PASJ: Publ. Astron. Soc. Japan 54, 471–479, 2002 June 25
c 2002. Astronomical Society of Japan.
Orbital Stability of a Protoplanet System under a Drag Force
Proportional to the Random Velocity
Kazunori I WASAKI , Hiroyuki E MORI , Kiyoshi NAKAZAWA, and Hidekazu TANAKA
Department of Earth and Planetary Sciences, Faculty of Science, Tokyo Institute of Technology, Meguro-ku, Tokyo 152-8551
[email protected]
(Received 2001 June 21; accepted 2002 March 11)
Abstract
We investigated the effects of a tidal interaction with a gas disk and the dynamical friction with a planetesimal
disk on the orbital instability of a protoplanet system. Both effects are expressed as the drag force, which is proportional to the random velocity of a protoplanet. We calculated numerically the orbits of 5 protoplanets with the
same separation distance under the drag-force effect and examined the orbital instability time under the drag force,
df
df
. We found that Tinst
can become much larger than the instability time under the drag-free condition, and that
Tinst
the onset of the orbital instability is prevented when the separation distance exceeds a critical value. We obtained a
relation between the critical separation distance and the surface density of the gas or planetesimal disk. By applying
this relation, we found that, for the formation of terrestrial planets from a protoplanet system with a typical orbital
separation (i.e., ∼ 10 Hill radii), the surface density of the nebular gas must be reduced to about one-thousandth
of that in the minimum-mass nebula model. Terrestrial planets would be formed after such a depletion of the solar
nebula.
Key words: Earth — instabilities — solar system: formation — Stokes-type drag force
1. Introduction
According to the standard scenario of planetary formation,
terrestrial planets are formed through multi-step accretion processes of planetesimals (e.g., Safronov 1972; Greenberg et al.
1978; Hayashi et al. 1985; Wetherill, Stewart 1989). In the
early stage of planetary formation, called the orderly growth
stage, planetesimals grow almost uniformly because the growth
rate is insensitive to their masses. However, after a gravitational focusing effect becomes important, a small number of
massive planetesimals grow preferentially, whereas the others
remain small. This growth mode is called “runaway growth”
(Wetherill, Stewart 1989; Kokubo, Ida 1996), through which a
handful of protoplanets (with masses of the order of the Moon)
are formed in the terrestrial region.
After that, we encounter a new stage, the oligarchic growth
stage, when the masses of the protoplanets become so large
that their mutual gravity plays an important role in a dynamical
system composed of the protoplanets and planetesimals. In this
stage, the masses of the protoplanets are regulated to be nearly
equal to each other, although the protoplanets grow dominantly
compared with the surrounding planetesimals (Kokubo, Ida
1998). As the growth of protoplanets proceeds, the planetesimals are gradually swept up by accretion onto the protoplanets
and, before long, the protoplanets cease to grow substantially.
At the end of the oligarchic growth stage, several tens of massive protoplanets are formed in the present terrestrial planet region and their mean mass of the protoplanets amounts to the
order of the Martian mass. They are distributed with a nearly
equal orbital separation, which is 5 to 10-times as large as the
Hill radius [see equation (1)] (Kokubo, Ida 1998, 2000).
Further growth to planets is generally conjectured as follows.
After the oligarchic growth stage, the orbital eccentricities
increase in a stochastic manner owing to a mutual gravitational
perturbation and, after a while, there happens to occur orbital
crossing between protoplanets to become capable of colliding
with each other. Through such collisions between protoplanets,
they continue to grow while increasing their mutual separation
distance (owing to a decrease in the number of protoplanets)
and reach the sizes of the present terrestrial planets with the
presently observed orbital separation (about 30 Hill radii or
more).
The orbital evolution of protoplanets after the oligarchic
growth stage was investigated by Chambers et al. (1996) for
the first time. They studied numerically the orbital behavior of
a multi-protoplanet system composed of 3 to 20 protoplanets
with equal mass (1 × 10−7 M , M being the solar mass), assuming that the protoplanets initially had null orbital eccentricities and inclinations, and were placed from 1 AU with the same
orbital separation distance, Ƌ 0 . Note that the separation distance is scaled by the Hill radius of two adjacent protoplanets,
rH , which is defined, for example, by
a10 + a20
rH =
h,
(1)
2
where a10 and a20 are the initial semimajor axes of the first and
second protoplanets. Furthermore, h is the reduced Hill radius,
given by
1
2m 3
h=
,
(2)
3 M
where m is the mass of the protoplanets. Through long-term
orbital calculations, they showed that the protoplanets always
experience an orbital instability (i.e., an orbital crossing) and
that the orbital instability time, Tinst , within which any two of
the protoplanets come close within one Hill radius for the first
472
K. Iwasaki et al.
time, can be expressed empirically by an exponential form of
Ƌ0 [see equation (14)].
