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Planetary migration - a review
Richard NelsonQueen Mary, University of London
Collaborators: Paul Cresswell (QMUL),Martyn Fogg (QMUL), Arnaud Pierens QMUL), Sebastien
Fromang (CEA), John Papaloizou (DAMTP)
Talk Outline • Type I migration in laminar discs• The role of corotation torques(non linear effects; planet traps; non isothermal effects)
• Multiple low mass planets • Protoplanets in turbulent discs:
Low mass planets Planetesimals
High mass planets
• Terrestrial planet formation during/after giant planet migration
• Conclusions and future directions
Low mass planets - type I migration
• Planet generates spiral waves in disc at Lindblad resonances
• Gravitational interaction between planet and spiral wakes causes exchange of angular momentum
• Wake in outer disc is dominant (pressure support shifts resonant locations) - drives inward migration
• Corotation torque generated by material in horseshoe region- exerts positive torque, but weaker thanLindblad torques
• Migration time scale ~ 70,000 yr for mp=10 Mearth
• Giant planet formation time 1 Myr
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Pollack et al. (1996)
Tanaka, Takeuchi & Ward (2002)• Low mass protoplanets migrate rapidly < 105 yr• Gas accretion onto solid core requires > 1 Myr - Difficult to form gas giant planets - Reducing dust opacity speeds up gas accretion but migration is always more rapid (e.g. Papaloizou & Nelson 2005, Lissauer et al 2006)
Evidence for type I migration• Short-period low mass planets:
20 planets with m sini < 40 Mearth
(e.g. HD69830 Lovis et al 2006, Gl581 Udry et al 2007, GJ436 Butler et al 2005)- but 45 candidates…
• Disc models agree T > 1500 K within 0.1AU- dust sublimates
• Mass of solids inside 1 AU~ 5 Earth masses for MMSN
• Type I migration does occur !- but probably more slowly than predicted by basic theory
…….10 Earth mass…….
Stopping/slowing type I migration• MHD Turbulence (see later)
• Planet-planet scattering (Cresswell & Nelson 2006) - migration stops if e > H/r
• Corotation torques may slow/stop planetmigration (Masset et al 2006)
• Planet enters cavity due to transition from ‘dead-zone’ to ‘live zone’ - planet trap (Masset et al. 2005)
• Corotation torque in optically thick discs(Pardekooper & Mellema 2007)
• Strong magnetic field (Terquem 2002; Fromang, Terquem & Nelson 2005)
• Opacity variations: sharp transition in density and temperature (Menou & Goodman 2002)
• Eccentric disks (Papaloizou 2002)
Can corotation torques slow type I migration ?
(Masset, D’Angelo & Kley 2006)
Basic idea - corotation region widens with planet mass and can boost corotation torqueFor = const. corotation torque can cause migration reversal for mp>10 ME
3D simulations with evolving planets
= constantH/r=0.05=0.005
= constantH/r=0.05=0.0
Questions: Dead-zones ? Does corotation torque operate in turbulent disc ?
m=10 Mearth
m=20 Mearth
m=30 Mearth
m=10 Mearth
m=20 Mearth
m=30 Mearth
Viscous discs Inviscid discs
Surface density transitions as planet traps
• Regions where surface density gradient is positive cause strongpositive corotation torque(Masset, Morbidelli & Crida 2005)
• Planets migrate into planet trapand migration is halted
Planet stops here
Gap formation by giant planet formsplanet trap for mp < 30 Mearth
Very low mass planets cannot formmean motion resonances withinterior giants (Pierens & Nelson 2008)
Corotation torques in optically thick discs
• Corotation torque can exceed Lindblad torques in optically thick discs(Paardekooper & Mellema 2007; Baruteau & Masset 2008; Pardekooper & Papaloizou 2008)
• Effect is due to warm gas being advected from inside to outside orbit of planet- and vice versa
• Pressure equilibrium leads to modification of density structure in horseshoe region
• High density region leads planet, low density region trails it - net positive torque - which saturates
Models with viscous heating and radiative cooling showsustained outward migration (Kley & Crida 2008)
Thermal time scale ~ horseshoe libration time scale
Now have a problem of rapid outward type I migration…
Low mass planetary swarms• Consider swarm consisting of
between 5 - 20 low mass interacting planets
• Question: can interaction within swarm maintain eccentric population and prevent type I migration ?
