# The X-ray and Radio Loud Fast Blue Optical Transient AT2020mrf:
Implications for an Emerging Class of Engine-Driven Massive Star Explosions

Yuhan Yao Division of Physics, Mathematics and Astronomy, California Institute
of Technology, Pasadena, CA 91125, USA  Anna Y. Q. Ho Department of Astronomy,
University of California, Berkeley, 501 Campbell Hall, Berkeley, CA, 94720,
USA  Miller Institute for Basic Research in Science, 468 Donner Lab, Berkeley,
CA 94720, USA  Pavel Medvedev Space Research Institute, Russian Academy of
Sciences, Profsoyuznaya ul. 84/32, Moscow, 117997, Russia  Nayana A. J. Indian
Institute of Astrophysics, II Block, Koramangala, Bangalore 560034, India
Daniel A. Perley Astrophysics Research Institute, Liverpool John Moores
University, IC2, Liverpool Science Park, 146 Brownlow Hill, Liverpool L3 5RF,
UK  S. R. Kulkarni Division of Physics, Mathematics and Astronomy, California
Institute of Technology, Pasadena, CA 91125, USA  Poonam Chandra National
Centre for Radio Astrophysics, Tata Institute of Fundamental Research, P.O.
Box 3, Pune, 411007, India  Sergey Sazonov  Space Research Institute, Russian
Academy of Sciences, Profsoyuznaya ul. 84/32, Moscow, 117997, Russia  Moscow
Institute of Physics and Technology, Institutsky per. 9, 141700 Dolgoprudny,
Russia  Marat Gilfanov  Space Research Institute, Russian Academy of Sciences,
Profsoyuznaya ul. 84/32, Moscow, 117997, Russia  Max-Planck-Institut für
Astrophysik, Karl-Schwarzschild-Str. 1, D-85741 Garching, Germany  Georgii
Khorunzhev  Space Research Institute, Russian Academy of Sciences,
Profsoyuznaya ul. 84/32, Moscow, 117997, Russia  David K. Khatami Department
of Astronomy, University of California, Berkeley, 501 Campbell Hall, Berkeley,
CA, 94720, USA  Rashid Sunyaev  Space Research Institute, Russian Academy of
Sciences, Profsoyuznaya ul. 84/32, Moscow, 117997, Russia  Max-Planck-Institut
für Astrophysik, Karl-Schwarzschild-Str. 1, D-85741 Garching, Germany

###### Abstract

We present AT2020mrf (SRGe J154754.2$+$443907), an extra-galactic ($z=0.1353$)
fast blue optical transient (FBOT) with a rise time of $t_{g,\rm rise}=3.7$
days and a peak luminosity of $M_{g,\rm peak}=-20.0$. Its optical spectrum
around peak shows a broad ($v\sim 0.1c$) emission feature on a blue continuum
($T\sim 2\times 10^{4}$ K), which bears a striking resemblance to AT2018cow.
Its bright radio emission ($\nu L_{\nu}=1.2\times 10^{39}\,{\rm erg\,s^{-1}}$;
$\nu_{\rm rest}=7.4$ GHz; 261 days) is similar to four other AT2018cow-like
events, and can be explained by synchrotron radiation from the interaction
between a sub-relativistic ($\gtrsim 0.07$–$0.08c$) forward shock and a dense
environment ($\dot{M}\lesssim 10^{-3}\,M_{\odot}\,{\rm yr^{-1}}$ for $v_{\rm
w}=10^{3}\,{\rm km\,s^{-1}}$). AT2020mrf occurs in a galaxy with $M_{\ast}\sim
10^{8}\,M_{\odot}$ and specific star formation rate $\sim 10^{-10}\,{\rm
yr^{-1}}$, supporting the idea that AT2018cow-like events are preferentially
hosted by dwarf galaxies. The X-ray luminosity of AT2020mrf is the highest
among FBOTs. At 35–37 days, SRG/eROSITA detected luminous ($L_{\rm X}\sim
2\times 10^{43}\,{\rm erg\,s^{-1}}$; 0.3–10 keV) X-ray emission. The X-ray
spectral shape ($f_{\nu}\propto\nu^{-0.8}$) and erratic intraday variability
are reminiscent of AT2018cow, but the luminosity is a factor of $\sim 20$
greater than AT2018cow. At 328 days, Chandra detected it at $L_{\rm X}\sim
10^{42}\,{\rm erg\,s^{-1}}$, which is $>200$ times more luminous than
AT2018cow and CSS161010. At the same time, the X-ray emission remains variable
on the timescale of $\sim 1$ day. We show that a central engine, probably a
millisecond magnetar or an accreting black hole, is required to power the
explosion. We predict the rates at which events like AT2018cow and AT2020mrf
will be detected by SRG and Einstein Probe.

X-ray transient sources (1852); Radio transient sources (2008); Supernovae
(1668); Core-collapse supernovae (304); High energy astrophysics (739); Sky
surveys (1464)

††software:  astropy (Astropy Collaboration et al., 2013), CASA (v5.6.1;
McMullin et al. 2007), CIAO (Fruscione et al., 2006), emcee (Foreman-Mackey et
al., 2013), LPipe (Perley, 2019), matplotlib (Hunter, 2007), pandas (McKinney,
2010), Prospector (Johnson et al., 2021), pyne2001 (Cordes & Lazio, 2002),
python-fsps (Foreman-Mackey et al., 2014), scipy (Virtanen et al., 2020)
††facilities:  CXO, XMM, PO:1.2m, Keck:I (LRIS), VLA, GMRT

##  1 Introduction

The past several years have shown that the landscape of massive-star death is
unexpectedly rich and diverse. Of particular interest is the group of “fast
blue optical transients” (FBOTs; Drout et al. 2014; Pursiainen et al. 2018).
As implied by the name, these events exhibit blue colors of $(g-r)<-0.2$ mag
at peak, and evolve faster than ordinary supernovae (SNe), with time above
half-maximum $t_{1/2}\lesssim 12$ days.

The earliest studies were stymied by the identification of FBOTs after the
transients had faded away. This situation has been rectified by cadenced wide-
field optical sky surveys such as the Zwicky Transient Facility (ZTF; Bellm et
al. 2019; Graham et al. 2019) and the Asteroid Terrestrial-impact Last Alert
System (ATLAS; Tonry et al. 2018), which enable real-time discovery and
spectroscopic classification. Ho et al. (2021a) recently identified three
distinct subtypes of FBOTs: (1) subluminous stripped-envelope SNe of type
Ib/IIb, (2) luminous interaction-powered SNe of type IIn/Ibn/Icn, and (3) the
most luminous ($-20\lesssim M_{\rm peak}\lesssim-22$) and short-duration
($t_{1/2}\lesssim 5$ days) events with properties similar to AT2018cow.

The nature of AT2018cow-like events remains the most mysterious. Following the
discovery of the prototype AT2018cow ($z=0.014$, Prentice et al. 2018), only
three analogs have been identified: AT2018lug ($z=0.271$, Ho et al. 2020),
CSS161010 ($z=0.034$; Coppejans et al. 2020), and AT2020xnd ($z=0.243$, Perley
et al. 2021). All of these events arose in low-mass star-forming galaxies,
which suggests a massive star origin and disfavors models invoking tidal
disruption by an intermediate-mass black hole (Perley et al., 2021). In the
radio and millimeter band, their high luminosities imply the existence of
dense circumstellar material (CSM), which points to significant mass-loss
prior to the explosion (Ho et al., 2019; Huang et al., 2019; Margutti et al.,
2019; Coppejans et al., 2020).

The X-ray luminosity of AT2018cow ($\sim 10^{43}\,{\rm erg\,s^{-1}}$) at early
times ($\lesssim 20$ days) is similar to that of long-duration gamma-ray
bursts (GRBs) (Rivera Sandoval et al., 2018). Its fast soft X-ray variability
suggests the existence of a central energy source (also called central
engine), and the relativistic reflection features seen in the hard X-ray
spectrum points to equatorial materials (Margutti et al., 2019). Probable
natures of the central engine include an accreting black hole, a rapidly
spinning magnetar, and an embedded internal shock (Margutti et al., 2019;
Pasham et al., 2021). Meawhile, AT2018cow’s late-time ($\sim 20$–45 days)
optical spectra are dominated by hydrogen and helium (Perley et al., 2019;
Margutti et al., 2019; Xiang et al., 2021), making it different from other
engine-powered massive stellar transients such as long GRBs and hydrogen-poor
super-luminous supernovae (i.e., SLSNe-I; see a recent review by Gal-Yam
2019), which are devoid of hydrogen and helium.

X-ray observations of AT2020xnd showed a luminosity consistent with that of
AT2018cow at 20–40 days (Bright et al., 2022; Ho et al., 2021b). Separately,
late-time ($\gtrsim 100$ day) observations of AT2018cow and CSS161010 showed
modest X-ray emission at $L_{\rm X}\approx{\rm few}\times 10^{39}\,{\rm
erg\,s^{-1}}$) (see Appendix A and Coppejans et al. 2020). AT2018cow-like
events are thus promising X-ray transients to be discovered by the eROSITA
(Predehl et al., 2021) and the Mikhail Pavlinsky ART-XC (Pavlinsky et al.,
2021) telescopes on board the Spektrum-Roentgen-Gamma (SRG) satellite (Sunyaev
et al., 2021).

AT2020mrf is an FBOT first detected by ZTF on 2020 June 12. On June 14, it was
also detected by ATLAS. On July 15, it was reported to the transient name
server (TNS) by the ATLAS team (Tonry et al., 2020). On June 17, an optical
spectrum obtained by the “Global SN Project” displayed a featureless blue
continuum. Burke et al. (2020) assigned a spectral type of “SN II”, and
tentatively associated it with a $z=0.059$ host galaxy ($109^{\prime\prime}$
offset). AT2020mrf was detected in the X-ray by SRG from 2020 July 21 to July
24 (§2.3), which made it a promising candidate AT2018cow analog and motivated
our follow up observations. Given that the SRG detection occurred $\sim 41$
days after the first optical detection, and we became aware of it even later,
our follow-up started in April 2021.

The paper is organized as follows. We outline optical, X-ray, and radio
observations, as well as analysis of AT2020mrf and its host galaxy
($z=0.1353$) in §2. We provide the forward shock and CSM properties in §3.1,
discuss possible power sources of the optical emission in §3.2, and present
host galaxy properties in §3.3. We summarize AT2020mrf’s key X-ray properties
and discuss the implication in the context of engine driven explosions similar
to AT2018cow in §3.4. We estimate the detection rates of events like AT2018cow
and AT2020mrf in current and upcoming X-ray all-sky surveys in §4. We give a
summary in §5.

UT time is used throughout the paper. We assume a cosmology of $\Omega_{\rm
M}=0.3$, $\Omega_{\Lambda}=0.7$, and $h=0.7$, implying a luminosity distance
to AT2020mrf of $D_{L}=637\,{\rm Mpc}$ and an angular-diameter distance of
$D_{\theta}=494\,{\rm Mpc}$. Optical magnitudes are reported in the AB system.
We use the Galactic extinction from Schlafly & Finkbeiner (2011) and the
extinction law from Cardelli et al. (1989). Coordinates are given in J2000.

##  2 Observations and Data Analysis

###  2.1 Optical Photometry

Figure 1: Optical (ZTF $gr$, ATLAS $co$) light curve of AT2020mrf (data points
are $>2.5\sigma$ detections, semi-transparent downward triangles are $3\sigma$
upper limits). The ‘S’ tick along the upper axis marks the epoch of
spectroscopy (§2.2). The solid orange line is a simple $r_{\rm ZTF}$-band
model fitted to data round maximum (see §3.2). The SRG eRASS2 scan duration is
marked by the vertical blue band (§2.3). The rest-frame equivalent light
curves of AT2018cow and AT2021csp are shown as dash-dotted lines (based on
blackbody parameters provided in Tab. 4 of Perley et al. 2019) and dotted
lines (based on Tab. 4 of Perley et al. 2022), respectively. The solid and
dashed black lines are observer-frame $r_{\rm ZTF}$-band light curves of
AT2018lug ($\lambda_{\rm rest}=5050$ Å) and AT2020xnd ($\lambda_{\rm
rest}=5165$ Å), respectively. Note that the apparent AB magnitude scale
pertains to AT2020mrf only — light curves of other objects are only shown in
absolute magnitude.  Figure 2: Optical spectrum of AT2020mrf, compared with
AT2018cow at similar phases (Perley et al., 2019). The dashed line is a
blackbody with $T=2\times 10^{4}$ K and $R=7.9\times 10^{14}$ cm.

