# DYNAMICAL POPULATION OF COMET RESERVOIRS

N. A. Kaib University of Oklahoma Kathryn Volk University of Arizona

###### Abstract

  * The Oort cloud and the scattered disk are the two primary reservoirs for long-period and short-period comets, respectively. In this review, we assess the known observational constraints on these reservoirs’ properties and their formation. In addition, we discuss how the early orbital evolution of the giant planets generated the modern scattered disk from the early, massive planetesimal disk and how $\sim$5% of this material was captured into the Oort cloud. We review how the Sun’s birth environment and dynamical history within the Milky Way alters the formation and modern structure of the Oort cloud. Finally, we assess how the coming decade’s anticipated observing campaigns may provide new insights into the formation and properties of the Oort cloud and scattered disk.

##  1 DISCERNING THE COMET RESERVOIRS

Comets make up one of the major populations of small bodies in the solar
system, and their passages near Earth and the Sun have been documented by
astronomers for millennia (_Kronk_ , 1999). However, these bodies are known to
have relatively short lifespans both dynamically, due to orbital instability,
and physically, due to the volatile loss and surface dust deposition that
occurs during each near-Sun perihelion passage (_Weissman_ , 1980; _Rickman et
al._ , 1990; _Levison and Duncan_ , 1994). As a result, it has long been
assumed that the short-lived modern population of comets are supplied from
more populous distant reservoirs whose members have much longer stability
timescales.

Until 1992, Pluto was the most distant known object in the solar system, and
the nature of comet reservoirs had to be inferred solely from the properties
of the more proximate but transient population of comets passing near Earth.
Historically, the catalog of known comets has been broadly divided into two
groups based on orbital period: short-period comets (SPCs) whose orbital
periods are less than 200 years, and long-period comets (LPCs) whose orbital
periods are greater than 200 years. It has long been recognized that the
orbital inclinations of comets with large periods are nearly isotropic, while
short-period comets have an inclination distribution strongly concentrated
toward prograde ecliptic orbits (e.g. _Everhart_ , 1972; _Marsden_ , 2009).
This fundamental difference in the orbital distribution of SPCs and LPCs has
fueled decades of dynamical studies attempting to understand how the supply of
comets in the inner solar solar system is replenished and what the source
reservoirs of this replenishment must look like (see, e.g., historical reviews
by _Davies et al._ 2008; _Fernández_ 2020).

###  1.1 An Oort Comet Cloud

Amongst the long-period comets, the distribution of the inverse semimajor axes
($1/a$) of the comets is a representation of their orbital energy
distribution, and _Oort_ (1950) noted an overabundance of comets with
$0<1/a<0.0001$ au-1, or $a>10^{4}$ au. _Oort_ (1950) posited that the comets
belonging to this overabundance (now known as the Oort spike) are bodies that
are making their first perihelion passage near the Sun, while long-period
comets with smaller semimajor axes are on subsequent passages, their orbital
energies having been drawn down via perturbations from Jupiter during previous
perihelion passages. Jovian perturbations can also increase the orbital
energies of new long-period comets during their initial perihelion passage,
but these comets are so weakly bound that this process permanently ejects them
from the solar system on hyperbolic orbits.

To explain a continuous supply of new long-period comets, _Oort_ (1950)
proposed that the solar system is surrounded by a vast cloud of $\sim$1011
comet-sized bodies extending to $\sim$105 au. In this scenario, LPCs passing
near Earth on extreme eccentricities comprise a tiny fraction of the bodies
within this cloud, whose orbits have semimajor axes of at least thousands of
au but whose eccentricities range all the way from 0 to 1. If this cloud were
dynamically inert, its high-eccentricity (low-perihelion) portion should have
been depleted long ago due to hyperbolic ejection events during perihelion
passages. However, at such large distances from the Sun, perturbations
external to the solar system become important. _Öpik_ (1932) noted that
impulses from passing stars will alter the perihelia (and therefore
eccentricities) of distant heliocentric orbits. This effect allows new Oort
cloud bodies to be brought into the inner solar system as previous long-period
comets are lost through ejection by the giant planets, namely Jupiter.
Moreover, these same perturbations will randomize the directions of orbital
angular momenta, leading to an isotropic distribution of LPCs and a spherical
structure to the cloud. Decades later it was realized that the tide of the
Milky Way’s disk (generated by the vertical distribution of matter within the
disk) will also provide perihelion and inclination shifts in Oort cloud bodies
that are, on average, larger than those due to passing stars (_Morris and
Muller_ , 1986; _Heisler and Tremaine_ , 1986). This perturbative force will
also steadily replenish the supply of LPCs from lower eccentricity Oort cloud
bodies.

###  1.2 The Edgeworth-Kuiper Belt and Scattered Disk

While the seminal work of _Oort_ (1950) outlined most of the broad features of
LPC dynamics and their distant reservoir, the origin of SPCs still remained a
puzzle due to their lower-inclination orbits. _Everhart_ (1972) argued that
the observed SPC population could be produced from multiple rounds of jovian
perturbations that lower the semimajor axes of low-inclination LPCs. However,
such a process appeared to be orders of magnitude too inefficient (_Joss_ ,
1973; _Kresak and Pittich_ , 1978) and would require an extremely massive Oort
cloud (_Fernandez_ , 1980b) to explain the abundance of SPCs.

Meanwhile, it had long been speculated that the falling surface density of
solid material with increasing heliocentric distance in the early outer solar
system would produce a belt of icy planetesimals that were never incorporated
into fully formed planets (_Edgeworth_ , 1943, 1949; _Kuiper_ , 1951). While
_Kuiper_ (1951) assumed that perturbations from Pluto (whose mass was then
thought to be comparable to Earth’s) had dispersed this belt into the Oort
cloud, _Edgeworth_ (1943) proposed that members of this belt occasionally
visit the inner solar system as comets. In another study, _Everhart_ (1977)
found that some Oort cloud bodies with perihelia near Neptune will eventually
be scattered into Jupiter-crossing orbits and evolve into SPCs. _Fernandez_
(1980b) realized that a belt of small bodies residing beyond Neptune would
naturally supply near-Neptune perihelia and could generate a population of
SPCs on its own. However, it was assumed that a reservoir completely comprised
of Neptune-encountering orbits would be quickly depleted over the age of the
solar system due to its inherent orbital instability.

_Duncan et al._ (1988) used numerical simulations to show that the
inclinations of Oort cloud bodies are approximately conserved as their
semimajor axes are decreased via encounters with the giant planets, and they
concluded that the Oort cloud cannot be the main source of SPCs. An additional
set of simulations showed that a low-inclination reservoir of Neptune-crossing
bodies generates a low-inclination population of SPCs similar to the observed
population, and these results were refined and confirmed in _Quinn et al._
(1990). Thus, it seemed that the population of SPCs required the existence of
a second reservoir of bodies in a belt beyond Neptune.

The mechanism that injected members of this belt into the comet population was
still unclear, however. Planetesimal-planetesimal scattering suggested by
_Fernandez_ (1980b) required such massive belt members that they would not
have evaded detection. _Torbett_ (1989) showed instead that chaos resulting
from the gravitational effects of the four giant planets could raise the
eccentricities of initially more circular orbits to slowly supply Neptune-
encountering bodies, and therefore comets, from this region. Direct
integrations showed that a similarly slow destabilization process takes place
near the boundaries of mean motion resonances with Neptune (_Duncan et al._ ,
1995). However, additional work in _Levison and Duncan_ (1997) found that most
regions of orbital space of the conventional Kuiper belt (low-eccentricity
orbits from $\sim$40-50 au) either destabilize quickly ($\lesssim$108 years)
or are stable over timescales longer than the age of the solar system.
However, _Duncan and Levison_ (1997) noted that a small subset ($\sim$5%) of
Kuiper belt objects that have a close encounter with Neptune will attain
$\sim$Gyr dynamical lifetimes. They achieve these long dynamical lifetimes by
occupying eccentric orbits whose perihelia have been temporarily lifted away
from Neptune, typically within mean motion resonances. With a dynamical
lifetime comparable to the solar system’s age, these “scattered disk” bodies
are a more efficient producer of SPCs compared to the classical Kuiper belt.
Thus, the subpopulation of trans-Neptunian objects (TNOs) known as the
scattered disk is widely thought to be the main source of SPCs.

####  1.2.1 Centaurs as a Comet-Reservoir Link

Before TNOs make it all the way into the inner solar system, they must pass
through the giant planet region as Centaurs, the intermediary small body
population connecting short-period comets and the scattered disk (e.g.,
_Duncan and Levison_ 1997; reviewed recently by _Peixinho et al._ 2020). The
exact orbital definition of Centaurs differs in the literature (e.g. _Gladman
et al._ , 2008; _Jewitt_ , 2009), but the general criteria is that they are
small bodies on giant-planet crossing orbits that are dynamically unstable on
timescales much shorter than the age of the solar system.

The first discovered object widely recognized as belonging to the new class of
objects later dubbed the Centaurs was (2060) Chiron (_Kowal and Gehrels_ ,
1977). It was quickly realized that Chiron’s orbit is dynamically unstable
(e.g. _Oikawa and Everhart_ , 1979; _Scholl_ , 1979). More general studies of
orbital stability in the giant planet region reveal that Centaurs have typical
dynamical lifetimes on the order of 1-10 Myr (e.g. _Gladman and Duncan_ ,
1990; _Holman and Wisdom_ , 1993; _Tiscareno and Malhotra_ , 2003; _Di Sisto
and Rossignoli_ , 2020). Centaurs thus are a transition population for comets
en-route from the transneptunian region (e.g. _Stern and Campins_ , 1996)
rather than a reservoir. However, not all Centaurs become comets. Centaurs
spend most of their time in the outer, ice giant region, with only about half
making it to the Jupiter-Saturn region and about a third evolving onto short-
period comet orbits (e.g., _Tiscareno and Malhotra_ , 2003; _Sarid et al._ ,
2019). The chapter by Fraser et al. in this volume discusses the evolution of
Centaurs into comets in more detail.

###  1.3 Bodies with Ambiguous Sources

There are a few groups of comets/comet predecessors that do not fall neatly
along the long- or short-period divide and are not easily produced by models
of Oort Cloud or scattered disk delivery. In addition to orbital period,
comets are often divided into subgroups based on their Tisserand parameter
with respect to Jupiter, $T_{J}$:

$T_{J}=\frac{a_{J}}{a}+2\sqrt{\frac{a}{a_{J}}(1-e^{2})}\cos{i},$ (1)

where $a_{J}$ and $a$ are the semimajor axes of Jupiter and the comet,
respectively, $e$ is the eccentricity of the comet’s orbit, and $i$ is its
orbital inclination. This parameter is related to a comet’s Jacobi constant,
which is a strictly conserved quantity in the restricted three-body problem
(Sun + circular Jupiter + comet) and is approximately conserved on 1000-year
timescales for SPCs in the real solar system (_Levison and Duncan_ 1994; see
also, e.g., _Murray and Dermott_ 1999 for a full discussion of the Jacobi and
Tisserand parameters). It has long been noted that the vast majority of short-
period comets have $T_{J}$ slightly below 3, indicating a tight dynamical
coupling with the gas giant (_Kresák_ , 1972), and _Levison and Duncan_ (1997)
demonstrated how dynamical transfer from the scattered disk generates this
tight distribution of $T_{J}$. Based on these results, _Levison_ (1996) argued
that the Tisserand parameter should be the main criterion upon which to base
comet classification. In particular, _Levison and Duncan_ (1997) argue that
comets with 2 $<T_{J}<$ 3 should be considered Jupiter-family comets and are
consistent with a Kuiper belt origin.

Figure 1: Semimajor axis vs perihelion distance for all Jupiter-family (JFC;
circles), Halley-type (HTC; triangles), and Encke-type (ETC; squares) comets
listed by JPL (queried from https://ssd.jpl.nasa.gov/sbdb_query.cgi on Aug.
22, 2021). The comets are color-coded by Tisserand parameter with respect to
Jupiter ($T_{J}$) to highlight the separations between the different dynamical
classes of short-period comets.

