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Formation of Terrestrial Planets from Planetesimals

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Formation of Terrestrial Planets from Planetesimals Formation of terrestrial planets from planetesimals Yuji MATSUMOTO ASIAA TIARA Summer School on Origins of the Solar System Outline - Basic dynamics of planetesimals - Runaway growth of planetesimals - Oligarchic growth of protoplanets - Giant impacts of protoplanets - Recent studies Minimum Mass Solar Nebula Model ice line https://www.rikanenpyo.jp/top/tokusyuu/toku2/index.html rocky icy planetesimals planetesimals rocky icy protoplanets protoplanets rocky planets gas giants Ice giants Minimum Mass Solar Nebula Model ice line Alma Partnership+’15 rocky icy planetesimals planetesimals Disk & planetesimals ・A planetary system forms from a protoplanetary disk. ・Planets/coresrocky are formed by accretion of planetesimals. icy protoplanets protoplanets- parent bodies of asteroids and comets - enough large to interact by their gravity —> building block of planet accretion (Safronovrocky planets 1969, Hayashigas 1981) giants Ice giants Minimum Mass Solar Nebula Model ice line How planetesimals grow up? rocky icy planetesimals planetesimals rocky icy protoplanets protoplanets rocky planets gas giants Ice giants Growth of planetesimals 1. Growth Mode 1 dM ∝ Mp M dt d M1 M1 1 dM1 1 dM2 = − (M1 > M2) dt ( M2 ) M2 ( M1 dt M2 dt ) orderly growth runaway growth d M d M 1 < 0 when p<0 1 > 0 when p>0 dt ( M2 ) dt ( M2 ) Growth of planetesimals 2. Growth Rate 1 dM ∝ Mp M dt dM v ≃ nσvrelm rel dt n: number density of bodies Target: σ: collisional cross-section M, R Field bodies: n, m vrel: relative velocity Growth of planetesimals 2. Growth Rate 1 dM ∝ Mp M dt dM v ≃ nσvrelm rel dt n: number density of bodies Target: σ: collisional cross-section M, R Field bodies: n, m vrel: relative velocity 1/2 v2 gravitational focusing R = R 1 + esc , GF 2 ( vrel ) RGF (vesc = 2GM/R) R v2 σ = πR2 = πR2 1 + esc . GF 2 ( vrel ) Growth of planetesimals 2. Growth Rate 1 dM ∝ Mp M dt dM v ≃ nσvrelm rel dt n: number density of bodies Target: σ: collisional cross-section M, R Field bodies: n, m vrel: relative velocity ρd Σd n = = , Σd: surface density of solid in disk m mhd hd: scale height (or vertical distribution) h v d ∼ rel , a vK vrel vrel −1 —> n ∝ vrel relative velocity ~ random velocity Growth of planetesimals 2. Growth Rate 1 dM ∝ Mp M dt dM v ≃ nσvrelm rel dt n: number density of bodies Target: σ: collisional cross-section M, R Field bodies: n, m vrel: relative velocity —> random velocity (vran) dM v2 ∝ R2 1 + esc 2 dt ( vran ) 1 dM v2 ⇔ ∝ M−1/3 1 + esc 2 M dt ( vran ) Growth of planetesimals 2. Growth Rate 1 dM 1 dM v2 p ⇔ ∝ M−1/3 1 + esc ∝ M 2 M dt M dt ( vran ) v2 1 - esc < 1, p = − < 0 : orderly growth 2 vran 3 v2 1 dM v2 - esc > 1, ∝ M−1/3 esc ∝ M1/3 v−2 2 2 ran vran M dt vran The growth mode is determined by vran! v2 Θ = esc : Safronov parameter (Safronov 1969 or 1972) 2 vran Runaway growth Runaway growth occurs!! N-body simulation, Kokubo&Ida(2000) Runaway growth Eccentricities of planetesimals increase. e determines vran. - at pericenter, v ≃ vK(1 + e) - at apocenter, v ≃ vK(1 − e) typically, vran ≃ evK (see Lissauer&Stewart1993) what determines e? Runaway growth occurs!! N-body simulation, Kokubo&Ida(2000) Viscous stirring 1. fundamental