Post-Newtonian N-body Codes. Sverre Aarseth. Institute of Astronomy, Cambridge

Transcription

1 Post-Newtonian N-body Codes Sverre Aarseth Institute of Astronomy, Cambridge Introduction N-body tools Three-body formulation PN implementations Numerical examples Discussion

2 Hermite Integration Taylor series for F and F (1) F = F 0 + F (1) 0 t F(2) 0 t F(3) 0 t3 F (1) = F (1) 0 + F (2) 0 t F(3) 0 t2 Prediction r j = (( 1 6 F(1) 0 δt j F 0)δt j + v 0)δt j +r 0 v j = (( 1 2 F(1) 0 δt j + F 0)δt j + v 0; δt j = t t 0 New forces F, F (1) Higher derivatives F (3) 0 = (2(F 0 F) + (F (1) 0 +F (1) )t) 6 t 3 F (2) 0 = ( 3(F 0 F) (2F (1) 0 +F (1) )t) 2 t 2 Corrector for i r i = 1 24 F(2) 0 t F(3) 0 t5 v i = 1 6 F(2) 0 t F(3) 0 t4

3 AC Neighbour Scheme Total force F(t) = n j=1 F j +F d (t) Prediction F(t) = F n +Ḟ d (t t 0 )+F d (t 0 ) Ḟ = Ḟ n +Ḟ d Time-scales t n << t d, n << N Neighbour sphere R new s = R old s ( ) np 1/3 n Neighbour selection r i r j < R s Derivative corrections F (2) ij, F(3) ij

4 Basic KS Relations New coordinates R = u 2 1+u 2 2+u 2 3+u 2 4 Time transformation Coordinate transformation Generalized Levi Civita matrix Inverse relations L(u) = dt = R dτ R = L(u) u u 1 u 2 u 3 u 4 u 2 u 1 u 4 u 3 u 3 u 4 u 1 u 2 u 4 u 3 u 2 u 1 R 1 = u 2 1 u 2 2 u 2 3+u 2 4 R 2 = 2(u 1 u 2 u 3 u 4 ) R 3 = 2(u 1 u 3 +u 2 u 4 ) R 4 = 0 Velocity Ṙ = 2Lu R, R = t = 2u u Equations of motion u = 1 2 hu+ 1 2 RLT P h = 2u L T P t = u u

5 KS Decision-Making Close encounter Time-step criterion Neighbour list search R cl = 4r h N C 1/3, t cl = β t k < t cl list all r 2 kj, t j < 2 t cl cl R 3 1/2 m Two-body selection R kl < R cl, Ṙ kl < 0 Dominant motion m k + m l R 2 kl > m k + m j R 2 kj KS initialization Initialization of c.m. F U, F U, τ & t (n) t r cm = m kr k + m l r l m k + m l Perturber search r p < 2m p m b γ min 1/3 a(1+e) Slow-down adjustment Termination test Delayed termination Final iteration Polynomial initialization γ < γ 0, τ κ t R > R 0, γ > γ T block t > t i τ from τ, τ,... and δt F j, Ḟ j, t j, j = k, l

6 Hermite KS with block-steps Predict u, u, h to highest order u pred = u+ τu + τ2 2 u + τ3 6 u + τ4 24 u 4 + τ 120 u pred = u + τu + τ2 2 u + τ3 6 u 4 + τ4 24 u 5, h pred = h+ τh + τ 2 h + τ 6 h + τ 24 h. Force evaluation P = L T uf, P = L T u F +rl T u F. (2) Derivatives for corrector u = 1 2 (hu+rp), u = 1 2 (h u+hu +r P +rp ), h = 2(u P), h = 2(u P +u P ). (3)

7 Fourth-order corrector (Hut, Makino & McMillan 1995) u corr = u old + τ 2 (u corr +u τ2 old ) 12 (u new u old ), u corr = u old + τ 2 (u new +u τ2 old ) 12 (u new u old ), h corr = h old + τ 2 (h new +h τ2 old ) 12 (h new h old ). (4) Velocity derivatives for prediction u 4 mid = u new u τ u 5 mid = 12 τ 2 old, ( u new +u old 2 u new u ) old. (5) τ Shifted from midpoint u 4 new = u 4 mid + τ 2 u 5 mid, u 5 new = u 5 mid, h new = h mid + τ h new = h 2 h mid, mid. (6)

8 Taylor series for physical time t = u u, t = 2(u u ), t = 2(u u +u u ), t 4 = 2(u u )+6(u u ), t 5 = 2(u u 4 +8(u u )+6(u u ), t 6 = 2(u u 5 +10(u u 4 +20(u u ). (7) Next look-up time t next = t+ 6 n=1 t n τ n. (8) n! Regularized time-step by inversion t+ 6 n=1 t n τ n n! = 0. (9) Newton-Raphson iteration to solve for τ x i+1 = x i f(x i) f (x i ).

9 Initial guess τ 1 = t bs t τ orig, with quantized time-step t bs (= 2 n t orig ). Commensurability condition mod (t sys, t bs ) = 0. Solve equation (9) to obtain τ bs.

10 Three-Body Regularization Initialconditions r i, p i, p 3 = (p 1 + p 2 ) Basic Hamiltonian µ k3 = m km 3 m k +m 3 H = 2 k=1 1 p 2 2µ k+ 1 p 1 p 2 m 1m 3 m 2m 3 m 1m 2 k3 m 3 R 1 R 2 R KS coordinate transformation Q 2 k = R k, (k = 1,2) Time transformation dt = R 1 R 2 dτ Regularized Hamiltonian Γ = R 1 R 2 (H E 0 ) Γ = 2 k=1 1 R 3 k P 2 k + 1 P T 8µ k3 16m 1A 1 A T 2P 2 3 m 1 m 3 R 2 m 2 m 3 R 1 m 1m 2 R 1 R 2 R 1 R 2 E 0R 1 R 2 Equations of motion dq k dτ = Γ P k, dp k dτ = Γ Q k Regular solutions: R 1 0 or R 2 0 Singularterm<regularterms: R 1 R 2 > max(r 1, R 2 )

11 Three-Body Transformations Coordinates & momenta q k = q k q 3, p k = p k Regularized coordinates (q 1 0) Q 1 = [ 1 2 ( q 1 +q 1 )] 1/2 Q 2 = 1 2 q 2/Q 1 Q 3 = 1 2 q 3/Q 1 Q 4 = 0 Regularized momenta Basic matrix A 1 = 2 P k = A k p k Q 1 Q 2 Q 3 Q 4 Q 2 Q 1 Q 4 Q 3 Q 3 Q 4 Q 1 Q 2 Q 4 Q 3 Q 2 Q 1 KS transformations Physical momenta q k = 1 2 A k T Q k p k = 1 4 A k T P k /R k Coordinates & momenta q 3 = 2 m k q k /M k=1 q k = q 3 +q k p k = p k p 3 = (p 1 +p 2 ) (k = 1,2)

12 PN Codes Two-body Hermite KS toy code alternative PN formulations Three-body AZ regularization dominant binary, sequential terms homework: Einstein shift Chain Algorithmic regularization compact sub-system sequential PN, unperturbed KS

13 6x10-8 Two-body KS 5x10-8 4x10-8 SEMI 3x10-8 2x10-8 1x x10-9 1x x10-8 2x x10-8 3x x10-8 4x x10-8 5x10-8 Time

14 V/C = V/C = ECC Time

15 0.006 V/C = V/C = SEMI Time

16 0.006 IPN = 1 IPN = SEMI Time

17

18 Initialization Two-body R < R cl, Ṙ < 0, KS Hierarchy B + S H, Stability test Triple chain a 1 (1 e 1 ) < 3a 0, ṙ < 0, T Four-body chain B +B, ṙ < 0, B-B Higher orders T +B, ṙ < 0, Q

19 Chain Initialization Unperturbed exit ω > Activate chain N ch = 2 or > 2 Alternative path perturbed KS search Selection chain members & c.m. Initialization c.m. force & chain vectors Synchronization block time-step Interface perturber list

20 Termination Hyperbolic KS R > R 0 or γ > Binary KS γ > 0.1, γ = (F 1 F 2 )R 2 m 1 +m 2 Collision KS S Triple d 3 > 20a, B +S Hierarchy a 1 (1 e 1 ) < Sa 0, γ > 0.01 ARchain R 1 << R 2 +R 3 B +2S

21 Relativistic Effects Einstein shift ω = 6πM ac 2 (1 e 2 ) GR radiation time-scale t GR = 5 c 5 g(e)a 4 64X(1+X)m 3 1 X = m 2, g(e) (1 e2 ) 7/2 m Speed of light & Schwarzschild radius c = V, R Sch = 2M c 2 Velocities ( M V = α R pc ) 1/2, v max = ( M a ) 1/2 ( ) 1/2 1+e 1 e PN conditions ω > , t GR < t, r < n R Sch

