# A LOOK at Outbursts of Comet C/2014 UN271 (Bernardinelli-Bernstein) Near 20
au

Michael S. P. Kelley Department of Astronomy, University of Maryland, College
Park, MD 20742-0001, USA<EMAIL_ADDRESS>Rosita Kokotanekova European
Southern Observatory, Karl-Schwarzschild-Str 2, 85748 Garching, Germany
Institute of Astronomy and National Astronomical Observatory, Bulgarian
Academy of Sciences, 72 Tsarigradsko shose Blvd., 1784 Sofia, Bulgaria  Carrie
E. Holt Department of Astronomy, University of Maryland, College Park, MD
20742-0001, USA  Silvia Protopapa Southwest Research Institute, 1050 Walnut
Street, Suite 300, Boulder, CO 80302, USA  Dennis Bodewits Physics Department,
Leach Science Center, Auburn University, Auburn, AL 36849, USA  Matthew M.
Knight Physics Department, United States Naval Academy, 572C Holloway Rd,
Annapolis, MD 21402, USA  Tim Lister Las Cumbres Observatory, 6740 Cortona
Drive Suite 102, Goleta, CA 93117, USA  Helen Usher The Open University,
Walton Hall, Milton Keynes, MK7 6AA, UK  Joseph Chatelain Las Cumbres
Observatory, 6740 Cortona Drive Suite 102, Goleta, CA 93117, USA  Edward Gomez
Las Cumbres Observatory, School of Physics and Astronomy, Cardiff University,
Queens Buildings, The Parade, Cardiff CF24 3AA, UK  Sarah Greenstreet
Department of Astronomy and the DIRAC Institute, University of Washington,
3910 15th Ave NE, Seattle, WA 98195, USA  Tony Angel Harlingten Observatory,
Observatorio Sierra Contraviesa, Cortijo El Cerezo, Torvizcon 18430, Granada,
Spain  Ben Wooding  St Mary’s Catholic Primary School, Llangewydd Road,
Bridgend, Wales, CF31 4JW, UK

(Accepted June 23, 2022)

###### Abstract

Cometary activity may be driven by ices with very low sublimation
temperatures, such as carbon monoxide ice, which can sublimate at distances
well beyond 20 au. This point is emphasized by the discovery of Oort cloud
comet C/2014 UN271 (Bernardinelli-Bernstein), and its observed activity out to
$\sim$26 au. Through observations of this comet’s optical brightness and
behavior, we can potentially discern the drivers of activity in the outer
solar system. We present a study of the activity of comet Bernardinelli-
Bernstein with broad-band optical photometry taken at 19–20 au from the Sun
(2021 June to 2022 February) as part of the LCO Outbursting Objects Key (LOOK)
Project. Our analysis shows that the comet’s optical brightness during this
period was initially dominated by cometary outbursts, stochastic events that
ejected $\sim 10^{7}$ to $\sim 10^{8}$ kg of material on short ($<1$ day)
timescales. We present evidence for three such outbursts occurring in 2021
June and September. The nominal nuclear volumes excavated by these events are
similar to the 10–100 m pit-shaped voids on the surfaces of short-period comet
nuclei, as imaged by spacecraft. Two out of three Oort cloud comets observed
at large pre-perihelion distances exhibit outburst behavior near 20 au,
suggesting such events may be common in this population. In addition,
quiescent CO-driven activity may account for the brightness of the comet in
2022 January to February, but that variations in the cometary active area
(i.e., the amount of sublimating ice) with heliocentric distance are also
possible.

Optical astronomy (1776) — Broad band photometry (184) — Long period comets
(933) — Coma dust (2159) — Comet surfaces (2161)

††journal: ApJL††facilities: LCOGT††software:  astropy (Astropy Collaboration
et al., 2018), sbpy (Mommert et al., 2019), astroquery (Ginsburg et al.,
2019), JPL Horizons (Giorgini et al., 1996), SEP (Barbary, 2016), DS9 (Joye &
Mandel, 2003), reproject (Robitaille et al., 2020), photutils (Bradley et al.,
2021), calviacat (Kelley & Lister, 2019), ccdproc (Craig et al., 2021), BANZAI
(McCully et al., 2018), fastrot (Van Selous & Kelley, 2021)

##  1 Introduction

Comet C/2014 UN271 (Bernardinelli-Bernstein) was discovered in a search of the
Dark Energy Survey data archive for new Solar System objects (Bernardinelli et
al., 2021). Originally designated as an asteroid, the object was found in data
from 2014 to 2018, at heliocentric distances, $r_{\mathrm{h}}$, of 29.0 to
23.7 au. Shortly after the discovery announcement in 2021 (Minor Planet
Center, 2021), new observations at $r_{\mathrm{h}}$=20 au found the object to
be extended and about –1.5 mag brighter than expected based on the 2014–2018
photometry, and the object was subsequently re-designated as a comet
(Kokotanekova et al., 2021a; Buzzi et al., 2021). The object was
serendipitously observed by the Transiting Exoplanet Survey Satellite (TESS)
in 2018 and 2020, showing direct evidence for activity as far out as 23 au,
and a lack of any periodic variability (Ridden-Harper et al., 2021; Farnham et
al., 2021; Bernardinelli et al., 2021). A re-analysis of the pre-discovery
data in the Dark Energy Survey and Panoramic Survey Telescope and Rapid
Response System (PanSTARRS) archives confirmed activity out to $\sim 26$ au
(Bernardinelli et al., 2021). The comet is notable for being discovered at
such a great distance (cf. Meech et al., 2017) and for its large size,
estimated to be 69$\pm$9 km in radius by Lellouch et al. (2022), and 60$\pm$7
to 69$\pm$8 km by Hui et al. (2022). It is the largest known Oort cloud comet
nucleus, and second to 95P/Chiron (Ruprecht et al., 2015) for the largest
cometary nucleus overall.

Little is known about cometary activity at $\sim$20 au. At such great
distances, activity is likely driven by the sublimation of volatiles with low
sublimation temperatures. The volatiles CO and CO2, found in abundance in the
cometary population (A’Hearn et al., 2012), have nominal sublimation
temperatures of 25 and 80 K, respectively, although some level of sublimation
can occur below these values (Meech & Svoren̆, 2004). The temperature of a
blackbody sphere is $T{\sim}60$ K at 20 au, which suggests CO sublimation can
persist out to many tens of au, as long as ices are available near the nuclear
surface. As an alternative to ice sublimation, water ice phase transitions may
also drive cometary activity, but they are unlikely to be major contributors
at such great distances (Jewitt et al., 2017). Aside from comet Bernardinelli-
Bernstein, the three comets with the most distantly observed activities are
C/1995 O1 (Hale-Bopp), active post-perihelion out to 27 au (Kramer et al.,
2014), C/2010 U3 (Boattini), active pre-perihelion at 26 au (Hui et al.,
2019), and C/2017 K2 (PanSTARRS), active pre-perihelion at 24 au (Hui et al.,
2018).

We present images and photometry of comet Bernardinelli-Bernstein taken as
part of the LCO Outbursting Objects Key Project (LOOK Project; Lister et al.,
submitted) using the Las Cumbres Observatory (LCO) global telescope network.
We characterize the coma color, lightcurve, and morphology, and show that the
comet’s activity near 20 au has varied substantially with time.

##  2 Observations and Reduction

Images of comet Bernardinelli-Bernstein were taken by our team and by school
children in Wales as part of the Comet Chasers educational outreach program
through the Faulkes Telescope Project. Observations were taken with LCO’s 1-m
robotic telescopes, specifically those located at: Siding Spring Observatory,
Australia; South African Astronomical Observatory, South Africa; and the Cerro
Tololo Inter-American Observatory, Chile. Each telescope is equipped with
identical Sinistro cameras, utilizing 4k$\times$4k CCDs with a nominal pixel
scale of 0$\farcs$39, and a suite of filters. For most observations, we used
the SDSS $g^{\prime}$ and $r^{\prime}$ filters, although $w$ (effectively
$g+r+i$), $V$, and $R$ filters were used in a few circumstances. LOOK Project
observing sequences were scheduled with the NEOExchange observation manager
(Lister et al., 2021). The telescopes tracked the proper motion of the comet,
and exposure times were typically 300s with 2 exposures per filter.

Images were processed by LCO’s BANZAI (Beautiful Algorithms to Normalize
Zillions of Astronomical Images) pipeline (McCully et al., 2018), which
includes bias, dark, and flat-field corrections, source extraction, and
astrometric calibration. Individual frame exposure times were short enough
that stars only trail by $\sim 0.66$–$0.71\arcsec$ in 300 s exposures. We used
the pipeline-produced source catalog to photometrically correct each image,
calibrating them to the PanSTARRS 1 (PS1) photometric system (Tonry et al.,
2012) using the ATLAS-RefCat2 photometric catalog (Tonry et al., 2018), and
LOOK Project derived color corrections. Details on the data reduction are
presented by Lister et al. (submitted). For this paper, we adopt the apparent
magnitude and effective wavelengths for the solar spectrum from Willmer
(2018): $g_{P1}=-26.54$ AB mag at 481.1 nm, and $r_{P1}=-26.93$ AB mag at
615.6 nm (PS1 system photometry is hereafter denoted $g$ and $r$). Sample
images are presented in Fig. 1.

Figure 1: Select LCO images of comet C/2014 UN271 (Bernardinelli-Bernstein).
Dates are as indicated, all images are through an $r^{\prime}$ filter, except
on 2021 June 22, which is $w$. All images are displayed with the same color
scale and are 1,320,000 km $\times$ 1,320,000 km in size (90″ on 2021 June
22). Celestial north is up, east to the left. Over this time period, the
heliocentric distance decreased from 21.78 to 19.28 au, the projected comet-
Sun vector increased from position angle 46° to 292°, and the projected
velocity vector increased from 144° to 153°.

Photometry of the comet was measured in co-added frames, grouped by telescope
and time. We chose an aperture size of 6$\farcs$0 (86,000 to 88,000 km at the
comet; background annulus 20–40″), and limited our analysis to images with
point source FWHMs $<$3″ and final photometric uncertainties $<0.15$ mag. We
have 179 photometric measurements in 96 distinct observational epochs. Our
first observation was on 2021 June 22 with the comet at a heliocentric
distance $r_{\mathrm{h}}$=20.18 au, a geocentric distance $\Delta$=20.20 au,
and a Sun-target-observer (phase) angle $\theta$=2.9°. Our last observation
presented in this paper was on 2022 February 23 at $r_{\mathrm{h}}$=19.28 au,
$\Delta$=19.70 au, and $\theta$=2.6°. The observational circumstances and
photometry are presented in Table 1.

Table 1: Observational circumstances and photometry.

