# APOPHIS - Effects of the 2029 Earth’s Encounter on the Surface and Nearby
Dynamics

G. Valvano1*  | O. C. Winter1*  | R. Sfair1*  | R. Machado Oliveira1*  | G. Borderes-Motta2*  | T. S. Moura1*  |
1 Grupo de Dinâmica Orbital e Planetologia, São Paulo State University, UNESP,
Guaratinguetá, CEP 12516-410, São Paulo, Brazil
2 Bioengineering and Aerospace Engineering Department, Universidad Carlos III
de Madrid, Leganés, 28911, Madrid, Spain E-mail<EMAIL_ADDRESS>mail<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>E-mail<EMAIL_ADDRESS><EMAIL_ADDRESS>
(Accepted xxx. Received xxx; in original form xxx)

###### Abstract

The 99942 Apophis close encounter with Earth in 2029 may provide information
about asteroid’s physical characteristics and measurements of Earth’s effects
on the asteroid surface. In this work, we analysed the surface and the nearby
dynamics of Apophis. The possible effects of its 2029 encounter on the surface
and environment vicinity are also analysed. We consider a 340 metres
polyhedron with a uniform density (1.29 g$\cdot$cm-3, 2.2 g$\cdot$cm-3 and 3.5
g$\cdot$cm-3). The slope angles are computed, as well their variation that
arises during the close approach. Such variation reaches 4∘ when low densities
are used in our simulations and reaches 2∘ when the density is high. The zero-
velocity curves, the equilibrium points, and their topological classification
are obtained. We found four external equilibrium points and two of them are
linearly stable. We also perform numerical simulations of bodies orbiting the
asteroid, taking into account the irregular gravitational field of Apophis and
two extra scenarios of perturbations: the solar radiation pressure and the
Earth’s perturbation during the close approach. The radiation pressure plays
an important role in the vicinity of the asteroid, only cm-sized particles
survived for the time of integration. For densities of 2.2 g$\cdot$cm-3 and
3.5 g$\cdot$cm-3, a region of 5 cm radius particles survived for 30 years of
the simulation, and for 1.29 g$\cdot$cm-3, only particles with 15 cm of radius
survived. The ejections and collisions are about 30-50 times larger when the
close encounter effect is added, but around 56-59% of particles still survive
the encounter.

###### keywords:

Celestial mechanics – Astrometry and celestial mechanics, minor planets,
asteroids, general – Planetary Systems, minor planets, asteroids – Planetary
Systems, planets and satellites: dynamical evolution and stability – Planetary
Systems

††pubyear: 2021††pagerange: APOPHIS - Effects of the 2029 Earth’s Encounter on
the Surface and Nearby Dynamics–APOPHIS - Effects of the 2029 Earth’s
Encounter on the Surface and Nearby Dynamics

##  1 Introduction

Upon its discovery on 2004 June at Kitt Peak by R.A. Tucker, D.J. Tholen, and
F. Bernardi (Garradd, 2004; Tucker et al., 2004; Smalley et al., 2004), 99942
Apophis (originally designated as 2004 MN4) has its orbit constantly monitored
since it was reported a high probability of collision with Earth of more than
2% (Chesley, 2005). Although this possibility was later discharged (Larsen &
Descour, 2004), other potential impact or approaches with Earth were reported
in the following years, and Apophis has been considered as a potentially
hazardous asteroid (PHA). The trajectory and the future close encounters
parameters were improved with ephemeris derived from radar observations in
2005-2006 from Arecibo radiotelescope (Giorgini et al., 2008). With this data,
they were able to reduce orbital uncertainties and infer a nominal distance
for the 2029 approach of about 6 Earth radii, predict a distance of 0.34 au
for the 2036 encounter, and find that an impact probability still remains
(Giorgini et al., 2008). The next approach will be the closest encounter with
the Earth and will occur on 13 April 2029. Apophis will pass at a distance
near to the geostationary orbit and a distance about one-tenth the distance
between the Earth and Moon.

The predictions for the pre-2029 orbit turn to be well-defined, however, for
the post-2029 orbit, the predictions are not well determined due to the
uncertainties caused by perturbations. The Yarkovsky effect is a considerable
source for the orbital uncertainties of Apophis’ orbit (Giorgini et al., 2008;
Farnocchia et al., 2013; Vokrouhlickỳ et al., 2015). Thus considering the
Yarkovsky effect and using astrometric data from 2004-2008, Farnocchia et al.
(2013) presented a new impact risk for Apophis. They infer an impact
probability of about $\sim$10-9 for the 2036 encounter and $\sim$10-6 for
2068.

Observations from the 2021 encounter provided measurements that improved the
fit for the orbit of Apophis and eliminate the possibility of impact for the
next 100 years. It led Apophis to be removed from the ESA’s asteroid Risk List
after remaining on this list for almost 17 years
111https://cneos.jpl.nasa.gov/sentry/removed.html.

The 2029 flyby may provide an opportunity to improve the 3D shape model,
investigate possible changes on the spin state, reshaping and effects of the
encounter on the surface. Scheeres et al. (2005) predicted that the
terrestrial torques caused by the encounter will change Apophis’ spin state
drastically, and consequently the Yarkovsky accelerations. Conversely, Souchay
et al. (2018) showed that the changes in the spin rate may be small, and the
larger effects may occur in the obliquity and precession in longitude. The
effects of a close approach with the Earth also could cause material
landslides and migration as is showed in Binzel et al. (2010). However, the
numerical simulations made for Yu et al. (2014) using soft-sphere code
implementation predicted that the effects of the tidal pertubations on the
Apophis’ surface may be small, but could produce small landslides.

Therefore, the closest approach of 2029 with the Earth may provide measures
through observations before, during, and after the encounter. The
observational data from these observations may improve some physical
characteristics, the understanding of the effects of the closest encounters
with the Earth, validate models about the material, and other possibilities.
In this work, we used the convex shape model from Pravec et al. (2014) to
analyse the surface and the nearby dynamics of the asteroid Apophis
considering the effects caused by the 2029 closest approach with the Earth.
The paper is composed of the following sections. In the next section, we
present the asteroid model used in this work, discussing its polyhedral shape
model, general characteristics and the 2029 encounter configuration. Section 3
introduces the gravitational potential and geopotential considering the
polyhedra method. We discuss physical features using the slope angle and its
variation due to the Earth perturbation. In Section 4, we explore the nearby
environment of Apophis by calculating the zero-velocity curves and equilibrium
points. We also present a set of numerical simulations of a disc of particles
around Apophis considering the gravitational field and two additional
scenarios of perturbations: the solar radiation pressure and, in Section 5,
the Earth perturbation on the 2029 trajectory. Finally, in the last section we
provide our final comments.

##  2 The asteroid model

Apophis was observed by the VLT (Very Large Telescope) and the data obtained
from a polarimetric observing campaign in 2006 resulted in an approximate
diameter of 270 metres for the asteroid (Cellino et al., 2006). In 2012 and
2013, Pravec et al. (2014) made a photometric observational campaign and
discovered that Apophis is a tumbling asteroid. They computed the Apophis’
tumbling spin state as a retrograde rotational period of 27.38 $\pm$ 0.07
hours and a precession period of 263 $\pm$ 6 hours.

