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A multi-mass velocity dispersion model of 47 Tucanae : no evidence for an intermediate-mass black hole Mann, Christopher Rhys 2018

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A Multi-mass Velocity DispersionModel of 47 TucanaeNo Evidence for an Intermediate-Mass Black HolebyChristopher Rhys MannB.A., The University of British Columbia, 2013B.Sc., The University of British Columbia, 2016A THESIS SUBMITTED IN PARTIAL FULFILLMENT OFTHE REQUIREMENTS FOR THE DEGREE OFMASTER OF SCIENCEinThe Faculty of Graduate and Postdoctoral Studies(Astronomy)THE UNIVERSITY OF BRITISH COLUMBIA(Vancouver)August 2018c© Christopher Rhys Mann 2018The following individuals certify that they have read, and recommend tothe Faculty of Graduate and Postdoctoral Studies for acceptance, a thesisentitled:A Multi-mass Velocity Dispersion Model of 47 Tucanae:No Evidence for an Intermediate-Mass Black Holesubmitted by Christopher Mann in partial fulfillment of the requirementsfor the degree of Master of Sciencein AstronomyExamining Committee:Harvey Richer, Professor of AstronomySupervisorJeremy Heyl, Professor of AstronomySupervisory Committee MemberiiAbstractIn this thesis we analyze stellar proper motions in the core of the globularcluster 47 Tucanae to explore the possibility of an intermediate-mass blackhole (IMBH) influence on the stellar dynamics. Our use of short-wavelengthphotometry affords us an unprecedentedly clear view of stellar motions intothe very centre of the crowded core, yielding proper motions for >50,000stars in the central 2 arcminutes. We model the velocity dispersion pro-file of the cluster using an isotropic Jeans model. The density distributionis taken as a central IMBH point mass added to a combination of Kingprofiles. We individually model the general low-mass cluster objects (mainsequence/giant stars), as well as the concentrated populations of heavy bi-nary systems and dark stellar remnants. Using un-binned likelihood model-fitting, we find that the inclusion of the concentrated populations in ourmodel plays a crucial role in fitting for an IMBH mass. Taking into accountall of these cluster sub-populations our model predicts an IMBH to clustermass ratio of 0.06% ± 0.13%. The concentrated binaries and stellar-massblack holes produce a sufficient enhancement to the velocity dispersion sig-nal in the core as to make an IMBH unnecessary to fit the observations. Weadditionally determine that a stellar-mass black hole retention fraction of&18% becomes incompatible with our kinematic observations for 47 Tuc.iiiLay SummaryA black hole occurs when an enormous amount of material is crushed into avery compact region. The gravity becomes so strong that not even light canescape. There is ample evidence for black holes of two classes: stellar massblack holes are a few times the mass of our sun, and supermassive blackholes are millions to billions of times heavier. Intermediate-mass black holes(IMBHs) seem to be missing from the picture.I am looking carefully at the stellar motions in a star cluster to deter-mine if such an IMBH is hidden there. I build a theoretical model of howfast objects in the core should be moving according to their gravitationalinteractions. If we observe the core objects moving faster than predicted,this could be evidence of an invisible IMBH. However, my findings suggeststhat there is no need to assume an IMBH in this cluster.ivPrefaceA streamlined version of this manuscript has been submitted for publicationin the Astrophysical Journal. The same article has been uploaded to theArXiv database. I was the primary author on the paper, and carried out allof the writing of both the journal submissions and this thesis manuscript.Portions of the introduction chapter are adapted from material frommy undergraduate thesis titled “Searching for an Intermediate-Mass BlackHole in 47-Tucanae using Kinematic Analysis” (2016) and completed at theUniversity of British Columbia.The entire project was conducted within the context of a research teamwho have been listed as co-authors on the publication manuscript. HarveyRicher, as my Master’s supervisor, was involved in the project from be-ginning to end, helping to guide the project in a direction that asked andanswered the most important scientific questions. Jeremy Heyl acted asa co-supervisor and technical consultant when designing the analysis codeand formulating the cluster models. Jay Anderson and Jason Kalirai cre-ated the photometric catalogues from which we derive stellar proper motions(Section 2.1), and completeness calculations carried out by Ryan Goldsburyas described in Goldsbury et al. (2016). Ilaria Caiazzo provided me withthe stellar isochrones used in characterizing the binary stars (Section 2.2).Swantje Mo¨hle and Alan Knee assisted in the initial stages of the project,helping to develop early versions of the code that would eventually modelthe dispersion profile. Holger Baumgardt provided me with access to hisN-body simulations of star clusters with and without central intermediatemass black holes.Figure 1.3 is reproduced with permission from its original authors andcopyright holder. Figure 1.4 needed to be removed for copyright restrictions.Permission for the use of this figure was granted by its original authors, buta substantial fee was required by the copyright holder. A description of thefigure and a link to the online source is provided.vTable of ContentsAbstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iiiLay Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ivPreface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vTable of Contents . . . . . . . . . . . . . . . . . . . . . . . . . . . . viList of Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viiiList of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ixAcknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . xDedication . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xi1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.1 Overview of Manuscript . . . . . . . . . . . . . . . . . . . . . 11.2 Globular Clusters . . . . . . . . . . . . . . . . . . . . . . . . 21.3 Globular Cluster 47 Tucanae . . . . . . . . . . . . . . . . . . 51.4 Black Holes: Stellar-Mass . . . . . . . . . . . . . . . . . . . . 71.5 Black Holes: Supermassive . . . . . . . . . . . . . . . . . . . 81.6 Black Holes: Intermediate-Mass . . . . . . . . . . . . . . . . 91.6.1 Background . . . . . . . . . . . . . . . . . . . . . . . 91.6.2 Proposed Methods of Formation . . . . . . . . . . . . 101.6.3 Methods of detection . . . . . . . . . . . . . . . . . . 131.6.4 Findings and controversies . . . . . . . . . . . . . . . 162 Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 192.1 Proper Motions . . . . . . . . . . . . . . . . . . . . . . . . . 192.2 Binary Population . . . . . . . . . . . . . . . . . . . . . . . . 202.3 Stellar Remnants . . . . . . . . . . . . . . . . . . . . . . . . 22viTable of Contents3 Velocity Dispersion Model . . . . . . . . . . . . . . . . . . . . 253.1 Jeans Model . . . . . . . . . . . . . . . . . . . . . . . . . . . 253.2 Treatment of Core Parameter a . . . . . . . . . . . . . . . . 263.3 Inclusion of Binary Populations . . . . . . . . . . . . . . . . 293.4 Inclusion of Dark Stellar Remnants . . . . . . . . . . . . . . 313.5 Binned vs. Un-binned Analysis . . . . . . . . . . . . . . . . . 324 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 354.1 Model Fitting Results . . . . . . . . . . . . . . . . . . . . . . 354.2 Re-sampling . . . . . . . . . . . . . . . . . . . . . . . . . . . 365 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 395.1 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 395.2 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45viiList of Tables1.1 Recent IMBH findings/limits . . . . . . . . . . . . . . . . . . 185.1 Exploring Retention Fraction . . . . . . . . . . . . . . . . . . 40viiiList of Figures1.1 Sample CMD of 47 Tuc describing features. . . . . . . . . . . 41.2 Map of the sky showing 47 Tuc and Galaxy. . . . . . . . . . . 71.3 Galaxy-BH empirical relations. . . . . . . . . . . . . . . . . . 101.4 Formation channels of SMBHs and IMBHs (removed due tocopyright restrictions). . . . . . . . . . . . . . . . . . . . . . . 132.1 CMD of 47 Tuc in visual/infrared filters to identify binarysystems. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 212.2 Initial mass function used to estimate dark stellar remnantcounts. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 243.1 Mean mass of stars within a given core distance. . . . . . . . 283.2 Completeness values for stars in our data set. . . . . . . . . . 293.3 Variability of IMBH mass fitting parameter with choice ofbinning scheme. . . . . . . . . . . . . . . . . . . . . . . . . . . 334.1 Velocity dispersion model curves fit to the proper motion data. 364.2 Log-likelihood values of data re-sampling process to test qual-ity of model. . . . . . . . . . . . . . . . . . . . . . . . . . . . 385.1 Cumulative distribution of stars in 47 Tuc’s core. No cuspdetected. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42ixAcknowledgementsI would like to acknowledge several people for their help during my Master’sprogram. To Harvey Richer who always knows the right science questionsto ask and has helped steer this work in the right direction. To Jeremy Heylwho has been a wellspring of technical advice, helping me work out a paththrough the various challenges I came across. To Javiera Parada and IlariaCaiazzo for being around to answer questions big and small as the projecttook shape and I learned how to be a graduate student.From external institutions, I greatly appreciate the thoughtful advicefrom Laura Watkins who initially suggested looking at multiple, differentlydistributed populations. This idea was not included in the initial concep-tion of the project, but became a central tenet as it evolved. Holger Baum-gardt had endless patience as I attempted many, many tests on his N-bodysimulations, asking for details and clarifications. Though not used in thefinal results of the research, a significant portion of the 2 year duration ofthis project involved his simulations. Finally, a big thank you to VincentHe´nault-Brunet for making the visit out to UBC and sharing his knowledgeof dynamics and mass modeling.xDedicationTo Tara.You told me you better have a ring on your finger before my Ph.D. I’m gladyou did.xiChapter 1Introduction1.1 Overview of ManuscriptIn recent years there have been increasing efforts to find and constrain themasses of a particular class of black hole (BH): the intermediate-mass blackhole (IMBH). Their very existence is fraught with controversy, largely dueto the difficulty of detecting a unique and unambiguous signature either indynamical or emission studies.Much of the motivation for the search for IMBHs arises from the ob-served empirical relationships between massive galaxies and their residentsupermassive black holes that may extend to globular clusters in some form.The fact that most proposed IMBH formation mechanisms require the densestellar environments found only in the cores of globular clusters lends en-couragement to the search in these regions. Confirmation of the existenceof IMBHs would help to illuminate unanswered questions about the forma-tion and growth histories of supermassive black holes found in galaxy cores.Overall, the