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Planetesimal DynamicsFormation of Terrestrial Planets from Planetesimals

106

105-6

107-8

Planetesimals

Protoplanets

yr

yr

yr

Terrestrial planets

Gas/Dust

..................................................................... .....................................................................

Protoplanetary disk

Eiichiro Kokubo 小久 保英 一 郎National Astronomical Observatory of Japan

OutlineBasic Dynamics of Planetesimals

(e.g., Stewart & Wetherill 1988; Ida 1990; Ida & Makino 1992a, b)

Runaway Growth of Planetesimals(e.g., Wetherill & Stewart 1989; Kokubo & Ida 1996)

Oligarchic Growth of Protoplanets(e.g., Kokubo & Ida 1998, 2002; Thommes+ 2003; Chambers 2006)

Giant Impacts of Protoplanets(e.g., Chambers & Wetherill 1998; Agnor+ 1999; Kominami & Ida 2002;

Raymond+ 2004; Kokubo+ 2006)

Formation of Terrestrial Planets: The Movie(Miura & Kokubo 2007)

Basic Hypotheses of Planet FormationDisk Hypothesis

• A planetary system forms from a light circumstellar disk(protoplanetary disk) that is a by-product of star formation.

• A protoplanetary disk consists of gas and dust.

Planetesimal Hypothesis• Planetesimals are formed from dust.• Solid planets are formed by accretion of planetesimals.• Gaseous planets are formed by gas accretion onto solid

planets (“core accretion” model).

(Safronov 1969; Hayashi+ 1985)

Terrestrial Planet Formation Scenario

106

105-6

107-8

Planetesimals

Protoplanets

yr

yr

yr

Terrestrial planets

Gas/Dust

..................................................................... .....................................................................

Protoplanetary disk

Act 1 Dust to planetesimals (gravitational instability/binary coagulation)Act 2 Planetesimals to protoplanets (runaway-oligarchic growth)Act 3 Protoplanets to terrestrial planets (giant impacts)

Terrestrial Planet Formation Scenario

106

105-6

107-8

Planetesimals

Protoplanets

yr

yr

yr

Terrestrial planets

Gas/Dust

..................................................................... .....................................................................

Protoplanetary disk

Act 1 Dust to planetesimals (gravitational instability/binary coagulation)Act 2 Planetesimals to protoplanets (runaway-oligarchic growth)Act 3 Protoplanets to terrestrial planets (giant impacts)

TerminologyRandom VelocityDeviation velocity from a non-inclined circular orbit

vran '√

e2 + i2vK

e : eccentricity, i : inclination, vK : Kepler velocity

σR ∝ σe, σz ∝ σi

Hill (Roche/Tidal) RadiusRadius of the potential well of an orbiting body

rH =

(m

3M�

)1/3

a

m : body mass, a : semimajor axis

Planetesimals

planetesimals

Surface density distribution

Σsolid = Σ1

( a

1 AU

)−αgcm−2

1 ≤ Σ1 ≤ 100, 1/2 ≤ α ≤ 5/2

standard protosolar disk: Σ1 ' 10, α = 3/2

Assumptions• no radial migration• perfect accretion

Equation of Motion

dvi

dt= −GM�

xi

|xi|3︸ ︷︷ ︸

solar gravity

+N∑

j 6=i

Gmjxj − xi

|xj − xi|3

︸ ︷︷ ︸

mutual interaction

+ fgas︸︷︷︸

gas drag

+ f col︸︷︷︸

collision effect

• solar gravity (dominant) ⇒ nearly Kepler orbits• mutual interaction ⇒ random velocity⇑• gas drag ⇒ random velocity⇓• collision ⇒ random velocity⇓

mutual interaction + gas drag ⇒ equilibrium random velocity

Two-Body Relaxation of PlanetesimalsElementary Process

• Two-body gravitational scattering

Viscous Stirring (Disk Heating)

• increase of random velocity vran (e and i)

Dynamical Friction

• equiparation of random energy mv2ran ∝ m(e2 + i2)

Viscous Stirring

• increase of e and i (σe > σi)• diffusion in a

Viscous Stirring

Two-body relaxation in a differentially rotating disk

• σe, σi ∝ t1/4 (two-body relaxation timescale)• σe/σi ' 2 (anisotropic velocity dispersion)

(Ida 1990; Ida, Kokubo &Makino 1993)

Dynamical Friction

• decrease of eM and iM (↔ increase of local e and i)• almost constant aM

Dynamical Friction

Two-body relaxation in a differentially rotating disk

• eM , iM → 0 (non-inclined circular orbit)(sufficient condition for runaway growth)

Growth Moded

dt

(M1

M2

)

=M1

M2

(1

M1

dM1

dt−

1

M2

dM2

dt

)

relative growth rate:1

M

dM

dt∝ Mp

runaway growthp<0 p>0

orderly growth

Collisional Cross-Section

R

Rgf

M

v

vesc

rel

Gravitational focusing

Rgf = R

(

1 +2GM

rv2rel

)1/2

= R

(

1 +v2esc

v2rel

)1/2

Collisional cross-section

Sgf = πR2gf = πR2

(

1 +v2esc

v2rel

)