process: two-body scattering v v change during close-scattering θ b ← Kepler problem! m M 2b G(M + m) tan θ = , b = (90 deg scattering) 2 90 2 (b/b90) − 1 v m m b/b Δv = v sin θ = 2 90 v ⊥ 2 M + m M + m 1 + (b/b90) m m 1 Δv = v(1 − cos θ) = − 2 v ∥ 2 M + m M + m 1 + (b/b90) Viscous stirring 2. N-body scattering: summation of two-body’s the amplitude of v distribution: ⟨Δv2⟩1/2 bmax 2 2 2 2 G nm ln Λ ⟨Δv ⟩ = Δv 2πnvbdb ∼ , Λ = bmax/b90 ∫0 v encounter in unit time viscous stirring (e.g., Ida 1990) v does not depend on M ! 2 2 2 e is determined by planetesimals (not larger one) ⟨Δv ⟩ ∼ e vK, 1 dM ∝ M1/3 v−2 ∝ M1/3, p>0, runaway growth occurs. M dt ran Runaway growth N-body simulation (3000 bodies) M get larger! Runaway body = protoplanet mean mass m is not changed much Kokubo&Ida(2000) How long does this stage last? Oligarchic growth - Protoplanets are formed locally - The inner protoplanets are quickly formed Kokubo&Ida(2002; 2012) Growth of protoplanets Runaway growth v2 v3 T = ∼ VS,m−m ⟨v2⟩ G2nm2 ln Λ v2 v3 T = ∼ VS,M−m 2 2 2 ⟨v ⟩ G nMM ln Λ M Growth of protoplanets Runaway growth v2 v3 T = ∼ VS,m−m ⟨v2⟩ G2nm2 ln Λ 2 2 nm > nMM M ≳ 100m (Ida & Makino 1993) 2 2 nm < nMM v2 v3 T = ∼ VS,M−m 2 2 2 ⟨v ⟩ G nMM ln Λ 1 n = M 2πaΔa × 2ai M Growth of protoplanets Runaway growth e of planetesimals is given by planetesimals, vram does not depends on M. 1 dM ∝ M1/3 v−2 ∝ M1/3 M dt ran Oligarchic growth e of planetesimals is given by a protoplanet, vram depends on M ! M Eccentricity in oligarchic growth Hill’s framework: framework for 3 body problem (e.g., Nakazawa & Ida 1988) m2 M* ≫ m1, m2, ej, ij ≪ 1, m1 |a1 - a2| ≪ a, M* - These bodies interact each other - The relative motion is smaller than that of the barycenter Hill radius: typical size of the m1-m2 interaction 1/3 M + m MaM + mam rH rH = , h = , ( 3M* ) ( M + m ) a Eccentricity in oligarchic growth Hill’s framework: framework for 3 body problem (e.g., Nakazawa & Ida 1988) m2 M* ≫ m1, m2, ej, ij ≪ 1, m1 |a1 - a2| ≪ a, M* The Jacobi integral (CJ): the conservable in restricted 3 body problem 2 2 2 1 e i 3 aj − ai 3 9 CJ = + − − + , 2 [( h ) ( h ) ] 8 ( rH ) r/rH 2 - planetesimals move according to its CJ - e, i~h - e increases as the protoplanet becomes larger Eccentricity in oligarchic growth 1023g 2 1/2 ⟨e ⟩ /hM M 2 1/2 ⟨i ⟩ /hM ⟨m⟩ e ~ h ~ (M/M )1/3 Kokubo&Ida(2000) M * 1 dM ∝ M1/3 v−2 ∝ M−1/3 M dt ran Oligarchic growth - protoplanet-protoplanet: 1 dM ∝ M1/3 v−2 ∝ M−1/3, M dt ran orderly growth, similar size protoplanets are formed! Kokubo&Ida(2000) Oligarchic growth - protoplanet-protoplanet: 1 dM ∝ M1/3 v−2 ∝ M−1/3, M dt ran orderly growth. similar size protoplanets are formed! - protoplanet-planetesimals: 1 dm ∝ m1/3 v−2 ∝ m1/3, m dt ran runaway growth continues. planetesimals keep small. Kokubo&Ida(2000) Oligarchic growth protoplanets - small e - similar orbital separation (b) (>~ 5rH) 5rH 5rH Kokubo&Ida(2000) Eccentricity of protoplanets - ⟨Δv2⟩1/2, the evolution of v amplitude: viscous stirring - ⟨Δv⟩, v change of each body: dynamical friction Eccentricity of protoplanets - ⟨Δv2⟩1/2, the evolution of v amplitude: viscous stirring - ⟨Δv⟩, v change of each body: dynamical friction two-body scattering 4πG2n M(m + M)ln Λ ⟨Δv⟩ ∼ − M v2 : massive body loses v largely ⟨Δv⟩ ∝ M(m + M) —> energy equipartition m M m v2 ∼ v2 ⇔ e ∼ e 2 m 2 M M M m Dynamical friction Kokubo&Ida(2012) e2 M T = ∼ T , DF de2/dt m relax Trelax : two-body relaxation time Due to (M/m) times two-body Even when eM > em, eM is damped. scattering, eM is damped. Oligarchic growth protoplanets - small e - similar orbital separation (b) (>~ 5rH) How b is determined? 