motivation to confirm IMBHs is strong, while the observationalevidence to support them is still inconclusive.In the course of this research, we conduct a kinematic search for thegravitational influence of a possible IMBH in the core of the globular cluster47 Tucanae (47 Tuc). We use proper motion analysis and a multi-massvelocity dispersion model to determine whether an IMBH is required toexplain the observed stellar motions. However, before discussing details ofthe analysis and findings, we lay some groundwork by describing conceptsand objects that are integral to the study and help put it into perspective.The remainder of Chapter 1 discusses the astronomical objects involvedin the project. It gives a general overview of globular clusters, and then aparticular description of the specific target: 47 Tuc. The chapter continueswith a discussion of the different classes of BHs, and where IMBHs fit inthe picture. Mass limits from some of the more recent IMBH studies arepresented, as well as some of the controversies involved. Chapter 2 describesthe data sets used in the analysis, and explains how the current proper mo-tion data allow for a much better view of the cluster core than has been11.2. Globular Clusterspreviously possible. It also shows how we estimate the populations of heav-ier objects from the data available. Chapter 3 contains most of the analysiscarried out in this study and it steps through how we build the velocity dis-persion model by considering populations of objects with different massesand distributions. It continues by explaining our use of an un-binned like-lihood fitting method to match the best model to the proper motion data.Our final results are presented in Chapter 4 including a re-sampling tech-nique used to verify the quality of our model. Finally Chapter 5 provides asummary and discussion of our entire process and findings.1.2 Globular ClustersThere are two general types of star clusters that are found within galaxies:open and globular clusters. While the focus of this thesis is only on globularclusters (and only one cluster in particular), we briefly describe open clustersfor the purpose of comparison.Open clusters are collections of gravitationally-bound stars that typicallycontain a few hundred to a few thousand stars. They are quite diffuse instellar concentration which causes them to exhibit low surface brightnessand are often relatively difficult to detect. The core radius (defined as thedistance from the centre where the surface brightness drops to 50% of itsmaximum value) of these objects are typically a few parsecs (1 parsec = 3.26light years = 3.086×1016 m). Their ages are generally measured to be a fewhundreds of millions of years. In the Milky Way, they are usually found nearthe plane of the disk and often surrounded by gas and dust. Open clustersare generally only weakly bound together, and therefore tend to becometidally disrupted and spread apart as they move through spiral arms or intothe central regions of the galaxy (Sparke & Gallagher, 2007).Globular clusters present a fairly stark contrast to open clusters. Assuggested by their name, they are spherically symmetric collections of starsand exhibit a much higher stellar density than their open cluster counter-parts. Globular clusters typically contain somewhere in the range of 104–106stars. They are some of the oldest objects in the galaxy, many with agesupwards of 10 billion years. Globular clusters contain very little gas, andtheir star formation ceased long ago. Because they are not producing newstars, all of the massive blue stars have died off, leaving behind only starsbelow a certain mass. As such, globular cluster stars are generally muchredder than disk stars. As generally goes hand-in-hand with very old stellarobjects, globular clusters exhibit a very low metallicity (proportion of ele-21.2. Globular Clustersments heavier than helium). Typical metallicity values for globular clustersare 1/3 to 1/300 of the solar composition. The reason older stars are moremetal-poor is because, at their time of formation, there had not been asmany supernovae to enrich the environment with heavy elements. As timegoes on, supernovae slowly raise the metallicity of star-forming regions asthey create and distribute metals. Globular clusters are distributed spher-ically about the the galactic centre rather than being confined to the disk.Their motions do not adhere the overall disk rotation of the Milky Way, butorbit in all directions in a galactic halo similar to elliptical galaxies (Sparke& Gallagher, 2007).Globular clusters are rich targets for study as they can provide informa-tion on many distinct astronomical and astrophysical topics. As mentionedpreviously, the effects of stellar evolution are starkly evident in a globu-lar cluster. A particularly useful tool for studying stellar populations andevolution is a colour-magnitude diagram (CMD). A CMD plots the colour(temperature) against the magnitude (brightness) for a population of stars.When this is done, a feature we call the main sequence indicates where starsare still fusing hydrogen in their cores. An example of a CMD can be seen inFigure 1.1. The long diagonal swath of points are stars on the main sequence.At the top (brightest), the stars fold back into the giant branch. These arestars that have recently run out of hydrogen fuel in their cores and are evolv-ing into a different stage of their lives, becoming giants. The main sequenceis also a mass sequence. Stars in the upper left (brighter/hotter) are themost massive. As you look towards the bottom right (fainter/cooler), thestars here are less massive. When looking at a colour-magnitude diagramof a globular cluster, there is a very distinct “turnoff” between the mainsequence and giant branch due to the narrow age range of all the clustermembers. As the cluster ages, the turnoff moves down the main sequenceas higher mass stars (which evolve more rapidly than lower mass stars) peelaway into the red giant branch on the right. The turn-off locations of mainsequences for clusters of different ages and metallicities help fine-tune modelsof stellar evolution, which are in turn used to inform ages and metallicitiesof newly studied stellar populations.31.2. Globular ClustersFigure 1.1: A sample CMD of 47 Tuc. Different features are labeled. A CMDis an extremely useful tool for studying stellar evolution. A star spends themajority of its life sitting on the main sequence, but evolves onto the giantbranch once it begins to deplete its hydrogen core. The giant branch will peeldown the main sequence over time since the hotter/brighter/more massivestars evolve faster than the cooler/fainter/less massive stars. Giant stars caneventually evolve through a phase called a “planetary nebula” where theyshed their outer layers, moving across the top of the CMD to the left. Theend result of this phase is to reveal a white dwarf which will steadily coolcausing it to slowly drop down the left side of the diagram. The location ofthe turnoff is a strong age indicator of the cluster.The distribution of globular clusters in the Milky Way helps to char-acterize the structure of our galaxy and our position within it. Findingthe centre of their spherical distribution provides an independent measureof the position of the galactic centre. This method largely avoids takingmeasurements within the galactic plane where the substantial gas and dustconcentrations interfere with measurements and produce large uncertainties.Their locations also help define the galactic halo where objects orbit in adisordered fashion more akin to elliptical galaxies than our spiral. Also, the41.3. Globular Cluster 47 Tucanaeenvironments within globular clusters may be favourable to look for exoticobjects that are not (easily) found elsewhere. Recently, studies have beenshowing increasing interest in, and support for, IMBHs in the cores of someclusters. Though much of the evidence is contested and no truly conclu-sive findings have been made to date, the search for one of these objects isthe topic of this manuscript and will be expanded upon in detail in latersections.1.3 Globular Cluster 47 TucanaeThe target globular cluster of this study is 47 Tucanae, also known asNGC 104. Its position on the sky is given by RA: 00h24m6s and Dec:−72◦04′53′′ (galactic coordinates: l = 305.9◦, b = −44.9◦ (McLaughlinet al., 2006)) and it lies at a distance of 4.69 ± 0.17 kpc (Woodley et al.,2012) kpc from our location in the galaxy. It is a reasonably bright objectwith a V-band magnitude of 3.95 and colour index B−V = 0.88. Its metal-licity is found to be [Fe/H] = −0.75, indicating that its iron-to-hydrogenratio is 10−0.75 or about 0.18 times that of our Sun (Hansen et al., 2013a).Figure 1.2 shows 47 Tuc’s location on the sky in relation to the rest ofthe galaxy. It is very close to the Small Magellanic Cloud (SMC, a smallsatellite galaxy orbiting the Milky Way), thought the two are not physicallyassociated as the SMC is much farther away.There are a few ways to measure the age of a globular cluster. Themost common method is to fit the observed main sequence turnoff withan isochrone. An isochrone tells you the shape of the CMD at a givencluster age and is generated through stellar evolution models. This methodgenerally produces ages of 10–13 billion years for 47 Tuc (Hansen et al.,2013a). However, there is a degeneracy between the age and the metallicityof a cluster that partially confounds this method. An analysis of 47 Tuc’swhite dwarf cooling sequence has produced an age of 9.9± 0.7 Gyr (to 95%confidence) (Hansen et al., 2013a). This method is more robust than mainsequence fitting, because it removes the metallicity–age covariance as heavyelements present in white dwarfs rapidly sink to the interior due to thestrong gravity (Hansen et al., 2013a).47 Tuc has also shown evidence for containing multiple stellar popula-tions. When viewed in certain photometric filters, distinct main sequences(and giant branches) lie parallel to one another in the CMD. There appearsto be a majority population (∼ 70% of the total stars) and a minority pop-ulation (∼ 30%). The chemical compositions of the two populations differ51.3. Globular Cluster 47 Tucanaeslightly from one another and reveal that the minority population was likelythe first generation of cluster stars. The majority population is a later gen-eration of stars, whose parent material was enriched by the evolution of thefirst generation (Milone et al., 2012). The different populations even behavedifferently from a dynamical standpoint. In splitting the main sequenceinto colour groups, the bluer populations show the largest proper motionanisotropy and are more centrally concentrated than the other, redder pop-ulations (Richer et al., 2013).Globular clusters are dynamically evolving systems. Of particular inter-est to the project at hand, is the phenomenon of mass segregation. Whenstars fly by one another and interact gravitationally, there is a tendencyto transfer energy such that all objects have similar kinetic energies. Thiscauses low-mass stars to move outwards in large orbits, and higher-massstars to sink towards the core on smaller orbits. As will be discussed inlater sections, the concentrated populations of heavy objects may play animportant role in the central dynamics. Heyl et al. (2015) have directlymeasured the diffusion of white dwarf stars in 47 Tuc due to gravitationalrelaxation. These white dwarfs have lost a sizeable fraction of their massrecently since they evolved from their main sequence phase. This causesthem to slowly diffuse outward into a distribution more similar to objects oftheir new, lower mass. The mass segregation of 47 Tuc has been fairly welldocumented (see, for example Parada et al., 2016; Goldsbury et al., 2013).All of this goes to show that 47 Tuc and globular clusters in general areanything but simple objects. They are superb laboratories to test modelsof stellar evolution and gravitational dynamics. Its proximity and appar-ent brightness has made 47 Tucanae one of the most well-studied globularclusters. A large number of observing programs between numerous observa-tories have provided photometric and spectroscopic data for many regions ofthe cluster in several wavelength bands. These have allowed for the study ofstellar populations, distributions, and compositions. Spectroscopic data andphotometry from different epochs allow for analysis of stellar motions thecluster, and provide insight into the dynamical processes occurring within.61.4. Black Holes: Stellar-MassFigure 1.2: An infrared map of the Milky Way. Labels indicate the locationsof 47 Tucanae, the Small Magellanic Cloud (SMC), and the Large MagellanicCloud (LMC). 