The 2012-2013 Apophis’ apparition also allowed the construction of a convex
shape model. The Apophis’ shape model is represented as an unscaled polyhedron
with 1014 vertices and 2024 triangular faces (Pravec et al., 2014). Brozović
et al. (2018) also provided a shape model for the asteroid Apophis, but they
used radar observations from Goldstone and Arecibo to improve the previous
model. This improved shape model is a 340 metres polyhedron with 2000 vertices
and 3996 faces.

From thermal infrared observations, Müller et al. (2014) reported an
approximated size of 375 metres for (99942) Apophis. Licandro et al. (2015)
also provided an estimated size for Apophis using this technique, however,
they combined their data with previous thermal observations. The resulting
size for Apophis ranges from 380 to 393 metres.

We adopted the Apophis’ diameter presented by Brozović et al. (2018), 340
metres, and the shape model provided by Pravec et al. (2014), hence the model
by Brozović et al. (2018) is not publicly available. We used the rotational
period of 27.38 hours with an obliquity of 180∘ to reproduce the Apophis’
retrograde rotation (Pravec et al., 2014) and do not consider the precession
period as it is 10 times larger than the rotational period.

(a)  Figure 1:  Geometric altitude map computed across the surface of Apophis
under different views. The sea-level (the smaller distance between the
geometric centre of the body and the barycentre among all triangular faces) is
130.54 metres.

The model for Apophis has an ellipsoid elongated format and tapered ends in
the equatorial regions, as shown in its geometric map (Fig. 1). To measure the
geometric altitude, we determine the distance between the geometric centre of
the body and the barycentre of each triangular face of the polyhedron. Then,
we identify the smaller distance between these points and set it as “sea-
level" (Scheeres et al., 2016). The sea-level is 130.54 metres and it is
almost half the value of the maximum distance calculated between the geometric
centre and the barycentre among all the faces.

Analysing the geometric map, we notice that the maximum values of the
geometric altitude are on the equatorial regions, while the minimum values are
on the poles, which means that these regions are at the sea-level. The Top and
Bottom views in Fig. 1 show the poles of the asteroid, and they shows that the
north pole (Top view) has a concentrated region of the smaller altitudes on
its middle. However, the south pole (Bottom view) presents a band of a larger
distribution of small altitudes.

Apophis has a grain density in the range 3.4-3.6 g$\cdot$cm-3 and a total
porosity with range of 4-62% because of its similar spectral characteristics
with LL ordinary chondrite (Binzel et al., 2009). This results in an
approximate bulk density of 1.29-3.46 g$\cdot$cm-3. As the composition and
some similarities suggest, Apophis could have a total porosity similar to the
asteroid (25143) Itokawa. So Binzel et al. (2009) adopted Apophis’ radius as
135 metres, density of 3.2 g$\cdot$cm-3 and Itokawa’s total porosity of 40% to
estimate Apophis’ mass as 2.0$\times$1010 kg. Dachwald et al. (2007) assumed a
spherical radius of 160 metres and a bulk density of 2.72 g$\cdot$cm-3 to
determine the Apophis’ mass as 4.67$\times$1010 kg. Müller et al. (2014) also
estimated a mass for Apophis, however they used a radius of 187.5 metres,
density of 3.2 g$\cdot$cm-3 and total porosity of 30-50%, resulting in a total
mass of 4.4-6.2$\times$1010 kg. Therefore, for this work, we assumed three
bulk densities of 1.29 g$\cdot$cm-3, 2.2 g$\cdot$cm-3 and 3.5 g$\cdot$cm-3.

Considering the density values adopted and the Apophis’ size provided from
Brozović et al. (2018), we constructed three models for the asteroid, one for
each density. As the mass is not a known parameter, we set the volume of the
asteroid as the same of a sphere with an equivalent radius of 170 metres. So,
we preserved the volume and the size and estimate the mass for each model
according to the density. The volume adopted was 0.0205 km3 and the three
calculated masses are 2.64$\times$1010 kg, 4.50$\times$1010 kg and
7.16$\times$1010 kg for the densities of 1.29 g$\cdot$cm-3, 2.2 g$\cdot$cm-3
and 3.5 g$\cdot$cm-3, respectively.

With the three models, we defined a coordinate system with the origin at the
object centre of mass and align the system axes to the axes of the principal
moment of inertia. The $x$, $y$, and $z$ axes correspond, respectively, to the
smallest, intermediate, and largest moments of inertia. This process was made
using the algorithm presented by Mirtich (1996) and it was assumed that
Apophis has a constant density and a uniform rotation about the largest moment
of inertia.

The values of the principal moment of inertia normalized by the mass are:

$I_{xx}/M=559.23\ \,\textrm{m}^{2},$ (1) $I_{yy}/M=897.72\ \,\textrm{m}^{2},$
(2) $I_{zz}/M=969.39\ \,\textrm{m}^{2}.$ (3)

From the principal moment of inertia, we calculated the second-order degree
terms $C_{20}$ and $C_{22}$ of the gravity expansion. The coefficient $J_{2}$
($-C_{20}$) represents how oblate Apophis is and $C_{22}$ how elongated. The
values are (MacMillan, 1958; Hu & Scheeres, 2004):

$C_{20}=-\frac{1}{2R_{n}^{2}}(2I_{zz}-I_{xx}-I_{yy})=-0.00837,$ (4)
$C_{22}=\frac{1}{4R_{n}^{2}}(I_{yy}-I_{xx})=0.00294.$ (5)

where the equivalent radius, $R_{equivalent}=$ 170 metres, was used for
normalization.

Note that the coefficient $C_{20}$ is of the same order of magnitude as the
coefficient $C_{22}$, but the absolute value of $C_{22}$ is smaller than
$J_{2}$. We also modeled Apophis’ shape as an ellipsoid with the parameters
$a$, $b$ and $c$ representing the ellipsoid semi axes of the asteroid. The
equivalent ellipsoid parameters for Apophis are:

$a=228.966631\ \,\textrm{m},$ (6) $b=159.026432\ \,\textrm{m},$ (7)
$c=139.797960\ \,\textrm{m}.$ (8)

Observe that $a$ is about 64% larger than the parameter $c$, evidencing the
elongated shape at the equator and flattened at the poles. So, considering the
harmonic coefficients and the equivalent ellipsoid parameters, the shape of
Apophis has a proportion of $5:3.5:3$ between the semi axes $a:b:c$,
respectively.

###  2.1 2029 Earth’s Encounter

On April 13 2029, Apophis will have the closest approach with the Earth at an
approximate distance of $\sim$38,000 km according to the JPL’s HORIZONS
ephemerides222https://ssd.jpl.nasa.gov/?horizons. The trajectory is shown in
Fig. 2, where the black dot represents the Earth, the green circle and line
illustrate the equatorial plane and Moon’s orbit, respectively. The filled and
the dashed blue lines represent the trajectory of the object above and below
the equatorial plane, respectively, while the arrow illustrates the direction
of the movement of Apophis. Note that Apophis approaches the Earth from left
to right and it takes 34.30 hours to enter and leave the Moon’s orbit. Apophis
traverses the equatorial plane from the bottom-up and just crosses the
equatorial plane near to the closest approach.

Since we do not know the exact orientation at the encounter, we investigate
several hypothetical orientations for the asteroid Apophis at the moment of
the encounter with the Earth in 2029 and see if this approach could change the
surface characteristics of the body (section 3). We also used the 2029
approach trajectory (Fig. 2) to compute the effects on the dynamical nearby
environment (section 4). This analysis may provide insights to the 2029
observational campaign, and eventual space missions designed to study the
asteroid.

(a)  Figure 2:  Apophis’ trajectory of the encounter with Earth in 2029 in the
$xoy$ plane. The black dot represents the Earth and, the green line and circle
represent the Moon’s orbit and the equatorial plane, respectively. The blue
line represents the trajectory above the equatorial plane and the blue dashed
line represents the trajectory below the equatorial plane. The arrow
represents the movement direction of Apophis.

##  3 Effects on the Surface

The geopotential is an effective way to measure the relative energy on the
body surface since it considers the gravitational and rotational potential
(Scheeres, 2012; Scheeres et al., 2016). It is possible to relate the
geopotential energy with the motion of a cohesionless particle by associating
the quantity of energy necessary to move this particle across the body
surface.

Assuming a constant angular velocity vector $\boldsymbol{\omega}$, the
reference frame centred at the centre of mass and the axes aligned to the
principal inertia axes for the asteroid Apophis, the expression for the
geopotential is (Scheeres et al., 2016):

$V(\boldsymbol{r})=-\frac{1}{2}\omega^{2}(x^{2}+y^{2})-U(\boldsymbol{r}),$ (9)

where $\boldsymbol{r}$ is the position vector of a massless particle in the
body-fixed frame and $U(\boldsymbol{r})$ the gravitation potential that is
given by the method of polyhedra.

To model the object we used the polyhedra method, that computes the
gravitational potential energy of an irregular body modeled by a uniform
density polyhedron with a given number of faces and vertices. Then, the
expression for the gravitational potential is (Werner & Scheeres, 1996):

$U=\frac{G\rho}{2}\sum_{e\in
edges}{\boldsymbol{r}_{e}}\cdotp{\boldsymbol{E}_{e}}\cdotp{\boldsymbol{r}_{e}}\cdotp
L_{e}-\frac{G\rho}{2}\sum_{f\in
faces}{\boldsymbol{r}_{f}}\cdotp{\boldsymbol{F}_{f}}\cdotp{\boldsymbol{r}_{f}}\cdotp\omega_{f},$
(10)

where $\rho$ is the density of Apophis, $G$ is the gravitational constant;
$\boldsymbol{r}_{f}$ and $\boldsymbol{r}_{e}$ represent, respectively, the
position vectors from a point in the gravitational field to any point in the
face $f$ and edge $e$ planes; $\boldsymbol{F}_{f}$ and $\boldsymbol{E}_{e}$
are the faces and edges tensors; $\omega_{f}$ and $L_{e}$ are the signed angle
viewed from the field point and the integration factor, respectively.

From the geopotential, Scheeres et al. (2016) derive the slope angle using its
gradient (see Section 3.1), this measure considers the effects on the surface
of the body, but we also can derive an approach for a particle in the nearby
environment of the asteroid.

Since it is considered that Apophis has a constant angular velocity
($\boldsymbol{\omega}$) about its axis of maximum moment of inertia, the
movement of a particle orbiting the nearby environment is described by
(Scheeres et al., 2016):

$\ddot{\boldsymbol{r}}+2{\boldsymbol{\omega}}\times\dot{\boldsymbol{r}}=-\frac{\partial
V}{\partial\boldsymbol{r}},$ (11)

where $\boldsymbol{r}$ is the position vector of the particle,
$\dot{\boldsymbol{r}}$ and $\ddot{\boldsymbol{r}}$ are its velocity and
acceleration vectors, respectively.

Equation 11 is time-invariant since the Apophis’ spin-rate is assumed to be a
constant value. Therefore this equation can be associated with a conserved
quantity, $C_{j}$, called “Jacobi constant". This constant is defined by
(Scheeres et al., 2016):

$C_{j}=\frac{1}{2}v^{2}+V(\boldsymbol{r}),$ (12)

where the magnitude of the velocity vector relative to the rotating frame of
the Apophis is denoted by $v$.

###  3.1 Slope

The slope angle is the supplement between the normal and total acceleration
vectors (Scheeres, 2012; Scheeres et al., 2016). The slope is an angle that
quantifies how inclined is a region on the Apophis’ surface with respect to
its local acceleration vector. Physically, slope assists in understanding the
movement of free particles on the Apophis surface.

The slope angle distribution across the Apophis’ surface for the larger and
smaller densities is shown in Fig. 3. The variation between the models is
small and the slope distribution is almost identical. The slope variation
amplitude goes from 35.81∘ for the smaller density, up to 36.16∘ for the
larger density, a difference of only 0.35∘.

(a) $\rho$ = 1.29 g$\cdot$cm-3

(b) $\rho$ = 3.5 g$\cdot$cm-3

Figure 3:  Slope angle maps across the surface of Apophis under different
views. The letters (a) and (b) represent, respectively, the slope distribution
considering the densities of 1.29 g$\cdot$cm-3 and 3.5 g$\cdot$cm-3.

For all three cases, the maximum slope angles are smaller than 37∘, and the
minimum angles are larger than 0.5∘. The maximum values of slope occur just in
the small regions on the equatorial extremities of Apophis (see front, left,
and bottom views in Fig. 3). The minimum values of the slope are distributed
preferentially at the central region of the body as we can see in the bottom
view in Fig. 3, the south pole has a strip of minimum values is passing
through the middle of the surface.

By definition, slope angles varies between 0∘ and 180∘. When it is larger than
90∘, the centrifugal force is larger than the gravitational force, causing
cohesionless particles to escape from the body’s surface. If the slope angle
is smaller than 90∘, the movement of the cohesionless particles can be
associated with the repose angle of a given material. The angle of repose for
geological material is about 35-40 degrees (Lambe & Whitman, 1969; Mitchell et
al., 1974; Al-Hashemi & Al-Amoudi, 2018).

As already referred, the maximum value of the slope considering the three
values of density is about 37∘. So we can see two regimes, one for the lower
limit value of the angle of repose, 35∘, and the other for the upper limit
value, 40∘. For the lower limit, the flow of the cohesionless particles may
occur, since we did not find a slope value higher than this limit.

(a)  Figure 4:  Slope angle map computed across the surface of Apophis with
the directions of the local acceleration tangent vectors and under different
views for the density of 2.2 g$\cdot$cm-3.

To understand the possible flow of free particles on Apophis’ surface, we
compute the directions of the tangential component of the local acceleration
vector (Scheeres, 2012; Scheeres et al., 2016). Figure 4 shows the tangential
acceleration vectors considering a density of 2.2 g$\cdot$cm-3.

As expected, the vectors are pointing to the regions where the slope is
smaller. The flow of loose material might occur from the small regions on the
equatorial extremities to the middle of the Apophis’ surface. The dark regions
in Fig. 4 represent the near-zero slopes, thus are stable resting areas. Those
regions are propitious regions to accumulate cohesionless particles.

(a) $\rho$ = 1.29 g$\cdot$cm-3

(b) $\rho$ = 3.5 g$\cdot$cm-3

Figure 5:  The $\Delta$slope angle maps across the surface of Apophis under
different views and considering the Earth perturbation in 2029 encounter. The
letters (a) and (b) represent, respectively, the slope distribution
considering the densities of 1.29 g$\cdot$cm-3 and 3.5 g$\cdot$cm-3.

