A
The Astrophysical Journal, 642:1131–1139, 2006 May 10
# 2006. The American Astronomical Society. All rights reserved. Printed in U.S.A.
FORMATION OF TERRESTRIAL PLANETS FROM PROTOPLANETS. I.
STATISTICS OF BASIC DYNAMICAL PROPERTIES
Eiichiro Kokubo
Division of Theoretical Astronomy, National Astronomical Observatory of Japan, Osawa, Mitaka,
Tokyo 181-8588, Japan; [email protected]
Junko Kominami
Department of Astronomy, University of Tokyo, Hongo, Bunkyo-Ku, Tokyo 113-0033, Japan
and
Shigeru Ida
Department of Earth and Planetary Sciences, Tokyo Institute of Technology, Ookayama, Meguro-Ku, Tokyo 152-8551, Japan
Received 2005 November 12; accepted 2006 January 13
ABSTRACT
The final stage of terrestrial planet formation is known as the giant impact stage, where protoplanets collide with
one another to form planets. As this process is stochastic, in order to clarify it, it is necessary to quantify it statistically.
We investigate this final assemblage of terrestrial planets from protoplanets using N-body simulations. As initial
conditions, we adopt the oligarchic growth model of protoplanets. We systematically change the surface density,
surface density profile, and orbital separation of the initial protoplanet system, and the bulk density of protoplanets,
while the initial system radial range is fixed at 0.5–1.5 AU. For each initial condition, we perform 20 runs, and from
their results we derive the statistical properties of the assembled planets. For the standard disk model, typically two
Earth-sized planets form in the terrestrial planet region. We show the dependences of the masses and orbital elements
of planets on the initial protoplanet system parameters and give their simple empirical fits. The number of planets
slowly decreases as the surface density of the initial protoplanets increases, while the masses of individual planets increase almost linearly. For a steeper surface density profile, large planets tend to form closer to the star. For the parameter ranges that we test, the basic structure of planetary systems depends only slightly on the initial distribution of
protoplanets and the bulk density as long as the total mass is fixed.
Subject headings: methods: n-body simulations — planets and satellites: formation
Online material: color figures
1. INTRODUCTION
disk–planet interaction (e.g., Ward 1993) and/or sweeping secular
resonance due to dispersal of the gas disk (e.g., Heppenheimer
1980). It is necessary to study statistical properties of assembled
planets to understand the basic dynamics of terrestrial planet formation, as the final assemblage process is very stochastic in nature. Unfortunately, the numerical investigation performed so far
does not incorporate a statistical approach but rather focuses
on case studies, since the simulation cost is very high. In order to
clarify the final stage quantitatively, it is necessary to perform
many simulations and derive the statistical properties of resulting
planets.
Recent observations of protoplanetary disks have revealed
that protoplanetary disks are diverse in mass and shape, with total masses ranging from 103 to 101 M ( Beckwith & Sargent
1996). To generalize the formation theory of terrestrial planet
formation, we need to study terrestrial planet formation from various protoplanetary disks. Wetherill (1996) first explored the effect
of the surface density of protoplanetary disks on terrestrial planet
formation, with surface densities of 0.5 and 1.5 times the standard
value, using a Monte Carlo method. Recently, Raymond et al.
(2005a) studied qualitatively the effect of the radial surface density profile on the properties of assembled planets.
Kokubo & Ida (2002) investigated the formation of protoplanets from various planetesimal disks. They varied the disk mass
and profile and investigated the properties of resulting protoplanets, confirming that the mass and growth timescale are consistent with the oligarchic growth model. Using these results, they
discussed the habitat segregation of planets, which describes the
The standard scenario of terrestrial planet formation consists
of three stages: (1) dust to planetesimals, (2) planetesimals to
protoplanets, and (3) protoplanets to planets. The first stage is the
formation of planetesimals from dust via gravitational instability
of the dust layer (e.g., Goldreich & Ward 1973) or binary coagulation of dust grains (e.g., Weidenschilling & Cuzzi 1993). During the second stage, planetesimals grow by collisions. The initial
growth mode is runaway growth, where larger planetesimals grow
faster than smaller ones (e.g., Wetherill & Stewart 1989; Kokubo
& Ida 1996). When the mass of protoplanets (runaway planetesimals) exceeds the critical mass, the growth mode shifts to oligarchic growth (Kokubo & Ida 1998, 2000). At the oligarchic growth
stage, protoplanets grow in an orderly mode while maintaining
orbital separation by orbital repulsion. As a result, at the end of the
second stage, protoplanets are formed with orbital separations
proportional to their Hill radii. The final stage is the assemblage of
planets from protoplanets, which is known as the giant impact
stage (e.g., Wetherill 1985).
There are several studies on the final stage of terrestrial planet
formation using N-body simulations (e.g., Chambers & Wetherill
1998; Agnor et al. 1999; Kominami & Ida 2002; Raymond et al.
2004; Nagasawa et al. 2005). Most of these are on terrestrial planet
formation in the solar system; in other words, the initial conditions
are similar to the minimum-mass disk model (Hayashi 1981).
