1. James Wadsley, Sijing Shen (McMaster), Lucio Mayer, Joachim Stadel (Zurich),Graeme Luftkin (Maryland), Tom Quinn (Washington) UWO Disk Workshop 2006 Simulating the Fragmentation of Proto-planetary disks

2. Overview Gravitational instability: a mechanism to produce structure in proto-planetary disks Instability  Spiral waves  energy/mass transport (self regulation?) Instability  Spiral waves  energy/mass transport (self regulation?) Non-linear spiral waves  fragmentation Non-linear spiral waves  fragmentation Fragmentation  rapid gas giant planets Fragmentation  rapid gas giant planets Alternative: Interplay between dust/planetesimals and gas structures dramatically modifies standard planetesimal picture Alternative: Interplay between dust/planetesimals and gas structures dramatically modifies standard planetesimal picture

3. Disk Stability Criterion Toomre Q parameter (gaseous disk) Toomre Q parameter (gaseous disk) Toomre’s Stability Criterion : Locally Q > 1 for a thin disk to be stable under axisymmetric perturbations (Toomre, 1964) Toomre’s Stability Criterion : Locally Q > 1 for a thin disk to be stable under axisymmetric perturbations (Toomre, 1964) Q MIN > 1.4 for a thin disk to be stable against fragmentation under non-axisymmetric perturbations (Papaloizou & Savonije 1991, Mayer et al., 2004) Q MIN > 1.4 for a thin disk to be stable against fragmentation under non-axisymmetric perturbations (Papaloizou & Savonije 1991, Mayer et al., 2004) C s Sound Speed κ Epicycle Freq. G Gravitational constant Σ Surface Density

4. Initial conditions Q < 1.5 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) constraints) Temperatures: < 100 K  ~ r  

5. Further Limitations (Or: was Q IC reasonable?) Are assumed initial conditions “natural” ? Are assumed initial conditions “natural” ? Effective cooling necessary for fragmentation: t cool < 3/Ω (  = 2, Rice et al 2003), t cool < 12/Ω (  = 7/5, Mayer et al) (cf. Rafikov 2005) Effective cooling necessary for fragmentation: t cool < 3/Ω (  = 2, Rice et al 2003), t cool < 12/Ω (  = 7/5, Mayer et al) (cf. Rafikov 2005) Self regulation: will viscous evolution saturate instabilities above Q ~ 1.5? Self regulation: will viscous evolution saturate instabilities above Q ~ 1.5? Ideal solution for proto-planetary disk sims: 1. Model collapse to form star+disk 2. Include full physics: MHD, radiation, dust, …

6. Different regime for GI: Early Proto-stellar Disks High disk/star mass ratio 1:1 High disk/star mass ratio 1:1 Lower surface density: optically thin Lower surface density: optically thin Radiative cooling effective Highly flared Large sizes ~ 2-3000 AU Quasi-static collapse Timescale ~ 10 5 years Yorke & Bodenheimer 1999

7. Self-consistent Disk 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

8. Extended Disks: Modified expectations for Toomre-Q Toomre’s stability criterion for gaseous disk (Toomre, 1964) Toomre’s stability criterion for gaseous disk (Toomre, 1964) Finite thickness lowers critical Q to ~ 0.75 Finite thickness lowers critical Q to ~ 0.75 (Kim et al. 2001) (Kim et al. 2001) Non-Axisymmetric modes Q ~< 1.1 C s Sound Speed κ Epicycle Freq. G Gravitational constant Σ Surface Density

9. 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

10. Resolution Matters…So Does Finite Thickness Isolated Disk Q = 1.3 Isolated Disk Q = 1.3 Spurious fragmentation No structure Must resolve both Jean Mass (Bate et al 1997) and Vertical structure 200,000 particles Finite disk thickness 2000 particles single layer 20000 particles, single layer

11. Resolution Matters…So Does Finite Thickness Collisions: Disk Q = 1.3 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) 200,000 particles,Finite disk thickness 2000 particles, single layer

12. Large proto-stellar disks Large cross-section for disk-disk and disk-star encounters Large cross-section for disk-disk and disk-star encounters Initially stable but very extended: Initially stable but very extended: Suffer massive perturbations in encounters Outcome? Outcome? … see talk by Sijing Shen on Friday

13. N-body Solver (Tree Method) and Smoothed Particle Hydrodynamics, Parallel N-body Solver (Tree Method) and Smoothed Particle Hydrodynamics, Parallel Physics: Gravity, Hydrodynamics, Atomic Chemistry (Radiative Heating, Cooling), *New*: Flux Limited Diffusion 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) Subgrid Physics: Star Formation, Supernova Feedback, Planetesimal Collisions (NB: NOT at the same time) Wadsley, Stadel & Quinn 2004 Controversy? Best to pour on some …

