Soft Disks: Proto-Planetary Disks in your Computer

1. Soft Disks: Proto-Planetary Disks in your Computer Garrelt Mellema

2. 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!

3. 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 << 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.

4. 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.

5. 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.

6. Proto-Planetary Disk Models • Gasdynamic simulations are used to study various processes in proto-planetary disks: • Jet collimation • Planet formation • Turbulence • Disk-Planet interaction

7. 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…

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

9. 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 > 107 years). Nowadays: problem solved…? (opacity, other changes). • Core collapse: • Scale problem, coupled to different physical regimes.

10. 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

11. 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

12. 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).

13. 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

14. 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/

15. 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 • SPHTREE (Ken Rice) • ParaSPH (Christoph Schäfer & Roland Speith)

16. Code Comparison Results Invisid Jupiter case

17. Code Comparison Results (2) Invisid Jupiter case

18. Code Comparison Results (3) Invisid Jupiter case

19. Comparison: Density Profiles L4 L5 Density profile along the planet’s orbit Density profile perpendicular to planet’s orbit

20. Comparison: Total Torques

21. 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%!

22. 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: λ ~ dust size.

23. 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!

24. Gas-Dust Disk Model • Planet: 0.1 MJ (no gap in gas!) • Dust:1.0 mm Paardekooper & Mellema (2004)

25. Dust Emission at λ=1 mm Gas and dust perfectly coupled 0.1 MJup at 5.2 AU, d=140pc, 12mas resolution (ALMA-like) With dust drift