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Simulating the Evolution of Stellar Clusters

?

Jarrod Hurley


Equation of Motion
N

= -G ri

б
j =1

m j (ri - rj ) 3 | ri - rj |



Direct integration = O(N3) cost 4N large -> 106 + density contrast + close encounters + binaries + long-lived system


4add softening parameter?
N

= -G ri

б
j =1

m j (ri - rj ) 2 2 (| ri - rj | + e )

3/2

l prevents force singularity -> "collisionless" lreduces relaxation and mass-segregation lnot accurate for relaxation dominated systems



4neighbour schemes to reduce cost -> N2 log(N) 4other methods: tree-code? Fokker-Planck?
require DE/E < 10
-5

approximate + realism -> cpu

4Direct N-body is preferred method


Regularization for close encounters and binaries 3D -> 4D + t transformation 4improves efficiency 4greater accuracy 4extended to three or more bodies


Hierarchical time steps for density contrast йh | F Dti = а к |F|
1/2 й |^ ~ ф Dtn = а Ї к n 1^ ~ 2Ї -1

h Є 0.02, n ё 40



4individual timesteps 4advance "block" of particles together 4facilitates sub-system search and works with ... Hermite Integration Scheme 44th order force polynomial 4more accurate and less memory ... but direct N-body still expensive


... N-body saved by

the GRAPE

l built by astrophysicists at University of Tokyo (1990 - )
[Makino, Kokubo & Taiji 1993]

l GRAvity piPE l "Newtonian" accelerator for the force calculation loop + prediction + neighbour list l special-purpose hardware with hardwired logic l GRAPE-4 available 1996 -> Gflops performance -> open clusters of 10,000+ stars l GRAPE-6 available 2001 -> Tflops for $50k -> small globular clusters


NBODY4 software
l includes stellar evolution
4 fitted formulae as opposed to "live" or tables 4 done in step with the dynamics

l and a binary evolution prescription
4tidal evolution, magnetic braking, gravitational radiation, wind accretion, RLOF: mass transfer, common-envelope, mergers

l and as much realism as possible
4perturbed orbits (hardening & break-up), chaotic orbits, exchanges, triple & higher-order subsystems, collisions, etc. ... regularization techniques + external tidal field + Hermite integration with GRAPE + block time-step algorithm


Everything you need to know ...
Gravitational N-body Simulations: Tools and Algorithms Sverre Aarseth, 2003, Cambridge University Press

or
From NBODY1 to NBODY6: The Growth of an Industry Sverre Aarseth, 1999, PASP, 111, 1333

also
The Gravitational Million-Body Problem Douglas Heggie & Piet Hut, 2003, Cambridge University Press


General Results: 1. The Effect of Binaries
(N = 30,000 models)

0% binaries

50% binaries

Inclusion of primordial binaries 4fraction of escaping stars increases by ~50% 4velocity of escaping stars increases by ~20% 4evidence for saturation of primordial binary effects above ~25%


2. WD Mass Fractions 4at t/trh = 10 f
b,0

fWD
0.15 0.17 0.17

0.4 0.1 0.0

4evaporation & binaries are important


3. Supra-Chandrasekhar DWD Merger Rate l 2 WDs, Mb > 1.44 Msun, Tgrav < 12 Gyr 410x expected (non-dynamical) merger events l Blame for enhancement shared equally between:
4 exchange interactions 4pre-DWD perturbations 4post-DWD perturbations

l Type Ia supernova? l AIC collapse to NS?
4 interesting either way


An Example Primordial Binary:
M1 = 6.9 Msun M2 = 3.1 Msun a = 4050 Rsun

After 60 Myr:
M1 = 6.3 on AGB e = 0.0 (tides) RLOF => CE M1 = 1.25 ONeWD

After 430 Myr:
M2 = 2.0 on AGB M1 = 1.30 (symbiotic) RLOF => CE M2 = 0.8 COWD a = 2500 Rsun

DWD with t

grav

= 1022 yr


and then ...

1.3

DWD 9100 d

0.8

14000 d, e=0.63

630 Myr

Resonant Exchange (few Myr)

4perturbed: 6000d, e=0.94 4CE + CE -> DWD (0.35d)

2.0 MS

0.8 WD

4GR -> merger after 10 Gyr M = 1.6 Msun


Comparison with Data: Simulation of M67 H 12,000 single stars (KTG1993 IMF: 0.1 - 50 Msun) H 12,000 binaries (q: uniform, e: thermal, a: flat-log, max 50 au) H Z = 0.02 H Circular orbit at Rgc = 8 kpc H Plummer Sphere in virial equilibrium
4M ~ 18700 Msun 4 Rt = 32 pc 4Trh ~ 200 Myr 4s ~ 3 km/s 4nc ~ 200 stars/pc3 46-7 Gyr lifetime 4 4-5 weeks of GRAPE-6 cpu


4show CMD movie (animated gif)


Colour-Magnitude Diagram Legend:
single main-sequence (MS) star, MS-MS binary single white dwarf (WD) WD-WD binary MS-WD binary

Upper-Right Panel:
Cumulative radial profiles of selected sub-populations (at current time): single MS stars

[

active CV

]

MS-MS binaries single giants

MS star in binary (non-MS or WD companion) single WDs Blue Straggler (BS) sub-giant, giant, or supergiant star naked Helium star WD in binary (non-MS or WD companion) Neutron star or Black Hole (only shown if in binary) number density of stars in the core e.g. BS-WD binary cluster mass as fraction of initial cluster mass (scales from 1 to 0)

Lower-Right Panel:
Evolution of selected cluster properties to the current time:



model at 4 Gyr = M67?
u Age = 4 Gyr (Vandenberg & Stetson 2004) (OCD: Mermilliod 1996) (Fan et al. 1996) 4 4 4 4 4 4

u Metallicity ~ Solar

u Binary fraction ~ 50%

u Mass ~ 1300 Msun in luminous stars within 10 pc (Fan et al. 1996) u Tidal radius ~ 15 pc (Bonatto & Bica 2005) (Fan et al. 1996)

u Half-mass radius ~ 2.5 pc



R h = 3.8 pc

2040 Msun

R t = 15 pc


R h = 2.6 pc

ML,10 = 1340 Msun

R t = 15 pc


M67 model

Bonatto & Bica (2003) - M67 observed


M67 model

Bonatto & Bica (2003) - M67 observed


M67 model

observed


Updated from Bonatto & Bica (2005)

Corresponding model stars


Software: Mackey & Gilmore 2003



Stellar Populations



Observed CMD
29 blue stragglers + 1/2 in binaries + Nbs/Nms,2to ~ 0.15 (high) + Rh,BS = 1.6 pc (cf. 2.5 pc) Simulation: 21 blue stragglers + 1/2 in binaries + Nbs/Nms,2to = 0.18 + Rh,BS = 1.1 pc 50%+ via dynamics


BS binary orbital parameters


l investigate other populations
46 RS CVn binaries (cf. Belloni, Verbunt & Mathieu 1998) 4 white dwarfs

l predictions for future observations
4e.g. BY Draconis X-ray binaries

l constrain initial conditions
4alternative binary period distributions ruled out by blue straggler analysis

l understand cluster evolution
4 luminosity functions -> mass segregation -> initial mass function? 4 core collapse 4binary "burning" 4 nucleosynthesis ... globular clusters with GRAPE-8



Some papers ...
A Complete N-body Model of the Old Open Cluster M67 Hurley, Pols, Aarseth & Tout, 2005, submitted White Dwarf Sequences in Dense Star Clusters Hurley & Shara, 2003, ApJ, 589, 179 Star Clusters as Type Ia Supernova Factories Shara & Hurley, 2002, ApJ, 571, 830

Collaborators

Sverre Aarseth Christopher Tout Onno Pols Rosemary Mardling Mike Shara


Hermite Integration: