particle-mesh codes and galactic disc dynamics
TRANSCRIPT
Particle-mesh codes and galactic disc
dynamics
Andreas Just
ARI @ ZAH
Heidelberg, Germany
SFB 881 - MPIA/HdA 15.5.2012 Andreas Just 2
Outline
Introduction
Superbox code
Applications
Speeding up
Introduction
Physics in galactic discs
Gravity
• Coupling of DM halo, disc, bulge, ISM, environment
• Instabilities: Jeans collapse, bars, spiral arms
• Resonances: energy and angular momentum transfer
Stellardynamics
• Violent relaxation: merger etc.
• Dynamical timescale short: equilibrium
• 2-body relaxation timescale long: collisionless
Interstellar medium
• Heating and cooling
• Pressure
• Turbulence
• Star formation and feedback
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Introduction
pure stellardynamical codes
Self-gravitation
• Integration time per timestep
• Direct N-Body: ~N²
• Tree: ~N logN
• FFT on uniform grids: ~N³ logN
• AMR: approximation on subgrids
Spatial resolution
• Direct N-body: d~n-1/3
• Mesh code: d~1/N
• Adaptive mesh code: d~f/N
Dynamics
• Timesteps dt
• Individual: a,da/dt,… neighbour list, regularization
• Global: dt~dmin/<v> for all particles/cells
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Grids
• Cartesian: N³ cells
• 3 levels of grids
• Comoving with density centre
Self-gravity
• Density in each cell
• FFT on each subgrid
Orbit integration
• Global timestep
• Interpolated leapfrog
Applications
• Well-defined structures
• Up to ~50 galaxies to interact
Particle mesh code Superbox
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Applications:
disc heating by satellite infall
Numerical heating reduced
• Force on grid
• Orbits of particles
Numerical thickening of disc
• Grids N=128³, dz=128pc
• ndisc=5·106
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Bien et al. 2012, subm. A&A
No numerical heating; no differential analysis needed as in Velazquez and White 1999 or Hayashi & Chiba 2006
Vertical profile well resolved in flattened grid
Applications (Tobias Brandt):
disc heating by satellite infall
Heating rate
• Msat=5.4·109 Msun
• Inclination i and eccentricity ε
• ~10 mergers to heat thin disc
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Massive satellites /DM clumps heat more efficient; Dynamical friction important for orbital evolution
Prograde merger Heat more efficient; Satellite after two approaches dissolved
Applications:
disc heating by satellite infall
Energy and angular momentum transfer
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Most energy and anglular momentum goes to halo (dynamical friction) Disc and bulge in deeper potential well after the merger Independent of inclination
Applications:
dynamical friction
Chandrasekhars formula
• Χ = fraction of fast stars
• Λ = (local scalelength L / resolution d/2)
Simulation
• Grids N=128³, particles n=107
• Outskirts of Plummer sphere
• L=R/5
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Just et al. 2011, MNRAS 411, 653
Effective resolution d/2; Global correlations of 2-body encounters (= gravitational wake) reproduced
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Spiral structure formation:
global unstable modes
Grand design spiral structure
• Correlations across the disc important
• Collective modes
• Growth rate
• Pattern speed
• Amplitude A(R,t) and shape (pitch angle i(R) )
disc model of Jalali & Hunter 2005
• 2-Dim analytic models
• rotation curve (total potential logarithmic with core)
• halo/bulge rigid
• disc density profile exponential (with same core)
• f0(E,Lz) given analytical with free parameter n
• increasing n=1, …, 6 decreases velocity dispersion
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Analytic disc models:
Spiral arm predictions
unstable modes for m=2,3,4
• m=2 primary, secondary
• CR=full; OLR dotted
bar-like mode spiral mode
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Analytic disc models:
Spiral arm patterns
unstable modes m=2,3,4
• m=3 and 4 (first found numerically!)
• CR=full; OLR dotted
m=3 m=4
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Superbox simulations:
The Superbox code (R. Bien, ARI)
collisisionless
• low artificial heating
• <5% in t=20tdyn for N=106 and 643 grids
good performance
• 1 week at local PC with N=13x106 and 2563 grids
cartesian grids
• no preference by grid geometry for low-m Fourier components
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Superbox simulations:
Models
A (stable)
• no evolution for ~10 rotations; independent on resolution
B (unstable)
• linear growth phase for resolution at
• extremly good central resolution
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Evolution of unstable modes:
Perturbations
Disc is unstable
• growing noise
• m=2 primary
• m=3 at larger r
• asymmetries from
superposition
Time unit = 8.5Myr
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Evolution of unstable modes:
Perturbations
Fourier analysis in radial rings
• 28 equally spaced rings
• m=1,2,3,4,5,6 at small time intervals
• Am(r,t), Θm(r,t) determined
• m=2: primary and secondary modes cannot be decoupled
• sum with different amplitudes, shapes and pattern speeds
• m=3: 3-armed spiral unique
• m=4: 4-armed spiral unique
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Evolution of unstable modes:
Comparison with analytic predictions
global structure
simulation
prediction
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Evolution of unstable modes:
Comparison with analytic predictions
central region
• minimum at centre
• phase jumps for m=4 reproduced
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Evolution of unstable modes:
global modes
exponential growth
• m=2,3,4: growth rate s>0
• s(r,t)=const.?
m=2: crosses m=3: triangles m=4: squares
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Evolution of unstable modes:
global modes
pattern speeds
• Ωp(r,t)=const.
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Evolution of unstable modes:
growth rates and pattern speeds
quantification
• t=40Myr
• In individual rings
• straight lines are theoretical values
growth rates pattern speeds
SFB 881 - MPIA/HdA 15.5.2012 Andreas Just 22
Evolution of unstable modes:
Comparison with analytic predictions
amplitudes
• only m=2 primary overplotted
Speed-up for larger grids:
parallelization and special hardware
Superbox-10
• Efficient speed-up of FFT
• Factor of ~2: FFTW
• Parallelization efficient
• for particle no. n<N³
• GPU version in progress
• Titan at ARI: 32 GPU cards
• New SFB cluster at Jülich
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Summary
Superbox code
• PM code free of numerical heating
• efficient for disc dynamics
• successful to reproduce unstable spiral arms growing out from noise
• Quantitative measurement of disc heating
Resolution
• N=1028³, d<20pc possible
• Particle number ndisc~107 needed
Impact of unstable modes?
• on dynamical heating by scattering or satellite infall
• On radial mixing of stellar populations
• Radial distribution of star formation rate