Dynamics of Galaxies: Practice. Frontiers in Numerical Gravitational Astrophysics July 3, 2008

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1 Dynamics of Galaxies: Practice Frontiers in Numerical Gravitational Astrophysics July 3, 2008

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3 Outline 1. Initial Conditions a) Jeans Theorem b) Exact solutions for spheres c) Approximations for disks 2. Analysis of N-body simulations a) Measurement b) Visualization techniques 3. Photo-dynamic modeling

4 Initial Conditions Galaxies close to equilibrium can be described by the timeindependent CBE: v f r Φ f v = 0 Possible approaches to generating initial conditions: 1. Solve time-independent CBE: Jeans theorem ansatz for equilibrium solution Draw initial coordinates ( r i, v i ) from f ( r, v) 2. Take moments of time-independent CBE: Solve equations for dispersion tensor σ αβ ( r) Draw initial velocities from Maxwellian distribution

5 Jeans Theorem An integral of motion is any function I( r, v) constant along orbits in a given potential Φ( r). which is The energy E( r, v) = 1 is always an integral. 2 v2 + Φ( r) Other integrals may or may not exist. Integrals are solutions of the time-independent CBE. Integrals come in two types, isolating and non-isolating; only isolating integrals constrain the shapes of orbits. In 3-D, a regular orbit has 3 independent isolating integrals. Jeans theorem states that in a potential where most orbits are regular, the DF of an equilibrium system may be expressed as a function of no more than 3 independent isolating integrals: f ( r, v) = f (E, I 1, I 2 )

6 Exact Solutions for Spheres In a spherical system, any equilibrium DF may be expressed as a function of the integrals E and J = r v. For simplicity, consider DFs which depend only on energy: Given the density profile f = f (E) Express the density as a function of the potential: ρ = ρ(φ). Then the unique DF is f (E) = 1 8π 2 ρ = ρ(r), solve the PE: dφ dr = G M(r) r 2 M(r) = d de Z 0 E Z r 0 4πq 2 dqρ(q) dφ dρ Φ E dφ Eddington (1916)

7 Known Distribution Functions Two models with known DFs useful in N-body simulations are: The Plummer (1911) model: ρ(r) = 3M 4π a 2 (r 2 + a 2 ) 5/2 Φ(r) = GM r2 + a 2 f (E) { ( E) 7/2, E < 0 0, E 0 The Hernquist (1990) model: ρ(r) = M a 2π r(r + a) 3 Φ(r) = GM r + a f (E) { 3sin 1 q+q 1 q 2 (1 2q 2 )(8q 4 8q 2 3) (1 q 2 ) 5/2, E < 0 0, E 0 q = ae GM

8 Tips and Techniques 1. To make an N-body realization of a system with an isotropic DF f = f (E), do for i = 1,...,N : select a uniform random number X i [0,M], and assign radius r i such that M(r i ) = X i at radius r i, pick speed by sampling v 2 f (Φ i + 1 2v 2 ) using a rejection technique (von Neumann 1951) v i 2. Eddington s formula can be solved numerically (Kazantzidis et al. 2004) even if ρ(φ) is only available numerically! f (E) = 1 8π 2 d de Z 0 E dφ dρ Φ E dφ 3. Anisotropic generalizations exist for DFs of the form f (E,J) = J 2α f 0 (E + J 2 /2r 2 a) (Cuddeford 1991)

9 Model Building. I Mass model: gamma profile (Dehnen 1993; Tremaine et al. 1994) with taper at large : r ρ(r) = Parameters: 3 γ 4π ρ b2 M a r γ (a + r) 4 γ, profie slope: γ = 3/2 total mass: M = 1.0 scale radius: a = taper radius: b = 4.0 r b r 2 e 2r/b, r > b half-mass radius: r 1/2 = 0.25 t 1/

10 N-body Potential Fit N-body code smooths density before computing potential: ρ( r) ρ s ( r) = (ρ s)( r) Plummer smoothing: s( r) = 3. Smoothing transforms cusp to constantdensity core; effect on mass profile is: ( M s (r) 1 + (3/2) κ/γ (ε/r) κ) γ/κ M(r) dφ s dr = G M s r 2 For ε = 0.005, get good fit to N-body potential with κ = π ε 2 ( r 2 +ε 2 ) 5/2 Compute potential due to smoothed mass:

11 Distributions and Dispersions Use Eddington s formula to construct DFs for: exact potential Φ (green) N-body potential Φ s (yellow) Use moment (Jeans) equations to find velocity dispersion σ(r) for: exact potential (blue) Φ N-body potential Φ s (red)

12 Equilibrium Tests 1. Construct realizations using both DF (E) and moment (J) techniques, and both exact and N-body potentials. N = 2 17 = Evolve using Tree code until t = 4. ε = t = 1/ Measure density profile ρ(r) at end. center on minimum of potential!

