# Fragmenting Active Asteroid 331P/Gibbs

David Jewitt1,2, Jing Li1 and Yoonyoung Kim3
1Department of Earth, Planetary and Space Sciences, UCLA, 595 Charles Young
Drive East, Los Angeles, CA 90095-1567
2Department of Physics and Astronomy, University of California at Los Angeles,
430 Portola Plaza, Box 951547, Los Angeles, CA 90095-1547
3Institut for Geophysik und Extraterrestrische Physik, Technische Universitat
Braunschweig, Mendelssohnstr. 3, 38106 Braunschweig, Germany<EMAIL_ADDRESS>
###### Abstract

We describe active asteroid 331P/Gibbs (2012 F5) using archival Hubble Space
Telescope data taken between 2015 and 2018. 331P is an outer main-belt active
asteroid with a long-lived debris trail that formed in 2011. Embedded in the
debris trail we identify 19 fragments with radii between 0.04 and 0.11 km
(albedo 0.05 assumed) containing about 1% of the mass of the primary nucleus.
The largest shows a photometric range ($\sim$1.5 magnitudes), a V-shaped
minimum and a two-peaked lightcurve period near 9 hours, consistent with a
symmetric contact binary (Drahus and Waniak 2016). Less convincing
explanations are that 331P-A is a monolithic, elongated splinter or that its
surface shows hemispheric 4:1 albedo variations. The debris trail is composed
of centimeter sized and larger particles ejected with characteristic 10 cm s-1
speeds following a size distribution with index $q$ = 3.7$\pm$0.1 to
4.1$\pm$0.2. The HST data show that earlier, ground-based measurements of the
nucleus were contaminated by near-nucleus debris, which cleared by 2015. We
find that the primary nucleus has effective radius 0.8$\pm$0.1 km and is in
rapid rotation (3.26$\pm$0.01 hour, range 0.25 magnitudes, minimum density
1600 kg m-3 if strengthless. The properties of 331P are consistent with a)
formation about 1.5 Myr ago by impact shattering of a precursor body
(Novakovic et al. 2014) b) spin-up by radiation torques to critical rotation
c) ejection of about 1% of the nucleus mass in mid-2011 by rotational
instability and d) subsequent evolution of the fragments and dispersal of the
debris by radiation pressure.

comets: general—asteroids: individual 2012 F5

††slugcomment: The Astronomical Journal, in press

##  1 INTRODUCTION

331P/Gibbs (2012 F5) (hereafter 331P) was discovered on UT 2012 March 22
(Gibbs et al. 2012). The object showed both the physical attributes of a comet
(in the form of a long, thin dust tail) and the orbital elements of an
asteroid. The orbital semimajor axis, eccentricity and inclination are 3.005
AU, 0.042 and 9.7°, respectively, corresponding to a Tisserand parameter with
respect to Jupiter, $T_{J}$ = 3.228. This is significantly above the nominal
$T_{J}$ = 3 dividing line separating asteroids from comets (Vaghi 1973), and
also above the practical $T_{J}$ = 3.08 line used to distinguish active
asteroids from short-period comets (Jewitt 2012, c.f. Hsieh and Haghighipour
2016). Numerical simulations indicate that 331P is dynamically stable on 109
year timescales (Stevenson et al. 2012). It is simply a main-belt asteroid
that ejects observable quantities of dust.

As a new member of the exciting and still poorly understood active asteroids
population, 331P attracted immediate observational attention. Moreno et al.
(2012) and Stevenson et al. (2012) used the position angle and morphology of
the tail to infer an impulsive origin on UT 2011 July 1$\pm$11 and UT 2011
July 7$\pm$20, respectively. As was the case with the impact-caused active
asteroid P/2010 A2 (Jewitt et al. 2010), the $\sim$8 month delay between the
release of the dust and the detection of the object is at least partly
explained by the small solar elongation (which reached a minimum $\sim$2° in
2011 August and remained $<$90° until 2011 December).

A statistically significant mini-cluster of eight asteroids dynamically
associated with 331P was identified by Novaković et al. (2014), suggesting
formation by impact fragmentation of a parent body. From the tight convergence
of the orbital elements within this cluster, these authors estimated a
separation age of only 1.5$\pm$0.1 Myr. 331P is small compared to the largest
cluster member, 20674 (1999 VT1), which has diameter $\sim$18 km (assuming the
same albedo), and the parent body may have been $\sim$24 km in diameter
(Novaković et al. 2014). Deep imaging observations taken in 2014 showed that
331P is rotating with a photometric range $\sim$0.2 magnitudes and a two-
peaked period $P=3.24\pm 0.01$ hour, and revealed four fragments embedded in
the dust tail (Drahus et al. 2015). The short rotational period of 331P
suggests a potential role for rotational breakup, instead of impact as the
origin of the comet-like appearance.

The available observations thus reveal 331P as an intriguing object. The
connection between its formation as an impact fragment $\sim$1.5 Myr ago and
the modern episode of mass loss is unclear. Did 331P experience another impact
to eject material in 2011, or is rotational instability to be implicated? The
outer-belt orbit of 331P is similar to the orbits of several active asteroids
in which the mass loss is likely due to sublimation, notably the prototype
main-belt comet 133P/Elst-Pizarro (7968) (Hsieh and Jewitt 2006), raising the
possibility that sublimation might also play a role in 331P. The ambiguities
suggested by these questions highlight the difficulties encountered in the
interpretation of the active asteroids as a population.

In this paper we examine archival Hubble Space Telescope (HST) observations
taken under General Observer programs 14192, 14475, 14798 and 15360 (PI: M.
Drahus) in order to characterize this intriguing object. These data have not
been published other than as a brief meeting abstract (Drahus and Waniak
2016). We first examine the physical properties of 331P and its ejecta as
revealed at HST resolution. In a separate work, we will analyse the dynamics
of the fragments.

##  2 OBSERVATIONS

###  2.1  Photometry and General Appearance

The WFC3 camera consists of two 2k$\times$4k charge coupled devices separated
by a 1.2″ gap (Dressel 2012). The image scale is 0.04″ pixel-1, giving a
Nyquist-sampled resolution near 0.08″, corresponding to $\sim$120 km at 2 AU
geocentric distance. The practical limits to HST point-source photometry are
set by cosmic rays, which pepper the detector in great abundance. Our
philosophy is to avoid heroic efforts to recover images photometrically
compromised by cosmic rays. We simply exclude severely compromised images from
further consideration and attempt only light editing (by interpolation of
surrounding pixels) of images showing minor cosmic ray and image defect
interference. A small minority of the images were so affected.

Limits to the photometry of diffuse sources are set by internally scattered
light from bright objects both inside and outside the field (c.f. Figure 1),
with particular impact on measurements of the diffuse trail. In addition to
scattered light, background stars and galaxies are smeared by strong
parallactic motion on solar system sources. When making composite images, we
experimented with different rejection schemes to reduce these background
variations. Smeared galaxies are particularly difficult to suppress, because
they are of large angular extent and frequently overlap in successive images.
Example residuals are evident in the diagonal streaks in Figure (1). In
several of the data sets, the residuals caused by scattered light and trailed
field objects are comparable to or brighter than the surface brightness of the
331P dust trail, making accurate measurement particularly difficult.

We relied on the use of small photometry apertures to reduce the impact of
both cosmic rays and background variations. For the discrete fragments, most
of which are faint and highly susceptible to cosmic ray and other background
uncertainties, we used a photometry aperture of 5 pixels (0.2″) radius with
sky determined from the median signal within a contiguous annulus extending to
4.2″. The use of a relatively large sky annulus was found experimentally to
give a more stable and accurate estimate of the local sky. For 331P itself, we
experimented with nested apertures 5, 7 and 9 pixels (0.20″, 0.28″and 0.36″)
in projected radius with sky subtraction from a contiguous annulus extending
to 4.2″. We find that the signal between 0.20″ and 0.36″ is about 8% of the
signal in the 0.2″ radius aperture, and that this signal fraction does not
vary with variations in the central magnitude. The sacrifice of a small
fraction of the total signal from each object is warranted by the improved
reduction of noise and rejection of interfering cosmic rays and image defects,
and does not affect the relative photometry between fragments. We have not
made a correction for the 8% reduction in photometrically estimated cross-
sections (4% in fragment radii); these corrections are, in any case,
inconsequential compared to the much larger uncertainties introduced by the
unknown albedos and phase functions of the target bodies. Our use of small
apertures is only possible because of the remarkable stability of the point-
spread function of HST (Dressel 2012).

