# Predicting asteroid material properties from a DART-like kinetic impact

Kathryn M. Kumamoto Lawrence Livermore National Laboratory, 7000 East Avenue,
Livermore, CA 94550  J. Michael Owen Lawrence Livermore National Laboratory,
7000 East Avenue, Livermore, CA 94550  Megan Bruck Syal  Lawrence Livermore
National Laboratory, 7000 East Avenue, Livermore, CA 94550  Jason Pearl
Lawrence Livermore National Laboratory, 7000 East Avenue, Livermore, CA 94550
Cody Raskin  Lawrence Livermore National Laboratory, 7000 East Avenue,
Livermore, CA 94550  Wendy K. Caldwell Los Alamos National Laboratory, P.O.
Box 1663, Los Alamos, NM 87545  Emma Rainey  Applied Physics Laboratory, Johns
Hopkins University, 11100 Johns Hopkins Road, Laurel, MD 20723  Angela Stickle
Applied Physics Laboratory, Johns Hopkins University, 11100 Johns Hopkins
Road, Laurel, MD 20723  R. Terik Daly  Applied Physics Laboratory, Johns
Hopkins University, 11100 Johns Hopkins Road, Laurel, MD 20723  Olivier
Barnouin Applied Physics Laboratory, Johns Hopkins University, 11100 Johns
Hopkins Road, Laurel, MD 20723

(Received August 12, 2022; Revised September 16, 2022; Accepted September 20,
2022)

###### Abstract

NASA’s Double Asteroid Redirection Test (DART) mission is the first full-scale
test of the kinetic impactor method for asteroid deflection, in which a
spacecraft intentionally impacts an asteroid to change its trajectory. DART
represents an important first step for planetary defense technology
demonstration, providing a realistic assessment of the effectiveness of the
kinetic impact approach on a near-Earth asteroid. The momentum imparted to the
asteroid is transferred from the impacting spacecraft and enhanced by the
momentum of material ejected from the impact site. However, the magnitude of
the ejecta contribution is dependent on the material properties of the target.
These properties, such as strength and shear modulus, are unknown for the DART
target asteroid, Dimorphos, as well as most asteroids since such properties
are difficult to characterize remotely.

This study examines how hydrocode simulations can be used to estimate material
properties from information available post-impact, specifically the asteroid
size and shape, the velocity and properties of the impacting spacecraft, and
the final velocity change imparted to the asteroid. Across $>$300 three-
dimensional simulations varying seven material parameters describing the
asteroid, we found many combinations of properties could reproduce a
particular asteroid velocity. Additional observations, such as asteroid mass
or crater size, are required to further constrain properties like asteroid
strength or outcomes like the momentum enhancement provided by impact ejecta.
Our results demonstrate the vital importance of having as much knowledge as
possible prior to an impact mission, with key material parameters being the
asteroid’s mass, porosity, strength, and elastic properties.

Asteroids (72) — Impact phenomena (779) — Near-Earth objects (1092)

††journal: the Planetary Science Journal

##  1 Introduction

Planetary defense preparedness depends upon the ability of space-faring
nations to deploy mature mitigation technologies in a timely manner in the
event of an Earth-impact emergency. As outlined in the United States’ National
Near Earth Object Preparedness Strategy and Action Plan (OSTP, 2018), further
development of deflection and disruption technologies is required before an
imminent threat arises. Kinetic impact is a relatively mature asteroid
deflection technology, owing to its simplicity: a spacecraft impacting at many
kilometers per second can deliver a significant momentum impulse to an
asteroid, through both its own mass and velocity and the additional “boost” of
momentum from escaping crater ejecta. As a result of launch vehicle mass
limitations, kinetic impact is typically not a viable strategy for shorter
warning time scenarios or asteroids that exceed a few hundred meters in
diameter (Dearborn & Miller, 2015; Dearborn et al., 2020). However, the range
of scenarios over which kinetic impact can be effective will partly depend
upon the momentum multiplier provided by escaping crater ejecta. For this
reason, in addition to demonstrating targeting capabilities and operational
readiness, it is desirable to conduct kinetic impact experiments on real
asteroids, in which the imparted change in momentum can be well quantified.

Asteroid characteristics that may contribute to uncertainty in an impulsive
deflection response include mass, strength, porosity, shape, internal
structure, spin state, and equation of state (e.g., Asphaug et al., 1998;
Bruck Syal et al., 2016; Feldhacker et al., 2017; Holsapple & Housen, 2012;
Jutzi & Michel, 2014; Raducan et al., 2019, 2020). Currently, a lack of direct
data for asteroid material properties, combined with the likely lack of
target-specific data on future threats and the observable diversity across the
population of Near-Earth Objects (NEOs), contributes to uncertainty in the
momentum multiplication from future kinetic impactor missions. While the
constitutive properties of meteorites can be carefully studied in Earth-based
laboratories (e.g., Cotto-Figueroa et al., 2016; Kimberley & Ramesh, 2011;
Moyano-Cambero et al., 2017; Flynn et al., 2018), these samples are biased
toward materials that survive atmospheric entry and impact. Additionally, the
rubble-pile structures of many NEOs include significant macroporosity,
producing bulk geotechnical properties that are likely distinct from the
properties of meteoritic hand samples. Both integrated experiments, such as a
kinetic impact test with a measurable deflection velocity, and focused science
experiments, such as in-situ or sample-return characterization, can provide
critical ground-truth information for asteroid material properties.

The Asteroid Impact and Deflection Assessment (AIDA) is a collaboration
between NASA and ESA to study the effects of impacting an asteroid, increasing
Earth’s preparedness for potentially hazardous asteroids (Cheng et al., 2015,
2016, 2018). It is composed of NASA’s Double Asteroid Redirection Test (DART)
mission and ESA’s Hera mission, as well as the Light Italian CubeSat for
Imaging of Asteroids (LICIACube; Dotto et al., 2021) carried by DART. In the
AIDA collaboration, the DART spacecraft acts as a kinetic impactor, LICIACube
follows a few minutes behind to image the ejecta cone soon after impact, and
Hera characterizes the asteroid a few years later. DART will impact Dimorphos,
the secondary asteroid of the binary system 65803 Didymos. One of the earliest
observable consequences of the DART impact will be the change in the orbital
period of Dimorphos around the primary asteroid Didymos, which will be
measurable from Earth-based telescopes. This change in asteroid velocity is
one of the key outcomes of a planetary defense deflection mission, for which
the goal would be to nudge the asteroid onto a trajectory that would not
impact Earth. DART represents an important planetary defense milestone, as it
will carry out Earth’s first asteroid deflection test. Interpretation of the
DART experiment’s results, through modeling and simulation comparison, is
essential for estimation of crater ejecta momentum and for the related
question of asteroid initial conditions at the impact site.

In support of the DART/AIDA mission, this study presents the results of an
inverse test designed to explore the predictive power of the limited
information likely available in the weeks following the DART impact. In this
test, the Applied Physics Laboratory at Johns Hopkins University (JHUAPL) and
Los Alamos National Laboratory (LANL) ran full impact simulations to act as
the “truth simulation.” The asteroid velocity change resulting from this
impact simulation was provided to a team at Lawrence Livermore National
Laboratory, along with information regarding the impacting spacecraft, a shape
model for the target asteroid, and the impact location on that target, but
without any information regarding the target asteroid’s material properties.
Using this limited dataset, we ran impact simulations covering a wide range of
material properties to find combinations of parameters that would satisfy the
provided velocity change. The inverse test can be viewed as an intensive
“dress rehearsal” for impact modelers to prepare for the actual DART
experiment, with the truth simulation taking the place of the actual DART
impact.

##  2 Information from the truth simulation

Figure 1: Example visualizations of the Spheral simulations. One quadrant is
clipped out of the simulation to show the interior structure of the asteroid.
(a) The setup for our simulations. The asteroid body is in grey, and the
impactor is in red. (b) Same as in (a) but zoomed in on the impactor. (c) The
results from simulation 11.09. The asteroid is colored by plastic strain, and
low-density ejecta is translucent grey. This simulation had a relatively
strong and porous asteroid body. (d) The results from simulation 22.05. Colors
are the same as in (c). This simulation had a relatively weak asteroid body
with low porosity.

For this inverse test, the target asteroid was modeled using a shape model of
the asteroid 25143 Itokawa (Fujiwara et al., 2006), shrunken such that the
long axis of the asteroid was $\sim$130 m, giving a total volume of
$\sim$280,000 m3 (Figure 1a). This shape model was constructed in Daly et al.
(2022) from images simulated to match those received from the DART mission,
including both the camera onboard DART (the Didymos Reconnaissance and
Asteroid Camera for Optical navigation, DRACO; Fletcher et al., 2018) and a
camera on LICIACube (the LICIACube Explorer Imaging for Asteroid, LEIA; Dotto
et al., 2021). Itokawa was chosen for this inverse test as it is an S-type
asteroid like Didymos (and as expected for Dimorphos) and may therefore have
similar surface attributes, topography, and characteristics. The aspect ratio
of Itokawa, however, is larger than that expected for Dimorphos, resulting in
the total volume of the shape model being $\sim$12% of that predicted for
Dimorphos.

The truth simulation of the impact was run using two hydrocodes: CTH (McGlaun
et al., 1990) at JHUAPL and Free LAGrange (FLAG; Burton, 1992, 1994a, 1994b)
at LANL. The material properties of the asteroid are presented in Table 1, but
parameters such as target strength and porosity were not provided to the
inverse modelers since the properties of Dimorphos will also be unknown prior
to impact. The spacecraft velocity vector, also used for the simulated DRACO
images to construct the shape model (Daly et al., 2022), was slightly oblique
to the surface of the target, with a mean angle of 9.5∘ between the spacecraft
velocity and the surface normal at the point of impact. Figures 1a and b
demonstrate the model setup, with the asteroid body in grey and the impactor
in red. Using a spacecraft with a mass of 612 kg and relative speed of 7.2
km/s, the impact caused the body’s orbital velocity to decrease by 1.972 $\pm$
0.07 cm/s in the truth simulation, reducing its orbital period around the
primary from 12 to 6 hours. For more details regarding the impact site and
impact orientation as well as additional information about the truth
simulation, please refer to the appendix.

