# Direct $N$-body simulations of satellite formation around small asteroids:
insights from DART’s encounter with the Didymos system

Harrison F. Agrusa Université Côte d’Azur, Observatoire de la Côte d’Azur,
CNRS, Laboratoire Lagrange, Nice, France  Department of Astronomy, University
of Maryland, College Park, MD 20742, USA  Yun Zhang Department of Climate and
Space Sciences and Engineering, University of Michigan, Ann Arbor, MI 48109,
USA  Derek C. Richardson Department of Astronomy, University of Maryland,
College Park, MD 20742, USA  Petr Pravec Astronomical Institute of the Academy
of Sciences of the Czech Republic, Fričova 298, Ondřejov, CZ 25165, Czech
Republic  Matija Ćuk SETI Institute, 339 N Bernardo Ave, Mountain View, CA
94043, USA  Patrick Michel Université Côte d’Azur, Observatoire de la Côte
d’Azur, CNRS, Laboratoire Lagrange, Nice, France  The University of Tokyo,
Department of Systems Innovation, School of Engineering, Tokyo 113-0033 ,
Japan  Ronald-Louis Ballouz Johns Hopkins University Applied Physics
Laboratory, Laurel, MD 20723, USA  Seth A. Jacobson Department of Earth and
Environmental Sciences, Michigan State University, East Lansing, MI 48824, USA
Daniel J. Scheeres Smead Department of Aerospace Engineering Sciences,
University of Colorado Boulder, Boulder, CO, USA  Kevin Walsh Southwest
Research Institute, Boulder, CO 80302, USA.  Olivier Barnouin  Johns Hopkins
University Applied Physics Laboratory, Laurel, MD 20723, USA  R. Terik Daly
Johns Hopkins University Applied Physics Laboratory, Laurel, MD 20723, USA
Eric Palmer  Planetary Science Institute, Tucson, AZ  Maurizio Pajola  INAF-
Osservatorio Astronomico di Padova, Padova, Italy  Alice Lucchetti  INAF-
Osservatorio Astronomico di Padova, Padova, Italy  Filippo Tusberti  INAF-
Osservatorio Astronomico di Padova, Padova, Italy  Joseph V. DeMartini
Department of Astronomy, University of Maryland, College Park, MD 20742, USA
Fabio Ferrari Department of Aerospace Science and Technology, Politecnico di
Milano, Milan, 20159, Italy  Alex J. Meyer Smead Department of Aerospace
Engineering Sciences, University of Colorado Boulder, Boulder, CO, USA  Sabina
D. Raducan Space Research and Planetary Sciences, Physics Institute,
University of Bern, Bern, 3012, Switzerland  Paul Sánchez Colorado Center for
Astrodynamics Research, University of Colorado Boulder, 3775 Discovery Dr,
Boulder, CO 80303, USA

(Received September 28, 2023; Revised December 18, 2023; Accepted January 10,
2024)

###### Abstract

We explore binary asteroid formation by spin-up and rotational disruption
considering the NASA DART mission’s encounter with the Didymos-Dimorphos
binary, which was the first small binary visited by a spacecraft. Using a
suite of $N$-body simulations, we follow the gravitational accumulation of a
satellite from meter-sized particles following a mass-shedding event from a
rapidly rotating primary. The satellite’s formation is chaotic, as it
undergoes a series of collisions, mergers, and close gravitational encounters
with other moonlets, leading to a wide range of outcomes in terms of the
satellite’s mass, shape, orbit, and rotation state. We find that a Dimorphos-
like satellite can form rapidly, in a matter of days, following a realistic
mass-shedding event in which only ${\sim}2-3\%$ of the primary’s mass is shed.
Satellites can form in synchronous rotation due to their formation near the
Roche limit. There is a strong preference for forming prolate (elongated)
satellites, although some simulations result in oblate spheroids like
Dimorphos. The distribution of simulated secondary shapes is broadly
consistent with other binary systems, measured through radar or lightcurves.
Unless Dimorphos’s shape is an outlier, and considering the observational bias
against lightcurve-based determination of secondary elongations for oblate
bodies, we suggest there could be a significant population of oblate
secondaries. If these satellites initially form with elongated shapes, a yet-
unidentified pathway is needed to explain how they become oblate. Finally, we
show that this chaotic formation pathway occasionally forms asteroid pairs and
stable triples, including co-orbital satellites and satellites in mean motion
resonances.

Asteroid dynamics (2210) — Asteroid satellites (2207) — Asteroid rotation
(2211) — Near-Earth objects (1092) — Small Solar System bodies (1469)

††journal: PSJ††software: pkdgrav (Richardson et al., 2000; Schwartz et al.,
2012; Zhang et al., 2017), REBOUND (Rein & Liu, 2012), REBOUNDx (Tamayo et
al., 2020), Persistence of Vision Raytracer (https://www.povray.org/), Alpha
Shape Toolbox (https://alphashape.readthedocs.io/), NumPy (Harris et al.,
2020), SciPy (Virtanen et al., 2020), Matplotlib (Caswell et al., 2020).

##  1 Introduction

Binaries make up ${\sim}15\%$ of the near-Earth asteroid (NEA) population
(Margot et al., 2002; Pravec et al., 2006). This fraction increases to
${\sim}65\%$ for fast-rotators greater than ${\sim}300$ m in diameter (Pravec
et al., 2006). Given the relatively short median dynamical lifetimes of NEAs
of about 10 Myr (Gladman et al., 2000), this high binary fraction implies an
efficient formation mechanism that can maintain a steady-state population or
formation before exiting the main asteroid belt. Among NEA binaries, the
primary component is typically small, less than ${\sim}5\text{ km}$ and with a
moderately sized secondary, usually less than ${\sim}60\%$ of the primary’s
diameter. Furthermore, the two bodies are typically on close orbits, separated
by less than ${\sim}10$ primary radii. Typically, the primary is rapidly
rotating and has a near-spherical shape with an equatorial bulge (Pravec et
al., 2006; Pravec & Harris, 2007; Benner et al., 2015). In addition, the
secondaries of these systems frequently have an elongated shape and are
typically in synchronous rotation when the secondary is on a close, near-
circular orbit (Pravec et al., 2016). The specific angular momentum of these
systems is often close to the angular momentum of a sphere having the same
total mass rotating at its critical disruption limit, indicating that the
creation mechanism of these binaries must be related to some sort of fission
or mass-shedding process (Pravec & Harris, 2007; Pravec et al., 2010).

The Yarkovsky–O’Keefe–Radzievskii–Paddack (YORP) effect is a process in which
the absorption and re-emission of solar radiation imparts a small torque on
irregularly-shaped bodies (Rubincam, 2000). The strength of the YORP effect is
highly dependent on the size and shape of the body and its heliocentric
distance, and is thought to be the dominant mechanism that spins-up small
asteroids leading to rotational disruption and the formation of a binary
(Vokrouhlický & Čapek, 2002; Bottke et al., 2002). For a review of YORP and
the formation of small binary asteroids, see Walsh & Jacobson (2015).

The recent Double Asteroid Redirection Test (DART) mission was the first
spacecraft to visit (albeit briefly) a small binary asteroid (Rivkin et al.,
2021). The primary, Didymos, has a flattened top-shape with a volume-
equivalent diameter of ${\sim}760$ m and a short spin period of 2.26 h (Naidu
et al., 2020; Daly et al., 2023b). The satellite, Dimorphos, has an
approximately circular orbit with a semimajor axis of ${\sim}3$ primary radii
(Naidu et al., 2022; Scheirich & Pravec, 2022). Dimorphos was thought to be in
synchronous rotation prior to DART’s kinetic impact (Richardson et al., 2022),
although this could not be determined directly. Both bodies have surfaces
consistent with being gravitational aggregates or “rubble piles” (Barnouin et
al., 2023), although any direct measurements of their interiors will have to
wait for the arrival of the European Space Agency’s Hera mission in 2027
(Michel et al., 2022). Dimorphos has a volume-equivalent diameter of
${\sim}150$ m, corresponding to a secondary-to-primary size ratio of
${\sim}0.2$. If Didymos and Dimorphos have equal bulk densities (which not
known), then they would have a mass ratio of ${\sim}0.01$. Dimorphos appears
to have a remarkably oblate shape, in which its semi-major axis, $a$, and
semi-intermediate axis, $b$, are nearly identical in length and roughly
${\sim}1.5$ times longer than its semi-minor axis, $c$ (i.e, principal
rotation axis) (Daly et al., 2023b). Additionally, the presence of rocks
perched on boulders with slopes no higher than ${\sim}35^{\circ}$ suggests
that Dimorphos’s surface may be cohesionless with a friction angle of
${\sim}35^{\circ}$ (Barnouin et al., 2023). A ${\sim}35^{\circ}$ friction
angle is also derived from an independent method based on the angularity of
boulders on Dimorphos’s surface (Robin et al., 2023). Although there could be
some strength at depth, some recent models of DART’s impact are consistent
with a cohesionless interior (Raducan et al., 2023).

The shape of Dimorphos came as a surprise, as previously observed systems
(either via radar or lightcurve) have found that the secondary tends to have a
more prolate, or elongated, shape in which $a>b>c$, where $a$, $b$, and $c$
are the body’s respective semimajor, semi-intermediate, and semi-minor axes.
Based on Dimorphos’s physical extents (which are well-fit to a uniform
ellipsoid), a preliminary estimate puts Dimorphos’s elongation at $a/b=1.02\pm
0.03$ (Daly et al., 2023b) which has since been updated to $a/b=1.06\pm 0.03$
(Daly et al., 2023a). Nevertheless, no previously published studies employing
lightcurve- or radar-based shape determination of similar binary systems have
found a satellite with $a/b\lesssim 1.1$ (Ostro et al., 2006; Naidu et al.,
2015; Becker et al., 2015; Pravec et al., 2016). The principal motivation for
this study was to understand how Dimorphos might have formed with its present
shape, but as we will show later, the results are broadly applicable to all
small binaries formed by rotational disruption.

In Sections 1.1 and 1.2, we summarize previous work on binary formation and
YORP spin-up. Then we introduce pkdgrav in Section 2 and show a full end-to-
end example simulation demonstrating the formation of a binary system via a
single YORP-driven mass-shedding event in Section 2.1. Then, based on this
simulation and constraints derived by DART, we introduce the simulation set-up
and initial conditions used for this study in Section 2.2. The results from
the numerical simulations are shown in Section 3 followed by conclusions in
Section 4.

###  1.1 Previous work

There is general agreement that many small binary asteroids are likely created
via YORP-induced spin-up in which a satellite forms when the primary exceeds
its critical spin limit. However, there is still some debate about precisely
how the binary system forms, which we address here.

Walsh et al. (2008) modeled the formation of a binary using a rubble-pile
asteroid model consisting of thousands of spherical particles in which angular
momentum is slowly added to the system as a proxy for YORP-induced spin-up.
Through the spin-up process, the primary reshapes, forming an equatorial
bulge, and the secondary is gradually built in orbit via gravitational
accumulation of material shed from the primary’s equator. This idea was
revisited in Walsh et al. (2012) with a broader simulation suite at higher
resolution, finding that that the resulting top-shaped primary and secondary
properties are broadly consistent with the observed population.

However, this model suffers from a few issues. First, it has been shown that
the magnitude and direction of the YORP torque is highly-sensitive to small
changes in the primary’s shape, meaning that each landslides, mass shedding,
and natural impacts could change the strength and direction of the YORP
effect. This could make YORP spin-up effectively a random walk process and may
significantly increase the amount of time required to build a secondary in
orbit (Statler, 2009; Cotto-Figueroa et al., 2015; Zhou et al., 2022). This
effect may significantly decrease the efficiency of such a formation process,
making a scenario in which the secondary forms from a single rotational
disruption event more attractive111Recent work shows that all twelve of the
available and reliable YORP detections show that that the asteroid’s spin rate
is increasing in time (Ďurech et al., 2023). This could due to YORP having an
underlying preference to increase the spin rates of asteroids rather than
decrease them (Golubov & Krugly, 2012). If there really is an underlying
preference for spin-up rather than spin-down, then the idea of stochastic YORP
does not present a significant issue to binary formation models.. On a related
note, Jacobson & Scheeres (2011) point out that a gradual process of satellite
formation suffers from the fact that single particles can escape the system
before the satellite is fully formed. The argument is that escape can occur
before the next shedding event because the system has positive free energy,
and the single (or multiple) shed particles would escape before the next YORP
cycle. Second, in the Walsh et al. (2008, 2012) works, the primary was
initialized in a hexagonal close-packed (HCP) arrangement and particle
contacts were handled using the Hard-sphere Discrete Element Method (HSDEM),
which is ill-suited to simulate particles undergoing persistent contact with
multiple other particles (see Murdoch et al., 2015; Sánchez & Scheeres, 2012,
2016). Although this model captures the general process of binary formation,
the HCP packing and HSDEM contact physics may affect the precise details of
the mass shedding and satellite formation.

Jacobson & Scheeres (2011) presented an alternative theory for binary
formation in which the secondary arises from a single large rotational
disruption event, dubbed “rotational fission” (Scheeres, 2007a). There seems
to be some confusion about these two ideas in the literature, and we attempt
to point out here that the “gravitational accumulation” idea of Walsh et al.
(2008) and the “rotational fission” idea of Jacobson & Scheeres (2011) are
similar, yet fundamentally distinct. Jacobson & Scheeres (2011) present a
model that follows the spin-orbit coupled dynamical evolution of a binary
system following a fission event. Their model assumes the initial rubble pile
consists of two ellipsoidal components (that will later become the primary and
secondary). Angular momentum is slowly added to the single body and at fast
spin rates, the long axes of the two constituent ellipsoids become aligned
while still at rest on one another, owing to this being the only stable
equilibrium configuration for two ellipsoids (Scheeres, 2007a). When the
critical spin limit is exceeded, the bodies then “fission” and begin to orbit
each other. Jacobson & Scheeres (2011) integrate the fully-coupled spin and
orbital evolution of the binary system, accounting for their nonspherical
gravitational potentials along with a treatment for tidal evolution. For mass
ratios $<0.2$, they argue that the post-fission system has positive free
energy (i.e., the sum of the bodies’ kinetic energies and the mutual
potential), meaning that there are one of two possible outcomes: the secondary
must either escape the system or it must fission itself to form (temporarily)
a triple system. These secondary fissions occur when the secondary is
gravitationally torqued by the primary such that it exceeds its stability
limit and is then split into two separate ellipsoids (whose shapes are
generated randomly). In most of their simulations the third body re-impacts
the primary, but it can also escape the system or re-impact the secondary.
This ongoing fission process, along with tidal evolution, will eventually lead
to a state with negative free energy, ultimately allowing the binary system to
be stable.

This model has the advantage that it only requires a single rotational
disruption event, thus avoiding the “stochastic YORP” problem. This model
broadly reproduces the characteristics of the NEA population, including the
existence of asteroid pairs. A follow-up study that includes a more
sophisticated asteroid population model shows that the asteroid fission theory
can reproduce many of the characteristics of binary asteroids (Jacobson et
al., 2016). Recently, the rotational fission model has been extended to
include 3D dynamics, and many of the conclusions still largely hold true
(Boldrin et al., 2016; Davis & Scheeres, 2020; Ho et al., 2022).

The Jacobson & Scheeres (2011) suffers from one potential weakness; in order
to allow for secondary fission events, which are necessary to achieve
stability, they invoke that the secondary itself is a rubble pile. This is of
course a sensible assumption, however, it presents a major issue; at the very
moment of the initial fission event, when the secondary (which is itself a
rubble pile!) detaches from the primary, it must tidally disrupt, as it is
lying within the primary’s Roche limit (Holsapple & Michel, 2006; Sharma,
2009) rather than continue to evolve as a coherent body. This dilemma can be
avoided in cases where the secondary’s bulk density is much higher than the
primary’s such that the Roche limit lies within the primary itself, or if the
secondary has a small amount of cohesion to prevent a tidal disruption (e.g.
Holsapple & Michel, 2008; Sánchez & Scheeres, 2016; Tardivel et al., 2018), or
if the secondary has a bilobate shape in which the neck region could fail
before the rest of the body (Hirabayashi & Scheeres, 2019), or some
combination of all three. However, neither of these three contingencies seem
viable for Dimorphos to form purely by fission if it is a rubble pile with
little-to-no cohesion and a friction angle of ${\sim}35^{\circ}$ (Barnouin et
al., 2023; Raducan et al., 2023). If such a body were to rotationally fission
from Didymos’s equator, it would immediately undergo tidal disruption (e.g.
Agrusa et al., 2022).

In this paper, we present an alternate formation path that incorporates the
ideas of both the Walsh et al. (2008) and Jacobson & Scheeres (2011) models.
In our model, the secondary forms via gravitational reaccumulation, similar to
the Walsh et al. (2008) theory, however, all the required mass is shed in a
single impulsive event, like Jacobson & Scheeres (2011). We will demonstrate
that this simple model of accumulation from debris shed from a single
rotational disruption event can lead to stable binary systems and secondaries
having a wide range of shapes.

Recently, Madeira et al. (2023) considered the gravitational accumulation of
Dimorphos from debris produced by Didymos using a 1D ring-satellite model
(HYDRORINGS) (Charnoz et al., 2010, 2011; Salmon et al., 2010). In this model,
material migrates outwards via viscous spreading until it reaches the Roche
limit, at which point it is immediately converted to a moonlet. Assuming the
initial debris ring is narrowly confined at the primary’s surface, they find
that it takes ${\sim}1$ yr for the ring to spread to the Roche limit, after
which Dimorphos would accrete rapidly, reaching ${\sim}90\%$ of its current
mass within days. This process requires ${\sim}25\%$ of Didymos’s mass to be
put into orbit to form a satellite with ${\sim}1\%$ of Didymos’s mass. Once
Dimorphos forms, they estimate that the ring would take thousands of years to
disappear. In terms of timescales and the required amount of mass, the
predictions of this model differ significantly than what we present in this
work, most likely due to different assumptions in the initial conditions and
necessary simplifications of the 1D model. A significant body of literature
suggests that a rubble pile undergoing rotational failure and mass shedding,
due to its nonzero friction and/or cohesion, will shed material onto much
wider initial orbits (e.g. Hirabayashi et al., 2015; Sánchez & Scheeres, 2016,
2020; Zhang et al., 2018; Hyodo & Sugiura, 2022) rather than within a narrowly
confined region, including several studies focused on Didymos itself (e.g., Yu
et al., 2018; Zhang et al., 2017, 2021). This means that satellites can start
forming much quicker, without having to wait for a ring to spread to the Roche
limit. Also, the Madeira et al. (2023) study assumes that all ring particles
are 1 m in diameter, whereas Dimorphos’s contains boulders at least as large
as 16 m (Pajola et al., 2023a). The presence of larger boulders would speed up
the accumulation process, due to their higher mass and larger collision cross
section. Finally, this model neglects gravity between moonlets, assuming that
they undergo perfect mergers whenever they enter their mutual Hill spheres. In
this work, we will show that this is often not the case, and that the merger
process of moonlets is highly chaotic, leading to a wide range of outcomes.

###  1.2 YORP spin-up and mass shedding

In this study, the considered binary formation scenario is based on the idea
that the progenitor body sheds substantial surface materials in a relatively
short timescale. However, this mass shedding failure behavior is not
unconditional. Prior theoretical work, relying on static stress analyses,
suggests that a homogeneous ellipsoidal body would first structurally fail
internally at its center instead of at the surface during spin up (Holsapple,
2010; Hirabayashi, 2015). This internal failure would lead to internal
deformation, suppressing surface landslides and mass shedding. Modeling of a
rubble-pile body’s spin-up using the soft-sphere discrete element method
(SSDEM) supports these static analyses, showing that homogeneous cohesionless
bodies could indeed fail through internal deformation (Sánchez & Scheeres,
2012; Hirabayashi et al., 2015) and homogeneous cohesive bodies could even
fail through global tensile disruption (Sánchez & Scheeres, 2016; Zhang et
al., 2018).

For mass shedding to occur on the surface, the body’s interior must be
mechanically stronger and/or denser than the exterior layer. Research by
Sánchez & Scheeres (2018) using SSDEM simulations on a spherical rubble pile
with an internal core occupying 70% of its radius demonstrates that the
interior needs to be 3 to 4 times stronger than the shell to prevent internal
failure. Similarly, Ferrari & Tanga (2022) use a polyhedron discrete element
method to show that a rigid core with a volume fraction exceeding 25% can also
facilitate mass shedding, as shown for the case of Didymos. By employing
higher particle resolution (${\sim}10^{5}$ particles) and a highly
polydisperse particle size distribution, the SSDEM spin-up simulations of
Zhang et al. (2017, 2021) reveal that the large particle size difference can
reduce the internal porosity and increase the mobility of surface regolith in
a rubble-pile. The resultant small scale heterogeneity could trigger surface
failure via mass shedding, even if mechanical interactions at the particle
level are uniform throughout the body. A similar effect is produced by the
interactions between non-spherical particles with irregular shape, where
geometrical interlocking mechanism adds mechanical strength to the inner
structure (Ferrari & Tanga, 2020). In all cases, maintaining some internal
shear strength and a low surface cohesion are crucial to initiate surface mass
shedding (Sánchez & Scheeres, 2020; Zhang et al., 2022).

Beyond considering the internal structure, the evolution of a rubble-pile body
after reaching its spin limit is also heavily influenced by numerous other
factors, such as shape and surface morphology (Hirabayashi & Scheeres, 2019;
Hirabayashi et al., 2020; Zhang et al., 2022). This complexity implies that
the surface failure conditions could vary across different bodies, which
determines why certain asteroids evolved into binary systems while others
didn’t, and could play some role in the apparent deficit of small binaries and
pairs among primitive asteroid types relative to silicate-rich bodies (Minker
& Carry, 2023). In Section 2.1, to validate the assumed binary formation
scenario for Didymos, we will present an SSDEM simulation example to showcase
the feasibility of Didymos’ mass shedding failure behavior under reasonable
conditions.

We finally note that the mass shedding doesn’t have to occur solely as a
result of YORP spin up. Once YORP spins up a body near its critical limit, a
natural impact could trigger the mass shedding event, for example. In fact,
this study is somewhat agnostic to the exact process causing the mass
shedding, we merely require that the mass shedding occur all at once such that
enough material is put into orbit to allow a rapid reaccretion process.

##  2 Methodology

We use pkdgrav, a gravitational $N$-body tree code to model the accumulation
of a rubble-pile satellite (Richardson et al., 2000; Stadel, 2001). Particle
contacts are handled using the soft-sphere discrete element method (SSDEM), in
which a Hooke’s Law spring provides a normal force between overlapping
particles (Schwartz et al., 2012). Parameters such as the spring constant and
coefficients of static, twisting, and rolling friction and cohesion can be
adjusted to represent desired material properties. We refer the reader to
Zhang et al. (2017) for a detailed description of the model. In short, $k_{N}$
and $k_{T}$ are the two spring constants that control the particle’s stiffness
in the normal and tangential directions respectively, $\epsilon_{N}$ and
$\epsilon_{T}$ are the coefficients of restitution in the normal and
tangential direction for controlling energy dissipation, $\mu_{S}$, $\mu_{R}$,
and $\mu_{T}$ are the coefficients of static, rolling, and twisting friction.
Finally, the shape parameter $\beta$ is used to approximate the non-spherical
nature of real particles by statistically defining their contact area,
enabling the calculation of the associated rotational resistance.

The normal spring constant ($k_{N}$) and timestep are selected such that
typical particle overlaps don’t exceed ${\sim}1\%$ of a particles radius and
that ${\sim}30$ timesteps take place over the course of a collision (Schwartz
et al., 2012). This ensures that particle contacts are resolved properly and
that particles do not undergo a nonphysical level of interpenetration. Based
on our typical particle mass, sizes, and collision speeds this corresponds to
a $k_{N}\sim 4\times 10^{6}\text{ kg s}^{-2}$ and a punishingly small timestep
of ${\sim}0.15$ s. Following common practice, we set $k_{T}=\frac{2}{7}k_{N}$
(Walton, 1995; Schwartz et al., 2012). In all simulations presented here, we
set the restitution coefficients to $\epsilon_{N}=\epsilon_{T}=0.55$, a
typical value for rocky material (Chau et al., 2002). $\mu_{R}$ and $\mu_{T}$
are set to 1.05 and 1.3, respectively, to represent the rough surfaces of
medium hardness rocks (Jiang et al., 2015), leaving $\mu_{S}$ and $\beta$ as
the only two parameters that are varied to achieve different friction angles.
These values and their corresponding friction angle are provided in Table 1.

###  2.1 Satellite formation via mass shedding

Here, we show an example simulation to test Didymos’s structural failure
behaviors at its spin limit. Didymos is modeled as a granular aggregate
composed of 87,635 spheres with radii ranging from about 4 to 16 m. The
particle size distribution follows a cumulative power-law distribution with an
exponent of $-2.5$, aligning with the boulder size frequency distribution with
similar radii found on Dimorphos, as observed in the images taken by the
camera onboard the DART spacecraft (Pajola et al., 2023a). The granular
aggregate was configured based on the pre-impact Didymos shape model (Barnouin
et al., 2023) to ensure accurate representation of Didymos. To allow the
initiation of surface mass shedding and maintain the stability of Didymos at
its current spin period of 2.26 h, the shape parameter $\beta$ is adopted to
be 0.8, and $\mu_{S}$ is taken to be 1.0, representing a material internal
friction angle of $\sim$40∘ (Zhang et al., 2022). This friction angle is
within the typical range of compacted dry sand, i.e., 33∘–43∘ (Bareither et
al., 2008). To resolve the quasi-static mechanical contact between particles,
we adopted a smaller timestep of $\sim$0.02 s and a larger $k_{N}{\sim}8\times
10^{7}$ kg s-2. The bulk density of the body is set to 2.7 g cm-3, consistent
with Didymos’ latest bulk density estimate constrained by the updated shape
model and the binary separation, i.e., $2.760\pm.130$ g cm-3 (Naidu et al.,
2023)222Due to uncertainties in the size and volume of Dimorphos, which has
some degeneracy in the body separation, there is significant uncertainty in
Didymos’s bulk density. The formal uncertainty of ${\pm}.130$ g cm-3 is likely
an underestimate and a realistic uncertainty might be larger.. Our numerical
investigation found that cohesion is no longer required to maintain the bulk
structural stability of Didymos at its current spin state for a bulk density
of $\gtrsim$2.7 g cm-3, and, therefore, the interparticle cohesive strength
was set to zero333We also performed simulations using a friction angle of
$35^{\circ}$ ($\mu_{S}=0.6$, $\beta=0.5$), and found that a bulk cohesive
strength of about 8 Pa is needed for the structural stability. The mass-
shedding structural failure behavior at faster spin is similar to the case
presented here.. All other parameters are the same as those introduced in the
previous section.

Didymos is quasi-statically spun-up (as a proxy for YORP) until structural
failure is detected, which we define by the body’s longest dimension changing
by more than 1% relative to the starting shape. Then the spin-up procedure is
halted and the granular system evolves purely under its own self gravity. The
results show that the primary structurally failed at a spin period of
${\sim}2.2596$ h, where it then shed surface particles from mid-to-low
latitudes, putting ${\sim}3\%$ of its total mass onto low inclination orbits.
Much of this material rapidly clumps into moonlets, and a Dimorphos-mass
satellite is formed within days following a series of moonlet mergers. As a
result of the conservation of angular momentum, the primary’s spin period
drops to ${\sim}2.5$ h by the end of the simulation due to mass shedding and
reshaping. Material that falls back onto Didymos preferentially lands on the
equator, which contributes to forming Didymos’s equatorial ridge, similar to
the process demonstrated by Hyodo & Sugiura (2022). The present-day shape of
Didymos is the subject of other ongoing and future studies (Barnouin et al.,
2023, Zhang et al., in prep).

Snapshots of this simulation are shown in Fig. 1 along with a time-series plot
of the satellite’s shape, mass, orbit, and attitude in Fig. 2. In this
simulation, we form an approximately Dimorphos-mass satellite in a matter of
only a few days. The satellite is also relatively oblate compared to the
measured shapes of other binary systems, having axis ratios of $a/b{\sim}1.15$
and $b/c{\sim}1.5$, based on the satellites dynamically equivalent equal-
volume ellipsoid (DEEVE)444For a body of mass $m$ and principal moments of
inertia $A$, $B$, $C$, its corresponding dynamically equivalent equal-volume
ellipsoid axis lengths $a$, $b$, $c$ are given by the following relations:
$A=\frac{m}{5}(b^{2}+c^{2}),B=\frac{m}{5}(a^{2}+c^{2}),C=\frac{m}{5}(a^{2}+b^{2})$..
The satellite is initially in synchronous rotation with a libration amplitude
of ${\sim}45^{\circ}$. However, this tidally-locked state is broken when the
satellite has a close encounter with a smaller moonlet, sending it inwards
where it undergoes a partial tidal disruption followed by an immediate merger
with an another moonlet. This sequence of events breaks the satellite’s
synchronous rotation state and happens to lead to a relatively oblate shape.
We encourage the reader to view the provided movie of this simulation.

This simulation robustly demonstrates that a Dimorphos-like satellite can form
rapidly from a single rotational disruption event. In addition, the
satellite’s series of close encounters and collisions with other moonlets
demonstrates that this process is highly chaotic. An infinitesimal change to
the initial conditions of this simulation could lead to a very different
outcome in terms of the satellite’s physical and dynamical properties.
However, due to the computational expense of these high resolution, full spin-
up-to-satellite-formation simulations, they are impractical for longer
simulations as well as studying outcomes statistically, which is the focus of
the rest of this study. Before proceeding to the rest of this study, we note
that the simulation presented in this subsection is a new result in its own
right, although its primary purpose is to motivate the initial conditions used
in the rest of this study. This simulation also highlights some key
differences between our study and recently published papers on binary
formation, which we address here.

Figure 1: An example of a satellite forming after mass shedding from a
Didymos-shaped primary. Particles are colored in white being shed from the
primary. The initial spiral arms are caused by mass shedding occurring at
localized regions rather than across the entire body. By ${\sim}0.5$ d, this
asymmetry largely smooths out in azimuth before moonlets begin forming. A
movie of this simulation is provided. Figure 2: A time series plot showing the
shape ($a/b$ and $b/c$, based on the satellite’s DEEVE), mass ratio
($M_{\text{B}}/M_{\text{A}}$), body separation
($r_{\text{orb}}/R_{\text{A}}$), and the attitude of the satellite formed via
mass shedding. Within only several days, a Dimorphos-mass satellite is formed.
The satellite is initially tidally locked to the primary (starting at
${\sim}1.3$ d), but then undergoes a partial tidal disruption followed by a
merger with a moonlet, causing it to break from synchronous rotation (at
${\sim}2.5$ d). The satellite has an oblate shape, similar to Dimorphos.
However, the satellite’s shape, mass, orbit and rotational state will continue
to change as it continues to accrete more material.