Apophis will have a close approach with the Earth in 2029 and will be affected
by time-varying forces. Thus, to evaluate the strongest possible effect on its
surface due to the Earth‘s gravity, we calculated a $\Delta$slope that
computes the slope variation caused due to the encounter at the closest
distance of the whole approach.

In order to identify the most extreme condition that may occur in 2029, we
assume a hypothetical configuration for the Apophis-Earth system. We set the
Earth on the equatorial plane of Apophis and the distance between the centre
of mass of Apophis and the Earth as the closest distance during the whole
encounter, and we calculate the slope on the Apophis’ surface considering its
geopotential including the Earth’s gravitational force on the closest
approach. Then, we rotate the shape model of Apophis all along the 360∘, so
that each Apophis’ orientation experiences the extreme approach condition.

The difference between the total acceleration vectors, with and without the
Earth’s gravity, leads to a slope that we call a delta slope. The delta slope
is the variation of the slope angle produced by the Earth’s gravitational
perturbation on the surface of Apophis at the extreme approach condition. The
$\Delta$slope map was also made over the rotation of 360∘ of the Apophis’
shape model, in order to compute the $\Delta$slope on each Apophis’
orientation.

Figure 5 shows the $\Delta$slope for the smaller and larger density of Apophis
(see the complete animation of the $\Delta$slope angle maps in the videos of
the complementary material). The top and bottom views are not shown since they
are not directly pointed to the Earth at the closest approach, as we defined
before. For the 1.29 g$\cdot$cm-3 density model, the slope variation was
smaller than 4∘, and for 3.5 g$\cdot$cm-3 density model was about 2∘. This
variation represents about 11% of the regular slope angle for the smaller
density model and 5.5% for the larger density model.

With the perturbation caused by the Earth, some slope angle values may exceed
the angle of repose. Ballouz et al. (2019) showed that a variation around 2∘
may sufficient to start a slow erosion process in regions with high-slope that
experiment larger perturbations. However, the perturbations due to the 2029
encounter will not trigger drastic reshaping on Apophis’ surface and shape.
The numerical simulations provided by Yu et al. (2014) revealed that the
effects caused by the Earth are small and just can cause local landslides.

##  4 Effects on the Nearby Stability

In this section, we present an investigation of the environment around
Apophis. The analysis of the variation of the Jacobi constant, through zero-
velocity curves, allows us to identify where the movement is permitted or not.
These curves delimit the external equilibrium points of the body, the location
and the topological classification of these points are presented. A set of
numerical simulations with a disc of particles is also presented in order to
analyse the stability around the asteroid considering the perturbation of its
own gravitational field and two additional perturbations: the solar radiation
pressure and the 2029 flyby scenario.

###  4.1 Equilibrium Points and Zero-Velocity Curves

To understand the stability in the vicinity of Apophis, we calculated the
zero-velocity curves and equilibrium points. The zero-velocity curves limit
the movement of a particle, delimiting where its movement is allowed or not.
This delimited movement depends on the Jacobi constant, Cj, and, it arises
from the inequality

$C_{j}-V(\boldsymbol{r})\geq 0,$ (13)

since the velocity term, $\frac{1}{2}v^{2}$, from equation 12 shall be always
positive. So, when $C_{j}<V(\boldsymbol{r})$ there will be forbidden regions
to the movement of the particle, considering that the inequality is infringed.
In the case of the regions where $C_{j}>V(\boldsymbol{r})$, there will be no a
priori limitation to the movement of the particle, since the inequality is
satisfied. When $C_{j}=V(\boldsymbol{r})$ we have the zero-velocity curves
that separate the regions between the permitted and prohibited movement.

Figure 6:  Zero-velocity curves contourplots in the $xoy$ plane considering
the mean density of 2.2 g$\cdot$cm-3. The colorbox represents the value of the
Jacobi constant.

Fig. 6 illustrates the zero-velocity curves countourplots in the $xoy$ plane
for the model with the mean density. The colour of the zero-velocity curves
changes as the Jacobi constant value changes. At a certain value of Cj, the
curves could form confined regions that will encompass an equilibrium point,
similar to a “banana” shape in Fig. 6. The region between these confined
curves also will include an equilibrium point.

The equilibrium points are critical points of geopotential where there is a
balance between the gravitational and centrifugal forces. Since the resulting
force is null at this points, the equilibrium points of the asteroid Apophis
are calculated by the solution of:

$\nabla V=0,$ (14)

where the $\nabla V$ means the gradient of the geopotential.

In general, there is no fixed number of solutions for equation 14, since the
solution depends on the body shape model, its density, and rotational period.
However, we could estimate the number of equilibrium points just by looking at
the zero-velocity curves of the body. The zero-velocity curves of Apophis show
it has at least four equilibrium points, two of them inside the curves with a
“banana” shape and two between the larger “banana” curves near to the $y=0$
line (Fig. 6).

The location of the equilibrium points also depends on the rotational period,
density, and shape of the asteroid. Once defined the shape model and the spin,
if we change its density, the location of the equilibrium points will change
as well. If the density decreases, the location of the equilibrium points will
be closer to the body due to the reduction of the gravitational force.
Conversely, if the density is increased, the equilibrium points will be more
distant from the body.

Figure 7:  Location of the equilibrium points in the $xoy$ plane for each
density. The point’s color represents the density of Apophis.

Considering the polyhedral model derived from Pravec et al. (2014) and the
densities of 1.29 g$\cdot$cm-3, 2.2 g$\cdot$cm-3, and 3.5 g$\cdot$cm-3, we
found five equilibrium points for each density, being one in the centre of the
body. The position of the equilibrium points are shown in table 1. We note
they are close to the equatorial plane, so Fig. 7 shows the projection of the
asteroid Apophis and the equilibrium points in the $xoy$ plane.

Note that the number of equilibrium points does not change, differently from
other systems Jiang & Baoyin (2018). Observe that although the model with a
larger density is almost three times larger than the model with a smaller
density, the difference in the radial distance between the equilibrium points
of the model with larger and smaller densities is about 300 metres.