The work presented here is similar to DART in terms of the data provided from
the truth simulation, that is, a shape model for the target asteroid, the
location of the impact, and many details regarding the impacting spacecraft.
These pieces of information will all be known prior to or developed soon after
the DART impact, and the goal of this work is to constrain the target
asteroid’s properties as much as possible from this limited dataset. In this
study, we do not include information about the ejecta cone, anticipated from
the LICIACube images (Dotto et al., 2021). Thus, we are modeling what can be
constrained from the minimum DART mission requirements. The exact numbers used
for the inverse test, however, are not the expected values for the actual DART
mission. For instance, the spacecraft used here is moving faster than expected
of the DART spacecraft (7.2 km/s here versus $\sim$6.1 km/s for the DART
impact; Stickle et al., accepted). The extra hard hit from the spacecraft
combined with the small volume of the shape model leads to a period change
given for the inverse test that is almost 50% of the orbital period of
Dimorphos, a value large enough to catastrophically affect the stability of
the binary system. In the actual DART impact, the velocity change for
Dimorphos is expected to be a few millimeters per second or less, rather than
centimeters per second. However, while the numbers and shape model used in
this study differ from those expected for the DART mission, all provided data
are representative of the types of information that will be available shortly
after the DART impact in September 2022, and future interventions on
potentially hazardous asteroids may require extreme deflections similar to
this study.

##  3 Spheral models

###  3.1 Spheral

We performed our impact calculations using Spheral, an open-source Adaptive
Smoothed Particle Hydrodynamics code maintained by Lawrence Livermore National
Laboratory (https://github.com/LLNL/spheral; Owen et al., 1998; Owen, 2010).
The simulation setup as well as two example simulations are shown in Figure 1.
Spheral’s elastic–perfectly plastic formulation and approach to modeling
material fracture is adapted from Benz & Asphaug (1994). We model fracture
using a tensor-based generalization of the statistical damage model of Grady &
Kipp (1980), which accounts for the decrease in tensile strength with
increasing volume. To model yield strength, we use a modified form of the of
the pressure-dependent approach of Collins et al. (2004) which accounts for
the increase in yield strength with confining pressure. To model porous
compaction, we use the strain-porosity model of Wünnemann et al. (2006) with
the thermal correction of Collins et al. (2011).

We simulate the impact in three spatial dimensions. The resolution of the
simulations is 10 cm at the impact point and decreases outward in spherical
shells, with spacing between particles increasing by a ratio of 1.01 in
successive shells. This resolution equates to 3 nodes per impactor radius.
Particle mass, volume, and smoothing length all vary consistent with this
initial graded particle distribution and the appropriate initial density. For
additional information regarding the Spheral models and material parameters
used in these models, please refer to the appendix.

We made a few simplifying assumptions to define the initial conditions of our
simulations. First, the impactor was modeled as a solid aluminum sphere
(Figure 1b). The asteroid was modeled as a single isomorphic, homogeneous,
microporous body of SiO2 using an SiO2 equation of state (EOS) from the
Livermore Equation of State (LEOS) database (Fritsch, 2016). The choice of
SiO2 as our material is not especially unusual as quartz is a well explored
material in mineral physics, though we also briefly examine the use of basalt,
granite, and pumice as asteroid materials, in addition to using both the ANEOS
(Thompson, 1990) and Tillotson EOS (Tillotson, 1962). Aside from the
simulations testing different materials, the density of fully solid material
(i.e., porosity = 0) was held constant between simulations at 2.65 g/cm3, the
density of SiO2 at ambient Earth conditions.

We varied seven variables describing the asteroid’s material properties: the
solid yield strength of intact and damaged material ($Y_{\text{s}0}$ and
$Y_{\text{d}0}$), the solid shear modulus of intact and damaged material
($G_{\text{s0}}$ and $G_{\text{d0}}$), the minimum pressure (i.e., maximum
tensile pressure) allowed in intact ($P_{\text{min}}$) and damaged
($P_{\text{d,min}}$) material, and the initial porosity of the starting
material ($\phi$). Properties for damaged materials were constrained to be
smaller in magnitude than their intact equivalents (e.g., $Y_{\text{d}0}$ $<$
$Y_{\text{s}0}$), but otherwise parameters were allowed to vary independently.
Specific ranges searched for each variable are in Table C.1.

###  3.2 Simulation metrics

There were three key results from each simulation: $\Delta V$ for the asteroid
in the orbital direction, the momentum enhancement factor describing the
contribution of the ejecta to asteroid momentum ($\beta$), and the crater
morphology. Ejecta was defined as material at least 1 m away from the asteroid
surface and moving faster than 5 cm/s away from the centroid of the asteroid
(Figure B.1; see appendix for more detail). $\Delta V$ was calculated from the
momentum of the impacting spacecraft ($\rho_{\text{s}}$), the momentum of the
ejecta ($\rho_{\text{e}}$), and the mass of the asteroid ($M_{\text{A}}$):

$\Delta
V=\frac{(\bm{\rho}_{\text{s}}+\bm{\rho}_{\text{e}})\cdot\bm{\hat{o}}}{M_{\text{A}}},$
(1)

in which the subscripts A, s, and e refer to the asteroid, spacecraft, and
ejecta, respectively. The unit vector in the orbital direction is denoted
$\bm{\hat{o}}$.

The momentum enhancement factor $\beta$ is a multiplicative factor applied to
the spacecraft momentum to express the contribution of impact ejecta to
asteroid momentum. In its simplest form, $\beta$ is the ratio of the momentum
of the asteroid after impact to the initial momentum of the impacting
spacecraft. In this study, $\beta$ was calculated using the definition for
momentum enhancement in the orbital direction described in Rivkin et al.
(2021):

$\beta=\frac{(M_{\text{A}}/M_{\text{s}})\Delta
V_{\text{A,o}}-\bm{V}_{\text{s},\perp\hat{n}}+V_{\text{s,n}}\bm{\epsilon}\cdot\bm{\hat{o}}}{V_{\text{s,n}}(\bm{\hat{n}}+\bm{\epsilon})\cdot\bm{\hat{o}}}.$
(2)

In this equation, $M_{\text{A}}$ is the mass of the asteroid, $M_{\text{s}}$
is the mass of the impacting spacecraft, $\bm{\hat{n}}$ is the surface normal
unit vector, and $\bm{\epsilon}$ is the offset vector between $\bm{\hat{n}}$
and the velocity vector of the ejecta. $\Delta V_{\text{A,o}}$ is the change
in the asteroid’s velocity in the orbital direction (simply referred to as
$\Delta V$ for the remainder of the paper). $V_{\text{s}}$ refers to the
velocity of the impacting spacecraft and is broken down into the magnitude of
velocity parallel to the surface normal for $V_{\text{s,n}}$ and the velocity
vector perpendicular to the surface normal (along the surface) for
$\bm{V}_{\text{s},\perp\hat{n}}$

The width ($W_{\text{c}}$) and depth ($D_{\text{c}}$) of the impact crater was
measured for successful simulations (i.e., those that produced a $\Delta V$ in
the range dictated by the truth simulation). Crater formation occurs over
times on the order of minutes to hours rather than the fractions of a second
to seconds typically run in hydrocodes. To predict the final crater as well as
possible from the final time steps of our successful simulations (from 0.6 s
up to 3.1 s, depending on the simulation), we applied a density constraint
such that material with a density less than 95% of the original porous density
of the asteroid was assumed to eventually leave the crater. We use this metric
as material in this state is typically under tension and in the process of
being evacuated from the growing transient crater. For relatively small
craters with diameters less than 20 m, crater sizes were calculated by fitting
the cratered region of the asteroid with a plane describing the asteroid
surface and a hyperboloid describing the crater (Figure B.3; Klein, 2013). For
larger craters, crater depth and diameter were measured by hand.

###  3.3 Efficiently covering search space using machine learning and
extrapolation

The large dimensional space associated with the seven input variables placed
constraints on our available computational time, requiring an efficient
methodology to adequately constrain the parameter space. Two main techniques
were used to mitigate the computational cost (additional details for both can
be found in the appendix). First, we used a machine learning decision-tree
algorithm to select parameter combinations. We initialized the tree with a
preliminary run of 40 simulations with randomized input parameters. For each
subsequent run, between 3 and 16 simulations were chosen from 10,000 possible
solutions, using the Mitchell’s Best Sampling algorithm (Mitchell, 1991) to
select the parameter combinations covering the largest parameter space in our
seven dimensions.

Second, while the ideal case would be to run all of our simulations until
stable values of $\beta$ and $\Delta V$ were reached, as well as stable crater
formation, the size of our dataset makes this approach prohibitively
computationally expensive. Thus, as the truth model provided a relatively
narrow target range for $\Delta V$ of $\pm$0.07 cm/s, simulations that clearly
overshot or undershot the velocity range were terminated prior to $\Delta V$
and $\beta$ stabilizing. The relationship between $\Delta V$ and time was
extrapolated using an exponential decay model, taking the infinite time limit
as the stable $\Delta V$ (see Figure D.2 for all simulation fits; see appendix
for more detail). For simulations with high porosity, a local maximum in
$\Delta V$ was often observed at small times (t $\sim$ 0.1–0.3 s). For the
purposes of extrapolation, this local maximum was fit phenomenologically by
adding a Gaussian peak to the exponential decay model. $\beta$ was
extrapolated in the same way, with or without an additional Gaussian peak
depending on the porosity in the simulation. Simulations with values of
$\Delta V$ close to the target $\Delta V$ were run until relatively stabilized
(i.e., with a small $\delta V/\delta t$).

##  4 Results

###  4.1 Trends across all simulations

We ran 338 simulations covering a wide range of input parameters. The results
of these simulations are summarized in Figure 3, the ranges of inputs and
outputs covered are listed in Table 1, and all inputs and deflection results
can be found in Table F.1. For clarity, we plot $\Delta V$ as its magnitude,
but all values of $\Delta V$ are negative in our simulations (i.e., the
asteroid orbital period decreases after impact).

