11institutetext:  European Southern Observatory, Karl-Schwarzschild-Strasse 2,
D-85748 Garching bei München, Germany 22institutetext: Institute for
Astronomy, 2680 Woodlawn Drive, Honolulu, HI 96822, USA 33institutetext:
University of NM - 1700 Lomas Blvd, NE. Suite 2200, Albuquerque, NM 87131-0001
USA 44institutetext: ESA SSA-NEO Coordination Centre, Largo Galileo Galilei, 1
00044 Frascati (RM), Italy 55institutetext: INAF - Osservatorio Astronomico di
Roma, Via Frascati, 33, 00040 Monte Porzio Catone (RM), Italy 66institutetext:
Indian Inst. Astrophys., II Block, Koramangala, Bangalor 560 034, India

# Disintegration of Active Asteroid P/2016 G1 (PANSTARRS)

Olivier R. Hainaut Disintegration of Active Asteroid P/2016 G1
(PANSTARRS)Disintegration of Active Asteroid P/2016 G1 (PANSTARRS) Jan T.
Kleyna Disintegration of Active Asteroid P/2016 G1 (PANSTARRS)Disintegration
of Active Asteroid P/2016 G1 (PANSTARRS) Karen J. Meech Disintegration of
Active Asteroid P/2016 G1 (PANSTARRS)Disintegration of Active Asteroid P/2016
G1 (PANSTARRS) Mark Boslough Disintegration of Active Asteroid P/2016 G1
(PANSTARRS)Disintegration of Active Asteroid P/2016 G1 (PANSTARRS) Marco
Micheli Disintegration of Active Asteroid P/2016 G1 (PANSTARRS)Disintegration
of Active Asteroid P/2016 G1 (PANSTARRS)Disintegration of Active Asteroid
P/2016 G1 (PANSTARRS)Disintegration of Active Asteroid P/2016 G1 (PANSTARRS)
Richard Wainscoat Disintegration of Active Asteroid P/2016 G1
(PANSTARRS)Disintegration of Active Asteroid P/2016 G1 (PANSTARRS) Marielle
Dela Cruz Disintegration of Active Asteroid P/2016 G1
(PANSTARRS)Disintegration of Active Asteroid P/2016 G1 (PANSTARRS) Jacqueline
V. Keane Disintegration of Active Asteroid P/2016 G1 (PANSTARRS)Disintegration
of Active Asteroid P/2016 G1 (PANSTARRS) Devendra K. Sahu Disintegration of
Active Asteroid P/2016 G1 (PANSTARRS)Disintegration of Active Asteroid P/2016
G1 (PANSTARRS) Bhuwan C. Bhatt Disintegration of Active Asteroid P/2016 G1
(PANSTARRS)Disintegration of Active Asteroid P/2016 G1 (PANSTARRS)

(Received 10/05/2019 /Accepted 01/07/2019)

We report on the catastrophic disintegration of P/2016 G1 (PANSTARRS), an
active asteroid, in April 2016. Deep images over three months show that object
is constituted by a central concentration of fragments surrounded by an
elongated coma, and presents previously unreported sharp arc-like and a narrow
linear features. The morphology and evolution of these characteristics
independently point toward a brief event on 2016 March 6. The arc and the
linear feature can be reproduced by large particles on a ring, moving at $\sim
2.5$ m s-1. The expansion of the ring defines a cone with a $\sim 40^{\circ}$
half-opening. We propose that the P/2016 G1 was hit by a small object which
caused its (partial or total) disruption, and that the ring corresponds to
large fragments ejected during the final stages of the crater formation.

###### Key Words.:

comets: general — comets: individual (C/2016 G1),

##  1 Introduction

P/2016 G1 was discovered by the Pan-STARRS1 (PS1) telescope on 2016 Apr. 1,
and the orbit ($a$ = 2.582 au, $e$ = 0.210, $i$ = 10.969∘, $T_{\rm Jup}$ =
3.367), suggested that this was possibly a main belt comet (Weryk et al.
2016). The object was approaching a perihelion of $q$ = 2.041 au on 2017 Jan.
26.24. Because the surrounding dust coma had a peculiar appearance in the PS1
images, we undertook an immediate campaign to follow this up with images from
the 3.6m Canada-France-Hawaii Telescope (CFHT). The first CFHT images,
obtained on 2016 Apr. 3 showed a structure to the south, 90 degrees from the
dust tail. Because of the unusual appearance, we dedicated time from our CFHT
PS1 discovery follow up program to image P/2016 G1 frequently during the dark
runs when the wide field imager was on the telescope. By mid-April, its
diffuse and core-less appearance made it clear that the object was undergoing
some sort of catastrophic disruption.

Hsieh et al. (2018) reported that P/2016 G1 is linked to the Adeona family,
which likely originated in a cratering event $\sim 700$ Myr ago (Benavidez et
al. 2012; Carruba et al. 2016; Milani et al. 2017). The Adeona family members
are typically C- and Ch-type, while the surrounding background objects are
predominantly of the S-type (Hsieh et al. 2018).

##  2 Observations & Data Reduction

###  2.1 Pan-STARRS1

With a well-known orbit and a brightness estimated to be above the PS1
limiting magnitude over most of its orbit, we conducted a search through the
PS1 database for pre-discovery images. On 2016 Mar. 7 the object is clearly
visible, apparently mostly stellar (see below for a more detailed assessment).
Photometry was obtained using PS1’s intrinsic calibration. A set of four
$w$-band (central wavelength $\lambda=6250$ Å, width 4416 Å) images on 2016
Jan. 10 shows nothing at all, with a limiting magnitude in the stack of
$r\sim$23\. This corresponds to a nucleus radius of 0.81 km for an albedo of
4% or 0.32 km for an albedo of 25%.

The position of P/2016 G1 was imaged 12 additional times in the PS1 data prior
to discovery, between 2011 Feb. 23 and 2015 Jan. 17 and nothing was visible
down to the limiting magnitudes shown in Table 1. The most constraining
observations (2012 Jun.) suggest that the object must have a radius
$R_{N}<0.4$ km for a 4% albedo or $R_{N}<0.2$ km for 25% albedo. For these
non-detections, a positional uncertainty region of 3-$\sigma$ was inspected.
These limits are thus valid under the assumption that the existing astrometric
arc is good and that a gravity-only solution models the orbit well enough over
5 years.

Figure 1:  Distribution of observations (see Table 1) of P/2016 G1 along its
orbit. The symbols indicate the apparitions, from aphelion (Q) to aphelion.
Perihelion is marked (q). Filled symbols show when the comet was detected,
open symbols where it was not detected. No observations were acquired during
the 2002 Jul. – 2006 Sep. apparition. The positions of the planet symbols
correspond to 2016 Jul.

Figure 2: [A] Images of P/2016 G1 between UT 2016 March 7 and July 4. All the
images are scaled to be 6$\times$104 km on a side. The Earth crossed the
object’s orbital plane on Apr. 13, so images around that date show the coma
extension above and below the plane. [B] Images of P/2016 G1 that have been
processed with unsharp masking to enhance the three clumps and the details
near the disintegrating body. Each image is 1.5$\times$104 km on a side. [C]
Contour plots on three dates showing the expansion of the southern arc and the
eastern linear feature (highlighted with red lines). Each panel is
$30^{\prime\prime}\times 20^{\prime\prime}$. The panel widths and pixel scales
are: April 3 (34,000 km, 227 km), April 14 (32,000 km, 231 km) and May 12
(29,000 km, 193 km). North is up, East is Left. The Sun and velocity vectors
are listed in Table 1.

###  2.2 Canada-France-Hawaii Telescope

Imaging data were obtained using the MegaCam wide-field imager which is an
array of 40 2048$\times$4612 pixel CCDs, with a plate scale of
0$\aas@@fstack{\prime\prime}$187 pixel-1. The epochs and circumstances of the
CFHT data are listed in Table 1. MegaCam is on the telescope for a period
centred on each new moon, and data are obtained through queue service
observing and are processed to remove the instrumental signature through the
Elixir pipeline111http://www.cfht.hawaii.edu/Instruments/Elixir.

The photometric calibration of the processed data accesses the Pan-STARRS
database or SDSS to provide a photometric zero point to each frame, using
published colour corrections (Tonry et al. 2012) to translate PS1 $g,r,i,z$
bands into SDSS or Johnson-Cousins systems. The object headers are used to
identify the target and download orbital elements from the Minor Planet
Center; the resulting object location is used to determine which object in the
frame corresponds to the target. In the final pass, Terapix tools (SExtractor,
Bertin & Arnouts 1996) are run to produce multi-aperture and automatic
aperture target photometry.

A first series of CFHT observations was obtained as an immediate follow-up to
the discovery, showing the dust features that constitute the core of the
following discussion. A second was acquired with the broad $gri$ filter on
2018 Dec. 12 and 31, over 1000 days after the first series. The object has
completely dispersed, showing no visible remain down to $gri=26$ ($5\sigma$
point-source, using the MegaCam Direct Imaging Exposure Time
Calculator222http://cfht.hawaii.edu/Instruments/Imaging/
MegaPrime/dietmegacam.html.

###  2.3 Himalayan Chandra Telescope (HCT)

Observations of P/2016 G1 were made on 2016 May 31 using the Himalayan Chandra
2.0-m telescope with the Himalaya Faint Object Spectrograph and Camera
(HFOSC). The instrument uses a SITe ST-002 CCD with a plate scale of
0$\aas@@fstack{\prime\prime}$296 per pixel. Data were obtained through Bessell
$R$-band filters under clear conditions with seeing of
$\sim$1$\aas@@fstack{\prime\prime}$6.

