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The globular clusters and halo of M31 Holland, Stephen 1997

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T H E G L O B U L A R C L U S T E R S A N D H A L O O F M31 By Stephen Holland B. Sc. (Physics &: Astronomy) University of Victoria (1989) M . Sc. (Astronomy) McMaster University (1991) A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF T H E REQUIREMENTS FOR T H E DEGREE OF D O C T O R OF PH I L O S O P H Y in T H E FACULTY OF GRADUATE STUDIES PHYSICS & ASTRONOMY We accept this thesis as conforming to the required standard T H E UNIVERSITY OF BRITISH COLUMBIA November 1997 © S t e p h e n Holland, 1997 In presenting this thesis in partial fulfilment of the requirements for an advanced degree at the University of British Columbia, I agree that the Library shall make it freely available for reference and study. I further agree that permission for extensive copying of this thesis for scholarly purposes may be granted by the head of my department or by his or her representatives. It is understood that copying or publication of this thesis for financial gain shall not be allowed without my written permission. Department of Physics & Astronomy 129-2219 Main Mall The University of British Columbia Vancouver, B. C , V6T 1Z4 Canada Date: Wovr. 3 , ! ? 9 7 Abstract The results of two studies of M31's globular cluster system are presented. One study used deep V- and / -band images of 24 M31 globular clusters taken with C F H T ' s H R C a m . This data is the best available ground-based data for determining structural parameters (such as core and tidal radii, ellipticities, and orientations) for M31's globular clusters. Two-dimensional M i c h i e - K i n g models were fit to each cluster and the results compared to the structural parameters of the Galactic clusters. There is no evidence for any difference between the mean tidal radii and ellipticities of globular clusters in the two galaxies. Core radii and half-mass radii are too strongly affected by seeing to allow a comparison between the two galaxies. The second study used deep HST W F C V- and /-band images of G302 and G312, two globular clusters in M31's halo, to obtain the deepest colour-magnitude diagrams of any M31 globular cluster. Structural parameters were determined for the two clusters and compared to those from the C F H T data. T ida l radii and ellipticities from the C F H T data agree with the more reliable results obtained from HST data. The core radii agree for seeings of <J 4 times the cluster's core radius. It is not possible to obtain reliable core parameters for the M31 clusters using ground-based telescopes unless adaptive optics are used to reduce the seeing to ^ (X ' l . However, the overall sizes and shapes of the clusters can easily be measured if F W H M ^ 1". Images of M31 clusters obtained from the ground have similar resolutions to those of Virgo globular clusters obtained with the HST's P C . Therefore detailed structural parameters can be obtained for globular clusters in the elliptical galaxies of the Virgo cluster. i i This thesis also presents the deepest colour-magnitude diagram for M31's halo. There is no evidence for young stars, or of a second parameter problem in M31's halo. The halo stars have [m/H] 0.6 with a spread of -2.0 £ [m/H] & -0.2. The M31 halo is ~ 8 times more metal-rich than the Galactic halo and ~ 4 times more metal-rich than the M31 globular cluster system. in Table of Contents Abstract ii List of Tables viii List of Figures x Publications xiii Acknowledgement xv 1 Introduction 1 1.1 Historical Background for the Andromeda Galaxy 1 1.2 The Halo of M31 4 1.3 Globular Clusters in M31 • 7 1.4 The Goals of this Thesis 12 2 The C F H T Data 14 2.1 Observations 14 2.2 Data Reduction 18 2.2.1 Preprocessing 18 2.2.2 Calibrating the Data 21 2.2.3 Background Subtraction 23 2.3 Seeing 24 iv 3 C F H T Resu l t s 29 3.1 Star Counts 29 3.2 Fitting Michie-King Models 33 3.2.1 The Theory Behind Michie-King Models 33 3.2.2 Fitting Methods 37 3.2.3 One-Dimensional Models 40 3.2.4 The Artificial Clusters . • . 46 3.2.5 Systematic Biases in Seeing Convolutions 48 3.2.6 Two-Dimensional Artificial Cluster Models . 51 3.2.7 Two-Dimensional Integrated Light Models 58 3.3 Properties of the M31 Globular Cluster System 61 4 T h e HST D a t a 69 4.1 Motivation for the HST Observations 69 4.2 Observations 71 4.3 Data Reductions 74 4.4 Artificial Star Tests 84 5 HST Observat ions of G 3 0 2 a n d G 3 1 2 91 5.1 The Colour-Magnitude Diagrams 91 5.1.1 Contamination 91 5.1.2 G302 95 5.1.3 G312 . . . '. 99 5.1.4 The Colour-Iron Abundance Relation 102 5.2 Luminosity Functions . 109 5.3 Structure 115 5.3.1 Colour Gradients 115 v 5.3.2 Ellipticities 117 5.3.3 Michie-King Models 122 5.4 Extended Stellar Halos 126 5.5 Mass Loss and the Orbit of G302 135 5.5.1 Mass Loss from G302 135 5.5.2 Mass Loss from G312 137 5.5.3 The Orbit of G302 138 5.6 A Comparison of the C F H T and HST Results 140 6 HST Observat ions of the H a l o of M 3 1 146 6.1 Colour-Magnitude Diagrams 146 6.1.1 Contamination in the Field 146 6.1.2 The Red-Giant Branch 151 6.1.3 The Horizontal Branch 159 6.2 The Halo Luminosity Function 162 7 G 1 8 5 : A Poten t ia l D o u b l e G l o b u l a r C l u s t e r 169 7.1 Multiple Globular Clusters 169 7.2 The HRCam Images 172 7.3 Probability that G185 and G185B are Line-of-Sight Objects 180 7.4 Colours 181 7.5 Structural Parameters 183 7.5.1 Cluster Concentrations 183 7.5.2 Ellipticities 185 7.6 Dynamical Considerations 191 7.6.1 The SIS Spectra 191 7.6.2 The Radial Velocities 197 vi 7.6.3 The Roche Limit of the System 199 7.7 Conclusions 202 8 C o n c l u s i o n s 203 8.1 The Globular Star Clusters of M31 203 8.2 The Halo of M31 206 8.3 Unanswered Questions 207 References 210 vii List of Tables 2.1 The C F H T fields 16 2.2 Photometric zero-points for the HRCam images 22 2.3 Seeing characteristics for the C F H T HRCam images 27 3.1 Best-fitting one-dimensional Michie-King models 44 3.2 Results from two-dimensional Michie-King models and star counts for G302 and G312 52 3.3 Two-dimensional Michie-King model fits to the C F H T data 59 3.4 Ellipticities and position angles for the C F H T data 60 4.1 Log of the HST observations 72 4.2 Aperture corrections for the HST data 76 4.3 Photometric uncertainties in the HST photometry of the halo stars. . . 80 4.4 Photometric uncertainties for G302 and G312 81 4.5 A sample of the stellar photometry for G302 and the surrounding fields. 81 4.6 A sample of the stellar photometry for G312 and the surrounding fields. 82 4.7 A sample of the stellar photometry for the globular cluster G302 82 4.8 A sample of the stellar photometry for the globular cluster G312 83 4.9 Magnitude shifts in the red-giant branch artificial star data 86 4.10 Magnitude shifts in the horizontal branch artificial star data 86 5.1 Properties of selected globular clusters in the M31 system 102 viii 5.2 V-band luminosity functions for G302 and G312 I l l 5.3 Ellipticities and position angles for G302 and G312 from the HST data. 117 5.4 Best-fitting Michie-King models for G302 and G312 using the HST data. 123 5.5 A comparison of core radii derived from ground-based and HST observations. 144 5.6 A comparison of tidal radii derived from ground-based and HST observa-tions 145 6.1 The observed V- and J-band halo luminosity functions 165 7.1 Log of the HRCam observations of G185 172 7.2 Integrated magnitudes for G185, G185B and vdB2 181 7.3 Two-dimensional Michie-King model fits to G185, G185B, and vdB2. . 183 7.4 Ellipticities and position angles for G185, G185B, and vdB2 185 7.5 Relative radial velocities for G185, G185B, and the background 197 ix List of Figures 2.1 The CFHT fields • 17 2.2 The seeing for the CFHT HRCam images 26 3.1 CFHT star counts for G302 and G312 32 3.2 An artificial globular cluster 47 3.3 Probability contours for the CFHT G302 /-band data . 54 3.4 Probability contours for the CFHT G302 V-band data . . . . . . . . . . 55 3.5 Probability contours for the CFHT G312 /-band data . . . . . . . . . . . 56 3.6 Probability contours for the CFHT G312 V-band data 57 3.7 Core radii for the M31 globular clusters 65 3.8 Tidal radii for the M31 globular clusters 66 3.9 Half-mass radii for the M31 globular clusters 67 3.10 Concentration vs. half-mass radius for isotropic King models 68 4.1 Aperture corrections for the HST images 77 4.2 Scatter in the HST artificial stars 88 4.3 The width of the red-giant branch . 89 4.4 Scatter in the horizontal branch photometry 90~ 5.1 HST CMD for G302 . . . . . 93 5.2 HST CMD for G312 94 5.3 Annular CMDs for G302 98 x 5.4 Annular CMDs for G312 101 5.5 The (Mi,(V-I)0) CMDs for G302 and G312 107 5.6 The [Fe/H]- (V- / ) 0 relation for M31 globular clusters 108 5.7 Luminosity functions for G302 and G312 113 5.8 Cumulative luminosity functions for G302 and G312 114 5.9 Colour profiles for G302 and G312 116 5.10 Ellipticity and position angle profiles for G302 120 5.11 Ellipticity and position angle profiles for G312 121 5.12 Surface brightness profiles for G302 124 5.13 Surface brightness profiles for G312 125 5.14 The extended stellar halo around G302 130 5.15 The (lack of an) extended stellar halo around G312 . . 131 5.16 Stellar density contours of the background fields 132 5.17 The C2 orientation statistics for G302 and G312 133 5.18 The cumulative probability distributions for £ 2 134 6.1 A CMD of the M31 halo near G302 149 6.2 A C M D of the M31 halo near G312 150 6.3 M31 halo CMD near G302 with fiducial red-giant branches 156 6.4 M31 halo CMD near G312 with fiducial red-giant branches 157 6.5 The metaUicity distribution in the M31 halo 158 6.6 Cumulative halo luminosity functions near G302 167 6.7 Cumulative halo luminosity functions near G312 168 7.1 The J-band image of the G185 field 175 7.2 The G185 field after background subtraction 176 7.3 The fit-and-subtract procedure for G185 and G185B 177 xi 7.4 The J-band surface brightness profiles 178 7.5 The IZ-band surface brightness profiles 179 7.6 Isophotal contours for G185 and G185B 188 7.7 EUipticity profiles for G185, G185B, and the PSF 189 7.8 Position angle profiles for G185, G185B, and the PSF 190 7.9 C F H T SIS spectrum of G185 193 7.10 C F H T SIS spectrum of G185B 194 7.11 Background spectrum near G185 195 7.12 Background spectrum near G185B 196 7.13 Roche lobes for 35 pc separation 200 7.14 Roche lobes for 14 pc separation 201 xii Publications Some of the material in this thesis has been previously published in the astronomical literature. Most of Chapter 5 appeared in Holland et al. (1997) [Astronomical Journal, 114, 1488]. Chapter 6 appeared in Holland et al. (1996) [Astronomical Journal, 112, 1035] and most of Chapter 7 appeared in Holland et al. (1995) [Astronomical Journal, 109, 2061]. A full list of my publications prior to the completion of this thesis is given below. • Holland, S., Fahlman, G. G., & Richer, H. B. (1997), Astronomical Journal, 114, 1488 • Richer, H. B., Fahlman, G. G., Ibata, R. A. , Pryor, C., BeU, R. A. , Bolte, M . , Mandushev, G. Harris, W. E . , Hesser, J. E . , Holland, S., Ivanans, N., Stetson, P. B., VandenBerg, D. A., k Wood, M . (1996), Astrophysical Journal, 484, 741 • Holland, S., Fahlman, G. G., &; Richer, H. B. (1996), Astronomical Journal, 112, 1035 • Richer, H. B., Harris, W. E . , Fahlman, G. G., Bell, R. A. , Bond, H. E . , Hesser, J . E . , Holland, S., Pryor, C , Stetson, P. B., &; van den Bergh, S. (1996) Astrophysical Journal, 463, 602 • Holland, S., Fahlman, G. G., & Richer, H. B. (1995), Astronomical Journal, 109, 2161 xiii • Couture, J . , Racine, R., Harris, W. E . , & Holland, S. (1995), Astronomical Journal, 1 0 9 , 2050 • Holland, S., & Harris, W. E. (1991), Astronomical Journal, 1 0 3 , 131 A list of conference presentations that I gave where preliminary results from this thesis were presented is given below. • HST WFPC2 Observations of Tidal Tads in Globular Clusters in M31 o 189th Meeting of the AAS at Toronto, Ontario, January 12-16, 1997. • The Double Cluster G185 in M31 o 185th Meeting of the AAS at Tucson, Arizona, January 8-12, 1995. • The Internal Structure of the M31 Globular Cluster G302 o 182nd Meeting of the AAS at UC Berkeley, California, June 6-10, 1993. o 24 t h Meeting of the C A S C A at the University of Victoria, B.C. , June 1-4, 1993. xiv Acknowledgement Regardless of what UBC's graduation requirements say a work like this can never be the product of just one person. Many people have contributed to this thesis in various ways and it would be impossible to list them all. I would, however, like to thank Greg Fahlman and Harvey Richer for their endless instruction, suggestions, and patience through-out my incarceration at UBC. I would also like to thank the entire crew of astronomers at U B C for sharing their knowledge of astronomy, and generally being a good group of people to work and play with. Thank go out to Peter Stetson for kindly making available his A L L F R A M E software, which proved so valuable in reducing the HST data presented here. I would like to thank Dave Bohlender and Christian Vanderriest for obtaining the SIS spectra of G185 and G185B. I would also like to thank Kent Ashman, Pat Cote, Pat Durrell, Rodrigo Ibata, and Sydney van den Bergh for many useful discussions and suggestions. A dissertation is not completely written in a vacuum so I would like to thank those people in the real world whom I saw far too little of over the past few years: Barry, the C H L managers, the CHL Widows' Club, Doug, Kelly & Gail, Kerry, Marc, Martha & Alan, Owen & Fiona, Phil & Jill, Tara, the von Schulmann clan, and the rest of the people who have kept life interesting over the past half-decade. And to my parents, for so many years of support and encouragement—thanks! Most importantly, I owe a very great deal to Kyla. Not only has she kept me somewhat sane, she has been been an inspiration and a star that outshines any mere globular cluster. Finally, I would like to thank Lewis Vending for the many balanced meals. xv Chapter 1 Introduction 1.1 Historical Background for the Andromeda Galaxy The Andromeda Galaxy is one of the few external galaxies that is visible to the naked eye and has been a part of humanity's store of myths and legends since antiquity. The earliest written reference to the galaxy was by the 4th-century Roman poet Rufus Festus Avienus, who alluded to the nebula in Andromeda in his translation of the Phaenonena of Aratus, a Greek poem describing the constellations, which is apparently based on a lost work of Eudoxus of Cnidos. Several hundred years later the Persian astronomer Abu I-Husain al-Sufi (903-986 A.D.) depicted the constellation of Andromeda with a lathka sahibiya or "nebulous spot" at the location of the Andromeda Galaxy. The earliest known telescopic observations of the Andromeda Nebula were by Simon Marius in 1612, Ismael Bullialdus in 1667, and Edmond Halley in 1715. Charles Messier catalogued the object as "M31" in his Catalogue of Nebulae and Stellar Clusters, originally published in 1774. After its recognition as a non-stellar object there was considerable debate over the nature of the Great Nebula in Andromeda. William Herschel described the nebula as a collection of millions of stars that was at a distance of no more than "two thousand 1 Chapter 1. Introduction 2 times the distance of Sirius" (de Vaucouleurs 1987 [40]). This would have placed M31 at a distance of only 5.3 kpc from Earth, but it is important to realize that Herschel's description is essentially that of what we would today consider a galaxy. The idea that ex-ternal galaxies might exist was first proposed by Thomas White in 1750 and by Immanuel Kant in 1755. However, these philosophers based their ideas on religious and philosoph-ical considerations, not on observational evidence or physical laws (Hoskin 1970 [74], Jones 1971 [81]). In 1847 George Bond observed dark "canals" in the Andromeda nebula. These were the first recorded observations of spiral arms in external galaxies, although the "Bond canals" were not recognized as the lanes between spiral arms until some forty years later. Bond also resolved approximately 1500 stars with a limiting magnitude of V ~ 15 in front of the nebula. The brightest super-giants in M31 are bright enough to have been detected in Bond's photographic plates. During the nineteenth, and early twentieth, centuries there was considerable debate as to whether M31 was a diffuse object within the Milky Way galaxy or a separate external galaxy. Support for the external galaxy hypothesis came from radial velocity measure-ments which showed that M31 was approaching us with a heliocentric radial velocity of vh ~ -300 km-s - 1 , and was rotating (e.g. Slipher 1913 [133], Pease 1918 [107]). These observations were difficult to explain if M31 was a small, near-by, nebula. In addition, spectroscopy of M31 (Scheinel 1898 [130]) revealed a Solar-type spectrum indicating that M31 was made up of a large number of stars similar to those found in the Solar neigh-bourhood, and not a diffuse gas as would be expected if the nebula were a star-forming region. On the other hand, in 1885 a "new star" was observed near the nucleus of M31. This star, which was named S Andromedae, reached a maximum visual magnitude of 6.7 then faded over several hundred days (see de Vaucouleurs & Corwin 1985 [41] for a complete light curve). S Andromedae's nova-like appearance was used to argue that M31 Chapter 1. Introduction 3 was located within the Milky Way galaxy. The debate over M31's distance was finally put to rest in 1923 when Edwin Hubble identified Cepheid variables in M31 and was able to use the Cepheid period-luminosity relation to derive a distance of 285 kpc (Hubble 1925 [75]). This demonstrated that M31 was not a part of the Milky Way galaxy but a separate stellar system located a considerable distance from our own. Detailed discussions of these early references to, and observations of, M31 can be found in Hodge (1992) [71], and de Vaucouleurs (1987) [40]. In the nineteen-forties Walter Baade took advantage of the wartime black-outs of Los Angeles to obtain deep photographic images of M31 to study a discrepancy be-tween the periods of the brightest Cepheids in Galactic globular clusters and the periods of Cepheids with comparable brightnesses in M31. This discrepancy eventually led to Baade's discovery that the Cepheids in the disc of M31 were Population I stars while those in the Galactic globular clusters were Population II stars and thus followed different period-luminosity relations. Applying the Population I period-luminosity relation to the Cepheids in M31 led to a revised distance of ~ 550 kpc for that galaxy, a discovery that doubled the estimated size of the Universe. Recently parallax measurements from the Hipparcos satellite have been published which have led to a further revision of the distance to M31, and thus the size of the Universe. If the prehminary Hipparcos results are confirmed by further studies then the new distance to M31 would be ~ 900 kpc: an increase of ~ 25% in the estimated size of the Universe (Feast & Catchpole 1997 [53]). Despite M31 being very near-by on the cosmological map there is still considerable uncertainty as to such fundamental parameters as its distance. The careful study of this galaxy can still teach us much about the fundamental nature of the Universe. Chapter 1. Introduction 4 1.2 T h e H a l o of M 3 1 The stellar populations in the halo of M31 provide a direct tracer of the star form-ation history, and the early evolution, of that galaxy. The earliest published colour-magnitude diagrams (CMDs) of the M31 halo were obtained by Crotts (1986) [34] and Mould & Kristian (1986) [101]. Mould & Kristian (1986) [101] studied a field ~ 7 kpc from the centre of M31 along the southeast minor axis and found a mean iron abund-ance of [Fe/H] ~ —0.6, comparable to that of the metal-rich Galactic globular cluster 47 Tuc. Subsequent ground-based studies by Pritchet & van den Bergh (1988) [113], Christian & Heasley (1991) [29], Davidge (1993) [37], Durrell et al. (1994) [48], and Couture et al. (1995) [31] produced CMDs reaching down to near the level of the hori-zontal branch stars. These studies confirmed that the M31 halo has a mean metallicity of [m/H] ~ —0.6, similar to that of 47 Tuc. This makes the M31 halo approximately eight times more metal-rich than the Galactic halo, which has [m/H] ~ —1.5. Several of these studies have found that the observed spread in colour across the red-giant branch is larger than what would be expected from photometric uncertainties, and could be due to an intrinsic spread in the metallicity of the halo of M31 (0.3 ^ t T [ m / H ] ^ 0.5). Couture et al. (1995) [31] undertook a deep V- and /-band study of the halo of M31 in the vicinity of five of M31's globular clusters. They found that the halo was dominated by a stellar population with a mean iron abundance comparable to that of 47 Tuc but that a small component of the red giants had iron abundances of between —1 and —1.5. This is consistent with Pritchet &; van den Bergh's (1987) [114] detection of RR Lyrae variables in the halo of M31. Recently Rich et al. (1996) [119] used the EST to obtain CMDs and luminosity functions of the metal-rich M31 globular cluster G l (— Mayall II) and the field near it. Chapter 1. Introduction 5 They found that the halo luminosity function for red-giants was steeper than Gl's red-giant luminosity function, and had a less pronounced horizontal branch. They compared their M31 halo luminosity function's to those of several Galactic globular clusters covering a range of metallicities and found that no single-metallicity luminosity function could reproduce the M31 halo's luminosity function. Deep C M D studies have the potential to provide direct information on the chemical composition and age of the halo of M31. This is important as there is some debate as to the ages of the globular clusters in the M31 system. There is considerable evidence that the halo of M31 is an old stellar system. First, integrated colours (Frogel et al. 1980 [55], Bohlin et al. 1993 [18]) and the morphologies of CMDs of individual clusters suggest that the M31 globular cluster system has an age comparable to that of the Galactic globular cluster system. Second, no extended asymptotic-giant branch stars have been detected in the halo or the bulge of M31 (Rich & Mighell 1995 [120]) as would be expected if an intermediate-age population was present. Third, the presence of RR Lyrae variables implies that at least part of the stellar population is old. Oh the other hand, measurements of H/3 enhancements (Burstein et al. 1984 [21], Tripicco 1989 [146]) in the spectra of M31 globular clusters, relative to spectra of Galactic globular clusters, have led to speculation that the M31 globular clusters could contain a higher fraction of main-sequence stars than the Galactic globular clusters do. This would imply that the M31 globular clusters are younger than their Galactic counterparts. It has recently been suggested (e.g. Faber 1995 [50]) that B.3 enhancements in red elliptical galaxies are due to the presence of a population of young main-sequence turn-off stars. It is, therefore, of great interest to obtain CMD-based determinations of the mix of ages and metallicities in the nearest population of stars resembling a "red elliptical" galaxy—the halo of M31. Ashman & Bird (1993) [4] have identified several distinct groups of globular clusters Chapter 1. Introduction 6 in the M31 system based on the positions and radial velocities of the clusters. These groups may be associated with sub-structure in the stellar halo of M31. This is sim-ilar to the situation in the Galactic halo where several globular clusters are believed to have originated in external galaxies that are orbiting, or have been accreted by, the Galaxy. Lin & Richer 1992 [86] show that four young globular clusters are located in the Magellanic Stream and thus may have been stripped from the Large Magellanic Cloud. Ibata et al. 1994 [77] found a dwarf galaxy crashing into the far side of the Milky Way. If accretion has played a significant role in adding material to the halo of M31 then stars from different accretion events could have different ages and metallicities resulting in the halo of M31 resembling a patchwork of different stellar populations. Any such clumps in the M31 halo will be seen in projection against the rest of the halo so a C M D will show features from both the underlying halo population and the population of the clump. If the halo of M31 is made up of accreted clumps then CMDs of the halo will show evidence of multiple populations. The exact mix of stellar populations will vary from one accreted clump to the next. The formation of galactic halos is a central problem in astronomy and has implic-ations for our understanding of galactic formation and evolution. Until recently it has been assumed that the spiral galaxies in the Local Group (M33, M31, and the Milky Way galaxy) formed at similar times out of the same proto-galactic material. However, evidence that the halo of M31 may have very different chemical abundances from those of the Galactic halo has opened up the possibility that M31 and the Galaxy have had very different formation histories, or chemical enrichment histories, or both. If this is the case then the assumption that all galaxies of a particular Hubble type have comparable stellar populations will have to be rethought. It should, however, be noted that the current estimates of the metallicity of the halo of M31 are dependent on the distance to M31. If the distance to M31 is increased then the halo metallicity will decrease. Chapter 1. Introduction 7 1.3 Globular Clusters in M31 Globular star clusters are self-gravitating collections of between 104 and 106 stars that are usually associated with galaxies, although there is evidence that some clusters of galaxies contain a population of "free" globular clusters that are associated with the potentials of clusters of galaxies as a whole, and not any individual galaxy (West et al. 1995 [155]). Almost all of the Galactic globular clusters are made up of stars with a single over-all chemical abundance suggesting that they formed in a single star-formation event1. Globular clusters typically have integrated magnitudes of —10' < My < —4, making globular cluster systems visible out to redshifts of z ~ 0.04, the approximate distance to the Great Wall galaxies. The shape of the globular cluster luminosity function has been assumed to be universal so it has been used as a distance indicator (see Harris 1991 [62] for a review). However, recent work has suggested that the shape of the globular cluster luminosity function may depend on the metauicity of the globular cluster system (Ashman et al. 1995 [3] ) and the details of the dynamical evolution of individual globular clusters (e.g. Okazaki & Tosa et al. 1995 [105], Murali & Weinberg 1996 [102]). Because of the wide-spread use of globular clusters to determine distances to external galaxies it is important to determine if globular clusters truly are the same from one galaxy to the next. This is best determined by studying the physical structures and stellar populations of globular clusters in nearby external galaxies. The nearest large globular cluster system outside the Milky Way Galaxy is that of the Andromeda Galaxy (= M31 = NGC 224 = U G C 00454). M31 is located at a distance of 725 kpc (/x0 = 24.3, van den Bergh 1991 [148])2 so individual stars in M31's 1 The most massive Galactic globular clusters, such as w Cen, do exhibit spreads in metallicity. This is probably due to the globular cluster being massive enough to retain debris from supernovae explosions for use in a second generation of star formation. 2 Recent parallax measurements from the Hipparcos satellite suggest that M31 may be at a distance of ~ 900 kpc (Feast & Catchpole 1997 [53]) while others (e.g. Madore & Freedman (1997) [93]) have Chapter 1. Introduction 8 globular clusters can be easily resolved with the Hubble Space Telescope (HST) and large ground-based telescopes at sites with sub-arcsecond seeing. M31's low inclination (i = 12°.5, Hodge 1992 [71]) means that many of its globular clusters are not superimposed against the disc of M31 making identification of globular clusters, and photometry of their stars, relatively straight-forward. M31 offers a unique laboratory to study the outer regions of globular clusters. Star counts are not reliable in the outer regions of Galactic globular clusters since the projected stellar densities of the clusters beyond distances of approximately half the tidal radius, are overwhelmed by random fluctuations in the background stellar number density (Innanen et al. 1983 [78]). However, M31's globular clusters have sufficiently small angular sizes (9 ~ 10" to 30") that both the cluster and the background can be imaged on a single large format CCD image. This eliminates the need to match photometric zero-points between the cluster and the background, which makes possible a more precise subtraction of the background light from the cluster light. Therefore, isophotal analysis is easier to perform on an M31 cluster than it is on a Galactic cluster. M31 is the nearest "grand design" spiral galaxy to our own and is often considered a twin of the Milky Way galaxy. M31 is the most massive member of the Local Group of galaxies and is approaching us at Vh = —297 km-s"1. The galaxy has ~ 300 globular clusters, approximately twice the number that our Galaxy has, but has a comparable globular cluster specific frequency3. For M31 5M31 = 0.7 ± 0.2 while for the Galaxy ^Gai = 0.5 ± 0.1 (Harris 1991 [62]). The globular cluster luminosity function is the same as for the Galaxy suggesting that both globular cluster systems have undergone similar overall dynamical evolutions. argued that the Hipparcos parallaxes are consistent with the accepted distance to M31 of ~ 725 kpc. In this thesis results are presented for both distance scales. 3Specific frequency, S , is essentially the number of globular clusters per unit luminosity of the parent galaxy. Chapter 1. Introduction 9 The first study of the internal structures of globular clusters in M31 was undertaken by Battistini et al. (1982) [7], who estimated core radii for ~ 120 clusters. Pritchet & van den Bergh (1984) [115] found that the surface brightness profile for G l (= May-all II)4 had an excess of light at large radii compared to the best-fitting seeing-convolved analytical King (1966) [82] model (King models, and their cousins Michie-King models, are discussed in detail in Section 3.2.1). They did find that G l was well fit by empirical King (1962) [83] models with core radii of rc <; 0.5 pc. Crampton et al. (1985) [33] used seeing-convolved King (1962) [83] models to derive a relation between the observed full-width at one-quarter maximum of the surface brightness profiles and true core ra-dius. They used this relation to estimate the core radii for nearly 500 M31 globular clusters. Their values, however, are systematically ~ 50% larger than those of Battis-tini et al. (1982) [7], despite the better seeing conditions of the Crampton et al. (1985) [33] data set. Bendinelli et al. (1990) [9] used ground-based data to produce seeing-deconvolved radial profiles for six bright globular clusters in M31. However, seeing and pixel scale limitations restricted them to resolutions of ~ 0'/3, insufficient to resolve the cores of the clusters. Still, their data suggested that M31 globular clusters have King-like profiles similar to those of Galactic globular clusters. The core structures of some of M31's globular clusters have been studied using the pre-refurbished HST. Bendinelli et al. (1993) [8] detected a power-law density cusp in G105 using HST's Faint Object Camera (FOC) images and a variety of image restoration and seeing deconvolution techniques. In addition, Fusi Pecci et al. (1994) [57] used similar methods to obtained half-width at half-maxima (HWHM) and half-light radii (r^) for thirteen M31 globular clusters from F O C images. They found HWHMs similar to the core radii of Galactic globular clusters. Their data, however could not be used to find the 4 T h e G-numbers used in this thesis are from Sargent et al. 1977 [127]. The prefix " K " is used in some of the astronomical literature but this thesis always uses the prefix " G " . Chapter 1. Introduction 10 tidal radii of these clusters as the pre-refurbishment FOC's point-spread function (PSF) overfilled the FOC's field of view. Cohen & Freeman (1991) [30] derived tidal radii for thirty M31 globular clusters by fitting seeing-convolved King (1962) [83] models. Although their fits to individual clusters were quite uncertain they did find a mean tidal radius for the M31 clusters— after adjustment for differences in galactic masses and rotation velocities-—that was very similar to that of the Milky Way clusters. Globular clusters do not exist in isolation but sit in the tidal field of a galaxy. Any stars that move beyond the tidal radius of a globular cluster will have velocity vectors similar to the velocity vector of the globular cluster. This can result in a globular cluster being surrounded by an extended halo of unbound stars which move in approximately the same direction as the globular cluster and have approximately the same velocity. This idea has been explored numerically by Oh & Lin (1992) [103] who predicted that globular clusters could be surrounded by extended halos of escaped stars which can persist for up to a Hubble time. Evidence for extended halos has been observed in some Galactic globular clusters by Grillmair et al. (1995) [59]. In addition, Grillmair et al. (1996, hereafter referred to as GAF) [58] have observed an excess of resolved and unresolved stars beyond the formal Michie-King tidal radii of several globular clusters in M31, as would be expected if extended halos were present. There has been some interest in determining the ellipticities of M31 globular clus-ters. Pritchet &: van den Bergh (1984) [115] measured an ellipticity of e = 0.22 for the region of G l with 12" ^ r <^  35". Spassova et al. (1988) [135] measured ellipticities for approximately two dozen globular clusters while a study by Lupton (1989) [90] suggested that the mean ellipticity measured in the inner 7 to 14 pc of an M31 globular cluster (e = 0.08) is indistinguishable from the mean ellipticity of Galactic globular clusters. In the outer 14 to 21 pc, however, the mean ellipticity of an M31 cluster is 0.11 ± 0.08. Chapter 1. Introduction 11 The mean elhpticity of the Galactic globular clusters is e = 0.08 ± 0.07 while the mean ellipticity of the globular clusters in the Large Magellanic Cloud e = 0.11 ± 0.07. The quoted uncertainties are the standard deviations of the distributions of eUipticities in each galaxy. Unfortunately Lupton does not provide information on the standard errors in the means for these quantities. However, he quotes results from two-tailed Wilcox-ian tests that suggest that the elhpticity distributions of the M31 and Galactic globular clusters are the same at the 99.6% confidence level. A similar test suggests that the dis-tributions for M31 and the Large Magellanic Cloud are the same at the 66% confidence level. Baev et al. (1997) [5] found systematic differences between the shapes of M31's disc and halo globular clusters. They found that the disc globular clusters are triaxial ellipsoids while the halo globular clusters are oblate or prolate spheroids, but cautioned that this is a preliminary result since the sample of halo globular clusters in their study is small compared to the sample of disc globular clusters. The first CMDs for globular clusters in M31 were for G l by Heasley et al. (1988) [65] and G219 (= Mayall IV) by Christian k Heasley (1991) [29]. Couture et al. (1995) [31] un-dertook a systematic study of five M31 globular clusters ( G i l , G319, G323, G327 = May-all VI, and G352 = Mayall V) with a range of iron abundances. Unfortunately none of these ground-based studies was able to reach the level of the horizontal branch at V ~ 25. The first CMDs constructed from HST data were for G l (Rich et al. 1996 [119]); G58, G105, G108, and G219 (Ajhar et al. 1996 [1]); and G280, G351, and Bo468 (Fusi Pecci et al. 1996 [56]). These CMDs were able to resolve stars one to two magnitudes below the red portion of the horizontal branch. Chapter 1. Introduction 12 1.4 The Goals of this Thesis The original goals of this work were to study the internal structures of the globular clusters in M31 and to compare them to the internal structures of the globular clusters in the Milky Way Galaxy. Globular clusters are believed to have formed at the same time as the Galactic and M31 halos, making them some of the oldest objects in the Universe. Therefore globular clusters can provide a tracer for the formation and dynamic evolution of their parent galaxies. In this work I will determine the core, tidal, and half-mass radii of approximately two dozen globular clusters in M31 and compare the distributions of these quantities with the distributions of these radii for the Galactic globular clusters. In addition, I will measure the projected ellipticities and the orientations of the apparent major axes of the M31 globular clusters and compare these to those of the Galactic globular clusters. The internal structures of globular clusters are determined by the stellar dynamics within the clusters, and by the tidal field that the clusters are located in. If the globular clusters' structural parameters turn out to be similar then this would suggest that the two galaxies have had similar dynamical histories. If they turn out to be significantly different then this would suggest that M31 and the Milky Way Galaxy, despite being morphologically similar, have lived very different lives. These differences would then have to be taken into account in any theory of galaxy formation. While this work was in progress I was able to obtain time on the HST to image two globular clusters in M31. This data was used to determine structural parameters of the globular clusters to a much higher degree of precision than could be done us-ing the ground-based data presented in this thesis. The structural parameters deter-mined using the space-based observations will be compared with those obtained using the ground-based observations in order to see if ground-based observations can return Chapter 1. Introduction 13 reliable structural parameters. Another goal for the HST data is to search for extended halos of escaped stars around G302 and G312. Such halos have been predicted by dynamical models of globular clusters, and have been observed around some Galactic globular clusters. The small apparent size of the M31 globular clusters make them ideal targets for searching for extended stellar halos. Finally, the HST data includes background fields in the halo of M31. This data will used to construct deep CMDs that could be used to study the stellar populations in the halo of M31. I will estimate the metalhcity distribution of stars in the halo of M31 and compare this with the metalhcity distribution in the Galactic halo and the M31 globular cluster system. Chapter 2 The CFHT Data 2.1 Observat ions The ground-based image data used in this thesis were obtained at the Canada-France-Hawai'i Telescope1 (CFHT) between August 16 and 20, 1990 by Greg Fahlman and Carol Christian. Deep V- and /-band photometry2 was obtained for fifteen fields in M31 that contained twenty-two globular clusters. Most of these fields were located roughly along the projected minor axis of M31 and extend from near the centre of that galaxy out to a projected distance of ~ 40 kpc. The minor axis fields were chosen such that they were spaced at approximately equal logarithmic intervals along the minor axis. In addition some of the fields are located off the minor axis in order to image bright globular clusters that did not he on the minor axis. The D A O / C F H T HRCam (McClure et al. 1989 [95]) with the SAIC 1 C C D detector was used to obtain the images. The HRCam is an adaptive-optics system that performs tip-tilt corrections to reduce the effects of seeing on the images. The SAIC 1 C C D chip has a low read-out noise, RN = 6.5 e~, and a gain of 1.6 e~ /ADU, making it well suited 1 The Canada-France-Hawai'i Telescope is operated by the National Research Council of Canada, le Centre National de la Recherche Scientifique de France, and the University of Hawai'i. 2 The V-band filter used here had a central wavelength of Ao = 5485 A and a bandwidth of A A = 908 A. The /-band filter used here had a central wavelength of Ao = 8325 A and a bandwidth of A A = 1975 A. 14 Chapter 2. The CFHT Data 15 for looking at low surface brightness objects such as the outer regions of partially resolved globular clusters. The image scale with this set-up is (X'131 per pixel and the C C D covers an area of ~ 2 • ' on the sky. Each of the four nights were photometric with the typical seeing being 6 ~ (K'7. Table 2.1 hsts the fields that were imaged. Some of these fields contain more than one globular cluster. The right ascensions and declinations are for the approximate centres of each field and were taken from Crampton et al. (1985) [33] and Huchra et al. (1991, hereafter referred to as HBK) [76]. N is the number of exposures taken in each filter. Figure 2.1 shows the locations of the fields relative to the disc of M31. Chapter 2. The CFHT Data 16 Field a 8 Date Filter Time RN Gain N (2000.0) (2000.0) (1990) (0 (ADU) 1 00h33m33^6 +39°31'20" Aug. 19/20 V 100.0 4.08 1.6 1 I 100.0 4.08 1.6 1 2 00h40m48*9 +41°11'31" Aug. 18/19 V 30.0 4.08 1.6 1 3 00h42m34M +41°13'30" Aug. 16/17 V 450.0 4.08 1.6 1 I 100.0 4.08 1.6 1 4 0 0 h 4 2 m 4 4 s 2 +41°14'56" Aug. 16/17 V 300.0 4.08 1.6 1 I 100.0 1.67 9.6 6 5 00h42m58*7 +41°08'52" Aug. 17/18 V 600.0 4.08 1.6 1 I 240.0 2.88 3.2 2 6 00h43m15!5 +41°06'32" Aug. 18/19 V 300.0 2.88 3.2 2 I 120.0 1.82 8.0 5 7 00h43m18*3 +41°10'55" Aug. 17/18 V 450.0 4.08 1.6 1 I 360.0 4.08 1.6 1 8 00h43m2E7 +41°15'27" Aug. 17/18 V 300.0 2.88 3.2 2 I 120.0 1.82 8.0 5 9 00h43m36*8 +41°07'30" Aug. 17/18 V 300.0 4.08 1.6 1 I 240.0 2.88 3.2 2 10 00h44m03=3 +41°04'28" Aug. 18/19 V 450.0 4.08 1.6 1 I 450.0 4.08 1.6 1 11 0 0 h 4 5 m l £ 0 +41°16'03" Aug. 16/17 V 450.0 4.08 1.6 1 I 450.0 4.08 1.6 1 12 00h45m25=2 +41°05'30" Aug. 16/17 V 1000.0 2.88 3.2 2 I 400.0 2.88 3.2 2 13 00h45m58*8 +40°42'32" Aug. 17-20 V 600.0 1.32 15.3 15 I 321.4 1.09 22.4 14 14 00 h50mlK0 +41°41'01" Aug. 18/19 V 600.0 2.36 4.8 3 I 300.0 2.36 4.8 3 15 00h51m33*9 +39°57'34" Aug. 18/19 V 450.0 4.08 1.6 1 Table 2.1: The C F H T fields. Chapter 2. The CFHT Data 17 a ( 2 0 0 0 ) Figure 2.1: This figure shows the approximate locations of the fields listed in Table 2.1. The ellipse shows the approximate size and orientation of the optical edge of the disc of M31 and the cross shows the approximate location of the centre of M31. Chapter 2. The CFHT Data 18 2.2 D a t a R e d u c t i o n 2.2.1 Preprocessing Bias subtraction and flat fielding was done for all images using standard C C D pre-processing methods (e.g. Walker 1990 [152]). First, an electronic pedestal voltage was subtracted from each image. Next, bias frames were created by taking several (~ 60) zero-second exposures with the shutter closed. A master bias was created by median-combining these images. The master bias frame was then subtracted from each image. The dark current for the SAIC 1 C C D chip is 0.2 e - per minute per pixel ( C F H T Ob-servers' Manual 1990 [26]). The longest exposure (1000 seconds) has a total dark count of 3.3 e~ per pixel, which corresponds to 2.08 A D U per pixel. Assuming a sky brightness of fiv = 21.1 mag per • " (CFHT Observers' Manual 1990 [26]) the contribution to the noise from the dark current will be only ~ 1.5% of the contribution from a uniform sky. The presence of unresolved stars in the disc and halo of M31 cause considerable surface brightness fluctuations in the sky which contribute to the noise in the images. These surface brightness fluctuations are large compared to the level of dark current in these exposures. Therefore the dark current can be ignored since its contribution to the total noise is small compared to the contribution from surface brightness fluctuations. An ideal C C D would have uniformly sensitive pixels. However, in real life variations in pixel-to-pixel sensitivity can be quite large. To compensate for this several images of blank regions of sky were taken during the morning and evening twilight periods. The twilight sky provides a near-uniform illumination of the CCD at similar wavelengths to those of images of the night sky. Unfortunately the twilight sky is not a truly uniform source of light. Stars, galaxies, and other objects are still present in the sky and they can rise above the noise during long exposures. This contamination can be reduced by Chapter 2. The CFHT Data 19 selecting regions of the sky that are devoid of bright objects and taking a large number of short exposures instead of a small number of long ones. The location on the sky of each exposure can be shifted by a few pixels to prevent underlying structure in the sky from illuminating the same pixel and thus appearing to be a flat-fielding artifact. Since the pixel sensitivity is wavelength dependent this process must be performed for each filter used. Several sky flats were obtained during either the morning or evening twilight periods on each night and these images were median-combined to produce a master flat-field frame for each filter. Each flat-field frame is normalized to a mean pixel value of one and then the data images were divided by the appropriate master flat-field frames. The next step is to remove cosmic rays from the images. A cosmic ray striking a pixel will create a high signal that will stand out above the pixel-to-pixel noise in the image. These pixels were identified as follows. First, each pixel is compared with its neighbours i n a 5 x 5 o r 7 x 7 comparison box. If the flux in that pixel exceeded the mean flux in the box of surrounding pixels by ~ 5<r, where a is the standard deviation in the sky in the comparison box, that pixel is flagged as a potential cosmic ray. Next, a plane is fit to the border pixels of the box around each cosmic ray candidate and subtracted. The mean flux in the background-subtracted box is then compared with the flux in the candidate pixel. If the ratio of these fluxes is less than ~ 0.05 then the pixel is flagged as a cosmic ray and replaced by the mean of its neighbours. The I R A F 3 implementation of this algorithm NOAO.IMRED.CCDRED.COSMICRAYS was used to remove cosmic rays from the data images. The box sizes and other parameters of the algorithm depend on the noise characteristics, seeing, background structure, and cosmic ray characteristics so they were interactively adjusted for each image. The number of cosmic ray hits depended 3Image Reduction and Analysis Facility ( IRAF) , a software system distributed by the National Optical Astronomy Observatories (NOAO). Chapter 2. The CFHT Data 20 on the exposure time of the images. On a typical 1000 second exposure the procedure described above found a few dozen cosmic rays. Multiple exposures were made of most of the fields. The deep exposures in each field were combined to improve their signal-to-noise properties in the following way. First, the individual frames were examined and those that were of significantly lower quality than the others (e.g. worse seeing, evidence of telescope motion, etc.) were discarded. The remaining images of each field were re-registered to a common coordinate system and then averaged using the appropriate IRAF routines. Table 2.1 lists the exposure times, noise characteristics, and number of C C D images combined for each field imaged in this study. The read-out noise and gain values have been adjusted for the number of frames that were combined to get a final image in each filter. Chapter 2. The CFHT Data 21 2.2.2 Calibrating the Data During the observing run several standard stars from the lists of Landolt (1983) [85], and the M92 Consortium fields of Stetson &; Harris (1988) [142], were observed. The standard star data was processed by Hodder (1995) [70] to obtain magnitude zero-points for the first three nights of the observing run. Unfortunately no standard stars were observed on the night of August 19/20 so these data were calibrated by calculating the zero-point offset between M31 halo stars observed in the G312 field on the night of August 18/19 (for which standard star observations were made) and the night of August 19/20 (for which no standard stars were observed). This was the only field observed on the night of August 19/20 that was also observed on one of the three photometric nights. The photometric zero-points for the night of August 19/20 are ~ 2cr brighter than the zero-points for the other three nights. This is probably due to the night of August 19/20 not being photometric and suggests that magnitudes determined on night 4 be treated with some caution. The relationship between the observed count rate, C , in A D U per second, and the calibrated surface brightness, u, in magnitudes per square arcsecond, is: tt = K 0 - 2.51og10(C) - 2.51og10(l/5'2) (2.1) where Ko is the zero-point, and S is the pixel scale (O'.'131/pixel for the data presented here). The zero-points, and their standard errors, for each night are listed in Table 2.2. Chapter 2. The CFHT Data 22 Night Date K0,v Koj 1 Aug. 16/17 24.023 ± 0.005 23.827 ± 0 . 0 2 7 2 Aug. 17/18 24.053 ± 0.019 23.910 ± 0.009 3 Aug. 18/19 24.040 ± 0.009 23.873 ± 0.034 4 Aug. 19/20 23.767 ± 0.019 23.668 ± 0 . 0 1 1 Table 2.2: Photometric zero-points for the HRCam images. Chapter 2. The CFHT Data 23 2.2.3 Background Subtraction The unresolved background light from the M31 halo and disc was removed by fitting and subtracting a surface on each frame. To prevent high-frequency contamination in the fitted background a single pass with DAOPHOT II (Stetson 1987 [141], 1991 [140], 1992 [139], Stetson et al. 1990 [143]) was made to identify and remove field stars from each image. In addition, galaxies and globular clusters were modeled and subtracted using the IRAF STSDAS.ISOPHOTE.ELLIPSE and STSDAS.ISOPHOTE.BMODEL tasks. The resulting images were visually inspected and regions with poor subtractions, and other cosmetic defects, were masked. Two-dimensional third-order splines were fit to most of the images to allow for large-scale variations in the background. For some fields subtracting a constant background was sufficient. Residual large-scale variations across each field after subtracting the fitted back-grounds were usually less than ~ 1% of the original background value. The fields located near the centre of M31 showed larger residuals than those located away from the galaxy. This is due to the large amount of small-scale galactic structure near the centre of M31. A visual examination of fields such as the G185 field or G177 reveals the presence of dust lanes and small-scale surface brightness fluctuations arising from unresolved stars along the line-of-sight in M31. Figures 7.1 and 7.2 in Chapter 7 show an example of the detailed structure that becomes visible in many of the fields once the large-scale structure in the unresolved light has been removed. Chapter 2. The CFHT Data 24 2.3 Seeing A typical Galactic globular cluster has a King (1966) [82] core radius of ~ 1 pc and and tidal radius of ~ 125 pc (although there is a large spread in these values from cluster to cluster). At the distance of M31 these sizes correspond to angles on the sky of ~ (Y/25 and ~ 35" respectively. Since the seeing discs for this data typically had FWHMs of ~ (X'7, the observed shapes and light distributions of the clusters were significantly affected by the seeing. In order to correct for this it is necessary to accurately determine the shape of the PSF and its dependence on position on the CCD image. Figure 2.2 shows that the star-to-star scatter in the stellar F W H M at the location of the cluster exceeds the systematic change in F W H M across the cluster by a factor of ~ 4. This suggests that the positional variations in the PSF can be ignored if only those stars located near the cluster are used to define the PSF. I found that a modified Moffatian (Moffat 1969 [100], Kormendy & McClure 1993 [84]): M(r) = Mo . r 1 + a (2.2) provided a good fit to the isolated bright stars on most of the frames. The central intensity is given by Mo = M(0). The parameter a is an estimate of the width of the intensity profile and is approximately given by 105 • A / r 0 (Walker 1987 [153]) where A is the wavelength of the light and r 0 is the coherence length of the atmospheric turbulence. The parameter (3 controls the sharpness of the intensity profile and is a measure of the spectrum of the sizes of the atmospheric turbulence cells that cause the seeing. 7 controls the size and shape of the core of the seeing profile. Moffatians were fitted to several stars in each image in order to find the mean seeing profile for each image. This functional form, however, assumes a circularly symmetric PSF—which was not the case for the C F H T data presented here. Typical ellipticities, Chapter 2. The CFHT Data 25 e = 1 — b/a, for the PSFs were 0.03 <; e <; 0.08. In principle the ellipticity can be treated by fitting a two-dimensional Moffatian to the stellar profiles. However the PSF derived using the DAOPHOT II PSF routine includes a table of residuals which will compensate for deviations in the PSFs away from the assumed functional form. This combination of analytical and empirical components enables DAOPHOT II to more accurately model the shape of the PSF than the purely analytical Moffatian description can. Therefore, this thesis uses the DAOPHOT II PSFs for seeing-convolutions. Since the seeing varies as a function of time and colour a separate PSF was determined for each cluster and each filter. Seeing parameters for each PSF are given in Table 2.3. If Moffatian parameters (a, 8, and -y) are given then the F W H M was computed from the one-dimensional Moffatian using F W H M = 2a(21^ - lfh. The ellipticities, e, and position angles, t90, were measured from the DAOPHOT II PSF. Through-out this thesis position angles are assumed to be measured in degrees east of north on the sky unless otherwise indicated. For some fields it was not possible to obtain a good fit using Mof-fatians. In these cases the F W H M was estimated directly from the surface brightness profiles of several stars in that field. The globular cluster V253 is object number 253 in Vetesnfk's (1962) [150] catalogue. Similarly the globular cluster Bol53 is object 153 in the Battistini et al. (1982) [7] catalogue, Nwhile the globular clusters vdB2 and vdB5 are from van den Bergh (1969) [149]. Chapter 2. The CFHT Data 26 0.8 S ^= CD > QJ !/3 o 0.7 h 0.6 FWHM = ( 4 . 9 7 ± l . l l ) x l 0 - 4 r + ( 0 . 6 4 ± 0 . 0 1 ) 50 100 Distance from HRCam Guide Star (") 150 Figure 2.2: The Gaussian FWHMs of the stellar profiles vary with the distance from the HRCam guide star. This figure shows the variation in F W H M for the /-band image of G302. The variations in this figure are typical of the variations seen during the entire C F H T observing run. G302 is located ~ 40" from the HRCam guide star and has a tidal radius of ~ 10". Since the systematic changes in the stellar F W H M over the region of the sky covered by the globular cluster is less than the scatter in the F W H M at a given distance from the guide star the PSF was assumed to be constant over the region of the C C D covered by the globular cluster. The solid line is the best-fitting straight line to the data. Chapter 2. The CFHT Data Field Filter a 7 F W H M e 0o G2 V 0"66 2.65 2 0V73 0.06 40° I 0"65 2.30 2 0"78 0.12 - 2 0 ° G70 V 0'.'68 0.05 39° G177 V 0"77 2.30 2 0'.'92 0.04 - 4 9 ° I 0"56 2.00 2 ff!72 0.08 - 6 4 ° G185 V 0'/69 0.03 - 8 ° I 0V72 0.20 - 6 4 ° G185B V 0'.'69 0.03 - 8 ° I 0772 0.20 - 6 4 ° G196 V 0"64 2.47 2 a'73 0.04 I 0"47 1.93 2 0'.'62 0.06 G208 V 0"83 2.21 2 l'.'Ol 0.03 I 0"42 1.75 2 &!59 0.07 G212 V l'/07 4.13 2 a;86 0.14 - 8 0 ° I 0"66 3.47 2 0'.'62 0.13 - 7 7 ° G213 V &!8Z 0.06 - 8 9 ° I 0'.'56 0.06 - 6 3 ° G215 V 1'.'07 4.13 2 0786 0.14 - 8 0 ° I 0"66 3.47 2 f/'62 0.13 - 7 7 ° G218 V l'/07 4.13 2 0'.'86 0.14 - 8 0 ° I 0"66 3.47 2 0'.'62 0.13 - 7 7 ° G222 V 0V41 2.06 2 0.04 I 0'.'40 1.75 2 0756 0.08 G231 V 0V58 1.85 2 0778 0.01 55° I Ql'.bQ 2.02 2 0/.'72 0.09 - 9 ° G233 V 0"58 1.85 2 0'.'78 0.01 55° I 0'.'56 2.02 2 C/72 0.09 - 9 ° G263 V 0"71 0.07 32° I C'71 0.10 5° G302 V (y.'58 1.90 2 ff!77 0.04 65° I a ;5i 2.15 2 ff!6Z 0.08 60° G312 V 0"56 2.51 2 0'.'64 0.03 90° I 2.40 2 0758 0.04 - 3 5 ° G352 V 0773 0.07 53° I 0770 0.10 30° G355 V 0778 0.04 - 7 2 ° Table 2.3: Seeing characteristics for the C F H T HRCam images. Chapter 2. The CFHT Data F i e l d F i l t e r a /3 7 F W H M e 00 B o l 5 3 V 0"83 2.21 2 1701 0.03 I 0"42 1.75 2 0759 0.07 v d B 2 V 0769 0.03 - 8 ° I 0772 0.20 - 6 4 ° v d B 5 V 07 77 2.30 2 0792 0.04 - 4 9 ° I a;56 2,00 2 0772 0.08 - 6 4 ° V 2 5 3 V l ' /07 4.13 2 0786 0.14 - 8 0 ° I 0'.'66 3.47 2 0762 0.13 - 7 7 ° 2.3 (cont inued) Chapter 3 C F H T Results 3.1 Star Counts Star counts were obtained in the outer regions of G302 and G312 using standard DAO-P H O T / A L L S T A R techniques. To aid in finding stars the integrated light from each cluster was modeled using the ARTDATA package in IRAF and subtracted from each image. Only-stars located at least 25 pixels (~ 3'.'3) from the cluster centre were used since stars nearer to the centre suffer from severe incompleteness problems. Subtracting the model cluster significantly altered the photometric properties of the stars located where the model was subtracted. However, since this study is primarily interested in the number density of stars in the outer regions of the clusters, not precise magnitudes of the individual stars, photometric accuracy is not required. To minimize contamination from cosmic rays, noise spikes, pixel defects, residuals from the subtraction of the unresolved light and other false detections, the V- and /-band star lists were matched and only objects that appeared in both bandpasses were considered to be stars. Due to the short exposure times for many of the HRCam images, stars could not be resolved in most of the clusters. The adaptive kernel smoothing techniques described by Merritt & Tremblay (1994) [96] were used to create smoothed one-dimensional surface density profiles from the G302 and G312 star count data. Adaptive kernel smoothing is done by placing a Gaussian of 29 Chapter 3. CFHT Results 30 standard deviation h at the location of each star then summing the contributions from all the Gaussians. The choice of the smoothing parameter, h, is somewhat arbitrary. Silverman (1982) [131] suggests using h = 1.06o-n-1/5 where n is the local number of data points and cr is the standard deviation of the data, for Gaussian-like distributions since this approximately corresponds to minimizing the integrated mean square error between the derived smoothed profile and the intrinsic profile that the data are drawn from. However, star counts in the outer regions of a globular cluster do not follow a Gaussian surface number density distribution so this prescription will underestimate h. Vio et al. (1994) [151] suggest using h = 0.9 min(cr, a;/1.34)n~1/'5, where x is the in-terquartile range of the distribution, since this is valid for a wider range of distributions than the Gaussian approximation is. Both of these schemes tend to provide too little smoothing resulting in significant noise in the smoothed profile. After some experiment-ation I adopted values of h roughly two to three times larger than those obtained using these data-based schemes. Figure 3.1 shows the smoothed projected surface density profiles for G302 and G312 with the mean stellar backgrounds subtracted. These profiles were obtained using the adaptive kernel smoothing procedure described above and were used to estimate the limiting radius of the globular clusters. Since the projected stellar densities in the outer regions of the globular clusters are similar to those in the halo of M31 the detailed shape of the smoothed star count profile will be strongly influenced by statistical fluctuations in the actual number of stars and by the details of the smoothing procedure. Therefore Figure 3.1 has only been used to estimate the location of the outer edge of each globular cluster. A comparison between the outer edge of the smoothed stellar density profiles and the outer edge of the integrated light profiles shows that both methods find the outer edges of the globular clusters to be at approximately the same locations. This suggests that both integrated light profiles and stellar density profiles provide consistent ways of Chapter 3. CFHT Results 31 locating the outer edges of globular clusters in M31. The dashed lines in Figure 3.1 show the Icr uncertainties in the smoothed star count profiles for each globular clusters. These uncertainties were estimated by generating 10,000 star fields using bootstrap resampling of the original stellar positions for each globular cluster. These data sets were smoothed in the same manner as the original data. The upper and lower dashed lines indicate the approximate uncertainty in the shapes of the smoothed star count profiles. The uncertainties are small when the stellar density of the cluster stars is greater than the stellar density of the background. For G302 the stellar density profiles give a Icr range for the limiting radius of 11'.'8 < Him < 1571. For G312 the limiting radius from Figure 3.1 is 9"2 < r H m < 1371 (also see Table 3.2). The structures seen at radii less than log10(reff) ~ 1.2 in each cluster are probably artifacts caused by small number statistics. The density enhancement at l°Sio(reff) ~ 1-1 is only significant at the Icr level and has a width approximately equal to the smoothing scale, h, at that radius. This suggests that this density enhancement is not real. Also, the integrated light profiles of each cluster are still dropping at log10(reff) ~ 1 suggesting that the stars in this density enhancement are still a part of the cluster. The density enhancements at log10(reff) ~ 1.3 He beyond the fitted Michie-King tidal radii of the globular clusters (see Tables 3.1, 3.2, and 5.4). It is not clear that these enhancements are anything other than a statistical effect caused by the small numbers of stars located at radii between ~ 10" and ~ 25". G312, for example, has only 45 stars between these radii, giving it a projected stellar density of S = 0.027 ± 0.004 stars/D", well within the Icr uncertainty of the background density, Sbkgd = 0 .026±0 .004 stars/D". Further, the widths of these density enhancements are approximately the same as the smoothing parameters used to generate the projected surface density profiles. However, these density enhancements are reminiscent of the halo of escaped stars predicted by Oh &; Lin (1992) [103]. This possibility will be explored in more detail in Chapter 5.4. Chapter 3. CFHT Results 32 Figure 3.1: Adaptive kernel smoothing of the star count data for the outer regions (r > 25") of G302 and G312 (solid lines). The error curves (dashed lines) were obtained from bootstrapping and represent the combined Icr uncertainties in the smoothing and in the background determinations. The stellar background densities are indicated by the arrows and filled circles represent the J-band integrated light profiles. The agreement between the limiting radii determined from star counts and the integrated light profiles can be seen. The turn-down at log10(reff) ~ 0.5 (reff ~ 3") on each profile is an artifact caused by truncating the star-count data at reff = 25 pixels. Chapter 3. CFHT Results 33 3.2 Fitting Michie-King Models 3.2.1 The Theory Behind Michie-King Models It is possible to use straight-forward dynamical models of globular clusters to predict the surface brightness profile that a cluster will have on the sky. In this thesis I am primarily interested in the overall structures of the globular clusters of M31, not in their detailed spatial and dynamical structures. Therefore I will use the models of Michie (1962) [97] and King (1966) [82] to describe their observed surface brightness profiles. To first-order accuracy globular clusters can be approximated as colhsionless spherical systems of stars moving under the influence of a smooth, self-generated, gravitational potential $ ( x , £ ) . At a given time, t, the phase-space density of the stars is given by / ( x , v ; £ ) . The phase-space density is simply the number of stars at position x with velocity x = v so it is physically meaningless to have / (x , v ; £ ) < 0. Each star is acceler-ated by the smooth underlying gravitational potential generated by the other stars so a star's acceleration is given by a = v = x = — V $ ( x , £ ) . If close encounters between stars are ignored then stars flow smoothly from one part of phase-space to another and the continuity equation holds: df df » df » . df , , dt ot fr[ oxi ^ ovi which is equivalent to ^ + v . V / - V * - ^ - = 0. (3.2) ot ov Equation 3.2 is called the colhsionless Boltzmann equation and is the fundamental equa-tion of stellar dynamics in the absence of star-to-star interactions. The three-dimensional space density at location x can be found by integrating / (x , v; t) with respect to velocity: Chapter 3. CFHT Results 34 p(x) = UJ f(x,v)47rv2dv. (3.3) |v|>0 A globular cluster does not extend to infinity but has a physical cut-off radius. Any stars whose velocities carry them beyond this radius will no longer be gravitationally bound to the cluster. Modifying the King (1966) [82] distribution function to include velocity anisotropy gives: /(r,v) = fcexp — — exp -r2vl\ f(V(r) - V(0))\ [._ f-v*\ „ _ f-v2 2r2a2 ) R I -a2 « P h r r - e x P 2cr2 / M 2cr2 (3.4) Here, Vir) is related to the total energy of a star by E = l/2v2 + V(r), the escape velocity for a star at a distance r from the centre of the cluster is ve = \/—2V, and cr is the one-dimensional velocity dispersion of the stars. The radial component of the velocity within the cluster is v, and the component of the velocity that is perpendicular to the radial direction in the cluster is v±. The anisotropy radius of the cluster is r a . Integrating over the velocities (up to the escape velocity) gives: p(W(r), ra) = AV2irka3 ew^~w° / A(V, ra) (e"" - e~w^) V 1 / 2 dn, (3.5) where W(r) = —V(r)/a2 and v — v2/2cr2. The dimensionless central potential is Wo = 1^(0). The anisotropy term, A(n,ra), is given by: exp(-r2/r2aV) r(r/ra)^ , A A(n,ra) = / exp [y ) dy. (3.6) where A(n,ra) goes to one as ra goes to infinity, i.e. the amount of anisotropy in the cluster's velocity dispersion tensor goes to zero as the anisotropy radius goes to infinity. A model with ra — +oo (all of the velocity dispersion is in the radial direction) is called Chapter 3. CFHT Results 35 an isotropic Michie-King model, or simply a King (1966) [82] model. A model with ra ^ +oo (some of the velocity dispersion is perpendicular to the radial direction in the cluster) is called an anisotropic Michie-King model. Since the spatial density is a function of the dimensionless central potential, Wo, and not directly of the radius, r, Equation 3.5 has to be solved using Poisson's Equation: V2W(R) = ^Go-2r2cP(W{R),ra), (3.7) where R = r/rc is the dimensionless radius and rc is the King core radius. The King core radius is defined to be the radius that satisfies — 4:ivGa~2r2p(0) = 9 and approximately corresponds to the half-power radius of the projected mass density of the cluster. Once the spatial density as a function of radius has been found it is then projected into the plane of the sky using: where rt is the King tidal radius of the globular cluster. This is the radius where the spatial density of the cluster drops, to zero, i.e. p(rt) = 0. Unfortunately the term "tidal radius" is somewhat confusing since, in Michie-King theory, rt is not determined by the tidal field of the gravitational potential well that the globular cluster sits in. The King tidal radius is simply the radius at which the cluster escape velocity is ve = 01. These models are based on the work of Michie (1963) [97] and King (1966) [82]. This simple approach results in model globular clusters that are similar to isothermal spheres out to near the tidal radius. In the outer regions of the cluster the spatial density rapidly drops to zero. Michie-King models are parameterized by their central potential and their anisotropy radius. These two parameters determine the shape of the surface 1 Through-out this thesis the term tidal radius will be used since this term is in standard use in the astronomical literature. Chapter 3. CFHT Results 36 density profile of the cluster. The models also contains two scaling parameters: the King core radius, and the central density. Such models have been found to be very effect-ive at reproducing the surface brightness and surface stellar number-density profiles of most Galactic globular clusters (e.g. Peterson & King 1975 [110], Alcaino 1977 [2], Har-ris & Racine 1979 [63], Madore 1980 [94], Djorgovski & King 1984 [45], Webink 1985 [154], Peterson & Reed 1987 [109], and Trager et al. 1993 [145]). In real life, however, globular clusters are not quite this simple. There are several physical effects which need to be considered in order to construct physically realistic dynamical models of globular clusters. Encounters between stars (which are most im-portant in the core of a cluster) can be taken into account by setting the right-hand side of Equation 3.2 equal to a collision term, T(/). This term denotes the rate of change of / due to stellar encounters. Since this thesis does not consider the detailed stellar interactions within a globular cluster I will assume that T(/) = 0, i.e. globular clusters are colhsionless systems. For details on solving Equation 3.2 using the Fokker-Planck approximation see §8.3 of Binney & Tremaine (1987) [17]. Gunn & Griffin (1979) [61] have extended the single-mass Michie-King model described above to incorporate real-istic stellar mass functions. Lupton & Gunn (1987) constructed rotating Michie-King models. For simplicity the single-mass, non-rotating Michie-King models are used in this thesis. Figures 3 and 9 of Gunn & Griffin (1979) [61] suggest that the tidal radii obtained by fitting single-mass Michie-King models are similar to the tidal radii obtained using multi-mass Michie-King models, but the detailed shape of the cluster profile (para-meterized by the central potential Wo) can be significantly different. Similarly, rotation is mainly important when fitting models to the velocity structure of a cluster, something which is not done in this thesis. Chapter 3. CFHT Results 37 3.2.2 Fitting Methods The problem of fitting a model globular cluster to CCD observations is a difficult one. Ideally a physically-realistic model would be used to generate a model cluster and this model would be compared directly with the observed data. A quantitative goodness-of-fit would be defined and the model parameters would be varied until the best-fitting model was found. This approach has several problems associated with it. 1. The light from stars in a globular cluster is smeared out by the PSF, which effect-ively makes the problem one of fitting a model to data that has been degraded by an instrumental response function (in this case the PSF). To make matters worse the observed PSF (which is measured from a C C D image) is an approximation of the true PSF (which is the actual blurring experianced by the stellar image due to seeing) and must be determined observationally. The observed PSF contains noise, which makes a direct deconvolution of the observed images undesirable. 2. All sources of noise must be identified and correctly parameterized in order for a goodness-of-fit statistic to be meaningful. The noise on a C C D image comes primarily from four sources. First, there is read-out noise and quantisation noise inherent in the C C D electronics. The amount of read-out noise is far greater than the amount of quantisation noise for the data used in this thesis. Second, there is Poisson noise from the source itself. For all but the shortest exposures in this thesis the Poisson noise dominates the total noise. Third, there is sky-noise, i.e. Poisson noise arising from scattered light in the sky. For C F H T the sky brightnesses are typically fiy = 21.1 mag per • " and fij = 19.2 mag per •" . Finally, there is Poisson noise from the background that the globular cluster is sitting upon. The contribution from the background varies greatly from field to field. For fields in the halo of M31 (e.g. G302) the background is negligible. However, for fields in the Chapter 3. CFHT Results 38 disc or bulge of M31 (e.g. G185) unresolved background light can be a significant source of noise. 