Disk fragmentation into giant planets: numerics and thermodynamics

1
Disk fragmentation into giant planets numerics
and thermodynamics
Lucio Mayer University
of Zurich/ETH Zurich
Thomas Quinn (Univ. of Washington), James Wadsley
(McMaster), Joachim Stadel
(Zurich), Graeme Lufkin (University of
Maryland), Artur Gawryszcak (Heidelberg), Annie
Mejia (Univ. of Washington)
2
Disk fragmentation

• Needs efficient cooling -- locally isothermal
  equation of state or
• cooling time orbital time -- gt disks 0.1 Mo
  with Toomre Q lt 1.4
• produce gravitationally bound long lasting clumps
  1 Mj or larger
• on eccentric orbits (Boss 2001, 2003 Mayer et
  al. 2002,2004 Rice
• et al. 2002, 2003 Gammie 2000) in a few hundred
  years
• Example uniformly growing disk, from 0.0085
  Mo to 0.085 in 1000 yr
• (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
• lt r 10 g/cm
• adiabatic EOS for r gt r
• Outer Tmin 30 K at
• t0 ( r motivated by
• radiative transfer
• calculations of
• Boss(2002))

- 4
-5
-10
3
3
Two main issues
(1) Numerics Is fragmentation a robust result
that does not depend on the numerical technique
employed? How well are different codes modeling
hydro and gravity? If local Jeans length not
adequately resolved both grid codes and SPH code
can produce spurious fragmentation (Truelove et
al. 1997, 1998 Bate Burkert 1997, Nelson 2006,
Klein et al. 2007). Artificial viscosity in SPH
can damp fragmentation (Mayer et al. 2004) (2)
Thermodynamics Is fragmentation still possible in
models that go beyond simple equations of
state/fixed cooling times by modeling directly
the balance between cooling and heating (i.e.
with radiative transfer)? Disagreement at the
moment Boss (2003, 2004, 2006) and Mayer et al.
(2006, 2007) say yes, Cai et al. (2006), Boley et
al. (2006) and Rafikov (2006) say unlikely. But
what if (2) combines with (1)?
4
Wengen code comparison
Aim test fragmentation of self-gravitating
gaseous disk with different numerical techniques
-- mainly SPH versus AMR By-product First AMR
simulations of self-gravitating protoplanetary
disks Strategy Initial Conditions SAME for all
codes. Start with particle realization of disk,
then interpolate onto grid with kernel masking
technique ---gt SPH and grids start with the same
built-in Poisson noise Type of ICs a disk with
Qmin lt 1 (fragmentation expected)
a disk with Qmin 1 (marginally unstable)
Disk evolved with locally
isothermal EOS
5
Participants
GASOLINE TreeSPH (Mayer, Quinn, Wadsley,
Stadel) (U. Zurich/UW
Seattle/McMaster U.) GADGET2 TreeSPH
Volker Springel MPA, Munich FLASH cartesian
AMR-PPM Artur Gawryszcak (FLASH group,
Univ. of Chicago) (Copernicus Astronomical
Institute
- Warsaw/MPIA Heidelberg) INDIANA
HYDRO Code Annie Mejia, Richard
Durisen, cylindrical static grid (w/art. visc.)
Aaron Boley (Indiana Univ) ENZO cartesian
AMR-PPM/Zeus Elizabeth Tasker,

Greg Bryant
(Columbia
University/UFD) HYDRA P3M-SPH
Frazer Pearce (U. Nottingham)
6
HI-RES LIS, Qmin lt 1, resolutions,
matched to have grid cell size SPH
smoothing/gravitational softening in ICs
Same ICs for all codes
GASOLINE (SPH) 106 particles
GADGET2 (SPH) 106 particles
Mayer et al., in prep.
FLASH 2 refinements, up to 10243 locally (no
art. viscosity)
Indiana code 512X1024X128
7
Initial disk with Qmin 1
t 0.5 Torb
GASOLINE, 200000 SPH particles
FLASH, 2 levels of ref. up to 10243, ref.
based on Jeans length (Truelove criterion
Truelove 1997)
t 1.5 Torb
Initial spatial resolution of SPH initial cell
size in AMR
8
t 2 Torb
Results widely different SPH fragments, AMR
does not However already at t 0.5 Torb maximum
spatial resolution of SPH higher than spatial
resolution of grid code (only 2 levels of
refinement, close to existing best fixed
grid simulations) .What if we increase the levels
of refinement?
9
GASOLINE 200000 particles
FLASH with 8th refinement levels
T 1.5 ORB
T2 ORB
T2.5 ORB
10
Density profile of first clump in AMR and SPH runs
11
Conclusions - I

• AMR produces results very close to SPH at
  comparable
• maximum spatial resolution. Indeed clumps denser
  in AMR
• runs (possibly because of reduced art. visc.)
• Initial grid resolution as important as number of
  refinements
• Truelove criterion is necessary but not
  sufficient.
• We find gt 8 cells per Jeans length are necessary
  to approach
• convergence during first few disk orbital times
• Low resolution can lead to spurious suppression
  of
• fragmentation.
• Exact convergence of results cannot be expected
  (especially
• over long timescales) owing to highly nonlinear
  evolution of
• the density field coupled with different
  estimates of the density
• in different codes. Seek statistical convergence
• (how to quantify this?).

