Soft Disks: Proto-Planetary Disks in your Soft Disks: Proto-Planetary Disks in your ComputerComputer

Garrelt MellemaGarrelt Mellema

Numerical ModelsNumerical Models

Reasons to use numerical models:Reasons to use numerical models:– Reproduce observations / fitting parametersReproduce observations / fitting parameters

Observations = radiation, so always requires radiative transfer of some Observations = radiation, so always requires radiative transfer of some sort.sort.

– ‘‘Experimental’ astronomy: understanding the physics of complex Experimental’ astronomy: understanding the physics of complex systems:systems: Disk structureDisk structure Planet-disk interactionPlanet-disk interaction Jet collimationJet collimation

Complex systems:Complex systems:– Gas (atoms, ions, molecules, electrons) / chemistryGas (atoms, ions, molecules, electrons) / chemistry– Dust (different sizes)Dust (different sizes)– Magnetic FieldsMagnetic Fields– PhotonsPhotons– Gravity (star, binary systems, planets)Gravity (star, binary systems, planets)

In principle we know how to calculate all of these!In principle we know how to calculate all of these!

Limitations of Numerical ModelsLimitations of Numerical Models

In practice one is limited by computational resources. To In practice one is limited by computational resources. To make calculations feasible one can resort to several make calculations feasible one can resort to several simplifications:simplifications:– Neglect parts of the physics. Can be done if their effects can be Neglect parts of the physics. Can be done if their effects can be

included in a simplified way, for exampleincluded in a simplified way, for example No magnetic fields, but assume a viscosity for the gasNo magnetic fields, but assume a viscosity for the gas No dust, but assume it is coupled perfectly to the gasNo dust, but assume it is coupled perfectly to the gas No radiation, assume that the gas is locally isothermalNo radiation, assume that the gas is locally isothermal

– Reduce to less than 3 dimensions, for exampleReduce to less than 3 dimensions, for example Work with surface density for thin disks (h << r)Work with surface density for thin disks (h << r) Assume cylindrical symmetry when studying vertical structure Assume cylindrical symmetry when studying vertical structure

For continuum processes, one also has to use an For continuum processes, one also has to use an (unphysical) discretization (mesh or grid). This implies a (unphysical) discretization (mesh or grid). This implies a finite dynamic range finite dynamic range DD: L/: L/ΔΔx. Typically x. Typically DD ~100-1000. ~100-1000.

Impact of LimitationsImpact of Limitations

As in the case of telescopes, one has to live with the As in the case of telescopes, one has to live with the limitations of the tools.limitations of the tools.

Looking back one can see in the (short) history of Looking back one can see in the (short) history of computational studies thatcomputational studies that– Often, adding more details, adds more details in the results Often, adding more details, adds more details in the results

(comparison to observations!), but does not change the basic (comparison to observations!), but does not change the basic results.results.

– But, in other cases, the added details change the basic results.But, in other cases, the added details change the basic results.– Increasing the dimensionality often makes a large difference, Increasing the dimensionality often makes a large difference,

especially when it comes to instabilities.especially when it comes to instabilities.

Numerical Gas DynamicsNumerical Gas Dynamics

The equations of gas dynamics are difficult to solve:The equations of gas dynamics are difficult to solve:– Five quantities (8 for magnetohydrodynamics) to solve for.Five quantities (8 for magnetohydrodynamics) to solve for.– Non-linear coupled differential equations.Non-linear coupled differential equations.– Allow discontinuous solutions (shocks, contact discontinuities).Allow discontinuous solutions (shocks, contact discontinuities).

Two basic approaches are used in astrophysicsTwo basic approaches are used in astrophysics– Grid-based codesGrid-based codes

Quantities defined on a mesh, nowadays often on an Quantities defined on a mesh, nowadays often on an adaptiveadaptive mesh. mesh. Good at discontinuities.Good at discontinuities. Limitations on Limitations on spatialspatial dynamic range: bad at following gravitational dynamic range: bad at following gravitational

collapse.collapse.

– Particle based codes (SPH, Smooth Particle Hydrodynamics)Particle based codes (SPH, Smooth Particle Hydrodynamics) Quantities associated with particles (representing fluid elements).Quantities associated with particles (representing fluid elements). Limitations on Limitations on massmass dynamic range. dynamic range. Good at gravitational collapse.Good at gravitational collapse. Bad at discontinuities.Bad at discontinuities.

Proto-Planetary Disk ModelsProto-Planetary Disk Models

Gasdynamic simulations are used to study various Gasdynamic simulations are used to study various processes in proto-planetary disks:processes in proto-planetary disks:– Jet collimationJet collimation– Planet formationPlanet formation– TurbulenceTurbulence– Disk-Planet interactionDisk-Planet interaction

Producing JetsProducing Jets

The collimation of jets & outflows is a classic astrophysical The collimation of jets & outflows is a classic astrophysical problem, and has been addressed with numerical problem, and has been addressed with numerical simulations.simulations.