Growth Rate

M mR

Test body: M,R, vesc

Field bodies: n, m

dM

dt' nπR2

(

1 +v2esc

v2rel

)

vrelm ⇒1

M

dM

dt∝ M

1

3 v−2ran

(

vrel ' vran, n ∝ v−1ran, vesc ∝ M1/3, R ∝ M1/3, vrel < vesc

)

Random velocity controls• the growth mode• the growth timescale

Runaway Growth of Planetesimals

(AU)

e

a

yr

yr

yr

(Kokubo & Ida 2000)

self-gravity of planetesimalsdominatesvran 6= f(M)

⇓

1

M

dM

dt∝ M

1

3 v−2ran ∝ M

1

3

runaway growth!

Runaway Growth of Planetesimals

(yr)t

(10

g)

M

,<m

>m

ax23

solid: Mmax, dashed: 〈m〉 (Kokubo & Ida 2000)

Oligarchic Growth of Protoplanetse

y

(AU)a

y

y

y

y

(Kokubo & Ida 2002)

Slowdown of runawayscattering of planetesimals by aprotoplanet with M >∼ 100m

vran ∝ rH ∝ M1/3

⇓

1

M

dM

dt∝ M

1

3 v−2ran ∝ M− 1

3

orderly growth!

Orbital repulsionorbital separation: b ' 10rH

(Kokubo & Ida 1998)

Protoplanets

protoplanets

Isolation mass

Miso ' 2πabΣsolid = 0.16

(

b̃

10

)3/2(Σ1

10

)3/2 ( a

1 AU

)(3/2)(2−α)M⊕

b : orbital separation, b̃ = b/rH

Growth time

Tgrow ' 3.2 × 105

(

b̃

10

)1/10(Σ1

10

)−9/10 ( a

1 AU

)(9α+16)/10yr

(Kokubo & Ida 2002)

Isolation Mass of Protoplanets

Standard protosolar diskΣ1 = 10, α = 3/2

Terrestrial Planet ZoneMiso ' 0.1M⊕

• large planets:impacts among protoplanets

• small planets:leftover protoplanets

heliocentric distance[AU]

prot

opla

net m

ass

[Ear

th m

ass]

Me

J

snow line

Ma

V E

S

U N

(Kokubo & Ida 2000)

Protoplanets to Terrestrial PlanetsGiant Impacts among Protoplanets

• Protoplanets gravitationally perturb each other to becomeorbitally unstable after gas dispersal

log Tinst ' c1(b/rH) + c2

(e.g., Chambers+ 1996; Yoshinaga+ 1999)

protoplanets

giant impacts

terrestrial planets

Timescale of Orbital Instability

Chambers+ (1996)

log Tinst ' c1(bini/rH) + c2

(Yoshinaga, Kokubo & Makino 1999)

Giant Impacts of Protoplanets

(Kokubo, Kominami & Ida 2006)

Total Mass-Planet MassΣ1 = 3(4), 10(©), 30(�), rin = 0.5AU, rout = 1.5, 2.0, 2.5, 3.0AU

〈M1〉: •, 〈M2〉: ◦

〈M1〉 ' 0.4Mtot, 〈M2〉 ' 0.3Mtot (global accretion!)(Kokubo & Ida 2009)

Spin Parametersdotted line: critical ω for rotational breakup

〈ω〉 ' ωcr = 3.3

(ρ

3gcm−3

)1/2

hr−1

dotted line: isotropic distribution

isotropic : ndε =1

2sin εdε

(Kokubo & Ida 2007)

Terrestrial PlanetsMass

• large planets: M ∝ Mtot

• small planets: leftover protoplanets

Orbital elements• e, i ' 0.1 (higher than the solar system values!)

Spin parameters• angular velocity: breakup velocity ωcr

• obliquity: isotropic distribution (ε ∼ 90◦)

Radial mixing• terrestrial planet zone wide

(Kokubo+ 2006; Kokubo & Ida 2007, 2009)

Important EffectsOligarchic Growth Stage

• Type I migration (e.g., Daisaka+ 2005; McNeil+ 2005)• Collisional disruption

Giant Impact Stage• Perturbation by gas giants (e.g., Chambers 2001)• Gravitational gas drag (e.g., Kominami & Ida 2002)• Dynamical friction from residual planetesimals (e.g., Agnor+

1999; O’Brien+ 2006)• Sweeping secular resonance due to gas disk dispersal

(e.g., Nagasawa+ 2005)• Hit-and-run collisions (Agnor & Asphang 2004; Kokubo & Genda

2009)

SummaryOrbital Dynamics

• Viscous stirring• Dynamical friction• Orbital instability

Accretionary Dynamics

• Runaway growth• Oligarchic growth• Giant impacts

Movie“Formation of Terrestrial Planets: The Movie”

Simulations:• Kokubo & Ida (2002)• Kokubo, Kominami & Ida (2006)• Kokubo & Ida (2007)• Genda, Kokubo & Ida (2009)

Visualization:• Miura & Kokubo (A 4D2U NAOJ Production)