We obtained public ZTF111https://ztfweb.ipac.caltech.edu/cgi-
bin/requestForcedPhotometry.cgi and ATLAS222https://fallingstar-
data.com/forcedphot/ forced photometry (Masci et al., 2019; Smith et al.,
2020) using the median position of all ZTF alerts (${\rm R.A.}=15^{\rm
h}47^{\rm m}54.17^{\rm s}$, ${\rm
decl.}=+44^{\circ}39^{\prime}07.34^{\prime\prime}$). The 1-day binned optical
light curve is shown in Figure 1. Following Whitesides et al. (2017) and Ho et
al. (2020), we compute absolute magnitude using

$\displaystyle M=m_{\rm obs}-5\,{\rm log_{10}}\left(\frac{D_{L}}{\rm
10\,pc}\right)+2.5\,{\rm log_{10}}(1+z)$ (1)

The last term in Equation (1) is a rough estimation of $K$-correction, and
introduces an error of 0.1 mag.

The first detection is $r=20.88\pm 0.17$, on 2020-06-12T06:14:12 (59012.2599
MJD) and the last non-detection is $o>21.73$, on 2020-06-11T10:12:13
(59011.4252 MJD). Therefore, we assume an explosion epoch of $t_{0}=59012.0$
MJD. Hereafter we use $\Delta t$ to denote rest-frame time with respect to
$t_{0}$. At $\Delta t=3.7$ days, AT2020mrf peaked at $M_{g}=-20.0$ mag.

###  2.2 Optical Spectroscopy

The transient spectrum333Available at https://www.wis-tns.org/object/2020mrf
was obtained on 2020 June 17 ($\Delta t=4.8$ days) with the FLOYDS-N
spectrograph on the 2 m Faulkes Telescope North (Burke et al., 2020). As shown
in Figure 2, the spectrum is similar to that of AT2018cow at similar phases —
a single broad feature at $\sim 5600$ Å was observed to span $\pm 600$ Å,
indicating a velocity of $0.1c$. The origin of this broad line in AT2018cow
remains an open question. Perley et al. (2019) note that although it is
vaguely reminiscent of the Fe II feature in Ic broad-line (Ic-BL) SNe around
peak (Galama et al., 1998), in SNe Ic-BL the blueshifted absorption trough
strengthens at later times, while in AT2018cow this line vanished at $\Delta
t\sim 8$ days. In terms of other AT2018cow-like objects, the peak-light
optical spectra of AT2018lug and AT2020xnd are consistent with being blue and
featureless (Ho et al., 2020; Perley et al., 2021), and that there exists no
published optical spectra of CSS161010.

A blackbody fit to AT2020mrf’s optical spectrum suggests a temperature of
$T\approx 2\times 10^{4}$ K and a radius of $R=7.9\times 10^{14}$ cm. This
temperature is typical of AT2018cow-like events.

###  2.3 Early-time X-rays: SRG

SRG is a space satellite at the L2 Lagrange point with a drafting rate of
$\approx 1^{\circ}\,{\rm\,day^{-1}}$. It is conducting eight all-sky surveys
from the beginning of 2020 to the end of 2023, with a cadence of 6 months.
Hereafter eRASS$n$ refers to the $n$’th eROSITA all-sky survey. SRG’s
rotational axis points toward the Sun, and the rotational period is 4 hours.
The eROSITA field-of-view (FoV) is 1 deg2. Therefore, during a single sky
survey, a particular region of the sky will be scanned by eROSITA at least
$\sim 6$ times ($\sim 1$ day), where each scan lasts for $\approx 40$ s (see
details in Sunyaev et al. 2021). AT2020mrf, at a relatively high ecliptic
latitude of $b_{\rm ecl}=61.9^{\circ}$, was scanned for $\approx 3$ days in
each all-sky survey.

Figure 3: HSC-SSP RGB false-color $g$/$i$/$z$ image centered at the ZTF
position of AT2020mrf (marked by the white crosshairs). AT2020mrf is an off-
nuclear source 0.50′′ offset from the host centroid. The position of the X-ray
transient detected by SRG/eROSITA is shown with a dotted circle, where the
radius represents the $4.28^{\prime\prime}$ uncertainty (98% confidence). The
more accurate positions provided by Chandra obsID 25050 and obsID 25064 are
shown with a dash-dotted circle and a dashed circle (1$\sigma$ confidence),
respectively.

During eRASS2 ($\Delta t\sim 36$ days), SRG/eROSITA discovered an X-ray
transient SRGe J154754.2$+$443907, with a 98% localization radius of
$4.28^{\prime\prime}$. SRGe J154754.2$+$443907 is only 0.56′′ from AT2020mrf
(see Figure 3), suggesting an association between the X-ray and the optical
transients. Figure 4 shows that the source exhibits significant variability —
the 0.2–2.2 keV count rate increased from $\approx 0.053\,{\rm count\,s^{-1}}$
($\Delta t\sim 35$ days) to $\approx 0.32\,{\rm count\,s^{-1}}$ ($\Delta t\sim
36$ days), and then decreased to $\approx 0.051\,{\rm count\,s^{-1}}$ ($\Delta
t\sim 37$ days).

Figure 4: eRASS2 light curve of AT2020mrf. Count rate uncertainties are
estimated using Gehrels’ approximation (Gehrels, 1986). The orange curve is a
fit to the data, generated using a Gaussian process model following procedures
laid out in Appendix B.4 of Yao et al. (2020).  Figure 5: eRASS2 spectrum of
AT2020mrf. We show the best-fit power-law model (dashed green line,
$f_{\nu}\propto\nu^{-0.8}$) and thermal plasma model (dotted brown line,
$k_{\rm B}T\sim 2.0$ keV) with $N_{\rm H}$ fixed at the Galactic value (see
Table 1).  Table 1: Modeling of the eRASS2 spectrum.

Component | Parameter | Power-law Model | Thermal Plasma Model
---|---|---|---
|  | (a) Fixed $N_{\rm H}$ | (b) Free $N_{\rm H}$ | (a) Fixed $N_{\rm H}$ | (b) Free $N_{\rm H}$
tbabs |  $N_{\rm H}$ ($10^{20}\,{\rm cm^{-2}}$) | 1.38 | $15.32^{+14.06}_{-10.70}$ | 1.38 | $9.04^{+7.10}_{-6.99}$
zpowerlw | $\Gamma$ | $1.81\pm 0.26$ | $2.79^{+1.00}_{-0.80}$ | … | …
|  $\rm norm_{pl}$ ($10^{-5}$) | $8.0_{-1.0}^{+1.1}$ | $14.8^{+11.8}_{-5.7}$ | … | …
apec | $k_{\rm B}T$ | … | … | $2.0_{-0.7}^{+1.9}$ | $1.0^{+1.1}_{-0.3}$
|  $\rm norm_{apec}$ ($10^{-4}$) | … | … | $3.8_{-0.7}^{+0.9}$ | $7.2_{-3.3}^{+5.2}$
cstat/dof | 25.94/35 | 24.09/34 | 24.80/35 | 23.60/34
Observed 0.3–10 keV flux ($10^{-13}\,{\rm erg\,s^{-1}\,cm^{-2}}$) | $3.90_{-1.00}^{+1.32}$ | $2.48_{-0.83}^{+0.51}$ | $2.40_{-0.75}^{+0.54}$ | $1.92_{-1.37}^{+0.15}$

Note. — $\rm norm_{pl}$ and $\rm norm_{apec}$ are the normalization parameters
in the model components (see the xspec documentation for units). Uncertainties
are represented by the 68% confidence intervals.

Figure 5 shows the average eRASS2 spectrum of AT2020mrf, which has been
grouped via ftgrouppha to have at least five counts per bin in the background
spectrum. We fit the 0.3–10 keV spectrum using xspec (12.11, Arnaud 1996) and
$C$-statistics. The data are modeled first with an absorbed power-law
(zpowerlw) and then with an absorbed thermal plasma (apec). For each model, we
first fix the column density at the Galactic value of $N_{\rm H}=1.38\times
10^{20}\,{\rm cm^{-2}}$ (Willingale et al., 2013), and then free this
parameter. The models with fixed $N_{\rm H}$ are shown in Table 1. The data do
not favor any particular model, since the cstat/dof ($C$-statistics divided by
degrees of freedom) values have small differences between the four fits.

Although we are not able to distinguish between the power-law and thermal
models using the eROSITA data, the optical/radio similarities between
AT2020mrf and AT2018cow (§2.1, §2.2, §3.1), and the non-thermal nature of
AT2018cow’s X-rays ($f_{\nu}\propto\nu^{-0.7}$, 36.5 days, 0.3–30 keV, see
Fig. 6 of Margutti et al. 2019) motivate us to adopt the power-law model in
the following discussion.

AT2020mrf was not detected in eRASS1, eRASS3 and eRASS4. Using the eROSITA
sensitivity maps, we calculate the 0.3–2.2 keV flux upper limits to be
$(1.12,1.35,1.54)\times 10^{-14}\,{\rm erg\,s^{-1}\,cm^{-2}}$ at the
confidence level likelihood of 6 ($\approx 2.8\sigma$).

###  2.4 Late-time X-rays: Chandra

We conducted deep X-ray observations of AT2020mrf with the Chandra X-ray
Observatory (Wilkes & Tucker, 2019) under a DDT program (PI Yao) on 2021 June
18 (22.0 ks, obsID 25050) and June 19 (19.8 ks, obsID 25064). We used the
Advanced CCD Imaging Spectrometer (ACIS; Garmire et al. 2003), with the aim
point on the back illuminated CCD S3. The data were reduced with the CIAO
package (v4.14).

In order to determine the astrometric shifts of Chandra images, we first ran
the CIAO tool wavdetect to obtain lists of positions for all sources in the
Chandra FoV. Wavelet scales of 1, 2, 4, and 8 pixels and a significance
threshold of $10^{-6}$ were used. A total of 8 and 12 X-ray sources were
detected in obsID 25050 and obsID 25064, respectively. We cross matched the
X-ray source lists with the Gaia EDR3 catalog (Gaia Collaboration et al.,
2021), using a radius of $2^{\prime\prime}$. This left two Chandra/Gaia
sources from both obsIDs. We define the astrometric shifts as the mean
difference in R.A. and decl. between the two matched sources. For obsID 25050,
$\delta{\rm R.A.}=-1.88\pm 0.42^{\prime\prime}$ and $\delta{\rm
decl.}=-0.58\pm 0.75^{\prime\prime}$; For obsID 25064, $\delta{\rm
R.A.}=-0.62\pm 0.27^{\prime\prime}$ and $\delta{\rm decl}=+0.61\pm
0.29^{\prime\prime}$.

Having applied the astrometric shifts, we found that an X-ray source at the
location of AT2020mrf was detected in both obsIDs. The position of the X-ray
source from obsID 25050 is ${\rm R.A.}=15^{\rm h}47^{\rm m}54.18^{\rm s}$,
${\rm decl.}=+44^{\circ}39^{\prime}07.83^{\prime\prime}$, with an astrometric
uncertainty of 1.47′′ from the residual offsets with the Gaia catalog; The
position of the X-ray source from obsID 25064 is ${\rm R.A.}=15^{\rm h}47^{\rm
m}54.18^{\rm s}$, ${\rm decl.}=+44^{\circ}39^{\prime}07.16^{\prime\prime}$,
with an astrometric uncertainty of 0.82′′ from the residual offsets with the
Gaia catalog. The Chandra positions are shown in Figure 3, which are more
accurate than the eROSITA position, and clearly associate the X-ray emission
with the ZTF position of AT2020mrf.

For each obsID, we extracted the source spectrum using a source region of
$r_{\rm src}=1.5^{\prime\prime}$ centered on the X-ray position determined by
wavdetect. A total of 30 and 10 counts (0.5–10 keV) were detected within the
source regions of obsID 25050 and obsID 25064, respectively. The background
spectrum was extracted using nearby source-free regions. The 0.5–10 keV net
count rate at 90% credible interval is $1.61^{+0.32}_{-0.28}\times
10^{-3}\,{\rm count\,s^{-1}}$ for obsID 25050, and $0.56^{+0.21}_{-0.17}\times
10^{-3}\,{\rm count\,s^{-1}}$ for obsID 25064, indicating that X-ray net count
rate has dropped by a factor of $2.9\pm 1.1$. Such a large flux decrease
reflects intrinsic X-ray variability.