We found useful observations of 331P taken in eight epochs spread over the
period from late 2015 to mid-2018, using a total of 30 HST orbits. We did not
find useful images from visits on UT 2018 May 4 and 5, and 2018 June 2,
despite the fact that the ephemeris was no less accurate on these dates, and
331P could not have been faint enough to elude detection by HST. Data from
these dates are not further discussed here. A journal of observations is given
in Table (1).

Figure (2) shows image composites from each epoch of observation, all rotated
to bring the central axis of the trail to the horizontal. The images have been
scaled to a common geocentric distance, $\Delta$ = 2.011 AU (i.e. the distance
of the first observation on UT 2015 December 25), and co-aligned on the
primary nucleus labeled 331P. The panels are variously affected by imperfect
removal of scattered light and trailed field objects, with the most glaring
residuals apparent on the right hand (western) side of the panel for UT 2017
Mar 08. Nevertheless, a consistent appearance is evident in the data from 2015
to 2018: 331P shows a bright primary nucleus, with a faint debris trail to the
west in which are peppered point source fragments. The images also show
ultrafaint diffuse material to the east of the primary.

Fragments 331P-A, -B and -C are prominent in each panel and are labeled in red
in order to guide the eye. Other fragments are identified only in the panel
where they are first apparent and discussed in Section 2.4. Note that the sky-
plane positions of the fragments in Figure (2) are affected by the changing
phase angle of observation so that a steady expansion from the primary cannot
be discerned.

###  2.2 Primary Nucleus

We computed the mean absolute magnitude of 331P from the apparent V-band
magnitude, $V$, using

$H=V-5\log_{10}(r_{H}\Delta)+2.5\log_{10}(\Phi(\alpha))$ (1)

where $r_{H}$ and $\Delta$ are the heliocentric and geocentric distances in
AU, respectively, and $\Phi(\alpha)$ is the phase function, equal to the ratio
of the brightness measured at phase angle $\alpha$ to that at $\alpha$ = 0.
The phase function is unmeasured in 331P and is not necessarily the same for
dust as for macroscopic objects. Based on measurements of asteroids we adopt
$2.5\log_{10}(\Phi(\alpha))=-0.04\alpha$ (Martikainen et al. 2021) and note
that a $\pm$50% error in this value would affect the derived absolute
magnitudes by up to $\pm$0.4 magnitudes. .

The absolute magnitudes are plotted as a function of the date of observation
in Figure (3), where we include both data from HST and from the literature.
Measurements taken in the R filter were converted to V using V-R = 0.4$\pm$0.1
(Novakovic et al. 2014). Measurements from the HST data have very small formal
error bars (typically $\pm$0.02 to 0.03 magnitudes) but somewhat larger
systematic errors (judged from the scatter to be of order 0.1 magnitudes)
owing to incomplete sampling of the rotational lightcurve (see below).

We adopt $H$ = 17.96$\pm$0.10 as our best estimate of the mid-light absolute
magnitude of the nucleus, indicated in the figure by a dashed line. Figure (3)
shows that the Stevenson et al. (2012) (labeled S12) and Novakovic et al.
(2014) (N14) absolute magnitudes are brighter than $H$ by about 2.0 and 0.6
magnitudes (factors of $\sim$6 and 1.7), respectively, presumably as a result
of near-nucleus dust contamination of the photometry. Even the measurements by
Drahus et al. (2015) (D15), taken in 2014 (three years after the release of
material), suffer dust contamination of $\sim$0.3 magnitudes ($\sim$30%).

The effective scattering cross-section, $C$ [km2], is computed from the
photometry using

$C=\frac{1.4\times 10^{6}}{p_{V}}10^{-0.4H}$ (2)

where $p_{V}$ is the (unmeasured) geometric albedo. Asteroids orbiting near 3
AU tend to be C-types with low albedos ($p_{V}\sim$ 0.05) as opposed to
S-types, which have brighter surfaces ($p_{V}\sim$ 0.2). In this regard, we
note that Stevenson et al. (2012) measured a nearly neutral optical color, B-R
= 1.03$\pm$0.04 (the color of the Sun is B-R = 0.99$\pm$0.02; Holmberg et al.
2006) as did Novakovic et al. (2014) with V-R = 0.40$\pm$0.14 (the solar color
is V-R = 0.35$\pm$0.01); both colors are consistent with C spectral type.
However, Figure (3) shows that both color determinations were made in the
presence of strong dust contamination, and so do not directly refer to the
nucleus.

To proceed, we assume that 331P is a C-type, with $p_{V}$ = 0.05. Then, with
$H$ = 17.96, Equation (2) gives $C$ = 1.8 km2 and the radius of an equal-area
circle is $r_{n}=(C/\pi)^{1/2}$, giving $r_{n}$ = 0.76$\pm$0.01 km. The cross-
section would be four times smaller, and the radius two times smaller, if the
albedo were instead $p_{V}$ = 0.2. Our estimate is small compared to the upper
limit, $r_{n}\lesssim$ 4.2 km, reported by Stevenson et al. (2012), because of
the effects of dust contamination in their early-time photometry.

We model the shape of the fading lightcurve in Figure (3) using the procedure
described in Jewitt et al. (2017). Briefly, particles released from the
nucleus at time $T_{0}$ are swept out of the photometry aperture on a
timescale that depends on the particle size. Small particles are accelerated
strongly by radiation pressure while large particles linger longer, such that
the average radius of particles remaining in the photometry aperture increases
even as the total cross-section in the aperture decreases. For a power-law
distribution of particle radii, in which the number of particles with radii
between $s$ and $s+ds$ is $n(s)ds=\Gamma s^{-q}ds$ ($\Gamma$ and $q>$ 3 are
constants), the time dependence of the cross-section may be written (Jewitt et
al. 2017)

$C(t)=C_{n}+K(t-T_{0})^{6-2q}$ (3)

where $C_{n}$ is the cross-section of the nucleus, $K$ is a constant and
$t\geq T_{0}$. Equation (3) is strictly valid provided that the initial
particle velocity is small compared to the velocity acquired under the action
of radiation pressure. (The Equation also does not apply when
$t-T_{0}\lesssim$ 2 months, because ejected fragments have then not yet
reached the edge of the photometry aperture. However, the first observations
have $t-T_{0}\sim$ 9 months, so this is not a serious limitation). We set
$T_{0}$ = 2011.5 and made a least squares fit of Equation (3) to the data,
finding $C_{n}$ = 1.81$\pm$0.18 km2 and $q=4.1\pm 0.2$. The fit is shown as a
solid line in Figure (3). $C_{n}$ gives $H$ = 17.98$\pm$0.04 and
$r_{n}=0.76\pm$0.02 km for $p_{V}$ = 0.05, in agreement with the
determinations above. In the following, we use $r_{n}$ = 0.8$\pm$0.1 km as a
working value. The derived value of $q$ is slightly steeper than $q=$ 3.7 to
4.0 obtained from the Monte Carlo models (Section 2.3) but, given the
heterogeneous nature of the photometric data (e.g. different filters and
different apertures were used to compile Figure 3), and the different
assumptions made in each method, the difference is not significant.

All HST observations show systematic, short-term brightness variability
indicative of rotation with a two-peaked period, $P$, near 3 hours. The
temporal sampling of HST (images are acquired within windows of about 40
minutes separated by gaps of about an hour) prevents continuous phase coverage
of the lightcurve in any given month. Observations from different years (Table
1) are, for the most part, too widely separated to be unambiguously phased
together. The exception is that data from UT 2015 December 25 and 28 (3 day
separation, corresponding to about 22 rotations) are close enough together
that we can meaningfully link the data to derive the rotation period, albeit
with aliases separated by about 0.014 day-1. Phase dispersion minimization
applied to the absolute magnitudes from these two dates gives comparably
acceptable, two-peaked periods $P$ = 3.170$\pm$0.010 hour and $P$ =
3.259$\pm$0.010 hour. The latter of these is close to the period deduced from
ground-based data, namely $P$ = 3.24$\pm$0.01 hour (Drahus et al. 2015). In
Figure (4) we show the data from UT 2015 December 25 and 28 phased to $P$ =
3.259 hour (yellow circles) compared to the lightcurve from Drahus et al.,
rescaled to the same period and shifted in $H$ using V-R = 0.4 to account for
the use of different filters. The phase offset between data from 2014 and 2015
is unknown. Figure (4) shows that the basic features of the lightcurve are
well reproduced, although with small differences. For example, the lightcurve
range, $\Delta V=0.25$ magnitude, is $\sim$0.05 magnitudes larger than in the
ground-based data. This likely reflects the combined effects of dilution of
the nucleus photometry in 2014 owing to near nucleus dust that had cleared by
2015 (Figure 3), and a year-to-year change in the viewing geometry with
respect to the projected spin-pole of 331P. A consistent but less precise
period, $P=3.25\pm 0.05$ hours, was determined from the HST image sequence on
UT 2017 February 13.

