planetary growth with collisional fragmentation and gas drag
TRANSCRIPT
Icarus 209 (2010) 836–847
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Icarus
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Planetary growth with collisional fragmentation and gas drag
Hiroshi Kobayashi a,*, Hidekazu Tanaka b, Alexander V. Krivov a, Satoshi Inaba c
a Astrophysical Institute and University Observatory, Friedrich Schiller University, Schillergaesschen 2-3, 07745 Jena, Germanyb Institute of Low Temperature Science, Hokkaido University, Kita-Ku Kita 19, Nishi 8, Sapporo 060-0819, Japanc School of International Liberal Studies, Waseda University, 1-6-1 Nishi-Waseda, Shinjuku-ku, Tokyo 169-8050, Japan
a r t i c l e i n f o
Article history:Received 25 February 2010Revised 23 April 2010Accepted 23 April 2010Available online 11 May 2010
Keywords:Planetary formationPlanetesimalsCollisional physicsOrigin, Solar SystemJovian planets
0019-1035/$ - see front matter � 2010 Elsevier Inc. Adoi:10.1016/j.icarus.2010.04.021
* Corresponding author. Fax: +49 3641 947 532.E-mail address: [email protected] (H. Ko
a b s t r a c t
As planetary embryos grow, gravitational stirring of planetesimals by embryos strongly enhances randomvelocities of planetesimals and makes collisions between planetesimals destructive. The resulting frag-ments are ground down by successive collisions. Eventually the smallest fragments are removed bythe inward drift due to gas drag. Therefore, the collisional disruption depletes the planetesimal diskand inhibits embryo growth. We provide analytical formulae for the final masses of planetary embryos,taking into account planetesimal depletion due to collisional disruption. Furthermore, we perform thestatistical simulations for embryo growth (which excellently reproduce results of direct N-body simula-tions if disruption is neglected). These analytical formulae are consistent with the outcome of our statis-tical simulations. Our results indicate that the final embryo mass at several AU in the minimum-masssolar nebula can reach about �0.1 Earth mass within 107 years. This brings another difficulty in formationof gas giant planets, which requires cores with �10 Earth masses for gas accretion. However, if the neb-ular disk is 10 times more massive than the minimum-mass solar nebula and the initial planetesimal sizeis larger than 100 km, as suggested by some models of planetesimal formation, the final embryo massreaches about 10 Earth masses at 3–4 AU. The enhancement of embryos’ collisional cross sections by theiratmosphere could further increase their final mass to form gas giant planets at 5–10 AU in the SolarSystem.
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1. Introduction
In the standard scenario of planetary formation, terrestrial plan-ets and cores of jovian planets are formed through the accretion ofplanetesimals with initial size of 10–100 km (e.g., Hayashi et al.,1985). This process called planetary accretion has been investi-gated by statistical simulations (e.g., Wetherill and Stewart,1993; Inaba et al., 2003; Kenyon and Bromley, 2004), by N-bodysimulations (e.g., Kokubo and Ida, 1996, 2002), and by the hybridmethod that combines an N-body simulation for large bodies calledplanetary embryos with a statistical simulation for small bodies(e.g., Bromley and Kenyon, 2006; Chambers, 2008). Assuming thateach collision of bodies leads to a perfect agglomeration, thegrowth of planetesimals can be accurately computed with N-bodycodes. Inaba et al. (2001) showed that the evolution of mass distri-bution and velocity dispersion of bodies calculated by N-body sim-ulation were reproduced by the statistical simulation applying thecollision rate and the velocity-dispersion-evolution rate based on
ll rights reserved.
bayashi).
the results of orbital integrations. Therefore, statistical simulationsare reliable unless planetary embryos collide with each other afterdeveloping a long-term orbital instability.
Planetary embryos initially form through the runaway growth(e.g., Wetherill and Stewart, 1989; Kokubo and Ida, 1996). Theembryos keep their orbital separations and grow through collisionswith surrounding planetesimals (Kokubo and Ida, 1998). At thisstage, planetesimals of almost initial size dominate the total massof bodies (surface density). When embryos reach about the mass ofMars, the velocity dispersion of planetesimals is increased by thegravitational scattering at the embryos. Accordingly, anothergrowth regime sets in, referred to as oligarchic growth. At thisstage, higher velocities of planetesimals cause their collisionalfragmentation. After a chain of successive destructive collisions, of-ten called ‘‘collision cascade”, bodies get smaller and smaller untilthey are removed by gas drag in protoplanetary disks or by radia-tion pressure and/or Poynting–Robertson drag in debris disks. As aresult, collision cascade decreases the surface density, which slowsdown the growth of planetary embryos (Inaba et al., 2003; Kenyonand Bromley, 2008).
The planetary core (embryo) exceeding the critical core mass aslarge as 10 Earth masses can no longer retain a hydrostatic enve-lope, resulting in the gas accretion and formation of the gas giant
H. Kobayashi et al. / Icarus 209 (2010) 836–847 837
planets (e.g., Mizuno et al., 1980; Bodenheimer and Pollack, 1986;Pollack et al., 1996; Ikoma et al., 2001). However, since the embryogrowth is hindered by the planetesimal depletion in collision cas-cade, the final embryo can hardly grow beyond the Mars mass andthus cannot form a gas giant (Kobayashi and Tanaka, 2010).
However, it is possible that fragmentation inhibits the embryogrowth to a lesser extent than ascertained before. Collisional frag-ments orbiting a central star drift inward by gas drag. The drifttime shortens as bodies become smaller by collision cascade, untilat a certain size their motion gets coupled with gas motion. Suchcoupled bodies have lower drift velocity and thus can survivearound planetary embryos for longer time. Kenyon and Bromley(2009) suggested that accretion of those coupled bodies by plane-tary embryos may promote further growth of the latter. In theirsimulations, they assumed that the coupled bodies no longer expe-rience an inward drift. However, the drift, although at a reducedrate, is important to determine the embryo mass gain due to theaccretion of coupled bodies.
This paper investigates the embryo growth taking into accountthe accretion of fragments resulting from the collision cascadebefore their removal by gas drag. To this end, we perform an ana-lytic study that extends the model of Kobayashi and Tanaka (2010)by including the removal by gas drag, as well as statistical simula-tions. In the numerical treatment, we do not neglect the drift ofcoupled bodies by gas drag. Instead, we take into account thatthe gas drag law changes for small bodies in the both analyticaland numerical procedure (e.g., Adachi et al., 1976). The goal is tofind out to what extent the collisional fragmentation combinedwith gas drag would affect the embryo growth and whether anembryo can reach the critical core mass.
We develop the analytic theory and derive the final embryomass with fragmentation and gas drag in Section 2. In Section 3,we check the formulae for the final mass against the statisticalnumerical simulations. Section 4 contains a summary and a discus-sion of our results.
2. Theoretical model
In this section, we summarize the oligarchic growth of plane-tary embryos and the surface density decline resulting from frag-mentation. Then we derive the general formulae for the finalmass through the oligarchic growth with fragmentation.
2.1. Disk and fragmentation model
We introduce a power-law disk model for the surface mass den-sity of solids and gas. The solid surface mass density is taken to be
Rs;0 ¼ ficeR1a
1 AU
� ��q
g=cm2; ð1Þ
where a is the distance from a central star, R1 is the reference sur-face density at 1 AU, q is the power-law index of the radial distribu-tion, and fice is the factor that represents the increase of soliddensity by ice condensation beyond the snow line aice at whichthe temperature equals the ice condensation temperature ’170 K.We set the gas surface density to
Rgas;0 ¼ fgasR1a
1 AU
� ��q
g=cm2; ð2Þ
where fgas is the gas–dust ratio. In the nominal case, we adoptfgas = 240 (Hayashi, 1981). If the disk is optically thin, the gas tem-perature is given by
T ¼ 280a
1 AU
� ��1=2 L�L�
� �1=4
K; ð3Þ
where L* and L� are luminosities of the central star and the Sun,respectively. For L* = L�, this yields aice = 2.7 AU. In the minimum-mass solar nebula (MMSN) model, R1 = 7 g/cm2, q = 3/2 and fice = 1(a < aice) and 4.2 (a > aice). However, since a large amount of smalldust is present even after completion of planetesimal formation,the disk is expected to be optically thick. This would make the tem-perature lower and its radial profile different (e.g., Kusaka et al.,1970). However, these effects would not drastically influence theembryo growth except for the location of the snow line.
