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, nonaxisymmetric 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 2002
Density map of
3D grid simulation
after few disk orbital
times
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
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 ---> 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
Simulations performed at
Tom Quinn
Univ. Washington
Pittsburgh
Ben Moore
Univ. Zurich
Supercomputing Center
Fabio Governato Univ. of Washington
& Zurich Zbox
Derek Richardson Univ. of Maryland
George Lake
Washington State
Jeff Gardner
Univ. of Pittsburgh
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
0.07 Mo
<M<0.125 Mo
Rin=4 AU
Rout=20 AU
~ r -3/2
(Weidenschilling 1979)
-14
10 g/cm 3
-8
10 g/cm3
-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 -1/2
T (>= 10 AU) =30-70 K
(see also Beckwith et al.
1990; D'Alessio et al. 2001)
Disk Evolution, Qmin ~1.75
Mayer et al. 2002
1 million particles, locally isothermal eq.of state , R=20 AU
Torb (10 AU) = 28 years
T=160 yr
T=350 yr
Disk Evolution, Qmin ~ 1.4
1 million particles, locally isothermal eq.of state, R=20 AU
Gravitationally
bound clumps,
106 times
denser than the
background
T=160 yr
T=350 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 (Romeo 1991, 1994)
lj < l*< lt
Jeans
length
Toomre
length
2
2
lt =4 G/W
(zero thickness disk)
Mj=r(vs/Gr)
6
(Mayer et
al. 2004)
3/2
For the same Q disks with lower temperature (lower masses) have a lower
3
fragmentation scale. From definition of Toomre mass, Mmax ~  ,from
5/4
Jeans mass, Mmin ~ T . In coldest models Saturn-sized clumps form.
Adiabatic versus Isothermal
Adiabatic with thermal energy equation ( = 1.4): cooling
only by decompression, heating by compression +
artificial viscosity (shocks).
P=( – 1)ru
-5
rmax~10 g/cm3
u=u(t)
Adiabatic after t ~ 160 yr
Locally Isothermal
T=350 yr
EOS switches to adiabatic when local density r>r*,r*~ 10 -10 g/cm 3
T=350 yr
(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 = 320 yr
T = 1900 yr (~ 70 orbital T = 4000 yr (~ 150 orbital
times at 10 AU)
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.
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 -5- 10 - 4Mo/yr)
Locally isothermal EOS for r<r*, outer Tmin = 35 K
For comparison disk model STARTING with 0.085 Mo and Tmin = 36 K
fragments.
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)
r>r* ~ 10
-10
g/c 3
m
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
Density
Temperature
Long lived clumps require
Tcool <~ Torb
See Mayer et al . (2003)
Tcool=0.5 Torb; =5/3
T=300 years
Snapshots of sims with different
Tcool, all after ~ 10 Torb (10 AU) ~
300 years
(Rice et al. 2003)
Tcool=0.8Torb; =7/5
Tcool=1.4 Torb; =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
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
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 T=200 yr
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.
Light disks, Md =0.012 Mo,
never fragment.
tcool = 1/2 torb In both cases
disks remain
cold enough to
support any
type of grain
For light disks
Same result
for tcool = 10
torb
T=200 yr
150 years
Unequal mass disks;
transient clump formation in
more massive disk, Mdisk ~ 0.1 Mo
Bottom line
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
- 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  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
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