11institutetext:  Department of Astronomy, University of Maryland, College
Park, MD, 20742, USA 22institutetext: Johns Hopkins University Applied Physics
Laboratory, Laurel, MD, 20723, USA 33institutetext: Smead Department of
Aerospace Engineering Sciences, University of Colorado Boulder, Boulder, CO
80303, USA 44institutetext: Royal Observatory of Belgium, 3 Avenue Circulaire,
1180 Brussels, Belgium 55institutetext: Université Côte d’Azur, Observatoire
de la Côte d’Azur, CNRS, Laboratoire Lagrange, Nice, France 66institutetext:
Department of Aerospace Engineering, Department of Geosciences, Auburn
University, Auburn, AL, USA

# Rotation-induced granular motion on the secondary component of binary
asteroids: Application to the DART impact on Dimorphos

H. F. Agrusa<EMAIL_ADDRESS>R. Ballouz 22 A. J. Meyer 33 E. Tasev 44
G. Noiset 44 Ö. Karatekin 44 P. Michel 55 D. C. Richardson 11 M. Hirabayashi
55

(Received June 30, 2022; accepted July 25, 2022)

###### Abstract

Context. NASA’s Double Asteroid Redirection Test (DART) mission will
kinetically impact Dimorphos, the secondary component of the Didymos binary
asteroid system, which will excite Dimorphos’s dynamical state and lead to
significant libration about the synchronous state and possibly chaotic non-
principal axis rotation. Although this particular outcome is human caused,
many other secondary components of binary systems are also prone to such
exotic spin states.

Aims. For a satellite in an excited spin state, the time-varying tidal and
rotational environment can lead to significant surface accelerations.
Depending on the circumstances, this mechanism may drive granular motion on
the surface of the secondary.

Methods. We modeled the dynamical evolution of a Didymos-like binary asteroid
system using a fully coupled, three-dimensional simulation code. Then, we
computed the time-varying gravitational and rotational accelerations felt over
the entire surface resulting from the secondary’s perturbed dynamical state.

Results. We find that an excited spin and orbit can induce large changes in
the effective surface slope, potentially triggering granular motion and
surface refreshment. However, for the case of the DART impact, this effect is
highly dependent on many unknowns, such as Dimorphos’s detailed shape, bulk
density, surface geology, and the momentum transferred. Aside from the Didymos
system and the DART mission, this effect also has important implications for
binary systems in general.

###### Key Words.:

Minor planets, asteroids: general – Celestial mechanics – Minor planets,
asteroids: individual: Didymos

##  1 Introduction

On September 26, 2022, NASA’s Double Asteroid Redirection Test (DART) mission
will kinetically deflect Dimorphos, the smaller component of the binary
asteroid 65803 Didymos, as a planetary defense demonstration test (Rivkin et
al., 2021). Prior to the impact, DART will deploy the Light Italian CubeSat
for Imaging of Asteroids (LICIACube), which will fly by the system to image
the initial phase of the cratering process as well as improve Dimorphos’s
shape determination (Dotto et al., 2021; Cheng et al., 2022). Following the
impact, the change in the mutual orbit period will be measured via ground-
based observations and used to infer the momentum enhancement factor, commonly
referred to as $\beta$ (Rivkin et al., 2021). Due to the contribution of
ejecta that exceeds the escape speed, $\beta$ is expected to exceed 1. Four
years after DART, the European Space Agency’s Hera mission will rendezvous
with Didymos to characterize the physical, dynamical, and compositional
properties of the system. Hera will also measure in detail the effects of the
DART impact, including the crater’s properties and the mass of Dimorphos,
allowing for a more precise determination of $\beta$ (Michel et al., 2022).

In addition to abruptly reducing the binary semimajor axis and orbit period,
the impact will also change the eccentricity and inclination (Cheng et al.,
2016). Due to a high degree of spin-orbit coupling, the dynamical evolution of
Dimorphos strongly depends on the initial conditions at the time of impact and
the body’s shape, which are currently unknown (Agrusa et al., 2020). Depending
on $\beta$ and Dimorphos’s shape, it is possible that Dimorphos may enter a
chaotic rotation state following the DART impact (Agrusa et al., 2021;
Richardson et al., 2022). Furthermore, numerical simulations that treat
Dimorphos as a rubble pile indicate that boulders may move on the surface,
depending on Dimorphos’s spin state, bulk shape, and material properties
(Agrusa et al., 2022). In this study, we take a closer look at the possibility
of post-impact surface motion on Dimorphos as a function of its complex spin
and orbital environment.

