1053
Progress of Theoretical Physics, Vol. 67, No.4, April 1982
The Gaseous Flow around a Protoplanet
In the Primordial Solar N ehula
Satoshi
MIKI
Department 0/ Electronic Engineering
Tokyo University 0/ Agriculture and Technology, Tokyo 184
(Received August 31, 1981)
Two dimensional stationary gaseous flow around a protoplanet in the primordial solar
nebula is studied numerically. When the protoplanet revolves with a small eccentricity, flow
pattern in the vicinity of the protoplanet is divided into three: Inside a belt which encloses the
orbit of the protoplanet and has a width of four times as large as the Hill radius, the inflow gas
is attracted by the protoplanet and returns baCk along the hairpin curve owing to the Coriolis
force. Out of this belt, the gas element passes by straightly. The gas element within the Hill
sphere of the protoplanet forms an atmosphere rotating in a prograde direction with smaller
velocity than the Keplerian velocity around the protoplanet. After a planetary mass has
exceeded one· tenth of the present Jupiter's mass, two wings of weak shock waves appear in the
pass-by flow region.
It is to be noticed that at a time of encounters with planetesimals, a gas element on the
hairpin flows changes its orbital radius around the Sun randomly and migrates in the solar
nebula radially. The migration time of the gas which makes up the mantle component of the
Jupiter is estimated to be about 2 x 10 7 yrs.
§ 1.
Introduction
Recently, formation processes of the planetary system in the presence of the
solar nebula have been investigated by many authors. These indicate that the
nebula gas played important roles on the growth of planets and their thermal
history: In the disk nebula which is formed around the Sun, solid particles
sediment owing to the gas friction and form a thin dust layer at the equatorial
plane of the disk. 1 ),2) This is followed by the fragmentation of the dust layer, that
is, the formation of planetesimals. 2 )-4) The gas drag force governs the random
motion of the planetesimals and, hence, their collisional growth. Eventually a
few number of proto planets with the mass of the order of 10 25 g is formed. 5 )
Hayashi, N akazawa and Adachi 6 ) have found that the growth time of protoplanets is determined, mainly, by the velocity of migration of the planetesimals
toward the Sun due to the gas drag effect. Furthermore, as shown by Mizun0 7 )
and others S ),9) the nebula gas surrounding the planetary core collapses onto the
surface to form a massive H2 and He envelope when the mass of the protoplanetary core exceeds some critical mass. This is the formation process of the
giant planets. Finally as long as the nebula gas exists, the protoplanet with the
mass greater than 10 26 g has its own dense primordial atmosphere, of which
s.
1054
Miki
blanketing effect keeps the protoplanet at a very high temperature, 10) and the
nebula gas dissolves into the melted protoplanet. l l )
At the same time, it is to be noticed that the protoplanets should have effect
on the nebula gas, that is, the gravities of protoplanets should disturb the motion
of the gaseous disk which primarily rotates circularly around the Sun. It is the
purpose of this paper to investigate numerically the disturbance of the gas motion
in the disk nebula by a protoplanet and to see the role of the disturbance to the
formation processes of the planetary system. Here we simply assume that the
fluid motion can be represented by the two dimensional steady flow in the orbital
plane of the protoplanet. A similar problem (but without shock waves) has been
studied semi-analytically by N akazawa and Hayashi,12) of which results are
consistent with ours.
In § 2 we describe basic equations governing the motion of the gaseous nebula
and in § 3 the numerical method is give!1. In § 4, the results obtained from the
numerical computations are presented. We will see that shock waves appear in
the nebula gas when the mass of the protoplanet exceeds some critical value.
The effect of the eccentric motion of the protoplanet is also examined.
A gas element encounters a number of protoplanets and its motion is disturbed by each encounter. Then, the orbital radius of the gas element will be
changed at random. The migration of the nebula gas due to the encounters with
protoplanets is discussed in § 5 in relation to the formation of giant planets.