####  1.3.1 Halley-type Comets

Figure 1 shows the current known population of SPCs, color coded by $T_{J}$;
it is clear that not all comets are confined in the tight cluster of
$T_{J}=2-3$. In particular, there exists a subgroup of SPCs with $T_{J}<2$,
which _Carusi et al._ (1987) designated Halley-type comets (or HTCs). The
driver for their lower $T_{J}$ values are their generally significantly larger
orbital inclinations (up to and including retrograde orbits) as well as
moderately larger semimajor axes. The ultimate source of these comets has been
debated, with _Levison et al._ (2001) suggesting that HTCs are consistent with
an origin from a flattened interior of the Oort cloud, _Levison et al._
(2006a) arguing for a scattered disk origin whose most distant members are
pushed inward via the Galactic tide (which also isoptropizes ecliptic
inclinations), and _Nesvorný et al._ (2017) arguing that HTCs are consistent
with returning LPCs from the Oort cloud whose semimajor axes have been drawn
down through repeated planetary encounters. _Fernández et al._ (2018) find
that HTC orbits can be produced from integrations of inactive Centaurs with
perihelia in the Jupiter-Saturn region, suggesting that the lack of observed
cometary activity could be due to a long residence time at these heliocentric
distances that depleted surface volatiles. While _Fernández et al._ (2018)
assume a scattered disk origin for all the Centaurs, perhaps this is also
consistent with the _Nesvorný et al._ (2017) returning LPC scenario.
Additional modeling and observations are likely necessary to test these
different proposed HTC origins.

####  1.3.2 Encke-type Comets

On the opposite side of the $T_{J}$ divide from the JFCs are the Encke-type
comets (ETCs); these comets typically have $T_{J}$ just above 3 with low
semimajor axes that slightly de-couple the comets from Jupiter compared to
JFCs (see Figure 1). _Levison et al._ (2006b) investigated the origins of
comet Encke and found that such orbits can be part of the standard model for
JFC evolution, but that it is a rare outcome that typically required
$1.5\times 10^{5}$ perihelion passages in the inner solar system to achieve.
This long timespan is problematic for producing active comets like the
observed ETCs because comets are expected to fade (either via disruption or
loss of volatiles) on timescales of order a few tens to a few hundred
perihelion passages (e.g. _Levison and Duncan_ , 1997; _Brasser and Schwamb_ ,
2015). _Levison et al._ (2006b) suggest that Encke could perhaps have built up
an inactive surface layer during this time which was then shed after a
dramatic drop in $q$, but it remains unclear whether this kind of evolution
can plausibly explain the entire ETC population. It is possible that the
production of ETCs depends on more than just purely gravitational dynamics.
Most dynamical models of how short period comets are supplied from their
source regions neglect nongravitational forces such as outgassing, but these
might be important for the ETCs. _Fernández et al._ (2002) found a higher rate
of ETC production when such forces were included in the model, though they
found they needed to be sustained over a relatively long timeframe. Future
models of ETC production with more realistic nongravitational forces would
likely help shed light on this question.

####  1.3.3 Highly Inclined Centaurs

While the majority of Centaurs have low-to-moderate inclinations consistent
with production from the scattered disk en-route to the JFC population, there
exist a number of very high-inclination and retrograde Centaurs and TNOs with
perihelia in the giant planet region (e.g. _Gladman et al._ , 2009; _Chen et
al._ , 2016). Much like the high-inclination HTCs, the origin of these highly
inclined objects is unclear. It has been proposed that such bodies could
originate from the Oort cloud (_Brasser et al._ , 2012b; _Kaib et al._ ,
2019). In this scenario, galactic perturbations drive the perihelia of Oort
cloud bodies (whose inclination distribution is nearly isotropic) into the
giant planet region at which point energy kicks from planetary perturbations
can draw these bodies’ semimajor axes to smaller values (_Emel’yanenko et al._
, 2005; _Kaib et al._ , 2009). An alternative explanation for high-inclination
centaurs is that they are signature of a distant undetected planet that
perturbs the inclinations of scattered bodies (_Gomes et al._ , 2015; _Batygin
and Brown_ , 2016b). However, such a mechanism may generate more high-
inclination centaurs than are actually observed (_Kaib et al._ , 2019) and has
other, observable consequences for the distribution of TNOs (_Shankman et al._
, 2017b). Another recently proposed origin for this population is outward
scattering of objects from the asteroid populations in the inner solar system,
which can produce very high-inclination and retrograde orbits (_Greenstreet et
al._ , 2020). This last hypothesis would have interesting (and testable)
compositional implications for the highest-inclination outer solar system
objects.

##  2 Observational Constraints On Reservoirs

###  2.1 Short-Period Comet Population

Figure 2 shows the current inventory of solar system objects with orbital
periods less than 200 years classified as comets by JPL. There are obviously
observational biases in this sample, but it highlights a few important
features of the SPC population. First, note the large spread of inclinations
for objects with $a\gtrsim 5$ au; many of the high-inclination objects in this
group are the HTCs discussed in Section 1.3.1, whose dominant source
population remains an active area of research. Second, note that the vast
majority of comets with semimajor axes near or inside Jupiter’s orbit (the
JFCs) have inclinations smaller than $\sim 30^{\circ}$. This concentration of
inclinations relatively close to the ecliptic plane was the first notable
evidence linking JFCs to the outer solar system’s TNO populations (as
discussed in Section 1.2), which have a similar inclination distribution.

Figure 2: Inclination vs semimajor axis distribution of observed cometary
objects with orbital periods shorter than 200 years (the standard definition
of SPCs; queried from JPL). The concentration of comets with low-$a$ and
low-$i$ orbits highlight the JFC population.

If we knew the exact number of SPCs in the inner solar system at the moment,
we could use our dynamical models of the delivery of these comets to constrain
the total number of comet-sized TNOs in the scattering population (the
dominant supplier of these comets). However, this extrapolation is complicated
by the fact that our observed sample of SPCs is biased toward active comets on
easy-to-observe orbits. The bias toward active comets is particularly
important because we have strong evidence that SPCs must ‘fade’ (i.e., either
become inactive and faint or be physically destroyed) on timescales much
shorter than their dynamical lifetimes. The inclination distribution of the
JFCs is the dominant evidence for this. Over their entire $\sim 0.5$ Myr
dynamical lifetimes in the inner solar system (_Levison and Duncan_ , 1994),
interactions with the planets tend to raise the orbital inclinations of JFCs
to values larger than those seen in the observed population. To get the
modeled JFC population to match the observed one in inclinations, _Brasser and
Wang_ (2015) estimate that comets could fade after as few as $\sim 50$
perihelion passages; _Nesvorný et al._ (2017) model fading as a size-dependent
process and estimate that small comets remain visible for a few hundred
perihelion passages (both models consider perihelion passages with $q<2.5$
au). Our models of comet fading are thus still quite uncertain, making it
difficult to extrapolate the current census of active JFCs into a total
population of active and inactive JFCs. Additionally, the very fact that
comets are active makes measuring their sizes more difficult because the gas
and dust coma obscures the comet nucleus. Thus measurements of the size
distribution of comets, necessary for connecting population estimates to their
source region, can vary (e.g. _Lowry et al._ , 2008; _Snodgrass et al._ ,
2011; _Fernández et al._ , 2013); this, combined with different assumptions
about the ratio of active to dormant comets, means estimates of the current
number of JFCs can vary by factors of several (see, e.g., discussion in
_Nesvorný et al._ 2017). These variable comet population estimates combined
with variations in orbital models for their scattering population reservoir
means that attempts to extrapolate from the number of observed JFCs to the
number of comet-sized objects in the TNO region can differ enormously.
Published estimates of the current population of the scattered disk based on
supplying the current number of JFCs include: $6\times 10^{8}$ (’comet-sized’;
_Duncan and Levison_ 1997), $\sim 10^{10}$ ($D>1.4$ km; _Emel’yanenko et al._
2004), $0.5-1.1\times 10^{9}$ ($D>1$ km; _Volk and Malhotra_ 2008), $\sim
6\times 10^{9}$ ($D>2.3$ km; _Brasser and Wang_ 2015), $4.4\times 10^{8}$
($D>2$ km; _Nesvorný et al._ 2017). While these estimates are not all for
identical size ranges, it is clear that they vary by at least an order of
magnitude.

###  2.2 Direct Observation of the Scattered Disk

Since the detection of the first scattering TNO (_Luu et al._ , 1997),
hundreds of additional members of this population and the closely related
high-$a$ resonant and detached populations have been detected. Our picture of
the high-$a$ TNO population has evolved as observations built up. It became
apparent fairly early that not all of the high-$a$ TNOs had perihelion
distances that brought them into direct gravitational range of Neptune;
observations of objects with high perihelion distances ($q\gtrsim 40$ au) led
to discussion of an ‘extended’ scattered disk (e.g. _Gladman et al._ , 2002)
or a dynamically ‘detached’ population (the current more commonly used
terminology; e.g., _Gladman et al._ 2008). The important role of Neptune’s
resonances in the scattering population was noted in early dynamical models
(_Levison and Duncan_ , 1997), and their potential role in creating the
detached population was also noted (e.g., _Gomes et al._ , 2005; _Gallardo_ ,
2006). As the orbits of distant TNOs have been measured more precisely, it is
also apparent that there are large, stable populations of objects in many of
Neptune’s distant resonances (e.g. _Gladman et al._ , 2012; _Pike et al._ ,
2015; _Volk et al._ , 2018) in addition to a large number of metastable
objects temporarily stuck in resonances on a variety of timescales (e.g.
_Lykawka and Mukai_ , 2007; _Bannister et al._ , 2016; _Holman et al._ ,
2018). Figure 3 shows the observed orbital distribution of TNOs in the
scattering, $a>50$ au resonant, and detached populations.

---
Figure 3: Observed distribution of scattering, detached, and large-$a$ ($a>50$
au) resonant TNOs in eccentricity (top panel) and inclination (bottom panel)
vs semimajor axis. The dashed lines in the top panel show constant perihelion
distances of 30 and 40 au for reference. We include only objects with known
dynamical classifications (we use the set of classified objects listed in
_Smullen and Volk_ 2020) to highlight the overlaps between these three
dynamical classes (see _Gladman et al._ 2008 for detailed definitions of these
classes).

It was apparent in the early TNO datasets that the higher-eccentricity TNOs
(including the scattering objects) have a much wider range of orbital
inclinations than the low-eccentricity ‘classical’ Kuiper belt objects, giving
the total TNO population a bimodal inclination distribution (_Brown_ , 2001).
This gave rise to a division of the TNOs into dynamically ‘cold’ and ‘hot’
components, with the cold population being found only in the classical belt.
The scattering, resonant, and detached populations all belong to the hot
population. This division has been strengthened by subsequent surveys of the
physical properties of TNOs. The photometric colors of hot and cold TNOs are
distinct (e.g. _Tegler et al._ , 2003; _Pike et al._ , 2017; _Schwamb et al._
, 2019), implying that they have different surface compositions, perhaps
reflecting different formation regions. The cold population also contains
significantly more binary objects than the hot population; the few observed
binary objects in the hot population tend to have a massive primary with small
satellites rather than the prevalent equal-sized binaries in the cold
population (see a recent review by _Noll et al._ 2020). There was also early
evidence that the hot population’s size distribution differed from the cold
population. _Bernstein et al._ (2004) found evidence for this difference,
finding significantly fewer hot population objects at faint magnitudes than
expected. Subsequent observational surveys have found additional evidence for
differences between the hot and cold populations and for a ‘turnover’ in the
magnitude/size distribution from a steep slope for the bright/large objects to
a shallower one at fainter/smaller objects (e.g., _Fraser and Kavelaars_ ,
2009; _Fuentes et al._ , 2009; _Fraser et al._ , 2010; _Shankman et al._ ,
2013; _Fraser et al._ , 2014; _Kavelaars et al._ , 2021); however the exact
nature and location of this change has been difficult to constrain due to the
strong observational biases and faintness of the TNOs. Recent results from the
Outer Solar System Origins Survey (OSSOS) have shown that the scattering
population of TNOs can be well fit with either a break or a ‘divot’ (an actual
decrease in the differential $H$ magnitude distribution rather than just a
change in slope) near $H_{r}\approx 8-8.5$ (e.g. _Shankman et al._ , 2013,
2017b; _Lawler et al._ , 2018b).

