# The western bulge of 162173 Ryugu formed as a result of a rotationally
driven deformation process

Masatoshi Hirabayashi  Auburn University, 211 Davis Hall, Auburn, Alabama,
36849, United States  Eri Tatsumi  University of Tokyo, 7-3-1 Hongo, Bunkyo-
ku, Tokyo 113-0033, Japan  Hideaki Miyamoto  University of Tokyo, 7-3-1 Hongo,
Bunkyo-ku, Tokyo 113-8656, Japan  Goro Komatsu  Università degli Studi “G.
d’Annunzio”, Viale Pindaro 42, 65127 Pescara, Italy  Seiji Sugita  University
of Tokyo, 7-3-1 Hongo, Bunkyo-ku, Tokyo 113-0033, Japan  Sei-ichiro Watanabe
Nagoya University, Furo-cho, Chikusa-ku, Nagoya, Aichi 464-8601, Japan  Daniel
J. Scheeres  University of Colorado Boulder, 429 UCB, Boulder, Colorado 80309
United States  Olivier S. Barnouin  Applied Physics Laboratory/Johns Hopkins
University, 11100 Johns Hopkins Road, Laurel, Maryland 20723, United States
Patrick Michel  Université Côte d’Azur, Observatoire de la Côte d’Azur, CNRS,
Laboratoire Lagrange, CS 34229, 06304 Nice Cedex 4, France  Chikatoshi Honda
University of Aizu, Ikki-machi, Aizuwakamatsu, Fukushima 965-8580, Japan
Tatsuhiro Michikami  Kindai University, 1 Takaya Umenobe, Higashi-Hiroshima,
Hiroshima 739-2116  Yuichiro Cho  University of Tokyo, 7-3-1 Hongo, Bunkyo-ku,
Tokyo 113-0033, Japan  Tomokatsu Morota  Nagoya University, Furo-cho, Chikusa-
ku, Nagoya, Aichi 464-8601, Japan  Naru Hirata  University of Aizu, Ikki-
machi, Aizuwakamatsu, Fukushima 965-8580, Japan  Naoyuki Hirata  Kobe
University, 1-1 Rokkodaicho, Nada, Kobe, Hyogo 657-8501, Japan  Naoya Sakatani
Institute of Space and Astronautical Science, Japan Aerospace Exploration
Agency, 3-1-1 Yoshinodai, Chuo-ku, Sagamihara, Kanagawa 252-5210, Japan
Stephen R. Schwartz  University of Arizona, 1629 E University Blvd, Tucson,
Arizona 85721, United States  Rie Honda  Kochi University, Akebono-Cho, Kochi,
Kochi 780-8520, Japan  Yasuhiro Yokota  Kochi University, Akebono-Cho, Kochi,
Kochi 780-8520, Japan  Institute of Space and Astronautical Science, Japan
Aerospace Exploration Agency, 3-1-1 Yoshinodai, Chuo-ku, Sagamihara, Kanagawa
252-5210, Japan  Shingo Kameda  Rikkyo University, 3-34-1 Nishi-
Ikebukuro,Toshima-ku, Tokyo 171-8501, Japan  Hidehiko Suzuki  Meiji
University, 1-1-1 Higashi-Mita, Tama-ku, Kawasaki, Kanagawa 214-8571, Japan
Toru Kouyama  National Institute of Advanced Industrial Science and
Technology, 1-1-1 Umezono, Tsukuba, Ibaraki 305-8560 Japan  Masahiko Hayakawa
Institute of Space and Astronautical Science, Japan Aerospace Exploration
Agency, 3-1-1 Yoshinodai, Chuo-ku, Sagamihara, Kanagawa 252-5210, Japan  Moe
Matsuoka  Institute of Space and Astronautical Science, Japan Aerospace
Exploration Agency, 3-1-1 Yoshinodai, Chuo-ku, Sagamihara, Kanagawa 252-5210,
Japan  Kazuo Yoshioka  University of Tokyo, 7-3-1 Hongo, Bunkyo-ku, Tokyo
113-0033, Japan  Kazunori Ogawa  Kobe University, 1-1 Rokkodaicho, Nada, Kobe,
Hyogo 657-8501, Japan  Hirotaka Sawada  Institute of Space and Astronautical
Science, Japan Aerospace Exploration Agency, 3-1-1 Yoshinodai, Chuo-ku,
Sagamihara, Kanagawa 252-5210, Japan  Makoto Yoshikawa  Institute of Space and
Astronautical Science, Japan Aerospace Exploration Agency, 3-1-1 Yoshinodai,
Chuo-ku, Sagamihara, Kanagawa 252-5210, Japan  Yuichi Tsuda  Institute of
Space and Astronautical Science, Japan Aerospace Exploration Agency, 3-1-1
Yoshinodai, Chuo-ku, Sagamihara, Kanagawa 252-5210, Japan