§ 2.
Basic equations
We assume, for simplicity, that a protoplanet revolves circularly around the
Sun in the same direction as that of the differentially rotating gaseous disk.
Moreover, we neglect the self gravity of the nebula gas and the time variation of
the mass of the protoplanet. Then, the equations of motion for a gas element
which moves under the gravities of both the Sun and the protoplanet are given
by*)
(2'1)
Here p isthe gas density, P the gas pressure, M the sum of the masses of the Sun
and the protoplanet, f1. the mass fraction of the protoplanet to the total, G the
gravitational constant, and rs and rp the vectors from the Sun and from the
protoplanet to the gas element, respectively. Moreover u is the velocity vector
of the gaseous element in the inertia frame_
We adopt a rotating frame with rectangular coordinates of (x, y) where the
*) As will be seen later we shall obtain the steady gaseous flow by solving the initial and boundary
value problem. Hence, the basic equations are written in the forms with time derivatives.
The Gaseous Flow around a Protoplanet in the Primordial Solar Nebula
1055
Fig. 1. The geometry of the problem in the
orbital plane. The symbols Sand P denote
the Sun and a protoplanet, respectively, and
Ll and L2 are so-called Lagrangian points.
The numerical boundaries, shown by the
dashed curves, and the y-axis are reduced to
the rectangle and the straight lines, respectively, under Hill's approximation. The
solid curves are the zero potential curves,
u(x, y)=O, and the hatched region near the
protoplanet is called the Hill sphere.
s
Sun is fixed to the x-axis and the origin coinsides with the position of the
protoplanet (see Fig. 1). Since we are interested in the gaseous motions in the
region near the protoplanet, let us normalize the units such that h = 1, w = 1 and
M = 1, where w = (GM/ap 3)1/2 and ap is the orbital radius of the protoplanet.
Furthermore, h is the Hill radius given by
h=apc with
(2 2)
c=(fJ./3)1/3.
0
We will also normalize the values of P and p by those far from the protoplanet, i.e., j5 = p/Po and 15 = p/Po. Thus, neglecting the small quantities, v 2 ,
compared to the Coriolis' term, we can rewrite Eq. (201) in the following form:
~~ +(v (1)v=-2vXno
!e
pr-2f1
15-
f1
U,
(2 3)
0
where v is the fluid velocity in the rotating frame and n is the unit rotational
vector perpendicular to the orbital plane. The effective potential, U, is given by
(2 4)
0
where UL is an integral constant and
(2 5)
0
As to the equation of state we adopt the polytropic relation with index y. In Eq.
(2 3), K is the ratio of the characteristic fluid velocity to the sound velocity of the
gas, Co = ( YPo/po )1/2, i.e.,
0
(2 6)
K=2hw/Co.
0
Since c is a quantity smaller than 0.03, we can expand Eq. (2 4) by a series
in powers of c. Neglecting the first and the higher order of c (this approximation
is called Hill's approximation 13» and setting to be UL = 3jc2 + 9/2, for convenience,
0
s.
1056
Miki
we have for U,
(2'7)
It is to be noticed that the potential given by Eq. (2' 7) does not depend on the
planetary mass f.L and has a symmetry with respect to the y-axis. Therefore,
only two parameters of the characteristic Mach number, K, and of the adiabatic
exponent, t, govern the features of the gaseous motion.
Finally, the equations of motion are supplemented by the well-known conservation of mass given by
ap + v /7 p
- -- - p-/7 v.
at
§ 3.
a)
(2'8)
Numerical method and subsidiary conditions
Numerical method
In order to find the steady gaseous flow, we will solve numerically the
equations given above as an initial value problem in the Eulerian scheme. That
is, the equations are solved under the given initial conditions and boundary
conditions, and the computations are continued until the a/at terms become small
enough. The steady flow, in fact, can be achieved after two or three Keplerian
periods of the protoplanet as long as suitable initial conditions are adopted.