All of these observations have led to the current view that the hot
populations, which are the progenitors of the short period comets, formed
closer to the sun than the current cold classical region. They formed in the
current giant planet region and were dispersed onto their current orbits
during the epoch of giant planet migration (discussed in Section 3). At
present, observational estimates of the number of comet-sized objects in the
scattering population are still fairly uncertain. In reflected sunlight, km-
sized bodies in the scattering population are far too faint for direct
detection even at perihelion. Comet-sized population estimates thus must be
extrapolated based on the size distribution of brighter bodies. But even the
population estimates for these brighter bodies have large uncertainties
because the observational biases in the scattering population are so strong
(see, e.g., discussion of observational biases in _Gladman and Volk_ 2021).
Briefly, TNOs with large semimajor axes and large eccentricities spend most of
their time at large heliocentric distances; they are typically only bright
enough to be detected near their perihelia and thus over only a small portion
of their orbits. Estimating the intrinsic number of scattering objects from
the observed sample of objects requires either weighting each observed object
by the inverse of its probability of detection based on its size and orbit (a
traditional ‘debiasing’ approach) or taking a forward-modeling approach and
applying observational biases to a model of the scattering population and
iterating that model until it gives you a match to the observed sample. The
accuracy of both approaches rely to a large extent on how well the biases of a
given observational survey are understood and measured as well as how well the
observed objects sample the orbital distribution of the scattering population
(this is discussed further in Section 4.1). _Adams et al._ (2014) provide
estimates of the scattered disk population based on debiasing the objects
detected by the Deep Ecliptic Survey (_Elliot et al._ , 2005); they estimate
that there are $\sim 10^{4}$ scattered objects with absolute magnitudes
$H>7.5$. Taking a forward modeling approach, _Lawler et al._ (2018b) provide
population estimates for the scattering population from OSSOS. They use the
_Kaib et al._ (2011) model of the intrinsic orbital distribution of the
scattering population (with a slightly modified inclination distribution to
better match the observed distribution, see _Shankman et al._ 2016) with a
variety of functional forms for the absolute magnitude distribution to find
acceptable matches to the observed sample. From this, _Lawler et al._ (2018b)
estimate that there are $(0.7-1)\times 10^{5}$ scattering objects with
$H_{r}>8.66$ (approximately $D>100$ km; range covers all of their $H$
magnitude models); extrapolating down to $H_{r}<12$ ($D\gtrsim 20$ km) yields
population estimates of $\sim(2-3)\times 10^{6}$ objects. These are our
current best estimates of the scattering population based on direct
observations; while these estimates will likely improve with additional
observations of the scattering population (see Section 4.1), the inherent
brightness limitations of TNO surveys will make it difficult to push the
direct constraints on the in-situ scattering population down to comet-sized
bodies.

Two avenues of current research show promise for connecting the observational
constraints on the size distribution of larger scattering objects to the
population of comet-sized ones: occultation surveys (discussed in Section 4.2)
and analysis of cratering records in the outer solar system. In the latter
case, data from the New Horizons mission provided a wealth of new data about
the distribution of impact craters on TNOs Pluto and Arrokoth (see, e.g.,
_Singer et al._ 2019; _Spencer et al._ 2020). Impact crater distributions do
not easily translate to direct constraints on the TNO size distribution. In
addition to some uncertainties in crater scaling laws, converting crater sizes
to impactor sizes requires knowing the impact speed distribution onto the
surface in question, and this distribution depends on the assumed model for
the orbital distributions and relative populations of the various dynamical
classes of TNOs (see discussion in _Morbidelli et al._ 2021); despite these
uncertainties, the crater distributions offer our strongest current
constraints on the distribution of the smallest TNOs. The scattering
population does not dominate the impact flux at either Pluto or Arrokoth
(_Greenstreet et al._ , 2015, 2019), so the crater size distribution does not
directly constrain its total population at small sizes. However the hot
classical population of TNOs (which is expected to have similar origins as the
scattering population; see Section 3.2) does contribute to the cratering on
both bodies, so the cratering record can offer insights into the size
distribution of the scattering population even if not the absolute numbers;
hopefully future detailed modeling can connect the implied combined TNO
impactor distributions to population estimates for the comet-sized portion of
the scattering population.

###  2.3 Direct Observation of Oort Cloud

With semimajor axes of thousands of au and perihelia decoupled from the solar
system’s planetary region, direct detection of Oort cloud objects is
exceptionally difficult. To date, the TNO 2015 TG387 (_Sheppard et al._ ,
2019) is the most likely candidate for a directly detected Oort cloud object
(outside of the LPCs and other lower perihelia bodies whose orbits have been
substantially modified by recent planetary interactions). With a perihelion of
65 au, it is clearly very decoupled from scattering events with the giant
planets. Its semimajor axis is 1170 au, meaning that it is sensitive to the
stellar and galactic tidal perturbations that influence the rest of the Oort
cloud (_Sheppard et al._ , 2019).

With a sample of just one detected object, it is extremely difficult to make
confident inferences about the nature of the Oort cloud. Given the steep
dependence of discovery probability on semimajor axis, we do not have strong
constraints on the semimajor axis distribution of the Oort cloud. In addition,
while 2015 TG387 is estimated to have a diameter of 300 km, one cannot
construct an accurate size distribution with a single object. Additional
direct detections will be required to develop confident constraints on the
Oort cloud’s properties.

###  2.4 Long-Period Comet Flux

Because of the extreme distance to the Oort cloud, the vast majority of
observational constraints come from observations of LPCs. In particular, the
flux of dynamically new ($a>10^{4}$ au) comets near Earth is one of the main
observational parameters to constrain the cloud. The reason that comets with
semimajor axes above $10^{4}$ au are considered dynamically new is that a
single passage through the gas giant region will typically deliver an energy
kick strong enough to either eject such a comet onto a hyperbolic orbit or
lower it to a much smaller semimajor axis.

For the last several decades, efforts have been made to measure the annual
flux of dynamically new comets with the idea that such measurements will
constrain the mass and population size of the Oort cloud. (A more massive or
more populous cloud will generate larger numbers of LPCs.) While quantifying
the flux, most works have sought to measure the number of new comets whose
absolute magnitude, $H_{T}$, is brighter than some value. Here, $H_{T}$ is
given by

$H_{T}=m-2.5n\log{r}-5\log{\Delta}$ (2)

where $m$ is the total apparent magnitude, $r$ is the distance to the Sun,
$\Delta$ is the distance to Earth, and $n$ is the comet’s photometric index,
or the power-law index defining the relationship between the comet’s
brightness and heliocentric distance. $H_{T}$ is the absolute magnitude of a
comet (nucleus and coma) when it is 1 au from the Earth and Sun. For a non-
active body $n=2$, but for a comet whose outgassing increases during a solar
approach $n$ is often higher. (If one assumes $n=4$, as is often done, the
absolute magnitude is noted as $H_{10}$.)

An oft-quoted metric is the flux of dynamically new comets per year with
perihelia below 4 au (interior to Jupiter’s orbit and potentially observable
from Earth) and whose $H_{T}$ is brighter than 11. _Everhart_ (1967) found
that $\sim$20 new LPCs pass within 4 au of the Sun per year, but this result
relied on the extrapolation of an uncertain comet magnitude distribution
derived from historical comet searches with selection biases that were
extremely difficult to quantify. Meanwhile, utilizing a greater number of
detections from modern surveys _Francis_ (2005) found a flux of only 2.9 per
year, and _Fouchard et al._ (2017) found a flux of 3.6 per year.

####  2.4.1 Population of Oort cloud

To estimate the total population of the Oort cloud, many past works have
examined the new LPC production rate within simulations of the Oort cloud and
scaled this value to the observed LPC flux. _Kaib and Quinn_ (2009) found
that, on average, 1 in every $\sim$1011 Oort cloud bodies passes through
perihelion within 4 au of the Sun per year. In contrast, _Fouchard et al._
(2014) found a lower rate of 1 in $5\times 10^{11}$, but this may have been
due to the higher level of central concentration in their Oort Cloud semimajor
axis distribution, which placed more bodies onto small semimajor axes with
lower comet production rates (_Kaib and Quinn_ , 2009). Taking a mean of the
two results and assuming a real flux of 3 new comets per year with $q<4$ au
and $H_{T}<11$, yields a total Oort cloud population of 7–8 $\times 10^{11}$
cometary bodies with $H_{T}<11$.

####  2.4.2 Mass of Oort cloud

$H_{T}$ is of course not an actual direct measure of a comet’s nucleus,
because a comet’s brightness is dominated by the coma when it is at 1 au. To
discuss the flux of comets of an actual size or mass, $H_{T}$ has often been
converted to a nuclear magnitude, which is then converted to a size/mass, but
such a conversion is fraught with uncertainty (e.g. _Bailey and Stagg_ , 1988;
_Weissman_ , 1996). Using mass-$H_{T}$ relationships derived from observations
of Halley’s Comet in combination with the long-period comet $H_{T}$
distribution found in _Everhart_ (1967), _Weissman_ (1996) determined the
average long-period comet mass to be 0.8–$4\times 10^{16}$ g. This would imply
an Oort cloud mass of 1–5 M⊕.

Alternatively, _Boe et al._ (2019) utilized Pan-STARRS1 long-period comet
detections and applied a universal comet sublimation model to convert each
cometary magnitude to nucleus radius. With this approach, they found that the
LPC size distribution is a broken power-law with a steep bright-end index
($r^{-4.6}$) transitioning to a shallow faint-end index ($r^{-1.5}$) near an
inferred comet radius of 2 km. This diminishes the relative contribution of
sub-km bodies to the mass of the Oort cloud, and _Boe et al._ (2019) estimated
a total Oort cloud mass of 0.5–2 M⊕ in bodies larger than 1 km in radius.

_Sosa and Fernández_ (2011) took yet a different approach and searched for a
relationship between the observed water sublimation and non-gravitational
forces altering comets’ orbits to find the masses of LPCs. Using this
relationship, they then derive one for the total visual magnitude and mass of
a comet:

$\log{R{\rm(km)}}=0.9-0.13H_{T}.$ (3)

While this method was a unique and independent attempt to measure LPC nuclei,
it relies upon several uncertain parameters, including an assumed bulk density
of comets and a conversion of visual magnitude into water sublimation rate.
The above mass-visual magnitude relationship was derived over a relatively
narrow range of absolute magnitudes of 5–9, corresponding to radii of 0.5–1.8
km according to Eq. 3. Applying this conversion relation to a broader number
of long-period comet discoveries, _Fernández and Sosa_ (2012) inferred a comet
size distribution comprised of three different power laws. However, extending
their bright-end power law to infinity implies an infinite mass to the Oort
cloud, so it must become steeper at some larger comet nucleus size beyond the
range they study ($r\sim$20 km).

####  2.4.3 Population Ratio of the Oort Cloud to the Scattered Disk

Historically, JFCs and LPCs have been used to estimate the population of
comet-sized bodies within the scattered disk and Oort cloud, respectively.
Taken at face value, these population estimates imply that there are nearly 3
orders of magnitude more such bodies in the Oort cloud than the scattered disk
(_Levison et al._ , 2010). However, these estimates still largely rely on
uncertain conversions of $H_{T}$ to comet nuclear magnitudes. _Brasser and
Morbidelli_ (2013) attempted to limit the uncertainty by only considering LPCs
and JFCs with inferred radii above 2.3 km, which largely avoided the
previously discussed breaks in the LPC size distribution. With this approach,
they found a substantially smaller Oort cloud-to-scattered disk population
ratio of 44${}^{+54}_{-34}$. In line with these results, _Nesvorný et al._
(2017) and _Vokrouhlický et al._ (2019) suggest that the sizes of LPC nuclei
may be overestimated and are typically just 300-400 m in diameter, rather than
the km sizes inferred in most works (e.g. _Francis_ , 2005; _Fernández and
Sosa_ , 2012; _Boe et al._ , 2019). This would also lower the population of
the Oort cloud and its ratio to the scattered disk (_Nesvorný et al._ , 2017).

##  3 BUILDING THE COMET RESERVOIRS

###  3.1 A Scattered Oort Cloud

####  3.1.1 A Rough Framework

Although the population of LPCs is very consistent with the existence of a
distant spherical Oort cloud, they do not tell us how this cloud was formed.
_Oort_ (1950) realized, however, that the same stellar perturbations that
produce LPCs from the Oort cloud can also explain the cloud’s formation. If
solid planetesimals are leftover after the formation of the planets, it is
inevitable that many will eventually undergo close encounters with the giant
planets, exchanging energy with the planets and driving a diffusion in the
planetesimals’ semimajor axes. As the planetesimals’ semimajor axes diffuse,
their perihelia effectively remain locked in the giant planet region, allowing
energy exchanges to occur during each perihelion passage. Left unchecked, this
process would eventually result in all of the planetesimals either being
ejected onto hyperbolic orbits or colliding with the Sun or a planet.

