simulating the evolution of stellar clusters...white dwarf sequences in dense star clusters hurley...
TRANSCRIPT
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Jarrod Hurley
Simulating the Evolution of Stellar Clusters
?
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Equation of Motion
†
˙ ̇ r i = -Gmj (ri - rj )| ri - rj |3j=1
N
Â
Direct integration = O(N3) cost 4N large -> 106
+ density contrast + close encounters + binaries + long-lived system
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†
˙ ̇ r i = -Gmj (ri - rj )
(| ri - rj |2 + e2 )3/2j=1
N
Â
4add softening parameter?
l prevents force singularity -> “collisionless”lreduces relaxation and mass-segregationlnot accurate for relaxation dominated systems
4neighbour schemes to reduce cost -> N2 log(N)4other methods: tree-code? Fokker-Planck?
require DE/E < 10-5approximate + realism -> cpu
4Direct N-body is preferred method
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Regularization for close encounters and binaries
3D -> 4D+ t transformation
4improves efficiency4greater accuracy4extended to three or more bodies
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Hierarchical time steps for density contrast
†
Dti =h | F || ˙ ̇ F |
Ê
Ë Á
ˆ
¯ ˜
1/2
Æ Dtn =12
Ê
Ë Á
ˆ
¯ ˜
n-1
h ª 0.02, n £ 40
4individual timesteps4advance “block” of particles together4facilitates sub-system search
and works with ...Hermite Integration Scheme
44th order force polynomial4more accurate and less memory
... but direct N-body still expensive
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... N-body saved by the GRAPE
l built by astrophysicists at University of Tokyo (1990 - )
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
[Makino, Kokubo & Taiji 1993]
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l includes stellar evolution
l and a binary evolution prescription
l and as much realism as possible
4fitted formulae as opposed to “live” or tables4done in step with the dynamics
4tidal evolution, magnetic braking, gravitational radiation, wind accretion, RLOF: mass transfer, common-envelope, mergers
4perturbed orbits (hardening & break-up), chaotic orbits, exchanges, triple & higher-order subsystems, collisions, etc. ... regularization techniques
NBODY4 software
+ external tidal field+ Hermite integration with GRAPE+ block time-step algorithm
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Everything you need to know ...
Gravitational N-body Simulations: Tools and AlgorithmsSverre Aarseth, 2003, Cambridge University Press
orFrom NBODY1 to NBODY6: The Growth of an IndustrySverre Aarseth, 1999, PASP, 111, 1333
alsoThe Gravitational Million-Body ProblemDouglas Heggie & Piet Hut, 2003, Cambridge University Press
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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%
General Results: 1. The Effect of Binaries(N = 30,000 models)
0% binaries
50% binaries
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4at t/trh = 10fb,0 fW D
0.4 0.15
0.1 0.17
0.0 0.17
2. WD Mass Fractions
4evaporation & binaries are important
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l 2 WDs, Mb > 1.44 Msun, Tgrav < 12 Gyr
4exchange interactions4pre-DWD perturbations4post-DWD perturbations
l Type Ia supernova?l AIC collapse to NS?
410x expected (non-dynamical) merger events
3. Supra-Chandrasekhar DWD Merger Rate
l Blame for enhancement shared equally between:
4interesting either way
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An Example
Primordial Binary:M1 = 6.9 MsunM2 = 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 AGBM1 = 1.30 (symbiotic)
RLOF => CE M2 = 0.8 COWD a = 2500 Rsun
DWD with tgrav = 1022 yr
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and then ...
ResonantExchange(few Myr) 4perturbed: 6000d, e=0.94
4CE + CE -> DWD (0.35d)
4GR -> merger after 10 Gyr M = 1.6 Msun
630 Myr
2.0 MS
1.3 0.8
0.8 WD
DWD9100 d
14000 d, e=0.63
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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 = 0.02H Circular orbit at Rgc = 8 kpcH Plummer Sphere in virial equilibrium
4M ~ 18700 Msun4Rt = 32 pc4Trh ~ 200 Myr4s ~ 3 km/s4nc ~ 200 stars/pc3
46-7 Gyr lifetime44-5 weeks of GRAPE-6 cpu
Comparison with Data: Simulation of M67
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4show CMD movie (animated gif)
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Colour-Magnitude Diagram Legend:single main-sequence (MS) star, MS-MS binary single white dwarf (WD)
WD-WD binary
MS-WD binary
MS star in binary (non-MS or WD companion)
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)
active CV[ ]
e.g. BS-WD binary
Upper-Right Panel:Cumulative radial profiles of selected sub-populations (at current time):
single MS stars
MS-MS binaries
single giants
single WDs
Lower-Right Panel:Evolution of selected cluster properties to the current time:
number density of stars in the core
cluster mass as fractionof initial cluster mass (scales from 1 to 0)
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model at 4 Gyr = M67?
u Age = 4 Gyr (Vandenberg & Stetson 2004) 4
u Metallicity ~ Solar (OCD: Mermilliod 1996) 4
u Binary fraction ~ 50% (Fan et al. 1996) 4
u Mass ~ 1300 Msun in luminous stars within 10 pc 4 (Fan et al. 1996)
u Tidal radius ~ 15 pc (Bonatto & Bica 2005) 4
u Half-mass radius ~ 2.5 pc (Fan et al. 1996) 4
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2040 MsunRh = 3.8 pc Rt = 15 pc
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Rh = 2.6 pc ML,10 = 1340 MsunRt = 15 pc
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Bonatto & Bica (2003) - M67 observedM67 model
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Bonatto & Bica (2003) - M67 observedM67 model
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M67 model observed
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Updated from Bonatto & Bica (2005)
Corresponding model stars
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Software:Mackey & Gilmore 2003
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Stellar Populations
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Observed CMD29 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
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BS binary orbital parameters
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46 RS CVn binaries (cf. Belloni, Verbunt & Mathieu 1998)4white dwarfs
l investigate other populations
l predictions for future observations
l constrain initial conditions
l understand cluster evolution
4e.g. BY Draconis X-ray binaries
4alternative binary period distributions ruled out by blue straggler analysis
4luminosity functions -> mass segregation -> initial mass function?4core collapse4binary “burning”4nucleosynthesis
... globular clusters with GRAPE-8
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Star Clusters as Type Ia Supernova FactoriesShara & Hurley, 2002, ApJ, 571, 830
White Dwarf Sequences in Dense Star ClustersHurley & Shara, 2003, ApJ, 589, 179
Some papers ...
Collaborators Sverre AarsethChristopher ToutOnno PolsRosemary MardlingMike Shara
A Complete N-body Model of the Old Open Cluster M67Hurley, Pols, Aarseth & Tout, 2005, submitted
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Hermite Integration: