Formation of Terrestrial Planets from Planetesimals

Formation of terrestrial planets from planetesimals

Yuji MATSUMOTO ASIAA

TIARA Summer School on Origins of the Solar System Outline

- Basic dynamics of planetesimals

- Runaway growth of planetesimals

- Oligarchic growth of protoplanets

- Giant impacts of protoplanets

- Recent studies Minimum Mass Solar Nebula Model ice line https://www.rikanenpyo.jp/top/tokusyuu/toku2/index.html

rocky icy planetesimals planetesimals

rocky icy protoplanets protoplanets

rocky planets gas giants Ice giants Minimum Mass Solar Nebula Model ice line Alma Partnership+’15

rocky icy planetesimals planetesimals Disk & planetesimals ・A planetary system forms from a protoplanetary disk.

・Planets/coresrocky are formed by accretion of planetesimals. icy protoplanets protoplanets- parent bodies of asteroids and comets - enough large to interact by their gravity —> building block of planet accretion

(Safronovrocky planets 1969, Hayashigas 1981) giants Ice giants Minimum Mass Solar Nebula Model ice line How planetesimals grow up?

rocky icy protoplanets protoplanets

rocky planets gas giants Ice giants Growth of planetesimals

1. Growth Mode 1 dM ∝ Mp M dt

d M1 M1 1 dM1 1 dM2 = − (M1 > M2) dt ( M2 ) M2 ( M1 dt M2 dt )

orderly growth runaway growth d M d M 1 < 0 when p<0 1 > 0 when p>0 dt ( M2 ) dt ( M2 ) Growth of planetesimals

2. Growth Rate 1 dM ∝ Mp M dt dM v ≃ nσvrelm rel dt n: number density of bodies Target: σ: collisional cross-section M, R Field bodies: n, m vrel: relative velocity Growth of planetesimals

2. Growth Rate 1 dM ∝ Mp M dt dM v ≃ nσvrelm rel dt n: number density of bodies Target: σ: collisional cross-section M, R Field bodies: n, m vrel: relative velocity 1/2 v2 gravitational focusing R = R 1 + esc , GF 2 ( vrel )

RGF (vesc = 2GM/R) R v2 σ = πR2 = πR2 1 + esc . GF 2 ( vrel ) Growth of planetesimals

2. Growth Rate 1 dM ∝ Mp M dt dM v ≃ nσvrelm rel dt n: number density of bodies Target: σ: collisional cross-section M, R Field bodies: n, m vrel: relative velocity ρd Σd n = = , Σd: surface density of solid in disk m mhd hd: scale height (or vertical distribution) h v d ∼ rel , a vK vrel vrel −1 —> n ∝ vrel relative velocity ~ random velocity Growth of planetesimals

2. Growth Rate 1 dM ∝ Mp M dt dM v ≃ nσvrelm rel dt n: number density of bodies Target: σ: collisional cross-section M, R Field bodies: n, m vrel: relative velocity —> random velocity (vran)

dM v2 ∝ R2 1 + esc 2 dt ( vran ) 1 dM v2 ⇔ ∝ M−1/3 1 + esc 2 M dt ( vran ) Growth of planetesimals

2. Growth Rate 1 dM 1 dM v2 p ⇔ ∝ M−1/3 1 + esc ∝ M 2 M dt M dt ( vran ) v2 1 - esc < 1, p = − < 0 : orderly growth 2 vran 3

v2 1 dM v2 - esc > 1, ∝ M−1/3 esc ∝ M1/3 v−2 2 2 ran vran M dt vran

The growth mode is determined by vran!

v2 Θ = esc : Safronov parameter (Safronov 1969 or 1972) 2 vran Runaway growth

Runaway growth occurs!!

N-body simulation, Kokubo&Ida(2000) Runaway growth

Eccentricities of planetesimals increase.

e determines vran.

- at pericenter, v ≃ vK(1 + e)

- at apocenter, v ≃ vK(1 − e)

typically, vran ≃ evK (see Lissauer&Stewart1993) what determines e? Runaway growth occurs!!