However, if planetesimals reach semimajor axes of thousands of au, the
perturbations from passing stars begin to become significant (e.g. _Rickman_ ,
1976). If stellar perturbations can torque the planetesimals’ perihelia beyond
the planetary region this would save these planetesimals from planetary
ejections onto hyperbolic orbits. These same perturbations would act to
isotropize the orbital planes of planetesimals and could give rise to a
distant spherical cloud of planetesimals similar to that inferred from LPC
observations.

Thus, _Oort_ (1950) hypothesized a basic framework for the formation of the
Oort cloud, but at the time, the efficiency of such a process was unknown.
While _Oort_ (1950) assumed that the planetesimals occupying the Oort cloud
originated between Mars and Jupiter, _Fernandez_ (1978) noted that the
probability that planetesimal-planet encounters result in a near-parabolic
orbit rather than a hyperbolic one increases with planet-Sun distance. As a
result, material scattering off of Uranus and Neptune should be more likely to
reach the Oort cloud than planetesimals closer to the Sun. This process was
numerically verified in _Fernandez_ (1980a), where particles were initiated on
near parabolic orbits with perihelia between 20–30 au. They were numerically
integrated as they scattered off of the giant planets while also being torqued
by passing stars for 109 years. In this time span, over 50% of bodies were
ejected while most survivors resided in a nearly isotropized cloud between 20
and 30 thousand au. Extrapolating from these results, _Fernandez_ (1980a)
concluded that Uranus and Neptune implanted $\sim$10% of the material they
scattered into the Oort cloud.

Another early attempt to model the construction of the Oort cloud through
these mechanisms was _Shoemaker and Wolfe_ (1984). _Shoemaker and Wolfe_
(1984) assumed the planetesimals were leftover from the formation of Uranus
and Neptune and employed a distribution of initially circular orbits between
15 and 35 au for their planetesimals. The evolution of these planetesimals’
orbits due to planetary encounters was modeled with a statistical approach
(_Opik_ , 1976), and the influence of passing stars was implemented as in
_Oort_ (1950). Largely in agreement with _Fernandez_ (1980a), _Shoemaker and
Wolfe_ (1984) found that $\sim$10% of planetesimals will avoid ejection or
collision with the Sun. Of those survivors, the vast majority ($\sim$90%) will
be retained on orbits with perihelia beyond the planets’ orbits and with
semimajor axes between 500 and 20,000 au. A smaller fraction ($\sim$10%) were
predicted to orbit with semimajor axes beyond 20,000 au, comprising the Oort
cloud that _Oort_ (1950) predicted to explain LPC production. Thus, the
survivors in the _Shoemaker and Wolfe_ (1984) work tended to orbit the Sun at
closer distances than the _Fernandez_ (1980a) work.

####  3.1.2 A Numerical Model

This process of Oort cloud formation was then revisited with still more direct
numerical techniques in _Duncan et al._ (1987). In this work, planetary
perturbations on planetesimals were still applied with a Monte Carlo approach
once per planetesimal orbit. However, the Monte Carlo sample was built from a
distribution of energy kicks compiled from direct integrations of
planetesimals through the giant planet region of the solar system. (For an
example of such a distribution, see Figure 5). As planetesimals reached large
($>$2000 au) semimajor axes, their orbits were directly integrated. During
these direct integrations, planetary energy kicks were still applied at each
perihelion passage, stellar perturbations consistent with the solar
neighborhood kinematics were applied via the impulse approximation, and a
vertical (with respect to the Galactic plane) tidal force due to the
distribution of matter within the Milky Way’s disk was included (_Heisler and
Tremaine_ , 1986).

The approach of _Duncan et al._ (1987) revealed the key timescales involved
that govern the formation of the Oort cloud. The semimajor axes of
planetesimals diffuse due to planetary energy kicks received near perihelion.
The timescale of this diffusion depends on the orbit’s perihelion (which
determines the giant planet that is primarily delivering the kicks) and the
current semimajor axis (which determines how often perihelion passages occur
as well as the orbit’s binding energy). As semimajor axis increases, the
diffusing timescale for further changes in semimajor axis decreases with
$a^{-1/2}$ for a fixed $q$. Meanwhile, a competing timescale is the typical
time for a significant change in perihelion due to perturbations from passing
stars and the Galactic tide. A torque large enough ($\Delta q\gtrsim 10$ au)
to potentially pull an orbit’s perihelion outside of the planetary region will
prevent future energy kicks from planetary encounters and halt diffusion in
semimajor axis. This second timescale also decreases with semimajor axis since
external perturbations become more significant as the distance from the Sun’s
gravitational pull increases, but it falls more steeply than the diffusion
timescale, or as $a^{-2}$.

Figure 4: Plot of key dynamical timescales as a function of particle semimajor
axis. $t_{q}$ is the timescale for a perihelion shift of 10 au (assuming
initial $q$ of 25 au) due to Galactic tidal torquing; $t_{D}$ is the diffusion
timescale for the semimajor axis to change by order of its present value due
to planetary energy kicks; $t_{r}$ is the timescale for the Galactic tide to
cycle an orbit from $q=25$ au to its maximum perihelion and back again; $P(a)$
is the orbital period; Horizontal line marks the age of the solar system.
Adapted from _Duncan et al._ (1987).

Both the timescale for semimajor axis diffusion due to planetary encounters
and the timescale for perihelion torques due to external perturbations are
plotted as a function of semimajor axis in Figure 4. To avoid a case where all
planetesimals are either ejected or collide with the sun or planets, the
timescale for a perihelion torquing must fall below the semimajor axis
diffusion timescale at some value of semimajor axis. The semimajor axis for
which the torquing timescale is shorter than the diffusion timescale depends
on the planetesimal’s initial perihelion distance because Jupiter and Saturn
provide larger energy kicks than Uranus and Neptune. For planetesimals being
scattered by Uranus and Neptune, the two timescales become comparable at
$a\simeq 5000$ au, and for planetesimals being scattered by Jupiter and
Saturn, they become comparable at $a\simeq 20,000$ au. These semimajor axis
values determine the inner edge (i.e., minimum $a$) of the Oort cloud,
interior to which objects’ perihelia cannot be changed by external
perturbations before they experience significant changes in semimajor axis;
objects interior to this $a$ boundary are part of the trans-Neptunian
scattering population rather than the Oort cloud. Exterior to this value,
objects’ perihelia can be lifted out of the planetary region, effectively
locking their semimajor axes. Thus, bodies gravitationally scattering off the
giant planets can enter the Oort cloud at or beyond the semimajor axis at
which the two timescales become comparable. There is also a natural maximum
semimajor axis to the Oort cloud that exists due to the tide of the Milky
Way’s disk, which overwhelms the Sun’s gravitational pull around 0.5–1 pc or
1–2$\times 10^{5}$ au (_Heisler and Tremaine_ , 1986).

With initial semimajor axes of 2000 au and bins of initial perihelia between 5
au and 35 au, _Duncan et al._ (1987) directly integrated $\sim$7000 particles
under the influence of the Sun, planets, passing stars, and the Galactic tide
for 4.5 Gyrs. At the end of this simulation, they found that a substantial
fraction of particles were trapped in the Oort cloud with perihelia beyond the
planets and semimajor axes between a few thousand au and $\sim$105 au.
However, the trapping fraction varied greatly depending on particles’ initial
perihelia. For those with perihelia of 5 au (near Jupiter), the trapping
fraction was only 2%, while the fraction approached 40% for those with
perihelia near Neptune (25–35 au). The reason for this is that the magnitude
of energy kicks particles receive during perihelion passage sensitively
depends on perihelion value (_Fernandez_ , 1978).

Figure 5: Typical energy change of a parabolic particle during a single
perihelion passage as a function of perihelion. Solid lines 1–6 mark 30∘
inclination bins from 0–30∘ to 150–180∘. The dashed line coincides with the
approximate width of the Oort cloud energy window for particles scattering off
Jupiter and Saturn. Adapted from _Fernandez_ (1981).

These energy kicks were first numerically measured by _Fernandez_ (1981), and
the typical energy kick per perihelion passage for different orbits is shown
in Figure 5. Objects passing near Jupiter and Saturn receive energy kicks more
than an order of magnitude larger than those passing near Uranus and Neptune.
As already stated, for objects with perihelia near Jupiter and Saturn, their
perihelia cannot be pulled from the planetary region until they attain
semimajor axis $\gtrsim$104 au. However, the entire orbital energy window of
the Oort cloud ($10^{4}\lesssim a\lesssim 10^{5}$ au) is smaller than the
typical energy kick these objects receive on a single perihelion passage.
Thus, most of them overshoot the Oort cloud and leave the solar system on
hyperbolic orbits. Since the opposite is true for objects scattering off
Uranus and Neptune (the weaker energy kicks of these planets are smaller than
the Oort cloud’s orbital energy window), Uranus and Neptune are much more
efficient at placing bodies into the Oort cloud. Within the populations of
particles trapped in the Oort cloud, _Duncan et al._ (1987) found that the
same perturbations (passing stars and the Galactic tide) that were responsible
for perihelion shifts were strong enough to complete isotropize orbital
inclinations and eccentricities beyond a semimajor axis of $\sim$5000 au.

Another prediction of the numerical simulations of _Duncan et al._ (1987) was
that the Oort cloud is strongly centrally concentrated, with 80% of bodies
orbiting on semimajor axes between 3000 and 20000 au and the remaining 20% of
bodies orbiting beyond $a>20000$ au. This finding was interpreted to imply
that only the outer 20% of the Oort cloud was capable of supplying observed
long-period comets. The reason for this is that the perihelion shift timescale
plotted in Figure 4 is longer than the orbital period for semimajor axes
$a\lesssim 20000$ au. Thus, objects on semimajor axes below 20000 au must
first make a perihelion passage near Jupiter and Saturn before they would be
capable of making a perihelion passage near Earth. However, as shown in Figure
5, the energy kick delivered during this first perihelion passage will
typically be powerful enough to either eject the particle on a hyperbolic
orbit or decrease its semimajor axis to a level where its perihelion evolution
is extremely slow.

####  3.1.3 Lower Central Concentration and Trapping Efficiency

From the above logic it was thought that LPCs only provided constraints on the
outer periphery of Oort cloud beyond 20,000 au, but the inner region could
contain large numbers of bodies as well (_Hills_ , 1981). _Hills_ (1981)
speculated that this inner portion could be $\sim$100 times more populous, yet
would only supply comets near Earth during “comet showers” coinciding with
rare, powerful stellar encounters (_Hills_ , 1981; _Hut et al._ , 1987). The
simulations of _Duncan et al._ (1987) suggested that these inner population
estimates were far too high.

Based on the high Oort cloud trapping efficiencies of objects gravitationally
scattering off Uranus and Neptune in _Duncan et al._ (1987), it was presumed
that tens of percent of leftover planetesimals may have been captured into the
Oort cloud after the giant planets formed. However, _Fernández_ (1997) noted
that many objects beginning on orbits near Uranus and Neptune evolve to
Jupiter- and Saturn-crossing orbits before being scattered to large semimajor
axes (_Fernandez and Ip_ , 1984) and that this process may lower the fraction
of material that is trapped in the Oort Cloud (_Fernandez and Ip_ , 1981).
Numerical work by _Dones et al._ (2004) confirmed this. _Dones et al._ (2004)
found that the actual fraction of initial bodies that are still trapped in the
modern Oort cloud was much more likely no more than $\sim$5%, and subsequent
works have pointed to even lower fractions of 2–3% (_Kaib and Quinn_ , 2008;
_Kaib et al._ , 2011). The reason for this difference from _Duncan et al._
(1987) was that _Dones et al._ (2004) initialized their particles on nearly
circular orbits between the giant planets, whereas _Duncan et al._ (1987) had
to begin their particles (due to computing limitations of the era) on highly
eccentric orbits with semimajor axes of 2000 au. The choice of _Duncan et al._
(1987) effectively locked the perihelia of their particles unless the Galactic
tide or passing stars altered them. On the other hand, the approach of _Dones
et al._ (2004) allowed the perihelia of bodies to significantly evolve before
they were scattered to eccentric orbits.

Figure 6: Evolution of a particle’s angular momentum and orbital energy over
time as it interacts with the giant planets. The energies and angular momenta
of the giant planets are marked with letters along the $y$-axis. Energy and
angular momentum units are such that $GM_{\odot}=1$ and distance is in au.
Adapted from _Fernandez and Ip_ (1984).

Through earlier numerical experiments of particles scattering off of the giant
planets done by _Fernandez and Ip_ (1984), it was found that particles
beginning on circular orbits near the ice giants often first evolve to the
Jupiter/Saturn region before being scattered to large semimajor axes. An
example this evolution is shown in Figure 6. This is the dynamical behavior
that lowered the Oort Cloud trapping efficiency so much in _Dones et al._
(2004). Although objects scattering off of Uranus or Neptune have a large
probability of becoming trapped in the Oort cloud, most of the bodies that
begin on circular orbits near these planets see their perihelia evolve toward
Jupiter and Saturn, and, as discussed above, these planets are much more
likely to eject objects than allow them to become trapped in the Oort cloud.
The diminished role of Uranus and Neptune in placing objects into the Oort
cloud also changes the cloud’s structure. Since objects scattered by Jupiter
and Saturn have little chance of becoming trapped in the Oort cloud until they
attain semimajor axes above 10–20 thousand au, the fraction of bodies retained
with $a<20,000$ au is nearly equal to that with $a>20,000$ au (_Dones et al._
, 2004). Thus, the inner region of the cloud, thought to be unconstrained by
observed long-period comets, was predicted to be comparably populous to the
outer region beyond 20,000 au.

###  3.2 Generating a Scattered Disk

The Oort cloud as described above is thought to be populated from an initially
dynamically cold planetesimal disk in the current giant planet region.
However, the existence of the SPCs revealed the need for an extension of this
disk past Neptune to provide a low-inclination source of these non-isotropic
comets (see Section 1.2). It was then realized that the dynamical erosion of
this disk that could feed the SPCs naturally leads to the formation of a
scattered disk of objects with semimajor axes inside the inner Oort cloud
boundary and perihelia near Neptune (_Torbett_ 1989; _Duncan and Levison_
1997; though as discussed below much of the scattering population is now
thought to originate from the initial planetesimal disk located interior to
Neptune’s current orbit). Such objects were observational confirmed amongst
the early TNO discoveries (_Luu et al._ , 1997).

However, as more TNOs were discovered, the complex dynamical structures
revealed in the classical, resonant, and scattering populations (recently
reviewed by _Gladman and Volk_ , 2021) showed that the giant planets did not
form on their current orbits. Early work on the interactions between the giant
planets and the initially massive planetesimal disk showed the energy and
angular momentum exchange can cause the planets’ orbits to migrate (_Fernandez
and Ip_ , 1984). Outward migration of Neptune was then shown to lead to
capture of objects into its external mean motion resonances. The large
eccentricity of Pluto’s 3:2 resonant orbit was potentially explained by
Neptune’s outward migration by $\sim 7$ au (_Malhotra_ , 1993, 1995). The
large populations of objects in many of Neptune’s resonances that have been
found in subsequent surveys has provided strong evidence of migration-related
resonant capture.

Our picture of the dynamical history of the giant planets has evolved
significantly since the early proposals of so-called smooth planetary
migration. To explain both the complex orbital distribution of the small
bodies in the outer solar system and the dynamical state of the giant planets’
orbits, more violent scenarios have been proposed. The Nice model of the giant
planets’ evolution was one such early model; _Tsiganis et al._ (2005)
suggested that Jupiter and Saturn crossed a strong mutual mean motion
resonance while migrating, triggering an epoch of large eccentricities in all
four giant planets’ orbits that led to planet-planet encounters and even a
swap in the order of the ice giants. This specific proposed resonance crossing
has since been shown to not satisfactorily reproduce the orbits of Jupiter and
Saturn (e.g., _Brasser et al._ , 2009), but the concept of excited planetary
eccentricities, potential planet-planet gravitational encounters, or even
ejection of an ice giant (see _Nesvorný_ 2018 for a recent review of the giant
planets’ dynamical history) hold promise for explaining some of the TNO
orbital distribution features.

The current picture of the giant planets’ dynamical history is that a mix of
violent and smooth migration likely took place (see, e.g., _Nesvorný_ 2018).
Their dynamical evolution dispersed the massive disk of planetesimals that
formed in the current giant planet region into the trans-Neptunian and Oort
cloud populations. A small portion of the observed TNOs (the cold classical
population on low-eccentricity, low-inclination orbits in the 42-48 au region)
represent objects that formed in-situ beyond Neptune in a low-mass density
disk (see, e.g., results from the New Horizons fly-by of the classical TNO
Arrokoth; _McKinnon et al._ 2020). The majority of the TNO populations (the
dynamically hot populations, including the scattering population) were
scattered outward from closer-in, more massive disk with an outer edge near 30
au. A small fraction of these scattered bodies survived in metastable orbits
within the modern TNO population. This same scattering process also populated
the Oort cloud.

It is clear from modeling that smooth migration into an extended massive disk
(one that extends out to $\sim 50$ au) significantly over-populates Neptune’s
resonances compared to the observed populations (e.g., _Hahn and Malhotra_ ,
2005). The other motivation for an end to the massive initial planetesimal
disk at $\sim$30 au is also, of course, that Neptune stopped migrating where
it did (i.e., ran out of fuel for migration; _Gomes et al._ 2004). But even
assuming a transition to a less massive disk at $\sim 30$ au, something
besides smooth migration is needed to produce the detailed distribution of
objects in the classical Kuiper belt as well as the distributions of objects
in the scattering, detached, and distant resonant/near-resonant populations.
The dynamical excitation from very violent scattering events (such as in the
original Nice model) can still be consistent with retaining the cold classical
population in certain circumstances (e.g., _Batygin and Brown_ , 2010; _Dawson
and Murray-Clay_ , 2012; _Gomes et al._ , 2018), though more recent models
investigated less dramatic instabilities. Smaller scale ‘jumps’ in Neptune’s
orbit caused, for example, by the ejection of an additional ice giant (see,
e.g., _Nesvorný_ 2011; _Nesvorný and Morbidelli_ 2012), might help explain the
‘kernel’ (an over-density of objects near 45 au noted by _Petit et al._ 2011)
in the cold classical region (_Nesvorný_ , 2015b); Figure 7 shows an example
of such an evolution for the giant planets.

Figure 7: One plausible version of the giant planets’ orbital histories from
_Nesvorný and Vokrouhlický_ (2016) (modified from their Figure 3). Semimajor
axis vs time for each planet is shown, with thinner dashed lines indicating
the perihelion and aphelion distances of each planet. This migration scenario
involves an additional ice giant that is ejected after $\sim 18$ Myr, causing
moderate eccentricity excitation and small jumps in the semimajor axes of
several planets, including Neptune (shown in the inset). Neptune slowly
migrates to its final location in the final $\sim 100$ Myr of the simulation.

In addition to planet-planet interactions, the number and mass of dwarf-
planets in the planetesimal disk affects how ‘grainy’ the planets’ migration
is; scattering events between

Figure 8:  An example simulated distribution of the TNO populations from
$a=50-100$ au from _Nesvorný et al._ (2016) (modified from their Figure 1)
produced from a giant planet history scenario similar to that shown in Figure
7 that also included grainy migration of the planets due to interactions with
Pluto-mass objects. Perihelion distance (top) and inclination (bottom) vs
semimajor axis are shown for the simulated particles that ended up in the
scattering population (red dots) and the high-$a$ detached and resonant
populations (black dots); a subset of the then-observed TNOs with $q>40$ au
are shown (blue diamonds) for comparison.

Neptune and these largest objects in the disk cause small but significant
jumps in Neptune’s semimajor axis that can alter the capture rate into
Neptune’s resonances (e.g, _Murray-Clay and Chiang_ , 2005; _Nesvorný and
Vokrouhlický_ , 2016). The speed of migration, likely determined by the mass
distribution of the planetesimal disk, also affects the expected TNO
distribution. There are models that predict that some period of slow
migration, particularly during the last stage of migration after any planet-
planet scattering has occurred, can help explain the high-perihelion objects
in the scattering/detached TNO population by allowing for secular-resonant
interactions to occur, which might also help excite the inclinations of the
hot populations (e.g. _Nesvorný_ , 2015a; _Lawler et al._ , 2019; _Fraser et
al._ , 2017). Adding graininess to this slow migration can help strand objects
out of the scattering/resonant population by dropping them out of resonances
at a high-$q$ portion of this secular evolution, building up a population of
near-resonant detached objects (e.g., _Kaib and Sheppard_ , 2016; _Lawler et
al._ , 2019). Figure 8 shows an example of the scattering/detached/resonant
populations from $a\simeq 50-100$ au that are expected for a planetary
migration scenario with a small instability followed by a long period of
grainy migration (from _Nesvorný and Vokrouhlický_ 2016). Observations of the
high-$a$, high-$q$ TNOs remain relatively sparse (as indicated by the
relatively small number of observed objects included in Figure 8) due to the
strong observational biases against detecting them, but future surveys will
help provide more robust observational constraints on models of the scattering
and detached populations (see Section 4).

###  3.3 Concurrent Oort Cloud and Kuiper Belt Formation

Over the past 3 decades, evidence has mounted that planetesimal scattering
drove the giant planets through substantial orbital evolution early in the
solar system’s history. However, Oort cloud formation, a product of this
scattering, was not simulated in concert with giant planet orbital evolution
until the work of _Brasser and Morbidelli_ (2013). In this work, the authors
replicated the post-instability behavior of the surviving giant planets within
an early iteration of the Nice Model (_Levison et al._ , 2008). Neptune and
Uranus began with eccentricities above 0.2 and damped to their modern values
while their semimajor axes migrated outward 2–3 au over the course of tens of
Myrs. At the initiation of this sequence, test particles were started with
semimajor axes between 29–34 au, eccentricities of 0.15, and coplanar
inclinations. It was this outer belt from which _Brasser and Morbidelli_
(2013) assumed the Oort cloud (and scattered disk) is formed.

After 4 Gyrs of evolution _Brasser and Morbidelli_ (2013) found an Oort cloud
trapping efficiency of $\sim$7%, modestly higher than _Dones et al._ (2004)
and more than double that of _Kaib and Quinn_ (2008). While giant planet
orbital evolution may have contributed to this, the confinement of test
particles to beyond $a\geq 29$ au may also have been responsible. No particles
were begun in the immediate vicinity of Jupiter and Saturn, which are very
inefficient Oort cloud populators, while the contributions of Uranus and
Neptune were maximized.

In addition to forming the Oort cloud, the simulations of _Brasser and
Morbidelli_ (2013) also generated a scattered disk. The population of this
scattered disk was $\sim$12 times smaller than that of the Oort cloud. This
result was in line with the $\sim$10:1 population ratios found in _Dones et
al._ (2004) and _Kaib and Quinn_ (2008). Meanwhile, comparisons of LPC flux
with the steady-state populations of JFCs have historically been interpreted
as implying a $\sim$100:1, or perhaps even higher ratio (see 2.4.3 as well as
_Levison et al._ , 2010).

The planetary orbital evolution of _Brasser and Morbidelli_ (2013) utilized an
early, violent version of the Nice model that was demonstrated to conflict
with numerous aspects of the modern solar system’s architecture (_Brasser et
al._ , 2009; _Batygin et al._ , 2011; _Dawson and Murray-Clay_ , 2012). As
discussed in Section 3.2, recent detailed comparisons of TNO surveys with
simulations of Kuiper belt formation appear to favor a less violent giant
planet evolution in which Neptune’s migration was relatively smooth from 24 to
30 au except for a brief interruption during the ejection of an additional ice
giant (_Nesvorný_ , 2011, 2015a, 2015b). Incorporating this later iteration of
early giant planet evolution, _Vokrouhlický et al._ (2019) again modeled the
formation of the Oort cloud. In this instance, particles were initiated on
circular orbits between 24–30 au while Uranus and Neptune migrated several au
to their modern semimajor axes over $\sim$100 Myr timescales. Although this
work focused on modeling the comet production from such a cloud, the authors
found that $\sim$6% of their bodies were trapped in the Oort cloud after 4.5
Gyrs of evolution, in line with the results of _Brasser and Morbidelli_
(2013), likely due to the backweighted distribution of initial test particle
orbits. In addition, _Vokrouhlický et al._ (2019) found that the inner Oort
cloud has nearly the same population size as the outer Oort cloud, although
they divided the inner and outer clouds at $a=15,000$ au.

For illustrative purposes, Figure 9 shows the distribution of particle
semimajor axes, eccentricities, and inclinations after 4 Gyrs of evolution in
an Oort cloud formation simulation. This simulation, taken from _Kaib et al._
(2019), forms an Oort cloud and scattered disk concurrently with very similar
parameters to the work of _Vokrouhlický et al._ (2019). As one can see, the
scattered disk generally has perihelia between $\sim$30 au and $\sim$45 au and
inclinations below $\sim$45∘ until semimajor axes above $\sim$103 au. At this
point, perturbations from the local Galaxy begin to isotropize inclinations
and detach perihelia from the planetary region.