We processed the data using our image reduction pipeline which bias-subtracts
and flattens the data, and then applies the Terapix tools to fit a precise WCS
to the frame. Following the removal of instrumental effects, the photometric
calibration proceeded in the same manner as for the CFHT data.

###  2.4 Observatory Archives and other publications

Additional pre-discovery images were found from the CFHT (2007 Feb. 16) and
the INT on La Palma (2000 Oct. 20), both with non-detections, both less
constraining than the 2012 Jun. images. For these older images, the positional
uncertainties were $\pm$45′′ and $\pm$3′, respectively.

Finally, the Canadian Astronomy Data Centre’s Solar System Object Image Search
(Gwyn et al. 2012) was used, and about 40 short exposures with the Cerro
Tololo Interamerican Observatory 4-meter telescope were identified. Based on
the 2016 Jan. 10 upper limit, and accounting for the geometry of the orbit,
these CTIO images would fail to reach the same limit by 1–2 mag.

The positions on the orbit for the dates for which we have observations
(including both detections and non-detections) are shown in Fig. 1. This shows
that we have tightly bracketed the period of apparent activity for this
current apparition, strongly suggesting that whatever caused the activity
began between 2016 Jan. 10 and 2016 Apr. 1 when P/2016 G1 was discovered, and
that the object did not present significant activity during previous
apparitions.

Moreno et al. (2016) have acquired images with the 10.4 m Gran Telescopio
Canarias on 2016 Apr. 21, May 29 and Jun. 08. The general appearance of their
images fits well with our CFHT images discussed below. They followed up with
two observations obtained with the Hubble Space Telescope on 2016 Jun. 28 and
Jul. 11 (Moreno et al. 2017), which focus on the central region of the
disrupted object. From these deep, high-resolution images, they estimate that
no fragment larger than $\sim 30$ m survived. They modelled the observations
using their Monte Carlo dust-tail code; their results are discussed in the
comparison with ours in Section 3.6.

Figure 3: Average surface brightness profile of P/2016 G1 compared to the
average profile for field stars as seen in the data obtained with the Pan-
STARRS1 telescope on 2016 March 7. The profiles for the field stars have been
normalised to the peak brightness for P/2016 G1. Although the images of the
comet appear stellar in the images, its profiles indicate that the object is
slightly –but significantly– extended.

##  3 Analysis & Results

###  3.1 Surface Brightness Profiles

The pre-discovery data on P/2016 G1 with the PS1 telescope on 2016 Mar. 7
consist of four images, showing the object with a stellar morphology. Only two
of the frames had the object clear of the chip gap boundary. The surface
brightness profile was computed in each of these frames for P/2016 G1 and for
three nearby field stars in order to assess whether there was any dust
surrounding the nucleus. We compute the radial flux profile using the median
per-pixel flux in circular annuli around the object, and convert the flux to
surface brightness using the zeropoint in the PS1 headers. Errors are obtained
using bootstrap re-sampling within each annulus. Figure 3 presents the average
normalised surface brightness profiles of both P/2016 G1 and the field stars,
and shows that P/2016 G1 is more extended. During each exposure, the object
moved about 0$\aas@@fstack{\prime\prime}$3, which is significantly less than
the seeing, so that trailing does not cause the flux excess seen in P/2016 G1.
We conclude that the object was already surrounded by dust on 2016 Mar. 7.

###  3.2 Description and nomenclature

Figure 2 shows a series of images of the object over time. The central
structure of the object is composed of 3 main concentrations of dust forming
an inverted “C” on the images (hereafter referred to as the three clumps), and
a much fainter concentration on the eastern side, leaving a central gap. This
structure is growing with time, preserving its appearance. The central
structure is bathed in a diffuse, elongated coma, that extends into a dust
tail westward from the central structure. The orientation of the coma and the
tail is rotating southward as the viewing geometry evolves.

Below the central structure, a small, narrow, well-defined arc extends toward
the South-South-East. It is clearly seen from Apr. 3 until May 12 and is still
detected on May 31. Directly eastward of the central structure, but not
pointing toward its centre, a faint linear feature is present. It can also be
seen until May 12. The arc and the linear features are marked on Fig. 2C.

###  3.3 Main Head Dust Feature

Figure 4: Size of the main head structure as a function of time. The line and
the horizontal error bar indicate the linear expansion fitted to the data and
the error on the origin, 2016 Mar. 6 $\pm$ 3 days.

The Earth crossed the comet’s orbital plane on 2016 Apr. 13. The images taken
around that date indicate that the complex head structure and the central coma
extends above and below the orbital plane, whose orientation corresponds on
the images to that of the tail feature (PA$=274^{\circ}$). The grains forming
the head structure must, therefore, have been ejected with a non-zero
velocity, with a non-zero component perpendicular to the orbital plane.

The size of the central structure was estimated measuring the distance between
the peak of the North and South concentrations. When the peak is not well
defined, the centre of the concentration was used instead. These angular
distances were converted to a linear distance in the plane of sky accounting
for the geocentric distance (and reported in Fig. 4). In the Mar. 7 image,
although the object is broader than stellar,the central structure is not
resolved yet. The diameter of the seeing disk is therefore used as an upper
limit. The size of the structure grew linearly with time. A linear regression
indicates that the growth would have started on 2016 Mar. 6 (JD=2457454) $\pm$
3 days, and is expanding with an average (plane of sky) velocity of
0.32$\pm$0.02 m s-1 (i.e. drifting from the centre of the original object at
half that speed). The Mar. 7 image would therefore have been obtained very
soon after the onset of the activity, the observed broadening of the image
could be caused by the cloud of fine dust expanding more rapidly than the
larger fragments.

###  3.4 Finson-Probstein Dust Modelling

The positions of the Earth, Comet and Sun in late May and in June show the
dust features of the tail with a geometry very favourable for a Finson-
Probstein (FP Finson & Probstein 1968; Farnham 1996) dust dynamical analysis.
FP modelling calculates the trajectories of an ensemble of dust grains of
different sizes, parameterised by $\beta$ (ratio between the radiation
pressure and the solar gravity) ejected from the surface of the nucleus at
different times, $\tau$, as acted upon by solar gravity and solar radiation
pressure. We used the FP approach to analyse the pattern of synchrones (loci
of the particles emitted at the same time) and syndynes (curves joining
particles with the same $\beta$) and the optical appearance of the comet’s
dust environment.

Figure 5: Syndynes (in green) and synchrones (in purple, labelled in days
before the observation date) for P/2016 G1 on 2016 May 31 and 2016 Jun. 8. The
thicker synchrone marks the brightest peak in the dust profile, and the
corresponding grain radii are marked in red. This fits with the disruption
taking place around March 6.

The main tail-like feature has a fairly sharp profile in azimuth as shown in
Fig. 5. The position angle of the peak was measured on the images; it is
marked as a thick line on the FP plots. The epoch of the corresponding
synchrone was obtained by comparing the position angles of synchrones
generated with a step of 1 day. The error on that epoch resulting from the
measurement uncertainty is $\pm$3 days. The azimuthal profile of the tail is
very roughly gaussian; converting the FWHM (measured using the same procedure
as above), this also results in a “$\sigma$” of $\pm$3 days. The broadening of
the tail is caused in part by the seeing (negligible far from the nucleus), by
the dispersion in emission velocity (which is neglected in the zero-velocity
FP method) and by the duration of the dust emission. The broadening caused by
the emission velocity can be estimated from the images obtained when the Earth
was close to the orbital plane (when the synchrones and syndynes degenerate
into a single line), around Apr. 13 (see Fig. 2): the width of the tail on
Apr. 14 is similar to that seen on Apr. 4–8, when the orbital plane was
observed at an angle. We conclude that the broadening of the tail is dominated
by the velocity dispersion of the grains (which occurs in all directions,
including perpendicular to the orbital plane), rather than purely by the
radiation pressure (which occurs only within the orbital plane).

The tail-like feature is therefore compatible with a burst of dust emission
centred on 2016 Mar. 4$\pm$3 days for the image from 2016 May 31, and 2016
Mar. 7$\pm$3 days for the image from 2016 Jun. 8. The emission profile is
compatible with a short, impulsive burst smeared out by a distribution of
initial velocities of the particles with $v<1$ m s-1 in random directions, or
a longer burst (increasing then decreasing over a few days) with no initial
velocity. Furthermore, dust is present in the areas of the image not covered
by the syndynes and synchrones, confirming that the dust must have been
emitted with some initial velocity. We therefore favour the short, impulsive
burst.

It is interesting that two independent measurements for the date of the onset
of disintegration: the FP estimates and the central feature growth are in
perfect agreement.

Assuming that the tail-like structure is dominated by dust emitted with zero
velocity, the spread of the dust along the tail is caused only by the
radiation pressure. Close to the head of the object, the actual velocity
dispersion of the grains will smear this relation, but further away, $\beta$
(the ratio between the radiation pressure and the solar gravitation for the
considered grain), $a$ (the grain radius) and $\rho$ (its density) are related
by:

$\beta=5.740\times 10^{-4}\times\frac{Q_{\rm pr}}{\rho a}.$ (1)

$Q_{\rm pr}$ is a radiation pressure efficiency coefficient, in the 1–2 range
for rocky and icy material. With this relation, we can estimate the
characteristic grain sizes in the images: with $\rho=3000$ kg m-3, and
$Q=1.05$, Eq. 1 gives $a=2\times 10^{-7}/\beta$ [m]. The loci of 10 to 200
$\mu$m grains are marked on Fig. 5. Closer to the head than the 100$\mu$m
mark, the blurring by the seeing and the non-zero velocity of the grains
forming the head structure prevent FP from making meaningful estimates of the
grain size beyond “they are large”, i.e. in the mm range or even larger.

