原始惑星系円盤の 自己重力的分裂の条件 高橋実道(京都大学) 塚本裕介、犬塚修一郎(名古屋大学) 円盤の分裂による天体形成 ○系外惑星の直接撮像 中心星からはなれた位置に 巨大惑星(M~10MJ, >20AU) 標準モデルで説明困難 ○原始惑星系円盤形成の数値計算 初期に重い円盤が形成 原始惑星系円盤の重力不安定性に よる分裂 巨大惑星形成? 連星系、褐色矮星形成にも関連。 原始惑星系円盤が分裂する条件は? (Carson et. al. 2013) 円盤の自己重力不安定性 自己重力不安定性 Toomre s Q parameter Q<1 不安定 1.Q.2 :音速 :エピサイクル振動数 :面密度 (local instability, Toomre 1969) 渦状腕が形成 (global instability Takahara 1976, 1978, Iye 1978) ● Shock 加熱 ● 重力トルクによる角運動量輸送 安定化 ガス降着 渦状腕による安定化に打ち勝ち分裂する条件は? 円盤の分裂条件 先行研究:冷却率に注目 Cooling time tcool = 数値計算 (Gammie 2001, Rice et al. 2005, 2014, Meru and Bate2012 等) Kepler time として冷却過程をモデル化 dE E 1 = , dt tcool 分裂するための冷却率の条件 < crit ⇠ 30 で分裂 (Meru and Bate 2012) 他の数値計算の結果と矛盾 例)Tsukamoto et al. 2015 : β < βcrit・分裂しない Machida et al. 2010 : 断熱・分裂する 冷却率に対する分裂条件では不十分 現実的な円盤分裂の条件を明らかにするために 数値計算を行い、分裂過程を解析 urface density, v vis isvelocity, EE is is internal energy perper unit area, P isP the vertically surface density, velocity, internal energy unit area, is the e thin disk approximation. We assume an ideal gas equationvertically of state, ressure, Φ is the gravitational potential, ΛC is the cooling rate per unit area. We pressure, Φ is the We gravitational potential, Λin the cooling rate per unit area. We C isthis pecific heat. adopt γ = 5/3 calculation. The temperature from the thin disk approximation. We assume an ideal gas equation of state, Φ from the thin disk approximation. We assume an ideal gas equation of state, (1.4) P = (γ − 1)E, Setup (FARGO) = (γ − 1)E, (1.4) 基礎方程式 P Pµm P = (γ − 1)E, (1.4) Open boundary H= 5/3 in this calculation. of specific heat. We adopt γ The temperature T = , (1.5) @⌃ he ratio of specific heat. We adopt γΣ = 5/3 in this calculation. The temperature k 連続の式 B + r · (⌃v) = 0 the ratio of specific heat. We adopt γ = 5/3 in this calculation. The rtemperature = 20, 1000AU @t ✓ kB isµm ◆H HPBoltzmann olecular weight, the constant and mH is (1.5) hydrogen µm P, @v T = (1.5) H ,PrP k=Bkµm Σ 運動方程式 ⌃ + vT· T= rv = ⌃r Σ ,is modeled asM (1.5) B Λ ⇤ = 0.5M = 2.34. The cooling rate follows (Hubeny, 1990; @t C kB Σ nhemolecular weight, kB kis the constant mhydrogen mean molecular weight, theBoltzmann Boltzmann constant andand mH is H is hydrogen B is the mean molecular weight, kB is the Boltzmann constantM and mH=is0.34 hydrogen , 0.38 04); 0.24M 0.28M P = ( 1)E 状態方程式 disk we adopt µ = 2.34. The cooling rate Λ is modeled as follows (Hubeny, 1990; pt µ = 2.34. The cooling rate ΛCC is modeled as follows (Hubeny, 1990; ✓ ◆ eoodman, we adopt µ = 2.34. The cooling rate Λ C is modeled as follows (Hubeny, 1990; 2004); @E r , 2004); 8 τ Goodman, 2004); 4 = Pr · v エネルギー +4 r ·T (Ev) Text = 150[K] ΛC = σ(T − ) (1.6) 8 τ @t 1 2 4 ext 4 1 2 1[AU] − Text ) τ1 2 + τ1√τ τ + (1.6) 方程式 3ΛC8 = 3 σ(T 84 2 44 44 τ + √ τ3 + 3 = σ(T − )Text 4) 1 3 2 ΛC = ΛCσ(T − T 3 1 ext 2+ 1 1 √ 2 3 τ τ + √ τ3 + 2 3 3 τ + 4 4 3 3 3/7 (1.6) (1.6) (Chiang and Goldreich 1997) = 5/3 the Stefan-Boltzmann constant, T is the equilibrium temperature due to the due to the Boltzmann constant, Textextis the equilibrium temperature 放射冷却・外部加熱 the the Stefan-Boltzmann constant, Text isoptical the equilibrium temperature due to the rom central star, and τ = κ Σ is the depth of the disk. The Rosseland R an-Boltzmann constant, Tis is (Hubeny the equilibrium temperature due to the extthe 1990) tral star, and τ = κ Σ optical depth of the disk. The Rosseland thegiven central τ = κR Σ is the optical depth of the disk. The Rosseland✓ 初期条件 ◆ yfrom κR is by star, and R Optical depth central star, and τ = κR Σ is the optical depth of the disk. The Rosseland r ity κ is given by R en by ! "2 12/7 ⌃/r exp T given by 2 −1 ! " κR =!