FORMATION AND COLLISIONAL EVOLUTION OF KUIPER
BELT OBJECTS
Scott J. Kenyon
Smithsonian Astrophysical Observatory
Benjamin C. Bromley
Department of Physics, University of Utah
David P. O’Brien
Planetary Science Institute
Donald R. Davis
Planetary Science Institute
This chapter summarizes analytic theory and numerical calculations for the formation and
collisional evolution of KBOs at 20–150 AU. We describe the main predictions of a baseline
self-stirring model and show how dynamical perturbations from a stellar flyby or stirring
by a giant planet modify the evolution. Although robust comparisons between observations
and theory require better KBO statistics and more comprehensive calculations, the data are
broadly consistent with KBO formation in a massive disk followed by substantial collisional
grinding and dynamical ejection. However, there are important problems reconciling the
results of coagulation and dynamical calculations. Contrasting our current understanding of
the evolution of KBOs and asteroids suggests that additional observational constraints, such
as the identification of more dynamical families of KBOs (like the 2003 EL61 family), would
provide additional information on the relative roles of collisional grinding and dynamical
ejection in the Kuiper Belt. The uncertainties also motivate calculations that combine collisional
and dynamical evolution, a ‘unified’ calculation that should give us a better picture of KBO
formation and evolution.
1. INTRODUCTION
Every year in the Galaxy, a star is born. Most stars
form in dense clusters of thousands of stars, as in the Orion
Nebula Cluster (Lada and Lada 2003; Slesnick et al. 2004).
Other stars form in small groups of 5–10 stars in loose asso-
ciations of hundreds of stars, as in the Taurus-Auriga clouds
(Gomez et al. 1993; Luhman 2006). Within these associa-
tions and clusters, most newly-formed massive stars are bi-
naries; lower mass stars are usually single (Lada 2006).
Large, optically thick circumstellar disks surround
nearly all newly-formed stars (Beckwith and Sargent 1996).
The disks have sizes of ∼ 100–200 AU, masses of ∼ 0.01-
0.1 M⊙, and luminosities of ∼ 0.2–1 L⋆, where L⋆ is the
luminosity of the central star. The masses and geometries
of these disks are remarkably similar to the properties of
the minimum mass solar nebula (MMSN), the disk required
for the planets in the solar system (Weidenschilling 1977b;
Hayashi 1981; Scholz et al. 2006).
As stars age, they lose their disks. For solar-type stars,
radiation from the opaque disk disappears in 1–10 Myr
(Haisch et al. 2001). Many older stars have optically thin
debris disks comparable in size to the opaque disks of
younger stars but with much smaller masses, . 1M⊕, and
luminosities, . 10−3 L⋆ (Chapter by Moro-Martin et al.).
The lifetime of this phase is uncertain. Some 100 Myr-old
stars have no obvious debris disk; a few 1–10 Gyr-old stars
have massive debris disks (Greaves 2005).
In the current picture, planets form during the transi-
tion from an optically thick protostellar disk to an opti-
cally thin debris disk. From the statistics of young stars in
molecular clouds, the timescale for this transition, ∼ 105
yr, is comparable to the timescales derived for the forma-
tion of planetesimals from dust grains (Weidenschilling
1977a; Youdin and Shu 2002; Dullemond and Dominik
2005) and for the formation of lunar-mass or larger
planets from planetesimals (Wetherill and Stewart 1993;
Weidenschilling et al. 1997a; Kokubo and Ida 2000; Nagasawa et al.
2005; Kenyon and Bromley 2006). Because the grains
in debris disks have short collision lifetimes, . 1 Myr,
compared to the ages of their parent stars, & 10 Myr,
high velocity collisions between larger objects must main-
tain the small grain population (Aumann et al. 1984;
Backman and Paresce 1993). The inferred dust produc-
tion rates for debris disks around 0.1–10 Gyr old stars,
∼ 1020 g yr−1, require an initial mass in 1 km ob-
http://arxiv.org/abs/0704.0259v1
jects, Mi ∼ 10–100 M⊕, comparable to the amount of
solids in the MMSN. Because significant long-term de-
bris production also demands gravitational stirring by
an ensemble of planets with radii of 500–1000 km or
larger (Kenyon and Bromley 2004a; Wyatt et al. 2005),
debris disks probably are newly-formed planetary sys-
tems (Aumann et al. 1984; Backman and Paresce 1993;
Artymowicz 1997; Kenyon and Bromley 2002b, 2004a,b).
KBOs provide a crucial test of this picture. With objects
ranging in size from 10–20 km to ∼ 1000 km, the size dis-
tribution of KBOs yields a key comparison with theoretical
calculations of planet formation (Davis and Farinella 1997;
Kenyon and Luu 1998, 1999a,b). Once KBOs have sizes
of 100–1000 km, collisional grinding, dynamical perturba-
tions by large planets and passing stars, and self-stirring by
small embedded planets produce features in the distribu-
tions of sizes and dynamical elements that observations can
probe in detail. Although complete calculations of KBO
formation and dynamical evolution are not available, these
calculations will eventually yield a better understanding of
planet formation at 20–100 AU.
The Kuiper belt also enables a vital link between the
solar system and other planetary systems. With an outer
radius of & 1000 AU (Sedna’s aphelion) and a current
mass of ∼ 0.1 M⊕ (Luu and Jewitt 2002; Bernstein et al.
2004, Cahpter by Petit et al.), the Kuiper belt has prop-
erties similar to those derived for the oldest debris disks
(Greaves et al. 2004; Wyatt et al. 2005). Understanding
planet formation in the Kuiper belt thus informs our inter-
pretation of evolutionary processes in other planetary sys-
tems.
This paper reviews applications of coagulation theory for
planet formation in the Kuiper belt. After a brief introduc-
tion to the theoretical background in §2, we describe results
from numerical simulations in §3, compare relevant KBO
observations with the results of numerical simulations in §4,
and contrast the properties of KBOs and asteroids in §5. We
conclude with a short summary in §6.
2. COAGULATION THEORY
Planet formation begins with dust grains suspended in
a gaseous circumstellar disk. Grains evolve into planets
in three steps. Collisions between grains produce larger
aggregates which decouple from the gas and settle into a
dense layer in the disk midplane. Continued growth of the
loosely bound aggregates leads to planetesimals, gravita-
tionally bound objects whose motions are relatively inde-
pendent of the gas. Collisions and mergers among the en-
semble of planetesimals form planets. Here, we briefly de-
scribe the physics of these stages and summarize analytic
results as a prelude to summaries of numerical simulations.
We begin with a prescription for the mass surface density
Σ of gas and dust in the disk. We use subscripts ‘d’ for the
dust and ‘g’ for the gas and adopt
Σd,g = Σ0d,0g
40 AU
, (1)
where a is the semimajor axis. In the MMSN, n =
3/2, Σ0d ≈ 0.1 g cm
−2, and Σ0g ≈ 5–10 g cm
(Weidenschilling 1977b; Hayashi 1981). For a disk with
an outer radius of 100 AU, the MMSN has a mass of
∼ 0.03 M⊙, which is comparable to the disk masses of
young stars in nearby star-forming regions (Natta et al.
2000; Scholz et al. 2006).
The dusty midplane forms quickly (Weidenschilling
1977a, 1980; Dullemond and Dominik 2005). For inter-
stellar grains with radii, r ∼ 0.01–0.1 µm, turbulent mixing
approximately balances settling due to the vertical compo-
nent of the star’s gravity. As grains collide and grow to r ∼
0.1–1 mm, they decouple from the turbulence and settle
into a thin layer in the disk midplane. The timescale for this
process is ∼ 103 yr at 1 AU and ∼ 105 yr at 40 AU.
The evolution of grains in the midplane is uncertain. Be-
cause the gas has some pressure support, it orbits the star
slightly more slowly than the Keplerian velocity. Thus,
orbiting dust grains feel a headwind that drags them to-
ward the central star (Adachi et al. 1976; Weidenschilling
1984; Tanaka and Ida 1999). For m-sized objects, the drag
timescale at 40 AU, ∼ 105 yr, is comparable to the growth
time. Thus, it is not clear whether grains can grow by di-
rect accretion to km sizes before the gas drags them into the
inner part of the disk.
Dynamical processes provide alternatives to random ag-
glomeration of grains. In ensembles of porous grains, gas
flow during disruptive collisions leads to planetesimal for-
mation by direct accretion (Wurm et al. 2004). Analytic es-
timates and numerical simulations indicate that grains with
r ∼ 1 cm are also easily trapped within vortices in the disk
(e.g. de la Fuente Marcos and Barge 2001; Inaba and Barge
2006). Large enhancements in the solid-to-gas ratio within
vortices allows accretion to overcome gas drag, enabling
formation of km-sized planetesimals in 104–105 yr.
If the dusty midplane is calm, it becomes thinner and
thinner until groups of particles overcome the local Jeans
criterion – where their self-gravity overcomes local orbital
shear – and ‘collapse’ into larger objects on the local dy-
namical timescale, ∼ 103 yr at 40 AU (Goldreich and Ward
1973; Youdin and Shu 2002; Tanga et al. 2004). This pro-
cess is a promising way to form planetesimals; however,
turbulence may prevent the instability (Weidenschilling
1995, 2003, 2006). Although the expected size of a col-
lapsed object is the Jeans wavelength, the range of plan-
etesimal sizes the instability produces is also uncertain.
Once planetesimals with r ∼ 1 km form, gravity dom-
inates gas dynamics. Long range gravitational interactions
exchange kinetic energy (dynamical friction) and angular
momentum (viscous stirring), redistributing orbital energy
and angular momentum among planetesimals. For 1 km
objects at 40 AU, the initial random velocities are compara-
ble to their escape velocities, ∼ 1 m s−1 (Weidenschilling
1980; Goldreich et al. 2004). The gravitational binding en-
ergy (for brevity, we use energy as a shorthand for specific
energy), Eg ∼ 10
4 erg g−1, is then comparable to the typ-
ical collision energy, Ec ∼ 10
4 erg g−1. Both energies are
smaller than the disruption energy – the collision energy
needed to remove half of the mass from the colliding pair
of objects – which is Q∗D ∼ 10
5–107 erg g−1 for icy mate-
rial (Davis et al. 1985; Benz and Asphaug 1999; Ryan et al.
1999; Michel et al. 2001; Leinhardt and Richardson 2002;
Giblin et al. 2004). Thus, collisions produce mergers in-
stead of debris.
Initially, small planetesimals grow slowly. For a large
ensemble of planetesimals, the collision rate is nσv, where
n is the number of planetesimals, σ is the cross-section,
and v is the relative velocity. The collision cross-section is
the geometric cross-section, πr2, scaled by the gravitational
focusing factor, fg,
σc ∼ πr
2fg ∼ πr
2(1 + β(vesc/evK)
2) , (2)
where e is the orbital eccentricity, vK is the orbital ve-
locity, vesc is the escape velocity of the merged pair
of planetesimals, and β ≈ 2.7 is a coefficient that ac-
counts for three-dimensional orbits in a rotating disk
(Greenzweig and Lissauer 1990; Spaute et al. 1991; Wetherill and Stewart
1993). Because evK ≈ vesc, gravitational focusing factors
are small and growth is slow and orderly (Safronov 1969).
The timescale for slow, orderly growth is
ts ≈ 30
1000 km
250 yr
0.1 g cm−2
Gyr ,
where P is the orbital period (Safronov 1969; Lissauer
1987; Goldreich et al. 2004).
As larger objects form, several processes damp parti-
cle random velocities and accelerate growth. For objects
with r ∼ 1–100 m, physical collisions reduce particle ran-
dom velocities (Ohtsuki 1992; Kenyon and Luu 1998). For
larger objects with r & 0.1 km, the smaller objects damp
the orbital eccentricity of larger particles through dynam-
ical friction (Wetherill and Stewart 1989; Kokubo and Ida
1995; Kenyon and Luu 1998). Viscous stirring by the large
objects excites the orbits of the small objects. For planetes-
imals with r ∼ 1 m to r ∼ 1 km, these processes occur
on short timescales, . 106 yr at 40 AU, and roughly bal-
ance when these objects have orbital eccentricity e ∼ 10−5.
In the case where gas drag is negligible, Goldreich et al.
(2004) derive a simple relation for the ratio of the eccentric-
ities of the large (‘l’) and the small (‘s’) objects in terms of
their surface densities Σl,s (see also Kokubo and Ida 2002;
Rafikov 2003c,b,d,a),
, (4)
with γ = 1/4 to 1/2. Initially, most of the mass is in small
objects. Thus Σl/Σs ≪ 1. For Σl/Σs ∼ 10
−3 to 10−2,
el/es ≈ 0.1–0.25. Because esvK ≪ vl,esc gravitational
focusing factors for large objects accreting small objects are
large. Runaway growth begins.
Runaway growth relies on positive feedback between
accretion and dynamical friction. Dynamical friction pro-
duces the largest fg for the largest objects, which grow
faster and faster relative to the smaller objects and contain
an ever growing fraction of the total mass. As they grow,
these protoplanets stir the planetesimals. The orbital ve-
locity dispersions of small objects gradually approach the
escape velocities of the protoplanets. With esvK ∼ vl,esc,
collision rates decline as runaway growth continues (eqs.
(2) and (4)). The protoplanets and leftover planetesimals
then enter the oligarchic phase, where the largest objects
– oligarchs – grow more slowly than they did as runaway
objects but still faster than the leftover planetesimals. The
timescale to reach oligarchic growth is (Lissauer 1987;
Goldreich et al. 2004)
to ≈ 30
250 yr
0.1 g cm−2
Myr , (5)
For the MMSN, to ∝ a
−3. Thus, collisional damping,
dynamical friction and gravitational focusing enhance the
growth rate by 3 orders of magnitude compared to orderly
growth.
Among the oligarchs, smaller oligarchs grow the fastest.
Each oligarch tries to accrete material in an annular ‘feed-
ing zone’ set by balancing the gravity of neighboring oli-
garchs. If an oligarch accretes all the mass in its feed-
ing zone, it reaches the ‘isolation mass’ (Lissauer 1987;
Kokubo and Ida 1998, 2002; Rafikov 2003a; Goldreich et al.
2004),
miso ≈ 28
40 AU
0.1 g cm−2
M⊕ . (6)
Each oligarch stirs up leftover planetesimals along its or-
bit. Smaller oligarchs orbit in regions with smaller Σl/Σs.
Thus, smaller oligarchs have larger gravitational focusing
factors (eqs. (2) and (4)) and grow faster than larger oli-
garchs (Kokubo and Ida 1998; Goldreich et al. 2004).
As oligarchs approach miso, they stir up the velocities
of the planetesimals to the disruption velocity. Instead of
mergers, collisions then yield smaller planetesimals and de-
bris. Continued disruptive collisions lead to a collisional
cascade, where leftover planetesimals are slowly ground to
dust (Dohnanyi 1969; Williams and Wetherill 1994). Ra-
diation pressure from the central star ejects dust grains
with r . 1–10 µm; Poynting-Robertson drag pulls larger
grains into the central star (Burns et al. 1979; Artymowicz
1988; Takeuchi and Artymowicz 2001). Eventually, plan-
etesimals are accreted by the oligarchs or ground to dust.
To evaluate the oligarch mass required for a disruptive
collision, we consider two planetesimals with equal mass
mp. The center-of-mass collision energy is
Qi = v
i /8 , (7)
where the impact velocity v2i = v
2+v2esc (Wetherill and Stewart
1993). The energy needed to remove half of the combined
mass of two colliding planetesimals is
Q∗D = Qb
+ ρQg
, (8)
1 3 5 7
log Radius (cm)
e = 0.001
e = 0.01
e = 0.1
Fig. 1.— Disruption energy, Q∗D, for icy objects. The solid
curve plots a typical result derived from numerical simulations of
collisions that include a detailed equation of state for crystalline
ice (Qb = 1.6 × 10
7 erg g−1, βb = −0.42, ρ = 1.5 g cm
−3, Qg =
1.5 erg cm−3, and βg = 1.25; Benz and Asphaug 1999). The other
curves plot results using Qb consistent with model fits to comet
breakups (βb ≈ 0; Qb ∼ 10
3 erg g−1, dashed curve; Qb ∼ 10
erg g−1, dot-dashed curve; Asphaug and Benz 1996). The dashed
horizontal lines indicate the center of mass collision energy (eq.
(7)) for equal mass objects with e = 0.001, 0.01, and 0.1. Col-
lisions between objects with Qi ≪ Q
D yield merged remnants;
collisions between objects with Qi ≫ Q
D produce debris.
where Qbr
βb is the bulk (tensile) component of the binding
energy and ρQgr
βg is the gravity component of the bind-
ing energy (Davis et al. 1985; Housen and Holsapple 1990,
1999; Holsapple 1994; Benz and Asphaug 1999). We adopt
v ≈ vesc,o, where vesc,o = (Gmo/ro)
1/2 is the escape ve-
locity of an oligarch with mass mo and radius ro. We define
the disruption mass md by deriving the oligarch mass where
Qi ≈ Q
D. For icy objects at 30 AU
md ∼ 3× 10
107 erg g−1
M⊕. (9)
Figure 1 illustrates the variation of Q∗D with radius for
several variants of eq. (8). For icy objects, detailed nu-
merical collision simulations yield Qb . 10
7 erg g−1,
−0.5 . βb . 0, ρ ≈ 1–2 g cm
−3, Qg ≈ 1–2 erg cm
and βg ≈ 1–2 (solid line in Fig. 1, Benz and Asphaug
1999, see also Chapter by Leinhardt et al.)). Models for
the breakup of comet Shoemaker-Levy 9 suggest a smaller
component of the bulk strength, Qb ∼ 10
3 erg g−1 (e.g.,
Asphaug and Benz 1996), which yields smaller disruption
energies for smaller objects (Fig. 1, dashed and dot-dashed
curves). Because nearly all models for collisional disrup-
tion yield similar results for objects with r & 1 km (e.g.,
Kenyon and Bromley 2004d), the disruption mass is fairly
independent of theoretical uncertainties once planetesimals
become large. For typical Q∗D ∼ 10
7–108 erg g−1 for 1–10
km objects (Fig. 1), leftover planetesimals start to disrupt
when oligarchs have radii, ro ∼ 200–500 km.
Once disruption commences, the final mass of an oli-
garch depends on the timescale for the collisional cascade
(Kenyon and Bromley 2004a,b,d; Leinhardt and Richardson
2005). If disruptive collisions produce dust grains much
faster than oligarchs accrete leftover planetesimals, oli-
garchs with mass mo cannot grow much larger than the
disruption radius (maximum oligarch mass mo,max ≈ md).
However, if oligarchs accrete grains and leftover planetes-
imals effectively, oligarchs reach the isolation mass before
collisions and radiation pressure remove material from the
disk (eq. (6); Goldreich et al. 2004). The relative rates of
accretion and disruption depend on the balance between
collisional damping and gas drag – which slow the colli-
sional cascade – and viscous stirring and dynamical friction
– which speed up the collisional cascade. Because deriv-
ing accurate rates for these processes requires numerical
simulations of planetesimal accretion, we now consider
simulations of planet formation in the Kuiper belt.
3. COAGULATION SIMULATIONS
3.1. Background
Safronov (1969) invented the current approach to plan-
etesimal accretion calculations. In his particle-in-a-box
method, planetesimals are a statistical ensemble of masses
with distributions of orbital eccentricities and inclinations
(Greenberg et al. 1978; Wetherill and Stewart 1989, 1993;
Spaute et al. 1991). This statistical approximation is es-
sential: N -body codes cannot follow the n ∼ 109–1012
1 km planetesimals required to build Pluto-mass or Earth-
mass planets. For large numbers of objects on fairly cir-
cular orbits (e.g., n & 104, r . 1000 km, and e . 0.1),
the method is also accurate. With a suitable prescription
for collision outcomes, solutions to the coagulation equa-
tion in the kinetic theory yield the evolution of n(m) with
arbitrarily small errors (e.g., Wetherill 1990; Lee 2000;
Malyshkin and Goodman 2001).
In addition to modeling planet growth, the statistical
approach provides a method for deriving the evolution of
orbital elements for large ensembles of planetesimals. If
we (i) assume the distributions of e and i for planetesimals
follow a Rayleigh distribution and (ii) treat their motions
as perturbations of a circular orbit, we can use the Fokker-
Planck equation to solve for small changes in the orbits
due to gas drag, gravitational interactions, and physical col-
lisions (Hornung et al. 1985; Wetherill and Stewart 1993;
Ohtsuki et al. 2002). Although the Fokker-Planck equation
cannot derive accurate orbital parameters for planetesimals
and oligarchs near massive planets, it yields accurate solu-
tions for the ensemble average e and i when orbital reso-
nances and other dynamical interactions are not important
(e.g., Wetherill and Stewart 1993; Weidenschilling et al.
1997a; Ohtsuki et al. 2002).
Several groups have implemented Safronov’s method for
calculations relevant to the outer solar system (Greenberg et al.
1984; Stern 1995, 2005; Stern and Colwell 1997a,b; Davis and Farinella
1997; Kenyon and Luu 1998, 1999a,b; Davis et al. 1999;
Kenyon and Bromley 2004a,d, 2005). These calculations
-3 -2 -1 0 1 2 3 4
log Radius (km)
0 Myr
40-47 AU
-3 -2 -1 0 1 2 3 4
log Radius (km)
Fig. 2.— Evolution of a multiannulus coagulation model with Σ = 0.12(ai/40 AU)−3/2 g cm−2. Left: cumulative mass distribution
at times indicated in the legend. Right: eccentricity distributions at t = 0 (light solid line), t = 10 Myr (filled circles), t = 100 Myr
(open boxes), t = 1 Gyr (filled triangles), and t = 5 Gyr (open diamonds). As large objects grow in the disk, they stir up the leftover
planetesimals to e ∼ 0.1. Disruptive collisions then deplete the population of 0.1–10 km planetesimals, which limits the growth of the
largest objects.
adopt a disk geometry and divide the disk into N concen-
tric annuli with radial width ∆ai at distances ai from the
central star. Each annulus is seeded with a set of planetes-
imals with masses mij , eccentricities eij , and inclinations
iij , where the index i refers to one of N annuli and the
index j refers to one of M mass batches within an annulus.
