# Modeling Early Clustering of Impact-induced Ejecta Particles Based on
Laboratory and Numerical Experiments

Kanon Nakazawa  Department of Earth and Planetary Science, Tokyo Institute of
Technology, Meguro, Tokyo 152-8551, Japan  Satoshi Okuzumi  Department of
Earth and Planetary Science, Tokyo Institute of Technology, Meguro, Tokyo
152-8551, Japan  Kosuke Kurosawa  Planetary Exploration Research Center, Chiba
Institute of Technology, Chiba 275-0016, Japan  Sunao Hasegawa  Institute of
Space and Astronautical Science, Japan Aerospace Exploration Agency, Kanagawa
252-5210, Japan Kanon Nakazawa<EMAIL_ADDRESS>
###### Abstract

A projectile impact onto a granular target produces an ejecta curtain with
heterogeneous material distribution. Understanding how the heterogeneous
pattern forms is potentially important for understanding how crater rays form.
Previous studies predicted that the pattern formation is induced by inelastic
collisions of ejecta particles in early stages of crater formation and is
terminated by the ejecta’s expanding motion. In this study, we test this
prediction based on a hyper-velocity impact experiment together with N-body
simulations where the trajectories of inelastically colliding granular
particles are calculated. Our laboratory experiment suggests that pattern
formation is already completed on a timescale comparable to the geometrical
expansion of the ejecta curtain, which is $\sim 10~{}{\rm\mu s}$ in our
experiment. Our simulations confirm the previous prediction that the
heterogeneous pattern grows through initial inelastic collisions of particle
clusters and subsequent geometric expansion with no further cluster
collisions. Furthermore, to better understand the two-stage evolution of the
mesh pattern, we construct a simple analytical model that assumes perfect
coalescence of particle clusters upon collision. The model shows that the
pattern formation is completed on the timescale of the system’s expansion
independently of the initial conditions. The model also reproduces the final
size of the clusters observed in our simulations as a function of the initial
conditions. It is known that particles in the target are ejected at lower
speeds with increased distance to the impact point. The difference in the
ejection speed of the particles may result in the evolution of the mesh
pattern into rays.

planets and satellites: surfaces — Moon

††software: REBOUND (Rein & Liu, 2012)

##  1 Introduction

Mutual collisions between planets, satellites, and minor bodies frequently
occur in the solar system. A hypervelocity impact of a small body onto a
larger body leads to the production of an excavation flow that eventually
opens a crater. The materials in the excavation flow launch from the outer
edge of a growing crater and form an inverted cone-like structure called an
ejecta curtain.

Impact craters are often accompanied by radial streaks of ejecta called crater
rays (e.g. Oberbeck, 1971, Melosh, 1989, Hawke et al., 2004). Crater rays
cause degradation of pre-existing craters, which may control crater
equilibrium found for small lunar craters (Minton et al., 2019). Understanding
how the morphology of crater rays is related to impact conditions may help us
better understand how the size frequency distribution of small craters on
planetary and lunar surfaces are determined.

While there are several studies characterizing crater rays on a variety of
planetary bodies (e.g. Trask & Guest, 1975, McEwen et al., 2005), only a few
studies have addressed the physical mechanisms responsible for crater ray
formation (e.g. Shuvalov, 2012, Kadono et al., 2015, 2020, Sabuwala et al.,
2018). Shuvalov (2012) propose that interaction of impact-induced shock waves
with pre-existing craters can produce crater rays.

More recently, Kadono et al. (2015, 2020) propose an alternative idea, which
we focus on in this study, that crater rays are formed by an enhanced contrast
in the particle number density in an ejecta curtain. They conducted impact
experiments onto granular targets and found that the resulting ejecta curtains
have mesh-like patterns. The observed mesh pattern simply expands with the
ejecta curtain and suggested that the pattern formation had already been
completed in an early stage of the ejecta curtain formation. They also
performed numerical calculations and suggested that the mesh patterns likely
result from inelastic collisions between granular particles. However, the
previous work did not address when exactly the pattern formation occurred in
the experiments and, more fundamentally, how the final size of the particle
clusters constituting the mesh pattern is related to the initial condition of
the ejecta curtain formation. A better constraint on the timing of pattern
formation is necessary for further investigating the processes of the pattern
formation through future experiments.

In this study, we focus on the very early phase of ejecta curtain formation
and study in detail how particle clustering proceeds and how it is terminated
in an expanding curtain. We present the results of a hyper-velocity impact
experiment where we observed the development of a mesh-like pattern in the
ejecta curtain on the timescale of the ejecta’s expansion (Section 2). We also
numerically simulate pattern formation of inelastically colliding granular
particles with expanding systematic velocities (Section 3). To interpret the
pattern formation observed in the simulations, we construct a simple
physically-based model for clustering in an expanding particle system (Section
4). We discuss our model’s validity and the evolution of the mesh pattern into
rays in Section 5. A conclusion is presented in section 6.

##  2 Laboratory Experiment

###  2.1 Method

The purpose of our impact experiment is to observe pattern formation in a
granular ejecta curtain immediately after an impact. Imaging the early phase
of crater formation using a video camera is generally hindered by self-
emission from high-temperature vapor plumes. To avoid this issue, we use a
monochromatic light source and a band pass filter, which enable us to image an
ejecta curtain as early as 10 $\rm\mu s$ after the hyper-velocity impact of a
millimeter-sized projectile. For comparison, Kadono et al. (2020) conducted
similar experiments but were only able to image the ejecta curtains $\sim
10~{}\rm ms$ after the impacts.

A schematic diagram of the experimental setup is shown in Figure 1. We used a
vertical two-stage light gas gun at Institute of Space and Astronautical
Science of Japan Aerospace Exploration Agency. A spherical polycarbonate
projectile with a diameter of 4.8 mm was accelerated and impacted on the
surface. We used zircon beads (FZB-400, the span of sizes is $30$ –
$63~{}\rm{\mu m}$) as granular targets. The median size of the beads was
$50~{}\rm{\mu m}$. The impact velocity was set to 6.4 km s-1. The experimental
chamber was evacuated down to 5 Pa prior to the shot. To cut off the
propellant gas mixing associated with projectile acceleration, we surrounded
the targets with a gas shield with only a narrow opening of several
centimeters for the projectile. A high-speed video camera (Shimadzu, HPV-X)
and a $640~{}\rm{nm}$ laser light source were used to observe the ejecta
curtain. The effects of self-luminous high-temperature plume on the curtain
observation was minimized by mounting a band-pass filter corresponding to the
laser wavelength in front of the camera lens. The spatial resolution was
$110~{}\rm{\mu m}$ per pixel. We obtained a total of 128 frames with a frame
rate of $1~{}\rm{\mu s}$. The frame rate is close to the characteristic time
for the projectile penetration. Thus, we could observe the curtain growth as
early as $\sim 10~{}{\rm\mu s}$ after the impact.

Figure 1: Illustration of the experimental setup.

###  2.2 Results

Figure 2: Snapshots showing the earliest phase of ejecta curtain formation
from the laboratory impact experiment. The projectile impacts onto the
granular target at $t=0$ (upper left panel). The red circles in the lower
panels show how a pattern on the ejecta curtain expands geometrically with
time.

Figure 2 shows the snapshots of the ejecta curtain within $100~{}\rm\mu s$
after the impact. The vertical and radial111We here adopt the cylindrical
coordinate system centered on the impact point. The radial direction refers to
the direction away from the vertical axis. velocities of the ejecta in this
early phase are estimated to be in the range of $\sim 100$–1000 $~{}\rm
m~{}s^{-1}$. Because the ejecta immediately after the impact have a radial
extent of $\sim 10~{}\rm mm$, they expand radially on a timescale of $\sim
10$–100 $\rm\mu s$. Below, we call the timescale defined in this way the
expansion timescale.

We find that the ejecta curtain exhibits a mesh-like pattern already at
$10~{}\rm{\mu s}$ after the impact (see the upper right panel of Figure 2).
Moreover, the snapshots at 76 and 91 $\rm\mu s$ after the impact show that the
pattern simply expands with the curtain (see the red circles in the lower
panels of Figure 2). Kadono et al. (2020) also observed the expansion of
ejecta curtain patterns in their similar experiments, but at $\sim 10~{}\rm
ms$ after impacts. Our experimental results presented here indicate that
pattern formation occurs and is completed much earlier, on a timescale
comparable to the expansion timescale of the ejecta curtain.

##  3 Numerical Simulations

In our laboratory experiment, we were unable to observe how the heterogeneous
pattern developed during the very early phase of the ejecta curtain formation.
To better understand this process, we carry out N-body simulations where we
calculate the motion and mutual inelastic collisions of granular particles in
an expanding curtain. We particularly focus on the evolution of the particles’
velocity dispersion and density pattern. To quantify the properties of the
particle pattern, we regard the pattern as a collection of particle clusters
and employ a cluster analysis.

