# Mass shedding activities of Asteroid (3200) Phaethon enhanced by its
rotation

Ryota Nakano  Department of Aerospace Engineering
Auburn University, 211 Davis Hall
Auburn, AL 36849, USA  Masatoshi Hirabayashi  Department of Aerospace
Engineering
Auburn University, 211 Davis Hall
Auburn, AL 36849, USA

###### Abstract

Asteroid (3200) Phaethon, a B-type asteroid, has been active during its
perihelion passages. This asteroid is considered to be a source of the Geminid
meteor stream. It is reported that this asteroid is spinning at a rotation
period of $3.60\ hr$ and has a top shape (an oblate body with an equatorial
ridge) with a mean equatorial diameter of $6.25\ km$. Here, we report that
Phaethon’s rotation state may be close to or above its critical rotation
period when the bulk density is $0.5\ -\ 1.5\ {g/cm^{3}}$ (a typical bulk
density of a B-type asteroid). We found that in this condition, the structure
of Phaethon is sensitive to failure unless the cohesive strength is ${\sim}50\
Pa\ -\ {\sim}260\ Pa$. This result implies that if there are some surface
processes driven by, for example, thermal waves, large-scaled deformation may
happen and cause mass shedding. From this interpretation, we propose the
processes that produced the Geminid meteor stream in the past and dust tails
recently. Phaethon initially rotated at a spin period shorter than the current
period. The magnitude of structural deformation at this stage was higher than
the present spin condition, and a large mass shedding event, i.e., the Geminid
meteor stream, occurred. After this deformation process, the body became more
oblate, and its spin slowed down. At this point, while the spin was high
enough for the body to have mass shedding events, the magnitude of these
events became small.

comets: general — meteorites, meteors, meteoroids — minor planets, asteroids:
general — minor planets, asteroids: individual (3200 Phaethon)

††journal: ApJL

##  1 Introduction

Asteroid (3200) Phaethon may be a source of the Geminid meteor stream
(Whipple, 1983; Gustafson, 1989; Williams & Wu, 1993) and has been observed
for a long time (Cochran & Barker, 1984; Chamberlin et al., 1996; Hsieh &
Jewitt, 2005; Wiegert et al., 2008). In 2009 (Jewitt & Li, 2010) and 2012 (Li
& Jewitt, 2013), dust tails from Phaethon were observed during the perihelion
passage, revealing that this asteroid is indeed an active asteroid. However,
the activities detected near the perihelion left a mystery of the production
of the Geminid meteor stream. The dust mass inferred from the observations is
${\sim}3\times 10^{5}\ kg$ (Li & Jewitt, 2013), which is much smaller than the
estimated mass of the Geminid meteor stream, ${10}^{12}-{10}^{13}\ kg$ (Hughes
& McBride, 1989; Jenniskens, 1994). Also, the estimated average mass-loss rate
of $3\ {kg}/s$ is too small to replenish the Geminid meteor stream mass
(Jewitt et al., 2013), if the dynamical lifetime of the Geminid meteor stream
is ${\sim}10^{3}\ yr$ (Gustafson, 1989; Ryabova, 2007). While most of the
proposed mechanisms were found incapable of producing dust tails, thermal
fracture and cracking due to dehydration in surface materials might be
reasonable processes to generate dust tails (Jewitt & Li, 2010; Jewitt, 2012;
Li & Jewitt, 2013).

Radar observations during the 2017 apparition revealed Phaethon’s shape.
Taylor et al. (2019) reported that Phaethon might be an oblate shape with an
equatorial ridge, or the so-called top shape. The equivalent diameter of this
asteroid was estimated to be $6\ km$ in diameter. The examples of top-shaped
asteroids are OSIRIS-REx’s target (101955) Bennu (Lauretta et al., 2019) and
Hayabusa2’s target (162173) Ryugu (Watanabe et al., 2019; Sugita et al., 2019;
Kitazato et al., 2019). Phaethon is currently spinning at a rotation period of
$3.60\ hr$ (Taylor et al., 2019). The radar albedo is reported to be the
lowest among the cataloged near-Earth asteroids (Taylor et al., 2019),
implying that its spectral type is consistent with a B-type (Taylor et al.,
2019), and thus the bulk density may be as low as ${\sim}1.0\ g/{cm}^{3}$
(Scheeres et al., 2019; Watanabe et al., 2019).

