Particle-mesh codes and galactic disc dynamics

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


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


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


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


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):


Heating rate

• Msat=5.4·109 Msun

• Inclination i and eccentricity ε

• ~10 mergers to heat thin disc


Massive satellites /DM clumps heat more efficient; Dynamical friction important for orbital evolution

Prograde merger Heat more efficient; Satellite after two approaches dissolved

Applications:


Energy and angular momentum transfer


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


Just et al. 2011, MNRAS 411, 653

Effective resolution d/2; Global correlations of 2-body encounters (= gravitational wake) reproduced


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


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


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


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


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


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



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



Comparison with analytic predictions

global structure

simulation

prediction




central region

• minimum at centre

• phase jumps for m=4 reproduced



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



global modes

pattern speeds

• Ωp(r,t)=const.



growth rates and pattern speeds

quantification

• t=40Myr

• In individual rings

• straight lines are theoretical values

growth rates pattern speeds




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



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

Documents

Particle-mesh codes and galactic disc dynamics