Some included giant planets outside the terrestrial planet region,
while others took into account the gravitational drag due to gas
1131
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KOKUBO, KOMINAMI, & IDA
regions for terrestrial, Jovian, and Uranian planets as a function of
disk parameters. Note that they discussed only the region in which
terrestrial planets form and not the number and physical and orbital properties of planets. The number of planets and their dynamical properties in the region remain open questions.
The goal of this paper is to clarify the statistical properties of
planets assembled by giant impacts among protoplanets. We
investigate the final stage from the protoplanet system formed by
oligarchic growth from systematically different planetesimal disks,
using N-body simulations. We derive the statistical properties of
the number, mass, and orbital elements of terrestrial planets from
results of 20 runs for each initial condition.
In x 2 we outline the initial conditions of protoplanets and the
numerical method. Our results are presented in x 3, where we
show the statistical properties of assembled planets and their dependences on the initial conditions of protoplanets. Section 4 is
devoted to a summary and discussion.
2. METHOD OF CALCULATION
We perform global N-body simulations of terrestrial planet
formation starting from various protoplanet systems. As the first
step, we consider gas-free cases without giant planets outside the
terrestrial planet region to clarify the basic dynamics.
2.1. Initial Conditions
We adopt protoplanet systems formed by oligarchic growth
from planetesimal disks whose surface density distributions are
given by
a g cm2 ;
ð1Þ
¼ 1
1 AU
where 1 is the reference surface density at 1 AU and is the
power-law index of the radial profile, with inner and outer cutoffs ain and aout. The standard disk model for the solar system
formation corresponds to a disk with 1 ’ 10 and ¼ 3/2.
The oligarchic growth model assumes that the orbital separation of adjacent protoplanets, b, is proportional to their Hill radius given by r H ¼ (2M /3M )1/3 a, where M and a are the mass
and semimajor axis of the protoplanets and M is the mass of the
central star. It is also assumed that the accretion efficiency is
100%. In other words, we assume that all planetesimals finally
turn into protoplanets in situ. Under these assumptions, the isolation mass of protoplanets is given by
Miso
b̃
’ 2ab ¼ 0:16
10
3=2 1
10
3=2 a (3=2)(2)
M ;
1 AU
ð2Þ
where b̃ is the orbital separation scaled by the Hill radius b̃ ¼
b/r H and M is Earth’s mass ( Kokubo & Ida 2002). Note that
the isolation mass increases with a for < 2, while it decreases
with increasing a for > 2.
We systematically change four system parameters: the reference surface density 1, disk profile , orbital separation b̃, and
bulk density of protoplanets, to investigate the dependences of
properties of assembled planets on these parameters. Because the
observationally inferred disk mass ranges from 0.1 to 10 times
the minimum-mass disk model, we consider disks with 1 ¼ 3,
10, 30, and 100. We also consider disks with flatter ( ¼ 1/2) and
steeper ( ¼ 5/2) radial surface density profiles, since the real
profile is unclear. These models may encompass the likely range
of actual values. We fix the disk range at ain ¼ 0:5 AU and aout ¼
Vol. 642
TABLE 1
Initial Conditions of Protoplanets
Model
1
b̃
(g cm3)
n
Mtot
(M)
j
( j )
ā
(AU )
1................
2................
3................
4................
5................
6................
7................
8................
9................
10..............
10
10
10
10
3
30
100
10
10
10
3/2
3/2
3/2
3/2
3/2
3/2
3/2
1/2
5/2
3/2
10
6
8
12
10
10
10
10
10
10
3.0
3.0
3.0
3.0
3.0
3.0
3.0
3.0
3.0
5.5
16
35
23
12
30
9
5
18
15
16
2.30
2.25
2.40
2.26
0.72
6.66
22.81
2.20
2.77
2.30
0.95
0.96
0.96
0.95
0.96
0.94
0.95
1.00
0.91
0.95
0.91
0.92
0.93
0.90
0.92
0.89
0.90
1.00
0.83
0.91
Note.—The unit of j is j ¼
Earth.
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
GM a , where a is the semimajor axis of
1:5 AU. The effect of disk width will be investigated in a separate
paper. We set M ¼ M .
The typical orbital separation of protoplanets is b̃ ’ 10 in
N-body simulations (e.g., Kokubo & Ida 2002). We test protoplanet systems with b̃ ¼ 6, 8, 10, and 12. In all protoplanet systems, the initial eccentricities e and inclinations i of protoplanets
are given by the Rayleigh distribution with dispersions he2 i1/2 ¼
2hi2 i1/2 ¼ 0:01(1 /10)1/2 (the units of i are radians). These dispersions are proportional to the reduced Hill radius of protoplanets, r H /a, with the isolation mass. The initial values of e and i
do not affect the results presented hereafter, since once the giant
impact stage begins, they are quickly raised due to scattering
among protoplanets. In most models, we set the bulk density of
protoplanets at ¼ 3 g cm3. This is set to the value of Earth,
5.5 g cm3, in one model, for comparison.
The initial conditions of protoplanet systems (1, , b̃, )
are summarized in Table 1 with their global system properties:
number of protoplanets n, total mass Mtot, specific angular momentum j, and mean semimajor axis ā ¼ j2 /(GM ). We refer to
the protoplanet system with 1 ¼ 10, ¼ 3/2, b̃ ¼ 10, and ¼
3:0 g cm3 (model 1) as the standard model, which corresponds
to the protoplanet system formed from a disk that is 50% more
massive than the minimum-mass disk. For each model, we perform 20 runs with different initial angular distributions of protoplanets. The angular distribution does not change the global
properties of protoplanet systems.