##### Comparison with Sugiura et al. (2021) and Hyodo & Sugiura (2022)

Recently, Sugiura et al. (2021) studied the shape evolution of a rubble pile
under YORP spinup using smoothed-particle hydrodynamics (SPH) simulations. In
contrast to DEM simulations where particles represent discrete objects (i.e.,
rocks, boulders, etc.), SPH is used to simulate continuum mechanics where the
particles sample local quantities such as density, internal energy, and
pressure. Sugiura et al. (2021) find that spin-up can result in a “top-shaped”
body for friction angles exceeding $70^{\circ}$. However, it is challenging to
compare this study to the result obtained due to differences in initial
conditions, material models, and spin-up procedure. First, the work of Sugiura
et al. (2021) starts from spherical shapes, whereas this simulation starts
from a Didymos-like shape. Second, these simulations consider friction angles
exceeding those of terrestrial and lunar granular material (typically $\sim
35{-}45^{\circ}$, e.g., Bareither et al., 2008; Al-Hashemi & Al-Amoudi, 2018;
Mitchell et al., 1972) as well as friction angle estimates of recently visited
asteroid surfaces (also on the order of ${\sim}35^{\circ}$, e.g., Fujiwara et
al., 2006; Watanabe et al., 2019; Barnouin et al., 2019, 2023; Robin et al.,
2023). This is because Sugiura et al. (2021) consider a single “effective
friction angle” that accounts for both friction and cohesion. Cohesion is
defined as the shear strength at zero pressure, where the shear strength of a
granular material can be written as $Y=\tan(\phi)p+c$, where $\phi$ is the
friction angle, $p$ is the confining pressure, and $c$ is the cohesion.
Sugiura et al. (2021) instead adopt a material mod4el of the form
$Y=\tan(\phi)p$ where $\phi$ is now an “effective friction angle” that
encompasses the effects due to cohesion, which justifies their choice to
explore values of $\phi$ that are significantly higher than real granular
materials. With this material model, we can see that as the confining pressure
goes to zero (i.e., near the surface), the shear strength goes to zero, which
effectively results in the material having zero cohesion. Therefore, we
suspect that the failure mechanisms observed by Sugiura et al. (2021) are a
result of the rubble pile implicitly having a cohesionless upper layer with a
stronger internal core. Ryugu and Bennu seem to have sub-Pa levels of cohesion
at their surfaces and potentially some strength at depth, so this implicit
assumption by Sugiura et al. (2021) isn’t unreasonable (Arakawa et al., 2020;
Jutzi et al., 2022; Walsh et al., 2022; Barnouin et al., 2019). However,
recent work by Zhang et al. (2022) demonstrates that rubble pile failure
mechanisms are highly sensitive to small changes in cohesion at a fixed
friction angle, and to small changes in friction at constant cohesion,
indicating that friction and cohesion cannot be combined into a single
parameter and should be treated separately.

Finally, the spin-up procedure used here differs from Sugiura et al. (2021) in
a critical way. Both methods apply a similar angular acceleration to the
rubble piles (${\sim}10^{-10}\text{ rad s}^{-2}$), which ensures that the
artificially induced Euler acceleration is always negligible compared to the
centrifugal acceleration. However, Sugiura et al. (2021) spin up the rubble
pile until 1$\%$ of its mass is put into orbit, whereas this study halts the
spin-up process at the moment of failure to ensure a realistic mass shedding
process. Although the total angular momentum added during the mass-shedding
process by Sugiura et al. (2021) is small, this can make a critical difference
for a rubble pile on the edge of stability. As a result, Sugiura et al. (2021)
typically find that ${\sim}10\%$ of the primary’s mass ends up going into
orbit following the spin-up process. Hyodo & Sugiura (2022) then track the
formation of moons after such a mass-shedding event, where they find similar
formation timescales to what is presented in this study. However, due to their
simulations starting with $10-20\%$ of the primary’s mass in a debris disk,
their simulations result in a system of satellites with a combined mass of
${\sim}5\%$ of the primary’s mass. Although many binary asteroids have
similarly high mass ratios (on the order of $5-10\%$), this model does not
produce binary systems with smaller satellites, such as the Didymos system,
where the mass ratio is ${\sim}1\%$ (Pravec et al., 2016, 2019).

##### Comparison with Madeira et al. (2023)

In comparison to the simplified 1D model of Madeira et al. (2023), there are
several key differences to highlight based on this simulation. Of course, this
model has the advantage of modeling the system for a much longer timescales
that what is possible with a direct $N$-body approach. In addition, their
model can provide useful insight into the formation of binaries. However, the
result presented here (as well as the result of Hyodo & Sugiura (2022))
demonstrates that the mass-shedding and accretion timescales are relatively
short, meaning that it may not be necessary to simplify the problem to one
dimension. In addition, the insight of the simplified model of Madeira et al.
(2023) is only useful to the extent to which it reflects the true nature of
the problem. Madeira et al. (2023) suppose an idealized disk in which only a
single satellite accretes from the Roche limit at a time. On the contrary,
this simulation demonstrates that following a realistic YORP-driven mass-
shedding event, multiple moonlets can form simultaneously before undergoing a
chaotic series of close encounters and mergers until one satellite remains,
which cannot be captured by a 1D model. It is not clear that mass shedding
would actually lead to a disk matching the assumptions of Madeira et al.
(2023).

Recently, Madeira & Charnoz (2024) extended their study, claiming that their
model explains the oblate shape of Dimorphos, the contact binary shape
Dinkinesh’s recently discovered satellite, Selam, and the prolate shapes of
other binary asteroids, where the key difference between these three outcomes
solely comes down to the mass-shedding timescale. Some caution should be
applied here until the model can address some key issues. First, the mass-
shedding timescales explored by Madeira & Charnoz (2024) are chosen
arbitrarily and need some physical justification, as there is currently no
connection between the initial conditions of their model and YORP processes or
the geophysical properties of the primary body. In addition, these 1D disk-
moon models do not directly account for the gravitational interactions and
subsequent mergers between moonlets. Rather, mergers are assumed to occur
anytime two moonlets enter one another’s Hill sphere. The shape outcome of
these mergers is not demonstrated either; rather a post-merger shape is
assumed based on simulations of mergers of Saturn’s inner moons (Leleu et al.,
2018). However, the geophysics and dynamics of mergers leading to the
formation of Saturn’s inner moons are very different than in the case of two
merging satellites of an S-type asteroid. For example, the moons Pan, Atlas,
and Prometheus have densities below $1\text{ g cm}^{-3}$ and orbit within or
near Saturn’s Roche limit while Dimorphos is thought to have a bulk density on
the order of ${\sim}2.4\text{ g cm}^{-3}$ and any mergers leading up to its
formation are thought to occur beyond the Roche limit, according to Madeira et
al. (2023). Any model that can explain the shapes of Dimorphos, Selam, other
binary satellites needs to self-consistently address the spin-up, mass-
shedding, accretion, and (potentially) merger processes. For example, the
shape of a post-merger satellite will depend on parameters such as its
density, friction angle, and cohesion. At the same time, these parameters are
intimately connected with the primary’s failure mechanisms and the mass
shedding process, which will then determine the initial conditions of any
orbiting debris. This is a challenging problem, and the work presented here
only attempts to address a small part of this process.

###  2.2 Simulation set up

The simulation shown in Figs. 1 and 2 are computational expensive requiring
${\sim}10^{5}$ particles, are limited to short integration times and are
impractical for determining statistical outcomes. Therefore, we use the above
result, along with known properties of the Didymos system, to inform a
simplified initial condition for a large suite of simulations. This allows us
to determine the properties of the resulting satellite from a statistical
point of view. It is important to note that the primary’s shape, density,
internal structure, and material properties will all play some role in
determining the initial conditions of the shed material (e.g., Sánchez &
Scheeres, 2020; Walsh et al., 2012; Zhang et al., 2022; McMahon et al., 2020).
Therefore, the simplified initial conditions presented here are intentionally
generic and may not be perfectly representative of a fully realistic scenario.
However, we will show this generic initial condition produces many of the
properties of both Dimorphos and other binary systems.

The primary was treated as a uniform oblate spheroid with semiaxes $a=b>c$,
with an equatorial radius of $a=b=400$ m and $a/c=1.5$, and a bulk density of
$2.4\text{ g cm}^{-3}$, similar to the best estimate for Didymos’s shape and
density at the time this study began (Daly et al., 2023b). In order to reduce
the total number of particles in the simulation, the primary is modeled as a
hollow shell of particles locked together into a rigid aggregate. In order for
the primary to have moments of inertia as if it were a solid body (which is
important for both gravity and collisions), we include a single point-mass
particle at the center, and then adjust the mass of the central particle and
the shell particles to achieve moments of inertia of an equivalent uniform
oblate spheroid555For a homogeneous spheroid, the moments of inertia can be
written in terms of its mass $M$ and its semi-axes $a,b,c$:
$A=\frac{M}{5}(b^{2}+c^{2}),B=\frac{M}{5}(a^{2}+c^{2}),C=\frac{M}{5}(a^{2}+b^{2})$..
In other words, the point-mass and surrounding spheroidal shell collectively
behave as if they are a single, solid body with the correct moments of
inertia. As shown in Fig. 3, the radius of the shell particles are over-
inflated to prevent small particles from falling through the cracks. This
approach enables a physically realistic solid-body primary without requiring
an excessive number of particles in the interior. The spin period of the
primary is initially set to 2.5 h in all simulations, approximately matching
the rotation period of the primary from Section 2.1 following its rotational
disruption.

The orbiting material is generated following a power-law size-frequency
distribution (SFD), with a power-law index of -3 and particle radii ranging
between 3 and 10 m. This SFD is informed by (but does not match exactly) the
observed boulder SFD on Dimorphos (Pajola et al., 2022, 2023b), as well as the
boulder SFD seen on other rubble pile asteroids (e.g., Dellagiustina et al.,
2019; Burke et al., 2021; Michikami et al., 2019; Michikami & Hagermann,
2021). Including the smallest boulders observed on Dimorphos is impractical
due to computational constraints; however, the smallest particles in the
simulation are 6 m in diameter, approximately the size of the two largest
boulders Bodhran Saxum (${\sim}6.1$ m) and Atabaque Saxum (${\sim}6.5$ m) near
the DART impact site (Daly et al., 2023b). Therefore, we consider this
particle resolution sufficient for modeling the formation and shape of a
Dimorphos-like satellite, as the particle sizes in these simulations approach
the actual sizes of individual large boulders on Dimorphos’s surface. Cohesion
between boulders is ignored in these simulations, as cohesion forces on rubble
piles likely arise from Van der Waals forces between fine-grained material
less than ${\lesssim}$10 $\mu$m (e.g., Sánchez & Scheeres, 2014). In a mass
shedding scenario, much of this fine-grained material (if present) would be
released when the mass is first shed, and when moonlets undergo collisions and
tidal disruptions. For a Dimorphos-like system, solar radiation pressure would
rapidly remove these fine grains (e.g., Ferrari et al., 2022). Therefore, we
might expect the cohesion of the secondary to be no higher than the primary’s
cohesion (if present). This may explain why Didymos’s surface requires a small
amount of cohesion to maintain its surface stability while Dimorphos does not
(Barnouin et al., 2023).

A disk of orbiting debris is placed in a circular region extending out to 1.5
times the equatorial radius of the primary, which approximately corresponds to
the effective Roche limit for a body with a friction angle of $35^{\circ}$
(Holsapple & Michel, 2006). The disk also has a finite thickness of 40 m
(roughly two diameters of the largest disk particle), to allow enough space to
initialize the required number of particles. The example simulation presented
in Section 1.2 demonstrates that mass shedding does not lead to a stable disk
but instead material clumps almost immediately to form moonlets. To
approximate this effect here, particles are purposefully not given any initial
velocity dispersion to ensure a disk that is gravitationally unstable (i.e., a
Toomre $Q<1$) so material will begin clumping immediately. All disk particles
have a density of $3.5\text{ g cm}^{-3}$, to match the grain density of L and
LL chondrites (Flynn et al., 2018), which are the best meteoritic analogues
for Didymos (de León et al., 2006; Dunn et al., 2013).

Each disk particle is placed on a circular orbit, accounting for the mass of
the primary and its oblateness, as well as the mass of all other enclosed disk
particles. Each disk particle’s mean motion (i.e., its initial orbital angular
velocity), can be written as a function of its radial distance $r_{i}$,

$n_{i}^{2}=\frac{GM_{\text{A}}}{r_{i}^{3}}\bigg{(}1+\frac{3}{2}J_{2}\bigg{(}\frac{R^{\text{eq}}_{\text{A}}}{r}\bigg{)}^{2}\bigg{)}+\frac{M_{\text{disk}}(<r_{i})}{r_{i}^{3}},$

where $M_{A}$, $R^{eq}_{A}$, and $J_{2}$ are the primary’s respective mass,
equatorial radius, and oblateness, $M(<r_{i})$ is the mass of all the disk
particles enclosed within particle $i$’s position, and $G$ is the
gravitational constant. Owing to the finite thickness of the disk, particles
are initialized with inclinations on the order of $2-3^{\circ}$. Particles are
generated in a symmetric disk to simplify the process of generating initial
conditions, which is not a realistic starting condition. However, we emphasize
that, owing to the disk being gravitationally unstable, collisions and close
encounters quickly excite the eccentricities and inclinations of the disk
particles, which we deomnstrate below rapidly leads to particle orbits
representative of the more realistic starting conditions following a mass-
shedding event, such as that shown in Fig. 1.

Any particles that reach a distance of 40 km ($100R^{eq}_{A}$) are
automatically deleted from the simulation. This boundary was set purely as a
precaution to prevent a single particle from being ejected from the system,
which could significantly slow down pkdgrav’s tree due to the single extremely
distant particle, and corresponds to ${\sim}2/3$ of the system’s Hill sphere
if it were located at 1 AU. We found that a small fraction of particles end up
reaching this boundary and are deleted. Typically only 1-2% of the disk’s
initial mass is ejected and we verified that the vast majority of the
particles that hit this boundary and were deleted were either on escape
trajectories ($e_{\text{orb}}>1$), or had an apoapse distance well outside the
Hill sphere and would have not returned to the binary system. Therefore,
deleting these particles has a negligible effect on the binary’s formation. In
a some cases, a large aggregate is ejected from the system forming an asteroid
pair and is discussed in Section 3.7.1.

A core motivation of this study was to demonstrate that a Dimorphos-sized
satellite could plausibly form via a single mass shedding event. Therefore,
most simulations simply vary the initial mass in the disk (i.e., number of
particles), however we also vary the friction angle of the material between
$29^{\circ}$ and $40^{\circ}$. For each set of parameters, we randomize the
initial locations of the particles to understand the chaotic nature of the
satellites formation. For each disk mass ($M_{\text{disk}}$) and friction
angle ($\phi$), we run a given number of ‘clone’ simulations, which are listed
in Fig. 1. In total, we run 120 simulations. 96 of the simulations have
$\phi=35^{\circ}$ and have an initial disk mass ranging between $0.02M_{A}$
and $0.04M_{A}$. These 96 simulations constitute the bulk of the results shown
in the following sections. With the remaining 24 simulations,
$M_{\text{disk}}$ is kept fixed at $0.03M_{A}$ and friction angles of
$29^{\circ}$, $32^{\circ}$, and $40^{\circ}$ are tested. All simulations were
limited to ${\sim}100$ d ($5.8\times 10^{7}$ steps) due to the high
computational cost imposed by the small timestep. Each simulation is assigned
a number between 1 and 120, and the result of each simulation is tabulated in
Table 3 in the Appendix.

Table 1: Number of simulations for each combination of $M_{\text{disk}}$ and
$\phi$, along with the static friction coefficient ($\mu_{S}$) and shape
parameter ($\beta$) used to achieve the given friction angle ($\phi$).

$M_{\text{disk}}/M_{\text{A}}$ | $\phi$ | $N_{\text{sims}}$ | $\mu_{S}$ | $\beta$
---|---|---|---|---
0.02 | $35^{\circ}$ | 32 | 0.6 | 0.5
0.03 | $35^{\circ}$ | 32 | 0.6 | 0.5
0.04 | $35^{\circ}$ | 32 | 0.6 | 0.5
0.03 | $29^{\circ}$ | 8 | 0.2 | 0.3
0.03 | $32^{\circ}$ | 8 | 0.4 | 0.4
0.03 | $40^{\circ}$ | 8 | 1.0 | 0.8

Note. — In all simulations, the primary consists of 2,371 particles, while the
number of debris particles varies between ${\sim}4,200$ and ${\sim}8,400$,
depending on the total mass in the disk. See Table 3 to see a full listing of
each simulation and its result.

Figure 3: A view of the initial conditions from the disk simulations. The
primary (gold colored particles) behaves as a single rigid body.

##  3 Results

###  3.1 An example case

First, we show a representative simulation to demonstrate the model. In Fig. 4
we show 8 snapshots of a simulation in which the
$M_{\text{disk}}=0.02M_{\text{A}}$ and $\phi=35^{\circ}$. Almost immediately,
particles begin clumping and forming short-lived spiral arms due to Keplerian
shear, as a result of the disk being gravitationally unstable (e.g. Kokubo et
al., 2000). A couple days later, several moonlets form and undergo a chaotic
series of close encounters and mergers until a single large satellite remains.
Interestingly, the satellite is born in a tidally-locked configuration with
the primary. The immediate tidal locking of the secondary is likely due to
several factors, including its low eccentricity and formation near the Roche
limit. In addition, all the mergers in this simulation occur rather gently
without significantly perturbing the secondary’s spin state. As we will see in
the next several subsections, the chaotic nature of the satellite’s formation
leads to a wide range out outcomes in terms of the satellite’s physical and
dynamical properties. Therefore, this simulation is not necessarily typical of
all cases.

