GIANT PLANET FORMATION VIA DISK INSTABILITY: SPH simulations Lucio Mayer (Zurich), Thomas Quinn (University of Washington), James Wadsley (McMaster University), Joachim Stadel (Zurich)
Disk instability: numerical simulations -Toomre parameter Q=Vsk/G When Q < 1 a (zero-thickness) gaseous disk is locally unstable to axisymmetric perturbations (from linear perturbation theory). For disk response to global, non-axisymmetric perturbations need numerical simulations. In general 1 < Q <2 interesting regime where m-armed spiral modes can grow(Laughlin & Bodenheimer 1994, Laughlin, Korchagin & Adams 1997, Pickett et al. 1998) A massive self-gravitating, keplerian disk with M ~0.1 Mo within 20 AU can become gravitationally unstable and fragment into Jupiter-sized clumps in the outer, cooler part (T ~ 50 K) after a few orbital times/hundreds of years(Boss 1998, 2001, 2002; Kuiper 1959; Cameron 1978).Initial Qmin < 1.5 Boss 2002 Density map of 3D grid simulation after few disk orbital times
Can clumps survive and collapse into protoplanets? Needvery high resolution to model gravity accurately at small scales and resolve huge density gradients + no restrictions on computational volume Mayer, Quinn, Wadsley & Stadel (Science, 2002): 3D TreeSPH simulations with up to 50 times more particles than previously done (Nelson & Benz 1998) SPH is spatially adaptive ---> very high dynamic range can be handled. Resolution high enough to resolve the local Jeans mass down to very small scales (Bate & Burkert 1997)
Cosmology and Hydrodyamics with -Conspirators: James Wadsley McMaster Univ. Joachim Stadel Univ. Zurich Tom Quinn Univ. Washington Ben Moore Univ. Zurich Fabio Governato Univ. of Washington Derek Richardson Univ. of Maryland George Lake Washington State Jeff Gardner Univ. of Pittsburgh Simulations performed at Pittsburgh Supercomputing Center & Zurich Zbox Multi Platform, Massively Parallel treecode + SPH, multi stepping, cooling, UV background, Star Formation, SN feedback . Santa Barbara tested. Several state-of-the art published calculations in cosmology, galactic dynamics and galaxy formation (Wadsley, Stadel & Quinn 2003).
Initial Conditions Rin=4 AU Rout=20 AU 0.07 Mo <M<0.125 Mo -3/2 ~ r (Weidenschilling 1979) -14 -8 3 3 10 g/cm 10 g/cm -3D axisymmetric nearly keplerian self-gravitating disk -Central star (usually 1 Mo) is a point mass and can wobble in response to the disk. -No inner/outer boundary conditions
Temperature profile Eq. profile from A. Boss (1996;1998) - uses 2D radiative transfer code for a disk irradiated by a solar-type star and heated by material infalling from the molecular envelope T (4 AU) = 500-1000 K for R > 5 AU T ~r T (>= 10 AU) =30-70 K (see also Beckwith et al. 1990; D'Alessio et al. 2001) -1/2
Disk Evolution, Qmin ~1.75 Mayer et al. 2002 1 million particles, locally isothermal eq.of state , R=20 AU T=160 yr T=350 yr Torb (10 AU) = 28 years
Disk Evolution, Qmin ~ 1.4 1 million particles, locally isothermal eq.of state, R=20 AU • Gravitationally • bound clumps, • 10 times • denser than the • background 6 T=350 yr T=160 yr
Scaling properties of disk fragmentation l*= characteristic scale at which clump formation occurs lt is corrected for finite disk thickness (pressure) and gravitational softening lj < l* < lt (Romeo 1991, 1994) Jeans length Toomre length (Mayer et al. 2004) 2 2 lt =4p GS /W (zero thickness disk) 3/2 Mj=pr(vs/Gr) 6 For the same Q disks with lower temperature (lower masses) have a lower fragmentation scale. From definition of Toomre mass, Mmax ~ S ,from Jeans mass, Mmin ~ T . In coldest models Saturn-sized clumps form. 3 5/4
Adiabatic versus Isothermal -5 3 max~10 Adiabatic with thermal energy equation ( = 1.4):cooling only by decompression, heating by compression + artificial viscosity (shocks). P=(g – 1)ru g/cm u=u(t) Adiabatic after t ~ 160 yr 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 flux-limited diffusion simulations by Boss 2002) Grav. bound clumps even with adiabatic switch
Adiabatic EOS since t=0 (Qmin ~ 1.4) N=200,000, T=250 yr T ~ 75 K Density Temperature (20 < T < 200 K) Clump formation in the spiral arms suppressed because of shock heating (no radiative cooling included). However temperature in the spiral arms only 50% higher than in isothermal case
Long term evolution Currently disk models are being extended by an order of magnitude in time (up to 10000 years) thanks to new faster gravity calculation 200.000 particles with switch to adiabatic (20 AU ) T = 4000 yr (~ 150 orbital times at 10 AU) T = 1900 yr (~ 70 orbital times at 10 AU) T = 320 yr 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.
