Evolution of a Dissipative Self-Gravitating Particle Disk Or Terrestrial Planet Formation Eiichiro Kokubo

Evolution of a Dissipative Self-Gravitating Particle Disk or Terrestrial Planet Formation

Eiichiro Kokubo National Astronomical Observatory of Japan Outline Sakagami-san and Me

Planetesimal Dynamics • Viscous stirring • Dynamical friction • Orbital repulsion Planetesimal Accretion • Runaway growth of planetesimals • Oligarchic growth of protoplanets • Giant impacts Introduction Terrestrial Planets Planets core (Fe/Ni) • Mercury, Venus, Earth, Mars Alias • rocky planets Orbital Radius • ≃ 0.4-1.5 AU (inner solar system) Mass • ∼ 0.1-1 M⊕ Composition • rock (mantle), iron (core) mantle (silicate)

Close-in super-Earths are most common! Semimajor Axis-Mass

Semimajor Axis–Mass

“two mass populations” Orbital Elements

Semimajor Axis–Eccentricity (•), Inclination (◦)

“nearly circular coplanar” Terrestrial Planet Formation

Protoplanetary disk Gas/Dust

6 10 yr Planetesimals ......

5-6 10 yr Protoplanets

7-8 10 yr Terrestrial planets

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) Planetesimal Disks Disk Properties • many-body (particulate) system • rotation • self-gravity • dissipation (collisions and accretion)

Planet Formation as Disk Evolution • evolution of a dissipative self-gravitating particulate disk • velocity and spatial evolution ↔ mass evolution Question

How does a dissipative self-gravitating particulate disk evolve? Planetesimal Dyanamics Question

How do particle orbits (a, e, i) evolve? Terminology

Random Velocity • deviation velocity from a non-inclined circular orbit

2 2 1/2 vran ≃ e + i vK σR ∝σe, σz ∝ σi

e : eccentricity, i : incination, vK : Kepler circular velocity

Hill (Roche/Tidal) Radius • radius of the potential well of an orbiting body 1 3 m / rH = a 3Mc

Mc : central body mass, m : orbiting body mass, a : semimajor axis Disk Properties

Dynamics • central gravity dominant (nearly Keplerian orbit) • differential rotation (shear velocity) • “collisional” system (evolution by two-body encounters)

Structure

• disk thickness ∝ velocity dispersion (σz ∝ σi) Equation of Motion

N dvi xi xj − xi = −GMc 3 + Gmj 3 + f gas + f col dt |xi| |xj − xi| Xj6=i gas drag collision effect central gravity mutual interaction |{z} |{z} | {z } | {z } • central gravity (dominant) ⇒ nearly Keplerian orbits • mutual interaction ⇒ random velocity⇑ • gas drag ⇒ random velocity⇓ • collision ⇒ random velocity⇓

mutual interaction + gas drag/collision ⇒ equilibrium random velocity Equation of Motion

Elementary Process • Two-body gravitational scattering

Chandrasekhar’s Two-Body Relaxation Time • Timescale to forget the initial orbit v2 1 v3 trelax ≡ 2 ≃ 2 = 2 2 dv /dt nπrg v lnΛ nπG m lnΛ

n: number density, rg: gravitational radius, lnΛ: Coulomb logarithm (Chandrasekhar 1949) Relaxation of Particulate Disks Viscous Stirring (Disk Heating)

• increase of random velocity vran (e and i)

Dynamical Friction 2 2 2 • equiparation of random energy mvran ∝ m(e + i ) Viscous Stirring

0.02 0.02

0.015 0.015

0.01 0.01

0.005 0.005

0 0

0.96 0.98 1 1.02 1.04 0.96 0.98 1 1.02 1.04

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

1/4 • σe,σi ∝ t (two-body relaxation timescale)

• σe/σi = σR/σz ≃ 2 (anisotropic velocity dispersion) Viscous Stirring

Elementary Process • two-body scattering: shear velocity → random velocity Timescale σ2 σ3 t ≡ ≃ ⇒ t ∝ σ4 ⇒ σ ∝ t1/4 VS dσ2/dt nπG2m2 lnΛ VS n ∝ (thickness)−1 ∝ σ−1 (Ida & Makino 1992; EK & Ida 1992)