Table 1: Ranges of input and output values for truth simulations and inverse
simulations

| Truth simulation | All inverse simulations | Simulation subset | Simulation subset
---|---|---|---|---
|  |  |  |  $\Delta V=-1.972\pm 0.07$ cm/s |  $\Delta V=-1.972\pm 0.07$ cm/s
| CTH | FLAG |  |  |  $M_{\text{A}}=599,758\pm 59,976$ kg
|  |  |  |  |  ($\phi=0.18\pm 0.08$)
$n$ | 1 | 1 | 338 | 37 | 12
|  |  | Minimum | Maximum | Minimum | Maximum | Minimum | Maximum
$Y_{\text{s}0}$ [MPa] | 1.00e+1 | 1.00e+1 | 1.00e$-$3 | 1.50e$+$2 | 5.01e$-$3 | 1.46e$+$2 | 5.01e$-$3 | 1.07e$+$2
$Y_{\text{d}0}$ [MPa] | — | — | 1.00e$-$5 | 1.00e$+$0 | 1.02e$-$3 | 9.52e$-$1 | 3.80e$-$3 | 6.92e$-$2
$G_{\text{s0}}$ [MPa] | —aaWhile an intact shear modulus was not set in the CTH simulation, the Poisson’s ratio was set to 0.25. | 2.90e+3 | 1.00e$+$1 | 1.00e$+$5 | 1.07e$+$1 | 1.00e$+$5 | 1.07e$+$1 | 1.02e$+$4
$G_{\text{d0}}$ [MPa] | — | — | 1.00e$-$1 | 1.00e$+$3 | 1.45e$-$1 | 9.91e$+$2 | 1.45e$-$1 | 9.55e$+$2
$\phi$ | 0.24 | 0.24 | 0.05 | 0.70 | 0.11 | 0.6 | 0.11 | 0.25
$-P_{\text{min}}$ [MPa] | — | — | 1.00e$-$5 | 1.00e$+$2 | 1.45e$-$5 | 8.13e$+$1 | 3.09e$-$5 | 8.13e$+$1
$-P_{\text{d,min}}$ [MPa] | — | — | 0 | 1.00e$-$4 | 0 | 9.55e$-$2 | 1.15e$-$7 | 9.55e$-$2
$\Delta V$ [cm/s] |  $-$2.02 |  $-$1.92 |  $-$0.89 |  $-$4.60bbDoes not include values for simulation with fluid-like behavior. |  $-$1.90 |  $-$2.04 |  $-$1.91 |  $-$2.04
$\beta$ | 3.3 | 3.5 | 1.4 | 6.9bbDoes not include values for simulation with fluid-like behavior. | 1.53 | 3.59 | 2.88 | 3.59
$W_{\text{c}}$ [m] | 9.2ccTransient crater measured at 0.1 s. | — | — | — | 11.3 | 47.0 | 19.2 | 47.0
$D_{\text{c}}$ [m] | 6.9ccTransient crater measured at 0.1 s. | — | — | — | 3.6 | 15.9 | 4.9 | 14.4

Note. — The truth simulation parameters were unknown to the inverse team when
inverse simulations were being run, but they are included here for ease of
comparison. The total inverse simulation population is broken down into
subsets meeting specific constraints. For each subset, the criteria for
selection are listed in the header and the corresponding range of material
properties are shown. Crater width and crater depth were only calculated for
simulations with the correct $\Delta V$, so ranges are not reported for the
whole simulation set.

Figure 2:  Figure 3: Results of all simulations. Final $\Delta V$ and $\beta$
are extrapolated to $t=\infty$. Simulations with $Y_{\text{s0}}>10$ MPa and
$G_{\text{s0}}>10$ GPa are colored in red and are more consistent with strong,
intact rock. Simulations with $Y_{\text{s0}}<10$ MPa and $G_{\text{s0}}<10$
GPa are colored in blue and are more consistent with weak, fractured rock or
sediment. Simulations in open grey circles have either a large $Y_{\text{s0}}$
and a low $G_{\text{s0}}$, or a low $Y_{\text{s0}}$ and a large
$G_{\text{s0}}$. Simulations with final $\Delta V$ values in the target range
are denoted with stars. Simulations with $P_{\text{d,min}}=0$ are plotted at
$10^{-7}$ MPa.

One of the largest individual effects on $\Delta V$ comes from porosity, with
larger values of $\Delta V$ associated with more porous targets (Figure 3).
This observation is predominantly a result of the effect of porosity on the
total initial mass of the asteroid, as asteroid volume and solid density
(i.e., density at 0% porosity) were constant between simulations. The momentum
of the impacting spacecraft is constant between simulations. Thus, neglecting
any effect of impact ejecta, conservation of momentum results in the velocity
change of the asteroid is larger when the asteroid has less mass. This effect
can be observed in the positive relationship between porosity and $\Delta V$,
particularly for high values of porosity.

Strength and shear modulus for both intact and damaged material have negative
correlations with $\Delta V$ (Figure 3), meaning that simulations with yield
strengths and shear moduli that more closely approximate fully intact rock are
more likely to result in small velocity changes. At the other extreme,
simulations that resemble weak dry sediment or sand are more likely to have
large velocity changes. Previous work examining the effects of yield strength
or cohesion on $\Delta V$ and $\beta$ also observed these correlations (e.g.,
Bruck Syal et al., 2016; Raducan et al., 2019).

$P_{\text{min}}$ and $P_{\text{d,min}}$ do not have strong effects on $\Delta
V$ across the simulations. There is essentially no correlation between the
magnitude of $P_{\text{min}}$ and $\Delta V$ and only a shallow positive
correlation between the magnitude of $P_{\text{d,min}}$ and $\Delta V$ across
six orders of magnitude (Figure 3). To quantify these relationships (or lack
thereof), we ran a Monte Carlo regression model, in which we fit a regression
to a subset of our dataset 10,000 times and took the average and standard
deviation of those 10,000 results. For the trends of both $P_{\text{min}}$ and
$P_{\text{d,min}}$ versus $\Delta V$, the standard deviation of the regression
results for the slope is larger than the average slope, suggesting that any
trend observed is statistically insignificant. The general lack of a trend
between $P_{\text{min}}$ and $P_{\text{d,min}}$ and the resulting deflection
suggests that tensile stresses in our simulations are not controlling the
overall deflection response of the asteroid.

As the mass of the asteroid is not constant between simulations, there is not
a linear relationship between $\Delta V$ and $\beta$ (Figure 5), but the two
are positively correlated through the general equation for $\beta$ (Eq. 2). In
fact, for most variables, the trends observed with $\Delta V$ and $\beta$ are
the same, i.e., both positive, both negative, or both essentially negligible
(Figure 3). The one exception is porosity. The correlation between $\phi$ and
$\Delta V$ is positive as high-porosity targets have lower mass. However,
there is a negative relationship between porosity and $\beta$. In more porous
targets, a larger portion of the kinetic energy of the impactor goes toward
compaction of the target material rather than the ejection of material. These
results are consistent with previous work examining the effects of porosity on
asteroid deflection (e.g., Bruck Syal et al., 2016; Raducan et al., 2019;
Stickle et al., 2017).

Figure 4:  Figure 5: Final $\Delta V$ versus $\beta$ for all simulations. Each
simulation is colored by initial porosity, $\phi$. The target $\Delta V$ of
$-$1.972 cm/s is indicated with a dashed line.

###  4.2 Defining realistic targets

Our dataset covers a large range of properties, and it is worth emphasizing
that we did little to control the realism of material parameter combinations.
While this methodology allows us to easily and effectively explore the
parameter space, some simulations are not predictive for real asteroids that
might be encountered. It is thus instructive to examine trends in subsets of
data that are more representative of real material.

####  4.2.1 Strong and weak targets

Figure 3 uses the intact asteroid strength and shear modulus to define
realistic combinations representing strong and weak target material.
Simulations shown in red in Figure 3 are a subset with large intact strength
and shear moduli more consistent with rock ($Y_{\text{s}0}$ $>$ 10 MPa,
$G_{\text{s0}}$ $>$ 10 GPa). Simulations in blue represent weaker material
more consistent with fractured rock or sediment ($Y_{\text{s}0}$ $<$ 10 MPa,
$G_{\text{s0}}$ $<$ 10 GPa). Finally, simulations in white are simulations in
which one intact parameter is large while the other intact parameter is small
(e.g., large $Y_{\text{s}0}$ and small $G_{\text{s0}}$), combinations which
are unlikely to occur. We do not include porosity in this comparison.

In both the strong and weak subsets, the trends between porosity and $\Delta
V$ are much clearer for intermediate and very porous materials ($\phi$ $>$
20%) than the trend for the whole dataset. For the same porosity, weak target
material generally results in a larger $\Delta V$ compared to stronger target
material, aligning with the overall trends observed for the full dataset
(Figure 3). Similarly, weak target material generally results in a larger
$\beta$ for the same porosity.

For the initially strong, more rock-like simulations (in red), there is not a
strong trend between $\Delta V$ and the variables for damaged material
($Y_{\text{d}0}$ and $G_{\text{d0}}$) (Figure 3). The lack of a trend is
particularly evident for $Y_{\text{d}0}$: the simulations in red at low
$Y_{\text{d}0}$ cover ranges of values for $\Delta V$ and $\beta$ similar to
simulations at high $Y_{\text{d}0}$, particularly for $Y_{\text{d}0}<10^{-1}$
MPa. In contrast, when $Y_{\text{s}0}$ and $G_{\text{s0}}$ are both small,
$\Delta V$ and the damaged variables ($Y_{\text{d}0}$ and $G_{\text{d}0}$)
have strong negative correlations. These relationships suggest that the ejecta
response for strong asteroid targets is controlled by the initial material
parameters ($Y_{\text{s}0}$ and $G_{\text{s0}}$), and the initial yielding of
asteroid material is the limiting step in ejecta formation. For initially weak
asteroid targets, on the other hand, the ejecta response is controlled by the
parameters for damaged material ($Y_{\text{d}0}$ and $G_{\text{d0}}$), and the
deformation of material that is already damaged controls the ejecta formation.

####  4.2.2 Realistic elastic parameters

We searched a large range of shear moduli, but we note that we did not
directly link the bulk modulus to the prescribed shear modulus. Instead, the
bulk modulus for each simulation was calculated from the SiO2 EOS. As a
result, the Poisson’s ratio ($\nu$) of our simulations varied substantially.
Simulations with low shear moduli had very large Poisson’s ratios, close to
the maximum value of 0.5. By comparison, porous rhyolitic lavas (up to $\phi$
= 0.5) also demonstrate Poisson’s ratios of up to $\sim$0.45 (e.g., Ji et al.,
2019; Mordensky et al., 2018). Vacuum-saturated beach sands demonstrate a
large range of Poisson’s ratios but have a maximum of 0.42 for typical grain
sizes of hundreds of micrometers to around a millimeter (Kimura, 2006).
Poisson’s ratios larger than $\sim$0.45 in geologic material, however, are
uncommon without introducing pore fluids (e.g., Kimura, 2006), which are
unrealistic for asteroid environments.

At the other extreme, simulations with extremely large shear moduli require
auxetic behavior with negative Poisson’s ratios. Auxetic materials generally
have very low densities resulting from their complex open structures (e.g.,
Alderson & Alderson, 2007), and they are exceedingly rare in geologic
materials. Alpha-cristobalite, a high-temperature polymorph of SiO2, is the
only known natural mineral with a negative Poisson’s ratio (Ji et al., 2018),
though some artificial pumice analogues have been reported to have negative
Poisson’s ratios as well (Wollner et al., 2018).