Approximating the shape of 331P by a prolate ellipsoid with semi-axes $a\times
b\times c$, with $a=b$, and $c/a\geq 1$, in rotation about a short axis, we
estimate the axis ratio from the lightcurve using $c/a=10^{0.4\Delta V}$.
Substituting $\Delta V$ = 0.25 magnitudes gives $c/a\sim$ 1.25. The
gravitational acceleration acting on the tip of the nucleus ellipsoid exceeds
centripetal acceleration provided the density exceeds a critical value

$\rho_{c}=\frac{3\pi}{GP^{2}}\left(\frac{c}{a}\right)^{2}.$ (4)

Substituting $P$ = 3.26 hours gives $\rho_{c}$ = 1600 kg m-3, which is of firm
significance only if 331P can be meaningfully represented as strengthless. For
comparison, the mean densities of C-type ($\rho\sim 1500\pm 500$ kg m-3) are
compatible with this value within the uncertainties (Hanus et al. 2017). With
$\rho$ = 1600 kg m-3 and $r_{n}$ = 0.8 km, the nucleus mass is $M_{n}=3\times
10^{12}$ kg. The gravitational escape velocity from a non-rotating sphere with
density $\rho$ = 1600 kg m-3 and radius $r_{n}$ = 0.8 km is $V_{e}$ = 0.8 m
s-1 but rotation may reduce the effective escape speed to near zero at the
tips of the long axis.

###  2.3 Debris Trail

The trail is the dominant morphological feature of 331P. Figure (5) shows the
position angle of the trail, $\theta_{PA}$, measured with respect to the
nucleus and plotted as a function of the true anomaly, $\nu$. The trail angle
and its uncertainty (typically $\pm$0.1°) were measured using determinations
of the location of peak brightness at different projected distances from the
primary nucleus. The position angle of the negative heliocentric velocity
vector, $\theta_{-V}$, obtained from JPL Horizons
(https://ssd.jpl.nasa.gov/horizons.cgi), is also shown in the figure as a
solid black line. There is generally excellent agreement between $\theta_{PA}$
and $\theta_{-V}$. On the other hand, there is no correspondence between
$\theta_{PA}$ and the projected antisolar vector, $\theta_{-\odot}$, which
undergoes much more dramatic excursions in the range 65°
$\leq\theta_{-\odot}\leq$ 169° during the period covered by the observations
(see Table 1). This confirms that the basic morphology of the trail reflects
the distribution of large particles confined to the orbit, as inferred by
Moreno et al. (2012) and Stevenson et al. (2012).

The dominance of large particles is consistent with the action of radiation
pressure, which truncates the size distribution by quickly sweeping small
particles away. The dispersal timescale is

$\tau_{rad}\sim\left(\frac{2\ell_{T}}{\beta g_{\odot}(r_{H})}\right)^{1/2}$
(5)

where $\ell_{T}$ is the length of the trail under consideration,
$g_{\odot}(r_{H})$ is the gravitational attraction to the Sun at distance
$r_{H}$, and $\beta$ is the dimensionless radiation pressure factor, defined
such that $\beta g_{\odot}(r_{H})$ is the particle acceleration. With a nearly
circular orbit at 3 AU, we write $g_{\odot}(3)=7\times 10^{-4}$ m s-2.
Parameter $\beta$ is a function of the wavelength, and of many unknown
particle parameters, including the composition, shape, size and porosity. For
spherical, uniform dielectric particles we take $\beta\sim 10^{-6}/s$, where
$s$ is the particle radius in meters (Bohren and Huffman 1983). To give an
example, the trail in 2015 December extends beyond the 160″ field of view of
WFC3, corresponding to $\ell_{T}\sim 2.4\times 10^{8}$ m, in the plane of the
sky, given the geocentric distance $\Delta$ = 2 AU. Substituting into Equation
(5) gives $\tau_{rad}\sim 28s^{1/2}$, with $\tau_{rad}$ in years. Setting
$\tau_{rad}$ = 4.5 years (the time elapsed between 2015 December and the
ejection in 2011 July), gives $s\sim$ 2.5 cm, meaning that smaller particles
have been swept away.

Particles ejected from the nucleus increase their distance from the orbit
plane in proportion to $t$, the time of flight, but are accelerated down the
trail by a distance proportional to $t^{2}$, as a result of the steady
acceleration by solar radiation pressure. Consequently, the trail width
perpendicular to the orbit plane, $W_{\perp}$, is expected to increase with
distance from the nucleus, $\ell_{T}$, as $W_{\perp}\propto\ell_{T}^{1/2}$.

We used the composite image from UT 2018 May 17 to test this expectation by
measuring the width of the dust trail as a function of distance along the
trail. Observations on this date were taken from the smallest angle (-1.5°)
between HST and the orbit plane (Table 1) and thus suffer least from the
effects of projection. The results are plotted in Figure (6). The widths were
determined by visual inspection from the FWHM of the perpendicular surface
brightness profile averaged over segments whose widths are indicated by
horizontal bars in the figure. The expected $W_{T}\propto\ell_{T}^{1/2}$
dependence provides an acceptable fit to the data, shown by the red line
least-squares fit in the figure.

The velocity vs width relation is (Jewitt et al. 2014a)

$V_{\perp}=\left(\frac{\beta g_{\odot}}{8\ell_{T}}\right)^{1/2}W_{\perp}$ (6)

in which $V_{\perp}$ is the ejection velocity perpendicular to the orbit
plane. We substitute $\beta=10^{-4}$ for centimeter-sized particles,
$g_{\odot}=7\times 10^{-4}$ m s-2 for the solar gravity at 3 AU, $W_{\perp}$ =
1700 km and $\ell_{T}=10^{5}$ km (Figure 6) to find $V_{\perp}\sim$ 2 cm s-1.
The result is approximate for two reasons; first, because the observations
were taken -1.5° from the orbit plane and thus suffer from projection and
second, because Equation (6) is strictly valid only when the particle speed
induced by radiation pressure is large compared to the ejection speed.
Nevertheless, the trail width measurements in Figure (6) are consistent with
an extremely small perpendicular velocity.

Figure (7) shows the surface brightness (black line) as a function of angular
distance from the primary nucleus along the axis of the trail using data
averaged from the independent profiles from UT 2015 December 25 and 28 to
improve the SNR. The surface brightness was computed within a region extending
$\pm$1.2″ from the axis, with sky background determined from the average of
adjacent strips of equal width centered 2.4″ above and below the trail. The
profile is binned to 0.4″ resolution along the trail direction in order to
improve the signal-to-noise ratio. Numerous trailed field stars and galaxies
contaminate the profiles, as do scattered light flares from bright objects
near the HST field and artifacts from imperfect flattening. The most egregious
of these are marked in Figure (7) by asterisks, $*$, and the region from 58″
to 77″ is omitted entirely owing to background light that could not be
adequately removed. The axial profile in the UT 2015 December data clearly
peaks about 40″ (58,000 km in the plane of the sky) behind the primary
nucleus.

We used a Monte Carlo model (Ishiguro et al. 2007) to reproduce the measured
surface brightness profile in Figure (7). The model considers the motions of
ejected particles under the action of radiation pressure and solar gravity.
While such a multi-parameter model cannot produce unique solutions, this
approach does allow us to limit the possible range of several important
particle parameters. For example, in gas drag acceleration, the initial
particle speed varies with the particle radius, $s$, as $V\propto s^{-1/2}$
(Whipple 1950). We find that no models with this velocity - size dependence
produce a convincing match to the data because the trail widths from faster
particles exceed the measured trail width, confirming a result from Moreno et
al. (2012). Models with ejection on UT 2011 July 1, low size-independent
velocity $V$ = 5 to 10 cm s-1, and a power-law size distribution with index
$q$ = 3.7, give a reasonable match to the data up to $\sim$20″ west of the
nucleus but, at larger angles, the measured surface brightness is lower than
in the $q$ = 3.7 model (Figure 7). This means that the largest particles,
which have traveled the smallest distance from the nucleus, are adequately
represented by $q$ = 3.7 but smaller particles, accelerated by radiation
pressure to larger distances, are not. We could find no single power law
distribution to improve the fit far west of the nucleus without losing the fit
nearer the nucleus. Therefore, we fitted a broken power law to the surface
brightness profile (Figure 7). We obtained an improved fit with a break at
$\beta_{crit}=10^{-5}$ (corresponding to $s\sim$ 10 cm particles) and $q$ =
4.0 for $\beta\leq\beta_{crit}$ ($s\geq$ 10 cm) and $q$ = 3.0 (blue line in
Figure 7) or even 2.5 (green line in the figure) for larger $\beta$ ($a<$ 10
cm). The physical significance of these fits is, of course, questionable,
since by adding a broken power law we are adding more free parameters to a
multi-parameter model. However, the results do broadly support the work by
Moreno et al. (2012), but using independent data that samples the debris four
years after launch. The results are also broadly compatible with the index
$q=4.1\pm 0.2$ deduced from the fit to fading lightcurve (Section 4 and
Equation 3).

In order to measure the integrated brightness of the trail we used a
3.2″$\times$48.0″ rectangular section with the long axis extending along the
trail to the west. More distant portions of the trail are very faint and
introduce excessive uncertainty owing to variations in the background. As a
measure of the background, we determined the median signal in two same-sized
flanking strips above and below the trail and subtracted their average.
Measurements on UT 2015 December 25 and 28 are close enough together in time
that the trail is unlikely to have changed, and so permit a measure of the
effect of interfering background sources and scattered light in the data. The
integrated trail magnitudes were V = 22.24 and 22.18 on December 25 and 28,
respectively, indicating agreement at the $\sim$6% level. The average of these
measurements, together with Equations 1 and 2, gives a dust scattering cross-
section $C=27\pm 3$ km2, again assuming $p_{V}$ = 0.05. This trail cross-
section is spread over the $\sim 3.3\times 10^{8}$ km2 projected area of the
measuring rectangle, corresponding to an average optical depth of only $\sim
6\times 10^{-9}$. This value lies within the range of optical depths reported
for the large-particle trails of short-period comets (Ishiguro et al. 2009).

Given that the trail is optically thin, its mass, $M$, can be written
$M\sim\rho\overline{a}C$, where $\overline{a}$ is the cross-section weighted
mean particle size. The trail rectangle length, $\ell_{T}$, is $\ell_{T}\sim
7\times 10^{4}$ km, and the observations were taken $\sim$4.5 years after the
particle release, giving $\tau_{rad}=1.4\times 10^{8}$ s. Substitution into
Equation (5) gives $\beta\sim 10^{-5}$, corresponding to a minimum particle
size in the measured portion of the trail of $\sim$10 cm. Strictly, $\ell_{T}$
is a lower limit to the visible length of the trail because of the effects of
projection, but we can confidently conclude that particles smaller than
centimeters in size will have already left the trail. With $\overline{a}$ =
0.1 m, we obtain a lower bound to the trail mass, $M\gtrsim 2\times 10^{8}$
kg. For comparison, Moreno et al. (2012) found $M\gtrsim 5\times 10^{8}$ kg
while Stevenson et al. (2012) found $M\gtrsim 5\times 10^{7}$ kg, both in data
from 2012. The ratio of the mass in the optically dominant particles relative
to the nucleus mass is $M/M_{n}\sim 10^{-4}$.

###  2.4 Fragments

The visibility of the fragments varies from panel to panel in Figure (2) both
because different numbers of images and total on-source integration time were
acquired in different orbits and because of the varying geometry of
observation (Table 1). For example, few fragments can be detected on UT 2016
February 13 and July 03 because of the larger geocentric distances and phase
angles on these dates, causing a geometrical dimming by $\sim$0.8 magnitude
relative to the observations from UT 2015 December 25. The composites from UT
2015 December 28 and 2018 May 17 are diminished by the smaller number of
images secured on these dates (Table 1).

Most of the fragments are detected in the first and deepest composite image
(from UT 2015 December 25) which benefits from both a large number of images
and a favorable geometry. 331P-A, -B and -C, as well as 331P itself, are
unambiguous in all panels and are marked in red in each panel of Figure (2) to
act as guideposts. Other fragments are marked when first noticed but are not
labeled in subsequent panels to avoid excessive clutter. For example, Fragment
Q is first noticed and labeled in UT 2017 February 13 and is visible again
(but not labeled) in data from UT 2018 May 17 and May 26 (and, possibly, UT
2018 July 03). Fragment S is slightly below Fragment A in Figure (2) when
first noticed (UT 2018 May 17) and is visible again in UT 2018 May 26 and July
03. Most of the fragments first detected after 2015 are close to the detection
threshold and could have been missed in UT 2015 December 25 as a result of
temporal brightness changes (due to rotation?), and in other visits because of
the changing limiting magnitude. 331P-R in UT 2018 May 17, for example, has no
convincing counterpart in earlier data, although it is faintly visible again
on May 26. Overall, the fragment trail is challenging to study because it is
spatially complex and temporally under-sampled (e.g. the long interval between
the high quality images from UT 2015 December 25 and 2017 March 08). Firm
linkages across the dataset await a detailed dynamical study to be undertaken
separately.

In Table (2) we list the mean absolute magnitudes of the fragments using data
from all eight visits. To minimize the effects of background light and,
especially, the interfering effects of imperfectly removed cosmic rays, we
present only photometry within a 5 pixel (0.2″) radius circular aperture, with
sky subtraction from a contiguous annulus extending to 105 pixels (4.2″). The
table lists the mean, $H_{5}$, and median, $H_{m}$, magnitudes showing that,
in most cases the two estimates of the central value are in good agreement.
The uncertainty on $H_{5}$ is just the standard error on the mean from
different orbits except that, for fragments measured in only one orbital
visit, we have set the uncertainty to 0.1 magnitudes. The last column of Table
(2) lists the fragment radius, computed from $H_{m}$ and Equation (2). The
uncertainties on $r_{n}$ are formal only, and underestimate the true
uncertainties because of the unknown albedos and, to a lesser extent, phase
functions of the fragments.

We first consider the HST photometry for 331P, 331P-A, -B and -C, the four
objects that were measured in all seven visits. The measurements shown in
Figure (8) are vertically offset for clarity. Lines in the figure show least-
squares fits to the photometry as a function of time and show that in no case
is there evidence for secular fading or brightening of the fragments across
the full 2.5 years of the HST data set. This is strong evidence that the
fragments are inert (asteroid-like) bodies, not releasing dust particles. The
lack of comet-like activity is further suggested by the absence of comae or
tails on the fragments, all of which are unresolved at HST resolution (0.08″
FWHM).

Other than 331P itself, only 331P-A is bright enough to permit the
determination of a meaningful rotational lightcurve. We compile photometric
observations using a 0.2″ radius aperture from UT 2015 December 25 and 28 and
UT 2017 February 13 in Figure 9. The three visits are so widely spaced (and
therefore so seriously aliased) that we cannot accurately phase the three
epochs together. Instead, we have arbitrarily shifted the time axes to match
that of the observations from UT 2015 December 25. The lightcurve shows a
large and stable range of $\Delta V\sim$ 1.3 to 1.5 magnitudes with a single
maximum and two clear minima separated by about 4.5 hours. If the lightcurve
is due to rotation (i.e. is two-peaked), Figure 9 suggests a period near 9
hours. The large photometric range and the V-shaped brightness minimum
(Lacerda and Jewitt 2007) are consistent with the lightcurve of a contact
binary, as noted by Drahus and Waniak (2016). If this is the case, we
represent the body by two ellipsoidal components, each with semi-axes $a\times
a\times c$ and $c\geq a$, touching nose-to-nose. The maximum and minimum
projected cross-sections, for a rotation axis in the plane of the sky, would
then be 2$\pi ac$ and $\pi a^{2}$, respectively, giving axis ratio

$\frac{c}{a}=\frac{10^{0.4\Delta V}}{2}.$ (7)

Substituting $\Delta V$ = 1.5 magnitudes gives $c/a$ = 2.0. Assuming $p_{V}$ =
0.05 and $H_{m}$ = 22.25 (Table 2), we estimate $a$ = 65 m, $c$ = 130 m such
that the overall structure would resemble a peanut four times longer than any
perpendicular dimension, rotating in about 9 hours. The net gravitational
attraction to the center would exceed the centripetal acceleration provided
the density exceeds a critical value,

$\rho_{c}=\frac{12\pi}{GP^{2}}\left(\frac{c}{a}\right)^{2},$ (8)

where $G=6.67\times 10^{-11}$ N kg-2 m2 is the gravitational constant.
Substituting $c/a$ = 2, $P$ = 9 hours, we find $\rho_{c}$ = 2150 kg m-3. This
value is slightly bigger than the limiting value for 331P (1600 kg m-3;
Equation 4), and than the mean density of C-types ($\rho\sim 1500\pm 500$ kg
m-3) but comparable to the mean density of S-type asteroids ($\rho\sim 2500\pm
500$ kg m-3) according to Hanus et al. 2017). However, the partial lightcurve
(Figure 9) and the approximate nature of this density estimate are not
sufficient to prove the contact binary hypothesis. Most importantly, even very
small values of the material cohesion can negate simplistic density estimates
like those above, based on its assumed absence (e.g. Holsapple and Housen
2019, Hirabayashi and Scheeres 2019).

