11institutetext:  1 Leiden Observatory, Leiden University, PO Box 9513, 2300
RA, Leiden, The Netherlands
2 Department of Physics and Astronomy, University of California, Los Angeles,
CA 90095, USA

# Oort cloud Ecology II: The chronology of the formation of the Oort cloud

Simon Portegies Zwart 11 Santiago Torres 1122 Maxwell X. Cai and Anthony G. A.
Brown 1111

Jan Hendrik Oort hypothesized the existence of a distant cloud of cometary
objects that orbit the Sun based on a spike in the reciprocal orbital
separation at $1/a\leavevmode\nobreak\ \
{\raise-2.15277pt\hbox{$\buildrel<\over{\sim}$}}\ \leavevmode\nobreak\
10^{-4}$ au-1. The Oort cloud is the source of long-period comets, but has not
been observed directly, and its origin remains theoretical. Theories on its
origin evoke a sequence of events that have been tested individually but never
as a consistent chronology.

We present a chronology of the formation and early evolution of the Oort
cloud, and test the sequence of events by simulating the formation process in
subsequent amalgamated steps. These simulations start with the Solar System
being born with planets and asteroids in a stellar cluster orbiting the
Galactic center. Upon ejection from its birth environment, we continue to
follow the evolution of the Solar System while it navigates the Galaxy as an
isolated planetary system.

We conclude that the range in semi-major axis between $\sim 100$ au and
several $\sim 10^{3}$ au still bears the signatures of the Sun being born in a
$\ {\raise-2.15277pt\hbox{$\buildrel>\over{\sim}$}}\ 1000$ ${M}_{\odot}$/pc3
star cluster, and that most of the outer Oort cloud formed after the Solar
System was ejected. The ejection of the Solar System, we argue, happened
between $\sim 20$ Myr and $50$ Myr after its birth.

Trailing and leading trails of asteroids and comets along the Sun’s orbit in
the Galactic potential are the by-product of the formation of the Oort cloud.
These arms are composed of material that became unbound from the Solar System
when the Oort cloud formed.

Today, the bulk of the material in the Oort cloud ($\sim 70$%) originates from
the region in the circumstellar disk that was located between $\sim 15$ au and
$\sim 35$ au, near the current location of the ice giants and the Centaur
family of asteroids. According to our simulations, this population is
eradicated if the ice-giant planets are born in orbital resonance. Planet
migration or chaotic orbital reorganization occurring while the Solar System
is still a cluster member is, according to our model, inconsistent with the
presence of the Oort cloud. About half the inner Oort cloud, between $100$ and
$10^{4}$ au, and a quarter of the material in the outer Oort cloud, $\
{\raise-2.15277pt\hbox{$\buildrel>\over{\sim}$}}\ 10^{4}$ au, could be non-
native to the Solar System but was captured from free-floating debris in the
cluster or from the circumstellar disk of other stars in the birth cluster.
Characterizing this population will help us to reconstruct the history of the
Solar System.

##  1 Introduction

The formation of the Oort cloud (Oort, 1950) requires a sequence of events on
temporal and spatial scales that span more than eight orders of magnitude,
from the Solar System (on scales of years and au) to the entire Galaxy (on
scales of Gyr and kpc). As a consequence, the formation of the Oort cloud was
not a simple happening; Occam’s razor does not seem to apply here.

Individual events that led to the formation of the Oort cloud have in the past
been modeled separately to explain specific features (see e.g., Hayashi et
al., 1985; Dones et al., 2004a; Higuchi et al., 2007; Lykawka & Mukai, 2008;
Leto et al., 2008; Paulech et al., 2010; Rickman, 2014; Dones et al., 2015;
Fouchard et al., 2018). However, the chain of events has never been tested as
a causal sequence. We present the results of computer simulations designed to
model the chronology of the formation of the Oort cloud in which individual
processes are included at their proper scales. Each phase is simulated
precisely and is connected to the subsequent phase of the cascade. By doing
so, we construct a consistent picture of the formation and early evolution of
the Oort cloud.

In our analysis, we assume that the Solar System formed, like most stars (Lada
& Lada, 2003; Adams, 2010), in a giant molecular cloud in which gas contracts
under its own gravity, and stars form with disks around them (Beckwith et al.,
1990; McKee & Ostriker, 2007; Gavagnin et al., 2017). The Solar System then
probably formed in a cluster of stars which interacted mutually before the Sun
escaped the cluster (Portegies Zwart et al., 2009). The circumstellar disk, a
leftover from the star-formation process, led to the coagulation of planets
(Kokubo & Ida, 1998, 2002; Kenyon & Bromley, 2006; Wyatt, 2008; Levison et
al., 2010b; Williams & Cieza, 2011; Emsenhuber et al., 2020a, b; Schlecker et
al., 2020) and a large number of planetesimals (Johansen et al., 2007;
Johansen & Lambrechts, 2017; Popovas et al., 2018; Johansen et al.,
2021)111For lack of better terminology, we use asteroids to indicate
planetesimals or comets. A glossary of terms is available in paper I
(Portegies Zwart, 2021), but see also (Gladman et al., 2008).. The presence of
nearby stars in the parent cluster will have affected the morphology of the
gaseous disk through tidal perturbations (Clarke & Pringle, 1993; Pfalzner et
al., 2005; Winter et al., 2018; Cuello et al., 2019), photo-evaporation
(Johnstone et al., 1998; Adams et al., 2004; Clarke, 2007), and stellar winds
(Offner & Arce, 2015).

The evolution of the circumstellar disk has profound consequences for the
newly formed planetary orbits (Laughlin & Adams, 1998; Breslau et al., 2014;
Vincke et al., 2015; Vincke & Pfalzner, 2016; Concha-Ramírez et al., 2021),
although it is currently difficult to quantify these effects. The formation of
the Oort cloud, however, seems somewhat problematic in this picture (but see
Levison et al., 2010a; Vokrouhlický et al., 2019, for a counter argument). We
argue that the majority of the Oort cloud formed after the Sun escaped the
cluster because perturbations of nearby stars would easily ionize an earlier
Oort cloud (see also Higuchi & Kokubo, 2015).

In the present paper, we present a study of the formation of the Oort cloud
from a numerical perspective. The chain of events that lead to the Oort cloud
can now be simulated in its entirety, although not fully self-consistently. We
revisit this problem because we feel slightly overwhelmed by the enormity of
the literature and the changing paradigm over the years. We hope to contribute
to a clearer picture of the processes that appear to be relevant to the
formation and evolution of the Oort cloud.

We perform a numerical investigation and show that the Oort cloud was not
formed in any straightforward manner, but resulted from a complex interplay
during the Sun’s infancy between the Sun itself and the neighboring stars,
free-floating debris, the Galactic tidal field, and planetary scattering. Each
of these components turn out to be important, although their relative
contributions are sometimes hard to quantify. The lack of a simple formation
mechanism, and a conspiracy of processes instead, make the Oort cloud unique.
A complicated story requires a complicated numerical approach that covers many
time and size scales. Apart from stellar evolution in the early star-cluster
phase, the physics is relatively simple. We deal with the fundamentally
chaotic nature of the underlying physics (see Miller, 1964; Goodman et al.,
1993) by repeating calculations with a different random seed. In this
discussion, we limit ourselves mainly to Newtonian physics (Newton, 1687). We
address each component of this calculation separately but knit the results
together to from a homogeneous narrative on the formation and evolution of the
Oort cloud as part of the Solar System.

The timescale of Oort-cloud formation is probably closely connected to the
birth environment of the Solar System (Brasser et al., 2012). If born in a
star cluster, as argued by Portegies Zwart (2009); Adams (2010); Parker
(2020); Pfalzner & Vincke (2020), with a characteristic size of $\sim 1$ pc
and with $\sim 2500$ siblings (Portegies Zwart, 2019), asteroids in wide and
highly elliptic orbits are vulnerable to being stripped from the Solar System
by the cluster potential or by passing stars (see also Nordlander et al.,
2017). Pfalzner & Vincke (2020) on the other hand, derive an even higher
density cluster with a density of up to $\sim 10^{5}$ ${M}_{\odot}$/pc3. If
formed too early in the lifetime of the Solar System, the outer parts of the
Oort would be lost due to stripping when a small change in relative velocity
$\delta v\equiv dv/v\ {\raise-2.15277pt\hbox{$\buildrel>\over{\sim}$}}\ {\cal
O}(10^{-4})$ is induced upon the asteroids (see Fig. 1). So long as the Sun is
a cluster member, asteroids with an eccentricity of $e\
{\raise-2.15277pt\hbox{$\buildrel>\over{\sim}$}}\ 0.98$ with a semi-major axis
of $a\ {\raise-2.15277pt\hbox{$\buildrel>\over{\sim}$}}\ 2400$ au ($a(1-e)\
{\raise-2.15277pt\hbox{$\buildrel>\over{\sim}$}}\ 50$ au) are easily lost.

In the following section, we discuss our current understanding of the outer
Solar System. We then describe the numerical setup and its ingredients in
Sect. 3. In Sect. 4.1 we discuss the simulation results, and describe a
holistic view of the evolution of the Sun and proto-Oort cloud in the Galaxy
in Sect. 4.2.1. The consequences of resonant planetary orbits are discussed in
Sect. 4.2.2. These arguments are supported by simulations presented in Table
6. To further explore the consequences of alternative models on the formation
of the Solar System, we include calculations adopting planetary orbits in
resonance. We conclude in Sect. 5.

##  2 Current understanding of the outer Solar System

We summarize our current understanding of the remote parts of the Solar
System, starting with the Kuiper-belt region, and subsequently move on to the
outer parts.

Four regimes in the outer parts of the Solar System are important for this
discussion, including: the dynamical class of trans-Neptunian objects (TNOs),
the parking zone222The parking zone is the region where asteroids (or dwarf
planets) in orbit around the Sun are not affected by the giant planets, and
are also hardly affected by the Galactic tidal field or an occasional
encounter with a field star (Portegies Zwart & Jílková, 2015; Vargya &
Sanderson, 2020). It was also called the inert zone by Saillenfest (2020) and
the forbidden region by Correa-Otto & Calandra (2019), the Hills cloud, and
the Oort cloud (for a review see Malhotra, 2019). The inner three regions
could have been populated and affected by encountering stars before the Sun
escaped the cluster. In that case, the presence and orbital distribution of
asteroids in the parking zone and Hills cloud could provide interesting
constraints on the dynamical evolution of the Sun in its birth cluster before
it was ejected (see also Moore et al., 2020).

###  2.1 The trans-Neptunian region and Kuiper belt

The phase-space structure in the Edgeworth-Kuiper belt is complex (Edgeworth,
1943; Kuiper, 1951). Close to the perturbing influence of Neptune (the red
curve in figure 1 marks the outer boundary of the conveyor belt333Area in
semi-major axis and eccentricity where an asteroid crosses the orbit of one or
more of the major planets, causing their orbits to drift to higher
eccentricity and larger semi-major axis while preserving pericenter distance.;
see the dotted line in figure 6) we find multiple families of Kuiper-belt
objects. The most striking might be the scattered disk (Brown, 2001; Luu &
Jewitt, 2002; Trujillo et al., 2000; Nesvorný, 2020), but there are other
exotic orbital characteristics observed in this region, including the warp (at
an inclination of $i=1^{\circ}.8^{+0^{\circ}.70}_{-0^{\circ}.4}$; Volk &
Malhotra, 2017), the mix of resonant families (Chiang et al., 2003), the broad
distribution in eccentricity and inclination of the Plutinos (Malhotra, 1993;
Brown, 2001), the orbital topology of the classical belt population (Elliot et
al., 2005), the outer edge at the 1:2 mean-motion resonance with Neptune
(Gomes et al., 2004), and the extent of the scattered disk (Gladman et al.,
2001). Apart from perhaps the population of almost circular objects in the
classical Kuiper belt (Tegler & Romanishin, 2000), many of these
characteristics can be explained at least qualitatively with some incarnation
of the Nice model (Gomes et al., 2005; Morbidelli et al., 2005; Tsiganis et
al., 2005; Levison et al., 2008, see § 2.5 for more on the Nice model). Some
of these features in the Kuiper belt originated $\
{\raise-2.15277pt\hbox{$\buildrel>\over{\sim}$}}\ 50$ Myr after the planets
formed (Nesvorný, 2020).

Here we demonstrate that it is possible to reconstruct part of the history of
the Solar System from the kinematics and phase-space distribution of the
orbiting bodies (see also Moore et al., 2020). Although we consider the
Edgeworth-Kuiper belt to be crucial to our understanding of the formation and
early evolution of the Solar System, the present paper focuses on the Oort
cloud and its formation.

###  2.2 Sedna, other trans-Neptunian objects in the parking zone, and the
Hills cloud

To date, several asteroids have been observed beyond the Kuiper cliff (at
$\sim 48$ au), between a few hundred and a few thousand astronomical units
(au) with a pericenter distance of $\
{\raise-2.15277pt\hbox{$\buildrel>\over{\sim}$}}\ 50$ au (Sheppard et al.,
2019; Alexandersen et al., 2019). These include the dwarf planet 90377 Sedna
(Brown et al., 2004), 2012 VP113 (Sheppard et al., 2014), and a dozen others.
These objects can be divided into two clusters, one population with their
argument of pericenter of $\sim 310^{\circ}$ (Trujillo & Sheppard, 2014), and
another at the opposite side (Sheppard et al., 2019). This small number of
known detached trans-Neptunian objects can, in part, be attributed to
observational selection effects because they are far (Alexandersen et al.,
2019), red (Brucker et al., 2009), and tend to have a relatively small albedo
(Rabinowitz et al., 2006).