Properties of clumps Color-coded velocity field shown. Clumps are: -in differential rotation, on coplanar orbits -flattened oblate spheroids with c/a ~ 0.7-0.9 -have rotation rates such that Vrot ~ 0.3-2 x Vrot (Jupiter) after contraction down to the mean density of Jupiter and assuming conservation of angular momentum -have a wide range of obliquities, from 2 to 180 degrees. Clump-clump and disk-clump J exchange. -temperatures 200-500 K
Initial Conditions: a growing disk Simulations starting with a disk already marginally unstable (Q ~ 1.3-1.4) are idealized. The disk will eventually approach such a state from a higher Q – will it eventually self-regulate itself and avoid fragmentation? 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) Locally isothermal EOS for r < r*, outer Tmin = 35 K For comparison disk model STARTING with 0.085Mo and Tmin = 36 K fragments. -5 - 4 Mayer et al. 2004
However, even the growing disk evolves isothermally up to the critical density threshold What if heating by shocks and PdV work was is not completely radiated away during disk growth? Need to follow heating and cooling self-consistently. Ideal goal is model with full 3D radiative transfer. Intermediate steps before: I) Volumetric cooling - disk cools at a fixed rate only dependent on radius.Tcool = A W(r) -1 (Rice et al. 2002) -10 r > r* ~ 10 g/cm 3 Cooling swtiched off when Rates and threshold consistent with lower res grid simulations with flux-limited diffusion or gray-Eddington approximation RT (Boss 2002; Johnson & Gammie 2003; Pickett et al. 2004)
Density Temperature Long lived clumps require Tcool <~ Torb SeeMayer et al . (2003) Tcool=0.5 Torb; g=5/3 Snapshots of sims with different Tcool, all after ~ 10 Torb (10 AU) ~ 300 years T=300 years (Rice et al. 2003) Tcool=0.8Torb; g=7/5 Tcool=1.4 Torb; g=7/5
Disk instability in binary systems • 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). Is fragmentation more or less efficient in binary sytems? T=10 Years T=150 years Set of 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). Initial orbit close to e.g. t Boo(Patience et al. 2003) T=250 years T=450 years
For Mdisk=0.1 Mo tides generate strong spiral shocks that suppress clump formation through heating the disk (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 and Tcool=1/3 Torb after 200 years In isolation after 200 years with Tcool=1/3 Torb Mayer et al., 2004 Nelson 2000
Intermediate mass disks, Md=0.05Mo stable in isolation, can fragment in binaries, but only for tcool <~ ½ torb. Fragmentation can occur because spiral shocks are weaker and heat the disk less. T=200 yr Light disks, Md =0.012 Mo, never fragment. In both cases disks remain cold enough to support any type of grain For light disks Same result for tcool = 10 torb tcool = 1/2 torb T=200 yr
150 years 200 years 0.1 Mo disks at a separation 2 times Bigger (120 AU) evolve similarly to isolated systems -> fragment for tcool ~ 0.5 torb Unequal mass disks; transient clump formation in more massive disk, Mdisk ~ 0.1 Mo Bottom line - if GPs form by disk instability then anti-correlation between binary separation and presence of planets - if GPs form by core-accretion no correlations with binarity (provided that Jupiters can form in a light disk, see Rice & Armitage 2003).
MANY OPEN QUESTIONS! -Can the disk cool efficiently by radiation/convection so that GI can actually proceed towards fragmentation? - What is the effect of turbulence on overdense regions? --can turbulence inhibit local collapse of clumps? - What is the effect of magnetorotational instability on the angular momentum/surface density evolution of the disk? --especially what one should expect as for the combined effect of GI and MRI? Is GI suppressed, enhanced or both depending on the situations? -Will protoplanets really contract down to giant-planet densities? Simulations limited by gravitational softening and lack of realistic radiation physics (just now flux limited diffusion included) -How does dust planetesimals respond to GI in the gaseous disk? --can GI help coagulation of planetesimals into large cores?
How to make realistic ICs? Simulating the formation of the protoplanetary disk+protostar system from the 3D collapse of a molecular cloud core with enough resolution to follow the gravitational instability in the disk. Use variable resolution to allow higher resolution in the central regions (where the disk assembles) and reduce computational cost
Collapse of a rotating 1 Mo molecular cloud core 0.5 million particles in total but inner 2000 AU effective resolution of a 2 million particles model. Use polytropic EOS with variable g to mimic change of gas opacity with density (Bate 1998) 0.05 pc 2000 AU
The inner ~ 100 AU Phase 1 – rapidly rotating bar unstable protostellar core T=0.02 Myr T=0.022 Myr
Phase II – bar fragmentation and merging of fragments T=0.025 Myr T=0.024 Myr
T=0.030 Myr Phase III – Formation of a binary system with protostars and protoplanetary disks Timesteps prohibitively small in the cores – maybe use sink particles? Need even higher mass and force resolution to follow Appropriately disk instability