Initial conditions for N-body/SPH simulations

Transcription

1 Initial conditions for N-body/SPH simulations Rubens Machado IAG / USP I Workshop de Computação Científica em Astronomia NAT - UnicSul, 02/06/2011

2 Outline 1 N-body simulations 2 Initial conditions 3 An example: dark matter halo with gas All models are wrong; some models are useful.

3 Part I N-body simulations

4 N-body simulations globular clusters galaxy clusters Heggie galactic dynamics structure formation Springel (2005)

5 N-body simulations F i = i j G m i m j r i r j 2 compute N 1 forces, for each of the N partices evolve positions and velocities from t to t + t

6 Direct summation is sometimes feasible special-purpose hardware GPUs GRAPE (GRAvity PipE)

7 N-body simulations tree code divide space in cells distant particles grouped together nearby particles taken into account in smaller cells Dehnen (2006)

8 N-body simulations octrees number of operations drastically reduced

9 Part II Initial Conditions

10 Initial conditions mass, pos, vel body m x y z v x v y v z 1. N

11 Initial conditions elliptical stellar disks gradually increasing particle masses during initial evolution

12 Initial conditions non-spherical dark matter haloes altering halo shape during the evolution: - shifting particle positions - (while re-scaling velocities to keep virial ratio)

13 Part III Creating initial conditions: an example

14 Initial conditions example dark matter gas

15 Initial conditions example density profiles

16 Initial conditions example Hernquist (1990) Density: ρ(r) = M h 2π Potential: Φ(r) = GM h r + a a r (r + a) 3 r 2 Cumulative mass: M(r) = M h (r + a) 2 : r(m) = M Mh a M 1 Mh

17 Initial conditions example cumulative mass function

18 Initial conditions example cumulative mass function

19 Initial conditions example cumulative mass function

20 Initial conditions example cumulative mass function

21 Initial conditions example Distribution Function: f ( r, v, t) d 3 r d 3 v f = f (x, y, z, v x, v y, v z ) f = f (E) f = ( ) v 2 f 2 + Φ( r )

22 Initial conditions example Eddington s formula: f (E) = 1 [ E 8π 2 0 d 2 ρ dψ 2 dψ + 1 ( ) ] dρ E Ψ E dψ Ψ=0 Hernquist distribution function: Mh 1 [ f (E) = 8π 3 a 5/2 2G (1 q 2 ) 5/2 3 sin 1 q + q(1 q 2 ) 1/2 (1 2q 2 ) (8q 4 8q 2 3) ] where q = a E GM h

23 Initial conditions example distribution function

24 Initial conditions example distribution function

25 Initial conditions example distribution function

26 Initial conditions: two-component total potential: Ψ T = Ψ 1 + Ψ 2 f 1 (E) = 1 E d 2 ρ 1 8π 2 0 dψ 2 T f 2 (E) = 1 E d 2 ρ 2 8π 2 0 dψ 2 T dψ T E ΨT dψ T E ΨT

27 Hydrodynamics Hydrostatic equilibrium: 1 ρ 1 P = Φ ρ k B d(ρt) = GM(r) µm H dr r 2 gas temperature profile T(r) = µm H k B 1 ρ(r) r ρ(r ) GM(r ) r 2 dr

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30 Cluster merger example gas density gas temperature

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33 Summary 1 given ρ(r), obtain Φ(r) and M(r) 2 randomly sample M(r) to obtain positions 3 given ρ(r) and Φ(r), obtain f (E) from Eddington s formula 4 von Neumann rejection of f (E) to set velocities 5 if more components, use total Φ(r) 6 gas temperature from hydrostatic equilibrium The purpose of computing is insight, not numbers :-) Computers only give you numbers, not insight :-( The purpose of computing numbers is not yet in sight :-/

34 Summary 1 given ρ(r), obtain Φ(r) and M(r) 2 randomly sample M(r) to obtain positions 3 given ρ(r) and Φ(r), obtain f (E) from Eddington s formula 4 von Neumann rejection of f (E) to set velocities 5 if more components, use total Φ(r) 6 gas temperature from hydrostatic equilibrium The purpose of computing is insight, not numbers :-) Computers only give you numbers, not insight :-( The purpose of computing numbers is not yet in sight :-/