E
The Astrophysical Journal, 581:666–680, 2002 December 10
# 2002. The American Astronomical Society. All rights reserved. Printed in U.S.A.
FORMATION OF PROTOPLANET SYSTEMS AND DIVERSITY OF PLANETARY SYSTEMS
Eiichiro Kokubo
Division of Theoretical Astrophysics, National Astronomical Observatory, Osawa, Mitaka, Tokyo, 181-8588, Japan;
[email protected]
and
Shigeru Ida
Department of Earth and Planetary Sciences, Tokyo Institute of Technology, Ookayama, Meguro-ku, Tokyo, 152-8551, Japan
Received 2002 May 14; accepted 2002 August 8
ABSTRACT
We investigate the formation of protoplanet systems from planetesimal disks by global (N ¼ 5000 and
10,000 and 0:5 AU < a < 1:5 AU, where N is the number of bodies and a is the distance from a central star)
N-body simulations of planetary accretion. For application to extrasolar planetary systems, we study the
wide variety of planetesimal disks of the surface mass density solid ¼ 1 ða=1 AUÞ g cm2 with 1 ¼ 1,
10, 100 and ¼ 1=2; 3=2; 5=2. The results are all consistent with the prediction from the ‘‘ oligarchic growth ’’
model. We derive how the growth timescale, the isolation (final) mass, and the orbital separation of protoplanets depend on the initial disk mass (1) and the initial disk profile (). The isolation mass increases in pro3=2
1=2
portion to 1 , while the number of protoplanets decreases in proportion to 1 . The isolation mass
ð3=2Þð2Þ
, which means it increases with a for < 2 while it decreases with a for > 2. The
depends on a as a
growth timescale increases with a but decreases with 1. Based on the oligarchic growth model and the conventional Jovian planet formation scenario, we discuss the diversity of planetary systems. Jovian planets can
form in the disk range where the contraction timescale of planetary atmosphere and the growth timescale of
protoplanets (cores) are shorter than the lifetime of the gas disk. We find that for the disk lifetime 108 yr,
several Jovian planets would form from massive disks with 1 e30 with Uranian planets outside the Jovian
planets. Only terrestrial and Uranian planets would form from light disks with 1 d3. Solar system–like
planetary systems would form from medium disks with 1 ’ 10.
Subject headings: planetary systems — planetary systems: formation — solar system: formation
On-line material: color figures
When the mass of a protoplanet exceeds a critical value,
runaway growth slows down (Ida & Makino 1993) and then
the growth mode shifts to oligarchic growth (Kokubo & Ida
1998), where among protoplanets larger bodies tend to grow
more slowly than smaller ones, while the mass ratios
between protoplanets and planetesimals still increase. As a
result, in the late stage, similar-sized protoplanets grow oligarchically, while most planetesimals remain small. In this
stage, orbital separations of protoplanets are kept wider
than about 5 Hill radii of the protoplanets through orbital
repulsion (Kokubo & Ida 1995). The final stage of terrestrial
planet formation would be giant impacts among the protoplanets (e.g., Chambers & Wetherill 1998; Agnor, Canup, &
Levison 1999; Kominami & Ida 2002), and that of Jovian
planet formation would be, according to the conventional
‘‘ core instability ’’ scenario, gas accretion onto cores (protoplanets) while accreting planetesimals (e.g., Mizuno 1980;
Bodenheimer & Pollack 1986; Pollack et al. 1996; Ikoma,
Nakazawa, & Emori 2000). Uranian planets would be failed
protoplanets that could not capture the massive gas envelope (e.g., Lissauer et al. 1995).
Most of the previous studies of planetary accretion
focused on solar system formation. In other words, the minimum-mass protoplanetary disk model was assumed in
which the disk mass Mdisk ’ 0:01–0.02 M (Weidenschilling
1977; Hayashi 1981) and the gas-to-dust ratio is the solar
abundance. This model contains the smallest amount of the
solid component (dust) that can reproduce the present solar
system. The distribution of planetesimals was then assumed
1. INTRODUCTION
In the standard scenario of solar system formation, the
building blocks of planets are kilometer-sized bodies called
planetesimals (e.g., Safronov 1969; Hayashi, Nakazawa, &
Nakagawa 1985). Planetesimals are formed from the solid
component (dust) in the solar nebula (a protoplanetary
disk). While orbiting the Sun, planetesimals accrete to form
terrestrial (rocky) and Uranian (icy) planets and cores of
Jovian (gaseous) planets. This process is called planetary
accretion. Planetary accretion is an important process of
planet formation that controls the basic structure of a planetary system and the formation timescale of planets.
In order to clarify solar system formation, many authors
investigated planetary accretion using a gasdynamic
approach (e.g., Greenberg et al. 1978; Wetherill & Stewart
1989; Weidenschilling et al. 1997) and N-body simulation
(e.g., Aarseth, Lin, & Palmer 1993; Kokubo & Ida 1996,
1998, 2000). The current understanding of planetary accretion is summarized as follows. In the early stage of planetary
accretion, a larger planetesimal grows faster than smaller
ones, resulting in the runaway growth of the largest planetesimal (Greenberg et al. 1978; Wetherill & Stewart 1989;
Kokubo & Ida 1996). This is because in a planetesimal system, the larger a planetesimal becomes, the larger its growth
rate becomes by gravitational focusing and dynamical
friction from smaller planetesimals. Protoplanets (runaway
planetesimals) are formed through runaway growth. As
protoplanets grow, they start to interact with one another.
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FORMATION OF PROTOPLANET SYSTEMS
to be consistent with the dust distribution. The observation
of protoplanetary disks confirmed that the typical mass of
protoplanetary disks is on the order of 0.01 M (e.g., Beckwith & Sargent 1996). However, it also revealed that the
disk mass ranges from 0.1 to 10 times the minimum-mass
disk model. The radial profile of the disk surface density can
also differ from that of the minimum-mass disk model.
Dust evolution in a gas disk was investigated in various
models of protoplanetary disks by Kornet, Stepinski, &
Rozyczka (2001). They numerically studied the formation
of a planetesimal swarm from dust under the dust-gas interaction and discussed the possible diversity of planetesimal
systems. We focus on the subsequent process, planetary
accretion from planetesimals, in the present paper.
In order to study the formation of extrasolar planetary
systems, we need to generalize the planetary accretion
theory. It is necessary to study planetary accretion in less
and more massive protoplanetary disks than the minimummass disk model. The variety of the mass distribution in
protoplanetary disks would potentially produce diverse
planetary systems. In particular, the number and location of
gas giant planets should depend on disk mass distribution
because the core mass depends on disk surface density and
gas giants form where the core mass exceeds some critical
value according to the core instability scenario. The recently
found extrasolar gas giants close to the central star or those
in eccentric orbits might be formed in much more massive
disks than the minimum-mass disk (e.g., Lin & Ida 1997).
In the previous N-body simulations of planetary accretion, Kokubo & Ida (1996, 1998, 2000) adopted a radially
narrow ring of planetesimals where the ratio of the ring
width Da to the ring radius a is Da=a 0:01–0.1. Kokubo &
Ida (1998, 2000) developed the oligarchic growth model
based on the result of these radially local N-body simulations. It is important to confirm the results of the local
N-body simulations by means of radially global N-body
simulations. The wide simulation width is necessary to see
the global interaction of protoplanets and the effect of the
radial distribution of planetesimals on planetary accretion.