Figure 9: Perihelia (upper panel) and inclinations (lower panel) vs semimajor
axes of particles at $t=4$ Gyrs in a simulation forming the Oort cloud and
scattered disk concurrently. Plot is generated from the “OC” simulation of
_Kaib et al._ (2019).

###  3.4 Influence of the Sun’s Local Environment

####  3.4.1 Solar Birth Cluster

While the structure of the scattered disk should be relatively insensitive to
the Sun’s local galactic environment, perturbations from this environment are
critical to the formation and evolution of the Oort Cloud, and variations in
this environment could impact the Oort Cloud’s formation. The numerical work
of _Duncan et al._ (1987) and _Dones et al._ (2004) assumed that the Sun’s
local galactic environment did not significantly change with time. The
strength of the Galactic tide was based on the local density of matter in the
Milky Way disk inferred from stellar motion (_Bahcall_ , 1984), and the
velocities and rates of passing stars were meant to mimic the population and
kinematics of the solar neighborhood (e.g. _García-Sánchez et al._ , 2001;
_Bailer-Jones et al._ , 2018).

This static galactic environment was, of course, an approximation, and there
are reasons to expect it to have changed significantly over the Sun’s
lifetime. Perhaps most significantly, it is well-known that most stars form
within clusters and spend some portion of their early lives within them (_Lada
and Lada_ , 2003). If this occurred for the Sun, the local stellar densities
during this cluster phase would have been orders of magnitude higher than the
modern solar neighborhood, and the tide of the cluster would have dwarfed the
current Galactic tide perturbing the modern solar system. _Gaidos_ (1995)
realized that these enhanced external perturbations could have dramatic
impacts on the formation of the Oort cloud, allowing objects to become trapped
within the Oort cloud at a much lower semimajor axis than predicted for a
galactic field environment. However, _Gaidos_ (1995) argued that such a comet
cloud would also be rapidly destroyed by the same cluster perturbations and
that little material should remain at these distances.

_Fernández_ (1997) further considered the scenario in which Oort cloud
formation begins within a stellar birth cluster. He noted that the heightened
perturbations of a cluster environment would dramatically widen the energy
window of the Oort cloud and potentially allow Jupiter and Saturn to play more
prominent roles in populating the cloud. Moreover, much of this material could
reside at hundreds of au to a few thousand au, well interior to the semimajor
axis range of the “classical” Oort cloud. Moreover, _Fernández_ (1997) argued
that the relatively short lifetimes of open and embedded clusters would have
allowed many of these bodies to remain trapped, contrary to the conclusions of
_Gaidos_ (1995).

The formation of the Oort cloud within a star cluster environment was then
first simulated in _Fernández and Brunini_ (2000). In this work, particles
were initiated on semimajor axes between 100–250 au with perihelia distributed
amongst the giant planets. The entire solar system was then immersed within a
star cluster environment that contained stars passing much more slowly (1
km/s) and closely than a Galactic field environment. The stellar densities of
clusters were varied between 10 and 100 stars/pc3 and linearly declined to
zero over 108 years. In addition, _Fernández and Brunini_ (2000) included
tidal torquing from a uniform-density spherical distribution of gas that
dispersed at 107 years meant to replicate the gaseous component of an embedded
cluster. They found that an inner comet cloud readily formed under such
conditions and the radial extent (semimajor axes from a couple hundred au to a
few thousand au) depended on the stellar density of the star cluster, with
denser clusters yielding more compact clouds. Moreover, as in _Duncan et al._
(1987), the fraction of material captured into this cloud depended sensitively
on particles’ initial perihelion. Those with perihelia in the Jupiter/Saturn
region only had trapping efficiencies of a few percent, whereas tens of
percent of material could be captured into the cloud for initial perihelia
near the ice giants. Finally, the relatively short cluster lifetimes allowed
these inner clouds to largely stay intact past cluster dissolution.

Oort cloud formation within a cluster was revisited by _Brasser et al._
(2006). This work specifically focused on the embedded cluster environment, in
which most stars appear to form (_Lada and Lada_ , 2003). The embedded cluster
environments employed within _Brasser et al._ (2006) had higher central
densities (in gas and stars) and shorter lifetimes than those used in
_Fernández and Brunini_ (2000). Due to the large energy window of the comet
cloud and the short cluster lifetimes, roughly 10% of material scattered by
Jupiter and Saturn could be caught in an inner comet cloud, whose range of
semimajor axes again depended on the assumed density of the cluster
environment (see Figure 10). Nearly all of this inner comet cloud material
would be retained until the present epoch and very little would be lost or
diffuse to the classical Oort cloud beyond $a>20,000$ au (_Brasser et al._ ,
2008).

Figure 10: Median position of bodies within the Oort cloud as a function of
the mean density of a cluster environment for a given Oort cloud formation
simulation. From _Brasser et al._ (2006).

Additional investigations by _Brasser et al._ (2007) studied the effect of the
gaseous component of the Solar Nebula during the formation of the Oort cloud
within a star cluster. For small km-sized bodies, aerodynamic drag prevented
objects from being scattered to large semimajor axes. Instead, they
concentrated into nearly circular orbits interior to Jupiter. Meanwhile,
objects with radii of tens of km were decoupled from the gas and freely
reached the Oort cloud. If the gaseous component of the Sun’s protoplanetary
disk outlived the Sun’s birth cluster, this suggested that the centrally
concentrated, cluster-generated comet cloud should only contain large
($r\gtrsim 10$ km) bodies.

_Brasser et al._ (2012a) repeated the embedded cluster scenario with improved
simulations that used updated, observationally motivated cluster profiles and
stellar velocity distributions. Furthermore, in this new work cluster stars
were able to gravitationally interact with one another. While centrally
concentrated Oort clouds still formed down to semimajor axes of several
hundred au, the formation efficiency dropped to 1–2% due to the Oort cloud
harassment that occurred with a heightened number of stellar interactions in
these new environments.

Complementing studies of Oort cloud formation within embedded clusters, _Kaib
and Quinn_ (2008) simulated the formation of the Oort cloud within longer-
lived ($t\simeq 10^{8}$ years) open cluster environments. Unlike the work of
_Fernández and Brunini_ (2000), particles were initiated on circular orbits.
As expected, this allowed particles near Uranus and Neptune to evolve to
Jupiter- and Saturn-crossing orbits before being scattered to large semimajor
axes. This led to lower ($<10$%) trapping efficiencies than predicted by
_Fernández and Brunini_ (2000). Owing to the clusters’ longer lifetimes and
the more prominent role of powerful stellar encounters compared to the gas-
rich environments of _Brasser et al._ (2006), the Oort clouds formed within
open clusters displayed lower overall trapping efficiencies ($\lesssim 5$%)
and a greater level of stochasticity in the degree of central cloud
concentration (which was largely dependent on the few most powerful stellar
encounters). Nonetheless, this work showed that an Oort cloud formed within an
open cluster should possess bodies on semimajor axes from several hundred to
several thousand au whose numbers are comparable to those in the classical
Oort cloud.

Discovered Objects as Possible Birth Cluster Relics

Beginning with the pioneering works of _Gaidos_ (1995) and _Fernández_ (1997),
it has been predicted that if the Sun formed within a star cluster, we should
expect the inner edge of the Oort cloud to potentially reside at a semimajor
axis of hundreds of au rather than the thousands of au suggested by models
that only consider the modern solar neighborhood. These predictions seem
extremely prescient, as in the early 2000’s TNO surveys began discovering
objects with semimajor axes of hundreds of au but perihelia that dynamically
decoupled them from planetary energy kicks (_Gladman et al._ , 2002). Although
some of these bodies may attain their high perihelia via Kozai-Lidov cycles
within Neptunian mean motion resonances, this mechanism cannot generate all
observed orbits (_Brasser and Schwamb_ , 2015; _Gomes et al._ , 2005; _Kozai_
, 1962; _Lidov_ , 1962). In particular, the objects Sedna and 2012 VP113 both
have perihelia beyond 70 au and semimajor axes beyond 250 au and are therefore
not susceptible to Kozai-Lidov cycling (_Brown et al._ , 2004; _Trujillo and
Sheppard_ , 2014).

The discoveries of such objects appear consistent with the existence of a
cluster-generated Oort cloud (_Morbidelli and Levison_ , 2004). However, other
explanations have been put forth, such as scattering off of an $\sim$Earth-
mass planetesimal or exchange with another young star system (_Gladman and
Chan_ , 2006; _Kenyon and Bromley_ , 2004; _Silsbee and Tremaine_ , 2018). Of
particular note, it has been posited that these detached TNOs with large
semimajor axes are the dynamical signature of a distant, yet-undetected planet
(_Batygin and Brown_ , 2016a). Such a planet has been shown to be capable of
aligning the poles and perihelion directions of an ensemble of such orbits,
and the detected orbits appear to display an anomalous anisotropy (_Batygin
and Brown_ , 2016a). However, the total sample size of objects remains small,
and the observing biases associated with combining the results of many TNO
surveys are complex (_Shankman et al._ , 2017a; _Kaib et al._ , 2019).
Detections from well-characterized surveys do not find evidence of significant
orbital clustering (_Shankman et al._ , 2017a; _Napier et al._ , 2021;
_Bernardinelli et al._ , 2021).

Oort Cloud Capture within a Birth Cluster

Another aspect of Oort cloud formation within a cluster environment concerns
the possibility of the Sun capturing exoplanetesimals, or bodies formed around
another star cluster member and subsequently ejected into intracluster space.
This possibility was first considered in _Eggers et al._ (1997) and then again
in _Zheng et al._ (1990). Both works found that forming our entire Oort cloud
through the capture of exoplanetesimals within a cluster is unlikley. Such a
scenario would require 1016 planetesimals per star, which would be difficult
to reconcile with the (at the time) non-detection of interstellar comets
passing through the solar system (_Zheng et al._ , 1990).

However, the capture probabilities used in those early works may have been
underestimates. _Levison et al._ (2010) directly simulated the dynamical
evolution and dissolution of embedded clusters in which each star was
surrounded by a disk of distant planetesimals. Over time, many planetesimals
became unbound from their stars-of-origin via cluster tides and star-star
encounters. A snapshot of one simulation is shown in Figure 11. In these
simulations, it was found that stars often captured other stars’
planetesimals. As stars and intracluster planetesimals escaped the potential
of the dissolving cluster, some star and planetesimal velocity vectors were
inevitably similar. The escape from the potential decreased relative
velocities even further, enabling capture. In this scenario, stars typically
captured a couple percent of their original planetesimal number through this
mechanism. Thus, the efficiency of such capture should generally not exceed
the efficiency of the solar system capturing its own scattered material
through the “classic” Oort cloud formation process. However, occasionally
direct exchange of planetesimals through a close star-star encounter led to
capture fractions over 10%. From these results, _Levison et al._ (2010) argued
that a significant fraction, and perhaps the majority, of bodies in the Oort
cloud could have originated around other stars. However, many of these bodies
were captured onto distant orbits with semimajor axes of tens of thousands of
au, and their subsequent evolution and survival within the Galaxy until the
present epoch was not modeled.

Figure 11: Snapshot of a simulation of Oort cloud capture within an embedded
cluster. Datapoints mark the Cartesian X and Y positions of simulation
particles. Yellow and red points are cluster stars, while green and blue
points mark planetesimals. Blue planetesimals become bound to the red cluster
star after cluster dissolution. From _Levison et al._ (2010).

####  3.4.2 Galactic Migration

Even after the Sun’s birth cluster disperses, there is ample reason to expect
significant changes in the Sun’s local galactic environment. Dynamical
simulations of spiral galaxies indicate that the galactocentric orbits of most
stars are not static with time. Instead, they often exchange angular momentum
with the spiral arms, typically within corotation resonances, whose locations
fluctuate as spiral wave pattern speed varies (_Sellwood and Binney_ , 2002;
_Roškar et al._ , 2008). On average, this transports disk stars outward over
time (although inward transport is also possible). Importantly, this process
does not necessarily increase the peculiar velocities of stars, and they can
remain on nearly circular orbits about the galactic center with modest
oscillations above and below the galactic midplane.

