# Deep optical observations contemporaneous with emission from the periodic
FRB 180916.J0158+65

Charles D. Kilpatrick Center for Interdisciplinary Exploration and Research in
Astrophysics (CIERA) and Department of Physics and Astronomy, Northwestern
University, Evanston, IL 60208, USA  Joseph N. Burchett  Astronomy Department,
Box 30001, Department 4500, New Mexico State University, Las Cruces, NM
88003-0001  David O. Jones  Department of Astronomy and Astrophysics,
University of California, Santa Cruz, CA 95064, USA  Ben Margalit  Astronomy
Department and Theoretical Astrophysics Center, University of California,
Berkeley, Berkeley, CA 94720, USA  Russet McMillan  Apache Point Observatory
and New Mexico State University, P.O. Box 59, Sunspot, NM, 88349-0059, USA
Wen-fai Fong  Center for Interdisciplinary Exploration and Research in
Astrophysics (CIERA) and Department of Physics and Astronomy, Northwestern
University, Evanston, IL 60208, USA  Kasper E. Heintz  Centre for Astrophysics
and Cosmology, Science Institute, University of Iceland, Dunhagi 5, 107
Reykjavík, Iceland  Nicolas Tejos  Instituto de Física, Pontificia Universidad
Católica de Valparaíso, Casilla 4059, Valparaíso, Chile  Alicia Rouco Escorial
Center for Interdisciplinary Exploration and Research in Astrophysics (CIERA)
and Department of Physics and Astronomy, Northwestern University, Evanston, IL
60208, USA

###### Abstract

We present deep Apache Point Observatory optical observations within seconds
of the outburst of the periodic fast radio burst (FRB) 180916.J0158+65
obtained on 3 September 2020. FRB 180916.J0158+65 is located in a nearby
spiral galaxy 150 Mpc away and has an “active phase” with a well-measured
period of approximately 16.3 days. Targeting the FRB at the peak of its
expected active phase and during a recent 30 minute observing window by the
Canadian Hydrogen Intensity Mapping Experiment (CHIME) in which a radio burst
was detected, we did not detect any transient optical emission at
$m_{i}\approx$24.7 mag (3$\sigma$) from 2.2 to 1938.1 seconds after the burst
arrival time in optical bands (corrected for dispersion). Comparing our
limiting magnitudes to models of a synchrotron maser formed in the circumburst
environment of FRB 180916+J0158.65, we constrain scenarios where the burst
energy was $>10^{44}$ erg and the circumburst density was $>$104 cm-3.

fixltx2e firstofone

##  1 Introduction

Fast radio bursts (FRBs) are millisecond timescale (Lorimer et al., 2007)
bursts of MHz–GHz radio emission from extragalactic sources (Thornton et al.,
2013; CHIME/FRB Collaboration et al., 2019; Cordes & Chatterjee, 2019; Petroff
et al., 2019a, and references therein). The detection of the first repeating
FRB 121102 (Spitler et al., 2014, 2016) enabled its precise localization in a
host galaxy at $z=0.193$ (Chatterjee et al., 2017). There may also be a non-
repeating population of FRBs (e.g., Ai et al., 2020; Hashimoto et al., 2020),
but the observed FRB rate and luminosity function suggests that most of these
sources must be repeating (whether or not they are observed to repeat, e.g.,
Ravi et al., 2019; Caleb et al., 2019). A handful of these FRBs with multiple
detections have been accurately localized and securely associated with host
galaxies, spanning a wide range of host types from star forming (FRB 121102
and FRB 190711; Chatterjee et al., 2017; Macquart et al., 2020; Li & Zhang,
2020; Kumar et al., 2020) to quiescent (FRB 180916 and FRB 180924 with a star-
formation rate $<2~{}M_{\odot}$ yr-1 at $z=0.3214$; Bannister et al., 2019;
Marcote et al., 2020) galaxies (see also Bhandari et al., 2018; Heintz et al.,
2020).

Several hypotheses have been proposed for FRB progenitor systems (Platts et
al., 2019), from eruptions on the surfaces of highly-magnetized neutron stars
(i.e., magnetars; Popov & Postnov, 2013; Lyubarsky, 2014; Kulkarni et al.,
2014; Katz, 2016; Beloborodov, 2017; Kumar et al., 2017; Metzger et al., 2017;
Wadiasingh & Timokhin, 2019; Beniamini et al., 2020) to accretion-induced
collapse of neutron stars into black holes (Falcke & Rezzolla, 2014). Notably,
repeating FRBs exhibit pulse widths $>$4$\sigma$ longer than those without
multiple detected bursts (CHIME/FRB Collaboration et al., 2019), suggesting
that there may be multiple progenitor channels. In addition, the recent
detection of an FRB-like event from the Galactic magnetar SGR 1935+2154
suggests that at least a subset of repeating, extragalactic FRBs originate
from magnetars (CHIME/FRB Collaboration et al., 2020a; Bochenek et al., 2020;
Lu et al., 2020; Margalit et al., 2020a). This interpretation is complicated
in part by evidence that FRB hosts lack a clear association to star formation
(although some studies suggest FRBs can be produced via magnetars that do not
trace star formation; see, e.g., Margalit et al., 2019; Safarzadeh et al.,
2020; Bochenek et al., 2020).

Precise localization of FRBs has also enabled both targeted and untargeted
follow up at wavelengths spanning from optical (Hardy et al., 2017; Bhandari
et al., 2018; Andreoni et al., 2020) to X-ray (Petroff et al., 2015; Scholz et
al., 2016; Pilia et al., 2020; Tavani et al., 2020; Scholz et al., 2020) to
gamma-ray wavelengths (Yamasaki et al., 2016; Best & Bazo, 2019; Cunningham et
al., 2019; Guidorzi et al., 2019, 2020), primarily for the well-localized,
repeating bursts FRB 121102 and FRB 180916. So far these searches have
revealed no potential counterparts for extragalactic FRBs (e.g., Chen et al.,
2020) apart from a 100 s duration gamma-ray transient detected at the
3.2$\sigma$ level by the Neil Gehrels Swift Observatory (Swift; Gehrels et
al., 2004) and contemporaneous with a burst from FRB 131104 (DeLaunay et al.,
2016), although the association may not be secure (Shannon & Ravi, 2017; Gao &
Zhang, 2017).