###  3.1 Method

Our simulations use the open-source N-body code REBOUND (Rein & Liu, 2012). We
use IAS15 (Rein & Spiegel, 2015) as an integrator. We approximate a portion of
an ejecta curtain as a thin rectangular particle sheet and treat the motion of
the ejecta particles in this portion in a two-dimensional Cartesian reference
frame ($x,y$) comoving with the particles’ center of mass, with the $x$ and
$y$ directions corresponding to the directions parallel and perpendicular to
the circumference of the cone-like curtain, respectively. We neglect the
gravity in this comoving frame assuming that the particles’ center of mass
follows a parabolic trajectory under the gravity. Because the particles in the
laboratory frame move away from the impact point, the particle sheet expands
in the $y$ direction. Including this expansion motion is essential here as we
aim to see how the expansion of the curtain terminates the growth of particle
clusters at late times. In our simulations, we mimic the expanding motion by
adding a velocity proportional to $x$ to the $x$ component of the initial
velocity of each particle (see below). In reality, the particle sheet would
also expand in the direction perpendicular to the curtain’s circumference if
particles ejected earlier have higher ejection speeds (see Section 5.2 for its
potential implications for crater ray formation). For those cases, the $x$
axis in our simulations should be regarded as the axis along which the
particle system expands faster.

The particles are initially placed in a square area of side length $L_{0}$.
Before we set the initial velocities of the particles, we allow the particles
to repel elastically in a periodic box to remove initial particle overlap.
Each particle (labeled by the subscript $j$) is given an initial velocity
consisting of random and expansion components,

$(v_{jx},v_{jy})_{t=0}=(v_{\mathrm{rand},jx}+v_{\mathrm{exp},j},v_{\mathrm{rand},jy})~{},$
(1) $v_{{\rm exp},j}=\Omega x_{j}~{},$ (2)

where $t$ is time, $v_{{\rm rand},jx}$ and $v_{{\rm rand},jy}$ are random
numbers, $v_{{\rm exp},j}$ is the initial expansion velocity, and $\Omega$ is
the rate of expansion. With this initial condition, the particles expand in
the $x$ direction and also mutually collide nearly isotropically. The random
velocity components are sampled from a uniform distribution ranging from
$-v_{\rm rand,max}$ to $v_{\rm rand,max}$, where $v_{\rm rand,max}$ is a
constant. Whenever two particles collide, we decompose their velocities into
the center-of-mass and relative velocity components and change the latter as

$v^{\prime}_{\rm rel,n}=-ev_{\rm rel,n},\qquad v^{\prime}_{\rm rel,t}=v_{\rm
rel,t},$ (3)

where $v_{\rm rel,n}$ and $v_{\rm rel,t}$ are the normal and tangential
components of the relative velocity before the collision, respectively,
$v^{\prime}_{\rm rel,n}$ and $v^{\prime}_{\rm rel,t}$ are those after the
collision, and $e$ is the coefficient of restitution. We ignore the rotation
of the particles by assuming that the particles exert no torque on each other
upon collision. In our simulations, we set $e$ as a constant. We carry out 10
independent simulation runs with 10 different sets of initial particle random
velocities but with the same initial particle configuration.

Table 1: Parameters Adopted in the Simulations Parameter | Value
---|---
Total number of particles $N_{\rm tot}$ | 30000
Particle radius $r$ | 50 $\rm{\mu m}$
Expansion rate $\Omega$ | $0.02~{}\rm{\mu s^{-1}}$
Maximum random speed $v_{\rm rand,max}$ | $40~{}\rm{m~{}s^{-1}}$
Restitution coefficient $e$ | 0.6
Initial box size $L_{0}$ | 20 mm

The values of the parameters adopted in the simulations are listed in Table 1.
All particles in the simulations are assumed to be spherical and equal-sized,
with radius $r$ being equal to those of the zircon beads used in the
experiment presented in Section 2. Because we exert no external force on
individual particles, our simulation results are independent of the particle
mass. For this reason, we adopt the code units in which the particle mass is
taken to be unity. The expansion rate $\Omega$ is set to $0.02~{}\rm{\mu
s}^{-1}$ so that the expansion velocity of the system, $\Omega
L_{0}/2=200~{}\rm m~{}s^{-1}$, is comparable to the horizontal velocity of the
ejecta curtain pattern observed in our laboratory experiment. The maximum
random velocity is taken to be 20% of the system expansion velocity. The
coefficient of restitution of our zircon particles are unknown, so we
arbitrarily take the default value of $e$ to be 0.6 so that the pattern formed
is visible enough to analyze. The validity of our choice of $e$ is discussed
in Section 5. Kadono et al. (2015) show that particles of a lower restitution
coefficient produce a clearer particle pattern through inelastic collisions.
For completeness, we also present simulations for $e=0.1$, 0.9, and 1.0.

The time step $\Delta t$ must be chosen to be smaller than the crossing times
of the particles so that every particle collision can be detected. The
particle crossing time is estimated as $2r/v_{\rm rel}\gtrsim r/v_{\rm
rand,max}\approx 1.3~{}\rm\mu s$, where $v_{\rm rel}$ is the relative velocity
of the particles. We therefore set $\Delta t=0.15~{}\rm{\mu s}$.

###  3.2 Data Analysis

Particles lose their random kinetic energies through inelastic collisions.
Since the particles also form clusters through the collisions, the random
energy of the system and the size of the clusters should be correlated with
each other. Below we describe how we quantify the two quantities.

As a diagnostic for the random motion of the particles, we define the root-
mean-squared (RMS) particle velocity as

$v_{\rm{RMS}}=\langle(v_{jx}-v_{{\rm exp},j})^{2}+v_{jy}^{2}\rangle^{1/2}~{},$
(4)

where the brackets represent an average over all particles (again, each
particle is labeled by $j$). To reduce statistical errors, we also take the
average of $v_{\rm RMS}$ over 10 runs. The masses of particle clusters are
measured using a cluster analysis. In this analysis, all particles in a given
snapshot of simulations are grouped into clusters. We adopt hierarchical
clustering in which pairs of clusters (initially individual particles) are
combined successively. Specifically, we use Ward’s method (Joe & Ward, 1963),
a commonly used hierarchical clustering algorithm that combines clusters so
that the variance of the particle positions within the clusters is minimized.
However, this algorithm alone does not tell us the optimal (i.e., most likely)
number of clusters. To determine it, we introduce a quantity called the
silhouette coefficient. For each particle $j$, its silhouette coefficient
$s_{j}$ is defined as (Rousseeuw, 1987)

$s_{j}=\frac{b_{j}-a_{j}}{\mathrm{max}\\{a_{j},b_{j}\\}}~{},$ (5)

where $a_{j}$ and $b_{j}$ are the average distances between particle $j$ and
the other particles inside and outside the the cluster, respectively, to which
it belongs. The higher the average silhouette coefficient $\langle
s_{j}\rangle$ is, the better the clusters are separated. For each snapshot of
particle distribution, we pick up the set of clusters that give the highest
$\langle s_{j}\rangle$, and calculate the mean and standard deviation of their
masses. We also take averages of the means and standard deviations over 10
simulation runs. The cluster analysis we employ is more naturally suited to
our problem than Fourier analysis as used by Kadono et al. (2015), because our
particle system has an open, expanding boundary and therefore the cluster
distribution within it is nonuniform.

###  3.3 Results

Figure 3: Snapshots of the particle distribution at times $t=$ 0, 15, 30, 60,
90, and 120 $\rm{\mu s}$ (top left, top right, middle left, middle right,
bottom left, and bottom right panels, respectively) obtained from a simulation
run (see Table 1 for the adopted parameter set)

.

Figure 3 shows snapshots of the particle distribution obtained from a single
simulation run. Particle clustering is visible already at $t=15~{}\rm{\mu s}$
(upper right panel). At $t>30~{}\rm{\mu s}$, the cluster pattern simply
expands in the $x$ direction rather than growing further (middle right, bottom
left, and bottom right panels), consistent with the result of our laboratory
experiment. The pattern is mesh-like, demonstrating that inelastic collisions
between particles at early times induce the formation of the mesh pattern in
ejecta curtains observed in laboratory experiments including ours (see Section
2). Because of the unidirectional expansion, the particle clusters
constituting the mesh pattern are elongated along the $x$ axis. If the
particles had higher expansion velocities in the $y$ direction, the resulting
mesh pattern would be more elongated along the $y$ axis.

Figure 4: Time evolution of the RMS particle velocity $v_{\rm{RMS}}$ from the
simulations. The thin gray lines are from 10 individual runs, while the thick
black line shows the average over the 10 runs.

Figure 4 shows the RMS random velocity $v_{\rm{RMS}}$ (Equation 4) averaged
over the 10 runs as a function of $t$. We find that $v_{\rm RMS}$ decreases at
$t<20~{}\rm{\mu s}$ when the mesh pattern is growing, and approaches to a
constant at later times when the pattern is expanding. This serves as another
piece of evidence for particles’ inelastic collisions being the cause of
pattern formation.

Figure 5: Particle-averaged silhouette coefficient $\langle s_{j}\rangle$ as a
function of the number of clusters ${\cal N}$ at different times from a single
simulation run. The cross symbols indicate the optimal number of clusters, at
which $\langle s_{j}\rangle$ is maximized. Figure 6: Snapshots showing the
particle cluster distribution at $t=0$ and $60~{}\rm\mu s$ (panels (a) and
(b), respectively) obtained by applying the cluster analysis to the particle
distributions shown in the top left and middle right panels of Figure 3. The
number of clusters ${\cal N}$ at $t=0$ and $60~{}\rm\mu s$ are $10285$ and
$1195$, respectively.

To illustrate how we determine the optimal number of clusters from the cluster
analysis, we plot in Figure 5 the particle-averaged silhouette coefficient
$\langle s_{j}\rangle$ as a function of the number of clusters $\mathcal{N}$
at different times for a single run. Here, the values of $\langle
s_{j}\rangle$ are calculated from $\mathcal{N}$ = 1 to 30000 for every 100
increment in $\mathcal{N}$. At all times, $\langle s_{j}\rangle({\cal N})$ has
a single maximum, allowing us to uniquely define the optimal number of
clusters as the number of clusters ${\mathcal{N}}(\langle
s_{j}\rangle_{\rm{max}})$ that gives the maximum silhouette coefficient.
Figure 6 illustrates how our cluster analysis works for the particular example
shown in Section 3 at $t=0$ and 60 $\rm\mu s$.

Table 2: Properties of the Optimal Clusters at Different Times $t$ (Averaged Over 10 Runs) $t~{}(\rm\mu s)$ | ${\mathcal{N}}(\langle s_{j}\rangle_{\rm{max}})$ | $\overline{N}$ | $\delta N$
---|---|---|---
0 | 10285 | 2.9 | 1.0
30 | 1265 | 23.7 | 8.7
60 | 1195 | 25.1 | 10.8
90 | 1575 | 19.0 | 8.9
120 | 1800 | 16.7 | 8.3
Figure 7:  Mean and standard deviation of the cluster size at different times
$t$ (circles and error bars, respectively) obtained from the cluster analysis
of the N-body simulations.

Table 2 lists the optimal cluster number ${\mathcal{N}}(\langle
s_{j}\rangle_{\rm{max}})$ averaged over 10 simulation runs, the corresponding
mean cluster size $\overline{N}=N_{\rm tot}/{\cal N}(\langle
s_{j}\rangle_{\rm{max}})$, and the standard deviation $\delta N$ of the
cluster size averaged over the 10 runs. Here, we search for
${\mathcal{N}}(\langle s_{j}\rangle_{\rm{max}})$ for individual runs in two
steps, first obtaining a rough estimate of $\langle s_{j}\rangle_{\rm max}$ by
computing $\langle s_{j}\rangle$ from $\mathcal{N}$ = 1 to 30000 for every 100
increment in $\mathcal{N}$, and second obtaining a more precise value of
$\langle s_{j}\rangle_{\rm max}$ by computing $\langle s_{j}\rangle$ for every
5 increment in $\mathcal{N}$ in the vicinity of ${\mathcal{N}}(\langle
s_{j}\rangle_{\rm{max}})$ from the first step. Figure 7 plots $\overline{N}$
and $\delta N$ versus $t$ given in Table 2. One can see that $\overline{N}$
increases in the first $30~{}\rm\mu s$, when the RMS particle random velocity
decreases (see Figure 4), indicating that our cluster analysis well captures
pattern formation through inelastic collisions. At later times, the cluster
mass plateaus out, reflecting the collisionless expansion of the system. Close
inspection shows that the derived cluster mass gradually decreases after
$t=60~{}\rm{\mu s}$. We interpret this as suggesting that our cluster analysis
does not fully resolve the clusters at $t\lesssim 60~{}\rm\mu s$.

Figure 8: Snapshots of the particle distribution at a time $t=$ 60 $\rm{\mu
s}$ from simulation run with $e=$ 0.1, 0.6, 0.9 and 1.0 (top left, top right,
bottom left, and bottom right panels, respectively). The parameters other than
$e$ are fixed as listed in Table 1.

Figure 8 compares the particle distribution at $t=60~{}{\rm\mu s}$ from
simulations of different values of $e$. We see that particle clusters become
more appreciable as $e$ decreases, again supporting the idea that the pattern
formation is driven by inelastic collisions. The observed trend is also
qualitatively consistent with the simulation results by Kadono et al. (2015,
their Figure 6).

##  4 Analytic Model

Our simulations presented in the previous section confirm the expectation by
Kadono et al. (2015, 2020) that the mesh pattern in ejecta evolves in two
stages, initial growth through inelastic particle collisions and subsequent
geometric expansion with no particle collision. However, it is yet to be
addressed when the initial growth stage is terminated and what initial
conditions set the final size of the clusters constituting the pattern. To
address these questions, we here construct an analytic model for the growth of
particle clusters in an expanding particle system.

###  4.1 Model Description

As discussed in the previous section, it is clear that inelastic particle
collisions are the cause of the cluster formation seen in our simulations.
Because particles constituting clusters experience inelastically collide with
each other frequently, we can expect that colliding clusters lose their
initial collision energy quickly. Based on this consideration, our analytic
model assumes that clusters coalesce perfectly whenever they collide. As
discussed in Section 5, this assumption is valid if the coefficient of
restitution of the particles is sufficiently low.

For the moment, we also assume the random velocity component dominates over
the non-isotropic velocity component arising from the system’s expansion
motion. This assumption breaks down at late times where the expansion motion
terminates the collisional evolution of the system. We account for this effect
at the end of this subsection. To make our model as simple as possible, we
approximate all clusters as equal-sized spheres and only consider their one-
dimensional motion.

####  4.1.1 Relationship Between the Cluster Velocity Dispersion and Mass

The assumptions made above gives a simple relationship between the cluster
velocity dispersion and cluster mass. Let the velocities of two colliding
clusters 1 and 2 be $V_{1}$ and $V_{2}$ and the velocity of the merged cluster
be $V^{\prime}$. Using the conservation of momentum law and the assumption
that the clusters are approximately equal in size, we have

$V^{\prime}=\frac{1}{2}(V_{1}+V_{2}).$ (6)

The assumption that the isotropic, random motion dominates the clusters’
velocities gives

$\begin{cases}\overline{V_{1}}=\overline{V_{2}}=0~{},\\\
\overline{V_{1}^{2}}=\overline{V_{2}^{2}}=\sigma^{2}~{},\\\
\overline{V_{1}V_{2}}=0,\end{cases}$ (7)

where $\sigma$ is the velocity dispersion of the clusters and the overlines
denote ensemble averages over all clusters. Using Equations (6) and (7), the
velocity dispersion of the clusters after a single collision,
$\sigma^{\prime}\equiv(\overline{V^{\prime 2}})^{1/2}$, can be written as

$\sigma^{\prime}=\frac{1}{\sqrt{2}}\sigma~{}.$ (8)

Equation (8) shows that the velocity dispersion of the clusters decreases by a
factor of $1/\sqrt{2}$ after every merging collision. Because the cluster mass
increases by a factor of 2 after every collision, one can relate $\sigma$ to
the cluster mass as

$\sigma(\overline{N})=\sigma_{0}\left(\frac{N_{0}}{\overline{N}}\right)^{1/2},$
(9)

where $\overline{N}$ is the average number of particles per cluster (which is
equal to the mean cluster mass in units of the monomer mass) and $\sigma_{0}$
and $N_{0}$ are the initial values of $\sigma$ and $\overline{N}$,
respectively.

####  4.1.2 Time Evolution of the Cluster Mass

The time evolution of $\overline{N}$ through cluster mergers can be described
by

$\frac{d\overline{N}}{dt}=\frac{\overline{N}}{\tau}~{},$ (10)

where $\tau$ is the mean collision time. For spherical clusters moving on a
two-dimensional space, $\tau$ can be estimate as

$\tau=\frac{1}{4\overline{R}n\sigma},$ (11)

where $\overline{R}$ and $n$ are the average radius and surface number density
of the clusters (the factor $4\overline{R}$ represents the collisional cross
section for equal-sized disks on a plane). Assuming that each two-dimensional
cluster is densely packed, $\overline{R}$ is related to $\overline{N}$ and the
particle radius $r$ as $\overline{N}=\pi\overline{R}^{2}/(\pi r^{2})$, or

$\overline{R}=r\overline{N}^{1/2}.$ (12)

Since the number of clusters in the system is $\approx N_{\rm
tot}/\overline{N}$, the cluster surface number density $n$ can be estimated as

$n=\frac{N_{\rm tot}}{L_{V}L_{H}\overline{N}},$ (13)

where $L_{H}$ and $L_{V}$ are the horizontal and vertical length of the
system. We assume that the vertical system size is approximately constant
whereas the horizontal system size increases at constant expansion rate
$\Omega$,

$L_{H}=L_{H_{0}}(1+\Omega t),$ (14)

with $L_{H_{0}}$ being the initial value of $L_{H}$. Substituting Equations
(9) and (11)–(14) into Equation (10), we obtain

$\frac{d\overline{N}}{dt}=\frac{\dot{N}_{0}}{1+\Omega t}~{},$ (15)

where $\dot{N}_{0}$ is a constant defined by

$\dot{N}_{0}=\frac{4rN_{0}^{1/2}N_{\rm tot}\sigma_{0}}{L_{V}L_{H_{0}}}~{}.$
(16)

The solution to eq. (15) is

$\overline{N}(t)=N_{0}+\frac{\dot{N}_{0}}{\Omega}\ln(1+\Omega t),$ (17)

where $N_{0}$ is the initial value of $\overline{N}$. The set of Equations (9)
and (15) predicts $\sigma$ and $\overline{N}$ as a function of $t$ under the
assumption that the expansion motion of the system has negligible effect on
the cluster collision velocity.