This stellar migration process obviously has large ramifications for our
understanding of spiral galaxies and their properties, but it has also raised
the prospect that the Sun may have undergone significant migration (likely
outward) during its lifetime in the Milky Way. If this were the case, this may
have significant implications for the formation of our Oort cloud. As the
Sun’s galactocentric distance changes, the mean density of its local
environment will also change, which will modulate the rate of stellar
encounters and the strength of the Galactic tide (_Matese et al._ , 1995).
_Brasser et al._ (2010) conducted simulations of Oort cloud formation at
various galactocentric distances of 2, 4, 6, 8, 14, and 20 kpc using an
analytic model for the Milky Way potential. Unsurprisingly, the different
environmental densities varied the Oort cloud inner edge (which was defined as
the 5th percentile in semimajor axis) from 30,000 au at $r=20$ kpc to 2,000 au
at $r=2$ kpc. In addition, the Oort cloud trapping efficiency varied between
2–4% and did not display a strong dependence on galactocentric distance.

However, the simulations of _Brasser et al._ (2010) still assumed a static
(local and global) galactic environment and tide for the each entire
simulation. In fact, stellar migration within galaxies implies that the local
environment and tide should be changing with time. The work of _Kaib et al._
(2011) addressed this by merging Oort cloud formation simulations with data
output from the evolution of a Milky Way analog spiral galaxy (_Roškar et al._
, 2008). Within the galactic simulation, an ensemble of 31 star particles were
selected that possessed Sun-like galactic kinematics and ages. The histories
of these particles were then backtraced throughout the simulation. These
particles displayed a wide variety of behaviors. On average, their
galactocentric distance varied by $\sim$5 kpc throughout their lifetimes, and
the closest approach to and furthest excursion from the Galactic center was 2
kpc and 13 kpc, respectively. At each galactic simulation time output, the
local density and galactic tidal field about each star particle was measured,
and with this data, a time-varying set of external perturbations (galactic
tide and passing field stars) was built for each star particle, or for each
example of a galactic dynamical history of a Sun-like star. These perturbation
sets were then used to model the formation of the Oort cloud for 31
hypothetical dynamical histories within the Milky Way.

_Kaib et al._ (2011) found that the galactic dynamical histories of their star
particles had a significant impact on Oort cloud formation, with the median
semimajor axis of the Oort cloud varying by nearly an order of magnitude
across the sample of star particles. Generally, the minimum galactocentric
distance attained, which typically determined the maximum local density of
matter inhabited, had the largest impact on the Oort cloud structure.
Excursions toward the Galactic center resulted in more centrally concentrated
clouds. Overall, Oort cloud formation models that do not account for the Sun’s
dynamical history within the Galaxy likely overestimate the cloud’s inner edge
and median semimajor axis.

In addition, the time at which a star attains its minimum galactocentric
distance affected the Oort cloud trapping efficiency. If the minimum
galactocentric distance was attained early in the system’s life, the trapping
efficiency was enhanced (to 3–4%), because the Oort cloud energy window was
widened early in the system’s history, and it was subjected to somewhat weaker
galactic perturbations later in time. Conversely, late excursions to small
galactocentric distances decreased Oort cloud trapping efficiency to 1–2%, as
the energy window of the Oort cloud remained small early in the system’s
history, and the late inward excursion drove the loss of weakly bound bodies.

####  3.4.3 Critical Roles of Stellar Passages

As illustrated in Figure 12, _Kaib et al._ (2011) also revealed that Oort
cloud structure contains significant variation even after the Sun’s potential
migration history is considered, especially with respect to the Oort cloud’s
inner edge. The main source of this scatter is due to the effects of stellar
passages. Since _Heisler and Tremaine_ (1986) showed the galactic tide to be a
more powerful mean perturbing force than stellar passages, works studying the
formation of the Oort cloud have often focused more on Galactic tidal effects
and less on stars. However, the conclusions of _Heisler and Tremaine_ (1986)
are only true on average. Passages of field stars are inherently stochastic
events that occur on very short ($<10^{5}$ years) timescales, so there can be
great variability in their significance from one epoch to another. Therefore,
over the course of 4.5 Gyrs, there must have been short periods where the
perturbation from stellar passages on the Oort cloud greatly exceeded the
perturbation from the Galactic tide. Because the time to be scattered into the
Oort cloud is much shorter than the age of the solar system, these brief,
stellar-dominated periods of perturbation are ultimately what sets the very
minimum semimajor axis at which the perihelia of bodies are torqued into the
Oort cloud.

Figure 12: Plot of maximum (squares, 95th percentile), median (crosses, 50th
percentile), and minimum (diamonds 5th percentile) semimajor axis of Oort
cloud as a function of the minimum galactocentric distance attained by the
solar analog within a galactic N-body simulation. Adapted from _Kaib et al._
(2011).

When accounting for the role of stellar passages setting the Oort cloud’s
inner edge, there is a large amount of stochasticity since it depends on the
few most powerful stellar passages over the Sun’s history. To estimate the
Oort cloud’s very inner edge, _Kaib et al._ (2011) examined orbits that had
only been weakly torqued from the planets ($60<q<100$ au) and measured the
tenth percentile of semimajor axis. With this parameterization, they found
that the Oort cloud’s inner edge can vary by a factor of several (even with a
fixed local galactic environment) due to the stochastic nature of stellar
passages. In fact, this inner edge was pushed interior to Sedna’s orbit in
$\sim$20% of their simulations, raising the prospect of whether a solar birth
cluster (or other alternative mechanism) is necessary to explain this orbit.
These effects of the most powerful stellar encounters can also be seen in the
smattering of low-semimajor-axis particles with detached perihelia in Figure
9. An additional analysis of the stellar encounters experienced by the Sun for
different galactic migration scenarios largely agreed with these results
(_Martínez-Barbosa et al._ , 2017).

Stellar encounters also play a critical role in maintaining the Oort cloud’s
orbital isotropy. The vertical term of the Galactic tide causes the orbital
elements of Oort cloud bodies to fluctuate in a very regular manner (_Heisler
and Tremaine_ , 1986). Consequently, left unperturbed, the Oort cloud would
exhaust the set of orbits that can attain very low ($q<15$ au) perihelion
under the action of the Galactic tide, since these bodies are generally lost
when they interact with Jupiter and Saturn. Before these regions of orbital
space are completely depleted by the Galactic tide, however, the randomizing
perturbations of stellar encounters replenish them (_Rickman et al._ , 2008).
Thus, although the Galactic tide delivers most Oort cloud bodies into the
planetary region as LPCs, stellar encounters are necessary to ensure that an
abundant supply of Oort cloud bodies remain within the phase space that yields
LPCs.

Recent Stellar Encounters with the Sun

As our census of the Sun’s nearby stellar neighbors has improved, numerous
works have searched for particularly close stellar passages in the recent past
as well as the short-term future ($\lesssim\pm$10 Myrs). Using Hipparcos data,
_García-Sánchez et al._ (2001) found several potential encounters within 1 pc
of the Sun within $\pm$10 Myrs. In particular, they noted that the K star
Gliese 710 should approach the Sun within 0.4 pc roughly 1.8 Myrs from now.
None of the anticipated encounters were projected to deliver significant
perturbations to the Oort cloud, but _Rickman et al._ (2012) noted that the
Hipparcos catalog was perhaps only $\sim$10–20% complete over this timeline
and uncertainties within astrometry and radial velocity prevented confident
periastron calculations for close stellar encounters (_García-Sánchez et al._
, 2001).

The ongoing Gaia mission is assembling the three-dimensional space velocities
of over 1 billion stars, four orders of magnitude greater than Hipparcos, and
its astrometric precision will be two orders of magnitude finer than
Hipparcos. _Rickman et al._ (2012) concluded that the final data release of
Gaia (anticipated in 2022) should reveal $\sim$90% of the stellar encounters
that have contributed to the flux of currently observable long-period comets.
With the second release of Gaia data, _Bailer-Jones et al._ (2018) found that
$\sim$20 stars per Myr approach the Sun within at least 1 pc. In addition,
with the improved astrometry of Gaia, they found that Gliese 710 has a 95%
probability of coming within 17,000 au of the Sun, and later refinements of
this star’s kinematics showed that it will come within 104 au 1.3 Myrs from
now (_de la Fuente Marcos and de la Fuente Marcos_ , 2020). The stellar
encounter flux of _Bailer-Jones et al._ (2018) implies that such a close
encounter should only be expected every $\sim$10 Myrs, and this is the closest
known stellar encounter with the Sun. The improved stellar radial velocities
coming with the third Gaia data release may still reveal other close passing
stars in the near future or recent past.

####  3.4.4 Comet Showers

Another well-studied effect that stellar encounters have on the Oort cloud
dynamics are comet showers. Typically, the only bodies in the Oort cloud that
can undergo a perihelion shift from beyond Saturn’s orbit to near Earth’s
orbit within a single revolution about the Sun have semimajor axes $\gtrsim
20,000$ au (_Heisler and Tremaine_ , 1986). However, as described in the last
section, the perturbative strength of field star encounters varies wildly over
time. During the most powerful stellar passages that have occurred in the
solar system’s history, the perturbations delivered on the Oort cloud are
temporarily strong enough to enable this perihelion shift for virtually any
semimajor axis in the Oort cloud. This led _Hills_ (1981) to argue that most
of the time, the Earth is only exposed to LPCs from the outer periphery of the
Oort cloud, but during powerful stellar passages, the Earth is temporarily
exposed to the entire Oort cloud, meaning that the flux of LPCs and potential
Earth impactors could increase greatly during these events, which are called
comet showers.

The potency of comet showers clearly depends on how well-populated the inner
($a\lesssim$ 20,000 au) portion of the Oort cloud is relative to the outer
cloud. _Hills_ (1981) assumed the inner Oort cloud could hold up to 100 times
as many bodies as the outer cloud, and under this assumption, the most
powerful stellar encounter of the past 500 Myrs would temporarily increase the
flux of comets near Earth by $\sim$3 orders of magnitude. _Hut et al._ (1987)
proposed that the impacts associated with such events could be responsible for
one or more of Earth’s mass extinctions. However, subsequent Oort cloud
formation models predicted a much more modest inner Oort cloud population that
is just 1–2 times larger than the outer Oort cloud (_Dones et al._ , 2004).

Since it was presumed that Jupiter and Saturn prevented the inner Oort cloud
from contributing to the known catalog of LPCs (the vast majority of which
have perihelia of a few au or less; _Marsden and Williams_ , 2008), estimates
of the inner Oort cloud population appeared to be completely model dependent.
However, _Kaib and Quinn_ (2009) showed that objects from the inner Oort cloud
can also evolve to become observable long-period comets. Illustrated in Figure
13, as the perihelia of inner Oort cloud bodies slowly march Sunward through
the outer solar system, some can incur a set of planetary energy kicks that
are too weak to eject them but sufficiently strong to inflate their semimajor
axes above $\sim$20,000 au. At this point, their perihelia can evolve rapidly
toward Earth in a single orbital period, allowing them to become discovered
with semimajor axis that obscure their inner Oort cloud source region.

Figure 13: Plot of the orbital evolution of an Oort cloud body as it enters
the solar system’s planetary region. Data points mark the semimajor axis vs
the perihelion each time the body enters the inner 50 au of the solar system.
The evolution begins at the star data point and ends at the square data point.
Letters mark the giant planet orbits along the $x$-axis. The shaded region
marks Oort cloud semimajor axes historically assumed to supply observed LPCs,
while unshaded semimajor axes had previously been assumed to be incapable of
producing known LPCs.

This dynamical pathway was found to be roughly half as efficient at producing
observable LPCs as the classic outer Oort cloud production (_Kaib and Quinn_ ,
2009). Thus, the current known population of LPCs implies that the inner Oort
cloud population cannot be much greater than $\sim$two times the number of
comets that had previously been estimated to occupy the outer Oort cloud.
Otherwise the comet flux would be higher than what is observed. This rather
modest inner Oort cloud population calls into question the significance of
comet showers’ relationship to mass extinctions.

##  4 FUTURE DEVELOPMENTS

###  4.1 Rubin Observatory

With a projected first light in the early to mid-2020s, the Vera C. Rubin
Observatory promises to revolutionize our understanding of the small body
populations of the solar system as it conducts the Legacy Survey of Space and
Time (LSST) over ten years (_Ivezić et al._ , 2019). This observatory and
survey is projected to discover tens of thousands of TNOs as well as provide
the most extensive and uniform set of comet observations and discoveries in
history. This will undoubtedly enhance our understanding of comets and their
reservoirs.

