1. evolution, core collapse, binaries, collisions
Star clusters:
evolution, core collapse, binaries, collisions
Simon Portegies Zwart
University of Amsterdam
Frontiers in Numerical Gravitational Astrophysics
2nd Course of the International School on Astrophysical Relativity
«John Archibald Wheeler»
Erice (Sicily), June 27-July 5, 2008

2. Odysseus and Sicilia

4. Binary frequencies in YoDeCs
NGC ~30% (Stolte etal 2004)
Open clusters ~50%
Globular clusters %-20%
Galactic disc % -1

5. How hard is hard Star cluster: 106 kT Planetary system: << 1kT
Hard binary: kT
1 AU binary: kT
Algol: kT
1AU (bh, bh): kT
AM-CVn: kT
(bh, bh) [merge in 1 year]: kT
(N=10^6, R=5pc)

6. Simulation #2

7. Simulation #2

8. Simulation #2

9. Simulation #2

10. Simulation #2

11. Low Mass X-ray Binaries
𝑁 𝐿𝑀𝑋𝐵 𝑀 𝐺𝑎𝑙 ≈ = 10 −9
𝑁 𝐿𝑀𝑋𝐵 𝑚 𝐺𝑙𝑜𝑏 ≈ = 10 −7

12. Formation of black holes
Supernova progenitors
M > 20–25 M⊙ => black holes (in 1–10 Myr)
assume mbh ~ 10 M⊙ for now
Scalo (1986) mass function, 0.1–100 M⊙
1/2200 stars have M > 25 M⊙
=> 500 black holes in a globular cluster
50 million black holes in the Galay

13. N= black hole

14. 10 𝑀 𝑠𝑢𝑛
𝑟 ℎ𝑎𝑙𝑓
𝑟 𝑐𝑜𝑟𝑒

15. Black hole Dynamics
(Kulkarni, Hut, & McMillan 1993; Sigurdsson & Hernquist 1993)
black holes sink to the center by dynamical friction:
mass segregation time scale ~ tRh/ m
[tRh = half-mass relaxation time ~ 0.1–1 Gyr, m = mbh / 〈m〉 ~ 10]
BH subsystem reaches approximate dynamical equilibrium with half-mass radius
rbh ~ m-1/2 rcore

16. Core collapse in dense star clusters
Makino
Time

17. Black hole dynamics (2)
mass stratification instability when (Spitzer 1987)
bh > c ⇒ Nc < m5/2 Nbh
BH binary formation time scale (Spitzer 1969)
B ~ Nbh tR,bh
 0 as the BH subsystem collapses
⇒ dynamical BH binary formation

18. N= black holes

19. Binary Dynamics binary interactions ⇒ binary hardening (Heggie 1975)
− median 〈DEb/Eb〉 ~ 20%
binaries ultimately recoil out of the cluster
E b,min ~ 36 W0 m kT
[3kT = 〈mv 2〉, W0 = 〈m〉 ∣0∣ kT ]
for m ~ 10, 〈m〉 ~ 0.5 M⊙, W0 ~ 5, find
E b,min ~ (0.1–1) m  103 kT

21. Gualandris & PZ 2004

22. Properties of ejected binaries
~40% of black holes ejected in the form of binaries
~10-4 N ejected binaries per cluster
ejection time scale ~ few Gyr (~ scaled cluster relaxation time)
N-body calibration of orbital properties of ejected binaries (for mbh ~ 10 M⊙):
Binding energies Eb have < Eb/kT < (for  = 10), roughly flat in log Eb
eccentricities e approximately thermal [p(e) = 2e]

24. Gravitational wave radiation
GR merger time scale (Peters 1964):
tmrg ≃ 140 (M⊙/m bh )3 (a /R⊙)4 (1 – e 2 )7/2 Myr
relate binary parameters to bulk cluster properties by
kT = 2Ekin/3N = –Epot/ 3N = G M 2/6Nrvir
⇒ Eb/kT = 3N (mbh/Mtot)2 (rvir/a) ⇒
Gravitational wave radiation
𝑡 𝑚𝑟𝑔 ≃150𝑀𝑦𝑟  𝑀 𝑠𝑢𝑛 𝑚 𝑏ℎ  3  𝑎 𝑅 𝑠𝑢𝑛  − 𝑒

25. LIGO Detection Volume Deff ≃ 123 m105/6 q1/2 (1 + q)-1/6 Mpc
effective de binary with primary mass 10 m10 M⊙ and mass ratio q is
Deff ≃ 123 m105/6 q1/2 (1 + q)-1/6 Mpc
= 109 m105/6 Mpc for q = 1
ℜns ~ yr-1
ℜbh ~ to 0.2 yr-1

26. Some Caveats black hole properties at birth mass range of progenitors
black hole mass spectrum (Fryer & Kalogera 2001)
supernova kicks? (White & van Paradijs)
Assumed 100% retention
XTE J

27. Black hole dynamics dynamics of BH subsystem with a broad mass range
Binary formation, massive BH binary dominates
most black holes ejected singly by binary BH
left with a single massive BH binary: 50–100 M⊙
subsequent binary hardening/merger by dissipative stellar encounters
growth of single BH (e.g. Miller & Hamilton 2002)?
may create/eject fewer BH binaries, but merger may be detectable to much greater distances ( Deff  m105/6 ), if it still lies in the LIGO band