Table 1: Coordinates and characteristic time of the equilibrium points of 99942 Apophis considering the three densities (1.29 g$\cdot$cm-3, 2.2 g$\cdot$cm-3, and 3.5 g$\cdot$cm-3) and their topological structures. Point |  $x$ (km) |  $y$ (km) |  $z$ (km) | Topological classification |  | Characteristic Time
---
(hours)
$\rho$ = 1.29 g$\cdot$cm-3
$E_{1}$ | 0.764033 | -0.014211 | -0.001470 | Saddle-Centre-Centre |  | 26.83192
---
26.97251
$E_{2}$ | -0.038260 | -0.752882 | 0.000598 | Centre-Centre-Centre |  | 27.28828
---
28.90402
88.39612
$E_{3}$ | -0.765131 | -0.007504 | -0.001544 | Saddle-Centre-Centre |  | 26.57328
---
26.90200
$E_{4}$ | -0.041584 | 0.752713 | 0.000675 | Centre-Centre-Centre |  | 27.30720
---
28.79664
90.94898
$\rho$ = 2.2 g$\cdot$cm-3
$E_{1}$ | 0.910268 | -0.013482 | -0.001033 | Saddle-Centre-Centre |  | 26.98514
---
27.08051
$E_{2}$ | -0.038718 | -0.900849 | 0.000424 | Centre-Centre-Centre |  | 27.31676
---
28.38482
107.46649
$E_{3}$ | -0.911036 | -0.007877 | -0.001078 | Saddle-Centre-Centre |  | 26.82109
---
27.04486
$E_{4}$ | -0.041482 | 0.900733 | 0.000470 | Centre-Centre-Centre |  | 27.32790
---
28.32686
110.03305
$\rho$ = 3.5 g$\cdot$cm-3
$E_{1}$ | 1.060747 | -0.012966 | -0.000760 | Saddle-Centre-Centre |  | 27.08273
---
27.15419
$E_{2}$ | -0.039031 | -1.052625 | 0.000314 | Centre-Centre-Centre |  | 27.33415
---
28.09159
126.74930
$E_{3}$ | -1.061310 | -0.008169 | -0.000789 | Saddle-Centre-Centre |  | 26.97445
---
27.13409
$E_{4}$ | -0.041388 | 1.052541 | 0.000343 | Centre-Centre-Centre |  | 27.34118
---
28.05680
129.32771

Applying the linearization method to the equations of motion, we analysed the
six eigenvalues of the characteristic equation for each point in order to
identify their topological classification (Jiang et al., 2014). Table 1 shows
the coordinates and the topological classification of each equilibrium point
for each density of Apophis. The odd indexed points, $E_{1}$ and $E_{3}$, are
classified as a Saddle-Center-Center, while the even points ($E_{2}$ and
$E_{4}$) are Center-Center-Center, implying that they are linearly stable
points.

According to Jiang et al. (2014) and Yu & Baoyin (2012), when analysing the
eigenvalues of the characteristic equation, we identified the existence of
three families of periodic orbits in the tangent space of the equilibrium
points $E_{2}$ and $E_{4}$, which have characteristic times or oscillation
periods of approximately 27.3 hr, 28.9 hr and 88.4 hr for $E_{2}$, and 27.3
hr, 28.8 hr and 90.9 hr for $E_{4}$, when we consider the density of 1.29
g$\cdot$cm-3. Only the period value of the third periodic orbit family
underwent a significant change, with an increase of about 21% and 43%,
respectively, for the Apophis intermediate and upper densities, when we
analysed the $E_{2}$ equilibrium point eigenvalues. The same conclusion can be
applied to point $E_{4}$. While for points $E_{1}$ and $E_{3}$, there are two
families of periodic orbits, which have periods of oscillation around 26.8 hr
and 26.9 for $E_{1}$, and 26.6 hr and 26.9 hr for $E_{3}$, and these values
undergo small changes, less than 30 minutes, for other Apophis densities
(Table 1).

Note that despite of changing the density of Apophis, the topological
classification of the equilibrium points $E_{2}$ and $E_{4}$ remains as
linearly stable. Thus, we performed numerical simulations of a disc of
particles encompassing the equilibrium points regions in order to identify
possible stable zones around them.

###  4.2 Stability Regions

Figure 8:  Variation of the parameters oblateness (blue), Earth and Sun tide
(black and green, respectively), the solar radiation pressure for a particle
with a radius of 1 cm (pink) and 15 cm (red) and the Apophis gravity (yellow)
as a function of the distance from Apophis. The shaded region indicates the
variation of these parameters between the pericentre and apocentre. The
vertical lines represent the limits of the radial location of the the
equilibrium points for the density model of 1.29 g$\cdot$cm-3 and 3.5
g$\cdot$cm-3. The equivalent radius, $R_{equivalent}$, of Apophis is 170
metres.

There have been some studies (Aljbaae et al., 2020, 2021; Lang, 2021)
investigating suitable orbits around Apophis for a spacecraft taking into
account the irregular gravitational field of the asteroid and the solar
radiation pressure for an area to mass ratio similar to the OSIRES-REX
spacecraft. Some stable orbits were found, but during the 2029 close
encounter, the majority of the orbits for a spacecraft suffer a collision or
ejection.

The small particles around Apophis may be subject to a plethora of forces
besides the gravitational potencial of the main body: the oblateness
coefficient add an extra gravitational pull and there are the tides by the Sun
and the Earth at the close encounter. If the grains are small they can also
experience the disturbance due to the solar radiation force.

To estimate which perturbation may be relevant, we computed adimensional
parameters that allow us to analyze the relative strength of each force (for a
detailed definition of the parameters, see Hamilton & Krivov (1996); Moura et
al. (2020)). Figure 8 shows the parameter strengths according to the distance
from Apophis: the solar tide (green), Earth’s gravity at the closest approach
distance (black), the solar radiation force for grains with radius of 1 cm
(pink) and 15 cm (red); it is also shown the oblateness effect (blue) and the
Apophis gravity (yellow). For each parameter the lines are calculated for the
nominal density (2.2 g$\cdot$cm-3) and distance equal to the semimajor axis,
while the shaded region corresponds to the variation of these parameters
between the pericentre and apocentre.

In Fig. 8 the vertical lines indicate the limits of the radial location of the
equilibrium points for the density model of 1.29 g$\cdot$cm-3 and 3.5
g$\cdot$cm-3. The Apophis gravity has the major magnitude among the other
perturbations, about seven orders of magnitude larger than the Sun tide and
five orders larger than the Earth tide at the closest approach distance. The
oblateness of Apophis as the Sun tide reaches the same magnitude at a distance
of about 9 radii of Apophis. The Earth tide is also lower than the oblateness
near the asteroid and they both are comparable in the region of the
equilibrium points. Beyond this distance the Earth tide dominates the dynamic
over the oblateness. So, the Sun tide will not contribute with major
perturbations to the system.

The solar radiation pressure for a particle with a 1 cm of radius gets a
larger order of magnitude than the oblateness at about 3 radii of Apophis,
what means that this particle will suffer a major influence of the solar
radiation pressure. However, for a particle with a 15 cm of radius, the solar
radiation pressure is about one order of magnitude smaller, and it is
equivalent to the oblateness near the region of the equilibrium points. Thus,
a particle of 15 cm is more likely to survive in the nearby environment.

In the current work we are concerned with natural objects, such as dust,
boulders, fragments or even larger bodies that might be orbiting around
Apophis. Therefore, aiming to identify the size and location of possible
stable regions around Apophis and/or around the equilibrium points, we
performed sets of numerical simulations of particles around these regions. We
numerically integrated a disc with 15,000 massless particles initially for 24
hours (in order to compare to the close encounter with the Earth in 2029
showed in Fig. 2) and subsequently for 30 years ($\sim$10,000 times the
rotation period of the asteroid).

The particles were distributed at an initial distance of 300 metres from the
asteroid centre of mass and different widths for each density model, since the
position of the equilibrium points changes as the density changes. For the
1.29 g$\cdot$cm-3, 2.2 g$\cdot$cm-3 and 3.5 g$\cdot$cm-3 densities, the final
distances were 1.0 km, 1.15 km, and 1.3 km, respectively. All the particles
were assumed to be initially with Keplerian angular velocity in the equatorial
plane and circular orbits.