Examining only the simulations with solid Poisson’s ratios similar to typical
geologic material (0–0.45), the trends we observe across simulations do not
change (Figure F.2). The ranges of material parameters for successful
simulations are similarly unaffected, aside from the solid shear modulus as it
is directly constrained by the Poisson’s ratio. Given the generally unchanged
results, we will consider all simulations, including those with extreme
Poisson’s ratios, in the remainder of the manuscript.

##  5 Predicting asteroid properties

###  5.1 Matching $\Delta V$

Thus far, we have discussed the trends observed across all our simulations.
However, the goal of the simulation efforts presented here is to find material
parameter combinations that produce a particular $\Delta V$, in this case
$-$1.972 cm/s. These solutions are shown as stars in Figure 3 and are white
lines in Figure 7. The minimum and maximum values for each variable for
successful simulations (i.e., those simulations with $\Delta V$ = $-$1.972
$\pm$ 0.07 cm/s) are reported in Table 1. Example visualizations of
simulations that produced the correct $\Delta V$ with very different material
parameters are shown in Figure 1c–d.

Figure 6:  Figure 7: Parallel coordinate plots of input parameters with
equivalent parameters in the truth simulations, as well as final $\Delta V$
and $\beta$. Simulation lines are colored by the difference between the
simulation $\Delta V$ and the target $\Delta V$, with blue being slower than
the truth simulation and red being faster than the truth simulation. White
stars indicate the values of the truth simulation. For $\phi$, the open star
indicates the original truth simulation value of 0.24 while the solid white
star indicates the porosity required for an equivalently massive asteroid in
our simulations. Plots are shown for (a) all simulations, (b) simulations with
initial asteroid mass within 10% of the truth simulation, and (c) simulations
with $Y_{\text{s0}}$ within one order of magnitude of the truth simulation.
Ticks and axis limits in (b) and (c) are the same as in (a).

Looking at material parameters in isolation, the successful simulations cover
a large range of properties. In line with the small effect of these variables
on $\Delta V$, both $P_{\text{min}}$ and $P_{\text{d,min}}$ for successful
simulations cover nearly the entire range of search space. Porosities for
successful simulations are in the range of 11% to 60%. This range covers
everything from roughly intact rock up to the porosities observed in rubble-
pile asteroids like Itokawa or 101955 Bennu (Walsh, 2018). The large range in
asteroid mass associated with these porosities leads to a range of values for
$\beta$ from 1.5 to 3.6.

Maximum values for $Y_{\text{s}0}$ and $G_{\text{s0}}$ for these simulations
are representative of strong rock ($Y_{\text{s}0}$ = 146 MPa, $G_{\text{s0}}$
= 100 GPa). Minimum values ($Y_{\text{s}0}$ = 174 kPa, $G_{\text{s0}}$ = 27
MPa) are similar to the cohesion and shear modulus for clay (Onur et al.,
2014). The strength and shear modulus for damaged material in successful
simulations are also similar to the properties of sediment. $Y_{\text{d}0}$
varies from 100 Pa to 71 kPa, covering a range similar to that expected for
lunar regolith (Holsapple & Housen, 2012) or the surface of the rubble-pile
asteroid 162173 Ryugu (Arakawa et al., 2020). With the exception of two
simulations with $G_{\text{d0}}$ of $\sim$500 kPa and one simulation with
$G_{\text{d0}}$ of $\sim$150 kPa, the shear moduli for damaged material range
from $\sim$30 MPa to $\sim$1 GPa, demonstrating elasticity similar to sand on
the low end of that range (Onur et al., 2014). Shear moduli of $\leq$500 kPa,
however, are quite small for geologic material, and nearly all the simulations
with similar shear moduli for damaged material result in velocity magnitudes
larger than the target $\Delta V$. One simulation with $G_{\text{d0}}$
$\sim$200 kPa, in addition to low strength and $G_{\text{s0}}$, even exhibited
fluid-like behavior (Figure E.2). While examining low values for
$G_{\text{d0}}$ was helpful for establishing the algorithmic search space,
this behavior is extremely unlikely for actual asteroid material.

###  5.2 Reproducing the truth simulation material parameters

The results from the truth simulation were defined using the average of
results from two different codes: CTH (McGlaun et al., 1990) and Free LAGrange
(FLAG; Burton, 1992, 1994a, 1994b). These simulations were run to a final time
of 0.1 s in CTH and 0.2 s in FLAG. Both simulations used a basalt EOS, SESAME
(Lyon & Johnson, 1992) for CTH or Mie-Grüneisen (Meyers, 1994) for FLAG, to
describe the asteroid material, with a yield strength of 10 MPa, a density of
2.855 g/cm3 for fully solid material, and a porosity of 24% (and thus a bulk
density of 2.170 g/cm3). The FLAG simulation explicitly set a shear modulus
for intact material of 2.9 GPa while the CTH simulation set a Poisson’s ratio
of 0.25. The average $\beta$ resulting from the CTH and FLAG truth simulations
was 3.4. These parameters are listed in Table 1, and additional detail about
the truth simulation codes can be found in the appendix. Figure 7 compares the
inverse simulations to the truth simulation in a parallel coordinate plot,
with the truth simulation parameters plotted as stars. (For a parallel
coordinate plot of all of the input parameters used in the Spheral
simulations, see Figure F.4 in the appendix.)

With the wide range of parameters covered by our successful simulations, the
truth simulation parameters fall inside the ranges described in the previous
section (Figure 7a). Indeed, several of our parameter combinations are quite
close to the truth simulation. However, solutions with similar parameters are
uncommon overall. For instance, values of $Y_{\text{s}0}$ within one order of
magnitude of the truth simulation value of 10 MPa are associated with both low
and high porosities, with higher porosities generally correlated with larger
shear moduli (Figure 7c). Looking at the inverse correlation, successful
simulations with low porosities, and thus asteroid masses similar to the truth
simulation, are associated with both high strengths in the range of MPa and
low strengths of a few kPa (Figure 7b).

Beyond the multi-solution degeneracy of the problem of asteroid deflection,
there are several potential explanations for why the parameter combination
from the truth simulation is uncommon in our inverse simulations. First, there
are inherent differences in the way different hydrocodes process impact
problems. Each code in this study utilizes a different discretization method:
CTH is an Eulerian finite-difference code, FLAG is an Arbitrary Lagrange-
Eulerian finite-volume code, and Spheral is a smoothed particle hydrodynamics
code. There are also other implementation aspects that differ between the
three codes. For example, once material is fully damaged in both the CTH and
FLAG simulations, it can no longer support any stress. Conversely, the Spheral
simulations explicitly set both a strength ($Y_{\text{d}0}$) and a shear
modulus ($G_{\text{d0}}$) for fully damaged material (see Owen et al.,
accepted, for additional details regarding the damage model in Spheral). Thus,
Spheral simulations with the same parameters describing intact material as the
truth simulation (i.e., $Y_{\text{s}0}$, $G_{\text{s0}}$, and $\phi$) could
have smaller values for $\Delta V$ and $\beta$ due to the contribution of non-
zero values of $Y_{\text{d}0}$ and $G_{\text{d0}}$. Even when great effort is
made to make sure hydrocodes are working on as identical a problem as possible
(e.g., by constraining material parameters, equations of state, and strength
models), there are still differences in final simulation results with
variations in $\beta$ of 15–20% between the different codes (Stickle et al.,
2020).

For the inverse test, we used the differences between codes to model the
imperfections of using our hydrocodes to simulate reality. As the truth
simulation was only a model itself, however, we can further examine the CTH,
FLAG, and Spheral simulations to pull apart some of the more minute
differences. For instance, in benchmarking tests using a basalt EOS to
describe the target, CTH predicts larger values of $\beta$ (and thus larger
values of $\Delta V$) than Spheral for targets with the same properties
(Stickle et al., 2020). Applying this difference to the inverse problem
described in this study suggests that replicating the truth simulation
parameters in Spheral would result in a smaller $\beta$ and $\Delta V$ than
the CTH results. Thus, in order to reproduce the correct $\Delta V$ with an
asteroid mass close to the truth simulation, more of our successful
simulations have lower values of $Y_{\text{s}0}$ (Figure 7b). Conversely, many
successful simulations with $Y_{\text{s}0}$ close to 10 MPa have higher
porosities than the truth simulation (Figure 7c).

Figure 8:  Figure 9: $\Delta V$ at $t=0.1$ s (purple) or $t=0.2$ s (orange)
versus $\Delta V$ at $t\sim\infty$. The blue line is 1:1.

We also consider the simulation time at which $\Delta V$ is measured or
estimated. The truth simulation reported $\Delta V$ at $t$ = 0.1 s and 0.2 s
after impact, while we extrapolate $\Delta V$ to $t\sim\infty$. If we compare
the $\Delta V$ at 0.1 s ($\Delta V_{0.1}$) with the extrapolated infinite-time
$\Delta V$ ($\Delta V_{\infty}$) for our simulations, the extrapolated $\Delta
V$ is nearly always larger than $\Delta V_{0.1}$ (Figure 9). This observation
also holds for $\Delta V$ measured at 0.2 s ($\Delta V_{0.2}$). The exceptions
are generally simulations for strong targets, where a local maximum is
observed in $\Delta V$ at early times. Since $\Delta V_{\infty}$ was larger
than $\Delta V_{0.1}$ for most simulations, simulations with $\Delta
V_{0.1}=-1.972\pm 0.07$ cm/s were generally shifted to weaker targets (i.e.,
targets with larger $\Delta V_{\infty}$) compared to simulations in which
$\Delta V_{\infty}$ was in the target velocity range.

An additional discrepancy between the truth simulation and our Spheral
simulations was the choice of EOS. Our choice of LEOS SiO2 for the asteroid
material does not match the SESAME or Mie-Grüneisen basalt EOS used in the
truth simulation. To briefly explore the effect of EOS and material choice, we
chose four of our successful simulations covering a range of material property
combinations. We reran these simulations using different choices for asteroid
material and EOS while keeping the seven input material properties (i.e.,
$Y_{\text{s}0}$, $Y_{\text{d}0}$, etc.) the same. The EOS/material
combinations we tested were Tillotson basalt, Tillotson granite, Tillotson
pumice, and ANEOS SiO2 (Thompson, 1990; Tillotson, 1962).

Figure 10:  Figure 11: Results of simulations examining the effect of
EOS/material choice showing (a) $\beta$ versus time, and (b–e) $\Delta V$
versus time. Lines are colored to group simulations with the same material
input parameters, and line type indicates the EOS and material choice. The
dashed black lines in plots (b) through (e) indicates the $\Delta V$ of the
truth simulation, with the dark grey covering $\pm$0.07 cm/s. Numbers in the
legend (e.g., 22.05) are simulation IDs (see Table F.1).