Theoretical models predict optical emission spanning faint luminosities of
$<10^{39}$ erg s-1 within seconds of the burst via synchrotron emission in the
circumburst medium of the progenitor (Metzger et al., 2019). However, the best
limits from optical follow up observations have only ruled out counterparts to
bursts down to relatively unconstraining luminosities of $\approx$1045 erg s-1
(Hardy et al., 2017) within milliseconds of a burst or $\approx$5$\times$1042
erg s-1 within minutes of a burst (Andreoni et al., 2020). Recent constraints
on the periodic activity window of the repeating FRB 180916.J0158+65
(hereafter FRB 180916; CHIME/FRB Collaboration et al. 2020b) offer a unique
opportunity to target emission from a FRB counterpart at non-radio
wavelengths. In addition to untargeted optical observations from the Zwicky
Transient Facility (Andreoni et al., 2020), Pilia et al. (2020) and Zampieri
et al. (2020) reported high-speed optical observations with the 1.2m Galileo
telescope during a burst that limit the Sloan $i$-band fluence of FRB 180916
to $<$0.151 Jy ms. Additional multi-band follow up of this source will further
benefit from the known 16.3 day period, suggesting that the counterpart is
highly active on this timescale (CHIME/FRB Collaboration et al., 2020b). The
relatively low redshift ($z=0.0337$ with an implied luminosity distance of
$D_{L}\approx 150$ Mpc; CHIME/FRB Collaboration et al., 2020b) means that any
observations will yield significantly deeper constraints than those for FRBs
previously targeted at optical wavelengths (e.g., FRB 121102; Hardy et al.,
2017; Bhandari et al., 2018).

Here we discuss targeted optical follow up of FRB 180916 with the Apache Point
Observatory (APO) 3.5m telescope. These observations were obtained
contemporaneously with observations from the Canadian Hydrogen Intensity
Mapping Experiment (CHIME) on 3 September 2020. During the observations, CHIME
detected a radio burst at a location and dispersion measure consistent with
previous bursts from FRB 180916 (CHIME/FRB Collaboration et al., 2020b;
Marcote et al., 2020). At the time of the burst and for 30 minutes thereafter,
we observed the FRB 180916 host galaxy with the APO telescope, but we did not
detect any transient optical emission at the FRB site. Based on the non-
detections, we place constraints on the allowed burst properties and
circumburst density in the synchrotron maser model. We discuss the timing and
details of our observations in Section 2. In Section 3, we analyze these
limits in the context of realistic optical counterparts to FRBs and discuss
the implications of these limits for future follow up efforts. We summarize
our findings in Section 4.

Figure 1: Upper row: $r$-band image starting from +2.2 s relative to the
dispersion-corrected burst arrival time, our $r$-band template from 13
September 2020, and the difference between the two frames. There is no
signature of transient emission at the FRB location (red circles). Middle row:
Same as the top row for our $g$-band image starting at +140.0 s. Bottom row:
Same as the top row for our $i$-band image starting at +329.5 s. Right: RGB
($igr$) image showing all of our stacked, post-burst data and centered on the
location of the burst.

##  2 Observations

We targeted FRB 180916 on 3 September 2020 with the APO 3.5m telescope,
mounted with the Astrophysical Research Consortium Telescope Imaging Camera
(ARCTIC; Huehnerhoff et al., 2016). Our observations began (i.e., the camera
shutter opened) at UTC 2020-09-03 11:05:39.7503. Each exposure sequence was a
$3\times 30.33$ s set of images in a single band (except for the third and
fourth set of $g$-band exposures, which consisted of 2 and 4 exposures,
respectively), following a $i\rightarrow r\rightarrow g$ pattern for 9 images
per pattern or 36 images over the full set of observations (Table 1). The
average time per exposure is 30.3 s over 2221.7 s for an observing efficiency
of $\approx$49%. In addition, we obtained follow up observations on 13
September 2020 in $gri$ bands with $3\times 100$ s exposures to use as
templates for comparison to the previous epoch. We show example $r$ and
$i$-band images obtained within $2$ min from the topocentric radio burst
arrival time in Figure 1.

The radio burst occurred during our first $gri$ sequence when the second
$r$-band image was exposing, implying that 32 out of 36 of our exposures
occurred around or after the burst arrival time. Based on the topocentric
burst time at 400 MHz provided by CHIME1112020-09-03T11:10:32.495964Z;
https://www.chime-frb.ca/repeaters/180916.J0158+65, the radio burst arrival
time was 6.908 s after the shutter opened and 23.442 s before the shutter
closed for this exposure. Thus including the second $r$-band exposure,
approximately $32\times 30.33$ s of cumulative exposure time was obtained
during or immediately after the time of burst. The exact time the shutter
opened and closed is given in Table 1 along with the relative time (in
seconds) from the radio burst arrival.

The time of arrival for an FRB is frequency ($\nu$) dependent following a
$\nu^{-2}$ dispersion law (Tanenbaum et al., 1968). Variation in the burst
arrival time is used to quantify the dispersion measure (DM) along the line of
sight to the burst, a value that correlates with the column of electrons to
the source and thus can be used to measure electron densities in the
intergalactic medium (Petroff et al., 2016; Shannon et al., 2018; Macquart &
Ekers, 2018; Petroff et al., 2019b; Macquart et al., 2020). The DM for the FRB
180916 burst on 3 September 2020 was $352.6\pm 3.2$ pc cm-3, consistent with
previous measurements toward the repeater (CHIME/FRB Collaboration et al.,
2020b; Marcote et al., 2020). Based on this measurement, we can estimate the
arrival time for the corresponding $r$-band optical emission from the FRB
observed at $400$ MHz, which would be earlier as it occurred at a higher
frequency. Following equation (1) in Cordes & Chatterjee (2019) (see also
Tanenbaum et al., 1968), we estimate that the burst arrival time in the
optical was 9.1 s earlier than at 400 MHz, implying an arrival time of UTC
2020-09-03 11:10:23.4, which we use throughout this paper as the reference
point for the “dispersion-corrected burst arrival time.”