We take the fragmentation model described by Kobayashi andTanaka (2010). Assuming that the fragmentation is scaled by theenergy, the total ejecta mass me produced by one collision betweenbodies with masses m1 and m2 is given by
me
m1 þm2¼ /
1þ /: ð4Þ
Here the scaled impact energy / is given bym1m2v2=2ðm1 þm2Þ2Q �D, where v is the collisional velocity betweenm1 and m2 and Q �D is the critical specific impact energy needed todisrupt the colliding bodies and eject 50% of their mass(me = (m1 + m2)/2). The value of Q �D is given by the larger of thetwo colliders (m1 and m2). Note that, since Kobayashi and Tanaka(2010) separately determine the ejecta mass from m1 and m2 intheir analysis, Eq. (4) is different from their definition.
The energy threshold is given by
Q �D ¼ Q 0sr
1 cm
� �bs
þ Q0gqr
1 cm
� �bg
þ Cgg2Gm
r; ð5Þ
where r and m are the radius and mass of a body and q is its density.Benz and Asphaug (1999) provide Q0s, bs, Q0g, and bg for r = 1–107 cm from the hydrodynamical simulation of the collisional dis-persion. The first term in the right-hand side of Eq. (5) controlsQ �D for r [ 104–105 cm and the second term describes Q �D of the lar-ger bodies. For r J 107 cm, Q �D is purely determined by the gravita-tional binding energy, being independent of material properties.The collisional simulation for the gravitational aggregates yieldsCgg ’ 10 (Stewart and Leinhardt, 2009).
2.2. Isolation mass
Planetary embryos no longer grow after they have accreted allplanetesimals within their feeding zones. The width of a feedingzone is equal to the orbital separation of the embryos,~bð2M=3M�Þ1=3a, where M is the embryo mass, M* is the mass of cen-tral star, ~b ’ 10 is a factor (Kokubo and Ida, 2002). Therefore, themaximum mass or ‘‘isolation mass” satisfy Miso ¼ 2pa2
ð2Miso=3M�Þ1=3~bRs. It can be expressed as
Miso ¼ 2:8~b
10
!3=2Rs;0
2:7 g=cm2
!3=2a
5 AU
� �3 M�
M�
� ��1=2
M�; ð6Þ
where M� is the Earth mass and M� is the solar mass. The planetaryembryo mass approaches the isolation mass if fragmentation is ig-nored (Kokubo and Ida, 2000, 2002). However, if fragmentation isincluded, the final embryo mass is expected to be smaller.
2.3. Planetary growth
2.3.1. Growth without accretion of fragmentsAt the runaway stage of planetary growth, the larger planetesi-
mals grow faster than smaller ones. The solid surface density atthe runaway growth stage is determined by relatively small plane-tesimals. At the later, oligarchic stage, planetary embryos becomemassive, start to gravitationally stir up the planetesimals and in-duce their collisional disruption. The resulting fragments arequickly removed by the inward drift due to gas drag. Thus the
838 H. Kobayashi et al. / Icarus 209 (2010) 836–847
fragmentation would reduce the surface density and hence the finalembryo mass. Here, taking into account the depletion of planetesi-mals due to the fragmentation, we estimate the final embryo mass.We neglect the accretion of fragments onto the embryos, whereasthe accretion of fragments will be considered in Section 2.3.2.
At the oligarchic stage, planetary embryos are distributed uni-formly due to their mutual gravitational interaction. Since theycannot collide with each other due to their large orbital separation,they grow slowly. If a planetary embryo with mass M collides withplanetesimals of mass m, its growth rate is given by
dMdt¼Z
dmmnsa2h2m;MhPcoliXK; ð7Þ
where nsdm is the surface number density of planetesimals withmass in the range of [m,m + dm], XK is the Kepler angular velocity,hm,M = [(m + M)/3M*]1/3 is the dimensionless reduced Hill radius be-tween m and M, and hPcoli is the dimensionless collision rate. We as-sume m�M, and Eq. (7) reduces to
dMdt¼ CaccRsa2h2
MhPcoliXK; ð8Þ
where Rs is the surface density of planetesimals, hM = (M/3M*)1/3,and Cacc is the correction factor on the order of unity. Because theeccentricity dispersion e* and inclination dispersion i* of planetesi-mals are much larger than hM for kilometer-sized or larger planetes-imals, the dimensionless collision rate hPcoli is given by (e.g., Inabaet al., 2001)
hPcoli ¼Ccol
eRh2M
e�2; ð9Þ
where Ccol ’ 36 for e* = 2i*. Here, eR ¼ ð9M�=4pqÞ1=3=a is the embryo
radius scaled by hMa and independent of the embryo mass. In thispaper, we do not take into account the enhancement of hPcoli dueto atmosphere which an embryo is expected to acquire. Such anenhancement may be efficient for M J 0.1M� (Inaba and Ikoma,2003). Note that embryos grow not only through collisions withswarm planetesimals but also with neighboring embryos becausetheir separation scaled by the Hill radius decreases by their growth.Furthermore, taking into account mass distribution of planetesi-mals, the actual accretion rate becomes larger. In this paper, we ap-ply Cacc ’ 1.5.
Since dM/dt / e*�2, we need to describe evolution of e*. Twomechanisms that affect e* are viscous stirring and damping bygas. The viscous stirring of planetary embryos raises the randomvelocities of planetesimals. The stirring rate of e*2 is given by
de�2
dt
!VS
¼ nMa2h4MhPVSiXK: ð10Þ
Since embryos are uniformly distributed with a separation21=3hMa~b, their surface number density nM is given by
nM ¼1
24=3p~bhMa2: ð11Þ
For e* = 2i*� hM, the dimensionless stirring rate hPVSi is given by(Ohtsuki et al., 2002)
hPVSi ¼CVSh2
M
e�2lnðK2 þ 1Þ; ð12Þ
where CVS ’ 40. Although ln(K2 + 1) weakly depends on e*, weassume ln(K2 + 1) ’ 3 for this analysis. On the other hand, e* isdamped by gas drag and the damping rate is given by
de�2
dt
!gas
¼ �Cgas
se�3; ð13Þ
where Cgas ’ 2.1 (e.g., Inaba et al., 2001) and
s ¼ mpr2CDqgasvK
: ð14Þ
Here, qgas is the gas density in the midplane, vK is the Keplerianvelocity, and CD is the dimensionless gas drag coefficient. The latteris defined as a factor that appears in the expression for the gas dragforce acting on a planetesimal with radius r: CDpr2qgasu
2/2 with ubeing the relative velocity between the planetesimal and gas. Forkilometer-sized or larger planetesimals, CD is a constant (Adachiet al., 1976). We assume that e* is determined by the equilibriumbetween the stirring and the damping (Eqs. (10) and (13)). Equatingthe stirring and damping rates, we obtain
e�
hM¼ CVS lnðK2 þ 1ÞXKs
24=3p~bCgas
" #1=5
: ð15Þ
We can estimate e* � (sXK)1/5hM. Since sXK� 1 for the planetesi-mals (Adachi et al., 1976), our assumption of e*� hM for the deriva-tion is valid. Inserting Eqs. (9) and (15) to Eq. (8), the growth rate isfound to be
dMdt¼ CaccRsa2Ccol
eRXK24=3p~bCgas
CVS lnðK2 þ 1ÞXKs
" #2=5
h2M : ð16Þ
The time evolution of a planetary embryo in the oligarchic growthmode, neglecting fragmentation, is summarized in Appendix A.