Anisotropic Velocity Dispersion

• σe/σi ∝ shear strength (cf. σR/σz ≃ 1.4 for the Galactic disk) (Ida, EK, & Makino 1993) Dynamical Friction

0.02 0.02

0.015 0.015

0.01 0.01

0.005 0.005

0 0

0.96 0.98 1 1.02 1.04 0.96 0.98 1 1.02 1.04

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

• almost constant aM Dynamical Friction

• eM , iM → 0 (non-inclined circular orbit) Dynamical Friction

Chandrasekhar’s Formula

A large particle with M and vM in a swarm of small particles with m and vm

2 1 dvM G Mmnm ∼ 3 vM dt vM

(vM >vm) (Chandrasekhar 1949) Application to a Particulate Disk 2 2 1 deM G Mmns,m G MΣ ∼ 3 3 3 ∼ 4 4 3 eM dt 2imaeM a Ω eM a Ω

(Σ = mns,m, vM ≃ eM aΩ, nm ≃ ns,m/2aiM , im

0.02 0.02

0.015 0.015

0.01 0.01

0.005 0.005

0 0

0.96 0.98 1 1.02 1.04 0.96 0.98 1 1.02 1.04

• expansion of orbital separation b: b =3rH → b ≃ 8rH

• keeping eM and iM small Orbital Repulsion Mechanism

b e

scattering dynamical friction

1. protoplanet-protoplanet scattering (eM , b ⇑)

2. dynamical friction from planetesimals (eM ⇓, aM constant)

b > 5rH ∼ (EK & Ida 1995) Summary

Two-Body Relaxation of Particulate Disks Disk Evolution: • Viscous stirring 1/4 – σe,σi ∝ t (⇐ disk) – σe/σi ≃ 2 (⇐ differential rotation) • Dynamical friction – e, i ∝ m−1/2 (⇐ energy equipartition) • Orbital repulsion – b ∼> 5rH (⇐ scattering, dynamical friction) All these elementary processes control the basic dynamics and structure of particulate disks! Planetesimal Accretion Question

How does particle mass distribution evolve by accretion? Planetesimals

planetesimals

Mass (size) • m ∼ 1018 g(r ∼ 1km) Surface density distribution −α a −2 Σsolid = Σ1 gcm 1au 1 ≤ Σ1 ≤ 100, 1/2 ≤ α ≤ 5/2

standard protosolar disk: Σ1 ≃ 10, α =3/2 (Hayashi 1981) Growth Mode

d M M 1 dM 1 dM 1 = 1 1 − 2 dt M2 M2 M1 dt M2 dt 1 dM relative growth rate: ∝ M p M dt

orderly growth runaway growth p<0 p>0 Collisional Cross-Section

vrel gravitational focusing Rgf R M

vesc Gravitational focusing 1/2 2 1/2 2GM vesc Rgf = R 1+ 2 = R 1+ 2 Rvrel vrel Collisional cross-section 2 2 2 vesc Sgf = πRgf = πR 1+ 2 vrel Growth Rate

Test body: M, R, vesc R Field bodies: n (number density), m M m

2 dM 2 vesc 1 dM 1/3 −2 ≃ nπR 1+ 2 vrelm ⇒ ∝ M vran dt vrel M dt

−1 1/3 1/3 vrel ≃ vran, n ∝ vran, vesc ∝ M , R ∝ M , vrel

self-gravity of planetesimals dominant for random velocity

yr vran =6 f(M)

e ⇓

1 dM 1/3 −2 1/3 ∝ M vran ∝ M yr M dt runaway growth!

a (AU) (EK & Ida 2000) Runaway Growth of Planetesimals 23 (10 g) max M ,

t (yr)

solid: Mmax, dashed: hmi (EK & Ida 2000) Runaway Growth of Planetesimals

dlog n 11 c ≃− dlog m 8 c n

m (10 23 g) dotted: 0 yr, dashed: 105 yr, solid: 2 × 105 yr (EK & Ida 2000) Oligarchic Growth of Protoplanets Σ1 = 10, α =3/2 Slowdown of runaway