Typically, these simulations the inner disk regions, and the Typically, these simulations the inner disk regions, and the disk is more of a ‘boundary condition’.disk is more of a ‘boundary condition’.

Simulations have been showing collimation for decades, Simulations have been showing collimation for decades, however there were always doubts as to the stability of however there were always doubts as to the stability of these flows, the flow evolution far away, etc.these flows, the flow evolution far away, etc.

There now appears to be a consensus that the jets are There now appears to be a consensus that the jets are magneto-centrifugally launched from a disk-wind, but many magneto-centrifugally launched from a disk-wind, but many open issues remain… open issues remain…

JetsJets

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

Planet FormationPlanet Formation

Two models for the formation of massive planetsTwo models for the formation of massive planets– Core accretion modelCore accretion model: slowish growth of planet from first : slowish growth of planet from first

planetesimals, then gas.planetesimals, then gas.– Core collapse modelCore collapse model: gravitational collapse of parts of a heavy : gravitational collapse of parts of a heavy

disk.disk.

Both have been studied numerically, with mixed Both have been studied numerically, with mixed successes.successes.

Core accretion:Core accretion:– Complex physics: sticking planetesimals, coupling to disk Complex physics: sticking planetesimals, coupling to disk

dynamics, accretion of gas (on solid). First models: too slow (tdynamics, accretion of gas (on solid). First models: too slow (t formation formation

> 10> 107 7 years). Nowadays: problem solved…? (opacity, other years). Nowadays: problem solved…? (opacity, other changes).changes).

Core collapse:Core collapse:– Scale problem, coupled to different physical regimes.Scale problem, coupled to different physical regimes.

Core Collapse SimulationCore Collapse Simulation

Mayer et al. 2002

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.

Magneto-Rotational InstabilityMagneto-Rotational Instability

Ionized disks are subject to the magneto-rotational Ionized disks are subject to the magneto-rotational instability (MRI), even if only slightly ionized.instability (MRI), even if only slightly ionized.

Simulations are the only way to evaluate whether MRI can Simulations are the only way to evaluate whether MRI can explain the disk ‘viscosity’ needed for accretion.explain the disk ‘viscosity’ needed for accretion.

Results are successful (Results are successful (αα ~ few times 10 ~ few times 10-3-3)), but note that , but note that many simulationsmany simulations– Are 2D or 2.5DAre 2D or 2.5D– Lack dynamic rangeLack dynamic range

Disk-Planet InteractionDisk-Planet Interaction

A planet embedded in a proto-planetary disk will interact A planet embedded in a proto-planetary disk will interact with it. The effects arewith it. The effects are– Gap opening (affecting accretion to the planet)Gap opening (affecting accretion to the planet)– Migration (due to angular momentum transfer with the disk)Migration (due to angular momentum transfer with the disk)

This problem has been studied extensively with This problem has been studied extensively with simulations. Most of the results are in 2D and for simulations. Most of the results are in 2D and for isothermal disks, often in in co-rotating coordinates.isothermal disks, often in in co-rotating coordinates.

2D simulations can be used if the Roche lobe of the planet 2D simulations can be used if the Roche lobe of the planet is either much smaller than the disk scale height (low mass is either much smaller than the disk scale height (low mass planets), or much larger (high mass planets).planets), or much larger (high mass planets).

Low mass planets do not open gaps (type I migration).Low mass planets do not open gaps (type I migration). High mass planets open gaps (type II migration).High mass planets open gaps (type II migration).

Disk-Planet Interaction: 2D/3DDisk-Planet Interaction: 2D/3D

Migration time Migration time against planet against planet mass (in stellar masses).mass (in stellar masses).

The lines indicate the analytical The lines indicate the analytical estimates for Type I and II estimates for Type I and II migration.migration.

2D: 2D: ◊◊ 3D: 3D: ●● The models follow mostly the The models follow mostly the

expected type I and type II expected type I and type II migration.migration.

The big difference occurs The big difference occurs around the transition between around the transition between the two: Roche lobe of planet is the two: Roche lobe of planet is approaching scale height of approaching scale height of disk.disk.

Mig

ratio

n tim

e

Type I

Type II

Planet-Disk Code ComparisonPlanet-Disk Code Comparison

Within the framework of the RTN Within the framework of the RTN Formation of Planetary Formation of Planetary SystemsSystems, a comparison of the results for a large range of , a comparison of the results for a large range of codes was made.codes was made.

Four standard problems (Jupiter/Neptune, inviscid/ Four standard problems (Jupiter/Neptune, inviscid/ viscosity) in 2D.viscosity) in 2D.

Seventeen codes.Seventeen codes. One of the first detailed code comparisons for a complex One of the first detailed code comparisons for a complex

astrophysical problem.astrophysical problem. Detailed results can be found at Detailed results can be found at

http://www.astro.su.se/groups/planets/comparison/http://www.astro.su.se/groups/planets/comparison/

Code OverviewCode Overview Upwind methods Upwind methods

– NIRVANA-GDANIRVANA-GDA (Gennaro D'Angelo) (Gennaro D'Angelo) – NIRVANA-GDNIRVANA-GD (Gerben Dirksen) (Gerben Dirksen) – NIRVANA-PCNIRVANA-PC (Paul Cresswell) (Paul Cresswell) – RH2DRH2D (Willy Kley) (Willy Kley) – GLOBALGLOBAL (Sebastien Fromang) (Sebastien Fromang) – FARGOFARGO (Frédéric Masset) (Frédéric Masset) – GENESISGENESIS (Arnaud Pierens) (Arnaud Pierens) – TRAMP van LeerTRAMP van Leer (Hubert Klahr) (Hubert Klahr)

High-order finite-difference methods High-order finite-difference methods – PencilPencil (Wladimir Lyra) (Wladimir Lyra)

Shock-capturing methods Shock-capturing methods – AMRAAMRA (Pawel Ciecielag & Tomasz Plewa) (Pawel Ciecielag & Tomasz Plewa) – Flash-AG (Artur Gawryszczak) Flash-AG (Artur Gawryszczak) – Flash-AP (Adam Peplinski) Flash-AP (Adam Peplinski) – TRAMP-PPM (Hubert Klahr) TRAMP-PPM (Hubert Klahr) – Rodeo (Sijme-Jan Paardekoper & Garrelt Mellema) Rodeo (Sijme-Jan Paardekoper & Garrelt Mellema) – JUPITER (Frédéric Masset) JUPITER (Frédéric Masset)

SPH methods SPH methods – SPHTREE (Ken Rice) SPHTREE (Ken Rice) – ParaSPH (Christoph Schäfer & Roland Speith) ParaSPH (Christoph Schäfer & Roland Speith)

Code Comparison ResultsCode Comparison Results

Invisid Jupiter case

Code Comparison Results (2)Code Comparison Results (2)

Invisid Jupiter case

Code Comparison Results (3)Code Comparison Results (3)

Invisid Jupiter case

Comparison: Density ProfilesComparison: Density Profiles

Density profile along the planet’s orbit Density profile perpendicular to planet’s orbit

L4 L5

Comparison: Total TorquesComparison: Total Torques

Code Comparison ConclusionsCode Comparison Conclusions

PPM codes in co-rotating coordinates show ‘ripples’.PPM codes in co-rotating coordinates show ‘ripples’. FLASH in cartesian coordinates does not reproduce the FLASH in cartesian coordinates does not reproduce the

gap structure well.gap structure well. SPH codes do not reproduce the gap structure well.SPH codes do not reproduce the gap structure well. Other codes (upwind & shock-capturing) roughly agree on Other codes (upwind & shock-capturing) roughly agree on

gap structure.gap structure. But: torques easily different by 50%!But: torques easily different by 50%!

Dust-Gas CouplingDust-Gas Coupling

Proto-planetary disks consist of dust and gas.Proto-planetary disks consist of dust and gas. Gas orbits at slightly sub-Keplerian velocities due to Gas orbits at slightly sub-Keplerian velocities due to

pressure gradient.pressure gradient. Dust wants to orbit at Keplerian velocity (no pressure), but Dust wants to orbit at Keplerian velocity (no pressure), but

feels the drag of the gas.feels the drag of the gas. Small dust particles (1-10Small dust particles (1-10μμm) m) couple well to the gas.couple well to the gas. Larger dust particles experience dust drift: gas-dust Larger dust particles experience dust drift: gas-dust

separation. Especially strong near gradients in gas separation. Especially strong near gradients in gas pressure.pressure.

Dust is observationally important: most of the emitted Dust is observationally important: most of the emitted radiation comes from dust.radiation comes from dust.

Rule of thumb: Rule of thumb: λλ ~ dust size. ~ dust size.

Dust Emission from Gas Disk ModelDust Emission from Gas Disk Model

Wolf et al. 2002Wolf et al. 2002

Jupiter-mass planet at 5.2 AUJupiter-mass planet at 5.2 AUImage at 0.7 mmImage at 0.7 mm

4 hour integration with ALMA4 hour integration with ALMA

Assumes perfect dust-gas coupling!Assumes perfect dust-gas coupling!

Gas-Dust Disk ModelGas-Dust Disk Model

Planet: 0.1 MPlanet: 0.1 MJJ

(no gap(no gap in gas! in gas!)) Dust:1.0 mmDust:1.0 mm

Paardekooper & Mellema (2004)Paardekooper & Mellema (2004)

Dust Emission at Dust Emission at λλ=1 mm=1 mm

Gas and dust perfectly Gas and dust perfectly coupledcoupled

0.1 MJup at 5.2 AU, d=140pc, 12mas resolution (ALMA-like)

With dust drift