####  4.1.3 “Freezeout” of the Cluster Pattern

For late times where the expansion motion of the system dominates over the
random motion of individual clusters, we expect that the clusters can no
longer grow collisionally. The cluster pattern then should “freeze out,” i.e.,
should simply expand self-similarly with time as observed in our experiment
and simulation. We expect the freezeout to occur when the collisional
timescale $\tau$ becomes longer than the expansion timescale
$\sim\Omega^{-1}$. We thus define the freezeout time $t_{\rm freeze}$ by

$\tau(t>t_{\rm freeze})>\frac{1}{\Omega}~{}.$ (18)

To account for the freezeout, we stop the evolution of $\sigma$ and
$\overline{N}$ stops at $t=t_{\rm freeze}$.

The model confirms the expectation from our laboratory expriment that the
cluster growth is completed on a timescale comparable to the expansion
timescale $1/\Omega$. By definition, the freezeout time satisfies
$\tau(t=t_{\rm freeze})=1/\Omega$. Using Equations (12)–(14), we rewrite
$\tau$ as

$\tau=\frac{L_{V}L_{H_{0}}(1+\Omega
t)}{4rN_{\rm{tot}}\sigma_{0}N_{0}^{1/2}}\overline{N}.$ (19)

For $\overline{N}\gg 1$, we have

$\overline{N}\approx\frac{\dot{N}_{0}}{\Omega}\ln(1+\Omega t),$ (20)

Substituting Equations (16) and (20) into Equation (19), we obtain

$\tau\approx\frac{(1+\Omega t)\ln(1+\Omega t)}{\Omega}.$ (21)

Therefore, the condition $\tau(t=t_{\rm freeze})=1/\Omega$ implies

$(1+\Omega t_{\rm{freeze}})\ln(1+\Omega t_{\rm{freeze}})\approx 1,$ (22)

from which we find $t_{\rm{freeze}}\approx 0.76/{\Omega}$. We note that the
freezeout time depends only on $\Omega$.

####  4.1.4 Final Cluster Mass and Velocity Dispersion as a Function of
Initial Conditions

From Equation (17) and $t_{\rm{freeze}}\approx 0.76/{\Omega}$, we can write
the final cluster size as a function of the initial cluster size and random
velocity dispersion as

$\overline{N}(t_{\rm freeze})\approx N_{0}+0.57\frac{\dot{N}_{0}}{\Omega}.$
(23)

Substituting Equation (23) into Equation (9), we obtain the final velocity
dispersion of the clusters as

$\sigma(t_{\mathrm{freeze}})\approx\sigma_{0}\left(\frac{N_{0}}{N_{0}+0.57\dot{N}_{0}/\Omega}\right)^{1/2}.$
(24)

###  4.2 Comparison with Simulations

Figure 9:  Cluster random velocity (left panel) and mean cluster size (right
panel) as a function of time $t$ from our simulation (solid lines; same as
Figures 4 and 7), compared with the prediction from our analytical model with
and without the freezeout of the cluster growth at $t>35~{}\rm{\mu s}$ (dashed
and dotted lines, respectively).

Now we test the analytic model derived above against our numerical
simulations. Figure 9 compare the cluster random velocity and mean cluster
size measured in the simulations with those predicted from the analytic model
(Equations (9) and (15), respectively) with and without the freezeout of the
clusters at $t>t_{\rm freeze}$. The parameters in the model are set to
$N_{0}=1$, $L_{V}=L_{H_{0}}=20~{}\rm mm$, and $\sigma_{0}=32.6~{}{\rm
m~{}s^{-1}}$. The value of $\sigma_{0}$ is taken to be close to the initial
RMS speed of the particles in the simulations. The assumed parameters give
$\dot{N}_{0}=0.49~{}\rm{\mu s^{-1}}$. The definition of the freezeout time
(Equation (18)) gives $t_{\rm{freeze}}=35~{}\rm{\mu s}$ for these parameter
choices.

We find that the analytic model neglecting the freezeout well reproduces the
evolution of $v_{\rm RMS}$ at least at early times $t\lesssim 30~{}\rm\mu s$.
This indicates that energy dissipation of particle clusters through perfect
coalescence explains the decrease of the random velocity. At $t\gtrsim
30~{}\rm\mu s$, the random velocity from the simulation decreases more slowly
than predicted from the model, suggesting that this model overestimates the
frequency of cluster collisions at late times.

The analytic model neglecting the freezeout reproduces the mean cluster size
in the simulations to within a factor of 2. The agreement is less good than
for the random velocity, in particular at $t\leq 60~{}\rm{\mu s}$, where
$\overline{N}$ from the simulation is considerably higher than predicted by
the model. This is likely because the clusters are so close that our cluster
analysis using the silhouette coefficient $s_{j}$ (Equation (5)) does not
fully resolve them. Density-based cluster analysis algorithms could mitigate
this issue, but would introduce additional free parameters (for example, the
DBSCAN algorithm (Ester et al., 1996) requires the detection radius and the
minimum number of cluster members). We defer further investigation of this
issue to future work. At $t\gtrsim 80~{}\rm\mu s$, the model predicts higher
$\overline{N}$ than observed in the simulation. This is another indication
that cluster collisions at late times are less frequent than expected by the
model.

Accounting for the freezeout of the clusters at $t>t_{\rm freeze}$
significantly improves the model predictions at late times (see the dotted
lines in Figure 9). Substituting the initial conditions of the simulation
(Table 1, $\sigma_{0}=32.6~{}\rm{m~{}s^{-1}}$) into Equations (23) and (24),
we obtain $\bar{N}(t_{\mathrm{freeze}})=14.9$,
$\sigma(t_{\mathrm{freeze}})=8.4~{}\mathrm{m~{}s^{-1}}$. These model
predictions are consistent with the final values from the simulation.

To summarize this section, we have shown that the initial growth of particle
clusters in an expanding particle system terminates when the collisional
timescale exceeds the expansion timescale $1/\Omega$. This occurs at $t_{\rm
freeze}\sim 1/\Omega$ after the clusters start growing. Our analytic model
quantitatively reproduces the evolution of the cluster mass and velocity
dispersion as a function of the initial conditions.

##  5 Discussion

###  5.1 Validity and Limitations of the Model

In this section, we discuss the validity and limitations of our analytic model
presented in Section 4.

Figure 10: Cluster random velocity as a function of time $t$ from simulations
for different values of the coefficient of restitution (solid lines), compared
with the prediction from the analytic model (dashed line, same as in the left
panel of Figure 9).

The analytic model assumes perfect coalescence of the clusters. It is natural
to expect that the model would not be applicable to granular materials with a
high coefficient of restitution $e$. To quantify the range of $e$ where the
model ramains valid, we reran simulations presented in Section 3 but adopting
different values of $e$. Figure 10 shows the time evolution of the random
velocity obtained from these simulations, compared with the prediction from
the model including the cluster freezeout. We find that the the simulation
results for $e<0.8$ deviate from the model prediction, suggesting that the
assumption of perfect coalescence is only valid for $e\lesssim 0.8$.

The next question then is whether the values of $e$ of real granular materials
falls into the range where our model remains valid. Measurements show that
$e=0.97$ for 3 mm glass beads and $e=0.87$ for 6 mm cellulose Foerster et al.
(1994), indicating that perfect cluster coalescence may not be a good
approximation for these millimeter-sized particles. However, for smaller
particles, cohesive forces can lead to a smaller coefficient of restitution
(Royer et al., 2009). There is also experimental evidence that collision
velocities exceeding $10~{}\rm m~{}s^{-1}$ cause a reduction of $e$ to below
0.9 (Labous et al., 1997). Therefore, we expect that the assumption of perfect
cluster coalescence is valid for a high-velocity impact onto a target made of
submillimeter-sized granules.

Another strong assumption in our model is that all ejecta particles are equal-
sized. Kadono et al. (2019) report that a target made of different sized
granules produces ejecta curtains with filament patterns. Extending our
analytic model to polydisperse granular targets will be the subject of future
work.

###  5.2 Evolution of the Mesh Pattern to Crater Rays

In this section, we discuss a possible evolution path from the mesh-like
pattern of an ejecta curtain to crater-rays frequently observed around flesh
craters.

Crater rays on planetary bodies have two important features: (1) the rays’
azimuthal pattern has a characteristic angle scale (Kadono et al., 2015); (2)
the sediments consisting of the rays are stretched in the radial direction
from the host craters’ center (e.g., McEwen et al., 2005; Kadono et al.,
2015). As already discussed by Kadono et al. (2015) and also demonstrated in
our experiment and simulations, mesh pattern formation through inelastic
particle collisions followed by the pattern’s geometric expansion naturally
explains the first feature of the crater rays.