Date | Year | Filter | Cal. | $r_{\mathrm{h}}$ | $\Delta$ | $\theta$ | Airmass | Seeing | $N_{\mathrm{images}}$ | $t_{\mathrm{exp}}$ | $m$ | $\sigma_{m}$ | $H_{r}$
---|---|---|---|---|---|---|---|---|---|---|---|---|---
(UTC) | (UTC) |  |  | (au) | (au) | (°) |  | (″) |  | (s) | (mag) | (mag) | (mag)
2021-06-22 04:13 | 2021.472 | $w$ | $r_{\mathrm{P1}}$ | 20.177 | 20.198 | 2.884 | 1.25 | 2.7 | 3 | 900 | 18.275 | 0.068 | 5.098
2021-09-08 02:33 | 2021.685 | $r^{\prime}$ | $r_{\mathrm{P1}}$ | 19.892 | 19.448 | 2.636 | 1.09 | 1.6 | 1 | 300 | 19.239 | 0.102 | 6.185
2021-09-09 22:04 | 2021.690 | $g^{\prime}$ | $g_{\mathrm{P1}}$ | 19.886 | 19.438 | 2.631 | 1.45 | 1.9 | 2 | 600 | 19.090 | 0.072 | 5.564
2021-09-18 02:09 | 2021.713 | $r^{\prime}$ | $r_{\mathrm{P1}}$ | 19.856 | 19.401 | 2.617 | 1.10 | 2.2 | 2 | 600 | 18.786 | 0.075 | 5.742
2021-09-23 05:03 | 2021.727 | $r^{\prime}$ | $r_{\mathrm{P1}}$ | 19.837 | 19.382 | 2.615 | 1.21 | 1.8 | 2 | 600 | 18.645 | 0.070 | 5.606
2021-10-07 21:40 | 2021.767 | $r^{\prime}$ | $r_{\mathrm{P1}}$ | 19.783 | 19.349 | 2.637 | 1.21 | 1.7 | 2 | 600 | 18.883 | 0.065 | 5.852
2021-11-05 19:45 | 2021.846 | $r^{\prime}$ | $r_{\mathrm{P1}}$ | 19.678 | 19.367 | 2.764 | 1.21 | 2.0 | 1 | 300 | 19.086 | 0.096 | 6.059
2021-12-10 19:21 | 2021.942 | $r^{\prime}$ | $r_{\mathrm{P1}}$ | 19.550 | 19.491 | 2.886 | 1.10 | 2.1 | 2 | 600 | 19.135 | 0.073 | 6.103
2022-01-11 12:57 | 2022.029 | $r^{\prime}$ | $r_{\mathrm{P1}}$ | 19.434 | 19.624 | 2.832 | 1.52 | 3.0 | 2 | 600 | 19.184 | 0.080 | 6.153
2022-02-14 01:28 | 2022.121 | $r^{\prime}$ | $r_{\mathrm{P1}}$ | 19.312 | 19.700 | 2.667 | 1.55 | 2.6 | 2 | 600 | 18.948 | 0.082 | 5.929
…

Note. — Table 1 is published in its entirety in the machine-readable format. A
portion is shown here for guidance regarding its form and content. Column
descriptions: (1) mid-time of the observation; (2) mid-time as fractional
year; (3) filter used in the observation; (4) PS1 filter to which the
photometry is calibrated; (5) heliocentric distance; (6) observer-target
distance; (7) Sun-target-observer (phase) angle; (8) calculated airmass of the
observation; (9) FWHM of point sources; (10) number of images; (11) total
exposure time; (12) apparent magnitude; (13) uncertainty on the apparent
magnitude; (14) absolute magnitude in the $r$-band ($g$-band photometry has
been scaled by the average color).

##  3 Results and Analysis

###  3.1 Color

Based on 70 photometry sets with contemporaneously obtained $g$\- and $r$-band
photometry, we find a weighted mean coma color of $g-r$=0.47$\pm$0.01 mag. All
color sets are consistent with the mean at the 1.7$\sigma$ level. There is no
evidence for a trend with time or brightness (Pearson and Spearman correlation
test p-values $\gg$0.05). The color of the coma corresponds to a spectral
gradient (A’Hearn et al., 1984) of $S_{g,r}=5.48\pm 0.05$% per 100 nm, where
subscripts denote the PS1 bandpasses upon which the calculation is based.

The comparison between the coma color of comet Bernardinelli-Bernstein and
those of other active long period comets (LPCs; Solontoi et al., 2012; Jewitt,
2015), is shown in Fig. 2. The comparison is made in terms of $V-R$ color for
convenience only. Using the relations reported by Tonry et al. (2012), the
$g-r$ color of comet Bernardinelli-Bernstein corresponds to $V-R=0.43\pm 0.02$
mag, which agrees within 2$\sigma$ of the mean color of active LPCs:
$V-R=0.47\pm 0.01$ mag (derived from data by Solontoi et al. 2012 and Jewitt
2015, and filter transformations by Ivezić et al. 2007). Therefore, comet
Bernardinelli-Bernstein is a nominal LPC in terms of coma color. For
comparison, the nucleus color of the comet measured by Bernardinelli et al.
(2021) is $V-R=0.40\pm 0.02$ mag (also shown in Fig. 2; color transformed
following Abbott et al. 2021). The nucleus and coma colors are consistent at
the 2$\sigma$ level, with the nucleus being potentially bluer than the coma.
This relationship between the colors of the coma and nucleus environments is
in agreement with the average colors reported by Jewitt (2015) for active and
inactive LPCs.

###  3.2 Brightness and Lightcurve

In our first image of the comet, the spatial distribution is relatively
compact, and a total brightness may be estimated. Within an 8$\farcs$9 radius
aperture (130,000 km), offset from the nucleus to make a tight fit around the
visible dust, we measure $r$=18.14$\pm 0.07$ mag. Converting this brightness
to an $r$-band absolute magnitude is done by scaling it to $r_{\mathrm{h}}$=1
au, $\Delta$=1 au, and to a phase angle of 0° using the Schleicher-Marcus
phase function for cometary dust (Schleicher & Bair, 2011), yielding
$H_{r}(1,1,0)=4.96$ mag.

Figure 2: $V-R$ color as a function of heliocentric distance,
$r_{\mathrm{h}}$, comparing the color of the coma of comet C/2014 UN271
(Bernardinelli-Bernstein) as measured in this work (square symbol) and that of
other active LPCs from Jewitt (2015) and Solontoi et al. (2012). The grey
rectangle defines the 3$\sigma$ limits around the average $V-R$ color of
active LPCs (0.47$\pm$0.01 mag), and the horizontal dashed line is the color
of the Sun. Color wise, comet Bernardinelli-Bernstein is a typical active LPC.
The color of the nucleus (Bernardinelli et al., 2021) is also shown for
comparison (diamond).

In Fig. 3, we present the lightcurve of the comet as absolute magnitude versus
time based on our full photometric data set. Initially, the comet’s intrinsic
brightness decreased with time, with the absolute magnitude in our photometric
aperture increasing from 5.10$\pm$0.07 mag (2021 June 22) to 6.11$\pm$0.04 mag
(2021 September 03–08). This fading period was followed by two brightening
events. The brightening events are likely outbursts, the first occurring
between 2021 September 08 02:33 and September 09 22:04 UTC (Kelley et al.,
2021), and the second between September 18 02:09 and September 23 04:58 UTC.