However, it should be noted that the protoplanet system set
up by Chambers et al. (1996) is highly idealized. Yoshinaga
et al. (1999) focused on a system of protoplanets with finite
initial random velocities (i.e., eccentricities and inclinations),
and found that in such a protoplanet system the orbital instability time is shortened by several orders of magnitude. Ito
and Tanikawa (1999, 2001) investigated the effect of secular
perturbations by the Jovian planets on the orbital instability of
the protoplanet system, and found that the instability time of
the system is reduced and deviates from relation (14) when the
separation distances are larger than about 14rH . Furthermore,
we must also question the assumption adopted in Chambers
et al. that protoplanets orbit in a “vacuum space”. Rather, it is
natural to think that orbits of the protoplanets are influenced,
not only by the mutual gravitational interaction, but also by
hydrodynamic gas drag due to nebular gas and/or dynamical
friction (i.e., gravitational interaction) originated by a swarm
of planetesimals.
Chambers and Wetherill (1998) pointed out that, just after the formation of planets from protoplanets, the orbital eccentricities of the planets are commonly about 10-times as
large as those of the present terrestrial type planets (e.g., Earth
and Venus) owing to their mutual gravitational scatterings.
Therefore, even after the formation of planets, a certain quantity of the nebular gas and/or planetesimals is needed around
them for damping the eccentricities to those on the present level
through interactions with the nebular and/or planetesimal disk.
Iwasaki et al. (2001) (which, hereafter, is referred to as
Paper I) studied a long-term orbital evolution of protoplanets
suffering the gas-drag force, which is proportional to the square
of the velocity, u, i.e.,
1
F = CD π r 2 ρ|u|u,
(3)
2
where ρ and r are the gaseous density and the radius of a
protoplanet, respectively, and CD is the drag coefficient (the
drag force given above will be called, hereafter, the Newtontype drag force, i.e., F ∝ u2 ). They considered five protoplanets with the same mass of 1 × 10−7 M , which are equally
spaced on initially circular and coplanar orbits (as those set
up in Chambers et al. ) as well as the solar nebular gas rotating
with a Keplerian circular velocity. Through a number of orbital
calculations, they discovered that under the gas-drag effect the
orbital instability time becomes enlarged in magnitude compared with that for the gas-free case [equation (14)], and that
the orbital instability does not occur substantially when the initial orbital separation, Ƌ 0 , is larger than the critical separation
distance, (Ƌ 0 )crit .
They also found that the critical separation distance,
(∆ã 0 )crit , can be written empirically as a function of the nebular gas density, ρ, and, obtained (∆ã 0 )crit 10 for the minimum
mass nebula model [ρ 1.0 × 10−9 g cm−3 at 1 AU given by
Hayashi (1981)]. Based on their results, they concluded that
there occurs an orbital instability (followed by a collision) in
a realistic protoplanet system with a typical orbital separation
of about 5–10 in units of the Hill radius (Kokubo, Ida 1998,
2000), even if the protoplanets move round in the solar nebula.
[Vol. 54,
The eccentricities of protoplanets are also depressed by the
tidal interaction with the nebular disk (Ward 1988; Artymowicz
1993). Furthermore, the dynamical friction due to a swarm of
planetesimals is important as a damping effect on the eccentricities of the protoplanets, if the eccentricities of the planetesimals are sufficiently suppressed by the nebular gas disk
through the gas-drag force (since the dynamical friction becomes effective with a decrease in the eccentricities of planetesimals). Unfortunately, the results obtained by Iwasaki et al.
(2001), mentioned above, cannot be applied to such cases. This
is because the dynamical friction and tidal interaction become
intense in proportion to the random velocity of a protoplanet
(Ida 1990; Artymowicz 1993), whereas the gas-drag force is
proportional to the square of the random velocity.
In the present paper, we concentrate on the orbital instability
of the protoplanet system under the action of dynamical friction by a swarm of planetesimals and the tidal interaction with
the nebular gas disk, both of which are proportional to the random velocity of a protoplanet. As for a protoplanet system,
we adopt the same initial conditions as those in Paper I, i.e.,
five protoplanets are placed with the same separation distances
from 1 AU and their initial eccentricities and inclinations are
set to be zero.
2.