Figure 4: An example of a simulation where the secondary is formed in
synchronous rotation. This is Disk 008 in Table 3, and has an initial disk
mass of $M_{\text{disk}}=0.02M_{\text{A}}$ and a friction angle of
$\phi=35^{\circ}$. A movie of this simulation is available. Figure 5: Time-
series plots of the example simulation from Fig. 4 (Disk 008). Starting from
the top, we plot the satellites DEEVE axis ratios, mass ratio, orbital
distance, and Euler angles, which describe its orientation in a frame rotating
with the orbit. In only a handful of days, the satellite reaches a near-
circular orbit at ${\sim}3R_{A}$ with a mass just under $0.01M_{A}$. The
rotation state of the satellite is excited, but tidally locked, given by all
three Euler angles being large but still well under $90^{\circ}$.

In order to compare whether the simplified initial condition of the disk is a
reasonable representation of the more realistic initial conditions following
mass shedding, as demonstrated in Section 2.1, we compare the distributions of
semimajor axis, eccentricity, and inclinations of orbiting debris at early
times. Fig. 6 compares the orbital elements of orbiting material between the
more realistic mass-shedding example from Fig. 1 and the simplified case shown
in Fig. 4. Owing to the gravitationally unstable disk and the role of
collisions, the orbits of disk particles are rapidly excited to wider orbits,
with higher eccentricities and inclinations, despite initially starting with
circular, nearly co-planar orbits. At least qualitatively, the distribution of
orbital elements several days into the simulation reasonably reflects the
orbits of post-mass-shedding debris (i.e., Fig. 1), although the more
realistic initial conditions have a slightly wider distribution in both
inclination and eccentricity.

Figure 6: A comparison of orbiting particles between the more realistic, full
spin-up simulation shown in Fig. 1 and the simplified initial conditions from
the simulation shown in Fig. 4. The orbital elements are plotted when moonlets
first begin forming. For the mass-shedding simulation, this corresponds to
${\sim}0.5$ d after mass is initially shed, while for the simplified disk,
this occurs around ${\sim}4$ d into the simulation. Although the distribution
of orbital elements between the two cases does not match perfectly, due to the
gravitationally unstable initial conditions of the simplified disk, we arrive
at a similar range in semimajor axis, eccentricities, and inclinations seen in
the more realistic simulation.

###  3.2 Satellite mass and density

In Fig. 7, we plot the secondary-to-primary mass ratio for all simulations
with $\phi=35^{\circ}$ in order to understand how the mass of the initial disk
determines the resulting satellite mass. Assuming the primary and secondary
have the same bulk density (which is approximately true in these simulations),
we also provide the size ratio
$\frac{D_{\text{B}}}{D_{\text{A}}}\approx\big{(}\frac{M_{\text{B}}}{M_{\text{A}}}\big{)}^{1/3}$
on the second y-axis. In the 96 simulations shown here, the satellite tends to
reach its final mass in the first few tens of days apart from a select few
special cases that undergo late mergers or disruptions. For context, the
Dimorphos-to-Didymos size ratio is ${\sim}0.2$ which corresponds to a mass
ratio ${\sim}0.01$ if the bodies have equal densities (Daly et al., 2023b).
Therefore, we find that an initial disk mass of only
${\sim}0.02-0.03M_{\text{A}}$ is capable of producing a Dimorphos-mass
satellite in only a matter of days. Although their study focuses on a regime
where significantly more mass and angular momentum is put into orbit, we find
that our formation timescale is broadly consistent with that found by (Hyodo &
Sugiura, 2022). Our result is orders of magnitude lower (both in time and
mass) than the calculation of Madeira et al. (2023) which requires 25% of
Didymos’s mass to be shed into a ring that will then take years to form a
Dimorphos-mass satellite. This disagreement likely stems from the different
approach to the problem, namely in the initial conditions. Our model starts
with a disk of particles initialized on much wider orbits, given the existing
body of literature on spin-up and mass shedding (e.g., Yu et al., 2019; Zhang
et al., 2018, 2022; Hyodo & Sugiura, 2022) and based on the spin-up example
provided in Section 2.1, whereas the Madeira et al. (2023) model supposes that
the orbiting debris starts in a narrowly confined region at the surface of the
primary and slowly spreads outwards, which substantially increases the
timescale for the satellite’s formation.

We plot a histogram of the bulk density of each satellite at the end of the
simulation in Fig. 8. The volume of the rubble pile is computed by the concave
hull (or “alpha shape”; (Edelsbrunner, 1995)) of the set of points defined by
the edges of the outermost spheres which make up the satellite. This method of
calculating bulk density provides high accuracy by providing a “tighter fit”
to the true shape of the rubble pile than other volume estimates such as the
volume of the convex hull or the dynamically equivalent ellipsoid. Since some
of the large void spaces could be filled with smaller boulders that are below
the resolution of the simulations, the measured bulk density here is only
notional and should be thought of as a soft lower limit.

Figure 7: The secondary-to-primary mass ratio over time, for all simulations
with $\phi=35^{\circ}$. The accretion of the satellite is highly efficient,
occurring in only a matter of days. The second y-axis shows the secondary-to-
primary size ratio, assuming that the two bodies have the same bulk density
(which is approximately true in these simulations). The spread in final mass
among disks with the same initial mass is due to the chaotic formation history
of each system. Note: most discontinuities in this plot are real and are due
to mergers or tidal disruptions of the satellites. However, there are a couple
discontinuities that result from the nearest-neighbor search algorithm
misidentifying the largest orbiting fragment. Figure 8: The bulk density of
all satellites with $\phi=35^{\circ}$ at the end of the simulations. The
density of all individual spheres is 3.5 g cm-3, so these rubble piles have
packing fractions of ${\sim}0.62$. Realistically, it is possible for the
“real” bulk density to be slightly higher than what is measured here, as small
particles which are below the resolution limit of these simulations would fill
in some void space.

###  3.3 Satellite orbit and rotational state

The majority of the simulations end with a single satellite on a near-circular
orbit having a semimajor axis between ${\sim}2.5$ and ${\sim}4$ primary radii.
Fig. 9 shows the final semimajor axis and eccentricity for the satellite in
the 96 simulations where $\phi=35^{\circ}$. Generally, we find that more
massive disks tend to produce a satellite on a wider, more eccentric orbit,
simply as a result of the disk having a larger initial angular momentum. This
is demonstrated in Fig. 9 where we show the mean $a_{\text{orb}}$ and
$e_{\text{orb}}$ along with their standard deviations for each of the 3
different disk mass cases.

For each satellite, we determine its rotation state based on the 1-2-3 Euler
angle set, consisting of the satellites roll, pitch, and yaw angles in a frame
rotating with its orbit (see Agrusa et al. (2021)). When the roll and pitch
angles are small, the yaw angle can be thought of as the classic planar
libration angle, i.e., the angle between the secondary’s long axis and the
line-of-centers. However, many of the satellites have undergone several
mergers or close gravitational encounters, and are in excited, non-planar
rotation states, necessitating the use of the Euler angle convention. In Fig.
10, we plot the maximum roll and yaw angle for each satellite over the final
10 days of the simulation. We see three distinct regions in this plot. If the
yaw angle stays below ${\lesssim}60^{\circ}$, we consider the satellite to be
in synchronous rotation, since its long axis stays pointed in the direction of
the primary. Out of the synchronous rotators, a minority are in “pure” (albeit
highly excited) synchronous rotation where the roll and pitch angles are also
${\lesssim}60^{\circ}$. Most of the synchronous rotators, however, are in a
rolling state about their long axis. In this so-called “barrel instability”,
the satellite remains on-average tidally locked to the primary although it
continues to roll about its long axis (Ćuk et al., 2021). This non-principal
axis rotation state within the 1:1 spin-orbit resonance could be long-lived as
this mode would dissipate inefficiently by tides. Since binary YORP (BYORP)
requires a synchronous secondary, this would also substantially weaken the
BYORP effect or prevent it entirely (Ćuk & Burns, 2005; Quillen et al., 2022).
Finally, we also see many satellites in an end-over-end tumbling state, where
their roll and yaw angles both reach $90^{\circ}$. Generally, the tumblers
have a higher eccentricity than the synchronous rotators as indicated in the
plot, however, the onset of chaotic tumbling also depends on the satellite’s
inertia ratios (Wisdom et al., 1984; Agrusa et al., 2021) as well as its
collision history.

Of course, it is no surprise that these satellites are never perfectly
tidally-locked, as these are short-term simulations in which the satellite has
undergone many collisions and mergers so its rotation state is naturally
excited. However, synchronization is the fastest-evolving tidal effect
(Goldreich & Sari, 2009), and we would therefore expect the satellite’s free
libration to damp on relatively short timescales, potentially within hundreds
of years since they are already in synchronous rotation (Meyer et al., 2023).
This may provide a simple explanation for why we observe so many synchronous
satellites (Pravec et al., 2016), without needing to invoke the more
complicated dynamical processes of rotational fission to achieve this
equilibrium (Jacobson & Scheeres, 2011). Instead, if the satellite forms via
accumulation of shed material, then it has a reasonable chance to form in or
near a tidally-locked state.

We are not aware of any studies that demonstrate a binary asteroid can
immediately form in a synchronous rotation state following rotational
disruption, although this seems like a relatively natural outcome. Formation
with synchronous rotation has been found in other studies on gravitational
accumulation near the Roche limit, such as the accretion of moonlets from
Saturn’s rings (Karjalainen & Salo, 2004) and from circumplanetary disks
(Hyodo et al., 2015).

Figure 9: The semimajor axis (in terms of the primary’s radius) and
eccentricity of the newly formed satellite, over all simulations with
$\phi=35^{\circ}$. The colors indicate the starting disk mass, and the three
points with error bars show the mean and $1\sigma$ standard deviation for the
three different disk masses. Although the variance is quite large, a larger
disk mass typically leads to a higher eccentricity and larger semimajor axis,
due to the disk having a higher angular momentum and more collisions that can
drive up the largest satellite’s eccentricity. Figure 10: The maximum roll and
yaw angles for each satellite, based on the final 10 days of the simulation.
There are three distinct post-formation rotation states for the satellites,
which are annotated on the plot to guide the reader’s eye. We consider the
satellite to be in synchronous rotation if its long axis remains aligned
towards the secondary (i.e., yaw $\lesssim 60^{\circ}$). The “barrel
instability” is a subset of the synchronous rotators where the secondary’s
long-axis stays largely aligned with the primary, yet it is able to roll about
its long axis like a barrel. Finally, a significant fraction of satellites
form in a tumbling state where all three Euler angles (roll, pitch, and yaw)
can reach $90^{\circ}$.

###  3.4 Satellite shape

While there is significant literature on binary asteroid formation, there are
no studies to our knowledge that directly model the expected shapes of the
satellite. Many studies consider the shape of the secondary, but do not model
that shape being formed directly (e.g. Jacobson & Scheeres, 2011; Davis &
Scheeres, 2020).

It is important to be clear with the definition of the satellite’s shape. In
this study, we define the shape of the satellite by its three principal axis
$a,b$ and $c$, which correspond to the body’s three principal moments $A,B$,
and $C$. We measure its axis lengths in two different ways. The first is
simply the physical extents of the body along these axes. The second measure
is the axis lengths of the dynamically equivalent equal-volume ellipsoid
(DEEVE). This is a uniform density ellipsoid having the same mass and moments
of inertia of the rubble pile. If the body has an approximately ellipsoidal
shape, then these two measures of its shape will match closely. Measuring the
body’s axis ratios by its DEEVE can be useful as they don’t fluctuate
significantly as a result of the motion of a single particle on the surface,
which is common as the satellite is forming. Therefore, we use the DEEVE axis
ratios in any time series plots as they are much less noisy. However, the
physical extent of the body tends to better represent the “true” shape of the
body and is most analogous to real life observations. In our simulations,
these two measures tend to differ significantly for highly irregular shapes
(high $a/b$ and/or $b/c$). This is demonstrated in Fig. 11, where the DEEVE
axis ratio tends to be quite larger than the physical extent axis ratio for
large axis ratios.

Figure 11: The DEEVE axis ratios of each simulated satellite compared to the
body’s axis ratios defined by its physical extents. Points lying under the
diagonal line have a DEEVE axis ratio (either $a/b$ or $b/c$) that is greater
than its physical extents and vice versa. Points lying on the line indicate
that the DEEVE shape is a good approximation of the physical shape. For
reference, we include the axis ratios of Dimorphos defined by its physical
extents and best-fit ellipsoid, demonstrating that Dimorphos’s shape is
exceptionally well-fit to an ellipsoid. Dimorphos’s $a/b$ and $b/c$ axis
lengths, including $1\sigma$ uncertainties, are indicated in blue and black,
respectively (Daly et al., 2023a)

Due to the discrepancy between the DEEVE-derived and extend-derived shapes
highlighted above, we compare both quantities to known secondary shapes for
thoroughness. In Fig. 12 we show both the DEEVE- and extent-derived semi-axis
ratios $a/b$ and $b/c$ at the end of each simulation with $\phi=35^{\circ}$.
We also include the shapes of the satellites of 66391 Moshup (formerly 1999
KW4), 2000 DP107, and 2001 SN263, which are the only publicly available radar-
derived shapes for the satellites of other binary asteroids (Ostro et al.,
2006; Naidu et al., 2015; Becker et al., 2015), as well as the axis ratios of
Dimorphos (Daly et al., 2023b, a). The updated shape of Dimorphos provided by
(Daly et al., 2023a) differs slightly from the initial assessment in (Daly et
al., 2023b), but the difference is small enough that we only plot the latest
values to avoid confusion. For each real asteroid system, we include $1\sigma$
uncertainties in $a/b$ and $b/c$, assuming that the reported uncertainties in
$a,b$ and $c$ from each respective paper are uncorrelated, which may not be
true. Therefore, the uncertainties should only be used to guide the reader’s
eye. The satellites of Moshup and DP107 are the best comparisons with
Dimorphos, as they are both S-type binaries, whereas SN263 is a C-type triple
(with large uncertainties in the satellite shapes). Fig. 12 demonstrates that,
generally speaking, these simulations produce satellites that are more
elongated (high $a/b$) and more flattened (high $b/c$ or $a/c$) than the
radar-derived secondaries. However, there are many cases that produce shapes
similar to radar-observed secondaries, given their uncertainties. Although no
satellites are produced within the $1\sigma$ uncertainty region of Dimorphos’s
shape, several simulations do come close. Dimorphos’s $b/c$ ratio lies
approximately in the middle of the simulated $b/c$ range. Although there is a
preference for elongated satellites, several simulations result in a low
elongation ($a/b\lesssim 1.1$), like Dimorphos. These simulations demonstrate
that more elongated shapes are preferred, although immediately forming a
Dimorphos-like shape by mass shedding is not implausible. Due to the
satellite’s accretion near the Roche limit, this strong preference for more
elongated shapes comes as no surprise and has been seen in analogous studies
(e.g. Porco et al., 2007; Hyodo et al., 2015). It is important to note that
the simulations here are run for only 100 days and there could be longer term
processes that will modify the satellite’s shapes (impacts, landslides, etc.),
so it is important to interpret any comparison between the simulated and
observed shapes with caution.

(a) DEEVE (b) Physical extents

Figure 12: The satellite’s shape, parameterized by its (a) DEEVE or (b)
physical axis ratios $a/b$ and $b/c$ for the 96 simulations with
$\phi=35^{\circ}$. The colored dots indicate the simulation results with
varying initial disk masses. The shape and $1\sigma$ uncertainty of Dimorphos,
along with the radar-derived shapes of the satellites of Moshup (1999 KW4),
DP107, and SN263 are shown for comparison (Daly et al., 2023a; Ostro et al.,
2006; Naidu et al., 2015; Becker et al., 2015).