Figure 6: Chandra spectrum of AT2020mrf at $\Delta t\approx 328$ days. The
data have been rebinned for visual clarity. The dashed line is the best-fit
model ($f_{\nu}\propto\nu^{+0.0}$). To account for the flux variation (see
text), the obsID 25064 data has been divided by 0.39.

We groupped the Chandra spectrum to at least one count per bin, and modeled
the data using $C$-statistics. We used a model of tbabs*zpowerlw, with $N_{\rm
H}$ fixed at the Galactic value. Since the count rate has significantly
decreased between the two obsIDs, we include a constant scaling factor
$\mathcal{C}$ between the two Chandra observations (Madsen et al., 2017), with
the constant for obsID 25050 ($\mathcal{C}_{1}$) fixed at 1. The result, with
$cstat/dof=32.25/34$, gives $\Gamma=1.00\pm 0.35$ and
$\mathcal{C}_{2}=0.39^{+0.17}_{-0.13}$, where uncertainties are represented by
the 68% confidence intervals. The best-fit model is shown in Figure 6.

Table 2: X-ray flux measurements of AT2020mrf.

$\Delta t$ | Telescope | Observed 0.3–10 keV flux
---|---|---
(days) |  | ($10^{-14}\,{\rm erg\,s^{-1}\,cm^{-2}}$)
$-127$ |  SRG/eRASS1 | $<2.93$
34.5–37.6 |  SRG/eRASS2 | $39.0_{-10.0}^{+13.2}$
192 |  SRG/eRASS3 | $<7.24$
327.4 | Chandra | $4.00_{-1.24}^{+0.68}$
328.2 | $1.57_{-0.49}^{+0.27}$
355 |  SRG/eRASS4 | $<8.26$

Note. — To convert the 0.3–2.2 keV eROSITA upper limits to 0.3–10 keV, we
assume the eRASS2 best-fit spectral model for the eRASS1 epoch, and the
Chandra spectral model for the eRASS3 and eRASS4 epochs.

Figure 7: X-ray emission of AT2020mrf, compared with AT2018cow (Rivera
Sandoval et al. 2018; Margutti et al. 2019, Appendix A), CSS161010 (Coppejans
et al., 2020), AT2020xnd (Bright et al., 2022; Ho et al., 2021b), cosmological
long GRBs (light blue solid lines; Appendix B), GRBs associated with SNe
(dashed grey lines; Kouveliotou et al. 2004; Tiengo et al. 2004; Campana et
al. 2006; Soderberg et al. 2006; Margutti et al. 2013), SLSNe-I (Levan et al.
2013; Margutti et al. 2018), the jetted TDE SwiftJ1644 (Mangano et al., 2016),
interacting SNe of type IIn (dashed-dotted green lines; Chandra et al. 2012,
2015; Dwarkadas et al. 2016; Katsuda et al. 2016) and type Ibn (dotted cyan
lines; Immler et al. 2008; Ofek et al. 2013), as well as normal CCSNe
(Dwarkadas & Gruszko, 2012). AT2020mrf is as luminous as cosmological GRBs.

The difference between the SRG and Chandra power-law indices is $\Gamma_{\rm
36\,d}-\Gamma_{\rm 328\,d}=0.81\pm 0.44$. Therefore, we conclude that a change
of $\Gamma$ is marginally detected at $1.9\sigma$. Table 2 summarizes the
0.3–10 keV fluxes.

Figure 7 compares the X-ray luminosity evolution of AT2020mrf with other types
of explosions. We further discuss this figure in §3.4.

###  2.5 Search for Prompt $\gamma$-rays

Given that cosmological long GRBs are the only type of massive-star explosion
with X-ray luminosities known to be comparable to AT2020mrf (see Figure 7), we
are motivated to search for bursts of prompt $\gamma$-rays between the last
ZTF non-detection and the first ZTF detection (§2.1). During this time
interval, only one burst was detected by the interplanetary network (IPN;
Hurley et al. 2010). The position of this burst (Sonbas et al., 2020) is
inconsistent with that of AT2020mrf. To obtain a constraint on the
$\gamma$-ray flux of AT2020mrf, we use the Konus instrument (Aptekar et al.,
1995) on the Wind spacecraft. Unlike other high energy telescopes on low Earth
orbit (LEO) spacecrafts (such as Swift/BAT and Fermi/GBM), Konus-Wind (KW)
continuously observe the whole sky without Earth blocking and with a very
stable background, thanks to its orbit around the L1 Lagrange point (see,
e.g., Tsvetkova et al. 2021). During the interval of interest, KW was taking
data (total duration of data gaps was $<1$% of the total time). Assuming a
typical long GRB spectrum444 The Band function with peak energy $E_{\rm
peak}=300$ keV, low-energy photon index $\alpha=-1$, and high energy photon
index $\beta=-2$ (Band et al., 1993).  and a timescale of 2.944 s, KW gives a
20–1000 keV upper limit of $<2\times 10^{-7}\,{\rm erg\,s^{-1}\,cm^{-2}}$.
This corresponds to an isotropic luminosity of $L_{\rm iso}<1.0\times
10^{49}\,{\rm erg\,s^{-1}}$, which strongly disfavors an on-axis classical GRB
(Frail et al., 2001).

###  2.6 Radio: VLA and uGMRT

Table 3: Radio observations of AT2020mrf.

Date | $\Delta t$ | Telescope/ | $\nu_{0}$ | $f_{\nu}$
---|---|---|---|---
in 2021 | (days) | Receiver | (GHz) | ($\mu$Jy)
Apr 2 | 259.5 | VLA/C | 4.30 | $254\pm 25$
4.94 | $234\pm 22$
5.51 | $330\pm 18$
6.49 | $327\pm 20$
7.06 | $336\pm 17$
7.70 | $349\pm 18$
Apr 6 | 262.9 | VLA/S | 3.00 | $165\pm 26$
VLA/X | 8.49 | $277\pm 23$
9.64 | $271\pm 20$
11.13 | $223\pm 17$
VLA/Ku | 12.78 | $213\pm 19$
14.32 | $189\pm 16$
16.62 | $153\pm 15$
VLA/K | 20.00 | $149\pm 8$
24.00 | $103\pm 8$
May 19 | 300.9 | uGMRT/B5 | 1.25 | $<45$
May 29 | 309.5 | VLA/S | 3.00 | $206\pm 48$
Aug 13 | 376.6 | uGMRT/B5 | 1.25 | $<105$
Sep 28 | 416.8 | uGRMT/B5 | 1.36 | $68\pm 15$
Sep 28–29 | 417.5 | VLA/S | 3.00 | $81\pm 10$
VLA/C | 6.00 | $87\pm 7$
VLA/X | 10.00 | $49\pm 8$
VLA/Ku | 13.55 | $65\pm 6$
16.62 | $51\pm 7$

Note. — $\nu_{0}$ is observed central frequency. $f_{\nu}$ is the observed
flux density values. Upper limits are $3\sigma$.

We began a monitoring program of AT2020mrf using the VLA (Perley et al., 2011)
under Program 21A-308 (PI Ho), and the upgraded Giant Metrewave Radio
Telescope (uGMRT; Swarup 1991; Gupta et al. 2017) under Program 40_077 (PI
Nayana). The data were analyzed following the standard radio continuum image
analysis procedures in the Common Astronomy Software Applications (CASA;
McMullin et al. 2007). The results are presented in Table 3. Incidentally,
AT2020mrf was not detected in the Karl G. Jansky Very Large Array Sky Survey
(VLASS, Lacy et al. 2020), which provides a 3-$\sigma$ upper limit of 0.42 mJy
at 2–4 GHz in March 2019. Hereafter radio flux density values have been
$K$-corrected and frequency values are reported in the rest-frame.
$K$-correction was performed following Condon & Matthews (2018), assuming a
steep synchrotron spectrum with a spectral index of $\beta=-1$
($f_{\nu}\propto\nu^{\beta}$).

Regarding data obtained within $(\Delta t/10)$ days as coeval, we model the
radio spectral energy distribution (SED) at $\Delta t\approx 261$ days and
$\Delta t\approx 417$ days with a broken power-law (Granot & Sari, 2002):

$\displaystyle L_{\nu}=L_{\nu\,\rm peak}\left[\left(\frac{\nu}{\nu_{\rm
peak}}\right)^{-s\beta_{1}}+\left(\frac{\nu}{\nu_{\rm
peak}}\right)^{-s\beta_{2}}\right]^{-1/s}$ (2)

where $\nu$ and $L_{\nu}$ are quantities in the object’s rest-frame,
$L_{\nu\,\rm peak}$ is the peak specific luminosity, $\nu_{\rm peak}$ is the
peak frequency, $\beta_{1}$ and $\beta_{2}$ are the asymptotic spectral
indices below and above the break, and $s$ is a smoothing parameter. We
perform the fit using the Markov chain Monte Carlo (MCMC) approach with emcee
(Foreman-Mackey et al., 2013). The reported uncertainties follow from the 68%
credible region.

Figure 8: Radio observations of AT2020mrf, overplotted with the best-fit
broken power-law models.

The best-fit models are shown in Figure 8. At $\Delta t\approx 261$ days,
$\nu_{\rm peak}=7.44_{-0.52}^{+0.44}$ GHz, $L_{\nu\,\rm
peak}=1.70_{-0.09}^{+0.23}\times 10^{29}\,{\rm erg\,s^{-1}\,Hz^{-1}}$,
$\beta_{1}=1.3^{+0.4}_{-0.2}$, and $\beta_{2}=-1.0\pm 0.1$. At $\Delta
t\approx 305$ days, the 1–4 GHz band probably remains below the broken
frequency, and the blue data in Figure 8 suggests $\beta_{1}>1.7$. At $\Delta
t\approx 417$ days, $\nu_{\rm peak}=4.82_{-1.18}^{+1.36}$ GHz, $L_{\nu\,\rm
peak}=4.33_{-0.34}^{+0.36}\times 10^{28}\,{\rm erg\,s^{-1}\,Hz^{-1}}$,
$\beta_{1}=0.4_{-0.2}^{+0.3}$, and $\beta_{2}=-0.5\pm 0.1$. Equation (2) does
not provide a decent description for the data.

The radio observations will further be discussed in §3.1

###  2.7 The Host Galaxy

####  2.7.1 Observations

Deep pre-explosion images of the target field are available in the Hyper
Suprime-Cam Subaru Strategic Program (HSC-SSP; Aihara et al. 2018) second
Public Data Release (PDR2; Aihara et al. 2019) and the Galaxy Evolution
Explorer (GALEX; Martin et al. 2005) UV imaging survey. As is shown in the
left panel of Figure 9, AT2020mrf is $0.50^{\prime\prime}$ offset from an
extended blue source (${\rm R.A.}=15^{\rm h}47^{\rm m}54.20^{\rm s}$, ${\rm
decl.}=+44^{\circ}39^{\prime}07.01^{\prime\prime}$), which is considered to be
the host galaxy. At the host redshift, the spacial offset corresponds to a
physical distance of 1.19 kpc. The photometry of the host is shown in Table 4.

Table 4: Observed photometry of the host galaxy.

Instrument | Band |  $\lambda_{\rm eff}$ (Å) | Magnitude
---|---|---|---
GALEX | FUV | 1528 | $>23.276$
GALEX | NUV | 2271 | $>23.579$
HSC | $g$ | 4755 | $23.282\pm 0.029$
HSC | $r$ | 6184 | $23.152\pm 0.046$
HSC | $i$ | 7661 | $22.635\pm 0.040$
HSC | $z$ | 8897 | $22.721\pm 0.079$
HSC | $y$ | 9762 | $22.359\pm 0.133$

Note. — The HSC Kron radius is $0.705^{\prime\prime}$. GALEX upper limits are
given in $3\sigma$.

