Planet Formation

1. Planet Formation Topic: Gravoturbulent planetesimal formation: Particle clustering, Streaming instability, Particle trapping Lecture by: C.P. Dullemond

2. Main idea: • If we can find a way to cluster the dust particles into small regions („dust clouds“) where the density of dust would be very large, then perhaps gravity could then take over and cause the dust cloud to gravitationally collapse (contract) to form a planetesimal. • Many ideas have been posed in the literature: • The classic Goldreich & Ward model • The streaming instability • Turbulent clustering through intermittency • Dust trapping in local pressure maxima (rings, vortices)

3. Disk gravitational instability and the Goldreich & Ward idea of planetesimal formation

4. Gravitational instability of a disk • When a protoplanetary disk becomes too massive compared to the central star, it can become gravitationally unstable. • The critical quantity is the „Toomre number“: • If Q>2 then the disk is stable • If 1<Q<2 then the disk develops huge spiral waves • If Q<1 then the disk fragments

5. Gravitational instability of a disk A perhaps more insight-giving way to write Q is: Where we estimated: Since for protoplanetary disks we typically roughly have we can say that a disk that is more than 2.5% of the stellar mass is gravitationally unstable, and if more than 5% then it is so unstable that it will fragment. This is a possible model of Gas Giant Planet formation! (but more on that later)

6. Example of GI in disk Gravitational instabilities are global processes. Their modeling requires global hydrodynamic disk models. Here is an example of how the disk behaves: First strong global spiral waves are formed. Then (if Q<1) gravitationally bound clumps form. Image: Quinn et al. From: http://www.psc.edu/science/quinn.html

7. The old Goldreich & Ward idea • In principle the same holds true for the dust subdisk: • In previous chapter we derived that big dust grains will gather into a thin dust layer near the equator: the dust subdisk. • Less turbulence leads to thinner subdisk, i.e. smaller ratio H/r, i.e. smaller Q, i.e. eventually Q<1, i.e. formation of clumps = planetesimals.

8. The old Goldreich & Ward idea Goldreich & Ward 1973

9. The old Goldreich & Ward idea Goldreich & Ward 1973

10. The old Goldreich & Ward idea Goldreich & Ward 1973

11. The old Goldreich & Ward idea Goldreich & Ward 1973

12. But thereis a problem... Cuzzi 1993; Johansen, Henning & Klahr 2006; Chiang 2008

13. But thereis a problem... Cuzzi 1993; Johansen, Henning & Klahr 2006; Chiang 2008 z ϕ Settlingleadstoρd>ρg

14. But thereis a problem... Cuzzi 1993; Johansen, Henning & Klahr 2006; Chiang 2008 z vgas<vK vlayer≈vK ϕ Gas in the disk is subkepler. But in the dust layer, where ρd>ρg, the pressure-per-mass is lower. This layer will move almost Kepler. This leads to shear between dust layer and gas above: KH instability!

15. But thereis a problem... Cuzzi 1993; Johansen, Henning & Klahr 2006; Chiang 2008 Settlingleadstoρd>ρg vlayer=vKepler>vgleadsto Kelvin-Helmholtz instability Induces turbulence, which prevents H from dropping too low, and thus stabilizes against gravitational instability.

16. The Streaming Instability of Youdin & Johansen (Clustering at large turbulent eddy scales)

17. Streaming instability Youdin & Goodman 2005; Johansen & Youdin 2007 Dust particles with St<1 will want to move with the gas, thus moving sub-Kepler. z vdust≈vgas<vK vdust≈vgas<vK vdust≈vgas<vK ϕ

18. Streaming instability Youdin & Goodman 2005; Johansen & Youdin 2007 Dust particles with St<1 will want to move with the gas, thus moving sub-Kepler. However, if they form a cluster, then this cluster acts as a kind of „large particle“ with St>1, thus wanting to move Kepler. Note: Cluster ≠ Aggregate! They don‘t stick, they just move together. z vcluster≈vK ϕ

19. Streaming instability Youdin & Goodman 2005; Johansen & Youdin 2007 If you have both clusters and individual particles, then the cluster will sweep up the individual particles. z vdust≈vgas<vK vdust≈vgas<vK vcluster≈vK vdust≈vgas<vK ϕ

20. Streaming instability Youdin & Goodman 2005; Johansen & Youdin 2007 If you have both clusters and individual particles, then the cluster will sweep up the individual particles. This leads to ever larger and compact clusters. The clusters will also ablate. Finding out the real „clustering factor“ requires massive numerical simulations. z ϕ

21. Streaming instability Youdin & Goodman 2005; Johansen & Youdin 2007 From: Johansen & Youdin 2007

22. Turbulent Particle Clustering by turbulent intermittency: Cuzzi & Hogan (Trapping at small turbulent eddy scales) Squires & Eaton 1991; Hogan & Cuzzi 2001; Pan et al. 2011

23. Particle clustering in turbulence Particles react most strongly to eddies with turn-over time equal to the stopping time. For St<<1 particles these are the sub-eddies, i.e. Eddies with For such small eddies, Coriolis forces are unimportant (exercise: why?). And therefore these eddies are never in geostrophic balance. Centrifugal forces thus always win and expel particles from the eddies, and collect them between the eddies.

