Planetesimal and dust dynamics in stratified disc models

1
Planetesimal and dust dynamics in stratified disc
models

• Richard Nelson
• Queen Mary, University of London
• Collaborator Sebastien Fromang (CEA Saclay)

2
Talk Outline

• Global, stratified turbulent disc models - a
  quick summary of results
• Planetesimal evolution in global discs
• Evolution of small dust particles
• Conclusions and future directions

3
Stratified disc models

• Model parameters
• H/R0.07 and H/R0.1 discs computed
• Locally isothermal equation of state
• 9 vertical scale heights in total (? 4.5)
• Variety of boundary conditions usedVertical
  direction - outflowRadial direction - outer
  buffer zone viscous outflow at inner edge
• p/4 and p/2 azimuthal wedges
• H/R0.07 (Nr, N?, N?) (460, 210,150)
• H/R0.1 (Nr, N?, N?) (360, 210, 120)
• Initial B-field B? with ?25
• (See Fromang Nelson 2006 for details)

4
Low resolution runs show long term decay of
activity Need gt 5 cells per ?max
alpha versus time
5
log(?)
log(vA)
Obtain a basic core-halo structureDense
MRI-unstable disc near midplane, surrounded by
magneticallydominant corona (see also Miller
Stone 2000 and other conference talks)
6
Midplane ?
log(?)
log(vA)
H/R0.1 model
7
Movie courtesy of Y.Fidali (CEA Saclay) Software
SDvision
8
Toroidal flux versus time
Initial toroidal magnetic flux is lost through
vertical boundariesPolarity and amplitude of
azimuthal flux oscillates by ? 10 of original
value in region z lt H - a poor mans
butterfly diagram
9
? versus time
Volume averaged ? value saturates at 3 - 4 x
10-3
10
? versus radius
Time averaged ? value shows factor 2 - 3
variation with radius Maxwell stress 3 - 4 x
Reynolds stress
11
Maxwell and Reynolds stress variation with
height- double peak structure is apparent
12
Magnetic field component ratiosField is
primarily toroidal at all heightsRadial and
vertical field components increase near disc
surface
13
Mach number
Mach number shows weak shocks in magnetically
dominant regionnear disc surface
14
Time averaged velocity componentsas function of
height
Pmag/P versus heightDisc is magnetically
dominated above 4 scale heights
15
Midplane density fluctuations in stratified
models 1/2 of thoseobtained in similar
cylindrical disc modelsThis may be important
for models of stochastic migration
16
Time averaged radial velocity gives good
agreement withstandard disc theory
17
Planetesimals in stratified discs
18

• Gas in a pressure supported disc orbits with
  sub-Keplerian velocity
• Solid bodies orbit with Keplerian velocity
• Planetesimals experience head wind
  (Weidenschilling 1977)
• Gas drag induces inward drift efficient
  eccentricity dampingConsider bodies ? 1m ?
  Stokes drag regime

19
We consider evolution of 25 1m, 10m, 100m and 1km
planetesimalssubject to gas drag and stochastic
gravitational forcing by discAim Calculate
velocity dispersion and diffusive migration
ratesfor bodies assumed to orbit at 5 AU.Disc
mass Md 0.02 M? between 2 - 16 AU.
20
A few remarks about planetary growth
As mentioned already, experiments unable to grow
bodies to sizes gt 1 m. For runaway growth
require planetesimal velocity dispersionto be
significantly smaller than escape velocity from
largest accretingobjects
For 10 km sized bodies with ?2 g/cm3 escape
velocity10 m/s
21
SPH simulations show that bodies with size
between 100 m - 1 kmcatastrophically disrupted
with collisional velocities in range 10 - 30 m/s
(Benz Asphaug 1999).
Movie from DerekRichardson
22
1 metre sized boulders
Semimajor axes versus time show rapid inward
migration - relatively unaffected by
turbulence?k?s 1 for a1 m in this disc model