In the present paper, we perform radially global N-body
simulations of planetary accretion with two major motivations: (1) to confirm the oligarchic growth model by global
N-body simulations and (2) to apply the oligarchic growth
model to various disk models. We adopt a planetesimal disk
with Da=a 1 and consider planetary accretion from not
only the minimum-mass disk model but also various disk
models. We investigate the dependence of the characteristics
of protoplanet systems on the disk models through global
N-body simulations along with analytical arguments.
Recently, Thommes, Duncan, & Levison (2002b)
applied the oligarchic growth model to the formation of
giant planets in the outer solar system. They developed a
semianalytical model of the evolution of planetesimals
and compared their model with global semi–N-body simulations of planetary accretion in which planetesimalplanetesimal interactions are neglected. Their simulations
are in agreement with their model for the growth timescale of protoplanets, but their masses tend to be smaller
than the prediction of their model, which may be due to
the edge effect of the simulation range. The migration of
planetesimals due to gas drag is considered in their
model. However, it is not taken into account for simplicity in the present study. We will discuss the effect of this
simplification later.
667
In x 2, we summarize oligarchic growth of protoplanets
(Kokubo & Ida 1998, 2000). The method of calculation is
presented in x 3. The results of N-body simulations are presented in xx 4 and 5. We discuss the diversity of planetary
systems based on the oligarchic growth model in x 6. Section
7 is devoted to summary and discussion.
2. OLIGARCHIC GROWTH OF PROTOPLANETS
In the oligarchic growth stage, protoplanets grow by
accreting planetesimals while gravitationally interacting
with one another. We summarize the oligarchic growth of
protoplanets in the standard disk model (Kokubo & Ida
1998, 2000) and then generalize it for general power-law
disk models.
2.1. Disk Model
We introduce the power-law disk model that consists of
solids (rock and ice) and gas. We express the solid surface
mass density as
a
g cm2 ;
ð1Þ
solid ¼ fice 1
1 AU
where a is the distance from a central star, 1 is the reference
surface density at 1 AU, is the power index of the radial
distribution, and fice is the ice factor that expresses the
increase of solids by ice condensation over the snow boundary asnow at which the disk temperature equals the ice condensation temperature ’170 K. The location of asnow
depends on the disk temperature profile. We adopt the temperature profile for an optically thin disk given by
1=2 1=4
a
L
2
K;
ð2Þ
T ’ 2:8 10
1 AU
L
where L is the luminosity of the central star (Hayashi
1981). For L ¼ L , we have asnow ’ 2:7 AU. In the minimum-mass disk model, 1 ’ 7, ¼ 3=2, and fice ¼ 1
ða < asnow Þ and 4.2 ða > asnow Þ (Hayashi 1981). Note that
the variation of L corresponds to the shift of asnow .
The amount of gas is proportional to that of solids with a
constant gas-to-dust ratio fgas ,
a
gas ¼ fgas 1
g cm2 :
ð3Þ
1 AU
We adopt fgas ¼ 240 (Hayashi 1981). For the temperature
profile equation (2), the density distribution of the gas disk
on the disk equator is given by
5=4
a
g cm3 ;
ð4Þ
gas ¼ 1
1 AU
where 1 is p
the
ffiffiffi reference density at 1 AU defined as
1 ¼ fgas 1 =ð H1 Þ and H1 is the scale height of the gas
disk at 1 AU. For L ¼ L , we have H1 ¼ 0:047 AU and
1 ¼ 2:0 109 ð fgas =240Þð1 =10Þ.
Kornet et al. (2001) studied the dust migration in a gas
disk, which leads to changing distribution of planetesimals.
We can apply the following discussion to their end state of
planetesimal distribution. In this case, the gas-to-dust ratio
should be changed according to dust migration.
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KOKUBO & IDA
2.2. Growth Timescale
In the oligarchic growth stage, protoplanets grow in a
swarm of planetesimals. The growth mode of protoplanets
is orderly growth where mass ratios of neighbor protoplanets tend to be unity. Still in this stage, however, the
mass ratios between protoplanets and planetesimals
increase. As a result, the mass distribution is divided into a
small number of large protoplanets and a large number of
small planetesimals.
The growth rate of a protoplanet with mass M and radius
R is given by (e.g., Ida & Makino 1993)
dM
2GMR
’ Csolid 2 2 ;
dt
he ia ð5Þ
he2 i1=2
where G is the gravitational constant,
is the rms
eccentricity of planetesimals, is the Kepler angular velocity, and C is an accretion acceleration factor of a few that
reflects the effects of the eccentricity and inclination distributions of planetesimals and the solar gravity (e.g., Ida &
Nakazawa 1988, 1989; Greenzweig & Lissauer 1992). The
growth timescale of protoplanets M=ðdM=dtÞ is given by
(e.g., Kokubo & Ida 2000)
1 1=3
1=3 p
C
M
Tgrow ’ 4:8 103 f 1 h~e2 i
2
1026 g
2 g cm3
1=2 1 solid
a
M 1=6
yr ; ð6Þ
1 AU
10 g cm2
M
where h~e2 i1=2 is the reduced rms eccentricity given by
h~e2 i1=2 ¼ he2 i1=2 =h and h is the reduced single Hill radius of
a protoplanet defined as
M 1=3
;
ð7Þ
h¼
3M
p is the material density of planetesimals, M is the mass of
the central star, and we introduced the radius enhancement
factor f (see x 3.3). In the oligarchic growth stage, he2 i1=2
increases in proportion to M 1=3 , which means that h~e2 i1=2 is
independent of M (Ida & Makino 1993), while in the
runaway growth stage, he2 i1=2 is independent of M, so
h~e2 i1=2 / M 1=3 (e.g., Kokubo & Ida 1996).
2.3. Orbital Separation
The orbital separation b of protoplanets is kept ~be5 by
orbital repulsion of adjacent protoplanets (Kokubo & Ida
1995), where ~
b ¼ b=rH and rH is the mutual Hill radius of
protoplanets given by
2M 1=3
a:
ð8Þ
rH ¼
3M
Orbital repulsion is a coupling effect of gravitational
scattering between protoplanets and dynamical friction
from planetesimals. In protoplanet-protoplanet (distant)
gravitational scattering, protoplanets tend to increase their
orbital eccentricities and inclinations and expand their orbital separation. After scattering, dynamical friction from
planetesimals damps the eccentricities and the inclinations.
As a result, protoplanets expand their orbital separation,
keeping nearly circular noninclined orbits.
Vol. 581
As protoplanets grow, the orbital separation measured
by the Hill radius decreases since rH / M 1=3 and the orbital
repulsion increases the orbital separation. This means that
there is a typical orbital separation realized in the oligarchic
growth stage. Kokubo & Ida (1998) found that the typical
orbital separation of protoplanets is ~b ’ 10 and it hardly
depends on the disk and protoplanet parameters according
to the f ¼ 4–6, gas-free calculations. Kokubo & Ida (2000)
confirmed this value with f ¼ 1 (real scale) calculations with
gas. In the following, we use ~b ¼ 10 for scaling of results.
2.4. Equilibrium Eccentricity
In the oligarchic growth stage, the random velocity
(eccentricity e and inclination i) of planetesimals is raised by
viscous stirring by protoplanets and is damped by gas drag.
The random velocity of planetesimals is estimated as
v ’ he2 i1=2 vK , where vK is the Kepler velocity. The timescale
of viscous stirring by protoplanets is given by (Ida &
Makino 1993)
TVS ’
G 2 n
v3
;
2
M M ln ð9Þ
where the surface number density of protoplanets is estimated as
nM ’
1
2ab2ai
ð10Þ
and ln is the Coulomb logarithm.
