Planetesimal formation in Planetesimal formation in turbulent protoplanetary discs turbulent protoplanetary discs Anders Johansen Planet formation Dust in MHD turbulence Planetesimal formation Zonal flows Analytical model Global models Anders Johansen Streaming and Leiden Observatory, Leiden University self-gravity Dead zones “Workshop on the Magnetorotational Instability in Protoplanetary Disks” (Kobe University, June 2009) Conclusions Collaborators: Andrew Youdin, Hubert Klahr, Wladimir Lyra, Mordecai-Mark Mac Low, Thomas Henning
Planet formation Planets form in protoplanetary discs from dust grains that collide and Planetesimal formation in stick together (planetesimal hypothesis of Safronov, 1969). turbulent protoplanetary From dust to planetesimals discs µ m → m: Contact forces in collisions cause sticking Anders Johansen m → km: ??? From planetesimals to protoplanets Planet formation km → 1,000 km: Gravity Dust in MHD From protoplanets to planets turbulence Terrestrial planets: Protoplanets collide Planetesimal Gas planets: Solid core attracts gaseous envelope formation Zonal flows Analytical model Global models → → → → Streaming and self-gravity Dead zones Conclusions
Planetesimals Planetesimal formation in turbulent Kilometer-sized objects massive enough protoplanetary discs to attract each other by gravity Anders (two-body encounters) Johansen Building blocks of planets Planet formation Formation: Dust in MHD µ m → cm: Dust grains collide and turbulence stick Planetesimal formation (Blum & Wurm 2000) Zonal flows cm → km: Sticking or gravitational William K. Hartmann Analytical instability model (Safronov 1969, Goldreich & Ward 1973, Weidenschilling & Global models Cuzzi 1993) Streaming and self-gravity Dynamics of turbulent gas important Dead zones for modelling dust grains and boulders Conclusions
Overview of planets Planetesimal formation in Protoplanetary discs turbulent protoplanetary discs Anders Johansen Dust grains Gas giants and Planet formation ice giants Dust in MHD turbulence Planetesimal formation Pebbles Zonal flows Terrestrial planets Analytical model Global models Streaming and self-gravity + Countless asteroids and Kuiper belt objects Dwarf planets Dead zones + Moons of giant planets Conclusions + More than 300 exoplanets
Particle dynamics Gas accelerates solid particles through drag force: Planetesimal formation in turbulent ∂ w ∂ t = . . . − 1 τ f ( w − u ) protoplanetary ❅ ■ discs ❅ ■ ❅ Anders ❅ ❅ Johansen Particle velocity Gas velocity Planet In the Epstein drag force regime, when the particle is much formation smaller than the mean free path of the gas molecules, the Dust in MHD turbulence friction time is (Weidenschilling 1977) a • : Particle radius Planetesimal formation τ f = a • ρ • ρ • : Material density c s : Sound speed Zonal flows c s ρ g ρ g : Gas density Analytical model Important nondimensional parameter in protoplanetary discs: Global models Streaming and self-gravity Ω K τ f ( Stokes number ) Dead zones At r = 5 AU we can approximately write a • / m ∼ 0 . 3 Ω K τ f . Conclusions
Diffusion-sedimentation equilibrium Planetesimal formation in Diffusion-sedimentation turbulent protoplanetary equilibrium: discs Anders Johansen � δ t H dust = Planet Ω K τ f H gas formation Dust in MHD H dust = scale height of dust-to-gas turbulence ratio Planetesimal formation H gas = scale height of gas Zonal flows Analytical δ t = turbulent diffusion coefficient, model like α -value Global models Streaming and Ω K τ f = Stokes number, proportional self-gravity to radius of solid particles Dead zones (Johansen & Klahr 2005) Conclusions
Diffusion coefficient Definition of Schmidt Planetesimal formation in number: Sc x turbulent Sc z protoplanetary Sc x ( L y =4) Sc = ν t / D t = α t /δ t discs 10 Sc z ( L y =4) Anders From the Johansen Sc scale-height of the Planet dust one can formation calculate the 1 Dust in MHD turbulence diffusion coefficient: Planetesimal 0.001 0.010 0.100 1.000 formation δ t = δ t ( H dust ) α Zonal flows Johansen & Klahr (2005): Sc z ≃ 1 . 5, Sc x ≃ 1 Analytical model (Turner et al. 2006: Sc z ≃ 1; Fromang & Papaloizou 2006: Sc z ≃ 3) Global models Carballido, Stone, & Pringle (2005): Sc x ≃ 10 Streaming and Johansen, Klahr, & Mee (2006): self-gravity The ratio between diffusion and viscosity depends on the Dead zones strength of an imposed magnetic field Conclusions