2.2. Orbital Integration
The orbits of protoplanets are calculated by numerically integrating the equation of motion of protoplanets,
N
X
xi xj
dvi
xi
¼ GM
GMj
;
3
dt
jxi j
jxi xj j3
j6¼i
ð3Þ
where x and v are the position and velocity of protoplanets, respectively. The first and second terms of the right-hand side of
equation (3) represent the gravity of the central star and the mutual gravitational interaction of protoplanets. For numerical integration, we use the modified Hermite scheme for planetary
N-body simulation ( Kokubo et al. 1998; Kokubo & Makino
2004) with the hierarchical time step ( Makino 1991). The simulations follow the evolution of protoplanet systems for 3 ; 108 yr
(1 ¼ 3 model), 2 ; 108 yr (1 ¼ 10 model), and 108 yr
(1 ¼ 30 and 100 models) until only a few planets remain. The
No. 2, 2006
FORMATION OF TERRESTRIAL PLANETS. I.
1133
TABLE 2
Numbers of Final Planets, Accretion Efficiency, and Timescale
1.............
2.............
3.............
4.............
5.............
6.............
7.............
8.............
9.............
10...........
hnM i
hni
Model
3.4
4.4
3.5
3.0
3.9
3.0
2.3
3.7
3.5
3.4
0.6
0.9
0.8
0.2
0.6
0.4
0.6
0.8
0.7
0.8
0.6
0.4
0.4
0.7
0.0
3.0 0.4
2.4 0.6
2.2 0.5
2.2 0.6
2.0 0.4
2.0
2.1
2.0
2.1
hna i
1.8
2.1
2.0
1.9
2.6
1.9
1.5
2.2
1.8
1.8
hTacc i
(108 yr)
h fa i
0.7
0.5
0.6
0.8
0.7
0.7
0.8
0.7
0.6
0.6
0.79
0.77
0.81
0.82
0.83
0.81
0.78
0.80
0.73
0.73
0.15
0.12
0.12
0.14
0.13
0.13
0.23
0.14
0.18
0.20
1.05
1.00
0.87
0.63
1.65
0.14
0.11
0.52
0.95
0.91
0.58
0.41
0.40
0.49
0.45
0.16
0.15
0.39
0.56
0.55
numerical integration is carried out on a dedicated Opteron/Athlon
cluster.
During the orbital integration, when two protoplanets contact,
we assume for simplicity’s sake that they always accrete. In accretion, the position and velocity of the center of mass are conserved.
Agnor & Asphaug (2004) estimated that more than half of all collisions between like-sized protoplanets do not result in accumulation into a larger protoplanet at the giant impact stage, and this
inefficiency lengthens the timescale of planet formation by a factor
of 2 or more, relative to perfect accretion. Note that the present
simulations correspond to the limiting case of efficient accretion.
Fig. 1.—Time evolution of the semimajor axes (solid lines) and pericenter and
apocenter distances (dotted lines) of planets for an example run of the standard model
(model 1). [See the electronic edition of the Journal for a color version of this figure.]
3.1. Standard Model
evolution of the semimajor axes and pericenter and apocenter
distances of planets for 108 yr. The snapshots of the system at
t ¼ 0, 106, 107, 108, and 2 ; 108 yr on the a-e and a-i planes are
shown in Figure 2. The number of planets in each panel is n ¼
16, 8, 5, 3, and 3 from t ¼ 0 to 2 ; 108 yr, respectively. As the
initial e and i of the protoplanets are relatively large, the protoplanet system quickly becomes unstable, and the giant impact
stage starts within a timescale of 105 yr. The giant impact stage
usually lasts for about 108 yr.
The numbers of the final planets are n ¼ 3 and nM ¼ na ¼ 2,
which is a typical outcome of the standard model. They are two
Earth-sized planets of M ¼ 1:1 M with a ¼ 0:59 AU and 1.0 M
with a ¼ 1:11 AU, and a small planet of M ¼ 0:15 M with a ¼
1:83 AU. For this system, fa ¼ 0:93 and Tacc ¼ 0:66 ; 108 yr. The
eccentricities and inclinations of the two Earth-sized planets are
e; i ¼ 0:04–0.05, while those of the small planet are e ’ 0:1 and
i ’ 0:04.
3.1.1. Example Run
3.1.2. Statistical Properties
The typical evolution of the standard protoplanet system
(model 1) is shown in Figures 1 and 2. Figure 1 shows the time
From the results of 20 runs, we find that the number of final
planets is hni ’ 3:4 0:6, while the number of Earth-sized
3. RESULTS
We first show the details of the standard model and describe
the processes common to all models. We then statistically investigate the dependences of the properties of assembled planets on
parameters of the protoplanet system. In this work, we focus on
the number of planets, n, the number of Earth-sized planets whose
mass is M > M /2, nM, and the number of planets in ain a aout , na. The in situ accretion efficiency is calculated as fa ¼
Ma /Mtot , where Ma is the total mass of planets in ain a aout .