κ10 [cm g ] (1.7) 2 r " disk T 10[K] 2 2 −1 "2 [cm κR =!κ10T g ] (1.7) −1 10[K] [cm22 g−1 r = 250[AU] T disk κ = κ ] =0.05 R 10 ng approximates result of Semenov et g al. (2003) in T ! 200 K, and almost (1.7) (1.7) κRto=theκ10 [cm ] 10[K] 10[K] (Semenov et assume al. ing approximates to the result of Semenov et 2003) al. (2003) TQ !⇠ 200 and almost ture of the disk is smaller than 200 K. We Text asinfollows (cf. 2K,Chiang ( Q&⌘ cs ⌦/(⇡G⌃) ) ature than 200 K. We assume T$ext as follows (cf. Chiang & 997); of the disk is smaller # mates to to thethe result etal. al.(2003) (2003) T 200 ! 200 K, and almost ! " oximates resultofofSemenov Semenov et in in T ! K, and almost 結果:分裂しない場合T ∼ Text c (T )Ω s ext epi ¯ Q == 0.34 Mdisk 0.28M πGΣini 大局的不安定による ¯!1 Q 渦状腕形成 ¯"2 Q Q ! 0.6 分裂せずに τ 準定常な構造 β 先行研究の分裂条件 Qcrit ∼ 0.6 1 tcool = −1 < 30Ω と比較 Q ! 0. and the scale height c /ΩT evaluated at the center of the spiral. The width of th ∼T s ext 渦状腕の冷却時間 τ β = tcool Ω < 30Ω −1 Qcrit ∼ 0.6 c (T )Ω ¯= Q πGΣ ¯!1 Q β τ Q ! 0.6 ¯"2 Q comparable to the scale height. The right panel of the second row shows the lin the spiral arm. The line mass of the spiral arm is approximately given by c2s /G. s ext epi 2 t=6285 line mass is smaller than yr the critical line mass 2cs /G, Optical the spiral isdepth supported by th 30 against the self-gravity in the directionini perpendicular to the spiral arm. 70 The left 65 25 arm and the velocity along the s the third row shows the pitch angle of the spiral 60 subtracted the azimuthal averaged rotation velocity. The right panel of the 20 55third r s 50 The the distribution of the Toomre’s Q parameter 15 at the center of the spiral arm. 45 value of Q is about 0.5. The left panel of the 10 bottom row shows the normalized n 40 35 norm time βnet , the normalized cooling time βcooling and the optical depth τ . the 5 30 cooling time βnet and the normalized cooling time βcooling are defined as follows; ¯"2 Q Q ! 0.6 β 0 βnet 25 0 20 40 60 80 100 120 E = Ω, s [AU] τ Normalized Cooling Time β βcooling = EΩ 3 ΛC ! 1 2 τ 4 + √1 τ 3 8σT 4 τ + 2 3 " . The right panel of the bottom row shows the epicycle frequency, angular frequency # Kepler frequency estimated star mass GM∗ /r3 . < critfrom ⇠ the 30 central を満たすが分裂しない。 結果:分裂する場合 Mdisk = 0.38 0.28M 大局的不安定による 渦状腕形成 渦状腕が分裂 分裂しない場合との違いは? 渦状腕中のQに注目 渦状腕のQ 分裂しない Q Mdisk =0.34 0.28M 1.1 1.05 1 0.95 0.9 0.85 0.8 0.75 0 Mdisk =0.38 0.28M Q 分裂する 1.6 1.5 1.4 1.3 1.2 1.1 1 0.9 0.8 0.7 0.6 0.5 0 20 50 40 60 80 100 120 s [AU] 100 150 s [AU] 200 250 T ∼ Text 渦状腕のQ c (T )Ω s ext epi ¯ 分裂しない πGΣini 1.1 ¯!1 Q Q Mdisk =0.34 0.28M Q= ¯"2 Q 渦状腕の分裂条件 Q ! 0.6 分裂する Q Mdisk =0.38 0.28M Q=0.6 1.05 1 0.95 0.9 0.85 0.8 0.75 0 1.6 1.5 1.4 1.3 1.2 1.1 1 0.9 0.8 0.7 0.6 0.5 0 20 50 40 60 80 100 120 s [AU] 100 150 s [AU] 200 250 Q ! 