N-body simulation, Kokubo&Ida(2000) Viscous stirring 1. fundamental process: two-body scattering v v change during close-scattering θ b ← Kepler problem! m M

2b G(M + m) tan θ = , b = (90 deg scattering) 2 90 2 (b/b90) − 1 v

m m b/b Δv = v sin θ = 2 90 v ⊥ 2 M + m M + m 1 + (b/b90) m m 1 Δv = v(1 − cos θ) = − 2 v ∥ 2 M + m M + m 1 + (b/b90) Viscous stirring 2. N-body scattering: summation of two-body’s

the amplitude of v distribution: ⟨Δv2⟩1/2

bmax 2 2 2 2 G nm ln Λ ⟨Δv ⟩ = Δv 2πnvbdb ∼ , Λ = bmax/b90 ∫0 v encounter in unit time viscous stirring (e.g., Ida 1990) v does not depend on M !

2 2 2 e is determined by planetesimals (not larger one) ⟨Δv ⟩ ∼ e vK,

1 dM ∝ M1/3 v−2 ∝ M1/3, p>0, runaway growth occurs. M dt ran Runaway growth N-body simulation (3000 bodies)

M get larger! Runaway body = protoplanet

mean mass m is not changed much

Kokubo&Ida(2000) How long does this stage last? Oligarchic growth

- Protoplanets are formed locally - The inner protoplanets are quickly formed

Kokubo&Ida(2002; 2012) Growth of protoplanets

Runaway growth

v2 v3 T = ∼ VS,m−m ⟨v2⟩ G2nm2 ln Λ

v2 v3 T = ∼ VS,M−m 2 2 2 ⟨v ⟩ G nMM ln Λ

M Growth of protoplanets

Runaway growth

v2 v3 T = ∼ VS,m−m ⟨v2⟩ G2nm2 ln Λ 2 2 nm > nMM M ≳ 100m (Ida & Makino 1993) 2 2 nm < nMM v2 v3 T = ∼ VS,M−m 2 2 2 ⟨v ⟩ G nMM ln Λ

1 n = M 2πaΔa × 2ai M Growth of protoplanets

Runaway growth

e of planetesimals is given by planetesimals,

vram does not depends on M. 1 dM ∝ M1/3 v−2 ∝ M1/3 M dt ran

Oligarchic growth

e of planetesimals is given by a protoplanet,

vram depends on M ! M Eccentricity in oligarchic growth

Hill’s framework: framework for 3 body problem (e.g., Nakazawa & Ida 1988)

m2 M* ≫ m1, m2,

ej, ij ≪ 1,

m1 |a1 - a2| ≪ a, M*

- These bodies interact each other - The relative motion is smaller than that of the barycenter

Hill radius: typical size of the m1-m2 interaction

1/3 M + m MaM + mam rH rH = , h = , ( 3M* ) ( M + m ) a Eccentricity in oligarchic growth

Hill’s framework: framework for 3 body problem (e.g., Nakazawa & Ida 1988)

m2 M* ≫ m1, m2,

ej, ij ≪ 1,

m1 |a1 - a2| ≪ a, M*

The Jacobi integral (CJ): the conservable in restricted 3 body problem

2 2 2 1 e i 3 aj − ai 3 9 CJ = + − − + , 2 [( h ) ( h ) ] 8 ( rH ) r/rH 2

- planetesimals move according to its CJ - e, i~h - e increases as the protoplanet becomes larger Eccentricity in oligarchic growth 1023g

2 1/2 ⟨e ⟩ /hM M

2 1/2 ⟨i ⟩ /hM ⟨m⟩

e ~ h ~ (M/M )1/3 Kokubo&Ida(2000) M * 1 dM ∝ M1/3 v−2 ∝ M−1/3 M dt ran Oligarchic growth

- protoplanet-protoplanet: 1 dM ∝ M1/3 v−2 ∝ M−1/3, M dt ran orderly growth,

similar size protoplanets are formed!

Kokubo&Ida(2000) Oligarchic growth

- protoplanet-protoplanet: 1 dM ∝ M1/3 v−2 ∝ M−1/3, M dt ran orderly growth. similar size protoplanets are formed!

- protoplanet-planetesimals:

1 dm ∝ m1/3 v−2 ∝ m1/3, m dt ran runaway growth continues. planetesimals keep small.

Kokubo&Ida(2000) Oligarchic growth

protoplanets

- small e - similar orbital separation (b)

(>~ 5rH)

5rH 5rH

Kokubo&Ida(2000) Eccentricity of protoplanets

- ⟨Δv2⟩1/2, the evolution of v amplitude: viscous stirring

- ⟨Δv⟩, v change of each body: dynamical friction Eccentricity of protoplanets

- ⟨Δv2⟩1/2, the evolution of v amplitude: viscous stirring

- ⟨Δv⟩, v change of each body: dynamical friction

two-body scattering

4πG2n M(m + M)ln Λ ⟨Δv⟩ ∼ − M v2

: massive body loses v largely ⟨Δv⟩ ∝ M(m + M) —> energy equipartition

m M m v2 ∼ v2 ⇔ e ∼ e 2 m 2 M M M m Dynamical friction

Kokubo&Ida(2012)

e2 M T = ∼ T , DF de2/dt m relax

Trelax : two-body relaxation time Due to (M/m) times two-body Even when eM > em, eM is damped. scattering, eM is damped. Oligarchic growth

protoplanets

- small e - similar orbital separation (b)

(>~ 5rH)

How b is determined?

5rH 5rH

Kokubo&Ida(2000) Orbital repulsion

Initial b=2rH

Protoplanet-protoplanet interaction. Scattering makes e and b larger. e/h 2 2 2 1 e i 3 b CJ = + − + ⋯ 2 [( h ) ( h ) ] 8 ( rH )

Protoplanet-planetesimal interaction.

eM is damped by dynamical friction. b spreads!

b becomes larger than 5rH.

a/rH Kokubo&Ida(1995) Collisions between protoplanets & planetesimals

Collision: two bodies are replaced by the barycenter body

m2, x2, v2 mG = m1 + m2,

m1x1 + m2x2 xG = , m1 + m2 mG, xG, vG m1, x1, v1 m1v1 + m2v2 vG = , m1 + m2 Collisions between protoplanets & planetesimals

m2, x2, v2 mG = m1 + m2,

mG, xG, vG m1, x1, v1

eccentricity vector: e ⃗ = (e cos(θ − ϖ), e sin(θ − ϖ))

ϖ : the longitude of pericenter conservative vector in Kepler problem Collisions between protoplanets & planetesimals

m2, x2, v2 mG = m1 + m2,

eccentricity vector: e ⃗ = (e cos(θ − ϖ), e sin(θ − ϖ)) Under the Hill’s approximations, m2 m2 m m e2 = 1 e2 + 2 e2 + 2 1 2 e e cos(ϖ − ϖ ), G 2 1 2 2 2 1 2 1 2 mG mG mG 1/2 random angle collision m2 m2 ⟨e ⟩ = 1 e2 + 2 e2 G 2 1 2 2 ( mG mG ) (Ohtsuki 1992) Collisions between protoplanets & planetesimals

m2, x2, v2 mG = m1 + m2,

1/2 m2 m2 ⟨e ⟩ = 1 e2 + 2 e2 G 2 1 2 2 ( mG mG )

planetesimal - planetesimal: m1~m2, e1~e2~e, ⟨eG⟩ ∼ e/ 2 2 2 protoplanet - planetesimal: m1e1 ~ m2e2 , ⟨eG⟩ ∼ e1 Oligarchic growth

- Almost all planetesimals are accreted - Inner protoplanets grow up faster

Kokubo&Ida(2002) Isolation mass of protoplanets

Isolation mass: Miso ≃ 2πabΣd the mass of all local dusts

b ∼ 10rH : orbital repulsion

−3/2 a Σ = 10 g cm−2 : 1.4×MMSN d ( 1 au )

3/2 3/4 −1/2 b a M* Miso ≃ 0.16 M⊕ ( 10rH ) ( 1 au ) ( M⊙ )

: about Mars size Isolation mass of protoplanets

3/2 3/4 −1/2 Isolation mass: b a M* Miso ≃ 0.16 M⊕ ( 10rH ) ( 1 au ) ( M⊙ ) M Timescale: t = grow dM/dt 2/15 27/10 1/3 mpl a M = 3.2 × 105 yr 23 ( 10 g ) ( 1 au ) ( 0.16M⊕ )

dM Σ v2 ≃ d πR2 esc v m ≃ nσvrelm 2 ram dt ( mvram /ΩK ) ( vram )

2GMR = CπΣ d 2 e avK Planetary cores disk mass

semimajor axis Kokubo&Ida(2002;2012) Minimum Mass Solar Nebula Model ice line

rocky icy Giant impacts protoplanets protoplanets

rocky planets gas giants Ice giants The orbital stability of protoplanets

˜ ˜ brH brH b˜rH b˜rH b˜ = 6 M ≃ 0.03M⊕

stability time: tinst = t(rij < rH)

tinst = 5.4 × 104 yr

apocenter semimajor axis pericenter unstable The orbital stability of protoplanets

˜ ˜ brH brH b˜rH b˜rH

- log tinst ∝ bini/rH (Chambers+1996) 106 - large 2 1/2 short ⟨eini⟩ , tinst 105 (e.g., Yoshinaga+1999, Zhou+2007) 104 - planets in resonances (Matsumoto+2012) 103 - e-damping (Iwasaki+2001; 2002) ○ 2 1/2 ⟨eini⟩ = 0 102 × 2 1/2 ⟨eini⟩ = 4h 10 2 4 6 8 10 Kokubo&Ida(2012) The orbital stability of protoplanets

˜ ˜ brH brH b˜rH b˜rH - e-damping (Iwasaki+2001; 2002) 8 te : e-damping timescale 7 upper limit Instability occurs after te > tinst. 6 - most planetesimals are removed inst t 5 - disk gas is depleted

log 4 tinst=te 3

1 4 5 6 7 8 b/rH Giant impact growth

Kokubo+(2006)

collisions

The giant impact stage lasts for ~108 yr. Two Earth-sized planets and a Mars-sized planet are formed! Giant impact growth

In giant impact stage, tinst grows up.

- n becomes smaller - b becomes larger

After tinst gets enough longer, planets are formed.

Ida & Lin (2010): population synthesis calibrated by N-body simulations by Kokubo + (2006). Giant impact growth

In giant impact stage, tinst grows up.

- n becomes smaller - b becomes larger

After tinst gets enough longer, planets are formed.

Ida & Lin (2010): population synthesis calibrated by N-body simulations by Kokubo + (2006). inst t tinst > 109 yr Planets by giant impacts

Kokubo + (2006) flled: e ○ scattered, : initial open: i high e larger planets ~10 collisions larger planets e <0.2

small planets are scattered orbital energy prevents from going too inward Planets by giant impacts

- close-scattering Kokubo + (2006) vesc eesc = flled: e vK scattered, 1/3 1/6 1/2 open: i M + M ρ a high e ≃ 0.28 k l −3 ( M⊕ ) ( 3 g cm ) ( 1 au ) once or twice close-scattering larger planets by larger bodies e <0.2 Planets by giant impacts

- close-scattering Kokubo + (2006) vesc eesc = flled: e vK scattered, 1/3 1/6 1/2 open: i M + M ρ a high e ≃ 0.28 k l −3 ( M⊕ ) ( 3 g cm ) ( 1 au ) once or twice close-scattering larger planets by larger bodies e <0.2

- collisions m2 m2 m m e2 = 1 e2 + 2 e2 + 2 1 2 e e cos(ϖ − ϖ ), G 2 1 2 2 2 1 2 1 2 mG mG mG

ϖ1 − ϖ2 in the last collision determines e! Planets by giant impacts

- collisions

m2 m2 m m e2 = 1 e2 + 2 e2 + 2 1 2 e e cos(ϖ − ϖ ), G 2 1 2 2 2 1 2 1 2 mG mG mG

Since protoplanets are isolated, a1 < a2,

∘ ϖ1 − ϖ2 = 0 is not collidable ϖ1 − ϖ2 = 180 is collidable

p2 p1 p2 p1

Matsumoto + (2015) Planets by giant impacts

- close-scattering Kokubo + (2006) vesc eesc = flled: e vK scattered, 1/3 1/6 1/2 open: i M + M ρ a high e ≃ 0.28 k l −3 ( M⊕ ) ( 3 g cm ) ( 1 au ) once or twice close-scattering larger planets by larger bodies e <0.2

