Soft Disks: Proto-Planetary Disks in your Computer

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Soft Disks Proto-Planetary Disks in your Computer

• Garrelt Mellema

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Numerical Models

• Reasons to use numerical models
• Reproduce observations / fitting parameters
• Observations radiation, so always requires
  radiative transfer of some sort.
• Experimental astronomy understanding the
  physics of complex systems
• Disk structure
• Planet-disk interaction
• Jet collimation
• Complex systems
• Gas (atoms, ions, molecules, electrons) /
  chemistry
• Dust (different sizes)
• Magnetic Fields
• Photons
• Gravity (star, binary systems, planets)
• In principle we know how to calculate all of
  these!

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Limitations of Numerical Models

• In practice one is limited by computational
  resources. To make calculations feasible one can
  resort to several simplifications
• Neglect parts of the physics. Can be done if
  their effects can be included in a simplified
  way, for example
• No magnetic fields, but assume a viscosity for
  the gas
• No dust, but assume it is coupled perfectly to
  the gas
• No radiation, assume that the gas is locally
  isothermal
• Reduce to less than 3 dimensions, for example
• Work with surface density for thin disks (h ltlt r)
• Assume cylindrical symmetry when studying
  vertical structure
• For continuum processes, one also has to use an
  (unphysical) discretization (mesh or grid). This
  implies a finite dynamic range D L/?x. Typically
  D 100-1000.

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Impact of Limitations

• As in the case of telescopes, one has to live
  with the limitations of the tools.
• Looking back one can see in the (short) history
  of computational studies that
• Often, adding more details, adds more details in
  the results (comparison to observations!), but
  does not change the basic results.
• But, in other cases, the added details change the
  basic results.
• Increasing the dimensionality often makes a large
  difference, especially when it comes to
  instabilities.

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Numerical Gas Dynamics

• The equations of gas dynamics are difficult to
  solve
• Five quantities (8 for magnetohydrodynamics) to
  solve for.
• Non-linear coupled differential equations.
• Allow discontinuous solutions (shocks, contact
  discontinuities).
• Two basic approaches are used in astrophysics
• Grid-based codes
• Quantities defined on a mesh, nowadays often on
  an adaptive mesh.
• Good at discontinuities.
• Limitations on spatial dynamic range bad at
  following gravitational collapse.
• Particle based codes (SPH, Smooth Particle
  Hydrodynamics)
• Quantities associated with particles
  (representing fluid elements).
• Limitations on mass dynamic range.
• Good at gravitational collapse.
• Bad at discontinuities.

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Proto-Planetary Disk Models

• Gasdynamic simulations are used to study various
  processes in proto-planetary disks
• Jet collimation
• Planet formation
• Turbulence
• Disk-Planet interaction

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Producing Jets

• The collimation of jets outflows is a classic
  astrophysical problem, and has been addressed
  with numerical simulations.
• Typically, these simulations the inner disk
  regions, and the disk is more of a boundary
  condition.
• Simulations have been showing collimation for
  decades, however there were always doubts as to
  the stability of these flows, the flow evolution
  far away, etc.
• There now appears to be a consensus that the jets
  are magneto-centrifugally launched from a
  disk-wind, but many open issues remain

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Jets
3D models by Kigure Shibata (2005). (note only
run for 2 inner-disk orbital perdiods)
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Planet Formation

• Two models for the formation of massive planets
• Core accretion model slowish growth of planet
  from first planetesimals, then gas.
• Core collapse model gravitational collapse of
  parts of a heavy disk.
• Both have been studied numerically, with mixed
  successes.
• Core accretion
• Complex physics sticking planetesimals, coupling
  to disk dynamics, accretion of gas (on solid).
  First models too slow (tformation gt 107 years).
  Nowadays problem solved? (opacity, other
  changes).
• Core collapse
• Scale problem, coupled to different physical
  regimes.