0.15 scattering of planetesimals by a protoplanet with M ∼> 100m 1/3 vran ∝ rH ∝ M

0.1 ⇓ 1 dM ∝ M 1/3v−2 ∝ M −1/3 M dt ran orderly growth! 0.05 (Ida & Makino 1993)

Orbital repulsion 0 orbital separation: b ≃ 10rH (EK & Ida 1998) 0.4 0.6 0.8 1 1.2 1.4 1.6

(EK & Ida 2002) Protoplanets

protoplanets Assumptions • no radial migration • 100% accretion efﬁciency Isolation mass 3/2 3/2 (3/2)(2−α) b Σ1 a Miso ≃ 2πabΣsolid =0.16 M⊕ 10rH 10 1 AU Growth time − − 2/5 1/10 9/10 (9α+16)/10 5 fgas b Σ1 a tgrow ≃ 3.2 × 10 yr 240 10rH 10 1 AU (EK & Ida 2002, 2012) Isolation Mass of Protoplanets

Standard protosolar disk

Σ1 = 10, α =3/2

Terrestrial planet zone

Miso ≃ 0.1M⊕ Final formation stage • large planets: impacts among protoplanets • small planets: leftover protoplanets

(EK & Ida 2000) Question

What is the ﬁnal state of disk evolution? Protoplanets to Terrestrial Planets

Giant Impacts among Protoplanets • Protoplanets gravitationally perturb each other to become 7 orbitally unstable after gas dispersal (tgas ∼< 10 yr)

log tinst ≃ c1(b/rH)+ c2 (e.g., Chambers+ 1996; Yoshinaga, EK & Makino 1999) protoplanets

giant impacts

terrestrial planets Timescale of Orbital Instability

log tinst ≃ c1(bini/rH)+ c2 (Yoshinaga, EK & Makino 1999) Example Runs

Σ1 = 10, α =3/2, b = 10rH r =0.1-0.3au r =0.5-1.5au

0.2 0.2

0.1 0.1

0 0

0.1 0.2 0.3 0 0.5 1 1.5 2

N : 24 → 5 N : 16 → 3 compact, dynamically cold sparse, dynamically hot System Parameters

Mass Distribution • most massive: M1/Mtot (0.51) • dispersion: σM /M¯ (0.85) Orbital Structure • mass-weighted orbital elements: haiM , heiM , hiiM (0.90au, 0.022, 0.034)

• mean orbital separation: ˜b = b/rH (43) • mean eccentricity: e˜ = ea/rH (10) • angular momentum deﬁcit (AMD): (0.0018)

M a 1 1 e2 cos i 2 2 j j √ j “ q j j ” j Mj (e + i )/2 D = P − − P j j (Hill’s approximation) Mj √aj ≃ Mj Pj Pj (solar system terrestrial planets) System Radius Dependence

2 1 2 Σ1 = 10, α =2, b = 10rH, e / =0.01, r = 0.05-0.15, 0.1-0.3, 0.2-0.6, 0.5-1.5 au h i

(EK+ in prep.) dlogh˜bi/dlogr ˜p ≃−0.3 Orbital Architecture by Giant Impacts

Key Parameter 1/3 9M∗ 1 • physical to Hill radius ratio: r˜ = r /r = p p H 4πρ a

Large r˜p Effects • relatively weak scattering and effective collisions → smaller e, less mobility → local accretion → dynamically cold compact system Final Conﬁguration

Σ1 = 10, 30, 100, α =3/2-5/2, b =5-15rH, r =0.05-0.15, 0.1-0.3, 0.2-0.6, 0.5-1.5au

he˜ﬁni increases with h˜bﬁni (with decreasing r˜p) ˜ dloghe˜i/dloghbi≃ 2 (EK+ in prep.) Summary

Planetesimal System • Dissipative self-gravitating particle disk Planetesimal Dynamics • Viscous stirring • Dynamical friction • Orbital repulsion Planetesimal Accretion • Runaway growth of planetesimals • Oligarchic growth of protoplanets • Giant impacts