With a functional form of the lightcurve, we can characterize the coma fading
and the September outburst strengths. The absolute magnitude within the 6″
radius aperture faded at a rate of $13.8\pm 0.4$ mmag $\textrm{day}^{-1}$
($\chi^{2}_{\nu}$=0.6, RMS=64 mmag) from 2021 June 22 to September 08. The
September 09 outburst relative strength is $-0.65\pm 0.07$ mag and the
absolute magnitude of the new material is $H_{r}=6.32$ mag. The September 23
outburst strength is derived by comparing pre- and post-outburst $r$-band
data, accounting for the linear decline in brightness: $\Delta m=-0.22\pm
0.07$ mag, $H_{r}=7.23$ mag. Temporal differences of the images indicate that
the photometric apertures at the time of the outburst discoveries include all
of the new ejecta. We find no evidence for any other outbursts during this
initial fading period. Each outburst is followed by a decline in coma
brightness likely due to expansion of the ejecta and any prior released
material beyond the photometric aperture.

Following the outbursts was a period of low-amplitude variability that started
in October, with a mean $H_{r}=5.98\pm 0.01$ mag but an RMS of 0.12 mag (19
$r$-band photometric data points between 2021 October 07 and 2022 February
23). The variability appears to be smooth with time, but gaps in the data
could hide small, $\sim-0.1$ mag outbursts. Monthly averaged absolute
magnitudes are $H_{r}=5.87\pm 0.02$, $6.06\pm 0.03$, $6.19\pm 0.05$, $6.06\pm
0.04$, and $5.94\pm 0.05$ mag, for 2021 October, November, December, 2022
January, and February, respectively. Some of the fading from October to
November may still be due to residual ejecta from the September outbursts, but
without a dynamical model or higher spatial resolution data, or some
assessment of possible quiescent activity, this interpretation is speculation.
We will return to this point in Section 4.2.

Figure 3: Absolute magnitude, $H_{r}(1,1,0)$, of comet C/2014 UN271
(Bernardinelli-Bernstein) measured within 6″ radius apertures from LOOK
Project photometry ($g$-band as circles, $r$-band as squares) versus time.
Data calibrated to the $g$-band have been scaled with the measured coma colors
to make an effective $r$-band data set. The brightness of the coma from
PanSTARRS 1 (PS1) data measured on 2019.66 UTC, using the same aperture size,
is marked with a dotted line, and the absolute magnitude of the nucleus is
given as a dashed line (Bernardinelli et al., 2021).

###  3.3 Dust Cross-sectional Area and Mass

With assumptions on the dust and/or ice grain properties, absolute magnitudes
may be converted into total cross-sectional area and mass. We initially assume
all grains have the same properties: effective radius $a_{\mathrm{eff}}$=10
µm, grain mass density of $\rho_{g}$=1.0 kg m-3, and $r$-band albedo
$A_{r}$=4%. The albedo is a nominal value that is similar to the nucleus
surface (Hui et al., 2022), and the grain mass density allows for porous dust
grains and/or the presence of water ice. Thus, the absolute magnitudes of the
total coma in June 2021 and the ejecta of the two outbursts in September 2021
($H_{r}$=$4.96\pm 0.07$, $6.32\pm 0.07$, and $7.23\pm 0.07$ mag) correspond to
cross-sectional areas $G$=$3.1\mbox{$\times 10^{5}$}$, $0.88\mbox{$\times
10^{5}$}$, and $0.38\mbox{$\times 10^{5}$}$ km2, and masses
$M$=$4.1\mbox{$\times 10^{8}$}$, $1.2\mbox{$\times 10^{8}$}$, and
$0.51\mbox{$\times 10^{8}$}$ kg, respectively (6% formal uncertainties on all
values). There is considerable uncertainty in the adopted grain parameters,
which greatly affects the mass estimates. The mass scales with the quantity
$a_{\mathrm{eff}}\,\rho_{g}/A_{p}$. Effective grain radius has the largest
effect, as sizes of 1 or 100 µm, or even a size distribution, could have been
chosen. We take $a_{\mathrm{eff}}=10$ µm as our nominal case (equivalent to a
differential size distribution $\propto a^{-3.5}$ for $a$ between 1 and 100
µm, see, e.g., Ishiguro et al. 2016a), and assume a factor of 10 in the mass
uncertainty (cf. Tubiana et al., 2015).

###  3.4 Morphological Evolution

The comet’s morphology changed during the initial fading period (2021 June
22–September 08). To quantitatively evaluate these changes, we median combined
$g^{\prime}$ and $r^{\prime}$ images taken over the time periods July 11–18,
August 03–08, and September 03–08. Figure 4 presents the results. The time
sequence suggests an expanding morphology. To emphasize the variation, we
divided each image by the previous image in the sequence, and show the results
in Fig. 4. The ratio confirms the expansion.

A projected dust expansion speed can be estimated using the co-added images in
Fig. 4. We generated a detection mask on each image, identifying the coma
using a 3$\times$3 pixel moving box where at least 1 pixel is more than
2$\sigma$ from the background. Assuming radial motion, the coma edge expands
at a rate of 48 to 84 m s-1 (approximately monotonically increasing from
$-$50° to +100° east of north) for the time period 2021 June 22 to July 14.
The expansion rate is non-linear with time in our data, falling to near 0 or
even negative values thereafter, at least for this edge detection method.
However, this approach does not track any one single feature, but instead the
edge of the detected coma, which depends on the image signal-to-noise ratio
and the actual surface brightness distribution of the coma. Furthermore, the
expansion may not be radial, and solar radiation pressure or Lorentz forces
may be accelerating the dust (e.g., Hui et al., 2019). At best we have
estimated the order of magnitude of the initial mean expansion speed, at worst
we have measured a lower limit to the fastest moving material. In addition, an
estimate of the slowest material may be made using the lightcurve. The slowest
ejecta takes at least 78 days to move 6″, thereby moving $\leq$13 m s-1 in
projection on the sky.

