From Planetesimals to Planets

Pre-Galactic Black Holes and ALMA

Gravitational collapse cloud core

Disk formation

Planetesimal formation, 1 m → 1 km tough

Agglomeration of planetesimals

Solar system

Growth of dust in disk; sticking

through van der Waals forces and/or (unstable) gravity

Equal mass/log. binE

qual

par

ticle

s/lo

g.

bin

Kernel Kij = <σv>ij = m

i+m

j

Many particles problem

Many particles needed to sample distribution! Very difficult to treat every collision separately

Kernels and growth

Linear kernel, No grouping With grouping

Kij = mi + mj

Run-away kernels

mass

Mas

s de

nsity

Particles m~1 dominate mass of system

Particles in tail will start runaway

High-m particles require more focus than low-m particles

High-m particles require more focus than low-m particles

Large grouping (low resolution)

Low/no grouping (high resolution)

Run-away kernels

Kij ~ (mass)β, β>1

particles i and j

E.g., product kernel; gravitational focussing: Kij=π(Ri+Rj)2 x

[vij+2G(mi+mj)/(Ri+Rj)vij]

Vesc=[2G(mi+mj)/(Ri+Rj)]1/2

At t=tR=1 the runaway particle

separates from the distribution → Kuiper Belt

[Wetherill (1990); Inaba et al. (1999); Malyshkin & Goodman (2001); Ormel & Spaans (2008); Ormel, Dullemond & Spaans, 2010]

Runaway time tR

Kij = m

i m

j, N=1020

Run-away to oligargic growth: roughly when MΣ_M~mΣ_m; from planetesimal self-stirring to proto-planet determining random velocities

km km

Dynamics in Solar System

Hill radius: RH=a(M/M*)1/3, VH=ΩRH

Hill radius is distance over which 3-body effects become important

In general, one has physical collisions, dynamical friction: 2-body momentum exchange that preserves random energy, and viscous stirring: energy extracted from or added to the Keplerian potential through 3-body effects

Dispersion-dominated: ~VH< W < Vesc (common)

Shear-dominated: W < ~VH

More Dynamics

Dynamical friction: Σ_M < Σ_m, planetesimal swarm dominates by mass and the orbit of the proto-planet is circularized by kinematically heating up the planetesimals (no physical collisions, only gravitational interactions, random energy preserved)

Viscous stirring: exchange of momentum can also be achieved by extracting from /adding to the Keplerian potential (random energy not preserved, three-body effect)

Brief period of run-away growth (dM/dt ~ M^4/3);

interplay between vescape and vHill of massive and satellite particles to oligarchic growth

(dM/dt~M^2/3)

Growth/Time (yr)

Gas drag effects, 1 AU

Fragmentation effects, 35 AU

Summary

Gravitational focussing important above 1 km; run-away → oligarchic

Gravitational stirring causes low-mass bodies to fragment, W > Vesc → in the oligarchic phase (re-)accretion of fragments is important

Sweep-up of dynamically cold fragments in the shear-dominated regime (fast growth), but in gas-rich systems particles suffer orbital decay

Gas planets form by accretion on rocky (~10 M_earth) cores Proto-planets clear out their surroundings (gap formation) Gravitational collapse of unstable disk still alternative