###  3.5 Dynamical Model of the Arc and Linear Feature

Figure 6: The positions of large particles emitted by the nucleus on 2016 Mar.
6, with velocities ranging from 0 to 3 m s-1 observed on 2017 May 12 are
marked with light grey dots in the plane of sky. The coordinates mark offsets
from the position of the nucleus for that time. For clarity, only a subset of
the particles is plotted. The particles matching the position of the 3 head
features are marked in blue. Those matching the arc are marked in red (the
ends of the feature are marked in magenta and black); the blob at the western
start of the linear feature (Fig. 2) is marked in orange.  Figure 7: For a
series of emission velocity, maps of the emission direction (in cometocentric
longitudes and latitudes) of the large particles matching the features
observed on May 12, using the same colour code as in Fig. 6. For any velocity,
a particle matching a feature always has a ”mirror particle”: one is emitted
toward the Earth, the other one away from the Earth. On the 2.375, 2.5 and
2.625 m s-1 panels, the position of a $40^{\circ}$ half-opening cone matching
best the feature is represented. The direction directly toward Earth is marked
by a black cross, away from Earth by a red cross, at $0^{\circ}$ latitude, and
$90^{\circ}$ and $270^{\circ}$ longitude respectively. Re-projecting these
maps onto concentric spheres, with the geometry of May 12, would lead to Fig.
6.

Figure 8: P/2016 G1 on 2016 Apr. 3, May 12 and 31. The green dots mark clumps
on the arc feature. The small red dots correspond to the synchrone
trajectories of small dust (submitted to radiation pressure) emitted from some
of the clumps on Mar. 6. The position of large particles emitted on Mar. 6 on
a cone with half-opening 40∘ and centred on (long., lat.) =
(68$\deg$,-15$\deg$) are over-plotted for velocities ranging from 1.625 to
2.750 m s-1. The solid line corresponds to 2.5 m s-1.

In order to further investigate the structure of the head, we simulated the
trajectory of large particles (i.e. not influenced by radiation pressure,
$\beta=0$), emitted from the nucleus’ position on 2016 Mar. 6 with velocity
values ranging from 0 to 3 m s-1, and directions covering the whole sphere
every 2.5∘ in longitude and latitude (cometocentric coordinates, $x$ and $y$
in the orbital plane), $y$ pointing toward the Earth). The position of each
particle was computed for 2016 Apr. 3, May 12, and May 31. The particles that
were located at the position of the 3 fragment clumps, the South-East arc and
the start of the linear feature are identified, as illustrated in Fig. 6.
Figure 7 shows the maps of the particles matching the features (in
cometocentric longitude and latitude at the time of the emission), for a set
of emission velocities.

The three main fragment clumps can be reproduced by particles emitted at any
velocity between 0.25 and 3 m s-1. However, for higher velocities, they must
be emitted almost exactly toward or away from Earth (so that the
foreshortening keeps them at their position near the origin), which is
unlikely. It is more likely that they were emitted at a random angle, i.e. at
velocities $<$ 1 m s-1.

The South-East Arc cannot have been emitted at velocities $<$ 2.25 m s-1: part
of the arc is not reproduced (some of the dots are missing). For velocities in
the 2.25 – 3 m s-1 range, the complete feature can be reproduced by particles
emitted at the same velocity. A cone with a half-opening of $40^{\circ}$,
whose axis points toward cometocentric coordinates (long., lat.) =
(68$\deg$,-15$\deg$), is found to reproduce well the arc feature and the start
of the linear feature, with a velocity of 2.5 m s-1: the arc and the clump at
the head of the linear features are on a ring corresponding to the 2.5 m
s-1slice of this same cone.

In order to test the validity of this interpretation, the position of the
particles was also computed for Apr. 3 and May 31, when the viewing geometry
was very different. As illustrated in Fig. 8, the same ring of dust matches
well the 3D position and evolution of the arc and the head of the linear
feature.

Adjusting the position and opening of the cone (by eye) gives an estimate of
the uncertainty on these parameters: $\pm 2.5^{\circ}$ in longitude and
latitude, and $\pm 5^{\circ}$ in opening. Changing the opening of the cone
also slightly modifies the velocities of the particles matching the features.
The date of the initial burst was also changed by 5, 10, 15 days. No effect is
noticeable for 5 day; a slight change of orientation and velocity is required
for the 10 and 15 day shifts. This therefore does not set additional
constraints on the time of the emission.

Using the FP formalism (Section 3.4), the trajectory of small dust grains
emitted together with the larger grains was calculated accounting for the
radiation pressure. The smaller grains form a line (synchrone) starting at the
large particle, and drifting away at an angle set by the emission date (while
the large particles move on a straight line in the image plane, from the
nucleus toward their observed position, the small particles start tangentially
to that line, and drift on a parabolic particle toward their observed
position). The eastern linear feature matches well the position of small
grains emitted at the same time, in the same direction and at the same
velocity as the clump at the head of the linear feature. They are highlighted
with red dots in the May 12 panel of Fig. 8. Similarly, various clumps in the
arc feature have similar “tails”, which together cause an enhancement of
surface brightness on the western side of the arc (two of them are highlighted
with red dots in the May 12 panel of Fig. 8). As the position angle of these
synchrone lines is set only by the time of the emission, the excellent match
between the eastern linear feature and the synchrone for a Mar. 4 emission
gives a very strong support to the hypothesis that it corresponds to small
grains emitted simultaneously and in the same direction as the big grains, and
further support the Mar. 4 emission date. It also rules out that the linear
feature would have been caused by a secondary disruption of a large grain at a
later date: the position angle of the corresponding synchrone would have been
different.

Overall, the southern arc and the head of the eastern linear feature are large
particles partially populating a ring. The arc covers at least a quarter of
the ring, possibly more, as its Northern part is lost in the main coma. A
single clump of large particles on the ring forms the tip of the eastern
linear feature. This ring grows in size with time, as the particles move away
from the emission spot at $(2.5\pm 0.1)$ m s-1, describing a cone with an
half-opening of 40∘.

It is possible that the sections of the ring that appear devoid of dust were
originally populated by small grains: solar radiation pressure would have
pushed them away from the field of view.

The ring corresponds to a very narrow velocity distribution: in particular,
the “walls” of the cone are not populated by dust grains, only a (partial)
ring.

We tried and failed to reproduce the arc and the eastern linear feature using
dust grains emitted along a great circle in cometocentric coordinates, as
would have been produced by an equatorial centrifugal ejection. We also tried
and failed to reproduce the observed features using grains with a broader
range of velocities: a velocity distribution around 2.5 m s-1, narrower than
0.125 m s-1, gives the best results.

###  3.6 The Moreno Model

Moreno et al. (2016) obtained data for P/2016 G1 on three nights between 2016
Apr. 20 and Jun. 8 and used Monte Carlo techniques to model the dust. They
obtained two additional epochs with the Hubble Space Telescope on 2016 Jun. 28
and Jul. 11 (Moreno et al. 2017).

Comparing the original dataset to model images generated using Monte-Carlo
emitted particles following a FP-like formalism, they infer that the dust
ejection began around 2016 February 10${}^{+10}_{-30}$, and decreased with a
half-Gaussian function with a half-width at half maximum (HWHM) of
$24^{+10}_{-7}$ d, corresponding to an emission of 1.7$\times$107 kg of dust,
of which some was preferentially ejected in the westward direction, consistent
with an impact aligned with the sun-asteroid vector. They inferred very low
dust velocities with average speeds around 0.8 m s-1 and grain sizes between
1$\mu$m to 1 cm in radius(the velocities in the original paper Moreno et al.
(2016) had a typo, corrected in an erratum Moreno et al. (2019)). The more
recent HST data are compatible with their original model.

Their analysis was focused on what we call here the central structure with the
3 main concentrations forming the inverted C on the images and the central
dust coma. From an FP analysis similar to that of Section 3.4, they set the
interval for the beginning of activity in their model from 2016 Jan. 31 to
Apr. 1.

Our estimate of the date of the peak of dust emission relies on two simple and
independent measurements: $i)$ the linear growth of the separation between the
condensations in the central structure, which agrees with the barely resolved
image obtained on Mar. 7 (Section 3.1), and $ii)$ the position angle of the
sharp cusp in the isophotes far from the origin, in Fig. 5. Measuring this
angle on Moreno’s Fig. 2, we get a very good agreement with our estimate. This
makes the simple assumption that far from the origin, the effect of the
radiation pressure dominates over an initial velocity. Moreno et al. run their
Monte-Carlo multi-parameter model to refine the combination of initial
velocity and radiation pressure, leading to a difference in emission time of
about a month.

Their analysis does not mention the southern arc, nor the eastern linear
features, whose geometry we find to strongly and independently confirm the
emission date provided by the orientation of the main tail and the growth of
the central structure, and to also support a very brief emission. This is
further confirmed by the very narrow eastern linear feature, that points
precisely at an impulsive emission on again the same date, Mar.6. Because of
the convergence of these independent arguments, because they are based on much
wider features than those Moreno et al. worked on, and because of the
simplicity of the underlying assumptions, we are confident that the disruption
occurred on 2016 Mar. 6$\pm$3 days.

