Planet Formation
Outline 1. Observations of planetary systems 2. Protoplanetary disks 3. Formation of planetesimals (km-scale bodies) 4. Formation of terrestrial and giant planets 5. Evolution and stability of planetary systems
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
Planetesimal formation Particle radius s, mass m, velocity relative to gas Δv Define friction time: t fric = m"v F D For particles smaller than the gas mean free path, drag force is described by Epstein law: F D = " 4# with v 3 $s2 v th %v th the mean thermal speed of the molecules Spherical dust particles, find: t fric = " m " s v th with ρ m the material density, ρ the gas density
" m s Substitute: t fric = 3 g cm-3 10 "4 cm 10 " "9 vg th cm -3 10 5 cm s = 3 s -1 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
Vertical settling Consider particle settling in a laminar disk Terminal velocity: m" 2 z = 4# 3 $s2 v th v settle v settle = $ m $ s v th " 2 z For micron sized particles at z ~ h at 1 AU: v settle ~ 0.1 cm s -1 t settle ~ 10 5 yr small relative velocities, but still settles on time scale short compared to disk lifetime Scale dependent v settle implies collisions between particles
Assume that all collisions lead to coagulation. Toy model: look at one particle, mass m, settling through background of smaller particles: dm dt = "s2 v settle f#(z) dz dt = $v settle f is dust to gas ratio (f ~ 0.01), model originally due to Safronov Simple coupled system to solve numerically
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 ~10 3 yr Too simple - but does show that if collisions lead to accretion then growth to small but macroscopic sizes is very quick
Intrinsic disk turbulence will oppose settling. Equate diffusion time across scale h to settling time: t diffuse = h 2 D ; t settle = h v settle Then, if D ~ ν: " = #e1 2 2 $ m s % 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
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)
Radial drift of solids Recall that gas rotated slightly slower than Keplerian orbital velocity (pressure support): $ v ",gas = v K 1# n c 2 s & 2 % v K ' ) ( 1 2 = v K ( 1#* ) 1 2 Consequences: dust particle (strong coupling to gas): - swept up, orbits at v = v φ,gas - unbalanced radial force - drifts in at terminal velocity large rock (100m) weak coupling: - orbits at v = v K - experiences head wind drag force - loses angular momentum, spirals in
General treatment after Weidenschilling (1977) Particle equations of motion: dv r dt = v 2 " r # $ 2 Kr # 1 t fric ( v r # v r,gas ) d dt ( rv " ) = # r t fric ( v " # v ",gas ) Assuming (reasonably) that inspiral is slow compared to the orbital time, can simplify to: v r = " #1 fricv r,gas #$v K #1 " fric + " fric where the dimensionless friction time is defined as: " fric # t fric $ K
v r = " #1 fricv r,gas #$v K #1 " fric + " fric Neglecting radial flow of the gas, maximum is evidently where τ fric =1 v r = " 1 2 #v K Critical point: η << 1 but not that small - η ~ 10-2 typical Since v K ~ 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 10 2 or 10 3 yr!
Actual particle size that corresponds to τ fric = 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
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)
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
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?
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?)
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
Stability analysis of a thin sheet of particles to gravitational instability yields a condition for instability: Q " #$ %G& s <1 where σ is the particle velocity dispersion and Σ p 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: m ~ "# 2 $ s ~ 3%10 18 g corresponds to a solid body with a radius of 5-10 km!
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
Stability to Kelvin-Helmholtz instability is determined by the Richardson number: Ri = " g zd ln# /dz dv $ /dz ( ) 2 with instability for Ri < 0.25 Applying to a dust layer of thickness h d estimate: # Ri " 0.25 h/r & % ( $ 0.05' )2 # h d /h & % ( $ 0.0375' For Goldreich-Ward to work we need h d ~ 10-4 h Appears to be a fatal obstable 2
Simulation: Chiang (2008)
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
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)
Johansen et al. (2007) find that collapse can occur in the flow
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
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
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
Are there circumstance under which planetesimals could not form? Is there an observational discriminant in the Solar System between these two models?