We calculate the accretion timescale Tacc defined as the duration of
collisions (the last collision time minus the first collision time). We
also obtain the mass and orbital characteristics of the largest and
second-largest planets. Their average values with 1 error are
summarized in Tables 2 and 3.
TABLE 3
Mass and Orbital Elements of Final Planets
Model
1........................
2........................
3........................
4........................
5........................
6........................
7........................
8........................
9........................
10......................
hM1 i
(M)
1.27
1.12
1.23
1.26
0.30
3.80
14.6
1.08
1.53
1.18
0.25
0.18
0.29
0.24
0.05
0.55
3.2
0.18
0.30
0.22
ha1 i
(AU)
0.75
0.73
0.73
0.80
0.88
0.79
0.70
0.91
0.66
0.67
0.20
0.21
0.24
0.21
0.26
0.20
0.22
0.25
0.18
0.19
he1 i
0.11
0.10
0.10
0.12
0.08
0.10
0.18
0.10
0.10
0.12
0.07
0.05
0.07
0.05
0.05
0.08
0.16
0.05
0.07
0.06
hi1 i
0.06
0.07
0.06
0.06
0.04
0.07
0.09
0.05
0.05
0.09
0.04
0.03
0.03
0.04
0.02
0.10
0.08
0.03
0.04
0.06
hM2 i
(M)
0.66
0.76
0.76
0.61
0.22
1.83
6.63
0.69
0.79
0.73
0.23
0.15
0.19
0.15
0.05
0.47
2.50
0.13
0.18
0.20
ha2 i
(AU )
1.12
0.84
1.11
0.90
0.80
1.18
2.31
1.02
0.96
1.08
0.53
0.34
0.40
0.45
0.32
0.53
2.20
0.43
0.40
0.33
he2 i
0.12
0.14
0.11
0.13
0.10
0.20
0.20
0.12
0.13
0.14
0.05
0.07
0.07
0.07
0.06
0.10
0.15
0.06
0.07
0.10
hi2 i
0.10
0.08
0.08
0.10
0.05
0.12
0.14
0.07
0.09
0.11
0.08
0.04
0.04
0.06
0.03
0.10
0.12
0.04
0.04
0.06
1134
KOKUBO, KOMINAMI, & IDA
Vol. 642
Fig. 2.—Snapshots of the system on the a-e (left) and a-i (right) planes at t ¼ 0, 106, 107, 108, and 2 ; 108 yr for the same run as in Fig. 1. The sizes of the circles
are proportional to the physical sizes of the planets.
planets is hnM i ’ 2:0 0:6, which means that the typical resulting system consists of two Earth-sized planets and a smaller
planet. In this model, we obtain hna i ’ 1:8 0:7. In other words,
one or two planets tend to form outside the initial distribution of
protoplanets. In most runs, these planets are smaller scattered
planets. Thus we obtain a high efficiency of h fa i ¼ 0:79 0:15.
The accretion timescale is hTacc i ¼ ð1:05 0:58Þ ; 108 yr. These
results are consistent with Agnor et al. (1999), whose initial conditions are the same as the standard model except for 1 ¼ 8.
The left and right panels of Figure 3 show the final planets on
the a-M and M–e, i planes for 20 runs. The largest planets tend to
cluster around a ¼ 0:8 AU, while the second-largest avoid the
same semimajor axis as the largest, shown as the gap around a ¼
ha1 i. Most of these are more massive than M /2. The mass of the
largest planet is hM1 i ’ 1:27 0:25 M, and its orbital elements
are ha1 i ’ 0:75 0:20 AU, he1 i ’ 0:11 0:07, and hi1 i ’
0:06 0:04. On the other hand, the second-largest planet has
hM2 i ’ 0:66 0:23 M, ha2 i ’ 1:12 0:53 AU, he2 i ’ 0:12 0:05, and hi2 i ’ 0:10 0:08. The dispersion of a2 is large, since
in some runs, the second-largest planet forms inside the largest
one, while in others it forms outside the largest. In this model, we
find a1 > a2 in seven runs.
Fig. 3.—All planets on the a-M (left) and M–e, i (right) planes formed in the 20 runs of the standard model (model 1). The symbols indicate the planets first
(circles), second (hexagons), third (squares), and fourth (triangles) highest in mass. The filled symbols are the final planets, and the open circles are the initial
protoplanets in the left panel. The filled and open symbols mean e and i in the right panel, respectively. [See the electronic edition of the Journal for a color version
of this figure.]
No. 2, 2006
FORMATION OF TERRESTRIAL PLANETS. I.
1135
Fig. 5.—Average numbers hni ( filled circles) and hna i (open circles) against
1 for models with 1 ¼ 3, 10, 30, and 100 (models 5, 1, 6, and 7), and the empirical fits for hni (solid line) and hna i (dotted line). [See the electronic edition of
the Journal for a color version of this figure.]
Fig. 4.—Average semimajor axes and masses of the largest ( filled symbols)
and second-largest (open symbols) planets for models with b̃ ¼ 6 (triangles),
8 (squares), 10 (circles), and 12 (hexagons; models 2, 3, 1, and 4) and ¼ 5:5 g
cm3 (pentagons; model 10). The error bars indicate 1 dispersion. [See the
electronic edition of the Journal for a color version of this figure.]