The gas drag timescale for a planetesimal with mass m
and radius r is given by
Tgas ’
2m
;
CD r2 gas u
ð11Þ
where CD is the drag coefficient and u is the relative velocity
between the planetesimal and gas (Adachi, Hayashi, &
Nakazawa 1976).
Equating the above two timescales, we obtain the equilibrium eccentricity as
2=15 ~ 1=5
1=15 p
b
m
2 1=2
h~e ieq ’ 5:6
23
3
10
10 g
2 g cm
1=5
1=5 1=5 gas
CD
a
;
1 AU
1
2 109 g cm3
ð12Þ
where we used u ’ v, he2 i1=2 ¼ 2hi2 i1=2 , and ln ¼ 3
weakly de(Stewart & Ida 2000). Note that h~e2 i1=2
eq
pends on m.
2.5. Isolation Mass
By extrapolating the oligarchic growth to the end of protoplanet accretion, we estimate the final mass or ‘‘ isolation
mass ’’ of protoplanets. For simplicity, the present oligarchic growth model assumes that planetary accretion proceeds locally; in other words, no considerable migration of
protoplanets and planetesimals occurs (for the application
of planetesimal migration to the oligarchic growth model,
see Thommes et al. 2002b). Given the orbital separation of
protoplanets, the oligarchic growth model predicts the
No. 1, 2002
FORMATION OF PROTOPLANET SYSTEMS
isolation mass as (Kokubo & Ida 2000)
Miso ’ 2absolid
!3=2
!3
!3=2
~
b
solid
a
¼ 0:16
10
1 AU
10 g cm2
1=2
M
M ;
M
ð13Þ
planet systems formed through oligarchic growth. We
discuss the diversity of protoplanet systems in x 5. We confirm the above oligarchic growth model expanded to the
power-law disk models by global N-body simulations in the
subsequent sections.
3. METHOD OF CALCULATION
where M is the Earth mass. In this case, the mutual Hill
radius of protoplanets with Miso is given by
!1=2
!1=2
~
b
solid
rH ’ 0:69 10
10
10 g cm2
2 a
M 1=2
AU :
1 AU
M
669
We perform global N-body simulations of planetary
accretion starting from various initial distributions of
planetesimals. For simplicity, we consider gas-free cases.
The effects of the neglect of the gas disk are discussed
in x 7.
2
3.1. Disk Model and Initial Conditions
ð14Þ
In the oligarchic growth mode, the number of protoplanets
in a certain radial range is inversely proportional to b ¼ ~brH
and thus rH .
2.6. Characteristics of Protoplanets for Power-Law
Disk Models
We apply the oligarchic growth model to the power-law
disk models given by equations (1) and (4). From equations
(6) and (12)–(14), the growth timescale of protoplanets, the
equilibrium eccentricity of planetesimals, the isolation mass
of protoplanets, and their Hill radius for the power-law disk
models are given by
Tgrow
!2
!1=3
!1
h~e2 i1=2
M
fice 1
’ 1:7 10 f
1026 g
6
10
1=2þ 1=6
a
M
yr ;
ð15Þ
1 AU
M
5 1
h~e2 i1=2
eq
!1=5
!1=15
!1=5
~
fgas
b
m
’ 5:6
10
1023 g
240
ð1=5Þþ1=20
1=5 1
a
;
1 AU
10
Miso ’ 0:16
~
b
10
!3=2
a
1 AU
fice 1
10
ð3=2Þð2Þ ~
b
10
ð16Þ
TABLE 1
Initial Conditions of Planetesimal Disks
!3=2
M
M
1=2
M ;
ð17Þ
!1=2
!1=2
fice 1
rH ’ 0:69 10
10
ð1=2Þð4Þ a
M 1=2
AU ;
1 AU
M
2
We adopt an axisymmetric surface density distribution of
a planetesimal disk given by equation (1) with inner and
outer cutoffs, ain and aout . We set ain ¼ 0:5 AU and
aout ¼ 1:5 AU in all models. In most models, planetesimals
have equal masses. In a model (model 4), we adopt the
power-law mass distribution, ndm / m dm with dynamic
range mmax =mmin ¼ 20. The number of bodies is N ¼ 5000
and 10,000. We set the internal density of planetesimals
p ¼ 2 g cm3.
The initial conditions of planetesimal disks are summarized in Table 1. In Table 1, ¼ 1 represents an
equal-mass distribution. In all the disk models, the initial
eccentricities and inclinations of planetesimals are given
by the Rayleigh distribution with dispersions he2 i1=2 ¼
2hi2 i1=2 ¼ 0:02 (Ida & Makino 1992a). We call the disk
model with 1 ¼ 10 and ¼ 3=2 the standard model,
which is similar to the minimum-mass disk model and
was often adopted in the previous N-body simulations.
Because observationally inferred disk mass ranges from
0.1 to 10 times the minimum-mass disk model, we calculate the cases with 1 ¼ 1 and 100. We also consider the
disks with flatter ( ¼ 12) and more centrally concentrated
( ¼ 5=2) radial density profiles.
For each model, we perform two runs with different random numbers for the initial distributions. Because the
results are similar for the two runs, we show the results of
one of the two runs in the following.
ð18Þ
where we used p ¼ 2 g cm3 , C ¼ 2, and CD ¼ 1. The
above estimates imply interesting characteristics of proto-
Model
1
N
Mdisk
(g)
f
1...............
2...............
3...............
4...............
5...............
6...............
7...............
8...............
1
10
10
10
100
’10
10
’10
3/2
3/2
3/2
3/2
3/2
1/2
3/2
5/2
1
1
1
5/2
1
1
1
1
10000
10000
10000
10000
10000
5000
5000
5000
1.5 1027
1.5 1028
1.5 1028
1.5 1028
1.5 1029
1.5 1028
1.5 1028
1.5 1028
10
10
6
6
6
6
6
6
Note.—Parameters 1, , , N, Mdisk , and f stand for the reference
surface density at 1 AU, the power index of radial distribution, the
power index of mass distribution, the number of planetesimals, the
total mass of the planetesimal disk, and the radius enhancement
factor, respectively.
670
KOKUBO & IDA
3.2. Orbital Integration
The orbits of planetesimals are calculated by numerically
integrating the equation of motion of planetesimals,
N
X
xi xj
dvi
xi
¼ GM
Gmj
;
3
dt
jxi j3
jx
i xj j
j6¼i
ð19Þ
where x and v are the position and velocity of planetesimals,
respectively. The right-hand side of equation (19) represents
the gravity of the central star and the mutual gravitational
interaction of planetesimals. We set M ¼ M .
For numerical integration, we use the modified Hermite
scheme for planetary N-body simulation (Kokubo, Yoshinaga, & Makino 1998) with the hierarchical time step
(Makino 1991). The most expensive part of the Hermite
scheme is the calculation of the mutual gravitational force
and its time derivative, whose cost increases in proportion
to N2. We calculate the force and its first time derivative by
directly summing up interactions of all pairs on the specialpurpose computer for N-body simulation, HARP/GRAPE
(Makino, Kokubo, & Taiji 1993; Makino et al. 1997).
Vol. 581
times the initial planetesimal mass form on nearly circular
noninclined orbits.
Protoplanet growth propagates from inner to outer disk.