As embryos grow, the random velocity of planetesimals in-creases, making collisions between planetesimals destructive.Fragments produced by the collisions get smaller and smaller bysuccessive collisions (collision cascade) until the smallest onesare brought inward by gas drag and are lost to the central star.Planetesimals with mass m dominate the surface density Rs duringthe oligarchic growth. Thus, the collision cascade induced by em-bryo growth reduces the surface density. The evolution of Rs dueto a collision cascade is given by (see Kobayashi and Tanaka,2010; Appendix B for a derivation)
dRs
dt¼ �R2
s ð2� aÞ2h0XK
m1=3
vðmÞ2
2Q �DðmÞ
!a�1
s123ðaÞ; ð17Þ
where h0 = 1.1q�2/3 and
s123ðaÞ ¼Z 1
0d/
/�a
1þ /� ln
�/ð1þ /Þ2
þ 12� b
" #/þ lnð1þ /Þ
( );
ð18Þ
where b is the power-law exponent of the mass distribution of ejec-ta yielded by one collision between m1 and m2 and � is a factor inthe expression for the mass of the largest ejecta: mL = �(m1 +m2)//(1 + /)2 with a use of the scaled impact energy /. For a colli-sion cascade, the mass distribution exponent a of fragments is givenby a = (11 + 3p)/(6 + 3p) for vðmÞ2=Q �DðmÞ / m�p and s123 is insensi-tive to constants � < 1 and b = 1–2 (Kobayashi and Tanaka, 2010).We set b = 5/3 and � = 0.2.
Dividing Eq. (16) by Eq. (17) and integrating, we obtain the finalembryo mass,
Mc ¼2a� 1
3
� �3=ð2a�1Þ a2CaccCcoleRm1=3
ð2� aÞ2h0s123
!3=ð2a�1Þ
v2K
2Q�D
� �3ð1�aÞ=ð2a�1Þ
3M�ð Þ�ð4�2aÞ=ð2a�1Þ lnRs;0
Rs
� �� �3=ð2a�1Þ
CVS lnðK2 þ 1ÞXKs24=3p~bCgas
" #�6a=5ð2a�1Þ
; ð19Þ
H. Kobayashi et al. / Icarus 209 (2010) 836–847 839
where v = e*vK. This equation is valid unless the surface density ismuch smaller than that of embryos, Rs�McnM. Therefore, an esti-mate of Rs for the final embryo mass is given by
Rs
Rs;0¼ CRs Mc
24=3p~ba2hMcRs;0
¼ CRs
Mc
Miso
� �2=3
; ð20Þ
where CRs � 1 is a constant. Generally, Mc should be smaller thanMiso.
We assume Q �D ¼ Q0gqrbg for the gravity regime and thusa = [11 + 3(bg � 2/15)]/[6 + 3(b � 2/15)]. For the minimum-masssolar nebula (R0 = 7 g/cm2 and q = 3/2), Eq. (19) can be re-writtenas
Mc ¼ 0:10lnðRs;0=RsÞ
4:5
� �1:21 a5 AU
� �0:63 m
4:2 1020 g
!0:48
M�
M�
� ��0:28 Q 0g
2:1 erg cm3=g2
� �0:89 q1 g=cm3
!1:85
M�; ð21Þ
where bg = 1.19 for ice. Here, we estimate ln (Rs,0/Rs) ’ 4.5 from Eq.(20) with the use of CRs ¼ 0:1 for Mc = 0.10M� and Miso = 2.8M�.With fragmentation, the final embryo mass given by Eq. (21) be-comes much smaller than the isolation mass, Eq. (6). As we willshow in Section 3 with the aid of statistical simulations, the plane-tesimal mass m is about 100m0, where m0 is the initial planetesimalmass. We assume m = 100m0 and will compare Mc with the embryomass obtained through the statistical simulation.
2.3.2. Growth with accretion of fragmentsFor the steady-state mass distribution of collision cascade (see
Appendix A), the surface density of small fragments is much lowerthan that of planetesimals and the accretion of fragments is insig-nificant for embryo growth. The collision cascade may terminate ata certain mass where destructive collisions no longer occur due tolow collisional velocities damped by a strong gas drag. The steady-state mass distribution is achieved if fragments at the low-massend of the cascade are quickly removed by the gas drag. However,when the Rs-decay time resulting from the collision cascade isshorter than the removal time of the fragments by gas drag, thetermination of the collision cascade yields a large amount of frag-ments at the low-mass end of the cascade and would dominate thesolid surface density. Here, we examine the termination of the col-lision cascade and estimate the embryo growth through accretionof such fragments.
For small fragments, the e*-damping rate resulting from the gasdrag is given by (Adachi et al., 1976)
de�
dt
� �gas;f¼ � e�g
s; ð22Þ
where g = (vK � vgas)/vK is the deviation of the gas rotation velocityvgas from the Keplerian velocity. For q = 3/2, g = 1.8 10�3 (a/1 AU)1/2. For small fragments, CD depends on u and e* becomes muchsmaller than g by the gas drag. Hence, the damping rate given by Eq.(22) is different from Eq. (13). Since the gas drag substantially dampse* of small fragments, hPVSi = 73 is independent of e* for e*� hM.Thus, the equilibrium condition between the stirring (Eqs. (10) and(11)) and the damping (Eq. (22)) gives
e�2 ¼ h3MhPVSisXK
24=3p~bg: ð23Þ
Smaller fragments have low e* because of low s.The collision cascade will no longer operate for such low e*. The
collisional energy between bodies of mass m is estimated to beme�2v2
K=4 and is much smaller than energy threshold 2mQ �D for
me = m at the low-mass end of the cascade. Therefore, at the low-mass end
e�2v2K ¼ CLQ �D; ð24Þ
where CL is a constant. As we will show, the surface density of frag-ments becomes higher at the low-mass end of the collision cascade.Our simulation yields CL ’ 1 at the low-mass end.
The surface density of the fragments decreases as a result of theradial drift by gas drag. From Eqs. (23) and (24), we estimate thescaled stopping time ~sstop of the fragments at the low-mass endof the cascade to be
~sstop ¼sXK
g¼ 24=3p~b
h3MhPVSi
CLQ �Dv2
K
; ð25Þ
’ 19CLa
5 AU
� � M0:1M�
� ��1 Q �D3:1 106 erg=g
!; ð26Þ
where Q �D ’ 3:1 106 erg=g for ice bodies with r = 10 m and we willderive M � 0.1M� in the following analysis. Since ~sstop � 1, the cou-pling between the fragments and gas is weak. Therefore, the driftvelocity of fragments is given by 2g2a/s (Adachi et al., 1976). Sinces / a3/4 for the fragments and Rs / a�q, the Rs-depletion rate is gi-ven by dRs/dt = �2(9/4 � q)g2Rs/s. Eliminating s of fragments atthe low-mass end of the cascade by Eq. (25), we get
dRs
dt¼ � 9
4� q
� �Rsh3
MhPVSiXKgv2K
21=3CLp~bQ �D: ð27Þ
Since hPcoli ’ 11:3ffiffiffiffieRp
is independent of M for e*� hM, integrationof Eq. (8) divided by Eq. (27) gives another formula for the final em-bryo mass,
Mf ¼27=3CL
~bpCacca2hPcolið3M�Þ1=3
3ð9=4� qÞgPVS
Q �Dv2
K
ðRs;0 � RsÞ" #3=4
; ð28Þ
where we set M = 0 at Rs = Rs,0. We recall that hPVSi = 73 andhPcoli ¼ 11:3
ffiffiffiffieRpin Eq. (28) are independent of Mf for e*� hM. In
addition, since Q �D for the strength regime is almost independentof the fragment mass m, we neglected the mass dependence of Q �Dfor the derivation of Eq. (28). Assuming Rs,0� Rs and q = 3/2, weobtain
Mf ¼ 0:14~b
10
!3=4a
5 AU
� �3=2 M�
M�
� �3=8 q1 g=cm3
!�1=8
Rs;0
2:7 g=cm2
� �3=4 Q �D3:1 106 erg=g
!3=4
M�: ð29Þ
The final mass given by Eq. (29) is also much smaller than the iso-lation mass, Eq. (6). Furthermore, Mf is comparable to Mc given byEq. (21). When planetesimals have a large Q �D because of a rigidmaterial (high Q0g) and/or a large size, Mc is larger than Mf. In theopposite case, Mf exceeds Mc. The final mass is supposed to be thelarger of the two, Mc and Mf. We apply Q �D for r = 10 m and will com-pare Mf with the embryo mass resulting from the statisticalsimulation.