We also compare our results to the lightcurve-derived shapes from Pravec et
al. (2016, 2019). Fig. 13 shows a histogram of the simulated $a/b$ ratio
compared to the dataset of Pravec et al. (2019) along with some new (not yet
published) secondaries (Pravec, personal communication). Here, we only include
lightcurve secondaries in which the primary is less than $20$ km in diameter
and the secondary-to-primary size ratio is less than 0.6, to ensure that we
are only comparing with binaries likely formed by YORP.

We find a remarkable agreement between the shapes of lightcurve secondaries
and those simulated in this work. Both data sets show that most satellites
have $a/b<1.6$, aside from a few outliers. The sharp drop-off in satellites
high $a/b$ has been attributed to chaotic dynamics of satellites with
$a/b\gtrapprox\sqrt{2}$ that ultimately destroys (or alters) them (Ćuk &
Nesvorný, 2010; Pravec et al., 2016). While this is a real dynamical effect,
we suggest that it may also be that secondaries simply don’t form with
extremely high elongations to begin with.

Interestingly, there are several secondaries with low elongations
($a/b\lesssim 1.1$) measured by Pravec. Naively, it appears as if the
simulations do a good job at matching this low elongation population. However,
it is important to note that lightcurves are heavily biased against detecting
satellites with a $a/b\lesssim 1.05$ (Pravec, personal communication). This
fact, combined with Dimorphos apparently having $a/b<1.1$ suggests the
existence of a significant population of secondaries with low $a/b$. If the
simulations presented here reasonably capture the formation of these
secondaries, we would expect the simulations to produce an overabundance of
low elongation secondaries compared to the observed population. This suggests
that either (1) there are other longer-term processes at play that could
reshape some satellites over time or (2), there is an effect not captured by
our model resulting from our assumptions or initial conditions.

Figure 13: Lightcurve-derived shapes are only sensitive to $a/b$, so we
construction histograms of $a/b$ from the simulations (defined by their
physical extents) to compare with $a/b$ of satellites derived from lightcurves
(Pravec et al., 2016, 2019). The y axis is not normalized since we are
comparing a similar number of real systems (87) and simulations (96).

###  3.5 Effect of friction angle

We conducted an additional set of simulations where the disk mass was held
constant at $M_{\text{disk}}=0.03M_{\text{A}}$ and the friction angle was
varied between $29^{\circ}$ and $40^{\circ}$. For each friction angle
($29^{\circ}$, $32^{\circ}$, and $40^{\circ}$), we conducted 8 simulations
with randomized initial conditions. With this smaller sample size, we are
dealing with small number statistics but can still draw a few basic
conclusions.

As the friction angle is decreased, there is significantly less variation
between simulations in general, including the satellite’s formation distance,
eccentricity, shape, and rotation state. In Fig.14a we show the satellite’s
final semimajor axis and eccentricity, along with an average and standard
deviation for each value of $\phi$. Although this is small number statistics,
there is a clear trend with friction; satellites with a lower friction angle
tend to form on a closer, more circular orbit. This is because the violent
processes such as collisions and close gravitational encounters that lead to
highly excited orbits tend to disrupt lower friction moonlets. This means that
the moonlets, which eventually merge to form the final satellite, are
necessarily on more circular orbits. Therefore, the final satellite will tend
to have a less excited orbit.

The axis ratios (based on the physical extents) are given in Fig. 14b. In
general, there is a smaller variation in the secondary’s shape as the friction
is reduced. In addition, a lower friction tends to result in a less flattened
object (i.e., more prolate). As the friction angle is lowered, the body is
forced to reshape itself until it is structurally stable, and bodies with
higher frictions are able to maintain a wider range of shapes. In the limit
that the friction angle were reduced to $0^{\circ}$, the satellite effectively
behaves like a fluid and would take on the shape of a Roche ellipsoid, at its
given orbital distance (Holsapple & Michel, 2006). This is not demonstrated
here, simply because a simulation with $\phi=0^{\circ}$ would take an
excessive amount of time to form a satellite.

A histogram of the secondary’s bulk density as a function of friction angle is
shown in Fig. 14c, where there is a clear trend. Bodies with a lower effective
friction are able to achieve a more efficient packing arrangement, thus having
less void space and a higher bulk density. At a grain density of 3.5 g cm-3,
this figure should be thought of as a lower limit for the secondary’s bulk
density. In reality, some void space could be filled with boulders that are
below the resolution of these simulations, so the “true” bulk density might be
be higher.

(a) Final orbits (b) Final shapes (extents)

(c) Bulk Density

Figure 14: Plots showing the effect of friction on the resulting satellite.
(a) The final semimajor axis and eccentricity from each simulation. (b) The
axis ratios of the satellite, based on its physical extents, as a function of
friction angle. (c) The bulk density of each satellite as a function of its
friction angle $\phi$. Generally, bodies with a lower effective friction are
able to achieve a more efficient packing and thus have a higher bulk density.

###  3.6 Matching the Shape of Dimorphos

Although these simulations demonstrate that immediately forming an oblate
satellite is relatively rare, we show two cases with a $\phi=35^{\circ}$ that
achieve shapes similar to Dimorphos. Although neither cases are a perfect
match, they do suggest that an oblate spheroid-like shape can plausibly be
formed from a single mass shedding event. In Fig. 15, we show the two
examples, both of which have an initial disk mass of
$M_{\text{disk}}=0.04M_{\text{A}}$. In the first example (Fig. 15a), the
satellite grows rapidly over the first several days until it undergoes a close
encounter with another satellite (not shown), which causes the satellite to
move inwards and undergo a partial tidal disruption event with the primary.
This torques the remaining mass onto a much wider, eccentric orbit and also
causes the satellite to rotate asynchronously yet maintain principal axis
rotation (as demonstrated by the roll and pitch angles remaining small and the
yaw angle circulating). As a result of the asynchronous rotation, the
satellite begins accreting material in an azimuthally symmetric manner, as it
has a different orientation at each periapse passage (where there is material
available to accrete). The satellite also undergoes a merger with the last
remaining moonlet at ${\sim}20$ d, which abruptly increases $b/c$. After the
merger, the satellite continues accreting on all sides, leading to a gradual
reduction in $a/b$ and increase in $b/c$, until there is almost no material
left to accrete. This process suggests that an oblate shape like that of
Dimorphos could plausibly be acquired if the satellite accretes material after
some dynamical process triggers an asynchronous rotation state.

In the second example (Fig. 15b), the satellite achieves its oblate shape
simply by undergoing two near-head-on mergers with other moonlets at early
times (around ${\sim}3.5$ d and ${\sim}7$ d, respectively). These mergers
serendipitously lead to a more oblate shape due to their favorable geometry.
In addition, the satellite’s relatively circular orbit, which is well outside
the Roche limit, prevents its shape from undergoing significant further
modification, despite its tumbling rotation state. Mergers among similar-sized
moonlets have been proposed to explain the oblate shapes of some of Saturn’s
small moons (Leleu et al., 2018).

In Fig. 16, we show renderings of these two examples in each body’s respective
body-fixed frame, using the same lighting conditions and viewing geometry as
the DART spacecraft on its approach to Dimorphos. Although these two examples
demonstrate that an oblate satellite can plausibly form by asynchronous
rotation during accretion or by lucky mergers, these events are relatively
rare. Out of over 100 simulations, Dimorphos-like shapes are only formed a few
percent of the time. Therefore, it is feasible that Dimorphos obtained its
shape through some other unexplored mechanism.

(a) Disk 069

(b) Disk 081

Figure 15: Two examples that result in a Dimorphos-like shape (i.e., low
$a/b$). (a) This satellite achieves an oblate shape due to a partial tidal
disruption torquing the remaining mass onto a wide, eccentric orbit. Then, due
to the body’s asynchronous rotation, it accretes material in an azimuthally
symmetric manner, as it sweeps through periapse with a different orientation
every orbit. (b) This satellite achieves an oblate shape due to serendipitous
mergers having a favorable geometry and is able to maintain its shape through
the rest of the simulation. Movies of these simulations are provided

(a) Disk 069
DEEVE: $a/b{\approx}1.15,b/c{\approx}1.69$
Extents: $a/b{\approx}1.14,b/c{\approx}1.49$

(b) Disk 081
DEEVE: $a/b{\approx}1.11$, $b/c{\approx}1.42$
Extents: $a/b{\approx}1.08$, $b/c{\approx}1.37$

(c) DRACO Image
DEEVE: $a/b{\approx}1.02$, $b/c{\approx}1.49$
Extents: $a/b{\approx}1.06$, $b/c{\approx}1.47$

Figure 16: A rendering of two satellites that have similar shapes to
Dimorphos, with the same lighting conditions and viewing geometry as DART’s
approach.

###  3.7 Unique outcomes

Due to the chaotic orbital and collisional evolution of the moonlets, some
simulations resulted in outcomes that were not expected at the conception of
this study. This includes the creation of asteroid pairs and triple systems.
To be clear, we do not claim that all or even a majority of pairs or triples
originate from a single mass shedding event, rather that this formation
mechanism is plausible for some systems.

####  3.7.1 Asteroid Pairs

An asteroid pair is where two unbound asteroids have similar enough orbital
elements (and colors/spectra; see Pravec et al. (2019), and references
therein) such that they can be traced back to a common origin. An asteroid
cluster is where multiple unbound bodies can be traced back to the same
system. Many pairs and clusters are thought to form either through collisions,
or through rotational fission (Vokrouhlický & Nesvorný, 2008; Pravec et al.,
2010, 2018). Because the dynamics of a newly formed binary system are chaotic
and have a positive free energy for mass ratios below ${\sim}0.2$, the
secondary can be ejected purely by the internal dynamics of the system
(Scheeres, 2007b; Jacobson & Scheeres, 2011). This phenomenon does an
excellent job at reproducing the observed correlation between the primary’s
rotation period and the mass ratio of the asteroid pair (Pravec et al., 2010,
2019). There is a distinct correlation between asteroid pair mass ratio and
the primary’s spin rate; pairs with a large mass ratio tend to have slower
primary rotation periods, presumably because more angular momentum was
transferred from the rapidly rotating primary to the secondary in order eject
it from the system.

Due to the azimuthal symmetry of the primary in these simulations, there is
very weak coupling between the primary’s rotation and the satellites orbit,
preventing these simulations from exploring this correlation. However, our
simulations demonstrate that pairs/clusters can also form via three-body
interactions between the moonlets during their formation. In all cases where
we form a successful pair or cluster, there is always a secondary still bound
to the primary. In other words, in any case where we form a pair, it is always
a “paired binary”.

As an example, we show a time-series plot of the semimajor axis and
eccentricity of the two largest satellites from Disk 040
($M_{\text{disk}}=0.03M_{A}$, $\phi=35^{\circ}$) in Fig. 17. In this case, the
smaller satellite ($M_{2}$) has a series of close encounters with the larger
satellite ($M_{1}$), which increase its semimajor axis and eccentricity until
its eccentricity eventually exceeds 1, placing it on an escape trajectory. The
particle then reaches the simulation boundary and is removed from the
simulation. In Fig. 18, we show snapshots of this simulation at the moment
when the $M_{2}$ is ejected. The two satellites have a close encounter at 48.3
d, where $M_{1}$ torques $M_{2}$ onto a hyperbolic orbit, ejecting it from the
system. Meanwhile, as a consequence of angular momentum conservation,
$M_{1}$’s eccentricity is raised (i.e., its periapse distance decreases)
causing it to have a close approach with the primary at 48.9 d. $M_{1}$ then
undergoes a partial tidal disruption, loosing some mass, reshaping, and
torquing it back onto a higher orbit safe from further tidal disruption. This
process also breaks $M_{1}$ from its previous synchronous rotation state.

Figure 17: An example of a paired binary being formed as the larger satellite
ejects the smaller satellite from the system. After an extremely close
encounter, the smaller satellite is ejected from the system
($e_{\text{orb}}>1$) before it hits the simulation boundary at
$100R_{\text{A}}^{\text{eq}}$ where it is then removed. This particular
simulation is Disk 040 ($M_{\text{disk}}/M_{\text{A}}=0.03$,
$\phi=35^{\circ}$) and a movie of this simulation is provided.

Figure 18: Simulation snapshots corresponding to Fig. 17 of the final close
encounter that ejects the smaller satellite from the system. When this occurs,
larger satellite’s periapse distance is lowered and its eccentricity is
raised. This leads to a partial tidal disruption where the remaining satellite
loses some mass and breaks from synchronous rotation. A movie of this
simulation is provided.

In Fig. 19a, we plot the size of each ejected fragment in relation to both the
primary and the secondary, from all simulations where $\phi=35^{\circ}$. It is
important to note that this figure excludes fragments consisting of one or two
particles, of which numerous are ejected from every simulation. Here, we only
show ejected fragments that contain 3 or more particles. Also, this figure
only includes fragments that have been ejected within the ${\sim}100$ d
pkdgrav integrations. However, as we will see in the next section, there are
several systems which still contain two satellites after 100 d that will be
ejected at a later time and would also count as a successful pair. In general,
we find that a larger initial mass shedding event tends to eject larger
fragments, as there is more material and angular momentum available. For
context, the smallest size ratio observed for an asteroid pair is roughly
${\sim}0.1$, so this is a size range that would be potentially discoverable
(Pravec et al., 2019). We also show the spin rate of each ejected fragment
(Fig. 19b), which demonstrates that most fragments are spinning relatively
fast, due to the high angular momentum nature of ejecting the fragment. The
observed population of asteroid pairs shows a much broader distribution of
spin rates of ejected secondaries, ranging between ${\sim}1-9$ d-1 (Pravec et
al., 2019). In addition, many of the pairs formed in these simulations are in
non-principal axis (NPA) rotation, whereas all observed secondaries of
asteroid pairs for which there is sufficient lightcurve coverage appear to be
in principal axis rotation. This suggests that these fragments undergo
significant spin evolution after being ejected (through damping and/or YORP)
or other mechanisms besides three body interactions are responsible for the
production of most asteroid pairs. The axis ratios are also shown in Fig. 19c
and are consistent with the axis ratios of observed pairs inferred from their
lightcurve amplitudes (Pravec et al., 2019). Finally, we also show the
eccentricity of the fragment before it is deleted from the simulation. The
vast majority are on hyperbolic (escape) trajectories, having
$e_{\text{orb}}>1$. However, some of these fragments are still bound to the
system. If these fragments were kept in the simulation, they would likely have
been ejected or collided during a future close encounter. If these are near-
Earth asteroids, these satellites would be lost anyways, having an apoapse
outside of the system’s Hill sphere. However, if these are main-belt asteroids
with larger Hill spheres, there is a chance that these objects deemed “pairs”
could remain bound with wide eccentric orbits.

For example, asteroid (3749) Balam is a main-belt triple with a fast-rotating
(2.8 h spin period), ${\sim}4$ km primary with a relatively close inner
satellite, separated by ${\sim}13$ km on a near-circular orbit (Pravec et al.,
2019). The outer satellite is on a wide orbit with an eccentricity between
0.35 and 0.77, (Merline et al., 2002; Vachier et al., 2012). The inner and
outer satellites have respective satellite-to-primary size ratios of
${\sim}0.46$ and ${\sim}0.24$ (Pravec et al., 2019), making them relatively
large. In addition, Balam is a member of an asteroid pair (Vokrouhlický,
2009), with backwards numerical integrations indicating that asteroid 2009
BR60 separated from Balam less than one million years ago (Vokrouhlický, 2009;
Pravec et al., 2019), making the Balam system very young. Due to its high
eccentricity, it has been suggested that Balam’s outer satellite could have
been formed as the result of a collision (Durda et al., 2004) or by capture
(Marchis et al., 2008), while the inner satellite is consistent with YORP
spin-up and rotational disruption due to the primary’s fast rotation
(Polishook et al., 2011). Since the simulations in this study considered much
more mild mass shedding events and removed these eccentric satellites, there
are no direct analogs to the Balam system in our study. However, our results
suggest that a plausible explanation for the Balam system could be formation
via a single, much more massive rotational disruption event. In such a
scenario, an inner satellite, an outer eccentric satellite, and an unbound
satellite could all be produced at the same time, although the details of such
a scenario require further study.