Observational evidence and theoretical arguments both indicate that chaotic
rotation is not uncommon for secondaries in tight binary systems (Pravec et
al., 2016; Ćuk et al., 2021; Seligman & Batygin, 2021; Quillen et al., 2022a),
and it is plausible that many synchronous secondaries have undergone some
level of chaotic rotation in their past or during their formation (Wisdom,
1987; Jacobson & Scheeres, 2011; Davis & Scheeres, 2020b). Therefore, the
methods and results presented here are also broadly applicable to the general
binary asteroid population.

##  2 Methods

Focusing on the DART impact, we first ran a simulation to capture the system’s
dynamics, from which the local slopes can be computed, in an approach
analogous to previous studies of dynamically triggered regolith motion (Yu et
al., 2014; Ballouz et al., 2019). In order to capture the coupled spin and
orbital motion of the secondary, we used the General Use Binary Asteroid
Simulator (gubas), an efficient rigid full two-body problem (F2BP) code (Davis
& Scheeres, 2020a, 2021). gubas has been benchmarked against other F2BP
simulation codes and has been used extensively to study the dynamics of
Didymos and other binary systems (Agrusa et al., 2020; Davis & Scheeres,
2020b; Meyer & Scheeres, 2021; Meyer et al., 2021). In accordance with
previous studies, the gubas simulations expand the gravitational potential of
the polyhedral shape models to degree and order 4 to adequately capture their
irregular gravity fields. All simulations presented herein were run for 1 yr
of integration time.

Table 1: Selected physical and dynamical parameters used for the simulated Didymos system, consistent with the current best estimates (Rivkin et al., 2021). The body diameters are the volume-equivalent spherical diameters. A synchronous spin state for Dimorphos is assumed, and we refer the reader to Richardson et al. (2022) for further discussion on this assumption. Parameter | Value
---|---
Primary bulk density ($\rho_{\text{P}}$) |  $2.2$ g cm-3
Secondary bulk densities ($\rho_{\text{S}}$) |  $[1.85,2.20,2.55]$ g cm-3
Primary mass ($M_{\text{P}}$) |  $5.47\times 10^{11}$ kg
Secondary masses ($M_{\text{S}}$) |  $[4.20,4.99,5.78]\times 10^{9}$ kg
Primary Diameter ($D_{\text{P}}$) |  $780$ m
Secondary Diameter ($D_{\text{S}}$) |  $164$ m
Initial body separation ($a_{\mathrm{orb}}$) |  $1200$ m
Initial Orbital Period ($P_{\mathrm{orb}}$) |  $11.92$ h
Primary Spin Period ($P_{\text{P}}$) |  $2.26$ h
Secondary Spin Period ($P_{\text{S}}$) |  $11.92$ h
Assumed DART Mass ($M_{\text{DART}}$) | 536 kg
Assumed DART Speed ($v_{\text{DART}}$) | 6.15 km/s

(a)  (b)  (c)

(d)

(e)

Figure 1:  Surface slope evolution as a function of Dimorphos’s dynamical
evolution. (a) Top-down view of the “Didymos-Squannit” system. From this view,
the spin and mutual orbit poles are pointing out of the page. (b) Side view.
(c) Surface slopes for a Squannit-shaped Dimorphos with a bulk density of
$\rho_{\text{S}}=2.2\text{ g cm}^{-3}$ in an idealized, relaxed dynamical
state. The black facet corresponds to the sub-Didymos point (at zero libration
amplitude) with a longitude and latitude of $\phi\approx\lambda\approx
0^{\circ}$. The white facet has a longitude and latitude of
$(\phi,\lambda)\approx(0^{\circ},45^{\circ})$ and corresponds to the time-
series plots in part (e). (d) Spin and orbital evolution for the Squannit-
shaped Dimorphos when $\beta=3\,(e=0.023)$. The Euler angles are the 1-2-3
Euler angle set (roll-pitch-yaw) expressed in the rotating orbital frame,
while the body spin rates are in the secondary’s body-fixed frame. (e) Slope
and surface accelerations on the white facet from part (c). The vertical
accelerations point along the facet’s surface normal and are generally
dominated by self-gravity. The horizontal accelerations are expressed as
magnitudes and point parallel to the surface. Initially, the Euler
acceleration is relatively small and the tides are the dominant time-varying
acceleration. After about 5 days, Dimorphos enters NPA rotation, and the Euler
accelerations become comparable to both the tidal and centrifugal
accelerations. We refer the reader to Appendix B for an identical plot showing
the full 365 d simulation.