As the equations of motion are antisymmetric with respect to the origin, we
have only to consider the half region where x 20. The region is divided into a
number of Eulerian cells (we usually use 18 (for x) X 39(for y) cells). Numerical
integrations of the equations have been performed by means of Fluid-In-Cell
method. 14 ) After the motion of the gaseous nebula becomes steady, we will
sometimes rezone the central region near the protoplanet into finer cells in order
to see detailed features of the gaseous motion inside the Hill sphere.
In order to eliminate numerical sound noises and to stabilize the numerical
computations, we add an artificial viscosity term, which is proportional to av/at,
to the right-hand side of Eq. (2'3). It is to be noticed that the artificial viscosity
term becomes small enough when the gaseous flow turns steady.
b)
Boundary conditions
We enclose the gas with the boundaries where y = ±6, x =5 and 0 numerically. We have found that out of this region the disturbance due to the gravity of
the protoplanet is so small that the gaseous flow is considered to maintain the
circular motion around the Sun, which is obtained by setting /7 p =0 in Eq. (2'3)
and rp ~ 1 in Eq. (2'7), and is given by
The Gaseous Flow around a Protoplanet in the Primordial Solar Nebula
{p,
Vx,
Vy}={l, 0, -3x/2}.
1057
(3-1)
Thus, the physical values on the boundaries, except for the y-axis, are fixed to
those given by Eq. (3- 1), and antisymmetric conditions are assumed on the y-axis.
When the computational region is rezoned, the values on the outer boundaries are
set to those of the solution before the rezoning.
Numerical inner boundary conditions are forced at the four innermost mesh
points, whose distances from the origin is about fifty times as large as the actual
radius of the protoplanet. At these points the pressure gradient of the gas is
almost balanced with the gravity of the protoplanet and, hence, the radial velocity
is small enough. 12 ) Thus, the slip condition is applied to these points. To see the
influence of the adopted inner boundary conditions, we have also calculated
models under other inner boundary conditions and found that the gaseous flow of
the region outside the Hill sphere was hardly affected even under the alternative
inner boundary conditions.
Values of K and r
As is mentioned in § 2, the gaseous flow is almost characterized by K. The
value of K, given by Eq. (2-6), is determined by the mass of the protoplanet and
its position from the Sun as well as the model of the solar nebula, for which we
adopt that obtained by Kusaka et al.I) We have K = 0.57 when a protoplanet has
the present terrestrial mass and is at the Earth's orbit. As is shown in Fig. 2, the
model with K = 0.57 represents not only the Earth's model but also the other
models. These models are called the Earth's type.
When the mass of the protoplanet becomes large enough, the value of K
c)
Jupiter
Saturn
•
pre-JupiteriMJ/IO)
pre-5cturn(Msl2)
Neptune
uranus +
•
Mars
Mercury
+
10-7 L....:--L_L---L--"'----,l-:-.--L--;!
0.2
1.0
10
o (o.u.l
50
Fig. 2. The values of K as a function of the
mass (measured in units of the solar mass)
and the position of the protoplanet. In the
figure, M, and Ms denote the masses of the
present Jupiter and Saturn, respectively. As
to the model of the solar nebula we adopt
that obtained by Kusaka et al. 1)
S. Miki
1058
exceeds unity and the shock waves are expected to appear in the nebula gas. For
the case of the proto-Jupiter which has the mass of one-tenth of the present Jovian
mass, (1/10)M], we have K=1.24. This model is named the proto-jupiter's
type.
In our case, the nebula gas is, mainly, composed of the hydrogen molecules
and helium atoms with the solar compositions. So, as for the value of r we have
used 1.425. In order to see the effect of the adopted value of r on the gaseous flow
we have also computed for the case r = 1 (isothermal). However, the results are
found to be almost the same as those in the adiabatic case except for the small
difference in the density distribution within the Hill sphere.
§ 4.