The semimajor axes of the third and fourth largest planets are
widely scattered over the initial protoplanet distribution, especially beyond aout. More planets end up outside aout than inside
ain, because for the same energy change, scattering outward is
easier than scattering inward, since the Kepler potential is steeper
for the inner part. The innermost and outermost planets have
a min ’ 0:4 AU and amax ’ 2:9 AU, respectively. In most runs,
the fourth planets are surviving protoplanets that experience no
collisions.
There are tendencies in the distributions of e and i for small
planets to have large e and i, as seen in the right panel of Figure 3.
These negative dependences of e and i on M would be the effect
of collisional damping and/or dynamical friction.
3.2. Dependences on System Parameters
We summarize how the statistical properties of assembled
planets are affected by the initial parameters of protoplanet
systems. We examine the effects of the initial orbital separation
of protoplanets, reference surface density, radial surface density
profile, bulk density, and total mass of protoplanets.
3.2.1. Orbital Separation
We vary the initial orbital separation from b̃ ¼ 6 to 12, keeping 1 and at the standard values (models 2, 3, 1, and 4). The
number of protoplanets decreases from n ¼ 35 with b̃ ¼ 6 to 12
with b̃ ¼ 12. The total mass Mtot ’ 2:3 M, the specific angular
momentum j ’ 0:95 j, and the mean semimajor axis ā ’
0:91 AU of the protoplanet system are almost the same among
these models. The small difference is due to the discrete number
of protoplanets.
We find that the initial orbital separation of protoplanets does
not substantially change the properties of the assembled planets
as long as b̃ ¼ 6–12. As seen in Tables 2 and 3 and Figure 4,
hnM i, hna i, h fa i, hTacc i, and the mass and orbital elements of the
largest and second-largest planets for the b̃ ¼ 6, 8, and 12 models are almost equivalent to those for the standard b̃ ¼ 10 model.
Only the number of planets hni, in other words, the number of
small scattered planets, is slightly larger for smaller b̃. This is because the probability of forming scattered planets increases with
the initial number of protoplanets. The accretion timescale has
no clear dependence on b̃. It is about 108 yr for all models.
This independence of the properties of the assembled planets
from the initial orbital separation suggests that it is not detailed
differences in protoplanet distribution but global disk properties
that control the final assemblage of planets.
3.2.2. Surface Density
The reference surface density at 1 AU is varied from 1 ¼ 3
to 100 (models 5, 1, 6, and 7). For the same and b̃, the total
mass of the initial protoplanets increases linearly from Mtot ¼
0:72 to 22.81 M, while their number decreases from n ¼ 30 to
(Kokubo & Ida 2002). In these
5, which is proportional to 1/2
1
models, j and ā are kept constant.
We find that the numbers of planets hni and hna i gradually decrease with increasing 1, as shown in Figure 5. For 1 ¼ 3–
100, we have hni ’ 3:9–2.4 and hna i ’ 2:6–1.5. In these models, we have the difference hni hna i ’ 1, which means that one
planet always forms outside the region of initial protoplanet
distribution. By the least-squares fit method, the dependences of
the numbers on 1 are approximated as
0:15
1
hni ’ 3:3
;
ð4Þ
10
0:14
1
hna i ’ 2:1
;
ð5Þ
10
which are also shown in Figure 5. These dependences on 1 are
weaker than that in the oligarchic growth model of protoplanets,
. We also find that Earth-sized planets do not form for
n / 1/2
1
1 ¼ 3. This means that there exists a minimum 1 for formation of Earth-sized planets, which we discuss later. For 1 ¼ 30
and 100, hnM i ¼ hni, since the mass of the initial protoplanets is
larger than M/2 in these models. Note that we find escapers in
two runs of the 1 ¼ 100 model. These are original protoplanets ejected due to strong scattering by large planets. In both
runs, the ejection of one planet results in formation of a stable
two-planet system. At the giant impact stage, the random velocity of planets reaches their surface escape velocity vesc (e.g.,
Safronov 1969). In the 1 ¼ 100 model, hMi1 ’ 15 M, for
which vesc can be as large as the circular velocity. In this case,
strong scattering often results in ejection.
1136
KOKUBO, KOMINAMI, & IDA
Fig. 6.—Same as Fig. 4, but for models with 1 ¼ 3 (triangles), 10 (circles),
30 (squares), and 100 (hexagons; models 5, 1, 6, and 7). [See the electronic
edition of the Journal for a color version of this figure.]
The accretion timescale decreases as 1 increases. It shortens
from 108 to 107 yr between 1 ¼ 10 and 30. This is because
for larger 1, the number of initial protoplanets is smaller, and
their masses are larger, and thus gravitational focusing is more
effective.
The average final planets for 1 ¼ 3–100 are shown on the
a-M plane in Figure 6. We find that the semimajor axes of
the largest planets are ha1 i ’ 0:8 AU for all 1 models, while the
dispersion of a2 increases with 1, which reflects strong scattering due to large planets. This independence of ha1 i from 1 suggests that it is the specific angular momentum of the system, j,
that determines the formation site of the first planet. Also due to
the same effect, he1 i, hi1 i, he2 i, and hi2 i increase with 1.