The contact binary interpretation cannot be regarded as unique. For example,
one possible alternative explanation for the 1.3 to 1.5 magnitude lightcurve
range is that 331P-A is a monolithic body with an axis ratio $c/a\sim$ 3 to 4.
Another possibility is that 331P-A is a sphere supporting a hemispheric
azimuthal albedo which varies by a factor of 3 to 4. We simply remark that
comparably extreme bodies are rare or unknown in the asteroid population. For
example, only 3 out of 4986 asteroid lightcurves in the compilation by Warner
et al. (2019) have $\Delta V\geq$ 1.4 magnitudes ($c/a\gtrsim$ 3.6:1) and
these might also be elongated contact binaries. Except for the pathological
case of Saturn’s satellite Iapetus (where synchronous rotation and an
external, anisotropic source of dust are responsible) 4:1 hemispheric albedo
variations are unknown in the solar system. Therefore, while alternative
explanations cannot be formally excluded, we regard them as less convincing
than the contact binary model. Such an object would presumably form by mutual
attraction of ejected fragments, as occurred in the gently disrupted
P/Shoemaker-Levy 9 (Movshovitz et al. 2012) and consistent with numerical
simulations showing complex and drawn-out interactions amongst the particles
ejected from rotationally unstable bodies (Walsh et al. 2012, Boldrin et al.
2016). Some numerical experiments suggest that asteroid contact binaries can
be produced in abundance by gravitational re-accumulation of fragments
following asteroid disruption (Campo Bagatin et al. 2020), as may be the case
for 331P-A.

The fragment size distribution is shown as a cumulative distribution in Figure
(10), with error bars in the latter figure assigned according to Poisson
statistics, and also schematically in Figure (11). Flattening of the
distribution at $r_{n}\lesssim$ 0.05 km reflects observational incompleteness
in the detection of small (faint) fragments. Lines of slope $q$ = 3, 4 and 5
are shown for reference. While noisy, the data are consistent with $q\sim 4$
to 4.5, slightly steeper than $q=3.7\pm 0.1$ to $q$ = 4.0 as inferred from
Monte-Carlo models of large particles in the trail (Moreno et al. 2012,
Section 2.3) and $q=4.1\pm 0.2$ found from the fading curve in Figure (3). If
the densities of the fragments are the same as the density of 331P, we can use
the radii in Table (2) to estimate the fractional mass in fragments as
$M/M_{n}\sim$ 0.01. This is 100$\times$ larger than the mass in diffuse
material (Section 2.3), consistent with a mass distribution dominated by the
largest bodies, even across the full cm to 100 m size range. We note that
Moreno et al. (2012) inferred a much larger $M/M_{n}\sim$ 0.2, in part because
they inferred a 331P nucleus radius of only $\sim$100 to 150 m, which our
nucleus photometry rules out.

In this regard, it is interesting to compare the fragments of 331P with the
carefully measured boulder size distributions of other asteroids, determined
from in-situ data. Boulders on the 0.5 km diameter, C-type asteroid 101955
Bennu follow a $q$ = 3.9$\pm$0.3 differential power law (Dellagiustina et al.
2018) similar to $q\sim$ 4 to 4.5 in 331P. Measurements of gravity data from
Bennu indicate that the surface boulders accurately reflect the size
distribution in the deep interior, and show that the largest “particle” inside
the body of Bennu is $\sim$ 150 m in size (Tricarico et al. 2021). This is
comparable to the $\sim$100 m equivalent spherical size of 331P-A. Given these
similarities, we speculate that the material ejected from 331P is just the
fragmentary matter from near the surface of 331P, destabilized and launched by
rotation.

The surface density of large boulders on Bennu is unpublished but on similar
object Ryugu, the density for boulders having radii $>$50 m is $\Sigma\sim$0.5
to 1 km-2 (Michikami and Hagermann 2021). Scaling to the surface area of 331P,
$A\sim 4\pi r_{n}^{2}\sim 8$ km2, we would expect $\Sigma A\sim$4 to 8 objects
of 50 m scale, when we nominally detect 15 objects with radii $>$50 m in 331P
(Table 2). The approximate agreement is consistent with the idea that most
331P fragments were pre-existing bodies (boulders) located on or near the
surface of 331P and liberated from it by a rotational instability.

##  3 DISCUSSION

###  3.1 Origin Scenario

We envision the following steps in the formation of 331P and its associated
fragments and debris trail.

Formation: 331P was liberated from its parent body (20674 (1999 VT1)) by
impact only $\tau_{C}\sim$1.5 Myr ago, as indicated by its membership in a
tightly clustered group of nine asteroids (Novaković et al. 2014). 331P
(radius $\sim$0.8 km) is the smallest of the known members of this group but
many smaller members presumably remain to be discovered (and, when they are,
should be examined for active asteroid status, since their spin-up timescales
will be shorter than that of 331P). The precursor body was the target of a
cratering collision, not a full blown disruption, and most of the parent body
mass remains in the 9 km radius cluster member 20674 (1999 VT1).

Spin-Up: Small size renders 331P susceptible to spin-up by radiation (YORP)
torques on timescales $\tau_{Y}\sim\tau_{C}$. Specifically, a fit to the
measured spin-up timescales of asteroids as a function of radius gives
$\tau_{Y}\sim 4r_{n}^{2}(r_{H}/3)^{2}$, where radius $r_{n}$ is expressed in
kilometers, $r_{H}$ is the semimajor axis in AU and $\tau_{Y}$ is in Myr, with
a wide dispersion due to shape and thermal differences amongst the asteroids
(Jewitt et al. 2015). Substituting $r_{n}$ = 0.8 km and $r_{H}$ = 3 AU for
331P gives $\tau_{Y}\sim$ 2 Myr, equal to $\tau_{C}$ within the considerable
uncertainties on both timescales. If interior ice was exposed by the
collisional ejection of 331P, then sublimation torques could also accelerate
the spin on a short timescale.

Rotational Instability: Rapid rotation (Figure 4, Drahus et al. 2015) and the
observed independence of the ejection velocity on particle size (Section 2.4,
also Moreno et al. 2012) are smoking guns for the role of rotational
instability. (Gas drag gives a velocity vs. size relation, $V\propto
s^{-1/2}$, which is incompatible with the data). Additionally, the observation
that the ejection velocities ($V\sim$ 5 to 10 cm s-1) are small compared to
the gravitational escape speed, $V_{e}\sim$ 80 cm s-1, is naturally explained
by rotational instability because, in this scenario, equatorial material is
effectively already travelling at the escape speed and only a very tiny
velocity increment is needed to induce escape.

The break-up of an aggregated body involves complex and incompletely
understood interactions between escaping fragments (Walsh et al. 2012, Davis
and Scheeres 2020, Hu et al. 2021). For example, material leaving radially at
5 cm s-1 takes $\sim$4.3 hours to travel one nucleus radius, during which time
the underlying source body has completed a full rotation, giving time for
torques from the aspherical primary to deflect the material. In numerical
simulations, interactions between escaping particles lead to energy
dissipation and can result in trapping in-orbit and even satellite formation
(Walsh et al. 2012, Hu et al. 2021). As a result of these complex
interactions, the estimated dynamical timescales in rotationally fissioned
bodies are model-dependent and, potentially, very long (e.g. days to years;
Boldrin et al. 2016). Some models show the formation of contact binaries in
the ejecta from low energy collisions (Campo Bagatin et al. 2020), an
intriguing result in view of the evidence that 331P-A is a contact binary. For
these reasons, it is presently difficult to relate the observations of 331P to
specific models which offer, at best, only statistical predictions of the
outcome.

The numbers of fragments projected east and west of 331P are not comparable,
showing that the ejection was not symmetric. We find 2 or 3 fragments to the
east of 331P and 16 or 17 to the west, rather than a more nearly 1:1 ratio
(Figure 2). This asymmetry suggests the approximately simultaneous release of
the fragments from a localized region of failure on the surface of 331P,
perhaps corresponding to one of the unstable tips of the rotating ellipsoid
representing the shape of the body. Alternatively, it is possible that a
single ejected body partially disintegrated into co-moving fragments during
the ejection. An analog might be the 160 m boulder “Otohime Saxum” on Ryugu,
whose fractured appearance (see Figure A5 in Yokota et al. 2021) suggests that
it would break up if it were to be disturbed. Still another possibility is
that the fragments initially exposed buried ice, leading to sublimation on
their sunward sides and a systematic non-gravitational acceleration away from
the Sun, as is commonly observed in the fragments of split comets (e.g.
Boehnhardt 2004). We regard this as not very likely because non-gravitational
acceleration varies as $1/r_{n}$, and the fragments of 331P are large compared
to those in typical split comets. Also, at 3 AU, the rate of sublimation of
water ice and the resulting recoil force are $\sim$16 times smaller than at 1
AU, where split comets are most commonly observed. Lastly, the HST images
provide no evidence for dust release from any fragment. However, we need
detailed dynamical studies to more firmly assess the possibility of non-
gravitational acceleration as the cause of the east vs. west asymmetry.