###  2.1 Simulation setup

In the F2BP simulations, the primary’s gravity is modeled using Didymos’s
radar-derived polyhedral shape model (Naidu et al., 2020). Dimorphos’s shape
is still unknown, so we used the radar shape model for Squannit, the secondary
component of the binary asteroid (66391) Moshup, scaled to the expected volume
of Dimorphos. Squannit is arguably the best available analog for Dimorphos.
Both the Didymos and Moshup systems are S types (Binzel et al., 2004; Dunn et
al., 2013) and have similar properties, including a fast-rotating primary with
a raised equatorial ridge and a tidally locked secondary component on a tight,
approximately circular orbit (Scheeres et al., 2006).111There are no
observations that show Dimorphos is spin locked, but circumstantial evidence
indicates that this is likely. We refer the reader to Richardson et al. (2022)
for a detailed discussion on this assumption. Squannit is the only currently
available secondary shape model for a near-Earth binary and contains
${\sim}2300$ facets (Ostro et al., 2006). Radar data tend to smooth and
flatten surface features, making the surface slope analysis presented here
somewhat conservative. When scaled to the dimensions of Dimorphos, Squannit’s
average facet has a surface area of ${\approx}38\text{ m}^{2}$. Schematics
showing the shape models for the primary and secondary are shown in Fig.
1(a-c).

We focused this short study on the role of $\beta$ and Dimorphos’s bulk
density ($\rho_{\text{S}}$) as they play a significant role in determining the
surface slope evolution of Dimorphos. The bulk density sets the mass and
therefore the self-gravity of the body, which has a significant effect on the
surface slope of a given shape model (Richardson & Bowling, 2014; Richardson
et al., 2019). For a fixed $\beta$, a smaller bulk density (i.e., lower mass)
will result in a larger perturbation to the mutual orbit, which can lead to
larger changes in surface slopes over time. We tested values of $\beta$ in the
range $0\leq\beta\leq 5$, in accordance with the best estimates from
hydrodynamic simulations of the DART impact (Raducan et al., 2022; Stickle et
al., 2022). Based on light curve and radar observations, the Didymos system is
expected to have a bulk density with $1\sigma$ uncertainties of
$\rho{\approx}2.2\pm 0.35\text{ g cm}^{-3}$ (Naidu et al., 2020; Rivkin et
al., 2021). Assuming Dimorphos has a bulk density within this range, we tested
values of $1.85,2.2$, and $2.55\text{ g cm}^{-3}$. It should be noted that the
reported uncertainties are for the bulk density of the entire system, which is
of course dominated by the primary, and it is certainly possible for Dimorphos
to have a bulk density outside of the range explored here (see the discussion
on Dimorphos’s density in Rivkin et al. (2021)). Table 1 provides the adopted
physical and dynamical parameters for this study.

First, the binary was given dynamically relaxed initial conditions (i.e., a
circular orbit with a synchronous secondary). Then, a change in velocity
($\Delta\@vec{v}$) was applied to the secondary’s instantaneous orbital
velocity consistent with a head-on DART impact and a given selection for
$\beta$ and $\rho_{\text{S}}$.222The DART impact will not be ideally head-on
and centered, but recent work indicates that these effects should be
negligible in terms of determining the system’s bulk dynamical properties
(Richardson et al., 2022). This $\Delta\@vec{v}$ reduces Dimorphos’s velocity,
causing the body to fall into a tighter, more eccentric
orbit.333$\Delta\@vec{v}$ is dependent on $\beta$, the impactor mass and
velocity, as well as the secondary’s mass. In a simplified scalar form, it can
be written as $\Delta v=-\beta M_{\text{DART}}v_{\text{DART}}/M_{\text{S}}$,
where the negative sign indicates that Dimorphos’s speed is reduced. Due to
the increased eccentricity, Dimorphos then begins librating and can also enter
a chaotic non-principal axis (NPA) rotation state at later times depending on
its shape. The attitude instability that leads to NPA rotation is driven by
intersections of various spin-orbit resonances among Dimorphos’s frequencies
of free libration, spin precession, nutation, and mean motion — more details
can be found in Agrusa et al. (2021). In results presented here, we give both
the value for $\beta$ and the corresponding binary eccentricity, $e$, in an
effort to make the results of this paper broadly applicable to other similar
binary systems. Due to the non-Keplerian nature of small binary systems, we
report $e$ as the geometric eccentricity, which is a function of the periapsis
$(r_{\text{p}})$ and apoapsis $(r_{\text{a}})$ distances:
$e=(r_{\text{a}}-r_{\text{p}})/(r_{\text{a}}+r_{\text{p}})$.