Figure 4: Top: LCO images of comet C/2014 UN271 (Bernardinelli-Bernstein).
Dates (2021) are as indicated. Except for the image on June 22, which is a
single night $w$-band average, images are median combined $g^{\prime}$ and
$r^{\prime}$ data taken over a range of dates: 2021 July 11–18, August 02–09,
and September 03–08. All images are displayed with the same photometric color
scale, and show an area of 880,000 km $\times$ 880,000 km (60″ on 2021 June
22). Bottom: The same images, but normalized by the previous image in the
sequence and smoothed with a $\sigma$=1-pix Gaussian filter. The expanding
morphology of the comet is apparent throughout this period.

Mean surface brightness ($S_{\nu}$) profiles as a function of distance to the
nucleus ($\rho$) show a morphological evolution of the initial coma (Fig. 5).
The mean profile of the comet flattens with time. Assuming a functional form
of $S_{\nu}\propto\rho^{k}$, the best-fit slopes between 3 to 10″ are
$k=-2.63\pm 0.06$, $-1.22\pm 0.02$, $-0.99\pm 0.02$, and $-0.90\pm 0.03$, for
the time periods 2021 June 22, July 11–18, August 02–09, and September 03–08,
respectively.

Figure 5: Surface brightness ($S_{\nu}$) profiles as a function of distance to
the nucleus ($\rho$) for the images presented in Fig. 4. The profile of the
comet is initially steep, but flattens with time. A nominal coma profile,
$\rho^{-1}$, is shown for reference. The mean seeing of these images is
1$\farcs$8\. Also shown are the monthly mean profiles from 2021 October to
2022 February, offset by $+1$ mag for clarity (mean seeing 2$\farcs$3).

Also included in Fig. 5 are mean radial profiles for images generated by
combining data month-by-month from 2021 October to 2022 February. At 3 to 10″,
the best-fit radial profile slopes are all steeper than $-1$: –1.39$\pm$0.04,
–1.26$\pm$0.06, –1.27$\pm$0.08, –1.71$\pm 0.06$ and –1.76$\pm 0.06$, for 2021
October, November, December, 2022 January, and February, respectively. Despite
the 50–120 day time period since the last outburst, the coma does not appear
to be in a steady-state, and even steepened as the comet brightened in 2022.

##  4 Discussion

We have characterized the lightcurve and morphology of comet C/2014 UN271
(Bernardinelli-Bernstein) from 2021 June 22 to 2022 February 23, in an effort
to help increase our knowledge of the activity of comets near 20 au. We now
synthesize the results to better understand the state of activity of this
particular comet, and compare them to previous work regarding its activity.

###  4.1 Outbursts Near 20 au

Following our first detection of the comet on 2021 June 22, the intrinsic
brightness faded, the morphology expanded, and the mean radial profile
flattened with time. Together, these are the signatures of material ejected by
a cometary outburst leaving the vicinity of the nucleus. In contrast, a
steady-state coma would have a surface brightness profile proportional to
$\rho^{-1}$. A steep initial slope, $k<-1$, indicates a rapid ejection of new
material into the unresolved source. The fading with time shows that the
driver of this new material had ceased, or at least rapidly decreased compared
to the timescale of dust expansion. That the radial profile flattened as the
brightness decreased in our photometric aperture is due to the expansion of
the material on the sky. The slope just before the first September outburst,
$k=-0.9$, suggests the coma might have been near a steady state at that time,
but due to the September outbursts, we could not study the evolution any
further to confirm this point (e.g., the profile may have continued to flatten
with the expansion of the material). Therefore, we conclude that the
appearance of the comet from 2021 June 22 to September 08 is dominated by an
outburst occurring before our first observation.

If these data were instead observations of a previously steady-state coma that
had suddenly ceased activity on our about 2021 June 22, then the radial
profile would have initially been near $-1$, rather than $<-2$, as observed.
Alternatively, if the comet activity had rapidly increased to establish a new
quiescent level by 2021 September, then an initially steep radial profile
could have been observed, trending toward –1 with time. However, in this case
the comet would have brightened, not faded with time.

Bernardinelli et al. (2021) measured the integrated brightness of the comet in
PanSTARRS 1 images taken on 2019 August 29 UTC at 22.6 au, 663 days before our
first image at 20.2 au. Their value, $H_{r}$=6.5 mag in a 6″ radius aperture,
compares favorably with our minimum brightness in 2021 September, $H_{r}$=6.1
mag. Thus, our suggestion that the radial profile at 3–10″ in September 2021
was trending toward a steady-state coma is possible. However, in 2021
November–2022 February, when the comet was at a similar brightness
($H_{r}$=6.2 to 5.9 mag), the radial profile slope was $-1.3$ to $-1.8$,
significantly steeper than $-1$. We suggest residual ejecta from the September
outbursts affect the spatial profile in November–December. Additional small
($\sim-0.1$ mag) outbursts may have caused the $\sim-0.25$ mag brightening
between 2021 December 16 and 2022 February 23. Validation of this hypothesis
would benefit from additional data that could establish the quiescent behavior
of the comet.

A comparison of the PS1 photometry from 2019 August to the total coma
brightness measured in our first image ($H_{r}=4.96$ mag) indicates that the
outburst preceding our initial observations of this comet was likely no
stronger than $\Delta m=-1.5$ mag (6″ radius aperture). Based on the rapid
evolution of the ejecta from 2021 June 22 to July 11–18 ($\sim 50$ m s-1
expansion speed), the outburst likely occurred in the month before our first
observation. In Section 3.3, we estimated the total mass of the observed dust
on 2021 June 22 to be $4\mbox{$\times 10^{8}$}$ kg, with uncertainties of a
factor of 10 due to unknown grain properties. Removing the PS1 measurement as
a possible ambient coma reduces the nominal mass to $3\mbox{$\times 10^{8}$}$
kg, which does not affect the order of magnitude estimate. In a summary of 17
outbursts over an $\sim 8$ year period, Ishiguro et al. (2016b) found 3 events
(18%) with cross-sectional areas similar to or larger than our nominal
estimate, indicating that this very large cometary nucleus at 20 au has
outbursts as large as any typical comet near 1 au. Outbursts of this scale
correspond to equivalent nuclear volumes of cylindrical pits with diameter
$D=80(m/10^{8}~{}\mathrm{kg})^{1/3}$ m (assuming a bulk nucleus density of 500
kg m-3, and a depth to diameter ratio of 0.5). The size of the excavated
region is only weakly affected by our large mass uncertainties; a factor of 10
uncertainty in mass corresponds to a factor of 2 uncertainty in diameter.
Therefore, such surface features would be comparable to the pits observed on
the surfaces of comets 9P/Tempel 1, 67P/Churyumov-Gerasimenko, and 81P/Wild 2
(Brownlee et al., 2004; Veverka et al., 2013; Vincent et al., 2015).