Figure 11: Illustration of a possible evolution path from the mesh pattern to
the observed ray. The mesh-like pattern formed by the inelastic collision is
stretched in the radial direction by the velocity contrast according to the
distance from the impact point, resulting in a crater ray-like pattern.

The second feature may also be explained if ejecta particles launched earlier
have higher launch speeds. As we mentioned in Section 3.3, if the expansion
associated with this velocity difference dominates over the expansion of the
curtain’s circumference, the particle mesh pattern should be stretched
vertically. Indeed, it is known that the launch speed of the ejecta particles
decreases with increasing distance from the impact point (e.g., Housen et al.,
1983). Therefore, we can envision the scenario in which the mesh pattern
forming in an ejecta curtain through initial particle clustering evolves into
a vertically elongated, crater ray-like pattern by the vertical expanding
motion as schematically illustrated in Figure 11. In our experiment, the
pattern evolution envisioned above was not observed due to the limited time
window of the high-speed imaging. On the other hand, such pattern evolution
was observed in the previous impact experiments conducted by Kadono et al.
(2015).

In the case of natural impacts occurring under low-gravity environments, such
as the lunar surface, the mesh pattern is more likely to evolve into rays than
in laboratory environments under the Earth’s gravity due to longer flight
times. Nevertheless, not all observed craters have rays around them. We expect
that this is because of degradation. On the Moon, the lifetime of rays is
estimated to be 1 Gyr, and most of the formation ages of lunar craters are
older than 1 Ga (McEwen et al., 1997). Therefore, rays are often not observed
in actual craters.

##  6 Summary and Conclusions

We have conducted laboratory and numerical experiments to study pattern
formation and evolution in an early phase of ejecta curtain formation. Our key
findings are summarized as follows.

  1. 1.

In our hyper-velocity impact experiments using a 10 mm-sized projectile, the
ejecta curtain already exhibited mesh-like patterns at $10~{}\rm{\mu s}$ after
the impact (Figure 2). This timescale is comparable to the timescale of the
ejecta’s expansion, suggesting that pattern formation is completed within the
expansion timescale.

  2. 2.

In the numerical simulations, we employed a cluster analysis to quantify
cluster formation in an expanding particle system. At early times, we find
that the particle random velocity decays and the mean cluster size increases
with time, confirming the prediction by Kadono et al. (2015) that the pattern
formation (particle clustering) is induced by inelastic particle collisions.
At later times, the cluster pattern simply expands with time, consistent with
the results of the previous laboratory experiments and ours.

  3. 3.

We have constructed an analytic model for pattern formation that assumes
perfect coalescence of clusters at early times and collisionless geometric
expansion of the clusters at later times (Section 4.1). The model confirms the
expectation from our laboratory experiment that the initial cluster growth
stage is completed on a timescale comparable to the expansion timescale of the
particle system (Section 4.1.3). Furthermore, the model reasonably reproduces
the final mass of the clusters seen in our simulations as a function of
initial conditions (Figure 9). Our model is valid as long as the coefficient
of restitution of the particles is 0.8 or smaller (Figure 10).

If the ejecta particles have a large positive vertical velocity gradient, the
mesh pattern forming through inelastic particle collisions would evolve into a
vertically elongated pattern. This may eventually form radially extended
crater rays upon deposition (Section 5.2 and Figure 11). The analytic model
presented in this study will allow us to better understand the pattern
formation in this process. However, our current model is limited to targets
made of monodisperse particles. Extension of the model to polydisperse
granular targets should be done in future work.

We thank Takanori Iwasawa for discussions that motivated this work. We
appreciate two anonymous referees for their careful reviews that helped
greatly improve the manuscript, and Dr. Edgard G. Rivera-Valentin for handling
the manuscript as an editor. This work was supported by JSPS KAKENHI Grant
Numbers JP20H01948, JP20H00182, JP19K03926, JP19K03941, JP18H05438, 17K18812
and by the Hypervelocity Impact Facility (former facility name: the Space
Plasma Laboratory), ISAS, JAXA.

## References

  * Ester et al. (1996) Ester, M., Kriegel, H.-P., Sander, J., & Xu, X. 1996, in Proceedings of the Second International Conference on Knowledge Discovery and Data Mining, KDD’96 (AAAI Press), 226â231
  * Foerster et al. (1994) Foerster, S. F., Louge, M. Y., Chang, H., & Allia, K. 1994, Physics of Fluids, 6, 1108, doi: 10.1063/1.868282
  * Hawke et al. (2004) Hawke, B., Blewett, D., Lucey, P., et al. 2004, Icarus, 170, 1 , doi: https://doi.org/10.1016/j.icarus.2004.02.013
  * Housen et al. (1983) Housen, K. R., Schmidt, R. M., & Holsapple, K. A. 1983, J. Geophys. Res., 88, 2485, doi: 10.1029/JB088iB03p02485
  * Joe & Ward (1963) Joe, H., & Ward, J. 1963, Journal of the American Statistical Association, 58, 236, doi: 10.1080/01621459.1963.10500845
  * Kadono et al. (2019) Kadono, T., Suetsugu, R., Arakawa, D., et al. 2019, The Astrophysical Journal, 880, L30, doi: 10.3847/2041-8213/ab303f
  * Kadono et al. (2020) Kadono, T., Suzuki, A. I., Matsumura, R., et al. 2020, Icarus, 339, 113590, doi: https://doi.org/10.1016/j.icarus.2019.113590
  * Kadono et al. (2015) Kadono, T., Suzuki, A., Wada, K., et al. 2015, Icarus, 250, 215 , doi: https://doi.org/10.1016/j.icarus.2014.11.030
  * Labous et al. (1997) Labous, L., Rosato, A. D., & Dave, R. N. 1997, Phys. Rev. E, 56, 5717, doi: 10.1103/PhysRevE.56.5717
  * McEwen et al. (1997) McEwen, A. S., Moore, J. M., & Shoemaker, E. M. 1997, Journal of Geophysical Research: Planets, 102, 9231, doi: https://doi.org/10.1029/97JE00114
  * McEwen et al. (2005) McEwen, A. S., Preblich, B. S., Turtle, E. P., et al. 2005, Icarus, 176, 351, doi: 10.1016/j.icarus.2005.02.009
  * Melosh (1989) Melosh, H. J. 1989, Impact cratering : a geologic process
  * Minton et al. (2019) Minton, D. A., Fassett, C. I., Hirabayashi, M., Howl, B. A., & Richardson, J. E. 2019, Icarus, 326, 63, doi: 10.1016/j.icarus.2019.02.021
  * Oberbeck (1971) Oberbeck, V. R. 1971, Moon, 2, 263, doi: 10.1007/BF00561880
  * Rein & Liu (2012) Rein, H., & Liu, S. F. 2012, A&A, 537, A128, doi: 10.1051/0004-6361/201118085
  * Rein & Spiegel (2015) Rein, H., & Spiegel, D. S. 2015, MNRAS, 446, 1424, doi: 10.1093/mnras/stu2164
  * Rousseeuw (1987) Rousseeuw, P. J. 1987, Journal of Computational and Applied Mathematics, 20, 53 , doi: https://doi.org/10.1016/0377-0427(87)90125-7
  * Royer et al. (2009) Royer, J., Evans, D., Oyarte GÃ¡lvez, L., et al. 2009, Nature, 459, 1110, doi: 10.1038/nature08115
  * Sabuwala et al. (2018) Sabuwala, T., Butcher, C., Gioia, G., & Chakraborty, P. 2018, Phys. Rev. Lett., 120, 264501, doi: 10.1103/PhysRevLett.120.264501
  * Shuvalov (2012) Shuvalov, V. 2012, Meteoritics and Planetary Science, 47, 262, doi: 10.1111/j.1945-5100.2011.01324.x
  * Trask & Guest (1975) Trask, N. J., & Guest, J. E. 1975, J. Geophys. Res., 80, 2461, doi: 10.1029/JB080i017p02461


# On collectively L-weakly compact sets of operators

Eduard Emelyanov1
$1.$ Sobolev Institute of Mathematics, Novosibirsk, 630090, Russia

###### Abstract

A set of bounded linear operators from a Banach space to a Banach lattice is
collectively L-weakly compact whenever union of images of the unit ball is
L-weakly compact. We extend the Meyer-Nieberg duality theorem to collectively
L-weakly compact sets of operators, study relations between these sets and
collectively almost limited sets, and discuss the domination problem for
collectively compact sets and collectively L-weakly compact sets.

Keywords: L-weakly compact set, almost limited set, Banach lattice, domination
problem

MSC 2020: 46B50, 47B01, 47B07, 47B65

##  1 Introduction

The theory of L-weakly compact sets and operators was developed by P. Meyer-
Nieberg [8]. Several versions of this theory were studied recently by many
authors (see [2, 6] and references therein). We extend the Meyer-Nieberg
duality theorem [8, Satz 3] and Theorem 2.6 of [5] to collectively L-weakly
compact sets of operators, and shortly discuss the domination problem for
collectively compact and collectively L-weakly compact sets.