The mass batches have mass spacing δ ≡ mj+1/mj . In
most calculations, δ ≈ 2; δ ≤ 1.4 is optimal (Ohtsuki et al.
1990; Wetherill and Stewart 1993; Kenyon and Luu 1998).
Once the geometry is set, the calculations solve a set of
coupled difference equations to derive the number of ob-
jects nij , and the orbital parameters, eij and iij , as func-
tions of time. Most studies allow fragmentation and ve-
locity evolution through gas drag, collisional damping, dy-
namical friction and viscous stirring. Because Q∗D sets the
maximum size mc,max of objects that participate in the
collisional cascade, the size distribution for objects with
m < mc,max depends on the fragmentation parameters (eq.
(8); Davis and Farinella 1997; Kenyon and Bromley 2004d;
Pan and Sari 2005). The size and velocity distributions of
the merger population with m > mc,max are established
during runaway growth and the early stages of oligarchic
growth. Accurate treatment of velocity evolution is impor-
tant for following runaway growth and thus deriving good
estimates for the growth times and the size and velocity dis-
tributions of oligarchs.
When a few oligarchs contain most of the mass, col-
lision rates depend on the orbital dynamics of individual
objects instead of ensemble averages. Safronov’s statisti-
cal approach then fails (e.g., Wetherill and Stewart 1993;
Weidenschilling et al. 1997b). Although N-body methods
can treat the evolution of the oligarchs, they cannot follow
the evolution of leftover planetesimals, where the statis-
tical approach remains valid (e.g., Weidenschilling et al.
1997b). Spaute et al. (1991) solve this problem by adding
a Monte Carlo treatment of binary interactions between
large objects to their multiannulus coagulation code.
Bromley and Kenyon (2006) describe a hybrid code, which
merges a direct N-body code with a multiannulus coagula-
tion code. Both codes have been applied to terrestrial planet
formation, but not to the Kuiper belt.
Current calculations cannot follow collisional growth ac-
curately in an entire planetary system. Although the six
order of magnitude change in formation timescales from
∼ 0.4 AU to 40 AU is a factor in this statement, most
modern supercomputers cannot finish calculations involv-
ing the entire disk with the required spatial resolution on
a reasonable timescale. For the Kuiper belt, it is possi-
ble to perform a suite of calculations in a disk extending
from 30–150 AU following 1 m and larger planetesimals.
These calculations yield robust results for the mass distri-
bution as a function of space and time and provide interest-
ing comparisons with observations. Although current cal-
culations do not include complete dynamical interactions
with the giant planets or passing stars ((see, for example,
Charnoz and Morbidelli 2007), sample calculations clearly
show the importance of external perturbations in treating
the collisional cascade. We begin with a discussion of
self-stirring calculations without interactions with the gi-
ant planets or passing stars and then describe results with
external perturbers.
3.2. Self-Stirring
To illustrate in situ KBO formation at 40–150 AU, we
consider a multiannulus calculation with an initial ensem-
ble of 1 m to 1 km planetesimals in a disk with Σ0d =
0.12 g cm−2. The planetesimals have initial radii of 1 m
to 1 km (with equal mass per logarithmic bin), e0 = 10
i0 = e0/2, mass density
1 ρ = 1.5 g cm−3 and fragmentation
parameters Qb = 10
3 erg g−1, Qg = 1.5 erg cm
−3, βb = 0,
1 Our choice of mass density is a compromise between pure ice (ρ = 1
g cm−3) and the measured density of Pluto (ρ ≈ 2 g cm−3 Null et al.
and βg = 1.25 (dashed curve in Fig. 1; Kenyon and Bromley
2004d, 2005). The gas density also follows a MMSN, with
Σg/Σd = 100 and a vertical scale height h = 0.1 r
(Kenyon and Hartmann 1987). The gas density is Σg ∝
e−t/tg , with tg = 10 Myr.
This calculation uses an updated version of the Bromley and Kenyon
(2006) code that includes a Richardson extrapolation pro-
cedure in the coagulation algorithm. As in the Eulerian
(Kenyon and Luu 1998) and fourth order Runge-Kutta
(Kenyon and Bromley 2002a,b) methods employed pre-
viously, this code provides robust numerical solutions to
kernels with analytic solutions (e.g., Ohtsuki et al. 1990;
Wetherill 1990) without producing the wavy size distribu-
tions described in other simulations with a low mass cutoff
(e.g. Campo Bagatin et al. 1994). Once the evolution of
large (r > 1 m) objects is complete, a separate code tracks
the evolution of lower mass objects and derives the dust
emission as a function of time.
Figure 2 shows the evolution of the mass and eccentric-
ity distributions at 40–47 AU for this calculation. During
the first few Myr, the largest objects grow slowly. Dynam-
ical friction damps the orbits of the largest objects; colli-
sional damping and gas drag circularize the orbits of the
smallest objects. This evolution erases many of the initial
conditions and enhances gravitational focusing by factors
of 10–1000. Runaway growth begins. A few (and some-
times only one) oligarchs then grow from r ∼ 10 km to
r ∼ 1000 km in ∼ 30 Myr at 40 AU and in ∼ 1 Gyr at
150 AU (see eq. (5)). Throughout runaway growth, dynam-
ical friction and viscous stirring raise the random velocities
of the leftover planetesimals to e ≈ 0.01–0.1 and i ≈ 2o–
4o (v ∼ 50–500 m s−1 at 40–47 AU; Figure 2; right panel).
Stirring reduces gravitational focusing factors and ends run-
away growth. The large oligarchs then grow slowly through
accretion of leftover planetesimals.
As oligarchs grow, collisions among planetesimals initi-
ate the collisional cascade. Disruptive collisions dramat-
ically reduce the population of 1–10 km objects, which
slows the growth of oligarchs and produces a significant de-
bris tail in the size distribution. In these calculations, dis-
ruptive collisions remove material from the disk faster than
oligarchs can accrete the debris. Thus, growth stalls and
produces ∼ 10–100 objects with maximum sizes rmax ∼
1000–3000 km at 40–50 AU (Stern and Colwell 1997a,b;
Kenyon and Bromley 2004d, 2005; Stern 2005).
Stochastic events lead to large dispersions in the growth
time for oligarchs, to (eq. (5)). In ensembles of 25–50 simu-
lations with identical starting conditions, an occasional oli-
garch will grow up to a factor of two faster than its neigh-
bors. This result occurs in simulations with δ = 1.4, 1.7, and
2.0, and thus seems independent of mass resolution. These
events occur in ∼ 25% of the simulations and lead to factor
of ∼ 2 variations in to (eq. (5)).
In addition to modest-sized icy planets, oligarchic
1993). The calculations are insensitive to factor of two variations in the
mass density of planetesimals.
6 7 8 9 10
log Time (yr)
R > 10 km (40-47 AU)
R > 100 km (40-47 AU)
Dust (40-150 AU)
Fig. 3.— Time evolution of the mass in KBOs and dust grains.
Solid line: dust mass (r . 1 mm) at 40–150 AU. Dashed (dot-
dashed) lines: total mass in small (large) KBOs at 40–47 AU.
growth generates copious amounts of dust (Figure 3). When
runaway growth begins, collisions produce small amounts
of dust from ‘cratering’ (see, for example Greenberg et al.
1978; Wetherill and Stewart 1993; Stern and Colwell 1997a,b;
Kenyon and Luu 1999a). Stirring by growing oligarchs
leads to ‘catastrophic’ collisions, where colliding planetes-
imals lose more than 50% of their initial mass. These dis-
ruptive collisions produce a spike in the dust production
rate that coincides with the formation of oligarchs with r &
200–300 km (eq. (9)). As the wave of runaway growth
propagates outward, stirring produces disruptive collisions
at ever larger heliocentric distances. The dust mass grows
in time and peaks at ∼ 1 Gyr, when oligarchs reach their
maximum mass at 150 AU. As the mass in leftover plan-
etesimals declines, Poynting-Robertson drag removes dust
faster than disruptive collisions produce it. The dust mass
then declines with time.
3.3. External Perturbation
Despite the efficiency of self-stirring models in remov-
ing leftover planetesimals from the disk, other mechanisms
must reduce the derived mass in KBOs to current ob-
servational limits. In self-stirring calculations at 40–50
AU, the typical mass in KBOs with r ∼ 30—1000 km
at 4–5 Gyr is a factor of 5–10 larger than currently ob-
served (Luu and Jewitt 2002, Chapter by Petit et al.). Un-
less Earth-mass or larger objects form in the Kuiper belt
(Chiang et al. 2007; Levison and Morbidelli 2007), exter-
nal perturbations must excite KBO orbits and enhance the
collisional cascade.
Two plausible sources of external perturbation can re-
duce the predicted KBO mass to the desired limits. Once
Neptune achieves its current mass and orbit, it stirs up
the orbits of KBOs at 35–50 AU (Levison and Duncan
1990; Holman and Wisdom 1993; Duncan et al. 1995;
Kuchner et al. 2002; Morbidelli et al. 2004). In ∼ 100 Myr
or less, Neptune removes nearly all KBOs with a . 37–38
AU. Beyond a ∼ 38 AU, some KBOs are trapped in or-
bital resonance with Neptune (Malhotra 1995, 1996); oth-
ers are ejected into the scattered disk (Duncan and Levison
1997). In addition to these processes, Neptune stir-
ring increases the effectiveness of the collisional cascade
(Kenyon and Bromley 2004d), which removes additional
mass from the population of 0.1–10 km KBOs and prevents
growth of larger KBOs.
Passing stars can also excite KBO orbits and en-
hance the collisional cascade. Although Neptune dy-
namically ejects scattered disk objects with perihelia
q . 36–37 AU (Morbidelli et al. 2004), objects with
q & 45–50 AU, such as Sedna and Eris, require an-
other scattering source. Without evidence for massive
planets at a & 50 AU (Morbidelli et al. 2002), a pass-
ing star is the most likely source of the large q for these
KBOs (Morbidelli and Levison 2004; Kenyon and Bromley
2004c).
Adams and Laughlin (2001, see also Chapter by Dun-
can et al.) examined the probability of encounters between
the young Sun and other stars. Most stars form in dense
clusters with estimated lifetimes of ∼ 100 Myr. To ac-
count for the abundance anomalies of radionuclides in so-
lar system meteorites (produced by supernovae in the clus-
ter) and for the stability of Neptune’s orbit at 30 AU, the
most likely solar birth cluster has ∼ 2000–4000 members,
a crossing time of ∼ 1 Myr, and a relaxation time of ∼ 10
Myr. The probability of a close encounter with a distance
of closest approach aclose is then ∼ 60% (aclose/160 AU)
(Kenyon and Bromley 2004c).
Because the dynamical interactions between KBOs in a
coagulation calculation and large objects like Neptune or
a passing star are complex, here we consider simple cal-
culations of each process. To illustrate the evolution of
KBOs after a stellar flyby, we consider a very close pass
with aclose = 160 AU (Kenyon and Bromley 2004c). This
co-rotating flyby produces objects with orbital parameters
similar to those of Sedna and Eris. For objects in the coag-
ulation calculation, the flyby produces an e distribution
eKBO =
0.025(a/30 AU)4 a < a0
0.5 a > a0
with a0 ≈ 65 AU (see Ida et al. 2000; Kenyon and Bromley
2004c; Kobayashi et al. 2005). This e distribution produces
a dramatic increase in the debris production rate throughout
the disk, which freezes the mass distribution of the largest
objects2. Thus, to produce an ensemble of KBOs with r &
300 km at 40–50 AU, the flyby must occur when the Sun
has an age t⊙ & 10–20 Myr (Figure 2). For t⊙ & 100 Myr,
2 The i distribution following a flyby depends on the relative orientations of
two planes, the orbital plane of KBOs and the plane of the trajectory of the
passing star. Here, we assume the flyby produces no change in i, which
simplifies the discussion without changing any of the results significantly.
the flyby is very unlikely. As a compromise between these
two estimates, we consider a flyby at t⊙ ∼ 50 Myr.
6 7 8 9 10
log Time (yr)
Flyby Model
40-47 AU
47-55 AU
66-83 AU
Fig. 4.— Evolution of Σ after a stellar flyby. After 50 Myr
of growth, the close pass excites KBOs to large e (eq. (10)) and
enhances the collisional cascade.
Figure 4 shows the evolution of the KBO surface den-
sity in three annuli as a function of time. At early times
(t . 50 Myr), KBOs grow in the standard way. After the
flyby, the disk suffers a dramatic loss of material. At 40–47
AU, the disk loses ∼ 90% (93%) of its initial mass in ∼ 1
Gyr (4.5 Gyr). At ∼ 50–80 AU, the collisional cascade re-
moves ∼ 90% (97%) of the initial mass in ∼ 500 Myr (4.5
Gyr). Beyond ∼ 80 AU, KBOs contain less than 1% of the
initial mass. Compared to self-stirring models, flybys that
produce Sedna-like orbits are a factor of 2–3 more efficient
at removing KBOs from the solar system.
To investigate the impact of Neptune on the collisional
cascade, we parameterize the growth of Neptune at 30 AU
as a simple function of time (Kenyon and Bromley 2004d)
MNep ≈
6× 1027 e(t−tN )/t1 g t < tN
6× 1027 g + C(t− t1) tN < t < t2
1.0335× 1029 g t > t2
where CNep = 1.947 × 10
21 g yr−1, tN = 50 Myr, t1 = 3
Myr, and t2 = 100 Myr. These choices enable a model Nep-
tune to reach a mass of 1 M⊕ in 50 Myr, when the largest
KBOs form at 40–50 AU, and reach its current mass in 100
Myr3. As Neptune approaches its final mass, its gravity stirs
up KBOs at 40–60 AU and increases their orbital eccentric-
ities to e ∼ 0.1–0.2 on short timescales. In the coagula-
tion model, distant planets produce negligible changes in i,
so self-stirring sets i in these calculations (Weidenschilling
1989). This evolution enhances debris production by a fac-
tor of 3–4, which effectively freezes the mass distribution
of 100–1000 km objects at 40–50 AU. By spreading the
3 This prescription is not intended as an accurate portrayal of Neptune for-
mation, but it provides a simple way to investigate how Neptune might stir
the Kuiper belt once massive KBOs form.
6 7 8 9 10
log Time (yr)
Neptune Stirring
SS: 40-47 AU
NS: 40-47 AU
NS: 47-55 AU
Fig. 5.— Evolution of Σ(KBO) in models with Neptune stirring.
Compared to self-stirring models (SS; dashed curve), stirring by
Neptune rapidly removes KBOs at 40–47 AU (NS; solid cruve)
and at 47–55 AU (NS; dot-dashed curve).
leftover planetesimals and the debris over a larger volume,
Neptune stirring limits the growth of the oligarchs and thus
reduces the total mass in KBOs.
Figure 5 shows the evolution of the surface density in
small and large KBOs in two annuli as a function of time.
At 40–55 AU, Neptune rapidly stirs up KBOs to e ∼ 0.1
when it reaches its current mass at ∼ 100 Myr. Large
collision velocities produce more debris, which is rapidly
ground to dust and removed from the system by radiation
pressure at early times and by Poynting-Robertson drag at
later times. Compared to self-stirring models, the change
in Σ is dramatic, with only ∼ 3% of the initial disk mass
remaining at ∼ 4.5 Gyr.
From these initial calculations, it is clear that exter-
nal perturbations dramatically reduce the mass of KBOs
in the disk (see also Charnoz and Morbidelli 2007). Fig-
ure 6 compares the mass distributions at 40–47 AU and at
4.5 Gyr for the self-stirring model in Figure 2 (solid line)
with results for the flyby (dot-dashed line) and Neptune stir-
ring (dashed line). Compared to the self-stirring model, the
close flyby reduces the mass in KBOs by ∼ 50%. Neptune
stirring reduces the KBO mass by almost a factor of 3 rel-
ative to the self-stirring model. For KBOs with r & 30–50
km, the predicted mass in KBOs with Neptune stirring is
within a factor of 2–3 of the current mass in KBOs.
These simple calculations for the stellar flyby and Nep-
tune stirring do not include dynamical depletion. In the stel-
lar flyby picture, the encounter removes nearly all KBOs
beyond a truncation radius, aT ∼ 48 (aclose / 160 AU)
AU (Kenyon and Bromley 2004c). Thus, a close pass with
aclose ∼ 160 AU can produce the observed outer edge of the
Kuiper belt at 48 AU. Although many objects with initial
a > aT are ejected from the Solar System, some are placed
on very elliptical, Sedna-like orbits4. In the Neptune stir-
4Levison et al. (2004) consider the impact of the flyby on the scattered disk
and Oort cloud. After analyzing a suite of numerical simulations, they
conclude that the flyby must occur before Neptune reaches its current orbit
-3 -2 -1 0 1 2 3 4
log Radius (km)
Initial state
Self-stirring
Flyby 
Neptune
40-47 AU
Fig. 6.— Mass distributions for evolution with self-stirring
(heavy solid line), stirring from a passing star (dot-dashed line),
and stirring from Neptune at 30 AU (dashed line). After 4.5 Gyr,
the mass in KBOs with r & 50 km is ∼ 5% (self-stirring), ∼
3.5% (flyby), and ∼ 2% (Neptune stirring) of the initial mass. The
number of objects with r & 1000 km is ∼ 100 (self-stirring), ∼
1 (flyby), and ∼ 10 (Neptune stirring). The largest object has
rmax ∼ 3000 km (self-stirring), rmax ∼ 500–1000 km (flyby),
and rmax ∼ 1000–2000 km (Neptune stirring).
ring model, dynamical interactions will eject some KBOs
at 40–47 AU. If the dynamical interactions that produce the
scattered disk reduce the mass in KBOs by a factor of 2 at
40–47 AU (e.g., Duncan et al. 1995; Kuchner et al. 2002),
the Neptune stirring model yields a KBO mass in reason-
ably good agreement with observed limits (for a different
opinion, see Charnoz and Morbidelli 2007).
3.4. Nice Model
Although in situ KBO models can explain the current
amount of mass in large KBOs, these calculations do not
address the orbits of the dynamical classes of KBOs. To ex-
plain the orbital architecture of the giant planets, the ‘Nice
group’ centered at Nice Observatory developed an inspired,
sophisticated picture of the dynamical evolution of the gi-
ant planets and a remnant planetesimal disk (Tsiganis et al.
2005; Morbidelli et al. 2005; Gomes et al. 2005, and refer-
ences therein). The system begins in an approximate equi-
librium, with the giant planets in a compact configuration
(Jupiter at 5.45 AU, Saturn at ∼ 8.2 AU, Neptune at ∼ 11.5
AU, and Uranus at ∼ 14.2 AU) and a massive planetesimal
disk at 15–30 AU. Dynamical interactions between the gi-
ant planets and the planetesimals lead to an instability when
Saturn crosses the 2:1 orbital resonance with Jupiter, which
results in a dramatic orbital migration of the gas giants and
the dynamical ejection of planetesimals into the Kuiper belt,
scattered disk, and the Oort cloud. Comparisons between
the end state of this evolution and the orbits of KBOs in
and begins the dynamical processes that populate the Oort cloud and the
scattered disk. If Neptune forms in situ in 1–10 Myr, then the flyby cannot
occur after massive KBOs form. If Neptune migrates to 30 AU after mas-
sive KBOs form, then a flyby can truncate the Kuiper belt without much
impact on the Oort cloud or the scattered disk.
6 7 8 9 10
log Time (yr)
20-25 AU
36-44 AU
66-83 AU
NICE Model
Fig. 7.— Evolution of Σ in a self-stirring model at 20–100 AU.
At 20–25 AU, it takes ∼ 5–10 Myr to form 1000 km objects. After
∼ 0.5–1 Gyr, there are ∼ 100 objects with r ∼ 1000–2000 km
and ∼ 105 objects with r ∼ 100–200 km at 20–30 AU. As these
objects grow, the collisional cascade removes ∼ 90% of the mass
in remnant planetesimals. The twin vertical dashed lines bracket
the time of the Late Heavy Bombardment at ∼ 300–600 Myr.
the ‘hot population’ and the scattered disk are encouraging
(Chapter by Morbidelli et al.).
Current theory cannot completely address the likelihood
of the initial state in the Nice model. Thommes et al. (1999,
2002) demonstrate that n-body simulations can produce a
compact configuration of gas giants, but did not consider
how fragmentation or interactions with low mass planetes-
imals affect the end state. O’Brien et al. (2005) show that
a disk of planetesimals has negligible collisional grinding
over 600 Myr if most of the mass is in large planetesimals
with r & 100 km. However, they did not address whether
this state is realizable starting from an ensemble of 1 km
and smaller planetesimals. In terrestrial planet simulations
starting with 1–10 km planetesimals, the collisional cascade
removes ∼ 25% of the initial rocky material in the disk
(Wetherill and Stewart 1993; Kenyon and Bromley 2004b).
Interactions between oligarchs and remnant planetesimals
are also important for setting the final mass and dynamical
state of the terrestrial planets (Bromley and Kenyon 2006;
Kenyon and Bromley 2006). Because complete hybrid cal-
culations of the giant planet region are currently compu-
tationally prohibitive, it is not possible to make a reliable
assessment of these issues for the formation of gas giant
planets.
Here, we consider the evolution of the planetesimal disk
outside the compact configuration of giant planets, where
standard coagulation calculations can follow the evolution
of many initial states for 1–5 Gyr in a reasonable amount of
computer time. Figure 7 shows the time evolution for the
surface density of planetesimals in three annuli from one
typical calculation at 20–25 AU (dot-dashed curve; Mi = 6
M⊕), 36–44 AU (dashed curve; Mi = 9 M⊕), and 66–83
AU (solid curve; Mi = 12 M⊕). Starting from the stan-
dard surface density profile (eq. 1), planetesimals at 20–25
AU grow to 1000 km sizes in a few Myr. Once the colli-
sional cascade begins, the surface density slowly declines
to ∼ 10% to 20% of its initial value at the time of the Late
Heavy Bombardment, when the Nice model predicts that
Saturn crosses the 2:1 orbital resonance with Jupiter.
