Gravitational instabilities in protostellar discs and the formation of planetesimals
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1 Gravitational instabilities in protostellar discs and the formation of planetesimals Giuseppe Lodato - Università degli Studi di Milano 17 February Bologna
2 Gravitational instabilities in protostellar discs Conditions for instability Dynamics of self-gravitating discs: Conditions for fragmentation/self-regulation Local vs global behaviour Planetesimal formation and evolution in spiral arms Self-regulated disc models and their application to planetesimal formation Self-gravitating discs with ALMA
3 Linear stability criterion Well known axisymmetric instability criterion: Q = c sκ πgσ < Q 1 Equivalent form of the instability criterion M disc (R) M H R Need the disc to be cold and/or massive What are the masses and aspect ratio in actual protostellar discs?
4 Linear stability criterion Well known axisymmetric instability criterion: Q = c sω πgσ < Q 1 Equivalent form of the instability criterion M disc (R) M H R Need the disc to be cold and/or massive What are the masses and aspect ratio in actual protostellar discs?
5 Are protostellar discs linearly unstable? Midplane temperature for irradiated discs (Chiang & Goldreich 1997, Chiang & Youdin 2009) gives: ( H R R 0.02 AU ) 2/7 Therefore H/R varies from 0.02 at 1AU to 0.06 at 100 AU Need disc masses of order 5% of the stellar mass to be unstable Protostellar disc masses difficult to measure (see Hartmann et al 2006)
6 Are protostellar discs linearly unstable? Disc masses in Taurus and Ophiucus by Andrews and Williams (2005, 2007) Clear trend to have smaller masses at later stages of evolution A substantial fraction of Class I (and even some Class II) objects expected to be unstable Disc masses might be underestimated significantly (Hartmann et al 2006) Uncertainties in dust opacities If density profile steep, most of the mass might be hidden in optically thick inner parts (Hartmann 2009) Class II
7 Are protostellar discs linearly unstable? Disc masses in Taurus and Ophiucus by Andrews and Williams (2005, 2007) Clear trend to have smaller masses at later stages of evolution A substantial fraction of Class I (and even some Class II) objects expected to be unstable Disc masses might be underestimated significantly (Hartmann et al 2006) Uncertainties in dust opacities If density profile steep, most of the mass might be hidden in optically thick inner parts (Hartmann 2009) Class II
8 Are protostellar discs linearly unstable? Disc masses in Taurus and Ophiucus by Andrews and Williams (2005, 2007) Clear trend to have smaller masses at later stages of evolution A substantial fraction of Class I (and even some Class II) objects expected to be unstable Disc masses might be underestimated significantly (Hartmann et al 2006) Uncertainties in dust opacities If density profile steep, most of the mass might be hidden in optically thick inner parts (Hartmann 2009) Class II
9 Are protostellar discs linearly unstable? Disc masses in Taurus and Ophiucus by Andrews and Williams (2005, 2007) Clear trend to have smaller masses at later stages of evolution A substantial fraction of Class I (and even some Class II) objects expected to be unstable Disc masses might be underestimated significantly (Hartmann et al 2006) Class I Uncertainties in dust opacities If density profile steep, most of the mass might be hidden in optically thick inner parts (Hartmann 2009)
10 Gravitational instabilities in protostellar discs Conditions for instability Dynamics of self-gravitating discs: Conditions for fragmentation/self-regulation Local vs global behaviour Planetesimal formation and evolution in spiral arms Self-regulated disc models and their application to planetesimal formation