In terms of duration, the analysis of Moreno et al. suggests that “[an] impact
would have induced a partial destruction of the asteroid,” causing the
anisotropic part of the dust emission, “with dust grains being emitted to
space nearly isotropically while the body is being torn apart,” that this
isotropic emission would have had a duration of $24^{+10}_{-7}$ d HWHM. Our
data directly support the impact-induced destruction of the asteroid, the arc
and linear feature being the signature of that impact. Concerning the broad
dust feature on which Moreno et al. focus their study, our interpretation is
that the broadening of the tail matches the dust velocity dispersion, and that
therefore it is compatible with a very short duration event, although we
cannot rule out (nor constrain) a longer duration for the final crumbling of
the asteroid.

###  3.7 Mass of the debris cloud

Figure 9: Top: radius of the dust grains as a function of the distance to the
centre of the object, using the FP scaling (see Sect. 3.4 and Fig. 5) for May
31. Bottom: the flux in the May 31 image is integrated over concentric rings
and converted into the number of particles (blue), and in mass (red), both in
each radial bin.

Because the exact grain size distribution present in each pixel of the image
is not known, and because the central structure can contain very large
particles (meter-sized or more), getting an evaluation of the mass of the
object is difficult. This is made worse by the fact that larger particles
contribute more to the mass, while smaller particles contribute more to the
flux. Below comes an attempt to get a rough estimate of this mass, based on
the May 31 image. We assume a dust density $\rho=3000$ kg m-3, and an albedo
$p=0.25$.

To get an order-of-magnitude estimate, one can consider a uniform size
distribution across the whole object (a power law with a $-3$ index, and a
cut-off at 1 cm, after Moreno et al. (2016)). This leads to a total mass in
the object of $1\times 10^{7}$ kg, in agreement with Moreno et al. who
obtained an ejected mass of $2\times 10^{7}$ kg with their detailed model.

As we consider that the dust release event was short, the dust distribution
(far from the position of the nucleus) is dominated by the radiation pressure:
smaller grains are pushed further away than larger ones. While this estimate
is also a simplification, it takes into account the fact that the dust is not
uniformly distributed. The grain size is set using the scaling between size
and distance from Fig. 5. The image is then integrated on concentric rings,
and the flux is converted in dust surface, then in the number of grain, then
in mass, as illustrated in Fig. 9. This results in a total mass for the object
$M_{N}=1.1\times 10^{9}$ kg. This estimate is still a lower limit, as larger
clumps could hide in the central concentration. The mass corresponds to a
spherical object with a radius $R_{N}\sim 44$ m, well below the limit of
detection of the pre-discovery images ($R_{N}<200$ m).

We don’t know whether the object was completely disrupted. Using their high-
resolution, deep HST images, Moreno et al. (2017) obtained an upper limit of
30 m to the fragments. The mass potentially contained in even a few of these
large fragments would completely dominate the mass of the dust. In what
follows we consider as a lower limit the mass of the dust (and the
corresponding radius $R_{N}>44$ m), and the upper limit from the pre-discovery
constraints ($R_{N}<200$ m). Most of the mass being located in the central
concentration, which we observed expanding at $v=0.16$ m s-1. The
corresponding kinetic energy is $K=1.4\times 10^{7}$ and $1.3\times 10^{9}$ J,
respectively. In order to reach this asymptotic velocity, the grains and
fragments had to overcome the (small) gravity potential of the object, with an
escape velocity $v_{e}=\sqrt{\frac{2GM}{r_{T}}}=0.06$ m s-1 and $0.26$ m s-1,
respectively. This implies that the disruption of the object required a total
energy at least $E=\frac{1}{2}M_{N}(v^{2}+v_{e}^{2})=1.6\times 10^{7}$ and
$4.7\times 10^{9}$ J, respectively. These are clearly lower limits that
consider a pure rubble pile, i.e. without and internal cohesion forces.

Considering that this energy was provided by an impactor, and that collision
happened at the median impact velocity in the main belt $v_{i}=5\,000$ m s-1,
this requires a tiny impactor of 1 kg, with a radius of 5 cm for the lower
limit on the object size, and 370 kg with a radius of 30 cm for the larger
object.

###  3.8 Summary of the analysis

Overall, our results are based on various independent measurements and
separate simple interpretation pointing toward the following scenario:

  * •

No image from before 2016 Mar. shows the object. These non-detections lead to
a radius $R_{N}<0.2$ km.

  * •

The onset of activity was on 2016 Mar. 6. Backward extrapolation of the
central structure gives an uncertainty of $\pm$3 days but the object was found
active on Mar. 7. This Mar. 6 date is confirmed by the following independent
measurements: orientation of the azimuthal peak of the main tail; position and
evolution of the southern arc and of the head of the linear feature;
orientation of the eastern linear feature.

  * •

The dust emission event had a very brief component (indicated by the narrow
eastern linear feature and the narrow ring of large particles populating the
southern arc), which can possibly have been followed by a longer emission of
dust from the central feature.

  * •

The central component of the object is composed of large dust grains moving at
low velocity ($0.16\pm 0.01$ m s-1 in the plane of sky) from the original
centre. It is surrounded by a cloud of smaller particles, also emitted at
similar low speed, and being pushed away by radiation pressure. The visible
tail is populated by particles in the 10–200 $\mu$m range. Larger particles
are certainly present, but confused into the central component; smaller
particles drifted away from the field of view. The broadening of the Mar. 7
image could be caused by the hyper-velocity material released during the
impact itself, while the slower material was released during the follow-up
disruption.

  * •

During the brief emission event, a group of particles was emitted on a cone
with a half-opening angle of $40\pm 5^{\circ}$, propagating at (2.5$\pm$0.1) m
s-1and forming part of a ring, with an arc covering over a quarter of the ring
plus a clump opposite to the arc (forming the southern arc and the head of the
eastern linear feature). The large particles were emitted together (in time,
direction and velocity) with smaller particles, which are seen drifting from
the ring under the radiation pressure. They form the eastern linear feature
and a faint westward extension to the southern arc. These features could not
have been reproduced by emission on a grand circle (matching, for instance, an
equatorial ejection).

##  4 Discussion: Impact & Disruption

From the observations, a major disruption occurred on P/2016 G1 on or around
2016 Mar. 6, disrupting –possibly destroying– the object, and releasing a ring
of large particles moving at $\sim 2.5$ m s-1, growing on a cone with a half-
opening angle of $\sim 40^{\circ}$. We consider the impact of a meter-scale
object hitting a 100-meter-scale asteroid at $\sim 5$ km s-1. All the hyper-
velocity ejecta which would be expanding at impact-scale speeds (in the km s-1
range) have moved degrees away from the position of the object at the time of
the first deep images (about one month after the impact). Radiation pressure
having further dispersed these grains, they are lost beyond the field-of-view
of the observations.

All that is left in the field-of-view are the spalls and large dis-aggregated
fragments of the objects that came off well after the impact, in the final
stages of the formation of the crater and after, together with the dust grains
released. Crater formation simulations (numerical and laboratory) show that
asymmetries in the filling of the ejection cone are common; this would explain
why the ring is not complete.

These grains ($>10\mu$m) are either the small end of a power-law size
distribution of the fragments, and/or the regolith that was dragged along with
bigger fragments. The bulk of the impactor’s energy is transferred into the
bulk of the body, causing its disruption into fragments that drift apart at
very low velocities.

Hydrodynamic simulations of impacts can provide information on the remnants,
in particular on their size distribution (see Durda et al. 2007, for a
systematic study of various parameters) and velocities. Simulations of
catastrophic impacts333 URLs: https://youtu.be/pqaqbfbevoo
https://youtu.be/e4HCTcQ-IWA show that the plate-shaped spall fragments,
corresponding to the final moments of the crater formation, are ejected with
different velocities and over a broad range of directions. However, the crater
formation process also produces wedge-shaped fragments, bounded by the
propagating radial cracks. These have a narrow distribution of slow velocity,
in the m s-1 range, and are axially isotropic about the point of impact. These
large wedges would become visible as they break apart and their surface area
increase. Based on these qualitative arguments, we suggest that the feature we
interpret as a ring of fragments corresponds to the left-over of these large
wedge-shaped fragments. Hydrodynamics simulations, beyond the scope of this
paper, will be used to quantitatively verify the plausibility of this
hypothesis. Interestingly, the velocity of ejection of these wedge-shaped
fragments is directly linked to the tensile strength of the object. The
tensile strength is one of the key structural characteristics of an asteroid,
but it is normally not accessible remotely. In the case of a rotationally
disrupted asteroid, the study of the fragmentation can cast some light on the
internal strength, but with some uncertainties caused by the unknown initial
spin of the object (see Hirabayashi et al. 2014, for an application to
rotationally disrupted P/2013 R3). A result of the ongoing simulation will,
therefore, be the first direct remote estimate of the tensile strength of an
asteroid, a measurement that would have implication for the mitigation
measures being investigated for planetary defence initiatives.

