Realistic Models of Star Cluster Evolution: Bridging the Gap Between Theory and Observations

Realistic models of star cluster evolution: Bridging the gap between theory and observations

Anna Catharina Sippel

Presented in fulﬁllment of the requirements of the degree of Doctor of Philosophy

August 21, 2014

Faculty of Science, Engeneering and Technology Swinburne University

i Abstract

This thesis focuses on the evolution of star clusters within the global picture of galaxy evolution, in particular the dynamical evolution of globular clusters and how this affects the way they are observed. The high density of globular clusters allows us to use them as laboratories of stellar evolution and stellar interactions: even though they are old, they are still dynamically active. As they are in general over 10 billion years old, they witnessed the formation and evolution of their host galaxy, and with observations we can only study them in their current, evolved state. In this thesis, such observations are complemented by direct, star-by-star (so-called direct N-body) simulations of cluster evolution and new methods for the analysis are presented to facilitate the comparison between theory and observations. I introduce star clusters in general, key information about star cluster and stellar evolution, as well as the approach of N-body modelling. The bulk of the thesis is about the impact of metallicity on cluster evolution: the chemical composition affects stellar evolution and hence mass-loss rates and remnant masses of cluster members. Globular cluster systems emerge in two sub-populations: metal-poor (appearing blue) and metal- rich (appearing red), with the red clusters more centrally concentrated within their host galaxy. In addition, the blue globular clusters are on average ∼ 20% larger in size than the red ones. I show that while clusters of different metallicity are structurally identical, an apparent size difference is found owing to the combined effects of metallicity and mass segregation. These findings are followed up with a study on the effects of metallicity on the evolution of cluster colour over time, with a particular focus to disentangle internal and external effects. An example is the impact of the preferential removal of low-mass stars in the cluster outskirts on cluster colour. I show that the cluster colour is driven by the giants in the cluster, and that in terms of colour, the cluster is rather unaware of the removal of low-mass stars. Before concluding the thesis, I show that two recently discovered black hole candidates in the Milky Way cluster M22, the first black holes detected in a Milky Way globular cluster, are no surprise from a dynamical point of view. While these findings can only be applied to other clusters with exceptionally large core radii like M22, I predict that the evaporation of remnant black holes is happening on a slower rate than previously thought. The model created and used for this study includes more than 250 000 stars and is currently the largest (and ultimately one of the most realistic) direct N-body model of a globular cluster. ii iii Acknowledgments

My thank goes to my fantastic supervisor Jarrod Hurley, without whom this project would not have been possible. Not only did the idea for this thesis come from him, but also he was very patient with me and my crazy ideas. He gave me the freedom to explore globular clusters from my point of view, travel, and carry out many outreach projects for which I am very grateful. I thank my other supervisors for their support, in particular Marie Martig, who manages to stay calm no matter how crazy life gets, to an extent that even rubs off on me, and George Hau, who welcomed me in Santiago. This thesis would not have been possible without the fantastic supercomputers at Swinburne. I am truly grateful to Gin Tan and Simon Forsayeth for their effort in keeping them up and running. Many thanks also go to my PhD brother Juan for endless discussions and group meetings at the bakery, which Guido and Luca later joined as well. Also thanks to Bil and Max for being the best office mates. With travels all around the world I am thankful to my amazing friends who have opened their homes to me: Feli and Jenny, Marie and Vincent, Christina and Michael, Felipe and Sheila with Rafa in Melbourne as well as Caro and Frick and Maria in Santiago and Anna back home. I am super happy to have met my chicas during the last four years: Becky, Feli, Christina, Marie, Julija, Maria, Rebekka, Amy, Cathy, Liz, Rebeca and of course Joanne and Anna, who has been there all along. The hours spent drinking coffee with you can’t be replaced by anything in the world! I couldn’t imagine my PhD without countless outreach projects, and I am happy I could show so many AstroTours at Swinburne. Mostly however I thank Joanne and Bill at the School of the Air in Alice Springs for “adopting” me as their Astronomer. I have joined many classes over web cam and am grateful that Scientists in Schools made a visit possible. To work with many students from all over the Northern Territory has been a truly enlightening experience. My thanks also go out to all the kids and students around the world that keep asking me amazing questions through various channels, you are truly inspiring! Finally, I want to thank my family for all their support in my life all around the globe and of course Javier for his endless encouragement, patience, help and simply being there. iv v Declaration

The work presented in this thesis has been carried out at the Centre for Astrophysics & Supercomputing at Swinburne University of Technology in Australia between 2010 and 2014. During this time, I spent the majority of 2013 as a research student at the European Southern Observatory in Santiago, Chile. This thesis contains no material that has been accepted for the award of any other degree or diploma and to the best of my knowledge, it contains no material previously published or written by another author, except where due reference is made in the text. The content of the chapters listed below has appeared in refereed journals or is in prepa- ration for submission. Minor alterations have been made to the published papers in order to maintain argument continuity and consistency of spelling and style. • Chapter 2 has been published as N-body models of globular clusters: metallicities, half-light radii and mass-to-light ratios, Sippel A. C., Hurley J. R., Madrid, J. P. and Harris, W. E., MNRAS, Vol. 427, Issue 1 p. 167-179 and can be retrieved under the following link: http://mnras.oxfordjournals.org/content/427/1/167

• Chapter 3 is submitted to MNRAS with the title Slicing and dicing globular clusters: dynamically evolved single stellar populations, Sippel A. C. & Hurley, J. R.

• Chapter 4 has been published as Multiple stellar-mass black holes in globular clusters: theoretical conﬁrmation, Sippel A. C & Hurley J. R., MNRAS Letters, Vol. 430, Issue 1, p. 30-34 and can be retrieved under the following link: http://mnrasl. oxfordjournals.org/content/430/1/L30 My contribution to these papers was as follows: I evolved and analyzed all N-models and wrote the manuscripts. My supervisor, Jarrod Hurley, provided the idea for Chapter 2 and Chapter 3 in particular, as well as continuous help, ideas and guidance. Juan Madrid and Bill Harris contributed with observational advice and discussions.

Anna Sippel Melbourne, Victoria, Australia 2014

Contents

Abstract i

Acknowledgments ii

Declaration iv

List of Figures viii

List of Tables x

1 Introduction 1 1.1 Globular Clusters ...... 1 1.1.1 Evolution of globular clusters ...... 5 1.1.2 Stellar evolution 101 ...... 11 1.1.3 Combining stellar and dynamical evolution in globular clusters . . . 13 1.1.4 Stellar populations in globular clusters ...... 14 1.2 Open Clusters ...... 16 1.3 Star cluster models with NBODY6 ...... 20 1.3.1 Development: Five decades from NBODY1 to NBODY7 ...... 20 1.3.2 Special Purpose Computers and Graphics Processing Units . . . . . 23 1.3.3 Initializing NBODY6 ...... 26 1.3.4 Alternatives to NBODY6 ...... 26 1.4 Comparing models and observations ...... 31 1.5 Outline ...... 32

2 Metallicity Effects on Globular Cluster Evolution 33 2.1 Globular cluster sizes ...... 33 2.2 Metallicity effects on stellar and star cluster evolution ...... 35 2.2.1 Stellar evolution of an entire population ...... 39 2.2.2 Size: projection effects vs. internal dynamics ...... 41 2.3 Simulation method & choice of parameters ...... 42 2.3.1 Binary fraction ...... 47 2.3.2 Treatment of remnants ...... 47 2.3.3 Models ...... 48 2.4 Evolution ...... 50 2.4.1 Binary systems ...... 51

vii viii Contents

2.4.2 Cluster size ...... 53 2.4.3 Surface brightness and half light radii ...... 57 2.4.4 Origin of the size diﬀerence and inﬂuence of remnants ...... 60 2.4.5 Mass-to-light ratio ...... 64 2.5 Discussion and conclusion ...... 64

3 Dynamically Evolved Single Stellar Populations 69 3.1 Introduction ...... 69 3.2 Simulation method & choice of parameters ...... 71 3.2.1 Colour ...... 72 3.3 Evolution ...... 73 3.3.1 General ...... 77 3.3.2 Integrated and dynamically evolved SSPs ...... 78 3.3.3 Resolved single stellar populations ...... 85 3.4 Conclusions ...... 95

4 Black Holes in Globular Clusters 99 4.1 Stellar mass black holes in globular clusters ...... 99 4.2 Simulation method & choice of parameters ...... 102 4.2.1 Velocity kicks ...... 102 4.2.2 Size and time scales: measurements ...... 103 4.3 Evolution ...... 104 4.3.1 Binary systems with black holes ...... 107 4.3.2 Comparison ...... 108 4.4 Conclusions ...... 111

5 Conclusions 113 5.1 Summary ...... 113 5.2 Future research and outlook ...... 114 5.2.1 Missing link to galaxy evolution ...... 114 5.2.2 The dynamical imprint of blue stragglers on cluster evolution . . . . 115 5.2.3 Initial conditions of star clusters ...... 116 5.2.4 Even larger realistic models ...... 118

Bibliography 132

A N-body core radius 133 List of Figures

1.1 Globular clusters in the Milky Way ...... 4 1.2 Cluster evolution: number, mass and binary fraction ...... 7 1.3 Cluster evolution: half-mass radius, velocity dispersion and half-mass relaxation time ...... 8 1.4 Cluster evolution: core radius, core members and core mass ...... 9 1.5 Hertzsprung-Russell diagram of a globular cluster ...... 15 1.6 Positions of Milky Way star clusters projected onto the galactic plane . . . 18 1.7 Open clusters in the Milky Way ...... 19 1.8 Increading N over time ...... 25 1.9 Numerical methods to study star cluster evolution ...... 28 1.10 Cluster view in three dimensions ...... 29 1.11 Observed GC model ...... 30

2.1 Size and galactocentric distance of the MW GC population ...... 38 2.2 Evolution of mass, light and mass-to-light ratio: stellar evolution ...... 40 2.3 Cluster outskirts in a tidal field ...... 45 2.4 Evolution of mass, light and mass-to-light ratio: dynamics ...... 46 2.5 Mass function of the dynamically evolved stellar population ...... 52 2.6 Langragian and core radii ...... 54 2.7 Surface brightness fit for an old cluster ...... 56 2.8 Half-light radius over time ...... 58 2.9 Radial profile for mass and luminosity ...... 61 2.10 Central luminosity ...... 63

3.1 Structural parameters ...... 74 3.2 Globular cluster radii ...... 76 3.3 Colour-magnitude diagram of a globular cluster population ...... 80 3.4 High metallicity CMD ...... 82 3.5 Intermediate metallicity CMD ...... 83 3.6 low metallicity CMD ...... 84 3.7 Hertzsprung Russell diagrams ...... 86 3.8 Cluster proﬁles ...... 87 3.9 Linking colour and velocity ...... 89 3.10 Mass-magnitude diagram ...... 90 3.11 Magnitude-distance diagram ...... 91

ix x List of Figures

3.12 Brightest stars and cluster colour ...... 93 3.13 Low-mass stars and cluster colour ...... 94

4.1 Location of black holes ...... 101 4.2 Size evolution of the cluster over time ...... 105 4.3 Cluster composition ...... 106 4.4 Mass-ecolution of the black hole population ...... 110

5.1 Cluster density and survival ...... 117

A.1 Sphere to estimate local density ...... 134 List of Tables

1.1 Parameters of Milky Way Globular Clusters ...... 3

2.1 Main sequence lifetimes for stars with diﬀerent metallicities ...... 37 2.2 Remnant masses at 12 Gyr ...... 49 2.3 Average cluster sizes ...... 59 2.4 Luminosity and mass-to-light ratios ...... 63

3.1 Model composition ...... 75

1 Introduction

One takes a random cluster of stars, puts them on a calculating machine as fast as possible, integrate the numerical equations of motions over an extended period of time, and watch how it evolves. - Sebastian von Hoerner 1960

Star clusters are collections of stars existing over a whole range of sizes, masses and ages. They orbit within the gravitational potential of their host galaxy and depending on their location, it is possible for them to live in a strong tidal ﬁeld close to the center of their galaxy or in near-isolation in its outskirts. Star clusters can be divided into two main groups: the more common and less massive open clusters and the old, majestic globular clusters. Star clusters are dynamically active and ideal test beds of stellar and dynamical evolution. Because of their dense cores, they provide convenient sites to study rare stellar types such as blue straggler stars or black hole binary systems. Since most stars form in clusters, they are regarded as the building blocks of galaxies.

1.1 Globular Clusters

Globular clusters (GCs) are self-gravitating, compact stellar systems that are fundamental components of all galaxies. Depending on their age, they are either well preserved remnants of the ﬁrst era of star formation (SF) in a galaxy or tracers of major SF events induced by, for example, mergers (Brodie & Strader, 2006). Average properties of GC populations are found to correlate with fundamental charac- teristics of their host galaxy. More luminous galaxies tend to feature larger GC populations (Brodie & Strader, 2006), of up to several thousands, in the case of giant elliptical galaxies (Peng et al., 2011), in contrast to ∼ 160 currently known in the Milky Way (MW; Harris

1 2 Chapter 1. Introduction

2010 at distances up to ∼ 145 kpc from the Galactic center (Laevens et al., 2014). The mass of the GC system of a galaxy has been shown to correlate with galaxy halo mass (Spitler & Forbes, 2009; Hudson et al., 2014), independent of galaxy morphology (Georgiev et al., 2010) and a connection between the number of GCs and the mass of the central black hole of a galaxy has been found (Burkert & Tremaine, 2010; Harris & Harris, 2011). Maschberger & Kroupa (2007) presented a method to derive the star formation history of a galaxy by its GC distribution under the assumption that all stars form in star clusters (SCs), with GCs representing the high-mass end of the SC mass function. Muratov & Gnedin (2010) estimate that at redshifts z > 3 more than 10% of the galactic stellar mass was locked in GCs compared to just 0.1% now, and that more than 20% of the SF in galaxies at z > 3 took place in GCs. This is in agreement with the current understanding of the life cycle of GCs: they are old and continuously dissolving (Wielen, 1988; Portegies Zwart et al., 1998). Internal dynamical interactions cause stars to be ejected, and more importantly, the tidal field of their host galaxy is successful in removing stars from the cluster outskirts (Odenkirchen et al., 2009; Küpper et al., 2010, 2012). In fact, SCs form over a wide range of masses with only the most massive ones, GCs, being able to survive beyond the Hubble time. A bimodal GC colour distribution is observed, especially for massive galaxies, com- prised of a blue and a red population (Zinn, 1985; Ashman & Zepf, 1992; Zepf & Ashman, 1993). While both, blue and red clusters, can be found in any environment within their host galaxy (i.e. bulge or halo), the distribution of red clusters is more centrally concentrated (Kinman, 1959; Brodie & Strader, 2006). Furthermore, this distinction extends to the dynamical properties of both populations, with the red clusters following the rotation of the galaxy and the blue clusters the non-rotation of the halo (Pota et al., 2013). Some galaxies have more than two populations: NGC4365 is a particularly well studied example that has three sub-populations (Puzia et al., 2002; Blom et al., 2012b) with distinct rotation properties (Blom et al., 2012a). The bimodality is driven by a difference in metallicity, with the red clusters being metal-rich and the blue clusters metal-poor, as confirmed spectroscopically (Usher et al. 2012 and references therein). Recent simulations show that the GC bimodality can naturally occur in a hierarchical galaxy assembly, with the red population formed in the main progenitor while the blue subpopulation is accreted from satellites and formed earlier (Tonini, 2013). 1.1. Globular Clusters 3

Name Rgc D rc rh log(ρ0) Mv 3 [kpc] [kpc] [arcmin] [arcmin] [L /pc ] ω Cen 6.4 5.2 2.37 5.00 3.15 −10.26 NGC 288 12.0 8.9 1.35 2.23 1.78 −6.75 M55 3.9 5.4 1.80 2.83 2.22 −7.57 NGC 6388 3.1 9.9 0.12 0.52 5.37 −9.41 M4 5.9 2.2 1.16 4.33 3.67 −7.19 M13 8.4 7.1 0.62 1.69 3.55 −8.55 M16 4.6 4.4 0.77 1.95 3.54 −7.48 M5 6.2 7.5 0.44 1.77 3.88 −8.81 47 Tuc 7.4 4.5 0.36 3.17 4.88 −9.42 NGC 6752 5.2 4.0 0.17 1.91 5.04 −7.73 M80 3.8 10.0 0.15 0.61 4.79 −8.23 M30 7.1 8.1 0.06 1.03 5.01 −7.45

Table 1.1 Parameters of the Milky Way GCs in Figure 1.1 extracted from Harris (2010). Listed are the galactocentric distance Rgc, distance from the Sun D, the core radius rc, the half-light radius rh, the central luminosity density ρ0 and absolute v-band magnitude Mv. 4 Chapter 1. Introduction

Figure 1.1 Mosaic of globular clusters in the Milky Way. From top left to bottom right: Omega Centauri (ω Cen), NGC288, M55, NGC6388, M4, M13, M10, M5, 47 Tucanae (47 Tuc), NGC6752, M80, and M30. See Table 1.1 for structural parameters of these clusters. Image credit: F. Ferraro (University of Bologna), NASA and ESA. 1.1. Globular Clusters 5

High resolution imaging, especially from the Hubble Space Telescope (HST), allows us to partially resolve GCs far beyond the local group to measure their sizes. This has been done for many galaxies (e.g. Kundu & Whitmore 1998; Larsen et al. 2001; Spitler et al. 2006; Madrid et al. 2009; Puzia et al. 2014) and entire clusters of galaxies, in particular the surveys of GC systems in the Coma and Virgo clusters have provided a wealth of information about their structural parameters such as mass, size, colour and metallicity (e.g. Jordánet al. 2005; Peng et al. 2011). With remarkable agreement, on average GC sizes are found to be 2 − 3 pc in projected half-light radius, with the blue clusters on average ∼ 20% larger than red clusters (see previous references, Jordán2004; Woodley & Gómez2010, and Chapter 2).

Such a trend had already been known for the MW with the GC sizes rh following the p relation rh ≈ Rgc, with Rgc the galactocentric distance. In the MW GCs are found in the halo, bulge and thick disk (with metal-poor clusters found predominantly in the halo), and orbital parameters are available as well (e.g. Dauphole et al. 1996). Both populations are thought to be of very similar age (VandenBerg et al., 2013), although age diﬀerences of up to 2 Gyr between metal-rich and metal-poor clusters have been suggested (Hansen et al., 2013). Several other cluster parameters such as their galactocentric distances and absolute magnitudes are summarized for a selection of MW GCs in Table 1.1, and in Fig. 1.1 images of those clusters are shown.

1.1.1 Evolution of globular clusters

The main parameters used to describe cluster evolution are the number of stars, the mass, half-mass radius and half-mass relaxation time (see Equation 1.2), and the core radius (see Appendix A). To illustrate this, Fig. 1.2 shows how the number of stellar systems N, cluster mass M and binary fraction bf evolve over time. Fig. 1.3 shows the evolution of the half-mass radius rhm, overall cluster velocity dispersion σv and half-mass relaxation time thr. Fig. 1.4 concentrates on the cluster core showing the core radius, number of systems in the core as well as the core mass. The lifetime of a GC depends on those parameters (as well as the external tidal ﬁeld), and can be divided in two main stages:

• The ﬁrst 6 2 Gyr are dominated by stellar evolution driven expansion (see Fig. 1.2)

and depending on metallicity, stars with masses > 2 M will end their life within this time and do so accompanied by heavy mass loss. The cluster reacts to this mass loss with an increase in size. 6 Chapter 1. Introduction

• After this time, the mass loss is dominated by the continued but slower loss of stars from the cluster. Stars can be ejected via few-body encounters, they can receive a velocity kick during a supernova explosion (see Chapter 4), or they can simply pass beyond the tidal boundary of the cluster. This is particularly eﬀective close to the disk of the external potential, noting that disk crossings occur at a frequency of the order of ∼ 100 Myr for inclined orbits at galactocentric distances comparable to the Sun.

• Simultaneously, the internal evolution of the cluster is driven by two-body relaxation. Given the high densities of GCs (see Table 1.1), stellar encounters are so common that they sort the distribution of stars over time: low-mass stars increase their velocity and travel towards the outskirts of the cluster, while high-mass stars sink to the bottom of the potential well. This process, following a trend towards equipartition of energies, is called mass segregation.

Two timescales are commonly used to describe the evolution of star clusters: the crossing time tcr and the half-mass relaxation time thr (Fig. 1.3). The former is deﬁned as: s r3 2r t = 10 hm ∝ hm (1.1) cr GM v

1 is of the order of only tcr ∼ 10 Myr , while the latter is expressed as:

1/2 3/2 N rhm thr = 0.138√ (1.2) mG ln Λ

(Spitzer, 1987) and is of the order of a few Gyr (see Figure 1.3). Here v the average velocity per star and m the mean stellar mass. ln Λ is the Coulomb logarithm, where 0.02N ≤ Λ ≤ 0.11N depending on the cluster mass function (Giersz & Heggie, 1994).

Generally, tcr thr tlife with tlife the cluster lifetime. The crossing time represents rapid interactions (as in a collisional systems) while the relaxation time represents the cumulative eﬀects of weak encounters (as in a collisionless system).

1 Using typical parameters such as rhm = 2.5 pc and v = 0.5 km/s. 1.1. Globular Clusters 7

1 0.8 a)

0 0.6

N/N 0.4 0.2 1 0.8 b) 0 0.6

M/M 0.4 0.2

6 5.5 [%]

f 5 b 4.5 c) 4 0 2 4 6 8 10 12 14 t [Gyr]

Figure 1.2 Evolution of a cluster with initially 100 000 stars, as described in detail in Chapter 2. Plotted are in a) the number of stellar systems N (scaled to the initial number of stars N0), in panel b) the cluster mass M scaled to the initial cluster mass M0, and c) the binary fraction bf = nb/(nb + ns), with nb the number of binary systems and ns the number of single stars at any given time. Stars are continuously lost due to the tidal ﬁeld, ejection following dynamical interactions and in some cases, velocity kicks during supernovae. 8 Chapter 1. Introduction

10 a) 9

[pc] 8 hm

r 7 6 4 b) 3 2 [km/s] v

σ 1

3 c) 2

[Gyr] 1 hr t 0 0 2 4 6 8 10 12 14 t [Gyr]

Figure 1.3 Same model as in Fig. 1.2, now showing in a) the half-mass radis rhm, in b) the overall cluster velocity dispersion σv, and in c) the half-mass relaxation time thr (see Eq. 1.2). 1.1. Globular Clusters 9

4 a) 3 2 [pc] c

r 1 0 ] 3 10 b) [10

c 5 n

] 0

sun 4 c) M 3 2 [10 c

m 0 0 2 4 6 8 10 12 14 t [Gyr]

Figure 1.4 Continuation of Figures 1.2 and 1.3. In a) the core radius rc is given (see also Appendix A), in panel b) the number of stars within the core and in c) the core mass are shown. 10 Chapter1.Introduction

Masssegregationandassociatedtwo-bodyprocessesleadtothecenterofthecluster beingdenselypopulatedbymassivestarsinacoreofdecreasingsize,i.e.,core-collapse, (Spitzer,1969).Thisisobservableasapower-lawwithoutacuspforthesurfacedensity distribution,oraconcentration c=log( rt/r c)>2.Here rtisthetidalradiusand rcthe coreradiusinaKingmodelfit(seeChapter2).IntheMilkyWay,21GCshave c > 2 andarehenceclassifiedascorecollapsed(Harris,2010). Initialcore-collapseisefficientlyhaltedbyheatingenergyprovidedfrombinarysystems (McMillanetal.,1990,1991).EventhoughobservedMW GCbinaryfractionsareonly around3 −10%(Hutetal.,1992;Miloneetal.,2012),theinteractionsbetweenbinaries andstarsinfew-bodyencountersfacilitatesthetighteningoforbits,hardeningthebinary (Heggie,1975).Suchhardbinarieshavebeenshowntohaltcore-collapseandquench masssegregation,andblackholebinariescanforminthecore(Mackeyetal.2008;Hurley &Shara2012;Sippel& Hurley2013;Breen& Heggie2013,andseeChapter4).As demonstratedinFig.1.2c),thefractionofbinarystarsinaclusterincreaseslateinthe clusterevolution,afteraninitialdecreasewhenanysoftprimordialbinariesarebroken- up 2. Ultimately,thegravothermalcatastrophy(Lynden-Bell& Wood,1968)willhaveits wayandtheclusterwillcontinueitsjourneytowardscore-collapse,makingejectionof starsorbinariesfromtheclusterveryeffective,aswellasthecontinuedformationoftight binaries.Eventually,theincreaseofkineticenergyinthecore,owingtobinaryencounters, leadsagaintoanexpansionofthecluster,whichisnowonacontinuoustracktowards evaporationanddissolution(seeFig.1.3at ∼ 13Gyr).Heggie& Giersz(2009)showed thatsemi-regularpost-corecollapseoscillationsontheorderof ∼100Myrcanoccur.

Thetime-scaleofcorecollapseisoftheorderofseveralrelaxationtimes thr ,forexample at ∼10 thr inthecaseofHurley&Shara(2012).Inthecaseofanidealizedsingle-mass clusterwhereallstarshaveanidenticalmass,corecollapseoccursat16 thr (Binney& Tremaine,2008). AclusterintheouterhalooftheMWcanliveinnear-isolation,whileaclusterinthe bulgeissubjecttoastrongtidalﬁeld.Inbothcases,owingtomasssegregation,low-mass starsaremorelikelyremovedfromtheclusterduetothetidalﬁeld(Baumgardt& Makino,

2Basedontheobservedbinaryfractionof3 10%formostMW GCs(Miloneetal.,2012)andthe factthatthebinaryfractiondoesnotchangesigniﬁcantlyoverseveralGyrofclusterevolution,aninitial orprimordialbinaryfractionof5%isusedinthemodelspresentedinthisthesis.Someofthesebinaries mightmightdisruptearlyon,whiledynamicalbinariesalsoform.Energeticanddynamicallyimportant binarysystemssuchasBH-BHbinariesareusuallynotprimordial,butformviaacascadeofinteractions startingfromaratherunspectacularMSorWDbinarysystem.ExamplesarementionedinChapter3 and4. 1.1. Globular Clusters 11

2003; Kruijssen & Lamers, 2008), an effect that is strong enough to alter a cluster’s current mass function (Kruijssen & Lamers, 2008). These stars are lost to the host galaxy via the two inner Lagrangian points and may form tidal tails, which are observed as clumps as the stars move along epicycles due to the coriolis and centrifugal forces (Capuzzo Dolcetta et al., 2005). Owing to the eccentricity of cluster orbits these over- and under-densities become distorted, which makes them difficult to observe, in particular at perigalacticon (Küpper et al., 2008, 2010, 2012). While core collapse is slowed by binaries, it is accelerated by the external tidal field and cluster rotation (e.g. Ernst et al. 2007).

1.1.2 Stellar evolution 101

Stars not only interact dynamically but also evolve (Russell 1914; Atkinson 1931; Harm & Schwarzchild 1955 and many more). Nuclear fusion of hydrogen into helium provides the radiation pressure necessary for a star not to collapse under its own weight, and a star spends most of its life in this delicate state of thermal and hydrostatic equilibrium on the main sequence (MS). Exhaustion of nuclear fuel stalls the delicate equilibrium at least temporarily and brings radical changes in the form of collapse in the core, which can be halted by the ignition of heavier elements (such as helium or carbon) if the star is massive enough to provide suﬃcient pressure and hence high enough core temperatures. If no further fuel can be burned, most stars will ultimately end their life as a remnant: either as a slowly cooling white dwarf (WD), a neutron star (NS), or as a black hole (BH) - depending on the star’s mass and metallicity. Less massive stars than our Sun can live longer than the current age of the Universe, while very massive stars only live on the order of a few million years before running out of fuel to burn. A stars position on the so-called Hertzsprung-Russell (HR) diagram, a relation between colour (or temperature) and magnitude, tells us about its evolutionary stage (see text below and Fig. 1.5).

For a low-mass star between 0.5 − 2.25 M , this implies a long lifetime of burning hydrogen in the core. Once this source is exhausted and the star fails to provide the necessary radiation pressure, its core contracts, increasing the temperature of the helium core and igniting a shell of hydrogen burning around it. The increased energy generation rate of the core causes the outer layers of the star to decouple and expand. This is accompanied by an increase in luminosity and a decrease in effective temperature, and we observe this stage as a red giant. The core continues to contract, becomes degenerate and the increased temperature eventually ignites helium. Burning of helium into carbon increases the core temperature dramatically, causing a so-called helium flash when it is no longer degenerate. The star has again reached a (shorter lived) state of equilibrium and 12 Chapter 1. Introduction moves off the red giant branch onto the horizontal branch.

The duration on the horizontal branch depends on the mass, and the exhaustion of available helium in the core will once again cause the core to contract. This contraction leads to a shell of now helium burning around the core, while the hydrogen burning shell has never ceased. This implies that the outer layers of the star expand and for a second time, the star becomes a red giant, now on the asymptotic giant branch. Nuclear burning is occurring in two shells: helium burning around the carbon core and a shell of hydrogen burning around the helium burning shell. Even though the hydrogen burning shell continues to produce helium, the helium burning shell can run out of fuel. The newly created helium can be ignited in ﬂashes, causing the star to thermally pulsate. During all this time, the star can shed its outer layers, expelling material in the form of stellar winds, and ultimately may be seen as a planetary nebula. The remaining hot core is visible as a hot blue star, running soon out of fuel to burn. Subsequent contraction is halted by electron degeneracy but the star will not reach suﬃcient temperatures and pressures for nuclear fusion again: it slowly cools as a helium white dwarf.

More massive stars, between 2.25 − 8 M , also spend the majority of their life on the main sequence burning hydrogen. Once the hydrogen is exhausted, a star in this mass range rapidly moves through the Hertzsprung gap. Core contraction ignites hydrogen shell burning on the giant branch and eventually a smooth transition ignites helium in the core, burning it into carbon and oxygen on the horizontal branch. Once helium is exhausted, the star will return to the giant branch, with a contracting helium core, a helium burning shell and a hydrogen burning shell, while the outer layers are blown away in stellar winds. The contracting carbon and oxygen core will only ignite when temperatures are so high that the core has become degenerate. The star either dies here as a carbon and oxygen white dwarf, or starts carbon fusion into neon. If carbon fusion starts, this will happen in a shell and produce a ﬂash, establishing another short lived period of equilibrium. Ultimately, also this fuel will run out and the star will die as a oxygen-neon white dwarf.

Stars > 8 M commence their life hot and blue and stay in this state during the entire period of hydrogen and helium burning. Only once helium is exhausted do they start contraction of the core and move onto the giant branch via the Hertzsprung gap. If the stars are massive enough, they will expand into blue super giants. Just as with the less massive stars, they go through several episodes of ignition and exhaustion of fuel in the core and shells, always followed by contraction and accompanied by heavy mass loss through stellar winds on the stars surface. However in such massive stars, this continues to the burning of oxygen, neon and silicon. The production of successively heavier elements 1.1. Globular Clusters 13 is only stopped once an iron core is established, the burning of which can not create - but requires - energy. The core contracts for a ﬁnal time, and depending on the core mass, this contraction can be halted by degenerate neutron pressure - creating a neutron star.

For core masses & 2.7 M the collapse cannot be halted and a stellar remnant black hole will be created.

1.1.3 Combining stellar and dynamical evolution in globular clusters

The fact that massive stars exhaust their fuel faster implies that stellar evolution sig- niﬁcantly alters the mass function of a GC at any given time. Coupled together with the dynamical interactions described in Section 1.1.1, a cluster is sorted such that heavy remnants (like NSs, oxygen-neon WDs or BHs) will sink towards the center of a potential well, while low-mass main sequence stars or white dwarfs migrate to the outskirts of the cluster. During the creation of binary systems, and in particular during the hardening of tight binary systems (whether they be primordial or dynamically formed), mass transfer from one component to the other becomes possible, and in extreme cases collisions might happen. Both are facilitated by the extreme densities in GCs compared to the solar neighborhood (see Table 1.1), and both can create exotic stars such as blue stragglers (BSs, see Fig. 1.5), illustrating the intimate connection of stellar and dynamical evolution in GCs. Accretion of mass onto a star will alter its evolution, and depending on the nature of the stars involved, the observable signal of stellar cannibalism diﬀers across the electromagnetic spectrum. For example:

• If the accreting star is an ordinary star such as our Sun, it can turn it into a blue straggler (Sandage, 1953), a bright and blue star only rarely observed outside a star cluster. Blue stragglers can form both via accretion and head-on collisions (Sills et al., 2005; Leigh et al., 2011, 2013; Simunovic & Puzia, 2014).

• If the binary is composed of a WD and a mass donor, typically a MS star, the system is called a cataclysmic variable (CV). In such a system, thermonuclear runaways can occur: Novae, which are short but violent bursts leading a star to increase its brightness by up to a factor of 105 (Shara, 1989) within a few days, only to then slowly decline over a period of a few months. Shara & Hurley (2006) have shown that the SC environment can accelerate CV evolution.

• Accretion of mass onto a neutron star results in strong X-ray emission. Those low- mass X-ray binaries (LMXB) are orders of magnitudes more abundant in GCs than 14 Chapter 1. Introduction

the rest of the Galaxy (Pooley, 2010). In addition, the number of those sources in GCs scales with the frequency of stellar encounters, pointing towards a dynamical origin of LMXBs (Pooley et al., 2003).

• The most extreme case is the transfer of mass onto a stellar remnant black hole, for which currently three candidates exist in Milky Way globular clusters (Strader et al., 2012; Chomiuk et al., 2013), all detected via faint emission in the radio part of the electromagnetic spectrum (see also Sippel & Hurley 2013 and Chapter 4).

Fig. 1.5 shows the colour-magnitude (CMD) or HR diagram of a cluster snapshot at the ages of 5 and 10 Gyr. Several features mentioned above can be seen, such as the MS, MS turnoﬀ, several stages of evolved stars (or giants), the WD cooling sequence and also BSs. Note that the MS appears double for low-mass stars. This is due to the fact that the minimum mass of 0.1 M per star results in a minimum mass of 0.2 M per binary system. For more details, Chapter 3 is devoted to the colour evolution of SSPs in GCs.

1.1.4 Stellar populations in globular clusters

To a first approximation, GCs are composed of single stellar populations (SSPs), meaning all stars formed at the same time out of identically enriched material. In the last few years, GCs with multiple populations have been found (Bedin et al., 2004; Piotto et al., 2007), and recent results indicate that an age spread of several 100 Myr is not the exception, but the norm in GCs in the Milky Way and Large Magellanic Cloud (LMC, see Gratton et al. 2012). The origin of these multiple populations is not yet fully understood, but it has been 6 shown that clusters with masses > 10 M can retain ejecta from stars on the asymptotic giant branch in their centers (e.g. Bekki 2011), which can explain some forms of enrichment as well as the so-called blue tilt (Strader & Smith, 2008). The blue tilt is a correlation between colour and integrated magnitude among the brighter metal-poor (blue) clusters of many galaxies, and it’s origin is thought to be either due to pre- or self-enrichment, or the possibility that clusters with multiple helium enriched populations could be remnants of nucleated dwarf galaxies (Renzini, 2008). The best fit model of this scenario, suggested by Strader & Smith (2008), is self-enrichment of the proto-cluster cloud (or in other words pre-enrichment, see Bailin & Harris 2009), in which star formation is governed by supernova feedback with an efficiency scaling with the proto-cluster cloud mass. More recently, Fensch et al. (2014) show that the broadened metallicity distributions, as in ω Cen in the Milky Way, can arise naturally from self-enrichment and do not require a dwarf galaxy progenitor. 1.1. Globular Clusters 15

3 5 Gyr 10 Gyr

sun 0

−1

log(L) L −2

−3

−4

−5 −0.1 0.2 0.5 0.8 −0.1 0.2 0.5 0.8 b−v b−v

Figure 1.5 Hertzsprung-Russell or colour-magnitude diagram of a globular cluster model at 5 Gyr and 10 Gyr of age. Ni = 100 000 stars are used in these models which are evolved according to the dynamical and stellar evolution algorithms in NBODY6 and described in this chapter. Noticeable features include the double main sequence created by binary systems, the white dwarf cooling sequence, the main sequence turnoﬀ as well as blue stragglers to the left to the main sequence turnoﬀ, in particular in the younger snapshot. 16 Chapter 1. Introduction

Given those uncertainties and the fact that the star formation timescale and the observed age spread of stars in GCs are orders of magnitude shorter than the lifetime of a cluster, individual GCs are approximated by a SSP in this work.

1.2 Open Clusters

Open clusters (OCs) are less massive and more loosely bound than GCs, implying shorter lifetimes of several hundred million years up to a few billion years. A common fate for OCs is that they will eventually disperse. In particular for clusters with low masses, the dissolution time can be as little as ∼ 10 Myr, as they can be disrupted in their early stages of evolution following an expansion caused by mass loss during the expulsion of gas not incorporated into stars (Lada & Lada, 2003). While it is not thought that this infant mortality is the fate of all low mass clusters (Bastian, 2011), many open questions remain about their dynamical state. Particularly, with escape velocities lower than the mean velocity of the member stars, many clusters are born to disperse. Remnant clusters with fewer than 20 remaining stars (de La Fuente Marcos, 1997) can form hierarchical high- order systems and may account for all observed high-order systems (Goodwin & Kroupa, 2005; Pavani & Bica, 2007). OCs are numerous, with over 3000 known in the solar neighborhood (see Fig. 1.6, Kharchenko et al. 2013), and new clusters are still being detected (Borissova et al., 2011, 2014). The projected distribution of Galactic OCs from the Kharchenko et al. (2013) OC catalog (which is complete to 1.8 kpc from the Sun) is shown in Fig. 1.6. The difficulty to detect new OCs, i.e. to disentangle them from the field-star population in a dusty environment, can be inferred from Fig. 1.7, where recently discovered OCs from the Vista Variable in the Via Lactea (VVV) survey (Minniti et al., 2010) are shown (Borissova et al., 2011). While GCs live anywhere within their host galaxy potential (bulge and halo), OCs are primarily found within the disk, and in particular spiral arms of the MW where star formation is ongoing. In general, star clusters form from interstellar clouds which follow a mass distribution n(m)dN ∼ M 2dM. However, this distribution changes during subsequent evolution owing to internal and external dynamical evolution, with a preferential dissolution of low mass clusters. Because it is more difficult to distinguish OCs from the fore- and back-ground con- tamination of nearby stars, in contrast to GCs, and because of their high binary fraction (assumed to be ≥ 50%, Sollima et al. 2010), and in many cases irregular shape, structural parameters for OCs are not defined as well as for GCs and uncertainties remain about 1.2. Open Clusters 17 their sizes and dynamical state. Considerable effort in this direction has been taken in the past with the result that several catalogs exist (Dias et al., 2002; Kharchenko et al., 2013). There is currently no indication for multiple populations in OCs as indications for a spread in metallicity, such as a red giant spread wide in colour, can be explained by the high binarity (e.g. Carraro et al. 2014). Given the lower numbers of stars in OCs compared to GCs, it is more feasible to evolve direct models. This has been done for several OCs such as the Pleiades, Praesepe and Hyades (Portegies Zwart et al., 2001), M67 (Hurley et al., 2001, 2005) and NGC188 (Geller & Mathieu, 2012; Geller et al., 2013). In both cases, the observed blue straggler population has been reproduced in the models, highlighting the fact that OCs are an ideal laboratory to study stellar and dynamical evolution combined. 18 Chapter 1. Introduction

0 Y [kpc] −10

−20

−30 −30 −20 −10 0 10 20 30 X [kpc]

Figure 1.6 Positions of Milky Way star clusters projected onto the galactic plane. Black dots represent open clusters (which due to their low mass can only be observed in the solar neighbourhood), grey dots are cluster remnants and red circles are globular clusters (following a distribution around the Galactic center, denoted with a blue plus sign). A grey dashed circle indicates a distance of 20 kpc from the Galactic center, and the dashed lines the position of the Sun. GCs extend beyond this scale, with the recently discovered MW GC PSO J174.0675−10.8774 at a heliocentric distance of ≈ 145 kpc the most distant one (Laevens et al., 2014). The data for this ﬁgure is taken from the Milky Way Star Cluster Catalogue by Kharchenko et al. (2013) available at http://heasarc.gsfc.nasa. gov/W3Browse/all/mwsc.html. 1.2. Open Clusters 19

Figure 1.7 Mosaic of new infrared detected open clusters in the Milky Way (Borissova et al., 2011). Image credit: ESO/J. Borissova. 20 Chapter 1. Introduction

1.3 Star cluster models with NBODY6

They key to be able to study the evolution of SCs as laboratories of stellar evolution as well as tracers of galaxy evolution is to evolve direct, star-by-star models including stellar evolution and an external tidal ﬁeld. The code of choice for this purpose is NBODY6 (Aarseth, 1999, 2003; Nitadori & Aarseth, 2012), a direct integrator following each star’s individual trajectory in space and evolution3 (for alternative options see Chapter 1.3.4). Given the direct nature of the code, each star “feels” the gravitational potential of every other star in the cluster and no smoothing or softening is applied: direct collisions between stars are possible and exotic stars such as blue stragglers can form. One cluster at a time is evolved within a given galactic potential. Stellar evolution and dynamical interaction start at t = 0, when Ni stars are already formed and any gas not incorporated into stars is assumed to have been expelled. At the heart of each N-body code lies the challenge to evaluate the force on each star according to Newton’s equations of motion

m m F~ = −G 1 2 ~r (1.3) 12 |~r|3

between two bodies with masses m1 and m2 with a relative position vector ~r = ~r1 − ~r2. This implies that to calculate the force alone, on the order of N(N −1)/2 partial diﬀerential equations need to be solved per time step just to obtain positions and velocities. During the development of direct N-body codes, at ﬁrst models were run with only around 20 equal-mass particles, while it is now possible to run models with over 1/4 million stars (see also Fig. 1.8). This advance has been possible only due to the continuous development of code and hardware.

1.3.1 Development: Five decades from NBODY1 to NBODY7

Holmberg (1940, 1941) studied gravitational encounters of two star clusters (then called nebulae) before it became possible to carry out the numerical integration of the force equations. He set up an experiment broadly consisting of a board equipped with light bulbs representing a projection of the interacting stars, and replaced mass with light where the brightness of a light bulb represented a particles mass. N-body simulations on computers were not introduced until von Hoerner (1960a, 1963) in Heidelberg, in a project later discontinued but simultaneously started and continued by Sverre Aarseth in Cambridge (UK), the main force behind the development of the most advanced direct

3 NBODY6 is publicly available at http://www.ast.cam.ac.uk/~sverre/web/pages/nbody.htm 1.3. Star cluster models with NBODY6 21

N-body code today. The development is outlined in detail in Aarseth (1999) and Aarseth (2003), and a few selected milestones along the way are listed below.

• The development of a random number generator for the setup of the initial distribution of stars within a given distribution (von Hoerner, 1957). • The availability of computing hardware and software, which was achieved with the introduction of the Siemens2002 machine4 and the introduction of the programming language formula translator, later known as FORTRAN, to communicate with the computer via punch cards. • NBODY1, including a polynomial fitting function for the force summation in third and later fourth order (Aarseth, 1963). Softening for collisionless systems (i.e. galaxy clusters) was included in order to reduce two-body relaxation and mass segregation. The polynomial fitting function facilitated the prediction of coordinates, which laid the path to use individual time steps. Stiefel & Kustaanheimo (1965) later developed a regularization technique to remove the singularity in the equations of motion of any two-body problem, now commonly known as KS-regularization, reducing the number of steps in the integration by transforming the Kepler orbit into a harmonic oscillator (Funato et al., 1996). • NBODY3, including three-body regularization for interactions of hard binaries with other single stars or binaries via the introduction of two coupled KS regularizations (Aarseth, 1972). This was later generalized into a global regularization scheme for all N(N −1)/2 interactions (Heggie, 1974). The Ahmad & Cohen neighbour scheme, splitting the total force acting on a particle into two parts, was also introduced. In this procedure, only the nearest N neighbours are directly integrated, separately adding the distant members and external effects, allowing also the use of different time steps for neighbours and distant particles (Ahmad & Cohen, 1973). • NBODY2 is a deviation into cosmology with the introduction of comoving coordinates to only integrate deviations from the Hubble flow (Aarseth, 1979). • Improvements towards more realistic star clusters with the inclusion of a tidal field (Hayli, 1970) and mass-loss for massive stars in form of supernova-like behavior (Wielen, 1972; Aarseth, 1973) • Chain regularization to treat strong triple and binary-binary interactions (Mikkola & Aarseth, 1990, 1993, 1996) • NBODY5, with a regularized Ahmad & Cohen scheme as well as Hermite integration (Makino, 1991). The former served as an opportunity to study binaries, as neighbour

4http://www.computermuseum-muenchen.de/computer/siemens/ 22 Chapter 1. Introduction

lists can be used to determine perturbers. The latter means that corrections of the higher order derivative are not directly required, implying that the forces from distant particles can be calculated more easily. The Hermite integration cleared the way for the introduction of special-purpose computers (see below), and was implemented with NBODY4. • The importance of stellar evolution was realized early on, and simple models with instantaneous (supernova-like) mass loss at the end of a stars lifetime were applied (“one-step stellar evolution” as in Aarseth & Woolf 1972 or Chernoff & Weinberg 1990 for Fokker-Planck models) until more sophisticated models incorporating mass- loss procedures due to stellar winds (Aarseth, 1996; Portegies Zwart et al., 1999) were available. This was (and still is) based on analytic fitting functions for synthetic stellar evolution (Eggleton et al., 1989) with binaries (Tout et al., 1997). Today, stellar evolution is implemented in NBODY6 and other N-body codes via the single stellar evolution (SSE) and binary stellar evolution (BSE) algorithms by Hurley et al. (2000, 2002), see Chapter 2 for more details. • Tidal circularization (Mardling, 1995a), hierarchical stability (Mardling, 1995b) and evolution of hierarchies (Mardling & Aarseth, 1999) • All these improvements were combined into NBODY4 and NBODY6 (Aarseth, 1999), where NBODY4 was made available for special purpose computers such as the HARP and GRAPE machines (see Chapter 1.3.2), and NBODY6 for ordinary workstations. Today, a version of NBODY6 running on a hybrid machine with graphics processing unit (GPU) is now available (Nitadori & Aarseth, 2012): the code used in this work. • NBODY7, an extension of NBODY6 including the relativistic treatment of, for example, binary black hole systems (Aarseth, 2012), using few-body regularization with post- Newtonian terms (Mikkola & Merritt, 2008). • Extensions such as McLuster (Küpper et al., 2011) are available to easily create input files, as well as a NBODY6 adapted for parallel supercomputers, NBODY6++ (Spurzem, 1999).

Twice it was possible to increase the number of stars in a model signiﬁcantly while reducing computation times: ﬁrst with the introduction of special purpose machines designed to evaluate the force equation of many stars over and over (in particular the HARP and GRAPE boards, see below) and more recently, the possibility to evolve star cluster models on conventional GPUs (see the dashed line in Fig. 1.8). 1.3. Star cluster models with NBODY6 23

1.3.2 Special Purpose Computers and Graphics Processing Units

GRAPE (GRAvity PipE) systems were the ﬁrst special-purpose computers developed speciﬁcally to simulate many-body problems using a pipeline architecture, i.e. hard-wiring the machine to solve a particular equation (Sugimoto et al., 1990). In terms of the force equation, this is applicable over a wide range of physical processes beyond star cluster models, including for example molecular or plasma dynamics. The GRAPE system works such that the GRAPE processors are connected via an interface to an ordinary UNIX workstation containing one or more central processing units (CPUs). This host computer manages the computing log with the knowledge of position and velocity of each star, up- dated at each time step (Makino et al., 1993). Similar to the GRAPE machine the HARP (Hermite AcceleRator Pipeline) computers were developed, hard-wired such that they were designed to use a Hermite integrator, implying that the derivative of the acceleration was calculated in addition to the acceleration (Makino et al., 1997). Given the specialization of the GRAPE and HARP cards to solely solve the force equations, the host computer also follows the computations beyond the force equation, such as for example stellar evolution. Of particular interest for star cluster simulations were the following machines5:

• GRAPE-1 and GRAPE-1A in 1989, where the GRAPE-1A included the “Aarseth suﬃx” to calculate the position at the next time step via extrapolation (Sugimoto et al., 1990)

• GRAPE-2 in 1990 with higher accuracy using ﬂoating point operations (Ito et al., 1991)

• HARP-1 and HARP-2 in 1993, modiﬁed to including Hermite integration, i.e. to calculate the acceleration as well as its derivative (Makino et al., 1993)

• GRAPE-4 in 1995, a parallel machine containing up to 1692 HARP chips for maximum speed (Taiji et al., 1996)

• GRAPE-6 in 19996, the successor of GRAPE-4 with up to 2048 chips (Makino et al., 2003), and GRAPE-6A (or micro-GRAPE), a single-card GRAPE-6 to be incorporated into the nodes of a parallel computer (Fukushige et al., 2005).

5See also http://nbodylab.interconnect.com/nbl_grape_history.html 6See also http://jun.artcompsci.org/grape6.html 24 Chapter 1. Introduction

These computers made it possible to evolve star cluster models with more than 10 000 stars, in particular the models by Baumgardt & Makino (2003) with 131 072 stars - corresponding to the maximum number of particles the GRAPE memory could hold at that time (Fukushige et al., 2005). The increase in number of stars per model is outlined in Fig. 1.8 covering over five decades of running direct N-body codes. Since the introduction of the GPU, models with ≥ 100 000 stars are “cheap” to evolve and not rare anymore. The most massive direct N-body models today have initially more than 200 000 stars (Hurley & Shara 2012; Sippel & Hurley 2013 and Heggie et al. 2014 in prep.). These are, together with other models run on special-purpose computers, above the dashed line in Fig. 1.8. On the other hand, the number of particles in collisionless systems have increased more rapidly, as is expected when detailed evolution of stars or binaries is not included. Ishiyama et al. (2012) have reported a one trillion body calculation on K computer7. The reason that the GRAPE/HARP machines worked so well is that the force equation is additive: in other words, it presents an embarrassingly parallel problem. Recently, the GPU provides a cheap alternative as it is parallel by nature (Gaburov et al., 2009) and used in general purpose computing. Designed for the acceleration of computer graphics (i.e. the calculation of colours red, green and blue in space), it can also handle the calculation of velocities in space. The hybrid approach of an ordinary workstation (i.e., CPU) plus parallel component for the force calculation is kept on, with the difference that even the dedicated GPU is not a special purpose component but commercially available, dropping the cost significantly. Today, we can efficiently run high-N GC models on a desktop computer containing several CPU cores and GPUs8, and more recently, GPU supercomputers have become available.

7http://www.aics.riken.jp/en/k-computer/about/ at RIKEN in Japan. 8In other words, a machine well suited for playing the latest and most hardware-demanding computer games. See http://gpgpu.org/ as a resource for General-Purpose computations on GPUs. 1.3. Star cluster models with NBODY6 25

106 Heggie in prep.

Sippel & Hurley (2013) Shara & Hurley (2006); Hurley & Shara (2012) 5 Baumgardt & Makino (2003) 10 Zonoozi et al. (2011)

Makino (1996)

Makino et al. (1993) 104 Dekel & Aarseth (1984) Spurzem & Aarseth (1996) Aarseth & Heggie (1993) Aarseth & Binney (1978) Aarseth et al. (1979) Heggie & Aarseth (1992) Particles 103 Gott et al. (1979) Terlevich (1987) # Aarseth (1972)

Aarseth (1969) Aarseth (1974)

2 Aarseth (1963) Wielen (1968) 10 Holmberg (1941)

von Hoerner (1963) von Hoerner (1960a) 101 1940 1950 1960 1970 1980 1990 2000 2010 2020 Year

Figure 1.8 Selected milestones in increasing the number of particles used in direct N-body simulations. See Dehnen & Read (2011) for an extension into high-N collisionless systems (in particular cosmology). The plot here simply focuses on the increase in N and does not give credit to many other milestones such as (but not limited to) the inclusion of binary systems or black holes. The gray dashed line indicates the change of technology from workstations to special purpose computers, for example GRAPE3 (Makino et al., 1993), GRAPE5 (Makino, 1996) or GRAPE6 (Baumgardt & Makino, 2003) and graphics processing units (Zonoozi et al., 2011; Sippel & Hurley, 2013). This ﬁgure is adapted from Heggie & Hut (2003); Hut (2010); Heggie (2011). 26 Chapter 1. Introduction

1.3.3 Initializing NBODY6

The evolution of any model commences with the initialization of the N stars, followed by an integration cycle for each time step. For the initialization, after global counters and variables are introduced, the input file prepared by the user is read. This file contains parameters and options, such as number of stars N and their initial mass function or the profile according to which they should be distributed, the binary fraction or cluster orbital parameters to name just a few. According to these parameters, the initial setup is calculated9. In practice, this means that masses for the N stars are drawn from the chosen initial mass function, and coordinates as well as velocities are assigned (again, just to name a few examples). All masses, positions and velocities are scaled to N-body units (see below), and scaling factors for the conversion to physical units are calculated. If they are indicated in the input file, an external tidal field, primordial binary distribution and stellar evolution parameters are designated. After this, the force polynomials are evaluated for the first time, and time-steps are defined. Close binaries (if any) are regularized as a last step of the initialization. Within the framework of NBODY6, so-called Hénonunits, formally known as standard N-body units, are used for calculations and output. This was suggested by Hénon(1971) (see also Heggie & Mathieu 1986) and are based on the following definitions10:

N X0 G = 1 ,M0 = m0,i = 1, and E0 = −0.25 , i=1 with M0 the initial cluster mass and m0,i the stellar masses, N0 the initial number of stars 2 and E0 the energy, implying m0 = 1/N0, σ = 0.5 for the mean square equilibrium velocity √ and a crossing time tcr = 2 2. The multiplication factors for conversion to physical units are given in the output of each NBODY6 run. See also Aarseth (2003) Chapter 7.4 for more details.

1.3.4 Alternatives to NBODY6

Alternatives to direct N-body codes or NBODY6 exist, each with their own advantages as well as disadvantages. To name just a few, these include tree codes scaling with N × log N (Barnes & Hut, 1986) and Monte-Carlo codes calibrated against direct N-body

9Alternatively, they can be provided by the user. McLuster (Küpper et al., 2011) provides a tool to create such files. 10From Aarseth (2003), Chapter 7.4: “The comparison problems were due to the theoreticians preference for considering a wide variety of scales and reporting the results in terms of natural variables, unlike observers who are constrained by reality.” 1.3. Star cluster models with NBODY6 27 simulations, in particular MOCCA11 (Giersz et al., 2008, 2013). EMACSS12 (Alexander & Gieles, 2012; Gieles et al., 2014; Alexander et al., 2014) is a code following the evolution of a global cluster using simple quasi-analytic models based on Hénon’sequations (Hénon, 1961, 1965). In Fig. 1.9 is an overview of alternatives to direct N-body codes. While all of those options produce essential results in a parameter space too large for direct N-body models, the direct approach is sometimes necessary, in particular for a detailed understanding of star-by-star interactions and their evolution. Wrappers such as STARLAB13 allow the computation of direct N-body models with a preliminary analysis of output. The N-body integrator within STARLAB is Kira (Portegies Zwart et al., 2001). Results from STARLAB have been found to agree well with NBODY4 and NBODY6 (Anders et al., 2009a, 2012). The AMUSE code14 provides a framework to link hydrodynamical codes with N-body and stellar evolution (McMillan et al., 2011). This multi-physics approach is also able to include gas in young open clusters (Portegies Zwart et al., 2013). On a different scale, an extension of NBODY6 called nbody6tt15 can read in a previously created general (and also time-variable) tidal tensor (Renaud et al., 2011). The most recent version of Sverre Aarseth’s N-body codes is NBODY7, which includes post-Newtonian treatment (Mikkola & Merritt, 2008) for relativistic encounters of black holes (Aarseth, 2012). While direct N-body codes are more computationally demanding and consequently slower than all other options shown in Fig. 1.9 they ultimately remain the most versatile and accurate option, and as such NBODY6 is the method of choice for this thesis.

11The MOnte Carlo Cluster simulAtor and see http://www.moccacode.net/. 12Evolve Me A Cluster of Stars, available via http://github.com/emacss 13STARLAB is developed by Piet Hut, Steve McMillan, Jun Makino and Simon Portegies Zwart, see http://www.sns.ias.edu/~starlab/starlab.html. 14http://amusecode.org/, standing for the Astrophysical Multipurpose Software Environment 15https://github.com/florentrenaud/nbody6tt/releases 28 Chapter 1. Introduction

GPU (Gaburov et al., 2009) GRAPE (Makino & direct sum- Taiji, 1998) mation parallel (Spurzem, N-body 1999)

tree code serial (Aarseth, 1999) rotating (Einsel & Spurzem, 1999) finite difference (Cohn, isotropic 1979) Fokker- finite difference (Taka- Planck hashi, 1995) anisotropic SC serial (Giersz, 1998) Monte hybrid (Giersz & Carlo parallel (Joshi et al., Spurzem, 2000) 2000) anisotropic (Louis & Spurzem, 1991) Fluid gas isotropic (Lynden-Bell & Eggleton, 1980) moment (Larson, 1970) evaporative (Chernoff, Scaling 1987)

Figure 1.9 Numerical methods to study star cluster evolution and alternatives to direct N-body codes (after lecture notes by Douglas Heggie, http://isima.ucsc.edu/ presentations/lectures/Heggie.pdf, slide 70, and see also Table 13.2 in Aarseth 2003). 1.3. Star cluster models with NBODY6 29

Figure 1.10 Three dimensional view of a GC model from Chapter 2 with N ≈ 30 000 stellar systems remaining after 12 Gyr of evolution. Single stars are in black and binaries in red, with the same symbol for all masses. For clarity, in this ﬁgure the axes are truncated at d ∼ 20 pc from the cluster center (this is not the case for the simulation). Projected onto the x, y and z-planes are the neutron stars (grey dots), the remaining black hole binary system (red cross), and half-mass radius rhm = 6.8 pc (black circle). The black hole binary has a total mass of ∼ 30 M (17.5 M and 12.9 M ) and is the third-closest system to the cluster center at the time of the snapshot. For reference, the half-mass radius of the binaries is ∼ 5.2 pc. See Fig. 1.11 for the same model in projection. 30 Chapter 1. Introduction

Figure 1.11 Same model and snapshot as in Fig. 1.10, using the stars v-band luminosity and projected along the x (top left), y (top right) and z-axis (bottom). 1.4. Comparing models and observations 31

1.4 Comparing models and observations

While it is nowadays computationally cheap to evolve realistic models of OCs (Hurley et al., 2005; Geller et al., 2013), their structural parameters and dynamical state are poorly understood. This is due to the reason that the clusters are spatially irregular, and it is observationally expensive to compute 3-d velocities of the member stars. In the near future, this will be dramatically improved when proper motions from Gaia16 become available, in combination with radial velocities from the ongoing Gaia-ESO survey17. The opposite is true for GCs: their structural parameters such as sizes and masses are well defined, as well as are some of their orbital parameters in the Milky Way and other nearby galaxies (Dauphole et al., 1996; Pota et al., 2013). Even radial velocities exist for some GCs in the Milky Way (e.g. Palomar 14, Jordi et al. 2009), but only with the most recent upgrades in computing hardware has it come into reach to create realistic models of GCs of moderate size and density (Zonoozi et al., 2011; Hurley & Shara, 2012; Sippel & Hurley, 2013; Zonoozi et al., 2014). The output of any N-body code allows us to analyze the dynamical state of a model using the entire information available in 3d-space: each star’s position, velocity, mass and evolutionary stage. Remnants such as WDs, NSs and BHs as well as low-mass stars are easily accounted for. In Fig. 1.10 is an example model where all stars and binary systems are marked, as well as the NSs and BHs in the center, shown in projection. In theory, one can rotate the coordinate system in Fig. 1.10 in any desired direction - but comparison to observations is somewhat limited. As an example, the half-mass radius is shown, which is not a direct observable. To be comparable to observations it is an advantage to look at the cluster light rather than mass, and to observe the cluster in projection. Fig. 1.11 is the same snapshot as in Fig. 1.10 projected along the x, y and z-axis, and shows more closely what an observer would see. Caution needs to be taken when comparing models and observations, as similar nomen- clature exists for different scaling parameters. This begins with the difference that observed clusters are measured in projection, while models are evaluated in three dimensions (see Fig. 1.10). Solely due to projection, a radius measured in 3d is expected to be a factor 4/3 larger (Fleck et al., 2006). While it is the convention to measure observed cluster sizes via the half-light or effective radius (see Chapter 2), in simulations half-mass, or

16Gaia is a space telescope for astrometry, launched in December 2013 by the European Space Agency. About one billion objects, mostly stars, will be targeted, see http://www.esa.int/Our_Activities/Space_ Science/Gaia_overview. 17A spectroscopic follow-up of 105 Gaia targets, see http://www.gaia-eso.eu/. 32 Chapter 1. Introduction more generally Jacobi radii are used. The latter are spheres enclosing a certain fraction of the cluster mass as measured from the center. Of particular interest in simulations is the so-called N-body core radius rc, which is a density weighted mean distance to the cluster center (see Appendix A and Fig. 1.4) and not related to the core radius obtained from King model ﬁts (King, 1962) in observations.

1.5 Outline

GCs, whether they be galactic or extra galactic, are proving to be important probes of galaxy formation and evolution. However, this has become a data-driven field which is in need of input from theory and modeling (Brodie & Strader, 2006). This thesis aims to help address this need by accounting for the dynamical evolution of stellar populations in GCs. To do so, we evolve direct N-body models as realistically as possible and use the output to study the clusters from a theoretical point of view: understanding the mass distribution allow us to know the dynamical state of a cluster. Using observational tools to analyze model clusters allows us to bridge the gap between models and observation. This thesis concentrates on using exactly this hybrid technique of observing cluster models to address questions that prove difficult to answer from observations where the corresponding dynamical information might not be available. In Chapter 2 models of different chemical composition are used to investigate the effect of metallicity on stellar evolution and hence mass loss on stars and the entire cluster. It is shown that an often implicitly assumed one-to-one correspondence of half-mass and half-light radius (Djorgov- ski, 1993) breaks down when taking the metallicity into account. Next, in Chapter 3 this is taken a step further when creating colour-magnitude diagrams for a single GC or a mock population of clusters, allowing us to study either cluster colour gradients or the colour evolution with time. It is intrinsic to this approach that the external tidal field is accurately taken into account, and hence dynamically evolved single stellar populations are presented. In Chapter 4, a larger N-body model is used in direct comparison to the MW GC M22 to investigate the possibility of multiple remnant BHs in its core. This thesis is concluded by final remarks in Chapter 5.

While globular clusters are old stellar populations, they are still dynamically active. 2 Metallicity Eﬀects on Globular Cluster Evolution

In the future, the method of analysis should be more focused on a comparison with observations. (...) The most important goal of such a comparison with the observations should be the attempt of a purely dynamical age determination of star clusters. - Sebastian von Hoerner 1963

2.1 Globular cluster sizes

Globular clusters (GCs) are substantial components of galaxies and found in populations of up to thousands in giant elliptical galaxies (Peng et al., 2011). The Milky-Way (MW) hosts a GC population of 157 confirmed clusters (Harris 1996, 2010 edition), with new clusters still being discovered (e.g. Minniti et al. 2011b). These clusters live within the bulge as well as the halo of the Galaxy and can - in contrast to star clusters beyond the Local Group - easily be resolved in ground-based observations. In general, the GC systems of galaxies tend to appear in two sub-populations: a blue and a red component (Zinn, 1985). Although the metallicity cannot be inferred directly from the cluster colour due to the age-metallicity degeneracy (Worthey, 1994), it has been well accepted that the blue clusters are metal-poor, whereas the red ones are metal-rich. Both sub-populations are old (e.g. Mar´ın-Franch et al. 2009), with a trend for the red clusters to be more centrally concentrated within their host galaxy’s potential than their blue counterparts (Kinman 1959; Brodie & Strader 2006, also see Fig. 2.1). The ability of the Hubble Space Telescope to partially resolve globular clusters even beyond the Local Group has lead to the finding that i) GCs have mean half-light radii rhl = 3 pc (Jordánet al., 2005) and ii) red clusters are on average ≈ 17−30% smaller than their metal poor counterparts (Kundu & Whitmore, 1998; Jordánet al., 2005; Woodley & Gómez,2010). Several explanations for this phenomena have been proposed: projection

33 34 Chapter 2. Metallicity Effects on Globular Cluster Evolution effects and the influence of the tidal field (Larsen et al., 2001) or a combined effect of mass segregation and the dependence of main-sequence lifetimes on metallicity (Jordán,2004; Jordánet al., 2005). Whether either of those effects are dominating or a combination of both can only be investigated through direct star cluster simulations where three- dimensional galactocentric distances are known and stellar evolution is included in the dynamical evolution of the cluster.

The effects of metallicity on the evolution of a single star manifests itself as a different rate of stellar evolution, which is accompanied by a different mass-loss rate and hence ultimately affects the stars lifetime and remnant mass (see Section 2.2). In general, low metallicity stars evolve faster along the main sequence than their high metallicity counterparts (Hurley et al., 2000). For a bound system such as a star cluster, the increased mass-loss rate can lead to a lower cluster mass and hence a lower escape velocity and. This in turn can produce a stronger increase in radius for the metal-poor cluster. At later stages, this might also lead to postponed core-collapse for the low metallicity cluster. Both effects could lead to a larger measured cluster size. A preliminary study along these lines has been carried out by Hurley et al. (2004) for open clusters. They showed that an increased escape rate for the metal-rich clusters owing to earlier core-collapse acts to cancel these effects resulting in only a 10% difference in cluster lifetime for metal-poor versus metal-rich cases - within the statistical noise of fluctuating results from one simulation to another. However, several aspects of our new simulations differ from this preliminary study. Among those are an adjusted binary fraction for GCs and an improved tidal field.

Most importantly we also use a higher initial number of stars Ni, bringing the N-body models into the GC regime. This ultimately leads to an increase in cluster lifetime and hence not necessarily core-collapse or depletion of stars within a Hubble time.

In this work, we make use of a set of star cluster simulations evolved with the direct N- body code NBODY6 (Aarseth, 1999, 2003) to study the effects of metallicity on star cluster dynamics, evolution and size (i.e. effective radius) to answer the question if metallicity alone could reproduce the observed size difference. We measure the sizes of these clusters along their evolutionary track with methods used both in observations and theory.

Recently Downing (2012) has published a set of Monte-Carlo models exploring the origin of the observed size difference between metal-rich and metal-poor GCs, which provides an excellent comparison for our work. This follows on from the N-body models of Schulman et al. (2012), who investigated the evolution of half-mass radius with metallicity in small-N clusters. Similarly to Downing (2012), we shall be careful to make a distinction between the actual size of a star cluster, represented by the half-mass radius (which we 2.2. Metallicity effects on stellar and star cluster evolution 35 shall denote as r50%, i.e. the 50% Lagrangian radius), and the observationally determined size (the half-light or effective radius, reff ). This paper is structured as follows. We introduce the differences in stellar evolution depending on metallicity in the next Section. In Section 2.3, we describe our simulation method and the models we have chosen to evolve. In Section 3.3, we analyze the evolution of cluster mass, binary fraction, luminosity, half-light radius and mass-to-light ratio which is followed by discussion and conclusions.

2.2 Metallicity eﬀects on stellar and star cluster evolution

The main sequence (MS) lifetime of a single star depends mainly on its mass (and hence luminosity), but also on its chemical composition: the metallicity Z (or [Fe/H]). Clayton (1968) shows that the MS lifetime can be represented as:

1 m/m tMS ∝ , (2.1) X L/L where m and L are a star’s mass and luminosity and X the hydrogen fraction. A star’s mass at given luminosity scales as: κ0.2 m ∝ 0 , (2.2) µ1.4 where κ0 is the central opacity and µ the mean molecular weight. The hydrogen fraction X and helium abundance Y can be set as a function of metallicity according to:

X = 0.76 − 3Z (2.3)

Y = 0.24 + 2Z (2.4) as in Pols et al. (1998). If Z is decreased, X increases while Y decreases slightly, leading to a marginally lower mean molecular weight:

2 µ ≈ . (2.5) 1 + 3X + 0.5Y

To ﬁrst order, it can be assumed that the central opacity is proportional to Z: κ0 ∝ Z (Clayton, 1968). Using this, in combination with Eqs. 2.1 and 2.2 we ﬁnd:

κ0.2X Z0.2X t ∝ 0 ≈ , (2.6) MS µ1.4 µ1.4 36 Chapter 2. Metallicity Eﬀects on Globular Cluster Evolution

0.2 0.2 with κ0 ∝ Z being the dominant term in this equation. A lower opacity implies less resistance for escaping photons from the hydrogen burning core and hence a higher luminosity and therefore a shorter lifetime (see also Table 2.1). For an extended discussion we refer to Clayton (1968). In the N-body models, we evolve stars according to the stellar evolution prescriptions of Hurley et al. (2000), which are based on the detailed models of Pols et al. (1998). These prescriptions are accurate for a wide range of metallicities and cover all phases of stellar evolution. This means stars are evolved from the zero-age main sequence up to and including the remnant phases: white dwarfs (WDs), neutron stars (NSs) and black holes (BHs). If necessary, the stellar evolutionary track evolves via the giant branch, core helium burning and thermally pulsating asymptotic giant branch. As shown by Hurley et al. (2000), the diﬀerence in MS lifetime is most prominent for low-mass stars and steadily decreases towards higher mass stars until M ≈ 8 M , where the high metallicity stars begin to have a shorter MS lifetime, although only marginally (and noting that model uncertainties are more prevalent at higher masses). This implies, that for clusters of the same age, the mass of the most massive MS star (and hence MS turnoﬀ mass mTO) is higher in a high-Z cluster. Examples for mTO are given in Table 2.1. It is not only the MS lifetime that is altered by the metallicity, but also the remnant mass. For initial masses less than 50 M our models give a maximum black hole mass mBH ≈ 28 M for metal-poor stars versus mBH ≈ 12 M for metal rich progenitors (Belczynski et al., 2006).

This trend is the same for all remnants: a 2 M progenitor with Z = 0.0001 will end life as a WD of mass m = 0.84 M , while a metal-rich counterpart with Z = 0.01 will have a

WD mass of m = 0.66 M . This occurs after ≈ 0.9 and 1.4 Gyr, respectively. Hence the remnant mass in a metal-poor cluster is always expected to be higher (see also Table 2.2). 2.2. Metallicity eﬀects on stellar and star cluster evolution 37

Table 2.1 Main sequence lifetimes for stars with different metallicities. Metallicity Z and [Fe/H] are in the first two columns, followed by the hydrogen (X) and helium (Y ) mass fraction (Eq. 2.3 and 2.4). Even though the mean molecular weight µ (Eq. 2.5) in column 5 is barely affected by the metallicity, different relative MS lifetimes tMS (column 6) are caused by a change in opacity for different metallicities according to Eq. 2.6 . The expected MS lifetimes up to the Hertzsprung Gap according to Hurley et al. (2000) for stars with initially 3, 1.5 and 0.8 M are given in the next three columns, followed by the MS turnover mass mTO in columns 10 and 11 at ages of 11 and 12 Gyr. We note that stars with Z = 0.001 and Z = 0.0001 evolve in a similar fashion compared to the metal rich case - hence Z = 0.001 is also a metal-poor case. This has already been noted by Hurley et al. (2004), as well as the fact that stars and clusters with Z = 0.01 evolve similar to solar metallicity Z = 0.02.

Z [Fe/H] X Y µ 0.0001 −2.3 0.76 0.24 0.59 0.001 −1.3 0.76 0.25 0.59 0.01 −0.3 0.74 0.26 0.60

Z tMS (Eq. 2.6) tMS(3 M ) tMS(1.5 M ) tMS(0.8 M ) 0.0001 0.25 0.29 Gyr 1.6 Gyr 13.5 Gyr 0.001 0.40 0.29 Gyr 1.7 Gyr 14 Gyr 0.01 0.60 0.35 Gyr 2.4 Gyr 21.7 Gyr

Z mTO (11 Gyr) mTO (12 Gyr) 0.0001 0.84 M 0.83 M 0.001 0.85 M 0.83 M 0.01 0.95 M 0.91 M 38 Chapter 2. Metallicity Eﬀects on Globular Cluster Evolution

15 [pc]

hl 10 r

0 100 101 102 d [kpc] gc

Figure 2.1 Size and galactocentric distance of the MW GC population (compiled from Harris 2010). Blue circles are used for metal-poor and red squares for metal-rich clusters, the distinction is made at [Fe/H]= −1.1. The solid black like denotes the size-distance p relation rhl ≈ dgc from van den Bergh et al. (1991). Metal-poor clusters tend to have larger galactocentric distances as well as larger sizes (half-light radii). The models used for this study are evolved at a galactocentric distance of 8.5 kpc, marked by the vertical dotted line. 2.2. Metallicity eﬀects on stellar and star cluster evolution 39

Since there is no strong evidence for an explicit metallicity dependence of the mass-loss rate of giants (Iben & Renzini, 1983; Carraro et al., 1996; Schr¨oder& Cuntz, 2005), generally mass-loss from the envelope during the giant branch phase and beyond is implemented according to Reimer’s law (formula of Kudritzki & Reimers 1978):

LR m˙ ∝ M yr−1 . (2.7) m

An implicit metallicity dependence exists as the evolution of the radius R and L depend on the mean molecular weight and hence Z, as mentioned earlier (e.g. Eq. 2.2). Exceptions apply for very massive stars, e.g. luminous blue variables with luminosity L > 4000 L , where the mass-loss is modeled according to:

0.5 −15 Z 0.81 1.24 0.16 −1 m˙ = 9.6 × 10 R L m M yr . (2.8) Z

This is Eq. 2 from Nieuwenhuijzen & de Jager (1990) but modified by the factor Z0.5 (Kudritzki et al., 1989). Note that mass-loss can also occur as a result of mass transfer - having ultimately the same effect of moving a star along the MS towards lower effective temperature and hence lower luminosity.