We hypothesize that the mass shedding activities of Phaethon may have been
enhanced by fast rotation, given recent work arguing that the equatorial
ridges of top-shaped asteroids were evolved by rotationally driven reshaping
(Walsh et al., 2008, 2012). We propose that structural failure on the surface
and/or inside the body triggered by fast rotation plays an important role in
mass shedding mechanism. This study provides better insights into the physical
activities of Phaethon to support DESTINY+, a planned flyby mission concept
led by the Japan Aerospace Exploration Agency (Arai et al., 2018).

##  2 Semi-analytical model for structural failure in a top-shaped body

Phaethon was reported to be a top-shaped body. Recent work has shown that the
global failure condition in uniformly rotating top shapes can be roughly
determined by assuming that they are triaxial ellipsoids (Hirabayashi, 2015).
For Phaethon, the shape is not well known at present while an effort in
analyzing it from radar observation data is being made (Taylor, personal
communication). Therefore, a simplified model that uses a triaxial ellipsoid
can still reasonably provide structural failure in this asteroid. We note that
the heterogeneity in structural failure may be critical once the detailed
shape of this asteroid is considered (Hirabayashi & Scheeres, 2019).

We consider that Phaethon has a mean equatorial diameter of $6.25\ km$ and an
oblateness (the ratio of semi-minor axis to semi-major axis) of $0.889$, which
is the same as Bennu’s (Barnouin et al., 2019). We note that Taylor et al.
(2019) did not specify the oblateness of Phaethon but implied that it would be
similar to that of Bennu. For the oblateness, we do not account for the semi-
intermediate axis to simplify the discussion. We assume that Phaethon’s
structure is uniform because the internal condition is unknown. We consider
three types of the bulk density, $0.5\ g/{cm}^{3}$, $1.0\ g/{cm}^{3}$, and
$1.5\ g/{cm}^{3}$, the averaged of which is consistent with that of a B-type
asteroid (Scheeres et al., 2019). Later, we denote the oblateness, the bulk
density, and the gravitational constant as $\epsilon$, $\rho$, and $G$,
respectively. Phaethon is assumed to be rotating along the maximum principal
axis. We define a three-dimensional Cartesian coordinate system such that the
$z$ axis is lined up along the rotation axis, and the $x$ and $y$ axes are
along the maximum and intermediate moment of inertia axes, respectively, on
the equatorial plane. Using these definitions, we compute how Phaethon
experiences structural failure at a give rotation period, $P$.

To determine the failure condition of Phaethon, we extend the technique by
Hirabayashi (2015), who only considered a sphere. We analyze when the stress
field in a given element reaches its yield condition. In this model, the
material behavior is assumed to be elastic-perfectly plastic, where a plastic
flow begins once the stress reaches its yield condition without material
hardening and softening. To describe such a material behavior, we use the
following material properties: Poisson’s ratio, $\nu$; Young’s modulus, $E$;
the angle of friction, $\phi$; and the cohesive strength, $Y$. It is worth
noting that the evolution of plastic deformation is a function of loading
paths. However, our purpose is not to track plastic deformation but to
determine what element would first have its plastic state in a quasi-static
condition where deformation is small enough that the time-variation is
negligible (Hirabayashi, 2015).

Although we follow the terminologies of the cohesive strength by Hirabayashi
(2015), we reintroduce them here to facilitate the following discussion. We
compute the minimum cohesive strength that can prevent structural failure of a
given element in an asteroid rotating at $P$. We call this strength ‘critical
cohesive strength’ and denote it as $Y^{*}$. On the other hand, we use ‘actual
cohesive strength’ as an assumed strength that an asteroid may have. We denote
this as $Y$. Also, the critical rotation period $P_{c}$ is the rotation period
at which a small particle on the equatorial surface of Phaethon gains the
centrifugal acceleration larger than the gravitational acceleration and is
lifted off from the surface.