This is because the growth timescale is smaller for small a
since the surface number density of planetesimals is higher
and the orbital period is smaller for small a as shown by
equation (15). For ¼ 3=2, Tgrow / a3 since h~e2 i1=2 / a1=2
in a gas-free case (eq. [21]). Planetesimal accretion is complete in the inner disk, while it is still on the way in the outer
disk. The eccentricities and inclinations of protoplanets are
kept small because of dynamical friction from planetesimals. The velocity dispersion of planetesimals is as large as
the surface escape velocity of protoplanets. The mass and
orbital separation of the protoplanet system are consistent
with the oligarchic growth model. The resultant characteristics of protoplanets in models 2, 3, 4, and 7
(1 ¼ 10; ¼ 3=2) are all similar. In the following, we
investigate the evolution of spatial, velocity, and mass
distributions of the system and the characteristics of protoplanets in model 3.
4.2. Spatial Distribution
3.3. Accretion
The evolution of the surface mass density distribution is
plotted in Figure 2. We found that the overall density profile
hardly changes in the simulation period. The bumpy structure of the profile is due to the discrete distribution of protoplanets and not important. This constancy of the density
profile is necessary for the oligarchic growth model to hold.
The reason for the almost time independent density distribution is that the diffusion timescale of the disk due to
gravitational scattering among planetesimals is much larger
than the growth timescale of protoplanets. Using the viscous stirring timescale of planetesimals, we estimate the
diffusion timescale as
!2
!2
!1
Da
he2 i1=2
m
10
Tdiff ’ 1:3 10
1 AU
1023 g
0:01
5=2
1 solid
a
yr ;
ð20Þ
1 AU
10 g cm2
We adopt an f-fold radius of bodies to accelerate the
accretion process and thus save calculation time. The use of
an f-fold radius changes the growth timescale but not the
growth mode as long as f is not too large (for details, see
Kokubo & Ida 1996). The timescale of planetary accretion
is reduced by a factor of f2 at most in gas-free cases. It
should be noted that the use of f affects the stage of giant
impacts among protoplanets, where the orbital elements of
planets formed by giant impacts relatively strongly depend
on f. In the present calculation, we do not follow this stage.
For simplicity, we assume that two planetesimals always
accrete when they contact. In accretion, the position and
velocity of the center of mass are conserved. The lack of collisional fragmentation or rebound seems to make no significant change in the growth mode of protoplanets (Wetherill
& Stewart 1993). Note that migration of fragments due to
gas drag may change the surface density profile, which we
discuss later.
It is worth noting that we should be careful when we use f
in the outer disk where planetesimals are loosely bound to
the system and thus their ejection due to scattering by protoplanets is effective. In this case, the radius enhancement
allows accretion to continue past the point where, in reality,
it would stall.
where Da is the width of the disk (for derivation, see the
Appendix). For small planetesimals that form the main
disk, Tdiff is much longer than Tgrow of protoplanets. Even if
f ¼ 1 is adopted and scattering by protoplanets is considered, Tdiff 4Tgrow still holds. Thus, the density profile
remains almost constant.
4. FORMATION OF PROTOPLANET SYSTEMS
4.3. Velocity Distribution
We investigate in detail the result of the standard disk
model (1 ¼ 10 and ¼ 3=2) from which the solar system
may form. With the global N-body simulations, we confirm
the oligarchic growth model constructed based on the
results of local N-body simulations.
The evolution of the rms eccentricity and inclination distributions of planetesimals is shown for four disk zones
(a < 0:75 AU, 0:75 AU < a < 1:0 AU, 1:0 AU < a < 1:25
AU, 1:25 AU < a) in Figure 3. The initial values are the
same for all the zones. Viscous stirring increases the eccentricity and the inclination, while dynamical friction reduces
those of large bodies. The inner disk is heated up first for the
same reason as the short growth timescale for the inner disk.
For me1025 g, as a result of dynamical friction, d loghe2 i1=2 =d log m ’ d loghi2 i1=2 =d log m ’ 12, which
means the equipartition of the random energy (e.g., Stewart
& Wetherill 1988; Ida & Makino 1992b). The ratio of eccentricity to inclination for planetesimals is always e=i ’ 2.
4.1. Overall Evolution
Figure 1 shows the snapshots of the system on the a-e and
a-i planes for model 3. Starting with 10,000 equal-mass
(1:5 1024 g) planetesimals, we have calculated the system
for 4 105 yr. We employed a sixfold radius of planetesimals. The number of bodies decreases from 10,000 to 333.
In 4 105 yr, 12 protoplanets whose mass is larger than 100
No. 1, 2002
FORMATION OF PROTOPLANET SYSTEMS
671
Fig. 1a
Fig. 1b
Fig. 1.—Snapshots of a planetesimal system on the (a) a-e plane and (b) a-i plane for model 3. The circles represent
planetesimals, and their radii are proportional to the radii of the planetesimals. The system initially consists of 10,000 equal-mass (1:5 1024 g) planetesimals
in 0.5 AU a 1:5 AU. The numbers of planetesimals are 2612 (5 104 yr), 1409 (105 yr), 673 (2 105 yr), and 333 (4 105 yr). The filled circles represent
protoplanets that are larger than 100 times the initial mass, and the length of the line on a protoplanet is 10rH .
This relaxation of the orbital elements is characteristic of
the Keplerian disk (Ida, Kokubo, & Makino 1993).
In a gas-free system, the random velocity of planetesimals
is determined mainly by viscous stirring. When the mass of
a protoplanet exceeds 100 times the mass of the typical
planetesimal, the viscous stirring by the protoplanet
becomes dominant (Ida & Makino 1993). In this case, the
random velocity of planetesimals becomes as large as the
surface escape velocity of the protoplanet vesc ’ eesc vK ,
where eesc is the escape eccentricity given by (e.g., Safronov
1969)
1=2
1=3 M
a
1=2
:
ð21Þ
eesc ’ 0:081f
1026 g
1 AU
For M ¼ 1027 g, a ¼ 1 AU, and f ¼ 6, eesc ’ 0:07
(~eesc ’ 13), which is consistent with the simulation.
Note that the eccentricities and inclinations of the protoplanets are always kept small. For growing protoplanets in
a planetesimal swarm, this is due to dynamical friction from
planetesimals. The relatively large orbital separations
~b ’ 10, and thus the weak gravitational interaction is the
reason for the isolated protoplanets (in longer timescales,
however, their eccentricities and inclinations can become
large because of long-term mutual perturbation).
4.4. Mass Distribution
The evolution of the mass distribution is plotted for the
four disk zones in Figure 4. The mass distribution first
relaxes to the power-law distribution with power index
d log nc =d log m ’ 1:5, where nc is the cumulative number
of bodies, which is characteristic of runaway growth
(Makino et al. 1998; Kokubo & Ida 2000). Then the slope of
the mass distribution gradually becomes gentle because in
this system there is no supply of small planetesimals. The
evolution of the mass distribution of large planetesimals or
protoplanets agrees with the result of the statistical growth
equation by Wetherill & Stewart (1993). The equivalence
between the results of the statistical approach and N-body
simulation is shown in detail in Inaba et al. (2001).
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KOKUBO & IDA
Vol. 581
It is clearly shown that the mass distribution is divided
into two components, small planetesimals with mass
m 1 10 1024 g and large protoplanets with mass
M 1027 g. This bimodal mass distribution is the result of
runaway and oligarchic growth of protoplanets.
