Lucio Mayer Zurich, Thomas Quinn University of

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GIANT PLANET FORMATION VIA DISK
INSTABILITY SPH simulations

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

2
Disk instability numerical simulations
-Toomre parameter QVsk/??G When Q lt 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 lt Q lt2
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 lt 1.5
Boss 2002 Density map of 3D grid simulation after
few disk orbital times
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Can clumps survive and collapse into
protoplanets?
Need very 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 ---gt very high
dynamic range can be handled. Resolution high
enough to resolve the local Jeans mass down to
very small scales (Bate Burkert 1997)
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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).
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Initial Conditions
Rin4 AU Rout20 AU
0.07 Mo ltMlt0.125 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
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Temperature profile
Eq. profile from A. Boss (19961998) - 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 gt 5 AU T r T (gt
10 AU) 30-70 K (see also Beckwith et al. 1990
D'Alessio et al. 2001)
-1/2
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Disk Evolution, Qmin 1.75
Mayer et al. 2002
1 million particles, locally isothermal eq.of
state , R20 AU
T160 yr
T350 yr
Torb (10 AU) 28 years
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Disk Evolution, Qmin 1.4
1 million particles, locally isothermal eq.of
state, R20 AU

• Gravitationally
• bound clumps,
• 10 times
• denser than the
• background

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T350 yr
T160 yr
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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 lt l lt lt
(Romeo 1991, 1994)
Jeans length
Toomre length
(Mayer et al. 2004)
2
2
lt 4p GS /W
(zero thickness disk)
3/2
Mjpr(vs/Gr)
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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
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Adiabatic versus Isothermal
-5
3
?max10?
Adiabatic with thermal energy equation (? 1.4)
cooling only by decompression, heating by
compression artificial viscosity (shocks). P(g
1)ru
g/cm
uu(t)
Adiabatic after t 160 yr
Locally Isothermal
T350 yr
T350 yr
-10
3
EOS switches to adiabatic when local density r gt
r , r 10
g/cm
(density threshold from flux-limited diffusion
simulations by Boss 2002)
Grav. bound clumps even with adiabatic switch
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Adiabatic EOS since t0 (Qmin 1.4)
N200,000, T250 yr
T 75 K
Density
Temperature (20 lt T lt 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
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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 lt 7 Mj. Orbits remain eccentric (e
0.1-0.3). Chaotic migration.
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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. -temperatu
res 200-500 K
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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 lt r, outer Tmin 35 K For comparison
disk model STARTING with 0.085 Mo and Tmin 36 K
fragments.
-5
- 4
Mayer et al. 2004
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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 gt 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)
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Density
Temperature
Long lived clumps require Tcool lt Torb
See Mayer et al . (2003)
Tcool0.5 Torb g5/3
Snapshots of sims with different Tcool, all after
10 Torb (10 AU) 300 years
T300 years
(Rice et al. 2003)
Tcool0.8Torb g7/5
Tcool1.4 Torb g7/5
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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?

T10 Years
T150 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)
T250 years
T450 years
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For Mdisk0.1 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
T150 years
T250 years
With companion and Tcool1/3 Torb after 200
years
In isolation after 200 years with Tcool1/3 Torb
Mayer et al., 2004
Nelson 2000
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Intermediate mass disks, Md0.05Mo stable in
isolation, can fragment in binaries, but only for
tcool lt ½ torb. Fragmentation can occur because
spiral shocks are weaker and heat the disk less.
T200 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
T200 yr
20
150 years
200 years
0.1 Mo disks at a separation 2 times Bigger (120
AU) evolve similarly to isolated systems -gt
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).
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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?
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How to make realistic ICs? Simulating the
formation of the protoplanetary diskprotostar
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
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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
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The inner 100 AU
Phase 1 rapidly rotating bar unstable
protostellar core
T0.02 Myr
T0.022 Myr
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Phase II bar fragmentation and merging of
fragments
T0.025 Myr
T0.024 Myr
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T0.030 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