Dispersal: The overall dispersal time of the trail particles is set by
radiation pressure and, for the larger bodies that are insensitive to
radiation pressure, by the initial velocity. As noted above, the dispersal
timescales for particles (Equation 5) are such that sub-centimeter particles
have already been swept from the observed portion of the trail. By comparison,
a large fragment launched at $V=0.1$ m s-1, would take a nominal $2\pi
r_{H}/V\sim 10^{6}$ years to circulate around the orbit, comparable to the age
of the asteroid cluster of which 331P is a member. Eventually, the ejected
fragments will dissolve into the background population of outer belt
asteroids, but not any time soon.

Conceivably, the observed fragmentation is only the most recent of numerous
such events in which mass and angular momentum are dumped, with 331P held at
the edge of rotational instability by the continued action of radiation
torques. Given a fractional mass loss $M/M_{n}\sim$ 1%, 331P could sustain
another $\sim$100 similar events.

###  3.2 Related Objects

At least three other active asteroids, P/2013 R3, 311P and Gault, are likely
products of rotational instability (Figure 12). The basic properties of these
objects are compared with 331P in Table (3) and briefly described here. P/2013
R3 showed a fragmented appearance with a dozen or more components enveloped in
a debris trail which had a peak cross-section $\sim$30 km2 (Jewitt et al.
2014b, 2018). Unlike 331P, P/2013 R3 lacked a dominant nucleus and consisted
instead of a set of closely-spaced ($\sim 10^{4}$ km), debris-enshrouded
bodies each $\lesssim$100 m in size, with a velocity dispersion amongst
fragments, $\Delta V\sim$ 30 cm s-1. The fragments continued to evolve with
separation times staggered over $\sim$5 months. The parent body of P/2013 R3
was estimated to be $r_{n}\lesssim$ 0.4 km in radius, therefore containing
$\lesssim$15% of the mass of 331P (equal densities assumed). Morphological
differences from 331P include the lack of a dominant nucleus and the
progressive disintegration of each observed fragment into smaller bodies. The
relative dustiness of P/2013 R3 results from it being observed with HST sooner
after the initial break-up (two to three months; Figure 12 of Jewitt et al.
2017) than was 331P (4.5 years) giving less time for radiation pressure
sweeping to remove the sub-centimeter debris. The lack of a dominant nucleus
suggests a more severe rotational instability than in 331P, in which the
entire body of the initial nucleus was consumed (Jewitt et al. 2014b,
Hirabayashi et al. 2014).

Active asteroids 311P/PANSTARRS (2013 P5) (Jewitt et al. 2013, 2018) and 6478
Gault (1988 JC1) (Jewitt et al. 2019, Kleyna et al. 2019) are two other bodies
likely experiencing rotational instability. Both showed multiple tail
structures indicating episodic mass loss, in the case of 311P over a nine-
month period (Jewitt et al. 2018) and in the case of 6478 Gault spread over at
least six years (Chandler et al. 2019). Rotational instability in 6478 Gault
is implicated by its short rotational period, 2.55$\pm$0.10 hours (Luu et al.
2021) while the rotational lighcurve of 311P has not been measured (Hainaut et
al. 2014, Jewitt et al. 2017). As judged by the fractional mass loss estimates
(Table 3), 311P and Gault experienced less severe mass loss events than 331P
which, in turn, was less severe than P/2013 R3. Mass loss from 311P, 6478
Gault and 331P may be more akin to avalanching (“surface shedding”;
Hirabayashi et al. 2015, Sanchez and Scheeres 2018) than to solid body break-
up seen in P/2013 R3.

Available data thus expose the rich diversity of outcomes resulting from the
rotational instability of asteroids, revealing this to be an exciting subject
and a promising area of investigation for planetary astronomers. The long-
lived and slowly spreading fragment chain makes 331P a particularly attractive
target for future spacecraft investigation. Even decades from now, the modest
separations between components would allow a single camera-equipped spacecraft
to first examine the excited rotational state of the parent body, then
investigate the geology of the detachment scar, and then travel down the trail
to each of the fragments in succession, including the putative contact binary
331P-A, all the while sampling the boulder distribution in the trail. Such a
mission would provide unprecedented insight into the physics of asteroid
rotational instability and fragment dispersal.

##  4 SUMMARY

We study active asteroid 331P/Gibbs using archival Hubble Space Telescope
data, finding that:

  1. 1.

The main component of 331P has a radius $r_{n}$ = 0.8$\pm$0.1 km (albedo 0.05
assumed), a rotation period $P=3.26\pm 0.01$ hour and a lightcurve range 0.25
magnitudes, indicating an axis ratio $\geq$ 1.25:1. The minimum density for a
strengthless nucleus with this period and shape to be stable against
centripetal loss from the equator is $\rho_{n}$ = 1600 kg m-3. Rapid rotation
raises the prospect that mass loss from 331P was aided and/or triggered by a
rotational instability.

  2. 2.

The debris trail consists of centimeter sized and larger particles, having
ejection speeds in the few to 10 cm s-1 range, dispersed by radiation
pressure, and distributed according to a power-law relation with differential
index $q=3.7\pm 0.1$ to $q=4.1\pm 0.2$ (radii $\gtrsim$ 1 cm).

  3. 3.

At least 19 discrete fragments are embedded within the debris trail. The
largest, 331P-A, is a 100 m scale body having a 1.3 to 1.5 magnitude (factor
of four) lightcurve range, deep and V-shaped lightcurve minima and a period
near 9 hours. These properties are consistent with a contact binary structure.

  4. 4.

Discrete fragments ejected from 331P contain about 0.01 of the mass of the
primary, and this mass is dominated by the largest fragments in the
distribution. The small fractional mass loss indicates a weak “mass shedding”
instability of loose surface materials, not a deep structural instability in
which the primary is disrupted.

We thank Pedro Lacerda, Jane Luu and the anonymous referee for comments on the
manuscript. Based on observations made with the NASA/ESA Hubble Space
Telescope, obtained from the data archive at the Space Telescope Science
Institute. STScI is operated by the Association of Universities for Research
in Astronomy, Inc. under NASA contract NAS 5-26555. Support for this work was
provided by NASA through grant number AR-16618 from the Space Telescope
Science Institute, which is operated by AURA, Inc., under NASA contract NAS
5-26555. Facilities: HST.

