# Dynamical origin of the Dwarf Planet Ceres 111Article accepted for
publication on

Rafael Ribeiro de Sousa  São Paulo State University, UNESP,Campus of
Guaratinguetá, Av. Dr. Ariberto Pereira da Cunha, 333 - Pedregulho,
Guaratinguetá - SP, 12516-410, Brazil  Alessandro Morbidelli  Laboratoire
Lagrange, UMR7293, Université Côte d’Azur, CNRS, Observatoire de la Côte
d’Azur, Boulevard de l’Observatoire, 06304 Nice Cedex 4, France  Rodney Gomes
Observatório Nacional, Rua General José Cristino 77, CEP 20921-400, Rio de
Janeiro, RJ, Brazil<EMAIL_ADDRESS>Ernesto Vieira Neto  São Paulo State
University, UNESP,Campus of Guaratinguetá, Av. Dr. Ariberto Pereira da Cunha,
333 - Pedregulho, Guaratinguetá - SP, 12516-410, Brazil  Andre Izidoro
Department of Physics and Astronomy 6100 MS 550, Rice University, Houston, TX
77005, USA  Department of Earth, Environmental and Planetary Sciences, 6100 MS
126, Rice University, Houston, TX 77005, USA  São Paulo State University,
UNESP,Campus of Guaratinguetá, Av. Dr. Ariberto Pereira da Cunha, 333 -
Pedregulho, Guaratinguetá - SP, 12516-410, Brazil  Abreuçon Atanasio Alves
São Paulo State University, UNESP,Campus of Guaratinguetá, Av. Dr. Ariberto
Pereira da Cunha, 333 - Pedregulho, Guaratinguetá - SP, 12516-410, Brazil

###### Abstract

The Dwarf Planet Ceres revealed the presence of ammonia and other unique
properties compared to other asteroids in the main belt which suggests that it
was not formed in situ. We model the early dynamical evolution of the outer
Solar System to study possible dynamical mechanisms to implant a Ceres-sized
planetesimal in the asteroid belt from the trans-Saturnian region. We
calculate that the fraction of the population of Ceres-sized planetesimals
that are captured in the asteroid belt is in the range of $2.8\times 10^{-5}$
to $1.2\times 10^{-3}$ depending on the initial location in the outer
planetesimal disk. The captured bodies have a 70% probability to have a
semimajor axis between $2.5$ and $3$ au, a 33% probability to have an
eccentricity smaller than 0.2 and a 45% probability to have an orbital
inclination smaller than 10∘. Assuming the existence of 3,600 Ceres-size
planetesimals in the inner part of the trans-Saturnian disk, consistent with
the estimate of Nesvorný & Vokrouhlický (2016) for the trans-Neptunian disk,
our estimated capture probability and a final 80% depletion of the asteroid
belt during the subsequent giant planet instability, lead to capture $\sim$ 1
Ceres in the asteroid belt, with a probability of 15%, 34%, and 51% to be
located in the inner, middle and outer belt respectively.

Ceres; Planetesimals; Planet-disk interactions; Solar System dynamical
evolution.

††journal: ICARUS

##  1 Introduction

The small-body populations in the Solar System provide constraints to unveil
the events that happened in its early history. The asteroids’ main belt is the
nearest and the best characterized population that we can use to test the
dynamical models for the evolution and formation of the planets. Ceres is the
largest object of the main belt (mean physical radius of 470 km) with many
interesting physical and chemical properties. Ceres comprises about $0.33\%$
of the total mass of the asteroid belt and has low density (2.2 g/cm3), low
albedo and a spectral reflectance similar to carbonaceous chondrites (McCord &
Sotin, 2005). Ceres orbits the Sun deep in the main belt, with a semimajor
axis of $2.767$ au, an eccentricity of $0.097$ and an inclination of about
9.73∘. The position of Ceres in the main belt is close to many C-types
asteroids, but Ceres is not a C-type (Bell et al., 1988; McKinnon, 2008).
Ceres is similar to G-type asteroids, which are all hydrated whereas C-types
show hydration features only in 60% of the cases (Britt et al., 1992) and is
rich in ammonia compounds that are typically not found in C-type asteroids and
carbonaceous chondrites (de Sanctis et al., 2015).

Ceres is in rotational hydrostatic equilibrium, at least partially
differentiated (Neumann et al., 2020) but it did not dynamically sculpt the
asteroid population in its neighborhood; thus, it is classified as a dwarf
planet (although it was considered a planet when it was discovered). The
interior structure of Ceres comprises a water-ice-rich mantle over a rock
core, with a bulk rock-ice fraction of 75%-25% respectively (Thomas et al.,
2005). All these properties, deduced from the observations of the Dawn mission
suggest that Ceres and the Kuiper belt share the same source location during
their accretion phase (Neumann et al., 2020). Indeed, the physical and
chemical properties of Ceres share many similarities with the largest Kuiper
Belt Objects (Triton, Eris and the Pluto-Charon binary system). In particular,
the discovery of ammonia on Ceres’ surface is a strong chemical evidence that
Ceres formed in the outer region of the Solar System (de Sanctis et al.,
2015). The presence of ammonia could mean that Ceres was formed far beyond the
water-snowline where ammonia is readily available. It is unlikely that
ammonia-rich material drifted from the outer Solar System and was accreted by
Ceres in situ (Vokrouhlický et al., 2016).

The so-called Nice model invokes a dynamical instability among the giant
planets that was triggered by their interactions with the primordial outer
planetesimal disk after the removal of gas from the system (Tsiganis et al.,
2005; Levison et al., 2011; Nesvorný & Morbidelli, 2012). McKinnon (2008,
2012) proposed that Ceres is a planetesimal captured from the primordial
trans-Neptunian disk during the events described in the Nice model. More
precisely, Ceres was dynamically scattered inwards during the reorganization
of the outer Solar System during the instability phase, then it was implanted
in a more massive primordial asteroid belt, and finally its orbit was
circularized due to dynamical friction. The dynamical justification of this
proposition comes from models showing KBOs and comets implanted in large
numbers into the Jovian Trojan clouds and in the outer asteroid belt
(Morbidelli et al., 2005; Levison et al., 2009).

Vokrouhlický et al. (2016) studied the capture of trans-Neptunian
planetesimals into the asteroid belt using a modified version of the Nice
model (Nesvorný & Morbidelli, 2012), which assumes that an additional giant
planet originally existed in the outer solar system (Jupiter, Saturn, Uranus,
Neptune and a Neptune-like body) and also a primordial disk containing
thousands of Pluto-mass bodies (Nesvorný & Vokrouhlický, 2016; Ashton et al.,
2021). They were one of the first to investigate how the interactions between
the fifth giant planet, Jupiter and planetesimals disk helped to implant
planetesimals into the asteroid belt, via temporary captures into the first-
order mean motion resonances with Jupiter, while others have repeated this
study with greater complexity (Quarles & Kaib, 2019; Clement et al., 2019;
Ribeiro de Sousa et al., 2019). Overall, the distribution of trans-Neptunian
planetesimals captured in the asteroid belt in the Vokrouhlický et al. (2016)
model reproduces the observed distribution of D and P-type asteroids better
than in previous models.