Also from the outer regions of the Solar System, the area in phase space where
Sedna and family are found is hard to reach, meaning that the asteroids in
this region cannot have come from either direction, the inner Solar System or
the periphery. This unreachable area in parameter space is called the parking
zone (Jílková et al., 2015), and is identified as such in figure 1. Asteroids
in this region keep their orbital parameters for a very long time, possibly
even as long as the age of the Solar System. We note that, if the asteroids in
the parking zone cannot have come from the inner Solar System, and they do not
have an origin from the periphery, they either formed in situ or were injected
there by other means. The several observed dwarf planets and asteroids (such
as Sedna) in this region preserve evidence for the dynamical history of the
Solar System (Kaib & Quinn, 2008; Jílková et al., 2015) and can be used to
constrain its origin.

These arguments led to the hypothesis that Sedna and family were captured from
another star in the parent cluster (Jílková et al., 2015). In this case, Sedna
and family would have been captured from the circumstellar debris disk around
another star. The population of asteroids in the parking zone may then form
the leftover evidence for such a captured population (Nurmi et al., 2002;
Morbidelli & Levison, 2004; Shankman et al., 2011; Napier et al., 2021a).

Such a capture then must have happened more than $4$ Gyr ago, and in that time
frame the influence of the ice-giant planets would have caused the orbits of
Sedna and family to change, in particular their arguments of pericenter should
have randomized. But their orbits are such that they pass the pericenter at
the same side of the Sun (Trujillo & Sheppard, 2014).

Based on the alignment of the arguments of pericenter of the Sedna family of
objects, Batygin & Brown (2016) and Batygin et al. (2019) argued in favor of
the existence of a planet in the inner Oort cloud. Such a planet in a wide
orbit or possibly even a stellar companion to the Sun could induce orbital
characteristics similar to those observed in Sedna and other detached trans-
Neptunian objects (Nesvorný, 2020), or even affect the orbits of asteroids in
the parking zone (Madigan & McCourt, 2016; Zderic & Madigan, 2020).

The main motivation for an extra planet disappeared, as the alignment of the
arguments of pericenter turn out to be an observational selection effect
(Sheppard et al., 2019; Napier et al., 2021b). Although a binary companion is
currently probably absent (Melott & Bambach, 2010; Hills, 1984; Hut, 1984),
the Sun might have had one in the past (Siraj & Loeb, 2020), particularly
because such wide stellar companions are rather common among other stars (Kaib
et al., 2013).

Further out, we find the hypothetical Hills cloud (Hills, 1981) between the
outer edge of the parking zone ($\
{\raise-2.15277pt\hbox{$\buildrel>\over{\sim}$}}\ 1\,000$ au) and the inner
edge of the Oort cloud ($\ {\raise-2.15277pt\hbox{$\buildrel<\over{\sim}$}}\
20\,000$ au). The origin of the objects in the Hills cloud is unclear, but the
population could be related to trans-Neptunian objects (Fernandez & Ip, 1981;
Dones et al., 2015). Hills (1981) estimated the total mass to exceed that of
the Oort cloud by as much as a factor of 100 (see however Dones et al., 2000,
who argue in favor for comparable populations). The majority of trans-
Neptunian objects have prograde orbits (Kavelaars et al., 2020) and isotropic
distributions in mean anomaly, longitude of the ascending node, and the
argument of perihelion (Bernardinelli et al., 2020). These support an inner
Solar System origin because asteroids scattered from the circumstellar disk
tend to have prograde orbits (Moore et al., 2020), whereas the orbits of
captured asteroids may well be retrograde, depending on the encounter that
introduced them into the Solar System (see also Hanse et al., 2018; Napier et
al., 2021a). The dynamical history of the Solar System in its birth cluster
therefore plays an important role in the formation and orbital topology of
asteroids in the parking zone and the Hills sphere (Parker, 2020). Because of
a lack of perturbing influences, signatures of a captured population in the
parking zone remain noticeable for much longer than in other parts of the
Solar System.

###  2.3 The Oort cloud

Discussions over the Öpik-Oort cloud began in 1932 by Öpik (Öpik, 1932) who
discussed the origin of nearby parabolic orbits in the Solar System, and in
1950 by Oort (Oort, 1950) with the discovery of a spike in the reciprocal
orbital separation $1/a\leavevmode\nobreak\ \
{\raise-2.15277pt\hbox{$\buildrel<\over{\sim}$}}\ \leavevmode\nobreak\
10^{-4}$ au-1 of observed comets. Oort (1950) argued that the long-period
comets originated from a region between 25 000 au and 200 000 au from the Sun.
Today, these estimates have not changed considerably (Correa-Otto & Calandra,
2019).

The outer limit is considered to coincide with the Hill radius (Hill, 1913) of
the Sun in the Galactic potential (Chebotarev, 1965). The origin and precise
location of the inner edge are less clear (Hills, 1981; Leto et al., 2008) and
are still debated (Brasser & Schwamb, 2015). What defines the transition
region between the Hills cloud and the Oort cloud also remains unclear.

The mass of the Oort cloud is estimated to range from $1.9$ M⊕ (Weissman,
1996) to $38$ M⊕ (Weissman, 1983). These estimates seem somewhat on the high
side when compared to those based on numerical simulations, which arrive at
$0.75\pm 0.25$ M⊕ (Brasser, 2008) to $1.0\pm 0.4$M⊕ (Fernández & Brunini,
2000).

With a typical comet-mass of a few times $10^{12}$ to $10^{14}$ kg (Rickman et
al., 1987; Sosa & Fernández, 2009, 2011) the Oort cloud contains ${\cal
O}(10^{12})$ comet-sized objects. Interestingly, this estimate is comparable
to Oort’s original estimate of $\sim 10^{11}$ objects (Oort, 1950), and to the
density of interstellar asteroids (Engelhardt et al., 2017; Portegies Zwart et
al., 2018; ’Oumuamua ISSI Team et al., 2019; Pfalzner et al., 2020).

###  2.4 The Formation and early evolution of the outer Solar System

Estimates of the formation timescale of the Oort cloud range from
instantaneously after the formation of the Sun (Fernández, 1997),
synchronously with Jupiter’s formation (Stevenson & Lunine, 1988; Fernández &
Brunini, 2000; Dones et al., 2004b), to after Jupiter formed and potentially
migrated (Shannon et al., 2019), to slow growth over several 100 Myr (Kaib &
Quinn, 2008; Nordlander et al., 2017), and even to gigayear timescales (Duncan
et al., 1987).

Two main scenarios are popular for the formation of the Oort cloud: One
stipulates that the cloud formed mainly by ejecting inner Solar System
material through planet–disk interactions (Dones et al., 2000, 2004a), and the
other concerns formation through two distinct processes: Local disk asteroids
are ejected into an inner region at $3000$–-$20\,000$ au, while the outer
region at $\ {\raise-2.15277pt\hbox{$\buildrel>\over{\sim}$}}\ 20\,000$ au is
mainly accreted from free-floating debris in the parent star cluster (Zheng et
al., 1990; Valtonen et al., 1992; Brasser et al., 2006, 2007; Brasser, 2008;
Levison et al., 2010a). Planets could also be captured in this way (Perets &
Kouwenhoven, 2012), possibly explaining the origin of a hypothetical exoplanet
in the outer parts of the Solar System (Mustill et al., 2016).

The relatively low mass in some of the numerically derived estimates stem from
the low efficiency, which ranges from $\sim 1.1$ % (Leto et al., 2008;
Morbidelli et al., 2009), $\sim 2$ % (Correa-Otto & Calandra, 2019) to $\sim
4$ % (Brasser et al., 2010), at which point disk material is launched into a
bound Oort cloud (Paulech et al., 2010); the majority of objects are expected
to escape the Solar System or hit the Sun. The giant planets could therefore
be insufficiently efficient to explain the currently anticipated mass of the
Oort cloud (Brasser et al., 2006; Kaib & Quinn, 2008; Brasser et al., 2012).

If the entire Oort cloud originates from the depletion of asteroids between
Uranus and Neptune (see also Fouchard et al., 2013), this region must have
been populated by objects with a total mass of $100$ to $3800$ MEarth
(Fouchard et al., 2014a, b). In a disk with a density profile of $\rho\propto
r^{-1.5}$ between 0.1 au and 45 au, about half the mass is between the orbits
of the ice giants. To supply the Oort cloud with sufficient material, the
proto-planetary disk must then have had a mass of $200$ to $7600$ M⊕ (or
equivalently $\sim 0.001$ to $0.02$ ${M}_{\odot}$) in comet-sized asteroids,
which is consistent with other estimates (Crida, 2009).

Given the low efficiency of the ice-giants to produce the Oort cloud, more
than 90% is likely to have come from elsewhere; such as the circumstellar
disks of other stars (Levison et al., 2010a), free-floating debris in the
parent cluster (Zheng et al., 1990; Neslušan, 2000), the capture of
interstellar objects (Valtonen & Innanen, 1982; Hands & Dehnen, 2020; Pfalzner
et al., 2021), or accretion from the circumstellar disks of other stars
(Fernández & Brunini, 2000; Levison et al., 2010a).

The orbits of captured asteroids could be rather distinct from those produced
by scattered asteroids from the proto-planetary disk (Hands et al., 2019;
Higuchi & Kokubo, 2020). In the Oort cloud, both populations will be affected
by the Galactic tidal field (Duncan et al., 1987; Fouchard et al., 2006b) and
probably mix on a timescale of a few hundred million years. Orbital
inclinations isotropize on this timescale, and eccentricities thermalize
(Fouchard et al., 2006a; Higuchi & Kokubo, 2015).

Once fully developed, the Oort cloud erodes by injecting comets into the inner
Solar System through Kaib-Quinn jumpers (Kaib & Quinn, 2008) or by a more
gradual process (sometimes referred to as creepers; Fouchard et al., 2014a,
2018), but also because of encounters with molecular clouds (Duncan et al.,
2011) and tidal interaction with the Galaxy (Heisler & Tremaine, 1986; Levison
et al., 2001; Gardner et al., 2011; Torres et al., 2019). These external
perturbations may be the main reason why comets are launched from the Oort
cloud into the inner Solar System. This process was suggested to lead to
periodic showers of comets (Davis et al., 1984; Gardner et al., 2011) causing
mass extinctions every $\sim 29.7$ Myr (Raup & Sepkoski, 1984; Rampino &
Prokoph, 2020). Such a wide companion was never found (Luhman, 2014), and re-
analysis of the data shows that there is no statistically significant evidence
for a periodicity in these mass extinctions (Patterson & Smith, 1987; Jetsu &
Pelt, 2000; Bailer-Jones, 2009).

###  2.5 The Nice model

Many of the model simulations above depend on some sort of chaotic
reorganization in the early Solar System. This could have happened during or
shortly after the formation of Jupiter (Li et al., 2006) up to about half a
gigayear after the planets formed (Neslušan et al., 2009; Leto et al., 2009).
This chaotic reorganization was introduced to explain the enhanced flux of
asteroids throughout the young Solar System (Morbidelli et al., 2005; Tsiganis
et al., 2005; Gomes et al., 2005), which could have lead to a peak in the flux
of lunar impactors (late-heavy bombardment is a commonly used term; Hartmann,
1965, 1966; Alvarez & Muller, 1984; Stöffler & Ryder, 2001; Neukum et al.,
2001; Ryder, 2002).

Follow-up analyses indicated that the impact period could be between 4.2 and
3.5 Gyr ago (Fritz et al., 2014; Zellner, 2017; Bottke & Norman, 2017), rather
than being a peak, rendering the need for a disturbance in the force in the
early Solar System unnecessary. Re-analysis of the lunar impact statistics by
Lineweaver (2010) indicated that a high flux of impactors is not needed to
explain the lunar cratering record. This study was further supported by the
re-analysis of the 267 Apollo samples on 40Ar/39Ar isotope ratios, where it
was demonstrated that the material on the moon arrived in a continuously
decreasing flux rather than a peak (Boehnke & Harrison, 2016, 2018).

The Nice model is successful in explaining several aspects of the Solar
System, including the orbital topology of the giant planets (Morbidelli et
al., 2007), the obliquity of Uranus (Wong et al., 2019), Trojan asteroids
(Emery et al., 2015), irregular moons (Bottke et al., 2010), some morphologies
of the Kuiper belt (Morbidelli & Nesvorný, 2020), and several other curious
orbital choreographies (Levison et al., 2008; Shannon et al., 2019). Although,
the Nice model seems to fail in explaining the global structure of the Oort
cloud (Fouchard et al. 2018, but see also Shannon et al. 2019, for competing
arguments based on the rocky comet C/2014 S3 (PANSTARRS)), it might be hard to
find a single alternative model that can explain so many phenomena in the
Solar System. One aspect of the Nice model that may be important for
explaining the phenomena of the Solar System is a period in which the orbits
tend to change chaotically (Thommes et al., 2008). Such a chaotic phase can be
initiated by planetary resonance —as in the Nice model— induced by a passing
star or by the last phase of evaporation of the circumstellar disk.

Several independently developed explanations exist, including models based on
internal dynamical processes (Duncan et al., 1995; Ida et al., 2000),
encounters with other stars (Ida et al., 2000; Kobayashi et al., 2005; Torres
et al., 2019), or a small molecular cloudlet in a wide orbit around the Sun
(Emel’yanenko, 2020). However, there is no global model that explains the
Solar System in unison.

This early phase in the evolution of the Solar System appears far from being
understood from first principles, and more research on this topic would help
to understand the apparent stability and inertness of the current Solar
System.

##  3 Methods, models, and simulations

This section explains the numerical methods, the initial conditions, and
simulations that we used to qualify and quantify the formation and early
evolution of the Oort cloud. Our calculations are not self-consistent because
the results were not produced in a single simulation, but the results are
melded together to form a consistent understanding.