These results provide a strong motivation to couple co-
agulation calculations with the dynamical simulations of the
Nice group (see also Charnoz and Morbidelli 2007). In the
Nice model, dynamical interactions with a massive plan-
etesimal disk are the ‘fuel’ for the dramatic migration of the
giant planets and the dynamical ejection of material into the
Kuiper belt and the scattered disk. If the mass in the plan-
etesimal disk declines by ∼ 80% as the orbits of the giant
planets evolve, the giant planets cannot migrate as dramati-
cally as in the Gomes et al. (2005) calculations. Increasing
the initial mass in the disk by a factor of 3–10 may allow
coagulation and the collisional cascade to produce a debris
disk capable of triggering the scattering events of the Nice
model.
3.5. A Caveat on the Collisional Cascade
Although many of the basic outcomes of oligarchic
growth and the collisional cascade are insensitive to the ini-
tial conditions and fragmentation parameters for the plan-
etesimals, several uncertainties in the collisional cascade
can modify the final mass in oligarchs and the distributions
of r and e. Because current computers do not allow coag-
ulation calculations that include the full range of sizes (1
µm to 104 km), published calculations have two pieces,
a solution for large objects (e.g., Kenyon and Bromley
2004a,b) and a separate solution for smaller objects (e.g.,
Krivov et al. 2006). Joining these solutions assumes that (i)
collision fragments continue to collide and fragment until
particles are removed by radiative processes and (ii) mu-
tual (destructive) collisions among the fragments are more
likely than mergers with much larger oligarchs. These as-
sumptions are reasonable but untested by numerical cal-
culations (Kenyon and Bromley 2002a). Thus, it may be
possible to halt or to slow the collisional cascade before
radiation pressure rapidly remove small grains with r ≈
1–100 µm.
In current coagulation calculations, forming massive oli-
garchs at 5–15 AU in a massive disk requires an ineffi-
cient collisional cascade. When the cascade is efficient,
the most massive oligarchs have m . 1 M⊕. Slowing
the cascade allows oligarchs to accrete planetesimals more
efficiently, which results in larger oligarchs that contain a
larger fraction of the initial mass. If collisional damping
is efficient, halting the cascade completely at sizes of ∼ 1
mm leads to rapid in situ formation of Uranus and Neptune
(Goldreich et al. 2004) and early stirring of KBOs at 40 AU.
There are two simple ways to slow the collisional cas-
cade. In simulations where the cascade continues to small
sizes, r ∼ 1–10 µm, the radial optical depth in small grains
is τs ∼ 0.1–1 at 30–50 AU (Kenyon and Bromley 2004a).
Lines-of-sight to the central star are not purely radial, so
this optical depth reduces radiation pressure and Poynting-
Robertson drag by small factors, ∼ e−0.2τs ∼ 10%–30%,
and has little impact on the evolution of the cascade. With
τs ∝ a
−s and s ∼ 1–2, however, the optical depth may re-
duce radiation forces significantly at smaller a. Slowing the
collisional cascade by factors of 2–3 could allow oligarchs
to accrete leftover planetesimals and smaller objects before
the cascade removes them.
Collisional damping and gas drag on small particles may
also slow the collisional cascade. For particles with large
ratios of surface area to volume, r . 0.1–10 cm, colli-
sions and the gas effectively damp e and i (Adachi et al.
1976; Goldreich et al. 2004) and roughly balance dynam-
ical friction and viscous stirring. Other interactions be-
tween small particles and the gas – such as photophoresis
(Wurm and Krauss 2006) – also damp particles randome
velocities and thus might help to slow the cascade. Both
collisions and interactions between the gas and the solids
are more effective at large volume density, so these pro-
cesses should be more important inside 30 AU than outside
30 AU. The relatively short lifetime of the gas, ∼ 3–10 Myr,
also favors more rapid growth inside 30 AU. If damping
maintains an equilibrium e ∼ 10−3 at a ∼ 20–30 AU, oli-
garchs can grow to the sizes, r & 2000 km, required in the
Nice model. Rapid growth at a ∼ 5–15 AU might produce
oligarchs with the isolation mass (r ∼ 10–30 R⊕; eq. 6)
and lead to the rapid formation of gas giants.
Testing these mechanisms for slowing the collisional
cascade requires coagulation calculations with accurate
treatments of collisional damping, gas drag, and optical
depth for particle radii r ∼ 1–10 µm to r ∼ 10000 km.
Although these calculations require factors of 4–6 more
computer time than published calculations, they are possi-
ble with multiannulus coagulation codes on modern parallel
computers.
3.6. Model Predictions
The main predictions derived from coagulation mod-
els are n(r), n(e), and n(i) as functions of a. The cu-
mulative number distribution consists of three power laws
(Kenyon and Bromley 2004d; Pan and Sari 2005)
n(r) =
−αd r ≤ r1
ni r1 ≤ r < r0
−αm r ≥ r0
The debris population at small sizes, r ≤ r1, always has
αd ≈ 3.5. The merger population at large sizes, r ≥ r0,
has αm ≈ 2.7–4. Because the collisional cascade robs
oligarchs of material, calculations with more stirring have
steeper size distributions. Thus, self-stirring calculations
with Qb & 10
5 erg g−1 (Qb . 10
3 erg g−1) typically yield
αm ≈ 2.7–3.3 (3.5–4). Models with a stellar flyby or stir-
ring by a nearby gas giant also favor large αm.
The transition radii for the power laws depend on the
fragmentation parameters (see Fig. 1; see also Pan and Sari
(2005)). For a typical e ∼ 0.01–0.1 in self-stirring models,
r0 ≈ r1 ≈ 1 km when Qb & 10
5 erg g−1. When Qb . 10
erg g−1, r1 ≈ 0.1 km and r0 ≈ 10–20 km. Thus the cal-
culations predict a robust correlation between the transition
radii and the power law exponents: large r0 andαm or small
r0 and αm.
Because gravitational stirring rates are larger than ac-
cretion rates, the predicted distributions of e and i at 4–5
Gyr depend solely on the total mass in oligarchs (see also
Goldreich et al. 2004). Small objects with r . r0 con-
tain a very small fraction of the mass and cannot stir them-
selves. Thus e and i are independent of r (Fig. 2). The
e and i for larger objects depends on the total mass in the
largest objects. In self-stirring models, dynamical friction
and viscous stirring between oligarchs and planetesimals
(during runaway growth) and among the ensemble of oli-
garchs (during oligarchic growth) set the distribution of e
for large objects with r & r0. In self-stirring models, vis-
cous stirring among oligarchs dominates dynamical friction
between oligarchs and leftover planetesimals, which leads
to a shallow relation between e and r, e ∝ r−γ with γ ≈
3/4. In the flyby and Neptune stirring models, stirring by
the external perturber dominates stirring among oligarchs.
This stirring yields a very shallow relation between e and r
with γ ≈ 0–0.25.
Other results depend little on the initial conditions and
the fragmentation parameters. In calculations with different
initial mass distributions, an order of magnitude range in e0,
and Qb = 10
0–107 erg g−1, βb = −0.5–0, Qg = 0.5–5 erg
cm−3, and βg ≥ 1.25, rmax and the amount of mass re-
moved by the collisional cascade vary by . 10% relative to
the evolution of the models shown in Figures 2–7. Because
collisional damping among 1 m to 1 km objects erases the
initial orbital distribution, the results do not depend on e0
and i0. Damping and dynamical friction also quickly erase
the initial mass distribution, which yields growth rates that
are insensitive to the initial mass distribution.
The insensitivity of rmax and mass removal to the frag-
mentation parameters depends on the rate of collisional dis-
ruption relative to the growth rate of oligarchs. Because
the collisional cascade starts when mo ∼ md (eq. (9)),
calculations with small Qb (Qb . 10
3 erg g−1) produce
large amounts of debris before calculations with large Qb
(Qb & 10
3 erg g−1). Thus, an effective collisional cascade
should yield lower mass oligarchs and more mass removal
when Qb is small. However, oligarchs with mo ∼ md still
have fairly large gravitational focusing factors and accrete
leftover planetesimals more rapidly than the cascade re-
moves them. As oligarchic growth continues, gravitational
focusing factors fall and collision disruptions increase. All
calculations then reach a point where the collisional cascade
removes leftover planetesimals more rapidly than oligarchs
can accrete them. As long as most planetesimals have r ∼
1–10 km, the timing of this epoch is more sensitive to grav-
itational focusing and the growth of oligarchs than the col-
lisional cascade and the fragmentation parameters. Thus,
rmax and the amount of mass processed through the colli-
sional cascade are relatively insensitive to the fragmentation
parameters.
4. Confronting KBO collision models with KBO data
Current data for KBOs provide two broad tests of coagu-
lation calculations. In each dynamical class, four measured
parameters test the general results of coagulation models
and provide ways to discriminate among the outcomes of
self-stirring and perturbed models. These parameters are
• rmax, the size of the largest object,
• αm, the slope of the size distribution for large KBOs
with r & 10 km,
• r0, the break radius, which measures the radius where
the size distribution makes the transition from a
merger population (r & r0) to a collisional popu-
lation (r . r0) as summarized in eq. (12), and
• Ml, the total mass in large KBOs.
For all KBOs, measurements of the dust mass allow tests of
the collisional cascade and link the Kuiper belt to observa-
tions of nearby debris disks. We begin with the discussion
of large KBOs and then compare the Kuiper belt with other
debris disks.
Table 1 summarizes the mass and size distribution pa-
rameters derived from recent surveys. To construct this
table, we used online data from the Minor Planet Center
(http://cfa−www.harvard.edu/iau/lists/MPLists.html)
for rmax (see also Levison and Stern 2001) and the results
of several detailed analyses for αm, rmax, and r0 (e.g.,
Bernstein et al. 2004; Elliot et al. 2005; Petit et al. 2006,
Chapter by Petit et al.). Because comprehensive KBO sur-
veys are challenging, the entries in the Table are incomplete
and sometimes uncertain. Nevertheless, these results pro-
vide some constraints on the calculations.
Current data provide clear evidence for physical differ-
ences among the dynamical classes. For classical KBOs
with a = 42–48 AU and q > 37 AU, the cold population
(i . 4o) has a steep size distribution with αm ≈ 3.5–4
and rmax ∼ 300–500 km. In contrast, the hot population
(i & 10o) has a shallow size distribution with αm ≈ 3 and
rmax ∼ 1000 km (Levison and Stern 2001). Both popu-
lations have relatively few objects with optical brightness
mR ≈ 27–27.5, which implies r0 ∼ 20–40 km for reason-
able albedo ∼ 0.04–0.07. The detached, resonant, and scat-
tered disk populations contain large objects with rmax ∼
1000 km. Although there are too few detached or scattered
disk objects to constrain αm or r0, data for the resonant
population are consistent with constraints derived for the
hot classical population, αm ≈ 3 and r0 ≈ 20–40 km.
The total mass in KBOs is a small fraction of the ∼
10–30 M⊕ of solid material in a MMSN from ∼ 35–50
AU (Gladman et al. 2001; Bernstein et al. 2004; Petit et al.
2006, see also Chapter by Petit et al.). The classical and res-
onant populations have Ml ≈ 0.01–0.1 M⊕ in KBOs with
TABLE 1. DATA FOR KBO SIZE DISTRIBUTION
KBO Class Ml (M⊕) rmax (km) r0 (km) qm
cold cl 0.01–0.05 400 20–40 km & 4
hot cl 0.01–0.05 1000 20–40 km 3–3.5
detached n/a 1500 n/a n/a
resonant 0.01–0.05 1000 20–40 km 3
scattered 0.1–0.3 700 n/a n/a
r & 10–20 km. The scattered disk may contain more mate-
rial, Ml ∼ 0.3 M⊕, but the constraints are not as robust as
for the classical and resonant KBOs.
These data are broadly inconsistent with the predictions
of self-stirring calculations with no external perturbers. Al-
though self-stirring models yield inclinations, i ≈ 2o–4o,
close to those observed in the cold, classical population, the
small rmax and large αm of this group suggest that an ex-
ternal dynamical perturbation – such as a stellar flyby or
stirring by Neptune – modified the evolution once rmax
reached ∼ 300–500 km. The observed break radius, r0 ∼
20–40 km, also agrees better with the r0 ∼ 10 km ex-
pected from Neptune stirring calculations than the r0 ∼ 1
km achieved in self-stirring models (Kenyon and Bromley
2004d; Pan and Sari 2005). Although a large rmax and
small αm for the resonant and hot, classical populations
agree reasonably well with self-stirring models, the ob-
served rmax ∼ 1000 km is much smaller than the rmax ∼
2000–3000 km typically achieved in self-stirring calcula-
tions (Figure 1). Both of these populations appear to have
large r0, which is also more consistent with Neptune stir-
ring models than with self-stirring models.
The small Ml for all populations provide additional ev-
idence against self-stirring models. In the most optimistic
scenario, where KBOs are easily broken, self-stirring mod-
els leave a factor of 5–10 more mass in large KBOs than
currently observed at 40–48 AU. Although models with
Neptune stirring leave a factor of 2–3 more mass in KBOs
at 40–48 AU than is currently observed, Neptune ejects
∼ half of the KBOs at 40–48 AU into the scattered disk
(e.g., Duncan et al. 1995; Kuchner et al. 2002). With an es-
timated mass of 2–3 times the mass in classical and reso-
nant KBOs, the scattered disk contains enough material to
bridge the difference between the KBO mass derived from
Neptune stirring models and the observed KBO mass.
The mass in KBO dust grains provides a final piece of
evidence against self-stirring models. From an analysis of
data from Pioneer 10 and 11, Landgraf et al. (2002) esti-
mate a dust production rate of ∼ 1015 g yr−1 in 0.01–2
mm particles at 40–50 AU. The timescale for Poynting-
Robertson drag to remove these grains from the Kuiper belt
is ∼ 10–100 Myr (Burns et al. 1979), which yields a mass
of ∼ 1022–1024 g. Figure 8 compares this dust mass with
masses derived from mid-IR and submm observations of
several nearby solar-type stars (Greaves et al. 1998, 2004;
Williams et al. 2004; Wyatt et al. 2005) and with predic-
tions from the self-stirring, flyby, and Neptune stirring mod-
els. The dust masses for nearby solar-type stars roughly fol-
5 6 7 8 9 10
log Time (yr)
Self-stirring
Flyby
Neptune
Fig. 8.— Evolution of mass in small dust grains (0.001–1
mm) for models with self-stirring (dot-dashed line), stirring from
a passing star (dashed line), and stirring from Neptune at 30 AU
(solid line) for Qb = 10
3 erg g−1. Calculations with smaller
(large) Qb produce more (less) dust at t . 50 Myr and some-
what more (less) dust at t & 100 Myr. At 1–5 Gyr, models with
Neptune stirring have less dust than self-stirring or flyby models.
The boxes show dust mass estimated for four nearby solar-type
stars (from left to right in age: HD 107146, ǫ Eri, η Crv, and τ
Cet; Greaves et al. 1998, 2004; Williams et al. 2004; Wyatt et al.
2005) and two estimates for the Kuiper belt (boxes connected by
solid line Landgraf et al. 2002).
low the predictions of self-stirring models and flyby models
with Qb ∼ 10
3 erg g−1. The mass of dust in the Kuiper
belt is 1–3 orders of magnitude smaller than predicted in
self-stirring models and is closer to the predictions of the
Neptune stirring models.
To combine the dynamical properties of KBOs with
these constraints, we rely on results from N -body simu-
lations that do not include collisional processing of small
objects (see Chapter by Morbidelli et al.). For simplic-
ity, we consider coagulation in the context of the Nice
model, which provides a solid framework for interpreting
the dynamics of the gas giants and the dynamical classes of
KBOs. In the Nice model, Saturn’s crossing of the 2:1 res-
onance with Jupiter initiates the dynamical instability that
populates the Kuiper belt. As Neptune approaches a ≈ 30
AU, it captures resonant KBOs, ejects KBOs into the scat-
tered disk and the Oort cloud, and excites the hot classical
KBOs. Although Neptune might reduce the number of cold,
classical KBOs formed roughly in situ beyond 30 AU, the
properties of these KBOs probably reflect conditions in the
Kuiper Belt when the instability began.
The Nice model requires several results from coagula-
tion calculations. Once giant planets form at 5–15 AU, col-
lisional growth must produce thousands of Pluto-mass ob-
jects at 20–30 AU. Unless the planetesimal disk was mas-
sive, growth of oligarchs must dominate collisional grind-
ing in this region of the disk. To produce the cold classical
population at ∼ 45 AU, collisions must produce 1–10 Pluto-
mass objects and then efficiently remove leftover planetesi-
mals. To match the data in Table 1, KBOs formed at 20–30
AU should have a shallower size distribution and a larger
rmax than those at 40–50 AU.
Some coagulation results are consistent with the trends
required in the Nice model. In current calculations, colli-
sional growth naturally yields smaller rmax and a steeper
size distribution at larger a. At 40–50 AU, Neptune-stirring
models produce a few Pluto-mass objects and many smaller
KBOs with e ∼ 0.1 and i ≈ 2o–4o. Although collisional
growth produces more Plutos at 15–30 AU than at 40–50
AU, collisional erosion removes material faster from the in-
ner disk than from the outer disk (Fig. 7). Thus, collisions
do not produce the thousands of Pluto-mass objects at 15–
30 AU required in the Nice model.
Reconciling this aspect of the Nice model with the coag-
ulation calculations requires a better understanding of the
physical processes that can slow or halt the collisional cas-
cade. Producing gas giants at 5–15 AU, thousands of Plutos
at 20–30 AU, and a few or no Plutos at 40–50 AU implies
that the outcome of coagulation changes markedly from 5
AU to 50 AU. If the collisional cascade can be halted as out-
lined in section §3.5, forming 5–10 M⊕ cores at 5–15 AU
is straightforward. Slowing the collisional cascade at 20–30
AU might yield a large population of Pluto mass objects at
20–30 AU. Because αm and rmax are well-correlated, bet-
ter constraints on the KBO size distributions coupled with
more robust coagulation calculations can test these aspects
of the Nice model in more detail.
To conclude this section, we consider constraints on the
Kuiper belt in the more traditional migration scenario of
Malhotra (1995), where Neptune forms at ∼ 20–25 AU
and slowly migrates to 30 AU. To investigate the relative
importance of collisional and dynamical depletion at 40–
50 AU, Charnoz and Morbidelli (2007) couple a collision
code with a dynamical code and derive the expected distri-
butions for size and orbital elements in the Kuiper belt, the
scattered disk, and the Oort cloud. Although collisional de-
pletion models can match the observations of KBOs, these
models are challenged to provide enough small objects into
the scattered disk and Oort cloud. Thus, the results suggest
that dynamical mechanisms dominate collisions in remov-
ing material from the Kuiper belt.
Although Charnoz and Morbidelli (2007) argue against
a dramatic change in collisional evolution from 15 AU to
40 AU, the current architecture of the solar system provides
good evidence for this possibility. In the MMSN, the ratio
of timescales to produce gas giant cores at 10 AU and at
25 AU is ξ = (25/10)3 ∼ 15. In the context of the Nice
model, formation of Saturn and Neptune at 8–11 AU in 5–
10 Myr thus implies formation of other gas giant cores at
20–25 AU in 50–150 Myr. If these cores had formed, they
would have consumed most of the icy planetesimals at 20–
30 AU, leaving little material behind to populate the outer
solar system when the giant planets migrate. The appar-
ent lack of gas giant core formation at 20–30 AU indicates
that the collisional cascade changed dramatically from 5–15
AU (where gas giant planets formed) to 20–30 AU (where
gas giant planets did not form). As outlined in §3.5, under-
standing the interaction of small particles with the gas and
the radiation field may provide important insights into the
evolution of oligarchic growth and thus into the formation
and structure of the solar system.
5. KBOs and Asteroids
In many ways, the Kuiper Belt is similar to the asteroid
belt. Both are populations of small bodies containing rela-
tively little mass compared to the rest of the Solar System;
the structure and dynamics of both populations have been
influenced significantly by the giant planets; and both have
been and continue to be significantly influenced by colli-
sions. Due to its relative proximity to Earth, however, there
are substantially more observational data available for the
asteroid belt than the Kuiper Belt. While the collisional and
dynamical evolution of the asteroid belt is certainly not a
solved problem, the abundance of constraints has allowed
for the development of reasonably consistent models. Here
we briefly describe what is currently understood about the
evolution of the asteroid belt, what insights that may give us
with regards to the evolution of the Kuiper Belt, and what
differences might exist in the evolution of the two popula-
tions.
It has long been recognized that the primordial as-
teroid belt must have contained hundreds or thousands
of times more mass than the current asteroid belt (e.g.
Lecar and Franklin 1973; Safronov 1979; Weidenschilling
1977c; Wetherill 1989). Reconstructing the initial mass
distribution of the Solar System from the current masses of
the planets and asteroids, for example, yields a pronounced
mass deficiency in the asteroid belt region relative to an oth-
erwise smooth distribution for the rest of the Solar System
(Weidenschilling 1977c). To accrete the asteroids on the
timescales inferred from meteoritic evidence would require
hundreds of times more mass than currently exists in the
main belt (Wetherill 1989).
In addition to its pronounced mass depletion, the aster-
oid belt is also strongly dynamically excited. The mean
proper eccentricity and inclination of asteroids larger than
∼50 km in diameter are 0.135 and 10.9o (from the cata-
log of Knežević and Milani (2003)), which are significantly
larger than can be explained by gravitational perturbations
amongst the asteroids or by simple gravitational pertur-
bations from the planets (Duncan 1994). The fact that
the different taxonomic types of asteroids (S-type, C-type,
etc.) are radially mixed somewhat throughout the main belt,
rather than confined to delineated zones, indicates that there
has been significant scattering in semimajor axis as well
(Gradie and Tedesco 1982).
Originally, a collisional origin was suggested for the
mass depletion in the asteroid belt (Chapman and Davis
1975). The difficulty of collisionally disrupting the largest
asteroids, coupled with the survival of the basaltic crust of
the ∼500-km diameter asteroid Vesta, however, suggest that
collisional grinding was not the cause of the mass deple-
tion (Davis et al. 1979, 1985, 1989, 1994; Wetherill 1989;
Durda and Dermott 1997; Durda et al. 1998; Bottke et al.
2005a,b; O’Brien and Greenberg 2005). In addition, col-
lisional processes alone could not fully explain both the dy-
namical excitation and the radial mixing observed in the as-
teroid belt, although Charnoz et al. (2001) suggest that col-
lisional diffusion may have contributed to its radial mixing.
Several dynamical mechanisms have been proposed
to explain the mass depletion, dynamical excitation and
radial mixing of the asteroid belt. As the solar nebula
dissipated, the changing gravitational potential acting on
Jupiter, Saturn, and the asteroids would lead to changes
in their precession rates and hence changes in the posi-
tions of secular resonances, which could ‘sweep’ through
the asteroid belt, exciting e and i, and coupled with gas
drag, could lead to semi-major axis mobility and the re-
moval of material from the belt (e.g., Heppenheimer 1980;
Ward 1981; Lemaitre and Dubru 1991; Lecar and Franklin
1997; Nagasawa et al. 2000, 2001, 2002). It has also been
suggested that sweeping secular resonances could lead to
orbital excitation in the Kuiper Belt (Nagasawa and Ida
2000). However, as reviewed by Petit et al. (2002) and
O’Brien et al. (2006), secular resonance sweeping is gen-
erally unable to simultaneously match the observed e and
i excitation in the asteroid belt, as well as its radial mix-
ing and mass depletion, for reasonable parameter choices
(especially in the context of the Nice Model).
Another possibility is that planetary embryos were able
to accrete in the asteroid belt (e.g., Wetherill 1992). The
fact that Jupiter’s ∼10 Earth-mass core was able to accrete
in our Solar System beyond the asteroid belt suggests that
embryos were almost certainly able to accrete in the aster-
oid belt, even accounting for the roughly 3-4× decrease in
the mass density of solid material inside the snow line. The
scattering of asteroids by those embryos, coupled with the
Jovian and Saturnian resonances in the asteroid belt, has
been shown to be able to reasonably reproduce the observed
e and i excitation in the belt as well as its radial mixing
and mass depletion (Petit et al. 2001, 2002; O’Brien et al.
2006). In the majority of simulations of this scenario by
both groups, the embryos are completely cleared from the
asteroid belt.
Thus, the observational evidence and theoretical mod-
els for the evolution of the asteroid belt strongly suggest
that dynamics, rather than collisions, dominated its mass
depletion. Collisions, however, have still played a key
role in sculpting the asteroid belt. Many dynamical fam-
ilies, clusterings in orbital element space, have been dis-
covered, giving evidence for ∼20 breakups of 100-km or
larger parent bodies over the history of the Solar System
(Bottke et al. 2005a,b). The large 500-km diameter asteroid
Vesta has a preserved basaltic crust with a single large im-
pact basin (McCord et al. 1970; Thomas et al. 1997). This
basin was formed by the impact of a roughly 40-km projec-
tile (Marzari et al. 1996; Asphaug 1997).
The size distribution of main-belt asteroids is known
or reasonably constrained through observational surveys
down to ∼1 km in diameter (e.g. Durda and Dermott 1997;
 0.1  1  10  100  1000
Diameter (km)
Asteroid and TNO Size Distributions
Bernstein (2004)
Fit to Data
 0.1  1  10  100  1000
Diameter (km)
Asteroid and TNO Size Distributions
Subaru
SKADS
Spacewatch
Cataloged Asteroids
Fig. 9.— Observational estimates of the main belt and TNO size
distributions. The pentagons (with dashed best-fit curve) show
the total TNO population as determined from the Bernstein et al.
(2004) HST survey, converted to approximate diameters assum-
ing an albedo of 0.04. Points with arrows are upper limits given
by non-detections. The solid line is the population of observed
asteroids, and open circles are from debiased Spacewatch main-
belt observations (Jedicke and Metcalfe 1998). These data, con-
verted to diameters, were provided by D. Durda. The two dashed
lines are extrapolations based on the Sloan Digital Sky Survey
(Ivezić et al. 2001) and the Subaru Sub-km Main Belt Aster-
oid Survey (Yoshida et al. 2003), and diamonds show the debi-
ased population estimate from the SKADS survey (Gladman et al.
2007). Error bars are left out of this plot for clarity. Note that the
TNO population is substantially more populous and massive, by
roughly a factor of 1000, than the asteroid population.
Jedicke and Metcalfe 1998; Ivezić et al. 2001; Yoshida et al.
2003; Gladman et al. 2007). Not surprisingly, the largest
uncertainties are at the smallest sizes, where good orbits
are often not available for the observed asteroids, which
makes the conversion to absolute magnitude and diame-
ter difficult (e.g., Ivezić et al. 2001; Yoshida et al. 2003).
Recent results from the Sub-Kilometer Asteroid Diame-
ter Survey (SKADS, Gladman et al. 2007), the first survey
since the Palomar-Leiden Survey designed to determine or-
bits as well as magnitudes of main-belt asteroids, suggest
that the asteroid magnitude-frequency distribution may be
well represented by a single power law in the range from
H=14.0 to 18.8, which corresponds to diameters of 0.7 to 7
km for an albedo of 0.11. These observational constraints
are shown in Fig. 9 alongside the determination of the TNO
size distribution from Bernstein et al. (2004).
While over some size ranges, the asteroid size distribu-
tion can be fit by a single power law, over the full range of
observed asteroid diameters from ∼1-1000 km, there are
multiple bumps or kinks in the size distribution (namely
around 10 and 100 km in diameter). The change in slope
of the size distribution around 100 km is due primarily to
the fact that asteroids larger than this are very difficult to
disrupt, and hence the size distribution of bodies larger than
100 km is likely primordial. The change in slope around
10 km has a different origin—such a structure is produced
as a result of a change in the strength properties of aster-
oids, namely the transition from when a body’s resistance
to disruption is dominated by material strength to when it is
dominated by self-gravity. This transition in strength prop-
erties occurs at a size much smaller than 10 km, but re-
sults in a structure that propagates to larger sizes (see, e.g.,
Durda et al. 1998; O’Brien and Greenberg 2003). The pres-
ence of this structure in the asteroid size distribution is con-
sistent with the asteroids being a collisionally-relaxed pop-
ulation, i.e. a population in which the size distribution has
reached an approximate steady state where collisional pro-
duction and collisional destruction of bodies in each size
range are in balance.
The collisional evolution of the asteroid belt has been
studied by many authors (e.g. Davis et al. 1985; Durda
1993; Davis et al. 1994; Durda and Dermott 1997; Durda et al.
1998; Campo Bagatin et al. 1993, 1994, 2001; Marzari et al.
1999). The most recent models of collisional evolution
of the asteroid belt incorporate aspects of dynamical evo-
lution as well, such as the removal of bodies by reso-
nances and the Yarkovsky effect, and the enhancement
in collisional activity during its massive primordial phase
(O’Brien and Greenberg 2005; Bottke et al. 2005a,b). In
particular, Bottke et al. (2005a) explicitly incorporate the
results of dynamical simulations of the excitation and clear-
ing of the main belt by embedded planetary embryos per-
formed by Petit et al. (2001). Such collisional/dynamical
models can be constrained by a wide range of observational
evidence such as the main belt size distribution, the num-
ber of observed asteroid families, the existence of Vesta’s
basaltic crust, and the cosmic ray exposure ages of ordi-
nary chondrite meteorites, which suggest that the lifetimes
of meter-scale stony bodies in the asteroid belt are on the
order of 10-20 Myr (Marti and Graf 1992).
One of the most significant implications of having an
early massive main belt, which was noted in early colli-
sional models (e.g. Chapman and Davis 1975) and recently
emphasized in the case of collisional evolution plus dynam-
ical depletion (e.g., Bottke et al. 2005b), is that the majority
of the collisional evolution of the asteroid belt occurred dur-
ing its early, massive phase, and there has been relatively lit-
tle change in the main-belt size distribution since then. The
current, wavy main-belt size distribution, then, is a ‘fossil’
from its first few hundred Myr of collisional and dynamical
evolution.
So how does the Kuiper Belt compare to the asteroid belt
in terms of its collisional and dynamical evolution? Evi-
dence and modeling for the asteroid belt suggest that dy-
namical depletion, rather than collisional erosion, was pri-
marily responsible for reducing the mass of the primordial
asteroid belt to its current level. In the case of the Kuiper
Belt, this is less clear. As shown in Sec. 3, collisional
erosion, especially when aided by stellar perturbations or
the formation of Neptune, can be very effective in remov-
ing mass. At the same time, dynamical models such as
the Nice Model result in the depletion of a large amount
of mass through purely dynamical means and are able to
match many observational constraints. Recent modeling
that couples both collisional fragmentation and dynami-
cal effects suggests that collisional erosion cannot play too
large of a role in removing mass from the Kuiper Belt, oth-
erwise the Scattered Disk and Oort Cloud would be too de-
pleted to explain the observed numbers of short- and long-
period comets (Charnoz and Morbidelli 2007). That model
currently does not include coagulation. Further modeling
work, which self-consistently integrates coagulation, colli-
sional fragmentation, and dynamical effects, is necessary to
fully constrain the relative contributions of collisional and
dynamical depletion in the Kuiper Belt.
We have noted that the asteroid belt has a collisionally-
relaxed size distribution that is not well-represented by a
single power law over all size ranges. Should we expect the
same for the Kuiper Belt size distribution, and is there evi-
dence to support this? The collision rate in the Kuiper Belt
should be roughly comparable to that in the asteroid belt,
with the larger number of KBOs offsetting their lower in-
trinsic collision probability (Davis and Farinella 1997), and
as noted earlier in this chapter, the primordial Kuiper Belt,
like the asteroid belt, would have been substantially more
massive than the current population. This suggests that
the Kuiper Belt should have experienced a degree of col-
lisional evolution roughly comparable to the asteroid belt,
and thus is likely to be collisionally relaxed like the asteroid
belt. Observational evidence thus far is not detailed enough
to say for sure if this is the case, although recent work
(Kenyon and Bromley 2004e; Pan and Sari 2005) suggests
that the observational estimate of the TNO size distribution
by Bernstein et al. (2004), shown in Fig. 9, is consistent
with a collisionally-relaxed population.
While the Kuiper Belt is likely to be collisionally re-
laxed, it is unlikely to mirror the exact shape of the aster-
oid belt size distribution. The shape of the size distribu-
tion is determined, in part, by the strength law Q∗D, which
is likely to differ somewhat between asteroids and KBOs.
This is due to the difference in composition between as-
teroids, which are primarily rock, and KBOs, which con-
tain a substantial amount of ice, as well as the difference
in collision velocity between the two populations. With a
mean velocity of ∼5 km/s (Bottke et al. 1994), collisions
between asteroids are well into the supersonic regime (rel-
ative to the sound speed in rock). For the Kuiper belt, col-
lision velocities are about a factor of 5 or more smaller
(Davis and Farinella 1997), such that collisions between
KBOs are close to the subsonic/supersonic transition. For
impacts occurring in these different velocity regimes, and
into different materials, Q∗D may differ significantly (e.g.,
Benz and Asphaug 1999).
The difference in collision velocity can influence the size
distribution in another way as well. With a mean collision
velocity of ∼5 km/s, a body of a given size in the aster-
oid belt can collisionally disrupt a significantly larger body.
Thus, transitions in the strength properties of asteroids can
lead to the formation of waves that propagate to larger sizes
and manifest themselves as changes in the slope of the size
distribution, as seen in Fig. 9. For the Kuiper belt, with col-
lision velocities that are about a factor of 5 or more smaller
than in the asteroid belt, the difference in size between a
given body and the largest body it is capable of disrupting
is much smaller than in the asteroid belt, and waves should
therefore be much less pronounced or non-existent in the
KBO size distribution (e.g., O’Brien and Greenberg 2003).
There is still likely to be a change in slope at the largest sizes
where the population transitions from being primordial to
being collisionally relaxed, and such a change appears in
the debiased observational data of Bernstein et al. (2004)
(shown here in Fig. 9), although recent observations sug-
gest that the change in slope may actually occur at smaller
magnitudes than found in that survey (Petit et al. 2006).
Is the size distribution of the Kuiper Belt likely to be a
‘fossil’ like the asteroid belt? The primordial Kuiper Belt
would have been substantially more massive than the cur-
rent population. Thus, regardless of whether the depletion
of its mass was primarily collisional or dynamical, colli-
sional evolution would have been more intense early on
and the majority of the collisional evolution would have oc-
curred early in its history. In either case, its current size
distribution could then be considered a fossil from that early
phase, although defining exactly when that early phase ends
and the size distribution becomes ‘fossilized’ is not equally
clear in both cases. In the case where the mass depletion
of the Kuiper Belt occurs entirely through collisions, there
would not necessarily be a well-defined point at which one
could say that the size distribution became fossilized, as the
collision rate would decay continuously with time. In the
case of dynamical depletion, where the mass would be re-
moved fairly rapidly as in the case of the Nice Model de-
scribed in Sec. 3.4, the collision rate would experience a
correspondingly rapid drop, and the size distribution could
be considered essentially fossilized after the dynamical de-
pletion event.
As noted earlier in this section, an important observable
manifestation of collisions in the asteroid belt is the for-
mation of families, i.e. groupings of asteroids with similar
orbits. Asteroid families are thought to be the fragments of
collisionally disrupted parent bodies. These were first rec-
ognized by Hirayama (1918) who found 3 families among
the 790 asteroids known at that time. The number increased
to 7 families by 1926 when there were 1025 known as-
teroids (Hirayama 1927). Today, there are over 350,000
known asteroids while the number of asteroid families has
grown to about thirty.
Given that the Kuiper Belt is likely a collisionally
evolved population, are there collisional families to be
found among these bodies? Families are expected to be
more difficult to recognize in the Kuiper Belt than in the
asteroid belt. Families are identified by finding statistically
significant clusters of asteroid orbit elements—mainly the
semi-major axis, eccentricity and inclination. The colli-
sional disruption of a parent bodies launches fragments
with speeds of perhaps a few hundred meters/sec relative
to the original target body. This ejection speed is small
compared with the orbital speeds of asteroids, hence the or-
bits of fragments differ by only small amounts from that of
the original target body and, more importantly, from each
other. Thus, the resulting clusters of fragments are easy to
identify.
However, in the Kuiper Belt, where ejection velocities
are likely to be about the same but orbital speeds are much
lower, collisional disruption produces a much greater dis-
persion in the orbital elements of fragments. This reduces
the density of the clustering of orbital elements and makes
the task of distinguishing family members from the back-
ground population much more difficult (Davis and Farinella
1997). To date, there are over 1000 KBOs known, many of
which have poorly-determined orbits or are in resonances
that would make the identification of a family difficult or
impossible. Chiang et al. (2003) applied lowest-order secu-
lar theory to 227 non-resonant KBOs with well-determined
orbits and found no convincing evidence for a dynamical
family. Recently, however, Brown et al. (2007) found evi-
dence for a single family with at least 5 members associated
with KBO 2003 EL61. This family was identified based on
the unique spectroscopic signature of its members, and con-
firmed by their clustered orbit elements.
Given the small numbers involved, it cannot be said
whether or not finding a single KBO family at this stage is
statistically that different from the original identification of
3 asteroid families when there were only 790 known aster-
oids (Hirayama 1918). However, the fact that the KBO fam-
ily associated with 2003 EL61 was first discovered spectro-
scopically, and its clustering in orbital elements was later
confirmed, while nearly all asteroid families were discov-
ered based on clusterings in orbital elements alone, suggests
that even if comparable numbers of KBO families and aster-
oid families do exist, the greater dispersion of KBO families
in orbital element space may make them more difficult to
identify unless there are spectroscopic signatures connect-
ing them as well.
Perhaps when the number of non-resonant KBOs with
good orbits approaches 1000, more populous Kuiper Belt
families will be identified, and as can be done now with
the asteroid belt, these KBO families can be used as con-
straints on the interior structures of their original parent
bodies as well as on the collisional and dynamical history
of the Kuiper Belt as a whole.
6. Concluding Remarks
Starting with a swarm of 1 m to 1 km planetesimals
at 20–150 AU, the growth of icy planets follows a stan-
dard pattern (Stern and Colwell 1997a,b; Kenyon and Luu
1998, 1999a,b; Kenyon and Bromley 2004a,d, 2005). Col-
lisional damping and dynamical friction lead to a short pe-
riod of runaway growth that produces 10–100 objects with
r ∼ 300–1000 km. As these objects grow, they stir the
orbits of leftover planetesimals up to the disruption veloc-
ity. Once disruptions begin, the collisional cascade grinds
leftover planetesimals into smaller objects faster than the
oligarchs can accrete them. Thus, the oligarchs always con-
tain a small fraction of the initial mass in solid material.
For self-stirring models, oligarchs contain ∼ 10% of the
initial mass. Stellar flybys and stirring by a nearby gas gi-
ant augment the collisional cascade and leave less mass in
oligarchs. The two examples in §3.3 suggest that a very
close flyby and stirring by Neptune leave ∼ 2% to 5% of
the initial mass in oligarchs with r ∼ 100–1000 km.
This evolution differs markedly from planetary growth
in the inner solar system. In ∼ 0.1–1 Myr at a few AU, run-
away growth produces massive oligarchs, m & 0.01M⊕,
that contain most of the initial solid mass in the disk. Aside
from a few giant impacts like those that might produce
the Moon (Hartmann and Davis 1975; Cameron and Ward
1976), collisions remove little mass from these objects. Al-
though the collisional cascade removes many leftover plan-
etesimals before oligarchs can accrete them, the lost ma-
terial is much less than half of the original solid mass
(Wetherill and Stewart 1993; Kenyon and Bromley 2004b).
For a & 40 AU, runaway growth leaves most of the mass in
0.1–10 km objects that are easily disrupted at modest colli-
sion velocities. In 4.5 Gyr, the collisional cascade removes
most of the initial disk mass inside 70–80 AU.
Together with numerical calculations of orbital dynam-
ics (Chapter by Morbidelli et al.), theory now gives us a
foundation for understanding the origin and evolution of
the Kuiper belt. Within a disk of planetesimals at 20–
100 AU, collisional growth naturally produces objects with
r ∼ 10–2000 km and a size distribution reasonably close
to that observed among KBOs. As KBOs form, migration
of the giant planets scatters KBOs into several dynamical
classes (Chapter by Morbidelli et al.). Once the giant plan-
ets achieve their current orbits, the collisional cascade re-
duces the total mass in KBOs to current levels and produces
the break in the size distribution at r ∼ 20–40 km. Contin-
ued dynamical scattering by the giant planets sculpts the
inner Kuiper belt and maintains the scattered disk.
New observations will allow us to test and to refine this
theoretical picture. Aside from better measures of αm,
rmax, and r0 among the dynamical classes, better limits on
the total mass and the size distribution of large KBOs with
a ∼ 50–100 AU should yield a clear discriminant among
theoretical models. In the Nice model, the Kuiper belt was
initially nearly empty outside of ∼ 50 AU. Thus, any KBOs
found with a ∼ 50–100 AU should have the collisional
and dynamical signatures of the scattered disk or detached
population. If some KBOs formed in situ at a & 50 AU,
their size distribution depends on collisional growth modi-
fied by self-stirring and stirring by ∼ 30 M⊕ of large KBOs
formed at 20–30 AU and scattered through the Kuiper belt
by the giant planets. From the calculations of Neptune stir-
ring (§3.3), stirring by scattered disk objects should yield a
size distribution markedly different from the size distribu-
tion of detached or scattered disk objects formed at 20–30
AU. Wide-angle surveys on 2–3 m class telescopes (e.g.,
Pan-Starrs) and deep probes with 8–10 m telescopes can
provide this test.
Information on smaller size scales – αd and r1 – place
additional constraints on the bulk properties (fragmentation
parameters) of KBOs and on the collisional cascade. In any
of the stirring models, there is a strong correlation between
r0, r1, and the fragmentation parameters. Thus, direct mea-
sures of r0 and r1 provide a clear test of KBO formation
calculations. At smaller sizes (r . 0.1 km), the slope of
the size distribution αd clearly tests the fragmentation al-
gorithm and the ability of the collisional cascade to remove
KBOs with r ∼ 1–10 km. Although the recent detection
of KBOs with r ≪ 1 km (Chang et al. 2006) may be an
instrumental artifact (Jones et al. 2006; Chang et al. 2007),
optical and X-ray occultations (e.g., TAOS) will eventually
yield these tests.
Finally, there is a clear need to combine coagulation
and dynamical calculations to produce a ‘unified’ picture
of planet formation at a & 20 AU. Charnoz and Morbidelli
(2007) provide a good start in this direction. Because the
collisional outcome is sensitive to internal and external dy-
namics, understanding the formation of the observed n(r),
n(e), and n(i) distributions in each KBO population re-
quires treating collisional evolution and dynamics together.
A combined approach should yield the sensitivity of αm,
rmax, and r0 to the local evolution and the timing of the
formation of giant planets, Neptune migration, and stellar
flybys. These calculations will also test how the dynami-
cal events depend on the evolution during oligarchic growth
and the collisional cascade. Coupled with new observations
of KBOs and of planets and debris disks in other planetary
systems, these calculations should give us a better under-
standing of the origin and evolution of KBOs and other ob-
jects in the outer solar system.
We thank S. Charnoz, S. Kortenkamp, A. Morbidelli,
and an anonymous reviewer for comments that consider-
ably improved the text. We acknowledge support from
the NASA Astrophysics Theory Program (grant NAG5-
13278; BCB & SJK), the NASA Planetary Geology and
Geophysics Program (grant NNX06AC50G; DPO), and the
JPL Institutional Computing and Information Services and
the NASA Directorates of Aeronautics Research, Science,
Exploration Systems, and Space Operations (BCB & SJK).
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This 2-column preprint was prepared with the AAS LATEX macros v5.2.
	INTRODUCTION
	COAGULATION THEORY
	COAGULATION SIMULATIONS
	Background
	Self-Stirring
	External Perturbation
	Nice Model
	A Caveat on the Collisional Cascade
	Model Predictions
	Confronting KBO collision models with KBO data
	KBOs and Asteroids
	Concluding Remarks

  This chapter summarizes analytic theory and numerical calculations for the
formation and collisional evolution of KBOs at 20--150 AU. We describe the main
predictions of a baseline self-stirring model and show how dynamical
perturbations from a stellar flyby or stirring by a giant planet modify the
evolution. Although robust comparisons between observations and theory require
better KBO statistics and more comprehensive calculations, the data are broadly
consistent with KBO formation in a massive disk followed by substantial
collisional grinding and dynamical ejection. However, there are important
problems reconciling the results of coagulation and dynamical calculations.