Vokrouhlický et al. (2016) also estimated the probability of Ceres being
captured from the trans-Neptunian planetesimal disk (between 23 and 30 au) in
the Nice model. Assuming that the trans-Neptunian disk once contained
3,600-14,000 Ceres-sized objects (Nesvorný & Vokrouhlický, 2016; Ashton et
al., 2021), Vokrouhlický et al. (2016) concluded that the probability that one
was captured into the asteroid belt is too low to be a viable explanation for
the origin of Ceres.

Raymond & Izidoro (2017a) also studied planetesimal capture into the asteroid
belt during the accretion phase of the giant planets, when gas was still
present. As a consequence of the combined effects of orbital destabilization,
gravitational scattering and aerodynamic gas drag, planetesimals from the
Jupiter-Saturn region could be implanted into the asteroid belt. These
implanted asteroids preferentially land into the outer main belt, and their
source region was 5 au-wide (from 4 to 9 au). Accounting for the migration of
Jupiter and Saturn from initially larger distances broadens the source region
of implanted objects up to 10 – 20 au. Raymond & Izidoro (2017a) identified
the implanted planetesimals with the C-type asteroids of the main belt,
because cosmochemical evidence (Kruijer et al., 2017) shows that these bodies
should have formed beyond Jupiter. However, other asteroid types (e.g. G, D,
P) could in principle also be implanted. Raymond & Izidoro (2017a) also
observed that the distribution of $D=1\,000$ km implanted planetesimals peaks
at around $2.7$ au. Thus, they claimed that Ceres was implanted during the
growth of Jupiter and Saturn. However, the amount of original planetesimals in
Jupiter-Saturn region is highly uncertain (Raymond & Nesvorny, 2020).

The goal of this paper is to investigate the possibility of capture of Ceres
from the Trans-Saturnian region, which has the advantage of drafting Ceres
from a farther heliocentric distance, where bodies are more likely to be
ammonia-rich. In contrast to Raymond & Izidoro (2017a), we study the capture
of Ceres-sized planetesimals after the full formation of Jupiter and Saturn
but during the growth of Uranus and Neptune. To do this, we use the model of
Izidoro et al. (2015a), described in Section 2, which is the most successful
one to reproduce the formation of these two planets and their large
obliquities. To study the possibility that Ceres is a refugee from the
primordial Trans-Saturnian disk we proceed as follows. First, we reproduce the
dynamical evolution of the planetary bodies simulated by Izidoro et al.
(2015a) and leading to the successful formation of Uranus and Neptune. Then,
we generate a planetesimal disk containing only objects of Ceres-sized
planetesimals. We calculate the fraction of Ceres-sized planetesimals that are
placed in orbits with a semimajor axis smaller than 3.5 au and study the
mechanism to implant these objects in a permanent stable orbit very close to
the current orbit of Ceres.

##  2 Methods

Izidoro et al. (2015a) proposed that the Uranus and Neptune formed in two
steps. By the time that Jupiter and Saturn had undergone rapid gas accretion
to become giant planets, pebble accretion had produced a system of
protoplanets of $\sim 5$ Earth masses, comparable in mass to the most-
frequently observed super-Earths (e.g., Mayor et al., 2011; Batalha et al.,
2013; Marcy et al., 2014; Wolfgang et al., 2016). These protoplanets migrated
inward in Type-I regime due to tidal torques from the gas disk (Goldreich &
Tremaine, 1980; Ward, 1986; Tanaka et al., 2002). While it is unclear whether
Jupiter and Saturn at this point would have been migrating inward, outward or
had roughly stationary orbits (Masset & Snellgrove, 2001; Morbidelli & Crida,
2007; Pierens & Nelson, 2008; Pierens & Raymond, 2011; Raymond & Morbidelli,
2014; Pierens et al., 2014), the gas giants’ migration was certainly slower
than that of the protoplanets for typical disk viscosities. The protoplanets
could not migrate through Jupiter and Saturn’s orbits; rather, the gas giants
acted as a migration barrier (Izidoro et al., 2015b). Protoplanets became
trapped in mean motion resonances with Saturn, forming a resonant chain. As
more protoplanets migrated inward, the resonant chain was destabilized,
leading to giant (obliquity-generating) collisions and to another phase of
migration into a new resonant chain. Izidoro et al. (2015a) showed that this
process typically produces 2-3 ice giants with masses comparable to those of
Uranus and Neptune in resonant chains that include Jupiter and Saturn.

In our previous work (de Sousa Ribeiro et al., 2020), we concentrated our
efforts using one of the best simulations from Izidoro et al. (2015a) to study
the dynamical sculpting of the primordial outer planetesimal disk during the
accretion of Uranus and Neptune from migrating protoplanets during the gas
disk phase. The de Sousa Ribeiro et al. (2020) model implemented the
interpolation of the orbital elements of all massive bodies (Jupiter, Saturn
and the protoplanets) into the Izidoro et al. (2015a) simulation using spline
functions. The interpolated evolution of the massive bodies was used to
simulate the planetesimal evolution including the effects of a gaseous
protoplanetary disk on the planetesimals (gas drag and dynamical friction;
Section 2.1).

In this work, we use the same approach as in de Sousa Ribeiro et al. (2020) to
model the capture process of Ceres because: i) Izidoro et al. (2015a) scenario
is the most successful model in explaining quantitatively the origin of Uranus
and Neptune and ii) The method of interpolation presented in de Sousa Ribeiro
et al. (2020) has the advantage of implementing the exact same evolution of
the giant planets and protoplanets of Izidoro et al. (2015a) in all
simulations (e.g varying the initial location of planetesimals). Most
importantly, the Izidoro et al. (2015a) simulation showed that some
protoplanets temporarily reached the inner region of the Solar System. The
history of these protoplanets may dynamically affect the capture process of a
Ceres-sized planetesimal. Note that the interpolation model for the evolution
of the protoplanets already accounts for the tidal forces between the
protoplanets and the gas disk (Izidoro et al., 2015a; de Sousa Ribeiro et al.,
2020).

###  2.1 Interactions of planetesimals with the gaseous protoplanetary disk

As in de Sousa Ribeiro et al. (2020) we consider two different effects of the
gas drag on planetesimals: the aerodynamic gas drag and the gas dynamical
friction.

The aerodynamic gas drag force on a particle moving in a gas disk environment
is expressed as a function of its shape, size, velocity and gas conditions. In
the particular case of a spherical body with radius $R$, the drag force is in
a direction opposite to the particle velocity and is expressed by:

$\vec{F}_{D}=-\frac{1}{2}C_{D}\pi R^{2}\rho_{g}v_{rel}\vec{v}_{rel},$ (1)

where $C_{D}$ is the drag coefficient and $\vec{v}_{rel}$ is the relative
velocity vector between the gas and the planetesimal. The drag coefficient for
a spherical object is a function of the Reynolds number ($R_{e}$), which is a
measure of the turbulence of the gas in the wake of a planetesimal, the Mach
Number ($M$), and Knudsen number ($K$). To evaluate the drag coefficient we
used the same approach of Brasser et al. (2007) where they estimated the
values of $M$, $K$ and $R_{e}$ as a function of planetesimal’s size and
velocity.

