1. Dynamics of planetesimal formation
Áron Süli
Eötvös University
Dept. of Astronomy

2. Overview
This talk reviews our current understanding of terrestrial planets formation.
Very short historical overview
Structure of the gas disk
The orbital velocity of the gas disk
The stages of the planetesimal and planet formation

3. Introduction The study of planet formation has a long history.
The idea that the Solar System formed from a rotating disk of gas and dust – the Nebula Hypothesis – dates back to Kant, Laplace in the 18th century.
A quantitative description of terrestrial planet formation was given by the late 1960s, when Viktor Safronov published his now classic monograph
The core accretion theory for gas giant planet formation were developed in the early 1980s.

4. Introduction
More recently, a wealth of new observations has led to renewed interest in the problem:
The most dramatic development has been the identification of extrasolar planets.
The discovery of protoplanetary disks and of the Solar System’s Kuiper Belt have been almost as influential in focusing theoretical attention on the initial conditions for planet formation
Geochemical and cosmochemical analyses performed with laboratory instruments produced data on the chemical and isotopic composition of the planets giving constraints on the chronology of their accretion and thermal evolution.
Remarkable increase in computer performance has allowed modelers to undertake increasingly realistic simulations of the dynamical process of terrestrial planet accretion.
Sufficient level of reliability to be integrated in a comprehensive view of the early evolution of the inner solar system.

5. Introduction
The formation of terrestrial planets from micron-sized dust particles requires growth through at least 12 orders of magnitude in size scale.
The formation is usually ordered in time and by the relevant different dominant physical processes involved.
Planetesimal formation: Planetesimals are defined as bodies that are large enough (R  km) that their orbital evolution is dominated by mutual gravitational interactions rather than aerodynamic drag.
Embryo formation: Embryos are between planetesimals and planets (100 < R < km). Giant planets probably form at roughly the same time as planetary embryos.
Terrestrial planet formation: after the disappearance of gas from, the embryos’ orbits become unstable, and their mutual collisions give birth to a small number of terrestrial planets
coupling to gas:
drag
collision (sticky)
Stage I
gravity
collision
Stage II
coupling to gas:
migration
gravity
collision
Stage III

6. Protoplanetary disk structure
Protoplanetary disks form because the diffuse gas has too much angular momentum to collapse directly to stellar densities
Disks survive as structures in quasi-equilibrium because its specific angular momentum increases with radius.
To accrete, angular momentum must be lost from, or redistributed within the disk gas, and this process requires time scales that are much longer than the orbital or dynamical time scale.
Since angular momentum transport is slow, it is assumed that the disk is static
This approximation suffices for a first study of the
temperature
density, and
composition profiles
of protoplanetary disks, which are critical inputs for models of planet formation.
Lets begin with the gas component of the disk

7. Protoplanetary disk: Vertical structure of the gas
vertical component of the
gravitational acceleration
vertical pressure gradient
Equation of
state
Approximation
mid-plane density
surface density
vertical scale height
orbital frequency

8. Protoplanetary disk: Orbital velocity of the gas
The radial velocity component of the gas can be determined from the momentum equation
This slight difference is in the heart of the so-called meter-size barrier problem

9. Stage I: from dust to planetesimals
Aerodynamic drag on solid particles
Aerodynamic drag is important to understand both the vertical and radial distribution of solid material

10. Stage I: from dust to planetesimals
Dust settling: dust grains sediment into a thin layer at the mid-plane of the disks
In the absence of turbulence micron-sized particles sediment out of the upper layers a time scale that is short compared to the disk lifetime.

11. Stage I: from dust to planetesimals
Previously both turbulence and growth via collision were negelcted.
Lets consider a single particle settling with coagulation in a laminar disk:

12. Stage I: from dust to planetesimals – turbulence
The sedimentation timescale and the volume density of grains near the mid-plane depend on the severity of turbulence in the disk gas, as strong turbulence inhibits sedimentation. Even in laminar disks, though, the volume density of grains in the mid-plane could not reach arbitrarily large values, because the sedimentation of an excessive amount of solids would itself generate turbulence in the disk via the so-called Kelvin-Helmoltz instability.

13. Stage I: from dust to planetesimals
Radial drift of solid particles
radial drift velocity

14. Stage I: from dust to planetesimals – m-size barrier problem
radial drift velocity
min timescale
barrier
Planetesimal formation must be rapid,
Radial redistribution of solids is very likely to occur
Radial drift with coagulation: relative velocity between particles of different sizes
-> coagulation. But the relative velocities are too large -> FRAGMENTATION

15. Stage I: from dust to planetesimals – an alternative
Planetesimals may form due to the collective gravity of massive swarms of small particles via the Goldreich-Ward mechanism, if they are concentrated at some locations, such as
local maxima of the gas density distribution
inter-vortex regions.
These models can explain the formation of planetesimals of size 100 km or larger without passing through intermediate small sizes, circumventing the meter-size barrier problem.
Evidence:
The size distribution of objects in the asteroid belt and in the Kuiper belt, where most of the mass is concentrated in 100km objects
The existence and the properties of binary Kuiper belt objects (have angular momenta too large to form single objects)

16. Stage II: from planetesimals to embryos

17. Stage II: from planetesimals to embryos

18. Stage II: from planetesimals to embryos
The detailed physics of the collision is neglected,
it is assumed that the bodies agglomerate into a
single object.
The collisional cross section is enhanced by the gravity of the bodies: this is called
gravitational focusing
Using
conservation of energy
conservation of angular momentum
the cross section for collision:

19. Stage II: from planetesimals to embryos
The initially more massive body grows faster than its
less massive cousin. This phenomenon is called
runaway growth

20. Stage II: from planetesimals to embryos
The growth rate of the embryos gets slower as the bodies increase in size.
The relative velocities are too high -> collisions between smaller bodies leads to disruption.
Only collisions between small and large bodies can lead to growth!
This phase is called oligarchic growth.

21. Stage II: from planetesimals to embryos
The maximum mass of a protoplanet, the isolation mass after the runaway and oligarhic growth can be estimated:
The body can accrete material only from within its feeding zone, which gives:
feeding zone

22. Stage II: from planetesimals to embryos – the giants

23. Stage III: from embryos to terrestrial planets
The final assembly of terrestrial planets starts once the oligarchs have depleted the planetesimal disk (dynamical friction can no longer maintain low e and i)
The largest bodies start to interact strongly, collide, and scatter smaller bodies across a significant radial extent of the disk. Numerical N-body simulations show that this final stage is by far the slowest, with large collisions continuing out to at least 100 Myr
The process is chaotic: identical initial conditions can give rise to a range of outcomes
The typical result:
2 to 4 planets are formed on well-separated and stable orbits, with eccentricities and inclinations that are comparable of the real planets
Quasi-tangent collisions of Mars-mass embryos onto the proto-planets are frequent. These collisions are expected to generate satellites (the formation of the Moon)
The accretion timescale in the simulations is ~ Myr, in general agreement with the timescale of Earth accretion deduced from radioactive chronometers
A small fraction of the original planetesimals typically remain in the asteroid belt on stable orbits

24. Stage III: from embryos to terrestrial planets

25. Thank you for your attention
Summary
Remarks: turbulance, collisions, migration, chemical composition, water, ...
the formation of terrestrial planets can be divided in three stages
the mechanisms responsible to create particles of cm scale are identified
the formation of planetesimals faces a sever problem: the meter-size barrier
the final assembly of terrestrial planets is well understood, and the outcomes statistically agree with the inner Solar System
the extraordinary diversity of the discovered planetary systems poses new challenges to the theories
Thank you for your attention