47 Tuc is not physically associated with the SMC, it simplylies along the same line of sight. Image credit to 2MASS (Skrutskie et al.,2006).1.4 Black Holes: Stellar-MassBefore discussing our search for an intermediate-mass black hole, we mustfirst put into perspective what “intermediate” is in relation to. A stellar-mass black hole (sBH) is the remnant object left over from the natural deathof a very massive star that has consumed all of its nuclear fuel. The for-mation process of these objects is relatively well known (Ryden & Peterson,2010). As massive stars evolve, the hydrogen in their cores is eventuallydepleted as fusion processes convert hydrogen into helium. The inert he-lium core contracts and raises temperatures high enough to begin fusion ofhydrogen in a shell around the helium core, resulting in a supergiant phase.Temperatures rise until the helium core itself begins to fuse into even heavierelements. The most massive stars have the required central temperaturesand pressures to repeat this “shell-burning” process, fusing heavier and heav-ier elements. Once the cycle reaches iron, the core ceases to produce energyas iron has the highest nuclear binding energy and fusion of these nucleirequire energy rather than produce it. Although it has ceased to produce71.5. Black Holes: Supermassiveenergy, the iron core is supported by the quantum effect of electron degen-eracy pressure. The outer shells continue to produce iron, increasing themass of the iron core. When the iron core exceeds the Chandrasekhar masslimit (∼ 1.4 M, where M is the mass of the Sun), the electron degen-eracy pressure can no longer support it and it destabilizes into a free-fall.The collapsing core is suddenly arrested as protons and electrons combineinto neutrons (and neutrinos) and neutron degeneracy pressure comes intoplay. In-falling matter rebounds off of the ultra-dense neutron core in anevent known as a supernova. Most supernova result in a central compactobject called a neutron star which is supported by the neutron degeneracypressure and exhibits an extraordinarily high density. There is, however, anupper mass limit known as the Tolman-Oppenheimer-Volkoff (TOV) limitbeyond which even neutron degeneracy pressure cannot support the neutronstar against its own gravity. Beyond the TOV limit the object collapses un-contested and forms a BH. The TOV limit is not precisely known, but isthought to be near 3 M which corresponds to a progenitor star mass of15–20 M (Bombaci, 1996; Spera & Mapelli, 2017). Stars that are bornabove this mass eventually produce BHs of several solar masses.One way to see these sBHs is to make observations of faint, massive, x-rayemitters in binary systems which have led to the classification of the “x-raybinary.” Such systems contain BHs in the mass range of ∼ 3–20 M (Mc-Clintock & Remillard, 2006; Casares, 2007). These types of detections have,until recently, been our best evidence for sBHs.Gravitational wave astronomy has since made its debut and provideddirect gravitational radiation detections of the in-spiral and merger of pairsof sBHs. Since early in 2016 the LIGO and Virgo observatories have reported5 merger events between sBHs ranging in mass from roughly 7 to 35 M(Abbott et al., 2016a,b, 2017b,c,d), as well as a merger event between twoneutron stars (Abbott et al., 2017a).1.5 Black Holes: SupermassiveSupermassive black holes (SMBHs) exist on a much grander scale. It hasbeen well established that SMBHs reside in the cores of most sizeable galax-ies, including our own Milky Way (Gillessen et al., 2009). These objectsspan the range of masses from about MBH = 106–109M. Unlike sBHs,it is not satisfactorily known how and when these gargantuan objects formand/or grow, though they appear to be a common feature of massive galax-ies. Uncertainty in the origin and evolution of these objects is part of the81.6. Black Holes: Intermediate-Massmotivation for the search for IMBHs. There are theoretical limits that canbe placed on the rate at which a BH can grow via accretion. For instance,the Eddington luminosity is reached when the radiation pressure generatedby the hot accreting material is strong enough to prevent the accretion ratefrom growing, thus limiting its own luminosity. The concept was first en-visaged for stellar luminosities, but has been increasingly applied to BHs.The Eddington limit becomes more complicated for the non-spherical ge-ometries of an accretion disk, but it is still difficult to explain how a sBHcould grow to billions of solar masses as early as we observe SMBHs in highredshift quasar studies (Heinzeller et al., 2007). A proposed solution is theassumption of IMBH seeds that then accrete at the Eddington limit (Lupiet al., 2016). Such objects, should they exist, provide a formation avenuefor SMBHs as well as bridge the enormous mass gap in the observed BHmass function.1.6 Black Holes: Intermediate-Mass1.6.1 BackgroundIMBHs are proposed to occupy the mass range between SMBHs and sBHs,roughly 102–105M. One of the many open questions surrounding IMBHs iswhether or not they follow the same empirical bulge-BH relations observedfor galactic supermassive BHs. These power-law relationships (MBH ∝ xβ)have been determined for x = σc, Mtot, Ltot (central velocity dispersion,cluster mass, cluster luminosity), where β is the best fit power law for eachquantity. Values for these power law relationships are measured to fall some-where near βσ ≈ 2.8, βM ≈ 0.8, and βL ≈ 0.9 (Lu¨tzgendorf et al., 2013;Ferrarese & Merritt, 2000). The reported values vary somewhat dependingon the fitting routines that were used to determine them, but the correlationis generally strong. By extrapolating any of these relationships down to thescale of a globular cluster using characteristic values for σc, Mtot, and Ltot,one could predict the presence of a black hole in the IMBH mass range (Lu& Kong, 2011) (see Figure 1.3).Such correlations suggest commensurate evolution and/or feedback be-tween the BH and its surroundings. IMBHs may follow the same or similarrelations with the properties of their host globular clusters. Similarities anddiscrepancies between the bulge-BH relations of SMBHs versus IMBHs mayshed light on how IMBHs themselves form, and if/how they are involved inthe growth of SMBHs (Mezcua, 2017).91.6. Black Holes: Intermediate-MassFigure 1.3: These plots show three galaxy-BH empirical relationships. It isnot totally clear yet if IMBHs will follow the same relations as the galac-tic SMBHs. Figure reproduced from the original version with permission(Figure 2 of Lu¨tzgendorf et al., 2013).1.6.2 Proposed Methods of FormationSeveral formation theories for IMBHs have been proposed, though none arewithout criticism. One major issue in explaining a SMBH is reconcilingthe growth rate required to reach a given mass by a certain age with aphysically plausible growth model. For BHs of mass . 103 M, growth viaaccretion of interstellar medium is too slow to be effective. More promising,is accretion of other stellar objects while in a crowded environment suchas a cluster core (Miller & Hamilton, 2002). Due to mass segregation, a101.6. Black Holes: Intermediate-MasssBH (heavier than most stellar objects) will sink into the core of the cluster,forming or joining binary systems. With multiple 3-body interactions theBH binary will tighten and lead to disruption and merger events, but therecoils of these interactions will eventually eject the BH from the clusterentirely (Sigurdsson & Hernquist, 1993). However, if the initial BH seedwas & 50 M, it would not be imparted enough recoil velocity to escape atypical cluster and could continue to grow indefinitely (Miller & Hamilton,2002). Many formation theories focus on how to generate this & 50 M seedBH.The simplest explanation may be that such a BH seed is the naturalproduct of the evolution and collapse of a very massive star. Numericalstudies of stars with mass & 30–40 M show that they may not detonatewith sufficient energy to eject the outer layers, thus retaining most of themain-sequence mass for the generation of the BH (Fryer, 1999). These verymassive stars may produce BHs close to the 50 M requirement.A variant on this same idea is the merger of massive stars in the coreof the cluster. If these stars sink to the core before evolving off the mainsequence, they may combine with one another creating even more massivestars (Portegies Zwart et al., 2004). These would collapse to form sizeableBHs that may also merge with each other or nearby stars and continue togrow.A third proposed mechanism for forming a & 50 M seed is throughthe consecutive mergers of smaller BHs. If 100’s to 1000’s of 10 M BHsform in the cluster and begin to sink to the core, interactions can occur thatmerge a minority of them to ∼ 20 M objects while the rest are ejected asmentioned previously. Some small few of these 20 M objects may mergeto beyond the ∼ 50 M threshold. While not impossible, this theory maybe less likely than the previous two as the majority of BHs are kicked fromthe cluster (Miller & Hamilton, 2002). There is a general consensus thatthe fraction of BHs that are ejected from globular clusters is fairly large,though there is a wide spread of estimates on this value. This will proveto be important for the study at hand and the retention fraction will bediscussed further in Section 2.3.It has been considered that IMBHs may form fairly instantaneously,rather than be built up over time. Early core collapse of the cluster isone proposed mechanism where this could occur (Quinlan & Shapiro, 1990;Gurkan et al., 2004; Freitag et al., 2006). In this process, massive stars sinkto the centre of the cluster early on, causing continuous and rapid mergersand disruptions. The violent interactions could result in a “very massivestar” of enormous proportion (several hundred M). The evolution of such111.6. Black Holes: Intermediate-Massan object is not well understood (Glebbeek et al., 2009), but it could possiblycollapse and directly form the IMBH (Konstantinidis et al., 2013). Over thenext few million years, accretion could grow the mass by nearly 100-fold(Vesperini