14. Qmin = 1.3 Qmin=1.5 Locally isothermal, T(r) N=1 million particles (Mayer, Quinn, Wadsley, Stadel 2002) 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=150 years T=250 years T=150 years T=250 years Back to Proto-planetary disks Start simple: Efficient Cooling  Isothermal

15. T=350 yr Grav. bound clumps persist with adiabatic transition  max~10  -5 g/cm 3 EOS switches to adiabatic when local density  ~ 10 Isothermal= cooling perfectly balances heating Adiabatic cooling by expansion, heating by compression / shocks -10 g/cm 3 (density threshold from simulations with radiation transport - Boss 2002) Locally Isothermal Adiabatic after t ~ 160 yr Optically thick, Inefficient cooling  Clumps Adiabatic

16. Models with shock heating, specified cooling timescale: Long lived clumps require Tcool <~ Torb Density Temperature Tcool=0.8Torb;  =7/5 Tcool=1.4 Torb;  =7/5 Snapshots of sims with different Tcool, all after ~ 10 Torb (10 AU) ~ 300 years T=300 years See Mayer, Wadsley et al. (2004, 2005) Rice et al. (2003), Lodato & Rice (2004) FRAGMENTATION NEEDS RAPID COOLING

17. Clump Properties Sensitive to surface density/temperature: Lighter & colder  smaller characteristic masses Clump mass distribution

18. 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. Clump Properties

19. 200.000 particles with switch to adiabatic 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. T = 4000 yr (~ 150 orbital times at 10 AU) Orbital Evolution

20. No rapid (< 10 4 years) planet migration in disk instability. No clear gap forms, slight outward migration PPV: Durisen, Boss, Mayer et al. 2006 Orbital Evolution

21. A growing, self-gravitating PP disk Disk started close to the critical Q are probably unrealistic We simulate a uniformly growing disk, initial mass ~ 0.0085 Mo becomes ~ 0.085 in 1000 years (constant growth rate ~ to accretion rate of protostellar objects from cloud cores, e.g. Yorke & Bodenheimer 1999; Boss & Hartmann 2002, dM/dt ~ 10 - 10 Mo/yr). Constant increase of particle masses. Locally isothermal EOS for  = 10 g/cm adiabatic EOS for  (Boss 2002). Outer Tmin = 30 K at t=0 -5 - 4 Mayer et al. 2004 Initial temperature profile from 2D radiative transfer simulations of protostellar disk formation (Boss 1998) -10 3

22. For Mdisk=0.1 Mo tides generate strong spiral shocks that suppress clump formation through heating the disk ( Mayer, Wadsley et al 2005) See also Nelson (2000): High temperatures problematic also for survival of water ice and core accretion Tmap T=150 years T=250 years With companion In isolation Binary Systems

23. -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 T=10 Years T=450 years T=200 years d=120 AU Prediction: giant planet formation less likely in tight binaries Consistent with recent survey (Eggenberger et al. 2005) Binary Systems

24. Gravitational Instability Code Disagreements: SPH vs. Grid Boundary Conditions Thermal Assumptions Gravitational Resolution vs. Hydro Radiative Transfer: Convection?

25. Wengen code comparison, HI-RES ISO sims Code Agreements: Same ICs for all codes Convergence on isothermal behaviour? GASOLINE (SPH)GADGET2 (SPH) Indiana code (fixed cyl. grid l with a.v) FLASH (PPM AMR, Cartesian grid) Mayer et al., in prep.

26. Fixed 3D eulerian grid (256x256x64). Fragmentation depends on struggle between pressure forces and gravitational forces ----> a sensible numerical simulation must be able to (1) model gravity accurately and (2) model realistically the balance between heating and cooling in the disk This is relevant for both (a) the formation of overdensities in the disk and (b) their transition into long-lasting gravitationally bound entities. Fixed-grid simulations can suffer from gravity resolution problems. (Pickett et al. 2003) Test (self-gravitating blob) shows azimuthal self-gravity significantly underestimated compared to analytical solution Achieving high gravitational force resolution is critical, clump formation involves both large and small scales!

27. 3D SPH simulations with radiative transfer With Graeme Lufkin (Maryland) - (Flux-limited) diffusion equation implemented as in Cleary & Monaghan (1999) for optically thick part of the disk (  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 (  < 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.

28. Disk grows to ~0.1 Mo over ~ 50 Torb 3D SIMULATIONS WITH RADIATIVE TRANSFER (flux-limited diffusion) (Mayer, Lufkin et al., 2006) -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?  = 2.4, RS = 1.4  = 2.4, RS = 0.8  = 2.4, RS = 1  = 2.7, RS= 1

29. 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). T=120 years 0.5 AU 4 T=160 years 0.5 AU Clump Colour = Temperature

30. Cells of vertical motions are: - intermittent - apparently associated with superadiabatic gradients (Schwarzschild criterion for convection) Evolution of vertical temperature profile of a cooling clump

31. Analysis complicated by: - shock bores that can also produce vertical motions and superadiabatic gradients (see also Cai 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!