###  2.1 Stress field computation

Similar to Hirabayashi et al. (2015) and Hirabayashi (2015), we apply a
technique by Dobrovolskis (1982) and Holsapple (2001) to provide an elastic
stress in a triaxial ellipsoid that is uniformly spinning at rotation period
of $P$. Here, while noting that the details are found in Dobrovolskis (1982),
we briefly introduce the formulation process. The displacement $u$ in
Cartesian coordinates can be expressed in terms of twelve unknown constants
$A$ through $L$:

$u_{x}=x\left[A+B\frac{x^{2}}{a^{2}}+C\frac{y^{2}}{b^{2}}+D\frac{z^{2}}{c^{2}}\right],$
(1)
$u_{y}=y\left[E+F\frac{x^{2}}{a^{2}}+G\frac{y^{2}}{b^{2}}+H\frac{z^{2}}{c^{2}}\right],$
(2)
$u_{z}=z\left[I+J\frac{x^{2}}{a^{2}}+K\frac{y^{2}}{b^{2}}+L\frac{z^{2}}{c^{2}}\right].$
(3)

where $a$, $b$, and $c$ $(a\geq b\geq c)$ are the principle semi-axes of the
triaxial ellipsoid. The strain is obtained by:

$\epsilon_{ij}=\frac{1}{2}\left(\frac{\partial u_{i}}{\partial
x_{j}}+\frac{\partial u_{j}}{\partial x_{i}}\right).$ (4)

Applying Hooke’s law, the stress tensor is also expressed in terms of $A$
through $L$:

$\sigma_{ij}=\lambda\epsilon_{kk}\delta_{ij}+2\mu\epsilon_{ij},$ (5)

where $\epsilon_{kk}=\epsilon_{xx}+\epsilon_{yy}+\epsilon_{zz}$, and
$\delta_{ij}$ is the Kronecker delta. $\lambda$ and $\mu$ are the Lame’s
constants obtained from

$\lambda=\frac{E\nu}{(1+\nu)(1-2\nu)},$ (6) $\mu=\frac{E}{2(1+\nu)}.$ (7)

To determine the twelve unknown constants, we must impose twelve linearly
independent relations. The stresses $\sigma_{ij}$ in a body in equilibrium
under body forces must satisfy the stress equilibrium equations:

$\frac{\partial}{\partial x_{j}}\sigma_{ji}=-\rho b_{i},$ (8)

where $b_{i}$ is the body force, which is driven by gravitational and
rotational effects in our problem. Equation (8) provides three constraints out
of the twelve required relations. Remaining nine relations are then imposed
from the traction free boundary condition:

$\sigma_{ij}n_{j}=0,$ (9)

where $n_{j}$ is the unit normal to the surface and given by:

$n_{x}=\frac{x}{a^{2}w},$ (10)

$n_{y}=\frac{y}{b^{2}w},$ (11)

$n_{z}=\frac{z}{c^{2}w},$ (12)

where

$w=\left(\frac{x^{2}}{a^{4}}+\frac{y^{2}}{b^{4}}+\frac{z^{2}}{c^{4}}\right)^{1/2}$
(13)

The twelve unknown constants are, hence, provided by twelve linearly
independent relations.

###  2.2 Structural failure condition

Once the stress field is obtained from the previous section, we use it to
determine the structural failure condition of a given element. Here, we apply
the Drucker-Prager yield criterion, a smooth approximation of the Mohr-Coulomb
yield criterion (Chen & Han, 1988):

$f=\alpha I_{1}+\sqrt{J_{2}}-s\leq 0.$ (14)

$I_{1}$ and $J_{2}$ are the stress invariants:

$I_{1}=\sigma_{1}+\sigma_{2}+\sigma_{3},$ (15)

$J_{2}=\frac{1}{6}\\{(\sigma_{1}-\sigma_{2})^{2}+(\sigma_{2}-\sigma_{3})^{2}+(\sigma_{3}-\sigma_{1})^{2}\\},$
(16)

where $\sigma_{i}$ ($i=1,2,3$) is the principal stress component, which can be
obtained from the derived stress field above. $\alpha$ and $s$ are material
constants and given by (Chen & Han, 1988):

$\alpha=\frac{2\sin{\phi}}{\sqrt{3}(3-\sin{\phi)}},$ (17)

$s=\frac{6Y\cos{\phi}}{\sqrt{3}(3-\sin{\phi)}}.$ (18)

From the equal condition of Equation (14), we obtain the following expression
for $Y^{*}$:

$Y^{*}=\frac{\sqrt{3}(3-\sin{\phi})}{6\cos{\phi}}\left(\alpha
I_{1}+\sqrt{J_{2}}\right).$ (19)

If Equation (19) becomes negative, we consider $Y^{*}$ to be $0\ Pa$, which
means that no strength is necessary for an element to keep the original shape.