## References

  * Boehnhardt (2004) Boehnhardt, H. 2004, Comets II, Comets II, M. C. Festou, H. U. Keller, and H. A. Weaver (eds.), University of Arizona Press, Tucson, 745 pp., p.301-316
  * Bohren, & Huffman, D. R. (1983) Bohren, C. F., & Huffman, D. R. 1983, Absorption and Scattering of Light by Small Particles (New York: Wiley)
  * Boldrin et al. (2016) Boldrin, L. A. G., Scheeres, D. J., & Winter, O. C. 2016, MNRAS, 461, 3982. doi:10.1093/mnras/stw1607
  * Burns & Safronov (1973) Burns, J. A. & Safronov, V. S. 1973, MNRAS, 165, 403. doi:10.1093/mnras/165.4.403
  * Campo Bagatin et al. (2020) Campo Bagatin, A., Alemañ, R. A., Benavidez, P. G., et al. 2020, Icarus, 339, 113603. doi:10.1016/j.icarus.2019.113603
  * Davis & Scheeres (2020) Davis, A. B. & Scheeres, D. J. 2020, Planetary Science Journal, 1, 25. doi:10.3847/PSJ/ab9a39
  * Dellagiustina et al. (2019) Dellagiustina, D. N., Emery, J. P., Golish, D. R., et al. 2019, Nature Astronomy, 3, 341. doi:10.1038/s41550-019-0731-1
  * Drahus et al. (2015) Drahus, M., Waniak, W., Tendulkar, S., et al. 2015, ApJ, 802, L8. doi:10.1088/2041-8205/802/1/L8
  * Drahus & Waniak (2016) Drahus, M. & Waniak, W. 2016, AAS DPS meeting #48, abstract id.522.03
  * Dressel (2012) Dressel, L. 2012, Wide Field Camera 3, HST Instrument Handbook, https://hst-docs.stsci.edu/wfc3ihb
  * Gibbs et al. (2012) Gibbs, A. R., Sato, H., Ryan, W. H., et al. 2012, Central Bureau Electronic Telegrams, 3069
  * Hanuš et al. (2017) Hanuš, J., Viikinkoski, M., Marchis, F., et al. 2017, A&A, 601, A114. doi:10.1051/0004-6361/201629956
  * Hainaut et al. (2014) Hainaut, O. R., Boehnhardt, H., Snodgrass, C., et al. 2014, A&A, 563, A75. doi:10.1051/0004-6361/201322864
  * Hartmann (1969) Hartmann, W. K. 1969, Icarus, 10, 201. doi:10.1016/0019-1035(69)90022-0
  * Hirabayashi et al. (2014) Hirabayashi, M., Scheeres, D. J., Sánchez, D. P., et al. 2014, ApJ, 789, L12. doi:10.1088/2041-8205/789/1/L12
  * Hirabayashi et al. (2015) Hirabayashi, M., Sánchez, D. P., & Scheeres, D. J. 2015, ApJ, 808, 63. doi:10.1088/0004-637X/808/1/63
  * Hirabayashi & Scheeres (2019) Hirabayashi, M. & Scheeres, D. J. 2019, Icarus, 317, 354. doi:10.1016/j.icarus.2018.08.003
  * Holmberg et al. (2006) Holmberg, J., Flynn, C., & Portinari, L. 2006, MNRAS, 367, 449. doi:10.1111/j.1365-2966.2005.09832.x
  * Holsapple & Housen (2019) Holsapple, K. A. & Housen, K. R. 2019, Planet. Space Sci., 179, 104724. doi:10.1016/j.pss.2019.104724
  * Hsieh & Jewitt (2006) Hsieh, H. H. & Jewitt, D. 2006, Science, 312, 561. doi:10.1126/science.1125150
  * Hsieh & Haghighipour (2016) Hsieh, H. H. & Haghighipour, N. 2016, Icarus, 277, 19. doi:10.1016/j.icarus.2016.04.043
  * Hu et al. (2021) Hu, S., Richardson, D. C., Zhang, Y., et al. 2021, MNRAS, 502, 5277. doi:10.1093/mnras/stab412
  * Ishiguro et al. (2007) Ishiguro, M., Sarugaku, Y., Ueno, M., et al. 2007, Icarus, 189, 169. doi:10.1016/j.icarus.2007.01.003
  * Ishiguro et al. (2009) Ishiguro, M., Sarugaku, Y., Nishihara, S., et al. 2009, Advances in Space Research, 43, 875. doi:10.1016/j.asr.2008.07.010
  * Jewitt et al. (2010) Jewitt, D., Weaver, H., Agarwal, J., et al. 2010, Nature, 467, 817. doi:10.1038/nature09456
  * Jewitt (2012) Jewitt, D. 2012, AJ, 143, 66. doi:10.1088/0004-6256/143/3/66
  * Jewitt et al. (2013) Jewitt, D., Agarwal, J., Weaver, H., et al. 2013, ApJ, 778, L21. doi:10.1088/2041-8205/778/1/L21
  * Jewitt et al. (2014) Jewitt, D., Ishiguro, M., Weaver, H., et al. 2014a, AJ, 147, 117. doi:10.1088/0004-6256/147/5/117
  * Jewitt et al. (2014) Jewitt, D., Agarwal, J., Li, J., et al. 2014b, ApJ, 784, L8. doi:10.1088/2041-8205/784/1/L8
  * Jewitt et al. (2015) Jewitt, D., Hsieh, H., & Agarwal, J. 2015, In Asteroids IV, Patrick Michel, Francesca E. DeMeo, and William F. Bottke (eds.), University of Arizona Press, Tucson, pp. 221.
  * Jewitt et al. (2017) Jewitt, D., Agarwal, J., Li, J., et al. 2017, AJ, 153, 223. doi:10.3847/1538-3881/aa6a57
  * Jewitt et al. (2019) Jewitt, D., Kim, Y., Luu, J., et al. 2019, ApJ, 876, L19. doi:10.3847/2041-8213/ab1be8
  * Kleyna et al. (2019) Kleyna, J. T., Hainaut, O. R., Meech, K. J., et al. 2019, ApJ, 874, L20. doi:10.3847/2041-8213/ab0f40
  * Lacerda & Jewitt (2007) Lacerda, P. & Jewitt, D. C. 2007, AJ, 133, 1393. doi:10.1086/511772
  * Luu et al. (2021) Luu, J. X., Jewitt, D. C., Mutchler, M., et al. 2021, ApJ, 910, L27. doi:10.3847/2041-8213/abedbc
  * Martikainen et al. (2021) Martikainen, J., Muinonen, K., Penttilä, A., et al. 2021, A&A, 649, A98. doi:10.1051/0004-6361/202039796
  * Michikami & Hagermann (2021) Michikami, T. & Hagermann, A. 2021, Icarus, 357, 114282. doi:10.1016/j.icarus.2020.114282
  * Moreno et al. (2012) Moreno, F., Licandro, J., & Cabrera-Lavers, A. 2012, ApJ, 761, L12. doi:10.1088/2041-8205/761/1/L12
  * Movshovitz et al. (2012) Movshovitz, N., Asphaug, E., & Korycansky, D. 2012, ApJ, 759, 93. doi:10.1088/0004-637X/759/2/93
  * Novaković et al. (2014) Novaković, B., Hsieh, H. H., Cellino, A., et al. 2014, Icarus, 231, 300. doi:10.1016/j.icarus.2013.12.019
  * Sánchez & Scheeres (2018) Sánchez, P. & Scheeres, D. J. 2018, Planet. Space Sci., 157, 39. doi:10.1016/j.pss.2018.04.001
  * Sanchez et al. (2019) Sanchez, J. A., Reddy, V., Thirouin, A., et al. 2019, ApJ, 881, L6. doi:10.3847/2041-8213/ab31ac
  * Stevenson et al. (2012) Stevenson, R., Kramer, E. A., Bauer, J. M., et al. 2012, ApJ, 759, 142. doi:10.1088/0004-637X/759/2/142
  * Tricarico et al. (2021) Tricarico, P., Scheeres, D. J., French, A. S., et al. 2021, Icarus, 370, 114665. doi:10.1016/j.icarus.2021.114665
  * Vaghi (1973) Vaghi, S. 1973, A&A, 29, 85
  * Walsh et al. (2012) Walsh, K. J., Richardson, D. C., & Michel, P. 2012, Icarus, 220, 514. doi:10.1016/j.icarus.2012.04.029
  * Warner et al. (2019) Warner, B.D, Harris, A.W., and Pravec, P. (2019). Asteroid Lightcurve Database (LCDB) Bundle V3.0, urn:nasa:pds:ast-lightcurve-database::3.0. NASA Planetary Data System, https://doi.org/10.26033/7h40-8s80.
  * Whipple (1950) Whipple, F. L. 1950, ApJ, 111, 375. doi:10.1086/145272
  * Yokota (2021) Yokota, Y., Honda, R., Tatsumi, E., et al. 2021, PSJ, 2, 177

Table 1: Observing Geometry  UT Date & Time | NaaNumber of images obtained on each date | ExpbbRepresentative integration time per image  |  $\nu$ccTrue anomaly, in degrees |  $r_{H}$ddHeliocentric distance, in AU  |  $\Delta$eeGeocentric distance, in AU  |  $\alpha$ffPhase angle, in degrees  |  $\theta_{-\odot}$ggPosition angle of projected anti-solar direction, in degrees  |  $\theta_{-V}$hhPosition angle of negative heliocentric velocity vector, in degrees |  $\delta_{\oplus}$iiAngle from orbital plane, in degrees
---|---|---|---|---|---|---|---|---|---
2015 Dec 25 09:33-14:58 | 24 | 368 | 39.3 | 2.906 | 2.011 | 9.7 | 64.7 | 269.0 | 3.9
2015 Dec 28 01:09-03:23 | 12 | 368 | 39.8 | 2.907 | 2.030 | 10.5 | 66.2 | 269.0 | 4.0
2016 Feb 13 03:47-09:16 | 24 | 368 | 49.3 | 2.920 | 2.573 | 19.4 | 78.3 | 269.6 | 3.6
2017 Feb 13 02:19-09:16 | 24 | 438 | 118.8 | 3.059 | 2.111 | 6.2 | 331.0 | 286.2 | 4.3
2017 Mar 08 16:13-23:00 | 25 | 438 | 123.0 | 3.068 | 2.109 | 5.8 | 70.4 | 286.7 | 3.4
2018 May 17 13:59-16:10 | 10 | 438 | 197.4 | 3.122 | 2.113 | 1.6 | 169.0 | 274.5 | -1.5
2018 May 26 09:19-16:17 | 20 | 438 | 198.9 | 3.121 | 2.126 | 4.3 | 125.4 | 274.9 | -2.2
2018 Jul 03 09:33-16:31 | 20 | 453 | 205.3 | 3.115 | 2.402 | 15.2 | 110.0 | 275.2 | -3.7