Variability in $\beta$ resulting from the choice of EOS and material for the
asteroid is 10–16% (Figure 11, Table F.2), consistent with previous
observations (Bruck Syal et al., 2016; Stickle et al., 2017). Variability in
$\Delta V$ is much larger (up to 26%), predominantly because of the varying
nonporous densities of the different EOS and material options. However, when
only considering the SiO2, basalt, and granite EOS options, which all have
similar nonporous densities, variability in $\Delta V$ is 8–14%, comparable to
the variability in $\beta$. The LEOS and ANEOS SiO2 results are similar to
each other both in terms of $\beta$ as well as $\Delta V$, and the Tillotson
EOS have similar values of $\beta$ (Figure 11). Interestingly, the different
asteroid EOS and material combinations are not consistent in their
distribution. For instance, using Tillotson granite to describe the asteroid
resulted in the smallest $\Delta V$ and $\beta$ for the two strongest cases
tested here (1-4 and 9-0), but ANEOS SiO2 had the smallest $\Delta V$ and
$\beta$ for the two weakest cases (22-5 and 11-9).

###  5.3 Constraints from Hera after impact

Additional constraints for the DART impact will be provided by Hera when it
reaches the Didymos system in 2026. Hera will measure the mass of Dimorphos,
with an expected accuracy of at least 10% (Michel et al., 2018). Figure 7b
demonstrates that we have three main populations of values for $Y_{\text{s}0}$
when constraining the mass of the target asteroid: tens of MPa, a few MPa, or
a few kPa. Remarkably, however, the shear modulus for all successful mass-
constrained simulations is within approximately half an order of magnitude of
the truth simulation. The ranges for different input variables of successful
solutions when constrained by asteroid mass are in Table 1.

In addition to mass, Hera will also measure the size of the impact crater left
by DART. The morphology of the residual crater after an impact is expected to
vary substantially with material properties (e.g., Holsapple & Housen, 2012),
and we observe a wide variety of crater sizes among successful simulations
(Figure 13, Table F.3). In general, both crater width and crater depth are
inversely correlated with $Y_{\text{s}0}$ (Figure 13a). The crater width in
the x direction and the crater width in the y direction are typically close,
though not the same. Small craters generally have slightly wider widths in the
y direction, likely due to the mild obliquity of the impact interacting with
the topography of the impact location. For larger craters, there are examples
with the width in the x direction being larger as well as examples with width
in the y direction being larger. This variation is at least in part a result
of the complicated crater shapes predicted for these simulations. In addition,
for a given $Y_{\text{s}0}$ (or narrow range of values for $Y_{\text{s}0}$),
crater width decreases with increasing porosity. The ratio of crater width to
crater depth is also correlated with porosity, with impacts into more porous
asteroids generating craters with lower aspect ratios (Figure 13b).

Figure 12:  Figure 13: (a) Crater size versus $Y_{\text{s0}}$ for successful
simulations. Solid symbols represent the crater estimated from the final
simulated time. Empty symbols represent the crater estimated at 0.1 s. The
truth simulation is shown in black. (b) The ratio of crater width to crater
depth versus $\phi$. Solid and empty symbols have the same interpretation as
in (a). The truth simulation is shown in black, with an empty symbol for the
24% porosity set for the basalt EOS of the truth simulation and a solid symbol
for the 18% porosity required for an SiO2 EOS to have the same bulk density.

The crater of the CTH truth simulation was measured at 0.1 s (the final time
step of that simulation) and had a width of 9.2 m and a depth of 6.9 m. While
the crater depth of the truth simulation falls directly on the general trend
observed in our simulations, the crater width of the truth simulation is
significantly smaller. This difference is observable both in the general trend
of crater width versus $Y_{\text{s}0}$ as well as crater size ratio versus
$\phi$ (Figure 13). The underestimation of crater size relative to the Spheral
simulations likely results from the transient nature of the crater in the CTH
simulation. When measuring crater size in the Spheral simulations, we
specifically attempted to predict a late-time crater shape by removing
material below a critical density after $\Delta V$ stabilized. For the
majority of crater shapes in our simulations, the change in width between the
transient crater and the predicted final crater is larger than the change in
depth. Thus, we expect that the predicted final crater in the truth simulation
would similarly demonstrate a large increase in crater width relative to the
transient with only a small increase in crater depth, bringing the truth
simulation crater size into relative alignment with the trends we observe.

Given the transient nature of the truth simulation crater, it is difficult to
use crater size as a precise constraint on our simulations (compared to the
easily quantified asteroid mass, for instance). In the interest of exploring
how crater size might narrow the range of possible material properties,
however, we can make a qualitative comparison. Thus, for successful
simulations, we examine the transient crater size at 0.1 s relative to the
predicted crater size. The results are plotted as empty symbols in Figure 13.

Unsurprisingly, the transient crater sizes at 0.1 s are all smaller than the
craters predicted from the final time steps of our simulations, and thus the
truth simulation lines up with the transient crater sizes quite well (Figure
13a). While there is a relationship between the transient crater size and
$Y_{\text{s}0}$, it is much weaker than the relationship between predicted
crater size and $Y_{\text{s}0}$. In fact, for simulations with asteroid masses
within 10% of the truth simulation, all the transient crater widths are within
15% of the truth simulation crater width. Thus, our results suggest that the
early transient crater size does not significantly improve constraints on the
material properties of the asteroid, but the final crater appears diagnostic
in conjunction with $\Delta V$.

###  5.4 Deflected or disrupted?

When post-impact asteroid velocities are as large as the ones discussed in
this study, the likelihood of disruption rather than deflection is an
additional important caveat. An asteroid velocity change of 10% of the escape
velocity is commonly considered a safe threshold for deflection (e.g.,
Dearborn & Miller, 2015), but the $\Delta V$ of the truth simulation is nearly
40% of the escape velocity estimated for Dimorphos (5 cm/s). The more porous
asteroids considered in this study would have even lower escape velocities,
making the truth simulation $\Delta V$ as high as 73% of escape velocity for
the most porous asteroids. The geometry of the inverse test, with the impact
vector perpendicular to the long axis of an asteroid with a large aspect
ratio, also increases the probability of the asteroid breaking into distinct
pieces.

Within the set of successful simulations, the possibility of disruption is
particularly stark for the largest craters, where the diameter of the crater
rim approaches the length of the minor axis of the model asteroid. In these
cases, a large proportion of the asteroid (up to more than 50%) is damaged,
with only minor regions of undamaged material a significant distance from the
impact site. When the strength of damaged material is low, the large regions
of damaged material are unlikely to remain coherent long after impact because
of minor variations in velocity across the asteroid body.

While a detailed discussion of disruption is beyond the scope of this study,
it is informative to examine the successful simulations for the probability of
disruption. We assume that an asteroid that is $>$25% damaged at the final
analyzed time step would be at risk of disruption as the damaged regions in
these simulations transect the entire diameter of the asteroid along the
impact axis. Among the successful simulations, six meet this criterion for
disruption risk. This set includes the four solutions with large craters and
low values of $Y_{\text{s}0}$. However, similarly large craters that occur at
larger values of $Y_{\text{s}0}$ have much smaller regions of damaged
material. In fact, there is no single parameter that determines the extent of
the damaged region. For instance, high porosities are generally associated
with smaller damaged regions in our simulations, and no simulation in our
disruption-risk set with a $\Delta V$ close to the truth simulation has a
porosity greater than 0.4. However, there isn’t a direct correlation between
porosity and damage extent. Similarly, while the set of successful simulations
at risk of disruption includes the four simulations with the lowest values of
$Y_{\text{s}0}$, two simulations with higher values of $Y_{\text{s}0}$,
including one with $Y_{\text{s}0}$ = 107 MPa, are also at high risk of
disruption. Additional work is necessary to establish disruption metrics,
including the effects of asteroid geometry and material properties on
disruption probability.

In terms of the DART mission, however, we should emphasize again that the
scenario presented in this test involves a significantly larger deflection
than we expect in the actual DART experiment. This discrepancy largely results
from the fact that the asteroid shape model used here is much smaller than the
best estimates for Dimorphos (Naidu et al., 2020). Based on the best available
models for Dimorphos, we do not expect disruption to occur in the DART impact,
though recent models of DART-like impacts into extremely low-strength targets
indicate the possibility of global resurfacing (Raducan & Jutzi, 2022).

##  6 Implications for DART and planetary defense

The results of this inverse test demonstrate the vitality of knowing as much
as possible about the asteroid target prior to attempting a kinetic
deflection, particularly the asteroid’s mass, porosity, strength, and elastic
properties. The results of this study describe a wide breadth of possible
asteroid responses when the target volume, target shape, target material,
impactor mass, impactor strength, and impactor velocity are defined. The
velocity changes for our asteroid simulations, modeled across the inverse test
as SiO2, vary from $-0.89$ cm/s to $-4.6$ cm/s depending on the properties
ascribed to the asteroid. Even when the mass of the target asteroid is
constrained to $\pm$10% of the truth simulation asteroid mass, the range of
velocity changes is still $-0.89$ cm/s to $-3.3$ cm/s. Similarly, for
successful simulations, with asteroid velocities close to the target $\Delta
V$, there are a wide range of properties, with strengths ranging from a few
kPa to more than more than 100 MPa and porosity ranging from 0.11 to 0.60.

Our inverse test results also demonstrate the need for robust extrapolations
for $\Delta V$ and $\beta$, or, at the very least, robust metrics for choosing
final values of $\Delta V$ and $\beta$. In the DART impact, $\Delta V$ will be
measured in the weeks following impact, and for theoretical planetary defense
missions, the efficacy of a kinetic deflection may be determined over months
to years. However, running hydrocodes out to even a few seconds after impact
can take days to weeks depending on the material parameters, problem scale,
simulation resolution, and available resources. The late-time behavior of a
body post-impact (e.g., a velocity change due to slow-moving ejecta) may be
critical when planning deflection missions.

Additional information will be available from the DART mission that was not
used in the inverse test. In addition to the mass estimate and crater
observations by Hera discussed above, LICIACube will follow a few minutes
behind the DART spacecraft to image the ejecta cone (Dotto et al., 2021). We
did not utilize information about the ejecta cone in the inverse test in order
to model what can be learned from the minimum DART mission requirements, but
the size and morphology of the ejecta cone may help constrain many properties.
In particular, the ejecta cone images may easily rule out extreme material
properties, when either a minuscule or massive amount of ejecta is produced.
In addition, the ejecta behavior may provide information about the shallow
subsurface makeup of Dimorphos. For instance, a large boulder directly below
the impact site would likely decrease the amount of ejecta produced.