To perform these simulations, we used an N-body integrator package called
$N$-$BoM$ (Moura et al., 2020; Winter et al., 2020). It considers the
gravitational potential of an irregular body as a mass concentration model,
MASCONS (Geissler et al., 1996). To reproduce the Apophis’ gravitational
potential field, we calculate the sum of the gravitational potential of all
the masses points as (Borderes-Motta & Winter, 2018)

$U(x,y,z)=\sum_{i=1}^{N}\frac{Gm}{r_{i}},$ (15)

where $N$ is the number of mass points (which is 20,457 in our model), $r_{i}$
is the distance mascon-particle, and $m$ is the mass of each mascon. As the
sum of all mascons is equal to the Apophis’ total mass, $M_{Apophis}=Nm$, the
mass of each mascon is different according to the density model.

We considered a particle as ejected if it reaches a distance larger than ten
times the distance between the Apophis’ centre of mass and the equilibrium
point, this is almost equivalent to half of the Hill radius (Hamilton & Burns,
1992) of Apophis with respect to the Sun. For the 24 hours simulations, an
additional criterion of ejection was considered since some particles may not
have sufficient time to exceed the ejection distance due to the short time of
the simulation. So we also considered as ejected those particles with positive
energy. In this way, a region is called stable where the particles remain at
the final integration time, obviously, without collision with the body or be
ejected.

(a) $\rho$ = 1.29 g$\cdot$cm-3

(b) $\rho$ = 2.2 g$\cdot$cm-3

(c) $\rho$ = 3.5 g$\cdot$cm-3

Figure 9:  Initial conditions of the 15 thousand particles around the asteroid
Apophis in the $xoy$ plane. The green dots represent the equilibrium points.
The black dots represent the particles that survived after the 24 hours of
integration and the blue dots the particles that collide with the asteroid.
The letters (a)-(c) represent, respectively, the densities of 1.29
g$\cdot$cm-3, 2.2 g$\cdot$cm-3 and 3.5 g$\cdot$cm-3.

(a) $\rho$ = 1.29 g$\cdot$cm-3

(b) $\rho$ = 3.5 g$\cdot$cm-3

Figure 10:  Initial conditions of the 15 thousand particles around the
asteroid Apophis in the $xoy$ plane. The green dots represent the equilibrium
points. The black dots represent the particles that survived after 30 years,
the blue and pink dots are the particles that collide with the asteroid and
are ejected from the system, respectively. The letters (a) and (b) represent,
respectively, the densities of 1.29 g$\cdot$cm-3 and 3.5 g$\cdot$cm-3.

Figure 9 shows the initial conditions of the 15,000 particles, colored
according to their final outcome at the end of the 24 hours of integration.
The blue dots represent the particles that collided with Apophis, while the
black dots indicate the particles that have survived the entire simulation. We
note that the distribution of the particles that collided with the surface is
similar among the three density models, and they are located in two
preferential regions closer to the surface of Apophis and near to the
equatorial extremities of the body, so the gravitational field of the
equatorial extremities is related to the cause of these collisions.

Most of the particles survived, but we can note a slight difference of the
collision percentage among the different density models. For the smaller
density, the percentage of particles that collided was about 1.4%, while for
the larger model was only 0.78%. Although these percentage of collisions, the
majority of the disc is stable for the whole time of integration. In section
5, we will simulate these same conditions adding the perturbation of the Earth
during the trajectory of the 2029 encounter (Fig. 2) to see its effects in the
nearby environment of Apophis.

Extending the simulation for 30 years, we obtain the results shown in Fig. 10.
The region of collided particles near the asteroid still remains the same for
the large density, but an increase of collided particles is noticed for the
smaller density. There is a random distribution of collided particles at the
proximities of the equilibrium points and a thin ring of collided particles at
about 500-600 metres from Apophis. Some of these particles remain in the
system for about 10 years and are chaotic, presenting a large orbital radius
variation and inclination.

The ejected particles of the larger density model are distributed in regions
involving the equilibrium points, similar to a zero-velocity curve. Note that
these particles are located in a limit region where the gravitational and
centrifugal forces of the central body exerts influence on them. This means
that outside these regions the Apophis influence is sufficiently weak to cause
perturbations on the particles, so they survived as we can note in Fig. 10b.
The same behavior is presented in the smaller density model but, since the
mass is smaller so the gravitational perturbation exerted on the particles,
the ejected regions are larger.

The survived particles are the majority for the larger density model, about
96% of particles survived for the 30 years simulation, while 3.1% were ejected
and 0.9% collided with Apophis. Meanwhile, for the smaller density model, 48%
were ejected or collided with Apophis and 52% survived for 30 years. For both
densities, the survived particles are located all over the disc, but there are
three preferentially concentrated regions of them (Fig. 10). The first one is
the region inner the equilibrium points due to the gravitational influence of
Apophis. The second one is around both linearly stable equilibrium points
$E_{2}$ and $E_{4}$. The third region is the region outside the influence of
the perturbation of Apophis near the ejected particles, as discussed before.
The smaller density model presents some agglomerate of survived particles
inside the region of ejected particles, these regions will be studied in
detail in future works.

Since the stability region of prograde orbits is about half the Hill radius
(Domingos et al., 2006; Hunter, 1967), we expand the radial distribution of
the particles around Apophis for the 24 hours simulations. For the smaller,
mean, and larger densities, the new radial distances, $r$, were 1.0 $<r<$ 7.5
km, 1.15 $<r<$ 9.0 km, and 1.3 $<r<$ 10.5 km, respectively.

All the 15,000 particles simulated for this new radial distance have survived
for the entire 24 hours simulations. The particles are distant from the
Apophis, so its irregular gravitational field is low and the system can be
interpreted as a two-body problem, thus the region is stable as expected.
Later we compare these simulations when the perturbation of the Earth during
the 2029 encounter is taken into account.

####  4.2.1 Solar Radiation Pressure

The previous simulations were performed considering only the gravitational
potential of Apophis. However, the solar radiation pressure may cause a
significant perturbation on the particles’ evolution. To identify the size of
the particles that may survive the effects of the solar radiation pressure, we
assume particles with different sizes and simulate the same conditions
described in section 4.2 adding the solar radiation pressure perturbation in
the system.

We have carried out numerical simulations with micrometric-sized particles,
and the entire ensamble did not survive for the 30 years for particles smaller
than 100 $\mu$m. This allowed us to set a lower limit for the particle radius
that could survive despite of the solar radiation disturbance. So, we proceed
the simulations with the $N$-$BoM$ package for particles with radius larger
than 100 $\mu$m. Considering the area to mass ratio ($A$) of the particle and
a solar constant in function of the solar luminosity and speed of light
($G^{*}=1\times 10^{17}$ kg$\cdot$m$\cdot$s-2), the radiation pressure
acceleration is given by (Scheeres & Marzari, 2002)

$\vec{a}_{srp}=\frac{AG^{*}(1+\eta)}{R^{3}}\vec{R},$ (16)

where $R$ is the module of the position vector Sun-particle ($\vec{R}$) and
$\eta$ is the reflectance of the particle, which is assumed to be a unit for a
totally reflective material. We do not consider the effects caused by
shadowing since they are not significant to produce a considerable change in
the surviving scenario, i.e particles that survive the entire simulation
without ejection or collision with Apophis.

