Simulating the Fragmentation of Proto-planetary disks UWO Disk Workshop 2006 James Wadsley, Sijing Shen (McMaster), Lucio Mayer, Joachim Stadel (Zurich) ,Graeme Luftkin (Maryland), Tom Quinn (Washington)
Overview Gravitational instability: a mechanism to produce structure in proto-planetary disks • Instability Spiral waves energy/mass transport (self regulation?) • Non-linear spiral waves fragmentation • Fragmentation rapid gas giant planets • Alternative: Interplay between dust/planetesimals and gas structures dramatically modifies standard planetesimal picture
Disk Stability Criterion • Toomre Q parameter (gaseous disk) • Toomre’s Stability Criterion : Locally Q > 1 for a thin disk to be stable under axisymmetric perturbations (Toomre, 1964) • QMIN > 1.4 for a thin disk to be stable against fragmentation under non-axisymmetric perturbations (Papaloizou & Savonije 1991, Mayer et al., 2004) Cs Sound Speed κ Epicycle Freq. G Gravitational constant Σ Surface Density
Initial conditions Q < 1.5 Requires a heavy, cold disk: Typically order 0.05-0.1 Msun ~ 10 times Minimum mass solar nebula (but within observational constraints) Temperatures: < 100 K -a ~ r (a=1-1.5)
Further Limitations(Or: was QIC reasonable?) • Are assumed initial conditions “natural” ? • Effective cooling necessary for fragmentation: tcool < 3/Ω ( = 2, Rice et al 2003), tcool < 12/Ω ( = 7/5, Mayer et al) (cf. Rafikov 2005) • Self regulation: will viscous evolution saturate instabilities above Q ~ 1.5? Ideal solution for proto-planetary disk sims: • Model collapse to form star+disk • Include full physics: MHD, radiation, dust, …
Different regime for GI:Early Proto-stellar Disks • High disk/star mass ratio 1:1 • Lower surface density: optically thin Radiative cooling effective Highly flared Large sizes ~ 2-3000 AU Quasi-static collapse Timescale ~ 105 years Yorke & Bodenheimer 1999
Self-consistentDisk Density Profile • Temperature structure assumed • Vertically hydrostatic (with self-gravity), Radially centrifugally balanced • Disk mass peaks ~ 500-1000 AU • Result close to the density profile of disk formation simulation by Yorke et al., 1999 Shen & Wadsley 2006
Extended Disks:Modified expectations for Toomre-Q • Toomre’s stability criterion for gaseous disk (Toomre, 1964) • Finite thickness lowers critical Q to ~ 0.75 (Kim et al. 2001) Non-Axisymmetric modes Q ~< 1.1 Cs Sound Speed κ Epicycle Freq. G Gravitational constant Σ Surface Density
Thickness of the Disk Minimum Q = 0.9 Minimum Q = 1.1 Minimum Q = 1.3 Note: Watkins et al 1998 Q ~ 2.4 More sphere than disk
Resolution Matters…So Does Finite Thickness 200,000 particles Finite disk thickness 2000 particles single layer 20000 particles, single layer • Isolated Disk Q = 1.3 Spurious fragmentation No structure Must resolve both Jean Mass (Bate et al 1997) and Vertical structure
Resolution Matters…So Does Finite Thickness 200,000 particles ,Finite disk thickness 2000 particles, single layer • Collisions: Disk Q = 1.3 Spurious fragmentation No structure In this regime, well resolved models with Q > 1.1 do not fragment, even in collisions Previous work (Watkins et al 1998) used flat disk assumptions, low resolution and see fragmentation even with Q ~ 2.4 (see also Lin et al 1998)
Large proto-stellar disks • Large cross-section for disk-disk and disk-star encounters • Initially stable but very extended: Suffer massive perturbations in encounters • Outcome? … see talk by Sijing Shen on Friday
Controversy? Best to pour on some … • N-body Solver (Tree Method) and Smoothed Particle Hydrodynamics, Parallel • Physics: Gravity, Hydrodynamics, Atomic Chemistry (Radiative Heating, Cooling), *New*: Flux Limited Diffusion • Subgrid Physics: Star Formation, Supernova Feedback, Planetesimal Collisions (NB: NOT at the same time) Wadsley, Stadel & Quinn 2004
Back to Proto-planetary disks Start simple: Efficient Cooling Isothermal Qmin = 1.3 Locally isothermal, T(r) N=1 million particles Qmin < 1.4 gives gravitationally bound clumps (Mayer, Quinn, Wadsley, Stadel 2002) Q threshold agrees with several other works, e.g. Pickett et al. 2000, Rice et al 2003, Johnson & Gammie (2003) T=250 years T=150 years Qmin=1.5 T=150 years T=250 years
Optically thick, Inefficient cooling Clumps Adiabatic -5 3 max~10 g/cm Adiabatic after t ~ 160 yr Isothermal= cooling perfectly balances heating Adiabatic cooling by expansion, heating by compression/shocks Locally Isothermal T=350 yr T=350 yr -10 3 EOS switches to adiabatic when local densityr > r* , r* ~ 10 g/cm (density threshold from simulations with radiation transport - Boss 2002) Grav. bound clumps persist with adiabatic transition
FRAGMENTATION NEEDS RAPID COOLING Density Temperature Models with shock heating, specified cooling timescale: Long lived clumps require Tcool <~ Torb Tcool=0.8Torb; g=7/5 See Mayer, Wadsley et al . (2004, 2005) Rice et al. (2003), Lodato & Rice (2004) T=300 years Snapshots of sims with different Tcool, all after ~ 10 Torb (10 AU) ~ 300 years Tcool=1.4 Torb; g=7/5
Clump Properties Sensitive to surface density/temperature: Lighter & colder smaller characteristic masses Clump mass distribution
Clump Properties Initial Eccentric Orbits Differential rotation, on coplanar orbits along disk midplane - Flattened oblate spheroids with c/a ~ 0.7-0.9 - Rotation: ~ 0.3-2 x Rotation of Jupiter after contraction down to the mean density of Jupiter, assuming conservation of angular momentum - Wide range of obliquities, from 2 to 180 degrees. Clump-clump and disk-clump J exchange.