##  3 Results

We investigate the critical cohesive strength based on the following assumed
parameters, $\nu=0.25$, $E=10^{7}\ Pa$, and $\phi=35\degree$, to make our
discussion consistent with earlier work (e.g., Hirabayashi & Scheeres 2015).
We note that Young’s modulus does not influence our stress field calculation
(Love, 2013), and the variations in Poisson’s ratio and the angle of friction
do not affect our results for geological materials significantly (Lambe &
Whitman, 1969; Hirabayashi & Scheeres, 2019). The current rotation period is
set to be $3.60\ hr$ (Taylor et al., 2019). We consider the bulk density to be
$0.5$, $1.0$, and $1.5\ g/{cm}^{3}$. Table 1 lists the parameters considered
in the current study.

We find that the failure mode varies with the bulk density. We plot the
distribution of $Y^{*}$ at the $x$-$z$ plane for different bulk densities in
Figure 1. Panels a, b, and c describe the bulk densities of $0.5$, $1.0$, and
$1.5\ g/{cm}^{3}$, respectively. In this range of the bulk density, the body
should have a cohesive strength to keep the original shape. For the case of
$\rho=0.5\ g/{cm}^{3}$, $P_{c}$ is found to be $4.83\ hr$. Therefore, the
current rotation period is shorter than $P_{c}$, indicating that materials
should be shed and highly sensitive to structural failure. Figure 1a shows
$Y^{*}>0$ everywhere except the pole region. $Y^{*}$ is higher in the interior
than on the surface, and its maximum value is $259\ Pa$ at the center. This
indicates that the interior is more sensitive to structural failure than the
surface. If the actual cohesive strength $Y$ is smaller than $Y^{*}$, the
central region structurally fails first. For the case of $\rho=1.0\
g/{cm}^{3}$, $P_{c}$ is found to be $3.42\ hr$ and is shorter than the current
period. However, the interior still exhibits high $Y^{*}$ in the major regions
(Panel b). The maximum value of $Y^{*}$ is $179\ Pa$ and is located at the
center. For the case of $\rho=1.5\ g/{cm}^{3}$, $P_{c}$ is found to be $2.79\
hr$. Unlike the other two cases, the interior has $Y^{*}=0$ in the most areas;
however, high $Y^{*}$ is still distributed beneath the surface (${\sim}50\
Pa$)(Panel c).

All these three cases show the sensitivity of Phaethon to structural failure.
This body needs a cohesive strength to keep the original condition without
shape deformation. However, if there is a trigger of reshaping, it is likely
that the deformation process would be enhanced by rotation, as seen from the
derived sensitivity. Because of the observed activities, the cohesive strength
of Phaethon is ${\sim}50\ Pa\ -\ {\sim}260\ Pa$, depending on the bulk
density. This range is consistent with that of observed small bodies (Scheeres
& Sánchez, 2018).

Table 1: Parameter settings Parameter | Symbol | Value | Units
---|---|---|---
Gravitational constant | G | 6.6738 ×10^-11 | m^3 ⋅kg^-1 ⋅s^-2
Semi-major axis | a | 3200 | m
Semi-minor axis | c | 2847 | m
Oblateness | ϵ | 0.889 | -
Current rotation period | P | 3.60 | hr
Bulk density | ρ | 0.5, 1.0, 1.5 | g ⋅cm^-3
Poisson’s ratio | ν | 0.25 | -
Elastic modulus | E | 10^7 | Pa
Friction angle | ϕ | 35 | deg
Figure 1: Distribution of $Y^{*}$ at the $x$-$z$ plane. The rotation period is
$3.60\ hr$. Panels a, b, and c describe bulk densities of $0.5$, $1.0$, and
$1.5\ g/{cm}^{3}$, respectively.