Throughout this paper vector spaces are real; operators are linear; letters
$X$ and $Y$ stand for Banach spaces; $E$ and $F$ for Banach lattices; $B_{X}$
for the closed unit ball of $X$; $I_{X}$ for the identity operator on $X$;
$\text{\rm L}(X,Y)$ (resp., $\text{\rm K}(X,Y)$, $\text{\rm W}(X,Y)$,
$\text{\rm Lim}(X,Y)$, $\text{\rm aLim}(X,Y)$) for the space of bounded
(resp., compact, weakly compact, limited, almost limited) operators from $X$
to $Y$;

$\text{\rm
absco}(A)=\Big{\\{}\sum\limits_{k=1}^{m}\alpha_{k}a_{k}:\sum\limits_{k=1}^{m}|\alpha_{k}|\leq
1\ \ \text{\rm and}\ \ a_{1},...,a_{m}\in A\Big{\\}}$

for the absolute convex hull of $A\subseteq X$; and $\text{\rm
sol}(A)=\bigcup\limits_{a\in A}[-|a|,|a|]$ for the solid hull of $A\subseteq
E$. We denote the uniform convergence of a sequence of functions $(g_{n})$ to
$g$ on a set $A$ by $g_{n}\rightrightarrows g(A)$.

A bounded subset $A$ of $Y$ is limited whenever $f_{n}\rightrightarrows 0(A)$
for every w∗-null sequence $(f_{n})$ in $Y^{\prime}$. J. Bourgain and J.
Diestel initiated the study of operators carrying bounded sets to limited ones
in [4], such operators are called limited. A decade ago J. X. Chen, Z. L.
Chen, and G. X. Ji introduced an almost limited set in a Banach lattice [5,
Definition 2.3], which is a disjoint version of the limited set, namely a
bounded $A\subseteq F$ is almost limited whenever $f_{n}\rightrightarrows
0(A)$ for every disjoint w∗-null sequence $(f_{n})$ in $F^{\prime}$.

The central notion of the present paper is due to P. Meyer-Nieberg [8,
pp.146-147]: a bounded $A\subseteq F$ is called L-weakly compact (briefly, is
an Lwc-set) whenever each disjoint sequence in $\text{\rm sol}(A)$ is norm-
null; an operator $T\in\text{\rm L}(X,F)$ is said to be L-weakly compact
(briefly, $T\in\text{\rm Lwc}(X,F)$) whenever $T(B_{X})$ is L-weakly compact.

Following P. M. Anselone and T. W. Palmer [3], we call a subset ${\cal T}$ of
$\text{\rm L}(X,Y)$ collectively (weakly) compact (collectively (almost)
limited, collectively Lwc) if the set ${\cal T}B_{X}=\bigcup\limits_{T\in{\cal
T}}T(B_{X})$ is relatively (weakly) compact (resp., (almost) limited, Lwc).
For convenience, we use bold letters to denote classes of collectively ${\cal
P}$-sets of operators acting from $X$ to $Y$, e.g., $\text{\bf K}(X,Y)$,
$\text{\bf W}(X,Y)$, $\text{\bf Lim}(X,Y)$, $\text{\bf aLim}(X,F)$. Note that
the collectively bounded sets of operators of $\text{\bf L}(X,Y)$ are exactly
the uniformly bounded sets.

More generally, collectively ${\cal C-P}$ sets of operators are defined for a
property ${\cal C}$ of subsets of $X$ and a property ${\cal C}$ of subsets of
$Y$ by saying that a subset ${\cal T}$ of $\text{\bf L}(X,Y)$ is collectively
${\cal C}-{\cal P}$ whenever ${\cal T}(C)=\bigcup\limits_{T\in{\cal T}}T(C)$
has the property ${\cal P}$ for each set $C$ with the property ${\cal C}$. For
example: collectively Gelfand – Phillips (resp., Bourgain – Diestel,
collectively limitedly L-weakly compact) sets $\text{\bf GP}(X,Y)$ (resp.,
$\text{\bf BD}(X,Y)$, $\text{\bf l-Lwc}(X,F)$) correspond to the property
${\cal C}$ to be limited and the property ${\cal P}$ to be relatively compact
(resp., relatively weakly compact, Lwc); collectively order bounded sets
$\text{\bf L}_{\text{\bf ob}}(E,F)$ correspond to the properties ${\cal C}$
and ${\cal P}$ to be order bounded; etc., (cf. [6, 2]). Clearly, each
collectively ${\cal C}-{\cal P}$ set consists of ${\cal C}-{\cal P}$
operators, for example, $T\in{\cal T}\in\text{\bf l-Lwc}(X,F)\Longrightarrow
T\in l-\text{\rm Lwc}(X,F)$.

For further unexplained notation and terminology, we refer to [1, 4, 5, 7, 9].

##  2 Main results

We need several technical lemmas, some of them might be already known. Since
we did not find appropriate references for them, we include their proofs.

######  Lemma 1.

The strong closure of the absolute convex hull of a collectively limited
(collectively almost limited, collectively weakly compact) set is collectively
limited (resp., collectively almost limited, collectively weakly compact).

###### Proof.

Let ${\cal T}\in\text{\bf Lim}(X,Y)$. Since $\overline{{\cal
T}}^{s}B_{X}\subseteq\overline{{\cal T}B_{X}}$, the strong closure
$\overline{{\cal T}}^{s}$ of ${\cal T}$ is collectively limited by [4,
Proposition]. The collectively almost limited case is similar, and the proof
for collectively weakly compact sets is a straightforward modification of the
proof of [3, Proposition 2.1]. ∎

The next lemma is a minor extension of [9, Corollary 3.6.4 ii)].

######  Lemma 2.

The absolute convex hull of every Lwc set is an Lwc set.

###### Proof.

Let $A$ be an Lwc set in $E$. By [1, Theorem 5.63], for proving that
$\text{\rm absco}(A)$ is an Lwc set it suffices to show that each disjoint
sequence in $B_{E^{\prime}}$ is uniformly null on $\text{\rm absco}(A)$. So,
let $(f_{n})$ be disjoint in $B_{E^{\prime}}$, and let $\varepsilon>0$.
Applying [1, Theorem 5.63], we have an $n_{\varepsilon}$ such that
$\sup\limits_{a\in A}|f_{n}(a)|\leq\varepsilon$ for all $n\geq
n_{\varepsilon}$. Take an arbitrary $x\in\text{\rm absco}(A)$, say
$x=\sum\limits_{k=1}^{m}\alpha_{k}a_{k}$, where
$\sum\limits_{k=1}^{m}|\alpha_{k}|\leq 1$ and $a_{1},...,a_{m}\in A$. Then

$|f_{n}(a)|=\Big{|}\sum\limits_{k=1}^{m}\alpha_{k}f_{n}(a_{k})\Big{|}\leq\sum\limits_{k=1}^{m}|\alpha_{k}||f_{n}(a_{k})|\leq\varepsilon\
\ \ \ \ (n\geq n_{\varepsilon}),$

and hence $f_{n}\rightrightarrows 0(\text{\rm absco}(A))$ as desired. ∎

######  Lemma 3.

The norm closure of every Lwc set is likewise an Lwc set.

###### Proof.

Let $A$ be an Lwc set in $E$, and let $(\bar{x}_{n})$ be a disjoint sequence
in the solid hull $\text{\rm sol}(\bar{A})$ of the norm closure $\bar{A}$ of
$A$. Suppose, in contrary, $(\bar{x}_{n})$ is not norm null. WLOG,
$\|\bar{x}_{n}\|\geq\varepsilon$ for some $\varepsilon>0$ and all $n$. For
every $n$, pick $\bar{a}_{n}\in\bar{A}$ such that
$|\bar{x}_{n}|\in[0,|\bar{a}_{n}|]$ and choose $a_{n}\in A$ satisfying
$\|\bar{a}_{n}-a_{n}\|\leq 2^{-n}$. Then, for all $n$,

$\||\bar{x}_{n}|\wedge|\bar{a}_{n}|\|-\||\bar{x}_{n}|\wedge|a_{n}|\|\leq\||\bar{x}_{n}|\wedge|\bar{a}_{n}|-|\bar{x}_{n}|\wedge|a_{n}|\|\leq\||\bar{a}_{n}|-|a_{n}|\|\leq\|\bar{a}_{n}-a_{n}\|\leq
2^{-n}.$

Since $(|\bar{x}_{n}|\wedge|a_{n}|)$ is disjoint in $\text{\rm sol}(A)$ then
$\||\bar{x}_{n}|\wedge|a_{n}|\|\to 0$ which is absurd as

$\||\bar{x}_{n}|\wedge|a_{n}|\|+2^{-n}\geq\||\bar{x}_{n}|\wedge|\bar{a}_{n}|\|=\||\bar{x}_{n}|\|=\|\bar{x}_{n}\|\geq\varepsilon$

for all $n$. The obtained contradiction completes the proof. ∎

Now, we are in the position to prove an Lwc-version of [3, Proposition 2.1].

######  Proposition 1.

The strong closure of absolute convex hull of a collectively Lwc set is
likewise collectively Lwc.

