PLANET FORMATION BY GRAVITATIONAL INSTABILITY? Kaitlin Kratter University of Arizona. Sagan Summer Workshop, July 2015

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1 PLANET FORMATION BY GRAVITATIONAL INSTABILITY? Kaitlin Kratter University of Arizona Sagan Summer Workshop, July 2015

2 PLANET FORMATION BY in GAS! GRAVITATIONAL INSTABILITY? Kaitlin Kratter University of Arizona Sagan Summer Workshop, July 2015

3 THE BOTTOM LINE Few disks appear to be massive enough to fragment (except perhaps at very early times) Those that do are more likely to produce more massive objects like brown dwarfs or m-stars Inward migration followed by tidal disruption most often leads to complete disruption, rather than mass reduction. But might assist in solid core formation

4 THE BOTTOM LINE Few disks appear to be massive enough to fragment (except perhaps at very early times) accretion disk studies were born in galactic / x-ray binary contexts, some of the standard assumptions Those that do are more likely to produce more made in these contexts are not well massive objects like brown dwarfs or m-stars suited to protostellar / protoplanetary disks. Inward migration followed by tidal disruption most 1. Thermodynamics are dominated often leads to complete disruption, rather than mass by stellar irradiation reduction. But might assist in solid core formation 2. H/R is not <<1

5 GLOSSARY Self-Gravity: the gravitational attraction of gas to itself is competitive with the central body Gravitational Instability: (GI) a linear, hydrodynamic instability that can arise in self-gravitating disks of gas, particles or both Fragment: a marginally bound gas clump that forms as the nonlinear outcome of GI Planet: depends on whom you ask

6 WHAT IS GRAVITATIONAL INSTABILITY? A hydrodynamic instability that arises in rotational;y supported disks when selfgravity wins out over pressure support on small scales, and stabilization due to shear on large scales Q = c s G = f M H M D r P Fg Ω Toomre s Q, rewritten this way can point us toward important, order of magnitude arguments

7 WHAT IS GRAVITATIONAL INSTABILITY? ^WHEN H<<R A hydrodynamic instability that arises in rotational;y supported disks when selfgravity wins out over pressure support on small scales, and stabilization due to shear on large scales P Fg Ω (! m (R)) 2 = c 2 s k 2 2 G k + apple 2, Q = c s G = f M H M D r Growth begins at Q=1 in the local approximation, for axisymmetric modes. since H<<R, Md<<M*

8 WHAT IS GRAVITATIONAL from Lau & Bertin 1978 INSTABILITY? ^WHEN H<R Fg P Ω Growth begins at Q>1 in the global approximation

9 Paardekooper 2012, Rice+2014 Kratter+2010 LOCAL VS GLOBAL GI H /R =0.01 These two scales are governed by a different dispersion relations, thus different modes H /R =0.4 H /R =0.10 grow at different values of Q H /R =0.2