4.5. Characteristics of Protoplanets
Fig. 2.—Radial profile of the surface mass density of planetesimals at
t ¼ 0 yr (solid line), 5 104 yr (dotted line), 105 yr (short-dashed line),
2 105 yr (long-dashed line), and 4 105 yr (dotted-dashed line) for
model 3.
The evolution of the protoplanet system on the a-M plane
is plotted in Figure 5. We found that 12 protoplanets of
about the predicted isolation mass M 0:1 M are formed
with the orbital separation ~b ¼ 13:7 2:5, which is almost
consistent with the oligarchic growth model.
The two curves in Figure 5 are analytical estimates of the
protoplanet growth for ~b ¼ 10 and 15 based on the oligarchic growth model. Using he2 i1=2 ¼ eesc , from equation (5)
we predict the growth of a protoplanet in the oligarchic
growth stage in a gas-free system as
"
solid
1=3
2 C
MðtÞ ’ M0 þ 32f
2
10 g cm2
3=2 #3
a
t
g;
ð22Þ
1 AU
1 yr
where M0 is the initial mass of the protoplanet. The
Fig. 3b
Fig. 3a
Fig. 3.—Snapshots of the rms (a) eccentricity and (b) inclination distributions against mass at t ¼ 5 104 , 105, 2 105 , and 4 105 yr for model 3; he2 i1=2
and hi2 i1=2 are calculated in four divided zones: a < 0:75 AU (circles), 0:75 AU < a < 1:0 AU ( pentagons), 1:0 AU < a < 1:25 AU (squares), and
1:25 AU < a (triangles).
No. 1, 2002
FORMATION OF PROTOPLANET SYSTEMS
Fig. 4.—Cumulative number of bodies is plotted against mass at
t ¼ 5 104 , 105, 2 105 , and 4 105 yr for model 3. The mass distribution
is calculated in four divided zones: a < 0:75 AU (solid line),
0:75 AU < a < 1:0 AU (dotted line), 1:0 AU < a < 1:25 AU (short-dashed
line), and 1:25 AU < a (long-dashed line). The initial number of bodies is
3037, 2656, 2223, and 2084 from inner to outer zones.
673
Fig. 5.—Mass of protoplanets with mass larger than 100 times the initial
mass is plotted against semimajor axis at t ¼ 5 104 , 105, 2 105 , and
4 105 yr for model 3. The two curves are the analytical estimates of the
protoplanet growth based on the oligarchic growth model for ~b ¼ 10 (solid
line) and 15 (dotted line).
5. DIVERSITY OF PROTOPLANET SYSTEMS
analytical estimate of the protoplanet growth is given by
M ¼ min½MðtÞ; Miso . The results of the N-body simulation
agree well with the analytical estimates with C ¼ 2. Protoplanets grow to the isolation mass, and the ‘‘ accretion
wave ’’ propagates from inner to outer disk.
Note that when M is small the analytical model overestimates the rms eccentricity of planetesimals and thus
underestimates the growth rate. This is because small
growing protoplanets cannot stir up neighbor planetesimals up to eesc within Tgrow since Tgrow of the small protoplanets is shorter than TVS . This effect leads to the fact
that the protoplanets in the N-body simulation grow
faster than those of the analytical model when they are
small.
We confirmed the oligarchic growth of protoplanets in
the standard disk model by the global simulation ranging
from 0.5 to 1.5 AU. The reason the oligarchic growth model
holds is that accretion proceeds locally; in other words, the
radial distribution of planetesimals is not greatly affected by
planetesimal dynamics and accretion.
In this section, a wide variety of power-law disk models
are adopted and the results of N-body simulations are compared with those predicted by the oligarchic growth model.
We investigate the dependence of characteristics of protoplanet systems on the disk mass (1) and the disk profile ().
5.1. Disk Mass Dependence
First we investigate the 1 dependence of protoplanet systems. The variation of 1 corresponds to the variation of
the disk mass when is fixed. Figures 6 and 7 show the
snapshots of the protoplanet systems on the a-e and a-M
planes for 1 ¼ 1 (model 1), 10 (model 3), and 100 (model
5). The power index of the radial distribution is fixed as
¼ 3=2. In Figure 7, the isolation mass of protoplanets for
~b ¼ 10 and ~b ¼ 15 is shown for 1 ¼ 1, 10, 100.
In the disk with 1 ¼ 100, at 105 yr, six large protoplanets
with M ’ 1 5 M are formed with the orbital separation
~b ¼ 11:5 1:7, which is in agreement with the oligarchic
growth model. Two more relatively small protoplanets are
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KOKUBO & IDA
Vol. 581
Fig. 7.—Same as Fig. 6 but on the a-M plane for protoplanets. The two
lines show the estimated isolation mass for ~b ¼ 10 (solid line) and 15 (dotted
line).
Fig. 6.—Snapshots of planetesimal systems on the a-e plane for model 1
(1 ¼ 1) at 6 105 yr, model 3 (1 ¼ 10) at 4 105 yr, and model 5
(1 ¼ 100) at 105 yr. The circles represent planetesimals, and their radii
are proportional to the radii of the planetesimals. The systems initially consist of 10,000 equal-mass [1:5 ð1023 1025 Þ g] planetesimals in 0.5
AU a 1:5 AU. Filled circles represent protoplanets that are larger than
100 times the initial mass, and the length of the line on a protoplanet is
10rH .
on the outside of the Figure 6 region, ða; eÞ ¼ ð1:48; 0:43Þ
and (1.73, 0.16), which is the result of close encounters
between protoplanets. The eccentricities of protoplanets in
the 1 ¼ 100 case are relatively highly pumped up because
the mass of protoplanets is relatively large and thus the
gravitational interaction is relatively strong. Note that the
predicted isolation mass of protoplanets around 1 AU is
several M , which may cause the onset of gas accretion
from the gas disk with the result that the protoplanets
become Jovian planets (e.g., Ikoma et al. 2000). We will discuss this case in the next section.
In the disk with 1 ¼ 1, at 6 105 yr, 31 protoplanets
b ¼ 15:1 5:1.
with M ’ 0:003 0:01 M are formed with ~
This orbital separation is a little bit larger than the typical
value ~
b ¼ 10. This difference of orbital separation is due to
protoplanet-protoplanet collisions. For protoplanet sysb ’ 10, and e i 0:01, the orbital
tems with M 1026 g, ~
instability timescale is on the order of 105 yr (Yoshinaga,
Kokubo, & Makino 1999). This means that, to some extent,
the protoplanet-protoplanet collisions already took place to
form larger protoplanets with larger orbital separation than
those predicted by the oligarchic growth model especially in
the inner disk. If protoplanets with separation ~b ¼ 10
collide, the resultant new separation for the merged protoplanet becomes ~b ’ 15, whose instability timescale is much
larger than the simulation period if e i 0:01.
It is clearly shown that the number of protoplanets
decreases with 1 while the mass of protoplanets increases
with 1. The oligarchic growth model predicts that the isola3=2
tion mass of protoplanets depends as Miso / 1 and the
1=2
1
(eqs. [17]
number of protoplanets nM / rH / 1
and [18]). The results of the N-body simulations are consistent with this prediction.
5.2. Disk Profile Dependence
Next we consider the -dependence of protoplanet systems. We compare three cases with ¼ 12 (model 6), 3/2
(model 7), and 5/2 (model 8) with the fixed disk mass
Mdisk ¼ 1:5 1028 g in between 0.5 and 1.5 AU. It should
be noted that in a real protoplanetary disk, is not a free
parameter but determined by some physical processes in the
disk. Because we do not know how the initial planetesimal
distribution is determined, it is of importance to explore the
-space.
