Planetesimal Formation
and Planet Coagulation
Protoplanetary Disks
disk mass ~ 0.001-0.1 stellar mass
Wilner et al. 00
200 AU
van Boekel et al. 03
Disk surfaces
at ~10 AU:
Growth to a few
microns
11.8 μm scattered light
McCabe,
Duchene, &
Ghez 03
Disk interior at ~100 AU:
Growth to a few cm
Fν
TW Hydra
max s
=
1 cm
max s
=
1 mm
van Boekel et al. 03
Calvet et al. 2002
Climbing the size ladder
τs
Grain stopping time tstop
Size
EC & Youdin 10
Annual Reviews
mvrel / Fdrag
Dimensionless stopping timeτs
ΩKepler tstop
Gas-particle entrainment
10
6
os = 1 tstop
104
10
2
10
0
10
assumes minimum-mass solar nebula
m
k
1
a=
v < cs
v
ï2
m
m
1
a=
ρ(cs − v)2
m
µ
1
a=
ï6
10ï8
10ï1
λmfp > s
m
1
=
a
10ï4
10
e.g., Epstein drag
1
10
R [AU]
ρ(cs + v)2
⇒ Fdrag ∼ ρcs v × πs2
102
A. Youdin
h
Grain growth
g
Sticking vstick ∼ 1 m/s for s ∼ 1 µm
δ
a2
∼
s
Hertz +
−→ vstick
surface tension
Terminal vterm
a
s
γ 5/6
∼ 4 1/3 1/2 5/6
E ρ s
ρ
Ωs
∼
ρg
vterm ∼ 1 m/s for s ∼ 10 cm
Sticking up to, but not beyond,
cm sizes
Chokshi et al. 93, Blum & Wurm 08
Radial drift from headwind
Ω2K r
−1 ∂P
Ω r ρg ∂r
GM∗
r2
GM∗
r2
2
∴ Ω < ΩK
Meter-sized
boulders
drift inward
from 1 AU
within 100 yr
Radial
Weidenschilling 77
Nakagawa, Sekiya, & Hayashi 86
Gravitational instability
Disk annulus can start to fragment if
Toomre Q ~ 1
!ρ" > ρToomre
M∗
∼
2πr3
Clump can resist tidal shear if
Roche unstable
ρ > ρRoche
3.5M∗
∼
r3
> 2 × 10−6 g cm−3
> 10−7 g cm−3
if Σg ∼ 2000 g cm−2 (minimum − mass solar nebula)
if Σd /Σg ∼ 10−2 (height − integrated solar metallicity)
Σd
Σg
then ρ ∼
+
hg
hd
∼ 3 × 10−9 g cm−3
if hd ∼ hg
Toomre unstable
Roche unstable
ρd /ρg ∼ 30
ρd /ρg ∼ 600
3
Toomre mass ∼ ρToomre λunstable
∼ 1019 g (10 km)
“Streaming” instability
= linear instability between
two fluids interacting
frictionally in a disk
growth rates peak
for τs 1
(marginally coupled bodies)
τs = 0.1
<ρd /ρg> = 1
Youdin & Goodman 05; Johansen & Youdin 07
Streaming instability: Physical interpretation
Jacquet et al. 2011
Relies strongly on marginally coupled bodies
Roche density
Bai & Stone 2011
Gravitational instability in the τs << 1 (small particle) limit
Self-gravity important when
ρ > ρToomre
> 10
−7
hg
M∗
∼
2πr3
g cm
g = −Ω2 z
hd
−3
at r = 1 AU
if Σg ∼ 2000 g cm−2 (minimum − mass solar nebula)
if Σd /Σg ∼ 10−2 (height − integrated solar metallicity)
Σd
Σg
then ρ ∼
+
hg
hd
∼ 3 × 10−9 g cm−3
if hd ∼ hg
∼ 10−7 g cm−3
if hd ∼ 5 × 10−4 hg
Can the settled “sublayer” achieve
Toomre density <ρd/ρg> 30?
Safronov 1969
Goldreich & Ward 1973
Kelvin-Helmholtz instability may limit dust settling
z
ϕ
Ω2K r
−1 ∂P
Ω2 r ρg ∂r
GM∗
r2
GM∗
r2
ρg
ρg + ρd
ρg
∴ Ω < ΩK
cs
∼ 25 m/s nearly independent of r
∆v ∼ cs
vK
Weidenschilling 1980
Necessary criterion for K-H instability in Cartesian shear flow:
1
g ∂ ln ρ/∂z
Richardson Ri ≡
< Ricrit =
2
(∂v/∂z)
4
=
2
ωBrunt
(∂v/∂z)2
1/∆z
∝
(1/∆z)2
z
ϕ
ρg
∆z
ρg + ρd
∝ ∆z
ρg
Numerical simulations of
dense midplanes
Initial conditions:
Spatially constant Ri
Sekiya 98
ρd
µ(z) ≡ (z)
ρg
=
!
1
1/(1 + µ0 )2 + (z/zd )2
where zd =
1/2
Ri
"1/2
−1
∆v
Ω
Input parameters:
(Ri, μ0)
(Ri, Σd/Σg)
Ri = 0.1
(μ0, Σd/Σg)
Ri = 1
Numerical simulations of
dense midplanes
EC 08; Lee, EC, Asay-Davis, and Barranco 10a
Gravitational Instability by Metal Enrichment
30
Unstable
Stable
μ0
Ri
|dvy /dz| ∼ ωBrunt
∼ (3/2)ΩK
(3/2)ΩK ! |dvy /dz|
(3/2)ΩK ∼ |dvy /dz|
Midplane dust-to-gas ratio
μ0
Bulk Disk Metallicity
Σd/Σg
Toomre-like densities possible for < 4x bulk solar metallicity,
or < 4x more mass than minimum-mass solar nebula
Lee et al. 10a
Gravitoturbulence ?
or
True Fragmentation ?