Contrasting our current understanding of the evolution of KBOs and asteroids
suggests that additional observational constraints, such as the identification
of more dynamical families of KBOs (like the 2003 EL61 family), would provide
additional information on the relative roles of collisional grinding and
dynamical ejection in the Kuiper Belt. The uncertainties also motivate
calculations that combine collisional and dynamical evolution, a `unified'
calculation that should give us a better picture of KBO formation and
evolution.

FORMATION AND COLLISIONAL EVOLUTION OF KUIPER
BELT OBJECTS
Scott J. Kenyon
Smithsonian Astrophysical Observatory
Benjamin C. Bromley
Department of Physics, University of Utah
David P. O’Brien
Planetary Science Institute
Donald R. Davis
Planetary Science Institute
This chapter summarizes analytic theory and numerical calculations for the formation and
collisional evolution of KBOs at 20–150 AU. We describe the main predictions of a baseline
self-stirring model and show how dynamical perturbations from a stellar flyby or stirring
by a giant planet modify the evolution. Although robust comparisons between observations
and theory require better KBO statistics and more comprehensive calculations, the data are
broadly consistent with KBO formation in a massive disk followed by substantial collisional
grinding and dynamical ejection. However, there are important problems reconciling the
results of coagulation and dynamical calculations. Contrasting our current understanding of
the evolution of KBOs and asteroids suggests that additional observational constraints, such
as the identification of more dynamical families of KBOs (like the 2003 EL61 family), would
provide additional information on the relative roles of collisional grinding and dynamical
ejection in the Kuiper Belt. The uncertainties also motivate calculations that combine collisional
and dynamical evolution, a ‘unified’ calculation that should give us a better picture of KBO
formation and evolution.
1. INTRODUCTION
Every year in the Galaxy, a star is born. Most stars
form in dense clusters of thousands of stars, as in the Orion
Nebula Cluster (Lada and Lada 2003; Slesnick et al. 2004).
Other stars form in small groups of 5–10 stars in loose asso-
ciations of hundreds of stars, as in the Taurus-Auriga clouds
(Gomez et al. 1993; Luhman 2006). Within these associa-
tions and clusters, most newly-formed massive stars are bi-
naries; lower mass stars are usually single (Lada 2006).
Large, optically thick circumstellar disks surround
nearly all newly-formed stars (Beckwith and Sargent 1996).
The disks have sizes of ∼ 100–200 AU, masses of ∼ 0.01-
0.1 M⊙, and luminosities of ∼ 0.2–1 L⋆, where L⋆ is the
luminosity of the central star. The masses and geometries
of these disks are remarkably similar to the properties of
the minimum mass solar nebula (MMSN), the disk required
for the planets in the solar system (Weidenschilling 1977b;
Hayashi 1981; Scholz et al. 2006).
As stars age, they lose their disks. For solar-type stars,
radiation from the opaque disk disappears in 1–10 Myr
(Haisch et al. 2001). Many older stars have optically thin
debris disks comparable in size to the opaque disks of
younger stars but with much smaller masses, . 1M⊕, and
luminosities, . 10−3 L⋆ (Chapter by Moro-Martin et al.).
The lifetime of this phase is uncertain. Some 100 Myr-old
stars have no obvious debris disk; a few 1–10 Gyr-old stars
have massive debris disks (Greaves 2005).
In the current picture, planets form during the transi-
tion from an optically thick protostellar disk to an opti-
cally thin debris disk. From the statistics of young stars in
molecular clouds, the timescale for this transition, ∼ 105
yr, is comparable to the timescales derived for the forma-
tion of planetesimals from dust grains (Weidenschilling
1977a; Youdin and Shu 2002; Dullemond and Dominik
2005) and for the formation of lunar-mass or larger
planets from planetesimals (Wetherill and Stewart 1993;
Weidenschilling et al. 1997a; Kokubo and Ida 2000; Nagasawa et al.
2005; Kenyon and Bromley 2006). Because the grains
in debris disks have short collision lifetimes, . 1 Myr,
compared to the ages of their parent stars, & 10 Myr,
high velocity collisions between larger objects must main-
tain the small grain population (Aumann et al. 1984;
Backman and Paresce 1993). The inferred dust produc-
tion rates for debris disks around 0.1–10 Gyr old stars,
∼ 1020 g yr−1, require an initial mass in 1 km ob-
http://arxiv.org/abs/0704.0259v1
jects, Mi ∼ 10–100 M⊕, comparable to the amount of
solids in the MMSN. Because significant long-term de-
bris production also demands gravitational stirring by
an ensemble of planets with radii of 500–1000 km or
larger (Kenyon and Bromley 2004a; Wyatt et al. 2005),
debris disks probably are newly-formed planetary sys-
tems (Aumann et al. 1984; Backman and Paresce 1993;
Artymowicz 1997; Kenyon and Bromley 2002b, 2004a,b).
KBOs provide a crucial test of this picture. With objects
ranging in size from 10–20 km to ∼ 1000 km, the size dis-
tribution of KBOs yields a key comparison with theoretical
calculations of planet formation (Davis and Farinella 1997;
Kenyon and Luu 1998, 1999a,b). Once KBOs have sizes
of 100–1000 km, collisional grinding, dynamical perturba-
tions by large planets and passing stars, and self-stirring by
small embedded planets produce features in the distribu-
tions of sizes and dynamical elements that observations can
probe in detail. Although complete calculations of KBO
formation and dynamical evolution are not available, these
calculations will eventually yield a better understanding of
planet formation at 20–100 AU.
The Kuiper belt also enables a vital link between the
solar system and other planetary systems. With an outer
radius of & 1000 AU (Sedna’s aphelion) and a current
mass of ∼ 0.1 M⊕ (Luu and Jewitt 2002; Bernstein et al.
2004, Cahpter by Petit et al.), the Kuiper belt has prop-
erties similar to those derived for the oldest debris disks
(Greaves et al. 2004; Wyatt et al. 2005). Understanding
planet formation in the Kuiper belt thus informs our inter-
pretation of evolutionary processes in other planetary sys-
tems.
This paper reviews applications of coagulation theory for
planet formation in the Kuiper belt. After a brief introduc-
tion to the theoretical background in §2, we describe results
from numerical simulations in §3, compare relevant KBO
observations with the results of numerical simulations in §4,
and contrast the properties of KBOs and asteroids in §5. We
conclude with a short summary in §6.
2. COAGULATION THEORY
Planet formation begins with dust grains suspended in
a gaseous circumstellar disk. Grains evolve into planets
in three steps. Collisions between grains produce larger
aggregates which decouple from the gas and settle into a
dense layer in the disk midplane. Continued growth of the
loosely bound aggregates leads to planetesimals, gravita-
tionally bound objects whose motions are relatively inde-
pendent of the gas. Collisions and mergers among the en-
semble of planetesimals form planets. Here, we briefly de-
scribe the physics of these stages and summarize analytic
results as a prelude to summaries of numerical simulations.
We begin with a prescription for the mass surface density
Σ of gas and dust in the disk. We use subscripts ‘d’ for the
dust and ‘g’ for the gas and adopt
Σd,g = Σ0d,0g
40 AU
, (1)
where a is the semimajor axis. In the MMSN, n =
3/2, Σ0d ≈ 0.1 g cm
−2, and Σ0g ≈ 5–10 g cm
(Weidenschilling 1977b; Hayashi 1981). For a disk with
an outer radius of 100 AU, the MMSN has a mass of
∼ 0.03 M⊙, which is comparable to the disk masses of
young stars in nearby star-forming regions (Natta et al.
2000; Scholz et al. 2006).
The dusty midplane forms quickly (Weidenschilling
1977a, 1980; Dullemond and Dominik 2005). For inter-
stellar grains with radii, r ∼ 0.01–0.1 µm, turbulent mixing
approximately balances settling due to the vertical compo-
nent of the star’s gravity. As grains collide and grow to r ∼
0.1–1 mm, they decouple from the turbulence and settle
into a thin layer in the disk midplane. The timescale for this
process is ∼ 103 yr at 1 AU and ∼ 105 yr at 40 AU.
The evolution of grains in the midplane is uncertain. Be-
cause the gas has some pressure support, it orbits the star
slightly more slowly than the Keplerian velocity. Thus,
orbiting dust grains feel a headwind that drags them to-
ward the central star (Adachi et al. 1976; Weidenschilling
1984; Tanaka and Ida 1999). For m-sized objects, the drag
timescale at 40 AU, ∼ 105 yr, is comparable to the growth
time. Thus, it is not clear whether grains can grow by di-
rect accretion to km sizes before the gas drags them into the
inner part of the disk.
Dynamical processes provide alternatives to random ag-
glomeration of grains. In ensembles of porous grains, gas
flow during disruptive collisions leads to planetesimal for-
mation by direct accretion (Wurm et al. 2004). Analytic es-
timates and numerical simulations indicate that grains with
r ∼ 1 cm are also easily trapped within vortices in the disk
(e.g. de la Fuente Marcos and Barge 2001; Inaba and Barge
2006). Large enhancements in the solid-to-gas ratio within
vortices allows accretion to overcome gas drag, enabling
formation of km-sized planetesimals in 104–105 yr.
If the dusty midplane is calm, it becomes thinner and
thinner until groups of particles overcome the local Jeans
criterion – where their self-gravity overcomes local orbital
shear – and ‘collapse’ into larger objects on the local dy-
namical timescale, ∼ 103 yr at 40 AU (Goldreich and Ward
1973; Youdin and Shu 2002; Tanga et al. 2004). This pro-
cess is a promising way to form planetesimals; however,
turbulence may prevent the instability (Weidenschilling
1995, 2003, 2006). Although the expected size of a col-
lapsed object is the Jeans wavelength, the range of plan-
etesimal sizes the instability produces is also uncertain.
Once planetesimals with r ∼ 1 km form, gravity dom-
inates gas dynamics. Long range gravitational interactions
exchange kinetic energy (dynamical friction) and angular
momentum (viscous stirring), redistributing orbital energy
and angular momentum among planetesimals. For 1 km
objects at 40 AU, the initial random velocities are compara-
ble to their escape velocities, ∼ 1 m s−1 (Weidenschilling
1980; Goldreich et al. 2004). The gravitational binding en-
ergy (for brevity, we use energy as a shorthand for specific
energy), Eg ∼ 10
4 erg g−1, is then comparable to the typ-
ical collision energy, Ec ∼ 10
4 erg g−1. Both energies are
smaller than the disruption energy – the collision energy
needed to remove half of the mass from the colliding pair
of objects – which is Q∗D ∼ 10
5–107 erg g−1 for icy mate-
rial (Davis et al. 1985; Benz and Asphaug 1999; Ryan et al.
1999; Michel et al. 2001; Leinhardt and Richardson 2002;
Giblin et al. 2004). Thus, collisions produce mergers in-
stead of debris.
Initially, small planetesimals grow slowly. For a large
ensemble of planetesimals, the collision rate is nσv, where
n is the number of planetesimals, σ is the cross-section,
and v is the relative velocity. The collision cross-section is
the geometric cross-section, πr2, scaled by the gravitational
focusing factor, fg,
σc ∼ πr
2fg ∼ πr
2(1 + β(vesc/evK)
2) , (2)
where e is the orbital eccentricity, vK is the orbital ve-
locity, vesc is the escape velocity of the merged pair
of planetesimals, and β ≈ 2.7 is a coefficient that ac-
counts for three-dimensional orbits in a rotating disk
(Greenzweig and Lissauer 1990; Spaute et al. 1991; Wetherill and Stewart
1993). Because evK ≈ vesc, gravitational focusing factors
are small and growth is slow and orderly (Safronov 1969).
The timescale for slow, orderly growth is
ts ≈ 30
1000 km
250 yr
0.1 g cm−2
Gyr ,
where P is the orbital period (Safronov 1969; Lissauer
1987; Goldreich et al. 2004).
As larger objects form, several processes damp parti-
cle random velocities and accelerate growth. For objects
with r ∼ 1–100 m, physical collisions reduce particle ran-
dom velocities (Ohtsuki 1992; Kenyon and Luu 1998). For
larger objects with r & 0.1 km, the smaller objects damp
the orbital eccentricity of larger particles through dynam-
ical friction (Wetherill and Stewart 1989; Kokubo and Ida
1995; Kenyon and Luu 1998). Viscous stirring by the large
objects excites the orbits of the small objects. For planetes-
imals with r ∼ 1 m to r ∼ 1 km, these processes occur
on short timescales, . 106 yr at 40 AU, and roughly bal-
ance when these objects have orbital eccentricity e ∼ 10−5.
In the case where gas drag is negligible, Goldreich et al.
(2004) derive a simple relation for the ratio of the eccentric-
ities of the large (‘l’) and the small (‘s’) objects in terms of
their surface densities Σl,s (see also Kokubo and Ida 2002;
Rafikov 2003c,b,d,a),
, (4)
with γ = 1/4 to 1/2. Initially, most of the mass is in small
objects. Thus Σl/Σs ≪ 1. For Σl/Σs ∼ 10
−3 to 10−2,
el/es ≈ 0.1–0.25. Because esvK ≪ vl,esc gravitational
focusing factors for large objects accreting small objects are
large. Runaway growth begins.
Runaway growth relies on positive feedback between
accretion and dynamical friction. Dynamical friction pro-
duces the largest fg for the largest objects, which grow
faster and faster relative to the smaller objects and contain
an ever growing fraction of the total mass. As they grow,
these protoplanets stir the planetesimals. The orbital ve-
locity dispersions of small objects gradually approach the
escape velocities of the protoplanets. With esvK ∼ vl,esc,
collision rates decline as runaway growth continues (eqs.
(2) and (4)). The protoplanets and leftover planetesimals
then enter the oligarchic phase, where the largest objects
– oligarchs – grow more slowly than they did as runaway
objects but still faster than the leftover planetesimals. The
timescale to reach oligarchic growth is (Lissauer 1987;
Goldreich et al. 2004)
to ≈ 30
250 yr
0.1 g cm−2
Myr , (5)
For the MMSN, to ∝ a
−3. Thus, collisional damping,
dynamical friction and gravitational focusing enhance the
growth rate by 3 orders of magnitude compared to orderly
growth.
Among the oligarchs, smaller oligarchs grow the fastest.
Each oligarch tries to accrete material in an annular ‘feed-
ing zone’ set by balancing the gravity of neighboring oli-
garchs. If an oligarch accretes all the mass in its feed-
ing zone, it reaches the ‘isolation mass’ (Lissauer 1987;
Kokubo and Ida 1998, 2002; Rafikov 2003a; Goldreich et al.
2004),
miso ≈ 28
40 AU
0.1 g cm−2
M⊕ . (6)
Each oligarch stirs up leftover planetesimals along its or-
bit. Smaller oligarchs orbit in regions with smaller Σl/Σs.
Thus, smaller oligarchs have larger gravitational focusing
factors (eqs. (2) and (4)) and grow faster than larger oli-
garchs (Kokubo and Ida 1998; Goldreich et al. 2004).
As oligarchs approach miso, they stir up the velocities
of the planetesimals to the disruption velocity. Instead of
mergers, collisions then yield smaller planetesimals and de-
bris. Continued disruptive collisions lead to a collisional
cascade, where leftover planetesimals are slowly ground to
dust (Dohnanyi 1969; Williams and Wetherill 1994). Ra-
diation pressure from the central star ejects dust grains
with r . 1–10 µm; Poynting-Robertson drag pulls larger
grains into the central star (Burns et al. 1979; Artymowicz
1988; Takeuchi and Artymowicz 2001). Eventually, plan-
etesimals are accreted by the oligarchs or ground to dust.
To evaluate the oligarch mass required for a disruptive
collision, we consider two planetesimals with equal mass
mp. The center-of-mass collision energy is
Qi = v
i /8 , (7)
where the impact velocity v2i = v
2+v2esc (Wetherill and Stewart
1993). The energy needed to remove half of the combined
mass of two colliding planetesimals is
Q∗D = Qb
+ ρQg
, (8)
1 3 5 7
log Radius (cm)
e = 0.001
e = 0.01
e = 0.1
Fig. 1.— Disruption energy, Q∗D, for icy objects. The solid
curve plots a typical result derived from numerical simulations of
collisions that include a detailed equation of state for crystalline
ice (Qb = 1.6 × 10
7 erg g−1, βb = −0.42, ρ = 1.5 g cm
−3, Qg =
1.5 erg cm−3, and βg = 1.25; Benz and Asphaug 1999). The other
curves plot results using Qb consistent with model fits to comet
breakups (βb ≈ 0; Qb ∼ 10
3 erg g−1, dashed curve; Qb ∼ 10
erg g−1, dot-dashed curve; Asphaug and Benz 1996). The dashed
horizontal lines indicate the center of mass collision energy (eq.
(7)) for equal mass objects with e = 0.001, 0.01, and 0.1. Col-
lisions between objects with Qi ≪ Q
D yield merged remnants;
collisions between objects with Qi ≫ Q
D produce debris.
where Qbr
βb is the bulk (tensile) component of the binding
energy and ρQgr
βg is the gravity component of the bind-
ing energy (Davis et al. 1985; Housen and Holsapple 1990,
1999; Holsapple 1994; Benz and Asphaug 1999). We adopt
v ≈ vesc,o, where vesc,o = (Gmo/ro)
1/2 is the escape ve-
locity of an oligarch with mass mo and radius ro. We define
the disruption mass md by deriving the oligarch mass where
Qi ≈ Q
D. For icy objects at 30 AU
md ∼ 3× 10
107 erg g−1
M⊕. (9)
Figure 1 illustrates the variation of Q∗D with radius for
several variants of eq. (8). For icy objects, detailed nu-
merical collision simulations yield Qb . 10
7 erg g−1,
−0.5 . βb . 0, ρ ≈ 1–2 g cm
−3, Qg ≈ 1–2 erg cm
and βg ≈ 1–2 (solid line in Fig. 1, Benz and Asphaug
1999, see also Chapter by Leinhardt et al.)). Models for
the breakup of comet Shoemaker-Levy 9 suggest a smaller
component of the bulk strength, Qb ∼ 10
3 erg g−1 (e.g.,
Asphaug and Benz 1996), which yields smaller disruption
energies for smaller objects (Fig. 1, dashed and dot-dashed
curves). Because nearly all models for collisional disrup-
tion yield similar results for objects with r & 1 km (e.g.,
Kenyon and Bromley 2004d), the disruption mass is fairly
independent of theoretical uncertainties once planetesimals
become large. For typical Q∗D ∼ 10
7–108 erg g−1 for 1–10
km objects (Fig. 1), leftover planetesimals start to disrupt
when oligarchs have radii, ro ∼ 200–500 km.
Once disruption commences, the final mass of an oli-
garch depends on the timescale for the collisional cascade
(Kenyon and Bromley 2004a,b,d; Leinhardt and Richardson
2005). If disruptive collisions produce dust grains much
faster than oligarchs accrete leftover planetesimals, oli-
garchs with mass mo cannot grow much larger than the
disruption radius (maximum oligarch mass mo,max ≈ md).
However, if oligarchs accrete grains and leftover planetes-
imals effectively, oligarchs reach the isolation mass before
collisions and radiation pressure remove material from the
disk (eq. (6); Goldreich et al. 2004). The relative rates of
accretion and disruption depend on the balance between
collisional damping and gas drag – which slow the colli-
sional cascade – and viscous stirring and dynamical friction
– which speed up the collisional cascade. Because deriv-
ing accurate rates for these processes requires numerical
simulations of planetesimal accretion, we now consider
simulations of planet formation in the Kuiper belt.
3. COAGULATION SIMULATIONS
3.1. Background
Safronov (1969) invented the current approach to plan-
etesimal accretion calculations. In his particle-in-a-box
method, planetesimals are a statistical ensemble of masses
with distributions of orbital eccentricities and inclinations
(Greenberg et al. 1978; Wetherill and Stewart 1989, 1993;
Spaute et al. 1991). This statistical approximation is es-
sential: N -body codes cannot follow the n ∼ 109–1012
1 km planetesimals required to build Pluto-mass or Earth-
mass planets. For large numbers of objects on fairly cir-
cular orbits (e.g., n & 104, r . 1000 km, and e . 0.1),
the method is also accurate. With a suitable prescription
for collision outcomes, solutions to the coagulation equa-
tion in the kinetic theory yield the evolution of n(m) with
arbitrarily small errors (e.g., Wetherill 1990; Lee 2000;
Malyshkin and Goodman 2001).
In addition to modeling planet growth, the statistical
approach provides a method for deriving the evolution of
orbital elements for large ensembles of planetesimals. If
we (i) assume the distributions of e and i for planetesimals
follow a Rayleigh distribution and (ii) treat their motions
as perturbations of a circular orbit, we can use the Fokker-
Planck equation to solve for small changes in the orbits
due to gas drag, gravitational interactions, and physical col-
lisions (Hornung et al. 1985; Wetherill and Stewart 1993;
Ohtsuki et al. 2002). Although the Fokker-Planck equation
cannot derive accurate orbital parameters for planetesimals
and oligarchs near massive planets, it yields accurate solu-
tions for the ensemble average e and i when orbital reso-
nances and other dynamical interactions are not important
(e.g., Wetherill and Stewart 1993; Weidenschilling et al.
1997a; Ohtsuki et al. 2002).
Several groups have implemented Safronov’s method for
calculations relevant to the outer solar system (Greenberg et al.
1984; Stern 1995, 2005; Stern and Colwell 1997a,b; Davis and Farinella
1997; Kenyon and Luu 1998, 1999a,b; Davis et al. 1999;
Kenyon and Bromley 2004a,d, 2005). These calculations
-3 -2 -1 0 1 2 3 4
log Radius (km)
0 Myr
40-47 AU
-3 -2 -1 0 1 2 3 4
log Radius (km)
Fig. 2.— Evolution of a multiannulus coagulation model with Σ = 0.12(ai/40 AU)−3/2 g cm−2. Left: cumulative mass distribution
at times indicated in the legend. Right: eccentricity distributions at t = 0 (light solid line), t = 10 Myr (filled circles), t = 100 Myr
(open boxes), t = 1 Gyr (filled triangles), and t = 5 Gyr (open diamonds). As large objects grow in the disk, they stir up the leftover
planetesimals to e ∼ 0.1. Disruptive collisions then deplete the population of 0.1–10 km planetesimals, which limits the growth of the
largest objects.
adopt a disk geometry and divide the disk into N concen-
tric annuli with radial width ∆ai at distances ai from the
central star. Each annulus is seeded with a set of planetes-
imals with masses mij , eccentricities eij , and inclinations
iij , where the index i refers to one of N annuli and the
index j refers to one of M mass batches within an annulus.
