Kobe 2013 Workshop
August 7-8

Critical Importance ofErosive Collisions inCollisional Cascade.
Hiroshi Kobayashi (Nagoya U.)• Collisional fragmentation in planetary accretion• Collisional cascade• Importance of erosive (cratering) collisions(See; Kobayashi & Tanaka 2010)

Planet Formation Model
105-6yr

Planetesimal Formation
105-7yr

Planetary embryo and core formation
106-7yrGas giant planet formation
107-9yr

Debris Disk109yr

Embryo Growth
Initial planetesimal:tion1010

ta

1km

en

11agm

at 3.2AUofr

0.10.1N

Embryo mass[M⊕]3xMMSN0.01
0.01

10310410510Time [yr]6107

3xMMSN1010

ag.

Initial planetesimal:

fr

111kmoat 3.2AU

NEmbryo mass[M⊕]Embryo Growth

0.1
0.1

Frag.

Fragmentation

0.01
0.01

10controls embryo
3104

105

10

Time [yr]6growth.
107

Collision Cascadem n(m)m̄eme n(m) m v n(m )dm

m

11/3 21(m̄ /m)

α=11/6# of collisionsm(r-3.5)(Dohnanyi 1969; Tanaka et al. 1996)m̄ /mmp= (11 + 3p)/(6 + 3p)(OʼBrien & Greenberg 2003; Kobayashi & Tanaka 2010)

Δmmass•Power-law mass distributionm(System mass in large bodies).
•The mass ﬂux is determined by me(mʼ/m).

Mass Flux
Mass conservation:mns (m)F (m)+=0tmm e = Me m t

ns(m): diﬀerential surface number density
F(m): mass ﬂux along the mass coordinate
mp v 22mt QD

For ns(m) = A m-α,F (m) =A2

K h0 m1132{Me ( ) ln Mm ( )total ejecta massscaled by target mass.

2v(m)2QD (m)[11d0Me ( )] ln [1Me ( )]}mean fragment mass
divided by target mass

Leinhardt & Stewart (2012)
m1:m2=1:40

110.8

KT

0.210model

0.50.5me / mt

Total Ejecta Mass/ Target Mass

Fragmentation Model

0.1
0.1
0.08
0.1

φ/(1+φ)1Qc/QD* = φ10

Important mass ratio
F(m,~y)/F(m)v2= 200, 2000
QD
• Collisions with small

0.06
0.04

Q<cQ

*
Dbodies determine
the ﬂux.

• Erosive collision
important.

0.02m̄
21 QD= 10 –10mv2

Q c > Q D*

010−10−8−6−410 10 10 10mass ratio, y−2100m̄e = 102–101m

Eﬃciency of erosive collisions
v2= 200, 2000
QDF’(m,y) / F(m)10.80.6Q

0.4<cQ

Qc > QD

*
D

10−10

are 10 times more
important forconstant v2/QD*.

• For gravity

0.20Erosive collisions•*−8−6−410 10 10 10y=m
2 / m1 ymassratio,

−2

100

regime, theeﬃciency is afactor 4-5.

Other Eﬀects• The power-law index does not changeby erosive collisions.

• Previous studies show that the powerlaw distribution we analytically derived
is modulated by a wavy structure.

• However, erosive collisions make thewavy structure smooth.
Erosive collisions also change the mass
distribution of bodies in collisional cascade.

Planet Formation• Runaway growth produces single
planetary embryo in each annulus,
while the total mass of leftoverplanetesimals surrounding the embryo
is much more massive.

• The embryo grow through collisional
accretion with planetesimals, until
planetesimals deplete by collisionalcascade.

Simulation
NumberFollow mass andvelocity evolution.

Mass(e.g., Wetherill & Stewart;
Inaba et al.; Weidenschilling et al.;Kenyon & Bromley)

Basic Equations(see HK, Tanaka, Krivov, Inaba 2010)

mns (m, a)
t

m
2

=

m

dm1

K

dm20m m1 me(hm1 ,m2 a)2 ns (m1 , a)ns (m2 , a) Pcol

(mm2 + m e )m1

K mns (m)dm2 (hm,m2 a)2 ns (m2 , a) Pcol
0+m1

mdm1K
mdm2 (m1 + m2 )f (m, m1 , m2 )0ns (m1 , a)ns (m2 , a)(hm1 ,m2 a)2 Pcol
1[amns (m, a)vdrift (m, a)],a a

de 2
dt
di 2
dt

==de 2dt
di 2dtgrav

de 2dtgrav

di 2
dt++gasde 2dtgas

di 2dt++me=
m1 + m21+= m1 m2 v 2 /2(m1 + m2 )2 QD,(m1 + m2 )f (m, m1 , m2 )coll,coll

=me
me

mmLb

for m < mL ,for mmL ,Mass & velocity evol.
eccentricity

inclination

Radius [cm]surface density[g/cm2]

Distance from the host star [AU]

size evolution

0Embryo Mass [M ]410 years

10−210−410−610−810310410510610m ns [g/cm ]021010−210−42

surface density

Time [year]

1cm

1m

1km

radius

103km

3.2AU

7100

Embryo Mass [M ]510 年後10−210−410−610−810310410510610m ns [g/cm ]021010−210−42

surface density

Time [year]

1cm1m

1km

radius

103km

3.2AU

7100

Embryo Mass [M ]610 years

10−210−410−610−810310410510610m ns [g/cm ]010−41cm1m

1km

radiusembryos−2

planetesimals

10fragments

2102

surface density

Time [year]

103km

3.2AU

710

Fragment size
log10 vr /vK

−14.7

⇤

−22*vr < QD

−31x106yr−4⌅

M−50.1M⇥1cm

12⇥ 18 ⇤

⌅

aQDm
3.2AU
6 erg/g
33.2
AU
3101m
1km 10 km

radiusFinal Mass
10 x MMSN

Fragmentation

withnoEmbryo Mass [M ]

10Myr

300
100Miso30
103.2AU

31Mc
Mf0.3
0.1
0.03
0.01

110100initial planetesimal radius [km](Kobayashi et al. 2010; 2011)

Summery• Collisional cascade:• The total ejecta mass from a single collision is mostimportant and the others are negligible.

• Erosive collisions mainly determine the mass ﬂux and thedepletion time of bodies.

• Planet formation:• Massive embryos are formed in a massive disk.
• Large planetesimals produce massive embryos but
via slow growth (and vice versa).

Application: Solar System
10MF

8=
2M

Saturnars=sarMM

82 WMF

1ε=MMtlenma=

862.
,upelt2n
8
1 W ma•Saturn49.
2=/2

•Mars

Mars

11

•Kobayashi+12εars

2•Kobayashi+11M

5•Jupiter

Jupiter &Jupiter
SaturnMF

Surface density
x [MMSN], x

Σ= x ΣMMSN(a/1AU)-1.5•Kobayashi&
27 10

20

100 200

InitialPlanetesimal
Radius [km]Planetesimal
radius[km]700Dauphas13

Solar System Formation

まとめ