Without a direct spectroscopic detection of a gas, we cannot determine with
certainty what drove the outbursts, or if they were mechanical or
thermophysical in nature (i.e., due to thermal cracking, or landslides). We
can likely rule out amorphous to crystalline transitions of water ice as
Jewitt et al. (2017) estimated that they most likely occur within 12.5 au from
the Sun for comets (outside of this distance crystallization timescales are
too long). Carbon monoxide ice is the most likely driver of activity and
outbursts at 20 au, being that it is both abundant in the cometary population
(Cochran et al., 2015) and volatile at this distance (Meech & Svoren̆, 2004).
While CO has been observed in distant comets (Wierzchos et al., 2017; Womack
et al., 2017; Yang et al., 2021), the relationship between CO and outbursts is
not clear, even in the best-observed outbursting comet 29P/Schwassmann-
Wachmann 1. Wierzchos & Womack (2020) could not identify a strong correlation
between their CO and optical lightcurves of comet 29P, and concluded that
increases in CO-outgassing are not always involved with the optically
identified outbursts.

###  4.2 Quiescent Activity and Drivers

Comet Bernardinelli-Bernstein was clearly active beyond $r_{\mathrm{h}}$=20 au
(Section 1). Persistent, distant activity of comets is typically attributed to
the sublimation of species more volatile than H2O, whose sublimation rates
drop off significantly for $r_{\mathrm{h}}$$>$2.5 au. Bernardinelli et al.
(2021) estimated the ice responsible for cometary activity, based on
photometry of the coma from $r_{\mathrm{h}}$=29 to 20 au and thermal models of
the nuclear surface. They concluded that the increase of brightness grew
rapidly with heliocentric distance, and was consistent with the sublimation of
CO2 ice. However, their data set included photometry from 2021 June, which we
have shown is substantially affected by an outburst that ejected
$10^{7}-10^{9}$ kg of material. This outburst brightened the coma within 6″ by
a factor of 3 to 4, as observed on June 22. This calls into question the
conclusion that CO2 is the responsible volatile, or at least requires a re-
analysis of the activity without using the 2021 data.

The analysis of Hubble Space Telescope data taken 2022 January 08 by Hui et
al. (2022) shows that the inner coma ($\rho<1$″) has logarithmic radial slopes
ranging from –1.7 to $\sim$–1.0 and is steepest in the direction of the Sun.
Remarking that a slope of –1.5 is expected in the sunward direction due to
radiation pressure, they conclude the observations are consistent with a
steady state coma. In our observations, the absolute magnitude of the comet on
2022 January 11 is consistent with the 2022 December average
($H_{r}$=6.15$\pm$0.08 mag and 6.19$\pm$0.05 mag, respectively). Therefore,
the brightening seen in 2022 likely occurs after this date and the conclusions
of Hui et al. (2022) are not inconsistent with our own. In fact, it may
indicate that our 2021 December photometry is dominated by a quiescent
activity state.

To assess the contribution of quiescent activity to our data, we have
developed an ad-hoc model with illustrative functional forms for each
contribution: three outbursts and a quiescent coma. All three outbursts are
considered to be step functions with an exponential fall off in dust cross-
section (i.e., magnitude increases linearly with time). For simplicity, the
decay timescales are identical for each event. For the quiescent activity we
take a standard approach in cometary astronomy by assuming the activity scales
as a power-law function of heliocentric distance: $r_{\mathrm{h}}$k. We
derived the power-law exponent from CO ice sublimation rates calculated with
the Cowan & A’Hearn (1979) ice sublimation model. Fits to the model CO
sublimation rates between 18 and 26 au yield slopes ranging from $-2.2$ for a
slowly rotating nucleus to $-2.4$ for a rapidly rotating nucleus (we assumed a
constant active area and a 5% albedo over all wavelengths; best-fit residuals
were $<1$%). The dust production rate is assumed to be directly proportional
to the CO sublimation rate, and scaled to the PS1 photometry from 2019 August
28. The outburst strengths and decay timescales are manually adjusted, and the
result is shown in Fig. 6.

The model lightcurve has poor agreement with the first outburst, either due to
our chosen outburst functional form (linear in magnitude space), or to the
amount of quiescent activity ($H=6.2$ mag on 2021 September 8). Here, an
outburst decay timescale of $\tau=24$ days is shown. Longer timescales would
better fit the July–August photometry but require a shallower heliocentric
distance slope for the quiescent activity in order to fit the early September
data points. However, the quiescent activity is already well below the 2022
January–February photometry ($\Delta H>0.1$ mag), and a shallower slope would
cause a greater disagreement. In addition, the assumption that the activity
has been growing since the PS1 measurement in 2019 could be incorrect, or that
some other effect is needed to match the 2022 data, such as the small
outbursts suggested in Section 4.1. Furthermore, large aperture photometry
from TESS by Bernardinelli et al. (2021) indicate 1.5$\pm$0.2 mag of
brightening from $r_{\mathrm{h}}$=23.8 to 21.2 au (2018 October to 2020
September), more consistent with activity proportional to
$r_{\mathrm{h}}^{-12}$ than $r_{\mathrm{h}}^{-2.2}$, or with additional pre-
discovery outbursts, which calls into question whether or not the PS1 data at
22.6 au are appropriate for establishing a quiescent activity level.
Altogether, the model suggests that CO ice driven sublimation is possible but
some other model assumption must be challenged to confirm this conclusion.

