Gravasco 2013

Stellar-mass black holes in a globular cluster

Douglas Heggie

University of Edinburgh, UKd.c.heggie@ed.ac.uk

Gravasco 2013

Motivation● Discovery of two stellar-mass BH in the Galactic globular cluster M22 (Strader+ 2012)● Strader talk at MODEST 12 (Kobe, Japan, August 2012) ● Total population ~5-100 (assuming accretion from white dwarf companion)● Theoretical prediction from 1993: expect 0 (or 1 or 2)● 1 BH in M62, 3 other GC* with no observable BH (Strader, pers. comm., 2013)

opticalradio

*M15, M19, and Terzan 5

● Other observational evidence: Variable X-ray sources (Shih+ 2010,Brassington+ 2010)

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OutlineIntroduction

● Observational motivation: M22● The theoretical problem● Numerical evidence

The tools for attacking the problem● Simulating star cluster evolution● N-body techniques● Monte Carlo

Application to M22● Fitting observational data● The expected BH population of M22

A revised theoretical formulation● The four stages of evolution● Mass segregation, core collapse, binary action● Balanced evolution● Energetics and escape of BH binaries● Evolution and depletion of the BH population● Interpretation of simulation data

Summary

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How many are created by normal stellar evolution?● About 0.3% by number● About 3% by mass ● Depends on IMF (initial mass function) of stars● See also talk by Ziosi (tomorrow) for dependence on metallicity

How many will remain at the present day?

● 80% of stars have escaped (H&G 2008), so 200?● But BH are centrally segregated, and so perhaps the number is much greater...● Did natal kicks eject most BH at birth?

May be similar to neutron star kicks (σ ~ 190km/s)May depend on primordial massOften parametrised as “retention fraction”

● What about dynamical kicks?

How many stellar-mass black holes do you expect to find?

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Dynamical kicks: The traditional viewTheory: Kulkarni, Hut & McMillan (1993), Sigurdsson & Hernquist (1993)

1. By two-body encounters (relaxation) BH try to achieve equipartition. The result isthat the BH mass-segregate.

2. Cluster is Spitzer-unstable (Spitzer 1969): BH subsystem cannot achieve equipartition3. BH subsystem forms a compact, almost isolated subsystem at centre of cluster4. 3-body encounters between BH occasionally cause escape. Timescale < 1Gyr.5. Thereafter, star cluster evolves without BH: essentially none (~ 1 or 2) at present day

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But ....

Two reasons why the time scale might be wrong. The BH subsystem is not isolated because

1. It is still in thermal contact with the low-mass stars● While the BH interactions are trying to heat the BH subsystem up, interactions

with the low-mass stars continue to cool it down.

2. It is in a deep potential well of the low-mass stars● Low-mass stars contribute ~90% of the depth of the potential well● This makes ejection of single and binary BH much harder

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Results from N-body simulations

1. Mackey+ 2008: N = 105, c = 200M

⊙pc-3, 100% retention of BH

Plot shows evolution of number of single and binary BH up to 10.6Gyr2. Merritt+ 2004: N = 103, 100% retention of BH

“Majority” ejected after ~5trh(0)

3. Aarseth 2012: N = 105, Rvir

= 1pc, ~10% retention fraction

~50% escape by 1Gyr4. Banerjee+ 2010: N ≤ 105, R

h ≤ 1pc, 50-100% retention fraction, no tide

~0 after 800Myr for 50% retention5. Heggie (unpub): N ≈ 4.9x105, R

h = 0.58 pc, ~50% retention fraction

leaving ~500, but 5 after 11.4Gyr6. Sippel & Hurley 2012: N = 2.5x105, R

h = 6.2 pc, 10% retention fraction

16 at 12 Gyr (see later)7. Hurley & Shara 2012: N = 2x105, R

h = 4.7 pc, retention fraction ?

4 at 11.5 Gyr

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Results from Monte Carlo simulations

NGC 6397 47 Tuc M4

● Numbers of stellar-mass BH against time● These models are designed to resemble stated cluster at the present day● All results from papers by Giersz & Heggie

Retention factor 100% Retention factor 100%Natal kicks, = 160/190 km/s

See also Morscher+ 2012: N = 3x105, rh = 2.44pc, retention fraction 86%. ~400 at 12 Gyr

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Clausen+ 2013: “We further assumed that nearly all BHs formed in the cluster were promptly ejected by self interaction, leaving 0, 1, or 2 BHs (Kulkarni et al. 1993; Sigurdsson & Hernquist 1993...”

Repetto+ 2012: “Kulkarni et al. 1993, Sigurdsson & Hernquist....form BH-BH binaries.Sequential dynamical interactions between these binaries and single BHs would lead to the ejection of the Bhs from the cluster in a timescale shorter than 10∼ 9

years.”Murphy+ 2011: “Stellar mass black holes are not included in these models ….This

central cluster of black holes would then form binaries that would then cause them to be ejected from the system a relatively short time (Kulkarni et al. 1993;Sigurdsson & Hernquist 1993).”

Heggie&Giersz 2008: “This shows that no stellar-mass black holes are expected to be present....The mechanisms which give rise to the ejection of black holes are well known (Kulkarni, Hut & McMillan 1993; Sigurdsson & Hernquist 1993;....”

Old ideas die slowly

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How many black holes should globular clusters contain?

● Theory is not to be trusted● The retention fraction is important● The result depends on the initial conditions, i.e. on the cluster

Black holes in M22

● We need initial conditions tuned to M22, and a simulation of its evolution● The retention fraction is an important unknown parameter

We need better theory (at least to understand the simulations)

Lessons from theory and simulations

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● N-body methods: NBODY6, starlab, AMUSE, ….

● Monte Carlo methods

How to simulate M22Giersz & H, 2013, MNRAS, sub

Computing time ∝ N3 Computing time ∝ N

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The time taken for a simulationN-body Monte Carlo

Based on simple theoretical scaling with N (number of stars) and R (radius of the cluster)

Data: Harris catalogue

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Stellar evolution: set by stellar evolution models

Two-body relaxation: time scale t

r N∝ 1/2R3/2 = N*1/2R*3/2. Therefore choose R* = R(N/N*)1/3 >> R

Not everything scales in the same way with N:Interactions of binariesThe escape speed (relevant for BH kicks)Stellar collisionsEscape time scale

Application to M22 (Sippel & Hurley 2013)

● N-body, N = 2.5x105, retention factor 11% ● t

rh and r

h/r

c close to M22 values at 12 Gyr

N-body models: scaling downThe idea: model a cluster with N stars by a model with N*<<N starsThe principle: get the time scale of the major evolutionary effects correct

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Monte Carlo code: the basic idea

Assume spherical symmetry.

Orbit of star characterisedby energy, E, and angularmomentum, J.

The code repeatedly alters E, J to mimic the effects ofgravitational encounters, using theory of relaxation.

Includes:● Stellar evolution● Binary evolution● Galactic tide● Binary dynamics

(binary-binary andbinary-single encounters)

Mirek Giersz

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Finding initial conditionsMethod 1. By trial and errorMethod 2. Automatically

Technique:● Specify initial conditions by N (number of stars), r

h, r

t, initial King parameter W

o,

three parameters specifying the piecewise power-law IMF. Call the parameter vector x● Devise a measure of goodness of fit of the evolved model (e.g. at 12Gyr) to the desired

cluster. This is on 2 lines, and involves the surface brightness profile, the velocity dispersion profile, the local luminosity function(s), pulsar accelerations, .... Call this A

● Optimise A(x) using the downhill simplex method (Nelder & Mead 1965, Press+ 1994)

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Finding initial conditions for M22

Fig. showing evolution of N, AFig. showing scatter plot of A,N

Best initial conditions (100% retention of BH)N = 7.8x105, r

h = 2.5pc, r

t = 102 pc, W

o= 2.9

IMF: power law index 0.9/2.7 below/above 0.67 M⊙pc

No technique yet for estimating confidence intervals

Convergence of N(0) Goodness of fit v. N(0)

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Example of fit

Surface brightness Velocity dispersion Luminosity function

(Deficient in bright stars)

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Conclusions:● Up to 100 stellar-mass BH may remain,

depending on assumption about natal kicksand high-mass IMF

● Up to two BH-WD binaries

Model No. of WD/BH binaries at 12 Gyr

100% retention 2

Fallback* 1

kick = 90 km/s 1 (at 10.5 Gyr)

*kick reduced by fraction of envelope mass which falls back onto the remnant

● with kick = 265 km/s no BH remain● model with kick = 90 km/s

not quite complete● with 100% retention: binaries are

2 BH/MS, 2WD/BH, 2 BH/BH

Evolution of the number of stellar-mass black holes (Full-size MC

models)

Cf Sippel & Hurley 2012 (scaled N-body model)● N-body, N = 2.5x105, retention factor 11% ● t

rh and r

h/r

c close to M22 values at 12 Gyr

● 16 BH at 12 Gyr (needs scaling)● 2 in BH-MS binary, 2 in BH-BH binary

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A revised theoretical view

References: Breen & H,MNRAS, 2013

Consider 2-component system: low-mass stars m

1, total mass M

1;

black holes m2, total mass M

2;

initially identically distributed in space

Consider regime m2 >> m

1 and M

2 << M

1

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Stage 1: Mass segregation, core collapseStage 2: binary formation, binary heating

1(a) Black holes mass-segregate(two-body encounters, tending toequipartition of kinetic energy)

1(b) Cluster is Spitzer-unstable (Spitzer 1969)BH subsystem cannot achieve equipartition,but keeps shrinking until it dominates thedensity at the centre.