et al., 2010).Finally, population III (Pop. III) stars (the first generation of stars pro-duced in the universe) have been proposed as possible progenitors of IMBHs(van der Marel, 2004; Whalen & Fryer, 2012; Madau & Rees, 2001). Pop. IIIstellar composition is completely absent of heavy elements, as their forma-tion predates any supernovae that would have produced such metals. Theextremely low metallicity of this first generation of stars allows them to forminto very massive objects (& 200 M) (Bromm et al., 1999). While the evo-lution of such stars is still poorly understood, the expected end productsof such stars are BHs of mass & 250 M (Heger & Woosley, 2002; Larson,2000; Schneider et al., 2000).IMBHs have been predicted for several decades (Wyller, 1970; Bahcall &Ostriker, 1975; Frank & Rees, 1976). These early considerations of IMBHspostulated methods by which they might be detected, though the observingtechnology at the time was insufficient to really test the predictions. De-spite devoted effort over the years, there is still no uncontested proof of theirexistence (Konstantinidis et al., 2013; Mezcua, 2017). However, studies andmeasurements have become detailed enough in recent years that ever-stricterlimits are being placed on the possible masses of globular cluster IMBHs (seeTable 1.1 for a list of recent IMBH limits and claims). Figure 1.4 diagram-matically shows how IMBHs might have played a role in the evolutionaryhistory of SMBHs, and how some of them may be leftover today.121.6. Black Holes: Intermediate-MassFigure 1.4 has been removed due to copyright restrictions.It was a schematic diagram of several proposed evolutionary historiesof SMBHs. The y-axis spanned time and the x-axis spanned BH mass.It displayed how differently formed IMBH seeds would form and growover time, producing the SMBHs we see today. The processes wouldallow some IMBHs to survive to the modern era.Original figure: (Figure 1 of Mezcua, 2017)(http://adsabs.harvard.edu/abs/2017IJMPD..2630021M)Figure 1.4: Schematic of some proposed formation channels of IMBHs andtheir involvement in the growth of SMBHs. On the y-axis, redshift is aproxy for time. The top of the plot is early in the universe’s history whilethe bottom is present day. The evolutionary processes of SMBHs are notexpected to have 100% efficiency, and therefore “leak” some IMBHs to thepresent day.1.6.3 Methods of detectionRegardless of how they form, several methods of detecting IMBHs in galaxiesand clusters have been developed. One such method is to measure and mapultra-luminous x-ray sources near the centres of clusters. A BH activelyaccreting material from a stellar neighbour or the local interstellar mediumcan become a strong x-ray and radio emitter. The power detected from thesex-ray sources can be upward of 2.8× 108L (Godet et al., 2011) (L is theluminosity of the Sun). Such high luminosities can be explained by super-131.6. Black Holes: Intermediate-MassEddington accretion onto a BH. The detected flux combined with accretionmodels place an upper limit on the mass of the accreting BH. This methodis frequently used to constrain sBHs in observed x-ray binaries as well asfor galactic SMBHs. The technique can be applied to IMBHs using the BHaccretion fundamental plane, an empirical correlation between an accretingBH’s radio flux, x-ray flux, and mass (Merloni et al., 2003). However, thisrelation has not yet been explicitly confirmed to continue into the IMBHmass regime. Many studies have made claims for IMBHs using these ultra-luminous x-ray sources (Binder et al., 2011; Sutton et al., 2012; Nyland et al.,2012), but as of yet, none of the evidence is indisputable (Konstantinidiset al., 2013).Our ability to look for the radio signatures of IMBHs in globular clusterswill be dramatically expanded when the Next Generation Very Large Arraycomes online in roughly a decade. A proposal by Wrobel et al. (2018) out-lines a plan to measure radio fluxes of many hundreds of globular clustersout to a distance of 25 Mpc. Reaching large numbers of globular clustersis important as it may be the case that only a small fraction of clusters arelikely to retain an IMBH even if they are commonly born with one (Fragioneet al., 2018).Kinematic study of stellar clusters is a common alternative approachfor central IMBH detection. One way to probe the gravitational environ-ment is to examine pulsar accelerations using measurements of their variousspin derivatives (e.g. Kızıltan et al., 2017; Gieles et al., 2018; Freire et al.,2017). There are, however, confounding factors such as the intrinsic spin-down of the pulsars and interactions with neighbouring stars. Of particularinterest to my thesis research are recent findings for 47 Tuc. Kızıltan et al.(2017) determined accelerations of millisecond pulsars (MSPs) in 47 Tuc andcompared them against N-body simulations with different masses of cen-tral IMBHs. Their analysis suggests that an IMBH of mass 2300+1500−850 M(0.30%+0.20%−0.12% of their total cluster mass) may be present in order to ex-plain their observed pulsar accelerations. We would like to note a majordeficiency with certain aspects of their method. The simulations used forcomparison with their pulsar data contained no significant mass in binaries.They included no primordial binary systems, leaving only those systemsthat formed dynamically. In a cluster with a short central relaxation timesuch as 47 Tuc, the binaries and other massive objects will be centrally con-centrated. As we demonstrate below, the presence of a concentrated andsubstantial mass distribution alone can mimic the dynamical effect of anIMBH, to say nothing of the dynamical heat source hardening binaries also141.6. Black Holes: Intermediate-Massprovide to the cluster in general. Including binaries is crucial for a clusterlike 47 Tuc whose binary mass fraction exceeds a few percent of the entirecluster mass. Kızıltan et al. (2017) also underestimated the neutron starmasses in their simulation, which they took as the canonical 1.4M (Baum-gardt, private correspondence). Most of the millisecond pulsars in 47 Tucare part of binary systems (15 of the known 25, see Freire et al. (2017)) andthese systems tend to have total dynamical masses upwards of 2 M. Thismass difference will affect their distribution in the cluster and must be takeninto account.Freire et al. (2017) conducted their own analysis of the 47 Tuc MSPs.In this case they used higher order derivatives of the MSP periods to findthe jerk as well as the accelerations experienced by the pulsars. They addi-tionally used a somewhat larger distance (4.69 kpc) than the Kızıltan et al.(2017) group which is more in line with recent assessments (e.g. Woodleyet al., 2012; Chen et al., 2018; Hansen et al., 2013b; Brogaard et al., 2017;Bogdanov et al., 2016). In their analysis they were able to account for theMSP kinematics without needing to invoke an IMBH in the cluster core.Perera et al. (2017) claim the timing solutions of a particular MSP inanother globular cluster (NGC 6624) indicates that it orbits a very largeIMBH (∼ 60, 000 M) on a highly eccentric orbit. Similarly, a study byPeuten et al. (2014) found some large negative period derivatives of NGC6624 accreting pulsars which led to the conclusion of non-luminous mass af-fecting the system. They consider the possibility of a > 19, 000 M IMBH,but favour the more likely scenario of a sizeable dark stellar remnant popula-tion driving up the mass-to-light ratio. In response, Gieles et al. (2018) hasemployed a more sophisticated multi-mass model of the cluster and foundthe pulsar accelerations to be easily explained without an IMBH. As willbe discussed in later sections of this manuscript, the proper choice for thecluster modelling can have a dramatic effect.Another kinematic approach to detecting an IMBH is to measure andmodel the velocity dispersion of stars in the core of the globular cluster.The dispersion is a measure of the spread in either proper motion or line-of-sight velocities at a given distance from the core. The shape of the velocitydispersion profile is determined by the underlying mass distribution and canbe used to probe for unseen massive objects. This technique is commonlyused in measuring galactic supermassive BHs (Chatzopoulos et al., 2015;Pagotto et al., 2017; Ahn et al., 2017; Walsh et al., 2013). For galacticstudies individual stars generally cannot be resolved and so gas-dynamicalmodels and spectral line widths are used to infer velocity dispersions. Similartechniques have produced upper limits for IMBHs in globular clusters where151.6. Black Holes: Intermediate-Massindividual stars can be measured either in line-of-sight velocity or propermotion. However, individual stellar measurements are typically hinderedby the crowding in the core, which is highly problematic as the core isthe region where an IMBH’s gravitational influence will be most prevalent.Using integrated-light spectroscopy, one can try to work around the visibilityissue in the crowded globular cluster cores by looking at the Doppler spreadof spectral lines. This method can be very useful, but also comes comes withits own limitations (see e.g. de Vita et al., 2017). From the IMBH findings inTable 1.1 it is clear that there is a wide range of estimates, even for studieswith the same target.As was introduced above, the unfolding era of gravitational astronomyis providing another avenue for the detection of BHs. In a very similarmanner to the already detected mergers between sBHs, it is expected thatadvanced gravitational wave detectors (e.g. Advanced LIGO and AdvancedVirgo, LISA) could make the first unambiguous confirmation of an IMBH ifone were to undergo what is termed an intermediate mass-ratio coalescence(IMRAC). With this next generation of instruments, a suitable IMRACshould be detectable for an IMBH with a mass of MBH & 100 M out toredshifts of z & 0.5 at 95% confidence (Haster et al., 2016). Searching thecurrent data from LIGO and Virgo for signs of a merging IMBH binary hasyielded no detections but was able place constraints on the occurrence rateof many classes of such events (Abbott et al., 2017e).1.6.4 Findings and controversiesEither from emission or kinematic studies, many clusters have had estimatesand upper mass limits placed on any potential IMBHs hidden in their cores,and there are often wide discrepancies between estimates. For emissionstudies there are numerous factors that contribute to this: unknowns inaccretion phenomena, local environmental variables, uncertainties in thefundamental plane relation, and beaming geometry to name a few. As thisthesis is a kinematic study, we will cover some of the controversies found indynamical research.Dynamical analysis studies suffer from their own discrepancies. For ex-ample, Noyola et al. 2008 conducted a velocity study on the globular clusterω Centauri using doppler shifts in stellar spectra to infer line-of-sight veloc-ities. They report finding a rise in the velocity dispersion profile towardsthe core consistent with a central 4 × 104 M IMBH. Anderson & van derMarel 2010 carry out a similar analysis using proper motions gathered fromHST data, but did not observe same signature. An additional study (van der161.6. Black Holes: Intermediate-MassMarel & Anderson 2010) placed an upper limit of only 1.2× 104 M on anycentral BH. The findings for IMBHs in other globular clusters are simi-larly uncertain, though ω Centauri is one of the most promising candidates.Lu¨tzgendorf et al. (2015) and Lanzoni et al. (2013) independently measuredifferent central velocity dispersion profiles for NGC 6388 and come to verydifferent conclusions regarding the mass of a central IMBH. Perera et al.