(a) Fragment size (b) Fragment spins

(c) Fragment shapes (d) Fragment eccentricity

Figure 19: Simulation results for all ejected fragments with
$\phi=35^{\circ}$. These plots ignore all fragments containing less than 3
particles, of which there are many. The different colors show the initial mass
of the disk. Only a minority of simulations eject fragments at all, but in
some cases, multiple fragments are ejected in a single simulation. (a) The
size of each ejected fragment as a function of both the fragment-to-primary
and fragment-to-secondary diameter ratio. Unsurprisingly, a more massive
initial disk ejects larger fragments. (b) A histogram of the spin rate of each
ejected fragment. (c) The shapes of each ejected fragment, defined by their
physical extents. (d) The eccentricity each fragment relative to the primary.
Most fragments are on escape trajectories ($e_{\text{orb}}>1$) while the
others have their apoapse outside of the systems Hill sphere if it were
located at 1 au.

####  3.7.2 Triples

In some cases, we find that 3-body interactions lead to the formation of
systems with two satellites on non-crossing orbits. However, these simulations
are run for less than 100 days, making it impossible to determine their
stability with pkdgrav alone. In rare cases where two satellites are still
remaining at the end of the pkdgrav simulation, we perform a brief test of the
system’s stability. We hand off the masses, positions, and velocities of the
primary and its satellites to the REBOUND $N$-body code (Rein & Liu, 2012;
Rein & Spiegel, 2015; Rein & Tamayo, 2017). These simulations are relatively
simple; the two satellites are treated as spheres but we include the effect of
the primary’s $J_{2}$ using the REBOUNDx package (Tamayo et al., 2020). As
these simulations do not include higher-order effects such as spin-orbit
coupling of the two satellites, tidal dissipation, BYORP, etc., we only
integrate the system for $10^{3}$ years and then check if both satellites
still remain. In reality, higher-order effects may strongly affect the
system’s evolution and stability. Additionally, the primary is largely
azimuthally symmetric, and a more realistic primary with a non-negligible
$C_{22}$ may make these close-in triple systems less stable. Here, we merely
demonstrate that the formation of a triple system from a single rotational
disruption event is plausible. In total, we find 11 triple systems after the
100 d pkdgrav simulations, of which 7 survive after further integration with
REBOUND.

We find that many of the stable triples achieved their stability due to
capture into mean motion resonances (MMRs) during the pkdgrav stage. While
moonlets are growing, their orbits are constantly perturbed by collisions,
mergers, and close encounters, so capture into a MMR is effectively random.
Out of the 7 stable triples, 5 are in MMRs, while two are not in resonance.

In three of the resonant cases, the two satellites form in a co-orbital
configuration (i.e., the 1:1 MMR) by co-accretion. When this occurs, the
smaller satellite occupies the $L_{4}$ or $L_{5}$ Lagrange point of the larger
body. Snapshots of the three cases are shown in Fig. 20. These systems satisfy
the condition for stability, in which
$\frac{M_{1}+M_{2}}{M+M_{1}+M_{2}}\lesssim 0.04$, where $M$ is the primary’s
mass and $M_{1}$ and $M_{2}$ are the two satellite masses (Murray & Dermott,
2000; Laughlin & Chambers, 2002; Sicardy, 2010). We also numerically test the
system’s stability using REBOUND and find that the smaller satellite remains
in stable libration about the Lagrange point for at least $10^{3}$ yr. We
restrict the REBOUND simulations to only $10^{3}$ yr, because we neglect tides
and non-gravitational forces which can tighten or loosen the tadpole orbits,
depending on the direction of migration (e.g., Fleming & Hamilton, 2000). We
find that co-orbital systems are only formed when the initial disk mass is
small ($M_{\text{disk}}/M_{\text{A}}\lesssim 0.03$), in agreement with the
study of Hyodo et al. (2015) on the formation of multiple satellites with
$N$-body simulations in circumplanetary disk.

We note that the idealized initial conditions of these simulations may
increase the likelihood of forming co-orbital satellites and that the
primary’s azimuthal symmetry likely contributes to their long term stability.
Here, we merely point out that forming a co-orbital system via a single mass
shedding event is possible, albeit unlikely. A future detection of a triple
asteroid with co-orbital satellites, however unlikely, could be explained by
co-accretion following a single mass shedding event.

Figure 20: Three cases where the co-orbital configuration of two satellites
remains stable after a $10^{3}$ yr integration with REBOUND. The top panels
show a view of the pkdgrav simulation after ${\sim}100$ d prior to handing off
to REBOUND, looking down from the primary’s spin pole. We encourage the reader
to view the provided movies of these simulations. The corresponding bottom
panels show the position of the two satellites in the rotating frame of the
larger satellite at each output over the $10^{3}$ yr REBOUND integration,
demonstrating that the smaller satellite resides in a Lagrange point (either
$L_{4}$ or $L_{5}$). In the left case (Disk 003), the smaller satellite
librates around the trailing Lagrange point ($L_{5}$) with an amplitude of
${\sim}7.2^{\circ}$, while in the middle and right cases (Disk 010 and Disk
108), the smaller satellite librates around the leading Lagrange point
($L_{4}$) with amplitudes of ${\sim}11.4^{\circ}$ and ${\sim}18.7^{\circ}$,
respectively. Movies of these threes simulations are provided.

The remaining two resonant cases are capture into the 2:1 and 5:2 MMRs. This
occurs after a satellite is first torqued onto a wider orbit
(${\gtrsim}4R_{A}$), while there is still material near the Roche limit
available to accrete. This allows a second body to grow closer in, which can
capture into a nearby MMR with the outer satellite while it is accreting. Once
the initial capture occurs, the resonance can further stabilize due to the
smooth growth of the inner satellite.

In Fig. 21 we show an example of capture into the 2:1 MMR, where the inner
satellite is the most massive. In Fig. 21a, we show a top-down view of the
triple system at the end of the 100 d pkdgrav integration along with a time-
series plot in Fig. 21b. Early on, both $M_{1}$ and $M_{2}$ grow rapidly until
they temporarily merge into a single body which is then quickly tidally
disrupted. The tidal disruption torques one large fragment (now $M_{2}$) onto
a wide orbit. Then, $M_{2}$ continues to grow and migrate outwards through a
series of mergers and close encounters with other moonlets. Meanwhile a new
inner satellite forms (now $M_{1}$) from one of the disrupted remnants and
migrates inwards. This process places $M_{1}$ and $M_{2}$ close enough to the
2:1 MMR that they capture. The resonance is then further stabilized by the
smooth growth of $M_{1}$, the inner satellite. We can confirm the existence of
the resonance due to the libration of the resonance argument
$\phi_{\text{res}}=2\lambda_{2}-\lambda_{1}-\varpi_{2}$, where $\lambda_{i}$
are the respective body’s mean longitudes and $\varpi_{2}$ is $M_{2}$’s
longitude of pericenter. We attribute the drift in the resonance argument to
the growth in the satellite masses and collisions, both of which make the
process non-adiabatic. We confirmed that the resonance argument continues to
librate when handed off to REBOUND.

We also find one example of capture into the 5:2 MMR, shown in Fig. 22.
Capture into such a high-order resonance came as a surprise and may have been
enabled by the high eccentricity of the outer satellite. If a triple asteroid
were to be observed in such a resonance, then this process should be studied
further.

We find two other long-term stable triples (disk 075, 085) with a more massive
outer satellite having period ratios close to 2:1 and 4:1 respectively,
although we confirm that are not in resonance. However, these systems have
well-separated satellites and thus remain long-term stable in the REBOUND
simulations.

Comparing our simulated triples to the known population is a problem of small
number statistics. There are very few known triple asteroids, none of which
are known to be in mean motion resonances. Out of the near-Earth asteroids,
there are three confirmed triples: 1994 CC, 2001 SN263, and 3122 Florence
(Brozovic et al., 2009; Nolan et al., 2008). Both CC and SN263 are well-
characterized, not near MMRs, and also have some mutual inclination, which as
been attributed a previous close planetary encounter (Brozovic et al., 2009;
Fang & Margot, 2012; Becker et al., 2015). However, no detailed dynamical
studies have been published for 3122 Florence. Preliminary estimates have put
the orbit periods of the two satellites at 22 h and 7 h (Brozovic et al.,
2018), which is not far from the 3:1 MMR, although a more comprehensive
analysis of the satellite orbits would be needed to confirm this. Despite our
simulations occasionally resulting in a stable triple in a mean-motion
resonance, a single mass shedding event is certainly not the only way to
achieve such a configuration. For example, a second mass-shedding event
occurring after the first satellite has migrated outwards, followed by
convergent migration (due to tides and/or BYORP) is an equally plausible
mechanism to form a MMR triple.

(a) Final frame from pkdgrav simulation

(b) Time series showing capture into 2:1 MMR.

Figure 21: An example of a stable triple system formed from a single mass
shedding event. (a) A snapshot of the system at the end of the simulation. (b)
A time-series plot showing the satellite mass, semimajor axis, eccentricity,
and resonance argument, $\phi_{\text{res}}$. Capture into the 2:1 MMR enables
the system to remain stable.

(a) Final frame from pkdgrav simulation

(b) Time series showing capture into 5:2 MMR

Figure 22: The two satellites capture into the 5:2 MMR around 15 d. Due to
$M_{2}$’s high eccentricity, the two satellites have a close encounter,
sending $M_{1}$ inwards where it undergoes a tidal disruption around 30 d,
which destroys both $M_{1}$ and the resonance. Around 60 d, a new inner
satellite begins growing and quickly captures again into the 5:2 resonance
again with the more massive outer satellite, this time remaining stable.

####  3.7.3 Simulation outcomes

In Fig. 23, we show the simulation outcomes for all simulations with
$\phi=35^{\circ}$, summarizing the resulting rotation state of the secondary
and whether the system formed a pair or triple. We consider the satellite to
be in synchronous rotation if its maximum yaw angle over the final 10 days of
the simulation is $<60^{\circ}$. If a synchronous rotator has a roll angle
$>60^{\circ}$ over the final 10 d, we also consider it to be in the “barrel
instability” (see Fig. 10). If a fragment consisting of 3 or more particles is
ejected from the system, we count the simulation as successfully forming a
pair. Triples are only counted if the system remains stable after further
integration with REBOUND. Each category is not exclusive, in the sense that a
triple could have also formed a pair, for example.

We find that a lower disk mass leads to a higher likelihood of the satellite
forming with synchronous rotation, likely due to fewer violent collisions and
mergers. For similar reasons, we find that more massive initial disks are more
likely to form asteroid pairs as there is more mass (and angular momentum)
that can launch fragments onto escape trajectories. Given that the final
outcomes are highly dependent on the initial angular momentum in the disk, we
caution against over-interpreting Fig. 23. A higher fidelity study with more
realistic initial conditions would provide better estimates for the fraction
of triples, pairs, and synchronous rotators that could plausibly form from a
single mass shedding event.

Figure 23: A summary of simulation outcomes showing the percentage of cases
that have a given outcome, considering only the 96 simulations where
$\phi=35^{\circ}$. A satellite is counted as a synchronous rotator if its yaw
angle remains $<60^{\circ}$ over the final 10 d of simulation. Similarly, a
satellite is considered to be in the barrel instability if its roll angle is
$>60^{\circ}$ over the final 10 d. These thresholds are set arbitrarily and
should only be considered notional.

##  4 Conclusions

We demonstrate that satellites can rapidly form by accretion following a
single mass shedding event. The formation time is relatively fast, with the
satellite reaching its terminal mass in a matter of days. Satellites tend to
form on near-circular orbits within ${\sim}4$ primary radii, although 3-body
interactions with other moonlets can sometimes put the final satellite onto
wider and more eccentric orbits. Due to the satellite’s accretion near the
Roche limit, the satellite tends to be prolate in shape and is oftentimes in
synchronous rotation. Many of these synchronous rotators, however, are in a
rolling state about their long axis while remaining tidally locked, which
would terminate or significantly weaken any BYORP evolution until this mode is
dissipated (Ćuk et al., 2021; Quillen et al., 2022).

The simulated secondary shapes are broadly consistent with the radar-derived
shapes of other asteroid satellites, given their uncertainties (Ostro et al.,
2006; Naidu et al., 2015; Becker et al., 2015). In addition, the distribution
of secondary elongations ($a/b$) agrees well with the lightcurve-derived data
set from Pravec et al. (2019). Our simulations rarely produce satellites with
$a/b\gtrsim 1.5$ in strong agreement with the observed population of
secondaries. Despite the strong preference to form elongated satellites,
oblate shapes like Dimorphos are occasionally formed in our simulations.
However, the fact that lightcurves are heavily biased against measuring
satellite shapes with low $a/b$, combined with Dimorphos’s low $a/b$ suggests
the existence of a significant population of oblate secondaries (unless
Dimorphos is an outlier). This implies that there may be a longer term, yet-
unidentified pathway that can reshape satellites to become oblate.

We tested four different friction angles
($29^{\circ},32^{\circ},35^{\circ},40^{\circ}$), finding only subtle
differences. Generally, higher friction satellites have more irregular shapes.
In addition, they are more resistant to disruption during close gravitational
encounters and collisions, allowing them to form on wider, more excited
orbits, whereas a lower friction satellite would disrupt and reaccrete during
this process. The most noticeable difference is that higher friction tends to
form satellites with lower bulk densities, due to their ability to maintain
larger interior void spaces.

The accretion process is highly chaotic, resulting in a wide range of
outcomes. In some cases, moonlets are ejected from the system via three-body
interactions, effectively forming asteroid pairs. We note that this process
does not explain the population of asteroid pairs in general, as there are
many other processes that can form pairs. Here, we merely demonstrate that
3-body interactions may play some role in the production of asteroid pairs.

In a few percent of cases, we form stable triple systems, including satellites
in mean motion resonances and co-orbital satellites. Although unlikely, a
future discover of a system with co-orbital satellites would be strong
evidence for formation from a single mass shedding event, as co-accretion
would naturally explain their configuration.

There are some caveats and limitations of our model. These simulations are
only run for 100 days due to their computational cost and therefore cannot
capture longer term effects. In addition, the primary is azimuthally symmetric
which may affect the likelihood of forming pairs and triples. Finally, this
study was originally focused solely on forming a Dimorphos-mass satellite,
therefore we only explored a narrow range of disk masses. Although the initial
conditions of these simulations were intentionally simplified, we would expect
the general results and trends to hold.

There are a wide range of parameters available to explore in future work, and
we list a few avenues to explore. With stronger spin-orbit coupling, an
asymmetric primary may change the fraction of systems that form asteroid
pairs, and would also allow the observed correlation between pair mass ratio
and primary spin rate to be explored. In addition, this may affect the
likelihood of forming a triple, as well as their long-term stability. In
addition, a larger disk mass should be explored to probe the formation of
systems with larger mass ratios such as Balam, and would also strongly affect
the production of unbound fragments. Rather than a single mass shedding event,
its plausible that these systems form from multiple events. If this is the
case, simulated additional mass shedding events after the satellite is
initially formed may change the distribution of secondary shapes.

The Hera mission will investigate the Didymos system for 6 months in early
2027 and will provide crucial information for binary formation models (Michel
et al., 2022). In particular, it will measure the mass of Dimorphos and its
internal properties, allowing us to check some of the predictions from our
modeling. Through the landing and possible bouncing of its CubeSats on
Dimorphos’ surface, we will be able to access some of the mechanical
parameters used in these simulations. However, since the shape of Dimorphos
may have changed as a result of the DART impact (Raducan et al., 2023), we may
not be able to put constraints on the formation models leading to either
oblate or prolate secondaries, although all efforts will be done to relate the
measured shape by Hera to the original one.

## Acknowledgments

This work was supported in part by the DART mission, NASA Contract
#80MSFC20D0004 to JHU/APL. H.F.A. was supported by the French government,
through the UCA J.E.D.I. Investments in the Future project managed by the
National Research Agency (ANR) with the reference number ANR-15-IDEX-01. P.P.
was supported by the Grant Agency of the Czech Republic, grant 20-04431S. P.
M. acknowledges funding support from the French space agency CNES and the
University of Tokyo. M.P., A.L., and F.T. were supported by the Italian Space
Agency (ASI) within the LICIACube project (ASI-INAF agreement n. 2019-31-HH.0)
and Hera project (ASI-INAF agreement n. 2022-8-HH.0). S.D.R. acknowledges
support from the Swiss National Science Foundation (project number
200021_207359). We thank Adriano Campo Bagatin, Kate Minker, and John
Wimarsson for helpful discussions and feedback. We are grateful to David
Minton and the anonymous referee, whose feedback substantially improved the
quality of this manuscript.