###  2.2 Computation of external accelerations

At each timestep, the gubas code outputs the full state of the system,
including the body locations, orientations, velocities, and spins, from which
the net surface accelerations of the secondary can be readily computed. The
net acceleration is evaluated at the center of each triangular facet (indexed
by $i$) of the shape model at each timestep (indexed by $t$) and can be
written as

$\mathbf{a}_{i,t}^{\text{net}}=\mathbf{a}_{i,t}^{\text{grav}}+\mathbf{a}_{i,t}^{\text{tides}}+\mathbf{a}_{i,t}^{\text{cent}}+\mathbf{a}_{i,t}^{\text{Euler}},$
(1)

where the vectors $\mathbf{a}_{i,t}^{\text{grav}}$,
$\mathbf{a}_{i,t}^{\text{tides}}$, $\mathbf{a}_{i,t}^{\text{cent}}$, and
$\mathbf{a}_{i,t}^{\text{Euler}}$ are the secondary’s self-gravity, the
primary’s tidal acceleration, the centrifugal acceleration, and Euler
acceleration, respectively. The Coriolis acceleration is neglected because
this study is focused on the conditions to trigger surface motion, rather than
details of the motion itself (Kim et al., 2021). The details of how each
respective acceleration was computed can be found in Appendix A. On each
facet, the surface slope is then defined as

$\theta_{i,t}=\mathbf{\hat{n}}_{i}\cdot\mathbf{\hat{a}}_{i,t}^{\text{net}},$
(2)

where $\mathbf{\hat{n}}_{i}$ is the surface normal and
$\mathbf{\hat{a}}_{i,t}^{\text{net}}=\frac{\mathbf{a}_{i,t}^{\text{net}}}{\lVert\mathbf{a}_{i,t}^{\text{net}}\rVert}$.

##  3 Results

###  3.1 A conceptual example

To demonstrate how the various acceleration components affect the surface
slope, we show time-series plots for a scenario in which $\beta{=}2$ and
$\rho_{\text{S}}{=}2.2\text{ g cm}^{-3}$ in Fig. 1. The initial slopes of the
secondary are shown in Fig. 1(c), and the post-impact spin and orbital
evolution is shown in Fig. 1(d). The slope and accelerations are shown in Fig.
1(e) for the facet shown in white in Fig. 1(c), which has a longitude and
latitude of $(\phi,\lambda)\approx(0^{\circ},45^{\circ}$). This particular
example was chosen to illustrate the relative importance of the various
accelerations considered here, as well as the sensitivity of the slope
evolution to the spin and orbit of Dimorphos.

The DART perturbation reduces the semimajor axis and increases the
eccentricity to $e{\sim}0.023$, the effect of which can be seen in the top
plot of Fig. 1(d). Through spin-orbit coupling, Dimorphos’s spin state is also
excited, and it begins librating while its spin rate oscillates. In only
${\sim}5\text{ d}$, the secondary becomes attitude unstable, as indicated by
the nonzero roll and pitch angles, although Dimorphos technically remains in
the 1:1 spin-orbit resonance (yaw angle $<90^{\circ}$). The influence of these
dynamical changes can be seen on the surface slope plot at the top of Fig.
1(e). At early times, changes in the surface slope are dominated by the tidal
acceleration. When Dimorphos enters slight NPA rotation, the centrifugal and
Euler accelerations become much more important, leading to abrupt and chaotic
surface slope changes.

###  3.2 Dependence on momentum enhancement ($\beta$)

Figure 2:  Time-series plots of the change in surface slope $(\Delta\theta$)
of each facet in the secondary shape model. Each line is colored based on its
initial surface slope $(\theta_{0})$. As $\beta$ (or $e$) increases, we see
much larger changes in surface slope. The bulk density is
$\rho_{\text{S}}=2.2\text{ g cm}^{-3}$. See Appendix C for equivalent plots
showing the full 365 d simulation and additional values for $\beta$.

Figure 3:  Maximum slope achieved after a $365\textrm{ d}$ simulation, with
arrows indicating the down-slope direction. See Appendix C for equivalent
plots for other values of $\beta$.

In Fig. 2 we show time-series plots of the change in surface slope
$(\Delta\theta=\theta(t)-\theta_{0})$ of each surface facet for $\beta=1$ and
$\beta=3$ with $\rho_{\text{S}}$ fixed at $2.2\text{ g cm}^{-3}$. The color of
each line corresponds to the slope at the start of the simulation,
$\theta_{0}$. When $\beta=1$, the orbit is not significantly perturbed. As
such, the tidal acceleration is weak and Dimorphos exhibits little NPA
rotation, resulting in small surface slope changes of $\Delta\theta\lessapprox
2^{\circ}$. When $\beta=3$, then the tidal environment becomes strong and
Dimorphos enters NPA after only ${\sim}5$ d, resulting in surface slope
changes as large as $\Delta\theta{\sim}10^{\circ}$.