On 2021 April 14 ($\Delta t=267.0$ days), we obtained a spectrum of the host
galaxy using the Low Resolution Imaging Spectrometer (LRIS; Oke et al. 1995)
on the Keck I 10 m telescope. We used the 560 dichroic, the 400/3400 grism on
the blue side, the 400/8500 grating on the red side, and the
$1^{\prime\prime}$ slit width. This setup gives a full-width half maximum
(FWHM) of $\approx 6.8$ Å. Exposure times were 3650 and 3400 s for the blue
and red cameras, respectively. The spectrum (upper panel of Figure 9) was
reduced and extracted using LPipe (Perley, 2019).

Figure 9:  Upper: Spectrum (corrected for Galactic extinction) of the host
galaxy, overplotted with HSC $griz$ photometry. Bottom: Zoomed-in regions of
the spectrum (black lines, corrected for Galactic extinction). The thick
orange lines show the best-fit ppxf model, which is a combination of the
stellar continuum (red lines) and emission lines (thin magenta lines). The
rest-frame wavelength range 4904–4920 Å is masked due to the large uncertainty
of $f_{\lambda}$ (comtamination by a sky line).

####  2.7.2 Analysis

In order to determine the redshift and emission line fluxes of the host, we
fit the Galactic extinction corrected LRIS spectrum with stellar population
models using the penalized pixel-fitting (ppxf) software (Cappellari &
Emsellem, 2004; Cappellari, 2017). We use the MILES library (${\rm
FWHM=2.5\,{\rm\AA}}$; Falcón-Barroso et al. 2011), and commonly observed
galaxy emission lines, including H$\alpha$, H$\beta$, H$\gamma$, [O II], [S
II], [O III], [O I], and [N II]. The [O I] $\lambda\lambda 6300$, 6364, [O
III] $\lambda\lambda 4959$, 5007 and [N II] $\lambda\lambda 6548,6583$
doublets are fixed at the theoretical flux ratio of 3.

The best-fit model suggests a redshift of $z=0.1353\pm 0.0002$. Zoom-in
portions of the spectrum around regions of emission lines are shown in the
bottom panel of Figure 9. The line fluxes are presented in Table 5. Note that
since the [O II] doublets are not resolved, the derived individual line fluxes
are not reliable, and we only report the total flux of the doublets.

Table 5: Galactic extinction corrected galaxy emission line fluxes.

Line | Flux ($10^{-18}\,{\rm erg\,s^{-1}\,cm^{-2}\,\AA^{-1}}$)
---|---
[O II] $\lambda\lambda 3726$, 3729 | $53.00\pm 6.09$
[Ne III] $\lambda 3869$ | $10.96\pm 1.97$
H$\gamma$ $\lambda 4340$ | $6.31\pm 1.55$
H$\beta$ $\lambda 4861$ | $9.81\pm 1.87$
[O III] $\lambda\lambda 4959$, 5007 | $11.67\pm 4.57$ (2.6$\sigma$)
[O I] $\lambda\lambda 6300$, 6364 | $1.59\pm 1.17$ (1.4$\sigma$)
H$\alpha$ $\lambda 6563$ | $22.82\pm 0.89$
[N II] $\lambda\lambda 6548,6583$ | $2.72\pm 1.71$ (1.6$\sigma$)
[S II] $\lambda 6716$ | $7.32\pm 2.01$
[S II] $\lambda 6731$ | $1.77\pm 1.26$ (1.4$\sigma$)

Note. — Marginally detected emission lines are indicated with the detection
significance shown in the parenthesis.

Table 6: Emission line ratios.

Definition | Value
---|---
[O III]$\lambda$5007/H$\beta$ | $0.90^{+0.78}_{-0.58}$
$\rm log${[O III]$\lambda$5007/H$\beta$} | $-0.05^{+0.27}_{-0.45}$
[N II]$\lambda 6583$/H$\alpha$ | $<0.18$
$\rm N2\equiv log${[N II]$\lambda 6583$/H$\alpha$} | $<-0.73$
$\rm O3N2\equiv log${[O III]$\lambda$5007/H$\beta$}$-$N2 | $<-1.71$
[S II]$\lambda\lambda 6716$,31/H$\alpha$ | $0.40\pm 0.17$
log{[S II]$\lambda\lambda 6716$,31/H$\alpha$} | $-0.40^{+0.16}_{-0.24}$
[O I]$\lambda 6300$/H$\alpha$ | $<0.12$
log{[O I]$\lambda 6300$/H$\alpha$} | $<-0.94$

Note. — Line ratios and their uncertainties are estimated using the 5th, 50th
and 95th percentiles of the MC simulations. When the 5th percentile value is
negative, we present the 95th percentile as an upper limit.

The calculated line ratios are given in Table 6. Uncertainties in line ratios
are calculated by performing $10^{4}$ Monte Carlo (MC) trials using the
measured flux uncertainties. Figure 10 shows the location of the host galaxy
on the Baldwin, Phillips, & Terlevich (BPT) diagrams (Baldwin et al., 1981).
Under the diagnostic definitions of Kewley et al. (2006), the host falls in
the region of star-forming galaxies.

Figure 10: The host galaxy of AT2020mrf on the BPT diagrams. The diagnostic
lines follow Fig. 4 of Kewley et al. (2006).

We measure the oxygen abundance using two metallicity indicators N2 and O3N2
(Pettini & Pagel, 2004), defined in Table 6. Using the calibration reported by
Marino et al. (2013), the gas-phase oxygen abundance is $<8.40\pm 0.16({\rm
sys})$ in the N2 scale, and $>8.17\pm 0.18({\rm sys})$ in the O3N2 scale.
Compared with the solar metallicity ($Z_{\odot}$) of $12+{\rm log}({\rm
O/H})=8.69$ (Asplund et al., 2009), our constraints suggest a metallicity of
$10^{-0.70}$–$10^{-0.13}Z_{\odot}$.

To obtain an estimate of the host galaxy total stellar mass ($M_{\ast}$), we
fit the host SED with flexible stellar population synthesis (FSPS; Conroy &
Wechsler 2009) models (Foreman-Mackey et al., 2014). We adopt a delayed
exponentially declining star-formation history (SFH) characterized by the
$e$-folding timescale $\tau_{\rm SFH}$, such that the time-dependent star-
formation rate $\psi_{\ast}(t)\propto te^{-(t/\tau_{\rm SFH})}$. The
Prospector package (Johnson et al., 2021) was used to run a Markov Chain Monte
Carlo (MCMC) sampler (Foreman-Mackey et al., 2013). We use log-uniform priors
for the following three parameters: $M_{\ast}$ in the range
[$10^{7}\,M_{\odot}$, $10^{9}\,M_{\odot}$], $\tau_{\rm SFH}$ in the range [0.1
Gyr, 100 Gyr], the metallicity ${{\rm log}(Z/Z_{\odot})}$ in the range $-0.70$
and $-0.13$, and the population age $t_{\rm age}$ in the range [0.1 Gyr, 12.5
Gyr]. Host galaxy extinction was included, with $E(B-V)_{\rm host}$ uniformly
sampled between 0 and 1. From the marginalized posterior probability functions
we obtain ${\rm log}(M_{\ast}/M_{\odot})=7.94_{-0.39}^{+0.22}$, ${\rm
log}(Z/Z_{\odot})=-0.46_{-0.17}^{+0.20}$, $\tau_{\rm
SFH}=11.6_{-10.0}^{+45.6}$ Gyr, $t_{\rm age}=1.82_{-1.50}^{+4.07}$ Gyr, and
$E(B-V)_{\rm host}=0.21_{-0.12}^{+0.10}$, where uncertainties are epresented
by the 68% confidence intervals.

Using the 90% confidence interval of the $M_{\ast}$ posterior probability
function and the mass–metallicity relation (MZR) of low-mass galaxies (Berg et
al., 2012), we infer that the typical log($Z/Z_{\odot}$) at the host mass
should be $-0.78^{+0.10}_{-0.16}\pm 0.15({\rm sys})$. The measured metallicity
is therefore on the high end of the distribution.

We convolve the observed LRIS spectrum with the HSC $i$-band filter and
compare the flux with the host photometry (Table 4), which suggests that 80.6%
of the total host flux is captured by the LRIS slit. Subsequently, we assume
the same fraction of total H$\alpha$ flux is captured by the slit and no host
extinction, and calculate the H$\alpha$ luminosity to be $L_{\rm
H\alpha}=(1.39\pm 0.05)\times 10^{39}\,{\rm erg\,s^{-1}}$. Using the Kennicutt
(1998) relation converted to a Chabrier initial mass function (Chabrier, 2003;
Madau & Dickinson, 2014), we infer a star formation rate (SFR) of $(6.93\pm
0.27)\times 10^{-3}\,M_{\odot}\,{\rm yr^{-1}}$. An extinction of $E(B-V)_{\rm
host}\sim 0.21$ will render the SFR higher by a factor of $\sim 1.5$.
Therefore, hereafter we adopt ${\rm SFR}=6.93^{+3.90}_{-0.27}\times
10^{-3}\,M_{\odot}\,{\rm yr^{-1}}$. The specific star formation rate is ${\rm
sSFR}\equiv{\rm SFR}/M_{\ast}=0.80^{+0.45}_{-0.03}\times 10^{-10}\,{\rm
yr^{-1}}$, where we only consider the uncertainty of SFR but exclude the
uncertainty of $M_{\ast}$.

##  3 Inferences and Discussion

###  3.1 A Mildly Relativistic Shock in a Dense Environment

####  3.1.1 Standard SSA Modeling

At $\Delta t\approx 261$ days, the observed spectral index of
$\beta_{2}\approx-1$ (§2.6) in the optically thin regime of the radio SED
motivates us to adopt the standard model given by Chevalier (1998), where the
electrons in the CSM are accelerated by the forward shock into a power-law
distribution of energy $N(E)=N_{0}E^{-p}$. We do not consider the alternative
of a relativistic Maxwellian electron-energy distribution, in which case we
expect a much steeper $\beta_{2}$ (see, e.g., Fig. 11 of Ho et al. 2021b) and
a shock speed of $v_{\rm sh}\gtrsim 0.2c$ (Margalit & Quataert, 2021). The
$v_{\rm sh}$ inferred from our observations is much slower (see below). We
note that the standard model might not be fully appropriate since the observed
spectral index of $\beta_{1}$ in the optically thick regime is much shallower
than the $\beta_{1}=2.5$ expected from SSA. We investigate the effects of CSM
inhomogeneity and scintillation in §3.1.2.

In the standard model of Chevalier (1998), the minimum electron energy is
$E_{\rm min}=511$ keV; the peak of the SED is governed by synchrotron self-
absorption (SSA) such that $\tau(\nu_{\rm peak})=1$; the radio emitting region
is approximated by a sphere with radius $R$ and volume filling factor $f$
(hereafter assumed to be 0.5); the magnetic energy density ($U_{B}\propto
B^{2}$) and the relativistic electron energy density ($U_{e}\propto N_{0}$)
are assumed to scale as the total (thermalized) post-shock energy density $U$,
such that $U_{B}=\epsilon_{B}U$ and $U_{e}=\epsilon_{e}U$.

We define $L_{\theta\nu}\equiv 4\pi D_{\theta}^{2}f_{\nu}=L_{\nu}/(1+z)^{4}$,
$L_{\theta\nu,29}\equiv L_{\theta\nu,\rm peak}/(10^{29}\,{\rm
erg\,s^{-1}\,Hz^{-1}})$ and $\nu_{5}\equiv\nu_{\rm peak}/(5\,\rm GHz)$, such
that

$\displaystyle R=$ $\displaystyle 7.1\times
10^{16}\left(\frac{\epsilon_{e}}{\epsilon_{B}}\right)^{-1/19}L_{\theta\nu,29}^{9/19}\nu_{5}^{-1}\,{\rm
cm}$ (3a) $\displaystyle B=$ $\displaystyle
0.36\left(\frac{\epsilon_{e}}{\epsilon_{B}}\right)^{-4/19}L_{\theta\nu,29}^{-2/19}\nu_{5}\,{\rm
G}$ (3b) $\displaystyle U=$ $\displaystyle 4.0\times
10^{48}\frac{1}{\epsilon_{B}}\left(\frac{\epsilon_{e}}{\epsilon_{B}}\right)^{-11/19}L_{\theta\nu,29}^{23/19}\nu_{5}^{-1}\,{\rm
erg}.$ (3c)

The upstream CSM density can be estimated under the conditions of strong
shocks and fully ionized hydrogen (see Eq. 16 of Ho et al. 2019):

$\displaystyle n_{e}=$ $\displaystyle
61\frac{1}{\epsilon_{B}}\left(\frac{\epsilon_{e}}{\epsilon_{B}}\right)^{-6/19}L_{\theta\nu,29}^{-22/19}\nu_{5}^{4}\left(\frac{\Delta
t}{\rm 100\,days}\right)^{2}\,{\rm cm^{-3}}$ (4)

Assuming that the CSM density profile is determined by a pre-explosion steady
wind with mass-loss rate $\dot{M}$ and velocity $v_{\rm w}$, we have (see Eq.
23555The normalization constant in Eq. 23 of Ho et al. 2019 is off by a factor
of $\sim 10$. Here we update the equation with the correct constant. of Ho et
al. 2019):

$\displaystyle\frac{\dot{M}}{v_{\rm w}}\left(\frac{1000\,{\rm
km\,s^{-1}}}{10^{-4}\,M_{\odot}\,{\rm yr^{-1}}}\right)=$ $\displaystyle
0.10\left(\frac{1}{\epsilon_{B}}\right)\left(\frac{\epsilon_{e}}{\epsilon_{B}}\right)^{-8/19}$
$\displaystyle\times L_{\theta\nu,29}^{-4/19}\nu_{5}^{2}\left(\frac{\Delta
t}{100\,{\rm days}}\right)^{2}$ (5)

We adopt $L_{\theta\nu\,{\rm peak}}=L_{\nu\,{\rm peak}}/(1+z)^{4}\approx
1.0\times 10^{29}\,{\rm erg\,s^{-1}\,Hz^{-1}}$ and $\nu_{\rm peak}\approx 7$
GHz at $\Delta t=261$ days. Assuming $\epsilon_{e}=\epsilon_{B}=1/3$, we have
$R\approx 5.1\times 10^{16}\,{\rm cm}$, $B\approx 0.50\,{\rm G}$, $U\approx
1.7\times 10^{49}\,{\rm erg}$, and $n_{e}\approx 3.5\times 10^{3}\,{\rm
cm^{-3}}$. Assuming $\epsilon_{e}=0.1$, $\epsilon_{B}=0.01$, we have $R\approx
4.6\times 10^{16}\,{\rm cm}$, $B\approx 0.31\,{\rm G}$, $U\approx 1.5\times
10^{50}\,{\rm erg}$, and $n_{e}\approx 5.6\times 10^{4}\,{\rm cm^{-3}}$. The
average shock velocity ($v_{\rm sh}=R/\Delta t$) is 0.07–0.08$c$, suggesting a
mildly relativistic shock. The derived $R$, $U$, $v_{\rm sh}$ should be taken
as upper limits, $B$, $n_{e}$, $\dot{M}/v_{\rm w}$ should be taken as upper
limits. See the discussion in §3.1.2.

Figure 11: Peak radio luminosity ($L_{\nu\,{\rm peak}}$) versus the product of
peak time and $\nu_{\rm peak}$. Under the assumptions of the standard SSA
model, dotted lines mark constant time-averaged velocity; Dashed lines mark
constant mass-loss rate ($\dot{M}_{-4}\equiv\dot{M}/(10^{-4}\,M_{\odot}\,{\rm
yr^{-1}})$) scaled to wind velocity ($v_{\rm w,3}\equiv v_{\rm
w}/(10^{3}\,{\rm km\,s^{-1}})$). The two panels show the results with
different assumptions of $\epsilon_{e}$ and $\epsilon_{B}$. The data of
AT2018cow-like objects and GRB-SNe are based on Fig. 9 of Ho et al. 2021b and
Fig. 3 of Nayana & Chandra (2021).

The upper panel of Figure 11 compares AT2020mrf with normal SNe (Bietenholz et
al., 2021), SNe associated with long GRBs, and four AT2018cow-like objects in
the literature. Note that all GRB-SNe are of type Ic-BL. The peak luminosity
of AT2020mrf is much greater than normal SNe and is in the same regime as
other AT2018cow-like objects. A physical interpretation is that the energy
divided by the shock radius ($U/R\propto L_{\theta\nu\,{\rm peak}}^{14/19}$)
is greater. This indicates a more efficient conversion/thermalization of
energy, which can come from a higher explosion energy or a higher ambient
density (Ho et al., 2019).

Moreover, we see that the CSM “surface density”
($n_{e}R^{2}\propto\dot{M}/v_{\rm w}$) of AT2020mrf at 261 days is similar to
AT2018cow at 22 days. At a similar shock radius of $R\sim 6\times 10^{16}$ cm,
the CSM number density of AT2018cow is $n_{e}<33\,{\rm cm^{-3}}$ (Nayana &
Chandra, 2021) — more than 100 times smaller than that in AT2020mrf. Since
$\dot{M}/v_{\rm w}$ generally decreases at later times (i.e., the density
profile is steeper than $n_{e}\propto r^{-2}$), the immediate environment of
AT2020mrf is probably denser than all other AT2018cow-like events.

####  3.1.2 CSM Inhomogeneity and Scintillation

Figure 12: Radio SEDs of AT2020mrf (solid markers), compared with the that of
AT2018cow at similar phases (hollow markers, interpolated from Table 1 of
Nayana & Chandra 2021). Dashed lines are the inhomogeneous SSA model fits to
the observations of AT2020mrf.  Figure 13: Color evolution of FBOTs. The upper
left panel shows four events associated with bright radio emission. The upper
right panel shows interacting SNe of type Icn, Ibn, and IIn. The lower panels
show type II SNe, as well as stripped envelope SNe of type IIb, Ib, Ic, and
Ic-BL.

The small values of $\beta_{1}$ and the flat-topped radio SEDs (Figure 8)
motivate us to assume an inhomogeneous CSM, which means that the distribution
of electrons or magnetic field strength varies within the synchrotron source
(Björnsson & Keshavarzi, 2017). In this model, between the standard SSA
optically thick $F_{\nu}\propto\nu^{5/2}$ regime and the optically thin
$F_{\nu}\propto\nu^{-(p+1)/2}$ regimes, there is a transition regime with a
spectral index of $0<\beta<2.5$. Since the measured $\beta_{1}$ remains below
$2.5$, we assume that the standard SSA optically thick regime is at
frequencies lower than our observations.

Following Chandra et al. (2019), we fit the full set of radio data with the
function

$\displaystyle L(\nu,t)=K_{1}\nu_{5}^{\beta}\left(\frac{\Delta t}{100\,{\rm
days}}\right)^{a}\left[1-{\rm exp}\left(-\tau_{\rm ssa}(\nu,t)\right)\right],$
(6)

where $\tau_{\rm ssa}$ is the SSA optical depth

$\displaystyle\tau_{\rm
ssa}(\nu,t)=K_{2}\nu_{5}^{-(\beta+\frac{p-1}{2})}\left(\frac{\Delta
t}{100\,{\rm days}}\right)^{-(a+b)}.$ (7)

The best-fit model is shown in Figure 12, with $K_{1}=5.4_{-4.4}^{+14.6}\times
10^{29}\,{\rm erg\,s^{-1}\,Hz^{-1}}$, $K_{2}=13_{-10}^{+53}$,
$\beta=1.6_{-0.6}^{+0.8}$, $p=3.3_{-0.3}^{+0.4}$, $a=-1.6_{-1.4}^{+1.8}$, and
$b=3.0_{-0.4}^{+0.3}$. Evidence of source inhomogeneities has also been found
in AT2018cow (Nayana & Chandra, 2021). With an inhomogeneous CSM, the $R$,
$U$, $v_{\rm sh}$ values derived in §3.1.1 should be taken as lower limits,
and $B$, $n_{e}$, $\dot{M}/v_{\rm w}$ should be taken as upper limits.

A few datapoints at $\Delta t>300$ days are not well fitted by the
inhomogeneous SSA model. We estimate the effects of interstellar scintillation
(ISS) to our radio observations using the NE2001 model (Cordes & Lazio, 2002)
of the Galactic distribution of free electrons. The transition frequency below
which strong scattering occurs is (Goodman, 1997):

$\displaystyle\nu_{\rm ss}=10.4({\rm SM_{-3.5}})^{6/17}d_{\rm
scr,kpc}^{5/17}\,{\rm GHz},$ (8)

where ${\rm SM}_{-3.5}\equiv{\rm SM}/(10^{-3.5}\,{\rm m}^{-20/3}\,{\rm kpc})$
is the scintillation measure, and $d_{\rm scr,kpc}$ is the distance to the
electron scattering screen in kpc. For the line of sight to AT2020mrf
(Galactic coordinates $l=71.339^{\circ}$, $b=50.806^{\circ}$), NE2001 predicts
$\nu_{\rm ss}=8.3$ GHz and ${\rm SM}_{-3.5}=0.53$, implying $d_{\rm
scr,kpc}=1.0$. This suggests that the 11.35 GHz “dip” (or 15–19 GHz “excess”)
cannot be explained by ISS.

AT2020mrf is subject to diffractive or refractive ISS if the source angular
size satisfies $\theta_{\rm s}<3.3\nu_{10}^{6/5}\,{\rm\mu as}$ or $\theta_{\rm
s}<2.0\nu_{10}^{-11/5}\,{\rm\mu as}$ (Goodman, 1997). We have shown that the
shock radius at times of our radio observations is $R\gtrsim 5\times 10^{16}$
cm, corresponding to $\theta_{\rm s}\gtrsim 6.8\,\mu{\rm as}$. Therefore, the
3.4 GHz “excess” at $\Delta t\approx 305$ days and the 1.5 GHz “excess” at
$\Delta t\approx 417$ days are likely caused by refractive ISS.

###  3.2 Properties of the Optical Emission

####  3.2.1 Rise and Decline Timescales

To constrain the optical evolution of AT2020mrf around maximum, we model the
multi-band photometry using a power-law rise and an exponential decay. For
simplicity we assume a blackbody SED and a single temperature for data at
$\Delta t<15$ days. The best-fit model in the $r_{\rm ZTF}$ band is shown as
the solid orange line in Figure 1.

To compare AT2020mrf with the sample of spectroscopically classified FBOTs
presented by Ho et al. (2021a), we calculate the time it takes for AT2020mrf
to rise from half-max to max ($t_{1/2,\rm rise}=2.4\pm 0.2$ days), and to
decline from max to half-max ($t_{1/2,\rm fade}=4.8\pm 0.2$ days). Its total
duration above half-max is $t_{1/2}=7.1_{-0.2}^{+0.3}$ days. On the $M_{\rm
peak}$ versus $t_{1/2}$ diagram (see, e.g., Fig. 1 of Ho et al. 2021a and Fig.
7 of Perley et al. 2022), AT2020mrf lies between previously studied AT2018cow-
like events ($t_{1/2}\lesssim 5$ days, $M_{\rm peak}\lesssim-20.5$) and
interacting SNe of type IIn/Ibn/Icn ($t_{1/2}\gtrsim 7$ days, $M_{\rm
peak}\gtrsim-20.0$).

####  3.2.2 Color Evolution

The $g-r$ color of AT2020mrf is $-0.34\pm 0.20$ mag at the day of discovery
($\Delta t\approx 0.25$ day), and reddens at later times. At $\Delta t\approx
6.4$, 11.7, and 23–28 days, the $g-r$ values are $-0.05\pm 0.06$ mag,
$-0.09\pm 0.14$ mag, and $0.05\pm 0.27$ mag, respectively. Assuming that the
optical SED can be modeled by a blackbody, the blackbody temperature ($T_{\rm
bb}$) decreases from $\sim 2\times 10^{4}$ K to $\sim 10^{4}$ K. Similar
cooling signatures have also been observed in AT2018lug, while both AT2018cow
and AT2020xnd remain blue post-peak.

Figure 13 compares the color evolution of AT2020mrf with other FBOTs. We have
included AT2018cow (Perley et al., 2019), AT2018lug (Ho et al., 2020),
AT2020xnd (Perley et al., 2021), the type Icn SNe 2019hgp (Gal-Yam et al.,
2021) and 2021csp (Perley et al., 2022), as well as the gold sample of 22
spectroscopically classified FBOTs presented by Ho et al. (2021a). The
calculated $g-r$ color has been corrected for Galactic extinction but assumes
no host reddening. As can be seen, the amount of $g-r$ increase observed in
AT2020mrf is closer to other multi-wavelength FBOTs and interacting SNe, but
smaller than events shown in the lower panels.

####  3.2.3 Possible Power Sources

Like many other FBOTs, the fast rise and luminous optical peak of AT2020mrf is
unlikely to be powered by radioactive 56Ni decay, which would require the
nickel mass $M_{\rm Ni}$ to be greater than the ejecta mass $M_{\rm ej}$ (see,
e.g., Fig. 1 of Kasen 2017). Possible emission mechanisms include shock
breakout (SBO) from extended CSM (Waxman & Katz, 2017), shock cooling emission
(SCE) from an extended envelope (Piro et al., 2021), continued interaction
between the SN ejecta and the CSM (Smith, 2017; Fox & Smith, 2019), and
reprocessing of X-ray/UV photons (potentially deposited by a central engine)
by dense outer ejecta (Margutti et al., 2019) or an optically think wind (Piro
& Lu, 2020). We do not attempt to distinguish between these scenarios due to a
lack of multi-wavelength observations at early times.

Figure 14: Bolometric light curve of AT2020mrf converted from ZTF photometry,
assuming $T_{\rm bb}=10^{4}$ K. Data at $<5$ days are shown as upward
triangles since the temperature at early time is $>10^{4}$ K. The $L_{\rm
bol}$ of SN1998bw (Galama et al., 1998) is shown for comparison. We show two
models of radioactivity powered SN in the photospheric phase (Valenti et al.,
2008; Lyman et al., 2016), adopting an opacity of $\kappa=0.07\,{\rm
cm^{2}\,g^{-1}}$ (typical for stripped envelope SNe; Taddia et al. 2018), and
a photospheric velocity of $v_{\rm phot}=2\times 10^{4}\,{\rm km\,s^{-1}}$
(typical for GRB-SNe; Modjaz et al. 2016).

Figure 15: Left: The host galaxy of AT2020mrf on the SFR–$M_{\ast}$ diagram,
compared to hosts of other AT2018cow-like events: AT2018cow itself (Perley et
al., 2019), AT2018lug (Ho et al., 2020), CSS161010 (Coppejans et al., 2020),
and AT2020xnd (Perley et al., 2021). For comparison, the grey contours show
the PTF/iPTF CCSNe host galaxy sample (Schulze et al., 2021), from $0.5\sigma$
to $2\sigma$ in steps of $0.5\sigma$. The light green and yellow bands show
the main sequence of star-forming galaxies at $0.02<z<0.085$ (Renzini & Peng,
2015) and $z\sim 1$ (Eq. 4 of Elbaz et al. 2007), respectively. Right: The
host galaxies of AT2018cow-like events and other massive star explosions on
the sSFR–$M_{\ast}$ diagram. Histograms show the normalized distribution of
18cow-like events (thick line, unfilled), DES RETs (thin line, unfilled), ZTF
FBOTs (dark filled), and SLSNe-I (light filled).

The decay rate of AT2020mrf is significantly slower than that of AT2018cow and
AT2020xnd (Figure 1). This is similar to the post-peak decay of AT2018lug,
which also slows down at $\Delta t\approx 6$–8 days (see Figure 1). The slower
decay can be caused either by the emergence of a radioactivity powered SN or
continued CSM interaction. Since the color evolution of AT2020mrf is most
similar to interacting SNe shown in the upper right panel of Figure 13, we
slightly favor the CSM interaction scenario. In Appendix C, we attempt to fit
the multi-band light curve using the one-zone SBO+SCE model presented by
Margalit (2021), but no satisfactory fit is obtained. However, given that the
CSM interaction model has many free parameters (e.g., anisotropy, radial
density structure), more detailed modeling would be required to determine
whether it is a viable emission mechanism.

Assuming $T_{\rm bb}=10^{4}$ K, the bolometric luminosity and blackbody radius
of AT2020mrf are shown in Figure 14. Although radioactivity is not required to
explain the optical emission, the light curve at $\Delta t\gtrsim 10$ days is
consistent with being dominated by nickel decay with $M_{\rm ej}\sim 1$–6
$M_{\odot}$ and $M_{\rm Ni}\sim 0.3$–0.4 $M_{\odot}$. Improved analytic
relations (compared to the “Arnett model” shown in Figure 14) have been
presented by Khatami & Kasen (2019). Adopting $L_{\rm peak}\approx 10^{42.8}$,
$t_{\rm peak}\approx 17$ days, and the dimensionless parameter $\beta\approx
1$, we use Eq. 21 of Khatami & Kasen (2019) to estimate $M_{\rm Ni}$, which
gives $M_{\rm Ni}\approx 0.26\,M_{\odot}$. In summary, the inferred $M_{\rm
ej}$ and $M_{\rm Ni}$ are broadly consistent with stripped envelope SNe of all
types (IIb, Ib, Ic, and Ic-BL; Drout et al. 2011; Taddia et al. 2018; Prentice
et al. 2019), but can not accommodate normal hydrogen-rich type II SNe (Meza &
Anderson, 2020; Afsariardchi et al., 2021).

###  3.3 A Dwarf Host Galaxy

Figure 15 shows the position of AT2020mrf on the SFR–$M_{\ast}$ and the
sSFR–$M_{\ast}$ diagrams (based on properties derived in §2.7.2). For
comparison, we also show the 28 FBOTs selected from ZTF (note that we excluded
the three 18cow-like events from the 31 objects in Tab. 17 of Ho et al.
2021a), the 49 rapidly evolving transients (RETs) from the dark energy survey
(DES) (Wiseman et al., 2020), and 18 PTF SLSNe-I from Perley et al. (2016a).
Compared with normal CCSNe (Schulze et al., 2021) and X-ray/radio-faint FBOTs,
the $M_{\ast}$ of AT2018cow-like events (a sample of five) is much smaller.
Indeed, all AT2018cow-like events are hosted by dwarf galaxies with
$M_{\ast}<2\times 10^{9}\,M_{\odot}$. This trend has been previously reported
by Perley et al. (2021), and argues for a massive star origin. Several types
of the most powerful explosions from massive stars are also preferentially
hosted by dwarf galaxies, including long GRBs (Vergani et al., 2015; Perley et
al., 2016b), hydrogen-poor SLSNe (Leloudas et al., 2015; Perley et al., 2016a;
Taggart & Perley, 2021), and SNe Ic-BL (Schulze et al., 2021).

Perley et al. (2021) have suggested that an elevated level of SFR or sSFR is
not a requirement for producing AT2018cow and similar explosions. The
properties of AT2020mrf’s host further support this suggestion. At
$M_{\ast}\sim 10^{8}\,M_{\odot}$, the SFR of AT2020mrf lies below the main-
sequence (MS) of local star-forming galaxies. Moreover, among the 369 PTF/iPTF
normal CCSNe hosted by galaxies with $M_{\ast}<2\times 10^{9}\,M_{\odot}$
(Schulze et al., 2021), the host galaxies of only 30 objects (8%) have ${\rm
sSFR}<8\times 10^{-11}\,{\rm yr^{-1}}$. This indicates that AT2020mrf does not
occur during a vigorous starburst, and that progenitor scenarios with a
slightly longer delay time than that of a typical CCSN are favored. Zapartas
et al. (2017) performed a population synthesis study of CCSNe, finding that a
prolonged delay time can be achieved by binary interactions, through common
envelope evolution, mass transfer episodes, and/or merging. Explosions driven
by the merging of a compact object with a massive star inside a common
envelope have indeed been proposed as promising channels for producing
AT2018cow-like events (Soker et al., 2019; Schrøder et al., 2020; Soker, 2022;
Metzger, 2022).

Among the five AT2018cow-like events, only AT2018lug lies above the local MS
of star-forming galaxies. For comparison, the majority (15/18) of SLSNe-I
presented by Perley et al. (2016a) lie above the local MS666 Compared with
AT2018cow-like events, the sample of SLSNe-I is at slightly higher redshifts
(the median is $z\sim 0.2$). We note that for $M_{\ast}\approx
10^{8}\,M_{\odot}$, the sSFR at $z\approx 0.2$ is only slightly ($\approx 0.2$
dex) higher than that at $z\approx 0$ (Speagle et al., 2014).. We perform a
two-sided Kolmogorov-Smirnov (K-S) test for the null hypothesis that the host
galaxy sSFR of SLSNe-I and AT2018cow-like events are drawn from the same
distribution. The returned $p$-value of 0.23 is too high to reject the null
hypothesis. A larger sample size is clearly needed to test if the host sSFR
between AT2018cow-like events and other powerful massive star explosions are
statistically different.

###  3.4 An Engine Driven Explosion

####  3.4.1 X-ray Properties

We have shown that the radio (§3.1) and early-time optical (§2.2, §3.2)
properties of AT2020mrf are similar to other AT2018cow-like events. Here we
summarize the key X-ray observables of AT2020mrf, and compare them with other
AT2018cow-like events.

At $\sim 36$ days, the mean 0.3–10 keV luminosity of AT2020mrf is $(1.9\pm
0.4)\times 10^{43}\,{\rm erg\,s^{-1}}$, a factor of $\sim 20$ brighter than
AT2018cow and AT2020xnd at similar phases (Figure 7). The best-fit powerlaw of
$f_{\nu}\propto\nu^{-0.8}$ (Figure 5) is similar to the 0.3–10 keV spectral
shape of AT2018cow and AT2020xnd (Margutti et al., 2019; Bright et al., 2022;
Ho et al., 2021b). From 34.5 to 37.6 days, the 0.2–2.2 keV flux varies by a
factor of $\approx 6$ on the timescale of $\approx 1$ day (Figure 4), similar
to the fast soft X-ray variability observed in AT2018cow at similar phases
(Figure 7).

At 328 days, the mean 0.3–10 keV luminosity of AT2020mrf is $\sim 1.4\times
10^{42}\,{\rm erg\,s^{-1}}$, which is $\sim 300$ times brighter than the upper
limit of CSS161010 at 291 days, and $\sim 200$ times brighter than AT2018cow
itself at 212 days. The spectrum of AT2020mrf has probably hardened to
$f_{\nu}\propto\nu^{0}$. From 327.4 to 328.2 days, the X-ray flux decreases by
a factor of $\sim 2.6$.

Among AT2018cow-like events, intraday X-ray variability has only been detected
in AT2018cow and AT20202mrf. This is probably because CSS161010, AT2018lug,
and AT2020xnd were not observed often enough to detect it. The isotropic
equivalent observed X-ray luminosity of AT2020mrf is as luminous as long GRBs.
The X-ray emission of long GRBs are produced by the afterglow synchrotron
radiation of electrons accelerated by a ultra-relativistic shock (Sari et al.,
1998). However, given the lack of a prompt $\gamma$-ray emission (§2.5) and
the sub-relativistic shock velocity (§3.1) observed in AT2020mrf, the nature
of its X-rays should be different from that of long GRBs.

As shown in Figure 7, in AT2018cow (and perhaps AT2020xnd), the 0.3–10 keV
light curve decay steepens from $L\propto t^{-1}$ ($t\lesssim 25$ days) to
$L\propto t^{-4}$ ($25\lesssim t\lesssim 100$ days). The overall decay shape
of AT2020mrf is consistent with a $L\propto t^{-1.3}$ power-law. However, we
can not rule out the existence of a steeper decay (see §3.4.3). Below we
discuss the physical origin of the X-ray emission associated with AT2020mrf.

Figure 16: SED of AT2020mrf. The dashed blue line shows an example synchrotron
spectrum one would expect at the epoch of the Chandra observation ($\Delta
t\approx 328$ days). Here we have assumed $B\sim 0.2$ G, and a cooling
frequency of $\nu_{c}=\gamma_{c}^{2}\nu_{g}\sim 3\times 10^{11}$ Hz, where
$\gamma_{c}=6\pi m_{e}c/(\sigma_{\rm T}B^{2}t)$ and $\nu_{g}=eB/(2\pi m_{e}c)$
(Sari et al., 1998; Granot & Sari, 2002). Therefore, the late-time X-ray
emission is much brighter than the synchrotron spectrum.

####  3.4.2 General Considerations

First, Figure 16 shows that the late-time X-ray luminosity of AT2020mrf is too
bright to be an extension of the radio synchrotron spectrum.