Table 2: Fragment PhotometryaaMean value of the absolute magnitude within a 5 pixel (0.2″) radius aperture, with the standard deviation on the mean Object |  $H_{5}$aaMean value of the absolute magnitude within a 5 pixel (0.2″) radius aperture, with the standard deviation on the mean |  $H_{m}$bbMedian value of the absolute magnitude obtained in the same way |  $N$ccNumber of images used |  $r_{n}$ddEffective radius, in km, computed from $H_{m}$, Equation (2) and $r_{n}=(C/\pi)^{1/2}$ [km]
---|---|---|---|---
Primary | 17.92 $\pm$ 0.03 | 17.96 | 7 | 0.771 $\pm$ 0.011
A | 22.31 $\pm$ 0.04 | 22.25 | 7 | 0.110 $\pm$ 0.002
B | 23.35 $\pm$ 0.08 | 23.31 | 7 | 0.067 $\pm$ 0.002
C | 23.13 $\pm$ 0.06 | 23.13 | 7 | 0.073 $\pm$ 0.002
D | 23.87 $\pm$ 0.17 | 23.95 | 3 | 0.050 $\pm$ 0.004
E | 23.18 $\pm$ 0.12 | 23.18 | 2 | 0.071 $\pm$ 0.004
F | 24.12 $\pm$ 0.07 | 24.12 | 2 | 0.046 $\pm$ 0.001
G | 23.75 $\pm$ 0.13 | 23.75 | 2 | 0.055 $\pm$ 0.003
H | 24.03 $\pm$ 0.32 | 24.03 | 2 | 0.048 $\pm$ 0.007
I | 23.76 $\pm$ 0.09 | 23.77 | 4 | 0.054 $\pm$ 0.002
J | 23.54 $\pm$ 0.10 | 23.55 | 6 | 0.060 $\pm$ 0.003
K | 23.77 $\pm$ 0.15 | 23.77 | 2 | 0.054 $\pm$ 0.004
L | 23.52 $\pm$ 0.10 | 23.52 | 2 | 0.061 $\pm$ 0.003
M | 23.58 $\pm$ 0.10 | 23.61 | 5 | 0.059 $\pm$ 0.003
N | 23.44 $\pm$ 0.11 | 23.45 | 4 | 0.063 $\pm$ 0.003
O | 23.85 $\pm$ 0.13 | 23.76 | 5 | 0.055 $\pm$ 0.003
P | 23.66 $\pm$ 0.12 | 23.66 | 2 | 0.057 $\pm$ 0.003
Q | 23.67 $\pm$ 0.10 | 23.62 | 3 | 0.058 $\pm$ 0.003
R | 24.43 $\pm$ 0.31 | 24.43 | 1 | 0.040 $\pm$ 0.006
S | 23.60 $\pm$ 0.20 | 23.70 | 1 | 0.057 $\pm$ 0.006

Table 3: Breakups Compared  Object |  $a$aaSemimajor axis |  $e$bbOrbital eccentricity |  $i$ccOrbital inclination, degree |  $r_{n}$ddEffective nucleus radius, km |  $P$eeNucleus rotation period, hour |  $\Delta M/M$ffFractional mass ejected |  $N$ggNumber of fragments detected | ReferencehhJ13 = Jewitt et al. 2013, J14 = Jewitt et al. 2014b, J17 = Jewitt et al. 2017, Jewitt et al. 2018, J19 = Jewitt et al. 2019, H14 = Hainaut et al. 2014, L21 = Luu et al. 2021, S19 = Sanchez et al. 2019
---|---|---|---|---|---|---|---|---
P/2013 R3 | 3.033 | 0.273 | 0.9 | 0.4? | ? |  $\sim$1 | 12 | J14, J17
331P | 3.005 | 0.042 | 9.7 | 0.8$\pm$0.1 | 3.26$\pm$0.01 | 3$\times 10^{-3}$ | 19 | This Work
311P | 2.189 | 0.116 | 5.0 | 0.19$\pm$0.03 |  $\geq$5.4 |  $10^{-4}$ to $10^{-3}$ | 0 | J13, H14, J18
6478 Gault | 2.306 | 0.193 | 22.8 | 2.0$\pm$0.4 | 2.55$\pm$0.10 | $\gtrsim 3\times 10^{-6}$ | 0 | J19, S19, L21

Figure 1: Composite full-frame (4k$\times$4k pixel) image from UT 2015
December 28 to show the impact of numerous imperfectly removed field stars and
galaxies, diffuse scattered light in the camera (S), various CCD defects (D)
and the chip gap. The cardinal directions are marked in the upper left, as are
the projected anti-solar direction (-Sun) and the negative heliocentric
velocity vector (-V). A 10″ scale bar is shown.

Figure 2: Composite of images scaled to $\Delta$ = 2.011 AU such that the
width of each panel corresponds to 92,000 km at the distance of 331P.
Fragments 331P-A, -B and -C are labeled in red in each panel. Other fragments
are identified only where they first appear. Diagonal streaks, especially
noticable in the UT 2018 May 17 panel, result from incompletely removed
trailed background objects. There is no correction for the changing phase
angle.

Figure 3: Absolute magnitude vs. date of observation for primary nucleus 331P
(yellow circles). Letters indicate measurements from the literature (S12 =
Stevenson et al. (2012), N14 = Novacovic et al. (2014), D15 = Drahus et al.
(2015)) color-corrected to the V filter. The downward pointing red arrow marks
2011 July 1, the nominal date of initiation of the activity in 331P, while the
dashed horizontal line shows the absolute magnitude of the primary nucleus in
the absence of dust. The black line is the best fit of the fading Equation
(3).

Figure 4: Primary lightcurve from HST data taken UT 2015 December 25 and 28
(yellow circles) compared with data from Drahus et al. (2015) taken in 2014.
The Drahus et al. data have been shifted in phase and absolute magnitude to
obtain the best match. The slight differences between the lightcurves reflect
a change in the observing geometry between 2014 and 2015 as well as greater
dilution of the lightcurve by near-nucleus dust in the 2014 data (see Figure
3).

Figure 5: Position angle of the trail (yellow circles) relative to the nucleus
as a function of the true anomaly. Statistical error bars on the measurements
are smaller than the plot symbols. The solid black line shows the position
angle of the projected negative velocity vector.

Figure 6: Full width at half maximum, $W_{\perp}$, measured perpendicular to
the trail axis as a function of distance from the primary, $\ell_{T}$.
Representative error bars equal to 15% of the FWHM are shown. A weighted
least-squares fit to a $W_{\perp}\propto\ell_{T}^{1/2}$ power law is shown as
a red line.

Figure 7: Measured V-band surface brightness along the trail axis (black
line). The primary nucleus is off-scale, with a core surface brightness 0.094
units. Lines show models for the brightness produced by radiation pressure
sweeping of (red line) a $q$ = 3.7 power law, (blue line) a broken power law
with $q$ = 3.0 and 4.0 and a break at $\beta_{crit}=10^{-5}$ and (green line)
a broken power law with $q$ = 2.5 and 4.0 and the same break $\beta_{crit}$.
Interfering stars and galaxies are marked “$*$”.

Figure 8: HST photometry of 331P and fragments 331P-A, -B and -C as a function
of the date of observation. The fragment magnitudes have been vertically
offset by $\Delta V$ for clarity, as indicated. There is no evidence for a
secular decline in the brightness of any object.

Figure 9: Lightcurve of 331P-A in data from UT 2015 December 25 and 28 and UT
2017 February 13. The times have been shifted arbitrarily to obtain the best
fit.

Figure 10: Cumulative distribution of fragment radii. Lines indicate
differential power-law indices 3, 4 and 5, as labeled.

Figure 11: Schematic diagram to show the relative sizes of 331P and its
fragments.

Figure 12: Four asteroids inferred to have been activated by rotation
instabilities, as observed by HST. 331P is from this work, P/2013 R3 from
Jewitt et al. (2014b), 311P from Jewitt et al. (2013) and Gault from Kleyna et
al. (2019). Each panel has North to the top, East to the left and is shown
with a 5000 km scale bar and arrows indicating the projected anti-solar vector
($-S$) and the negative heliocentric velocity vector ($-V$).