The mass batches have mass spacing δ ≡ mj+1/mj . In
most calculations, δ ≈ 2; δ ≤ 1.4 is optimal (Ohtsuki et al.
1990; Wetherill and Stewart 1993; Kenyon and Luu 1998).
Once the geometry is set, the calculations solve a set of
coupled difference equations to derive the number of ob-
jects nij , and the orbital parameters, eij and iij , as func-
tions of time. Most studies allow fragmentation and ve-
locity evolution through gas drag, collisional damping, dy-
namical friction and viscous stirring. Because Q∗D sets the
maximum size mc,max of objects that participate in the
collisional cascade, the size distribution for objects with
m < mc,max depends on the fragmentation parameters (eq.
(8); Davis and Farinella 1997; Kenyon and Bromley 2004d;
Pan and Sari 2005). The size and velocity distributions of
the merger population with m > mc,max are established
during runaway growth and the early stages of oligarchic
growth. Accurate treatment of velocity evolution is impor-
tant for following runaway growth and thus deriving good
estimates for the growth times and the size and velocity dis-
tributions of oligarchs.
When a few oligarchs contain most of the mass, col-
lision rates depend on the orbital dynamics of individual
objects instead of ensemble averages. Safronov’s statisti-
cal approach then fails (e.g., Wetherill and Stewart 1993;
Weidenschilling et al. 1997b). Although N-body methods
can treat the evolution of the oligarchs, they cannot follow
the evolution of leftover planetesimals, where the statis-
tical approach remains valid (e.g., Weidenschilling et al.
1997b). Spaute et al. (1991) solve this problem by adding
a Monte Carlo treatment of binary interactions between
large objects to their multiannulus coagulation code.
Bromley and Kenyon (2006) describe a hybrid code, which
merges a direct N-body code with a multiannulus coagula-
tion code. Both codes have been applied to terrestrial planet
formation, but not to the Kuiper belt.
Current calculations cannot follow collisional growth ac-
curately in an entire planetary system. Although the six
order of magnitude change in formation timescales from
∼ 0.4 AU to 40 AU is a factor in this statement, most
modern supercomputers cannot finish calculations involv-
ing the entire disk with the required spatial resolution on
a reasonable timescale. For the Kuiper belt, it is possi-
ble to perform a suite of calculations in a disk extending
from 30–150 AU following 1 m and larger planetesimals.
These calculations yield robust results for the mass distri-
bution as a function of space and time and provide interest-
ing comparisons with observations. Although current cal-
culations do not include complete dynamical interactions
with the giant planets or passing stars ((see, for example,
Charnoz and Morbidelli 2007), sample calculations clearly
show the importance of external perturbations in treating
the collisional cascade. We begin with a discussion of
self-stirring calculations without interactions with the gi-
ant planets or passing stars and then describe results with
external perturbers.
3.2. Self-Stirring
To illustrate in situ KBO formation at 40–150 AU, we
consider a multiannulus calculation with an initial ensem-
ble of 1 m to 1 km planetesimals in a disk with Σ0d =
0.12 g cm−2. The planetesimals have initial radii of 1 m
to 1 km (with equal mass per logarithmic bin), e0 = 10
i0 = e0/2, mass density
1 ρ = 1.5 g cm−3 and fragmentation
parameters Qb = 10
3 erg g−1, Qg = 1.5 erg cm
−3, βb = 0,
1 Our choice of mass density is a compromise between pure ice (ρ = 1
g cm−3) and the measured density of Pluto (ρ ≈ 2 g cm−3 Null et al.
and βg = 1.25 (dashed curve in Fig. 1; Kenyon and Bromley
2004d, 2005). The gas density also follows a MMSN, with
Σg/Σd = 100 and a vertical scale height h = 0.1 r
(Kenyon and Hartmann 1987). The gas density is Σg ∝
e−t/tg , with tg = 10 Myr.
This calculation uses an updated version of the Bromley and Kenyon
(2006) code that includes a Richardson extrapolation pro-
cedure in the coagulation algorithm. As in the Eulerian
(Kenyon and Luu 1998) and fourth order Runge-Kutta
(Kenyon and Bromley 2002a,b) methods employed pre-
viously, this code provides robust numerical solutions to
kernels with analytic solutions (e.g., Ohtsuki et al. 1990;
Wetherill 1990) without producing the wavy size distribu-
tions described in other simulations with a low mass cutoff
(e.g. Campo Bagatin et al. 1994). Once the evolution of
large (r > 1 m) objects is complete, a separate code tracks
the evolution of lower mass objects and derives the dust
emission as a function of time.
Figure 2 shows the evolution of the mass and eccentric-
ity distributions at 40–47 AU for this calculation. During
the first few Myr, the largest objects grow slowly. Dynam-
ical friction damps the orbits of the largest objects; colli-
sional damping and gas drag circularize the orbits of the
smallest objects. This evolution erases many of the initial
conditions and enhances gravitational focusing by factors
of 10–1000. Runaway growth begins. A few (and some-
times only one) oligarchs then grow from r ∼ 10 km to
r ∼ 1000 km in ∼ 30 Myr at 40 AU and in ∼ 1 Gyr at
150 AU (see eq. (5)). Throughout runaway growth, dynam-
ical friction and viscous stirring raise the random velocities
of the leftover planetesimals to e ≈ 0.01–0.1 and i ≈ 2o–
4o (v ∼ 50–500 m s−1 at 40–47 AU; Figure 2; right panel).
Stirring reduces gravitational focusing factors and ends run-
away growth. The large oligarchs then grow slowly through
accretion of leftover planetesimals.
As oligarchs grow, collisions among planetesimals initi-
ate the collisional cascade. Disruptive collisions dramat-
ically reduce the population of 1–10 km objects, which
slows the growth of oligarchs and produces a significant de-
bris tail in the size distribution. In these calculations, dis-
ruptive collisions remove material from the disk faster than
oligarchs can accrete the debris. Thus, growth stalls and
produces ∼ 10–100 objects with maximum sizes rmax ∼
1000–3000 km at 40–50 AU (Stern and Colwell 1997a,b;
Kenyon and Bromley 2004d, 2005; Stern 2005).
Stochastic events lead to large dispersions in the growth
time for oligarchs, to (eq. (5)). In ensembles of 25–50 simu-
lations with identical starting conditions, an occasional oli-
garch will grow up to a factor of two faster than its neigh-
bors. This result occurs in simulations with δ = 1.4, 1.7, and
2.0, and thus seems independent of mass resolution. These
events occur in ∼ 25% of the simulations and lead to factor
of ∼ 2 variations in to (eq. (5)).
In addition to modest-sized icy planets, oligarchic
1993). The calculations are insensitive to factor of two variations in the
mass density of planetesimals.
6 7 8 9 10
log Time (yr)
R > 10 km (40-47 AU)
R > 100 km (40-47 AU)
Dust (40-150 AU)
Fig. 3.— Time evolution of the mass in KBOs and dust grains.
Solid line: dust mass (r . 1 mm) at 40–150 AU. Dashed (dot-
dashed) lines: total mass in small (large) KBOs at 40–47 AU.
growth generates copious amounts of dust (Figure 3). When
runaway growth begins, collisions produce small amounts
of dust from ‘cratering’ (see, for example Greenberg et al.
1978; Wetherill and Stewart 1993; Stern and Colwell 1997a,b;
Kenyon and Luu 1999a). Stirring by growing oligarchs
leads to ‘catastrophic’ collisions, where colliding planetes-
imals lose more than 50% of their initial mass. These dis-
ruptive collisions produce a spike in the dust production
rate that coincides with the formation of oligarchs with r &
200–300 km (eq. (9)). As the wave of runaway growth
propagates outward, stirring produces disruptive collisions
at ever larger heliocentric distances. The dust mass grows
in time and peaks at ∼ 1 Gyr, when oligarchs reach their
maximum mass at 150 AU. As the mass in leftover plan-
etesimals declines, Poynting-Robertson drag removes dust
faster than disruptive collisions produce it. The dust mass
then declines with time.
3.3. External Perturbation
Despite the efficiency of self-stirring models in remov-
ing leftover planetesimals from the disk, other mechanisms
must reduce the derived mass in KBOs to current ob-
servational limits. In self-stirring calculations at 40–50
AU, the typical mass in KBOs with r ∼ 30—1000 km
at 4–5 Gyr is a factor of 5–10 larger than currently ob-
served (Luu and Jewitt 2002, Chapter by Petit et al.). Un-
less Earth-mass or larger objects form in the Kuiper belt
(Chiang et al. 2007; Levison and Morbidelli 2007), exter-
nal perturbations must excite KBO orbits and enhance the
collisional cascade.
Two plausible sources of external perturbation can re-
duce the predicted KBO mass to the desired limits. Once
Neptune achieves its current mass and orbit, it stirs up
the orbits of KBOs at 35–50 AU (Levison and Duncan
1990; Holman and Wisdom 1993; Duncan et al. 1995;
Kuchner et al. 2002; Morbidelli et al. 2004). In ∼ 100 Myr
or less, Neptune removes nearly all KBOs with a . 37–38
AU. Beyond a ∼ 38 AU, some KBOs are trapped in or-
bital resonance with Neptune (Malhotra 1995, 1996); oth-
ers are ejected into the scattered disk (Duncan and Levison
1997). In addition to these processes, Neptune stir-
ring increases the effectiveness of the collisional cascade
(Kenyon and Bromley 2004d), which removes additional
mass from the population of 0.1–10 km KBOs and prevents
growth of larger KBOs.
Passing stars can also excite KBO orbits and en-
hance the collisional cascade. Although Neptune dy-
namically ejects scattered disk objects with perihelia
q . 36–37 AU (Morbidelli et al. 2004), objects with
q & 45–50 AU, such as Sedna and Eris, require an-
other scattering source. Without evidence for massive
planets at a & 50 AU (Morbidelli et al. 2002), a pass-
ing star is the most likely source of the large q for these
KBOs (Morbidelli and Levison 2004; Kenyon and Bromley
2004c).
Adams and Laughlin (2001, see also Chapter by Dun-
can et al.) examined the probability of encounters between
the young Sun and other stars. Most stars form in dense
clusters with estimated lifetimes of ∼ 100 Myr. To ac-
count for the abundance anomalies of radionuclides in so-
lar system meteorites (produced by supernovae in the clus-
ter) and for the stability of Neptune’s orbit at 30 AU, the
most likely solar birth cluster has ∼ 2000–4000 members,
a crossing time of ∼ 1 Myr, and a relaxation time of ∼ 10
Myr. The probability of a close encounter with a distance
of closest approach aclose is then ∼ 60% (aclose/160 AU)
(Kenyon and Bromley 2004c).
Because the dynamical interactions between KBOs in a
coagulation calculation and large objects like Neptune or
a passing star are complex, here we consider simple cal-
culations of each process. To illustrate the evolution of
KBOs after a stellar flyby, we consider a very close pass
with aclose = 160 AU (Kenyon and Bromley 2004c). This
co-rotating flyby produces objects with orbital parameters
similar to those of Sedna and Eris. For objects in the coag-
ulation calculation, the flyby produces an e distribution
eKBO =
0.025(a/30 AU)4 a < a0
0.5 a > a0
with a0 ≈ 65 AU (see Ida et al. 2000; Kenyon and Bromley
2004c; Kobayashi et al. 2005). This e distribution produces
a dramatic increase in the debris production rate throughout
the disk, which freezes the mass distribution of the largest
objects2. Thus, to produce an ensemble of KBOs with r &
300 km at 40–50 AU, the flyby must occur when the Sun
has an age t⊙ & 10–20 Myr (Figure 2). For t⊙ & 100 Myr,
2 The i distribution following a flyby depends on the relative orientations of
two planes, the orbital plane of KBOs and the plane of the trajectory of the
passing star. Here, we assume the flyby produces no change in i, which
simplifies the discussion without changing any of the results significantly.
the flyby is very unlikely. As a compromise between these
two estimates, we consider a flyby at t⊙ ∼ 50 Myr.
6 7 8 9 10
log Time (yr)
Flyby Model
40-47 AU
47-55 AU
66-83 AU
Fig. 4.— Evolution of Σ after a stellar flyby. After 50 Myr
of growth, the close pass excites KBOs to large e (eq. (10)) and
enhances the collisional cascade.
Figure 4 shows the evolution of the KBO surface den-
sity in three annuli as a function of time. At early times
(t . 50 Myr), KBOs grow in the standard way. After the
flyby, the disk suffers a dramatic loss of material. At 40–47
AU, the disk loses ∼ 90% (93%) of its initial mass in ∼ 1
Gyr (4.5 Gyr). At ∼ 50–80 AU, the collisional cascade re-
moves ∼ 90% (97%) of the initial mass in ∼ 500 Myr (4.5
Gyr). Beyond ∼ 80 AU, KBOs contain less than 1% of the
initial mass. Compared to self-stirring models, flybys that
produce Sedna-like orbits are a factor of 2–3 more efficient
at removing KBOs from the solar system.
To investigate the impact of Neptune on the collisional
cascade, we parameterize the growth of Neptune at 30 AU
as a simple function of time (Kenyon and Bromley 2004d)
MNep ≈
6× 1027 e(t−tN )/t1 g t < tN
6× 1027 g + C(t− t1) tN < t < t2
1.0335× 1029 g t > t2
where CNep = 1.947 × 10
21 g yr−1, tN = 50 Myr, t1 = 3
Myr, and t2 = 100 Myr. These choices enable a model Nep-
tune to reach a mass of 1 M⊕ in 50 Myr, when the largest
KBOs form at 40–50 AU, and reach its current mass in 100
Myr3. As Neptune approaches its final mass, its gravity stirs
up KBOs at 40–60 AU and increases their orbital eccentric-
ities to e ∼ 0.1–0.2 on short timescales. In the coagula-
tion model, distant planets produce negligible changes in i,
so self-stirring sets i in these calculations (Weidenschilling
1989). This evolution enhances debris production by a fac-
tor of 3–4, which effectively freezes the mass distribution
of 100–1000 km objects at 40–50 AU. By spreading the
3 This prescription is not intended as an accurate portrayal of Neptune for-
mation, but it provides a simple way to investigate how Neptune might stir
the Kuiper belt once massive KBOs form.
6 7 8 9 10
log Time (yr)
Neptune Stirring
SS: 40-47 AU
NS: 40-47 AU
NS: 47-55 AU
Fig. 5.— Evolution of Σ(KBO) in models with Neptune stirring.
Compared to self-stirring models (SS; dashed curve), stirring by
Neptune rapidly removes KBOs at 40–47 AU (NS; solid cruve)
and at 47–55 AU (NS; dot-dashed curve).
leftover planetesimals and the debris over a larger volume,
Neptune stirring limits the growth of the oligarchs and thus
reduces the total mass in KBOs.
Figure 5 shows the evolution of the surface density in
small and large KBOs in two annuli as a function of time.
At 40–55 AU, Neptune rapidly stirs up KBOs to e ∼ 0.1
when it reaches its current mass at ∼ 100 Myr. Large
collision velocities produce more debris, which is rapidly
ground to dust and removed from the system by radiation
pressure at early times and by Poynting-Robertson drag at
later times. Compared to self-stirring models, the change
in Σ is dramatic, with only ∼ 3% of the initial disk mass
remaining at ∼ 4.5 Gyr.
From these initial calculations, it is clear that exter-
nal perturbations dramatically reduce the mass of KBOs
in the disk (see also Charnoz and Morbidelli 2007). Fig-
ure 6 compares the mass distributions at 40–47 AU and at
4.5 Gyr for the self-stirring model in Figure 2 (solid line)
with results for the flyby (dot-dashed line) and Neptune stir-
ring (dashed line). Compared to the self-stirring model, the
close flyby reduces the mass in KBOs by ∼ 50%. Neptune
stirring reduces the KBO mass by almost a factor of 3 rel-
ative to the self-stirring model. For KBOs with r & 30–50
km, the predicted mass in KBOs with Neptune stirring is
within a factor of 2–3 of the current mass in KBOs.
These simple calculations for the stellar flyby and Nep-
tune stirring do not include dynamical depletion. In the stel-
lar flyby picture, the encounter removes nearly all KBOs
beyond a truncation radius, aT ∼ 48 (aclose / 160 AU)
AU (Kenyon and Bromley 2004c). Thus, a close pass with
aclose ∼ 160 AU can produce the observed outer edge of the
Kuiper belt at 48 AU. Although many objects with initial
a > aT are ejected from the Solar System, some are placed
on very elliptical, Sedna-like orbits4. In the Neptune stir-
4Levison et al. (2004) consider the impact of the flyby on the scattered disk
and Oort cloud. After analyzing a suite of numerical simulations, they
conclude that the flyby must occur before Neptune reaches its current orbit
-3 -2 -1 0 1 2 3 4
log Radius (km)
Initial state
Self-stirring
Flyby 
Neptune
40-47 AU
Fig. 6.— Mass distributions for evolution with self-stirring
(heavy solid line), stirring from a passing star (dot-dashed line),
and stirring from Neptune at 30 AU (dashed line). After 4.5 Gyr,
the mass in KBOs with r & 50 km is ∼ 5% (self-stirring), ∼
3.5% (flyby), and ∼ 2% (Neptune stirring) of the initial mass. The
number of objects with r & 1000 km is ∼ 100 (self-stirring), ∼
1 (flyby), and ∼ 10 (Neptune stirring). The largest object has
rmax ∼ 3000 km (self-stirring), rmax ∼ 500–1000 km (flyby),
and rmax ∼ 1000–2000 km (Neptune stirring).
ring model, dynamical interactions will eject some KBOs
at 40–47 AU. If the dynamical interactions that produce the
scattered disk reduce the mass in KBOs by a factor of 2 at
40–47 AU (e.g., Duncan et al. 1995; Kuchner et al. 2002),
the Neptune stirring model yields a KBO mass in reason-
ably good agreement with observed limits (for a different
opinion, see Charnoz and Morbidelli 2007).
3.4. Nice Model
Although in situ KBO models can explain the current
amount of mass in large KBOs, these calculations do not
address the orbits of the dynamical classes of KBOs. To ex-
plain the orbital architecture of the giant planets, the ‘Nice
group’ centered at Nice Observatory developed an inspired,
sophisticated picture of the dynamical evolution of the gi-
ant planets and a remnant planetesimal disk (Tsiganis et al.
2005; Morbidelli et al. 2005; Gomes et al. 2005, and refer-
ences therein). The system begins in an approximate equi-
librium, with the giant planets in a compact configuration
(Jupiter at 5.45 AU, Saturn at ∼ 8.2 AU, Neptune at ∼ 11.5
AU, and Uranus at ∼ 14.2 AU) and a massive planetesimal
disk at 15–30 AU. Dynamical interactions between the gi-
ant planets and the planetesimals lead to an instability when
Saturn crosses the 2:1 orbital resonance with Jupiter, which
results in a dramatic orbital migration of the gas giants and
the dynamical ejection of planetesimals into the Kuiper belt,
scattered disk, and the Oort cloud. Comparisons between
the end state of this evolution and the orbits of KBOs in
and begins the dynamical processes that populate the Oort cloud and the
scattered disk. If Neptune forms in situ in 1–10 Myr, then the flyby cannot
occur after massive KBOs form. If Neptune migrates to 30 AU after mas-
sive KBOs form, then a flyby can truncate the Kuiper belt without much
impact on the Oort cloud or the scattered disk.
6 7 8 9 10
log Time (yr)
20-25 AU
36-44 AU
66-83 AU
NICE Model
Fig. 7.— Evolution of Σ in a self-stirring model at 20–100 AU.
At 20–25 AU, it takes ∼ 5–10 Myr to form 1000 km objects. After
∼ 0.5–1 Gyr, there are ∼ 100 objects with r ∼ 1000–2000 km
and ∼ 105 objects with r ∼ 100–200 km at 20–30 AU. As these
objects grow, the collisional cascade removes ∼ 90% of the mass
in remnant planetesimals. The twin vertical dashed lines bracket
the time of the Late Heavy Bombardment at ∼ 300–600 Myr.
the ‘hot population’ and the scattered disk are encouraging
(Chapter by Morbidelli et al.).
Current theory cannot completely address the likelihood
of the initial state in the Nice model. Thommes et al. (1999,
2002) demonstrate that n-body simulations can produce a
compact configuration of gas giants, but did not consider
how fragmentation or interactions with low mass planetes-
imals affect the end state. O’Brien et al. (2005) show that
a disk of planetesimals has negligible collisional grinding
over 600 Myr if most of the mass is in large planetesimals
with r & 100 km. However, they did not address whether
this state is realizable starting from an ensemble of 1 km
and smaller planetesimals. In terrestrial planet simulations
starting with 1–10 km planetesimals, the collisional cascade
removes ∼ 25% of the initial rocky material in the disk
(Wetherill and Stewart 1993; Kenyon and Bromley 2004b).
Interactions between oligarchs and remnant planetesimals
are also important for setting the final mass and dynamical
state of the terrestrial planets (Bromley and Kenyon 2006;
Kenyon and Bromley 2006). Because complete hybrid cal-
culations of the giant planet region are currently compu-
tationally prohibitive, it is not possible to make a reliable
assessment of these issues for the formation of gas giant
planets.
Here, we consider the evolution of the planetesimal disk
outside the compact configuration of giant planets, where
standard coagulation calculations can follow the evolution
of many initial states for 1–5 Gyr in a reasonable amount of
computer time. Figure 7 shows the time evolution for the
surface density of planetesimals in three annuli from one
typical calculation at 20–25 AU (dot-dashed curve; Mi = 6
M⊕), 36–44 AU (dashed curve; Mi = 9 M⊕), and 66–83
AU (solid curve; Mi = 12 M⊕). Starting from the stan-
dard surface density profile (eq. 1), planetesimals at 20–25
AU grow to 1000 km sizes in a few Myr. Once the colli-
sional cascade begins, the surface density slowly declines
to ∼ 10% to 20% of its initial value at the time of the Late
Heavy Bombardment, when the Nice model predicts that
Saturn crosses the 2:1 orbital resonance with Jupiter.
