Stellar-mass black holes in a globular cluster

Transcription

1 Stellar-mass black holes in a globular cluster Douglas Heggie University of Edinburgh, UK d.c.heggie@ed.ac.uk

2 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) optical radio Other observational evidence: Variable X-ray sources (Shih+ 2010, Brassington+ 2010) * M15, M19, and Terzan 5

3 Outline Introduction 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

4 How many stellar-mass black holes do you expect to find? 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 mass Often parametrised as retention fraction What about dynamical kicks?

5 Dynamical kicks: The traditional view Theory: Kulkarni, Hut & McMillan (1993), Sigurdsson & Hernquist (1993) 1. By two-body encounters (relaxation) BH try to achieve equipartition. The result is that the BH mass-segregate. 2. Cluster is Spitzer-unstable (Spitzer 1969): BH subsystem cannot achieve equipartition 3. BH subsystem forms a compact, almost isolated subsystem at centre of cluster 4. 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

6 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

7 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 10.6Gyr 2. 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 1Gyr 4. Banerjee+ 2010: N 105, Rh 1pc, % retention fraction, no tide ~0 after 800Myr for 50% retention 5. Heggie (unpub): N 4.9x105, Rh = 0.58 pc, ~50% retention fraction leaving ~500, but 5 after 11.4Gyr 6. Sippel & Hurley 2012: N = 2.5x105, Rh = 6.2 pc, 10% retention fraction 16 at 12 Gyr (see later) 7. Hurley & Shara 2012: N = 2x105, Rh = 4.7 pc, retention fraction? 4 at 11.5 Gyr

8 Results from Monte Carlo simulations 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 M4 NGC Tuc Retention factor 100% Natal kicks, = 160/190 km/s Retention factor 100% See also Morscher+ 2012: N = 3x105, rh = 2.44pc, retention fraction 86%. ~400 at 12 Gyr

9 Old ideas die slowly 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 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 109 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;...

10 Lessons from theory and simulations 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)

11 How to simulate M22 Giersz & H, 2013, MNRAS, sub N-body methods: NBODY6, starlab, AMUSE,. Monte Carlo methods Computing time N3 Computing time N

12 The time taken for a simulation N-body Monte Carlo Based on simple theoretical scaling with N (number of stars) and R (radius of the cluster) Data: Harris catalogue

13 N-body models: scaling down The idea: model a cluster with N stars by a model with N*<<N stars The principle: get the time scale of the major evolutionary effects correct Stellar evolution: set by stellar evolution models Two-body relaxation: time scale tr N1/2R3/2 = 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 binaries The escape speed (relevant for BH kicks) Stellar collisions Escape time scale Application to M22 (Sippel & Hurley 2013) N-body, N = 2.5x105, retention factor 11% trh and rh/rc close to M22 values at 12 Gyr

14 Monte Carlo code: the basic idea Assume spherical symmetry. Orbit of star characterised by energy, E, and angular momentum, J. The code repeatedly alters E, J to mimic the effects of gravitational encounters, using theory of relaxation. Includes: Stellar evolution Binary evolution Galactic tide Binary dynamics (binary-binary and binary-single encounters) Mirek Giersz

15 Finding initial conditions Method 1. By trial and error Method 2. Automatically Technique: Specify initial conditions by N (number of stars), r, r, initial King parameter W, h t 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)

16 Finding initial conditions for M22 Fig. showing evolution of N, A Fig. showing scatter plot of A,N Convergence of N(0) Goodness of fit v. N(0) Best initial conditions (100% retention of BH) N = 7.8x105, rh = 2.5pc, rt = 102 pc, Wo= 2.9 IMF: power law index 0.9/2.7 below/above 0.67 M pc No technique yet for estimating confidence intervals

17 Example of fit Surface brightness Velocity dispersion Luminosity function (Deficient in bright stars)

18 Evolution of the number of stellarmass black holes (Full-size MC models) Model 100% retention Fallback* kick = 90 km/s No. of WD/BH binaries at 12 Gyr (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 Conclusions: Up to 100 stellar-mass BH may remain, depending on assumption about natal kicks and high-mass IMF Up to two BH-WD binaries Cf Sippel & Hurley 2012 (scaled N-body model) N-body, N = 2.5x105, retention factor 11% t and rh/rc close to M22 values at 12 Gyr rh 16 BH at 12 Gyr (needs scaling) 2 in BH-MS binary, 2 in BH-BH binary

19 A revised theoretical view References: Breen & H,MNRAS, 2013 Consider 2-component system: low-mass stars m1, total mass M1; black holes m2, total mass M2; initially identically distributed in space Consider regime m2 >> m1 and M2 << M1

20 Stage 1: Mass segregation, core collapse Stage 2: binary formation, binary heating 1(a) Black holes mass-segregate (two-body encounters, tending to equipartition of kinetic energy) 1(b) Cluster is Spitzer-unstable (Spitzer 1969) BH subsystem cannot achieve equipartition, but keeps shrinking until it dominates the density at the centre. 1(c)The BH subsystem goes into core collapse (Hénon 1961, von Hoerner 1963), forming very high central density. 2(a)Three-body interactions become important, forming three-body binaries 2(b) Formation of binaries, and their interaction with single 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)

21 Stage 3: Balanced evolution Hénon s Principle Energy created BH low-mass stars Hénon 1961, 1975 Energy flows through The core adjusts to produce the BH subsystem energy required at the half-mass radius Energy transferred to low-mass stars If the core produces too much energy, the core expands, suppressing energy production. Similarly if it produces too little. Energy flows through low-mass stars The core of the BH subsystem produces the energy, but the low-mass stars (which dominate at the half-mass radius) determine how much is produced. half-mass radius of whole system cf Eddington s insight into stellar structure

22 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 M BH φ

23 The rate of escape of BH The rate of heating E M BH φ is determined by the energy flux at the half-mass radius E E t rh where E is total energy of entire system, trh is the half-mass relaxation time. Therefore E M BH φ t rh is dominated by the low-mass stars. Thus the rate of loss of BH is determined by the properties of the whole system, which is dominated by low-mass stars.

24 Stage 4: Exhaustion of BH energy source Using Hénon's Principle core and half-mass radii of BH subsystem can be estimated: RcBH/RhBH (40/NBH)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 NBH < 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

25 Interpretation of N-body data Stage 1. Mass segregation of BH: t < 250; core collapse in BH subsystem Stage 2. BH binary formation and heating, initiating expansion Stage 3. Balanced evolution: to t ~ 5000 Stage 4. Exhaustion of the BH subsystem; recollapse of the core of the light stars to enhance the energy generation by residual BH binary Stage 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, MBH/M = 0.02 Plots show core radius against time

26 Illustration: Monte Carlo model of M4 This model also has stellar evolution and a population of primordial binaries Stage 1. Mass segregation of BH; core collapse in BH subsystem at t ~ 15Myr Stage 2. BH binary formation and heating Stage 3. Balanced evolution (assisted by mass loss from stellar evolution) to ~ 5Gyr Stage 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 ~ 8Gyr Stage 6: Balanced evolution powered by primordial binaries (unstable) from t ~ 8Gyr to the present day Time (Myr) Model of Heggie & Giersz 2008 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

27 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)