FLASH
GASOLINE
T 5 Torb
12
3D SPH simulations with radiative transfer
With Graeme Lufkin (Maryland), Annie Meija (UW),
Thomas Quinn (UW)

• - (Flux-limited) diffusion equation implemented
  as in Cleary Monaghan
• (1999) for optically thick part of the disk (t gt
  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 DAlessio et al. 2001,
  same as those used by
• Durisens group)
• Optically thin disk boundary cools as a blackbody
  (t lt 1, radiative
• efficiency adjustable as a parameter). fa0 for
  particles with t gt 2/3,
• fa1/2-1 for particles at edges. fa determined by
  looking within edge
• detection angle (only free parameter)
• -Shock heating included via quadratic b term in
  standard Monaghan
• artificial viscosity. No external irradiation
  no back scattering of photons

13
Growing disk with flux-limited diffusion
Mayer et al, ApJ, in press
60 AU
- Fragmentation less likely than with fixed EOS
or cooling time orbital time (see also Rafikov
2006) - Only disks with M gt 0.15 Mo fragment
into permanent clumps at about 15 AU (while Boss
(2004,2005) finds fragmentation at lt 10 AU in lt
0.1 Mo disks) Movie disk growing from 0.02 Mo
to 0.15 Mo over 50 Torb
14
How does the disk
midplane cool? Transport of thermal energy away
from the midplane Timescale radiative diffusion
time ( krh2/c) x gas pressure/radiation
pressure 104 years, i.e. gt 100 Torb Gas
pressure/radiation pressure dominant, 104
because gravitational instability source of
strong compression (see also Pickett et al.
2000Cai et al. 2005). Changing opacity by a
factor of 50 has no effect because pressure term
dominates (confirms Boss 2003). Disk midplane
cannot cool efficiently by radiation
15
A turbulent disk midplane
(1) Upwellings/downwelling with typical speeds
0.1 Km/s (orbital velocities 1 km/s at 10 AU)
in overdense regions (2) transport the heat from
the midplane to the atmosphere in 30 years
Torb at 10 AU
Turbulent cells are - intermittent -
associated with superadiabatic entropy gradients
as in convection (Schwarzschild criterion for
convection verified)
0.5 AU
T120 years
16
Varying simulations parameters
(Mayer et al., ApJ, in press)

RSemitting area/geometric surface area

• For conservative cases
• (RS 1) one needs M gt
• 0.15 Mo for fragmentation
• to happen
• Mean molecular weight
• changed in late phase, after
• strong spiral shocks form
• Fragmentation sensitive
• to
• mean molecular weight
• efficiency of radiative
• cooling at optically thin
• boundary

m2.4, RS 1.5
m3, RS 0.8
m2.7, RS 1
m2.7, RS 1.2
17
Solids and gas in a GI active disk
How do dust/planetesimals respond to GI in the
gaseous disk? - Rapid migration (faster than
orbital time) of cm-m sized particles towards
overdensities (pressure maxima) due to gas drag
(Durisen et al. 2004, 2007 Rice et al.
2004,2006 Mejia, Quinn Mayer, 2006 in
preparation). Locally dust-to-gas ratio increases
by more than a factor of 10 To explore in next
few years can GI speed up coagulation of
planetesimals into a core? ? hybrid planet
formation model (Durisen et al. 2007, Rice
et al. 2006)
50 cm particles
18

• Varying molecular weight and gas metallicity
• Fragmentation insensitive to opacity but higher
  mean
• molecular weight favors fragmentation.
• How can one get changes in m?
• Example ice grains (30 of dust grains)
  vaporized in spiral
• shocks because T gt 200 K ---gt m 2.5 instead of
  2.38 for H20 mixed
• with H2, He if gas solar and dust/gas ratio
  pumped up by a factor of
• 10 in spiral arms (Haghighipour Boss 2003, Rice
  et al. 2005, 2006
• Mejia, Quinn Mayer 2006)
• If gas 3 times solar (assuming metallicity dust
  content) same
• mixture gives m 2.85 -?higher metallicity
  should favour GI.
• In the optically thin layer cooling more
  efficient with higher
• metallicity (see Cai et al. 2006) ----gt steeper T
  gradient
• ----gt favours convection---gt favours
  fragmentation

19
Conclusions - II

• Gravitational instability can form giant planets.
  Main caveat is keeping the cooling time short
  enough as the instability begins. RT simulations
  suggest a non-radiative mechanism (convection?)
  might do the job but need disk mass gt 0.15 Mo
  very efficient cooling at the optically-thin
  disk boundary. However conditions might
• relax with as the resolution of simulations
  increases.
• Planets formed by GI (also with RT) are large
  (1-10 Mj) and have eccentric orbits. Mass
  depends on disk properties (density,temperature)
  which set most unstable wavelength ---gt hard to
  form small planets (lt 1 Mj) for reasonable disk
  parameters
• It is not necessarily true that disk
  instability is insensitive to gas metallicity.
  Dependence of results on molecular weight and
  cooling rate at the optically thin surface point
  to increased formation efficiency with higher gas
  metallicity.
• These effects still needs to be robustly
  quantified,
• Collapse of protoplanets formed by GI down to gas
  giant densities still needs
• to be explored. Much higher dust-to-gas ratio
  expected at the location of clumps
• -----gt giant planet formed by GI could be metal
  enriched compared to the disk.
• Need 3D simulation of gas and dust with
  time-dependent chemestry (e.g changes in
  molecular weight/composition due to shocks
  etc..).
• Hi-res 3D SPH runs combined with semi-analytical
  impact code of Mordasini, Alibert Benz as
  first attempt to treat dyamics of cm-m size
  bodies inside clumps/spiral arms

20

• Cells of vertical motions are
• - intermittent
• associated with superadiabatic entropy gradients
• (Schwarzschild criterion for convection verified)

Evolution of vertical temperature profile of a
cooling clump
21
Molecular weight should temporarily increase in
the spiral shocks, while stay constant
outside In current simulations m varies globally
instead of locally (tests necessary to have
stable solution across boundaries with different
equation of state, SPH often bad in presence of
sharp gradients) However change is late enough
that there is little radial mixing (a few percent
in mass) between intra-arm and inter-arm regions
before clump formation ? instability local in
nature!
22
A higher molecular weight can favour
fragmentation in two ways (1)Reduces
compressional heating, since PdV
rkBT/mdV (2)Increases surface cooling because L
m4 Tests show that first effect dominates.
This supports the idea that intensity
compressional/shock heating plays a key role in
unstable disks (Pickett et al. 2003 Cai et al.
2005)
23
OUTLINE

• FRAGMENTATION WITH DIFFERENT 3D NUMERICAL
• METHODS SPH vs. AMR
• 3D SPH SIMULATIONS WITH RADIATIVE TRANSFER
• GRAVITATIONAL INSTABILITY IN BINARY SYSTEMS
• -- implications for both disk instability and
  core-accretion