2.2.1 Stellar evolution of an entire population

To quantify the effects of stellar evolution on a non-dynamical population, we evolve 105 000 stars together through stellar evolution alone (Hurley et al., 2000). This means that dynamical effects such as the influence of the galactic tidal field as well as the intrinsic N-body evolution within the cluster are ignored. The set-up of the initial masses of this population is identical to our N-body models introduced in Section 2.3, where the dynamical evolution is fully incorporated. In Fig. 2.2, the mass, luminosity and mass- to-light ratio evolution of this model is illustrated for the three metallicities Z = 0.01, Z = 0.001 and Z = 0.0001. At the Hubble time, ≈ 30% of the initial stellar mass is lost purely due to stellar evolution and only ≈ 50% of the initial mass in MS stars is still remaining (in agreement with Baumgardt & Makino 2003). The overall mass of the low-Z population stays higher throughout, while the mass contained in MS stars is always higher in the high-Z population, as expected due to the higher MS turnoff mass. 40 Chapter 2. Metallicity Effects on Globular Cluster Evolution

1 106 0.8

0.6 0 5

L 10

M/M 0.4

0.2 Z=0.01 Z=0.001 104 Z=0.0001 0 0 5 10 0 5 10 t [Gyr] t [Gyr] 101

100 M/L

10−1

0 5 10 t [Gyr]

Figure 2.2 Population of 105 000 stars, corresponding to the set-up for our N-body models, evolved with only stellar evolution (i.e. no dynamical interactions). The metallicities are Z = 0.0001 (dashed blue), Z = 0.001 (solid green) and Z = 0.01 (dashed-dotted red). Left panel: The upper set of lines is the mass (scaled by the initial mass) of the entire population of stars (including all remnants) while the lower set of lines only includes stars on the main sequence. This illustrates that the mass contained in MS stars is always higher for metal-rich clusters and that the overall mass evolution depends critically on the treatment of remnants. Middle panel: Corresponding luminosity evolution (in units of L ). In this case the treatment of remnants is not crucial. Note that even though the metal-rich model contains more stars on the MS, the metal-poor models have a higher luminosity. This is in agreement with Eq. 2.2 and implies that the higher luminosity of low-Z stars is outweighing the fact that the metal-poor clusters have a lower MS turnoﬀ mass at any given time than the metal-rich cluster (see Table 2.1). Note that both metal- poor cases are expected to evolve in a similar fashion (also see Table 2.1), however some variation is caused depending on the number of bright stars at any given time. Right panel: Resulting mass to light ratio M/L (in units of M /L ). As expected from the luminosity evolution, the mass-to-light ratio is higher for the metal-rich model. 2.2. Metallicity eﬀects on stellar and star cluster evolution 41

The luminosity (actually calculated as the V-band luminosity) drops by an order of magnitude within the ﬁrst ≈ 2 Gyr and roughly another magnitude over the next 10

Gyr of evolution. We see that even though a high-Z cluster will have a higher mTO, the luminosity of a metal-poor cluster remains 1.5 − 2 times higher throughout the entire evolution - based on stellar evolution alone. This implies that the increased brightness of low-Z stars is outweighing the higher number of MS stars in the high-Z case. As expected from the evolution of mass and luminosity in this non-dynamical model, the mass-to-light ratio M/L is predicted to be higher by nearly a factor two for a metal- rich cluster. In a dynamically evolved model with a tidal ﬁeld, the mass-to light ratios are likely to be modiﬁed as preferentially low-mass stars are lost from the outskirts of the cluster (Baumgardt & Makino, 2003). Low-mass MS stars are faint and have a high mass-to-light ratio. We will compare Fig. 2.2 to our dynamical models in Section 2.4.5.

2.2.2 Size: projection eﬀects vs. internal dynamics

For the MW, van den Bergh et al. (1991) found that the GC half-light radius rhl can be p related to the galactocentric distance dgc via rhl ≈ dgc (see Fig. 2.1). As the MW is the only galaxy where three-dimensional galactocentric distances are available, one has to rely on projected distances for extragalactic GC systems. Studies of extragalactic GC systems have shown that red and blue clusters are found to have different spatial distributions within the potential of their host galaxy: red clusters are distributed closer to the centre of the galaxy and subject to a stronger influence of the tidal field than the blue clusters (Brodie & Strader, 2006). A size difference ranging from 17% (Jordánet al., 2005) to 30% (Woodley et al., 2008) for the blue and red population has been found in numerous studies. Several scenarios have been proposed for the origin of this effect: projection effects and/or the effect of stellar evolution in combination with mass segregation, which we describe below. In addition, the possibility of different initial conditions during cluster formation have been proposed (Harris, 2009) as well as different initial mass functions for metal-poor or metal-rich clusters (Strader et al., 2009, 2011). Larsen & Brodie (2003) found that projection effects may account for the observed size difference of red and blue GCs, if the GC distribution flattens out near the centre of the galaxy (e.g. King profile) and there is a steep relation between cluster size r and galactocentric distance dgc. However, this is not the case for either more centrally peaked distributions or shallower r − dgc relations. Spitler et al. (2006) find in agreement with Larsen & Brodie (2003) that projection effects could explain the observed size difference in the Sombrero galaxy. A size gradient for GCs is found for small but not large (projected) 42 Chapter 2. Metallicity Effects on Globular Cluster Evolution galactocentric distances. In contrast to this, Jordán(2004) has found that the combined effects of mass segregation and MS lifetime lead to a size difference of low vs. high metallicity clusters. Under the assumption that the average half-mass radius does not depend on metallicity, the observed light-profiles were modeled with Michie-King multi-mass models and stellar isochrones leading to the result that a size difference of the observed magnitude arises naturally, with the metal-rich model having a half-light radius ∼ 14% smaller than its metal-poor counterpart. The reasoning for this originates from the different speed in stellar evolution of stars with different Z implying that the light profile of a high Z cluster can appear more concentrated. Unfortunately, in this approach the interplay between stellar dynamics and evolution was not considered. We note that Jordán(2004) assumes the average half-mass radius to be independent of [Fe/H] - an assumption pointing to a universality in the formation and evolution of GCs. As part of the ACS Virgo Cluster Survey (Côtéet al., 2004) the sizes of thousands of globular clusters belonging to 100 early type ellipticals in Virgo were measured (Jordán et al., 2005). They find in agreement with previous studies that the average half-light radius depends on the color of the GCs, with red GCs being ≈ 17% smaller than their blue counterparts. This size difference was proposed to originate from the effects of mass- segregation and metallicity, hence intrinsic cluster mechanisms as in Jordán(2004). The arguments given above show that it is necessary to know the three dimensional galactocentric distance of GCs to their host galaxy to fully understand and disentangle the influence of the environment and metallicity on GC evolution. To be able to distinguish between those effects, we focus on metal-poor and metal-rich clusters at the same location, i.e. where both coexist. In the MW, 16 GCs are located in the region between 7 ≤ dgc ≤

10 kpc with a mean size of rhl = 3.95 pc. Four of those are metal-rich ([Fe/H]> 1.1) and

12 metal-poor. Thus we chose a galactocentric distance of dgc = 8.5 kpc for our models.

2.3 Simulation method & choice of parameters

We use the direct N-body code NBODY6 (Aarseth, 1999, 2003) to construct and evolve our models. This state-of-the art N-body code takes advantage of the possibility to carry out such simulations on a graphics processing unit (GPU) coupled together with conventional central processing units (Nitadori & Aarseth, 2012). The simulations were carried out on Tesla S1070 graphics cards at Swinburne University. We use a Kroupa initial mass function (IMF: Kroupa et al. 1993) within the limits 0.1 2.3. Simulation method & choice of parameters 43 to 50 M to populate our cluster model with stars. The beginning t = 0 for the simulation corresponds to the zero-age MS and no gas is included in the models. The simulations start with Ni = 100 000 stellar systems, including a primordial binary frequency of 5% (see Section 2.3.1). These stars are initially distributed following a Plummer density proﬁle

" #−5/2 3M r 2 ρ(r) = 3 1 + (2.9) 4πRsc Rsc

(Plummer, 1911; Aarseth et al., 1974) where M is the cluster mass and Rsc is a scale radius (see below). As the Plummer profile formally extends out to infinite radius, a cut-off at ten times the half-mass radius is applied to avoid rare cases of stars at large distances (Aarseth, 2003). The individual initial positions and velocities are then assigned such that the cluster is in virial equilibrium. The cluster is subject to a constant, MW-like tidal field consisting of three components: a point-mass bulge, an extended smooth disc (Miyamoto & Nagai, 1975), and a dark 10 10 matter halo. We use Mb = 1.5 × 10 M and Md = 5 × 10 M for bulge and disc mass, respectively (Xue et al., 2008). The scale parameters for the Miyamoto disc are a = 4 kpc (disc scale length) and b = 0.5 kpc (galactic thickness). Formally the disk extends to infinity but with this choice of parameters the strength has dropped to less than 0.1% of the central value at a distance of 40 kpc. The dark matter halo follows a logarithmic profile 2 2 2 0.5 Φ ∝ v0 ln(d + b ) (Aarseth, 2003). Here d is the distance from the galactic centre at any given time, and b is constrained such that the combined mass of the bulge, disk and halo give an orbital velocity of v0 = 220 km/s at a galactocentric distance of dgc = 8.5 kpc.

As mentioned earlier, we choose to place our clusters in an orbit at dgc ≈ 8.5 kpc to match an environment where red and blue clusters coexist within the MW (see Fig. 2.1). The orbit is inclined ≈ 22 deg to the galactic disc reaching a maximum height of z ≈ 3 kpc above the galactic plane. The apogalacticon is 8.8 and perigalacticon 8.2 kpc with orbital period of ≈ 0.2 Gyr (see Fig. 2.3). We chose a mid eccentricity to not start with extreme cases. The inclination results in a maximum z = 3 kpc, which is typical for many MW clusters (Dauphole et al., 1996). During the lifetime of a cluster, stars are naturally lost due to dynamical relaxation, evolution and disc-shocking events. The tidal radius of a cluster in the Milky Way potential described above can be approximated as:

GM 1/3 r ' (2.10) t 2Ω2

(Küpper et al., 2010), where Ω is the angular velocity of the cluster orbit and G is the 44 Chapter 2. Metallicity Effects on Globular Cluster Evolution gravitational constant. Calculated at apogalacticon gives a rt = 52 pc, which we take as our initial value. This is adjusted as the cluster evolves according to the factor M 1/3. Stars are only removed from the cluster once their distance from the cluster centre exceeds twice the tidal radius. Gieles et al. (2011) have expressed the impact of the galactic tidal field 5 on a cluster by quantifying a boundary Mlim < 10 M × 4 kpc/Rgc below which clusters are tidally affected, whilst more massive clusters are tidally unaffected. The clusters in this study fall below this limit and hence are tidally limited. Within the framework of NBODY6, the only remaining parameter is the scale radius

Rsc, which sets the initial cluster size or density and acts as a conversion factor between physical and N-body units. It is an ongoing debate as to how extended GCs are when they are born. Recently, it has been pointed out that GCs could be the remnants of much bigger stellar structures such as the nuclei of accreted dwarf galaxies (Freeman, 1993; B¨oker, 2008; Forbes & Bridges, 2010). In general, shortly after stellar nuclear fusion is ignited within a proto-cluster, the cluster it is expected to increase it’s size as the remaining gas not incorporated into stars during star formation is ejected from the cluster. So far, globular cluster sizes at this early stage cannot be determined through observations. We choose

Rsc = 8, corresponding to an initial three-dimensional half-mass radius of r50% ≈ 6.2 pc. The half-mass radius evolves to ≈ 7 pc at the Hubble time, but is a three dimensional quantity and hence a smaller half-mass radius by 25% would be expected when measuring projected radii in two dimensions (Fleck et al., 2006). This places our models within the size range of observed clusters in the MW at dgc ∼ 8.5 kpc (see Fig. 2.1) as well as in the large and small Magellanic Clouds (Mackey et al., 2008). 2.3. Simulation method & choice of parameters 45

270 4

240 2

out 210 0 M

180 −2 z [kpc]

150 −4 11 11.5 12 12.5 13 t [Gyr]

Figure 2.3 Mass in the outskirts of the cluster (black solid line) and height z above the galactic plane (dashed grey line). Mout is deﬁned as the mass between one and two tidal radii. The galactic disc corresponds to z = 0. Equivalent behaviour is observed for all models and metallicities. Approximately 30 M are lost at every disc crossing. 46 Chapter 2. Metallicity Eﬀects on Globular Cluster Evolution

1 106 0.8

0.6 0 105 L

M/M 0.4

0.2 104

0 0 5 10 0 5 10 t [Gyr] t [Gyr] 101

100 M/L

10−1

0 5 10 t [Gyr]

Figure 2.4 Mass-loss rate for a cluster model dynamically evolved (including stellar evolution) - compared to Fig. 2.2 without dynamical interactions. Left Panel: The solid line denotes the overall cluster mass (note that in contrast to Fig. 2.2 the high-Z population is no longer less massive than the low-Z populations). The dashed line is the mass contained in MS stars and the dotted line is the mass contribution from WDs. Middle panel: Corresponding luminosity evolution. Although most apparent up to ≈ 7 Gyr, the metal- poor clusters stay more luminous throughout the entire evolution. As exected the overall luminosity is lower than in the non-dynamical case (Fig. 2.2). Right panel: Resulting mass-to-light ratio. Within the noise the values are equivalent to Fig. 2.2. Hence the overall dynamical evolution has an impact on the mass of the cluster, but little eﬀect on M/L. 2.3. Simulation method & choice of parameters 47

2.3.1 Binary fraction

All models used in this study are evolved with the same number of initial stellar systems,

Ni = 100 000, incorporating a binary fraction

Nb bf = = 0.05 . (2.11) Ns + Nb

This translates into Ns = 95 000 single stars and Nb = 5000 binary systems, therefore 105 000 stars in total. Some of these primordial binary systems may be disrupted early on, while new binaries form during the cluster evolution due to two- or three-body interactions. Open cluster studies found in the literature are usually evolved with binary fractions of 0.2 − 0.5 (Hurley et al., 2004, 2005; Trenti et al., 2007), as observations find higher binary fractions in these clusters (e.g. Montgomery et al. 1993; Richer et al. 1998 for M67). Much lower binary fractions are observed in GCs: Milone et al. (2012) have measured the binary fractions of 59 GCs in the MW and commonly find values around bf ≈ 0.05. Binary systems in GC models have proven to be important from a dynamical point of view: even a small binary fraction in the core can be sufficient to heat the cluster core enough to postpone core-collapse significantly (Hut et al., 1992; Heggie et al., 2006). Within NBODY6, standard binary evolution is treated according to the binary algorithm of Hurley et al. (2002) where circularization of eccentric orbits as well as angular momentum loss mechanisms are modeled. Wind accretion from one binary component to the other is possible as well as mass transfer when either star fills its Roche lobe. Stable hierarchical three- and four-body systems are detected and evolved (Mardling & Aarseth, 2001), with single-binary and binary-binary encounters followed directly. This allows for the replacement of one member of a binary by an incoming star, formation of binaries in few-body encounters and direct collisions (Kochanek, 1992), often leading to the formation of exotic stars such as blue stragglers. Nearby stars can perturb binary systems and cause chaotic orbits (Mardling & Aarseth, 2001).

2.3.2 Treatment of remnants

Neutron stars are assumed to be subject to a velocity kick arising from asymmetries during their formation through core-collapse supernovae, with observations of NSs indicating a vast range of velocities from several up to hundreds of km/s. Such velocities are easily in excess of a typical GC escape velocity and, in combination with observations of substantial NS populations in GCs, is known as the neutron-star retention problem (Pfahl et al., 2002). Indeed, X-ray sources (e.g. Woodley et al. 2008 in the case of NGC 5128) and 48 Chapter 2. Metallicity Eﬀects on Globular Cluster Evolution milli-second pulsars (Bogdanov et al. 2011 in the core of NGC 6626) indicate that NSs and BHs are common and even BH-BH binaries may exist. In N-body simulations, several diﬀerent methods to assign velocity kicks to NS or BH remnants have been used in the past. Baumgardt & Makino (2003) simply retain all NSs. With their IMF not reaching masses higher than 15 M , the number of NSs is not excessive and no BHs form. Mackey et al. (2007) retain all stellar-mass remnant BHs whilst using an IMF up to 100 M . In contrast to this, Zonoozi et al. (2011) retain no NSs or BHs. Hurley & Mackey (2010) use a Gaussian velocity kick distribution peaked at ≈ 190 km/s for both NSs and BHs, where the formation of a BH-BH binary is later observed to postpone core-collapse.

In this study, we adopt an intermediate approach by choosing vk at random from a ﬂat kick distribution in the range 0 − 100 km/s and assigning this to NSs and BHs at p their birth. Because of the low escape velocity ve = 2GM/r ≈ 4.7 km/s at the half- mass radius, or ve ≈ 2.8 km/s at the tidal radius (both at a cluster age of 500 Myr), this reproduces a retention fraction of ≈ 5% (Pfahl et al., 2002). We use the same algorithm to assign kick velocities to BHs at their formation (Repetto et al., 2012). We note that the metallicity inﬂuences the mass of the remnants. In our model, the maximum BH mass is ≈ 30 M for metal-poor progenitors and ≈ 10 M for their metal-rich counterparts (Hurley et al., 2000; Belczynski et al., 2010).

2.3.3 Models

We evolve three sets of models a), b) and c) with identical set-up apart from the random number seed for the initial particle distributions. Each set consists of three models with metallicities Z = 0.0001, Z = 0.001 and Z = 0.01 (see Table 2.2), i.e. low, intermediate and high metallicity. GCs in the MW are found within the metallicity range −2.37 ≤ [Fe/H] ≤ 0 (Harris 1996). The intermediate metallicity case Z = 0.001 of this study already corresponds to a metal-poor cluster in the MW (and also other galaxies). The low- Z case Z = 0.0001 is an example from the metal-poor end of the metallicity distribution. We expect these two low-metallicity clusters to exhibit similar evolution to each other (e.g. the MS turnoff masses agree fairly well: see Table 2.1) but distinct from the high- metallicity case. This has previously been noted by Hurley et al. (2004). All models are evolved up to 14 Gyr, while we concentrate our analysis at typical GC age of 12 Gyr (Hansen et al., 2007). 2.3. Simulation method & choice of parameters 49 , N , number of stars , mass locked in WDs BH M N MS M NS 2523 2 3 1831 2 2 1626 3 24 0 2 24 2 27 1 N 3 3 3 3 3 3 3 3 3 10 10 10 10 10 10 10 10 10 × × × × × × × × × WD M 6.10 5.42 6.14 5.51 4.38 6.13 5.50 4.44 4.32 3 3 3 3 3 4 4 3 4 10 10 10 10 10 10 10 10 10 MS × × × × × × × × × M MS N = 0 and various parameters at 12 Gyr: mass f t as well as the total mass contained in MS stars MS N b N 35958 0.060634654 27948 0.0595 9 .57 27036 9 .39 35699 0.059734787 27626 0.0595 9 .48 27025 9 .40 35017 0.061536149 28229 0.058534804 1 .04 28065 0.0606 9 .66 27015 9 .43 35680 0.0604 27975 1 .03 34596 0.0628 27896 1 .03 4 4 4 4 4 4 4 4 4 10 10 10 10 10 10 10 10 10 × × × × × × × × × (12 Gyr) 1.56 1.50 1.57 1.51 1.59 1.51 1.52 1.45 1.49 M 4 4 4 4 4 4 4 4 4 10 10 10 10 10 10 10 10 10 0 × × × × × × × × × M (Eq. 2.11), number of MS stars f b −2.3−1.3 6.42 6.42 −2.3 6.43 −0.3−2.3 6.42 6.36 −1.3 6.43 −1.3 6.36 −0.3 6.43 −0.3 6.36 Z [Fe/H] 0.01 0.01 0.01 and the number of NSs and BHs, all at 12 Gyr. The variation in the initial cluster mass arises from the difference in random .0001 0.001 0.001 0.001 0 0.0001 0.0001 WD able 2.2 Metallicities and initial masses for all models at c) a) b) T binary fraction seed when drawing starssame-mass from progenitors the and IMF. in Note addition the more consistently stars higher have WD already mass turned for off metal-poor the MS clusters: into WD WDs. masses are higher for M 50 Chapter 2. Metallicity Effects on Globular Cluster Evolution

2.4 Evolution

During cluster evolution, stars are lost in three ways: i) an increase of velocity during two-body encounters (evaporation) or ejection following three or four-body encounters, ii) velocity kicks owing to SN explosions and iii) tidal stripping and disc shocking (i.e. the influence of the tidal field of the host galaxy). These three effects cannot be completely disentangled: the former two might bring a star close to or even beyond the tidal boundary rt (Eq. 2.10) such that when crossing the galactic disc, stars are easily removed from the system. The periodicity of this event is ≈ 100 Myr and causes the number of stars within the cluster envelope between one and two tidal radii to continually fluctuate between

180 − 240 M , with ≈ 30 M lost each time the disc is crossed (see Fig. 2.3). The evolution of a star cluster is linked to the two-body relaxation time:

s 0.14N r3 t = 50% (2.12) rh ln(0.4N) GM

(Spitzer & Hart 1971; Binney & Tremaine 2008, see also Hurley et al. 2001). For our models, the relaxation time is highest at ≈ 2.5 Gyr when trh corresponds approximately to the cluster lifetime at that point. The relaxation time then decreases to roughly one Gyr at 12 Gyr of cluster age. The half-mass relaxation timescale is not significantly affected by the metallicity: variations up to 10% occur. This means that the clusters of different metallicity are dynamically of similar age, which is not the case from a stellar evolution point of view. As can be seen in Table 2.2, all three metallicity models have similar mass and number of stars at 12 Gyr, whereas the distribution of mass among MS stars and remnants differs (the metal-poor cluster containing more mass in remnants). Fig. 2.4 is a reproduction of Fig. 2.2, but now for our full N-body models. The three panels are again mass, luminosity and M/L. In Fig. 2.2 we only considered the mass-loss owing to stellar evolution, causing ≈ 40% mass-loss up to 12 Gyr. For Fig. 2.4 the dynamical interactions are taken into account, resulting in an additional mass-loss of the same order, leaving a cluster mass of ≈ 25% after 12 Gyr. Hence the three effects i)-iii) mentioned above are together responsible for approximately half the mass-loss of the cluster, while stellar evolution alone is responsible for the rest of the mass-loss. It can also be seen in Fig. 2.4 that nearly 40% of the mass at 12 Gyr is contained in WDs. All stars are split into their relevant stellar populations in Fig. 2.5 for further illustration, also at a cluster age of 12 Gyr. While stars below 0.5 M exclusively are on the main sequence, the contribution of WDs is significant for higher masses, causing a second peak in the mass 2.4. Evolution 51 function at ≈ 0.6 M . There is always more (luminous) mass contained in MS stars in the metal-rich cluster, which is expected from the higher MS turnoff mass (see Table 2.1 and Fig. 2.5), while the low-Z clusters are more luminous in spite of this. See Section 2.4.5 for further details of the evolution of luminosity and mass-to-light ratio. Even though the MS turnoff is higher for metal-rich clusters, the number of MS stars is not always the highest (see Table 2.2, column 7). The fluctuation of NMS is mainly due to the fact that the number of low-mass stars with m ≤ 0.2 M varies depending on metallicity: the mass in those stars is typically 5 − 10% higher for Z = 0.0001 than for Z = 0.001 or Z = 0.01. While statistical noise may be responsible for some of the fluctuations, the lowest-Z cluster also has the highest mass and hence a slightly higher escape velocity.

2.4.1 Binary systems

The binary fraction of initially 0.05 slightly increases to ≈ 0.06 at 12 Gyr for any metallicity or model (see Table 2.2), where the binary fraction for the high-Z model is always slightly higher than for low metallicities. While some of the initial systems may easily disrupt, others form during few-body encounters. Hard binaries (Heggie, 1975; Hut et al., 1992) have been shown to successfully halt core-collapse over large periods of time and BH binary systems in particular can heat the core substantially. With an initial mass function up to

50 M and the inclusion of stellar evolution, black hole remnants will form early on in the cluster evolution. While most BHs get ejected almost immediately (e.g. Section 2.3.2), remaining BHs will sink towards the centre of the cluster owing to mass segregation. While doing so, they may become part of a binary or triple system, breaking up a previously existing binary system. Once BHs are part of binary systems, BH-BH binaries can easily form in a further encounter through exchange interactions. All BH-BH binaries in this stydy are dynamical binaries, having formed through such few-body interactions. This also means each component in a BH binary is the remnant of a high-mass MS progenitor that was either a single star or born in a binary system that later disrupted. 52 Chapter 2. Metallicity Eﬀects on Globular Cluster Evolution

3000 Z=0.01 Z=0.001 Z=0.0001

2000 N

1000

0 0 0.5 1 0 0.5 1 0 0.5 1 m m m

Figure 2.5 Mass function of the dynamically evolved stellar population at 12 Gyr for Z = 0.01 (left panel), Z = 0.001 (middle panel) and Z = 0.0001 (right panel). Here we focus on model set b) but the behaviour is similar for all sets. The grey area is the entire population of stars, the thin black line the remaining stars on the main sequence and the dashed line the contribution of white dwarfs (peaked at ≈ 0.6 M ). For stars with M ≤ 0.5M the population is made entirely out of MS stars. For metal-poor clusters, the MS turnoﬀ is noticeably smaller (see also Table 2.1). The number of NSs and BHs is insigniﬁcant compared to MS stars and WDs. 2.4. Evolution 53

2.4.2 Cluster size

Owing to the cumulative effects of mass-loss, two-body relaxation and the influence of the tidal field, the models are expected to go through an initial expansion, followed by contraction. In Fig. 2.6 this is shown by means of the three-dimensional half-mass radius r50%. We find no metallicity dependence on the half-mass radius. Moving further inwards, we look at the 10% Lagrangian radius r10% (middle panels of Fig. 2.6) and the N-body core radius rc (bottom panel). Small differences in r10% are evident for the different models, noting that this inner radius is susceptible to the actions of highly energetic binaries in the core, even so, the evolution of r10% remains fairly steady. The N-body core radius rc is similar in size to the 10% Lagrangian radius, however we see that rc is heavily fluctuating when BHs, BH-BH binaries or otherwise energetic binary systems are present (all of which are more likely to reside in the central regions owing to mass segregation). The N-body core radius is not to be confused with an observational King-core radius, as the N-body core radius is a density weighted mean distance to the cluster centre (not taking luminosity into account). In the procedure of calculating rc, the mean density (in terms of mass) of the six neighboring stars is calculated for each star (Casertano & Hut, 1985), introducing a large bias towards stars in the neighborhood of BHs: a BH might be up to 28 M , a binary BH up to twice as much, while a MS is less massive than two solar masses after one Gyr of cluster evolution. We note that the N-body core radius is consistently less fluctuating at high metallicity than in the lower metallicity cases. This is not a sampling effect. Instead, it results from remnant masses being lower for the high-Z population. With BH masses only up to 10 M , the density contrast around stars will be less steep. BH-BH binaries can mimic core-collapse (Fig. 2.6) when indeed just a subsystem of stars is responsible for this effect. In addition, peaks in the core radius can (but don’t have to be) closely correlated with highly-energetic binary systems. As an example, the drop in rc for the low-Z model b) in Figure 2.6 (middle panel) at 2 Gyr is caused by a short-period binary composed of two carbon-oxygen white dwarfs of masses 0.7 and 0.8 M . At t = 1.85 Gyr, the two WDs merged and the product was subsequently ejected from the cluster. The maximum binding 2 energy before coalescence is 141 M /AU. This is followed by another dip in rc at 2.6 Gyr, when the core radius shrinks to 0.72 pc. At this point, more than 50% of the core-mass is contained in BHs and a BH-BH binary forms. 54 Chapter 2. Metallicity Effects on Globular Cluster Evolution

10 8 6

[pc] 4 a) b) c) 50% 2 0 4 3

[pc] r 2 10% 1 0 4 3 [pc] r

c 2 r 1 0 0 5 10 0 5 10 0 5 10 t [Gyr] t [Gyr] t [Gyr]

Figure 2.6 Evolution of lagrangian radii and core radius with models a) on the left, b) in the middle panel and c) on the right. Top: half-mass radius. Slight size differences between the models occur, however this is not primarely related to metallicity. Size differences are originating from few-body encounters and high-energy binary systems in the centre of the cluster and are enhanced in the 10% lagragian radius (middle). Bottom: The N-body core radius rc. Short-term effects on the core radius are often linked to high-energetic binaries or the presence of BHs in the core, which can severely impact the evolution of rc. 2.4. Evolution 55

In the low-Z model of set a), the N-body core radius drops by more than factor of two to 1.4 pc at 5 Gyr. This is caused by a chain of reactions involving four remnant BHs

(out of ten present at that time). The masses of the four BHs are 27, 26, 14 and 11 M , respectively. Initially, the least massive BH is ejected from this four-body subsystem, and leaves the cluster. The remaining three form a short-lived triple-system which ends with a BH-BH binary and a single BH being ejected from the core as a result from enhanced velocities obtained in the interaction. This implies that four of the most massive components are lost from the core within a time frame of only 40 Myr. We conclude that the metallicity has no effect on the half-mass radius or other scaling parameters based on cluster mass. However as Figs. 2.2 and 2.4 already indicate - the metallicity influences the overall luminosity of GCs with high-Z clusters being fainter than metal-poor clusters. To explore this possibility in more detail, we measure the half-light of effective radius reff by fitting King (1966) models to our clusters - analogous to sizes are measured from observations. We illustrate this method in Section 2.4.3. 56 Chapter 2. Metallicity Effects on Globular Cluster Evolution

0 [pc]

−15 −30

2 1 ) v 0 −1 log(I −2 r =4.06 r =4.28 r =3.78 eff eff eff −3 yz plane xz plane xy plane 0.1 1 10 0.1 1 10 0.1 1 10 r [pc] r [pc] r [pc]

Figure 2.7 Example fits for a cluster at the age of 13 Gyr. The three panels denote the same cluster at the same time, projected along the x−, y− and z− axis. The corresponding snapshot is printed above. Each snapshot is fitted individually. The measured data points for the surface brightness profile are denoted by black sqares with poisson error, the black line is the gridfit King66 fit. The resulting effective radius reff is indicated by the red dotted line. 2.4. Evolution 57

2.4.3 Surface brightness and half light radii

Among other properties, the output of NBODY6 incorporates the mass, luminosity and radius for each star. This means effective temperatures can easily be calculated and this data can be cross convolved with stellar atmosphere model calculations (Kurucz, 1979) to obtain Johnson V-band magnitudes. We project this data in a two dimensional image and slightly smooth it with a Gaussian filter (see Fig. 2.7 for an example of a cluster at the age of 13 Gyr). This means the light of each star is conserved, but not contained within one single pixel, which implies that the starlight can be divided between consecutive bins when creating a surface brightness profile, which is crucial in cases of very bright stars. For each model, at each snapshot three such images are obtained by using the degree of freedom to project in either the x, y or z direction (in theory multiple projections are possible, see Noyola & Baumgardt 2011) and a surface brightness profile is obtained separately for each projected snapshot (Fig. 2.7). For simplicity, we assume a background of zero. We chose to fit King (1966) models as they have shown to be a robust solution to fit GCs. Another option would be Wilson (1975) models, having a greater sensitivity in the outer regions of the cluster (McLaughlin et al., 2008). However, in this work we are not investigating tidal fluctuations but the overall cluster evolution, which the King models are well suited for. Since there is no analytical solution for the surface density of this model, a grid of model fits has to be pre-calculated. We utilize the gridfit code (McLaughlin et al., 2008) where this has been done. Each snapshot is fitted three times according to the three different projections along the x-, y- and z-axes, as illustrated in Fig. 2.7. Obvious bad fits are rejected from further analysis (note that no bright stars have been masked for fitting). For each given time, the final effective radius is the mean along all three projections. The result is plotted in Fig. 2.8 over the entire evolution of the cluster. Similar to the half-mass radius, an initial expansion when mass-loss is dominated by stellar evolution winds from massive stars in the core is followed by a contraction when the mass-loss is dominated from the cluster boundary. Yet there are differences in comparison to the half-mass radius: Firstly, reff is approximately half as large as the half-mass radius. As r50% is a three dimensional quantity, reff is expected to be only 3/4 as large simply due to projection effects. A size difference further to this implies that the luminosity alters the measured cluster size. Secondly, there is a clear effect of the metallicity on the reff evolution of the clusters: the metal-poor clusters are consistently observed to be larger than their metal-rich counterparts. 58 Chapter 2. Metallicity Effects on Globular Cluster Evolution

t [Gyr] t [Gyr] t [Gyr] 0 2 4 6 8 10 12 0 2 4 6 8 10 12 0 2 4 6 8 10 12 8

[pc] 4 eff r 2 a) b) c) 0

6 5

[pc] 4 eff

r 3 2 a) b) c) 11 12 13 11 12 13 11 12 13 t [Gyr] t [Gyr] t [Gyr]

Figure 2.8 Half-light or effective radius reff from King (1966) model fits using gridfit (McLaughlin et al., 2008). In the top panels, the overall evolution of the half-light radius is indicated for all sets of models: a) on the left, b) in the middle and c) on the right. Of greatest interest is the data at late times, which are highlighted below. Average cluster sizes for each metallicity are calculated for the intervalls 10.25 − 11 Gyr, 11 − 11.75 Gyr, 11.75 − 12.5 Gyr and 12.5 − 13.25 Gyr, using blue squares for the low-Z, green diamonds for intermediate and red circles for the high-Z case. It is clearly seen, that the metal-poor cluster snapshots (blue) have a larger observed half-light radius than the metal-rich (red) snapshots. The average sizes are summarized in Table 2.3. 2.4. Evolution 59

Table 2.3 Average cluster sizes measured for all sets for the intervals t10 = 10.25 − 11 Gyr, t11 = 11−11.75 Gyr, t12 = 11.75−12.5 Gyr and t13 = 12.5−13.25 Gyr. In the bottom line the size diﬀerence ∆r = rb − rr is given for the corresponding time interval, where rb is the average cluster size observed for blue, metal-poor and rr for red, metal-rich clusters. The overall size diﬀerence for all ages is 17%.

t10 t11 t12 t13 Z = 0.01 4.30 pc 4.08 pc 3.85 pc 3.82 pc Z = 0.001 4.82 pc 4.81 pc 4.61 pc 4.39 pc Z = 0.0001 5.01 pc 4.75 pc 4.64 pc 4.31 pc ∆ r 16.5% 16.4% 20.5% 12.6% 60 Chapter 2. Metallicity Eﬀects on Globular Cluster Evolution

Also in Fig. 2.8 we highlight the time window of 10 − 13 Gyr which is of most signifi- cance for old GCs. The data is averaged over δt = 750 Myr windows: t10 = 10.25−11 Gyr, t11 = 11 − 11.75 Gyr, t12 = 11.75 − 12.5 Gyr and t13 = 12.5 − 13.25 Gyr. The results are summarized in Table 2.3 and combined give an overall size difference of ≈ 17% between red and blue GCs. If split into sets, the difference is 19, 21 and 10% for sets a), b) and c), respectively. This result implies that the observed size difference between the metal-poor and metal-rich GC sub-populations can (at least partly) be explained by the effects of metallicity.

2.4.4 Origin of the size diﬀerence and inﬂuence of remnants

We observe no size difference with metallicity for the clusters when measuring the size by means of the mass distribution, e.g. half-mass radius. This indicates that the clusters are structurally identical, and different mass-loss rates depending on metallicity are not causing the cluster size to change appreciably. Also, the overall mass and mass segregation are not largely affected by metallicity: a higher MS turnoff mass for the metal-rich cluster is compensated by a lower remnant mass, two effects almost canceling each other out. In Fig. 2.9 i) we show the typical radial profile of the average stellar mass for the three different metallicities at a late age. The models are in good agreement, showing no significant variation with Z. However we find size differences of up to 20% when measuring the cluster size by means of the stellar luminosity. The reason for this is two-fold. Firstly, less massive remnants in the high-Z cluster free more space in the core for MS and giant stars, i.e. luminous matter, steepening the luminosity profile in the central regions. This is evident in Fig. 2.9 ii) which plots the radial profile of the average luminosity per radial region. The second factor can also be clearly seen in the same figure: even though low-Z clusters have a lower MS turnoff mass, the luminosity of MS stars of identical masses is higher in the low-Z case. This results in the low-Z clusters appearing brighter beyond the centre, with the differences beyond two parsecs being significant in relation to the errorbars, as shown in Fig. 2.9 ii). Combined, these effects result in a larger cluster appearance for the metal-poor clusters. To reinforce this we show in Fig. 2.10 the luminosity within the 10% Lagrangian radius normalized by the total luminosity, as a function of time. Here we see that the metal-rich cluster consistently has a greater central concentration of luminous matter. 2.4. Evolution 61

0.8 0.2 0.7 i) ii) 0.6 0.15 0.5 0.4 0.1 0.3 0.2 0.05 0.1 0 0 1 10 1 10 r [pc] r [pc]

0.8 0.2 0.7 iii) iv) 0.6 0.15 0.5 0.4 0.1 0.3 0.2 0.05 0.1 0 0 1 10 1 10 r [pc] r [pc]

Figure 2.9 Radial profiles of the average stellar mass (panel i) and average luminosity (panel ii)) for different metallicities. This is for model b) at an age of ≈ 11.5 Gyr, averaged over ten consecutive snapshots (covering about 130 Myr). The shaded regions indicate the errors involved, calculated as the standard deviation from the mean within those ten snapshots. In calculating the average mass all stars and remnants are taken into account, while for the average luminosity only stars not yet in the remnant phase are taken into account (e.g. only luminous stars). There is a general trend for the luminosity distribution to be steeper in the high metallicity case. However beyond the core of the cluster, the metal-poor cluster has a higher average luminosity. The panels iii) and iv) are a repeat of the panels on the left, but for a set of models without NS or BH remnants. While the overall evolution for these models is similar to the other models in this study, we do not observe a significant size difference. 62 Chapter 2. Metallicity Effects on Globular Cluster Evolution

The fact that low-Z stars are brighter for a given mass than their metal-rich counterparts, will be the case independent of a different treatment for NSs and BHs. However, different NS and BH abundances might affect the surface brightness profile by altering the central concentration of luminous stars. We have evolved an additional set of models where NSs and BHs receive a larger kick at formation, resulting in neither sub-population being present in the cluster after a few hundred Myr of cluster evolution (with the exception of the rare case that a NS may form via a WD-WD merger). In contrast to the previous models that contain NSs and BHs, this causes the luminosity profiles for different metallicity clusters to be nearly identical (see the far right panel of Fig. 2.9). This is no surprise: the remnant mass depends on metallicity and removing the remnants erases some of the metallicity effects. This is in excellent agreement with the findings by Down- ing (2012), where significant half-light radii differences are measured with Monte Carlo models (utilizing the same stellar evolution prescription Hurley et al. 2000) only when BHs are retained in the cluster. While our model clusters are smaller than those of Downing (2012), and we only retain a few BHs compared to hundreds in their study, we find the same effect already with very few BHs present, with a contribution also from the NSs that are present. 2.4. Evolution 63

0.25

0.2

0.15 )/L(tot) 0.1 10%

L(r 0.05

0 0 2 4 6 8 10 12 t [Gyr]

Figure 2.10 Evolution of the luminosity contained withing the 10% Lagrangian radius normalized by the total luminosity at that time, for model set b). The high-Z cluster (dotted red) has a higher concentration of light within r10% than the metal-poor models.

Table 2.4 Luminosity L and mass-to-light ratio M/L for stars with diﬀerent masses and metallicities. For given mass, the luminosity increases with metallicity, causing M/L to decrease.

0.1 M 0.5 M 0.8 M Z = 0.01 0.001 L 0.04 L 0.32 L 100 M /L 12.5 M /L 2.5 M /L Z = 0.001 0.0013 L 0.06 L 0.5 L 77 M /L 8.2 M /L 1.6 M /L Z = 0.0001 0.0015 L 0.07 L 0.56 L 66 M /L 7 M /L 1.4 M /L 64 Chapter 2. Metallicity Eﬀects on Globular Cluster Evolution

2.4.5 Mass-to-light ratio

In Section 2.2.1 we have already mentioned the mass-to-light ratio M/L for a stellar population evolved purely with stellar evolution, but no dynamical interaction (see Fig. 2.2, right panel). The higher overall luminosity for metal-poor populations implies a lower M/L ratio: the mass-to-light ratio increases with increasing metallicity. The same trend has previously been observed by e.g. Anders et al. (2009b) where GALEV models were computed based on the models of Baumgardt & Makino (2003). In Fig. 2.4 we repeated the same analysis as in Fig. 2.2, but now for our N-body models. We chose model set b) as an illustrative case, but all three sets are equivalent. The evolution of mass for all metallicities is nearly identical (Fig. 2.4), whereas the metal-poor cluster has a slightly higher overall mass while the metal rich cluster has a slightly higher MS mass. The overall luminosity is evolving in a similar fashion as in the non-dynamical model, but a factor of two lower owing to the loss of stars. Metallicity differences in L are obvious especially for t < 6 Gyr, but continue up to 13 Gyr. The dynamical evolution introduces selective effects on the evolution of M/L as low-mass main sequence stars are preferentially lost from the outskirts of the cluster (Baumgardt & Makino, 2003). Those low-mass stars have a high M/L. White dwarfs also have relatively low average mass compared to stars in the central regions. Thus they are candidates to be lost and have a mass-to-light ratio approaching infinity. As a general rule, losing a low-mass MS star or a white dwarf will decrease the mass-to-light ratio (see Table 2.4). There is an additional effect arising from metallicity differences to consider: for any given mass at a certain time, the luminosity of the metal-poor star will be higher than for a metal-rich star and hence the low-Z star will have a lower M/L. This implies that escaping metal-rich stars will cause a larger decrease of M/L. In other words: the mass-to-light ratio will be more affected by the loss of low-mass stars in a high-Z cluster. While this is in agreement with the models by Baumgardt & Makino (2003) and Anders et al. (2009b), it is in disagreement with the observed mass-to-light ratios of metal-rich clusters in M31 (Strader et al., 2009, 2011). Strader et al. (2011) have suggested different initial mass functions for red GCs, which has not been tested here.

2.5 Discussion and conclusion

We have measured the sizes of GC models with different metallicity, evolved with the direct N-body code NBODY6. All clusters start their evolution with 105 000 stars and a 4 mass of ≈ 6 × 10 M . We find no size differences with metallicity when measuring sizes 2.5. Discussion and conclusion 65 by means of the half-mass radius or other mass-weighted radii, with the exception that lower remnant masses for high-Z stars cause the N-body core radius to fluctuate less. This indicates, that there is no structural difference between clusters of low and high metallicity. Even though the mass-loss rates of low-Z stars are higher, especially in the initial stages of evolution, a consequently lower escape velocity and higher average remnant mass cancels this effect, leading to no overall size difference. In accordance with this, we also find that the number of stars and cluster mass remaining at a particular time do not vary noticeably with the metallicity of the cluster.

Schulman et al. (2012) evolved N-body models starting with N = 8 192 stars and different metallicities to find a size difference between metal-poor and metal-rich clusters, in terms of the half-mass radius. This is in disagreement with our results and those of the Monte Carlo models of Downing (2012). The Schulman et al. (2012) models were evolved with some softening so that the effects of close binaries were not included. They were evolved to a dynamical age of 5 trh which translated to physical ages in the range of 100 − 500 Myr for the small-N models. The claim is that the results should be applicable to larger clusters, including GCs, because the impact of different stellar evolution and mass-loss histories at various Z will not depend on N, and also because they performed models in the range of 1 024 to 16 384 stars that showed similar half-mass radius evolution. We would counter that as the MS lifetime of a MS turn-off star changes with age and the half-mass relaxation timescale of a cluster varies with N, it is not at all obvious that the interplay between stellar evolution and cluster dynamics will scale in a straightforward manner. Indeed, our models here and the open cluster models of Hurley et al. (2004) with N ∼ 30 000, both show that the half-mass radius of metal-rich models can be smaller than that of the metal-poor models at early times (see Fig. 2.6) but that the difference is erased or even reversed later in the evolution. Factors including different core-collapse times, the stellar evolution of low-mass stars as a function of metallicity (particularly for globular clusters with ages of 10 Gyr or more) and different remnant masses need to be taken into account to gain the full picture. Furthermore, statistical fluctuations are generally prevalent in small-N simulations and it can be necessary to average the results of many instances to establish true behavior (e.g. Küpper et al. 2008). Our models presented here are at the lower edge of the GC mass function but even for these we would suggest that larger models again are desired before making any final judgment about the size measurements of GCs in general. However, our agreement with the large-scale Monte Carlo models of Downing (2012), performed with 5 × 105 stars, on the issue of half-mass radius variation (or non-variation) with metallicity is reassuring. 66 Chapter 2. Metallicity Effects on Globular Cluster Evolution

In contrast to the evolution of the half-mass radius, we find that the half-light (or effective) radius does vary with metallicity. We find that blue, metal-poor clusters can appear on average 17% larger than red, metal-rich clusters, with even larger differences possible when comparing individual models. This is in agreement with observations of extra-galactic GC systems, where size differences of 17 − 30% (Larsen et al., 2001; Jordán et al., 2005; Woodley & Gómez, 2010) have been found. It is also in agreement with the Monte Carlo models of Downing (2012). Indeed, our N-body models and these Monte Carlo models provide excellent independent validation of the main result – that the observed size differences in GCs are likely caused by the interplay of stellar evolution and mass segregation. Stellar evolution causes low-Z stars to be brighter than their high-Z counterparts while mass segregation causes the most massive remnants to sink to the centre. Successively more massive remnants in low-Z clusters leads to a steeper surface brightness profile for high-Z clusters. The overall mass segregation is similar for metal- poor and metal-rich clusters but more effective in the luminous stars for high-Z clusters owing to a higher main-sequence turnoff mass. This is in excellent agreement with the predictions of Jordán(2004) using multi-mass Mitchie-King models to estimate the size difference between blue and red GCs, finding a difference of 14% due to the combined effect of mass-segregation and stellar evolution.

The apparent size difference does have a dependence on the treatment of remnants. When ejecting all NSs and BHs, no significant size difference (half-light radius) is found, partly owing to the fact that one of the variations with metallicity (remnant masses) has been negated. When we retain ≈ 5% of the NSs and BHs arising from the primordial population, our results are in general agreement with the Downing (2012) models that retained large numbers of BHs. While there are uncertainties in the retention fractions for NSs and BHs, there are also uncertainties for the masses of remnant BHs. We have used the stellar evolution wing mass-loss prescriptions from Hurley et al. (2000), while improved, Z dependent mass-loss rates are now available (Vink et al., 2001). However, the resulting differences for BH masses are most apparent for stars above 40 M (Belczynski et al., 2010), while just a few stars are drawn from this mass range in the models presented here.

The average size difference of 17% implies that blue GCs do indeed appear larger as a result of metallicity effects. Since this is at the lower end of what is found in observations, other causes (such as projection effects) can also be expected to play a role. In the future we plan to extend our study by performing additional N-body simulations that explore parameters such as larger N, smaller initial size and differing initial density profiles, as well as different cluster orbits, to further understand the effects of cluster evolution and 2.5. Discussion and conclusion 67 environment on measured sizes. Our spread of individual measurements in Fig. 2.8 can be compared to extragalactic studies of GC systems as well as in the Milky Way, in which half-light radii of GCs are found to be distributed between 1 to 8 pc (e.g. Larsen & Brodie 2003 Fig. 4, Spitler et al. 2006 Fig. 19, Madrid et al. 2009 Fig. 10). Since clusters of different masses and at different galactocentric distances are included in the observational samples, a larger scatter is expected than for our models (which currently give values between 2 − 6 pc). We would expect the model spread to increase when we extend our study to include a range of cluster parameters. In addition to the half-light radius, we have also analyzed the evolution of the mass- to-light ratio. When comparing cluster models evolved purely through stellar (but no dynamical) evolution with the thorough N-body models, there is little change in M/L. As seen before in Baumgardt & Makino (2003), we find that M/L increases with time, where dynamical interactions lead to a decrease in M/L as low-mass stars (carrying a high mass-to-light ratio) are preferentially lost from the cluster. The decrease in overall cluster luminosity with time results in an increase of the mass-to-light ratio. The results in this paper have in the meantime been confirmed by observations of extragalactic GC systems (Usher et al., 2013).

3 Dynamically Evolved Single Stellar Populations

A severe nuisance turned up: the creation of close pairs or “double” stars, slowing down the calculation considerably by short time steps. Dancing around for 10 − 400 orbits until their ﬁnal divorce. - Sebastian von Hoerner 2001

3.1 Introduction

Globular clusters (GCs) are substantial components of galaxies of different morphologies from dwarf spheroidals to giant ellipticals. They usually exist in two populations, with one appearing red and the other blue (Zinn, 1985; Zepf & Ashman, 1993; Ostrov et al., 1993; Larsen et al., 2001). This colour bimodality is thought to originate from a bimodality in metallicity, with blue clusters being metal-poor and red clusters more metal-rich. The bimodality is spectroscopically confirmed (Usher et al., 2012; Larsen et al., 2012) and universal for galaxies across all mass scales (Brodie & Strader, 2006), with the exception of some galaxies with even three populations, such as NGC 4365 (Puzia et al., 2002; Blom et al., 2012b). Apart from colour and metallicity, the blue and red populations differ in that the red clusters are distributed more closely to the center of the galaxy, and that the blue clusters are on average ≈ 20% larger in size than the red ones (Larsen et al., 2001; Jordánet al., 2005; Madrid et al., 2009; Masters et al., 2010). This size difference originates from a combined effect of metallicity and mass segregation (Jordán, 2004; Jordán et al., 2005; Downing, 2012) and is, at least partially, just an apparent effect observable without structural differences (Sippel et al., 2012). In other words, the size difference can be explained by the internal formation and evolution of GCs (Harris, 2009; Webb et al., 2012; Usher et al., 2013) with no need for an external origin. While the origin of the two populations of GCs is still somewhat debated, it has recently been shown that this bimodality is a natural outcome of the hierarchical clustering scenario

69 70 Chapter 3. Dynamically Evolved Single Stellar Populations of galaxies by simply assuming that GCs form with the same primordial metallicity as their host galaxy (Tonini, 2013), which is supported by findings that GCs carry a chemical record of the galaxies they were formed in (Pota et al., 2013). This implies that GCs are fossil records of galaxies, and ideal tracers of both star formation histories and galaxy assembly. It is also in agreement with the fact that GCs, both blue and red, are old (Mar´ın-Franch et al., 2009). To a first approximation, GCs are composed of single stellar populations (SSPs). GCs with multiple populations have been found (Bedin et al., 2004; Piotto et al., 2007), and recent results indicate that an age spread of several 100 Myr is not the exception but the norm in GCs in the Milky Way and Large Magellanic Cloud (LMC, see Gratton et al. 2012). Given the fact that the star formation timescale and the observed age spread of stars in GCs are orders of magnitudes shorter than the lifetime of a cluster, GCs are approximated by one SSP in this work. To estimate spectroscopic properties of stellar populations, evolutionary synthesis models predict the integrated properties of the SSPs at any chosen age. Such models include STARBURST99 (Leitherer et al., 1999), GALAXEV (Bruzual & Charlot, 2003), GALEV (Kotulla et al., 2009), the Maraston models (Maraston, 2005) or PARSEC isochrones (Bressan et al., 2012), to name just a few. In general, spectra are assigned to the stars within one or multiple populations, weighted according to the initial mass function (IMF) of these populations and integrated along the isochrone. This assumes a mass function evolved from the former IMF and implies that the population of stars is aged, but not dynamically evolved. While this is not an issue for large systems of stars such as an entire galaxy, star clusters are affected by the tidal field of their host galaxy (Baumgardt & Makino, 2003) and dynamical processes (Sippel et al., 2012). While the importance of mass-loss and the disruption of clusters in a tidal field is well-studied from a theoretical point of view (Baumgardt & Makino, 2003; Lamers et al., 2005; Gieles et al., 2006, 2007, 2011; Madrid et al., 2012; Webb et al., 2014), the effect of the tidal field on the photometric evolution is not as certain. Owing to mass-segregation, the tidal field removes predominantly low-mass stars from the cluster, while stars across the whole mass range can be ejected through dynamical interactions. This has been shown to alter the cluster mass function (Baumgardt & Makino, 2003). Efforts to combine mass loss and colour evolution, and therefore the impact of the former on the latter, have been made by Lamers et al. (2006); Kruijssen & Lamers (2008) and Anders et al. (2009b). They have improved the approach used to estimate the colour evolution of GCs by deriving an analytic description of the evolving mass function (based 3.2. Simulation method & choice of parameters 71 on the direct N-body models by Baumgardt & Makino 2003), to which GALEV models are applied. With this approach they find a colour shift towards blue for intermediate ages, and a reddening for very old clusters with ages ≥ 80%tdis, where tdis is the dissolution time, corresponding to only 5% of the initial cluster members left in the N-body model. While this evolving mass function takes the removal of low-mass stars owing to weak dynamical interactions into account, it is an analytic description based on N-body models rather than a fully direct approach. In the work presented here, we take the analysis one step further and calculate the luminosity of each individual star within a direct GC N-body model which takes both stellar and dynamical evolution into account. These luminosities can be integrated to derive cluster colours, which we use to describe the colour evolution of this dynamically evolved single stellar population (see Section 3.3.2). As the colour (and location) of each single star is known, we also analyze the distribution of stellar types and colours within a resolved GC and differentiate between internal and external effects (see Section 3.3.3).

3.2 Simulation method & choice of parameters

We evolve realistic models of GCs with the direct N-body code NBODY6 (Aarseth, 1999, 2003; Nitadori & Aarseth, 2012). With this code we are able to evolve one cluster at a time within an external tidal ﬁeld, while the stars evolve and simultaneously interact through the gravitational force they exert on each other. Below we give a summary of the setup of the models presented here, which are identical to those in Sippel et al. (2012), where a more detailed description of the method as well as choice of parameters can be found.

We draw N = 100 000 stars from a Kroupa et al. (1993) IMF within 0.1 to 50 M , with a binary fraction bf = Nb/(Ns +Nb) = 0.05 (as commonly found in GCs, e.g. Milone et al. 2012). The stars are initially distributed following a Plummer density profile (Plummer, 1911; Aarseth, 1974) and the initial cluster size is set to a three dimensional half-mass radius rhm = 6.2 pc. The individual initial positions and velocities are assigned such that the cluster is in virial equilibrium. The cluster is subject to a constant, Milky Way-like tidal field consisting of a point- mass bulge, an extended smooth disk (Miyamoto & Nagai, 1975) and a dark matter halo following a logarithmic potential (Aarseth, 2003). The clusters orbit at a galactocentric distance dgc ≈ 8.5 kpc, matching an environment where metal-poor and metal-rich GCs co-exist within the Milky Way (MW, Harris 1996). An initial tidal radius rt = 52 pc is 2 1/3 given by the equation rt ' (GM/2Ω ) (Küpper et al., 2010), where M is the cluster 72 Chapter 3. Dynamically Evolved Single Stellar Populations mass, G is the gravitational constant and Ω the angular orbital velocity of the cluster. The tidal radius is adjusted to the mass loss by changing M 1/3 accordingly.

The single stellar evolution (SSE) incorporated in NBODY6 is via the prescriptions of Hurley et al. (2000), which are based on the detailed models of Pols et al. (1998). They are accurate for a wide range of metallicities and cover all phases of stellar evolution from the zero-age main-sequence (MS) up to and including the remnant phases, namely main sequence, Hertzsprung gap and subgiants, ﬁrst-ascent red giants, horizontal branch and helium-burning giants, early asymptotic giants and red supergiants, thermally pulsating asymptotic giants, naked helium main sequence, naked helium (sub) giants and giants, helium white dwarfs (WDs), carbon/oxygen WDs, oxygen/neon WDs, neutron stars (NSs) and black holes (BHs). In this work, we refer to giants or evolved stars to stars on any stage of stellar evolution between the main sequence and the relevant remnant phase. Within SSE, the mass-loss rate of giants is implemented according to Reimer’s law (Kudritzki & Reimers, 1978). Since both NSs (Bogdanov et al., 2011) and BHs (Strader et al., 2012; Chomiuk et al., 2013) are found in GCs, we retain a moderate amount of these remnants, taking into account that a signiﬁcant fraction of NSs are expected to receive a velocity kick during the super nova, caused by asymmetries in the shell just before collapse (Janka, 2013). To −1 do so, we apply a random velocity kick between vk = 0 − 100 km s for NSs and BHs at their birth, treating NSs and BHs equally (Repetto et al., 2012). At the time these −1 remnants form, the escape velocity of the cluster at the half-mass radius is ve ≈ 5 km s , implying a retention fraction of ≈ 5%. As an extension to SSE, binary star evolution (BSE) is included in NBODY6 by utilizing the BSE algorithm described in Hurley et al. (2002), which builds on the earlier work of Tout et al. (1997).

3.2.1 Colour

The metallicity of stars aﬀects both, their luminosity and colour. For stars of identical mass, a lower metallicity implies a lower mean molecular weight and opacity (Clayton 1968; Hurley et al. 2000; Sippel et al. 2012), resulting in slightly hotter (bluer) and brighter stars. On the other hand, metals in the atmosphere absorb preferentially shorter wavelengths, causing metal-rich stars to be redder.

Within NBODY6, stars are evolved with SSE (see above). At any given time along the evolutionary track, mass m, radius r and V-band luminosity L (among other properties) 3.3. Evolution 73 are known for each star. We use this information to calculate eﬀective temperatures

log Teﬀ = 3.762 + 0.25 × log L − 0.5 × log r (3.1) and surface gravity

log g = log m + 4 × log Teff − log L − 10.6071 (3.2) for each star individually. These values are cross-convolved with the stellar atmosphere calculations of Castelli & Kurucz (2003), combined with Bergeron et al. (2001) for WDs, to obtain bolometric corrections, colour indices, and hence absolute magnitudes. To make this work in principle comparable to both the deep Hubble Space Telescope (HST) imaging of MW GCs (e.g. Richer et al. 2013) and extragalactic studies of GC systems where magnitudes and colours for large numbers of clusters can be measured (e.g. Jordánet al. 2005; Peng et al. 2009), we chose the Advanced Camera for Surveys (ACS) filter system in the three wave bands B = F555W, V = F606W and I = F814W. Note that within SSE and BSE, mass-loss can also occur not only as a result of stellar evolution, but also owing to close interactions between stars. For example, mass transfer within short-period binary systems when the primary star fills its Roche-lobe, wind accretion from one to the other binary component and even stellar collisions/mergers. This implies that the treatment of asymptotic giant branch stars (as mentioned above), as well as blue straggler stars (BSs), is intrinsic to our approach.

3.3 Evolution

We evolve three models with the metallicities Z = 0.0001 ([Fe/H] ≈ −2/3), Z = 0.001 ([Fe/H] ≈ −1.3) and Z = 0.01 ([Fe/H] ≈ −0.3). The ﬁrst two correspond to metal-poor, blue GCs (or Population II stars) and have similar main sequence lifetimes (Hurley et al., 2005; Sippel et al., 2012), while the latter represents a red, metal-rich cluster (composed of population I stars) with longer MS lifetimes, especially for low-mass stars (see Table 1 in Sippel et al. 2012 for more details). 74 Chapter 3. Dynamically Evolved Single Stellar Populations d) 10 5 t [Gyr] , b) average mass per star 0 M 3 2 1 0

2.5 1.5 0.5

[Gyr] t c) 10 .001: a) total mass = 0 5 t [Gyr] Z 0 4 3 2

3.5 2.5

[km/s] σ . There is no significant difference between the models of different hr t b) 10 5 t [Gyr] 0 0.5 0.4 0.6

0.55 0.45

sun

m [M m ] , and d) half-mass relaxation time v a) σ 10 5 t [Gyr]

7 6 5 4 3 2 1

sun

] M M [10 M 3.1 Evolution of structural parameters for a representative cluster with 4 c) velocity dispersion Figure m, metallicity. 3.3. Evolution 75 WD m WD 74237880 0.73 0.78 6479 0.68 N ] 3 [×10 5.4 6.1 4.38 WD M G m G N 290 0.93 169 0.81 103 0.79 ] 3 [×10 269 137 81.5 G M MS m MS N 28229 0.37 27036 0.35 27948 0.34 ] 3 [×10 9.4 10.4 9.57 MS M m N 35017 0.43 34654 0.43 35958 0.44 ] 4 [×10 1.48 1.51 1.50 .

M M Z 0.01 0.001 0.0001 able 3.1 Composition of the models at 12 Gyr. Given are the total mass, number of stars and average mass per star for all cluster T members (columns 2-4), themasses main are sequence in stars only (columns 5-7), giants (columns 8-10) and white dwarfs (columns 11-13). All 76 Chapter 3. Dynamically Evolved Single Stellar Populations

10 a) b) c)

r [pc] 4

0 0 5 10 0 5 10 0 5 10 t [Gyr] t [Gyr] t [Gyr]

Figure 3.2 Half-mass radius rhm (dashed black line), N-body core radius rc (solid black line) and the Lagrangian radius containing 10% of the mass (red dotted line) for models of diﬀerent metallicity from high to low, left to right. Strong ﬂuctuations in the core radius are associated with BH binary encounters and often ejections. 3.3. Evolution 77

3.3.1 General

All models start their evolution with N0 = 105 000 stars at the zero-age MS. The chosen 4 IMF (see Section 3.2) dictates that this corresponds to an initial mass M0 ≈ 6.4×10 M , and an average mass m ≈ 0.6 M per star. An overview of the evolution of a typical model is shown in Fig. 3.1. The early phase of the cluster evolution is dominated by rapid mass-loss from fast evolving massive stars as well as two-body relaxation. We see that the cluster mass decreases by a large fraction initially, and then decreases steadily until ∼ 1/4 of the initial mass remains at a cluster age of 12 Gyr (Fig. 3.1a). Stellar evolution causes the average mass per star to decrease drastically in the first ∼ 2 − 3 Gyr. After this time, mass-loss from stellar evolution is only moderate, while the cluster preferentially loses low-mass stars from the outskirts, causing m to increase from ∼ 2 − 3 Gyr onward (Fig. 3.1b). This is caused by the tendency towards energy equipartition which drives mass-segregation, efficiently moving massive stars towards the cluster center and low- mass stars towards the outskirts. m increases more rapidly for the low-Z cluster, though just at an insignificant level. Concurringly, evolution and relaxation decreases the overall cluster velocity dispersion with time (Fig. 3.1c). The dynamical age of a GC is characterized by the number of half-mass relaxation 1/2 3/2 times thr ∝ N × rhm that have passed since formation. thr is on the order of a few Gyr (see Fig. 3.1d), first increasing with the half-mass radius and then decreasing. Hence the clusters are less than 10 thr old at the Hubble time and in addition, not yet core collapsed.

This can also be seen in Fig. 3.2, where characteristic radii are given: rhm, the N-body core radius rc (which is a density weighted mean distance to the cluster center), and the 10% Lagrangian radius. The mass loss at the beginning of the simulation, described above, causes the cluster to initially expand, followed by a subsequent contraction. Oscillations of rc as typically observed in core collapse (e.g. Hurley & Shara 2012) do not occur yet.

However, we see that sometimes rc varies dramatically or jumps, usually during BH binary encounters and ejections of BHs. Next we list a few selected such events. In the high-Z cluster (Fig. 3.2a), at 10.35 Gyr, a BH-BH binary gets ejected from the cluster (the components have masses of 11.2 M and 10.7 M ). For the intermediate-Z cluster (Fig. 3.2b) , similar encounters are more frequent. The

ﬂuctuations of rc between ≈ 2−2.5 Gyr are caused by the presence of two BH-BH binaries, the ﬁrst with components bh1 and bh2 of masses 25.7 M and 21.4 M , and the second binary with components bh3 and bh4 of masses 22.9 M and 24.7 M . Just before 2.5 Gyr these binaries interact with each other, creating the new binary bh3 + bh1 and ejecting bh4, while bh2 remains in the cluster. At ≈ 3.4 Gyr, bh1 and bh3 separate during the 78 Chapter 3. Dynamically Evolved Single Stellar Populations interaction with yet another BH, bh5, which forms a binary with bh1. Both, this binary and bh3 are ejected immediately, marking the last escape of a BH from the cluster. A new BH-BH binary forms shortly afterwards, and following several interactions and exchanges with other stars or remnants over the next ∼ 3 Gyr, a BH-BH binary with masses 12.9 M and 17.5 M forms at 7.7 Gyr and stays rather undisturbed until beyond 13 Gyr.

In the case of the low-Z cluster (Fig. 3.2c), a BH-BH binary with masses 22.1 M and

26.7 M is ejected at ∼ 5.5 Gyr. Shortly after, another BH-BH binary with components bh1 and bh2 and masses 16.4 M and 13.7 M forms. In an interaction with with yet another BH (bh3, 7.9 M ) at ≈ 11.6 Gyr, the system bh2 + bh3 forms while bh1 is ejected immediately, followed by the binary just a few Myr after. While BHs remain in the cluster, no other BH-BH binary forms. The sudden loss of mass and energy from the core in the BH interaction events causes the core to expand in each case and the fluctuating nature of the core to be damped, for a short time at least, as described in Hurley & Shara (2012). All considered, we find no significant structural dissimilarities for the clusters with different metallicity. Note that we have previously used the same models to show that the metallicity does not affect the cluster size or structure in terms of mass, i.e., clusters of different metallicity can have different apparent half-light radii without different half- mass radii (Sippel et al., 2012). This apparent difference in size is caused by the combined effect of stellar evolution and mass segregation (see also Jordán2004). The composition of the cluster at 12 Gyr, such as the number of stars, WDs, and their average masses, is summarized in Table 3.1. To first order, the cluster mass is composed of 1/3 WDs and 2/3 MS stars, where the mass fraction in WDs, as well as their number and average mass, increases with decreasing metallicity.

3.3.2 Integrated and dynamically evolved SSPs

Following the description in Section 3.2.1, we calculate the magnitudes of each single star in the cluster from evolution snapshots taken every ∆t ≈ 15 Myr. These are converted to luminosity and integrated over all cluster members to obtain the colour and magnitude of the entire cluster, i.e., a dynamically evolved SSP. The results are shown in Fig. 3.3, where integrated cluster colours are plotted (hence each dot represents an entire cluster, not a star). In particular, we show the evolution of a GC mock population, where only one cluster is used for each metallicity, and this cluster is evolved from 2 Gyr to 12.25 Gyr and sampled every ≈ 15 Myr. While we observe a large scatter of ∼ 0.3 mag in cluster colour with time, three distinct ‘populations’ are observed for the three diﬀerent metallicities, in particular with the red high-Z cluster distinct from the blue low-Z and green intermediate- 3.3. Evolution 79

Z clusters, which is in agreement with the fact that the stars in our blue and green clusters are evolving similarly (see Hurley et al. 2005; Sippel et al. 2012). The distinct colour evolution for each metallicity is also illustrated by the median colour calculated in 1 Gyr intervals (marked by squares). For comparison, we show the colour of the PARSEC1 SSP models with matching metallicity (Bressan et al., 2012). The PARSEC tables give the integrated absolute magnitudes of clusters scaled to 1 M , which can be converted to an entire cluster by adding the term −2.5 ∗ log10(M0), with M0 the initial cluster mass. We chose not to use the entire cluster mass, but the mass of all stars which are not expected to form NSs or BHs when viewed at t = 0. In other words we use M = P m, I m

1http://stev.oapd.inaf.it/cmd 80 Chapter 3. Dynamically Evolved Single Stellar Populations

−7.5

−7 2−3

3−4 −6.5 4−5 2−3

5−6 3−4 −6 6−7 4−5

B 7−8 8−9 5−6 −5.5 6−7 9−10 7−8 10−11 8−9 −5 11−12 9−10 10−11

−4.5 11−12

−4 0.6 0.8 1 1.2 1.4 B−I

Figure 3.3 Colour-magnitude diagram for a mock-population of extragalactic GCs. Each dot represents the integrated colour and magnitudes of the entire cluster sampled every ≈ 15 Myr, evolved from 2 Gyr (brighter) to 12.25 Gyr (fainter). This is represented by the blue (low-Z), green (intermediate-Z) and orange (high-Z) dots. For intervals of 1 Gyr the median values are represented by blue, green and red squares, respectively. For comparison, the evolution of a PARSEC SSP with corresponding metallicity (Bressan et al., 2012) is shown as solid lines (the PARSEC models are limited to a maximum age of 12.25 Gyr, which explains the upper limit used for the models in this figure). While the scatter between different snapshots can be large, the distinct populations of different metallicity can be seen. 3.3. Evolution 81

To investigate the origin of the fluctuation in colour between consecutive snapshots and the discrepancy between the N-body and SSP models, Fig. 3.3 is repeated in Fig. 3.4 to 3.6 (left panels) for each metallicity individually. A distinction is made between the integrated colour of all cluster members (dots), giants (diamonds) and MS-stars only (line). As mentioned earlier, we consider any evolved star that has left the MS and not yet reached the remnant phase to be a giant. The integrated colour-magnitude relation for the MS stars only in the N-body model is parallel to the relation predicted from the SSP models by Bressan et al. (2012), however the N-body model is shifted towards fainter and bluer colours, as expected when excluding evolved stars. Virtually no scatter is observed and the distribution is smooth. The integrated colour of the entire cluster as well as of all giant cluster members however follows a scattered distribution from bright to faint at, to first order, constant colour and turning bluer from ≈ 10 Gyr onward. This shows that the colour contribution of the giants significantly alters the cluster colour and is alone responsible for the fluctuation in colour, as previously shown by Fouesneau & Lancon (2010). The turn of our models towards the blue with respect to the SSP models is in agreement with Kruijssen & Lamers (2008). To analyze the origin of this turn in colour towards the blue, and to isolate the effect of the removal of low-mass stars during cluster evolution (Baumgardt & Makino, 2003), we differentiate between the entire cluster colour and the integrated colour of all stars within and outside rhm, the 3d half-mass radius. Choosing the 3d radius implies that any effect (if found) would be enhanced compared to the same analysis in projection. The colour in those two radial bins is shown in Fig. 3.4 to 3.6 (right panels), where the integrated cluster colour (grey dots), colour within (orange diamonds) and outside (purple squares) the half-mass radius is given in the top panel. The median colour in 1 Gyr age bins is given with larger symbols, however we observe no significant difference. For reference, the half-mass radius is in the middle panel, and the number of giants within and outside rhm in the bottom panel. While we find more giants in the cluster center as expected owing to mass segregation, we find a continuous and significant fraction of giants outside the half-mass radius. In the next section we argue we argue that these stars significantly alter the cluster colour in the outskirts such that it stays roughly constant, even though being significantly fainter than the cluster center. 82 Chapter 3. Dynamically Evolved Single Stellar Populations

−7.5

−7

−6.5 2−3

−6 3−4 4−5 5−6 −5.5 6−7 7−8 B 8−9 −5 9−10 10−11 −4.5 11−12

−4

−3.5

0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 B−I

1.4 1.3 1.2 1.1 1 B−I 0.9 0.8 0.7 0.6

10 9

[pc] 8 hm

r 7

300 200 # 100 0 0 2 4 6 8 10 12 14 t [Gyr]

Figure 3.4 Left: Same as Fig. 3.3, only the high-Z case. Dots represent the integrated colour and magnitude for the entire cluster, sampled every ≈ 15 Myr from 2 Gyr to 12.25 Gyr. The diamonds are the same integrated cluster colour, taking only giants into account, while the coloured line in approximately the bottom-left quadrant represents the integrated colour and magnitude of the cluster taking only the main sequence stars into account. For comparison, the grey line is the integrated PARSEC isochrone from Bressan et al. (2012). Note that the integrated giant colour is clearly oﬀset from the MS cluster colour at cluster ages ≥ 10 Gyr. Right: Cluster colour (top), half-mass radius (middle) and giant population over time (bottom), again for high metallicity. In the top panel, the grey dots represent the entire cluster colour, orange diamonds the stars with d∗ ≤ rhm, and purple diamonds the stars with d∗ > rhm. The evolution of rhm is in the middle panel, and in the bottom the number of giants with d ≤ rhm (orange), and with d > rhm (purple). 3.3. Evolution 83

−7.5

−7 2−3

−6.5 3−4 4−5 −6 5−6 6−7 −5.5 7−8 8−9 B 9−10 −5 10−11 11−12

−4.5

−4

−3.5

0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 B−I

1.4 1.3 1.2 1.1 1 B−I 0.9 0.8 0.7 0.6

10 t [Gyr] 9

[pc] 8 hm

r 7

300 200 # 100 0 0 2 4 6 8 10 12 14 t [Gyr]

Figure 3.5 Same as Fig. 3.4, but for intermediate-Z. The signiﬁcantly lower number of giants than in Fig. 3.4 causes the ﬂuctuation of integrated giant colour, and hence cluster colour to decrease drastically for high ages in a way such that the contribution from MS stars and giants to the magnitude and colour are comparable. 84 Chapter 3. Dynamically Evolved Single Stellar Populations

−7.5

−7 2−3

3−4 −6.5 4−5 5−6 −6 6−7 7−8 −5.5 8−9 9−10 B 10−11 −5 11−12

−4.5

−4

−3.5

0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 B−I

1.4 1.3 1.2 1.1 1 B−I 0.9 0.8 0.7 0.6

10 9

[pc] 8 hm

r 7

300 200 # 100 0 0 2 4 6 8 10 12 14 t [Gyr]

Figure 3.6 Same as Fig. 3.4 and 3.5, but for low-Z. The evolution is similar to the intermediate Z case in Fig. 3.5, as expected from stellar evolution (Hurley et al., 2005). 3.3. Evolution 85

3.3.3 Resolved single stellar populations

In this section, instead of integrating the cluster colour we now focus on the resolved cluster and the individual contribution of each star to the cluster magnitude and colour. In Fig. 3.7 the colour-magnitude diagrams (CMDs) for the clusters of all three metallicities are given for snapshots at the ages of 9, 11 and 13 Gyr. Note that no errors are added in these figures. Features to note are the double MS for low-mass stars caused by binaries, a direct artifact of the simulation setup with minimum mass of 0.1 M per star and hence a minimum of 0.2 M per binary. The main sequence turnoff, with several blue stragglers, can be seen, as well as the giant and horizontal branches and the WD cooling sequence. The evolved stars cover a wide range of colour while always staying bright, in particular for low-Z, with giants as red as low-mass MS stars. The horizontal branch is significantly bluer in the lower-Z models, explaining the stronger turn towards blue for the overall cluster colours as observed in Fig. 3.4 to 3.6. The MS turnoff mass is mTO = 0.83 M for low and intermediate Z, and mTO = 0.91 M for high Z, at an age of 12 Gyr. In panels a) of Fig. 3.8 we show the B- and I-band magnitude, the colour in panels b) and the number and mass density in panels c) and d). This is an average of 15 snapshots around 12 Gyr. We sample every ≈ 15 Myr, hence this is an average over ≈ 200 Myr to reduce fluctuations. It is of no surprise that the clusters are brighter in I-band as the brightest stars are always red (as in Fig. 3.7). We also see that the radial colour profile fluctuates heavily, such that no gradient in colour can be seen2.

2The lack of a colour gradient is in agreement with observations of MW clusters where, if any, only few clusters have colour gradients (Chun & Freeman, 1979; Pastoriza et al., 1986; Bica et al., 1998), potentially caused by dust in the cluster center (Angeletti et al., 1981a,b, 1982). Here, we do not take such dust into account. A colour gradient in 47 Tuc has been attributed to a concentration of evolved stars and and a reddening of the main sequence in the central region, potentially caused by an abundance gradient of carbon, oxygen and nitrogen in the early stages of cluster formation (Chun, 1991). In contrast, we use a uniform metallicity for our entire cluster. 86 Chapter 3. Dynamically Evolved Single Stellar Populations

Figure 3.7 Colour-magnitude diagrams of cluster snapshots at 9 Gyr (left), 11 Gyr (middle) and 13 Gyr (right panel) for Z = 0.01 (upper panels), Z = 0.001 (middle panels) and Z = 0.0001 (lower panels) models. No errors are added, and the double MS for faint stars and broadening of the MS for brighter stars is purely due to the binary population in the clusters. Stars in the region between the WD cooling sequence in the bottom left and the main sequence are binary systems of a newly formed WD and a faint MS star, where for a short amount of time the WD is still brighter than the MS companion. Just to the left of the MS turnoﬀ, several blue straggler stars which formed during the cluster evolution can be seen. 3.3. Evolution 87

−0.25 3 2 1.1 b) 2 c) 1 −0.2 a) d) ]

1 2 ] 0 2 −0.15 1 0 /pc

2 −1

−1 sun −0.1 −2

B−I 0.9 ) [#/pc −2 ) [M mag/pc −3 −0.05 −3

0.8 log( ρ −4 −4 0 log( ρ B −5 −5 I 0.05 0.7 −6 −6 0 0.5 1 1.5 0 0.5 1 1.5 0 0.5 1 1.5 0 0.5 1 1.5 log(d) [pc] log(d) [pc] log(d) [pc] log(d) [pc] −0.25 3 2 1.1 b) 2 c) 1 −0.2 a) d) ]

1 3 ] 0 3 −0.15 1 0 /pc

3 −1

−1 sun −0.1 −2

B−I 0.9 ) [#/pc −2 ) [M mag/pc −3 −0.05 −3

0.8 log( ρ −4 −4 0 log( ρ B −5 −5 I 0.05 0.7 −6 −6 0 0.5 1 1.5 0 0.5 1 1.5 0 0.5 1 1.5 0 0.5 1 1.5 log(d) [pc] log(d) [pc] log(d) [pc] log(d) [pc]

Figure 3.8 We show in a) the B- and I- magnitude profiles, in b) the B − I colour profile in black, as well as the colour for just MS stars (grey line), in c) the number density and in d) the mass density. In the top panels this is done in projection, in the bottom panels in three dimensions. In the number and mass density profiles mass segregation is clearly visible. The black line represents NSs and BHs, in mangenta are giants, WDs in cyan and all stars in blue. The grey lines are for selected MS stars of masses m ≤ 0.8 M in intervals of 0.1 M down to 0.5 M . No population is predominant anywhere, although the NS/BH population is obviously strongly peaked towards the cluster centre. Given the fluctuations from one snapshot to the next, we average 15 snapshots separated by ≈ 15 Myr around 12 Gyr. 88 Chapter 3. Dynamically Evolved Single Stellar Populations

Taking only MS stars into account, the cluster turns redder in the outskirts, both in 3d and in projection. Looking at the density of both, stars and mass, in 3d and projection the most massive components, BHs and NSs, are the most centrally concentrated populations, followed by the WDs. Note that the mass and number density of giants are lower than those of low-mass MS stars, even in the core, to the extent that the slopes of giants and MS stars remain similar throughout the cluster (see also Fig. 3.11 for more on the stellar distributions). To illustrate the interplay between cluster dynamics and the colours of stars, we link the colour, magnitude, and velocity of each stellar system in Fig. 3.9. Presented is one intermediate-Z snapshot at 13 Gyr, with the dots in the CMD coded by the velocity of each star (the center of mass velocity in the case of binaries). While there are always exceptions, there is a general trend for fainter, low-mass stars to have higher velocities than higher mass stars close to, or beyond the MS turnoff and binaries (which are effectively higher mass stars in the center of mass system). While they are distributed all over the cluster, there is a trend for brighter and more massive stars being more centrally concentrated (see also Fig. 3.11), implying that recently formed WDs at the top of the WD cooling curve have lower velocities than older WDs. This is not unexpected, as they will have been gone through mass-loss recently. To further illustrate this point, we show in Fig. 3.10 the relations between magnitude and mass for the same snapshot as before, where binaries are separated into their individual components. Four MS stars are visible beyond the MS turnoff, these are BSs. It can be seen that, while certain populations can be limited to a narrow mass-range (e.g. the horizontal branch), they can be distributed over a wide range in brightness. Not only their brightness is spread, but as expected from a lack of colour gradient in the inner and outer radial bins in Fig. 3.4-3.6, also their distances d from the cluster center (measured in 3d). Fig. 3.11 shows that each grouping (or evolution type) of stars covers a wide range of distances. In particular, MS stars spread out to the tidal boundary at ∼ 50 pc, while WDs are slightly more concentrated, which is expected owing to mass segregation: m ∼ 0.35 M and mWD ∼ 0.73 M (see Table 3.1 for more details at 12 Gyr). In addition, recently formed WDs have only just lost mass when transitioning from the phase of an evolved star to the WD sequence, and are hence on a journey to a position at larger distances than when they were still giants (for the same reason, young WDs have higher velocities than older ones, see below and Fig. 3.9). Stars in the Hertzsprung gap are distributed as far out as ∼ 20 pc (or almost 3rhm), as well as are stars on the giant or horizontal branches. 3.3. Evolution 89

−4 4 −2 3.5 0

2 3

4 2.5 6 I 2 8

10 1.5

12 1 14 0.5 16

−1 −0.5 0 0.5 1 1.5 2 2.5 3 B−I

Figure 3.9 Colour-magnitude diagram for the stars in one cluster snapshot of intermediate metallicity at 13 Gyr. The colour of the dots is coded with velocity, where red dots represent higher-velocity cluster members and blue dots lower velocity stars. For binary systems the center of mass velocity is used. It can be seen that, although exceptions are present, giants and binaries have lower velocities, as expected owing to mass segregation. Recently formed WDs at the brighter end of the WD cooling curve have lower velocities than older WDs, which is expected as they still have a position in the cluster that corresponds to their mass before the most recent mass loss during WD formation. 90 Chapter 3. Dynamically Evolved Single Stellar Populations

−5

5 I

0.2 0.4 0.6 0.8 1 1.2 1.4 m [M ] sun

Figure 3.10 Mass-magnitude diagram for a snapshot of the model with Z = 0.001 at 13 Gyr. MS stars are orange dots, WDs blue dots, the Hertzsprung gap is represented by bright green stars, ﬁrst ascent red giants are marked by a dark green plus signs, the horizontal branch by pink triangles and the early asymptotic giants or red supergiants are the two purple stars. Four blue stragglers can be seen as main sequences stars at high masses. Note the narrow mass range for giants and the distinct MS, with the exception of two outliers and four BSs as well as three evolved BSs in the top right quadrant. 3.3. Evolution 91

Figure 3.11 Magnitude-distance relation for the same snapshot as in Fig. 3.10, with the same symbols. The plot is cut at magI = 17. Note the spatial distribution of several populations of stars with a large spread across all distances d to the cluster centre. Here, the distance is measured in three dimension, see Fig. 3.8 for a comparison between 3d and projection. At the time of this snapshot, rhm ≈ 7.8 pc. 92 Chapter 3. Dynamically Evolved Single Stellar Populations

We find that the overall cluster colour varies strongly depending on the distribution of the brightest stars as well as the colour of those stars. While the brightest stars are mass-segregated towards the cluster center, they are by no means limited to that location. To test the susceptibility of the cluster to the removal or addition of certain stars, we carry out the following two experiments in Fig. 3.12 and 3.13 either removing or adding stars. First, in Fig. 3.12 we integrate the cluster colour while removing one star after the other, starting from the brightest star. It can be seen that the removal of the first few (i.e., brightest) stars leads to a significant change in cluster colour. In particular, by just removing the first three stars the cluster turns bluer by ∆(B − I) > 0.1 mag. Since the brightest stars are red (Fig. 3.7), this turns the cluster bluer at first. This illustrates that the brightest stars, whose presence and location fluctuates over time, significantly alter the cluster colour. The host galaxy of a GC removes stars from the cluster over the tidal boundary: of the initial 105 000 stars in the model, only 30 461 remain at 13 Gyr. Owing to stellar evolution these will be preferentially low-mass stars. Reversing the previous exercise, we now integrate the cluster colour at 13 Gyr while adding stars. Starting from the low-mass end, we add each star three more times to the cluster. Adding three times all stars up to 0.3 M , the cluster ends up with ∼ 100 000 stars. As can be seen in Fig. 3.13, the cluster colour is not sensitive to the addition of low mass stars: if stars up to 0.3 M (approximately the mean cluster mass) are added, the colour changes by ≤ 0.015 mag. Hence Fig. 3.12 and 3.13 further illustrate that the cluster colour is highly responsive to the presence of bright stars, while remaining relatively unaware of the addition (or subtraction) of low-mass stars. 3.3. Evolution 93

1.4 1.3 1.2 m=0.8 m=0.7 1.1 m=0.815 m=0.75 1 B−I 0.9 0.8 0.7 0.6

−6 −5.5 −5 −4.5 −4 −3.5 −3 I

Figure 3.12 To test the contribution of the brightest stars to the overall cluster colour, we use the integrated cluster colour and calculate it by steadily removing one star after the other, starting from the brightest. The initial cluster colour is given by the dashed black line, and the sequence commences on the left of the figure. With black vertical lines we mark when the first MS star with mass 0.815, 0.8, 0.75 and 0.7 M is removed. From the knee at I ≈ −4.7 towards the right the horizontal branch is removed star by star, at 0.815 M the MS turnoff is marked. 94 Chapter 3. Dynamically Evolved Single Stellar Populations

1.1

m=0.4

1.05 m=0.5 m=0.3

m=0.6 m=0.25

1 m=0.9 m=0.7 m=0.2 B−I m=0.8 m=0.15

0.95

0.9 −7.2 −7 −6.8 −6.6 −6.4 −6.2 −6 −5.8 −5.6 I 1.02 m=0.6 1.01 m=0.5

1 m=0.4 0.99 m=0.3

B−I 0.98 m=0.25 m=0.2 0.97 m=0.15

0.96

0.95 −6 −5.95 −5.9 −5.85 −5.8 −5.75 I

Figure 3.13 Top panel: Following a similar approach as in Fig. 3.12, we test the change of cluster colour by adding stars, starting from the low-mass end (hence the sequence commences on the right). To test an extreme case, each star is added three times, and we mark when the first MS star of 0.15, 0.2, 0.25, 0.3, 0.4, 0.5, 0.6, 0.7, 0.8, and 0.9 M is added. From the right, MS stars are added, around the mark m = 0.6 M the low-mass sequence of the WD cooling curve is added (see Fig. 3.10), followed by the addition of the horizontal branch (loose dots around I ≈ −7), again WDs around the mark of m = 0.7 M , the high-mass end of the MS up to just over 0.8 M and finally the remaining giants and BSs. While WDs of such high masses are in the cluster, their number is low and hence their contribution insignificant. Jumps or gaps occur whenever either a large number of stars with the same mass are added (e.g., WDs, as their mass function peaks at ∼ 0.6 M ), or when one single star is sufficiently bright to change the cluster brightness on its own (i.e., when the horizontal branch stars are added). Bottom panel: As above, but zoomed to the low-mass end of the distribution. When adding a significant amount of low-mass stars, the cluster colour changes only by a small amount. There are 11154 stars with m < 0.3 M in the cluster at that time. 3.4. Conclusions 95

3.4 Conclusions

We have used direct N-body models of different metallicity to study the effect of metallicity and cluster evolution on the evolution of cluster colour. The colours are calculated for each star individually, implying that giants and blue straggler stars are automatically included in this analysis. More important, when presenting integrated colours we take the contribution from every star into account, as would be the case for observations of extragalactic GCs. This method allows us to analyze the colour evolution of a dynamically integrated single stellar population, and we find that:

1. The cluster appearance fluctuates over short timescales, owing to the stochastic presence of stars during various stages of giant evolution. This is in agreement with previous findings by Fouesneau & Lancon (2010) and Fouesneau et al. (2012) and implies that caution needs to be taken into account when transforming observed cluster colours and magnitudes into mass and age (in agreement with Anders et al. 2009b). A different approach was taken by Zhuang et al. (2014) who remove giants (and hence stochastic fluctuations) from their models altogether.

2. Even though the clusters are mass-segregated (see Fig. 3.8), no population of stars is predominant in any location in the cluster (see Fig. 3.11), with the result that no signiﬁcant colour gradient is found, even at distances larger than several half-mass radii.

3. Our models represent low-density GCs in a moderate tidal field, implying that they are more susceptible to the external tidal field than a denser GC might be. Yet we find no significant change in cluster colour when changing the contribution of low-mass stars.

4. The integrated B − I does not follow the predictions from SSP models. Instead, the colour is rather constant over time, as its evolution is dominated by the colour of the giants in the cluster. Only taking MS stars of the N-body model into account, the colour evolves at a similar slope as the SSP model. This suggests that the removal of low-mass MS stars does not alter the cluster colour noticeably, which instead is dominated by the evolved stars. As the giants become bluer from 10 Gyr onward, so does the entire cluster.

The last point indicates that while a selective mass-loss owing to mass segregation and the external tidal field is present (increasing the average mass per star, as in Fig. 3.1), we 96 Chapter 3. Dynamically Evolved Single Stellar Populations observe no great change in colour: the evolution of the cluster B − I is driven by evolved stars rather than MS stars. Even though giants are more centrally concentrated than MS stars due to mass segregation (Fig. 3.11), they are distributed over the entire cluster, leading to the observation of a luminosity, but not necessarily colour, gradient. We note that the observed effects may be less strong for more massive clusters, when stochasticity becomes less of an issue. However we still predict the cluster colour to be driven by giants rather than MS stars. Furthermore, we cannot exclude that a stronger tidal force, as for clusters closer to the center of the galaxy, may have an influence on cluster colour just before dissolution. This is what Lamers et al. (2006); Kruijssen & Lamers (2008) and Anders et al. (2009b) find, in that the colour changes again when only less than 5% of the initial cluster members remain. However, several Gyr before the cluster dissolution, i.e. when 5% of the initial members remain, we find a similar colour behavior to that reported by these authors. While they attribute this change to the stripping of low mass stars from the cluster by the tidal field, we find that is not necessarily the case. As we have already mentioned, our results strongly suggest that the cluster colour is dominated by the giants and not the MS stars. The results by Lamers et al. (2006); Kruijssen & Lamers (2008); Anders et al. (2009b) are based on the models by Baumgardt & Makino (2003), which use a slightly different IMF and external tidal field, but differing initial number of stars and density. Selecting the models with galactocentric distances comparable to ours from the Baumgardt & Makino

(2003) series, only the two models on a circular orbit with 131 072 stars and either W0 = 5.0 or W0 = 7.0 have dissolution times similar to our models, hence only for those models the combination of stellar evolution and dynamical age agrees with our models. It is therefore plausible that the Baumgardt & Makino (2003) models experience more severe mass loss owing to the tidal field than we observe. We note that giant evolution is quite sensitive to the chosen stellar evolution models, including the treatment of stellar winds. For the limited number of cluster parameters taken into account in this study, we conclude that the cluster colours are not strongly affected by a moderate external tidal field but driven by the fluctuating contribution of giants in the cluster. While this effect is most likely less strong for more massive clusters – possibly mitigated by resolved observations of individual clusters where bright stars can be excluded – we consider it sensible to argue that caution needs to be taken into account when using single stellar population models to transform observed cluster properties (e.g. metallicity, magnitude and colour) into mass. 3.4. Conclusions 97

In the future it would be useful to repeat this study but taking into account more massive

(i.e. higher N0) and denser clusters (which may reach core collapse), most importantly however, also with diﬀerent IMFs (motivated by e.g. Strader et al. 2009, 2012; Zaritsky et al. 2012) and multiple populations (Gratton et al., 2012).

4 Black Holes in Globular Clusters: Theoretical Conﬁrmation

Furthermore, at the moment it is not clear how many stars have to be in a cluster such that the previously obtained results can still be applied. - Sebastian von Hoerner, 1963

4.1 Stellar mass black holes in globular clusters

Stellar-mass black holes (BHs) are formed at the endpoints of stellar evolution of very massive stars. Depending on the chemical composition or metallicity of a star, a supernova resulting in either a neutron star or a black hole is the ultimate fate for most stars above

≈ 7 − 8 M (Pols et al., 1998). Tens or hundreds of such stellar-remnant BHs are expected to form in globular clusters (GCs), however at the current stage it is still not entirely clear how many will receive a velocity kick at formation and get ejected immediately, and how many BHs might be removed during the subsequent dyamical evolution of the cluster. While several extragalactic GCs containing BHs are known at the current stage (e.g. Maccarone et al. 2011), none had been conﬁrmed in a Milky Way GC until the recent discovery of two such stellar- mass BHs in M22 (NGC 6656, Strader et al. 2012). Since only BHs currently undergoing observable accretion can be detected via radio or X-ray emission and Strader et al. (2012) estimate that 2 − 40% of BHs are expected to become members of binary systems with observable accretion over 10 Gyr, it is possible that a total population of ≈ 5 − 100 BHs exists in M22. Theoretical predictions in the past have lead to the assumption that well before a cluster age of 12 Gyr all but up to four (possibly all) BHs should be ejected from the cluster core (Sigurdsson & Hernquist, 1993), or similarly that nearly all BHs should be

99 100 Chapter 4. Black Holes in Globular Clusters ejected, with the possibility of remaining BHs capturing normal stars to form low-mass X-ray binaries in low-density environments (Kulkarni et al., 1993). Strader et al. (2012) claim their observations to be in contrast to such prior theoretical predictions. However it was not possible to test this in direct N-body models comparable to clusters like M22 until very recently as such models are computationally expensive. We note that multiple BHs have already been found to remain in N-body models with smaller numbers of stars (e.g. Mackey et al. 2007, 2008) as well as in Monte Carlos simulations of globular cluster evolution (e.g. Downing et al. 2010; Downing 2012), while BHs are ejected from the cluster in N-body models by Banerjee et al. (2010) where high numbers of BHs were added initially. Based on the recent ﬁndings by Repetto et al. (2012) suggesting that BHs receive similar velocity kicks as neutron stars (NSs) upon formation, and the observations that not all NSs are expected to receive a high kick (Pfahl et al., 2002), we present a direct N-body model for an intermediate-mass globular cluster containing 262 500 stars in total and incorporating a retention fraction of ≈ 10% for both BHs and NSs. The cluster is evolved using NBODY6 (Aarseth, 1999, 2003) in a Milky Way-like gravitational potential ﬁeld. While the model presented in this paper is by no means an attempt at a direct model of M22, it is representative owing to comparable dynamical and absolute ages. We evolve the model up to an age of 20 Gyr, and focus our analysis at the age of 12 Gyr: the estimated age for many globular clusters in the Milky Way including M22 (Salaris & Weiss, 2002). At this stage, ≈ 180 000 stars are still retained within the cluster (including 16 BHs), implying that this is the largest direct N-body model of a globular cluster currently available (cf. Baumgardt & Makino 2003; Hurley & Shara 2012). 4.1. Stellar mass black holes in globular clusters 101

Figure 4.1 Location of BHs (red crosses) at 12 Gyr of cluster age, projected in front of the cluster to increase visibility. 102 Chapter 4. Black Holes in Globular Clusters

4.2 Simulation method & choice of parameters

We use the direct N-body code NBODY6 (Aarseth, 1999, 2003; Nitadori & Aarseth, 2012) to evolve our model and chose a set-up similar to the clusters presented in Sippel et al.

(2012). We evolve a star cluster with Ni = 250 000 stellar systems and an initial binary fraction of bf = 0.05, implying that the initial numbers of stars is in fact N = 262 500 including 25 000 stars in 12 500 binary systems. The stars are drawn from the initial mass function presented by Kroupa et al. (1993) within the limits 0.1 ≤ m ≤ 50 M and distributed according to a Plummer density profile (Plummer, 1911; Aarseth et al., 1974). 5 The initial cluster mass adds up to Mi = 1.6×10 M . The conversion from N-body units to physical scale sizes leaves the scale radius as a free parameter which is set to give an initial half-mass radius of r50% = 6.2 pc. The cluster is orbiting a Milky Way like galactic potential consisting of three components: a point-mass bulge, an extended smooth disc (Miyamoto & Nagai, 1975) and a logarithmic dark matter halo (Aarseth, 2003). We chose a galactocentric distance of dgc = 8.5 kpc. From King model fits (King, 1966) we find an initial tidal radius of rt ≈ 50 pc for the N-body model, while M22 has a smaller tidal radius of rt = 27 pc (Mackey & van den Bergh, 2005) owing to the smaller galactocentric distance of dgc = 4.9 pc. Within the Milky Way, globular clusters across the whole range of metallicities exist around the galactocentric distance used here and we chose a low to intermediate metallicity Z= 0.001 (corresponding to [Fe/H]≈ −1.3). This metallicity is close to that of M22 [Fe/H]= −1.7 and stars within this metallicity range evolve in an almost identical fashion (as illustrated in Hurley et al. 2000, 2004; Sippel et al. 2012). Stars are evolved according to stellar and binary evolution algorithms (Hurley et al., 2000, 2002) implying that stellar collisions may occur and stellar remnants such as white dwarfs (WDs), NSs and BHs form.

BHs form from progenitors with masses greater than 20 M , where the resulting remnants are within the mass range of 4−30 M (Hurley et al., 2000; Belczynski et al., 2010). These massive stars evolve rapidly with most BHs forming during the ﬁrst ≈ 10 Myr of cluster evolution.

4.2.1 Velocity kicks

It is well established that some NSs will receive velocity kicks upon formation resulting from asymmetries in the stars surface before collapse (Pfahl et al., 2002). To mimic a neutron star retention fraction similar to indications from observations of ≈ 10% (c.f. Pfahl et al. 2002; Pfahl 2003) we adopt a random ﬂat kick distribution between 0 − 100 km/s. 4.2. Simulation method & choice of parameters 103

Recent indications suggest that BHs should receive similar kicks to NSs (Repetto et al., 2012), as such we apply an identical kick distribution to BHs. For our model, the escape p velocity is vesc = 2GM/r50% = 14 km/s at the very beginning of cluster evolution (M is the cluster mass, r50% the half-mass radius and G the gravitational constant). At an age of 200 Myr, by which time all BHs and NSs will have safely formed, the escape velocity is 4 10.9 km/s owing to r50% = 8.5 pc and M = 11.8 × 10 M .

We note that in general, for a supernova occurring in a tight binary system, the kick velocity can be signiﬁcantly above the cluster escape velocity and still result in one or both binary components being retained, with the binary loosened or broken up. Seven BHs form while part of a binary system, all of them being disrupted either immediately or within the next ≈ 10 Myr. Ultimately, this means that out of ≈ 450 stellar remnant BHs formed through stellar evolution, 48 are retained beyond the ﬁrst 200 Myr, i.e. are not immediately ejected from the cluster via the velocity kick process. Within less than one Gyr, they have settled into the cluster core owing to mass-segregation and have a steadily decreasing half-mass radius of ≈ 1 pc. The spatial distribution of BHs within the model cluster at 12 Gyr is illustrated in Fig. 4.1 (see also Fig. 4.2).

4.2.2 Size and time scales: measurements

The sizes of N-body star cluster models are usually measured by either the half-mass radius or the core radius. Both quantities are measured in three dimensions, in contrast to observational size scales. The half-mass radius r50% is simply the 50% Lagrangian radius, however the N-body core radius is a density weighted average distance from the cluster centre (von Hoerner, 1960b, 1963; Casertano & Hut, 1985; Aarseth, 2003) and hence not comparable to the observational core radius resulting from e.g. King model fits to the surface brightness profile. The procedure to calculate both observable half-light as well as core radii is similar to that described in Sippel et al. (2012). In particular, we cross convolve the NBODY6 output of mass, luminosity and radius for each single star with stellar atmosphere model calculations (Kurucz, 1979) to obtain V-band magnitudes. Surface brightness profiles are produced by taking the average over projection along the x−, y− and z−axes; and we use gridfit (McLaughlin et al., 2008) to fit a King model to the cluster. From projection effects alone, the half-mass radius is ≈ 1.3 times larger than the half-light radius. 104 Chapter 4. Black Holes in Globular Clusters

4.3 Evolution

The first few Gyr of cluster evolution are dominated by heavy mass loss arising from stellar evolution as well as dynamical relaxation that results in an initial expansion of the cluster to a half-mass radius of up to r50% ≈ 11 pc (Fig. 4.2). The cluster then slowly starts to contract to r50% = 10.6 pc and an N-body core radius rtc = 3.1 pc, both three- dimensional quantities, at 12 Gyr. In terms of physical size, the model has a half light radius of rh = 6.8 pc at 12 Gyr and we find an observational core radius of roc = 3.6 pc. The half-mass relaxation time at 12 Gyr is ≈ 2.1 Gyr, where we utilize the half-light radius to to calculate this number as suggested by Djorgovski (1993) and Harris (1996). Owing to the influence of the external galactic tidal field as well as dynamical interactions, stars are continuously lost from the cluster, with ≈ 180 000 stars remaining at 12 Gyr. This corresponds to ≈ 68% with respect to the initial number of stars and a total 4 mass of M = 6.7 × 10 M (≈ 42% of the initial mass). Globular clusters are efficient in equalizing the distribution of energy, causing the most massive stars or remnants to sink to the centre, resulting in mass segregation. Even though our model is segregated after 12 Gyr, the average mass within the innermost pc is only

0.7 M (compared to an overall average of 0.38 M ). Simultaneously, heating of the core from either BHs or NSs can be expected to produce signatures such as an expanded core (Mackey et al., 2007, 2008), where both effects can influence the surface brightness profile and are measurable by means of the core radius (King, 1966). M22 has a large core radius compared to other globular clusters in the Milky Way, indicating that a heating source may be present. While our model does not enter the phase of core collapse during its evolution, the subsystem of stars contained within the innermost 2 pc (dynamically dominated by BHs) is entering a phase of core-collapse at ≈ 1 Gyr of cluster age, indicated by the drop and fluctuation in the core radius (Fig. 4.2). This results from the fact that the BH population has a much smaller half-mass relaxation time than the cluster as a whole, resulting the the BHs segregating to the cluster centre. The half-mass radius of the entire BH population is ≈ 0.2 pc and smaller than the 0.1% Lagrangian radius (see Fig. 4.2).

At 12 Gyr, the total mass contained in BHs is ≈ 190 M corresponding to only ≈ 0.3% of the overall cluster mass but ≈ 32% of the mass within the central parsec. This already indicates that BHs are an important component to the contribution of mass in the cluster centre. Indeed we ﬁnd that at 12 Gyr, the innermost parsec (measured in three dimensions) is composed equally of main sequence stars, white dwarfs, and BHs (Fig. 4.3). 4.3. Evolution 105

101

100 r [pc]

0 2 4 6 8 10 12 t [Gyr]

Figure 4.2 Size evolution of the cluster over time, from top to bottom these are half-mass r50% (dashed), N-body core radius rc (dotted), further Lagrangian radii r1%, r0.5% and r0.1% as well as the half-mass radius for the BH population (red), always smaller than even the 0.1% Lagrangian radius. 106 Chapter 4. Black Holes in Globular Clusters

103

] 10 sun M [M 101 m m MS m wd m BH 0 10 0 2 4 6 8 10 12 t [Gyr]

Figure 4.3 Contribution of stars to the central parsec of the cluster, i.e. d < 1 pc, where the distance d is again measured in three dimensions. It can be seen that at 12 Gyr, MS stars as well as WDs and BHs are contributing roughly 1/3 of the dynamical mass each, while the contribution of other stars (giants or neutron stars) is minimal (i.e. ≤ 10 M for both together). The ratio is diﬀerent when looking at the number of stars in each category: at 12 Gyr, MS stars count for roughly 2/3 of the stars within the innermost pc, while WDs make up ≈ 1/3 of the stars. The dip in BH mass within the central region at ≈ 5.5 Gyr is caused by an energetic interaction within a close subsystem of three BHs, two of them previously in a BH-BH binary. One BH is ejected out of the cluster, while the BH-BH binary receives a recoil velocity, ejecting it momentarily beyond the central parsec. The loss of kinetic energy causes an expansion of the cluster core, however the BH-BH binary quickly sinks back to the centre. 4.3. Evolution 107

For the metallicity of both M22 and our model the main sequence turnoﬀ mass is

0.83 M at 12 Gyr, implying that most white dwarfs (with a peak in their mass distribution at 0.6 M ) will be less massive than the main sequence turnoﬀ, while all neutron stars and black holes will be more massive. In total, 245 M is contained in the innermost pc in 352 stars (compared to 2 322 stars or 1 263 M in projection), and this central density increases to 528 stars or 423 M at an age of 20 Gyr (or 2 582 stars and 1 616 M in projection).

4.3.1 Binary systems with black holes

Binary formation and disruption, as well as hardening of binary systems, are common processes in globular clusters. However it is easier to form new binary systems via exchange interactions than creating them in two-body encounters (Heggie, 1975; Heggie & Hut, 2003). A BH can be part of a binary system where the other component can be at any stellar evolution stage. In our simulation, the ﬁrst dynamical binary system including a

BH forms at ≈ 380 Myr (a 25.3 M BH and a 0.2 M MS star). At 12 Gyr, the initial binary fraction of bf = 0.05 is slightly decreased to bf = 0.045 (with 7 850 binary systems present). In M22, one or potentially both observed BHs are in a binary with a low-mass main sequence star, or possibly a WD (Strader et al., 2012). In our model, we find that at around 12 Gyr, two BHs are in a long-period binary system with a main sequence star each, where neither of those systems are undergoing mass transfer. Specifically, system a) consists of a BH with mass 6.2 M and a main sequence star with mass 0.8 M while system b) consists of a BH with mass 6.4 M and a main sequence star of 0.4 M . Just before 12 Gyr a three-body encounter occurs involving system b) and a main sequence star with mass 0.7 M , replacing the lighter main sequence star and forming again a binary system consisting of a BH and MS star. A BH-BH binary system is also present at around 12 Gyr, where the components of this binary have masses of 22 M each. Just after 12 Gyr, during a three-body encounter with another BH with 13 M , this binary gets disrupted, whilst the individual components of the three-body system stay retained within the cluster. A BH-BH binary in a new combination forms quickly and the total number of 16 BHs is stable around the 12 Gyr timeframe, as illustrated in Fig. 4.4. Dynamical interactions like these are what can ultimately cause the depletion of all but one BH or a tight BH-BH binary (which might ultimately merge, Aarseth 2012), from the cluster. However it is not certain over which timescale the evaporation of BHs from globular clusters should take place. This depletion timescale is dependent on random encounters, necessary to eject BHs, while 108 Chapter 4. Black Holes in Globular Clusters the frequency of such encounters depends on cluster parameters such as the core density (Pooley et al., 2003). We conclude that 12 Gyr is not enough time to deplete all or almost all BHs. In fact we have evolved the model up to 20 Gyr, and even at this late stage, 10 BHs remain in the cluster. Five of those are in a binary system with either a main sequence star, white dwarf or another BH. We also find a BH-NS system with a total mass of ≈ 18 M at ≈ 18 Gyr in a stable configuration for ≈ 0.6 Gyr, at which stage the NS gets replaced by a BH with mass 7.1 M in a three-body encounter. Clausen et al. (2013) have recently presented a study regarding BH-NS mergers in GCs and we conclude that dynamical interactions don’t efficiently produce BH-NS binaries in GCs.

4.3.2 Comparison

While a direct model approach for M22 is still out of reach from a computational point of view, the N-body model presented in this work is comparable in many aspects. While our model has a half-mass relaxation time of trh = 2.1 Gyr, the correspinding value is trh = 1.7 Gyr for M22 (Djorgovski, 1993; Harris, 1996). M22 has a half-light rh = 3.1 and core radius roc = 1.2 pc (Harris, 1996), implying that the N-body model is less dense than M22. We argue that the higher central density of M22 does not severly change our predictions for stellar remnant BHs in globular clusters with large core radii since examples of denser model clusters with BHs exist in the literature. Most prominent is the N-body model by Hurley & Shara (2012) consisting of 200 000 stars initially, evolved at a galactocentric distance of 3.9 kpc and with an initial half-mass radius of 4.7 pc. The strong gravitational field of the host galaxy at this distance has an impact on the cluster by stripping off stars in the outskirts, resulting in only ≈ 35 000 stars remaining at 12 Gyr. However, the half-mass relaxation time of the Hurley & Shara (2012) model is much below that of M22 resulting in the model reaching the end of its initial phase of core-collapse at 11.5 Gyr, greatly exceeding the dynamical age of M22. At this stage, the core radius is roc = 0.84 pc with four BHs still present (two of those in a BH-BH binary). This is the case even though the central density is significantly enhanced at this time (just over 3 000 stars in the innermost parsec, or ≈ 5 600 in projection). Even though the black holes ultimately get ejected during the evolution subsequent to core-collapse, we can safely conclude that BHs in the core of M22 can be expected, as the density of M22 lies in between the Hurley & Shara (2012) model and the N-body model presented in this work – however much closer to the new model here. 4.3. Evolution 109

30 25 a)

] 20

sun 15 10 m [M

5 max mean 0 45 b) 30 N 15 0 0 2 4 6 8 10 12 t [Gyr]

Figure 4.4 Evolution of the mass of the entire black hole population a) and the number of BHs within the cluster b). In the upper panel, the solid black line denotes the maximum mass of the BHs still retained in the cluster, while the blue-dashed line denotes the mean mass of the BH population at a given time. The number of BHs is decreasing b) from 48 at 200 Myr to 16 BHs at 12 Gyr of cluster age. A yellow background denotes the presence of one BH-BH binary, while orange means two such systems are present, and white none. Note that the disruption of both BH-BH binary systems at ≈ 6 and ≈ 11 Gyr is accompanied by the ejection of the most massive black hole contained in the cluster. The disruption of one BH-BH binary is also accompanied by the ejection of one BH at ≈ 2 and ≈ 7.5 Gyr. 110 Chapter 4. Black Holes in Globular Clusters

Other examples of N-body cluster models with BHs currently exist in the literature (Mackey et al. 2007, 2008; Hurley & Mackey 2010 just to name a few), as well as Monte Carlo models of more massive globular clusters at various densities (Downing et al., 2010; Downing, 2012; Morscher et al., 2013). In addition we note that models involving mergers of stellar remnant black holes as well as well as intermediate mass BHs have been carried out in the past (e.g. Portegies Zwart & McMillan 2002; Baumgardt et al. 2004; O’Leary et al. 2006; Aarseth 2012).

4.4 Conclusions

We conclude that all three findings of Strader et al. (2012): a) the observation of two stellar remnant black holes in M22 which are b) potentially in binary systems with main sequence stars and that c) 5 − 100 stellar remnant black holes might be present in a cluster such as M22 – are in excellent agreement with our direct N-body model. A key point in this analysis is that M22 has a large core radius compared to the average for the globular cluster population of the Milky Way, and is not dynamically old. We conclude that multiple stellar remnant black holes in the core of such clusters can exist even beyond the current time. Furthermore, in the N-body model presented here we find that 1/3 of the BHs retained in the cluster upon formation remain in the core beyond the Hubble time. With further advances in computing hardware we expect these results to be confirmed by even larger models with an expanded set of initial parameters in the near future. In the meantime, several other BH candidates have been found, such as in M62 (Chomiuk et al., 2013) and in 47 Tuc (Strader et al. in prep.). 5 Conclusions

The mathematical description of these areas is partially quite complicated and is even more so, the more exact one wants to be. The exact description of the system as a whole, with the time evolution, is probably hopelessly complicated. - Sebastian von Hoerner 1960

5.1 Summary

Since each chapter of this thesis is concluded with its own summary, I present just the key results here:

• Stellar evolution is an important ingredient in star cluster evolution, and the differences in mass-loss rates and brightness in low- versus high-metallicity stars are sufficient to let globular clusters of different metallicity appear with a size difference of almost 20%, while remaining structurally identical.

• The colour of globular clusters is not dominated by the loss of low-mass stars, but by the population of evolved stars in the cluster, when all stars are considered in the analysis. While the giants are centrally concentrated owing to mass segregation, they are still present throughout the entire cluster, which results in a luminosity but not colour gradient throughout the cluster.

• From a dynamical point of view globular clusters, and those with extended cores in particular, can host a substantial black hole population. Even though black holes are ejected from clusters via few-body encounters, the evaporation is slower than previously thought. I used a direct N-body model of a cluster with initially 262 500 stars and compared it to the Milky Way cluster M22, where two black hole candidates were recently discovered.

111 112 Chapter 5. Conclusions

Each of the projects presented here was only possible by analyzing the models from a diﬀerent point of view than traditionally done: the clusters were analyzed in a parameter space including the stellar light, and not only mass.

5.2 Future research and outlook

Following up on the ﬁndings of this thesis, in combination with questions raised by observations and models carried out by others during the last few years, I have found several questions still open on the subject of the evolution of star clusters. In the next paragraphs I outline a few ideas for the future.

5.2.1 Missing link to galaxy evolution

Given that GCs live and evolve within a galaxy, they are also subject to the long-term evolution of the external tidal field: the shape and strength of this tidal field can change over time, for example via accretion of satellites or mergers with neighbouring galaxies. This implies that clusters will lose stars to the external gravitational potential at different rates, and prompts the question of if, and how, it impacts the stream of stars lost to the galaxy. Following the internal evolution of GCs within a cosmological simulation (which keeps track of the evolution of the galaxy and hence the tidal field) would require simulations covering more orders of magnitude in space (and time) than computationally possible today. Because of this, star clusters evolved with NBODY6 are living in an idealized and static external tidal field. Renaud et al. (2011) implemented a two-step approach to make the inclusion of a varying tidal field in GC simulations possible. First, the output of a simulation on a cosmological scale is used to extract the tidal tensor of a galaxy (over time). In the second step, this tidal field information is used as an input in NBODY6tt (Renaud et al., 2011), a modified version of NBODY6, to evolve a GC model. The success of this approach has been demonstrated in the case of young clusters in the merging Antennae galaxies (Renaud et al., 2011; Renaud & Gieles, 2013), and it is of great interest to repeat such a study for old GCs in the Milky Way or neighbouring galaxies. The reason for this is two-fold: First, it will enable us to study the impact that the accretion of a globular cluster (from one galaxy to another galaxy) has on its evolution. This will at the same time help answering the question of wheter or not the entire populations of globular clusters might be accreted during mergers (Tonini 2013 and references therein), and give us an insight into the merger history of our own Milky Way galaxy. Second, a wealth of data from the 5.2. Future research and outlook 113

Gaia-ESO survey1, spectroscopically mapping > 105 stars in the Galaxy, will profoundly enhance our understanding of such stars in velocity-space. Combined with astrometry from the Gaia spacecraft2, this data will be a powerful tool to better understand the life cycles of open clusters as well as the poorly understood cluster outskirts: both surveys will help us to better understand the transition region between a cluster and its host galaxy. Even then, interpreting the data will be delicate, and a better understanding of cluster dissolution processes from models will help our understanding (e.g. K¨upper et al. 2010, 2011, 2012).

5.2.2 The dynamical imprint of blue stragglers on cluster evolution

Given the extreme stellar densities of GCs compared to the solar neighborhood, mass transfer between stars, or in extreme cases even stellar collisions are common. Such processes can create blue straggler stars (BSs). This makes it possible to identify them in a colour-magnitude diagram (Sandage, 1953) as a unique product of combined stellar and dynamical evolution. Their formation mechanisms via both collision and transfer of mass via Roche-lobe overﬂow are quite well understood (Sills & Bailyn, 1999; Sills et al., 2005), as well as it is possible to determine their mode of formation from observations (Sills et al., 2000; Simunovic & Puzia, 2014). More recently, observations of the distribution of stars on the asymptotic giant branch, or BSs and binary systems in GCs in general, indicate that their distribution changes over time and might provide an independent indicator of the dynamical age of a GC (Ferraro et al. 2012; Beccari et al. 2013). As explained in this thesis, and in particular in Chapter 3, mass segregation moves more massive stars to the center, yet they can be found towards the cluster outskirts as well. At the same time, giants lose mass on short timescales. Fully understanding the change of location of evolved stars and BSs over time will providing such an independent indicator of age, which is of great importance as it is diﬃcult to determine the dynamical age of a GC through observations (De Marchi et al., 2007). Hence it will be an advantage to complement observations with dynamical models, which enable us to know the exact dynamical age of a GC at any time, as well as true three-dimensional coordinates of BSs or giants (which are only known in projection by observations).

1http://www.gaia-eso.eu/ 2http://sci.esa.int/gaia/ 114 Chapter 5. Conclusions

5.2.3 Initial conditions of star clusters

I have carried out preliminary tests evolving small star clusters of different initial density to investigate which densities lead to the longest cluster survival (see Fig. 5.1). This is a crucial factor in the rate of cluster survival and I find, in agreement with Gieles & Baumgardt (2008), that denser clusters are better protected against the forces of the external tidal field but that self-disruption via few-body encounters balances this effect. This would help understand the natural birth conditions of SCs we observe today, and more importantly, better constrain the concept of infant mortality: are most (or all) stars formed in clusters, with the majority of those disrupting over short timescales (< 10 Myr, Lada & Lada 2003), or is the fraction of stars forming in clusters overestimated (Bastian, 2011)? It will also help to better understand the dynamical states of the wealth of open clusters being detected by the VVV survey3 (Minniti et al., 2011a; Borissova et al., 2011).

Star clusters inevitably go through mass segregation during their evolution, making low-mass stars in the cluster outskirts more vulnerable to the impact of the tidal field (as mentioned in Chapters 2 and 3). This implies that it is difficult to constrain the IMF via observations today. While the work presented in this thesis used an identical IMF (following a Kroupa 2002 distribution) for different metallicities, it is currently not clear if this assumption is correct (Harris 2009; Strader et al. 2009; Zaritsky et al. 2012). Repeating the projects from this thesis with a differing IMF will help us to better understand the mass-to-light ratios of GCs.

3Vista Variables in the Via Lactea (VVV) survey, http://mwm.astro.puc.cl/mw/index.php/VVV_ Survey:About 5.2. Future research and outlook 115

Figure 5.1 Clusters with different initial half-mass radii at the beginning of the simulation (zero-age main sequence, left panel) and evolved to 10 Gyr (right panel). Initial half-mass radii are indicated in the top-right corner of each snapshot and marked with red circles. The least dense cluster is most severely affected by the tidal field and dissolved during the integration time. However the most massive cluster at this time is not the initially densest one, indicating that self-evaporation due to few-body encounters is accelerated. In this example, the cluster with an initial half-mass radius of 1.6 pc (top right corner) is most massive at 10 Gyr. 116 Chapter 5. Conclusions

5.2.4 Even larger realistic models

Creating globular cluster models that are as realistic as possible is only feasible by using state-of-the-art computing hardware. The ultimate goal over the coming years will be to create a direct model of Milky Way GCs. The direct model of the low-density clusters Palomar 14 and Palomar 4 (Zonoozi et al., 2011, 2014), and models comparable to NGC6397 and M22 (Hurley & Shara, 2012; Sippel & Hurley, 2013), bring this within reach for the ﬁrst time. Soon a direct model of M4 with almost 500 000 initial stars will become available (Heggie et al. in prep.), all of which is leading towards the ﬁrst direct N-body model of a massive Milky Way GC and will bring us one step closer to solve the million-body problem4.

GPU supercomputers are more and more frequent, making it easier to evolve realistic GC models. Observations of GCs with new instruments, such as from Gaia or MUSE5, will provide the data to better understand GCs from the faint low-mass stars in the cluster outskirts to the black hole remnants, or possible intermediate mass black holes, in the centers of GCs.

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In simulations, a definition of the core radius rc differing from the commonly used core radius from King-model fits of observations (King, 1962) has emerged. This is motivated by the fact that fifty years ago, when the first N-body simulations were carried out by von Hoerner (1960a, 1963), the particle number was limited to only N = 25 equal mass particles

(mi = m), implying that the cluster was neither symmetrical, nor had a stable center. It follows then that the centre of mass could deviate significantly over a short period of time, hence the cluster centre had to be determined thoroughly at each time step. von Hoerner (1960a) first introduced the idea of a density weighted centre in N-body simulations. This has been further developed by Casertano & Hut (1985) and Aarseth (2003). Here I give a summary. von Hoerner (1963) chose to define the cluster centre as the position with highest local density. Let ~ri be a star’s position and ri,3 the distance to its third closest neighbour. For each star, a local density ρi,3 up to the third-closest neighbor is assigned, where

3 3.5 m ρi,3 = 3 (A.1) 4π ri,3

(see Fig. A.1). The total mass 3.5 m within this sphere is due to the fact that the central star i within the sphere, as well as the ﬁrst and second closest stars are counted, plus half the mass of the third closest star at distance ri,3 (note that all stars have the same mass).

The density centre ~rd = (xc, yc, zc) of the cluster is now deﬁned by the density-weighted average of the coordinates of all stars, deﬁned as

P ρ (x , y , z ) (x , y , z ) = i,3 i i i (A.2) c c c PN i=1 ρi,3

(Eq. 5 and 6 in von Hoerner 1963). This new density centre is deviating by dc = p 2 2 2 xc + yc + zc from the original zero point. Finally, a density radius Rρ, which is the

131 132 Appendix A. N-body core radius

2 1 i

r3 3

Figure A.1 Two dimensional projection of the sphere used to estimate the local density for star i (adapted from von Hoerner 1963) .

average distance of each star to the density centre ~rd weighted by the local density ρi,3, is calculated as: P ρi,3|~ri − ~rd| Rρ = P . (A.3) ρi,3 This idea has been slightly modified by Casertano & Hut (1985) to adjust for larger N and to achieve higher accuracy without an unnecessary increase in computation time. First, the definition of the local density (Eq. A.1) is adjusted such that it is now calculated to the 6th closest star at a distance ri,6. Second, the individual masses of the stars were not required to be equal anymore, which implies that Eq. A.1 is modified as

P5 3 k=1 mk ρi,6 = 3 (A.4) 4π ri,6 where the mass mi of the central star, as well as the mass of the 6th star m6, are no longer included. Two additional adjustments are made by Aarseth (2003) to avoid extensive computational cost with growing N. First, the individual components in Eq. A.3, which are ∝ N 2, are not evaluated for all N particles anymore but just for the innermost N/5 cluster members, using a reduced membership N∗ = N/5. This density radius determination converges to the same result as including the outer particles, and has been found to agree particularly well when squaring the individual components in Eq. A.3. This new density radius Rρ was thus renamed as the core radius

1/2 "PN∗ 2 # i=1 (ρi,6|~ri − ~rd|) rc = (A.5) PN∗ 2 i=1 ρi,6 as described in Eq. 15.4 of Aarseth (2003).