We reduced all ARCTIC data using a custom-built pipeline based on the photpipe
imaging and photometry package (Rest et al., 2005; Kilpatrick et al., 2018).
Each frame was corrected for bias and flat-fielded using bias and sky flat
field frames obtained in the same instrumental configuration. We registered
the images using 2MASS astrometric standards (Skrutskie et al., 2006) observed
in the field of each image. Finally, we performed point spread function (PSF)
photometry using DoPhot (Schechter et al., 1993) and calibrated the $gri$ data
using PS1 DR2 standard stars in each image (Flewelling et al., 2016). We
subtracted the observations taken on 13 September 2020 from all 3 September
2020 observations using HOTPANTS (Becker, 2015) to perform PSF convolution and
difference imaging and then estimated the 3$\sigma$ limiting magnitude at the
position of FRB 180916 (from Marcote et al., 2020) with fake star injection.
Thus the limits we derive can be interpreted as the maximum average in-band
specific flux integrated over the 30.33 s window for each exposure or the
entire observation window for the limits in the stacked data. As shown in
Table 1, the typical limiting magnitude of each individual frame was
$\approx$24.5 mag in $gri$ bands or $\approx$26.0 mag in the stacked frames.

##  3 Constraining the FRB Emission Mechanism

The timeline for our exposures immediately after the radio burst is shown in
Figure 2. We estimate based on the 9.1 s dispersive delay of the radio
emission that the FRB occurred approximately 2.2 s before the shutter opened
for our second $r$-band exposure. Thus our closest and most constraining image
of FRB 180916 covered roughly 2.2 to 30.5 seconds relative to the burst
arrival time. Our full, post-burst data set covers approximately +2.2 s to
+1938.1 s relative to the dispersion-corrected burst arrival time in $gri$
(Table 1). We also give the limiting magnitudes for times before the expected
burst arrival time, but in the context of the models below, we do not consider
these data as no optical emission is predicted.

We give the 3$\sigma$ limiting magnitude in AB magnitudes both for a source at
the location of FRB 180916 in each image and in absolute magnitudes after
correcting for the distance modulus and Milky Way foreground extinction
$A_{V}=2.767$ mag (from Schlafly & Finkbeiner, 2011). We assume the redshift
$z=0.0337\pm 0.0002$ derived for the host galaxy of FRB 180916 in Marcote et
al. (2020) along with Planck Collaboration et al. (2016) cosmology, from which
we derive a luminosity distance of $D_{L}=153\pm 1$ Mpc or a distance modulus
of $\mu=35.92\pm 0.02$ mag.

We compare these limits to predictions of the synchrotron maser model as shown
in Metzger et al. (2019); Margalit et al. (2020b) (but also see Lyubarsky,
2014; Beloborodov, 2017, 2020, for slightly different optical predictions,
especially at times comparable to the burst duration when the total luminosity
may be larger). In the Metzger et al. (2019) formalism for this model, the
radio burst originates in a shock from a radially-expanding plasmoid launched
from a central engine (e.g., a magnetar). The relativistic plasmoid may be
decelerated by surrounding material in the immediate environment of the
engine. If this material is sufficiently magnetized, synchrotron maser
emission will be produced (Plotnikov & Sironi, 2019). In the model of Metzger
et al. (2019); Margalit et al. (2020b) (and first proposed by Beloborodov
2017) the surrounding upstream material is baryon-loaded ejecta expelled in
previous flaring activity of the magnetar.

One prediction of this model (see Section 4 in Metzger et al., 2019) is that
there should be a broadband (incoherent) synchrotron afterglow that will
accompany and follow the FRB. On timescales similar to the FRB duration, this
afterglow will peak in hard X-rays/gamma-rays, but it can subsequently cascade
through optical bands on timescales of minutes post-burst. Assuming a plasmoid
ejection event with energy $E_{\mathrm{flare}}$ that lasts for a duration
$\delta t$, and a fractional magnetization $\sigma$ in the material upstream
from the forward shock, the peak frequency $\nu_{\mathrm{syn}}$ of this
synchrotron afterglow will vary with time $t$ from the burst event
approximately as (following equations (56)-(57) in Metzger et al., 2019)

$\displaystyle h\nu_{\mathrm{syn}}(t_{\mathrm{dec}})$ $\displaystyle=$
$\displaystyle
57~{}\mathrm{MeV}\left(\frac{\sigma}{0.1}\right)^{1/2}\left(\frac{E}{10^{43}~{}\mathrm{erg}}\right)^{1/2}\left(\frac{\delta
t}{10^{-3}~{}\mathrm{s}}\right)^{-3/2}$ (1) $\displaystyle
h\nu_{\mathrm{syn}}$ $\displaystyle=$
$\displaystyle\begin{cases}h\nu_{\mathrm{syn}}(t_{\mathrm{dec}})\left(\frac{t}{t_{\mathrm{dec}}}\right)^{-1},&t<t_{\mathrm{dec}}\\\
h\nu_{\mathrm{syn}}(t_{\mathrm{dec}})\left(\frac{t}{t_{\mathrm{dec}}}\right)^{-3/2},&t>t_{\mathrm{dec}}\end{cases}$
(2)

where $t_{\mathrm{dec}}\approx\delta t\approx 10^{-4}~{}\text{s}$, which is a
fiducial parameter and can be set to the observed duration of the burst. The
peak synchrotron frequency cascades down to the synchrotron cooling frequency
($\nu_{c}$), which depends on properties of the circumburst material.
Motivated by constraints on the engine of FRB121102 (Margalit & Metzger,
2018), Metzger et al. (2019) considered previously ejected baryonic-shells as
the circumburst material, and parameterized $\nu_{c}$ in terms of the velocity
of the ejected material ($\beta=v/c$), the average rate at which this material
is injected into the surrounding medium ($\dot{M}$), and the characteristic
time between ejection events ($\Delta T$). Following equation (60) in Metzger
et al. (2019), the cooling frequency is

