How do we model each process of planet formation? How do results depend on the model parameters?

Transcription

1 How do we model each process of planet formation? How do results depend on the model parameters? Planetary Population Synthesis: The Predictive Power of Planet Formation Theory, Ringberg, Nov 29, 2010

2 Core accretion model - sequential processes of different physics core accretion > 5-10M gas envelope contraction >100M runaway gas accretion & gap formation population synthesis protoplanetary disk: H/He gas (99wt%) + dust grains (1wt%) planetesimals coagulation of planetesimals type I migration terrestrial planets cores gas accretion onto cores gas giants orbital instability type II migration Newton Press

3 Population synthesis model (2004a,b,2005,2008a,b,2010), Mordasini et al. (2009a,b) Combine individual planet formation processes to predict distributions of exoplanets explain existing data, predict future observations, & constrain a theoretical model for each process -- link theory and observation derive semi-analytical formulas for individual processes integrate equations of planetary growth/migration + add stochastic changes The formulas must be based on detailed simulations to reflect essential physics, while it must be simple enough... dm dt Determinis(c = M +ΔM embyo + M τ planetesimal τ gas da dt = a + Δa scatt/coll τ migration de dt = Δe scatt/coll Stochas(c > Monte Carlo

4 Example of the integrations disk gas giant impacts disk edge evolution type-ii migration type-i migration gas accretion onto a core dm dt = M +ΔM embyo + M τ planetesimal τ gas da dt = a + Δa scatt/coll τ migration final state gas giant type-i migration planetesimal accretion rocky planets icy planets 0.6 sec on Mac air

5 disk model self-similar sol. at <~10AU + photo-evaporation disk surface density gas Σ gas = f g exp(-t /τ dep ) 1100 r 1AU planetesimlas Σ pl = f d η ice 10 r 1AU ( ) 1 g/cm 2 ( ) 1.5 g/cm 2 MMSN constant α initial [Fe/H] = 0 & τ dep = 3 Mys for today s results a ice = 2.7(M * /M ) 2 AU f g η ice self-similar sol. for α = y 105 y 10 7 y Σ gas 1/r 1AU r 10AU r [AU]

6 disk model disk surface density gas Σ gas = f g exp(-t /τ dep ) 1100 r 1AU planetesimlas Σ pl = f d η ice 10 r 1AU ( ) 1 g/cm 2 ( ) 1.5 g/cm 2 f g

7 disk model disk surface density gas Σ gas = f g exp(-t /τ dep ) 1100 r 1AU planetesimlas Σ pl = f d η ice 10 r 1AU ( ) 1 g/cm 2 ( ) 1.5 g/cm 2 f g Christoph: slightly massive

8 disk model disk surface density gas Σ gas = f g exp(-t /τ dep ) 1100 r 1AU planetesimlas Σ pl = f d η ice 10 r 1AU ( ) 1 g/cm 2 ( ) 1.5 g/cm 2 f d : global evolution due to accretion of multiple embryos f d =Σ pl /Σ pl,mmsn 0yr 10 4 yr 10 6 yr (2010) a [AU]

9 seed embryos Δa inner regions: feeding zone of isolation mass outer regions: feeding zone of embryo s mass at 10 9 y multiple-generation next-generation seeds are set when previous ones have migrated, if Σ pl remains

10 f d =Σ pl /Σ pl,mmsn Planetesimal Accretion Particle-in-a-box formula based on oligarchic growth model τ planetsimal v 2 πgσ pl RΩ K v e: embryo s steering, damping due to gas drag and collision Σ pl : global evolution due to accretion of embryos - Σ pl can have been depleted by preceding embryos migrating cores do not significantly grow different from Christoph s 0yr 10 4 yr 10 6 yr a [AU] (2010) dm dt = v R M +ΔM embyo + M τ planetesimal τ gas da dt = a + Δa scatt/coll τ migration de dt = Δe scatt/coll

11 Planetesimal Accretion planetesimal accretion can be accelerated (smaller field planetesimals, capture by core s atmosphere, ) gas giants & ice giants: more abundant [ limited by timescale] rocky planets: not significantly affected [ limited by isolation] close-in super-earths: anti-correlated with gas giants 3 times faster nominal 3 times slower gaseous icy

12 massive disks: form massive multiple jupiters destroy SEs medium-mass disks: retain Super-Earths Disk mass [MMSN] >100M rocky, 1-20M icy, 1-20M jupiter worlds super-earth/ neptune/jupiter mixed worlds sub-earth/subneptune worlds

13 Onset: Mc,hydro 10 Gas Accretion onto a Planet dm /dt 10 6 M /y 0.25 f atm f MMSN 0.25 M dm/dt : calculated at every timestep Ikoma et al. (2000) f atm =f MMSN Gas accretion rate: KH contraction M disk depletion gas supply by disk accretion τ sup M 3πΣgν years Ikoma et al.(2000) gap formation viscous: M = vis aω 40ν M * thermal: r H = 1.5h different from Christoph s M /τ KH KH contraction M c,hydro M /τ sup gap formation M vis planet mass M th

14 Gas Accretion onto a Planet slower gas accretion (new opacity table) faster gas accretion (lower dust-to-gas ratio of gas envelope) slower case: more consistent gas giants distribution with obs. but no gas giant formation for MMSN 10 times faster nominal 10 times slower