Method of Orbital Calculations
2.1. Basic Equations
We numerically calculated the orbits of n protoplanets (in
our present case, n = 5) revolving around the Sun, while including both the mutual gravity and the drag force which is
proportional to the random velocity of a protoplanet. Thus, in
a coordinate system where the Sun is at the origin, the motion
of the i-th protoplanet is governed by the following equation:
xi − xj
xi
−
Gmj
3
|xi |
|xi − xj |3
n
ẍi = −GM
j =i
−
n
j
Gmj
xj
Fi
− ,
3
|xj |
mi
(4)
where mi and xi are the mass and the position vector of the
i-th protoplanet, respectively, and G is the gravitational constant. Furthermore, Fi shows the drag force acting on the i-th
protoplanet, and is given by
mi
ui ,
(5)
Fi =
τD
where ui denotes the relative velocity vector of the i-th protoplanet from the Keplerian circular velocity, vK (i.e., the random
velocity of the protoplanet):
ui = ẋi − vK .
(6)
In equation (5), τD is the characteristic time constant. Adachi
et al. (1976) investigated the effect of the drag force obeying
equation (5) (which, hereafter, will be called the Stokes-type
drag force, i.e., F ∝ u) on the motion of a planetesimal orbiting
around the Sun. Through the orbital average of a planetesimal,
they obtained an analytical formula of the time variation of the
orbital eccentricity, e, i.e.,
No. 3]
Orbital Stability of a Protoplanet System
1 de
1
=− .
e dt
τD
(7)
Thus, τD represents the damping time of the eccentricity.
In planetary accretion, as mentioned in section 1, there are
two processes of eccentricity damping, described by equation (7); that is, the dynamical friction by a planetesimal swarm
and the tidal interaction with a nebular disk. The expression
for τD have been obtained for each damping process. Ida
(1990) performed many orbital calculations of encounters between two planetesimals for various initial phases under the
solar gravitational field. As a result, he obtained the damping
rate of the eccentricity of a planetesimal due to the dynamical
friction of a swarm of planetesimals. By applying his results to
a protoplanet embedded in a swarm of planetesimals based on
the minimum-mass nebula model (Hayashi 1981), τD is written
as (Ida 1990)
4 13
vm
m
2
τD = 4 × 10
3hp vK
1 × 10−7 M
−1
a − 12
σd
×
TK ,
(8)
1 AU
7.1g cm−2
where σd is the surface density of planetesimals at 1 AU and
TK is the Keplerian period at 1 AU. In the above equation, hp
denotes the reduced Hill radius between the protoplanet and a
planetesimal, and is given by hp = 2−1/3 h [see equation (2)],
since the planetesimal mass is much smaller than the protoplanet mass. In the post-oligarchic growth stage considered in
the present paper, the surface density of planetesimals is expected to be depleted from its initial value, which makes the
above damping time longer. Note that we assume the surface density, σ , varies with the distance from the Sun, a, as
σ ∝ a −3/2 according to Hayashi (1981). It should also be noted
that the above equation is valid only when the random velocity
of a planetesimal, vm , satisfies
1
2
+ im2 ) 2 vK ≥ 3hp vK ,
vm = (em
(9)
where em and im are the means of the eccentricity and inclination of planetesimals, respectively. The above assumption is always satisfied in our present case, since planetesimals
have large random velocities because of gravitational scatterings with massive protoplanets (Ida, Makino 1993).
Ward (1988) and Artymowicz (1993) calculated the tidal
torque via density waves on a body embedded in a gaseous
disk without a gap, and estimated the damping rate of the eccentricity of the body. According to their results, for a protoplanet with small eccentricities (evK cs where cs is the
sound velocity) embedded in the gas nebula, τD is expressed
by (Artymowicz 1993)
4 −1 cs
m
a − 12
3
τD = 7 × 10
0.03vK
1 × 10−7 M
1 AU
−1
σg
×
TK ,
(10)
1.7 × 103 g cm−2
where σg denotes the surface density of the nebular gas at 1 AU
(we assume again that σ ∝ a −3/2 ).
473
2.2. Initial Condition
In order to see the orbital evolution of protoplanets under
the influence of the Stokes-type drag force, we numerically integrated equation (4). We considered five protoplanets with the
same mass of 1 × 10−7 M , which is about one-third of the
Martian mass, and their initial configuration in orbits was set
up in the same way as those in Paper I [and also Chambers et al.
(1996)]:
(a) The semimajor axis of the innermost protoplanet, a1 , is
1 AU,
(b) The five protoplanets with null orbital eccentricity and
inclination are placed with an equal orbital separation,
Ƌ 0 ,
and
(c) The azimuthal angles of positions of five protoplanets
are selected randomly, but satisfying that the difference
between the position angles of any two adjacent protoplanets is more than 20◦ .
It should be noted that, in assumption (b), the orbital separation is scaled by the Hill radius [see equation (1)]. Thus, the
actual separation distances in the outer pairs of protoplanets are
slightly larger compared with those in the inner pairs.