3. Numerical simulations
3.1. Method
Many authors attacked the problem of the planetary growthfrom planetesimals to embryos by N-body simulations (e.g.,Kokubo and Ida, 1996, 2002), the statistical method (Wetherilland Stewart, 1993; Kenyon and Bromley, 2004), and the hybridmethod (Kenyon and Bromley, 2008; Chambers, 2008). We apply
840 H. Kobayashi et al. / Icarus 209 (2010) 836–847
the statistical method developed by Inaba et al. (1999, 2001),which we briefly explain now. The mass distribution of bodies inorbit around a central star evolves through mutual collisions, tak-ing into account gravitational focusing of colliding bodies (see In-aba et al. (2001) for detailed expressions). Along with the massevolution, velocity dispersion changes by gravitational perturba-tions, collisional damping, and gas drag. Equations for the massdistribution of bodies are integrated simultaneously with equa-tions for their velocity distribution.
The time evolution of the differential surface number densityns(m) at a distance a is given by
@mnsðm; aÞ@t
¼ m2
XK
Z m
0dm1
Z 1
m�m1�me
dm2
ðhm1 ;m2 aÞ2nsðm1; aÞnsðm2; aÞhPcoli dðm�m1 �m2 þmeÞ
�XKmnsðmÞZ 1
0dm2ðhm;m2 aÞ2nsðm2; aÞhPcoli
þ @
@mXK
Z 1
mdm1
Z m1
0dm2ðhm1 ;m2 aÞ2nsðm1; aÞ
nsðm2; aÞhPcoliðm1 þm2Þf ðm;m1;m2Þ
� 1a@
@a½amnsðm; aÞvdriftðm; aÞ; ð30Þ
where d(x) is the delta function, (m1 + m2)f(m,m1,m2) is the mass offragments less than m produced by a collision between m1 and m2,and me is the total mass of the fragments (given by Eq. (4) in ourfragmentation model). We apply the dimensionless collisional ratehPcoli which Inaba et al. (2001) provide as a function of m1, m2,and e* and i* by compiling previous studies. Eq. (30) describes themass transport in the two-dimensional space composed of massand distance (radial direction). The first and second terms in theright-hand side of Eq. (30) represent the mass transport along themass coordinate caused by coagulation and the third term does thatdue to fragmentation. The fourth term describes the mass transportalong the radial coordinate due to the drift of bodies. We calculatethe transport on a grid of mass and radial bins.
Assuming a power-law mass distribution of fragments,(m1 + m2) f(m,m1,m2) is given by
ðm1 þm2Þf ðm;m1;m2Þ ¼me
mmL
� ��bfor m < mL;
me for m P mL;
8<: ð31Þ
where we recall the mass of the largest ejecta mL = �(m1 + m2)//(1 + /)2 and the total ejecta mass me given by Eq. (4).
The drift velocity is characterized by the dimensionless param-eter ~sstop, where the scaled stopping time ~sstop is given by aX2
Ks=u.The relative velocity u between a body and gas is equal to(e* + i* + g)aXK. For ~sstop K 1 and ~sstop � 1, the drift velocity canbe written as (Adachi et al., 1976; Inaba et al., 2001)
vdriftðm;aÞ ¼� 2gaXK
~sstop
~s2stop
1þ~s2stop
for ~sstop K1;
�2 ags
½2Eð3=4ÞþKð3=4Þ29p2 e�2þ 4
p2 i�2þg2n o1=2
for ~sstop� 1;
8><>:ð32Þ
where E and K are the elliptic integrals. Both regimes in Eq. (32) canbe combined into
vdriftðm; aÞ ¼ �2ags
~s2stop
1þ ~s2stop
½2Eð3=4Þ þ Kð3=4Þ2
9p2 e�2 þ 4p2 i�2 þ g2
( )1=2
: ð33Þ
Indeed, if ~sstop K 1, then e* and i* are damped by the strong gas dragto e*� g and i*� g, and Eq. (33) reduces to the first of Eq. (32). If~sstop � 1, Eq. (33) simply coincides with the second of Eq. (32).
Since we treat small fragments, we consider three gas-drag re-gimes: quadratic, Stokes, and Epstein ones. The gas drag coeffi-cients CD for these regimes are given by (Adachi et al., 1976)
CD;quad ¼ 0:5; CD;Stokes ¼ 24=Re; CD;Epstein ¼ 16=K Re; ð34Þ
where K = 1.7 ((10�9 g/cm2)/qgasr) is the Knudsen number andRe = 4.4(u/c)K�1 is the Reynolds number with c being the soundvelocity of gas (Adachi et al., 1976). All three regimes can be de-scribed together by adopting
CD ¼ CD;quad þ1
CD;Stokesþ 1
CD;Epstein
� ��1
: ð35Þ
Since the mean collision rate is a function of e* and i*, the evo-lution of e* and i* should be calculated precisely. We consider grav-itational perturbations from other bodies, collisional damping, andgas drag. Their net effect can be calculated as a square ofdispersions:
de�2
dt¼ de�2
dt
!grav
þ de�2
dt
!gas
þ de�2
dt
!coll
; ð36Þ
di�2
dt¼ di�2
dt
!grav
þ di�2
dt
!gas
þ di�2
dt
!coll
; ð37Þ
where terms with subscripts ‘‘grav”, ‘‘gas”, and ‘‘coll” indicate thetime variation due to gravitational perturbations, gas drag, and col-lisions, respectively. These terms are provided by Inaba et al. (2001).Note that for the collisional damping, both fragments and mergersresulting from a collision have the velocity dispersion at the gravitycenter of colliding bodies, as described in their Eq. (30). We followthe planetary growth by simultaneously integrating Eqs. (30), (36),and (37).
Kokubo and Ida (1998) and Weidenschilling et al. (1997)showed that the radial separation of orbits of runaway bodiesformed in a swarm of planetesimals is about 10 times their mutualHill radius. Since dynamical friction from the field planetesimalsmakes orbits of the runaway bodies nearly circular and coplanar,the orbital crossing of these runaway bodies never occurs beforethe gas is dispersed (Iwasaki et al., 2001). Therefore, they cannotcollide with each other. In our simulation, when the bodies reacha certain mass mrun such that the sum of their mutual Hill radiiequals the radial-bin width divided by ~b, the bodies are regardedas ‘‘runaway bodies” which do not have any collisions and dynam-ical interaction due to close encounters among them, following In-aba et al. (2001). We set ~b ¼ 10. As the bodies grow, theirseparation measured in the Hill radii decreases and these bodiescan no longer be treated as ‘‘runaway bodies”. Therefore, mrun in-creases during the simulation and then the number of the ‘‘run-away bodies” decreases.