##  4 Discussions

Generation of dust tails at present

During perihelion passage in 2009 and 2012, the observations of Phaethon’s
dust tails revealed that this asteroid was an active asteroid (Jewitt & Li,
2010; Li & Jewitt, 2013). While the detailed mechanisms are not well known,
thermal waves may be one of plausible explanations for the reported mass
shedding activities (Li & Jewitt, 2013).

At the current rotation period of $3.60\ hr$, the body is sensitive to
structural failure regardless of the bulk density and thus needs cohesive
strength to maintain its shape. The derived cohesive strength of Phaethon is
less than ${\sim}50\ Pa\ -\ {\sim}260\ Pa$, which is consistent with that of
small bodies ranging up to a few hundred pascals (Scheeres & Sánchez, 2018).
We interpret this sensitivity as a potential enhancement of mass shedding. If
there is a trigger of reshaping even at small scales, the structure of Phaeton
would be perturbed, leading to rotationally driven reshaping at larger scales.
Such a trigger can be thermal waves in thin surface layers (Jewitt & Li, 2010;
Jewitt, 2012; Li & Jewitt, 2013). Micrometeoroid impacts or other processes
may also be possible, as seen on Bennu (Lauretta et al., 2019), although
thermal waves, again, are more consistent to explain the activities of
Phaethon around its perihelion passage (Li & Jewitt, 2013).

Possible source of the Geminid meteor stream

We expect that rotationally induced structural failure makes Phaethon more
oblate, i.e., $\epsilon$ becomes lower (Hirabayashi, 2015). Because the
angular momentum is constant during deformation, Phaethon may have been less
oblate and rotated faster at an earlier stage before it had large deformation.
Figure 2 shows the dependence of $P$ on $\epsilon$ and $P_{c}$ for different
bulk densities. We find that if Phaethon is less oblate, the rotation period
is shorter, and thus $Y^{*}$ should be higher at a shorter rotation period. If
Phaethon is a sphere ($\epsilon=1.0$; $a=b=c$), which is the shape condition
that is the least affected by rotation, the rotation period should become
$P=3.38\ hr$. Figure 3 shows the distribution of $Y^{*}$ for this case.
Similar features observed in Figure 1 are evident in this case; however,
$Y^{*}$ is higher, implying that Phaethon should have been more sensitive to
structural failure at a shorter rotation period. Therefore, the failure mode
may become severer, and more materials should be shed significantly at a
shorter rotation period.

Figure 2: Rotation period as a function of the oblateness (the blue solid
line). $P=3.60\ hr$ when $\epsilon=0.889$ and $P=3.38\ hr$ when
$\epsilon=1.0$. The dotted lines indicate $P_{c}$ for different bulk
densities: the red, green, and magenta show bulk densities of $0.5$, $1.0$,
and $1.5\ g/{cm}^{3}$, respectively.

Figure 3: Distribution of $Y^{*}$ at the $x$-$z$ plane. The rotation period is
$3.38\ hr$. Panels a, b, and c describe bulk densities of $0.5$, $1.0$, and
$1.5\ g/{cm}^{3}$, respectively.

From these results, we propose a possible evolution scenario of Phaethon
(Figure 4). Phaethon was originally less oblate and spinning at a shorter
rotation period than the current period. This stage is before the Geminid
meteor stream was generated. Possible initiation processes such as thermal
waves during Phaethon’s apparition passages triggered reshaping, and
rotational deformation enhanced this reshaping process significantly. Because
the rotation period at this stage was closer to or above $P_{c}$, the
reshaping process caused mass shedding at large scale, which became a source
of the Geminid meteor stream. Thus, the current oblate shape maybe a remnant
of this large deformation event. When the oblateness evolved, the rotation of
Phaethon slowed down. However, the structure was still sensitive to failure.
When there is similar perturbation such as thermal waves at present,
rotationally driven failure can be triggered; however, because the centrifugal
effect is less significant, the magnitude of mass shedding is less intense
than that in the past.

