Soft Disks: Proto-Planetary Disks in your Computer

Soft Disks: Proto-Planetary Disks in your
Garrelt Mellema
Numerical Models
 Reasons to use numerical models:
– Reproduce observations / fitting parameters
 Observations = radiation, so always requires radiative transfer of some
– ‘Experimental’ astronomy: understanding the physics of complex
 Disk structure
 Planet-disk interaction
 Jet collimation
 Complex systems:
Gas (atoms, ions, molecules, electrons) / chemistry
Dust (different sizes)
Magnetic Fields
Gravity (star, binary systems, planets)
 In principle we know how to calculate all of these!
Limitations of Numerical Models
 In practice one is limited by computational resources. To
make calculations feasible one can resort to several
– 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 << 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/Δx. Typically D ~100-1000.
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
– 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.
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
– 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.
Proto-Planetary Disk Models
 Gasdynamic simulations are used to study various
processes in proto-planetary disks:
Jet collimation
Planet formation
Disk-Planet interaction
Producing Jets
 The collimation of jets & outflows is a classic astrophysical
problem, and has been addressed with numerical
 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…
3D models by Kigure & Shibata (2005).
(note: only run for 2 inner-disk orbital perdiods)
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
 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 > 107 years). Nowadays: problem solved…? (opacity, other
 Core collapse:
– Scale problem, coupled to different physical regimes.
Core Collapse Simulation
SPH Simulation (3D)
• Problems:
1) Isothermal equation of
state not valid after
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
Mayer et al. 2002
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 (α ~ few times 10-3), but note that
many simulations
– Are 2D or 2.5D
– Lack dynamic range
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).
 Migration time  against planet
mass (in stellar masses).
 The lines indicate the analytical
estimates for Type I and II
 2D: ◊ 3D: ●
 The models follow mostly the
expected type I and type II
 The big difference occurs
around the transition between
the two: Roche lobe of planet is
approaching scale height of
Migration time
Disk-Planet Interaction: 2D/3D
Type I
Type II
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
Code Overview
Upwind methods
High-order finite-difference methods
Pencil (Wladimir Lyra)
Shock-capturing 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)
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
SPHTREE (Ken Rice)
ParaSPH (Christoph Schäfer & Roland Speith)
Code Comparison Results
Invisid Jupiter case
Code Comparison Results (2)
Invisid Jupiter case
Code Comparison Results (3)
Invisid Jupiter case
Comparison: Density Profiles
Density profile along the planet’s orbit
Density profile perpendicular to planet’s orbit
Comparison: Total Torques
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%!
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
 Dust is observationally important: most of the emitted
radiation comes from dust.
 Rule of thumb: λ ~ dust size.
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!
Gas-Dust Disk Model
Paardekooper & Mellema (2004)
 Planet: 0.1 MJ
(no gap in gas!)
 Dust:1.0 mm
Dust Emission at λ=1 mm
0.1 MJup at 5.2 AU, d=140pc, 12mas resolution (ALMA-like)
Gas and dust perfectly
With dust drift

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