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

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

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

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

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

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 = 10-3 10 6 y 105 y 10 7 y Σ gas 1/r 1AU r 10AU r [AU]

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 0.1 1 10

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 0.1 1 10 Christoph: slightly massive

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]

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

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

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

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

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

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

τ migi = 1 C 1 τ migi,tanaka = artificial scaling factor Type I migration 5 10 4 1 M 1 C 1 f g Tanaka et al. (2002) 1 C 1 (3.2 +1.1α 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

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

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 5 10 5 C 2 0.1 1-1 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

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

Planet-Planet Dynamical Interactions

Effects of Dynamical Interaction giant impacts ejection resonant trapping disk gas

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)

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

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:

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

2 2 a [AU] 1.5 1 ~ 0.2M ~ 1M a [AU] 1.5 1 0.5 0.5 0 0 10 7 2x10 7 3x10 7 0 2x10 7 6x10 7 10 8 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

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)

Theory Theory Observation Observation

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,...)

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))

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)