Scattered and captured asteroids (see Sect. 3.2)
---
Designation |  Simulation A.
Simulations |  Two simulations of the Solar System with planets, and scattered and captured asteroids in the Milky way potential.
Star |  Single 1 ${M}_{\odot}$ star in the smooth potential of the Milky Way galaxy.
Planets |  Four planets per star in a circular disk, with Jupiter, Saturn, Uranus, and Neptune in circular orbits with semi-major axes of $5.1$, $9.5$, $19.2$, and $30.1$ au, respectively.
Asteroids |  One simulation with 2000 asteroids (from planetary system # 157 of sect. 3.3.2 (see also Torres et al., 2020a). One simulation with 1888 asteroids (from the most probable encounter, which, according to Jílková et al. (2015), resulted in a population of Sedna-like objects.
Numerics |  Huayno (Pelupessy et al., 2012) coupled with the Galactic model (see Table 4) via bridge (Portegies Zwart et al., 2020) in AMUSE (Portegies Zwart & McMillan, 2018).
Computer |  run on Cartesius, and ALICE using GPU.
Duration |  The simulations started at an age of $\sim 100$ Myr. By that time, the Solar System has left the parent cluster, and continues for $1$ Gyr.
Table 1: Simulations of the Solar System with captured and scattered asteroids
in the Galactic potential. Star cluster simulation (see Sect. 3.3)
---
Designation |  Simulation B.
Simulations |  Two-hundred simulations with 200 asteroids per star, and 24 simulations with 2000 asteroids per star in a cluster where stars interact with the asteroids.
Stars |  Two-thousand stars with a Kroupa (2001) mass function between 0.08 and 100 ${M}_{\odot}$in R=1pc virialized Plummer sphere (Plummer, 1911). No primordial binaries, no residual gas.
Planets |  Four planets per star with circular orbits. Each star has the same planetary systems in a plane, but each planetary system has a different random orientation in space. In one model (extended), the planets Jupiter, Saturn, Uranus, and Neptune have initial semi-major axes $5.1$, $9.5$, $19.2$, and $30.1$ au, respectively. In the other model (compact), Jupiter, Saturn, Neptune, and Uranus have initial semi-major axes of $5.5$, $8.1$, $11.5$, and $14.2$ au.
Asteroids |  In a disk between 40 au and 1000 au in the extended model and between 16 au and 400 au in the compact model.
Numerics |  NBODY6++GPU (Wang et al., 2015) and REBOUND using IAS15 integrator (Rein & Spiegel, 2015) coupled via the LonelyPlanets scheme (Cai et al., 2019) in AMUSE (Portegies Zwart & McMillan, 2018).
Computer |  Run on a GPU-equipped Little-Green-Machine-II (LGM-II) computer.
Duration |  The simulations start with all stars on the zero-age main sequence with four planets in the adopted initial orbits. The calculation is continued for 100 Myr.
Table 2: Simulations of the Solar System in its birth star cluster using the
LonelyPlanets approach (see sect. 3.3.1). The resulting orbital distribution
of asteroids from the Sun-representative star is used in this work to further
explore the evolution of the distribution of asteroids in the Oort cloud.
These simulations do not lead to a reliable representation of abducted
asteroids because the lonely planet approach does not allow exchange
interactions.

We start with a discussion on the software framework, the Astrophysics
Multipurpose Software Environment (AMUSE; see Sect. 3.1), which was used for
the simulations presented here. We discuss each of the following processes:

  * A

The evolution of the circumstellar disk of an isolated Solar System (see Sect.
3.2 and Table 1).

The calculation includes the four giant planets orbiting the Sun together with
a disk of mass-less particles. The entire setting is integrated in the
potential of the Galaxy, starting with the approximate position in the Galaxy
where the Sun was $1$ Gyr ago, and lasts for 1 Gyr.

  * B

The evolution of the Solar System in its birth cluster (Sect. 3.3 and Table
2).

The arguments here are based on two quite distinct calculations: first a
simulation of 2000 stars in a virialized cluster with a half-mass radius of
$\sim 1$ pc. We study the evolution of debris disks under the influence of the
encounters with neighboring stars. In a second calculation, we study the
exchange efficiency and orbital characteristics of the circumstellar disks
that were perturbed in encounters between two stars. These two simulations are
used to understand the orbital parameters of various populations of asteroids:

    1. 1.

The captured asteroids and their migration towards the Oort cloud (see Sect.
3.3.2),

    2. 2.

and the scattered-disk asteroids along the conveyor belt (see Sect. 3.3.3).

  * C

Transport of asteroids along the conveyor belt and eccentricity damping by the
tidal field once the Oort cloud is reached (see Sect. 3.4 and Table 3).

The formation of the Oort cloud is supported by simulations of the population
of asteroids in the conveyor-belt region. These calculations include the four
giant planets (with various orbital configurations) and the Galaxy’s tidal
field. Both are necessary to study the transport of asteroids from the
planetary region to the Oort cloud. The eccentricity and semi-major axis of
the asteroids increases when evolving along the conveyor belt. This process is
driven by the major planets. Once the pericenter of the orbits of the asteroid
detaches from the major planets, the Galactic tidal field dampens the
eccentricity of the orbits.

figure 1 presents a schematic overview of the evolution of the Oort cloud.
Asteroids near Jupiter and Saturn are vulnerable to being ejected on a
relatively short timescale ($\
{\raise-2.15277pt\hbox{$\buildrel<\over{\sim}$}}\ 10$ Myr), whereas asteroids
originally formed between Uranus and Neptune, and the scattered and captured
populations (green arrows), tend to reach the Oort cloud on a much longer
timescale of $\sim 100$ Myr. All asteroids that eventually reach the Oort
cloud pass through the narrow neck at an eccentricity of $\
{\raise-2.15277pt\hbox{$\buildrel>\over{\sim}$}}\ 0.998$ at a semi-major axis
of around $10^{4}$ au, after which they are subjected to eccentricity damping
by interacting with the Galactic tidal field (green arrows to the right in
figure 1). figure 2 presents a superposition of simulation data $\sim 100$ Myr
after the Solar System escaped its birth star cluster. This data originates
from multiple calculations that simulate various aspects of the formation of
the Oort cloud.

Figure 1: Orbital migration of asteroids and how they end-up in the Oort
cloud. Several of the thin curves are also plotted in figure 2. The four major
planets are indicated with their current semi-major axis and eccentricity as
blue dots. The initial circumstellar disk is presented in five colors,
depending on the final destination of the asteroids populating the different
disk sections. The arrows indicate the movement of asteroids originating from
the disk or captured from other stars. The timescales presented near the
arrows give an estimate of the timescale of migration. The colors indicated in
the legend give the original inner disk (magenta), which mainly migrates away
from resonant orbits (Delbo’ et al., 2017) (four important resonances are
indicated with thin vertical dotted lines). Ogre indicates those asteroids
that are ejected from the Solar System on a relatively short timescale ($\
{\raise-2.15277pt\hbox{$\buildrel<\over{\sim}$}}\ 10$ Myr). Light blue
indicates the kernel and the resonant Kuiper belt. The light green and dark
green curves show the migration patterns of the asteroids that eventually
reach the Oort cloud. These objects migrate from a semi-major axis of a few
tens of au to beyond $10^{4}$ au through a narrow neck of high eccentricity.
The asteroids in the dark green area require external support from a stellar
encounter to be able to migrate to the Oort cloud. This migration is more
visible in figure 2, where the eccentricity (y-axis) is in logarithmic units.
We note that the closer an asteroid is to the Hill radius (vertical red dotted
line to the right), the quicker its orbit will be circularized by the Galactic
tidal field (see also figure 7, where this is illustrated). The gray, pink,
and brown curves indicate where the relative velocity kick imparted at
apocenter by the Galactic tidal field to an asteroid is $\delta v=10^{-8}$,
$10^{-5}$, and $10^{-3}$ of the orbital velocity, respectively. The purple
dashed curve indicates the orbital separation and eccentricity for which the
tidal eccentricity damping timescale is equal to the orbital period (see
Duncan et al., 1987).  Figure 2: Phase-space distribution of asteroids around
the Sun $\sim 100$ Myr after escape from its parent cluster. The shaded region
presents a kernel-density estimation of the simulation results, representing
various families of objects. We adopted a non-parametric Gaussian kernel
density estimator with a symmetric bandwidth of $0.02$ (Scott, 1992). At this
moment, part of the Oort cloud is already in place (to the right), but the
formation process is still ongoing. The colored diagonal curves (from top left
to bottom right) indicate orbits that cross those of the giant planets (see
also figure 1). The dashed burgundy colored curve to the right indicates where
the orbital period, $P_{\rm orb}$, is equal to the eccentricity damping-
diffusion timescale by the Galactic tidal field ($t_{\rm diff}$, Eq.5 of
Duncan et al., 1987). The solid black curves indicate the Galaxy’s perturbing
influence in terms of the velocity-kick imparted to an object. Relative
velocity perturbations of $\delta v=10^{-3}$ (right), $10^{-5}$, and $10^{-8}$
(left) are indicated. The little area to the right of Neptune’s influence
(between the red diagonal curve and the black curve indicating a perturbation
$\delta v=10^{-8}$) corresponds to the Kuiper-belt kernel distribution. The
parking zone is indicated in red. The Oort cloud is to the right of the
rightmost solid black curve (labeled as $\delta v=10^{-3}$) and the burgundy
colored dashed curve. The Hills cloud is to the left of this dividing line
(indicated in black). In our simulations, the Hills cloud at low eccentricity
($e\ {\raise-2.15277pt\hbox{$\buildrel<\over{\sim}$}}\ 0.95$) is mostly empty,
but at higher eccentricity (near the bottom of the figure) its population is
substantial, in particular along the conveyor belt. The captured and scattered
asteroids are indicated in white contours. Locally, at the extreme, both
populations have comparable phase-space density. At the age of 100 Myr, a
considerable fraction of the native disk population has already reached the
Oort cloud, or is on its way there through the conveyor belt. Some captured
asteroids are currently migrating along the conveyor belt and a few have
already reached the Oort cloud. However, the majority of the scattered and
captured asteroids are in the parking zone between $\sim 100$ au and $\sim
1000$ au, where they will stay for the duration of the simulation. Jupiter and
Saturn eject asteroids along the conveyor belt into escaping orbits (indicated
in orange). Uranus and Neptune eject asteroids on a timescale considerably
longer ($\ {\raise-2.15277pt\hbox{$\buildrel>\over{\sim}$}}\ 100$ Myr) than
that for Jupiter and Saturn ($\
{\raise-2.15277pt\hbox{$\buildrel<\over{\sim}$}}\ 10$ Myr), allowing these
asteroids to be circularized by the Galactic tidal field (also in orange).
This is also visible in the lower kernel density along the Jupiter–Saturn
conveyor belt in comparison with the Uranus–Neptune conveyor belt. Eventually,
the latter asteroids become members of the Oort cloud. The red dotted curve
indicates the Hill radius of the Sun in orbit around the Galactic center, here
at about 0.65 pc. The thin dotted diagonal curves indicate pericenter
distances of 1 au and 1 R⊙. Comets from the Oort cloud may enter the inner
Solar System (to the far bottom right and below the 1 au curve).

###  3.1 The Astrophysics Multipurpose Software Environment

The calculations in this study are performed using the Astrophysics
Multipurpose Software Environment (Portegies Zwart & McMillan, 2018). AMUSE is
a modular language-independent framework for homogeneously interconnecting a
wide variety of astrophysical simulation codes. It is built on public
community codes that solve gravitational dynamics, hydrodynamics, stellar
evolution, and radiative transport using scripts that do not require
recompilation. The framework adopts Noah’s Arc philosophy, meaning that it has
incorporated at least two codes that solve the same physics (Portegies Zwart
et al., 2009).

Most calculations are carried out using a combination of symplectic, direct,
N-body test-particle integration and the tidal field of the Galaxy. For the
former two, we use Huayno, which adopts the recursive Hamiltonian splitting
strategy (much like bridge; see Fujii et al., 2007; Portegies Zwart et al.,
2020) to generate a symplectic integrator that conserves momentum to machine
precision (down to $10^{-14}$ in normalized units; Pelupessy et al., 2012).
Additional calculations were performed using the ABIE code (Cai, 2018),
NBODY6++GPU (Wang et al., 2015), and REBOUND (Rein & Liu, 2012) using the
IAS15 scheme (Rein & Spiegel, 2015). These methods are combined using the
AMUSE framework in the LonelyPlanets approach (Cai et al., 2019, see also
Sect. 3.3.1).

Where appropriate, we include stellar mass loss in our calculations using the
SeBa binary population synthesis package (Portegies Zwart & Verbunt, 1996;
Toonen et al., 2016). For the Galaxy, we take a slowly varying potential of
the bar, bulge, spiral arms, disk, and halo into account. The parameters for
the Galaxy model are listed in Table 4 (see also Martínez-Barbosa et al.,
2015, 2016, 2017).

Simulation of the migration of asteroids from the conveyor belt to the Oort
cloud (see Sect. 3.4).
---
Designation |  Simulation C.
simulations |  Nine simulations with various initial realizations, but geared toward populating the Oort cloud from the conveyor belt.
Planets |  Four planets to each star in a circular disk, with Jupiter, Saturn, Uranus, and Neptune in orbits with semi-major axes of $5.1$, $9.5$, $19.2$, and $30.1$ au, respectively.
Asteroids |  $10^{4}$ asteroids per run distributed in the ecliptic plane in the conveyor belt on eccentric orbits of between 0.1 and 0.9 and with a pericenter distance of between 5.1 au and 30.1 au (see also Duncan et al., 1987). Four simulations with the ecliptic plane in the Galactic plane and five simulations with an inclination of $60^{\circ}$ to the Galactic plane.
Numerics |  Huayno (Pelupessy et al., 2012) coupled with the Galactic model (see Table 4) via bridge (Portegies Zwart et al., 2020) in AMUSE (Portegies Zwart & McMillan, 2018). Seven simulations were conducted with standard precision of the Huayno integrator, and two simulations at twice the precision (by reducing the time-step parameter $\eta$ from 0.02 to 0.01).
Computer |  Run on ALICE using a GPU.
Duration |  Start at an age of $\sim 100$ Myr (once the Solar System has left its birth cluster, and continued for 1 Gyr ($\sim 1$ Gyr after the Sun leaves the cluster; see sect. 3.3.4).
Table 3: Simulation of the Solar System with asteroids in the conveyor belt.
These simulations are performed with a populated conveyor belt. The initial
conditions for the asteroids are therefore not the result of earlier
calculations. These conditions were selected to mimic the phase in which
asteroids are launched onto the conveyor belt. As this process may last for up
to some 100 Myr, we shorten this by generating initial conditions when the
asteroids are already in the conveyor-belt region. These simulations are
mainly used to study the eccentricity-damping process of the Galactic tidal
field and the phase-space distribution of asteroids in the Oort cloud. Table
4: Model parameters of the Milky Way.

Axisymmetric component
---
Mass of the bulge ($M_{\mathrm{b}}$) |  $1.4\times 10^{10}$ M⊙ |
Scale length bulge ($b_{\mathrm{1}}$) |  $0.3873$ kpc |
Disk mass ($M_{\mathrm{d}}$) |  $8.56\times 10^{10}$ M⊙ |
Scale length 1 disk ($a_{\mathrm{2}}$) |  $5.31$ kpc | 1)
Scale length 2 disk ($b_{\mathrm{2}}$) |  $0.25$ kpc |
Halo mass ($M_{\mathrm{h}}$) |  $1.07\times 10^{11}$ M⊙ |
Scale length halo ($a_{\mathrm{3}}$) | 12 kpc |
Central Bar
Pattern speed ($\Omega_{\mathrm{bar}}$) |  $40$ km s-1 kpc-1 | 2)
Mass ($M_{\mathrm{bar}}$) |  $9.8\times 10^{9}$ M⊙ | 4)
Semi-major axis ($a$) |  $3.1$ kpc | 5)
Axis ratio ($b/a$) | $0.37$ | 5)
Vertical axis ($c$) | 1 kpc | 6)
Present-day orientation | $20^{\circ}$ | 3)
Spiral arms |
Pattern speed ($\Omega_{\mathrm{sp}}$) |  $20$ km s-1 kpc-1 | 2)
Number of spiral arms ($m$) | $2$ | 7)
Amplitude ($A_{\mathrm{sp}}$) |  $3.9\times 10^{7}$ M⊙ kpc-3 | 4)
Pitch angle ($i$) | $15.5^{\circ}$ | 4)
Scale length ($R_{\mathrm{{\Sigma}}}$) |  $2.6$ kpc | 7)
Scale height ($H$) |  $0.3$ kpc | 7)
Present-day orientation | $20^{\circ}$ | 5)

References: 1) Allen (1973); 2) Gerhard (2011);
3) Romero-Gómez et al. (2011); 4) Jílková et al. (2012);
5) Martínez-Barbosa et al. (2015); 6) Monari et al. (2014);
7) Drimmel (2000); 8) Jurić et al. (2008)

The galactic tidal-field code is coupled to the various N-body codes using the
augmented non-linear response propagation pattern (called rotating bridge in
Martínez-Barbosa et al., 2017). The interaction time-step between the Galactic
tidal field and the Solar System was $100$ yr. This time step is sufficient
because the period of the asteroids in such wide orbits exceeds 0.1 Myr.

Figure 3: Galactic distribution of asteroids 1 Gyr after the Sun escapes its
birth cluster (still $\sim 3.6$ Gyr in the past). The colored background
represents the adopted Galaxy potential: bottom left, top left, and bottom
right views show the various Cartesian coordinates (the bar is not shown). The
red and green dots show the bound and unbound asteroids. The starting position
of the Sun is also indicated. The top-right corner panel shows a magnified
view of 6 pc by 6 pc around the Sun. The outer circle shows the Hill radius at
$\sim 0.65$ pc, and the inner circle represents the inner edge of the Oort
cloud at about $\sim 0.21$ pc (at $\delta v=10^{-3}$).

###  3.2 Evolution of the circumstellar disk of an isolated Solar System

The simulations presented in this section are designed to help us to
understand the consequences of evolution of the newly born Solar System with a
disk of planets and asteroids as part of the Galactic tidal field. The Solar
System is initialized in the Galactic potential at $-8.4$ kpc along the x-axis
and $17$ pc above the Galactic plane. The initial velocity of the Solar System
was $v_{\odot}=(11.352,233.105,7.41)$ km/s, which follows an almost circular
orbit around the Galactic center (see figure 3). From this position and
velocity we calculate the position of the Sun in the Galaxy backwards for $1$
Gyr, which is where our calculations start. The location of the Solar System 1
Gyr ago was $(-4.451,7.796,53.99)$ kpc, with a velocity of
$(204.1,101.3,5.742)$ km/s.

We initialized the Solar System with the four giant planets, Jupiter, Saturn,
Uranus, and Neptune, as they appear in their orbit on 1 January 2020
(2020-01-01T00:00:00.000 UTC), according to the JPL Horizon database.444see
https://ssd.jpl.nasa.gov/?horizons_doc (We note here that the choice of the
initial epoch should not considerably effect either the qualitative or the
quantitative statistical results. However, for the orbits of individual
asteroids, the precise initial realization is important; see Sect. 4.4.2 for a
brief discussion). The ecliptic was initialized with an angle of $60^{\circ}$
to the Galactic plane. The initial conditions for these simulations are
presented in Table 1. A wider range of initial realizations was tried,
including changing the orbital phases, and separations of the planets and the
parameters for the debris disk (see Sect. 4.1). Varying the orbital phases
does not result in qualitative differences, but the results are relatively
sensitive to changes in the initial orbital separations of the planets. After
initializing the planets, we place a disk of test particles in the ecliptic
plane around the Sun. The disk was generated using the routine
ProtoPlanetaryDisk in the AMUSE framework with a Toomre-Q parameter of $25$
(Toomre, 1964). Most simulations are performed with disks of between 16 au and
35 au, but other ranges were explored with a minimum of 3 au and an upper
limit of 1000 au.

In the simulations, Jupiter starts to populate the resonance regions (such as
the asteroids of the Hilda-family; in figure 1 the purple arrows to the bottom
left illustrates how they migrate) and launches asteroids into the conveyor
belt (light green and black curves) directly from the start of the simulation.
This process happens very early after the formation of the Solar System which
may then still be a cluster member. After $1$ Myr, $43.4$% of the asteroids
born between $3$ and $15$ au are still in resonant orbits, and $39.4$ % have
escaped the Solar System (see also Pirani et al., 2018, 2019). Asteroids
initially on wider orbits are not affected momentously within this short time-
frame. figure 4 illustrates this process by plotting the orbit of a test
particle integrated with the Solar System and the tidal field of the Galaxy.

Figure 4: First 2 Myr of the orbital evolution of an asteroid born in a
circular orbit in 4:5 mean-motion resonance with Jupiter. Repeated
interactions with the giant planets (colored circles: blue, orange, green, and
red for Jupiter, Saturn, Uranus, and Neptune, respectively) launch the
asteroid along the conveyor belt into the Oort cloud until it reaches the Hill
sphere at a semi-major axis of $\
{\raise-2.15277pt\hbox{$\buildrel>\over{\sim}$}}\ 14\,000$ au with
eccentricity $\ {\raise-2.15277pt\hbox{$\buildrel>\over{\sim}$}}\ 0.995$. The
inset to the lower left gives the first 0.1 Myr of evolution, and the inset to
the top right gives the first 0.3 Myr. Each frame is on a different scale.

On a longer timescale, based on the simulations listed in Table 3 and Table 6,
the ice-giant planets transport asteroids along the conveyor belt into the
Oort cloud, where they slowly circularize and isotropize. After $\sim 100$
Myr, the orbital distribution of asteroids is similar to that of the present
epoch; the stable resonant regions near the giant planets are populated, and
the non-resonant orbits as well as the conveyor belt are depleted. Only $1.4$
% of the asteroids born between $15$ and $50$ au are still in the disk after
100 Myr and $59.2$% have become unbound from the Solar System. The remainder
of the asteroids born between $1$5 and $50$ au have migrated to the Oort cloud
at this point in time.

The giant planets only scatter the asteroids in inclination by a few degrees
(see also Di Sisto & Rossignoli, 2020). By the time the asteroids reach the
inner Oort cloud, their inclinations are still not much larger than a few tens
of degrees around the ecliptic plane. Even asteroids that are scattered along
the conveyor belt to the Oort cloud preserve a relatively low inclination with
respect to the ecliptic plane (see also Duncan et al., 1987; Correa-Otto &
Calandra, 2019). The inclination distribution only isotropizes once the
Galactic tidal field starts to dominate the orbital evolution of the asteroids
(see also Fouchard et al., 2018). Although the majority of the asteroids
escape the Solar System while being kicked out by the planets, the number that
remain bound is sufficient to explain the richness of the predicted population
of objects in the Oort cloud.

Nevertheless, the schematic view presented above is rather idealized because
we assume that the Solar System was born as a single star orbiting in the
Galactic potential. In the following section, we relax this assumption and
study the consequences of the Sun being born in a young stellar cluster. It
turns out that being born in a clustered environment has profound consequences
for the efficiency of the formation of the Oort cloud.

###  3.3 The evolution of the Solar System in its birth cluster

For the simulations presented in this section, the Solar System is initialized
as a member of a star cluster (see also Torres et al., 2020a; Stock et al.,
2020; Veras et al., 2020). Two series of simulations were performed: one in
which the Sun has four giant planets in a compact configuration, and the other
that shows a more extended configuration (see Table 2). Simulations are
performed with $2000$ asteroids in circular orbits of between 16 au and 35 au
for the compact, and another series with asteroids in circular orbits of
between 40 and 1000 au for the extended configuration (see also Torres et al.,
2020a). An additional set of 32 simulations (not listed in Table 2) was
performed in which the Solar System, with a disk of 1000 asteroids,
experiences a single close encounter at a distance of 225 au or 400 au with a
relative velocity of 1 km/s. Here we varied the impact angle of the
encountering star from $0^{\circ}$ (in the ecliptic plane) to $30^{\circ}$,
$60^{\circ}$, and $90^{\circ}$ (perpendicular to the ecliptic).

In these models, we ignore the inner disk (within $40$ au for the extended
models and within $16$ au for the compact models). This is motivated by the
fragility of the inner Solar System. Any encounter that would perturb the
inner region would probably leave the Solar System unrecognizable today.
Therefore, there seems to be no particular reason to include the inner disk in
the calculations. Each simulation was performed up to 100 Myr. This timescale
is smaller than the cluster lifetime, but is sufficient to support the
conclusions of this paper.

The cluster in those simulations is built using NBODY6++GPU (Wang et al.,
2015), while we use REBOUND (Rein & Liu, 2012) to integrate the planetesimals
(using the IAS15 integration scheme from Rein & Spiegel, 2015). The
simulations are carried out using the LonelyPlanets approach designed in Cai
et al. (2017, 2018, 2019).

####  3.3.1 Simulating planetary systems in a dense star cluster: the
LonelyPlanets approach

In the LonelyPlanets module in AMUSE , we first evolve a star cluster without
planets or asteroids for $100$ Myr using NBODY6++GPU (Wang et al., 2015). This
calculation includes the $N$-body dynamics of the stars, stellar evolution,
and the interaction with the Galactic tidal field (as described above).
Clusters are born instantaneously without residual gas and with stars from a
mass function on the zero-age main sequence distributed in a virialized
Plummer sphere. We store masses, positions, and velocities on the five nearest
neighbors for each star during the calculation at 1000-year time intervals.
Interactions with a single nearest star provided satisfactory statistics for
the strongest encounters (Glaser et al., 2020), but in our opinion lagged in
weaker perturbations. We therefore include the nearest five stars.

In the second pass through the data, each stored encounter is treated as a
scattering experiment lasting 1000 years. After each scattering experiment
with five perturbing single stars and one star with planets, we continue with
the new set of five perturbing stars while carrying the planetary system over
from one scattering experiment to the next, $1000$ years later. This works as
follows. The masses, positions, and velocities of the five perturbing stars
are recovered from file, and the target star with planets and test particles
as asteroids is integrated together with the perturbers (see Table 2). After
$1000$ years of integration, we recover the next set of five perturbers from
file and integrate them as a new experiment together with the target star,
planets, and asteroids. We note that these five nearest neighbors may be
different stars from one snapshot to the next ($1000$ years later). The
presence of planets in the scatter experiment leads to small perturbations of
the encountering stars, which affects their orbits. Therefore, even if a
subset of perturbing stars is identical from one snapshot to the next, the
positions and velocities of these perturbers does not necessarily have to be
identical at the end of a scatter experiment and the beginning of the next
snapshot because the encounter database was built without planets. Our
approach is therefore inconsistent.

The encounter between the five perturbing stars and the one star with planets
and asteroids is calculated using REBOUND (Rein & Liu, 2012). This process is
repeated for each encounter for the duration of the star-cluster simulation
($100$ Myr). figure 5 presents an illustration of this method with the six
interacting stars identified.

In the following two sections, we discuss two populations: the captured and
the scattered populations of asteroids. The amount of mass (or the number of
asteroids) that is transferred is similar to the number of asteroids at the
periphery of the disk, which is unbound or scattered into highly eccentric and
inclined orbits (Jílková et al., 2015).

Figure 5: Illustration of the extended Solar System model (not to scale). The
gray shaded region represents the debris disk with an outer radius of 1000 au.
The dotted red circle indicates the boundary of the neighboring sphere, beyond
which we ignore the influence of perturbations by passing stars. Gray symbols
represent the stars in the cluster that are ignored. The colored star symbols
(P1, P2, P3, P4, and P5) represent the five nearest stars that perturb the
Solar System at a given time. The arrows indicate the direction of motion of
the perturbers (figure from Torres et al., 2020a).