15 τ migi = 1 C 1 τ migi,tanaka = artificial scaling factor Type I migration M 1 C 1 f g Tanaka et al. (2002) 1 C 1 ( α Σ ) M 1 a 1AU 3/ 2 M * M 1 years M * Σ gas a 2 c s v K 2 Ω K 1 [α Σ = dlog Σ g /dlogr] now working on Paardekooper s formula Migration is halted at the disk inner edge (f g =0) eccentricity trap (see later) dm dt = M +ΔM embyo + M τ planetesimal τ gas da dt = a + Δa scatt/coll τ migration de dt = Δe scatt/coll

16 Type I migration Tanaka et al. formula (C 1 =1): much fewer gas giants than obs. (although more abundant than (2008) due to allowance of core merging during migration) C 1 <~0.1 or other formula (e.g., Paardekooper s) is needed. nominal C 1 =0.1 C 1 =0.3 Tanaka et al. C 1 =1

17 Type II migration Onset: gap formation M > M vis (no significant change if M > M th is used) Disk-dominated case - Migration with disk accretion τ migii = τ diff = (2/3)a 2 /ν Planet-dominated case - large M or small disk mass angular momentum flux through a disk J = 2πrΣ g v r (advective) + 3πΣ g r 2 vω K (viscous) at r m (v r = 0), J m = 3πΣ g,m r m 2 v m Ω K,m planetary migration rate (da /dt) (1/2)MΩ K a(da /dt) = C 2 J m - C 2 (<1) = (residual J )/(total J ) nominal case : C 2 = 0.1. τ migii = 1 C 2 (1/2)MΩ K a 2 3πΣ g,m r m 2 v m Ω K,m C M f g,m M J α 10 3 Lin & Papaloizou 1 a 1AU 1/ 2 yrs dm dt = M +ΔM embyo + M τ planetesimal τ gas da dt = a + Δa scatt/coll τ migration de dt = Δe scatt/coll v r v r r m r

18 Type II migration C 2 -dependence is very weak ( regulated by τ diff for M<~10C 2 M J ) nominal C 2 =0.1 & α=10-3 C 2 =0.3 one-sided C 2 =1

19 Planet-Planet Dynamical Interactions

20 Effects of Dynamical Interaction giant impacts ejection resonant trapping disk gas

21 Resonant Trapping - before gas depletion - determine trapped or not Shiraishi & Ida (2008), (2010) distant perturbation differential type I & II migrations + r H -expansion if trapped set Δa = 5r H N-body simulation for a convoy of migrating embryos for M=M J, Δa =5r H 2:1 or 3:2 if not trapped collision (embryo) or ejection/scattering (gas giant) eccentricity trap is applied for resonantly trapped bodies Ogihara et al. (2010, ApJ)

22 weakened e-damping excitation of e orbit crossing ceased by ejection or collision giant-giant or giant-embryo mostly ejection embryo-embryo collision model orbital crossing, scattering & collision 1) evaluate t cross (Zhou & Lin 2007) & Δe (v esc /v K ) 2) choose ejection or collision 3) set up remaining bodies or mergers using conservations of energy & Laplace-Runge-Lenz vector 4) go back to 1) until t cross > t system (2010, ApJ) Orbital Crossing - after gas depletion - 3/18 t cross

23 2 giants case 3/18 Final e of a non-ejected body µ=m 1 /M * =M 2 /M * β=m i /(M 1 +M 2 ) N-body: Ford & Rasio (2008) Monte Calro:

24 M=M J, a 0 =5.0, 7.25, 9.0AU ( 3 giants case Final e, q(=a(1-e)) & a of non-ejected bodies [M 1 =M 2 =M 3 =M J ] e e a [AU] q [AU] tidal cicularization no tide N-body: Nagasawa et al. (2008) ~ a week/200runs on a PC a [AU] Monte Carlo: ~ 0.02sec/1000runs on a PC

25 2 2 a [AU] ~ 0.2M ~ 1M a [AU] x10 7 3x x10 7 6x t [yr] t [yr] Monte Carlo: (2010, ApJ) < 0.1sec/run on a PC N-body : Kokubo et al. (2006) ~ a few days/run on a PC

26 final largest bodies 20 runs each M [M ] N-body Kokubo et al. (2006) 10xMMSN MMSN Monte Carlo 0.1xMMSN eccentricity semimajor axis [AU] (2010, ApJ)

27 Theory Theory Observation Observation

28 Remaining Uncertainties Disk model MRI dead zones, ice line initial radial distribution of planetesimals inner edge Type I migration effect of entropy gradient transition to type II (non-linear regime of type I) Gas accretion onto a core opacity of gas envelope truncation due to gap opening Jupiter/Saturn-like systems accretion of an outer core under the effect of an inner gas giant type II migration of two giants in a co-gap Secular perturbations (secular resonance, Kozai,...)

29

30 Resonant Trapping embryo-embryo 1) equilibrium Δa (orbital separation between trapped bodies): distant perturbation vs. differential type I migration 2) Δa < 3.5r H collision Δa > 3.5r H * trapped at 5r H, considering migrating convoy * migration is halted at disk inner edge (eccentricity trap; Ogihara, Duncan, Ida (2010))

31 Resonant Trapping giant-embryo distant perturbation vs. (differential type I & II migrations + r H expansion of a giant) Shiraishi & Ida (2008) * penetration ejection or close scattering * trapping at 5r H (for M=M J, Δa =5r H 2:1 or 3:2)

32 How do we model each process of planet formation? How do results depend on the model parameters? Planetary Population Synthesis: The Predictive Power of Planet Formation Theory, Ringberg, Nov 29, 2010