10 LOCAL VS GLOBAL GI 1978ApJ L Stochastic disc fragmentation 3295 H /R =0.4 These two H /R = Q <. (94) scales are 3 In other words, keeping the temperature fixed, we only need an ingoverned aof a factor of 3 over the background Q crease in surfaceby density 1 state to form a clump that can resist the shear. Once formed, these p different clumps in general contract on a cooling time-scale (Kratter Mwill disk & Murray-Clay,2011). Their survival depends mainly on if they J = 6m can resist dispersion Mthe? weak shocks that sweep around in gravitoturbulence. Since shock heating is very localized, this makes fragmentation a stochastic process: there will be a large spread in clump survival relations, thus times, until the first lucky clump survives long enough for collapse to proceed. It should be noted that the condition given by equadifferent modes and P.J. Armitage tion (93) is not necessary if the cooling time-scale is comparable to dynamical time-scale. If cooling acts on a dynamical time-scale, growthe at there is nodifferent time for the clump to shear apart before it collapses. H /R =0.10 H /R =0.2 We have observed fragmentation up to β = 7, more than twice values Qtime-scale found by Gammie (2001). The corthe critical of cooling where R0 is the radial distance to the central star and M is its mass. This condition can be recast in terms of the local value of Q: Higher resolution Figure 1. Final state of a full 10 million particle simulation using smoothed cooling with βcool = 8. At this stage (after 6.5 outer rotation periods) the6.disc has settled into in a quasi-steady andsurface there isdensity, no evidence of igure Surface density, terms of thestate initial on a loga- We find that increasing the resolution by a factor of 2 (N x = N y = 2048) leads to easier fragmentation at higher values of β. As an example, we show in Fig. 7 four simulations at β = 9, differing only in the phase (not magnitude) of the initial noise. Two of the discs fragment, one at "t 500 and other at "t 750. The other two discs maintain a steady gravitoturbulent state for the full length of the simulation. This nicely illustrates the stochastic nature of disc fragmentation at high values of β: only in two out of four simulations does a clump survive for long enough for collapse to proceed. It is expected that if the simulations would be continued, Paardekooper 2012, Rice+2014 Downloaded from by g responding maximum value of the stress is α max For larger values of β, the disc remained in a steady, gravitoturbulent state for "t < 1000, with values of α that agree well with equation (3). Kratter+2010

11 From GI to Fragmentation Fragmentation occurs when the instability does not saturate in the linear phase. Saturation typically occurs in one of two ways: mode-mode coupling thermal feedback

12 MODE-MODE COUPLING Interaction of multiple growing modes saturates the amplitude of density perturbations surface density may never get high enough *locally* for collapse, because global modes are triggered at higher Q. global modes provide very efficient angular momentum transport may be important where MRI / disk winds fail FIG. 6.ÈGlobal Fourier amplitudes, C ÈC, plotted as a function of time 1 4 Laughlin, Korchagin, Adams 1996

13 THERMAL FEEDBACK: THE COOLING TIME CRITERION waves/spiral arms generate subsonic shocks, which heat the gas. too much heating, Q>1, self-regulated GI is possible too little heating, Q<1, fragmentation c = 1 c2 s T 4 f( ) since the instability grows on a dynamical time, cooling on a similar timescale is too fast to stave off fragmentation.

14 HOW FAST? WHAT IS Physically: set by optical depth due to dust. It depends on the (effective) EOS ( ) Probably between 8-15 for protoplanetary disks with =7/5 Numerical modelling required

15 Gammie, 2001 Saturation vs fragmentation Kratter+2010a Fragments also occur on scales ~H Most power in gravito-turbulence occurs on scales 1<H<10

16 WHAT ARE THE RIGHT PARAMETERS FOR PROTOSTELLAR AND PROTOPLANETARY DISKS?

17 Kratter & Lodato, in prep H<R accretional heating only including irradiation Conditions in a massive, protostellar disk around a sun-like star T 4 mid = 3 8 f( R)F acc + T 4 h, + T 4 ex F acc = 3 8 Ṁ 2

18 Kratter & Lodato, in prep H<R Conditions in a massive disk around a sun-like star corresponding values of Q and cooling time, between AU it is close enough to give rise to GI

19 H<<R Kratter & Lodato, in prep Conditions for measured Class I disks around sun-like stars Disks that are low enough in mass to operate in the local regime are typically too hot to suffer from GI.

20 HOW DID WE GET HERE? Disk with Q>1 Q = c s G = f M H M D r The outer regions of protostellar disks are dominated by stellar irradiation, which fixes the temperature.

21 HOW DID WE GET HERE? Disk with Q>1 Q = c s G = f M H M D r add mass, raise surface density The outer regions of protostellar disks are dominated by stellar irradiation, which fixes the temperature.

22 HOW DID WE GET HERE? Disk with Q>1 Q = c s G = f M H M D r cool the disk down add mass, raise surface density The outer regions of protostellar disks are dominated by stellar irradiation, which fixes the temperature.