While we cannot strongly constrain whether the generation of the Geminid
meteor stream was a single event or episodic, our study gives some hints of
large-scaled mass shedding processes as a source of the Geminid meteor stream.
Because the generation of the Geminid meteor stream may have occurred within
the last $1\ ka$ (Gustafson, 1989; Williams & Wu, 1993; Ryabova, 2007), the
large-scaled reshaping and mass shedding processes may have occurred in this
timescale. These processes should be more rapid and intense to be completed
(within the last $1\ ka$) than the YORP driven evolution, which may take
${\sim}1\ Ma$ based on earlier work (Čapek & Vokrouhlický, 2004; Bottke et
al., 2006). Given this short timescale, a possible explanation of the
deformation mode may be internal failure, which can provide large-scaled
deformation and thus mass shedding at fast rotation (Hirabayashi, 2015).

We note that the total mass of the Geminid meteor stream is about $2.5\%$ of
that of Phaethon. If a mass shedding even that can produce the same magnitude
of the Geminid meteor stream occurs every $1ka$, the lifetime of Phaethon
would only be ${\sim}40\ ka$, which may be much shorter than that predicted by
the orbital evolution, ${\sim}26\ Ma$ (De León et al., 2010). While there is
no decisive evidence, the YORP effect may give some insights into this
discrepancy. The YORP evolution timescale of Phaethon may become longer by
many different factors such as the mass, shape, surface composition, and
stochastic evolution (Bottke et al., 2015). Furthermore, its highly eccentric
orbit, $e=0.890$, with small perihelion distance of 0.14 AU (JPL Small-Body
Database) may give strong variations in solar radiation acting on Phaethon.
Thus, it may be possible that Phaethon has stochastically spun up by the YORP
effect for the entire orbital age, ${\sim}26Ma$ (De León et al., 2010), and it
recently experienced large-scaled mass shedding that formed the Geminid meteor
stream. Then, a large mass shedding event that produced the Geminid meteor
stream may have decelerated Phaethon’s spin, but the spin state after this
event may have been still high enough to have some mass shedding events at
small level, similar to what we observed in 2009 and 2012 (Jewitt & Li, 2010;
Jewitt, 2012; Li & Jewitt, 2013). To fully address the detailed timescale of
the rotationally driven reshaping process is beyond the scope of this study;
we leave this problem as future work.

Finally, We assumed that Phaethon currently has a top shape having Bennu’s
oblateness, $\epsilon=0.889$, by following Taylor et al. (2019). To check if
this setting is consistent with other top-shaped asteroids, we consider six
top-shaped near-earth asteroids: Ryugu (Watanabe et al., 2019), 1994 KW4
(Ostro et al., 2006), 2008 EV5 (Busch et al., 2011), 1994 CC (Brozović et al.,
2011), 2001 SN263 (Becker et al., 2015), and 2000 DP107 (Naidu et al., 2015).
We find the range of $\epsilon$ to be $0.873$ – $0.968$, and thus Bennu’s
$\epsilon$ is within this range. We conduct the same analyses for these
objects and found trends similar to Bennu’s. The variation in the oblateness
at this magnitude does not affect the results of $Y^{*}$ significantly, which
is less than 12%. Therefore, we conclude that our oblateness setting is
meaningful to capture a possible scenario of Phaethon’s activities.

Figure 4: Possible evolution scenario of Phaethon. Phaethon has a highly
eccentric orbit ($e=0.890$) with small perihelion distance of 0.14 AU. At an
earlier stage, Phaethon was less oblate and rotating faster, leading to mass
shedding at larger scale. For both stages, perturbation such as thermal waves
may be a possible trigger for mass shedding.

Potential issues

We finally address issues of our analysis approach. This study explored
Phaethon’s rotationally induced structural failure by modeling Phaethon as a
triaxial ellipsoid. We did not account for local topographic features;
therefore, our semi-analytical model does not capture local deformations which
may differ from global deformations (Hirabayashi & Scheeres, 2019; Hirabayashi
et al., 2019). However, Hirabayashi (2015), who compared an analytical
solution and a FEM solution, found that there was no significant variation
between the two. Therefore, we conclude that our semi-analytical model can
provide solutions with reasonable accuracy. It is our future work to perform
FEM analysis using a shape model to investigate the local deformations.

Another issue to be addressed is that our model does not assert the detailed
failure mode. While numerous research studies have been undertaken, it is
still not well determined how top shape asteroids deform. Hirabayashi (2015)
showed that depending on the internal structure and the bulk density, the
failure mode may become different – either surface processing or internal
deformation. However, because our semi-analytical model can only describe a
homogeneous structure, we cannot infer how Phaethon’s deformation is
controlled by the heterogeneity. Therefore, the detailed failure mode cannot
be specified. If Phaethon’s heterogeneous structure is revealed, we need to
employ different approach (i.e. Hirabayashi et al. 2015).