• Answer: No !• Outcomes:
Initial burst of gravitational scatteringCollisons (~ 1 per run)“Stacked” mean motion resonancesInward migration in lockstepExotic planet configurations:Horseshoe and tadpole systems(sometimes in MMR with each other)
Coorbital planets stable even duringsignificant mass growth - giant coorbital planets canremain stable(Cresswell &Nelson 2008) .
May be detected byCOROT, KEPLER,or RV surveys
High mass protoplanets• When planets grow to ~ Jovian mass they open gaps:
(i) The waves they excite become shock waves when RHill > H
(ii) Planet tidal torques exceed viscous torques
• Inward migration occurs on viscous evolution time scale of the disk
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• Inward migration occurs on time scale of ~ few x 105 year • Jovian mass planets remain on ~ circular orbits• Heavier planets migrate more slowly than viscous rate due to their inertia• A 1 MJ planet accretes additional 2 – 3 MJ during migration time of ~ few x 105 yr
Eccentricity Evolution• Disc interaction can cause both
growth and damping of e due tointeraction at ELRs, CRs, and COLRs
• For Mp > 5 MJup can get disc eccentricity growth- planet eccentricity growth ?(Kley & Dirksen 2005)
• Most simulations show e dampingfor Jovian mass planets- but D’Angelo et al (2006) findmost e growth
• Origin of exoplanet eccentricities:planet-planet scattering ?(Rasio & Ford; Papaloizou & Terquem 2003; Laughlin & Adams 2004; Juric & Tremaine 2007)
Evidence for type II migration
• Existence of short period planets (Hot Jupiters)
• Resonant multiplanet systems: GJ876 – 2:1 HD82943 – 2:1 55 Cnc – 3:1 HD73526 - 2:1 HD128311 - 2:1
Planets in turbulent discs
0/ <Ω dRd
• Magnetorotational instability vigorous turbulence in discs (Balbus & Hawley 1991; Hawley, Gammie & Balbus 1996, etc…)
• Necessary ingredients: (i) Weak magnetic field (ii) (iii) Sufficient ionisation: X(e-) ~ 10-12
(iv) Rem > 100
Dust free disc ~ 50 % of matter turbulent Dusty disc ~ 1 % of matter turbulent
Ilgner & Nelson (2006a,b,c)
Obtain a basic core-halo structure:Dense MRI-unstable disc near midplane, surrounded by magneticallydominant corona (see also Miller & Stone 2000)
Stratified disc models• H/R=0.07 and H/R=0.1 discs computed
• Locally isothermal equation of state• ~ 9 vertical scale heights
πφ
4BB
mrT = φδρδ vvT rR .=
=
PTT mR−=
Stratified global model
H/R=0.1, mp=10 mearth
Nr x N x N = 464 x 280 x 1200
Local view – turbulent fluctuations ≥ spiral wakes
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• Planet in laminar disc shows expected inward migration
• Planet in turbulent discundergoes stochastic migration
T ~ 20 x type I torque for mp=10 Earth mass ttype 1 ~ 400 tcorr
T ~ 200 x type I torque for mp=20 Earth massttype 1 ~ 40,000 tcorr
Run times currently achievable ~ 200 orbits
tTT
+>=<
Can treat stochastic migration asa signal to noise problem(assume linear superposition oftype I + stochastic torques)
Calculate time scale over whichtype I torque dominates random walk
Results show exampleswhere stochastic torques (and migration)contain long-termsignal…
This is apparently due to persistent featuresdeveloping in the flow(transient and looselydefined vortices)
This example:mp=1 EarthH/R=0.1
This example:mp=10 EarthH/R=0.07
Planetesimals in stratified, turbulent discs
• Gas in pressure supported disc orbits with sub-Keplerian velocity• Solid bodies orbit with Keplerian velocity Planetesimals experience head wind (Weidenschilling 1977) • Gas drag induces inward drift & efficient eccentricity damping
Consider evolution of 1m, 10m, 100m and 1km planetesimalssubject to gas drag and stochastic gravitational forcing
Aim: Calculate velocity dispersion assuming bodies orbit at ~ 5 AU
For runaway growth require planetesimal velocity dispersionto be much smaller than escape velocity from largest accretingobjects:
For 10 km sized bodies with ρ=2 g/cm3 escape velocity=10 m/s
Collisional break-up occurs for impact velocities 15 - 30 m/sfor bodies in size range 100m - 1km (Benz & Asphaug 1999)
1m-sized bodies stronglycoupled to gas.Velocity dispersion ~ turbulent velocities
10m bodies have<v> ~ few x 10 m/s - gas drag efficient atdamping random velocities
100m - 1km sizedbodies excited by turbulent density fluctuations<v> ~ 50-100 m/s
Larger planetesimals prevented from undergoing runaway growthPlanetesimal-planetesimal collisions likely to lead to break-up
Need dead-zones to form planets rapidly ? Or leap-frog this phase with gravitational instability ? Or can a relatively small number of bodies avoid catastrophic collisions and grow ?