Ratio of gas drag force disc gravityis
typically 5 - 6 ?k?s 1 Evolution is gas
drag dominated for1 metre sized bodies
23
1 metre sized bodies strongly coupledto the gas.
Gas drag forces rapidly drive increased velocity
dispersion Planetesimal velocity dispersion
gas velocity dispersion (disc gravity assists in
providing small increase above ltvggt) Expect
planetesimals to have similarvelocities locally
due to dominance ofgas drag forces - but this
needs to be checked
24
10 metre sized boulders
Gas drag causes inward drift of 10 msized bodies
on time scale of 1000 yr
Ratio of drag force disc gravity
0.05Fluctuating forces due to disc
gravitybegin to dominate over gas drag
fluctuations
25
Velocity fluctuations few x 10 m/s.Reduction
in magntiude of gas dragmeans that boulders
respond to theorbit-averaged drag force - this
damps velocity fluctuations whichare driven by
gravitational fluctuations- this is a sweet spot
for minimisingvelocity fluctuations
Velocity fluctuations disruption velocities of
20 m/s reported by Benz Asphaug (1999)
26
100 metre sized planetesimals
Gas drag migration time for 100 m sized
planetesimals gt 50,000 yearsSome planetesimals
undergo a randomwalk.Others show periods of
non-stochasticsustained inward/outward migration
Disc gravity dominates significantly overgas
drag forces
27
Weaker gas drag allows gravitationalforce
fluctuations to drive velocitydispersion up to
100 m/s- significantly in excess of
disruptionvelocity ( 10 m/s).
Velocities significantly in excess of escape
velocity from 10 km sized bodies? runaway
growth of planetesimalsnot possible in this
environment
28
1 km sized planetesimals
Migration results similar to 100 m
bodiesStochastically migrating
planetesimalsSome showing monotonic migration
oversignificant time periods
Drag force gravitational force ratio 5 x 10-5
29
Velocity dispersion is gt 100 m/s.Collisions
will be highly disruptiveOnly 200 km sized
bodies will be ableto undergo runaway growth in
thisenvironment
Bodies experiencing stochastic forcinghave
velocity dispersions 100 m/s.Bodies
experiencing periods of monotonic migration have
increasedvelocity dispersion 300 m/s.
30
A closer look at the orbital evolution of 1 km
sized planetesimals
Stochastically migrating planetesimals experience
rapidly varying gravitational torques as expected
31
Planetesimals which experience periods of
monotonic migration experiencetorques with long
term variations superposed on the stochastic
componentThis indicates the presence of
persistent features in the flow
providinglong-term torques on embedded bodies
32
In this simulation, a near - coorbital
anticyclonic vortex persists for 100 orbital
periods and provides long-term torques on nearby
planetesimalsSimilar features observed in
numerous runs with embedded planets
33
This examplemp1 EarthH/R0.1
34
This example mp10 EarthH/R0.07
35
Origin of these vortices not yet fully
understood.They appear as loosely defined
structures associated with local regionsof
anticyclonic vorticity - hence the associated
positive density perturbationThey appear to form
and break apart intermittently for durations of
30 - 100 orbits before dissolving into the
background flow. A possible origin is stochastic
accumulation of vorticity perturbations induced
bythe MHD turbulence which achieve a threshold
for longevity.Or vortex growth at radial
pressure maxima in the disc via the PPI (RWI).
36
Long-term orbital evolution of planetesimals -
stochastic case only
r.m.s. torque fluctuations Trms 6 x
10-5Angular momentum diffusion coefficient DJ
(Trms) 2 ?corr (Johnson, Goodman, Menou
2006)Stochastic migration time tmig (?J)2 /
DJ
Fluctuating torque correlation time ?corr
0.5 orbits (5 years at 5 AU) Time for
planetesimals to stochastically migrate from 5
AU to 1 AU 6.6 Myr 3 AU
to 1 AU 4.4 Myr
37
Vertical settling of dust grains
38
The consequence of turbulence for dust settling
Gravitational force drag force promotes
vertical settling MHD turbulence opposes
vertical settling
39
The model
Fromang Nelson (2006)

• The disk
• ??0(R0/R)1/2
• EQS locally isothermal
• TT0(R0/R) ? HH0(R/R0)
• Gas vertical profile Gaussian
• H/R0.1
• The computational domain
• R ? 1,8
• Vertical direction 9 scaleheights in total
• Density ratio between midplane and surface
  4.10-5
• Azimuthal direction 0,?/2

40
Coupling between gas and solids

• ??s lt 1 ? strong coupling (Garaud et al 2004)
• ??s 1 ? weak coupling

41
Disk properties
alpha
Radial profile for alpha lt?gt 5?10-3 - 1?10-2
42
III. Dust
43
An example

• Dust treated as a fluid (P0)
• Coupling between gas and solids via gas drag

Turbulent disc
Laminar disc
44
??s10-2,10-3,10-4
Consider a disk with Md 0.01 M and Rd 300 AU
45
Steady state

• Is turbulence required?
• How does it affect the vertical dust profile ?

46
Is turbulence required?
??s10-3
??s10-2
??s10-4
MHD
HYDRO t30 orbits
Even for small particles, ??s is large in the
disk upper layers (Dullemond Dominik 2004) ?
dust/gas decouple
47
Settling timescales
Fdragmp?2z ? vsett?2z?s
tsettz/vsett1/(?2?s)
(?2?s)mid10-4, zH ? tsett965 orbits z2H ?
tsett215 orbits z3H ? tsett17
orbits z4H ? tsett0.5 orbits
48
Three models for ?d
(Dubrulle et al. 1995, Schrapler Henning 2004,
Dullemond Dominik 2004, Pinte et al. 2008,
Fromang Papaloizou 2006, Fromang Nelson 2009)
49
Three models for Ddust and ?d
(Fromang Nelson 2009)
Model 2a Ddust is constant
Model 2b Ddust?vz2?corr

• No analytical solution
• ?vz measured in the simulations
• ?corr0.15 orbits (measured in the simulations)

50
The case ??s10-2
Simulation data
Gaussian profile
Ddustconst
Ddust?vz2?corr
? Good agreement with all the models ?
Hd0.3H strong settling limit dust profile
Gaussian Hd ? a-0.5
51
The case ??s10-3
Simulation data
Gaussian profile
Ddustconst
Ddust?vz2?corr
52
The case ??s10-4
Simulation data
Gaussian profile
Ddustconst
Ddust?vz2?corr

• ? Disagreement between the models
• Best agreement with model D?vz2?0

53
Hd ?a?

• Hd?a-0.2
• Pinte et al. (2008) Hd?a-0.05 for IM Lupi

54
Conclusions future work

• For fully turbulent discs, velocity dispersion
  of 1 km sized planetesimals ltvgt ? 100 m/s ?
  collisional breakup and quenching of runaway
  growth
• Macroscopic bodies experience orbital diffusion
  on AU scales within expected disc life
  timesMay be possible to use solar system data
  to constrain strength ofmidplane turbulence and
  size of the dead zone (e.g. gradients inasteroid
  belt)
• Non homogeneous turbulence leads to height
  varying diffusion coefficientand enhanced dust
  suspension in disc upper layersCollaboration
  with J.-C. Augereau (LAOG), J.Oloffson (LAOG)to
  put dynamical model output into radiative
  transfer codes forcomparison with data ?
  possible way of measuring turbulent
  velocitiesfrom disc observations