The gaseous dynamical friction (GDF) effect arises from the fact that a
perturber moving on a straight line in a uniform gaseous medium generates a
wake, which in turn exerts a gravitational torque on the perturber. The
aerodynamic drag force is dominant for small planetesimals and scales as the
square of the radius of the planetesimals. The GDF is dominant for
intermediate sizes planetesimals and scales with the sixth power of the
planetesimal radius. Grishin & Perets (2015) showed that for planetesimals in
the mass range of $m\sim{10}^{21}-{10}^{25}$ g the GDF is possibly dominant
over gas-drag for some gas density conditions. The GDF force is given by:

$\vec{F}_{GDF}=-\frac{4\pi G^{2}m^{2}\rho_{g}}{v_{rel}^{3}}\vec{v}_{rel}I(M),$
(2)

$I(M)$ is a dimensionless factor depending on the Mach number ($M$) and it is
given by:

$I(M)=\\\ \frac{1}{2}\ln\left(\frac{1+M}{1-M}\right)-M,\quad for\>M<1$ (3)

$I(M)=\\\
\frac{1}{2}\ln\left(1-\frac{1}{M^{2}}\right)+\ln\left(\frac{v_{rel}\>t}{R}\right),\quad
for\>M>1$ (4)

where the GDF may be important to investigate the capture of Ceres ($m$
$\sim{10}^{24}$ g). For the disk gas properties, we used those adopted in
Izidoro et al. (2015a), which had been calculated from a hydrodynamical
simulation, starting from a disk with initial surface density
$\Sigma(r)=1\,000$ g/cm2 at 1 au, and exhibits a double gap around the orbits
of Jupiter and Saturn. We assume that this surface density decays uniformly
over time as $\exp(-t/\tau_{gas})$ with $\tau_{gas}$ = 5 My. The aspect ratio
of the disk is assumed to be:

$h=\frac{H(r)}{r}=0.033r^{0.25}\ ,$ (5)

where $H(r)$ is the pressure scale height at radius $r$.

In this work, our preliminary numerical test showed that the GDF force is not
more effective than aerodynamic gas drag. We decided nevertheless to keep both
the gas drag and the GDF effect for completeness.

We used the same version of the N-body REBOUND code (Rein & Liu, 2012; Rein &
Spiegel, 2015; Rein & Tamayo, 2015) modified by de Sousa Ribeiro et al. (2020)
to perform the N-body simulations including the effects cited above as
additional forces. In our simulations, the planetesimals were assumed to be
non-interacting with each other.

###  2.2 Evolution of planets and planetary embryos

The actual migration history of Jupiter and Saturn is not well known. Some
works claimed that a Jupiter ending at 5 au could only have formed from a seed
initially located beyond 20 au (Bitsch et al., 2015). de Sousa Ribeiro et al.
(2020) investigated the role of Jupiter migration in the planetesimal disk
evolution and concluded that a long-range inward migration of Jupiter from 15
au or beyond would over-excite the inclinations of Kuiper belt objects with
respect to what is presently observed. Thus, we adopt Jupiter at 5 au in a
fixed orbit. A short range migration of Jupiter should not affect
significantly the results.

In this work, we follow the Izidoro et al. (2015a) model for the origin of
Uranus and Neptune, where in this model Jupiter and Saturn are in a 3:2
resonance between Jupiter and Saturn (Nesvorný & Morbidelli, 2012). However,
there are other planetary configurations that have been considered in the
recent literature. For example, Clement et al. (2021) (a) and (b) showed that
a primordial 2:1 resonance between Jupiter and Saturn is advantageous in terms
of its ability to consistently replicate the current Jupiter and Saturn
eccentricities and the possibility that the giant planets’ primordial orbits
were eccentric.

Because the simulation of Izidoro et al. (2015a) has Jupiter at 3.5 au, we
rescaled the semimajor axes of all bodies by a factor of $5/3.5$. The radial
surface density profile of the gas is also rescaled to keep the gaps centered
on the giant planets’ orbits. More specifically, given the rescale factor
$R(t)$ on semimajor axes, the surface density at the heliocentric distance r
is computed as:

$\frac{\Sigma\left(\frac{r}{R(t)}\right)}{(R(t))^{2}},$ (6)

where $\Sigma$ is the surface gas density.

Figure 1 illustrates our reproduction of the Izidoro et al. (2015a) simulation
using the interpolation scheme and scaling the initial semimajor axes. We show
in panel (a) the initial semimajor axis and eccentricities of the planetary
system (all initially zero) which contains Jupiter (at $5.0$ au), Saturn (at
$6.55$ au), and a collection of $11$ protoplanets distributed with semimajor
axis in a range from $8$ to $40$ au with masses between $3$ to $9$
$M_{\oplus}$.

In Panels (b) and (c), we show snapshots of the semimajor axis and
eccentricity of Jupiter, Saturn, and the protoplanets at two different times:
$1\times 10^{5}$ yr and $1\times 10^{6}$ yr respectively. At these times we
can see that most protoplanets migrated inward, although they were blocked by
Jupiter and Saturn remaining beyond the giant planets’ orbits. Most of these
protoplanets will grow more massive through mergers. However some protoplanets
can enter the inner region of the Solar System ($a<5$ au) for a short period
of time due to scattering by Jupiter. We define these protoplanets as
invaders.

The invaders are particularly interesting objects because they may affect the
planetesimals capture process in the inner Solar System. The gray lines in the
Panels (b) and (c) of Figure 1 show the boundary in (a,e) space beyond which
the planetesimals are intersected by the invader’s trajectory. These lines
cross the mean motion resonances (MMRs) with Jupiter (the 3:1, 5:2, and 2:1
MMRs) are presented in panels (b) and (c) of Figure 1 by vertical black lines.
In Panel (d) of Figure 1, we show the final semimajor axis and eccentricity of
Jupiter, Saturn, Uranus, Neptune and of a fifth comparable-mass planet at
$5.1$ Myr, the end of the protoplanetary gas disk lifetime.

Observing the pericenter evolution of the protoplanets, showed in Panel (e) of
Figure 1, we note that the invaders enter deeply into the inner Solar System,
down to a distance between 1.3 au to 3 au during the simulation time. These
invaders can potentially have close encounters with the planetesimals in the
inner Solar System.

Figure 1: Panels (a) - (d) are semimajor axis vs. eccentricity snapshots of
the system with Jupiter, Saturn and a collection of protoplanets of the
considered Izidoro et al. (2015a) simulation. The 3:1, 5:2 and 2:1 mean motion
resonance with Jupiter are presented by vertical lines. The gray lines
represent the boundaries in (a,e) space within which planetesimals are
intersected by an invader. Panel (e) represents a complete evolution of
planets (black lines) and protoplanets (magenta lines) pericenters in the same
simulation.

###  2.3 Distribution of planetesimals

The large size of Ceres suggests that it has not been altered significantly
during its collisional evolution (McCord & Sotin, 2005). Thus, we assume that
Ceres formed in the trans-Saturnian region and dynamically evolved keeping its
current size and density. The size and density of Ceres are very important due
to the effects of the gas disk.

We used $1,000$ km sized planetesimals to build a disk distributed beyond the
orbits of the giant planets, throughout the region occupied by the
protoplanets (8 au to 40 au). We divided this region into three different
reservoirs constrained by chemical models that explain the surface colors of
observable Kuiper Belt objects (Brown et al., 2011).

Brown et al. (2011) suggested that planetesimals are characterized by strong
heliocentric-distance gradients in surface composition due to ultraviolet and
particle irradiation. This process is called volatile evaporation and could
explain the surface colors that are seen today in the objects of the Kuiper
Belt. Based on this model, objects as large as Ceres can always retain on its
surface H2O, CO2 and H2S. Between 20-30 au the planetesimals can retain also:
HCN, C2H6, CH3OH, while more distant planetesimals (from 30-50 au) can also
have NH3.

Adapted from Figure 1 of Brown et al. (2011), Figure 2 shows our initial
distribution of planetesimals. The planetesimals are differentiated by color
(black, red, and blue) based on their initial heliocentric distances and the
reservoir of molecular species that are expected to be present at their
surface (green colored text).

Ceres has ammonia on its surface but this is not an indication that Ceres
could have formed only in the blue region. Moreover, Ceres spent at least
almost 4.5 Gy in the asteroid belt, so it _should_ have lost by irraditation
its surface ammonia, regardless of where it formed. It may be possible that
ammonia is rejuvenated by outgassing from the interior or it reacted
chemically to form more refractory ammoniated salts. In any of these cases, if
this happened in the asteroid belt it could have happened anywhere in the
disk. So, Ceres does not need to have been formed beyond 30 au. Instead, to
accrete ammonia in the first place, Ceres should have formed beyond the
ammonia snowline ($\sim 90$K). In an early disk (Bitsch et al., 2015) this
could be 7-10 au. In a late disk the ammonia snowline can go into the asteroid
belt, but at that time Jupiter was already formed and blocking the drift of
ammonia-rich dust, so it is safe to assume that Ceres formed beyond Jupiter
and maybe even beyond Saturn. In summary, Ceres can be any object from the
black, red or even blue region in Figure 2.

Figure 2: Initial distribution for planetesimals of 1 000 km size in our
simulations. The planetesimal are represented by black, red and blue circles.
The text in green colors report the molecular species that are expected to
survive on the surface of the planetesimals in these regions. Adapted by Brown
et al. (2011).

As initial conditions, we assume that in the black and red regions there are 5
000 planetesimals each, in the blue region there are 10 000 planetesimals.
Notice that these numbers are not related to the actual number of Ceres-size
objects expected in the disk. A large number of objects is considered to have
reliable statistical results on the probability of implantation in the
asteroid belt. The resulting probabilities are then mutiplied by the number of
Ceres-size objects provided by the size distribution of the original
planetesimal disk of Nesvorný & Vokrouhlický (2016). The latter was determined
from constraints on the Kuiper belt structure and apply to the planetesimal
disk in the 20-30 au region after the formation of Uranus and Neptune. We
assume here for simplicity that the same size distribution applies in the
10-20 au region and in the 30-40 au region namely in each region of Fig. 2. In
particular in this work we do not just check that the implantation of a Ceres-
size object from the trans-Saturnian disk into the asteroid belt is possible,
but we also wish to verify that the implantation of 1 object is consistent
with the number of Ceres-size objects expected in the trans-Saturnian disks.
We acknowledge that the latter is quite uncertain, given that we extrapolate
the Nesvorný & Vokrouhlický (2016) results to the whole disk. We reiterate
that the planetesimals are non-interacting particles. The eccentricities and
inclinations of the planetesimals are initially chosen to be $10^{-3}$. Their
argument of pericenter and longitude of ascending node are randomly
initialized between 0∘ and 360∘.

###  2.4 Ceres Candidates Runs

We monitor the orbital evolution of each planetesimal and record the orbital
state of planetesimals that are scattered inward (reaching temporarily
$2<a<3.5$ au). We call these planetesimals Ceres candidates and determine the
fraction of planetesimals they represent from each source reservoir (see
Section 3.1). Then, from the analysis of the dynamical evolution of Ceres
candidates, we study the mechanisms (potential scattering off one of the
invaders, resonances with the giant planets and gas drag) that may insert
these planetesimals in stable orbits close to the current Ceres’ orbit. Many
works have shown that most of the Solar System is chaotic (Laskar, 1990; Lecar
et al., 2001; Batygin et al., 2015). The influence of the invaders make the
inner Solar System even more chaotic. Therefore, we make clones of the Ceres
candidates to measure the probability that the temporary capture is turned
into a permanent capture by some irreversible process (e.g. gas drag, GDF or
the encounter with an invader). To obtain the implantation statistics, we
multiply the fraction of planetesimals that become Ceres candidates by the
fraction of clones that are permanently trapped in the inner Solar System. The
cloned bodies were given a small change in the original velocity of the parent
body (fractionally changed by $10^{-6}$), where a similar procedure was used
in Brasser & Morbidelli (2013).

##  3 Results

In this section we present results of a simulation from which we assess the
number and the dynamical state of planetesimals that become Ceres candidates.
The initial setup of our numerical simulations and the prescribed interactions
are described in Section 2.

###  3.1 Fate of Planetesimals

Figures 3 and 4 show snapshots of the simulation where planetesimals were
initialized in the black, red, and blue regions. The sculpting process of the
planetesimal disk is detailed in de Sousa Ribeiro et al. (2020). During the
dynamical evolution of the planetary embryos, the planetesimal population is
depleted by close encounters and collisions with protoplanets or either
Jupiter or Saturn.

Figure 3: Panels (a) - (f) show eccentricity/semimajor axis snapshots of the
dynamical evolution of the planetesimals. The color of a planetesimal is
associated to its initial location in the black, red, or blue disk regions.
The color of the protoplanets is associated to the color scale representing
the mass. Jupiter and Saturn are in black. The vertical lines are 3:1, 5:2 and
2:1 MMR with Jupiter.

A large fraction of the planetesimal population persists in the outer region
of the Solar System (a $>$ 20 au). Only a small fraction of the planetesimal
population is scattered inward by close encounters with the planets and
reaches the inner Solar System. The orbits of some of these planetesimals have
a semimajor axis within the asteroid-belt region but eccentricities that are
large enough to bring them close to Jupiter’s trajectory. Therefore, they are
not on permanent stable orbits (see Section 3.2 to see the mechanisms to get
permanent stable orbits).

A higher fraction of the planetesimal population starting in the black region
becomes Ceres candidates, compared to the planetesimal populations starting in
either of the two other reservoirs (See Table 1). These fractions are 11.11%,
4.26% and 0.28% for the black, red, and blue populations respectively. The
planetesimals in the red and blue regions need to reach the black region via
close encounters with the embryos, before having a chance to be sent into the
inner Solar System by encounters with Saturn and Jupiter. This explains their
smaller implantation probability compared to the objects native from the black
portion of the disk.