(2017) and Gieles et al. (2018) draw different conclusions regarding the mil-lisecond pulsars in NGC 6624 with the former requiring a 60, 000 M IMBHto fit their model and the latter needing no IMBH when using a more sophis-ticated multi-mass cluster model. 47 Tuc is no exception to the controversy.Kızıltan et al. (2017) recently claimed a 2300 M IMBH whereas my ownanalysis presented here as well as that of Freire et al. (2017) find no evidenceto support an IMBH. For a comprehensive review of the history, findings,and background of IMBHs, see Mezcua (2017).Throughout this paper we will be referring to different components andsub-populations of 47 Tuc with the following abbreviations in the text andquantity subscripts: intermediate-mass black hole (IMBH), low-mass clusterobjects (Cl), binaries (bin), white dwarfs (WD), neutron stars (NS), andstellar-mass black holes (sBH).171.6. Black Holes: Intermediate-MassTable 1.1: Recent IMBH findings/limitsTarget Method MIMBH[M]ref.ω Cen kinematic <12,000 (1)ω Cen kinematic 40,000 (2)ω Cen kinematic lower thanref. (2)†(3)NGC 6388 kinematic <2000 (4)NGC 6388 kinematic 28,000 (5)NGC 6388 x-ray 1500 (6)NGC 6388 x-ray <600*<1200*(7)47 Tuc Pulsar timing 2300 (8)47 Tuc Pulsar timing 0 (9)NGC 6535 comparativesimulationpresence‡ (10)ULX-7 (M51) x-ray <1600*<35,000*(11)M15 Radio VLBI <500 (12)Molec. cloudCO-0.40-0.22Radio, gas dy-namics100,000 (13)NGC 6624 Pulsar timing >7500 (14)NGC 6624 Pulsar timing 0 (15)G1 (M31) kinematic 17,000 (16)G1 (M31) comparativesimulation0 (17)Sources: (1) van der Marel & Anderson (2010), (2) Noyola et al. (2008), (3)Zocchi et al. (2017), (4) Lanzoni et al. (2013), (5) Lu¨tzgendorf et al. (2015),(6) Cseh et al. (2010), (7) Bozzo et al. (2011), (8) Kızıltan et al. (2017), (9)Freire et al. (2017), (10) Askar et al. (2017), (11) Earnshaw et al. (2016),(12) Kirsten & Vlemmings (2012), (13) Oka et al. (2017), (14) Perera et al.(2017), (15) Gieles et al. (2018), (16) Gebhardt et al. (2005), (17) Baum-gardt et al. (2003).† Anisotropy considerations are unquantified, but suggest reference (2) esti-mate is too high.* These estimates reflect different techniques and assumptions within thereferenced study.‡ Comparison with simulations suggest the presence of an IMBH, but thereare no reported constraints on mass.18Chapter 2Data2.1 Proper MotionsOur proper motion data come from two epochs of Hubble Space Telescope(HST) observations. HST Program GO-12971 (PI: Richer) observed 47 Tucover 10 orbits early in 2013, imaging the core in the F225W and F336Wfilters using Wide Field Camera 3. These data were used in conjunctionwith those of an earlier epoch; GO-9443 (PI: King) imaged the core in 2002using the ACS High-Resolution Channel in the F475W filter. We match thestars between the F336W and F475W images, providing a baseline of ∼ 11years over which to calculate proper motions.It is worth taking a moment to note the benefits of using such blue/ultravioletfilters. 47 Tuc has historically been imaged at longer wavelengths and, insuch filters, visibility in the core is hampered both by crowding and by thedominating brightness of the many red giant stars located there. Thesegiants saturate a portion of the detector in any moderately deep image,bleeding outwards and reducing the ability to accurately measure their stel-lar neighbours. Our short wavelength filters act to suppress the light fromthe cool red giants and allow for an unimpeded deep exposure right in thecluster’s core. The shorter wavelengths also improve the diffraction limitof every star in the frame, reducing the degree of crowding. For a de-tailed description of the observations and completeness correction process,see Goldsbury et al. (2016).As a result of this extremely clear view into the core, we achieve qualityproper motion measurements of over 50,000 stars including 12 within thecentral arcsecond, and 100 within the central three arcseconds. Measure-ments this close to the cluster core have been extremely difficult in the pastand yet these are exactly the stars that probe the region of greatest potentialIMBH influence. A dynamical estimate (Peebles, 1972) sets the BH’s radiusof influence (rI) to berI =GMIMBHσ2c, (2.1)where σc is the central velocity dispersion of the cluster. Adopting a distance192.2. Binary Populationto 47 Tuc of 4.69 kpc (Woodley et al., 2012) and measuring the centralvelocity dispersion to be σc ≈ 15 kms−1, we find the sphere of influence ofa 2000 M BH in 47 Tuc to have a radius of rI ≈ 1.7′′. It is the central fewarcseconds where an IMBH has its most pronounced effect, so visibility inthis region is critical.2.2 Binary PopulationWhen looking for a dispersion signature in the core, it is important to sep-arately account for populations of dynamically massive objects. Relaxationeffects cause more massive objects to concentrate in the core while the lessmassive ones diffuse outward. This segregation will prove to be importantin the context of velocity dispersion and will be discussed in later sections.For the purpose of our analysis, we make the distinction between high-mass(& 1.0 M) and low-mass (. 1.0 M) cluster objects. For the low-masscategory we group together the low-mass binaries, low-mass WDs, main se-quence stars, and giants as a single composite population. In the high-masscategory we individually model the populations of heavy binary systems,heavy WDs, NSs, and sBHs. These higher-mass objects are more concen-trated than the typical cluster object and have the potential to affect thevelocity dispersion deep into the core, mimicking the effects of an IMBH.202.2. Binary Population0.4 0.6 0.8 1.0 1.2 1.4F606W-F814W1718192021222324F606W0.86 M¯0.86 M¯0.85 M¯0.80 M¯0.70 M¯0.60 M¯0.50 M¯0.40 M¯0.30 M¯Fiducialflow=0.0265fmid=0.0098fhigh=0.0039Main Sequence   0.5<q<0.7   0.7<q<0.9      q>0.9Figure 2.1: A colour-magnitude diagram of 47 Tuc stars in visual/infraredfilters. In these filters the binary population is adequately separated fromthe main sequence. We use a fiducial main sequence line and isochrones toestimate where the binary sequence would lie for different mass ratio (q)values. Binaries with q & 0.5 are distinct from the main sequence. Tomeasure the binary fraction in each q range, we draw a box bounded bythese isochrones and compare binary counts against main sequence starswithin a certain colour range.First we make an observationally motivated estimate of the binary pop-ulation in 47 Tuc. On a CMD the binary sequence lies above the mainsequence (brighter and redder), and the extent of this offset depends on themass ratio (q) of the binary pair. A system with mass ratio of q = 1 has twoidentical stars and simply twice the flux in every filter, leading to a vertical212.3. Stellar Remnantsrise of ∼0.75 magnitudes and no shift in colour. Systems with q < 1 willbe brightened by a lesser amount and reddened to a small degree dependingon the specific q value. While the UV filters mentioned previously provideexcellent positional information and allow for high quality proper motiondetermination, they unfortunately are not ideal for identifying binary sys-tems. In the UV CMD the binary population is largely blended with thethick main sequence, likely due to multiple populations with differing com-positions (Richer et al., 2013). Only those binaries with a mass ratio close tounity stand significantly apart from the main sequence, and even those havesubstantial contamination. Figure 2.1 shows a CMD of 47 Tuc from anotherdata set using the redder filters F606W and F814W (Sarajedini et al., 2007,GO-10775). These optical/infrared data cover a similar field of view as theUV data (central 100′′) and provide a clearer distinction between the mainsequence and the binary sequence, allowing us to distinguish systems downto q ≈ 0.5. Of the hidden binaries that have q < 0.5, the large majority ofthem will have masses < 1.0 M. Their masses will cause them to followthe distributions of similar mass main sequence stars and are thus left to beincluded as part of the general low-mass population.For a more refined characterization of the binaries, we look at the binaryfraction in three different q ranges. These are displayed in Figure 2.1, andthe specific values will be used in computing the binary mass distributionin Section 3.3.We create the isochrones used to calculate stellar masses using MESAstellar evolution models (Modules for Experiments in Stellar Astrophysics,Paxton et al., 2011, 2013, 2015). The fluxes in each filter were determinedusing Phoenix atmospheres1, by Baraffe et al. (2015) and Allard (2016).2.3 Stellar RemnantsIn addition to the concentrated binary systems, 47 Tuc will harbour a col-lection of dark stellar remnants in the form of heavy WDs, NSs, and sBHs.Most of these objects are virtually impossible to directly detect, but wecan infer their parameters based on an initial mass function (IMF) for thecluster.In Figure 2.2 we have plotted the observed stellar mass function of thevisual/infrared data as blue points. Overlaid as a dashed red line is a Kroupa(2001) IMF normalized to the heaviest observed objects. The Kroupa IMFfollows a broken power law, N ∝ M−α, where α = 0.3, 1.3, 2.3 in the mass1https://phoenix.ens-lyon.fr/Grids/BT-Settl/CIFIST2011_2015/222.3. Stellar Remnantsranges M < 0.08, 0.08 < M < 0.5 and M > 0.5 M, respectively. Thedeficit of low-mass stars between the data and IMF is presumably due tothe preferential loss of low-mass stars from our field of view (segregation)and from the cluster as a whole (evaporation). Baumgardt & Sollima (2017)verify this strong depletion of low-mass stars, and yet the present-day massfunction agrees well with an evolved Kroupa IMF (Baumgardt & Hilker,2018). We make the assumption that the highest mass stars still observedin the cluster (∼ 0.85 M) remain at their natal abundance and use thisIMF to estimate how many progenitor objects there were at different masses.This is, in several ways, a conservative approach to estimating the numberof stellar remnants. The Kroupa IMF is among the steepest mass functions,and assuming some fraction of the 0.85 M stars have been lost would onlyserve to raise the IMF and increase the remnant estimates.As mentioned previously, we are interested in objects that have a massabove 1.0 M, and would therefore be distributed more centrally than mainsequence stars. The red region of Figure 2.2 marks progenitor stars thatwould form WDs of < 1.0 M. These WDs would relax into a distributioncomparable to the main sequence stars. The blue region marks progenitorsthat produced WDs of mass 1.0 M < M < 1.4 M (Cummings et al.,2016). The green region indicates the more massive progenitors that formNSs, and the purple region marks the progenitors that produce sBHs of upto 10 M (Spera & Mapelli, 2017). The number of objects in each colouredmass range is determined by integrating the mass function. We cut offthe IMF beyond M = 50 M because the BHs above 10 M tend to becompletely lost during cluster evolution (Morscher et al., 2015).We have to consider that natal kicks imparted upon formation as wellas gravitational encounters within the cluster will cause the loss of NSs andsBHs, however