Constant Ri profiles
yield infinite density
with self-gravity
Local enrichments of metallicity
Particle
Settling
Radial pileups
d
Radial Drift
Pileups
Photoevaporated
Gas
Particle
Settling
Dusty
Midplan
UV
Youdin & EC 04
Modeled metallicities
Guillot et al. 06
Toomre density requires:
Super-solar bulk metallicities
(< 4 x; Σd/Σg = 0.05)
and/or
Disk masses >
minimum-mass nebula (< 4 x)
Johnson et al. 10
Elliptical motion = Guiding center + Epicycle
Ω = guiding center (azimuthal) frequency
κ = epicyclic (radial) frequency
“Kepler degeneracy”: κ = Ω
le κ
c
/
y
u
c
i ~
p
e ze
si u/Ω
=
u = epicyclic velocity
= small body random velocity
particle
Massive disks of small bodies
big bodies:
(Goldreich, Lithwick & Sari 04, ARAA)
R, Σ, v ! u
without gravitational focussing:
tacc
ρR
M
∼
∼
∼ 1012 yr
σΩ
Ṁ
small bodies:
with gravitational focussing:
ρR
ΣΩ
!
u
vesc
"4
viscous stirring of small by big
tacc
∼
ρR
∼
σΩ
!
u
vesc
"2
s, σ, u
∼ 107 yr
ρs
σΩ
collisional damping of small by small
Extra Slides
Grain growth: Too fast
No turbulence
Sticks too well
Problem persists even if
• grains are fractal
• monomers are nonspherical
Dullemond & Dominik 05
Turbulence
Proposed solution:
Replenishment of micron-sized
grains (near-IR opacity) by
fragmentation
For small particles well coupled to gas (τs << 1):
1.
Is the Richardson criterion a good predictor of stability?
• Doesn’t formally apply because flow is 3D and rotational
• Brunt vs. vertical shear vs. Coriolis vs. Kepler shear
• Coriolis is destabilizing
• Kepler shear is stabilizing
2.
How does maximum dust density ρd depend on
bulk (height-integrated) metallicity Σd/Σg?
• Disk metallicity may be supersolar
• Host stars of extrasolar planets tend to be metal-rich
• Planets themselves are metal-rich
Ishitsu & Sekiya 03; Gomez & Ostriker 05; Johnson et al. 10; Guillot et al. 06
Well-coupled gas and dust in a shearing box
! ∂P "
∂vx
−1 ∂P
∂x 0
2
+ v · ∇vx =
+ 2Ω0 vy + 2qΩ0 x −
∂t
ρ +ρd ∂x
ρ + ρd
∂vy
−1 ∂P
+ v · ∇vy =
− 2Ω0 vx
∂t
ρ +ρd ∂y
y
∂vz
−1 ∂P
+ v · ∇vz =
− Ω20 z
∂t
ρ +ρd ∂z
anelastic
∂ρ
+ ∇ · (ρv)
=
0
0
∂t
x
∂ρd
+ ∇ · (ρd v) = 0
∂t
∂ε
+ v · ∇ε = −(P + ε)∇ · v
∂t
P = (γ − 1)ε
x
y
Barranco 09
Code limitations (so far):
1.
No self-gravity
• Use Toomre density as guide for onset of gravitational
instability
2.
Dust and gas are perfectly coupled
•
Restricted to studying stability of given initial conditions
Grain growth
h
g
mg ∼ Fdrag (v)
µs3 Ω2 h ∼ ρg s2 cs v
µ
−→ Terminal v ∼ Ωs (bigger is faster)
ρg
d
ρd
3
2
Accretion (µs ) ∼ ρd vs −→ ṡ ∼
v
dt
µ
(faster is bigger)
−→ Exponential growth s ∼ s0 exp(ρd Ωt/ρg ) (fastest growth in inner disk)
Since t ∼ h/v −→ s ∼ s0 exp(Σd /µs)
}
s0 ∼ 1 µm
µ ∼ 1 g cm−3
Σd ∼ 10 g cm−2
s ∼ 1 cm
t ∼ 100 yr
v ∼ 1 m/s
Safronov 1969
Relaxing constant Ri: Settling from arbitrary initial conditions
1D + 3D
Lee et al. 10b
Finding the marginally stable state
Marginally stable states
Evidence for
constant Ri
<Ri>crit
≈ 0.5
<μ>
<Ri>
<Ri>crit
≈ 0.25
Superlinear relation
between
midplane density
and
bulk metallicity
<μ0>
(Σd/Σg)>1
Grain growth
vcrit ∼ 1 m/s for s ∼ 1 µm
Repulsion (elastic modulus E)
δ
Stress σ ∼ E∇ξ ∼ E
a
mv
σ∼
(δ/v) × a2
Repulsive force FR ∼ σa2 ∼ µ3/5 E 2/5 s2 v 6/5
δ
2
a
∼
s
Repulsive energy UR ∼ FR δ
Adhesion (surface tension γ )
Binding energy UB ∼ γa2
a
s
UR = UB −→ vcrit
γ 5/6
∼ 4 1/3 1/2 5/6
E µ s
Hertz 1882, Chokshi et al. 93
Ri
Coriolis
2Ω
−1
destabilizing
Cabot 84
Gomez & Ostriker 05
Kepler
radial shear
3 −1
Ω
2
stabilizing
Ishitsu & Sekiya 03
EC 08
Barranco 09
}
Rotational Effects
Brunt
oscillation
<
Ω
−1
stabilizing
Vertical
shear
<
Ω
−1
destabilizing