Second, inverse-Compton (IC) scattering of the radiation field (i.e.,
UV/optical photons) by electrons accelerated in the forward shock is found to
be the main early-time ($t\lesssim 40$ days) X-ray emission mechanism for SNe
Ib/c exploding in low-density environments (Fransson et al., 1996; Kamble et
al., 2016). The ratio of IC to synchrotron radiation losses is $P_{\rm
IC}/P_{\rm syn}=u_{\rm rad}/u_{B}$, where $u_{\rm rad}$ is the energy density
in seed photons, and $u_{B}=U_{B}/(4\pi R_{\rm sh}^{3}/3)$. To first order,
$P_{\rm IC}/P_{\rm syn}\sim L_{\rm X}/L_{\rm radio}$. At $\Delta t\approx 36$
days, the bolometric luminosity of the optical transient is $L_{\rm bol}\sim
10^{42.3}\,{\rm erg\,s^{-1}}$ (see Figure 14). Assuming $v_{\rm sh}\sim
0.07$–0.08$c$ (§3.1.1), the shock radius is $R_{\rm sh}\sim 7\times 10^{15}$
cm. Therefore $u_{\rm rad}/L_{\rm X}=L_{\rm bol}/(4\pi cR_{\rm sh}^{2})/L_{\rm
X}\sim(0.1\,{\rm erg\,cm^{-3}})/(2\times 10^{43}\,{\rm erg\,s^{-1}})\sim
5.4\times 10^{-45}\,{\rm s\,cm^{-3}}$. Assuming that the standard SSA model
applies at $\Delta t\approx 36$ days777This assumption will not be accurate if
$v_{\rm sh}\gtrsim 0.2c$, at which condition we expect thermal electrons to
contribute significantly to the synchrotron spectrum (Ho et al., 2021b;
Margalit & Quataert, 2021)., from Equation (3c) we have $u_{B}/L_{\rm
radio}\sim 8\times
10^{-31}L_{\theta\nu,29}^{4/19}(\epsilon_{e}/\epsilon_{B})^{-11/19}\nu_{100}^{-1}>2\times
10^{-31}\,{\rm s\,cm^{-3}}$, where we have assumed that the early-time
synchrotron emission peaks at $\sim 100$ GHz and $L_{\theta\nu}>10^{29}\,{\rm
erg\,s^{-1}\,Hz^{-1}}$. Therefore, $u_{\rm rad}/L_{\rm X}\ll u_{B}/L_{\rm
radio}$, and IC is not likely to be the dominant mechanism for the X-ray
emission. At $\Delta t\approx 328$ days, the observed X-ray spectral shape of
$f_{\nu}\propto\nu^{0}$ is too hard to be consistent with IC.

Finally, X-rays from most normal CCSNe and interacting SNe have been
successfully modeled by thermal bremsstrahlung from supernova reverse-shock-
heated ejecta or the forward-shock-heated CSM (Chevalier & Fransson, 1994;
Dwarkadas & Gruszko, 2012). The shortest variability timescale expected from
clumpy CSM encountered by a forward shock is much slower — $\Delta t/t=v_{\rm
sh}/c\sim 0.1$ (see Section 3.3.1 of Margutti et al. 2019). In contrast, the
X-ray relative variability and flux contrast are $\Delta t/t\approx 0.03$,
$\Delta F/F\approx 2.5$ at $t\approx 36$ days and $\Delta t/t\lesssim 0.003$,
$\Delta F/F\approx 1$ at $t\approx 328$ days.

Some previous studies have interpreted AT2018cow as the tidal disruption of a
white dwarf or star by an IMBH (Kuin et al., 2019; Perley et al., 2019). Since
the observed early-time non-thermal X-ray spectrum and fast variability are
not consistent with observations of thermal X-ray loud TDEs (Sazonov et al.,
2021), the X-rays are thought to be powered by a jet similar to that observed
in the jetted TDE SwiftJ1644 (Burrows et al., 2011; Bloom et al., 2011).
However, for AT2020mrf and AT2018cow-like events in general, the TDE scenario
is disfavored by the dense environment (§3.1) and the host properties (§3.3).

Therefore, the most natural origin of the X-rays in AT2020mrf is a central
compact object — either a neutron star (§3.4.4) or a black hole (§3.4.3) —
formed in a massive star explosion. Since the UV/optical luminosity of
AT2020mrf remains much lower than $L_{\rm X}$ throughout the evolution, we can
assume that the central engine luminosity $L_{\rm e}$ is mostly tracked by
$L_{\rm X}$. The engine timescale is set by the duration of the X-ray emission
$t_{\rm e}>328$ days. The total energy release in the X-ray is $E_{\rm
e}>(2\times 10^{43}\,{\rm erg\,s^{-1}})\times(36\,{\rm days})+(10^{42}\,{\rm
erg\,s^{-1}})\times[(328-36)\,{\rm days}]=9\times 10^{49}\,{\rm erg}$.

####  3.4.3 Stellar Mass Black Hole Engine

The engine of AT2020mrf can be a stellar mass BH, where X-rays are powered by
accretion. The isotropic equivalent luminosity of $10^{42}$–$10^{43}\,{\rm
erg\,s^{-1}}$ corresponds to an Eddington ratio of $L_{\rm engine}/L_{\rm
Edd}>10^{4}$–$10^{3}$ for a $10\,M_{\odot}$ BH, suggesting that the emission
is likely beamed.

In the case of a failed explosion, $t_{\rm e}$ is determined by the free-fall
of the stellar envelope (Quataert & Kasen, 2012; Fernández et al., 2018):

$\displaystyle t_{\rm ff}=\frac{\pi
r^{3/2}}{(2GM_{\star})^{1/2}}=706\left(\frac{r}{10^{14}\,{\rm
cm}}\right)^{3/2}\left(\frac{M_{\star}}{10\,M_{\odot}}\right)^{-1/2}\,{\rm
day}.$ (9)

In order to power AT2020mrf’s X-ray emission out to 328 days, a weakly bound
red supergiant (RSG) progenitor with $r>6\times 10^{13}\,{\rm cm}$ is
required. The amount of mass around the disk circularization radius is much
smaller than that in the stellar envelope, and the fast X-ray variability is
related to the change of angular momentum in the accreting material (Quataert
et al., 2019).

In the case of a successful explosion, the accretion is supplied by fallback
of bound material (Dexter & Kasen, 2013). In compact progenitors such as blue
supergiants (BSGs), a reverse shock decelerates the inner layers of the
ejecta, resulting in enhanced fallback mass (Zhang et al., 2008). The fast
X-ray variability might be caused by disk instability since the viscous time
is much shorter than the fallback time. The temporal coverage of our X-ray
data is poor. It is possible that $L_{\rm e}$ decays shallower than $t^{-1.3}$
initially, followed by a steeper decay (e.g. $L_{\rm e}\propto t^{-5/3}$) due
to fallback. This might be consistent with a range of SN energies, with lower
energies corresponding to later transition times between an early less steep
light curve to a later steeper fallback light curve (Quataert & Kasen, 2012).

####  3.4.4 Millisecond Magnetar Engine

Another speculation is that the engine of AT2020mrf is a young magnetar (i.e.,
an extremely magnetized neutron star), where $L_{\rm e}$ is primarily provided
by rotational energy loss due to spindown. For a neutron star with a spin
period of $P_{\rm ms}\equiv P/(1\,{\rm ms})$ and a mass of $1.4\,M_{\odot}$,
the rotational energy is $E_{\rm rot}\approx 2.5\times 10^{52}P_{\rm
ms}^{-2}\,{\rm erg}$ (Kasen & Bildsten, 2010; Kasen, 2017). The spin period
required to power $E_{\rm e}$ is thus $P\lesssim 17$ ms. If the NS has a
radius of $10$ km and a magnetic field of $B_{14}\equiv B/(10^{14}\,{\rm G})$,
the characteristic spindown timescale is $t_{\rm spindown}\approx
0.5B_{14}^{-2}P_{\rm ms}^{2}$ day. The luminosity extracted from spindown is
roughly constant when $t\lesssim t_{\rm spindown}$, and decays as $L_{\rm
e}\propto t^{-2}$ afterwards. Extrapolating the Chandra detection back to the
SRG luminosity suggests that the transition occurs at $\sim 73$ days, which
implies $B\lesssim 1.4\times 10^{14}$ G. This is similar to the $B$ field
required to power AT2018cow inferred by Margutti et al. (2019).

In this scenario, X-rays are generated in a “nebula” region of
electron/positron pairs and radiation inflated by a relativistic wind behind
the SN ejecta (Vurm & Metzger, 2021). Additional energy injection by fallback
accretion widens the parameter space of magnetar birth properties, and
predicts a late-time light curve decay shallower or steeper than $L_{\rm
e}\propto t^{-2}$ (Metzger et al., 2018). The day-timescale X-ray variability
can be accounted for by magnetically driven mini-outbursts.

##  4 The Detection Rate in X-ray Surveys

Table 7: The detection rates ($\dot{N}_{\rm det}$ in yr-1) of events similar
to AT2018cow and AT2020mrf in X-ray surveys, under three different assumptions
of the event volumetric rates ($\mathcal{R}$ in ${\rm Gpc^{-3}\,yr^{-1}}$).

Survey | $f_{-13}$ | $D_{\rm max}$ |  $\dot{N}_{\rm det}$ if $\mathcal{R}=2.1$ |  $\dot{N}_{\rm det}$ if $\mathcal{R}=70$ |  $\dot{N}_{\rm det}$ if $\mathcal{R}=420$
---|---|---|---|---|---
SRG/eROSITA | 1.8 | 373 | 0.080 | 2.7 | 16
964 | 1.7 | 57 | 340
Einstein Probe | 20 | 112 | 0.012 | 0.41 | 2.5
289 | 0.21 | 7.1 | 43

Note. — $D_{\rm max}$ is given in Mpc. The values with yellow background
assume an X-ray light curve shape similar to AT2018cow. The values given in
boldface assume a conservative light curve shape similar to AT2020mrf, and
therefore the derived $\dot{N}_{\rm det}$ should be taken as lower limits.

AT2020mrf is the first multi-wavelength FBOT identified from X-ray surveys.
This motivates us to estimate the rate of such events in present and future
X-ray surveys. The core collapse SN rate is $R=7\times 10^{4}\,{\rm
Gpc^{-3}\,yr^{-1}}$ (Li et al., 2011). The birthrate of 18cow-like events
estimated by ZTF is $\mathcal{R}=3\times 10^{-5}$–$6\times 10^{-3}R$ (Ho et
al., 2021a), or 2.1–$420\,{\rm Gpc^{-3}\,yr^{-1}}$.

Here we assume that a multi-wavelength FBOT has an X-ray light curve either
similar to AT2018cow itself or similar to AT2020mrf. We approximate the 0.3–10
keV X-ray luminosity of AT2018cow as a plateau with a luminosity of $L_{\rm
X,p0}=3\times 10^{42}\,{\rm erg\,s^{-1}}$ and a duration of $t_{\rm X,p0}=30$
days (Figure 7). The light curve shape of AT2020mrf is less well constrained.
For simplicity, we assume a conservative shape consisting of two plateaus,
with $L_{\rm X,p1}=2\times 10^{43}\,{\rm erg\,s^{-1}}$, $t_{\rm X,p1}=36$
days, $L_{\rm X,p2}=1\times 10^{42}\,{\rm erg\,s^{-1}}$, and $t_{\rm
X,p2}=350$ days.

The transient detection rate is

$\displaystyle\dot{N}_{\rm det}=\frac{\Omega}{3}D_{\rm
max}^{3}\mathcal{R}\cdot p_{\rm s}$ (10)

where $\Omega$ is the solid angle of the surveyed area ($\Omega=4\pi$ for an
all-sky survey), $D_{\rm max}$ is the maximum distance out to which the source
can be detected, and $p_{\rm s}$ is the probability that the transient is “on”
when being scanned by the X-ray survey. If the survey cadence is shorter than
the transient duration, $p_{\rm s}=1$. Setting a survey flux threshold of
$f_{\rm thre}=10^{-13}f_{-13}\,{\rm erg\,cm^{-2}\,s^{-1}}$, we have $4\pi
D_{\rm max}^{2}f_{\rm thre}=L_{\rm X,p}$.