###### Proof.

Let ${\cal T}\in\text{\bf Lwc}(X,F)$. Since $\overline{\text{\rm absco}({\cal
T})}^{s}B_{X}\subseteq\overline{\text{\rm absco}({\cal T})B_{X}}$,
$\overline{\text{\rm absco}({\cal T})}^{s}$ is collectively Lwc by Lemmas 2
and 3. ∎

The following result provides collectively (almost) limited, weakly compact,
and Lwc versions of [3, Propositions 2.2]. It involves integrals of operator
valued functions. Suppose $K_{\xi}(t)\in\text{\rm L}(X,F)$ for
$t\in[a,b]\subset\mathbb{R}$ and $\xi$ in an index set $\Xi$. Below, under the
integral $\int_{a}^{b}K_{\xi}(t)dt$ we understand the strong or norm limit of
the approximating sums $\sum_{k=1}^{m}K_{\xi}(t_{k}^{\prime})(t_{k}-t_{k-1})$.

######  Proposition 2.

The following statements hold:

  1. (i)

If $\\{K_{\xi}(t):\xi\in\Xi,a\leq t\leq b\\}$ is collectively (almost) limited
(collectively weakly compact) in $\text{\rm L}(X,Y)$ then
$\Big{\\{}\int_{a}^{b}K_{\xi}(t)dt:\xi\in\Xi\Big{\\}}$ is likewise
collectively (almost) limited (resp., collectively weakly compact).

  2. (ii)

If $\\{K_{\xi}(t):\xi\in\Xi,a\leq t\leq b\\}$ is a collectively Lwc subset of
$\text{\rm L}(X,F)$ then
$\Big{\\{}\int_{a}^{b}K_{\xi}(t)dt:\xi\in\Xi\Big{\\}}$ is collectively Lwc.

###### Proof.

(i) follows from Lemma 1, whereas (ii) is a consequence of Proposition 1. ∎

Each element of a collectively (almost) limited (collectively weakly compact,
collectively Lwc) set is an (almost) limited (resp., weakly compact, Lwc)
operator. The converse fails as can be seen by considering the set of all
normalized rank one operators in $\ell^{1}$. However, similarly to [3, Theorem
2.5] we have:

######  Theorem 1.

Every totally bounded set of (almost) limited (weakly compact, Lwc) operators
is collectively (almost) limited (resp., collectively weakly compact,
collectively Lwc).

###### Proof.

We include the proof only for Lwc operators. Remaining cases are left for the
reader. So, let ${\cal T}\subseteq\text{\rm Lwc}(X,F)$ be totally bounded. By
the equivalence $i)\Longleftrightarrow iii)$ of [9, Corollary 3.6.4], we
obtain that ${\cal T}B_{X}$ is an Lwc set. Now, the applying of Proposition 1
completes the proof. ∎

The converse of Theorem 1 is false. The following two examples are motivated
by [3, Example 2.6].

######  Example 1.

Let $T_{n}\in\text{\rm L}(\ell^{1})$ be defined by $T_{n}x=x_{n}e_{1}$, where
$x=\sum\limits_{k=1}^{\infty}x_{k}e_{k}\in\ell^{1}$ and $e_{k}$ is the k-th
standard unit vector in $\ell^{1}$. Clearly, the set
$\\{T_{n}\\}_{n=1}^{\infty}$ is collectively Lwc (compact, weakly compact, and
limited). But $\\{T_{n}\\}_{n=1}^{\infty}$ is not totally bounded, for
$\|T_{n}-T_{m}\|=2$ with $n\neq m$.

The adjoint set $\\{T^{\prime}_{n}\\}_{n=1}^{\infty}\subset\text{\rm
L}(\ell^{\infty})$ is collectively limited, yet not collectively compact.
Indeed,

$(T^{\prime}_{n}f)x=f(T_{n}x)=f(x_{n}e_{1})=x_{n}f(e_{1})=x_{n}f_{1}\ \ \ \ \
(f\in\ell^{\infty},x\in\ell^{1}),$

and hence $T^{\prime}_{n}f=f_{1}e_{n}$. Thus,
$\bigcup\limits_{n\in\mathbb{N}}T^{\prime}_{n}\bigl{(}B_{\ell^{\infty}}\bigl{)}=\bigcup\limits_{n\in\mathbb{N}}[-e_{n},e_{n}]\subset
c_{0}$. Therefore, $\\{T^{\prime}_{n}\\}_{n=1}^{\infty}$ is collectively
limited due to Phillip’s lemma (cf. [1, Theorem 4.67]). Since $e_{n}\in
T^{\prime}_{n}B_{\ell^{\infty}}$ for every $n$, the set
$\\{T^{\prime}_{n}\\}_{n=1}^{\infty}$ is not collectively compact.

Note that, since the set $\\{T_{n}\\}_{n=1}^{\infty}$ is collectively Lwc then
$\\{T^{\prime}_{n}\\}_{n=1}^{\infty}$ is collectively Mwc by Theorem 2. Thus,
$\\{T^{\prime}_{n}\\}_{n=1}^{\infty}$ is collectively weakly compact by
Corollary 1.

Now, let $S_{n}\in\text{\rm L}(c_{0})$ be defined similarly by
$S_{n}x=x_{n}e_{1}$. Again, $\\{S_{n}\\}_{n=1}^{\infty}$ is collectively
compact. However, in this case
$\\{S^{\prime}_{n}\\}_{n=1}^{\infty}\subset\text{\rm L}(\ell^{1})$ is not
collectively weakly compact, since
$\\{e_{n}\\}_{n=1}^{\infty}\subset\bigcup\limits_{n\in\mathbb{N}}[-e_{n},e_{n}]=\bigcup\limits_{n\in\mathbb{N}}S^{\prime}_{n}\bigl{(}B_{\ell^{1}}\bigl{)}$
and the subset $\\{e_{n}\\}_{n=1}^{\infty}$ of $\ell^{1}$ is not relatively
weakly compact.

######  Example 2.

Let $\dim(F^{a})=\infty$, where $F^{a}$ is the order continuous part of $F$.
Take a disjoint normalized sequence $(u_{n})$ in $F^{a}_{+}$ and define rank
one operators $T_{n}:\ell^{\infty}\to F$ by $T_{n}x=x_{n}u_{n}$. Every $T_{n}$
is an Lwc operator since $T_{n}(\ell^{\infty})\subseteq F^{a}$. However,
${\cal T}=\\{T_{n}\\}_{n=1}^{\infty}\notin\text{\bf Lwc}(\ell^{\infty},F)$.
Indeed, for the bounded set
$A=\\{e_{n}\\}_{n=1}^{\infty}\subset\ell^{\infty}$, the set ${\cal
T}A=\\{0\\}\cup\\{u_{n}\\}_{n=1}^{\infty}$ is not Lwc.

By the Meyer-Nieberg result [8, Satz 3], the L- and M-weakly compact operators
are in duality to each other (cf. [9, Proposition 3.6.11], [1, Theorem 5.64]).
We need the following collective analogue of the notion of Mwc operators.

######  Definition 1.

A subset ${\cal T}$ of $\text{\rm L}(E,Y)$ is said to be collectively M-weakly
compact (briefly, collectively Mwc) if
$\lim\limits_{n\to\infty}\sup\limits_{T\in{\cal T}}\|Tx_{n}\|=0$ holds for
every norm bounded disjoint sequence $(x_{n})$ of $E$.

The following result is a generalization of the Meyer-Nieberg duality theorem
to collectively L(M)-weakly compact sets.

######  Theorem 2.

The following statements hold:

  1. (i)

A subset ${\cal T}$ of $\text{\rm L}(X,F)$ is collectively Lwc iff its adjoint
${\cal T}^{\prime}=\\{T^{\prime}:T\in{\cal T}\\}$ is collectively Mwc.

  2. (ii)

A subset ${\cal T}$ of $\text{\rm L}(E,Y)$ is collectively Mwc iff its adjoint
is collectively Lwc.

###### Proof.

(i) Let ${\cal T}\subseteq\text{\rm L}(X,F)$. Then ${\cal T}$ is collectively
Lwc iff every disjoint sequence of $\text{\rm
sol}\left(\bigcup\limits_{T\in{\cal T}}T(B_{X})\right)$ is uniformly null on
$B_{F^{\prime}}$ iff $f_{n}\rightrightarrows 0\left(\bigcup\limits_{T\in{\cal
T}}T(B_{X})\right)$ for every disjoint $(f_{n})$ in $B_{F^{\prime}}$ by [1,
Theorem 5.63] iff $\lim\limits_{n\to\infty}\sup\limits_{T\in{\cal
T}}\|T^{\prime}f_{n}\|=0$ (or, equivalently,
$\lim\limits_{n\to\infty}\sup\limits_{S\in{\cal T}^{\prime}}\|Sf_{n}\|=0$)
holds for every disjoint $(f_{n})$ of $B_{F^{\prime}}$ iff ${\cal T}^{\prime}$
is collectively Mwc.