###### Acknowledgements.

We are very grateful for the detailed review and comments provided by the
referee, Fernando Moreno. KJM, JK, and JVK acknowledge support through an
award from the National Science Foundation AST1413736. RJW acknowledges
support by NASA under grant NNX14AM74G. This paper is based on observations
obtained with MegaPrime/MegaCam, a joint project of CFHT and CEA/DAPNIA, at
the Canada-France-Hawaii Telescope (CFHT) which is operated by the National
Research Council (NRC) of Canada, the Institut National des Sciences de
l’Univers of the Centre National de la Recherche Scientifique of France, and
the University of Hawaii. Data were acquired using the PS1 System operated by
the PS1 Science Consortium (PS1SC) and its member institutions. We thank the
staff of IAO, Hanle and CREST, Hosakote, that made these observations
possible. IAO and CREST are operated by the Indian Institute for Astrophysics,
Bangalore. The Pan-STARRS1 Surveys (PS1) have been made possible by
contributions from PS1SC member Institutions and NASA through Grant
NNX08AR22G, the NSF under Grant No. AST-123886, the Univ. of MD, and Eotvos
Lorand Univ. We would like to thank Detlef Koschny (ESA) and Mikael Granvik
for their helpful discussion on impact cratering mechanics and disruption.
This research was supported by the Munich Institute for Astro- and Particle
Physics (MIAPP) of the DFG cluster of excellence “Origin and Structure of the
Universe”.

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Table 1: Observations

Archive data |  |  |  |  |  |
---|---|---|---|---|---|---
UT Date | Tela | Mid-JDb | Filt | #c | Expd |  $r$e |  $\Delta$e |  $\alpha$e | PA-⊙f | PA-vf | TAf |  $m$h |  $R_{4\%}$i |  $R_{25\%}$i |
2000-10-20 | INT | 1838.45737 | z |  |  | 2.163 | 1.351 | 19.38 | 74.6 | 255.9 | 46.9 |  $>$21.03 | 0.74 | 0.30 |
2007-02-16 | CFHT | 4147.93142 | i |  |  | 3.043 | 2.111 | 7.44 | 319.2 | 284.2 | -153.7 |  $>$21.53 | 1.03 | 0.41 |
2011-02-23 | PS1 | 5615.91806 |  $r_{P1}$,$i_{P1}$ |  $2$,$2$ | 170 | 3.084 | 2.111 | 3.99 | 25.4 | 284.3 | -161.5 |  $>$20.5 | 1.58 | 0.63 |
2011-05-25 | PS1 | 5706.78070 | $w_{P1}$ | 4 | 180 | 2.993 | 2.899 | 19.71 | 110.6 | 282.6 | -146.2 |  $>$21.5 | 1.78 | 0.71 |
2012-06-09 | PS1 | 6087.93886 | $i_{P1}$ | 2 | 90 | 2.232 | 1.242 | 7.73 | 159.6 | 268.5 | -59.6 |  $>$22 | 0.36 | 0.14 |
2012-06-13 | PS1 | 6091.86993 | $w_{P1}$ | 6 | 270 | 2.224 | 1.242 | 8.97 | 147.7 | 268.3 | -58.3 |  $>$22 | 0.37 | 0.15 |
2013-10-28 | PS1 | 6594.13022 | $z_{P1}$ | 2 | 60 | 2.599 | 1.734 | 13.22 | 274.8 | 269.3 | 103.8 |  $>$21 | 1.03 | 0.41 |
2013-11-04 | PS1 | 6601.05904 |  $g_{P1}$,$r_{P1}$ |  $2$,$2$ | 166 | 2.615 | 1.703 | 10.61 | 279.9 | 268.7 | 105.4 |  $>$21 | 0.97 | 0.39 |
2013-11-28 | PS1 | 6624.92394 | $w_{P1}$ | 4 | 180 | 2.667 | 1.693 | 4.23 | 0.9 | 267.3 | 110.6 |  $>$22 | 0.55 | 0.22 |
2014-11-18 | PS1 | 6980.07870 | $w_{P1}$ | 4 | 180 | 3.123 | 2.994 | 18.45 | 289.3 | 283.0 | 173.8 |  $>$22 | 1.49 | 0.59 |
2014-11-28 | PS1 | 6990.11418 | $w_{P1}$ | 4 | 180 | 3.125 | 2.853 | 18.28 | 291.4 | 282.9 | 175.4 |  $>$22 | 1.41 | 0.56 |
2014-11-30 | PS1 | 6992.05683 | $w_{P1}$ | 4 | 180 | 3.126 | 2.825 | 18.19 | 291.8 | 282.8 | 175.7 |  $>$22 | 1.40 | 0.56 |
2014-12-02 | PS1 | 6994.09527 | $w_{P1}$ | 4 | 180 | 3.126 | 2.797 | 18.09 | 292.2 | 282.8 | 176.1 |  $>$22 | 1.38 | 0.55 |
2015-01-17 | PS1 | 7040.03093 | $w_{P1}$ | 2 | 90 | 3.126 | 2.263 | 10.16 | 311.2 | 283.0 | -176.7 |  $>$22 | 0.96 | 0.39 |
2016-01-10 | PS1 | 7398.14409 | $w_{P1}$ | 4 | 180 | 2.699 | 2.904 | 19.79 | 287.9 | 277.1 | -113.9 |  $>$23 | 0.81 | 0.32 |
2016-03-07 | PS1 | 7455.11400 | $w_{P1}$ | 3 | 135 | 2.575 | 2.028 | 20.89 | 282.8 | 273.9 | -101.2 | 18.99$\pm$0.06 |  |  |
New data |  |  |  |  |  |
UT Date | Tela | Mid-JDb | Filt | #c | Expd |  $r$e |  $\Delta$e |  $\alpha$e | PA-⊙f | PA-vf | TAf | $m_{10k}^{j}$ | $m_{7.5k}^{j}$ | $m_{5k}^{j}$ | $m_{2.5k}^{j}$
2016-04-03 | CFHT | 7482.06559 | w | 3 | 540 | 2.513 | 1.678 | 15.400 | 278.2 | 274.0 | -94.8 | 18.806$\pm$0.002 | 18.982$\pm$0.002 | 19.424$\pm$0.002 | 20.688$\pm$0.003
2016-04-04 | CFHT | 7483.04336 | w | 3 | 540 | 2.511 | 1.668 | 15.104 | 277.9 | 274.0 | -94.5 | 18.793$\pm$0.002 | 18.975$\pm$0.002 | 19.457$\pm$0.002 | 20.758$\pm$0.003
2016-04-05 | CFHT | 7484.05387 | w | 3 | 540 | 2.509 | 1.657 | 14.792 | 277.6 | 274.1 | -94.3 | 18.789$\pm$0.002 | 18.973$\pm$0.002 | 19.450$\pm$0.002 | 20.737$\pm$0.003
2016-04-08 | CFHT | 7487.04616 | w | 6 | 1080 | 2.502 | 1.626 | 13.827 | 276.6 | 274.1 | -93.6 |  |  |  |
2016-04-14 | CFHT | 7493.05300 | w | 5 | 600 | 2.488 | 1.569 | 11.712 | 273.9 | 274.3 | -92.1 | 18.721$\pm$0.002 | 19.005$\pm$0.002 | 19.622$\pm$0.002 | 21.012$\pm$0.004
2016-04-29 | CFHT | 7507.96100 | w | 3 | 360 | 2.454 | 1.465 | 5.838 | 255.1 | 274.6 | -88.3 | 18.556$\pm$0.002 | 18.945$\pm$0.003 | 19.642$\pm$0.003 | 21.078$\pm$0.006
2016-05-10 | CFHT | 7518.94939 | w | 3 | 360 | 2.429 | 1.425 | 3.386 | 189.6 | 274.7 | -85.4 |  |  |  |
2016-05-11 | CFHT | 7519.98075 | w | 5 | 600 | 2.426 | 1.423 | 3.511 | 181.3 | 274.7 | -85.2 | 18.490$\pm$0.002 | 18.901$\pm$0.003 | 19.618$\pm$0.003 | 21.080$\pm$0.006
2016-05-12 | CFHT | 7521.00253 | w | 7 | 840 | 2.424 | 1.421 | 3.703 | 173.7 | 274.7 | -84.9 | 18.502$\pm$0.002 | 18.903$\pm$0.000 | 19.612$\pm$0.002 | 21.079$\pm$0.004
2016-05-31 | CFHT | 7539.96859 | w | 5 | 600 | 2.381 | 1.435 | 11.317 | 125.4 | 274.2 | -79.8 | 18.915$\pm$0.003 | 19.329$\pm$0.003 | 20.048$\pm$0.004 | 21.545$\pm$0.007
2016-05-31 | HCT | 7540.29466 | rc | 5 | 625 | 2.381 | 1.435 | 11.317 | 125.4 | 274.2 | -79.8 | 18.940$\pm$0.015 | 19.342$\pm$0.014 | 20.032$\pm$0.019 | 21.449$\pm$0.064
2016-06-05 | CFHT | 7544.95929 | w | 5 | 600 | 2.369 | 1.452 | 13.431 | 122.2 | 274.0 | -78.4 | 19.095$\pm$0.003 | 19.486$\pm$0.003 | 20.164$\pm$0.004 | 21.627$\pm$0.009
2016-06-08 | CFHT | 7547.96720 | w | 7 | 840 | 2.363 | 1.465 | 14.649 | 120.7 | 273.8 | -77.6 |  |  |  |
2016-06-27 | CFHT | 7566.97056 | w | 1 | 120 | 2.321 | 1.583 | 21.039 | 114.6 | 272.9 | -72.2 | 19.789$\pm$0.013 | 20.175$\pm$0.015 | 20.858$\pm$0.018 | 22.190$\pm$0.031
2016-06-29 | CFHT | 7568.96143 | w | 15 | 1800 | 2.316 | 1.599 | 21.567 | 114.1 | 272.8 | -71.7 | 19.940$\pm$0.008 | 20.315$\pm$0.009 | 20.990$\pm$0.011 | 22.232$\pm$0.018
2016-07-01 | CFHT | 7570.82249 | w | 10 | 1200 | 2.312 | 1.613 | 22.036 | 113.6 | 272.8 | -71.1 |  |  |  |
2016-07-03 | CFHT | 7572.84000 | w | 4 | 480 | 2.308 | 1.629 | 22.518 | 113.2 | 272.7 | -70.5 | 19.974$\pm$0.006 | 20.379$\pm$0.007 | 21.072$\pm$0.008 | 22.420$\pm$0.014
2016-07-04 | CFHT | 7573.89747 | w | 7 | 840 | 2.305 | 1.638 | 22.760 | 112.9 | 272.7 | -70.2 | 19.930$\pm$0.004 | 20.338$\pm$0.005 | 21.026$\pm$0.006 | 22.384$\pm$0.010
2018-12-12 | CFHT | 8465.139 | w | 5 | 1200 | 3.110 | 2.515 | 16.1 | 295.3 | 281.8 | 168.3 | $>25$ |  |  |
2018-12-31 | CFHT | 8484.122 | w | 9 | 2160 | 3.118 | 2.316 | 12.2 | 304.1 | 281.8 | 171.3 | $>26$ |  |  |

