Hamiltonian chaos and dark matter

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1 Hamiltonian chaos and dark matter Martin D. Weinberg UMass Astronomy July 15, 2013 Monterey 07/15/13 slide 1

2 Goals Goals Bars Post-modern dynamics galactic dynamics Progress report A new approach Influence of DM on galaxy structure why still challenging? Example: perturbation theory and chaos DM implications The future... Monterey 07/15/13 slide 2

3 In the beginning... Goals Bars Dark matter was implied by rotation curves in 1970 s Motivated the Cold Dark Matter cosmongony (Primack et al. 1984). But: = Nothing agreed with CDM on small scales. = Disagreement given a title that often ended in crisis The overcooling crisis The satellite crisis The angular momentum crisis The rotation curve conspiracy The cusp/core problem The bar slow-down problem Monterey 07/15/13 slide 3

4 The bar crisis 50% of spiral galaxies are barred From morphology: bars are rapidly rotating Bars slow down (due to dynamical friction) in a DM halo Can t have dark matter in bar s vicinity! MOND? [Debattista & Sellwood 2000] Buta & Crocker 1993 Monterey 07/15/13 slide 4

5 The bar crisis 50% of spiral galaxies are barred From morphology: bars are rapidly rotating Bars slow down (due to dynamical friction) in a DM halo Can t have dark matter in bar s vicinity! MOND? [Debattista & Sellwood 2000] Debattista & Sellwood 2000 Monterey 07/15/13 slide 4

6 Bars and pseudo-bulges Two proposed ways to grow bars Dynamical instability (simulations, 1970 onwards) Secular growth (Polyachenko & Polyacheko) Triggered by encounter Cored simulation: buckling andm = 2 instability Even/odd vertical power [from Petersen, Weinberg, Katz (2013)] Monterey 07/15/13 slide 5

7 Bars and pseudo-bulges Two proposed ways to grow bars Secular bar needs DM halo to grow... Dynamical instability (simulations, 1970 onwards) Secular growth (Polyachenko & Polyacheko) Triggered by encounter Cored simulation: buckling andm = 2 instability Even/odd vertical power [from Petersen, Weinberg, Katz (2013)] Monterey 07/15/13 slide 5

8 Bars and pseudo-bulges Two proposed ways to grow bars Dynamical instability (simulations, 1970 onwards) Secular growth (Polyachenko & Polyacheko) Halos suppress instability but promote secular growth Provide an angular momentum sink Resonant transfer Depends on DM phase-space density and gradient Transfer weak in a core The bar crisis is subtle Monterey 07/15/13 slide 5

9 Bars and pseudo-bulges Pseudo-bulge formation is stronger in cored profile Indirect DM diagnostic? Predictions in progress... Cuspy Monterey 07/15/13 slide 6

10 Bars and pseudo-bulges Pseudo-bulge formation is stronger in cored profile Indirect DM diagnostic? Predictions in progress... Cored Monterey 07/15/13 slide 6

11 In the beginning... is HARD How to move forward? Perturbation theory The KAM theorem Chaos example Dark matter was implied by rotation curves in 1970 s Motivated the Cold Dark Matter cosmongony (Primack et al. 1984). But: = Nothing agreed with CDM on small scales. = Disagreement given a title that often ended in crisis The overcooling crisis The satellite crisis The angular momentum crisis The rotation curve conspiracy The cusp/core problem The bar slow-down problem Monterey 07/15/13 slide 7

12 I contend... is HARD How to move forward? Perturbation theory The KAM theorem Chaos example that most of these crises are the result of inattention to dynamical mechanism Corollary 1: Halos are NOT crystalline spheres; if dark matter dominates the gravity, it takes part in galaxy evolution! Corollary 2: if you can t see in the simulation it s not true is an error-prone world view Monterey 07/15/13 slide 8

13 Why galactic dynamics is HARD is HARD How to move forward? Perturbation theory The KAM theorem Chaos example The N-body crisis Difficult to analyze and understand Images are crude summary statistics Simulation parameters are chosen by economics Dark matter only; baryons added later Regimes are difficult to identify Simulations replace collisionless with collisional systems Softening reduces noise on small but not large scales Example: Secular evolution Resonant transfer Requires many particles in near resonance Insufficient particle numbers: resonance has no effect Monterey 07/15/13 slide 9