24
Gravitational Instability fragmentation?
A disk with M 0.1 Mo within 20 AU
may fragment into Jupiter-sized clumps in the
outer, cooler part (T 50 K) after a few orbital
times/hundreds of years Short timescale
attractive, no problem with disk lifetimes of
million years (Boss 1997, 1998, 2001, 2002
Kuiper 1959 Cameron 1978). Initial Qmin lt 1.5.
A massive, cold disk is expected a few 10
years after the onset of molecular core
collapse into diskprotostar, somewhere in
between Class 0 and Class 1 Objects (e.g. Yorke
Bodenheimer (1999).
Boss 2001 3D eulerian simulation
Initial Qmin lt 1.5 QminWvs/pGS
Density Map Grid 512x128x64

4
25
Evolution of protoplanets
200.000 particles with switch to adiabatic (20 AU
)
T 4000 yr ( 150 orbital times at 10 AU)
T 320 yr
T 1900 yr ( 70 orbital times at 10 AU)
Merging drastically reduces the number of
clumps. Only a few remain after 500 yr, with
masses 2 Mj lt 7 Mj. Orbits remain eccentric (e
0.1-0.3).
26
Fragmentation depends on struggle between
pressure forces and gravitational forces ----gt a
sensible numerical simulation must be able to (1)
model gravity accurately over wide range of
scales 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.
Fixed 3D eulerian grid (256x256x64).
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!
(Pickett et al. 2003)
27
Increasing refinement in FLASH
With more refinement levels overdensities become
stronger (always fulfill Truelove (1997)
criterion, gt 4 cells per Jeans length)
28
Fragmentation achieved as refinement levels
increase Truelove criterion (necessary to avoid
spurious fragmentation) was always satisfied yet
results change with increasing resolution. A
stronger criterion, gt 8 cells per Jeans length,
seems more robust
29
INCREASING MASS RESOLUTION IN SPH
160000 particles
1.3 million particles
Clump transient
Clump permanent
30
SMALL VARIATIONS OF ICs Mdisk 2
Mdisk 2 - GASOLINE
Standard run - GASOLINE
Mdisk 2 FLASH
Mdisk2 Hires GASOLINE
31
Fragmentation and thermodynamics. I
Boss (1998, 2000) if the disk starts with
sufficienltly low Q (lt 1.2) fragmentation
occurs for any the equation of state (locally
isothermal or adiabatic). Does not include shock
heating (adiabaticisentropic) Pickett et al.
(2000, 2001, 2003), Nelson et al. (2000)
locally isothermal EOS considerably enhances
instabilities and clump formation compared to
adiabatic conditions. Include shock
heating! Shock/compressional heating is work
done by gravity ? disk tends to self-regulate
into a hotter, more stable state.
32
FRAGMENTATION NEEDS RAPID COOLING
Temperature
Density
Long lived clumps require Tcool lt Torb
See Mayer et al . (2003, 2005), Rice et al.
(2002, 2003, 2005)
Snapshots of sims with different Tcool, all after
10 Torb (10 AU) 300 years
Tcool0.8Torb g7/5
T300 years
Tcool1.4 Torb g7/5
33

• 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 local!

34
Disk instability in binary systems

• About 15 of known extrasolar planets are in
  binary systems (Eggenberger et al. 2004, 2005
  Patience et al. 2003) and targeted surveys are on
  the way (e.g. the Geneva Group).

T10 Years
T150 years
T200 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)
d120 AU
T250 years
T450 years
Prediction giant planet formation less likely in
tight binaries Consistent with recent survey
(Eggenberger et al. 2005)
35
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
In isolation
Mayer et al., 2005
Nelson 2000
36
Light disks, Md 0.012 Mo never fragment in
isolation or in binaries. They also do not heat
up in neither case no matter the cooling time,
can support ice grains ----gt core-accretion can
proceed
For light disks Same result for tcool 10 torb
tcool 1/2 torb
T200 yr
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 Alibert et al. 2005,
2006).
37
Mayer, Boss Nelson 2007
No RT, Tcool 0.5 Torb
With RT
Md0.05 Mo
Binary disks with radiative transfer and masses
0.05 Mo do not produce clumps. Net cooling
shock heating cooling less efficient than
assuming Tcool 0.5 Torb Confirms trend of
isolated disk simulations with RT only very
massive disks gt 0.12 Mo can fragment
38
Tidal torques in binaries GI -----gt steeper
disk density profile
--Important to take into account for
planet formation in general --Increased overall
surface density might partially counteract grain
vaporization due to shock heating. Core-accretion
might still proceed locally due to higher overall
surface density
39
GI in binary systems favoured or suppressed?

• --Nelson (2000) and Mayer et al. (2005) conclude
  that GI is suppressed
• in binaries. Confirmed by new runs with radiative
  transfer. Lodato Rice
• (2006) find similar results for fast disk
  encounters
• --Boss (2006) finds GI is promoted in binarys
  systems
• Why the difference?
• Different initial disk masses/temperatures
• (2) Boss (2006) does not use artificial
  viscosity, so less heating in shocks
• - shock heating crucial for suppression of GI in
  Mayer et al. (2005).
• See Mayer, Boss Nelson (2007) for extended
  discussion

