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. bin Equal particles/log. bin Kernel K ij = 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 K ij = m i + m j

Run-away kernels mass Mass density Particles m~1 dominate mass of system Particles in tail will start runaway High-m particles require more focus than low-m particles Large grouping (low resolution)‏ Low/no grouping (high resolution)‏

Run-away kernels K ij ~ (mass) β, β>1 particles i and j  E.g., product kernel; gravitational focussing: K ij =π(R i +R j ) 2 x [v ij +2G(m i +m j )/(R i +R j )v ij ]  V esc =[2G(m i +m j )/(R i +R j )] 1/2  At t=t R =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 t R K ij = m i m j, N=10 20

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

Dynamics in Solar System Hill radius: R H =a(M/M * ) 1/3, V H =ΩR H 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: ~V H < W < V esc (common) Shear-dominated: W < ~V H

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 v escape and v Hill 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 > V esc → 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