The results of Fig. 2 highlight the strong temporal dependence of the surface
slopes. The surface slope evolution is also spatially dependent, as
demonstrated by Fig. 3. These plots show the maximum slope achieved over the
same simulations shown in Fig. 2. The arrows on the plot indicate the down-
slope direction. These plots suggest that the highest slopes are achieved in
regions that start off with a high slope. For this particular shape and
assuming loose regolith covering the surface, we would expect most motion near
the equator and mid-latitudes, and very little, if any, near the poles. This
spatial dependence may have implications for inferred crater ages in different
regions of Dimorphos’s surface. In addition, the spatial dependence on the
surface slope evolution could be leveraged to distinguish between causes of
surface refreshment. For example, we might expect surface motion triggered by
the re-accretion of impact ejecta to occur over much of Dimorphos’s surface,
while tidal and rotationally induced surface motion may be restricted to
regions that can achieve high slopes. We refer the reader to Appendix C for
additional plots that show the surface slopes for other values of $\beta$.

###  3.3 Dependence on the bulk density ($\rho_{\text{S}}$)

Figure 4:  Maximum slope as a function of $\beta$ and $\rho_{\text{S}}$. The
slope over a given simulation increases with $\beta$ due to the spin and orbit
of Dimorphos being more excited. Lower densities achieve higher slopes due to
the higher orbital eccentricity for a given $\beta$, in addition to a weaker
self-gravity in relation to the tidal and rotational accelerations.

The surface slopes of a given shape are highly dependent on the body’s bulk
density (Richardson & Bowling, 2014; Susorney & Teanby, 2022). It sets the
mass and self-gravity, which partially determine the initial slope of each
facet. On a related note, a low density means that the self-gravity is weaker,
making the accelerations due to tides and rotation stronger in comparison and
in turn allowing larger slope changes. Finally, a low density (i.e., a low
mass) means a higher eccentricity (and shorter periapsis distance) for a fixed
value of $\beta$. Therefore, a lower density will result in a more perturbed
orbit, in which the tidal and rotational accelerations play an increasingly
important role. For these reasons, the possibility and magnitude of any
granular motion will by highly dependent on Dimorphos’s bulk density.

We see precisely this result in Fig. 4, which shows the maximum surface slope
achieved as a function of $\rho_{\text{S}}$ and $\beta$. The color of the dots
indicates the eccentricity of the particular orbit, which depends on both
$\beta$ and $\rho_{\text{S}}$. We see that the surface slopes increase
dramatically as a function of $\beta$, especially for
$\rho_{\text{S}}=1.85\text{ g cm}^{-3}$, reaching ${\sim}40^{\circ}$ for high
$\beta$ due to the higher eccentricity and resulting in stronger tidal and
rotational forces.

##  4 Discussion

If Dimorphos’s surface has an angle of repose of ${\sim}35^{\circ}$, similar
to that reported at Ryugu and Bennu (Watanabe et al., 2019; Barnouin et al.,
2022), then we would expect significant landslides and shape changes in cases
where $\theta$ exceeds this value. For the Dimorphos shape used in this study,
this would only occur for lower densities and high $\beta$ values. Without
knowing the true shape of Dimorphos, however, it is impossible to say with
certainty how probable any post-impact surface motion is. The aim of this
paper is to demonstrate the plausibility of any dynamics-induced granular
motion or shape change, and this topic will be revisited once Dimorphos’s true
shape is known.

Recent work focused on surface refreshment on Mars’s moon Phobos indicates
that a time-varying $\Delta\theta$ of only a few degrees can lead to a gradual
creep motion of granular material, without the slope ever exceeding the formal
angle of repose. Ballouz et al. (2019) combined dynamical modeling, granular
physics, and geologic mapping of color units to demonstrate that regions of
combined high values of $\theta$ and $\Delta\theta$ coincide with Phobos’s
blue units. This work indicated an active surface-refreshing process that
could excavate pristine un-weathered material. Depending on Dimorphos’s
geophysical properties, it may be plausible that a similar creep motion
process will occur following the DART impact. We note that surface refreshment
could be currently ongoing, if Dimorphos is already in an NPA rotation state
as predicted by Quillen et al. (2022a).

It is also important to consider that both $\beta$ and $\rho_{\text{S}}$ could
lie outside the range explored in this paper. Of course, Dimorphos’s real
shape and surface geology are also unknown, so the results presented here are
illustrative and meant to highlight the range of post-impact possibilities.
After DART’s impact, this phenomenon can be explored with higher fidelity,
incorporating the initial shape model and surface geology obtained with DART
and LICIACube imagery. When Hera arrives, its optical instruments and
CubeSats, especially the Juventas CubeSat and its onboard GRAvimeter for small
Solar System bodies (GRASS) instrument, will measure the dynamical slopes as
one of its science objectives (Michel et al., 2018; Karatekin et al., 2021;
Ritter et al., 2021). The seismic pulse delivered by the DART impact may
significantly alter surface features on Dimorphos (Quillen et al., 2022b;
Thomas & Robinson, 2005). We also note that the global shape of Dimorphos may
also be immediately altered by the DART impact itself (Raducan & Jutzi, 2022).
In addition to affecting the system dynamics (Nakano et al., 2022), these
processes will create a unique challenge in discerning the various surface
refreshment mechanisms upon Hera’s arrival.