As an alternative to slowly increasing sublimation driven mass loss from CO,
we present a second model for the quiescent activity in Fig. 6. Here, we leave
the activity as constant since the PS1 measurement, then add a linearly
increasing enhancement starting in early October. This model has a much better
agreement with the data. The outburst decay timescale, $\tau=40$ days, is long
enough for the lightcurve to be nearly linear in July–August, consistent with
the data. Moreover, the local minimum in brightness in 2021 December is
reproduced, along with the brightening trend in 2022. We have no particular
physical motivation for this model, other than the possibility that the
outbursts signaled the onset of a new active area, perhaps due to the
propagation of heat into the subsurface as the comet approaches the Sun.

Taken together, our illustrative models in Fig. 6 indicate that the lightcurve
is outburst dominated from 2021 June through September, and likely quiescent
activity dominated by 2021 December. Observations with the Hubble Space
Telescope in 2022 January support this conclusion, and the early assessment
that the comet’s activity was driven by CO2 sublimation, which partially
relied on outburst-dominated photometry from 2021 June, is unlikely and needs
revision.

Figure 6: Ad-hoc model lightcurves (absolute magnitude, $H(1,1,0)$ versus
time) based on illustrative functional forms for the outbursts and quiescent
activity of comet C/2014 UN271 (Bernardinelli-Bernstein). LOOK Project
photometry of the comet is shown as gray circles ($g$ has been scaled to $r$
as in Fig. 4). Outbursts follow an exponential decay in dust cross-section,
with the same decay timescale, $\tau$, for all events within a panel.
Quiescent activity models are scaled to match the PS1 photometry from 2019
August 28. Top: Quiescent activity model based on model CO ice sublimation
rates, $Z$, parameterized as $Z\propto r_{\mathrm{h}}^{-2.2}$. Bottom:
Quiescent activity model based on constant activity since 2019, and a linearly
increasing enhancement starting at 2021.76 (2021 October 5). Together the
models indicate that the lightcurve is outburst dominated but transitions to
quiescent activity dominated by 2021 December.

##  5 Conclusions

We examined the intrinsic brightness of comet C/2014 UN271 (Bernardinelli-
Bernstein) as part of a study of this comet’s distant activity. Our data set
was comprised of $g^{\prime}$\- and $r^{\prime}$-band images from the LOOK
Project, including data from the Comet Chasers outreach program, over the time
period from the discovery of activity on 2021 June 22 to 2022 February 23.
Within 6″ radius apertures, the comet had brightened by $\Delta m=$–1.5 mag
between the last small-aperture photometric measurements in 2019 August from
the PanSTARRS 1 survey (Bernardinelli et al., 2021) and the first LOOK Project
photometric data on 2021 June 22. The absolute magnitude faded with time at a
rate of 13.8$\pm$0.4 mmag $\textrm{day}^{-1}$, up until two small outbursts on
our about 2021 September 09 and 18 ($\Delta m=-0.65$ and $-0.22$ mag). The two
smaller outbursts occurred while material from the June outburst was still
expanding and leaving the aperture, i.e the September outbursts happened
before the quiescent coma state was reached. These events were immediately
followed by a slight fading in intrinsic brightness ($+0.26$ mag), then a
period of low-amplitude variability ($H_{r}$=5.9 to 6.2 mag), up to the end of
our data set. Radial profiles of the coma suggest the variability may be
caused by small, $\sim-0.1$ mag outbursts, although no direct photometric
evidence (e.g., a short-scale discontinuity in the lightcurve) is seen. A
comparison of our data to the Hubble Space Telescope observations on 2022
January 08 (Hui et al., 2022) suggests our photometry in 2021 December was
dominated by the quiescent coma, and an ad-hoc analysis of our lightcurve
confirms this point. Ejecta from the first (prior to 2021 June 22) outburst
expanded with speeds of at least 50 m s-1 in the plane of the sky, and the
slowest moving ejecta no faster than 13 m s-1. Throughout the observed period,
the color of the comet was constant (mean $g-r=0.47\pm 0.01$ mag) and
consistent with the long-period comet population.

A pre-perihelion outburst ($\Delta m\sim-0.5$ mag) near $r_{\mathrm{h}}\sim
20$ au was also observed at comet C/2010 U3 (Boattini) by Hui et al. (2019).
It may be that pre-perihelion outbursts are common for Oort cloud comets in
the outer Solar System, although none have been reported for the third comet
observed at these distances: comet C/2017 K2 (Meech et al., 2017; Jewitt et
al., 2017; Hui et al., 2018). The nuclei of comets Bernardinelli-Bernstein and
Boattini are likely complex geological worlds, and the cause(s) of these
events may be difficult to discern at the present time. However, further study
of cometary activity and outbursts at large heliocentric distances will be
beneficial to understanding the long-term surface evolution (e.g., pit
formation) of these comets and how it relates to the paucity of Oort cloud
comet discoveries (i.e., the long-period comet fading phenomenon; Vokrouhlický
et al., 2019; Kaib, 2022). Furthermore, combining future observations with the
pre-2021 data, or observations with higher signal-to-noise ratios and finer
spatial resolving powers may reveal more about the quiescent activity in this
particular comet (e.g., Kokotanekova et al., 2021b; Hui et al., 2022).

We appreciated the helpful comments from the manuscript referee and the
scientific editor. Support for this work was provided by the NASA Solar System
Observations program (80NSSC20K0673). HU, TA, and BW were supported by a UK
Science and Technology Facilities Council (STFC) “Spark Award” grant for the
Comet Chasers school outreach program.

This work makes use of observations from the Las Cumbres Observatory global
telescope network as part of the LCO Outbursting Objects Key (LOOK) Project
(KEY2020B-009), and the Comet Chasers school outreach program
(FTPEPO2014A-004). The Comet Chasers program is part of the Faulkes Telescope
Project, which is partly funded by the Dill Faulkes Educational Trust. Pupils
from the following Welsh schools made the observations: St Mary’s Catholic
Primary School Bridgend, Montgomery Church in Wales School, White Rose Primary
School New Tredegar, and Idris Davies School Rhymney.