40
How to make realistic protoplanetary
disk ICs? Simulate the formation of the
protostellar disk protostar system from the 3D
collapse of a rotating molecular cloud core with
enough resolution to follow the gravitational
instability in the disk Make SPH more adaptive
higher mass/spatial resolution in the center of
the core which will form the bulk of the disk
(by variable mass resolution setup or particle
splitting)
41
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
Grav. force resolution 0.3 AU
Mayer et al., in prep.
42
The inner 100 AU
Phase 1 rapidly rotating bar unstable
protostellar core
T0.02 Myr
T0.022 Myr
43
Phase II bar fragmentation and merging of
fragments
T0.025 Myr
T0.024 Myr
44
T0.035 Myr
Phase III Formation of a binary system
with protostellar cores and protostellar disks.
Mdisk 0.6 Mstar after one binary orbit (1000
years)
Need even higher mass and force resolution to
follow disk instability
45
Simulations shows gravitational torques in
protostellar disks strong, dM/dt 3-5 x 10
Mo/yr. Accretion rates from the cloud onto the
disks instead significantly lower lt 10 Mo/yr.
At this rate the 0.1 Mo will be accreted onto
the star in lt 10 years
- 5
- 5
5
Protostellar disk (M gtgt M) transitions into
protoplanetary disk (M lt M) thanks to
gravitational torques
46
Properties of clumps
Color-coded velocity field shown. Clumps
are -in differential rotation, on coplanar
orbits along disk midplane -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
47
Disk instability
)predicts short formation timescales, good with
disk lifetimes )naturally produces big planets
and eccentric orbits )migration is not a problem
because planets grow in mass so quickly that they
decouple from the disk -)hard to produce a small
Neptune-sized planet -)terrestrial planets and
giant planets form differently -)no clear
relation with stellar metallicity, but new
calculations show it might arise through via the
molecular weight -)not clear how to explain that
SS giants have envelopes with 30-40 Mearth of
solids (e.g. Jupiter and Saturn) --- did
metal enrichment (relative to solar metallicity
gas) occurred after formation?
48
Disk instability
)predicts short formation timescales, good with
disk lifetimes )naturally produces big planets
and eccentric orbits )migration is not a problem
because planets grow in mass so quickly that they
decouple from the disk -)hard to produce a small
Neptune-sized planet -)terrestrial planets and
giant planets form differently -)no clear
relation with stellar metallicity, but new
calculations show it might arise through the
strength of pressure gradients which regulate
fragmentation -)not clear how to explain that SS
giants have envelopes with 30-40 Mearth of solids
(e.g. Jupiter and Saturn) --- did
metal Enrichment (relative to solar metallicity
gas) occurred after formation?
49
Surviving clumps evolve on time-dependent
eccentric orbits (see also Lufkin et al. 2003)
Ecc 0.1-0.3
Clump mass function after T350 years (no more
fragmentation). Evolves towards few
big protoplanets due to clump merging and
accretion of gas
50
GASOLINE (SPH)
T90 yr
T160 yr
FLASH (AMR)
Arthur Gawrysczak (Warsaw)
51
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 ?Univ. of Zurich
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).
52
Gas giants from core accretion

Solid core forms first, reaches a critical mass
and rapidly accretes a gaseous envelope. An R
density distribution of gas and dust is
followed, Mgas 100xMdust in mass. Spherically
symmetric gas accretion, no angular momentum, no
self-consistent disk dynamics
-3/2
Alibert et al. 2004
Pollack et al. 1996
Timescale dependent on S of solids, opacity of
gaseous envelope, motion of solid cores in the
disk (migration) 3-10 million years required to
form Jupiter (Bodenheimer et al. 2000Hubickyj et
al. 2003)
53
Preliminary results with FLD
- self-gravitating bound clumps still form,
although less fragmentation compared to models
with isothermaladiabatic EOS or cooling
dependent on orbital time. - episodic upwellings
and downwellings of gas observed fragmentation
occurs, midplane can become cooler than
atmosphere. Are we seeing thermal convection
(e.g. Boss 2003)? - very sensitive to molecular
weight (sets strength of pressure gradients in
spiral shocks, perhaps controls
convective instability by setting P(z) for a
given T(z)). 0.1 Mo disk fragments only with mgt
2.2 (m2 - pure H2-used in non-FLD simulations).
Is this related to observed correlation with
star metallicity? - little difference if
opacities changed by a factor of 10 (see also
Boss 2002), shock heating always dominates
radiative cooling.
54
Light disks, Md 0.012 Mo never fragment in
isolation or in binaries. They also do not heat
up in neither case no matter the cooling time,
can support ice grains ----gt core-accretion can
proceed
For light disks Same result for tcool 10 torb
tcool 1/2 torb
T200 yr
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).
55
Transition from protostellar to protoplanetary
disk need mechanism that causes rapid (in lt 10
-10 years, disk lifetime!) angular momentum
loss. - MRI (Balbus Hawley 1991), powerful in
principle, needs small seed magnetic field yet
probably only in the inner disk ( 1 AU) or in
the upper layers where ionization can be high
enough. - gravitational torques (e.g. cause by
spiral instabilities) (e.g. Yorke et al. 1999) in
principle everywhere, but are they strong
enough?
5
6
Note disk photoevaporation very important
only in OB associations, and always not important
inside 10-20 AU (e.g. Matsuyama et al. 2003)
56
The zBox can process 1012 operations per second
and transfer information to the server at 109
bits/second. This is about 1 of a human brain.
It also consumes and dissipates 1000 times as
much power.
Institute of Theoretical Physics, Univ. Zurich
zBox (design by J.Stadel) 288 AMD MP2200
processors, 144 Gigs ram, 10 Terabyte
disk Compact, easy to cool and maintain Very fast
Dolphin/SCI interconnects - 4 Gbit/s, microsecond
latency A teraflop computer for 500,000
(250,000 with MBit) Roughly one cubic meter, one
ton and requires 40kilowatts of power
57
Fragmentation depends on struggle between
pressure forces and gravitational forces ----gt 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.
Fixed 3D eulerian grid (256x256x64).
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!
(Pickett et al. 2003)
58
-- Schwarzschild criterion for convection
(dlogT/dlogz)/(dlogP/dlogz) gt 0.286 satisfied at
regions where upwellings and downwellings
are seen (also Boss 2003) But criterion derived
for stars, how about a rotating disk? -- Vertical
gas motions occur on scales gtgt SPH smoothing
length, at least this is comfortable!
T(z) at the location of overdensity ---gt midplane
first hotter than photosphere (white line) then
becomes colder than photosphere (red line) if
long lasting clump forms.
59
3D SPH with radiative transfer
with Graeme Lufkin (U.Maryland), Annie Meija
(UW), Thomas Quinn (UW)
- (Flux-limited) diffusion equation implemented
as in Cleary Monaghan (1999) for optically
thick part of the disk (t gt 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 DAlessio et al. 2001,
same as those used by Pickett, Durisen, Meija
collaborators) - Optically thin disk boundary
cools as a blackbody. - No stellar irradiation
(although it might be not important if
outer disks shadowed by flared structure, see
DAlessio et al. 2001) - No incoming radiation
at atmosphere, but tunable blackbody
efficiency. - Shock heating included via
quadratic b term in standard Monaghan artificial
viscosity.
60
Gas giants from core accretion

Solid core forms first, reaches a critical mass
and rapidly accretes a gaseous envelope. An R
density distribution of gas and dust is
followed, Mgas 100xMdust in mass. Spherically
symmetric gas accretion, no angular momentum, no
self-consistent disk dynamics
-3/2
Alibert et al. 2004
Pollack et al. 1996
Timescale dependent on S of solids, opacity of
gaseous envelope, motion of solid cores in the
disk (migration) 3-10 million years required to
form Jupiter (Bodenheimer et al. 2000Hubickyj et
al. 2003)
61
Planets form out of a
protoplanetary disk of gas and dust in rotation
around the central star. Dust (initial
size 0.1-1?) settles into the disk
plane due to gas drag. 1 - Dust grains
grow by coagulation until they reach the size of
big rocks ( 1 km) also called
planetesimals this phase is vaguely known,
chemical/structural properties of materials
important (laboratory) 2- Planetesimals
grow further by collisional agglomeration
only gravity matters ---gt process can be
simulated by N-body algorithms (Lissauer
1993 Kokubo Ida 1966, 1998, 2000
Richardson et al. 1999 Thommes, Levinson
Duncan 2001) 3- Largest planetesimals
undergo (runaway, oligarchic) growth into
planets Reach the mass of a few Earth
masses in 1-100 Myr depending on local
surface density of solids and velocity
distribution. At r gt 5 AU accretion
faster than at 1 AU (Earth) because 3 times more
solids (ice grains). If a substantial
gas reservoir is still present this can
be accreted and a gas giant like Jupiter will
result (Pollack et al. 1996)
Conventional planet formation
62
Problems
Formation of a 10 Earth masses solid core plus
accretion of a gaseous envelope requires
several Myr

• Disk lifetimes estimated from observations range
  from
• 0.1 Myr (in dense star forming regions) to a few
  Myr
• (e.g Briceno et al. 2000)
• Disk-planet interaction (gravitational torques)
  causes planet
• migration, typically inward (Lin Papaloizou
  1986). Unless
• mass of the planet larger than 0.3 Mjup (in a
  disk with
• 0.01- 0.02 Mo) migration towards the star in
  less than 0.1
• Myr (Bate et al. 2002 Nelson Benz 2003 but
  see talk by
• Ed Thommes)
• Current models of Jupiters interior indicate a
  core mass
• 0-10 Mearth (Saumon Guillot 2004)

63
GIANT PLANET FORMATION VIA DISK
INSTABILITY

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

64
Evolution of Q profiles LIS simulations
Disks in next slides - different initial Qmin
because of different initial mass (0.1 or 0.08
Mo) same Tmin65 K (only right model fragments
Qmin falls below 1). Q(r) is azimuthally
averaged.
T0, T160 years, T200 years
65
Initial Conditions a growing disk
Simulations starting with a disk already
marginally unstable (Q 1.3-1.4) can abruptly
amplify spurious modes present in the ICs (e.g.
m1 mode at inner boundary, see Nelson Benz,
1998, edge modes). If disk starts slowly
approaches critical Qmin will it eventually
self-regulate 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). Constant increase
of particle masses. Locally isothermal EOS for r
lt r, outer Tmin 30 K. For comparison a disk
model STARTING with Mdisk0.085 Mo and Tmin
40 K also fragments.
-5
- 4
Mayer et al. 2004
66
Dependence on resolution
Locally isothermal simulations after 350 years
6
N200.000
N10
Fragmentation enhanced at higher resolution, more
clumps form
67
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)
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
68
M0.12 Mo, Qmin (t0) 1.3
Density
Density
T100 years
T160 years
Temperature (lt 400 K)
Density
T 300 years
T300 years
69
Clump formation
increases with resolution
Criterion for boundness 2Eth U lt 0 Shown
results for the same disk model run with 1
million particles (red) and 200000 particles
(blue)
70
Problem 1Disks last only a few million years
Observables that trace disks disappear when stars
older than few million years. Maybe not enough
time to form giants in current models of
core-accretion..
Briceno et al. 1999
Stellar age gt 3 Myr
Stellar age lt 3 Myr
71
Problem 3orbital migration
Numerical simulations of a core embedded in a 2D
or 3D gaseous protoplanetary disk
Bryden 1999
Masset 2002
Spiral density waves excited by the planet
extract its orbital angular momentum---gt a 10
Mearth core sinks towards the star in less than
0.1 Myr ltlt gas accretion timescale in
core-accretion model! Only a full grown Jupiter
mass planet can avoid sinking by carving a gap in
the disk --? quick formation time needed
72
Problem 2Do gas giants have a solid core ?
Not clear if solid core exists for Jupiter
(bounds 0-10 Mearth), small core for Saturn (5-10
Mearth). Core-accretion models require Mcore gt 10
Mearth (Alibert et al. 2004)

Only outer atmosphere directly
probed by space probes! For internal structure
just constrained modeling. Models assume
hydrostatic equilbrium. Density profile
constrained by moments of the .
gravitational field measured by space probes
(Voyager, Galileo). Eq.of state P(r) has
to be assumed outcome strongly dependent on
poorly understood EOS of H at P gt 1 3
Mbar for metallic hydrogen (Saumon Guillot
2004).
Gaseous molecular envelope
Liquid molecular hydrogen
Metallic hydrogen
Solid core?
73
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74
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
75
Physical scales -Jeans mass - Toomre
mass Artificial scales -gravitational
softening length - SPH smoothing length
- artificial viscosity length (h h2 for
Monaghan standard)
76

• Problems clumps transient features, not tidally
  stable or lost outside the imposed inner/outer
  disk boundaries (Boss 2001, 2002 Pickett et al.
  2000a,b).
• Low resolution causing artificial fragmentation
  can also be an issue (Nelson et al. 1998, 2002
  Laughlin Bodenheimer 1994 Boss 2003).
• But low resolution in the gravity (cell size in
  grid codes) can also suppress clump formation!

Fixed 3D eulerian grid (256x256x64)
Azimuthal self-gravity significantly
underestimated compared to analytical solution
(Pickett et al. 2003) Achieving high resolution
in the gravitational force is critical
77
Covered 15 orders of magnitude in density and
0.1 Myr of evolution so far
20000 AU 0.05 pc
2000 AU
100 AU
1 AU
Mayer et al., in prep.
78
Planet formation

• A good theory of planet formation must explain
  both the Solar
• System and the extrasolar planets
• Detected extrasolar planets are gas giants giant
  planets are
• now the only ground where to test a 'global'
  theory until we
• find terrestrial planets outside the SS.
• Many of the extrasolar planets discovered have
  orbits and
• masses very different from the planets of the
  Solar System.
• However this results from an observational bias
  last Sept. a
• Jupiter mass planet at 5 AU from the central
  star has been
• discovered (Marcy et al. 2002) and on Jan. 16th a
  Neptunian-size
• planet (Fischer et al. 2003).

79
Surviving clumps evolve on time dependent
eccentric orbits (see also Lufkin et al. 2003)
Ecc 0.1-0.3
Clump mass function after T350 years (no more
fragmentation). Evolves towards the 3
protoplanets configuration due to clump merging
and accretion of gas
80
N-Body simulations of planetesimal
accumulation
(Richardson, Quinn, Stadel Lake 2000)
Runaway growth
Initial Conditions 1 million planetesimals, size
100 km, power law profile between 1 and 3.8
AU Accretion of an Earth mass planet possible
in 30 million years
81
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82
Several models with 1.1 ltQ lt2, varying Mdisk,
M,T. Here only two representative cases shown.
See Mayer et al. (2004, ApJ, in press) for
details
QVs?/??G
-3/2
?? r
0
T (Rgt 10 AU) 60 K
(Boss 2000,2001)
83
LIFETIME OF CIRCUMSTELLAR DISKS
In actively star forming regions like the Orion
nebula disks are ablated by strong UV flux from
bright stars and evaporate in 0.1 Myr. About
80 of the star formation of our galaxy occurs
in OB associations like these (Bally et al.
1998, Briceno et al. 2001)
In less violent star formation environments
circumstellar disks show a longer lifetime, a
few Myr (Haida, Laisch Haida 2001). However
age based on observations of dust or CO.
Direct observations of H2 (most important
constituent) are hard (no dipole, only quadrupole
rotational transitions observable but transition
rate low). Maybe lt 1 Mj present after 10 Myr
(Thi et al. 2000)