3. Globular clusters are not inherently smooth sources of light. The azimuthal place-ment of stars in a sphericaly symmetric globular cluster is stochastic. Since the number of stars on the upper red-giant branch and the asymptotic-giant branch is small it is possible for these bright stars not to be uniformly distributed at a given radius. The brightest stars in M31 globular clusters can be resolved in the C F H T images. While Michie-King models describe the radial dependence of the surface number density of stars they say nothing about the positions of individual stars within a globular cluster. So, the stochastic placement of the resolved red giants adds an additional component of noise to the C C D image of the globular cluster. Observed surface brightness profiles are dependent on both the underlying surface brightness profile of the object and on the seeing, regardless of whether the seeing is dominated by the atmosphere, as in ground-based observations, or the optical system of the telescope, as is the case for HST observations. For ground-based imaging, seeing effects can significantly alter the observed profiles of extended objects that only subtend a few arcsecond on the sky (e.g. Schweizer 1979 [129]). If the PSF is well defined then it can be used to restore the image. Unfortunately image restoration can be very sensitive to small errors in the PSF and can be biased by noise in the images. The Lucy-Richardson (Richardson 1972 [121], Lucy 1974 [89]) algorithm, for example, will only preserve flux if there is no read-noise present in the image. A second approach involves modeling the data in some way and finding the model that gives the best fit to the observations. This approach requires that the form of the PSF, and underlying model that describes the source, be well known. The model is Chapter 3. CFHT Results 39 convolved with the seeing and compared in some way to the original data. These tech-niques avoid the enhancement of noise that can occur in image restoration but requires a priori knowledge of the underlying model that describes the data. Michie-King models (Michie 1963 [97], King 1966 [82]) and their extensions (e.g. Gunn & Griffin 1979 [61], Lupton & Gunn 1987 [91]) have been found to give reasonable fits to the surface bright-ness profiles and stellar density profiles of roughly three-quarters of the Galactic globular clusters. Since the Michie-King parameterisation has been successfully used for so many Galactic globular clusters (see Trager et al. 1993 [145] for a recent compilation) it is a natural choice for use with the M31 globular clusters. Michie-King models have been used purely to facilitate the comparison of the shapes of the surface brightness profiles of the M31 globular clusters with those of the Galactic globular clusters. The fits presented in this thesis should not be interpreted as being representative of the dynamical states of the clusters in question. For simplicity I have chosen to use single-mass Michie-King models instead of the more physically realistic multi-mass models of Gunn & Griffin (1979) [61]. A variety of different methods were used to determine the best-fitting Michie-King parameters for the globular clusters that were observed with the C F H T . These methods are described in the following sections. Chapter 3. CFHT Results 40 3.2.3 One-Dimensional Models Tidal radii were determined by fitting one-dimensional Michie-King models to the ob-served V- and /-band surface brightness profiles of each globular cluster. Surface bright-ness profiles for each globular cluster were obtained using the IRAF (Version 2.10.1) im-plementation of the STSDAS task ISOPHOTE.ELLIPSE. This task uses Jedrzejewski's (1987) [80] isophote-fitting algorithm to fit elliptical isophotes to the observed isophotes. This al-gorithm determines the centres, elhpticities, orientations, and intensities of a series of ellipses logarithmically spaced between ~ 0/.'131 (~ 1 pixel) and ~ 20"0 (~ 150 pixels). The spacing between successive ellipses varies from significantly less than the size of the seeing disc ( F W H M ~ 0"7) near the centre of each globular cluster to several times the size of the seeing disc beyond the tidal radius of the globular cluster. The properties of the ellipses are not independent of each other for radii of less than approximately eight times the size of the seeing disc (e.g. Schweizer 1979 [129]). Since the spacing between ellipses is logarithmic the degree of correlation will vary with the distance from the centre of the globular cluster. There is an additional correlation caused by the overlap in the subsets of pixels being used to compute successive best-fitting ellipses. In light of these effects the uncertainties in the surface brightnesses, elhpticities, and orientations of the isophotal ellipses returned by ELLIPSE were taken to be guides to the reliability of the individual values relative to the values for the other fitted ellipses for that globular clus-ter. They were not treated as statistically rigorous uncertainty estimates. In the inner regions of each globular cluster (refr ^ 2//0) the large gradient in surface brightness, and the small number of pixels, make the fitted ellipses less reliable than those fit further out. Finally, the one-dimensional surface brightness profiles for each globular cluster along the effective radius axis of the globular cluster were extracted. Chapter 3. CFHT Results 41 CERN's MINUIT (Version 94.1) function minimisation package was used to fit one-dimensional seeing-convolved Michie-King models to the V- and /-band surface bright-ness profiles of each cluster. The fitting was done by simultaneously solving for the two Michie-King parameters: central potential, Wo, and the anisotropy radius, ra; as well as the two scaling parameters: core radius, rc, and central surface brightness, a0. Once the best-fitting model was found the tidal radius, rt, and the concentration, c = log 1 0 (r t /r c ) , were computed. After some experimentation a weighting scheme of Wi = ( l /<7; ) 2 , where o~i is the root-mean-square (RMS) scatter in the isophotal intensity about the i'th fitted isophote, was adopted. This weighting scheme was tested by fitting Michie-King models to a series of profiles created by drawing points from an unconvolved Michie-King distri-bution. Good agreement was found between the input and the fitted models. The exact form of the weighting did not significantly affect the values of the best-fit parameters provided that the weighting assigned to each data point was reasonably indicative of the reliability of that point relative to the other points in the profile. Each Michie-King model was convolved with the appropriate one-dimensional seeing function. For the purposes of the convolution the seeing profile was calculated out to a radius where M(r)/M(0) = 10 - 8 . For Moffatian profiles with 3 = 2.5 this corresponds to r ~ 35 x F W H M . For profiles with 3 = 1.5 the limiting radius of the calculated seeing is r ~ 303 x F W H M . The uncertainty in the shape of the seeing profile has a significant effect on the results of the fitting. A series of convolutions of Michie-King models with Moffatian seeing profiles in which the Moffatian seeing parameters were allowed to vary by up to ± 3 C T from their observed values were performed. This resulted in variations of up to ± 2 0 % in the fitted values of r c and ± 5 % in rt. Table 3.1 gives the best-fit parameters for isotropic Michie-King models. Typically anisotropic Galactic globular clusters have anisotropy radii of between approximately 5 and 10 times their core radii, indicating that anisotropy becomes important only in the Chapter 3. CFHT Results 42 outer regions of a globular cluster. Unfortunately the C F H T data is highly uncertain at large radii. I attempted to fit anisotropic Michie-King models to the profiles of several of the globular clusters in Table 3.1 but found that the resulting fits were no better than, and often worse than, the best fits with isotropic Michie-King models. The existence of anisotropy in any of the M31 globular clusters in the C F H T data set can not be ruled out, but there is no convincing evidence for it from fitting one-dimensional Michie-King models. Reduced chi-square-fitting is an objective way to identify a single best-fitting model for each data set. However, since the surface brightness data points were not independent, the xt value for a fit could not be used to rigorously estimate the reliability or uniqueness of that fit. The fits to most of the globular clusters looked reasonable to the eye, although some clusters were not well fit. In general the clusters that were well fit by Michie-King models were those with similar Michie-King parameters in both the V- and /-bands. Globular clusters with parameters that differ by more than ~ 20% between the V- and /-bands were usually poorly fit by Michie-King models. Approximately 20% of Galactic globular clusters have collapsed cores (e.g. Djorgov-ski &: King 1986 [44]) characterized by central power-law cusps in their surface brightness distributions. To investigate the possibility that some of the M31 globular clusters have central power-law cusps I modified the fitting software to allow the simultaneous fitting of a Michie-King model with a central power-law to a surface brightness profile. This involved the introduction of three additional free parameters: the power-law slope, the radius at which the power-law profile switches to a Michie-King profile, and, to avoid infinite projected density at r = 0, an inner cut-off radius interior to which the surface brightness profile is assumed to be constant. These three power-law parameters, and the core radius, are determined by the shape of the inner few pixels of the surface brightness profile, which are dominated by the seeing. I found that, although the outer regions could Chapter 3. CFHT Results 43 be well fit with combined Michie-King-power-law models, Central power-law cusps could not be reliably identified for profiles where the seeing F W H M is similar to rc. Therefore I chose not to attempt to fit central power-laws to the M31 globular cluster data. Chapter 3. CFHT Results 44 Cluster Filter Mo) Wo rc ra/rc n c G2 V 16.50 ± 0 . 0 2 8.2 0723 +oo 17736 1.9 16.50 ± 0 . 0 2 8.2 0723 690 17739 1.9 I 15.85 ± 0 . 0 1 7.9 0726 +oo 16780 1.8 15.85 ± 0 . 0 1 7.9 0726 367 16781 1.8 G70 V 16.98 ± 0 . 0 1 7.2 0732 +oo 12704 1.6 G177 V 16.99 ± 0 . 0 1 5.9 0737 +oo 6"06 1.2 I 17.88 ± 0 . 0 3 6.6 0728 +oo 7717 1.4 G185 V 15.08 ± 0 . 0 1 5.0 0765 +oo 6"93 1.0 I 14.41 ± 0.03 5.4 0758 +oo 7753 1.1 G185B V 18.51 ± 0 . 1 0 2.1 0759 ± o o 1798 0.5 I 17.20 ± 0.10 1.4 0772 ± o o 1781 0.4 G196 V 19.07 ± 0 . 5 0 5.9 0769 ± o o 12/.'05 1.2 I 17.80 ± 0 . 0 1 6.2 0"73 +oo 14786 1.3 G208 V 18.41 ± 0.01 3.6 0786 ± o o 5701 0.8 I 16.82 ± 0 . 0 1 4.8 0765 ± o o 6733 1.0 G212 V 18.52 ± 0 . 1 0 6.1 0734 +oo 6772 1.3 I 18.14 ± 0 . 1 0 5.6 0"45 +oo 6"52 1.2 G213 V 15.67 ± 0 . 0 1 5.5 0738 ± o o 5715 1.1 I 14.17 ± 0 . 0 1 6.6 0727 +oo 6791 1.4 G215 V 16.87 ± 0 . 1 0 8.7 O'.'IO ± o o 11725 2.0 I 16.48 ± 0.25 7.5 0721 ± o o 10741 1.7 G218 V 17.31 ± 0 . 0 5 8.9 0715 ± o o 18w47 2.1 I 16.27 ± 0 . 2 5 7.8 0728 ± o o 16712 1.8 G222 V 15.80 ± 0 . 0 1 7.2 0725 ± o o 9"90 1.6 I 14.74 ± 0 . 0 1 7.2 0725 ± o o 9"92 1.6 G231 V 17.29 ± 0 . 0 1 5.3 0756 +oo 7706 1.1 I 16.20 ± 0 . 0 1 5.5 0751 ± o o 7701 1.1 G233 V 16.43 ± 0 . 0 1 7.2 0735 ± o o 13"44 1.6 I 15.18 ± 0 . 0 1 7.5 0729 +oo 13791 1.7 G263 V 16.54 ± 0 . 0 1 7.3 0732 ± o o 13760 1.6 I 15.68 ± 0 . 0 1 6.6 0742 +oo 10"70 1.4 G302 V 16.48 ± 0 . 0 1 7.6 0731 +oo 15742 1.7 16.48 ± 0 . 0 1 7.6 0731 487 15743 1.7 I 15.33 ± 0 . 0 3 7.5 0732 ± o o 15"48 1.7 15.33 ± 0 . 0 3 7.5 0732 649 15748 1.7 Table 3.1: Best-fitting one-dimensional Michie-King models. Chapter 3. CFHT Results 45 Cluster Filter Mo) W0 rc ra/rc n c G312 V 16.70 ± 0 . 0 2 7.5 0"29 ± o o 13'.'92 1.7 16.70 ± 0 . 0 2 7.5 OV29 1552 13"92 1.7 I 15.48 ± 0 . 0 1 7.4 a'29 ± o o 12781 1.6 15.49 ± 0.01 7.4 0'.'29 21 13779 1.7 G352 : v • 17.23 ± 0.01 6.4 C/41 ± o o 9719 1.4 i 15.99 ± 0 . 0 1 6.9 0'.'32 ± c o 9779 1.5 G355 V 17.41 ± 0 . 0 1 13.0 0702 ± o o 16738 2.9 Bol53 V 17.09 ± 0 . 0 1 4.5 0'.'42 ± o o 3"59 0.9 I 15.17 ± 0 . 0 1 5.7 0727 +oo 4716 1.2 vdB 2 V. 15.52 ± 0 . 0 1 4.3 0'.'47 ± o o 3"64 0.9 I 14.07 ± 0 . 0 3 5.1 Qf!S8 ± o o 4734 1.1 vdB 5 V 17.42 ± 0 . 0 1 5.7 0741 ± o o 6723 1.2 I 16.08 ± 0 . 0 3 7.0 0?!29 ± o o 9747 1.5 V253 V 18.26 ± 0.10 13.0 0702 +oo 19725 2.9 I 16.85 ± 0.25 8.2 0719 +oo 14742 1.9 3.1 (continued) Chapter 3. CFHT Results 46 3.2.4 The Artificial Clusters In order to investigate the accuracy of fitting one-dimensional Michie-King profiles to the data, and to investigate the fitting algorithm's stability against stochastic effects in the locations of stars within the cluster and the field that the cluster sits upon, I created a series of artificial clusters, measured their surface brightness profiles, and fit Michie-King models in the manner described above. The artificial clusters were constructed by randomly drawing stellar coordinates from a Michie-King profile, and magnitudes from a realistic globular cluster luminosity function. Experiments with a variety of luminosity functions based on those of Galactic globular clusters showed that the observed surface brightness profiles of the artificial clusters did not depend on small changes in the luminosity function such as those caused by metallicity or age spreads similar to those observed in Galactic globular clusters. Therefore, a luminosity function based on that of 47 Tuc (Hesser et al. 1987 [68]) was adopted. The biggest unknown in the adopted luminosity function is the distribution of horizontal branch stars in M31's globular clusters. Galactic globular clusters exhibit a wide range of horizontal branch morphologies that appear to be correlated with metallicity and some as-of-yet unknown second parameter (e.g. Carney et al. 1991 [30]). While there is no direct evidence for a second parameter problem in the M31 globular clusters, spectral studies suggest that the two systems have similar, but not identical, stellar populations (e.g. Burstein et al. 1984 [22], Tripicco 1989 [146], Covino et al. 1994 [32]) so it is not inconceivable that a second parameter may be active in the M31 globular cluster system. For simplicity I have assumed that stars are uniformly distributed along the horizontal part of the horizontal branch and ignored any complications introduced by the instability strip and the extreme blue tail of the horizontal branch. The faint end of the luminosity function was truncated at the magnitude where each star contributed one count to the Chapter 3. CFHT Results 47 image over the exposure time of the image. The IRAF task NOAO.ARTDATA.MKOBJECT and the DAOPHOT II PSF were used to add these artificial stars to a background region on the image containing the M31 globular cluster in question. Stars were added until the total flux in the added stars matched that of the appropriate real cluster, then Poisson and read-out noise was added. An example of an artificial cluster is shown in Figure 3.2. Elliptical isophotes were fit to these artificial clusters in exactly the same manner as they were to the real clusters and the resulting surface brightness profiles were compared with that of the real cluster using a xt statistic. Since this modeling procedure is physically similar to what is happening to the light from the real cluster, and since the artificial clusters were placed on background regions of the original images, any systematic effects affecting the real clusters will also affect the artificial ones. Figure 3.2: The left-hand figure shows the /-band image of G312 while the right-hand image shows an artificial cluster with W0 = 7.7, rc = 0723 (= 1.76 pixels), and e = 0.0. The artificial cluster is made up of 116,790 stars with 20.5 < / < 29.9. It has been placed on an M31 halo field located ~ 35" south of G312. Chapter 3. CFHT Results 48 3.2.5 Systematic Biases in Seeing Convolutions It is not obvious that convolving a one-dimensional seeing profile with a one-dimensional surface brightness model is physically realistic. The effects of seeing on light from an astronomical source with a given intrinsic shape is usually treated as distortions in the wavefront of the light caused by turbulent cells in the Earth's atmosphere. Pressure and temperature differences between adjacent turbulent cells result in the index of refraction varying from one cell to the next which introduces distortions in the wavefront of the light. An alternative way of thinking of the problem, which is useful when thinking about image reconstruction, is to follow the paths of individual photons from the source to the detector. Each photon is perturbed from its original path by atmospheric effects, and by the telescope's optics. Further, the exact location of the photon when it strikes the C C D detector is not known due to the finite size of the detector's pixels. All three of these effects need to be taken into account to accurately reconstruct the original light source from the image that is observed at the detector. The observed distribution of light on the CCD, 0(x), is related to the intrinsic dis-tribution from the source, I(y) by 0(x) = / / J(y) P(x | y) dy (3.9) where P(x | y) is the conditional probability that a photon from a location y on the sky will be observed at location x on the CCD. In other words, P(x | y) is a function that describes the path of a photon through the atmosphere, and the telescopes optics. If P(x [ y) is independent of position on the C C D , and the pixel size is small compared to the spatial scale on which the flux from the source varies, then P(x | y) = P(x — y), making Equation 3.9 a two-dimensional convolution integral. The observed surface brightness profile is obtained by taking a slice along the effective Chapter 3. CFHT Results 49 radius axis2 of the light distribution on the CCD image. However, the analysis described in Chapter 3.2.3 involved first taking one-dimensional profiles of a Michie-King model and the seeing function, convolving them, then comparing the resulting profile to the observed one-dimensional profile. The difference between these two approaches can be illustrated using a Gaussian model cluster of ellipticity ec: C(r, 9) oc exp | and a Gaussian PSF of ellipticity ep: cos2(c9) sin2(t9) + (3.10) cos2(t9) sin2(69) (3.11) P(r, 9) oc exp f — -r2 V L "p,x "p,y where <rx and o~y are the standard deviations of the two-dimensional Gaussians, and both ellipses have the same orientation. The surface brightness profiles of each are obtained by extracting the radial profiles along their effective radius axes then convolving these one-dimensional profiles to get e^ff 5 i ( r d r ) < x e x H " t < » ( i - ^ ) + ' ^ ( i - £ p ) j - ( 3 1 2 ) On the other hand, convolving C(r, 9) and P(r, 9) directly then extracting the radial profile gives: / S2(ref[) oc exp (3.13) which differs in shape from Si(reff). Although this example is for a special case (one in which C(r, 9) has the same orientation as P(r, 9)) it demonstrates that the two methods 2 The effective radius, reg-, of an ellipse with semi-major and semi-minor axes a and 6 respectively is defined by 7rr eff 2 = nab. The effective radius axis is either of the two axes along which the edge of the ellipse intersects the edge of a circle of radius reg. Chapter 3. CFHT Results 50 of treating seeing-convolution are not equivalent. In general S i and S2 will have the same shape only if C(r, 9) and P(r, 9) have the same ellipticities and orientations. For realistic PSFs and cluster profiles it is usually not possible to perform the con-volution and extraction of radial slices analytically so numerical methods must be used. To investigate the effects of seeing on realistic globular clusters I constructed a grid of 400 circularly symmetric artificial clusters with identical Michie-King parameters (W0 = 7.4, rc = 0'.'236 (= 1.8 pixels), rt = 1C/493 (= 80 pixels), and c = 1.65) and fit seeing-convolved Michie-King models to their radial surface brightness profiles. This process compares one-dimensional profiles, but the convolutions are applied to the two-dimensional distributions. The parameters returned by these fits were Wo = 7.31 ± 0.08, r~c = 07303 ± 07008 (= 2.31 ± 0.06 pixels), Tt = 12"682 ± 07487 .(= 96.81 ± 3 . 7 2 ) pixels, and c = 1.62 ± 0.02 (the uncertainties are standard deviations). Fitting one-dimensional Michie-King models results in rc being overestimated by ~ 28% while the concentration of the cluster is underestimated by ~ 2%. This suggests that fitting one-dimensional models to surface brightness profiles gives a reasonable estimate of the overall shape of the surface brightness profile but seriously overestimates the radial scale. Because of this the fitted one-dimensional core and tidal radii in Table 3.1 should be treated as upper limits to their true values. Unfortunately the discrepancies between the fitted and intrinsic core and tidal radii will vary with the seeing quality, the signal-to-noise ratio of the images, and the intrinsic shape of the cluster, so a simple correction factor can not be used to go from the fit-ted one-dimensional Michie-King parameters to the intrinsic Michie-King parameters. Therefore, seeing must be treated as a two-dimensional phenomenon when modeling surface brightness profiles. Chapter 3. CFHT Results 51 3.2.6 Two-Dimensional Artificial Cluster Models In order to estimate the Michie-King parameters of an M31 globular cluster without using one-dimensional convolutions a grid of artificial clusters with Michie-King parameters based on those obtained by fitting a one-dimensional Michie-King model to the radial profile of the globular cluster was constructed. This procedure was tested on the V-and /-band images of G302 and G312. The adopted grid for G302 and G312 had W0 between 6.5 and 8.9, and rc between CK'13 and (X'33. The artificial clusters were circularly symmetric and had no velocity anisotropy (i.e. ra = -foo). For each artificial cluster a surface brightness profile was derived and the xt between it and the surface brightness profile of the real globular cluster in question was calculated. To convert the xt statistic into a confidence level a set of 100 artificial clusters with W0 = 7.4 and rc = 0'.'24 were generated and the xt between each of these artificial clusters and the original artificial cluster was computed. Contour plots showing the probability of fit for G302 and G312 are shown in Fig-ures 3.3, 3.4, 3.5, and 3.6. The jumpiness in the contours is due to modeling the clusters as a set of individual stars rather than a smooth light distribution. Since the stars are placed randomly in the artificial cluster (within the limitations imposed by the model surface brightness profile, elhpticity, and orientation) models with identical input para-meters can return significantly different observed surface brightness profiles. As can be seen in Figures 3.3, 3.4, 3.5, and 3.6, these simulations do not give a unique set of Michie-King parameters for any of the clusters under investigation. Instead the clusters can be modeled with a range of Michie-King models of similar tidal radii. The large uncertainties in the core radius are due to the shape of the core being smeared out by the seeing, and to the large size of the pixels relative to the scale of the PSF. In addition, the light in the core is dominated by a small number of stars near the top of the red-giant Chapter 3. CFHT Results 52 branch and may contain significant contributions from field stars that are superimposed on the core. Much of the scatter in rc may be due to the random placement of these bright stars within the core. Fortunately star count data (see Chapter 3.1) can provide an additional constraint on the tidal radius. By superimposing the upper and lower limits on the tidal radius, as determined from star count data, it is possible to identify the most probable Michie-King parameters and .estimate the uncertainties in them. Table 3.2 lists these parameters. The limiting radii in Table 3.2 are those obtained from the star-counts and represent the radii at which the observed projected stellar densities drop sharply towards zero. The ranges given for each parameter in Table 3.2 represent the range of values allowed given the la uncertainties in the best-fit artificial clusters and star-count data. Cluster Filter Wo rc c G302 V 7.4-8.0 0721-0'.'24 1178-1571 1.8 I 7.6-8.0 0721-0725 1178-1571 1.8 G312 V 7.5-8.5 0"13-0"20 972-13'.'1 1.8 I 7.1-7.7 0720-0"26 972-1371 1.7 Table 3.2: Results from two-dimensional Michie-King models and star counts for G302 and G312. A comparison of the core and tidal radii for G302 and G312 derived using the one-dimensional method (see Table 3.1) and those derived using the two-dimensional methods (see Table 3.2) shows that fitting one-dimensional models leads to systematic errors in the Michie-King parameters. The core radii obtained from one-dimensional fitting are overestimated by ~ 33%. The tidal radii obtained by fitting one-dimensional models are overestimated by ~ 25%. The core and tidal radii for G302 and G312 were also determined using HST WFPC2 images in the V- and /-bands (see Chapter 5). The core and tidal radii obtained by fitting Michie-King models to this HST data (see Table 5.4) Chapter 3. CFHT Results 53 are in agreement with those obtained from fitting two-dimensional artificial cluster models to the C F H T data (see Table 3.2). This strongly suggests that it is possible to obtain reliable core and tidal radii of globular clusters in M31 from ground-based observations if the seeing is good enough that individual stars in the outer regions of the clusters can be resolved. The resolved stars are needed in order to use star counts to constrain the tidal radii of the globular clusters. Chapter 3. CFHT Results 54 Figure 3.3: Contour plots showing the probability that the observed surface brightness profile can be obtained from an artificial cluster with a given set of Michie-King pa-rameters. The solid lines represent the upper and lower cut-offs determined from the star-count data (see Chapter 3.1). The combination of artificial cluster comparisons and the limiting radii determined from the star-counts provide strong constraints on the range of models that can reproduce the observations. Chapter 3. CFHT Results Chapter 3. CFHT Results 56 Figure 3.5: Probability contours for the C F H T G312 /-band data. Chapter 3. CFHT Results Figure 3.6: Probability contours for the C F H T G312 V-band data. Chapter 3. CFHT Results 58 3.2.7 Two-Dimensional Integrated Light Models Unfortunately, the amount of exposure time required to resolve individual stars in M31 globular clusters is large. Further, clusters that were projected onto the disc, or the nuclear regions, of M31 are seen against a sea of surface brightness fluctuations that look like stars. This makes the identification of stars, even in deep exposures, uncertain. Because of these two effects it was not possible to use the forward approach of comparing artificial globular clusters to the data to determine the Michie-King parameters for most of the globular clusters in the CFHT data set. In order to salvage something useful from the situation I fit two-dimensional Michie-King models of the integrated light from a globular cluster directly to the background-subtracted images of each globular cluster. Modeling the integrated light of the cluster should work well on images with poorer seeing or shorter exposure times since individual stars will not be resolved. This approach will result in an integrated light model being a good approximation to the CCD image of the globular cluster. For globular clusters such as G302 and G312, where individual stars are well-resolved in the clusters' outer regions, modeling the integrated light will not be as successful as building an artificial cluster from artificial stars. The two-dimensional isotropic Michie-King models were fit in the same manner as described in Chapter 3.2.3. The following parameters were solved for simultaneously: Wo, rc, total magnitude, e, and 80. Once the best-fit was determined the three-dimensional half-mass radius, r ,^ was computed and the model cluster was subtracted from the data image. The subtracted image was examined by eye to obtain an estimate of the quality of the subtraction, which was used to set a flag indicating the overall reliability of the fit. The results of directly fitting the two-dimensional integrated light Michie-King models to each globular cluster are presented in Tables 3.3 and 3.4. The fit quality flags are: 1 = good subtraction, reliable fit; 2 = fair subtraction, uncertain fit; 3 = poor subtraction, Chapter 3. CFHT Results unreliable fit; 4 = globular cluster is partially resolved into stars. 59 Cluster ^M31 Wo n rh Flag (-pc) V I V I V I V I V I G2 31.6 8.47 8.48 0711 O'/ll 10743 10719 1746 1747 1 1 G185A 0.3 6.33 6.10 0'.'38 0739 8731 7713 0"39 0"36 1 1 G185B 0.3 1.46 0.61 0755 0789 1740 1732 0711 0"07 1 1 G196 1.5 7.23 6.28 0748 0753 6707 11718 0756 0"38 1 1 G208 1.3 4.98 4.34 0759 0765 6719 5" 17 0726 0722 1 1 G212 1.6 6.08 4.81 0731 0741 5778 4702 0736 0725 1 1 G213 2.2 3.80 7.09 0747 0716 3"00 5"79 0/.'20 0"52 4 4 G215 1.7 7.98 6.64 0709 0719 5781 5"03 0"95 0743 2 2 G218 1.9 7.65 6.69 0716 0725 8726 6781 0773 0"44 2 2 G222 1.9 7.85 8.02 0713 0711 7"77 7"50 0786 0"99 3 3 G231 2.7 6.53 6.54 0755 0729 13755 7"34 0741 0741 3 3 G233 2.8 7.94 8.42 0736 0711 10761 9791 0"92 r/40 3 3 G302 7.6 8.02 7.83 0717 0720 11796 11781 0799 0784 4 4 G312 10.8 8.32 7.91 0712 0716 10736 10"25 1728 0"90 4 4 Bol53 1.2 9.73 8.14 0"03 0707 5"34 4790 3W64 1710 2 2 vdB2 0.2 7.86 7.43 0709 a"12 5771 5741 0786 0763. 1 1 V253 1.7 7.29 6.39 0711 0719 4748 4740 0758 0739 1 1 Table 3.3: Two-dimensional Michie-King model fits to the C F H T data. The structural parameters obtained by fitting integrated light models to G302 and G312 are in agreement with those obtained by fitting artificial clusters to the data and with the structural parameters derived from the HST data. This suggests that the Michie-King parameters, and the elhpticities in Tables 3.3 and 3.4 are reasonable estimates of the true values. Chapter 3. CFHT Results 60 Cluster #M3i Magnitude e 0o Flag (kpc) V I V I V I V I G2 31.6 15.39 um 0 000 0.000 1 1 G185A 0.3 14.23 13.04 0 080 0.183 26?1 17?4 1 1 G185B 0.3 17.81 16.69 0 167 0.244 76?6 77?8 1 1 G196 1.5 16.93 15.95 0 191 0.286 1 1 G208 1.3 16.61 15.42 0 000. 0.000 . . . . 1 1 G212 1.6 17.45 16.69 0 143 0.164 -56? 1 -75?1 1 1 G213 2.2 13.41 0.045 63?9 4 4 G215 1.7 16.18 16.39 0 079 0.258 -37?5 51?5 2 2 G218 1.9 16.31 15.45 0 052 0.165 -40?1 60?4 2 2 G222 1.9 15.14 13.95 0 104 0.149 3 3 G231 2.7 15.86 14.88 0 043 0.000 -75? 3 3 3 G233 2.8 15.14 14.14 0 184 0.000 -55? 3 3 3 G302 .7.6 15.06 14.09 0 205 0.198 -83? 5 -81?8 4 4 G312 10.8 15.65 14.53 0 043 0.088 78?0 53?4 4 4 Bol53 1.2 16.02 14.74 0 050 0.188 . . . . 2 2 vdB2 0.2 14.56 13.37 0 083 0.042 82?4 45?4 1 1 V253 1.7 17.68 16.39 0 156 0.258 37?4 42?9 1 1 Table 3.4: Ellipticities and position angles for the C F H T data. Chapter 3. CFHT Results 61 3.3 Properties of the M31 Globular Cluster System Figures 3.7, 3.8, and 3.9 compare the distributions of core, tidal, and half-mass radii of the M31 globular clusters studied in this thesis with the distributions of those same quantities for the Galactic globular cluster system. In all three figures the core, tidal, and half-mass radii for the Galactic globular clusters have been scaled to the apparent sizes they would have at 725 kpc (u-o = 24.3 kpc). The core, half-mass, and tidal radii will, to some degree, be dependent on the strength of the galactic potential. Heggie & Ramamani (1995) [66] investigated the effects of a galactic tidal field on King models and found that, while tidal truncation can lead to triaxial globular clusters that are ~ 50% larger than the Michie-King models described in Chapter 3.2.1, the cores of globular clusters are almost unaffected by tidal truncation. The fitted half-mass radii in Table 3.3 are small compared to the size of the cluster so it is reasonable to assume that, to the accuracy of the data, the strength of the galactic potential is not affecting the half-mass radii. The mean core radius of the M31 globular clusters is (r c)M3i = 0728 ± 0705 where the mean core radius of the Milky Way globular clusters is (rc)MW = 0728 ± 0702. The quoted uncertainties are the standard errors in the mean. While the means are the same Figure 3.7 shows that the distributions of core radii are quite different. M31 appears to lack clusters with small (r c £ 071) core radii. Chapter 5.6 shows that, due to the effects of seeing, the core radii measured for M31 globular clusters and listed in Table 3.3 should be considered to be upper limits on the true core radii of the M31 globular clusters. This would explain the apparent lack of globular clusters with small core radii in M31. In light of this there is no point making a more detailed comparison between the core radii Chapter 3. CFHT Results 62 of the globular clusters in the two galaxies. The mean half-mass radius for the M31 globular clusters is (T*/I)M3I = 0 /.'75±0 //14 while the mean half-mass radius for the Milky Way globular clusters is ( r ^ M W = l'-'29 ± 0'.'06. The half-mass radii for the Galactic globular clusters were determined by constructing a series of isotropic King models with central potentials between Wo = 1 and W0 = 12 and computing the concentration and half-mass radius for each model. The one-to-one correspondence between these two quantities in an isotropic King model is shown in Figure 3.10. Half-mass radii for the Galactic globular clusters were found by spline-interpolating the published concentration parameters on this diagram. An examination of Figure 3.9 shows that the shapes of the distributions of half-mass radii are similar in the two galaxies. However, the mean half-mass radius is ~ 40% smaller in M31 than it is in the Milky Way Galaxy. This difference between the two mean values is significant at more than the 99.99% confidence level (a ~ 3.5<r difference). If a distance of 900 kpc is adopted for M31 the new mean half-mass radius for the Milky Way globular clusters (after being scaled to the distance of M31) is (r^MW = 1'.'15±0'.'06, corresponding to a difference at the 99% confidence level (a ~ 2.5a difference). This suggests that the difference between the two mean half-mass radii is not due to the uncertainty in the distance to M31. Since the half-mass radius is computed from the best-fitting Michie-King model, and the fitted core radii for the M31 globular clusters are only upper limits on the true core radii of those clusters, the difference in the mean half-mass radii is probably an artifact of the model-fitting process and the uncertainties in the data, and not a real difference in the structural properties of the globular clusters in the two galaxies. Since the half-mass radius increases when the concentration increases any systematic uncertainty in determining the concentration of a globular cluster will result in a systematic uncertainty in the half-mass radius. As discussed above, and in Chapter 5.6, the core radii for the Chapter 3. CFHT Results 63 M31 globular clusters are upper limits on the true core radii while the tidal radii for the M31 globular clusters are reasonable estimates of the true tidal radii. This will result in the concentration (c = log10(rt/rc)) being systematically underestimated, which will lead to the half-mass radii of the M31 globular clusters being systematically underestimated. The tidal radii will be affected by the gravitational potential of the parent galaxy (see e.g. Innanen et al. 1983 [78], Heggie & Ramamani 1995 [66]). In order to compare the tidal radii of the globular clusters in M31 with the tidal radii of the globular clusters in the Milky Way Galaxy it is necessary to correct for the differences in the masses of the two galaxies. For a spherical logarithmic potential the tidal radius of a globular cluster is given by: where Mc\, is the mass of the globular cluster, Rp is the perigalactic distance of the globular cluster's orbit, VTOt is the amplitude of the flat portion of the galactic rotation velocity curve, G is the Newtonian gravitational constant, and g(e) is a slowly varying function of the orbital eccentricity, e = (Ra — Rp)/(Ra + Rp). Ra is the apogalactic distance of the globular cluster's orbit. The function g(e) varies slowly with eccentricity has values of g(0) = 1 for circular orbits, and g(0.Q) ~ 2 for orbits with eccentricities similar to those of the Galactic globular clusters. The globular clusters in the Milky Way Galaxy have a mean normalized tidal radius of ( r t / - ^ c i / 3 ) G a i = 0.46 pc/M1^3. This value was taken from Cohen & Freeman (1991) [30] who assumed a mass-to-light ratio of T y = Mv/Ly = 2 for Galactic globular clusters. They hmited their sample of Galactic globular clusters to those with My < —7.1 mag (the bright half of the Galactic globular cluster distribution). The faintest M31 globular cluster in the sample presented in this thesis has V = 17.45 (excluding G185B) which (3.14) Chapter 3. CFHT Results 64 corresponds to My = —6.85. Therefore, both samples of globular clusters have similar l imit ing magnitudes. 1 /3 The data in Tables 3.3 and 3.4 gives a mean normalized tidal radius of (rt/A4cl )M3I = 1 /3 0.35 ± 0.03 pc/M^ for the M31 globular clusters assuming a distance modulus of fi0 = 24.3 and a mass-to-light ratio of Ty = 2 (Dubath k Gri l lmair 1997 [47] find T = 1.9 ± 0.4). The rotation velocities for the two galaxies are VTOt = 220 k m - s - 1 for the Galaxy and VTOt = 265 km-s" 1 for M31 (Roberts 1966 [123]). The ratio of these is ( K o t , M 3 i / K o t , G a i ) 2 / ' 3 = 1-132. Mul t ip ly ing the mean normalized t idal radius by this ratio gives a scaled, mean normalized tidal radius of (rtjM.