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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)