The results of the work presented here have the following implications, in the
context of the DART and Hera missions as well as binary asteroids in general:

Granular motion and surface changes. Through images and infrared measurements,
Hera may identify refreshed areas of Dimorphos’s surface exposed by dynamics-
induced surface motion. Furthermore, a comparison of images taken by DART and
Hera may be used to identify surface features that have moved or changed
during the four years between the missions. If there is long-term boulder
motion on the surface, Hera may detect the motion of boulders over the course
of its six-month mission lifetime. Furthermore, this effect may noticeably
alter the system’s dynamics (Brack & McMahon, 2019).

Crater degradation. Impact craters (both natural craters and DART’s crater)
may degrade at different rates based on their location on the surface as
surface slope changes are spatially dependent. This may have important
implications for understanding crater morphology and the surface age of
Dimorphos, a challenge that does not usually require consideration for single
asteroids due to their quasi-static spin states (Sugita et al., 2019; Walsh et
al., 2019; Richardson et al., 2020).

Tidal dissipation. Granular surface motion may affect tidal dissipation in two
ways. First, any material undergoing surface motion will dissipate energy
through friction, potentially enhancing dissipation beyond what is assumed
from traditional tidal theories (Goldreich & Sari, 2009; Nimmo & Matsuyama,
2019). Second, granular motion will change Dimorphos’s mass distribution and,
therefore, its gravitational potential. This mechanism could subtly remove
energy from the system, an effect not captured by simplified tidal treatments.

Binary formation and evolution. One proposed scenario of binary formation
assumes the secondary forms through a spin-up fission event driven by the
Yarkovsky–O’Keefe–Radzievskii–Paddack (YORP) effect and initially orbits
chaotically. At some later time, the secondary must fission a second time,
forming a short-lived triple system and liberating excess free energy in order
to enter a stable, synchronous spin state (Jacobson & Scheeres, 2011). Given
the results presented herein, we might expect landslides on the surface well
before a secondary fission event. This process may dissipate energy and
reshape the secondary, allowing for synchronous rotation without the need to
invoke additional fissions. Furthermore, if all secondaries undergo chaotic
rotation at some point, then we might expect the population to have broadly
similar shapes. However, this would largely depend on the relative timescales
for tidal locking and surface refreshment, as well as other competing slope-
altering processes such as meteorite impacts. In any case, rotation-driven
surface motion, shape change, and energy dissipation may be important effects
that should be accounted for in any binary asteroid formation scenario.

##  5 Conclusions

In this paper we have shown that perturbed post-impact spin and orbital
dynamics may lead to significant fluctuations in Dimorphos’s surface slopes.
Depending on Dimorphos’s shape, bulk density, surface geology, and $\beta$, we
predict that this may trigger long-lived granular motion on the surface. The
implications for dynamics-driven granular motion include a refreshment of
Dimorphos’s surface, impact crater degradation, and enhanced tidal
dissipation. Understanding these effects will help guide and interpret the
measurements Hera will obtain on Dimorphos’s surface and interior. In
addition, this effect may have implications for the formation and evolution of
small binary systems in general.

Thanks to this initial study, post-impact granular motion will be explored
more closely and with higher fidelity when Dimorphos’s shape model first
becomes available. Future work includes directly modeling granular motion on
the surface in addition to coupling that motion back to the resulting
dynamical evolution.

###### Acknowledgements.

We kindly thank the anonymous referee, whose insightful comments significantly
improved the manuscript. This work was supported in part by the DART mission,
NASA Contract #80MSFC20D0004 to JHU/APL. E.T., G.N., O.K., and P.M.
acknowledge support from the European Union’s Horizon 2020 research and
innovation program under grant agreement No. 870377 (project NEO-MAPP). E.T.,
G.N., and O.K. acknowledge support by Belgian Federal Science Policy (BELSPO)
through the through the ESA/PRODEX Program. P.M. acknowledges support from ESA
and from the French Space Agency CNES. The simulations herein were carried out
on The University of Maryland Astronomy Department’s YORP cluster,
administered by the Center for Theory and Computation.

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##  Appendix A Calculation of surface accelerations and slopes

We provide additional details for how exactly the accelerations were computed
over the surface of the secondary shape model. On each facet, all
accelerations were evaluated at the midpoint (i.e., the center) of the given
facet.