\{Z)M3\ = 0.40 ± 0.03 p c / A ^ Q 3 . This is within la of the Galactic value. This result agrees with Cohen k Freeman's (1991) [30] scaled value of 0.42 for the M31 globular clusters. Comparing the tidal radii in the two globular cluster systems depends on the assumed distance to M31 . If the distance modulus of M31 is increased to the Hipparcos value of fio = 24.77 (a distance of 900 kpc) (Feast k Catchpole 1997 [53]) then the scaled, mean normalized tidal radius for the M31 globular clusters in this sample becomes (rt/.M*/3)M31 = 0.43 ± 0.04 pc / 'M 1 Q . This is a slightly better agreement with the Galactic value than is obtained with a distance modulus of uo = 24.3. The mean projected ellipticity of the M31 globular clusters presented here is (e)M3i = 0.12 ± 0.04. Lupton (1989) [90] obtained ( e ) M s i = 0.12 ± 0.06 for the outer (4" to 6") regions of 18 M31 globular clusters. His paper quotes mean projected ellipticities of 0.08 ± 0.06 for the Galactic globular clusters and 0.11 ± 0.07 for those in the Large Magellanic Cloud. Chapter 3. CFHT Results 65 40 30 ^ 20 10 0 i 1 r n 1 r i 1 r i 1 r Milky Way M31 0 n 1 r _ N 140 MW ^ M 3 1 = 1 ? I 0.2 0.4 0.6 r (arcsec) 0.8 Figure 3.7: This figure shows core radii derived from the C F H T HRCam images. The solid histogram represents the M31 globular clusters while the dashed histogram repres-ents the Galactic globular clusters. The core radii for the Galactic globular clusters have been scaled to a distance of 725 kpc. Chapter 3. CFHT Results 66 20 | 1 1 r 15 ^ 10 0 "i 1 1 r i 1 1 r Milky Way NMW M31 140 ^ M 3 1 = I? _ T i 0 10 15 r t (arcsec) Figure 3.8: This figure shows tidal radii derived from the C F H T HRCam images. The solid histogram represents the M31 globular clusters while the dashed histogram repres-ents the Galactic globular clusters. The tidal radii for the Galactic globular clusters have been scaled to a distance of 725 kpc. Chapter 3. CFHT Results 67 50 40 h 30 h 20 10 0 I 1 1 1 r Milky Way M31 i 1 1 r J V „ = 140 r 1 J I I LL 1 L l I I I LL 0 1 2 r h (arcsec) Figure 3.9: This figure shows half-mass radii derived from the C F H T HRCam images. The solid histogram represents the M31 globular clusters while the dashed histogram represents the Galactic globular clusters. The half-mass radii for the Galactic globular clusters have been scaled to a distance of 725 kpc. Chapter 3. CFHT Results 68 100 - i 1 i 1 1 1 1 i i I i i i r 80 60 40 20 0 I • I • * • L J I I L J I L 0 1 2 c = log 1 0 (r t /r c ) Figure 3.10: This figure shows the one-to-one relationship between the concentration of an isotropic single-mass King model and its three-dimensional half-mass radius. Chapter 4 The HST Data 4.1 Motivation for the HST Observations While this work was in progress a corrective optics package was successfully installed on the HST. Space-based observations of M31's globular clusters have several advantages over ground-based observations1. In light of this I applied for time to use the iJST's WFPC2 to obtain high-resolution images of two globular clusters in M31. The main motivations for doing this were as follows. First, individual stars in M31 globular clusters can be resolved on WFPC2 images to within ~ 2" of the centre of the globular cluster. In the C F H T observations individual stars can only be resolved in the deepest exposures and only stars in the outer regions of the globular clusters can be resolved. Therefore, HST images can be used to construct CMDs that reach below the level of the horizontal branch. This allows stellar population studies to be done to determine ages and metallicities. Second, the stellar PSF on W F C images has a F W H M of ~ 0715. This is ~ 5 times smaller than the size of the seeing disc in the C F H T data. The improved resolution of 1 Recent advances in adaptive optics for astronomical telescopes have made ground-based observations of globular clusters in M31 comparable to HST observations. The Adaptive Optics Bonnette at the CFHT (for example) can obtain diffraction-limited near-infrared images of the cores of M31 globular clusters. 69 Chapter 4. The HST Data 70 the WFPC2 means that seeing has a much smaller effect on the cluster images than it does for the ground-based data. Therefore structural parameters can be determined more reliably from HST data. By comparing the observed structural parameters between HST observations of a globular cluster and the C F H T observations of the same cluster it will be possible to directly determine how reliable the ground-based observations are. Third, the ratio of cluster size to PSF size (resolution ratio) for the C F H T images of the M31 globular clusters presented in this thesis is ~ f/.'7/lO" = 0.07. This is similar to the resolution ratio (~ O'.'05/l" = 0.05) for globular clusters in the Virgo cluster of galax-ies when observed with the Planetary Camera (or the up-coming Advanced Camera). If a comparison of the structural parameters for the M31 globular clusters determined from the ground-based data with those determined from the space-based data suggests that ground-based observations can give reliable results then it will be possible to use the HST to obtain detailed structural information of globular clusters out to approximately the distance of M87 (~ 17 Mpc). This would allow the structures of globular clusters in elliptical and cD galaxies to be studied to determine how the gravitational potential of the parent galaxy influences the structure of a globular cluster. Four, HST images will be deep enough (Vum ~ 27) to resolve enough stars beyond the King tidal radius of a cluster to search for extended halos of unbound stars around the globular cluster. The lower resolution of the C F H T images presented in this thesis makes it impossible to observe sufficient stars to identify such a halo from the ground. Chapter 4. The HST Data 71 4.2 Observations I received eight orbits of HST2 time. I obtained deep V- and J-band images of two bright globular clusters, G302 (a 2 0 0o.o = 00h45m25=2, £200o.o = +41°05'30") and G312 (ctzooo.o = 00h45m58=8, cT2ooo.o = +40°42'32") in the halo of M31. Deep V- and /-band HRCam images of these two clusters are also available (see Chapter 2). The HST data were obtained using the HST in cycle 5 program #5609. The WFPC2 operated at a temperature of — 88°C and with a gain setting of 7 e~ /ADU. Table 4.1 lists the exposures obtained in each filter. Exposure times were based on the central surface brightnesses of the two globular clusters that were determined from the C F H T observations (see Table 3.1 in Chapter 3) and chosen to avoid saturating the cores of the globular clusters. The total exposure times were 4320 seconds in the F555W (WFPC2 broadband V) filter and 4060 seconds in the F814W (WFPC2 broadband I) filter for each field. The data was preprocessed through the standard STScI pipeline for WFPC2 images. Known bad pixels were masked, and geometric corrections were applied using standard techniques. G302 is located 32'.1 from the centre of M31 (at a projected distance of RM3I = 6.8 kpc), approximately along the southeast minor axis. G312 is located 49'8 from the centre of M31 (RM3I = 10.5 kpc), also approximately along the southeast minor axis. G302 and G312 are located in fields 12 and 13 respectively of Figure 2.1. These globular clusters were selected for observation with the HST because there were deep C F H T images available for each cluster (see Chapter 2.1). This will allow me to make a direct comparison between the structural parameters obtained using ground-based data and that obtained using the superior resolution of the HST. It will then be possible to identify systematic differences between the results obtained from the two data set. G302 and G312 2 The N A S A / E S A Hubble Space Telescope and the Space Telescope Science Institute are operated by the Association of Universities for Research in Astronomy, Inc., under N A S A contract NAS5-26555. Chapter 4. The HST Data 72 Field Date Filter Exposure (1995) (s) G302 Nov. 5 F555W 8 X 500 2 X 160 F814W 7 X 500 1 X 400 1 X 160 G312 Oct. 31 F555W 8 X 500 2 X 160 F814W 7 X 500 1 X 400 1 X 160 Table 4.1: Log of the HST observations. are bright, isolated clusters located in the halo of M31 so the amount of contamination from stars in the disc of M31 is small. Each globular cluster was centred on the WF3 CCD. The globular clusters were located on the WF3 CCD instead of taking advantage of the higher spatial resolution offered by the smaller pixels of the PC CCD for three reasons: 1. The tidal radii of G302 and G312, as determined by fitting Michie-King models to the surface brightness profiles of G302 and G312 derived from the HRCam observations taken at the C F H T , are rt ~ 10" (see Chapter 3). Therefore each globular cluster would occupy a significant fraction of the total field of view of the PC, leaving insufficient area left to make reliable determinations of the spatial variations in the background stellar density. 2. Grillmair et al. (1995) [59] found evidence for extended stellar halos, extending to ~ 2 to 3 times the tidal radius, in several Galactic globular clusters. Recently, G A F found weak evidence for stellar density enhancements out to ~ 2rt in some M31 Chapter 4. The HST Data 73 globular clusters. Centring the globular clusters on the WF3 C C D ensured that as much as possible of any extended halos would be located on the same C C D as the globular cluster itself. This avoided possible systematic effects that may occur if surface brightness data and stellar number density data is transferred between the PC and W F C CCDs, with their different pixel scales and sensitivities. 3. Locating the globular clusters at the centre of the WF3 C C D allowed the use of the WF2 and WF4 CCDs to probe the stellar populations in the halo of M31. These results are presented in Chapter 6. Having two background fields for each glob-ular cluster field made it possible to differentiate between stellar number-density enhancements that are due to statistical fluctuations in the star counts from M31 halo stars and those that are due to physical structures, such as extended halos, that are related to the globular clusters. If the globular clusters had been centred on the PC C C D then all three W F C CCDs would have been available to image the background. However, if this had been done then any extended stellar halos around the globular clusters might have overfilled the PC's field of view. This would have resulted in cluster stars contaminating the background fields. Chapter 4. The HST Data 74 4 . 3 Data Reductions Photometry was performed on stars imaged by the WFPC2's Wide Field CCDs using the D A O P H O T I l / A L L F R A M E software (Stetson 1987 [141], Stetson 1994 [138]). All nineteen images for each field were re-registered to a common coordinate system and median-combined. This served to eliminate cosmic-ray hits and to increase the signal-to-noise ratio in the data allowing fainter stars to be detected. Cosmic rays strike each C C D at a rate of ~ 20 per second so on a 500 second exposure ~ 1.7% of the pixels are affected by cosmic rays. In order to eliminated large-scale gradients in the images due to unresolved light from the globular clusters (the gradient in the hght from the halo of M31 was ^ 1% in both the G302 and G312 fields) a square median-filter was run over the combined image to produce a map of the large-scale gradients. This smoothed image was then subtracted from the combined image and a constant sky added back to produce a median-subtracted image. The size of the square median-filter was ~ 5 times the stellar full-width at half-maximum (FWHM) to avoid smoothing out structure in the stellar images (e.g. Stetson & Harris 1988 [142]). Subtracting the median-filtered image resulted in a reasonably clean subtraction of the integrated hght from each globular cluster to within ~ 25 pixels (~ 2"5 ~ 9 pc) of the centre of each globular cluster. This process is the digital equivalent of unsharp masking. The D A O P H O T FIND routine, with a F IND threshold of 7.5<7sky, was used to detect peaks on the median-subtracted image. Experiments with different F IND thresholds showed that 7.5<xsky excluded most of the mis-identifications of noise spikes and cosmic-ray events as stars at faint magnitudes while including almost all of the stars down to near the photometric limit of the data. F I N D was able to reliably detect stars to within ~ 25 pixels (~ 2"5) of the centre of each globular cluster. Artificial star tests showed that Chapter 4. The HST Data 75 detections closer to the centres of the globular clusters than this were usually spurious. The resulting list of stellar candidates was used as input, to A L L F R A M E . A L L F R A M E does simultaneous PSF fitting on all the original images (not the combined median-subtracted image) to preserve the photometric properties of the data. The PSFs used (Stetson 1996 [137]) were two-dimensional Moffatians with j3 — 1.5, 7 = 2 and a look-up table of residuals. The PSFs varied quadratically over each C C D to account for variations in the form of the PSF from one part of the CCD to another. Only stars that appeared in at least seven frames in each filter were considered to be real stars. Tests with different minimum numbers of frames showed that requiring stars to be found on at least seven frames provided a clean C M D without significantly affecting the limiting magnitude of the C M D . Aperture corrections were obtained for each C C D in the following manner. First a set of ~ 50 bright, isolated stars was selected on each CCD and all the remaining stars were subtracted from these images. Next, the total magnitude in an aperture with a radius of 0'/5 (Holtzman et al. 1995b [73]) was measured for each star. Aperture corrections were defined in the sense Av = v&p — VPSF for each of the isolated stars. We discarded any star with a combined photometric uncertainty of y<r\v + ^PSF > 0.14, which corresponds to a signal-to-noise ratio of ~ 10 for each of the aperture and PSF magnitudes. The size of the aperture correction should be independent of magnitude so a plot of vap vs fpsF (see Figure 4.1) should yield a slope of unity with a zero-point offset corresponding to the value of the aperture correction. However, the W F CCDs suffer from considerable cosmic ray contamination that may bias the observed aperture magnitudes towards brighter magnitudes. In addition the presence of background galaxies and undetected faint stars can bias the vAP - VPSF relation away from a slope of unity. To correct for this a plot of the PSF-magnitudes as a function of the aperture-magnitudes for each W F C C D was made. Stars that biased the best-fitting straight line away from a slope of unity were Chapter 4. The HST Data 76 interactively discarded until the slope approached unity and discarding additional stars failed to drive the slope closer to unity. The aperture correction was then computed from the sample of stars that survived this culling process. Table 4.2 lists the adopted aperture corrections for the WF2 and WF4 CCDs. Figure 4.1 shows the residuals for the /-band aperture corrections to the WF2 C C D in the G312 field. I was unable to obtain reliable aperture corrections for the PC images of the G312 field due to a lack of bright stars in this field. Photometry in this field exhibited an unusually high degree of scatter despite the stellar images appearing to be normal to the eye. As a result the PC data in the G312 field is not included in any of the analysis presented here. Field CCD Filter (ap - PSF) Slope N G302 PC F555W -0.0226 ± 0.0355 0.992 12 F814W -0.0222 ± 0.0218 1.009 29 WF2 F555W +0.0184 ± 0.0216 1.065 25 F814W +0.0374 ± 0.0099 1.017 80 WF3 F555W -0.0124 ± 0.0101 1.003 87 F814W +0.0150 ± 0.0063 1.018 217 WF4 F555W +0.0333 ± 0.0119 1.011 90 F814W +0.0297 ± 0.0053 1.005 252 G312 PC F555W F814W WF2 F555W -0.0344 ± 0.0144 1.008 186 F814W +0.0405 ± 0.0087 1.034 119 WF3 F555W -0.0198 + 0.0081 1.018 101 F814W -0.0030 + 0.0013 1.015 125 WF4 F555W +0.0513 + 0.0165 1.001 85 F814W +0.0333 + 0.0102 1.027 89 Table 4.2: Aperture corrections for the HST data. The photometry was calibrated to standard Johnson-Cousins V- and /-band mag-nitudes using the transformations of Holtzman et al. (1995b) [73]. I adopted a charge Chapter 4. The HST Data 77 20 19 in OL, 18 17 0.4 _ 0.2 2 0 ro SH -0.2 -0.4 i 1 1 r i 1 1 r i 1 1 r Slope = 1.034 ± 0.012 G312 Field, WF2 CCD, I-Band (i a p - W> = +0.0405 ± 0.0087 N = 119 J I I L J I I L 17 18 19 20 ap Figure 4.1: The upper panel shows the aperture-magnitude vs PSF-magnitude relation for the J-band photometry for the WF2 CCD in the G312 field. Stars were discarded until the slope of this relation approached unity. The lower panel shows the residuals ((*ap — *PSF) — A i , where Ai — (iap —^PSF))- Stars that failed the culling process described in Chapter 4.3 have not been plotted. Chapter 4. The HST Data 78 transfer efficiency correction of 2% (Holtzman et al. 1995a [72]) for all the WFPC2 im-ages. The reddening from the Milky Way galaxy in the direction of G302 and G312 is EB-v = 0.08 ± 0.02 (Burstein & Heiles 1982 [23]). The EB_V reddening was conver-ted to a Ev-i reddening using the relationship of Bessell & Brett (1988) [13]: Ey-i = 1.25EB~V = 0 .10±0.03 . The interstellar extinctions were taken from Tables 12A and 12B of Holtzman et al. (1995) [73]. These extinction values are dependent on the stellar spec-trum. For a K5 giant the extinctions were Ay = 0.320 and Aj = 0.190. The standard distance modulus for M31 is UQ = (m — M ) 0 = 24 .3±0 .1 (van den Bergh 1991 [148]). This value agrees well with other M31 distance moduli obtained using Popu-lation II distance indicators. Pritchet Sz van den Bergh (1987) [114] found u = 24 .34±0.15 from RR Lyrae variables in the halo of M31 while Christian & Heasley (1991) [29] found fi ~ 24.3 from the brightest red giants in M31 globular clusters. Studies of Population I distance indicators such as Cepheids (e.g. Freedman &: Madore 1990 [54]) and carbon stars (e.g. Brewer et al. 1995 [19]) find u ~ 24.4, slightly larger than the Population II value. The lower distance modulus was adopted since the halo and globular clusters of M31 appear to consist of Population II stars. As this work was being completed, parallax measurements from the Hipparcos satellite were published which have led to a revised estimate of the distance to M31. The new distance modulus of fio = 24.77 ± 0.11 (Feast & Catchpole 1997 [53]), or 900 ± 45 kpc, is a 25% increase over the distance ad-opted in this work. However, other researchers (e.g. Madore &; Freedman 1997 [93]) have argued that the Hipparcos data supports the old distance scale so the question of M31's distance has still not been answered. A L L F R A M E ' s x statistic represents the ratio of the observed pixel-to-pixel scatter in the PSF fitting residuals to the expected amount of scatter given the noise properties of the data. Values of x that are significantly greater than unity indicate that a stellar image is not well fit by the stellar PSF and thus may not be a star. In light of this any Chapter 4. The HST Data 79 stars with a final % value greater than two were discarded. In order to eliminate stars with poorly determined magnitudes an interactive approach based on the uncertainty in the calibrated magnitude was used. A plot of uncertainty vs magnitude was made and a locus defined that corresponded to the expected photometric scatter at a given magnitude. Any stars that were judged to have a significantly greater uncertainty than the typical value for that magnitude were discarded. Visual examinations of the CMDs and the W F C images suggested that this culling processes did not remove any legitimate stars. It should be stressed that this culling was done solely on the basis of the photometric uncertainty of each star, not on the star's location on the C M D or the C C D . Mean photometric uncertainties for the entire culled data set are listed in Table 4.3. The mean photometric uncertainties, based on the formal photometric uncertainties returned by D A O P H O T I l / A L L F R A M E , are listed in Table 4.4. The entire data set of photometry is published on Volume 9 of the AAS CD-Rom series. The first ten lines of the complete set of calibrated stellar photometry for G302 and its surrounding fields is listed in Table 4.5 while a sample of the photometry for G312 and its surrounding fields (with the exception of the PC field) is listed in Table 4.6. Column (1) gives the C C D that the star was observed on. Column (2) gives an identification number for each star. The identification numbers are only unique for stars on a given C C D . Columns (3) and (4) give the location of the star on the C C D in pixel coordinates. Column (5) is the V-band magnitude, column (6) the lcr uncertainty in the V-band magnitude, and column (7) is the value of the D A O P H O T \ statistic. Columns (8) to (10) are the same as columns (5) to (7) except they give values for the /-band. Tables 4.7 and 4.8 list a sample of the photometry for stars within 10" of the centre of G302 and G312. This data is the same as for Tables 4.5 and 4.6 except the C C D is always WF3, so a C C D name is not listed in the tables and an extra column, giving r, the distance of the star from the centre of the globular cluster, has been added. Mean photometric Chapter 4. The HST Data 80 G302F G312F G302F G312F V <Jy <Tv I aI aI 20 25 0.035 0 033 20.25 0.033 0 028 20 75 0.033 0 038 20.75 0.028 0 029 21 25 0 031 21.25 0.030 0 028 21 75 0.050 0 035 21.75 0.031 0 031 22 25 0.043 0 038 22.25 0.034 0 034 22 75 0.039 0 042 22.7.5 0.041 0 040 23 25 0.045 0 045 23.25 0.049 0 048 23 75 0.053 0 053 23.75 0.061 0 060 24 25 0.062 0 061 24.25 0.075 0 073 24 75 0.076 0 076 24.75 0.100 0 100 25 25 0.091 0 090 25.25 0.142 0 144 25 75 0.121 0 122 25.75 0.208 0 205 26 25 0.165 0 166 26.25 0.299 0 304 26 75 0.235 0 241 26.75 0.463 0 459 27 25 0.344 0 346 27.25 0.752 0 677 27 75 0.547 0 492 27.75 ... Table 4.3: Photometric uncertainties in the HST photometry of the halo stars. uncertainties for the entire culled data set are listed in Table 4.3. The mean photometric uncertainties for the globular cluster stars are listed in Table 4.4. Chapter 4. The HST Data G302 G312 G302 G312 V <TV I 07 20.25 20 25 0 031 20.75 20 75 0 026 0 023 21.25 21 25 0 031 0 028 21.75 0 038 21 75 0 036 0 033 22.25 0 037 22 25 0 038 0 039 22.75 0 043 0 046 22 75 0 046 0 043 23.25 0 049 0 049 23 25 0 051 0 049 23.75 0 055 0 051 23 75 0 067 0 063 24.25 0 065 0 062 24 25 0 081 0 077 24.75 0 079 0 077 24 75 0 118 0 099 25.25 0 094 0 090 25 25 0 145 0 153 25.75 0 129 0 121 25 75 0 217 0 222 26.25 0 176 0 182 26 25 0 333 0 314 26.75 0 262 0 271 26 75 0 570 0 457 27.25 0 356 0 310 27 25 27.75 27 75 28.25 28 25 Table 4.4: Photometric uncertainties for G302 and G312. C C D ID X Y V av Xv I cri Xi WF3 892 64.328 212 348 17.032 0 082 1 000 16.294 0 071 1 000 WF2 3440 466.457 631 376 17.628 0 084 0 744 16.365 0 077 0 727 WF3 2583 437.139 413 836 18.195 0 065 1 341 17.356 0 059 1 551 WF3 2606 439.952 415 701 18.454 0 063 1 462 17.642 0 064 1 830 WF3 2489 438.016 410 038 18.854 0 095 1 700 17.995 0 057 1 240 WF4 4532 300.305 699 237 19.152 0 031 0 739 17.418 0 030 0 877 WF3 2474 441.016 410 137 19.183 0 111 1 823 17.977 0 073 1 819 WF2 150 321.521 80 650 19.311 0 036 0 657 16.489 0 089 0 941 WF2 2719 310.787 523 757 19.669 0 070 1 499 18.432 0 063 1 959 WF3 2554 442.354 412 389 19.741 0 101 1 325 18.639 0 076 1 507 Table 4.5: A sample of the stellar photometry for G302 and the surrounding fields. Chapter 4. The HST Data 82 C C D ID X Y V o-v Xv I o-i Xi WF3 503 689.043 346.030 17.369 0.050 0.827 16.723 0.025 0.758 WF2 1198 252.106 634.779 17.758 0.057 0.745 16.875 0.055 0.830 WF4 537 650.877 360.807 17.934 0.051 0.982 17.217 0.026 0.801 WF3 802 417.826 417.893 17.983 0.060 1.509 17.020 0.061 1.704 WF4 869 310.743 507.150 18.569 0.028 0.630 17.479 0.026 0.789 WF3 811 419.875 419.020 18.662 0.078 1.841 17.422 0.053 1.494 WF3 1654 229.986 766.329 18.743 0.035 0.907 18.116 0.019 0.601 WF3 412 167.937 301.246 18.896 0.034 0.877 16.521 0.030 0.756 WF3 770 415.495 417.557 19.926 0.055 1.014 18.819 0.065 1.407 WF3 1451 676.379 661.313 20.225 0.038 0.948 18.614 0.025 0.736 Table 4.6: A sample of the stellar photometry for G312 and the surrounding fields. ID X Y r V o-v Xv I o-i Xi 2583 437.139 413.836 07230 18.195 0.065 1.341 17.356 0.059 1.551 2606 439.952 415.701 07332 18.454 0.063 1.462 17.642 0.064 1.830 2489 438.016 410.038 07265 18.854 0.095 1.700 17.995 0.057 1.240 2474 441.016 410.137 07308 19.183 0.111 1.823 17.977 0.073 1.819 2554 442.354 412.389 07332 19.741 0.101 1.325 18.639 0.076 1.507 2532 429.789 412.006 07921 20.376 0.079 1.847 19.356 0.067 1.961 2420 440.762 402.723 07989 20.812 0.071 1.629 19.648 0.063 1.828 2553 428.372 414.756 17084 20.947 0.058 1.355 19.735 0.043 1.256 2460 418.943 405.789 27108 21.148 0.027 0.679 19.992 0.029 0.922 2456 427.305 405.329 17368 21.241 0.048 1.071 19.853 0.046 1.304 Table 4.7: A sample of the stellar photometry for the globular cluster G302. Chapter 4. The HST Data I D X Y r V Xv I o-i Xi 802 417.826 417.893 0"039 17.983 0.060 1.509 17.020 0.061 1.704 811 419.875 419.020 07198 18.662 0.078 1.841 17.422 0.053 1.494 770 415.495 417.557 07274 19.926 0.055 1.014 18.819 0.065 1.407 866 417.573 427.651 07972 22.019 0.086 1.941 20.250 0.045 1.217 783 429.730 414.665 17191 22.086 0.056 1.330 20.499 0.050 1.467 803 405.224 418.070 17294 22.180 0.053 1.224 20.275 0.048 1.413 841 396.894 424.414 27221 22.310 0.073 1.267 20.965 0.059 1.244 871 412.104 427.591 17141 22.355 0.072 1.544 21.183 0.066 1.812 915 418.244 434.008 17603 22.645 0.079 1.777 21.687 0.068 1.789 796 434.824 416.729 17658 22.676 0.041 0.920 21.468 0.035 0.953 Tab le 4.8: A sample of the stel lar p h o t o m e t r y for the g lobular c lus ter G 3 1 2 . Chapter 4. The HST Data 84 4.4 Artificial Star Tests In order to determine the effects of crowding, and the efficiency of the star-finding and photometry techniques, on the photometry a series of artificial star tests were performed. The following procedure was used to generate and recover artificial stars: 1. For each field a series of artificial stars with (V,(V —/)) colours based on the ob-served C M D and luminosity function of the M31 halo stars was generated. 2. The artificial stars were added on to each of the original images. The total number of artificial stars added in each field was ~ 10% of the total number of stars in that field. 3. The entire data reduction process for the photometry that was described in Chap-ter 4.3 (FIND/ALLFRAME/culling/matching) was performed on this new set of im-ages. The resulting photometry was then matched against the original list of arti-ficial stars for that field. 4. This process was repeated until the number of recovered artificial stars was approx-imately the same as the number of real stars found on that CCD. This process was performed for each CCD field. Figure 4.2 shows the relationship between the input magnitudes of the artificial stars and their recovered magnitudes for a typical set of stars, in this case using the WF2 field near the globular cluster G312. Artificial stars with magnitude differences of more than ±AV = 1 mag have been omitted. This figure plots the magnitude difference as a function of the recovered magnitude, not the input magnitude, of the artificial stars. This was done because the observed magnitudes of stars in the real data set are recovered magnitudes, not the true (or input) magnitudes of the real stars. Therefore, Figure 4.2 Chapter 4. The HST Data 85 can be used to directly determine the uncertainty in a star's observed magnitude. An examination of this figure suggests that the distribution of artificial horizontal branch stars is different from the distribution of artificial red-giant branch stars. To test if the two distributions are the same I performed a Kolmogorov-Smirnov (KS) test. The results showed that the distributions of recovered magnitudes for the artificial red-giant branch and horizontal branch stars are the same at the 98.892% confidence level in the V-band and at the 99.999% confidence level in the J-band. This suggests that the null hypothesis (that the distributions of the two types of stars are the same) can not be rejected. Therefore there is no evidence that the apparent excess scatter in the magnitudes of the horizontal branch artificial stars is anything other than an artifact of the large number of artificial horizontal branch stars. Table 4.9 lists the systematic shifts in V and (V — I)0 as well as the degree of scatter as a function of magnitude for the artificial red-giant branch stars of G312. Table 4.10 lists these quantities for the G312 artificial horizontal branch stars. In these tables the notation [x] indicates the median value of the individual Xi values. The columns are (1) the boundaries of the bin, (2) the median recovered magnitude, [V]; (3) the median photometric uncertainty in the recovered magnitudes (from D A O P H O T ) , [cy]; (4) the median magnitude difference, [Sy] = [Recovered — Knput ] ; (5) the amount of scatter present in the artificial stars in that bin, ay = 8y — [8v] ', columns (6) to (9) are the same quantities for the (V — I)0 colour; and (10) is the number of artificial stars in that bin. These quantities are identical to those described by Stetson &z Harris (1988) [142] except for a, which is defined in non-parametric terms instead of in Gaussian terms. There is a slight tendency for the reduction process to underestimate the brightnesses of, stars near the photometric limit of the data (see column 4 of Table 4.9). However, this systematic shift is less than the internal scatter in the photometry and is smaller than ~ 0.05 mag for red-giant branch stars at the level of the horizontal branch. Therefore, Chapter 4. The HST Data 86 V [VI Wv\ [Sv] <7y {(v-i)0] [*(V-/)J i8(V-I)n] N 22.75 22.947 0.058 -0.008 0.039 1.536 0.074 0.002 0.049 2 23.25 23.361 0.065 -0.030 0.061 1.382 0.082 -0.013 0.028 19 23.75 23.800 0.064 -0.006 0.056 1.237 0.089 -0.010 0.062 31 24.25 24.262 0.081 0.024 0.055 1.081 0.102 0.009 0.064 55 24.75 24.781 0.087 0.023 0.085 1.007 0.115 0.028 0.087 78 25.25 25.251 0.104 0.041 0.098 0.918 0.138 -0.006 0.094 110 25.75 25.752 0.128 0.043 0.176 0.890 0.177 -0.009 0.187 116 26.25 26.235 0.179 0.068 0.277 0.881 0.244 0.004 0.192 88 26.75 26.654 0.232 0.138 0.412 0.873 0.319 0.042 0.184 33 27.25 27.076 0.308 0.347 0.052 0.826 0.415 -0.011 0.169 3 Table 4.9: Magnitude shifts in the red-giant branch artificial star data. V [V] [cry] [8y] by [{V-I)0] [<T(V-I)a] [5(V-I)J *(V-I), N 24.25 24.384 0.073 -0.744 0.100 0.623 0.106 0.099 0.159 12 24.75 24.911 0.086 -0.194 0.123 0.429 0.134 0.010 0.181 94 25.25 25.182 0.102 0.055 0.090 0.537 0.153 -0.020 0.136 823 25.75 25.615 0.214 0.486 0.200 0.945 0.236 0.400 0.293 12 26.25 26.054 0.246 0.890 0.000 1.569 0.274 0.856 0.000 1 Table 4.10: Magnitude shifts in the horizontal branch artificial star data. the systematic uncertainties in the photometric reductions do not affect the morphology of the CMDs above V ~ 26. Figure 4.3 shows a CMD created from a set of artificial red-giant branch stars added to the G312 WF3 image. The horizontal branch stars have been omitted in order to show the increase in the scatter in the recovered colours at fainter magnitudes. The amount of scatter seen at a particular magnitude is comparable to the amount of scatter seen in the C M D of G312, which is believed to be made up of stars with a single metalhcity. If the stellar populations in M31's globular clusters are analogous to the stellar popu-lations in the Galactic globular clusters then the observed spread in G312's red-giant Chapter 4. The HST Data 87 branch is primarily due to photometric uncertainties. Figure 4.3 shows that photometric uncertainties are sufficient to explain all the observed width of the red-giant branch of G312. Table 4.10 and Figure 4.4 show that horizontal branch stars, because of the large colour spread in the horizontal branch, can suffer badly from systematic differences be-tween their input and recovered magnitudes. Blue horizontal branch stars are scattered by as much 0.5 mag in (V —J)0 while red horizontal branch stars have a tendency to be recovered redder and fainter than they really are. This tends to move red horizontal branch stars into the lower half of the red-giant branch resulting in an underestimate of the total number of red horizontal branch stars in the globular cluster. These results show that stars from the red and blue sides of the horizontal branch can be scattered into the RR Lyrae gap making it very difficult to distinguish RR Lyrae candidates from other horizontal branch stars that have simply scattered into the RR Lyrae gap. RR Lyrae stars are variable stars with periods of ~ 10 to ~ 20 hours. They can, in principle, be identified simply by constructing a light curve for each star and looking for variability. Unfortunately this approach will not work for the HST data used in this thesis since the observations occurred over approximately three hours, which is only ~ 15% to ~ 30% of the period of the RR Lyraes. Since the photometric uncertainties at the level of the RR Lyraes (V ~ 25) are uy ~ 0.09 mag and the variability of an RR Lyrae star is AV ~ 0.5 mag it was not possible to determine if any of the stars in the RR Lyrae gap were varying in magnitude over the course of the observations. Chapter 4. The HST Data 88 T r i i i i i i l i i i i i i i i 22 24 26 28 Recovered Magnitude Figure 4.2: This figure shows the scatter in the recovered magnitudes of the artificial stars added to the WF2 background field ~ 1' south of the globular cluster G312. The upper (a) panel shows the observed magnitude shifts for the V-band photometry while the lower (b) panel shows the observed magnitude shifts in the J-band. Open circles denote red-giant branch stars while crosses denote horizontal branch stars. Only stars with A mag < 1 mag are shown. Negative magnitude shifts indicate that the star was recovered brighter than it was input while positive shifts indicate the star was recovered fainter than it was input. The apparent excess scatter in the horizontal branch artificial stars is an artifact of the large number of artificial horizontal branch stars (see Chapter 4.4). Chapter 4. The HST Data 89 22 L Artificial Stars 24 26 28 i i i i i i i i i i i i i i i i i i i i i i i i i i i i i i i i i i i i i i i i - (a) I I I I I I I I I I I l l l I I I I I G312 (b) 1 • i • i i i i i i i i i I i i i i i ' i i i 1 0 1 2 3-1 0 1 2 3 Figure 4.3: The left-hand panel (a) shows a CMD for a set of artificial stars added to the G312 WF3 field. Large solid circles represent the input colours and magnitudes while small solid squares represent the recovered photometry. For clarity, only red-giant branch stars are shown in this figure. The right-hand panel (b) shows the observed C M D for red giants in G312. The spread in the recovered colours in the artificial stars is approximately the same as the spread in colour observed in the G312 C M D suggesting that the observed width of the red-giant branch is due to photometric uncertainties and is not intrinsic to G312. Chapter 4. The HST Data 90 24.5 Horizontal Branch 25 ^25.5 h 26 26.5 0 (v - /)„ Figure 4.4: The large solid circles show the input artificial horizontal branch. The hori-zontal branch has been divided into distinct red and blue parts to aid in the identification of scatter from each component. Small solid squares show the recovered colours and mag-nitudes of the artificial stars. Chapter 5 HST Observations of G302 and G312 5.1 The Colour—Magnitude Diagrams 5.1.1 Contamination There are four possible sources of contamination in the CMDs of the globular clusters. The first three—Galactic halo stars, background galaxies, and M31 disc stars—are dis-cussed in Chapter 6.1.1. By scaling those results to the area on the sky covered by each globular cluster I estimate that at most 15 ± 4 of the objects with V < 24 in each of Figures 5.1 (G302) and 5.2 (G312) are due to contamination from these three sources. The fourth source of contamination is the halo of M31 itself. To estimate the number of halo stars that should be visible in the CMD for each globular cluster I computed the stellar surface densities in the WF2 and WF4 CCDs for each field. These CCDs were centred ~ 60" from the centre of the WF3 CCD (and thus the globular clusters). Since G302 and G312 have tidal radii of - 10" the WF2 and WF4 CCDs will not suffer from contamination from globular cluster stars. The mean background surface density for M31 halo stars with V < 24 near G302 is S b k g d = 0.0654 ± 0.0024 stars/H", corresponding to 91 Chapter 5. HST Observations of G302 and G312 92 20 ± 5 halo stars in the red-giant branch for G302. Near G312 the background surface density is Sbkgd = 0 . 