0.6 2 2 Ω th plitude ofπGM surface density of the spiral is small. Thus we 4R define ”渦状腕”の自己重力不安定性 c region wherec the surface density is 0.85 times peak surface densit L ω ˜ = k˜2 − 2 2 s 2 ˜ ˜ ˜ ˜ ˜ [K0 (k/2)L ( k/2) + K ( k/2)L ( k/2)] k + −1 1 0 τ epi 2 s [ /(L/cs )]2 (振動数)2 回転する細いリングとする arm is larger 渦状腕 than 20∼AU since the temperature is high and the s e two parameters 2 β (Line mass ne mass of the分散関係 spiral arm is comparable to c GML = s /G and ) the Q value f≡ 2 , s larger30than 0.75. Thus the resultcsthat this spiral does not fra Q ∼ 0.6 crit 25 terion for the fragmentation Q ! 0.6. Since the temperature of th 2RΩepi 20 Q=1 l ≡ . 最大成長波長 2L cs ∼ βnet ! 3 in satisfied in t rate is also large. As the result, βcooling 15 10 cansuggests rewrite normalized dispersion cooling relation安定 also that the normalized time itself is not related Q=0.6 5 gmentation. 0 2 2 2 2 ˜ ˜ ˜ ˜ ˜ ˜ ω ˜ = k − πf [K0 (k/2)LQ=0.4 −1 (k/2) + K1 (k/2)L0 (k/2)]k + l . -5 不安定 0 1 2 3 4 5 6 7 kLthe surface density of the ring as Gaussian sume that the structure of Inner radius 波数(L : リングの幅) is 0.5R. Then Toomre’s Q parameter is simulations performed in this chapter, thel surface density of the in cs Ω Q= =√ . after the calculations start. πGΣ Thus results 8πf of our simulations [ /(L/cs )]2 (振動数)2 s both spiral arms satisfy condition for the cooling time suggested by Mes arms. Therefore, the the axisymmetric mode cannot grow in the Q ! 0.6 2 2 4R Ωspir plitude of surface density of the spiral is small. Thus we define th πGM 2). This result suggests that the condition for the Q parameter in the L epi 2non-axisymmetric 2 2 ˜ ˜ ˜ ˜ ˜ mode may grow in the spiral arms. Sinc ω ˜ = k˜ −”渦状腕”の自己重力不安定性 [K ( k/2)L ( k/2) + K ( k/2)L ( k/2)] k + 0 −1 1 0 2 eregion essential thanc2sthe for the cooling c where the condition surface density is 0.85time. times peak surface densit s τ arms are small (! 0.2), we adopt the conditions for the instab ∼AU 回転する細いリングとする arm is larger 渦状腕 than 20 since the temperature is high and the s Chapter 1 Condition for the fragmentation of d e two parameters e critical Q value is roughly estimated about ! 0.6 and the mo 2 β (Line mass ne of the分散関係 spiralon arm is comparable GML to = cs /G and ) the Q value .4 mass Dependence the opacity f≡ , analogy with filament ( three times the width of the ring by 2 csthat this spiral does not fra scooling largerrate than 0.75. Thus the result 30 offormed the disks depends on arm the opacity. Since the most of the reg agments are in the spiral of the model 1, but are not fo Q ∼ 0.6 crit Appendix the conditi 25 A). The spiral arm in model 1 satisfies −1 terion for the fragmentation Q ! 0.6. Since the temperature of th 2RΩ epi s are optically thick, the cooling rate is proportional to κ . Thus the radiati 20 Q=1 l ≡ . 2L 最大成長波長 5 does not satisfy the condition. We test the condition by c rate is also large. As the result, β ∼ β ! 3 in satisfied in t fficient when κ is small. The result of the model 1, 8, 9, 10, and 12 sugges s cooling net 15 est unstable wavelength of the gravitational instability derived from f can them fragment andcase the others do not安定 fragment. Then we fin mentation in the where thecooling opacity is small andisthe radiation 10occurs rewrite normalized dispersion relation also suggests that the normalized time itself not related 2 1/2 c linear stability analysis 2π(c /Ω)Q/(1 + (1 − Q ) ) is longer tha s Q=0.6 5other parameters are not changed. Figure 1.8 shows the structu ent if the agmentation of the spiral arms is valid for the all spiral arms gmentation. arms. Therefore, the axisymmetric mode cannot grow in the a 0 2 2 2 spiral 2 ˜ ˜ ˜ ˜ ˜ ˜ l arm in ω dose not fragment. In the spiral arm, Q > 0.6 is ˜the=model k −8,πfwhich [K0 (k/2)L ( k/2) + K ( k/2)L ( k/2)] k + l . −1 1 0 Q=0.4 -5 nd Figure 1.7, the normalized cooling times β is smaller 不安定 cool d, non-axisymmetric mode may grow in the spiral arms. Since the s the result that spiral3 arm4 is not is consistent with our criterio 0 1this 2 5 fragment 6 7 自己重力不安定性が成長する条件 ms the condition for thethe cooling time Mo arms are small (! 0.2), wethe adopt conditions for instability kL sume that the structure surface density of 1the ring as Gaussian the satisfy opacity is larger thanof the opacity of the model bysuggested athe degree ofby magni Inner radius normalized cooling time βcoolQ∼parameter βnet " 10.the In the case theinopacity is he critical Q value is roughly estimated about ! 0.6where and the most 渦状腕が長さ suggests that the condition for2L Q parameter the uns spi で is 0.5R. Then Toomre’s is l arms cannot shrink enoughoftothe satisfy the Qanalogy < 0.6 in with the region a few(Inutsu times la ut three times the width ring by filament the condition for the cooling time. simulations performed in this chapter, thel surface density of the in cΩ 数値計算の結果と一致 s is important for the fragmentation of width. In this sense the efficient cooling = in model = √1 satisfies . e Appendix A). The spiral Q arm the condition, an calculations start. Thus results of our simulations 円盤の分裂条件 渦状腕の自己重力不安定条件 8πfworks. , after and ourthe criterion is consistent withπGΣ the previous Opacityに対する依存性 Chapter 4 Condition for the fragmentation of di 円盤質量 大 50 Adiabatic 45 Σ100 [g cm-2] 40 35 Fragmentation 分裂 30 25 20 15 円盤質量 小 10 0.0001 冷却しやすい No fragmentation 分裂しない 0.001 0.01 0.1 κ10 [cm2 g-1] 1 冷却しにくい e 4.13: Classification of the simulation results on the κ10 −Σ100 plane. Filled circles Opacityに対する依存性 Chapter 4 Condition for the fragmentation of di 円盤質量 大 50 Adiabatic 45 Σ100 [g cm-2] 40 35 30 25 20 15 円盤質量 小 Fragmentation 分裂 10 0.0001 冷却しやすい 冷却率しにくいほど 分裂に必要な質量 大 No fragmentation 分裂しない 0.001 0.01 0.1 κ10 [cm2 g-1] 1 冷却しにくい e 4.13: Classification of the simulation results on the κ10 −Σ100 plane. Filled circles Opacityに対する依存性 Chapter 4 Condition for the fragmentation of di 円盤質量 大 50 Adiabatic 45 Σ100 [g cm-2] 40 35 30 25 20 15 円盤質量 小 Fragmentation 分裂 T ∼ Text c (T )Ω s ext epi ¯ Q= πGΣini ¯!1 Q 10 0.0001 冷却率しにくいほど 分裂に必要な質量 大 No fragmentation 分裂しない ¯ " 20.01 0.001 0.1 1 Q 断熱まで含めた広いパラメータ領域で κ10 [cm2 g-1] 冷却しにくい 冷却しやすい 分裂条件は Q ! 0.6 で与えられる。 e 4.13: Classification of the simulation results on the κ10 −Σ100 plane. Filled circles まとめ • 原始惑星系円盤の自己重力不安定性による分裂は、連星系、 褐色矮星、ガス惑星形成メカニズムの候補として重要。 • これまで、円盤が分裂する条件として円盤の冷却率が重要だ と考えられてきが、この条件は他の数値計算結果と矛盾す る。 • 自己重力円盤の数値計算を行い円盤の分裂過程を詳細に解析 した。円盤が分裂する条件は円盤に形成された渦状腕中で Q<0.6で与られることがわかった。この結果はリングの線形解 析の結果と一致する。 • Opacity が大きく冷却しにくい円盤ほど分裂に必要な質量は大 きい。断熱まで含めた広いパラメータで分裂条件はQ<0.6で与 られる。
© Copyright 2024