$h\nu_{c}=9~{}\mathrm{keV}\left(\frac{\sigma}{0.1}\right)^{-3/2}\left(\frac{\beta}{0.5}\right)^{3}\left(\frac{\dot{M}}{10^{21}~{}\text{g
s}^{-1}}\right)^{-1}\\\
\left(\frac{t}{10^{-3}~{}\text{s}}\right)^{-1/2}\left(\frac{\Delta
T}{10^{5}~{}\text{s}}\right)^{2}.$ (3)

Critically, the average period $\Delta T$ between bursts, although directly
observable and constrained for FRB 180916 as 16.3 days (CHIME/FRB
Collaboration et al., 2020b), simply depends on the circumburst density under
the assumption that the surrounding medium is filled by ions from previous
mass ejection events. In Equation 3, we assume that this medium is
characterized by a series of discrete ion shells with a number density of ions
in the surrounding medium $n_{\mathrm{ext}}\propto(\Delta T)^{-2}$ where the
density profile with radius $r$ from the source of the burst is
$n_{\mathrm{ext}}\propto r^{-k}$. For the discrete shells case, we adopt a
$k=0$ following the prescription in Margalit et al. (2020b).

We find that this circumburst density profile provides a natural model for the
environment of a source with episodic ejections, but this profile need not be
the case if the FRB progenitor erupts inside of a steady wind ($k=2$), a low-
density, ambient medium, or homologously expanding shells of ejecta from a
supernova. These density profiles would imply significantly different optical
evolution than predicted here, which in general is less constraining as the
circumburst density at the radius of the shock drops significantly on
timescales much longer than the burst duration. In addition, although the
radio burst itself would probe the density of the circumburst medium out to a
radius $r_{\mathrm{dec}}\approx 2\Gamma^{2}ct_{\mathrm{dec}}$ ($\approx
10^{12}$–$10^{13}$ cm for $t_{\mathrm{dec}}=10^{-3}$ s and a Lorentz factor
$\Gamma$; Metzger et al., 2019; Margalit et al., 2020b), we assume that this
density profile continues outward for several decades in distance because most
of the optical emission is produced at time $t_{\rm syn}\gg t_{\rm dec}$ when
the FRB is emitted (see Equation 7 below). This assumption may hold true if
the medium is filled via the continuous ejection of material from a magnetar,
but it is a caveat to the following analysis.

Throughout the rest of this paper, we transform $\Delta T$ in (i.e., in
Equation 3) to $n_{\mathrm{ext}}$, representing the circumburst density at a
radius from the progenitor $r_{\mathrm{dec}}=2\Gamma^{2}ct_{\mathrm{dec}}$,
following equation (32) in Metzger et al. (2019) such that

$h\nu_{c}=2.3~{}\mathrm{keV}\left(\frac{\sigma}{0.1}\right)^{-3/2}\left(\frac{n_{\mathrm{ext}}}{10^{3}~{}\mathrm{cm}^{-3}}\right)^{-1}\left(\frac{t}{10^{-3}~{}\mathrm{s}}\right)^{-1/2}.$
(4)

The optical light curve is luminous until after the time when the peak
synchrotron frequency is equal to the cooling frequency, at which point
cooling is no longer efficient. Although the light curve will briefly continue
to rise at $t^{1/2}$ in this regime, as $\nu_{\mathrm{syn}}$ drops below the
observing frequency ($\approx$3.6–7.5$\times$1014 Hz for $gri$) the optical
luminosity will decline exponentially. We include this cutoff in our light
curves by rescaling the optical luminosity by
$\exp(-(\nu/\nu_{\mathrm{syn}}-1))$ when $\nu>\nu_{\mathrm{syn}}$. Overall, we
use equations (63)-(64) in Metzger et al. (2019) to model the peak luminosity
($L_{\mathrm{pk}}$) and specific luminosity ($L_{\nu}$) of the optical light
curve at $t>t_{\rm dec}$ as

$\displaystyle L_{\mathrm{pk}}$ $\displaystyle=$ $\displaystyle
10^{45}~{}\text{erg
s}^{-1}\left(\frac{E_{\mathrm{flare}}}{10^{43}~{}\text{erg}}\right)\left(\frac{t}{10^{-3}~{}\text{s}}\right)^{-1}$
(5) $\displaystyle\nu L_{\nu}$ $\displaystyle=$
$\displaystyle\begin{cases}L_{\mathrm{pk}}\left(\frac{\nu}{\nu_{c}}\right)^{4/3}\left(\frac{\nu_{c}}{\nu_{syn}}\right)^{1/2},&\nu<\nu_{\mathrm{c}}\\\
L_{\mathrm{pk}}\left(\frac{\nu}{\nu_{\mathrm{syn}}}\right)^{1/2},&\nu_{c}<\nu<\nu_{\mathrm{syn}}.\end{cases}$
(6)

We note that in Figure 2 the light curve begins to decline when
$\nu=\nu_{\mathrm{syn}}$, which occurs at

$t_{\mathrm{syn}}=82.6~{}\mathrm{s}\left(\frac{\lambda}{5000~{}\mathrm{\AA}}\right)^{2/3}\left(\frac{\sigma}{0.1}\right)^{1/3}\left(\frac{E_{\mathrm{flare}}}{10^{43}~{}\text{erg}}\right)^{1/3}$
(7)

where $\lambda$ is the observed wavelength. Thus for $r$-band ($\approx$6231
Å), the timescale for $E_{\mathrm{flare}}=10^{45}$ erg and $\sigma=0.3$ (as in
Metzger et al., 2019) is $\approx$640 s, or about $1/3$ of our observation
window. This is also significantly longer than any individual observation,
implying that our limits for the full set or some subset of our observations
are much more constraining than limits from individual exposures.