Orbital Evolution 200.000 particles with switch to adiabatic T = 4000 yr (~ 150 orbital times at 10 AU) T = 320 yr T = 1900 yr (~ 70 orbital times at 10 AU) Merging drastically reduces the number of clumps. Only three remain after ~ 500 yr, with masses 2Mj < 7 Mj. Orbits remain eccentric (e ~ 0.1-0.3). “Chaotic” migration.
Orbital Evolution No rapid (< 104 years) planet migration in disk instability. No clear gap forms, slight outward migration PPV: Durisen, Boss, Mayer et al. 2006
Binary Systems T=250 years T=150 years With companion In isolation Tmap For Mdisk=0.1 Mo tides generate strong spiral shocks that suppress clump formation through heating the disk ( Mayer, Wadsley et al 2005) See alsoNelson (2000):High temperatures problematic also for survival of water ice and core accretion
About 15% of known extrasolar planets are in binary systems (Eggenberger et al. 2004; Patience et al. 2003)and targeted surveys are on the way (e.g. the Geneva Group). • Runs with different • cooling times, orbit with • ecc ~ 0.1, mean sep. 60 AU. • In massive disks • (M~ 0.1Mo) clump formation • does not occur even with • Tcool as short as ~ 1/3 • Torb (shown here). • Fragmentation needs • d >~ 100 AU • t Binary Systems T=10 Years T=450 years d=120 AU T=200 years Prediction: giant planet formation less likely in tight binaries Consistent with recent survey (Eggenberger et al. 2005)
Gravitational InstabilityCode Disagreements: SPH vs. Grid Boundary Conditions Thermal Assumptions Gravitational Resolution vs. Hydro Radiative Transfer: Convection?
Wengen code comparison, HI-RES ISO sims Code Agreements: Same ICs for all codes Convergence on isothermal behaviour? GASOLINE (SPH) GADGET2 (SPH) Mayer et al., in prep. Indiana code (fixed cyl. grid l with a.v) FLASH (PPM AMR, Cartesian grid)
3D SPH simulations with radiative transfer With Graeme Lufkin (Maryland) • - (Flux-limited) diffusion equation implemented as inCleary & Monaghan • (1999)for optically thick part of the disk (t > 1). Flux limiter as in • Bodenheimer et al. (1990). • - Complete set of (Rosseland mean) dust grain opacities for grain • sizes up to 1 mm(from D’Alessio et al. 2001, same as those used by • Pickett, Durisen, Meija & collaborators) • Optically thin disk boundary cools as a blackbody (t < 2/3, depth • of radiating zone and radiative efficiency adjustable) • No external irradiation from central star/neighboring stars and no back • scattering of emitted photons (crudely mimic these by changing radiative • efficiency at the boundary) • - Shock heating included via standard Monaghan artificial viscosity.
3D SIMULATIONS WITH RADIATIVE TRANSFER (flux-limited diffusion) (Mayer, Lufkin et al., 2006) Disk grows to ~0.1 Mo over ~ 50 Torb -Fragmentation less likely than in ISO+ADI Simulations. Need more massive disk, M > 0.12 Mo instead of M > 0.08 Mo as in Boss (2004) -Fragmentation sensitive on (a) molecular weight (controls strength of pressure gradient in spiral shocks) And (b) efficiency of radiative losses in optically-thin region Link with frequency/metallicity relation of extrasolar planets? m=2.4, RS= 1.4 m=2.4, RS= 1 m=2.4, RS= 0.8 m=2.7, RS= 1
Convective cooling? From Mayer et al. (2004) and Rice et al. (2002)we know that Tcool ~ 1-2 Torb at the midplane to overcome shock heating and lead to fragmentation. But timescale for cooling by vertical radiation transport too long at overdensities ~ 10 years! Boss (2002):disks cooled by convection. In our FLD simulation up/downwelling with typical speeds ~ 0.1 Km/s (orbital velocities ~ 1 km/s at 10 AU), enough to transport the heat from the midplane to the upper layers (disk scale height ~ 2 AU) in ~ 30 years (Torb at ~ 10 AU). 4 0.5 AU 0.5 AU T=120 years T=160 years Clump Colour = Temperature
Cells of vertical motions are: • - intermittent • apparently associated with superadiabatic gradients • (Schwarzschild criterion for convection) Evolution of vertical temperature profile of a cooling clump
Analysis complicated by: • shock bores that can also produce vertical motions and • superadiabatic gradients (see alsoCai et al. 2005) • concurrent 3D accretion flow towards overdensities) • Accretion of nearby colder, optically-thin regions could also • promote growth of clumps. Cooling highly inhomogeneous!