With an expected limiting magnitude of $\sim 24.5$ covering all of the
southern sky and the northern sky near the ecliptic plane, LSST’s observations
of the TNO populations will dwarf the present TNO catalog. To date, surveys
reaching those limiting magnitudes have only covered small portions of the sky
(e.g., the $\sim 120$ deg2 Outer Solar System Origins Survey; _Bannister et
al._ 2018), with all-sky surveys more typically limited to magnitudes brighter
than $\sim 22$ (e.g., PanStarrs; _Chambers et al._ 2016). LSST’s combined wide
sky coverage and deep limiting magnitude will dramatically increase the number
of observed TNOs in the scattering and detached populations. This is critical
to improving our models of both the intrinsic orbital and size distributions
of these populations. Current constraints on these populations have been
limited by the fact that that these TNOs spend essentially all of their time
at distances where they are un-observable. This limits the ability to directly
de-bias observations (by weighting the observed objects by observability) to
build a picture of the intrinsic population because the small current sample
sizes make it likely we are missing examples from key regions of their orbital
distributions. Likewise, forward modeling via the use of survey simulators to
test models of the intrinsic scattering/detached TNO population against
observations (see, e.g., _Lawler et al._ 2018a) must be done carefully because
portions of the model phase space will be invisible, and thus un-testable. The
much larger observational sample expected from LSST, and the fact that it will
come from a coherent survey with (hopefully) understood biases will lead to
more detailed models of the scattering and detached populations. The very wide
latitude range of the observations (especially in the southern sky) will also
provide strong new constraints on the inclination distribution of the TNOs.
This will provide better tests of solar system formation and giant planet
migration scenarios, which will also inform ideas of where different kinds of
comets formed. The improved size distribution measurements along with improved
orbital distributions will also yield higher-fidelity models of the rate at
which comets are supplied to the inner solar system and perhaps help inform
models of cometary physical evolution.

Rubin comet observations will dramatically improve our observational
constraints on the Oort cloud. At present, our understanding of how cometary
absolute magnitudes ($H_{T}$) relate to nuclear magnitudes is extremely
uncertain. For each comet discovery, LSST will possess perhaps 50 separate
temporally spaced observations made with the same instrument. This will allow
us to precisely measure the photometric index of each comet as well as observe
the comet at epochs when its brightness is less coma-dominated. In this
manner, we can hope to measure nuclear magnitudes much more precisely and
therefore estimate the size distribution of comets much more precisely. This
in turn will enable much better mass estimates of the Oort cloud.

In addition, the Rubin observatory will build up a catalog of LPCs with larger
perihelia than most of those that have been previously discovered. This offers
a new dynamical window to the Oort cloud because in order for a LPC to attain
a perihelion within the terrestrial planet region ($q\lesssim 5$ au), it must
either already have possessed a large ($a\gtrsim 20,000$ au) semimajor axis,
or it must have attained one through previous encounter(s) with the giant
planets at larger perihelia (see Figure 12). Thus, the true semimajor axis
distribution of Oort cloud bodies is obscured within our current sample of
comets. As the sample of observed cometary perihelia approach and even exceed
10 au, the semimajor axis distribution of LPCs should change, and these
changes will depend on the orbital distribution of bodies within the Oort
cloud (_Silsbee and Tremaine_ , 2018).

Rubin’s anticipated TNO discoveries will also be immensely helpful in
furthering our understanding of the Oort cloud. As discussed previously,
highly inclined Centaurs are a class of objects that potentially originate
from the Oort cloud. Their larger sizes and more distant perihelia offer a
different set of constraints on the Oort cloud. Currently, $\sim$20 such
bodies are known with perihelia beyond Saturn’s orbit and inclinations over
45∘, and these have been detected with an amalgam of different surveys, making
it difficult to account for observational biases within the current catalog.
We should expect hundreds of such objects detected by the single LSST survey,
giving us a less biased catalog of highly inclined centaurs, making it more
straightforward to infer their ultimate dynamical source.

LSST will also be capable of detecting Sedna-like bodies out to distances of
over 100 au (_Ivezić et al._ , 2019). Thus, our sample of TNOs with large
semimajor axes and perihelia very decoupled will increase by many fold. This
will help us discern whether the Sun’s cluster environment, a distant planet,
or some other mechanism placed these objects on their current orbits.

###  4.2 Occultation Searches

Observing $\sim$km-sized objects in-situ within the scattered disk or Oort
cloud is not viable due to their extreme faintness. Moreover, the uncertainty
of converting cometary brightness into nucleus size leaves our estimates of
the population of $\sim$km-sized, or comet-sized, bodies within these
reservoirs poorly constrained. While the crater size distribution observed by
the New Horizons mission may offer some clues for the scattered disk (_Singer
et al._ 2019; _Morbidelli et al._ 2021; see also Section 2.2), stellar
occultations may be the only tool to detect sizeable numbers of km-sized
bodies as they orbit within the scattered disk and Oort cloud (_Bailey_ ,
1976; _Nihei et al._ , 2007; _Ortiz et al._ , 2020). When such objects pass in
front of a star, the star’s brightness will dim for a fraction of a second
(typically, for a km-sized body at least tens of au from Earth).

At present only a handful of serendipitous stellar outer solar system
occultations have been observed. Using high-cadence stellar observations taken
with HST’s Fine Guidance Sensors, _Schlichting et al._ (2009) announced the
discovery of a stellar occultation of a $r\simeq 500$ m object at $\sim$45 au.
With this single detection, they concluded that there was a deficiency of sub-
km bodies within the Kuiper belt (and potentially the scattered disk) compared
to a simple extrapolation of the power-law size distribution observed for TNO
sizes above 100-km diameters. Analyzing additional HST guidance sensor data,
_Schlichting et al._ (2012) discovered a second occultation by a similarly
sized TNO at 40 au, which confirmed their previous conclusions about a
deficiency of sub-km TNOs relative to an extrapolation of the $\sim$100-km TNO
size distribution. The Taiwanese-American Occultation Survey (TAOS) also
searched for such events by monitoring $\sim$500 stars at 5 Hz cadence for 7
years (_Zhang et al._ , 2013). Although no occultation events were discovered,
this non-detection placed an upper limit on the slope of the TNO size
distribution below 90 km. A second generation survey is set to begin
operations in 2022 (TAOS-II) that will monitor over 104 stars at 20 Hz cadence
(_Lehner et al._ , 2012). This occultation survey has a potential rate
detection that is 100 times greater than its predecessor and will of course
provide much tighter constraints on the TNO size distribution between
$0.5\lesssim r\lesssim 30$ km. Although these constraints will largely be
supplied by the classical Kuiper belt, they can likely be extrapolated to the
scattered disk.

Owing to its much larger average distance, occultation events by km-sized Oort
cloud objects will generate much smaller decreases in stellar brightness
(_Nihei et al._ , 2007). This may place the detection of km-sized Oort cloud
occultations beyond the realm of ground-based surveys (_Nihei et al._ , 2007).

##  5 Summary

New short-period comet and long-period comets are steadily produced from the
large, distant reservoirs of the scattered disk and the Oort cloud,
respectively. The scattered disk was generated early in the solar system’s
history via the dispersal of a massive primordial belt of planetesimals
exterior to the formation region of the giant planets. This dispersal occurred
when Neptune migrated outward by several au and the giant planets likely
passed through an orbital instability (e.g. _Malhotra_ , 1993; _Nesvorný_ ,
2018). Less than one percent of these bodies were captured into the modern TNO
populations, and an even smaller fraction now reside in the scattered disk.
While some of these distant TNOs are actively scattering off of the giant
planets today (the scattering TNOs, a fraction of which are destined to become
JFCs), many are stored in mean motion resonances with Neptune and/or on orbits
whose perihelia are dynamically detached from the planets. These detached
orbits become unstable on timescales comparable to the solar system’s age and
seed the production of new scattering objects and JFCs. Our best current
observational constraints imply a population of $\sim 10^{5}$ scattering TNOs
larger than $\sim 100$ km in diameter (_Lawler et al._ , 2018b).
Extrapolations down to comet-sized objects are still very uncertain; models of
the delivery of short period comets imply a few times $10^{8}$ scattering TNOs
larger than $\sim 2$ km in diameter (_Nesvorny et al._ , 2017).

Meanwhile, up to $\sim$5% percent of the dispersed primordial belt bodies are
captured into the Oort cloud at semimajor axes of $\sim$103-5 au (_Dones et
al._ , 2004). Most of this capture occurs during the first Gyr of the solar
system’s history. As the giant planets scatter these primordial belt bodies to
larger semimajor axes, perturbations from passing field stars and the Galactic
tide torque the perihelia of some of these bodies out of the planetary region,
ending the scattering process and (at least temporarily) saving them from
ejection. Although most dispersed bodies ultimately scatter off of Jupiter,
this planet is more prone to ejecting bodies than emplacing them into the Oort
cloud because the total orbital energy window of the Oort cloud is smaller
than the typical energy kick delivered by Jupiter during a single encounter
(_Fernandez and Ip_ , 1984; _Duncan et al._ , 1987). Neptune and Uranus are
much more efficient populators of the Oort cloud but are less likely to retain
dynamical control of bodies throughout the scattering process.

Simulations of the formation of the Oort cloud and scattered disk suggest that
$\sim$10 times as many bodies should reside in the Oort cloud than in the
scattering population (_Dones et al._ , 2004). However, LPC observations
suggest that the Oort cloud contains 7–8 $\times 10^{11}$ comet-size bodies
(yielding an uncertain mass of 0.5–5 M⊕), which is a factor of $\sim$50–1000
times larger than scattered disk population estimates (_Brasser and
Morbidelli_ , 2013; _Levison et al._ , 2010). This potentially represents a
major discrepancy between formation models and comet observations, but this
population ratio estimate relies on a highly uncertain conversion of cometary
magnitude to nucleus size.

Simulations have shown that the structure of the Oort cloud is highly
sensitive to the Sun’s dynamical history within the Galaxy, in particular its
stellar birth cluster and the degree of radial migration it has undergone
within the Milky Way (_Fernández_ , 1997; _Kaib et al._ , 2011). Both of these
effects can push the Oort cloud’s range of semimajor axes closer to the Sun,
potentially as close as several hundred au. The exact inner edge of the Oort
cloud is also very dependent the handful of closest stellar encounters that
have occurred over the Sun’s history, and the process of setting the inner
edge is therefore quite stochastic (_Kaib et al._ , 2011). While it has been
previously speculated that the number of comet-sized bodies in the interior
20,000 au of the Oort cloud could be many times the number that are exterior
to 20,000 au, we now know that LPCs can be generated from Oort cloud semimajor
axes as small as $\sim$5000 au (_Kaib and Quinn_ , 2009). This finding places
an upper limit of $\sim$1012 comet-sized bodies orbiting beyond $\sim$5000 au
in the Oort cloud.

To date, much of our understanding of the population and orbital structure of
the Oort cloud relies upon observations of comets passing near Earth. In the
coming decade, the Vera C. Rubin Observatory will increase the inventory of
small solar system bodies by many fold (_Ivezić et al._ , 2019). Its
discoveries will include a sample of comets discovered and observed at large
($\gtrsim$10 au) perihelion and heliocentric distance that will have
experienced less thermal processing (by the Sun) and dynamical processing (by
Jupiter) compared to the historical comet catalog. In addition, we expect LSST
to increase the inventory of observed TNOs by an order of magnitude, including
discovering many more objects at high ($\gtrsim$ 45∘) inclinations and with
large ($\sim$40–100 au) perihelia. Finally, we anticipate the scope and
sensitivity of stellar occultation searches to advance in the coming decade,
and this should provide new, comet-independent constraints on the number of
km-sized bodies in the scattered disk (_Lehner et al._ , 2012). However, a
similar effort for the Oort cloud may require a space-based campaign (_Nihei
et al._ , 2007). These anticipated discoveries will provide new windows into
the population and structure of the scattered disk and the Oort cloud, whose
properties offer clues to the giant planet’s orbital evolution as well as the
Sun’s dynamical history.

Acknowledgments.

We thank reviewer Julio Fernández and another anonymous reviewer for comments
and suggestions that greatly improved this review. NAK thanks the Department
of Astronomy at Case Western Reserve University for hosting him as a visitor
during this work’s preparation. NAK acknowledges support from NSF CAREER award
1846388 and NASA Emerging Worlds grant 80NSSC18K0600. KV acknowledges support
from NSF (grant AST-1824869) and NASA (grants 80NSSC19K0785 and
80NSSC21K0376).

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