####  3.3.2 The influence of the cluster on the circumstellar disk

When a star is a young cluster member, mutual encounters affect the
circumstellar disks (Vincke et al., 2015; Vincke & Pfalzner, 2016). However,
most simulation studies address the effects of one star on the disk of another
star resulting in the deformation of the disks (Breslau et al., 2014;
Rawiraswattana et al., 2016; Xiang-Gruess, 2016; Breslau et al., 2017). A star
in a cluster is exposed to multiple encounters, each having a subsequent
effect on the disk’s morphology and mass. Only a few studies take these
multiple interactions into account (Jiménez-Torres, 2020; Torres et al.,
2020a).

Moreover, mutual interactions also lead to the transport of material from one
star to another (see also Korycansky & Papaloizou, 1995). We address both
processes separately. Here we describe the scattering of the native disk due
to stellar encounters. In sect. 3.3.3, we discuss asteroids that are abducted
from the disk of another star.

In the calculations with LonelyPlanets, we follow the dynamical evolution of
the debris disk and planets around each star individually (see previous
paragraph). The disks are only resolved in the second pass through the data.
As a result, disks do not interact mutually, but they are affected by multiple
encounters with other stars.

Each star experiences multiple interactions with other stars. We focus on one
particular star in this simulation: the 1 ${M}_{\odot}$ star # 157 from Torres
et al. (2020, their figure 2). We picked this star because the orbital
topology of its scattered population resembles the Kuiper belt of the Solar
System. The initial inner edge of the disk in this calculation is $40$ au.
This seems rather large, but a strong perturbation of the more inner region
would have had considerable repercussions for the entire planetary system,
contrary to what is observed in the Solar System.

In figure 6, we present the orbital distribution of the scattered disk
population (green dots). The small black lines from each dot point in the
direction of the evolution of its orbit over the next 300 Myr. The population
near the inner part of the disk ($a\
{\raise-2.15277pt\hbox{$\buildrel<\over{\sim}$}}\ 50$ au) is affected by the
outer planets, causing an increase in their eccentricity. The population
between $50$ and $500$ au is hardly affected by either the planets or the
other stars, and is hardly affected over the lifetime of the Solar System,
until the Sun evolves beyond the main sequence and sheds its outer layers
(this process was studied extensively by Veras et al., 2011; Veras & Tout,
2012; Veras et al., 2014; Zink et al., 2020).

Once the star and surviving disk leave the parent cluster, $\sim 4.2$ % of its
native outer-disk asteroids have been scattered through interactions with
other stars directly into the conveyor belt region. The majority of these
asteroids move on a timescale of $\sim 100$ Myr along the conveyor belt until
they reach the Oort cloud (once they cross the curve $t_{\rm diff}=P_{\rm
orb}$, or the curve of $\delta v=10^{-3}$). This process is illustrated in
figure 6 with the long lines that start at the asteroids and point to their
orbital parameters 300 Myr later in time (in the direction of the Oort cloud).
We note that in figure 6 the curve for $\delta v=10^{-5}$ is given, which
roughly corresponds to the inner edge of the Hills cloud. The difference
between the $\delta v=10^{-5}$ and $\delta v=10^{-3}$ is also illustrated in
Figs. 1 and 4.

After $\sim 100$ Myr, some asteroids of $\
{\raise-2.15277pt\hbox{$\buildrel>\over{\sim}$}}\ 2.5$ Neptune-Hill radii from
the orbit of Neptune are still in the process of ascending the conveyor belt
(see the thin dotted curve to the right of the red curve in figure 6). This
small population of asteroids within the conveyor belt does not migrate to the
Oort cloud. Di Sisto & Rossignoli (2020) found a similar population of
lingering asteroids, and the migration process in their calculations takes
$\sim 68$ Myr, which is comparable to the $\sim 100$ Myr in the calculations
presented here. Also, Fernández et al. (2004) studied the migration rate of
scattered trans-Neptunian objects to the Oort cloud and argue that eventually,
after 5 Gyr, half the scattered disk settles in the Oort cloud.

####  3.3.3 The captured asteroids

Apart from scattering of the circumstellar disk, stellar encounters also lead
to the capture of asteroids. The majority are deposited in the parking zone,
possibly in highly inclined orbits (Jílková et al., 2016). Those that enter
the conveyor belt migrate towards the Oort cloud or are ejected. The fraction
of captured asteroids that enter the conveyor belt depends on the details of
the encounter, and the disk parameters of the encountered star. Although the
numbers vary strongly depending on the parameters of the encounter, the
typical fraction of captured asteroids is comparable to the fraction of
asteroids lost, at least in an equal-mass encounter (see also Breslau et al.,
2017). Encounters in which the encountering star meets the prograde orbiting
debris disk are most effective in producing unbound asteroids (Pfalzner et
al., 2021). The population of captured asteroids may therefore have a high
proportion of retrograde orbits compared to those ejected from the inner Solar
System.

We included a population of captured asteroids in our analysis. However, the
calculations using the LonelyPlanets approach are not suitable for acquiring a
census of their orbital distributions (Cai et al., 2018) because once an
asteroid becomes unbound from its parent star it is lost from the simulation.
Instead of taking the results of system # 157 from Torres et al. (2020a), as
we did for the scattered disk, we adopt the results of Jílková et al. (2015)
for the captured asteroids. These latter authors performed $N$-body
calculations to study the mutual stellar encounter that could explain the
orbit of the dwarf planet Sedna. They argue that Sedna was captured from the
circumstellar disk of another star. This must have happened while both stars
were members of the same parent cluster. The captured population from Jílková
et al. (2015) reproduces Sedna’s orbit as a captured object.

At present, Sedna is orbiting in the parking zone well outside the conveyor
belt and is not expected to reach the Oort cloud. From a Galactic point of
view, Sedna is relatively close to the Solar System. Asteroids within $\
{\raise-2.15277pt\hbox{$\buildrel<\over{\sim}$}}\ 1000$ au of the parent star
are hardly affected by the tidal field of the Galaxy. The parking zone is
therefore composed of a roughly equal number of scattered and captured
asteroids. The orbits of asteroids in the parking zone are no expected to
affected by either the planets or by the Galactic field. Frozen in time, this
population may bear information about the mechanism that brought it there.

We introduced the captured population into the Solar System and integrated it
as test particles together with the giant planets and the background potential
of the Galaxy over a timescale of 300 Myr555The timescale of 300 Myr was
selected empirically. The migration timescale for highly inclined asteroids
turns out to be about 100 Myr (see Sect. 3.3.2). We therefore had to integrate
for at least this time-span. The calculations, however, are sufficiently
expensive to not continue them for too long.. The simulations are summarized
in Table 1. The resulting orbital parameters are presented in figure 6.

During integration, the captured asteroids that were introduced in the
conveyor belt region have been driven by the giant planets into the Oort cloud
region. The timescale on which they reach the Oort cloud depends on their
relative inclination to the ecliptic, and on the pericenter distance. Highly
inclined asteroids, for example, may have their closest approach to the Sun
far from the perturbing influence of the giant planets. Asteroids orbiting in
the ecliptic plane are more strongly affected by the giant planets, and are
scattered into the conveyor belt on a shorter timescale.

Figure 6: Orbital distribution of the scattered (green dots) and captured
(brown) asteroids. These conditions originate from the scatter and capture
calculations, but form the initial conditions for subsequent calculations. The
former are scattered from the circumstellar debris disk by a close encounter
with another star in the parent cluster. The latter population is captured
from another star with which the Sun interacted in its parent cluster. The
simulations for both populations are described in Table 1. The small thin
black lines from each dot-point point in the direction in which the orbit
migrates over 300 Myr (we adopted 300 Myr here for the presentation, although
we realize that adopting 100 Myr would have been a more systematic choice).
These data were acquired by integrating the Solar System in the tidal field of
the Galaxy. The asteroids that orbit the outer and inner parking zones are
hardly affected by the Galactic tidal field. Those near the conveyor belt are
strongly affected by the giant planets and some eventually reach the Oort
cloud. The thin black dotted curve is parallel to the 2.5 Hill-radii to the
right of Neptune’s pericenter influence (solid red curve), and indicates the
extent to which the planet still affects the orbits of the asteroid.
Simulation parameters are listed in Table 1.

####  3.3.4 When did the Sun escape the parent cluster?

Here, we assume that stars form instantaneously in random positions from a
point-symmetric Plummer (1911) potential in virial equilibrium. Planets also
form instantaneously in almost circular orbits in a randomly oriented ecliptic
plane.

These are strong (and, in our view, rather preposterous) assumptions, and some
of our results are affected by these choices. However, relaxing any of these
assumptions will open up parameter space and lead to a dramatic increase in
the number of calculations required to acquire reliable uncertainties on the
simulation results. One improvement would be to take the hydrodynamical
collapse of the molecular cloud into account, including the processes of star
formation, stellar feedback, disk evolution, and planet formation. However,
these calculations are somewhat elaborate (Calura et al., 2020; Fukushima &
Yajima, 2021).

Even though the processes discussed in this paper are fundamental, the
timescales and efficiencies of the various processes are affected by the
initial conditions of the simulations (Adams et al., 2006). The timescale over
which, and the order in which, the giant planets form are important for the
presented model and the efficiency at which asteroids are injected into the
Oort cloud. The conclusion that asteroids were ejected early on by the giant
planets while the Sun was a cluster member remains largely unaffected. The
Oort cloud will be prevented from forming so long as the Sun is a member of a
star cluster (Higuchi & Kokubo, 2015). Only after the Sun has left the birth
cluster is it possible to keep asteroids bound in orbits as wide as the
current Oort cloud up to the Hill radius in the Galactic potential.

It is not trivial to constrain the moment at which the Sun escaped its birth
cluster. In the first $\ {\raise-2.15277pt\hbox{$\buildrel<\over{\sim}$}}\ 10$
Myr, the cluster is expected to have a relatively high density $\
{\raise-2.15277pt\hbox{$\buildrel>\over{\sim}$}}\ 10^{3}$ ${M}_{\odot}$/pc-3
(Lüghausen et al., 2012; Martínez-Barbosa et al., 2016; Pfalzner & Vincke,
2020; Pfalzner et al., 2020). After that, the cluster expansion is driven by
stellar winds, radiation feedback, and supernovae until it eventually
dissolves in the Galaxy’s tidal field. This latter phase in which the cluster
dynamics is dominated by two-body relaxation and tidal stripping lasts for
approximately 100 Myr (Adams et al., 2006; Stock et al., 2020). While the
cluster expands with time, the local density gradually decreases, and the
encounter rate drops (Baumgardt & Kroupa, 2007; Cai et al., 2017). These time
scales are all short compared to the formation timescale for Jupiter of $\
{\raise-2.15277pt\hbox{$\buildrel<\over{\sim}$}}\ 4$ Myr (Kruijer et al.,
2017; Movshovitz et al., 2010; D’Angelo et al., 2021).

The Solar System may have survived this first dense phase rather unharmed,
leaving the cluster at a later stage, possibly around the cluster’s half-life
timescale. According to our calculations (but also see Kaib & Quinn, 2008), it
seems more difficult to grow a rich Oort cloud if the Sun stays in the parent
cluster for $\ {\raise-2.15277pt\hbox{$\buildrel>\over{\sim}$}}\ 100$ Myr,
because this corresponds to the timescale on which the ice-giant planets eject
their local asteroids into the conveyor belt. We therefore argue that the
Solar System might have been ejected from the parent cluster within $\sim 20$
or $50$ Myr after birth.

The effect of a nearby encounter on the cold Kuiper belt may not have had a
long-lasting impact, as this population might have regrown over time (Astakhov
et al., 2005; Punzo et al., 2014; Moore et al., 2020). Other signatures of the
difference between a few strong encounters and extended exposure to relatively
weak perturbations are hard to quantify. We did not explore the long-term
survival of the Solar System in its birth cluster and its possible
consequences for the outer edge of the planetary disk. It would be interesting
to study this aspect, but these calculations are elaborate, and the parameter
space is extended.

###  3.4 Eccentricity damping of the asteroids in the Oort cloud

While the gas-giant and ice-giant planets launch asteroids further into the
Oort cloud, the Galactic tidal field gradually becomes a stronger perturber.
Eventually, when asteroids cross the purple-dashed curve in figure 6 (far to
the right, but more visible in Figs. 1 and 2) their orbits become strongly
affected by the Galactic tidal field through von Zeipel-Lidov-Kozai resonance
(von Zeipel, 1910; Lidov, 1962; Kozai, 1962); (see Ito & Ohtsuka, 2019, for a
historical overview on the terminology). This leads to damping of the
eccentricity (as is also demonstrated in Vokrouhlický et al., 2019). In our
simulations, this proceeds through a random walk in semi-major axis and
eccentricity but directed towards lower eccentricities. In figure 7 we
illustrate this process with three asteroids from one of our simulations.
These particles start in the narrow neck between 1000 au and 3000 au at an
eccentricity $\ {\raise-2.15277pt\hbox{$\buildrel>\over{\sim}$}}\ 0.998$ while
they are kicked by one of the giant planets along the conveyor belt into the
Oort cloud region. The tidal field of the Galaxy subsequently reduces the
orbital eccentricity of these asteroids. This eccentricity-damping process
proceeds as a random walk in eccentricity and orbital separation. The
influence of the Galactic tidal field near the apocenter of the asteroid may
cause the orbital eccentricity to increase as well as decrease, whereas the
semi-major axis is less strongly affected. The global trend is a reduction in
eccentricity. Asteroids further in the Oort cloud are more strongly affected
by the Galactic tidal field, and therefore have a shorter circularization
timescale. Not all asteroids circularize to the low eccentricities of the
three examples presented figure 7. Eventually, after a few hundred million
years, we find empirically that the cumulative probability density function
for the eccentricity approaches $f(e)\propto(1-e^{2})^{-2/9}$. Figure 8
presents the evolution of the eccentricity distribution of the Oort cloud
(four lower curves). Table 3 presents the conditions for the simulations that
support this section.