High mass planets in turbulent discs
• mp=30 mearth accretes gas andforms gap
• Migrates inward on viscous time scale ~ 105 yr
• Gas accretion rate enhanced due to magnetic torques
Terrestrial Planet Formation During Giant
Planet Migration• N-body simulations performed (Fogg & Nelson 2005, 2006, 2007)
• Initial conditions: inner disk of planetesimals+protoplanets undergoing different stages of `oligarchic growth’ within a viscously evolving gas disc
• Giant planet is introduced which migrates through inner planet-forming disc
• General outcomes:(i) massive terrestrial planets can form interior to migrating giant(ii) significant outer disk forms from scattered planetesimals and embryos(iii) water-rich terrestrial planets can form in outer disc
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Continued accretion in the scattered disk. Initial condition.
Continued accretion in the scattered disk. t + 1 Myr.
Continued accretion in the scattered disk. t + 6 Myr.
Compositional Mixing. Before.
Compositional Mixing. After.
Ocean PlanetsOcean Planetspredictedpredicted
A case where an inner super-earth forms…
Conclusions and Future Directions
• Low mass planets migrate rapidly in laminar discs- but this remains an active research area
• Multiple low mass planet systems display:inward resonant migration, horseshoe and trojan systems - observable by COROT or KEPLER ?
• Turbulence modifies type I migration and may prevent large-scale inward migration for some planets
• Turbulence increases velocity dispersion of planetesimals and may lead to destructive collisions and quenching of runaway growth
• Stochastic forces experienced by planets in vertically stratified discs lower in amplitude due to finite disc thickness - work in progress
• Water-rich terrestrial planets probably form in “hot Jupiter” systems
GJ876: Already known to have 2 planets in 2:1 resonance Velocity residuals showed periodic variation - 3rd planet with mass ~ 7.5 Earth masses
Gl436 - 22 Earth mass transiting planet lightcurve suggests it is just like Neptune & Uranus
Gl581 - Short-period 5 Earth mass planet detected by radial velocity
System Age = 1.5 Myr:
t = 80,000 years
t = 20,000 years
154,700 years
Power spectrum – shows torques have temporal variations ~ run time of simulations
Stochastic torques may overcome type I torques over significant time scales for some planetsRequire longer simulations…
Torque distributions σ
tTT
+>=<
Naïve application suggests inward migration should be obtained for mp=10
Planetesimals in laminar discs• Gas in pressure supported
disc orbits with sub-Keplerian velocity• Solid bodies orbit with Keplerian velocity Planetesimals experience head wind (Weidenschilling 1977) • Gas drag induces inward drift & efficient eccentricity damping
Planetesimals in turbulent discs
• Evolution of 100 - 1000 planetesimals calculated to examine inward drift and velocity dispersion
• Planetesimal treated as particlesthat experience gas drag and gravitational force due to disc andcentral star
• Sizes: 1m – 1km
1 metre sized planetesimals
1 metre sized objects migrate inward within ~ 30 orbits (300 years)1m sized boulders become trapped in long-lived vortices - about 50% of particles trappedTight coupling to gas causes large eccentricities – potentially destructivevelocity dispersion ?Neighbouring planetesimals appear to be on very similar orbits
Mp = 10 Earth masses
Mp = 1 Earth mass
10 metre sized planetesimals
• Most 10 m size boulders drift inward on time scale of ~ few thousand years – a few drift in more slowly• Velocity dispersion remains quite small – coupling too weak to allow individual fluctuations in gas velocity to determine velocity dispersion • Danger of destructive collisions: e=0.01 <v> ~ 0.12 km/s at 5 AU• Icy 10 m sized bodies fragment with <v> ~ 20 m/s (Benz & Asphaug 1999)
1 km sized planetesimals
• Results similar to 100 metre sized objects
• Large velocity dispersion prevents runaway growth of planetesimals to form planetary embryos
Stopping type II migration
• Fortuitous disk removal – form planets late on ?