11 Non linear evolution of GI Investigated numerically in the last decade by several authors (Laughlin & Bodenheimer 1994, Laughlin et al 1998, Pickett et al 2000, Boss 2000, Gammie 2001, Mayer et al 2002, Lodato & Rice 2004, 2005, Mejia et al 2005, Boley et al 2006) Early simulations used an isothermal or polytropic equation of state (Laughlin & Bodenheimer 1994, Mayer et al 2002) Starting from Gammie (2001) it has become clear that the evolution is strongly dependent on the cooling time t cool Introduce a cooling parameter as the ratio of cooling to dynamical timescale β = t cool Ω
12 Thermal self-regulation of GI Role of cooling time clear if one thinks at the form of the stability parameter Q Q = c sκ πgσ T 1/2 Development of the instability feeds energy back onto the equilibrium and stabilizes the disc Works as an effective thermostat for the disc Expect the disc to stay close to marginal stability Q ~ 1 In order for thermostat to work, cooling time must not be faster than instability growth time Expect a change in behaviour for β ~ a few
13 Short cooling time: fragmentation Fragmentation occurs for cooling times shorter than β ~ 3-5 (depending on numerical setup, ratio of specific heats, etc... Gammie 2001, Rice, Lodato & Armitage 2005, Clarke, Harper-Clark & Lodato 2007) Simulation by Peter Cossins β = 4 If cooling strongly temperature dependent (e.g. close to opacity gap) fragmentation might be easier (Johnson & Gammie 2003, Cossins, Lodato & Clarke 2009b)
14 Short cooling time: fragmentation Exact value heavily debated recently!! Meru & Bate (2010, 2011) Fragmentation occurs for cooling times shorter than β ~ 3-5 Lodato & Clarke (2011) (depending on numerical setup, ratio of specific heats, etc... Gammie 2001, Rice, Lodato & Armitage 2005, Clarke, Harper-Clark & Lodato 2007) Simulation by Peter Cossins β = 4 If cooling strongly temperature dependent (e.g. close to opacity gap) fragmentation might be easier (Johnson & Gammie 2003, Cossins, Lodato & Clarke 2009b)
15 Short cooling time: fragmentation Exact value heavily debated recently!! Meru & Bate (2010, 2011) Fragmentation occurs for cooling times shorter than β ~ 3-5 Lodato & Clarke (2011) (depending on numerical setup, ratio of specific heats, etc... Gammie 2001, Rice, Lodato & Armitage 2005, Clarke, Harper-Clark & Lodato 2007) Simulation by Peter Cossins β = 4 If cooling strongly temperature dependent (e.g. close to opacity gap) fragmentation might be easier (Johnson & Gammie 2003, Cossins, Lodato & Clarke 2009b)
16 Long cooling time: self-regulation Lodato & Rice (2004) Simulation by Peter Cossins β = 6
17 Long cooling time: self-regulation Lodato & Rice (2004) Simulation by Peter Cossins β = 6
18 Long cooling time: self-regulation Lodato & Rice (2004) Simulation by Peter Cossins β = 6
19 Long cooling time: self-regulation Lodato & Rice (2004) Simulation by Peter Cossins β = 6
20 Thermal saturation of GI Cossins, Lodato & Clarke 2009 Self-regulation is established through thermal saturation of the spiral waves. Amplitude of density perturbation must be related to cooling rate We find that: Σ Σ 1 β Natural if consider that energy content of waves is proportional to the square of the perturbed fields
21 Spectrum of excited modes Many modes excited at the various radii (roughly on the local H scale)
22 Spectrum of excited modes Many modes excited at the various radii (roughly on the local H scale)
23 Sonic condition for spiral waves Cossins, Lodato & Clarke 2009 We have computed the pattern speed of the underlying spiral structure and its Mach number The Doppler-shifted Mach number is very close to unity, independently on radius, cooling rate, and disc mass. Density jump for almost sonic shocks also directly leads to Σ Σ 1 β
24 Local vs global behaviour Lodato & Rice 2004 Can the evolution of self-gravitating discs be described within the standard, local, α-like prescription? Can compute gravitational + Reynolds stresses directly from simulations and compare with expectations from standard α-theory (LR04, see also Boley et al. 2006) Lodato & Rice 2004 Boley et al. 2006
25 Local vs global behaviour Cossins, Lodato & Clarke 2009 Can the evolution of self-gravitating discs be described within the standard, local, α-like prescription? Described in detail by Balbus & Papaloizou (1999), recently discussed extensively by Cossins et al (2009) Relation between energy and angular momentum densities in a density wave E = Ω p L E = Ω L p Relation between power and stress due to local (viscous) processes E ν = Ω L ν If density waves dissipate far from co-rotation, behaviour is non-local
26 Local vs global behaviour Cossins, Lodato & Clarke 2009 Degree of non-locality can be measured by ξ = Ω Ω p Ω Sonic condition for wave dissipation also tells us something about this: ξ c s v φ = H R To the extent that the disc is thin (H<<R), global behaviour should be negligible Possible to construct local, viscous models of disc evolution (Clarke 2009, Rafikov 2009)
27 Threshold for fragmentation: convergence issues Recent work has cast doubts on convergence of critical cooling time for fragmentation as obtained by SPH (Meru & Bate 2010, 2011) Two different results: 1. Meru & Bate (2010): location of fragments depends on surface density profile! (Very difficult to understand...) In particular, critical value of β depends on m(r) = Σ(R)R2 M 2. Meru & Bate (2011): critical value of β keeps increasing with increasing number of particles
28 Threshold for fragmentation: convergence issues Lodato & Clarke (2011): both effects are due to resolution! Indeed, h H = η m(r) ( 2q π 2 Q 2 N ) 1/3
29 Threshold for fragmentation: convergence issues Lodato & Clarke (2011): both effects are due to resolution! Indeed, h H = η m(r) ( 2q π 2 Q 2 N ) 1/3 Smoothing length/ Thickness
30 Threshold for fragmentation: convergence issues Numerical par. = 1.2 Lodato & Clarke (2011): both effects are due to resolution! Indeed, h H = η m(r) ( 2q π 2 Q 2 N ) 1/3 Smoothing length/ Thickness
31 Threshold for fragmentation: convergence issues Numerical par. = 1.2 Lodato & Clarke (2011): both effects are due to resolution! Indeed, h H = η m(r) ( 2q π 2 Q 2 N ) 1/3 Smoothing length/ Thickness Disc/star mass ratio
32 Threshold for fragmentation: convergence issues Numerical par. = 1.2 Lodato & Clarke (2011): both effects are due to resolution! Indeed, h H = Smoothing length/ Thickness η m(r) ( 2q π 2 Q 2 N Disc/star mass ratio ) 1/3 Number of SPH particles
33 Threshold for fragmentation: convergence issues h H = η m(r) ( 2q π 2 Q 2 N ) 1/ Simple interpretation if Frag. inhibited for insufficient resolution Resolution depends on β 6 4 β res 2 H h m(r)n 1/3 Resolved 2 More details in Lodato & Clarke (2011)
34 Threshold for fragmentation: convergence issues h H = η m(r) ( 2q π 2 Q 2 N ) 1/3 100 Simple interpretation if Frag. inhibited for insufficient resolution Resolution depends on β 10 β res 2 H h m(r)n 1/3 Resolved More details in Lodato & Clarke (2011) 1
35 Threshold for fragmentation: convergence issues h H = η m(r) ( 2q π 2 Q 2 N ) 1/3 100 Convergence? Simple interpretation if Frag. inhibited for insufficient resolution Resolution depends on β 10 β res 2 H h m(r)n 1/3 Resolved More details in Lodato & Clarke (2011) 1
36 Threshold for fragmentation: convergence issues Origin of cooling rate dependent fragmentation: Artificial heating? Resolution criterion corresponds to artificial viscosity providing a fixed fraction (5%) of the cooling rate. Smearing of the density peak? Astrophysical consequences: slight adjustment in location of fragmentation in protostellar discs No major effects on local/global behaviour
37 Gravitational instabilities in protostellar discs Conditions for instability Dynamics of self-gravitating discs: Conditions for fragmentation/self-regulation Local vs global behaviour Planetesimal formation and evolution in spiral arms Self-regulated disc models and their application to planetesimal formation
38 Planetesimal formation The first step of the core accretion model for planet formation How to form km-sized planetesimals? Combined effect of local pressure maxima, drag force, and dust selfgravity The so-called streaming instability (Youdin & Johansen) works only at rather large metallicities Alternatives?