Figure 7: Orbital evolution of three asteroids for 100 Myr in semi-major axis
and eccentricity. Each starts from the conveyor belt (to the top left) until
it circularizes. These asteroids are launched into the conveyor belt by a
giant planet. This process is illustrated in figure 4. Once the Galactic tidal
field starts to dominate the orbital evolution of the asteroids (to the right
of the red dashed curve), it detaches them from the influence of the inner
planets. The Galactic tidal field subsequently drives the eccentricity
evolution of the asteroid until it circularizes. The timescale of the
circularization process is illustrated with the color bar. Asteroids further
in the Oort cloud are more strongly affected by the Galactic tidal field, and
therefore have a shorter circularization timescale. Not all asteroids
circularize to the low eccentricities of the three examples presented here.
The eventual eccentricity distribution of the Oort cloud objects is presented
in figure 8. The vertical dotted curve to the right indicates the Hill radius
of the Solar System in the Galactic potential.  Figure 8: Cumulative
distribution of the eccentricity at various stages of the evolution of the
eventual Oort-cloud population of asteroids. The dotted and dashed curves are
presented to guide the eye. The dotted curve gives the probability density
function for the thermal distribution in eccentricity ($f(e)\propto e^{2}$).
The dashed curve gives an eccentricity probability density function
$f(e)\propto(1-e^{2})^{-2/9}$, which matches the eventual Oort cloud
distribution at 1 Gyr. The top four curves give the eccentricity distribution
of the initial disk around the Sun (see Table 5), at the ages of $1$ Myr, $3$
Myr, and $10$ Myr (red curve). The purple curve through the middle of the
figure near the dotted curve gives the distribution of the two populations of
scattered disk objects and the captured asteroids (see also figure 6 and Table
1). Both the scattered and captured populations are comparable in number. The
bottom four curves give the distribution of the Oort cloud asteroids, from
bottom to top immediately after the asteroids have entered the Oort cloud
(indicated with “initial Oort cloud”), followed by the distribution at ages of
100 Myr, 500 Myr, and 1 Gyr (as indicated, see also Table 3). This figure
contains data from simulations for which the parameters are listed in Table 5
(top curves), 1 (middle purple curve), and 3 (bottom set of curves).

Asteroids in the disk start with almost circular orbits (uppermost blue curve
in figure 8). In time, this distribution becomes more skewed to higher
eccentricity, mainly due to the injection of objects into the conveyor belt
(orange curve). The scattered and captured asteroid populations have, upon
capture, a steeper eccentricity distribution (purple curve), which approaches
the thermal distribution (dotted curve).

A fraction of the disk particles together with some of the scattered disk and
captured asteroids migrate further along the conveyor belt until their
eccentricity distribution resembles the lowest (red) curve in figure 8. This
latter distribution continues to evolve over time until it resembles the
purple curve (indicated with the Oort cloud at 1 Gyr). Further evolution of
the Oort cloud is slow, on a timescale of $\
{\raise-2.15277pt\hbox{$\buildrel>\over{\sim}$}}\ 1$ Gyr, and is dominated by
evaporation. By counting the number of objects that re-enter the inner Solar
System, we derive a rate of $\sim 2$–$6\times 10^{-12}$ comets per asteroid in
the Oort cloud per year that find their way back into the inner Solar System
within 10 au (see Sect. 4.2.3). This leads to one or two comet arrivals in the
inner Solar System per year, which is consistent with earlier estimates (Byl,
1986; Heisler & Tremaine, 1986; Gardner et al., 2011; Fouchard et al., 2020).

###  3.5 Summary of the chain of simulations

Now that we have briefly discussed each of the simulations, we can construct a
more holistic view of the formation and early evolution of the Oort cloud.
Here, we summarize this model and in the following section we discuss the
consequences.

Asteroids in the conveyor belt cross the orbit of the giant planets. Each time
this happens, they receive a small kick causing their orbits to drift with
constant pericenter distance to higher eccentricity and larger semi-major axis
(illustrated in figure 4, but see also Duncan et al., 1987). Jupiter ejects
its nearby asteroids along this conveyor belt (indicated in figure 2) on a
timescale of a few million years.

Once the eccentricity of an asteroid is $\
{\raise-2.15277pt\hbox{$\buildrel>\over{\sim}$}}\ 0.998$ and its semi-major
axis is $\ {\raise-2.15277pt\hbox{$\buildrel>\over{\sim}$}}\ 20\,000$ au, the
Galactic tidal field starts to dominate the orbital evolution of the asteroid
at apocenter (Higuchi et al., 2007, see Fig 2). As soon as the pericenter
distance exceeds the semi-major axis of the giant planet, the eccentricity of
the asteroids’ orbit continues to be reduced. By this time, the orbital period
is $\ {\raise-2.15277pt\hbox{$\buildrel>\over{\sim}$}}\ 3$ Myr, and the Galaxy
starts damping the asteroid’s eccentricity and randomizing the inclination
(Higuchi et al., 2007) on a timescale of $\sim 100$ Myr (Duncan et al., 1987;
Di Sisto & Rossignoli, 2020).

Eventually this process leads to the Oort cloud. The distance travelled by an
asteroid into the Oort cloud depends on the last interaction with the planets
before its orbit detaches (see figure 7). Once near apocenter, the Galactic
tidal field reduces the eccentricity of the orbit, causing the pericenter
distance to increase.

Our calculations did not last a sufficient length of time to circularize the
entire population, but the reduction in eccentricity converged in $\sim 1$ Gyr
to $f(e)\propto(1-e^{2})^{-2/9}$. We therefore argue that the Oort cloud still
hosts a considerable fraction of asteroids in relatively high-eccentricity
orbits. This distribution in eccentricity is consistent with the one found in
Higuchi & Kokubo (2015) at an age of $500$ Myr to $1$ Gyr. These latter
authors further argue that the eccentricity distribution in the Oort cloud
thermalizes in $\sim 5$ Gyr. We do not observe such a thermalization of the
eccentricity distribution. The structure of the Oort cloud can then also
inform us about what happened in the early Solar System (see also Fouchard et
al., 2018).

The orbital distribution of the giant planets is not very important in this
evolutionary sequence, except that more massive planets, upon their last
interaction, can inject asteroids in wider orbits with a higher eccentricity.

The large mass and short orbital period of Jupiter clears the local asteroid
field in as little as a few million years (Duncan et al., 1987; Fouchard et
al., 2014b). Therefore, the Oort cloud cannot have formed from asteroids
ejected by Jupiter or Saturn, because their ejection timescale is shorter ($\
{\raise-2.15277pt\hbox{$\buildrel<\over{\sim}$}}\ 10$ Myr) than the expected
cluster lifetime of $\ {\raise-2.15277pt\hbox{$\buildrel>\over{\sim}$}}\ 100$
Myr (Portegies Zwart et al., 2010). In Sect. 3.3.4 we conclude that the Sun
escaped the parent cluster even before that time, that is, in $20-50$ Myr. In
the first million years after their formation, the giant planets launch $\
{\raise-2.15277pt\hbox{$\buildrel>\over{\sim}$}}\ 80.3$ % of the asteroids
between 3 and 15 au onto the conveyor belt, and $57.0$ % of these escape the
Solar System within the next million years. Within $\sim 10$ Myr, all the
asteroids born near Jupiter and Saturn are either deposited in resonances or
escape the Solar System. Jupiter and Saturn therefore cannot have contributed
much to the formation of the Oort cloud (this was also concluded by Torres et
al., 2020a, b). However, the details depend on the relative timing of the
various events and on the mass distribution within the disc. Constraining
those more precisely would require more extended and sophisticated self-
consistent simulations.

In our simulations, we ignore the star, planet, and asteroid-formation
processes: all are born instantaneously. We have not explored parameter-space
exhaustively, but argue that the details regarding the precise moment and the
orbits in which the giant planets form are relevant only to second order
because they affect the timescale for the formation of the Oort cloud but not
the fundamental process.

The time scale over which asteroids are ejected by the planets is sufficiently
long that an uncertainty of few million years in the planet-formation process
(Kruijer et al., 2017) does not affect the efficiency of the ejection process.
So long as the Solar System is a member of a star cluster, a sequence of
distinct processes drive the formation of the Oort cloud.

##  4 Results and Discussion

###  4.1 The birth environment of the Solar System

The birth environment of the Solar System plays a major role in the formation
of the Oort cloud: It prevents the gas-giant planets from forming the Oort
cloud, but at the same time stimulates its formation by scattering the outer
regions of the circumstellar disk and by introducing new asteroids by
capturing them from other stars, or from the interstellar free-floating
population (see also Zheng et al., 1990).

The Oort cloud has, according to this view, only a minor contribution from the
scattered asteroids originating from the gas-giant planets, Jupiter and
Saturn. Apart from those asteroids parked in resonant orbits, these planets
cause asteroids to escape the Solar System and become interstellar free-
floating objects (see also Torres, 2020; Torres et al., 2020a).

In contrast to Jupiter and Saturn, the ice giants, Uranus and Neptune, are
more favorable in producing the Oort cloud (see also Leto et al., 2008;
Paulech et al., 2010). They carry out the same process of ejecting asteroids,
but do this, because their lower mass and wider orbits, on a longer timescale
and with smaller impulsive changes in eccentricity and semi-major axis of the
asteroids (see also Duncan et al., 1987; Fernández et al., 2004; Correa-Otto &
Calandra, 2019).

The longer timescale on which the ice-giants planets eject asteroids is
important for the formation of the Oort cloud because so long as the Solar
System is a member of the parent cluster, asteroids in wide orbits are easily
lost from the Solar System (Nordlander et al., 2017). This process of asteroid
stripping due to encounters with nearby stars is particularly effective if the
asteroid’s orbit is wide and highly eccentric. After a few tens of millions of
years to $\sim 500$ Myr, once the Solar System has escaped the parent cluster,
such asteroids can remain bound.

The wide and highly eccentric orbits of new arrivals in the Oort cloud can
subsequently be perturbed at apocenter by the Galactic tidal field, leading to
a reduction in their eccentricity through von Zeipel-Lidov-Kozai resonance. As
a consequence, the pericenter distance of these asteroids increases, which
brings them outside the influence of the planets and detaches them from the
conveyor belt. While the orbital eccentricity of the asteroids continues to
decay due to the Galactic tidal field, they slowly, over a timescale of $\sim
100$ Myr, form a distribution of very wide but relatively low-eccentricity
orbits The distribution has to shape of the tidal lobe of the Solar System in
the Galactic potential. The size of the Oort cloud is confined on the inner
side to the distance where the Galaxy starts to influence the orbits of the
asteroids (at a relative velocity change of $\delta v\
{\raise-2.15277pt\hbox{$\buildrel>\over{\sim}$}}\ 10^{-3}$), and at the side
furthest from the Sun in the Galactic potential by the surface of the Hill
sphere (illustrated in Figs. 1 and 5).

According to Leto et al. (2008), the number density in the Oort cloud is
proportional to $n\propto r^{-3.53}$, whereas we find a considerably more
complex structure. In the Hills cloud and Oort cloud, we find $n\propto
r^{-2.25}$, with an overall flatter slope of $-2.57$ for the entire range from
100 au to the Hills radius (see figure 9). The slope of the density profile of
$\sim-2.25$ between $2\times 10^{4}$ au and $10^{5}$ au is somewhat shallower
than the slope of $\sim-4.0$ by Higuchi & Kokubo (2015), that of $\sim-3.35$
found by Vokrouhlický et al. (2019), or that of $\sim-3.0$ by Leto et al.
(2008). Higuchi & Kokubo (2015) argue that this slope is independent of the
mechanism that brings asteroids to the Oort cloud, but that does not explain
the differences in the density profile between these different studies. Part
of this structure may be the result of orbital evolution of the Solar System
and its possible migration in the Galaxy (Rickman et al., 2008; Kaib et al.,
2011a, b), but another part may be the result of planetary migration (Fouchard
et al., 2018).

Figure 9: Density profile of the initial circumstellar debris disk (blue, to
the top left) and the eventual Oort cloud, $\sim 1$ Gyr after the Solar System
left the parent cluster (orange). The red dotted line to the right indicates
the formal Hill radius of the Sun in its orbit around the Galactic center. The
thin-solid black-dotted curve shows the initial density profile of the
circumstellar disk of $\propto r^{-2.5}$ (from simulations of Table 3). The
thin dotted curve shows the density profile $\propto r^{-3.5}$ which,
according to Leto et al. (2008), resembles the slope of the density profile of
the Oort cloud. In our calculations, the distribution is somewhat shallower,
but not as shallow as found by Higuchi & Kokubo (2015) who argue in favor of
$\propto r^{-2.0}$, which is presented here with green dashes. The small tail
of asteroids beyond the Solar System Hill radius in the Galactic tidal field
has contributions of a minority of asteroids that remain bound to the Sun, and
those that slowly move away.

###  4.2 Results

####  4.2.1 The Sun in the Galaxy

A small fraction ($\sim 0.59\pm 0.1$ %) of the Oort cloud population on the
far side of the Hill surface remains bound to the Sun (see figure 9). At an
age of 1 Gyr in our simulations, this population has an eccentricity of
$\langle e\rangle=0.80\pm 0.23$ with an inclination of $\langle
i\rangle=96^{\circ}\pm 42$. These orbital parameters are similar to those of
the Oort cloud asteroids within the Sun’s Hill surface, and they are
consistent with the population presented in figure 7 of Vokrouhlický et al.
(2019). The objects at the far side of the Hill surface (beyond $\sim 200000$
au), are not expected to stay there long as they are ionized from the Solar
System by close stellar passages in the Galactic field (Correa-Otto &
Calandra, 2019).