As for the characteristic time constant, τD , we consider the
following four cases:
τD = 1.0 × 103 , 3.0 × 103 , 9.0 × 103 , and 2.7 × 104 TK .(11)
For each value of τD , we calculated orbits numerically while
changing the separation distance, Ƌ 0 , from 3.6 to 6.8 with a
step of 0.2:
Ƌ0 = 3.6, 3.8, 4.0, . . . , 6.6, and 6.8 .
(12)
For an arbitrary set of τD and ∆ã 0 , we made orbital simulations
for ten cases with different phases in position angles according
to assumption (c).
We numerically integrated the orbits of protoplanets using
a 4th-order P (EC)n Hermite scheme (Makino, Aarseth 1992;
Kokubo et al. 1998; Kokubo, Makino 1998), which is quite the
same as discussed in Paper I.
3.
Effect of the Drag Force on the Orbital Stability
3.1. Instability Time
Through a number of orbital calculations, we found the ordf
, under the Stokes-type drag force.
bital instability time, Tinst
Here, we adopt the same criterion for the occurrence of an orbital instability as those in Paper I; that is, a first approach of
df
two protoplanets within one Hill radius. We expect that Tinst
becomes larger than the instability time for the case without any
drag force, since the drag force suppresses the eccentricities of
the protoplanets. Thus, in order to save computational time, we
stopped our orbital calculations when the evolutionary time, t,
exceeds the cut-off time, Tstop , which is given by
Tstop = 200 × Tinst .
(13)
In the above equation, Tinst denotes the instability time for the
cases without a drag force, which was found empirically in
Paper I:
474
K. Iwasaki et al.
[Vol. 54,
df , as a function of Ƌ . The four panels correspond
Fig. 1. Instability time for the cases where the protoplanets suffer from the Stokes-type drag force, Tinst
0
to the cases of (a) τD = 1.0 × 103 TK , (b) τD = 3.0 × 103 TK , (c) τD = 9.0 × 103 TK , and (d) τD = 2.7 × 104 TK . In all of the panels, the solid lines represent
Tinst , given by equation (14), and the upper dotted lines show the cut-off time of calculations, Tstop (= 200 × Tinst ). The two vertical dotted lines denote
the boundaries of the transition zone.
log10 (Tinst /TK ) = 0.78∆ã0 − 0.15.
(14)
In figures 1a to d, we plot the orbital instability time of the
df
protoplanet system, Tinst
, against the initial separation distance,
∆ã 0 , for the four cases of τD = 1.0 × 103 TK , τD = 3.0 × 103 TK ,
τD = 9.0 × 103 TK , and τD = 2.7 × 104 TK . In these figures, the
results for ten cases with different initial phases are plotted for
each ∆ã0 . For any τD , we can readily see that the protoplanet
system is stabilized owing to the drag force when Ƌ0 is large
enough. It is convenient for understanding the stability of a system in a quantitative manner to divide the abscissa of figure 1
into three zones according to the orbital behaviors of the ten
cases with the same ∆ã0 , i.e., the “unstable” zone, the “transition” zone, and the “stable” zone (Paper I). In the unstable
zone (e.g., ∆ã0 < 5.4 for τD = 9.0 × 103 TK ), all ten cases with
the same Ƌ0 experience the orbital instabilities within a time
nearly equal to Tinst . On the contrary, in the stable zone (e.g.,
∆ã0 ≥ 5.8 for τD = 9.0 × 103 TK ), all cases reach the cut-off
time of the orbital calculations, Tstop , without the occurrence of
an orbital instability. The above two zones are separated by the
transition zone (e.g., 5.4 ≤ ∆ã 0 < 5.8 for τD = 9.0 × 103 TK ),
in which the protoplanet system suffers an orbital instability in
some cases and not in other cases.
The two vertical dotted lines in each panel of figure 1 denote
No. 3]
Orbital Stability of a Protoplanet System
475
Fig. 2. Orbital behaviors of a protoplanet system for the case of (τD , ∆ã 0 ) = (9.0 × 103 TK , 5.6). The left-hand panel shows the time evolution of the root
mean square of the eccentricities of the five protoplanets. The case without the drag force (Ƌ 0 = 5.6) is also shown. The onset of the orbital instability
is suppressed for the case with the drag force. In the right-hand panel, the separation distances, Ƌ, between adjacent protoplanets are shown. A rapid
orbital repulsion is observed at t 5 × 104 yr.
the positions of the boundaries of the transition zone, that is,
(Ƌ 0 )1 (the left dotted line) and (Ƌ 0 )2 (the right dotted line).