Since our goal was to determine the order of magnitude of particle’s size that
could survive in the system despite the perturbation due to the solar
radiation force, we simulate a set of particles with radius ($r_{p}$) in a
discrete distribution of 1 cm, 5 cm, 10 cm, and 15 cm. The particles have the
same density of Apophis considering each density model (1.29 g$\cdot$cm-3, 2.2
g$\cdot$cm-3 or 3.5 g$\cdot$cm-3).

(a) $r_{p}$ = 15 cm and $\rho$ = 1.29 g$\cdot$cm-3

(b) $r_{p}$ = 5 cm and $\rho$ = 2.2 g$\cdot$cm-3

(c) $r_{p}$ = 5 cm and $\rho$ = 3.5 g$\cdot$cm-3

Figure 11:  Initial conditions of the 15 thousand particles around the
equilibrium points of the asteroid Apophis in the $xoy$ plane considering the
perturbation of the solar radiation pressure. The green dots represent the
equilibrium points. The black dots represent the particles that survived after
the 24 hours of integration and the blue dots the particles that collide with
the asteroid. The letters (a)-(c) represent, respectively, the densities of
1.29 g$\cdot$cm-3, 2.2 g$\cdot$cm-3 and 3.5 g$\cdot$cm-3. For the densities of
2.2 g$\cdot$cm-3 and 3.5 g$\cdot$cm-3 the radius of the particles is 5 cm and
for the density of 1.29 g$\cdot$cm-3, 15 cm.

For the lower density, Figure 11a shows the initial conditions of the
particles where the blue dots represent the collided particles and the black
dots the survivors. For the sizes of 1 cm and 5 cm, no particles survived, and
for the radius of 10 cm, just one particle survived. Then, we simulated
particles with radius of 15 cm, and for this, about 3% of the 15,000 particles
survived 30 years. We notice that the survived particles are concentrated in
three small regions near the inner edge of the disc and, consequently, close
to Apophis.

For the density models of 2.2 g$\cdot$cm-3 and 3.5 g$\cdot$cm-3, there are no
survivors for 1 cm particles, but for particles with radius of 5 cm it has
about 3.5% of survivors for the mean density model and almost 18% for the
larger one. The small regions with surviving particles for the mean density
model are similar to the regions of the smaller model, also presenting three
defined regions (Fig. 11b). However, the larger density model has a single
ring of survived particles (Fig. 11c). Note that we can clearly see the same
two preferential regions of collisions close to the Apophis’ surface existent
in the 24 hours simulations without the Earth’s perturbation (see Fig. 9).
Thus, just particles with cm-sized survive for 30 years of simulations
considering the solar radiation pressure.

(a)

(b)

(c)

(d)

(e)

(f)

Figure 12:  Three different selected particles among the simulations with
solar radiation pressure representing the different orbital radius behavior.
The letters (a)-(f) represent the orbital radius and $z$ coordinate for three
survived selected particles. The letters (a), (b), (e) and (f) represent the
plot for two selected particles with a density of 3.5 g$\cdot$cm-3. The
letters (c) and (d) represent the plot for a selected particle with a density
of 2.2 g$\cdot$cm-3.

Analysing the particles that survived for the entire simulation, we note two
different predominant behaviours of particles’ orbital radius. The first and
most common behaviour is a radial oscillation, that could happen either in a
large or small scale as exemplified in Figures 12e and 12a. The small radial
oscillation occurs preferentially for the densities models of 1.29
g$\cdot$cm-3 and 2.2 g$\cdot$cm-3 since the initial conditions of the survived
particles are near the Apophis’ surface (Figs. 11a and 11b). Thus the orbit of
these particles could not have a large amplitude, since it would implicate in
large eccentricity, a pericentre closer to the surface, and consequently in a
collision with Apophis. The small radial oscillation also occurs for the
density of 3.5 g$\cdot$cm-3, but the frequency of large radial oscillation is
larger (Fig. 11c). Since there is a large region of survived particles and
most of them have initial conditions more distant from the body, the orbit of
these particles can obtain larger amplitudes and eccentricities without
collision with Apophis.

The second behavior is an abrupt radial variation and occurs for the three
densities models. The irregular gravitational potential of the body could
produce a variation of eccentricity. When the eccentricity increases the
particle collides with Apophis and when it decreases, it produces the decays
shown in Figure 12c. Despite different behavior, the orbital radius of all the
particles that survived are restricted to the minimum value of 300 metres,
approximately, which is expected since an inferior value could result in a
collision with the body’s surface. The $z$ coordinate also has a variation
(Figs. 12a, 12b and 12f), implying the change in the orbits’ inclination. The
amplitude of the orbital inclination is related to the amplitude of the
orbital variation, thus orbits with larger eccentricity allow larger
inclinations.

##  5 Instability Due to the Earth’s Encounter

To understand how the Earth will affect the environment around Apophis in the
2029 encounter, we simulate the initial condition described in section 4.2 for
the disc encompassing the equilibrium points and the expanded disc for 24
hours adding the perturbation of the Earth and using the trajectory shown in
Fig. 2. Those particles we simulate are at least cm-sized, otherwise they are
removed by the solar radiation pressure.

To evaluate the largest possible perturbation of the Earth, we positioned
Apophis in such a way that its orbital plane coincides with its equatorial
plane of the Earth. Then, we interpolate this trajectory in the $N$-$BoM$
integrator as an additional external force and simulated the 30 thousand
particles around Apophis for 24 hours.

(a)

(b)

(c)

(d)

(e)

(f)

(g)

(h)

(i)

Figure 13:  Initial conditions of the 15 thousand particles around the
equilibrium points of the asteroid Apophis in the 2029 approach in the $xoy$
plane. The green dots represent the equilibrium points. The black dots
represent the particles that survived after the 24 hours of integration and
the blue and pink dots the particles that collide and ejected, respectively,
with the asteroid. The letters (a)-(c), (d)-(f) and (g)-(i) represent,
respectively, the densities of 1.29 g$\cdot$cm-3, 2.2 g$\cdot$cm-3 and 3.5
g$\cdot$cm-3.

The initial positions of the 15,000 particles around the equilibrium points in
the encounter are shown in Fig. 13 for the three densities models. The larger
density model has about 56% of surviving particles, while 16% and 28% collided
and ejected, respectively. On the other hand, the smaller density model has
about 59% of surviving particles, while 15% collided and 26% were ejected by
the Earth perturbation.

Without the Earth perturbation, almost 99% of the particles survive for 24
hours (section 4.2). So, the encounter causes a large change in the Apophis
nearby environment as the number of surviving particles decreases
approximately 40% and 44% for the smaller and larger model of density,
respectively. The number of collisions due to the approach increase about 19
times for the smaller model of density and 11 times for the larger model. The
ejection of the particles did not occur for the previous simulation, however,
considering the 2029 approach 26-28% of the particles were ejected.

(a)

(b)

Figure 14:  Two-body orbital energy of two different particles in the 24 hours
simulation considering the 2029 encounter with the Earth. The orange line
denotes the time that the closest approach occurs and the black line
represents the line of zero energy. The letters (a) and (b) represent the
particle that was ejected due to the Earth perturbation and the surviving one,
respectively.