###### Abstract

162173 Ryugu, the target of Hayabusa2, has a round shape with an equatorial
ridge, which is known as a spinning top-shape. A strong centrifugal force is a
likely contributor to Ryugu’s top-shaped features. Observations by Optical
Navigation Camera onboard Hayabusa2 show a unique longitudinal variation in
geomorphology; the western side of this asteroid, later called the western
bulge, has a smooth surface and a sharp equatorial ridge, compared to the
other side. Here, we propose a structural deformation process that generated
the western bulge. Applying the mission-derived shape model, we employ a
finite element model technique to analyze the locations that experience
structural failure within the present shape. Assuming that materials are
uniformly distributed, our model shows the longitudinal variation in
structurally failed regions when the spin period is shorter than $\sim 3.75$
h. Ryugu is structurally intact in the subsurface region of the western bulge
while other regions are sensitive to structural failure. We infer that this
variation is indicative of the deformation process that occurred in the past,
and the western bulge is more relaxed structurally than the other region. Our
analysis also shows that this deformation process might occur at a spin period
between $\sim 3.5$ h and $\sim 3.0$ h, providing the cohesive strength ranging
between $\sim 4$ Pa and $\sim 10$ Pa.

minor planets, asteroids: individual (162173 Ryugu)

††journal: ApJ Letters

##  1 Introduction

When an asteroid has a round shape with equatorial ridges, this asteroid is
called a spinning top-shaped asteroid. Ground radar observations have shown
that spinning top-shapes are common in the solar system (Benner et al., 2015).
162173 Ryugu (Watanabe et al., 2019) and 101955 Bennu (Scheeres et al., 2016,
2019) are examples of spinning top-shapes. Such asteroids may have natural
satellites (e.g. Ostro et al., 2006; Michel et al., 2016; Naidu et al., 2016;
Brozovic et al., 2011; Fang & Margot, 2011; Becker et al., 2015). Importantly,
a spinning top-shape is found to be independent of material composition
(Benner et al., 2015).

The formation of a spinning top-shaped asteroid may result from a
reaccumulation process after catastrophic disruption (Michel et al., 2018,
2019) and/or quasi-static spin-up of a spheroidal body driven by
micrometeorite impacts or a solar radiation pressure-driven torque, which is
the so-called YORP effect (Rubincam, 2000). Regarding the quasi-static spin-up
process, there are two spinning top-shape formation scenarios depending on the
internal strength distribution: surface mass movement if there is a strong
interior covered by a weak layer (Walsh et al., 2008, 2012) and internal
deformation if there is uniform structure (Hirabayashi & Scheeres, 2014;
Hirabayashi et al., 2015; Zhang et al., 2017; Hirabayashi & Scheeres, 2018).
Note that even if the rotational period is relatively long, the uniform
structure may also cause surface failure (Hirabayashi, 2015). All these
processes make the shape settle into the minimum potential configuration
(Scheeres, 2015; Walsh, 2018).

Ryugu is the target of the Hayabusa2 mission led by JAXA (Watanabe et al.,
2019). Figure 1 describes images taken by the telescope camera, which is part
of the Optical Navigation Camera (ONC) system, onboard the spacecraft (Kameda
et al., 2017; Suzuki et al., 2018; Tatsumi et al., 2018). This asteroid has
the common shape features of a spinning top-shaped asteroid, and its spin axis
is perpendicular to the equatorial ridge (Watanabe et al., 2019). The
currently reported spin period is 7.63262 h (Watanabe et al., 2019), which is
much longer than the $\sim 2.3$-h spin barrier, the typical spin limit of
small rubble pile asteroids (Pravec et al., 2008). The bulk density was
estimated as $\sim$1.19 g cm-3 from the orbital motion of the Hayabusa2
spacecraft (Watanabe et al., 2019).

The surface condition of Ryugu was observed to be longitudinally divided into
the eastern and western regions (Sugita et al., 2019). These areas are
distinguished by Tokoyo and Horai Fossae, a trough system that is widely
placed in the southern hemisphere and possibly spreads towards the northern
hemisphere (Figure 1). The western region (160 deg E - 70 deg W), later known
as the western bulge, is apparently less affected by crater bombardment and
thus is smoother than the other side (Sugita et al., 2019). Furthermore, while
Ryugu’s spinning top-shape is axisymmetric in general; however, it is not
strictly so (Figure 1). The ridge angle, which defines an angle between the
surfaces in the northern and southern hemispheres at the equatorial ridge, or
Ryujin Dorsum, varies when seen from different views. The ridge angle of the
western bulge is 95 deg (Panel d in Figure 1) and is sharper than that of
other sides, which is 105 deg (Panels a through c in Figure 1). This feature
was also discussed in Watanabe et al. (2019) in detail.

This study tests a hypothesis that this longitudinal dichotomy results from
shape deformation when Ryugu was rotating with a short rotational period in
the past. Following is the outline of this paper. First, we employ the finite
element model (FEM) analysis (Hirabayashi & Scheeres, 2018) by assuming that
the material distribution is uniform, which is compatible with observations
(Sugita et al., 2019; Watanabe et al., 2019). Second, we use a dynamical
analysis model to determine at what spin period Ryugu would have experienced
the shape deformation process that caused the western bulge. Our work extends
the structural analysis by Watanabe et al. (2019), who only focused on the
general internal deformation process at spin periods of 3.75 hr and 3.5 hr, to
analyze the deformation process that implies the formation of the western
bulge at fast rotation.

Figure 1: Ryugu seen from the Hayabusa2 spacecraft at different views. a. 0
deg (image: hyb2_onc_20180630_074839_tvf_l2a). b. 90 deg East (image:
hyb2_onc_20180630_140543_tvf_l2a). c. 180 deg (image:
hyb2_onc_20180630_120959_tvf_l2a). d. 90 deg West (image:
hyb2_onc_20180630_101759_tvf_l2a). The yellow circles indicate the western
bulge. The red regions are Tokoyo and Horai Fossae. The ridge angles are
determined by only considering the sunlit sides and by averaging the slopes at
the middle latitudes. Image credit: JAXA/UTokyo/Kochi U./Rikkyo U./Nagoya
U./Chiba Ins. Tech/Meiji U./U. Aizu/AIST.

##  2 How did Ryugu deform at a short spin period?

We employ a FEM analysis (Hirabayashi et al., 2016; Hirabayashi & Scheeres,
2018) to investigate the structural failure mode of Ryugu at a given spin
period. This technique is the same as that done by Watanabe et al. (2019). In
this work, we use the shape model developed from the stereophotoclinometry
(SPC) technique (Watanabe et al., 2019). The version used is
SHAPE_SPC_3M_v20180731.obj in which there are 1,579,015 vertices and 3,145,728
facets (Watanabe et al., 2019). To avoid computational intensity, we reduce
the size of the shape model by using the Quadric Edge Collapse Decimation
technique available in Meshlab (http://www.meshlab.net/). The newly derived
shape model consists of 3,072 facets and 1,538 vertices.

We use a FEM solver available in ANSYS Mechanical APDL (18.1) under the
license owned by Auburn University’s Samuel Ginn College of Engineering. We
develop a three-dimensional, 10-node FEM mesh for the current shape of Ryugu
by using TetGen, an open tetrahedron-mesh generator (Si, 2015), and a mesh-
refining tool in ANSYS (Hirabayashi & Scheeres, 2018). The generated FEM mesh
consists of 8,308 elements and 15,038 nodes. This model assumes that spin-up
acceleration is so small that dynamic change in stress is negligible. Thus, we
solve the equilibrium equation of structure:

$\displaystyle\frac{\partial\sigma_{ij}}{\partial x_{j}}+\rho b_{i}=0.$ (1)

where $\rho$ is the bulk density, $\sigma_{ij}$ is the stress component
($i,j=1,2,3$), and $b_{j}$ is the body force. Also, we use the Einstein
notation in this equation. For elastic deformation, we use a linear-elastic
rule, i.e., Hookean deformation. To describe inelastic deformation, we assume
a plastic flow to be associated with a yield criterion where the behavior of
materials becomes inelastic (the associate flow rule) (Hirabayashi et al.,
2016). A plastic flow is given as a function of a partial derivative of a
yield criterion with respect to stress (Chen & Han, 1988). To model the yield
condition of regolith in Ryugu, we use the Drucker-Prager yield criterion,
which depends on the cohesive strength and the angle of friction. In this
work, the angle of friction is fixed at 35∘, which is a typical value for
regolith and soil materials (Lambe & Whitman, 1969). On the other hand, we
consider the cohesive strength to be a free parameter to investigate the
failure mode at a given spin period (Hirabayashi & Scheeres, 2018). The bulk
density is fixed at 1.2 g cm-3.

We introduce two critical parameters in our analysis. The first parameter is
the smallest cohesive strength that can hold the present shape, which is later
known as the minimum cohesive strength (Hirabayashi & Scheeres, 2018). This
parameter is an ideal value that could induce structural failure somewhere in
Ryugu’s body. Thus, if the actual cohesive strength is lower than the minimum
cohesive strength, structural failure should happen, causing deformation. We
describe the minimum cohesive strength as a function of the spin period, which
is called a failure mode diagram (FMD) (Hirabayashi & Scheeres, 2018). The
second parameter is the stress ratio, a ratio of the stress state to the yield
stress. If an element has a stress ratio reaching unity, it fails
structurally.

We compute the failure mode of the present shape at nine spin periods: 2.5 h,
3.0 h, 3.5 h, 3.75 h, 4.0 h, 5.0 h, 6.0 h, 7.6 h, and 8.0 h. Note that
Watanabe et al. (2019) only introduced the cases at spin periods of 3.5 h and
3.75 h. Figure 2 shows the FMD of Ryugu. At a spin period longer than 3.75 h,
inelastic deformation may appear due to shear stress under compression if
materials are nearly cohesionless. Such sensitive regions depend on local
topography and are located in small regions on the surface (Figure 2). Thus,
we rule out the scenario that inelastic deformation at slow rotation produced
the western bulge. At fast rotation, on the other hand, because the
centrifugal force plays a dominant role in the stress distribution, Ryugu
needs high cohesive strength to hold the present shape. Failed regions start
to appear in the interior of the body when the spin period is shorter than
3.75 hr (Watanabe et al., 2019). Also, the minimum cohesive strength
monotonically increases as the spin period becomes short (Figure 2).

Figure 2: Failure mode diagram of Ryugu. The $x$ axis is the spin period,
while the $y$ axis is the minimum cohesive strength. The dashed line indicates
the boundary between compression-dominant deformation and tension-dominant
deformation at a spin period of 3.75 h. The shaded region indicates that Ryugu
cannot structurally exist because the cohesive strength is below the minimum
cohesive strength. The top right panel describes the failure mode of Ryugu at
a spin period of 7.6 hr. The green and yellow regions are structurally intact
and failed areas, respectively.

In this paper, to discuss the deformation process that made the western bulge,
we employ the following approach. Our technique only accounts for the present
shape and provides the failure mode that this shape experiences, meaning that
we do not directly show the failure mode that the precursor shape had. To
infer the deformation process in the past, we search for structurally intact
points in the present shape. When a deformation process occurs at a given spin
period, the deformed area may settle into a configuration that the potential
of the shape becomes minimum (Scheeres, 2015; Walsh, 2018). Therefore, when
Ryugu encounters a fast-rotation condition once again, only the elements that
did not experience structural failure previously become sensitive to failure.
The originally failed regions, on the other hand, may not experience
structural failure because they are structurally relaxed.

If the body is axisymmetric, the deformation process should be symmetric
because the centrifugal and gravity forces are the main loadings. If an
asteroid is usually irregular in shape, the failure mode becomes complex and
asymmetric (Hirabayashi & Scheeres, 2018). Our FEM results show that the
structurally intact region corresponds to the western bulge. Figure 3 gives
the failed regions viewed from the spin axis direction. Lines $AA^{\prime}$,
$BB^{\prime}$, and $CC^{\prime}$ are the cross sections at three different
longitudes. Lines $AA^{\prime}$ and $BB^{\prime}$ cross the western bulge
while Line $CC^{\prime}$ does not. While the equatorial plane and surface
widely experience structural failure at spin periods of both 3.75 h and 3.5 h,
the structurally intact regions appear at longitudes between 120∘ E and 90∘ W.
We note that the 3.75-h spin period gives wider failed regions than the 3.5-hr
spin period because a transition between compression and tension happens at a
spin period of $\sim 3.75$ h, causing structurally failed regions to be larger
(Hirabayashi, 2015).

We also show the failed regions on the vertical cross sections (Figure 4).
Along $AA^{\prime}$ and $BB^{\prime}$, structurally intact regions spread in
the subsurface of the western bulge. On the other hand, the distribution of
the failed region along $CC^{\prime}$ is almost symmetric along the spin axis
direction. The FEM-derived minimum cohesive strength at spin periods of 3.75 h
and 3.5 h are 1.75 Pa and 4 Pa, respectively (Watanabe et al., 2019). Our
analysis indicates that the existence of the intact region beneath the western
bulge cannot be explained by symmetric deformation. We propose that this
results from a deformation process that made the western bulge.

Figure 3: Distributions of the failed regions (yellow) at spin periods of 3.75
h (top) and 3.5 h (bottom). The left, middle, and right columns show the top
surface, the bottom surface, and the cross-section, respectively. The circle
with a dashed line corresponds to the western bulge (Sugita et al., 2019).
Lines $AA^{\prime}$, $BB^{\prime}$, and $CC^{\prime}$ define the cross-
sections, which are cut through longitudes of 0∘ and 180∘, 60∘ E and 120∘ W,
and 120∘ E and 60∘ W. Figure 4: Distributions of the failed regions at spin
periods of 3.75 h (Panels a through c) and 3.5 h (Panels d through f). Panels
a and d show the failure mode on $AA^{\prime}$, which is defined in Figure 3.
Similarly, Panels b and e indicate the failure mode on $BB^{\prime}$, and
Panels c and f describe that on $CC^{\prime}$. The dotted lines show the
structurally intact regions that appear asymmetrically in the subsurface of
the western bulge. Note that Panels c and f are given in Watanabe et al.
(2019).

##  3 How fast did Ryugu spin?

In the previous section, we argued that the western bulge might result from a
deformation process. In this section, we analyze the rotational state that
triggered this deformation process. During the deformation occurs at fast
spin, fluidized materials with weak mechanical strength (Melosh, 2011) may
reach the equatorial region. If the centrifugal force is strong, such
materials should be lofted from there. Because the centrifugal force is
proportional to the radial distance from an asteroid’s center of mass, this
material shedding process can create a circular ridge (Scheeres et al., 2006),
which supports the observed circularity of Ryugu (Watanabe et al., 2019).

To determine this spin period condition, we compute the locations of the
dynamical equilibrium points at which the total force is balanced in the
rotating frame. If these points touch the surface, fluidized materials should
be lofted from there. The force balance condition is described as:

$\displaystyle{\mbox{\boldmath{$f$}}}_{g}({\mbox{\boldmath{$x$}}})+{\mbox{\boldmath{$f$}}}_{c}({\mbox{\boldmath{$x$}}})=\mbox{\boldmath{$0$}},$
(2)

where ${\mbox{\boldmath{$f$}}}_{g}$ is the gravity force, and
${\mbox{\boldmath{$f$}}}_{c}$ is the centrifugal force. We compute the gravity
force vector by using the algorithm developed by Werner & Scheeres (1997) and
a refined shape model that consists of 3,072 facets and 1,538 vertices. These
force terms are functions of the dynamical equilibrium points, $x$. We
numerically compute the dynamical equilibrium points by using Newton’s
implicit equation solver (e.g. Press et al., 2007).

We also compute the zero-velocity curves, which show the pseudo-energy levels
that correlate with the locations of the dynamical equilibrium points (Murray
& Dermott, 2008). Energetically, these points become either saddle points or
local extrema. Seen from Ryugu’s center of mass, the region outside the
dynamical equilibrium points may experience strong centrifugal forces, while
the gravity is dominant in the area within these points. For spinning top-
shaped asteroids, there are usually more than four dynamical equilibrium
points because of their axisymmetric shapes (Scheeres et al., 2006).

Similar to the FEM analysis, we fix the bulk density at 1.2 g cm-3. Figure 5
shows the locations of the dynamical equilibrium points and the zero velocity
curves at spin periods of 7.6 h and 3.5 h. The zero-velocity curves are drawn
at pseudo-energy levels near the dynamical equilibrium points to enhance the
energy conditions at these points. We find that at a spin period of $3.5$ h,
12 dynamical equilibrium points appear and are about to touch the surface. At
this spin condition, the zero velocity curves are circular, the size of which
is consistent with Ryujin Dorsum.

We also give constraints on the cohesive strength of Ryugu. Structural failure
should occur in Ryugu at a spin period at 3.5 h if the cohesive strength is
$\sim 4$ Pa (see Figure 2). This spin period condition may be an upper bound.
When Ryugu deforms, its moment of inertia changes due to deformation. If this
asteroid rotates at a given spin period before deformation, it should change
its spin period after deformation in a manner that the angular momentum is
conserved. Because of the rotationally induced deformation mode of a spinning
top-shaped asteroid (e.g. Hirabayashi & Scheeres, 2014), the moment of inertia
increases as deformation continues. Thus, before Ryugu’s precursor deforms to
become the current shape at the 3.5-h spin period, it may have a shorter spin
period than that period.

To consider a lower bound of the spin period of deformation, we conduct the
following thought experiment. Once Ryugu’s precursor structurally failed at a
spin period shorter than 3.5 hr, the shape eventually settled into the current
configuration at the 3.5-hr spin period. Assuming that it was spinning along
the maximum moment of inertia axis, we conservatively choose a sphere that has
the smallest moment of inertia to be the shape of the precursor body. Here,
the spin period of this object before deformation is considered to be the
lower bound. We assume that the total mass and volume are constant, ignoring
the amount of shedding materials. Because the mean radius of Ryugu is 448.4 m
(Watanabe et al., 2019), we obtain this spin period as 3.0 h and the minimum
cohesive strength as $\sim$10 Pa (Figure 2). We conclude that the cohesive
strength of Ryugu ranges between $\sim 4$ Pa and $\sim 10$ Pa.

Figure 5: Locations of the dynamical equilibrium points. a shows the case at a
spin period of 7.6 h. b describes the case of a spin period of 3.5 h. The red
points indicate the shape of Ryugu. The star makers are the locations of the
dynamical equilibrium points. The contour outside the shape describes the
zero-velocity curves. The red dots describe the shape of Ryugu viewed from the
spin axis. The dashed line indicates the location of the western bulge.

##  4 Discussion

We showed that the subsurface structure of the western bulge is not sensitive
to structural failure at present, implying that this region is likely to be
relaxed structurally. We proposed that Ryugu experienced a deformation process
that generated the western bulge. This process might occur only on the side of
the bulge. Such an asymmetric deformation process was numerically predicted
for a spheroidal object (Sánchez & Scheeres, 2016), who pointed out that even
small structural heterogeneity can cause an asymmetric deformation process,
although theoretical studies missed this point (e.g., Holsapple, 2010;
Hirabayashi, 2015). Our hypothesis explains the east-west geomorphological
dichotomy. The smooth, less-cratered terrain on the western bulge can be
generated due to mass movement. Also, the deformation process can produce void
space in the interior, followed by a material infilling process into it, which
explains the formation of Tokoyo and Horai Fossae.

While our discussions were focused on rotationally induced failure, we note
alternative possibilities. First, impact craters may contribute to deformation
processes in Ryugu. However, we rule out this possibility as a direct
contributor because this does not produce the circularity of Ryujin Dorsum.
Second, a head-on collision of two similar-sized objects may produce the east-
west dichotomy (Leleu et al., 2018). However, although the contact point
should be highly compressed and have a ridge-like feature (Leleu et al.,
2018), Ryugu does not have such a feature. Instead, we observe depression
features, Horai and Tokoyo Fossae.

We note that our work does not determine whether internal deformation (e.g.
Hirabayashi & Scheeres, 2014; Scheeres et al., 2016; Zhang et al., 2017;
Hirabayashi & Scheeres, 2018) or surface mass movement without internal
deformation (e.g. Walsh et al., 2008, 2012; Zhang et al., 2017) is the primary
cause of the considered asymmetric deformation process. If the internal
structure is strong, Ryugu may have surface mass movement. On the other hand,
if the interior is uniform, this asteroid may have internal deformation. We,
however, rule out large density inhomogeneity between the western bulge and
other regions because the center of mass of Ryugu derived by Hayabusa2
corresponds to numerical prediction based on the mission derived shape model
with constant density within error (Watanabe et al., 2019). It is also
possible that Ryugu may have complex deformation processes if its rubble pile
structure possesses internal discontinuities that formed during the
reaccumulation process after catastrophic disruption of its parent body.

## Acknowledgments

M.H. thanks support from Auburn University and acknowledges ANSYS Mechanical
APDL (18.1) under the license owned by Auburn’s Samuel Ginn College of
Engineering. P.M. acknowledges support from the French space agency CNES as
well as from Academies of Excellence: Complex systems and Space, environment,
risk and resilience, part of the IDEX JEDI of the Université Côte d’Azur in
connexion with its Center For Planetary Origins. The ONC has been developed
under the leadership of JAXA with collaboration with Univ. of Tokyo, Kochi
Univ., Rikkyo Univ. Nagoya Univ., Chiba Inst. of Technology, Meiji Univ.,
Univ. of Aizu, and AIST and contracted contribution of NEC corporation. This
study is supported by the JSPS core-to-core program, “International Network of
Planetary Sciences.” The FEM results for the 3.5 hr and 3.75 hr periods are
from Watanabe et al. (2019) and available at
http://hdl.handle.net/11200/49368.

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