Planetesimal Formation and Planet Coagulation

Transcription

1 Planetesimal Formation and Planet Coagulation

2 Protoplanetary Disks disk mass ~ stellar mass Wilner et al AU

3 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 TW Hydra F max s = 1 cm max s = 1 mm van Boekel et al. 03 Calvet et al. 2002

4 Climbing the size ladder s Size EC & Youdin 10 Annual Reviews Grain stopping time Dimensionless stopping time s

5 Gas-particle entrainment s = t stop assumes minimum-mass solar nebula a = 1 km a = 1 m a = 1 mm a = 1 µm R [AU] A. Youdin e.g., Epstein drag ρ(c s v) 2 λ mfp >s v<c s v ρ(c s + v) 2 F drag ρc s v πs 2

6 Grain growth h g Sticking v stick 1m/s fors 1 µm a δ a2 s s Hertz + γ 5/6 v stick 4 surface tension E 1/3 ρ 1/2 s 5/6 Terminal v term ρ ρ g Ωs 1m/s fors 10 cm Sticking up to, but not beyond, cm sizes Chokshi et al. 93, Blum & Wurm 08

7 Radial drift from headwind Ω 2 Kr Ω 2 r 1 P ρ g r GM r 2 GM r 2 Ω < Ω K Meter-sized boulders drift inward from 1 AU within 100 yr Radial Weidenschilling 77 Nakagawa, Sekiya, & Hayashi 86

8 Gravitational instability Disk annulus can start to fragment if Toomre Q ~ 1 ρ >ρ Toomre M 2πr 3 > 10 7 gcm 3 Clump can resist tidal shear if Roche unstable ρ>ρ Roche 3.5M r 3 > gcm 3 if Σ g 2000 g cm 2 (minimum mass solar nebula) if Σ d /Σ g 10 2 (height integrated solar metallicity) then ρ Σ g + Σ d h g h d gcm 3 if h d h g Toomre unstable ρ d /ρ g 30 Toomre mass ρ Toomre λ 3 unstable g (10 km) Roche unstable ρ d /ρ g 600

9 Streaming instability = linear instability between two fluids interacting frictionally in a disk growth rates peak for (marginally coupled bodies) = 0.1 < Youdin & Goodman 05; Johansen & Youdin 07

10 Streaming instability: Physical interpretation Jacquet et al Relies strongly on marginally coupled bodies Roche density Bai & Stone 2011

11 Gravitational instability in the s << 1 (small particle) limit Self-gravity important when ρ>ρ Toomre M 2πr 3 > 10 7 gcm 3 at r =1AU h g h d g = Ω 2 z if Σ g 2000 g cm 2 (minimum mass solar nebula) if Σ d /Σ g 10 2 (height integrated solar metallicity) then ρ Σ g h g + Σ d h d gcm gcm 3 if h d h g if h d h g Can the settled sublayer achieve Toomre density < Safronov 1969 Goldreich & Ward 1973

12 Kelvin-Helmholtz instability may limit dust settling z ϕ ρ g Ω 2 Kr Ω 2 r 1 P ρ g r ρ g + ρ d GM r 2 GM r 2 ρ g Ω < Ω K v c s c s v K 25 m/s nearly independent of r Weidenschilling 1980

13 Necessary criterion for K-H instability in Cartesian shear flow: Richardson g ln ρ/ z Ri <Ri ( v/ z) 2 crit = 1 4 ωbrunt 2 = ( v/ z) 2 z ϕ ρ g z 1/ z (1/ z) 2 ρ g + ρ d z ρ g

14 Numerical simulations of dense midplanes Initial conditions: Spatially constant Ri Sekiya 98 µ(z) ρ d ρ g (z) = [ 1 1/(1 + µ 0 ) 2 +(z/z d ) 2 where z d = Ri1/2 v Ω Input parameters: (Ri, 0 ) ] 1/2 1 Ri = 0.1 Ri = 1 (Ri, Σ d /Σ g ) 0, Σ d /Σ g )