5rH 5rH Kokubo&Ida(2000) Orbital repulsion Initial b=2rH Protoplanet-protoplanet interaction. Scattering makes e and b larger. e/h 2 2 2 1 e i 3 b CJ = + − + ⋯ 2 [( h ) ( h ) ] 8 ( rH ) Protoplanet-planetesimal interaction. eM is damped by dynamical friction. b spreads! b becomes larger than 5rH. a/rH Kokubo&Ida(1995) Collisions between protoplanets & planetesimals Collision: two bodies are replaced by the barycenter body m2, x2, v2 mG = m1 + m2, m1x1 + m2x2 xG = , m1 + m2 mG, xG, vG m1, x1, v1 m1v1 + m2v2 vG = , m1 + m2 Collisions between protoplanets & planetesimals Collision: two bodies are replaced by the barycenter body m2, x2, v2 mG = m1 + m2, mG, xG, vG m1, x1, v1 eccentricity vector: e ⃗ = (e cos(θ − ϖ), e sin(θ − ϖ)) ϖ : the longitude of pericenter conservative vector in Kepler problem Collisions between protoplanets & planetesimals Collision: two bodies are replaced by the barycenter body m2, x2, v2 mG = m1 + m2, mG, xG, vG m1, x1, v1 eccentricity vector: e ⃗ = (e cos(θ − ϖ), e sin(θ − ϖ)) Under the Hill’s approximations, m2 m2 m m e2 = 1 e2 + 2 e2 + 2 1 2 e e cos(ϖ − ϖ ), G 2 1 2 2 2 1 2 1 2 mG mG mG 1/2 random angle collision m2 m2 ⟨e ⟩ = 1 e2 + 2 e2 G 2 1 2 2 ( mG mG ) (Ohtsuki 1992) Collisions between protoplanets & planetesimals Collision: two bodies are replaced by the barycenter body m2, x2, v2 mG = m1 + m2, mG, xG, vG m1, x1, v1 1/2 m2 m2 ⟨e ⟩ = 1 e2 + 2 e2 G 2 1 2 2 ( mG mG ) planetesimal - planetesimal: m1~m2, e1~e2~e, ⟨eG⟩ ∼ e/ 2 2 2 protoplanet - planetesimal: m1e1 ~ m2e2 , ⟨eG⟩ ∼ e1 Oligarchic growth - Almost all planetesimals are accreted - Inner protoplanets grow up faster Kokubo&Ida(2002) Isolation mass of protoplanets Isolation mass: Miso ≃ 2πabΣd the mass of all local dusts b ∼ 10rH : orbital repulsion −3/2 a Σ = 10 g cm−2 : 1.4×MMSN d ( 1 au ) 3/2 3/4 −1/2 b a M* Miso ≃ 0.16 M⊕ ( 10rH ) ( 1 au ) ( M⊙ ) : about Mars size Isolation mass of protoplanets 3/2 3/4 −1/2 Isolation mass: b a M* Miso ≃ 0.16 M⊕ ( 10rH ) ( 1 au ) ( M⊙ ) M Timescale: t = grow dM/dt 2/15 27/10 1/3 mpl a M = 3.2 × 105 yr 23 ( 10 g ) ( 1 au ) ( 0.16M⊕ ) dM Σ v2 ≃ d πR2 esc v m ≃ nσvrelm 2 ram dt ( mvram /ΩK ) ( vram ) 2GMR = CπΣ d 2 e avK Planetary cores disk mass semimajor axis Kokubo&Ida(2002;2012) Minimum Mass Solar Nebula Model ice line rocky icy planetesimals planetesimals rocky icy Giant impacts protoplanets protoplanets rocky planets gas giants Ice giants The orbital stability of protoplanets ˜ ˜ brH brH b˜rH b˜rH b˜ = 6 M ≃ 0.03M⊕ stability time: tinst = t(rij < rH) tinst = 5.4 × 104 yr apocenter semimajor axis pericenter unstable The orbital stability of protoplanets ˜ ˜ brH brH b˜rH b˜rH - log tinst ∝ bini/rH (Chambers+1996) 106 - large 2 1/2 short ⟨eini⟩ , tinst 105 (e.g., Yoshinaga+1999, Zhou+2007) 104 - planets in resonances (Matsumoto+2012) 103 - e-damping (Iwasaki+2001; 2002) ○ 2 1/2 ⟨eini⟩ = 0 102 × 2 1/2 ⟨eini⟩ = 4h 10 2 4 6 8 10 Kokubo&Ida(2012) The orbital stability of protoplanets ˜ ˜ brH brH b˜rH b˜rH - e-damping (Iwasaki+2001; 2002) 8 te : e-damping timescale 7 upper limit Instability occurs after te > tinst.
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