84
Planetesimal accumulation
Outcome of a binary collision depends on Safronov
number ?(ve /2vrel ) ve2G(m1 m2)/(r1
r2) escape speed ? 1 --gt planetesimals
stick to each other (collisions non elastic) ??lt1
--gt planetesimals bounce (and maybe fragment
depending on density) Analytical methods
(Safronov 1969 Wetherill 1980, 1993) treat
planetesimals as 'particles in a box' following
the kinetic theory of gases. Assuming that the
bodies are of nearly equal mass one can compute
the evolution of the velocity distribution f(v)
(for an initial distribution of orbits) as a
result of collisions. One finds a Maxwellian
distribution and can calculate a mean vrel ---gt
dM/dt ?vrel?R 1 2? 100 milllion years
to grow an Earth mass body within 1 AU in a
minimum mass protosolar nebula.
½
2
2
2
-3/2
2
Minimum mass nebulaS R , S (5 AU) 5 g/cm
85
ACCRETION OR FRAGMENTATION?
Formation of giant planets must obviously occur
before the disk is dissipated. The
core-accretion mechanism requires a timescale
longer than a few million years longer than
typical disk lifetimes Either giant planet
formation is very rare (but we know it is not!)
or formation occurs by other means. A massive
self-gravitating, gaseous disk with M 0.1 M
within 20 AU can become gravitationally unstable
and form clumps in the outer, cooler part after
only a few hundred years (Cameron 1978 Boss
1998, 2001, 2002). However, clumps are typically
transient features Dissolved by tides or are
lost outside the imposed disk Boundaries (Boss
2001, 2002 Pickett et al. 2000a,b). How clump
formation is affected by resolution in the
simulations is also an issue (Nelson et al. 1998,
2002, Laughlin Bodenheimer 1994).
86
A region in a keplerian disk of gas might
collapse if
QVs?/??G lt 1 If 1ltQlt2 global non-axisymmetric
instabilities (e.g. spirals) can arise
(Laughlin, Korchagin Adams 1997). Highly
overdense regions within the rings/spiral arms
might fragment (Cameron 1978).
A "minimum" protosolar nebula model, Mdisk
0.01 M, would result in Q gt 2 everywhere (Boss
1994) gt need more massive disk for low Q
How massive are the observed
circumstellar disks?
Recent models of circumstellar disks spectra
(dust is observed) with radiative transfer
(D'Alessio et al. 2001A,b) find M 0.05-0.1 Mo
within 30-50 AU for a gas/dust ratio 100 --gt
with the low temperature of disks (around 30 K
at 10 AU from the star) Q 1-1.5 Rgt 5 AU.
Massive disks are also required by the core
accretion model otherwise timescale totally
inconsistent with obs.!
87
Can overdensities collapse into
protoplanets?
A growing density perturbation needs to overcome
stellar and disk tides (Ft(r,t) GM(r,t)/r ) to
keep collapsing. Problem complex, need
simulation with high resolution to resolve a
wide range of densities and the time dependent
gravitational field. Mayer, Quinn, Wadsley
Stadel (2002) 3D SPH (smoothed particle
hydrodynamics) simulations with nearly 100 times
more resolution (particles) than ever used
before. SPH is spatially adaptive ---gt
arbitrarily high densities can be reached
(Gingold Monaghan 1979) the only limitation
being computing power (large Npart needed)
3
88
An equation of state
Gravity is resisted by pressure need to model
gas pressure realistically to follow the
collapse. In principle one should solve for
explicit heating and cooling (both from PdV work
and radiation, and with radiative transfer). To
simplify thermodynamics one can fix an equation
of state (assuming gas to be H2) PK??
Ideal gas ????1 isothermal max. rad.
cooling ??? 7/5
adiabatic (H2) no rad. cooling
??
Any actual behaviour will likely fall between
these two extremes (Boss 1997, 1998) Locally
isothermal hypothesis supported by Tcool Torb
for typical disks in absence of external heating
sources (Boss 2001). However eventual overdense
regions would absorb most of the radiation
(optically thick) and behave nearly adiabatically
89
Disk Thermodynamics
Boss (1998) if the disk starts with
sufficienltly low Q ( 1) instabilities develop
for any the equation of state. Pickett et al.
(2000a,b) locally isothermal assumption considera
bly enhances instabilities and clump
formation compared to adiabatic or isentropic
conditions. Clumps are ejected with free outer
boundary. Boss(2002) includes radiative
transfer in the diffusion approximation --gt
instability pattern and clump formation are
substantially similar to locally isothermal case
for Qmin 1.4-1.5
RT
LI
T350 years
Boss 2002
90
Cosmology and Hydrodyamics with
-Conspirators James Wadsley McMaster
Univ. Joachim Stadel Univ. Zurich Tom
Quinn Univ. Washington Fabio
Governato Univ. Washington Ben Moore
Univ. Zurich Derek Richardson Univ. of
Maryland George Lake ISB/U.
Washington
Simulations performed at Pittsburgh
SupercomputingCenter
Multi Platform, Massively Parallel treecode
SPH, multi stepping, cooling, UV background,
Star Formation, SN feedback . Santa Barbara
tested. Several fundamental works
accomplished in cosmology, galactic dynamics and
galaxy formation.
91
Initial Conditions
Rin4 AU Rout20 AU
0.07 M ltMdisklt0.125 M
-3/2
?? r
(Boss 2001)
-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 time dependent disk potential. -No
inner/outer boundary conditions
92
Temperature profile
Eq. profile from A. Boss (19961998) - uses 3D
radiative transfer code treating the
interaction between disk, star and molecular
envelope
T (4 AU) 500 K for R gt 5 AU T r T (gt 10
AU) 50 K (see also Beckwith et al. 1990
D'Alessio et al. 2001)
-1/2
93
Several models with 1.1 ltQ lt2, varying Mdisk,
M,T. Will show here two representative cases.
QVs?/??G VsVs(T) ???(M) ???(M)
-3/2
?? r
T (4 AU) 500 K T r T (gt 10 AU) 50 K
-1/2
(Boss 2000,2001)
94
Disk Evolution, Qmin 1.75
1 million particles, locally isothermal eq.of
state , R20 AU
T160 yr
T350 yr
Torb (10 AU) 28 years
95
Disk Evolution, Qmin 1.4
1 million particles, locally isothermal eq.of
state, R20 AU
T350 yr
T160 yr
96
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97
Adiabatic versus Isothermal
-5
3
?max10?
g/cm
Adiabatic EOS (? 1.4) cooling only by
decompression, heating by compression
artificial viscosity (shocks)
T350 yr
T350 yr
EOS switches to adiabatic when local density
becomes gt 10 times higher than the initial
value.
Long-lived clumps occur whether EOS changed
or not
98
Evolution of clumps density
After 3 orbital times mass is typically Mjup
and Jeans mass (within 2 softening lengths) is
0.1 Mj --gt Mclump gtgt Mjeans
99
Long term evolution
200.000 particles, adiabatic after T 200 yr
T 320 yr
T 850 yr
Merging drastically reduces the number of
clumps. Only three remain after 500 yr, with
final masses 2Mj lt 5.5 Mj
100
Orbital evolution of clumps, 200000
particles simulation
Clump mass function after 350 years in hi-res sim
101
Where are the rocky cores? In the
fragmentation model rocky cores can form by
collisional growth and settling of dust grains
by gas drag while the envelope contracts to
planetary densities, in 1000 yr (Boss 1997b)
- before the central temperature becomes high
enough (Tc gt 1000 K, in 10 yr) (Bodenheimer et
al. 1980) for iron and silicate grains to
evaporate. New models of the interior of
Jupiter suggest small core masses, anywhere
between 0 and 10 Earth masses (Guillot, Gautier
Hubbard 1997 Guillot 1999) - consistent with
the fragmentation model (Mcore gt 6 Mearth in
Jupiter) - maybe new problem for the
core-accretion model (Mcore gt 10 Mearth needed
for substantial gas accretion). Higher-than-solar
metallicity in Jupiter atmosphere by
pressure-driven planetesimals near gaseous
overdensities (Haghighipour Boss 2002)
5

102
How does clump formation depend on the initial
conditions? A realistic disk will not start with
Q close to critical but might reachit as it
cools to lower temperatures (Pickett et al.
2000). Can weaker instabilities in a disk with
Q higher than critical ( 1.4) prevent any
clump formation by redistributing the mass and
raising Q above the regime of clump formation?
Spiral instabilities transport mass inward and
angular momentum outwards the disk expands
lowering its surface density in the cold outer
region (Lin Pringle 1989 Laughlin
Bodenheimer 1994 Laughlin, Korchagin Adams
1997) Even with very efficient cooling T gt 20 K
(temperature of the parent molecular cloud)
----gt if ? decreases Q might never become
low enough to start fragmentation
103
Disk with initial Q 2 (Tmin 100 K) is
evolved isothermally for 350 years (gt 10 orbital
times at R10 AU) and then cooled at the rate
of 0.2 K/year
Clump formation starts when Tmin lt 42 K
! (after 650 years)
104
Detection of planets at R 50-100 AU would
support the fragmentation model. Coronographic
techniques will give use the answer in a few
years (Luhman Jayahwardhana 2002)
105
Substructure and complex features form quickly in
the fragmentation model. Observations of
spirals, lumps and gaps in disks around
young stars (ltlt 1 million years) will thus
provide constraints to planet formation
106
Two-way planet formation? what
is good and what is bad for terrestrial
planets.....
Even if giant planets can form quickly by grav.
instability, terrestrial planets probably still
form later by planetesimal accumulation because
1)disk too hot for instability at R lt 5 AU and
2)terrestrial planet mostly rocky

• Would an early Jupiter stir too much the orbits
  of small planetesimals
• and prevent them from coagulating in
  planetary-sized bodies?
• Is that what happened in the asteroid belt in the
  Solar System?
• Can Jupiters act as a regulating mechanism for
  planetesimal accretion
• in a way favourable for the survival of
  terrestrial planets by limiting
• impacts with asteroids?