These results provide a strong motivation to couple co-
agulation calculations with the dynamical simulations of the
Nice group (see also Charnoz and Morbidelli 2007). In the
Nice model, dynamical interactions with a massive plan-
etesimal disk are the ‘fuel’ for the dramatic migration of the
giant planets and the dynamical ejection of material into the
Kuiper belt and the scattered disk. If the mass in the plan-
etesimal disk declines by ∼ 80% as the orbits of the giant
planets evolve, the giant planets cannot migrate as dramati-
cally as in the Gomes et al. (2005) calculations. Increasing
the initial mass in the disk by a factor of 3–10 may allow
coagulation and the collisional cascade to produce a debris
disk capable of triggering the scattering events of the Nice
model.
3.5. A Caveat on the Collisional Cascade
Although many of the basic outcomes of oligarchic
growth and the collisional cascade are insensitive to the ini-
tial conditions and fragmentation parameters for the plan-
etesimals, several uncertainties in the collisional cascade
can modify the final mass in oligarchs and the distributions
of r and e. Because current computers do not allow coag-
ulation calculations that include the full range of sizes (1
µm to 104 km), published calculations have two pieces,
a solution for large objects (e.g., Kenyon and Bromley
2004a,b) and a separate solution for smaller objects (e.g.,
Krivov et al. 2006). Joining these solutions assumes that (i)
collision fragments continue to collide and fragment until
particles are removed by radiative processes and (ii) mu-
tual (destructive) collisions among the fragments are more
likely than mergers with much larger oligarchs. These as-
sumptions are reasonable but untested by numerical cal-
culations (Kenyon and Bromley 2002a). Thus, it may be
possible to halt or to slow the collisional cascade before
radiation pressure rapidly remove small grains with r ≈
1–100 µm.
In current coagulation calculations, forming massive oli-
garchs at 5–15 AU in a massive disk requires an ineffi-
cient collisional cascade. When the cascade is efficient,
the most massive oligarchs have m . 1 M⊕. Slowing
the cascade allows oligarchs to accrete planetesimals more
efficiently, which results in larger oligarchs that contain a
larger fraction of the initial mass. If collisional damping
is efficient, halting the cascade completely at sizes of ∼ 1
mm leads to rapid in situ formation of Uranus and Neptune
(Goldreich et al. 2004) and early stirring of KBOs at 40 AU.
There are two simple ways to slow the collisional cas-
cade. In simulations where the cascade continues to small
sizes, r ∼ 1–10 µm, the radial optical depth in small grains
is τs ∼ 0.1–1 at 30–50 AU (Kenyon and Bromley 2004a).
Lines-of-sight to the central star are not purely radial, so
this optical depth reduces radiation pressure and Poynting-
Robertson drag by small factors, ∼ e−0.2τs ∼ 10%–30%,
and has little impact on the evolution of the cascade. With
τs ∝ a
−s and s ∼ 1–2, however, the optical depth may re-
duce radiation forces significantly at smaller a. Slowing the
collisional cascade by factors of 2–3 could allow oligarchs
to accrete leftover planetesimals and smaller objects before
the cascade removes them.
Collisional damping and gas drag on small particles may
also slow the collisional cascade. For particles with large
ratios of surface area to volume, r . 0.1–10 cm, colli-
sions and the gas effectively damp e and i (Adachi et al.
1976; Goldreich et al. 2004) and roughly balance dynam-
ical friction and viscous stirring. Other interactions be-
tween small particles and the gas – such as photophoresis
(Wurm and Krauss 2006) – also damp particles randome
velocities and thus might help to slow the cascade. Both
collisions and interactions between the gas and the solids
are more effective at large volume density, so these pro-
cesses should be more important inside 30 AU than outside
30 AU. The relatively short lifetime of the gas, ∼ 3–10 Myr,
also favors more rapid growth inside 30 AU. If damping
maintains an equilibrium e ∼ 10−3 at a ∼ 20–30 AU, oli-
garchs can grow to the sizes, r & 2000 km, required in the
Nice model. Rapid growth at a ∼ 5–15 AU might produce
oligarchs with the isolation mass (r ∼ 10–30 R⊕; eq. 6)
and lead to the rapid formation of gas giants.
Testing these mechanisms for slowing the collisional
cascade requires coagulation calculations with accurate
treatments of collisional damping, gas drag, and optical
depth for particle radii r ∼ 1–10 µm to r ∼ 10000 km.
Although these calculations require factors of 4–6 more
computer time than published calculations, they are possi-
ble with multiannulus coagulation codes on modern parallel
computers.
3.6. Model Predictions
The main predictions derived from coagulation mod-
els are n(r), n(e), and n(i) as functions of a. The cu-
mulative number distribution consists of three power laws
(Kenyon and Bromley 2004d; Pan and Sari 2005)
n(r) =
−αd r ≤ r1
ni r1 ≤ r < r0
−αm r ≥ r0
The debris population at small sizes, r ≤ r1, always has
αd ≈ 3.5. The merger population at large sizes, r ≥ r0,
has αm ≈ 2.7–4. Because the collisional cascade robs
oligarchs of material, calculations with more stirring have
steeper size distributions. Thus, self-stirring calculations
with Qb & 10
5 erg g−1 (Qb . 10
3 erg g−1) typically yield
αm ≈ 2.7–3.3 (3.5–4). Models with a stellar flyby or stir-
ring by a nearby gas giant also favor large αm.
The transition radii for the power laws depend on the
fragmentation parameters (see Fig. 1; see also Pan and Sari
(2005)). For a typical e ∼ 0.01–0.1 in self-stirring models,
r0 ≈ r1 ≈ 1 km when Qb & 10
5 erg g−1. When Qb . 10
erg g−1, r1 ≈ 0.1 km and r0 ≈ 10–20 km. Thus the cal-
culations predict a robust correlation between the transition
radii and the power law exponents: large r0 andαm or small
r0 and αm.
Because gravitational stirring rates are larger than ac-
cretion rates, the predicted distributions of e and i at 4–5
Gyr depend solely on the total mass in oligarchs (see also
Goldreich et al. 2004). Small objects with r . r0 con-
tain a very small fraction of the mass and cannot stir them-
selves. Thus e and i are independent of r (Fig. 2). The
e and i for larger objects depends on the total mass in the
largest objects. In self-stirring models, dynamical friction
and viscous stirring between oligarchs and planetesimals
(during runaway growth) and among the ensemble of oli-
garchs (during oligarchic growth) set the distribution of e
for large objects with r & r0. In self-stirring models, vis-
cous stirring among oligarchs dominates dynamical friction
between oligarchs and leftover planetesimals, which leads
to a shallow relation between e and r, e ∝ r−γ with γ ≈
3/4. In the flyby and Neptune stirring models, stirring by
the external perturber dominates stirring among oligarchs.
This stirring yields a very shallow relation between e and r
with γ ≈ 0–0.25.
Other results depend little on the initial conditions and
the fragmentation parameters. In calculations with different
initial mass distributions, an order of magnitude range in e0,
and Qb = 10
0–107 erg g−1, βb = −0.5–0, Qg = 0.5–5 erg
cm−3, and βg ≥ 1.25, rmax and the amount of mass re-
moved by the collisional cascade vary by . 10% relative to
the evolution of the models shown in Figures 2–7. Because
collisional damping among 1 m to 1 km objects erases the
initial orbital distribution, the results do not depend on e0
and i0. Damping and dynamical friction also quickly erase
the initial mass distribution, which yields growth rates that
are insensitive to the initial mass distribution.
The insensitivity of rmax and mass removal to the frag-
mentation parameters depends on the rate of collisional dis-
ruption relative to the growth rate of oligarchs. Because
the collisional cascade starts when mo ∼ md (eq. (9)),
calculations with small Qb (Qb . 10
3 erg g−1) produce
large amounts of debris before calculations with large Qb
(Qb & 10
3 erg g−1). Thus, an effective collisional cascade
should yield lower mass oligarchs and more mass removal
when Qb is small. However, oligarchs with mo ∼ md still
have fairly large gravitational focusing factors and accrete
leftover planetesimals more rapidly than the cascade re-
moves them. As oligarchic growth continues, gravitational
focusing factors fall and collision disruptions increase. All
calculations then reach a point where the collisional cascade
removes leftover planetesimals more rapidly than oligarchs
can accrete them. As long as most planetesimals have r ∼
1–10 km, the timing of this epoch is more sensitive to grav-
itational focusing and the growth of oligarchs than the col-
lisional cascade and the fragmentation parameters. Thus,
rmax and the amount of mass processed through the colli-
sional cascade are relatively insensitive to the fragmentation
parameters.
4. Confronting KBO collision models with KBO data
Current data for KBOs provide two broad tests of coagu-
lation calculations. In each dynamical class, four measured
parameters test the general results of coagulation models
and provide ways to discriminate among the outcomes of
self-stirring and perturbed models. These parameters are
• rmax, the size of the largest object,
• αm, the slope of the size distribution for large KBOs
with r & 10 km,
• r0, the break radius, which measures the radius where
the size distribution makes the transition from a
merger population (r & r0) to a collisional popu-
lation (r . r0) as summarized in eq. (12), and
• Ml, the total mass in large KBOs.
For all KBOs, measurements of the dust mass allow tests of
the collisional cascade and link the Kuiper belt to observa-
tions of nearby debris disks. We begin with the discussion
of large KBOs and then compare the Kuiper belt with other
debris disks.
Table 1 summarizes the mass and size distribution pa-
rameters derived from recent surveys. To construct this
table, we used online data from the Minor Planet Center
(http://cfa−www.harvard.edu/iau/lists/MPLists.html)
for rmax (see also Levison and Stern 2001) and the results
of several detailed analyses for αm, rmax, and r0 (e.g.,
Bernstein et al. 2004; Elliot et al. 2005; Petit et al. 2006,
Chapter by Petit et al.). Because comprehensive KBO sur-
veys are challenging, the entries in the Table are incomplete
and sometimes uncertain. Nevertheless, these results pro-
vide some constraints on the calculations.
Current data provide clear evidence for physical differ-
ences among the dynamical classes. For classical KBOs
with a = 42–48 AU and q > 37 AU, the cold population
(i . 4o) has a steep size distribution with αm ≈ 3.5–4
and rmax ∼ 300–500 km. In contrast, the hot population
(i & 10o) has a shallow size distribution with αm ≈ 3 and
rmax ∼ 1000 km (Levison and Stern 2001). Both popu-
lations have relatively few objects with optical brightness
mR ≈ 27–27.5, which implies r0 ∼ 20–40 km for reason-
able albedo ∼ 0.04–0.07. The detached, resonant, and scat-
tered disk populations contain large objects with rmax ∼
1000 km. Although there are too few detached or scattered
disk objects to constrain αm or r0, data for the resonant
population are consistent with constraints derived for the
hot classical population, αm ≈ 3 and r0 ≈ 20–40 km.
The total mass in KBOs is a small fraction of the ∼
10–30 M⊕ of solid material in a MMSN from ∼ 35–50
AU (Gladman et al. 2001; Bernstein et al. 2004; Petit et al.
2006, see also Chapter by Petit et al.). The classical and res-
onant populations have Ml ≈ 0.01–0.1 M⊕ in KBOs with
TABLE 1. DATA FOR KBO SIZE DISTRIBUTION
KBO Class Ml (M⊕) rmax (km) r0 (km) qm
cold cl 0.01–0.05 400 20–40 km & 4
hot cl 0.01–0.05 1000 20–40 km 3–3.5
detached n/a 1500 n/a n/a
resonant 0.01–0.05 1000 20–40 km 3
scattered 0.1–0.3 700 n/a n/a
r & 10–20 km. The scattered disk may contain more mate-
rial, Ml ∼ 0.3 M⊕, but the constraints are not as robust as
for the classical and resonant KBOs.
These data are broadly inconsistent with the predictions
of self-stirring calculations with no external perturbers. Al-
though self-stirring models yield inclinations, i ≈ 2o–4o,
close to those observed in the cold, classical population, the
small rmax and large αm of this group suggest that an ex-
ternal dynamical perturbation – such as a stellar flyby or
stirring by Neptune – modified the evolution once rmax
reached ∼ 300–500 km. The observed break radius, r0 ∼
20–40 km, also agrees better with the r0 ∼ 10 km ex-
pected from Neptune stirring calculations than the r0 ∼ 1
km achieved in self-stirring models (Kenyon and Bromley
2004d; Pan and Sari 2005). Although a large rmax and
small αm for the resonant and hot, classical populations
agree reasonably well with self-stirring models, the ob-
served rmax ∼ 1000 km is much smaller than the rmax ∼
2000–3000 km typically achieved in self-stirring calcula-
tions (Figure 1). Both of these populations appear to have
large r0, which is also more consistent with Neptune stir-
ring models than with self-stirring models.
The small Ml for all populations provide additional ev-
idence against self-stirring models. In the most optimistic
scenario, where KBOs are easily broken, self-stirring mod-
els leave a factor of 5–10 more mass in large KBOs than
currently observed at 40–48 AU. Although models with
Neptune stirring leave a factor of 2–3 more mass in KBOs
at 40–48 AU than is currently observed, Neptune ejects
∼ half of the KBOs at 40–48 AU into the scattered disk
(e.g., Duncan et al. 1995; Kuchner et al. 2002). With an es-
timated mass of 2–3 times the mass in classical and reso-
nant KBOs, the scattered disk contains enough material to
bridge the difference between the KBO mass derived from
Neptune stirring models and the observed KBO mass.
The mass in KBO dust grains provides a final piece of
evidence against self-stirring models. From an analysis of
data from Pioneer 10 and 11, Landgraf et al. (2002) esti-
mate a dust production rate of ∼ 1015 g yr−1 in 0.01–2
mm particles at 40–50 AU. The timescale for Poynting-
Robertson drag to remove these grains from the Kuiper belt
is ∼ 10–100 Myr (Burns et al. 1979), which yields a mass
of ∼ 1022–1024 g. Figure 8 compares this dust mass with
masses derived from mid-IR and submm observations of
several nearby solar-type stars (Greaves et al. 1998, 2004;
Williams et al. 2004; Wyatt et al. 2005) and with predic-
tions from the self-stirring, flyby, and Neptune stirring mod-
els. The dust masses for nearby solar-type stars roughly fol-
5 6 7 8 9 10
log Time (yr)
Self-stirring
Flyby
Neptune
Fig. 8.— Evolution of mass in small dust grains (0.001–1
mm) for models with self-stirring (dot-dashed line), stirring from
a passing star (dashed line), and stirring from Neptune at 30 AU
(solid line) for Qb = 10
3 erg g−1. Calculations with smaller
(large) Qb produce more (less) dust at t . 50 Myr and some-
what more (less) dust at t & 100 Myr. At 1–5 Gyr, models with
Neptune stirring have less dust than self-stirring or flyby models.
The boxes show dust mass estimated for four nearby solar-type
stars (from left to right in age: HD 107146, ǫ Eri, η Crv, and τ
Cet; Greaves et al. 1998, 2004; Williams et al. 2004; Wyatt et al.
2005) and two estimates for the Kuiper belt (boxes connected by
solid line Landgraf et al. 2002).
low the predictions of self-stirring models and flyby models
with Qb ∼ 10
3 erg g−1. The mass of dust in the Kuiper
belt is 1–3 orders of magnitude smaller than predicted in
self-stirring models and is closer to the predictions of the
Neptune stirring models.
To combine the dynamical properties of KBOs with
these constraints, we rely on results from N -body simu-
lations that do not include collisional processing of small
objects (see Chapter by Morbidelli et al.). For simplic-
ity, we consider coagulation in the context of the Nice
model, which provides a solid framework for interpreting
the dynamics of the gas giants and the dynamical classes of
KBOs. In the Nice model, Saturn’s crossing of the 2:1 res-
onance with Jupiter initiates the dynamical instability that
populates the Kuiper belt. As Neptune approaches a ≈ 30
AU, it captures resonant KBOs, ejects KBOs into the scat-
tered disk and the Oort cloud, and excites the hot classical
KBOs. Although Neptune might reduce the number of cold,
classical KBOs formed roughly in situ beyond 30 AU, the
properties of these KBOs probably reflect conditions in the
Kuiper Belt when the instability began.
The Nice model requires several results from coagula-
tion calculations. Once giant planets form at 5–15 AU, col-
lisional growth must produce thousands of Pluto-mass ob-
jects at 20–30 AU. Unless the planetesimal disk was mas-
sive, growth of oligarchs must dominate collisional grind-
ing in this region of the disk. To produce the cold classical
population at ∼ 45 AU, collisions must produce 1–10 Pluto-
mass objects and then efficiently remove leftover planetesi-
mals. To match the data in Table 1, KBOs formed at 20–30
AU should have a shallower size distribution and a larger
rmax than those at 40–50 AU.
Some coagulation results are consistent with the trends
required in the Nice model. In current calculations, colli-
sional growth naturally yields smaller rmax and a steeper
size distribution at larger a. At 40–50 AU, Neptune-stirring
models produce a few Pluto-mass objects and many smaller
KBOs with e ∼ 0.1 and i ≈ 2o–4o. Although collisional
growth produces more Plutos at 15–30 AU than at 40–50
AU, collisional erosion removes material faster from the in-
ner disk than from the outer disk (Fig. 7). Thus, collisions
do not produce the thousands of Pluto-mass objects at 15–
30 AU required in the Nice model.
Reconciling this aspect of the Nice model with the coag-
ulation calculations requires a better understanding of the
physical processes that can slow or halt the collisional cas-
cade. Producing gas giants at 5–15 AU, thousands of Plutos
at 20–30 AU, and a few or no Plutos at 40–50 AU implies
that the outcome of coagulation changes markedly from 5
AU to 50 AU. If the collisional cascade can be halted as out-
lined in section §3.5, forming 5–10 M⊕ cores at 5–15 AU
is straightforward. Slowing the collisional cascade at 20–30
AU might yield a large population of Pluto mass objects at
20–30 AU. Because αm and rmax are well-correlated, bet-
ter constraints on the KBO size distributions coupled with
more robust coagulation calculations can test these aspects
of the Nice model in more detail.
To conclude this section, we consider constraints on the
Kuiper belt in the more traditional migration scenario of
Malhotra (1995), where Neptune forms at ∼ 20–25 AU
and slowly migrates to 30 AU. To investigate the relative
importance of collisional and dynamical depletion at 40–
50 AU, Charnoz and Morbidelli (2007) couple a collision
code with a dynamical code and derive the expected distri-
butions for size and orbital elements in the Kuiper belt, the
scattered disk, and the Oort cloud. Although collisional de-
pletion models can match the observations of KBOs, these
models are challenged to provide enough small objects into
the scattered disk and Oort cloud. Thus, the results suggest
that dynamical mechanisms dominate collisions in remov-
ing material from the Kuiper belt.
Although Charnoz and Morbidelli (2007) argue against
a dramatic change in collisional evolution from 15 AU to
40 AU, the current architecture of the solar system provides
good evidence for this possibility. In the MMSN, the ratio
of timescales to produce gas giant cores at 10 AU and at
25 AU is ξ = (25/10)3 ∼ 15. In the context of the Nice
model, formation of Saturn and Neptune at 8–11 AU in 5–
10 Myr thus implies formation of other gas giant cores at
20–25 AU in 50–150 Myr. If these cores had formed, they
would have consumed most of the icy planetesimals at 20–
30 AU, leaving little material behind to populate the outer
solar system when the giant planets migrate. The appar-
ent lack of gas giant core formation at 20–30 AU indicates
that the collisional cascade changed dramatically from 5–15
AU (where gas giant planets formed) to 20–30 AU (where
gas giant planets did not form). As outlined in §3.5, under-
standing the interaction of small particles with the gas and
the radiation field may provide important insights into the
evolution of oligarchic growth and thus into the formation
and structure of the solar system.
5. KBOs and Asteroids
In many ways, the Kuiper Belt is similar to the asteroid
belt. Both are populations of small bodies containing rela-
tively little mass compared to the rest of the Solar System;
the structure and dynamics of both populations have been
influenced significantly by the giant planets; and both have
been and continue to be significantly influenced by colli-
sions. Due to its relative proximity to Earth, however, there
are substantially more observational data available for the
asteroid belt than the Kuiper Belt. While the collisional and
dynamical evolution of the asteroid belt is certainly not a
solved problem, the abundance of constraints has allowed
for the development of reasonably consistent models. Here
we briefly describe what is currently understood about the
evolution of the asteroid belt, what insights that may give us
with regards to the evolution of the Kuiper Belt, and what
differences might exist in the evolution of the two popula-
tions.
It has long been recognized that the primordial as-
teroid belt must have contained hundreds or thousands
of times more mass than the current asteroid belt (e.g.
Lecar and Franklin 1973; Safronov 1979; Weidenschilling
1977c; Wetherill 1989). Reconstructing the initial mass
distribution of the Solar System from the current masses of
the planets and asteroids, for example, yields a pronounced
mass deficiency in the asteroid belt region relative to an oth-
erwise smooth distribution for the rest of the Solar System
(Weidenschilling 1977c). To accrete the asteroids on the
timescales inferred from meteoritic evidence would require
hundreds of times more mass than currently exists in the
main belt (Wetherill 1989).
In addition to its pronounced mass depletion, the aster-
oid belt is also strongly dynamically excited. The mean
proper eccentricity and inclination of asteroids larger than
∼50 km in diameter are 0.135 and 10.9o (from the cata-
log of Knežević and Milani (2003)), which are significantly
larger than can be explained by gravitational perturbations
amongst the asteroids or by simple gravitational pertur-
bations from the planets (Duncan 1994). The fact that
the different taxonomic types of asteroids (S-type, C-type,
etc.) are radially mixed somewhat throughout the main belt,
rather than confined to delineated zones, indicates that there
has been significant scattering in semimajor axis as well
(Gradie and Tedesco 1982).
Originally, a collisional origin was suggested for the
mass depletion in the asteroid belt (Chapman and Davis
1975). The difficulty of collisionally disrupting the largest
asteroids, coupled with the survival of the basaltic crust of
the ∼500-km diameter asteroid Vesta, however, suggest that
collisional grinding was not the cause of the mass deple-
tion (Davis et al. 1979, 1985, 1989, 1994; Wetherill 1989;
Durda and Dermott 1997; Durda et al. 1998; Bottke et al.
