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Planet Formation
XIII Ciclo de Cursos Especiais
Outline
1.
2.
3.
4.
5.
Observations of planetary systems
Protoplanetary disks
Formation of planetesimals (km-scale bodies)
Formation of terrestrial and giant planets
Evolution and stability of planetary systems
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Routes to planet / planetesimal formation
1. Collisionless collapse - gravitational physics
• dark matter halos
• Goldreich-Ward mechanism for planetesimal formation
2. Collisional (gas) collapse - gravity + hydrodynamics
• Jeans analysis of stability
• basis of star formation
• possible role in giant planet formation?
3. Collisional growth via coagulation - surface physics,
+ gravity (sometimes)
• described by theory due to Smoluchowski (1913)
• describes terrestrial planet formation, dust aggregation,
perhaps planetesimal formation
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Planetesimal formation
Particle radius s, mass m, velocity relative to gas Δv
Define friction time: t fric
m"v
=
FD
For particles smaller than the gas mean free path, drag
force is described
by Epstein law:
!
4# 2
with vth the mean thermal
FD = "
$s v th %v
speed of the molecules
3
Spherical dust particles, find: t fric =
!
"m s
" v th
…with ρm the material density, ρ the gas density
!
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Substitute:
t fric
fric
"3m gscm-3 10"4 cm
= "9
=3s
-3
5
-1
" v gth cm 10 cm s
10
Small particles are tightly coupled to the gas
!
Growth is affected / controlled by turbulence mediated
via aerodynamic drag
Define planetesimals as the smallest bodies for which
mutual gravitational interactions dominate over aerodynamic
drag forces
Occurs for ~10 km scale under typical conditions… building
these bodies from (initially) micron-sized dust is the first
stage of planet formation
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Vertical settling
Consider particle settling in a laminar disk
4# 2
$s v th v settle
3
$m s 2
=
"z
$ v th
m"2 z =
Terminal velocity:
v settle
For micron sized particles at z ~ h at 1 AU:
! cm s-1
v settle ~ 0.1
t settle ~ 10 5 yr
…small relative velocities, but
still settles on time scale short
compared to disk lifetime
Scale dependent vsettle implies collisions between particles
!
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Assume that all collisions lead to coagulation. Toy model:
look at one particle, mass m, settling through background
of smaller particles:
dm
= "s2 v settle f#(z)
dt
dz
= $v settle
dt
f is dust to gas ratio (f ~ 0.01), model
originally due to Safronov
Simple coupled system to solve numerically
!
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Assume that all collisions lead to coagulation. Toy model:
look at one particle, mass m, settling through background
of smaller particles:
For 1 AU conditions,
growth to s ~ 1 mm
and sedimentation to
midplane on time scale
of ~103 yr
Too simple - but does
show that if collisions
lead to accretion then
growth to small but
macroscopic sizes is
very quick
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Intrinsic disk turbulence will oppose settling. Equate diffusion
time across scale h to settling time:
t diffuse
h2
h
= ; t settle =
D
v settle
#e1 2 $ m s
Then, if D ~ ν: " =
2 %
!
Estimate (very crude!) that turbulence with α ~ 10-2 will
!
stir up small
particles efficiently, but particles with s > 1 mm
or so ought to start settling to the disk midplane
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Dullemond
& Dominik
(2005)
Coagulation models that include both growth and impacts
that lead to fragmentation yield growth but sustain a
population of small grains (necessary observationally
to match observations of YSOs)
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Radial drift of solids
Recall that gas rotated slightly slower than Keplerian orbital
velocity (pressure support):
2 '1 2
$
c
12
v" ,gas = v K &1# n 2s ) = v K (1# *)
vK (
%
Consequences:
• dust!particle (strong coupling to gas):
- swept up, orbits at v = vφ,gas
- unbalanced radial force
- drifts in at terminal velocity
• large rock (100m)… weak coupling:
- orbits at v = vK
- experiences head wind drag force
- loses angular momentum, spirals in
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General treatment after Weidenschilling (1977)
Particle equations of motion:
2
dv r v"
1
= # $2K r #
v r # v r,gas )
(
dt
r
t fric
d
r
rv" ) = #
v" # v" ,gas )
(
(
dt
t fric
Assuming (reasonably) that inspiral is slow compared to the
orbital time, can simplify to:
!
#1
fric r,gas
" v # $v K
vr =
" fric + " #1
fric
!
where the dimensionless friction
time is defined as:
" fric # t fric $K
!
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" #1
fric v r,gas # $v K
vr =
" fric + " #1
fric
Neglecting radial flow of the gas, maximum is evidently
where τfric=1
1
!
v r = " #v K
2
Critical point: η << 1 but not that small - η ~ 10-2 typical
Since vK ~ 30!km s-1 at 1 AU, predict extremely fast radial
drift… up to 100 m s-1 at peak
Unless something else happens, rocks will drift into the
star on time scale of 102 or 103 yr!