Moreover, the predicted luminosity is comparable to or brighter than our
limits in the $10^{45}$ erg case, with the peak occurring around this time at
$\nu L_{\nu}=8\times 10^{40}$ erg s-1. We also note that higher densities will
result in significantly more luminous bursts with $\nu L_{\nu}\propto
n_{\mathrm{ext}}^{5/6}$ as we are always in the regime where the optical
frequency is below the cooling frequency.

Figure 2: Optical light curves of the synchrotron afterglow from a magnetar-
driven burst as described in Section 3. We compare the predicted specific
luminosity in $\nu L_{\nu}$ for $r$ band and two sets of model light curves
corresponding to injected energies $E=10^{45}$ erg (solid lines) and
$E=10^{44}$ erg (dashed lines) in $g$ (green), $r$ (red), and $i$ (orange) to
the limits derived from our optical data. Otherwise, the models assume our
fiducial parameters as given in Section 3 (see also figure 8 in Metzger et
al., 2019) and the median circumburst density ($n_{\mathrm{ext}}\approx 2000$
cm-3) in Margalit et al. (2020b). The model is predicted to decline rapidly at
the time $t_{\mathrm{syn}}$ where the peak of the synchrotron spectrum
cascades below the observing frequency.

Assuming the synchrotron maser model with the same fiducial parameters given
above, we consider varying the energy scale of the burst
$\log(E_{\mathrm{flare}}/\text{erg})\in[41.2,47.4]$ and circumburst ion
density $\log(n_{\mathrm{ext}}/\text{cm}^{-3})\in[1,6.2]$. We then model the
total in-band emission for our $gri$ observations by calculating the average
specific luminosity integrated over the time of each observation relative to
the burst arrival time and averaged over frequencies corresponding to the
filter response function. We convert this value to a predicted apparent
magnitude assuming the distance modulus and foreground extinction given above.
For each model in our grid, if the computed apparent magnitude is brighter
than any of our 3$\sigma$ limiting magnitudes given in Table 1 (including the
3$\sigma$ limits for the stacked, post-burst imaging), we consider that model
ruled out as shown through the greyed-out region from Figure 3.

This model assumes that both the circumburst and interstellar host extinction
are negligible in the context of likely optical counterparts. In an ion-rich
medium the optical depth would be dominated by electron scattering with
$\tau\approx 10^{-7}$ following equation (33) in Metzger et al. (2019).

The interstellar burst host extinction could be dominated by dust with no
indication in the radio signal, and indeed, FRB 180916 appears coincident with
a small enhancement in the optical emission from its host galaxy (Marcote et
al., 2020; Tendulkar et al., 2020). This finding suggests that FRB 180916 is
located within or near a star-forming region (it is 250 pc from a young
stellar clump based on a H$\alpha$ detection in Hubble Space Telescope
imaging; Tendulkar et al., 2020) where there could be excess gas and dust
obscuring the optical counterpart (similar to, e.g., stripped-envelope SNe,
which evolve from very massive, young stars and thus are observed with high
average interstellar host extinction near their birth environments;
Stritzinger et al., 2018).

A constraint on the total extinction in the FRB 180916 host galaxy comes from
the DM observed toward this source, which is known to be on average
$\approx$350 pc cm-3 and 352.6$\pm$3.2 pc cm-3 for the 3 September 2020 burst.
From this value, we account for Milky Way interstellar dispersion adopting
171.7 pc cm-3 on the line-of-sight toward FRB 180916 (using the NE2001 model
of Cordes & Lazio, 2002) as well as 50 pc cm-3 for the Milky Way halo,
although this latter value ranges from 30–80 pc cm-3 (e.g. Prochaska & Zheng,
2019; Platts et al., 2020). To account for the intergalactic DM, we adopt the
relation in Macquart et al. (2020) at $z=0.0337$, which gives $56\pm 20$ pc
cm-3. Multiplying the residual DM by $1+z$, we obtain a source-frame DM host-
galaxy contribution in the line-of-sight to FRB 180916 of $75\pm 30$ pc cm-3.
This is equivalent to a hydrogen column density of $N_{H}=2.3\pm 1.5\times
10^{21}$ cm-2 following the locally-derived relation in He et al. (2013).
Finally, this column yields $A_{V}\approx 1.0\pm 0.6$ mag following Güver &
Özel (2009), but we acknowledge a significant systematic uncertainty on this
value. Fitting the dust content in the FRB 180916 host yields a much lower
dust content of $E(B-V)=0.12$ mag (Heintz et al., 2020), which implies
$A_{V}=0.4$ mag assuming $R_{V}=3.1$. Although this value is not a line-of-
sight probe similar to the DM, it is nominally consistent with the lower bound
of our interstellar host extinction estimate. Therefore, we conservatively
adopt $A_{V}=0.50$ mag with $R_{V}=3.1$ to model our observations below and
assume a Cardelli et al. (1989) reddening relation, implying that $A_{g}=0.53$
mag, $A_{r}=0.37$ mag, and $A_{i}=0.27$ mag due to interstellar dust in the
host of FRB 180916.