Some of the ejections occur by the energy criterion, as the Apophis approaches
the Earth the orbital energy of the particles increase, and some of them
remain positive for the rest of the simulation. Figure 14 shows the energy of
two different particles, a particle that survived and an ejected one. The
orange line represents the moment that occurs the closest distance between
Apophis and the Earth and the black line represents the line of zero energy.
The energy of the particle that was ejected is shown in Fig. 14a, observe that
near the closest encounter with the Earth the orbital energy increases to
approximately 0.0025 m${}^{2}\cdot$s-2 and after the encounter, it decreases
to 0.0005 m${}^{2}\cdot$s-2 until the end of the simulation. Similar behaviour
occurs for the particle that survived the 2029 approach (Fig. 14b), but after
the encounter, the particle energy decreases to $-$0.0005 m${}^{2}\cdot$s-2.

(a)

(b)

(c)

(d)

(e)

(f)

(g)

(h)

(i)

Figure 15:  Initial conditions of the 15 thousand particles of the expanded
ring in the 2029 approach in the $xoy$ plane. The green dots represent the
equilibrium points. The black dots represent the particles that survived after
the 24 hours of integration and the blue and pink dots the particles that
collided with the asteroid and were ejected, respectively. The letters (a)-(c)
represent, respectively, the densities of 1.29 g$\cdot$cm-3, 2.2 g$\cdot$cm-3
and 3.5 g$\cdot$cm-3.

For the expanded radial distribution the number of ejections for the regular
case (section 4.2) was null, but for the encounter simulation was high (Fig.
15). About 75% and 77% of the particles were ejected for the smaller and
larger model of density, respectively, being 1.3% of this ejection by the
energy criterion for the smaller and larger models. The percentage of survived
particles was approximately 24.2% for the model with the smaller density,
while for the larger one was about 22.5%. The number of collisions was small,
about 0.8% and 0.5% for the smaller and larger model of density.

The 2029 approach with the Earth shown to be significant in the nearby
environment of Apophis for both disc cases. For the disc around the
equilibrium points, the Earth’s perturbation causes a high number of
ejections, but also almost twice the value of collisions due to the
gravitational field of this region being high. On the other hand for the
expanded disc, the encounter causes a massive number of ejections and a
smaller number of collisions and surviving particles.

##  6 Final Comments

In this study, we provided the exploration of the effects of the 2029 Earth’s
encounter on the surface and nearby dynamics of the asteroid Apophis. Firstly,
we briefly discuss the Apophis’ shape model (Pravec et al., 2014), its
physical properties as the density and size discordance (Binzel et al., 2009;
Müller et al., 2014; Licandro et al., 2015; Brozović et al., 2018), and the
2029 Earth’s encounter and its trajectory. Then, we define gravitational
potential by the polyhedra method (Werner & Scheeres, 1996) and the
geopotential on the Apophis’ surface (Scheeres, 2012; Scheeres et al., 2016).

We presented the slope angle and its implications on the body’s surface. The
slope computed on Apophis’ surface was about 36∘ for the three density models
with small variations among the models, a difference of only 0.35∘ between the
smaller and larger density model. To comprehend the possible effect caused by
the 2029 encounter, we calculated the variation of the slope angle produced by
the Earth’s gravitational perturbation called $\Delta$slope. The $\Delta$slope
was smaller than 4∘ and 2∘ for the density of 1.29 g$\cdot$cm-3 and 3.5
g$\cdot$cm-3, respectively. Those variations may cause migration of
cohesionless particles since they can reach the value of the repose angle of
geological materials (35∘-40∘) in some regions (Lambe & Whitman, 1969;
Mitchell et al., 1974; Al-Hashemi & Al-Amoudi, 2018). A variation of about 2∘
may generate a slow erosion process in those regions with high-slope (Ballouz
et al., 2019), but numerical simulations have shown that the perturbation of
the Earth in the 2029 flyby may create just local landslides (Yu et al.,
2014).

Next, we analysed the zero-velocity curves and computed the equilibrium points
of the body. Apophis has four external equilibrium points and two of them are
topologically classified as Centre-Centre-Centre for the three density models,
implying that they are linearly stable points (Jiang et al., 2014). Thus, we
performed a set of numerical simulations of a disc of 15,000 particles in the
nearby environment of Apophis. The first set of simulations was performed
considering just the gravitational perturbation of Apophis for a period of 24
hours and 30 years. For a period of 24 hours, the majority of the ring
encompassing the equilibrium points is stable with just 0.78-1.4% of collision
and no ejection, while for the expanded disc all particles survived. For the
period of 30 years, 96% of the particles survived for the larger density
model, 0.9% collided and 3.1% ejected in a region that has limited the
influence of Apophis, creating an external disc of survived particles (Fig.
10b). The same phenomenon happens for the smaller density, however, 52% of the
particles survived, 4% collided, and 44% ejected. For the density of 1.29
g$\cdot$cm-3, we found interesting regions of survived particles that are
related to resonance regions and will be studied in future works.

We added the perturbation of the solar radiation pressure at the system and
simulate the same initial conditions of the previous set of simulations, but
now just for the period of 30 years. With those simulations, we identify that
just centimetres particles survive for the whole period. For the density model
of 2.2 g$\cdot$cm-3 and 3.5 g$\cdot$cm-3, we compute a significant amount of
surviving particles with a radius of 5 cm and for the density model of 1.29
g$\cdot$cm-3, just particles with radius of 15 cm survived, considering our
set of discrete sizes of particles.

In addition, we performed simulations considering the 2029 encounter and the
Earth’s perturbation but removing the solar radiation pressure. Since the
solar radiation pressure removes small particles, for this set of simulations
we implicitly assume that all of the particles must have a centimetre size of
about 5-15 cm. At the end of the 24 hours simulation for the disc encompassing
the equilibrium points, we compute a survival percentage of 56% for the larger
density model and 59% for the smaller one. However, for the expanded disc, the
majority of the particles were ejected and just 22.5-24.2% of particles
survived. In general, the 2029 approach shows that the Earth perturbation is
significant in the nearby environment of Apophis, but a considerable number of
particles still remain at the end of the encounter.

All the analyses presented in this study, whose purpose was to identify the
possible effects on Apophis’ surface and nearby environment due to the 2029
flyby, may contribute to the 2029 observational campaign and also to validate
our results.

## Acknowledgements

This study was financed in part by the Coordenação de Aperfeiçoamento de
Pessoal de Nível Superior - Brasil (CAPES) - Finance Code 001, Fundação de
Amparo à Pesquisa do Estado de São Paulo (FAPESP) - Proc. 2016/24561-0 and
Proc. 2019/23963-5, Conselho Nacional de Desenvolvimento Científico e
Tecnológico (CNPq) - Proc. 305210/2018-1.

## ORCID iDs

G. Valvano *  | https://orcid.org/0000-0002-7905-1788
O. C. Winter *  | https://orcid.org/0000-0002-4901-3289
R. Sfair *  | https://orcid.org/0000-0002-4939-013X
R. Machado Oliveira *  | https://orcid.org/0000-0002-6875-0508
G. Borderes-Motta *  | https://orcid.org/0000-0002-4680-8414
T. S. Moura *  | https://orcid.org/0000-0002-3991-8738

## Data availability

The data underlying this article will be shared on reasonable request to the
corresponding authors.

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