Steve McMillan Department of Physics Drexel University - - PowerPoint PPT Presentation

• ✁

• Steve McMillan

Department of Physics Drexel University

compact binary formation scenarios

mass transfer needed to create close systems

stellar winds

hard to make black-hole binaries

(old) estimates of raw merger rates: Rns

2–4 × 10−7 h 3 yr−1 Mpc−3 Rbh

2 × 10−9 h 3 yr−1 Mpc−3

[h ≡ H0 / 100 km s−1 Mpc−1]

(Narayan et al. 1991; Phinney 1991; Tutukov & Yungelson 1993)

LIGO-I: Deff ~ 20 Mpc for 1.4 M

NS binaries, 100 Mpc for 10 M

BH binaries, so (h = 0.65): rns

2–4 × 10−3 yr−1 rbh

2 × 10−3 yr−1

advanced LIGO: rates up by factor of

100 – 1000

alternative scenario:

create black hole binaries by dynamical processes in star clusters

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

7.1 × 10−4 of stars have M > 20 M

4.5 × 10−4 of stars have M > 25 M

for N stars, expect ~ 6 × 10−4 N black holes

(Kulkarni, Hut, & McMillan 1993; Sigurdsson & Hernquist 1993)

Black holes sink to the center by dynamical friction: mass segregation time scale

tRh / µ

[tRh = half-mass relaxation time

0.1–1 Gyr, µ = mbh /

m

10]

Black hole subsystem reaches approximate dynamical equilibrium with half-mass radius rbh

µ−1/2 rc [rc = cluster core radius]

Cluster core collapse: ρc

rc

−2

Nc

−2

rbh /rc

µ (Nbh / Nc)3

µ5/2 (Nbh / Nc )

Mass stratification instability (Spitzer 1987) when ρbh > ρc

Nc < µ5/2 Nbh

Black-hole binary formation time scale (Spitzer 1969) τB

Nbh tR,bh

as the BH subsystem collapses

dynamical BH binary formation

Binary interactions

binary hardening (Heggie 1975) − median

∆Eb/Eb

20%

Binaries ultimately recoil out of the cluster E b,min

36 W0 µ kT [3kT =

mv 2

W0 =

m

φ0

kT ]

For µ

10,

m

0.5 M

, W0

5, E b,min

(0.1–1) µ × 103 kT

(Portegies Zwart & McMillan 2000)

40% of black holes ejected in the form of binaries

10−4 N ejected binaries per cluster

ejection time scale

few Gyr

Distribution of orbital properties (for mbh

10 M

)

binding energies Eb have 103 < Eb/kT < 104, roughly flat in log Eb

eccentricities e approximately thermal [p(e) = 2e]

GR merger time scale (Peters 1965) tmrg ≈ 150 (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)

tmrg ≈ 3000 m-4 µ10

5 (M6/R5)-4 (Eb/103 kT)-4 (1 – e 2 )7/2 Gyr

(µ10 = mbh/10M

, M6 = Mtot/106 M

, R5 = rvir/5 pc, m = Mtot/NM

)

dp / d log t mrg

tmrg ≈ 3000 m−4 µ10

5 (M6/R5)−4 Gyr (Eb/103 kT)−4 (1 – e 2 )7/2

τ ≡ log10 tmrg = log10 T0 – 4 log10 (Eb/103 kT) + 7/2 log10 (1 – e 2 ) uniform in [~0, log µ]

uniform in [0, 1]

dp / d τ [τ ≡ log10 t mrg] 3.2 (tmrg /T0 ) 2/7 0.25 [1 − (tmrg /T0 ) 2/7] 10-4 T0 T0 (m = 0.4)

Average over distributions in Eb and e

peak tmrg at

0.3 m−4 µ10 (M6 /R5 )−4 Gyr

• pen clusters

R5

0.2, M6

0.02

105 Gyr

globular clusters (take m

0.4) R5

1, M6

1

10 Gyr

nuclear clusters R5

0.1, M6

0.1

10 Gyr

Specific cluster frequency (van den Bergh 1984)