###  A.1 Gravitational accelerations

On a given facet, $i$, and at a given time, $t$, the two gravitational
accelerations felt on the surface are due to self-gravity,
$\mathbf{a}_{i,t}^{\text{grav}}$, and the tidal acceleration due to the
presence of the primary, $\mathbf{a}_{i,t}^{\text{tides}}$.

The self-gravity was computed using an algorithm identical to that presented
in Werner & Scheeres (1997). This method computes the exact gravitational
acceleration due to a polyhedral shape model with uniform density. Though the
calculation is somewhat computationally expensive, it only needs to be done
once, as we assume that Dimorphos’s global shape does not undergo significant
change. At each facet, $\mathbf{a}_{i,t}^{\text{grav}}$ was computed in
Dimorphos’s body-fixed frame.

Unlike the self-gravity, the tidal acceleration must be computed at every
facet at every timestep. Therefore, we turned to MacCullagh’s formula to
approximate the tides to save computational costs while still capturing
effects due to Didymos’s irregular shape. MacCullagh’s formula is written as
(MacCullagh 1844a, b; Murray & Dermott 2000)

$V=\frac{GM}{r}-\frac{GM}{2r^{5}}f(A,B,C,x,y,z),$ (3)

where $G$ is the gravitational constant, $M$ is the body mass, $r$ is the
distance from the body’s barycenter to the external field point, $A$, $B$, and
$C$ are the body’s principal moments of inertia, $x$, $y$, and $z$ are the
coordinates of the external field point measured in the primary body-fixed
frame, and $f$ is defined as

$f(A,B,C,x,y,z)=(B+C-2A)x^{2}+(C+A-2B)y^{2}+(A+B-2C)z^{2}.$ (4)

The gravitational acceleration can be calculated by taking partial derivatives
of $V$ with respect to $x$, $y$, and $z$ (see Chapter 5 of Murray & Dermott
(2000)):

$\displaystyle a_{x}$ $\displaystyle=-\frac{\partial V}{\partial
x}=-\frac{GMx}{r^{3}}+\frac{G(B+C-2A)x}{r^{5}}-\frac{5Gx}{2r^{7}}f(A,B,C,x,y,z)$
(5) $\displaystyle a_{y}$ $\displaystyle=-\frac{\partial V}{\partial
y}=-\frac{GMy}{r^{3}}+\frac{G(A+C-2B)y}{r^{5}}-\frac{5Gy}{2r^{7}}f(A,B,C,x,y,z)$
(6) $\displaystyle a_{z}$ $\displaystyle=-\frac{\partial V}{\partial
z}=-\frac{GMz}{r^{3}}+\frac{G(A+B-2C)z}{r^{5}}-\frac{5Gz}{2r^{7}}f(A,B,C,x,y,z).$
(7)

In order to calculate the net gravitational acceleration felt at a point on
Dimorphos’s surface due to Didymos, we took the difference between the
acceleration evaluated at a given surface point and the acceleration evaluated
at Dimorphos’s barycenter. This acceleration vector was computed in the
primary’s body-fixed frame before being rotated into the secondary’s body-
fixed frame.

###  A.2 Rotational accelerations

The centrifugal acceleration at a given point on Dimorphos’s surface denoted
with the index $i$ can be written as

$\mathbf{a}_{i,t}^{cent}=(\mathbf{\Omega}_{t}\times\mathbf{r}_{i})\times\mathbf{\Omega}_{t},$
(8)

where $\mathbf{\Omega}_{t}$ is the spin angular velocity vector of Dimorphos
at a given time and $\mathbf{r_{i}}$ is the position vector of the surface
point, coordinated in Dimorphos’s body-fixed frame. Since Dimorphos’s spin
rate is time varying, we also account for the Euler acceleration,

$\mathbf{a}_{i,t}^{\text{Euler}}=\mathbf{r}_{i}\times\frac{d\mathbf{\Omega}_{t}}{dt}.$
(9)

Since gubas does not directly output the time derivative of the secondary’s
spin, we calculated it in post-processing with a fourth-order central finite-
difference scheme. At a 60 second timestep, the fourth-order approximation
sufficiently approximates $\frac{d\mathbf{\Omega}}{dt}$.

##  Appendix B Longer-term spin-orbit and surface slope evolution for the test
case

Figure 5 shows plots identical to those in Fig. 1 but with the time span
increased to 365 d in order to show the longer-term evolution of the dynamics
and slope. These plots indicate that the dominant mechanism for changing the
surface slope is the NPA rotation of the secondary. We also see that Dimorphos
is able to enter the “barrel instability,” a unique spin state where the
secondary remains tidally locked despite rolling about its long axis, as
indicated by the roll angle hitting $180^{\circ}$ (Ćuk et al. 2021). It seems
that this does not significantly affect the surface slopes, however.