Notes: a: Telescope; b: Exposure Mid Julian date-2450000; c: Number of
exposures; d: Total exposure time, [s]; e: Heliocentric, geocentric distance
[AU], Solar phase angle, [degrees]; f: Position of the extended Sun-target
radius vector (anti-solar direction) and negative of the heliocentric velocity
vector (dust tail orientation) as seen in the plane of the sky measured east
from the north celestial pole; g: True anomaly, [degrees]; h: Limiting mag for
non-detection, or total magnitude; i: Inferred nucleus radius [km] assuming an
albedo of 4% or 25%; j: Magnitude through aperture diameters of 10000, 7500,
5000, and 2500 km.


2^{k}c_{0}(\log r)^{1/2}\Delta_{r}$}\Bigg{)}$
$\displaystyle\leq\sum_{k=k_{0}+1}^{\infty}2^{kd}r^{2d}C_{44}\exp\left(-\frac{C_{45}}{\sigma(r+2^{k}c_{0}(\log
r)^{1/2}\Delta_{r})}\left(t\sigma_{r}+\frac{c_{4}c_{0}\mu}{4}2^{k}(\log
r)^{1/2}\Delta_{r}\right)\right)$ $\displaystyle\leq
c_{5}\exp\left(-\frac{c_{6}(t\sigma_{r}+r)}{\sigma(2r)}\right)$ (6.5)
$\displaystyle\leq c_{2}e^{-c_{3}t}.$

The last inequality uses $t\leq h(r)/\sigma_{r}$.

We now deal with the remaining case $|y^{*}|\leq c_{0}(\log
r)^{1/2}\Delta_{r}$. For $C_{26}$ from Theorem 1.3, we take
${\color[rgb]{1,0,0}K}=\sqrt{t/C_{26}(d-1)}$ to be specified and subdivide
$H_{[r,r+2]}$ into blocks

${\color[rgb]{1,0,0}Y_{\mathbf{m}}}=[r,r+2]\times\prod_{i=1}^{d-1}[(m_{i}-1)K\Delta_{r},m_{i}K\Delta_{r}],\quad\mathbf{m}\in\mathbb{Z}^{d-1}.$

Let ${\color[rgb]{1,0,0}z_{\mathbf{m}}}$ denote the center point of
$Y_{\mathbf{m}}$ and let

${\color[rgb]{1,0,0}\mathfrak{M}}=\Big{\\{}\mathbf{m}\in\mathbb{Z}^{d-1}:0\notin
Y_{\mathbf{m}},\,\exists y\in Y_{\mathbf{m}}\text{ with }|y^{*}|\leq
c_{0}(\log r)^{1/2}\Delta_{r}\Big{\\}}.$

Given $\mathbf{m}\in\mathfrak{M}$, there exists a cylinder
$\mathcal{C}_{\mathbf{m}}$, with axis containing $\Pi_{0z_{\mathbf{m}}}$,
radius $2\sqrt{d-1}K\Delta_{r}$, and length $2r$, which contains $\Lambda_{r}$
and $Y_{\mathbf{m}}$. See Figure 11. Further,
$d_{\infty}(Y_{\mathbf{m}},[r,r+2]\times\\{0\\})$ is a positive integer
multiple of $K\Delta_{r}$, and for $x\in\Lambda_{r}$ and $y\in Y_{\mathbf{m}}$
satisfying $d_{\infty}(Y_{\mathbf{m}},[r,r+2]\times\\{0\\})=jK\Delta_{r}$ for
some $j$, we have using (4.20) and Lemma 3.6(i) that

$|y-x|\geq
r+\frac{(jK\Delta_{r})^{2}}{3r}=r+\frac{(jK)^{2}}{3}\sigma_{r}\quad\text{and
hence}\quad h(|y-x|)\geq h(r)+\frac{\mu(jK)^{2}}{6}\sigma_{r}.$

Figure 11.  The block $\Lambda_{r}$ in $H_{0}$ (“fattened” slightly to
thickness 2), the similar block $Y_{\mathbf{m}}$ of $H_{r}$, and the cylinder
$\mathcal{C}_{\mathbf{m}}$ containing both, for $\mathbf{m}=(0,1)$. The gray
blocks in $H_{r}$ are those with $j=1$.

The definition of $K$ says $t=C_{26}(d-1)K^{2}$, so by rotational invariance
we can apply Theorem 1.3 (for downward deviations) to the cylinder
$\mathcal{C}_{\mathbf{m}}$ and obtain

$\displaystyle P$ $\displaystyle\Big{(}T(x,y)\leq h(r)-t\sigma_{r}\text{ for
some $x\in\Lambda_{r}\cap\mathbb{V}$ and $y\in Y_{\mathbf{m}}$}\Big{)}$
$\displaystyle\leq P\left(T(x,y)\leq
h(|y-x|)-\left(t+\frac{\mu(jK)^{2}}{6}\right)\sigma_{r}\text{ for some
$x\in\Lambda_{r}\cap\mathbb{V}$ and $y\in Y_{\mathbf{m}}$}\right)$ (6.6)
$\displaystyle\leq
C_{27}\exp\left(-C_{28}\left(t+\frac{\mu(jK)^{2}}{6}\right)\right).$

Summing over $j\leq{\color[rgb]{1,0,0}j_{0}}=\max\\{j:jK\Delta_{r}\leq
c_{0}(\log r)^{1/2}\Delta_{r}\\}$ gives

$\displaystyle P$ $\displaystyle\Big{(}T(x,y)\leq h(r)-t\sigma_{r}\text{ for
some $x\in\Lambda_{r}\cap\mathbb{V},y\in H_{[r,r+2]}$ with $|y^{*}|\leq
c_{0}(\log r)^{1/2}\Delta_{r}$}\Big{)}$ $\displaystyle\leq P\Big{(}T(x,y)\leq
h(r)-t\sigma_{r}\text{ for some $x\in\Lambda_{r}\cap\mathbb{V},y\in
Y_{\mathbf{m}}$, and $\mathbf{m}\in\mathfrak{M}$}$ $\displaystyle\hskip
56.9055pt\text{with $d_{\infty}(Y_{\mathbf{m}},[r,r+2]\times\\{0\\})\leq
j_{0}K\Delta_{r}$}\Big{)}$
$\displaystyle\leq\sum_{j=0}^{j_{0}}P\bigg{(}T(x,y)\leq h(r)-t\sigma_{r}\text{
for some $x\in\Lambda_{r}\cap\mathbb{V},y\in Y_{\mathbf{m}}$, and
$\mathbf{m}\in\mathfrak{M}$}$ $\displaystyle\hskip 56.9055pt\text{with
$d_{\infty}(Y_{\mathbf{m}},[r,r+2]\times\\{0\\})=jK\Delta_{r}$}\bigg{)}$
$\displaystyle\leq\sum_{j=0}^{j_{0}}(2j+2)^{d-1}C_{27}\exp\left(-C_{28}\left(t+\frac{\mu(jK)^{2}}{6}\right)\right)$
(6.7) $\displaystyle\leq c_{7}e^{-c_{8}t}.$

With (6), (6), and (6) this completes the proof of Lemma 6.1. ∎

Define the “small box”

${\color[rgb]{1,0,0}\mathcal{G}_{r}}=[0,r]\times[-\Delta_{r},\Delta_{r}]^{d-1}.$

We now tile a region of $\mathbb{R}^{d}$ with translates of $\mathcal{G}_{r}$.
Fix $k,K$ to be specified and let $j={\color[rgb]{1,0,0}j(r,k,K)}$ be the
least $j$ for which $(2j+1)\Delta_{r}\geq K\Delta(kr)$. Define the “big box”

${\color[rgb]{1,0,0}\mathcal{G}^{+}}=[0,kr]\times[-(2j+1)\Delta_{r},(2j+1)\Delta_{r}]^{d-1},$

so the width of $\mathcal{G}^{+}$ is near $2K\Delta(kr)$. For
$\mathbf{m}\in\mathbb{Z}^{d}$ let
$\mathcal{G}_{r,\mathbf{m}},\Lambda_{r,\mathbf{m}}$ be
$\mathcal{G}_{r},\Lambda_{r}$ translated by
$(rm_{1},2\Delta_{r}\mathbf{m}^{*})$, and let
${\color[rgb]{1,0,0}\mathcal{M}}=\\{\mathbf{m}\in\mathbb{Z}^{d}:\mathcal{G}_{r,\mathbf{m}}\subset\mathcal{G}^{+}\\}$;
see Figure 12. Then the number of small boxes comprising $\mathcal{G}^{+}$ is