The values of (∆ã0 )1 and (∆ã 0 )2 for each τD are shown in
table 1. Here, we should note that, for any τD , the transition
zone is very narrow [(∆ã0 )2 − (∆ã0 )1 = 0.4], compared with
the unstable zone and the stable zone. This means that the system is completely stabilized as soon as Ƌ 0 exceeds a narrow
threshold which corresponds to the transition zone. We also see
that the position of the transition zone is shifted toward a direction of large Ƌ 0 (in other words, the unstable zone spreads
out) as τD increases. The reason is as follows: According to
the increase in τD (i.e., the decrease in the intensity of the drag
force), the drag force does not work efficiently as the damping
mechanism of eccentricities [see equation (7)] and, as a result,
the system loses the stabilizing function.
In figure 2, we show the orbital evolution of the protoplanet
system with (τD , ∆ã 0 ) = (9.0 × 103 TK , 5.6), which is a typical example of the system stabilized by the drag force. The
left panel of figure 2 shows the time variation of the root mean
square of eccentricities, e2 1/2 , of the five protoplanets under
the drag force, together with that for the drag-free case with
the same value of Ƌ 0 . In the case without the drag force,
e2 1/2 increases rapidly near t = 5 × 104 yr and, soon after, the
protoplanet system is visited by a catastrophic orbital crossing. In the case with the drag force, we also observe a rapid
increase in the eccentricities of the protoplanets at almost the
same time. But, in this case, the rapid increase in e2 1/2 is
followed by gradual decrease caused by the drag force and, as
a result, an orbital instability is prevented through damping of
the eccentricities. Such a behavior of the protoplanet system
is quite the same as that seen in Paper I where we consider
the Newton-type drag force (such an event is called a “falsely
unstable event”).
A rapid decrease in eccentricities (i.e., a rapid energy dissipation) leads to a rapid orbital repulsion of protoplanets, as
Table 1. Boundaries of the transition zone, (Ƌ0 )1 and (Ƌ 0 )2 .
τD /103 TK
1.0
3.0
9.0
27
(Ƌ0 )1
(Ƌ0 )2
4.6
5.0
5.0
5.4
5.4
5.8
6.2
6.6
pointed out by our previous study. In the right panel of figure 2,
we show the time evolution of the separation distances normalized by the Hill radius, Ƌ, of all pairs of two adjacent protoplanets. The orbits of the protoplanets change suddenly just
after the falsely unstable events and settle into a stable configuration with small eccentricities. As a result, the normalized
separation distance, Ƌ, between the second and third protoplanets expands by about 0.2 before and after the event, while
those of the other pairs keep almost constant (in other words,
the third and the outer two protoplanets move outward coherently, whereas the innermost protoplanet does inward with the
second protoplanet).
3.2. Comparison between the Two Drag Laws
As shown in the last part of the previous subsection, the
qualitative properties of the stabilizing mechanism are common between the Stokes-type (present study) and Newton-type
(Paper I) drag forces. However, the degrees of stabilization are
rather different in both cases. In order to see this, we compared
the orbital behaviors under the two kinds of drag forces, while
keeping the intensities of the drag force at the same level. The
intensity of the drag force was properly evaluated by the characteristic damping time of the eccentricity, Tdamp . Here, Tdamp
is defined as the time interval during which the eccentricity of
a protoplanet decreases by a factor of 1/e owing to the drag
force, i.e.,
476
K. Iwasaki et al.
[Vol. 54,
This is due to the difference in the e-dependence of the
damping rate of the eccentricity. For the case of the Newtontype drag force, where the damping rate of the eccentricity is
proportional to e, the damping of the eccentricity becomes effective only at the stage when the eccentricity of a protoplanet
is large or, in other words, at the stage just before the occurrence of orbital crossing. On the other hand, for the case of
the Stokes-type drag force, the damping rate is independent of
e, and hence, the damping effect works constantly regardless
of the magnitude of the eccentricity. Thus, a protoplanet system suffering from the Stokes-type drag force is more strongly
stabilized than a system of the Newton-type drag force.
3.3. Critical Separation Distance
Fig. 3. Time evolution of the root mean square of the eccentricities,
e2 1/2 , of the protoplanet systems suffering from the Stokes-type drag
force (solid line) and from the Newton-type drag force (dotted line). In
both cases, Tdamp and ∆ã 0 are set to be 1.7 × 103 TK and 5.6, respectively.
Tdamp = e
de .
dt (15)
For the Stokes-type drag force given by equation (5), Tdamp is
simply equal to τD . For the Newton-type drag force, however,
Tdamp depends on the eccentricity, itself. For our present purpose, it is natural to define the damping time by the use of the
eccentricity at the moment of the onset of the orbital crossing,
ecross , that is,
τg
N
Tdamp
=
,
(16)
0.77ecross
where τg is a time constant depending on the planetary radius
and mass as well as the gaseous density (see Paper I; the superscript N denotes “the Newton-type”). In the above equation,
ecross is given by
1
ecross = Ƌ0 h.
