円盤における固体の衝突成長と移動
奥住聡（東工大地惑）(c) Koji Wada

1. Basic Concepts of Global Dust Evolution(particle motion, collision outcome)2. Observational Tests

Global Dust Evolution in Disks
Dust (<μm)

Protoplanetary D

isk

Planets (<Mm)

Planetesimals (>km)
Sticking+ Instabilities?

mutual gravity• How does planet formation begin?
• How does disk opacity evolve?
2Global Dust Evolution in Disks
Particle Size
Coagulation(Growth)DUST
Radial Drift& Diffusion

Fragmentation

Distancefrom Star3

Theoretical Modeling
Collision (Coag/Frag)

Radial Drift• K: “Collision Kernel”

Depends on collision model and particle
dynamics (relative velocity)• vr : Radial Drift Velocity

Depends on gas disk structure

Collision & Disk Physics
Global Dust & Opacity Evolution

Observation4

Dust Growth:
Basic Concepts

Key Ingredients
Particle Size
Coagulation(Growth)DUST
Radial Drift

Fragmentation

Distancefrom Star6

Particle Dynamics in Disks● Motion in laminar disks- Vertical settling, radial drift, azimuthal drift● Motion in turbulence- Driven by velocity perturbation (drag force)(for planetesimals, density perturbation becomes more important)Blum (2004)7

Radial Drift● Radial pressure gradient changes gas’s orbital velocity- Normally dP/dr < 0 ➡ Sub-Keplerian- “Pressure bumps”: local maxima of P (see Taki’s talk)● When dP/dr<0,gas acts as “headwind” on dust
➡ dust loses angular momentum➡ dust drifts inward

Hdniw
daeGasDustradial drift(Adachi et al. 1976;Weidenschilling 1977)
Disk (face-on view)

8Radial Drift●:Dimensionlessstopping time (a.k.a. Stokesnumber)● Drift fastest when St = 1( m-size boulder at 1 AU,cm-size pebble at 100 AU )(Adachi et al. 1976;Weidenschilling 1977)
9Effect of Disk Turbulence● Gas’s random motion drives
particle’s velocity
Analytic (Youdin & Lithwick ‘07; Ormel & Cuzzi ‘07)
Numerical (Carballido et al. ‘10, 11)●A● Relative velocity Δv highest
when St = 1 (particle velocities
cancel out when St < 1)(e.g., Ormel & Cuzzi 2007)

Based on Ormel & Cuzzi (2007)10

Particle Collision Velocity
Reaches ~50m/s (~180km/h) when St ~ 1weakly turbulent

strongly turbulent

11Collision Outcome
StickingFragmentation

Dominik & Tielens (1997)

Δv = vfrag

Collision Velocity Δv● How does vfrag depend on particle properties?
● Does Δv in disks exceed vfrag ?
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Particle Sticking from Microphysics● Microscopic particles stick via surface tension
contactarea(c) Koji WadaChokshi et al. (1993)● Binding energy per contact area
γ: surface tension
𝓔: Young modulus
a0: particle (monomer) radius(Johnson et al. 1971; See also Chokshi et al. 1993, Dominik Tielens 1997)13

Fragmentation Threshold
Eimp = 40NEbreak● Aggregate disrupted if impact
energy Eimp is higher thanN : total # of monomers(Dominik & Tielens 1997; Wada et al. 2009)↑mass loss

● Since Eimp,crit ∝ mass, criticalvelocity vfrag is independent ofaggregate mass

↓mass gain

Wada et al. (2009)14

Fragmentation Velocity
H2O ICEγ = 100 mJ/m2
𝓔 = 7 GPaρmat = 1 g/cc

ROCK
γ = 25 mJ/m2
𝓔 = 54 GPaρmat = 3 g/cc

(See Ueda’s poster for organic-coated particles)

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Comparison with Lab ExperimentsRoughly speaking, theory is consistent with experiments(perhaps underestimates vfrag)

Lab Exp.