To place our final limiting magnitudes in context, we consider the full range
of circumburst densities corresponding to FRB 180916 in Margalit et al.
(2020b). In general, the energy of the burst $E_{\mathrm{flare}}$ can be
estimated from the equivalent isotropic energy in a single radio burst
($E_{\mathrm{radio}}$) following Margalit et al. (2020a, b) as

$\frac{E_{\mathrm{radio}}}{E_{\mathrm{flare}}}\approx 8.6\times
10^{-3}f_{e}f_{\xi}\left(\frac{\nu_{\mathrm{obs}}~{}t_{\mathrm{FRB}}}{1~{}\mathrm{GHz~{}ms}}\right)^{-1/5}$
(8)

where the observation frequency $\nu_{\mathrm{obs}}$ and the total burst
duration $t_{\mathrm{FRB}}$ are known from radio observations, the ratio of
electron to ion number densities in the upstream medium $f_{e}=0.5$ is
assumed, and the synchrotron maser efficiency is $f_{\xi}\approx 10^{-3}$
(Plotnikov & Sironi, 2019). We do not currently know the fluence ($S_{\nu}$)
or duration of the 3 September 2020 radio burst from FRB 180916, and so we
assume that it followed the distribution from CHIME/FRB Collaboration et al.
(2020b), with $E_{\mathrm{radio}}=4\pi D^{2}\nu S_{\nu}$ at 400 MHz of
(1.1–39.8)$\times$1037 erg and $t_{\mathrm{FRB}}$ of $\approx$0.6–8.6 ms
(corresponding to $E_{\mathrm{flare}}=0.04$–1.45$\times 10^{43}$ erg).
Assuming an average $E_{\mathrm{radio}}=6.8\times 10^{37}$ erg and
$t_{\mathrm{FRB}}=3.7$ ms for bursts with well-measured fluence and duration
in CHIME/FRB Collaboration et al. (2020b), we estimate that the average energy
per burst is $E_{\mathrm{flare}}=2.5\times 10^{42}$ erg for FRB 180916 as
shown in Figure 3.

This places the average optical counterpart well outside the range of
detectability for our observations and the model parameters above (noting the
greyed-out region in Figure 3 and assuming FRB 180916 parameters -red error
bars- from Margalit et al., 2020b). From the expected moment of dispersion-
corrected burst arrival at optical wavelengths, the timescale for an optical
light curve with $E_{\mathrm{flare}}=2.5\times 10^{42}$ erg and $\sigma=0.3$
is $t_{\mathrm{syn}}\approx 87$ s, implying that most of our imaging after
this point is not very constraining in the context of likely optical
counterparts.

Moreover, for a circumburst density $n_{\mathrm{ext}}=2000$ cm-3 (roughly the
median value for FRB 180916 in Margalit et al., 2020b), the burst would have
an optical luminosity $\nu L_{\nu}\approx 6\times 10^{38}$ erg s-1
($M$$\approx$$-8$ mag or $m$$\approx$$28$ mag at 150 Mpc) on the timescale
$t_{\mathrm{syn}}$. This is below the threshold of detectability for nearly
all optical telescopes, even assuming infinite integration time. For high-
speed optical imagers that can observe on the timescale of tens of ms around a
burst (e.g., the 2.4m Thai National Telescope/ULTRASPEC with
$m_{\mathrm{limit}}$$\approx$$16.8$ mag over 70 ms; Hardy et al., 2017), the
detection threshold is shallower by several orders of magnitude. Thus to
detect a burst at optical wavelengths with a light curve similar to those
above, a large aperture telescope and an anomalously energetic burst with
$E_{\mathrm{flare}}>10^{44}$ erg would be needed (following our limits and the
maximum densities inferred for FRB 180916 in Figure 3).

On the other hand, if the source were in a highly active state with
significantly larger energies and shorter timescale between bursts, this might
boost the circumburst density and the corresponding optical signal and
potentially place the counterpart within the range of detectability. The
shortest timescales between bursts for FRB 180916 are only 0.5 milliseconds
(observed on 19 Dec 2019; CHIME/FRB Collaboration et al., 2020b), or
$\approx$100 s in cases where $\Delta T/t_{\mathrm{FRB}}\gg 1$. Taking $\Delta
T$=100–1.4$\times$106 s as the full range of burst periods and assuming an
extremely energetic burst with $E_{\mathrm{flare}}=$1044 erg, the circumburst
density might exceed $10^{4}$ cm-3 (as with FRB 121102, the burst with the
highest inferred circumburst density in Margalit et al., 2020b). Based on the
assumed $n_{\mathrm{ext}}\propto\Delta T^{-2}$ and $L_{\mathrm{pk}}\propto
n_{\mathrm{ext}}^{5/6}$ scaling above, this would require a change in average
period of at least a factor of 5 or $\Delta T\approx 3.3$ days.

The FRB may have outbursts on timescales significantly shorter than 3.3 days
(CHIME/FRB Collaboration et al., 2020b), although it is unclear whether these
bursts would have similar flare energies or lead to a significantly denser
circumburst medium in the magnetar-driven synchrotron maser model we adopt
above. Moreover, the $k=0$ density profile discussed above may not be
representative at all projected radii if the burst properties change
significantly with time.

Thus while it is unclear if FRB 180916 can briefly enter the parameter range
we rule out in Figure 3, we would only require moderately deeper imaging or
more frequent and energetic bursts to detect the optical counterpart. Finally,
detailed analysis of the 3 September 2020 radio burst detected by CHIME will
provide a direct constraint on the circumburst density (following equation (8)
in Margalit et al., 2020b).

For future optical follow up efforts, this implies that in the context of the
baryonic-shell version of the synchrotron maser model, the most promising
search strategies will be to target FRB 180916 with a 8–10 m telescopes when
it is in an active state and likely to have one or more bursts. On the other
hand, if the burst profile is significantly more luminous on short timescales
after the burst (e.g., following the models of Beloborodov, 2020), high-speed
cameras such as ULTRASPEC (Hardy et al., 2017) on large aperture telescopes
will yield the strongest constraints on potential optical counterparts.
Moreover, if the burst period decreases and the circumburst density is
temporarily enhanced, the optical counterpart may be bright such that prompt
and/or long-lived counterparts are detectable.

Figure 3: Flare energy ($E_{\mathrm{flare}}$) and circumburst density
($n_{\mathrm{ext}}$) parameter space for FRBs for Metzger et al. (2019)
synchrotron maser light curves as described in Section 3. We show the range
inferred for individual bursts as error bars (from Margalit et al. 2020a) and
FRB 180916 bursts shown in red. We demonstrate the energy-density parameter
range that we can rule out for the 3 September 2020 burst of FRB 180916 and
using our optical limits as a grey region.