Moreover, Inaba et al. (2001) set the number of collisions duringa numerical time interval to be an integer using a random numbergenerator and hence keep the number of bodies an integer. Ourprocedure is different. Instead of dealing with the number ofbodies, we treat the mathematical expectation N of the numberof bodies with mass larger than m. When N is much lower thanunity, the bodies may not yet exist. We introduce a certain criticalnumber Nc (a ‘‘threshold”) to get rid of such bodies. If N(mc) = Nc,we neglect collisions with bodies larger than mc and dynamicalinteractions with them. The value of Nc should be of the order ofunity, because we can say that bodies are not yet born for N� 1,whereas choosing a high Nc would delay the embryo growth.Bodies with masses ranging from mrun to mc are treated as‘‘runaway bodies”. The mass range of the ‘‘runaway bodies” tends
1
10
100
1000
1
10
100
1000
1 10 100 10001
10
100
1000
(a) 1 105 yr
(b) 2 105 yr
(c) 4 10 5 yr
m / m0
Cum
ulat
ive
Num
ber
Fig. 1. Evolution of the mass distribution of bodies with m0 = 2 1023 g, q = 2 g/cm2, and e�0 ¼ 2i�0 ¼ 0:002 at 1 AU. The initial mass distribution is shown with thedotted curve in panel (a). Collisional fragmentation is neglected. Gray dashed linesrepresent the results of N-body simulation shown in Fig. 4 of Inaba et al. (2001).
0.001
0.01
0.1
0.001
0.01
0.1
1 10 100 1000
0.001
0.01
0.1
5 yr
5 yr
(a) 1 10
(b) 2 10
(c) 4 10 5 yr
m / m0
e* , i*
Fig. 2. Evolution of the velocity dispersion of bodies for the same conditions as inFig. 1. The solid and dotted lines indicate e* and i*, respectively. Circles and trianglesrepresent e* and i* calculated by N-body simulation shown in Fig. 4 of Inaba et al.(2001), respectively.
3
6
9
96 7 8 6 7 8 9
log r [cm]
(a) 3.2AU (d) 9.0AU
A 104yrB 105yr
C 106yr
D 4 106yr
AB C D
H. Kobayashi et al. / Icarus 209 (2010) 836–847 841
to extend during the simulation for low Nc, although being keptsmall in N-body simulation (Kokubo and Ida, 2002). We alwaysstart the simulation with Nc = 0.1 and Nc may change from 0.1 to10 during the simulation to keep the small mass range of the ‘‘run-away bodies”. As shown in Figs. 1 and 2, our simulation with such achoice of Nc reproduces the mass and velocity-dispersion distribu-tions obtained from the N-body simulation, presented in Figs. 4and 5 of Inaba et al. (2001). Where about 100 runs of the statisticalcode by Inaba et al. (2001) are required to reproduce the N-bodysimulation, our simulation does the same with only one run.
0
0
3
6
9
0
3
6
9
18 20 22 24 26 28 18 20 22 24 26 28
log m [g]
log
n c
(b) 4.5AU
(c) 6.4AU
(e) 13AU
(f) 18AU
A BC
D
A BC
D
A
B C D
AB
C
D
Fig. 3. The mass distribution of bodies at 104 (A), 105 (B), 106 (C), 4 106 (D) years,neglecting fragmentation. Different panels correspond to different radial annuli.
Table 1Material properties.
Q (erg/g) b Q (erg cm3/g2) b C q (g/cm3)
3.2. Embryo growth
3.2.1. Without fragmentationWe calculate the evolution of the number and velocity disper-
sion of bodies by summing up the time variations coming fromall mass and radial bins and by integrating them over time. Weintegrate Eq. (30) through the fourth order of Runge–Kutta methodfor the mass evolution and Eqs. (36) and (37) through the linearmethod for e* and i* evolution. We set the mass ratio betweenadjacent mass bins to 1.2, which we found sufficient to reproduceN-body simulations (see Figs. 1 and 2). Fig. 3 shows the resultsignoring fragmentation (Q �D ¼ 1) with a set of six concentricannuli at 3.2, 4.5, 6.4, 9.0, 13, and 18 AU for M* = M�, R1 = 7 g/cm2, and q = 3/2. We consider planetesimals with initial mass ofm0 = 4.2 1018 g (radius of r0 = 10 km for q = 1 g/cm3) and velocitydispersion given by e* = 2i* = (2m0/M*)1/3. The planetesimals are as-sumed to be composed of ice whose physical parameters are listedin Table 1. We artificially apply the gas surface density evolution inthe form Rgas = Rgas,0exp(�t/Tgas,dep), where Tgas,dep is the gasdepletion timescale.1 Here, we set Tgas,dep = 107 years.
Fig. 3a shows the mass distribution of bodies at 3.2 AU from104 years to 4 106 years. After 105 years, the mass distributionof small bodies (m [ 1024 g) is a single power law, which is
0s s 0g g gg
Ice 7.0 107 �0.45 2.1 1.19 9 1Silicate 3.5 107 �0.38 0.3 1.36 9 3
1 Assuming a constant Rgas gives almost the same results for the final embryo massbecause we consider time spans t 6 Tgas,dep.
,
0
10
20
0
10
20
20
82 4 6 2 4 6 8
log
n c
log r [cm]
(a) 3.2AU
(b) 4.5AU
(c) 6.4AU
(d) 9.0AU
(e) 13AU
(f) 18AU
A 104yr
B 105yr
C 106yr
D 4 106yr
A B
C
D
A
B C
D
A
B C
D
842 H. Kobayashi et al. / Icarus 209 (2010) 836–847
consistent with that of the runaway growth, nc /m�5/3 (Makinoet al., 1998). At the same time, the oligarchic growth results inthe flatter distribution for large bodies (m J 1024 g). Such largebodies have almost a single mass (�3 1026 g at 106 years and�2 1027 g at 4 106 years). As the growth of the large bodiesproceeds, the random velocities of small bodies exceed the escapevelocity in the entire mass range of small bodies (m [ 1024 g) be-cause of the viscous stirring by large bodies. Then, the mass dis-tribution starts to depart from this power law. Collisions betweensmall bodies at such velocities lead to a flat mass distribution forsmall bodies (m [ 1021 g at 106 years and m [ 1022 g at4 106 years) and a steep one for large bodies. This means thatplanetesimals dominating the surface density become larger after105 years.
The velocity dispersion evolution (e* and i*) at 3.2 AU is shownin Fig. 4. At 104 years, e* and i* are proportional to m�1/2, beingdetermined by the dynamical friction. At 105 years, e* and i* form [ 1022 g are independent of mass, whereas e* and i* of largerbodies still preserve an m�1/2 dependence. The viscous stirring bylarge bodies dominates e* and i* of small bodies. In this case, e* isgiven by Eq. (A1). After 106 years, the velocity dispersion of smallbodies (m [ 1022 g at 106 years and m [ 1025 g at 4 106 years)reaches the equilibrium between the stirring and the gas drag. Inthis case, e* is given by Eq. (15). However, bodies with intermediatemass do not yet have the velocity dispersion in the equilibrium.The velocity dispersion of bodies dominating the surface densityis still given by Eq. (A1). The growth of large bodies are describedby Mn given by Eq. (A3) in 105–107 years (see Appendix A).
In the outer region, the mass distribution and velocity disper-sion evolve in a similar way but in a longer timescale. At4 106 years, the bodies with m J 1024 g grow through the oli-garchic mode at a [ 9 AU. Beyond 9 AU, the largest bodies havenot yet reached 1024 g by that time.