The general results of this study, e.g., the trends identified between the
deflection magnitude and asteroid material parameters as well as the
degeneracy associated with predicting material properties from a single
observation, will be applicable to the DART impact regardless of the nature of
the target body, Dimorphos. However, the precise numbers used and generated in
this study are not expected in the DART impact. For instance, previous work
shows that using a sphere as an impactor leads to a larger $\Delta V$ than
would be expected from the DART spacecraft, with a sphere overestimating
$\Delta V$ by up to 25% (Owen et al., accepted; Raducan et al., 2022). Another
over-simplification in the models presented here is modeling the internal
structure of the asteroid as homogeneous: rubble-pile structures can have
important effects on ejecta production (e.g., Graninger et al., 2021; Stickle
et al., 2017). Finally, the Itokawa shape model in this study was small
relative to the prediction for Dimorphos, leading to larger deflections than
predicted for DART. Thus, the expected risk for disruption or destabilization
of the orbit of Dimorphos around Didymos is low.

While the miniature Itokawa simulated for the inverse test is likely small
compared to Dimorphos, it is large enough to potentially require mitigation
efforts were it on an impact trajectory with Earth. Depending on the albedo of
this theoretical Dimorphos-sized hazardous object, the warning time in such a
scenario could be short, requiring a deflection of the magnitude used for this
inverse test and thus a non-negligible risk of disruption. The results of this
study suggest that, in addition to understanding the effects of asteroid
material properties, the role of asteroid geometry and impact location may be
critical in analyzing disruption risk.

At the moment, significant uncertainties remain in the possible assemblages of
material properties for hazardous asteroids, and, apart from computational
studies, we know little about how these properties will affect kinetic
deflection outcomes. DART will provide an essential first glimpse at this
technology’s potential, but its applicability across the diverse population of
Near-Earth Asteroids will remain uncertain without future reconnaissance and
mitigation demonstration missions. As emphasized in the 2023-2032 Planetary
Science and Astrobiology Decadal Survey (2022), the ability to demonstrate
rapid reconnaissance of asteroids will provide an important link between
discovery and design of optimal mitigation missions. The computational study
presented here provides further support for the importance of pre-mitigation
characterization data on specific asteroid targets, while reinforcing the need
to characterize the geotechnical properties of more Near-Earth Asteroids in
general. The more information we have on likely impact scenarios prior to a
potential deflection, the more likely a mission to defend the planet will be
successful.

This work was supported by the DART mission, NASA Contract No. 80MSFC20D0004.
Lawrence Livermore National Laboratory is operated by Lawrence Livermore
National Security, LLC, for the U.S. Department of Energy, National Nuclear
Security Administration under Contract DE-AC52-07NA27344. LLNL-JRNL-837536.
This work was also supported in part by the Advanced Simulation and Computing
(ASC)–Verification and Validation (V&V) program and ASC–Integrated Codes (IC)
program at Los Alamos National Laboratory and a Chick Keller Postdoctoral
Fellowship in Earth and Space Science from the Center for Space and Earth
Science (CSES) at Loa Alamos National Laboratory. Los Alamos National
Laboratory, an affirmative action/equal opportunity employer, is operated by
Triad National Security, LLC, for the National Nuclear Security Administration
of the U.S. Department of Energy under contract 89233218NCA000001. LA-
UR-22-28492

##  Appendix A The truth simulation

The shape model used in this study to represent Dimorphos was a miniature
Itokawa, with a total volume of 276,386 m3 and a resolution of 2.31 m per
facet. The model was provided in a reference frame such that the positive X
direction was pointed away from the Didymos primary asteroid and the positive
Y direction was parallel to the orbital velocity of Dimorphos (Figure A.2). In
this reference frame, the impact location was at (5.57$\times$10-5, 0.029025,
$-$0.01614) km, placing the impact location on a crater wall. The spacecraft
velocity at impact was ($-$0.012, $-$6.314, 3.511) km/s. For the Spheral
simulations, we rotated the simulation setup such that the spacecraft velocity
was oriented in the negative Z direction for convenience; the Spheral
reference frame is shown in Figure 1.

Figure A.1:  Figure A.2: Initial reference frame for this study, showing the
shrunken Itokawa shape model and indicating the impact velocity vector (in
red), the impact location (in blue), the vector toward the Didymos primary
asteroid, and the vector of the asteroid’s orbital velocity.

###  A.1 CTH

CTH is a two-step Eulerian shock physics hydrocode developed and maintained by
Sandia National Laboratories (McGlaun et al., 1990). The CTH truth simulation
uses a SESAME (Lyon & Johnson, 1992) aluminum EOS to describe the impactor and
a SESAME basalt EOS to describe the asteroid. The impactor has a radius of
36.51 cm, a mass of 612 kg, a von Mises strength of 275 MPa, and a tensile
fracture strength of 310 MPa. The behavior of the asteroid is represented
using a von Mises strength model with a strength of 10 MPa, a tensile fracture
strength of 10 MPa, and a Johnson-Cook damage model with failure at a strain
of 0.05. Porosity is modeled using a pressure-alpha model using parameters
from Jutzi et al. (2008). The resolution of the simulation is set to 3 cells
per projectile radius (cppr) on a flat mesh.

###  A.2 FLAG

FLAG (Free LAGrange) is a multiphysics Arbitrary Lagrangian-Eulerian code
maintained by Los Alamos National Laboratory (Burton, 1992, 1994a, 1994b).
FLAG has been verified and validated for impact crater and high-velocity
impacts and has been used to study planetary science applications (Caldwell et
al., 2018; Caldwell, 2019; Caldwell et al., 2020, 2021). The FLAG truth
simulation uses a Mie-Grüneisen EOS for both the aluminum impactor as well as
the basalt asteroid. Porosity is described with a pressure-alpha model,
artificial viscosity with a von Neumann Richtmeyer formulation with $q2$ = 1.3
and $q1$ = 0.3, and damage with a Johnson-Cook model with $d1$ = 0.05 (Johnson
& Cook, 1985). A Tipton closure model is used to handle mixed-material zones.
The resolution of the simulation ranged from 7.3 cm (5 cppr) at the impact
site to 1 m (0.37 cppr) away from the impact.

##  Appendix B Spheral material models and metrics: Additional information

###  B.1 Material models

We use a 5th order B-spline kernel and our adaptive smoothing length algorithm
maintains roughly 4 radial neighbors spanning the extent of the kernel (Owen,
2010). This equates to roughly 268 neighbors in three dimensions. We use
Monaghan-Gingold (Monaghan & Gingold, 1983) artificial viscosity with linear
coefficient, $Cl$=1.5, and quadratic coefficient, $Cq$=0.75. Strength and
shear modulus are modified by the porosity as $n=n_{0}(1-\phi)$, with $n$
being either the strength or shear modulus of intact or damaged material. The
coefficient of internal friction is set to 1.2 for intact material and 0.6 for
damaged material.

In our strain-porosity model, compaction is treated as elastic up to a strain
of $–1.88\times 10^{-4}$, beyond which the compaction is modeled as
exponential with exponential compaction factor $\kappa$=0.9. In our damage
model, our flaw distribution is governed by the proportional and exponential
Weibull constants, $k$ and $m$ respectively. Flaws become active and propagate
damaging the material once their activation strain is exceeded. In this paper,
we set $k$ and $m$ equal to $5.00\times 10^{24}$ and 9.0, respectively. The
activation strain is calculated using Spheral’s “Pseudo Plastic Strain”
option. This involves using the time-derivative of deviatoric stress and the
shear modulus to integrate the activation strain in time. Additional details
regarding the damage model and the parameters used can be found in Owen et al.
(accepted).

The impactor is modeled as a solid aluminum sphere with no porosity and a
radius of 37.8 cm (calculated from the density of solid aluminum at 2.7 g/cm3
and an impactor mass of 612 kg). Most simulations in this paper use a LEOS
(Livermore Equation of State) aluminum EOS to describe the impactor and a LEOS
SiO2 EOS to describe the asteroid (Fritsch, 2016). Simulations testing EOS
choice use either a LEOS aluminum EOS or a Tillotson aluminum EOS to describe
the impactor (the choice of impactor EOS had little effect on the impact
simulation). Asteroid EOS and material combinations for these simulations are
ANEOS SiO2 (Thompson, 1990), Tillotson basalt, Tillotson granite, and
Tillotson pumice (Tillotson, 1962). Parameters used in the Tillotson EOS for
different phases can be found in Table B.1.

Table B.1: Parameters used in Tillotson EOS

Parameter | Material
---|---
Basalt | Granite | Pumice
$\rho_{0}$ [g/cm3] | 2.700 | 2.680 | 2.327
a | 0.5 | 0.5 | 0.5
b | 1.5 | 1.3 | 1.5
A [dyne/cm2] | 2.67e11 | 1.80e11 | 2.67e11
B [dyne/cm2] | 2.67e11 | 1.80e11 | 2.67e11
$\alpha$ | 5.0 | 5.0 | 5.0
$\beta_{\text{EOS}}$ | 5.0 | 5.0 | 5.0
$\varepsilon_{0}$ [erg/g] | 4.87e12 | 1.60e11 | 4.87e12
$\varepsilon_{\text{liquid}}$ [erg/g] | 4.72e10 | 3.50e10 | 4.72e10
$\varepsilon_{\text{vapor}}$ [erg/g] | 1.82e11 | 1.80e11 | 1.82e11

Note. — $\beta_{\text{EOS}}$ is a parameter in the Tillotson EOS (Tillotson,
1962) and is entirely separate from the momentum enhancement factor $\beta$
discussed in this paper.

###  B.2 Ejecta definition

Many hydrocodes define ejecta using two filters: velocity and spatial
location. The first filter checks that the velocity of the material is greater
than the escape velocity for the target being struck. The second filter is a
plane above the asteroid surface with a normal parallel to the spacecraft
velocity vector. Material above the plane moving faster than the escape
velocity is considered ejecta and contributes to the ejecta momentum (Figure
B.1a).

Figure B.1: (a) Example simulation exhibiting the difference in ejecta
definition. Material in grey is considered part of the asteroid and material
in light red is considered ejecta in both definitions. The material in dark
red is considered ejecta only by our definition using the asteroid surface.
(b) $\Delta V$ convergence with the old z-plane definition (black) and the new
surface definition (red).

We used a planar filter for ejecta at the outset of this study. However, the
geometry of the shrunken Itokawa asteroid meant that a portion of the early
ejecta traveled at a shallow angle relative to the spacecraft velocity vector.
This trajectory meant this material was not accounted for using the typical
ejecta definition (Figure B.1a). For our simulations, therefore, we
incorporated a filter parallel to the asteroid surface, recalculating the
results of previous simulations. Material moving faster than the escape
velocity and located at least 1 m above the original asteroid surface was
labeled as ejecta. In addition to capturing fast-moving low-angle ejecta, this
ejecta definition had the added benefit of converging to a stable $\Delta V$
and $\beta$ more quickly (Figure B.1b). We expect the magnitude of this
benefit to be directly related to the geometry of the asteroid and the angle
of impact relative to the asteroid surface orientation.