##  4 Conclusions

We presented APO 3.5m/ARCTIC observations of FRB 180916 around the time of a
fast radio burst. Comparing to models of synchrotron maser emission
corresponding to the broadband, relatively long-lived counterpart to a radio
burst, we find:

  1. 1.

Our observations are constraining for synchrotron maser emission in cases
where the energy per burst is larger than $\approx 10^{44}$ erg and the
circumburst density is greater than $10^{4}$ cm-3.

  2. 2.

Comparing to previous constraints on the FRB 180916 burst energy from Margalit
et al. (2020a), our limits are not constraining for the predicted burst
parameters. However, if the circumburst density is temporarily enhanced when
the FRB progenitor is highly active and multiple discrete bursts occur (e.g.,
on 19 Dec 2019 and 4 Feb 2020; CHIME/FRB Collaboration et al., 2020b), the
predicted optical light curve could exceed the magnitude limit achievable by
large aperture telescopes.

  3. 3.

Future optical follow up efforts that target emission similar to the
synchrotron maser model will benefit from joint optical and radio observations
of FRB 180916 longer than $30$ min and during its active phase when multiple
bursts are likely to occur.

We thank J. X. Prochaska for helpful comments on this manuscript. C.D.K.
acknowledges support through a NASA grant in support of Hubble Space Telescope
program AR-16136. K.E.H. acknowledges support by a Project Grant (162948–051)
from The Icelandic Research Fund. W.F. acknowledges support by the National
Science Foundation under grant Nos. AST-1814782 and AST-1909358. B.M. is
supported by NASA through the NASA Hubble Fellowship grant #HST-
HF2-51412.001-A awarded by the Space Telescope Science Institute, which is
operated by the Association of Universities for Research in Astronomy, Inc.,
for NASA, under contract NAS5-26555. Based on observations obtained with the
Apache Point Observatory 3.5-meter telescope, which is owned and operated by
the Astrophysical Research Consortium.