$|\mathcal{M}|=k(2j+1)^{d-1}\leq
2k\left(\frac{K\Delta(kr)}{\Delta_{r}}\right)^{d-1}\leq
c_{0}K^{d-1}k^{1+(d-1)\chi_{2}},$

and we have $\mathcal{G}^{+}\supset G_{kr,K}\cap H_{[0,kr]}$. Let
$\mathcal{E}_{\mathbf{m}}(t)$ be the event in (6.1) translated to
$\mathcal{G}_{r,\mathbf{m}}$, that is, we replace $\Lambda_{r}$ with
$\Lambda_{r,\mathbf{m}}$ and $H_{r}^{+}$ with $H_{(m_{1}+1)r}^{+}$. Fix $C$ to
be specified; when $\omega\in\mathcal{E}_{\mathbf{m}}(C\log k)$ we say
$\Lambda_{r,\mathbf{m}}$ is a _fast source_. Let
${\color[rgb]{1,0,0}\mathcal{L}(r)}=(h(r)-r\mu)/\sigma_{r}$, so

(6.8) $0\leq\mathcal{L}(r)\leq C_{46}\log r$

by Proposition 3.3.

Figure 12.  The big box $\mathcal{G}^{+}$, of size
$kr\times(4j+2)\Delta_{r}\approx kr\times K\Delta(kr)$, here with $k=5,j=2$.
Each small box $\mathcal{G}_{r,\mathbf{m}}$ has size $r\times 2\Delta_{r}$.
$\Lambda_{r,(3,1)}$ is a fast source if some path like the dashed one is
“sufficiently fast” relative to $h(r)$.

We now consider the passage time between the ends of the big box. If
$\Gamma_{xy}\subset G_{kr,K}$ then $\Gamma_{0,kre_{1}}$ must intersect at
least $k$ of the regions $\Lambda_{r,\mathbf{m}}$, one for each $0\leq
m_{1}<k$. If there are no fast sources, this means
$T(0,kre_{1})>kr\mu+k\mathcal{L}(r)\sigma_{r}-C\sigma_{r}k\log k$. Provided
$K$ is large, it then follows from Lemmas 5.1 and 6.1 that

$\displaystyle P\Big{(}T(0,kre_{1})\leq
kr\mu+k\mathcal{L}(r)\sigma_{r}-C\sigma_{r}k\log k\Big{)}$ $\displaystyle\leq
P\Big{(}\Gamma_{0,kre_{1}}\not\subset
G_{kr,K}\Big{)}+P\Big{(}\cup_{\mathbf{m}\in\mathcal{M}}\mathcal{E}_{\mathbf{m}}(C\log
k)\Big{)}$ $\displaystyle\leq
C_{75}e^{-C_{76}K^{2}}+C_{81}|\mathcal{M}|e^{-C_{82}C\log k}$ (6.9)
$\displaystyle\leq
C_{75}e^{-C_{76}K^{2}}+c_{1}K^{d-1}k^{1+(d-1)\chi_{2}}e^{-C_{82}C\log k}.$

By taking $K$ large and then $C$ large, we can make the right side of (6) less
than 1/2 for all $k\geq 3$. On the other hand, from (1.11) for large $c>1$ we
have

$P\Big{(}T(0,kre_{1})\leq
kr\mu+\mathcal{L}(kr)\sigma(kr)+c\sigma(kr)\Big{)}\geq
1-C_{24}e^{-C_{25}c}>\frac{1}{2}.$

From this and (6) it follows that

(6.10) $(\mathcal{L}(kr)+c)\sigma(kr)>k(\mathcal{L}(r)-C\log
k)\sigma_{r}\quad\text{for all $r$ large and }k\geq 3.$

Fix $k\geq 3$ large enough so $k^{1-\chi_{2}}\geq 3C_{23}c$, with $C_{23}$
from (1.10), and, to get a contradiction, suppose there exists $r$ arbitrarily
large with $\mathcal{L}(r)\geq 2C\log k$. By (1.10) and (6.10) we then have

$\mathcal{L}(kr)\geq\frac{\sigma_{r}}{\sigma(kr)}\frac{k\mathcal{L}(r)}{2}-c\geq\frac{k^{1-\chi_{2}}}{3C_{23}}\mathcal{L}(r)\geq
c\mathcal{L}(r).$

Then iteration and (6.8) give that for all $n\geq 1$, $C_{46}(n\log k+\log
r)\geq\mathcal{L}(k^{n}r)\geq c^{n}\mathcal{L}(r)$. Since this is false for
large $n$, we must have $\mathcal{L}(r)<2C\log k$ for all large $r$, so
$\mathcal{L}(\cdot)$ is bounded, which completes the proof of Theorem 1.5, and
thus (3) of Remark 1.7.

##  7\. Proof of Theorem 1.3—upward deviations.

We move on to (4) of Remark 1.7. The proof is similar to the LPP proof for
$d=2$ in ([8] Proposition 10.1). As with downward deviations, we need only
consider $x,y$ as in (4.18). Let $\mathcal{H}_{xy}$ be as in the
downward–deviations proof, and

${\color[rgb]{1,0,0}\mathcal{H}_{xy}^{ter}}=\\{\text{all terminal hyperplanes
in }\mathcal{H}_{xy}\\},$

so $\mathcal{H}_{xy}^{ter}=\\{H_{s_{1}},\dots,H_{s_{n}}\\}$ for some
$n,s_{1},\dots,s_{n}$ depending on $(x,y)$. Note that $\mathcal{H}_{xy}^{ter}$
does not depend on $\Gamma_{xy}$; the $j$–terminal hyperplane at each end of
$[\hat{x}_{1},\hat{y}_{1}]$ is just the second–closest $j$th–scale hyperplane
to the endpoint. We form a marked PG path

${\color[rgb]{1,0,0}\Omega_{xy}:\hat{x}=u^{0}\to\cdots\to u^{n+1}=\hat{y}}$

by taking $u^{i}$ as the closest point to
${\color[rgb]{1,0,0}w^{i}}=\Pi_{\hat{x}\hat{y}}\cap H_{s_{i}}$ in the
$j$th–scale grid, when $H_{s_{i}}$ is a $j$–terminal hyperplane; this means
$|u^{i}-w^{i}|\leq\sqrt{d-1}\beta^{j}\Delta_{r}$ and $n\leq 2j_{1}$ (from
(4.16).) Thus the CG approximation gets coarser as we move away from the
endpoints of the interval. See Figure 3.

We proceed generally as in Step 8 of the proof for downward deviations. Let
${\color[rgb]{1,0,0}w_{\perp}^{i}}$ be the orthogonal projection of $u^{i}$
into $\Pi_{\hat{x}\hat{y}}$. We use the terminology $j$th–scale link, final
link, and macroscopic link from there. For $j\leq j_{1}$, for a $j$th–scale
link $(u^{i-1},u^{i})$ we have

(7.1)
$\displaystyle|u^{i}-u^{i-1}|\leq|w_{\perp}^{i}-w_{\perp}^{i-1}|+\frac{(|u^{i-1}-w_{\perp}^{i-1}|+|u^{i}-w_{\perp}^{i}|)^{2}}{\delta^{j}r}\leq|w_{\perp}^{i}-w_{\perp}^{i-1}|+\frac{2(d-1)\beta^{2j}\Delta_{r}^{2}}{\delta^{j}r}.$

Further, the angle $\alpha$ between $e_{1}$ and $\hat{y}-\hat{x}$ satisfies
$\tan\alpha\leq 2K\Delta_{r}/\epsilon r$ and therefore

(7.2) $|w^{i}-w_{\perp}^{i}|\leq c_{0}\frac{K\Delta_{r}}{\epsilon
r}|u^{i}-w^{i}|\leq c_{1}\frac{K\beta^{j}}{\epsilon}\sigma_{r},$

and similarly for $i-1$ in place of $i$. We have also
$\delta^{j}r/2\leq(u^{i}-u^{i-1})_{1}\leq 2\delta^{j}r$ so from (4.20) and the
bound on $\alpha$ we also get

$|w^{i}-w^{i-1}|-(w^{i}-w^{i-1})_{1}=\frac{(w^{i}-w^{i-1})_{1}}{(\hat{y}-\hat{x})_{1}}\Big{(}|\hat{y}-\hat{x}|-(\hat{y}-\hat{x})_{1}\Big{)}\leq\frac{2\delta^{j}}{\epsilon}\frac{2K^{2}}{\epsilon}\sigma_{r}.$

Combining this with (4.13), (7.1), and (7.2) we see that

$|u^{i}-u^{i-1}|\leq(u^{i}-u^{i-1})_{1}+\left(\frac{2(d-1)\beta^{2j}}{\delta^{j}}+\frac{2c_{1}K\beta^{j}}{\epsilon}+\frac{4\delta^{j}K^{2}}{\epsilon^{2}}\right)\sigma_{r}\leq(u^{i}-u^{i-1})_{1}+\frac{8\delta^{j}K^{2}}{\epsilon^{2}}\sigma_{r}$

and therefore by Theorem 1.5, using (4.13) and the assumption $t\geq
C_{26}K^{2}$,

