Lucio Mayer (Zurich), Thomas Quinn (University of Washington), James Wadsley (McMaster University), Joachim Stadel (Zurich)

GIANT PLANET FORMATION VIA DISK INSTABILITY: SPH simulations

Disk instability: numerical simulations

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

-Toomre parameter Q=VsGWhen 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)

Boss 2002Density map of3D grid simulationafter few disk orbitaltimes

Can clumps survive and collapse into protoplanets?

Need very high resolution to model gravity accuratelyat 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)

-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

Cosmology and Hydrodyamics with

Multi Platform, Massively Parallel treecode + SPH, multi stepping, cooling, UV background, Star Formation, SN feedback .Santa Barbara tested. Several state-of-the art published calculationsin cosmology, galactic dynamics and galaxy formation (Wadsley,Stadel & Quinn 2003).

Simulations performed atPittsburgh Supercomputing Center& Zurich Zbox

Initial Conditions

~ r-3/2

-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

Rin=4 AURout=20 AU

10-14

g/cm 3 10-8

g/cm 3

0.07 Mo<M<0.125 Mo

(Weidenschilling 1979)

Temperature profile

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

Eq. profile from A. Boss (1996;1998) - uses 2D radiative transfer code for a disk irradiated bya solar-type star andheated by materialinfalling from themolecular envelope

Disk Evolution, Qmin ~1.75

1 million particles, locally isothermal eq.of state , R=20 AU

T=160 yr T=350 yrTorb (10 AU) = 28 years

Mayer et al. 2002

Disk Evolution, Qmin ~ 1.4

1 million particles, locally isothermal eq.of state, R=20 AU

T=160 yr T=350 yr

Gravitationallybound clumps,10 times denser than the background

6

Scaling properties of disk fragmentation

*= characteristic scale at which clump formation occurs

j < < t t is corrected for finite disk thickness (pressure) and gravitational softening

For the same Q disks with lower temperature (lower masses) have a lowerfragmentation scale. From definition of Toomre mass, Mmax ~ ,from Jeans mass, Mmin ~ T . In coldest models Saturn-sized clumps form.

(Mayer etal. 2004)

5/4

Jeans length

(Romeo 1991, 1994)

Toomrelength

3

t =4 G/2 2

Mj=vs/G6

3/2

(zero thickness disk)

Adiabatic versus Isothermal

T=350 yr T=350 yr

Grav. bound clumps even with adiabatic switch

max~10 -5

g/cm3

EOS switches to adiabatic when local density ~ 10

Adiabatic with thermal energy equation ( = 1.4): cooling only by decompression, heating by compression + artificial viscosity (shocks).P=( – 1)u

-10 g/cm 3

(density threshold from flux-limited diffusion simulations by Boss 2002)

u=u(t)Locally Isothermal Adiabatic after t ~ 160 yr

Adiabatic EOS since t=0 (Qmin ~ 1.4)

Density Temperature (20 < T < 200 K)

T ~ 75 K

N=200,000, T=250 yr

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

200.000 particles with switch to adiabatic (20 AU )

T = 320 yr T = 1900 yr (~ 70 orbitaltimes at 10 AU)

Merging drastically reduces the number of clumps. Only three remain after ~ 500 yr, with masses 2Mj < 7 Mj. Orbitsremain eccentric (e ~ 0.1-0.3). “Chaotic” migration.

Currently disk models are being extended by an order of magnitudein time (up to 10000 years) thanks to new faster gravity calculation

T = 4000 yr (~ 150 orbitaltimes at 10 AU)

Properties of clumpsColor-coded velocity field shown.

Clumps are:

-in differential rotation, on coplanar orbits

-flattened oblate spheroids withc/a ~ 0.7-0.9

-have rotation rates such that Vrot~ 0.3-2 x Vrot (Jupiter) aftercontraction 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-clumpand disk-clump J exchange.

-temperatures 200-500 K

Initial Conditions: a growing diskSimulations 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 , outer Tmin = 35 KFor comparison disk model STARTING with 0.085 Mo and Tmin = 36 K fragments.

-5 - 4

Mayer et al. 2004

However, even the growing disk evolves isothermally up to the critical density thresholdWhat 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 (r) -1

~ 10

-10 g/cm

3

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)

Cooling swtiched off when

(Rice et al. 2002)

Long lived clumps require Tcool <~ Torb

Density Temperature

Tcool=0.5 Torb; =5/3

Tcool=0.8Torb; =7/5

Tcool=1.4 Torb; =7/5

Snapshots of sims with differentTcool, all after ~ 10 Torb (10 AU) ~300 years

T=300 years

See Mayer et al . (2003)

(Rice et al. 2003)

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

T=250 years T=450 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)

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

Mayer et al.,2004 Nelson 2000

Tmap

T=150 years T=250 years

With companion and Tcool=1/3 Torb after 200 years

In isolation after 200years with Tcool=1/3 Torb

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.

T=200 yr

tcool = 1/2 torb In both casesdisks remain cold enough to support any type of grain

For light disksSame resultfor tcool = 10 torb

Unequal mass disks;transient clump formation inmore massive disk, Mdisk ~ 0.1 Mo

0.1 Mo disks at a separation 2 timesBigger (120 AU) evolve similarlyto isolated systems -> fragmentfor tcool ~ 0.5 torb

150 years 200 years

Bottom line

- if GPs form by disk instability then anti-correlation between binaryseparation and presence of planets- if GPs form by core-accretion no correlations with binarity (providedthat Jupiters can form in a light disk, see Rice & Armitage 2003).

-Can the disk cool efficiently by radiation/convection so that GIcan 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 theangular momentum/surface density evolution of the disk?--especially what one should expect as for the combinedeffect of GI and MRI? Is GI suppressed, enhanced orboth depending on the situations?

-Will protoplanets really contract down to giant-planet densities?Simulations limited by gravitational softening and lack of realisticradiation 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?

MANY OPEN QUESTIONS!

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 resolutionin 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 resolutionof a 2 million particles model. Use polytropic EOS with variable 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=0.024 Myr T=0.025 Myr

Phase III – Formationof a binary system withprotostars and protoplanetarydisks

Timesteps prohibitivelysmall in the cores – maybeuse sink particles?

Need even higher mass and force resolution to follow Appropriately disk instability

T=0.030 Myr