0 1 7 4 ± 0.0012 stars/D", corresponding to 5 ± 1 stars in the red-giant branch of G312. These estimates are reasonably consistent with the numbers of objects in the upper halves of Figures 5.1 and 5.2 that do not fall on either the red-giant branch or the asymptotic-giant branch of each globular cluster. On the basis of these calculations I expect that no more than ~ 7% of the objects in the G302 C M D are not members of that globular cluster and no more than ~ 16% of the objects in the G312 CMD are not members of that globular cluster. These numbers are consistent with the number of discrepant objects in each C M D so I conclude that the morphologies of the CMDs in Figures 5.1 and 5.2 are real and not due to contamination. Chapter 5. HST Observations of G302 and G312 93 i 1 1 1 1 1 1 1 1 1 1 i i i | i i i i -I [Fe/H] = -2.17 -1.91 -1.58 -1.54 -1.29 -0.71 22 24 26 28 • m m * • . • ' <>. . J l_ J L J I I L G302 /i = 24.3 Ey_j = 0.10 I I I I L -1 0 (V-I), Figure 5.1: This figure shows the CMD for G302. Only stars between 275 < r < 10" are plotted to minimize contamination from stars in the halo of M31 and from unreliable photometry due to the extreme crowding in the central regions of G302. The distinct blue horizontal branch is consistent with a metal-poor stellar population. The fiducial sequences are of Galactic globular clusters and were taken from DCA. Chapter 5. HST Observations of G302 and G312 94 22 26 28 1 r - i i | i i i i | i i i i _ [Fe/H] = -2.17 -1.91 -1.54 -1.29 III r '' * "* | i i i i -0.71 -0.40 _ • * — *\\u / * / • * — ' K • G312 /j, = 24.3 _ • • * . . • • — • • • i i i i 1 i i i i 1 i i i i E^_7 = 0.10 _ 1 i i i i I I I I I I I I I I I I I I I I 1 1 1 1 1 ' - 1 0 1 2 3 (V-I)o Figure 5.2: This figure shows the C M D for G312. As in Figure 5.1 only stars between 2'-'5 < r < 10" are plotted. The fiducial sequences are from D C A with the exception of the [Fe/H] = -0.40 fiducial which is from Bergbusch & VandenBerg (1992) [11]. The lack of a blue horizontal branch is consistent with G312 being metal-rich. Chapter 5. HST Observations of G302 and G312 95 5.1.2 G302 Figure 5.1 shows the C M D for the globular cluster G302. Stars located between 2"5 < r < 10" from the centre of the cluster are shown. The r = 10" cut-off approxim-ately corresponds to the tidal radius of the globular cluster while the r = 2'.'5 cut-off is the point where extreme crowding conditions make it impossible to obtain reliable stellar photometry. To estimate the metalhcity of G302 I took the dereddened fiducial red-giant branch sequences for several Galactic globular clusters (from Da Costa & Ar-mandroff 1990 [36], hereafter referred to as DC A), shifted them to the distance of M31 (/Xo = 24.3), and overlaid them onto Figure 5.1. The lack of a turn-over at the top of the G302 red-giant branch is consistent with a metal-poor population with an iron abundance of [Fe/H] ~ —1.5 to —2.2. I interpolated between the D C A fiducial sequences to obtain [Fe/H] = —1.85 ± 0.12 for G302, where the uncertainty is one-third of the spread in the red-giant branch and corresponds to approximately one standard deviation if I assume the spread in the red-giant branch is due to normally distributed errors in the photome-try. This estimate of the iron abundance for G302 agrees with spectroscopic estimates by HBK, [Fe/H] = - 1 . 7 6 ± 0 . 1 8 , and de Freitas Pacheco (1997) [39], [Fe/H] = - 1 . 8 0 ± 0 . 2 . If the Hipparcos distance modulus for M31 (/io = 2 4 . 7 7 ± 0 . 1 1 , Feast &c Catchpole 1997 [53]) is adopted the iron abundance for G302 becomes [Fe/H] = —2.2. This is ~ 2.5<r less than the spectroscopic values (which are independent of distance). Asymptotic-giant branch stars in G302 are clearly present to the blue of the low-metalhcity fiducial sequences. Unfortunately the asymptotic-giant branch blends with the blue edge of the red-giant branch so it is not possible to unambiguously distinguish asymptotic-giant branch stars from red-giant branch stars. This is primarily due to photometric scatter (oy_j ~ 0.06 at V — 23) and, to a lesser degree, contamination from stars in the halo of M31. Chapter 5. HST Observations of G302 and G312 96 No stars brighter than the tip of the red-giant branch are present in the data. The lack of such super-luminous stars supports the hypothesis that globular clusters in M31 have ages similar to those of the Galactic globular clusters. Spectral line blanketing in cool giants can cause very red giants to appear fainter than bluer, less evolved, red-giant branch stars. This can result in some massive evolved stars appearing to have V — I col-ours considerably redder than the rest of the globular cluster stars. Artificial star tests suggest that all the stars brighter than the horizontal branch were found so it is unlikely that any super-luminous asymptotic-giant branch stars in the outer regions of G302 have been missed. However, these stars, being more evolved, will have slightly more massive progenitors than the rest of the red- and asymptotic-giant stars. It is, then, possible that mass segregation will have concentrated them in the inner r = 2//5 of the globular cluster making them undetectable with our data. To date no super-luminous asymptotic-giant stars have been detected in any M31 globular cluster (e.g. see Rich et al. 1996 [119], Fusi Pecci et al. 1996 [56], Jablonka et al. 1997 [79] for CMDs obtained using HST obser-vations). This strongly supports the idea that the M31 globular clusters have comparable ages to those of the Galactic globular clusters. It should be emphasized, however, that although there is no evidence in the data for G302 having an age that is any different from the ages of the Galactic globular clusters, it is possible that super-luminous stars are present in the inner regions of G302. Further evidence that G302 is an old globular cluster with an age similar to that of the Galactic globular clusters comes from de Freitas Pacheco (1997) [39]. That study used integrated spectra and single-population stellar population models to estimate the ages of twelve M31 globular clusters including G302. The result was a mean age of 15 ± 2.8 Gyr for those globular clusters and an age of 17.0 ± 2.9 Gyr for G302. The weighted mean magnitude of the stars in the RR Lyrae gap is VRR = 24.93 ± 0 . 0 9 (standard deviation). Using this, the derived iron abundance of [Fe/H] = —1.85, and Chapter 5. HST Observations of G302 and G312 97 MV(RR) = 0.20[Fe/H] + 0.98 (5.1) (Chaboyer et al. 1996 [28]), gives a distance modulus of fi0 = 24.32 ± 0.09 for G302. Chaboyer et al. (1997) [27] have derived a new relation, My (RR) = (0.23 ± 0.04) ([Fe/H] + 1.9) + (0.39 ± 0.08 ±g;J|) , (5.2) between the iron abundance and absolute magnitude of RR Lyrae stars which utilizes the Hipparcos parallaxes. Equation 5.2 gives a distance modulus of /io = 24.53 ± 0.08 JZolil f ° r G302, placing G302 ~ 100 kpc in front of M31 if M31 is 900 kpc from the Earth. This is inconsistent with the H B K radial velocity and the orbital parameters from Chapter 5.5.3. In order to identify the outer limiting radius of G302, and to search for any radial dependence in the morphology of the various branches of the C M D , I constructed CMDs in four annuli centred on the cluster. These CMDs are shown in Figure 5.3. The fiducial sequences plotted in Figure 5.3 are those of NGC 6397 ([Fe/H] = -1.91) and 47 Tuc ([Fe/H] = -0.71) taken from DCA. For r £ 8" the N G C 6397 fiducial provides a good fit to the G302 red-giant branch. The globular cluster appears to end at a radius somewhere between 8" and 11", in agreement with what is found by fitting Michie-King models (see Chapter 5.3). Beyond 11" the red-giant branch of the metal-rich halo population is clearly visible. The apparent increase in the number of blue horizontal branch stars in the inner regions of G302 is an illusion. A KS test shows that the radial distribution of blue horizontal branch stars is the same as that of the red-giant branch stars at the 99.91% confidence level. This indicates that the null hypothesis (that there is no difference between the radial distributions of the blue and red horizontal branch stars) can not be rejected. Therefore, there is no evidence in the data that there is a difference in the radial distribution of these two types of stars. Chapter 5. HST Observations of G302 and G312 98 i i i i I i i i i I i i i i I i i i i I - 2".5 d r < 5" N G C 6397 22 h 24 26 ^ 28 22 24 26 28 I I I I Tuc (a)-L 8" ^ r < 11" N G C 6397 47 Tuc I I I I I I I I I I I I I I I I I I I 1 I I I I I I I I I I I I I I I I I I I I I I I 5" ^ r < 8" ' NGF6397 47 Tuc (b). 11" < r < 14" N G C 6397 47 Tuc (d). i i i i 1 i i i i 1 i i i i I i i i i I i i 1 0 1 2 3 - 1 0 (v-i)0 1 2 3 Figure 5.3: This figure shows CMDs for four annuli centred on G302. The fiducial sequences are from DC A. Chapter 5. HST Observations of G302 and G312 99 5 . 1 . 3 G 3 1 2 Figure 5.2 shows the CMD for the globular cluster G312. As for G302 only stars be-tween 2"5 < r < 10" are shown since both star counts and the surface brightness pro-files suggest that the Michie-King tidal radius of G312 occurs at ~ 10". Interpolat-ing between the fiducial giant branches suggests that G312 has an iron abundance of [Fe/H] = —0.56 ± 0.03, comparable to the mean iron abundance of stars in the halo of M31. This is in broad agreement with the spectroscopic iron abundance determined by HBK of [Fe/H] = —0.70 ± 0.35. The small number of red giants falling under the metal-poor fiducial sequences is consistent with the expected degree of contamination (see Chapter 5.1.1) from disc and metal-poor halo stars in M31 (see Chapter 6.1.2 for details on the metallicity distribution of halo stars in M31). If the Hipparcos distance to M31 is adopted then the iron abundance of G312 becomes [Fe/H] = —0.7, consistent with the spectroscopically determined iron abundance. The horizontal branch of G312 is a red clump located at V ~ 25 and slightly to the blue of the red-giant branch. This is consistent with G312 being a metal-rich globular cluster. No horizontal branch stars bluer than (V —7)0 = 0.6 are present on the CMD so it is not possible to directly determine the magnitude of the horizontal branch in the R R Lyrae gap. Therefore, I estimated V R R by taking the weighted mean magnitude of the red clump stars ( VHB = 25.09 ± 0.09) and added a correction of A V ^ f f = 0.08 (see Sarajedini et al. 1995 [126], and Ajhar et al. 1996 [1]) to get V R R . This gives V R R = 25.17 ± 0.09 for G312. Using the iron abundance derived from the shape of the red-giant branch, and the Chaboyer et al. (1996) [28] relation (Equation 5.1), gives a distance modulus of u-0 = 24.30 ± 0.09 for G312. If the Chaboyer et al. (1997) [27] relation (Equation 5.2, which is based on Hipparcos parallax data) is used then the distance modulus for G312 becomes /XQ = 24.47 ± 0.08 l^is-Chapter 5. HST Observations of G302 and G312 100 Since so few halo stars are expected in the upper red-giant branch of G312 the ob-served width of the red-giant branch provides a good estimate of the true photometric uncertainties in our observations. Photometric and spectroscopic evidence strongly sug-gests that all stars in a single Galactic globular clusters1 have the same metallicity and there is no evidence to suggest that this should be different for globular clusters in M31. The mean width of the red-giant branch is o~v-i ~ 0.08, consistent with what is pre-dicted by the artificial star simulations (see Chapter 4.4). Therefore I believe that the width of the red-giant branch in Figure 5.2 is typical of a single-metalhcity population for the W F C observations. This figure can be compared to Figures 6.3 and 6.4 to show that the large range of metallicities seen in the halo of M31 is not merely an artifact of photometric uncertainties along the' halo red-giant branch. Figure 5.4 shows no radial dependence in the morphology of the G312 C M D . 1Some of the most massive Galactic globular clusters, such as w Cen and M22, show evidence for having a range of metallicities. However, these two Galactic globular clusters are ~ 10 times more massive than G312 is. Chapter 5. HST Observations of G302 and G312 101 Figure 5.4: This figure shows CMDs for four annuli centred on G312. Since G312 has a metallicity similar to the mean metallicity of the M31 halo stars in the line of sight near G312 it is not possible to determine the limiting radius of G312 in this figure. The fiducial sequences are from DC A. Chapter 5. HST Observations of G302 and G312 102 5.1.4 The Colour—Iron Abundance Relation Figure 5.5 shows the CMDs for G302 and G312 after converting the calibrated magnitudes to absolute /-band magnitudes. This was done to determine the colour of the red-giant branch for each globular cluster at Mj = —3, hereafter referred to as (V — I)0_3. DCA found a strong relationship between iron abundance and (V — / ) 0 _ 3 for Galactic globular clusters with —2.2 < [Fe/H] < —0.7. Figure 5.6a shows the relationship between ( V - J ) 0 3 and [Fe/H]s for G302 and G312 and several other M31 globular clusters with published CMDs. The published data has been adjusted to a distance modulus of /x0 = 24.3. The [Fe/H]s values were all taken from HBK so the (V — /) 0 _ 3 colour and iron abundance are determined completely independently of each other for each globular cluster. Colour, iron abundance, and reddening data for each globular cluster is listed in Table 5.1 along with the source of the CMD, the projected distance of each globular cluster from the centre of M31 (-RM3I)> and Y, the projected distance of each globular cluster from the major axis of M31 with positive Y being on the northwest side of the major axis of M31. Cluster (V - I)o,s [Fe/H]s [ F e / H ] C M D Ev-i •RM31 Y Reference G l 1.47 ± 0 . 0 6 -1.08 ± 0 . 0 9 -0.65 ± 0 . 1 0 0.10 ± 0 . 0 3 152.'3 +29.'l [119] G i l 1.22 ± 0 . 0 4 -1.89 ± 0 . 1 7 -1 .7 ± 0.20 0.10 ± 0 . 0 3 75.'7 +43.'6 [31] G58 1.80 ± 0 . 0 5 -0.57 ± 0 . 1 5 -0.57 ± 0 . 1 5 0.14 ± 0.04 28.'2 +27.'3 [1] G105 1.31 ± 0 . 0 2 -1.49 ± 0 . 1 7 -1.49 ± 0 . 1 7 0.08 ± 0.02 64.'8 -29.'9 [1] G108 1.62 ± 0 . 0 5 -0.94 ± 0 . 2 7 -0.80 ± 0 . 1 0 0.15 ± 0 . 0 4 20.'8 + 19.'7 [1] G219 1.20 ± 0 . 0 2 -1.83 ± 0 . 2 2 -2.04 ± 0 . 2 2 0.08 ± 0.02 87.'2 -58.'8 [1] G302 1.24 ± 0 . 0 4 -1.76 ± 0 . 1 8 -1.85 ± 0.12 0.10 ± 0.03 32.'1 -30.'4 §5.1.2 G312 1.88 ± 0 . 0 7 -0.70 ± 0 . 3 5 -0.56 ± 0 . 0 3 0.10 ± 0.03 49.'8 -49.'7 §5.1.3 G319 1.70 ± 0 . 1 7 -0.66 ± 0 . 2 2 -0.6 ± 0 . 9 0 0.10 ± 0.03 72.'1 -69.'0 [31] G323 1.17 ± 0 . 0 4 -1.96 ± 0 . 2 9 -2 .0 ± 0 . 2 0 0.10 ± 0.03 53.'8 -53.'8 [31] G327 1.31 ± 0 . 0 6 -1.78 ± 0 . 1 1 -1.3 ± 0 . 3 0 0.10 ± 0.03 99.'7 + 19.'9 [31] G352 1.84 ± 0 . 0 6 -0.85 ± 0 . 3 3 -0.5 ± 0 . 1 0 0.10 ± 0 . 0 3 87.'1 -49.'6 [31] Table 5.1: Properties of selected globular clusters in the M31 system. Chapter 5. HST Observations of G302 and G312 103 Since several of the M31 globular clusters in Figure 5.6a have iron abundances at least as high as that of 47 Tuc the D C A relationship was extended to higher iron abun-dances. Extending this relationship to globular clusters more metal-rich than 47 Tuc ([Fe/H] = —0.71) is somewhat difficult since the red-giant branch in the ((V — J ) 0 ,Mj ) plane becomes asymptotically flat as the iron abundance increases. This makes it diffi-cult to define (V — I)0_3 since, as [Fe/H] approaches the Solar value, this point occurs in the horizontal portion of the red-giant branch. In order to extend the relationship to higher iron abundances I determined (V — / ) 0 _ 3 for the metal-rich Galactic globular cluster NGC 6553 using the (I,(V-I)) CMD of Ortolani et al. (1990) [104]. I as-sumed a distance modulus of UQ = 13.35 (Guarnieri et al. 1995 [60]), a reddening of EB-v = 0.78 (Bico & Alloin 1986 [16]), and an iron abundance of [Fe/H] = -0.29 (Zinn & West 1984 [156]). Using this data I found ( V - J ) 0 _ 8 = 2.4 ± 0.2 for NGC 6553. This point lies just below the region where the red-giant branch becomes horizontal so there is a significant uncertainty in the value of (V — I)0 _ 3 for NGC 6553 solely due to the width of the red-giant branch in (V —1)0 at M j = —3. I extrapolated the D C A relation to [Fe/H] = -0.29 simply by connecting the [Fe/H] = -0.71 end of the D C A relation to the location of NGC 6552. The resulting relation is: [Fe/H] = -15.16 + 17.0(V - /)„__, - A.9(V - I)l_s (5.3) for -2.2 < [Fe/H] < -0.7 (from DCA) and [Fe/H] = -1.36 + 0.44(V - I)0_3 (5.4) for —0.7 < [Fe/H] < —0.29 (my extension). It should be stressed that this extension to high iron abundances is an estimate and is intended only to provide a reasonable estimate of what happens to the DCA relation at high iron abundances. Chapter 5. HST Observations of G302 and G312 104 If the spectroscopic iron abundances ([Fe/H]s) are adopted then nine of the twelve M31 globular clusters are redder than predicted by the extended D C A relation. This discrepancy is present at all iron abundances but is most noticeable for the metal-rich globular clusters. The individual uncertainties in [Fe/H] s for all the globular clusters except G l and G327 are large enough to account for the difference between their loca-tions in Figure 5.6a and the predictions of the extended D C A relation. A recent paper by de Freitas Pacheco (1997) [39] gives iron abundances for twelve globular clusters in the M31 system. Most of his iron abundances are higher than the HBK values by ~0.2 dex. An examination of the CMDs for each globular cluster shows that several globu-lar clusters have red-giant branches that are significantly redder and natter than would be expected from their spectroscopic iron abundances. This is particularly noticeable for the more metal-rich globular clusters. To test if errors in the spectroscopically de-termined iron abundance determinations are sufficient to account for the discrepancies between the globular clusters and the extended D C A relation in Figure 5.6a I replotted the globular clusters using iron abundances determined from the shapes of the red-giant branches. This data is shown in Figure 5.6b. All of the globular clusters now fall within their uncertainties of the extended D C A relationship with the exception of G108, which has a (V — I)0 _ 3 value nearly 3<r redder than its predicted value. Alternately the iron abundance Ajhar et al. (1996) [1] derived from Gl08's CMD could be in error. However, their Figure 20 shows the upper portion of the red-giant branch of G108 to be not quite as flat than that of 47 Tuc, indicating that G108 has slightly less iron than 47 Tuc does. It would be difficult, using their data, to fit a fiducial to the G108 red-giant branch that is flatter (and thus more iron rich) than the 47 Tuc fiducial. Therefore it is unlikely that the estimated iron abundance of G108 is in error. One possible explanation for the red colour of G108 is that the H I in M31 extends ~ 15 kpc beyond the outer edge of the optical disc and past the location of G108. Chapter 5. HST Observations of G302 and G312 105 Cuillandre et al. (1997) [35] found dust beyond the optical limit of the M31 disc, and that the distribution of this dust correlates with the distribution of H I beyond the disc. Since G108 is located within the extended disc of H I around M31 it is possible that G108's (V — J) 0 _ 3 colour may be due to internal reddening in the extended disc of M31. Using the H I maps of Emerson (1974) [49], and the relationships between H I column density and reddening given in Cuillandre et al. (1997) [35], I find an excess reddening of Ey-i — 0.14 ± 0 . 0 2 . However, an examination of the C M D for G108 (Figure 20 of Ajhar et al. 1996 [1]) suggests that internal reddening from M31 does not exceed Ey-i ~ 0.05. This discrepancy may be due to the patchy nature of the H I distribution in the outer regions of M31's extended disc. Brinks k Bajaja (1986) [20] found many small regions in the H I distribution in M31 where the H I column densities were significantly less than the H I column densities in the surrounding column densities. These holes were typically a few arcminutes in diameter, smaller than the beam-size used to construct the Emerson (1974) [49] H I maps. It is possible that G108 (which has an angular diameter of ~ 0.'4) is located in a hole in the H I distribution that is smaller than the resolution of the H I maps. If this is the case then the Emerson (1974) [49] H I maps will lead to an overestimate of the reddening toward G108 and my estimate of the excess reddening towards G108 would not be inconsistent with the published C M D for that globular cluster. If the Hipparcos distance modulus (fiQ = 24 .77±0.11) of Feast k Catchpole (1997) [53] is adopted then the metal-rich globular clusters in Figure 5.6 would be in better agree-ment with the extended D C A relation. However, the metal-poor globular clusters would have (V —1)0 _ 3 colours that are bluer than would be expected if the distance modulus of the M31 globular cluster system is fio = 24.3 ± 0.1. This would result in the metal-poor globular clusters lying on the left of the D C A relation, implying that the spectroscop-ically determined iron abundances for the metal-poor M31 globular clusters have been Chapter 5. HST Observations of G302 and G312 106 systematically overestimated. Since iron abundances are more difficult to measure in spectra of metal-rich globular clusters it is more likely that the [Fe/H] values for the metal-rich globular clusters have been underestimated than it is that the [Fe/H] values for the metal-poor globular clusters have been overestimated. Chapter 5. HST Observations of G302 and G312 107 - 4 1 i i i i 1 i i i i 1 i i i i 1 i i i i 1 ~G302 • — • - J T T ' I 1 | 1 1 1 1 | 1 1 1 1 | 1 1 1 1 | "G312 •> • • • -• • • • — - 2 • v . . i" •• • •Jf. 0 £ v.. :,-*£>^  •• • - $ .v.:* • • • • • V | . 2 . . . v * • • . _ 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 - 1 0 1 2 3 - 1 0 1 2 3 (V-I)0 (V-I)Q Figure 5.5: This figure shows the (Mj, (V - / )„ ) CMDs for G302 and G312. A distance modulus of fi0 = 24.3, a reddening of Ey-i — 0.1, and an extinction Ai = 0.19 was assumed as discussed in Chapter 4.3. Chapter 5. HST Observations of G302 and G312 108 0 - 0.5 h OT K -1 1.5 -2 CD 0 - 0.5 Q s 1 fe-1.5 -2 i 1 1 r i 1 1 r i 1 1 r _ - -* (a) + J I I L J I I L (b) J L 1.5 2 (V ~ J)o.- 3 2.5 Figure 5.6: The open .circles represent the data for G302 and G312 from Chapter 5. The solid circle is data from Rich et al. (1996) [119], the solid squares are data from Couture et al. (1995) [31], and the solid triangles are data from Ajhar et al. (1996) [1]. The horizontal error bars only include the uncertainty due to the width of the red-giant branch, not any uncertainties in the reddening or distance modulus of the individual globular clusters. The solid line shows the DCA relationship between iron abundance and the (V — I)0 _ 3 colour while the dashed line shows my extension of this relationship to include the metal-rich Galactic globular cluster NGC 6553 (indicated with a "*"). Spectroscopic [Fe/H] values from H B K are plotted in panel (a) and [Fe/H] values derived from the CMDs are plotted in panel (b). Chapter 5. HST Observations of G302 and G312 109 5.2 Luminosity Functions The raw, n(V), and completeness corrected, <f>(V), V-band stellar luminosity functions for G302 and G312 are presented in Table 5.2. The raw luminosity functions include all stars within 2"5 < r < 12"5 of the centre of each globular cluster. The background luminosity function for each globular cluster was obtained by taking the luminosity functions for the M31 halo near each globular cluster and scaling them to the area covered by each globular cluster. This scaled background luminosity function was subtracted from the raw luminosity function before the effects of incompleteness were considered. Subtracting the raw background luminosity function assumes that the completeness corrections i n the W F 2 and W F 4 fields are the same as in the W F 3 field. A n examination of Figure 5.7 shows that the completeness corrections are small for stars with V ^ 25. Therefore, using the raw background luminosity function, instead of the completeness-corrected background luminosity function, wi l l not alter the results. The raw and completeness corrected luminosity functions are related by n(V) = P(j)(V) where P is the finding-probability matrix. P describes the probability of a star with a true magnitude of Vm being recovered with a magnitude of VTec. Artif icial star tests suggest that completeness corrections need to be applied to the raw globular cluster and background luminosity functions. These corrections are very small for V <; 25, but artificial star tests suggest that bin-jumping (the tendency for stars with input magnitudes in one magnitude bin to be recovered in a different magnitude bin) becomes significant at approximately the level of the horizontal branch and needs to be compensated for in order to accurately reconstruct the luminosity function. For G302 the finding-probability matrix is: Chapter 5. HST Observations of G302 and G312 110 21.75 22.25 22.75 23.25 23.75 24.25 24.75 25.25 25.75 26.25 26.75 27.25 21.75 (l 000 0.000 0.000 0 000 0.000 0.000 0.000 0.000 0.000 0 000 0 000 0 000^ 22.25 0 000 1.000 0.000 0 000 0.000 0.000 0.000 0.000 0.000 0 000 0 000 0 000 22.75 0 000 0.000 1.000 0 000 0.000 0.000 0.000 0.000 0.000 0 000 0 000 0 000 23.25 0 000 0.000 0.000 1 000 0.071 0.000 0.000 0.000 0.000 0 000 0 000 0 000 23.75 0 000 0.000 0.000 0 000 0.857 0.000 0.000 0.000 0.000 0 000 0 000 0 000 24.25 0 000 0.000 0.000 0 000 0.000 0.615 0.032 0.000 0.000 0 000 0 000 0 000 24.75 0 000 0.000 0.000 0 000 0.000 0.000 0.548 0:023 0.007 0 000 0 000 0 000 25.25 0 000 0.000 0.000 0 000 0.000 0.000 0.065 0.526 0.099 0 000 0 000 0 000 25.75 0 000 0.000 0.000 0 000 0.000 0.000 0.000 0.053 0.447 0 094 0 013 0 000 26.25 0 000 0.000 0.000 0 000 0.000 0.000 0.000 0.000 0.057 0 170 0 077 0 000 26.75 0 000 0.000 0.000 0 000 0.000 0.000 0.000 0.000 0.000 0 132 0 026 0 014 27.25 1° 000 0.000 0.000 0 000 0.000 0.000 0.000 0.000 0.000 0 000 0 013 0 027 j while for G312 WF3 C C D the finding-probability matrix is: 22 75 23 25 23 75 24 25 24 75 25 25 25 75 26 25 26 75 27 25 22.75 23.25 23.75 24.25 24.75 25.25 25.75 26.25 26.75 1.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 1.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.600 0.100 0.067 0.000 0.000 0-000 0.000 0.000 0.000 0.000 0.800 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.667 0.014 0.014 0.000 0.000 0.000 0.000 0.000 0.000 0.067 0.500 0.014 0.000 0.000 0.000 0.000 6.000 0.000 0.000 0.081 0.301 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.055 0.125 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.038 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 27.25 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.010 The rows of P are the input V-band magnitudes-of the artificial stars and the are the recovered F-band magnitudes of the stars. columns Chapter 5. HST Observations of G302 and G312 111 G302 G312 V n(V) <KV) n(V) 21.75 3 2.719 0 0.000 22.25 17 16.184 0 0.000 22.75 26 22.139 2 0.725 23.25 32 20.397 19 16.635 23.75 46 34.960 12 4.148 24.25 58 61.228 25 26.724 24.75 125 166.622 48 57.009 25.25 116 43.772 85 139.459 25.75 62 0.000 36 61.921 26.25 50 142.707 26 157.769 26.75 25 0.000 13 313.156 27.25 4 0.000 3 216.386 Table 5.2: V-band luminosity functions for G302 and G312. Given P there is no ideal method for reconstructing a stellar luminosity function from incomplete star count data. For a recent review of the problem see Fahlman (1993) [51]. Ideally the true luminosity function, 4>(V), can be obtained from the observed luminosity function, n(V), using <f>(V) = P _ 1 n ( V ) as described in Drukier et al. (1988) [46] and Mighell (1990) [98]. Unfortunately many faint artificial stars are lost in the noise and thus not recovered at all. This results in some artificial stars essentially bin-jumping off the faint end of the finding-probability matrix so there is insufficient information to reliably invert P. An alternative to computing P _ 1 is to use a model luminosity function, <j>'(V), and solve the forward equation n'(V) = P(j)'(V). The estimated luminosity function can then be adjusted until A n = n — n' is minimized. This approach will not accurately reconstruct the faint end of the luminosity function since P does not contain sufficient information at faint magnitudes, due to bin-jumping by faint stars, to do this. However, unlike the inversion method, the forward approach will work if P is nearly singular and Chapter 5. HST Observations of G302 and G312 112 thus allows an estimate of the completeness corrected luminosity function to be made from incomplete information. The drawbacks to the forward approach are that there is no obvious way to compute the uncertainties in the resulting luminosity function, and that some a priori information is needed about the form of the true luminosity function. This forward method will give different results depending on the initial guess at 4>(V) and the scheme used to adjust <A' during the iteration. I found that for the HST data presented in this thesis the luminosity function is not dependent on the initial guess at the form of the luminosity function when V <J 25 (approximately the level of the horizontal branch). I adjusted <j>'{V) using #(V) = ^ ( V ) + An(V) and iterated n'(V) = P<f>'(V) until A n was minimized. Figure 5.7 shows the observed and completeness corrected, background subtracted lu-minosity functions for G302 and G312. These luminosity functions are reliable for V ^ 25. Figure 5.8a compares the cumulative luminosity function for G302 with that for the metal-poor ([Fe/H] = -1.65) Galactic globular cluster M13 (Simoda & Kimura 1968 [132]) while Figure 5.8b compares the cumulative globular cluster for G312 to a theoretical lu-minosity function for an old (io = 14 Gyr) globular cluster with [Fe/H] = —0.47 and [O/Fe] = +0.23 from Bergbusch & VandenBerg (1992) [11]. Stellar evolution models do not show any relationship between the slope of the luminosity function of the upper red-giant branch and either the iron abundance or the age of the stars (Bergbusch & Van-denBerg 1992 [11]) so Figure 5.8 can not be used to determine the ages or the iron abun-dances of G302 or G312. There is a general agreement between the shape of the G312 luminosity function and the shape of the theoretical metal-rich luminosity function, as well as between the shape of the G302 luminosity function and the metal-poor compari-son luminosity function, which suggests that the star counts are not seriously incomplete down to the level of the horizontal branch (V ~ 25). Chapter 5. HST Observations of G302 and G312 113 1 — i — I — i — i — i I i i i I i i r 3 0 0 L G302 n — i — i — i — i — i — i — I — i i r (a) G312 (b)_ 2 0 0 -100 0 zu 3 0 0 h 4{v) -n(V) 2 0 0 -Y 100 ( c ) ~_ MV) Y 1(d) 0 b i J L_ J I I I I 1 1 L 22 24 26 28 22 24 26 28 V V Figure 5.7: The upper panels, (a) and (b), show the observed luminosity functions for G302 and G312 respectively. Only stars between 2//5 < r < 1275 were used to con-struct these luminosity functions. The lower panels, (c) and (d) show the complete-ness-corrected luminosity functions with the completeness-corrected background lumi-nosity function subtracted (solid lines) and the observed luminosity functions (dashed lines). The completeness corrected luminosity functions become unreliable below the level of the horizontal branch (indicated with an arrow). Chapter 5. HST Observations of G302 and G312 114 i 1 r T 1 r 3 h 2 so £ 1 h (a) M13 "i 1 ^ -"i • • • • 0 3 - (b) B & V 2 1 0 J I L [Fe/H] = -1.65 — [Fe/H] = -0.47 J J I L J I L 22 24 26 V 28 Figure 5.8: The upper panel (a) compares the cumulative luminosity function for G302 (solid circles) to the cumulative luminosity function of the metal-poor Galactic globular cluster M13. Panel (b) compares the cumulative luminosity function for G312 with a theoretical luminosity function for an old metal-rich globular cluster. The comparison luminosity functions have been scaled so that they contains the same number of stars at V = 24 as the M31 globular cluster luminosity functions do. Chapter 5. HST Observations of G302 and G312 115 5 . 3 Structure 5.3.1 Colour Gradients The mean colour for r e f f < 275 for G312 is ( ( V - / ) 0 ) = 1.07 while for G302 ( ( V - J ) 0 ) = 0.83. These colours are consistent with G302 having a metal-poor stellar population and G312 having a metal-rich stellar population. Figure 5.9 shows the dereddened (V — I)Q colour profiles of G302 and G312. These profiles were obtained by subtracting the /-band isopliotal surface brightness profile from the V-band profile then dereddening the resulting profile. The error bars represent the Icr standard error in the colour. The. inner portion of the G302 profile is truncated at '"eff ~ 071 (= 1 pixel) since the innermost pixels (r < 1 pixel) of the W F C images.of G302 were shghtly saturated. Individual stars can be resolved to within reg CK 275 of the centre of the globular cluster and the isophotes become less certain when the light becomes resolved into individual stars. Artificial star tests suggest that the large scatter in the colour profiles in the outer regions of G302 and G312 represent stochastic star placement within the globular clusters, not intrinsic colour gradients. In the inner 275 the colour gradients are d(V-I)0/drei? = -0.028 ± 0.007 for G302 and d(V-I)0/dreS = —0.090 ± 0 . 0 0 7 for G312. To determine if these colour gradients are due to chance 10,000 . bootstrap resamplings of the colour data for each globular cluster were performed and the colour gradients for each resampled data set were computed. This analysis gave a 2.1255% chance that the observed colour gradient (for re^ < 275) in G302 is due to chance and a 0.0005% chance that the observed colour gradient (for reff < 275) in G312 is due to chance. Therefore the existence of a radial colour gradient in G312 can not be rejected based on the results of Monte-Carlo simulations. However, these simulations only weakly (at the 2ACT level) suggest that there is a radial colour gradient in G302. Chapter 5. HST Observations of G302 and G312 M i l l i i i i i i i i o 1—1 1 1 t • ^ -0.5 - • -0 1 (a) G302 i i i i 111 i i i i i i 11 i i o •—1 1 —1—t I I l l | l I I i II 11 i i 1 0.5 • — 0 1 (b) G312 1 0.1 1 10 r e f f (arcseconds) Figure 5.9: This figure shows the integrated colour profiles for G302 and G312. Chapter 5. HST Observations of G302 and G312 117 5.3.2 Ellipticities Figures 5.10 and 5.11 show the elhpticity and orientation profiles for G302 and G312 respectively. The error bars in Figures 5.10 and 5.11 should be treated as guides to the reliability of the individual measurements and not taken as statistically rigorous estimates of the uncertainties in each point. In order to estimate the overall projected elhpticity of each globular cluster the weighted mean elhpticities, I and position angles, 60 (in degrees east of north on the sky) were determined using the values for the individual isophotes between reff = 1" and reff = 10". The results are presented in Table 5.3. The uncertainties are standard errors in the mean and N is the number of isophotes used to compute e and 0o. Cluster Filter e do N G302 V 0.194 ± 0.011 - 7 1 ° ± 2° 24 I 0.195 ± 0 . 0 1 2 - 8 7 ° ± 2° 24 G312 V 0.072 ± 0 . 0 1 5 77° ± 5° 24 I 0.058 ± 0.014 77° ± 6° 24 Table 5.3: Elhpticities and position angles for G302 and G312 from the HST data. The projected major axis of G302 points approximately towards the centre of M31. The elhpticity of each globular cluster is the same in the /-band as it is in the V-band but the projected major axis of G302 in the V-band is located 16° ± 3° west of the projected major axis in the /-band. This shift in orientation between the two colours is visible at all radii in G302 and is likely due to a small number of giant stars dominating the hght from the globular cluster. This data shows that the projected major axis of G302 in the V-band is oriented 20° farther west from the V-band orientation obtained by Lupton (1989) [90] (6>0 = 0?0±0°6) but that Lupton's orientation is only 4° east of my /-band orientation. This supports the idea that the observed difference between the / -Chapter 5. HST Observations of G302 and G312 118 and V-band orientations is due to the different numbers of resolved stars in the different bands and data sets. The elhpticity that I measure (e = 0.195 ± 0.012) is somewhat higher than that found by Lupton (1989) [90] (e = 0.15 ± 0.02). Figure 5.10 shows an apparent decrease in the elhpticity of G302 between re$ ~ 1" and reff ~ 10". The slope of e(reff) = mreff + bis m — —0.0381 ± 0.0383. In order to determine if this slope represents a real decrease in elhpticity with radius I performed 10,000 bootstrap resamplings of the elhpticity data between 1" < re^ < 10" and computed the slope for each set of bootstrapped elhpticities. I found that there was a 22.34% chance of observing a slope at least as great as the slope seen in the actual data simply by chance. Therefore, I conclude that there is no evidence for a change in elhpticity with radius between reff = 1" and reff = 10" in G302. It is not possible to say if the apparent increase in elhpticity between reff ~ 0"3 and reff ~ 1"0 is real or due to the presence of a small number of unresolved bright stars located within a few core radii of the centre of G302. Since the stellar F W H M is ~ 0"2 the chance location of a single unresolved bright star within (¥.'3 of the centre of G302 along the projected minor axis of the globular cluster could reduce the elhpticity of the fitted isophotes. G312 has a projected elhpticity of e = 0.065 but there is considerably more scatter in the fitted position angles (the standard deviation is a ~ 27°) than there is in the G302 position angles (cr ~ 10°). This suggests that the observed projected elhpticity of G312 is primarily due to the stochastic placement of stars within G312, variations in the stellar background of the M31 halo, and the intrinsic precision of the ELLIPSE algorithm. Running the ELLIPSE task on a series of circular artificial Michie-King globular clusters that had been convolved with the WF3 PSF suggests that seeing alone can produce elhpticities of between 0.01 and 0.02 at radii of between approximately 0"5 and 2". This agrees with the fluctuations in elhpticity seen in Figure 5.11 at these radii. While seeing has very little effect on the observed elhpticity beyond reff ~ 2", artificial star tests Chapter 5. HST Observations of G302 and G312 119 ind ica t e tha t s tochast ic star p lacement can in t roduce an unce r t a in ty i n the e l l i p t i c i t y of be tween ± 0 . 