大質量星の重力崩壊と重力波 東大 総合文化 関口 雄一郎 & 柴田 大 §1 Introduction 回転星コアの重力崩壊 – Long GRBs の中心動力源の候補(BH formation) Cf, collapsar model (Woosley 1993) – 重力波源の候補 重力崩壊後のコア(NS形成) – 初期角運動量が小、微分回転の度合い小 軸対称崩壊 → 2D計算 バウンス、原始中性子星の振動に伴う重力波 – 初期角運動量が大、微分回転の度合い大 非軸対称変形の可能性 → 3D計算の必要性 より強い重力波源 §2 Numerical Implementation アインシュタイン方程式 – ADMによる3+1分解 – 中村・柴田による最定式化(Shibata and Nakamura 1995) – Cartoon method (Alcubierre et al. 2001) 相対論的数値流体力学 – High resolution shock capturing scheme (e.g. Font 2003) 座標条件 – Approximate maximal slicing condition (Shibata 1999) – Dynamical gauge (shift) condition (Shibata 2003) Parametric Equations of state – 系統的なパラメータスタディが可能 – (Janka et al. 1993; Zweger and Muller 1997; Dimmelmeir et al. 2002) §3 初期条件と状態方程式 初期条件 – 大質量回転星の鉄コア – 4 / 3 ポリトロープの回転平衡形状でモデル化 – 1.0 1010 g/cm3 c M / M 2.0 3.0 q cJ 0 1.1 2 GM – コアの回転則 (Komatsu et al. 1989) ut u d2 (0 ) 微分回転パラメータ A d / e , 1.0, 0.5, 0.1 A , rigid rotation e : radius at equator §3 初期条件と状態方程式 Parametric equations of state P Ppoly Pth Ppoly K1 ( nuc ) K2 2 ( nuc ) 4 / 3 光分解反応、電子捕獲反応により不安定化 1 2 2.0 核力によって状態方程式が硬くなる Pth (th 1) th 衝撃波加熱の効果 EOSのパラメータ : (1, 2 , nuc , th ) – 球対称ポリトロープの最大質量が M max NS 1.6M ほぼ同一になるように設定 – 簡単のため th 1 とおいた §3 初期条件と状態方程式 EOSのパラメータ(まとめ) (1, 2 , nuc / 14 ), th 1 14 1014 g/cm3 EOS-a : (1.32, 2.25, 2.0) EOS-b : (1.30,2.5,2.0) EOS-d : (1.28,2.75,2.0) M max NS 1.6M §4 軸対称崩壊 同一の初期条件でも、状態方程式の違いにより崩 壊ダイナミクスは大きく異なる 重力波波形、スペクトルに反映 EOS-a : (1, 2 , nuc / 14 ) (1.32, 2.25, 2.0), th 1.32 EOS-b : (1, 2 , nuc / 14 ) (1.3, 2.5, 2.0), th 1.3 EOS-d : (1, 2 , nuc / 14 ) (1.28, 2.75, 2.0), th 1.28 §4.1 軸対称崩壊 - EOS-a ダイナミクス- EOS-a M 2.5M q 1.0 rig. rot. §4.1 軸対称崩壊 - EOS-a - EOS-a M 2.5M q 1.0 rig. rot. §4.1 軸対称崩壊 - EOS-a BH formation - EOS-a M 2.7M q 1.0 rig. rot. §4.1 軸対称崩壊 - EOS-a 重力波波形- EOS-a M 2.5M q 1.0 rig. rot. BH形成 h 3 10 20 I zz I xx 10 kpc 2 sin 1000 cm r §4.1 軸対称崩壊 - EOS-a スペクトル- EOS-a M 2.5M q 1.0 rig. rot. 1/ 2 dE / df heff 5 1020 46 10 erg/Hz 10kpc 20 h ~ 5 10 at r 10kpc eff r §4.2 軸対称崩壊 - EOS-b ダイナミクス- EOS-b M 2.5M q 1.0 rig. rot. §4.2 軸対称崩壊 - EOS-b BH formation - EOS-b M 2.7M q 1.0 rig. rot. §4.2 軸対称崩壊 - EOS-b 重力波波形- EOS-b M 2.5M q 1.0 rig. rot. Note: M core M Ch では h 3 10 20 I zz I xx 10 kpc 2 sin 1000 cm r hpeak 600 §4.2 軸対称崩壊 - EOS-b スペクトル- EOS-b M 2.5M q 1.0 rig. rot. 1/ 2 dE / df heff 5 1020 46 10 erg/Hz 10kpc 20 h ~ 5 10 at r 10kpc eff r §4.3 軸対称崩壊 - EOS-d ダイナミクス- EOS-d M 2.5M q 1.0 rig. rot. §4.3 軸対称崩壊 - EOS-d 重力波波形- EOS-d M 2.5M q 1.0 rig. rot. h 3 10 20 I zz I xx 10 kpc 2 sin 1000 cm r §4.3 軸対称崩壊 - EOS-d スペクトル- EOS-d M 2.5M 幾つかの振動 モード q 1.0 rig. rot. §4.4 軸対称崩壊 - EOS依存性 - 核密度に達する前のEOSが重要 核密度前に圧力大幅減 (Γ1小) – ⇒中心だけ先に暴走的に重力崩壊 – ⇒ショック形成時期の中性子星の質量が小さい – ⇒重力波の振幅が小さくなる 圧力減少幅が小さい場合 (Γ1~4/3) – ⇒ホモロガスな崩壊 – ⇒ショック形成時期の中性子星の質量が大きい – ⇒重力波の振幅大 §4.4 軸対称崩壊 - EOS依存性 - Γ2 依存性 (Γ2 小) – 原始中性子星の状態 方程式柔らか – バウンス時の overshooting 大 – 急な密度勾配が形成さ れる(インナーコアが大 きい場合、回転速い場 合) – 非対称衝撃波 – 衝撃波強 §4.5 軸対称崩壊 hSB~11hrms 10kpc 軸対称崩壊 実質的 感度 §5 非軸対称変形 セキュラー不安定性 – T/W>0.14 の場合に起こる(Cf. Shibata and Karino 04) 動的不安定性 – T/W>0.27の場合に起こる – 微分回転の度合いが大きい場合に(T/W<0.1でも)起こる (Shibata et al. 02) 動的不安定性の条件が満足されるような重力崩壊 があるか? – ⇒ 系統的研究例なし – ⇒ Shibata and YS : GR simulation 05(PRD) §5 非軸対称変形 答え:あることはあるが、限定されている – 初期に高速回転かつ微分回転の度合いが大き いことが必要不可欠 (回転軸と表面の角速度の比>100) A=0.1 – 初期に圧力減少の度合いが大きいことも必要 Γ1<~1.28 §5 非軸対称変形 -Dynamical Instability- T/W の初期値と最大値の関係 動的不安定性 成長の候補 0.27 最大値 微分回転の 初期条件 セキュラー不安定 性成長の候補 0.14 初期に剛体回転 初期値 YS and Shibata 05 §5 非軸対称変形 - Dynamical Instability - A=0.1 のモデル T /W 0.27 (T/W)_init increases 動的不安定性成 長の候補 EOS-d T / W |init 0.0127 M 2.5M A 0.1 強微分回転 モデル EOS-d T / W |init 0.0177 M 2.5M A 0.1 強微分回転 モデル §5 非軸対称変形 - Dynamical Instability - EOS-d T / W |init 0.0127 Gauge inv. M 2.5M A 0.1 Quadrupole formula h 10 21 Axi. Sym. の~10 倍以上の振幅 R , 10Mpc 18 R , 10kpc 0.31km r 3 10 0.1km r §5 非軸対称変形 - Dynamical Instability - EOS-d T / W |init 0.0177 Gauge inv. M 2.5M A 0.1 Quadrupole formula h 10 21 R , 10Mpc 18 R , 10kpc 0.31km r 3 10 0.1km r §5 非軸対称変形 - Dynamical Instability - ~ 1 kHz EOS-d T / W |init 0.0177, 0.0127 M 2.5M A 0.1 1/ 2 heff 2 10 22 dE / df 1047 erg/Hz 10Mpc 18 h ~ 3 10 at r 10kpc eff r §5 非軸対称変形 - Dynamical Instability - EOS-d T / W |init 0.0124 Gauge inv. M 1.5M A 0.1 Quadrupole formula h 10 21 R , 10Mpc 18 R , 10kpc 0.31km r 3 10 0.1km r §5 非軸対称変形 - Dynamical Instability - 非軸対称変形 により、Axi. Sym. の~10 倍以上の振幅 M 1.5M M 2.5M §6 まとめ ① 回転遅い、微分回転の程度小 – ⇒軸対称重力崩壊 – h , 2 5 1020 @10kpc, f 0.5-2kHz ② 回転速く、かつ微分回転の程度大 (稀かもしれない) – ⇒ダイナミカルに非軸対称変形 – h, 2 3 1019 @10kpc, f 1kHz ③ ある程度回転が速い; T / W 0.14 – – ⇒セキュラー不安定性? 不定性が多い これに関する詳細な研究は必要 §6 まとめ 2 非軸対称不 安定 @10kpc 1 軸対称崩壊 実質的 感度 hSB~11hrms §6 セキュラー不安定性の可能性 初期に適度に早く回転していて、かつ適度に差動回 転していれば、重力崩壊後T/W>0.14とはなりうる 成長時間:~1s以上で、T/Wが小さいほど長い – – ⇒現実的には、磁場や粘性が存在 ⇒成長する前に、成長が押さえられる?? 原始中性子星形成直後には物質が取り巻く – ⇒角運動量輸送が効いて、非軸対称性がすぐになくな る? 不定性は大きい A criterion for prompt black hole formation §7 A criterion for prompt black hole formation a : (1.32, 2.25, 2.0) b : (1.30,2.5,2.0) mass c : (1.30, 2.22,1.0) d : (1.28,2.75,2.0) ■ : BH for all EOS ☆ : BH for EOS-b (-d) × : BH for EOS-a □ : NS for all EOS rig. rot. Angular momentum §7 A criterion for prompt black hole formation a : (1.32, 2.25, 2.0) b : (1.30,2.5,2.0) c : (1.30, 2.22,1.0) d : (1.28,2.75,2.0) ■ : BH for all EOS ☆ : BH for EOS-b (-d) × : BH for EOS-a □ : NS for all EOS rig. rot. §8 Black hole formation - Dependence on EOS - EOS-a : (1, 2 , nuc / 14 ) (1.32, 2.25, 2.0), th 1.32 M 2.4M q 0.66 rig. rot. §8 Black hole formation - Dependence on EOS - EOS-a M 2.7M q 1.0 rig. rot. §8 Black hole formation - Dependence on EOS - EOS-a : (1, 2 , nuc / 14 ) (1.32, 2.25, 2.0), th 1.32 A black hole is formed directly without any distinct bounce Mass of the inner is larger than the maximum allowed NS mass density Minner core M NS max No shock propagates Pth is small BH is more liable to be formed 1 GM / c2 R §8 Black hole formation - Dependence on EOS - EOS-b M 2.4M q 0.66 rig. rot. §8 Black hole formation - Dependence on EOS - EOS-b M 2.7M q 1.0 rig. rot. §8 Black hole formation - Dependence on EOS - EOS-b : (1, 2 , nuc / 14 ) (1.3, 2.5, 2.0), th 1.3 EOS-d : (1, 2 , nuc / 14 ) (1.28, 2.75, 2.0), th 1.28 Inner cores experience a bounce before BH formation Minner core M NS max Shocks propagate outward Pth contributes much Threshold mass is larger than for the cases with EOS-a §8 Black hole formation - Dependence on EOS - Dependence on : For larger – Mass of the inner core at bounce is larger – Shocks heat less fraction of the core – Degree of overshooting at bounce is larger Dependence on 2 : For smaller 2 – Equation of state for proto-neutron star is softer – Degree of overshooting is larger Larger inner core mass BHs are more liable to form promptly Larger degree of overshooting – Shocks are stronger if it were generated and propagate – Compactness at maximum compression is larger §8 A criterion for prompt black hole formation a : (1.32, 2.25, 2.0) b : (1.30,2.5,2.0) c : (1.30, 2.22,1.0) d : (1.28,2.75,2.0) ■ : BH for all EOS ☆ : BH for EOS-b (-d) × : BH for EOS-a □ : NS for all EOS §8 Black hole formation - Dependence on EOS - EOS-b : (1, 2 , nuc / 14 ) (1.3, 2.5, 2.0), th 1.3 EOS-c : (1, 2 , nuc / 14 ) (1.3, 2.22,1.0), th 1.3 BHs are less likely to be formed for EOS-c Smaller bounce density i.e. smaller surface pressure Larger bounce Shocks are stronger for EOS-c Threshold mass is larger §8 Black hole formation - Effect of shock - Maximum mass of cold spherical polytrope M max NS 1.6M Critical mass for spherical models Mcrit sphe 2.1 2.3M Thermal effects increase the threshold mass by 20 ~40 % §8 Black hole formation - Effects of rotation - Rotational effects (i) supply additional pressure (ii) Reduce the amount of matter falling into inner core Threshold mass for rotating models may be written as (Shibata (2000) PThP 104, 325) Mcrit rot M crit sphe Crot q 2 Rotational effects increases the threshold mass at most by 17 ~ 20 % §8 Black hole formation - Effect of differential rotation - Degree of dif. rot. Angular momentum distribution inside the core The threshold for BH formation is located between these curves Mcrit BH q 0.89, NS, rigid q 0.79, BH, rigid q 0.63, NS, A 0.5 q 0.54, BH, A 0.5 The inner region which is responsible for black hole formation “rotates” more rapidly §10 Summary and Discussion Rotating stellar core collapse to a black hole – Thermal effects (in particular shock) increase the threshold mass by 20 ~ 40 % – Rotational effects increase the threshold at most by 17 ~ 20 % – These effects depend sensitively of the equations of state Direct black hole formation and fallback-induced collapse – Differential rotation further increases the threshold Bipolar explosion and funnel structure Possibility for onset of dynamical nonaxisymmetric instabilities – Unlikely to occur for rigidly and moderately diff. rot. cases More realistic setting (EOS, neutrino treatment) Black hole + Disk system (BH excision technique) § Notes ・ Definition of q(j) and m(j) J ( j) q ( j ) m ( j )2 ・ Difference between EOS-b and EOS-d EOS-d : Smaller inner core → Larger fraction of the core undergoes the shock heating EOS-b : Core bounce and Γth are lager → shock itself is stronger These effects may cancel each other §Notes EOS which is stiff in subnuclear density – Coherent collapse and larger inner core mass EOS which is soft in supernuclear density – Large degree of overshooting and stored energy A large iron core mass These lead to ….. Steep density gradient along rotational axis Much smaller density in front of the “pole” of the shock surface Stronger shock generation and more rapid propagation of shock waves along rotational axis § Prediction of mass of disk ・ Consider the innermost stable circular orbit (ISCO) around a formed BH Fluid elements of j jISCO will fall into the black hole ・ If jISCO increases as a result of accretion, more fluid elements fall into the BH ・ Thus, if jISCO has a maximum, the dynamical growth of BH terminates Mass of the formed disk will be < 10% of the initial mass § General feature of the collapse Infall phase : – Inner core : collapses coherently at subsonic velocity – Outer core : collapses at quasi-free-fall and supersonic velocity Bounce phase : – – – – Sudden stiffening of EOS decelerates the inner core The inner core overshoots its equilibrium state (a) Mass of the inner core is very large → collapse to a black hole (b) mass of the inner core is not too large → inner core bounces Part of stored internal energy at bounce is released The shock wave is generated at the outer edge of the inner core Ringdown phase : – The inner core damps via its PdV works – (a) Shock is strong enough → a neutron star is formed – (b) Shock is not strong enough → fallback induced collapse to BH § Black hole formation - Effect of differential rotation - As the degree of differential rotation increases, a black hole is less likely to form q 0.89, NS, rigid q 0.79, BH, rigid q 0.79, NS, A 1.0 q 0.70, BH, A 1.0 The inner region which is responsible to black hole formation “rotates” more rapidly
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