the retention fraction in globular clusters is not a well con-strained quantity. Simulations tend to produce values anywhere in the rangeof about 5−50% (Baumgardt & Sollima, 2017; Morscher et al., 2015; Moody& Sigurdsson, 2009; Mackey et al., 2008). We adopt a fairly conservativeestimate of 10% retention for NSs and sBHs, but we demonstrate later thatthis choice is well motivated.232.3. Stellar Remnants1.0 0.5 0.0 0.5 1.0 1.5log(M/M¯)012345678log(dN/dM) [M−1 ¯]0.85M¯5M¯8M¯25M¯50M¯Integrated number of objects: 165009 128495 6619 6025 1058  Data (Completness Corrected)Kroupa IMFFigure 2.2: A Kroupa initial mass function is used to estimate the numberof progenitor objects that form remnants of various masses. The red, blue,green, and purple regions respectively count progenitor objects that wouldcreate low-mass WDs (< 1 M), high-mass WDs (> 1 M), NSs, and sBHs.For the species above 1 M we use these counts to infer their populationparameters and include them in the velocity dispersion model. The largenumber of predicted low-mass WD progenitors does agree with observedWD counts when one considers the WD birth rate, time a WD takes to coolbelow detection thresholds, and the age of the cluster (Goldsbury et al.,2016).24Chapter 3Velocity Dispersion Model3.1 Jeans ModelIn order to determine whether or not our observations are consistent with acentral IMBH, we create a theoretical velocity dispersion model that includesthe contributions of all the non-IMBH cluster objects as well as an IMBHmass parameter that affects the dispersion signal in the core. We use a3D version of the King density model (King, 1962) as a starting point indetermining the shape of the dispersion profile. The King density profile,ρ(r) = K[1(r2 + a2)3/2− 1(r2t + a2)3/2], (r ≤ rt), (3.1)describes how the density (ρ) falls off with 3D radius (r) as we move awayfrom the core. The density drops to zero at a prescribed tidal radius (rt)which is set by the Galactic potential. The profile scales with K, a constantthat encodes the total mass contained in the distribution and the parametera describes an effective core radius. For radii interior to a the distributionis largely flat.The density profile and the associated enclosed mass as a function ofradius, M(r), can be used in the isotropic Jeans equation,ddr(ρσ2) = −ρdΦdr= −ρGr2M(r), (3.2)to isolate the square of the velocity dispersion σ2(r). Here Φ and G arethe gravitational potential and constant, respectively. We use the isotropicversion as the level of anisotropy is low in the core region of 47 Tuc whereour proper motion data occur (Heyl et al., 2017; Watkins et al., 2015; Belliniet al., 2017). The squared velocity dispersion then takes the form,σ2(r) = − Gρ(r)∫ r0ρ(r)M(r)r2dr. (3.3)This, of course, is the dispersion profile for 3D radius r. Once σ2(r) is ob-tained, it is integrated against the density along the line of sight to produce253.2. Treatment of Core Parameter athe projected squared velocity dispersion,σ2p(R) =∫∞R σ2(r)ρ(r)r(r2 −R2)−1/2dr∫∞R ρ(r)r(r2 −R2)−1/2dr=MTF (R),(3.4)where R is the on-sky projected radius from the centre. In the second linewe have simply parcelled the equation into two parts: a function that holdsthe R dependence, and a scaling term out front. F (R) will have units of[(velocity)2/mass] and describe how the dispersion falls off with R. TheMT term is the total mass of the distribution in question and scales themagnitude of the dispersion profile. It arises from M(r) in Equation 3.3which involves an integral over the density profile and a coefficient that canbe written in terms of MT . We adopt this format for ease of writing separatecomponent contributions later on in Section 3.4.The operations between Equations 3.2–3.4 are fully generic manipula-tions of the Jeans equation and make no specifications about the densityprofile. In our full density distribution, we combine different King mod-els with individual total masses and core radii to capture the unique dy-namical contributions of the various cluster populations. For an analogousprocedure using Gaussian distributions and with detailed formalism, seeEmsellem et al. (1994) and Cappellari (2008). Before we apply the Jeansequation directly to our cluster model, there are a few things to consider.3.2 Treatment of Core Parameter aThe parameter a in Equation 3.1 acts as a characteristic core radius. This isnot necessarily a global value as objects of different mass will be distributedwith different core radii. To correct for the inevitability that our tracer stars(those for which we have proper motions) are not perfectly representativeof the true mass distribution, we make a distinction between two differentcore radius parameters. In the right-hand side of Equation 3.2, the factor−dΦ/dr = −GM(r)/r2 generates the force acting on each particle. ThisM(r) must therefore reflect the true underlying mass distribution of thecluster, whether those objects are detected in the data or not. Conversely,the ρ terms of Equation 3.2 describe the distribution of tracer particles whosemotions we are observing. These tracers need not be the same objects thatare generating the potential. We therefore calculate ρ(r) using the measuredcore radius of the stars for which we have proper motions, aPM, and calculatethe potential generating M(r) term using a more globally representative263.2. Treatment of Core Parameter acluster parameter, aCl, which is left as a fitting parameter. We fit a Kingdistribution to our tracer stars and find aPM = 36.0′′.Because our tracer stars (with completeness corrections) are reasonablyrepresentative of the total cluster population we expect aCl to be similar toaPM.To quantify this we can look at the mean masses of stars in different lo-cations. In Figure 3.1 we see that the mean mass of stars interior to a givenradius (Rlim) is higher for stars in the core, as expected by mass segrega-tion. The more massive stars should have correspondingly lower velocitiesbased on the equipartition of kinetic energy that drives mass segregation.The difference, however, is only a matter of 0.04 M between the innermoststars and the mean of the whole cluster, so the effect will be small (< 10%).Additionally we can look at the completeness of our catalogue. Figure 3.2shows a visualization of the completeness of our sample, displaying the char-acteristic trends of the completeness dropping with distance from the coreand with fainter magnitudes. However, our images retain high completenessthrough most of their sensitive magnitudes. 97% of the catalogue objectshave completeness values of greater than 0.5. The process used to calculatethe completeness of the data set is explained in detail in Goldsbury et al.(2016).The binary and stellar remnant populations that we discuss next will alsohave their own distribution parameters. Due to the differing concentrations,the functional form of the velocity dispersion in Equation 3.4 now becomesF (R|a) where a specifies the mass distribution in that particular population.273.2. Treatment of Core Parameter aFigure 3.1: Our set of proper motion data contains 12 stars in the central1′′, 100 within 3′′, 1115 within 10′′, and 8402 within 30′′. The value at theouter regions reflects the mean mass of all ∼ 50, 000 stars in our catalogue.The overall difference in mean mass of the central stars compared to themean mass of all the stars is only 0.04 M.283.3. Inclusion of Binary PopulationsFigure 3.2: The general trend of any cluster’s completeness is to drop off atsmall radii (crowding issues) and faint magnitude (sensitivity issues). We seeboth these trends here, but the completeness stays high for the vast majorityof the stars. 97% of the stars have completeness > 0.5. The scattered pointspoints that look like their own population are the white dwarfs which havemuch more similar magnitudes in these filters than the main sequence stars.3.3 Inclusion of Binary PopulationsIf there is no IMBH in the mass distribution, we expect the velocity disper-sion to flatten off in the core as M(r) → 0. However, a central BH pointmass causes M(r) to remain positive for any r > 0 and produces a risingslope into the cluster core. A concentrated population of objects in the clus-ter can also cause the dispersion profile to rise farther into the core beforeflattening out, mimicking the central effects of a BH. When testing for anIMBH it is therefore very important to take careful consideration of pop-ulations more centrally concentrated than the typical main sequence star,which may cause the dispersion to rise deeper into the core.For this reason we model the binary population separately from the gen-293.3. Inclusion of Binary Populationseral lower-mass cluster stars. However, the binary systems are not a singlehomogeneous population in terms of their mass. For a given primary mass(i.e. position along the main sequence) systems with high mass ratios aremore massive than those with low q-values, and for a given mass ratio, bi-naries at the top of the main sequence are more massive than those belowthem. The higher mass systems will have correspondingly small distributionparameters. We therefore break the binary contribution into components toaccount for binary sub-populations with different distributions. The expres-sion for the squared velocity dispersion of the binary stars becomes,σ2p(R) =∑j,αM j,αbinF (R|aj,αbin). (3.5)The index j signifies different mass bins down the main sequence in therange of 0.85–0.55M. The index α specifies one of the three mass ratioranges seen in Figure 2.1: lower (0.5 < q < 0.7), moderate (0.7 < q < 0.9),or high (q > 0.9). The mass of binaries in each j-bin and α-range (M j,αbin) iscalculated as,M j,αbin =fα(AMSAbin)(1 + 〈q〉α)(M jMSMobs)MCl. (3.6)The first item, fα, is the observed binary fraction for a given mass ratiorange (α). These values are determined by counting stars in a section ofthe CMD as seen in Figure 2.1. We assume that these fractional valueshold true for the entire main sequence. The A factors are the proportion ofthe population that is visible within our 100′′ field of view according to thedistribution parameter for that population. Therefore, the term AMS/Abinmodifies the observed binary fraction to describe the entire cluster, includingobjects beyond our image field. It is necessary because the binary fractionwill drop away from the core since the binary systems are more concentratedthan the main sequence stars. The factor (1 + 〈q〉α) accounts for the massof the secondary star in the binary system where 〈q〉α is the midpoint ofthe mass ratio range α. At the end of Equation 3.6, the quantity M jMSis the observed mass in main sequence bin j, obtained by counting starsand consulting the isochrone. It is divided by Mobs, the total observedmass of the main sequence, to determine the mass proportion of the entirecluster our selection j corresponds to. Finally, multiplying by MCl scalesthis proportion to the fitted dynamical cluster mass.Because of the shape and the width of the main sequence and giantbranch in Figure 2.1, the binary stars cannot be identified for all masses303.4. Inclusion of Dark Stellar Remnantsand thus, their distributions cannot be measured directly. We make use ofthe results of Goldsbury et al. (2013) to determine the aj,αbin parameter ofEquation 3.5. In order to quantify the mass segregation in 54 globular clus-ters (including 47 Tuc), Goldsbury et al. (2013) measured the distributionsof various stellar groups, determining a power-law relation between the ob-jects’ mass and their distribution parameter (Rc in their paper works out tobe equivalent to a in ours). We use this power-law to determine aj,αbin giventhat the mass of the binary system is the mass of the primary times thefactor (1 + 〈q〉α).3.4 Inclusion of Dark Stellar RemnantsThe dark stellar remnants are included in a much simpler manner. Eachof the heavy WD, NS, and sBH populations are given a single distributionspecified by their total mass and core radius parameter. The heavy WDs areassumed to have a mean mass of 1.2 M, the mid-point of the consideredmass range (1 M to the Chandrasekhar limit 1.4 M). The NSs are allconservatively estimated to have masses of 1.4 M, and the sBHs are allassumed to have masses of 10 M (Morscher et al., 2015; Cummings et al.,2016; Spera & Mapelli, 2017). The populations’ overall masses are simplygiven by the predicted number of objects multiplied by their mass. Thenumber of objects in each of Figure 2.2’s coloured mass ranges is determinedby integrating the mass function. This value is then corrected to includeeverything beyond the data’s 100′′ field of view. This correction processis analogous to the treatment used for binaries within Equation 3.6. Withestimates of the remnant masses we can infer their distributions and includethem in our velocity dispersion model using the Goldsbury et al. (2013)mass segregation power law as was done for the binary systems. However,the sBHs and NSs will suffer additional losses, largely due to natal kicks fromtheir progenitor supernovae. We assume a moderate value of 10% retentionfor NSs and sBHs. The projected squared velocity dispersion contributionfrom the dark stellar remnant populations becomesσ2p,k(R) =MkF (R|ak), (3.7)where k is an index that refers to the dark stellar remnant in question (WD,NS, sBH). The full and final expression for the squared velocity dispersion313.5. Binned vs. Un-binned Analysiscomes together as:σ2p(R) = MIMBHF (R)+MClF (R|aCl)+∑j,αM j,αbinF (R|aj,αbin)+∑kMkF (R|ak).(3.8)We have now built a σ2p(R) distribution up from 47 Tuc’s different masscomponents. Using measured binary fractions and inferred distributions weaccount for the individual contributions of binary sub-populations that havediffering distributions and total masses. An initial mass function normal-ized to observed cluster stars allows us to estimate how many dark stellarremnants remain in the cluster. The fit parameters MCl and aCl describethe gross shape of the dispersion curve (its vertical scale and the radius atwhich it falls off), while the concentrated binaries and remnants bring upthe dispersion value towards the core. Any central rise observed in the datathat is not due to the concentrated populations is taken up by the IMBH fitparameter MIMBH.All of this put together produces a radial profile for the projected velocitydispersion that depends on three parameters: central IMBH mass, total clus-ter mass, and cluster core radius. The rest of the inputs are measured or in-ferred quantities. The projected dispersion profile, σp(R|MIMBH,MCl, aCl),is now in a form that can be fit to the data and the optimal values for the3 input parameters found.3.5 Binned vs. Un-binned AnalysisWith a theoretical velocity dispersion model now built, we are in a positionto find the set of the 3 parameters (MIMBH, MCl, and aCl) that best fit theobservations. This is most simply and commonly done by radially binningthe data, finding the velocity dispersion in each bin, then finding a least-squares fit of the model to the data.We must, however, be careful in trusting the results of a binned dataset. Any binning scheme has some degree of arbitrariness that may influ-ence the result in unpredictable ways. At the very least, introducing binswill smear out the radius-dependent signal over a larger range of radii. Atworst, the choice of binning system may unintentionally amplify or suppressa particular result.323.5. Binned vs. Un-binned AnalysisAs a test of the influence of bin choice, we set up a logarithmicallyspaced binning scheme then steadily change the number (and thus the size)of the bins. Figure 3.3 shows how the binned-fit MIMBH parameter fluctuatesas we slowly scale the bin sizes. The fluctuations are both pronounced andunsystematic. To bypass the issue altogether, we use the following un-binnedlikelihood maximization technique to find the best-fit dispersion model.5 10 15 20 25 30 35No. of Bins1000500050010001500MBH [M¯]Figure 3.3: A radial binning scheme is chosen that follows a logarithmicspacing out to the limit of the UV data (120′′). When the number of bins(and thus, the size of the bins) is changed, the binned-fit MIMBH parameterfluctuates dramatically and unpredictably. The other fit parameters (notshown) follow similar variations. We note this unreliability as a justifica-tion for using an un-binned likelihood analysis in determining best-fit modelparameters. Note: A negative IMBH mass simply means the non-IMBHcomponents of the model produce a higher dispersion in the core than thedata indicate. The negative mass is an attempt by the optimizer to bringthe dispersion value down to improve the fit.333.5. Binned vs. Un-binned AnalysisAgain, under the assumption of isotropy in the core for 47 Tuc, we as-sume the observed velocity components (vx, vy) of each star are samplesfrom Gaussian-distributed populations. The widths of these Gaussian pop-ulations are estimated by the model dispersion value at each star’s projecteddistance from the core, broadened by the uncertainty in the velocity mea-surements. We find the set of MIMBH, MCl, and aCl parameters for σp(R)that maximize the global log-likelihood function,lnL =∑i− v2x,i2(σ2p,i + δ2x,i)− v2y,i2(σ2p,i + δ2y,i)− 12ln(σ2p,i + δ2x,i)−12ln(σ2p,i + δ2y,i).(3.9)Here σp,i = σp(Ri|MIMBH,MCl, aCl) is the model dispersion value at theprojected radius of star i, and δx,i and δy,i are the measurement uncertain-ties in the velocity components of star i that arises from our photometricdata. This likelihood estimator is equivalent to Equation 8 of Walker et al.(2006), but has been adapted for 2D proper motions and a continuous σ2p(R)rather than 1D radial velocities and binned dispersion values. Furthermore,we ignore the constant offset introduced to the lnL value by Gaussian nor-malization. For a full derivation and explanation of the entire un-binnedlikelihood technique, see Goldsbury et al. (2016). We employ this un-binnedsystem to avoid any unwanted bias introduced through arbitrarily choosinga binning system to apply to the data. The results of the fit are plotted inFigure 4.1.34Chapter 4Results4.1 Model Fitting ResultsHaving built a velocity dispersion model that includes measured binary frac-tions, inferred dark stellar remnants, and a NS and sBH retention frac-tion of 10%, we obtain the following best-fit parameter values: MIMBH =840 ± 1700 M, MCl = (1.38 ± 0.06) × 106 M, aCl = 43.7 ± 1.5′′ (un-certainties are determined through bootstrapping). Of perhaps more valuefor comparison purposes is the MIMBH/MCl ratio of 0.06% ± 0.13%. Theratio avoids scaling discrepancies in cluster distance and total mass. Thedispersion model described by these parameters is shown in Figure 4.1, plot-ted over binned velocity dispersion data. The binned data are included forvisualization purposes only and the fit was made using the un-binned likeli-hood maximization described in the previous section. Of concentrated heavypopulations, the binaries and sBHs play the dominating roles on the cen-tral dispersion rise. The binary population contain a large overall mass andtheir distribution causes the dispersion to continue rising into ∼ 10′′. ThesBHs, while comprising a lesser total mass, are much more concentratedand continue this dispersion rise into ∼ 2′′. The combined effect leaves littleroom for any substantial IMBH which would drive the dispersion above theobservations.We note that throughout this paper the reported velocities (and thusthe fitted masses that scale the dispersion curves) are dependent on thechoice of distance to the cluster. We have adopted a distance of d = 4.69kpc from Woodley et al. (2012) who fit spectral energy distributions of 47Tuc’s WDs. Additionally, our proper motion data only extends to 120′′,a small fraction of the tidal radius, rt ≈ 42′ (Harris, 1996, 2010 edition).This means we are really only fitting the central region of the cluster. Ourmass estimates involve integrals of the density distributions and thereforeassume the density beyond 120′′ follows a perfect King distribution out tothe tidal radius. There may be substantial deviation in the tails which maycontribute to our fairly large cluster mass determination. However, Giersz& Heggie (2011) model a cluster mass of 0.9×106 M and note that perhaps354.2. Re-sampling34% consists of stellar remnants. In light of the different masses betweenstudies, the MIMBH/MCl ratio may be a better way to compare results.100 101 102R [arcsec]024681012141618σp [kms−1]All contributionsAll except IMBHCluster (<1M¯)Binaries (>1M¯)sBHs (10M¯)WDs (>1M¯)NSs (1.4M¯)Figure 4.1: The solid red line shows the best-fitting (un-binned) velocitydispersion model. Model parameters are given in Section 4.1. Note thatthe black points display binned data and are included for visualization pur-poses only. The model was fit using an un-binned likelihood maximization.Dashed lines show how the different components (cluster stars, binaries,sBHs, WDs, and NSs) contribute to the overall velocity dispersion profilewith the individual components being added in quadrature. We see herethat the sBHs and binaries produce a dispersion rise in the core that isnearly adequate to explain the observations. The un-binned likelihood max-imization requires only a very minor IMBH contribution to fit the propermotion data.4.2 Re-samplingWe employ a system of statistical re-sampling in order to assess the efficacyof our modelling process. This re-sampling involves generating new vx and364.2. Re-samplingvy velocity components for each star. These velocities are drawn randomlyfrom Gaussian distributions whose widths are determined by the dispersionvalue at the radius of each data point, broadened by a measurement errorof the velocity. Keeping the original positions for each star ensures theirradial distribution remains unchanged. Once all the stars have had theirvelocities re-sampled according to the dispersion model, a best-fit parameterset is determined for this generated data. The parameters are recorded alongwith the log-likelihood value (Equation 3.9) of the newly generated velocitiesmeasured against the original data’s best fit σp(R|MIMBH,MCl, aCl). Thisprocess is repeated many times over to build distributions of the modelparameters and log-likelihood values.We use this process to address whether or not our model is faithfully de-scribing the data. Being able to find a set of best-fit model parameters doesnot necessarily mean one has accurately modelled the data. For example,any set of curved data will have a straight line that fits it best, but that doesnot mean a straight line is a good description of those data. To determinethe quality of our model, we re-sample stellar velocities based on the disper-sion parameters that best fit our data. Figure 4.2 shows the distribution oflog-likelihood values generated when we draw these samples many times. Ascan be seen, the line indicating the original data’s log-likelihood value fallsvery near the middle of the distribution. Such close agreement indicates ourmodel is a good description of the data. We would expect to draw data fromour best-fitting model that differs at least as much as the real data do fromthat model 87.3% of the time. Artificial velocity data generated from ourmodel really do look like the cluster’s velocity data.We similarly verify that the distribution of MIMBH values of the gener-ated data are centred around the input model’s MIMBH value. This indicatesour model can extract the input IMBH mass without systematic bias in ei-ther direction.374.2. Re-sampling311000 310500 310000 309500lnL0500100015002000250030003500CountsOriginal data lnLµ,±1σ,±2σ,±3σFigure 4.2: Presented here is a histogram of the log-likelihood (lnL) valuesfor the best-fit curves of re-sampled data sets. Here the re-sampled dataare drawn from the original data best-fit dispersion curve. All fits used un-binned likelihood maximization. Fits to the re-sampled data are in closeagreement with our fit to the original data, indicating the model is a gooddescription. We would expect a deviation of at least this magnitude in 87.3%of cases.38Chapter 5Conclusion5.1 SummaryOur use of UV imaging data has allowed us to glean velocity informationfrom stars right into the very centre of the globular cluster 47 Tucanae.Historically, crowding has been a limiting factor in IMBH velocity dispersionsearches as it severely reduces the numbers of stars measurable at verysmall projected radius. Resolution and accuracy in this region are of keyimportance in trying to estimate the mass of a central IMBH through thismethod. With visibility right in to the cluster core, we are able to probe theregion of the cluster where an IMBH’s influence would be most pronounced.Beginning with a King model, we used the isotropic Jeans equationto build a velocity dispersion profile with the added influence of a cen-tral IMBH. The dispersion profile was built with contributions from severalsub-components: a central IMBH point mass, a collective group of low-masscluster objects (. 1.0 M), and individual contributions from concentratedpopulations such as the binary systems, heavy white dwarfs, neutron stars,and stellar-mass black holes.To avoid the known and unknown biases of binning our data, we employan un-binned likelihood analysis when fitting our model. This techniqueweighs the combined likelihoods of each individual star’s velocities beingdrawn from a given dispersion model. It prevents the radial smearing-outthat occurs when data are binned together, and avoids the unpredictablefluctuations that result from the arbitrary choice of a binning scheme.To determine how good a fit our dispersion model is to the data, werandomly re-sample the velocity components of our stars based on the best-fit dispersion curve. The log-likelihood of the real data compared to thebest-fit model is in very good agreement with the distribution of re-sampledlog-likelihood values, indicating our model does a good job of describing thedata.Our analysis produces a best-fit value of MIMBH = 840 ± 1700 M fora central IMBH in the core of 47 Tucanae, giving a MIMBH/MCl ratio of0.06%± 0.13%. This value is consistent with zero, unlike the recent results395.1. SummaryTable 5.1: Exploring Retention FractionRetention (%) MIMBH [M] MCl [M] MIMBH/MCl0 4348 1344642 0.00325 2587 1363820 0.001910 838 1384564 0.000612 134 1393112 0.000115 -893 1407125 -0.000620 -2602 1431780 -0.001825 -4285 1458876 -0.0029Assuming different retention fractions of the sBH population produces arange of IMBH estimates. Typical 1σ error bars on the MIMBH values are±1700 M. Retention of 7− 18% is the approximate 1σ range around zeroIMBH. A very low retention fraction is required to produce a significantIMBH detection. Considering that our mass function is already conservativein its sBH estimate, these results suggest that a retention & 18% may notbe possible in 47 Tuc. The Kızıltan et al. (2017) mass fraction of 0.30%requires a sBH retention very near zero.of Kızıltan et al. (2017) who found a MIMBH/MCl ratio of 0.30%+0.20%−0.12%,however, our results do agree at the high and low ends of our respective 1σuncertainties. We emphasize that the proper characterization of the binarypopulation and sBHs can have a marked effect on the IMBH estimate andtheir omission will likely bias that estimate high. We explored a range ofsBH retention rates to determine the effect it can have on the final fittedIMBH mass. We observe that in order to find the MIMBH/MCl fractionconcluded by Kızıltan et al. (2017) we would need to assume essentially all10 M sBHs are ejected from the cluster. Retaining only ∼ 12% raisesthe central dispersion enough for our model to require no IMBH at all, andpushing the retention beyond ∼ 18% creates higher dispersion in the corethan is observed, requiring a ∼ 1σ significant negative IMBH mass, aclearly unphysical situation. Table 5.1 displays the final fit results underthese different retention assumptions.405.2. Discussion5.2 DiscussionThe velocity dispersion profile investigated in this study does not supportan IMBH detection for 47 Tuc unless one posits a very low sBH retention(. 7%). The concentrated populations of binary stars and dark stellarremnants alone are enough to explain the central velocity dispersion.Beyond studying the velocity dispersion, there are several independentavenues of kinematic analysis available to collectively probe for the presenceof an IMBH. In some cases they provide contradictory results, but should becollectively considered. As mentioned previously, the Kızıltan et al. (2017)study produced a positive IMBH result based on accelerations of millisec-ond pulsars in 47 Tuc. There is numerical evidence that the presence of anIMBH should cause a weak central cusp in the density/surface brightnessprofile (Baumgardt et al., 2005; Noyola & Baumgardt, 2011), however otherwork has shown that an observed cusp may also be caused by a state changein the degree of core collapse (Trenti et al., 2010; Vesperini & Trenti, 2010),and is not sufficient to infer an IMBH. Either way, we find no evidence fora central cusp in the density distribution. Figure 5.1 shows the cumulativedistribution of objects (corrected for incompleteness) as a function of pro-jected distance from the core. This distribution is compared against an R2relation, which would indicate a perfectly flat density profile. By eye thereappears to be a mild deviation from R2 in the centre, but the difference isvery small (low end of the logarithmic scale) and includes very few objects(. 10). The deviation is not statistically significant and we cannot rejectthe null hypothesis (i.e. that the data are drawn from an R2 distribution)to any reasonable significance until we look beyond ∼ 20′′ where the clusterdensity begins to fall off. Additionally, we find no sign of a small populationof high velocity stars. If the IMBH were part of a binary system, 3-bodyinteractions should periodically eject high velocity stars into the cluster onlargely radial orbits. Our data do not show any stars with velocities beyond∼ 60 kms−1 while N-body simulations including central IMBHs show a smallpopulation of stars with velocities up to ∼ 80 kms−1 (Baumgardt, privatecorrespondence).415.2. Discussion10-1 100 101 102R [arcsec]10-1100101102103104105106No. of starsFigure 5.1: The cumulative distribution of stars (incompleteness corrected)as a function of distance from the core (R). The dashed red line is an R2relation, indicative of a flat density distribution. The apparent deviation inthe core from the R2 line is not statistically significant as the deviation issmall and the number of stars is low in this region. Significant deviation(> 1σ) only occurs beyond a radius of ∼ 20′′ where the cluster density nolonger remains flat. We find similar results when we relocate the cluster’scentroid within the uncertainty range of Goldsbury et al. (2010).There are many contrary findings concerning IMBHs in globular clusters.Of particular note are the conflicting results of Lu¨tzgendorf et al. (2015) andLanzoni et al. (2013) on NGC 6388, van der Marel & Anderson (2010) andNoyola et al. (2008) on ω Centauri, and Perera et al. (2017) and Gieles et al.(2018) on NGC 6624. There is clearly more work to be done in charac-terizing and evaluating the shortcomings and strengths of different IMBHdetection methods and we could benefit from more studies that evaluate the425.2. Discussionaccuracy and biases of these techniques. For example, de Vita et al. (2017)provide insight on how using integrated light spectroscopy might affect re-sults versus using velocity measurements of individual objects. Simulationscan be extremely useful with their ability to explicitly incorporate a centralIMBH of known mass to determine its observable effects, but the expensivecomputation limitations require simplified physics and assumptions aboutinitial conditions and short-timescale dynamics. Including an appropriatebinary population is one aspect that suffers in this regard.In the course of our study, we have found that building a careful massmodel of the cluster that includes concentrated heavy populations has animportant bearing on an inferred IMBH presence. Due to their concentrateddistribution and substantial total mass, binary systems and sBHs cause thevelocity dispersion to continue rising deeper into the core than would beexpected from lower mass cluster stars, mimicking the central slope producedby an IMBH. As a result, we find no need to invoke an IMBH to explainedthe observed velocity dispersion unless the core has a sBH retention fractionnear zero.Consideration of the sBH retention fraction leads to an unanticipatedadditional result of this study. We employ a conservative initial mass func-tion to estimate the number of sBHs, and still find that a retention of & 18%becomes incompatible with our dispersion observations. Above this fractionour model requires a significantly negative IMBH mass in an attempt todrop the dispersion values in the core, which is clearly unphysical. For 47Tuc it seems that we can place a constraint on the sBH retention to be atmost 18%.It should be noted that a King density distribution that has isotropicstellar velocities is not a self-consistent dynamical model once an IMBHpoint mass is added in the core. We use an isotropic model because thereis no evidence for anisotropy in the core, and we use a King model becausethere is no evidence for a central density cusp. The IMBH parameter allowsus to capture any additional rise in the central dispersion that is observed.Limiting the input profiles in such a way makes it easier to detect an IMBHinfluence by keeping the number of fit parameters low. Even so, the modelrequires very little “extra” dispersion in the core to fit the observations.We have shown here that proper motion dispersion analysis can prove tobe a useful tool in the search for IMBHs, even in the crowded environmentsof a globular cluster core. With the right choice of photometric filters wecan overcome the crowding issues and glean a multitude of data in theregion most sensitive to an IMBH’s influence. Our inclusion of concentratedpopulations in our mass model is sufficient to explain the observed velocity435.2. Discussiondispersion in 47 Tuc’s core. 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