- collisions m2 m2 m m e2 = 1 e2 + 2 e2 + 2 1 2 e e cos(ϖ − ϖ ), G 2 1 2 2 2 1 2 1 2 mG mG mG

ϖ1 − ϖ2 in the last collision determines e! e < 0.2 planets can be explained by opposite pericenter collisions. Planets by giant impacts

Σ@1au=3(△), 10(○), 30() flled: M1, open: M2 g/cm2 M1 ~ 0.5 Mtot ain = 0.5 au

aout = 1.5, 2.0, 2.5, 3.0 au

global accretion

M2 ~ 0.3 Mtot

Kokubo + (2006) The formation of Terrestrial planets

- initial condition: * isolation mass protoplanets −3/2 a * standard values: Σ = 10 g cm−2, ain = 0.5 au, aout = 1.5 au, d ( 1 au ) - results: ⟨e1⟩ = 0.11 ± 0.07

⟨e2⟩ = 0.12 ± 0.05

E V ⟨e⊕⟩ ∼ ⟨eV⟩ ∼ 0.03

* Shallow disk? * e-damping?

Kokubo + (2006) Giant impacts

based on Genda+(2012) https://www.youtube.com/watch?v=Y9aN59dsUUc Giant impacts Genda+(2012) vimp = 1.3vesc

0.1M⊕

merging Giant impacts Genda+(2012) vimp = 1.3vesc

0.1M⊕

merging

vimp = 1.5vesc hit-and-run Giant impacts

M2 vimp θ

Msys = M1 + M2

Genda+(2015) Head-on (θ=0) and high vimp collisions are destructive Giant impact fragments

- late veneer? - e is damped by GIFs? Genda+(2017) - the origin of debris disk? Giant impact fragments SPH + N-body simulation - initial condition

16 MMars protoplanets

total mass: 2.3M⊕

- SPH simulation 104 particles

single collision: ~0.01Msys

Genda+(2017) Ejecta in runaway and oligarchic growth

Dusts are ejected in collisions melting dusts: chondrules pebble accretion

accreted by a protoplanet and planetesimals

Mch, cum

Macc, pr Macc, pl

Wakita, Matsumoto+(2017) (also Johnson+ 2015, Kobayshi+2010) Matsumoto+(2017) Numerical costs

- runaway growth stage

t~ 105 yr, the onset of runway growth: R~150 km (Kobayashi+2016, @1 au)

N~ (M⊕/mpl)~105 , (locally)

- oligarchic growth stage large N and long t simulation

t~ 106 yr, method: GRAPE, tree, parallel code: LIPAD, PKDGRAV, REBOUND, FDPS

- giant impact stage t~ 107- 8 yr, N~ 10 - 100 (N> 104 , if GIFs are considered) small (or large) N and long t simulation method: vector computation? Large N-body simulation

Kominami+(2016): Direct N-body simulation Using K super computer, 82,362 planetesimals for ~5×104 yr Global simulation of runaway growth. ice line mpl =1024 g, R ≃ 430 km Summary: dynamic processes

- runaway growth stage

vrel determines (dM/dt)/M . 1 dM 1/3 epl : viscous stirring between planetesimals, ∝ M M dt t~ 105 yr - oligarchic growth stage 1 dM epl : viscous stirring by a protoplanet, ∝ M−1/3 M dt epr : dynamical fction, epr ≪ epl t~ 106 yr

- giant impact stage

tinst evolution t~ 107-8 yr Movie: from planetesimals to planets

https://www.youtube.com/watch?v=5h0fzy2aUes

Evidence of giant impacts : Moon formation Giant impact Disk formation Accretion (e.g., Canup 2004)

Ultra-high-resolution SPH simulation (108 particles)

1h 2h 3h 4h 5h

6h 7h 8h 9h 10h

Disk formed Hosono+(2017) Evidence of giant impacts : Moon formation Giant impact Disk formation Accretion (e.g., Canup 2004) N-body simulation (107 particles)

Sasaki & Hosono(2018) Problems

- Small e,i of Earth and Venus

- Small Mars problem

- Depletion of asteroids

- No close-in planets

- Jupiter formation

- Water delivery

- Age of terrestrial planets

(also see Raymond+ 2009)