23 HOW DID WE GET HERE? Disk with Q>1 Q = c s G = f M H M D r cool the disk down add mass, raise surface density The outer regions of protostellar disks are dominated by stellar irradiation, which fixes the temperature.

24 HOW DID WE GET HERE? Disk with Q>1 Q = c s G = f M H M D r cool the disk down add mass, raise surface density The outer regions of protostellar disks are dominated by stellar irradiation, which fixes the temperature. only under very special circumstances can the disk reach instability by getting colder, rather than by adding mass

25 normalized accretion radius = Ṁin M d 0.1 Low Mass Stars High Mass Stars 0.01 Singles multiples / fragments binary twins? Kratter et al 2010 normalized accretion rate = ṀG c 3 s

26 Fragment mass, zeroth order M frag = 2 H 2 M iso /M isanumericalconstant M iso 4πf H ΣR H r. numericalconstantrep R H = r(m iso /3M ) 1/3, If disks are driven unstable by infall, it can be challenging to avoid isolation mass H/R Kratter et al 2010b Q

27 WHY DO FRAGMENTS GROW IN DISKS WITH INFALL?

28 WHY DO FRAGMENTS GROW IN DISKS WITH INFALL? Ṁ in

29 WHY DO FRAGMENTS GROW IN DISKS WITH INFALL? Ṁ in

30 WHY DO FRAGMENTS GROW IN DISKS WITH INFALL? Ṁ in R hill

31 WHY DO FRAGMENTS GROW IN DISKS WITH INFALL? Ṁ in R hill

32 WHY DO FRAGMENTS GROW IN DISKS WITH INFALL? Ṁ in R hill

33 WHY DO FRAGMENTS GROW IN DISKS WITH INFALL? Ṁ in R hill

34 First order: how massive are fragments? Initial mass estimates all scale with H 2 Fragments that are not disrupted can also easily grow from the parent disk Kratter & Lodato, in prep, with data from Kratter+2010,Boley +2010,Forgan & Rice 2013, Young & Clarke 2015

35 First order: how massive are fragments? Initial mass estimates all scale with H 2 Fragments that are not disrupted can also easily grow from the parent disk Kratter & Lodato, in prep, with data from Kratter+2010,Boley +2010,Forgan & Rice 2013, Young & Clarke 2015

36 Choose your own GI adventure! local local (cold, low mass) or global (hotter, higher mass)? global slow or fast slow or fast cooling? cooling? fast slow fast slow

37 Choose your own GI adventure! local local (cold, low mass) or global (hotter, higher mass)? global slow or fast slow or fast cooling? cooling? fast slow fast slow probably a young, compact Class 0

38 Choose your own GI adventure! local local (cold, low mass) or global (hotter, higher mass)? global slow slow or fast cooling? fast slow probably a young, compact Class 0 slow or fast cooling? fast fragment! but destruction/ binary formation likely

39 Choose your own GI adventure! local local (cold, low mass) or global (hotter, higher mass)? global slow slow or fast cooling? fast fragment with extreme mass ratio! you might be a shadowed PPD or an AGN slow probably a young, compact Class 0 slow or fast cooling? fast fragment! but destruction/ binary formation likely

40 Choose your own GI adventure! local local (cold, low mass) or global (hotter, higher mass)? global slow slow or fast cooling? self-regulated GI you might be an AGN! fast fragment with extreme mass ratio! you might be a shadowed PPD or an AGN slow probably a young, compact Class 0 slow or fast cooling? fast fragment! but destruction/ binary formation likely

41 Choose your own GI adventure! local local (cold, low mass) or global (hotter, higher mass)? global slow slow or fast cooling? self-regulated GI you might be an AGN! fast infall fragment with extreme mass ratio! you might be a shadowed PPD or an AGN slow probably a young, compact Class 0 slow or fast cooling? fast fragment! but destruction/ binary formation likely

42 ON THE NEXT INSTALLMENT.FATE OF FRAGMENTS Disruption on few dynamical times Partial or complete disruption due to inward migration Direct collapse / continued growth GI population synthesis, and other ways to make wide orbit planets