Asteroids ejected from the Solar System become free-floating interstellar
objects. This latter population forms a trail of asteroidal objects in the
Galaxy along the orbit of the parent star (Correa-Otto & Calandra, 2019;
Torres et al., 2020a; Tutukov, Dremova, & Dremov, 2020; Portegies Zwart,
2021). As the asteroids have a low velocity relative to the orbital speed of
the Solar System, they continue to follow the same orbit as the Sun (see
figure 3). Long after they have become unbound, these asteroids still orbit in
a cloud around the Solar System distributed along extended trailing and
leading arms (see also Tutukov, Sizova, & Vereshchagin, 2021). Eventually,
these trails may be perturbed and scattered by passing molecular clouds or
other stars, scattering their orbits and widening their phase-space
distribution (see Martínez-Barbosa et al., 2016; Pfalzner et al., 2020; Torres
et al., 2020a).

####  4.2.2 Consequences of planets in resonant orbits

We performed several additional simulations to study the consequences of
planet migration in the early Solar System. In Tables 5 and 6 we present an
overview of these calculations. In addition to the currently observed
distribution of orbits for the giant planets, we also simulated a more compact
configuration and a resonant configuration. The main problem in exploring
these initial conditions lies in the enormity of the initial parameter space
to be covered; our parameter-space coverage is far from complete. We therefore
selected a few cases which have been of some interest in the past (see e.g.,
Wyatt, 2003; Gomes et al., 2005; Malhotra, 1998; Crida et al., 2007; Brunini,
2019).

In one of these cases, we initiate a Nice-like planetary orbital distribution
and a disk between 3 au and 50 au. In that configuration, Jupiter was placed
on a circular orbit at 5 au in the ecliptic plane, and the subsequent planets
in 2:1 mean-motion resonances with the one closer to the star. We also tried
other configurations (see Table 3). The main conclusion, namely that the gas-
giant planets (Jupiter and Saturn) eject asteroids on a timescale shorter ($\
{\raise-2.15277pt\hbox{$\buildrel<\over{\sim}$}}\ 10$ Myr) than the expected
survival timescale in the parent cluster ($\
{\raise-2.15277pt\hbox{$\buildrel>\over{\sim}$}}\ 100$ Myr), also holds for
these initial conditions.

Changes in the orbital distribution of the giant planets affect the location
and efficiency of the conveyor belt, and the timescale on which asteroids
migrate along the conveyor belt. In our initial resonant configuration, the
entire disk, up to the orbit of the outermost planet, at around 15 au, is
cleared within 1 Myr. The area slightly outside the outermost planet, up to a
distance of 20 au from the star, is cleared within a few million years. By the
time the Solar System leaves the parent cluster, there are too few asteroids
left to populate the Oort cloud.

Pirani et al. (2019) also simulated the early Solar System including the
migration of the giant planets. Although these latter authors focused on the
inner Solar System and the populations of resonant and Trojan orbits, their
conclusions are consistent with the results of the calculations presented
here. Planets in an initial resonant configuration are excluded based on the
consequence that the majority of asteroids would be lost well before the Sun
escapes the parent cluster. This also accounts for Uranus and Neptune; if they
are born in (near) resonance, the chaotic reorganization of the Solar System
would eject most of the asteroids in the disk on a timescale shorter than the
expected lifetime of the Sun in the parent star cluster. Asteroids outside the
outermost planet would be affected by resonant reorganization of the planetary
orbits, but these asteroids are not likely to migrate into the conveyor belt,
and therefore are not expected to arrive in the Oort cloud. Fully resonant
initial conditions can therefore be omitted based on the existence of the Oort
cloud, except for the part of the Oort cloud formed from material that was
accreted from other stars or from free-floating debris.

Even if a resonant configuration in the ice-giant planets could lead to a
dramatic change in the distribution of orbital parameters after $\sim 100$
Myr, as is advertised in the original Nice model (Gomes et al., 2005;
Morbidelli et al., 2005; Tsiganis et al., 2005), the number of asteroids has
already reduced below the number needed to explain the Oort cloud. We
therefore argue that a system in which the planets are born close to
resonance, or in which planets migrate in the first few million years since
formation (such as advertised in the grand-tack model; Walsh et al., 2011), is
unlikely to lead to the formation of an Oort cloud. Planetary migration after
the Oort cloud formed has no further consequence for the Oort cloud, and we
cannot exclude such evolution (Valtonen & Innanen, 1982). Nevertheless, again,
resonant initial conditions for the planets cannot be excluded if the majority
of the Oort cloud material was accreted.

Planets born in orbital resonance (see sect. 4.2.2)
---
Designation |  Simulation D.
Simulations |  One simulation with planets (in various configurations), asteroids, and Galactic tidal field.
Star |  Single 1 ${M}_{\odot}$ with four giant planets and 10000 asteroids in the smooth potential of the Milky Way galaxy.
Planets |  The four planets have almost circular orbits ($e<0.006$) in the plane ($i<0.1^{\circ}$) with Jupiter, Saturn, Neptune, and Uranus in orbits with semi-major axes of $5.1$, $9.5$, $19.2$, and $30.1$ au respectively. The ecliptic was inclined by $60^{\circ}$ to the Galactic plane. We performed two more simulations with initial semi-major axes of $3.75$, $5.953$, $9,449$, and $15.0$ au, and with semi-major axes of $5.5$, $8.1$, $11.5$, and $14.2$ au for the four giant planets.
Asteroids |  Ten-thousand asteroids in circular orbits between 3 au and 50 au in a thick disk (Toomre-Q parameter of $25$ Toomre, 1964) in the plane of the planets.
Numerics |  Huayno (Pelupessy et al., 2012) coupled to the Galactic model; see Table 4 via bridge (Portegies Zwart et al., 2020) in AMUSE (Portegies Zwart & McMillan, 2018).
Computer |  Run on LGM-II with GPU
Duration |  Simulation performed for a duration of 1 Gyr.
Table 5: Simulations of a hypothetical young Solar System with planets in
various orbital configurations. Planets born in orbit around an isolated star
(see Sects. 3.2 and 4.2.2)
---
Designation |  Simulations E.
Simulations |  Twelve simulations to study the internal planetary dynamics and their response to the local asteroids.
Star |  Single 1 ${M}_{\odot}$ with 4 planets, 2000 asteroids.
Planets |  Four planets in almost circular orbits ($e<0.006$) in the plane ($i<0.1^{\circ}$) with Jupiter, Saturn, Neptune, and Uranus in orbits with semi-major axes of $5.1$, $9.5$, $19.2$, and $30.1$ au. We performed two more simulations with initial semi-major axes of $3.75$, $5.953$, $9,449$, and $15.0$ au.
Asteroids |  Two-thousand asteroids in circular orbits in a thick $Q=25$ disk in the ecliptic plane of the planets between 3 au and 50 au.
Numerics |  Huayno (Pelupessy et al., 2012)
Computer |  Run on a 192 core Intel-Xeon workstation.
Duration |  Simulation performed for a duration of 50 Myr.
Table 6: Simulations of the young Solar System.

####  4.2.3 Retrograde and re-entry orbits

We measure the fraction of asteroids in retrograde orbits. At the age of $300$
Myr, well after the Sun escaped the cluster, $8.1\pm 0.3$% of the scattered
asteroids ($a\ {\raise-2.15277pt\hbox{$\buildrel>\over{\sim}$}}\ 100$ au) have
retrograde orbits, and for the compact resonant initial conditions this
fraction is $7.1\pm 0.5$%. In comparison, $71.9\pm 0.3$% of the captured
asteroids ($a\ {\raise-2.15277pt\hbox{$\buildrel>\over{\sim}$}}\ 100$ au) have
retrograde orbits. However, this latter fraction depends notable on the
encounter parameters and the number may vary considerably for other
simulations. Once well along the conveyor belt (at $a(1-e)\
{\raise-2.15277pt\hbox{$\buildrel>\over{\sim}$}}\ 50$ au and eccentricity $\
{\raise-2.15277pt\hbox{$\buildrel>\over{\sim}$}}\ 0.998$), about $57\pm 1$ %
of the asteroids have retrograde orbits. This large fraction of retrograde
orbits in the conveyor belt indicates the moment that the Oort cloud becomes
spherical and isotropic. By this time, the orbits have also reached their
terminal distribution in eccentricity (which is not thermal; see figure 8).

Because of interaction with the tidal field, a small fraction of asteroids
return to the inner Solar System after having spent some time in the Oort
cloud. We find $0.6\pm 0.1$% of the asteroids that were launched from the
conveyor belt into the Oort cloud return into the inner 10 au of the Solar
System on a timescale of 1 Gyr. This results in a re-entry rate of $2$ to
$6\times 10^{-12}$ yr-1. The re-entry rate for approaching the Sun to within 1
au is an order of magnitude smaller. The observed rate of observable comets in
the Solar System, measured between 110 BC and 1970, was remarkably constant at
a rate of $0.86\pm 0.067$ per year (Licht, 1999). A larger rate of $6.4\times
10^{-12}$ per year was derived by Dones et al. (2004a), but they accounted for
perturbations in the Oort cloud due to passing stars, which we neglected in
our calculations. With our derived re-entry rate of comets within 1 au of the
Sun of $\sim 6\times 10^{-13}$ yr-1, and a total number of $\sim 10^{12}$
cometary bodies in the Oort cloud, we derive a rate of about two new comets
every 3 yr. Even when taking the tidal field and passing stars into account,
our derived rate is somewhat on the low side compared to observations. The
presence of a planet in the Oort cloud would pose an interesting possibility
in increasing the cometary influx rate (Fouchard et al., 2014a; Ito & Higuchi,
2020; Batygin & Brown, 2021).

###  4.3 Interstellar comets

According to the simulations presented here, the Solar System is a copious
polluter of interstellar space (Torres et al., 2019; Pfalzner et al., 2021);
interstellar comets produced in this way could have characteristics similar to
’Oumuamua or Borisov (Higuchi & Kokubo, 2020). This pollution predominantly
happens in four rather distinct phases in the evolution of the Solar System:
(A) while being a cluster member, when encounters between stars cause debris
from the circumstellar disk to escape; (B) while the giant planets (Jupiter
and Saturn) eject most of the asteroids within their gravitational influence;
(C) after the Sun ascapes the cluster, and the ice giant planets (Uranus and
Neptune) start kicking out asteroids; and (D) eventually due to the Sun’s
copious mass loss while ascending the asymptotic giant branch and evolving
into a white dwarf, causing asteroids to become unbound (Veras et al., 2011;
Veras & Tout, 2012; Veras et al., 2014, 2020).

According to Cai et al. (2019) $20$–$80$% (with an average of $50$ %) of the
circumstellar material survives the first $100$ Myr of its evolution in the
parent cluster. The majority of this mass is lost through encounters with
other stars. The amount of material lost from the Solar System in the
simulation presented here falls within this range, meaning that the
circumstellar disk has lost about half of its mass due to interactions with
other stars in the parent cluster, or about $100$ M⊕ to $3000$ M⊕. Each of the
other processes results in a mass loss of roughly $20$%. A small fraction of
the ejected asteroids acquire bound orbits in the Oort cloud.

With an estimated mass of the Oort cloud of $\sim 10^{13}$ kg/comet, and a
Hill radius of $\sim 0.65$ pc, we arrive at a density of $\sim 10^{14}$
kg/pc3. If the Solar System had ejected $\sim 90$ % of its asteroids into
interstellar space rather than forming an Oort cloud, interstellar space would
have an average density of a few times $\sim 10^{14}$ kg/pc3. This estimate is
consistent with earlier estimates for the interstellar density of interstellar
objects (Portegies Zwart et al., 2018; Do et al., 2018; ’Oumuamua ISSI Team et
al., 2019)

Regretfully, tracking a single interstellar interloper back to its origin is
hindered by the uncertainties in the orbital parameters and the positions of
stars in the past. Such tracking is limited to at most a few tens of millions
of years (Zhang, 2018). However, if multiple objects were found to have the
same origin, tracking the orbit back in time could lead to a more precise
determination of the original launching point (Portegies Zwart, 2021).
However, according to our calculations, most objects will have been ejected at
an early stage while planets are still forming and migrating (see also Pirani
et al., 2019). Possibly the star is even still a cluster member, in which case
internal scattering with other stars will make it hard to trace the
interstellar object back to a single star.

###  4.4 Discussion

####  4.4.1 Interpretations and caveats in the numerical approach

Numerical simulations are always affected by the choice of initial conditions
and by numerical errors; either from the discretization of the underlying
differential equations or by the exponential growth of round-off in the least
significant digit. There is also the possibility of glitches or ”bugs” in the
simulations.

These uncertainties and errors become a particular potential cause of concern
in chaotic systems. The outer parts of the Solar System, for example, have an
estimated Lyapunov timescale ranging from $\
{\raise-2.15277pt\hbox{$\buildrel<\over{\sim}$}}\ 20$ Myr (Applegate et al.,
1986; Sussman & Wisdom, 1988; Laskar, 1990; Sussman & Wisdom, 1992; Murray &
Holman, 1999; Farrés et al., 2013) to as much as 50 Myr (Laskar et al., 2011)
or even exceeding the age of the Solar System (Grazier et al., 1999; Varadi et
al., 2003).

Choices made in the initial conditions have considerable consequences for the
numerical results and their interpretation. At the same time, numerical errors
will also have repercussions for the conclusions in Section 5. Similar caveats
in numerical work were acknowledged by Charles Babbage (1791-1871), the
inventor of the difference and the analytical engines. Aware of these and
other limitations, we describe the caveats in our approach in this section and
discuss how to mitigate them.

####  4.4.2 Integration, symplecticity, and numerical errors

Newton’s equations of motion are intrinsically chaotic. Planetary systems are
therefore also chaotic, as in the Solar System (Lecar et al., 2001; Valtonen
et al., 2004). Chaos in the outer parts of the Solar System manifests itself
as phase-bound or stable chaos (Milani & Nobili, 1992; Saillenfest et al.,
2019) and although it affects the stability of the Solar System (Todorović et
al., 2020), it will not lead to a global unstable configuration (Tanikawa &
Ito, 2007). Chaos in the Solar System is driven by resonance overlap (Murray &
Holman, 1999, 2001). Even the Oort cloud is chaotic (Saillenfest et al.,
2019).