Figure 7 shows the masses of the largest and second-largest
planets against 1 and their empirical fits. Contrary to hni and
hna i, hM1 i and hM2 i increase with 1. This reflects the fact that
the number of planets in stable orbits in a certain region decreases as their masses increase, due to the self-excitation of eccentricities by mutual gravitational interaction. The dependences
of hM1 i and hM2 i on 1 are almost linear. We obtain the fitting
formulas for hM1 i and hM2 i as
1:1
1
hM1 i ’ 1:2
M ;
ð6Þ
10
0:97
1
hM2 i ’ 0:68
M :
ð7Þ
10
These dependences are also weaker than that in the oligarchic
growth model of protoplanets, M / 3/2
1 . Despite the wide range
of 1, the efficiency h fa i remains almost constant because of these
almost linear dependences. The result that larger 1 forms fewer
planets with larger mass is qualitatively consistent with Wetherill
(1996) and Raymond et al. (2005b).
The almost linear mass dependence on 1 means that in these
models, the feeding zone a of a planet is not proportional to its
Hill radius or escape velocity (/M 1/3), with which we have M /
3/2
1 , but almost constant (independent of M). Actually, we find
Vol. 642
Fig. 7.—Average masses of the largest hM1 i ( filled circles) and second-largest
hM2 i (open circles) planets against 1 for models with 1 ¼ 3, 10, 30, and 100
(models 5, 1, 6, and 7), and the empirical fits for hM1 i (solid line) and hM2 i (dotted
line). [See the electronic edition of the Journal for a color version of this figure.]
that for the largest planets ha1 i ’ 0:5–0.6 AU for all 1 models, whose outer edge is as large as aout, since ha1 i ’ 0:8 AU.
The feeding zone here is defined as the initial radial extent of the
protoplanets that finally form a specific planet. This wide feeding
zone is a result of the radial diffusion due to scattering and collisions among planets. As the accretion takes place globally in
the giant impact stage, the mass of planets may change with the
radial extent of the initial protoplanet system. We will investigate
the mass dependence on 1 and the initial system size of protoplanets in detail in a separate paper.
Fig. 8.—Same as Fig. 4, but for the models with ¼ 1/2 (triangles), 3/2
(circles), and 5/2 (squares; models 8, 1, and 9). [See the electronic edition of the
Journal for a color version of this figure.]
No. 2, 2006
FORMATION OF TERRESTRIAL PLANETS. I.
1137
Fig. 10.—Average numbers hni ( filled symbols) and hna i (open symbols)
against Mtot for the standard model (circles; model 1) and models with 1 ¼ 3, 30,
and 100 (squares; models 5, 6, and 7) and ¼ 1/2 and 5/2 (triangles; models 8 and
9), and the empirical fits for hni (solid line) and hna i (dotted line). [See the electronic edition of the Journal for a color version of this figure.]
Fig. 9.—Average semimajor axis of the largest planet ha1 i against ā for
models with ¼ 1/2 (triangle), 3/2 (circle), and 5/2 (square; models 8, 1, and
9), and the empirical fit (solid line). [See the electronic edition of the Journal for
a color version of this figure.]
3.2.3. Radial Density Profile
We consider the effect of the radial surface density profile of
¼ 1/2 5/2 on the properties of assembled planets (models 8,
1, and 9). For ¼ 1/2 5/2, we have n ¼ 18–15 and Mtot ¼
2:20–2:77 M. In addition, j and ā decrease with increasing ,
which have been kept almost constant so far.
It is found that although the disk profile is varied in the range
¼ 1/2 5/2, the numbers of assembled planets, hni, hnM i, and
hna i, the accretion efficiency h fa i, and the accretion timescale
hTacc i are not significantly affected. Figure 8 shows the largest
and second-largest planets on the a-M plane for ¼ 1/2, 3/2,
and 5/2. It is clearly shown that the semimajor axis of the largest
planet ha1 i decreases as increases, while its mass hM1 i increases
with . The semimajor axis shifts inward with , because with
large , the mass is concentrated in the inner part of the system,
and thus larger planets tend to form there. In other words, ha1 i increases with j and ā. In Figure 9, ha1 i is plotted against ā with its
empirical fit. We find that ha1 i relatively strongly depends on ā,
while ha2 i barely depends on ā. By the least-squares fit method,
we obtain the fitting formula
1:7
ā
AU:
ð8Þ
ha1 i ’ 0:90
1 AU
The slightly smaller h fa i for ¼ 5/2 is a result of this effect. In a
few runs of the ¼ 5/2 model, a1 or a2 is inside ain, which leads
to small fa. It is also found that the eccentricities and inclinations
of the planets are almost equal in all models. The masses of
the planets hM1 i and hM2 i increase with , since the total mass
of the initial protoplanet system increases with for the same 1.
The effect of the total mass of protoplanets is discussed below.
These results are consistent with Raymond et al. (2005a) in
the sense that for steeper profiles, more massive planets form,
and their positions are closer to the star.
3.2.4. Bulk Density
The effect of the bulk density of protoplanets is tested by comparing results of models with ¼ 3:0 and 5.5 g cm3 (models 1
and 10). In the ¼ 5:5 g cm3 model, the radii of planets are 18%
smaller than those in the ¼ 3:0 g cm3 model. In these models,
the global properties of the initial protoplanet systems are identical.