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Core Collapse Simulation

• SPH Simulation (3D)
• Problems
• 1) Isothermal equation of state not valid after
  collapse.
• 2) Long term stability of the fragments.
• 3) Role of shocks

Attempts to do this problem with grid-based codes
have mostly revealed problems with resolving
gravitational collapse.
Mayer et al. 2002
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Magneto-Rotational Instability

• Ionized disks are subject to the
  magneto-rotational instability (MRI), even if
  only slightly ionized.
• Simulations are the only way to evaluate whether
  MRI can explain the disk viscosity needed for
  accretion.
• Results are successful (a few times 10-3), but
  note that many simulations
• Are 2D or 2.5D
• Lack dynamic range

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Disk-Planet Interaction

• A planet embedded in a proto-planetary disk will
  interact with it. The effects are
• Gap opening (affecting accretion to the planet)
• Migration (due to angular momentum transfer with
  the disk)
• This problem has been studied extensively with
  simulations. Most of the results are in 2D and
  for isothermal disks, often in in co-rotating
  coordinates.
• 2D simulations can be used if the Roche lobe of
  the planet is either much smaller than the disk
  scale height (low mass planets), or much larger
  (high mass planets).
• Low mass planets do not open gaps (type I
  migration).
• High mass planets open gaps (type II migration).

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Disk-Planet Interaction 2D/3D

• Migration time ? against planet mass (in stellar
  masses).
• The lines indicate the analytical estimates for
  Type I and II migration.
• 2D ? 3D ?
• The models follow mostly the expected type I and
  type II migration.
• The big difference occurs around the transition
  between the two Roche lobe of planet is
  approaching scale height of disk.

Type I
Migration time
Type II
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Planet-Disk Code Comparison

• Within the framework of the RTN Formation of
  Planetary Systems, a comparison of the results
  for a large range of codes was made.
• Four standard problems (Jupiter/Neptune,
  inviscid/ viscosity) in 2D.
• Seventeen codes.
• One of the first detailed code comparisons for a
  complex astrophysical problem.
• Detailed results can be found at
  http//www.astro.su.se/groups/planets/comparison/

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Code Overview

• Upwind methods
• NIRVANA-GDA (Gennaro D'Angelo)
• NIRVANA-GD (Gerben Dirksen)
• NIRVANA-PC (Paul Cresswell)
• RH2D (Willy Kley)
• GLOBAL (Sebastien Fromang)
• FARGO (Frédéric Masset)
• GENESIS (Arnaud Pierens)
• TRAMP van Leer (Hubert Klahr)
• High-order finite-difference methods
• Pencil (Wladimir Lyra)
• Shock-capturing methods
• AMRA (Pawel Ciecielag Tomasz Plewa)
• Flash-AG (Artur Gawryszczak)
• Flash-AP (Adam Peplinski)
• TRAMP-PPM (Hubert Klahr)
• Rodeo (Sijme-Jan Paardekoper Garrelt Mellema)
• JUPITER (Frédéric Masset)
• SPH methods

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Code Comparison Results
Invisid Jupiter case
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Code Comparison Results (2)
Invisid Jupiter case
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Code Comparison Results (3)
Invisid Jupiter case
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Comparison Density Profiles
L4
L5
Density profile along the planets orbit
Density profile perpendicular to planets orbit
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Comparison Total Torques
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Code Comparison Conclusions

• PPM codes in co-rotating coordinates show
  ripples.
• FLASH in cartesian coordinates does not reproduce
  the gap structure well.
• SPH codes do not reproduce the gap structure
  well.
• Other codes (upwind shock-capturing) roughly
  agree on gap structure.
• But torques easily different by 50!

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Dust-Gas Coupling

• Proto-planetary disks consist of dust and gas.
• Gas orbits at slightly sub-Keplerian velocities
  due to pressure gradient.
• Dust wants to orbit at Keplerian velocity (no
  pressure), but feels the drag of the gas.
• Small dust particles (1-10µm) couple well to the
  gas.
• Larger dust particles experience dust drift
  gas-dust separation. Especially strong near
  gradients in gas pressure.
• Dust is observationally important most of the
  emitted radiation comes from dust.
• Rule of thumb ? dust size.

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Dust Emission from Gas Disk Model
Wolf et al. 2002
Jupiter-mass planet at 5.2 AU Image at 0.7 mm
4 hour integration with ALMA
Assumes perfect dust-gas coupling!
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Gas-Dust Disk Model
Paardekooper Mellema (2004)

• Planet 0.1 MJ
• (no gap in gas!)
• Dust1.0 mm

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Dust Emission at ?1 mm
0.1 MJup at 5.2 AU, d140pc, 12mas resolution
(ALMA-like)

• Gas and dust perfectly coupled

With dust drift