###  B.3 Crater definition

Figure B.2:  Figure B.3: Example of crater fit for simulation 11.09. (a)
Original surface of critical density. (b) Plane/hyperboloid fit to the
surface.

Crater morphology was described for simulations that produced the correct
$\Delta V$. Small craters ($<$20 m in diameter) were fit using a hyperboloid
to describe the crater itself and a plane to describe the asteroid surface.
The crater depth was defined as the distance between the plane and the
hyperboloid below the plane. Crater diameter was defined as the average of the
two axes of the ellipse formed by the intersection of the plane and the
hyperboloid (Klein, 2013). While the craters in the simulations were not
always hyperbolic in morphology, the crater depth and diameter fit using this
method were accurate for many different crater geometries (Figure B.3). For
large craters with diameters greater than 20 m, the volume of excavated
material was large enough to invalidate the planar assumption for the asteroid
surface. In these cases, crater size was measured manually from two
perpendicular cross-sections.

##  Appendix C Using machine learning to examine the search space

For the first 138 simulations, $Y_{\text{s}0}$ and $G_{\text{s0}}$ were set to
cover ranges more equivalent to intact or partially fractured rock (Table
C.1). For next 120 simulations, however, we focused on weaker materials more
similar to sand or clay. The ranges for $P_{\text{min}}$, $P_{\text{d,min}}$,
and $\phi$ did not vary across these first 258 simulations. The final 80
simulations included information on asteroid mass, thus constraining porosity.
Other variables in these final simulations were allowed to vary across the
full ranges explored in this study.

We set up a machine learning decision tree algorithm to direct the parameter
choices tested in this study, using the seven material parameters as inputs
and $\Delta V$ as the output. The algorithm was initially seeded with a set of
40 simulations with randomized parameter combinations (Scan 1). Ten successive
scans of 3–16 simulations each were run for the “intact rock” parameters, nine
scans of 8–-16 simulations each were run for “fractured rock” parameters, and
five scans of 16 simulations each were run for the mass-constrained
parameters.

For each scan, the algorithm generated between 5,000 and 10,000 possible
parameter combinations that could potentially produce the correct $\Delta V$
using a decision tree with a tree depth of 34. We then selected the
simulations for that scan using Mitchell’s Best Sampling algorithm (Mitchell,
1991). This selection algorithm uses a k-d tree to find the simulations that
are the furthest apart in the seven-dimensional input space. Once the
simulations for each scan were run and we were satisfied with the
extrapolations for $\Delta V$, the results were added to the training set for
the decision tree algorithm before choosing material parameter combinations
for the next scan.

We note that we changed the ejecta definition used to calculate $\Delta V$
after Scan 17. Details regarding the two ejecta definitions are described in
Section B.2. All reported values for $\Delta V$ and $\beta$ are calculated
using the new ejecta definition, but the decision tree algorithm we utilized
likely would not reproduce the same material parameter sets tested in this
study.

Table C.1: Variable ranges searched for simulations

Variable | Strong intact rock (n = 138) | Weak fractured rock (n = 120) | Mass-constrained (n = 80)
---|---|---|---
$Y_{\text{s}0}$ [MPa] |  $10^{0}-10^{2.2}$ (linear) |  $10^{-3}-10^{0}$ (logarithmic) |  $10^{-3}-10^{2.2}$ (logarithmic)
$Y_{\text{d}0}$ [MPa] |  $10^{-4}-10^{0}$ (linear) |  $10^{-5}-10^{-1}$ (logarithmic) |  $10^{-5}-10^{0}$ (logarithmic)
$G_{\text{s0}}$ [GPa] |  $10^{-1}-10^{2}$ (linear) |  $10^{-2}-10^{1}$ (logarithmic) |  $10^{-2}-10^{2}$ (logarithmic)
$G_{\text{d0}}$ [GPa] |  $10^{-3}-10^{0}$ (linear) |  $10^{-4}-10^{0}$ (logarithmic) |  $10^{-4}-10^{3}$ (logarithmic)
$-P_{\text{min}}$ [GPa] |  $10^{-8}-10^{-1}$ (logarithmic) |  $10^{-8}-10^{-1}$ (logarithmic) |  $10^{-8}-10^{-1}$ (logarithmic)
$-P_{\text{d,min}}$ [GPa] |  $10^{-10}-10^{-4}$ (logarithmic) |  $10^{-10}-10^{-4}$ (logarithmic) |  $10^{-10}-10^{-4}$ (logarithmic)
$\phi$ |  $0.05-0.70$ (linear) |  $0.05-0.70$ (linear) |  $0.10-0.26$ (linear)

Note. — Whether the search was conducted in linear or logarithmic space is
noted for each variable in each simulation set. $P_{\text{min}}$ and
$P_{\text{d,min}}$ are tensile pressures.

##  Appendix D Extrapolations to large times

The large parameter space examined in this study necessitates running many
simulations to adequately sample the multi-dimensional space. Thus, due to
computational constraints, simulations could not always be run until the
system equilibrated. For the purposes of the machine learning algorithm, we
fit curves of $\Delta V$ versus time with an exponential decay model. A
Gaussian was added to phenomenologically describe the local maximum observed
at early times. The curve was fit as:

$\Delta
V=C_{1}\times[1-\text{exp}(-C_{2}(t+C_{3}))]+C_{4}\times\text{exp}\left[-\frac{(t-C_{5})^{2}}{2C_{6}}\right]$
(D1)

where $t$ is time and $C_{1}$ through $C_{6}$ are fitting constants. $\Delta
V$ at $t_{\infty}$ is equal to $C_{1}$. Figure D.2 depicts the fits to all of
the simulations run for the inverse test, demonstrating how we extracted
extrapolated values for $\Delta V$ and $\beta$ from simulations that do not
run very long. For simulations that significantly undershot or overshot the
target $\Delta V$ (the blue bar), in particular, these extrapolations gave us
more realistic velocity magnitudes to input into the machine learning
algorithm.

Figure D.1:  Figure D.2: Extrapolations for $\Delta V$ for all simulations,
with time in seconds on the X-axis and the magnitude of $\Delta V$ in
centimeters per second on the Y-axis. Solid lines show the data from each
simulation, and dotted lines in the same color show the fit of the
extrapolation function to those data. The blue bar at the center of each plot
shows the target $\Delta V$ of 1.972 $\pm$ 0.07 cm/s.

The extrapolations we’ve used here are limited in scope and likely predict
convergence at earlier times than expected for true system equilibrium. Even
simulations run out to times of $>$3 s do not capture the slow-moving ejecta
that may leave the system at late times. The escape velocity of Dimorphos is
estimated to be 5 cm/s. Since our ejecta definition required material to be at
least 1 m above the asteroid surface before it would be calculated as ejecta,
the slowest-moving proto-ejecta particles would take at least 20 s to pass
this boundary. This would take weeks of computation time on hundreds of
processors and would not be usable for the quick turnarounds desired for
planetary defense missions, including DART.

That said, while the velocity of this ejecta will clearly be much slower than
the ejecta analyzed, significant mass may still escape the asteroid and
contribute to a momentum change. For instance, in one simulation of a weak
asteroid (22-05), the projected final crater was $\sim$40 m wide and excavated
$\sim$15 m down, with $\sim$6% of the asteroid’s mass predicted to eventually
escape. This is dramatically different from the $\sim$0.46% of mass that had
escaped to this point, despite the comparatively flat relationship between
$\Delta V$ and time. Thus, the overall magnitude of the contribution of this
late-stage ejecta must be analyzed in future studies.

##  Appendix E Extremes in behavior

While the exponential decay model used to describe the late-time behavior of
our simulations works well for most of our parameter combinations, this is not
true for all simulations. Simulations of weak asteroid targets with
significant ejecta coming from locations other than the impact site were
especially poorly fit by the exponential decay model. The extra ejecta
typically decreases $\Delta V$ and $\beta$ in the orbital direction with time,
resulting in a lack of convergence with time. For these simulations, reported
values for $\Delta V$ and $\beta$ are for the final time step (commonly $>$2
s), but they are expected to be lower.

One simulation is an extreme outlier, with nearly fluid-like behavior (Figure
E.2). Many of the parameters for this simulation are well within the ranges
describing successful simulations, except for the damaged shear modulus
($G_{\text{d0}}$ = 174 kPa). This modulus is even smaller than low-density
aerogel (e.g., Scherer et al., 1995) and as such is unsuitable for describing
a realistic asteroid response.

Figure E.1:  Figure E.2: Fluid-like behavior in simulation 16.05. Asteroid
material is colored according to its velocity in the Z direction (with white
material moving upward at or faster than 1 m/s). Material with a density below
0.5 g/cm3 is shown in translucent black. One quarter of the simulation
visualization is clipped out to show the velocity fields interior to the
asteroid.