(17)
2
In the following, we compare the orbital behaviors of the protoplanetary systems with the Stokes-type and the Newton-type
drag forces of the same intensity in the sense that
N
Tdamp
= τD .
(18)
Figure 3 shows the evolutionary behaviors in the root mean
square of the eccentricities of the five protoplanets for both
cases of the Stokes-type drag force and the Newton-type drag
force. In both cases, Tdamp and Ƌ 0 are commonly set to be
1.7 × 103 TK and 5.6, respectively. For the case of the Newtontype drag force, e2 1/2 increases abruptly at t 4.0 × 104 yr
and the system is visited by the orbital crossing. For the
alternative case, e2 1/2 also increases rapidly at almost the
same time as in the previous case. But, soon, e2 1/2 gradually decreases because of the drag force and, as a result, the
protoplanetary system is prevented from undergoing orbital
crossing.
In subsection 3.1, we find that a protoplanet system under the
Stokes-type drag force experiences no orbital instability when
Ƌ0 exceeds a certain critical level. In order to obtain a simple
measure by which we can judge the stability of a protoplanetary system with an arbitrary intensity of the drag force, we introduce a critical separation distance, (Ƌ 0 )crit , which is quite
the same as that in Paper I, i.e.,
(Ƌ0 )1 + (Ƌ0 )2
,
(19)
2
where (Ƌ 0 )1 and (Ƌ 0 )2 are the boundary values of Ƌ 0 of
the transition zone. In figure 4, we plot (Ƌ 0 )crit as a function
of the damping time of the eccentricity, Tdamp , for the cases
of the Stokes-type drag force (this study) and the Newton-type
drag force (Paper I). In both cases, (Ƌ 0 )crit increases almost
in proportion to log10 Tdamp with the same inclination, although
the values of (Ƌ 0 )crit are different between the two kinds of
drag forces.
The relation between (Ƌ 0 )crit and Tdamp was also estimated
based on the following theoretical consideration. As shown
in subsection 3.1, a protoplanet system with a small Ƌ 0
[Ƌ 0 < (Ƌ 0 )crit ] experiences an orbital instability, since the
stirring time of the eccentricity due to the mutual gravity between protoplanets is shorter than the damping time of the
eccentricity due to the drag force, Tdamp . On the contrary, a
system with a large Ƌ 0 [(Ƌ 0 )crit < Ƌ 0 ] is stabilized owing
to the drag force, since Tdamp is shorter than the stirring time
due to the mutual gravity. Thus, it is natural to consider that
the stirring time of the eccentricity due to the mutual gravity
would balance with Tdamp at Ƌ 0 (Ƌ 0 )crit . As pointed out
in Paper I, the stirring time of the eccentricity due to the mutual gravity is proportional to the orbital instability time for the
drag-free case, Tinst . Therefore, when Ƌ 0 = (Ƌ 0 )crit , the following relation should be realized:
(Ƌ0 )crit =
Tdamp = CTinst ,
(20)
where C is a numerical constant. Using the expression of Tinst
[equation (14)], we obtain the relation between (Ƌ 0 )crit and
Tdamp , i.e.,
(Ƌ 0 )crit = 1.28 log10 [Tdamp /(CTK )] + 0.192.
(21)
The solid lines in figure 4 show the above theoretical relations for C = 1, 0.3, and 0.1. From this figure, we can see that
relation (21) for C = 0.3 fits the critical separation distances,
(Ƌ0 )crit , which were found numerically in the present study,
No. 3]
Orbital Stability of a Protoplanet System
477
1 × 10−7 M , were placed with the same separation distance,
Ƌ 0 , and their initial orbits were circular and coplanar. Our
obtained results are summarized as follows:
Fig. 4. Critical separation distances, (Ƌ 0 )crit , versus Tdamp . The filled
circles show the results of (Ƌ 0 )crit found by numerical calculations
for the case of the Stokes-type drag force, the open circles are for the
case of the Newton-type drag force, and the vertical bars denote the
widths of the transition zone [(Ƌ 0 )1 < Ƌ 0 < (Ƌ 0 )2 ]. The results of
Paper I (open circles) are also shown for a comparison. The solid lines
are the critical values evaluated by relation (21) for C = 1, 0.3, and 0.1.
but it does not for C = 1. This reflects the fact that, in an actual evolution of a protoplanet system, the eccentricities of the
protoplanets increase not uniformly, but abruptly, in a relatively
short period (compared with the orbital instability time, Tinst )
just before the orbital crossing. Namely, the stirring time of
the eccentricity due to the mutual gravity should be taken to
be smaller than Tinst ; in other words, the value of C should be
smaller than 1.
As already shown in Paper I (and, as seen from figure 4), we
have a similar relation between (Ƌ 0 )crit and Tdamp for a system
with the Newton-type drag force although the smaller numerical coefficient C should be chosen, i.e., C = 0.1. Remember
that the damping effect of the eccentricities due to the Stokestype drag force works effectively over the whole evolutionary
stage of a system, whereas the effect due to the Newton-type
drag force becomes effective only in the later stage. This is a
reason why a smaller value of C must be chosen for the cases
of the Newton-type drag force compared with that for the case
of the Stokes-type drag force.
4. Summary and Conclusion
In the present study, we investigated the orbital instability of
a protoplanet system suffering from a drag force proportional
to the random velocity of a protoplanet, by numerically calculating the orbits of the protoplanets, while including both the
mutual gravity between them and the drag force. In our numerical simulations, a protoplanet system was set up in the same
way as in Paper I, i.e., five protoplanets with the same mass,
(1) Under the action of a drag force proportional to the random velocity of a protoplanet, the occurrence of an orbital instability of a protoplanet system is prevented,
more or less, through damping of the eccentricities of
the protoplanets.
(2) The stabilizing effect of the drag force becomes remarkable when the initial separation distance, Ƌ 0 , exceeds
the critical separation distance, (Ƌ 0 )crit , and, as a result,
df
the orbital instability time under the drag force, Tinst
, becomes more than 200-times as long as the orbital instability time under the drag-free condition, Tinst . In practice, a protoplanet system experiences no orbital instabilities substantially under such conditions.
(3) A drag force which is proportional to the random velocity of a protoplanet (the Stokes-type drag force) stabilizes a protoplanet system more strongly than the drag
force which is proportional to the square of the random
velocity (the Newton-type drag force which was investigated in Paper I). Actually, for the same value of the
damping time of the eccentricity, Tdamp , (Ƌ 0 )crit under
the action of the Stokes-type drag force is small compared with that under the action of the Newton-type drag
force.
(4) The value of (Ƌ 0 )crit depends on the damping time of
the eccentricity due to the drag force, τD , and is given
empirically by the following relation:
(∆ã 0 )crit = 1.3 log10 (τD /TK ) + 0.86.
(22)
Readers may feel anxious about the generality of our results
mentioned above, since we considered an ideal protoplanet system composed of 5 protoplanets with the same mass and the
same radial spacings, which move initially along circular and
coplanar orbits. According to Chambers et al. (1996), even in
a gas-free system with dispersions of the masses and spacings
of the protoplanets, the orbital instability time changes hardly
from Tinst given by equation (14). They confirmed that the orbital instability time is almost constant independently of the
number of constituent protoplanets, N , as long as N is equal to
or greater than 5. Thus, we can say that our results mentioned
above are rather general as far as the initial eccentricities and
inclinations of protoplanets are confined within a small level,
say, of the order of h.
We now will apply our results to the problem of the formation of terrestrial planets. First, we consider the dynamical friction due to a swarm of planetesimals as a process of eccentricity damping. In order to obtain the value of τD , we must know
the mean of the eccentricities of planetesimals, em [see equation (8)]. Ida and Makino (1993) showed that, in the presence
of massive protoplanets, the eccentricities of planetesimals are
determined by the balance between the stirring effect of the
eccentricity due to the gravitational scattering with the protoplanets and the damping effect due to the gas-drag force by
the solar nebular gas, in which the gaseous density, ρ, varies
depending on the semimajor axis, a, as ρ ∝ a −11/4 , i.e.,
478
K. Iwasaki et al.
em = 7
mp
1023 g
181 a 274
1 AU
ρg
1 × 10−9 g cm−3
− 16
h,(23)
where ρg is the gaseous density of the nebula at 1 AU and mp
is the typical mass of planetesimals. Assuming that im = em /2
and using equation (8), the damping time, τD , due to the dynamical friction with planetesimals is evaluated as 4 × 104 TK .
In the evaluation, we assume that ρg is equal to that of the minimum mass nebula model (= 1 × 10−9 g cm−3 ) and the surface
density of planetesimals is assumed to be half as much as that
in the minimum-mass nebula model, namely we consider the
situation that a half of the solid mass accretes onto the protoplanets. For the evaluated value of τD , (∆ã 0 )crit is obtained
from equation (22) as follows:
(Ƌ 0 )crit = 6.8.
(24)
Secondly, we consider eccentricity damping due to the tidal interaction. For the case of the tidal torque due to the density
wave excited in the nebular gas disk, τD is found from equation (10). Based on the minimum mass nebula model, where
cs 0.03vK at 1 AU (Hayashi 1981), (Ƌ 0 )crit is readily evaluated as [see equation (22)]
(Ƌ 0 )crit = 5.9.
(25)
Note that (Ƌ 0 )crit given by equation (24) or (25) is very small
compared with that evaluated in Paper I [i.e., (Ƌ0 )crit 10].
From the difference in (Ƌ 0 )crit between equations (25) and
(24) or the difference in τD , it is found that the damping due to
the tidal interaction is much more effective than that due to the
dynamical friction.
According to Kokubo and Ida (1998, 2000), the separation distance, Ƌ0 , between protoplanets, which are formed
through oligarchic growth, is typically 10 or so. Based on their
result, we concluded in Paper I that a protoplanet system orbiting in the standard solar nebula would experience an orbital
instability, since we have (Ƌ0 )crit 10 for the case of the
Newton-type gas-drag. However, if we take fully account of
the effect of the gravitational interaction (i.e., the dynamical
friction and the tidal interaction) with the gas and/or planetesimal disk, (Ƌ 0 )crit becomes 5.9, as shown above. This critical
value is too small for the occurrence of an orbital instability of
a protoplanet system with a typical orbital separation. Namely,
protoplanets could not grow to the present planets.
As mentioned above, a protoplanet system will not experience an orbital instability when the surface density of the nebula is as large as that in the minimum-mass solar nebula model.
However, considering the present appearance of our solar system without a gas disk, the nebular gas disk must gradually be
depleted. As the surface density of the nebular gas disk decreases, the effect of the tidal interaction becomes weak (i.e.,
τD increases) and the value of (∆ã 0 )crit increases. Actually,
(Ƌ 0 )crit becomes about 10, which is a typical orbital separation of protoplanets, for the tidal interaction with the gas nebula, when the surface density of the nebular gas decreases to
about one-thousandth as large as that of the minimum mass
solar nebula. Then, the protoplanet system will be visited by
an orbital instability and protoplanets will start to collide and
accrete.
[Vol. 54,
As can be seen from equations (8) and (23), the depletion
of the nebula also increases the eccentricities of planetesimals
and makes the dynamical friction ineffective. Thus, as for
the dynamical friction with planetesimals, the difficulty of a
small (Ƌ 0 )crit would be overcome by reducing the surface
density (and also the density) of the nebular gas. For the
above-mentioned surface gas density (which is one-thousandth
as large as that in the minimum-mass solar nebula model),
(Ƌ 0 )crit becomes as large as about 10 without reducing the
surface density of planetesimals. If the surface density of the
planetesimals decreases, (Ƌ 0 )crit will become larger than 10.
This obtained value of the surface density for the orbital instability is consistent with the results of recent studies concerning the formation of terrestrial planets. Recently, Kominami
and Ida (2001) investigated the accretional process of protoplanets by a direct N -body simulation which included the effect of the tidal interaction with the nebular gas as a drag force.
They found that in order for protoplanets to grow to Earth-size
planets, and in order for their eccentricities to be damped to
the present values of the terrestrial planets, the plausible surface density is 1 × 10−3 to 1 × 10−4 times as large as that
in the minimum-mass solar nebula model. Agnor and Ward
(2002) also examined the eccentricity damping of the terrestrial planets (i.e., Mercury, Venus, Earth, and Mars), while taking account of secular perturbations between the planets. Their
results also showed that an appropriate value of the surface
density is 1 × 10−3 to 1 × 10−4 times as large as that in the
minimum-mass solar nebula model. From the coincidence of
these two results, we can say that, in the presence of such a
depleted gas nebula, both of the two difficulties for the formation of the terrestrial-type planets (i.e., the onset of the orbital
instability) and the decrease in the eccentricities of the planets,
would be overcome.
Finally, we discuss the validity of extrapolating our results to
the cases with such a small surface density of the gas disk. A
nebular gas disk with an extremely small surface density is expected to be strongly disturbed and heated up by protoplanets
through gravitational interactions and, as a result, the damping
effect on the eccentricities of the protoplanets (which is also
brought about by the gravitational interactions) may become
ineffective. However, concerning the gas disk with the abovementioned surface density (i.e., one-thousandth as large as
that in the minimum-mass nebula model), the influence on the
gas disk by protoplanets would be negligibly weak. Actually,
the angular momentum transfered to the gas disk √
by damping of the eccentricity, e, of a protoplanet ( e2 m GM a)
is small by several orders of magnitude compared with the angular momentum of the corresponding part of the disk around
the protoplanet (i.e., the ring region with a width of 10 Hill
radii). Furthermore, the temperature of the gas disk would be
hardly affected owing to the gravitational interactions with the
protoplanets, since the thermal energy injected into the disk by
protoplanets is immediately carried away by radiation.
The authors thank S. Ida for his valuable advice and continuous encouragement. This work was supported by the Grantin-Aid of the Japanese Ministry of Education, Culture, Sports,
Science and Technology (Nos. 11101002 and 12640405).
No. 3]
Orbital Stability of a Protoplanet System
479
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