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1 A 99% MS-Free Presentation

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3 Galactic Dynamics (Binney & Tremaine 1987, 2008) Dynamics of Galaxies (Bertin 2000) Dynamical Evolution of Globular Clusters (Spitzer 1987) The Gravitational Million-Body Problem (Heggie & Hut 2003) II ( 2008) N ( )

4 ( )

5 φ e r/λ r λ : λ fm λ fm λ = λ eff / λ =

6 d2 x N i dt 2 = x j x i d 2 x Gm j x j x i 3 dt 2 j=1,j i = Gm x x 3 + f p

7 ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )

8 d 2 x i dt 2 = N j=1,j i Gm j x j x i x j x i 3 3N 2 O(N 2 )

9 f(x, v, t) f f + v f Φ t v = 0 2 Φ = 4πGρ ρ = m fdv ( )

10 1 f(x, v, t) w = (x, v) ẇ = (ẋ, v) = (v, Φ) ( ) f t + 6 i=1 (fẇ i ) w i = 0 (f ) df dt = f f + v f Φ t v = 0

11 N (m 0) f = δ(r r(t), v v(t))

12 2 f(x, v, t) = c S ( 1 ) n n = ( f x, f v, f t ) C = n C = 0 ( v, dφ ) dx, 1 (S ) S (dx, dv, dt) C dx v = dv dφ/dx = dt 1 dx dt = v, dv dt = dφ dx

13 f f t = 0 f Φ f 0 ( )

14 Φ x v I E 3 J ( ) d I I(x, v) = v I Φ dt v = 0

15 f f I k 0

16 ρ = m 1 r 2 d dr f(e, J) f f(e) ( r 2 dφ dr ( ) 1 2 v2 + Φ dv ) = 4πGρ 1. f v ρ(φ) 2. ρ Φ(r)

17 King f { f e E e E e E 0 if E < E 0 0 if E > E 0 Plummer E 0 : f ( E) 7/2

18 Plummer f ( E) 7/2, ρ (c 2 + r 2 ) 5/2 N = 1024

19 0 : ν t + (νv i) = 0 x i ν = fdv, v i = 1 fv i dv ν 1 : v j ν v j t + νv v j i = ν Φ (νσ2 ij ) x i x j x i σ 2 ij = (v i v i )(v j v j ) = v i v j v i v j (1 )

20 ρ t + (ρv i) = 0 x i ρ v j t + ρv v j i = ρ Φ p x i x j x i

21 d(νv 2 r ) dr + ν r ( )] [2vr 2 vθ 2 + v2 φ = ν dφ dr d(νv 2 r ) dr = ν dφ dr = ν GM(r) r 2 M(r) = rv2 r G ( dln ν dln r + dln v2 r dln r ) [(ν(r), v 2 r(r)] M(r)

22 1 : x k (ν ρ ) 1 2 T jk = 1 2 d 2 I jk dt 2 = 2T jk + Π jk + W jk I jk = ρx j x i dx ρ v j v k dx, Π jk = 1 2 ρσ 2 jkdx W jk = 1 2 G ρ(x)ρ(x ) (x j x j)(x k x k) x x 3 dx dx (1 )

23 K = 1 2 2K + W = 0 ρ(x)v 2 dx, W = 1 2 ρ(x)ρ(x 1 ) x x dxdx E = K + W = K = 1 2 W 1/2

24 : M v 2 M v 2 + W = 0 v 2 = W M = GM r g 0.4 GM r h : r g = GM 2 W : r h 0.4r g M v2 r h 0.4G ( v 2, r h ) M

25 f f + v f Φ t v = f t coll f t : 2 coll : : 2 2

26 2? ( ) N ( )

27 2 V b θ (m,v = 0,n) (m = 0,v) v = v sin θ = 2v 2 v 2 = bmax b min b/b (b/b 0 ) 2, b 0 = Gm v 2 v 2 2πnvbdb G2 nm 2 ln Λ v Λ = b max b min b max :, b min : 90

28 T relax v2 v 2 v 3 G 2 nm 2 ln Λ N ln N T cross T cross = R v ( v 2 GNm/R) N T cross [yr] T relax [yr]

29 (m f, v f, n f ) (m t, v t ) V = v f v t tan θ = 2b (b/b 0 ) 2 1, b 0 = G(m t + m f ) V 2 v = v = m f m t + m f V sin θ = 2V m f m t + m f V (1 cos θ) = 2V m f m t + m f b/b (b/b 0 ) 2 m f m t + m f (b/b 0 ) 2

30 v f = 0 ( ) v = 0 v = 4πG2 n f m f (m t + m f ) ln Λ V 2 v 2 = 8πG2 n f m 2 f ln Λ V v 2 = 4πG2 n f m 2 f V

31 1 ( ) v = 4πG2 n f m f (m t + m f ) ln Λ V 2 (n f m f )

32 2 v 2 = 8πG2 n f m 2 f ln Λ V v 2 = 4πG2 n f m 2 f V

33 2 2 T relax v 2 v 2 σ 3 = 0.34 G 2 ρm ln Λ ( σ = ln Λ 10kms 1 ) 3 ( ) 1 ( ) 1 ρ m 10 3 M pc 3 [yr] M T half relax = 0.14N ln(0.4n) r 3 h GM ( M ln(0.4λ) 10 5 M ) 1/2 ( m M ) 1 ( ) 3/2 rh [yr] pc

34 T half relax > 2 ρ σ T relax 2 Λ (b min ) PM 2 Λ (b min ) 2!

35 = ( 1) (2 ) ( ) = log( ) 7 15

36 bit 0 1 ( )

37 2 : (7 ) (7 ) = (1 ) 2 : =

38 dx dt = f(x, t), x(t 0) = x 0 x x i+1 = x i + f(x i, t i ) t x i+1 x i t t i t i+1 t

39 m x(t i+1 ) x i+1 = O( t m+1 ) x(t i+1 ):, x i+1 : O( t m+1 ) t t = O( tm )

40 N dv i dt = N j=1,j i Gm j x j x i x j x i 3 dx i dt = v i O(N 2 )

42 v i+1/2 = v i + a i t 2 x i+1 = x i + v i+1/2 t v i+1 = v i+1/2 + a i+1 t 2 2

43 v i+1 = v i + a i t + 1 2ȧi t 2 x i+1 = x i + v i t a i t 2 ȧ i = a i+1 a i t 2

44 1. v 0 a 0 v 1/2 2. x 0 v 1/2 x 1 3. x 1 a 1 4. v 1/2 a 1 v

45 : d 2 x dt 2 = x ( ) ( ) ( ) 2 ( )

46 4 a ȧ

47 x p = x 0 + v 0 t + a 0 2 t2 + ȧ0 6 t3 v p = v 0 + a 0 t + ȧ0 2 t2 x c = x p + a(2) 0 24 t4 + a(3) t5 v c = v p + a(2) 0 6 t3 + a(3) 0 24 t4

48 3 a a t 0 t t 1 t (t 0, a 0, ȧ 0 ), (t 1, a 1, ȧ 1 ) a(t) = a 0 + ȧ 0 t + a(2) 0 2 t2 + a(3) 0 a (2) 0 = 6(a 0 a 1 ) t(4ȧ 0 + 2ȧ 1 ) t 2 a (3) 0 = 12(a 0 a 1 ) + 6 t(ȧ 0 + ȧ 1 ) t 3 6 t3

49 1. x 0 v 0 a 0 ȧ 0 x p v p 2. x p v p a 1 ȧ 1 3. a 0 ȧ 0 a 1 ȧ 1 a (2) 0 a(3) 0 4. x p v p a (2) 0 a(3) 0 x c v c 5. 1

51 1. t i + t i

52 !

53 PM(Particle-Mesh)(FFT) P 3 M(Particle-Particle Particle-Mesh)

54 O(N 2 ) O(N log N) : l d < θ l: d: θ:

55 2

56 GRAPE GRAPE Host Computer Position etc. Force etc. GRAPE Orbital Integration etc. Data Transfer Force Calculation GRAPE x j x ij Σ( f ) j ij x y j x i y ij Σ( f ) j ij y y i z j z ij Σ( f ij ) z j z i m j.. ε i 2 2 x ij + 2 y ij 2 z ij ε i r ij 2 r ij 2 r ij 3 m j /r 3 ij r ij 3

57 GRAPE LSI YEAR GRAPE-5 GRAPE-6 MDM (MDGRAPE-2 + WINE-2) PROGRAPE-1 Arbitrary Interaction GRAPE 94 GRAPE-4 HARP-2 MD-GRAPE SPH GRAPE-3 HARP-1 GRAPE-2A WINE-1 Ewald HARP-1 ȧ HARP-2 89 GRAPE-1A GRAPE-1 Limited Accuracy GRAPE-2 Full Accuracy Arbitrary Central Force

58 1 2

4 19

4 19 I / 19 8 1 4 19 : : f(e, J), f(e) Phase mixing Landau Damping, violent relaxation : 2 2 : ( ) http://antwrp.gsfc.nasa.gov/apod/ap950917.html ( ) http://www-astro.physics.ox.ac.uk/~wjs/apm_grey.gif