15 Numerical simulations of dense midplanes EC 08; Lee, EC, Asay-Davis, and Barranco 10a

16 Gravitational Instability by Metal Enrichment 30 Ri dv y /dz ω Brunt (3/2)Ω K (3/2)Ω K dv y /dz 0 Unstable Stable (3/2)Ω K dv y /dz Midplane dust-to-gas ratio 0 Bulk Disk Metallicity Σ d /Σ g Lee et al. 10a Toomre-like densities possible for < 4x bulk solar metallicity, or < 4x more mass than minimum-mass solar nebula

17 Gravitoturbulence? or True Fragmentation? Constant Ri profiles yield infinite density with self-gravity

18 Local enrichments of metallicity Radial pileups Radial Drift Pileups d Photoevaporated Gas UV Youdin & EC 04

19 Modeled metallicities Guillot et al. 06 Toomre density requires: Super-solar bulk metallicities (< 4 x; Σ d /Σ g = 0.05) and/or Disk masses > minimum-mass nebula (< 4 x) Johnson et al. 10

20 epicycle size ~ u/κ = u/ω Elliptical motion = Guiding center + Epicycle Ω = guiding center (azimuthal) frequency κ = epicyclic (radial) frequency Kepler degeneracy : κ = Ω u = epicyclic velocity = small body random velocity particle

21 Massive disks of small bodies (Goldreich, Lithwick & Sari 04, ARAA) big bodies: R, Σ,v u without gravitational focussing: t acc Ṁ M ρr σω 1012 yr small bodies: s, σ, u with gravitational focussing: ρr ΣΩ ( u v esc ) 4 viscous stirring of small by big t acc ρr σω ( u v esc ) yr ρs σω collisional damping of small by small

22 Extra Slides

23 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

24 For small particles well coupled to gas ( s << 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 ρ d depend on bulk (height-integrated) metallicity Σ d /Σ g? 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

25 Well-coupled gas and dust in a shearing box v x t + v v x = v y t + v v y = v z t + v v z = ρ t ε t anelastic + (ρv) =0 0 1 ρ +ρ d 1 ρ +ρ d 1 ρ +ρ d + v ε = (P + ε) v P =(γ 1)ε P x +2Ω 0v y P y 2Ω 0v x P z Ω2 0z +2qΩ 2 0x ρ d t + (ρ dv) =0 ( P x ρ + ρ d y y x ) 0 x Barranco 09

26 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

27 Grain growth mg F drag (v) µs 3 Ω 2 h ρ g s 2 c s v h g Terminal v µ ρ g Ωs (bigger is faster) Accretion d dt (µs3 ) ρ d vs 2 ṡ ρ d µ v (faster is bigger) Exponential growth s s 0 exp(ρ d Ωt/ρ g ) (fastest growth in inner disk) Since t h/v s s 0 exp(σ d /µs) s 0 1 µm 2} s 1cm µ 1gcm 3 t 100 yr Σ d 10 g cm v 1m/s Safronov 1969

28 Relaxing constant Ri: Settling from arbitrary initial conditions 1D + 3D Lee et al. 10b Finding the marginally stable state

29 Marginally stable states Evidence for constant Ri <Ri> crit 0.25 Superlinear relation between midplane density and bulk metallicity <Ri> <Ri> crit Σ d /Σ g

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32 Grain growth v crit 1m/s fors 1 µm Repulsion (elastic modulus E) δ a2 s Stress σ E ξ E δ a mv σ (δ/v) a 2 Repulsive force F R σa 2 µ 3/5 E 2/5 s 2 v 6/5 Repulsive energy U R F R δ Adhesion (surface tension γ ) a s Binding energy U B γa 2 U R = U B v crit 4 γ 5/6 E 1/3 µ 1/2 s 5/6 Hertz 1882, Chokshi et al. 93

33 Rotational Effects Coriolis Kepler radial shear Brunt oscillation 2Ω Ω Ω destabilizing Cabot 84 Gomez & Ostriker stabilizing Ishitsu & Sekiya 03 EC 08 Barranco 09 Vertical shear <}Ri < Ω stabilizing destabilizing