##  Appendix F Additional supplemental figures and results

Figure F.1:  Figure F.2: Trends across all simulation results. This figure
contains the same simulation results as Figure 3, but points are colored in
red for Poisson’s ratios between 0 and 0.45 and black for lower or higher
Poisson’s ratios. Stars indicate simulations with extrapolated values of
$\Delta V$ in the success range. Figure F.3:  Figure F.4: Parallel coordinate
plots showing all input variables of the inverse simulations, as well as final
$\Delta V$ and $\beta$. Simulation lines are colored by the difference between
the simulation $\Delta V$ and the target $\Delta V$, with blue being slower
than the truth simulation and red being faster than the truth simulation.
White stars indicate the values of the truth simulation. For $\phi$, the open
star indicates the original truth simulation value of 0.24 while the solid
white star indicates the porosity required for an equivalently massive
asteroid in our simulations. Plots are shown for (a) all simulations, (b)
simulations with initial asteroid mass within 10% of the truth simulation, and
(c) simulations with $Y_{\text{s0}}$ within one order of magnitude of the
truth simulation. Ticks and axis limits in (b) and (c) are the same as in (a).
Table F.1: Inverse simulation inputs and results

Sim ID | Processors | Run time | Sim time | $Y_{\text{s0}}$ | $Y_{\text{d0}}$ | $G_{\text{s0}}$ | $G_{\text{d0}}$ | $\phi$ | $P_{\text{min}}$ | $P_{\text{dmin}}$ |  | $\beta_{t=0.1}$ | $\beta_{t=\text{max}}$ | $\beta_{t=\infty}$ | $\Delta V_{t=0.1}$ | $\Delta V_{t=0.2}$ | $\Delta V_{t=\text{max}}$ | $\Delta V_{t=\infty}$
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---
|  | [hr] | [s] | [MPa] | [MPa] | [MPa] | [MPa] |  | [MPa] | [MPa] |  |  |  |  | [cm/s] | [cm/s] | [cm/s] | [cm/s]
1.00 | 144 | 54.2 | 0.8038 | 81 | 0.785 | 4530 | 169 | 0.67 |  $-$5.87e$-$4 | 0 |  | 1.7958 | 1.5308 | 1.5355 |  $-$2.5920 |  $-$2.5576 |  $-$2.2568 |  $-$2.2603
1.01 | 144 | 38.5 | 0.2768 | 64 | 0.985 | 100000 | 380 | 0.35 |  $-$8.38e$-$4 | 0 |  | 1.5411 | 1.4957 | 1.4951 |  $-$1.1353 |  $-$1.1170 |  $-$1.1113 |  $-$1.1112
1.02 | 144 | 86.1 | 0.7703 | 113 | 0.375 | 52200 | 331 | 0.52 |  $-$2.49e$-$4 | 0 |  | 1.8247 | 1.8548 | 1.8571 |  $-$1.7902 |  $-$1.8212 |  $-$1.8288 |  $-$1.7982
1.03 | 144 | 38.3 | 0.4156 | 119 | 0.99 | 28400 | 83 | 0.26 |  $-$3.20e$-$4 | 0 |  | 1.6932 | 1.6082 | 1.6092 |  $-$1.0871 |  $-$1.0557 |  $-$1.0490 |  $-$1.0489
1.04 | 144 | 76.2 | 0.6123 | 117 | 0.952 | 76000 | 953 | 0.6 |  $-$5.42e$-$4 | 0 |  | 1.6247 | 1.5831 | 1.5835 |  $-$1.9458 |  $-$1.9361 |  $-$1.9021 |  $-$1.9035
1.05 | 144 | 38.1 | 0.2900 | 118 | 0.716 | 88000 | 297 | 0.41 |  $-$6.62e$-$4 | 0 |  | 1.5656 | 1.5805 | 1.5818 |  $-$1.2697 |  $-$1.2788 |  $-$1.2858 |  $-$1.2869
1.06 | 144 | 38.6 | 0.5289 | 38 | 0.859 | 16400 | 875 | 0.56 |  $-$2.49e$-$4 | 0 |  | 1.7659 | 1.5177 | 1.5199 |  $-$1.8900 |  $-$1.7730 |  $-$1.6496 |  $-$1.6507
1.07 | 144 | 38.8 | 0.3272 | 50 | 0.976 | 64000 | 36 | 0.28 |  $-$5.48e$-$5 | 0 |  | 1.7115 | 1.6557 | 1.6572 |  $-$1.1276 |  $-$1.1065 |  $-$1.0993 |  $-$1.1000
1.08 | 144 | 85.0 | 0.8513 | 132 | 0.286 | 40200 | 336 | 0.52 |  $-$2.59e$-$4 | 0 |  | 1.8740 | 1.9276 | 1.9279 |  $-$1.8328 |  $-$1.8841 |  $-$1.8897 |  $-$1.8900
1.09 | 144 | 19.3 | 0.1623 | 108 | 0.58 | 70100 | 101 | 0.37 |  $-$5.18e$-$4 | 0 |  | 1.6870 | 1.6756 | 1.6639 |  $-$1.2733 |  |  $-$1.2663 |  $-$1.2572
1.10 | 144 | 38.7 | 0.4465 | 131 | 0.897 | 22400 | 622 | 0.4 |  $-$6.71e$-$4 | 0 |  | 1.5759 | 1.4229 | 1.4213 |  $-$1.2519 |  $-$1.1772 |  $-$1.1301 |  $-$1.1312
1.11 | 144 | 19.0 | 0.1516 | 143 | 0.602 | 94000 | 887 | 0.63 |  $-$2.43e$-$5 | 0 |  | 1.6997 | 1.7196 | 1.7269 |  $-$2.1948 |  |  $-$2.2199 |  $-$2.2256
1.12 | 144 | 38.8 | 0.5089 | 118 | 0.896 | 10400 | 541 | 0.18 |  $-$5.96e$-$4 | 0 |  | 1.6795 | 1.5270 | 1.5303 |  $-$0.9689 |  $-$0.9117 |  $-$0.8908 |  $-$0.8935
1.13 | 144 | 19.3 | 0.1743 | 103 | 0.686 | 58100 | 352 | 0.43 |  $-$5.73e$-$4 | 0 |  | 1.6232 | 1.5940 | 1.5833 |  $-$1.3563 |  |  $-$1.3342 |  $-$1.3242
1.14 | 144 | 38.4 | 0.3587 | 69 | 0.871 | 46100 | 935 | 0.4 |  $-$3.25e$-$4 | 0 |  | 1.5851 | 1.4934 | 1.4949 |  $-$1.2613 |  $-$1.2154 |  $-$1.1939 |  $-$1.1951

Note. — This table is published in its entirety in the machine-readable
format. A portion is shown here for guidance regarding its form and content.
The simulation ID column is repeated here for clarity. The number of computer
processors used as well as the computer run time required to reach the final
simulation time are reported for each simulation. $\beta$ is reported at
$t=0.1$ s, $t=t_{\text{max}}$, and the extrapolated $t=t_{\infty}$. $\Delta V$
is reported at $t=0.1$ s, $t=0.2$ s, $t=t_{\text{max}}$, and the extrapolated
$t=t_{\infty}$.

Table F.2: Results of simulations comparing EOS/material choice

Original sim ID: | 1.04 | 9.00 | 11.09 | 22.05
---|---|---|---|---
EOS | Material | $\Delta V$ | $\beta$ | $\Delta V$ | $\beta$ | $\Delta V$ | $\beta$ | $\Delta V$ | $\beta$
[cm/s] |  | [cm/s] |  | [cm/s] |  | [cm/s] |
LEOS | SiO2 |  $-$1.90 | 1.58 |  $-$1.97 | 2.14 |  $-$1.92 | 2.55 |  $-$1.97 | 3.72
ANEOS | SiO2 |  $-$1.94 | 1.63 |  $-$1.98 | 2.18 |  $-$1.80 | 2.40 |  $-$1.93 | 3.65
Tillotson | basalt |  $-$2.01 | 1.73 |  $-$2.06 | 2.30 |  $-$2.07 | 2.82 |  $-$2.07 | 4.00
Tillotson | granite |  $-$1.84 | 1.55 |  $-$1.89 | 2.08 |  $-$1.96 | 2.64 |  $-$2.10 | 4.03
Tillotson | pumice | — | — |  $-$2.31 | 2.21 |  $-$2.32 | 2.71 |  $-$2.32 | 3.90

Note. — Tillotson pumice not run for simulation 1.04. All values of $\Delta V$
and $\beta$ in this table are extrapolated to $t\sim\infty$, following the
method used in the rest of this study.

Table F.3: Transient and final predicted crater dimensions for successful simulations Sim ID | Transient crater (0.1 s) | Final predicted crater
---|---|---
x [cm] | y [cm] | z [cm] | x [cm] | y [cm] | z [cm]
1.04aaFinal predicted crater measured algorithmically. | 825 | 801 | 438 | 1148 | 1211 | 373
2.04aaFinal predicted crater measured algorithmically. | 754 | 719 | 438 | 1268 | 1338 | 508
3.13aaFinal predicted crater measured algorithmically. | 650 | 688 | 375 | 1124 | 1212 | 430
4.01aaFinal predicted crater measured algorithmically. | 568 | 672 | 375 | 1058 | 1158 | 408
6.08aaFinal predicted crater measured algorithmically. | 694 | 719 | 406 | 1228 | 1307 | 466
6.14aaFinal predicted crater measured algorithmically. | 696 | 766 | 391 | 1315 | 1401 | 525
8.03aaFinal predicted crater measured algorithmically. | 681 | 813 | 438 | 1298 | 1371 | 476
8.04aaFinal predicted crater measured algorithmically. | 617 | 642 | 375 | 1049 | 1151 | 397
8.08aaFinal predicted crater measured algorithmically. | 678 | 719 | 438 | 1178 | 1257 | 477
8.11aaFinal predicted crater measured algorithmically. | 754 | 781 | 406 | 1387 | 1476 | 575
9.00aaFinal predicted crater measured algorithmically. | 581 | 688 | 359 | 1142 | 1179 | 446
10.00aaFinal predicted crater measured algorithmically. | 661 | 719 | 391 | 1216 | 1258 | 472
11.01aaFinal predicted crater measured algorithmically. | 604 | 674 | 359 | 1091 | 1168 | 360
11.09aaFinal predicted crater measured algorithmically. | 754 | 751 | 375 | 1339 | 1428 | 573
12.04aaFinal predicted crater measured algorithmically. | 946 | 907 | 484 | 1927 | 2133 | 701
13.13 | 994 | 969 | 531 | 2709 | 2566 | 1148
15.01 | 2343 | 2205 | 750 | 3909 | 3079 | 1586
16.03 | 1072 | 1032 | 563 | 2954 | 2830 | 1406
17.02 | 1041 | 1001 | 594 | 2610 | 2425 | 711
17.04 | 1056 | 954 | 563 | 2502 | 2440 | 984
17.07 | 1087 | 1001 | 594 | 2556 | 2676 | 773
19.06 | 994 | 954 | 500 | 2462 | 2345 | 734
20.00 | 1041 | 1017 | 547 | 2787 | 2611 | 1305
20.05 | 1154 | 1095 | 656 | 2574 | 2579 | 1281
20.14 | 1058 | 1017 | 625 | 2541 | 2548 | 1289
21.00 | 994 | 987 | 438 | 3107 | 3410 | 906
21.03 | 979 | 907 | 453 | 2849 | 2814 | 828
21.06 | 981 | 1016 | 500 | 2508 | 2612 | 742
21.12 | 1043 | 969 | 469 | 4108 | 4175 | 1375
22.04 | 992 | 1000 | 563 | 3776 | 3504 | 1422
22.05 | 1092 | 985 | 438 | 4344 | 5049 | 1438
23.07 | 886 | 876 | 438 | 1937 | 1909 | 492
23.15 | 997 | 984 | 453 | 4114 | 5128 | 1094
25.01 | 992 | 970 | 438 | 3016 | 3300 | 727
25.04 | 944 | 939 | 438 | 3304 | 3096 | 648
25.13 | 1025 | 985 | 469 | 4459 | 4315 | 1164
25.15 | 1041 | 969 | 500 | 3463 | 3047 | 1242

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