2005a,b; O’Brien and Greenberg 2005). In addition, col-
lisional processes alone could not fully explain both the dy-
namical excitation and the radial mixing observed in the as-
teroid belt, although Charnoz et al. (2001) suggest that col-
lisional diffusion may have contributed to its radial mixing.
Several dynamical mechanisms have been proposed
to explain the mass depletion, dynamical excitation and
radial mixing of the asteroid belt. As the solar nebula
dissipated, the changing gravitational potential acting on
Jupiter, Saturn, and the asteroids would lead to changes
in their precession rates and hence changes in the posi-
tions of secular resonances, which could ‘sweep’ through
the asteroid belt, exciting e and i, and coupled with gas
drag, could lead to semi-major axis mobility and the re-
moval of material from the belt (e.g., Heppenheimer 1980;
Ward 1981; Lemaitre and Dubru 1991; Lecar and Franklin
1997; Nagasawa et al. 2000, 2001, 2002). It has also been
suggested that sweeping secular resonances could lead to
orbital excitation in the Kuiper Belt (Nagasawa and Ida
2000). However, as reviewed by Petit et al. (2002) and
O’Brien et al. (2006), secular resonance sweeping is gen-
erally unable to simultaneously match the observed e and
i excitation in the asteroid belt, as well as its radial mix-
ing and mass depletion, for reasonable parameter choices
(especially in the context of the Nice Model).
Another possibility is that planetary embryos were able
to accrete in the asteroid belt (e.g., Wetherill 1992). The
fact that Jupiter’s ∼10 Earth-mass core was able to accrete
in our Solar System beyond the asteroid belt suggests that
embryos were almost certainly able to accrete in the aster-
oid belt, even accounting for the roughly 3-4× decrease in
the mass density of solid material inside the snow line. The
scattering of asteroids by those embryos, coupled with the
Jovian and Saturnian resonances in the asteroid belt, has
been shown to be able to reasonably reproduce the observed
e and i excitation in the belt as well as its radial mixing
and mass depletion (Petit et al. 2001, 2002; O’Brien et al.
2006). In the majority of simulations of this scenario by
both groups, the embryos are completely cleared from the
asteroid belt.
Thus, the observational evidence and theoretical mod-
els for the evolution of the asteroid belt strongly suggest
that dynamics, rather than collisions, dominated its mass
depletion. Collisions, however, have still played a key
role in sculpting the asteroid belt. Many dynamical fam-
ilies, clusterings in orbital element space, have been dis-
covered, giving evidence for ∼20 breakups of 100-km or
larger parent bodies over the history of the Solar System
(Bottke et al. 2005a,b). The large 500-km diameter asteroid
Vesta has a preserved basaltic crust with a single large im-
pact basin (McCord et al. 1970; Thomas et al. 1997). This
basin was formed by the impact of a roughly 40-km projec-
tile (Marzari et al. 1996; Asphaug 1997).
The size distribution of main-belt asteroids is known
or reasonably constrained through observational surveys
down to ∼1 km in diameter (e.g. Durda and Dermott 1997;
 0.1  1  10  100  1000
Diameter (km)
Asteroid and TNO Size Distributions
Bernstein (2004)
Fit to Data
 0.1  1  10  100  1000
Diameter (km)
Asteroid and TNO Size Distributions
Subaru
SKADS
Spacewatch
Cataloged Asteroids
Fig. 9.— Observational estimates of the main belt and TNO size
distributions. The pentagons (with dashed best-fit curve) show
the total TNO population as determined from the Bernstein et al.
(2004) HST survey, converted to approximate diameters assum-
ing an albedo of 0.04. Points with arrows are upper limits given
by non-detections. The solid line is the population of observed
asteroids, and open circles are from debiased Spacewatch main-
belt observations (Jedicke and Metcalfe 1998). These data, con-
verted to diameters, were provided by D. Durda. The two dashed
lines are extrapolations based on the Sloan Digital Sky Survey
(Ivezić et al. 2001) and the Subaru Sub-km Main Belt Aster-
oid Survey (Yoshida et al. 2003), and diamonds show the debi-
ased population estimate from the SKADS survey (Gladman et al.
2007). Error bars are left out of this plot for clarity. Note that the
TNO population is substantially more populous and massive, by
roughly a factor of 1000, than the asteroid population.
Jedicke and Metcalfe 1998; Ivezić et al. 2001; Yoshida et al.
2003; Gladman et al. 2007). Not surprisingly, the largest
uncertainties are at the smallest sizes, where good orbits
are often not available for the observed asteroids, which
makes the conversion to absolute magnitude and diame-
ter difficult (e.g., Ivezić et al. 2001; Yoshida et al. 2003).
Recent results from the Sub-Kilometer Asteroid Diame-
ter Survey (SKADS, Gladman et al. 2007), the first survey
since the Palomar-Leiden Survey designed to determine or-
bits as well as magnitudes of main-belt asteroids, suggest
that the asteroid magnitude-frequency distribution may be
well represented by a single power law in the range from
H=14.0 to 18.8, which corresponds to diameters of 0.7 to 7
km for an albedo of 0.11. These observational constraints
are shown in Fig. 9 alongside the determination of the TNO
size distribution from Bernstein et al. (2004).
While over some size ranges, the asteroid size distribu-
tion can be fit by a single power law, over the full range of
observed asteroid diameters from ∼1-1000 km, there are
multiple bumps or kinks in the size distribution (namely
around 10 and 100 km in diameter). The change in slope
of the size distribution around 100 km is due primarily to
the fact that asteroids larger than this are very difficult to
disrupt, and hence the size distribution of bodies larger than
100 km is likely primordial. The change in slope around
10 km has a different origin—such a structure is produced
as a result of a change in the strength properties of aster-
oids, namely the transition from when a body’s resistance
to disruption is dominated by material strength to when it is
dominated by self-gravity. This transition in strength prop-
erties occurs at a size much smaller than 10 km, but re-
sults in a structure that propagates to larger sizes (see, e.g.,
Durda et al. 1998; O’Brien and Greenberg 2003). The pres-
ence of this structure in the asteroid size distribution is con-
sistent with the asteroids being a collisionally-relaxed pop-
ulation, i.e. a population in which the size distribution has
reached an approximate steady state where collisional pro-
duction and collisional destruction of bodies in each size
range are in balance.
The collisional evolution of the asteroid belt has been
studied by many authors (e.g. Davis et al. 1985; Durda
1993; Davis et al. 1994; Durda and Dermott 1997; Durda et al.
1998; Campo Bagatin et al. 1993, 1994, 2001; Marzari et al.
1999). The most recent models of collisional evolution
of the asteroid belt incorporate aspects of dynamical evo-
lution as well, such as the removal of bodies by reso-
nances and the Yarkovsky effect, and the enhancement
in collisional activity during its massive primordial phase
(O’Brien and Greenberg 2005; Bottke et al. 2005a,b). In
particular, Bottke et al. (2005a) explicitly incorporate the
results of dynamical simulations of the excitation and clear-
ing of the main belt by embedded planetary embryos per-
formed by Petit et al. (2001). Such collisional/dynamical
models can be constrained by a wide range of observational
evidence such as the main belt size distribution, the num-
ber of observed asteroid families, the existence of Vesta’s
basaltic crust, and the cosmic ray exposure ages of ordi-
nary chondrite meteorites, which suggest that the lifetimes
of meter-scale stony bodies in the asteroid belt are on the
order of 10-20 Myr (Marti and Graf 1992).
One of the most significant implications of having an
early massive main belt, which was noted in early colli-
sional models (e.g. Chapman and Davis 1975) and recently
emphasized in the case of collisional evolution plus dynam-
ical depletion (e.g., Bottke et al. 2005b), is that the majority
of the collisional evolution of the asteroid belt occurred dur-
ing its early, massive phase, and there has been relatively lit-
tle change in the main-belt size distribution since then. The
current, wavy main-belt size distribution, then, is a ‘fossil’
from its first few hundred Myr of collisional and dynamical
evolution.
So how does the Kuiper Belt compare to the asteroid belt
in terms of its collisional and dynamical evolution? Evi-
dence and modeling for the asteroid belt suggest that dy-
namical depletion, rather than collisional erosion, was pri-
marily responsible for reducing the mass of the primordial
asteroid belt to its current level. In the case of the Kuiper
Belt, this is less clear. As shown in Sec. 3, collisional
erosion, especially when aided by stellar perturbations or
the formation of Neptune, can be very effective in remov-
ing mass. At the same time, dynamical models such as
the Nice Model result in the depletion of a large amount
of mass through purely dynamical means and are able to
match many observational constraints. Recent modeling
that couples both collisional fragmentation and dynami-
cal effects suggests that collisional erosion cannot play too
large of a role in removing mass from the Kuiper Belt, oth-
erwise the Scattered Disk and Oort Cloud would be too de-
pleted to explain the observed numbers of short- and long-
period comets (Charnoz and Morbidelli 2007). That model
currently does not include coagulation. Further modeling
work, which self-consistently integrates coagulation, colli-
sional fragmentation, and dynamical effects, is necessary to
fully constrain the relative contributions of collisional and
dynamical depletion in the Kuiper Belt.
We have noted that the asteroid belt has a collisionally-
relaxed size distribution that is not well-represented by a
single power law over all size ranges. Should we expect the
same for the Kuiper Belt size distribution, and is there evi-
dence to support this? The collision rate in the Kuiper Belt
should be roughly comparable to that in the asteroid belt,
with the larger number of KBOs offsetting their lower in-
trinsic collision probability (Davis and Farinella 1997), and
as noted earlier in this chapter, the primordial Kuiper Belt,
like the asteroid belt, would have been substantially more
massive than the current population. This suggests that
the Kuiper Belt should have experienced a degree of col-
lisional evolution roughly comparable to the asteroid belt,
and thus is likely to be collisionally relaxed like the asteroid
belt. Observational evidence thus far is not detailed enough
to say for sure if this is the case, although recent work
(Kenyon and Bromley 2004e; Pan and Sari 2005) suggests
that the observational estimate of the TNO size distribution
by Bernstein et al. (2004), shown in Fig. 9, is consistent
with a collisionally-relaxed population.
While the Kuiper Belt is likely to be collisionally re-
laxed, it is unlikely to mirror the exact shape of the aster-
oid belt size distribution. The shape of the size distribu-
tion is determined, in part, by the strength law Q∗D, which
is likely to differ somewhat between asteroids and KBOs.
This is due to the difference in composition between as-
teroids, which are primarily rock, and KBOs, which con-
tain a substantial amount of ice, as well as the difference
in collision velocity between the two populations. With a
mean velocity of ∼5 km/s (Bottke et al. 1994), collisions
between asteroids are well into the supersonic regime (rel-
ative to the sound speed in rock). For the Kuiper belt, col-
lision velocities are about a factor of 5 or more smaller
(Davis and Farinella 1997), such that collisions between
KBOs are close to the subsonic/supersonic transition. For
impacts occurring in these different velocity regimes, and
into different materials, Q∗D may differ significantly (e.g.,
Benz and Asphaug 1999).
The difference in collision velocity can influence the size
distribution in another way as well. With a mean collision
velocity of ∼5 km/s, a body of a given size in the aster-
oid belt can collisionally disrupt a significantly larger body.
Thus, transitions in the strength properties of asteroids can
lead to the formation of waves that propagate to larger sizes
and manifest themselves as changes in the slope of the size
distribution, as seen in Fig. 9. For the Kuiper belt, with col-
lision velocities that are about a factor of 5 or more smaller
than in the asteroid belt, the difference in size between a
given body and the largest body it is capable of disrupting
is much smaller than in the asteroid belt, and waves should
therefore be much less pronounced or non-existent in the
KBO size distribution (e.g., O’Brien and Greenberg 2003).
There is still likely to be a change in slope at the largest sizes
where the population transitions from being primordial to
being collisionally relaxed, and such a change appears in
the debiased observational data of Bernstein et al. (2004)
(shown here in Fig. 9), although recent observations sug-
gest that the change in slope may actually occur at smaller
magnitudes than found in that survey (Petit et al. 2006).
Is the size distribution of the Kuiper Belt likely to be a
‘fossil’ like the asteroid belt? The primordial Kuiper Belt
would have been substantially more massive than the cur-
rent population. Thus, regardless of whether the depletion
of its mass was primarily collisional or dynamical, colli-
sional evolution would have been more intense early on
and the majority of the collisional evolution would have oc-
curred early in its history. In either case, its current size
distribution could then be considered a fossil from that early
phase, although defining exactly when that early phase ends
and the size distribution becomes ‘fossilized’ is not equally
clear in both cases. In the case where the mass depletion
of the Kuiper Belt occurs entirely through collisions, there
would not necessarily be a well-defined point at which one
could say that the size distribution became fossilized, as the
collision rate would decay continuously with time. In the
case of dynamical depletion, where the mass would be re-
moved fairly rapidly as in the case of the Nice Model de-
scribed in Sec. 3.4, the collision rate would experience a
correspondingly rapid drop, and the size distribution could
be considered essentially fossilized after the dynamical de-
pletion event.
As noted earlier in this section, an important observable
manifestation of collisions in the asteroid belt is the for-
mation of families, i.e. groupings of asteroids with similar
orbits. Asteroid families are thought to be the fragments of
collisionally disrupted parent bodies. These were first rec-
ognized by Hirayama (1918) who found 3 families among
the 790 asteroids known at that time. The number increased
to 7 families by 1926 when there were 1025 known as-
teroids (Hirayama 1927). Today, there are over 350,000
known asteroids while the number of asteroid families has
grown to about thirty.
Given that the Kuiper Belt is likely a collisionally
evolved population, are there collisional families to be
found among these bodies? Families are expected to be
more difficult to recognize in the Kuiper Belt than in the
asteroid belt. Families are identified by finding statistically
significant clusters of asteroid orbit elements—mainly the
semi-major axis, eccentricity and inclination. The colli-
sional disruption of a parent bodies launches fragments
with speeds of perhaps a few hundred meters/sec relative
to the original target body. This ejection speed is small
compared with the orbital speeds of asteroids, hence the or-
bits of fragments differ by only small amounts from that of
the original target body and, more importantly, from each
other. Thus, the resulting clusters of fragments are easy to
identify.
However, in the Kuiper Belt, where ejection velocities
are likely to be about the same but orbital speeds are much
lower, collisional disruption produces a much greater dis-
persion in the orbital elements of fragments. This reduces
the density of the clustering of orbital elements and makes
the task of distinguishing family members from the back-
ground population much more difficult (Davis and Farinella
1997). To date, there are over 1000 KBOs known, many of
which have poorly-determined orbits or are in resonances
that would make the identification of a family difficult or
impossible. Chiang et al. (2003) applied lowest-order secu-
lar theory to 227 non-resonant KBOs with well-determined
orbits and found no convincing evidence for a dynamical
family. Recently, however, Brown et al. (2007) found evi-
dence for a single family with at least 5 members associated
with KBO 2003 EL61. This family was identified based on
the unique spectroscopic signature of its members, and con-
firmed by their clustered orbit elements.
Given the small numbers involved, it cannot be said
whether or not finding a single KBO family at this stage is
statistically that different from the original identification of
3 asteroid families when there were only 790 known aster-
oids (Hirayama 1918). However, the fact that the KBO fam-
ily associated with 2003 EL61 was first discovered spectro-
scopically, and its clustering in orbital elements was later
confirmed, while nearly all asteroid families were discov-
ered based on clusterings in orbital elements alone, suggests
that even if comparable numbers of KBO families and aster-
oid families do exist, the greater dispersion of KBO families
in orbital element space may make them more difficult to
identify unless there are spectroscopic signatures connect-
ing them as well.
Perhaps when the number of non-resonant KBOs with
good orbits approaches 1000, more populous Kuiper Belt
families will be identified, and as can be done now with
the asteroid belt, these KBO families can be used as con-
straints on the interior structures of their original parent
bodies as well as on the collisional and dynamical history
of the Kuiper Belt as a whole.
6. Concluding Remarks
Starting with a swarm of 1 m to 1 km planetesimals
at 20–150 AU, the growth of icy planets follows a stan-
dard pattern (Stern and Colwell 1997a,b; Kenyon and Luu
1998, 1999a,b; Kenyon and Bromley 2004a,d, 2005). Col-
lisional damping and dynamical friction lead to a short pe-
riod of runaway growth that produces 10–100 objects with
r ∼ 300–1000 km. As these objects grow, they stir the
orbits of leftover planetesimals up to the disruption veloc-
ity. Once disruptions begin, the collisional cascade grinds
leftover planetesimals into smaller objects faster than the
oligarchs can accrete them. Thus, the oligarchs always con-
tain a small fraction of the initial mass in solid material.
For self-stirring models, oligarchs contain ∼ 10% of the
initial mass. Stellar flybys and stirring by a nearby gas gi-
ant augment the collisional cascade and leave less mass in
oligarchs. The two examples in §3.3 suggest that a very
close flyby and stirring by Neptune leave ∼ 2% to 5% of
the initial mass in oligarchs with r ∼ 100–1000 km.
This evolution differs markedly from planetary growth
in the inner solar system. In ∼ 0.1–1 Myr at a few AU, run-
away growth produces massive oligarchs, m & 0.01M⊕,
that contain most of the initial solid mass in the disk. Aside
from a few giant impacts like those that might produce
the Moon (Hartmann and Davis 1975; Cameron and Ward
1976), collisions remove little mass from these objects. Al-
though the collisional cascade removes many leftover plan-
etesimals before oligarchs can accrete them, the lost ma-
terial is much less than half of the original solid mass
(Wetherill and Stewart 1993; Kenyon and Bromley 2004b).
For a & 40 AU, runaway growth leaves most of the mass in
0.1–10 km objects that are easily disrupted at modest colli-
sion velocities. In 4.5 Gyr, the collisional cascade removes
most of the initial disk mass inside 70–80 AU.
Together with numerical calculations of orbital dynam-
ics (Chapter by Morbidelli et al.), theory now gives us a
foundation for understanding the origin and evolution of
the Kuiper belt. Within a disk of planetesimals at 20–
100 AU, collisional growth naturally produces objects with
r ∼ 10–2000 km and a size distribution reasonably close
to that observed among KBOs. As KBOs form, migration
of the giant planets scatters KBOs into several dynamical
classes (Chapter by Morbidelli et al.). Once the giant plan-
ets achieve their current orbits, the collisional cascade re-
duces the total mass in KBOs to current levels and produces
the break in the size distribution at r ∼ 20–40 km. Contin-
ued dynamical scattering by the giant planets sculpts the
inner Kuiper belt and maintains the scattered disk.
New observations will allow us to test and to refine this
theoretical picture. Aside from better measures of αm,
rmax, and r0 among the dynamical classes, better limits on
the total mass and the size distribution of large KBOs with
a ∼ 50–100 AU should yield a clear discriminant among
theoretical models. In the Nice model, the Kuiper belt was
initially nearly empty outside of ∼ 50 AU. Thus, any KBOs
found with a ∼ 50–100 AU should have the collisional
and dynamical signatures of the scattered disk or detached
population. If some KBOs formed in situ at a & 50 AU,
their size distribution depends on collisional growth modi-
fied by self-stirring and stirring by ∼ 30 M⊕ of large KBOs
formed at 20–30 AU and scattered through the Kuiper belt
by the giant planets. From the calculations of Neptune stir-
ring (§3.3), stirring by scattered disk objects should yield a
size distribution markedly different from the size distribu-
tion of detached or scattered disk objects formed at 20–30
AU. Wide-angle surveys on 2–3 m class telescopes (e.g.,
Pan-Starrs) and deep probes with 8–10 m telescopes can
provide this test.
Information on smaller size scales – αd and r1 – place
additional constraints on the bulk properties (fragmentation
parameters) of KBOs and on the collisional cascade. In any
of the stirring models, there is a strong correlation between
r0, r1, and the fragmentation parameters. Thus, direct mea-
sures of r0 and r1 provide a clear test of KBO formation
calculations. At smaller sizes (r . 0.1 km), the slope of
the size distribution αd clearly tests the fragmentation al-
gorithm and the ability of the collisional cascade to remove
KBOs with r ∼ 1–10 km. Although the recent detection
of KBOs with r ≪ 1 km (Chang et al. 2006) may be an
instrumental artifact (Jones et al. 2006; Chang et al. 2007),
optical and X-ray occultations (e.g., TAOS) will eventually
yield these tests.
Finally, there is a clear need to combine coagulation
and dynamical calculations to produce a ‘unified’ picture
of planet formation at a & 20 AU. Charnoz and Morbidelli
(2007) provide a good start in this direction. Because the
collisional outcome is sensitive to internal and external dy-
namics, understanding the formation of the observed n(r),
n(e), and n(i) distributions in each KBO population re-
quires treating collisional evolution and dynamics together.
A combined approach should yield the sensitivity of αm,
rmax, and r0 to the local evolution and the timing of the
formation of giant planets, Neptune migration, and stellar
flybys. These calculations will also test how the dynami-
cal events depend on the evolution during oligarchic growth
and the collisional cascade. Coupled with new observations
of KBOs and of planets and debris disks in other planetary
systems, these calculations should give us a better under-
standing of the origin and evolution of KBOs and other ob-
jects in the outer solar system.
We thank S. Charnoz, S. Kortenkamp, A. Morbidelli,
and an anonymous reviewer for comments that consider-
ably improved the text. We acknowledge support from
the NASA Astrophysics Theory Program (grant NAG5-
13278; BCB & SJK), the NASA Planetary Geology and
Geophysics Program (grant NNX06AC50G; DPO), and the
JPL Institutional Computing and Information Services and
the NASA Directorates of Aeronautics Research, Science,
Exploration Systems, and Space Operations (BCB & SJK).
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This 2-column preprint was prepared with the AAS LATEX macros v5.2.
	INTRODUCTION
	COAGULATION THEORY
	COAGULATION SIMULATIONS
	Background
	Self-Stirring
	External Perturbation
	Nice Model
	A Caveat on the Collisional Cascade
	Model Predictions
	Confronting KBO collision models with KBO data
	KBOs and Asteroids
	Concluding Remarks