Numerical results
The model of the Earth's type (K = 0.57)
In Fig. 3, the flow pattern is shown for the case K = 0.57. This shows that the
gaseous flow is divided into three regions, i.e., the hairpin flow region, the passby flow region and the atmospheric region. A gaseous element which comes
close to a protoplanet from a far distance along the circular orbit of x = Xl, where
Osxls2, returns back along the hairpin curve owing to the Coriolis force and
leaves the protoplanet to approach the orbit X = - Xl. This means that the gases
of the outer belt (Osx s2) and of the inner belt (-2sx sO) are exchanged with
a)
y
Fig. 3. Steady flow pattern for the case of Earth's
type with K=O.57. Two parabolic curves
denote the zero potential curves and cross
over the x·axis at x = ± L The arrows show
the velocity vectors of the gas element and
solid curves denote the equidensity contours
to which the attached number is the values of
p/Po.
The Gaseous Flow around a Protoplanet in the Primordial Solar Nebula
1059
each other.
On the other hand, we have a different flow pattern in the case of a gaseous
element which revolves circularly around the Sun with the orbital radius XI
greater than 2. In this case, the gaseous element passes by the planet nearly
straightly and flows away again into the stream of the circular orbit with the
same orbital radius as that before encounter. The stream lines in the hairpin
region and pass-by region are very similar to the orbits obtained for the case of
particle motion under the gravities of both the Sun and the protoplanet. 15 ) This
is because in the region outside the Hill sphere the pressure force of the gas does
not play an important role, since the density distribution of the nebula gas is
almost flat.
The stream lines which have started from the upper boundary at X = ± 2 are
led to the stagnation point on the x-axis at x = ±0.7. Inside a sphere whose
radius is 0.7, the stream lines make closed circles and the gases form the proper
atmosphere of the protoplanet. We will discuss the proper atmosphere later in
more detail.
b)
The model of proto-fupiter's type (K = 1.24)
The results are shown in Fig. 4. The flow pattern of this model is again divided into the three regions which are similar to those for the model of Earth's
type. Particularly, the width of the hairpin flow region is about four times as
large as the Hill radius in spite of the different mass of the protoplanet and of the
different value of the characteristic Mach number, K, from those of Earth's type.
However, there is a significant difference between the cases of Earth's type
and of the present. This is an appearance of a pair of shock waves. As seen
from Fig. 4(a), the tongues, in which the gas density is higher than that of the
surwundings, extend from the Lagrangian points Land L2. The fronts of the
tongues, where the density increases rapidly, are the shock waves. These shock
waves as well as the tongues extend widely from x = 1 (or -1) to x = 3 (or - 3).
The shock waves are not strong as seen from the density enhancement. The
appearance of the shock waves is also found to be general for K greater than
unity.
In Fig. 4(b), we can see the flow pattern in detail near the protoplanet. As
pointed out earlier, inside a sphere whose radius is 0.7, the gas revolves circularly
around the protoplanet progressively and form the proper atmosphere, which is
distinguished from the solar nebula. Apparently, prograde circular motion is not
always realized in invicid gaseous flow. But, generally speaking, prograde
motion is preferable energetically so that the gaseous flow in the proper atmosphere can be concluded to be circularily prograde motion except for the very
close region to the protoplanet. Moreover the density distribution of the proper
atmosphere is almost determined by the static equilibrium. Since the fluid
s.
1060
Miki
y
11
f f
!I
I
I
I
I
f
f
f
f
f
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f
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(a)
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,,
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'A\ '\ '\ '\ ,
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't
t-t-+-;.~-'l
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Z ! I ! ! ,,' r , , , ,
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Fig. 4. Steady flow pattern for the case of proto·Jupiter type with K=1.24. In (a),
regions with a density higher than 1.414 are marked by dots, in order to clarify
the position of the shock waves. In (b), detailed features near the protoplanet
obtained as a result of rezoning.