We find that all properties of the assembled planets are barely
affected by , as seen in Tables 2 and 3 and Figure 4. This result
justifies the use of a constant bulk density during accretion,
which in reality may increase with the mass of planets.
This independence of the results from the bulk density holds
as long as the collision timescale is larger than the timescale
of gravitational relaxation, which is common in the giant impact
stage. As the collision timescale increases with , for an extremely
small , the situation changes. If is small enough to make the
collision timescale smaller than the timescale of gravitational relaxation, the initial conditions of protoplanets do not relax during
accretion and affect the assemblage of planets.
3.2.5. Total Mass
From the dependences on b̃, 1, and , we find that an important parameter of the initial protoplanet system for the number and mass of planets is the total mass of protoplanets, Mtot.
The effects of 1 and on the numbers and masses of planets are
unified using Mtot.
Figure 10 shows hni and hna i against Mtot for all 1 and models and their empirical fits. Both of them decrease with increasing Mtot, although their dependences are weak. By the leastsquares fit method, we obtain their fitting formulas as
Mtot 0:15
;
2 M
Mtot 0:14
hna i ’ 2:1
;
2 M
hni ’ 3:5
ð9Þ
ð10Þ
whose dependences on Mtot are the same as those on 1.
In Figure 11, hM1 i and hM2 i are plotted against Mtot for all 1
and models with their empirical fits. It is clearly shown that
both hM1 i and hM2 i increase almost linearly with Mtot. By the
least-squares fit method, the fitting formulas are given by
Mtot 1:1
M ;
ð11Þ
hM1 i ’ 1:0
2 M
Mtot 0:98
hM2 i ’ 0:60
M ;
ð12Þ
2 M
1138
KOKUBO, KOMINAMI, & IDA
Vol. 642
Fig. 11.—Average masses of the largest hM1 i ( filled symbols) and secondlargest hM2 i (open symbols) planets against Mtot for the standard model (circles; model 1) and models with 1 ¼ 3, 30, and 100 (squares; models 5, 6, and
7) and ¼ 1/2 and 5/2 (triangles; models 8 and 9), and the empirical fits for
hM1 i (solid line) and hM2 i (dotted line). [See the electronic edition of the
Journal for a color version of this figure.]
Fig. 12.—Same as left panel of Fig. 3, but for the model with 1 ¼ 9, ¼ 1/2,
b̃ ¼ 10, and ¼ 3:0 g cm3. The average values with 1 errors are plotted for the
largest and second-largest planets. The large open symbols indicate the terrestrial
planets in the solar system. [See the electronic edition of the Journal for a color
version of this figure.]
which can be reduced to the more simple forms hM1 i ’ 0:5Mtot
and hM2 i ’ 0:3Mtot , respectively.
5. The bulk density of protoplanets does not affect the numbers and properties of assembled planets for ¼ 3:0–5.5 g cm3.
6. The total mass of protoplanets Mtot is an important parameter for the number and mass of planets. The numbers hni and
hna i have weak negative dependence on Mtot, while the masses
hM1 i and hM2 i increase with Mtot almost linearly.
4. SUMMARY AND DISCUSSION
We have investigated the basic dynamical properties of the
terrestrial planets assembled from protoplanets through giant
impacts. The radial range of the initial protoplanet distribution
has been fixed at ain ¼ 0:5 AU and aout ¼ 1:5 AU. The number
of planets n, the number of Earth-sized planets nM, the number of
planets in between ain and aout, na, the accretion efficiency fa, the
accretion timescale Tacc, and the masses and orbital elements of
planets have been studied statistically with a number of N-body
simulations. The obtained statistical properties are the following:
1. The numbers of planets formed from the standard protoplanet system (1 ¼ 10, ¼ 3/2, b̃ ¼ 10, and ¼ 3:0 g cm3)
are hni ’ 3 and hnM i ’ hna i ’ 2. The na planets contain about
80% of the system mass on average. The accretion timescale is
about 108 yr. The masses of the largest and second-largest
planets are hM1 i ’ 1:3 M and hM2 i ’ 0:7 M. The largest
planets tend to form around ha1 i ’ 0:8 AU, while a2 is widely
scattered in the initial protoplanet region. Their eccentricities
and inclinations are 0.1. A small planet typically forms at a <
ain or a > aout with larger e and i.
2. The initial orbital separation of protoplanets b̃ barely affects the final assemblage as long as b̃ ¼ 6–12.
3. The numbers hni and hna i slowly decrease as the reference
surface density at 1 AU, 1, increases, while the masses hM1 i
and hM2 i increase with 1 almost linearly for 1 ¼ 3–100. The
semimajor axis ha1 i only slightly depends on 1, while the eccentricity and inclination increase with 1. The accretion timescale decreases as 1 increases.
4. The numbers of planets and their masses and orbital elements are almost independent of the radial surface density profile
for ¼ 1/2 5/2, except hM1 i and ha1 i. The mass hM1 i increases with , while its semimajor axis ha1 i decreases with
increasing .