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Actual particle size that corresponds to τfric = 1 depends
upon disk conditions - but s ~ 1 m is typical…
To grow from small scales to planetesimals via pairwise
collisions cannot avoid passing through this stage
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Conclude: growth to planetesimals must be very rapid,
or likely that most of the mass of solids would
be lost to the inner disk, evaporated and flow
into star
From Solar System evidence, must be able to form
planetesimals across range of disk radii (< 1 AU - 50 AU)
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Note: have ignored a lot of detail
regarding turbulence
Detailed models (e.g. Dominik
et al. 2007) confirm that large
collision velocities are expected
for m-scale bodies within the
disk
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Coagulation theory for planetesimal formation
Simplest theory: form planetesimals from sequence of
pairwise collisions leading to coagulation, starting
from dust
Toy model (similar to the vertical settling model) suffices
to show that the “optical depth” to collisions during radial
drift is plausibly > 1… so enough collisions for this to work
despite the rapid inspiral time
BUT - are collisions at high relative velocities accretional
or destructive?
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Lab experiments (reviewed by Dominik et al. 2007) suggest
that the composition of the bodies is critical:
• solid bodies rebound / shatter at Δv ~ 1 m s-1
• very porous aggregates can continue to gain mass
at Δv > 10 m s-1 (is this good enough?)
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Goldreich-Ward theory for planetesimal formation
Classical mechanism:
Particle settling leads to
formation of a dense layer
of solid bodies at the disk
midplane
Gravitational instability in
the particle layer results
in prompt collapse into
km-scale bodies
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Stability analysis of a thin sheet of particles to gravitational
instability yields a condition for instability:
#$
Q"
<1
%G&s
…where σ is the particle velocity dispersion and Σp the
surface density of particles. Most unstable scale:
2
2
#
!
"~
G$s
Instability at 1 AU in a disk with a particle density of 10 g cm-3
would form solid bodies of mass:
!
…corresponds to a solid body
2
18
m ~ "# $ s ~ 3 %10 g
with a radius of 5-10 km!
!
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The simple Goldreich-Ward mechanism fails, due to the
inevitability of self-excited turbulence
…shear between the particle-rich midplane and the gas-rich
disk above is unstable to Kelvin-Helmholtz instability
Resulting turbulence prevents the formation of the unstable
layer in the first place, even if the background disk is laminar
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Stability to Kelvin-Helmholtz instability is determined by the
Richardson number:
Ri = "
gz d ln # /dz
(dv
$
/dz)
2
…with instability for Ri < 0.25
!
Applying to a dust layer of thickness hd estimate:
# h/r &)2 # hd /h & 2
Ri " 0.25%
( %
(
$ 0.05 ' $ 0.0375 '
For Goldreich-Ward to work we need hd ~ 10-4 h
! be a fatal obstable
Appears to
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Simulation: Chiang (2008)
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New research directions
Variant of Goldreich-Ward can still work, if other instabilities
result in local clumping of the solid bodies
Enhances local density of solids, so layer does not need
to be so cold (low σ) at the onset of instability
Requires turbulence, so turbulence not a barrier…
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Gas
Particles
Local clumping of particles at pressure maxima within
self-gravitating gas disks
Same physics as radial drift, but now applied locally
Rice et al. (2005)
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Johansen et al.
(2007) find that
collapse can
occur in the flow
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How do planetesimals form?
Formation of planetesimals involves:
• aerodynamic forces (well understood)
• turbulence (disk, 2 fluid)
• material properties (porosity, sticking efficiency)
Don’t have a fully satisfactory model for planetesimal
formation - even at first order (does turbulence help or
hinder the process?)
Probably the least well known stage of planet formation
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What do we know?
• empirically, planetesimals must form across a wide
range of disk radii - can’t appeal to unique
locations within the disk
• theoretically, planetesimal formation must be rapid
• early stages: micron-sized up to mm or cm-sized
seem straightforward - growth occurs via
direct collisions that lead to sticking
• particle settling is inhibited by disk turbulence - rules
out original Goldreich-Ward proposal
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Planetesimal formation via coagulation
Plausible mechanism if collisions at high velocity (at
least 10 m s-1 and maybe up to 100 m s-1) lead to
net growth of the target bodies
Need: more lab experiments, better knowledge of
composition of particles as they grow
Planetesimal formation via gravitational collapse
Plausible if other instabilities generate very large
local overdensities of solid material
Need: simulations to confirm such instabilities
exist in presence of intrinsic turbulence,
non-linear outcome
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Are there circumstance under which planetesimals could
not form?
Is there an observational discriminant in the Solar System
between these two models?
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