On average, every 0.5 yr, SRG/eROSITA samples the same region of the sky over
$\sim 12$ passes within $\sim 2$ days. For a single event, somewhere on the
sky, with an X-ray light curve shape similar to AT2018cow, the probability of
being imaged by SRG during its X-ray active phase is $p_{\rm
s0}=2\times(t_{\rm X,p0}+2)/365=0.175$. For a light curve shape similar to
AT2020mrf, $p_{\rm s1}=2\times(t_{\rm X,p1}+2)/365=0.208$, and $p_{\rm s2}\sim
1$. The sensitivity of an eROSITA single sky survey is $\approx 2.5\times
10^{-14}\,{\rm erg\,s^{-1}\,cm^{-2}}$ (see Fig. 17 of Sunyaev et al. 2021). In
reality, to be selected as a transient by eRASS$n$ ($n>1$), the source needs
to exceed the eRASS1 sensitivity limit by a factor of $\approx 7$. Therefore,
the flux threshold is $f_{-13}\approx 1.8$.

Einstein Probe (EP) is a lobster-eye telescope for monitoring the X-ray sky
(Yuan et al., 2018) to be launched at the end of 2022. With an orbital period
of 97 min, the entire sky can be covered over three successive orbits. Here we
assume that its Wide-field X-ray telescope (WXT) is 2 orders of magnitude more
sensitive than the Monitor of All-sky X-ray Image (MAXI) mission888From slide
#32 of https://sites.astro.caltech.edu/~srk/XC/Notes/EP_20200923.pdf. MAXI has
a transient triggering threshold of 8 mCrab for 4 days (Negoro et al., 2016),
leading us to assume $f_{-13}\approx 20$ for EP.

The calculated detection rates in eROSITA and EP are summarized in Table 7.
The rate of similar events in present and future millimeter transient surveys
is given by Ho et al. (2021b) and Eftekhari et al. (2021).

##  5 Summary

We report multi-wavelength observations of AT2020mrf, the fifth member of the
class of AT2018cow-like events (i.e., FBOTs with luminous multi-wavelength
counterparts). Among the four 18cow-like events ever detected in the X-ray
(i.e., AT2018cow, CSS161010, AT2020xnd, AT2020mrf), AT2020mrf is the most
luminous object, exhibiting day-timescale X-ray variability both at early
($\approx 36$ days) and late times ($\approx 328$ days), with a luminosity
between $10^{42}$ and ${\rm few}\times 10^{43}\,{\rm erg\,s^{-1}}$.
Previously, the only object showing evidence of a NS/BH central engine was
AT2018cow (Margutti et al., 2019; Pasham et al., 2021). Here we show that a
compact object — a young millisecond magnetar or an accreting black hole — is
required to be the central energy source of AT2020mrf (see §3.4).

AT2020mrf also provides accumulating evidence to show that AT2018cow-like
events form another class of engine-driven massive star explosions, after long
GRBs and SLSNe-I. Intriguingly, all three classes of events are preferentially
hosted by dwarf galaxies. Given the MZR (Gallazzi et al., 2005; Berg et al.,
2012; Kirby et al., 2013), low metallicity probably plays an important role in
the formation of such exotic explosions by reducing angular momentum loss of
their progenitors (Kudritzki & Puls, 2000). Local environment studies with
integral-field unit (IFU) observations (e.g., Lyman et al. 2020) and high
spatial resolution images (e.g., with the Hubble Space Telescope) can further
illuminate the nature of their progenitors.

Although AT2018cow, AT2018lug, and AT2020xnd are FBOTs with $-20.5<M_{g,\rm
peak}<-21.5$ and $t_{1/2}<5$ day, the optical light curve of AT2020mrf is of
lower peak luminosity ($M_{g,\rm peak}=-20$) and slower evolution timescale
($t_{1/2}=7$ days). This should guide searches of such events in optical wide
field surveys to be more agnostic of the light curve decay rate. Real-time
identification of FBOTs and comprehensive spectroscopic follow up observations
are necessary to distinguish between different emission mechanisms: shock
interaction with extended CSM, radioactivity, or wind reprocessing (§3.2.3).
The discovery of X-ray emission in AT2020mrf also showcases how X-ray surveys
such as SRG can be essential in the identification of multi-wavelength FBOTs.

Once identified, millimeter and radio follow-up observations are needed to
reveal the CSM density as a function of distance to the progenitor, which
contains information about the mass-loss history (§3.1). X-ray light curves
provide diagnostics for the nature of the power source (§3.4), while broad-
band X-ray spectroscopy can constrain the evolution of the geometry of the
material closest to the central engine (Margutti et al., 2019). Given the
late-time X-ray detections of AT2018cow at $\Delta t\approx 212$ days
(Appendix A) and of AT2020mrf at $\Delta t\approx 328$ days (§2.4), future
Chandra observations of these two objects may further constrain the timescales
of their central engines.

Acknowledgements – We thank Patrick Slane for allocating DD time on Chandra.
We thank the staff of Chandra, VLA, Keck, and GMRT that made these
observations possible. We thank Jim Fuller, Mansi Kasliwal, Wenbin Lu, Tony
Piro, and Eliot Quataert for helpful discussions. We thank the anonymous
referee for constructive comments and suggestions. Y.Y. thanks Eric Burns for
discussion about IPN, and Dmitry Svinkin for providing information about
Konus-WIND.

Support for this work was provided by the National Aeronautics and Space
Administration (NASA) through Chandra Award Number DD1-22133X issued by the
Chandra X-ray Observatory Center, which is operated by the Smithsonian
Astrophysical Observatory for and on behalf of NASA under contract NAS8-03060.

Y.Y. acknowledges support by the Heising-Simons Foundation. Nayana A.J. would
like to acknowledge DST-INSPIRE Faculty Fellowship (IFA20-PH-259) for
supporting this research. P.C. acknowledges support of the Department of
Atomic Energy, Government of India, under the project no.
12-R&D-TFR-5.02-0700. P.M., M.G., S.S., G.K. and R.S. acknowledge the partial
support of this research by grant 21-12-00343 from the Russian Science
Foundation.

GMRT is run by the National Centre for Radio Astrophysics of the Tata
Institute of Fundamental Research.

This work is based on observations with the eROSITA telescope on board the SRG
observatory. The SRG observatory was built by Roskosmos in the interests of
the Russian Academy of Sciences represented by its Space Research Institute
(IKI) in the framework of the Russian Federal Space Program, with the
participation of the Deutsches Zentrum für Luft- und Raumfahrt (DLR). The
SRG/eROSITA X-ray telescope was built by a consortium of German Institutes led
by MPE, and supported by DLR. The SRG spacecraft was designed, built, launched
and is operated by the Lavochkin Association and its subcontractors. The
science data are downlinked via the Deep Space Network Antennae in Bear Lakes,
Ussurijsk, and Baykonur, funded by Roskosmos. The eROSITA data used in this
work were processed using the eSASS software system developed by the German
eROSITA consortium and proprietary data reduction and analysis software
developed by the Russian eROSITA Consortium.

The ZTF forced-photometry service was funded under the Heising-Simons
Foundation grant #12540303 (PI: Graham). This work has made use of data from
the ATLAS project, which is primarily funded to search for near earth
asteroids through NASA grants NN12AR55G, 80NSSC18K0284, and 80NSSC18K1575;
byproducts of the NEO search include images and catalogs from the survey area.
This work was partially funded by Kepler/K2 grant J1944/80NSSC19K0112 and HST
GO-15889, and STFC grants ST/T000198/1 and ST/S006109/1. The ATLAS science
products have been made possible through the contributions of the University
of Hawaii Institute for Astronomy, the Queen’s University Belfast, the Space
Telescope Science Institute, the South African Astronomical Observatory, and
The Millennium Institute of Astrophysics (MAS), Chile.

##  Appendix A XMM-Newton Late-time Detection of AT2018cow

Figure 17: XMM-Newton/pn 0.3–10 keV image centered on AT2018cow, obtained at
$\Delta t=212$ days. The solid circle is the source region with $r_{\rm
src}=20^{\prime\prime}$, and the dashed circle is the background region with
$r_{\rm bkg}=30^{\prime\prime}$.

AT2018cow was observed by XMM-Newton/EPIC on three epochs (PI Margutti) at
rest-frame 29.6, 78.1, and 211.8 days since explosion. The first two epochs
yielded clear X-ray detections, which have been reported by Margutti et al.
(2019). Pasham et al. (2021) analyzed the 0.25–2.5 keV EPIC/MOS1 data of the
third epoch, and reported a non detection. Here we analyze the third epoch
EPIC/pn data to derive the flux (or upper limit) in 0.3–10 keV, which is
important to be compared with the late-time X-ray detection of AT2020mrf. The
pn instrument generally has better sensitivity than MOS1 and MOS2.

We reduced the pn data using the XMM-Newton Science Analysis System (SAS) and
relevant calibration files. Events were filtered with the conditions
PATTERN<=4 and (FLAG&0xfb0825)==0. We removed high background time windows and
retained 43178 s good times among the total exposure time of of 53163 s.
Following Margutti et al. (2019), we extracted the source using a circular
region with a radius of $r_{\rm src}=20^{\prime\prime}$ to avoid contamination
from a nearby source located $36.8^{\prime\prime}$ southwest form AT2018cow.
The background is extracted from a source-free circular region with a radius
of $r_{\rm bkg}=30^{\prime\prime}$ on the same CCD (see Figure 17).

Figure 18: Dense CSM shock breakout and cooling model fit to the multi-band
light curve of AT2020mrf. The maximum a posteriori model is shown via solid
lines.

The average count rate of the source is $0.00486\,{\rm count\,s^{-1}}$. The
average count rate of the background (multiplied by $r_{\rm src}^{2}/r_{\rm
bkg}^{2}$ to match the area of the source region) is $0.00360\,{\rm
count\,s^{-1}}$. Therefore, AT2018cow is detected at a (Gaussian equivalent)
confidence limit of $4.2\sigma$. Assuming an absorbed power-law model with
$\Gamma\approx 2$ and $N_{\rm H}\approx 7\times 10^{20}\,{\rm cm^{-2}}$, the
0.3–10 keV flux is $\sim 1.6\times 10^{-14}\,{\rm erg\,s^{-1}\,cm^{-2}}$,
corresponding to a luminosity of $\sim 7\times 10^{39}\,{\rm erg\,s^{-1}}$.

##  Appendix B A Sample of GRB X-ray Light Curves

The sample of GRB light curves shown in Figure 7 is collected as follows. We
start with the list of GRBs given by the Swift GRB
Table999https://swift.gsfc.nasa.gov/archive/grb_table/fullview/.. Next, we
retain the 339 long GRBs ($T_{90}>2\,{\rm s}$) with reported redshifts. After
that, we require the last Swift/XRT detection to be at $[(t-T_{0})/(1+z)]>20$
days, where $T_{0}$ is the GRB trigger time. This step selects 12 events,
including GRB171205A ($z=0.0368$), GRB190829A ($z=0.078$), GRB180728A
($z=0.12$), GRB161219B ($z=0.15$), GRB130427A ($z=0.34$), GRB061021
($z=0.35$), GRB091127 ($z=0.49$), GRB060729 ($z=0.54$), GRB090618 ($z=0.54$),
GRB090424 ($z=0.54$), GRB080411 ($z=1.0$), and GRB100814A ($z=1.4$). We
supplement the XRT light curves with deep late-time X-ray observations
reported in the literature (Grupe et al., 2010; De Pasquale et al., 2017).

##  Appendix C Modeling the Optical Light Curve with CSM SBO+SCE

Figure 19: Corner plot showing the posterior constraints on the model
parameters. Each parameter is marginalized over $\sigma_{0}$.

For simplicity, we adopt the one-zone model presented in Appendix A of
Margalit (2021) to fit the optical light curve of AT2020mrf. Following Yao et
al. (2019), we add a constant additional variance $\sigma_{0}^{2}$ to each of
the measurement variance $\sigma_{i}^{2}$ to account for systematic
uncertainties. The multi-band light curves are parameterized using five free
parameters: $t_{0}$, $t_{\rm dyn}$, $t_{\rm a}$, $\beta$, and $E_{0}$ (see
Table 1 of Margalit 2021 for the definitions of these variables). The best-fit
model is shown in Figure 18, and the posterior distribution is shown in Figure
19.

We are not able to obtain a decent fit to the observed light curves. This is
due to the fact that in the CSM shock breakout and cooling model, the light
curve decay can not be significantly slower than the rise, making it
difficulty to reproduce the “flux excess” observed at $\Delta t\sim$15–35
days.

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