Simulations and analysis were performed on the ASTRA cluster administered by
the Center for Theory and Computation, part of the Department of Astronomy at
the University of Maryland and on Mésocentre SIGAMM hosted at the Observatoire
de la Côte d’Azur.

##  Appendix A Notation

For reference, Table 2 lists the notation used in this paper.

Table 2: Notation table

Variable | Definition
---|---
$M_{\text{A}}$ | Primary’s mass
$M_{\text{B}}$ | Secondary’s massa
$M_{\text{disk}}$ | Initial mass of disk placed in orbit around primary
$R_{\text{A}}$, $D_{\text{A}}$ | Primary’s radius and diameterb
$R_{\text{A}}^{\text{eq}}$ | Primary’s equatorial radius
$R_{\text{B}}$, $D_{\text{B}}$ | Secondary’s radius and diameterb
$a$ | Secondary’s semimajor (longest) axis lengthc
$b$ | Secondary’s semi-intermediate axisc
$c$ | Secondary’s semi-minor axis (i.e., principal spin axis)c
$r_{\text{orb}}$ | Instantaneous binary orbital separation
$a_{\text{orb}}$ | Semimajor axisd
$e_{\text{orb}}$ | Orbital eccentricityd
$P_{\text{orb}}$ | Orbital periodd
$\theta$ | Libration anglee
$\theta_{1},\theta_{2},\theta_{3}$ | The roll, pitch, and yaw anglesf
$\phi$ | Friction angle

Note. — a$M_{1}$ and $M_{2}$ are used when there is more than one satellite.
bBased on the body’s volume-equivalent sphere. cThis length is measured either
based on the body’s physical extents, or by the axis lengths of the
corresponding dynamically equivalent equal-volume ellipsoid. dKeplerian
element, based on the instantaneous body positions and velocities. eThe angle
between the secondary’s long axis and the line connecting the body centers
fThe 1-2-3 Euler angle set coordinated in the secondary’s rotating orbital
frame (see Agrusa et al. (2021)).

##  Appendix B Simulation Results

Table 3 shows the results of each simulation in this study.

Table 3: Tabulated simulation outcomes for each simulation.

ID | $\frac{M_{\text{disk}}}{M_{\text{A}}}$ | $\phi$ | $M_{\text{B}}/M_{\text{A}}$ | $D_{\text{B}}/D_{\text{A}}$ |  $a/b$, $b/c$ |  $a/b$, $b/c$ | $\frac{a_{\text{orb}}}{R_{\text{A}}}$ | $e_{\text{orb}}$ | $P_{\text{orb}}$ | $\theta_{\text{max}}$ | $N_{\text{frag}}$ | Stable?
---|---|---|---|---|---|---|---|---|---|---|---|---
|  | [∘] |  | (DEEVE) | (DEEVE) | (Extents) |  |  | [H] | [∘] |  |
001 | 0.02 | $35^{\circ}$ | 0.0081 | 0.206 | 1.299, 1.384 | 1.29, 1.279 | 2.774 | 0.0407 | 9.8 | 26.1 | 0 |
002 | 0.02 | $35^{\circ}$ | 0.0069 | 0.197 | 1.241, 1.242 | 1.237, 1.248 | 3.177 | 0.0387 | 12.02 | 51.8 | 0 |
003 | 0.02 | $35^{\circ}$ | 0.0069 | 0.198 | 1.501, 1.656 | 1.474, 1.464 | 2.872 | 0.0403 | 10.33 | 12.7 | 0 |
|  |  | 0.0028 | 0.146 | 1.673, 1.298 | 1.612, 1.213 | 2.889 | 0.0371 | 10.44 | 12.7 |  | Yes
004 | 0.02 | $35^{\circ}$ | 0.0074 | 0.203 | 1.849, 1.177 | 1.793, 1.131 | 2.706 | 0.0603 | 9.45 | 18 | 0 |
|  |  | 0.0011 | 0.106 | 1.115, 1.178 | 1.09, 1.197 | 39.539 | 0.9324 | 529.15 | 88.8 |  | No
005 | 0.02 | $35^{\circ}$ | 0.0062 | 0.19 | 1.452, 1.156 | 1.447, 1.165 | 3.86 | 0.0126 | 16.1 | 89.3 | 0 |
006 | 0.02 | $35^{\circ}$ | 0.0099 | 0.226 | 1.308, 1.853 | 1.268, 1.606 | 2.873 | 0.0232 | 10.32 | 48 | 0 |
007 | 0.02 | $35^{\circ}$ | 0.0082 | 0.208 | 1.326, 1.503 | 1.305, 1.409 | 3.346 | 0.0296 | 12.98 | 16.2 | 0 |
008 | 0.02 | $35^{\circ}$ | 0.0085 | 0.21 | 1.318, 1.202 | 1.267, 1.177 | 3.155 | 0.0345 | 11.88 | 16.6 | 0 |
009 | 0.02 | $35^{\circ}$ | 0.0087 | 0.215 | 1.623, 1.422 | 1.51, 1.334 | 3.081 | 0.0134 | 11.47 | 23.5 | 0 |
010 | 0.02 | $35^{\circ}$ | 0.0076 | 0.202 | 1.411, 1.504 | 1.254, 1.506 | 2.743 | 0.0689 | 9.64 | 22.5 | 0 |
|  |  | 0.0014 | 0.116 | 1.723, 1.06 | 1.511, 1.127 | 2.756 | 0.0426 | 9.74 | 20.8 |  | Yes
011 | 0.02 | $35^{\circ}$ | 0.0089 | 0.214 | 1.557, 1.133 | 1.294, 1.146 | 2.87 | 0.0175 | 10.31 | 17.6 | 0 |
012 | 0.02 | $35^{\circ}$ | 0.0078 | 0.204 | 1.535, 1.241 | 1.414, 1.251 | 3.32 | 0.0361 | 12.83 | 18.6 | 0 |
013 | 0.02 | $35^{\circ}$ | 0.0091 | 0.215 | 1.576, 1.183 | 1.419, 1.24 | 3.033 | 0.0775 | 11.2 | 30.3 | 0 |
014 | 0.02 | $35^{\circ}$ | 0.0058 | 0.184 | 1.188, 1.204 | 1.164, 1.168 | 2.639 | 0.0584 | 9.1 | 89.1 | 0 |
|  |  | 0.0027 | 0.143 | 1.207, 1.042 | 1.28, 1.005 | 4.029 | 0.1254 | 17.2 | 90 |  | Yes
015 | 0.02 | $35^{\circ}$ | 0.0081 | 0.206 | 1.217, 1.148 | 1.154, 1.11 | 3.474 | 0.0782 | 13.74 | 89.5 | 0 |
016 | 0.02 | $35^{\circ}$ | 0.0092 | 0.217 | 1.473, 1.743 | 1.342, 1.701 | 2.807 | 0.0429 | 9.97 | 12.5 | 0 |
017 | 0.02 | $35^{\circ}$ | 0.0069 | 0.196 | 1.39, 1.191 | 1.322, 1.161 | 3.654 | 0.0315 | 14.82 | 87.9 | 0 |
018 | 0.02 | $35^{\circ}$ | 0.0069 | 0.197 | 1.316, 1.149 | 1.228, 1.163 | 3.459 | 0.0833 | 13.65 | 90 | 0 |
019 | 0.02 | $35^{\circ}$ | 0.0072 | 0.2 | 1.78, 1.231 | 1.585, 1.161 | 3.4 | 0.0233 | 13.31 | 49.1 | 0 |
020 | 0.02 | $35^{\circ}$ | 0.0089 | 0.215 | 1.356, 1.676 | 1.336, 1.586 | 2.77 | 0.0556 | 9.78 | 21.3 | 1 |
021 | 0.02 | $35^{\circ}$ | 0.0045 | 0.172 | 1.445, 1.261 | 1.411, 1.141 | 4.54 | 0.2352 | 20.56 | 89.9 | 0 |
|  |  | 0.0025 | 0.137 | 1.509, 1.049 | 1.446, 0.976 | 2.551 | 0.0716 | 8.67 | 62.5 |  | Yes
022 | 0.02 | $35^{\circ}$ | 0.0068 | 0.194 | 1.293, 1.201 | 1.228, 1.23 | 3.741 | 0.1173 | 15.36 | 89.9 | 0 |
023 | 0.02 | $35^{\circ}$ | 0.0072 | 0.2 | 1.523, 1.207 | 1.382, 1.156 | 3.545 | 0.0188 | 14.16 | 25.7 | 0 |
024 | 0.02 | $35^{\circ}$ | 0.0076 | 0.203 | 1.669, 1.121 | 1.559, 1.128 | 3.405 | 0.0221 | 13.33 | 12.7 | 0 |
025 | 0.02 | $35^{\circ}$ | 0.0094 | 0.218 | 1.411, 1.723 | 1.385, 1.658 | 2.875 | 0.0467 | 10.33 | 7.5 | 0 |
026 | 0.02 | $35^{\circ}$ | 0.0087 | 0.218 | 1.609, 1.329 | 1.411, 1.34 | 3.061 | 0.036 | 11.35 | 87.5 | 0 |
027 | 0.02 | $35^{\circ}$ | 0.0082 | 0.208 | 1.583, 1.251 | 1.474, 1.171 | 2.708 | 0.0516 | 9.45 | 11.7 | 1 |
028 | 0.02 | $35^{\circ}$ | 0.0092 | 0.216 | 1.436, 1.638 | 1.414, 1.441 | 2.653 | 0.0196 | 9.16 | 36.8 | 0 |
029 | 0.02 | $35^{\circ}$ | 0.0068 | 0.195 | 1.188, 1.192 | 1.127, 1.24 | 3.432 | 0.0261 | 13.5 | 35.1 | 1 |
030 | 0.02 | $35^{\circ}$ | 0.0081 | 0.206 | 1.522, 1.484 | 1.447, 1.472 | 2.623 | 0.0692 | 9.01 | 47.6 | 0 |
031 | 0.02 | $35^{\circ}$ | 0.0084 | 0.211 | 1.718, 1.331 | 1.615, 1.304 | 3.074 | 0.0624 | 11.43 | 27.1 | 1 |
032 | 0.02 | $35^{\circ}$ | 0.0071 | 0.198 | 1.445, 1.357 | 1.377, 1.335 | 3.531 | 0.0395 | 14.08 | 25.4 | 0 |
033 | 0.03 | $35^{\circ}$ | 0.013 | 0.242 | 1.411, 1.318 | 1.348, 1.309 | 3.263 | 0.0277 | 12.47 | 35.9 | 1 |
034 | 0.03 | $35^{\circ}$ | 0.0137 | 0.246 | 1.726, 1.381 | 1.509, 1.348 | 3.465 | 0.0274 | 13.65 | 88.9 | 0 |
035 | 0.03 | $35^{\circ}$ | 0.0125 | 0.238 | 1.228, 1.445 | 1.134, 1.389 | 3.974 | 0.0068 | 16.77 | 89.9 | 0 |
036 | 0.03 | $35^{\circ}$ | 0.0139 | 0.249 | 1.568, 1.335 | 1.468, 1.22 | 3.136 | 0.0399 | 11.75 | 35 | 1 |
037 | 0.03 | $35^{\circ}$ | 0.0123 | 0.238 | 1.317, 1.208 | 1.337, 1.08 | 4.089 | 0.0486 | 17.51 | 89.6 | 0 |
038 | 0.03 | $35^{\circ}$ | 0.0151 | 0.254 | 1.515, 1.172 | 1.473, 1.156 | 3.256 | 0.0391 | 12.42 | 44.7 | 1 |
039 | 0.03 | $35^{\circ}$ | 0.0135 | 0.245 | 1.16, 1.539 | 1.155, 1.405 | 4.014 | 0.2034 | 17.02 | 89.6 | 1 |
040 | 0.03 | $35^{\circ}$ | 0.0099 | 0.221 | 1.249, 1.252 | 1.205, 1.223 | 3.37 | 0.1597 | 13.11 | 89.2 | 1 |
041 | 0.03 | $35^{\circ}$ | 0.0159 | 0.261 | 1.15, 1.968 | 1.095, 1.79 | 3.054 | 0.0233 | 11.28 | 56.3 | 0 |
042 | 0.03 | $35^{\circ}$ | 0.0136 | 0.246 | 1.198, 1.427 | 1.179, 1.383 | 3.677 | 0.0952 | 14.92 | 88.1 | 0 |
043 | 0.03 | $35^{\circ}$ | 0.0136 | 0.245 | 1.191, 1.511 | 1.206, 1.381 | 2.55 | 0.0647 | 8.61 | 89.6 | 2 |
044 | 0.03 | $35^{\circ}$ | 0.0146 | 0.253 | 1.449, 1.355 | 1.377, 1.245 | 3.535 | 0.0377 | 14.05 | 77.6 | 0 |
045 | 0.03 | $35^{\circ}$ | 0.0122 | 0.237 | 1.499, 1.455 | 1.38, 1.378 | 2.575 | 0.0651 | 8.75 | 22 | 0 |
|  |  | 0.0026 | 0.142 | 1.522, 1.188 | 1.297, 1.238 | 17.675 | 0.7959 | 158.06 | 89.1 |  | No
046 | 0.03 | $35^{\circ}$ | 0.0134 | 0.245 | 1.573, 1.135 | 1.498, 1.129 | 3.684 | 0.0363 | 14.96 | 46.2 | 0 |
047 | 0.03 | $35^{\circ}$ | 0.0128 | 0.241 | 1.206, 1.561 | 1.172, 1.423 | 3.84 | 0.1239 | 15.93 | 89 | 0 |
048 | 0.03 | $35^{\circ}$ | 0.0119 | 0.235 | 1.255, 1.232 | 1.241, 1.211 | 4.033 | 0.0693 | 17.15 | 89.5 | 0 |
049 | 0.03 | $35^{\circ}$ | 0.0156 | 0.259 | 1.435, 1.751 | 1.419, 1.648 | 3.154 | 0.0192 | 11.84 | 18.9 | 0 |
050 | 0.03 | $35^{\circ}$ | 0.014 | 0.248 | 1.148, 1.184 | 1.082, 1.269 | 3.56 | 0.0325 | 14.21 | 89.7 | 0 |
051 | 0.03 | $35^{\circ}$ | 0.0121 | 0.237 | 1.225, 1.54 | 1.208, 1.384 | 3.993 | 0.0192 | 16.89 | 77.1 | 1 |
052 | 0.03 | $35^{\circ}$ | 0.0133 | 0.245 | 1.467, 1.308 | 1.42, 1.243 | 3.656 | 0.0343 | 14.79 | 89.7 | 0 |
053 | 0.03 | $35^{\circ}$ | 0.015 | 0.255 | 1.462, 1.737 | 1.348, 1.58 | 3.218 | 0.0697 | 12.2 | 28.6 | 0 |
054 | 0.03 | $35^{\circ}$ | 0.0147 | 0.252 | 1.269, 1.963 | 1.225, 1.83 | 2.546 | 0.014 | 8.59 | 32.7 | 3 |
055 | 0.03 | $35^{\circ}$ | 0.0142 | 0.25 | 1.066, 1.912 | 1.055, 1.719 | 3.521 | 0.0544 | 13.98 | 88.4 | 2 |
056 | 0.03 | $35^{\circ}$ | 0.0147 | 0.252 | 1.544, 1.443 | 1.449, 1.319 | 3.415 | 0.021 | 13.34 | 45.6 | 0 |
057 | 0.03 | $35^{\circ}$ | 0.0121 | 0.238 | 1.359, 1.324 | 1.342, 1.175 | 4.075 | 0.0521 | 17.41 | 88 | 0 |
|  |  | 0.0019 | 0.129 | 1.386, 1.091 | 1.288, 1.1 | 2.397 | 0.0143 | 7.89 | 46.2 |  | Yes
058 | 0.03 | $35^{\circ}$ | 0.0119 | 0.235 | 1.112, 1.259 | 1.074, 1.181 | 3.98 | 0.045 | 16.81 | 89.7 | 0 |
059 | 0.03 | $35^{\circ}$ | 0.0112 | 0.23 | 1.133, 1.13 | 1.1, 1.124 | 4.31 | 0.129 | 18.95 | 89.9 | 0 |
060 | 0.03 | $35^{\circ}$ | 0.0145 | 0.252 | 1.347, 1.779 | 1.331, 1.529 | 3.374 | 0.0415 | 13.1 | 53.1 | 0 |
061 | 0.03 | $35^{\circ}$ | 0.0148 | 0.254 | 1.182, 1.693 | 1.151, 1.581 | 3.361 | 0.029 | 13.03 | 53 | 1 |
062 | 0.03 | $35^{\circ}$ | 0.0153 | 0.257 | 1.149, 1.389 | 1.106, 1.324 | 3.269 | 0.0187 | 12.49 | 30 | 0 |
063 | 0.03 | $35^{\circ}$ | 0.0158 | 0.26 | 1.625, 1.71 | 1.479, 1.639 | 3.098 | 0.0468 | 11.52 | 12.9 | 0 |
064 | 0.03 | $35^{\circ}$ | 0.0144 | 0.252 | 1.552, 1.306 | 1.528, 1.27 | 3.493 | 0.1292 | 13.8 | 89.6 | 0 |
065 | 0.04 | $35^{\circ}$ | 0.0216 | 0.288 | 1.309, 2.051 | 1.26, 1.881 | 3.282 | 0.0721 | 12.53 | 57 | 0 |
066 | 0.04 | $35^{\circ}$ | 0.0192 | 0.277 | 1.328, 1.215 | 1.236, 1.178 | 3.863 | 0.0762 | 16.02 | 75.5 | 1 |
067 | 0.04 | $35^{\circ}$ | 0.0225 | 0.292 | 1.372, 1.972 | 1.337, 1.839 | 3.056 | 0.0118 | 11.26 | 22.7 | 0 |
068 | 0.04 | $35^{\circ}$ | 0.0174 | 0.267 | 1.288, 1.366 | 1.26, 1.326 | 4.359 | 0.1232 | 19.22 | 89.9 | 0 |
069 | 0.04 | $35^{\circ}$ | 0.0176 | 0.27 | 1.153, 1.686 | 1.137, 1.492 | 4.549 | 0.3122 | 20.49 | 89.8 | 0 |
070 | 0.04 | $35^{\circ}$ | 0.0209 | 0.285 | 1.109, 1.408 | 1.156, 1.325 | 3.627 | 0.1367 | 14.56 | 89.8 | 2 |
071 | 0.04 | $35^{\circ}$ | 0.021 | 0.284 | 1.289, 1.64 | 1.275, 1.496 | 3.119 | 0.0458 | 11.61 | 51.4 | 2 |
072 | 0.04 | $35^{\circ}$ | 0.0215 | 0.29 | 1.142, 1.768 | 1.133, 1.634 | 3.336 | 0.0675 | 12.84 | 27.2 | 0 |
073 | 0.04 | $35^{\circ}$ | 0.0199 | 0.279 | 1.413, 1.143 | 1.302, 1.159 | 3.619 | 0.0251 | 14.52 | 89.3 | 0 |
074 | 0.04 | $35^{\circ}$ | 0.0172 | 0.266 | 1.395, 1.657 | 1.342, 1.541 | 4.115 | 0.0788 | 17.64 | 89.8 | 1 |
075 | 0.04 | $35^{\circ}$ | 0.0172 | 0.266 | 1.355, 1.094 | 1.314, 1.09 | 3.456 | 0.0287 | 13.57 | 38.2 | 3 |
076 | 0.04 | $35^{\circ}$ | 0.0219 | 0.292 | 1.356, 1.901 | 1.208, 1.762 | 3.043 | 0.0631 | 11.18 | 41.2 | 0 |
077 | 0.04 | $35^{\circ}$ | 0.0221 | 0.292 | 1.816, 1.46 | 1.709, 1.289 | 3.037 | 0.0187 | 11.15 | 26.1 | 0 |
078 | 0.04 | $35^{\circ}$ | 0.0206 | 0.284 | 1.708, 1.191 | 1.612, 1.139 | 3.3 | 0.0697 | 12.64 | 63.9 | 1 |
079 | 0.04 | $35^{\circ}$ | 0.0196 | 0.28 | 1.209, 1.992 | 1.187, 1.781 | 3.944 | 0.0686 | 16.52 | 26.1 | 1 |
080 | 0.04 | $35^{\circ}$ | 0.0218 | 0.29 | 1.566, 1.827 | 1.516, 1.69 | 3.193 | 0.0948 | 12.02 | 25.2 | 0 |
081 | 0.04 | $35^{\circ}$ | 0.0216 | 0.289 | 1.114, 1.422 | 1.075, 1.367 | 3.311 | 0.0385 | 12.69 | 90 | 0 |
082 | 0.04 | $35^{\circ}$ | 0.0187 | 0.275 | 1.704, 1.195 | 1.484, 1.236 | 3.995 | 0.0806 | 16.85 | 87.2 | 0 |
083 | 0.04 | $35^{\circ}$ | 0.0164 | 0.26 | 1.164, 1.306 | 1.149, 1.235 | 2.718 | 0.0535 | 9.47 | 84.9 | 1 |
084 | 0.04 | $35^{\circ}$ | 0.0205 | 0.283 | 1.671, 1.424 | 1.501, 1.36 | 3.704 | 0.0179 | 15.03 | 51.7 | 0 |
085 | 0.04 | $35^{\circ}$ | 0.0095 | 0.218 | 1.089, 1.38 | 1.067, 1.339 | 6.324 | 0.1577 | 33.73 | 89.3 | 5 |
|  |  | 0.005 | 0.176 | 1.666, 1.364 | 1.736, 1.235 | 2.488 | 0.0612 | 8.34 | 12.2 |  | Yes
086 | 0.04 | $35^{\circ}$ | 0.0208 | 0.286 | 1.116, 1.994 | 1.12, 1.864 | 3.705 | 0.0962 | 15.04 | 89.4 | 0 |
087 | 0.04 | $35^{\circ}$ | 0.0193 | 0.276 | 1.365, 1.445 | 1.243, 1.462 | 3.83 | 0.07 | 15.82 | 59.8 | 0 |
088 | 0.04 | $35^{\circ}$ | 0.0183 | 0.273 | 1.894, 1.232 | 1.744, 1.131 | 3.983 | 0.038 | 16.78 | 42.1 | 0 |
089 | 0.04 | $35^{\circ}$ | 0.0192 | 0.279 | 2.228, 1.143 | 1.877, 1.215 | 3.513 | 0.0379 | 13.89 | 26.9 | 0 |
090 | 0.04 | $35^{\circ}$ | 0.02 | 0.285 | 1.101, 1.443 | 1.063, 1.411 | 3.665 | 0.0486 | 14.8 | 89.9 | 0 |
091 | 0.04 | $35^{\circ}$ | 0.0188 | 0.274 | 1.589, 1.147 | 1.49, 1.12 | 3.941 | 0.0211 | 16.52 | 39.4 | 0 |
092 | 0.04 | $35^{\circ}$ | 0.0183 | 0.271 | 1.405, 1.363 | 1.349, 1.302 | 4.023 | 0.0136 | 17.03 | 40.4 | 1 |
093 | 0.04 | $35^{\circ}$ | 0.0196 | 0.277 | 1.189, 1.655 | 1.157, 1.542 | 3.862 | 0.1091 | 16.01 | 89.6 | 0 |
094 | 0.04 | $35^{\circ}$ | 0.0202 | 0.281 | 1.359, 1.386 | 1.292, 1.33 | 3.607 | 0.0639 | 14.45 | 54.6 | 1 |
095 | 0.04 | $35^{\circ}$ | 0.0194 | 0.277 | 1.285, 1.443 | 1.202, 1.414 | 3.943 | 0.1736 | 16.52 | 89.6 | 1 |
096 | 0.04 | $35^{\circ}$ | 0.0189 | 0.279 | 1.683, 1.485 | 1.535, 1.317 | 4.18 | 0.2117 | 18.04 | 89.6 | 0 |
097 | 0.03 | $29^{\circ}$ | 0.0131 | 0.24 | 1.396, 1.174 | 1.355, 1.131 | 3.322 | 0.0081 | 12.81 | 53.3 | 0 |
098 | 0.03 | $29^{\circ}$ | 0.0136 | 0.243 | 1.357, 1.111 | 1.33, 1.117 | 3.221 | 0.0308 | 12.23 | 48.9 | 0 |
099 | 0.03 | $29^{\circ}$ | 0.0145 | 0.249 | 1.416, 1.507 | 1.324, 1.434 | 3.076 | 0.0167 | 11.41 | 17.1 | 0 |
100 | 0.03 | $29^{\circ}$ | 0.0139 | 0.244 | 1.268, 1.564 | 1.229, 1.456 | 3.108 | 0.0823 | 11.59 | 20.6 | 0 |
101 | 0.03 | $29^{\circ}$ | 0.0143 | 0.247 | 1.365, 1.266 | 1.232, 1.302 | 2.998 | 0.0562 | 10.98 | 31 | 0 |
102 | 0.03 | $29^{\circ}$ | 0.0148 | 0.249 | 1.312, 1.164 | 1.184, 1.159 | 3.133 | 0.0491 | 11.72 | 38.2 | 0 |
103 | 0.03 | $29^{\circ}$ | 0.014 | 0.245 | 1.428, 1.46 | 1.299, 1.465 | 3.072 | 0.0488 | 11.39 | 21.1 | 0 |
104 | 0.03 | $29^{\circ}$ | 0.0129 | 0.238 | 1.355, 1.126 | 1.348, 1.144 | 3.447 | 0.015 | 13.54 | 55 | 0 |
105 | 0.03 | $32^{\circ}$ | 0.0152 | 0.255 | 1.501, 1.743 | 1.43, 1.681 | 2.943 | 0.0891 | 10.67 | 54.4 | 2 |
106 | 0.03 | $32^{\circ}$ | 0.0145 | 0.252 | 1.444, 1.174 | 1.33, 1.161 | 3.233 | 0.0266 | 12.29 | 89.7 | 1 |
107 | 0.03 | $32^{\circ}$ | 0.0113 | 0.231 | 1.265, 1.22 | 1.228, 1.159 | 3.919 | 0.024 | 16.43 | 81.5 | 0 |
108 | 0.03 | $32^{\circ}$ | 0.0136 | 0.245 | 1.614, 1.323 | 1.507, 1.277 | 3.049 | 0.0412 | 11.26 | 27.1 | 0 |
|  |  | 0.0023 | 0.137 | 1.176, 1.23 | 1.155, 1.153 | 3.003 | 0.0491 | 11.07 | 89.6 |  | Yes
109 | 0.03 | $32^{\circ}$ | 0.0158 | 0.258 | 1.386, 1.74 | 1.301, 1.621 | 2.99 | 0.033 | 10.93 | 37 | 0 |
110 | 0.03 | $32^{\circ}$ | 0.015 | 0.253 | 1.386, 1.721 | 1.353, 1.588 | 3.007 | 0.0274 | 11.02 | 20.7 | 1 |
111 | 0.03 | $32^{\circ}$ | 0.015 | 0.253 | 1.241, 1.735 | 1.179, 1.646 | 3.193 | 0.0288 | 12.06 | 13 | 0 |
112 | 0.03 | $32^{\circ}$ | 0.0154 | 0.257 | 1.277, 1.607 | 1.197, 1.594 | 2.98 | 0.0565 | 10.87 | 52.3 | 0 |
113 | 0.03 | $40^{\circ}$ | 0.015 | 0.257 | 1.439, 1.685 | 1.376, 1.462 | 3.662 | 0.0502 | 14.81 | 54.3 | 0 |
114 | 0.03 | $40^{\circ}$ | 0.0116 | 0.244 | 1.639, 1.219 | 1.383, 1.274 | 4.732 | 0.2459 | 21.8 | 89.5 | 1 |
115 | 0.03 | $40^{\circ}$ | 0.0155 | 0.26 | 1.417, 1.924 | 1.336, 1.67 | 2.867 | 0.0663 | 10.26 | 36.8 | 0 |
|  |  | 0.0013 | 0.115 | 1.11, 1.089 | 1.007, 1.062 | 40.748 | 0.9502 | 553.62 | 89.9 |  | No
116 | 0.03 | $40^{\circ}$ | 0.0156 | 0.261 | 1.307, 1.741 | 1.267, 1.608 | 3.507 | 0.0484 | 13.88 | 89.7 | 1 |
117 | 0.03 | $40^{\circ}$ | 0.0134 | 0.249 | 1.509, 1.162 | 1.421, 1.102 | 3.944 | 0.0356 | 16.57 | 89.3 | 0 |
118 | 0.03 | $40^{\circ}$ | 0.0074 | 0.2 | 1.051, 1.412 | 1.047, 1.307 | 2.387 | 0.0722 | 7.83 | 89.5 | 3 |
|  |  | 0.007 | 0.199 | 1.08, 1.492 | 1.04, 1.444 | 7.467 | 0.5582 | 43.31 | 89.7 |  | No
119 | 0.03 | $40^{\circ}$ | 0.0149 | 0.257 | 1.113, 1.231 | 1.08, 1.143 | 3.857 | 0.0408 | 16.02 | 89.9 | 0 |
120 | 0.03 | $40^{\circ}$ | 0.0174 | 0.274 | 1.494, 2.091 | 1.396, 1.963 | 3.197 | 0.0396 | 12.07 | 20.5 | 0 |

