Internal Dynamics of Globular Clusters

Douglas HeggieUniversity of Edinburgh

UK

Bologna 21 April 2016 Star Clusters as Cosmic Laboratories for Astrophysics and Fundamental Physics – MODEST 16

Italian Astro-quiz

● le gigante rosse

– Red giants

● le nane bianche

– White dwarfs

● le vedove nere

– Black widows

● le vagabonde blu

– Blue stragglers

With thanks to E. Lapenna and E. Vesperini for gender correction, and A.L. Varri for the initial inspiration.

Topics

● Simulating dynamical evolution

– N-body, Monte Carlo, Fokker-Planck, gas, EMACSS

● Understanding dynamical evolution

– Two-body relaxation and core collapse

– Equipartition and mass segregation

– Mass segregation instability

– Energy generation

– Gravothermal oscillations

● Application I: Stellar-mass black holes

● Application II: Interpretation of evolving models

– N4372

– 47 Tuc

– M4

Simulating dynamical evolutionTechnique Pros Cons Examples of specific clusters

N-body Gold standardFreely available

Takes months/yrsScaling tricky

Pal 14, Pal 4 (Hasani Zonoozi+ 2011, 2014)M4 (H 2014), N4372 (Wang+ 2016)

Monte Carlo Takes day(s)Freely available (MOCCA)

No rotationTides trickyNeeds ongoing checking with N-body

M4, N6397, 47 Tuc, M22 (Giersz & H 2008, 2009, 2011, 2014)

Fokker-Planck SimpleNo noise

Slowed by binaries, MF

M15 (Phinney 1993, Murphy+ 2011, ….)47 Tuc (Behler+ 2003)N6624 (Grabhorn+ 1992)M71, N6397 (Drukier[+] 1992, 1995)

Gas Even simplerNo noise

Slowed by binaries, MF

M3 (Angeletti+ 1980)

Synthetic Takes msecFreely available (EMACSS)

Global values only

M4, 47 Tuc, N6397, M22, ω Cen, Pal 14,

Pal 4, G1 And (Pijloo+ 2015)

Two-body relaxation● Causes outward flow of energy (Hénon 1961)

● If there is no centrally-concentrated energy source, this energy requirement tends to cause core collapse in finite time ( Lynden-Bell & Eggleton 1980)

(Left) Evolving 3D density distribution in core collapseLarson 1970

● “Hard” binaries can act as an energy source (Hills 1975, Heggie 1975)● If there is a centrally-concentratedenergy source, the core may adjustso that energy generation comes into balance with this requirement inpost-collapse evolution (Hénon 1975)

● rate of energy generation dE/dt ~ |E|/t

r

● whole cluster expands

Equipartition and mass segregation● In systems with stars of different masses, two-body relaxation tends towards

equipartition, i.e. m1<v12> = m2<v2

2> (i.e. equal temperatures T )

● In a cluster potential well, slow-moving stars sink and speed up, and so

– There is mass segregation (leading to core collapse)

– The tendency to equipartition is thwarted, partially

– Numerous references in recent years have quantified these effects

(Above) Temperature and density profiles in a 5-component Fokker-Planck model (Inagaki & Saslaw 1985)

t = 0 During core collapse

Mass-segregation instability

● Spitzer's (1969) mass-segregation instability theory gives a condition on the mass function for equipartition to be achievable

● but see Watters+ 2000 for an empirical, improved criterion (based on MC simulations);

Individual mass ratio

Tota

l m

ass r

atio

Energy Generation● Binary formation

– Centrally concentrated (∝ cube of number density)

– “3-body binaries”, but ≥ 3 stars involved (Tanikawa+ 2013, 2014; Geller & Leigh 2015)

● Binary evolution

– Primordial binaries (which become centrally concentrated by mass segregation)

– “3-body” binaries centrally concentrated at formation

● Mass-loss induced by stellar evolution

– May be centrally concentrated by mass segregation of progenitor (either primordial or dynamically induced segregation)

● Energy generation and escape

– If mass Δm > 0 is lost in core, where potential is Φc < 0, energy generated is -Δm.Φc .