107
Conclusions
-Results at very high resolution converge
towards clump formation for marginally unstable
disks (Qmin lt 1.5) -Clumps contract to very
high densities they become self-gravitating,
tidally stable protoplanets in lt 500
yr -Dynamical evolution of clumps complex, both
merging and interactions with other clumps
and tidal torques by the non-axisymmetric disk
play a role. -Masses and orbital eccentricities
after several orbital times are remarkably
similar to those of observed extrasolar planets

108
Clump formation still happens for a realistic
temperature even when the disk undergoes a
prolonged phase of 'moderate' spiral
instabilities This is because the surface
density at 10-15 AU is barely altered
?
R(AU)
109
In less violent star formation environments
circumstellar disks show a longer lifetime, a
few Myr (Haida, Laisch Haida 2001)
Hillenbrand 2002 age is taken as the average of
the stars in the cluster biggest data
compilation so far.
Direct observations of H2 are difficult, maybe
lt 1 Mj present after 10 Myr (Thi et al.
2000)
110
Minor bodies of the Solar System
Most of the asteroids between Mars and Jupiter,
material is very sparse, only 0.001 Mearth
between 3.1 and 4 AU --gt Jupiter likely damped
accretion
Comets are icy asteroids coming from beyond
Pluto's orbit
111
Resolution and noise
N20.000
N200.000
Q1.9
S0.06 AU
N106
112
Balancing softening with N
N20.000
N20.000
Q1.9
N200.000
S0.6 AU
S0.06 AU
S0.06 AU
113
Varying T or Mdisk (N200K, LI)
Same Tmin (50 K), different Mdisk --gt diff. Qmin
Same Mdisk, different Tmin and Q min(1.4)
Similar Qmin (1.9), different Mdisk and Tmin
114
A case with stronger instability
(LIEOS, Qmin1.4)
N200000, s0.06 AU
M0.5 Mo, Mdisk0.07 Mo
M1 Mo, Mdisk0.1 Mo
T100 yr
T200 yr
T200 yr
Spiral instabilities are stronger around less
massive stars, likely because swing amplification
is enhanced
115
Swing amplification in disk models
Strong swing amplification requires X lt3
(Binney Tremaine 1987)
116
THE FRAGMENTATION MODEL
A protostellar disk massive and cool enough to
undergo strong gravitational instabilities might
fragment into gaseous clumps that would then
contract to protoplanetary densities (Kuiper
1951 Cameron 1978). A disk will be locally
unstable to axisymmetric instabilities
if QVs?/??G lt 1 VsVs(T) ???(M)
???(M) Weaker conditions might apply for
non-axisymmetric instabilities. The
fragmentation model would solve the timescale
problem gravitational instabilities develop on
the orbital timescale of disks (100 -1000
yr). Core accretion can occur work only in a
fairly massive disk - formation by fragmentation
likely occur in such a disk, fast enough to be
dominant formation mechanism
117
In planetesimal models surface densities gt 15
g/cm are likely required to form a 10 Earth
mass core by runaway growth (Thommes, Duncan
Levison, 2001). This is larger by a factor of
5 than the density in the "minimum" solar nebula
model, where massive cores could still form by
gradual accumulation, but only in gt 10 yr
(Lissauer 1993).
2
7
In addition, accretion of the gaseous envelope by
cores might require an even larger density (by a
factor of 5) to avoid hydrodynamic blow off of
the envelope itself (Wuchterl, Guillot Lissauer
2000) which would produce only Uranus and
Neptune-like planets. Disks with a gaseous mass
as large as 0.1 Mo are therefore realistic (Boss
2000).
118
Resolution in N-Body/SPH simulations
Resolution is limited by the number of particles.
Each particle represents a parcel of gas and
carries the "average" physical properties of
its neighboring region whose size changes with
the evolution of the density field. Gravity
softened on a fixed small scale to reduce
two-body relaxation and balance the granularity
of the particle representation. - gravity can be
artificially weaker than pressure if softening
is too large - noise can amplify modes if
softening is too small artificial fragmentation
can arise
(e.g. Bate Burkert 1997)
119
Simulations of Disk Evolution
Grid codes and smoothed-particle-hydrodynamics (SP
H) codes are used to study the evolution of disk
models resulting from protocloud collapse.
In grid codes the fluid equations for a
self-gravitating gaseous nearly keplerian disk
are solved on grid points. The Poisson
equation for the gravitational potential is
solved using multipole expansion techniques or
FFT solvers. Best at resolving discontinuities
like shocks and low levels of numerical noise
in the initial conditions compared to particle
based methods (Nelson et al. 1998). However,
with fixed grids (e.g. Boss 1998, 2001) the
resolution is fixed in a given patch of the disk
(hard to follow localized overdensities) and
expansion of the system due to outward
transport of angular momentum (expected with
gravitational instabilities, e.g. Laughlin
Bodenheimer 1994) is not followed .
120
SPH codes solve the fluid equations using a
lagrangian approach the values of the
hydrodynamical variables are carried by particles
with a given mass that represent fluid
element.s. The hydrodynamical variables are
calculated interpolating overa given number of
neighboring particles (Monaghan 1979, 1981).
Gravity is usually solved using a tree algorithm
that also