This research has made use of data and services provided by the International
Astronomical Union’s Minor Planet Center.

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# On the number of constituents of
products of characters

Maria Loukaki  Department of Applied Mathematics
University of Crete
Knosou Av. 74 GR-71409 Heraklion-Crete GREECE<EMAIL_ADDRESS>and
Alexander Moretó  Departament d’Àlgebra
Universitat de València
46100 Burjassot. València SPAIN<EMAIL_ADDRESS>
###### Abstract.

It has been conjectured (see [1]) that if the number of distinct irreducible
constituents of the product of two faithful irreducible characters of a finite
$p$-group, for $p\geq 5$, is bigger than $(p+1)/2$, then it is at least $p$.
We give a counterexample to this conjecture.

The second author was supported by the FEDER, the Spanish Ministerio de
Ciencia y Educación, grants MTM2004-04665 and MTM2004-06067-C02-01, the
Generalitat Valenciana and the Programa Ramón y Cajal.
Key words: Products of faithful irreducible characters
2000 Mathematics Subject Classification Primary 20C15
Received by the editors April 12, 2005 and, in revised form, January 9, 2006

In this note, $p$ will stand for a prime number. Let $\varphi$ and $\psi$ be
faithful irreducible characters of a finite $p$-group $P$. What can be said
about the number of different irreducible constituents of the product
$\varphi\psi$? At first sight, it does not seem reasonable to expect strong
restrictions for the possible values of this number. However, in [1] it was
proved that if the number of constituents of this product is bigger than one,
then it is at least $(p+1)/2$.

It seems reasonable to ask what further restrictions can be found. In p. 237
of [1] it was conjectured that if the number of constituents of the product of
two faithful irreducible characters of a finite $p$-group, for $p\geq 5$, is
bigger than $(p+1)/2$, then it is at least $p$. The theorem that follows shows
that this is not true.

######  Theorem.

Let $P=C_{p}\wr C_{p}$ for $p\geq 5$. There exist
$\varphi,\psi\in\operatorname{Irr}(P)$ faithful such that $\varphi\psi$ has
exactly $p-1$ distinct irreducible constituents.

###### Proof.

Write $P=CA$, where $A$ is the base group, which is elementary abelian of
order $p^{p}$. It is clear that the non-linear characters of $P$ are induced
from characters of $A$. In particular, they have degree $p$. Fix any non-
principal character $\lambda\in\operatorname{Irr}(C_{p})$. Then any character
of $A$ can be written in the form
$\nu=\lambda^{i_{1}}\times\cdots\times\lambda^{i_{p}}$ for some integers
$i_{j}=0,\dots,p-1$. Thus, we can identify the character $\nu$ with the
$p$-tuple $(i_{1},\dots,i_{p})$. It is clear that
$\nu^{G}\in\operatorname{Irr}(G)$ if and only if not all the $i_{j}$’s are
equal.

We have that $Z(P)=\\{(x,\dots,x)\mid x\in C_{p}\\}$ is the unique minimal
normal subgroup of $P$. Also, if $\nu^{G}\in\operatorname{Irr}(G)$, it follows
from Lemma 5.11 of [2] that $\nu^{G}$ is faithful if and only if
$(x,\dots,x)\not\in\operatorname{Ker}\nu^{G}$. Notice that if
$\lambda(x)=\varepsilon$ for a primitive $p$th root of unity, then

$\nu^{G}(x,\dots,x)=p\varepsilon^{i_{1}+\cdots+i_{p}}.$

Thus $\nu^{G}$ is faithful if and only if $\sum_{j=1}^{p}i_{j}\not\equiv
0\pmod{p}$.

Consider the characters of $A$ associated to the $p$-tuples $(1,0,0,\dots,0)$
and $(1,1,0,\dots,0)$. They induce faithful irreducible characters of $P$,
$\varphi$ and $\psi$ respectively. We claim that $\varphi\psi$ has $p-1$
distinct irreducible constituents.

The character $\varphi_{A}$ decomposes as the sum of the characters associated
to $(1,0,0,\dots,0),(0,1,0,\dots,0),(0,0,1,\dots,0),\dots,(0,0,0,\dots,1)$. We
can argue similarly with $\psi_{A}$. The product of two characters of $A$
corresponds to the componentwise sum of the associated $p$-tuples and two
characters of $A$ are $P$-conjugate if and only we can go from the $p$-tuple
associated to one of the characters to the other by a cyclic permutation of
the components. Now, it is easy to see that the number of constituents of the
character $\varphi\psi$ is the number of characters of $P$ lying over the
characters of $A$ corresponding to the $p$-tuples
$(2,1,0,0,\dots,0),(1,2,0,0,\cdots,0),(1,1,1,0,\dots,0),(1,1,0,1,\dots,0),\dots,\\\
(1,1,0,0,\dots,1)$. Here we have $p$ different $p$-tuples. The third of these
$p$-tuples and the last one correspond to $P$-conjugate characters of $A$, so
they induce the same character of $P$. It is easy to see that no other pair of
$p$-tuples are conjugate. The claim follows. ∎

We have been unable to find any example where the number of constituents of
the product of two faithful irreducible characters of a $p$-group has more
than $(p+1)/2$ distinct irreducible constituents but less than $p-1$. So the
following modification of the conjecture could still be true.

######  Question.

Let $\varphi$ and $\psi$ be faithful irreducible characters of a finite
$p$-group $P$. Assume that $\varphi\psi$ has more than $(p+1)/2$ distinct
irreducible constituents. Does it necessarily have at least $p-1$ irreducible
constituents?

## References

  * [1] E. Adan-Bante, Products of characters and finite $p$-groups, J. Algebra 277 (2004), 236–255.
  * [2] M. Isaacs, Character Theory of Finite Groups, Dover, New York, 1994.