(a)

(b)

Figure 5:  Spin, orbit, and surface slope evolution for the Squannit-shaped
Dimorphos when $\beta=3\,(e=0.023)$. (a) Body separation, Euler angles, and
body spin of Dimorphos. (b) Slope and surface accelerations on a facet near
$(\phi,\lambda)\approx(0^{\circ},45^{\circ})$. These plots are identical to
those of Fig. 1, coming from the same simulation, except they show a longer
time duration to highlight how the evolving spin and orbital motion of
Dimorphos influences the accelerations felt on the surface. Dimorphos’s NPA
rotation (as indicated by the roll and pitch angles or spin about the A and B
axes) leads to large increases in the centrifugal and Euler accelerations that
are capable of driving large surface slope changes.

##  Appendix C Additional surface slope plots

Here we provide supplemental plots of Dimorphos’s spin and orbit, as well as
its surface slope evolution, extending out to the full 365 days. The plots
below are only for the nominal bulk density of $\rho_{\text{S}}=2.2\text{ g
cm}^{-3}$. When $\beta=1$ (Fig. 6), the orbital eccentricity remains
relatively low, keeping Dimorphos in a stable rotation state, which results in
small changes to the surface slopes. When $\beta$ is increased to 2 (Fig. 7),
Dimorphos becomes attitude unstable. Due to increased NPA rotation, we see
much larger changes to the surface slopes. The slopes also vary chaotically
since Dimorphos’s spin state is chaotic. When $\beta=3$ (Fig. 8), Dimorphos is
not only in NPA rotation, but it also enters the barrel instability,
characterized by rotation about its long axis. As $\beta$ increases further
(Figs. 9 and 10), Dimorphos’s spin and orbit are increasingly perturbed,
leading to larger tidal and rotational accelerations that result in larger
changes in the surface slopes.

(a)

(b)  (c)

Figure 6:  The spin, orbit, and surface slope evolution for $\beta=1$ and
$\rho_{\text{S}}=2.2\textrm{ g cm}^{-3}$ over 365 days. (a) Body separation,
Euler angles, and body spin of Dimorphos. (b) Corresponding change in slope
($\Delta\theta$) over time for each surface facet, colored by the starting
slope ($\theta_{0}$) of that facet. (c) Maximum surface slope achieved on each
facet over the full 365 d simulation, with arrows pointing in the down-slope
direction.

(a)

(b)  (c)

Figure 7:  The spin, orbit, and surface slope evolution for $\beta=2$ and
$\rho_{\text{S}}=2.2\textrm{ g cm}^{-3}$ over 365 days. (a) Body separation,
Euler angles, and body spin of Dimorphos. (b) Corresponding change in slope
($\Delta\theta$) over time for each surface facet, colored by the starting
slope ($\theta_{0}$) of that facet. The spikes in $\Delta\theta$ correspond to
periods of increased NPA rotation of Dimorphos. (c) Maximum surface slope
achieved on each facet over the full 365 d simulation, with arrows pointing in
the down-slope direction.

(a)

(b)  (c)

Figure 8:  The spin, orbit, and surface slope evolution for $\beta=3$ and
$\rho_{\text{S}}=2.2\textrm{ g cm}^{-3}$ over 365 days. (a) Body separation,
Euler angles, and body spin of Dimorphos. (b) Corresponding change in slope
($\Delta\theta$) over time for each surface facet, colored by the starting
slope ($\theta_{0}$) of that facet. The surface slope evolution is dominated
by NPA rotation. (c) Maximum surface slope achieved on each facet over the
full 365 d simulation, with arrows pointing in the down-slope direction.

(a)

(b)  (c)

Figure 9:  The spin, orbit, and surface slope evolution for $\beta=4$ and
$\rho_{\text{S}}=2.2\textrm{ g cm}^{-3}$ over 365 days. (a) Body separation,
Euler angles, and body spin of Dimorphos. (b) Corresponding change in slope
($\Delta\theta$) over time for each surface facet, colored by the starting
slope ($\theta_{0}$) of that facet. The surface slope evolution is dominated
by NPA rotation. (c) Maximum surface slope achieved on each facet over the
full 365 d simulation, with arrows pointing in the down-slope direction.

(a)

(b)  (c)

Figure 10:  The spin, orbit, and surface slope evolution for $\beta=5$ and
$\rho_{\text{S}}=2.2\textrm{ g cm}^{-3}$ over 365 days. (a) Body separation,
Euler angles, and body spin of Dimorphos. (b) Corresponding change in slope
($\Delta\theta$) over time for each surface facet, colored by the starting
slope ($\theta_{0}$) of that facet. The surface slope evolution is dominated
by NPA rotation. (c) Maximum surface slope achieved on each facet over the
full 365 d simulation, with arrows pointing in the down-slope direction.