39 Evolution of solids in selfgravitating discs (Rice, Lodato et al 2004, 2006) Effects of gas drag on solid particles is to induce fast migration towards pressure maxima. In a laminar disc this produces a fast inward migration of meter-sized particles (Weidenschilling 1977) v r = v Ωt s +1/Ωt s v c2 s v K ln ρ ln R
40 Evolution of solids in selfgravitating discs (Rice, Lodato et al 2004, 2006) Effects of gas drag on solid particles is to induce fast migration towards pressure maxima. In a laminar disc this produces a fast inward migration of meter-sized particles (Weidenschilling 1977) Radial velocity of solids v r = v Ωt s +1/Ωt s v c2 s v K ln ρ ln R
41 Evolution of solids in selfgravitating discs (Rice, Lodato et al 2004, 2006) Effects of gas drag on solid particles is to induce fast migration towards pressure maxima. In a laminar disc this produces a fast inward migration of meter-sized particles (Weidenschilling 1977) Radial velocity of solids v r = v Ωt s +1/Ωt s Stopping time v c2 s v K ln ρ ln R
42 Evolution of solids in selfgravitating discs (Rice, Lodato et al 2004, 2006) Effects of gas drag on solid particles is to induce fast migration towards pressure maxima. In a laminar disc this produces a fast inward migration of meter-sized particles (Weidenschilling 1977) Radial velocity of solids v r = v Ωt s +1/Ωt s Stopping time Difference between gas velocity and keplerian v c2 s v K ln ρ ln R
43 Evolution of solids in selfgravitating discs (Rice, Lodato et al 2004, 2006) Pressure maxima in spiral structure efficient trap for meter sized objects (see also Haghighipour & Boss 2003, Durisen et al 2005). Run SPH simulations of a two component system (gas + solids)
44 Evolution of solids in selfgravitating discs (Rice, Lodato et al 2004, 2006) Pressure maxima in spiral structure efficient trap for meter sized objects (see also Haghighipour & Boss 2003, Durisen et al 2005). Run SPH simulations of a two component system (gas + solids) Gas
45 Evolution of solids in selfgravitating discs (Rice, Lodato et al 2004, 2006) Pressure maxima in spiral structure efficient trap for meter sized objects (see also Haghighipour & Boss 2003, Durisen et al 2005). Run SPH simulations of a two component system (gas + solids) 50 cm Gas
46 Evolution of solids in selfgravitating discs (Rice, Lodato et al 2004, 2006) Pressure maxima in spiral structure efficient trap for meter sized objects (see also Haghighipour & Boss 2003, Durisen et al 2005). Run SPH simulations of a two component system (gas + solids) 50 cm Gas 10 m
47 Solid agglomeration in pressure maxima (Rice, Lodato et al 2004, 2006) Density of meter sized objects enhanced by up to two orders of magnitude Rice et al 2004 Density becomes high enough to become comparable to Roche density 10 m Gravitational collapse of solids is possible Confirmed through additional simulations including the solids self-gravity (Rice et al. 2005) 50 cm Resulting planetesimals mass expected to be high (but difficult to measure from simulations)
48 Evolution of large bodies in a selfgravitating disc (Britsch, Lodato, Clarke 2008) What is the dynamics of large (decoupled) bodies in a self-gravitating disc? Simulations of km-103 km sized bodies Similar to the case of planetesimals in an MRI turbulent disc (Nelson 2005), they undergo strong stochastic migration Reach average eccentricities of order e ~ 0.07 Significantly reduce ability of further growth Induce potentially disruptive collisions
49 Gravitational instabilities in protostellar discs Conditions for instability Dynamics of self-gravitating discs: Conditions for fragmentation/self-regulation Local vs global behaviour Planetesimal formation and evolution in spiral arms Self-regulated disc models and their application to planetesimal formation
50 Planetesimals in self-gravitating discs Particle traps in spiral arms are an effective way of producing large solid bodies in the disc: Resulting planetesimal mass quite large Dynamically stirred population of planetesimals Expected to occur in early phases of star formation (<~ 1Myr) Is this process limited to some specific radial range in the disc? Note: Rice et al. used an idealized cooling function leading to a rather large amplitude spiral Σ/Σ 0.1 Need a detailed model of self-gravitating discs with realistic cooling