The Gaseous Flow around a Protoplanet in the Primordial Solar Nebula
1061
velocities of the atmospheric gases are less than one-fourth of Keplerian motion
around the protoplanet except for the innermost two or three mesh points where
the coarse mesh gives larger velocities because of the numerical inaccuracy.
c)
Models with non-zero eccentricity
A protoplanet has an eccentricity of about 0.05 6 ) as well as the present
planets. Here we examine the effect of the eccentric motion of a protoplanet on
the gaseous flow near the protoplanet.
Let ap be the semi-maior axis and e be the eccentricity of the orbit of a
protoplanet. For simplicity, we consider the gaseous flows only when the
protoplanet is at the perihelion, a/ = ape 1 - e). Moreover we assume the steady
gaseous flows, though there exists the time variation of the gravitational force
due to the eccentric motion of the protoplanet. This assumption is justified when
the time scale of the gravity variation, i.e., the Keplerian time, TK, is longer than
that of the propagation of the sound wave over the distance of about 4h, i.e.,
or
(4°1)
which is satisfied for our values of K. We adopt a rotating frame with the
rotational velocity of the protoplanet at the perihelion point. By means of the
same procedure as that given in § 2, we have a modified effective potential, instead
of Eq. (207), i.e.,
(4 2)
0
where
Xe=-e/3c.
(4 3)
0
For the outer boundary conditions Eq. (301) is simply modified as
{p,
Vx,
vy}={l, 0, -3(x-xe)/2}.
(4°4)
Substituting U' in place of U in Eq. (2 3), we have calculated the steady flows of
the nebula gas by the same procedure as earlier. In this case, however, the flow
is not antisymmetric to the y-axis. Hence we enclose the fluid within the four
numerical walls of x = ± 5 and y = ± 6 and on the boundaries the physical values
are put to be those given by Eq. (4°4).
The flow patterns of the model of proto-jupiter's type (K = l.24) with Xe =-1
are shown in Fig. 5(a). The features of flow pattern are essentially the same as
those obtained in the case with e=O besides the fact that the flows are almost
antisymmetric to the point (x, y) = (Xe, 0). That is, the gaseous flows are divided
into the same three regions and a pair of shock waves also appear. The width
of the hairpin flow region is again about four times as large as the Hill radius.
0
S. Miki
1062
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l-;-;-1-1-f-l,---<',~~--1
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\:
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Flow pattern for the cases of proto-Jupiter type with the eccentricities with
(a) and Ie=-2 (b). A protoplanet is assumed to be at the perihelion.
For comparison, the zero potential curves for the model with e=O are also
displayed by the solid curves. Even in the case with Ie = -1 we can see a pair
of shock waves in the pass-by flow. But for the case with Ie = -2, a single shock
wave can only be observed and the hairpin flow disappears.
Ie=-l
The Gaseous Flow around a Protoplanet in the Primordial Solar Nebula
1063
Thus, we can find that the gas near the perihelion within an outer belt (-l:Sx
:S 1) is exchanged with that of the inner belt (- 3:S x :S -1).
For the case with Xe = -2, the model has very different flow patterns from
those of the above-mentioned models, as shown in Fig. 5(b). In this case, the
surface of the Hill sphere is exposed to the strong gas flow and the dynamical
pressure, pu 2 , at x =0 is as large as the static pressure at one-fourth of the Hill
radius, which can be estimated from the results of the model of proto-jupiter's
type with e = O. Thus, the atmospheric flow region will be suppressed strongly
and in our computation where the fine resolution cannot be expected, in fact, the
proper atmospheric region disappears. At the same time, the hairpin region can
be hardly observed and only single shock wave, which includes the protoplanet,
appears in the pass-by region.