13 Equilibrium Tests t = 0

14 Equilibrium Tests t =

15 Equilibrium Tests t = 0.25

16 Equilibrium Tests t = 1

17 Equilibrium Tests t = 4

18 Equilibrium Tests t = 4 t = 0

19 Equilibrium Tests t = 4 t =

20 Equilibrium Tests t = 4 t = 0.25

21 Equilibrium Tests t = 4 t = 1

22 Equilibrium Tests t = 4 t = 4

23 Convergence Test. I Head-on parabolic collision of two γ = 1 models (Hernquist 1990). Repeat with N = 2 17,2 18,...,2 21 bodies; do results converge?

24 Convergence: Central Trajectories

25 Individual Trajectories Salt one of the models with N test = 1024 test particles, using same initial test distribution for all 5 experiments.

26 Convergence: Individual Trajectories

27 Gamma-Model Mergers Parabolic collision of two γ = 2 models (Jaffe 1983), with pericentric separation r p = 0.5 = 2r 1/2.

28 Gamma-Model Mergers Parabolic collision of two γ = 2 models (Jaffe 1983), with pericentric separation r p = 0.5 = 2r 1/2. Wide view of 1 st & 2 nd pass.

29 Gamma-Model Mergers Parabolic collision of two γ = 2 models (Jaffe 1983), with pericentric separation r p = 0.5 = 2r 1/2. Wide view of 1 st & 2 nd pass. 8 x zoom of 3 rd pass to merger.

30 Measuring Phase-Space Density For an isotropic system, let D(lnr,lnv) be the density on the (lnr,lnv) plane. t = 58 Then the coarse-grain phasespace density is f (r,v) = D(lnr,lnv) 16π 2 r 3 v 3 t This is useful even if the DF depends on other integrals besides the energy.

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32 Merger-Induced Phase-Space Structure

33 Embed a bulge in a dark halo. 1. Bulge is tapered γ = 1 model. Model Building. II M b = a b = 0.02 b b = 4 2. Halo is Navarro et al. (1996) model with fast taper (Springel & White 1999): M h (a h ) 1 4π(ln(2) 1 ρ h (r) = 2 ) r(r + a h ) 2, r b h ρ b β h h r β e r/a h, r > b h ( ρ and ρ h (r) h(r) are continuous at r = b h ) M h (a h ) = 0.16 a h = 0.25 b h = total halo mass: M h = 1

34 Model Building. IIa Add an exponential (de Vaucouleurs 1959) disk with a sech 2 vertical profile (van der Kruit & Searle 1981): ρ d (R,z) = M d 4πa 2 d z e R/a d sec 2 (z/z d ) d M d = a d = 1/12 z d = To compute approx. N-body potentials, we need a spherically-averaged disk profile: M d (r) = M d (1 e r/a d ) (1 + r/a d )

35 N-body Potential Fit Define profiles for total mass and spheroid: M mass (r) = M b (r) + M d (r) + M h (r) M sphr (r) = M b (r) + M h (r) Approximate smoothed mass profiles: ( x) M x,s (r) 1 + (3/2) κx/γ ( ε/r) κ γ/κx Mx (r) (NB: x specifies component; let ε ε to improve fit to potential.) Compute potential and compare to N-body result using ε = ; get good fit wth ε = κ mass = κ sphr = 2.025

36 Distribution Functions Given density profiles ρ b (r) and ρ h (r) for the bulge and halo, use Eddington s formula to construct DFs in equilibrium with potential generated by M mass,s (r) : where the functions G(x) exp( 1 2 x2 ) f b = f b (E) f h = f h (E) For the disk, make do with an approximate DF expressed using dispersions σ α (R) and mean rotation velocity v(r). ( ) ( ) ( ) f d (R,z,v R,v φ,v z ) ρ d (R,z)H v R H vφ v(r) G vz σ R (R) σ φ (R) σ z (R) H (x) exp( 1 2 (x/c)2 1 R and c is set so dxx2 H R 4 (x/c)4 ) (x) = dxh (x).

37 Disk Model Vertical dispersion from isothermal sheet (Binney & Tremaine1987): σ z (R) 2 = πgz d Σ d (R) Radial dispersion: σ R (R) µ(r)σ z (R) µ 2 Circular velocity: v 2 c(r) = G M sphr,s(r) R + R dφ d,s dr Moment equations give azimuthal dispersion and mean velocity.

38 Equilibrium Test 1. Construct realizations of bulge, disk, and halo DFs using: N b = N d = N h = Evolve using Tree code until t = 4. ε = t = 1/ Plot each component on (logr,v r ) and (E, E/E) planes.

39 t = 0

40 t =

41 t = 0.25

42 t = 1

43 t = 4

44 What is the inclination and argument of each disk? In what direction are we looking? How much time has elapsed since pericenter? How close was the passage? What are the length and velocity scale factors?

45 A Good Fit

46 A Model of the Mice

47 Thank You!