The results of the models with Mtot ’ 2 M are consistent
with the present solar system in the sense that two Earth-sized
planets (Earth and Venus analogs) form in 0:5 AU P a P 1:5 AU
and the mass of the second-largest planet is 50%–80% of that of
the largest one. However, their eccentricities and inclinations are
an order of magnitude larger than the proper eccentricities and inclinations of the present terrestrial planets. To achieve the small
eccentricities and inclinations, some damping mechanism after
the giant impact stage is necessary. The possible mechanisms include damping by gravitational drag (dynamical friction) from a
dissipating gas disk (Kominami & Ida 2002) and the residual
planetesimal disk (Agnor et al. 1999). It should be noted that in
most runs of Mtot ’ 2 M models, a1 and/or a2 fall into the habitable zone for a G-type main-sequence star, 0:7 AU P a P 1:5 AU
(Kasting et al. 1993).
The position of the largest planet is a decreasing function of or ā. To have the largest planet (Earth analog) around 1 AU from
a protoplanet system with ain ¼ 0:5 AU and aout ¼ 1:5 AU, a
shallower profile is preferable. In the range of the initial conditions that we explored in the present paper, one of the best initial
conditions for protoplanets to reproduce the solar system analog
would be the protoplanet system with Mtot ’ 2 M (1 ’ 9) and
’ 1/2. Figure 12 shows the final planets and the average
values for the largest and second-largest planets on the a-M plane
for 20 runs of this model, together with the terrestrial planets in
the solar system. The numbers of planets are hni ¼ 3:8 0:6,
hnM i ¼ 1:9 0:5, and hna i ¼ 2:5 0:6, and h fa i ¼ 0:85 0:11 and hTacc i ¼ (0:87 0:54) ; 108 yr. The masses and orbital
elements of the largest and second-largest planets are hM1 i ¼
0:92 0:15 M, ha1 i ¼ 0:92 0:27 AU, he1 i ¼ 0:10 0:05,
No. 2, 2006
FORMATION OF TERRESTRIAL PLANETS. I.
and hi1 i ¼ 0:04 0:03; and hM2 i ¼ 0:62 0:12 M, ha2 i ¼
0:97 0:34 AU, he2 i ¼ 0:10 0:07, and hi2 i ¼ 0:07 0:05.
We find that Earth is in the cluster of the largest planets and is
within 1 of the a1 and M1 distributions, while Venus is located
within 1 of the a2 distribution and near the top edge of the M2
distribution. In most runs, the third planet around 0:5 AU P a P
1:5 AU is several times larger than Mars, where the sum of M1 and
M2 is smaller than that of Earth and Venus. The Mercury analog is
formed by inward scattering of a protoplanet. This may suggest
that depletion of planetesimals inside the Mercury orbit is one
possibility to explain the formation site of Mercury. The terrestrial
planet system for the solar system may correspond to a case in
which most of the system mass is concentrated in two large
planets.
Using the habitat segregation pattern of terrestrial, Jovian, and
Uranian planets against parameters of the protoplanetary disk,
Kokubo & Ida (2002) discussed the idea that for the disk with
1 k 30 and ¼ 3/2, it would be difficult to harbor a stable terrestrial planet system, since the Jovian planet region extends inward beyond the snow line, a ’ 3 AU. The Jovian planets close
to the terrestrial planet region can easily destroy the terrestrial
planet system by strong gravitational perturbation. This argument suggests that there may exist a maximum 1, 1,max, that
Agnor, C., & Asphaug, E. 2004, ApJ, 613, L157
Agnor, C. B., Canup, R. M., & Levison, H. F. 1999, Icarus, 142, 219
Beckwith, S. V. W., & Sargent, A. I. 1996, Nature, 383, 139
Chambers, J. E., & Wetherill, G. W. 1998, Icarus, 136, 304
Goldreich, P., & Ward, W. R. 1973, ApJ, 183, 1051
Hayashi, C. 1981, Prog. Theor. Phys. Suppl., 70, 35
Heppenheimer, T. A. 1980, Icarus, 41, 76
Kasting, J. F., Whitmire, D. P., & Reynolds, R. T. 1993, Icarus, 101, 108
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———. 1998, Icarus, 131, 171
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1139
allows the formation of a stable terrestrial planet system. Then
we can roughly estimate the maximum mass of terrestrial planets formed from the 1;max ’ 30 disk by equation (6), which is
Mmax ’ 4 M. We can also estimate the minimum 1, 1,min,
that allows the formation of Earth-sized planets (M k M /2).
From equation (6), we obtain 1;min ’ 5. In summary, we obtain
the conditions for a disk that potentially produces Earth-sized
planets in stable orbits as 5 P 1 P 30 for ¼ 3/2.
In the present paper, we fixed the radial width of the initial
distribution of protoplanets. In future work, we will relax this
condition and probe the effect of the radial width to further generalize the model of terrestrial planet formation. We will also address the nature of giant impacts and resulting planetary spin.
We thank referee Sean N. Raymond for valuable comments.
This research was partially supported by MEXT (Ministry of
Education, Culture, Sports, Science and Technology), Japan, the
Grant-in-Aid for Scientific Research on Priority Areas, ‘‘Development of Extra-Solar Planetary Science,’’ and the Special
Coordination Fund for Promoting Science and Technology,
‘‘GRAPE-DR Project.’’
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