e+14 | 136.78 | 7.56 | 1
0.14797 | MeerLICHT | $i$ | 3.919e+14 | 169.05 | 26.47 | 1
0.15050 | MeerLICHT | $z$ | 3.276e+14 | 275.44 | 48.20 | 1
0.16412 | MeerLICHT | $q$ | 5.169e+14 | 157.05 | 21.70 | 1
0.16650 | MeerLICHT | $q$ | 5.169e+14 | 141.91 | 27.45 | 1
0.16775 | MeerLICHT | $g$ | 6.246e+14 | 119.13 | 24.14 | 1
0.913 | MeerLICHT | $r$ | 4.789e+14 | 25.12 | 4.40 | 1
0.915 | MeerLICHT | $q$ | 5.169e+14 | 24.21 | 1.56 | 1
0.915 | MeerLICHT | $g$ | 6.246e+14 | 16.75 | 3.70 | 1
0.918 | MeerLICHT | $i$ | 3.919e+14 | 41.31 | 8.37 | 1
1.018 | MeerLICHT | $r$ | 4.789e+14 | 21.09 | 4.66 | 1
1.018 | MeerLICHT | $q$ | 5.169e+14 | 23.12 | 1.28 | 1
1.020 | MeerLICHT | $g$ | 6.246e+14 | 17.70 | 2.77 | 1
1.023 | MeerLICHT | $i$ | 3.919e+14 | 32.21 | 8.31 | 1
2.03 | MeerLICHT | $q$ | 5.169e+14 | 7.52 | 0.83 | 1
3.00 | MeerLICHT | $q$ | 5.169e+14 | 3.53 | 0.68 | 1
0.00277 |  Swift/UVOT | $white$ | 7.488e+14 | 16.29 | 4.80 | 1
0.13848 |  Swift/UVOT | $v$ | 5.511e+14 | 120.23 | 19.93 | 1
0.20686 |  Swift/UVOT | $b$ | 6.848e+14 | 92.90 | 7.70 | 1
0.27287 |  Swift/UVOT | $uvm2$ | 1.319e+15 | 13.43 | 2.60 | 1
0.340 |  Swift/UVOT | $uvw1$ | 1.115e+15 | 16.00 | 2.21 | 1
0.349 |  Swift/UVOT | $u$ | 8.583e+14 | 47.43 | 3.93 | 1
0.408 |  Swift/UVOT | $uvw2$ | 1.401e+15 | 5.15 | 1.23 | 1
0.418 |  Swift/UVOT | $v$ | 5.511e+14 | 87.10 | 12.03 | 1
0.560 |  Swift/UVOT | $b$ | 6.848e+14 | 36.31 | 7.69 | 1
0.606 |  Swift/UVOT | $uvm2$ | 1.319e+15 | 3.53 | 1.40 | 1
0.650 |  Swift/UVOT | $uvw1$ | 1.115e+15 | 5.30 | 1.37 | 1
0.694 |  Swift/UVOT | $u$ | 8.583e+14 | 13.80 | 3.94 | 1
0.747 |  Swift/UVOT | $uvw2$ | 1.401e+15 | 3.40 | 1.13 | 0
0.757 |  Swift/UVOT | $v$ | 5.511e+14 | 27.04 | 11.21 | 1
0.802 |  Swift/UVOT | $b$ | 6.848e+14 | 21.09 | 7.19 | 1
2.79 |  Swift/UVOT | $uvm2$ | 1.319e+15 | 1.91 | 0.64 | 0
2.90 |  Swift/UVOT | $uvw1$ | 1.115e+15 | 2.25 | 0.75 | 0
3.20 |  Swift/UVOT | $u$ | 8.583e+14 | 7.24 | 2.41 | 0
3.21 |  Swift/UVOT | $b$ | 6.848e+14 | 15.42 | 5.14 | 0
3.21 |  Swift/UVOT | $white$ | 7.488e+14 | 3.56 | 1.19 | 0
3.21 |  Swift/UVOT | $v$ | 5.511e+14 | 35.32 | 11.77 | 0
3.71 |  Swift/UVOT | $uvw2$ | 1.401e+15 | 1.79 | 0.60 | 0
0.18419 | GROND | $g^{\prime}$ | 6.536e+14 | 116.96 | 1.08 | 1
0.18419 | GROND | $r^{\prime}$ | 4.820e+14 | 148.60 | 1.37 | 1
0.18419 | GROND | $i^{\prime}$ | 3.924e+14 | 188.81 | 1.74 | 1
0.18419 | GROND | $z^{\prime}$ | 3.335e+14 | 227.00 | 2.09 | 1
0.18437 | GROND | $J$ | 2.418e+14 | 307.63 | 5.67 | 1
0.18437 | GROND | $H$ | 1.820e+14 | 428.57 | 7.89 | 1
0.18437 | GROND | $K$ | 1.381e+14 | 554.66 | 15.33 | 1
1.225 | GROND | $g^{\prime}$ | 6.536e+14 | 13.68 | 0.13 | 1
1.225 | GROND | $r^{\prime}$ | 4.820e+14 | 18.20 | 0.17 | 1
1.225 | GROND | $i^{\prime}$ | 3.924e+14 | 22.91 | 0.42 | 1
1.225 | GROND | $z^{\prime}$ | 3.335e+14 | 27.29 | 0.50 | 1
1.225 | GROND | $J$ | 2.418e+14 | 46.99 | 2.60 | 1
1.225 | GROND | $H$ | 1.820e+14 | 57.02 | 5.78 | 1
1.225 | GROND | $K$ | 1.381e+14 | 58.62 | 17.82 | 1
2.21 | GROND | $g^{\prime}$ | 6.536e+14 | 5.11 | 0.09 | 1
2.21 | GROND | $r^{\prime}$ | 4.820e+14 | 6.25 | 0.17 | 1
2.21 | GROND | $i^{\prime}$ | 3.924e+14 | 7.24 | 0.40 | 1
2.21 | GROND | $z^{\prime}$ | 3.335e+14 | 9.64 | 0.53 | 1
2.21 | GROND | $J$ | 2.418e+14 | 20.14 | 2.78 | 1
2.21 | GROND | $H$ | 1.820e+14 | 22.29 | 5.75 | 1
2.21 | GROND | $K$ | 1.381e+14 | 59.71 | 19.90 | 0
5.25 | GROND | $g^{\prime}$ | 6.536e+14 | 1.51 | 0.14 | 1
5.25 | GROND | $r^{\prime}$ | 4.820e+14 | 1.45 | 0.17 | 1
5.25 | GROND | $i^{\prime}$ | 3.924e+14 | 1.14 | 0.38 | 0
5.25 | GROND | $z^{\prime}$ | 3.335e+14 | 1.57 | 0.52 | 0
5.25 | GROND | $J$ | 2.418e+14 | 8.32 | 2.77 | 0
5.25 | GROND | $H$ | 1.820e+14 | 15.56 | 5.19 | 0
5.25 | GROND | $K$ | 1.381e+14 | 49.21 | 16.40 | 0
285.0 | GROND | $g^{\prime}$ | 6.536e+14 | 0.40 | 0.07 | 1
285.0 | GROND | $r^{\prime}$ | 4.820e+14 | 0.48 | 0.09 | 1
285.0 | GROND | $i^{\prime}$ | 3.924e+14 | 0.76 | 0.25 | 0
285.0 | GROND | $J$ | 2.418e+14 | 6.31 | 2.10 | 0
285.0 | GROND | $H$ | 1.820e+14 | 10.97 | 3.66 | 0
285.0 | GROND | $K$ | 1.381e+14 | 27.54 | 9.18 | 0
0.385 | KAIT | $clear$ | 4.722e+14 | 104.72 | 4.82 | 1
1.183 | VLT | $r$ | 4.830e+14 | 18.54 | 0.34 | 1
1.186 | VLT | $g$ | 6.394e+14 | 13.18 | 0.24 | 1
1.189 | VLT | $z$ | 3.124e+14 | 28.84 | 0.80 | 1
2.09 | NOT | $r$ | 4.830e+14 | 6.79 | 0.25 | 1
2.10 | NOT | $z$ | 3.124e+14 | 12.94 | 1.07 | 1
6.08 | NOT | $r$ | 4.830e+14 | 1.05 | 0.18 | 1
10.8 | MeerKAT | L | 1.4e+09 | 42.0 | 14.0 | 0
18.2 | VLA | C | 6.0e+09 | 136.0 | 32.0 | 1
18.2 | VLA | X | 1.0e+10 | 182.0 | 8.0 | 1
34.1 | MeerKAT | L | 1.4e+09 | 41.1 | 13.7 | 0
34.2 | VLA | C | 6.0e+09 | 140.0 | 10.0 | 1
34.2 | VLA | X | 1.0e+10 | 199.0 | 9.0 | 1
59.7 | MeerKAT | L | 1.4e+09 | 40.2 | 13.4 | 0
67.1 | VLA | C | 6.0e+09 | 102.0 | 9.0 | 1
67.1 | VLA | X | 1.0e+10 | 50.0 | 15.0 | 1
118.0 | VLA | C | 6.0e+09 | 66.0 | 11.0 | 1
118.0 | VLA | X | 1.0e+10 | 37.0 | 7.0 | 1
Table 5: Continued. 101010All times are relative to the Swift/BAT trigger
time. X-ray, optical and radio data are separated by horizontal lines.
Detections are all at least at the 3$\sigma$ level except for the first
UVOT/$white$ detection, which was at the 2.3$\sigma$ level. Optical and radio
upper limits are at the 3$\sigma$ level.