Note. — Column $M_{\text{disk}}/M_{\text{A}}$ is the initial disk-to-primary
mass ratio, $\phi$ is the material’s friction angle. The rest of the columns
are the simulation outcome: the resulting secondary-to-primary mass ratio
($M_{\text{B}}/M_{\text{A}}$), the diameter ratio $D_{\text{B}}/D\text{A}$,
the secondary’s axis ratios $a/b$ and $b/c$ (both DEEVE and extent-derived),
the secondary’s semimajor axis normalized to the primary’s volume-equivalent
radius ($a_{\text{orb}}/R_{\text{A}}$), the secondary’s eccentricity
($e_{\text{orb}}$), orbit period ($P_{\text{orb}}$). $\theta_{\text{max}}$ is
the secondary’s maximum libration angle over the final 10 days of the
simulation. If this number is close to 90∘, then the secondary in an
asynchronous or tumbling rotation state. $N_{\text{frag}}$ is the number of
fragments ejected from the system. In cases where there is a second satellite,
we include a second row in the table containing the parameters of the
satellite. The final column specifies whether that satellite was stable after
the handoff to REBOUND. If the satellite was unstable, it was either ejected
from the system, or collided with the primary or secondary.

##  Appendix C Final renderings

Here we show renderings of a subset of the simulations to give the reader a
sense of the variance in satellite shape between similar simulations and as a
function of the friction angle. All renderings are cases where
$M_{\text{disk}}=0.03M_{A}$ and we show 8 cases for each value of friction
($\phi$). Each frame is rendered from a view looking down from the body’s
principal rotation axis.

$\phi=29^{\circ}$:

$\phi=32^{\circ}$:

$\phi=35^{\circ}$

$\phi=40^{\circ}$

## References

  * Agrusa et al. (2022) Agrusa, H. F., Ferrari, F., Zhang, Y., Richardson, D. C., & Michel, P. 2022, The Planetary Science Journal, 3, 158, doi: 10.3847/psj/ac76c1
  * Agrusa et al. (2021) Agrusa, H. F., Gkolias, I., Tsiganis, K., et al. 2021, Icarus, 370, 114624, doi: 10.1016/j.icarus.2021.114624
  * Al-Hashemi & Al-Amoudi (2018) Al-Hashemi, H. M. B., & Al-Amoudi, O. S. B. 2018, Powder Technology, 330, 397, doi: https://doi.org/10.1016/j.powtec.2018.02.003
  * Arakawa et al. (2020) Arakawa, M., Saiki, T., Wada, K., et al. 2020, Science, 368, 67, doi: 10.1126/science.aaz1701