Th ↑eo↓ ry16

Implications for Dust Evolution in Disks

Dust Composition Matters!
● Rocky aggregates very likely
fragment in disks (unless a0 < 10nmor Δv is somehow small)● Icy aggregates may survive
fragmentation✴ Rocky and icy planetesimals
may form in different ways

✴ Snow region may be opticallythinner than silicate region

17Particle Growth Timescale
SIMPLE MODEL:● Forget fragmentation (particles always stick)● Forget size distribution (single-size at each r)● Just compute growth timescale

nd: # density of dust particles
σcoll : collision cross section
Δv: particle relative velocity
18Particle Growth Timescale
FURTHER SIMPLIFICATION:● Turbulence dominates Δv● Vertical settling balances with turbulent diffusion● Epstein’s drag law applies (particle size < gas m.f.p.)● St < 1 (consider small particles)

Takeuchi & Lin (2005); Brauer et al. (2008)

Size-independent tgrow

Exponential growth19

Particle Size vs. Disk Age

a0: initial particle sizeFor Σd/Σg = 0.01,● t~1kyr: 1cm pebbles at 1AU● t~1Myr: 1cm pebbles at 100AU

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The Radial Drift Barrier● In typical disks, marginallydecoupled (St=1) particles drift
on timescale ~ 100/ΩK● Full 2D simulations show radialdrift does takes over growth➡ “Radial Drift Barrier”
Brauer et al. (2008)● But porous dust particles
break through the barrier

Okuzumi et al. (2012); see Kataoka’s talk21

Theory vs. Observation

Millimeter Photometric Survey
Ricci et al. (2010abc); Rommen et al.(2007,09,10); Ubach et al.(2012)

mm emission predominantly fromcold, optically thin outer disk
➡ ● Flux ~ Dust Mass
● Slope ~ Dust Size23

Disk Opacity vs. Dust Size
←Rayleigha = λ/2π(x = 1)

Geometric Optics→1mm opacity
per gram of dust

Opacity slope

β ~ βISM ~ 2β<2Ricci et al. (2010a) [see also Miyake & Nakagawa (1993)]
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Millimeter Slope vs. Flux
For opt. thin disk,

21 β0Testi et al., PPVI Review Talk
[Data from Ricci et al.(2010abc); Rommen et al.(2007,09,10); Ubach et al.(2012)]25

Model with Coagulation andFragmentation (no Radial Drift)

Mass per log size

higher vfrag

higher Σdlower αturb

higher Σg
0.1μm

1mm

0.1μm

1mm
Birnstiel et al. (2011)26

Model with Coagulation andFragmentation (no Radial Drift)
Fails to explain disks with low α1-3mm & low F1mm

Observation

Birnstiel, Ricci, et al. (2010)27

With Coagulation, Fragmentation,
and Radial Drift
Big dust removed while smaller dust remains➡ Increasing α1-3mm with decreasing F1mm

TheoryPinilla, Birnstiel, et al. (2012)28

With Coagulation, Fragmentation,
Radial Drift, and Pressure Bumps
Pinilla, Birnstiel, et al. (2012)

A=0.3, vfrag=10m/s

A: relative amplitudeof Σ bumps
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With Coagulation, Fragmentation,Radial Drift, and Pressure Bumps“Including both drift & bumpis better than neglecting bothdrift & bumps”

TheoryPinilla, Birnstiel, et al. (2012)30

Simplest Isn’t Best ?
Let’s forget all complexities.
- No fragmentation(particles are somehow sticky)- No radial drift(many strong pressure bumps,
or just theorists are wrong)- Single size at each r :31

Simplest Isn’t Best ?
Single-size pure coagulation:- Σg (r) ∝ r−1.5 (truncated at 150AU), Σd/Σg = 0.01
- a0 = 0.1µm<1Myr ×0.5MMSNÊÊ
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Simplest Isn’t Best ?
Single-size pure coagulation:合うじゃんÊÊÊÊÊÊ‡‡‡‡‡‡ÏÏÏÏÏÏÚÚÚÚÚÙÙÙÚÙÊÊ‡ÏÊ‡ÏÚ‡ÏÚÙ‡ÏÚÏÚÙÙÚÙ

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Things Not Discussed Today• Transitional Disks- Coag-frag-drift models reproduce infrared dip in SED, butnot cavity seen in mm imaging (Birnstiel et al. ‘12)

• Radial variation of β

- decreasing β with decreasing r (Isella et al. ‘10, Perez et al. ‘12)
- Observations consistent with drift-limited model (Perez etal. ‘12)

• Lopsided disks- Strong m=1 asymmetry in mm continuum- Dust trap in vortices? (Birnstiel et al. ’13; Lyra & Lin ’13)
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Summary● Dust collision velocity reaches ~ 50 m/s● Collision outcome strongly depends on materials(icy aggregates are much stickier than rocky ones)➡ Snow line may be visible (τ may change there)● W/o drift & fragmentation barriers, dust growthtimescale is ~ (dust/gas ratio)*(Keplerian timescale)● Comparison with observation just started (mostly mm)● Some models (but not only one) reproduce observation➡ Further test needed (with imaging, different wavelengths)➡ Further improvement on dust model may also be needed(e.g., porosity evolution)
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