Figure 8 shows the protoplanet mass against semimajor
b ¼ 10
axis for t ¼ 2 105 yr with the isolation mass for ~
and 15. As the oligarchic growth model predicts, the
results show that the mass of protoplanets depends on a
as M / að3=2Þð2Þ , with ~b ¼ 10–15. Note that in the outer
disk, accretion is still on the way. The oligarchic growth
model means that if the initial disk is centrally condensed
( > 2) Miso decreases with a and if the initial disk has a
flatter distribution ( < 2) Miso increases with a. In the
standard model ¼ 3=2, we have larger protoplanets on
larger semimajor axes. The radial profile index is an
important factor that determines the mass distribution of
No. 1, 2002
FORMATION OF PROTOPLANET SYSTEMS
675
planetary systems formed from the diverse protoplanet
systems.
6. DIVERSITY OF PLANETARY SYSTEMS
The oligarchic growth model describes the growth of
solid (rocky and icy) protoplanets. Based on the oligarchic
growth model together with the gas giant planet formation
scenario, we discuss the possible diversity of planetary
systems. The gas giant planet is a dominant member of
planetary systems that controls the basic architecture of
planetary systems.
We use the oligarchic growth model for the power-law
disks, equations (15), (16), and (17), and assume that the
orbital separation of protoplanets is scaled by the Hill
radius and ~b ¼ 10. For simplicity, we also assume that all
the solid materials are finally incorporated into planets in
situ. Thus, in the following discussion, the isolation mass
and the growth timescale of protoplanets are the maximum
and the minimum values. We discuss the effect of planetesimal migration in x 7. The disk parameters are set as
fice ¼ 4:2, fgas ¼ 240, and asnow ¼ 2:7 AU. We adopt planetesimals with m ¼ 1023 g and set M ¼ M .
6.1. Formation of Planets from Protoplanets
Fig. 8.—Mass of protoplanets plotted against semimajor axis at
t ¼ 2 105 yr for models 6 ( ¼ 1=2), 7 ( ¼ 3=2), and 8 ( ¼ 5=2) with
the isolation mass for ~
b ¼ 10 (solid line) and 15 (dotted line).
protoplanets, which leads to formation of diverse planetary systems as will be shown in the next section.
In summary, we confirmed that oligarchic growth holds
for the power-law disk models with 1 ¼ 1 100 and
¼ 1=2 5=2. The dependence of protoplanet systems on 1
and is schematically illustrated in Figure 9. Given the initial distribution of planetesimals, we can immediately
obtain the protoplanet systems that are formed from the
planetesimals based on the oligarchic growth model. The
variation of 1 and leads to diversity of protoplanet systems. In the next section, we discuss possible diversity of
α
2
a
a
a
a
Σ1
Fig. 9.—Schematic illustration of the diversity of protoplanet systems on
the 1- plane. Four examples of protoplanet systems are illustrated. [See
the electronic edition of the Journal for a color version of this figure.]
Protoplanets form through oligarchic growth in the gas
disk. The fate of protoplanets depends on their mass and
growth timescale and the lifetime of the gas disk. From the
observation of protoplanetary disks, the lifetime of protoplanetary disks Tdisk is estimated as 106–107 yr (Strom,
Edwards, & Skrutski 1993; Zuckerman, Forveille, & Kastner 1995) and the amount of the gas necessary for a Jovian
planet would remain for a few times longer than the estimated disk lifetime (Thi et al. 2001).
The protoplanet system with ~b ¼ 10 is orbitally stable as
long as 0.1%–1% of the gas disk remains because the gas
disk–planet interaction damps the eccentricities of protoplanets (Iwasaki et al. 2002). In other words, in the gas disk, no
accretion among protoplanets occurs. Thus, we can use the
isolation mass and the growth timescale for the isolation
mass to consider the final stage of planet formation.
A gas giant or Jovian planet is formed by gas accretion
onto a protoplanet (solid core) by the gravity of the protoplanet. In the core instability scenario, the onset of gas
accretion occurs when the mass of a core exceeds a critical
mass Mcr ’ 5 15 M (e.g., Mizuno 1980; Bodenheimer &
Pollack 1986; Pollack et al. 1996). Note that the cores of
Jovian planets are not formed by accretion among protoplanets in the gas disk as mentioned above.
In the standard disk model, terrestrial planets form
through giant impacts among protoplanets whose mass is
smaller than the critical core mass after most of the gas disk
is depleted (e.g., Kominami & Ida 2002). Uranian planets
would be protoplanets themselves whose growth timescale
is longer than the lifetime of the gas disk in their vicinity
(e.g., Kokubo & Ida 2000). In the following, we consider the
diversity of planetary systems according to this concept of
final planet formation.
6.2. Conditions for Gas Giant Planet Formation
Gas accretion begins when the growth timescale of a protoplanet becomes longer than the contraction timescale of
the planetary atmosphere (Ikoma et al. 2000). Planetesimal
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KOKUBO & IDA
Vol. 581
accretion plays the role of heat source to support the planetary atmosphere. At the end of the oligarchic growth stage,
planetesimal accretion almost ceases and the growth timescale becomes formally infinity. Then gas accretion begins.
The contraction timescale Tcont is given by
8
Miso 5=2
9
Tcont 10 10
yr ;
ð23Þ
M
which decreases rapidly with the isolation mass of protoplanets (Ikoma et al. 2000). If Miso is relatively small, gas accretion proceeds slowly. However, gas accretion becomes
increasingly rapid as the total planetary mass increases (e.g.,
Pollack et al. 1996; Ikoma et al. 2000).
The lifetime of protoplanetary disks leads to two conditions for the formation of gas giant planets: (1) the contraction timescale must be shorter than the lifetime of the disk:
Tcont < Tdisk ;
ð24Þ
which is equivalent to the conventional condition that
Miso > Mcr , where we define critical core mass as the minimum mass satisfying Tcont < Tdisk ; and (2) in order to capture the gas, protoplanet growth must be completed before
the depletion of the gas:
Tgrow < Tdisk :
Fig. 10a
ð25Þ
In other words, only protoplanets with Tcont < Tdisk
(Miso > Mcr ) and Tgrow < Tdisk can become gas giants. This
means that there is the limited range of semimajor axis for
gas giant planet formation.
We adopt Tcont ¼ 108 ðMiso =M Þ5=2 yr and scale results
with Tdisk ¼ 108 yr in the following discussion. These values
are favorable for gas giant planet formation. The qualitative
feature of the following discussion, however, hardly changes
as long as Tcont and Tdisk are in reasonable ranges.
6.3. Habitat Segregation of Planets
In Figure 10, we plot the isolation mass of protoplanets
against the semimajor axis for 1 ¼ 1, 10, 100 and
¼ 3=2; 5=2. As discussed in the previous section, Miso
increases with a for ¼ 3=2, while Miso decreases with a for
¼ 5=2. The isolation mass jumps at the snow border. Note
that asnow sets a rough boundary between terrestrial (rocky)
and Uranian (icy) planets.
Substituting equations (16) and (17) into equation (15),
we obtain the growth timescale of protoplanets with Miso in
the gas disk
~ 1=10 9=10
b
1
1=2
Tgrow ’ 3:2 105 fice
10
10
ð9þ16Þ=10
a
yr :
ð26Þ
1 AU
In Figure 11, Tgrow and Tcont for Miso are drawn against the
semimajor axis for 1 ¼ 1, 10, 100 and ¼ 3=2, 5/2. The
dependence of the contraction timescale on the semimajor
axis for < 2 is qualitatively different from that for > 2.