From these calculations, we can conclude that the results obtained for the
case of the models with e=O can also be applied to the models with the eccentricity, as long as we are concerned with models with IXel:S 1. For the model of the
proto-jupiter's type with the mass of (l/10)M], the restriction, Ixel:S1, indicates
that e:S 0.095, which will be satisfied sufficiently as well as in the case of the
present Jupiter. Thus, the proto-Jupiter can be expected to have a pair of stable
shock waves during the whole period of the revolution.
d)
Summary of the numerical results
From our numerical calculations mentioned above, we can summarize the
features of the gaseous flow encountering the protoplanet as follows.
(l) Flow pattern in the vicinity of the protoplanet is divided into three; that is,
the pass-by flow, the hairpin flow and the proper atmospheric flow.
(2) The width of the hairpin flow region is almost independent of both the
protoplanetary mass and the value of K and is about four times as large as the
Hill radius.
(3) A pair of weak shock waves appears in the pass-by flow region when K is
greater than unity.
(4) In the atmospheric region, the gases revolve circularly and progressively
around the protoplanet. The velocity is less than one-fourth of that of the
Keplerian motion ar~:)Und the protoplanet.
(5) The above characters are also valid even in the case where the protoplanet
has a finite eccentricity, as long as we have e:S3E.
§ 5.
Migration of the nebula gas
The orbit of a gaseous element in the hairpin flow region is found to be
changed by the gravities of protoplanets as is shown in Fig. 4(a). If there exists
a density gradient in radial direction or a density deficient region near the orbit
S. Miki
1064
of a giant planet, the nebula gas is expected to migrate over the interplanetary
space by the successive encounters with protoplanets. We will discuss the
migration of the gas of solar nebula through the random encounters with
protoplanets, which can be treated as a diffusion process.
We assume that the mass of a proto-Jovian planet is already more than five
times as great as that of the Earth, and the surrounding gases within the' Hill
sphere have collapsed to the protoplanetary surface. 9 ) Hence, the nebula gas
depletes near the orbit of the protoplanet. Besides the protoplanet, a large
number of planetesimals rotate around the Sun in the solar nebula near the orbit
of the proto planet considered here. Let n( m, e) be the number density of
planetesimals with mass, m, and eccentricity, e. We assume that n( m, e) has the
following form:
n( m, e)dedm= f( m)g( e)ededm .
(5'1)
As for f(m), we adopt the power low spectrum according to Nakagawa,S)
no ( m
f(m)= { m2
)-312
~2
(5'2)
otherwise
where no is a normalization factor determined later and ml and m2 are the
minimum and the maximum mass of planetesimals, respectively (we assume
m2'Pmd. For g( e), we use the normal distribution, that is,
g(e)=
<e~>m
2
2
exp[-e /<e >m],
(5'3)
where the root-mean-square, <e 2>M2, is given byS)
(5'4)
In the above, m is the mean mass of the planetesimals and 7J is a constant
depending upon the distance from the Sun and the gas density.
We can obtain the normarization factor, no, from the surface density of the
solar nebula, Ps, that is,
f
mn( m, e)dedm=
l:.;z ==
q ,
(5'5)
where t is the mass fraction of condensable materials to the total and Hz is the
scale height of the orbits of planetesimals, which is connected with the inclination
and is given by Hz = a< e 2 >M; / /2 (a being the representative distance from the Sun
of the region considered here). Thus, we have no = q/2m2.
N ow, we estimate the collision time between a planetesimal and a gas
The Gaseous Flow around a Protoplanet in the Primordial Solar Nebula
1065
element. Here it should be noticed that all of planetesimals cannot contribute to
the migration of the nebula gas, as shown in § 4, but planetesimals with small
eccentricities of e:S3.s can disturb the gas, of which fraction, Y, is given by
(5·6)
Hereafter, Y is assumed to be independent of the mass m of the planetesimal and
is replaced by the value for the case m = m2. It will be justified because of a small
mass dependence (.s/<e 2 >M 2cx m l !15). Then, the mean collision time of a gas
element with planetesimals with masses between m and m + dm is given by
(5'7)
As seen in § 4 the orbital radius of the gaseous element lS changed by the
encounter with a planetesimal with an orbital radius of ap, when the conditions
1h :Sia- api :S2h. Here we exclude the region of ia- api:S 1h because of its small
contribution to the migration of the gas. In other words, a gas element can
encounter a planetesimal existing within the two rings which have the orbital
radii of a ± l.5h and the rectangular section of h x 28, where 8 is the height of the
Hill sphere and is equal to 0.6h (see Fig. 6). Furthermore the last parenthesized
term of Eq. (5'7) represents the mean relative velocity between the gas element
and the planetesimal and VK is the Keplerian velocity at the orbital radius a.