22 Post-Newtonian Terms Equation of motion d 2 r dt = M [ ( 1+A) r ] 2 r 2 r +Bv First-order precession M = m 1 +m 2, η = m 1m 2 M 2 A 1 = 2(2+η) M r (1+3η)v ηṙ2 B 1 = 2(2 η)ṙ Higher-order precession A 2 =..., B 2 =..., A 3 =..., B 3 =... Gravitational radiation A 5/2 = 8 ( ) 17M 5 ηm r ṙ 3r +3v2 (3 Mr ) +v2 B 5/2 = 8 5 ηm r Total GR perturbation P GR = M c 2 r 2 [ (A 1 + A 2 c + A 5/2 2 c )r 3 r + (B 1 + B 2 c + B ] 5/2 2 c )v 3 GR spin dominant BH ARC energy change E = m 1m 2 M (P GR +P dyn ) vdt

23 Decision-Making Increasing GR effect t GR < 500, 50, 1 IPN = 1, 2, 3 GR time-scale T z = 150a4 M 3 (1 e2 ) 7/2 Myr Graduated PN w > (1,10,100) 10 4 IPN = 1, 2, 3 Coalescence R < 4R Sch = 8M c 2 Alternative merging IPN = 3, N = 2, N p = 0 IPN 2, a(1 e) < R Sch IPN = 3, N = 3, a 1 (1 e 1 ) > 100a Unperturbed KS Tidal disruption t GR < 500 & w > , γ < KS or CHAIN

24 PN Elements Energy ǫ b = ǫ 0 + ǫ 1 c 2 + ǫ 2 c 4 + ǫ 3 c 6, a = M 2ǫ b ǫ 0 = 1 2 V 2 M R, η = m 1m 2 M 2 ǫ 1 = 1 2 M R +3 8 (1 3η)V ( (3+η)V 2 +ηṙ 2) M R Angular momentum J = J 0 (1+f 1 /c 2 +f 2 /c 4 ) Eccentricity e 2 = (1 J2 Ma ) Lenz vector e = V R V/M R/R Periapse advance ω = 6πM c 2 a(1 e 2 ) Kozai-Lidov cycles T Kozai = T2 out T in ( ) 1+qout q out (1 e 2 out )3/2 g(e in,ω in,ψ)

25 Unperturbed GR Orbit Compact objects max (K 1,K 2) > 12 Derivatives ȧ = 64M 1M 2 (M 1 +M 2 ) 5c 5 a 3 (1 e 2 ) 7/2 ė = 304eM 1M 2 (M 1 +M 2 ) g(e) 15c 5 a 4 (1 e 2 ) 5/2 Einstein shift ω = 6π(M 1+M 2 ) ac 2 (1 e 2 ) KS rotation New elements θ = ω t T K, rotate by θ/2 a 1 = a+ȧ t, e 1 = max (e+ė t,0) Energy updating E = µ(h old H new ) Modify KS orbit Shrink, e = const; decrease, H = const Validity Unperturbed KS < ω < γ <

26 Membership Reduction escape R 1 or R n 1 > R, ṙ > 0 Double KS n = 4, two well separated pairs Injection γ > & r p < R, ṙ p < 0 Retention n = 2, IPN > 0, ω > Perturbers massive binary, N p 3 10 Perturbation γ 2m p M bh max ( ) 3 Rk d p

27 PN1 and PN Energy Time

28 e-06 Unperturbed PN 7.144e e e e e-06 A e e e e e e Time

29 Energy Corrections Unperturbed KS < ω < Perturbed KS P 2.5 & Ṗ 2.5 added to F & Ḟ Energy derivatives Ė = P v, Ë = P v+ṗ v Hermite method h = 1 2 (Ė 0 +Ė) t (Ë 0 Ë) t 2 Energy loss µ h Conservation E ch = E sub E E tot = E N +E ch

30 1 ECC

31 Tidal Disruption ( mbh ) 1/3 Classical condition r t = r m Case A Two-body regularization Corrections Case B Mass transfer Ejection Updating Initialization Pericentre test: a(1 e) < r t Transform to apocentre θ = π/2 Mass transfer m = 0.1m Escape velocity kick KS elements and energy Chain regularization m = 0.1m Energy corrections Chain membership and tables Chain c.m. polynomials Requirements Effects Special cases Astrophysics Observations Large eccentricity Mass segregation & Kozai-Lidov WD & AGB cores Instantaneous accretion Light flickering & emission lines

32 0.378 Model ECC Time

33 3.625e-08 Model e e e-08 SEMI 3.605e e e e Time

34 0.001 Model BH e-05 A 1e-06 1e-07 1e Time

35 Model BH8 1e-05 1e-06 a 1e-07 1e-08 1e-09 1e Time

36 1000 Model BH TGR e-05 1e-06 1e Time

37 Discussion IMF Mass outcome upper limit, Z BH from BSE PN simulations speed of light c = V Dynamics binary shrinkage & Kozai-Lidov Slingshot Retention Depletion Observations Astropysics fast escapers velocity kicks ejections of BH & BBH BH + S binaries, hyper-velocities tidal disruption & BH accretion