(ii) Let ${\cal T}\subseteq\text{\rm L}(E,Y)$. ${\cal T}$ is collectively Mwc
iff $\lim\limits_{n\to\infty}\sup\limits_{T\in{\cal T}}\|Tx_{n}\|=0$ for every
disjoint $(x_{n})$ in $B_{E}$ iff $x_{n}\rightrightarrows
0\left(\bigcup\limits_{T\in{\cal T}}T^{\prime}(B_{Y^{\prime}})\right)$ for
every disjoint $(x_{n})$ in $B_{E}$, and by [1, Theorem 5.63], iff every
disjoint sequence of $\text{\rm sol}\left(\bigcup\limits_{T\in{\cal
T}}T^{\prime}(B_{Y^{\prime}})\right)$ is norm null, iff ${\cal T}^{\prime}$ is
collectively Lwc. ∎

The following corollary generalizes another Meyer-Nieberg result (cf. [1,
Theorem 5.61]) to collectively L(M)-weakly compact sets.

######  Corollary 1.

Every collectively Lwc (Mwc) set is collectively weakly compact.

###### Proof.

It is a consequence of [9, Proposition 3.6.5] for collectively Lwc sets. Now,
for collectively Mwc sets, the result follows from Theorem 2. ∎

Next, we present a collective version of [5, Theorem 2.6].

######  Theorem 3.

For every $X$ and $F$, $\text{\bf Lwc}(X,F)\subseteq\text{\bf aLim}(X,F)$.
Furthermore, the following conditions are equivalent.

  1. (i)

$F$ has order continuous norm.

  2. (ii)

For every $X$, $\text{\bf aLim}(X,F)\subseteq\text{\bf Lwc}(X,F)$.

  3. (iii)

$\text{\bf aLim}(F)\subseteq\text{\bf Lwc}(F)$.

###### Proof.

Let ${\cal T}$ be a collectively Lwc subset of $\text{\rm L}(X,F)$. By Theorem
2, ${\cal T}^{\prime}\subseteq\text{\rm L}(F^{\prime},X^{\prime})$ is
collectively Mwc. Then, for every disjoint $(f_{n})$ in $B_{F^{\prime}}$,

$\lim\limits_{n\to\infty}\sup\\{|f_{n}(Tx)|:x\in B_{X},T\in{\cal
T}\\}=\lim\limits_{n\to\infty}\sup\limits_{T\in{\cal
T}}\|T^{\prime}f_{n}\|=\lim\limits_{n\to\infty}\sup\limits_{S\in{\cal
T}^{\prime}}\|Sf_{n}\|=0,$

and hence ${\cal T}B_{X}=\bigcup\limits_{T\in{\cal T}}T(B_{X})$ is almost
limited, i.e. ${\cal T}$ is collectively almost limited.

(i)$\Longrightarrow$(ii) Let ${\cal T}\in\text{\bf aLim}(X,F)$. Since the set
${\cal T}B_{X}=\bigcup\limits_{T\in{\cal T}}T(B_{X})$ is almost limited, it is
an Lwc set by [5, Theorem 2.6], and hence ${\cal T}$ is collectively Lwc.

(ii)$\Longrightarrow$(iii) It is obvious.

(iii)$\Longrightarrow$(i) Now, assume that every collectively almost limited
subset of $\text{\rm L}(F)$ is collectively Lwc. Then each almost limited
operator is Lwc, and hence the norm in $F$ is order continuous by [5, Theorem
2.6]. ∎

We turn our discussion to the domination problem for collectively compact
sets.

######  Definition 2.

Let ${\cal S}$ and ${\cal T}$ be two subsets of $\text{\rm L}_{+}(E,F)$. Then
${\cal S}$ is dominated by ${\cal T}$ if, for each $S\in{\cal S}$ there exists
$T\in{\cal T}$ such that $S\leq T$.

The proofs of the following two propositions are straightforward modifications
of proofs of [1, Theorem 5.11] and [1, Lemma 5.12] respectively.

######  Proposition 3.

Let ${\cal S}\subset\text{\rm L}_{+}(E,F)$ be dominated by $\\{T\\}$ for some
$T\in\text{\rm L}_{+}(E,F)$. If $T[0,x]$ is $|\sigma|(F,F^{\prime})$-totally
bounded for each $x\in E_{+}$, then $\bigcup\limits_{S\in{\cal S}}S[0,x]$ is
likewise $|\sigma|(F,F^{\prime})$-totally bounded for each $x\in E_{+}$.

######  Proposition 4.

Let ${\cal S}\subset\text{\rm L}_{+}(E,F)$ be dominated by $\\{T\\}$ for some
$T\in\text{\rm L}_{+}(E,F)$. If $T(A)$ is a norm totally bounded set, then for
each $\varepsilon>0$ there exists some $u\in F_{+}$ such that
$\sup\limits_{S\in{\cal S}}\bigl{\|}(Sx-u)^{+}\bigl{\|}\leq\varepsilon$ holds
for all $x\in A$.

The author does not know whether or not a single operator $T$ in Proposition 3
(in Proposition 4) can be replaced by a set ${\cal T}\subset\text{\rm
L}_{+}(E,F)$ dominating ${\cal S}$ such that $\bigcup\limits_{T\in{\cal
T}}T[0,x]$ is $|\sigma|(F,F^{\prime})$-totally bounded (resp.,
$\bigcup\limits_{T\in{\cal T}}T(A)$ is norm totally bounded).

Replacing [1, Theorem 5.11] by Proposition 3 and [1, Lemma 5.12] by
Proposition 4 in the proof of [1, Theorem 5.13] gives a collective extension
of the Aliprantis–Burkinshaw theorem as follows.

######  Theorem 4.

Let ${\cal S}\subset\text{\rm L}_{+}(E)$ be dominated by $\\{T\\}$ for some
$T\in\text{\rm K}_{+}(E)$, then the set $\\{S^{3}:S\in{\cal S}\\}$ is
collectively compact.

A similar modification of the proof of [1, Theorem 5.20] gives the following
collective extension of the Dodds–Fremlin theorem.

######  Theorem 5.

Let the norms in $E^{\prime}$ and in $F$ be order continuous. If ${\cal
S}\subset\text{\rm L}_{+}(E,F)$ is dominated by $\\{T\\}$ for some
$T\in\text{\rm K}_{+}(E,F)$, then ${\cal S}\in\text{\bf K}(E,F)$.

The author does not know whether or not the operator $T\in\text{\rm
K}_{+}(E,F)$ in Theorems 4 and 5 can be replaced by a collectively compact set
${\cal T}\subset\text{\rm L}_{+}(E,F)$ dominating ${\cal S}$.

Finally, we mention briefly the domination problem for collectively Lwc sets.
Whereas collective extensions of the Aliprantis–Burkinshaw theorem [1, Theorem
5.11] and of the Dodds–Fremlin theorem [1, Theorem 5.20] are not sufficiently
clear yet, the domination problem for collectively Lwc sets is much easier and
has the complete solution, as follows.

######  Theorem 6.

Let ${\cal S}\subset\text{\rm L}_{+}(E,F)$ be dominated by ${\cal
T}\subset\text{\rm L}_{+}(E,F)$.

  1. (i)

If ${\cal T}$ is collectively Lwc, then ${\cal S}$ is collectively Lwc.

  2. (ii)

If ${\cal T}$ is collectively Mwc, then ${\cal S}$ is collectively Mwc.

###### Proof.

(i) Let ${\cal T}$ be a collectively Lwc set. Since ${\cal S}$ is dominated by
${\cal T}$, then $Sx\in\bigcup\limits_{T\in{\cal T}}T(B^{+}_{E})$ for each
$x\in B^{+}_{E}$. As $\bigcup\limits_{S\in{\cal
S}}S(B^{+}_{E})\subseteq\text{\rm sol}\left(\bigcup\limits_{T\in{\cal
T}}T(B_{E})\right)$, and $\bigcup\limits_{T\in{\cal T}}T(B_{E})$ is an Lwc
subset of $F$, therefore, $\bigcup\limits_{S\in{\cal S}}S(B^{+}_{E})$, and
hence $\bigcup\limits_{S\in{\cal S}}S(B_{E})$ is also Lwc by [9, Corollary
3.6.4].

(ii) Apply Theorem 2. ∎

## References

  * [1] Aliprantis, C.D., Burkinshaw, O.: Positive operators. Springer, Berlin (2006)
  * [2] Alpay, S., Emelyanov, E., Gorokhova, S.: Duality and norm completeness in the classes of limitedly Lwc and Dunford–Pettis Lwc operators. Turk. J. Math. 48(1), 267–278 (2024)
  * [3] Anselone, P.M., Palmer, T.W.: Collectively compact sets of linear operators. Pacific J. Math. 25(3), 417–422 (1968)
  * [4] Bourgain, J., Diestel, J.: Limited operators and strict cosingularity. Math. Nachr. 119, 55–58 (1984)
  * [5] Chen, J.X., Chen, Z.L., Ji, G.X.: Almost limited sets in Banach lattices. J. Math. Anal. Appl. 412(1), 547–553 (2014)
  * [6] Emelyanov, E., Gorokhova, S.: Operators affiliated to Banach lattice properties and their enveloping norms. Turk. J. Math. 47(6), 1659–1673 (2023)
  * [7] Kusraev, A.G.: Dominated Operators. Kluwer, Dordrecht (2000)
  * [8] Meyer-Nieberg, P.: Über klassen schwach kompakter Operatoren in Banachverbänden. Math. Z. 138, 145–159 (1974)
  * [9] Meyer-Nieberg, P.: Banach lattices. Universitext. Springer, Berlin (1991)