Assuming the mean free path of gas migration to be of 3h, we have a
diffusion constant, D, which is given by
130
m2
<e 2>1/2
Y t;psa2(~)5/3L
m2
3M0
'T'
1 K
,
(5'8)
where ml/m2 is neglected compared with unity. Using this, we can write down,
simply, the migration time Llt during which the nebula gas migrates radially over
the distance, Lla, due to the successive encounters with planetesimals; that is, we
z
-=-2.:..:.h_ _
x
Fig. 6. Cross section of the rings (hatched region) within which a gas element is returned
back along the hairpin curves by the encounter with a protopJanet, P.
1066
S. Miki
have for Llt,
Llt= (LlfJ)2 .
(5·9)
Now, we estimate the migration time for the Jupiter's region. As for the
maximum mass, we adopt m2 = 1 X 10 26 g, which is the order of the mass of the
Galileian satellites. Almost all of the present planets have an eccentricity of the
order of 0.05. Planetesimals, considered here, which have the masses of the order
of 10 26 g or less and rotate around the Sun in the nebula gas, have an eccentricity
smaller than that of the present planets, say, 0.02. When we put to be 0.02 for
<e 2 >M;, then we have Y =0.14. As for the value of ps, we adopt 15% of the value
given by Kusaka et al.1) at the Jupiter region and half the values of the Saturn and
Uranus regions, because new models for the core masses of the giant planets are
given by Mizuno 7) and Slattery.16) For S, we use Cameron's value; =0.017.
Using the above values for the parameters, we can obtain the diffusion
constant. During the period of 2 x 107 yrs, within which the nebula gas is dissipated due to the solar wind and/or solar UV radiation, 17),18) the nebula gas at the
Jupiter region can migrate over the region of Lla/a""'-0.20. This indicates that the
gas, which is needed to form the Jupiter (whose mass is about 300 times that of
the Earth), can be supplied over the region between 6.3 a.u. and 4.3 a.u.
For the cases of Saturn's and Uranus' regions, the migration lengths, Lla/a,
within a time interval of 2 X 107 yrs, are about 0.12 and 0.07, respectively. Thus,
for these regions the gas migration mechanism due to the encounters with
planetesimals does not work fully and the nebula gas will be dissipated by the
solar wind and/or solar UV before the gas is migrated efficiently to these
planetary regions.
Finally, we will emphasize that the migration time, given by Eq. (5·n
depends sharply on the mean eccentricity of planetesimals. Hence, in order to
estimate more quantitatively the mass of nebula gas which is migrated to be
captured by the proto-Jupiter, statistical behaviors of the motion of planetesimals
must be accurately studied.
s
Acknowledgements
The author would like to acknowledge the suggestion of the problem, critical
discussions and continuous encouragement of Professor C. Hayashi. He is also
grateful to Dr. K. N akazawa for helpful discussions and the critical reading of
manuscripts, to Dr. M. Itoh for continuous encouragement and to Dr. Y. Nakagawa and Dr. H. Mizuno for helpful discussions. Numerical computations were
performed at the Data Processing Center of Kyoto University. A part of this
work was supported by a Grant-in-Aid for Scientific Research (454047) and a
The Gaseous Flow around a Protoplanet in the Primordial Solar Nebula
1067
Grant-in-Aid for special Project Research (511409) of the Ministry of Education,
Science and Culture.
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