There still remain many unknowns regarding Phaethon’s physical properties. We
will further elaborate our approach to give constraints on the activity of
Phaethon. Concurrently, further constraints are vitally important to support
DESTINY+.

RN and MH acknowledge support from NASA/Solar System Workings
(NNH17ZDA001N/80NSSC19K0548) and Auburn University/Intramural Grant Program.

## References

  * Arai et al. (2018) Arai, T., Kobayashi, M., Ishibashi, K., & Yoshida, F. 2018, in 49th LPSC
  * Barnouin et al. (2019) Barnouin, O. S., Daly, M. G., Palmer, E. E., et al. 2019, Nature Geoscience, 12, 247, doi: 10.1038/s41561-019-0330-x
  * Becker et al. (2015) Becker, T. M., Howell, E. S., Nolan, M. C., et al. 2015, Icarus, 248, 499, doi: 10.1016/j.icarus.2014.10.048
  * Bottke et al. (2006) Bottke, W. F., Vokrouhlický, D., Rubincam, D. P., & Nesvorný, D. 2006, Annual Review of Earth and Planetary Sciences, 34, 157, doi: 10.1146/annurev.earth.34.031405.125154
  * Bottke et al. (2015) Bottke, W. F., Vokrouhlický, D., Walsh, K. J., et al. 2015, Icarus, 247, 191, doi: 10.1016/j.icarus.2014.09.046
  * Brozović et al. (2011) Brozović, M., Benner, L. A., Taylor, P. A., et al. 2011, Icarus, 216, 241, doi: 10.1016/j.icarus.2011.09.002
  * Busch et al. (2011) Busch, M. W., Ostro, S. J., Benner, L. A., et al. 2011, Icarus, 212, 649, doi: 10.1016/j.icarus.2011.01.013
  * Čapek & Vokrouhlický (2004) Čapek, D., & Vokrouhlický, D. 2004, Icarus, 172, 526, doi: 10.1016/j.icarus.2004.07.003
  * Chamberlin et al. (1996) Chamberlin, A. B., McFadden, L. A., Schulz, R., Schleicher, D. G., & Bus, S. J. 1996, Icarus, 119, 173, doi: 10.1006/icar.1996.0009
  * Chen & Han (1988) Chen, W.-F., & Han, D.-J. 1988, Plasticity for Structural Engineers (2007, J. Ross Publishing) (Springer-Verlag)
  * Cochran & Barker (1984) Cochran, A. L., & Barker, E. S. 1984, Icarus, 59, 296, doi: 10.1016/0019-1035(84)90029-0
  * De León et al. (2010) De León, J., Campins, H., Tsiganis, K., Morbidelli, A., & Licandro, J. 2010, Astronomy and Astrophysics, 513, 1, doi: 10.1051/0004-6361/200913609
  * Dobrovolskis (1982) Dobrovolskis, A. R. 1982, Icarus, 52, 136, doi: 10.1016/0019-1035(82)90174-9
  * Gustafson (1989) Gustafson, B. A. S. 1989, Astronomy and Astrophysics, 225, 533
  * Hirabayashi (2015) Hirabayashi, M. 2015, Monthly Notices of the Royal Astronomical Society, 454, 2249, doi: 10.1093/mnras/stv2017
  * Hirabayashi et al. (2015) Hirabayashi, M., Sánchez, D. P., & Scheeres, D. J. 2015, Astrophysical Journal, 808, 63, doi: 10.1088/0004-637X/808/1/63
  * Hirabayashi & Scheeres (2015) Hirabayashi, M., & Scheeres, D. J. 2015, Astrophysical Journal Letters, 798, doi: 10.1088/2041-8205/798/1/L8
  * Hirabayashi & Scheeres (2019) —. 2019, Icarus, 317, 354, doi: 10.1016/j.icarus.2018.08.003
  * Hirabayashi et al. (2019) Hirabayashi, M., Tatsumi, E., Miyamoto, H., et al. 2019, The Astrophysical Journal Letters, 874, L10
  * Holsapple (2001) Holsapple, K. A. 2001, Icarus, 154, 432, doi: 10.1006/icar.2001.6683
  * Hsieh & Jewitt (2005) Hsieh, H. H., & Jewitt, D. 2005, The Astrophysical Journal, 624, 1093, doi: 10.1086/429250
  * Hughes & McBride (1989) Hughes, D. W., & McBride, N. 1989, Monthly Notices of the Royal Astronomical Society, 240, 73, doi: https://doi.org/10.1093/mnras/240.1.73
  * Jenniskens (1994) Jenniskens, P. 1994, Astronomy and Astrophysics, 287, 990
  * Jewitt (2012) Jewitt, D. 2012, Astronomical Journal, 143, doi: 10.1088/0004-6256/143/3/66
  * Jewitt & Li (2010) Jewitt, D., & Li, J. 2010, Astronomical Journal, 140, 1519, doi: 10.1088/0004-6256/140/5/1519
  * Jewitt et al. (2013) Jewitt, D., Li, J., & Agarwal, J. 2013, Astrophysical Journal Letters, 771, 1, doi: 10.1088/2041-8205/771/2/L36
  * Kitazato et al. (2019) Kitazato, K., Milliken, R. E., Iwata, T., et al. 2019, Science, 364, 272, doi: 10.1126/science.aav7432
  * Lambe & Whitman (1969) Lambe, W., & Whitman, R. 1969, Soil Mechanics By Lambe and Whitman.pdf (Wiley)
  * Lauretta et al. (2019) Lauretta, D. S., Hergenrother, C. W., Chesley, S. R., & Leonard, J. M. 2019, Science, 366, doi: 10.1126/science.aay3544.1
  * Li & Jewitt (2013) Li, J., & Jewitt, D. 2013, Astronomical Journal, 145, doi: 10.1088/0004-6256/145/6/154
  * Love (2013) Love, A. E. H. 2013, A treatise on the mathematical theory of elasticity (Cambridge university press)
  * Naidu et al. (2015) Naidu, S. P., Margot, J. L., Taylor, P. A., et al. 2015, Astronomical Journal, 150, 54, doi: 10.1088/0004-6256/150/2/54
  * Ostro et al. (2006) Ostro, S. J., Margot, J. L., Benner, L. A. M., Giorgini, J. D., & Scheeres, D. J. 2006, Nature, 314, 1276
  * Ryabova (2007) Ryabova, G. O. 2007, Monthly Notices of the Royal Astronomical Society, 375, 1371, doi: 10.1111/j.1365-2966.2007.11392.x
  * Scheeres & Sánchez (2018) Scheeres, D. J., & Sánchez, P. 2018, Progress in Earth and Planetary Science, 5, doi: 10.1186/s40645-018-0182-9
  * Scheeres et al. (2019) Scheeres, D. J., McMahon, J. W., French, A. S., et al. 2019, Nature Astronomy, 3, 352, doi: 10.1038/s41550-019-0721-3
  * Sugita et al. (2019) Sugita, S., Honda, R., Morota, T., et al. 2019, Science, 364, doi: 10.1126/science.aaw0422
  * Taylor et al. (2019) Taylor, P. A., Rivera-Valentín, E. G., Benner, L. A., et al. 2019, Planetary and Space Science, 167, 1, doi: 10.1016/j.pss.2019.01.009
  * Walsh et al. (2008) Walsh, K. J., Richardson, D. C., & Michel, P. 2008, Nature, 454, 188, doi: 10.1038/nature07078
  * Walsh et al. (2012) —. 2012, Icarus, 220, 514, doi: 10.1016/j.icarus.2012.04.029
  * Watanabe et al. (2019) Watanabe, S., Hirabayashi, M., Hirata, N., et al. 2019, Science, 272, eaav8032, doi: 10.1126/science.aav8032
  * Whipple (1983) Whipple, F. 1983, International Astronomical Union Circular, 3881
  * Wiegert et al. (2008) Wiegert, P. A., Houde, M., & Peng, R. 2008, Icarus, 194, 843, doi: 10.1016/j.icarus.2007.12.013
  * Williams & Wu (1993) Williams, I. P., & Wu, Z. 1993, Monthly Notices of the Royal Astronomical Society