Figure 4:  Same as Fig. 3, but illustrating the evolution in semi major axis vs. inclination.  Table 1:  | Black region | Red region | Blue region
---|---|---|---
$N_{total}$ | $5,000$ | $5,000$ | $10,000$
$N_{temp}$ | 587 | 213 | 28
$C_{EM}$ | $0.111$ | $0.0426$ | $0.0028$
Table 1: Statistics of planetesimals that temporarily evolved into the inner Solar System ($a<3.5$ au). $N_{total}$ is the number of initial of planetesimals for each region. $N_{temp}$ and $C_{EM}$ are the number and fraction ($\frac{N_{temp}}{N_{total}}$) of planetesimals that became Ceres candidates respectively. Table 3:  | Stable orbits
---|---
$N_{eff}$ | 294,087
$N_{ab}$ | 3012
$N_{inner}$ | 800
$N_{central}$ | 897
$N_{outer}$ | 1315
Table 2: Statistics of planetesimals implanted in orbits in the asteroid belt. $N_{eff}$ is the total effective number of planetesimals (number of Ceres candidates times the number of clones for each candidate) represented in our simulations. $N_{ab}$ is the number of planetesimals implanted in the asteroid belt region (2.1 $<$ a $<$ 3.25 au, e $<$ 0.4 and i $<$ 40 degrees). We used a broad asteroid belt region because some high-e or high-i objects can then be pushed within the current asteroid belt boundary during the subsequent dynamical instability of the giant planets (Deienno et al., 2016). $N_{inner}$, $N_{central}$ and $N_{outer}$ are the number of planetesimals implanted in the inner asteroid belt region ($2.1<a<2.5$ au), central asteroid belt region between the 3:1 MMR and 5:2 MMR ($2.5<a<2.82$ au) and outer main belt region ($2.82<a<3.25$ au) between the 5:2 MMR and 2:1 MMR, respectively. Table 5:  | Black region | Red region | Blue region
---|---|---|---
Asteroid belt region | $1.20\times 10^{-3}$ | $4.36\times 10^{-4}$ | $2.87\times 10^{-5}$
Inner Main belt region | $3.19\times 10^{-4}$ | $1.16\times 10^{-4}$ | $7.62\times 10^{-6}$
Central Asteroid belt region | $3.58\times 10^{-4}$ | $1.30\times 10^{-4}$ | $8.54\times 10^{-6}$
Outer Asteroid belt region | $5.25\times 10^{-4}$ | $1.90\times 10^{-4}$ | $1.25\times 10^{-5}$
Table 3: The capture efficiency in the asteroid belt per Ceres candidate. Here, the asteroid belt region is the region with 2.1 $<$ a $<$ 3.25 au, e $<$ 0.4 and i $<$ 40 degrees. The inner, central and outer asteroid belt regions are respectively the regions with $2.1<a<2.5$ au, $2.5<a<2.82$ au and $2.82<a<3.25$ au, e $<$ 0.4 and i $<$ 40 degrees. Table 7:  | Black region | Red region | Blue region
---|---|---|---
Asteroid belt region | 4-17 | 1-6 | 0.10-0.40
Inner Main belt region | 1-4 | 0.41-1 | 0.02-0.10
Central Asteroid belt region | 1-5 | 0.46-2 | 0.03-0.11
Outer Asteroid belt region | 2-7 | 0.68-2 | 0.04-0.17
Table 4: Probability that a Ceres-size planetesimal is implanted in the
asteroid belt in our scenario. When the probability is larger than 1, more
than one implanted object is expected. Figure 5: Panels (a) and (b) show, for
the planetesimals started in the black region, the orbital elements achieved
when they are first recorded as Ceres candidates. The color represents the
time that each planetesimal subsequently spent as a Ceres candidate. The
yellow and blue boxes show respectively the eccentricity and inclination
evolution of a single Ceres candidate. The black and gray curve represent the
boundary in (a,e) space above which a planetesimal crosses Jupiter’s orbit and
crosses four different invaders’ orbits, respectively.

Figure 5 shows the osculating elements (semimajor axis, eccentricity, and
inclination) of the planetesimals when they first become Ceres candidates.
Note that there is a pile up of objects near 3.5 au. This is mostly due to a
sampling bias, given that the bodies are plotted when they first enter in the
Ceres candidate region, whose outer boundary is 3.5 au. Figure 5 (a) shows
that most planetesimals are in/close to the scattering region of Jupiter,
where the black curve represents the boundary in (a,e) space above which a
planetesimal crosses Jupiter’s orbit. Gray curves represent the boundary in
(a,e) space above which a planetesimal crosses four different invaders’
orbits. Many of the planetesimals spend less than 500 thousand years as a
Ceres candidate. However, there are a few exceptional planetesimals that spend
1 Myr or more with a semimajor axis in the asteroid belt region. Therefore,
there are two populations of Ceres candidates: those which remain in the
scattering regime (and are therefore short-lived) and those who manage to
decouple from Jupiter due to resonance capture, encounters with invaders, or
due to gas drag.

Figure 5 (b) shows that the inclinations of Ceres candidates have a very
excited distribution, with some planetesimals even presenting a retrograde
orbit (inclinations larger than 90∘).

We did not find any correlation between the orbits trapped in mean motion
resonances (MMR) with Jupiter and their initial location in the trans-
Saturnian disk. This is a consequence of the chaotic evolution of the system.
For the same reason, the distribution of the osculating elements of the
planetesimals from the red and blue regions of the disk are very similar to
the those of the planetesimals originating from the black region, although
their number is much smaller.

###  3.2 Dynamical mechanisms for the implantation of Ceres-sized planetesimal

We explore a scenario to implant a Ceres-sized planetesimals stably near the
real Ceres’ orbit from the outer region of the Solar System. A scheme of the
scenario is presented in Figure 6. The scenario starts with a fast radial
mixing in the position of the planetesimals in the outer planetesimal disk
(Fig. 6 (a)). The radial mixing is caused by a fast scattering process of the
protoplanets in the first thousand years (Fig. 3 (b)). Once close to Saturn
(even if originally coming from the outer disk as the blue planetesimal on
which Fig. 6 (b) focuses on) a planetesimal can be injected into the inner
Solar System by encounters with the giant planets.

When the scattered planetesimals become Ceres candidates their orbital
eccentricities are typically too large and their orbits are not dynamically
stable because they cross the orbits of the giant planets. However, three
mechanisms can drop the eccentricities of the planetesimals, allowing them to
avoid further close encounters with Jupiter: (i) mean motion resonances with
Jupiter coupled with the Kozai resonance, (ii) close encounters with some
invaders, and (iii) the effect of the gas drag. We discuss each of these
processes below.

(i) Once a planetesimal is scattered into a mean motion resonance, it is
phase-protected from further close-encounters with Jupiter for some time,
where secular effects can drop the planetesimal’s eccentricity. The Kozai
resonance within a mean motion resonances is a particularly strong secular
effect that can drop the eccentricities of planetesimals with initially
moderate to large inclinations (Raymond & Nesvorny, 2020).

(ii) When a planetesimal encounters an invader with a random phase, the
planetesimal’s eccentricity can either increase or decrease. If the
eccentricity decreases the planetesimal can reach an orbit that does not come
close to Jupiter at aphelion, and which can become stable when the invader is
removed.

(iii) Gas drag and gaseous dynamical friction are a functions of the density
of the gaseous disk, the orbit and the size of the planetesimal. In our
numerical simulations, the GDF is too weak to drop the eccentricities of these
objects. However, the aerodynamic gas drag can be sufficient to drop the
eccentricities and the inclinations of the planetesimals particularly once
they become protected from planetary encounters (e.g. in a resonance with
Jupiter or after the removal of the invaders).

These three mechanisms do not necessarily act disjointly. We have seen in
Figure 7 that some planetesimals are placed in mean motion resonances with
Jupiter where their dynamical lifetime is enhanced, giving more time to
mechanisms (ii) and (iii) to act, in addition to secular effects. Thus, the
second phase of our scenario is the capturing of planetesimals in mean motion
resonance with Jupiter.

The mechanisms (i)-(iii) discussed above can chaotically diffuse the orbits of
Ceres-like candidates. Thus, chaotic diffusion is the last part of the
scenario (Fig. 6 (d)) to place a Ceres candidate onto an orbit close enough to
the current orbit of Ceres. Obviously, once the gas is removed from the
protoplanetary disk, all invaders are removed and Ceres-candidates are
extracted from the original MMR, mechanisms (i)-(iii) stop and the capture on
a Ceres-like orbit becomes permanent.

To demonstrate the effectiveness of chaotic diffusion of the orbits of the
Ceres’ candidates, we performed simulations cloning each Ceres candidate 500
times. Figure 7 shows the evolution of the semimajor axis, eccentricity and
inclination of Jupiter, Saturn (red and blue lines respectively), of the
protoplanets (purple lines), of a Ceres candidate (black line), and of 50 of
its clones (pink lines). As a result of the chaotic diffusion the orbital
evolution of the clones is different from that of the original Ceres candidate
placing some of them in orbits that reach a stable state in the end. In green
color, we represent a planetesimal that reaches an orbit very close to the
current orbit of Ceres (black star point in Figure 7).

In order to make a statistical analysis of the efficiency of capture of
planetesimals in the asteroid belt region, we focus on the black planetesimals
(planetesimals that started in the region between 10 to 20 au) because they
have more objects becoming Ceres candidates (about 587 planetesimals), but
they are a representative also of the dynamical evolution for the red and blue
planetesimals once the latter start to be scattered by the giant planets. The
cloning process gives us a total number of 294,087 planetesimals, which is
equivalent to having initially a total of 2.5 million planetesimals in the
black portion of the disk for the purpose of statistics. Figures 8 (a) and (b)
show the eccentricity, inclination and semimajor axis of the giant planets and
the cloned planetesimals (pink points), at the end of the gas disk phase. Note
that many of them will not be stable on a long timescale because their
eccentricities and inclinations exceed the boundaries of our extended asteroid
belt (e = 0.4, i = 40∘). Note that implanted planetesimals cover the whole
asteroid belt. Some planetesimals can reach orbits close to the current orbits
of Ceres.

The cumulative normalized distribution of the semimajor axes, eccentricities
and inclinations of the cloned planetesimals at the end of our simulations are
shown in Panels (a), (b) and (c) by red curves in Figure 9. Note that 40% of
the cloned planetesimals that survived untill the end of the simulation, have
a semimajor axis between 2.5 and 3 au, 25% have an eccentricity smaller than
0.2 and about 45% have an orbital inclination smaller than 10∘. If we consider
only planetesimals captured in the region with 2.0 $<$ a $<$ 3 au, e $<$ 0.4
and I $<$ 40∘, the cumulative normalized distribution of a, e and I are shown
in Panels (a), (b) and (c) by the black curves. In this case, note that 70% of
the planetesimals captured in this region have semimajor axis between 2.5 and
3 au, 33% have an eccentricity smaller than 0.2 and 45% have an inclination
orbit smaller than 10∘.

Figure 6: Sketch of the different steps of the scenario envisioned to implant
a Ceres-sized planetesimals in the asteroid belt (see text to more details).
Figure 7: Panels (a), (b) and (c) show the evolution of the semimajor axis,
eccentricity and inclination of Jupiter (red line), Saturn (blue line),
protoplanets (purple lines), the original implanted planetesimal (black line)
and of its clones (pink and green lines). The green line represents a clone
planetesimal that reaches an orbit very close to the current orbit of Ceres
(black point). The error bars denote the range of eccentricity and inclination
that Ceres spans over long timescales (Vaillant et al., 2019). Figure 8:
Panels (a) and (b) show the eccentricity, inclination and semimajor axis of
the cloned planetesimals at the end of the gas phase. The error bars denote
the range of eccentricity and inclination that Ceres spans over long
timescales (Vaillant et al., 2019). Figure 9: Panels (a), (b) and (c) show the
cumulative distribution of semimajor axis, eccentricity and inclination of the
implanted planetesimals. The red color and black color stand for the
cumulative distribution at the end of the gas phase for the inner Solar System
region (a $<$ 3 au) and for the region with semimajor axis between 2 and 3.0
au, eccentricity smaller than 0.4 and inclination smaller than 40∘,
respectively.

###  3.3 Capture Results in the asteroid belt region

We evaluate which fraction of Ceres candidates are permanently stored in the
asteroid belt in our simulations. Similarly to Vokrouhlický et al. (2016), we
use the mass of the disk and size distribution proposed in Nesvorný &
Vokrouhlický (2016) to estimate the probability of a Ceres permanent capture
at the end of the gas disk phase.

The capture efficiency ($C_{E}$), or probability, per planetesimal is
calculated using:

$C_{E}=r_{region}\times C_{EM},$ (7)

where $r_{region}$ is the ratio between the number of planetesimals implanted
in stable orbits in a particular region of the asteroid belt and the total
number of cloned planetesimals while $C_{EM}$ is the fraction of disk
planetesimals that became Ceres candidate and have been cloned.

Of the total number of 294,087 clones, we found 3012 bodies captured in stable
orbits in the asteroid belt between 2.1 and 3.25 au and eccentricity smaller
than 0.4. Thus, the capture efficiency in the asteroid belt per Ceres
candidate is $\frac{3012}{294,087}$. We assume that this probability is the
same whatever the original provenance in the trans-Saturnian disk. Because the
probability that a planetesimal starting from the black portion of the disk
becomes a Ceres candidate is $1.11\times 10^{-1}$, the probability of
implantation in the asteroid belt from this portion of the disk is
$C_{E}=\frac{3012}{294,087}\times 1.11\times 10^{-1}=1.20\times 10^{-3}$.
Repeating the calculation for planetesimals in the red and blue portions of
the disk (see Table 1 for the corresponding values of $C_{EM}$) we find
$4.36\times 10^{-4}$ and $2.87\times 10^{-5}$ respectively. The probability of
capture in the inner asteroid belt (a $<$ 2.5 au) is 8/30 of the total.

Assuming that there were 3600–14,000 Ceres-sized bodies in each of the three
considered portions of the protoplanetary disk, and using the capture
efficiencies for each region, the probabilities to implant a Ceres-like
planetesimals in the asteroid belt are: 4-17, 1-6, 0.10-0.40 from the black,
red and blue portions of the disk respectively. A probability larger than 1
means that more than one captured object is expected. How these probabilities
are partitioned between the inner, middle and outer parts of the belt is shown
in Table 4.

##  4 Discussion and Conclusions

The results of our numerical simulations support the idea that some Ceres-
sized planetesimals might have been implanted from the trans-Saturnian region
into the asteroid belt by a combination of close encounters with giant
planets, with other protoplanets, gas drag, chaotic diffusion and mean motion
resonances with Jupiter. In this section we discuss how our result fits within
a broader context. We describe how the scenario proposed here can fit a
narrative proposed for the asteroid belt evolution: during the gas phase
(Section 4.1), after the gas phase (Section 4.2) and the total mass implanted
in the asteroid belt (Section 4.3). We conclude with a discussion of the
limitations of our work in Section 4.4.

###  4.1 Asteroid belt evolution during the gas phase

Ceres and the other planetesimals were probably formed during the first five
million years after the formation of the first solids in the inner Solar
System (CAIs (Bouvier & Wadhwa, 2010)). The quick formation of planetesimals
can be explained by the radial pile-up of dust, reaching a sufficient
concentration to trigger the streaming instability (Youdin & Goodman, 2005;
Johansen & Lambrechts, 2017; Squire & Hopkins, 2018). The streaming
instability is expected to form planetesimals with most of the mass
concentrated in large objects (hundreds of km in size) and, in addition, the
largest planetesimals can continue to grow by accreting other planetesimals
(Kokubo & Ida, 2000) or radially drifting dust (Lambrechts & Johansen, 2012).

The first phase of the Asteroid Belt evolution happens during the gaseous disk
phase. Jupiter and Saturn formed close to the asteroid belt. While the S-type
asteroids probably formed in-situ, the C-type planetesimals were implanted
from the Jupiter-Saturn region by the combined effects of giant planet growth,
orbital destabilization, gravitational scattering and aerodynamic gas drag
(Raymond & Izidoro, 2017a; Raymond & Nesvorny, 2020).

While considering the Raymond et al. scenario for the origin of C-type
asteroids, we propose another possible scenario for the implantation of Ceres
in the asteroid belt. We remind the reader that Ceres is not C-type and is
rich in ammonia compounds that are typically not found in C-type asteroids and
carbonaceous chondrites. This motivates us to look for an alternative
mechanism capable of implanting Ceres from farther out in the disk than the
original source region of C-type asteroids.

In our model, the implantation of a Ceres-size planetesimal started at the
time when Jupiter and Saturn became giant planets and a system of protoplanets
of about 5 $M_{\oplus}$ formed beyond Saturn orbit. The migration of the
protoplanets blocked by Jupiter and Saturn produced by mutual collisions a set
of ice giant planets with Neptune-like masses ($m\sim 15M_{\oplus}$) (Izidoro
et al., 2015a). In this work, we show that 11% of the total population of
Ceres-size objects initially between 10 and 20 au is temporarily trapped in
the asteroid belt as a Ceres candidate, due to the combined scattering from
the protoplanets and the giant planets. A smaller fraction (0.28%) is
implanted from 30 to 50 au region. This fraction is not zero, thanks to the
partial radial mixing of planetesimals in the trans-Saturnian disk due to
close encounters with the migrating protoplanets.

In order to improve statistics and calculate the probability that a Ceres-like
planetesimal is permanently implanted into the asteroid belt region, we clone
500 times all the Ceres candidates. We show that the temporary population of
Ceres-size planetesimals can reach stable orbits by three different
mechanisms: capture into mean motion resonances with Jupiter, scattering off
protoplanets temporarily crossing the inner solar system and gas drag. Some
permanently implanted Ceres-size planetesimals can be found close to the
current orbit of Ceres.

We find that a fraction of about 1% (3012 planetesimals) of the Ceres
candidates survives and reaches the asteroid belt region defined by
$2.1<a<3.25$ au and eccentricity smaller than 0.4 (Table 2). We also determine
the capture efficiencies separately for each region of the asteroid belt and
for each natal region in the trans-Saturnian planetesimal disk (black, red,
and blue regions; Table 3). The capture efficiency in the asteroid belt region
per disk of planetesimals varies from $2.85\times 10^{-5}$ to $1.20\times
10^{-3}$ for the outer and inner parts of the trans-Saturnian disk,
respectively.

We use the mass of the planetesimal disk and its size distribution proposed in
Nesvorný & Vokrouhlický (2016) to estimate the probability that Ceres was
captured by this process. We assume that the primordial disk beyond Saturn
contained $3600-14000$ Ceres-sized planetesimals (Nesvorný & Vokrouhlický,
2016). Combining these numbers with our implantation probabilities, (see Table
4), we find that about 4-17 Ceres-size objects could have been implanted in
the asteroid belt if they started in the inner region of the disk. Instead,
the probability to implant a Ceres-sized objects from the outer region of the
disk (a $>$30 au) drops to 10-40% (in addition it is unlikely that the disk
beyond 30 au contained thousands of Ceres-size objects given that the cold
Kuiper belt contains none). Therefore, we argue that Ceres is more likely to
come from the inner region of the trans-Saturnian disk, where the number of
Ceres-sized objects implanted in the asteroid belt can be greater than 1. If
we look at our normalized and cumulative orbital distribution of the implanted
planetesimals, we find that 33% of the implanted planetesimals have
eccentricity smaller than 0.2 and about 45% of them have a probability of
having an inclination smaller than 10 degrees. We also found that 70% of the
implanted planetesimals have a semimajor axis between 2.5 and 3 au.

Looking at how many Ceres-sized planetesimals are implanted in other regions
that do not have a Ceres-like object today, for instance the inner and outer
asteroid belt region, we find 1-6 and 2-10 Ceres-like objects, respectively
from red and black regions. Therefore, in our scenario, it is more probable to
have Ceres-like planetesimals in the outer asteroid belt region than in the
central asteroid belt region. We discuss a possible solution of this
discrepancy in Sect. 4.2.

###  4.2 Asteroid belt evolution after the gas phase

Now we turn our discussion to the asteroid belt’s dynamical evolution after
the disappearance of the gas disk. During this period, some planetesimals may
be implanted from the terrestrial planet region scattered by the population of
Mars-mass protoplanets which formed the terrestrial planets (Raymond et al.,
2006; Morbidelli et al., 2012; Raymond & Izidoro, 2017b). However, some
planetesimals may have been implanted from the trans-Neptunian region during
the giant planet instability (Levison et al., 2009; Vokrouhlický et al.,
2016). The planetesimals implanted from the trans-Neptunian region during the
instability phase are usually identified with P/D type asteroids, because
these are the types that dominate in the Jupiter Trojan population, which is
expected to have been implanted at the same time from the same source region
(Nesvorný et al., 2013).