NGC = SN 10–0.4 (Mv + 15)

galaxy density Mv SN GC density type

[10-3 h3 Mpc-3]

[h3 Mpc-3] E–S0 3.49

• 20.7

10 6.65 Sa–c 9.00

• 19.5

3.0 1.73 Blue E 1.87

• 19.6

14 1.81

GC number density φGC ≈ 10 h 3 Mpc−3

Galactic globular cluster parameters log Mtot (M

) = 5.5

0.5 log rvir (pc) = 0.5

0.3

combine merger time scales with net globular cluster density

merger rate per unit volume of black-hole binaries formed in globular clusters RGC

6 × 10−8 h 3 yr−1 Mpc−1

Effective distance for LIGO-I detection of the inspiral of a black-hole binary with primary mass 10 µ10 M

and mass ratio q is Deff ≈ 123 µ10

5/6 q1/2 (1 + q)−1/6 Mpc

= 109 µ10

5/6 Mpc for q = 1

LIGO-I detection rate rGC

• 0.3 h 3 yr−1
• 0.09 yr−1 for h = 0.65

advanced LIGO: rates up by

• 100 – 1000

Mtot

104 – 105 M

, rvir

0.1 – 0.5 pc

Numbers not well known, but

Dutra & Bica (2000) find 58 candidates within ∼600 pc (in projection) of the Galactic Center

Portegies Zwart et al. (2001) find that most clusters may be undetectable for most of their lifetimes

Suppose SN comparable to value for globulars

RGC

5 × 10−8 h 3 yr −1 Mpc−1

comparable to the globular cluster rate

continuous formation!

black hole properties

cluster formation history

initial cluster parameters

cluster dynamics in an external field − large exponents in uncertain quantities! tmax

0.3 m−4 µ10 (M6 /R5 )−4 Gyr

up to

70k stars, Scalo mass function, 0.01–100 M

0 to

20% binaries, contact to few tens of A.U.

tidally limited cluster, dissolution time

few Gyr

initial mass

3 × 104 M

, virial radius

10 pc

no BH kicks/scaled BH kicks

“ maximal” BH mass = CO core mass

50 black holes formed in first few tens of Myr

typical masses (M

): 47, 32, 29, 19, 17, 16, ..., <10

black hole mass spectrum (e.g. Fryer & Kalogera 2001)

relation to progenitor mass

effect of metallicity

black hole kick velocities

• riginal analysis assumed 100% BH retention

what is expected kick velocity distribution?

“ scaled down” neutron star kicks?

Fryer & Kalogera (2001)

black hole mass spectrum (e.g. Fryer & Kalogera 2001)

relation to progenitor mass

effect of metallicity

black hole kick velocities

• riginal analysis assumed 100% BH retention

what is expected kick velocity distribution?

“ scaled down” neutron star kicks?

dynamics of BH subsystem with a broad mass range

binary formation (massive BH binary dominates?)

black hole ejection (40 of 50 in 500 Myr)

BH binary ejection—can we eject any? (1–2 in this run)

may create/eject fewer BH binaries, but may be visible to much greater distances ( Deff ∝ µ10

5/6 )

globular cluster formation history

early/extended/continuous/starburst

globular cluster masses and radii at birth

current clusters smaller and more massive in past

but most clusters dissolved long ago

cluster mass function

affects numbers and properties of black holes

also affects cluster survival—more black holes mean cluster is more likely to disrupt

29 M

22 M

47 M

t [Myr]

16 M

23 M

do black holes get kicks?

what is the black hole mass function?

how is this affected by metallicity?

how does it affect the black hole dynamics?

can we distinguish dynamically formed black-hole binaries from those formed by binary evolution?

what is the cluster formation rate?

what were the initial cluster parameters?