Looking Under The Hood: An N-body Mechanic Tutorial

Kelly Holley-BockelmannCenter for Gravitational Wave Physics and

Department of Astronomy, Penn State

From Millennium simulation

N-body simulations can track structures as they assemble within a cosmological volume

Matthias Steinmetz

matter = 0.35 +/- 0.1 baryon= 0.045 +/- 0.15 dark energy = 0.65 +/- 0.1

Or how galaxies harrass

one another at late times

within a galaxy cluster

How an individual galaxy responds to perturbations

And how well the dark matter couples to the galaxy’s evolution

They can also provide a window into the arrangement of the dark matter within a halo

Volker Springel

5123 particles in 4803 Mpc –then zoom in and repopulate

…which totally neglects a large set of n-body simulations

• cosmological simulations

• gas

• solar system/ few body

• direct n-body simulations

• black holes with ‘realistic’ gravity

• non-equilibrium experiments --like mergers or collapse

If you’re interested in this stuff, we can do another tutorial later

We’ll concentrate on equilibrium galaxies

Collisionless Stellar Systems

Tc ~ Gh

1 Crossing time

Tr ~ Tc N

8 ln NRelaxation time

Tc << Tr

(NOT ‘collision time’)

Tc=106 years

Tr=109 years

N ~ 106 stars

Tc=108 years

Tr>1010 years

N ~ 1011 stars

The Collisionless Boltzmann Equation

Let the total mass in stars within the phase space volume d3r d3v at time t be:

f(r,v,t) d3r d3v

Distribution function

f t

f r

v

f+ V

• - • = 0Then: Completely describes the evolution of the system.

2 = 4G f d3vWhere: But..impractical to solve in 3-d via finite difference methods

Another approach: N-body realization

Build model at some initial time t0 with N particles such that the probability of a particle at r,v is proportional to f(r,v,t0)Evolve this N-body realization according to very simple equations of motion:

dri

dt= vi dt

dvi= - r=ri

Problem: Assume 109 particles distributed with 1 kpc resolution in a 1 Mpc cube. If we want to evolve a system for 100 timesteps, that’s 1020 calculations! If each one is ~ 10 Floating Point Operations, that’s 1021 FLOPs. At supercomputer speed of 1013 operations/second, that’s a 3 year calculation…sucks to be me.

N-body simulations are limited by:

• numerical errors

round-off error

‘truncation’ error in timestep

dynamic range effects

discreteness

bugs

• sub-grid physics

• oversimplification of the model

Early N-body Mechanic

To consider simultaneously all these causes of motion and to define these motions by exact laws allowing of convenient calculation exceeds, unless I am mistaken, the forces of the entire human intellect.

--Newton

• Force of Gravity 1/r2, as does light intensity

• Holmberg built special purpose computer using 74 light bulbs (37 for each of two colliding galaxies)

• They were on ruled graph paper, each was unscrewed in turn and replaced with a 4-sided photometer to measure the force. Positions and velocities were updated

• Digital computers didn’t integrate 74 particles for 25 years!

Collisions of “light bulb” galaxies

Describe initial conditions as adistribution of massive particles

Calculate forces between them efficiently

follow their motions in time

Modern N-body Simulations

First Step: Build a Galaxy

Most studies need a model in dynamical equilibrium –

Defined by a potential-density pair such that the

distribution function is time-independent.

Let’s go through the exercise of making a very simple galaxy model, the Plummer sphere (polytrope of index 5)

Building an Equilibrium Plummer Sphere

(r) = - G M( r2 + a2)1/2

1

Assume spherical symmetry, so that (r)=(r ), (r )=(r ), and that the velocity distribution is isotropic. In general, spherical symmetry in position space can still allow a cylindrical symmetry in velocity space – i.e. the distribution function is a function of energy *and* angular momentum. However, by assuming isotropy, we assert that:

ƒ(E, J)=f(E)

Getting the density:To find the density, just solve Poisson’s equation:

(r) = 3 M a2

4 (r2 + a2)5/2Now we just throw particles down randomly according to this density – how?

Introducing the cumulative mass distribution:

m(r ) = 4r2(r ) dr0

r

So m(r )/M is a number between 0 and 1…looks like a good form for a random number!

Laying down the positions

So, we can pick a random number between 0 and 1 to get the fractional mass, but what we really want is the *position* at that mass --- r(m) rather than m(r )

m(r ) = r3(r2+a)-3/2

r(m ) = a (rand-2/3 –1)-

1/2

Laying down the velocities

We’re going to assign random isotropic velocities so that they obey the distribution function f(r,v). So, first we gotta determine f(r,v) from (r) and (r )…

Recall that f(r,v) dr dv is the mass within a volume element dr dv

Recall, too that we can write f(E)=f(r,v) in a spherical isotropic system, so f(E) dr dv is the mass within a volume element dr dv

Ahem, notice it’s not f(E) dE, like I bet you were tempted to say

(r ) = f(E) 4v2dv

0

vescDensity is essentially a 3-d projection of the 6-d phase space

Laying down the velocities (con’t)

where vesc = 2 (r2 +1)-1/4

(r ) = 25/2 f() (-)1/2 d0

And inverting this gives you the distribution function:

f() =1 d2 d 1 d

8 d2 - d =0

Laying down the velocities (con’t)

f(r,v)dr dv =384 2

7m()7/2 r2v2drdv

Probability g(v) to have velocity v at position r

()7/2v2 dv

change variables q = v/vesc: g(q)=(1-q2)7/2 q2

Rejection technique, choose q such that it’s weighted by g(q)

dg(q)dq

= q (2- 9q2) (1- q2)5/2 0.092

Choose random variables such that q {0,1} and g(q) {0,0.1}

Ta-da! – spherical, isotropic system in equilibrium

• disks and triaxial spheriods

• multicomponent systems

• anisotropy

• increased resolution – multimass

• black holes

• non-equilibrium experiments

• Moderate Triaxiality (0.25 < T < 0.75)

• Cuspy initial inner density profile

• Variable black hole masses

To resolve the BH, a multimass scheme: m(E,J) rp

where rp = (r vt)2

E+ M/a

To resolve the dynamics: • N=10,000,000

• multiple timestep, 4th order Hermite integrator

Example: Building BH-Embedded Triaxial Models

N-body on the cheap

Calculating pair-wise Newtonian gravitational forces by direct summation is an N2 problem – 1010 stars and 10HUGE dark matter particles

Newton was a smart guy

Reduce the calculation to O(N) by writing the potential as a basis expansion

(r ) = Anlm nlm ( r ) , ( r) = Anlm nlm ( r)

nlm = Cn () 4 Ylm (,) rl

r( 1+r)2l+3

2l+32

2

nl

nlm nlm

nlm = - rl

( 1+r)2l+1 Cn () 4 Ylm (,)

2l+32

Anlm = mk nlmrk,k,k*k

N-body on the cheap: Hardware version

• Grape-6 – special purpose computer to directly calculate newtonian gravity between particles

• 32 chips/board , 64 boards 63 Tflops!

• particles are set into SRAM, then injected into a pipeline where acceleration is calculated

N-body on the cheap: Treecode version

Assume that at large distances, the gravitational force of many particles in a cell can be approximated by the force from the center of mass of the cell.

• Divide cube into 8 subcubes

• if zero particles in subcube, delete

• if subcube contains more than 1 body, recursively subdivde

O(N logN) calculation

Integrators

• Leapfrog – second order (accuracy in position/energy

increases by at least a factor of 100 if t cut by 10…truncation error t2)

vn+1/2 = vn + ½ t a(rn)

rn+1 = rn +t vn+1/2

vn+1 = vn+1/2 + ½ t a(rn+1)

Single timestep vs. individual timestepsSymplectic time-reversible -- built-in energy conservation

Galaxy/cosmological simulations can tolerate a lot more positional, energy errors than solar system sims potential fluctuations cause bigger errors

n-1 n-1/2 n n+1/2 n+1 n+3/2 r, a r, a r, a v v v

Integrators, con’t• Hermite integrator 4th order -- all about the ‘jerk’ r , the highest derivative of r that requires only 1 pass through the particles (see also snap,

crackle and pop…)

j = G j=1 ji

N

Mj

vji 3(rji vji)• rji

rji3 rji

5

where vji = vj-vi

where aji = aj-ai

aji = Gk=1 kj

NMk

rkj3

rkj k=1 ki

NMk

rki3

rki

ri+1 = ½ (vi+vi+1)t +1/12 (ai – a i+1) t2

vi+1 = ½ (ai+ai+1)t +1/12 (ji – j i+1) t2

junks = G j=1 ji

N

Mj

aji

rji3

…which totally neglects a large set of n-body simulations

• cosmological simulations

• gas

• solar system/ few body

• direct n-body simulations

• black holes with ‘realistic’ gravity

• non-equilibrium experiments --like mergers or collapse

If you’re interested in this stuff, we can do another tutorial later

We’ll concentrate on equilibrium galaxies