0
10
5 10 15 20 25 5 10 15 20 25
log m [g]
A B CD
AB
C
D
3.2.2. With fragmentationWe now take into account fragmentation, using Q �D in Eq. (5).
Kenyon and Bromley (2009) suggested that the small bodiescoupled with gas ([1 m) affect the embryo growth. To treat the
−5
−4
−3
−2
−1
0
18 20 22 24 26 28
6 7 8 9
104 yr105 yr
106 yr
4 106yr
log m [g]
log
(e* , i
* )
log r [cm]
Fig. 4. The velocity-dispersion distribution at 3.2 AU after 104, 105, 106, 4 106
years of evolution. Fragmentation is neglected. The solid and dotted lines indicate e*
and i*, respectively.
coupled bodies, we set the smallest mass to be 4.2 g (radius of1 cm for q = 1 g/cm3).
The mass distribution and velocity evolution that we calculatedwith fragmentation are shown in Figs. 5 and 6. During the first105 years, bodies grow through mutual collisions. Fragments(m [ m0) are not yet numerous because of the low velocity disper-sion (Fig. 6). At t = 106 years, the runaway growth occurs inside6.4 AU. Similar to the case without fragmentation, the cumulativenumber nc is almost proportional to m�5/3 for m J 1020 � 100m0
and the distribution of large bodies with m J 1024 g is flat. Since
Fig. 5. Same as Fig. 3, but with fragmentation.
−7
−6
−5
−4
−3
−2
−1
0
5 10 15 20 25
2 4 6 8
104 yr 105 yr
106 yr
4 106yr
log m [g]
log
(e* , i
* )
log r [cm]
Fig. 6. Same as Fig. 4, but with fragmentation.
103 104 105 106 10710−8
10−6
10−4
10−2
100
Time [year]
Em
bryo
Mas
s [M
]
Mc
Mf
Fig. 7. Evolution of the embryo mass at 3.2 AU with fragmentation (black solid line)and without fragmentation (gray solid line). Dashed lines indicate Mc (Eq. (19);black) and Mf (Eq. (28); gray).
0.8 3 7 100.001
0.003
0.01
0.03
0.1
0.3
1
3
10
30
Distance [AU]
Em
bryo
Mas
s [M
]
Snow
Lin
e
Mc
Miso
Mf
1
Fig. 8. The embryo mass at 107 years for initial planetesimal mass of 4.2 1018 gwith fragmentation (filled circles) and without fragmentation (open squares).Dashed lines indicate Mc (Eq. (19); black) and Mf (Eq. (20); gray). Gray solid lineshows Miso (Eq. (6)) for comparison. The vertical line represents the snow linea = aice = 2.7 AU.
103 104 105 106 10710−8
10−6
10−4
10−2
100
Time [year]
Em
bryo
Mas
s [M
]
Mc
Mf
Fig. 9. Same as Fig. 7, but at 1 AU.
H. Kobayashi et al. / Icarus 209 (2010) 836–847 843
the velocity dispersion of planetesimals �e*vK exceeds their escapevelocity for m [ 1024 g, planetesimals dominating the surface den-sity become larger after 106 years. In contrast to the case withoutfragmentation, the mass of bodies dominating the surface densityceases at about 100m0, owing to a high collisional velocitye�vK J
ffiffiffiffiffiffiffiQ �D
p. The slope of nc for m � 1011–1020 g (r = 0.1–10 km)
is nearly �(5 + 3p)/(6 + 3p), which is typical of a collision cascadefor v2=Q �D / m�p (Kobayashi and Tanaka, 2010). The downwardmass flux along the mass coordinate is constant with mass in thecollision cascade. For m [ 1011 g, the velocity dispersions (e* andi*) are effectively damped by the gas drag in the Stokes regime(see Fig. 6). Because the collisional energy � e�2v2
K
is lower than
the energy threshold Q �D
for a destructive collision, the down-ward mass flux becomes insignificant compared to that in the col-lision cascade. Therefore, the number of the bodies increasesaround 1011g where e�2v2
K � Q �D. Bodies with m � 105 g (r � 1 m)are most efficiently removed by the radial drift and smaller bodies(m� 105 g or r� 1 m) are coupled with gas. As a result, the massdistribution becomes wavy (see Fig. 5). The surface density ofbodies with e�2v2
K � Q �D is much higher than that of the coupledbodies. However, planetesimals with �100m0 dominate the sur-face density. At 4 106 years, the oligarchic growth is inhibitedby fragmentation inside 5 AU, where the number of fragments de-creases compared to that at 106 years.
Fig. 7 shows the embryo-mass2 evolution at 3.2 AU. At the run-away growth phase (the first [105 years), the embryo mass growsas an exponential function of time. The growth rate becomes muchlower in the oligarchic mode at 105–106 years. The embryos exceedthat for the case without fragmentation. They are substantially fedby small fragments whose velocities are damped by gas drag andare low (see Fig. 6), resulting in their rapid growth. However, theygrow at a sluggish pace at about 4 105 years because the surfacedensity of planetesimals is decreased by collision cascade followedby removal of small bodies by gas drag. As a result, the final embryomass is reduced by fragmentation. In this case, since Mc and Mf arecomparable, the embryo mass is determined by both effects. Evenso, the final embryo mass is consistent with Mc and Mf.
Fig. 8 presents the embryo mass at 107 years as a function ofdistance, including the results for three additional annuli at 1,
2 Here, embryo mass represents the average mass of the ‘‘runaway bodies”.
3 The results come from two simulations: one in a < 2.7 AU and the other ina > 2.7 AU.
1.4, and 2.0 AU.3 In simulations at a 6 2.7 AU, we use silicate bodies(Table 1) instead of icy bodies. Decline of the planetesimal surfacedensity caused by fragmentation results in embryos smaller thanthose for the case without fragmentation inside 6 AU. The massesreach only 0.01–0.1M� in 107 years. The embryo masses ata [ 1.5 AU and a = 2.7–5 AU are consistent with Mc or Mf. Insidea [ 1.5 AU, the final mass is determined by the largest of the twomasses, Mc and Mf (see Fig. 9). For m0 = 4.2 1015 g (r0 = 1 km),the embryo mass is determined by Mf because of Mf > Mc (seeFig. 10). In addition, if both Mc and Mf are smaller than Mn in Eq.(A3), the fragmentation is nearly negligible and the embryo massis given by Mn (see Appendix A for an analysis of embryo growthwithout fragmentation).
103 104 105 106 10710−8
10−6
10−4
10−2
100
Time [year]
Em
bryo
Mas
s [M
]
Mc
Mf
Fig. 10. Same as Fig. 7, but for m0 = 4.2 1015 g (r0 = 1 km).
1016 1018 1020 10220.001
0.003
0.01
0.03
0.1
0.3
1
3
10
30105 106 107
initial planetesimal mass [g]
Em
bryo
Mas
s [M
]
initial planetesimal radii [cm]
Mc
Mf
Miso
Fig. 11. The embryo mass at 3.2 AU at 107 years for the minimum-mass solarnebula model (R1 = 7 g/cm2 and q = 3/2) as a function of the initial planetesimalmass. The symbols and lines are same as Fig. 8.
1016 1018 1020 10220.01
0.03
0.1
0.3
1
3
10
30
100
300105 106 107
initial planetesimal mass [g]
Em
bryo
Mas
s [M
]
initial planetesimal radii [cm]
Mc
Mf
Miso
Fig. 12. Same as Fig. 11, but for R1 = 70 g/cm2.
844 H. Kobayashi et al. / Icarus 209 (2010) 836–847
Since the initial mass of planetesimals depends on their forma-tion process, which is not well understood, it is worthwhile inves-tigating the embryo mass dependence on the initial planetesimalmass. Fig. 11 shows the dependence of the embryo masses on m0
at 3.2 AU for the minimum-mass solar nebula after 10 millionyears. If the initial mass is smaller than �1017 g, the embryo massis determined by accretion of small fragments. For larger m0,growth is halted by the collision cascade. The embryo mass in-creases with m0 because the timescale of the surface density de-cline due to collision cascade is longer for larger planetesimals,strengthened by self-gravity (large Q �D). If m0 J 1021 g, the embryogrows without substantial fragmentation. For a massive disk(R1 = 70 g/cm2), the embryo mass can reach the Earth mass. In thiscase, Mc and Mf determine the embryo mass because of the shortgrowth time for r0 = 1–100 km (see Fig. 12).
