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: 2–4 × 10 − 7 h 3 yr − 1 Mpc − 3 R ns 2 × 10 − 9 h 3 yr − 1 Mpc − 3 R bh [ h � ≡ H 0 / 100 km s − 1 Mpc − 1 ] (Narayan et al. 1991; Phinney 1991; Tutukov & Yungelson 1993)

✟ ✡ ☞ ✠ ✟ ☛ ✌ ☛ ✠ ✡ ✠ ✟ ✟ LIGO-I: D eff ~ 20 Mpc for 1.4 M NS binaries, 100 Mpc for 10 M BH binaries, so ( h = 0.65): 2–4 × 10 − 3 yr − 1 r ns 2 × 10 − 3 yr − 1 r bh advanced LIGO: rates up by factor of 100 – 1000 alternative scenario: create black hole binaries by dynamical processes in star clusters

✍ ✑ ✑ ✏ ✔ ✑ ✎ ✓ ✒ ✑ ✏ ✎ ✍ ✑ Supernova progenitors ⇒ black holes (in 1–10 Myr) M > 20–25 M assume m bh 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: t Rh / µ mass segregation time scale 0.1–1 Gyr, µ = m bh / [ t Rh = half-mass relaxation time 10] m Black hole subsystem reaches approximate dynamical equilibrium with half-mass radius µ −1 /2 r c r bh [ r c = cluster core radius] − 2 − 2 Cluster core collapse: ρ c N c r c µ ( N bh / N c ) 3 µ 5/2 ( N bh / N c ) r bh / r c

✤ ★ ✩ ✪ ✘ ★ ✥ ✩ ✧ ✦ ✦ ✥ ✤ ✧ Mass stratification instability (Spitzer 1987) when ρ bh > ρ c N c < µ 5/2 N bh Black-hole binary formation time scale (Spitzer 1969) τ B N bh t R,bh 0 as the BH subsystem collapses dynamical BH binary formation

✫ ✳ ✱ ✲ ✳ ✫ ✱ ✯ ✴ ✯ ✴ ✬ ✯ ✯ ✵ ✮ ✰ ✯ ✭ ✫ ✬ ✫ ✰ ✯ ✮ ✬ Binary interactions binary hardening (Heggie 1975) ∆ E b / E b − median 20% Binaries ultimately recoil out of the cluster 36 W 0 µ kT E b,min φ 0 [ 3 kT = kT ] mv 2 W 0 = m For µ 10, m 0.5 M , W 0 5, (0.1–1) µ × 10 3 kT E b,min

✶ ✹ ✹ ✹ ✺ ✸ ✷ ✶ ✸ ✸ ✸ ✶ ✶ ✷ ✹ (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 m bh 10 M ) binding energies E b have 10 3 < E b / kT < 10 4 , roughly flat in log E b eccentricities e approximately thermal [ p(e) = 2 e ]

✻ ✽ ✽ ✽ ✽ ✾ ✾ ✻ ✽ ✼ ✻ ✻ ✻ ✼ ✻ ✻ ✻ GR merger time scale (Peters 1965) t mrg ≈ 150 ( M / m bh ) 3 ( a /R ) 4 (1 – e 2 ) 7/2 Myr Relate binary parameters to bulk cluster properties by kT = 2 E kin /3 N = – E pot / 3 N = G M 2 /6 Nr vir E b / kT = 3 N ( m bh / M tot ) 2 ( r vir / a ) t mrg ≈ 3000 m -4 µ 10 5 ( M 6 / R 5 ) -4 ( E b /10 3 kT ) -4 (1 – e 2 ) 7/2 Gyr ( µ 10 = m bh /10 M , M 6 = M tot /10 6 M , R 5 = r vir /5 pc, m = M tot / NM )

dp / d log t mrg

✿ t mrg ≈ 3000 m − 4 µ 10 5 ( M 6 / R 5 ) − 4 Gyr ( E b /10 3 kT ) − 4 (1 – e 2 ) 7/2 τ ≡ log 10 t mrg = log 10 T 0 – 4 log 10 ( E b /10 3 kT ) + 7/2 log 10 (1 – e 2 ) uniform in [~0, log µ ] uniform in [0, 1]

10 -4 T 0 0.25 [1 − ( t mrg / T 0 ) 2/7 ] dp / d τ 3.2 ( t mrg / T 0 ) 2/7 T 0 ( m = 0.4) [τ ≡ log 10 t mrg ]

❀ ❂ ❃ ❃ ❃ ❃ ❂ ❃ ❀ ❄ ❂ ❃ ❄ ❃ ❃ ❂ ❁ ❄ Average over distributions in E b and e 0.3 m − 4 µ 10 ( M 6 / R 5 ) − 4 Gyr peak t mrg at open clusters 0.2, M 6 0.02 10 5 Gyr R 5 globular clusters (take m 0.4 ) R 5 1, M 6 1 10 Gyr nuclear clusters R 5 0.1, M 6 0.1 10 Gyr

❁ ❂ Specific cluster frequency (van den Bergh 1984) N GC = S N 10 –0.4 (M v + 15) galaxy density M v S N GC density [ h 3 Mpc -3 ] type [10 -3 h 3 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 φ GC ≈ 10 h 3 Mpc − 3

❅ ❇ ❉ ❈ ❅ ❇ ❆ Galactic globular cluster parameters log M tot ( M ) = 5.5 0.5 log r vir (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 6 × 10 − 8 h 3 yr − 1 Mpc − 1 R GC

❊ ❆ ● ❋ ● ❈ ● ❅ ❋ ❊ Effective distance for LIGO-I detection of the inspiral of a black-hole binary with primary mass 10 µ 10 M and mass ratio q is 5/6 q 1/2 (1 + q ) − 1/6 Mpc D eff ≈ 123 µ 10 = 109 µ 10 5/6 Mpc for q = 1 LIGO-I detection rate 0.3 h 3 yr − 1 r GC 0.09 yr − 1 for h = 0.65 advanced LIGO: rates up by 100 – 1000

❅ ■ ■ ■ ❍ ❈ ❅ ■ ❍ ❅ ❍ ❆ 10 4 – 10 5 M 0.1 – 0.5 pc M tot , r vir 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 S N comparable to value for globulars 5 × 10 − 8 h 3 yr − 1 Mpc − 1 R GC 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! 0.3 m − 4 µ 10 ( M 6 / R 5 ) − 4 Gyr t max

P ❲ ❜ ❛ ❵ ❴ ❫ ❪ ❭ ❬ ❩ ◗ ❳ ❨ ◗ ❙ P ❘ ◗ ❯ ❚ ❱ ◗ 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 3 × 10 4 M initial mass , 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 original analysis assumed 100% BH retention what is expected kick velocity distribution? “ scaled down” neutron star kicks?

Fryer & Kalogera (2001)

❦ ❦ ❧ ♠ ♥ ♦ ♣ q r black hole mass spectrum (e.g. Fryer & Kalogera 2001) relation to progenitor mass effect of metallicity black hole kick velocities original analysis assumed 100% BH retention what is expected kick velocity distribution? “ scaled down” neutron star kicks?

s ② ✉ s t ✉ ✈ ✇ t ① ③ 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 ( D eff ∝ µ 10 5/6 )

④ ❶ ⑤ ⑥ ⑦ ⑧ ④ ⑨ ❷ ⑩ 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

❸ ❸ ❹ ❺ ❻ 29 M 22 M 47 M t [Myr]

❼ ❼ ❽ ❾ 16 M 23 M

❿ ➅ ❿ ❿ ➀ ➁ ➂ ❿ ➃ ➆ ➄ 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?