24. Particle clustering in turbulence Mathematical derivation (following Pan et al. 2011): Start from the known equation for the acceleration of a dust particle: Assume τstop<<τeddy, so that We can then replace with in the above equation and obtain (after rewriting slightly):

25. Particle clustering in turbulence Here the time derivative of the gas velocity is the comoving one: At scales much below the largest eddy scale, the gas behaves nearly incompressibly, so let us assume: If we now take the divergene of the particle velocity (eq above): Using Einstein‘s summation convection we can write:

26. Particle clustering in turbulence One can show that this can be rewritten as: Where ω is the vorticity, ω2 is called the enstrophy: and sij is the strain tensor: Conclusion: particles diverge from regions of high enstrophy and converge into regions of high strain.

27. Particle clustering in turbulence Particles diverge from regions of high enstrophy and converge into regions of high strain. This happens faster for larger particles (as long as they have stopping time smaller than the eddy turn-over time. high enstrophy high enstrophy high strain high enstrophy high enstrophy

28. Particle clustering in turbulence DEMO Very simple model (yes, you may do this at home! ;-)): Periodic 2D box with exactly 4 „eddies“ given by the formula: 1 y The angles φ and ψ are random angles that are randomly changed between 0 and 2π every eddy turnover time. 0 0 1 x Now insert particles with some stopping time and see what happens

29. Particle clustering in turbulence DEMO

30. Multiplicator model of Hogan & Cuzzi Idea: As eddy splits, dust is unevenly distributed. By repeating this again and again (multiplicator) you can get (rare) excessive clusters. These may exceed Roche density and become a planetesimal. Figure: Chambers, Icarus 208 (2010) 505–517 Note: Still controversial

31. Which amount of clustering can trigger gravitational contraction? The Roche density

32. Roche density • A cluster of matter can only remain gravitationally bound if it lies within its own Hill radius. If not, the tidal forces by the Sun will shear the cluster apart. • Now replace: and

33. Roche density • When R<RHill the cluster is gravitationally sufficiently bound to withstand the shear (tides of the Sun): • For Sun: ρ*=1.4 g cm-3 and R*=6.96x1010 cm, we get

34. Roche density • What is the closest distance a comet (ρ≈1g/cm3) can get to the Sun without being sheared apart? Assume that a comet is only bound by its own gravity. • Answer: About 1.6 R*. • Example: Comet Lovejoy

35. Relation Roche density & Toomre Q • We now have 2 criteria for gravitational collapse: • Toomre: Q<1 • Roche: ρ>ρRoche • Which is which? • First work this out for a gas disk: • Q<1 leads to: • ...which is very close to the Roche condition (only a factor of ~6 lower critical density).

36. Difficulty of gravoturbulent PF • The gas must have density much lower than the Roche density, otherwise the gas disk itself would become unstable. • The dust must cluster to a density well above the Roche density to start the contraction that produces a planetesimal. • Therefore: the dust must cluster to densities well above the gas density! • Normally the dust-gas feedback (friction) will work against this. • Unless the streaming instability is triggered (for that we need a long-lived gas pressure gradient).

37. Particle trapping in global pressure bumps

38. Particles move toward pressure peak P(r) r Whipple 1972; Barge& Sommeria 1995; Klahr & Henning 1997; Kretke & Lin 2008; Dzyurkevich et al. 2010; Kato et al. 2010; Johansen et al. 2009; Garaud 2007

39. New Paradigm: Particle Trapping Barge & Sommeria; Klahr & Henning; Johansen et al.; Kretke & Lin; Kato et al.; Dzyurkevich et al.

40. New Paradigm: Particle Trapping Barge & Sommeria; Klahr & Henning; Johansen et al.; Kretke & Lin; Kato et al.; Dzyurkevich et al.

41. New Paradigm: Particle Trapping Barge & Sommeria; Klahr & Henning; Johansen et al.; Kretke & Lin; Kato et al.; Dzyurkevich et al.

42. Presumably we already observe them... LkHa 330 SR 21 HD 135344B Brown et al. 2009

43. Particle trapping in local pressure bumps (anticyclonic vortices)

44. Vortices as pressure traps Simple model of a vortex Remember the local equations of motion of a parcel of gas in the corotating frame: Let us, for the sake of simplicity, ignore the 3(GM/r^3)x term, because it makes the solution much harder, and is not critical for the basic mechanism. The forces are the pressure gradient:

45. Vortices as pressure traps Simple model of a vortex Let us assume that the gas flows in concentric circles with angular frequency ω: Let us also assume as Ansatz that the pressure goes as: Inserting this into the above equations, and replacing for simplicity ρ with the average ρ0, yields two identical equations, namely:

46. Vortices as pressure traps Simple model of a vortex One can see that for one gets meaning that the pressure profile in the vortex has a local maximum: Anticyclonic vortices have a pressure maximum, while cyclonic vortices have a pressure minimum. Anicyclonic vortices can only exist because of the Coriolis forces (Ω>0).

47. Vortices as pressure traps Anti-cyclonicvortex: Dustgetstrapped Cyclonicvortex: Dustgetsexpelled Star Barge & Sommeria (1995), Klahr & Henning (1997)

48. Presumably we also already see these! Thanks to ALMA! HD142527 IRS 48 Van der Marel et al. 2013 Casassus et al. 2013