Table 1: APO 3.5m/ARCTIC Observations of FRB 180916 Shutter Open (Epoch)aaAll times are UTC on 2020-09-03. The relative epoch is given in seconds compared with the dispersion-corrected burst arrival time (UTC 2020-09-03 11:10:23.4) as described in Section 2. | Shutter Closed (Epoch)aaAll times are UTC on 2020-09-03. The relative epoch is given in seconds compared with the dispersion-corrected burst arrival time (UTC 2020-09-03 11:10:23.4) as described in Section 2. | Cumulative Exposure | Filter |  $\alpha$bbPointing center of our ARCTIC observation. Note that ARCTIC is a 2048$\times$2048 imager with $\approx$0.232′′ pixels, for a 7.92′$\times$7.92′ field of view. |  $\delta$bbPointing center of our ARCTIC observation. Note that ARCTIC is a 2048$\times$2048 imager with $\approx$0.232′′ pixels, for a 7.92′$\times$7.92′ field of view. |  $m_{\mathrm{lim}}$cc3$\sigma$ apparent limiting magnitude at the location of FRB 180916 averaged over the entire exposure. |  $M_{\mathrm{lim}}$dd3$\sigma$ absolute limiting magnitude accounting for a distance modulus $\mu=35.92$ mag (Section 3) and foreground extinction from Schlafly & Finkbeiner (2011). The values in this table assume no interstellar host extinction, but we adopt $A_{g}=0.53$ mag, $A_{r}=0.37$ mag, and $A_{i}=0.27$ mag following the discussion in Section 3.
---|---|---|---|---|---|---|---
(s) | (s) | (s) |  | (J2000) | (J2000) | (AB mag) | (AB mag)
11:05:39.7503Z (-283.6) | 11:06:10.0854Z (-253.3) | 30.335 | $i$ | 01:57:58.831 | +65:43:05.13 | 24.87 |  $-$12.77
11:07:05.7701Z (-197.6) | 11:07:36.1048Z (-167.3) | 30.335 | $i$ | 01:57:57.938 | +65:43:05.02 | 24.75 |  $-$12.88
11:08:20.3947Z (-123.0) | 11:08:50.7298Z (-92.7) | 30.335 | $i$ | 01:57:57.972 | +65:43:00.04 | 24.82 |  $-$12.81
11:09:27.4379Z (-56.0) | 11:09:57.7672Z (-25.6) | 30.329 | $r$ | 01:57:58.777 | +65:43:05.05 | 24.86 |  $-$13.36
11:10:25.5880Z (2.2) | 11:10:55.9377Z ( 32.5) | 30.350 | $r$ | 01:57:57.966 | +65:43:05.14 | 24.87 |  $-$13.35
11:11:31.1064Z (67.7) | 11:12:01.4421Z ( 98.0) | 30.336 | $r$ | 01:57:57.937 | +65:42:59.95 | 25.02 |  $-$13.21
11:12:43.4415Z (140.0) | 11:13:13.7769Z (170.4) | 30.335 | $g$ | 01:57:58.739 | +65:43:05.26 | 24.60 |  $-$14.65
11:13:41.9013Z (198.5) | 11:14:12.2416Z (228.8) | 30.340 | $g$ | 01:57:57.941 | +65:43:05.20 | 24.62 |  $-$14.63
11:14:39.9834Z (256.6) | 11:15:10.3232Z (286.9) | 30.340 | $g$ | 01:57:57.935 | +65:43:00.17 | 24.42 |  $-$14.83
11:15:52.9075Z (329.5) | 11:16:23.2385Z (359.8) | 30.331 | $i$ | 01:57:58.825 | +65:43:04.73 | 25.00 |  $-$12.64
11:16:51.1111Z (387.7) | 11:17:21.4439Z (418.0) | 30.333 | $i$ | 01:57:57.990 | +65:43:04.33 | 24.94 |  $-$12.69
11:17:50.0614Z (446.7) | 11:18:20.4010Z (477.0) | 30.340 | $i$ | 01:57:57.972 | +65:42:59.66 | 24.79 |  $-$12.84
11:18:56.7009Z (513.3) | 11:19:27.0366Z (543.6) | 30.336 | $r$ | 01:57:58.771 | +65:43:04.94 | 24.85 |  $-$13.38
11:19:55.0051Z (571.6) | 11:20:25.3396Z (601.9) | 30.334 | $r$ | 01:57:57.993 | +65:43:05.02 | 24.80 |  $-$13.43
11:20:53.2300Z (629.8) | 11:21:23.5680Z (660.2) | 30.338 | $r$ | 01:57:57.947 | +65:42:59.99 | 24.96 |  $-$13.27
11:22:03.6474Z (700.2) | 11:22:33.9856Z (730.6) | 30.338 | $g$ | 01:57:58.721 | +65:43:05.29 | 24.58 |  $-$14.67
11:23:01.7765Z (758.4) | 11:23:32.1159Z (788.7) | 30.339 | $g$ | 01:57:57.961 | +65:43:05.07 | 24.51 |  $-$14.75
11:23:59.3999Z (816.0) | 11:24:29.7394Z (846.3) | 30.339 | $g$ | 01:57:57.981 | +65:43:00.28 | 24.56 |  $-$14.69
11:25:12.1161Z (888.7) | 11:25:42.4525Z (919.1) | 30.336 | $i$ | 01:57:58.801 | +65:43:04.87 | 25.02 |  $-$12.62
11:26:10.6413Z (947.2) | 11:26:40.9768Z (977.6) | 30.335 | $i$ | 01:57:57.963 | +65:43:04.80 | 25.02 |  $-$12.61
11:27:08.5896Z (1005.2) | 11:27:38.8951Z (1035.5) | 30.306 | $i$ | 01:57:57.980 | +65:42:59.81 | 24.89 |  $-$12.75
11:28:16.3330Z (1072.9) | 11:28:46.6618Z (1103.3) | 30.329 | $r$ | 01:57:58.774 | +65:43:04.93 | 25.03 |  $-$13.20
11:29:13.7910Z (1130.4) | 11:29:44.1262Z (1160.7) | 30.335 | $r$ | 01:57:57.969 | +65:43:04.86 | 25.09 |  $-$13.14
11:30:12.4674Z (1189.1) | 11:30:42.8033Z (1219.4) | 30.336 | $r$ | 01:57:57.988 | +65:43:00.04 | 25.08 |  $-$13.15
11:31:20.9344Z (1257.5) | 11:31:51.2694Z (1287.9) | 30.335 | $g$ | 01:57:58.758 | +65:43:05.19 | 24.64 |  $-$14.61
11:32:20.6130Z (1317.2) | 11:32:50.9150Z (1347.5) | 30.302 | $g$ | 01:57:57.943 | +65:43:05.17 | 24.65 |  $-$14.60
11:33:18.7291Z (1375.3) | 11:33:49.0651Z (1405.7) | 30.336 | $i$ | 01:57:57.969 | +65:43:00.09 | 22.73 |  $-$14.90
11:34:16.2548Z (1432.9) | 11:34:46.5853Z (1463.2) | 30.331 | $i$ | 01:57:58.808 | +65:43:04.75 | 24.77 |  $-$12.86
11:35:13.6288Z (1490.2) | 11:35:43.9768Z (1520.6) | 30.348 | $i$ | 01:57:57.987 | +65:43:04.90 | 24.83 |  $-$12.80
11:36:25.2157Z (1561.8) | 11:36:55.5512Z (1592.2) | 30.335 | $r$ | 01:57:57.957 | +65:42:59.96 | 25.57 |  $-$12.66
11:37:23.8992Z (1620.5) | 11:37:54.1358Z (1650.7) | 30.237 | $r$ | 01:57:58.743 | +65:43:05.36 | 24.97 |  $-$13.26
11:38:21.0860Z (1677.7) | 11:38:51.4219Z (1708.0) | 30.336 | $r$ | 01:57:57.974 | +65:43:04.88 | 24.88 |  $-$13.35
11:39:19.1340Z (1735.7) | 11:39:49.4752Z (1766.1) | 30.341 | $g$ | 01:57:58.003 | +65:43:00.10 | 25.99 |  $-$13.26
11:40:16.5898Z (1793.2) | 11:40:46.9242Z (1823.5) | 30.334 | $g$ | 01:57:58.753 | +65:43:05.38 | 24.40 |  $-$14.86
11:41:13.7692Z (1850.4) | 11:41:44.1046Z (1880.7) | 30.335 | $g$ | 01:57:57.950 | +65:43:05.31 | 24.45 |  $-$14.81
11:42:11.2244Z (1907.8) | 11:42:41.5377Z (1938.1) | 30.313 | $g$ | 01:57:57.934 | +65:43:05.21 | 24.39 |  $-$14.86
11:10:25.5880Z (2.2) | 11:38:51.4219Z (1708.0) | 364.10eeStacked exposure for all imaging after the expected optical arrival time of the 3 September 2020 burst for FRB 180916. The limiting magnitude is 3$\sigma$ calculated empirically in the stacked frame as described in Section 3. | $r$ | 01:57:57.877 | +65:43:05.08 | 25.83 |  $-$12.40
11:12:43.4415Z (140.0) | 11:42:41.5377Z (1938.1) | 364.10eeStacked exposure for all imaging after the expected optical arrival time of the 3 September 2020 burst for FRB 180916. The limiting magnitude is 3$\sigma$ calculated empirically in the stacked frame as described in Section 3. | $g$ | 01:57:57.877 | +65:43:05.08 | 25.84 |  $-$13.41
11:15:52.9075Z (329.5) | 11:35:43.9768Z (1520.6) | 273.08eeStacked exposure for all imaging after the expected optical arrival time of the 3 September 2020 burst for FRB 180916. The limiting magnitude is 3$\sigma$ calculated empirically in the stacked frame as described in Section 3. | $i$ | 01:57:57.877 | +65:43:05.08 | 25.70 |  $-$11.94

Facilities: APO 3.5m (ARCTIC)

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