4. Summary and discussion
In this paper, we studied analytically and numerically thegrowth of planetary embryos in the framework of the standard sce-nario. We took into account that embryos growing in the oligarchicmode pump up relative velocities of planetesimals, which causestheir collisional fragmentation. We also considered the fate ofsmallest fragments of the resulting collision cascade, namely theirinward drag in the ambient gas and possible accretion by nascentembryos.
Taking into account fragmentation, we have analytically de-rived the final mass of a planetary embryo in Section 2 in twocases: the mass Mc of an embryo growing through the accretionof planetesimals which are removed by collision cascade and themass Mf in the case where an embryo grows through collisionswith fragments which are removed by gas drag. In Section 3, weshowed that the final embryo mass obtained in numerical simula-tions can be reproduced by the larger of the two analytic estimates,MAX(Mc,Mf). If the embryo mass Mn of Eq. (A3) for the case with-out fragmentation is smaller than MAX(Mc,Mf), fragmentation isnegligible and the final mass is given by Mn. Altogether, our analyt-ical formulae provide a good estimate of the final mass of planetaryembryos.
The gas giant planet formation via core accretion requires thesolid core as large as �10M� (e.g., Ikoma et al., 2001). Kenyonand Bromley (2009) suggested that the planetary embryo may fur-ther grow by the accretion of bodies coupled with gas because thecoupled bodies are no longer involved in collision cascade. Takinginto account three regimes of gas drag (see Eq. (35)), we found thatthe collision cascade halts at bodies having v2 � Q �D, i.e. at sizesthat are larger than the critical coupling size corresponding to~sstop ¼ 1. As a result, coupled bodies should be produced only in lit-tle amounts and can hardly contribute to planetary growth, despitetheir low velocity dispersion in the laminar disks considered here.
In this paper, we derive the final embryo masses, assuming auniform distribution of solid bodies. However, the planetary em-bryos can open gaps in a solid disk, which would affect the accre-tion of fragments on the embryo and their removal by the gas drag.Tanaka and Ida (1999) derived a condition for the gap openingaround a migrating embryo. Replacing the embryo’s migration
H. Kobayashi et al. / Icarus 209 (2010) 836–847 845
time by the drift time s/2g2 for fragments, we can obtain a criterionfor an embryo with mass M to open a gap in the solid disk fromtheir Eq. (3.6),
sXKh2M
4pg2 P 0:81ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1þ 0:45ðsXK=2pÞ2=3
qþ 1
� �2
; ð38Þ
where s is given by Eq. (14) for fragments surrounding the embryos.Since Eq. (38) requires sXK/2p� 1 for M J 0.1M�, the critical frag-ment radius rg above which Eq. (38) is satisfied is given by
rg ’ 1:0 102 M0:1M�
� ��1=3� a5 AU
�3=8
m; ð39Þ
where we adopt q = 3/2 and CD = CC, Stokes. Levison et al. (2010) per-formed N-body simulations involving fragments and foundrg � 30 m for M P M�, which is consistent with Eq. (39). On theother hand, as we have shown, the solid surface density is reducedby the elimination of fragments at the low-mass end of collisioncascade due to gas drag. In addition, the embryo grows throughthe accretion of the fragments for small initial planetesimals. Theradius re of the fragments is estimated by Eq. (25) as
re ’ 5:0�
M0:1M�
��1=2� a5 AU
�1=2
m: ð40Þ
Eqs. (39) and (40) give re� rg. This means that the collision cascadequickly grinds planetesimals down to the size much less than rg. Asa result, the gap formation in the solid disk composed of fragmentswith the radius re does not occur and the embryo growth would notbe influenced.
Considering the range of initial planetesimal radii r0 = 1–100 km, and the range of disk surface densities R1 � 7–70 g/cm2
at 3.2 AU (1–10 MMSN), our results suggest that the fragmenta-tion averts the planetary growth at M � 0.1–10M� at several AU.This is consistent with the results of planetary growth simulationsbeyond 30 AU by Kenyon and Bromley (2008). The cores approach-ing the critical mass of 10M� can only form at a = 3–4 AU, onlyfrom planetesimals that have r0 � 100 km initially, and only indensest nebula with R1 � 70 g/cm2. Thus it appears problematicto explain formation of giant planets at 5–10 AU, such as Jupiterand Saturn in the Solar System, even for R1 � 70 g/cm2 andr0 � 100 km.
An effect that could help further is the enhancement of the em-bryo’s collisional cross section due to a tenuous atmosphere ac-quired by the embryo in the ambient gas. Inaba and Ikoma(2003) and Inaba et al. (2003) have shown that a core with massexceeding about 0.1M� could grow by that effect to the critical coremass. As found here, the embryo masses can indeed be about0.1M� under the minimum mass solar nebula conditions, at leastfor an initial planetesimal radius greater than 10 km.
On any account, our results strongly suggest that increasing thedisk surface density and/or the initial embryo size helps forming lar-ger cores. This brings up the question whether the values that weneed to grow the embryo to the critical mass, 10 MMSN andr0 � 100 km – or less, if the atmospheric effect is taken intoaccount – appear plausible. As far as the disk density is concerned,the answer is probably yes. 10 MMSN is close to the gravitationalstability limit, and all densities below this limit cannot be ruled out.For instance, Desch (2007) and Crida (2009) point out that the MMSNprofile is inconsistent with the ‘‘Nice model” (Gomes et al., 2004) andshould be replaced with another surface density profile, whichwould possibly imply larger surface densities at several AU.
We now consider the initial size of planetesimals. The mecha-nisms of planetesimal formation are highly debated but, despiteintensive effort, remain fairly unknown. Classical models in whichdust smoothly grows to planetesimals with r0 � 1 km face the‘‘meter barrier” problems: first, meter-sized objects should be lost
to the central star as a result of gas drag (Weidenschilling, 1977;Brauer et al., 2008), and second, further agglomeration of meter-sized objects upon collision is problematic (Blum and Wurm,2008). Accordingly, in the last years, competing scenarios weresuggested that circumvent these barriers, such that the ‘‘primaryaccretion” mechanism proposed by Cuzzi et al. (2008) and ‘‘gravi-turbulent” formation triggered by transient zones of high pressure(Johansen et al., 2006) or by streaming instabilities (Johansen et al.,2007). These models all imply rapid formation of rather largeplanetesimals in the r0 � 100–1000 km range. Support for thesescenarios may come from the analysis of left-over planetesimalsin the Solar System. Morbidelli et al. (2009), for instance, suggestthat the initial planetesimals should be larger than 100 km toreproduce the mass distribution of asteroids in the main belt.
Our last remark is related to the Type I migration of bodies dueto interaction with gas (e.g., Tanaka et al., 2002). The planetary em-bryo grows in the runaway mode followed by the oligarchic one.The timescale of the runaway growth is proportional to the initialsize of bodies. In addition, the embryos rapidly grow through col-lisions with small fragments for small r0 even in the oligarchicgrowth. The embryo eventually grows to a larger mass if it is largeinitially, while it forms earlier if it starts with a smaller mass.Therefore, initially small planetesimals seem to be preferred forthe core formation prior to their removal due to Type I migration.However, this is only valid for laminar disks. Turbulence may helpprevent the loss of growing embryos to Type I migration (e.g.,Laughlin et al., 2004). So, the alternative scenarios of planetesimalformation that all require turbulent disks may help here, too.