51 Local models of self-regulated protostellar discs If transport is local (cf. Cossins et al 2009), then in thermal equilibrium (and absent other sources of heating, e.g. irradiation): α = γ(γ 1) 1 Ωt cool Possible to construct models of self-regulated discs (Q~1), where viscosity is related to cooling time (Clarke 2009, Rafikov 2009) Identify various possible regimes for self-gravitating protostellar discs
52 Local models of self-regulated protostellar discs Clarke 2009, Cossins, Lodato & Clarke 2010, Rafikov 2009 Note: such models generally have a steep density profile in the inner disc, e.g.: Σ R 9/4 Most of the mass might be hidden in the optically thick innermost few AU (cf. Hartmann 2009)
53 Local models of self-regulated protostellar discs Clarke 2009, Cossins, Lodato & Clarke 2010, Rafikov 2009 Note: such models generally have a steep density profile in the inner disc, e.g.: Σ R 9/4 Class I Most of the mass might be hidden in the optically thick innermost few AU (cf. Hartmann 2009)
54 Local models of self-regulated protostellar discs Clarke 2009, Cossins, Lodato & Clarke 2010, Rafikov 2009 Note: such models generally have a steep density profile in the inner disc, e.g.: Σ R 9/4 Class I Most of the mass might be hidden in the optically thick innermost few AU (cf. Hartmann 2009) Class II - T Tauri
55 Where do planetesimals form? Clarke & Lodato (2009) Check at which radii spiral arms concentration can produce large solid bodies Under what conditions is solid concentration effective? In the presence of a density perturbation on a scale λ: Migration time is: v r v φ v K c2 s t mig = λ v r ( λ H v K ( R λ ) 2 ( Σ Σ ) Σ Σ ) 1 1 Ω For a self-gravitating spiral structure λ ~ H, and hence tmig is comparable to structure lifetime for Σ Σ >f 0.1
56 Where do planetesimals form? Clarke & Lodato (2009) Planetesimal formation through this process occurs at 30AU < R < 50 AU Roughly coincident with the location of the Kuiper belt Some evidence for a large inner hole in debris disc systems (Currie et al. 2008), based on the apparent increase of debris disc brightness at late ages ~ Myrs Rapid production of large bodies in the outer disc may preserve sub-mm emission in the T Tauri phase (Takeuchi, Clarke & Lin 2005)
57 Where do planetesimals form? Clarke & Lodato (2009) Planetesimal formation through this process occurs at 30AU < R < 50 AU Roughly coincident with the location of the Kuiper belt Some evidence for a large inner hole in debris disc systems (Currie et al. 2008), based on the apparent increase of debris disc brightness at late ages ~ Myrs Rapid production of large bodies in the outer disc may preserve sub-mm emission in the T Tauri phase (Takeuchi, Clarke & Lin 2005)
58 Self-gravitating discs with ALMA Cossins, Lodato & Testi (2010) ALMA (Atacama Large Millimeter Array) will be the largest telescope operating at millimeter wavelengths 66 antennas with a maximum baseline of 18km Unprecedented angular resolution (down to ~ 1mas) First science in 2012 At mm wavelengths disc emission optically thin --> possible to detect non-axisymmetric, spiral structures?
59 Self-gravitating discs with ALMA Cossins, Lodato & Testi (2010) Start from one of our simulations, e.g. M* = 1M Sun, M disc = 0.2M * Place disc at 140pc (in Taurus) or at 50 pc (distance to TW Hya) Assume a standard opacity law
60 Self-gravitating discs with ALMA Cossins, Lodato & Testi (2010) Start from one of our simulations, e.g. M* = 1M Sun, M disc = 0.2M * Place disc at 140pc (in Taurus) or at 50 pc (distance to TW Hya) Assume a standard opacity law
61 Self-gravitating discs with ALMA Cossins, Lodato & Testi (2010)
62 Self-gravitating discs with ALMA Cossins, Lodato & Testi (2010) Simulated ALMA images at 50 pc
63 Self-gravitating discs with ALMA Cossins, Lodato & Testi (2010)
64 Self-gravitating discs with ALMA Cossins, Lodato & Testi (2010) Simulated ALMA images at 140 pc
65 Conclusions Class I discs are likely to be gravitationally unstable Self-regulated evolution of GI leads to sustained angular momentum transport for ~ 1 Myr, bringing the disc into the T Tauri phase Density waves dissipate when they become sonic Induced transport is local IF disc is sufficiently thin Spirals induce rapid formation of planetesimal in an annular region at large distances (30-50 AU): possibly consistent with observations of debris discs and the Kuiper belt ALMA could be able to detect such discs very soon!
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