• Overlap gaps of ‘Jupiter’ and ‘Saturn’ ?• Roche lobe overflow – only works close-in• Magnetospheric cavity – only works close-in• Switch viscosity off ? – difficult to explain observed distribution of exoplanets – or observed accretion rates onto T Tauri stars
Planets in turbulent discs
0/ <Ω dRd
• Magnetorotational instability vigorous turbulence in discs (Balbus & Hawley 1991; Hawley, Gammie & Balbus 1996, etc…)
• Necessary ingredients: (i) Weak magnetic field (ii) (iii) Sufficient ionisation: X(e-) ~ 10-12
(iv) Rem > 100
Dust free disc ~ 50 % of matter turbulent Dusty disc ~ 3 % of matter turbulent
Ilgner & Nelson (2006a)
πφ
4BB
mrT = φδρδ vvT rR .=
=
PTT mR−=
Low mass planets
• Consider orbital evolution of: mp=1, 3, 5, 10, 30 Earth mass planets
• Question: what is effect of turbulence on type I migration ?
(Nelson & Papaloizou 2004; Nelson 2005; Laughlin, Adams & Steinaker 2004)
Fluctuating torques – suggest stochastic migration
Mp = 10 Earth masses
Mp = 10 Earth masses
Power spectrum – shows torques have temporal variations ~ run time of simulations
Stochastic torques may overcome type I torques over significant time scales for some planetsRequire longer simulations…
Torque distributions σ
tTT
+>=<
Naïve application suggests inward migration should be obtained for mp=10
Planetesimals in laminar discs• Gas in pressure supported
disc orbits with sub-Keplerian velocity• Solid bodies orbit with Keplerian velocity Planetesimals experience head wind (Weidenschilling 1977) • Gas drag induces inward drift & efficient eccentricity damping
Planetesimals in turbulent discs
• Evolution of 100 - 1000 planetesimals calculated to examine inward drift and velocity dispersion
• Planetesimal treated as particlesthat experience gas drag and gravitational force due to disc andcentral star
• Sizes: 1m – 1km
1 metre sized planetesimals
1 metre sized objects migrate inward within ~ 30 orbits (300 years)1m sized boulders become trapped in long-lived vortices - about 50% of particles trappedTight coupling to gas causes large eccentricities – potentially destructivevelocity dispersion ?Neighbouring planetesimals appear to be on very similar orbits
10 metre sized planetesimals
• Most 10 m size boulders drift inward on time scale of ~ few thousand years – a few drift in more slowly• Velocity dispersion remains quite small – coupling too weak to allow individual fluctuations in gas velocity to determine velocity dispersion • Danger of destructive collisions: e=0.01 <v> ~ 0.12 km/s at 5 AU• Icy 10 m sized bodies fragment with <v> ~ 20 m/s (Benz & Asphaug 1999)
100 metre sized planetesimals
• 100 m sized objects dominated by fluctuations in disc gravity• Instead of inward drift undergo `random walk’ on time scales ~ 100 orbits• Icy 100m sized objects fragment if <v> ~ 14 m/s• <v> ~ 0.24 km/s for e=0.02 at 5 AU • Destructive collisions likely as neighbouring orbits randomly orientated
1 km sized planetesimals
• Results similar to 100 metre sized objects
• Large velocity dispersion prevents runaway growth of planetesimals to form planetary embryos
Migration in optically thick discs• Corotation torque can
exceed Lindblad torques inoptically thick discs(Paardekooper & Mellema 2007)
• Effect is due to warm gasbeing advected from insideto outside orbit of planetand vice versa
• Pressure equilibrium leads to modification of density structure
• High density region leads planet, low density region trails it - positive torque