Such chaos hinders the integration of the equations of motion for the Solar
System; it renders these simulations notoriously unreliable for individual
orbits. Possibly statistically they are trustworthy, meaning that the
integrated bodies preserve the same phase space, but this has never been
tested (for a discussion see Portegies Zwart & Boekholt, 2018).

In this paper and many others, we rely on the statistical ensemble, averaging
by numerical integration. It is widely suggested that, in these simulations,
the conservation of energy is sufficient to preserve the final orbital
parameter space. Symplectic integration methods are developed with precisely
this objective, to preserve energy over secular timescales. Therefore, these
methods are popular for integrating planetary systems (Ito & Tanikawa, 2002),
but not for integrating star clusters or galaxies.

Although details in our calculations will depend on precise initial
realization of the giant planets, the general results remain unaffected. We
tested the effect of rather drastic variations of the initial conditions and
the effect of subtle changes (such as the epoch of the Solar System), but we
did not explore these systematically. The various simulation codes to perform
this study are made public via the AMUSE framework, and we encourage others to
continue exploring the parameter space.

##  5 Summary and conclusions

In our simulations, shortly after their birth, Jupiter and Saturn start to
eject local asteroids from the circumstellar disk along the “conveyor belt”.
Asteroids in the conveyor belt cross the orbit of the giant planets near their
pericenter. Each time this happens, they receive a small kick causing their
orbits to drift to higher eccentricity and larger semi-major axis while
preserving pericenter distance. This process has been studied for asteroids
whose pericenter is near Jupiter (Duncan et al., 1987; Fernández, 1997). We
confirm that Jupiter, with its relatively high mass and short orbital period,
clears the local asteroid field in a few million years (Fernández, 1997;
Pirani et al., 2019). In figure 4, we illustrate this process by presenting
one calculation where a single asteroid is ejected along the conveyor belt to
reach the Oort-cloud region in a few million years. However, for the ice
giants with their smaller mass and wider orbits, the process takes up to $\sim
100$ Myr (Fernandez, 1981; Fouchard et al., 2013; Di Sisto & Rossignoli,
2020), and if the asteroid has an inclined orbit, the process may last a
gigayear.

While a member of a star cluster, an asteroid in a wide $\
{\raise-2.15277pt\hbox{$\buildrel>\over{\sim}$}}\ 10^{3}$ au orbit is easily
lost from the Solar System. A small perturbation in relative velocity of
$\delta v\equiv dv/v\geq{\cal O}(10^{-4})$ is sufficient to unbind an
asteroid. It turns out that asteroids on an eccentric orbit $\geq 0.98$ and
with a semi-major axis of $\ {\raise-2.15277pt\hbox{$\buildrel>\over{\sim}$}}\
2400$ au are vulnerable to being stripped from the Solar System. The period of
such a wide orbit ($\ {\raise-2.15277pt\hbox{$\buildrel>\over{\sim}$}}\ 0.1$
Myr) is comparable to the mean encounter time between the Sun and another star
in the parent cluster (assuming that the encountering star induces a similar
perturbation in relative velocity). In the first million years after their
formation, the giant planets launch $\sim 80$ % of the asteroids between 3 and
15 au onto the conveyor belt (see Sect. 3.5). Once there, the eccentricity and
orbital separation of the asteroids rapidly increases.

The majority of the asteroids ($\sim 57$ %) are moving along trajectories that
become so wide that they are stripped within a few orbits, and within $\sim
10$ Myr most of the asteroids born near Jupiter and Saturn are either parked
in resonant orbits (such as the Hilda family of asteroids) or their orbits
have become so wide that they are easily stripped by passing stars in the
parent cluster, and escape in about $20$ to $50$ Myr. The Oort cloud therefore
cannot have formed in the first 10 Myr by Jupiter and Saturn ejecting
asteroids because their ejection timescale is smaller than the timescale on
which the Solar System is expected to be ejected from the parent cluster ($\
{\raise-2.15277pt\hbox{$\buildrel<\over{\sim}$}}\ 100$ Myr). This statement is
unaffected by the few-million-years formation timescale for the giant planets.

Before escaping the cluster, stars in our simulations experience multiple
encounters. The dynamic signature of multiple encounters on the outer regions
of a planetary system is distinctly different from that of a single strong
encounter (Hands et al., 2019). The outer parts of the Solar System have
signatures of both: A single strong encounter with another star can explain
the Kuiper-cliff (Punzo et al., 2014) and the orbit of Sedna (Shankman et al.,
2011; Jílková et al., 2015), but the complex distribution of orbital
parameters in the scattered Kuiper-belt beyond $\sim 45$ au (Kobayashi et al.,
2005) seems best explained with a series of relatively weak encounters (see
Brasser et al., 2012; Breslau et al., 2017; Hands et al., 2019; Moore et al.,
2020, and Sect. 3.3.2). We therefore argue that the Solar System was affected
by multiple encounters in its parent cluster, until a strong encounter caused
it to escape.

In our simulations, multiple interactions lead to $\sim 5$ % of the asteroids
in the outer disk beyond 45 au being scattered into the conveyor belt region
(see Sect. 3.3.2), but the Sun also captures material from the disks of the
stars it encounters. The efficiencies at which material is transported from
one star to another is rather symmetric (Bhandare et al., 2016). The scattered
and captured populations are comparable in number, but populate different
regions in orbital parameter-space around the Sun.

In a reconstruction of the encounter that brought Sedna into the Solar System
as a captured asteroid from the disk of another star, $\sim 28$% of the
captured asteroids are injected directly into the conveyor belt region (see
figure 1 of Jílková et al., 2015, and Sect. 3.5). The high inclination of the
scattered and captured populations compared to the native disk causes them to
interact less efficiently with the giant planets. These asteroids reach the
Oort cloud on a timescale of $\sim 1$ Gyr. Because of this long timescale,
these objects remain in the parking zone of the Solar System where they are
protected against ejection caused by passing stars. Once the Sun is isolated,
asteroids in the conveyor belt escape the Solar System only if their apocenter
distance exceeds the Hill radius of the Solar System in the Galactic
potential, at $\sim 0.65$ pc, or $\delta v\
{\raise-2.15277pt\hbox{$\buildrel>\over{\sim}$}}\ 0.1$.

Before an asteroid reaches such a wide orbit, it is subject to eccentricity
damping by the Galactic tidal field via von Zeipel-Lidov-Kozai oscillations
(von Zeipel, 1910; Lidov, 1962; Kozai, 1962). The relatively high inclination
of the ecliptic to the Galactic plane (of $\sim 60^{\circ}$) helps to
circularize their orbits and randomize their inclinations (Higuchi et al.,
2007). The von Zeipel-Lidov-Kozai process causes the pericenter of an
asteroid’s orbit to detach from the planetary region (Higuchi et al., 2007),
preventing it from being kicked out by a planet. Both processes, namely the
ejection by the giant planets and the eccentricity damping by the tidal field,
are essential ingredients that operate on comparable timescales. Once the
Galactic tidal field reduces the eccentricity below $0.998$ and the semi-major
axis $\ {\raise-2.15277pt\hbox{$\buildrel>\over{\sim}$}}\ 20\,000$ au
($a(1-e)\ {\raise-2.15277pt\hbox{$\buildrel>\over{\sim}$}}\ 50$ au, to the
right of the dashed curve in figure 2), the driving force that pumps an
asteroid’s orbit further into the conveyor belt switches off. By this time,
the asteroid is detached from the inner Solar System.

The ejection of the native scattered, and captured asteroids along the
conveyor belt, and their subsequent eccentricity damping by the Galactic tidal
field lead to the formation of the Oort cloud some 100 Myr after the Sun
escaped the parent cluster (Kaib & Quinn, 2008). The effects of passing stars
in the parent cluster and planet–asteroid interactions together with the
Galactic tidal field are essential ingredients that contribute to defining the
orbital structure of the outer Kuiper belt, the Hill surface, and the Oort
cloud. In figure 2, we illustrate the orbital parameter space at this instance
(see also Sect. 3.5).

The conveyor belt beyond the ice-giant planets is further depleted in the next
few $100$ Myr, resulting in the clearance of the asteroids beyond Saturn’s
orbit and the provision of material to the Oort cloud. This could explain the
dearth of Centaurs in the Solar System without requiring a chaotic
reorganization of the ice-giant planets (Fouchard et al., 2014b).

Based on our calculations, $\sim 5$ % of the asteroids in the Oort cloud
originate from the outer disk beyond $\sim 45$ au. About one-third ($\sim
30$%) are captured from another star, and the rest originate from the
circumstellar disk between $\sim 15$ au and $45$ au. The majority of scattered
and captured asteroids never make it to the Oort cloud, but linger around in
the parking zone, between the two solid curves indicated with $\delta v\sim
10^{-8}$ and $\delta v\sim 10^{-5}$ in Figs. 2 and 6, or scape the Solar
System. These captured objects should still be there and can be identified
based on their unorthodox orbits and composition. The difference in the color
of Kuiper belt and Oort cloud objects indicates that they are not related
(Jewitt, 2002). Once in the Oort cloud, the origin of an asteroid can be
established by studying its kinematics, because the phase-mixing driven by the
chaotic process that injected it into the Oort cloud manifests itself on a
timescale longer than the Solar System’s lifetime. The outer Solar System
origin of these asteroids can possibly be established by spectral analysis,
much in the same way as C/2019 Q4 (Borisov) was analyzed using the OSIRIS (de
León et al., 2019) and MUSE (Bannister et al., 2020) instruments. One
remarkable finding is the depletion of C2 (Opitom et al., 2019) and NH2 in
this object (Bannister et al., 2020).

The scenario for the formation of the Oort cloud presented here is relatively
insensitive to the details of the orbits of the outer planets. Asteroids can
be launched into the conveyor belt to reach the Oort cloud from the current
orbital distribution of the giant planets, or from a chaotic migration of the
ice-giant planets. However, according to our simulations, if any chaotic
reorganization or migration of the giant planets happens before the Sun
escapes the parent cluster, the Oort cloud contains considerably fewer objects
because most asteroids escape; in addition, the orbital distribution of
objects in the Oort cloud would differ (Shannon et al., 2019). We continued
our calculations for 1 Gyr, but already after a few hundred million years the
Oort cloud is fully formed, and the eccentricity distribution approaches a
probability distribution of $f(e)\propto(1-e^{2})^{-2/9}$ (see sect. 3.3.3)
with a density distribution $\propto r^{-3}$ (see sect. 4.1).

By this time, the Solar System is surrounded by a cloud of unbound asteroids
which co-move along the Sun’s orbit in the Galaxy. This co-moving group
contains $\ {\raise-2.15277pt\hbox{$\buildrel>\over{\sim}$}}\ 10^{11}$ comet-
mass asteroids, most of which originated from the inner Solar System. This
unbound population is illustrated in figure 3. If other stars have Oort clouds
of their own, the Solar System moves through a sea of Oort-cloud objects that
originally belonged to other stars (see also Torres et al., 2020a; Portegies
Zwart, 2021).

The Oort cloud gradually evaporates with a half-life of about 1 Gyr due to the
tidal field’s kinematic heating and passing stars. Lost objects become free-
floating asteroids in interstellar space (Torres et al., 2019), much like
’Oumuamua and Borisov.

Interaction with the Galactic tides also causes asteroids to be launched into
the inner Solar System, where they can be observed as comets. In our
simulations, the rate at which Oort cloud objects re-enter the inner Solar
System (to within 1 au) is $0.2$—$0.6$ per year (see Sect. 4.2.3), which is
somewhat lower than the empirical estimate of $0.86\pm 0.07$ per year (Licht,
1999). Nevertheless, here, we ignored the possible ionizing effect of passing
stars in the Galactic potential.

## Public data

The source code, input files, and simulation data for this manuscript are
available at 10.6084/m9.figshare.13214471. A tutorial on how to run the
various codes in AMUSE is available at https://github.com/spzwart/AMUSE-
Tutorial.

## Acknowledgments

It is a pleasure to thank Diptajyoti Mukherjee, Fransisca Concha-Ramírez,
$\ln(a)$ Sellentin, Julia Wasala, Eiichiro Kokubo, Ramon Brasser and Tim de
Zeeuw for discussions. We also thank the anonymous referee(s) for valuable
comments on on the manuscript. We are grateful for the support of the Mexican
National Council for Science and Technology (CONACYT) grant #291004-410780.
This work was performed using resources provided by the Academic Leiden
Interdisciplinary Cluster Environment (ALICE), LGM-II (NWO grant #
621.016.701) and the Dutch national e-infrastructure with the use of the Dutch
national supercomputer Cartesius, and the support of SURF Cooperative.

### Energy consumption of this calculation

Being concerned about the polluting influence of our science (Burtscher et
al., 2020; Portegies Zwart, 2020) we would like to raise awareness of the
environmental impact of our calculations. We run AMUSE for about 40000 single-
core CPU hours. This results in about 2MWh of electricity http://green-
algorithms.org/) being consumed by the Dutch National supercomputer. With our
estimate of the proportion of green electricity used, this process produces
$\sim 530$ kg CO2, which is comparable to driving a car from Leiden to Kars
(about 4000km). However, the same data were used in additional research, and
we argue that less than half the CO2 produced in these calculations should be
attributed to this paper.

### Software used for this study

This work would have been impossible without the following public open source
packages and libraries: Python (van Rossum, 1995), matplotlib (Hunter, 2007),
numpy (Oliphant, 2006), MPI (Gropp et al., 1996; Gropp, 2002), ABIE (Cai,
2018, see https://github.com/MovingPlanetsAround/ABIE), NBODY6++GPU (Wang et
al., 2015) and REBOUND (Rein & Liu, 2012), SeBa (Portegies Zwart & Verbunt,
1996; Toonen et al., 2016), and AMUSE (Portegies Zwart et al., 2018, available
for download at https://amusecode.org).

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