– Rate of energy generation is of order Φc.dM/dt

– If evolution is balanced (Hénon 1975) then Φc.d(Bettwieser & Sugimoto

M/dt ~ dE/dt ~ |E|/trh

– Balanced evolution may be unstable to gravothermal oscillations (Bettwieser & Sugimoto 1983)

Core bounce with primordial binaries

● Core collapse creates high

density

● Enhances binary interactions

and energy production

● Can reverse core collapse

(core “bounce”)

● But is it a spike?

(Above) Cumulative number of collisions, and core Radius, in a full N-body model of M4 from 10 to 12 Gyr. In this simulation (Heggie 2014) there is no sudden increase in the collision rate at core collapse (around 10.9Gyr)

S. Bannerjee commented to me that the formation rate of blue stragglers may beobscured (in the figure on right) by numerous collisions which do not lead toobservable blue stragglers. But a check showed that, while the number of BS (as recorded by NBODY6) is about 1/3 of the total number of collisions, they show a very similar trend with time, and no signature of core collapse

Application IStellar-mass black holes

1. Mass segregation of progenitors and BH (if not ejected by natal kicks, and if of sufficient total mass to be Spitzer unstable)

2. BH segregation feeds energy to other stars (Merritt+ 2004), but BH themselves lose energy, which drives them to core collapse (Mackey+ 2008)

3. “Three-body” binary BH form and provide energy. This flux

1. Prevents further collapse of the BH system

2. Provides energy required for balanced evolution and expansion of the entire cluster (Breen & H 2013)

Exhaustion of the BH subsystem

● Energy production by BH subsystem requires loss of BH at a rate dNBH/dt ~ E/(mBHΦc trh) [slide 8]

● Φc dominated by rest of the cluster

● ⇒Escape rate of BH determined by the rest of the cluster

● When NBH falls to ~ 50, they can no longer provide

sufficient energy

● Core collapse of the entire cluster....

● ...until an alternative energy source (e.g. primordial binaries) leads to balanced evolution

Application IIInterpretation of evolving models

Cluster N4372 47 Tuc M4

Source Wang+ 2016 Giersz & H 2011 H 2014, 2015

Technique N-body Monte Carlo N-body

M0[M

⊙] ~5x105 1.64x106 3.5x105

Rh0

[pc] 7.6 1.91 0.58

fb0

0.05 0.022 0.07

Trh0

[Gyr] ~7 0.7 0.12

NGC 4372

Long time scales

Not much dynamical ejection of BH

Example of central BH cluster in GC with long relaxation time

(1st) Core collapse Balanced evolution powered by ?“3-body” BH binaries

Expansion ?due to SE mass-loss

47 Tuc

Corecollapse Expansion driven by

BH binaries and Stellar evolution

Dimishing energy production by BH binaries & SE leads to very slow (relative) core collapse (after many more Gyr).Primordial binaries slow the decline

< 50 BH cannot sustain balanced evolution (based on simplified models in Breen & H 2013)

Natal kicks

M4

Mass segregation

Onset of s.e.

Expansiondue to s.e.

Core collapseof BH

Gravothermaloscillations inBH subsystem

Numericalfaults

(Second) core collapseOnset of GTO

● Short time scales● After a few Gyr, the BH cannot

sustain the energy flow throughthe cluster

● Core collapse ensues● Post-collapse evolution

● sustained by primordial binaries● includes gravothermal oscillations

Evolution of globular clusters:What I have not talked about

● Tidal evolution – M. Gieles

● Rotation, anisotropy – A.L. Varri

● MSP – B. Prager

● CV – D. Belloni

● IMBH – M. Giersz

● Collision products – M. Mapelli

● Planetary systems – M.B.N. Kouwenhoven

● MSP – E. Vesperini

These are examples of other talks which deal with these dynamical problems

Summary

“There is nothing here that has not been seen before, and for that I apologize.... But as abstruse details accumulate, the simple truths are often lost sight of....”

Ivan King

Proc IAU 246 (2007)Ivan King in Tokyo ~1995