Acknowledgments
We thank Glen R. Stewart and an anonymous referee for theirthorough reviews of this paper. H.K. and A.V.K. are grateful to Mar-tin Ilgner for helpful comments.
Appendix A. Planetary growth without fragmentation
When the growth timescale Tgrow of embryos is much longerthan the e*-damping timescale Tgas by gas drag, e* settles to theequilibrium between the stirring and the gas drag and is givenby Eq. (15). However, the growth timescale at the beginning of oli-garchic growth is relatively short. In this case, e* cannot reach theequilibrium for Tgrow� Tgas. Using these two limits of e*, we willderive the embryo-mass evolution as follows.
In the non-equilibrium case (Tgrow� Tgas), e* increases as Mgrows. Then, combining the growth rate (Eqs. (8) and (9)) withthe stirring rate (Eq. (10) with Eqs. (11) and (12)), we get
e� ¼ 3CVS lnðK2 þ 1ÞM4=3
210=3pCacc~ba2Rs
~RCcolð3M�Þ1=3
" #1=2
; ðA1Þ
where we assume that the initial embryo mass and eccentricity dis-persion are much smaller than M and e*, respectively. Inserting Eq.(A1) into Eq. (8) with Eq. (9) there results
dMdt¼
210=3p~b CaccRsCcola2eR� �2XK
9M�CVS lnðK2 þ 1Þ: ðA2Þ
If the embryo grows with this rate for sufficiently long time, it is va-lid use M = 0 at t = 0 as the initial condition. In this case, the embryomass Mn at time t is given by
Mn ¼210=3p~b CaccRsCcola2eR� �2
XKt
9M�CVS lnðK2 þ 1Þ; ðA3Þ
which is independent of m.
846 H. Kobayashi et al. / Icarus 209 (2010) 836–847
In the equilibrium case (Tgrow� Tgas), the growth rate of M is gi-ven by Eq. (16). Also we consider that the embryo mass becomesmuch larger than the initial mass. Integrating Eq. (16) over timewith M = 0 at t = 0, we have
Me ¼CaccRa2Ccol
eRXK
3ð3M�Þ2=3
" #324=3p~bCgas
CVS lnðK2 þ 1ÞXKs
" #6=5
t3: ðA4Þ
The embryo mass is proportional to m2/5 from the m dependence ofs. The growth time of embryo mass Me is similar to that estimatedby Kokubo and Ida (2002).
Fig. 13 shows the embryo growth at 3.2 AU. The runawaygrowth switches to oligarchic growth after 105 years. The embryomass is consistent with Mn given by Eq. (A3) in 105–107 years.On contrary, the embryo mass does not agree with Me given byEq. (A4). That is because e* of the representative planetesimal,which dominates the surface density, does not reach the equilib-rium. As seen in Fig. 3, the representative planetesimal grows be-cause its random velocity is higher than its surface escapevelocity. Although e* reaches the equilibrium for small bodies, thatof the representative planetesimal cannot do due to the growth.Since the representative planetesimal growth is caused by the per-fect sticking even for a high velocity, the growth came from theartificial treatment. In practice, the growth halts at �100m0 whenfragmentation is included, because the impact energy exceeds Q �D.In this case, e* of the representative planetesimals reach the equi-librium and the embryo mass is determined by Mc or Mf because ofthe active fragmentation.
Appendix B. Mass depletion due to collision cascade
Here, we derive the surface-density depletion rate due to colli-sion cascade, following Kobayashi and Tanaka (2010). Focusing onthe mass transport by fragmentation, we neglect terms in theright-hand side of Eq. (30) except for the third one:
@mns
@t¼ @
@mXK
Z 1
mdm1
Z m1
0dm2ðhm1 ;m2 aÞ2nsðm1; aÞ
nsðm2; aÞhPcolim1 ;m2ðm1 þm2Þf ðm;m1;m2Þ: ðB1Þ
Collisional velocities exceed the surface escape velocity of collidingbodies in collision cascade, resulting in ðhm1 ;m2 aÞ2hPcoli ¼ ð17:3=
103 104 105 106 10710−8
10−6
10−4
10−2
100
Time [year]
Em
bryo
Mas
s [M
] Mn
Me
Fig. 13. Evolution of the embryo mass at 3.2 AU, neglecting fragmentation. Thedashed lines indicate Mn (Eq. (A3); black) and Me (Eq. (A4); gray).
2pÞðr1 þ r2Þ2 ¼ h0m2=31 ½1þ ðm2=m1Þ1=32. Here, r1 and r2 are the radii
of m1 and m2, respectively. Since collisions with m1�m2 mainlylead to the mass transport along the mass coordinate in collisioncascade, the f function given by Eq. (31) becomes
f ðm;m1;m2Þ ¼m
m1
� �2�b/
1þ/�/
ð1þ/Þ2
h ib�2for m
m1< �/ð1þ/Þ2
;
/1þ/ for m
m1P �/ð1þ/Þ2
:
8><>: ðB2Þ
Furthermore, ðhm1 ;m2 aÞ2hPcoli¼h0m2=31 and /¼ðvðm1Þ2=2Q �Dðm1ÞÞðm2=
m1Þ. Introducing the dimensionless variable x = m/m1, we can re-write Eq. (B1) as
@mns
@t¼ @
@mA2XKh0m
113�2a v2ðmÞ
2Q �DðmÞ
� �a�1
Z v2ðm1Þ=2Q�Dðm1Þ
0d/
/1�a
1þ /
Z �/=ð1þ/Þ2
0dxx2a�8=3þða�1Þp�b
"
�/ð1þ /Þ2
!b�2
þZ 1
�/=ð1þ/Þ2dxx2a�14=3þða�1Þp
35; ðB3Þ
where ns = Am�a and v2ðmÞ=Q �DðmÞ / m�p. The upper integrationlimit over / in Eq. (B3) is different from Eq. (31) of Kobayashi andTanaka (2010), which comes from the different definition of f. De-spite that, Eq. (17) is consistent with Kobayashi and Tanaka becauseof a insignificant contribution of collisions with / > v2ðm1Þ=2Q �Dðm1Þ to the integral over / in their Eq. (31). Since v2ðm1Þ=2Q �Dðm1Þ � 1, we change the upper limit to1. We also take into ac-count that v2ðmÞ=Q �DðmÞ / m�p and that @m ns/@t = 0 in a steady-state collision cascade. Then, Eq. (B3) gives a = (11 + 3p)/(6 + 3p).Note that Eqs. (31), (34), (B2), and (B3) of Kobayashi and Tanaka(2010) contain mistakes: their power-law exponents of m shouldbe 11/3 � 2a, as in Eq. (B3).
In the oligarchic growth, since planetary embryos sufficientlyincrease the collisional velocity of the swarm planetesimals, colli-sion cascade depletes the swarm. Smaller bodies quickly reach thesteady state of collision cascade. Therefore, integrating Eq. (B3)over mass in the steady-state collision cascade (a = (11 + 3p)/(6 + 3p)), we obtain
@Rs
@t¼ �A2m11=3�2a
max XKh0v2ðmmaxÞ
2Q �DðmmaxÞ
� �a�1
Z 1
0d/
/1�a
1þ /
Z �/=ð1þ/Þ2
0dxx1�b �/
ð1þ /Þ2
!b�224
þZ 1
�/=ð1þ/Þ2dxx�1
#; ðB4Þ
where mmax is the mass of largest planetesimals in collision cascade.Integrating over x on the right-hand side of Eq. (B4), and usingRs ¼ Am2�a
max=ð2� aÞ, we obtain Eq. (17). In Eq. (17), we convert mmax
to m, according to the definition in Section 2. Moreover, Eq. (17) in-cludes the additional terms due to the mass transport of the rem-nant according to Kobayashi and Tanaka, although the transportterms are much smaller than others.4
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