The role of the Schmidt number Planetesimal formation in turbulent protoplanetary discs Anders Safronov (1969): Johansen Dust grains coagulate and gradually decouple from the gas Planet formation Sediment to form a thin mid-plane layer in the disc Dust in MHD turbulence Planetesimals form by continued coagulation or Planetesimal self-gravity (or combination) in dense mid-plane layer formation Zonal flows HOWEVER: Analytical model MRI-driven turbulence very efficient at diffusing dust Global models Streaming and self-gravity Dead zones Need to look at how larger particles react to turbulence Conclusions
Dust nomenclature Planetesimal formation in My suggestion for naming solid particles (not official): turbulent protoplanetary discs Diameter Name Anders Johansen < 1 mm Dust Planet formation Dust in MHD 1 mm Sand turbulence Planetesimal formation 1 cm Pebble, gravel Zonal flows Analytical model 10 cm Cobble, rock Global models Streaming and self-gravity Dead zones > 1 m Boulder Conclusions
Radial drift Planetesimal Balance between drag force and head wind gives radial drift formation in turbulent speed (Weidenschilling 1977) protoplanetary discs 2 Anders v drift = − Ω K τ f + ( Ω K τ f ) − 1 η v K Johansen Planet for Epstein drag law (solids smaller than gas mean free path). formation Dust in MHD MMSN at r =5 AU turbulence 10 2 MMSN η from Cuzzi et al. Planetesimal formation 1993 Zonal flows 10 1 Maximum drift speed of 50 v drift [m/s] Analytical model m/s Stokes drag 10 0 Global models Drift time-scale of 50-100 Streaming and orbits for solids of 30 cm in Epstein drag self-gravity 10 −1 10 −3 10 −2 10 −1 10 0 10 1 radius at 5 AU, but 1 cm at Dead zones a [m] 100 AU Conclusions
Boulders in turbulence Planetesimal Johansen, Klahr, & Henning (2006): formation in turbulent 2,000,000 boulders moving in magnetorotational turbulence protoplanetary discs Anders Johansen Planet formation Dust in MHD turbulence Planetesimal formation Zonal flows Analytical model Global models Streaming and self-gravity Dead zones Conclusions
Gas density bumps Planetesimal formation in turbulent protoplanetary discs Anders Johansen Planet formation Dust in MHD turbulence Planetesimal formation Zonal flows Analytical Strong correlation between high gas density and high model particle density (Johansen, Klahr, & Henning 2006) Global models Streaming and Solid particles are caught in gas overdensities self-gravity (Whipple 1972, Klahr & Lin 2001, Haghighipour & Boss 2003) Dead zones Gravoturbulent formation of planetesimals Conclusions
Gas density bumps Planetesimal formation in turbulent protoplanetary discs Anders Johansen Planet formation Dust in MHD turbulence Planetesimal formation Zonal flows Analytical Strong correlation between high gas density and high model particle density (Johansen, Klahr, & Henning 2006) Global models Streaming and Solid particles are caught in gas overdensities self-gravity (Whipple 1972, Klahr & Lin 2001, Haghighipour & Boss 2003) Dead zones Gravoturbulent formation of planetesimals Conclusions
Pressure gradient trapping Planetesimal formation in turbulent protoplanetary discs Anders Outer edge: Johansen Gas sub-Keplerian. Particles forced by gas drag to move Planet formation inwards. Dust in MHD Inner edge: turbulence Gas super-Keplerian. Particles forced by gas drag to move Planetesimal formation outwards. Zonal flows Analytical model Global models Streaming and self-gravity Dead zones Conclusions
Global models Fromang & Nelson (2005): Planetesimal formation in Dust concentrates in long-lived vortex turbulent protoplanetary discs Dust density (5 cm and 25 cm): Anders Johansen Planet formation Dust in MHD turbulence Planetesimal formation Gas density and vorticity ( ω z ): Zonal flows Analytical model Global models Streaming and self-gravity Dead zones Conclusions
Increasing box size Planetesimal Stratified shearing box simulations with increasing box size formation in turbulent 100 1.05 100 protoplanetary 80 80 discs 60 Σ g /< Σ g > 60 t / T orb t / T orb 1.00 Anders 40 40 Johansen 20 20 0 0.95 0 −0.5 0.0 0.0 0.0 0.5 −1.0 −0.5 0.0 0.0 0.0 0.5 1.0 Planet x / H x / H formation 100 80 Dust in MHD 60 t / T orb turbulence 40 Planetesimal 20 formation 0 −2.5 −2.0 −1.5 −1.0 −0.5 0.0 0.5 1.0 1.5 2.0 2.5 x / H Zonal flows Orbital advection algorithm with Pencil Code Analytical model (Fourier interpolate the Keplerian advection term) Global models No spurious density depression in box centre Streaming and self-gravity (Johnson et al. 2008) Pressure bumps of few percent amplitude appear and Dead zones Plot by T. Sano reappear at time-scales of many orbits Conclusions
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