0 1 and ± 0 . 0 5 . In l ight of this there is no evidence for G 3 1 2 be ing e l l i p t i c a l i n the p lane of the sky. It is , however, possible tha t tha t G 3 1 2 is e longated a long the l ine-of-sight . Chapter 5. HST Observations of G302 and G312 120 i — i — r T ~ r CD 15 20-25 0.3 0.2 0.1 0 300 200 100 0 T 1—i—i— I I I I 1 1 1 I I M | G302 . ' / -Band • V-Band'o degrees E of N JXD<D<M> i -1 M M M o . i 10 r ,. (arcseconds) Figure 5.10: This figure shows the calibrated surface brightness profiles, p, ellipticity profiles, e, and orientation profiles, 0o, for G302 in the V- and /-bands. Chapter 5. HST Observations of G302 and G312 121 CD 1 5 2 0 2 5 0 . 6 0 . 4 0 . 2 0 3 0 0 2 0 0 1 0 0 0 i - B a n d • l V-Band ° 1 I I I 1 i — i — i i i i I 1—i—i—i i i i •• ••••«. 1 1 I—I—I I I I | G 3 1 2 __ degrees E of N i i i i i i J I L i - - I 4_U_ 0 . 1 1 0 r „ (arcseconds) Figure 5.11: This figure shows the cahbrated surface brightness profiles, elhpticity pro-files, and orientation profiles for G312 in the V- and /-bands. Chapter 5. HST Observations of G302 and G312 122 5.3.3 Michie-King Models The tidal radii were determined by fitting Michie-King models to the observed surface brightness profiles of each globular cluster. The unresolved background light was meas-ured by masking out a circle with a radius of 300 pixels (~ 3r t) centred on the globular cluster then fitting a plane to the remaining area on the WF3 C C D . Subtracting a plane resulted in residual variations of less than 1% across the WF3 C C D for both the G302 and G312 fields. There is no evidence for intensity or colour gradients in the unresolved background light. Surface brightness profiles for each globular cluster were obtained in the same way as is described in Chapter 3.2.3. The higher resolution of the HST data meant that isophotes could be reliably fit to within reff = Of.'l of the centre of each globular cluster, allowing the core structure to be studied to a much higher degree of precision than is possible with the C F H T data described in Chapter 2. CERN's MINUIT (Version 94.1) was used to fit seeing-convolved one-dimensional Michie-King models to the data in the same manner as described in Chapter 3.2.3. Each Michie-King model was convolved with the PSF for the WF3 C C D at the location of centre of the globular cluster. The central potential, Wo, King core radius, r c , tidal radius, rt, concentration, c = rt/rc, anisotropy radius, ra, and half-mass radius, rh, were determined for G302 and G312 in the V- and /-bands. Table 5.4 gives the parameters of the best-fitting Michie-King models for each globular cluster. The V- and /-band surface brightness profiles, along with the best-fitting Michie-King models, are shown in Figures 5.12 and 5.13. The profiles for both globular clusters are well fit by isotropic Michie-King models out to ~ 5". Beyond this distance from the centre of the globular cluster there is an excess of light over what is predicted by isotropic (ra = -foe) Michie-King models. In order to test if the excess light is due to velocity anisotropy in the Chapter 5. HST Observations of G302 and G312 123 globular clusters anisotropic Michie-King models were fit to each globular cluster. The results are listed in Table 5.4. In all cases the anisotropic models yield slightly larger xl values than the corresponding isotropic models, which suggests that the isotropic models are formally better fits to the data than the anisotropic models are. Furthermore, the best-fitting anisotropy radii are greater than the fitted tidal radii for each globular cluster, which is physically meaningless. Therefore, I conclude that the overabundance of hght at large radii is not due to velocity anisotropy. This supports the hypothesis that these globular clusters are surrounded by extended halos of unbounded stars. Cluster Filter Wo rc rt c ra rh V G302 V 7.56 0720 9792 1.70 +oo 0752 1.827 49 7.61 0720 10712 1.71 282740 0754 1.855 48 I 7.65 0720 10749 1.73 +oo 0756 0.846 51 7.47 0721 9"80 1.67 20706 0749 0.906 48 G312 V 7.57 0719 9751 1.70 +oo 0753 0.839 53 7.48 0720 9716 1.67 492713 0749 0.882 52 I 7.46 0721 9764 1.67 + C O 0749 0.458 53 7.47 0721 9770 1.67 381788 0749 0.467 52 Table 5.4: Best-fitting Michie-King models for G302 and G312 using the HST data. MlNTJIT returns formal uncertainties of ~ 5 to 10% on the Michie-King parameters. However, MINUIT was generally unable to compute a fully-accurate covariance matrix for the fitted parameters so MINUIT's uncertainty estimates are not reliable. Therefore, the uncertainties in the fits were estimated by generating a series of artificial globular clusters and fitting seeing-convolved Michie-King models to them in exactly the same way as was done for the real data. It was found that MINUIT tended to underestimate the uncertainties in the fits by a factor of between 1.5 and 2. This suggests the true l c uncertainties in the parameters are between ~10 and 15% of the best-fit values of the parameters. Chapter 5. HST Observations of G302 and G312 124 T 1—I I I I I | I I I I I I I I r e f f (arcsec) Figure 5.12: The solid points represent the observed surface brightness profiles of G302 while the open triangles in the inner O'.'l indicate the surface brightnesses corresponding to saturation for the WF3 CCD (these points were not used in the fits). The solid lines are the best-fitting isotropic Michie-King models and the dashed lines are the best-fitting anisotropic models. The arrows indicate the tidal radii of the isotropic models. Chapter 5. HST Observations of G302 and G312 125 15 h 25 15 ^ 20 25 n 1—i i i i i i | 1 1—i i i i i 11 i i i i i i i 11 i r G312 V -Band J isotropic model (r a = +°°) anisotropic model (r a = 492'/13)^ +-+ + - T -G312 7-Band J isotropic model (r a = + °°) _.anisotropic model..(ra = 381'/88)\ i i 111 J I I—L J L 0.01 0.1 1 r e f f (arcsec) 10 Figure 5.13: The solid points represent the observed surface brightness profiles of G312. The solid lines are the best-fitting isotropic Michie-King models and the dashed lines are the best-fitting anisotropic models. The arrows indicate the tidal radii for the isotropic models. Chapter 5. HST Observations of G302 and G312 126 5.4 Extended Stellar Halos G A F have observed an excess of resolved and unresolved stars beyond the formal Michie-King tidal radii of several globular clusters in M31, as would be expected if stars that have been stripped, or evaporated, from these globular clusters have remained in extended halos around those globular clusters. Another test for extended stellar halos is to identify an asymmetrical overdensity of stars in two dimensions beyond the formal tidal radius. This has been done for some Galactic globular clusters (see Grillmair et al.1995 GF95) using multiple Schmidt photographic plates for each globular cluster. However, it has not been attempted for any of the M31 globular clusters. In M31 the globular cluster, and the background, can be imaged on a single CCD. This eliminates possible systematic effects arising from comparing star counts across multiple fields. I have attempted to find such enhancements beyond the formal isotropic and anisotropic tidal radii of G302 and G312. To search for extended halos around G302 and G312 I examined the two-dimensional distribution of stars beyond the tidal radii of each globular cluster. The total background stellar surface densities around each globular cluster were small (Sbkgd = 0.6191 ± 0.0099 stars/D" for G302 and S b k g d = 0.1688 ± 0.0052 stars/D" for G312). The stellar positions were rebinned into "super-pixels" in order to ensure that there were enough stars in each bin that the Poisson fluctuations in a bin would be small compared to the total number of stars in that bin. In order to compute the optimum binning size I assumed that all the stars on the WF3 CCD were part of the globular cluster and that the globular cluster could be approximated by a Gaussian distribution of stars. It can be shown (Heald 1984 [64]) that for any Gaussian distribution the bin size, Sx, which maximizes the signal-to-noise ratio in each bin is given by Sx ~ (20/N)l^5cr where cr is the standard deviation of the Gaussian and N is the size of the sample. This technique provides Chapter 5. HST Observations of G302 and G312 127 a balance between the need to keep the bin sizes small compared to the width of the Gaussian and the need to have a large number of data-points in each bin to reduce Poisson noise in the bin. Since I am searching for overdensities of stars beyond the Michie-King tidal radii of the globular clusters the assumption that globular cluster can be approximated by a Gaussian will lead to an underestimate of the bin size needed to maximize the signal-to-noise. After some experimentation I found that bin sizes of 32 pixels (~ 3'.'2), approximately two to three times the computed optimal size for Gaussians, provided the best signal-to-noise ratio for the data. I then smoothed the resulting binned data by convolving it with a unit Gaussian with a dispersion equal to the bin size (cr = 3'/2). The resulting stellar number density distribution for G302 is shown in Figure 5.14 and the stellar number density distribution for G312 is shown in Figure 5.15. Figure 5.14 shows an asymmetric overdensity of stars around G302 extending to at least twice the Michie-King tidal radius. The coherence of the isodensity contours that He beyond the tidal radius, but inside the background contour, suggest that observed overdensities are not an artifact of the binning, smoothing, or contouring processes. The surface brightness profile of G302 (see Figure 5.12) light beyond the tidal radius (rt ~ 10") of the globular cluster is in agreement with the asymmetric halo of stars seen in Figure 5.14. The surface brightness profile of G312, however, is consistent with there being no light beyond the tidal radius (rt ~ 10"). This is reflected in the lack of an extended halo of stars in Figure 5.15. The fact that an overdensity of stars is seen beyond the tidal radius of G302, but not beyond the tidal radius of G312, suggests that the overdensity is real and not an artifact of the analysis. In order to check the reality of the observed halo surrounding G302 I carried out the contouring procedure on the star counts from the WF2 and WF4 images in the G302 and G312 fields. Figure 5.16 shows that no structures of comparable size to the halo around G302 are seen in any of the four background fields. To further test whether or not a Chapter 5. HST Observations of G302 and G312 128 random distribution of stars could give rise to coherent structures that could be mistaken for an extended halo I constructed a series of random star fields with number densities comparable to those found in the G302 and G312 background fields. None of these artificial star fields showed evidence for coherent sub-structure. In addition I used Michie-King surface density profiles to place artificial stars onto randomly generated star fields to see if the presence of a globular cluster would bias the binning/smoothing/contouring process in favour of finding structure beyond the tidal radius when none was really present. None of the Monte-Carlo images showed any evidence for extended halos. In order to quantify the orientation of the halo around G302 I computed the second moment of the distribution of stars beyond the tidal radius (r t ~ 10") determined from fitting Michie-King models to G302 (see Chapter 5.3.3). Only stars between 10" and 35" from the centre of the globular cluster were used to ensure that neither the globular cluster, nor the edges of the CCD, biased the sample. I computed the statistic £ 2 = \Jj? J2{Li Vi where TV is the number of stars beyond the tidal radius and yi is the distance from the i t h star to an arbitrary axis of symmetry for the globular cluster. This statistic corresponds to the root-mean-square distance between the stars and the axis of symmetry so the value of £ 2 will be at its minimum when the assumed axis of symmetry coincides with the true major axis of the distribution. I computed £2 for angles of symmetry between 0° and 180°. This data is shown in Figure 5.17. A second statistic, 77 = G.max — C2,min, gives a measure of the degree of symmetry in the distribution of stars about the centre of the globular cluster. The greater the value of 77, the greater the deviation from a circular distribution of stars. For a perfectly circular distribution (2 would be the same for all angles, resulting in 77 = 0. In order to determine the probability of getting the observed value of 77 by chance from a random distribution of stars I generated 10,000 sets of stellar coordinates and determined 77 for each set. The stellar coordinates were randomly drawn from a circular distribution of stars with the same size and number density as the Chapter 5. HST Observations of G302 and G312 129 field around G302. This data formed the cumulative probability distributions shown in Figure 5.18. For G302 this analysis yields v — 7.4826, which occurs at a position angle for the major axis of the extended halo of 83° (= —97°) east of north. The Monte-Carlo tests indicate that there is only a 1.84% chance of obtaining this value of n in a random star field. This corresponds to the extended halo being real at the 2.5<r level. In contrast, the WF2 and WF4 fields near G302 yielded n values of 3.5530 and 4.6618 corresponding to 38.47% and 20.62% chances respectively of occurring by chance. The orientation of the extended halo around G302 is 10° to the west of the projected major axis of G302 as observed in the V-band and 26° west of the projected major axis for the globular cluster as observed in the J-band. In light of the lack of coherent overdensities in any of the background images or simulated images, the results of the moment analysis, and the observed overdensity of stars beyond the tidal radius of the best-fitting Michie-King model, I believe that the extended asymmetric halo around G302 that is seen in Figure 5.14 is a real feature of G302. Since the probability of observing a given value of r\ by chance depends on the num-ber density of stars on the image, I performed a separate series of 10,000 Monte-Carlo simulations for the G312 data. The resulting probability distribution indicated that the observed value of T] = 12.9960 for the stars beyond the tidal radius G312 in the WF3 field had a 9.96% chance of occurring in a randomly distributed set of stars. The WF2 and WF4 fields around G312 have rj values of 9.6342 and 9.1801 respectively, corresponding to 27.65% and 30.73% probabilities of occurring at random. From this, and the lack of a significant excess of stars beyond the tidal radius of the best-fitting Michie-King models, suggests that G312 does not exhibit any evidence for having an extended halo of stars. However, it is possible that such a halo does exist and is aligned along the line-of-sight. Chapter 5. HST Observations of G302 and G312 130 0 20 40 60 80 X (arcsec) Figure 5.14: The solid ellipse is the fitted Michie-King tidal radius for G302. The dashed contour represents the surface density of stars in the M31 halo at the location of G302 (= S b k g d = 0.6196 stars/D'). The solid contours are S = 0.8,1.2 stars/D' and the dotted contour is £ = 0.4 stars/D". The outermost contours are strongly influenced by the size of the WF3 C C D . The contours inside the fitted tidal radii have not been plotted for clarity. The arrow points towards the centre of M31. Chapter 5. HST Observations of G302 and G312 131 0 20 40 60 80 X (arcsec) Figure 5.15: The solid circle represents the fitted Michie-King tidal radii of G312. The dashed contour represents the surface density of stars in the M31 halo at the location of G312 (= S b k g d = 0.1688 stars/D'). The solid contours are £ = 0.4,0.8,1.2,1.6 stars/D'. The innermost contours have not been plotted for clarity. Chapter 5. HST Observations of G302 and G312 132 0 20 40 60 800 20 40 60 80 X (arcsec) Figure 5.16: This figure shows isodensity contours for the WF2 and WF4 fields for G302 and G312. The contour intervals are the same as for Figures 5.14 and 5.15. Chapter 5. HST Observations of G302 and G312 133 2 0 0 n—i—[-• i i i i | i i i i | i i i r) = 10 - / • _ . = 7.4826 1 i 2,max a 2,min i | i i i G302 1 P(TJ) = 1.84% -190 60 = 82°7 ^ — CV2 180 170 i i i i i i I i i i i i i i i 1 i i i 2 0 0 H 1 p 1 1 1 1 | 1 1 1 1 | l l I TI = t0 12.9960 / a 2,max a 2,min i i- i i G312 1 P(r]) = 9.96% : 190 — 0O = 110°9 — CV2 - -180 — ^ ^ ^ ^ ^ 170 i i I i i i i l i i i i i i i i i i i i i 100 150 2 0 0 2 5 0 6 (degrees E of N) Figure 5.17: The upper panel shows the value of ( 2 as a function of position angle for G302 while the lower panel shows ( 2 as a function of position angle for G312. Chapter 5. HST Observations of G302 and G312 134 1 0.8 i i i i | , , , i | 1 1 G 3 0 2 : 0.6 • — 0.4 — \ 77 = 7 . 4 8 6 2 — 0.2 0 — ( a ) ~: i i i i 1 i i i 1 1 1 1 1 \ I I I I I I I 1 L_ — 0 5 10 Figure 5.18: This figure shows the cumulative probability distributions for observing a particular value of 77 by chance if the observed distribution of stars was drawn from a circularly symmetric distribution of stars. The two distributions are different because the fields around G302 and G312 contain different numbers of stars. Chapter 5. HST Observations of G302 and G312 135 5.5 Mass Loss and the Orbit of G302 5.5.1 Mass Loss from G302 Figures 5.12 and 5.14 show that there is an excess of both light and stars beyond the tidal radius of the best-fitting Michie-King model for G302. By computing the amount of mass beyond the tidal radius the mass-loss rate for G302 can be estimated. The V-band surface brightness profile for G302 was converted to a projected mass profile using a mean mass-to-light ratio of Tv = 1.9 Solar units (from Dubath & Grillmair 1997 [47]). The total mass inside the Michie-King tidal radius (rt = 10") was M = (8.89 ± 0.06) x 105 MQ. The total mass of G302 was estimated by applying the mass-to-light ratio to the total integrated V-band magnitude of G302 (V = 14.90, from HBK). This gave a mass of M o t - (9.52 ± 0 . 2 4 ) x 105M§ which gives a total mass of M = (0.63 ± 0 . 2 5 ) x 1 O 5 - M 0 beyond the formal tidal radius of G302. Mass-segregation within a globular cluster can result in. Yy increasing with radius so the total mass beyond the tidal radius that is derived here will be a lower limit.on the true mass in the extended halo around G302. If G302 has an age of t0 = 14 Gyr, and a constant rate of mass loss is assumed, then the mass-loss rate required for the observed amount of mass to escape beyond the Michie-King tidal radius is M' — 4500 ± 1800 MQ/Gyr. The relaxation time at the half-mass radius for a globular cluster is given by (Spitzer 1987 [136].): M1/2r3/2 ^ = 8 - 9 3 3 YW^'\M,)U0ANty (5'5) where M C \ is the total mass of the globular cluster in Solar masses, is the half-mass radius of the globular cluster in pc, (M+) is the mean mass of the stars in the globular cluster in Solar masses, and N+ = A4ci/{M+) is the estimated number of stars in the Chapter 5. HST Observations of G302 and G312 136 globular cluster. Following Djorgovski (1993) [43] I adopted (M*) = 1/3MQ. The half-mass radius obtained by fitting isotropic Michie-King models to G302 is rh = 1.9 ± 0 . 1 pc which gives a half-mass relaxation time of tTih = 0.49 ± 0.04 Gyr. This yields a projected escape rate of r = (2.3 ± 0.9) X 10~3 per half-mass relaxation time. Converting observed surface brightnesses to a deprojected mass profile involves solving an Abel integral which contains the radial derivative of the surface brightness distribution. Since the surface brightness data for G302 contains noise this inversion is inherently un-stable. Therefore, I chose to project the theoretical escape rates of Oh &; Lin (1992) [103] into the plane of the sky before comparing them to the observed escape rate. The Oh & Lin (1992) [103] evaporation rates for isotropic globular clusters with ages of io ~ 30 ,^/j in the Galactic potential then become ~ 10 - 3 to 10~2 per relaxation time. This is comparable to the escape rate inferred from the data. This result further strength-ens my claim to have detected an extended halo of unbound stars around G302. Chapter 5. HST Observations of G302 and G312 137 5.5.2 Mass Loss from G312 Using the methods described above I find that G312 has a mass of Ad = (0.60 ± 4.70) x 1O4A40 beyond its fitted isotropic Michie-King tidal radius (rt — 9'/55). The large uncertainty in the amount of mass beyond the tidal radius is due to the large uncertainties in the surface brightness profile at large radii. Figure 5.15 does not show any evidence for an extended halo of stars around G312 so it is likely that the hght seen beyond the fitted tidal radius in Figure 5.13 is due to low-level variations in the unresolved background hght. Alternately, it is possible that the use of multi-mass Michie-King models would result in a larger tidal radius. If I assume that the observed excess of hght in Figure 5.13 is due to an extended halo of stars then the mass-loss rate required is M = 430 ± 3360 A ^ 0 / G y r , approximately 10% of the mass-loss rate from G302. G312 has a half-mass relaxation time of tr>h = 0.29 ± 0 . 0 2 Gyr so the escape rate per relaxation time is r = (0.38 ± 2.95) x 10 - 3 . While this is consistent with Oh & Lin's (1992) [103] theoretical values the large uncertainties in the photometry near the tidal radius of G312, and thus the large uncertainty in the computed escape rate, suggests that the observed escape rate is consistent with there being no observable mass-loss from G312. Chapter 5. HST Observations of G302 and G312 138 5.5.3 The Orbit of G302 H B K measured a heliocentric radial velocity of —8 ± 32 km-s - 1 for G302. Correcting for Rubin k Ford's (1970) [124] 21 cm velocity for M31 ( v M 3 i = -297 km-s"1) gives G302 a radial velocity of v — +289 ± 32 km-s - 1 relative to the centre of M31. This is one of the fastest radial velocities observed for a globular cluster in M31 which suggests that almost all of G302's space velocity is along the line-of-sight. This can only occur when G302 is near its perigalactic passage. Since very little of G302's space velocity is in the plane of the sky, and G302 is near its perigalacticon, G302 must be at approximately the same distance from the Earth as the centre of M31. This means that the distance measured on the sky between G302 and the centre of M31 is approximately the true separation; so the perigalactic distance for G302 is 9P ~ 32'1 corresponding to dp ~ 6.77 kpc assuming a distance modulus of u0 = 24.3 for M31. In order to constrain the orbit of G302 a three-component model for the M31 potential was used. The disc was modeled by a Miyamoto k Nagai (1975) [99] potential, the bulge by a Hernquist (1990) [67] potential, and the halo by a spherical logarithmic potential: $disc = , =, (5.6) — GMbulge / r T \ $ b u l g e = j (5.7) r + c $haio = V ^ X r 2 + d2) + ln(<7). (5.8) I adopted M ^ = 2.0 x l O n M 0 , Mhvige = 6.8 x 1010MQ, a = 7.8 kpc, b = 0.31 kpc, c = 0.84 kpc, d = 14.4 kpc, and Vhaio = 128 km-s - 1 . These parameters were chosen by scaling the Galactic values to the observed mass and diameter of M31. The constant, C, in Equation 5.8 contains the units required to make the equation dimensionally correct. Chapter 5. HST Observations of G302 and G312 139 Using this model, and the observed radial velocity and perigalactic distance of G302 I found that the orbit of G302 has an eccentricity of e = (da — dp)/(da + dp) = 0.65 where da is the apogalactic distance (in kpc) for G302. This value assumes that all of the space velocity of G302 is along the line-of-sight and gives G302 an apogalactic distance of 6a = 2?5 or da = 31.5 kpc If 40% of G302's space velocity is assumed to be in the plane of the sky then the orbital parameters become e = 0.71 and da = 37.0. Since G302's radial velocity is faster than 95% of the radial velocities of other M31 globular clusters it is unlikely that more than ~ 5% of G302's space velocity is tangential to the line-of-sight. If this is the case then the orbital eccentricity for G302 is 0.65 ^ e <; 0.66 and the apogalactic distance is da — 31.5 kpc. The period of G302's orbit was defined to be twice the time required for the globular cluster to move from its perigalacticon to its apogalacticon. If all of the globular cluster's space velocity is along the line-of-sight then the orbital period is P = 0.42 Gyr. If 40% of the space velocity is tangential to the line-of-sight then P = 0.52 Gyr. Chapter 5. HST Observations of G302 and G312 140 5.6 A Comparison of the C F H T and HST Results There have been several attempts made to determine core and tidal radii of globular clusters in M31 using data from ground-based telescopes. To investigate how successful these attempts have been I have compared the structural parameters obtained from the ground-based results presented in Chapter 3 to those derived from the HST data presented in this chapter. At present there are seventeen globular clustersin M31 that have had their structural parameters derived from HST data; six of these have used the post-refurbishment WFPC2 while the rest used F O C data taken before the installation of the Corrective Optics Space Telescope Axial Replacement (COSTAR) package. The F O C results are from Fusi Pecci et al. (1994) [57]. They used three image restora-tion techniques to correct for the pre-refurbished FOC's PSF and measured the H W H M of the deconvolved cluster surface brightness profiles. The H W H M is approximately equal to the King, core radius. Table 5.5 contains a list the H W H M for each globular cluster obtained by averaging the HWHMs obtained from each of their three deconvo-lution methods. The typical standard deviation between the H W H M estimates for each globular cluster is a ~ 0.03. The PC results for four of the globular clusters (G58, G105, G108, and G219) are from G A F while the data for G302 and G312 were obtained with the W F C and are presented in this thesis. The structural parameters obtained using ground-based data are taken from Battis-tini et al. (1982) [7], Crampton et al.(1985) [33], Cohen & Freeman (1991) [30], and this thesis. Battistini et al. (1982) [7] fit a Moffatian with (3-2 and 7 = 2 to each M31 globular cluster then defined a quantity W\/n (= the half-width of the Moffatian profile at one-quarter of the central intensity = the Moffatian parameter a when /? = 7 = 2). They then scaled the surface brightness profiles of several Galactic globular clusters to Chapter 5. HST Observations of G302 and G312 141 the distance of M31 and used these to derive a relation between W1/4 and rc. Cramp-ton et al. (1985) [33] used a similar technique to derive core radii for 509 globular cluster candidates in the M31 system. Davoust k Prugniel (1990) [38] derived a relation be-tween the observed H W H M and the King core radius by convolving King models with their seeing profiles. They used this relation to estimate core radii for several clusters in the M31 system. Cohen k Freeman (1991) [30] fit seeing-convolved two-dimensional King (1962) [83] profiles to 30 globular clusters to determine their core and tidal radii. Table 5.5 lists the pubhshed core radii for all the M31 globular clusters that have been observed with the HST. The core radii have been converted to arcseconds. Similarly Table 5.6 lists the pubhshed tidal radii for M31 globular clusters that have been observed with the HST. The core radii derived for G302 and G312 using the C F H T data are within 30% of the core radii for these globular clusters that were measured using the HST data. The C F H T data for G2 gives a core radius that is 2.7 times larger than the value derived by Fusi Pecci et al. (1994) [57] by deconvolving pre-refurbishment F O C images. It is tempting to argue that the complexity of the pre-refurbishment F O C PSF, and the uncertainties inherent in non-linear deconvolution, make the Fusi Pecci et al. (1994) [57] core radius unreliable. However, the C F H T images of G2 have exposures of only 100 seconds, and there are two bright stars located near G2, so it is more likely that the core radius that was derived for G2 in Chapter 3 is in error. Further support for the Fusi Pecci et al. (1994) [57] value comes from noticing that the pre-refurbishment and post-refurbishment core radius for G58 are in reasonable agreement with each other. The discrepancy between the pre- and post-refurbishment core radii for G105 is probably due to the presence of a collapsed core in that globular cluster (Bendinelh et al. 1993 [8]). The core radii obtained for G108 and G312 by Cohen k Freeman (1991) [30] are four to six times larger than those obtained for the same clusters by other groups. This Chapter 5. HST Observations of G302 and G312 142 is probably due to differences in the methods of determining the core radius and ac-counting for seeing. Battistini et al. (1982) [7] and Crampton et al. (1985) [33] derived their core radii indirectly from the observed half-width of the cluster at one-quarter of its maximum intensity and compared these values to seeing-convolved King models. Co-hen & Freeman (1991) [30] fit seeing-convolved King (1962) [83] models directly to the portion of the one-dimensional surface brightness profile between 1'.'3 and 2/.'7 from the centre of the cluster. This means that the Cohen &: Freeman (1991) [30] fits do not include the cores of the clusters, where the effects of seeing are strongest. Therefore these results are given less weight in this thesis than the Battistini et al. (1982) [7] and Crampton et al. (1985) [33] results are. The core radii for G64 and G78 obtained by Davoust & Prugniel (1990) [38] are two to four times larger than those obtained by Battistini et al. (1982) [7] and Cramp-ton et al. (1985) [33]. They used a similar technique to what Battistini et al. (1982) [7] and Crampton et al. (1985) [33] used to determine core radii but their study suffered from a large pixel size (0"324) and poor seeing (particularly for G64). Therefore the Da-voust <fc Prugniel (1990) [38] results will be given less weight in the following discussion. If the Cohen & Freeman (1991) [30] data and the Davoust & Prugniel (1990) [38] data is discarded then the pubhshed ground-based measurements of core radii for M31 globular clusters have overestimated the core radii of most M31 globular clusters by factors of approximately two to ten. This is due to the core radius being smaller than the radius of the seeing disc for the ground-based observations. However, the core radii presented in Chapter 3 of this thesis agree with core radii obtained from HST observations. Table 5.5 shows that ground-based observations can give reasonable estimates of the core radius only if seeing is compensated for in a physically realistic manner and if the core radius is at least - 0.25 times the F W H M of the seeing disc (G58, G108, G302, and G312). This suggests that ground-based observations can not obtain reliable core radii for the Chapter 5. HST Observations of G302 and G312 143 globular clusters in the M31 system unless some sort of adaptive optics system is used to reduce the seeing to ^ O/.'l. An examination of Table 5.6 shows that the tidal radii measured from images taken with ground-based telescopes are similar to those measured from images taken with the HST. The tidal radii that were derived for G302 and G312 using the C F H T data presented in Chapter 3 are within <~10% of the tidal radii that were derived using HST data. Assuming a distance of 725 kpc for the M31 globular clusters gives tidal radii of ~ 35 to ~ 50 pc, comparable to the tidal radii of typical Galactic globular clusters. This suggests that ground-based telescopes, operating under conditions of sub-arcsecond seeing, can obtain reliable tidal radii for M31 globular clusters Chapter 5. HST Observations of G302 and G312 144 Name Ground-Based F O C H W H M f PC2 W F C G2 G i l G58 0719 0723 G64 0735 0726 G78 0751 0762 G105 G108 0725 0733 G219 G244 0"22 0736 G272 0738 0768 G302 0729 0729 G305 0720 G312 G319 G322 0723 G352 Bo289 1711 0791 1702 0784 0"11 0718 0714 0703 0707 0714 0704 0708 0705 0703 0704 0704 0705 0704 0707 0703 0717 0709 0725 0"87 0720 0720 a - Battistini et al.(1982) [7] b - Crampton et al.(l985) [33] c - Davoust & Prugniel (1990) [38] d - Cohen & Freeman (1991) [30] e - this work f - Fusi Pecci et al.(1994) [57] g - GriUmair et al.(1996) [58] h - this work Table 5.5: A comparison of core radii derived from ground-based and HST observations. Chapter 5. HST Observations of G302 and G312 145 Name Ground-Based PC2 W F C rf G58 9764 G105 9"64 G108 18727 12V32 G219 13V93 G302 11789 10721 G312 9"86 10730 9"58 d - Cohen & Freeman (.1991) [30] e - this work g - Grillmair et al.(1996) [58] h - this work Table 5.6: A comparison of tidal radii derived from ground-based and JfST observations. Chapter 6 HST Observations of the Halo of M31 6.1 Colour—Magnitude Diagrams 6.1.1 Contamination in the Field Figures 6.1 and 6.2 show the (I,(V-I)0) CMDs for the halo fields around the M31 globular clusters G302 and G312 respectively. No culling has been applied to these CMDs beyond that done to identify real stellar images, as described in Chapter 4.3. The CMDs have been dereddened and corrected for interstellar absorption as described in Chapter 4.3. The CMDs of the M31 halo contain contamination from Galactic halo stars. Galactic M-dwarfs have colours of (V —/) ~ +1.5 to +3 so they will appear to the red of the red-giant branch while main-sequence stars and horizontal branch stars in the Galaxy will appear to the blue of M31's red-giant branch. Star count models of the Galaxy in the direction of M31, from Ratnatunga & Bahcall (1985) [116], suggest that there will be ~ 14 ± 4 stars redder than the red giants and ~ 1 ± 1 stars bluer than the red giants between 21 < V < 27 in each of the W F C fields. Galactic field star contamination, 146 Chapter 6. HST Observations of the Halo of M31 147 therefore, contributes only ~ 0.2% to the total number of stars observed in the G302 halo field and ~ 0.7% to the G312 halo field. The blue end of the horizontal branch is sparsely populated, but only 1 ± 1 foreground stars are expected on the blue side of the red-giant branch, so uncertainties due to photometric scatter will dominate over the effects of contamination. The small number of Galactic stars expected in the W F C fields means that foreground contamination is not a problem. Some faint background galaxies may have been mis-identified as stars in the D A O -P H O T / A L L F R A M E reductions. The deep galaxy counts of Smail et al. (1995) [134] suggest that 19 ± 2 background galaxies with 20 < I < 24 and 168 ± 2 0 galaxies with 24 < / < 27 should be located in each of the two fields. Typical galaxies have colours of (V — I) ~ 1.0 so background galaxies are indistinguishable, by colour, from stars in a C M D . The small expected number of galaxies with / < 24 suggests that background galaxies are not making a significant contribution to the morphology of the red-giant branch. The blue end of the horizontal branch is ~ 1 mag bluer than typical background galaxies so the blue horizontal branch is not significantly contaminated by galaxies. The galaxy counts also suggest that for / > 24 background galaxies make up at most 2.9% ± 0.3% of the objects on the G302 field CMD. In the G312 field background galaxies can account for up to 10.4% ± 1.2% of the objects on the CMD. D A O P H O T / A L L F R A M E will discard objects that are not morphologically similar to the stellar PSF for the frame in question so many of the faint background galaxies will already have been discarded. Therefore mis-identified background galaxies are not significantly biasing the distribution of stars on the CMDs. A third potential source of contamination is the disk of M31 itself. Mould & Krist-ian (1986) [101] used the surface photometry of de Vaucouleurs (1958) [42] to estimate that the disk-to-halo ratio in a field ~ 35' from the centre of M31 along the southeast minor axis was <; 0.014. Pritchet & van den Bergh (1988) [113] compared star counts in Chapter 6. HST Observations of the Halo of M31 148 a field 40' from the centre of M31 along the southeast minor axis with various models and found that even with a thick disk component the disk-to-halo ratio would be ^ 0.04. Hodder (1995) [70] adapted the Bahcall &: Soneira (1984) [6] Galaxy model to provide star counts and colour distributions for an external spiral galaxy. He used this model to estimate a disk-to-halo number-density ratio of ~ 0.1 in a field around G302 and ~ 0.03 in a field around G312, although these ratios are somewhat dependent on the details of the disk and halo models. Taking the worst-case scenario for disk contamination (that of Hodder 1995 [70]) leads to the conclusion that less than 10% of the stars in the G302 field, and less than 3% in the G312 field, are due to contamination from the disk of M31. On the basis of these calculations at most ~ 13% of the objects in each field are background galaxies or stars that are not members of the M31 halo. Therefore, the overall morphologies in the CMDs represent real features in the halo population of M31. The exception is the group of blue objects visible ~ 1 mag above the horizontal branch in the G302 field (see Figure 6.1). A visual examination of the individual C C D frames of the G302 field showed that approximately half of these objects fall on or near background galaxies, saturated stars, or pixels that have been flagged as bad for whatever reasons. The remaining ~ 10 bright blue objects may be nucleated dwarf galaxies, blends of stars, or stars with a large photometric uncertainties. Chapter 6. HST Observations of the Halo of M31 149 18 I — i — i — i — i — I — i — i — i — i — I — i i r i—r i — i — r 20 -22 24 26 G302 Field i i = 24.3 E v _j = 0.10 28 J I I L J I I L J I I L I I l l J L -1 0 1 2 (V-Do Figure 6.1: This figure shows the CMD for the M31 halo field around the globular cluster G302. All the stellar sources in the WF2 and WF4 images are shown. No attempt has been made to remove foreground or background contamination. The dashed hne shows the location where the photometric uncertainties are <T(v-i) — 0.2. This also corresponds to the approximate location where the data becomes incomplete. The extinction quoted is the value that was used to cahbrate the photometry (see Holtzman et al. 1995b [73]). Chapter 6. HST Observations of the Halo of M31 150 1 8 i — i — i — i — i — i — i — i — i — i — i — i — i — i — i — i — i — i — 1 i l 1 1 1 r 2 0 2 2 2 4 'MX. 2 6 G 3 1 2 Field / X = 2 4 . 