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From Planetesimals to Planets
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
Many particles problem
Kernel Kij = <σv>ij = mi+mj
Equal mass/log. bin

Many particles needed to sample distribution!
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Very difficult to treat every collision separately
Kernels and growth
Linear kernel, No grouping
With grouping
Kij = mi + mj
Run-away kernels
Particles m~1 dominate
mass of system
Large grouping
(low resolution)
High-m particles
require more focus
than low-m particles
Particles in tail will
start runaway
mass
Low/no grouping
(high resolution)
Run-away kernels
Kij = mi mj, N=1020
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
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
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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 3body effects

Dispersion-dominated: ~VH< W < Vesc (common)

Shear-dominated: W < ~VH
More Dynamics
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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 sheardominated 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