$\displaystyle h(|u^{i}-u^{i-1}|)$ $\displaystyle\leq
h((u^{i}-u^{i-1})_{1})+\frac{16\mu\delta^{j}K^{2}}{\epsilon^{2}}\sigma_{r}\leq\mu(u^{i}-u^{i-1})_{1}+2C_{33}\sigma(2\delta^{j}r)+\frac{16\mu\delta^{j}K^{2}}{\epsilon^{2}}\sigma_{r}$
(7.3)
$\displaystyle\leq\mu(u^{i}-u^{i-1})_{1}+\frac{c_{2}\delta^{j\chi_{1}}K^{2}}{\epsilon^{2}}\sigma_{r}\leq\mu(u^{i}-u^{i-1})_{1}+\frac{\lambda^{j}}{2}t\sigma_{r}.$

It follows using Lemma 3.2 that

$\displaystyle
P\Big{(}\hat{T}(u^{i-1},u^{i})-\mu(u^{i}-u^{i-1})_{1}\geq\lambda^{j}t\sigma_{r}\Big{)}$
$\displaystyle\leq
P\left(\hat{T}(u^{i-1},u^{i})-h(|u^{i}-u^{i-1}|)\geq\frac{\lambda^{j}}{2}t\sigma_{r}\right)$
$\displaystyle\leq
C_{44}\exp\left(-C_{45}\frac{\lambda^{j}t\sigma_{r}}{\sigma(2\delta^{j}r)}\right)$
(7.4) $\displaystyle\leq
C_{44}\exp\left(-c_{3}\left(\frac{\lambda}{\delta^{\chi_{1}}}\right)^{j}t\right).$

The same is valid for a final link, with $j=j_{1}$. For macroscopic links it
is valid with $\sigma(2\delta^{j}r)$ replaced by $\sigma_{r}$, giving

(7.5) $P\Big{(}\hat{T}(u^{i-1},u^{i})-\mu(u^{i}-u^{i-1})_{1}\geq\lambda
t\sigma_{r}\Big{)}\leq C_{44}e^{-c_{4}\lambda t}.$

The numbers of possible links inside $G_{r}(K)$ are as follows:

  * (i)

$j$th–scale links: at most $c_{5}K^{d-1}/\delta^{2j}\beta^{2j(d-1)}$,

  * (ii)

final links: at most $c_{6}K^{d-1}r^{d}$,

  * (iii)

macroscopic links: at most $c_{7}K^{d-1}/\delta^{2}\beta^{2(d-1)}$.

Since $t\geq C_{26}K^{2}$, provided $C_{26}$ is taken large enough it follows
from these bounds and (4.14), (7), (7.5) that

$\displaystyle P$ $\displaystyle\Big{(}\text{for some $\hat{x},\hat{y}\in
G_{r}(K)$ and $j\leq j_{1}$, there is a $j$th--scale link $(v,w)$ in
$\Omega_{xy}$}$ $\displaystyle\qquad\text{satisfying
}\hat{T}(v,w)-\mu(w-v)_{1}\geq\lambda^{j}t\sigma_{r}\Big{)}\leq
c_{8}\exp\left(-c_{10}\left(\frac{\lambda}{\delta^{\chi_{1}}}\right)^{j}t\right),$
$\displaystyle P$ $\displaystyle\Big{(}\text{for some $\hat{x},\hat{y}\in
G_{r}(K)$ there is a final link $(v,w)$ in $\Omega_{xy}$ }$
$\displaystyle\qquad\text{satisfying
}\hat{T}(v,w)-\mu(w-v)_{1}\geq\lambda^{j_{1}}t\sigma_{r}\Big{)}\leq
c_{8}\exp\left(-c_{10}\left(\frac{\lambda}{\delta^{\chi_{1}}}\right)^{j_{1}}t\right),$
$\displaystyle P$ $\displaystyle\Big{(}\text{for some $\hat{x},\hat{y}\in
G_{r}(K)$ the macroscopic link $(v,w)$ in $\Omega_{xy}$ }$ (7.6)
$\displaystyle\qquad\text{satisfies }\hat{T}(v,w)-\mu(w-v)_{1}\geq\lambda
t\sigma_{r}\Big{)}\leq c_{8}e^{-c_{10}\lambda t}.$

If $\omega$ is not in any of the events in (7), and $\omega\notin
J^{(0)}(c_{29})$, then for all $x,y$ as in (1.12),

$\displaystyle T(x,y)=T(\hat{x},\hat{y})$
$\displaystyle\leq\sum_{i=1}^{n+1}\hat{T}(u^{i-1},u^{i})+c_{29}n\log r$
$\displaystyle\leq\mu(y-x)_{1}+c_{29}j_{1}\log
r+(\lambda+\lambda^{j_{1}})t\sigma_{r}+\sum_{j=1}^{j_{1}}\lambda^{j}t\sigma_{r}$
$\displaystyle\leq h(|y-x|)+\frac{2\lambda}{1-\lambda}t\sigma_{r}.$

Taking $\lambda<1/3$ it follows that

$P\Big{(}T(x,y)-ET(x,y)\geq t\sigma_{r}\text{ for some $x,y\in G_{r}(K)$ with
}|y-x|\geq\epsilon r\Big{)}\leq c_{10}e^{-c_{11}t},$

which completes the proof of Theorem 1.3 for downward deviations.

##  8\. Proof of Theorem 1.6 and Lemma 1.2.

We finish with (5) of Remark 1.7 by proving Theorem 1.6. In the proof in
section 5 for fixed $x,y$, transverse wandering cases (i)–(iii) were dealt
with uniformly over $x,y$, so we need only consider the case there:

(iv) $s\leq c_{0}(\log r)^{1/2}$ and there exist $x,y,u\in\mathbb{V}$ with

(8.1) $x,y\in G_{r}(K),\quad u\notin G_{r,s},\quad d(u,G_{r,s})\leq
2\quad|y-x|>r_{0},\quad u\in\Gamma_{xy},\quad u_{1}\in[x_{1},y_{1}].$

See the dashed line in Figure 10. As in section 5, this means that, for
$C_{56}$ from Lemma 3.6,

$|u-x|+|y-u|-|y-x|\geq s^{2}\sigma_{r},\quad\text{so}\quad
h(|u-x|)+h(|y-u|)-h(|y-x|)\geq\frac{\mu s^{2}}{2}\sigma_{r}-C_{56}$

while

$T(x,u)+T(u,y)-T(x,y)=0$

so by Theorem 1.3 and Remark 1.4,

$\displaystyle P\Big{(}(iv)\text{ holds}\Big{)}$ $\displaystyle\leq
P\left(\text{there exist $v,w\in G_{r}(2s)$ with
}\big{|}T(v,w)-h(|w-v|)\big{|}\geq\frac{\mu s^{2}}{7}\sigma_{r}\right)$ (8.2)
$\displaystyle\leq C_{36}e^{-c_{1}s^{2}}.$

Together with cases (i)–(iii) in section 5, this completes the proof of
Theorem 1.6.

###### Proof of Lemma 1.2.

Fix $M$ (large) and $\epsilon>0$, and define

${\color[rgb]{1,0,0}f(r)}=\log\rho(e^{r})-\chi
r,\quad{\color[rgb]{1,0,0}\beta_{k}}=\sup\\{f(r):2^{k-1}M<r\leq 2^{k}M\\},$

so $f(r)=o(r)$ and hence $\beta_{k}=o(2^{k})$. Letting

${\color[rgb]{1,0,0}a_{k}}=\frac{(2^{k-1}+2^{k})M}{2},$

we define $\tilde{f}$ as follows. First let $\tilde{f}\equiv\beta_{1}$ on
$(0,a_{1}]$. Then for those $k\geq 1$ with $\beta_{k}\geq\beta_{k+1}$, define
$\tilde{f}$ on $(a_{k},a_{k+1}]$ by

${\color[rgb]{1,0,0}\tilde{f}(t)}=\begin{cases}\beta_{k}&\text{if
}t\in(a_{k},2^{k}M]\\\ \beta_{k+1}&\text{if }t=a_{k+1}\\\ \text{linear on
}[2^{k}M,a_{k+1}].\end{cases}$

For $k\geq 1$ with $\beta_{k}<\beta_{k+1}$, define $\tilde{f}$ on
$(a_{k},a_{k+1}]$ by

${\color[rgb]{1,0,0}\tilde{f}(t)}=\begin{cases}\beta_{k+1}&\text{if
}t\in[2^{k}M,a_{k+1}]\\\ \beta_{k}&\text{if }t=a_{k}\\\ \text{linear on
}[a_{k},2^{k}M].\end{cases}$

Since $\beta_{k}=o(2^{k})$ it is easily seen that if we take $M$ sufficiently
large, then the slope of the piecewise–linear function $\tilde{f}$ is never
more than $\epsilon$, and the slope at $r$ approaches 0 as $r\to\infty$.
Defining $\tilde{\rho}(r)$ by

$\log{\color[rgb]{1,0,0}\tilde{\rho}(e^{r})}=\tilde{f}(r)+\chi r$

it follows that for all $s\geq r>1$,

$\frac{\log\tilde{\rho}(s)-\log\tilde{\rho}(r)}{\log s-\log
r}=\frac{\tilde{f}(\log s)-\tilde{f}(\log r)}{\log s-\log
r}+\chi\in[\chi-\epsilon,\chi+\epsilon],$

and the sublinearly powerlike property follows. ∎

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