1(c)The BH subsystem goes into core collapse(Hénon 1961, von Hoerner 1963), formingvery high central density.

2(a)Three-body interactions become important,forming “three-body binaries”

2(b) Formation of binaries, and their interaction withsingle BH, are exothermic processes. They feed energy to the core of the BH subsystem.

What happens next? It depends on whether you think the BH are an isolated system, or not.If it is isolated, the BH subsystem evaporates on its own short relaxation time scale (Kulkarni, Hut & McMillan (1993), Sigurdsson & Hernquist (1993)

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Stage 3: Balanced evolution

BH

low-mass stars

half-mass radiusof whole system

Energycreated

Energy flows throughBH subsystem

Energy transferred to low-mass stars

Energy flows throughlow-mass stars

Hénon’s PrincipleHénon 1961, 1975

“The core adjusts to produce the energy required at the half-mass radius”

cf Eddington’s insight into stellar structure

If the core produces too much energy,the core expands, suppressing energyproduction. Similarly if it produces toolittle.

The core of the BH subsystem producesthe energy, but the low-mass stars (whichdominate at the half-mass radius)determine how much is produced.

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How much energy does a BH produce?

1. Binaries are formed in interactions between three single BH

2. Interactions between binary and single BH cause escape of BH

3. Each escape of a BH gives energy mBH to the cluster,

where (>0) is the depth of the potential well

4. Therefore the heating rate is proportional to the rate of escape of BH:

E∼−MBH φ

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The rate of escape of BH

E∼−MBH φ

The rate of heating

is determined by the energy flux at the half-mass radius

where E is total energy of entire system, trh

is the half-mass relaxation time.

Therefore

is dominated by the low-mass stars. Thus the rate of loss of BH isdetermined by the properties of the whole system, which is dominatedby low-mass stars.

E∼−Et rh

MBH∼E

φ trh

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Stage 4: Exhaustion of BH energy source● Using “Hénon's Principle” core and half-mass radii of BH subsystem

can be estimated:R

cBH/R

hBH ∝ (40/N

BH)2/3

● If this formula predicts RcBH

> RhBH

it means that the BH subsystem can

no longer produce the required energy – there are too few BH left. Condition is roughly N

BH < 40

● Then the core of low-mass stars starts to contract (to increase the energy generated in interactions between the BH and low-mass stars)

● Eventually (almost) all BH have escaped, and then the system behaves like a system of low-mass stars only

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Interpretation of N-body dataStage 1. Mass segregation of BH: t < 250; core collapse in BH subsystemStage 2. BH binary formation and heating, initiating expansion Stage 3. Balanced evolution: to t ~ 5000Stage 4. Exhaustion of the BH subsystem; recollapse of the core of the light

stars to enhance the energy generation by residual BH binaryStage 5. Core collapse of the light stars (not shown)Stage 6. Balanced expansion powered by binary activity in the light stars (not shown)

N=64K, mBH

/<m> = 20, M

BH/M

= 0.02

Plots show core radius against time

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Illustration: Monte Carlo model of M4

Time (Myr)

Model of Heggie & Giersz 2008

Stage 1. Mass segregation of BH; core collapse in BH subsystem at t ~ 15MyrStage 2. BH binary formation and heatingStage 3. Balanced evolution (assisted by mass loss from stellar evolution) to ~ 5GyrStage 4. Exhaustion of BH energy production; collapse of low-mass components to enhance energy production Stage 5: Core collapse of the low-mass components t ~ 8GyrStage 6: Balanced evolution powered by primordial binaries (unstable) from t ~ 8Gyr to the present day

● It is no coincidence that the last BH is lost at the time of “core collapse”● Suggests stellar-mass BH to be found in clusters with large core radii

This model also has stellar evolution and a population of primordial binaries

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Summary

1. BH subsystems survive for a few Gyr and maybe a Hubble time depending on the cluster and assumptions on kicks

2. Mass loss and spatial evolution of BH subsystem may be understood on the basis of simple but quantitative arguments

But we have ignored the second generation problem (see talk later by Mastrobuono-Battisti)