3 E v _ ! = 0 . 1 0 2 8 i I l I I I I I I I I l l I I I I I -I 1 1 1 1 L - 1 0 1 2 ( V - D o Figure 6.2: This figure is the same as Figure 6.1 but shows the C M D for the M31 halo field around the globular cluster G312. Chapter 6. HST Observations of the Halo of M31 151 6.1.2 The Red-Giant Branch The red-giant branches in the G302 and G312 fields are morphologically similar except for a clump of stars at I ~ 23 in the red-giant branch of the G302 field (see Figure 6.1). There is no apparent corresponding clump visible in the G312 field (see Figure 6.2). This clump is similar to the clumps of stars seen approximately one magnitude above the horizontal branch in several metal-rich Galactic globular clusters (Rich et al. (1997) [118]). In order to check if this clump is a real feature of the red-giant branch I determined the differential luminosity function for each red-giant branch for 22 < V < 24.5 using a non-parametric histogram. A second luminosity function was constructed where the clump was removed by interpolating the differential luminosity function across the region containing the clump. The resulting "clumpless" luminosity function was compared to the observed luminosity function using a KS test. The results of this test implied that the density enhancement at V ~ 24.25 is significant at less than the 5.0 x 10 _ 6% confidence level. This suggests that the null hypothesis (that the clump in the red-giant branch is not real) can not be rejected. Peacock's (1983) [106] two-dimensional KS test was applied to the entire red-giant branch above V = 24.5 to determine if the distribution of stars within the red-giant branch is the same in G302 and G312. The KS test results said that the red-giant branch stars in the G302 and G312 fields were drawn from the same distributions at the 91% confidence level. The same tests found that the distributions of stars on the blue sides of the upper red-giant branches (V < 24, (V —1)0 < 1.6) were the same at the 85% confidence level and that the red sides of the upper red-giant branches (V < 24, (V —J)0 > 1.6) were the same at the 93% confidence level. This is weakly suggestive that the two fields have different stellar populations. However, the small number of stars in the red-giant branch of G312 (335 stars with V < 24.5) and the contamination from Galactic Chapter 6. HST Observations of the Halo of M31 152 halo stars, background galaxies, and the disc of M31 (see Chapter 6.1.1) makes a direct comparison between the distribution of stars within the red-giant branches of the two halo fields difficult. Therefore, the weak evidence for a difference in stellar populations in the two locations should be treated with some caution. Figures 6.3 and 6.4 show the (V,(V-I)0) CMDs for the M31 halo fields around G302 and G312 with a series of red-giant branch fiducial sequences for Galactic globular clusters of different iron abundances added. The three metal-poor fiducial sequences are the red-giant branch ridge lines for the Galactic globular clusters M15 ([Fe/H] = —2.19), NGC 1851 ([Fe/H] = -1.29), and 47 Tuc ([Fe/H] = -0.71) taken from D C A . The two metal rich fiducials are the t0 = 13.8 Gyr isochrones of Bertelh et al. (1994) [15] with [m/H] = —0.4 and [m/H] = 0.0. The horizontal branch fiducial sequence is that of M54 ([m/H] = -1.42) from Sarajedini & Lay den (1995) [125]. The spread in metallicity in the red giants is clearly visible from the locations of the fiducial sequences. The observed distribution of stars along the red end of the red-giant branch is consistent with a metal-rich population while the blue side of the red-giant branch is consistent with a metal-poor population. However, photometry of 47 Tuc (Lee 1977 [87]) shows that ~ 15% of the stars ~ 1 mag brighter than the horizontal branch are evolved asymptotic giant branch stars. If this ratio holds for the halo of M31 then many of the stars blueward of the [Fe/H] = —1.29 fiducial sequence will be asymptotic-giant branch stars. The majority of the stars, however, have [Fe/H] > —0.7, consistent with recent studies (see references in Chapter 1.2) which have found metallicities of [m/H] 0.6 for the halo of M31. To estimate the contribution to the width of the red-giant branch from the finite depth of the halo it was assumed that the projected surface density distribution of stars in the M31 halo follows a de Vaucouleurs R1?4 law: Chapter 6. HST Observations of t i e Halo of M31 153 log 1 0 ( s ( i2 ) /S ( i2e /a , ) ) ( R 1/4 = -3.3307 Re/as - 1 (6.1) with an effective radius of Re = 1.3 kpc and an axial ratio of cts = 0.55, in accordance with Pritchet & van den Bergh (1994) [112]. The volume density, p(r), can be obtained from S(-R), through Abel's integral, as follows: where R is the projected distance of the field from the centre of M31 and r is the true The density distributions along the hnes-of-sight at projected distances of 32' and 50' from the centre of M31 were computed. The spread due to the depth of the halo has a half-power width in the V-band of ~ 0.02 mag at RMSI = 32' (the G302 field) and ~ 0.03 mag at i?M3i — 50' (the G312 field). Changing the effective radius and axial ratio over the ranges found by Hodder (1995) [70] in his study of the structure of M31 does not significantly alter these results. A change of a few hundredths of a magnitude along any of the D C A fiducial red-giant branches corresponds to a negligible change in colour which suggests that the depth of the M31 halo does not contribute significantly to the observed width of the red-giant branch. The mean photometric uncertainties in the data are cry-i — 0.05 near the tip of the red-giant branch (J ~ 20) and increase to av-i — 0.10 at the level of the horizontal branch, yet the observed spread in colour along the red-giant branch is ~ 0.5 mag at the level of the horizontal branch and ~ 2 mag near I ~ 20. These spreads are too large to be due to photometric uncertainties or the depth of the M31 halo. The turn-over in the red-giant branch near V = 23 is due to the increased opacity from molecular bands as giants expand and become cooler. For shell-hydrogen burning stars with [m/H] ^ — 1 (6.2) galactocentric distance of the volume element p(r) from the centre of M31. Chapter 6. HST Observations of the Halo of M31 154 the molecular opacity in the star's atmosphere can become high enough to reduce the luminosity of the star enough that these stars become fainter as they ascend the red-giant branch. However, Figures 6.3 and 6.4 shows that this flattening will not explain all of the structure in the red-giant branch. Therefore a portion of the width of the red-giant branch is due to an intrinsic spread in the metallicity of the M31 halo stars. In order to estimate the metallicity distribution of red-giant branch stars metal-licity values for each star were interpolated based on the fiducial sequences in Fig-ures 6.3 and 6.4 and the theoretical isochrone of Bertelli et al. (1994) [15] with [m/H] = +0.4 (not plotted). It was assumed that the age of the M31 halo is t0 — 14 Gyr, compara-ble with the age of the Galactic halo globular cluster system (e.g. Richer et al. 1996 [122]). The morphology of the red-giant branch is not sensitive to changes in age of a few Gyr so the exact age adopted for the red-giant branch isochrones is not critical. Figure 6.5 shows the probability density distributions of metallicity ([m/H]) for red-giant branch stars with I < 23 in the G302 and G312 fields. Based on the photometric uncertainties in the data at the level of the red-giant branch, and the uncertainties inherent in matching theoretical isochrones to observational data, the metallicity estimates for the red-giant branch stars have uncertainties of <T[m/H] ~ 0.25. A KS test shows that the two metallicity distributions differ at less than the 5.0 x 10_ 6% confidence level, strongly suggesting that the stellar populations are the same in both fields. The G302 field distribution has a peak at [m/H] = -0.6 and a F W H M of 1.3 dex while the G312 field distribution has a peak at [m/H] = -0.7 and a F W H M of 1.6 dex. These FWHMs are overestimates of the true metallicity distribution since the probability density distribution function shown in Fig-ure 6.5 is essentially a convolution of the data with a unit Gaussian that has a dispersion equal to the uncertainty in a typical measurement (cr[ m /H] = 0.25). The intrinsic spread in the metallicity can be estimated by deconvolving this Gaussian from the probability density distribution function. This gives an intrinsic F W H M of 1.2 dex for the G302 field Chapter 6. HST Observations of the Halo of M31 155 and 1.5 dex for the G 3 1 2 field. The halo stars are clearly more metal-rich than, and have a slightly greater spread in metallicity than, the M 3 1 globular cluster system. The halo metallicity distribution is clearly asymmetric with an extended metal-poor tail. This is partly due to the presence of asymptotic-giant branch stars on the blue edge of the red-giant branch and partly due to the metallicities less than [m/H] = —2.19 being extrapolated from the higher-metallicity fiducial sequences. Therefore the shape of the metallicity distribution beyond [m/H] ~ —2 should be regarded with caution. There is a sharp cut-off in the halo metallicity distribution at approximately Solar metallicity. The metal-rich tail of the metallicity distribution drops to zero quite rapidly between [m/H] ~ —0.2 and [m/H} ~ +0.2. A visual examination of the locations of the [m/H] = —0.4 and [m/H] = 0.0 isochrones in Figures 6,3 and 6.4 shows the most metal-rich red-giant branch stars having [m/H] ~ —0.2 suggesting that the extra-Solar metallicity tail in Figure 6.5 is an artifact of the uncertainties in the metallicity determination. Chapter 6. HST Observations of the Halo of M31 156 Figure 6.3: This figure shows the CMD for stars images on the WF2 and the WF4 CCDs in the G302 field. The fiducial sequences are described in Chapter 6.1.2. Chapter 6. HST Observations of the Halo of M31 157 Figure 6.4: This figure shows the CMD for stars images on the WF2 and the WF4 CCDs in the G312 field. The fiducial sequences are described in Chapter 6.1.2. Chapter 6. HST Observations of the Halo of M31 158 [m/H] Figure 6.5: The probability density distribution for the metallicity of the red giants in the field around G302 (solid line), the field around G312 (dashed line), and the M31 globular cluster system (dotted line). For each star a unit Gaussian with a standard deviation of cr = 0.25 dex, the estimated uncertainty in the metalhcity determination for an individual star, was generated. For the globular clusters the <7[Fe/H] values from HBK were used. The normalized sum of these Gaussians is a non-parametric histogram of the probability density, P([m/H]), distribution. The sharp edge at the metal-rich end the distribution ([m/H] ~ 0) suggests that there is a well-defined upper limit to the metalhcity of stars in these fields. The slow decline on the metal-poor edge suggests that the metal-poor stars cover a range of met alii cities. A portion of the metal-poor tail arises from the confusion of asymptotic-giant branch stars with metal-poor red-giant branch stars. Chapter 6. HST Observations of the Halo of M31 159 6.1.3 The Horizontal Branch The horizontal branch stars in both fields are concentrated in a red clump between 0.5 < (V-I)0 < 1.0 with a handful of blue stars between 0.0 (V~I)0 < 0.5. The red clump is consistent with what is seen in metal-rich Galactic globular clusters such as 47 Tuc while the small number of blue horizontal branch stars is indicative of a metal-poor population. However, an age spread of ~ 5 Gyr in the halo population could produce a horizontal branch morphology similar to what a metalhcity spread of ~ 1 dex would produce. This is the classic "second parameter" problem (see van den Bergh 1993 [147], Lee et al. 1994 [88], and Chaboyer et al. 1996 [28] for recent reviews) affecting the colours of horizontal branch stars. The lack of any extended asymptotic-giant branch candidates in the data, or in any other pubhshed M31 halo fields, suggests that a young population (^ 5 Gyr) is not present. However, it does not rule out the possibility that the M31 halo may contain stars as young as ~ 10 Gyr. The morphology of the horizontal branch in each field can be parameterized by count-ing the number of red and blue horizontal branch stars, and computing the horizontal-branch-ratio = (NB — NR)/(NB + Ny + NR) where NB is the number of horizontal branch stars bluer than the instability strip, Ny is the number of stars in the instability strip, and NR is the number of horizontal branch stars redder than the instability strip. Since photometric scatter is quite large (a^y-i) — 0.16) at the level of the instability strip, and the instability strip has a width of only ~ 0.25 mag, I did not attempt to identify RR Lyrae candidates. Instead the horizontal branch was split at (V — I)0 — 0.5 so the ratio (NB ~ NR)/(NB + NR) could be calculated. The G302 field has a horizontal-branch-ratio of —0.91 ± 0.12 (Poisson uncertainty) while the G312 field has a horizontal-branch-ratio of —0.92 ± 0.26 suggesting that both fields have the same horizontal branch morphology. These ratios are lower limits on the actual horizontal-branch-ratios since the Chapter 6. HST Observations of the Halo of M31 160 blue horizontal branch extends to below the cr^v-i) = 0.2 photometric limit suggesting that incompleteness may be causing a seriously underestimate of NB. The theoretical horizontal branches of Lee et al. (1994) [88] were used to estimate the expected values of the horizontal branch type given the metalhcity ratios determined in Chapter 6.2. The predicted horizontal-branch-ratios were —0.6 ± 0 . 3 in the G302 field and —0.1 ± 0 . 1 in the G312 field, somewhat more positive than the values measured from the CMDs. This sug-gests that either the blue horizontal branch suffers from a high degree of incompleteness, or that there is a second parameter contributing to the morphology of the horizontal branch in the halo of M31. Detailed incompleteness tests suggest that the results are due to incompleteness in the blue end of the horizontal branch (see Figure 4.4). The morphology of the horizontal branch in the halo of M31 is broad agreement with the metalhcity distribution determined from the red-giant branch morphology but further work is needed to determine if the lack of blue horizontal branch stars (relative to the metalhcity spread observed in the red-giant branch stars) is due to photometric limits in the data or a second parameter effect1. The (V, (V" —/)„) CMDs (Figures 6.3 and 6.4) show that the horizontal branch is not horizontal in the V-band. To estimate the spread in the magnitudes of the horizontal branch stars the horizontal branch was spht into bins with widths of L\[V — I) = 0.1 and fit a sloping Gaussian to the distribution of stars in the V-band in each bin. The mean width of these Gaussians was oy = 0.14 ± 0.01 (standard error) in the G302 field and ay = 0.12 ± 0.03 in the G312 field. The mean photometric uncertainty at the level of the red horizontal branch is &v = 0.086. Subtracting this quadratically from the observed spread in the F-band magnitude of the horizontal branch gives intrinsic spreads of a v = 0.11 ± 0 . 0 1 in the G302 field and ay = 0.08 ±0.03 in the G312 field. The 1Ideally V- and /-band photometry reaching to the main-sequence turn-off in the M31 halo (V ~ 29) would be needed to do a high-precision photometric study of the horizontal branch morphology. Such a study would require at least forty orbits of HST time per field using the W F P C 2 . Chapter 6. HST Observations of the Halo of M31 161 finite depth of the halo of M31 also introduces a small broadening in the V-band (see Chapter 6.1.2) of the horizontal branch. However, this broadening is negligible (~ 3%) compared with the broadening introduced by photometric uncertainties so it was ignored. The resulting distribution of V-band magnitudes was converted to iron abundances using the relation of Chaboyer et al. (1996) [28], see below. The derived mean iron abundance is [Fe/H] = -0.5 ± 0.6 (standard deviation) in the G302 field and [Fe/H] = -0.5 ± 0.4 in the G312 field. This is consistent with the iron-abundance distribution seen in the red giants. The relationship between iron abundance and horizontal branch magnitude is very sensitive the adopted value of the distance modulus. An error of only 0.02 magnitudes in the distance modulus will result in an uncertainty of ~ 0.1 dex in metallicity. Therefore, the metallicity distribution estimated from the red-giant branch is a better indicator of the mean metallicity of the M31 halo than the metallicity distribution obtained from the horizontal branch. However, it should be noted that the metal-poor edge of the horizontal branch met alii city distribution ([Fe/H] — 3o"[Fe/H]) occurs at [Fe/H] ~ —2, in rough agreement with the metallicity distribution of red-giant branch stars. The lack of horizontal branch stars with [Fe/H] ^ —2 supports the conclusion that the metal-poor tail of the red-giant branch metallicity distribution is due to calibration uncertainties and contamination of the blue side of the red-giant branch by asymptotic-giant branch stars. The metal-rich edge of the horizontal branch metallicity distribution ([Fe/H] + 3cr[Fe/H] occurs at [Fe/H] ~ +1. The red-giant branches show no evidence for stars with [Fe/H] ^ —0.2 so the metal-rich tail in the horizontal branch metallicity distribution is most likely an artifact of fitting a Gaussian to a non-Gaussian distribution. In general, red horizontal branches have a well defined faint edge but photometric uncertainties and contamination from red giants make it impossible to identify this faint edge in the data. Therefore the metal-rich tail of the horizontal branch metallicity distribution is spurious. Chapter 6. HST Observations of the Halo of M31 162 6.2 The Halo Luminosity Function Artificial star tests (see Chapter 4.4) indicate that the photometry is complete to approx-imately the level of the horizontal branch (V ~ 25) in both the G302 and G312 fields. This is consistent with the results of Rich et al. (1996) [119] who found that star-counts in the halo of M31 obtained using WFPC2 data were complete to at least V ~ 26. The raw V- and J-band luminosity functions in each field are tabulated in Table 6.1. The n(V) values are the number of stars in each field per 0.2 magnitude bin. The total area of each field is ~ 3.2 • ' after masking and bad pixels are taken into account. Figure 6.6 shows the luminosity function for the G302 field compared with luminosity functions for the metal-rich M31 globular cluster G l (Rich et al. 1996 [119]), a theoretical lumi-nosity function for an old (t0 = 14 Gyr) metal-rich ([Fe/H] = -0.47, [O/Fe] = +0.23) population (Bergbusch & VandenBerg 1992 [11]), the metal-rich Galactic globular clus-ter 47 Tuc (Hesser et al. 1987 [68]), and the metal-poor Galactic globular cluster M13 (Simoda & Kimura 1968 [132]). Figure 6.7 show the luminosity function for the G312 field and the same comparison luminosity functions. Bergbusch k VandenBerg (1992) [11] calculated a series of oxygen-enhanced stellar models for old stars and used these models to investigate the relationships between the shape of the luminosity function of the red-giant branch and various astrophysical pa-rameters. They found that neither iron abundance, nor age, has any detectable affect on the slope of the upper portion of the red-giant branch. Similarly, in a review of the literature, Renzini k Fusi Pecci (1988) [117] found that luminosity functions of red giants are much more sensitive to details of stellar structure than they are to the stellar ages. The discrepancies between the luminosity function of the halo of M31 and the various comparison luminosity functions is probably due to a combination of incompleteness, Chapter 6. HST Observations of the Halo of M31 163 small number statistics, and confusion between red giants, asymptotic giants, and ho-rizontal branch stars. The artificial star tests described in Chapter 4.4 suggest that incompleteness is not a problem for V ^ 25. To see if the shape of the M31 halo lu-minosity function changes significantly if the size of the magnitude bins are altered the stars were rebinned in 0.1, 0.25, and 0.5 magnitude bins. The comparison luminosity functions were compared to the rebinned M31 data using a KS test. In all cases the KS tests showed that the M31 halo luminosity function and the comparison luminosity function were the same at less than the 50% confidence level, suggesting that the M31 halo luminosity function is different from the comparison luminosity functions. In light of this, it is unlikely that the differences between the M31 halo data and the comparison data are merely due to statistical uncertainties in the data or the binning process. Lee et al. (1977) [87] found that ~ 15% of the stars in 47 Tuc that are more than ~ 1 mag brighter than the horizontal branch are shell-helium burning stars on the asymptotic-giant branch. Since 47 Tuc has a similar metallicity and age as the stars in the halo of M31 near G302 and G312, it is reasonable to assume that a similar fraction of the stars brighter than the horizontal branch are asymptotic-giant branch stars and not part of the red-giant branch. Hesser & Hartwick (1977) [69] obtained separate luminosity functions for the red giants and the asymptotic giants of 47 Tuc. There is no significant difference in the slopes of these two luminosity functions, but stars on the asymptotic-giant branch had more of a tendency to be grouped in magnitude than stars on the red-giant branch did. If the ratio of red giants to asymptotic giants is the same at all magnitudes above the horizontal branch then including asymptotic giants in the halo luminosity function should not affect its shape. However, if the luminosity function of the asymptotic-giant branch is clumpy then confusion between the two giant branches can alter the observed shape of the luminosity function. Evidence for this can be found in Rich et al. (1997) [118]. They observe clumps in the asymptotic-giant branches for four metal-rich Galactic globular Chapter 6. HST Observations of the Halo of M31 164 clusters that do not appear to correspond to clumps in the red-giant branches of these clusters. The large spread in metalhcity in the halo of M31 makes it very difficult to distinguish between asymptotic-giant branch stars and stars on the upper red-giant branch. Since most of the observed width is due to the real spread in metalhcity further improvement in photometric accuracy will not help in identifying helium shell-burning stars in the halo of M31. Chapter 6. HST Observations of the Halo of M31 165 n(V) n(V) n{I) n(I) V G302F G312F I G302F G312F 20.00 0 0 20.00 2 0 20.20 2 0 20.20 4 2 20.40 1 1 20.40 8 4 20.60 1 0 20.60 17 9 20.80 0 0 20.80 35 5 21.00 1 1 21.00 42 9 21.20 0 0 21.20 53 12 21.40 0 1 . 21.40 • 49 9 21.60 2 1 21.60 46 9 21.80 4 3 21.80 49 11 22.00 9 3 22.00 71 25 22.20 8 6 22.20 69 18 22.40 20 5 22.40 113 28 22.60 29 10 22.60 89 28 22.80 45 8 22.80 121 30 23.00 65 10 23.00 154 42 23.20 78 20 23.20 194 52 23.40 106 36 23.40 174 45 23.60 123 25 23.60 225 44 23.80 180 46 23.80 304 68 •24.00 168 52 24.00 555 138 24.20 222 52 24.20 1204 275 •24.40 217 52 24.40 987 237 24.60 263 58 24.60 586 145 24,80 478 115 24.80 418 106 25.00 953 259 25.00 388 110 25.20 1260 275 25.20 357 115 25.40 674 144 25.40 346 110 25.60 476 122 25.60 319 96 25.80 436 125 25.80 290 .102 Table 6.1: The observed V- and /-band halo luminosity functions. Chapter 6. HST Observations of the Halo of M31 166 n(V) n(V) n(J) n(J) V G302F G312F I G302F G312F 26.00 419 126 26.00 254 90 26.20 398 129 26.20 185 78 26.40 355 98 26.40 81 43 26.60 300 101 26.60 43 23 26.80 257 86 26.80 13 5 27.00 150 77 27.00 5 4 27.20 97 40 27.20 0 1 27.40 40 31 27.40 1 0 27.60 14 11 27.60 0 0 27.80 6 2 27.80 0 0 28.00 1 1 28.00 0 0 6.1 (continued) Chapter 6. HST Observations of the Halo of M31 167 1—i—i—I i i i I i i i — I — r n — I I I I — i — I — i — r — i i r~i i i r — i r Figure 6.6: This figure compares the cumulative luminosity function for the M31 halo field near G302 (solid lines) with four comparison luminosity functions (dashed lines): (a) the G l luminosity function, (b) a theoretical luminosity function (B & V) with [Fe/H] = -0.47, [O/Fe] = +0.23 and t0 = 14 Gyr, (c) the 47 Tuc luminosity function, and (d) the M13 luminosity function. The faint cut-off magnitude was set to V = 25.5 because the photometry is reasonably complete down to approximately this point (see Chapter 6.2). Figure 6.7: This figure compares the cumulative luminosity function for the M31 halo field near G312 with the four comparison luminosity functions used in Figure 6.6. Chapter 7 G185: A Potential Double Globular Cluster 7.1 Multiple Globular Clusters When the HRCam images of G185 were being examined it became apparent that an object ~ 4" northwest of G185 had an extended surface brightness profile resembling that of a globular cluster. No multiple globular clusters have been observed in the Galaxy or in any large extra-Galactic system, although some multiple globular clusters have been found in the Magellanic Clouds. Binary globular star clusters are dynamically very fragile objects so it is not surprising that none have been observed in the Galaxy. Innanen et al. (1972) [78] have used numer-ical simulations to show that a binary globular cluster with a separation of less than ~ 50 pc, and a periGalacticon of less than ~ 1 kpc, would not survive a single Galactic orbit. Two globular clusters that are near enough to each other to avoid being separated by the Galactic tidal field will be too close to avoid being disrupted by the mutual tidal field of the clusters. Alternately, if the inter-cluster separation is large enough that mutual gravitation is insufficient to cause the two clusters to merge into a single object then the effects of the Galactic tidal field would be great enough to cause the two clusters to 169 Chapter 7. G185: A Potential Double Globular Cluster 170 move away from each other. Galactic globular clusters are among the oldest objects in the Galaxy so a primordial binary globular cluster will have undergone many Galactic orbits and thus could not survive, as a binary cluster, to the present day. The mass of M31 is approximately twice that of the Galaxy so it is even less likely that a binary globular cluster could survive in M31. A transient binary globular cluster, however, may be observed if two clusters passed near enough to interact with each other, although such an encounter would likely destroy the two clusters since a typical encounter time-scale is longer than the time required for stars in one cluster to respond to the other cluster's gravitational field. There is evidence that star clusters form in groups where the individual clusters are not gravitationally bound to each other. Lynga, & Wramdemark (1984) [92] cite similarities in metallicities, stellar content, and stellar ages to argue that several multiple open clusters in the direction of the Gould Belt have a common origin. Such systems, however, are probably not dynamically bound binary clusters but simply clusters that have a common origin and are in the process of either separating or merging. The Galactic open clusters h & x Persei are one possible such pair. This scenario is unlikely among the M31 globular clusters since that system appears to have an age comparable with that of the Galactic globular cluster system (Frogel et al. 1980 [55], Bohlin et al. 1993 [18]). The Large Magellanic Cloud (LMC), however, presents a more promising environment for binary clusters. The tidal field of the L M C is weak enough that a binary globular cluster could survive indefinitely (Innanen et al. 1972 [78]) without experiencing fatal dis-ruption. Bhatia & Hatzidimitriou (1988) [14] have catalogued 69 pairs of stellar clusters in the L M C with inter-cluster separations of less than 13" (~ 3.16 pc) and use statistical arguments to show that only about half of these pairs are due to chance. Radial velocity studies have shown that the individual components of some of these multiple objects are interacting with each other while colour-magnitude studies show that several of these Chapter 7. G185: A Potential Double Globular Cluster 171 multiple clusters are young compared to the ages of Galactic globular clusters. Because of their young age the L M C multiple clusters may not be true dynamically bound multiple cluster systems but rather clusters sharing common origins that are currently parting company. Chapter 7. G185: A Potential Double Globular Cluster 172 7.2 The HRCam Images The C F H T data (see Chapter 2) contained several Johnson V- and Cousins /-band im-ages of a 2' x 2' field near the core of M31 that includes G185 (a2ooo.o = 00h42m44!!2, <Wo = +41°14'28") and vdB2 (van den Bergh 1969 [149]) (a2ooo.o = 00h42m41U, £2000.0 = +41°15'26"). These images were all taken on the night of August 16/17, 1990 and were pre-processed in the manner described in Chapter 2.2. A summary of the observations of the G185 field is given in Table 7.1. Frame Filter UT-7 Exposure Airmass F W H M (0 83922 V 12:15:21.7 200 1.109 0769 83924 I 12:24:54.1 100 1.099 0768 83925 I 12:31:03.4 100 1.094 0783 83926 I 12:34:32.7 100 1.091 0772 83927 I 12:38:06.0 100 1.089 0768 83928 I 12:41:37.4 100 1.086 0769 83929 I 12:45:09.7 100 1.084 0770 Table 7.1: Log of the HRCam observations of G185. The unresolved background light was removed by fitting and subtracting a two-dimensional cubic spline as described in Chapter 2.2.3. The residual variations in the background were approximately 1%, not including small-scale variations due to dust lanes and surface brightness fluctuations arising from unresolved stars in the bulge and halo of M31. Figure 7.1 shows the /-band image of the field containing the three globular clus-ters. Figure 7.2 shows the same field with the unresolved light from M31 removed. Dust lanes and surface brightness fluctuations are clearly visible. Primini et al. (1993) [111] report two ROSAT X-ray sources located within 0.'5 of G185 (their sources 39 and 46) but find no evidence for an X-ray source in G185. There are no unusual features on either Chapter 7. G185: A Potential Double Globular Cluster 173 the G185 V- or J-band images within the error ellipses of the X-ray sources. The IRAF implementation of the STSDAS task ELLIPSE was used to fit elliptical iso-photes to each cluster. Surface brightness profiles were then extracted along the effective radius axes of each cluster. The two objects were fitted iteratively as follows. First, the fainter component (G185B) was fitted and and subtracted from the original image. Next, the brighter (G185) component was fitted using the image with the G185B component subtracted. The G185 component was then subtracted from the original image and a new fit made to the G185B component. This cycle was repeated until the fitted ellipses for the two components were stable from one iteration to the next. Two or three iterations were sufficient for both the V- and J-band images. Figure 7.3 shows the subtractions of each component of G185 from the original image. Describing the stellar PSF for each image was somewhat difficult since there were very few bright, non-saturated stars in the field. I attempted to fit both one-dimensional Moffatians and one-dimensional multi-Gaussians (Bendinelh et al. 1987 [10]) to the stars on each frame but found that the wings of the PSFs (beyond ~ l'/25) were not well fit by either functional form. Therefore, DAOPHOT II was used to define a PSF in each band. I created a set of noiseless bright artificial stars and fit elliptical isophotes to these in the same manner as was done for the globular clusters. This worked well out to ~ 1725 but beyond this the uncertainties in the intensities of the fitted ellipses became larger than the PSF intensities. The difficulty in describing the PSFs beyond ~ 1725 is due to small-scale (a few pixels) fluctuations in the image background. Since the program field is located only ~ 360 pc from the centre of M31 there are large numbers of unresolved bulge and halo stars in the line of sight to each pixel. The PSF stars are sitting atop these large surface brightness fluctuations which in turn contribute to the shape of the PSF. In the core of the PSF the stellar luminosity is large enough to dominate the shape Chapter 7. G185: A Potential Double Globular Cluster 174 of the PSF, however in the wings of the PSF surface brightness fluctuations in the field can significantly influence the shape of the PSF. The large signal-to-noise ratio of the /-band data results in larger surface brightness fluctuations in the /-band than in the V-band making the wings of the PSF harder to measure in / than in V. This can be seen in the wings of the PSF profiles in Figures 7.4 and 7.5. The observed surface brightness profiles for each cluster and the PSFs are shown in Figures 7.4 and 7.5. G185B is clearly more extended than the PSF suggesting that it is not merely a foreground star. However, the outermost points of the surface brightness profile of G185B actually crosses the profile of the PSF. This, and the extreme truncation of the cluster profile, is likely an artifact of the uncertainties in measuring the surface brightness profiles of G185B and the PSF. G185B has only ~ 3% of the luminosity of G185 or vdB2. This results in the uncertainties in the surface brightness profile of G185B being correspondingly larger than the uncertainties in the profiles of G185 and vdB2 at similar fractions of the central surface brightness. Further, fitting and subtracting the unresolved background light from M31, and removing G185 from the images, results in additional uncertainties in the surface brightness measurements. Chapter 7. G185: A Potential Double Globular Cluster 175 Figure 7.1: This figure shows the J-band image of G185 (centre), G185B (just below and to the right of G185), and vdB2 (upper right). The centre of M31 is located to the right of the frame. The top of the image is oriented 84° east of north. Chapter 7. G185: A Potential Double Globular Cluster 176 F i g u r e 7.2: T h i s figure is the same as F i g u r e 7.1 except the unresolved l ight f rom the bulge of M 3 1 has been sub t rac ted l eav ing dust lanes and surface brightness fluctuations v is ib le . N o t i c e the surface brightness fluctuations are greater on the r igh t -hand side of the image ( towards the centre of M 3 1 ) . Chapter 7. G185: A Potential Double Globular Cluster 177 Figure 7.3: This figure shows G185 and G185B at various stages of the iterative fit-and-subtract procedure described in Chapter 7.2. (1) shows the two clusters G185 and G185B. (2) shows the same field with G185B subtracted while (3) shows the field with only G185 subtracted. (4) shows both clusters removed from the field. Chapter 7. G185: A Potential Double Globular Cluster 178 log 1 0( r e f f ) ["] Figure 7.4: The normalized observed surface brightness profiles of each cluster and the PSF in the /-band. G185 is denoted by open squares, G185B by filled triangles, vdB2 by open circles and the PSF by the solid line. The noise in the wings of the /-band PSF is due the higher signal-to-noise ratio of the /-band image causing larger surface brightness fluctuations in the background. The error bars show the approximate uncertainties in the surface brightness profiles of G185 and vdB2 at the given intensity levels. G185B is nearly four magnitudes fainter than the other two clusters so its uncertainties will be greater than indicated by the error bars To estimate the uncertainties in the G185B profiles shift the given error bars up by approximately 1.5 units in log10(5'). Chapter 7. G185: A Potential Double Globular Cluster 179 Figure 7.5: This figure is the same as Figure 7.4 except is shows V-band surface bright-ness profiles. G185B is nearly four magnitudes fainter than the other two clusters so its uncertainties will be greater than indicated by the error bars. To estimate the uncer-tainties in the G185B profiles shift the given error bars up by approximately 1.5 units in l°gio(S)-Chapter 7. G185: A Potential Double Globular Cluster 180 7.3 Probability that G185 and G185B are Line-of-Sight Objects The probability that two globular clusters 1'.7 from the core of M31 will have an observed separation of 4" by chance is estimated as follows. Crampton et al. (1988) [33] find that the projected density of the M31 globular cluster system drops as .ft1/4 with projected distance from the centre of M31. This density relation gives a probability of such a chance alignment on a 2' x 2' field as 0.002. There are, however, -~ 5.34 such fields within an annulus of width 2' centred 1'.7 from the centre of M31. Therefore, the probability that a randomly selected field 1'.7 from the centre of M31 will contain such a pair.of globular clusters is ~ 0.002 x 5.34 = 0.01. This calculation assumes that there is no angular dependence in the distribution of globular clusters in M31. In reality the small number density of globular clusters in this annulus will result in Poisson fluctuations in the number of clusters observed in different fields located at the same distance from the centre of M31. To account for this the method of Bhatia & Hatzidimitriou (1988) [14] was used to estimate the probability that a 2' x 2' field containing three globular clusters will contain two globular clusters separated by 4" or less. This method gives a probability of 0.016 for a single field. The probability of finding such a pair within the annulus, then, is 0.09 ± 0.03. A chi-square test indicates that this is significant at the 99% (2.5<x) confidence level, however the small sample size (three clusters) suggests that this result not be given too much weight. Chapter 7. G185: A Potential Double Globular Cluster 181 7.4 Colours The total integrated magnitudes for G185, G185B, and vdB2 were determined using aperture photometry with uncertainties estimated from fluctuations in the flat portion of the curve of growth for each cluster. To test this method of determining the uncertainties in the integrated magnitudes, a series of artificial globular clusters were constructed by adding artificial stars to the HRCam images using realistic luminosity and radial density profiles. The luminosity function used was that of 47 Tuc (Hesser et al. 1987 [68]) with the horizontal branch stars distributed uniformly over the horizontal part of the horizontal branch. In practice the exact form of the luminosity function did not affect the results. The radial stellar density profiles used were generated from the Michie-King models that were fit to the clusters (see Chapter 7.5.1). From these artificial globular clusters the uncertainties in the aperture magnitudes were estimated to be less than 0.1 magnitudes. The total magnitudes for each cluster were computed by integrating under the observed surface brightness profiles. The two methods of measuring magnitudes agreed to within la for all three clusters. The total calibrated aperture magnitudes are listed in Table 7.2. No corrections for reddening have been applied to these values. Cluster I (V-I) G185 13.05 ± 0 . 