The probability of capturing Ceres from the trans-Neptunian region during the
giant planet instability was calculated by Vokrouhlický et al. (2016). They
also considered that the primordial disk beyond Neptune between 23 and 30 au
may have had 3600 - 14000 Ceres-sized planetesimals. However, their
computations showed that the capture probability in this phase is only
$3\times 10^{-7}$ per disk planetesimal, so that the probability to capture a
Ceres-like object is only is 0.1 to 0.4%. They concluded that this probability
is too low to support the capture of Ceres from the trans-Neptunian region
during the giant planet instability phase. This makes sense, because capture
typically requires sufficient gas drag after the transport, which is
nonexistent once the primordial gas dissipates.

The giant planet instability can also partially deplete the asteroid belt,
including the pre-implanted objects. The depletion of the asteroid belt
depends on the asteroid belt initial orbital distribution and on the specific
evolution of the planets. Roig & Nesvorný (2015) and Nesvorný et al. (2017)
studied the asteroid belt evolution during the giant planet instability.
Nesvorný et al. (2017) found that about 20% of the original asteroids
(starting at 2 to 3.5 au) would survive in the main belt after the giant
planet instability (i.e. and 80% depletion). Clement et al. (2019) tested the
possibility that the giant planet orbital instability depleted more
substantially the main belt, leading to its small current mass and orbital
structure. In some cases Clement et al. (2019) found depletion factors for the
main belt larger than 99%. Therefore, the depletion factor of the asteroid
belt during the instability is still very uncertain.

The number of Ceres-sized planetesimals implanted in our scenario (Table 4) is
consistent with depletion factors during the giant planet instability of
75-94%. If larger than 94%, our model does not expect that an implanted Ceres-
size planetesimal survives in the asteroid belt today. If less than 75%, it
predicts too many Ceres-sized planetesimals surviving in the asteroid belt.
Thus, our scenario is consistent with the depletion factors found in Roig &
Nesvorný (2015) but it is not consistent with the extreme depletion factor
found in Clement et al. (2019).

Notice that the dynamical depletion of the asteroid belt during the giant
planet instability is typically stronger in the outer belt than in the central
belt. Thus, our result that the trapping probability in the outer belt is the
highest may not be in contradiction with the observation that the only
surviving Ceres-like body is in the central belt.

###  4.3 Total mass implanted in the Asteroid Belt

Because there is gas in our simulations, the implantation efficiency is
probably higher for smaller planetesimals, due to the effect of gas drag.
However, assuming for simplicity that that the implantation efficiency is size
independent and assuming that the trans-Saturnian disk comprised at least 30
Earth masses (this is the mass needed in the trans-Neptunian disk in the Nice
model), we estimate that 3–12$\times 10^{-3}$ Earth masses of trans-Saturnian
planetesimals should have been implanted into the asteroid belt. This is much
more than the total mass of the asteroid belt.

The problem arises from the fact that Ceres comprises $1/3$ of the total mass
of the asteroid belt, which is much larger than the fraction of the disk mass
in Ceres-like planetesimals according to the size-frequency distribution (SFD)
considered by Nesvorný & Vokrouhlický (2016) and observed today in the hot
Kuiper belt population. Thus, a model that implants in the asteroid belt the
correct number of Ceres-like planetesimls necessarily implants a too much in
total mass, whereas a model that implants an acceptable total mass necessarily
implants an insufficient number of Ceres-like planetesimals.

The simplest possible solution to this conundrum seems to be that Ceres was
derived from a region of the disk that had a top-heavy SFD, with a Ceres-
mass/total-mass ratio more similar to that of the asteroid belt than to that
of the Kuiper belt. We speculate that this SFD characterized the planetesimals
in the vicinity of Saturn, whereas the SFD of Nesvorný & Vokrouhlický (2016)
characterized the more distant disk which, as we have seen, has a very small
probability to implant objects in the asteroid belt.

###  4.4 Limitations of our work

As with any numerical study, our simulations are a simplified and idealized
version of reality. One main limitation is that our results are based on a
single simulation from Izidoro et al. (2015a). Nevertheless, this simulation
represents, to our knowledge, the most successful one in reproducing the
origin of Uranus and Neptune as well as of a third ice-giant planet, required
in the most recent versions of the Nice model. However, the implantation of
objects into the Asteroid Belt also depends on the scattering process caused
by the protoplanets. The scattering process, therefore, depends on how long
the protoplanets stay in the disk, the number of them (in particular of the
invaders) and the migration of the whole chain. Thus, other formation
histories of the ice giant planets, even within the broad Izidoro et al.
(2015a) model, could possibly lead to different results in terms of efficiency
of implantation of objects in the main belt. In particular, we did not
consider a case where no protoplanet is temporarily scattered onto an orbit
interior to Jupiter’s (invader), because the production of invaders is a
characteristic feature of the Izidoro et al. (2015a) model. Another factor
that could be important is the density of the gas that could change the role
of gas drag and hence the probabilities to capture objects in the inner Solar
System. To be consistent with the planetary evolution of Izidoro et al.
(2015a), we used only one profile for the gas density. The total number of
Ceres-like bodies we assumed in the trans-Saturnian disk is inspired by the
work of Nesvorný & Vokrouhlický (2016), which is valid only for the trans-
Neptunian region at the end of the planet formation phase, so it also suffers
from large uncertainties. Finally, we did not consider the terrestrial planets
formation in our simulations, which is reasonable, in first approximation.
However, there might have been protoplanets forming in the inner Solar System
during the period of time we considered. These local protoplanets might also
have affected the implantation probability and the survival in the Asteroid
belt region. Another caveat in our study is the fact that we have only one
real Ceres in the asteroid belt, therefore it is subject to small number
statistics. Ceres could have followed a special track in the dynamical
evolution of the Solar System. We are considering here that if a model gives
less than 4.6% of probability (which in statistics is like 2.0 $\sigma$ for a
normal probability distribution), we must look for another model that explains
the existence of the object with reasonable probability.

## Acknowledgments

We thank the anonymous referee for the valuable comments andsuggestions that
helped improve the quality of this paper. This work was possible thanks to the
scholarship granted from the Brazilian Federal Agency for Support and
Evaluation of Graduate Education (CAPES), in the scope of the Program CAPES-
PrInt, process number 88887.310463/2018-00, Mobility number
88887.572647/2020-00 88887.468205/2019-00. This research was supported in part
by the São Paulo Research Foundation (FAPESP) through the computational
resources provided by the Center for Scientific Computing (NCC/GridUNESP) of
the São Paulo State University (UNESP). Ribeiro acknowledges support provided
by grants No. 2016/24561-0, São Paulo Research Foundation (FAPESP). A. Izidoro
acknowledges NASA grant 80NSSC18K0828 (to Rajdeep Dasgputa) for financial
support during preparation and submission of the work. A. Izidoro acknowledges
support from the Welch Foundation grant No. C-2035-20200401. A. Izidoro thanks
also support from CNPq via process 313998/2018-3 and CAPES/PrInt/UNESP during
initial development of this project.

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