14 Why galactic dynamics is HARD is HARD How to move forward? Perturbation theory The KAM theorem Chaos example The N-body crisis Difficult to analyze and understand Images are crude summary statistics Simulation parameters are chosen by economics Dark matter only; baryons added later Regimes are difficult to identify Simulations replace collisionless with collisional systems Softening reduces noise on small but not large scales N = N = N = N = energy (E) vs. scaled angular momentumκ Monterey 07/15/13 slide 9

15 Why galactic dynamics is HARD is HARD How to move forward? Perturbation theory The KAM theorem Chaos example The N-body crisis Difficult to analyze and understand Images are crude summary statistics Simulation parameters are chosen by economics Dark matter only; baryons added later Regimes are difficult to identify Simulations replace collisionless with collisional systems Softening reduces noise on small but not large scales Example: Secular evolution Resonant transfer Requires many particles in near resonance Insufficient particle numbers: resonance has no effect Dynamical chaos: overlapping of resonances Monterey 07/15/13 slide 9

16 Why galactic dynamics is HARD is HARD How to move forward? Perturbation theory The KAM theorem Chaos example Analytic galactic dynamics has stalled out... why? Time scales Not in the time-asymptotic regime τ evol τ dyn (unlike solar system dynamics, tokamaks, accelerators) Not in the impulsive approximation easy (Effectively) infinite degrees of freedom = Many orbits in a galaxy are irregular Common lore: irregularity is not important What does chaos do to galaxy evolution? Monterey 07/15/13 slide 10

17 How to move forward? is HARD How to move forward? Perturbation theory The KAM theorem Chaos example Perturbation theory is very hard, owing to d.o.f. N-body is difficult to interpret and verify What to do? My answer: combination of hybrid perturbation theory and numerical computation... work toward mutual confirmation and understanding Monterey 07/15/13 slide 11

18 Modern dynamics in a nutshell is HARD How to move forward? Perturbation theory The KAM theorem Chaos example Hamiltonian Equations of motion: q = ṗ = H = H 0 (p,q) H 0 (p,q) p H 0 (p,q) q Monterey 07/15/13 slide 12

19 Modern dynamics in a nutshell is HARD How to move forward? Perturbation theory The KAM theorem Chaos example Solve H-J equation to get action-angle variables Equations of motion: ẇ = İ = ( ) S H 0 q,q H 0 (I) I H 0(I) w + S t = 0 = H = H 0(I) = Ω = constant =0 Actions remain constant Angles change linearly in time These are rosette orbits! Monterey 07/15/13 slide 12

20 Modern dynamics in a nutshell is HARD How to move forward? Perturbation theory The KAM theorem Chaos example Add a perturbation Equations of motion: H = H 0 (I)+ǫV 1 (p,q) ẇ = Ω +ǫ V 1(p,q) I İ = ǫ V 1(p,q) w Perturbation may be expanded in series indexed by rosette orbits Monterey 07/15/13 slide 12

21 Modern dynamics in a nutshell is HARD How to move forward? Perturbation theory The KAM theorem Chaos example Expand the perturbation in a Fourier series Equations of motion: H = H 0 (I)+ǫ l ẇ = Ω +ǫ l U 1l (I)e i(l w mωt) U 1 (I) e (il w mωt) I İ = ǫ l U 1 (I)ile i(l w mωt) Monterey 07/15/13 slide 12

22 Modern dynamics in a nutshell is HARD How to move forward? Perturbation theory The KAM theorem Chaos example Expand the perturbation in a Fourier series Equations of motion: H = H 0 (I)+ǫ l ẇ = Ω +ǫ l U 1l (I)e i(l w mωt) U 1 (I) e (il w mωt) I İ = ǫ l U 1 (I)ile i(l w mωt) Choose one particular terml where: l Ω mω 0 Monterey 07/15/13 slide 12

23 Modern dynamics in a nutshell is HARD How to move forward? Perturbation theory The KAM theorem Chaos example Expand the perturbation in a Fourier series Equations of motion: H = H 0 (I)+ǫU 1l (I)e i(l w mωt) ẇ = Ω +ǫ U 1l (I) e (il w mωt) I İ = ǫu 1l (I)il e i(l w mωt) Choose one particular terml where: l Ω mω 0 Then, averaging over time: only one term remains... = Equations of a non-linear pendulum! = This gives bar evolution, spiral arms, etc.... Monterey 07/15/13 slide 12

24 Modern dynamics in a nutshell is HARD How to move forward? Perturbation theory The KAM theorem Chaos example Expand the perturbation in a Fourier series Equations of motion: H = H 0 (I)+ǫ l ẇ = Ω +ǫ l U 1l (I)e i(l w mωt) U 1 (I) e (il w mωt) I İ = ǫ l U 1 (I)ile i(l w mωt) Alternatively: keep all the terms. Then, attempt to solve for new actions and angles using H-J: ( ) S H q,q + S t = 0 = KAM theorem Monterey 07/15/13 slide 12

25 The KAM theorem is HARD How to move forward? Perturbation theory The KAM theorem Chaos example Kolmogorov (1954), Arnold (1963), Moser (1962) Original regular orbit (torus) may not exist Can nearly always find action-angle pair close to the original Regularity and chaos are interleaved Regions of regularity diminish in size with increasing strength Do this by solving the H-J equation with the perturbation This can be done iteratively Quadratic convergence Can use the same technique numerically for real-world problems!! Monterey 07/15/13 slide 13

26 Chaos example is HARD How to move forward? Perturbation theory The KAM theorem Chaos example Suppose we keep two terms... After a big bunch of algebra... equations of motion are that of forced parametric resonance: one pendulum tries to force another at a nearby multiple of frequencies This result: chaos! Poincaré SOS From Luo, 2001, Nonlin. Dyn., 26, Monterey 07/15/13 slide 14

27 Chaos example is HARD How to move forward? Perturbation theory The KAM theorem Chaos example Suppose we keep two terms... After a big bunch of algebra... equations of motion are that of forced parametric resonance: one pendulum tries to force another at a nearby multiple of frequencies This result: chaos! This is the generic situation in galaxies. How chaotic are galaxies? What influences the measure of this chaos? Monterey 07/15/13 slide 14

28 How we study chaos? Study of chaos Experiments Model parameters Application to bars 1. Standard approach to this problem: Poincaré surface-of-section: define a potential integrate orbits SALI/GALI (e.g. Minos & Athanassoula) 2. NEW APPROACH: use H-J to determine which tori are ceasing to exist everywhere in phase space. Advantages: Constructive: Know actions and trajectories for regular orbits Know which actions (tori) are no longer Can be used iteratively in time to understand evolution Predict general features of chaos in disks, bulges and halos Monterey 07/15/13 slide 15

29 Numerical experiments Study of chaos Experiments Model parameters Application to bars Orbits are regular in potentials separable in conic coordinates: Spheres, spheroids and ellipsoids, cylindrical disks... Let s make trouble! 1. Add a halo to the disk; consider bar (or strong spiral arms) Disk orbits are no longer regular (broken symmetry) What happens to the disk? 2. Add a disk to a halo Halo orbits are no longer regular What does this do to the halo structure? Monterey 07/15/13 slide 16

30 Model parameters Study of chaos Experiments Model parameters Application to bars Consider NFW halos with and without cores, exponential disks ρ halo (r) (r+r c ) 1 (r+r a ) 2 Σ disk (R) e R/A No core r a = 6.67A orc = 15 Monterey 07/15/13 slide 17

31 Model parameters Study of chaos Experiments Model parameters Application to bars Consider NFW halos with and without cores, exponential disks ρ halo (r) (r+r c ) 1 (r+r a ) 2 Σ disk (R) e R/A r c = A r a = 6.67A orc = 15 Monterey 07/15/13 slide 17

32 Model parameters Study of chaos Experiments Model parameters Application to bars Consider NFW halos with and without cores, exponential disks ρ halo (r) (r+r c ) 1 (r+r a ) 2 Σ disk (R) e R/A Consider bars of varying strength (quadrupole) Amplitude = 0.4 (in disk units) U 22 (r) r 2 /[1+(r/b 5 ) α ] 5/α Monterey 07/15/13 slide 17

33 Application to bars NO halo, 50% bar Study of chaos Experiments Model parameters Application to bars Circular orbits Radial orbits Monterey 07/15/13 slide 18

34 Application to bars NO halo, 50% bar Study of chaos Experiments Model parameters Application to bars Resonances: (l 1,l 2 ): l 1 Ω r +l 2 Ω φ = 2Ω p Monterey 07/15/13 slide 18

35 Application to bars NO halo, maximum bar Study of chaos Experiments Model parameters Application to bars Bar lives here Resonances: (l 1,l 2 ): l 1 Ω r +l 2 Ω φ = 2Ω p Monterey 07/15/13 slide 18

36 Application to bars 3X halo Study of chaos Experiments Model parameters Application to bars Resonances: (l 1,l 2 ): l 1 Ω r +l 2 Ω φ = 2Ω p Monterey 07/15/13 slide 19

37 Application to bars 10X halo Study of chaos Experiments Model parameters Application to bars Resonances: (l 1,l 2 ): l 1 Ω r +l 2 Ω φ = 2Ω p Monterey 07/15/13 slide 19

38 Application to bars 10X halo with core Study of chaos Experiments Model parameters Application to bars Resonances: (l 1,l 2 ): l 1 Ω r +l 2 Ω φ = 2Ω p Monterey 07/15/13 slide 20

39 Application to bars 10X halo with core, 75% of maximum Study of chaos Experiments Model parameters Application to bars Resonances: (l 1,l 2 ): l 1 Ω r +l 2 Ω φ = 2Ω p Monterey 07/15/13 slide 20

40 Application to bars Study of chaos Experiments Model parameters Application to bars Conclusions: Strong bar without halo is on the verge of fully chaotic Dark matter halo dramatically reduces the measure of chaotic orbits Why? Increase in dynamic range of frequencies DM may be required to have strong bars Raises the question: is strength of bar limited by chaos? Monterey 07/15/13 slide 21

41 Application to bulges Study of chaos Experiments Model parameters Application to bars Thickened Kuzmin disk (20%) and Kuzman halo (Hunter 2005) Compute Poincaré surface of section The result: chaos Monterey 07/15/13 slide 22

42 Application to bulges Study of chaos Experiments Model parameters Application to bars Miyamoto-Nagai disk (10%) and NFW halo Compute numerical H-J The result: chaos The bulge region is mostly irregular in both models; new shape? Ask: will inner DM halo remain cuspy with all those irregular orbits? Monterey 07/15/13 slide 22

43 Application to bulges Study of chaos Experiments Model parameters Application to bars points: Inner galaxy/bulge region is fully chaotic! Raises the questions Do bulges result from chaos? Will a central DM cusp simply diffuse away? This technique can be extended to galaxy evolution generally... Numerically intensive but no more so than an N-body simulation Monterey 07/15/13 slide 23

44 Take-away messages The future Dark-matter halos take part in dynamical evolution not only passively influence kinematics Galaxy structure/morphology may be indirect DM indicators Chaos: bulge creating/bar self-limiting Stochastic web; e.g. radial migration May be used to constrain DM profiles Provides yet another mechanism to destroy cusps different under influence of DM and alternative gravity... No ang. mom. transport & bar amplification The chaos in a pure disk will be different Precision kinematics and photometry should discriminate Monterey 07/15/13 slide 24

45 Galaxy evolution horizon The future Compute chaotic diffusion rates and their role in shaping galaxies Very little work has been done here! One exception: mode hopping/radial migration Post KAM chaos and numerical noise are NOT the same Need to know: are numerical simulations up to the task? Monterey 07/15/13 slide 25

46 The future The End Monterey 07/15/13 slide 26

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