0 3 1.19 ± 0 . 0 3 G185B 16.87 ± 0 . 1 0 1.09 ± 0 . 1 4 vdB2 13.35 ± 0 . 0 3 1.17 ± 0 . 0 3 Table 7.2: Integrated magnitudes for G185, G185B and vdB2. Burstein & Heiles (1984) [22] quote an external reddening of EB-V = 0 .080±0 .003 for M31. Using EV-i = l.2hEB-v (Taylor 1986 [144], Fahlman et al. 1989 [52]), and ignoring internal reddening within M31, gives Ey-i = 0.10 ± 0.01 so all three globular clusters Chapter 7. G185: A Potential Double Globular Cluster 182 have dereddened colours that are consistent with the V — Io colours of the Galactic glob-ular clusters (e.g. Peterson 1993 [108]). The V-I colours of G185 and G185B suggest that they are old objects similar in age to the rest of the M31 globular cluster system. Therefore it is unlikely that G185 and G185B are a pair of young clusters that formed together and are currently separating. The field around G185 shows a large number of dust clouds which obscure the bulge of M31. The fact that none of the three clusters exhibits an unusual degree of reddening suggests that they all Ue on the near side of the centre of M31 and may be situated in front of the majority of the material in M31. Chapter 7. G185: A Potential Double Globular Cluster 183 7.5 Structural Parameters 7.5.1 Cluster Concentrations The procedure described in Chapter 3.2.6 was used to estimate the central potentials, core and tidal radii, and elhpticities of G185, G185B, and vdB2. Table 7.3 lists Michie-King parameters for each cluster and bandpass derived using this two-dimensional modeling. The quoted uncertainties were estimated as follows. A series of artificial globular clusters with Wo = 5, rc = 075, rt = 5'.'35, and e = 0.0 were constructed. Two-dimensional models were then fit to these clusters. The scatter in the recovered parameters was taken to be indicative of the uncertainties in the best-fitting models. This, however, does not include uncertainties due to the model PSF. I was unable to constrain the Michie-King parameters of G185B as well as was done for the other two clusters. This is reflected in the larger uncertainties quoted for G185B in Table 7.3. Cluster Filter Wo rc rt c G185 V 5.2 ± 0 . 4 0750 ± 0'.'08 6709 ± 0'.'58 1.07 ± 0.09 I 5.5 ± 0 . 3 0"46 ± 0'.'03 6726 ± 1709 1.14 ± 0 . 0 7 G185B V 3.0 ± 1.0 0"40 ± 0"13 1777 ± C'30 0.67 ± 0 . 1 7 I 3.0 ± 1.0 0739 ± 0713 1770 ± C/25 0.67 ± 0 . 1 7 vdB2 V 6.5 ± 0 . 3 0'.'13 ± 0'.'02 5"90 ± 0"67 1.39 ± 0 . 0 9 I 7.0 ± 0 . 3 0"17 ± 0'.'02 5"90 ± 0"67 1.53 ± 0 . 0 9 Table 7.3: Two-dimensional Michie-King model fits to G185, G185B, and vdB2. G185B appears to be a very loose cluster with a concentration of 0.67 ± 0.17. This is less than that of the majority of the Galactic globular clusters. In fact, only a handful of outer halo Galactic clusters have similar concentrations (Trager et al. 1993 [145]). This, and the fact that G185B does not appear to be unusually reddened, argues that G185B is not situated near the centre of M31 and may be located well in front of M31. G185B Chapter 7. G185: A Potential Double Globular Cluster 184 has an apparent tidal radius approximately three-and-a-half times smaller than that of G185. For this difference in size to be a purely geometric effect G185B would have to be located at ~3.5 times the distance of G185 meaning that either G185 or G185B is not a member of the M31 globular cluster system. Huchra et al. (1991) [76] give a radial velocity for G185 of v = —185 ± 2 5 km-s - 1 relative to M31, consistent with G185 being a member of the M31 globular cluster system. Since G185 contributes ~ 97% of the total light this radial velocity will be a reasonable estimate of the radial velocity of G185. The colour of G185B is not unduly red for an old globular cluster so it is unlikely that G185B is being viewed through the bulge of M31. This suggests that G185B is a member of the M31 globular cluster system and is intrinsically smaller than G185. The fitted core radius for G185 is somewhat smaller than that found by Cohen k Free-man (1991) [30]. I found a core radius of 0746 ± 0"04 (= 1.6 ± 0.1 pc) and a tidal radius of 6'.'13 ± 0"69 (= 21.5 ± 2.4 pc) for G185. Cohen k Freeman (1991) [30] found rc = 2.9 pc and rt = 20 pc. The difference in core radius is probably due to the improved seeing conditions of the observations. It may also be due to the identification of G185B as a separate object. Chapter 7. G185: A Potential Double Globular Cluster 185 7.5.2 Ellipticities Figure 7.6 shows observed isophotes and some fitted isophotal ellipses for G185 and G185B. Observed weighted mean ellipticities, e, and position angles, 80, were calculated for each cluster based on the fitted isophotes. "These are presented in Table 7.4 along with the number, N, of fitted ellipses used to compute these values. Figures 7.7 and 7.8 show the radial variations of these quantities for G185, G185B and the PSF. Position angles are measured from north to east on the sky. The quoted uncertainties are due to the variation in the fitted position angle from the centre to the outer edge of each cluster, and to the estimated uncertainties in orienting the CCD images with respect to north. Large uncertainties are indicative of a large radial variation in position angle. Figures 7.7 and 7.8 show that radial changes in the ellipticity and orientation of the PSF are echoed in the shape of G185B suggesting that seeing is the dominant effect in determining the shape of G185B. . Cluster Filter e ' • 0o - N G185 V • 0.034 ± 0.004 - 8 7 ? 1 ± 3 8 ? 7 29 I 0.105 ± 0 . 0 0 5 - 7 3 ? 8 ± 4?3 34 G185B V 0.115 ± 0 . 0 1 6 - 1 4 ? 4 ± 14?5 16 I 0.108 ± 0 . 0 1 2 -52?8 ± 23?8 19 vdB2 V 0.027 ± 0 . 0 0 2 -8?0 ± 15?8 26 I 0.036 ± 0 . 0 0 1 - 7 8 ? 9 ± 5?2 29 PSF V 0.029 ± 0.002 -7?8 ± 33? 4 24 I 0.199 ± 0 . 0 0 6 - 6 4 ? 1 ± 3 1 ? 5 23 Table 7.4: Ellipticities and position angles for G185, G185B, and vdB2. The observed ellipticity of a semi-resolved globular cluster is dominated by the shape of the PSF out to ~ 8 times the seeing F W H M . This corresponds to 5'.'7 in the /-band and 6'.'9 in the V-band. However Figures 7.7 and 7.8 show that ellipses could only be fit Chapter 7. G185: A Potential Double Globular Cluster 186 out to ~ 5" for the best defined cluster. This, and the similarities between the elhpticities of the clusters and the PSFs, suggests that the observed projected elhpticities of G185 and G185B are due to seeing effects. To test this conclusion a series of artificial globular clusters were built based on G185 but with elhpticities between 0.00 and 0.21. These were placed on the /-band data frame and elliptical isophotes fitted to them as was done for the real globular cluster data. The recovered /-band elhpticities at a given location on the image showed no trend with input elhpticity and had a mean of e = 0.036 ± 0.004 (standard deviation). Artificial clusters were added to three locations on the original data frame and found that the recovered mean elhpticities and position angles varied significantly with position. Recovered elhpticities were between 0.029 and 0.041 while recovered position angles varied by up to 60°. Positional dependence is due to the presence of large pixel-to-pixel surface brightness fluctuations in the unresolved hght from the bulge of M31. These fluctuations are blurred by the seeing so that they have the same shape and orientation as the PSF. The luminosity of a fluctuation depends on the number of stars in the line of sight and thus its location on the frame relative the centre of M31. Further, the PSF on an HRCam image is known to vary with position on the frame so the size and shape of the surface brightness fluctuations will also vary with position. This would account for the variation in recovered elhpticity and position angle with location on the frame and partially explain why the three clusters are observed to have different elhpticities and position angles. A further reason for cluster-to-cluster elhpticity variations is that the hght from each cluster is dominated by a small number of stars near the tip of the red giant branch which will result in small-number statistics dominating the observed shapes of the clusters. This effect should be more noticeable in small clusters and may explain the large observed elhpticity of G185B. This conclusion is supported by the fact that the orientation of G185B is significantly different in each colour. The fact that the V- and Chapter 7. G185: A Potential Double Globular Cluster 187 /-band ellipticities of G185B are similar but the orientations are different suggests that G185B is not simply a multiple-star system1. The apparent elongation of G185 has the cluster pointing 13? 1 west of G185B and ~ 75? west of the centre of M31. However, G185 is elongated in approximately the same direction that the PSF is oriented, and G185B is not distorted in the direction of G185. The observed V-band shape of G185 were reproducible with artificial clusters of zero elhpticity but the observed elhpticity in the /-band was not. It is, therefore, uncertain if G185 is actually oriented towards G185B or the observed elongation is merely due to the seeing and background fluctuations. As noted in Chapter 7.2 the surface brightness fluctuations in the /-band are greater than those in the V-band so they would be expected to cause a greater distortion in the observed shape of G185 in the /-band image. x It can be argued that a globular cluster is nothing more than a large multiple star system, but in this thesis I will use the traditional definition of the term multiple-star system. Chapter 7. G185: A Potential Double Globular Cluster 188 10 20 30 40 50 60 70 80 90 100 10.. 20 ,.30 4 0 , „ 5 0 . . 60_ 70.. .80 90 100_ Figure 7.6: This figure shows a contour plot of the J-band image of G185 and G185B. The solid contours show surface brightnesses spaced between uj = 14.5 and fii = 20.0 while the dashed contours show fitted isophotal ellipses. G185B is located 4" northwest of G185. The centre of M31 is ~ 6° west of north (the top of the plot is points towards 84° east of north). Chapter 7. G185: A Potential Double Globular Cluster 189 0.5 0.4 0.3 0.2 0.1 0 1 1 1 1 r 1 o "i 1 1 r "1 1 1 1—q G 1 8 5 - A <t>. l tD(D©a>® O CD © <D O 4> c|> <J> • <}> 1 cd I 0.5 0.4 I 0.3 I 0.2 0.1 0 -] 1 1 1 1 1 1 1 1 1 1 r G 1 8 5 - B I - |—i—i—i— i—|— i— i— i i | FI i i i i I i i i i I i i i i I i—i—i—i—1—i—i—i—i— 0 0.5 0.4 r-0.3 p-0.2 0.1 o h n 1 1 r 0 PSF <D(t> _l I 1_ i 1 1 r I l i i i I I I i I I 1 1 L. j i I i_ 2 3 r.„ (") Figure 7.7: /-band ellipticity profiles of G185, G185B, and the PSF. Only points where E L L I P S E (see Chapter 7.2) was able to successfully fit an elliptical isophote are shown. The increase in elhpticity in the inner 0'.'5 is probably an artifact of the fitting algorithm and seeing effects. I was unable to reliably measure elhpticities or position angles in the outer regions of the clusters since Poisson noise and surface brightness fluctuations dominate the signal there. Chapter 7. G185: A Potential Double Globular Cluster 190 5 0 E-0 - 5 0 i 1 1 1 1 1 1 r G 1 8 5 - A -| 1 1 1 r 0 O m CD (D tlO cu 5 0 0 - 5 0 _ L I L p i i i i | i i i i | - i — i — i — i — | — i — T T ^ T : L G 1 8 5 - B IIII —©- -e- IIII - ^fWt' 1 i i i i 1 i i i i 1 i i i i 1 i l l l " 5 0 0 - 5 0 P S F I I ! IA I I i I I I I I L I 1 1 1 1 1 1 1 1 1 L 0 2 3 re„ (") Figure 7.8: /-band position angle profiles of G185, G185B, and the PSF. The same caveats apply as did for Figure 7.7. The position angles are measured in degrees east on the sky from north. Since these profiles are all measured within the inner regions of the clusters (where seeing dominates the observed shape, see Chapter 7.5.2) the large variations in #o for G185B are probably due to the seeing and not intrinsic to the cluster. Chapter 7. G185: A Potential Double Globular Cluster 191 7.6 Dynamical Considerations 7.6.1 The SIS Spectra In order to further check if G185 and G185B were associated with each other, long-slit spectra were obtained for each object. The spectra were taken by Christian Vanderriest on the night of August 31, 1995 using the SIS at the C F H T . The spectra were taken using the R300 grism and a slit width of 1'.'2. The R300 grism is centred on a wavelength of A = 7307 A and has a dispersion of 1.0 A /15 / t m pixel. The Loral 3 C C D was used since it has a gain of 1.45 e _ 1 / A D U . The spectra consisted of one 120 second exposure of G185 and one 600 second exposure of G185B. Both spectral have signal-to-noise ratios of ~ 5 per pixel. Bias subtraction and flat fielding were done in the same manner as for the C F H T image data (see Chapter 2.1). The spectra were extracted using the N O A O . O N E D S P E C tasks in IRAF. Since the SIS was used in long-slit mode the central rows of the spectrum will be dominated by light from the core of the globular cluster while the outer rows will contain a higher fraction of background light from the nucleus of M31. To compensate for this the central five rows of each spectrum were extracted and used to produce summed spectra of G185 and G185B. The outer five rows of each spectrum were extracted and summed to produce background spectra for G185 and G185B. The SIS slit was aligned perpendicular to the line formed by connecting the centres of the two globular clusters. This was done to minimize the amount of contamination from the other globular cluster. The spectra were calibrated by taking a 6.0 second exposure of the C F H T neon and argon calibration arcs. This spectrum was pre-processed and summed in exactly the same way as the program spectra. The arc lines were identified using the IRAF task N O A O . O N E D S P E C . I D E N T I F Y . Figures 7.9 and 7.10 show the calibrated spectra for G185 Chapter 7. G185: A Potential Double Globular Cluster 192 and G185B. Figures 7.11 and 7.12 show the spectra of the M31 nucleus near each globular cluster. The spectral features near A ~ 7600 A and A ~ 6900 A are the TeUuric oxygen bands. Chapter 7. G185: A Potential Double Globular Cluster 193 Figure 7.9: This figure shows the spectrum for G185. This spectrum is the sum of the five rows centred on the centre of G185. Chapter 7. G185: A Potential Double Globular Cluster 194 Figure 7.10: This figure shows the spectrum for G185B. This spectrum is the sum of the five rows centred on the centre of G185B. Chapter 7. G185: A Potential Double Globular Cluster 195 Figure 7.11: This figure shows the spectrum of the background. This spectrum is the sum of the five rows on each edge of the original G185 spectrum. Chapter 7. G185: A Potential Double Globular Cluster 196 Figure 7.12: This figure shows the spectrum of the background. This spectrum is the sum of the five rows on each edge of the original G185B spectrum. Chapter 7. G185: A Potential Double Globular Cluster 197 7.6.2 The Radial Velocities The radial velocity of G185B relative to G185 was determined by cross-correlating the spectrum of G185B with the spectrum of G185 using the NOAO.RV.FXCOR task in IRAF. Cross-correlating the entire G185 and G185B spectra yields a relative radial velocity of Av = +41.5 ± 98.6 km-s - 1 . This velocity difference is define as Av = VQIS5 — ^ G I S S B -The uncertainty in the velocity difference is the standard deviation of the best-fitting Gaussian to the cross-correlation peak. Cross-correlating the two background spectra gives Av = -11.4 ± 79.4 km-s - 1 . Unfortunately the spectra of the globular clusters contain significant contamination from the nucleus of M31 and absorption lines in Earth's atmosphere. The most obvious of these are the Telluric A and B blends at A ~ 7600 A and A ~ 6900 A respectively. Unfortunately the low signal-to-noise ratio of these spectra, and the strengths of the atmospheric features makes it difficult to separate features from globular cluster from features from the background and atmosphere. In order to prevent the cross-correlation algorithm from locking onto the atmospheric features the spectra were split into three regions: (1) 6400 A to 6875 A, (2) 7100 A to 7500 A, and (3) 7850 A to 8200 A. These regions were chosen to exclude the Telluric A and B bands. The relative radial velocities returned by cross correlating the G185 and G185B spectra in these three wavelength regions are given in Table 7.5. Region AA AvGC A^bkgd (A) (km-s- 1) (km-s"1) 1 6400 - 6875 +4.6 ± 53.1 -58.0 ± 154.7 2 7100 - 7500 +28.9 + 164.1 +23.1+ 31.3 3 7850 - 8200 +11.4+ 15.6 +20.0 + 104.7 Table 7.5: Relative radial velocities for G185, G185B, and the background. Chapter 7. G185: A Potential Double Globular Cluster 198 The radial velocity difference between the two background fields should be zero. How-ever, the two spectra were taken 45 minutes apart. Since the detailed absorption spectrum of the Earth's atmosphere can change on time scales of only a few minutes it is likely that the difference in Avbkgd in the three spectral regions are due to temporal variations in the atmospheric lines. If this is the case then some portion of the variations in the A I > G C values obtained from the three spectral regions will also be due to the temporal variations in the atmospheric absorption lines. Therefore, it is not possible to draw any conclusion about the radial velocity difference between G185 and G185B with this data. Chapter 7. G185: A Potential Double Globular Cluster 199 7.6.3 The Roche Limit of the System The mean radial velocity of a globular cluster in the M31 system relative to M31 is v = 125 km-s - 1 (Huchra et al. 1991 [76]) which implies an encounter time-scale of at least 105 to 106 years, assuming a nearly head-on encounter (i.e. the geometry that produces the fastest possible encounter). This minimum encounter time is similar to the crossing time for a star in a globular cluster so interacting clusters will have sufficient time to be spatially distorted by each others' tidal fields. I have calculated the shapes of the Roche lobes for G185 and G185B assuming that the two clusters are interacting and that they can be modeled by point masses. In reality the two clusters are not true point sources, but since the half-mass radii of the Michie-King models fitted to them are comparable to their fitted core radii, this assumption is reasonable. The fitted half-mass radii for G185 and G185B respectively are l'/05 ± 0707 and 0749 ± 0716. Figure 7.13 shows that the cut-offs imposed by the tidal field of M31 are approximately the same as the cut-offs imposed by the mutual tidal fields of the two clusters if G185 and G185B have a true separation of ~ 35 pc (~ 10"). If the true separation is greater than this then the shapes of the clusters will be determined by the tidal field of M31. If, however, the true separation is smaller then inter-cluster tidal forces will have a significant effect on the internal dynamics of each cluster. Figure 7.14 shows that if the true separation is equal to the observed projected separation (14 pc ~ 4") then the cut-offs imposed by their mutual tidal fields are closer to the clusters' centres than the tidal cut-offs imposed by M31. Since hght is observed beyond these Roche hmits the clusters are either not interacting or have not had time to dynamically respond to each others' tidal fields. Chapter 7. G185: A Potential Double Globular Cluster 200 w XI ti o o cu w o ca 10 0 10 -i 1 r ~i 1 r Separation = 10 (") J I L . -10 _l I I U _l I I L. X (arcseconds) (a) 10 Figure 7.13: This figure shows the Roche lobes for the two clusters computed assuming the clusters are interacting and can be approximated by point masses (see Chapter 7.6.3 for a justification of the latter assumption). The solid lines show the Roche lobes while the dashed lines are the fitted /-band Michie-King tidal radii from Table 7.3. The dotted lines show the outermost point of the observed surface brightness profile for each cluster. The two clusters are assumed to have a true separation of 35 pc (~ 10"). Chapter 7. G185: A Potential Double Globular Cluster 201 - 1 0 0 10 X (arcseconds) F i g u r e 7.14: T h i s figure is the same as F i g u r e 7.13 except the two clusters are assumed to have a t rue separa t ion of 14 pc ( ~ 4") , the same as the observed p ro jec ted separa t ion be tween G 1 8 5 and G 1 8 5 B . Chapter 7. G185: A Potential Double Globular Cluster 202 7.7 Conclusions G185 and G185B appear to be two separate globular star clusters in M31. There is no direct evidence that the two clusters are interacting. There is no evidence of tidal disruption in either clusters although the /-band ellipticity of G185 is greater than can be accounted for due to seeing effects and G185 is approximately oriented towards G185B. Chapter 8 Conclusions 8.1 The Globular Star Clusters of M31 This thesis presents the results of two studies of the globular clusters in the M31 system. The first study was done using V- and J-band images obtained with the HRCam at the C F H T . The second study made use of V- and J-band HST WFPC2 images of two globular clusters in M31. Two-dimensional, seeing-convolved Michie-King models were fit to seventeen globular clusters in M31. The fitted tidal radii are the same as those of Galactic globular clusters. This confirms the work of Cohen & Freeman (1991) [30], who compared the tidal radii of thirty M31 globular clusters with the tidal radii of the Galactic globular clusters. I also find that the projected elhpticities of the M31 globular clusters are consistent with the elhpticities of the Galactic globular clusters, confirming the work of Lupton (1989) [90]. The core radii and half-mass radii of globular clusters in the two galaxies appear to be similar, but seeing effects make it impossible to obtain accurate values for these quantities using the C F H T HRCam data. This similarity in structural parameters suggests that the two globular cluster systems formed in a similar way and have had similar dynamical histories. M31 is structurally very similar to the Milky Way Galaxy so globular clusters in both galaxies will experience 203 Chapter 8. Conclusions 204 similar tidal fields and similar amounts of bulge- and disc-shocking. In light of this it is not surprising that the globular clusters appear to be structurally identical in a mean sense. The HST WFPG2 photometry was used to construct deep (V ~ 27) CMDs for two globular clusters in the halo of M31. Both globular clusters appear to have a single old population of stars similar to what is found in Galactic globular clusters. The shape of the red-giant branch for G302 gives an iron abundance of [Fe/H] = —1.85±0.12, in agreement with the published values obtained using spectroscopy. G312 has an iron abundance of [Fe/H] = —0.56 ± 0.03, which is somewhat more metal-rich than the spectroscopically determined value. Neither globular cluster shows any indication that there is a second parameter acting upon their horizontal branch morphologies. Both globular clusters have Michie-King tidal radii of rt ~ 10", core radii of rc ~ 0"2, central concentrations of c ~ 1.7, and half-mass radii of rn — 075. There is no evidence for velocity anisotropy in either G302 or G312. G302 has a colour of (V — I)0 = 0.83 while the colour of G312 is (V — I)0 = 1.07. Both of these colours are consistent with the integrated colours of Galactic globular clusters with similar metallicities. G302 has a projected ellipticity of e = 0.195 with the major axis oriented approxi-mately towards the centre of M31. This globular cluster has an excess of light beyond its formal tidal radius that is not consistent with either an isotropic or an anisotropic Michie-King model. The two-dimensional distribution of stars around G302 is consistent with the presence of an extended halo of unbound stars extending to two to three times the formal tidal radius from the cluster. G312, on the other hand, has an ellipticity of e ~ 0 and neither the integrated light nor star counts show any evidence for an extended stellar halo. It is possible that such a halo does exist for G312 but is oriented along the line of sight. I have estimated the projected mass-loss rate from G302 to be M = 4500 ± 1800 MQ Chapter 8. Conclusions 205 per Gyr which corresponds to a projected escape rate of r = (2.3 ± 0.9) x 10~3 per half-mass relaxation time. The projected escape rate from G312 is r = (0.38 ± 2.95) x 10~3 per half-mass relaxation time. These are consistent with the escape rates predicted by Oh & Lin (1992) [103]. The large photometric uncertainties near the tidal radius of G312 makes the escape rate for this globular cluster much less reliable than that for G302. A comparison of published structural parameters obtained using HST and ground-based data shows that ground-based observations (with sub-arcsecond seeing) are suf-ficient to obtain reliable King tidal radii for M31 globular clusters. Ground-based ob-servations can reliably determine the core radii if the core radii are at least 0.25 times the F W H M of the seeing disc. This requires seeing of F W H M ^ 071 in order to reli-ably measure a core radius of ^ 0703 (~ 0.1 pc) in an M31 globular cluster. Therefore, ground-based determinations of core radii are not, in general, reliable, except to establish broad concentration classes. At present reliable tidal radii have been obtained for ~ 50 globular clusters in M31, but reliable core radii have only been obtained for ~ 20 of these. The HST is scheduled to obtain V- and /-band WFPC2 images of twenty M31 globular clusters during Cycles 6 and 7. Since the globular clusters in the two galaxies appear to have identical distri-butions of structural parameters there it is not critical to obtain more images to study the core structures of these clusters. On the other hand, M31's globular clusters offer ideal laboratories to study the structure of globular clusters near their King tidal radii. This thesis presents evidence for an extended halo of unbound stars around G302. Ten to twenty orbits with the HST would be needed to resolve stars down to Vu m ~ 28, which would increase the number of stars that could be photometered in an extended halo by approximately a factor of two. This would greatly improve the statistical significance of stellar density measurements beyond the formal King tidal radius and allow the shape of an extended halo to be determined to greater accuracy than was possible in this study. Chapter 8. Conclusions 206 8.2 The Halo of M31 This thesis presents the results of a photometric study of the steUar populations in two fields in the halo of M31. This study used HST WFPC2 V- and J-band photometry to show that the stellar halo of M31 has a mean metalhcity of [m/H] ~ —0.6 with a spread of —2 ^ [m/H] <^  —0.2, comparable to the metallicity determinations made using ground-based observations. [m/H] ~ —0.6 is ~ 15 times more metal-rich than the halo of the Galaxy ([m/H] ~ —1.8). This result is primarily based on a comparison of the morphology of the halo red-giant branch to fiducial sequences of Galactic globular clusters and theoretical red-giant branch isochrones. As such the metalhcity estimate is distance dependent. If M31 lies at a greater distance the mean metalhcity will decrease. Therefore, it is important that distance-independent estimates of the met alii cities of stars in the halo of M31 be made. The halo horizontal branch consists primarily of red-clump stars, similar to what is seen in 47 Tuc, and a small number of blue stars. The horizontal branch morphology ratio is (NB — NR)/(NB + NR) ~ —0.9, suggesting that there is a metal-poor popula-tion present in the halo of M31 that could give rise to the RR Lyrae variables seen by Pritchet & van den Bergh (1987) [114]. The derived helium abundance is Y ~ 0.20 to 0.27, similar to that in Galactic globular clusters. There is no conclusive evidence for a difference in stellar population between the G302 field and the G312 field. The metalhcity distribution of the red giants indicates that there is no significant difference in the ratio of metal-rich to metal-poor stars in the two fields. Deep photometry will be needed in a large number of fields in the M31 halo to determine how (or if) metalhcity varies with position in the halo of M31. Chapter 8. Conclusions 207 8.3 Unanswered Questions There are still many unanswered questions about the nature of M31. In this section I will list a few projects that will add to our knowledge of that galaxy. 1. The metallicity distribution in the halo of M31 is still not well known. All of the attempts to determine the metallicity distribution to date have involved comparing CMDs of the upper red-giant branch of M31's halo to fiducial red-giant branches of known metallicity. This approach will work provided that the photometric uncer-tainties are small and that the distance to M31 is well known. However, the recent Hipparcos parallaxes have led to an uncertainty of ± ~ 0.25 mag in M31's distance modulus, which calls into question the existing distance-dependent metallicity es-timates for the halo of M31. The results presented in Chapter 6.1.2 assumed a distance modulus of UQ = 24.3 ± 0.1, which was the accepted distance modulus when this work was started. If M31 is actually more distant then the mean metal-licity of the halo will decrease. A better way of determining the metallicity of the halo of M31 would be to obtain deep narrow-band photometry of a large number of upper red-giant branch stars. The Stromgren v, b, and y filters can be used to obtain a metallicity index, m i = (v — b) — (b — y), that is independent of the distance to M31. Such observations would be possible with the approximately two nights of observing with UHCam at the C F H T . 2. Spectroscopic iron abundances exist for most of the M31 globular clusters. These can be used to estimate the distances to the M31 globular clusters if good-quality (V,(V — I)) photometry of the upper red-giant branch is available. A series of fiducial red-giant branches of known iron abundances can be overlaid on the cluster's Chapter 8. Conclusions 208 CMD and the distance modulus can be adjusted until the red-giant branch is at its best agreement with the spectroscopically-determined iron abundance. This technique would require that the reddening towards the globular cluster be well-known. However, since M31 lies at a high Galactic latitude, and most of the M31 globular clusters that have pubhshed CMDs lie well beyond the edge of the disc of M31 the uncertainties in the reddening towards these clusters are small. 3. Hodder (1995) [70] found a field in the northeast quadrant of the halo of M31 that contains two to three times more stars than would be expected using a version of the Bahcal &; Soneira (1984) [6] Galaxy model that he adapted for use in M31. Most of the studies of the stellar populations in the halo of M31 have been done in the northeast quadrant of M31 to avoid contamination from M32 and N G C 205. Further, because of the small sizes (~ a few arcminutes square) of CCDs in the past all pubhshed CCD studies of the halo of M31 have sampled the halo of M31 and not mapped it in its entirety. I have obtained UHCam images of two 0?5 x 0?5 fields in the halo of M31. The first field is centred on the field containing Hodder's (1995) [70] overdensity while the second is located on the opposite side of the major axis of M31. A comparison of the star counts and the CMDs in these two fields will tell us if the overdensity is due to the structure of M31 (e.g. a flattened halo, or a thick disc), or a clump in the northeast quadrant of the halo. 4. At present there are nine globular clusters in M31 which have CMDs that reach to the horizontal branch. The mean l/-band magnitudes of the stars in the RR Lyrae gaps of these clusters have been measured by Fusi Pecci et al. (1996) [56] and in Chapter 5.1. Combining these VRR values with spectroscopic iron abundances and the [Fe/H]-M^(RR) relation for RR Lyraes gives the distance modulus to the globular cluster in question, i^o = VRR — ct[Fe/H] — /3. Unfortunately there is no Chapter 8. Conclusions 209 consensus as to what the values of a and (5 are (e.g. Carney et al. (1992) [24], Chaboyer et al. (1996) [28], Chaboyer et al. (1997) [27]). Pubhshed disagreement in the values of these parameters can give results that differ by ± ~ 0.2 mag. A second problem is that the photometric uncertainties at the level of the RR Lyrae gap (V ~ 25) are ay ~ 0.1. Over the next two years twenty additional globular clusters will be imaged with the HST. This should provide a large enough database to obtain a mean distance to the M31 globular cluster system to an accuracy of ± ~ 0.02 mag, not counting uncertainties in the [Fe/H]-Mv(RR) relation. References [1] Ajhar, E . A . , Gri l lmair , C . J . , Lauer, T . R. , Baum, W . A . , Faber, S. 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