5=2
15=4 ð15=4Þð2Þ
a
. Thus, Tcont
For Miso , Tcont / Miso / 1
decreases with a for ¼ 3=2, while Tcont increases with a for
¼ 5=2. This difference leads to different formation ranges
for Jovian planets.
In the following, we focus on the ¼ 3=2 case, which has
the same radial profile as the standard disk model. The
Fig. 10b
Fig. 10.—Isolation mass of protoplanets with ~b ¼ 10 against the semimajor axis for disks with 1 ¼ 1, 10, and 100 for (a) ¼ 3=2 and (b)
¼ 5=2. The discontinuity at asnow ¼ 2:7 AU (snow border) is due to ice
condensation.
application of a similar discussion to other disks is,
however, straightforward. For < 2, the condition
Tcont < Tdisk sets the inner boundary for gas giant formation. The condition Tgrow < Tdisk draws the outer boundary.
From equation (24), we obtain the inner boundary as
amin
gas
’
2
12fice
Tdisk
108 yr
!8=15
~b
10
!2
1
10
!2
AU :
ð27Þ
No. 1, 2002
FORMATION OF PROTOPLANET SYSTEMS
677
max
Fig. 12.—Habitat segregation of planets based on amin
gas , agas , and the
snow border asnow for disks with Tdisk ¼ 107 yr (dotted line) and 108 yr (solid
line) against 1 for ¼ 3=2 and ~b ¼ 10.
Fig. 11a
1 in Figure 12. The territories of terrestrial, Jovian, and
Uranian planets are determined by the following rule:
max
Jovian planets can form in amin
gas dadagas , and for terrestrial
and Uranian planets, adasnow and aeasnow except for the
Jovian planet zone, respectively. This is the basic habitat
segregation of planets for < 2 disks and consistent with
the present solar system. The short disk lifetime narrows the
Jovian territory as shown in equations (27) and (28). The
qualitative segregation pattern, however, is independent of
the disk lifetime.
In the standard disk model 1 ’ 10, it is difficult to form
the cores of Jovian planets in the solar system within
Tdisk ¼ 107 yr. For that, a massive (1 50) disk is
required, which is consistent with Thommes et al. (2002b).
It is also possible to form them in a disk with Tdisk ¼ 108 yr
and 1 10. The increase of the disk lifetime has the same
effect on the habitat segregation of planets as the increase of
the disk mass. In the rest of this section, as an example, we
investigate the Tdisk ¼ 108 yr case in detail.
Fig. 11b
Fig. 11.—Growth timescale of the isolation mass protoplanet with
~
b ¼ 10 and the contraction timescale of the planetary atmosphere against
the semimajor axis for 1 ¼ 1 (dotted line), 10 (solid line), and 100 (dashed
line) with (a) ¼ 3=2 and (b) ¼ 5=2. For ¼ 3=2, d log Tgrow =d log a > 0
and d log Tcont =d log a < 0, and for ¼ 5=2, d log Tgrow =d log a
> d log Tcont =d log a > 0.
The outer boundary is obtained from equation (25) as
!2=59
!20=59
!18=59
~
b
Tdisk
1
10=59
max
agas ’ 5:5fice
AU :
10
108 yr
10
ð28Þ
max
We draw habitat segregation of planets based on amin
gas , agas ,
and the snow border asnow for Tdisk ¼ 107 and 108 yr against
6.4. Disk Mass Dependence
We discuss the possible diversity of planetary systems
using the above habitat segregation of planets. The observation of protoplanetary disks by Beckwith & Sargent (1996)
revealed that the protoplanetary disk has a wide range of
mass, 0.1–10 times the mass of the minimum-mass disk
model. We investigate the cases 1 ’ 1, 10, 100 with
Tdisk ¼ 108 yr.
6.4.1. Light Disk (1 ’ 1)
In light disks with 1 ’ 1, we have Miso ’ 0:05 M at 1
AU and 0:3 M at 10 AU. In this case, to form gas giants,
the lifetime of the gas disk must be on the order of 109 yr,
which is far longer than the estimated lifetime of the gas
disk. Therefore, gas giants would not form at all in this case.
Figure 12 shows that light disks with 1 d3 would not have
any gas giants. Note that even if giant impacts among
678
KOKUBO & IDA
protoplanets form more massive planets than Mcr , they cannot become gas giants since giant impacts are possible only
after the depletion of most of the gas (Iwasaki et al. 2002;
Kominami & Ida 2002). A planetary system formed from
the light disk would consist of many relatively small solid
planets, terrestrial planets inside the snow border, and
Uranian planets outside the snow border.
6.4.2. Massive Disk (1 ’ 100)
For the disk as massive as 1 ’ 100, Miso ’ 5 M at 1
AU, which is large enough for gas accretion within Tdisk .
Gas giants can form in the inner disk (a 1 AU). Furthermore, in the massive disks, the growth timescale of protoplanets is so short that Tgrow < Tdisk even at large a.
Therefore, several gas giants would form in relatively massive disks with 1 e30. Uranian planets would form outside
the Jovian planets. We will discuss the massive disk case in
relation to the origin of observed extrasolar planets in more
detail below.
6.4.3. Medium (Standard) Disk (1 ’ 10)
In the disk with 1 ’ 10, a planetary system similar to the
solar system is expected. In this disk, gas giants can form
only in the limited range beyond the snow border. This
range depends on Tdisk . For Tdisk 108 yr, one or two gas
giants may form between the snow border and about 10
AU. In this case, we have terrestrial planets, Jovian planets,
and Uranian planets from inner to outer system.
In Figure 13, we schematically summarize the predicted
diversity of planetary systems produced by the disk mass
variation for disks with < 2.
It should be noted that in the oligarchic growth model we
assumed the accretion in the gas disk. However, by definition, Tgrow of Uranian planets beyond the Jovian planet
zone exceeds Tdisk . After the dispersal of the gas disk, the
random velocity of planetesimals is pumped up as high as
the escape velocity of protoplanets. This high random velocity makes the accretion process slow and inefficient and thus
Tgrow longer. This accretion inefficiency is a severe problem
Mdisk
T cont <Tdisk
Tgrow<Tdisk
Vol. 581
for the formation of Uranian planets in the solar system
(Levison & Stewart 2001). One possible solution to this
problem is that Uranian planets form in the Jovian
planet region and are subsequently transported outward
(Thommes, Duncan, & Levison 1999, 2002a).
6.5. Origin of Extrasolar Planets
The disk mass dependence of planetary systems suggests
that the number of Jovian planets increases with the disk
mass. However, initially formed Jovian planet systems
would not be the final configuration of planetary systems
since planetary systems with more than three giant planets
may not be stable systems in the long term (e.g., Chambers,
Wetherill, & Boss 1996; Marzari & Weidenschilling 2002). A
planetary system of several gas giants may become orbitally
unstable against long-term mutual perturbations. After the
ejection of some planets or merging, orbitally stable planets
in eccentric orbits would remain, which may correspond to
observed extrasolar planets in eccentric orbits (Rasio &
Ford 1996; Weidenschilling & Marzari 1996; Lin & Ida
1997; Marzari & Weidenschilling 2002). In addition, interactions between gas giants and a residual relatively massive
gas disk may lead to significant orbital decay to a central star
(e.g., Lin & Papaloizou 1993), which may correspond to
extrasolar planets with short orbital periods (hot Jupiters)
such as 51 Peg b (Lin, Bodenheimer, & Richardson 1996).
If an extremely massive disk with 1 e200
(Mdisk e0:3 M for ¼ 3=2) is considered, Figure 12 suggests that in situ formation of hot Jupiters at a 0:05 AU
such as 51 Peg b, And b, etc., may be possible. However,
dust particles may be evaporated at a 0:05 AU in the
disk, which inhibits planetesimal formation, and/or ultraviolet and X-ray radiation from a T Tauri star may strip the
gas envelope of a young gas giant (Lin et al. 1996). Hence,
the migration model may be favored for hot Jupiter
formation.
On the other hand, in situ formation of extrasolar planets
in circular orbits around a ’ 0:2 AU such as CrB b and
HD 192263 b is likely to occur in relatively massive disks
with 1 e100 (Mdisk e0:15 M ). The inhibition processes
for in situ formation for hot Jupiters do not apply to this
case. It is difficult for the migration (Lin et al. 1996) or the
slingshot model (Rasio & Ford 1996) to explain planets in
circular orbits at a ’ 0:2 AU because tidal interaction or
the magnetic field of a host star, which circularizes orbits,
may be weak there. In situ formation in relatively massive
disks may be most promising.
7. SUMMARY AND DISCUSSION
a
Fig. 13.—Schematic illustration of the diversity of planetary systems
against the initial disk mass for < 2. The left large circles stand for central
stars. The double circles (cores with envelopes) are Jovian planets, and the
others are terrestrial and Uranian planets. [See the electronic edition of the
Journal for a color version of this figure.]
Terrestrial and Uranian planets and solid cores of
Jovian planets form through accretion of planetesimals. In
planetary accretion, oligarchic growth of protoplanets is a
key process that controls the basic structure of planetary
systems.
We confirmed that the oligarchic growth model generally
holds in the wide variety of planetesimal disks
solid ¼ 1 ða=1 AUÞ g cm2 with 1 ¼ 1, 10, 100 and
¼ 1=2; 3=2; 5=2 by performing global N-body simulations. We derived how the characteristics of protoplanet
systems depend on the initial disk mass (1) and the initial
disk profile (). The oligarchic growth model gives the
growth timescale and the isolation mass as equations (15)
and (17), respectively, which are in good agreement with the
No. 1, 2002
FORMATION OF PROTOPLANET SYSTEMS
results of the N-body simulations. The isolation mass
3=2
increases in proportion to 1 , and the number of proto1=2
planets decreases in proportion to 1 . Because the isolað3=2Þð2Þ
, for < 2 it increases
tion mass depends on a as a
with a while it decreases with a for > 2. The growth timescale increases with a but decreases with 1. These characteristics of protoplanets provide a base for considering the
diversity of planetary systems.
Based on the oligarchic growth model and the gas giant
planet formation scenario, we discussed the diversity of
planetary systems. Gas giant planets can form only in the
disk range where the contraction timescale of the atmosphere of a core and the growth timescale of the core are
shorter than the lifetime of the gas disk. The contraction
timescale decreases with the isolation mass as
5=2
Tcont / Miso , and the growth timescale increases with
the isolation mass and the semimajor axis as
1=3
Tgrow / Miso aþ1=2 . We derived the disk range where
Jovian planets can form as a function of the disk parameters. For example, for ¼ 3=2 and Tdisk 108 yr, several
Jovian planets would form from massive disks with 1 e30
with Uranian planets outside the Jovian planets. Only terrestrial and Uranian planets would form from light disks
with 1 d3. Solar system–like planetary systems would
form from medium disks with 1 ’ 10 (a more massive disk
with 1 50 is required for Tdisk 107 yr).
This diversity of planetary systems suggests that there
exist many extrasolar terrestrial planets. The present detection probability of extrasolar planets around solar-type
stars is a few percent. Because of the observational selection
effect, most extrasolar planets so far discovered have relatively small semimajor axes and large masses (e.g., Marcy,
Cochran, & Mayor 2000). These extrasolar planetary systems might correspond to the planetary systems formed
from massive disks. A massive disk may form several giant
planets. The long-term orbital instability of the planetary
system and/or the (type II) migration of the giant planets
due to gas disk–planet interaction may lead to a system similar to those of the extrasolar planets (Lin & Papaloizou
679
1993). The other disks with smaller mass, which are the
majority of protoplanetary disks (Beckwith & Sargent
1996), may form terrestrial planets if planet formation proceeds in a similar way as described here.
In the present discussion, potentially important processes due to the gas disk are not considered, namely,
(type I) migration of protoplanets and migration of planetesimals due to gas drag. The migration of protoplanets
due to tidal interaction with the gas disk (e.g., Goldreich
& Tremaine 1980; Ward 1986, 1997; Papaloizou & Larwood 2000) would modify the oligarchic growth of protoplanets. Protoplanets can be lost, spiraling into the
central star. On the other hand, Tanaka & Ida (1999)
showed the possibility that the inward migration of a
protoplanet accelerates the growth of the protoplanet
since planetesimals with low random velocity are supplied
as the protoplanet migrates. The migration speed of protoplanets is, however, still uncertain at present (Tanaka,
Takeuchi, & Ward 2002). Another gas disk–planet interaction, the damping of e and i of protoplanets, does not
affect the oligarchic growth. It strengthens the effects of
dynamical friction from small planetesimals so that the
isolation of protoplanets is reinforced. Small planetesimals also migrate because of gas drag. In particular, fragmentation of planetesimals at collisions and subsequent
migration of fragments due to gas drag may change the
initial spatial distribution of planetesimals. In fact, the
migration of small planetesimals reduces the available
disk mass for planetary accretion and thus leads to
smaller masses for protoplanets than in the migrationless
case as shown by Thommes et al. (2002b).
It is possible to extend the oligarchic growth model to
take into account the migration processes of protoplanets
and planetesimals. Before integrating the effects of the
migration into the oligarchic growth model, however, we
need to clarify each process in more detail. We propose the
present discussion on the diversity of planetary systems as a
piece of framework to consider possible planetary systems
in the universe.
APPENDIX
DIFFUSION OF A PLANETESIMAL DISK BY SELF-GRAVITY
In a planetesimal disk, the random velocity of planetesimals is given by v ’ he2 i1=2 vK . From the conservation of the Jacobi
energy, we have
db2 4 d½ðeaÞ2 þ ðiaÞ2 4 a2 ðe2 þ i2 Þ
’
’
;
3
dt
3
TVS
dt
ðA1Þ
where b is the orbital separation of planetesimals and TVS is the viscous stirring (relaxation) timescale due to planetesimalplanetesimal gravitational interactions given by
v3
;
TVS ’ pffiffiffi
2G2 nm m2 ln ðA2Þ
where nm is the surface number density of planetesimals (e.g., Ida 1990). Using equation (A2), we obtain the diffusion
timescale
Tdiff ðDaÞ2
db2 =dt
’ 1:3 1010
Da
1 AU
!2
he2 i1=2
0:01
!2
m
1023 g
!1
solid
10 g cm2
!1
a
1 AU
!5=2
where we used nm ’ solid =ð2hi2 i1=2 amÞ, he2 i1=2 ¼ 2hi2 i1=2 , and ln ¼ 3 (Stewart & Ida 2000).
M
M
!3=2
yr ;
ðA3Þ
680
KOKUBO & IDA
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